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EL'IE\'IER
STATISTICS OF LINEAR POLYMERS IN DISORDERED MEDIA
Bikas K. Chakrabarti EDI TOR
Statistics of Linear Polymers in Disordered Media
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Statistics of Linear Polymers in Disordered Media
Edited by
Bikas K. Chakrabarti Saha Institute of Nuclear Physics Kolkata, India
2005
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PREFACE With the mapping of the partition function graphs of then-vector magnetic model in the n --+ 0 limit as the self-avoiding walks, the conformational statistics of linear polymers was clearly understood in early 1970s. Various models of disordered solids, percolation model in particular, were also established by late seventies. Subsequently, investigations on the statistics of linear polymers or of self-avoiding walks in, say, porous medium or disordered lattices were started in early 1980s. lnspite of the brilliant ideas forwarded and extensive studies made, the problem is not yet completely solved in its generality. This intriguing and important problem has remained since a topic of vigorous and active research. This book intends to offer the readers a first hand and extensive review of the various aspects of the problem, written by the experts in the respective fields. S. M. Bhattacharjee has reviewed the success in dealing with the directed polymers in random medium and has also discussed the problem of unzipping of a pair of directed chains where disorder appears along the chains. A. J. Guttmann has reviewed the series studies of self-avoiding walk statistics for various constrained and random geometries, including the problem of unzipping of the DNA chains. V. Blavats'ka, C. von Ferber, R. Folk and Yu. Holovatch has reviewed extensively the field theoretic and real space renormalization group studies for the problem, including the effects of correlated disorder. D. Dhar and Y. Singh have reviewed most of the exact results for self-avoiding walks on different non-random fractals, including various simplex lattices, employing the real space renormalization group technique. A. Ordemann, M. Porto and H. E. Roman have reviewed the extensive numerical studies on self-avoiding walks on various deterministic and random fractals; percolation clusters in particular. In absence of strict self-avoiding restriction, the analogy of the problem with that of quantum particles in disorder, and the consequent localization of the polymers in random media, have been reviewed by Y. Y. Goldschmidt andY. Shiferaw. P. Bhattacharyya and A. Chatterjee have reviewed the properties of various optimal and most probable (self-avoiding in general) paths on randomly disordered lattices, including the statistics of the Travelling Salesman Problem on dilute lattices. Finally, G. D. J. Phillies has given an extensive overview of the experimental studies on polymer diffusion in random environments of the solutions. We earnestly hope, the contents of the book will provide a valuable guide for researchers in statistical physics of polymers and will surely induce further research and advances towards a complete understanding of the problem. I am grateful to all the contributors for their wonderful contributions and cooperations. I am thankful to Arnab Chatterjee for his help in the compilation of the book.
Bikas K. Chakrabarti Theoretical Condensed Matter Physics Division, and Centre for Applied Mathematics & Computational Science Saha Institute of Nuclear Physics, Kolkata 700064, India. January 2005.
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Contents Polymers in random media: An introduction B. K. Chakrabarti
1
Directed polymers and randomness SM&~oc~~
Self-avoiding walks in constrained and random geometries: Series studies A. J. Guttmann
9
59
Renormalization group approaches to polymers in disordered media V. Blavats'ka, C. von Ferber, R. Folk and Yu. Holovatch
103
Linear and branched polymers on fractals D. Dhar· and Y. Singh
149
Self-avoiding walks on deterministic and random fractals: Numerical results A. Ordemann, M. Porto and H. E. Roman
195
Localization of polymers in random media: Analogy with quantum particles in disorder Y. Y. Goldschmidt and Y. Shiferaw 235 Geometric properties of optimal and most probable paths on randomly disordered lattices P. Bhattacharyya and A. Chatterjee 271 Phenomenology of polymer single-chain diffusion in solution G. D. J. Phillies
305
Index
357
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Statistics of Linear Polymers in Disordered Media Edited by Bikas K. Chakrabarti © 2005 Elsevier B.V. All rights reserved.
Polymers in random media: An introduction Bikas K. Chakrabarti a aTheoretical Condensed Matter Physics Division, and Centre for Applied Mathematics & Computational Science, Saha Institute of Nuclear Physics, 1/ AF Bidhannagar, Kolkata 700064, India. email:
[email protected] We introduce first the (lattice) Self-Avoiding Walk (SAW) model of polymer chains, their critical statistics and the criteria indicating effects of lattice disoder on the critical behaviour. Prominent indications for the effect of disorder on the SAW statistics are then discussed. Next, some mean field and scaling arguments are discussed for the SAW statistics in disordered medium; percolating lattice in particular. 1. POLYMER STATISTICS AND SAW MODEL
Linear polymers are long flexible molecular chains [1] whose building blocks are the monomers. The chain flexibility arises when the chains are dissolved in solvents. The chains are completely flexible in good solvent. To study the conformational properties of polymer in good solvent, e.g., to estimate the variation of average radius of gyration or the end-to-end distance with the chain length, we use a lattice model of linear polymers where the polymer is viewed as a walk on a lattice: the monomer size is represented by the lattice constant and the size of the polymer chain by the walk length. A random walk can easily capture the flexibility of the chain. However, such a walk can cross itself or may trace back the same path. For polymer chains, the steric hindrence induces monomermonomer excluded volume restriction which invalidates this model. Hence a realistic model of a polymer chain in a good solvent is a random self-avioding walk (SAW) on a (translationally invariant) lattice. These SAWs are random walks without self-intersection or crossing; as such they are the self-avoiding subset of random walks. The statistics of the SAW model [2] of linear polymers is quite well studied. The generator of the self-avoiding walk statistics or the distribution function GN(r), which represents the number of N-stepped SAW configurations with end-to-end distance r, is not Gaussian as in the case ofrandom walk (for random walk GN(r) '""exp[-r 2 /N]). From the distribution function GN(r), one can obtain the asymptotic behaviour of various moments. A brief summary is as follows: The statistics of SAWs are characterised by the connectivity constant /-l (G N = I:r G N(r) '""/-lN NY-l ), which is nonuniversal and depends on the lattice types, and the universal exponents like the radius of gyration exponent v (R'fv = I:r r 2 GN(r)/GN'"" N 2v), which depend only on the lattice dimension d. Extensive numerical studies give, /-l = /-1° ~ 2.638, 4.151 and 4.684 for square, triangular and simple 1
B. K. Chakrabarti
2
cubic lattices respectively [3]. Various theoretical and numerical studies give the value of v v~ = 3/4, ~ 0.592 and 1/2 (and 'Y = 'Y~ = 43/32, ~ 1.17 and 1) ford= 2, 3 and 4 respectively; here the superscript 0 stands for pure lattice [4]. Ford 2: 4, the (statistical) effect of the excluded volume fluctuation disappear and self-avoiding walk and random walk belongs to the same universality class; the upper critical dimensionality (for the SAW statistics) is four [2]. In the limit of high temperature the effective interactions between the monomers arise mostly from the excluded volume considerations and the random SAW model, discussed above, is quite successful in capturing the universal behaviour of the conformational statistics. With lowering of the temperature, or in poor solvents, the effect of monomermonomer attraction grows and the polymer radius shrinks. The changes in the conformational statistics of linear polymers, with lowering of temperature from a high temperature limit, have also been studied extensively [1,2]. At the 11-point temperature, the two body excluded volume term is exactly cancelled by the growing attractive interactions and the statistics is governed by the higher order excluded volume terms. This point has been identified [2] as the tricritical point. The 11-points for the lattice SAW model have been estimated and (like the connectivity constant /-1) they depend on the lattices. At this particular temperature, a crossover occurs from the high temperature SAW statistics to a tricritical (11-point) statistics. The size exponent is then given by the 11-point exponent value v = v2 = 4/7 [4] and 1/2 in d = 2 and 3 respectively: with the upper critical dimension for 11-point statistics equal to three [2]. Below this tricritical temperature the attractive force dominates over the repulsive (en tropic) term and induces the chain to collapse (with collapsed polymer size exponent v = v~ = 1/d). We summarised so far the statistics of SAWs on regular lattices.
=
2. SAW STATISTICS IN DISORDERED MEDIUM We consider in this book the problem of polymer chain statistics in a disordered (say, porous) medium. If the porous medium is modelled by a percolating lattice [5], we can consider the following problem: let the bonds (sites) of a lattice be randomly occupied with concentration p (2: Pc; the percolation threshold); the SAWs are then allowed to have their steps only on the occupied bonds (through the occupied sites). We address the following questions [6,7]: does the lattice irregularity (of the dilute lattice) affect the SAW statistics? We expect /-lor II to vary with the lattice occupation concentration p: 1-l(P) < /-1° and O(p) < 11° for p < 1 where superscript 0 refers to that for pure lattice (p = 1). In particular, the values of 1-l(Pc) and O(pc) on various lattices are very significant lattice statistical quantities: 1-l(Pc) (> 1) and O(pc) (> 0) values signify the nature of ramification [5] of the percolation clusters (see e.g., [7]). We will discuss if the size exponents (vs or vo) are affected by the lattice (configurational) fluctuations: if vs(P) is different from v~ or if vo is different from v2 for p < 1. This question arises naturally from the application of the Harris criterion [8] to the n-vector model in the n--+ 0 limit [2], when the partition function graphs are all SAWs. A naive application of the criterion to the SAWs suggested [6] a possible disorder induced crossover in the critical behaviour of SAW statistics for any amount of disorder (p < 1). A modified
Introduction
3
analysis [9] of course indicated that a tricky cancellation of the disorder induced crossover occurs at n--+ 0 limit and the SAW statistics remains unchanged for 1 2: p > Pc· However, since at Pc the dimensionality of the fractal percolating lattice is different from that of the Euclidean lattice, and as the size exponents are determined by the dimensionality, we expect vfc and v~c to be different from those (v~ and v~ respectively) of pure lattice. Let us now look into the initial indication of the effect of lattice disorders on SAW statistics. Harris [8] and Fisher [10] gave heuristic arguments which suggested that the critical behaviour of a system would be affected by the presence of disorder (quenched and annealed respectively), if the internal energy fluctuation or specific heat of the (pure) system diverges (with positive specific heat exponent o:). For quenched disorder, the Harris criterion indicates only the possibility of a crossover for systems with diverging specific heat but it cannot be extended to indicate the new critical behaviour. For annealed disorder, the arguments by Fisher gives also the nature of the new (Fisher renormalised) critical behaviour. These findings, using these arguments, had later been supported using renormalisation group techniques.
2.1. Quenched impurity: Harris criterion Here, as in the percolating systems discussed earlier, the impurities are not in the same thermal bath as the ordering system; rather they are quencehed to zero temperature. The mean square fluctuation in the disorder concentration in a typical volume element ~d (~ denoting the thermal correlation length of the ordering system) is then
(1) Here p is not critical; p > > Pc· The change in internal energy due to this fluctuation in interaction strength corresponds to a fluctuation 6.Tc in local transition temperature Tc given by
(2) However, since~~ I6.TI-v, where 6.T = IT- Tci/Tc, we can express 6.T as'"" ~- 1 /v. For a sharp transition the local fluctuations in Tc is required to be much smaller than this critical temperature interval 6.T, and consequently we need 6.Tc << 6.T, or d/2 > v- 1 , or (2- dv) < 0. Using the hyperscaling relation o: = 2- dv, the above condition becomes o: < 0; indicating nontrivial effect (a new different sharp transition with negative value of o:, or a smeared transition) due to quenched disorder for (pure) systems with o: > 0 [8]. Using the Flory estimate v = v~ = 3/(2 +d), as discussed in the next section, and the hyperscaling relation o:~ = 2 - dv~, we get o:~ = (4- d)/ ( d + 2) > 0 for d < 4 for SAWs on pure lattices. This would indicate [6] a disorder induced crossover in the SAW statistics for any amount of disorder at p > Pc· However, this indication is not quite correct for the n-vector model in the n --+ 0 limit [9]. For a general n-vector model, a similar estimate of transition temperature fluctuation would give ~-df).Tc ~ n6.p, since the Hamiltonian contains a sum over all the n components. This gives 6.Tc ~ n~-d/ 2 , compared to the usual 6.T ~ ~- 1 /v, mentioned above. Hence, in the n --+ 0 (SAW) limit, 6.Tc << 6.T always, irrespective of the o: = 2 - dv exponent value. This modified Harris criterion, thus, does not indicate any effect of lattice disorder on SAW statistics (for p > Pc) as the
B. K. Chakrabarti
4
critical temperature fluctuation never exceeds the temperature interval of the correlation length in an n-vector model in the n --+ 0 limit, although the specific heat exponent of the model is positive.
2.2. Annealed impurity: Fisher renormalisation Since the impurities here are in the same thermal bath as the ordering system, one can consider a total Hamiltonian of the system including the impurity. Let the impurity coupling be denoted by A. For such annealed systems, Fisher assumed [10] F(t:,.T, A)
~
F(t:,.T*(t:,.T, A)),
(3)
2 where F denotes the free energy of the system with F(t:,.T*) "" 1/::,.T* 1 -", o: denoting the specific heat of the pure system, and the renormalised temperature interval /::,.T* is an analytic function of its argument /::,.T and A. The impurity concentration p is then given by p "" aFiaA "" (8FI8T*)(8T* loA). Assuming now analytic constraints over p, both sides of the above expression (3) can be expanded and the lowest order terms in /::,.T and /::,.T* on both sides may be compared to give (since both /::,.T and /::,.T* are much less than unity):
(4) and 1/::,.T*I = 1/::,.TI foro:< 0. Usual critical behaviour of the annealed system, expressed in terms of /::,.T*, gives in effect the Fisher renormalised exponents o:R = -o:l(1 - o:), f3R = /31(1 - o:), 'YR = 'YI(1- o:) and vRI(1- o:) for the impure system when o: > 0 for the pure system. These renormalisation of exponents keeps the scaling and hyperscaling relation o: + 2(3 + 'Y = 2 and o: = 2 - dv unchanged. Using o:~ = (4- d)l(2 +d) for SAWs, using the Flory value for v~ = 31(2 +d), one gets vR = vf ~ 3l2(d -1) for the Fisher renormalised polymer size exponent (in the annealed random medium), which is often unacceptable (e.g., vf > 1 ford= 2!) [7].
3. ESTIMATES OF SAW SIZE EXPONENT ON PERCOLATION CLUSTER The above considerations indicate that SAW statistics may have different critical behaviours in disoderd media. We now try to estimate the SAW size exponent v using some approximate theories.
3.1. Flory theory The Flory theory [1,2] estimates the functional form of the free energy F(R) of a chain of size N and average radius of gyration or end-to-end distance R. For this, it proceeds as follows: For N monomers in a volume of average radius R, the average monomer concentration c = Nl Rd. Now the repulsive energy due to monomer-monomer interaction is proportional to c2 , and the total repulsive energy Frep ~ c2 Rd "" N 2 I Rd. From entropic consideration, the elastic energy Fe 1 "" R 2 IN. Hence, total energy F(R) "" (N 2 I Rd) + (R 2 IN). Minimising the total energy F with respect toR, we get Rd+ 2 ""N 3 • This gives R "" Nv• with Vs = v~ = 3l(d + 2). This estimate for v~ (= 314, 315 and 112 for d = 2, 3 and 4 respectively) is remarkably accurate [2] compared to the exact
Introduction
5
value (for d = 2) and best estimates [4,5]. This also gives de = 4 for the upper critical dimension of SAW statistics, where the entropy term Fe 1 wins over the excluded volume term Frep· One can easily see that even a naive replacement of d in this Flory formula by the appropriate fractal dimension ( < d) of the percolation clusters indicate a significant change in the SAW statistics; namely the polymer size exponent vfc on the percolation cluster clearly changes (vfc > v~) [11]. The above method can be extended to the 11-point, where the above two-body repulsion (O(c 2 )) is exactly balanced by the attractive interaction and three-body interaction (0 (c3 )) comes into play. Hence, here F (R) '"" (N 3 I R 2d) + (R 2 IN). Again from energy minimum condition we get R'"" Nve, v0 v2 = 2l(d + 1). This correctly gives the upper critical dimension de= 3 (where v2 = 112) for the 11-point statistics. However, the value v2 = 213 c::: 0.67 in d = 2 is far from the real value (v2 = 417 c::: 0.57 [4]). There have been several attempts to extend the above Flory theory to correct for its failure for the v0 estimate (in d = 2) and for estimating Vs and vo on fractals. We give here a simple and elegant one (cf. [7]) which starts with a functional estimate of the radius of gyration distribution P(R) '""exp[-F(R)], instead of that for the free energy F(R). The form of the distribution function P(R) of the polymer radius of gyration is given by
=
(5) where (3, 5 > 0. The proportionality factors are omitted for simplicity. This form of P(R) ensures that the probability of getting the polymer size R outside the bound Na :S R :S Nb is exponetially small. In fact, the above form of P(R) assumes that it decays exponentially to zero as R crosses the above bounds. The distribution function is maximum at the most probable size RN when dP(R)IdRIR=RN = 0. If we express the most probable size RN '""Nv, then we get, v =(a+ ""b)l(1 +""),where""= 51/3. Let us first consider the regular lattices: (i) For SAWs in a pure lattice in the high temperature limit (random SAW limit), N 1 1d :S R :S N. In this case the two-body interaction term in the Flory free eneregy (F(R) '"" lnP(R)) is ensured by choosing (3 = d. Similarly the elastic term in F(R) will come by choosing 5 = 2. In the SAW limit, therefore, we get v = v~ = 3l(d + 2), the normal Flory estimate for SAW size exponent. (ii) At the II- point, the appropriate bound was suggested to be (cf. [7]) N 1/d :S R :S Nv~. Here the three-body term in F(R) is ensured by the choice (3 = 2d and the elastic term again comes from the choice 5 = 2. One thus gets v = v2 = (d + 5)l[(d + 2)(d + 1)]. It gives v2 = 7112 c::: 0.58, compared to the exact value 417 '"" 0.57 in d = 2. This d = 2 result for v2 (and also the F(R)) is the same as that obtained from various screening considerations. Also, v2 = 112 when three-body term vanishes (becomes independent of N) at d 2': dec::: 2.4 in this approxmation [7]. We now consider the percolation clusters: (i) Here in the high temperature limit, the bound on R is N 1/dB :S R :S N 1ldmin, where ds is the percolation backbone dimension and dmin is the shortest chemical path dimension [5]. "" (= 51/3) should incorporate here the spectral (random walk) dimension of the percolation cluster for the elastic energy term and is given by "" = dwdminl ds (dw - dmin), where dw is the random walk dimension on the percolation cluster. One then gets v = vfc = (dmin + ""ds)ldsdmin(1 + ""). The same expression has also been reproduced by
B. K. Chakrabarti
6
others using quite different methods (see [7]). This gives the values of vfc ~ 0. 77, 0.66 and 1/2 in d = 2, d = 3 and d 2: 6 respectively. (ii) For SAWs at the 11-point on the percolation cluster, the appropriate bound is assumed to be (cf. [7]) N 1idB :::;_ R :::;_ Nvfc. The values of"" corresponding to the three- body interaction gives v = vGc = (1 + ""dsvfc)fds(2 + "")· This gives vGc(2: v2) ~ 0.68, 0.61 and 1/2 in d = 2, d = 3 and d 2: 6 respectively. It is to be noted that here also the upper critical dimensionality for the 11-point on the percolation cluster shifts to de = 6. 3.2. Scaling theory We consider now a scaling theory of SAW statistics on percolation cluster. This indeed gives the (exact) result that vfc 2: v~ ford< 6 and that vfc = 1/2 at d 2: de = 6. Since there is no new fixed point for SAWs on p > Pe cluster [7], we can write RN '""A(p)Nv~, where the enhancement factor A(p) = 1 for p = 1 and A(p) diverges asp approaches Pe· This factor comes because of the swelling of the SAW with lattce dilution (keeping Vs unchanged at the pure lattice value v0). So, for p > Pe, we assume
(6) with CJ > 0. For p just below Pe the SAWs are restricted on the 'incipient infinite cluster' [5], which will force the SAW to move on the cluster within its boundary. Hence there should be no N dependence on RN and RN will be of the order of the average size of the cluster for such N values. So, for p < Pe, SAWs are possible until
where ~P is the percolation correlation length and vp is the corresponding exponent. One can now combine the above limiting results for different regions of p into one consistent theory by constructing a suitable scaling function of the general form (cf. [7]) RN
rv
NvfcY(Nx(p- Pe)).
(8)
The scaling function Y(z) (rv z-(J for z > 0 and'"" z-vp for z < 0; z = Nx(p- Pe) ) is assumed to be asymptotically defined with power law variations. This gives RN '"" Nvfc at p = Pe· From the fitting of the scaling form with the assumed forms for the abovementioned limits, we get x = vfc jvp and CJ = (vfc- v~)jx. Since CJ 2: 0, vfc 2: v~. Of course, since the percolation cluster backbone dimension ds is less than two in d = 4 (ds = 2 for d 2: 6 only [5]), where the fractal dimension ( (v~)- 1 ) of the ordinary SAW becomes equal to two, they cannot be accomodated on the percolation fractal until they swell, suggesting vfc > v~ at least for d = 4 and indicating generally the same for d < 6. An independent estimate of CJ can be made by node-link-blob model [5] of the percolating cluster and this in turn can give the estimate of vfc (from the above relation). For p > Pe, in the 'nodes and links' model the infinite percolation cluster (without dangling ends) is a super-lattice where one dimensional channels or 'links' (with occasional blobs in it) meet at the crossing points or 'nodes'. The nodes of the superlattice are separated by a 'crow flying' distance (rv O(~p), ~p rv IP- Pei-Vp denoting the percolation correlaion length) and also by a chemical distance L (the length of the quasi-one dimensional links), where it is assumed that L (2: ~p) diverges near Pe i.e., L '"" (p - Pe)-(p. Hence, with
Introduction
7
an appropriate scaling of the variable RN as RN I~P and number of steps N as NIL, the relation RN rv NV~ becomes RN rv (~pL -v~)N-v~ This relation, when recast in the form (6) appropriate for p > Pc, gives CJ = Vp - (pv~ . For d < 6, when the blob exists, (p depends on the appropriate choice of chemical length L. At d 2': 6, blobs do not exist and (p = 1 [5]. Also vP = 112 for d 2': 6 where v~c = v~ = 112. Hence v~c > v~ for d < 6 and v~c = v~ = 112 for d 2': 6. This gives the upper critical dimension for SAW statistics on percolation cluster to be six. 0
4. OUTLOOK
Here we have very briefly introduced the (lattice) SAW model of linear polymers, their configurational statistics and the (lattice) percolation model of disodered media. Approximate mean field-like and scaling arguments have been forwarded to indicate that the SAW critical behaviour on disordered lattices, percolating lattice in particular, could be significantly different from those of SAWs on pure lattices. More careful analysis, as we will see in the following chapters, show even more subtle effects of disoder on the polymer conformation statistics. Also, as we will see, such effects are not necessarily confined only to the cases of extreme disorder like percolating fractals. REFERENCES
1. P. Flory, Statistics of Chain Molecules, Cornell Univ. Press, Ithaca, NY, 1969. 2. P. G. de Gennes, Scaling Concepts in Polymer Physics, Cornell University Press, Ithaca, NY, 1979. 3. See e.g., A. J. Guttmann, in Phase Transition and Critical Phenomena, vol. 13, Eds. C. Domb and J. L. Lebowitz, Academic Press, London, 1989, p. 1. 4. B. Nienhuis, Phys. Rev. Lett. 49 (1982) 1062; B. Duplantier and H. Saleur, Phys. Rev. Lett. 59 (1987) 539. 5. See e.g., D. Stauffer and A. Aharony, Introduction to Percolation Theory, 2nd edition, Taylor and Francis, London, 1992. 6. B. K. Chakrabarti and J. Kertesz, Z. Phys B 44 (1981) 221. 7. K. Barat and B. K. Chakrabarti, Phys. Rep. 258 (1995) 377. 8. A. B. Harris, J. Phys. C 7 (1974) 1671. 9. A. B. Harris, Z. Phys. B. 49 (1983) 347. 10. M. E. Fisher, Phys. Rev. 176 (1968) 257. 11. K. Kremer, Z. Phys. B 45 (1981) 149.
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Statistics of Linear Polymers in Disordered Media Edited by Bikas K. Chakrabarti © 2005 Elsevier B.V. All rights reserved.
Directed polymers and randomness Somendra M. Bhattacharjeea alnstitute of Physics, Bhubaneswar 751 005, India email:
[email protected] The effects of two types of randomness on the behaviour of directed polymers are discussed in this chapter. The first part deals with the effect of randomness in medium so that a directed polymer feels a random external potential. The second part deals with the RANI model of two directed polymers with heterogeneity along the chain such that the interaction is random. The random medium problem is better understood compared to the RANI model. 1. DIRECTED POLYMERS
A long flexible elastic string, to be called a polymer, has several features of a critical system, showing power law behaviours without much fine tuning [1,2]. An important quantity for a polymer is its size or the spatial extent as the length N becomes large. For a translationally invariant system with one end (z = 0) fixed at origin, the average position at z = N is zero but the size is given by the rms value
(1) with v = 1/2, for the free case. Similar power laws can be defined in other properties also. In general, such exponents like the size exponent v define the polymer universality class and these depend only on a few basic elements of the polymer. In addition to the geometric properties, the usual thermodynamic quantities, e.g. free energy (or energy at temperature T = 0), entropy etc., are also important, especially if one wants to study phase transitions. Power laws generally imply absence of any length scale in the problem. Given a microscopic Hamiltonian with its own small length scales like the bond length, interaction range etc, power laws occur only in the long distance limit (large N) for thermally averaged quantities which require summing over all possible configurations. As a result, in the long distance limit, these power laws become insensitive to minute details at the microscopic level, a feature that may be exploited to choose appropriate simplified models to describe a polymer. In thermal equilibrium, the Boltzmann distribution ultimately determines the macroscopic behaviour. In most cases one may avoid the issue of probability distribution but instead may concentrate only on the first few, may be the first two, moments or cumulants. For example, one needs to know the average energy, entropy etc and the various response 9
10
S. M. Bhattacharjee
functions which come from the width of the distribution. Thermodynamic descriptions do not generally go beyond that. In random physical systems, one encounters an extra problem. If the randomness is not thermalizable ("quenched"), any quantity of interest becomes realization dependent. As a result, an additional disorder averaging has to be done over and above the usual thermal averaging for each realization. It is therefore necessary to know if and how this extra averaging alters the behaviour of the system. Critical-like systems [3] become a natural choice for studying the effect of quenched randomness because it is hoped that at least some of the properties would be insensitive to the minute details of the randomness. Since for a critical system, the influence of the randomness on a long scale is what matters, it transpires that the critical behaviour will be affected if the disorder is a relevant variable. In the renormalization group language, a coupling is classified as relevant, irrelevant or marginal, if, with increasing length scale, it grows, decays or remains invariant, because the contribution of a relevant quantity cannot be ignored at long distances even if it is numerically small to start with. With relevant disorder, there is the obvious possibility of a change in the critical properties (e.g. new set of critical exponents). More complex situations may include emergence of new or extra length scales. One may recall that approach to criticality, most often, is described by a diverging length scale. Developing a description of the system in terms of this large length scale only goes by the name of scaling theory. Emergence of any new or extra length scale would then alter the corresponding scaling description. In case it is possible to change the nature of disorder from relevant to irrelevant (say by changing temperature), then a phase transition would occur which will have no counterpart in the pure problem. For the disorder-dominated phase, on a large scale, there are possibilities of rare events (see Appendix A) which necessitates a distinction between the average value and the typical (e.g. most probable) value. In such situations higher moments of the quantity concerned become important. These are some of the aspects that make disorder problems important, interesting and difficult.
z=N
(~
z=~ Figure 1. (a) A random walk in d dimensions with z as the variable along the contour of the polymer i.e. giving the location of the monomers. (b) Directed polymer on a square lattice. A polymer as of (a) can be drawn in d + 1 dimensions. This is like a path of a quantum particle in nonrelativistic quantum mechanics. (c) A situation where both the transverse space (r) and z are continuous. (d) The directed polymers on a hierarchical lattice. Three generations are shown for 4 bonds. (e) A general motif of 2b bonds.
Directed polymers and randomness
11
The problem of a polymer in a random medium was initiated by Chakrabarti and Kertesz [4,5] with the application of the Harris criterion. This problem has enriched our overall understanding of polymers and random systems in general, but still a complete understanding remains elusive. Not surprisingly, the look out for simpler problems that capture the basic essence of the original complex system gained momentum. In this context, directed polymers played a very crucial role. Let us define the problem here. Consider a polymer where each monomer sees a different, independent, identically distributed random potentials. Geometrically this can be achieved if the monomers live in separate spaces. One way to get that is to consider the polymer to be a d + 1 dimensional string with the monomers in d dimensional planes but connected together in the extra dimension. As shown in Fig. 1, this is a polymer which is directed in one particular direction. Hence the name directed polymer [6-9]. For a directed polymer, the size would now refer to the size in the transverse d-directions and so Eq. (1) refers to the transverse size as the length in the special z-direction increases For long enough chains, it is this size that matters and enters the scaling description. The significance of directed polymer lies in the fact that the pure system is very well understood and exactly solvable in all dimensions while the random problem can be attacked in several different ways, a luxury not affordable in most situations. Two types of randomness can arise in the context of directed polymers. One type would involve imposition of a random external potential (random medium problem). In the second type, the interaction (between say two chains) is random (RANI model). In the random medium problem, the random potential would like to have a realization dependent ground state which may not match with the zero-field state. In the RANI model, the randomness in the interaction may lead to a change in the phase transition behaviour shown by the polymers. These two classes are discussed separately.
2. OUTLINE We first consider the random medium problem and then the random interaction (RANI) model. In both cases, disorder turns out to be marginally relevant though at two different dimensions. The quantities of interest in a disordered system are introduced in Sec. 3. For the random medium problem, the question of relevance of disorder, the size exponent v and the free energy fluctuation exponent (} are discussed in Sees. 4 and 5. Sec 4 deals with the renormalization group (RG) for the moments of the partition function and also the special Bethe ansatz results for the 1 + 1 dimensional problem. A Flory approach and scaling ideas are also summarized there. Sec. 5 deals with the RG approach for the free energy via the Kardar-Parisi-Zhang equation. The behaviour of the overlap especially near the transition to strong disorder phase may be found in Sec. 6 . We briefly mention the unzipping behaviour in presence of a force and summarize some recent results for the pure case in Sec. 7. More on unzipping of a two chain system may be found in Sec. 9.4. These results and scaling arguments are then used to visualize the nature of the ground state in Sec. 8. Various aspects of the RANI model can be found in Sees. 9 and 10. The question of relevance, and annealed versus quenched disorder in multi chain system are analyzed in Sec. 9. The two different types of randomness or heterogeneity on hierarchical lattices are studied in Sec. 10. In the last part of this section, one may
S. M. Bhattacharjee
12
find the general validity and extension of the Harris criterion as applied to this polymer problem. Various technical issues are delegated to the Appendixes. An example of rare events is given in App. A. A short review of the pure polymer behaviour can be found in App. B. The issue of self-averaging and some recent results about it for disordered systems are surveyed in App. C. The renormalization group approach to polymers as used in Sec. 4.2.1 is spelt out in App. D in the minimal subtraction scheme with dimensional regularization. The momentum shell RG approach for the free energy is dealt with in App. E. All the possible flow diagrams are reviewed in App. F. A short introduction to the transfer matrix approach for the directed polymer problem is given in App. G. The exact RG for the RANI model can be found in App. H.
On Notation and convention: • To avoid proliferation of symbols, we reserve the symbol f to denote an arbitrary or unspecified function, not necessarily same everywhere. • By a sample or a realization we would mean one particular arrangement of the random parameters over the whole system. For a thermodynamic (infinitely large) system the sample space is also infinite.
• Sample averaging is denoted by [... Jav while thermal averaging is denoted by ( ... ). • The Boltzmann constant is set, most often, to one, k 8
=
1.
• "Disorder" and "randomness" will be used interchangeably.
3. HAMILTONIAN AND RANDOMNESS By definition, a directed polymer is defined in D dimensions out of which one direction is special that represents the sense of direction of the polymer. It is then useful to show that explicitly by writing D = d + 1 where d is the dimension of the transverse space. One may consider various possible models of the underlying space as shown in Fig. 1. • One may consider a lattice (square lattice in the Fig. 1b) with the polymer as a random walk on the lattice with a bias in the diagonal z-direction, never taking a step in the -z direction. The length of the polymer is then the number of steps on the lattice. • Instead of a lattice model, one may consider continuum where both the space and the z-direction are continuous as shown in Fig. 1c. The polymer itself may consist of small rods or bonds whose length provides us with a small scale cut-off. In many situations, it helps to consider polymers without any intrinsic small scale cutoff. • Quite often it is useful to consider very special lattices, e.g. hierarchical lattices as shown in Figs. 1d, and 1e, because of the possibility of exact analysis. Here one starts with a bond and then replaces the bond iteratively by a predetermined motif ("diamond" of 2b bonds) and the process can be iterated ad infinitum. This is a lattice with a well-defined dimension (see Sec. 10) but unfortunately without any proper Euclidean distance. Consequently geometric properties are not useful here.
13
Directed polymers and randomness
The effective dimension of the lattice is detr = (ln 2b) / ln 2, ifthere are 2b bonds per motif. A directed polymer can be taken as a random walk on this lattice starting from say the bottom point, going up, and ending at the top end.
3.1. Pure case Taking the polymer as an elastic string, one may define a Hamiltonian
(2) which gives a normalized probability distribution of the position vector r at length z from the end at (0, 0) P( r, z )
1
= (27rz)d/2
e-r2 /2z
(Kd/kBT = 1).
(3)
Here k 8 is the Boltzmann constant and T is the temperature. One can even write down the distribution for any two intermediate points (r;, z;) and (r 1 , z f) as
(4) For the lattice random walk, there is no "energy" and the elastic Hamiltonian of Eq. (2) just simulates the entropic effect at non-zero temperatures. One needs to look at the lattice problem in case one is interested in low or zero temperature behavior. A recapitulation of a few properties of polymers is done in Appendix B. For a polymer of length N the probability distribution gives
(Kd/kBT = 1)
(5)
so that the transverse size of the polymer is given by
(6) The power law growth of the size of a polymer as the length increases is a reflection of the absence of any "length scale" in the Hamiltonian. 3.2. Random medium Let us now put this polymer in a random medium. In the lattice model of Fig. 1, each site has an independent random energy and the total energy of the lattice polymer is the sum of the energies of the sites visited. In continuum, the Hamiltonian can be written as
H = Ho
+
1N dz Jdr TJ(r, z) c5(r(z)- r) = H + 1N dz TJ(r(z), z) 0
(7)
where TJ(r, z) is an identical, independent Gaussian distributed random variable with zero mean and variance ~ > 0,
[TJ(r, z)Jav = 0, [TJ(r, z) TJ(r', z')Jav =
~c5(r-
r')c5(z- z').
(8)
14
S. M. Bhattacharjee
The averaging over TJ is to be called sample averaging, denoted by [... Jav (as opposed to thermal averaging, denoted by ( ... )). With this distribution of random energies, we see [HJav = H 0 and so the average Hamiltonian is not of much use. Disorder averaging of sample dependent thermal averages is to be called quenched averaging while disorder averaging done at the partition function level is to be called annealed averaging. We shall consider the situation with one end point (z = 0) fixed. Otherwise, the polymer may drift in the medium to locate the best possible region that would minimize its free energy. Such a case, eventhough formally quenched in nature, is tantamount to an annealed case.
3.2.1. Partition function The partition function for a polymer in a random medium or potential is given by
(9) This is a symbolic notation ("path integral") to denote sum over all configurations and is better treated as a continuum limit of a well-defined lattice partition function
z = I.:: e-f3.,(r,z)
(10)
paths
where the sum is over all possible paths of N steps starting from r = 0 at z = 0. It often helps to define the partition function such that Z( { TJ = 0}) = 1 to avoid problems of going to the continuum limit (see Eq. (3)). This is done by dividing (or normalizing) Z by Z 0 = J-lN, Z 0 being the partition function of the free walker with 1-l as the connectivity constant (= 2 for Fig. 1b). Let us define the free energy F
= -TlnZ,
(11)
for a polymer of length N where the end point at z energy may be defined as
F(r, N) = -Tln Z(r, N), when the end at z
=
=
N is free. A more restricted free
(12)
N is at r.
3.3. Unzipping and response Quite often it is useful to force a system to change its overall configuration by applying an external field The response function then tells us about the rigidity of the system against such external perturbations. E.g., a magnetic field may be applied on a magnet and the magnetic susceptibility is the corresponding response function. A similar applied force for directed polymer is an unzipping force or a pulling force applied at one end (see Fig. 2). We call such a force an unzipping force because of its role in unzipping of DNA-type double stranded polymers [10]. There could two different ensembles. One is a fixed force ensemble where one applies a force at the free end z = N and studies the change in the size and shape of the polymer or its response. The position of the end point
15
Direct.ed polymers and randomness
is given by r = - ToF(g, N)fog. The second is the fixed distance ensemble where the free end is at point r and then what is the force required to maintain it at that point. Using the constrained free enPxgy, we may write g = - T\1 F(r, N). (Here we used the same notation F to denote the free energies of the two ensembles. The arguments and the context would distinguish the two.) The two ensembles behave differently in a disordered system.1 .
Figure 2. A directed polymer with an unzipping force.
Figure 3. A blob picture of the polymer under a force. Though drawn as sphere, the z-direction is elongated with isotropy in the transverse direction.
If we consider the response of a directed polymer to the unzipping force, the response function comes from the Hamiltonian
lg2N 2 Kd .
(13)
The disorder is Gaussian-distributed as in Eq. (8). The general response function for the force is (14)
with i,j representing the components. It is known in statistical mechanics that the response of a system in equilibrium is determined by the fluctuations. 1 The inequivalence of the two ensembles is known also for pure case if the force is applied at some intermediate point [11)
S. M. Bhattacharjee
16
3.3.1. Exact result on response: pure like By a redefinition of the variables and using the c5-correlation of the disorder in the z direction, we have g2N
[lnZ(g)Jav
= [lnZ(g = O)Jav + Kd,
(15)
from which it follows that
TN
(16)
Cr= Kd'
as one would expect in a pure system, Eq. (3). And there are no higher order correlations. Two things played important roles in getting this surprising pure-like result: (i) The disorder correlation has a statistical translational invariance coming from the delta function in the z-coordinate, and (ii) the quadratic nature of the Hamiltonian. If disorder had any correlation along the length of the polymer, Eq. (16) will not be valid. The significance of the result is that the conventional thermal fluctuation, averaged over randomness, superficially does not say much about the effect of disorder. We shall see later that this innocuous result however contains important information.
3.3.2. Quantities of interest Let us list some of the quantities which are of interest for a disordered system. • A random system needs to be described by the probability distribution of various physical quantities or by the averages and moments (over realizations). The moments are useful, especially in absence of full information on the probability distribution and also for characterization of the properties of the polymer. Since there is no unique partition function, one of the important probability distributions would be of the partition function, P(Z). Any quantity of interest needs to be averaged over such a distribution. Similarly the probability distribution P(F) of the free energy is also of interest. The thermodynamic behaviour is obtained from [FJav· In case the probability distribution (over the realizations) of a thermal averaged quantity X becomes very sharp, especially in the large size limit, one may avoid doing the disorder averaging. This may happen for extensive quantities because of additivity over subsamples. Such quantities are called self-averaging. Certain aspects of self-averaging is discussed in Appendix C. • The first thing to determine is the relevance of disorder. To do so, we may write 2
[lnZJav
= [ln{[ZJav + (Z- [ZJav)}lav = ln[ZJav + [Z Ja[
j}ZJ~v + ....
2 Z av
(17)
This shows the importance of the variance of the partition function. If the variance remains small, in the limit N ---+ oo, then the polymer can be described by the average partition function which is more or less like a pure problem. Otherwise not. We see that the relevance of the disorder may be inferred from the nature of the variance of the partition function.
17
Directed polymers and randomness • For the partition function we may use the simple identity
(18) where [(ln z)m]~~ are the cumulants. In contrast to Eq. (17), it is now the fluctuations of the free energy that become important. One may introduce a scaling behaviour, namely
(19) defining a new exponent 0. Obviously, for a pure problem (} = 0. If higher order fluctuations (or cumulants) do not require any new exponent, then it is fair to expect [(ln z)m]~~ rv Nmo. This free energy fluctuation exponent is one of the new quantities required to describe the random system. • A simpler form of Eq. (18) is the basis of the replica approach for disordered systems, namely [ ln Z] av
zn]
[ = n-+0 lim
-1 av
n
(20)
.
so that to compute the average free energy we may consider a case of n-replicas of the original system or after averaging, ann-polymer problem with extra interactions induced by the disorder though an n --+ 0 limit is to be taken at the end. A few possible paths to take the limit for long chains are shown in Fig. 4.
computer /numerical (c)
~ ~path(b) 1/N
Analytical approach (a)
~
n Figure 4. Paths for replica approach
Nontrivial results are expected if and only if the origin in Fig. 4 is a singular point so that the limits n --+ 0 and N --+ oo become non-interchangeable. In other words, the n and N dependences should be coupled so that the appropriate path is a scaling path like (b) in the figure 4.
18
S. M. Bhattacharjee • If we demand that ln [Znlav is proportional toN for large N, then, apart from the extensive term (ex nN), there will be corrections which may be assumed to involve a scaling variable x = nN°. For x --+ oo, F(x) rv x 11° so that (21) This is for path (a) of Fig. 4. In contrast, if we taken--+ 0 for finite N, path (c), a Taylor series expansion gives (22) Eqs. 21,22 can be used to calculate(}, the free energy fluctuation exponent. • So far as the geometrical properties are concerned, we first note the lack of translational invariance for a particular realization of disorder and therefore (r) =/= 0, but on averaging over randomness, translational invariance will be restored statistically and so [(r) lav has to be zero. One may therefore consider the size of the polymer by the "disorder" correlations
(23) We have already seen in Eq. (16) that the usual correlation function is the thermal correlation of the pure problem and has no signature of the disorder. Of course, the disordered averaged probability distribution P(r, N) = [Z(r, N)/ drZ(r, N)Jav is also of importance.
J
• For the pure case (.0. = 0), there is no "energy", only configurational entropy of the polymer. But with .0. =/= 0, there may be one or more lowest energy states. The nature of the ground state is an important issue for any disordered problem. For the lattice problem, the energy is the sum of N (--+ oo) random energies of the sites visited. But it is the minimization over a connected path that makes the problem difficult. If the low temperature behaviour of the polymer is same as that at T = 0, the fluctuation of the ground-state energy will also be described by the exponent (} of Eq. (19). Same is true for the size exponent also. Such a situation requires that the disorder dominated phase is controlled by the "zero temperature" fixed point. • The problem we face in a disordered system is that there is no well defined ground state - the ground state is sample dependent. There is therefore no predefined external field that will force the system into its ground state (e.g., a magnetic field in a ferromagnetic problem). This is a generic problem for any random system. But, suppose, we put in an extra fictitious (ghost) polymer and let it choose the best path. Now we put in the actual polymer in the same random medium but with a weak attraction v with the ghost. At T = 0, this polymer will then sit on top of the ghost. In absence of any interaction (v --+ 0 ) , the polymer would go over the ghost in any case if there is a unique ground state, not otherwise. At non-zero temperatures there will be high energy excursions and how close to the ground state
Directed polymers and randomness
19
we are will be determined by the number of common points of the two polymers. This is called overlap which may be quantitatively defined as
(24) for a given sample i and then one has to average over the disorder samples, q
=[q;Jav·
In case of a repulsive interaction, the situation will be different. If there is more than one ground state, the two chains will occupy two different paths and there is no energetic incentive to collapse on top of each other when the repulsion v ---+ 0+. In such a scenario, the overlap q(v---+ 0+) #- q(v---+ 0-). Conversely, a situation like this for the overlap would indicate presence of degenerate ground states. For a unique ground state, the second chain would follow a nearby excited path with certain amount of overlaps with the ground state. A little reflection shows that overlap is associated with the second moment of the partition function.
4.
RELEVANCE OF DISORDER Let us first see if disorder is at all relevant.
4.1. Annealed average: low temperature problem If the disorder is irrelevant, then the average partition function [Zlav (annealed average) is expected to give the thermodynamic behaviour. However this extra averaging of the partition function may lead to a violation of the laws of thermodynamics, questioning the validity of the annealed averaging. This happens for directed polymers. With a Gaussian distribution for the random energy, from Eq. (10),
[ZJav = exp({3 2 D.N/2) exp(Nlnf.l),
(25)
= -T(ln f.l+f3 2 b./2). The entropy obtained from this partition function (S = -oF/oT) by definition has to be positive which requires T 2: TA = ~· Annealed
so that F/N
averaging will not work at very low temperatures. This does not necessarily mean that something like a phase transition has to happen, because this problem will occur for any disordered system, even finite in size. However, for the directed polymer problem, this does signal a phase transition, though the proof comes from other analysis.
4.2. Moments of Z The moments can be written as the partition function of an n-polymer problem with an extra interaction induced by the disorder. Starting from H as given by Eq. (7), and averaging over the Gaussian distribution of Eq. (8), we have
where Hn
(26)
20
S. M. Bhattacharjee
This particular form can be understood in terms of two polymers, to which one may restrict if the interest lies only in the second cumulant of Z. These two polymers start from the same point and do their random walk as they take further steps. If there is a site which is energetically favourable, both the polymers would like to be there. The effect is like an attractive interaction between the two polymers - an interaction induced by the randomness. For the many polymer problem (for the n-th moment), there is nothing beyond two polymer interaction. This has to do with the nature of correlation of disorder. 4.2.1. Bound state: two polymer problem and RG For the second moment, we have a two polymer problem. The analogy with quantum mechanics tells us that for d < 2, any binding potential can form a bound state but a critical strength is required for a bound state for d > 2. In the polymer language, this means that any small disorder will change the behaviour of the free (pure) chain for d < 2 (disorder is always relevant) but for d > de = 2, if (3tl < ((3ll)c, the chain remains pure-like (disorder is irrelevant). Actually in higher dimensions (d > 2) the delta-function potential needs to be regularized appropriately (e.g., by a "spherical" well). Such cases are better treated by renormalization group (RG) which also helps in making the definitions of relevance/irrelevance more precise. We discuss this below [12]. In short, the second term (fluctuation in partition function) in the expansion of Eq. (17) cannot be ignored if d < 2 or if (3tl is sufficiently large for d > 2. This signals a disorder dominated phase for all disorders in low dimensions or at low temperatures (strong disorder) in higher dimensions. 4.2.2. Expansion in potential We do an expansion in the interaction potential and just look at the first contributing term. The full series can of course be treated exactly. On averaging, the first order terms drop out, yielding
This is the first order term if Eq. (26) is used. A diagrammatic representation is often helpful in book-keeping as shown in Fig. 5. Some details of the renormalization group approach is given in Appendix D. 4.2.3. Reunion For this two polymer problem, the interaction contributes whenever there is a meeting or reunion of the two polymers at a site [13]. At the order we are considering, there is only one reunion but this reunion can take place anywhere along the chain and anywhere in the transverse direction. The second order term as shown on the right side of Fig. 5 can be thought of as two walkers starting from origin have a reunion anywhere, thereby forming a loop. The probability that two walkers starting from origin would meet at r at z is given by P 2 (r, z) (Eq. (3)) so that a reunion anywhere is given by a space integral of this probability which gives
nz
=I
drP 2 (r,z)
= (47rz)-w, with
w= d/2.
(28)
21
Directed polymers and randomness
Figure 5. Renormalization of the interaction. The two polymers are represented by the two lines of different thickness and an intersection represents an interaction. A heavy circle on the left hand side represents the effective interaction that is to be used for renormalization. This exponent \ll will be called the reunion exponent. The occurrence of a power law is again to be noted. The eventual renormalization group approach hinges on this power law behaviour. It should be noted that the value of the reunion exponent above is that of simple Gaussian chains. This need not be the case with interaction. For example for repulsive interaction between two directed polymers, \ll = 3/2 in d = 1 but RN rv N- 1 (ln N)- 2 in d = 2. The Gaussian behaviour is recovered in d > 2 [13]. 4.2.4. Divergences The contribution in Eq. (27) at the next one loop order (0(.0. 2 )) as shown in Fig. 5 involves, apart from some constants, an integral over the reunion behaviour given in Eq. (28). This integral in the limit N ---+ oo is N
1 a
1 dz - zd/2
rv
al-d/2
for d > 2 but "'
Nl-d/
2
for d < 2.
(29)
For a finite cut-off, as is usually the case, the integral is finite for d > 2, and therefore [(Z- [ZJav) 2 Jav ~ 0({3 2 .0.). This however is not the case ford < 2 with d = 2 as a borderline case. The divergence we find for d < 2 comes from the large N behaviour of the probability distribution and is therefore ignorant of the details at the microscopic level. In other words, a lattice model will also show this divergence in low dimensions. This forms the basis of a rigorous analysis done in Ref. [14], but we pursue a renormalization group approach here. 4.2.5. RG flows The problem of divergence we face here is ideal for a renormalization group approach. Let us introduce an arbitrary length scale L in the transverse direction and define a dimensionless "running" coupling constant
u(L)=(f3 3 K.0.)U,
E=2-d.
(30)
The purpose behind this length scale is to choose a tunable scale at which we may probe the system. One may then study the RG flow of the coupling constant as the scale L is changed.
S. M. Bhattacharjee
22
In the dimensional regularization scheme we adopt here (see Appendix D for details) the divergence seen in Eq. 29 are handled by analytic continuation in d. The problem of convergence of the integral then appears as singularities at specific d. One then tries to remove these divergences in E by absorbing them in the coupling constants, thereby renormalizing the coupling. This in a sense takes care of reunions at small scales to define the effective coupling on a longer scale. One then has to rescale the system to preserve the original length scales. With this rescaling, one ends up with a description on a coarser scale with small scale fluctuation effects getting absorbed in the redefined parameters. The fact that the process can be implemented without any need of additional parameters is linked to renormalizability of the Hamiltonian. The effect of renormalization is best expressed by the variation of the parameters or coupling constants with length scale. This is called a flow equation. For the problem in hand, the eventual flow equation is du L dL
= (2- d)u + u2 ,
(31)
where the first term follows from the definition of u while the second one is from the loop. The magnitude of the coefficient of the u 2 term is not very crucial because at this order, this coefficient can be absorbed in the definition of the u itself. What matters is the sign of the u 2 term. General cases are discussed in Appendix F. d.;;2
)(
u=O
.
d>2
• .. u=O u=lel (a)
---·-d=l
u=O
(b)
Figure 6. RG Fixed points for u. (a) Based on the second moment of the partition function. (b) Based on the KPZ equation. Arrows show the flow of u. E = 2- d. General flow diagrams are discussed in App. F. Since u emanates from the variance of the disorder distribution, it cannot be negative. We therefore need to concentrate only on the u 2: 0 part with the initial condition of u(L = a) = u 0 . What we see is that for d < 2, the flow on the positive axis goes to infinity indicating a strong disorder phase for any amount of disorder provided we look at long enough length scale. An estimate of this length scale may be obtained from the nature of divergence for a given u 0 . An integration of the flow equation gives L rv u~/(2-d) (d < 2), a crossover length beyond which the effect of the disorder is appreciable. For d > 2, there is a fixed point at u* = lEI where E = 2- d. For u < u*, the disorder strength goes to zero and one recovers a "pure" -like behaviour. This is a weak disorder limit. But, if u > u* the disorder is relevant. Based on the fixed point analysis, we conclude, as already mentioned, that disorder can be relevant depending on the dimensions we are in (i.e. the value of d) and temperature or strength of disorder. In particular, one finds
Directed polymers and randomness
23
1. A disorder-dominated or strong disorder phase for all temperatures for d :::; 2. 2. A disorder dominated or strong disorder phase at low temperatures for d > 2.
For d > 2, one sees a phase transition by changing the strength of the disorder or equivalently temperature for a given .0.. This is an example of a phase transition induced by disorder which cannot exist in a pure case. It is to be noted that the phase transition (the unstable O(IEI) fixed point) occurs because of the positive u 2 term in the flow equation ofEq. (30). At d = 2, u is marginal (no L dependence) but renormalization effects lead to an eventual growth. Such parameters are called marginally relevant. Any marginally relevant variable will produce an unstable fixed point, and hence a phase transition, in dimensions higher than the dimension in which it is marginal. A general statement can then be made: Disorder is expected to produce a new phase transition if it is marginally relevant at its critical dimension. The new phase transition (a critical point) is to be characterized by its own set of exponents. An important quantity is the length scale behaviour. The flow equation around the fixed point for d > 2 shows that one may define a diverging "length-scale" associated with the critical point as ~
rv
lu- u*l-(,
with (
= 1/12- dl.
(32)
In the critical dimension d = 2, there are log corrections. In the weak disorder phase where the disorder is irrelevant, [ln ZJav f'::! ln [ZJav' and therefore one may put a bound on the transition temperature Tc for a lattice model as Tc 2: TA as defined below Eq. (25). Attempts were made to determine (by numerical methods and verify the RG prediction. However, the results from specific heat [15] and size calculations [16] agree neither with each other nor with the RG result of Eq. (32). This remains an open problem. We come back to the strong disorder phase in Sec. 5.6.
4.3. Bethe ansatz and (} For the directed polymer problem, a mapping to a quantum problem helps in the evaluation of [znJav at least in d = 1. For a Gaussian distributed, delta-correlated disorder, [Znlav corresponds to the partition function of ann-polymer system with the Hamiltonian given by Eq. (26). Noting the similarity with the quantum Hamiltonian with z playing the role of imaginary time, finding N- 1 ln [znJav for N ---+ oo is equivalent to finding the ground state energy E of a quantum system of n particles. This problem can be solved exactly only in one dimension (d = 1) using the Bethe ansatz [23,24]. This gives the ground state energy as
E = -K(n- n 3 ) in d = 1,
(33)
which gives 1
(} =
3'
(34)
from Eq. (21) As we shall see below, this implies v = 2/3 so that the polymer has swollen far beyond the random walk or Gaussian behaviour. What looks surprising in
24
S. M. Bhattacharjee
this approach is that there is no "variance" (2nd cumulant) contribution. It is just not possible to have a probability distribution whose variance vanishes identically. This is a conspiracy of the N --+ oo limit inherent in the quantum mapping and the value of the exponent(} that suppressed the second cumulant contribution (see Eq. (21)). 4.3.1. Flory approach Using the quantum analogy, we may try to estimate the ground state energy in a simple minded dimensionally correct calculation based on the assumption of only one length scale. Such an approach generally goes by the generic name of "Flory approach". The elastic energy is like the kinetic energy of quantum particles which try to delocalize the polymers (random walk) while the attractive potential tries to keep the polymer together. For n polymers there are n( n - 1) 12 interactions. We take the large n limit so that if the particles are bound in a region of size R, the energy is (using dimensionally correct form with R as the only length scale)
(35) which on minimization gives E rv n 3 consistent with the Bethe ansatz solution. At this point we see the problem of the replica approach if the limit is taken too soon. Since our interest is eventually in n --+ 0, we could have used in this argument the linear term of the combinatorics. That would have made energy "extensive" with respect to the number of particle and replaced the disorder-induced attraction by a repulsion (note the negative sign). The end result would however have no n 3 dependence. This is a real danger and any replica calculation has to watch out of these pitfalls. Quite strangely we see that the correct answer comes by taking n --+ oo first and then n --+ 0 or, probably better to say, by staying along the "attractive part" of the interaction only. 4.3.2. Confinement energy Suppose we confine the polymer in a tube of diameter D. This is like the localization length argument used to justify the energy in the quantum formulation. The polymer in the random medium won't feel the wall until its size is comparable to that, D rv N 0 which gives the length at which the polymer feels the wall. Elastic energy of a blob is D 2 I N 0 . But because of the tube, the polymer will be stretched in the tube direction. One may then consider the polymer as consisting of free N I N 0 blobs aligning with the force, so that the energy is
(36) This gives the known form 11 D 2 used in the quantum analogy, Eq. (35), (and consistent with dimensional analysis) but for v = 213, this gives 11 D. A cross-check of this comes from the energy of a blob. Each blob has the fluctuation energy Ng and so free energy per unit length FIN"' NtiNo "'D- 2(l-v)/v.
25
Directed polymers and randomness
5. ANALYSIS OF FREE ENERGY: SPECIALTY OF DIRECTED POLYMER Another unique feature of this directed polymer problem is that there is a way to study the average free energy and implement RG directly for the free energy bypassing the n ---+ 0 problem of the replica analysis completely, giving an independent way of checking the results of replica approach.
5.1. Free energy and the KPZ equation For a polymer, the partition function satisfies a diffusion or Schrodinger-like equation. This equation can be transformed to an equation for the free energy F(r, z) = - T ln Z (r, z). This is the free energy of a polymer whose end point at z is fixed at r. To maintain the distance fixed at r a force is required which is given by g = - '\7 F. If we want to increase the length of a polymer by one unit, we need to release the constraint at the previous layer (think of a lattice). The change in free energy would then depend on the force at that point, and of course the random energy at the new occupied site. The change oF(r)/oz being a scalar can then depend only on the two scalars '\J. g and g2 . A direct derivation of the differential equation for the free energy shows that these are the three terms required. The differential equation, now known as the Kardar-Parisi-Zhang equation [7], is
oF T 1 az= 2K'\l F- 2K('\JF) 2
2
)
+ry(r,z.
(37)
If we can solve this exact equation and average over the random energy 7], we get all the results we want. One may also write down the equation for the force in this "fixed distance" ensemble as
og
r
2
1
oz = 2K '\7 g- 2Kg. '\lg + '\lry(r, z).
(38)
This equation is known as the Burgers equation.
5.2. Free energy of extension: pure like We want to know the free energy cost in pulling a polymer of length N from origin (where the other end is fixed) to a position r. For the pure case, the free energy follows from Eq. (3) (with K inserted) as
F(r, N)- F(O, N)
d
=2K
T
2
N. (pure)
The probability distribution for the free energy can be obtained by choosing g in Eq. (13) as.
P(F(rN)) = P(F(O)
+ ~~Iv).
(39)
= K drN / N (40)
One then gets the surprising result,
[F(r, N)- F(O, N)Jav =
d
2K
T
2
N. (disorder)
( 41)
S. M. Bhattacharjee
26
Therefore, on the average the stretching of a chain is pure-like (elastic) with the same elastic constant though the fluctuation is anomalous (0 =/= 0). This is analogous to the pure-like result for the correlation function, Eq. (16). These results have a far reaching consequence that in a renormalization group procedure, the elastic constant must remain an invariant (independent of length scale). As we shall see, this invariance condition puts a constraint on v and (}, making only one independent.
5.3. RG of the KPZ equation To analyze the nonlinear KPZ equation, an RG procedure may be adopted. This RG is based on treating the nonlinear term in an iterative manner by starting from the linear equation. This is a bit unusual because here we are not starting with a "Gaussian" polymer problem, rather, a formal linear equation[25] that does not necessarily represent a polymer. Leaving aside such peculiarity, one may implement the coarse-graining of RG to see how the couplings change with length scale. 5.3.1. Scale transformation and an important relation Under a scale transformation x ---+ bx, z ---+ b1 1v z, and F ---+ b01v F, the randomness transforms like .0. ---+ b-d-(l/v) .0.. This transformation done on Eq. (41) shows that for K to be an invariant (no b-dependence) we must have,
(} + 1 = 2v.
(42)
This is trivially valid for the Gaussian pure polymer problem but gives a relation between the free energy fluctuation and the size of the polymer. This is borne out by the intuitive picture we develop below. This relation gives the size exponent v = 2/3 in d = 1. The equation in terms of the transformed variables is then oF
oz
= I____b(l-2v)/v \72 F 2K
__1_b(0-2v+l)/v (V' F)2
2K
+ b(l-dv-20)/v TJ(r
z) '
(43)
The b-dependent factors can be absorbed to define new parameters, except for K. The temperature however gets renormalized as T ---+ T b(l- 2v)/v. Its flow is described by the flow equation fJT
L~
uL
1- 2v
= - - T (to leading order)
(44)
1/
For v > 1/2, T(L) ---+ 0. The disorder dominated phase is therefore equivalent to a zero temperature problem. In other words, the fluctuation in the ground state energy and ground state configurations dominate the behaviour at low temperatures in situations with v > 1/2. It is this renormalization that was missing in Sec 4.2.1.
5.4. RG flow equation The nonlinearity (or g2) contributes further to the renormalization of the temperature through the appropriate dimensionless variable u = (K.0./T 3 )L 2 -d (same as in Eq. (30). As in the previous section, the important flow equation is for this parameter u (upto constant factors). The renormalization of temperature acquires subtle d-dependence that
Directed polymers and randomness
27
introduces a new element in the flow equation. Some details may be found in Appendix E. We quote the flow equation here as
du L dL = (2- d) u
2d- 3
+ 2d u 2 .
(45)
General cases of such a flow equation are discussed in Appendix F. One immediately sees a major difference with the flow equation Eq. (31) for d = 1. Because of the change of sign of the quadratic term in Eq. (45), there is now a fixed point ford= 1 as shown in Fig. 6(b). This flow equation does not behave properly in a range d E [1.5, 2) but that is more of a problem of implementation of RG than directed polymer per se, and so, may be ignored here. Note also that no extra information can be obtained from Eq. (45) for d 2: 2 other than what we have obtained so far in Sec. 4, namely the existence of a critical point. This approach however has the advantage of getting the renormalization of temperature by u. Some details of this RG is given in Appendix E. Eq. (44) then gets modified to (46) If we now demand scale invariance at a fixed point of u, we can determine the exponent v. At d=1, the stable fixed point u* = 2 from Eq. (45) then gives the exact exponents at d=1 2
1
1/=-0=3' 3'
(47)
in agreement with the Bethe ansatz results mentioned above. To get (} the exponent relation Eq. (42) (from invariance of K) has been used. The RG results for d = 1 are expected to be exact. 5.5. Critical point for d > 2 Ford> 2, the unstable fixed point is u* = O(IEI), (d = 2- E). This gives at the critical point v = 1/2 + 0(E 2 ) indicating the possibility of
v = 1/2, and (} = 0
( 48)
to be exact. One may argue [26] for (} = 0 in the following way. At T = Tc, thermal fluctuation enables the polymer to get out of the trap set by the random potential ("ground state"). Just above the critical point, on a scale determined by the correlation length of the critical point, the random potential scaling is set by b0 with the value of(} at Tc. As T --+ Tc+, the length scale diverges and therefore the relevant energy scale would also grow with the same exponent. However a critical point implies the energy scale to be O(Tc) which is finite. These can be reconciled if (} = 0 at the critical point. One then gets v = 1/2. Though this is the same as that of a Gaussian polymer, we shall see later that the polymer has extra non-Gaussian features.
28
S. M. Bhattacharjee
5.6. Strong disorder phase ford 2: 2 Unlike the strong disorder phase at d=l, the absence of any fixed point for the strong disorder phase for d > 2 in this approach forbids quantitative results about the phase itself. In addition, the behaviour of the strong disorder phase in d = 1 can be obtained by various methods. Unfortunately, there are few concrete results in higher dimensions d 2: 2. Most reliable values of the exponents come from various numerical approaches based on the KPZ equation. A recent estimate for d = 2 is [17] 1/v = 1.67 ± 0.025, and (} = 0.229
± 0.05.
(49)
Numerical studies indicate that v decreases as d increases. A question arises about the existence of an upper critical dimension d = ducn such that v = 1/2 for d > dvcD· For example, higher loop contributions in the RG of Sec. 4.2.5 show singularities at d = 4 which could indicate d = 4 as another critical dimension. Various analytical approaches [6,18,19] suggest ducn = 4, or even nonintegral dimension [20]. But numerical simulations and other arguments [21,22,9] suggest ducD = oo. This issue is yet to be resolved. The fact that the size exponent v (often called wandering exponent) is different from 1/2 has important implications in various applications, especially for flux lines in superconductors. For example, confinement of a flux line in presence of many other flux lines would lead to a steric repulsion[27] (similar to the confinement energy in Sec. 4.3.2) and the interaction of the vortices may lead to an attractive fluctuation induced (van der Waals type) interaction[28].
6. OVERLAP In a replica approach, occupancy of different ground states may be achieved by "replica symmetry breaking" (i.e. various replicas occupying various states) but the difficulty arises from the n --+ 0 limit. In the case of directed polymer, we have argued that the degenerate states occur only rarely and therefore the effect of "replica symmetry breaking" if any has to be very small. This is why the Bethe ansatz gave correct results without invoking replica symmetry breaking. The method to compute the overlap was developed by Mukherji [29]. By introducing a repulsive potential v dzc5(r 1 (z) - r 2(z)) for the two polymers in the same random medium,
J
H =
~1N dz [(8r~;z))2 +(8r~;z))2] + 1N dzry(rl(z),z) +
1N dz ry(r2(z), z) + 1N dz vo 8(r 2(z)). 1
(50)
The free energy F(r 1 (z),r 2(z),z,v) satisfies a modified KPZ type equation
~~
=
L (2: '
"V7F-
2 ~("V;F)
2 + ry(r;,
z)) + vc5(r
1 -
r 2 ).
(51)
which can be studied by RG. The mutual interaction has no effect on the single chain behaviour but the interaction gets renormalized. The flow equation for the dimensionless
29
Directed polymers and randomness
parameter u of Eq. (45) remains the same. The exponent relation of Eq. (42) also remains valid. The interaction gets renormalized as
ov
L fJL
(1 -
(}
u)
= -v- - d + 2" v,
(52)
where v is in a dimensionless form. For the pure problem ((} = 0, v = 1/2) this reduces to the expected flow equation of Eq. (31) for repulsive interaction (u--+ -u). For overlap one needs only the first order term because we need v --+ 0. The overlap can be written in a polymer-type scaling form q = N'E f(vN-4>v), where ~ = (} - ¢v - 1. The above RG equation for v shows that the exponent ~ = 0 at the stable fixed point for u of Fig 6(b) at d = 1. However, ~ < 0 at the transition point for d > 2. This means that the overlap vanishes at the transition point from the strong disorder side as q rv IT- Tcii'EI(_ This approach to overlap can be extended to m-chain overlaps also, which show a nonlinear dependence on m at the transition point [30]. This suggests that eventhough the size exponent is v = 1/2 Gaussian like, there is more intricate structure than the pure Gaussian chain. Overlaps of directed polymers on trees have been considered in Ref. [31]. A case of cross-correlation of randomness (each polymer seeing a different noise) has been considered by Basu in Ref. [32].
7. UNZIPPING: PURE CASE Unzipping was first considered in the context of DNA [10]. However the same ideas play a role here. Let us consider a pure case of a directed polymer with one end fixed at origin and with an attractive interaction with a line at r(z) = 0 (instead of being in a random medium). The Hamiltonian for a delta-function interaction can be written as
H=~K1N dz(f)r) 2 o f)z
2
-v1N dzc5(r(z))-g·1N dz or, o
o
f)z
(53)
very similar in form with Eq. (13) except here we have an attractive interaction instead of a random medium. For the zero force case, there is a critical unbinding transition at v = Vc. For d :::; 2, Vc = 0. The pulling force would like to align the polymer in the direction of the force while the interaction would like to keep the polymer attached to the rod. At zero temperature the unzipping transition takes place at a force where the binding energy is compensated by the force term. Upto a geometric factor a, this is given by Nv = agN. At nonzero temperature, the entropic effects are to be taken into account, which may be done by using the quantum analogy. The problem can be mapped on to a quantum Hamiltonian, albeit non-hermitian, for a particle of co-ordinate r (54) in units of n(= ksT) = 1 and mass = 1, with p as momentum. For long chains N--+ oo the free energy is the ground-state energy of this non-hermitian Hamiltonian. A phase
30
S. M. Bhattacharjee
transition takes place whenever the ground state is degenerate. The analysis done in Ref. [10] shows that if the ground state energy ( i.e. the binding energy of the polymer per unit length) is E 0 , then the critical force is given by
(55) where the v-dependences of E 0 close to vc, for general d, is used. In fact if the bound state has extensive entropy, then there is a possibility of a re-entrance at low temperatures (see Sec. 9.4. This however is not possible in this case in hand.
8. NATURE OF GROUND STATES AND EXCITATIONS Powered by the quantitative estimates of the free energy fluctuation and size exponents, we now try to generate a physical picture. 8.1. Rare events We have seen that there is a low temperature region (in lower dimensions for all T) where randomness results in a new phase but the response to an unzipping force is the same as for the pure system. For the pure case as N ---+ oo the width of P(rN, N) increases. Hence the increase of Cr with N. With randomness, for T ---+ 0 we need to look for the minimum energy path. Let us suppose that there is a unique ground state, i.e. E(r N) or F(r N) is a minimum for a particular path. This tells us that as the temperature is changed, T still low, the polymer explores the nearby region so that the probability distribution gains some width which is determined by the thermal length. Susceptibility would be the width of the distribution and this is independent of N. This cannot satisfy the relation given by Eq. (42). If we invoke the the unzipping argument, then we need to exert a force exceeding the critical force to take the polymer out of the bound state and so the response to a small force (g ---+ 0) would be insignificant. The situation will not be any better even on averaging over the random samples if every sample has a unique ground state. However, it may happen that most of the samples have unique ground-states but once in a while (rare samples) there is more than one ground state which happens to be far away from each other. Suppose there are such rare samples, whose probabilities decay as N-", where the paths are separated by Nv, then the contribution to the fluctuation from these samples would be N 2v-". In case r;, = 2v - 1, we get back the exact result. The relation of Eq. (42) tells us r;, = 0. The rare events control the free energy fluctuation. From the unzipping point of view, the threshold in such rare cases is at zero force because a small force can take a polymer from one ground state to another one, gaining energy in the process. Following Ref. [33], one may argue that the gain in energy from the force should be similar to the energy fluctuation. Assuming a scaling of the force, g rv gNu, then §N(J NV rv N° which gives (J = (}- 1/ = 1/3 in d = 1. This argument implies that if the average stretching is proportional to g and to N, i.e., [ < r > lav rv gN, then one should get a linear plot if [ < r > lav/ Nv is plotted against gNv-O. The surprising feature is the sample dependence of such a plot. For a directed polymer, these quantities can be obtained by a transfer matrix calculation which is exact for a given sample and finite N. Some details on the transfer matrix approach are given in Appendix G. As shown in Fig.
31
Directed polymers a.nd randomness
7, one sees steps with an overall linear dependence. The susceptibility within a flat step is zero as seen in the plot of the fluctuations.
16 4
1024 - 1024 ..............
~
2
12
N-
1024 1024 2048 2048
-.............. ---
II )(
'ii
v
0
~
)(
v
N
-2
8
><
:!..
4
-4 -6
-4
-2
2
4
6
-4
·2
0
g
2
4
6
Ll/3
Figure 7. (a) Plot of < r > fN" vs gN"- 6 ford= 1. Two different values of N and in different realizations of disorder. (b) Corresponding fluctuation in position. From Ref. (34]. What we see here is that though the average behaviour is the same as that of the pure system, the underlying phenomenon is completely different; the average thermal response is determined by the rare samples that have widely separated degenerate ground state and the probability of such states also decays as a power law of the chain length. This picture also shows the ensemble dependence. What we have discussed is the fixed force ensemble. In the fixed distance ensemble, if we keep the end point fixed at r and try to determine the force to maintain it at that point, then by definition, the force comes from a small displacement around r . Such small displacements will never lead to the big jumps that ultimately contribute to the average susceptibility. This difference in behaviour in the two ensembles is one of the important features of quenched randomness. In a given sample elastic energy "' r 2 f N "' N 2"- 1 _ The pinning energy would also grow with length say as N 9. We see, 0 = 2v - 1 = 8. One way to say this is that the sample to sample fluctuation and the energy scale for a given sample are the same. These results can now be combined for an image of the minimum energy paths. If the end points at z = N are separated by r « N", the paths remain separated (p_ach path exploring an independent. disordered region) until they join at !:lz "' r 11", after which they follow the same path. If the end points are separated by a dist.ance r ~ N" , the two paths explore independent regions and they need not meet. This picture (Fig.8) is often alluded to as the river-basin network. 8.2. Probability distribution For a pure polymer, the probability distribution of the end point is Gaussian but it need not be so for the disordered case. One way to explore the probability distribution is to study the response of the polymer as we take it out of its optimal or average position, e.g.
32
S. M. Bhattacharjee
Figure 8. Various paths for various locations of the end point.
by applying a force. In a previous section we used the fixed distance ensemble where the end point was kept fixed and we looked at the force g required to maintain that distance (see Eq. (38)). Here we consider the conjugate fixed force ensemble. 8.2.1. Response to a force Let us apply a force that tries to pull the end of the polymer beyond the equilibrium valuer ,. . ., R 0 • In equilibrium, the average size R or extension by the force can be expressed in a scaling form
(56) This is because for zero force one should get back the unperturbed size while the force term may enter only in a dimensionless form in the above equation where the quantities available are the size R 0 and the thermal energy. For small g, linearity in g is expected. This requirement gives R
=
Ro
R 0 Tg
(k 8
= 1),
(57)
R is not proportional to N if v -=f. 112 (Ro ,. . ., Nv). The polymer acts as a spring with T I R6 as the effective spring constant. 8.2.2. Scaling approach Let us try to develop a physical picture and the corresponding algebraic description (called a scaling theory). The polymer in absence of any force has some shape of characteristic size R 0 . The force stretches it in a way that it breaks up into blobs of size ~9 = T I g. For size < ~9 the polymer looks like a chain without any force and these blobs, connected linearly by geometry, act as a "new" polymer to respond to the force by aligning along it. We now have two scales R 0 and ~9 = T I g. A dimensionless form is then
R6 (Ro) rv-g T
R=Rof~g
(58)
33
Directed polymers and randomness Now each blob is of length N 9 so that ~9 expect
R
= N; and there are N / N 9 blobs. We therefore
= : ~g = N~:-1/v = N ( ~) (1-v)/v
(59)
g
This gives a susceptibility X = oR/ og ,. . ., g(l- 2 v)fv
.
8.2.3. Probability Distribution Let us try to get the susceptibility of Eq. (59) in another way. Let us assume that the probability distribution for large R is
P(r),....., exp (-(r/R 0 ) 6 ).
(60)
The entropy is given by S(r) = -lnP. The free energy in presence of a force which stretches the polymer to the tail region is given by
F
= T(r/ R 0 ) 6 -
(61)
gr.
This on minimization gives -
T
g--
)6-1
( r
Ro
-
(62)
Ro
By equating this form with Eq. (59), we get
8=-11-v
(63)
and
P(R) ,. . ., exp -
[ (; 0)
1/(1-v)]
(64)
•
For v = 1/2 we do get back the Gaussian distribution. The above analysis, done routinely for polymers, relies on the fact that there is only one length scale in the problem, namely, the size of the polymer. If we are entitled to do the same for the disorder problem, namely only one scale, R 0 ,....., Nv, matters, then the blob picture goes through in toto. The chain breaks up into "blobs" and the blobs align as dictated by the force. Each blob is independent and the polymer inside a blob is exploring its environment like a directed polymer pinned at one end. The probability distribution is therefore given by Eq. (64) which for d = 1 is
P(r) "'exp( -lrl 3 /N 2 )
(d = 1).
(65)
=
If we use the relation !::..F F(x,N)- F(O,N) ,. . ., x 2 /N, then the above probability distribution can be mapped to the distribution of the free energy as
P(F)
rv exp [- (
I!::..FI)1/2(1-v)] NO
rv exp
( I!::..FI3/2) - N1/3
(d = 1).
This has been tested numerically [6]. See, e.g. Ref. [35] for more recent work.
(66)
34
S. M. Bhattacharjee
9. RANDOM INTERACTION - RANI MODEL So far we have been considering the problem of random medium. A different situation arises if there is randomness in the interaction of polymers. This is the RANI model [36,37]. Consider the problem of two directed polymers interacting with a short range interaction as in Eq. (51) but take v to be random. Such problems are of interest, especially in the context of DNA where the base sequence provides heterogeneity along the chain. In this DNA context, the randomness can be taken to be dependent only on the z coordinate and not on others like the transverse position r. It can be written as Hint=
1N
dz Vo [1
+ b(z)]
8(r1(z)- r2(z)),
(67)
where the randomness in introduced through b(z). We take uncorrelated disorder with a Gaussian distribution (68) This would correspond to uncorrelated base sequence of a DNA problem. The full Hamiltonian can be written as H
=
21 1
N
dz
iJr1 Z ; )) 0
[
(
Or2( 2+ ( --az-1) 2] + Z
1 N
dz vo [1
+ b(z)]
V(r12(z)).
(69)
where ri(z) is the d-dimensional position vector of a point of chain i at a contour length z, and r 12 (z) = r 1 (z)- r 2 (z ). Though written for a general potential V(r), we shall consider only short-range interaction that, when convenient, can be replaced by a 8-function. 9.1. Annealed case: two chain As expected, the average partition function (annealed averaging) would correspond to a pure-type problem. This however is not the case always as we see in Sec. 9.3 for more than two chains. A straightforward averaging of Z = J Dr 1 Dr 2 exp( -H) using the probability distribution of Eq. (68) defines an effective Hamiltonian 1letr such that
[ZJav =
J
(70)
Dr1 Dr2 exp(-1f.etr),
and it is given by 1letr
=
~ 1N dz
t (iJr~;z))
2
+ Vo
1N
dz V(r12(z))-
v526.
1N
dz V2(r12(z)).
(71)
It appears from the above expression of the effective Hamiltonian that an attraction is generated between the two chains. Now, since any short range potential under renormalization maps onto a 8 function potential, we can take the "minimal" effective Hamiltonian for (Z) as
(72)
Directed polymers and randomness
35
where ii 0 is the reduced coupling constant which takes care of the attraction described earlier. We believe that the large length scale properties as described by Eq. (72) is same as that of Eq. (71). If necessary, we can restrict the strength of the disorder so that ii0 , which represents the effective coupling between the two chains, is positive (i.e. repulsive interaction). Now the problem reduces to a relatively simple situation where the two chains interact with a pure 8-function interaction with a reduced coupling constant ii0 . The solution of this pure problem is known [12] as discussed in the Appendix D.
Figure 9. (a) Inter-replica interaction in the RANI model. The dotted wavy line indicates the "r" -type interaction between the pairs (1,2) and (3,4). (b) indicates a loop formed by the disorder induced interaction.
9.2. Marginal Relevance of disorder When we consider the second cumulant of the partition function, we require four chains. On averaging over the disorder, a new "inter-replica" interaction is generated that couples the original chains with the replica, (73) with r0 = v5~ and 3, 4 representing the two new chains, and r;j(z) = r ; - rj. This is a special four chain interaction in the sense that this interaction favours a contact for chains 3 and 4 at z if chains 1 and 2 also enjoy a contact there, though not necessarily at the same point in the transverse space (see Fig. 9). In addition to this four chain interaction, the disorder also creates an effective two body attraction that changes the starting or bare interaction. So far as the effective two body interaction is concerned its effect on the long length scale is given by the RG flows of Eq. (31) except that u can now be negative also. Assuming that we are at the critical point of this two body interaction, e.g. for low dimensions, at u* = 0, the effect of the disorder can be obtained from the RG flow of r 0 • Defining the dimensionless coupling constant r0 = r0 L 2'' where E1 = 1- d with r(L) as the dimensionless running coupling constant, the RG equation is given by [36,37]
dr 2 LdL =2(b+r ).
(74)
There are two fixed points (i) r = 0, and (ii) r* = -E1. A nontrivial fixed point becomes important for E1 < 0, i.e., ford> 1. See Appendix H for details.
S. M. Bhattacharjee
36
We see certain similarities of disorder or randomness becoming marginally relevant at some dimension: de = 2 for the random medium problem, but de = 1 for the RANI problem. A new fixed point emerges above this critical dimension. For the random medium problem, this implies the existence of a new phase and a disorder induced phase transition, but for the random interaction problem it defines a new type of critical behaviour. These results based on the exact RG analysis [36,37] were later on also recovered from a dynamic renormalization group study [38]. 9.3. Annealed case: three and four chains With pairwise random interaction, the RANI problem can be formulated for more than two chains also. In such cases, even the annealed averaging problem is not just the equivalent pure type problem [37]. The Hamiltonian for a four chain system is given by
H
=
1 {N
2 Jo
4
dz ~
( ()
( ) ) 2
r~zz
{N
+ Jo
4
dzv 0 (1
+ b(z)) ~ 8(rij(z)),
(75)
i<j
where rij = ri(z)- rj(z). After averaging of the partition function, using the Gaussian distribution of b(z), one gets the following effective Hamiltonian
1letr
=
1 {N
2 Jo
4
dz ~
(or(z))2
Tz
{N
+ v0 Jo
4
dz
~ 8(rij(z)) i<j
L 8(rij)8(rjk)- 2v~ll 1N dz i,j,k
0
t
8(rij)8(rkt).
(76)
i,j,k,l=l
i<j
i<j
The remarkable feature of the effective Hamiltonian is that there are now two new types of attractive interaction one of which involves three chains, while the other one couples four chains together, similar to the 2-chain quenched problem. If we take a three chain system, the corresponding effective Hamiltonian will involve (76). There is now only the three chain term but no four chain interaction of Eq. the possibility of a disorder induced multicritical behavior. The four chain attractive interaction is marginal at d = 1 and so is the three chain interaction. The presence of these two marginal operators (at d = 1) however remain decoupled mainly because for number of chains involved) do not directed polymers, higher order interactions (order renormalize lower order interactions. This has already been seen in the overlap problem for the random medium case in Sec. 6. Therefore the resulting renormalization of the two new couplings are independent of each other, and, in fact they are the same by virtue of the nature of the interaction. Because of the four body interaction, we expect a disorder induced criticality as for the two chain quenched case, but here this happens for a real four chain system - no replica is involved.
=
9.4. Unzipping If we consider a pure problem, there will be an unzipping transition as described in Sec. 7 above. The transition is first order in nature. In this particular case there is no thermal unbinding in one dimension but if we forbid crossing of the polymers, then a
Directed polymers and randomness
37
thermal unbinding is possible. On a lattice this model can be solved exactly with a phase boundary given by [39] gc(T)=Tcosh- 1
1 -1 ) . ( )1- exp(v/T)- 1 + expv/T
(77)
as shown in Fig. 10. One of the noteworthy features of the phase diagram is the reentrance in the low temperature region [39,40]. At low temperatures, the unzipped chains are stretched by the force so that on a lattice their entropy is exponentially suppressed. Though one gains energy by unzipping, one loses both the binding energy and the entropy of the bound chain. If ln J.lB is the entropy per site of the bound state in the ground-state then, the transition occurs at gc = v +T ln J.lB with a positive slope. In the case of a single chain interacting with a rod, the bound state has no entropy (J.ls = 1) and therefore no re-entrance. This argument also shows that the transition is first order.
g 0.6 Zipped
0.2 OL-~--~--~--~~--~--~
0
Tlv
2
3.5
Figure 10. Unzipping phase diagram (exact) for two mutually avoiding directed polymers with an attractive interaction v in 1 + 1 dimensions. Note the reentrance at low temperatures. From Ref. [39].
It has been argued that the unzipping transition for the quenched averaged RANI case is second order [41]. However for real DNA, it is not the quenched averages that matter. There is strong ensemble dependence and sample to sample variation. This has been exploited to identify point mutants by a comparison of the unzipping force in a fixed distance ensemble [42]. An experimental determination of the unzipping phase boundary for a real DNA has been reported in Ref. [43]
10. HIERARCHICAL LATTICE Useful information can be obtained from models amenable to exact analysis even if they look artificial. Real space renormalization group approach can be handled in an exact fashion for a class of tailor-made lattices called hierarchical lattice. Such lattices are constructed in a recursive fashion as shown in Fig. 1. The problem of a directed polymer in a random medium on hierarchical lattices has been considered in Ref. [44,45]. Here we consider the RANI problem on hierarchical lattices. As already noted, the effective
S. M. Bhattacharjee
38
dimensionality is detr = (ln 2b) / ln 2 for a motif of 2b bonds. Two different situations can arise, the polymers interacting on shared bonds [46] or on contacts at vertices [47]. Two cases are different. Aperiodic variation of the interaction has also been considered [48]. 10.1. Randomness on sites Let us now consider interaction on the vertices. The problem is different from the bond case because by construction there are sites with large degrees. In other words, all sites are not equivalent, unlike the bond case where all bonds are on the same footing. Let Z 11 be the partition function of a given realization of randomness and let S 11 = bL~-l the number of single-chain configurations, at the 11th generation. Here L 11 = 211 is the length of a directed polymer. We define Z 11 (n) = [Z~lav/s;n, to factor out the free chain entropy. We call Z 11 (n) the moments. For a given temperature, there is a critical value n = nc('il) below which all moments are in their high temperature phase, in the thermodynamic limit of course (over bar indicating disorder average). In this limit, Z 11 ( n) approaches a fixed point value for n < nc, whereas, for n > nc, the moments diverge but with a finite "free energy" density f 11 (n) = (nL11 )- 1 ln Zw The approach to the thermodynamic limit can be written generically as
Z 11 (n) 1 1n = Z(n) 11n
+ Bz(n)L;1/J + ...
(78)
where Z(n) is the thermodynamic limit (11 ~ oo) and Bz(n) is the amplitude of the finite size correction. For a given realization of disorder, the partition function can be written as (see Fig.1)
(79) The first term originates from the configurations that require the two directed polymers to meet at a vertex. In contrast, the second term counts the "no encounter" cases. There are no energy costs at the two end points. The Boltzmann weight is random and for a 2 Gaussian distribution of energy, ym = ym • The moments of the partition function can be written as n
Z 11+l(n)
= b-n L
Pnmz;(m)
with Pnm
= (:) (b-
1)n-mym,
(80)
m=O
and the initial condition Z 0 ( n) = 1 for all the moments because there is no interaction in the zeroth generation (one single bond). By iterating the recursion relations, the moments can be calculated exactly to arbitrary precision from which Bz(n) can be estimated. One finds that Bz(n) blows up as
(81) with r = 0.71 ± 0.02. This exponent is independent of temperature but depends on b. As mentioned, in the site version there are special sites with large degree. Whether such vertices lead to a Griffiths' type singularity needs to explored. Griffiths singularity in the context of random interactions has recently been discussed in Ref. [49].
Directed polymers and randomness
39
10.2. Bond randomness: Problem with Harris criterion According to the Harris criterion, disorder is irrelevant at a critical point if the pure specific heat exponent a < 0. We show that a more general criterion is required for the directed polymer problem [46]. We place two interacting polymers on a hierarchical lattice. They start from one end (bottom) and meet at the other end (top). See Fig. 1. There is an attractive interaction -v( v > 0) if a bond of the lattice is shared by two polymers. As in real space, here also the polymers undergo a binding unbinding transition forb> 2. Randomness is introduced by allowing the interaction energy to be random on each and every bond. The first model, model A, has independent random energy on all the 2b bonds. The randomness in the second model, model B, is taken only along the longitudinal direction so that equivalent bonds on all directed paths have identical random energy. Model B is a hierarchical lattice version of the continuum RANI model. The pure problem can be solved easily by a Real space renormalization approach where one needs only the renormalization of the Boltzmann factory = exp(v/T). Let Yn, Zn, and En be the renormalized weight, partition function and energy at the nth generation. By decimating the diamonds the recursion relations are given by
Yn-1 Zn+l
+ b- 1)/b, (z; + b- 1)/b,
En+l
2 Z?;En --b Zn+l
(y;
(82) (83) (84)
The two fixed points of the quadratic recursion relation, Eq. (82), are 1 and b-1 of which the larger one is the unstable fixed point representing the transition point. Since y > 1 there is a transition at Yc = b- 1 only for b > 2. The other fixed point y* = 1 corresponds to the high temperature limit. The length scale exponent (, and the specific heat exponent, a, can be obtained from Eqs. (82)-(84) as ln 2 (=ln[2(b-1)/b]' anda= 2 -(.
(85)
Note that a < 0 for b < 2 + v'2. It is clear that hyperscaling holds good with d = 1 and not the effective dimension detr of the lattice. It is also gratifying to observe that the same v and a describe the finite size scaling form of EnFor the disordered case, the recursion relation for the Boltzmann weight can be written as (86) where y~ij) is the Boltzmann weight in the nth generation for the disorder on the upper (j = 1) or lower (j = 2) part of the ith branch as in Fig. 1. To understand the effect of the disorder at the pure critical point we introduce a small disorder y~ij) = Yc + E~j). The average of the disorder, [slav' acts like the temperature as it measures the deviation from the pure critical point. The second moment is the measure of disorder. In principle, one
40
S. M. Bhattacharjee
should look at the variance of E , but at the pure critical point the variance would be the same as the second moment. Since our motivation is to find the flow of the disorder at the pure fixed point, we need only study the first two moments, starting from a finite and small variance. The crossover exponent for the disorder is defined through the homogeneity of the singular part of the free energy in terms of the scaling fields J-1 1 (temperature) and J-1 2 (disorder). Under decimation, the free energy behaves
(87) defining 4> which can be obtained from the RG recursion relations for the first two moments. This crossover exponent determines the relevance of disorder at the critical point and can be computed exactly for both models A and B. One finds a striking difference between the two models as ln[2(b- 1) 2 /b2 ]
ln(2y~/b 3 )
4>
= ln[2(b- 1)/b]
(model A),
and 4>
= ln[ 2 (b _ 1)/b] (model B).
(88)
For model A, 4> is negative for all b > 2, implying irrelevance of disorder and 4> -=/- a but it is equal to 2- detrV, while for model B, 4> =a and not 2- deffV. Since the randomness in model B is highly correlated, the Harris criterion is less expected to be valid here as opposed to model A. but it turns out to be so. In order to construct a general framework for checking the validity of the Harris criterion, we start with the Taylor expansion of the recursion relation of Eq. (86),
E = g.(b)(El
+ E2 + ... EN)+ O(s;) + ... ,
(N = 2b)
(89)
which defines g.(b). Simple arguments show that g.(b) determines a whose positivity is guaranteed if g.(b) > v'2N- 1 Now, suppose that the bonds are grouped inn sets with N; bonds in the ith group such that the members of a set have the same randomness. Obviously 2:: N; = N. Starting with a narrow distribution, the relevance of the disorder at the pure transition then requires
(90) Hence, the Harris criterion holds good if either
g.(b) >max [ v'2 N,
JL:1 Nl ] , or g.(b) < mm. [v'2 N, JL:1 Nl ] ·
(91)
For model A, N = 2b and N; = 1 Vi, while for model B, n = 2 with N; = b. If the disordered models are classified by ± according to the sign of a, and I or R according to irrelevance or relevance of disorder, then the Harris criterion suggests the existence of only two classes (+R) and (-I). On the other hand, the above inequalities allow special classes like (+I) and (-R) where the Harris criterion fails. Model A is in the (+I) class forb> 2 + ,J2. Model B is in either the (+R) or (-I) class depending on b. It is possible to construct models that would belong to any of the four possible classes, especially (-R) [46].
Directed polymers and randomness
41
11. SUMMARY The behaviour of a directed polymer in a random medium in 1+1 dimensions seems to be well understood. There is a strong disordered phase at all temperatures for d < 2. For d > 2 renormalization group analysis shows a phase transition from a low temperature strong disordered phase to a weak disorder, pure-like phase. There are rare configurations with degenerate widely separated ground states, giving a contribution to "overlap", and strong sample dependent response to an unzipping force. The RANI model remains less understood compared to the random medium problem. Exact renormalization analysis establish the marginal relevance of the disorder at d = 1, indicating a disorder dominated unbinding transition in d 2: 1. Several features including a generalization of the Harris criterion for this criticality via relevant disorder and aspects of unzipping have been discussed. In both cases of random medium and random interaction, many issues still remain open. Acknowledgments: The author thanks Rajeev Kapri and Soumen Roy for many useful comments on the manuscript. APPENDIX A. Typical vs. average Consider a random variable x that takes two values
X1 = ea..JN and X2 = ef3N, {3 > 1,
(92a)
with probabilities P1 = 1- e-N, and p2 =e-N.
(92b)
In the limit N ~ oo, the average value [xlav ~ e(f3-l)N while the typical or most probable value is X= xl with probability 1. On the other hand [lnxlav ~ affi in the same limit, showing that [ lnxJav is determined by the typical value of the variable while the moments are controlled by the rare events. Note that this peculiarity disappears if x has a smooth probability distribution in the sense of no special or rare events. B. Pure polymers The universality of the "Gaussian" polymer is actually a consequence of the Central limit theorem. Suppose we construct a flexible polymer from bonds with independent probability distribution 'lj;(r) for a bond vector r. The end-to-end distance is given by R = 2::; r; so that the probability density of R is
P(R) =
In I IT'
dr; 1/J(r;)D(L r;- R)
' dr; 1/J(r;) exp(ik · r;) exp( -ik · R)dk
(93)
'
~[~(k)]N exp( -ik · R)dk,
(94)
S. M. Bhattacharjee
42
where ,b(k) is the Fourier transform of 1/J(r). For a symmetric distribution with finite variance, ln ,b(k) ~ 1- Ak 2 /2 ... , which on integration over k leads to a Gaussian distribution. This is valid for a lattice model also. With the probability distribution of Eq. (3), the entropy in a fixed distance ensemble can be written as
S(r) = S(O)-
1 r2
2 6' R
which gives F(r)
= F(O) +
1 Tr 2
2 R6.
(95)
This identifies an effective spring constant for the polymer, namely 3T/2R6. This spring like behaviour is purely an entropic effect. An important and general point is to be noted. The macroscopic quantity involves an "external" parameter like r which is scaled by Ro the characteristic long-length scale size of the polymer. That the long distance behaviour can be described by a single length scale is the basis of "scaling" approach to polymers. Another approach to scaling is to study the changes in the properties of a polymer as the microscopic variables are scaled. E.g., if we make a scale change, r ~ br and z ~ b11v z, the Hamiltonian of Eq. (2) remains invariant if v = 1/2. Under such a transformation, the size behaves as
Ro(N)
= b- 1 Ro(b 11v N),
(96)
so that by choosing b = N-v, R 0 ,....., Nv, i.e., the size exponent becomes v = 1/2. In presence of interaction or disorder, v may not be obtained so simply and then renormalization group methods become useful. B.l. Scaling approach in presence of a force If the polymer is now pulled with a force g, keeping the end at z = 0 fixed, the polymer would align on the average with the force. The polymer can be thought of as consisting of blobs within each of which a polymer can be considered as free from the force whereas the blobs as unit form a still coarse grained model that shows stretching. This is shown schematically in Fig. 3 and is used in Sec. 8.2.1. This description is called a "blob picture". This picture essentially depends on the scaling idea that Ro is the only relevant scale for the macroscopic description. This can definitely be justified at the Gaussian level. The partition function with the unzipping force can be written as Z =
J
drP(r, N) exp(f3g. r),
(97)
where P(r, N) is Eq. (3). The Gaussian integral can be done (keeping T explicitly in Eq. (3)) to obtain
I< r >I=
R2
,J g,
(98)
which is consistent with the idea of an effective spring constant of the polymer derived after Eq. (95). A scaling approach to derive Eq. (98) would as follows. Let R 9 be the characteristic size of the polymer in presence of the force. Then, from a dimensional analysis point of
Directed polymers and randomness
43
view, this is similar to the zero force size R 0 • From the nature of the force term, g is dimensionally like T divided by a length scale. Only lengthscale in the problem is R 0 • Hence a dimensionally correct form is
R9 "'R 0 f(gR 0 IT).
(99)
Note the absence of any microscopic scale in the above form. All microscopies go in R 0 . For a linear law at small force we require f(x) "' x (x ~ 0) giving back Eq. (98). One may rephrase this by saying that the force has a characteristic size ~9 "'T I g. If the polymer is confined in a tube of diameter D then the dimensionless variable is R 0 I D. This will appear in the form for change in entropy or in "confinement energy". This is used in Sec. 4.3.2. C. Self-averaging Let us build a large system by adding blocks A, B, C, D, ... systematically so that an extensive quantity is a sum over its values on individual blocks. In case this averaging over blocks leads to a very sharp probability distribution, then no further disorder averaging is warranted, i.e., any large sample would show the average behaviour. A quantity with this property is often called self-averaging. This may not be the case if the distribution is broad especially in the sense discussed in App. A. A self-averaging quantity has the advantage that one may study one realization of a large enough system without any need of further disorder or sample averaging. For numerical simulations, the statistics of a non-self-averaging quantity cannot be improved by increasing the number of realizations. To be quantitative, let us choose a quantity M which is extensive meaning M = N m where m is the "density" or per particle value. This is based on the additive property over subsystems M = 2:: M;. For a random system we better write M = M(N, {Q}), with {Q} representing all the random variables. To recover thermodynamics, we want [Mlav to be proportional to N for N ~ oo. Now, if it so happens that for large N
M(N, {Q})
~
Nmd,
(100)
with md independent of the explicit random variables, then M is said to have the self averaging property. Note that no averaging has been done in Eq. (100). One way to guarantee this self-averaging is to have a probability distribution (101) This is equivalent to the statement that the sum over a large number of subsystems gives the average value, something that would be expected in case the central limit theorem (CLT) is applicable. This generally is the case if quantities like M for the sub-blocks are independent and uncorrelated random variables. For many critical systems CLT may not be applicable and self-averaging is not selfevident. A practical procedure for testing self-averaging behaviour of a quantity X is to study the fluctuations CYJv = [X 2 Jav - [XJ~v and then check if Rx,N
IJ2
N]2 ~ 0, as N ~ oo. X av
= [
(102)
S.M. Bhattacharjee
44
A quantity is not self-averaging if the corresponding R does not decay to zero. The central limit theorem would suggest Rx,N "' N- 1 , while a decay of Rx slower than this would be termed as '"weakly" self-averaging. We may then classify a quantity X, based on the large N behaviour, as follows: Rx,N
rv rv rv
constant N- 1 N-P with 0 < p < 1
non self-averaging strongly self-averaging =? weakly self-averaging.
=? =?
Recent renormalization group arguments seem to suggest that if disorder is relevant then at the new (disorder-dominated) critical point thermodynamic quantities are not self-averaging [50]. The arguments leading to this extremely significant prediction of non self-averaging nature of critical quantities can be summarized as follows. Let£;= IT- Tc(i, N)I/Tc be a sample dependent reduced temperature where Tc(i, N) is a pseudo-critical temperature of sample i of N sites with Tc as the ensemble averaged critical temperature in the N ~ oo limit. In terms of this temperature, a critical quantity X is expected to show a sample dependent finite size scaling form (103) where p characterizes the behaviour of [XJav at Tc, ( being the length scale exponent. (E.g. p = '/ / d( when X is the magnetic susceptibility X· This is plausible because the critical region sets in when the size of the system is comparable to the correlation volume ~d which diverges as IT- Tc 1-(. The RG approach validates this hypothesis of Eq. (103), especially the absence of any extra anomalous dimension in powers of N for Rx. Incidentally, this hypothesis, Eq. (103), excludes rare events that may lead to Griffiths' singularity. Using this scaling form, the relative variance Rx at the critical point or in the critical region can be determined as (104)
where [(8Tc) 2 ] is the sample average variance of the pseudo-critical temperature. A finite size scaling form for Rx can also be written down, but it is not required here. A random system can have several temperature scales, namely (Tc(N) - Tc) and (TTc) "' ~- 1 /(, in addition to the shift in the transition temperature itself. For a system with relevant disorder, all these scales should behave in the same way so that typical fluctuations in the pseudo-critical temperature is set by the correlation volume (~d). In the finite size scaling limit ~d"' N, and then (relevant disorder)
(105)
An immediate consequence of this is that Rx approaches a constant as N ~ oo indicating complete absence of self-averaging at the critical point in a random system. For a pure type critical point (irrelevant disorder) a < 0 where a is the specific heat exponent (i.e. c"' IT- Tel-") of the pure system. In this case the fluctuation in Tc is set by the size, i.e. (irrelevant disorder)
(106)
45
Directed polymers and randomness
so that, by using the hyperscaling relation 2- a= d(, one gets 0 < p = la/(1 < 1 where Rx "'N-P. Hence all critical quantities in this case are weakly self-averaging. Moreover, it is the same power law involving a and (, for every critical quantity X no matter what its critical exponent is. These predictions have been verified for various types relevant and irrelevant disorders [51-53]. Exception to such non self-averaging behaviour with relevant disorder occurs if the Tc distribution approaches a delta function for large lattices. In such a situation, one gets back strong self-averaging behaviour [54].
D. Details of RG for polymers: Dimensional regularization Some details of the renormalization group approach for polymers as done in Sec. 4.2.1 are given here [12]. We consider the problem of two interacting directed polymers and study the second virial coefficient. The second virial coefficient is related to the two-chain partition function with all the ends free. Dimensional regularization is to be used here. For this appendix we take a simpler form of the Hamiltonian given be Eq. (26) as
H2 =
d {N
2 Jo
{N L (Or;)2 oz - vo Jo dz 8(r (z)- r (z)), 2
dz
1
(107)
2
' where v 0 is the bare or starting interaction strength. By introducing an arbitrary length scale L (may be the scale chosen to study the system or in momentum shell approach, this is the cutoff length), we may define the dimensionless variables u 0 = v0 U/(27r)dl 2 and N = N/ L 2 , withE= 2- d. For long distance properties, we want L to be large. By definition, the second virial coefficient comes from the connected two chain partition function with all the ends free. An expansion in terms of the coupling constant would involve polymer configurations as shown in Fig. 5. This is incidentally identical to Eq. ( 27). Each line represents the probability of free polymer going from r, z to r1 , z, 1
2
1 1 1 ( (r -r) ) G(r 'z lr, z) = [27r(z1- z)]d/2 exp - 2(z1- z)
(108)
as in Eq. (4). A crossing of the lines in a diagram represents an interaction at (r, z) which can take place anywhere requiring an integration over rand z E (0, N). Since G is normalized, the spatial integration over the free end points lead to unity and so the dangling legs of the diagrams do not require any evaluation. One needs to do only the loop integrals. For example, the one loop diagram shown in Fig. 11 corresponds to
1N 1z' dz
1
xI
dz
I I dr
dr1 G(rzlr 10)
1 11 dr G 2(r z lrz)
I
dr 2 G(rzlr 20)
I
11 dl\ G(i\Nir z )
I
11 df: 2 G(r 2Nir z ), (109)
which after integrations over the end coordinates r 1 , r 2 , r 1 , r 2 reduces to an integral of the type
1 1 II z'
N
dz
1
dz
dr
1 2 11 dr G (r z lrz)
(110)
46
S. M. Bhattacharjee
(ijNR
(~N)
(r',z') (r,z) (rpO)
(r2 ,0)
Figure 11. The one loop diagram for two polymers. The labels denote the position and the z of the points. There are integrations over all these free coordinates.
so that integrations over the space coordinates lead to integrals of the type given by Eqs.28, 29. Using d as a continuous variable, one can write N
1
dz z-'I!
Nl-'I!
= --
0
1- w
(111)
with W = d/2. This form is now valid for all d so that the loop integral may be written as, besides other constant factors, Nl-d/2
2NV~ 2 --
2-d
(112)
with a simple pole at d = 2. V is the total volume. One then identifies d = 2 as the special dimension. Using the above form with the singularity at d = 2, we may write the second virial coefficient as
(113) where a is a constant, and we used N
(114) Substitute in the expression for A 2 , and choose D's to cancel the poles. Zu is called a multiplicative renormalization factor. We adopt the minimal subtraction scheme where the D's are chosen to subtract the poles and only the poles. The calculation is order by order and so, to one loop order, one cannot determine D 2 , which involves u 3 (two loop term ).
47
Directed polymers and randomness Choosing D 1 it follows that
= -1/E, we see that the divergence is removed to 0(1/E). Upto this order,
u2 uo=u--+ ...
(115)
E
Since this absorbs all the divergences, we do not have to renormalize N. The divergencefree quantity is A 2 R(L, u, N) (R for renormalized; expressed in terms of u). Now, A 2 should not depend on L because L is put in by hand. This is ensured by demanding that
L dA2 = O dL '
(116)
where the factor of L in front has been put in for a dimensionless derivative operator. Written in a long form
[L :L
+
fJ(u) :u] A2R(L, u, N) = 0,
with
fJ(u) = L ~~.
(117)
Note that A 2 ,R( u) = A 2 ( u 0 ), though, in general, additional renormalization factors (multiplicative and/or additive) may be needed. By definition, fJ(u) tells us how the renormalized u changes with the length scale and is called the beta function in RG. Some more algebra gives
R( ) =Lou = ~ Louo = ~ = (olnu0 ) f.' u oL OUo oL EUoOUo E ou
-1
(118)
so that the variation of the coupling constant with scale L is given by (119) The factor of 27r can be absorbed in the definition of u, as have often done. This equation, Eq. 119 is called a renormalization group flow equation with the initial condition u = u 0 for some L = L 0 . It is analytic in E so that various dimensions can be handled with this equation. Initial condition may be taken as u = u 0 for some L = L 0 • In this particular case the beta function is exact to all orders. The zeros of the beta function are called fixed points which can be of two types, stable or unstable. For this particular case, the fixed points are u* = 0 and u* = -E. If we start with a very small u, the flow equation Ldu/dL R::! w shows a growth of u with LifE > 0, i.e., if d < 2. This means u is a relevant variable at the noninteracting point. For the disordered system it translates to relevance of disorder at the pure fixed point. One gets a nontrivial fixed point at u* = -27rE where fJ(u) = 0 which is an unstable fixed point if E < 0. This unstable fixed point for d > 2 represents the disorder induced critical point. For the disorder problem, u < 0 is not meaningful, but for the interaction problem as in the RANI model, full range of u is allowed. The nontrivial fixed point for d < 2 is a stable fixed point and it describes the phase of repulsive polymers (fermion like). There is a
48
S. M. Bhattacharjee
critical binding-unbinding transition for d > 2 for pure attractive short range interaction. The unbinding transition is at zero potential for d :::; 2 For the transition one can define a characteristic length ~ so that at scales> ~' the critical effects can be ignored. If we start with an initial value u 0 = u* + ~ 0 , then one may say this crossover happens at some arbitrarily chosen value of ~u = 1 say where ~u is the renormalized deviation from the fixed point. One may determine this by linearization as d~u
L dL
=IE I ~u,
(120)
which gives ~u = ~o(L/ £ 0 )1<1. If we take ~ 0 << 1 as a measure of the deviation from the critical point (like T- Tc), then a small deviation grows and goes over either to the stable f.p. for the unbound phase or to -oo for the bound phase depending on the starting initial sign. Setting L = ~, we then get (121) Since the beta function is exact, we have obtained the exact correlation length scale exponent for the binding-unbinding critical transition. At the critical dimension, d = 2, the flow equation is
L d~u dL
= (~ )2 u
(122)
'
which gives ~ ,. . ., exp(1/ ~ 0 ). This exponential dependence of~ on ~ 0 accounts for the divergence of v at E = 0.
E. RG of the KPZ equation: Momentum shell technique We discuss the momentum-shell RG approach to the KPZ equation oF
oz
=
T 1 2 2 2K"'il F- 2K("'ilF) +'IJ(r,z).
(123)
which is Eq. (37). More details on this equation may be found in Ref. [55]. The idea is to get the behaviour of the parameters of the differential equation in the long distance limit. The three parameters are T, K and ~' of which K remains invariant so that 2v = 8 + 1, Eq. (42), is satisfied. Note that if the nonlinear term (i.e. the force term in the fixed distance ensemble) is absent, the differential equation becomes linear which can be solved exactly[25]. The RG scheme uses this exact solution for an iterative approach to tackle the nonlinear term. In contrast to the RG in the polymer approach of App. D, here the starting point is not a Gaussian polymer but a linear equation which need not represent any polymer. It is convenient to work in the Fourier space
F(x,z)=
j
oo
-oo
-dw
271"
1
k
ddk ( )dF(k,w)= 271"
11 q
w
F(k,w)
(124)
49
Directed polymers and randomness
where A is an upper cutoff in k-space (related to a real space short distance cutoff). The long chain, long distance limit corresponds tow, k ~ 0. The KPZ equation can be written in the following form 1 F(k,w)=Go(k,w)'IJ(k,w)+- G 0 (k,w) 2K
fj
}q
q·(k-q)F(q,w-S1)F(k-q,w), (125)
w
which suggests an iterative scheme. In this equation 8F
Go(k,w)
= D'l](k,w)
1 Dk 2
-
(126)
iw'
for the linear equation with D = T /2k. The effect of the nonlinear or the second term of the right hand side of Eq. (125) is to change G 0 to G (such that F = G'T]). From this c-l' the coefficient of k2 in the k, w ~ 0 limit would give the effective temperature of the problem. G may be written as G(k,w) =
G 0 (k,w) +
~~G~(k,w)
xG 0 (qFor w = 0 and k
~
kw
2, 2 -
if (
q-
S1) G 0 (q +
~)
· (q+
~)
k·(q+ k
kw
w
2 , 2 + S1) G 0 (q + 2 ,- 2 -
~) S1).
(127)
0, the above equation simplifies to
c- 1 (k, o) = G 01 (k, o) ( 1- u d ~
2
j
dq qd- 3 )
(128)
,
where (129) is the dimensionless coupling constant in this problem. Note that this is the same as in App. D. The power of q in the integral follows from power counting while the prefactor (d- 2) comes from the angular contributions of the dot products. In writing this form, the d-dependent solid angle contribution has been absorbed in the definition of u. The nature of ultraviolet (small distance) divergence (as A~ oo) is apparent in the integral of Eq. (128). If in the residual integration in Eq. (128), we perform a thin-shell integration between A(1 - 81) < q < A and set A = 1, the effective temperature for the left-over long wavelength part is given by
r< = T (1 -
u
d~ 2 M) .
(130)
On rescaling x ~ bx, we have k ~ (1- M)k where b = exp(M) temperature is then
t = b(l-2v)fv r<
R:;
r< ( 1 + 8l 1 -v 2v)
.
R::!
1 +81. The renormalized
(131)
S.M. Bhattacharjee
50 Combining Eqs. 131 and 130, one gets (M LoT
aL
= ( 1 - 2v
+
v
=8L/L)
2- d ) T 4d u ·
(132)
as quoted in Eq. (46). The renormalization of the disorder strength can be obtained from the definition
[F*(k, w) F(k, w)Jav
=
2 Li G(k, w) G( -k, -w).
(133)
By using Eqs. 125 and 128 for F and G and following the same iterative procedure as above, the flow equation for ~ can be determined as (134) Combining these two, one gets the flow equation for u
du
LdL
2d- 3 2d
= (2- d)u+ - - u2
'
(135)
as quoted in Eq. (45). The extra d- 2 factor in the flow equation forT leads to the stable fixed point at d = 1. If the d-dependent u term in the flow ofT in Eq. (132) is ignored, the resulting flow equation for u would be similar to Eq. (118) upto a constant factor for u. As already pointed out the coefficient of the quadratic term in the flow equation can always be scaled to 1 by a redefinition of u. There is a major difference between the RG of App. D and the RG done here. In App. D, E = 2- dis a small parameter and used as such in various expansions, though ultimately the equations remain valid for a wider range. Here however there is no small parameter and so the approximation cannot be controlled by choosing small E.
F. Various flow diagrams We discuss the various possibilities of the flows for an RG flow equation represented by
du L dL
2
= EU + cu'
(136)
where u(L) is the running coupling constant at length scale L. The equation has two fixed points (fp), u* = 0, -E/c. For convenience, let us call the u = 0 case as the "free" problem so that the u* = 0 fp corresponds to the free case. This fixed point is to be called the trivial fixed point while the nonzero one is the nontrivial fp. The behaviour of the coupling constant with length scale is determined by the signs of the two constants E and c, and the initial value u(L 0 ) at L = L 0 . If E > 0, then u is a relevant variable at the free fixed point. while u is irrelevant there if E < 0. Special situations correspond to E = 0 for which u is a marginal variable. The possibilities we need to consider are
51
Directed polymers and randomness
marginal relevance
I E=O, C>OI IB: £<0,
C>O
U:O
c
I
U=IEI
I A: U=-E
£>0, C>O I
u:O
-------------------4------------------E 011111(
....
U:O
I C: £<0, C
1111(
U=E
I D: £>0, C
marginal irrelevance Figure 12. The various types of flow diagrams depending on E and c, the coefficients of the linear and the quadratic terms of the flow equation given by Eq. (136). Solid bullet (•) represents a stable fixed point while the star (*) represents an unstable fixed point.
52
S. M. Bhattacharjee 1. A: E > 0, c > 0. Here the nontrivial fixed point is negative and stable. There is no fixed point for u > 0. A bare u < 0 would then be equivalent to a state described by the nontrivial fixed point. For the random medium problem, u > 0 and so a relevant u flows to large values. The resulting state cannot be described in this approach. If u represents the interaction between two polymers, then u < 0 in Sec. 4.2.5 or App. D represents a repulsive interaction while u > 0 is for attraction. Hence one gets a stable nontrivial phase with a repulsive interaction in dimensions E > 0 in region A. For one dimension (d = 1), one may associate this fixed point with a fermion (or hard core boson)-like behaviour.
2. B: E < 0, c > 0. Change in stability of the fixed points. Here the nontrivial fixed point is on the positive side and the trivial or the free fixed point is stable. A system with negative u would behave on a long scale like a free system and so also for small values of u > 0. The nontrivial fixed point now represents a critical point so that a phase transition can be obtained by tuning u. Large values of u correspond to a different phase not accessible by this RG flow equation because the flow goes to infinity. 3. C: E < 0, c < 0. The situation is similar to Region B except that the phase transition is now at a negative value of u and large negative u phase is not accessible. All positive u values are equivalent to the free case (asymptotically free). 4. D: E > 0, c < 0. Here we see the free system is unstable while a positive u case is described by the nontrivial fixed point. 5. E = 0. This is the c-axis, representing the marginal case. The second order leads to a growth of u > 0 if c > 0. In this case, u is a marginally relevant variable while for c < 0 it is a marginally irrelevant variable. There is no fixed point to describe the system. However, the general trend is that if for E = 0, there is a marginally relevant variable, then that variable leads to a critical behaviour (phase transition) forE< 0. The various possibilities are summarized in Fig. 12. In all these cases, if the nontrivial fixed point is stable, then it represents a "critical" phase with characteristic exponents, while if it is unstable it represents a critical type phase transition with its own characteristic exponents. The reunion behaviour in Sec. 4.2.3 and the reunion exponents[13] are examples of nontrivial exponents at a stable fixed point. The unstable fixed point will be associated with a diverging length scale with an exponent ( = 1/[c[ as in Eq. (32). Regions A and C are related by u -+ -u but others are distinct. For the polymer problem, as d (dimensionality) changes, the nature of flow in the partition function approach (Sec. 4.2.5 or App. D) goes from region A to B. One sees a new criticality developing for higher d (d = 2 - E) via a marginally relevant variable. The RG flow for the RANI model (Sec. 9 or App. H) also belongs to this type. For the KPZ approach (Sec. 5.6 or App. E) one goes from region D for d = 1 to region B.
Directed polymers and randomness
53
G. On transfer matrix The directed nature of the directed polymer problem makes it amenable to a transfer matrix approach. This is another feather in the cap of the directed polymer problem. If we know the partition function at a point (x, z), we may construct the partition function for the z + 1th step because it is completely determined by the information available at the z step. The exact form of the transfer matrix would depend on the particular geometry used.
(x,z+l) (a)
A
(x-l,z)
(x,z+l)
(b)
(x+l,z)
ffi
(x-l,z) (x,z)
(x+l,z)
Figure 13. Lines show the possible steps from step z to z+ 1 on a squarre lattice. Backward steps are not allowed for directed polymers. (1) Lattice oriented by 45 degrees. The x and z axes are along the diagonals of the square lattice. (b) Axes are along the two directions of the square lattice but the polymer can take steps along the diagonals of the unit cell also.
If we consider the geometry shown in Fig. 1b, where the x and z axes are along the diagonals of the square lattice, then the partition function satisfies (see Fig. 13a)
Z(x, z + 1) =
e-1J(x,z+l)/T
[Z(x- 1, z)
+ Z(x + 1, z)].
(137)
For the standard geometry (Fig. 13b) with x and z axes along the principal directions of the lattice but if we allow diagonal steps, then the partition function is given by
Z(x, z + 1) =
e-1J(x,z+l)/T
[Z(x, z)
+ 1Z(x- 1, z) + 1Z(x + 1, z)],
(138)
where an extra 1 factor has been introduced to provide appropriate elasticity to the polymer. One may set 1 = 1 for a fully flexible polymer. The initial condition is
Z(x, 0) = Ox,o,
(139)
where Oa,b is the Kronecker delta. The partition function for a chain of length N starting from (0, 0) can be obtained by iterating this equation. For a given realization of the randomness TJ, the partition function (and therefore any physical quantity) can be calculated exactly for finite N. For quenched averaging one has to average over various realizations, and this is where the exactness of the approach gets lost. As the length N of the polymer increases, the span of x also increases linearly so that for N -+ oo one has to study an infinitely large matrix. This allows the possibility of phase transitions in a seemingly one dimensional problem. For numerical analysis, special
54
S. M. Bhattacharjee
care needs to be taken to keep track of the rapid growth of the partition function as the length increases. For T -+ 0, the problem reduces to determination of the ground state energy. One may take the limit E(x,N) = -limr--toTlnZ(x,N), but a direct approach is also possible. For geometry of Fig. 13a, the energy can be obtained from
E(x, z + 1) = min(E(x- 1, z)
+ ry(x, z + 1), E(x + 1, z) + ry(x, z + 1)),
(140)
so that the globally minimum energy path is
E(N) = minE(x,N).
(141)
X
Though we are considering the square lattice (in 1 + 1 dimensions), generalization to other lattices and higher dimensions are straightforward. Similarly, one may consider cases with random energies on the bonds instead of vertices. For the overlap problem of Sec. 6, the partition function would satisfy in 1+1 dimensions
Z(xl' X2, z + 1) =
e-(1)(x,,z+l)+1J(X 2 ,z+l))/T X
LL
Z(x1
+ p,x2 + q, z) [(1- Ox
1 ,x 2 )
+ e-vfTox,x
2 ],
(142)
p=±l q=±l
where the last term involving v is the Boltzmann factor for the interaction on contact (x 1 = x 2 ). If v is taken as a random quantity chosen from a predetermined distribution, then the same transfer matrix can be used to treat the RANI model also.
H. RG for the RANI model We show that the flow equation Eq. (74) is exact in the minimal subtraction scheme using dimensional regularization. More details may be found in Ref. [36,37]. The Hamiltonian needed for [Z2 J~v is
1i
11N0 =2
4
2
--a;- - ro-1N 0 dz o(rl2(z)) o(r34(z)),
dz ~ (8r;(z))
(143)
where we have set the effective two body interaction to zero. For the cumulant, one need only consider the "connected" partition function for this Hamiltonian. It is advantageous to consider the Laplace transform of the N-dependent partition function as (144) the Laplace conjugate variable being s. An expansion in the coupling constant r 0 can be arranged like a ladder ("time-ordered" diagrams) as shown in Fig. 14 The individual pairs of chains are represented by thick lines. The horizontal wiggly lines in these diagrams stand for r0 . Such a representation is possible because the o function in H 2 , Eq. (73), forces the members of a pair to have the same r, z coordinates. Each chain is described by the free distribution ("propagator")
Directed polymers and randomness
55
(a)
Figure 14. (a) The contributing ladder diagrams for [Z2 J~v· A thick line corresponds to a pair of chains. A wiggly line stands for an fo factor in the evaluation of the diagrams. There are one and two loops in diagrams (b) and (c) respectively. Divergences arise from loop integrations.
G(rf- r;, ZJ- z;) = [27r(ZJ- z;)]-d/ 2 exp[-(r 1 - r;) 2 /2(zf- z;)] with end points (rf, ZJ) and (r;, z;). In conformity with the current usage we use the word "propagator" for the lines. Two chains are therefore described by G 2 (r, z)
= (47rz )-d/2 G(r, z/2).
(145)
This G 2 is the propagator for the thick lines. At each wiggly line, connecting four chains (all four having the same chain length z), there are two integrations over the spatial coordinates of the two separate pairs of chains (thick lines). The loops formed out of the wiggly lines are only responsible for the divergence at d = 1. In order to trace the algebraic origin of the singularity, note that, by very nature of the interaction, the spatial integrations associated with the two thick lines are independent of each other. Each section of the thick lines, with z1 , z2 as the end points, in a loop formed with the wiggly lines, contributes (z1 - z2 )-df2 from the identity in Eq. (145). Since the interaction demands same z for the two thick lines, the z integrals involve (z 1 - z2)-d type factors whose Laplace transform would contribute f(1 - d) with pole at d = 1. The two independent spatial coordinates which are left out after the successive use of the normalization J dr G(r, z) = 1, lead to a V 2 factor (total volume) for each diagram. The convolution nature of the z integrals, thanks to the time ordering, leads to a simple product of the individual Laplace transforms of the integrands, resulting in a geometric series for Z. As an example we consider the two loop diagram of Fig. 14( c). After integration over the free ends, we are left with the following (146) Here the subscripts denote the successive points along the thick lines while the two sets of chains are distinguished by the prime. In the Laplace space this becomes
(t:' = 1- d).
(147)
This can be generalized to arbitrary orders since only ladder type diagrams are involved.
56
S. M. Bhattacharjee
r0
Defining the dimensionless coupling constant r 0 through an arbitrary length scale L as = r 0 L 2'(47r)-d, E = 1- d, we write the series for Z to all orders in r 0 as (148)
It is clear from the above expression that there is a divergence at d = 1 at each order (> 1). A renormalization through minimal subtraction would require absorption of the poles in E1 through (149) with an = 2:.-::;=l an,pCP and r as the renormalized coupling constant. In such a scheme, an,p(P =/= n) terms are required to take care of the subleading divergences. The geometric series of Eq. (148) guarantees that the removal of the leading poles is sufficient to remove the subleading ones. All the divergences can be absorbed by the choice ap = (-E')-P which can be obtained by an explicit order by order calculation. The f3 function is therefore exact to all orders in perturbation series and is given by (150) There are two fixed points: (i) r = 0 and (ii) r* = -E1 • The bare coupling constant r 0 which originates from v5~, where ~' the variance of the distribution, is strictly positive, requires a positive r. Therefore, the nontrivial fixed point for d < 1 in negative r is unphysical. It however moves to the physical domain for d > 1. Exactly at d = 1, E = 0, r grows with length L as
r(L) = r(O) [1 + 2r(O) ln
7]
-I ,
(151)
r(O) being the coupling at length £ 0 . Hence, the disorder is marginally relevant. For d > 1, there exists an unstable nontrivial fixed point at r =I E I which separates two distinct regimes of disorder. If we start with a strong enough disorder, on the right side of the fixed point, it increases with length scale, going beyond the perturbative regime. This is the strong disorder phase. On the other hand, the left side of the fixed point is the weak disorder regime, since r flows to zero (the stable fixed point).
REFERENCES 1. 2. 3.
P. de Gennes, Scaling Concepts in Polymer Physics, Cornell Univ. Press, Ithaca, 1979. M. Doi and S. F. Edwards, Theory of Polymer Dynamics, Clarendon Press, Oxford, 1988. S. M. Bhattacharjee, Critical Phenomena: An Introduction from a Modern Perspective in "Field Theoretic Methods in Condensed Matter Physics", lOP Publishing, Bristol, 2003/ Hindusthan Publishing, Delhi, 2001 (for India); cond-mat/0011011
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B. K. Chakrabarti and J. Kertesz, Z. Phys. B 44, (1981) 221. K. Barat and B. K. Chakrabarti, Phys. Rep. 258 (1995) 377. T. Halpin-Healy and Y. C. Zhang, Phys. Rep. 254 (1995) 215 . M. Kardar, G. Parisi andY. Zhang, Phys. Rev. Lett. 56 (1986) 889. D. A. Huse, C. L. Henley and D. S. Fisher, Phys. Rev. Lett. 55 (1985) 2924. D. S. Fisher and D. A. Huse, Phys. Rev. B 43 (1991) 10 728. S. M. Bhattacharjee, J. Phys. A 33 (2000) L423. (cond-mat/9912297) R. Kapri, S. M. Bhattacharjee and F. Seno, Phys. Rev. Lett. 93 (2004) 248102. J. J. Rajasekaran and S.M. Bhattacharjee, J. Phys. A 24 (1991) L371. S. Mukherji and S. M. Bhattacharjee, J. Phys. A 26 L1139 (1993); Phys. Rev. E 48 (1993) 3427. 14. J. Z. Imbrie and T. Spencer, J. Stat. Phys. 52 (1988) 609. 15. B. Derrida and 0. Golinelli, Phys. Rev. A 41 (1990) 4160. 16. J. M. Kim et al., Phys. Rev. A 44 (1991) R4782. 17. L. Giada, A. Giacometti and M. Rossi, Phys. Rev. E 65 (2002) 036134. 18. M. Lassig and H. Kinzelbach, Phys. Rev. Lett. 78 (1997) 903. 19. J. K. Bhattacharjee, J. Phys. A, 31 (1998) L93. 20. T. Blum and A. J. McKane, Phys. Rev. E 52 (1995) 4741. 21. E. Marinari, A. Pagnani, G. Parisi and Z. Racz, Phys. Rev. E 65 (2002) 026136. 22. C. Castellano, M. Marsili, L. Pietronero, Phys. Rev. Lett. 80 (1998) 3527. 23. For Bethe ansatz, see e.g. J. Ziman, Models of Disorder, Cambridge Univ. Press, Cambridge, 1979. 24. M. Kardar, Nucl. Phys. B 290 (1987) 582. 25. S. F. Edwards and D. R. Wilkinson, Proc. R. Soc. London A 381 (1982) 17. 26. C. A. Doty and J. M. Kosterlitz, Phys. Rev. Lett. 69 (1992) 1979. 27. T. Nattermann and R. Lipowsky, Phys. Rev. Lett. 61 (1988) 2508. 28. S. Mukherji and T. Nattermann, Phys. Rev. Lett. 79 (1997) 139. 29. S. Mukherji, Phys. Rev. E 50 (1994) R2407. 30. S. Mukherji and S. M. Bhattacharjee, Phys. Rev. B 53 (1996) R6002. 31. B. Derrida and H. Spohn, J. Stat. Phys. 51 (1988) 817. 32. A. Basu, Phys. Rev. E 62 (2000) 4675. 33. M. Mezard, J. Phys. France 51 (1990) 1831. 34. Rajeev Kapri, unpublished report. (2003). 35. M. Praehofer and H. Spohn, J. Stat. Phys. 115 (2004) 255. 36. S. M. Bhattacharjee and S. Mukherji, Phys. Rev. Lett. 70 (1993) 49. 37. S. Mukherji and S. M. Bhattacharjee, Phys. Rev. E 48 (1993) 3483. 38. H. Kallabis and M. Lassig, Phys. Rev. Lett. 75 (1995) 1578. 39. D. Marenduzzo, S. M. Bhattacharjee, A. Maritan, E. Orlandini and F. Seno, Phys. Rev. Lett. 88 (2002) 028102. 40. D. Marenduzzo, A. Trovato, and A. Maritan, Phys. Rev. E 64 (2001) 031901. 41. D. K. Lubensky and D. R. Nelson, Phys. Rev. Lett. 85, (2000) 1572. 42. S. M. Bhattacharjee and D. Marenduzzo, J. Phys. A 35 (2002) L349. 43. C. Danilowicz et al. Phys. Rev. Lett. 93 (2004) 078101. 44. J. Cook and B. Derrida, J. Stat. Phys. 57 (1989) 89. 45. M. Kardar and E. Medina, J. Stat. Phys. 71 (1993) 967.
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46. 47. 48. 49. 50. 51.
S. Mukherji and S. M. Bhattacharjee, Phys. Rev. E 52 (1995) 1930. S. M. Bhattacharjee and S. Mukherji, Phys. Rev. E 52 (1995) 4860. T. A. S. Haddad, R. F. S. Andrade and S. R. Salinas, J. Phys. A: 37 (2004) 1499. Y. Kafri and D. Mukamel. Phys. Rev. Lett. 91 (2003) 055502. A. Aharony and A. B. Harris, Phys. Rev. Lett. 77 (1996) 3700. S. Wiseman and E. Domany, Phys. Rev. E 52 (1995) 3469; Phys. Rev. E 58 (1998) 2938. 0. Dillmann, W. Janke and K. Binder, J. Stat. Phys., 92 (1998) 57. M. I. Marques and J. A. Gonzalo, Phys. Rev. E 60 (1999) 2394; 65 (2003) 057104. S. Roy and S. M. Bhattacharjee, cond-mat/0409012. A.-L. Barabasi and H. E. Stanley, Fractal Concepts in Surface Growth, Cambridge Univ. Press, Cambridge, 1995.
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Statistics of Linear Polymers in Disordered Media Edited by Bikas K. Chakrabarti © 2005 Elsevier B.V. All rights reserved.
Self-avoiding walks in constrained and random geometries: Series studies Anthony J. Guttmann a a Department
of Mathematics and Statistics, The University of Melbourne, Victoria, 3010, Australia ARC Centre of Excellence for the Mathematics and Statistics of Complex Systems. The self-avoiding walk is an excellent representation of an isolated polymer chain in dilute solution, as well as being a classical combinatorial problem. It lends itself to modelling many other situations by modifying the nature of monomer-monomer and other interactions. Randomness can be introduced either in the surrounding environment, or within the walk itself. In this chapter we discuss a variety of series studies of random selfavoiding walks, with both types of randomness, as well as series studies of self-avoiding walks in constrained geometries. In the latter case, one can consider interactions between the walk and the wall of the confining geometry, as well as inter-monomer interactions. A variety of interesting physical phenomena can be so modelled, and we discuss series studies of a variety of chemical and biological phenomena. These include the denaturation of DNA, the "pulling" of a macromolecule, such as DNA, away from an adsorbing surface, and the collapse transition of a vesicle. 1. INTRODUCTION
This topic is sufficiently large that an entire book could be devoted to it, and still not give complete coverage. Reviews by Barat and Chakrabarti [1] on SAW on random lattices, and by Soteros and Whittington [2] on random copolymers are individually of comparable length to this chapter, yet discuss just two of several topics addressed here. Thus this chapter consists of a (hopefully representative) selection of topics that inevitably reflects both the expertise and personal taste of the author. Knowledge of the fundamentals of the theory of SAW, and of series generation and analysis are assumed. A good introduction to these can be found in [3], where the problem of SAW in a constrained geometry is also briefly discussed. We will initially discuss SAW in random geometries, and then SAW in constrained environments. In the former case, many of the studies have used methods other than series analysis, typically Monte Carlo, and so our discussion of this topic will be briefer than that of SAW in constrained geometries. For SAW in random geometries, the randomness can be in the environment, for example SAW on a regular lattice with randomly missing sites or bonds. We discuss this situation in the next section. Alternatively, the randomness can be in the walk itself, which is typically the case with copolymers, in which the SAW comprises more than one type of monomer, each having different interaction properties, and with the monomers randomly 59
60
A. J. Guttmann
distributed. We discuss this situation in section 3. We subsequently consider the problem of SAW in restricted geometries, confined to a wedge, a strip, a slab, a square or a cube. We then discuss SAW on non-regular lattices, such as the Penrose and Ammann-Beenker tilings. Finally, we discuss variants of the SAW designed to model a variety of biological phenomena, notably DNA denaturation, the collapse of a vesicle, and the pulling of a macromolecule from an adsorbing surface by an applied force normal to the surface. While the primary focus of this chapter is on series methods, it has been necessary to make reference to other approximation techniques, such as Monte Carlo methods and renormalization group methods in order to assess the reliability of certain results. It has also been necessary to provide some background to a number of the problems discussed, as these are unlikely to be familiar to many readers. Finally, certain rigorous results have been discussed, as it is necessary to focus on what is rigorously known, what is exactly known (but not rigorously), and what is conjectural. To put these into focus, the existence of the connective constant, or growth constant, of a SAW is rigorously known. The value of the critical exponent v = 3/4 for SAW on a regular two-dimensional lattice is exactly known, where v is the critical exponent characterising the growth of the mean square end-to-end distance of ann-step SAW. The corresponding quantity for three-dimensional SAW is known only numerically, and hence approximately. We will first very briefly review the properties of SAW on a regular, hypercubic lattice zd, in particular those on the square lattice Z 2 . For walks on an infinite lattice, it is generally accepted [4] that the number of such walks of length n, equivalent up to a translation, grow as Cn "' const.J.Lnn"f-l, with metric properties, such as mean-square radius of gyration or mean-square end-to-end distance growing as (R 2 )n "' const.n 2 v, where 1 = 43/32 and v = 3/4. The growth constant JL is lattice dependent, and for the honeycomb lattice is believed to be J.lhoney = J(2 + J2) [5], while for the square lattice it is not known exactly, but is indistinguishable numerically from the unique positive root of the equation 13x 4 - 7x 2 - 581 = 0 [6]. In three dimensions the growth constants and exponents are known only approximately, and are not believed to be algebraic (the growth constants) or rational (the exponents). We denote the SAW generating function by C(x) := I:n CnXn, and it will be useful to define a second generating function for those SAW which start at the origin (0, 0) and end at a given point (u, v), as G(O, 0; u, v). In terms of this generating function, the mass m is defined [4] to be the rate of decay of G along a coordinate axis, m:= lim -logG(O,O;n,O)_ n-+oo n
(1)
where we have assumed isotropy, so that the choice of co-ordinate direction is irrelevant.
2. SELF-AVOIDING WALKS IN RANDOM GEOMETRIES Consider a SAW on a lattice in which some sites are missing. One can remove sites at random, such that a fraction p of sites is removed, and study the behaviour of SAW averaged over an ensemble of such lattices. Thus one obtains the properties of the SAW averaged over a uniform distribution of such lattices. This is the quenched situation.
SAW in constrained and random geometries
61
Brief reflection allows one to conjecture the effect of this site dilution. Firstly, it is clear that the problem now becomes one of SAW on a percolation cluster. For p > Pe there is no infinite cluster, and hence all SAW are of finite length. For p < Pe, the average number of neighbours of a lattice site is modified from q, the coordination number of the undiluted lattice, to q(1 - p). Thus one would expect the growth constant JL also to be reduced, and JL(p) to be a monotone decreasing function of p for 0 < p < Pe· What is less clear is how the critical exponents behave. As discussed above, for the undiluted lattice one has (R 2 )n "' const.n2 v in the limit as n becomes large. The exponent v is given phenomenologically by the Flory formula, v = 3/(d+2) which seems to be exact for d = 2 and d =de= 4 (ignoring logarithmic corrections at d =de= 4). The Flory formula prediction is about 2% high at d = 3. Above four dimensions the exponent "sticks" at the value 1/2, as the SAW constraint becomes increasingly irrelevant. Field theory, again for the undiluted lattice [7], gives v = 1/2 + t:/16 + 15c2 /512 + ... , where E = 4- d. Naive application of the Harris criterion [8] would suggest that for Pe > p > 0 the exponents would be modified. That is to say, the slightest level of disorder would be sufficient to change the exponent. Loosely speaking, the Harris criterion says that the universality class of a system is affected by the presence of disorder if the specific heat exponent a of the pure system is positive. For both two- and three-dimensional SAW this is the case. For d = 2, a = 1/2, while for d = 3, a ::::; 0.236. However, as SAW are given by the N -+ 0 limit of the O(N) model, a modified Harris criterion applies [8]. In the usual Harris criterion argument, one can relate the concentration induced change in transition temperature to the correlation length, and then using the hyperscaling relation dv = 2- a deduce that if a > 0, the exponent v will change with impurity concentration p > 0, no matter how small. However, in the N -+ 0 limit, the transition temperature change does not depend on a, but only on the dimensionality of the system. Thus weak disorder is not expected to modify the universality class. However, at p = Pe we are in a regime of strong disorder, and one expects a change in universality class. This situation was first discussed within the framework of the epsilon expansion by Meir and Harris [9] in 1989, and demonstrated numerically by Grassberger [10] in 1993. The epsilon expansion has recently been extended by von Ferber et al. [11], who found that at p = Pe one has Ve = 1/2 + t:/42 + 110c2 /21 3 + ... where now E = 6- d. The appropriate Flory formula [12] is now Ve = 3/(dp+2), where dp = d-j3pjvp and /3p and Vp are the appropriate percolation exponents. In two dimensions we have /3p = 5/36 and Vp = 4/3, hence dp = 91/48 and Ve = 144/187 = 0.7700 .... In [9] the values 0.76 ± 0.08, 0.67 ± 0.04, 0.63 ± 0.02 and 0.54 ± 0.02 were obtained for d = 2, 3, 4, 5 respectively, based on the analysis of short series. Grass berger [10] obtained v = 0. 783 ± 0.003 by Monte Carlo analysis, while Lam [13] estimated Ve = 0.81 ± 0.03. The most recent series work, [14] finds Ve = 0. 778 ± 0.015 and Ve = 0. 787 ± 0.010 from two different analyses of the two-dimensional case, and Ve = 0.662 ± 0.006 in the three-dimensional case. These are consistent with recent Monte Carlo analyses, and also with the epsilon-expansion given above, which evaluates to Ve = 0. 785 .... and Ve = 0.678 ... for two- and three-dimensions respectively. Thus all numerical work and theory is in reasonable agreement, and this situation can be considered to be well understood.
62
A. J. Guttmann
3. RANDOM COPOLYMERS For the problem of random copolymers, unlike the situation discussed in the previous section, the randomness is now not in the medium, but in the SAW itself. One of the simpler cases is when there are two types of monomer, say A and B. Typically one has a fraction p of A-type monomers, and a fraction (1- p) of B-type monomers. One usually assumes that the monomers are randomly distributed and constrained only by the value of p. An excellent contemporary review of this topic can be found in [2]. More generally, one can consider the case with k types of monomer, denoted m 1 , ... , mk, where the state of the polymer, modelled by an n-step SAW, is given by ann-tuple a= a 1 , ... , an, where a; E {m 1 , ... , mk}. The values of a; are taken from some distribution, appropriately chosen to model the problem at hand. In [2] three representative situations are discussed. The first is adsorption of a copolymer onto a surface, the second is the localization of a copolymer at an interface between two immiscible liquids, and the third is the temperature- or solventinduced coil-ball collapse of a copolymer. We discuss these three representative problems below. We will consider the case of an isolated SAW, of length n. Two types of averaging are usually considered, corresponding to the quenched and annealed cases respectively. In the quenched case, the average is taken of the logarithm of the partition function over the entire distribution, so that the free energy is
:Fq
:=
lim
n-+oo
~(log Zn), n
(2)
while in the annealed case . 1 Fa := hm -log(Zn)· (3) n-+oo n The averages are taken over all a. It is noteworthy that the annealed free energy is an upper bound on the quenched free energy. This is useful as the former is usually easier to calculate. Better approximations are often provided by the Morita [15] approximation and the replica method. We refer readers to [2] and references therein for further details. We will now consider the three problems referred to above. Random copolymer adsorption refers to the situation in which a copolymer (most simply described as one with two types of monomer, say A and B) is adsorbed onto a surface. Consider an n-step SAW, with a proportion p of A type monomers, and a proportion 1 - p of B type monomers in a half-space. In two dimensions we are therefore referring to that section of ~} with y 2': 0. Let the origin of the SAW lie in the line y = 0. Finally, the energy comes from the A monomers in the line y = 0. That is to say, only the A monomers are attracted to the surface. There is no surface interaction with B monomers. For those SAW, with cardinality c~(vA[a) with VA vertices in y = 0, and monomer distribution as described above given by a, where a; = 1 if the ith vertex is A and zero otherwise, the partition function is given by
(4) For technical reasons one usually insists that the last step of the SAW also lies in the line y = 0, so that we are studying self-avoiding loops, rather than SAW. The reason for this
SAW in constrained and random geometries
63
restriction is that it allows concatenation of such walks, and hence application of standard tools like sub-additivity, which are so ubiquitous in existence proofs of free-energies and the like. What makes this problem so difficult-and indeed most copolymer problems difficultis the necessity to take the average over all possible 2n distributions of monomers on the polymer. Nevertheless, some exact enumeration studies of this problem have been conducted. Martin [16] estimated the location of the transition using exact enumeration methods. An open question is the value of the crossover exponent, ¢, which describes the shape of the free-energy near the adsorption critical temperature. The conclusion from the above study, and others, is that the difference between this exponent and its homopolymer counterpart, if it exists, is too small to be detectable by any current numerical studies. The second problem of considerable interest is that of localization of a copolymer. Consider a mixture of two immiscible liquids such as oil and water. Then consider a copolymer, modelled as usual by a SAW, with two kinds of monomer, distributed uniformly and independently, and with origin at the interface between the two liquids. Let A monomers be energetically attracted to the water phase and B monomers to the oil phase. At high temperatures, the SAW will sit entirely in the liquid that gives rise to the lowest energy configuration. At low temperatures however, the SAW will try to orient itself with A monomers in the water and B monomers in oil. It will thus cross the interface frequently. This is the so called localized phase, as in it the SAW is largely constrained to the vicinity of the interface. As the temperature is raised, one expects a phase transition to a delocalized phase, heralded by a singularity in the quenched free energy. The model can also be generalized to include an interaction energy for monomers in the interface. James et al. [17] and Martinet al. [18] studied the localization problem for the simplecubic lattice Z 3 , where the plane z = 0 defines the interface. Most of their studies are devoted to establishing rigorous results, of the existence and convexity type, and determining qualitatively the shape of the phase diagram. The third problem we will consider is the collapse of copolymers. In the homopolymer case (where there is only one type of monomer), the collapse transition is known to take place at the () temperature, brought about by sufficiently strong attractive interactions between nearest neighbour monomers that are not joined by a path of the SAW. In that case, the exponent v that characterises metric properties such as the mean-square endto-end distance and the radius of gyration, which are characterised by the exponent v, changes abruptly to v8 at the() temperature, and then to Vc = 1/d at lower temperatures, when the SAW collapses into ad-dimensional ball. In the copolymer case, the simplest realistic model is as usual to have A monomers with probability p and B monomers with probability 1 - p. Then instead of just considering interactions between nearest neighbour contacts, as in the homopolymer case, we associate different energies to AA, BE and AB contacts. Typically, one allows like monomers to attract and unlike monomers to repel, or vice versa. In the former case we have a model of hydrophilic and hydrophobic monomers, and in the latter case we have a model of a highly screened Coulomb system. Golding and Kantor [19] and Kantor and Kardar [20] studied a version of this problem by both exact enumeration and Monte Carlo methods. In the former situation described above, they find a collapse transition much like the transition discussed above in the
64
A. J. Guttmann
homopolymer case, which holds irrespective of the ratio of type A to type B monomers. In the latter situation, modelling a screened Coulomb system, the behaviour depends on the ratio of the two types of monomers. If this is around 1, so that the two types of monomer have similar cardinality, the system has no nett charge, and there is a collapse transition. But if the monomer ratio is substantially different from 1, the repulsive interactions between the monomers that are in excess prevent any collapse, and the SAW remains in its coiled state at all temperatures. A later enumeration study by Monari and Stella [21] considered the system with equal numbers of A and B monomers and a Coulombic interaction energy. They found that at the () temperature, the radius of gyration exponent v and the crossover exponent ¢ were indistinguishable from their homopolymer values in both two- and three-dimensions. Many features of this fascinating problem have not been mentioned here, but can be found in the comprehensive and up-to-date review of random copolymers in [2]. In particular, many of the rigorous results and the numerical results obtained both by other approximation methods and by Monte Carlo methods which have not been discussed here can be found there
4. SAW IN WEDGES In this section we consider the effect of confining a SAW to a wedge, formed by the intersection of two (non-parallel) d- !-dimensional planar surfaces in ad-dimensional hypercubic lattice, with co-ordinates labelled by {XJ [o = 1, 2, ... , d}. The surfaces intersect in ad- 2 dimensional surface perpendicular to both x 1 and x 2 , with wedge angle o:. It is not difficult to show that the growth constant JL remains unchanged for most "obvious" geometries, such as the semi-infinite lattice, or indeed any positive wedge angle. A much more subtle result was proved by Hammersley and Whittington [22] who considered SAW confined to a subset zd(J) of the d-dimensional hypercubic lattice, such that the co-ordinates (x 1 ,x 2 , ... ,xd) of each walk vertex satisfy X;?. 0 and 0::; xk::; A(x 1 ) for k = 2, 3, ... , d. They showed that the growth constant remains unchanged provided that fk(x)-+ oo as x-+ oo. In this problem, either end of a walk can be in the bulk, at a surface, or at the wedge "edge". This gives rise to a number of possible generating functions, and we now develop the theory in terms of the general N-vector model, and take the limit N-+ 0 in order to recover the SAW situation. Expressed as a problem in the framework of the O(N) model, the Hamiltonian is
1i = -K
L
a;. a j -
Lo
L ail)- Ll L ail)- L2 L ail)
(5)
where a; is anN-dimensional unit vector with components (ai 13 l, f3 = 1, 2, ... , N) and £ 0, 1 £ 1 and £ 2 are bulk, surface and edge fields respectively, all parallel to ai ). The first sum is over nearest-neighbour pairs, the second is over all spins, the third is over all spins in a surface, and the fourth is over all edge spins. The edge magnetization is the expectation value of an edge spin:
(6)
SAW in constrained and random geometries
65
Differentiation with respect to the three fields £ 0 , £ 1 and £ 2 allows for three susceptibilities to be naturally defined, viz:
(7)
x21 "' const.C"~ X22 "'
21
,
(8)
const.C"~ 22 ,
(9)
where t
= T /Tc -
1. If we now write the free energy in the form
(10) where V is the 'volume' of the system, A is the 'area' of the surface, and L is the 'length' of the edge, then ib, is and ie denote the bulk, surface and edge free energies respectively. Standard scaling theory informs us that in this case, with three fields, the singular part of the free-energy will scale as:
\ Yl \ Y2 ) /\ 9o, /\ 91, /\ 92 f sing -- /\\ -di (\Yo
(11)
where y0 , y 1 and y2 are the renormalization group eigenvalues associated with the three fields. The three free energies referred to above can be written in terms of the canonical scaling form as
ib "'t 2-"1J!b(hCY0 "),
(12)
is"' t2-"''llb(hrYo", h1 CY1 "), ie "'t 2-<>ewb(hCY0 ", h1CY1 ", h2CY 2 "),
(13) (14)
where y0 , y 1 and y2 are the bulk, surface and edge scaling indices, and
(2- a)= dv, (2- as)= (d -1)v, and (2- ae) = (d- 2)v.
(15)
All susceptibilities now follow by taking the appropriate derivatives, so that
(16) (17) (18) (19) (20) (21)
A. J. Guttmann
66
and by performing these differentiations, we immediately find 1
= I/(2Yo - d)
(22)
= v(yo + Y1 - d + 1) Ill = v (2y1 - d + 1)
(23)
+ Y2 - d + 2) 121 = v(y1 + Y2 - d + 2) 122 = v(2y2- d + 2).
(25)
11
12
=
(24)
v(yo
(26)
(27)
These results can be combined to produce the scaling law
= 1 + v,
211 - 1u
(28)
which was first derived by Barber [23], and the additional scaling laws,
= 1 + 2v,
212 - 122
(29)
and 2121 - 122
= 1u + v,
(30)
first given by Guttmann and Torrie [24]. The susceptibilities can also be defined in terms of correlation functions, which will be useful in taking the N -+ 0 limit of the O(N) model. Define the correlation function
(31) between a spin at the edge (r' = 0) and a spin at (p, x 1, x 2 ) where pis a (d-2)-dimensional vector with components x 3, x 4 , ••• , xd. First, note that the correlation function will depend on the orientation of the vector (p, x 1, x 2 ) even near Tc. Denote its magnitude by r, its orientation within the (x 1,x 2) plane by() (clearly 0::::; ()::::;a), where a is the wedge angle, and its orientation within the surface plane by ¢. For T > Tc we have C(r, B, ¢) where
f
"'r-+oo
f(r, B, ¢) exp[-r/.;e,q,(t)],
decays more slowly than the exponential term. At T
C(r, B, ¢)
"'r-+oo
A(B, ¢)/rd- 2 -~e,¢,
(32)
= Tc, (33)
which leads to the special cases ry2 = TJ for (B > 0,¢ > 0), ry2,1 for (B = 0,¢ > 0), and ry2,2 for (B = 0,¢ = 0). Clearly ry2 ::::; ry21 ::::; ry22 as there are more possible paths leading to ry2 than the others, and similarly for the last inequality. A typical susceptibility is X2
=L p
00
00
L' L C(r' = O;p,x ,x 1
x,=O x2=0
2)
(34)
SAW in constrained and random geometries
67
with a similar expression holding for the other susceptibilities. (The prime is to remind one of the restriction 1r S: a). Following an argument of Sarma, given in the appendix to a paper by Daoud et al. [25], we see that x2 is the generating function for SAW in the wedge, with one end tethered to the edge, and no restrictions on the other end. x21 is given by the generating function for SAW with one edge tethered to the edge, and the other end in the surface, while x22 is given by the generating function for SAW with both edges tethered to the edge. Replacing sums by integrals in the above expression, it is straightforward to show that 12
= v(2- TJ2),
1'12 /22
(35)
= v(1- TJd, = - VTJ22 ·
(36) (37)
Defining TJoo = TJ, and TJ;o = TJ; these expressions can be combined to give T/pq
= (T/pp + T/qq) /2,
a result first obtained by Cardy [26] from field-theoretical arguments. Based on extensive enumerations in wedges of various angles, Guttmann and Torrie [24] found compelling numerical evidence for SAW in two-dimensional wedges of angle a, that y2 = -57r/8a. If one recalls that Yo= 91/48 and y 1 = 3/8 were already known, it follows that all exponents for the edge problem follow. Cardy subsequently verified this conjecture for y 2 (a), again from field theoretical arguments. For three-dimensional SAW, the enumerations in [24]led to the conjecture y 2 (a) = 1/2- 0.85(2)/a. Both these results can be compared to the mean-field theory result, yf!F = 1- d/2- 1rja. Note that the mean-field result agrees with the enumeration results in both the constant term and the form of the angular dependence. An interesting and physically significant generalisation of this problem can be achieved by associating a fugacity with those monomers lying in the surface, or surfaces, in the case of a wedge geometry. If the fugacity is sufficiently attractive then at some critical temperature the SAW becomes adsorbed onto the surface. In an interesting paper, Hammersley, Torrie and Whittington [27] considered the simpler case of a single surface (corresponding to a = 1r) on the hypercubic lattice and showed that there must be a phase transition. In [28], Batchelor et al. studied the problem of two-dimensional SAW, and more general SAW networks in which the surface is allowed to assume a variety of boundary conditions on either side of the wedge. That is to say, there is a different fugacity associated with SAW contacts in one surface to that in the other surface. As a result of this study, they were able to give conjectured exponents for an arbitrary polymer network-that is to say, not just SAW, but stars-connected to the surface of an arbitrary wedge (in two dimensions) where the surface is allowed to have general mixed boundary conditions.
5. SELF-AVOIDING WALKS IN STRIPS AND SLABS Consider SAW in a strip of width L on the square lattice. This is essentially a onedimensional problem. Indeed, the column-to-column transfer matrix is finite, and hence by a well-known theorem [29] the generating function is rational. The "critical exponent"
68
A. J. Guttmann
is 1, corresponding to a simple pole, and the number of SAW of length n, which we denote cn(L) "'const.JL(L)n, where the growth constant (sometimes called the connective constant, though this latter term is also applied to the logarithm of JL by some authors) is a strictly monotone increasing function of L [22], and limn--+oo JL(L) = JL, the growth constant in the (two-dimensional) bulk case. The generating functions have been investigated for small values of L in [30]. Daoud and de Gennes [31] has given a scaling argument for the L dependence of JL(L), which leads to logJL -logJL(L) "'DwL~lfv, where v = 3/4 as usual denotes the exponent that characterises the mean-square end-to-end distance of an n-step SAW, and Dw is an amplitude. An interesting point occurs if we consider self-avoiding polygons, similarly constrained. An n-step self-avoiding polygon (SAP) is a subset of (n- 1)-step SAW; those with adjacent end-points. Clearly, joining the end-points results in an n-step SAP. By identical arguments we see that the generating function of SAP in a strip is rational. The "critical exponent" is 1, corresponding to a simple pole, and the number of SAP of perimeter n, which we denote Pn(L) "' const.J.Lp(L)n, where the growth constant is a monotone increasing function of L, and limL--+oo J.lp(L) = JL. However, [32] J.lp(L) < JL(L), even though these two quantities have the same limit as L becomes infinite. This follows from Kesten's pattern theorem [32], which, crudely speaking, says that if a SAW pattern in one geometry can occur at least three times, and it cannot occur in a different geometry, then SAW in the former class are exponentially more numerous than those in the latter. Soteros and Whittington [32] displayed such a pattern for SAW in strips of finite widths, hence the above result follows. As a similar scaling Ansatz applies for SAP, viz: log JL -log JL(L) "' DpL ~lfv, it follows that Dw =/= Dp. We can make use of existing results in the literature to quantify this effect. In [30] the growth constants for SAW and SAP in a strip of width L are given, for L ::; 6. Using these results, and Daoud and de Gennes's scaling form above, we find Dw/ Dp ::::; 0.52. For a slab geometry, corresponding to SAW constrained between two parallel planes, which is essentially a two-dimensional problem, there is no similar pattern, and one has the same growth constant for SAW and SAP confined between two planes separated by a distance L [33]. The reason for this difference is, loosely speaking, that it is easy to devise "blocking" patterns for SAW in strips, but not in slabs, where the extra dimension allows the SAW to avoid the blockage. The generating function is no longer rational in this geometry, but the strictly monotone increasing property of the growth constant with separation L still prevails [22].
6. SELF-AVOIDING WALKS IN SQUARES AND CUBES Another interesting constrained SAW problem is that of SAW confined to lie entirely within a square, or rectangle. We will considering in detail the problem of self-avoiding walks on a subset of the square lattice~}, though several of the theorems we give apply to the hypercubic lattice, and so hold for zd, d 2': 2. We are interested in a restricted class of square lattice SAW which start at the origin (0,0), end at (L,L), and are entirely contained in the square [O,L] x [O,L]. A fugacity x is associated with each step of the walk. Historically, this problem seems to have led two largely independent lives. One as a problem in combinatorics (in which case the fugacity
SAW in constrained and random geometries
69
has been implicitly set to x = 1), and one in the statistical mechanics literature where the behaviour as a function of fugacity x has been of considerable interest, as there is a fugacity dependent phase transition. The problem seems to have first been discussed by Abbott and Hanson [34] in 1978, many of whose results and methods are still the most powerful today. A key question considered both then and now, is the number of distinct SAW on the constrained lattice, and their growth as a function of the size of the lattice. Let Cn ( L) denote the number of n-step SAW which start at the origin (0, 0), end at (L, L) and are entirely contained in the square [0, L] X [0, L]. Further, let CL(x) := I:n cn(L)xn. Then C£(1) is the number of distinct walks from the origin to the diagonally opposite corner of an L x L lattice. In [34], and independently in [35] it was proved that C£(1) "' const.>-P. The value of >. is not known, though bounds and estimates have been given in [34~36]. In the statistical mechanics literature, the problem appears to have been introduced by Whittington and Guttmann [35] in 1990, who were particularly interested in the phase transition that takes place as one varies the fugacity associated with the walk length. At a critical value, Xc the average walk length of a path on an L x L lattice changes from growing as L to growing as £ 2 . In [35] the critical fugacity was proved to be :::0: 1/ JL, and conjectured to be Xc = 1/ JL, and in [37] the conjecture was proved. In [34] the more general problem of SAW constrained to an L x M lattice was considered, where the analogous question was asked: how many non-self-intersecting paths are there from (0, 0) to (L, M). If one denotes the number of such paths by CL,M, it is clear that, for M finite, the paths can be generated by a finite dimensional transfer matrix, and hence that the generating function is rational. Indeed, in [34] it was proved that (38)
(where here we have corrected a typographical error). It follows that C£, 2
"'
const.>.~L,
where >. 2 = ~ = 1.81735 .... In [36] two further problems which can be seen as generalisations of the stated problem were considered. Firstly, they considered the problem in which SAWs are allowed to start anywhere on the left edge of the square and terminate anywhere on the right edge; so these are walks spanning the rectangle from left to right. Secondly, they considered the problem in which there may be several independent SAW, each SAW starting and ending on the perimeter of the square. The SAW are not allowed to take steps along the edges of the perimeter. Such walks partition the rectangle into distinct regions and by colouring the regions alternately black and white one gets a cow-patch pattern. Each problem is illustrated in fig. 1. Following the work in [35], Madras in [37] proved a number of theorems. In fact, most of Madras's results were proved for the more general d-dimensional hypercubic lattice, but here we will quote them in the more restricted two-dimensional setting. Let (39)
70
A. J. Guttmann
..... ... .
....
----
.
................. .. .
--- -
.......... . ............................. ... ... ... ... ... ... ... ... . . . . . . . . . . . ..-- ••• •m ........ ... ... ... ........ t ---- .. ---.... . . ..·---. ... ... ... ... ----·.. -.- -------.. -------' . . .·---- . ---- ---- .·---- ....... . .. .. -.. .. .. . . . . . . . .. .. .. .. .. .. . . . . .. .. .. .. .. .. ..·---- ----·----·---. . - - --- -. .... •. . . . . .
LH _,
,,
. . . . . . . .... FR .. •
•
•
•
•
0
•
:........:.... :....0 ........:.... : •
•
0
•
Figure 1. An example of a SAW configuration crossing a square (left panel), spanning a square from left to right (middle panel) and a cow-patch (right panel).
(40) Theorem 1. {i) The limit {:19) exists and is finite for 0 ::S x ::S 1/IJ., and is infinite for x > 1/IJ.. We have 0 < .X 1 (x) < 1 for 0 < x < 1/IJ. and .>. 1 (1/mu) = 1. {ii) The limit {40) exists and is finite for all x > 0. We have >.2(x) = 1 for 0 < x < 1/ 1J. and >.2(x) > 1 for x > 1/JL. The average length of a walk is defined to be
(41) n
n
In the next theorem , we use the notation a ::::::: b to mean that there exist two positive constants C 1 and C2 such that C1 b ::S a ::S C 2 b.
Theorem 2. As L-+ oo, we have (n(x)}L::::::: L for 0 < x < l/!J. and (n(x)}L::::::: I} foT
X> 1/JL. The situation at x = 1/Jt is unknown. Compelling numerical evidence is given in [36] that in fact (n(1/ IJ.)) L is proportional to V lv , in accordance with an intuitive suggestion in [37).
Theorem 3. For x > 0, define ft(x) = log.X 1 (x) and h(x) = log.X 2 (x) . {i) The function !I is a strictly incTeasing, negative-valued convex function of logx JoT 0 < x < 1/IJ., and JI(x)::::::: - rn as x-+ 1/!J.-. (m is the mass, defined above.) {ii) The function h is a strictly incTeasing, convex function of log x for x > 1/IJ., and satisfies 0 < fz(x) :::; log!J. + log x. Some, but not all of the above results were previously proved in [35), but these three theorems elegantly capture all that is rigorously known.
•
SAW in constrained and random geometries
71
In [36] a highly efficient algorithm for enumerating such walks is given. The time required to obtain the number of walks on L x M rectangles grows exponentially with M and linearly with L. The algorithm is easily generalised to include a step fugacity x. The generalisation to spanning walks is also quite simple. The generalisation to cow-patch patterns is more complicated. Graphs can now have many separate components. Using this algorithm, they calculated Cn ( L) for all n for L :S 17. In addition, they computed Cls(1) and C19(1). I
As noted, it has been proved [34,35] that limM--too c~ = ). exists. From this it is reasonable to expect, but not a logical consequence, that RM = CM+l,M+l/CM,M "'>.2 M. If so, the generating function R(x) = I:M RMxM has a radius of convergence Xc = 1/ >. 2 , which they estimated using differential approximants [38]. For the crossing problem they found Xc = 0.32858(5), for the spanning problem Xc = 0.3282(6) and for the cow-patch problem Xc = 0.328574(2). It can be proved that these three problems have the same growth constant, so taking the most precise estimate, they obtained>.= 1.744550(5). As well as this estimate, it is possible to obtain rigorous bounds. In [34] it was proved that I
Theorem 4. For each fixed M, limL--+oo C t,X£ = AM exists. I
It similarly follows that limM--too C~ =A exists, which was proved rather differently '
...M__
in [35]. In [34] the useful bound ). > ). tJ+ 1 is proved. The above evaluation of >. 2 immediately yields >. > 1.4892 .... Based on exact enumeration, Bousquet-Melou et al. [36] obtained the exact generating functions G3 (x), G 4 (x) and G 5 (x). From these, they found the following values: >. 3 = 1.76331 ... , >. 4 = 1.75146 ... , and >. 5 = 1.74875 ... from which they obtained the bound ). > 1.59321 .... 6.1. Asymptotics Bousquet-Melou et al. [36] also considered the exact and asymptotic results for walks crossing a square of length 2£ + 2K. Recall that the shortest walk is of length 2£. For K = 0 the number of such SAW is just Cf). These are so-called ballot numbers. This result is obvious, as there are 2£ steps in the path, of which L must be in the positive x (and of course positive y) direction. Note that this has the asymptotic expansion (42) For K = 1 they proved that the number of paths is given by 2£ (}!2). This result has the asymptotic expansion
L4L 33 y'"l;ff( 2 - 4L
1345
+ 64£2
23835 - 512£3
(43)
+ ... ).
ForK= 2 they proved that the number of paths is given by 2(2£)1
~'::---'-.:-:-(48
L!(L + 4)!
+ 90L + 8£2 -
28£3 -3£ 4
+ 4£5 + L 6 ) - 4
'
(44)
A. J. Guttmann
72 which has the asymptotic expansion £ 2 4£
,;r:rr(2 -
For K
49 4£
2913
(45)
+ 64£2 + ... ).
= 3 they found, numerically, that
L 3 4L (4
49 6L
and for K
= 4 they found the corresponding result
£ 4 4£ 2
11
,ff;ff
3-
+
1931 ± 1 64£2
)
+ ... '
,;r:rr (3 + 4£ + ... ).
(46)
(47)
They also gave an heuristic argument for the general form of the leading term in the asymptotic expansion for the case K = k. The asymptotic behaviour is dominated by the 2 term ~ ( They argue that this holds for k = o(£ 113 ). Hamiltonian walks (which are necessarily SAW) can only exist on 2L x 2L lattices. For lattices with an odd number of bonds, one site must be missed. A Hamiltonian walk is 2 of length 4L(L + 1) on a 2£ x 2L lattice. The number of such walks grows as T 4£ , where in [36] it is estimated that T ~ 1.472. This is about 20% less than>., the growth constant for all paths. In [34] it is proved that 2113 :::; T :::; 12 114 . Numerically, 1.260 :::; T :::; 1.861. While undoubtedly true, it has not yet been proved that >. > T. It follows from the three theorems due to Madras, cited above, which hold for general dimensionality d 2: 2, that most of the qualitative features described above for the twodimensional lattice also hold in higher dimension. Having discussed SAW in random and restricted geometries above, we turn now to other types of randomness, by considering SAW on quasi-random lattices. In exploring the features that change a universality class of a SAW model, it is well known that dimensionality is an important feature, but within a given dimensionality, one can also ask what features determine the universality class? It is known that a change of lattice, among the usual Euclidean Archimedian lattices has no effect on the universality class. In the next two sections we first investigate the effect of a change from a regular to a quasiperiodic lattice, and then in the following section we look at the more drastic change in which the space has negative curvature, and investigate the SAW model on the Poincare disk. In summary, we find no evidence of a change of universality class in the former case, but a definite change in the latter case.
;t.
7. SELF-AVOIDING WALKS ON QUASIPERIODIC TILINGS Quasiperiodic tilings are most widely used for the description of quasi-crystals. With appropriate atomic decorations of the vertices, they serve as structure models which explain physical properties of quasi crystals [39]. From a theoretical point of view, they are idealisations of real substances on which the usual models of statistical physics like the Ising model may be studied [40-42]. Quasiperiodic tilings arose before the discovery of quasi-crystals, however, more as an object of aesthetic interest in geometry [43,44].
SAW in constrained and random geometries
73
From a combinatorial point of view, they provide an interesting example of a nonperiodic yet structured graph where typical problems of combinatorics like the counting of objects on a lattice become more complex. This is fundamentally different from counting problems on semi-regular lattices, where the underlying translational invariance is still present [45], and also from self-similar graphs, where the self-similarity allows for the solvability of some counting problems. In the counting problem of n-step SAW on a quasiperiodic tiling, the number depends on the chosen starting point. On a regular lattice, this phenomenon does not occur due to translation invariance. Questions arise, such as if the universal properties of the walks like critical exponents [4] are changed for quasiperiodic tilings, and how different ways of counting affect asymptotic properties. The first question has been investigated for self-avoiding walks (SAWs) and self-avoiding polygons (SAPs) in [46]. Extrapolation of exact enumeration data for a number of quasiperiodic tilings [47,40] indicates that the critical exponents 1 for SAWs and a for SAPs are consistent with the corresponding values on regular lattices a= 1/2 and 1 = 43/32. In [48], the related problem of SAWs on Penrose random tilings [49] has been studied by Monte Carlo simulations, indicating the same mean square displacement exponent as for regular lattices. The studies [47,40] suffered however from strong finite-size effects due to the relatively short series data available. This is mainly due to the fact that the finite-lattice method [50], being the most successful method known to date for walk enumeration on regular lattices [51,52], cannot easily be applied to this problem, and so to date the exponentially slower method of direct counting has been used. Another reason for the pronounced finite-size behaviour is that the number of walks or loops depends on the chosen starting point. One might suspect that suitable averaging over different starting points reduces these effects, leading to behaviour comparable to the square lattice case. In [46] a thorough study of this topic was made by utilising three different methods of counting. Whereas the first one depends on a chosen vertex of the tiling, the last two are averages over the whole tiling. • Fixed origin walks. The number of n-step SAW emanating from a given vertex are counted. This number depends on the chosen vertex. • Mean number of walks. Each n-step SAW is weighted by the probability that it occurs in the infinite tiling. For tilings with quasi-crystallographic k-fold symmetries, these probabilities are numbers in the underlying module Z[e 2rri/k]. This leads to a generating function with non-integer coefficients. • Total number of walks. All translationally inequivalent n-step SAW are counted. These may occur anywhere in the lattice. This number is bigger than the number of fixed origin walks, and by definition, takes into account vertices over the whole tiling.
For self-avoiding polygons, one must distinguish between different fillings of the same loop. In [46] they employed the second and the third method of counting for this problem. The third method has another combinatorial interpretation: The total number of n-step polygons is the number of n-step loops on the tiling, where one counts all possible ways of filling the interior. This may be denoted a random polygon. For SAPs, the second method
A. J. Guttmann
74
has been implemented previously [40] in order to obtain the high temperature expansion of the Ising model. The mean number of SAPs up to length 2n = 18 has been determined on the Ammann-Beenker tiling [53,54] and on the rhombic Penrose tiling [43,55]. Rogers, Richard and Guttmann [46] counted SAWs and SAPs on the Ammann-Beenker tiling and the rhombic Penrose tiling and compared different counting schemes, thereby extending and generalising the previous approaches to counting SAWs [47] and SAPs [40]. Generally speaking, averaging reduces oscillation of data due to finite size effects, providing improved estimates for critical points and critical exponents. Within numerical accuracy, they find it plausible that SAWs on the Ammann-Beenker and on the rhombic Penrose tilings have the same exponents as on the (regular) square lattice. The data for the total number of walks (polygons) gives a different exponent, reflecting the fact that the number of patches grows quadratically with the patch size, in contrast to the regular lattice case [56,57]. The limited SAP data did not allow them to draw decisive conclusions about exponents. As it is not at all clear how one generates SAW and SAP on these lattices, we describe this in the following subsection. 7.1. Graph generation Quasiperiodic tilings in JRd may be obtained by projecting certain subsets of lattices from a higher-dimensional space !Rn into JRd. This is described by a cut-and-project scheme, summarised in the following diagram:
11"11 Ell ::: JRd
+--
u
1-0
£11
7r.L
E
= IRn
--+
E.l :::' JRm
u
/.dense
u
L lattice
= 1r11(L)
W polytope
It consists of a Euclidean vector space E, together with orthogonal projections 1r11 and 1r.l· The vector spaces E11 = 1r11 (E) and E .l = 1r.l (E) are called direct and internal space, respectively. Let L C E be a lattice. The projections are such that 1r11IL is one-to-one and 1r.l ( L) is dense in E .l (or dense in some subspace of E .l). Let W C E.l be a polytope (or a finite union of polytopes). The set W is also called the acceptance window. The set of tiling vertices A(W) is defined by
A(W)
= {xll E L11l x ELand X.i E W}.
(48)
The edges of the tiling are defined by the following rule: The tiling vertices 1r11 (x) and 1r11 (y) are adjacent iff the lattice vectors x and y are adjacent. For the Ammann-Beenker tiling [53,54], one has n = 4 and d = m = 2. The lattice is L = Z 4 . The projections 1r11 and 1r.l are defined as follows. For x E !Rn, we set XII
X.i
cos¥ cos~) X 4 sin¥- sin¥ ' cos Qz!: cos¥ cos~) 4 · 9rr X. sin~ sin Qz!: Slll4 4 4
cos~ sin~
G G
(49)
SA lV in constrained and ramlom geometries
75
The acceptance window W C JR'" is a regular octagon with unit side length centred at the origin, having edges perpendicular to the axes. A typical patch is shown in Figure 2.
Figure 2. A patch of the Armnann-Beenker tiling.
L
For the rhombic Penrose tiling [43,55], one has n = 5, d = 2 and m = 4. The lattice is The projections 7Tt: and 7T.L are, for x E IR", defined by
= Z5 •
xu =
X .L
cos 2511" cos 45" cos G7r cos~) 5 sin 25,. sin 45"" sin 65"" sins; x, cos 15,. cos fu! cos 12 '~~" 5 x. sin 4,. sin 8" sin 1 ~,. sin 18,. 5 5 5 5 1 1 1 1
G G
(50)
='~)
The acceptance window lV C Rm is made up of four regular pentagons in the planes x .1.3 = 0, 1, 2, 3. The pentagons in the 0 and 3 x 3 -planes have unit side length and t.he othf'..rs have side length 2 cos~- Each pentagon is centred at XJ.l = 0, x.12 = 0. Pentagons 0 and 2 have an edge crossing the positive x.1 1 axis at right angles while pentagons 1 and 3 are rotated through ~- A typical patch is shown in Figure 3. Note that a more natural embedding of the rhombic Penrose tiling is the root lattice A 4 , see also [58), but Z 5 is more convenient for computations. Moreover, the Ammann-
76
A . .J. Guttmann
Figure 3. A patch of the rhombic Penrose tiling.
Bccnker tiling and the rhombic Penrose tiling may be alternatively defined by inflation rule.s for their prototiles (59,60].
SA nr in constrained and random geometries
77
7 .2. Enumeration The number Cn of translationally inequivalent n-step walks on a regular lattice is clearly independent of the choice of origin vertex. Hence this series is representative of the entire lattice. The enumeration of SAWs on non-periodic tilings introduces complications to the interpretation of the generating function C(x) = Ln>o enxn . This is because the possible origin vertices produce an infinite range of different series C(x). Each origin produces a different series which is representative only of that vertex's immediate neighbourhood in the tiling. As described above, in [46] Rogers ct al. adopted three different approaches to enumerating SAWs on quasiperiodic tilings, viz: • Fixed origin walks. • Mean number of walks. • Total nurnber of walks.
7.3. Fixed origin walks Take a random selection of origin vertices x E L (if x .L ¢ W the vertex is not a suitable choice and is ignored). For each suitable origin, generate the neighbourhood of the vertex, including all vertices up to some Euclidean distance N away. Two such neighbourhoods are shown in Figure 2 and Figure 3. Enumerate all SAWs from the origin up to length n in the neighbourhood using backtracking [61]. This takes time proportional to the number of walks Cn .
X.L
= (0, 0)
X .L
= (1 , 0)
X.L
= (1 - ./{, ./{)
X .L
= (1- J2,0)
Figure 4. The actual Arnmann-Beenker neighbourhoods chosen for the enumerations.
If each of the series in Table 1 and Table 2 showed lattice consistent properties, it would be a good indication that these properties belong to the entire tiling.
A . .J. Guttmann
78
n
0 1 2 3 4 5 6 7 8 9 10 11
12 13 14 15 16 17 18 19 20 21 22 23 24 25
X .i = (1, 0) 1 1 8 3 16 13 48 34 144 108 448 292 1088 952 3680 2458 9584 7746 28336 21348 82960 61478 225408 177230 657536 495808 1834768 1412152 5140752 3985706 14584112 11125408 40222672 31617786 114683280 87149372 313146848 248799302 896810944 680172768 2437468000 1943692238 6958267152 5303535884 18981078176 15086983820 53728620912 41295324398 147472084608 116624466842 413887940176
x .1.
= (0,0)
X.i
= (1- 4, ~)
X.i
1 4 12 46 108 374 976 3042 8330 24556 68376 197820 554108 1576464 4400920 12531794 34541864 98846548 270221012 773046904 2109562128 6011045200 16431248782 46538635588 127704810544
= (1- .12,0) 1
5 16 42 152 388 1194 3412 9678 27218 79150 217562 628996 1741464 4968606 1:H24682 39209054 107503768 306845714 8-10463852 2386875508 6548653714 18500898140 50883461478 142927122532
Table 1 The number of n-step fixed origin SAWs for various starting points x.1 in the AmmannBeenker tiling, with starting point coordinates (x, y) given in the internal space.
X .i
=
(0, 0, 1)
X .i
=
(0, 0, 0)
X .i
=
(0.5, 0.5, 1)
X.i
=
(0.25, 0.5, 0)
Figure 5. The actual rhombic Penrose neighbourhoods chosen for the enumerations.
SAW in constrained and random geometries
79
n Xj_ = (0, 0, 1) X_L = (0, 0, 0) Xj_ = (0.5, 0.5, 1) X_L = (0.25, 0.5, 0) 1 1 1 1 0 4 1 5 5 5 12 2 20 10 14 40 40 46 3 50 130 130 4 112 160 394 450 310 406 5 1170 1140 1177 938 6 3416 7 4000 2680 3316 9723 9480 8 9360 7866 27312 9 32910 23150 27356 10 72520 77747 66150 76090 11 262250 196980 224102 215924 12 619460 555290 615549 545062 13 2050500 1646990 1812802 1698548 14 5052310 4292010 4869786 4411293 15 15828550 13403280 14455725 13367278 16 41103090 33637420 38524509 35243859 17 121759470 106779600 114089288 105117832 18 331072990 265198150 304434061 279216083 19 840669610 894584372 825140032 937563530 20 2642381430 2092703550 2399386239 2199738033 21 7227151280 6573888100 6988332717 6459329037 22 20931973090 16491425740 18844561759 17267339059 23 55793302330 51185968460 54473434666 50419312152 24 164764171030 129673789110 147471723662 135162732506 Table 2 The number of fixed origin n-step SAWs for various starting points X1_ in the rhombic Penrose tiling, with starting point coordinates (x, y, z) given in the internal space.
A. J. Guttmann
80
Figure 6. Examples of ll'0
...
W 7 for a particular walk on the Arnrnann-Beenker tiling.
7.4. Mean number of walks Given that a pair of vertices from A(W) are adjacent if and only if they arc adjacent in L , one sees that the neighbours of a vertex with image XJ. can be found by sequentially adding the E J. image of all possible edges in L to XJ. and testing if the new points lie in W. If it docs the adjacent vertex exists in A(W). By recursively checking all possible neighbours of a vertex, all possible walks on the lattices will be found. Given an origin vertex x 0 in A(W), we know its E J. image x~ must lie somewhere in lV, i.e. x~ E W 0 = W (Wn is the region x~ can lie in, given our knowledge of then steps in the walk). If one takes a step s (with projection s.l. onto E J.) to a possible adjacent vertex x 1 , then one knows x}. = x~ + s J. . Furthermore if x 1 E A(W) is true, then x}. E W. Hence W 1 = (W n (lV 0 + SJ.)) - S _L . The probability that the step sis possible from a random x 0 is given by the ratio between the areas of H' 1 and w·. Extending this to a walk of length n with steps si, i = 1 . .. n, Wk = (W n (wk-J + L~=l si))- L~= l si, the probability of the walk exist.ing is the ratio of the areas of wn and lf'. For example, consider the shaded Wi in Figure 6, for the particular walk in the Ammann-Beenker tiling which steps west, south, south-west, north-west, west, north then north . For the Ammann-Bcenker tiling, these probabilities are of the form a+ b).., where )., = 1 + V2 and a, b E Q. Adding the self-avoiding constraint and summing the probabilities results in the expected number of SAWs beginning at a random origin. This process leads to mean numbers of the form a+ bT, where 7 = (1 + ...;5)/2 is the golden ratio, and a, b E Z. The rhombic Penrose tiling also allows steps in t.en directions, more than the AmmannBcenkcr tiling's eight. These facts combine to allow greater length series to be computed on the Ammann-Beenker tiling.
SAW in constrained and random geometries
81
Ammann-Beenker n rhombic Penrose 1 1 0 1 4 4 52-16.\ 2 62 -307 80-16.\ 3 -4 + 287 444-134.\ 4 914-4887 1280-380.\ 5 -820 + 7327 4492-1430.\ 13842 -78947 6 10848-3248.\ 7 -17732 + 128607 60988-21700.\ 173876 -1019887 8 89800-27036.\ 9 -255784 + 1737207 10 643248-237732.\ 1923078-11439887 11 979776-324200.\ -3149856 + 20731927 12 5486960-2043420.\ 19566548-117347067 13 10785736-3819788.\ -34951044 + 226129927 14 45253532-16927618.\ 192557132-1161512747 15 110294592-40576780.\ -366912524 + 2348039047 16 375796808-141368464.\ 17 1058437232-398339560.\ 18 3259350860-1238175678.\ 19 9526156024-3632872284.\ 20 29127575440-11192322668.\ 21 81536068712-31337365980.\ 22 259724099656-100797073134.\ Table 3 The mean number of n-step SAWs for the Ammann-Beenker tiling and the rhombic Penrose tiling, where A = 1 + v'2 and 7 = (1 + v'5) /2.
7 .5. Total number of walks Investigating all possible walks as in the mean number of walks method, Rogers et al. [46] counted instead the number of non-zero contributions to the mean value. This counts the number of translationally inequivalent walks with wn having positive area or, equivalently, the number of translationally inequivalent walks which may occur anywhere in the tiling.
A. J. Guttmann
82
Penrose n Ammann- Beenker 1 1 0 1 10 8 2 56 90 3 288 560 4 2800 1280 12060 5344 5 20288 48520 6 182000 74192 7 260336 8 658300 892800 2282400 9 10 2976512 7749440 11 9828256 25634920 12 31758112 83615140 13 101847216 268113660 14 322240144 850895040 15 1012048208 2668534600 16 3147031584 17 9732815728 18 29852932384 19 91182029360 20 276695822928 21 836719766336 22 2516664888416 Table 4 The total number of n-step SAWs for the Ammann-Beenker tiling and the rhombic Penrose tiling.
7.6. Self-avoiding polygons A self-avoiding polygon is equivalent to a self-avoiding walk in which the two degree one vertices are adjacent. In the enumeration of self-avoiding polygons, one must distinguish between polygons having, up to a translation, the same boundary but different fillings of the interior. SAPs may be enumerated in the same manner as SAWs. For the computation of occurrence probabilities of self-avoiding polygon patches, all vertices of the patch were taken into account. If the self-avoiding walk patch has n vertices xi E A(W), where i = 1, ... , n, the acceptance domain is wn = n;(W- x~), and the occurrence probability is given by the ratio between the areas of wn and W, see also [40]. The various sequences were analysed using standard methods of asymptotic analysis of power series expansions as described in [38]. For self-avoiding walks and polygons, it is easy to prove that the limit liiDn-+oo(cn) 1/n exists by use of concatenation arguments [4]. We assume the usual asymptotic growth of the sequence coefficients Cn, viz:
(n-+oo,O<E::=;l).
(51)
SAW in constrained and random geometries
83
mean number total number mean number total number n 4 10 2 4 8 4 48 80 8 8 12.\ 384 108-487 840 6 240-647 6480 800-272.\ 2960 8 49760 10 2840-880.\ 21600 6192-33647 170256 25584-132487 394080 12 28152-9984.\ 14 47712-9884.\ 1322048 179200-953407 3087140 869600-299392.\ 10194720 162976-54407 24020160 16 18 215712+294408.\ 183529440 79960896 2704140-10675807 20 14980920-3730840.\ 618248240 22 152588920-47048100.\ 4726263168 Table 5 The mean number of n-step SAPs and the total number of SAPs for the Ammann-Beenker tiling (first two columns) and for the rhombic Penrose tiling (last two columns), where A = 1 + v'2 and 7 = (1 + J5) /2.
On the square lattice, there is overwhelming evidence [52] of the above asymptotic behaviour with exponent 1 = 43/32. There is however no proof of this assumption. The above assumption results in the following asymptotic growth of the ratios r n Cn 1 'rn=-=Cn-1 Xc
-1-o)]
[1+--+0(n 1 - 1 n
(n--+ oo,O <
E
S 1),
(52)
which may be used to extrapolate numerical estimates of Xc and 1· Whereas it has been proved for the square lattice that the limit limn-+oo en/ Cn- 2 exists and coincides with x;;- 2 [62], a similar statement for the ratios r n is not known. Indeed, for some lattices, counterexamples are known [63]. Fig. 7 shows a plot of the ratios r n against 1/n for a typical fixed origin AmmannBeenker walk (full circles) and for the ratios of the mean numbers of Ammann-Beenker walks (large empty circles). Notice that the fixed origin data suffers from dramatic fluctuations, which are smoothed out by averaging, but are still larger than the corresponding square lattice data [52], which is shown in small circles. The oscillating behaviour of the mean number of walks data is due to an additional singularity of the sequence generating function at x = -xc, which, for the case of the square lattice, is well understood due to anti-ferromagnetic ordering [52]. To obtain estimates of Xc and 1, Rogers et al. [46] used the standard method described in [38] and first mapped away the singularity on the negative real axis by an Euler transform and then used Neville-Aitken series extrapolation. They also used the method of differential approximants (DAs) [38]. The underlying idea is to fit a linear differential equation with polynomial coefficients to the generating function of the sequence, truncated at some order n 0 . In Table 6 the results for the DA analysis are listed. The estimate using the data for the mean number of walks yields the most precise estimates, which are, however, one order of magnitude in accuracy worse than the corresponding estimates for square lattice SAWs, given such a series of equal length.
A. J. Guttmann
84
2.95
• •
2.9 2.85 2.8 2.75
2.7
••• •
0 • 0
~F:Pooo
0 0
0
0
0
0
·~··0
•••• •
• 0 0
i
0 0
o
2.65
•
2.6 0.05
0.1
0.15
0.2
Figure 7. Ratio plot of en/en- I against 1/n for fixed origin (full circles) and mean number (large empty circles) Ammann-Beenker SAWs. The square lattice SAW data is plotted in small circles for comparison.
mean no. (0,0) (1- ~.~) (1,0) (1 - ../2,0) 0.36414(18) 0.3659(14) 0.3644(13) 0.3644(21) 0.3657(16) 1.46(17) 1.325(19) 1.45(16) I 1.30(14) 1.32(17) 'Y Table 6 Estimates of Xc and 'Y for Ammann-Beenker SAWs. Numbers in brackets denote the uncertainty in the last two digits. Xc
An analysis of the total number of SAWs on the Ammann-Bcenker tiling using first order DAs yielded Xc = 0.3647(33) and 'Y = 3.14(37). Whereas the critical point estimate is consistent with the previous analysis, the exponf'.nt estimate deviates from the value of 43/32 = 1.34375 for fixed origin SA\Vs or the mean number of SAWs. This phenomenon reflects the fact that the number of Arnmann-Beenker patches of radius r grows asymptotically as r 2 • {lvlore generally, for aperiodic Dclone sets in JRd described by a primitive substitution matrix, the number N(r) of patches of radius r grows like N(r) ~ rd [57,56].) Since the SA\V has fractal dimension 4/:J, we expect an asymptotic increase of the number of SAWs by n 2"', where v = :J/4. Thus /'= 43/32 + 2v = 2.84:W5. The above estimate is consistent with this value. The analysis of SAP data followed the same linf'~'>. However, the estimates suffered from large finite size errors due to the low number (11) of available coefficients. First. order differential approximauts for the mean number of SAPs yielded Xc = 0.3688(41). For the critical exponent o: = 2 + f' , one expects by universality that o: = l/2, being the believed exact value for the square lattice (and numerically confirmed to very high precision [64]}. Due to lack of data it is not possible to give estimates of critical exponents. An analysis of the total number of SAPs on the Ammann-Beenker tiling using first order DAs yields Xc = 0.3587(15).
SAW in constrained and random geometries
85
The above analysis has also been applied to the rhombic Penrose tiling data. That data displays qualitatively the same finite size behaviour as the Ammann-Beenker tiling data. In Table 7 estimates of Xc and 1 obtained by analysing first order differential approximants are given. (0.5, 0.5, 1) mean no. (0, 0, 0) (0,0, 1) (0.25, 0.5, 0) 0.36322(29) 0.3621(12) 0.3613(16) 0.36248(83) 0.36347(51) 1.28(14) 1.387(62) 1.333(26) 1.19(22) 1.303(83) 1 Table 7 Estimates of Xc and 1 for Penrose SAWs. Numbers in brackets denote the uncertainty in the last two digits. Xc
An analysis of the total number of SAWs on the rhombic Penrose tiling using first order DAs yielded Xc = 0.3638(31) and 1 = 2.77(21). The estimate of the critical point is consistent with the estimates from the other methods of counting. For the critical exponent, one again expects a value of 1 = 43/32 + 2v = 2.84375, which agrees with the extrapolation within numerical accuracy. For the analysis of SAP data, only 9 series coefficients are available. First order differential approximants for the mean number of SAPs yielded Xc = 0.372(11). An analysis of the total number of SAPs on the rhombic Penrose tiling using first order DAs yielded Xc = 0.3590(22). Again, due to lack of data, it is not possible to obtain reasonable estimates for the critical exponent a. In summary, it turns out that averaging with respect to the occurrence probability in the whole tiling leads to the best estimates of the critical parameters, whereas data produced by fixing an origin leads to strong finite size oscillations. The results support the universality hypothesis that the critical exponents appear to be the same as for the square lattice, within confidence limits. For the total number of walks (polygons) a new exponent was found, reflecting the polynomial complexity of the number of patches of the underlying tiling. On a mathematically rigorous level, it may be possible to show the equality of the critical points for SAPs and SAWs on quasiperiodic tilings by appropriately modifying the existing proofs for the hypercubic lattice [68]. Furthermore, it would be interesting to carry out an analysis to determine if random walk behaviour can be proved for dimensions greater than four [4].
8. SELF-AVOIDING WALKS AND POLYGONS ON HYPERBOLIC LATTICES In two Euclidean dimensions, if we require that every site and every bond of a lattice be equivalent, there are only three possible choices. The square, triangular or honeycomb lattice. On the Poincare disk, a space with negative curvature, there are infinitely many possibilities. We wish to construct lattices with equilateral faces and given co-ordination number. The formal geometric construction is succinctly given by Rietman, Nienhuis and Oitmaa
A. J. Guttmann
86
[65] and we quote: "These lattices can be embedded in a homogeneous two dimensional space with constant Ricci curvature. When the curvature is positive, this is the sphere S 2 • After Reimann projection z = X + iy = ei.P tan~ this becomes the plane with the metric g"v(x, y) = (l~;..y. When the curvature R = 0 we simply have the Euclidean plane with the metric g"v(x, y) = 8"nu· When the curvature is negative we may choose the hyperboloid embedded in a 2 +!-dimensional Minkowski space, given by
x2
+ y2 -
z2 = - ~
4'
z
>~
- 2
(53)
This hyperboloid can be parameterized with coordinates 1j; E [0, oo), ¢ E [0, 211-): X = sinh 1j; cos ¢/2, Y = sinh 1j; sin ¢/2, Y = cosh 1/;, and in terms of z = x + iy = ei.P tan* it becomes the circular disc lzl < 1 with metric g"v(x, y) = (l~;:z) 2 • The geodesics of this metric are circular arcs (in the fiat metric) orthogonal to the circle at infinity, lzl = 1. The distance between two points P 1 and P2 is given by
(54) Note that one is not restricted to the unit disk, though that is the geometry utilised by Swierczak and Guttmann [66] in their study of SAW on the hyperbolic lattices. A conformal mapping would enable geometries other than the unit disk to be considered. Hyperbolic lattices are characterised by two integers, both greater than 2. One is the number of neighbours of each vertex, v, and the other is the number of sides of each face, denoted f. Lattices are described by the ordered pair (!, v ). Another, equivalent, view is that the lattice consists of f-gons, with v of them meeting at each vertex. Such a lattice can be embedded in the sphere, plane or hyperboloid whenever (v- 2)(!- 2) is, respectively, less than 4, equal to 4 and greater than 4. Thus the familiar Euclidean triangular, square and hexagonal lattices are, in this notation, the (3, 6), the (4, 4) and the (6, 3) lattices respectively. There are five possible lattices on the sphere, corresponding to the five Platonic solids. For example, the (3, 3) lattice is the tetrahedron. The construction of the hyperlattices is rather subtle. As the boundary of a finite section of the lattice contains a finite fraction of the vertices in the finite section, the usual concept of thermodynamic limit becomes rather suspect. In fact, as we increase the "size" of the lattice, the surface to volume ratio remains non-zero, unlike Euclidean lattices, where this quantity vanishes. This is reminiscent of a Cayley tree, and from this observation alone it should not be too surprising that one observes "mean-field" type exponents, as this is the usual manifestation of this "infinite-dimensional" characteristic. From an algebraic viewpoint, the symmetry group of the lattice is generated by two rotational elements. One is a rotation about a given lattice point by 21rjv, and the other is a rotation about the centre of an adjacent face by 27r /f. The group generated by these two operations is a subgroup of the group of isometries of the unit disc in the complex plane, notably the projective group PSU(l, 1) = SU(l, 1)/{I, -I}. Further details of the group, and group action, can be found in [65]. In [66] both SAW and SAP on the {3, 7}, the {5, 5} and the {6, 3} lattices were investigated. In addition, an irregular hyperbolic lattice, denoted T(2, 3, 7) was studied. This
SAW in constrained and random geometries
87
last lattice is a triangulation of the { 8, 4} lattice, in which each octagon is triangulated with 16 triangles, having a common vertex at the centre of the octagon. Enumeration was done by backtracking. In backtracking algorithms on Euclidean lattices, the only problem is speed. Memory demands are minimal. As the algorithm is of exponential complexity, it is that which limits progress. When enumerating on hyperbolic lattices, one has the additional problem of exponential growth in memory requirements. Two strategies to limit this problem are discussed in [66]. For all lattices, series analysis yielded a simple pole as the singularity of the SAW generating function. This is of course different to the Euclidean situation, where the corresponding exponent 1 = ~· Furthermore, the growth constant for SAW and SAPs for a given lattice is different. Surprisingly, the exponent for SAP appears to be the same as for Euclidean lattices, that is a = ~· For the {6, 3} lattice it was possible to obtain an exact solution. This lattice has a tree-like dual structure, which accounts for the solvability. For SAP on this lattice the generating function was found to be
P(x) =
1 - 2x 4
-
·
\
1(1 - 4x 4 )
2x
= x 5 + 2x 9 + 5x 13 + ....
This is immediately recognisable as the generating function for Catalan numbers, Cn = n~l Thus the connective constant is /-tp = 1/ v'2,, and a = 1/2. For SAW the generating function is of a similar form, viz:
e:).
C(x)
=
A(x)
+ B(x)y'(1- 4x 4 ) x 6 C(x)
4
= 1 + 3x + 6x 2 + 12x3 + 24x + ... ,
where A(x), B(x) and C(x) are polynomials of degree 12, 8 and 5 respectively. The growth constant is in this case given by a zero of the denominator polynomial, rather than by the branch point of the numerator, as was the case for SAPs. In [66] the value Xc = 1/ p, = 0.513839377 ... is given, which, as can be seen, is greater than the corresponding result for SAP. The origin of the simple pole singularity is also clear. For the sake of completeness, we remark that in [65] similar results for the Ising model on a hyperbolic lattice were obtained, while in [67] considerable progress is made in understanding many of the strange properties of SAW, and indeed percolation, on such lattices.
9. BIOLOGICAL MODELS: SAW AS MODELS OF DNA By constraining self avoiding walks in various geometries, and introducing interactions by appropriate fugacities, such SAW turn out to be useful in modelling a surprising variety of biological situations. In this section we will consider just three. One of the first, historically, is the modelling of the denaturation of DNA. When a solution of DNA is heated, the double stranded molecules denature into single strands. In this process, "looping out" of AT rich regions of the DNA segments first occurs, followed eventually by separation of the two strands as the paired GC segments denature. This denaturation process corresponds to a phase transition [69]. A simple model of this DNA denaturation transition was introduced in 1966 by Poland and Scheraga [70,71] (hereinafter referred to as PS) and refined by Fisher [72,73]. The model consists of an alternating sequence (chain) of straight paths and loops, which
88
A. J. Guttmann
idealize denaturing DNA, as a sequence of double stranded and single stranded molecules. An attractive energy is associated with paths. Interactions between the different parts of a chain and, more generally, all details regarding real DNA such as chemical composition, stiffness or torsion, are ignored. It was found that the phase transition is determined by the critical exponent c of the underlying loop class. Due to the tractability of the problem of random loops, that version of the problem was initially studied by PS [71]. The model displays a continuous phase transition in both two and three dimensions. It was argued by Fisher [72] that replacing random loops by self-avoiding loops, suggested as a more realistic representation accounting for excluded volume effects within each loop, sharpens the transition, but does not change its order. We shall investigate this particular model in a little more detail below. Another early example of a biological problem modelled by SAW-or in this case SAP, was the study of vesicles by Leibler and Fisher, [74], Fisher et al. [75] and Banavar et al. [76]. A vesicle is a biological object such as a blood cell whose behaviour is mediated by pressure. If the internal pressure exceeds the external pressure the cell will be inflated, whereas in the opposite situation it will be collapsed. Clearly, there will be a phase transition that occurs at a critical value of the pressure. Early series studies were largely restricted to two dimensions, the cell was modelled by a two-dimensional SAP, and the pressure induced phase transition was mediated by associating a fugacity with the enclosed area. Thus by varying this fugacity, polygon shapes of minimal area, which are long and thin, could be achieved, as could those of maximal area, which are effectively square. We will consider this problem in a little more detail below. Our third and final example is the use of SAW to model the micromanipulation of polymer molecules, particularly DNA, attached to a surface. In this situation, optical tweezers [77,78] are used to pull the adsorbed biological molecule from the surface. This force is applied perpendicular to the adsorbing surface and will favour desorption. It is reasonable to expect some sort of a phase transition. At low levels of the force, the polymer remains adsorbed, but at higher levels it will be desorbed. There will be a temperature dependent force fc(T) between these two states. The shape of the force-temperature curve is of considerable interest, and can be considered a phase boundary in the T - f plane. This can be modelled by a SAW, tethered to a wall, with a fugacity associated with nearest-neighbour bonds, subject to a force perpendicular to the wall, as shown in the figure below.
9.1. The DNA denaturation transition With the advent of efficient computers, it has recently been possible to simulate analytically intractable models extending the PS class, which are assumed to be more realistic representations of the biological problem. One of these is a model of two self-avoiding and mutually avoiding walks, with an attractive interaction between the different walks at corresponding positions in each walk [79-82]. The model exhibits a first order phase transition in two and three dimensions. The critical properties of the model are described by an exponent c' related to the loop length distribution [83-85,80-82], see also Fisher's review article [73]. For PS models, this exponent coincides with the loop class exponent c if 1 < c < 2. Within a refined model, where different binding energies for base pairs and stiffness are taken into account, the exponent c' seems to be largely independent of
89
SAW in constrained and random geometries . ........ ................. .. .. .. .. . .. .. . . ...
Figure 8. A SAW model of a polymer subject
to
an elongational force.
the specific DNA sequence and of the stiffness of paired walk segments corresponding to double stranded DNA parts [80]. There are, however, no simulations of melting curves for known DNA sequences which are compared to experimental curves for this model. An approximate analytic derivation of the exponent related to this model was given by Kafri et al. [83- 85] using the theory of polymer networks. 'They estimated the excluded volume effect arising from the int<~raetion between a single loop and two attad1ed walks. This approach (refined recently [81]) yields an approximation of the loop length distribution exponent d, which agrees well with simulation results of interacting self-avoiding walk pairs [80- 82]. There is a recent debate about the relevance of this approximation to real DNA [86]. The polymer network approach, as initiated previously [83], led to a number of related applications [84,85,81,87,88]. Some care needs to be taken in studying the literature on this problem, which may be misleading. This is discussed in considerable detail in Richard and Guttmann [89]. Loop classes discussed in the early approaches [71,72] are classes of rooted loops and lead to chains which are not self-avoiding. This seems unsatisfactory from a biological point of view, since real DNA is self-avoiding. Secondly, the common view holds that PS models with self-avoiding loops cannot display a first order transition in two or three dimensions. In fact, this view led to extending the PS class [79,83- 85] in order to find a model with a first order transition. However this view is incorrect, as demonstrated [89] by a self-avoiding PS model with self-avoiding loops. Thirdly, the two exponents c and d, extracted from different expressions as described above, are used in the literature without distinction, although there arc subtle differen(:es. In [79] a SAW model ofthis transition was introduced. Canso et al. [79] considered pairs of SAW on the simple cubic lattice, with a common origin, which are allowed to overlap only at the same monomer position along each chain. That is to say, if one numbers the ·monomers from the common origin 0, 1, 2, .. . i, .. . n, the two chains are mutually and individually self-avoiding except possibly where site i of the first monomer coincides with site i of the second monomer. Such overlaps are encouraged by a fugacity t:, associated with such contacts. As the temperature increases, there is a transition temperature Tm above
A. J. Guttmann
90
which the entropic advantage in breaking such bonds overcomes the attractive energy. From extensive simulations, it was concluded [79] that the transition is first order, as the energy density at the transition was found to be discontinuous. 9.2. Poland-Scheraga models: general formalism Consider a discrete model, defined on the hypercubic lattice zd. Double stranded DNA segments are modeled by paths, and single stranded DNA segments are modeled by loops on the edges of the lattice. Each loop is assumed to have two marked vertices to indicate where paths are attached. Any alternating sequence of paths and (marked) loops is called a chain. A PS model consists of all chains obtained by concatenation of paths and loops from a given path class and a given loop class, where the initial segment and the final segment of a chain are both paths. Note that, in general, such chains are not self-avoiding, in contrast to real DNA. Self-avoidance may be violated by paths, by loops, or by the way segments are concatenated. A chain is called segment-avoiding if there are no overlaps, i.e., every two non-neighboring segments have no vertex in common, and every two neighboring segments have exactly the marked vertex in common. We call a PS model self-avoiding if paths and loops are self-avoiding and if all chains of the model are segment-avoiding. The requirement of self-avoidance restricts the admissible path classes and loop classes. A simple subclass of self-avoiding PS models are directed PS models: We call a chain directed if there is a preferred direction such that the order of the chain segments induces the same order on the vertex coordinates (w.r.t. the preferred direction), for each pair of vertices taken from two different chain segments. Such chains are then segment-avoiding. We call a PS model directed if paths, loops and chains are directed. For a given PS model, let Zm,n denote the number of chain configurations with m contacts and length n. The generating function is defined by
(55) m,n
where the fugacity xis conjugate to the chain length n. The Boltzmann factor w = e-EfkT takes into account the attractive interaction (achieved by setting the energy E < 0) between bonds. T is the temperature, and k is Boltzmann's constant. Over the relevant temperature range 0 < T < oo, we have oo > w > 1. Note that there is no interaction between different segments in a chain. The generating function Z(x, w) can be expressed in terms of the generating functions for paths V(x) and loops U(x). These are 00
V(x) =
L bnxn, n=O
00
U(x) =
LP2nXn,
(56)
n=l
where bn is the number of paths of length n, and p 2 n is the corresponding number of loops. Due to the chain structure, one gets a geometric series in V(wx)U(x),
~ n V(wx) Z(x,w) = 1- U( XV ) (WX ) = ~Zn(w)x. n=l
(57)
91
SAW in constrained and random geometries
Since we want to analyze phase transitions of the model, which can only occur in the infinite system, we define the free energy of the model as .
1
f(w) = hm -logZn(w) = -logxc(w), n-+oo n
(58)
where, for fixed w, xc(w) is the radius of convergence of Z(x, w). Concatenation arguments and supermultiplicative inequalities can be used to show that the free energy exists [90]. We will alternatively investigate properties of the free energy in terms of the generating functions for paths and loops. It is instructive to consider an exactly solvable model, which is provided by the following fully directed model. 9.3. Fully directed walks and loops This directed model consists of fully directed walks for the paths in the chain. These are clearly self-avoiding, and only take steps in positive directions. The corresponding loops are staircase polygons, which consist of two fully directed walks, which do not intersect or touch, but have a common starting point and end point. Paths are attached to these points. We distinguish the two strands of a loop. In d = 2, the generating functions for paths and marked loops are [90] 1
V(x) = 1- 2x'
U(x) = 1- 2x- v'1- 4x.
(59)
U(x) is twice the generating function of staircase polygons. The loop class exponent is c = 3/2. The generating functions V and U have critical points xv = 1/2, xu = 1/4 and We = 1, i.e., a phase transition occurs at T = oo. (If the empty path would not be allowed, a phase transition would occur at a finite temperature). The free energy f(w) is given by
2(w-1? ) 1 ( )1+(w-1)2-1 f(w)=og
(1::::; w < oo).
(60)
The fraction of shared bonds follows as
O(w) = __33!!...._ )1 + (w- 1)2- 1- (w- 1)2/2 w-1 1+(w-1)2-J1+(w-1)2
(1:S:w
which approaches zero linearly in w- 1. The asymptotic behavior of Zn(w) about w is given by
(61)
=1 (62)
uniformly in w, where ha(x) = )rr + axe(ax) (1 + erf( ax)) and erf(x) is the error function. As shown in [89], this PS model of fully directed walks and loops is exactly solvable in arbitrary dimension. The phase transition is found to be first order for d ~ 6, to have a continuous phase transition in dimensions 2 ::::; d ::::; 5, and to occur at finite temperature in d ~ 4 only. 2
A. J. Guttmann
92
Since the critical behavior of PS models is essentially determined by the properties of loops, PS [70, 71], and later Fisher [72], were led to consider various loop classes (together with straight paths for the double stranded segments). Whereas PS analyzed loop classes derived from random walks, Fisher considered loop classes derived from self-avoiding walks. 9.4. Loops and walks An oriented, rooted loop of length n is a walk of length n - 1, whose starting point and end point are lattice nearest neighbors. We identify such loops if they have the same shape, i.e., if they are equal up to a translation, possibly followed by a change of orientation. These objects we call unrooted, unoriented loops, or simply loops. Each loop of length n has at most 2n corresponding walks. If the walks are self-avoiding, each loop has exactly 2n corresponding walks. The number of loops of length n is denoted by Pn· For example, for self-avoiding loops on 7!} we have p 4 = 1 and p 6 = 2. For a given class of walks, the above description defines the corresponding (unmarked) loop class. Within a chain structure, two paths are attached to each loop. Different choices for attachment positions increase the number of (marked) loop configurations to Pn 2: Pn- If we assume the DNA condition that the two paths attached to a loop bisect it into pieces of equal length, then the number of possible attachments of two paths to a loop of length n is less than or equal to 2n. (We distinguish the two strands of a marked loop). PS and Fisher consider classes of oriented rooted loops. Self-avoiding oriented rooted loops can be interpreted as loops with 2n possible attachment positions of paths to a loop of length n. A similar interpretation for oriented rooted random loops is not obvious. We stress that these loop classes result in chains which are not segment-avoiding, as paths will intersect the loops. Both models cannot therefore represent real (self-avoiding) DNA. 9.5. Oriented rooted random loops The first simple example of loops, discussed by PS [71], is that of oriented rooted random loops derived from random walks. Random walks on zd have the generating function Vd(x) = 1/(1- 2dx). The asymptotic behavior of the number of oriented rooted random loops of length 2n is given [4] (Appendix A) by
(n-+ oo).
(63)
This implies no phase transition in d = 2 and a continuous phase transition in d = 3. The PS results led to the question [72] whether accounting for excluded volume effects within a loop increases the loop class exponent c, which might change the order of the phase transition. This led to considering self-avoiding loops, which are loops derived from self-avoiding walks. By definition, the self-avoiding loop class fully accounts for excluded volume interactions within a loop. In d = 3, self-avoiding loops of length n 2: 24 may be knotted. Fisher considered oriented rooted loops fi2n = 4np2 n. Their loop class exponent c, fi2 n "' Bp,Jn-c, is related to the mean square displacement exponent v of self-avoiding walks by the hyperscaling relation c = dv. For unknotted self-avoiding loops, which is the preferable model from a biological point of view, it has been proved [32] that the exponential growth constant is strictly less than that of all self-avoiding loops, while the exponent (if it exists) is expected to coincide with that of all self-avoiding loops.
SAW in constrained and random geometries
93
Fisher concluded that the above values of the loop class exponent c imply a continuous phase transition in d = 2 and d = 3. In [89] it is shown that the phase transition condition is satisfied in both d = 2 and in d = 3 for the case when the double stranded segments are treated as straight paths. Note that in d = 2, self-avoiding walks as paths will result in no phase transition. The previously discussed PS models are not fully self-avoiding, in that the chains are not mutually avoiding. Using SAW for walks, a PS model with segment-avoiding chains may be defined as follows: Take only those walks with extremal first and last vertex. That is to say, if v(O), v(1), ... , v(n) are the vertices of ann-step walk v, this walk is taken as a path iff vx(O) < vx(i) < vx(n) for all 1 < i < n. Such walks are bridges [4] (Sec. 1.2), whose last step is in the x-direction. For (unmarked) loops, take loops derived from the walks, that is, SAP. Marking of the loops may be achieved in different ways. To this end, consider for a given loop the sets At (Ar) of vertices of smallest (largest) x-coordinate. We distinguish four different types of marking: complete marking (with DNA constraint), where we mark a loop at all vertex pairs from At and Ar (whenever the DNA condition is satisfied), and unique marking (with DNA constraint), where we only mark a loop at a single vertex pair, for example the bottom vertex and the top vertex in a lexicographic ordering (if they satisfy the DNA condition). Unique marking would then imply Pn = 2pn (we distinguish the two strands of a marked loop) and hence increase the previous exponents by one. Hence, such a PS model displays a first order transition with c = 5/2 in d = 2 and with c = 2.7631(18) in d = 3. Unique marking with the DNA constraint results in Pn :<:; 2pn- If we assume that the exponential growth constant for marked self-avoiding loops is given by the usual SAP growth constant, this implies a critical exponent c greater or equal to the model with unique marking, i.e., a first order phase transition in d = 2 and d = 3. For complete marking, we have 2pn :<:; Pn :<:; 2npn, which rules out a decrease of c by more than one. However we expect that the number of possible markings is of order 1 as n -+ oo, and so cremains unchanged, and the model again displays a first order phase transition. Similar considerations apply for the case of complete marking with DNA constraint. In summary, as we have shown in brief, and as shown in more detail in [89], a selfavoiding PS model (with unique marking, with self-avoiding bridges and (unrooted) selfavoiding loops) as defined above yields a first order phase transition in both d = 2 and d = 3. For the presumably more realistic model of pairs of interacting self-avoiding walks [79,83~85,80~82], the results of [89] suggest an interpretation of excluded volume effects, which complements the common one [83~85]. The self-avoiding PS models discussed above correctly account for excluded volume effects within a loop, but overestimate excluded volume effects between different segments of a chain, due to their directed chain structure. Since this leads to a first order phase transition in d = 2 and d = 3, one can conclude that the relaxation of excluded volume effects between different segments of the chain does not change the nature of the transition. The simpler, directed model displays a first order phase transition in d = 2, but such models seem to be of limited relevance to the biological problem due to their directed structure and other limitations. In conclusion, the question of the mechanisms applying in real DNA which are respon-
94
A . J. Guttmann
sible for the denaturation process and which explain multistep behavior as observed in melting curves, are still far from being satisfactorily answered in our opinion, though models based on SAW and SAP still seem to be the most appropriate way to study this important problem.
9.6. Vesicles collapse In modelling vesicle collapse by SAP, let Pm(n) be the number of SAP per site on an infinite lattice, with perimeter m enclosing area n. In [75] it was proved that the free energy lim
m-+oo
2._ PmnQ" m log"'"""' L...,; 1
:= K(q)
(64)
n
exists and is finite for all values of the fugacity q :::::; 1. Furt.her, K(q) is log-convex and continuous for these values of y and is infinite for y > 1. In terms of the natural two-variable generating function
(65) x ,q
it was further proved that for q < 1, P(x,q) converges for x < e-~<(q) , while for q > 1, P(x, q) converges only for x = 0. The expected phase diagram is shown in figure 9 below .
• ()
0
1
q
Figure 9. The phase diagram showing the phase boundary Xc(q).
In the region below the phase boundary, the polygons are ramified objects, closely resembling branched polymers. That is to say, they are collapsed and string-like. As q
SAW in constrained and random geometries
95
approaches unity, they fill out more, and become less string-like. At q = 1 one has pure SAP. For q > 1 the polygons become "fat", and approximate squares, with their average area scaling as the square of their perimeter. In [75] rigorous upper and lower bounds to the shape of the phase boundary were found, and the locus of the actual phase boundary was found numerically from extrapolation of SAP enumerations by area and perimeter. Iri the extended phase q = 1, the mean area of polygons (a)m of perimeter m grows asymptotically like m 312 , whereas it grows like m in the deflated phase q < 1. It can be shown that in the limit q ---+ 0 the generating function is dominated by polygons of minimal area. Since for SAPs these polygons may be viewed as branched polymers, the phase q < 1 is also referred to as the branched polymer phase. This change of asymptotic behaviour is reflected in the singular behaviour of the perimeter and area generating function. Typically, the line q = 1 is a line of finite essential singularities for x < Xc. The line xc(q), where P(x, q) is singular for q < 1, is typically a line of logarithmic singularities. For branched polymers in the continuum limit, the logarithmic singularity has been proved recently in [91]. Of special interest is the point (xc, 1) where these two lines of singularities meet. The behaviour of the singular part of the perimeter and area generating function about (xc, 1) is expected to take the special form
P(x, q) "-' p(reg)(x, q)
+ (1- q) 8 F((xc- x)(1- qt<~>),
(x, q) ---+ (x;:-, 1-),
(66)
where F(s) is a scaling function of combined arguments= (xc- x)(1- q)-<1>, commonly assumed to be regular at the origin, and B and ¢ are critical exponents. The singular behaviour about q = 1 at the critical point Xc is then given by p(sing)(xc, 1) "' (1-q) 8 F(O). This scaling assumption implies an asymptotic expansion of the scaling function of the form 00
F(s)
fk s(k-B)Ir/>"
=L
(67)
k=O
The leading asymptotic behaviour characterises the singularity of the perimeter generating function via P(x, 1) "'fo(xc- x)-'Y, where 8 + ¢7 = 0. The first singularity of F(s) on the negative axis determines the singularity along the curve xc(q). The locus on the axis (say at s = sc) determines the line Xc( q) "' Xc - sc(1 - q) near q = 1, which meets the line q = 1 vertically for ¢ < 1. In [76] a similar model was studied analytically using a mapping onto a gauge model. As in other studies, they found the critical behaviour to be governed by a branched polymer fixed point. More recently, Richard, Guttmann and Jensen [92] have given a very persuasive conjecture as to the exact nature of the scaling function at the bicritical point (x~, 1). In their work, it was natural to work with rooted SAP, and in that case the conjectured form of the scaling function was found to be
Xc d . ( -1f (2E ) ~3 s) F (r )(s) = --logA1 0 1f!J
ds
Xc
(68)
A. J. Guttmann
96
with exponents B = 1/3 and ¢ = 2/3. The conjectured form of the scaling function is then obtained by integration and is
(7f
1 F(s) = --logAi - (2E0 )32 s )
(69)
Xc
1f(J
with exponents B = 1 and ¢ = 2/3. The parameters for the square lattice are !7 = 2 and Xc = 0.379052277757(5). The parameters for the hexagonal lattice are !7 = 2 and Xc = 1/ V2 + V2 (known exactly from the work of Nienhuis [5]) and for the triangular lattice 17 = 1 and Xc = 0.24091757 45(3). Further details of this important calculation can be found in [93]. 9. 7. Macromolecular desorption from a surface As briefly described above, in this situation we are applying a force perpendicular to an adsorbing surface to which a polymer chain is attached. At low temperature, surface attraction dominates, but at high temperatures entropy dominates, and the polymer is free of the surface. The temperature dependent force needed to extend the polymer is calculated. Let the polymer have N monomers, of which n lie in the surface. (In two dimensions the "surface" is a line). Let cN(n, z) be the number of such SAW whose endpoint is at perpendicular distance z from the surface. The model may be described by the partition function
(70) n,z
where w = e-
97
SA\-V in constrained and random geomet1·ies
f Desorbed
Adsorbed
T
Figure 10. A reentrant force-temperature curve.
V.8
Fmcc
Figure 11. The "stick-release" behaviour of the force-extension curve for walks of various lengths al T = 0.1. Lengths increase from the left- to the right-hand curves.
A . .1. Guttmann
98
0.8
.c:
~0.6
.3
" ~
~ 0.-l
<
0 o~~~o~2--~-0~ A--~~o.o--~~u.-.~~-L--L-~
Fot"ce
Figure 12. The "stick-release" behaviour of the force-extension curve for 50 step walks at temperatures. Temperatures increase from t.he left- to t.he right-hand curves.
cliff~>rent
dimensions, to N = 50 Jensen and Guttmann [97] obtained clear evidence of a reentrant force-temperature diagram even in two dimensions. The "stick-release" behaviour of SA\V pulled from a surface is also clearly shown in that study. In figures 11 and 12 the average length is plotted against applied force. The series of plateaus becotne smoothed out as the length of the walk increases at fixed temperature, and also as the temperature is raised for SAW of fixed lengt,h. lt can be seen from the above selection of series studies that. SAW models are capable of shedding light on a wide variety of biological problems. The range of problems that can be addressed in this way seems to be limited only by the imagination of the researchers.
CONCLUSION ln this d1apter we have discussed a reprP.sent.at.ive, but. far from exhaustive, range of problems which can be usefully modelled by SA\V in a random or rest.ricted environment, and which can be successfully studied by series methods. In most cases other approximation techniques are also applicable, and as a general rule one cau be much more confident of eonclusions drawn from approximate studies from several different methods-provided, of course, that they are self-consistent. The methods of series gCiwrat.ion and analysis is particularly powerful in determining lattice dependent properties. For example, the growth constants of SAW on various lattices is most accurately determined by series methods. Universal properties, such as critical exponents seem to be equally well determined by high-class Monte Carlo work or high class series work. In two dimensions, where finite-lattice methods (FL!vl) can be applied [50,64], series methods are extremely powerful, and likely to be better than all but the most exhaustive Monte Carlo calculations. Where interactions become very complicated, or of longer range, so that FLM type methods are difficult or impossible
SAW in constrained and random geometries
99
to apply, Monte Carlo methods are often superior. In all situations however, it is useful to seek cross validation from both methods, as well as field theoretical techniques, when applicable. As can be seen from the above examples, there is a huge range of problems to which SAW can be applied, and the outcome determined by series analysis. In this chapter we have hopefully indicated the ingenuity of many authors in such applications, and it is clear that many more interesting applications will be made in the future.
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B. Griinbaum Band G. C. Shephard, Tilings and Patterns, Freeman, New York, 1987. R. Sedgewick, Algorithms in C++, 2nd ed., Addison-Wesley, Reading, 1992. H. Kesten, J. Math. Phys., 4 (1963) 960. J. M. Hammersley, Phys. Rev., 118 (1960) 656. I. Jensen, J. Phys. A: Math. Gen., 33 (2000) 3533. R. Reitman, B. Nienhuis and J. Oitmaa, J. Phys. A: Math. Gen., 25, (1992) 6577. E. Swierczak and A. J. Guttmann, J. Phys. A: Math. Gen., 29 (1996) 7485. N. Madras and C. Chris Wu, to appear in Combinatorics, Probability, and Computing, 2005. 68. J. M. Hammersley, Math. Proc. Cambridge Phil. Soc., 57 (1961) 516. 69. R. M. Wartell and A. S. Benight, Phys. Rep., 126 (1985) 67. 70. D. Poland and H. A. Scheraga, J. Chern. Phys., 45 (1966) 1456. 71. D. Poland and H. A. Scheraga, J. Chern. Phys., 45 (1966) 1464. 72. M. E. Fisher, J. Chern. Phys., 45 (1966) 1469. 73. M. E. Fisher, J. Stat. Phys., 34 (1984) 667. 74. S. Leibler, R. R. P. Singh and M. E. Fisher, Phys. Rev. Lett., 59 (1987) 1989. 75. M. E. Fisher, A. J. Guttmann and S. G. Whittington, J. Phys. A: Math. Gen., 24 (1991) 3095. 76. J. A. Banavar, A. Maritan and A. Stella, Phys. Rev. A, 43 (1991) 5752. 77. K. Svoboda and S. M. Block, Ann. Rev. Biophys. Biomol. Struct., 23 (1994) 247. 78. A. Ashkin, Proc. Natl. Acad. Sci. USA, 94 (1997) 4853. 79. M. S. Causo, B. Coluzzi and P. Grassberger, Phys. Rev. E, 62 (2000) 3958. 80. E. Carlon, E. Orlandini and A. L. Stella, Phys. Rev. Lett., 88 (2002) 198101. 81. M. Baiesi, E. Carlon and A. L. Stella, Phys. Rev. E, 66 (2002) 21804. 82. M. Baiesi, E. Carlon, Y. Kafri, D. Mukamel, E. Orlandini and A. L. Stella, Phys. Rev. E, 67 (2002) 21911. 83. Y. Kafri, D. Mukamel and L. Peliti, Phys. Rev. Lett., 85 (2000) 4988. 84. Y. Kafri, D. Mukamel and L. Peliti, Eur. Phys. J. B, 27 (2001) 135. 85. Y. Kafri, D. Mukamel and L. Peliti, Physica A, 306 (2002) 39. 86. A. Hanke and R. Metzler, Phys. Rev. Lett., 90 (2003) 159801. 87. M. Baiesi, E. Carlon, E. Orlandini and A. L. Stella, Eur. Phys. J. B, 29 (2002) 129. 88. A. Hanke and R. Metzler, Biophysical Journal, 85 (2003) 167. 89. C. Richard and A. J. Guttmann, J. Stat. Phys., 115 (3/4) (2004) 925. 90. E. J. Janse van Rensburg, The Statistical Mechanics of Interacting Walks, Polygons, Animals and Vesicles, Oxford University Press, New York, 2000. 91. D. C. Brydges and J. Z. Imbrie, Ann. Math., 158 (2003) 1019. 92. C. Richard A. J. Guttmann and I. Jensen, J. Phys. A: Math. Gen., 34 (2001) L495. 93. C. Richard, J. Stat. Phys., 108 Nos. 3/4, (2002) 459. 94. D. Marenduzzo, A. Maritan, A. Rosa and F. Seno, Phys. Rev. Lett., 90 (2003) 088301. 95. P. K. Mishra, S. Kumar andY. Singh, Europhys. Lett., (to appear) (2004). 96. E. Orlandini, M. Tesi and S. G. Whittington, J. Phys. A: Math. Gen., 34 (2004) 1535. 97. I. Jensen and A. J. Guttmann, Stretching of a polymer adsorbed on a surface, (in preparation).
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Statistics of Linear Polymers in Disordered Media Edited by Bikas K. Chakrabarti © 2005 Elsevier B.V. All rights reserved.
Renormalization group approaches to polymers in disordered media V. Blavats'kaa, C. von Ferberb, R. Folkc and Yu. Holovatcha,cd aInstitute for Condensed Matter Physics, National Academy of Sciences of Ukraine, 1 Svientsitski Str. Lviv, UA~79011 Ukraine bTheoretical Polymer Physics, Hermann-Herder-Str. 3, 79104 Freiburg University, Germany cinstitut fiir Theoretische Physik, Johannes Kepler Universitat Linz, Linz, A~4040, Austria divan Franko National University of Lviv, 12 Dragomanov Str. UA~79005 Lviv, Ukraine This chapter focuses on the universal properties of the static critical behaviour of polymer systems with different types of complex structural disorder, in particular long-rangecorrelated defects ofrandom orientation and defects defined by bond percolation. Universal properties oflong flexible polymer chains in a solvent are well described by self-avoiding walks (SAWs) on a regular lattice. In the same way SAWs on disordered lattices serve as a model for polymer solutions in disordered media. To approach the universal content the field-theoretical renormalization group (RG) has been extraordinarily successful. We elaborate how the RG is extended to disordered systems focussing on uncorrelated and long-range correlated disorder as well as disorder that acts to confine the polymer to a percolation system. We review different field theoretical approaches as well as real space renormalization· group results. 1. INTRODUCTION AND PHENOMENA
The transfer of ideas from the theory of critical phenomena to polymer science has led to considerable progress in modelling polymers and understanding their universal properties since P. G. de Gennes seminal work (in 1972) [1] provided a first link. In the theory of critical phenomena one is interested in the peculiarities of the behavior in the vicinity of the critical temperature Tc > 0. The critical temperature essentially depends on microscopic details of the model. However, asymptotically close to Tc the thermodynamic and correlation functions for very different systems may depend on the thermodynamic parameters by the same power-like functions primarily characterized by their critical exponents (scaling laws). In contrast to the critical temperature itself, the values of the critical exponents do not depend on any of the microscopic details of the system and are determined only by the global properties of the model such as the (lattice) dimension d, the dimension m and other symmetry properties of the order parameter. Due to the fact that the critical exponents may be identical for systems with very different 103
V. Blavats'ka, C. von Ferber, R. Folk and Yu. Holovatch
104
microscopic nature, the critical exponents are called universal. Subsequently, if the critical properties of two systems are described by the same set of scaling functions and an identical set of critical exponents they are said to belong to the same universality class. The concept of universality has found its reflection in polymers, which appeared to belong to the universality class of the O(m)-symetrical spin model in the formal limit m ---+ 0. Furthermore, if one is interested in the behavior of polymers influenced by structural disorder, one of the central questions that arises concerns the possible change of the universality class under the influence of disorder. As far as scaling properties of polymers are well described by the formalism of critical phenomena, it is natural to study them by the renormalization group (RG) approach to get a quantitative description. The field-theoretical RG method has originally been developed to investigate quantum field theory problems [2]. The application of the field-theoretical RG in the theory of critical phenomena is based on the formal similarity of statistical averages at the critical point with the expectation values in quantum field theory; this is especially apparent for the perturbation theory expansion using the Feynman diagram technique [3-6]. This powerful approach is successfully applied in the study of polymer models [7]. The complementary real-space RG approach has been developed in parallel starting from the famous Kadanoff's block transformation [8] and decimation procedure. Here, we are concerned with long flexible polymer chains dissolved in a good solvent. A good solvent is defined as one, in which it is energetically more favourable for the monomers of the polymer to be surrounded by molecules of the solvent than by other monomers. As a consequence, there exists around each monomer a region (the excluded volume) in which the chance of finding another monomer is very small. It turns out, that such diluted polymer solutions have physical properties that are independent of most of the details of the chemical microstructure of the chains and of the solvent. It is well established that the universal scaling properties of long flexible polymer chains in a good solvent are perfectly described within a model of self-avoiding walks (SAWs) on a regular lattice [9]. In this model, the monomer size is represented by the lattice constant, and the size of the polymer chain by the walk length. The trajectory consists of N steps, performed towards the nearest neighbor site, taking into account that the trajectory is not allowed to cross itself. The impossibility of the trajectory to cross itself reflects the excluded volume effect. As an example, the mean square end-to-end distance R and the number of different space configurations ZN of a SAW with N steps obey in the asymptotic limit of long chains N---+ oo the scaling behavior [1,7,9]:
(1) where z is non-universal fugacity. Although the microscopic structure of the bonds is different for different types of lattices, the exponents v and 7 are universal, depending only on the space dimensionality d. In three dimensions d = 3 the exponents read [11] v = 0.5882 ± 0.0011 and 7 = 1.1596 ± 0.0020. A surprisingly good approximation for the critical exponent v of SAW in general dimension d is given by the Flory formula [10]: 3 d+2
V=--.
(2)
Renormalization group approaches to polymers in disordered media At d
=
1 one has a completely stretched chain with v
=
1. At d
105
=
2 the exact result
(v = 3/4) [13] is obtained. The upper critical dimension is d = 4, above which the polymer behaves as a random walker. The values of the universal exponents for SAWs on d - dimensional regular lattices have also been calculated by the methods of exact enumerations and Monte Carlo simulations. In particular, at the space dimension d = 3 in the frames of field-theoretical renormalization group approach one has (v = 0.5882±0.0011 [11]) and Monte Carlo simulation gives (v = 0.592 ± 0.003 [12]), both values being in a good agreement. A question of great interest is the influence of disorder in the medium on the universality class of dissolved flexible polymers, namely: are the universal exponents (1) in this case the same as in the pure case? The question of how linear polymers behave in disordered media is not only interesting from a theoretical point of view, but is also relevant for understanding transport properties of polymer chains in porous media, such as an oil recovery, gel electrophoresis, gel permeation chromatography, etc. [14]. Let us note, that there are two classes of problems, dealing with disorder, namely those with annealed and quenched disorder. In the first case, the different realizations of disorder are averaged simultaneously with a thermodynamical averaging over different conformations of the SAW. In the present review, we however focus on the case of quenched disorder [15], which is introduced such that it is not in thermodynamic equilibrium with the unperturbed system. The quantities of physical interest must then first be calculated for a particular configuration of disorder, followed by the average over all configurations of disorder. The question of the change in scaling behavior of SAWs in disordered medium can be reformulated in the frames of the field theory in the form: how does the disorder influence the critical behavior of the m-component model in the limit m ---+ 0? Numerous MC simulations [16~21], exact enumerations [22~28], and theoretical studies [29~39] which have been published since the early 80-ies [14], lead to the conclusion that there are the following regimes for the scaling of a SAW on a disordered lattice: (i) weak uncorrelated disorder, when the concentration p of bonds allowed for the random walker is higher than the percolation concentration Pc (ii) weak, but long-range correlated disorder, and (iii) strong disorder, directly at p = Pc· By further diluting the lattice to p < Pc no macroscopically connected cluster, "percolation cluster", remains and the lattice becomes disconnected. In regime (i) the scaling law (1) is valid with the same exponent v for the diluted lattice independent of p, whereas in cases (ii) and (iii) the scaling law (1) holds with a new exponent vP -1- v. For magnetic systems, the effect of weak quenched uncorrelated point-like disorder on the critical behavior is usually predicted by the Harris criterion [40]: disorder changes the critical exponents only if the specific heat critical exponent apure of the pure (undiluted) system is positive: O!pure
= 2- dvpure > 0,
(3)
Vpure being the correlation length critical exponent of the pure system. This was confirmed by numerous theoretical and experimental studies (see [41,42]). The straightforward application of the Harris criterion to the statics of SAWs indicates, that the critical exponents should be modified in the presence of any amount of lattice
106
V. Blavats'ka, C. von Ferber, R. Folk and Yu. Holovatch
disorder, as far as a for SAW on the pure lattice is positive. However, the renormalization group study [29] contradicts this prediction. The Harris criterion was then modified [43] for the quenched disordered m-vector model in the m ---+ 0 limit and leads to the conclusion, that the critical behavior of SAWs is not affected by weak uncorrelated lattice disorder. Since this point is of central interest in this review, let us consider it in more details. Let 7 denote a SAW configuration of N steps starting at some site i and R; the square of the end-to-end vector. Then for the mean square end-to-end distance of the trajectory one has:
L:R;
(R2) =
_"~_
2::1
,.._,
N2v
(4)
.
'Y
Following the analytic considerations of Harris [43]. we denote by [... ]p the average over all random configurations of the lattice, where every site is occupied with probability p and empty with probability 1- p. The SAWs are allowed to have their steps only on the occupies sites. For every such a configuration C, let us denote 7 E C for configurations 7 of N-step SAWs that start at a given site i and exists on C. The mean square end-to-end distance of the SAWs that start at point i reads:
L:P(C) C
L:P(C) C
L
r;
7EC
L
L:r; 'Y
1
L C:7EC
P(C) (5)
7EC
here P(C) is the probability of the realization of the configuration C. In case the distribution of occupied sites is uncorrelated, every SAW with N steps can be realized with probability pN. So, the last equality can be rewritten:
(6) 'Y
'Y
and the expression obtained is the same as for the pure (non-diluted) lattice. The only effect of the disorder lies in the change of the possibility of formation of each SAW with N steps, whereas the scaling properties of the realized SAW are not influenced. In the present review, we turn our attention to the three different classes of quenched (non-equilibrium) disorder, which from one side can be treated within the RG scheme and from the other side allow for a physical realization: 1) non-correlated point-like defects, 2) long-range correlated defects with a correlation function, governed by a power law "'r-a at large distances, where a is some constant 1 , 3) the case when the concentration of dilution is exactly at the percolation threshold. While the influence of short- and longrange correlated disorder can be studied by related field-theoretical models with an upper critical dimension dupper = 4, the percolation problem leads to a different field theory 1 This type of disorder allows to describe so-called extended impurities: a = d - 2 corresponds to the presence in the system of straight lines of impurities of random direction, whereas a = d -1 gives random planes of defects.
Renormalization group approaches to polymers in disordered media
107
with dupper = 6. Let us note, that in the case of m-component systems with long-rangecorrelated disorder the Harris criterion is modified: for a < d the disorder is relevant, if the correlation length critical exponent of the pure system obeys v < 2/ a. For a 2: d the original Harris criterion (3) remains valid. Again, this criterion cannot be applied directly to the polymer case, due to the peculiarities of the polymer limit. The set-up of the paper is as follows. In the following section we present the fieldtheoretical description of the polymer model, introduce different types of structural disorder into this model and present an introduction to real space renormalization. Section 3 reviews different treatments of these systems by field theoretical and real space RG approaches to analyze the scaling properties and to estimate the critical exponents. In the present review we focus on static properties of polymers. RG treatments of the impact of disorder on polymer dynamics may be found e.g. in [44~48].
2. MODELING THE SYSTEM 2.1. Polymers: Field theory with zero-components Let us consider the model that we use to describe a single polymer chain in solution. In a first discrete version we describe a configuration of the polymer by a set of positions of segment endpoints ri:
Configuration{ r 1, ... , r N} E
JRdx N.
Its statistical weight (Boltzmann factor) with the Hamiltonian H divided by the product of Boltzmann constant k 8 and temperature T will be given by [ k 1TH] expB
= exp{- 41£2 ~ L..)ri- rj_ 1 ) 2 -
d
~
(3£ 0 ~
0 j=l
od( ri- rj)}.
(7)
i#j=l
The first term describes the chain connectivity, the parameter £0 governs the mean square segment length. The second term describes the excluded volume interaction forbidding two segment end points to take the same position in space. The parameter (3 = 1/kT gives the strength of this interaction. The third parameter in our model is the chain length or number of segments N. The partition function Z is then calculated as an integral over all configurations of the polymer divided by the system volume n, thus dividing out identical configurations just translated in space:
Z(N) =
n1 IN
g
1 driexp[-k TH{ri}].
(8)
8
This will give us the 'number of configurations' of the polymer, which obey the scaling relation (1). We will do our investigations by mapping the polymer model to a renormalizable field theory making use of well developed formalisms (see [1,9,7] for example). To this end we make use of a continuous version of our model as proposed by Edwards [49,50]. The configuration of polymer is given by a path r(s) in d- dimensional space JRd parameterized by a variable 0 :::; s :::; S that has the dimension of a surface. The Hamiltonian H is then given by 1
[
1
rs
dr(s)
2
1
ksTHr]=4Jo ds(~) +6uo
I
d
2
drp(r),
(9)
V. Blavats'ka, C. von Ferber, R. Folk and Yu. Holovatch
108
where u0 is the excluded volume parameter with densities p(r) = J0 dsod(r- r(s)) . In this formalism the partition function is calculated as a functional integral: 8
Z(S)
=I
D[r(s)] exp{- k~TH[r]}.
(10)
Here the symbol D[r(s)] includes normalization such that Z(S) = 1 for u = 0. To make the exponential of o-functions in (10) and the functional integral well-defined in the bare theory a cutoff s 0 is introduced such that all simultaneous integrals of any variables sand s' on the same chain are cut off by Is-s' I> s0 . Let us note here that the continuous chain model (9) may be understood as a limit of a model of discrete self-avoiding walks, when the length of each step is decreasing £0 -+ 0 while the number of steps N is increasing keeping the 'Gaussian surface' S = N£6 fixed. The continuous chain model (10) can be mapped onto a corresponding field theory by a Laplace transform in the Gaussian surface variables S to conjugate chemical potential ("mass variables") f-L [9]:
(11) The Laplace-transformed partition function Z(t-L) can be expressed as them= 0 limit of the functional integral over vector fields ;J with m components { ¢ 1 , ... , ¢m} :
Z(t-L)
=I
D[¢(r)]exp{-1l[¢]} lm=O·
(12)
The Landau-Ginzburg-Wilson Hamiltonian 1£[¢] of this ¢ 4 -theory then reads (13) here ¢;2 ( r) = 2::;': 1 ( ¢; )2. Note that this theory is symmetric under 0 (m) transformations of the m-component vectors ¢. The limit m = 0 in (13) results in the cancellation of some special types of diagrams contributing to the perturbation theory expansions, which contains closed loops and thus are proportional to m. The same field-theoretical representation may be obtained starting from the quite different type of lattice model known as one of the basic models in the theory of magnetic systems. We present it here since for what follows it serves to introduce different types of disorder in the polymer system. Let us consider a simple (hyper) cubic lattice of dimension d, and to each site prescribe a m-component vector S(r) with a fixed length (for convenience one usually sets ISI 2 = m). Imposing a pair interaction with the energy proportional to the scalar products between pairs of spins, this defines the Stanley model (also known as the O(m) symmetric model). The Hamiltonian of this model reads [77]:
1l = -
L
JS(r) · S(r'),
(14)
where the summation is over nearest neighbor sites r, r' and J parameterizes the interaction between spins S(r) and S(r'), located at sites rand r'.
Renormalization group approaches to polymers in disordered media
109
The effective field-theoretical Hamiltonian of the model (14), obtained via special Stratonovich-Hubbard transformation, passing from the discrete system to the continuous field theory, is the same as (13). The relevant global properties of a microscopic model is represented by the structure of the effective Hamiltonian. In this case, f-Lo is a squared bare mass proportional to the temperature distance to the critical point, u 0 > 0 is a bare coupling constant. Note that the effective Hamiltonian (13) preserves the O(m) symmetry of the Stanley model (14). As far as the Stanley model is in this sense equivalent to the O(m) symmetric ¢ 4 theory, the analytic continuation m---+ 0 of this model again leads to the polymer limit. 2.2. Randomness: symmetry consideration To introduce different types of disorder in the polymer system, we start from the mcomponent model (14). The presence of impurities can be modeled by a class of site~ diluted models [51]. One considers some fraction of the sites to be occupied with some defects. The site-diluted Stanley model is introduced by the Hamiltonian:
1l = -
L JcrCr'S(r)S(r'),
(15)
(r,r') where the occupation numbers Cr are introduced, which equall when the site r is occupied and 0 when it is empty. 2.2.1. Uncorrelated disorder Let us consider the case when the values Cr in (15) are non-correlated random values distributed according to the probability distribution: (16) Cr
with: if Cr if Cr
=1 = 0.
(17)
Here p is the spin concentration and 1 - p the concentration of impurities. The pure non-diluted lattice corresponds to the case p = 1. To explain possible generalizations of the model (15) with site occupation distribution (16) let us note, that the first two moments of the distribution determine the critical behavior of the model. For the distribution (16) one gets:
(cr) = p,
(18)
g(lr- f'l) = (crcr')- (cr) 2 = p(l- p)o(r- r')
(19)
where (... ) means averaging with the distribution function (16), and o(r - r') is Kronecker's delta. Let us introduce the notation v0 = -p(l - p) for the following. As far we consider the case of quenched disorder, the free energy of the system is obtained by averaging the logarithm of the partition function Z over the disorder distribution [15]; this amounts to use so-called replica trick [52] writing the logarithm in the
V. Blavats'ka, C. von Ferber, R. Folk and Yu. Holovatch
110
form of the following limit: ln Z
zn -1
= lim - - - . n-+0
n
(20)
While the powers of Z can only be evaluated for integer values of n, analytical continuation in n is assumed to perform the limit n --+ 0. As a result one ends up [53] with the effective Hamiltonian:
Here, Greek indices denote replicas. Comparing this expression with the Hamiltonian of the pure model (13), we notice an additional interaction of the order parameter field: the last term presents the effective interaction between different replicas. The coupling constant v0 is proportional to the variance of the disorder distribution. The coupling u 0 must be positive, otherwise the pure system undergoes a 1st order transition. For final results the replica limit n --+ 0 has to be taken. There exists a second way to obtain the effective Hamiltonian (21). A weak quenched disorder term can be introduced directly into the effective Hamiltonian (13). The presence of non~magnetic impurities in a microscopic model (15) manifests itself in fluctuations of the local temperature of the phase transition. Introducing 'ljJ = 'lj;(r) as the field of local critical temperature fluctuations, one obtains the effective disordered Hamiltonian [53]: (22) The Hamiltonian (22) depends on a number of macroscopic parameters that describe the specific configuration of the field 'lj;(r). On the other hand, the observables should not depend on the specific realization of the random field 'ljJ and are to be averaged over the possible configurations of 'ljJ [15]. In particular, the singular contribution to the free energy of the diluted quenched Stanley model (15) can be written in the form of a functional integral: F ex
J
D['lj;(r)]P['Ij;(r)]lnZ['Ij;(r)],
(23)
where the configuration~dependent partition function Z['lj;(r)] is the normalizing factor of the Gibbs distribution with effective Hamiltonian (22); P['lj;] defines a probability distribution of the field 'lj;(r). Introducing n replicas of the model (22) and taking that 'ljJ obeys a Gaussian distribution:
(24) with w2 being the dispersion parameter, one again ends up with the effective Hamiltonian (21).
Renormalization group approaches to polymers in disordered media
111
The model (21) is interesting in the polymer limit m ---+ 0. In this case it can be interpreted as a model for SAWs in disordered media. Note that such a limit is not trivial. As noted by Kim [62], once the double limit m, n ---+ 0 has been taken, both u 0 and v 0 terms are of the same symmetry, and an effective Hamiltonian with one coupling u0 u0 - v0 of the O(mn = 0) symmetry results. This leads to the conclusion that weak quenched uncorrelated disorder does not change the universal critical properties of SAWs. These results were confirmed by numerical [16-19,26,63,64] and analytical methods
=
[38,39]. 2.2.2. Long-range correlated disorder In the above case of the diluted d-dimensional m-component spin model the disorder is correlated according to (16). Another type of disorder was proposed in the work of Weinrib and Halperin [65], with a pair correlation function of defects that decays at large distances If- f'l according to a power law:
(25) where a is some constant. This model describes so called extended impurities in the system. In particular, the correlation function (25) with a = d- 1 describes randomly oriented impurity lines, while planar defects correspond to a = d - 2. In magnets, longrange-correlated disorder may be present in the form of continuously distributed dislocations and disclinations, so-called extended structural defects. These defects may have the form of lines or planes of random orientation [69] or may form some sponge-like fractal objects, which are considered as aggregation clusters [70]. The particular case of systems with dislocation lines or planes of parallel orientation, is investigated in Ref. [73]. The Fourier transform of the correlation function (25) at small k has the form [65]: (26) where v 0 and w 0 are some constants. Let us recall, that for the case of point-like noncorrelated defects the correlation function (19) reads g(lf- r'l) = v0 o(lr- f'l), so its Fourier transform for small k behaves as:
g(k)rvvo.
(27)
Comparing (26) and (27), we can see that the case a = d describes point-like disorder. Applying the replica method in order to average the free energy over different configurations of quenched disorder one finds the effective Hamiltonian of the m-vector model with long-range-correlated disorder [65]:
Here, the replica interaction vertex g(x) is the correlation function with Fourier image
(26). Passing to the· Fourier image in (28) and taking into account (26), an effective Hamiltonian results that contains three bare couplings u 0 , v0 , w 0 . For a > d the w 0 -term is
112
V. Blavats'ka, C. von Ferber, R. Folk and Yu. Holovatch
irrelevant in the RG sense and one obtains an effective Hamiltonian of a quenched diluted (short-range correlated) m-vector model (21) with two couplings u0 , v0 . For a < d we have, in addition to the momentum-independent couplings, the momentum dependent one w 0 ka-d. Note that g( k) is positively definite being the Fourier image of the correlation function. From here one gets w 0 2: 0 at small k. A one-loop approximation for the model was given in Ref. [65] using an expansion in E = 4-d, r5 = 4-a. A new long-range-correlated fixed point was found with a correlationlength exponent v = 2/a and it was argued, that this new scaling relation is exact and also holds in higher order approximations. This result was questioned recently in Refs. [66,67], where the static and dynamic properties of 3d systems with long-range-correlated disorder were studied by means of the massive field-theoretical RG approach in a 2-loop approximation, for different fixed values of the correlation parameter, 2 :S:: a :S:: 3. The f3 and 'Y functions in the two-loop approximation were calculated as an expansion series in renormalized vertices u, v and w. The study revealed the existence of a stable random fixed point with u* i 0, v* i 0, w* i 0 in the whole region of the parameter a. The obtained exponents v for various a however violate the supposed exact relation v = 2/ a of ref. [65]. The authors assign difference to a more accurate field-theoretical description using higher order approximations for the 3d system directly together with methods of series summation. Recently, Ballesteros and Parisi [68] presented Monte-Carlo simulations of the diluted Ising model in three dimensions with extended defects in the form of lines of parallel orientation, confirming that the simulated critical exponents for the Ising model agree fairly well with theoretical predictions of Weinrib and Halperin. We are interested in the polymer limit m -+ 0 of the model (28) interpreting it as a model for SAWs in disordered media. Note, that such a limit is not trivial, as explained in the last section. In Refs. [71, 72] the asymptotic behavior of SAWs in long-range-correlated disorder of type (25) is investigated and found to be governed by a set of critical exponents which are different from that of the pure case. In ref. [48] an environment of a quenched configuration of a semi-dilute polymer solution is introduced as a special case of long-range correlated disorder for polymer dynamics. For the statics this environment is shown to be equivalent to an annealed one, i.e. without impact.
2.3. Geometry: Percolation system 2.3.1. Percolation clusters and the Potts model Like the SAW, the problem of percolation can also be treated as geometrical critical phenomenon [74]. To introduce the percolation problem, let us consider a regular hypercubic d-dimensionallattice, where either the sites or bonds are occupied with probability p. In percolation one asks questions concerning the connectivity of occupied bonds. Sets of mutually connected bonds form cluster. One can then ask what is the probability that there is a cluster spanning from the one end of the lattice to the opposite end; in the thermodynamic limit (the number of sites goes to infinity) this spanning cluster becomes the infinite cluster. An important quantity in percolation theory is the percolation probability P(p), which gives the probability that given site belongs to the infinite cluster. One can show that there exists a critical value Pc (also called the percolation threshold) such that P(p) is
Renormalization group approaches to polymers in disordered media
113
= 2[99] d = 3[98] d = 4[98] d = 5[98] 2.51 ± 0.02 3.05 ± 0.05 3.69 ± 0.02 4 91/49 Table 1 Fractal dimension of percolation clusters in dimensions d = 2, ... , 6. d
df
zero when p :S Pc· For p > Pc, it obeys the following scaling law, approaching Pc from above: (29) where f3pc is an universal critical exponent. Because of this property, we can consider P(p) is an order parameter. The correlation length of the percolation lattice (or connectedness length) is defined as follows:
(30) j
where Pii is the probability for two sites i and j at the distance r;j = lfi- i'j I to belong to the infinite cluster; Vpc is the critical exponent. Let us note, that the infinite percolation cluster itself is a fractal with fractal dimension dependent on d: d 1 (d) = d- f3pc (d) jvpc( d) [74]. The estimates for the fractal dimensions of the percolation cluster are given in Table 1. Like SAWs that are related to the O(m)-model, percolation is connected to the qstate Potts model. The q-state Potts model [86] is a spin model whose high temperature expansion (when q -+ 1) corresponds to diagrams that are percolation configurations. In the Potts model to the each site x of a regular lattice a spin variable ax corresponds which can be in q possible states (q = 2 corresponds to the Ising case). Interactions favor nearest neighbors which are in the same state. The effective Hamiltonian for the model reads: (31)
with J an interaction constant and r5 the Kronecker symbol. The thermodynamical critical exponents of the magnetization f3 and the correlation length v of this model in the limit q -+ 1 correspond to the geometrical critical exponents of the percolation problem (29),(30). Consider the behavior of the m-component spin model on the disordered lattice with dilution near the percolation threshold and at low temperatures. As p decreases, also the critical temperature Tc(P) decreases and reaches zero at the percolation point Pc· At T = 0 the critical properties are determined by the properties of the connected spin clusters, i.e. this is a percolation problem. It has been argued [78] that the point p = Pc, T = 0 should be viewed as an multicritical point, here the thermal correlation length and correlation length of the percolation cluster diverge simultaneously. The studies of Ising model [91]
V. Blavats'ka, C. von Ferber, R. Folk and Yu. Holovatch
114
and general xy-model [100] near the percolation threshold have shown, that such systems are characterized by a multicritical spectrum of the critical exponents v;, defined as v; = Vpcf¢;, where Vpc is the critical exponent of the correlation length of the percolation lattice (30) and ¢; is a set of the crossover exponents [100].
2.3.2. Modified Flory theories Let us turn our attention to the scaling of a SAW on a disordered lattice with strong disorder, directly at p = PPC· The Flory formula (2), giving a surprisingly good estimation for the critical exponent v of SAWs on pure lattice, has tempted a number of authors to search for generalizations to determine the exponent Vp of SAWs on the percolation cluster [16]. The most direct one is: 3 v =-p
2 +d/
(32)
Here, d 1 is the fractal dimension of the percolation cluster. This first simple generalization is in a good agreement with results of Monte Carlo simulation [16-21] and exact enumerations [22-28] (see the Table 1). However, there exist a number of other attempts to derive Flory-like formulas for SAWs on fractals. For instance one may write the free energy F of theN-step SAW as the sum of energetic and entropic terms [34] as: N2
F =a-d-+ [-lnP(R, N)], R!
(33)
where R is the end-to-end distance for N-step SAW, and P(R, N) is the probability for a random walker to reach this distance after theN steps on the fractal. In general P(R, N) for large R has an exponential form:
P( R, N) ~ exp[ -b( Rdw / N)<],
(34)
where dw is the random walk dimension on the fractal, (is some constant. The minimization of Eq. (33) with respect to R with Eq. (34) yields the Flory formula:
(35) Rammal et al. [31] suggested that ( = 1, which leads to the Flory formula: Vp
3
= -,---,df +dw
(36)
Aharony [34] on the other hand proposed that
(37) where
dmin
is the fractal dimension of the minimum path on a fractal.
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115
Another kind of Flory-type formula is suggested in [30], where it was argued that the spectral dimension d 8 of the fractal percolation cluster must be an intrinsic property:
(38) In this problem there is one point to clarify. If one believes that the SAW can only live on the backbone of percolation cluster (otherwise it will be trapped at the dangling ends) [59,35,34,31], one should use in the expressions above the corresponding fractal characteristics (d7, d:) for the backbone. A more sophisticated expression was proposed in [34]: (39) where df is called the spreading or the connectivity dimension of the backbone of the percolation fractal. A more general version of Eq. (35) with the additional condition (37) was derived in [36], assuming for the probability distribution P(N, R):
(40) representing the requirement, that for small R the free energy of a SAW is dominated by a term (N / RdB )<'', which may be thought as being the repulsive energy between distant basic units, and for largeR by a term (RjN 1fd='n) 6 , which represents a configurational entropic term. Here, dmin is the fractal dimension of the shortest SAW on percolation cluster, a and r5 are unknown positive exponents. For the critical exponent Vp the following expression results: (41) The mean-field result for this ratio is given in terms of the random walk dimension d~: (42)
Another possible expression for vp was proposed in [32], using geometrical considerations for the node-link picture of the backbone of the percolation cluster. For the diluted lattice just above the percolation threshold the lattice can be viewed as a collection of nodes which are connected by links, thought of as random paths. Two important lengths enter this picture. One is the distance between nodes which is of order of the percolation correlation length (30), the other important length is the length L of the random path between nodes which is given by a separate exponent ( through: (43)
V. Blavats'ka, C. von Ferber, R. Folk and Yu. Holovatch
116
If the number of steps N of SAWs is larger than L, a dilute lattice would behave as a normal d-dimensional lattice. We call this the "normal" regime. In the normal regime, (R2 ) is still given by (1) with the critical exponent v. If N < L, the diluted lattice would behave anomalously like a fractal and we call this regime the anomalous one, here (R 2 ) scales with new exponent vP. Considering the crossover for N"' L,one obtains [32]: Vpc
(44)
Vp=-
(
where vpc is the correlation length critical exponent of the percolation transition. The Flory-like theories [30-37] offer good approximations for vp(d) in a wide range of d (see Table 2).
d
VSAW FL, [30] [31] [32] [33] [34] [35] [36] MC, [16] [17] [18] [19] [20] [21] EE, [22] [23] [23] [24] [25] [26] [27] [27] RS, [60] [30] RG, [22] (133)
2 3/4 0.778 0.69(1) 0.770 0.76 0.75-0.76 0.77
'.':::. VSAW '.':::. VSAW
3 0.5882(11) 0.662 0.57(2) 0.70(3) 0.656 0.65 0.64-0.66 0.66 '.':::. 2/3 0.612(10) 0.605(10)
4 1/2 0.593 0.49(3) 0.63 0.57 0.58 0.57-0.59 0.62
0.55-0.57 0.56
0.62-0.63 0.67(4)
0.56-0.57 0.63(2)
0.54(2)
0.548 0.595
0.524 0.536
5 1/2 0.543 0.56 0.52
6 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2
0.77(1) 0.783(3) 0.76(8) 0.81(3) 0.745(10) 0.745(20) 0.770(5) 0.778(15) 0.787(10) 0.767 0.778 0.595 0.785
0.635(10) 0.65(1) 0.640(15) 0.660(5) 0.66(1) 0.662(6) 0.724 0.571 0.678
1/2 1/2
Table 2 The exponent vP for a SAW on a percolation cluster in several dimensions. FL: Flory-like theories, EE: exact enumerations, RS, RG: real-space and field-theoretic RG. The first line shows VsAw for SAW on the regular lattice (d = 2 [13], d = 3 [11]).
Renormalization group approaches to polymers in disordered media
117
2.3.3. Monte Carlo analysis Kremer [16] was the first to perform Monte Carlo study of SAWs statistics on randomcite diamond 3-dimensional lattice. His study indicates no change in the exponent v for weak dilution, but for p close to Pc observes a higher value Vp ~ 2/3. The data show a prefactor in the scaling law for the mean-square end-to-end distance (1) which increases with increasing dilution. Right at the percolation threshold Pc, a crossover to a larger exponent 1 > Vp > v occurs. The trivial upper bound corresponds to the case of the absolutely stretched chain. This crossover (for large N) is described by the following scaling assumption: (45) where y is a crossover exponent, and the scaling function
f
has the form:
x -+ oo· const: x-+ 0.'
xv-vp
f(x)-+
{
For p > Pc the remaining lattice is built up by the "fractal" regions of extension the "normal" region. The extension ~ diverges in the percolation limit as:
~
and
(46) If the end-to-end distance of a SAW is greater than the correlation length R »~'we find the behavior of a SAW on a normal lattice. On the other hand, if R «~we would expect the behavior of a SAW on the infinite percolation cluster. The crossover takes place when both quantities are almost equal:
Nl/p
ex~
ex
IP- Pcl-vpc
=
IP- Pel Vp ~ 1 we find y = Vpcfvp.
-+ N.
Comparing this with (45) In Ref. [30] it was noted, that one expects to have two different critical exponents, depending on whether one averages the SAW configurations only on the infinite percolation cluster at Pc or on all clusters. If the SAW is averaged only on the infinite percolation cluster (infinite cluster, IC), we expect to have:
(47) whereas if the SAWs are averaged on the percolation system of all clusters (AC), we may anticipate that (48)
where v 1c
i
v AC and these exponents are universal. Moreover, we should have:
>
>
Vpure,
VJC
V AC
d::::: 4,
(49)
since the fractal structure of the clusters should enhance the self-avoiding effect. The position-space renormalization-group (PSRG) approach was used in [30] to estimate v1c in two and three dimensions.
V. Blavats'ka, C. von Ferber, R. Folk and Yu. Holovatch
118
More recently, Kim [35] claimed, based on a scaling argument, that the inequality of the type (49) is invalid and v 1c = v AC should hold. It was also argued, that that the critical behavior of the IC averages is also similar to that on the backbone (BB) of an infinite cluster, suggesting: d::::: 4.
(50)
Both equalities are based on the following argument. The infinite cluster generated near Pc in general contains many dead ends and "dangling loops" that are connected to the backbone via a single path. Since the walk that visits such bonds is eventually terminated after a finite number of steps, the walks that survive in the limit N -+ oo are those confined to the backbone. With Eq. (50) and the inequality:
(51) where dmin is the dimension of the shortest SAWs between two points on the infinite cluster and the fractal dimension of the backbone, Kim claimed that both v AC and v 1c should be larger than Vpure because otherwise the last inequality breaks down for d 2:: 4 since dr < 2. The MC simulation of Kremer [16] has been referred to as the only numerical estimate for Vp for a number of years, until Lee et al. [17] performed simulations for SAWs on square and cubic lattices at dilution very close to the percolation threshold. Their results for the SAWs critical exponent v1c rather surprisingly indicate that the exponent v1c even at Pc is close to the pure result. Furthermore, numerical uncertainties in the [16] were indicated; namely, it was noted, that the chains used in that simulation are probably not long enough to estimate the critical exponents. The validity of either (49) or (50) has been tested numerically for the first time in [19]. Here, the critical behavior of the AC and BB averages of SAWs on two-dimensional site-percolation for a square lattice has been studied via Monte Carlo simulations. It was found, that v AC is very close to the pure lattice value, while VBB = 0, 77 ± 0.01 which is about 3% larger than the full lattice value, concluding that the relation VBB > v1c holds. In conclusion, it was claimed, that exponent vp, governing the scaling behavior of SAWs on the percolation cluster is close to the pure lattice value and, if any, the difference would be very small and virtually unobservable in Monte Carlo simulations. In Ref. [26] a detailed numerical study of enumerated SAWs on (Monte Carlo generated) random site diluted lattices in two and three dimensions was carried out. The nonsatisfactory results of MC simulations were explained in particular with difficulties in realizing the SAW trajectory on the percolation cluster. The percolation fractal contains self-similar regions connected by linearly or singly connected links, and MC diffusion of SAWs through these links is extremely difficult. This forces the MC generated SAWs on the percolation cluster to be confined (localized) practically to a some region of the percolation cluster of a size determined by the lattice size and the particular realization. As a result, this leads to incorrect estimates for the exponent of the mean-square end-toend distance of SAWs on the percolation cluster.
d7
Renormalization group approaches to polymers in disordered media
119
2.3.4. Effective Hamiltonian The field theory for SAWs on the percolation cluster developed in Ref. [22] supports an upper critical dimension dup = 6. The calculation of vP was presented to the first order of perturbation theory, however the numerical estimates obtained from this result are in poor agreement with the numbers observed by other means. In particular, they lead to estimate that vP c::o v in d = 3. Recently this investigation has been extended to the second order in perturbation theory [101], which leads to the qualitative estimates of critical exponents in good agreement with numerical studies and Flory-like theories. To obtain the effective Hamiltonian, describing the behavior of SAWs on the percolation cluster, we make use of results from [91,22,101]. Let us consider them-vector spin model on a diluted lattice with bond randomness, described by the Hamiltonian: H =-
L
(52)
Jcrr'S(r)S(r'),
where the sum is over nearest neighbor pairs r, r' and Crr' are independent bivalued random variables, that indicate whether a given bond between the sites r and r' is present or not, and take on the value 1 and 0 with probabilities p and 1-p respectively. We are interested in the properties of the phase transition in such a model for bond concentration close to Pc and thus the critical temperature is very close to zero, this allows us to interpret Eq. (52) as a model for self-avoiding walks on the percolation cluster. To perform the average of the free energy over the quenched disorder, the replica method is applied and for the n-replicated partition function we obtain:
II (1-p+pe llf f=
-
zn=Tr
S"'(r)S"'(r')
a=!
)=Tre-1m,
(53)
=I rr:=l ITr dS"(r)b'(IS"(r)l-vm)( ... ), which defines an equivalent Hamil-
with Tr( ... ) tonian 1i through -{31{
=
Performing the expansion of the logarithm in this expression one finds after some transformations: n
oo
-f31i=Z::
1
+ L:at21 L l=O
et1=l i1=l
l=O
n
00
m
{(La L 2:S~ 1 (r)S~ 1 (r') m
L
S~ 1 (r)S~ 1 (r')S~ 2 (r)S~2 (r')
cq,a:2=l i1,i2=l cq
+ ... + L:att1 l=O
n
L
m
L
et1 , ... ,ett=l i1 , ... ,it=l
cq < ...
S~ 1 (r)S~ 1 (r') ... S4'(r)S4'(r')) + ... }
(54)
V. Blavats'ka, C. von Ferber, R. Folk and Yu. Holovatch
120
where the first three coefficients a; read
ao = -ln(1- p), _ exp(-2f3J)(pln2- 2p(1- p)) _ _ p_exp(-f3J) 2(ln2)2(1- p)2 1- p ln2 exp -2f3J p (ln )2 _ p + O(exp( -4f3J)). a2 = 4 2 1 a1
-
+
O( exp
(- f3J)) 4
'
(55)
Introducing notations for the products of spin components on the same site, as proposed in [91]:
so.J, ... p.'(r) 1-l,·•·,'tt
=So. (x) ... so.•(r)
a 1 < ... < a 1 ,
1
Zl
'tt
'
(56)
the Hamiltonian (54) can be written as a quadratic form: -{31{
n m = -1 "'(ao + Mt) " L.....,; L.....,;'
2
n
" L.....,;'
i1, ... ,it=l
t=l
eti, ... ,a:t=l
1 1 " , ... ,o.,(r)So. , ... ,o.,(r') L.....,;' So. 'tl, ... ;tt 'tl, ... ;tt
(57)
O.J< ...
2.:::
here, we have introduced the notation M 1 = 1 a;ti. This expression can be converted to a continuous field theory using the standard Stratonovich-Hubbard transformation and introducing fields ¢~1,::::;~n (r), conjugate to the products of spin components (56). The final field-theoretical effective Hamiltonian of this model then reads [101]: 1{
2:: L:(q2 + rt)
= -2
(t)
+ ~~
ifJ,i[z,if3 (t)
(l)
(58)
(s)
where the notations have been introduced ¢(tl (q') for ¢~1,::::;~' ( q') and m
L(. ..)stands for L L (t)
t
n
L (... );
(59)
il, ... ,it=l Ctl, ... ,ett=l
O.J< ...
+ M1 , r0 = 1 - ln(1 - p), w 0 = 1 is the bare coupling constant. We thus have a theory of interacting fields ¢1 (k) with different masses r 1 • The expression for the effective Hamiltonian (58) served as a basis of RG analysis, performed in the work [22] in the one-loop order and in [101] in two-loop order calculations. r1 = r0
2.4. Real space renormalization group approach Real-space RG (RSRG), sometimes also called position- or configuration- space RG allows to complement the usual momentum space RG by a more direct method. Generally, no Hamiltonian is needed and the method is especially suited for geometrically posed problems of connectivity as they are given for polymers and percolation [102,103].
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Renormalization group approaches to polymers in disordered media
z
z z
z'
z ---+
z
z z
z
I
z
Figure 1. Elementary cell of a square lattice and its reduction.
We introduce its application to polymers following Family [104]. For SAWs on a square lattice one may divide the lattice into cells like the one shown in Fig.l. A SAW starting at the origin of this cell crosses the cell either horizontally or vertically. This fact may be described by a reduced cell. Assigning a fugacity z for each step of the SAW the renormalized fugacity z' on the bonds of the reduced cell is calculated by summation of the contributions of the corresponding SAW configurations: (60) This defines an RG transformation renormalizing polymer segments of extension 2L to monomers of extension L and weight z'. At the nontrivial fixed point z* of (60) we may then calculate the correlation length exponent v by (61) where here AL = 2 is the rescaling factor of the lattice spacing and Az In two dimensions this results in z*
= 0.4656 and
v
= dR(z)jdzlz=z'.
= 0. 7152
For the corresponding three-dimensional L 3 cell the recursion relation is for L
(62)
= 2 [104] (63)
with a fixed point z* = 0.2973 and v = 0.587. This result is surprisingly close to the Flory result of v = 0.6 and the high-order RG result of v = 0.588. However, there is no systematic way of improving this result. Increasing for instance the cell size to L = 3 the exponent becomes v = 0.5814 [105]. Its success and the simple way of implementation has led to a number of RSRG studies of SAWs on such lattices with disorder. The application of the RSRG method to the percolation problem may be illustrated by the renormalization of the same Kadanoff-cell given in Fig. 1. Following Ref.[102] we will now be interested in the probability of the left and right borders of the cell to be connected. As shown in Fig. 2 the cell is in this respect equivalent to a Wheatstone bridge configuration which represents the contribution of the backbone. Assigning a probability
V. Blavats'ka, C. von Ferber, R. Folk and Yu. Holovatch
122
Figure 2. Connectivity between peripheral sites of the elementary cell and equivalence to the Wheatstone bridge.
p to each bond the renormalization step for the probability of the endpoints i, j of the
renormalized bond to be connected is described by [106]
(64) With p* = 1/2 we have Ap = 13/8. This approach now allows to study the behavior of the mean number of bonds in the minimal and maximal paths along the backbone. For the Wheatstone bridge these are [102]
(65) The scaling behavior of these quantities near the percolation threshold Pc is then described by an exponent (x via (66) With the renormalizations p' the reduction one has
= R(p) and x' = AxX under the change of scale implied by (67)
Here, we have x' = p* at p = p* for both Amax = 17/8 which result in [102] (min
= 1.55 and
(max
= 1.835.
Xminfmax
and thus the factors
Amin
= 17/8 and (68)
To treat SAWs on the percolation cluster we will have to use recursion relations involving both the bond probability as well as a fugacity for the polymer. Work on this problem is reviewed in section 3.5.
Renormalization group approaches to polymers in disordered media
123
3. TREATING THE THEORY BY RENORMALIZATION GROUP METHODS 3.1. The general principles In this subsection, we give a brief account of the main relations of the field-theoretical renormalization group (RG) formalism that can be evaluated in different variants. The object of investigation are one particle irreducible vertex functions r~L,N), defined as averages of N order parameter fields ¢ and L insertions of the type ¢ 2 as follows:
8(2.:: k;+ 2.:: Pi )r~L,N) ({ k }; {p }; t-t5; {Ao} )=fAa ei(k;R;+PjTj l x (¢ 2 h) ... ¢ 2 (rL)¢(Rl)· . .¢(RN ))~?V dd R1 ... dd RNddr1 ... ddr£.
(69)
here {p} and { k} are sets of external momenta, A0 is a cutoff and ( ... ) stands for averaging with the corresponding effective Hamiltonian, {-\ 0 } stands for the set of bare couplings of the effective Hamiltonian of the system: for instance, in the case of a model with point-like uncorrelated disorder, which is described by the effective Hamiltonian (21), one has two couplings: u 0 and v0 . An intrinsic feature of the functions (69) is their divergence in the asymptotic limit A0 -+ oo. In order to control the divergences and to map the divergent mathematical objects to convergent physical quantities one performs a rearrangement of the series for the vertex functions. Renormalization factors Zq, and Zq,z for the fields are introduced defining the renormalized function f~,N): (70) Zq, and Zq,z are constructed such that rW·L) has no divergences order by order in perturbation theory. In this procedure the mass t-to and couplings {-\ 0 } are renormalized to ft and {A}. Here we describe in more detail two effective schemes, that are used in the context of the model, considered in our review: the massive and minimal subtraction schemes.
3.1.1. Massive scheme In the massive scheme [80,81], the renormalization is performed at non non-zero mass and zero external momenta, passing from the initial parameters of the model f.to, { .\ 0 } to the renormalized mass tt and coupling constants {A}. The regularization scheme suggested in (70) is not unique. To define the regularization scheme completely one imposes renormalization conditions for the renormalized vertex functions [4,80]. For theories, where the interaction is represented by ¢ 4-term in the Hamiltonian (e.g., (21)), these conditions read: 2 r~· l(k, -k; t-t 2, {A}) lk=o d (o,2) ( . 2 {A}) Ik=O dk2 f R k, -k, ft '
({
r~:;; k }; t-t 2, {A}) l{k;}=O 2 f~' l(p; k, -k; t-t 2, {A})Ik=p=o
f.t2,
(71)
1,
(72)
tt4-d Aj,
(73)
1.
(74)
124
V. Blavats'ka, C. von Ferber, R. Folk and Yu. Holovatch
In the case of long-range-correlated disorder we have another global parameter a along with d, and for the renormalization of the coupling w 0 in Eq. (28) one imposes [67]:
(75) with a renormalized coupling w. For the theory, where the interaction is represented by ¢ 3-term in the Hamiltonian (e.g., (58)), instead of (73) one has: (76) The scaling properties of the system asymptotically close to the critical point are expressed by the homogeneous Callan-Symanzik equation [4,80] for the renormalized vertex function:
with the coefficients defining the RG flow of the couplings (f)-functions) and the critical exponents ('!'-functions) 'Yq, and "fq,z:
iJ.d{>.})
o>.;
oZq,
_
oZq,z
= olnt-tl{.\o},!-'o> 'Yq, = olnt-tl{.\o},!-'o> "'/q,2 = -oln){.\o},l-'o
(78)
where Zq,2 = Zq,2Zq, and the sum over i spans over all couplings A;. The fixed points {>. *} of the RG transformation are given by the solutions of the system of equations:
;3.\; ({>.*}) = 0,
i
= 1, 2, ....
(79)
A stable fixed point, corresponding to the critical point of the system, is defined as a fixed point where all the eigenvalues {w;} of the stability matrix: (80) possess positive real parts. In such a point, the function 'Yq, determines the value of the pair correlation function critical exponent 17:
(81) and the correlation length critical exponent v is determined as: (82) All other critical exponents may be obtained from familiar scaling relations. For example, the susceptibility exponent 'Y is given by: 'Y
= v(2- 17).
(83)
Renormalization group approaches to polymers in disordered media
125
3.1.2. Minimal subtraction scheme In Refs. [82,83] the elegant method of renormalization at zero mass and non-zero external momenta was developed, which avoids the additional renormalization conditions. Here, the vertex functions are analytically continued in the dimensional parameter d leading to a so called dimensionally regularized theory, where the cutoff A0 in 69 can be removed. As one approaches the critical dimension (de = 4 in the case of ¢ 4-theory and de = 6 for ¢ 3 -theory), from below, the terms in perturbation series for vertex functions develop poles in E = de - d, which correspond to logarithmic ultraviolet divergences. In the dimensional renormalization scheme, the coupling constants of the given model { .\0 } and renormalization factors Zq, and Zq,2 are written as series in the renormalized couplings {A}, which for k couplings are defined as follows: 00
Ao; = .\;[1 + Zq,
1+
Zq,2 = 1 +
00
L ... L alJ, ... ,tk>-il ... ,\~k],
00
00
h=l
lk=l
00
00
h=l
lk=l
L ... L L ... L
blJ, ... ,lk>-il ... _\~k
ClJ, ... ,lk>-il ... _\~k
The coefficients alJ, ... ,lk' blJ, ... ,lk' and ClJ, ... ,lk are minimal Laurent series in are constructed such as to cancel in the renormalized vertex functions
r~·Nl ({ q}; {P }; {>-})
= z;2z;/2-Lr~L,Nl ({ q}; {P }; {>-o})
(84) E
=de- d which
(85)
all poles in E of the bare functions r~L,N) in every order of {A}. The renormalized vertex functions obey the renormalization group equation
Here, T is a rescaling parameter, which defines the scale of the external momenta in the minimal subtraction scheme. In the same way as in the Callan-Symanzik equation (77) the coefficients in (86) define the renormalization group functions:
8.\; oZq, _ 8Zq,2 ;3>., ({A}) = o ln T I{>.o} ' 'Yq, = [) ln T I{>.a} ' 'Y 2 = - [) ln ) {>.o} ·
(87)
The fixed points {A*} are again defined as common zeroes of all f)-functions, and the critical exponents are defined following Eqs. (81), (82). 3.1.3. Resummation of asymptotic series The ;3- and "!-functions are calculated perturbatively as series in the couplings A;. The order of the expansion corresponds to the number of loops in the diagrammatic Feynman representation of the vertex functions (69). However, due to the (asymptotic) divergence of the RG functions series [84], one cannot directly derive reliable physical information
126
V. Blavats'ka, C. von Ferber, R. Folk and Yu. Holovatch
from these expressions, the fact that is familiar to the theory of critical phenomena. If the nature of the divergence is such that the series are asymptotic, then the situation is, at least in principle, controllable: in this case a good estimate for the sum of the series is obtained by either keeping an optimally chosen number of low order terms ("optimal truncation") or by applying an appropriate resummation procedure. When resumming a series one should note that the values and the accuracy of the resummed results will depend on the procedure of resummation. Thus the analysis and appropriate selection of an adapted resummation procedure is crucial to obtain reliable results. However, there remains the principal question about the Borel summability of the perturbation theory series for a given model. Up to now a proof of summability exists only for the ¢4 theory with one coupling [85]. However, the field theoretical RG series for models with several couplings are generally analyzed as if they are asymptotically. Nevertheless, there exists strong evidence of possible Borel non-summability of the series obtained for disordered models [87-89]. The Pade-Borel resummation technique [90] is performed as follows: • Given the initial function in the form of truncated series in coupling u the Borelimage is constructed:
~
~a;u
i
==>
i=l
~a;(ut)i. 1
~-.-- ,
i=l
(88)
L
• this Borel-image is extrapolated by a rational approximant
[M/N]
=
[MjN] (ut);
(89)
here, [M/N] stands for the quotient of two polynomials in ut; M is the order of the numerator and N is that of the denominator; • the resummed function is then obtained by taking the inverse Borel image:
sre•(x)
=
l)Q dtexp( -t) [M/N] (ut).
(90)
The Pade-Borel procedure can be optimized by introducing an additional fit parameter p to the Borel transformation. Substituting the factorial i! by the Euler gamma-function f(i + p + 1) and inserting an additional factor tP into the integral (90), one defines the
Pade-Borel-Leroy resummation procedure. In order to suit the resummation procedure (88)-(90) for functions that depend on several variables one should change the first step (88): for example, for the two-variable case one defines the Borel image by [92]: ""'
~a;,jU
.. 't,J
i
i ""'a;,j(ut)i(vt)i v ==> ~ ('l+ J. ')I . .. 't,J
(91)
The rational approximation (89) can be performed then either in the dummy variable tor in the variables u, v. This defines two approaches that one can use to resum the functions
Renormalization group approaches to polymers in disordered media
127
of several variables. In the first case the resummation procedure is referred to as the PadeBorel resummation for resolvent series [94]. The application of the Chisholm approximants [93] which are the generalization of Pade-approximants to the many-variable case is necessary in the second case. A Chisholm approximant can be defined as a ratio of two polynomials both in variables u and v, of degree M and N such that the first terms of its expansion are equal to those of the function which is approximated. Again, the resummation is performed in eq. (90) replacing the Pade with the Chisholm approximant. This method will be referred below as the Chisholm-Bore! resummation. It is obvious that an initial sum can be resummed in different ways. Apart from the Leroy fit parameter p mentioned above some arbitrariness arises from the different types of rational approximants one may construct. For instance, within the two-loop approximation the method of Pade-Borel resummation of a resolvent series can be done using either the [0/2] or the [1/1] approximants. The Chisholm-Borel approximation implies even more arbitrariness and demands a careful analysis of the approximants to be chosen. In the resummation schemes described above for series in several variables the variables are treated as "equal in rights". One may also relax this condition as in the recently proposed method of subsequent resummation, developed in the context of the d = 0dimensional disordered Ising model in Ref. [89]. Given the RG function f(u,v) as a power series in u,v (for the model (21), u is the original coupling of the undiluted system and v is the variance of the quenched disorder) one rewrites it as a series in the variable v:
(92) n
k
n
with coefficients that are in turn series in the variable u and are to be resummed in advance by any one-variable method:
An(u)
=L
Ck,nUk.
(93)
k
The main result of Ref. [89] is that the expansions of the coefficients (93) and the resulting series (92) at fixed u are Borel summable (for ad= 0 dimensional system, however). This suggests to analyze the RG expansions of the d = 3 disordered models by first performing a Pade-Borel resummations of the corresponding series for the coefficients in one coupling and then, using the computed coefficients, resumming the series in the other subsequent coupling(s). 3.2. Unperturbed case Let us illustrate, how the renormalization scheme works, considering the m-component model in the limit m -+ 0 on the pure (undiluted) lattice with the effective Hamiltonian (13). Here, we have only one coupling constant u 0 , and the situation is rather simple. In the so-called one-loop approximation, corresponding to the first order of perturbation
V. Blavats'ka, C. von Ferber, R. Folk and Yu. Holovatch
128
theory in the renormalized coupling u, the massive scheme RG functions read [4]:
f3u (u)
-EU (
1-
~uh)
,
(94)
"!q,(u)
(95)
where E = 4 - d is the deviation from the upper critical dimension and the one-loop integral reads: 11=
I
ddq
(96)
(q2+1)2'
The next step is to find and analyze the stable fixed points (see (79) for definition) of the given /3-function. One gets: • The Gaussian fixed point u* • the pure SAW: u*
i
= 0, which is stable at d 2: 4;
0, stable at d < 4.
The critical exponent v at Gaussian fixed point reads v = 1/2, which corresponds to the fact that SAWs at dimensions above the upper critical one behave like RWs (simple random walks). There are essentially two ways to proceed in order to obtain the qualitative characteristics of the critical behavior of the model. The first is to substitute the loop integral in equations (94), (95) by its c:-expansion: (97) which finally leads to obtaining the expressions for critical exponents (via Eq. (81), (82)) as expansions in parameter c:. Another scheme is to consider the polynomials in Eq. (94), (95) and evaluate the integral for fixed dimension d. In the first case, one finds for SAWs on a pure regular lattice:
1
Vpure
E
= 2 + 16'
(98)
Currently, the RG function of the pure m-component model are known in the 6-loop approximation in fixed d-scheme [96], leading in three dimensions to the estimate [11] Vpure = 0.5882 ± 0.0011. Critical exponents have also been calculated in the form of c:-expansions up to five loops [97]. The estimate for the exponent v in this case in three dimensions reads [11]: Vpure = 0.5878 ± 0.0011. 3.3. Uncorrelated disorder Let us consider the properties of the critical behavior of the model (21). Following the renormalization procedure the expressions for the RG functions are obtained as series in the renormalized couplings u and v. The massive RG functions of the model with uncorrelated disorder are known up to six-loop expansions [95] up to date. However, already the one-loop analysis of the model in the polymer limit has led to contradictory results concerning the existence of the stable fixed point in the system and thus of a
129
Renormalization group approaches to polymers in disordered media
phase transition [29]. Let us analyze the fixed point structure of the !'-functions of the m = 0-model, obtained in the massive scheme within the one-loop approximation. They read:
(u - (~u2 + 2uv) I
1) ,
(99)
f'v(u,v) = (v - Gv + ~uv) I
1) ,
(100)
!'u (n,v)
=
-E:
2
-€
with I 1 given by (96) . One obtains ·1 fixed points, sec Fig. 3,a: • Gaussian (G) : u
= 0, v = 0 corresponds to ideal model without interaction;
• polymer (Heisenberg) point(H): • unphysical (U):
u i= 0, v = 0 turns out to be unstable;
u = 0, vi= 0 is stable;
• mixed (random) point (R) : u i= 0, vi= 0 unphysical and unstable.
b)
a)
G
j; II
d >4
II
II
/
(:
Figure 3. a) The lines of fixed points of the SAWs on a lattice with point-like uncorrclatcd disorder (sec the text) . b) Fixed points in a 4 + f. expansion for d > 4 according t.o Le Doussal and Machta [39] Let us recall, that only the region u > 0, v > 0 is physically accessible in this problem. Thus, the situation is uncertain concerning the critical behavior of the system - there no physical stable fixed point, governing the universality class of the system. In Rcf.[39] it was argued that while the upper critical dimension for the problem of polymers in disorder may be larger than de = 4 one may analytically continue the f.expansion from d = 4 - f. to a 4 + f.-expansion. For the fixed points this has a dramatic effect (sec Fig.3b) . In the 4 + f. case the unphysical fixed point (U) is mapped to D in the physical region. It is interpreted by these authors as a strong disorder fixed point. Also
V. Blavats'ka, C. von Ferber, R. Folk and Yu. Holovatch
130
the random fixed point R is mapped to R in the physical region where it is interpreted as a multicritical fixed point. However, as it has been noticed in Ref. [62], in the double limit m, n-+ 0 the terms with u 0 and v0 become of the same symmetry, joining these terms one passes to an effective Hamiltonian with only one coupling constant of the O(mn = 0)-symmetry. This can be proved for the expressions of the one-particle irreducible vertex functions f(N,L). Following Kim [62] we consider the so-called faithful representation of the familiar Feynman diagrams [4], representing the vertexes with coupling constants u 0 , v0 , which describe the interaction of the fields diagrammatically, as is shown in Fig. 4. Here
Figure 4. Interacting vertexes in diagrammatic representation. The Latin indexes denote field components, the Greek indexes denote replicas.
= 1, ... , m are the field component indices. The diagrammatic representation of the two-point function r)~ (k) of the one-loop contributions is shown in Fig.5. Here, a solid line represents a propagator k 2 + tt5, loops correspond to the integration over an internal moment. We call disconnected any loop that is not connected to external lines only via propagator lines. If the diagram has a disconnected loop, the corresponding replica and field indexes are summed over and give contributions that are proportional to m and n, as it is shown in the figure. The analytical expression for the function f;a( 2 l(f) corresponding to Fig.5, in the massive scheme reads: i, j
(2)(~)
_
f;" k - k
2
uo -vo -vo) + Mo2 + ( 4m uo ! + 8 4! + 4nm4f + 84! 4
( ) I tto ,
(101)
here I(p, 0 ) is the one-loop integral:
I(tto) =
I
(q2
dif + Mo) .
(102)
In the double limit m, n -+ 0 the diagrams with closed disconnected loops have zero contributions, and the expression (101) has the form: (2) ( ~)
ria
k
8 (Uo - Vo ) I ( fto ) . = k 2 + Mo2 + 4f
(103)
Similarly, in all orders of the expansion the vertex function diagrams, that contain closed disconnected loops with contributions proportional to m and mn, disappear in the
Renormalization group approaches to polymers in disordered media
131
'L;;bmn
... uo ,
.········.Vo•. +
:
~-vo
-----+
~.Q..- Vo
,_
.......
_.~..;_
Figure 5. The one-loop contributions to the one-particle irreducible vertex function r~~ (k) in the diagrammatical representation.
double limit m, n -+ 0. Thus the corresponding contributions become symmetric in u 0 and v 0 at all orders. In this way one can turn to the problem with only one coupling u0 = u 0 - v0 already at the level of the effective Hamiltonian (21). These considerations [62] solve the question of universality class of SAWs in media with uncorrelated weak disorder. Passing to the problem with one coupling u one finds that fixed points are replaced by lines (Fig. 3). One has the line of the stable fixed points (containing points Hand U) and non-stable (containing G and R). The universal behavior of the system is defined by the fixed point of the pure model H. In this way the RG confirms, that weak point-like uncorrelated disorder does not change the universality class of polymers. 3.4. Long-range-correlated disorder The effective Hamiltonian (29) is the starting point for a study of the polymer limit m -+ 0 of the weakly diluted Stanley model with long-range-correlated disorder. Imposing the renormalization conditions of the massive scheme in the one-loop approximation leads to the following expressions for the RG functions [72]:
(104)
(105) (106) Here, besides the one-loop integral h given by (96) we get three additional one-loop integrals, that depend on the space dimension d and the correlation parameter a:
12=
I
dqqa-d (q2+1)2' 13=
I
dqq2(a-d) 0 (q2+1)2' 14=[)k2
[I
dqqa-d ] [q+kj2+1 k2=o·
(107)
Note that in contrast to the usual ¢ 4 theory the 'Yq, function in Eq. (106) is nonzero already at the one-loop level. This is due to the k-dependence of the integral 14 in Eq. (107). Similarly as explained in subsection 3.2, one can proceed either by considering the polynomials in Eqs. (104), (105) for fixed a, d and look for the solution of the fixed
V. Blavats'ka, C. von Ferber, R. Folk and Yu. Holovatch
132
point equations (it is easy to check that these one-loop equations do not have any stable accessible fixed points ford< 4), or by evaluating these equations in a double expansion in c = 4- d and o = 4- a as proposed by Weinrib and Halperin [65]. Substituting the loop integrals in Eqs. (104)-(106) by their expansion in c = 4- d and o = 4- a:
0)
0-
1( c) 1 ( 1 ( 28 - c) 1 ( c) h = ~ 1 -2 'h = 8 1 -2 'h = 28- c 1 - - 2 - ' 14 = 8 -2- ' one obtains the 3 fixed points given in the table 3.4. The following conclusions may be
Fixed Point u* w* W1 w2 -E: Gaussian (G) 0 0 -0 Pure SAW (P) c c c/2- o 0 2.5" 2 - 4co + 882} Long-range (LR) (e-Jl 48 ± Vc ijl €-0 Table 3 Fixed points and stability matrix eigenvalues in the first order of the c, o- expansion [72].
-1;
He-
drawn from these first order results: Three distinct accessible fixed points are found to be stable in different regions of the a, d-plane: the Gaussian (G) fixed point, the pure (P) SAW fixed point and the long-range (LR) disorder SAW fixed point. The corresponding regions in the a, d-plane are marked by I, II and III in Fig. 6. In the region IV no stable fixed point is accessible. For the correlation length critical exponent of the SAW, one finds distinct values //pure for the pure fixed point and VLR for the long-range fixed point. Taking into account that the accessible values of the couplings are u > 0, w > 0, one finds that the long-range stable fixed point is accessible only for o < c < 28, or d < a < 2 + d/2, a region where power counting shows that the disorder is irrelevant. In this sense the region III for the stability of the LR fixed point is unphysical. Formally, the first order results for d < 4 read: 1/ _
-
{
= 1/2 + c /16, o < c/2, = 1/2 + 0/8, c/2 < o
//pure
VLR
(108)
The first line in (108) recovers (98), whereas the second line brings about a new scaling law. Thus, in this linear approximation the asymptotic behavior of polymers is governed by a distinct exponent VLR in the region III of the parameter plane a, d. However, the region where the LR fixed point is found appears to be unphysical. Something similar happens if the c, o-expansion is applied to study models of m-vector magnets with long-range-correlated quenched disorder [65,71]: also in the case of magnets, as well as for polymers the first order c, o-expansion leads to a controversial phase diagram. In order to obtain a clear picture and more reliable information, one should proceed to higher order calculations. Here we pass to the analysis performed for fixed values of parameters d = 3, a. To investigate the peculiarities of the critical behavior in the 2-loop approximation one may
Renormalization group approaches to polymers in disordered media
133
d 4
IV
3 2
II
1
2
3
4
a
Figure 6. The critical behavior of a polymer in a medium with long-range-correlated disorder in different regions of the d, a-plane as predicted by the first order c, o-expansion [72]. Region I corresponds to the Gaussian random walk behavior, in the region II scaling behavior is the same as in the medium without disorder, in region III the "long-range" fixed point LR is stable and the scaling laws for polymers are altered, in region IV no accessible stable fixed points appear; this may be interpreted as the collapse of the chain.
make use of them-+ 0 limit of the appropriate m-vector model (28), investigated recently [66,67]. Starting from the two-loop expressions of Ref. [67] for the RG functions of the Stanley model with long-range-correlated disorder and making use of the symmetry arguments [62, 72] in the polymer limit m = 0 one finds the following expressions for the d = 3 RG functions of the model described by the effective Hamiltonian (28) [71]:
f3u(u,w) (109)
f3w(u,w)
-(4- a)w- (h(a)- h(a))w 2 + u; 23
216
!'q,(u,w) =
2
1
+ b10 (a)w 3 -
2
u w + 4:bl2(a)uw ,
~h(a)w + 1 ~ 8 u 2 + c1 (a)w 2 - ~c2 (a)uw,
(110) (111) (112)
Here, the coefficients J;(a) are expressed in terms of the one-loop integrals in Eq. (107), b;(a) and c;(a) originate from the two-loop integrals and are tabulated in Ref. [67] for
V. Blavats'ka, C. von Ferber, R. Folk and Yu. Holovatch
134
···--.)
.-~-~
j :t:
1
_./
fG
-·
P _./__....----·--
0 · ·-•.•______,........--··
Figure 7. The lint=>~'> of zeroes of the 3d ,6-functions (109), (110) resummed by the Chisholm-Borel method at a= 2.9. The dashed line corresponds to f:Ju = 0, the solid lines depict f:Jw = 0. The intersections of the dashed and solid lines give three fixed points shown by filled circles at u* = 0, w* = 0 (G), u• = 1.63, w* = 0 (P), and u• = 4.13, w • = 1.47 (LR) . The fixed point LR is stable.
d = 3 and different values of the parameter a in the range 2 :::; a :::; 3. The series are normalized by a standard change of variables u --t ~ h, w --t ~I1, so that the coefficients of the terms u, u 2 in f:Ju become 1 in modulus. As discussed before, the question about the summability of the series in Eqs. (109) - (112) is open. In Ref. [71] various kinds of resummation techniques have been applied in order to obtain reliable quantitative results for the system under consideration and to check the stability of these results. First, a simple two-variable Chisholm-13orel resummation technique was employed, which turns out to be the most effective one for the given problem. In addition to the familiar fixed points G and P describing Gaussian chains and polymers (SAWs) on regular lattices, a stable long-range fixed point LR for polymers in long-range-correlated disorder was found. Fig. 7 depicts the lines of zeroes of the resummed ,6-functions (109) , (110) at a = 2.9 in the u , w-plane in the region of interest are depicted. The intersections of these curves correspond to the fixed points. The corresponding values of the stable fixed point coordinates and the stability matrix eigcnvalu<>.s for different values of the correlation parameter a < 3 are given in Table 3.'1. To calculate the critical exponents the same resummation technique was applied. The numerical values for v, 'Y and '17 arc listed in Table 3.4 for a = 2.3, . . . , 2.9. Note, that for a = 3, which eorresponds to short-range-correlated point-like defects, the interactions u and w become of the same symmetry, so one can pass to one coupling (u - w) and reproduce the well-known values of the critical exponents for the pure SAW model. The numerical values corresponding to those listed in Table 3.4 in this case read: u* = 1.63, v = 0.59, 'Y = 1.17, '17 = 0.02, w = 0.64. As departing from the value a = 3 downward to 2 one notices a major increase of the value of the coupling u, so the results are more reliable for a close to 3. At some value n = U.marg the LR fixed point becomes unstable. To verify the results, also the method of subsequent resummation was applied to the
Renormalization group approaches to polymers in disordered media a
u*
w*
l/
"(
TJ
135
w1,2
2.9 4.13 1.47 0.64 1.25 0.04 0.25 ± 0.62 i 2.8 4.73 1.68 0.64 1.26 0.04 0.22 ± 0.76 i 2.7 5.31 1.81 0.65 1.28 0.03 0.18 ± 0.89 i 2.6 5.89 1.87 0.66 1.29 0.03 0.15 ± 0.99 i 2.5 6.48 1.89 0.66 1.31 0.02 0.11 ± 1.09 i 2.4 7.10 1.87 0.67 1.33 0.01 0.07 ± 1.18 i 2.3 7.76 1.84 0.68 1.36 0.01 0.03 ± 1.26 i Table 4 Stable fixed point of the 3d 2-loop ,6-functions, resummed by the Chisholm-Borel method, the corresponding critical exponents and the stability matrix eigenvalues at various values of a.
RG functions (109), (110). Here, the summation was carried out first in the coupling u and subsequently in w. Again, the presence of stable fixed point LR for amarg ~ a < d was confirmed. The results may be summarized and interpreted as follows: (i) A new stable fixed point (LR) for polymers in long-range-correlated disorder is found for d = 3, a < d, leading to critical exponents that are different from those of the pure model; (ii) There is a marginal value amarg for the parameter a, below which the stable fixed point is absent, indicating a chain collapse of the polymer for disorder that is stronger correlated. (iii) The critical exponent v increases with decreasing parameter a, like in the Weinrib and Halperin case. But the relation v = 2/ a does not hold. Physically this means that for weak long-range-correlated disorder (a > amarg) the polymer coil swells with increasing correlation of the disorder. The self avoiding path of the polymer has to take larger deviations to avoid the defects of the medium. 3.5. RG theory for SAWs on the percolation cluster In the following we elaborate the renormalization group scheme for the theory of SAWs on the percolation cluster as described by the effective Hamiltonian developed in section 2.4. Extending the ideas of Meir and Harris [22] in this respect we refer to this as the MH-model. The motivation for this model is to calculate the average of a logarithm- as usual for a quenched average-
F(p, K) = L)'Yii log G;i (K)]P
(113)
j
with logG;j(K) the generating function of SAWs with a fugacity K per step between the sites i and j. The sites are connected if "/ij = 1 (else "/ij = 0). The average over bond occupation is denoted by [·]p· As usual this can be done in terms of kth moments in the
V. Blavats'ka, C. von Ferber, R. Folk and Yu. Holovatch
136 limit k ---+ 0. MH derive that G(k)(p,K)
=
L ['-y;jG7j(K)]p
(114)
j
Xp
+ kF(p, K) + O(k 2 )
(115)
where the susceptibility Xp is given by the k = 0 moment. In terms of a spin model this means that the MH model seeks to calculate the scaling behavior of the moments G(k)(p, T)
=
L [('"Y;jSl(i)Sl(j))~]p
(116)
j
in the polymer limit with zero component spin fields S(i) averaging over the disorder at the critical concentration p = Pc and over spin configurations Or at temperature T. While the ¢ 4 -theory discussed in section 3.3 does not provide such averages it is essential that these can be performed in the framework of the MH model. With the effective Hamiltonian derived in section 2.4 it turns out that the moments G(k) correspond to the propagators of this theory with masses rk that reflect the fact that there is a distinct critical point associated to each moment, i.e. there is no multicritical point as in the spin models with finite numbers of components and as suggested by the d > 4 interpretation of the ¢ 4 polymer theory [39] in sect. 3.3. In extracting the scaling behavior of the moments G(k) or equivalently of the masses rk the central quantities will be the terms linear in k in an expansion in k as suggested by Eq. (115). As far as the MH-model is a ¢ 3 field theory the diagrams and integrals that appear are identical to those of the standard ¢3 field theory of the Q = 1-Potts model. However, we have to carefully deal with the overhead of spin component and replica indices and the algebra that results from their combinatorics. In the following we review how this is worked out in the two loop approximation in Ref. [107]. Only those ¢3 cubic terms are allowed in the perturbative expansion of the theory for which any pair of indices (a;, (3;) on the contributing factors appear exactly twice. In terms of the diagrams where each line carries the indices of the corresponding field ¢ any pair of indices that "flows" into an interaction vertex must flow out on another line. Furthermore, the polymer limit of m = 0 components for the (3 indices implies that diagrams in which some indices (3; flow on closed loops i.e. do not appear on external lines, have vanishing contributions. As noted above we are interested in the k ---+ 0 limit of the algebra that is generated by replica and component index summation together with the combinatorics of the mass parameters rk. It is a somewhat tedious but straightforward matter to show that with respect to the evaluation of the standard ¢ 3 theory (see [108]) there is no change to the contributions to the vertex functions f( 3 ) which controls the renormalization of the coupling w 0 and the inverse propagator f( 2 l in this limit. However, the contributions to the vertex functions f(l, 2 ) with a mass insertion develops a characteristic behavior in linear and higher orders of k as we demonstrate in the following. Let us for simplicity consider the one loop contribution in detail. The one loop diagrams and their integrals contributing to f( 2) and f(l, 2) are labeled A 1 and A~ respectively in Fig. 8. For any particular choice of the indices on the in- and outgoing lines of diagram A 1 its contribution is the sum over alllabellings of the inner lines that are compatible with the above rules. Fork label pairs on the outer
Renormalization group approaches to polymers in disordered media
137
Figure 8. Graphs of one- and two- loop contributions to ¢3 - theory. Graphs A;: contributions to f( 2l, Graphs B;: contributions to f( 3l, Graphs A~: contributions to f(l, 2) .The insertion is marked by a dot. Graphs A' and B in the same column have identical contributions [108].
lines we have thus to sum over the possibilities to distribute those among the inner lines. Representing the masses rk by an expansion rk = Lt Utk 1 , this combinatorics gives for the first order contribution to f( 2 ) (p, rk) = 1 + w5I:l,k(P, u;) + ...
(117) Evaluating this for renormalized critical mass u; = 0 and taking the limit k --+ 0 one recovers the standard first order contribution to f( 2l. The same applies to the mass insertion generated by the derivative -88uo . However, we are interested in the behavior of terms of linear and possibly higher order in k. So we evaluate the general derivative
(118) which defines the first order contribution to f~1 ' l. We expand f~ 1 ' l again in powers of k 2
r(l,2J f
= r(l,2J + kr(l,2J + k2r(l,2J +··· f,O f,l f,2
2
(119)
r6\)2
) we recover the standard mass insertion while the one loop approximation Then, with 2 to l is the, term considered by MH. The generalized two loop contribution to the bare vertex function may be written as
ri\
(120)
V. Blavats'ka, C. von Ferber, R. Folk and Yu. Holovatch
138
with the integrals A~ corresponding to the diagrams given in Fig. 8 and coefficients a~;J derived from the combinatorics of distributing indices among the lines of the diagrams. This is calculated in terms of the sums [107]
I: k-1
(1)
af
s=1
e)s l
(121)
~G)~
(2)
af
e)
2~
(3)
k-1
af
(4)
af
(5)
af
8
~ s
(
~
) k-s
;
e) ~
8
l{)r+t()s-r+tfh-s-r+t
(123)
~G) ~e~s)l
(124)
~~G) k~1 e~s)l
(125)
1
2~
(6)
af
(122)
(;)rf
k-1
e) 8
~ s
(
;
~
) k-s
e) ~
8
(r
+ t) 1()r+t()s-r+t()k-(s-r+t)
(126) (127)
Here, the {}-factors with ()x = 1 for x > 0 else ()x = 0 control that only those terms contribute which have at least one index pair on each line. The k-expansions of these series a~q) = a~;6 + ka~;{ + k 2 a~;~ + ... then define the coefficients a~;J. For our considerations we need
(128) (129)
( -1, 1, 4, 1, 1/2, 1) (-1/2,1/3,2,2/3,1/6,1/4)
no,o
to calculate the two-loop approximation of r6~6 l, ri~i l. The minimal renormalization of these by Z¢2-factors as explained in section 3.1.2 2
z¢
2
r(1,2J -
,f f,f
-
2
r(1,2J
f,f,R
(130)
would the define exponents ry~~ of the percolation system for £ = 0 and of the polymer for£= 1. However, it turns out that such a multiplicative renormalization does not exist 12 for ri i ). We have thus to resort to an additive renormalization procedure by taking into accou~t the coupling between the different orders in k. i.e. the k-linear contributions of r~1 ' 2 l. With
a 2 ,1
= (1/4, -5/18,-1,-5/18,-5/36, -3/8)
(131)
we find that the k-linear term of the two-loop linear combination r(1,2J ( c1,1 1,1
+ c2,1 r(1,2l) 2,1
(132)
Renormalization group approaches to polymers in disordered media
139
1 + c2,J/2 and can now be renormalized multiplicatively by a Z-factor Z~·. 1 if c1, 1 c2 ,1 = 1/2 while prP.scrving the one-loop result [101 ,107]. The {:1-function and the -y~ exponcnt arc determined by the minimal subtraction scheme (see sect. 3.1.2) for the standard ,P3-theory. With the renormalizing Zq,•,rfactors for the expressions in eqs.(130), (132) this the defines corresponding exponents "'~• according to cq. (87). We find the standard result for -y~~l and [101]
-y~~ = -e/7 - 604e 2 /9261,
(133)
Together with the standard result for -y~ and the scaling relation v - 1 = 2 - -y~ - '"'fq,• thif'> brings about the correlation length exponent for the polymer in the percolation system [101] (134) recovering the :'vlH one-loop result and extending it to the second loop order. As far as the coefficients in this truncated series are small we may evaluate it without resummat.ion. The numerical values of vp at fixed space dimension is then obtained by direct substitution vp(d = 2) = 0.785, vp(d = 3) = 0.678, vp(d = 4) = 0.595, which is in good agreement with results of MC simulations, exact enumerations and Flory-like formulae.
0.8 0.75
\
•
\
0.7 c.
:;>
\
0.65 0.6
•,
\
0.55 0.5 2
2.5
3
3.5
4
4.5
5
5.5
\
\.
'·
6
d
Figure 9. The correlation exponent vp. Bold line: (133), thin line: one-loop result [22], filled boxes: Flory result vp = 3/(dpc + 2) with dpc from [109]. Exponents for the shortest and longest SAW on percolation cluster [llO] are shown by dotted lines.
From the physical point of view, this result for the exponent Vp together with the data of EE and Flory-like theories predicts a swelling of a polymer coil on the percolation
V. Blavats'ka, C. von Ferber, R. Folk and Yu. Holovatch
140
cluster with respect to the pure lattice: 1/p > VsAw for d = 2 - 5. At d = 3 the formula (133) predicts a 13% increase of 1/p with respect to VsAw which is larger than at d = 2 (5%) and should be more easily observed by current state-of-art simulations. Given that even at d = 2 we are in nice agreement with MC and EE data and the reliability of the perturbative RG results increases with d, this number calls for verification in MC experiments of similar accuracy. The validity of the MH model has been questioned in Ref.[39] by an argument which in our present formulation reads that the one loop contributions to r~ 1/) for higher £indicate the instability of the fixed point with respect to each parameter u1 .' These authors explain that it remains to be shown that the instability with respect to u 1 with £ > 1 does not influence the renormalization of the 8l8u 1-insertion. To understand how this may be resolved in the present approach it is instructive to consider the insertion corresponding to a derivative in u 2 . For this case one finds in the same way as before a multiplicative renormalization for the k 2-term of [107] ( c2,2
r(l,2J 2,2
+ c3,2 r(l,2l) 3,2
with coefficients a-factors
c 22
(135)
= 1 + c 32 l2 and
c 32
= 31104
using the following results for the
( -314,11118,3,17118,11136, 518), (318, -19136,-312,-11136,-19172, -9116).
(136) (137)
This way we may define a corresponding exponent [101] (138) related to the scaling behavior of higher order cumulants. The corresponding calculations for I OUt-insertions are straight forward. These lead to the following conclusions. With respect to multiplicative renormalization and thus to the RG flow we have instead of 2 rk = u 0 + u 1 k + u 2 k + ... the generic expansion
a
(139) For this expansion one has multiplicative renormalization at the two-loop level for the 8l8vrinsertions decoupling the flow of the v1 for different £. Explicitly,this means that the renormalization of the I ovl-insertion is independent from that of I OVf for £ > 1. It may be speculated that this scheme can be continued to higher loop orders [107]. The different series parameterized by the Vf would then correspond to the fact that instead of a multicritical point we expect to find a series of critical points in this problem [113,22] generating a multifractal spectrum of exponents ry~~.
a
a
3.6. Real space RG for polymers on diluted lattices Iterating the RSRG transformation e.g. for the Wheatstone bridge as described in section 2.4 one effectively operates on a hierarchical lattice, see Fig. 10. One may then investigate the influence of disorder on models defined on such lattices by RSRG methods.
Renormalization group approacl1es to polymers in disordered media
141
Figure 10. Iterating the Wheatstone bridge renormalization
<>~ Figure 11. Diamond lattice cell and generalization to b branches
With respect to the Harris criterion it has been shown explicitly that while disorder is always relevant if the specific heat exponent. is positive a > 0, it may also be relevant for some negative a [112,111], in particular for lattice cells in which not all bonds are equivalent. In a first attempt. to study polymers on disordered lattices by RSRG methods [111) Derrida and Griffith constrain their investigation to directed polymers simplifying the lattice to the so-called diamond hierarchical lattice and its generalization to b branches, sec Fig. 11. The latter model can be analytically continued to non-integer band allows for a perturbative solution in terms of an expansion in c = b - 1. The authors calculate the ground state energy and the growth of its fluctuation with polymer size L which increases like Dw with an exponent w. In Re£.[115) Cook and Derrida extend this study to finite temperature where the renormalization acts on energy or partition function distributions. The temperature enters only the initial distribution of the partition function. Using higher order approximations a high temperature phase is identified for b > 2 where the energy distribution approaches a delta function as the number n of renormalization iterations increases. In a separate low temperature phase the width of this distribution grows with n . However, for the transition temperature Tc between these phases only an upper bound could be constructed. The main advantage of the RSRG method for the problem of polymers or SAW on disordered lattices is that it provid<'..s means to study the crossover between different universal scaling behavior in terms of an RG flow. In the following we first introduce the RSRG flow for this problem as proposed by Meir and Harris [22)(MH). \Ve then proceed to discuss the critique brought about on this approach by Le Doussal and :Vlachta [39)(DM). We follow [22,116] in defining the generating function
G;j(K) =
L CNij(K)KN
(140)
N
of SAWs between sites i and jon the lattice, where CNi; denotes the number of SAWs
V. Blavats'ka, C. von Ferber, R. Folk and Yu. Holovatch
142
between i and j and K is the fugacity for each step. Indicating if sites i and j are connected by the diluted lattice or not by /'ij = 1 or 0, the expansion of the log-average of G;j(K) around K = K 0 reads ~
[f';i log GiJ(Ko)]P +
K-K0
Ko
-
N;i(Ko)
LN CN;j(K)N KN] /'ij LN CN;j(K)KN p
/'ij [
(141) (142)
Here, K 0 is chosen such that the first term on the right hand side of eq.(141) vanishes. The correlation length exponent lip for the SAW on the diluted lattice is the defined by the power law scaling for the end-to-end distance R;j between the sites i and j: (143) For the Wheatstone bridge log Gij of the renormalized bond is equal to log K. recursion relations for log K and p read [22] p'
p'logK'
2p2 + 2p3 - 5p4 + 2p5 4p2 log K + 6p3 log K + p 4(log 2- 14log K + 4log(K + 1)) +p5(6logK- 3log(K + 1))
The (144) (145)
where the first relation is the standard one for the Wheatstone bridge [106]. The flow diagram that results from these recursion relations is depicted in Fig. 12. The fixed points marked A (p* = 1, K* = 0.366), B (p* = 1/2, K* = 0.788), and A (p* = 0, K* = oo) are interpreted to correspond to SAWs on the pure lattice(A), at the percolation threshold (B), and on lattice animals (C). The corresponding crossover exponents ¢together with the 2D-percolation correlation exponent Vpc = 4/3 give the SAW correlation exponent lip = 1/pc/¢. At the fixed point B this reads ¢ = 1.682. The RG flow in Fig. 12 also brings about two more fixed points D and E which are interpreted as corresponding to minimal and maximal SAWs with rPmin = 1.55 and rPmax = 1.835. An additional fixed point C 1 (p* = 0, K* = 1) is found in Fig. 12 which apparently was overlooked by MH. In a similar approach on the same lattice DM derive recursive relations involving distributions instead of the constant fugacity K on each bond visited by a SAW which corresponds to a bimodal distribution. This relation for the distribution is (146) Note that also eq. (145) can be decoupled from (144) in terms of the variable pK:
p' K' = 2p2K 2 + 2p3K 3
(147)
corresponding to all Z; = pK. DM criticize the approach of MH which projects back to the bimodal distribution after each renormalization step. The full RG flow derived by DM from these equations is depicted in Fig. 12a. Here, there is one more parameter, namely the variance .6.(log Z) of the bond energy distribution which grows with the size of the SAW according to
RenormaJization group approaclws to polymers in disordered media
c'l :::::::::
a)
0. .. . ....
. , ..... .. ,,,,,,,,, , ,,,,,,, , , ,,, ,,,,,, ,, ,, ,,,,,, , ,,,,, ,, , ~
,,,,,. """"" ,,,,, ,,,, K + l f'"'""' ,.""""" ,,,, c . . ,, . . ,, , , ""'"'fl ,,, ,,, ,,,, ,,,,, K
o.e
--
" , , , , ,, t
0.6
b)
143
(logZ} J l + ~ logZ
D
B
, , ,,,,,, ,
I I
• ••
•..
B. • • ,,
I, ,t
1 11'1111 11
1111 1 1111 111111111
0.2 f''""""'' '
:::::~~:: . A
p - p ..
,,,,,.,.,. ~ , .
,,,,,, ,. ,, ,.,.,. ,. ..... . I I I I I I I I
[ 0
.,
~,
,,,
.:'-~---·. .
0,
04
' ' \ \ \ 11 11 \ • •
. . ... , ..... " . . . E Oli
~--
-·
~-=-·
0_8
,,
1
Figure 12. a)RSRG flow derived from Eqs. (144) and (145) [22]. b) Full RSRG flow derived from Eqs. (144) and {146); the critical manifold is shaded [39].
~(log Z)
""' Dw. It appears that the plane .6(log Z) = 0 where the MH flow was defined is unstable. Instead two strong disorder fixed points SPc and S1 appear at infinite b. (log Z) and the fixed points 5 and E corresponding to the maximal and minimal SAWs in this picture are also shifted to infinite .6(log Z). D~1 derive exact values for their corresponding (-exponents that arc related to v via 1/v = 1/( - w. These values, ( min = log2flog(17/8) and (max = log 2/log(39/16) are again in accordance with previous numerical results [117]. At the strong disorder fixed point SPc at percolation p = Pc however, D~I find ( = 0.850 which is very close to ( = 0.848 found at the fixed point B of the MH flow diagram. We conclude that apparently the Mil RSRG approach oversimplified the situation. However, a systemat ic scheme for the strong disorder fixed points that \vould include an upper critical dimension de = 6 and in particular a field theoretic model that could reproduce the complete RG flow remain to be found. Acknowledgments: R.F. and Yu.H. acknowledge t.he Austrian Fonds zur Forderung wisscnschaftlicher Forschung for support under project N 16574-PHY.
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V. Blavats'ka, C. von Ferber, R. Folk and Yu. Holovatch
D. Staufer Z. Phys. B 22 (1975) 161. T. C. Lubensky, Pys. Rev. B 15 (1977) 311. G. Parisi, Proceedings of the Cargrese Summer School- 1973. unpublished. G. Parisi, J. Stat. Phys. 23 (1980) 49. G. 't Hooft and M. Veltman, Nucl. Phys. B 44 (1972) 189. G. 't Hooft, Nucl. Phys. B 61 (1973) 455. G. H. Hardy, Divergent Series, Oxford, 1948. J.-P. Eckmann, J. Magnen and R. Seneor, Commun. Math. Phys. 39 (1975) 251; J. S. Feldman and K. Osterwalder, Ann. Phys. 97 (1976) 80; J. Magnen and R. Seneor, Commun. Math. Phys. 56 (1977) 237; J.-P. Eckmann and H. Epstein, Commun. Math. Phys. 68 (1979) 245. 86. F. Wu, Rev. Mod. Phys. 54 (1982) 235. 87. R. B. Griffiths, Phys. Rev. Lett. 23 (1969) 17. 88. A. J. Bray, T. McCarthy, M.A. Moore, J.D. Reger, and A. P. Young, Phys. Rev. B 36 (1987) 2212; A. J. McKane, Phys. Rev. B 49 (1994) 12003. 89. G. Alvarez, V. Martin-Mayor, and J. J. Ruiz-Lorenzo, J. Phys. A 33 (2000) 841. 90. G. A. Baker, B. G. Nickel, M. S. Green, and D. I. Meiron, Phys. Rev. Lett. 36 (1976) 1351; G. A. Baker, B. G. Nickel, and D. I. Meiron, Phys. Rev. B 17 (1978) 1365. 91. M. I. Stephen and G. S. Grest, Phys. Rev. Lett. 38 (1977) 567. 92. The construction of the Borel-image for the functions of more than two variables is performed similarly to the procedure of the Eq.(91). 93. J. S. R. Chisholm, Math. Comp. 27 (1973) 841; G. A. Baker Jn., P. Graves-Morris, Pade Approximants, Addison-Wesley, Reading, MA, 1981. 94. P. J. S. Watson, J. Phys. A 7 (1974) L167. 95. J. M. Carmona, A. Pelissetto, and E. Vicari, Phys. Rev. B 61 (2000) 15136. 96. S.A. Antonenko and A.l. Sokolov, Phys. Rev. B 51 (1995) 1894. 97. H. Kleinert, J. Neu, V. Schulte-Frohlinde, K.G. Chetyrkin and S.A. Larin, Phys. Lett. B 272 (1991) 39; H. Kleinert, J. Neu, V. Schulte-Frohlinde, K.G. Chetyrkin and S.A. Larin, Phys. Lett. B 319 (1993) 545 (erratum). 98. S. Grassberger, R. I. Cohen, J. Phys. A 19 (1986) 1681. 99. S. Havlin, D. Ben Abraham, Adv. Phys. 36 (1987) 155. 100A. B. Harris and T. C. Lubensky, Phys. Rev. B 35 (1987) 6964. 101.C. von Ferber, V. Blavatska, R. Folk, and Yu. Holovatch, Phys. Rev. E 70 (2004) 035104(R). 102D. C. Hong and H. E. Stanley, J. Phys. A 16 (1983) 2525 103B. E. Stanley, P. Reynolds, S. Redner, and F. Family, 1982, in Real-Space Renormalization, edited by. T. W. Burkhardt and J. M. J. van Leeuwen (Springer-Verlag, Berlin). 104F. Family, J. Phys. A 13 (1980) L325 105.F. Family, J. Physique 42 (1981) 189 106P. J. Reynolds, H. E. Stanley, and W. Klein, J. Phys. C 10 (1977) L167 107.C. von Ferber, V. Blavatska, R. Folk, and Yu. Holovatch, in preparation (2005). 108D. J. Amit, J. Phys. A 9 (1976) 1441. 109B. Nienhuis, J. Phys. A 15 (1982) 199; P. N. Strenski, Phys. Rev. Lett. 66 (1991) 1330; N. Jan, D. C. Hong, and H. E. Stanley, J. Phys. A 18 (1985) L935.
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110B. K. Janssen, Z. Phys. B 58 (1985) 311; J. Cardy and P. Grassberger, J.Phys. A 18 (1985) L267; H. K. Janssen and 0. Stenull, Phys. Rev. E 61 (2000) 4821. 111B. Derrida, H. Dickinson, and J. Yeomans, J. Phys. A 18 (1985) L53. 112D. Andelman and A. N. Berker, Phys. Rev. B 29 (1984) 2630. 113B. Derrida, Phys. Rep. 103 (1984) 29. 114B. Derrida and R. B. Griffith, Europhys. Lett. 8 (1989) 111. 115J. Cook and B. Derrida, J. Stat. Phys. 57 (1989) 89. 116.Y. Meir. Thesis. Tel Aviv University. (1988) 117 P. Sen and P. Ray, J. Stat. Phys. 59 (1990) 1581
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Statistics of Linear Polymers in Disordered Media Edited by Bikas K. Chakrabarti © 2005 Elsevier B.V. All rights reserved.
Linear and branched polymers on fractals Deepak Dhara and Yashwant Singh b a Department of Theoretical Physics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India bDepartment of Physics, Banaras Hindu University, Varanasi, U.P. 221005, India. This is a pedagogical review of the subject of linear polymers on deterministic finitely ramified fractals. For these, one can determine the critical properties exactly by real-space renormalization group technique. We show how this is used to determine the critical exponents of self-avoiding walks on different fractals. The behavior of critical exponents for the n-simplex lattice in the limit of large n is determined. We study self-avoiding walks when the fractal dimension of the underlying lattice is just below 2. We then consider the case of linear polymers with attractive interactions, which on some fractals leads to a collapse transition. The fractals also provide a setting where the adsorption of a linear chain near on attractive substrate surface and the zipping-unzipping transition of two mutually interacting chains can be studied analytically. We also discuss briefly the critical properties of branched polymers on fractals.
1. INTRODUCTION The problem of self-avoiding walks ( SAWs) on lattices provides perhaps the simplest geometrical model of equilibrium critical phenomena ( i.e., non-trivial power-laws in the behavior of different quantities in a system in thermal equilibrium). The two other familiar examples of geometrical models showing phase transitions, the percolation problem and a system of hard particles (spheres or rods), both involve more complex geometrical structures. The model of SAWs captures the important macroscopic features of polymers in solution, and is closely related to other models of phase-transitions in statistical physics like the Ising model [1], and can also be seen as the n ---+ 0 limit of the n-vector model [2]. Given the many technological applications of polymers, and the importance of SAWs as a model of critical phenomena, it is not surprizing that the model has attracted a lot of attention in the last sixty years. Several good reviews are available summarizing our current understanding of this problem [3]. The SAW problem is clearly trivial in one dimension. In spite of the large number of papers related to this problem, an exact solution of the problem has not been possible so far, for any non-trivial case. In two dimensions, the exact value of the growth constant is 149
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known for the hexagonal lattice [4], and has been conjectured for the square lattice from large-order exact series expansions [5]. The exact value of the critical exponents in two dimensions is known from conformal field theory [6]. Even for dimensions d > 4, where the critical exponents are known to take mean-field values [7], a full solution has not been possible. Given the paucity of exact solutions in this area, it seems reasonable to look for some artificially constructed graphs, e.g. fractals, for which an exact solution can be found. This solution then, can be considered as an approximation to the original problem. The advantage of this approach, over other ad-hoc approximations like the Flory approximation, is that one is assured of well-behavior requirements like the convexity of the free energy, and avoids problems like getting two different values for a quantity ( e.g. the pressure for hard-sphere systems), if one calculates it in two different ways within the same approximation. Another motivation for the study of SAWs on fractals comes from the need to understand how the critical exponents depend on the dimension and the topological structure of underlying space. In the formal techniques like c-expansions, one treats the dimension of space d as a parameter that can be varied continuously, but such techniques involve a formal analytical continuation of various perturbative field theory integrals of the type f ddk .. to all real values of d [8]. However, this formal approach does not give any simple answer to the question, "What is the Ising model in 3.99 dimensions ?". On the other hand, fractals are explicitly constructed spaces of non-integer dimension, and one can construct fractals with dimension close to any given real number d. One can thus study how critical exponents change by changing the geometry of the underlying space. Interestingly, the results from the formal c-expansion techniques do not match with those from explicitly constructed fractals. The main reason seems to be the assumption of translational in variance in the former, while the explicitly constructed fractals do not have this property. In fact, this assumption seems to be problematic for spaces with non-integer dimensions, and leads to pathologies such as non-positive measure [9]. For a longer discussion of these issues, see [10]. A third reason for the study of SAWs on fractals is that these provide excellent pedagogical examples of application of renormalization group techniques to the determination of critical exponents. If one tries to implement the renormalization transformation to some system like the Ising model in two dimensions, one eventually generates an infinity of additional multi-spin or longer range couplings, which are presumably irrelevant, and are neglected to get renormalization equations in terms of a finite number of variables. The justification given for doing this usually involves too much 'handwaving' for a discerning student. In case of fractals, one gets exact renormalization equations in terms of a finite number of coupling constants. One can then study these in detail, and explicitly work out their trivial and non-trivial fixed points, stable and unstable directions, basins of attractions of fixed points, critical exponents in terms of eigenvalues of the linearized transformation etc., and learn the use ofrenormalization group techniques for determining critical properties. Finally, one can treat the fractal lattice as a simple model for a disordered substrate on which the polymer is adsorbed, and use the exact results found for polymers on fractals to develop some understanding about real experimental systems. But for this, this article
Linear and branched polymers on fractals
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(b)
Figure 1. (a) The Bethe lattice of coordination number 3. (b) A Husimi cactus made up of triangles.
would not have been included in this volume. The simplest of graphs for which the SAW problem is analytically tractable is the Bethe lattice, for which the exact solution is trivial, as the graph has no loops. Only slightly more complicated is the case of Husimi cactus graphs, which are like the Bethe lattice on the large scale, but have small loops [Fig. 1]. These do change some properties like the connectivity constant for the walks, but the large-scale properties of walks on such lattices are the same as on the Bethe lattice. In this chapter, we will discuss fractal graphs which have loops of arbitrarily large size, and for which the critical exponents are different from the mean-field values. This chapter is organized as follows: In section 2, we start with the precise definitions of the fractals we study. We also define the SAW generating functions, their annealed and quenched averages and the critical exponents of self-avoiding walks. In section 3, we discuss the renormalization group treatment of the SAWs on the 3-simplex lattice in detail. We explicitly construct the recursion equations, and show how the critical exponents of SAWs can be determined from an analysis of the linearized renormalization group transformation near the fixed point. The dense phase of the polymer, and the fluctuations of the number of rooted walks are also discussed briefly. In section 4, we show how this treatment is generalized to other fractals where one has more complicated recursion equations than the simplest case of 3-simplex lattice.. We also discuss the behavior of critical exponents on the n-simplex in the limit of large n. In section 5, we discuss the SAWs on the Given-Mandelbrot family of fractals, and study the behavior of critical exponents of walks when the fractal dimension of the lattice tends to 2 from below. We use finite-size scaling theory to determine how the structure of the renormalization equations depends on the parameter b defining the fractals. We develop a perturbative expansion for the critical exponents valid for large b when the fractal dimension of the lattice is just below 2. In section 6, we discuss the collapse transition of linear polymers with attractive self-interaction, and the tricritical 11-point. We also discuss other intermediate 'semicompact' phases that is seen on some fractals. In section 7, we show that much of our
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(a)
(b)
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Figure 2. The recursive construction of then-simplex fractal for n = 5. (a) The first order graph (b) the graph of a (r + 1) order triangle, formed by joining n r- th order graphs, shown as shaded blobs here (c) the second order graph.
treatment of linear polymers can be extended to branched polymers. One can determine the critical exponents (} for average number of branched polymer configurations per site, and v for the mean polymer size using the RSRG techniques. Interestingly, one finds that for G Mb fractals with b > 2, the average number of branched polymers per site increases as exp(an + bn'I/J), with 0 < 'ljJ < 1, and the leading form of correction to the exponential growth is not a power-law correction. This stretched exponential form arises from the presence of favorable and unfavorable regions on the fractal lattice. In section 8, we discuss a polymer with self-attraction near an attractive surface. In this case, there is a competition between the collapse transition in the bulk of the fractal, and the tendency of the polymer to stick to the surface. We discuss the qualitative features of the phase-diagram, and critical exponents. The phase-diagram is somewhat complicated when the next layer interaction is included, and provides a good pedagogical example for the study of higher order multi-critical points. One can study even more complicated systems. Section 9 contains a discussion of two mutually interacting linear chains, as a model of the zipping -unzipping transition in the double-stranded DNA molecules. Section 10 contains some concluding remarks, and we mention some open problems which deserve further study.
2. DEFINITIONS We start by giving definitions of the family of fractals that we study in this article. All these fractals have a finite ramification number, and are defined recursively. The first family we shall discuss is the n-simplex family, defined for all positive integers n > 2. Then= 3 case was first defined by Nelson and Fisher [11], who called it truncated tetrahedron lattice. The construction was generalized for arbitrary n in [12]. The recursive construction of the graph of the fractal is shown in Fig. 2. The first order graph is a single vertex with n bonds. In general, the r-th order graph will have nr-l vertices, and
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Figure 3. The recursive construction of the Given-Mandelbrot fractal for b = 3. (a) The graph of first order triangle. (b) the graph of a (r + 1) order triangle, formed by joining b(b + 1)/2 r-th order triangles shown as shaded triangles here (c) graph of the 2nd order triangle.
(nr + n) /2 bonds. Of these (nr - n) /2 are internal bonds, and there are n external bonds, that are used to connect this graph to other vertices to form bigger graphs. To form a graph of (r + 1)-th order, we taken graphs of r-th order, and join them to each other by connecting a dangling bond of each to a dangling bond of the other r-order subgraphs. There is one dangling bond left in each of the subgraphs, and n bonds altogether. These form the dangling bonds of the (r + 1)-th order graph. We let r tend to infinity to get an infinite graph. The fractal dimension Dn of this graph is easily seen to be log n/ log 2. It was shown in [12] that the spectral dimension dn of this graph is 2log n/ log(n + 2). The second family we shall consider is the Given-Mandelbrot family of fractals [13], defined in Fig. 3. Members of the family are characterized by an integer b, with 2 ::; b < oo. We start with a graph with three vertices and three edges forming a triangle. This is called the first order triangle. To construct the graph of the (r + 1)-th order triangle, we join together graphs of b(b + 1)/2 triangles of r-th order, (i.e. identify corner vertices) as shown in the figure, to form an equilateral triangle with base which is b times longer. We shall call the resulting graph the G Mb fractal. It is easy to see that the number of edges in the graph of the r-th order triangle is 3[b(b+ 1)/2]"- 1 . The fractal dimension is Db= log[b(b+ 1)/2]/logb. Forb= 2,3,4 .. , these values are 1.5849, 1.6309, 1.6609 ... respectively. The spectral dimension db of the graph can also be calculated exactly for general b [14]. The values of db forb= 2 to 10 are listed in [15]. For large b, db tends to 2, and the leading correction to its limiting value is given by db~ 2 -log log b/ log b [16]. In fact, following Hilfer and Blumen [15], one can define a general fractal family of Sierpinski -like fractals, to be called H B(b, d) family here, such that the the n-simplex corresponds to HB(2, n- 1), and the GMb fractal corresponds to HB(b, 2). Here the basic unit is a (d+ I)-simplex graph, and one makes the (r + 1)-th order graph by making a simplex of b layers of the r-th order graphs. The construction is illustrated in Fig. 4 for
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the fractal HB(3 , 3).
Figure 4. First two stages in the iterative construction of the Hilfer-Blurnen fractal H B(3, 3). This has a fractal dimension d = log 10/ log3.
The third family of fractals we shall use is a generalization of the modified rectangular lattice [12). Here the first order graph is a set of 2d vertices forming a d-dimensional hypercube. In general, the r-th order graph has 2r+d-l vertices in the shape of a cuboid, with each corner vertex of the cuboid having an extra dangling bond to connect to outside. The graph of an (r + 1)-th order cuboid is formed by taking two r-th order cuboids, and joining the opposite faces by 2d-l bonds. The direction of the face that is selected for this is changed cyclically at the next order, so that the lengths of different sides of the cuboid are within a factor 2 of each other. It is easily seen that the fractal dimension of this graph is d, and it can be shown that the spectral dimension is 2(1 - 2- d). The construc:tion is illustrated in Fig. 5 for the case d = 2. The determination of the generating function for the linear polymers on these fractals follows the treatment of [17,18). Let Pn(N ) be the number of distinct simple polygons of perimeter n on a graph with N total number of sites in the graph, different translations of the polymer being counted as distinct. [The use of same symbol n for the n-simplex and n-stepped walk should not cause any confusion, as it. is clear from context which is intended.) For large N, this number increases linearly with N . We then define Pn as the average number of polygons of perimeter n per site by
. Pn(N) Pn = LzmN-. 00 ~;
(1)
and 00
P(x)
=L
Pnxn,
(2)
n =>3
For large n, Pn varies as J.tnno- a, where JL is a constant, and a is a critical exponent for the walks. This corre$ponds to P(x)"' {1 - J.tX) 2-o, for x tending to 1/ J.t from below.
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(a)
D
(b)
(c)
I I
I I
I I
I I
11
11
I I
I I
Figure 5. The recursive construction of the modified rectangular lattice for d = 2. (a) The graph of first order square. (b) the (r + 1)- th order graph, formed by joining 2 r- th order graphs shown as shaded rectangles here (c) graph of the 5th order rectangle.
Note that P(x) is a finite-degree polynomial in x for the graphs like the Husimi cactus (Fig. 1). We similarly define Cn, the average number of open walks of length n, as the average over all positions of the root, of the number of n-stepped self-avoiding walks. For large n, this varies as p.nn'~-1, which defines the critical exponent 'Y· The corresponding generating function 00
C(x)
=
L
Cnxn
(3)
n=l
varies as (1- xp.)-'~ as x tends to 1/ p. from below. We also define the critical exponent v for SAWs by the relation that the average end-to-end distance for walks oflength n varies as n" for large n. Here in the context of disordered systems, we imagine that the polymer can freely move over all space, and the averages calculated correspond to annealed averages, as we are averaging the partition function of SAWs over different positions of the root. The logarithm of the number of polymer configurations is the entropy, and one can also define the equivalent of quenched averages, where one averages, not the numbers of walks, but logarithms of numbers of rooted walks, over different positions of the root. These are more difficult to determine. We will indicate how these can be handled in sec. 3.
3. RENORMALIZATION EQUATIONS FOR THE 3-SIMPLEX FRACTAL The procedure to determine the critical behavior of SAWs on fractals is simplest to illustrate using the 3-simplex as an example. We will keep the presentation informal (hopefully not inaccurate). Readers who prefer a more formal approach, may consult [19]. We discuss the calculation of annealed averages first.
3.1. Calculation of annealed averages Consider one r-th order triangular subgraph of the infinite order graph. It is connected to the rest of the lattice by only three bonds. Our aim to sum over different configurations of the SAW on this subgraph, with a weight x for each step of the walk. These can be divided into four classes, as shown in Fig. 6. Here A(r) is the sum over all configurations
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of the walk within the r-th order triangle, that enters the triangle from a specified corner, and with one endpoint inside the triangle. B(r) is the sum over all configurations of walk within the triangle that enters and leaves the triangle from specified corners. In C(r), the walk enters and leaves the subgraph from specified vertices, and reenters afterwards from the third corner, and has one endpoint inside the triangle. fl(r) consists of sum over configurations that have both endpoints of the walk inside the triangle, but part of the walk is outside the triangle. We can write down the values of these for r = 1 immediately. AC 1l
= v;i, BC 1l = x, C(ll = DC 1l = 0.
(4)
We define the order of a closed polygon on the infinite 3-simplex graph to be r, if the polygon is completely contained inside an r-th order triangle, but not inside any (r -1)-th order triangle. It is easy to see that sum over all r-th order polygons within one such 3 triangle is B(r-l) . The number of sites in the (r + 1)-th order triangle is 3r, hence we get
P(x) =
f
3-r B(rl3.
(5)
r=l
The sum over open walks can be expressed similarly
C(x) =
00
L
3-r[3A(r) 2 + 3B(r) A(r) 2
+ 3B(r) 2 D(rl].
(6)
r=l
It is straight forward to write down the recursion equations for these weights A(r+l), B(r+l), C(r+l) and D(r+l) in terms of the values at order r. For example, Fig. 7 shows the only two possible ways one can construct a polymer configuration of type B. It is easy to verify that the resulting equations are [17]
(7) (8) cCr+lJ
= AB 2 + C(3B 2 );
D(r+ll
= (A 2 + 2A 2 B + 4ABC + 6BC2 ) + D(2B + 3B 2 );
(9) (10)
where we have dropped the superscripts (r) in the right-hand side of the equations to simplify notation. It is straightforward to determine the analytical behavior of the generating functions P(x) and C(x) from the equations (4)-(10). We note that the equation for B(r+l) depends only on B(r), and does not involve the other variables. The recursion equation (8) has two trivial fixed points B* = 0, and B* = oo, and a non-trivial fixed point B* given by B* = (v'5 -1)/2. For B(o) = x < B*, the variables B(r) decrease with r under iteration, and tend to zero for large r. For B(o) > B*, B(r) increases with r, and diverges to infinity, and the series for P(x) and C(x) diverge. This implies that Xc
= B* = 1/{L = [y'5 -1]/2.
(11)
Linear and branched polymers on fractals
s
157
c
o
Figure 6. Restricted partition functions for the 3-simplex lattice.
s
Figure 7. Graphical representation of terms that contribute to the recursion equation for B(r+l) for the 3-simplex lattice.
which shows that the growth constant p. of SAWs on this lattice is the golden mean. Using Eq. (8), we see that P(x) satisfies the functional equation
(12) From this equation, we get P(x = B*) = p.- 3 /2. For x near B*, we can linearize the recursion equation. We write x = B*- b, and Q(b) = P(B*)- P(B*- b) giving 1
Q( b) ~ p.- 2 b + 3Q( (2 + p.- 2 )b).
(13)
If we assume that the singular part of Q( b) varies as b2 -", this gives us
log3
Q
= 2- log (2 + p.- 2) '::::' 0.7342.
(14)
The recursion equations for A(r+lJ and c
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0 +
++
+
I
-1
i
#'\\
+
c
a...
f
-2
+
0>
+
0
t+
\
+
::: + + +
-3 +
+ +
+ +
v
+ +
+
+ +
-4
-5 10
20
80
40
160
320
n Figure 8. Plot of the difference between the exact value of log Pn and the asymptotic form showing log-periodic oscillations with n for the 3-simplex lattice.
we have clearly ro =
log(1/b) log(2 + w 2 ).
(16)
,----....,.=--'--'--'--:::-:-
For r < r 0 , in the recursion equations for
A(r)
and C(r), we can put
B(r)
= B*, giving
A(r+l) ) _ ( ( cCr+l)
(17)
This implies that A(r) and C(r) vary as A~, where A+ is the larger eigenvalue of the 2 x 2 matrix in Eq. (17). For r > r 0 , A(r) ~ A(rol, and C(r) ~ 0, then, in Eq. (6), the leading contribution comes from the term r = r 0 , giving (18) with log(A~/3)
1
= log (2 + p.- 2)
~ 1.3752.
The typical size of the polymer for x log2
v
= log( 2 + p.- 2 ) ~ 0.7986.
(19) :0::
Xc-
b is 2ro, which varies as (1/b)", with (20)
Linear and branched polymers on fractals
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It is quite straightforward to generate the exact series P(x) using Eq.(5) with symbolic manipulation programs like Mathematica. Just keeping terms up to r = 7, we get a series for P(x) exact up to x 383 ( the coefficient of this term has more than 70 digits!). As seen above, the singular part of P(x) varies as (1 - xJJ.) 2-a for x near p.- 1 its nth Taylor coefficient must vary as p.nna- 3 . However, in general, the solution of Eq.(13) allows an additive term which is periodic function of 1ogb with period log(2 + p.- 2 ). In Fig. 8, we have plotted the difference ~log Pn between the exact value of log Pn and the asymptotic value log(p.nna- 3 ). We clearly see the log-periodic oscillations. Such logperiodic oscillations are seen in many other systems showing discrete scale-invariance [20]. The existence of these oscillations makes it very difficult to estimate critical exponents from extraplations of exact series expansions using only a few terms [21]. For the fugacity x > Xc, the linear polymer fills the available space with finite density. Since the logarithm of the single loop partition function for the r-th order triangle is now proportional to the number of sites in the triangle, we define the free energy per site in the dense phase as .
log B(r)
f(x) = r-too hm - -1- . 3r-
(21)
Then, from Eq.(8), it follows that f(x) satisfies the equation (22) The density of the polymer p(x) in the dense phase is given by
d
) 2 + 3x )p(x 2 + x 3 . 3 X+ 1
p(x ) = x-d f(x) = ( X
(23)
Iterating this equation, we get
(24) From Eq. (22), it can be seen that as x tends to Xc from above, the polymer density decreases as (x- xc) 1 -a. Equivalently, for small densities p, the entropy per monomer p.(p) c::: p.(O)- Kp'~", where K is some constant.
3.2. Quenched averages We note that the 3-simplex lattice is not homogenous, and different sites are not equivalent. In the context of disordered systems, the calculation of P(x) and C(x) are examples of annealed averages. We now show how one can calculate "quenched averages" in this problem. We define Pn(S) as the number of polygons of perimeter n that pass through a given site S ( these are called rooted polygons, one could also study rooted open walks), and ask how does it vary with S. What is its average value, variance etcetra? A quantity of particular interest is the average value of log Pn(S). A good understanding of these for the regular fractals is prerequisite for understanding the more complicated case of random
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(b)
(a)
Figure 9. (a) labelling of different sites on the 3-simplex (b) definition of weights for rooted graphs
quenched disorder. We mention briefly some very recent results about the behavior of fluctuations of Pn(S) with S [22]. To describe rooted polygons, we first need to set up a coordinate system to describe different sites on the fractal. A simple and natural choice for this is shown in fig. 9. A point on the r-th order triangle is labelled by a string of (r-1) characters, e.g. 0122201 .... Each character takes one of three values 0, 1 or 2. The leftmost character specifies in which of the three sub-triangles the point lies (0, 1 and 2 for the top, left and right subtriangle respectively). The next character specifies placement in the (r - I)-order sub-triangle, and so on. The restricted partition functions for the rooted polygons are also defined in 9. Here B~r) (x, S) is the sum over walks on the r-th order triangle that go through the left and right corners of the triangle, and also visit the site characterized by string S inside the triangle. Other weights are defined similarly. A site in the (r + 1)-th order will be characterized by a string of r characters. Hence, a site characterized by stringS at the r-th stage will be characterized by one of the strings OS, lS or 2S. The recursion relations in this case are linear, and can be written in the matrix form
(25) where
(26) and we have suppressed the superscript (r) on B in the matrices. The generating function for rooted polygons is given by 00
P(x,S)
= "LB~r)(x,Sr)B(rl r=l
2
,
(27)
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Linear and branched polymers on fractals
,. -;:: ,:-_.., '"'"""'
-8
,'~·
.
.."
, ..... ,'_··
"",
\.
,'
".-' :
...
n
Figure 10. A plot of the average value of log Pn(S)p.-n as a function of n. The uppermost and the lowermost curves are theoretically derived upper and lower bounds to this number over different positions of the root. The dashed and dotted lines show the annealed and quenched average values respectively.
where Sr is the string consisting of the last (r - 1) characters of the position of S. If we ignore the constraint that the walk has to pass through S, we get an upper bound on the number of such walks, B~r) (x, S) ::; B(r) (x), where inequality between polynomials is understood to imply inequality for the coefficient of each power of x. This implies that for all sites S, 00
P(x, S) :S
L
3
B(r) .
(28)
r=O
If we write the right-hand-side as .Fu(x), then .Fu(x) satisfies the equation
.Fu(x) = x 3 + .Fu(x 2 + x 3 ).
(29)
This equation should be compared with Eq.(13), and differs from it only in the absence of the factor 1/3. This functional equation has the fixed point at x* = 1/ p., and linear analysis near the fixed point shows that .Fu(x) diverges as -log(1- xp.) for x tending to 1/ p. from below. This implies that the coefficient of xn in the Taylor expansion of .Fu varies as p.n /n for large n. Thus,
Pn(S) :S K p.n /n, for all n, and all S,
(30)
where K is some constant. Sumedha and Dhar [22] also derive a lower bound Pn(S) 2:: K 1 p.nn-b-l, for all n and all S, where b = 2log p./ log(2 + p.- 2 ) ~ 0.92717, and K 1 is some constant. For a randomly chosen S, the problem reduces to a random product of noncommuting matrices M 0 , M 1 and M 2 . The expected value of the logarithm of a
D. Dhar and Y. Singh
162
A(r)
~r)
F (r)
Figure 11. Different restricted partition functions for the 4-simplex lattice. The shaded squares denote graphs of r-th order 4-simplex, of which only the external bonds are shown.
product of r independent random matrices varies linearly with r for larger. This implies that for large n, the quenched average< logPn(S) >varies as nlog{L- (2- aq)logn. The numerical estimate of aq is approximately 0. 729 ± .001, which is just a bit less than the annealed value 0. 7342. In figure 10, we have plotted the numerical values of the upper and lower bounds to log(Pn(S)f.L-n), and the annealed and quenched averages log < Pn(S){L-n >, and < log(Pn(S)f.L-n) >. It should be noted that the lower bound is the best possible, in the sense that for each value of n, there is a finite density of roots that saturate the bound. The upper bound is not always optimal. Also, whether the annealed and quenched averages are very close or less so appears to be an oscillatory function of logn.
4. RECURSION EQUATIONS FOR OTHER FRACTALS We now briefly indicate how the real-space renormalization group technique of the previous section can be extended to other fractals.
4.1. n-simplex fractals for n > 3 The case of n-sim plex fractal for higher n is a straight forward extension of the technique used for the 3-simplex. The case n = 4 was studied in [17], and the treatment was extended to n = 5 and 6 in [23]. For higher n, the number of restricted partition functions we have to define to generate a closed set of recursion equations is larger. Here the permutation symmetry between the corner vertices of the n-simplex graph is very useful in reducing the number of variables needed. For n = 4 and 5, we need two functions A(r) and B(r) to generate the closed loops generating function P(x). For the open walks generating function C(x), four more variables are needed. Their definition is shown in Fig. 11. They will be denoted by cCrJ, D(rJ, E(rJ and F(rJ. The starting values of these weights are A(l)
= x;
(31)
The recursion relations for these weights are written by constructing graphs by all possible
Linear and branched polymers on fractals
163
ways. This leads to [17] A(r+l)
= A2 + 2A3 + 2A 4 + 4A 3 B + 6A 2 B 2 ,
(32)
B(r+l)
= A4 + 4A 3 B + 22B4 ,
(33)
and similar equations for the other variables (omitted here). The expression for P(x) in this case is 3
P(x) = f(4A(r) + 3A(r)\
(34)
r=l There is a similar but more complicated expression for the generating function for open walks. We note that recursions of A(r+l) and B(r+l) depend only on A(r) and B(r). For any given value of x, we can determine P(x) numerically by explicitly iterating the recursion equations. One finds Xc
= 0.4373,
{L
= 2.2866,
(35)
and the corresponding non-trivial fixed point is (A*, B*) = (0.4264, 0.04998). Linearization of Eqs. (32) and (33) around this fixed point gives largest eigenvalue A1 = 2.7965. We can express the end-to-end distance exponent v in terms of A1 using arguments similar to the case of 3-simplex. We get v
log2 og /\1
= -l \ = 0.6740,
(36)
and specific heat exponent
a= 2- vD 4 = 0.6519.
(37)
To find 7, we consider the recursion equations for C(r+l) and D(r+l), which are linear in C(r) and D(r), and diagonalizing the 2 x 2 matrix gives the eigenvalue A+ = 4.2069, which corresponds to 1 = log[(A~)/4]/log[A 1 ] = 1.4461. For the 5-simplex lattice, the closed loop generating function can be found in terms of variables A (r) and B(r), defined as before. In this case, the recursion equations are more complicated [23]. We list these here to show how the complexity of the polynomials rises rapidly with increasing n:
A (r+ll
= 132(B5 + AB 4 )+A 2 (1 + 18B2 +96B3 )+ A3 (3+ 12B+ 78B 2 )+A 4 (6+30B)+6A 5 ,(38) (39)
We omit the other equations. Starting with (AC 1l, B( 1 l) = (x, 0), one finds a non-trivial fixed point for x = Xc = 0.336017, which corresponds to p. = 1/xc = 2.97603 and the fixed point (A*, B*) = (0.3265, 0.02791). Linearization of the recursion relations Eq. (38) and (39) around this fixed point leads to one eigenvalue A1 = 3.13199 greater than unity. Using this value of A1 one gets v = 0.6049 and a = 0.5955. The exponent 1 is found from the recursion relations for the variables corresponding to a single end point. The largest eigenvalue A+ of the matrix which characterizes the linear
D. Dhar and Y. Singh
164
transformation of the recursion relations of these functions is A+ = 5.24398 corresponding to 1 = 1.4875. For the 6-simplex lattice, even for the polygon generating function, we need three restricted partition functions A (r), B(r) and C(r), corresponding the cases where the walk enters and exits the r-th order subgraph once, twice or three times respectively. For the open walks, we need six more variables. For details, consult [23]. In this case, the nontivial fixed point is found to be (A*,B*,C*) = (0.262352,0.017588,0.0007011).
(40)
This fixed point is reached if we start with x = Xc = 0.27166, which corresponds to the connectivity constant f-1 = 3.68107. The linearization of recursion relations about this fixed point yields only one eigenvalue, A1 = 3.4965, higher than unity giving v = 0.5537 and a = 0.5686. By diagonalizing the matrix corresponding to the linear recursions for weights corresponding to one end point, one gets the largest eigenvalue A+ = 6.26709. This gives 1 = 1.500094 for the 6-simplex lattice.
4.2. n-simplex lattice in the limit of large n One can explicitly determine the critical exponents for a few more values of n using a computer to generate the recursion equations. However, this soon becomes very timeconsuming. If n is very large, some simplifications occur, and one can determine the leading behavior of the critical exponents of the SAW problem on the n-simplex in this limit [24]. We discuss this below. We start by noting that as n increases, the probability of occurrence of loops in a random walk on the graph decreases, and as the loops become rarer, the random walk without self-exclusion should approximate the properties of the SAW. In particular, we would expect that the growth constant for SAWs on the n-simplex should be approximately equal to n - 1. Let us denote, as before, the restricted partition functions for the r-th order lattice corresponding to polygon generating function by A(r), B(r), C(r) ... , corresponding to configurations where the SAW enters (and exits) the graph once, twice, thrice ... respectively. Then for the fixed point corresponding to the swollen state, we 2 would have A*~ 1/n. Since B(r)::::; A(rl , it is at most of order 1/n2 , and similarly, C* is 3 at most of order 1/ n . Thus, B*, C*, ..... approach zero faster than A* as n is increased. It is straight forward to write down the recursion equation for A(r+l) if we neglect the terms involving B, C, etc. There is only one configuration of walk corresponding to the term A2 (we drop the superscript of A(r) for convenience), but (n- 2) terms of type A 3 , and (n- 2)(n- 3) terms of type A4 etc. This gives us the recursion equation A(r+l) ~ A2 + (n- 2)A 3
+ (n-
2)(n- 3)A4 + .......
+ (n-
2)!An.
(41)
When A is of order 1/n, each of these terms is of order 1/n2 , and the sum needs to be evaluated with some care. In [23], it was noted that, with only a small error when A is near A*, this series can be rewritten as
An(n- 2)![1 + l.A- 1 + (1/2!)A- 2 + ... + (1/(n- 2)!)A-n+ 2 ] (n- 2)! An exp(1/A)
(42) (43)
Linear and branched polymers on fractals
165
One can then study the asymptotic behavior of this recursion equation for large n. Here we shall use a different approach. If we change the variable from A(r) to E(r) using the relation 1
E(r)
n
fo
= - exp(-).
A(r)
(44)
and use the fact that, for n r
r2
j=l
2n
» r»
1,
ll(l- j/n) ~ exp(--),
(45)
the series of Eq.(35) can be written as 1
A(r+l) ~ ( - )
n
r2
E(r)r
L:exp(-- -). n2 r=2 fo 2n
(46)
Substituting x for r / fo and replacing the summation over r for fo » E(r) » 1 by integration over x from -oo to oo (for E » 1, the lower limit can be extended to -oo with negligible error) we get A(r+l) ~
-
~
(-V,c,_·")
n3/2 exp
(r)2 2 .
(-E-)
(47)
From this equation, we see that the nontrivial fixed point is given by E(r)
=
E*
~ Jlogn.
(48)
Let us now look at the equation for alone, we get B(r+l)
~ A4 + 2(n-
4)A
5
B(r+l).
Again keeping only the terms involving A's
+ 3(n- 4)(n- 5)A 6 + ...
(49)
Using the same approximation as before, we get B(r+l)
~ (~) ~rexp(~- _C_) ~ vf27rlogn -
n4 ~
fo
2n -
n5/2
'
(50)
where we have used the approximate fixed point value of E* from Eq.(48). The fact that B(r+l) decreases faster than n- 2 justifies neglecting these terms in determining the asymptotic behavior of the recursions. The value of the derivative of the linearized recursion equation for A at the nontrivial fixed point is (51)
This implies that the critical exponent v is given by v
~
2log 2 [ log log n - -- 1 1ogn 1ogn
. + h1gher
] order terms .
(52)
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166
We note that the correction term to Eq.(41) involving B(r) in leading order are of the form 6A(r+l)
= 2A 3 B(n- 2)(n- 3) + 4A 4 B(n- 2)(n- 3)(n- 4) + ....
(53)
Using the estimate Eq.(50) for B(r), we see that near the fixed point, the error in Eq.(41) 6A(r+l) is of order n- 312 • This implies that the fractional error in the value of v using Eq.(41) is of order n- 112 • Thus Eq.(41) is much more accurate than Eq.(52), where the error is of order 1/ (log n). To calculate "f, one has to consider configurations with one endpoint of walk inside the graph. Again, keeping only terms involving powers of A, the recursion relation for the weight of configurations with the walk entering the graph once ( call it X) is found to be X(r+l) ~ X[1
+ (n-
1)A*
+ (n -1)(n- 2)A* 2 + .... + (n-
1)!A*(n-ll].
(54)
Using the arguments as before, this can be evaluated using the fixed point value of A*, giving X(r+l) ~X Jnlogn.
(55)
It follows that, for large n, 1 ~ 2 [1 -
loglogn log n
+
. ] h1gher order terms .
(56)
Note that the critical exponents do not take mean-field values, even when the fractal dimension of the lattice becomes greater than 4. This is clearly because of the special structure of the n-simplex lattice, where, even though the fractal dimension becomes greater than 4 for large n, the spectral dimension remains below 2, and probability of intersection of paths of random walks remains large. Also that the non-analytical dependence of the type 10f0~~n in the critical exponents on the lattice cannot be obtained from E-expansion like power-series expansions in deviation of dimensionality from some reference value. 4.3. Modified rectangular lattice This lattice is interesting as its fractal dimension, and the spectral dimension are both rational numbers. Also, one can get its graph by selectively deleting some bonds from the graph of d-dimensional hypercubicallattice. Since the (r + 1)-th order graph is formed by taking only two smaller graphs, the recursion equations involve only a small number of configurations, and are easy to write down. But the number of variables needed is larger, as the symmetry of the graph is lower than that of the H B(b, d) family. The behavior of SAWs on the d = 2 lattice was studied in [17]. The recursion equation for a polygon is written in terms of five restricted generating functions (see Fig. 12) by constructing graphs by all possible ways [17]. This gives
B(1 +D), B 2 +2DE,
B(r+l)
= A2 + C2,
E(r+l) = D2.
c(r+l)
= 2AC, (57)
167
Linear and branched polymers on fractals
cCr)
E(r)
Figure 12. Diagrams representing the restricted partition functions for the various ways the polymer can cross the rth order rectangle.
The starting values of these recursions are (58) The polygon number generating function P(x) is given by [17] x4
P(x) =
oo (B(r))2
4 + L[2r+2].
(59)
r=l
Numerical iteration of these equations gives occurs at (A*, B*, C*, D*, E*)
Xc
= 0.5914 and f-1 = 1.6909. The fixed point
= (0.5000, 0.4201, 0.4124, 0.1902, 0.0362),
(60)
giving one eigenvalue A1 = 1.6839 greater than one. Using this eigenvalue one finds v = log 2/2log A1 = 0.66503 and a = 0.6699. A similar analysis for the open-walk configurations gives the critical exponent 1 = 1.4403.
5. SELF-AVOIDING WALKS ON FRACTALS WITH DIMENSION 2-
E
In the preceding sections we studied the properties of SAWs on several different fractals. For the calculation of critical exponents of SAWs on still other fractals, see for example [25]. It would appear that for each fractal, one has to write down the polynomial recursion equations, and calculate the values of critical exponents using the technique outlined. There is no simple expresion for the critical exponents as a function of the geometrical properties of the fractal, ( an improved Flory formula?) that would allow one to predict these without doing the full calculation. Unfortunately, this level of understanding of the problem is still not achieved. The best way to understand such systematics is to study the variation of exponents across a family of fractals as some property of the fractal is changed. For the n-simplex family studied earlier, the fractal and spectral dimensions did not tend to the same value for large n. We now discuss variation of critical exponents for SAWs on the GMb fractals family as the parameter b is varied from 2 to infinity. For large b, both the fractal
D. Dhar and Y. Singh
168
and spectral dimensions of the fractal tend to 2 where the properties of walks are wellunderstood. Such a study makes contact with the formal E-expansion technique which has played an important role in development of renormalization theory of critical phenomena. The self-avoiding walks for the G M 2 fractal have the same exponents as for the 3simplex. For small values of b, it is straight forward to generate the explicit recursion equations on a computer, and determine the critical exponents. These were worked out by Elezovic et al [26] for b=2 to 8, and these studies were extended to b = 9 by Bubanja et al [27]. The general form of recursion equations for the function A(r), B(r), C(r) and D(r) ( definition of these is same as in Fig. 6) are of the form B(r+l) A(r+l) ] [ C(r+l)
Jb(B(rl), [
Pu(B(r))
P12(B(r)) ] [
P21 (B(rl)
P22(B(r))
A(r) ] C(r) .
(61)
where fb, Pu, P12 , P21 and P22 are polynomials of B(r), whose exact form depends on b. From analysis of these equations, it was found that the critical exponent v for SAWs takes the values 0.7986, 0.7936, 0.7884, 0.7840, 0.7803, 0.7772,0.7744 and 0.77218 as b is varied from 2 to 9. It thus seems to converge to the Euclidean value v2 d = 3/4. However, the value of the critical exponent 1 changes from 1.3752, 1.4407, 1.4832, 1.5171, 1.5467 and 1.5738 as b is varied from 2 to 7, and seems to diverge away from the known exact two dimensional value 7 2d = 43/32 ~ 1.344. The variation of these exponents with b for large b was explained in [28] using the finite size scaling theory. For large b, the growth constant of SAWs on the b-fractal would be close to the critical value in two dimensions. It is convenient to change variables from B(r) to a variable that is proportional to the departure from criticality in these systems. We write (62) Then B(r+l) is related to the properties of SAWs that traverse an equilateral triangle of side b from one corner to another. From finite-size scaling theory, this would be expected to be a function of a single variable Eb 1 1v: K fb(B) ~ - exp[g(Ebl/v)], (63) ba where K is some constant, and a is an exponent. From the conformal field theory [6,29], the scaling dimension of a spin at the corner of a wedge of angle 1r /3 is known, and that implies that a=15/4. The scaling function g(x) has to have the following properties: (i) It is a monotonically increasing convex function of x. We can set g(O) = 0, by redefining K. (ii) ForE< 0, fb should decrease exponentially with b. This implies that g(x) ~ -lxlv for x « -1. (iii) For fixed E > 0, fb should vary as exp(b 2 ) for large b. Thus, we must have (64) From Eqs. (61), it is easy to see that the fixed point value of E for large b is given by g(EiYiv) ~ alogb. Using Eq.(64), this gives *~
Eb-
(
alogb K1 )
l/2v
b-l/v
.
(65)
169
Linear and branched polymers on fractals The derivative of B(r+l) with respect to
A1(b) =
B(r)
at the fixed point is found to be
2
bl/v~! l•=•i: ~ K2bljv(logb) ~;; 1 ,
(66)
where K 2 = 2va(Kda) 112 v is a constant. Expressing the critical exponent v(b) in terms of A1 (b), we get 1 1 2v - 1log log b v(b)=;+~ 1ogb+ ... ,
(
67
)
where the dots represent terms of order Io~b· It is interesting to note that the leading correction term to the asymptotic value of v is negative. Thus this analysis predicts that variation of v with b is not monotonic. In particular, vb should be less than v for large enough b. This rather striking prediction of the finite-size scaling analysis given above was checked by Milosevic and Zivic using numerical Monte-carlo renormalization group studies [30,31], and they found that this happens forb~ 27. We can similarly determine the leading correction to the susceptibility exponent 1· It is known that in two dimensional critical theory, a 3-leg vertex has a higher scaling dimension than a 1-leg vertex. This implies that at the critical point E = 0, the ratio C(r) /A(r) will tend to zero as a power of b, and hence C(r) can be ignored in determining the leading correction. Then the value of the exponent 1 is determined by the value of Pu at the fixed pointE= E*(b). We can now write the scaling ansatz for this variable: (68) where K 3 is a constant, cis a critical exponent, and h(x) is a scaling function. ForE < 0, it is known that as b-+ oo, Pu varies as c 1164 [29,32]. This implies that we must have h(x) ~ ( -1/64) logx, and c = 1/48. Now, for large x, the functions h(x) and g(x) should increase in a similar way. Then using Eq.(63), we can write Pu as (69) For large x, both h(x) and g(x) increase as K 1 x 2 v, but the leading dependence is the same exactly. Thus, for large x, h(x)- g(x) varies at most as a sublinear power of log b. Thus, we get . log Pu (Eb, b) l1m = c+ a. log b
(70)
b-+oo
As 7(b) =
log[b~i]i]J/ 1ogb, this implies that
lim 7(b) = 2(c +a- 1)v
b-+oo
=
133/32.
(71)
Using the known scaling exponents for the dense polymer problem in two dimensions, it was shown in [28] that the leading correction to the large b limiting value of 1 is given by
"~(b) = 133 - 321 log log b + higher order terms. '
32
128 1ogb
(72)
170
D. Dhar and Y. Singh
We note that the leading correction to asymptotic value of the exponent is proportional to 2- db. The next correction term is of order 1/ log b, and is proportional to 2- Db, where Db = 2 - E is fractal dimension. These are like the E-expansions, except that there are several inequivalent definitions of dimension for fractals. Thus there are several E's, and the exponents on fractals may require a multi-variable E-expansion. Interestingly, at higher orders, corrections to scaling to the finite-size scaling functions f(x), g(x) etc. would give corrections to the exponents of the type 1/bt::.. These are of the type exp( -6/E), and such correction terms are not calculable within the conventional E-expansions framework. The reason the critical exponents in the large b limit do not tend to the Euclidean value may be understood as a crossover effect. For large b, the space looks Euclidean at length scales smaller than b, and the effective polymer exponents for E > 1/b 11v would be near the Euclidean values. However, forE« 1/b 11v, the polymer has to go through the constrictions, and the asymptotic value of exponents for large polymers can, and do, take different values.
6. COLLAPSE TRANSITION IN POLYMERS WITH SELF-ATTRACTION We have seen that the qualitative features of the behavior of linear polymers with no interaction other than the excluded volume interaction is well-modelled by SAWs on fractals. Now we will like to show that polymers on fractals can also be used to understand more complicated behavior of polymers like the transition from the swollen state to compact globular state transition that is observed in dilute polymer solutions as the temperature is lowered [33]. In order to model this, we have to include the attractive interaction between different parts of the polymer when they are close by, but not overlapping. The simplest lattice model for this phenomena associates an energy -Eu for each pair of nearest-neighbor lattice points occupied by the polymer that are not consecutive along the walk. The equilibrium properties of such a self-attracting SAW can be described using the grand partition function G(x, u);
G(x,u) =
L
f2(N,Nu)xNuN",
(73)
N,Nu
where f2(N, Nu) is the number of different configurations per site of a self-avoiding polygon of N steps and energy -NuEu, u = exp(tJEu), xis the fugacity per step of the chain, and ;3 is the inverse temperature. For small ;3, the effect of Eu can be ignored, and the typical size of polymer varies as Nv, where the exponent v takes the value for SAWs. This is called the swollen state of the polymer. For large ;3, the polymer folds up like a tangled ball of yarn, in order to minimize the energy - NuEu. In this phase, called the collapsed phase, the typical size of polymer varies as N 11d in d dimensions. In the limit when the number N of monomer units goes to infinity, there is sharp transition from the swollen to collapsed phase at a critical value of u. This transition is described as a critical phenomena analogous to a tricritical point for magnetic system [2]. For large polymers, the average gyration radius R at the transition behaves as RN "' Nve where the exponent vo is intermediate between the value v for swollen state and the value Vc = 1/ D for compact globule on a lattice of fractal dimension D.
Linear and branched polymers on fractals
171
6.1. Self-interacting polymer on the 3-simplex lattice The properties of a polymer chain with self-attraction on a fractal were first studied by Klein and Seitz [34]. They used the self-avoiding walks on the Sierpinski gasket, which is the b = 2 member of the Given-Mandelbrot family. We consider below the case of 3-simplex, which is somewhat simpler to treat.
Figure 13. Restricted partition functions for the self-attracting walk for the 3-simplex.
To calculate G(x, u) for the 3-simplex lattice, we define restricted partition functions Bbr) and Bir), for walks that cross the r-th order triangle, as shown in Fig. 13. Bbr) is the sum of weight of all walks that enter an r-th order triangle of the 3-simplex from one corner and leave from another corner, but do not visit the third corner. Bir) is the sum of weights of walks that enter and leave the r-order triangle, and also visit the third corner of the triangle. Then it is easy to see that these weights satisfy the recursion equations
Bbr+l) = [Eo+ B1] 2 + Bo[Bg + 2BoBl + Biu], Bir+l) = Bl[Bg + 2BoBl
+ BiuJ.
(74)
(75)
The generating function for all loops is given by the formula 00
P(x, u)
= ~)Bbr) + Birlp3-r.
(76)
r=l
We start with the initial weights
Bb l = 0; Bi l = x. 1
1
(77)
These variables tend to the trivial fixed point B 0 = B; = 0, if the starting value of x is less than a critical value xc(u), and to the fixed point B 0 = B; = oo, if x > xc(u). For x = xc(u), Bir) tends to zero for larger, and the Eq. (74) reduces to Eq.(8). The critical properties of the chain are continuous functions of u, and there is no phase transition as a function of u. We note that here the recursion equations involve the interaction parameter u. This complicates the analysis. Consider a modified problem where the interaction -U occurs only between the nearest neighbor bonds in the same 2-nd order triangle, and not otherwise. One would expect the qualitative behavior of the system very similar, but now, the recursion equations do not involve u. In fact, they are the same as the case without self-attraction [Eq.(8)]. This observation is helpful in studying the collapse on other fractals.
D. Dhar and Y. Singh
172
A
Figure 14. Different fixed points for the 4-simplex lattice. The non-trivial fixed points labelled C,T and S correspond to the collapsed, the tricritical B-point, and the swollen state.
6.2. Collapse transition on n > 3 -simplex lattices The collapse transition on the 4-simplex lattice was first studied in [35]. We restrict the attractive interaction to nearest neighbors within the same 2nd order simplex. Then, the recursion equations Eq. (32-33) describe the collapse transition also, if we use the starting weights and
(78)
The renormalization group flows of the variables A and B in this case is shown in Fig. 14. There is the trivial fixed point A* = B* = 0, which is reached if x < Xc(u). If x > Xc( u), the fixed point A* = B* = oo is reached. The two-dimensional space of possible initial conditions (A (l), B( 1l) is divided into the basins of attraction of these two fixed points. The common boundary of these basins is a line. This line is an invariant sub-manifold of the renormalization flows (i.e. points starting on the line remain on the line). On this line we have three fixed points : 1. The fixed point (0.4264, .04998) corresponds to the swollen state. This is reached for u < Uc ':::'. 3.106074 and x = Xc(u). This is denoted by Sin Fig. 14. If we start at a point near this fixed point on the critical line, the renormalization flows are towards this fixed point.
2. The fixed point (A*, B*) = (0, 22- 113 ) is reached when u > U 0 • It gives one relevant eigenvalues>.= 4 corresponding to Vc = 1/ D 4 = 1/2. In this phase, the polymer fills the available space with a finite density of monomers. This fixed point is denoted by C in Fig. 14. This is also attractive for points starting nearby on the critical line. 3. The fixed point (1/3, 1/3), denoted by T in Fig. 14, is obtained for u = U 0 • On the critical line x = Xc( u), this unstable fixed point lies between the fixed points
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corresponding to the swollen and the collapsed phases. The matrix corresponding to linearized renormalization transformation at this point has two eigenvalues greater than one; .\ 1 = 3. 7037 and .\2 = 2.2222. The third relevant eigenvalue corresponds to the renormalization of fugacity of end points (>.+in Eq. (18)). Thus this is a tricritical point. at this point, Vc = ln 2/ ln .\ 1 = 0.5239. The singular part of the free energy varies as (u - u 0 ) 2 -a with ln .\1 a = 2 - - - = 0.36027. ln .\2
(79)
One can similarly study then= 5 simplex lattice [24], with the interactions are confined to the internal bonds of the second order graph. It is found that just like the n = 3 case, there is no collapsed phase. One can understand this by noting that the ground state configuration corresponds to a walk that visits all the 5 sites of the 2nd order subgraph. There are many configurations corresponding to the minimum energy, and most of these are extended. Even when there is attractive interaction between all pairs of neighbors, the energy-cost of pulling a polymer confined to an r-th order subgraph to something that is spread over a subgraph of order (r + 1) is finite. As there is a large entropy associated with the place where this break occurs, this makes the collapsed phase unstable to such breaks, and brings the collapse transition temperature to zero. In a similar way, we can study the 6-simplex. Here we have coupled polynomial recursion equations of degree 6 for the three variables A (r), B(r) and C(r). For the details of the recursion equations, see [24]. Again, we have two trivial attractive fixed points corresponding to (A*, B*, C*) equal to (0, 0, 0) or (oo, oo, oo ). There is a two-dimensional critical manifold that marks the boundary of the basins of attraction of these fixed points. For the renormalization flows on this two-dimensional critical surface, there are two attractive fixed points: The fixed point (A*, B*, C*) = (0.262352, 0,017586, 0.000701) corresponds to the swollen state as discussed in Sec. 4.1. The fixed point (A*, B*, C*) = (0, 0, 0.071329) is reached for all u > uc( = 3.4999847) at x = X 0 • The largest eigenvalue for this case is >. = 6 corresponding Vc = 1/ D 6 = 0.3869. This describes the collapsed phase of the polymers. The common boundary of basins of attraction of these two fixed points is a 1- dimensional line, which is also an invariant submanifold for the renormalization flows. This line has one attractive fixed point: The fixed point (0.12949,0.09572,0.05344) is obtained for u = Uc and x = Xc and has two eigenvalues greater than one; .\ 1 = 5.4492 and .\2 = 1.9049. This is a tricritical point with exponent Vt = 0.4088 and a= 2 -ln .AI/ .\2 = 0.6309. The crossover exponent¢ at the tricritical point is¢= vt/vth = 0.3801, where the exponent Vth controls the divergence of thermal correlation and is defined as Vth = ln 2/ ln .\2 = 1.0755. There are two other fixed points (0.254037, 0.022159, 0.07098) and (0.2000, 0.0666, 0.0666) on the line separating the basins of attraction of the fixed points corresponding to the collapsed and the swollen phases. These are purely repulsive, and cannot be reached starting with our choice of initial condition. These correspond to higher order multicritical points( tetra-critical).
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We note that the collapse transition on fractal lattices corresponds to a new fixed point, intermediate between swollen (SAW) phase and the collapsed phase and cannot be viewed as a perturbation of the Gaussian fixed point describing random walks. 6.3. Some unusual phases The nature of possible collapsed phases of linear polymers on fractals with a finite ramification number depends strongly on the geometrical constraints imposed by the connectivity properties of the the fractal, much more so than in the extended phase. As we already noted, for then-simplex lattice with n odd, a collapsed phase with a finite density of the polymer is not found. For the modified rectangular lattice, the behavior of linear polymer with self-attraction was studied in [35]. Here the number of variables needed to describe the polymer with self-attraction becomes rather large. For example, to describe closed loops it is necessary to introduce additional weights Al'), Bl'), A~r), . .. etc., where the subscript 1(2) indicates the number of extra corner sites of the rth -order are visited, a total of eleven variables. To describe open chains, we would need 17 additional variables, making a total of 28 variables- a rather formidable number. However, as in case of n-simplex lattices many of these variables are irrelevant and may be set equal to zero. Equivalently, we study collapse when the attractive interaction is restricted to the first order rectangles. If we restrict ourselves to the analysis of closed polygons, we require only the five variables defined earlier in section 4.3. Interestingly, one finds three different phases of the polymer, depending on the value of the attraction parameter u. For small u < uc1 = 3.2023, the polymer exists in the swollen phase, with typical size R "' Nv, and v = 0.6650. For large u > u 02 = 3.2341, it exists in a compact phase of finite density, with R "' N 112 . However, between the swollen and compact phases, for uc1 < u < U 02 , one finds a 'rod-like phase', where the shape of the polymer is highly anisotropic. In the x-direction, the average extent of polymer increases as N, and in the other direction it remains finite. At u = uc1, there is a non-trivial fixed point of the recursion equations of period 2. Linear analysis of the renormalization equation about this fixed point gives v = 0.80503. At u = U 02 , one gets v = 1/2, same as in the collapsed phase, with possible logarithmic corrections. There is a logarithmic singularity in the specific heat also. The reason why the rod-like phase is found in this case is not clear : obviously the anisotropy of the lattice is responsible for it, but it does not matter so much in other( swollen or collapsed ) phases, where the anisotropy is of the usual type ( the ratio of average diameter of an N-stepped walks in the x- and ydirections is a finite number). Another different type of phase, labelled 'semi-compact' phase was found by Knezevic and Vannimenus in their analysis of the collapse transition on the HB(3, 3) fractal [37]. In this case, the connectivity of the graph is such that a linear polymer cannot fill the available space with a finite density. For large value of the attraction strength u the polymer shrinks into a "semicompact" state. In this phase, the average monomer density tends to zero for large polymers. The recursion relations found for restricted generating functions A(r) and B(r) where A(r) counts the number of configurations when the polymer goes once through the rth_ order gasket while B(r) counts the configurations where the polymer goes twice through the gasket, are rather complicated [37], but for the large u regime, the equations are
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dominated by only a single term in each polynomial. For large order of iteration, the recursion equations in this phase may be approximated by (80) These can be linearized by taking the logarithm of both sides, which shows that for large r, log.4(rl and logB(r) vary a.c; >.~ ,where A± = (11 ± J73)/2. They will both diverge to + oo, or to - oo depending upon the coefficient of proportionality being positive or negative ( both are of same sign for the largest eigenvalue). Thus, this constant must be proportional to distance from the critical line x- Xc(u). If Xc - x = o« Xc, number of iterations before the deviation becomes of order 1 is r 0 ~ log(1/6)/log>.+.
(81)
As the size of the polymer is approximately 2ro,...., (lfoY', this corresponds to v' = 0.48195. The fractal dimension of the chain in this phase is, 1/v' = 2.07491 a value just a bit less than the fractal dimension of the latticeD = 2.09590. For x = Xc, A(r) decreases to zero as exp( -a>.~) where a is some constant, and H(r ) increases to infinity as an inverse power of .4
Figure 15. Schematic representation of the different phases of a linear chain with selfattraction on the Sierpinski gasket with at most two visits per site allowed: (a) the swollen phase (b) branched polymer phase with branches made of the doubled-up chain (c) the collapsed phase
In all the cases discussed so far a restriction on the walks that a lattice bond or a lattice site visited onee cannot be visited again has been imposed. One can relax this condition and allow the walk to revisit a site already visited so long as no bond is traversed twice. A consistent set of visited bonds is called a trail, if different sequences in which the same set of bonds bonds may be visited arc considered equivalent (38]. Orlandini ct al [39] considered self-attracting SAWs on a 2d Sierpinski gasket, with this constraint, and showed that this model displays a new multicritical point corresponding to a collapse from linear into branched polymer in which the polymer becomes a randomly doubled-up chain ( see fig. 15), which is followed by a further transition into compact globule. In Fig. 15, we have drawn sr.hematically what the different phases look like ( drawn here for
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the Euclidean 2-dimensional space). The universality class of the two new multicritical points corresponding to the phase transition points is different from the tricritical points discussed so far.
7. BRANCHED POLYMERS The models of branched polymers is related to other important problems in statistical mechanics, such as lattice animals, percolation [40] and the Lee-Yang edge singularity [41]. The study of branched polymers on fractal lattices are therefore very instructive as one can analyze the system in detail. The number of different branched polymers made of N monomers and their average gyration radius are expected to grow as p,N N- 0 and Nv, respectively for large N; (} and v are critical exponents, whose values are different from those from that of linear polymers. If we allow loops in the cluster, we get the model of lattice animals, which corresponds to the p ---+ 0 limit of the percolation problem. These are known to be in the same universality class as branched polymers. If in addition to the excluded volume interaction, one has an attractive nearest-neighbor interaction between the monomers, the branched polymers can undergo a collapse transition, just like the linear polymers. Near the collapse point, the free energy per monomer shows singular critical behavior. Knezevic and Vannimenus realized that real-space renormalization method for studying linear polymers on fractals can be extended directly to the case of branched polymers [42,44]. They considered asymptotic properties of branched polymers with attractive selfinteraction on fractal lattices, restricting the attractive interactions to bonds within first order units of the fractal lattices. We summarize their findings here.
7.1. The 3-simplex lattice The macroscopic thermodynamic quantities of interest can be obtained from the grandpartition function G(x, u), whose definition is same as Eq.(73), with fl(N, Nu) now defined as the average number of unrooted branched polymers with N bonds, and Nu nearest neighbor bonds, the average being taken over different positions of the polymer on the fractal. This can be determined in terms of the r-th order restricted partition functions defined in Fig. 16. The closed set of recursion equations involve six restricted generating functions as described in Fig 16 are easily written down [42]: A(r+ll B(r+ll
= A[1 + 2B + 2B2 ] + 2B 2 C + F[B 2 + A2 + 2BD], = B 2 + B 3 + F[4BC + 2AB] + F 2 [B + D],
(82)
and similar equations for other variables. G(x,u) can be seen as a sum of terms of the same general form as Eq. (5) and (6). The singular behavior of the sum can be analyzed by looking at the fixed points of the recursion equations. There are two trivial fixed points with all variables zero or infinite. For any given value of u, we have to tune the initial value of x to a critical value Xc(u) to get a non-trivial fixed point. The analysis of the recursion equations lead to following three different nontrivial fixed points:
1. For all u < Uc = 5.5, and x = Xc( u), the recursion equations lead to a non-trivial fixed point (A*, B*, .. .) corresponding to the swollen state with large scale properties
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Figure 16. Diagrams representing the six restricted generating functions for branched polymers on the two dimensional Sierpinski gasket. C(r) corresponds, for instance, to configurations where a part of the polymer joins two vertices of an r 1h order triangle while one of its ends penetrates through the third vertex. The diagram on the right shows a term B(r)C(r) F(r) contributes to B(r+l).
same as for the random-animal problem (u=1). All the fixed point values A*, B*, ... at this fixed point are non-zero. Linear analysis of the recursion equations about this fixed point shows that the generating function G(x,u) has a power-law singularity when x = Xc(u) and the critical exponents are v = 0.71655 and(}= 0.5328. 2. For u > U0 , and x = Xc(u), the polymer is in the collapsed phase. The corresponding fixed point has A* = B* = D* = 0. F(r) tends to zero, and C(r) F(r) and E(r) F(r) 2 tend to a finite limiting value. The largest value of the linearized renormalization transformation matrix is 3, corresponding to Vc = 1/ D 3 = 0.63093. This dense phase of branched polymers is in the same universality class as the spanning trees. For the spanning trees, one can also define a chemical distance exponent z, by the relation f "" Rz, where f is the distance along bonds between two randomly picked points on the polymer, and R is the euclidean distance. This was calculated in [43], and shown that z = log[(20 + v'205)/15]/log2 c::: 1.1939. 3. The collapse transition occurs at u = U 0 • At this fixed point, all the values A*, B*, ... are non-zero. This is a tricritical fixed point, with two eigenvalues larger than one. The exponent v1 = 0.63250 is very close to V 0 • The exponent a is negative, a = -4.0269, showing that the singularity in the specific heat at u = Uc is very weak. The closeness of Vt to Vc has also been found on square lattice where an accurate transfer - matrix study [45] of the collapse of branched polymer gives Vt = 0.509 ± 0.003 which is very close to the compact value 1/2. This suggests that the phenomenon is not accidental and should have a more general explanation. The problem of linear polymers is recovered if all terms containing the functions E and F one suppressed in Eqs.(7.1) and (7.2). The truncated equations have only one fixed
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point, corresponding to the swollen phase with v = 0.7986 and there is no collapsed phase (see Section 3).
7.2. The 4-simplex lattice The closed recursion equations in this case involve eleven restricted generating functions. The number of polymer configurations to consider is large, and Knezevic and Vannimenus [44] used computer enumeration to sort them out. We omit the details. The analysis found three non-trivial fixed points that describe the large-scale behavior of self-interacting branched polymers. 1. For u < Uc R::J 2 the random - animal fixed point corresponding to the swollen phase of the polymer is reached. Linearizing around the fixed point one finds only one relevant eigenvalues .\ 1 = 3.14069. With this eigenvalue one finds v = log 2/ log >. 1 = 0.60566, and(}= 0.75667. 2. For u > Uc one gets a fixed point corresponding to the collapsed state in which polymer occupies all vertices of the lattice. The relevant eigenvalue found in this case is >. = 4 which gives Vc = 1/2. 3. For u = Uc the fixed point reached represents a tricritical point as it has two eigenvalues greater than 1: >. 1 = 3.94050 and >. 2 = 1.32094. The exponent Vt = log 2/ log .\ 1 = 0.50546. This value is very close to the value Vc = 1/2 of the collapsed phase. The exponent a= -2.8267 is negative but less negative than for the 3-simplex lattice.
7.3. Other fractals Knezevic and Vannimenus also studied the branched polymer problem on the GM3 fractal [42,44]. In this case, they found the unexpected result that unlike the case on the GM2 fractal (for which the behavior is the same as on then= 3 simplex), for the b = 3 case, the number of animals of size n grows as p,n exp(Kn!J;), where 0 < 1/J < 1. This corresponds to an essential singularity in the generating function of branched polymers G(x, U = 1) "'exp( lxc~xiP), with p = 1/J/(1 -1/J). The analysis of Knezevic and Vannimenus has recently been extended to all b, and one finds that forb 2: 3, the average number of animals per site behaves as p,(b)n exp(Kn!J;), where the values of the the singularity exponent 1/J, and the size exponent v can be determined exactly [46]. 8. SURFACE ADSORPTION It is well known that a long flexible polymer in a good solvent can form a self-similar adsorbed layer near an attractive wall at the critical temperature Ta. Using the correspondence between an adsorbed polymer chain and the model of ferromagnets with n-vector spins in the limit n--+ 0 with a free surface, it has been shown that the adsorption point Ta corresponds to a tricritical point and in its proximity a crossover regime is observed. In particular, the mean number of monomers, M, at the surface is shown to behave as [47]
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M
forT< Ta; forT= Ta; forT> Ta.
179
(83)
A good model for this phenomenon is a SAW on some lattice with an absorbing surface (boundary). Every site on surface visited by the polymer contributes an energy -E8 • This model has widely been studied on various lattices and via a number of techniques that include exact enumeration [48,49], Monte Carlo [50], transfer matrix [51], renormalization group [52] etc. For a 2-d Euclidean lattice, exact value of ¢ found from conformal field theory is 1/2 [53]. Bouchaud and Vannimenus [54] were the first to apply RSRG techniques on fractals to study the linear polymers near an attractive substrate. They showed that the known phenomenology of the adsorption -desorption transition is well-reproduced on fractals, and different critical exponents can be evaluated exactly. The values of the exponent ¢ for HB(2, 2) and HB(2, 3) fractals were found to be 0.5915 and 0.7481 respectively. They also showed that for a container of fractal dimension D and adsorbing surface d" ¢ has lower and upper bounds; 1- (D- ds )v -< '~-'A.< ds. D
(84)
8.1. Surface adsorption in polymers with self-attraction Polymer chains with self attraction near an attractive surface can exhibit a rich variety of phases, characterized by many different universality domain of critical behavior [57]. This is due to competition between the two interactions. At the intersection of two tricritical surfaces, one corresponding to the B-transition and another to the adsorptiondesorption transion, one can expect higher order critical points. Understanding of the different phases that are possible and understanding and classifying the multicritical points that can exist appear difficult on standard Euclidean lattices[47]. It is here fractal lattices have been particularly helpful [58]. The values of critical exponents found are, of course, different for different fractals, but the general features of the phase-diagrams remain the same. Investigating the problem on fractals helps us understanding the problem in real experimental systems. We consider a linear polymer chain on a truncated n-simplex lattice and make one boundary surface of it attractive. We treat one of the edges of the fractal container as an attractive surface and associate an energy -E8 < 0 with each site on it occupied by the polymer, and an energy Et > 0, with each occupied site that is at a distance 1 from the occupied surface i.e. on the layer adjacent to the surface, and an energy - Eu for nearest-neighbor bond between monomers not consecutive along the chain. Then, to each N-step walk having Ns steps along the surface, Nt steps lying on the layer adjacent to the surface and Nu number of nearest neighbors a weight xN wN,tN'uNu is assigned, where w = exp(j3E5 ), and t = exp( -f3Et). The grand partition function for this system is given by G(x,w,t,u)=
(85)
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where fl(N, N 8 , Nt, Nu) is the number of different configurations per site of a SAW of N steps, rooted at a specified site on the attractive surface, with given values of Ns, Nt and Nu. We may put Et = 0 (or t = 1) for simplicity, but a non-zero value seems to give rise to interesting behavior, and is also important to many physically realizable cases. The attractive interaction between monomers is restricted, as in preceding sections, to bonds within the lowest order subgraphs of the fractal lattices. As shown in section 6.2, polymers in the n-simplex container show a collapsed phase only for even values of n. Thus, the 4- and 5-simplex lattices exhibit contrasting behavior and represent two different scenario which may arise in real systems. In the case of 5simplex lattice, there is no collapse transition possible in the bulk, but its 4-simplex surface can show a collapsed phase. In the case of 4-simplex lattice, on the other hand, there is a collapsed globule phase in the bulk but no collapse possible in the surface-adsorbed polymer. We can therefore have situations in which the bulk acts as a poor-solvent medium, while the surface acts as a good solvent medium or the opposite case of surface being poor-solvent medium and the bulk good solvent. However, the situation in which both bulk and surface show collapsed phases can not be modelled by polymers on an n-simplex fractal. 8.2. The 4-simplex lattice
S
(r)
C
(r)
Figure 17. The restricted partition functions for the rth order 4-simplex with attractive interactions at one boundary surface. The internal structure of the 4-simplex is not shown. The shaded triangle reprsents the attractive surface.
For the 4-simplex lattice the grand partition function of Eq. (85) can be written in terms of the five restricted partition functions shown in Fig 17. The shaded regions in the figure represent the surface. Out of five configurations, two (A(r) and B(rl) represent the sum of weights of configurations of the polymer chain within one r-th order subgraph away from the surface, and the remaining three (S(r), C(r) , and E(r)) represent the surface functions. The recursion relations for these restricted partition functions can easily be
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3 .. ----
.,
LIO
1111
,
..."'
...
.,
0<
,,. LOO
""'
"'
...
..,
-
----.
""
•n
·-:
,.,
... ...
""'
...
Figure 18. Sections of the w - u - t pha1;e diagram for two different values of t for the -1-simplcx lattice: (a) t = 0.2 (b) t= 0.5. Regions marked by AS, DS and DC represent, respectively, the adsorbed polymer in swollen state, dcsorbcd polymer in swollen and collapsed (globular) state. The collapse transition between the DC and DS phases is denoted by the dashed line. The special adsorption line is indicated by full line and part of it by dotted line. The point where B-line meets with the adsorption line is a multicritical point. The dotted part of the adsorption line indicates the region of coexistence of adsorbed SAW and the (bulk} globule phase.
written (sec [58] for details) . The equations for A(r .J) and B (r +l) do not get affected by the presence of the surface interactions. In this case, we have three phases possible: the dcsorbed swollcn(DS), the desorbed collapsed (DC) and the adsorbed swollen (AS) phases. The fixed points corresponding to t.he df'.sorbed pha.<>P.s have S* = C* = E* = 0, and A*, B * equal to the value for the 4-simplcx at the S or C fixed point ( section 6.2). The fixed points corresponding to the polymer adsorbed on the surface, on the other hand, have A* = B* = C* = E * = 0 and S* equal to the value 11- 1 corresponding to the swollen pha.se fixed point on the 3simplcx surface ( section 3.1). The phase boundaries, determined numerically by finding the basins of attraction of these fixed points arc shown in Fig. 18. If we start at any point on the boundary between the DS and AS phases, under renormalization, we flow to the symmetrical fixed point S* = C* = A*, B* = E *, with A*, B* having the value corresponding to the point S in Fig. 14. Linearizing the recursion equations ncar this fixed point two eigenvalues>. = 2.7965 (corresponding t.o the swollen bulk state) and Aq, = 2.1583 greater than one [58] are found . This point is the expected symmetrical special absorption point which describes the polymer at the desorption transition. A simple calculation gives the crossover exponent ¢ = 0.7481 and a= 0.6653. ln Fig 18, the full line shows intersection of this surface with the surface t = constant. The bulk transition between DC and OS phases is described by the fixed point s· = = E* = 0, and A* and B* have the value {1/3,1/3) corresponding to the point Tin Fig. 14. This boundary has the equation u = uc, and the value Uc does not depend on w or t.
c·
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The points on the boundary between the DC and the AS phases are found to fall in the basin of attraction of two different fixed points: F P 1 = (A* = S* = C* = 0, B* = E* = 0.3568) and F P 2 = (A* = C* = E* = 0, B* = 0.3568, S* = 0.61803). Correspondingly, we see two different behaviors of various quantities as we cross the AS to DC phase boundary. The first fixed point FP1 is reached for all points on the AS-DC surface with t greater than a critical value t*, and for large enough u even for t < t*. This has two relevant eigenvalues .\ 1 = 4, and .\2 = 3. The fixed point F P 2 corresponds to a coexistence between the adsorbed SAW and the free collapsed globule phase. In Fig 18(a), the line corresponding to this fixed point is shown by dotted line. This point is reached if t < t*, and u is greater than, but near U 0 • The line where the three phases meet is also an invariant manifold for the renormalization group flows. We find three fixed points on this line: 1. Fort< t*(= 0.34115 ... ) the fixed point (A*,B*,S*,C*,E*) = (~, ~' 0.4477, 0.4528, 0.0815) is reached. The linearized equations have three repulsive directions with eigenvalues AsM = 2.2715, .\ 1 = 3.7037 and .\ 2 = 2.2222. The values .\1 and .\2 are the same as found for the bulk (}point (see section 6.2). 2. Fort> t* the fixed point (i, ~' 0, 0, 0.3693) is found. Again we find three eigenvalues greater than one, where AsM = 3 and the other two .\ 1 and .\ 2 are the same as those given above. 3. Fort = t* we find the symmetric" disordered and collapsed" fixed point ( ~, ~, ~, ~' ~). This fixed point has four eigenvalues greater than one. These values are AsM, = 2. 7620 and AsM2 = 1.4964... and the other two are >. 1 and .\2 as given above. The first two of these are tetracritical points, and the third point is an even higher order multicritical point ( pentacritical). We now discuss the behavior of phase boundaries. When t < t* = 0.34115, the critical line w = w 0 (u, t) which is the phase boundary between the DS and AS phases, is almost linear with positive slope. Beyond the multicritical point, slope of the line separating the AS and the DC phases rises rather sharply. In a region specified by Uc < u < U 01 (the value of u 01 depends on t) we have the coexistence between the adsorbed SAW and the collapsed globule phase. This region is shown in Fig 18 by a dotted line. For u > U 01 (t) the line w = w* (u, t) becomes almost flat. The value of u 01 ( t) decreases as t is increased and becomes equal to that of Uc at t = t* = 0.34115. At t = t* the multicritical point becomes a symmetric "desorbed and collapsed" pentacritical point having four eigenvalues greater than one. For t > t* the critical line w = w* (u, t) has a different shape than for t < t*. The line appears to have a maximum at u :::; U 0 • It drops rather sharply (see Fig. 18(b) for t=0.5) in contrast to the case oft < t* at the multicritical point. Furthermore, the line w = w*(u, t) for u > Uc separating the bulk collapsed and adsorbed phases shows the decreasing tendency as u is increased. The two tetracriticallines on the three-dimensional u- w- t phase space meet at a pentacritical point [58]. Along one of these tetracritical lines, the adsorbed (swollen) polymer coexists with both the desorbed polymer and the desorbed globule, and the other line is the common boundary of three critical surfaces of a continuous transition.
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The behavior of the special adsorption line w = w*(u, t) described above can be understood from contribution of different coexisting polymer configurations to the bulk and surface free energies. When both the adsorbed and desorbed phases are in swollen state, the adsorption line has same nature in the w - u plane for all values of t, although the slope of the line decreases as t is increased. At t = 1, the adsorption takes place at w = 1 and the adsorption line in the w - u plane has a zero slope. This is due to the fact that at t = 1 and w = 1 the surface is just a part of the bulk lattice. As t is increased, w has to be increased to have adsorption, and since u in such a situation favors the bulk phase we have to increase the surface attraction to counteract this tendency. In the other extreme, i.e. when u > > U 0 , the adsorption line has a zero slope. Here the coexisting polymer configurations are those given by B and E in Fig. 17. The free energies due to these two configurations balance each other at all u values and therefore the line remains insensitive to the value of u. It is only in the neighborhood of the special B-point that the line becomes sensitive to the value of t and u. When t < t*, the surface layer is strongly repulsive and prohibits the occurrence of the E configurations in the neighborhood of the B-point. The adsorbed state is still given by the configurationS, although the bulk is in the globular compact phase. Thus to balance the free energy w has to be increased. However, at t > t* the surface is only moderately repulsive and therefore at certain value of w the polymer configuration given by E is formed. Thus a lower value of w is needed to balance the bulk free energy at the special B-point. A casual look at Fig. 18(b) may give the impression of the existence of a re-entrant adsorbed phase as u is increased. One should, however, realize that these figures are merely a projection on the w-u plane of three dimensional figures in which the third dimension is given by t. The value of crossover exponent ¢ for the 4-simplex lattice is 0.7481 equal to the value found for HB(2, 3). 8.3. The 5-simplex lattice The grand partition function of Eq. (85) for 5- simplex lattice is written in terms of six restricted partition functions shown in Fig 19. Out of six configurations two (A(r) and B(rl) represent the sums of weights of configurations of the polymer chain within one r-th order subgraph away from surface, and the remaining four (C(r), S(r), E(r) and F(rl) represent the surface functions. As in the case of the 4-simplex, the recursion relations for A (r) and B(r) do not include other variables. In this case, we have three phases: The adsorbed swollen (AS), the adsorbed collapsed (AC), and the desorbed swollen (DS). It is straight forward to write down the fixed points corresponding to these phases from the known fixed points for SAWs on the 4- and 5simplexs. The basins of attraction of these fixed points are shown in Fig. 20. For the DS phase, we have A*, B* taking the value for the swollen phase of the 5simplex (section 6.2), with C*,S*,E* and F* zero. In the AS and AC phases, we have A*, B*, C* and E* zero, and S*, F* has the value of corresponding the fixed points S and C respectively in Fig. 14 (A*, B* in the terminology of section 6.2). The critical surface separating the DS and AS phases is a two-dimensional surface in the 3-dimensional parameter space (w, u, t). All points on this surface, under renormalization, flow to the "symmetrical" fixed point (S* = C* = A*, E* = F* = B*), with values of
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C
(r)
E(r)
p(r)
Figure 19. Restricted partition functions for the rth order 5-simplex. Here represent the bulk generating functions for the polymer chain and S(r), C(r), represent the surface functions.
A(r) E(r)
and B(r) and F(r)
A*, B* same as for the DS fixed point. The linearization of recursion relations about this fixed point gives two eigenvalues .\ 1 = 3.1319 and .\ 2 = 2.5858 greater than one. The line we( u, t) is therefore a tricriticalline. The crossover exponent ¢ = 0.8321. The AS and AC phases are separated by the critical B-snrface w = we(n, t) (see Fig 20). For w » we(u, t), this surface tends to the surface u = Ue = 3.316074 in agreement with the critical value of u for the collapse transition in the bulk of the 4-simplex lattice. However, in this case, this B-surface never meets the AS-DS phase boundary. It is shown in [58] that at w ?: we( u, t) the B-line bends and approaches very slowly the special adsorption line we( u, t) as the value of u is increased. Even for u = 200 the two lines have not merged. The whole AS-AC boudary corresponds to the basin of attraction of a single fixed point, and the parameter t has no qualitative effect on the phase diagram.
8.4. Surface adsorption of a SAW in dimensions 2- E We consider the family of G Mb fractals and evaluate the value of the exponent ¢ and examine its behavior as b is varied from 2 to oo. The energies Es and Et are defined as before. We put Eu = 0 for simplicity. A walk is called a surface walk (and assigned configuration S) when it enters through one corner of the surface and leaves from the other. A bulk walk represented by B has no step on the surface or on the bonds connecting the surface with the bulk. A walk which enters through one corner of the surface and ends up in the bulk is assigned configuration C. The generating functions for these walks can be written as B(r)(x) s(rl(x,
w, t)
c(r)(x,w,
t)
LB(r)XN,
L s(r)(N, M, R)xN wMtR, L c(r) ( N, M, R)xN WM tR.
(86)
Linear and branched polymers on fractals
185
I.e()
1.150
..
AS
.'
.'
'•
'' '
'·"O
AC
''
- - - - _ --_ -::._-~-::..-=---== /
'
OS
I 1- 2 0 - f - - --,.
7.00
~ 00
9.00
11-00
u
Figure 20. A section of t.he w - u - t phase diagram for the 5-simplex at a typical value oft = 0.5. For other values oft, the phase boundaries shift, but the qualitative behavior is the same.
Here x, as before, is the fugacity associated with each visited site ofthe lattice, B(rl(x) is the number of distinct configurations of the SAW which joins two vertices of the rth order of the fractal in the bulk and N is the number of sitC's visited by SAW, M and R represent respectively, the number of visited sites of the lattice which lie on the surface and on a layer adjacent to the surface. Summations in Eqs.{85) are on the repeated indices and S(rl(N, M, R) and C (rl(N, M, R) repre.sent number of configurations of respective walks of the r-th order gasket. The definition of parameters w and t is same as in sec. 8.1. For b = 2 the following three non-trivial fixed points arc found:
= 0) corresponds to the bulk state with v = 0.7986. For x = xc(w) the fixed point is reached for all w < wc(t). I\ote that wc(t) is a function oft; fort = 0.5 the value of wc{0.5)=1.1118.
1. The fixed point (B* = 0.61803, S* = C*
2. The fixed point (B* = c· = 0, s· = 1) is reached for all w > Wc(t) and X < Xc(w) . This represents the adsorbed state for the polymer chain with v = 1, as expected for a SAW adsorbed on a line. 3. The fixed point (B* = s· = c· = 0.61803) is obtained for w = Wc(t). The linearization of Eqs.{85) about this fixed point yields two eigenvalues greater than one, i.e. >.s = 1.6709 and >.b = 2.3819. We identify this as a tricritical point. The equality of B*, S* here shows that at this point the attraction at the surface, and repulsion at the next layer exactly compensate each other. The crossover exponent if>= 11og"b 0S~· = 0.5915 At the point of adsorption transition the leading singular behavior of free-energy density is given as f(T) "' (1~- T)2-u Thus the "specific heat"
D. Dhar and Y. Singh
186
exponent a at the tricritical point is related to ¢ as a = 2 - ~ for b = 2 the value of a = 0.3094. It is straightforward to extend this method to other members of this family. However, as the value of b increases the number of possible configurations of different walks increase rapidly. For 2 :::; b :::; 9 [53,54], the value of ¢ found are 0.5915, 0.5573, 0.5305, 0.5089, 0.4908, 0.4753, 0.4617 and 0.4497. The value of¢ decreases as b increases and becomes lower than the Euclidean value of 1/2 at b = 6. Zivic et al [55] have used Monte Carlo renormalization-group (MCRG) method to obtain the value of ¢ for 2 :::; b :::; 100 and found that lower bound of Eq(84) is violated forb= 12. The limit b-+ oo was analyzed by Kumar et al [53] using the finite size scaling theory (see sec. 5) and it was found that in this limit
cp(b) = 1 + v(b)(1 -Db) [1-
(2v(b)-1)loglogb 2 log b
+
1] terms of order . log b
(87)
i·
As forb-+ oo v(b) = ~'we get ¢(oo) = The first correction to¢ term to finite b is proportional to 2- db, similar to that found for 7(b) in Eq(71). Since 2 v(~)-l is positive for v = ~' the first correction term in Eq(86) is negative which, when multiplied by (1 - Db)v(b) makes a positive contribution to cp(b). This implies that cp(b) approaches to value monotonically as b is increased. Similar to the behavior of exponent "Y for SAWs discussed in section 5, we find ¢(b) does not converge to the Euclidean value in the limit b-+ oo. It has, however, been noted in [53] that the adsorbing surface of a fractal container is similar to that of a penetrable surface of a regular lattice in which case ¢ = 1 - v should be satisfied [56, 57].
i
9. INTERACTING WALKS So far we were concerned with a single linear flexible polymer chain and studied its conformational properties in different environments. We now show how the critical behavior of two interacting long flexible linear polymer chains can also be studied using a lattice model of two interacting walks. Depending on the solvent quality and the attractive interactions between intrachain and interchain monomers a system of two interacting long flexible polymer chains can acquire different configurations. The chain may be in a state of interpenetration in which the chains intermingle in such a way that they cannot be distinguished from each other. Or, the two chains may get zipped together in such a way that they lie side by side as in a double stranded DNA. It may also be possible, particularly at high temperatures, that the two chains get separated from each other without any overlap. By varying the temperature or tuning the interaction the system can be transformed from one state to another. The point at which the zipped-unzipped transition takes place is a tricritical point and in its proximity a crossover is observed. In the asymptotic limit the mean number of monomers M in contact with each other at the tricritical point is assumed to behave as M ex NY,
(88)
Linear and branched polymers on fractals
187
where N is the total number of monomers in a chain and y is the contact exponent. A lattice model of two interacting walks on the n-simplex lattices has been developed in [61-63]. Two different situations were considered: In one, the two walks have no mutual exclusion, and a lattice bond can be occupied by one step of each walks. This model has been referred to as a model of two interacting crossed walks (TICWs). In the second model, there is mutual exclusion between chains and a lattice bond can at most be occupied by a step of either walks. This model is called a model of two interacting walks (TIWs). In a general model of two interacting walks one can associate an energy Eb for each bond occupied by a step of both walks. The model of TICWs corresponds to Eb = 0 and that of TIWs to Eb = oo. In the TIWs the two chains cannot be at the same site because of mutual exclusion, but there is lowering of energy if they occupy nearest neighbour sites. The strength of both the inter- and intra-chain monomer interactions depend on the solvent and chemical nature of the monomers. Let the two chains or walks be referred to as P 1 and P 2 . The generating functions for this system can be written as ~ ~
XN!UR1XN2UR2UR 3 I I 2 2 3 '
(89)
all walks
where N 1 ,x 1 ,u 1 and R 1 (N2 ,x 2 ,u 2 and R 2 ) refer, respectively, to the number of steps, fugacity weight attached to each step, the Boltzmann factor associated with the attractive interaction between monomers and the number of pairs of nearest neighbors in the chain P 1 (P2 ). R 3 is the number of pairs of monomers of different walks occupying the nearest neighbor lattice sites and u 3 denotes the Boltzmann factor associated with the attractive interaction between monomers of P 1 and P 2 . Since the individual chain can be either swollen, collapsed or at the tricritical (B) point the variables (x 1 , u 1 ) and (x 2 , u 2 ) can be taken to be known (see section 6.2). Therefore, Eq.(89) has only u 3 as independent variable. With these simplifications it has become possible to evaluate the restricted generating functions for the n-simplex lattices. From the generating function of Eq.(89) one can calculate the average number of monomers of the two chains which are in contact (nearest neighbor) with each other from the relation
(R3) = U3 alogG. au3
(90)
The RSRG transformation has been used in [60, 61] for n-simplex lattices to solve the model exactly and calculate the phase diagram and the value of y for the different conditions of the solvent. Since the topological structure of a 3-simplex lattice is such that it can not have two SAWs on it, the model of TIWs was solved on the 4,5 and 6 simplex lattices. These lattices can have two interacting SAWs, with possible self-attraction also, without the walks crossing each other at any lattice point. On the 3- and 4- simplex lattices TICWs model in a condition in which both chains are in the swollen state has been solved in [61].
9.1. TIW's on the 4-simplex lattice To describe the two walks on the 4-simplex lattice we need five restricted partition functions. Two of these, A(r) and B(r) defined in Fig 11 correspond to the walk P 1 and
D. Dhar and Y. Singh
188 700 100-
.
Figure 21. The state of a system of two polymer chains in a non-selective solvent on the 4-simplcx lattice in the x 3 (= -JXIX2), u 3 plane. Lines SS and CC represent the trieritical lines of the zipped state of two chains each in the swollen state and the interpenetration state of the chains in a compact globule phase, respectively. Point TT , at which these lines meet, represents a transition point from a segregated to an interpenetrated state of the chains each at its B point. For a given value of x 3 which corresponds to the swollen state of both chains, the chains are in interpenetrated state when the value of u3 is less than the value given by the line SS. For the value of x3 corresponding to the chains in their compact globule state or at their B points, the two chains are in the segregated state for all values of u.3 less than the value given by the line CC or point TT. We also show the x-u phase diagram of a single chain for comparison's sake. :"Jote that the line CC overlaps with line C, TT with point T.
identical functions C(r} and D(r) corrf>Bpond to walk P2 • The restricted partition function £(r) represents the configurations where walks P 1 and P2 occupy neighboring sites and is sum of weights of configurations in whi<:h walks Pt and P2 enter and exit the r-th order subgraph once each. The two corner vertices of the subgraph arc occupied by the walk P 1 and the other two corner vertices by walk P2 • The recursion relation for the generating function £ (r) involves both A(rl, B(r} and C(rl, D(r} and is \vritten as[60] , (91) As shown in sec. 6.2, a polymer chain on a 4-simplex lattice can he in any one of t.he states of the swollen, compact globule and the B-point and is described in the asymptotic limit by the fixed points (0.4294 .. , 0.04998 .. ), (0, 22-l ) and (!, !), respectively. The fixed points corresponding to a swollen state is reached for all values of u 1 (or u 2 ) < Uc=3.31607 at x 1 (or x 2)=xc. The value of Xc is a function of the interaction u 1 {or u 2 ). The end to end distance for a chain of N-monomcrs in this state varies as Nv with v=0.7294 .... The fixed point corresponding to the compact globule state is reached for all values of u 1 (or u 2 ) > Uc (i.e. at low temperatures) at x 1 (or x 2)=xc(ui), where i=l or 2. In a compact
Linear and branched polymers on fractals
189
"" '·~o':: ,.--::o':-:ro,...---o,._n--o..J...u.· --:'o':: ,.--:o.'::a--:o'-...,--'o.n .,.~
Figure 22. The phase diagram of a system of two chains in a selective solvent on the 4-simplex lattice in the x 3 ( = JXIX2), 1L3 plane. Lines SC and ST respectively, represent the interpenetration states of two chains when one chain is in a swollen state and other is in compact globule phase or at its f) point. Line CT corresponds to the configuration of interpenetration of the chains when one in the compact globule phase and the other is at its fJ-point. When the value of u 3 is less than that given by corresponding linP.s, the two chains are segregated from each other.
globule state the polymer chain has a finite density of the monomer per site when N---t oo. At u 1 (or u 2 )=1lc=3.31607 and x 1 (or x 2 )= xc(1Lc)=0.22913 ... the fJ-point state is reached. In a system of two chemically different polymer chains, we may have six independent combinations of the individual chains which we indicate by SS, CC, TT, SC, ST and CT, where letters S, C and T now stand for the swollen, compact globule and fJpoint states. The recursion relation {91) is solved using the fixed points (A*, n•) and (C• , D*) corresponding to the given states of the chain P 1 and P2 and the starting weight. E <1l = (x 1 x 2 ) 2 u~. The solution leads to two fixed points denoted as E; and E; for each combination of states of the individual chains. In table I the values of these fixed points, relevant eigenvalues and the contact exponents y at the tricritical points are listed (62] . Figs. 19 and 20 show the state oftwo chains in the x 3 (= JXIX2), u 3 plane respectively, for non-selective and selective solvents. Lines SS and CC in Fig. 19 represlmt the tricriticallines of the zipped states of two chains each in the swollen state and the interpenetrating state of the chains each in a compact globule phase, respectively. Point TT, at which these lines meet, represents a transition point from a segregated to an interpenetrated state of the chains each at its f) point. For a given value of x 3 which corresponds to a swollen state of both chains, the chains are in the interpenetrated state when the value of u 3 is less than the value given by the line SS. For the values of x 3 , corresponding to the chains in their compact globule state or at their fJ-point, the two chains arc in a segregated state for all values of u 3 less
D. Dlmr and Y. Singh
190 7-0
.. .
1-0
'
5-0
,
\
~;., "-0
\ _______ _
l-0
5·7-o
·----.,
I
'•, ---
1.0 L._,--'1'-,-_---L'.,--~':------=':'.,---~·:---:::--'=',-~--'----=''-
0.0
o.os
o.1o
o.1s
o.20
o:rs
o30
035
"3...r.;;; . "I
Figure 23. The phase diagram representing the configurations of a system of two chains on the 5-simplex lattice in the x 3 ( = JXiX2), v.3 plane. Line S S, as in Fig 22, represents the zipped state of the chains. For all values of u 3 lf'$S than the value given by line SS, the two chains intermingle with each other. LineS represents the swollen state of a chain.
than the values given by line CC or point TT. The x, v. phase diagram of a single chain is also given in the figure for the comparison sake. The state of two interacting chains in a selective solvent on the 4-simplex lattice is shown in Fig 20. Lines SC and ST, respectively, represent the interpenetration state of two chains when one is in the compact globule phase or at its 0-point. Line CT corresponds to the configuration of the interpenetration of the chains when one is in the compact globule phase and the other is at its 0-point. \'11lwn the value of v.3 is less than given by the corresponding lin£>$, the two chains are segregated from each other. The individual chains configuration remains unchanged whether they are segregated or intermingled. 9.2. TIW's on the 5-simplex lattice For the 5-simplex lattice we also need five restricted partition functions to describe the generating functions of two walks; two corresponding to walk P 1 , two corresponding to walk P2 and E (r) which represents the configurations in which walks P1 and P2 enter and exit the r-th order subgraph once each and may occupy neighboring sites. The recursion relation for E (r) in this case is lengthy[60] and is therefore not reproduced here. As already shown in section 6.2 a polymer chain always remains in a swollen state for all values of int.eraction, (self-) attraction on a 5-simplex lattice. This state is characterized by the fixed point (0.3265 ... , 0.0279 ...). In this case we therefore have only one combination, i.e. SS of chains. Using these values of fixed points for single chains, the recursion relation for the partition function E(r) of two interacting chains (see [62]) is solved and fixed points E; = 0.0279 or Ei = 0.2713 are found. The fixed point E; is found for all values of v. 3 < v.3c(x 3 ). At u 3 = u 3c(x 3 ) fixed pointE; is found.
t:"-1
f:;•
Table 1 Values of fixed points, relevant eigenvalues, and the contact exponent y at the critical point for the 4-simplex lattice. The swollen, compact globule, and the tricritical configuration of each chain is indicated by letters S, C, and T respectively. A1 is the largest eigenvalue of the system, and A; refers to the two chain configuration point.
C1l
...,
~ ~
:::;
Q.,
~ ~
:::;
g.. State of individual chains
ss cc
TT
sc
ST TC
C1l Q.,
E*s 0.04998 0.0 0.01484 0.0 0.02687 0.0
E* ' 0.6125 22-1/3
1/3 0.4880 0.4294 0.3680
A; 2.6420 2.5440 2.2222 2.8520 2.8526 2.5740
A1 2.7965 4.0 3.7037 4.0 3.7037 4.0
y
"o
0.9447 0.6735 0.6098 0.7559 0.6311 0.6820
§
0
C1l
~
0
:::;
~
~
g_ ~
Table 2 Values of fixed points, G;, H;, I;, c;, H; and I;. The labeling is the same as in Table 1, but for 6-simplex. State of ind. chains
ss cc
TT
sc
ST CT
G*s
H*s
I*s
G*
H*
I*
A;
A1
y
0.0175 0.0 5.4x1o- 4 0.0 0.0028 0.0
0.0007 0.0 5.4x 10- 4 0.0 0.0003 0.0
0.0007 0.0 5.4x10- 4 0.0 0.001 0.0
0.1406 0.0 0.0957 6.75x1o-s 0.229 0.0060
0.0147 0.0713 0.0535 0.1049 0.1047 0.0714
0.0147 0.0713 0.0535 0.0711 0.0776 0.0806
2.2054 4.2201 3.2225 5.0529 5.2434 4.4019
3.4965 0.6318 5.4492 6.0 5.4492 6.0
0.6318 0.7902 0.6902 0.9041 0.9659 0.8271
'
'
'
>--'
(0
>--'
192
D. Dhar and Y. Singh
In Fig. 21, U3 is plotted as a function of x 3 = .Jx 1cX 2c· The interpenetrated state is found for all values of u 3 which lie below lines SS at a given x 3. When the values of u 3 reaches a value given by the line S S, the two chains are zipped together. Line S indicates the critical values of fugacity Xc and the self-attraction u 1 (or u 2 ) of a polymer chain.
9.3. TIW's on the 6-simplex lattice In this case one needs nine generating functions[60]; three corresponding to chain P1 , three corresponding to chain P 2 and the remaining three denoted as G(r), H(r), and J(r) represent, respectively, the sum of weights of configurations in which walks n and p2 enter and exit the r-th order subgraph once each, P1 twice and P2 once, and n once and P2 twice. Since the 6-simplex lattice exhibits the collapse transition there are six independent combinations of single chain states, similar to the case of 4-simplex lattice. The fixed points corresponding to single chains found in section 6.2 have been used to find the fixed points of the recursion relation for the restricted partition functions for G, H, and I with suitable starting weights [62]. The values of fixed points, corresponding eigenvalues and contact exponents are given in table II. The qualitative features of the phase diagram found for this lattice are the same as that of the 4-simplex lattice. Though the model of TIW's discussed above ignores the effects of one chain on the critical behavior of the other chains, it provides a qualitative description of the phase diagram of systems of two polymer chains in a solution which may have different qualities for different chains and may serve as a starting point for more thorough investigation of segregation and entanglements in a real systems. The phase diagrams plotted here are in the plane x3(= yXIX2), u 3. Plot of phase diagrams in a three-dimensional u 1 , u 2 , u 3 space is expected to give more informations about the states the two interacting chains.
10. CONCLUDING REMARKS It seems fair to say that the study of linear and branched polymers on fractals has been very useful in developing a better understanding of the critical behavior of polymers. One can verify the general qualitative features of the polymers on fractals which are very often similar to that in real experimental systems. One has a good deal of freedom in selecting the details of the fractal, and this can be used to find one that represents best the local geometry of the space. The exact values of the critical exponents do depend on the details of the fractal. But what is more important is that one can handle the different interactions in the problem, between different monomers, with the substrate, or with a different chain consistently and satisfactorily in way that allows exact calculation. Of course there are many unsolved problems, and possible directions for further research in this area. The most interesting problem would be to try to extend these exact solutions to some fractals with infinite ramification index. There are some studies of statistical physics models of interacting degrees of freedom on Sierpinski carpets, using Monte Carlo simulations, or approximate renormalization group using bond-moving, or other ad-hoc approximations. An exactly soluble case would be very instructive here. Determination of the exact scaling functions, not just the critical exponents, for these problems would be instructive: for example, the exact functional form of the scaling function of the probability distribution of the end-to-end distance of the SAW's on these fractal, or the periodic function of Fig. 8. For a numerical study of the former, see [64].
Linear and branched polymers on fractals
193
Solving the SAW problem with quenched disorder is another interesting question. For the 3-simplex fractal, this corresponds to making the variables B(r) random variables, and one has to determine the probability distribution of this variable for large r. We have discussed only the equilibrium properties of polymers. Of course, in many real systems, the time scales for equilibriation can be very large. It is thus of interest to study non-equilibrium properties of statistical mechanical systems on fractals. A simple prototype is the study of kinetic Ising model on fractals. Closer to our interests here, one can study, say, the reptation motion of a polymer on the fractal substrate. This seems to be a rather good first model of motion of a polymer in gels. Acknowledgements: It is a pleasure to thank Sumedha for a careful reading of the manuscript, and her many constructive suggestions for an improved presentation.
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25. J. R. Melrose, J. Phys. A : Math. and Gen. 18 (1985), L17; and references cited therein. 26. S. Elezovic, M. Knezevic and S. Milosevic, J. Phys. A 20 (1987) 1215. 27. V. Bubanja and M. Knezevic, cited in [30]. 28. D. Dhar, J. Phys. (Paris), 49 ( 1988) 397. 29. J. Cardy and S. Redner, J. Phys A: Math and Gen. A 17 (1984) L933. 30. S. Milosevic and I Zivic, J. Phys A 24 (1991) L833. 31. I. Zivic and S. Milosevic, J. Phys. A 26 (1993) 3393. 32. A. J. Guttmann and G. M. Torrie, J. Phys. A: Math and Gen 17 (1984) 3541. 33. K. Tanaka, Sci. Am. 244(1) (1981) 110; J. des Cloiseaux and G. Jannink; Polymers in Solutions, Clarendon Press, Oxford, 1990. 34. D. J. Klein and W. A. Seitz, J. Physique Lett. 45 (1984) L241. 35. D. Dhar and J. Vannimenus, J. Phys. A: Math. Gen.20, (1987) 199. 36. S. Kumar and Y. Singh, Phys. Rev. A 42 (1990) 7151. 37. M. Knezevic and J. Vannimenus, J. Phys. A: Math.Gen. 20 (1987) L969. 38. A. Malakis, J. Phys. A: Math. Gen. 9 (1976) 1283. 39. E. Orlandini, F. Seno, A. L. Stella and M. C. Tesi Phys. Rev. Lett. 68 (1992) 488. 40. A. B. Harris and T. C. Lubensky, Phys. Rev. B 24(1981) 2656; T. C. Lubensky and J. Isaacson, Phys. Rev. Lett. 41(1978) 829, Phys. Rev.A 20 (1979) 2130. 41. G. Parisi and N. Sourlas, Phys. Rev. Lett. 46 (1981) 871; D. C. Bridges and J. Z. Imbrie, Ann. Math. 158 (2003) 1019. 42. M. Knezevic and J. Vannimenus, Phys. Rev. Lett. 56 (1986) 1591. 43. D. Dhar and A. Dhar, Phys. Rev. E 55 (1997) R 2093. 44. M. Knezevic and J. Vannimenus, Phys. Rev. B 35 (1987) 4988. 45. B. Derrida and H. Herrmann, J. Phys.(Paris) 44 (1983) 1365. 46. D. Dhar, (2004), submitted for publication. 47. K. De'Bell and T. Lookman, Rev. Mod. Phys. 65 (1993) 87. 48. R. Rajesh, D. Dhar, D. Giri, S. Kumar andY. Singh, Phys. Rev. E 65 (2002) 056124. 49. Y. Singh, S. Kumar and D. Giri, J. Phys. A: Math. Gen 32, L407, (1999); 34 (2001) L67. 50. P. Grassberger and R. Hegger, Phys. Rev. E 51 (1995) 2674. 51. A. R. Veal, J.M. Yeomans and G. J. Jug, J. Phys. A: Math. Gen 24 (1991) 827. 52. S. Kumar and Y. Singh, Physica A 229 (1996) 61. 53. T. W. Burkhardt, E. Eisenriegler and I. Guim, Nucl. Phys. B 316 (1989) 559. 54. E. Bouchaud and J. Vannimenus, J. Physique 50 (1989) 2931. 55. S. Kumar, Y. Singh and D. Dhar, J. Phys. A:Math. Gen. 26 (1993) 4835. 56. V. Bubanja, M. Knezevic and J. Vannimenus, J. Stat. Phys. 71 (1993) 1. 57. I. Zivic, S. Milosevic and H.E. Stanley, Phys. Rev. E 49 (1994) 636. 58. T. Ischinabe, J. Chern. Phys. 80 (1984) 1318. 59. Y. Singh, S. Kumar and D. Giri, Pramana J. of Phys. 53 (1999) 37. 60. S. Kumar and Y. Singh, Phys. Rev. E 48 (1993) 734. 61. S. Kumar andY. Singh, J. Phys. A:Math and Gen. 26 (1993) L987. 62. S. Kumar andY. Singh, Phys. Rev. E 51 (1995) 579. 63. S. Kumar andY. Singh, J. Stat. Phys. 89 (1997) 981. 64. A. Ordemann, M. Porto, and H. E. Roman, Phys. Rev. E 65 (2002) 021107.
Statistics of Linear Polymers in Disordered Media Edited by Bikas K. Chakrabarti © 2005 Elsevier B.V. All rights reserved.
Self-avoiding walks on deterministic and random fractals: Numerical results Anke Ordemanna, Markus Portob and H. Eduardo Romanc alnstitut fiir Physiologie, Philipps-Universitiit Marburg, Deutschhausstr. 1, 35032 Marburg, Germany blnstitut fiir Festkorperphysik, Technische Universitiit Darmstadt, Hochschulstr. 8, 64289 Darmstadt, Germany cDipartimento di Fisica, Universita di Milano-Bicocca, Piazza della Scienza 3, 20126 Milano, Italy Numerical techniques for studying statistical properties of self-avoiding random walks (SAWs) on deterministic and random fractal substrates are reviewed. To this end, numerical algorithms are discussed for the generation of SAW configurations of N steps, which are based on Monte Carlo methods and the exact enumeration technique. The advantages and disadvantages of both approaches are highlighted in connection to the different substrates and the statistical quantities of interest. Applying these methods, the scaling behavior (N-dependence) of such static quantities as the total number of SAWs, their end-to-end distance, and the associated probability distribution functions, as well as a possible multifractal behavior, are studied. Scaling forms known for SAWs on regular lattices seem to remain valid also on deterministic and random fractals, while the corresponding relations between the scaling exponents need to be modified in some cases. A prominent example of the latter is the case of percolation at criticality. 1. INTRODUCTION
Self-avoiding walks (SAWs) constitute the simplest, yet non-trivial model for studying the static behavior of a linear polymer embedded in a good solvent. Such a polymer is a chain-like array of N + 1 monomers rigidly connected to each other, in which the only residual interaction between non-consecutive monomers is a (short-ranged) monomer core repulsion [1-6]. Despite the intrinsic difficulties arising from the emergence of long-range monomermonomer correlations along such linear chains, many exact results are known (see Chapter 1 by Chakrabarti). If, in addition to this complexity, one aims to study the static behavior of SAWs on substrates displaying self-similarity, one is soon faced with models whose exact solutions become even harder to obtain (possible approaches such as renormalization group analysis and series expansions are discussed in the other chapters of this book). In this more general context, therefore, numerical investigations become a valuable approach to address such issues in a quantitative fashion. Nonetheless, the dif195
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ferent numerical schemes can also be applied to situations where exact results are known in order to test the validity and accuracy both of the analytical results as well as the numerical schemes. In studying SAWs on a given substrate (irrespectively of being regular or fractal), the typical observables characterizing their static behavior are: (i) the number of configurations CN of SAWs of N steps/ (ii) the probability distribution function P(riN), where P(riN) dr is the probability of finding an N-step SAW having an Euclidean end-to-end distance between r and r + dr, and (iii) the root-mean square Euclidean end-to-end dis00 tance r(N), which is related to P(riN) by r(N) = f 0 r P(riN) dr. In the case of fractal substrates, one has to distinguish between two main subclasses of structures, namely deterministic and random fractals. Within the class of deterministic fractals, one additionally has a subdivision in finitely and infinitely ramified fractals. Here, (either finite or infinite) ramification refers to the number of cut operations which are required to disconnect any given subset of the structure, the upper limit of which is independent of the chosen subset [7,8]. An example of a finitely ramified structure is the Sierpinski triangular lattice, whereas the Sierpinski square lattice is an example of an infinitely ramified structure. See Figs. 2( a) and 6 in Section 4 for the respective sketches of these structures in d = 2. In the case of random fractals, the paradigm and most commonly studied model is percolation (see also Chapter 1 by Chakrabarti). It is important to note that non-trivial modifications of the scaling behavior of SAWs on percolation are believed to occur only at the percolation threshold [5]. Indeed, it is only at criticality, that the infinite critical (the so-called incipient) percolation cluster spans self-similar structures on all length scales. The long-standing and, still, largely unresolved question concerning the characteristics of SAWs is: Does the statistical behavior of SAWs change in the presence of (annealed or quenched) structural disorder? And, if so, which observables do change and how? Do these changes depend on the type of disorder and the kind of self-similarity of the substrate? In the above quest, 'quenched' structural disorder refers to the procedure that for a given observable, the average is first taken for a fixed realization of the structural disorder, and only afterwards an average is performed over the different realizations of the structural disorder. This scenario has to be distinguished from the case of 'annealed' structural disorder where both averages are performed simultaneously [9,10]. 2 In many numerical studies the two types of averages are performed simultaneously, which implicitly leads to an incomplete set of possible configurations per structure, thereby yielding typical rather than mean values for the observables considered (see Section 3). This Chapter is organized as follows: Section 2 revisits general definitions of quantities of interest, for both deterministic and random fractals, and introduces the notation used. Section 3 contains the numerical techniques such as Monte Carlo (MC) methods and the exact enumeration (EE) technique as well as data analysis schemes. Section 4 provides a discussion of SAWs on deterministic fractals, specifically the Sierpinski square and 1
Note that we use a notation where N refers to the number of steps, not to the actual number of monomers, which then is N + 1. Other authors use the alternative notation calling N the number of monomers. This difference becomes of course irrelevant in the thermodynamic limit N -+ oo. 2 Note that, due to historical reasons, what is called 'quenched disorder' is sometimes also referred to as 'annealed average' in the literature, whereas 'annealed disorder' is denoted as 'quenched average'.
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triangular lattices, whereas Section 5 deals with SAWs on random fractals, modelled by percolation at criticality. Similarities and differences of the behavior of SAWs on these and related structures are discussed in Section 6. Technical aspects of the numerical methods are relegated to the Appendices.
2. DEFINITIONS AND GENERALITIES 2.1. SAWs on regular lattices revisited It is useful to briefly revisit the behavior of SAWs on regular lattices, mainly for reminding the reader about the quantities of interest and the notation employed (see also Chapter 1 by Chakrabarti). The number CN of all possible N-step SAW configurations displays the following scaling behavior
(1) governed by the so-called enhancement exponent 'T and the effective coordination number (effective connectivity) p,. The latter is, due to the self-avoiding condition, smaller than the coordination number f..Llatt of the lattice, p, :::; f..Llatt· The exponent 'T ~ 1 describes how much the number of SAW configurations is enhanced over the bare number p,N, counteracting the decrease of the coordination number. The enhancement factor N-r-l is absent in dimensions above the upper critical dimension, d ~ de = 4 (i.e. 'T = 1), suggesting that such high dimensional spaces render the self-avoidance condition irrelevant. Note that 'T is a universal exponent since it depends only on the dimensionality d of the lattice, whereas p, is not universal, as it also depends on the type oflattice considered, for instance whether it is a square or a hexagonal lattice. The mean end-to-end distance r(N) of all N-step SAW configurations scales as
(2) where the exponent Vr is universal and traditionally called the Flory exponent [11], being a measure of the spatial spread of the SAWs. For d = 2, is has been recently proven by Bueter [12] that Vr = 3/4, whereas Vr = 1/2 at the upper critical dimension de= 4. For d = 3, it has been proven that 7/12 :::; Vr :::; 2/3 by the same author [13],3 and a recent numerical value (obtained by a Monte Carlo method) yields Vr = 0.58758 ± 0.00007 [15]. This dependence of Vr on the spatial dimension d means that SAWs become more compact as d increases. More information about the spatial shape of SAWs is contained in the probability distribution function P(riN), where P(riN) dr is the probability that an N-step SAW 00 has an Euclidean end-to-end distance between r and r + dr, so that J0 P(riN) dr = 1. The scaling behavior of P(riN) is given by [1,3]
P(riN) = 3
~ Fr
c;vr) ,
(3)
Note that there is a revised version of the preprint [13] proving the above weaker statement 7/12 :S: Vr :S: 2/3 for d = 3. The previous version made the much stronger statement Vr = 7/12 for d = 3, as cited by us in Ref. [14].
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where the scaling function Fr(x) behaves asymptotically as for x for x
« »
1 1
(4)
Similarly to the exponent 'T and vn the exponents gr, gf, and 8r are believed to be universal, and depend upon each other (see Chapter 1 by Chakrabarti) by the Fisher relation [16] 1 br=-1- Vr'
(5)
by the des Cloizeaux relation [17] r
gl
'T- 1
= --,
(6)
Vr
and by the McKennzie-Moore relation [18]
(7) The exponent 8r describes the steepness of the exponential decay of the probability to observe very elongated SAW configurations, with a larger value indicating the existance of fewer elongated configurations. On the other hand, gr measures the power-law decay of the probability of compact SAW configurations to occur (or, more precisely, to have small end-to-end distances), where a small or zero value for gr indicates the abundance of such compact SAW configurations. Lastly, the exponent gf characterizes the probability of SAW configurations being neither elongated nor compact, with a medium end-to-enddistance. Note that until quite recently, most numerical analyses did not distinguish between the two different scaling regimes described by grand gf, rather a single exponent gr was used for both regimes (mainly due to the fact that the actual values can be very similar and could not be distinguished due to limitations in numerical accuracy). 2.2. The substrate intrinsic metric: The £-space For any type of substrate, being it either an ordered or a disordered one, it is convenient to numerically measure distances, say, end-to-end SAWs distances, in a metric which takes into account the topology of the structural connecting paths. This so-called topological or chemical metric (where the space in which it is defined is referred to as £-space) is the 'natural' metric of the structure, in which the distance between two substrate points equals the length of the shortest path on the structure connecting them. An example illustrating the difference between the £-distance and the standard Euclidean distance (in r-space) for a disordered structure is shown in Fig. 1. For regular lattices and the deterministic fractal substrates considered here, as well as for percolation above criticality beyond the correlation length, the Euclidean and the topological metrics are simply proportional to each other. 4 In contrast, for percolation at criticality these two metrics scale differently as [19,20],
(8)
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199
Figure 1. The relation between the two distance metrics, the Euclidean distance r and the topological distance exemplified using two sites (marked in white) on a disordered substrate (a percolation cluster, shown in grey).
e,
with a fractal dimension dmin > I. denoted as the fractal dimension of the minimal path. Eq. (8) means that, on average, the topological distance e between two sites separated by an Euclidean distance r increases with r proportional to rd"''" · For ordered substrates as well as deterministic fractal substrates, one trivially has dmin = 1. 4 For both dm;n = 1 and (particularly) dmin > 1, the use of the f-metric might seem superfluous and to make the analysis complicated without need. However, it is of considerable advantage in both cases to measure distances in rather than in r-space in numerical studies, as the data shows in general much less fluctuations and hence displays a 'smoother' scaling behavior with f than with r [22]. l'<evertheless, both measurements remain fully equivalent. A very simple example showing the relation between and r-space is given by the mass !vJ of the substrate, which scales as
e-
e-
(9) with the fractal dimensions dt and d, being related by dt = dr/dmin (usually dr is called de in the fractal literature, but for the sake of consistency with our notation we refer here tor-space, one t.o this quantity as d,) . :VIore generally, to convert a quantity f from has to calculate the convolution
e-
J 00
J(r)
=
fdt-
1
1l>(elr) f(e)de,
(10)
T
where ll>(elr) de is the probability that for a given r, a distance between e and e+de is observed, and is normalized as j~"" [de- 1 11>(elr) dt = 1. Note that there are some subtleties concerning the lower integration limit in Eq. (10) , see Subsection 5.2. Because of the abovementioned reduction of fluctuations in l-space, it is easier to obtain a good estimate for the probability distribution P(eiN) of the topological end-to-end 4
Detcrministic fractals usually have a linear scaling relation between the two mctriccs (i.e. dmin = 1), but there exists exceptions to this rule, see Ref. [21) for an example.
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200
distance£ of SAWs of N steps, rather than the related quantity P(riN). As for Euclidean distances, one derives from the former quantity the mean topological end-to-end distance 00 C(N) = J0 £ P(CIN) d£, which displays a scaling behavior
(11) where the univeral exponent ve is related to
Vr
by
(12) The scaling behavior of P(CIN) itself, with
P(CIN)
J0
00
P(CIN) d£ = 1, is similar to P(riN),
=~Fe (:"t) ,
(13)
with the scaling function being given by for x for x
« »
1 1
(14)
Comparing Eqs. (4) and (14), note that the dimension din the former represents the substrate mass dimension, which in that case coincides with the embedding spatial dimension. When considering the probability distribution function P(riN) for the more general case of fractals, the exponent d becomes the substrate's mass fractal dimension dr. The exponents gf, g~, and be are believed to be universal, and gf depends on its counterpart in r-space as
e
gr
gl =-d.'
(15)
mm
whereas be is given by 1
be=--. 1- Ve
(16)
A relation between g~ and g; is not known for any fractal substrate.
3. NUMERICAL SCHEMES When studying SAWs on fractal substrates, the complexity arising from the statistics of the SAWs becoming entangled with the self-similarity of the substrate causes that many analytical techniques developed for SAWs on regular lattices are no longer applicable. In these cases, but also where analytical results are available, numerical methods are very valuable tools. As far as SAWs are concerned, one distinguishes mainly two classes of numerical algorithms, i.e. Monte Carlo (MC) methods and the exact enumeration (EE) technique. There exists a wide variety of MC methods, and in Subsection 3.1 we will concentrate
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on a few most used ones. All these different MC methods share the advantage that one can evaluate very long chains (presently up to 106 steps in d = 2 and 6.4 x 105 steps in d = 3 [23]), so that finite length effects are strongly diminished. This benefit has to be confronted with the general disadvantage of MC methods, namely that certain 'atypical' SAW configurations are either under-represented in the ensemble or not sampled at all, so that one obtains typical rather than mean values for the observables (although certain reweighting techniques have been applied which are able to lessen this problem). The incomplete sampling of SAW configurations for a given realization of the substrate is also called 'annealed averaging' and becomes particularly disadvantageous when multifractality is present (see Subsection 5.3). In this respect, the clear advantage of the EE technique is that by definition all possible SAW configurations are considered, so that there is no bias in the ensemble and true mean values are obtained for the observables. The disadvantage of the EE technique is that, due to the enormous amount of SAW configurations occuring even for short SAWs, severe limitations are imposed on the maximally possible chain length (presently, up to 59 steps (71 steps when restricting the analysis to the number of SAW configurations) in d = 2 [24] and 26 steps in d = 3 [25] have been considered on regular lattices). This constraint makes it difficult to extract the asymptotic behavior so that one has to apply sophisticated evaluation techniques (see Subsection 3.3). Irrespective of the actual techniques employed, with any numerical algorithm which includes a (pseudo) random number generator (e.g. MC algorithms for SAW generation or the Leath growth [26] and Hoshen-Kopelman algorithm [27] to construct a percolation cluster), caution has to be exercised that the sequence of random numbers does not show harmful correlations, as these can significantly affect the statistics and hence the obtained observables [28]. 3.1. Monte Carlo methods During the years of studying SAWs on regular lattices, a whole zoo of MC methods has developed. These diverse algorithms can be classified according to 'static,' 'quasistatic,' and 'dynamic' MC methods [29] (the name 'dynamic MC methods' should not be mistaken with true polymer dynamics, it just refers to the fact that monomers and bonds are moved in these schemes to generate new SAW configurations). These classes can be further divided, the class static MC methods contains (i) simple sampling and its variants, (ii) the Rosenbluth-Rosenbluth algorithm and its variants, as well as (iii) dimerization and its higher order generalizations. The class quasi-static MC methods summarizes (i) variants of quasi-static sampling, (ii) diverse enrichment schemes, and (iii) incomplete enumeration. Last but not least, the most abundant class is the one of dynamic MC methods, which can be sorted according to whether the moves (i) are N-conserving or N-changing, (ii) are end point-conserving or end point-changing, and (iii) are local or non-local. All the dynamic class MC methods need a starting SAW configuration to operate on, which one usually generates by using either a static or quasi-static method. In all these diverse MC methods, irrespective of the class, a central issue is the correct weighting of the different SAW configurations observed. In extreme cases, some methods have to be carefully checked whether certain SAW configurations can not occur at all, thus even breaking ergodicity. Due to the large diversity of MC methods, we cannot discuss them here in detail, the
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interested reader is refered to the very exhaustive review by Sokal [29]. Nonetheless, in the present context, it is important to stress that there are certain MC methods which have been successfully applied to study SAWs on fractal substrates, yet there are others which are not suitable (or inefficient) in this case. The applicable ones are simple sampling (belonging to the static class) [30,31], incomplete enumeration (belonging to the quasi-static class) [10,32], and reptation (belonging to the dynamic class) [30]. The Pivot algorithms (belonging to the dynamic class), which are the most efficient family of algorithms to study SAWs on regular lattices [23,33,34], are suitable neither for deterministic nor for random fractals, as essentially all Pivot moves are rejected due to the constraints by the fractal substrate. Nevertheless, one should note that Pivot algorithms have been successfully applied to SAWs on regular lattices to whose sites energy terms are assigned (SAWs on energy landscapes) [35]. Here, Pivot moves are rejected when the self-avoidance condition is broken or when the over-all energy of the SAW is higher than a certain cut-off. To some extent, these systems interpolate between SAWs on regular lattices (flat energy landscape) and SAWs on disordered substrates (very rough energy landscape), and Pivot algorithms are successful as long as the energy landscape is not too rough with respect to the cut-off. 3.2. The exact enumeration technique There are several cases where the inherent incomplete sampling of SAW configurations when using MC methods is disadvantageous, particularly when multifractality is present. In these cases, the EE technique has the clear advantage that by definition all possible SAW configurations up to a certain number of steps are considered (see Appendix A). The major drawback of the EE technique for SAW studies is the severe limit in the possible number of steps. Hence, it is important to exploit the symmetry of the substrate to increase the maximum number of steps. However, such symmetries generally do not exist even for deterministic fractals, due to the lack of translation and rotational invariances of the substrate. Although some selected starting sites for the SAWs on deterministic fractals do have some symmetry that one might be able to exploit, due to the nonisotropic environment, an average over several starting sites is required in order to lower the strong inherent fluctuations (see Appendix B). In such an average, those starting sites are the majority for which no symmetry can be exploited. In contrast to these additional difficulties, fractal substrates have the advantage of being usually 'thinner' on all scales, because the mass fractal dimension is smaller than the embedding spatial dimension. Thus, the fractal substrates provide only a diluted space for the SAWs and consequently the total number of configurations (and therefore the computation time) is considerably reduced with respect to the regular lattice counterpart. Overall, the virtue of the EE is that the configurational space is fully explored providing the highest accuracy for the given SAW length. Furthermore, the full set of configurations is essential to address questions such as for instance whether certain quantities are multifractal or not. Studies show that, despite the shorter SAW lengths, one can still draw accurate conclusions about the asymptotic behavior of SAWs when using sophisticated data analysis techniques. 3.3. Data analysis techniques In scaling analysis one is interested in the behavior of the observables in the thermodynamic limit, i.e. when N --+ oo. The simulations, however, can only be done at finite
Self-avoiding walks on deterministic and random fractals: Numerical results
203
N (which is even rather small in the case of the EE technique) and one has to apply sophisticated finite-size scaling schemes. One of the simplest techniques is the so-called successive slopes method, in which the local slope of the quantity of interest versus N (usually plotted in log-log scale) is obtained numerically and plotted versus 1/N. The thermodynamic limit is then determined as the limit of 1/N --+ 0. For examples of the successive slopes method see Figs. 3, 4, 7, 8, and 14. More elaborated methods are the ratio method and Pade and differential approximants [36-38]. A method specific to a problem occuring when studying PDFs of SAW end-to-end distances is to separate the power-law scaling from the exponential one. Here, Wittkop et al. [32] have proposed a rescaling ansatz, which has been further improved [39] to account for the two different power-law scaling regimes (see Appendix C).
4. SAWs ON DETERMINISTIC FRACTALS Let us consider the case of deterministic fractals first, i.e. self-similar substrates which can be constructed according to deterministic rules. Prominent examples are Sierpinski triangular or square lattices, also called gasket or carpet (in d = 2) and sponge (in d = 3), respectively, Mandelbrot-Given fractals, which are models for the backbone of the incipient percolation cluster, and hierachicallattices (see for instance the overview in Ref. [21]). In this chapter, however, we restrict the discussion to the Sierpinski triangular and square lattice for brevity. Renormalisation group (RG) techniques have been applied to several finitely ramified structures, so that results are available for some deterministic fractals including Sierpinski triangular lattices [40-47] (for a comprehensive discussion see Ref. [48]). For infinitely ramified structures, there is no RG result available and one has to rely on numerically evaluating SAWs on these fractals (note, however, the study of Taguchi [49] of SAWs on Sierpinski square lattices). Nonetheless, even in the former case when RG results are available, it is instructive to apply numerical schemes as mentioned in the Introduction.
4.1. SAWs on finitely ramified substrates: Sierpinski triangular lattices This subsection is devoted to the Sierpinski gasket (d = 2) and its corresponding sponge ( d = 3), further on called Sierpinski triangular lattices. This fractal is characterized by a mass fractal dimension ds = In( d + 1) /In 2, which depends on the embedding spatial dimension d (see Fig. 2 for examples in d = 2 and d = 3). Note that for Sierpinski lattices in general, the Euclidean distance r between two lattice sites scales as the topological distance £, £ ""' r, so that there is only one mass fractal dimension ds, M ""' rds ""' f!ds. Concerning SAWs on Sierpinski triangular lattices, the observables which are usually studied are: The number of configurations CX, of SAWs of N steps, which scales as C~
= Ds p.~ N 15 - 1
for N
»
1,
(17)
defining the coordination number f.J.s and the enhancement exponent ')'s; the probability distribution Ps( £IN) of the topological end-to-end distances £ of SAWs of N steps (normalized as J Ps(CIN) d£ = 1), Ps(CIN)
= ~ Fs
(:vs)
(18)
A. Ordemann, M. Porto and H. E. Roman
204
(a)
Figure 2. Example of the Sierpinski triangular lattice in (a) d = 2 and (b) d = 3, both after two iterations. The sites available to the SAW are shown as disks (d = 2) and cubes (d = 3) and the inter-connecting bonds as lines.
with for x « 1 for x ~ 1'
(19)
defining the exponents g~, g~, and 8s; the mean topological end-to-end distance C(N) (derived from Ps(£1N)), which displays a scaling behavior
C(N)
rv
Nvs
(20)
with the exponent Vs. Concerning the above exponents, it is generally assumed that 1 bs=-1- vs
(21)
holds, resembling the analogous one for regular lattices, Eq. (5). It has been suggested [50] that additionally a des Cloizeaux relation, Eq. (6), is valid, s /'S- 1 gl = - - . vs
(22)
Both relations are supported by numerical calculations (see below and Table 4). A relation for g~ is not known, and a naive straightforward conversion of the McKennzie-Moore relation [18] derived for regular lattices, Eq. (7), does not lead to consistent results. Renormalization group studies of SAWs on Sierpinski triangular lattices using different techniques have been reported in Refs. [40-46], leading to the exponent vs = 0.7986 in
Self-avoiding walks on deterministic and random fractals: Numerical results
··a
102
£
~
0.4 -------,-.-.-----;
~ 0.2
I
::1.
0
"'
~
205
10
0.05
0.1
1
(a)
102
i:: ~
::1.
"'
~
"'EJ ----~-- ...-;-.-- ..
0.4
~ 0.2
I
0
0.05
0.1
0.15
10 1
~
,_
(b)
100
10 1 N
Figure 3. Total number of SAWs of N steps (CN,s) on the Sierpinski triangular lattice (adapted from Ref. [50]): Determination of f..Ls and ')'s, by plotting (CN,s) f..L-N versus N in double-logarithmic form, for different values of f..L· The accepted value for f..Ls is obtained when (CN,s) f..L-N displays a satisfactory power-law, and the associated slope yields/'s -1. The plots correspond to: (a) d = 2, for f..L = f..Ls = 2.29 (circles), 2.24 (diamonds), and 2.34 (squares). The dashed line is a fit yielding/'s= 1.35. It has been obtained from the successive slopes shown in the inset. The continuous line represents the RG result [40-43], /'s- 1 ~ 0.3752. (b) d = 3, f..L = f..Ls = 3.82 (circles), 3.72 (diamonds), and 3.92 (squares). The successive slopes shown in the inset yield /'s = 1.41 (note that the second analysis method discussed in the main text yields /'s = 1.43), the value is represented by the dashed line. The continuous line represents the RG result [40], /'s- 1 ~ 0.4461.
d = 2 and Vs = 0.7294 in d = 3 (note that in Ref. [46], Vs = 0.67402 has been determined in d = 3). For the exponent ')'s, the RG results give 1.3752 and 1.4461 in d = 2 and d = 3 [40,43], respectively, while for f..L the values 2.288 and 3.815 have been reported [4143,47]. Note also the interesting numerical results for f..Ls and /'s of SAWs on a family of Sierpinski triangular lattices which asymptotically approaches the regular triangular lattice [51] (see also subsequent Comment [52]). As the recent EE technique study of SAWs on Sierpinski triangular lattices in Ref. [50] provides an accurate and complete set of numerical estimates for the above exponents in both d = 2 and d = 3, we discuss these results in the following. To determine the values of the exponents, first, fractal substrates of sufficient size have been constructed and then all possible SAW chains up to N = 30 in d = 2 and
206
~
~
1.20
A. Ordemann, M. Porto and H. E. Roman
1.0
0.8
__,
:> 0.6
--:- .-
'-"'
•• ••
0.4
0
(l(N))
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0.1
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-
0.4
(f(N))
-
•
-
0
0.05
0.1
0.15
1/N
(b)
10°
~--~_._.~~~----~~~
100 N
Figure 4. Mean end-to-end chemical distance (l(N)) for SAWs on the Sierpinski triangular lattice as a function of the step length N in (a) d = 2 and (b) d = 3 (adapted from Ref. [50]). The continuous lines correspond to the values of Vs ~ 0.7986 (d = 2) and 0.67402 (d = 3) as obtained from RG calculations [43]. The dashed lines display the results obtained by successive slopes (see insets), yielding Vs ~ 0.78 (d = 2) and 0.66
(d=3). N = 20 in d = 3 originating from certain starting sites have been enumerated by the EE technique (see Appendix A). To avoid spurious lattice effects caused by the nonequivalence of possible starting sites (first investigated in Ref. [53] for random walks on Sierpinski triangular lattices), an additional average over a selected set of starting sites has been done, indicated in the following by ( · ) (see Appendix B). The expected behavior from Eq. (17) has been analyzed in two different ways (the reason for this is discussed at the end of this Subsection): The first one is based on studying the quantity (CN,s) f..i.-N as a function of N, for different values of f..i., as shown in Fig. 3 for d = 2 and d = 3. The accepted value f..1. = f..l.s yields the 'best' power-law dependence of (CN,s) f..i.-N versus N, for large N. The second method consists in a direct fit of the function CN,s =As f..i.~ N-rs-l in the form
ln(CN,s)
lnAs
-'-::N~-'- = -N-
+
I n f..l.s
+
(
)InN /'s - 1 -N-
(23)
with suitable values of the fit parameters As, ')'s, and f..l.s (not shown for the case discussed here, see Fig. 11 for a corresponding example for SAWs on the backbone of the incipient
Self-avoiding walks on deterministic and random fractals: Numerical results
100 ~
10-2
~
~ 10-4
""'
10-6
207
/-\
(a)
~~]'. ~
S?~
::SI
~--~~ ~
•
10 ,
•
•
\
~
\
1~
\
0.1
0.2
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~
~
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··~
/
~
\
~11 ··1: ----,/1 i
10-4
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\
~ ._. 10- 1
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10-6
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10-1
Figure 5. The PDF (Ps(£, N)) of the end-to-end chemical distance £ for fixed number of steps N on the Sierpinski triangular lattice, plotted as £ ( Ps (£, N)) versus £/ N" 5 in (a) d = 2 with N = 30 and (b) d = 3 with N = 20, using the values vs obtained in Fig. 4 (adapted from Ref. [50]). The continuous lines represent the theoretical value g~ + ds obtained using Eq. (22) for the RG values reported in Table 4. The dashed lines in (a) and (b) represent fits of the data, in the regimes £ « N"5 and £ » N"5 , according to Eqs. (18) and (19). An alternative determination of the exponent g~ is illustrated in the insets, where the quantity £(Ps(£,N))/(C/N"5 )ds is plotted versus C/N"5 • The exponent g~ is obtained from the slope of the ansatz £(Ps(£,N))/(C/N"5 )ds rv (£/N"s)Y'f for£« N"5 , yielding g~ = 0.44 (d = 2) and 0.65 (d = 3).
percolation cluster). Both methods yield consistent values for f.J.s and ')'s, which are summarized in Table 4. The behavior of (C(N)) as a function of N is shown in Fig. 4 for both d = 2 and d = 3, from which the value of the exponent vs has been obtained using successive slopes. Both the results for Vs as well as f.J.s and /'s are in very good agreement with the known values from RG calculations, in support of the approach based on the EE calculation of short chain lengths. The RG results have been obtained partly by using the fact that Sierpinski lattices belong to the same universality class as the truncated n-simplex lattices [41,54], so the universal exponents v and I' are expected to be identical, but the non-universal f..1. is different (see also Chapter 5 by Dhar and Singh). The PDF (Ps(£1N)) for both d = 2 and d = 3 is shown in Fig. 5. Note that, despite
208
A. Ordemann, M. Porto and H. E. Roman
of the fact that the average has been performed over different starting sites, few small irregularities are still present. In d = 2, these irregularities occur around £ = 17 (i.e. for f!/N" 5 ~ 1.2), whereas in d = 3 they occur around£= 9 and£= 13. The irregularities (also visible as a kind of oscillation in the inset of Figs. 3 and 4) are due to the structure of the underlying Sierpinski lattices, as at relative distances b.£ + 1 = 5, 9, 13, and 17, etc., the SAW enters a new substructure after passing the points b.£ = 4, 8, 12, 16, etc., where two substructures merge together, providing a bottleneck to the SAW. If the calculations were performed by considering only a single starting site for the chains, located for instance at a 'vertex' of the lattice, the resulting oscillations would completely dominate the PDF making it very difficult to extract the scaling exponents reliably. To further attenuate the irregularities being still visible in Fig. 5, an exponentially increasing number of starting sites would be needed. The values for g~, g~, and 8s (the latter two using the numerical procedure discussed in Appendix C) shown in Fig. 5 are reported in Table 4. Within the presently available accuracy, it seems that the relation between bs and vs, Eq. (21), holds. The verification of Eq. (22) is somewhat more delicate, and therefore it is crucial to determine the values of g~, vs, and (in particular) /'s as precisely as possible (this being the reason for the two methods of analysis to determine /'s discussed above). By doing this, the des Cloizeaux relation, Eq. (22), has been found to hold. Contrarily to the case of g~, a theoretical estimation for g~ is still lacking. For d-dimensional regular lattices, it is well known that g2 is given by the McKennzie-Moore relation [18], Eq. (7). Unfortunately, a naive straightforward conversion of Eq. (7) to the Sierpinski lattice does not lead to consistent results. 4.2. SAWs on infinitely ramified structures: Sierpinski square lattices This Subsection deals with the Sierpinski carpet (d = 2) and its corresponding sponge (d = 3), further on called Sierpinski square lattices (an example in d = 2 is shown in Fig. 6). Note that in this Subsection, we use the identical notation as in the previous Subsection discussing Sierpinski triangular lattices, avoiding a reiteration of Eqs. (17) to (22). Nonetheless, a major difference between the two types of Sierpinski structures is the ramification, as Sierpinski triangular lattices are finitely ramified, whereas Sierpinski square lattices are infinitely ramified. The infinitely ramification renders the application of RG technique elusive and has drastic consequences on the actual values obtained numerically for the exponents. The Sierpinski square lattices' mass fractal dimension is given by ds = In( nd- k) /Inn. In contrast to the Sierpinski triangular lattice, there are different types of Sierpinski square lattices with different mass fractal dimensions. The type of the Sierpinski square lattice depends on the choice of two parameters (n, k), where k indicates the number of elements being removed out of a square or cube of linear size n. Following the majority of studies, the results discussed below are mainly for (n, k) = (3, 1) in d = 2 and (n, k) = (3, 7) in d = 3. However, even after specifying the two parameters nand k, there is additionally the choice of the so-called lacunarity, which indicates how symmetric the removal occurs: The smallest lacunarity indicates the most symmetric configuration, whereas more asymmetric cases yield a larger lacunarity. Unlike the mass fractal dimension ds which depends only on (n, k) but not on the lacunarity, the exponents describing the scaling behavior of
Self-avoiding walks on deterministic and random fractals: Numerical results
209
Figure 6. Example of Sierpinski square lattice in d = 2 with (n, k) = (3, 1) and minimal lacunarity, after the second iteration. The sites available to the SAW are shown as black squares and the inter-connecting links as lines. Note that the number n counts the bonds in the present case, whereas in several studies in the literature the numbers n counts the sites. This means that the SAW monomers live on the nodes in the present case, whereas otherwise they are located in the center of the (white) squares formed by the bonds.
SAWs on Sierpinski square lattices do depend on both (n, k) and the lacunarity (for instance, the exponent vs is known to increase for increasing lacunarity [49,55]). Also here, the discussion below follows the majority in the literature and is restricted to the most symmetric configuration with the smallest lacunarity, see Fig. 6. Note that for Sierpinski lattices in general, the Euclidean distance r between two lattice sites scales as the topological distance £, /! ,. . ., r, so that there is only one mass fractal dimension ds, M ,. . ., rds ,....., f!ds. Due to the fact that the exponents for SAWs on Sierpinski square lattices depend on both (n, k) and lacunarity, it is difficult to compare numerical results from different studies with each other, as in most cases the substrates differ. Using the bond-movingtype RG method to study SAWs on nine different Sierpinski square lattices in d = 2, Taguchi [49] determined the exponent vs, reporting values between 0.798 and 0.880. For five of these substrates, Reis and Riera [55] used a series expansion technique in d = 2, quoting considerable smaller values for vs, between 0.7 ± 0.5 to 0.82 ± 0.09. 5 They also determined the exponents 'Ys and J-Ls for SAWs on those substrates, reporting for 'Ys values between 1.26 ± 0.37 and 1.38 ± 0.07, as well as values between 2.183 and 2.619 for f-Ls· Apart from that, the MC study in Ref. [56] gives the estimate Vs = 0.7535 in d = 2 for
(n, k) = (3, 1). When performing numerical simulations of SAWs on the Sierpinski square lattice, an average over different starting sites for the SAWs is required to weaken the strong lattice effects typical of Sierpinski structures, similarly to the simulations for Sierpinski triangular 5 Note
that for SAWs on some Sierpinski square lattices in d = 2 characterised by low lacunarity, the values reported for vs are smaller than the one for the regular square lattice.
A. Ordemann, M. Porto and H. E. Roman
210 102 <: I
-6"'
:::..
-
1 1
06 I
0.4
.;t:. §::
0.2
10 1
.
••• --~..•---------~~
0
0.05
0.1
<:;!
1/N
~0~--~
F
-
- - - -€1- -B----rr ·l"l1Joo~
(a)
~ l--·-··-.--.J -j!?.. 0.2
0
005
01
-~
015
~
1/N
---~-~-~&~
--(b)
N
Figure 7. Total number of SAWs of N steps (CN,s) on Sierpinski square lattices, plotted identically to Fig. 3 (adapted from Ref. [14]). The plots correspond to: (a) d = 2, for J-L = J-Ls = 2.515 (circles), 2.415 (diamonds), and 2.615 (squares). The value 'Ys = 1.24 is obtained from the successive slopes (inset), shown as dashed line (note that a second analysis method discussed in the main text yields 'Ys = 1.22). (b) d = 3, f-L = J-Ls = 4.26 (circles), 4.16 (diamonds), and 4.36 (squares). The successive slopes (inset) yield 'Ys = 1.36, being represented by the dashed line.
lattice (see Appendix B). The results for (n, k) = (3, 1) in d = 2 and (n, k) = (3, 7) in d = 3 discussed in following, which provide an accurate and complete set of estimates for the above exponents in both d = 2 and d = 3, have been obtained using the EE technique (see Appendix A) of SAWs up to N = 30 in d = 2 and uptoN= 20 in d = 3 [14]. Analogously to the analysis in the previous section, two methods are applied to obtain J-Ls and 'Ys (see Fig. 7 for one method), yielding again consistent values, being summarized in Table 4. For SAWs on regular lattices and on the incipient percolation cluster (see Eq. (30) below) the coordination numbers /-Liatt and J-L(p) are related by [57] J-L(P)
= P /-Llatt,
(24)
where p ::2: Pc is the percolation occupation probability (note that Eq. (24) has some subtleties at p = Pc, see Subsection 5.3). One might argue that the validity of Eq. (24) has its roots in the fact that for SAWs on the incipient percolation cluster, as compared to SAWs on regular lattices, the reduced connectivity of the percolation cluster quantified by
Self-avoiding walks on deterministic and random fractals: Numerical results 1.2
§::
_ _., . . ···v 1
0.8
~
0.4 0
(l(N))
. 0.05
/
/
/
••
••
··a
(a)
/
e'/
~ ~
0.6
.
---~~ .......-·-·-
> 04
. 0
(l(N))
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.....
.....
........
/
0.1
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...
211
0.15
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/
....-"'
...
...........
(b) N
Figure 8. Mean end-to-end chemical distance (C(N)) for SAWs on Sierpinski square lattices as a function of the step length N in (a) d = 2 and (b) d = 3 (adapted from Ref. [14]). The dashed lines display the results obtained by successive slopes (inset), yielding Vs = 0.75 (d = 2) and 0.58 (d = 3).
the probability pis the dominant issue. It is interesting to note a possible relation between the coordination numbers Miatt and P,s for SAWs on regular and on Sierpinski square lattices, respectively. To show this, one might argue that it is not the connectivity of the underlying substrate which is important, but rather the spatial restrictions quantified by the fractal dimension ds of the Sierpinski square lattice, i.e. Ms
ds
= dMiatt·
(25)
This conjecture is obeyed quite well in d = 2 for a range of values of (n, k) (for instance taken from Ref. [55]) as well as in d = 3 for (n, k) = (3, 7) [14], suggesting that the above relation might hold generally for SAWs on Sierpinski square lattices. Eq. (25) holds in any case for the limit of regular square lattices obtained by taking simultaneously n --+ oo and k --+ 0, as then ds(n --+ oo, k --+ 0) --+ d and P,s(n --+ oo, k --+ 0) --+ J.liatt (when simultaneously n --+ oo and k --+ 0, in particular the symmetric case studied here corresponds asymptotically to regular lattices [58]). The important question whether Eq. (25) holds in identical or similar form also for SAWs on other deterministic fractals is not yet answered. Note that a relation similar to Eq. (25) does not hold for the Sierpinski triangular lattice in d = 2, presumably due to the bottlenecks in the structure rendering it
A. Ordemann, M. Porto and H. E. Roman
212
100
(a)
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-
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-
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~
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0.4
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Figure 9. The PDF (Ps(£, N)) of the end-to-end chemical distance £. for fixed number of steps N (circles for odd N, diamonds for even N) on the Sierpinski square lattice, for (a) d = 2 and N = 29, 30 and (b) d = 3 and N = 19, 20, plotted identically to Fig. 5 (adapted from Ref. [14]). The dashed lines in (a) and (b) represent fits of the data, in the regimes£. « N" 8 and£.» N" 8 , according to Eqs. (18) and (19). A more accurate determination of the exponent g~ is illustrated in the insets, plotted identically to Fig. 5, yielding g~ = 0.54 (d = 2) and 0.16 (d = 3).
very different from its regular lattice counterpart. Additionally, there is no regular lattice counterpart of the Sierpinski triangular lattice in d = 3. The analysis of (C(N)) is shown in Fig. 8. It should be noted that the numerical values obtained for vs can hardly be distinguished from the values known for the exponent v for SAWs on regular lattices. The analysis of (Ps(£1N)) is shown in Fig. 9. For the Sierpinski square lattices (as well as for other square lattices such as the regular one) one encounters the difficulty that for SAWs of odd/even lengths N only odd/even end-to-end distances£. can occur due to the lattice topology. Therefore, Fig. 9 shows the results for both N = 29 and N = 30 in d = 2 and both N = 19 and N = 20 in d = 3. Note that when comparing the estimates for the exponents from different studies, not only the numerical technique employed for the generation of SAW configurations matters but also the data analysis technique used. For example, the series expansion investigation of SAWs on a substrate with (n, k) = (3, 1) in d = 2 for N ::; 18 reported J-ls = 2.502±0.003
Self-avoiding walks on deterministic and random fractals: Numerical results
213
and Is= 1.35±0.05 [55], using a different method of data analysis than applied in Ref. [14]. When reanalysing the values of SAW configurations given in Ref. [55], Table IV, using the data analysis methods applied in Ref. [14] for consistency, it results in a slightly larger estimate of J-ls = 2.51 ± 0.02 and a somewhat smaller Is = 1.28 ± 0.05, which are however consistent with both the ones originally reported in Ref. [55] as well as those determined in Ref. [14].
5. SAWs ON RANDOM FRACTALS: PERCOLATION After having discussed the behavior of SAWs on deterministic fractals, we move on to the second major topic of this chapter, namely the numerical study of SAWs on random fractals, the latter modelled by percolation. As non-trivial changes in the exponents characterizing the structure of SAWs on the incipient percolation cluster (and as a consequence on its backbone) are only expected at criticality [5], i.e. for probability p of available sites being p = Pc, the following discussion is restricted to this case. A summary of exponents and fractal dimensions characterizing critical percolation is given in Table 1. For brevity, we will only consider the case of d = 2 and d = 3. First, the proposal of using the backbone of the incipient percolation cluster instead of the whole cluster as the (effective) substrate for the SAW is discussed. Further on, the absence of self-averaging for the minimum topological distance l!min for a certain Euclidean distance r is addressed. In this context, the advantages of studying the observables in the so-called £-space (see Section 2.2) will become more evident. When dealing with the number of possible SAW configurations on the backbone, the appearance of multifractality is outlined. Finally, it is shown that the numerical estimates for the exponents characterizing the PDFs of the end-to-end distances agree with a generalization of the des Cloizeaux relation for regular lattices, Eq. (6). Renormalization group (RG) techniques are applicable to SAWs on percolation and have
Table 1 Critical exponents and fractal dimensions of the incipient percolation cluster, for spatial dimensions d = 2, 3, and d ~ 6. Note that usually dr is called dr in the fractal literature, but for the sake of consistency with our notation we refer here to this quantity as dr. quantity d= 2 d= 3 d>6 dr 91/48 1 2.524 ± 0.008 2 43 4 5 dg 1.6767 ± 0.0006 1.84 ± 0.02 23 6 4 dmin 1.1306 ± 0.0003 1.374 ± 0.004 23 d~ 1.6432 ± 0.0008 7 1.87 ± 0.03 5 23 d~ 1.446 ± 0.001 8 1.36 ± 0.02 5 13 1 2 /3perc 5/36 0.417 ± 0.003 19 1 2 Vperc 4/3 0.875 ± 0.008 1/2 9 1
6
2
7 Monte
Exact results, Ref. [59]. Monte Carlo simulations, Ref. [60]. 3 Refs. [61,62]. 4 Monte Carlo simulations, Ref. [63]. 5 Monte Carlo simulations, Ref. [64].
Monte Carlo simulations, Ref. [65]. Carlo simulations, Ref. [66] 8 Relation d~ = d~ /dmin· 9 Exact result, Ref. [67].
214
A. Ordemann, M. Porto and H. E. Roman
been applied already quite early by Meir and Harris [68], although it remained unclear why their results for the exponents were not in agreement with those obtained from numerical studies. However, von Ferber et al. very recently reconsidered the issue [69] and argue that their RG result gives an accurate description in the whole range of dimensions 2 ::; d ::; 6, in accordance with known MC and EE numerical data (see also Chapter 4 by Blavatska et al. ). As the recent EE technique study of SAWs on the backbone of the incipient percolation cluster in Ref. [39] provides an accurate and complete set of estimates for the above exponents in both d = 2 and d = 3 in £-space and r-space, we discuss these results in detail in the following Subsections. Results from other numerical studies are compared when applicable.
5.1. Backbone For numerical studies of SAWs on percolation, especially when using the EE technique (see Appendix A), it is advisable to use the backbone instead of the whole percolation cluster as the substrate for the SAWs (see Fig. 10 for an example of a percolation cluster's backbone). This is due to the fact that there is convincing evidence that the asymptotic behavior of SAWs is strongly dominated by the SAW configurations which are restricted to the backbone, rendering the dead ends in the percolation structure irrelevant. Solely considering the backbone has furthermore the advantage that the substrate is 'thinner', i.e. the fractal dimension d~ is smaller, d~ < dg. Therefore, the numerical simulations using the EE technique become computationally more efficient since SAWs with a larger maximum chain length N can be considered. For MC methods, it can be computationally either more or less efficient to consider solely the backbone, depending on the actual algorithm. The reduction of the critical percolation cluster to its backbone can be done
••••
•• • r.:AJ • • • II •r.=: :;,
••
tsJ I ::J•
••r.:..::-::J • L.:=:J
• • II II
•
Figure 10. Example of a percolation cluster on the square lattice (full squares) and its corresponding backbone between the sites S and A (S being the 'seed' of the Leath growth algorithm [26] which has generated the cluster, and A being a randomly chosen site on the last grown shell; adapted from Ref. [39]).
Self-avoiding walks on deterministic and random fractals: Numerical results
215
using the so-called 'burning algorithm' [64,70]. 5.2. Absence of self-averaging In contrast to regular lattices and (most) deterministic fractals, for which £- and rspaces are essentially identical, there exists a non-trivial scaling relation between both metrics for percolation, i.e. on average£ rv rdm;n (where the symbol indicating the average has been omitted for notation simplicity), with dmin > 1 [19,20] (see Table 1). Despite of this somewhat odd behavior, the topological distance £ has the important advantage that numerical data is found to show less fluctuations when measured as a function of distance £rather than distance r (which is also related to the absence of self-averaging, see below). Furthermore, many efficient algorithms (such as the Leath growth algorithm [26]) for generating a percolation cluster at criticality are such that the cluster is guaranteed, after each growth step, to be 'complete' in £- yet not in r-space. Therefore, more accurate estimates for critical exponents in r-space can be obtained by studying the corresponding quantities in £-space and transforming them back to r-space. As a consequence of this non-trivial scaling, the minimum topological distance £min between two sites at Euclidean distance r observed among Nconf cluster configurations is not only a function of r, but does depend on Nconf as well [71]. Obviously, the shortest possible minimum topological distance Emin(r, Nconr) is r. Among a finite number Nconf of cluster configurations, a configuration which indeed has Emin(r, Nconr) = r exists, on average, at least once if r is smaller than a critical distance rc(Nconr) [71], Tc (
N
) _ ln /-llatt + ln Nconf conf ln 1/Pc ,
(26)
where J-liatt denotes the connectivity of the underlying lattice. For r > rc(Nconf ), the shortest minimum topological distance found among Nconf cluster configurations is proportional to rdm;n, which together leads to £min(r, Nconf)
= {rCt'min r d
. ffilU
for r < rc(Nconf) for r > rc(Nconr) '
(27)
with a prefactor O:min = [rc(Nconr)]dm;n- 1 • This dependence of the shortest mmtmum topological distance on the number of cluster configurations Nconf considered has severe consequences on all quantities measured in r-space. These observables become dependent on Nconf and therefore may not be self-averaging [22,71,72]. This becomes clear when looking at the transformation from £- to r-space, co
f(r)
=
j £dt-l B(£1r) f(£) d£
(28)
£min ( T ,Nconf)
(with B( £1r) d£ being the probability of observing a topological distance between£ and£+ 00 d£ for two sites on the backbone at Euclidean distance r, normalized as J0 £dt-l B(£1r) d£ = 1 [64]). Here, the lower integration limit depends on Emin(r, Nconr) and hence on Nconf· As a consequence, one has to be careful that, for a given Euclidean distance r, one averages over a sufficient number of cluster configurations (which increases exponentially with r) to ensure a convergent result [39].
A. Ordemann, M. Porto and H. E. Roman
216
5.3. Number of SAW configurations and multifractality Due to the disordered structure of the incipient percolation cluster, the total number CN,B of SAW configurations that exists on a single backbone originating from a given site fluctuates strongly among different backbone configurations. As in general for such multiplicative random processes, a multifractal behavior is expected. To establish this, the fluctuations have to be characterized by studying the different moments q of the total 11 number of SAW configurations, (C'Jv,B) q, averaged over many backbone configurations. One denotes a quantity multifractal if the scaling with q is non-trivial (meaning non-linear in q, see below). The fact that CN,B displays a multifractal behavior has been shown by numerical studies [39] and is also indicated by very recent RG results [69]. Note that a related study of 'ideal' chains (i.e. chains which can intersect themselves) on percolation at criticality has shown non-trivial dependences on q as well [73]. The presence of multifractality means inherently the existence of rare events playing a dominant role in any averaging procedure. Although reweighting techniques have been applied in MC methods, these results inevitably yield typical values (corresponding to the q = 0 moment or 'annealed' average, 6 see Section 3), which will differ from the true mean values (corresponding to the q = 1 moment or 'quenched' average) determined by averaging within the EE technique. This problem affects both non-universal quantities such as p, and universal quantities such as I· It can be understood by noting that EE calculations yield by definition the whole ensemble (quenched average in the present notation), whereas MC simulations intrinsically sample only a small subset of all possible configurations, omitting rare configurations (annealed average). As a consequence, which is quite often overlooked in literature, it is not possible to directly compare averages from MC and EE. However, a meaningful comparison is possible as EE results can be (and should be!) analyzed with respect to the different moments q, so that one can compare the MC data with the q = 0 moment data of EE. Nevertheless, it is important to note that EE data can suffer from considering an insufficient number of substrates configurations, so that a detailed analysis of numerical data is necessary to confirm that the set of substrate configurations is large enough (see Appendix D). For the moment q of the number of SAW configurations on the backbone of the incipient percolation cluster, the scaling ICq \
N,B
)1/q = D
nN N"~q-1 q rq
(29)
holds, where P,q are the generalized effective coordination numbers of the backbone and /q the generalized enhancement exponents. Results for different values of q are shown in Figs. ll(a) and ll(b) for square and simple cubic lattice, respectively [39]. The values of P,q and /q are displayed in Fig. 12 for d = 2, clearly displaying a non-linear dependence on q, reminiscent of a multifractal behavior. For large negative values of q, backbone substrates with a small CN,B are strongly emphasized in the averaging procedure. Such small number of SAW configurations are observed on (rare) backbone substrates with an almost linear shape, so that /-lq ---+ 1 and /q ---+ 1 for q ---+ -oo. On the contrary, for large values of q the averaging procedure singles out such backbone substrates which can host 6 Note that the q = 0 moment of A corresponds to a logarithmic average of A, i.e. limq--+O (Aq)lfq = exp (log A).
Self-avoiding walks on deterministic and random fractals: Numerical results
217
0.8 (a) 0.6 0.4 ..-----. ~
::::-
---;:p ~;;.,;
2-
~
..::
0.2 ~
0.5 0.4 0.3 5
10
15
20
25
30
35
40
N
Figure 11. Generalized moments ( C'Jv,B) of the total number C N,B of SAW configurations
11
on the backbone of the critical percolation cluster, plotted as N- 1ln [( C'Jv,B) q] versus N (adapted from Ref. [74]). (a) d = 2 for q = 2, 1, 0.5, 0, -0.5, -1 and -2 (from top to bottom); and (b) d = 3 for q = 1 (top) and 0 (bottom). The continuous lines are the best fits based on Eq. (29). Some representative values for /q are /o = 1.456 ± 0.005 and 1 1 = 1.565 ± 0.005 (d = 2) and /o = 1.317 ± 0.005 and 1 1 = 1.462 ± 0.005 (d = 3).
a large number of SAW configurations. Since these backbone substrates are the most compact ones, /-lq and /q are strongly increased with respect to q = 1. As already noted above, the analysis of the different moments of EE data is able to resolve earlier controversies regarding the values for both p, and 1 obtained from MC and EE simulations of SAWs on percolation. For the square lattice, for example, the values p,(EE) = 1.53 ± 0.05 [76] and 1(EE) = 1.33 ± 0.02 [77] have been obtained from EE calculations, while from MC simulations the values J.t(MC) = 1.459 ± 0.003 and 1(MC) = 1.31 ± 0.03 have been determined [57]. Calculating the moments q from EE data, one gets p, 1 = 1.565±0.005, 1 1 = 1.34±0.05 and p, 0 = 1.456±0.005, /o = 1.26±0.05, corresponding to the previous EE and MC results, respectively (see Fig. 11). Resolving the discrepancy concerning the value(s) of p, has allowed to confirm an old prediction by Woo and Lee [57], which was originally suggested in the form p, = Pc /-liatt, where /-liatt is the effective coordination number of SAWs on the underlying regular lattice. This relation could not be confirmed because of the conflicting (MC and EE) values
A. Ordemann, M. Porto and H. E. Roman
218
2.5 2.0 1.5 1.0
,'"[ ' ~ l..··· 1.4
1.2
-~
-2
0
0.0 • -10
~
...
.
0.5 Yq-1
"--+
/
2
••••• •••
• -5
0
•••
l!
!
I I
!
y-1---+
10
5
q
Figure 12. The effective coordination numbers P,q and enhancement exponents lq versus q for -10 ::; q ::; 10 of SAW configurations on the backbone of the critical percolation cluster in d = 2 (adapted from Ref. [74]). The values for p, and 1 on regular square lattice are marked by arrows, clearly showing that limq-+oo lq is larger than 1 on regular square lattice. The inset shows /-lq versus q for -2 :S q :S 2 in d = 2, in good agreement with the theoretical result P,q = p, 0 (1 + qCJ5/2) (continuous line), suggested for lql ---+ 0, with J-to = 1.456 and CJo = 0.45 [75].
obtained for p,, yet the relation /-ll
= Pc /-l!att
(30)
is very accurately fulfilled (so that the /-lq which is important here is the one for q = 1), with Pc = 0.5927460 for the square [78] and Pc = 0.311605 for the simple cubic lattice [63] (for a relation between the coordination numbers for SAWs on regular lattices and Sierpinski square lattices see Eq. (25)). 5.4. Scaling relations and probability distribution functions Here, we discuss the scaling behavior of the distribution functions of the end-to-end distances, analogously to Section 4. For SAWs on backbone substrates, the results are presented both in £- and r-space to allow for a direct comparison. The quantities are hence (PB(£,N)) and (PB(r,N)), averaged over many backbone configurations, where PB(£, N) d£ is the probability that after N steps the topological end-to-end distance of a SAW on a single backbone is between£. and £+d£, and PB(r, N) dr is the respective quantity in r-space. These distribution functions obey scaling forms similar to the one valid on regular lattices, Eq. (3), and deterministic fractals, Eq. (18), with the corresponding scaling exponents [22]. Explicitly, one has
(31) with the scaling function for x for x
« »
1 1 '
(32)
Self-avoiding walks on deterministic and random fractals: Numerical results 10 1
~
(a)
-·- -·
10-2
~
s
.... ~
-- ..- ~· .:--
~
219
~ I
10-5
""
10-8 100
10-1
f/Nvl
10 1 (b)
····~\\
A_--t-•--~----
10-2
~ -S
s...
••
••• t
10-5
t
10-8 100
10-1
r/Nv,
Figure 13. (a) The PDF (PB(£,N)) of the end-to-end chemical distance£ and (b) the PDF ( PB (r, N)) of the end-to-end Euclidean distance r, for SAWs on the percolation backbone of fixed number of steps N = 39 (diamonds) and N = 40 (circles) in d = 2 (adapted from Ref. [39]). The dashed and full lines represent fits of the data, in the regimes£« N"t,r and£» N"t,r, according to Eqs. (31),(33) and Eqs. (4),(14).
and (PB(r,N))
= ~FM;,_,J,
(33)
with for x « 1 for x » 1
(34)
Both distribution functions are normalized according to J0 ( PB (£, N)) d£ = 1 and J000 ( PB (r, N)) dr = 1. The exponents df and d~ are the mass fractal dimensions of the backbone and are reported in Table 1. The exponents V£ and Vr are related to each other by Vr = v£/dmin, Eq. (12), whereas one has gr = gf dmin [22], Eq. (15). It is accepted that relations directly analogous to the Fisher relation [16], Eq. (5), hold, 00
lir
1
=1- Vr
and
lig
1
=1- Vg'
(35)
A. Ordemann, M. Porto and H. E. Roman
220
Vt
(£(N))
0 . 9 0 E •J . 0.85 0.00
0.04 1/N
•
•
0.08
• • •
N
Figure 14. The mean chemical end-to-end distance (£(N)) versus N for SAWs on the percolation backbone in d = 3 (adapted from Ref. [39]). The value of V£ is obtained by successive slopes (inset), yielding V£ = 0.910 ± 0.005.
whereas such a direct analogous does neither exist of the des Cloizeaux relation [17], Eq. (6), nor of the McKennzie-Moore relation [18], Eq. (7) (however, concerning a 'generalized des Cloizeaux relation' see the next Subsection). The mean topological end-to-end distance (£(N)) and the root mean square Euclidean end-to-end distance (r(N)) (derived from (PB(£,N)) and (PB(r,N))) scale with N as
(36) and
(37) respectively. The first average --: is performed over all SAW configurations on a single backbone, and the second average ( · ) is afterwards carried out over many backbone configurations. The numerical results for the distribution functions in d = 2 are shown in Fig. 13 in both£- and r-space (note the much smoother behavior of the PDF in£- than in r-space). The exponents gf and gr have been estimated directly from the slope of f~ and fi3 in the double logarithmic plots. Since gf and gr are related by gr = gf dmin, Eq. (15), a more precise estimate for gr can be derived from the estimate for gf. The determination of g~ and g; is more difficult, since both exponents occur in the non-dominant part and are masked by the exponential. Therefore, it is necessary to use a slightly more involved numerical procedure discussed in Appendix C. The numerical results obtained in Ref. [39] for gr and g; are reported in Table 4. The numerical results for v"' which have been obtained using the successive slopes technique to determine V£ and calculating Vr using Eq. (12), are displayed in Table 4. As an example, Fig. 14 shows the determination of V£ in d = 3. Results for vr from other numerical studies are presented in Table 2. The difference in the results can in
Self-avoiding walks on deterministic and random fractals: Numerical results
221
Table 2 Overview of the critical exponent Vr for SAWs on various percolation structures as substrate (vreg = 3/4 is the value for regular lattices in d = 2). technique d=2 d=3 1 MC R::J Vreg 0.612 ± 0.010 1 2 MC R::J Vreg 0.605 ± 0.010 2 MC 0.77 ± 0.01 3 MC 0.783 ± 0.003 4 0.67 ± 0.045 EE 0. 76 ± 0.085 6 EE 0.81 ± 0.03 EE 0.78±0.01 7 0.65 ± 0.01 7 8 EE 0.745 ± 0.020 0.640 ± 0.015 8 9 EE 0. 770 ± 0.020 • EE 0.770 ± 0.005 10 0.660 ± 0.005 10 EE 0.787 ± O.QlOn 0.662 ± 0.006n 1
Incipient percolation cluster [79] (using Hoshen-Kopelman algorithm [27]). Incipient percolation cluster [31] (using Hoshen-Kopelman algorithm [27]). 3 Incipient percolation backbone [57] (using Hoshen-Kopelman algorithm [27]). 4 Critical percolation cluster (using Leath algorithm [26]) with incomplete enumeration [10]. 5 Critical percolation cluster [68]. 6 Incipient percolation cluster [76] (see also subsequent Comment [77]). 7 Incipient percolation cluster [9]. 8 All critical percolation cluster [80] (using Hoshen-Kopelman algorithm [27]). 9 Incipient percolation cluster [80] (using Hoshen-Kopelman algorithm [27]). 10 Incipient percolation cluster [81] (using Hoshen-Kopelman algorithm [27]). 11 Incipient percolation backbone [74] (using Leath [26] and 'burning' algorithm [64]). 2
most cases be explained by looking at the details of how a certain numerical method is implemented and what kind of substrate is used, e.g. all (finite and infinite) critical percolation clusters, only the incipient percolation cluster, the backbone of the incipient percolation cluster, etc. Further important issues in the different numerical procedures are the choice of starting sites, a sufficient and unbiased ensemble of SAW configurations, a proper account of disordered substrate, etc. (concerning the latter see Appendix D). A comprehensive discriminating discussion of these differences can be found in Ref. [5].
5.5. 'Generalized des Cloizeaux' relation According to Eqs. (33) and (34)
(PB(r, N))
1( r ) r N"r
g) +d~
rv-
for r
«
N"r
(38)
yields the probability density that both ends of the SAW get close to each other, and the question arises what is the value of the exponent in the case of percolation. Numerical does not obey a des Cloizeaux relation of the form, = (1 1 results indicate that 1)/ Vr [74]. The presence of the subindex 1 for ')'1 is expected since by definition (PB (r, N) ), obtained by averaging over all possible SAW configurations, corresponds to the q = 1 moment in the multifractallanguage.
gr
gr
gr
222
A. Ordemann, M. Porto and H. E. Roman
The discrepancy between the numerical results and the above des Cloizeaux relation has been resolved by a careful consideration of the scale-invariant effects of disorder associated to critical percolation clusters [74]. In the following, we present a brief derivation of the so-called 'generalized des Cloizeaux' relation which is able to explain the numerical results satisfactorily. The way to attack the problem has its origin in a simple yet appealing idea of de Gennes [1], in which the probability density P(r =a, N) that (on a regular lattice) the end-to-end distance r of the SAW is equal to the lattice constant a can be related to the corresponding number of SAW configurations CN(a), divided by the total number of configurations, CN. In the case of percolation, one has (in an obvious notation)
(39) Now, for percolation one assumes for (CN,B(a
= 1)) the more general scaling form [74] (40)
where p, 1 represents the (q = 1-multifractal moment of the) effective coordination number of the substrate, and the factor HB contains the above mentioned effects of the selfsimilar disordered substrate. On regular d-dimensionallattices, the fractal dimension d~ in Eq. (40) is replaced by d. To determine the scaling behavior of HB, one resorts to the probability P00 that an arbitrary site belongs to the infinite percolation cluster. For concentrations p above the percolation threshold Pc, the latter reads
(41) where ~ is the correlation length and Vperc and /3perc are standard critical percolation exponents (see Table 1). If one assumes that the mean size R of the SAW is R < ~' then the probability that both ends of the SAW are located on the 'infinite cluster' is given by HB rv R-f!pmfvpecc. This yields, (42)
from which, by comparison with Eq. (38), the generalized des Cloizeaux relation follows, (43) This generalized des Cloizeaux relation is in very good agreement with numerically obtained values [7 4]. The second term in Eq. (43) has its origin in the (self-similar) disordered nature of the backbone of critical percolation clusters and is expected to be absent on deterministic fractals such as the Sierpinski lattice. EE results support this conclusion as shown in Section 4 for both triangular and square Sierpinski lattices.
Self-avoiding walks on deterministic and random fractals: Numerical results
223
6. DISCUSSION 6.1. Methods: EE vs MC In this chapter, we have presented the two major classes of numerical methods used to study SAWs on deterministic and random fractals, namely Monte Carlo (MC) methods and the exact enumeration (EE) technique. Both classes have their advantages and disadvantages, so that the question which is the 'best' method strongly depends on the substrate as well as on the question at hand. If one is interested in the case of deterministic fractals, all exponents v, g1 , g 2 , r5, 1, and p, can be determined reasonably well by means of both MC and EE. This is mainly due to the fact that SAWs on most deterministic fractals do not show multifractal behavior. Here, the advantages (long chains) and disadvantages (bias towards typical configurations) of MC on the one hand, and the advantages (no bias towards typical configurations) and disadvantages (only short chains) of EE on the other hand roughly counterbalance each other. An important point concerning the determination of 1 and p, is that the number of SAW configurations are absolute in EE, but they can only be determined as ratios in MC. For data analysis, one has to take into account the systematic fluctuations caused by the structure of the substrate, for instance odd/even fluctuations for SAWs on regular lattices and percolation, which affect ratios stronger than absolute numbers. Hence, when doing MC simulations, one has to study the ratio CN/CN- 2 rather than CN/CN-l· This becomes even worse for Sierpinski lattices, for which a whole hierarchy of fluctuations corresponding to the 'magic' distances in the structure are observed. In the case of random fractals, where percolation being the most prominent example, one encounters the problem that some exponents might show a multifractal scaling. Presently, it is accepted that v, g1 , g 2 , and r5 are not affected by the presence of multifractality and that they can be determined by both MC and EE. Unlike the latter exponents, it has been observed that those characterizing the configurational space, such as 1 and p,, do depend on the moment q considered. As a consequence, using MC methods which are only able to deliver the q = 0 moment are disadvantageous. This is due to the fact that the presence of multifractality and of rare configurations inevitably occur together. These rare configurations are inherently missed in MC methods, so that only typical values rather than mean ones are determined. Consequently, only EE is able to disentangle the different contributions determining the averages. One might loosely say that from EE data one can get (short chain) MC data, but not vice versa. 6.2. Results: Deterministic vs random fractals Here, we have discussed SAWs on three different substrates, comparing them with the results obtained for SAWs on regular lattices. The first two substrates, Sierpinski triangular and square lattices, are representative members of the class of deterministic fractals, whereas percolation is the standard model for random fractals. Deterministic fractals consist of two distinct subgroups, which are the finitely (Sierpinski triangular) and infinitely (Sierpinski square) ramified fractals. As we have reported above, the behavior of SAWs is drastically affected by the underlying substrate. Nevertheless, certain similarities emerge: SAWs on Sierpinski triangular and square lattices display a kind of intermediate behavior between SAWs on the corresponding regular lattices and on percolation. Here, the spatial characteristics of SAWs on Sierpinski square lattices are closer to the ones observed for
A. Ordemann, M. Porto and H. E. Roman
224
Table 3 Dimension dependence for v, g1 , g 2 , and p,, where .j,. indicates a decrease of the given exponent from d = 2 to d = 3 (presumably persisting for higher dimensions), whereas t indicates the opposite trend. Note that p, is not universal. quantity regular sq Sierpinski sq Sierpinski tr percolation sq v .!.!.!.j,. gl .!.!t t g2 .!.!t t I 1-l
.!t
t t
t t
.!.!-
regular lattices, while SAWs on Sierpinski triangular lattices are more similar to their counterparts on percolation. An indication for this pattern are the exponents characterizing the structure, v, g1 , and g 2 , and their trend with increasing spatial dimensionality, see Table 3. Note that such patterns of different behaviors, depending on the underlying structure, is found for various other models as well (see, for instance, Ref. [82] for a study of Ising models on various substrates). {i) Structural exponents: v, g1 , and g2 . The exponents describing the structure of SAWs on Sierpinski square lattices behave like their regular counterparts, i.e. vs(sq) and v are merely identical for both d = 2 and d = 3, and g~(sq) and g1 as well as g~(sq) and g 2 show the same trend (decrease) with increasing spatial dimension. Furthermore, in both cases the same 'ordering' of the exponents occurs: g1 < g2 and g~(sq) < g~(sq) for d = 2, but g1 > g 2 and g~(sq) > g~(sq) for d = 3. This has to be distinguished from the behavior of the corresponding exponents for SAWs on Sierpinski triangular lattices and on the incipient percolation cluster. Here, Vr and vs(tr) are both larger than v. This indicates that the fractal nature of the underlying lattice is strong enough to affect the
Table 4 Overview of the critical exponents and other quantities for SAWs on various substrates. The column 'regular sq/tr' shows the values for regular square/triangular and simple cubic lattices (note that the value for p, is non-universal and depends on the lattice type), 'Sierpinski sq' shows the values for the Sierpinski square lattices, 'Sierpinski tr' shows the values for Sierpinski triangular lattices, whereas 'percolation sq' shows the values for the (square lattice) incipient percolation cluster. Note that no sub-/superscripts are given in the leftmost column, so that for example the row 'v' shows v for regular lattices, Vs for Sierpinski square lattices, v~ for Sierpinski triangular lattices, and Vr for the incipient percolation cluster (for percolation, the values of p, and 1 refer to the particular exactenumeration values p, 1 and 1 1 [39], whereas v, g1 , and g 2 refer to the exponents in Euclidean metric, v"' gr, and g;, respectively). Wherever exact values are known, approximate numerical values are given in square brackets to ease comparison. For the fractal lattices shown in the three rightmost columns, available analytical estimates are given in round brackets to allow an estimation of the accuracy of the numerical values.
~
o"
(;)
}'(')
"'
'0
c+
::;· t:! 0 t:!
'0
'"'
('!)
< ::;·
~
w '0
"'
oq
~
quantity 2D dr 3D 2D Vr 3D 2D g]_ 3D 2D g; 3D 2D ljr 3D 2D sq 2D tr J-l 3D sq 3D tr 2D I 3D
regular sqjtr 2 3 3/45 [= 0.75] 0.58758 ± 0.000076 11/24 12 [~ 0.458] 0.268 12 15 5/8 [= 0.625] 0.255 15 416 16 12/5 [= 2.4] 2.63815852927(1) 18 4.15096 ± 0.00036 19 4.68404 ± 0.00009 20
Sierpinski sq ln 8/ ln 3 1 [~ 1.893] ln20/ln3 1 [~ 2.727] 0.75 ± 0.05 1 0.58 ± 0.03 7 0.54 ± 0.03 7 0.16 ± 0.05 7 1.41 ± 0.08 1 0.10 ± 0.05 7 3.73 ± 0.30 1 (4 17 ) 2.65 ± 0.50 7 (2.38 17 ) 2.515 ± 0.015 7 (2.499 21 )
43/32 24 [= 1.34375] 1.1575 ± 0.0006 25
1.23 ± 0.04 1 1.36 ± 0.03 7
by d~ = ln(nd- k)flnn. 2 0btained by d~' = ln(d + 1)/ln 2. 3 Refs. [59,83]. 4 Ref. [60]. 5 Ref. [11]. 6 Ref. [15]. 7 Ref. [14]. 8 Ref. [50]. 9 Refs. [40-43]. 1 0btained
4.26 ± 0.02 7 (4.258 21 )
10
Sierpinski tr ln 3/ ln 22 [~ 1.585] 22 0.78 ± 0.03 8 (0.7986 9) 0.66 ± 0.04 8 (0.674 10 ) 0.44 ± 0.05 8 (0.47 13 ) 0.65 ± 0.08 8 (0.662 13 ) 2.34 ± 2.6 ± 5.1 ± 0.2 8 3.0 ± 0.3 8
0.10 8 0.4 8 (4.965 11 ) (3.068 17)
percolation sq 91/48 3 [~ 1.896] 2.524 ± 0.008 4 0.787 ± O.OlOu 0.662 ± 0.006ll 0.55 ± 0.06ll (0.54 14 ) 0.92 ± 0.08ll (0.916 14 ) 1.56 ± 0.20ll 2.6 ± 0.2ll 4.85 ± 0.20ll (4.695 17 ) 3.1 ± 0.2 11 (2.960 17 ) 1.565 ± 0.005ll
2.29 ± O.Ql 8 (2.28803 22 ) 3.82 ± 0.02 8 (3.815 23 ) 1.36 ± 0.03 8 (1.3752 9) 1.42 ± 0.048 (1.4461 26 )
Refs. [44--46]. 11 Ref. [39] 12 Ref. [17] and Eq. (6). 13 Ref. [50] and Eq. (22). 14 Ref. [74] and Eq. (43). 15 Ref. [18]. 16 Ref. [16]. 17 0btained by Eq. (21) and its analogs. 18 Refs. [84,85].
19
U)
......
(1)
"i'>
~
~.
& t:!
o-q
~
~ 0 t:! Q..
~
(1)
8t:;· ~·
r;·
~
1.462 ± 0.005ll
t:! Q..
1.34 ± 0.05ll 1.29 ± 0.05ll
t:! Q.. 0
....
Ref. [86]. 20 Ref. [87]. 21 0btained by Eq. (25). 22 Refs. [41-43]. 23 Ref. [41]. 24 Ref. [59]. 25 Ref. [88]. 26 Ref. [40].
~
t:l
~ ~
g_
~ 2: t:l
>=:: (1)
::!. (')
e:..
....
(g
E.
&j
226
A. Ordemann, M. Porto and H. E. Roman
spatial structure of the SAWs such that the scaling behavior changes. Correspondingly, g~(tr) and g'i_ as well as g~(tr) and g?j_ show the same trend (increase) with increasing spatial dimension. Additionally, the same 'ordering' of the exponents occurs: g]_ < g?j_ and g~(tr) < g~(tr) for both d = 2 and d = 3. {ii) Configuration space: 1 and p,. The dimensional dependence of the enhancement exponent 1 is governed by the existence or absence of an upper critical dimension de. One might argue that, for finite critical dimension (de = 4 for SAWs on regular lattices and de = 6 for SAWs on percolation), the value of 1 should decrease with increasing spatial dimension, reaching the (mean-field) value of 1 at de. It is important to note that these two mean-field behaviors occur for different reasons. For regular lattices, 1 = 1 for d = de = 4, as the effect of self-avoidance of SAWs becomes negligible at the upper critical dimension and SAWs behave like standard random walks, for which 1 = 1. For the incipient percolation cluster, the underlying substrate governing the behavior of the SAWs, which is the percolation backbone, gets less and less compact with increasing spatial dimension, and the topological dimension df of the backbone reaches the value df = 1 at the upper critical dimension de = 6, being characteristic of a linear structure. This linear character of the underlying structure makes the self-avoidance of SAWs irrelevant, so that also here 1 1 = 1. In contrast, an exponent 1 increasing with increasing spatial dimension, as for SAWs on Sierpinski triangular and square lattices, might indicate the absence of, or an infinite critical dimension. Note that in Ref. [89] it was predicted, for SAWs on Sierpinski triangular lattices, that ls(tr) ---+ 1.618 for d ---+ oo, which seems to point towards the existence of an upper critical dimension de (although it might still be that de= oo). The behavior of the coordination number p, with spatial dimension is simpler, since it follows the trend of the substrate's one. The value of p, increases with spatial dimension for regular lattices and Sierpinski triangular and square lattices, while p, decreases with spatial dimension for percolation, the latter in accordance with the fact that the backbone becomes eventually linear. {iii) Des Cloizeaux relations. For SAWs on regular lattices, a relation between the structural exponents v and g 1 , and the configurational exponent 1, Eq. (6), has been derived by des Cloizeaux [17]. As far as deterministic fractals are concerned, it has been found that the equivalent relation is well obeyed by SAWs on Sierpinski triangular lattices, Eq. (22), whereas an analogous relation for SAWs on Sierpinski square lattices does not hold. The latter might be due to the intermediate or competing behavior of SAWs on Sierpinski square lattices: One obtains for the naive generalisation [ls(sq) - 1]/vs(sq) the values 0.32 and 0.62 in d = 2 and d = 3, respectively, which increase with spatial dimension d. These values have to be compared to g~(sq) = 0.54 and g~(sq) = 0.16 in d = 2 and d = 3, respectively, which decrease with d. It is presently not clear why this competing behavior occurs, and whether and how the des Cloizeaux relation, Eq. (6), can be generalised to SAWs on Sierpinski square lattices. On the incipient percolation cluster, being a paradigm example of random fractals, SAWs do not obey the standard des Cloizeaux relation, in the form of Eq. (6). Rather, numerical results are consistent with the generalized relation, Eq. (43) [74]. The latter is based on two main features, one is the underlying multifractal nature of SAWs on such random fractals, reflected by the presence of the first-moment 1 1 , and the second the effects of the structural disorder on the probability that the end-to-end SAW distance
Self-avoiding walks on deterministic and random fractals: Numerical results
227
vanishes, responsible for an additional term in the des Cloizeaux relation. As we have learned from the present review, the interplay between the self-avoiding condition of a linear chain and the self-similar nature of the substrate on which the chain is located displays a very rich behavior. The resulting 'compound' random process attains its maximum complexity in the ca.c;e of a random substrate modelled by percolation, responsible for the occurrence of multifractality in the statistical behavior of SAWs on such disordered structures. It is important to note that some a..c;pects of possible multifractality of SAWs on percolation cluster have not been studied yet. Already in the much simpler system of random walks on percolation cluster, that is walks that can intersect themselves [90], these quantities show a rich multifractal behavior. The extension of such a study to SAWs on percolation is only at a preliminar stage, and further numerical investigations are required before general conclusions can be drawn.
APPENDIX A. Exact enumeration technique The exact enumeration (EE) technique allows to enumerate and evaluate all SAW configurations on a given substrate. There exist several reali;.:ations of the EE technique which differ in details, the method described here is the one by Grassberger /91]. The central idea of EE is to generate all SAW chains of length N - 1 from all next neighbors of the starting site by, for each next neighbor, (i) blocking the neighbor site under consideration, (ii) generating all chains of length N - 2 from all its unblocked next neighbors, and (iii) unblocking the neighbor site under consideration (sec Fig. 15 for a simple example of EE on a small percolation cluster). This proc:edurc is usually performed recursively.
(a)
(b)
(c)
(d)
(c)
(f)
(g)
(h)
(i)
(i)
Figure 15. Example of exact enumeration of SAVv's in a constrained geometry. The starting site is marked by a framed small black square, the temporarily blocked sites by a black square, and the remaining freely available sites by a gray square. The shown order of evaluating nearest neighbors is up, right, down, and left.
A. Ordcmann, lvf. Porto and II. E. Roman
228
A central site of the substrate is chosen as starting site. However, it is important to ensure that the SAW chains of length N never reach the border of the substrate to avoid spurios boundary effects. For a percolation cluster grown by the Leath algorithm (26J to a sufficient size (at least N grown shells), one usually choosf>~'> the seed oft.he Leath growth as starting site of the EE. The choice of the starting site is somewhat more delicate in the case of substrates for which its sites are not equivalent, such as for the Sierpinski lattices. In these cases, it is advisable to carry out EE for several distinguished starting sites (see the next Subsection) and perform an average of the quamities of interest.
B. Averaging over several starting sites In case one applif>.s the EE technique on substrates for which the sites arc not equivalent, such as for Sierpinski lattices, an average over different distinct starting sites is required. These starting sites should be chosen such that the set contains as many different 'classes' of sites as possible, in numbers which reflect their relative occurence in the structure. To obtain the results shown in Section 4, for example for the Sierpinski triangular lattice in d = 2, an average was performed over all 15 sites of the Sierpinski triangular lattice after the second iteration (sec Fig. 16{a)). Ford= 3, alllO sites of the Sierpinski sponge after the first iteration were used. To ensure that a SAW chain of a certain length N originating from the starting sites is not. able to reach the border of the substrate, it is advisable to generate the whole Sierpinski structure by expanding the lattice around the starting sites in a circular fashion. For the Sierpinski triangular lattice in d = 2, the first steps of this expansion are shown in Fig. 16(b)-(d), where the last structure is capable of hosting SAW chains, originating from the starting sites, up to a length of N = 4 (i.e. 5 monomers). Accordingly, for the Sierpinski square lattiec, an average over 16 starting sitr.s in d = 2 and 64 starting sites in d = 3 was performed. These starting sites corrf>.spond to the substrate sites of the initiator of the fractal (i.e. of the zeroth iteration step) in both cases.
(a)
(b)
(c)
(d)
Figure 16. Sketch of the scheme how to select a proper set of starting points for the EE in d = 2. (a) Sierpinski triangular lattice after the second iteration, all black circles are starting points. Around this gray shaded triangle the Sicrpinski lattice is expanded in a clockwise fashion, (b) to the upper right, (c) downwards, and (d) to the upper left.
Self-avoiding walks on deterministic and random fractals: Numerical results
229
C. Improved Wittkop procedure A numerical procedure which allows for a more accurate determination of the exponents describing the scaling form of the PDFs has been proposed by Wittkop et al. [32] (cf. Ref. [92]). It is done by separating the power-law from the exponential regime in the PDF. This procedure has been improved to account for the two distinct power-law scaling regimes [39]. In the following, the scheme is illustrated for the case ofregular lattices (i.e. dr =d), thus the exponents are refered to as Vn gr, g;, and 8r. The distribution function Eq. (3) can be written as
P(riN)
=
fW Fr r
(-r-) N"r
(44)
with S1 = 271' in d = 2 and S1 = 471' in d = 3. The scaling function Fr(x) is defined as
x91 +d
Fr(x)
= { x9~+d exp [-bx0r]
for x < z for x > z '
(45)
where 6r = 11(1- vr), Eq. (5). The actual value of the crossover z is determined from the numerical results. The constants B and b can be obtained from the normalization 00 00 condition J0 P(riN) dr = 1 and from the second moment J0 r 2 P(riN) dr = N 2"r. Upon integration, one gets the exact relations (46) where r(u, z) is the incomplete Gamma function, and 1
S1B { !jr b(g~+d+2)/or
r
(gr 2
+d+2 !jr
)
'bzor
z9)+d+2 }
+ gr + d + 2 = 1.
Thus, by plotting the distribution function in the case x > z as y
(47)
= b(g~+d)/or (S1Bt 1 r
P(riN) exp [ (blforr I N"r )or] versus b for r I N"r in a double logarithmic plot, the exponent 1
g; can be read off from the relation y rv xg~+d and adjusted until the above relations Eqs. (46) and (47) are satisfied. This method yields much more accurate results than by directly fitting the distribution function itself. The accuracy of the result can be assessed by plotting y = -ln [b(g~+d)/Or (S1B)-l r P(riN) (bl/Or r I N"r r(g~+d)] = b (r I N"r )0r versus b1 for r I N"r in a double logarithmic plot, from which the exponent 6r can be determined and compared with the expected value 8r = 11(1- vr), Eq. (5). D. Generalized averaging procedure To obtain an estimate of whether the ensemble£ of substrate configurations considered is sufficiently large in order to get convergent results, the data has to be analyzed using a generalized averaging procedure as follows: The total ensemble £ containing ntot substrate configurations is divided into subsets £; containing neff configurations each. The
230
A. Ordemann, M. Porto and H. E. Roman
generalized average is then defined as (48) The obtained results (CN , E)(q) may depend sensitively on the different substrate conneff figurations and therefore may display strong fluctuations, indicating that the system is not self-averaging. In order to smooth out these fluctuations, a second average has to be performed, which is a logarithmic average over the ntotfneff subsets [71], (49) In Eq. (49), the limiting case neff -+ 1 corresponds to the limit q-+ 0, while the usual average (cf. Eq. (1)) is recovered when neff = ntot· A dependence of the coordination numbers /-lq,neff and the enhancement exponents /'q,neff on neff indicates that the given ensemble is too small to obtain the asymptotic values. If, on the contrary, the ensemble of substrate configurations is sufficiently large, then f.lq,neff and /'q,neff no longer depend on neff·
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S. Milosevic, I. Zivic, and S. Elezovic-Hadzic, Phys. Rev. E 61 (2000) 2141. L. Acedo and S.B. Yuste, Phys. Rev. E 63 (2000) 011105. D. Dhar, J. Math. Phys. 18 (1977) 577. F.D.D. Aarao Reis and R. Riera, J. Stat. Phys. 71 (1983) 453. Y.-M. Zhu, Z.-H. Lu, and Y. Wei, Phys. Lett. A 177 (1993) 279. K.Y. Woo and S.B. Lee, Phys. Rev. A 44 (1991) 999. Y. Gefen, Y. Meir, B. Mandelbrot, and A. Aharony, Phys. Rev. Lett. 50 (1983) 145. B. Nienhuis, J. Phys. A 15 (1982) 199. P.N. Strenski, R.M. Bradley, and J.M. Debierre, Phys. Rev. Lett. 66 (1991) 133. D. Stauffer and A. Aharony, Introduction to Percolation Theory, Taylor & Francis, London, 1992. 62. A. Bunde and S. Havlin (eds.), Fractals and Disordered Systems, 2nd ed., Springer, Heidelberg, 1996. 63. P. Grassberger, J. Phys. A 25 (1992) 5867. 64. M. Porto, A. Bunde, S. Havlin, and H.E. Roman, Phys. Rev. E 56 (1997) 1667. 65. P. Grassberger, J. Phys. A 32 (1999) 6233. 66. P. Grassberger, Physica A 262 (1999) 251. 67. J.W. Essam, Rep. Prog. Phys. 43 (1980) 833. 68. Y. Meir and A.B. Harris, Phys. Rev. Lett. 63 (1989) 2819. 69. C. von Ferber, V. Blavats'ka, R. Folk, and Yu. Holovatch, Phys. Rev. E 70 (2004) 035104(R). 70. H.J. Herrmann, D.C. Hong, and H.E. Stanley, J. Phys. A 17 (1984) L261. 71. A. Bunde and J. Drager, Phys. Rev. E 52 (1995) 53. 72. A. Bunde and J. Drager, Physica A 202 (1994) 371. 73. A. Giacometti and A. Maritan, Phys. Rev. E 49 (1994) 227. 74. A. Ordemann, M. Porto, H.E. Roman, and S. Havlin, Phys. Rev. E 63 (2001) 020104(R). 75. H.E. Roman, A. Ordemann, M. Porto, A. Bunde, and S. Havlin, Phil. Mag. B 77 (1998) 1357. 76. P.M. Lam, J. Phys. A 23 (1990) L831. 77. H. Nakanishi and S.B. Lee, J. Phys. A 24 (1991) 1355. 78. R.M. Ziff, Phys. Rev. Lett. 72 (1994) 1942. 79. S.B. Lee and H. Nakanishi, Phys. Rev. Lett. 61 (1988) 2022. 80. C. Vanderzande and A. Komoda, Phys. Rev. A 45 (1992) 5335. 81. M.D. Rintoul, J. Moon, and H. Nakanishi, Phys. Rev. E 49 (1994) 2790. 82. J.M. Carmona, U.M.B. Marconi, J.J. Ruiz-Lorenzo, and A. Tarancon, Phys. Rev. B 58 (1998) 14387. 83. M.P.M. den Nijs, J. Phys. 12 (1979) 1857. 84. A.J. Guttmann and J. Wang, J. Phys. A 24 (1991) 3107. 85. I. Jensen and A.J. Guttmann, J. Phys. A 32 (1999) 4867. 86. A.J. Guttmann, T.R. Osborn, and A.D. Sokal, J. Phys. A 19 (1986) 2591. 87. D. McDonald, S. Joseph, D.L. Hunter, L.L. Mosely, N. Jan, and A.J. Guttmann, J. Phys. A 35 (2002) 1501. 88. S. Caracciolo, M.S. Causo, and A. Pelissetto, Phys. Rev. E 57 (1998) 1215. 89. S. Kumar, Y. Singh, andY. Joshi, J. Phys. A 23 (1990) 2987.
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Statistics of Linear Polymers in Disordered Media Edited by Bikas K. Chakrabarti © 2005 Elsevier B.V. All rights reserved.
Localization of polymers in random media: Analogy with quantum particles in disorder Yadin Y. Goldschmidt a university
a
and Yohannes Shiferaw b
of Pittsburgh, Department of Physics, Pittsburgh, Pennsylvannia 15260.
bUniversity of California, Department of Medicine and Cardiology, Los Angeles, CA 90095-1679. In this chapter we review the rich behavior of polymer chains embedded in a quenched random environment. We first consider the problem of a Gaussian chain free to move in a random potential with short-ranged correlations. We derive the equilibrium conformation of the chain using a replica variational ansatz, and highlight the crucial role of the system's volume. A mapping is established to that of a quantum particle in a random potential, and the phenomenon of localization is explained in terms of the dominance of localized tail states of the Schrodinger equation. We also give a physical interpretation of the 1-step replica-symmetry-breaking solution, and elucidate the connection with the statistics of localized tail states. We proceeded to discuss the more realistic case of a chain embedded in a sea of hard obstacles. Here, we show that the chain size exhibits a rich scaling behavior, which depends critically on the volume of the system. In particular, we show that a medium of hard obstacles can be approximated as a Gaussian random potential only for small system sizes. For larger sizes a completely different scaling behavior emerges. Finally we consider the case of a polymer with self-avoiding (excluded volume) interactions. In this case it is found that when disorder is present, the polymer attains a conformation consisting of blobs connected by straight segments. Using Flory type free energy arguments we analyze the statistics of these conformational shapes, and show the existence of a localization-delocalization transition as a function of the strength of the self-avoiding interaction. 1. INTRODUCTION
Polymers are very long chain-like macromolecules that play an important role in a wide variety of physical systems. Many of the materials that we encounter in our every day lives, such as plastics and rubber, are essentially a mesh like structure of polymers. In biology, polymers also play a central role, as many biological molecules, such as DNA, have a long chain-like structure. In this chapter we focus on the equilibrium statistical mechanics of a single polymer chain immersed in a quenched random medium. For example, a chain that is free to move in a porous material, such as a sponge or a complicated gel network. This problem is relevant to a number of technologically important processes, such as filtration [1,2], gel permeation chromatography [3,4], and the transport of polymers through porous 235
236
Y. Y. Goldschmidt and Y. Shiferaw
membranes [5]. Here, we will not address these practical issues, but rather, give an overview of the theoretical aspects of the problem. In this chapter we present analytical and numerical results on the conformational statistics of polymers in quenched random media. We first consider the problem of a Gaussian polymer chain in a random potential characterized by short ranged correlations. This simplified mathematical model of a real polymer is tackled analytically using the replica method, and also via a mapping to the equivalent problem of a quantum particle in a random potential. Here, we will focus on the statistical properties of a long polymer chain, where the free-energy landscape is complex and possesses many meta-stable states. Using the path integral mapping between the partition function of a Gaussian polymer chain and the imaginary time Schrodinger equation, we show that the glassy phase can be understood in terms of the localized eigenfunctions of the Schrodinger equation with a random potential. Furthermore, we show that the glassy behavior can also be described analytically by a one-step replica-symmetry-breaking (RSB) solution. We explore the connection between the replica solution and the eigenfunctions of the Schrodinger equation, and show that the one-step RSB solution can be interpreted in terms of the dominance of localized tail states. We proceed to investigate the more realistic case of a chain immersed in a sea of hard obstacles that are randomly distributed in space. Prior to our work it was often assumed in the literature that that this problem is equivalent to the random potential problem. Using Flory type free energy arguments we elucidate the similarities and differences between this case and that of a random potential with short ranged correlations. In particular, we show that the dependence of the polymer size on chain length can exhibit three possible scaling regimes depending on the system size. Finally, we introduce a more realistic model that includes a self-avoiding interaction between monomers of the polymer chain. We show that in the limit of a very long chain, and when the self-avoiding interaction is weak, the equilibrium chain conformation consists of many blobs with connecting segments. These blobs are situated in regions of low average potential, in the random potential case, or in a region of low density of obstacles in the random obstacles case. We also show that as the strength of the selfavoiding interaction is increased relative to the strength of the random potential, the polymer chain undergoes a localization-delocalization transition, where the chain is no longer bound to a particular region of the medium but can easily wander around under the influence of a small perturbation.
2. THE STATISTICS OF A GAUSSIAN CHAIN IN A RANDOM POTENTIAL 2.1. Path integral formalism A polymer is a collection of molecules, called monomers, which interact with each other to form a long flexible chain. For example, a typical polymer like polyethylene consists of a chain of roughly 105 CH2 molecules. The large number of monomers allows for a statistical description of macroscopic properties which are independent of details on the monomer scale. The simplest mathematical model of a polymer chain is referred to as the Gaussian chain [6]. In this model the polymer is described by a position d-dimensional
Polymers in random media: Quantum analogy
237
vector R(u), where u is a continuous variable which satisfies 0 < u < L, and which runs along the contour of a chain with length L. The probability of finding a conformation R(u) is given by P[R(u)] = N exp [-
2~2 1£
du (
~~u)
rJ ,
(1)
where N is a normalization constant, and where b is the average bond length. In this description the self-avoiding interaction of the chain with itself is not taken into acount, and in the absence of an external potential the conformation of the chain resembles the trajectory of a random walker with a mean step b. Thus, the average end-to-end distance of the chain satisfies
(2) The conformational statistics of a polymer chain will change if the chain is placed in a random environment. We can model the effect of the random environment by introducing an interaction energy E;nt[R(u)] =
1L
(3)
du V(R(u)),
where V(R) is the potential energy of a monomer at the position R due to the environment. The potential function V(R) will depend on the type of random medium that is being studied, and will be discussed in more detail later. The conformational probability is now
2~2 1£ du ( d~~u)) ~ 1L du V(R(u))] , 2
P[R(u)] = N exp [-
-
(4)
where we multiplied the free Gaussian conformational probability with the Boltzmann factor exp (-~Eint[R(u)]), with~= 1/kT. Given the conformational probability we can we write the partition sum (Green's function) for the paths of length L that go from R toR' as R(L)=R'
Z(R,R'; L) = {
[dR(u)]exp( -~H),
(5)
JR(O)=R
where
(6) and where M = d/ (~b 2 ). All the statistical properties of the polymer chain can be derived from this partition sum. In this chapter we will also consider the effects of the volume of the random medium i.e. the system size. In order to incorporate this effect in the partition function we include a harmonic term
V (R(u))--+ V (R(u))
+ ~R 2 (u),
(7)
Y. Y. Goldschmidt and Y. Shiferaw
238
where the coefficient fl, will be a measure of the available volume to the polymer (A larger fl, implies a smaller system volume). The confining well is also important to ensure that the model is well defined, since it turns out that certain equilibrium properties of the polymer diverge in the infinite volume limit (fl,---+ 0). A quenched random medium, such as a rough surface or a frozen gel network, is a complex structure that can in principle be modelled by a complicated potential function V (R). However, we will not be interested in the physical properties of a polymer chain immersed in a specific environment, but rather in an ensemble of similar environments. Hence, we will have to specify instead the probability distribution of the random potential V(R). Here, we will consider random potentials that are taken from a Gaussian distribution defined by
(V(R)) = 0, (V(R)V(R')) = f ((R- R'?)
(8)
In particular we will consider a correlation function of the form
f ((R- R'?)) = (1r~;)d/ 2 exp ( -(R- R') 2 /e) ,
(9)
where g determines the strength of the disorder and the parameter ~ conveniently controls the correlation range of the random potential. Here, we will consider only the case of short-range correlations, where ~ is much smaller than the system size. All the statistical properties of the polymer will depend on the partition sum. For instance, the average end-to-end distance of a polymer chain that is free to move is given by (
R 2 (L)) = F
(f dRdR'(RR')2Z(R,R';L)) I dRdR' Z(R, L) ' R';
(10)
where the over-bar stands for the average of the ratio over realizations of the random potential. This average is referred to as a quenched average, as opposed to an annealed average, where the numerator and denominator are averaged independently. In some previous studies it has been argued that for the mean end-to-end distance, as defined in Eq. 10, one can replace the quenched average by the more analytically tractable annealed average. However, this replacement can be justified only when the system size is strictly infinite, since only in that limit can the polymer sample all of space and find the most favorable potential well that will be similar to its environment in the annealed case. The main problem with this approach is that in practice we always deal with finite-size systems, and it is not always easy to assess how big the system size has to be so that the annealed average is a good approximation to the quenched average. In addition, the time it takes the chain to sample a large volume is exceedingly long and unreachable over a reasonable experimental time. Before ending this section we would like to point out the relationship between the statistical properties of a polymer chain and a variety of other physical problems. The partition sum, as written in Eq. (5), is in the form of a path integral over all possible chain conformations. Since many physical problems can be formulated in terms of path integrals, we can map the problem of a Gaussian polymer in a random media to a wide
Polymers in random media: Quantum analogy Polymer chain
Quantum particle
Flux line
u
T
z
(monomer label)
(imaginary time)
(plane label)
f3 L
1/n
f3 Lz
n/3
(chain length)
d/ f3b
2
239
(distance along z-direction) m (mass)
E£
(b=bond length) (line tension) Table 1 The relationship between different physical systems.
variety of seemingly unrelated problems. First, we can map the partition sum of a polymer chain to the density matrix of a quantum particle. The mapping [7,8] is given by
f3 ---+
1/n,
L ---+ f3n.
(11)
The density matrix of a quantum particle at inverse temperature f3 is then related to the partition function of the polymer as
p(R,R';/3) = Z(R,R';L = f3n,f3 =
1/n).
(12)
The monomer label u is now interpreted as the Trotter (imaginary) time, and M as the mass of the quantum particle. The density matrix is relevant to the equilibrium statistical mechanics of a quantum particle, such as an electron in a dirty metal. The polymer partition sum for a Gaussian chain in random potential can also be mapped into the partition sum of a flux-line in type-II superconductors in the presence of columnar disorder [9,10]. Here, R(u) is the transverse displacement of the flux-line (or vortex). The variable u is interpreted as the distance along the z-axis (direction of the magnetic field), and the variable M corresponds to the line tension of the flux-line. Thus a flux-line in three dimensions propagating along the z-direction maps into a polymer in two spatial dimensions (its projection onto the plane), and the columnar disorder maps into point disorder. Although, in the case of several flux-lines there is a repulsive electromagnetic interaction among them which is not present for polymer chains. In Table 1 we have summarized the relationships between a polymer chain, a quantum particle, and a fluxline in a superconductor. Finally, we can also map the polymer problem to the problem of diffusion in a random catalytic environment [11,12]. This process describes, for instance, auto-catalytic chemical reactions in a disordered background, or the spreading of a population with a growth rate that depends on local random conditions. If we make the replacements 2
1 M f3 ---+ D , f3V(R) ---+ -U(R) , L ---+ t
(13)
where D is the diffusion constant, and U(R) is the growth rate (or reaction rate) at position R, and tis the time. Then Z(R', R; t) gives the concentration of the constituents at position R' at time t, given a delta function concentration concentrated at R at time t = 0.
Y. Y. Goldschmidt and Y. Shiferaw
240
2.2. Flory arguments A remarkably fruitful approach to the statistics of polymers is via simple and intuitive free energy arguments. This approach was used with great success by Cates and Ball [13] to derive the essential scaling properties of a polymer in a random potential. Here, we reproduce their beautiful intuitive arguments in order to elucidate the effect of a finite volume on the behavior of an untethered free chain in a random potential. First, Cates and Ball argue that a Gaussian chain situated in an infinite random medium is always collapsed in the long-chain limit. Their argument goes as follows: Consider a white-noise random potential V(x) of zero mean whose probability distribution is
P(V(x)) rx g- 1/ 2 exp( -V 2 j2g).
(14)
If we now coarse-grain the medium and denote by V the average value of the potential over some region of volume n, then the coarse-grained potential will have the distribution 2
Po(V) rx (gjn)- 112 exp( -nV j2g).
(15)
Now, consider a polymer chain situated in the random potential, and assume that it shrinks into a volume n corresponding to a place where the mean potential V takes on a lower value than usual. In this situation the free energy of the chain is crudely estimated to be ( neglecting all numerical factors):
F(n, V)
2
= L/R2 +LV+ nV j2g.
(16)
Here, Lis the length of the chain (number of monomers), R is the radius of gyration (or end-to-end distance) and the volume n is related toR via n "'Rd in d-spatial dimensions. The first term on the r.h.s. is an estimate of the free energy of a long chain confined to a region of size R in the absence of an external potential (see e.g. [14], Eq. 1.12). The second term is just the potential energy of the chain in the random potential of strength V. The third term arises from the chance of incurring a random potential of strength V. The quantity ln P(V) gives an associated effective entropy for the system. Minimizing this free energy over both V and n determines the lowest free energy configuration. Minimizing with respect to V yields V = -Lgjn. Substituting in F gives: F(R) =
!:__ - L2g R2
2Rd.
(17)
This shows that for any d 2: 2, F ---+ -oo as R ---+ 0. Thus, the mean size of the chain is zero, or in the presence of a cutoff, the size of one monomer i.e. R "'1,
d
(18)
2: 2.
For d < 2, the free energy has a minimum for
R"' (Lg) 1f(d- 2)
,
d<2,
(19)
which in the long chain limit (L 2: 1/g) cuts off again at R "' 1. These results are the same as those for the case of an annealed potential that is able to adjust locally to
Polymers in random media: Quantum analogy
241
lower the free energy of the system. The reason is that for an infinite system containing a finite (even though long) chain, space can be divided into regions containing different realizations of the potential, and the chain can sample all of these to find an environment arbitrarily similar to that which would occur in the annealed situation. These results stand in contrast to the replica calculation of Edwards and Muthukumar (EM) [15], who found that for a long chain R "'
d
g-1/(4-d)'
<4
(20)
when g 21(4 -d) L ---+ oo , whereas R "' £11 2 when g 21(4 -d) L ---+ 0. Note that the result (20) is independent of L as opposed to Eq. (19). To reconcile the two apparently different results, Cates and Ball argue that the quenched case is different from the annealed case only for the case when the medium has a finite volume V. In a finite box, arbitrarily deep potential minima are not present. Instead the most negative V averaged over a region of volume n « v occupied by the chain, is approximately (keeping only leading terms in the volume V) given by solving the equation (the l.h.s. of which represents the area under the tail of the distribution)
v
J
-oo
dy Po.(Y) ~
n V,
(21)
which yields
V=-Jgl~V·
(22)
This expression when plugged into Eq. (16) leads to (note that the last term in (16) just becomes a constant independent of R)
F(R)
=
__!:__ -
R2
LJ g ln V. Rd
(23)
When this free energy is minimized with respect to R it gives rise to
R"' (g ln V)- 1/( 4 -d),
d<4 .
(24)
Using this value the binding energy per monomer becomes Ubind/ L
= (g ln V) 21 (4 -d).
(25)
For the polymer to be localized its total binding energy has to be greater than the translational entropy ln V, which is always satisfied for 2 < d < 4 when Lor V are large (and ford= 2 when Lis large). 2.3. The relation to the localization of a quantum particle In order understand the conformational statistics of a Gaussian chain in a random potential, we map the partition sum to an imaginary time Schrodinger equation. This mapping (see Ref. [7] Eqs. (3.12)-(3.18)) is given by R(L)=R
Z(R, R'; L) =
1
R(O)=R'
[dR(u)]exp ( -;JH[R(u)]) = (Riexp( -;JLH)IR'),
(26)
Y. Y. Goldschmidt and Y. Shiferaw
242 where ' H
()2 1 2M/3 8R2
p, ' 2
2
'
= ---2 , - + -R + V(R).
(27)
So for a given realization of the random potential the polymer partition sum can be expressed as a matrix element of the imaginary-time evolution operator. The matrix elements can be expanded in eigenfunctions of the Hamiltonian operator to yield 00
(Riexp(-f3LH)IR')
=L
exp(-f3LEm)m(R)~(R'),
(28)
m=O
where (29) The Schrodinger equation with a random potential is a well known problem that has been intensely studied for a long time [16-19]. The main property that we will use is that when V(R) has short range correlations (i.e. correlation length is shorter than any other length scale in the problem), and if the system size is infinite, then in any dimension all eigenstates with energy below a critical energy EM ( referred to as the mobility edge) are exponentially localized in the form
(30) Here Rm is the localization center of the mth state, and £m is the localization length of that state. The localization length satisfies 1/£m = /3J2MIEml for Em« 0, i.e. deep in the tail region. For E > EM extended states exist when d > 2. For d = 1, 2 there is no mobility edge and all states are exponentially localized. The states with energies E > EM are called extended since they are no longer localized but are spread over a finite fraction of the system. Also, it is known that the eigenvalues of the localized states are discrete, while the eigenvalues of the extended states form a continuum. For finite system size, or if p, -1- 0 in the Hamiltonian given in Eq. (27), the above discussion has to be modified. First, the eigenfunctions are always discrete in any dimensions. But even in one dimension as the energy increases the width of the localized states eventually becomes comparable to the system size and thus a localized particle of that energy can go from one end of the sample to the other. Thus the distinction between localized and extended states becomes blurred for a finite system at energies much above the ground state. Nevertheless, there will still be a qualitative difference between the low energy tail states and the higher energy states with large localization lengths. All the physical properties of the polymer chain can be expressed in terms of the eigenstates of Schrodinger equation. For instance we can write the end-to-end distance for a given realization of the random potential as
Polymers in random media: Quantum analogy
243
where am= J dRm(R), and where Ov refers to a configurational average for the case of a fixed realization ofrandom potential. When Lis large enough so that (E 1 -E9 s)L >> 1, where E 1 is the eigenvalue of the first excited state and E 98 is the ground state eigenvalue, then only the ground state contributes. In this case we have (32)
where 98 (R) is the ground state eigenfunction. It can be shown that the ground state wave function is positive definite and so in the large L limit (R~(L)) 1 / 2 can be interpreted as the width of the ground state eigenfunction. Assuming the ground state has the form given in Eq. (30), we can write (R}(L))v = 2d(d+ 1)£;8 , where £98 = £0 is the localization length of the ground state. Upon averaging over all realizations of the random potential we get that (R~(L)) = 2d(d + 1)£~" and so the quenched average of the end-to-end distance, in the long chain limit, is proportional to the square of the average localization length of the ground state eigenfunction.
2.4. Localized eigenstates and glassy behavior Using the path integral mapping we can evaluate the partition sum by solving the discretized Schrodinger equation. In d = 1 this can be accomplished by simply diagonalizing an N x N Hamiltonian matrix, with lattice spacing 6.. = S/N, where S is the system size. Details of the numerical procedure are given in [20]. Using the lattice computation, we explore the connection between the eigenstates of the Schrodinger equation and the physical properties of the polymer chain. Here, we focus on the probability distribution defined as P(R,L)
= Z(R,R,L)/
f
Z(R,R,L)dR
(33)
which can be interpreted as the probability of finding a closed polymer chain of length L which passes through the point R (for a given realization of the random potential). We consider this probability distribution since it gives the most direct connection between the chain properties and the eigenfunctions of the Schrodinger equation. In Fig. 1 we plot P(R, L) vs. R for four different chain lengths. We also include a plot of the random potential sample that is used. From the plot we can see clearly that as the chain length is increased the probability distribution tends to localize around a few valleys of the random potential landscape. As L is increased further there is only one peak as the chain finds the most favorable position. These results can be explained in terms of the eigenfunctions of the Schrodinger equation using the expansion (34) m
Which shows that as L is increased the localized tail states dominate the partition sum until only the ground state remains when (E1 - E 98 )L > > 1. The dominance of tail states for long polymer chains leads to large sample-to-sample variations in measured physical quantities. This is because the partition sum, for a given realization of the random potential, is dominated by a few favorable conformations which
Y. Y . Goldschmidt and Y . Sl1iferaw
244
Figure 1. Plot of P( R, L) vs R for four values of L. The bottom most curve is the random pot.ential sample that is used. From bottom to top we use L = .3, 1, 10, 20. The model parameters arc M = 1/2, g = 25, f3 = l, and J.l = 0.01. We use a lattice of size 8=40 with !:l. = 0.2. The random potential is modeled by generating N numbers {V{(i)}i=I,... ,N which satisfy (\l{(i)l~(i + l)} ex exp( -b..2 l 2 Je), where~= 1/.../2.
are strongly sample dependent. In Fig. 2.4 we plot the relative sample-to-sample fluctuat.ions of the end-to-end distance (RHL)}, as a function of chain length. The relative sample-to-sample fluctuation is defined as b..pf(Rj.. (L)} with
D.p
=
2) 1/ 2 2 , ( (RHL)} - (R}(L)}
(35)
and where the averaging is done over many realizations of the random potential. From the plot it is clear that as the length of the chain is increased the sample-to-sample fluctuations increases dramatically at a chain length Lc "' 0.5. Large sample-to-sample fluctuations in measured physical quantities are typical of glassy systems, which arc a broad class of systems characterized by rugged energy landscapes [21]. This is precisely the case here, where the free energy of a long chain will be dominated by conformations that reside in the deep valleys of the random potential landscape. The main new result here is that the emergence of glassy characteristics of a polymer chain in random media can be traced to the dominance of localized tail states of the Schrodingcr equation. 2.5. Averaging over disorder: The replica trick The numerical results in the previous section reveal that a polymer in a random potential can be viewed as a glassy system. In order to develop this point of view, we apply the replica method, which has been successfully applied in the study of glassy systems [21]. The replica approach was invented in order to compute quenched averagP~'> such as (R}(L)}, as given in Eq. (10), which is difficult since it entails averaging a ratio of the
Polymers in random media: Quantum analogy
245
02
0.1
Figure 2. Plot of 6.F/(R}(L)} vs. L. The parameters are the same as in Fig. 1 and we average over 1000 random potential realizations.
partition sum. The starting point is the formal identity - 1 ln(Z) = lim - - ,
zn
n-+0
(36)
n
which can be used to calculate the quenched free-energy
-F =
--
zn-1
-kT ln(Z) = -kT lim--, (37} n-+0 n where the order of the averaging and taking the n --t 0 limits has been interchanged, where n is a positive hopefully with no ill ramifications. The average of a term like integer, is easy to implement by creating n-identical copies of the system, referred to as replicas, and then averaging over the random potential. This process of averaging results in interactions between different replicas. The resulting analytical expression is then continued analytically to n = 0. Introducing n-copies of the system and computing it is straight forward to show that the average end-to-end distance can be formally written as
zn,
zn
(R2 (L)} = r I n:=l dRa n:=l dR'a(Rl- R~) Zn( {Ra}. {R;.}; L) ,.. nl!!b I n:=l dR, n:=l dR'aZn( {Ra}. {R'a}; L) • 2
(38)
where
Zn( {Ra}. {R~}; L) =
1
R., ( L)=R~
n
Il[dRa] exp( -IJHn), R.. (O)=R.. a=l
(39)
and where
(40)
Y. Y. Goldschmidt and Y. Shiferaw
246
Thus, the averaged equilibrium properties of the polymer can be extracted from an nbody problem by taking then ---+ 0 limit at the end. This limit has to be taken with care, by solving the problem analytically for general n, before taking the limit of n ---+ 0. 2.6. The Replica Variational Approach In order to compute quenched averages of the polymer chain we will have to solve the n-body replicated partition sum given in Eq. (40). This path integral cannot be evaluated analytically and a variational approach has been used in Refs. [22,8] to make further progress. The procedure is to follow the work of Feynman [7] and others [23,24] and model Hn by a solvable trial Hamiltonian hn which is determined by the stationarity of the variational free energy
(41) Following [22] we use a quadratic trial Hamiltonian of the form
(42) where the matrix elements Pab are the variational parameters. The physical motivation for this ansatz is that the replica-replica interaction in the original Hamiltonian is modeled by a quadratic interaction which can be different for different replica pairs. Also, the form of the quadratic interaction was chosen specifically [22] to preserve the translational invariance of the original Hamiltonian. Apart from the J-t-dependent term, (or alternatively in the limit J-l---+ 0), the original Hamiltonian Hn given in Eq. (40) is invariant under the transformation
Ra(u)---+ Ra(u)
+ C,
a= 1, · · · , n
(43)
where C is a constant vector. This reflects the fact that in the infinite system, without a confining harmonic term, after averaging over the random potential the interaction is translational invariant. A good variational Hamiltonian must be one that can preserve this translational symmetry. Thus the variational ansatz must be rich enough to implement the relation (43). The ansatz chosen by EM [15] violated this translational invariance for any q -1- 0, where q denoted their single variational parameter. This imposed an unphysical origin on the system when none existed in the infinite volume limit that they considered. It was actually shown in Ref. [22] that if the variational ansatz is rich enough the translation invariance actually emerges from the variational extremization even if not assumed explicitly for the trial Hamiltonian. We will now assume that in the n ---+ 0 limit the matrix Pab can be parameterized according to the 1-step replica symmetry breaking scheme of Parisi [24]. In the 1-step RSB scheme, the matrix Pab can be parameterized as (p, p(x)) with
p(x) = { Po
P1
0 <X<
Xc
Xc <X<
1,
(44)
Polymers in random media: Quantum analogy
247
m
m
Figure 3. Illustration of first step replica symmetry breaking. The limits n ---+ 0 and m ---+ Xc have to be taken with 0 < Xc < 1.
and where x is Parisi's replica index. See Figure(3) for an illustration. Calculating the variational free energy using Eq. (41) it could be expressed [22] in the limit n---+ 0 as a function of the four variational parameters. i.e. F = F(fJ,Po,P 1 ,xc)· The stationarity of the free energy yields four coupled non-linear equations for the four variational parameters. In the limit of small fl, (large volume) and long chains (L large) these equations were solved analytically. Denoting by A the combination
A=
fl,-
p + (1 -
xc)pl
+ XcPo,
(45)
we find from the analytical variational solution
(46) From this final result we can obtain the radius of gyration which can be shown to be proportional to A- 1/ 4 . Recalling also that llnfl,l rx ln V, we find (47) a result which coincides with the prediction of Cates and Ball for the case of a finite volume. It could also be shown that a replica-symmetric ansatz for the matrix p yields a result consistent with the annealed average of the disorder where the chain collapses in the limit of L ---+ oo. For non-asymptotic values of fl, and L the non-linear stationarity equations equations could be solved numerically [20] using a standard iterative method [25]. We found that for a given set of parameters there is a chain length Lc (which depends on the strength of the disorder) such that for 0 < L < Lc there is only a replica symmetric solution. This is the case when the variational parameters satisfy Xc = 1 and s0 = s 1 . For L > Lc there is still a replica symmetric solution but we also find an additional replica symmetry breaking solution. So in this regime we find an additional solution such that 0 < Xc < 1
248
Y. Y. Goldschmidt and Y. Shiferaw
.·····
.... aaaa:lt:aaiSISIJ:SI:J:IIIUESIIII:II:I:I:IJ:II:
Figure 4. Plot of (R}.(L)} vs. L. The dotted line is generated by averaging over 10000 samples, and error bars arc found by computing the standard deviation of 10 sets of 1000 samples. The dashed line is the RS solution, and the solid line is the RSB solution.
and s 0 =I s 1 • In order to decide which solution correctly describes the physics in that regime we compare their respective predictions to the lattice computation of (R~- (L)}. In Fig. 4 we plot the mean squared displacement (R~.. {I-)} vs. L for a given set of parameters. We plot this quantity using the lattice result, and also using the two predictions of the variational method. Note that in the labels of the plots the average over the disorder is denoted by a second set of brackets rather than an over-bar. For L below Lc ~ 0. 73 there is only a RS solution whir.h is very close to the lattice prediction. For L greater than Lc the RS and R.SB solutions are different and it is clear that the RSB solution is closer to the lattice result. We can see that the end-to-end distance saturatcJi at a constant value as L increases. This behavior is correctly predicted by the RSB solution but not by the RS solution. So far we have seen that the glassy characteristics of a polymer in a short range correlated random potential is closely related to the dominance of low energy eigenfunctions. We also know that the variational solution possesses a RS solution for L < Lc and an RSB solution for L > Lc. It is well known that replica symmetry breaking is typically associated with glassy behavior, and for our model we show that the onset of HSB is precisely when the system begins to exhibit glassy behavior. The variational parameter that best reveals the transition between RS and RSB is the break point Xc. If Xc = 1 then there is only an RS solution, and if 0 < Xc < 1 then that corresponds to an RSB solution. In Fig. 5 we plot Xc vs. L using the same parameters that were used in Fig. 4. We can see that onset of the RSB solution is at Lc ~ .73, after which we find that the break point decreases like Xc (X 1/ L. If we compare this result to the plot of ~F/(R}.(L)} vs L in Fig. 2.4, we see that near Lc ~ 0.5 the sample-to-sample fluctuations begin to rise rapidly. This result provides strong evidence that when the RSB solution is valid the polymer chain docs indeed exhibit glassy behavior.
Polymers in random media: Quantum analogy
249
12
..
0 .8
0.6
Figure 5. Plot of Xc vs. L. The parameters are the same as those used in Fig. 4.
2.7. Physical interpretation of the 1-step RSB solution In this section we study the physical interpretation of the replica symmetry breaking solution. Our purpose is to see if the underlying physical picture predicted by 1-step RSB is indeed consistent with the presence of exponentially localized eigenstates. The following analysis is valid for a very long polymer (large L) when the system becomes glassy. We begin by evaluating the replicated partition sum defined as R.a (L )= R.a
Zn({Ra}) =
1
n
Il!dRo]exp(- J)hn),
R.a{O)=R.a
(48)
a= l
where hn is the quadratic trial Hamiltonian in Eq. (42). Since hn is quadratic the path integrals can be evaluated analytically and t.he final result can we written in the form Zn({Ra}) = const. x exp (
-~ ~Q;b1 Ra · Rb).
(49)
The details of this calculation along with the relationship between the matrices Qab and are given [20]. Now, since Pub was parameterized according to the 1-step RSB scheme, it implies that Qab can also be parameterized in the same way. Mezard and Parisi [24] discuss the interpretation of a representation of the form (49) for the case of directed polymers. In particular they show how to deduce the structure of the probability distribution
Pab
Pv(R)
= Zv(R, L)/
f
dRZv(R, L),
(50)
which is the probability of finding a polymer loop that passP$ through R for a given realization (which we denote by V) of the the random potential. Here Zv(R, L) is just the partition sum Z(R, R; L) as given in Eq. (5) . This probability is related to the replicated partition function given in Eq. (48) by
Pv(R) = IimjdR2···dR,. Zn({Ra})R,=R· n - >0
(51)
Y. Y. Goldschmidt and Y. Shiferaw
250
Mezard and Parisi's analysis has to be adapted for the case ofreal (non-directed) polymers of length L in a random potential which is independent of time. The changes will be pointed out below. If Qab is parameterized by { ij, q(x)} such that
q(x)
= { qo q!
<
X Xc X> Xc,
(52)
one proceeds to obtain Pv(R) by the following procedure: 1. For each sample ( a realization of the random potential) generate a random variable R 0 which is picked from the distribution P(Ro)
= (27r:o)d/2 exp (- ~~) ·
(53)
2. Consider a set of "states" labeled by the index a whose physical meaning will be elucidated shortly. Each of these states is characterized by a weight Wa and a position variable Ra. Given R 0 , the variables Ra are an infinite set of uncorrelated random variables distributed according to
(54) The distribution of weights will be discussed below. 3. Given these "states" for a given sample, The probability distribution Pv(R) for that sample has the form (55) The weights Wa are given in terms of some "free energy" variables fa:
W _ a-
exp( -f3fa) "E-rexp(-f3f.r)"
(56)
These free energy variables are chosen from an exponential distribution
(57) where J is an upper cutoff. What is the meaning of these variables in the present case? To determine the weights Wa we compare Eq. (55) to the eigenfunction expansion given in Eq. (34). From Eq. (34) together with Eq. (50) it becomes clear that
(58) where
A _ a-
exp( -f3LEa)
2::-r exp( -f3LE--y)"
(59)
251
Polymers in random media: Quantum analogy Comparing Eq. (55) and Eq. (59) it becomes obvious that Wa 2
a (R) rx exp
(
= Aa and
R")2) .
- (RC ) 2 q- q1
(60)
Hence, the "states" labeled by a are in our case the actual eigenstates of the imaginary time Schrodinger equation. These are localized tail states centered at position Ra with an associated "weight" Wa. Thus the 1-step RSB solution approximates the tail states by a fixed Gaussian form. The width of these Gaussians, denoted by w 0 , give an estimate of the size of the polymer chain i.e. w 0 "'Rp. In the limit of large Land small f1, it can be shown to leading order [22] that w5 "' d/ (2/3 y'IM), where >, =
d4/(4-d) (/32 M)(4+d)/(4-d) ( ll 1)4/(4-d) (27r )2d/(4-d) g n fl,
(61)
So in terms of the disorder strength and the system size the chain size scales like
Rp rx (gllnfl,l)-1/(4-d) rx (glnV)-1/(4-d).
(62)
It should be emphasized that the subtle dependence on the volume of the system is a direct consequence of replica symmetry breaking. In fact, as shown in Fig. 4 the replica symmetric solution does not correctly describe the size of the polymer chain, since it fails to capture the dominance of localized tail states. We now consider the distribution P(Ra) given in Eq. (54). This is just the distribution for the localization centers Ra for a given value of R 0 . Hence, we can calculate the average distance between the localized states for a given sample. We find that the width w of the Gaussian P(Ra) satisfies w 2 = d(q 1 - q0 ). For small f1, and large Lit is straight forward to show that that w 2 R::! d/ ((3 fl,Lxc), where the break point Xc is given by Xc
) -1/(4-d) 1 ( dd-2 _ _ 2(3d+4 Md ll ld-2 L (27r)dg nf.J,
= _
(63)
2.8. Density of states and the 1-step RSB solution To develop the analogy further we first notice that the free energies fa are equal to LEa. This make sense if we think of lEal as representing the binding energy per monomer, and thus fa = LEa represent the total energy of the chain. These arguments lead us to expect that within the 1-step RSB scheme, the energy variables Ea are independent random variables taken from an exponential distribution:
(64) withE being some energy scale determined by the upper cutoff of the tail region. We will now argue that the distribution given above is just the expected distribution of groundstate energies i.e. the probability of finding the lowest energy level to have energy E. We first review some very basic results of extreme value statistics as presented in Ref. [26]. Given K independent and identically distributed random variables E;, pulled from a distribution of the form
-
A
o
P(E) = lEI" exp( -BIEI ),
(65)
Y. Y. Goldschmidt and Y. Shiferaw
252
the probability that the lowest of the K energies is E (for E -+ -oo and K -+ oo) is given by
P(E) ex exp [BoiEcl 8- 1E] ,
(66)
where
Ec = -
cog~K)) r;a
(67)
The value of Ec, the lowest energy expected to be attained in K trials, is easily obtained from
l
Ee
-oo
dEF(E) ~
1/ K.
(68)
The reason why we chose a distribution of the form given in Eq. (65) is that in d = 1 the probability F(E) is known to have that form exactly for the case of delta correlated random potentials (see [27]). Ford> 1 Lifshits [19] argued that the form given by Eq. (65) is also valid. Our goal now is to see if the distribution Eq. (64) derived using the 1-step RSB solution is indeed consistent with the distribution Eq. (66) predicted using extreme value statistics. Comparing Eq. (64) and Eq. (66) we find that for consistency the break point should satisfy
(69) Notice that the 1/ L behavior of Xc is exactly the same as was found analytically for large L in Ref. [22] and numerically for any L > Lc in the present work. We can go further by using the fact that the number of energy levels K, within a fixed energy interval is directly proportional to the system size, which in our formulation is effectively determined by f-L. Assuming log(K) <X llog(J-L) I and comparing to the approximate solution for Xc in Eq. (63) we find that o = (4- d)/2 and B <X 1/g. Now F(E) is just proportional to the density of states p(E), and it is known exactly in one dimension. Indeed, when d = 1, o = 3/2 and B <X 1/g. For 2 _::; d < 4, o agrees with the result derived by Lifshits [19]. Hence, the exponent o and the disorder dependence of B is correctly predicted by the 1-step RSB solution. The above results show that the 1-step RSB solution correctly predicts some important features of the eigenvalue distribution. More importantly, we have shown that the 1-step RSB solution can be interpreted in terms of the eigenstates of the Schrodinger equation with a random potential. However, there are differences and these reveal the limitations of the 1-step RSB solution. For example, all the localized states are approximated by the same Gaussian profile when in fact the localization lengths should increase with energy.
3. LOCALIZATION OF POLYMERS IN A MEDIUM WITH FIXED RANDOM OBSTACLES In this section we discuss the static properties of a Gaussian polymer chain without excluded volume interactions that is confined to a medium populated with quenched
Polymers in random media: Quantum analogy
253
random obstacles. It is important to distinguish between the following two important cases that have been discussed in the literature: 1. A Gaussian random potential with short range correlations. 2. Random obstacles which prevent the chain from visiting certain sites. Numerical simulations performed in three dimensions were restricted, to our knowledge, only to the case ofrandom obstacles [28-30]. On the other hand, extensive analytical work using the replica variational approach and Flory type free energy arguments, has been done for the case of a Gaussian random potential [13,15,20,22,11]. The case of a bounded (saturated) random potential was also addressed in Ref. [13]. It was not clear to us to what extent these theoretical investigations could be applied to the case of infinitely strong random obstacles placed randomly in the medium, as simulated numerically. This motivated us to investigate this problem in detail [31]. The results indicated that only in a special case (a small value of the embedding volume) the two problems mentioned above are similar, but otherwise they are quite different. We will assume that the obstacles are infinitely strong-they totally exclude the chain from visiting a given site occupied by an obstacle. Each obstacle is taken to be a block of volume ad, where d is the number of spatial dimensions and where a is the linear dimension of the block. We take for simplicity the polymer bond length b to be approximately equal to a. Thus, a will be the small length scale in the problem, and we will measure all distances in units of a. The obstacles are placed on the sites of a cubic lattice with lattice spacing a. We denote by x the probability that any given lattice site is occupied by an obstacle (block). Our main results will concern the case of small x, in particular x < Xc, where Xc refers to the percolation threshold (xc = 0.3116 for a cubic lattice in d = 3), but we will also comment on the case of a larger concentration of obstacles. We denote by V the total volume of the system. Assume that the chain is occupying a spherical region (lacuna) of volume rl rv Rd. In this region the actual volume fraction of obstacles will be denoted by x. It is crucial to realize that although the average number of obstacles per site is fixed by x, the actual number of obstacle in a small region of volume rv Rd is a fluctuating quantity which occurs with probability b(Rdx; Rd, x), where
(70) denotes the binomial probability distribution. In the limit where the system has infinite volume V the free energy for the chain is given by
F(R,x)
= -Lln(z) + ~2 + Lx -ln[b(Rdx; Rd,x)],
(71)
All these terms originate from entropy F = - T S where for simplicity we take T = 1 since the temperature does not play a significant role here with respect to the results. The first term originates from the entropy of a free chain in d-dimensions where z is the coordination number which for a cubic lattice is equal to 2d. The second term originates from the entropy of confinement in a cavity of radius R. The third term is the entropy loss due to obstacles. This linear dependence is justified in Ref. [31] and in the Appendix
Y. Y. Goldschmidt and Y. Shiferaw
254
therein. The forth term represents an entropy given by the logarithm of the probability to have a region of size 0 with Ox obstacles. This free energy, valid for V-+ oo, is called the annealed free energy since when the polymer can sample the entire space it is the same as the random potential adjusting itself to the polymer configuration. The free energy has to be minimized (the entropy maximized) with respect to Rand x. The most favorable value of xis 0. Since b(O; Rd, x) = (1- x)Rd, we find L
= -Lln(z) + R 2 -
F(R)
(72)
Rdln(1- x).
This free energy has now to be minimized with respect to R to yield Rm,annealed
L ) ( lln(1- x)l
rv
1/(d+2)
(73)
Thus the size of the chain grows with L, but with an exponent smaller than 1/2, the free chain exponent. So far we discussed the case of an infinite volume V. In a finite volume we find that the so called quenched and annealed case differ, at least when the volume is not too big. We actually find that there are three regions as a function of the size of the system volume V. First, if V < V 1 ':::J_ exp(x-(d- 2ll 2/(1- x)), it is unlikely for a chain of volume 0 rv Rd to find a region which is totally free of obstacles. Thus x does not vanish in this regime. To proceed further we must use an approximation to the binomial distribution b(Ox; 0, x). If 0 is not too small we can approximate the binomial distribution by a normal distribution [32] 2
b(Ox; 0, x)
i=:j
O(x-x) ) (27r0x(1- x))- 112 exp ( ( ) . 2x 1- x
(74)
This approximation is good provided Ox» 1 and 0(1- x) » 1. We verified that these conditions are indeed met in our case when x is small. In a finite volume V, the lowest expected value of x, to be denoted by Xm, can be found from the tail of the distribution
1
xm ,
dxexp ( - 0(x-x)2) 2~
0
0
':::J_ - ,
v
(75)
which gives
,
Xm ':::J_ X -
where we put y
JxylnV
-------w-,
=1-
(76)
x. The free energy becomes
F1 (R) = -Lln(z)
+ RL2 + Lx- L JxylnV -------w-·
(77)
The last term in the annealed free-energy is missing since it is negligible for large L when R is independent of L. Minimizing F(R) with respect to R we find
R ml
rv
(xy ln V) -1/(4-d)
(78)
Polymers in random media: Quantum analogy
255
and
XmJ
= x- (xy ln V) 2/( 4 -d).
(79)
The result for the radius of gyration of the chain, as represented by Rm1 is the same result as for the case of the Gaussian distributed random potential, but with the strength g replaced by x(1- x). The polymer in this case is localized and its size is independent of L for large L. As V grows Rm decreases until eventually Xm vanishes. This happens when V = V 1 ~ exp(x-(d- 2ll 2 y- 1 ). For V > V 1 , Rm is no longer given by Rm1 , but rather by the solution of XmiJ = 0. It is the largest region free of obstacles expected to be found in a volume V. Rather than using the normal approximation we can estimate Rm directly from the relation
(1- x)n ~ with rl
rv
R'/;,.
Rmn
rv
n;v,
Solving for
(80)
Rm
) ln V ( lln(1- x)l
we obtain
I/d
(81)
The polymer is still localized but the dependence on x and on ln V has changed. In this region which we call region II the free energy is given by lnV
Fn=-Lln(z)+aL ( lln( 1 -x)l
)-
2
/d
,
(82)
where some undetermined constant a is introduced for later convenience. As V grows in region II, Rmii continues to grow until it reaches the annealed value given above. This happens when
(83) to leading order in x, which is enormous for large L. For V > V 2 we have the third region in which Rmni = Rm,annealed and it grows like £l/(d+ 2). We can thus summarize the behavior of the end-to-end distance as the function of the system's volume as follows: Region I V < V 1 ~ exp(x-(d- 2 )/ 2 )
Rm1
rv
(x ln V)-r/( 4 -d)
(84)
Region II
Rmn region III
rv
) ln V ( lln(1- x)l
I/d
(85)
Y. Y. Goldschmidt and Y. Shiferaw
256
L
)
I/(d+2)
(86)
Rmrv ( lln(1-x)l
The behavior in region II can be deduced from known results of the density of states for a quantum particle in the presence of obstacles (repulsive impurities). In that case [19] the density of states is given by (when the obstacles are placed on a lattice) p(E) rv exp(-clln(1- x)IE-d/ 2 ), E > 0
(87)
with c being some dimension dependent constant and xis the density of impurities. Note that p(E) vanishes for E < 0. We can estimate the lowest energy in a finite volume V from the integral
lEe
dEp(E)
~ 1/V,
(88)
and find ln V
Ecrv ( lln(1-x)l
) - 2 /d
(89)
'
and thus the localization length is given by _
f!crviEcl
112
rv
(
ln V lln(1-x)l )
1/d
(90)
We now make some remarks on the validity of the spherical droplet approximation. The shape of a long polymer chain is determined by the regions of the random medium that have a lower than average number of obstacles. For V > V1 these regions are essentially free of obstacles. The probability of finding such empty regions depends only on its volume and not its shape. However given regions of varying shapes and equal volumes, it will be entropically more favorable for a long polymer chain to reside in a region whose shape is closest to a sphere. This is because the confinement entropy is maximized for a sphere over other shapes of the same volume. The argument is equivalent to that proposed by Lifshits [19] in the context of electron localization and is shown rigorously by Luttinger [33]. For V < V1 the relevant regions contain a small number of obstacles but we believe that the same argument should roughly hold and deviations from a spherical shape will be small or irrelevant. We have compared our analytical results with numerical simulations performed by Dayantis et al. [30], and also comment on the relation to earlier simulations done by Baumgartner and Muthukumar [28]. Dayantis et al. carried out simulations of free chains (random-flight walks) confined to cubes of various linear dimensions 6 - 20, in units of the lattice constant. These chains can intersect freely and lie on a cubic lattice. They introduced random obstacles with concentrations x = 0, 0.1, 0.2 and 0.3. The length of the chains vary between 18 - 98 steps. They also simulated self-avoiding chains that we will not discuss here. They measured the quenched entropy, the end-to-end distance, and also the radius of gyration which is a closely related quantity. Unfortunately, these
257
Polymers in random media: Quantum analogy
-0.8
"
-1.0
... -1.2
...
""
•
'J
~-1.4
(0
c
-
+
I
...J
CQ
-1.8
E
-2.0
...
• ' : t
-1.6
18
"... •
38
I
98
58
.I
I
' I
-2.2
l ... I
-2.4
t
•
-2.6 2.5
3.0
3.5
In ( In VI lln(1-x)
4.0
4.5
I)
Figure 6. A plot of ln(-S/L +ln6) vs. ln(lnV/Iln(1- x)l)- The labels are marked according to the chain length.
authors did not have a theoretical framework to analyze their data, and thus could not make it collapse in any meaningful way. We show below how it is possible to fit the data nicely to our analytical results. Even for x = 0.1, the value that we get for V1 is about 33 which is an order of magnitude smaller than the the smallest volume used in their simulation, which is 216 for a cube of side 6. Hence we expect to be in region II. To check the agreement with our analytical results we show in Figure 6 a plot of ln(-S/L + ln6) vs. ln(lnV/Iln(1- x)l) where S is the entropy measured in the simulations and V = B 3 for a box of side B. Recall that F = -S and Eq. (82) predicts a straight line with slope -2/3. The best fit is obtained for a slope of -0.72 ± 0.05, which is in excellent agreement with our analytical results in region II. In order to analyze the simulation results for the end-to-end distance and radius of gyration we have to introduce some additional compensation for the results obtained in the previous section. First we must realize that Eq. (81) is valid only when the number of steps (monomers) is very large. In the simulations they used chains of varying lengths whose size did not yet reach asymptotia. Hence, we introduce a correction factor
(91) which interpolates between the size of a free chain as L -+ 0 and the value of Rm from Eq. (81) as L-+ oo. The second correction we have to implement arises when the expected value of the chain is not much smaller than the size of the confining box. Even for a free chain confined to
258
Y. Y. Goldschmidt and Y. Shiferaw
a box of side B with no obstacles present, the end-to-end distance is not simply R = £ 112 for £ 112 < B and R = B for larger L. We have to take into account the fact that the length of the chain has a Gaussian distribution about its expected value, and the tail of the Gaussian is cut off by the presence of the box (this is for the absorbing boundary conditions that is used in the simulations). Thus, for the case of no obstacles (x = 0), The measured end-to-end distance should approximately be R~
=
which gives
B 1
R2
1B
R2
dR R 2 exp(--)/ dR exp(--), -B 2L -B 2L
Rc = .JLh(B/.JL)
h(x)=
(
1-
(92)
with
f£ xy'2exp(-x /2) Y:;;: erf(x/ 2)
1/2
2
)
(93)
This indeed gives good agreement with the measured values in the no obstacle case. For the obstacle case we thus have to introduce these two corrections in subsequent order: (94) where Rm = Rmn as given by Eq. (81). (A constant of proportionality of 1.8 has been introduced on the rhs of Eq. (81) to obtain a good fit). In Figure 7 we show a comparison of the simulation results for the end-to-end distance with the calculated results as given by Eq. (81) and Eq. (94). The agreement seems remarkable, since all the data collapses to a straight line with a slope close to 1. Dayantis et al. emphasize that they did not consider concentrations of obstacles above the percolation threshold, which is at Xc = 0.3116 for a simple cubic lattice. The reason is that above the percolation threshold the medium of random obstacles begins to form disconnected islands free of obstacles. Thus, in their simulation the polymer chain will only sample a limited fraction of the volume available. What happens is that effectively the volume available for the chain is not the total volume of the cube but rather the volume of the disconnected region it occupies. For most realizations of the random medium this effective volume will be smaller than the value V1 , which is the limit of region I of the last section. In that case one expects the end-to-end distance to scale like x- 1 as given in Eq. (78) instead of like x- 113 as given by Eq. (81). Baumgartner and Muthukumar's simulation was for both below the percolation threshold and also above it (x = 0.4 and 0.5). However, they only estimate the exponent above the percolation threshold, and find it to be about -1. They do not estimate the exponent for x below the percolation threshold, which appears from their data to scale with a much smaller exponent. Thus, it seems likely that the reason these authors report a behavior corresponding to region I, even though their box is quite large, is because the effective volume is small for the cases for which they exceed the percolation threshold.
259
Polymers in random media: Quantum analogy
Y18 &38
es8
•98
,. Y.
•
5 Rcalculated
Figure 7. A plot of the observed vs. calculated end-to-end distance
4. LOCALIZATION AND DELOCALIZATION OF POLYMERS WITH SELFAVOIDING INTERACTION IN THE PRESENCE OF DISORDER In the previous sections an ideal (Gaussian) chain has been used, which corresponds approximately to the experimental situation at the so called 8-temperature when the solvent effectively screens the self-avoiding interaction of the chain. In some early papers [28,11,34] there was an attempt to include the effects of the self-avoiding interaction of the polymer. These attempts were far from complete. For example in Ref. [11] it was assumed that the conformation of the polymer consists of one spherical blob and it was argued that a quenched random potential is irrelevant for a very long chain when a self-avoiding interaction is present. In Ref. [34] analytical results were obtained for annealed disorder, and simulations were performed for strictly self-avoiding walks. Ref. [28] presents numerical evidence for a size transition of the polymer as a function of the relative strength of the disorder and the self-avoiding interaction. The simulations were carried out for a random distribution of hard obstacles with a concentration exceeding the percolation threshold. In a recent paper [35] we tried to shed more light on this important problem. We have made use mainly of Flory-type arguments, and considered both the case of a Gaussian random potential and the case of randomly placed obstacles. Note that a polymer with self-avoiding interactions cannot be mapped into a quantum particle at a finite temperature in a simple manner, because for a quantum particle there is no impediment to return at a later time (or Trotter time) to a position it visited previously. An important point to keep in mind is the strength of the excluded volume interaction. If one considers a strictly self-avoiding walk on a lattice (SAW) corresponding to a non-self-
260
Y. Y. Goldschmidt and Y. Shiferaw
intersecting chain, then the strength of the Edwards parameter v [6] is fixed at 0(1) x a 3 where a is the step size (or monomer size) and depends only on the type of lattice. On the other hand one can consider a Domb-Joyce model [36] where there is a finite penalty for self overlapping of polymer segments, and then the strength of v can be varied substantially and reduced continuously to zero. The interplay between the strength of the self-avoiding interaction and the strength of the disorder can then be investigated to a larger extent. Experimentally the Edwards parameter is given approximately [14] by v = a 3 (1 - 2x), where x is the Flory interaction parameter, which depends on the chemical properties of the polymer and the solvent, and on the temperature (and pressure). It takes the value 1/2 at the 8-point. The case x = 0 corresponds to a solvent that is very similar to the monomer. In general good solvents have low x whereas poor solvents have high x resulting in v being negative. In the following we will restrict ourselves to the case of positive v, which leads to the more interesting and non-trivial results. We now revisit the meaning of the term localization as applied to polymers in a random medium. Although some authors connect the compact size of the chain when L -+ oo with the notion of localization, this is actually not so. The compact size should be viewed as a separate feature from the notion of localization. Recall that for a Gaussian chain in an uncorrelated Gaussian random potential of variance g the chain has typical size
RF ex (g ln V)-l/( 4 -dl,
(95)
and the binding energy per monomer is given approximately by Ubind/ L
rv -
(g ln V) 2/( 4 -d).
(96)
To insure localization, the binding energy of a chain Ubind has to exceed the translational entropy ln V. From Eq. (96) this amounts to the condition ln V < L(g ln V) 2/( 4 -d),
(97)
which holds for any 2 :::; d < 4 when L is large enough (for 2 < d < 4 and any fixed L, the condition can be satisfied for large enough V) . This condition assures that the polymer will stay confined at a given location and will not, under a some small perturbation move to a different location. Thus repeating an experiment or a simulation with the same fixed realization of the disorder, but with different initial conditions, will result in finding the polymer situated at the same region of the sample as in a previous experiment, provided of course one waits enough time (which can be enormous) for the system to reach equilibrium. We observe that this condition is satisfied for large enough L provided the binding energy per monomer is positive. Another interpretation of the inequality given above in the context of equilibrium statistical mechanics is that the partition sum is dominated by the term involving the ground state as opposed to the contribution of the multitude of positive energy extended states. The contribution of these states is proportional to the volume of the system and thus the inequality above results from the condition exp(-LE0 ) > V.
(98)
What we will see in the following sections is that in the presence of a self-avoiding interaction, a localization-delocalization transition occurs when varying the strength of the the self-avoiding interaction for a fixed amount of disorder or alternatively upon varying the strength of the disorder for a fixed value of the self-avoiding interaction.
261
Polymers in random media: Quantum analogy
4.1. A self avoiding chain in a random potential Consider first the case of a random potential with a Gaussian distribution. For simplicity, the discussion in the rest of this section be limited to three spatial dimensions (d = 3). Recall that in the case when there is no self-avoiding interactions the optimal size of a chain Rm is found by minimizing the free energy F in Eq. 23. This yields Rn,rv
1
1 - G(V)'
(99)
=-
g log(V)
where we defined the volume dependent disorder strength by G(V) this result in F we obtain L Fm = -3R;,. -
R:j
= g ln(V).
Substituting
-G(V )2 L .
(100)
We see that - F m/ L is the binding energy per monomer, and it is strictly positive, so the polymer is localized. In what follows we will assume that g is small enough so that G « 1 for the given system volume, hence Rm » 1, and the chain is not totally collapsed unless
v-+ 00. We now add a self avoiding interaction and assume first that it is small, i.e. v << g, or at least v < g. If the chain is still localized in the same well, which we will see momentarily not to hold when L is large, then L F=--L R2m
g log(V)
R3
m
+ v£2 R3
"-J
rv
-G(V)2 L
+ vG(V)3 £2 .
(101)
m
Here, besides assuming that v is small we assume for the moment that L is not too big so the last term in the free energy, resulting from the self-avoiding interaction, is small enough so one does not have to take into account the change in Rm due to the presence of v. If we plot F vs. L, we see that it is lowest when 1
L
(102)
= Lm = 2vG(V)"
Thus if vg
«
1, we have Lm
»
1, and
Fm = - G(V). 4v
(103)
For L = 2Lm the free energy vanishes and for larger L it eventually increases fast like £ 2. We can now verify that if L does not exceed Lm then the approximation used above, assuming that Rm does not change appreciably from its ideal chain value, is justified. If we differentiate the above F in Eq. (101) with respect to Rm we find (104) again omitting coustauts of order uuity. Thus the correction vG L evaluated at L is of order unity, and we can still use the value Rm rv 1/G.
= Lm
262
Y. Y. Goldschmidt and Y. Shiferaw
Figure 8. Conformation of a chain consisting of many "blobs" with connecting segments as L increases. Initially the first blob "overflows" and there is "hopping" to another deep minimum. There is a competition between the length of the connecting segment and the probability to find a deep minimum nearby.
For larger L the approximation seems to break down, but fortunately what happens is that since the free energy increases when L exceeds Lm, it is energetically favorable for part of the chain to jump into a distant well. Even though there is a cost for the polymer segment between the wells one still gains in the overall free energy from the binding energy in the wells. Thus the picture that emerges is that as L increases, the chain divides itself into separate blobs with connecting segments. In each blob the number of monomer does not exceed Lm, which is the optimal value for that well. The idea is depicted in Figure 8. To be more specific we will now construct a model for the free energy of the chain. The first blob will be located in the deepest minimum in the total volume, whose depth is roughly given by -G(V) 2 per monomer, where V is the total volume of the system. Subsequent blobs will reside in the most favorable well within a range Y, which has to be taken self-consistently as the length of the jump. Thus within a range Y the chain is likely to find a potential minimum of depth -G(Y 3 ) 2 • The farther you jump, the deeper well you are likely to find. Thus assuming for simplicity that the jumps are roughly of equal size, and there are K blobs in addition to the initial blob, the free energy of the chain will be given roughly by
263
Polymers in random media: Quantum analogy
with L- L 0 1 3 K = - - , L0 = G(V), G(Y) = 3gln(Y). w+m 2v
(106)
We defined w to be the number of monomers in each blob, and m to be the number of monomers in each connecting segment. The term Y 2lm results from the "stretching" entropy of the segment and miY 2 from the confinement entropy. The term vm 2 IY 3 represents the self-avoiding interaction for the connecting segments. L 0 is the number of monomers in the initial blob whose free energy was taken care of separately. It is evident that when L is very large we can neglect the free energy of the first blob and also take K i=:j Ll(w + m). Thus we find for the free energy per monomer 2
(Y f( w, m, Y) = _ F(w, m, L Y; L) ""'_1_ "' w+m m
2
+ ym2 + vm y3
_
w G(Y3)2
+ vw 2G(Y3)3)
.
(107) This function has to be minimized with respect to w, m and Y to find the parameters giving rise to its lowest value. For the connecting pieces of the chain we did not include a contribution from the random potential since it is expected to average out to zero for these parts. To gain some feeling into the behavior of this function and the values of the parameters which minimize it, we display in Table I the value of the parameters and free energy per monomer for g = 0.05 and various values of v, as obtained from a minimization procedure. The delocalization transition is the point where f changes sign from negative to positive, as discussed earlier. Actually, to be more precise, the delocalization transition occurs when f = - (ln V) I L for finite L, when the translational entropy starts to exceed the binding energy. In the limit of large L we can say that the transition is at f = 0. We observe that the delocalization transition occurs at v = 0.0478 which is close to the value of g. We also observe that m « w for v « g and m » w near the transition. Also for small v, m rv Y, whereas near the transition m rv Y 2. If we compare the value of w from Table I with the value of Lm = 1l(2vG(Y3)) we find that w is smaller than Lm in the entire range. The ratio w I Lm varies from rv 0.4 to 1 as v changes from 10-5 to 0.048. Thus the assumption we have made previously concerning this ratio is justified a posteriori. Luckily it was possible to solve the minimization equations analytically almost entirely in both the limits v « g, and near the transition when f i=:j 0. Details of the solutions are given in the Appendix of Ref. [35]. Here we only display the results: A. The case v «g. The parameters are given by
y = ~(ln y -1)-lf2(ln y v
m
=
w
=
+ 3)-3/2,
2 1 vgln y(ln Y -1)- (ln Y
3
2
3vg 1n
y (ln Y
+ 3)-I,
+ 3)- 1
f = -9g2(ln Y) 2(ln Y- 1)(ln Y + 3)- 1 .
(108) (109)
(110) (111)
Y. Y. Goldschmidt and Y. Shiferaw
264
y v 0.00001 2206 0.0001 346 0.001 60 0.01 13.7 0.02 11.1 12.8 0.03 22 0.04 35.6 0.045 0.0478 69.4 0.048 77 Table 2 Parameters and
m w 2413 16178 534 2580 148 461 66 97 69.2 60.8 105.5 41.9 302.6 26.8 720.1 20.7 2342.4 16.5 16 2800
f -0.835249 -0.421023 -0.164673 -0.0370883 -0.0156444 -0.0056033 -0.0009992 -0.0001776 0 0.00000725
free energy for the case g=0.05
The first equation can be easily solved numerically for Y for a given value of v and the result substituted in the other equations. Very good agreement is achieved with Table I for small values of v. B. Solution near the delocalization transition. Let us define the parameter 4v 3g
(112)
t>,= - .
In terms of this parameter we have
Y=
__!__ VK,
(1+~)2,
(113)
K,
m=-1-(1+~)3, v2 ,.,2 ,.,
(114)
K, 1 - -=-=:-- 8v 2 ln Y - 2vG(Y 3 )"
w-
(115)
The transition point is obtained by solving the equation
1
2 1 +~ + "' +]:_(1-2ln( ( +~))) "' 1 + V1 + "'2 "' ,j3fi,.,2
=0.
(116)
Once the solution "'cis determined for a given g, then the transition point Vc is determined from Vc = 3g,.,c/4. For g in the range 0.01-0.2 we find that "'c is a number of order unity (varies from 1. 7 to 0.96 as g changes in that range). This means that Vc is quite close to g. Once Vc is known, all the parameters Y, m and w at the transition are determined by the solution above. For g = 0.05 we get "'c = 1.2744, and thus Vc = 0.0478 in excellent agreement with the minimization results from Table 2.
265
Polymers in random media: Quantum analogy
T R
1 Figure 9. A typical chain conformation when v ~g. The dark regions are regions of low average potential. Only short segments of the chain are situated in these regions.
For v > Vc the chain is delocalized. The above expression for the free energy may no longPx he accurate, but the general picture is clear. There will be very few monomers in the low regions of the potential, and the chain will behave very much like an ordinary chain with a self-avoiding interaction in the absence of a random potential. Any little perturbation can cause the chain to move to a different location in the medium (see Fig. (9)) . For small values of v when the chain is localized, we still expect its size to grow like that of a self-avoiding walk. Thus we expect roughly
(117) since Y is the step size, and K is the number of steps. But sinee K large L, we find Rg'""
y
(m + w)0.6
Lo.6
.
~
L/(m + w) for (118)
Thus the chain behaves as a self avoiding walk with an effective step size per monomer given by aeff _ Y a (m + w) 0 ·6 ·
(119)
From Table 2 it becomes clear that the effective monomer size changes from a value of 6 for v = 0.00001 to a value of'"" 0.66 at the transition. The reason for the large value
Y. Y. Goldschmidt and Y. Shiferaw
266
of the effective monomer size at very small v is that the chains makes long jumps to take advantage of deep wells of the random potential, very much like anchored chains in a random potential that make sub-ballistic jumps [13]. This is also the reason why the number of monomers w in each well is somewhat less than Lm , since a sufficient number of monomers need to be used for the connecting segments. For v » Vc in the delocalized phase we expect the effective monomer size to be about 1, since the chain behaves almost like an ordinary self-avoiding chain with the random potential not playing any significant role. Thus the chain is expected to have its smallest size in the vicinity of the transition. It is important to notice that the discussion above is in the limit for very large L. If L :5 1/(2vG(V)), then in the localized phase when v « Vc the polymer will be confined to a single well and will appear compact, even though it will not remain so for large L. This may explain why in simulations that were done typically with L :::; 320 [28] the delocalization transition appeared as a transition from a compact to a non-compact state of the polymer. We should also note that for an annealed random potential there is a transition from a collapsed state into an ordinary self-avoiding chain as v increases through the point v = g [11]. This is because the annealed free energy reads F(R)
=
L g£2 R2 - Ji3
v£2
R2
+ Ji3 + y·
(120)
For v < g the fourth term is negligible and the free energy is lowest when R-+ 0 (when L is large). For v > g the first term is negligible, and the radius of gyration grows like
(121) 4.2. A self avoiding chain in a sea of hard obstacles We now turn to the case of hard obstacles. This case of was discussed at length in the previous section in the absence of a self-avoiding interaction. We will make use of the results applicable to three spatial dimensions. Three different behaviors were identified as a function of the system's volume. Region I is defined when the system's volume V < V1 rv exp(cr/ y!X), where c 1 is a constant of order unity. Here 0 < x < 1 is the average concentration of obstacles per site (total number of obstacles divided by total number of sites). We also assume that x is less than the percolation threshold (xc = 0.3116 for a cubic lattice), so sites occupied by obstacles don't percolate. In Region I we recall from the previous section that in the absence of a self-avoiding interaction, the free energy per site for a chain situated in a spherical region of volume R 3 in three dimensions is given by
FJ/ L = -ln(z)
+ 1/ R 2 + x
(122)
where x is the actual concentration of obstacles in that region, whose minimal expected value in a system of total volume V is Xm '="'
x- Vx~3v.
(123)
The binding energy per monomer inside the blob resulting from the lower concentration of obstacles in this region is given by x- x, since it is equal the entropy gain from a lower
Polymers in random media: Quantum analogy
267
concentration as compared to the average (background) concentration x. The chain is "sucked" towards regions with low concentration of obstacles since it can maximize its entropy there, and these regions of space act like the negative potential regions of the Gaussian random potential. Thus in order for the free energy per monomer to reflect correctly the binding energy of the chain inside the blob, both the constant x of the background and the constant term -ln(z), which is always there regardless of the chain's position, have to be subtracted. The relevant free energy per monomer situated in the blob (which is equal to minus the binding energy) is given by
f = I
_.!.__ _
R2
JxlnV R3 .
(124)
This result coincides with Eq. (23) upon the substitution g -+ x. Thus all the results of the previous section carry on to Region I with this simple substitution . Therefore we are going to discuss the situation when the system's volume is greater than V1 (Region II). In this case the many blob picture still holds, where the blobs are now situated in regions free of obstacles (with x = 0 ) whose size is determined again by the distance Y of the jump which is also assumed to satisfy Y 3 2: V1 ( an assumption which will be justified a posteriori). In this case the farther the jump, it is more likely for the chain to find a larger space empty of obstacles, which will reduce further its confinement entropy w / R 2 and also the self-avoiding energy ( but there is a cost resulting from the connecting segments and the constraint of the total length being fixed). The blob size is given by Rmn = 1/Go (the largest expected empty region in a volume Y 3 ) with [31] (125) Thus the free energy per monomer of a chain consisting of a number of blobs with connecting parts, each of length Y, is given by
where again the constant x, which is independent of m, w and Y, has been subtracted. This assures that the delocalization transition again occurs at f = 0 and not at f = x. The results of a numerical minimization of this free energy is displayed in Table 3 for the case of x = 0.1. The transition occurs between v = 0.041 and 0.042. Again we could find analytically almost the entire solution both for v ~ x and at the transition. The solution is given in the Appendix. There we show that the transition occurs at v = 0.04142. We observe that as for the case of a Gaussian random potential the ratio w/m changes from» 1 to« 1 as v approaches the transition from below. The values of Y are seen to be consistent with the assumption Y 3 2: V1 . We also checked that the free energy from Table II is lower than what one would obtain by constraining Y to be in Region I, i.e. Y 3 < V1 . Finally for V > V2 , where V2 ~ exp (x 2 /(d+ 2 l£d/(d+ 2l) (Region III), we expect the behavior of the chain to stay the same. This is because for jumps within a volume V2 the situation reverts to the previously discussed scenario, since the effective volume of interest
Y. Y. Goldschmidt andY. Shiferaw
268 v
y 552 127 38 23 32 59 194 248 357
m
w
0.00001 2206 72092 0.0001 9135 565 0.001 206 1241 230 207 0.01 0.02 536 137 0.03 1737 120 0.04 13915 131 21327 134 0.041 0.042 39310 144 Table 3 Free energy and parameters for
f -0.062 -0.051 -0.034 -0.0077 -0.0019 -0.00035 -0.0000099 -0.0000023 0.0000024
a polymer in a sea of blockers when V > V1 .
which determines the statistics of the free spaces is of order Y 3 and we don't expect Y to be that large. An important point to note is that if one performs a simulation with strict self-avoidingwalks on a diluted lattice (with x < 1), one has v rv 1, and hence one will always be in the delocalized phase and will not see any localization effects [37]. A few words are in order about the case of an annealed potential. This case has been already investigated in the literature [34], and we will review it briefly. The free energy in the annealed case reads (ford= 3) (127) The second term represents the entropy cost of a fluctuation in the density of obstacles that creates an appropriate spherical region of diameter rv R. It was assumed that the chain occupies a spherical volume, or at least deviations from a spherical shape are not large [34]. In the case v = 0 one obtains by minimizing the free energy that R rv (L/x) 115 , a well known result. For v > 0 the first term is irrelevant (and so is the "stretching term" of the form R 2 / L) and one finds that R rv ( v /x) 113 £ 1/ 3 • In d dimension, the size scales like L 1fd when v > 0, which is larger than the £ 1/(d+ 2) dependence in the v = 0 case. There is no indication for a phase transition in these arguments, although some authors [34] speculate that it breaks down for large v and a transition to a Flory £ 3/(d+ 2l dependence takes place.
5. CONCLUSIONS In this chapter we have demonstrated the rich behavior of polymer chains embedded in a quenched random environment. As a starting point, we considered the problem of a Gaussian chain free to move in a random potential with short-ranged correlations. We derived the equilibrium conformation of the chain using a replica variational ansatz, and highlighted the crucial role of the system's volume. A mapping was established to that of a quantum particle in a random potential, and the phenomenon of localization was explained in terms of the dominance of localized tail states of the Schrodinger equation.
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We also gave a physical interpretation of the 1-step replica-symmetry-breaking solution, and elucidated the connection with the statistics of localized tail states. Our concusions support the heuristic arrguments of Cates and Ball, but it starts with the microscopic model. We then proceeded to discuss the more realistic case of a chain embedded in a sea of hard obstacles. Here, we showed that the chain size exhibits a rich scaling behavior, which depends critically on the volume of the system. In particular, we showed that a medium of hard obstacles can be approximated as a Gaussian random potential only for small system sizes. For larger sizes a completely different scaling behavior emerges. Finally we considered the case of a polymer with self-avoiding (excluded volume) interactions. In this case it was found that when disorder is present, the polymer attains the shape like that of a pearl necklace, with blobs connected by straight segments. Using Flory type free energy arguments we analyzed the statistics of these conformational shapes, and showed the existence of localization-delocalization transition as a function of the strength of the self-avoiding interaction. The work described in this chapter is concerned with static (equilibrium) properties of polymers in random media. There is a lot of theoretical work still to be done related to the dynamics, and especially nonequilibrium properties of polymers in random media. This is also of practical importance, for example for the separation of chain of different length or mass like DNA molecules under the effect of an applied force when embedded in a random medium like a gel [38]. Acknowledgements: This work was supported by the US Department of Energy (DOE), Grant No. DE-FG02-98ER45686. REFERENCES
1. G. Guillot, L. Leger and F. Rondelez, Macromolecules 18 (1985) 2531. 2. M.T. Bishop, K.H. Langley, and F. Karasz, Phys. Rev. Lett. 57 (1986) 1741. 3. L. Liu, P. Li, and S.A. Asher, Nature (London) 397, 141 (1999); L. Liu, P. Li, and S.A. Asher, J. Am. Chern. Soc. 121 (1999) 4040. 4. J. Rousseau, G. Drouin, and G.W. Slater, Phys. Rev. Lett. 79 (1997) 1945. 5. D.S. Cannell and F. Rondelez, Macromolecules 13 (1980) 1599. 6. M. Doi and S. F. Edwards, The Theory of Polymer Dynamics, Oxford University Press, Oxford, 1986. 7. R. P. Feynman, Statistical Mechanics: A Set of Lectures, Benjamin, New York, 1972. 8. Y. Y. Goldschmidt, Phys. Rev. E 53 (1996) 343. 9. D. R. Nelson and V. M. Vinokur, Phys. Rev. B 48 (1993) 13060. 10. Y. Y. Goldschmidt, Phys. Rev. B 56 (1997) 2800. 11. T. Nattermann and W. Renz, Phys. Rev. A 40 (1989) 4675. 12. W. Ebeling, A. Engel, B. Esser and R. Feistel, J. Stat. Phys. 37 (1984) 369. 13. M. E. Cates and C. Ball, J. Phys. (France) 89 (1988) 2435. 14. P. G. de Gennes, Scaling Concepts in Polymer Physics, Cornell Univ. Press, Ithaca, 1979. 15. S. F. Edwards and M. Muthukumar, J. Chern. Phys. 89 (1988) 2435.
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16. P. W. Anderson, Phys. Rev. 109 (1958) 1492. 17. B. Souillard, in Chance and Matter, Proceedings of the Les Bouches Lectures, 1986, Eds. J. Souletie, J. Vannismenus and R. Stora, North-Holand, Amsterdam, 1987. 18. J. Frohlich and T. Spencer, Commun. Math. Phys. 88 (1983) 151. 19. I. M. Lifshitz, S.A. Gredeskul, and L.A. Patur, Introduction to the Theory of Disordered Systems, Wiley, NY, 1988; I. M. Lifshits, adv. Phys. 13 (1964) 483. 20. Y. Shiferaw andY. Y. Goldschmidt, Phys. Rev. E 63 (2001) 051803. 21. M. Mezard, G. Parisi and M. A. Virasoro, Spin Glass Theory and Beyond, World Scientific, Singapore, 1987. 22. Y. Y. Goldschmidt, Phys. Rev. E 61 (2000) 1729. 23. E. I. Shakhnovich and A.M. Gutin, J. Phys. A 22 (1989) 1647. 24. M. Mezard and G. Parisi, J. Phys. I France 1 (1991) 809. 25. W. H. Press, et al., Numerical Recipes in Fortran, 2nd ed., Cambridge Univ. Press, 1992. 26. J. P. Bouchaud and M. Mezard, J. Phys. A: Math. Gen, 30 (1997) 7997. 27. B. Halperin, Phys. Rev. 139 (1965) A104. 28. A. Baumgartner and M. Muthukumar, J. Chern. Phys. 87, 3082 (1987); see also review chapter by these authors in Advances in Chemical Physics (vol. XCIV) Polymeric Systems, I. Prigogine and S. A. Rice editors, John Wiley & Sons, Inc., New York, 1996, and references therein. 29. K. Leung and D. Chandler, J. Chern. Phys. 102 (1995) 1405 ; D. Wu, K. Hui, D. Chandler, J. Chern. Phys. 96 (1992) 835. 30. J. Dayantis, M.J.M. Abadie, and M.R.L. Abadie, Computational and Theoretical Polymer Science, Vol. 8 (1998) 273. 31. Y.Y. Goldschmdit andY. Shiferaw, Eur. Phys. J. B 25 (2002) 351. 32. W. Feller, An Introduction to Probability and its Applications, Wiley, New York, 1963. 33. J. M. Luttinger in Path Integrals and their Applications in Quantum, Statistical, and Solid State Physics, G. J. Papadopoulos and J. T. Devreese eds., Plenum Press, New York, 1978. 34. J. D. Huneycutt and D. Thirumalai, J. Chern. Phys. 90 (1989) 4542. 35. Y. Y. Goldschmidt andY. Shiferaw, Eur. Phys. J. B. 32 (2003) 87. 36. C. Domb and G. S. Joyce, J. Phys. C 5 (1972) 956. 37. K. Barat and B. K. Chakrabarti, Phys. Rep. 258 (1995) 377. 38. Jean-Louis Viovy, Rev. Mod. Phys. 72 (2000) 813.
Statistics of Linear Polymers in Disordered Media Edited by Bikas K. Chakrabarti © 2005 Elsevier B.V. All rights reserved.
Geometric properties of optimal and most probable paths on randomly disordered lattices Pratip Bhattacharyyaa and Arnab Chatterjeea aTheoretical Condensed Matter Physics Division, Saha Institute of Nuclear Physics, 1/ AF Bidhannagar, Kolkata 700 064, India.
This chapter reviews the geometry of optimal paths and most probable paths, expressed as the relation between the length and the end-to-end distance, on lattices with randomly disordered bond energies and the methods of determining them. Optimal paths are defined at zero temperature as the paths of minimum energy and most probable paths are defined at finite temperatures as the paths of minimum free energy. Optimal paths are always selfavoiding. While directed optimal paths are always self-affine, irrespective of the strength of the disorder, non-directed optimal paths are self-affine only when the bond energies are weakly disordered and these have self-similar forms when the bond enrgies are strongly disordered. Most probable paths undergo a phase transition at a finite temperature: the paths conform to the condition of minimum energy in the low temperature phase and to the condition of maximum entropy in the high temperature phase. For the most probable directed paths the phase transition occurs in 3 + 1 and higher dimensions whereas for the most probable non-directed paths the phase transition occurs in all dimensions. The review ends with a discussion on the classic optimization problem of the shortest path of a travelling salesman through a set of randomly chosen sites on a lattice.
1. INTRODUCTION
Paths are conceived as lines in a geometric space. In a simple way, the geometry of a path is expressed as the relation between its length and its end-to-end distance. An abstract path assumes a concrete form when it is associated with a physical object, for example, the trail of a moving particle, the course of a river, the configuration of a linear polymer and the contour of a domain. Optimal paths and most probable paths are two classes of such concrete forms. While optimal paths are defined for zero temperature and are always self-avoiding, most probable paths are defined in the thermodynamic sense for all temperatures and may have self-intersections. This is a review of the geometric properties of optimal and most probable paths on lattices with randomly disordered bonds; it contains a brief description of the methods used to study these paths and the main characteristics of their forms. 271
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P. Bhattacharyya and A. Chatterjee
1.1. The notion of the optimal path The notion of the optimal path is derived from the notion of the shortest geometrical path between two points by generalizing the latter to a form that satisfies a condition of minimum weight. The shortest geometrical path between two points in a given space is the path along which the distance between them is minimum; it is the optimal path between the two points when the weight of the path is defined as its length. In Euclidean space the shortest geometrical path between two points is the straight line connecting them; on the surface of a sphere it is the smaller arc of the geodesic connecting the two points; whereas on a graph it is the minimum set of edges from one point to the other. If the weight is a quantity other than the length of the path, the optimal path is not necessarily the one which is geometrically the shortest, for example, when the weight of a path is defined as the time or the energy required by an object to move along it from one end to the other. The weight of a path is a measure of the total interaction between the object on the path and the medium in which the path is embedded. In free space there is no cause of physical interaction and the geometrical length of the path serves as its weight. The length of the path also serves as its weight when the space is occupied by a homogeneous medium, since, in a homogeneous medium the interaction is the same at all points and quantities such as the time and the energy of transport, or, the energy of remaining adsorbed are directly proportional to the length of the path. However, in a disordered medium, the path of least time or of minimum energy does not usually coincide with the path of minimum length and the quantity to be considered as the weight is determined by the specific requirement of the physical problem. A physical medium is represented on a lattice by assigning a value of energy to each of its bonds; it represents the energy of interaction between the object at the location of the bond and the medium. The energy of a path is therefore the sum of the energies of all its constituent bonds. If all the bonds are assigned the same value of energy, the lattice represents a homogeneous medium and paths of the same length have the same energy. A disordered medium is simulated by assigning different values of energy to the lattice bonds; paths of the same length no longer have the same energy and hence arises the problem of determining the optimal path. This also illustrates the difference between the shortest geometrical path and the optimal path, defined as the path of minimum energy, between two sites since the bonds of the lattice may be assigned energy values such that a path, other than the shortest path, between the two sites has the least energy. This happens almost always when the bond energies are randomly disordered. The disorder is random if the bond energies are uncorrelated:
(1)
where Eij is the energy of the bond between two neighbouring sites Ai and Aj on the lattice and the angular brackets denote the average over all bonds of the lattice. Most of this review considers lattices with randomly disordered bond energies as the media in which the geometric properties of optimal and most probable paths are studied.
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273
1.2. The notion of most probable paths At finite temperatures the notion of the optimal path is replaced by the notion of most probable paths . When the optimal path is defined as the path of least energy it is also the most probable path, in the thermodynamic sense, at zero temperature. However at a finite temperature two competing factors decide the course of the most probable paths in thermodynamic equilibrium: while the paths tend to follow the course of minimum energy, they also tend to follow the course of maximum entropy. Consequently the most probable paths are the paths of minimum free energy-the free energy of a path is now considered as its weight. On a lattice with homogeneous bond energies paths with equal lengths are equally probable only when the end-to-end distances are equal. However, on a lattice with disordered bond energies, the probabilities of the paths differ even for those that are equal in both the lengths and the end-to-end distances. 1.3. Geometric description of paths Every path in a given space is characterized by two geometric quantities: the length and the end-to-end distance. The length L of a path is the distance measured along the path from one end to the other, while the end-to-end distance R is measured directly through the space in which the path is embedded. Both the distances L and R are measured by using the same metric. The ones commonly used are the Euclidean metric, the Cartesian metric and the link metric. The Euclidean metric is used mainly for measuring distances in continuum space; using this metric the distance between two points with coordinates (1) , x (2) , ... , x (D)) an d x -= ( x (1) , , x (2) , ... , x (D)) m . a D - d"1mens10na . l space 1s . x 1 -= ( x 1 2 1 1 2 1 2 2 given by:
(2) In the Cartesian metric the distance between the two points is given by: D
lx12l
=
L Hd)- x~d)l,
(3)
d=1
which makes it suitable for measuring distances on a lattice. The link metric is used for graphs (that includes all kinds of lattices) where the distance along any path between two vertices is given by the number of edges of the graph from one point to the other along that path. The element of length in the link metric is a unit edge of the graph. The relation between the length and the end-to-end distance of a path determines its universality class. This relation is usually in the form of a power-law
(4) where the exponent (portrays the extent of wandering of one end of the path from other. Many of the paths that occur in natural or that are generated by artificial processes belong to three major classes: (a) The simplest of these is the universality class of ordinary random walks on regular lattices [1], characterised by the relation: Rex £1/2
(5)
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in all spatial dimensions. (b) The universality class of self-avoiding random walks on regular lattices have the property
(6) in two dimensions [2-5]; in three dimensions ( is approximately 3/5 whereas in four and higher dimensions the value of (is equal to 1/2 [2] and the universality class of self-avoiding random walks merges with that of ordinary random walks. However, on irregular lattices, such as fractal lattices, the exponent ( for random walks, ordinary or self-avoiding, deviate from its regular value; this point is addressed in detail in the other chapters of this volume. (c) Paths in the universality class of directed walks (or, directed polymers) have end-toend distances that are directly proportional to the respective lengths [6], whatever be the dimension of the embedding space:
Rex L.
(7)
The values of the exponent in Eqs. (5), (6) and (7) imply that, for a fixed path-length L, wandering is minimum for paths that are in the universality class of ordinary random walks (( = 1/2) and maximum for paths that are in the universality class of directed walks (( = 1). On the contrary, for a given end-to-end distance, paths in the universality class of ordinary random walks are much longer than those in the universality class of directed polymers. There are two kinds of approaches to the problem of determining the relation between the length Land the end-to-end distance R of optimal paths. In the first kind of approach the end-to-end distances of the optimal paths are to be determined as a function of their preassigned lengths. The second kind of approach is the converse of the first: lengths of the optimal paths are to be determined as a function of their preassigned end-to-end distances (for example, when both the endpoints are preassigned). The first kind of approach appears in the study of linear polymers (or self-avoiding walks) [7,8] whereas the determination of first passage times [9] is an example of the second kind. At zero temperature the optimal paths in both cases are found exactly; consequently Land Rare measured directly and the relation between them is determined by observing the variation of the one with the other. Unlike the case of optimal paths at zero temperature, the most probable paths at finite temperatures are not unique and therefore cannot be determined exactly. Their geometric properties are determined by the method of statistical mechanics that begins with the construction of the partition function. Similar to the case of optimal paths there are two kinds of approaches: either the mean (or the most probable) end-toend distance is calculated as a function of the path-length or, conversely, the mean (or the most probable) path-length is calculated as a function of the end-to-end distance. Studies of directed paths usually follow the first kind of approach, whereas studies of non-directed paths use both kinds of approaches.
2. DIRECTED OPTIMAL PATHS AT ZERO TEMPERATURE 2.1. Construction of directed paths The common notion of a directed path from a point A to a point B is depicted in graph theory: it is represented by an ordered pair {A, B} of end-points. Similarly a directed
Optimal and most probable paths
275
path through several points is represented by the set of points arranged in the order in which they appear on the path. This is a topological representation. In physical contexts a directed path is defined with respect to a longitudinal direction, with the implication that the direction opposite to it is forbidden. The embedding space for the directed path is designated D + 1 dimensional; one of the dimensions contains the longitudinal direction whereas the other D dimensions are transverse to it. Every element of length along a directed path will have a component in the longitudinal direction. It incorporates the topological notion since the existence of a directed path from A to B ensures that there will be none directed from B to A. The position of a point in the D + 1 dimensional space, with respect to a chosen origin, is denoted by (x, t) where x = (x 1 , x 2 , ... , x D) are the coordinates of the point in the transverse dimensions and t is its coordinate in the longitudinal direction. In the simplest construction of directed paths in D+ 1 dimensions, the embedding space is one of the 2D+l parts into which the D + 1 dimensional hypercubic lattice is divided by the coordinate planes through the origin. The chosen part of the lattice is oriented to put the origin at the apex. The longitudinal direction is assumed to be along the diagonal of the lattice pointing downward from the apex. Consequently all bonds of this oriented lattice are constrained to point downward; from each site of the lattice there are D + 1 downward bonds. The trail of a walk on the lattice along a sequence of downward bonds defines a directed path. By construction every directd path is self-avoiding. Each lattice bond is assumed to be of unit length so that all lengths measured with the Cartesian metric are in terms of the number of lattice bonds (hence equivalent to the lengths measured with the link metric). If this part of the lattice is further truncated by a plane that intersects the coordinate axes at a distance L from the origin, the remaining segment has the geometry of D + 1 dimensional delta. Figure 1 shows a 1 + 1 dimensional lattice with the delta geometry. With the Cartesian metric all directed paths from the apex to the base of the delta have a common feature: all of them are of the same length L and the end-to-end distance is always equal to the length:
(8)
R=L.
On the D + 1 dimensional delta there are (D + 1)£ distinct directed paths of length L from the apex to the base; for example, in 1 + 1 dimensions there are 2£ directed paths of length L with L + 1 distinct endpoints on the base of the delta. Directed paths such as these that have a fixed length L are called directed polymers [6]. On the lattice delta directed polymers are characterised by the relation of Eq. 8. When the bond energies are randomly disordered, the directed paths on the delta are called directed polymers in random media (DPRM) . Studies on DPRM usually assume that the randomly disordered bond energies are distributed uniformly on the unit interval: p(c)
= 1,
0 < c:::; 1
(9)
or follow the normal distribution with zero mean and unit standard deviation: p(c)
= (1/J2;) exp( -10 2 /2),
(10)
both of which represent weak disorder. The directed path of least energy from the apex of the delta to a given point on the base is a locally optimal path. Among all the locally
P. Bhattacharyya and A. Chatterjee
276
optimal paths from the apex to the base the one for which the energy is again the least (i.e., the best of the local optima) is the globally optimal path on the delta. The problem of determining the relation between L and R of the globally optimal path is trivial, since it is given by Eq. (8) for all directed paths. The problem becomes nontrivial if, instead of seeking the overall end-to-end distance R in D + 1 dimensions, only the transverse end-to-end distance of the globally optimal path (i.e., the separation between its ends in the transverse dimensions) is sought as a function of the path-length L.
2.2. Determination of the directed optimal path: the transfer matrix method for zero temperature A directed locally optimal path of length L is determined by using a transfer matrix method [10] which requires time that increases with the length of the path only as a plynomial in L. For simplicity the method is described in 1 + 1 dimensions, referring to Figure 1. The position of a lattice site on the delta is denoted by (x, t) where x and tare the coordinates of the site in the transverse and the longitudinal direction respectively. Since the apex of the delta is the origin of the lattice its position is denoted by (0, 0). In this particular orientation of the lattice the coordinates, x and t, of any site are either both even or both odd; for any given value of t, the transverse coordinate x assumes alternate integer values from -t to t. Moreover the longitudinal coordinate t also represents the length of a directed path from (0, 0) to (x, t). The transfer matrix method is based on the fact that the directed optimal path from (0, 0) to (x, t + 1) must contain the directed optimal path from (0, 0) to either (x -1, t) or (x + 1, t) since the site (x, t + 1) can be reached from the two sites (x- 1, t) and (x + 1, t) only. The optimal path to (x, t + 1) is obtained either from the optimal path to (x -1, t) by appending the bond from (x - 1, t) to (x, t + 1) or from that to (x + 1, t) by appending the bond from (x + 1, t) to (x, t + 1), selecting the combination that results in the path of lower energy. If E(x, t) denotes the energy of the directed optimal path from (0, 0) to (x, t) and c( x - 1, t) and c( x + 1, t) are the energies of the bonds that are directed towards the site (x, t + 1) from the left and from the right respectively, the transfer matrix method is expressed by the following recurrence relation:
E(x, t + 1) =min [E(x- 1, t)
+ c-(x- 1, t), E(x + 1, t) + c-(x + 1, t)].
(11)
All locally optimal paths directed from the apex to the base of the delta and their energies are determined by applying the above recurrence relation systematically to all points (x, t) from t = 1 to t = L with the initial condition
E(O,O) = 0.
(12)
For each of the sites on the left and the right edges of the delta there is only one directed path from the apex. Hence, for these boundary sites, Eq. (11) reduces to
E( -t, t) E(t, t)
E( -t + 1, t- 1) + c-( -t + 1, t- 1), E(t- 1, t- 1) + c-(t- 1, t- 1).
(13) (14)
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Optimal and most probable paths
I
t
1
A
Figure 1. The delta geometry in 1 + 1 dimensions obtained hy orienting a quadrant of the square lattice so that the origin (0, 0) lies at the apex 0. The longitudinal direction is chosen along the vertically downward diagonal of the square lattice, thus allowing paths that are always directed downward along the lattice bonds. The bold line shows a directed path from the apex 0 to a point A, with coordinates (x, t) , on the base of the delta. Since the site A can only be reached from the sites B and C, the directed optimal path from 0 to A must contain the direeted optimal path from 0 to either B or C.
The globally optimal path of length L is hence identified as the one with the energy E(L) = min [E(x, £)]. X
(15)
This method is extended to higher dimensions by using multiple transverse coordinates.
2.3. Geometric features of the globally optimal path Every directed path of length L in the D + 1 dimensional delta of linear size L has one end fixed at the apex while the other end lies on the base. The longitudial coordinate of the end on the base is the same for all these paths and is given by t = L. Therefore the difference in the positions of the ends on the base is entirely contained in their transverse coordinates x = (x 1 , x 2 , • .. , xv). The transverse end-to-end distance of a path from the apex to a point (x, L) on the base, using the Cartesian metric, is given by: D
lxl = :E lx;l .
(16)
i=l
Since, by construction, all directed paths from the apex of the delta to the base satis(y Eq. (7) (or, the special case of Eq. (8) if the Cartesian metric is used) irrespective of their energies, a further geometrical condition is required to characterize the globally
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P. Bhatta.charyya and A. Chatterjee
optimal path. This necessary condition is provided by the relation between the length of the globally optimal path and the transverse distance between its ends. The value of lxl of the globally optimal path is determined from the position of its ends obtained by the transfer matrix method and the mean value lxl and averaged over a large sample of configurations of the disordered bond energies (denoted by JXi). By varying the size of the delta, JXi of the globally optimal path is determined as a function of the length L; the relation shows that the nature of the globally optimal path is self-affine [10,12]: (17)
where the exponent (J. portrays the wandering in the transverse directions. This relation also appears in nonequilibrium surface growth models in which the surface profiles are equivalent to directed optimal paths and the dynamic exponent for characterising the surface growth is equivalent to the transverse wandering exponent (J. of the optimal path [11,12]. In the most common case of directed paths on a randomly disordered lattice, the uncorrelated bond enegies are drawn from the uniform or the normal distribution (Eq. 9 and Eq. 10). In 1 + 1 dimensions the exponent (J. is found to be exactly 2/3 [10,11,13]. Early estimates of the exponent in 2 + 1 and 3 + 1 dimensions [12] suggested that (J. = 2/3 for all dimensions, but later estimates of (J. obtained from computer simulations of surface growth models showed that its value decreases with an increase in the number of dimensions of the embedding space. Subsequently a simple formula was conjectured [14] for the exponent in D + 1 dimensions: (J. = (D + 3)/[2(D + 2)]. The formula gives the correct value of the exponent for D = 1 and for D-+ oo (where, due to the entropy, the projection of the directed path on the transverse dimensions has the form of an ordinary random walk). However further numerical studies in 2 + 1 and 3 + 1 dimensions appear to disprove this conjecture; recent numerical estimates available are (J. = 0.622 ± 0.001 in 2 + 1 dimensions, (J. = 0.5929 ± 0.0006 in 3 + 1 dimensions and (J. = 0.573 ± 0.001 in 4 + 1 dimensions [15]. Since (J. > 1/2 in all finite dimensions, the transverse wandering of the globally optimal path is superdiffusive; the disorder in the bond energies appear as a kind of noise that makes the globally optimal path to wander farther than the wandering that can occur due to thermal noise alone ((J. = 1/2 for directed random walks). Details of the theoretical analysis of the wandering of the DPRM are discussed in the second chapter by Bhattacharjee in this book. If the random bond energies are drawn instead from a power-law distribution, the value of (J. is found to vary with the power of the distribution [16,17]. Further, if the bond energies are correlated in the transverse directions (which represents a medium with correlated disorder ) while being uncorrelated in the longitudinal direction, the transverse wandering exponent assumes anomalous values [18]. When the correlation is short-ranged, the value of (J. in 1 + 1 dimensions is found to be approximately the same as its value ((J. = 2/3) in the case of uncorrelated Gaussian disorder [19]. For sufficiently long-ranged correlations (J. reaches the limiting value of unity in 1 + 1 dimensions, where the globally optimal path appears like the domain interface in the two dimensional random field Ising model [20]; this is in contrast with the case ofrandom Gaussian disorder where the globally optimal path in 1 + 1 dimensions appears like the domain interface in the two dimensional random bond Ising model [10].
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Eq. (17) describes only the average geometric form of the globally optimal path; it gives the most probable position of the end of the go bally optimal path on the base of the delta as a function of its length. Deviations from the most probable position is depicted in the probability distribution P (x, L) of the position of its end on the base of the delta over the ensemble of the disordered configurations of bond energies. The probability distribution was determined by the transfer matrix method. In 1 + 1 dimensions, with bond energies drawn randomly from a normal distribution, it was observed that [21]
p
(x, L)
ex exp [-
(lxl I L213 f]'
(18)
where the exponent'/ determines the form of the distribution. Numerical data from tranfer matrix studies show that the probability distribution changes from a nearly normal form (i.e., '/ ~ 2) to a clearly non-normal form at lxl I £ 213 ~ 1 [21], whereas its asymptotic form is theoretically expected to have'/= 3 [13]. 2.4. Geometric features of the locally optimal paths Beside the properties of the globally optimal path, the locally optimal paths too have remarkable geometric features. All the locally optimal paths from the apex to the base of the D + 1 dimensional delta form a hierarchical tree that bears a morphological resemblance to the directed patterns found in certain natural systems such as river deltas [12]. Furthermore the locally optimal paths are found never to intersect and the tree was found to be ultrametric [12]. The proof of the non-intersecting property is based on the uniqueness of the directed optimal path between any two sites on the lattice delta. If the bond energies are drawn randomly from an infinite set of values such as a Gaussian distribution, even within a finite interval of it, the optimal path between any two sites on the lattice will be unique; however, if the bond energies are drawn from a finite set of values, there will be a finite probability of having degenerate optimal paths between any two points. Consider two locally optimal paths, C1 and C2 , from the apex 0 to the sites A 1 and A2 respectively on the base of the delta. Let C1 and C2 intersect at the site B in between their ends. Since C1 is the optimal path from 0 to A1 and B is a site on it, it must be the optimal path from 0 to B plus the optimal path from B to A1 • Similarly C2 is the optimal path from 0 to B plus the optimal path from B to A2 • Since the optimal path from 0 to B is unique, the segments of C1 and C2 from 0 to B must be identical. Therefore C1 and C2 do not intersect but only branch out from the site B towards A1 and A2 respectively. The ultrametricity of the tree is a direct consequence of the non-intersecting property of the locally optimal paths. In ultrametric space, any three points A1 , A2 and A 3 satisfy the inequality [22]:
(19) where U(Ai, Aj) denotes the ultrametric distance between the points Ai and Ai. Now consider three sites A1 , A2 and A 3 on the base of the delta. Since these three sites are distinct, the locally optimal paths from the apex of the delta to each of them are at least partly distinct (schematically shown for 1 + 1 dimensions in Figure 2). Let it be that all the three locally optimal paths follow a common course from the apex 0 to a site B
P. Bhattacharyya and A. Chatterjee
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0
Figure 2. An illustration of the ultrametric tree formed by locally optimal paths from the apex to the base of a 1 + 1 dimensional delta. Three non-intersecting paths are drawn to schematically represent the directed optimal paths from 0 to the sites A1 , A2 and A3. The ultrametric distances between these three sites on the base have the relation U(A 1 ,A2) = U(A3,Al) > U(A2,A3) which is the same as the relation in Eq. (22).
on the delta. At the site B let the optimal path to A 1 branch out while the other two continue to remain on a common course. Farther down the delta let the optimal paths to A2 and A 3 branch out at a site C. On the tree formed by these three locally optimal paths, the length z(Ai, A 1) for which the locally optimal paths from the apex to the sites Ai and A 1 are distinct works like the ultrametric distance between Ai and A{
z(A1, A2)
= z(A3, A1)
L- ts,
(20)
z(A2, A3)
L- tc.
(21)
Since the site Cis assumed to be lower down the delta than the site B, tc > t 8 . Therefore the lengths z(Ai, A 1) are related by: (22)
which conforms to the property of an ultrametric space (Eq. 19). Since A 1 , A2 and A 3 are any three sites on the base of the delta, Eq. (22) proves that the tree of locally optimal paths on the delta is ultrametric. More details of the structure of the tree are found in the studies on the branching of locally optimal paths from the globally optimal path. A variable h is defined complementary to the longitudinal coordinate: h = L - t; a site on the delta that is at a distance t
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below the apex is at a height h above the base. It was found that the branching probability Pbranch at any site on the globally optimal path is inversely proportional to its height h from the base [23]: Pbranch (h) ex 1/ h.
(23)
Consequently the number Nbranch of branches that come out of the segment of the globally optimal path between a particular site on it and the base was observed to increase logarithmically with the height of that site [23]: Nbranch(h)
=
hh Pbranch(h')dh' ex lnh.
(24)
Both of these observations were found to be valid only for 1 ~ h < L/4, i.e., near the base of the delta. The width W of the globally optimal path at a site B on it is defined as the number of sites on the base of the delta that are connected from the site B by directed locally optimal paths [23]; W is thus equal to the number of locally optimal paths through B, from the apex of the delta to its base. In 1 + 1 dimensions, with random bond energies from a Gaussian distribution, the width of the globally optimal path at a height h (i.e., the height h of a site B on the path) above the base of the delta of linear size L was found to be [23]
W(h) ex h 2/ 3 for 1 ~ h < L/2.
(25)
The exponent appearing in Eq. (25) is the same as the transverse wandering exponent of the globally optimal path in Eq. (17). The pattern of the locally optimal paths thus reflects the geometry of the globally optimal path. Perfectly directed optimal paths hardly occur in nature. However the universality class of directed polymers (Eq. 7) is widespread because there are many instances of nondirected optimal paths in which the segments going against the longitudinal direction are insignificant over large distances.
3. MOST PROBABLE DIRECTED PATHS AT FINITE TEMPERATURES The study of directed paths at finite temperatures is developed from their partition function. For the convenience of formulation, the embedding space is considered to be the delta segment of the D + 1 dimensional hypercubic lattice described in the previous section, with the longitudinal direction pointing from the apex to the base of the delta along the body diagonal of the hypercube. The linear size L of the delta is the number of lattice bonds in a directed path from the apex to the base. The energy Ec(x,t) of a directed path C(x, t) from the apex (0, 0) to a site (x, t) on the delta is the sum of the energies of its constituent lattice bonds. The partition function Z(x, t) of the ensemble of directed paths {C(x, t)} from (0, 0) to the site (x, t), at a temperature T, is the sum of their Boltzmann weights:
Zr(x, t) =
L C(x,t)
exp [-Ec(x,t)/kT],
(26)
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282
where k is the Boltzmann constant. Hence the partition function of the ensemble of all the directed paths of length L, from the apex to the base of the delta, is given by
Zr(L) = LZr(x,L).
(27)
X
At the temperature T the free energy of directed paths from the apex to a particular site (x, L) on the base of the delta is given by:
Fr(x,L) = -kTlnZr(x,L),
(28)
whereas that of directed paths from the apex to any site on the base of the delta is given by:
Fr(L) = -kTlnZr(L).
(29)
As T--+ 0 the Boltzmann weight of the locally optimal path from (0, 0) to (x, t) and that of the globally optimal path from the apex to the base of the delta are the only terms that dominate the sums for Zr(x, t) and Zr(L) respectively. However at a finite temperature other paths in the ensemble too contribute significantly to the partition function and the paths that dominate are the those of minimum free energy, called the most probable paths. For a given path-length the quantity that can be determined is the most probable or the mean end-to-end distance. For a directed path, whether it is at zero temperatute or at a finite temperature, the overall end-to-end distance is known by definition to be linearly proportional to its length (Eq. 7); therefore, the quantity that is determined as a function of the path-length is the most probable or the mean value of the transverse end-to-end distance lxl defined in Eq. (16). For directed paths at a finite temperature from the apex to the base of the delta, lxl has the most probable value when Zr(x, L) is maximum (since, from Eq. (28), the free energy is minimum) while the thermal average of lxl provides its mean value:
(lxl) = 2:x lxl Zr (x, L).
(30)
2:x Zr (x, L)
3.1. The transfer matrix method for finite temperatures The transfer matrix method for calculating the partition function of directed paths on the delta at a finite temperature is expressed in terms of a recurrence based on the fact that the Boltzmann weight of any path is the product of the Boltzmann weights of its parts. The method considers a directed path from the apex to any site on the delta as formed of two segments: the last bond on the path and the rest of the path. For the simplicity of expression, the 1 + 1 dimensional case is described, referring to Figure 1. The partition function Zr(x, t) of directed paths from the apex (0, 0) to a site (x, t) on the 1 + 1 dimensional delta is calculated recursively by the relation [25,26,28]
Zr(x, t
+ 1) =
Zr(x- 1, t) exp[-c(x- 1, t)/T]
+
Zr(x + 1, t) exp[-c-(x + 1, t)/T]
(31)
where c-(x - 1, t) and c-(x + 1, t) are the energies of the bonds that are directed towards the site (x, t + 1) from the left and from the right respectively. Eq. (31) may be extended to higher dimensions by using more than one transverse coordinate.
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283
3.2. Phase transition at a finite temperature Consider the lattice delta with disordered bond energies Eij assigned in an uncorrelated manner: (c:ijEi'j') = 2f6ii'6jj'· At zero temperature the most probable directed path from the apex to the base of the delta is the minimum energy path; it is the globally optimal path from the apex to the base of the delta. Its form is determined by the disordered configuration of bond energies and is described by the geometric relation (lxl) ex £(1_ (Eq. 17). This is also the form of the most probable paths at low temperatures and defines the low temperature phase. At infinite temperature the form of the most probable paths is determined by the entropy and the disorder in the bond energies has no effect; consequently it takes the form of a random walk with (x) = 0; this is the high temperature phase. The transition between the two phases is studied by using a dimensionless quantity g(T, L) defined in Ref. [27] as a function of the temperature T and path-length L:
g(T,L) = (x(T,L)) 2 /(x 2 (T,L)).
(32)
2
Since (x) = (x 2 ) at T = 0, the low temperature phase is defined by g = 1, whereas, since (x) = 0 at T = oo, the high temperature phase is defined by g = 0. At high temperatures the asymptotic expression of g(T,L), to the first order in f/T 2 , in D + 1 dimensions appears as [27]:
g(D+l)(T, L) "' ~2
£(2-D)/2
(33)
which indicates that 2 + 1 is a critical dimension. According to the above equation, in less than 2 + 1 dimensions the quantity g increases with the path-length L while in more than 2 + 1 dimensions g --+ 0 as L --+ oo. Therefore the high temperature phase is unstable below the critical dimension and stable above it: in 1 + 1 dimensions the high temperature phase always crosses over to the low temperature phase, while in 3 + 1 dimensions a transition occurs between the two stable phases at a finite temperature. In 2 + 1 dimensions Eq. (33) shows no dependence of g on the path-length L; to study the stability of the phases in 2 + 1 dimensions it is necessary to consider the second order correction in r /T2 :
(34) where K is a positive constant and £ 0 is a cutoff path-length. It shows that g grows logarithmically with L and therefore the high temperature phase is unstable and eventually crosses over to the low temperature phase . The above theoretical analysis was supplemented by a detailed numerical study of directed paths at finite temperatures [27] using the transfer matrix method described in the previous subsection. The thermal averages (x) and (x 2 ) for the directed paths from the apex to the base of the delta were calculated using the partition function and the quantity g was evaluated as a function of the temperature T and path-length L in 1 + 1, 2 + 1 and 3 + 1 dimensions [27]. The numerical data showed that in 1 + 1 dimension
(35)
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284
in complete agreement with the theoretical expectation (Eq. 33). This result also indicated that the path-length Lx for observing the crossover from the unstable high temperature phase to the stable low temperature phase increases algebraically with temperature: (36) With this temperature dependence of the crossover length the data for g versus the pathlength L at different temperatures were found to collapse to the scaling form: g(T, L) = g' ( L/ Lx (T)) which further confirmed the result of Eq. (36). In 2 + 1 dimensions the numerical data for g as a function of L at different values of the temperature T were found to collapse when T 2 g was plotted against [ln(L/ L0 )]/T 2 using a suitable value of the cutoff path-length £ 0 • In this case logarithm of the crossover length was found to increase algebraically with the temperature:
J
(37) in agreement with Eq. (34). The crossover length in 2 + 1 dimensions is therefore very large at high temperatures; for path-lengths that are not sufficiently large, this creates a misleading impression of a stable high temperature phase [26,27]. In 3 + 1 dimensions the graphs of g(T, L) as a function of temperature for different path-lengths L were found to cross one another at a particular temperature Tc. It implies that the most probable paths undergo a phase transition at Tc. Near Tc, g(T, L) assumes a finite-size scaling form
(38) where ¢ is the critical exponent of the longitudinal correlation length . From the collapse in the numerical data for different values of L [27], the value of the exponent was found to be
¢ = 4.0 ± 0.7.
(39)
At high temperatures T > Tc it was observed that [27] (40)
that agrees with the theoretical result for 3 + 1 dimensions obtained from Eq. (33).
3.3. Probability distribution of the endpoint position For a directed path of length L from the apex to the base of the delta, the probability of finding its end on the base at the site (x, L), at a temperature T, is given by: P(x L)
'
=
Zr(x, L)
2:x Zr(x, L)
(41)
where Zr(x, L) is the partition function defined in Eq. (26). Numerical studies using the transfer matrix method at finite temperatures show that the disorder-averaged probability distribution of the position of the end of directed paths of length L on the base of the 1 + 1
Optimal and most probable paths
285
dimensional delta, having uncorrelated bond energies drawn from a Gaussian distribution with zero mean and unit standard deviation, has the scaling form [28]: for 0.25 :::;
lxl/ £21 3 :::; 1.2
for
lxl/ £21
(42) 2.0:::;
3
with an irregular transition region 1.2 < lxl/ £21 3 < 2.0. Since the average geometric features of directed paths at all finite temperatures in 1 + 1 dimensions are the same as those of the optimal path at zero temperature, the result of Ref. [28] expressed in Eq. (42) confirms the theoretical prediction of Ref. [13] that the asymptotic form of the probability distribution (i.e., for lxl » £21 3 ) at T = 0 is non-Gaussian. At finite temperatures the quenched and annealed averages over the configurations of disordered bond energies are not the same. Consequently ln P(x, L) # ln P(x, L), where the former quantity is annealed-averaged and the latter is quenched-averaged. Theoretical studies [29,30] supplemented by numerical results obtained by the transfer matrix method [31] show that the disorder average of ln P(x, L) has the scaling form ln P
(x, L) ex -lxl 2 /L
(43)
while its second cumulant was found to be (44) For lxl > £21 3 the second cumulant was eventually observed to become constant due to the fact that two directed paths that are once separated by a distance greater than £21 3 do not interfere subsequently and therefore their correlation saturates [30].
4. NON-DIRECTED OPTIMAL PATHS AT ZERO TEMPERATURE 4.1. Optimal paths on weakly disordered lattices A weakly disordered medium is represented on a lattice by assigning bond energies drawn randomly from a bounded, or effectively bounded, distribution. Commonly used are the uniform distribution within a finite interval of energy values and the gaussian distribution with finite half-width. 4.1.1. Determination of the optimal path: transfer matrix method The transfer matrix method for finding the non-directed optimal path and its energy at zero temperature between two sites on a D-dimensional hypercubic lattice is similar to the method used for finding the directed optimal path and its energy in D + 1 dimensions. However there is a significant difference: for directed paths, including the optimal path, between the two sites the length is determined by the end-to-end distance: L = R (Eq. 8), whereas for the non-directed optimal path only the end-to-end distance is known; consequently, the procedure for finding the directed optimal path in D + 1 dimensions considers only those directed paths between the two sites that have length L = R, whereas
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82 81
A
83
84
0
X
Figure 3. A non-directed path between the origin 0(0,0) and a site A(x1,x2) on a square lattice. To reach A the path must pass through one of its four neighbouring sites B1, B2, B3 or B 4 • This fact leads to a recurrence relation (Eq. 45) for the energy of the optimal path between 0 and A and forms the basis of the transfer matrix method for determining the non-directed optimal path and its energy.
the procedure for finding the non-directed optimal path in D dimensions needs to consider all paths of lengths L 2': R between the two sites. Determination of the optimal path and its energy is thus simpler in the directed case than in the non-directed case. On the D-dimensional hypercubic lattice a site 0 is chosen as the origin. Since the paths are defined to be along the bonds of the lattice, any site A with coordinates x can be reached through its 2D neighbouring sites only, i.e., only (x 1 , x 2 , •.• , xn) through the sites that are connected to it by a bond (illustrated for the square lattice in Figure 3). Therefore the optimal path between the origin 0 and the site at xis composed of the optimal path between 0 and a neighbour of the site x plus the bond between the site x and that neighbour, such that its energy E(x) satisfies the recurrence relation:
=
E(x) =min [E(x + o) 0
+ c-(x + 8)],
(45)
where the c-(x + o) denotes the energy of the bond between the site x and its neighbouring site x + o. The transfer matrix method for determining the non-directed optimal path is formulated by augmenting the original D-dimensional space of the lattice by an
Optimal and most probable paths
287
extra dimension. The extra dimension contains a longitudinal direction so that the augmented space is the D + 1 dimensional hypercubic lattice. The purpose of introducing the extra dimension is to view the problem of determining the non-directed optimal path on the original lattice as the problem of determining a directed optimal path on the augmented lattice by using the length of the non-directed paths as the longitudinal coordinate: a non-directed path of length L between the origin and a site with the position x = (x 1 , x 2 , ... , xn) on the D-dimensional hypercubic lattice is uniquely represented by a directed path from the origin to the site (x, L) on the D + 1 dimensional augmented lattice. The end-to-end distance R of a non-directed path is given by R = ~~ 1 lxil which is identical to the transverse end-to-end distance of the corresponding directed path (Eq. 16). The directed optimal path from the origin to the site (x, L) and its energy E(x, L) is determined for all values of L by the transfer matrix method described in section 2.2; for the D + 1 dimensional augmented lattice its is expressed as: E(x, L + 1) =min [E(x + 0
o, L) + c(x + 8)].
(46)
The non-directed optimal path between the origin and the site x on the D-dimensional lattice is therefore identified as the path with the energy E(x)
= L:L?_R min [E(x, £)].
(47)
4.1.2. Determination of the optimal path: Dijkstra's algorithm Dijkstra's algorithm for finding the shortest geometrical path from a particular site to each of the other sites on a connected graph (for example, a lattice with each site connected to each of its nearest neighbours by a bond) can be used to determine the optimal paths by replacing the length of each bond in the algorithm with the energy of interaction assigned to it [33]. The procedure considers the set V of all the sites of the graph by dividing it into two complementary parts: the set S of those sites to which the optimal paths have been found, and the set V- S which contains the remaining sites. In the beginningS is empty; in the end, after the application of the algorithm is complete, S is equal to the set V. A site, denoted by A 0 , is given as the source from which the optimal paths to all other sites are to be determined. The energy of the optimal path from the source Ao to a site Ai is denoted by Ei and in the algorithm it appears as the energy of the site Ai when Ai is inS. The site that precedes Ai on the optimal path is denoted by Ai-l and is called the predecessor of Ai· The algorithm systematically determines the prdecessor Ai-l of each site Ai on the optimal path from A 0 , along with its energy Ei; the optimal path between A 0 and Ai is reconstructed by joining the successive predecessors from Ai to Ao. The algorithm has three main steps: initialization, sorting and relaxation, of which the latter two are repeated in a loop until the optimal paths to all the sites have been found. The initialization procedure sets E 0 = 0 for the source A 0 and Ei = oo for all other sites; it also sets S to empty and assumes that initially none of the sites has a predecessor. The sorting procedure locates the site Ai with the minimum Ei in the set V- S. Ai is then transferred to the set S and Ei is the energy of the optimal path between A 0 and Ai; among those remaining in V- S the sites that are connected to Ai by a bond (the neighbours of Ai) are relaxed. The relaxing procedure updates the energy Ei of each site
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that is connected to A;: if Ej > Ei + Eij, where Eij is the energy of the bond connecting and Aj, the site Aj is assigned a new value of energy, Ej Ei + Eij, and it has a new predecessor Aj_ 1 = Ai; otherwise Ej and Aj_ 1 remain unaltered. The time required by Dijkstra's algorithm to determine the optimal paths from the given source to all sites on the graph is O(IVI 2 ). This algorithm, however, has a drawback: the original algorithm was developed for working with distances-which are non-negative quantities-since it aimed at finding the shortest geometrical path; consequently the algorithm adapted for finding the optimal paths works with non-negative bond energies only. Aj Ai
=
4.1.3. Geometrical features of the optimal path in weak disorder The non-directed optimal paths between the sites on a lattice with randomly disordered bond energies were found to belong to the same universality class as that of directed optimal paths when the path-lengths are large [32,33]; their lengths Land the corresponding end-to-end distances R follw the asymptotic linear relation R"' L (Eq. 7). This result implies that for large L (or large R) the non-directed regions on the optimal paths are insignificant and the optimal paths appear as self-affine. To compare the two forms, the transfer matrix method was used to determine both the directed and the non-directed optimal paths between two opposite vertices of hypercubic lattices in two to six spatial dimensions [32]. Further Dijkstra's algorithm was used to determine the directed and the non-directed global optimal paths from the apex to the base of the lattice delta of various linear sizes in two and three dimensions, with the disordered bond energies distributed uniformly [33]. In all the cases studied the non-directed optimal path was found to have very few non-directed segments and its transverse end-to-end distance showed the same relation to its length (with same value of the exponent (J.) as observed in the case of the directed optimal path (Eq. 17). Therefore it was concluded that in weak disorder there is no significant difference in the asymptotic geometric forms of the directed and the non-directed optimal paths. However, over small lengths, these have different forms: while the directed optimal paths are still self-affine, the non-directed optimal paths have a self-similar form which crosses over to the self-affine form for large path-lengths; studies on this crossover is discussed in the end of the next subsection. 4.2. Optimal paths on strongly disordered lattices The feature that defines a strongly disordered medium is a wide range of values that are assumed by the disordered physical property. Porous rocks having a wide range of pore diameters and electrical networks containing a wide range of conductances are examples of natural and artificial media respectively that are strongly disordered. On a lattice a strongly disordered medium is simulated by assigning bond energies drawn randomly from a wide, in principle unbounded, distribution. Since the bond energies are widely distributed, the largest of the bond energies on any path is expected to be overwhelmingly larger than the rest and, therefore, the energy of the path (i.e., the sum of the bond energies along the path) is expected to be almost equal to the largest bond energy on it. Consequently the energy Ec of a path C in the case of strong disorder is formally defined as the largest bond energy on it:
Ec =max cc
(48)
Optimal and most probable paths
289
where cc denotes the energies of the lattice bonds that constitute the path C. The optimal path between two sites A 1 and A2 on the lattice is defined as the path with minimum energy E(A 1 ,A 2 ) in the set of all the paths C(A 1 ,A 2 ) between them: (49) The definition of the energy of a path given in Eq. (48) implies that the space of the optimal path energies has ultrametric structure: for any three sites A 1 , A2 and A 3 on the lattice
(50) since the largest bond energy on the optimal path between A 1 and A2 must be less than the largest bond energy on any other path between them, say, the path formed of the optimal path between A 1 and a site A 3 plus the optimal path between A 3 and A2 ; the equality in the above relation appears for the case when the site A 3 is on the optimal path between A 1 and A2 • In order to represent strong disorder, the bond energies ought to be drawn from a distribution that provides high energy values frequently enough to be present on any path on the lattice, yet sufficiently few so that the energy of a single bond can dominate the energy of the entire path. The distribution
P (log 10 c:) = constant
(51)
of the bond energies c: was found to satisfy these requirements [40].
4.2.1. Determination of the optimal path: the percolation algorithm The definition of the energy of a path given in Eq. (48) allows a simple method for determining the optimal path in strong disorder [34]. Consider two sites 0 and A on the lattice (say, the D-dimensional hypercubic lattice) where the bond energies are strongly disordered and initially every pair of neighbouring sites is connected by a bond. The bonds are removed one by one in descending order of their energy values (i.e., starting from the bond with the largest energy) such that the connection between 0 and A is preserved; if the removal of a particular bond destroys the connection, it is left in its place and the procedure is continued by removing the next bond in the order. When no more bonds can be removed only a single path between 0 and A remains on the lattice; this is the optimal path (by the definition in Eq. 49) between the two sites 0 and A. This method works only when the bond energies are strongly disordered such that the energy of a path on the lattice follows the definition of Eq. (48). It does not work in the case of weak disorder where the energy of a path is defined as the sum of all the bond energies on it. However, the methods used for determining the optimal paths on a weakly disordered lattice, i.e., the transfer matrix method and Dijkstra's algorithm, also can also be used in the case of strong disorder. The above procedure is equivalent to a new kind of percolation approach [34]: Bonds are removed randomly from the lattice, one by one, irrespective of their energies, leaving in place only those bonds that are found necessary to preserve the connection between
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P. Bhattacharyya and A. Chatterjee
the sites 0 and A; if the removal of a particular bond destroys the connection, it is left in its place and an alternative bond is removed. When no more bonds can be removed there is only a single path between 0 and A on the lattice; the procedure leaves no bond outside the path and no branches on the path. This form of percolation is significantly different from the approach in ordinary percolation [35] in which the bonds are removed randomly only until the connection between between 0 and A is to be broken for the first time. Since the bonds of the lattice are assigned energy values chosen randomly from a given distribution, removing the bonds in any particular order of their energy values is equivalent to removing the bonds from randomly chosen positions on the lattice irrespective of the energy values assigned to them and hence the procedure for finding the optimal path is equivalent to this percolation algorithm.
4.2.2. Geometrical features of the optimal path in strong disorder Optimal paths between two sites on a strongly disordered lattice are found to be selfsimilar fractals [34,36,37]: the optimal path-length L scales with the end-to-end distance R as (52)
where Dapt is the fractal dimension of the optimal path. Numerical studies using the procedure described above show that Dapt = 1.22 ± 0.01 on the square lattice (D = 2), Dapt = 1.42 ± 0.02 on the simple cubic lattice (D = 3), Dapt = 1.59 ± 0.02 on the four dimensional hypercubic lattice (D = 4) [37]. Optimal paths in the presence of strong disorder is found to belong to the universality class of the shortest paths on self-organised critical clusters in invasion percolation with and without trapping [38]. The length L of the shortest paths on percolation clusters scales with the end-to-end distance R as (53)
where Dmin is the fractal dimension of the shortest path. For invasion percolation clusters on the square (D = 2) and the simple cubic (D = 3) lattices the values of Dmin were found to be similar to the corresponding values of Dapt [38]; however the shortest path on ordinary percolation clusters belong to a different universality class with Dmin = 1.130 ± 0.002 forD = 2, Dmin = 1.374 ± 0.004 for D = 3 and Dmin = 1.61 ± 0.02 for D = 4 [39].
4.3. Crossover from self-similar to self-affine form When the distribution of bond energies on the lattice is intermediate between the weak and the strong limits it appears to be strongly disordred over small distances and weakly disordered over large distances. This happens because on a sufficiently short path large bond energies are infrequent enough for the largest one on the path to dominate the total energy of the path whereas on a suffiently long path large bond energies occur so frequently that the energy of a single bond can no more dominate the total energy of the path. Consequently, on a lattice with a wide but bounded distribution of bond energies the optimal paths are self-similar over small lengths and self-affine over large lengths [40]. To study the relation between the form of the optimal paths and the strength of disorder
Optimal and most probable paths
291
the distribution of bond energies used for generating strong disorder was truncated at a lower bound cmin and an upper bound cmax:
p (log 10 c)
=
0 for c < cmin constant for cmin:::; c :S: cmax { 0 for cmax < c
(54)
With bond energies drawn randomly from the interval (cmin, cmax), optimal paths on the square and the simple cubic lattices were determined by using Dijkstra's algorithm and their geometric properties, the path-length L and radius of gyration R (equivalent to the end-to-end distance), were measured. The numerical data showed the existence of a crossover path-length Lx such that for L « Lx the optimal paths had self-similar form: R ex Ul Dapt, whereas for L » Lx the optimal paths had self-affine form: R ex L. The crossover path-length was found to depend on the bounds of the bond energy distribution in the following way [40]:
(55) where r;, = 1.60 ± 0.03 in both the square (D = 2) and the simple cubic (D = 3) lattices which suggests that the value of r;, may be independent of the dimension of the lattice. Using the crossover length as the only relevant lengthscale for the optimal paths, the following scaling hypothesis was formulated [40]:
(56) where F(u) is a scaling function with the properties F(u) ex u 1/Dapt for u « 1 and F(u) ex u for u » 1. Numerical data, obtained by measuring Rand L of the optimal paths, for different values of log 10 (cmax/cmin) were found to collapse onto a single curve when R and L were rescaled to R/ L~ Dapt and L/ Lx respectively, thus confirming the scaling formulation [40].
5. MOST PROBABLE NON-DIRECTED PATHS AT FINITE TEMPERATURES While the most probable directed paths undergo a phase transition at a finite temperature only in 3 + 1 and higher dimensions, the most probable non-directed paths undergo a phase transition in all spatial dimensions owing to the greater entropy available to the paths in the latter case [41,42]. Though the existence of the phase transition has been shown for a regular homogeneous lattice, it is expected to be observed for lattices with disordered bond energies too. Consider two sites separated by a distance R on a D-dimensional hypercubic lattice with homogeneous bond energies c. Using the Cartesian metric R is measured as:
(57)
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292
where Rd is the component of the vector R (directed from any one of the two sites to the other) in the d-th dimension. The length L of any path connecting these two points is measured as: (58) where Ld is the number of lattice bonds along the path in the d-th dimension. The paths of minimum energy connecting the two points are the paths of shortest length between the two points: minL= R.
(59)
At a finite temperature T the statistical weight of a bond of energy E along any path is exp(-s/kT). Therefore the partition function for paths of length Land end-to-end distance R is
Zr(L, R) = N(L, R) exp( -Ls/kT),
(60)
and the partition function for all paths with an end-to-end distance Rat the temperature Tis 00
Zr(R) =
L Zr(L, R). L=R
(61)
The number N(L, R) of distinct paths of length Lon the lattice, connecting two points separated by a distance R is given by:
N(L R) - "' '
-
L..,
{Ld}
L!
(62)
TIDd=1 (Ld-Rd)' (Ld+Rd)' 2 · 2 ·
where the summation indices { Ld} are subject to the conditions of Eq. (58) and Eq. (59). For large path-lengths L, the partition function in Eq. (60) reduces to the expression:
Zr(L, R)"' (
1
1r)D_ 112
2
[ (D- "21) lnL + (L + D)(ln2 + lnD)- 2DL R
exp -
2
Ls] . (63) - kT
The free energy -kT ln Zr(L, R) at a temperature T is minimum when the path-length assumes the most probable value L for that temperature; this leads to the condition: 2 [(ln2 + lnD)-
k~] L2 -
(2D
-1)L+ DR2 =
0,
(64)
with the solution
L(R, T) =
1- v1- 4(2~=:1)2f1 (t-1) ~) £
2D-1
k
(
Tc-
T
R2
(65)
Optimal and most probable paths
293
where E
kT = ln 2 + ln D.
(66)
c
In Eq. (65) only the negative sign of the square-root is admissible in order to ensure that L is always positive. At low temperatures, i.e. T < Tc, Eq. (65) is expressed as:
(67) where R (T)
=
X
2D -1
2V2J5
[~ (~k
T
]__)]-1/2
(68)
Tc
is a crossover end-to-end distance. ForT< Tc the most probable paths therefore undergo a crossover in its form, from paths that have the feature of random walks:
L """ _D __ R 2
for R
2D-1
«
Rx
(69)
to paths that have the feature of directed walks:
~ L""'
~ 1 -1 - (T
2
Tc
)-1/2 R
for R
»
Rx.
(70)
Since the crossover distance diverges as T ---+ Tc from below the asymptotic form (i.e., for large R) of the most probable paths undergo a transition from the directed walk phase for T < Tc to the random walk phase at Tc. For high temperatures, i.e. T > Tc, the most probable path-length in Eq. (65) is expresed as:
L(r,T) =
2DR~ax [ 1 - ~ 1 - ( R )'] 2D -1 Rmax
(71)
where
Rmax(T) =
-1 [E ( 1 1)]-l/ 2V215 k Tc- T
2D
2
(72)
is a cutoff end-to-end distance. For small end-to-end distances the most probable paths have the random walk feature, since Eq. (71) shows that: 2 L""'_D_R D_ 1 2
f or R
«
R max·
(73)
However Eq. (71) also shows that for end-to-end distances R > Rmax(T), at any temperature T > Tc, a real and finite value of the most probable path-length does not exist.
P. Bhattacharyya and A. Chatterjee
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The expression for the critical temperature Tc, given in Eq. (66), shows that it gets lower with increasing dimension of the lattice because the larger the dimension, the greater the entropy of the paths, and therefore, the lower is the noise (given as the temperature) required to produce the random walk nature. For a lattice with disordered bond energies, the transition temperature is expected to be lower than the value for the homogeneous case, since disorder in the bond energies appear as additional noise that makes the most probable paths wander more between their ends to seek lower free energy configurations.
6. UNIFIED PICTURE OF OPTIMAL PATHS A scheme for unifying the different forms of optimal paths-directed and non-directed, self-similar and self-affine-was proposed in Ref. [43]. The bond energies were distributed on the lattice with a density function
(c: _ a)I//3-I p(c:) =
1;01
(74)
where a ::0: 0 and -oo ::; ;0 ::; oo are the parameters for controlling directedness and disorder respectively. The bond energies are required to satisfy the conditions 0 < E- a ::; 1 for ;0 > 0 and E - a > 1 for ;0 < 0. For ;0 = 0 there is no disorder in the bond energies and all directed paths have the same energy. Consequently the optimal paths have the self-affine form of directed random walks on a regular homogeneous lattice with the transverse wandering exponent (j_ = 1/2 [44]. The parameter range 0 < ;0 < oo represents weak disorder in which the optimal paths, both directed (a > 0) and non-directed (a = 0), are self-affine and belong to the universality class of directed optimal paths on randomly disordered lattices. For a = 0, the limit ;0 ---+ +oo represents strong disorder and the optimal paths are non-directed and self-similar; this is also the case for a ::0: 0 in the limit ;0 ---+ -oo. However in the region -oo < ;0 < 0, the form of the optimal paths are nonuniversal, and depends on the value of the parameter ;0. With these findings the phase diagram of optimal paths on the (a, ;0)-plane was constructed in [43] . Besides being studied on randomly disordered lattices, optimal paths were also studied on randomly disordered networks. Since it is beyond the scope of this chapter it is not discussed here; an account of their geometric properties is given in a recent review [45].
7. THE TRAVELLING SALESMAN PROBLEM ON A LATTICE 7.1. The standard traveling salesman problem A classic problem of optimal paths is the travelling salesman problem (TSP) : given a set of cities and a metric for the intercity distances, the salesman needs to find the shortest closed path (contour) through all the cities. It is a problem of combinatorial optimization and is known to be nondeterministic polynomial complete (NP complete) [46,47]. The optimal contour is necessarily a self-avoiding loop which is a special case of self-avoiding paths discussed in this book. The travelling salesman problem is normally defined on a continuum space with N cities represented by N points chosen randomly within a finite volume. The quantity
Optimal and most probable paths
295
to be optimised is the mean travel distance per city, denoted by [E(N) in the Euclidean metric and lc(N) in the Cartesian metric. Since the mean nearest neighbour distance among N points chosen randomly in a fixed volume of the D-dimensional Euclidean space scales as 1/NfJ for large N (see Appendix B) so does the optimal travel distance per city; hence the quantity to be determined in the travelling salesman problem is the optimal value of the normalised travel distance per city, defined as S1E = [ENl/D in the Euclidean metric and S1c = [0 N 11D in the Cartesian metric, such that the quantities S1E and S1c are independent of ithe number of cities N. In the two dimensional Euclidean space approximate analytic bounds for the value of S1 have been estimated to be 1/2 < S1E < 0.92 [1,48] and 2/7r < Slc < [49]. The lower bound for for S1E is estimated from the expression of the mean nearest neighbour distance among N random points in the unit square [1,50] while the upper bound is calculated by dividing the area into strips of width W and minimising S1E with respect to W [48,51]. The bounds for S1c are calculated from those of S1E by using an approximate relation between [E and lc wich can be derived as follows [49]: Consider two successive cities on the contour of the travelling saleman and assume that they are separated by the distance lE =~in the Eucldean metric and lc = lxl + IYI in the Cartesian metric; if the angular position of the one city with respect to the other is B such that x = lE cos B and y = lE sin B, the Cartesian distance may be written as lc = lE(I cos Bl +I sin Bl); assuming that the contour is rotationally symmetric, all possible orientations B of the straight line joining the two cities are equally likely; therefore, the mean distance in the two metrics are related by lc = 2ZE(I cosBI) = (4/7r)lE, suggesting that S1c = (4/7r)S1E.
[413
7.2. TSP on randomly diluted lattices To define the travelling salesman problem on a dilute lattice, for example the square lattice, the cities are represented by randomly occupied lattice sites, the fraction of occupied sites being denoted by p. Since the mean optimal travel distance varies as 1/ ..;p, the normalized travel distance per city is defined as S1E = [E..;P in the Euclidean metric and S1c = lc..;P in the Cartesian metric. The one-dimensional problem is trivial: any directed tour is the optimal path. In two-dimensions a directed version of the problem on a lattice was studied in [52] . This is a simple model in which the cities (occupied sites) are distributed randomly, with concentration p on the square lattice of linear size W (W lattice bonds on each side). The travelling salesman is forbidden to move opposite to one fixed lattice direction (say, the upward direction in Figure 4). In order to visit all the cities, the salesman visits every city of each horizontal layer, starting from the first, till the last city in that layer is reached; then the salesman moves down to the next layer and proceeds in the same manner. This ensures that none of the lattice bonds are retraced and the travel is only along the lattice bonds. When all the cities have been visited in the last layer, the salesman returns to the origin (this return must however take the forbidden direction). The non-degeneracy of the Cartesian-type path traced by the directed salesman makes the problem exactly solvable, compared to the case of the nondirected travelling salesman, where the degeneracy of the optimal contour (similar to the ground state of spin glass) makes the problem of finding globally optimal contour extremely difficult [46,47]. The contour length of the salesman is composed of (i) the sum L 1 of the horizontal
296
P. Bhattacharyya and A. Chatterjee
~--------vv--------~ 1-E-----
VV- m ---~
Figure 4. A typical example of a square lattice of width W = 10 containing 21 occupied sites or cities (filled circles), distributed randomly with concentration p = 0.21. The vertically downward arrow indicates the direction of travel. The bold line shows a path followed by the directed travelling salesman.
distances travelled between the extreme cities in each layer, (ii) the sum L 2 of the vertical distances travelled for moving from each layer to the next, and (iii) the sum L 3 of the extra length travelled horizontally in each layer due to the mismatch of the positions of the extreme cities of successive layers. These three components are given by [52]: n=W
L1
=
Wp 2
L
qm(m + 1)(W- m),
(75)
n=O
(76) (77) where q = 1- p. The expression for L 1 is obtained from the fact that in a layer with two extreme sites separated by distance n lattice bonds there are m = W - n vacant sites (each vacant with probability q), on the outer side of the two extreme occupied sites and these m vacant sites can be arranged in m + 1 ways. The term 2W in L 2 is due to the downward travel for moving from each layer to the next and the final upward travel from
Optimal and most probable paths
297
the last layer to the first. Therefore, for q < 1 (or, p > 0), the total contour length is given by [52] W 2 [(1- qW+l)- (W + 1)(1- q)qW+l] +W[2(1- q)- 2q 2 (1- qW+l)/(1- q) +(W + 2)(W + 1)(1- q)qW+I + 2(W + 2)qW+ 2 +2q/(1- q2 )],
(78)
which, in the case of large W, reduces to (79) This expression is correct for all p > 0 (more specifically for (1- p)w ---+ 0). For p = 1, the contour length is L = W 2 + 2W which is also the exact solution for the nondirected problem. For the ordinary TSP, a mean field argument indicates that the mean distance of travellc between two cities on a two-dimensional lattice is of the order of 1/ ..;P while for the directed travelling salesman problem, Zc = L/(pW 2 ) = 1/p + (1/W)[2(1- 2q 2 )/(1q2 )p], where an extra factor of 1/ ..;P appears in the leading term (which becomes more dominant asp---+ 0), with a 1/W-order correction term. In further studies of the TSP on a randomly diluted lattice [49] the expression for Z0 (p), the average length of the shortest path per occupied site was derived. For a triangular lattice, Zc (p) is unity and has a correction of order of (1-p )5 . In the case of a square lattice, retaining only first order terms, the optimal paths can be derived from the dynamics of a model of a one-dimensional gas of kinks and anti-kinks. With the Cartesian metric, it was found that Zc(P) ::; 1 + 0 ((1- p) 312 ); a constructive upper bound valid for all p
[413.
was also calculated: Zc(P) ::; J4/(3p) asp ---+ 0, which implies that S1c ::; These inequalities suggest that S1c should have a monotonic variation from unity for p = 1 to a constant for p ---+ 0. The problem of the travelling salesman on the lattice reduces to the standard TSP on the continuum in the limit p ---+ 0, where the problem is known to be NP-hard. The different approaches for studying the travelling salesman problem is a vast subject and only a few important results are mentioned below to supplement the above discussion on the problem on lattices. The normalized optimal contour length per city was calculated by using the universality of the scaling the n-th neighbour distance with the number N of cities where the cities are represented by a set of N points chosen randomly in a unit volume of the D-dimensional hypercube with toroidal boundary conditions [53]: S1E = 0.7120 ± 0.0002 in D = 2 and S1E = 0.6979 ± 0.0002 in D = 3. The mean-field approach [54] in the limit N ---+ oo gives
(80) from which it is found that SlijfF = 0.7251 in D = 2 [54]. Using an enumeration technique, a branch and bound algorithm [55], the optimal contours for N ::; 100 were determined in
P. Bhattacharyya and A. Chatterjee
298
D = 2 [56] and by extrapolating the measured of the variation of S1E and S1c with p for large systems it was found that S1E ~ 0.73 and S1c ~ 0.93 asp---+ 0 [56] which indicated that S1c/S1E ~ 4/7r asp---+ 0 as derived in [49].
APPENDIX A. Calculation of the upper bound for S1 Consider the travelling salesman problem on an infinite plane (D = 2) where cities are distributed with a concentration of N per unit area. Consider two successive cities on the travelling salesman's path that lies within an infinite strip of width W on the plane; these two cities are referred as the last city visited and the next city to be visited. The distance measured from the last city visited along the axis of the strip is denoted by x and that measured across the strip is denoted by y. The probability that there is no city within a distance x from the one last visited is 1 - NW x R> exp(- NW x) and the probability that the next city exists in the interval between x and x + dx is NW dx. Similarly the probability that there is no city within a strip of width y from the last visited city is 1 - y /W and the probability that the next city exists within the strip between y and y + dy is dy/W. Therefore the mean distance between two successive cities on the strip is given by (cf. [48,51]) [E
=2
!
x=oo 1y=W 2 x=O y=O vx
+ y 2 NWdx
exp(-NWx)
d
~
(
1-
~
)
(81)
in the Euclidean metric and
1
x=oo 1y=W y=O
lc = 2 x=O
(lxl + IYI)
dy ( y ) NWdx exp(-NWx) W 1- W .
(82)
= y /W
the above expressions
in the Cartesian metric. Substituting u = NW x and v appear as
21:: 1:: N~ 21:: 1:: N~ 00
[E
=
and
00
lc =
Using by
S1E
1
v'u 2 + N 2 W 4 v 2 exp( -u) (1- v) du dv
1
2
( u + NW v) exp( -u) (1- v) du dv.
(83)
(84)
W for W ,;N, the dimensionless quantities S1E = [E,jN and S1c = [0 ,;Fi are given 1
= _:_
J.u=oo 1v= W u=O v=O
Vu2 + W4v 2exp(-u) (1- v) du dv
(85)
and 1
Slc = _:_ /,u=oo 1v= (u W u=O
v=O
+ lV 2 v)
exp(-u) (1- v) du dv.
From the variations of S1E and S1c with were obtained for W = 1.73 [48,51].
W the optimal values
(86) S1E ~ 0.92 and S1c ~ 1.15
Optimal and most probable paths
299
B. Calculation of the lower bound for S1 The mean nearest (or, first) neighbour distance between N random points in a given volume (say, the unit volume) gives the lowest possible bound for the normalised mean travel distance S1E per point on the optimal contour through the N points.
B.l. A heuristic approach to the mean n-th neighbour distance Consider random geometrical points, i.e., points with uncorrelated positions, distributed uniformly in aD-dimensional Euclidean space, with a density of N points per unit volume. A point is said to be the n-th neighbour of another (the reference point) if there are exactly n - 1 other points that are closer to the latter than the former. The quantity to be determined is the mean distance (r~Dl(N)) between any point and its n-th neighbour, n< N. The following is a heuristic approach [50] to the mean first neighbour distance. Consider a typical unit volume of the space, say, in the form of a hypersphere or a hypercube containing exactly N random points. Let this unit volume be divided into N equal parts. Since the N random points are distributed uniformly over the unit volume each part is expected to contain just one of these. The mean distance (.riD) (N)) between any point and its first neighbour is naively given by the linear extent of each part. Since the volume of each part is 1/N, it is expected that
(87) The above result for the first neighbour is extended to the n-th neighbour by the following heuristic argument [50]. A point is chosen as the reference and its n-th neighbour, n < N, is located. The expected distance between them is ( r~D) ( N)). Keeping these two points fixed the number of points in the unit volume is changed toN a by adding or removing points at random; the factor a is arbitrary to the extent that N a and na are natural numbers. Since the distribution of points is uniform, the hypersphere that had originally enclosed n points is now expected to contain na points. Therefore, what was originally the n-th neighbour of the reference point is now expected to be the na-th neighbour for which the expected distance from the reference point is now (rna (Na)). Since the two points under consideration are fixed, so is the distance between them. Consequently (88)
The above relation is approximate as the change in the density of points does not always convert the n-th neighbour of the reference point to exactly its na-th neighbour. Now take a= 1/n, so that (89)
which shows that the mean n-th neighbour distance for a set of N random points distributed uniformly is approximately given by the mean first neighbour distance for a depleted set of N / n random points in the same volume. The above relation is derived for such values of n that divide N exactly; however this approximate relation may be used
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P. Bhattacharyya and A. Chatterjee
for any value of n when N » n, by replacing N / n with the integer nearest to it. Using the expression of (r 1 (N)) from Eq. (87) (90) which, therefore, requires as a necessary condition that N » n. For n = 1 the above result provides the necessary scaling of the mean optimal travel distance per city with the number N of cities in the travelling salesman problem. The results of Eq. (87) and Eq. (90) are extremely crude approximations; however these provide a rough picture of the mean n-th neighbour distance. Heuristic constructions such as this may be usefull in more complicated cases where the distribution of the points is inhomogeneous.
B.2. Rigorous calculation of the mean n-th neighbour distance Consider the system of random points described in Appendix Bl. Assuming a certain random point as the reference there will be N - 1 other random points within a typical D-dimensional hypersphere of unit volume with the reference point at its center. For a given reference point the probability of finding its n-th neighbour (n < N) at a distance between r n and r n + dr n from it is given by the probability that out of the N - 1 random points (other than the reference point) distributed uniformly within the hypersphere of unit volume, exactly n - 1 points lie within a concentric hypersphere of radius Tn and at least one of the remaining N- n points lie within the shell of internal radius Tn and thickness drn: (91) where
(92) is the volume of the D-dimensional hypersphere of radius r n centered at the reference point. Ignoring differentials of order higher than the first (q = 1) in Eq. (91): (93) Since the n-th neighbour (n < N) must certainly lie within a unit volume centered at the reference point, its mean distance from the reference point is given by:
(94) where R is the radius of the D-dimensional hypersphere of unit volume:
(95)
Optimal and most probable paths
301
Changing the variable of integration in Eq. (94) from radius to volume (by the relation of Eq. (92)) and using the probability distribution of Eq. (93) the exact result for the mean n-th neighbour distance is obtained [50]:
[r
(Q2 + 1)]1/D ( 7r 1; 2
~~~
)
(N- n)
f
V,:'+( 1/dl- 1 (1- Vnt-n- dVn 1
[r(li+1)r/D ( N-1) ( 1 ) 7r 1; 2 n- 1 (N-n)B n+ D,N-n
[r (li + 1) 12 1r /
t
D
r (n + {J) r(n)
f( N) r
(N + -t)"
(96)
Here B(x, y) is the beta function and f(z) is the complete gamma function which are related by the formula: B(x, y) = f(x) f(y)/f(x + y). An alternative derivation of the above result is found in [57]. For large density N of points, using Stirling's approximation for the gamma function: r (N + 1/ D) /f (N) "'N 11D, Eq. (96) reduces to the following asymptotic form: (DJ
(rn (N))
rv
[r(li+1)tD r(n+i) ( 1 )1/D 7r1/2 f(n) N
(97)
If the neighbour index n is also large (but n < N), the above asymptotic expression of the mean n-th neighbour distance further reduces to:
(98) The above equation shows that the expression of (rn(N)) obtained by heuristic means (Eq. 90) has the correct asymptotic dependence on N and n. For n = 1 in Eq. (97), (ri2l(N)),...., ~N- 1 1 2 and (ri 3l(N)),...., (Dffl~l13 N- 1 1 3 as found by Chandrasekhar [1].
REFERENCES 1. S. Chandrasekhar, Rev. Mod. Phys. 15 (1943) 1. 2. P-G. de Gennes, Scaling Concepts in Polymer Physics, Cornell University Press, Ithaca, 1979. 3. B. Nienhuis, Phys. Rev. Lett. 49 (1982) 1062. 4. B. Nienhuis, J. Stat. Phys. 34 (1984) 731. 5. L. Peliti, Phys. Rep. 103 (1984) 225. 6. T. Halpin-Healy andY-C. Zhang, Phys. Rep. 254 (1995) 215. 7. D. S. McKenzie, Phys. Rep. 27 (1976) 35. 8. K. Barat and B. K. Chakrabarti, Phys. Rep. 258 (1995) 377. 9. J. W. Haus and K. W. Kehr, Phys. Rep. 150 (1987) 263.
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Statistics of Linear Polymers in Disordered Media Edited by Bikas K. Chakrabarti © 2005 Elsevier B.V. All rights reserved.
Phenomenology of polymer single-chain diffusion in solution George D. J. Phillies a aDepartment of Physics, Worcester Polytechnic Institute, Worcester, MA 01609 The literature on self-diffusion of polymers in solution, and on tracer diffusion of probe polymers through solutions of matrix polymers, is systematically reviewed. Virtually the entirety of the published experimental data on the concentration dependence of polymer self~ and probe- diffusion is represented well by a single functional form. This form is the stretched exponential exp( -acv), where c is polymer concentration, a is a scaling prefactor, and v is a scaling exponent. Correlations of the scaling parameters with polymer molecular weight, concentration, and size are examined. a increases markedly with polymer molecular weight, namely a ~ Mx for x R> 1. v is R> 0.5 for large polymers (M larger than 400 kDa or so), but increases toward 1.0 or so at smaller M. Scaling parameters for the diffusion of star polymers do not differ markedly from scaling parameters for the diffusion of linear chains of equal size. 1. INTRODUCTION
This chapter treats the phenomenology for polymer single-chain motion (self- and tracer-diffusion) in solutions. We examine how the self diffusion coefficient depends on polymer concentration c, polymer molecular weight M, and solvent quality. We further examine how rapidly a probe polymer, molecular weight P, diffuses through a solutions of a matrix polymer, molecular weight M, at various matrix and probe concentrations. A considerable part of this Chapter is condensed from a longer review article available on the Cornell Physics Archives [1]. In particular, the extensive numerical tables on individual systems, which are of narrow interest and are represented graphically in Section VI, have been omitted here. Polymer melts and mutual diffusion are largely beyond the scope of this review. Reviews treating melts and theory include Graessley [2], Tirrell [3], Pearson [4], Skolnick [5], Lodge [6], and McLeish [7]. Recent papers by Schweizer and collaborators [8-10] and Skolnick [11], while actually research papers, include such thorough literature references as to be useful as reviews. Section 2 presents nomenclature and a theoretical background. Sections 3 and 4 review, respectively, (i) self-diffusion of polymers in solution and (ii) probe diffusion through polymer solutions. Section 5 notes other experimental papers that do not lend themselves to our analytic approach. Section 6 treats experimental studies on polymers in porous media and true gels. Section 7 treats universal phenomenological features in our analysis. Section 8 summarizes conclusions. 305
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2. THEORETICAL BACKGROUND This Section presents a consistent nomenclature for the diffusion coefficients and how these diffusion coefficients are measured. A short description is given of literature on diffusion in multimacrocomponent solutions, which provides a basis for interpreting experiments. A sketch of classes of models for polymer dynamics is presented. Proposed classes of phenomenological models are identified. A short sketch is made of alternative theoretical models that treat part or all of polymer dynamics. 2.1. Nomenclature This section sets out a consistent nomenclature for the diffusion coefficients. In a solution containing a solvent and one macromolecular species, there are two physically distinct diffusion coefficients. The two-particle or mutual diffusion coefficient Dm describes via Fick's law
f(r, t)
= Dm ~c(r, t)
(1)
the relaxation of a concentration gradient. Here J and c are the diffusion current and macromolecule concentration at position and time (r, t). The single-particle or selfdiffusion coefficient D 8 describes the diffusion of a single macromolecule through a uniform solution of elsewise identical macromolecules. In a simple macromolecule solution, quasi-elastic light scattering spectroscopy (QELSS) measures Dm [12,13]. Pulsed-field-gradient nuclear magnetic resonance (PFGNMR) measures D 8 • Most other experimental approaches for studying single-chain motion actually examine ternary solutions of a solvent and two distinct macrocomponents. In such ternary solutions, the relaxation of concentration fluctuations has at least two relaxation times, each describing a coupled mode involving both macrocomponents. If one macrocomponent, the probe, is dilute (the other macrocomponent, the matrix, may be either dilute or concentrated), the probe species' diffusion is governed by a single-particle diffusion coefficient, namely the probe diffusion coefficient Dp. If the matrix species and solvent are isorefractive, the matrix species scatters next to no light. If furthermore the probe species dominates light scattering by the solution (solvent scattering may be an issue), the diffusion coefficient measured by QELSS is [13,14] Dp. A variety of physical techniques have been used to measure probe diffusion in polymer solutions. In Fluorescence Recovery After Photobleaching (FRAP), an intense pulse of light is used to bleach fluorescent labels attached to macromolecules in small regions of the solution. In Forced Rayleigh Scattering (FRS), a similar light pulse generates an index of refraction grating in solution. Much weaker laser illumination then monitors fluorescence recovery, or decay of the index of refraction grating, as bleached and unbleached molecules diffuse in and out of the photomodified grating regions. If the probe species is dilute, the time dependence of the recovery or decay profile is determined by Dp [14]. In fluorescence correlation spectroscopy (FCS), one tracks the temporal evolution of fluctuations in the number of fluorescent molecules in a small volume of space. The diffusion coefficient measured by fluorescence correlation spectroscopy varies from Dp to Dm as the fraction of macromolecules that bear fluorescent tags is varied from small to large [14,15]. When is a probe species dilute? Dp is the probe diffusion coefficient at near-zero probe concentration. If Dp is substantially independent of cp, no extrapolation cp ---+ 0 is needed.
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If Dp depends significantly on cp, extrapolation to cP ---+ 0 must be performed. The initial slope of the dependence of Dp on probe concentration, and the slope's dependence on matrix concentration, have been measured in some systems and should be accessible to theoretical analysis. In this review: If the probe and matrix polymers differ appreciably in molecular weight or chemical nature, the phrase probe diffusion coefficient is applied. If the probe and matrix polymers differ primarily in that the probes are labelled, the phrase self diffusion coefficient is applied. The tracer diffusion coefficient is a single-particle diffusion coefficient, including both the self and probe diffusion coefficients as special cases. The interdiffusion and cooperative diffusion coefficients characterize the relaxation times in a ternary system in which neither macrocomponent is dilute.
2.2. General Theory of Diffusion in Multimacrocomponent Solutions What is the basis for using spectroscopic methods such as QELSS, FRAP, FRS, or FCS to measure D 8 or Dp? In each case physical theory links the spectroscopic correlation function to fluctuations in microscopic variables that describe the liquid. For QELSS, the spectrum is determined [16] by the field correlation function g(!l(q, t) of the scattered light, which arises from the scattering molecules via
(2) where i and j label two of theN scatterers, r;(t) is the position of scatterer i at timet, aj is the scattering cross-section of particle j, and q is the scattering vector. If the scattering particles are dilute (non-scattering particles may be concentrated), correlations between their positions vanish. The correlation function for probe scattering becomes
g~)(q, T) = (
t a7
(3)
exp[iq ·Clr;(T)]),
in which llr;(T) = r;(t + T)- r;(t)). Three general approaches lead to values for g~)(q,T). First, scatterers can be treated microscopically as objects having hydrodynamic and direct, e.g., excluded volume, interactions, and dg~)(q, T)/dT can be calculated. Second, because aq(t) 'L~ exp[iq · (r;(t)] is the qth spatial Fourier component of the scatterer density, semicontinuum hydrodynamics and the Onsager regression hypothesis predict the average temporal evolution of fluctuations. Third, scatterers could be assumed to perform simple Brownian motion. Light scattering spectra of non-ideal single-component macromolecule solutions, including direct interactions and hydrodynamic interactions at the Oseen level, were initially calculated by Altenberger and Deutsch [17]. Calculations of light scattering spectra of non-dilute many-component macromolecule solutions soon followed [12,13,18], reflecting earlier lines of indexKirkwood-Riseman model Kirkwood, et al. [19]. If neither macromolecular species is dilute, the light scattering spectrum obtained by QELSS is predicted [12,13,18] to contain two relaxation modes, even if one macromolecule species scatters no light. If the tagged macromolecules are dilute, fluorescence correlation spectroscopy and equivalent techniques measure the self-diffusion coefficient of the tagged probes [14].
=
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Jones [20] used a Smoluchowski approach to examine interacting spherical polymers. Jones predicted that, if one polymer species is dilute and labelled, the measured diffusion coefficient from QELSS is determined only by hydrodynamic interactions of the tagged polymers and their untagged matrix neighbors, and is the single-particle diffusion coefficient. The hydrodynamic approach culminated in analyses of Carter, et al. [21] and Phillies [22] of mutual and tracer diffusion coefficients, including hydrodynamic and direct interactions and reference frame issues. Semicontinuum treatments of polymer diffusion were made by Benmouna, Cohen and others: In particular: In 1987, Benmouna, et al. [23] calculated dynamic scattering from a ternary polymer solution, including excluded volume terms with a Flory interaction parameter but ignoring hydrodynamic interactions. An assumption that the hydrodynamic cross-diffusion tensor vanishes cannot be true simultaneously in the solvent- and volumefixed reference frames [19], leading to complications in the physical meaning of statements that hydrodynamics has been neglected. Subsequently, Benmouna, et al. [23] examined two species identical except for their scattering cross-sections, showing, consistent with microscopic treatments, that if the matrix species scatters weakly and is dilute, a mode describing single-chain motion is dominant. The same physical model was then extended to treat solutions of copolymers [24] and homopolymer: copolymer mixtures [25]. Foley and Cohen [26] analyzed concentration fluctuations in polymer: polymer: solvent mixtures, using an ornate Flory-Huggins form for the thermodynamic free energy of the mixture and neglecting hydrodynamic interactions. The case of a solvent isorefractive with the matrix polymer was treated. If both polymer species were nondilute, the relaxation spectrum was predicted to have three relaxation times. Roby and Joanny [27] improved the model of Benmouna, et al. [23] by incorporating interchain hydrodynamic interactions. At elevated concentrations, reptation dynamics were assumed, approximating the solution as a polymer melt in which mesoscopic polymersolvent blobs are effective monomers. Hammouda [28] repeated the calculation, removing the restriction that the system contained equal amounts of two species having the same molecular weight, and analyzing tagged-tracer experiments. Wang [29] analyzed a ternary polymer:polymer:solvent solution, while assuming that the cross terms in the mobility matrix vanish. Wang found a range of interesting parameters that follow from light scattering spectra and his model. In addition to the tracer: isorefractive matrix case, Wang examined the special case "zero average contrast", in which fluctuations in the the total polymer concentration at fixed composition scatter no light, showing that the spectrum of a zero-average-contrast system is almost always bimodal. Treatments using a Brownian motion description begin with Berne and Pecora [16], who treat a solution of dilute, noninteracting Brownian particles, for which from the Central Limit Theorem the probability distribution for the displacements of a particle is
G 8 (llR, t)
=
(4)
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the mean-square particle displacement being
(5) Combining eqs 3, 4, and 5, one finds
(6) For Brownian particles in low-viscosity small-molecule solvents, the Stokes-Einstein equation D=
(7)
kBT 61rTJR
(in which k8 , T, T), and R are Boltzmann's constant, the absolute temperature, the viscosity, and the particle radius) is generally accurate. Two ad hoc extensions of this form are encountered for probes in polymer solutions: First, one may formally define a microviscosity T)" via
and compare TJ" with the macroscopic solution and solvent viscosities, as has been done with data on mesoscopic globular probe particles [30] in polymer solutions. Second, relaxation spectra are not always single exponentials. No matter what functional form g~)(q, T) has, one may formally say that
(9) which defines D(T) as the time-dependent diffusion coefficient, leading to a nominal frequency-dependent diffusion coefficient
D(w) =
l)() dT exp(-iwT)D(T),
(10)
and frequency-dependent microviscosity
(11) The second extension has serious physical difficulties. From eqs 5, 6, and 9, and reflection symmetry, the extension assumes (12)
=L:f=
Equation 12 would be correct if sequential random changes in aq 1 exp[iq·rj] were described by a Gaussian random process. Brownian motion, with particle displacements described by the Langevin equation, indeed does generate a Gaussian random process for aq, leading to eq 12. For extensive details, see ref [16]. However, the above analysis errs in that it omits the consequences ofDoob's First Theorem [31]. Doob treated the correlation
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function of random variables such as aq. If aq is a Gaussian random process, then eq 12 is correct, but it is then an inescapable consequence of Doob's First Theorem that the relaxation spectrum g(ll (q, T) is a single exponential. Conversely, if the spectrum is not a single exponential, then from Doob's theorem (13) Berne and Pecora's text [16] is sometimes incorrectly cited as asserting that eq 12 is uniformly correct for light-scattering spectra. The analysis in Berne and Pecora [16] correctly obtains eq 12. However, this analysis refers the special case of a system in which particle displacements are governed by the simple Langevin equation. In these systems, particle displacements in successive moments are uncorrelated. However, in a viscoelastic system, the polymer solution has a memory; particle displacements in successive moments are no longer uncorrelated. With respect to probes in polymer solutions, Berne and Pecora's analysis is only applicable at times much longer than any viscoelastic relaxation times. If particle motion were observed over times shorter than the viscoelastic relaxation times: Particle displacements in successive moments would be correlated. Equation 12 would not be correct. log[g(ll (q, T)] / q2 would not be proportional to the mean-square particle displacement during T. With respect to polymer dynamics, the interest in eqs 9-12 has been the short time regime in which viscoelastic effects are apparent in D(T), but in the short time regime eq 12 does not describe probe motion in solution.
2.3. Phenomenological Forms for D(c, M) This chapter will compare experimental measurements of D 8 and Dp with c, P, and M dependences predicted by various models. Most models can be grouped into two major classes, differing in the functional forms that they predict for D.(c), namely a scaling (power) law and a stretched-exponential law. Both forms follow [32] from the same theoretical renormalization-group treatment, depending on the location of the supporting fixed point. The analysis below will examine whether either form describes experiment. Note that the forms are incompatible. Data described by a power law cannot be described with a stretched exponential, and vice versa, other than as a tangential approximant. (1) In scaling-law models [33], Ds and Dp follow power laws
(14) where 'Y and x are scaling exponents, and D 1 is a scaling prefactor. Some models have a restricted range of applicability, while others are only to be true asymptotically. Some models for melts derive a power law for D8 (M) and predict 'Y· For polymer solutions, more typically a power law is postulated, and 'Y and x are calculated. Many scaling-type models propose a transition concentration Ct between a lower concentration dilute regime and a higher concentration semidilute or entangled regime. The models often supply the functional dependence of Ct on M, without numerical coefficients, so a transition might or might not be found at unit value for Ct. Correspondingly, finding an apparent transition at, say, 2ct rather than Ct does not disprove a scaling model, because the numerical prefactors are not known with any precision. Also, none of these models treats the crossover in enough detail to predict whether observed crossovers should
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be sharp or broad. The transition concentration to the semidilute regime is the overlap concentration c*, formally the c* = N /V at which 47r R~N/ (3V) = 1. Here N is the number of polymers in a volume V and R9 is the radius of gyration. c* is often calculated from the intrinsic viscosity via c* = n/[1]] for some n E (1, 4). The transition concentration for entanglement, denoted ce, is sometimes obtained from a log-log plot of viscosity against concentration by extrapolating an assumed low-concentration linear behavior and an assumed higher-concentration power-law behavior (c"' for, e.g., x = 4) to an intermediate concentration Ce at which they agree. Alternatively, Ce is taken to demark the onset of viscous recovery. (2) In exponential models [34,35], D 8 follows an exponential or stretched exponential (15)
Here Do is the limiting diffusion coefficient, a is a scaling prefactor, and v is a scaling exponent. If the probe and matrix molecular weights P and M differ, an elaborated form
(16) is applied. Here a, ry, and 8 are additional scaling exponents, Do represents Dp in the limit of zero matrix concentration of a hypothetical probe having unit molecular weight, and p-a describes the dependence of the zero-concentration Dp on probe molecular weight. In derivations [34-40] of the stretched-exponential models, functional forms and numerical values for exponents and pre-factors were calculated, using various approximations (in particular that chain motion is adequately approximated by whole-body translation and rotation) that are not obviously appropriate if P and M differ greatly. Some exponential models [41,42] also include a transition between a lower-concentration regime, in which a stretched-exponential concentration dependence is found, and a higher-concentration regime, in which transport coefficients show power-law concentration dependences. This transition sometimes appears in viscosity data [41-43], typically at some c[T]] > 35, but is very rarely found with D 8 or Dp· Models of polymer dynamics are also partitioned by their assumptions as to the dominant forces in solution, these assumptions being totally independent of the assumed concentration dependence. In some models, excluded-volume forces (topological constraints) dominate, while hydrodynamic interactions dress the monomer diffusion coefficient. In other models, hydrodynamic interactions dominate, while chain-crossing constraints are secondary. Experimentally, D 8 ( c) is directly accessible, but the intermolecular forces can at best only be inferred from numerical coefficients D 1 , a, and so forth.
2.4. Alternative Models for Polymer Dynamics A wide variety of theoretical models have been proposed as treatments of polymer dynamics in solutions and melts. A full review of all these models is too lengthy for this Chapter. This section sketches classes of model, noting reviews and entry points into the theoretical literature. Much literature involves extensions of the reptationjtube model of deGennes, Doi, and Edwards. The deGennes model [44] was originally proposed to describe a linear polymer chain diffusing in the presence of fixed obstacles, such as those presented by a covalentlycrosslinked gel. In the original model, the chains of the gel are rigidly locked in place,
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so a diffusing probe chain cannot displace them. The probe is instead constrained to diffuse back and forth, parallel to its own curvilinear length. Hydrodynamic interactions ("back-flow effects") were not included, because " .. .in a gel, we expect that back flow around a moving object decays exponentially within a finite distance ... ", Debye [45] being cited as validating this expectation. A closing two-paragraph section "Conjectures on Polymer Melts" observes that the relaxation time calculated from the model scales as M 3 , while the relation time in polymer melts was known to scale as M 3 .4, but that "The transposition of our results (for one chain in a fixed gel) to a system of many mobile chains is extremely delicate ... ". Doi and Edwards' results are summarized by their monograph [46]. The authors begin with the single-chain models of Zimm [47] and Rouse [48] and then treat nondilute solutions, chains moving through fixed lattices, and finally present their tube model for concentrated and melt systems. Discussions of concentration dependences in nondilute solutions are confined to a few pages, in which arguments based on scaling transformations are used to compute exponents for power-law concentration dependences of the zero-shear viscosity and other variables. Doi and Edwards, in presenting these results, emphasize as caveats ' ... whether dynamic scaling holds for systems dominated by the topological interaction is still a matter of debate ... " and "The concentration dependence of TJo is delicate since the friction factor ~ depends on concentration ... ". Several authors have presented calculations based on entanglement/tube models, in which excluded volume or chain noncrossing constraints are assumed to dominate, hydrodynamic interactions are neglected or used to dress the single-bead diffusion coefficient, and scaling-law (Ds "' ex) behavior for the diffusion coefficients is assumed or derived under certain assumptions. Almost all of these calculations refer only to the melt. However, Hess [49], using the Mori formalism, assumptions about the interchain forces and their correlations that are appropriate for reptation, and disincluding solvent mediated (hydrodynamic) forces) obtained x ~ 7/4 for solution systems. Schaeffer [50] proposes an alternative scaling treatment of semidilute (low-concentration, heavily-overlapped) polymer solutions. Shaeffer noted that his model was not an extension of the Doi-Edwards model. In his model, solution structure is based on the Schaeffer-Joanny-Pincus [51] model, with a hydrodynamic screening length that differs from the tube radius, leading to x in the range 1. 75-3. Schaeffer notes that in some circumstances the hydrodynamic screening length in his model is larger than the tube radius, implying that a diffusing chain would be obliged, as it moved, to drag along its tube. Schweizer and collaborators have elaborated an extensive mode-coupling model of polymer dynamics [52-54]. The model does not make obvious assumptions about the nature of polymer motion or the presence or absence of particular long-lived dynamic structures, e.g., tubes; it yields a set of generalized Langevin equations and associated memory functions. Somewhat realistic assumptions are made for the equilibrium structure of the solutions. Extensive calculations were made of the molecular weight dependences for probe diffusion in melts, often leading by calculation rather than assumption to power-law behaviors for various transport coefficients. However, as presented in the papers noted here, the model is applicable to melts rather than solutions: Momentum variables have been completely suppressed, so there are no hydrodynamic interactions. Readers should recall that 'hydrodynamic' interactions usually refer to interactions that are solvent-mediated.
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Here, however, all momentum variables have been suppressed, so hydrodynamic interactions between polymer chains, that would have been mediated by momentum transport by the viscoelastic fluid that is a polymer melt, have equally been omitted. The tube model rests upon several assumptions, in particular that the tube lifetime is meaningfully long, and that the walls of a tube are capable of blocking the motion of the chain enclosed within it: 1) The first assumption, improved upon in more recent calculations, requires that the tube lifetime is at least as long as the time required for the probe chain to translate out of the tube. Because the tube is composed of other polymer strands, it is not obviously consistent to propose that the tube lasts so long. Extensive calculations to relax or validate this assumption have been made. 2) The second, far more fragile, yet unexamined assumption of the tube models is the assertion that when a chain segment, moving transversely, collides with a segment of a second chain, it is brought to a stop. This assumption was credible for deGennes original model of a chain diffusing in a crosslinked gel-the gel chains are connected to the container walls. In an on-lattice computer simulation, the second assumption is valid: There is no mechanism whereby which a moving chain can force an obstructing chain from its path. However, in solution the chains forming the walls of the tube are not infinitely more massive than the probe chain, nor do they have drag coefficients that are arbitrarily larger than the drag coefficient of the probe chain. Why, then, should they instantly bring the probe chain to a halt when they collide? A graphic image for the alternative outcome, namely drag-along dynamics, is provided by a pot of noodles heating on a stove. The water has extensive convective flow. The noodles move transverse to their length, but do not obstruct each other's movements. If one chain moves sidewise, its neighbors are very nearly certain to move in parallel. In the polymer solution, the chains play the role of the noodles, and hydrodynamic interactions between nearby chains play the role of the convecting, heated water. An entirely different approach to calculating polymer dynamics is presented by K. L. Ngai and collaborators [55], who begin by treating the mathematical structure of the relaxations. Ngai and many collaborators [56-62] propose that relaxations of wide classes of nonlinearly coupled modes have certain common features. In particular, at short times individual modes relax as exp( -t/r0 ), but after some time tc the exponentials follow stretched-exponential forms exp( -(t/r) 1-n), in which T and 7 0 are relaxation times, and n measures the strength of the nonlinear intermode couplings. A requirement that the two relaxations have a continuous first derivative constrains the relative values of f3 and the three time parameters. Simulations [56] on a simple model have found the behaviors predicted by the coupling model. Ngai's model has been extensively applied to polymer dynamics [57-62]. The theoretical treatment most closely related to this review is Ngai, et al.'s comparison of self-diffusion of linear, 3-armed, and 12-armed star polymers in solution [59]. In this paper, Ngai and collaborators examine diffusion of polystyrene probes through linear polyvinylmethylether [59] matrix solutions. If one considers f-armed star polymers (where a linear chain has f = 2), Ngai, et al., show for fixed arm molecular weight Ma that, as f is increased, DP in a given matrix polymer solution should fall, the extent of the fall being larger as matrix concentration is increased. They propose that this result shows a dependence of DP on
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diffusant architecture. However, if one increases f at fixed Ma, the total molecular weight of the probe is also increased. Is the significant change in the polymer chain the increase in f or the increase in M? Teraoka, et al. [63] treat theoretically the dynamics of polymers moving through true tubes, namely holes in random porous media, such as porous vycor glasses. The random pores had a representative average distance over which they are straight, and a mean distance between branch points. For pore diameters smaller than the polymer radius of gyration, the partition coefficient between the porous volume and the bulk medium was calculated. Increasing the chain rigidity makes it easier for chains to be found in pores, until the rigidity is so large that the chains cannot conform to the bends in the tubes. The authors also calculated, as a function of time, the mean-square displacement of a representative bead. Yet another model for random walks in porous media is presented by Chakrabarti and colleagues, who examine self-avoiding walks on lattices, only a randomly chosen subset of whose points are accessible to the walks. The scaling relationship between the end-to-end distance and the chain length was obtained [64]. The effect of varying the fraction of points accessible to a walk is subject to renormalization group, analytic, and numerical analysis [65]. For a walk of this sort of given chain length and radius of gyration, Chakrabarti, et al. [66] consider the long-ranged (hydrodynamic) interaction on top of the Rouse model, and compute a self-diffusion coefficient for the chain. In this model, hydrodynamic interactions are primarily important between beads of the same chain, interchain hydrodynamics being taken to approximately cancel. Lattice occupation by neighboring chains changes the chain diameter of a representative chain. However, when the representative chain diffuses the occupied sites are by implication carried along with the representative chain, rather than blocking its motion as would be the case in tube models. Ref. [66] obtains scaling expressions for the dependence of the diffusion coefficient on the ratio of the percolation correlation length to the polymer radius and on the fraction of accessible sites (effectively, the matrix concentration). On the other hand, when Flory-type arguments relating to the polymer free energy and the percolation disorder were applied to the same walks, Ds was predicted [66] to depend on chain length Nand distance in concentration c from the percolation threshold Cp via a stretched exponential exp(- N 7 (c- cpY) with exponents 1 and v related to the polymer size and percolation exponents, respectively. The model of Chakrabarti, et al. [66] is one of the few polymer diffusion models to predict (correctly, as seen below) that Ds is related to c and M via a stretched exponential, not a power law. The indexKirkwood-Riseman model Kirkwood-Riseman model [67] treats the dynamics of a single polymer chain, representing the chain as a set of point hydrodynamic beads interacting via hydrodynamically neutral springs ("bonds") and via low-frequency solvent-mediated hydrodynamic forces as represented by the Oseen tensor. In principle, the model should naturally be extensible to solutions. Calculations of the effect of intermolecular hydrodynamics on viscosity include works by Riseman and Ullman [68], and Saito [69,70]. Peterson and Fixman [71] have proposed that a correct calculation of the concentration dependence of TJ either must treat correctly the boundary stresses or in some clever way avoid the issue: Peterson and Fixman ingeniously avoid the issue by simultaneously calculating the average stress tensor and the average velocity gradient. Saito found the second-order concentration correction to the viscosity of a polymer solution,
Polymer diffusion in solution
315
but concluded that this calculation, as made with the Oseen hydrodynamic interaction, led to improper integrals, and proposed an alternative treatment of the fundamental hydrodynamics to avoid this difficulty. Some authors have proposed to avoid the improper integrals by asserting that hydrodynamic interactions are screened in polymer solutions, but the definitive calculations of Altenberger, et al. [72] and of Freed and Perico [73] show there is no hydrodynamic screening, at low frequencies, if polymer chains are free to move with respect to each other. This author and collaborators [35-37] have applied the indexKirkwood-Riseman model Kirkwood-Riseman model to treat the concentration dependence of Ds in non-dilute polymer solutions, leading to the hydrodynamic scaling model for polymer transport coefficients. In the hydrodynamic scaling model, long and short range hydrodynamic forces between polymer chains are taken to dominate the dynamics, so that if one chain moves, its neighbors move in similar ways. While tubes of the sorts described by tube models might exist as static structures, the dynamic effects of topological entanglements are taken to be negligible. Readers should recall that 'entangled', as a descriptor of polymer solutions, refers historically to certain viscoelastic properties, such as elastic recoil and the presence of a single extremely long 'terminal' relaxation, with no presumption as to the physical mechanism that leads to the macroscopic viscoelastic effects. The hydrodynamic scaling model in its present development approximates a polymer coil as a cloud of beads with a center of mass, orientation, and internal coordinates whose fast relaxations are neglected. Interchain interactions are described by the bead-bead Oseen tensor, averaged over the beads in both chains, with the consequence that a polymer chain that translates and rotates with respect to the unperturbed solvent introduces fresh translations and rotations with respect to the solvent in neighboring chains. Calculations are made in the low-frequency limit in which the net force and torque on the beads of each chain must sum to zero. Each neighboring chain induces additional flows in the liquid, ad infinitum, leading to a multiple-scattering description of hydrodynamic interactions between chains. The multiple-scattering description has been carried out through fourth order [39], this being the lowest order that tests the renormalization group extension of the calculation. Standard averages over multi-chain distribution functions lead from hydrodynamic interaction tensors to series expansions for the concentration dependence of D 8 • Selfsimilarity arguments or Altenberger's [32] renormalization-group expansion lead from the series expansion to stretched exponentials including eq. 16. On applying a shear field with zero net applied force (so boundary issues do not arise) a corresponding expression for the low-shear viscosity is [38] obtained. The potentially theoretically underevaluated parameter in the expressions for TJ(c) and Ds(c), as independently calculated and determined from experiment, are in reasonable agreement [38]. If short-range hydrodynamic interactions are included in the calculation, comparison is possible between the excess drags experienced in nondilute polymer solutions by a representative polymer bead and by a solvent molecule. The calculated drag excesses are unequal, showing that the concentration dependence of Ds of a small-molecule solute or solvent molecule does not [40] give the concentration-dependent part of the "monomer friction coefficient". The cited review articles [2-7], viewed chronologically, in additional to many other excellent features, are notable for an increasing though not perfectly regular tendency to
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focus more and more on melts, and less and less on solutions. Lodge, et al. [6] make a very careful examination of selected measurements of Ds, carefully examine the expansion of tube radius with dilution, compare with the molecular weights, radii of gyration, and concentrations for which there were at time of that writing Ds(c) measurements, and conclude " ... As argued in the text, it is unlikely that reptation is significant in the semidilute regime, whereas in melts it is often the dominant mode. There has not [GP: in 1990!] been a thorough examination of the concentration regime of 10-100% polymer, particularly for diffusion measurements ... " Lodge, et al's concern was asking whether or not linear polymer chains reptate under at least some circumstances, so their review rationally excluded detailed and thorough examination of concentration-molecular weight regimes in which reptation should not occur. The following parts of this chapter largely treat experiments on diffusion by neutral polymers in good and theta solvents, there being little published data on polyelectrolytes. We advance by systematically comparing eqs. 14-16 with the literature via non-linear least-squares fits, minimizing the fractional mean-square difference between the data and each fitting function. In some cases, one or more potentially free parameters were held constant ("frozen") during the fitting process.
3. SELF-DIFFUSION This Section treats the true self-diffusion coefficient D 8 • Results appear alphabetically by first author, showing for each paper the data (points) and fits to stretched-exponentials (lines). Brown, et al. [74] measured Ds from PFGMR for dextran (Mn = 44 kDa, Mw = 64.2 kDa) in water, for c up to 250 g/L, as seen in Fig. 1. Brown and the same collaborators [75] also measured Ds of of narrow molecular weight distribution (Mw/Mn E (1.02, 1.20)) polyethylene oxides in water, as seen in Fig. 2. Polymer molecular weights were in the range 73-661 kDa; polymer concentrations reached 70 g/l. Data were fit both to a pure (v = 1) and a stretched exponential inc; the simple exponentials (which gives excellent fits) are shown. Callaghan and Pinder [76-78] used PFGNMR to study Ds of 2, 110, 233, and 350 kDa polystyrenes, 1.06 ::::; Mw/Mn ::::; 1.10, in CCl4 (Fig. 3) and C6 D6 (Fig. 4). Lowconcentration measurements were only made in CC1 4 • Stretched exponentials describe D s (c) well in all cases, with a tending to increase while v tends to decrease with increasing M. Deschamps, et al. [79] used FRS to study Ds of polystyrenes in cyclopentane near the theta point, for M of 262, 657, and 861 kDa, and concentrations 1 ::::; c ::::; 240 g/L, at 1.1 ~ Mw/Mn ~ 1.3, as seen in Figure 5. Ds of the 657 kDa polymer was not reported at small c; its fit was made by interpolating Do and v from the 262 and 861 kDa polymers and computing a. The change from good to near-theta solvent conditions does not impair the quality of the fit to stretched exponentials in c. Fleischer [80] used PFGNMR to observe self-diffusion of 125 kDa polystyrene, Mw/ Mn ~ 1.02, in toluene for concentrations 80-320 g/L or, equivalently 0.5ce ::::; c ::::; 2ce, estimating Ce from rheological data [81]. Fleischer's data agree well with eq. 15. Fleischer found that the incoherent dynamic structure factor Sinc(t) has only a single fast relaxation, even
Polymer diffusion in solution
317
-ff!.
E
~
o"'
10
100
c (g/L)
Figure 1. Ds of 64.2 kDa dextran: water from Brown, et al. [74].
10
100
c (g/L)
Figure 2. Ds of (top to bottom) 73, 148, 278, and 661 kDa polyethylene oxides in water, from Brown, et al. [75].
when the QELSS spectrum has a dominant slow mode. Fleischer's observations on Sinc(t) closely resemble Zero and Ware's [82] data on FRAP of poly-L-lysine solutions at small salt concentration I. The slow mode dominated under small-c, small-/ conditions that enhance 'glassy' behavior. The appearance and dominance of the slow mode had little effect on D 8 , which is hard to understand if the slow mode is caused by the formation of large numbers of long-lived physical aggregates, but is readily understood if the slow mode reflects long-lived dynamic structures. Giebel, et al. [85] report Ds from PFGMR of 15, 530, and 730 kDa polydimethylsiloxanes in toluene, including data of Skirda, et al. [86], for cup to 900 g/L. As seen in Fig. 6, at each M a stretched exponential inc describes Ds(c) well. Hadgraft, et al. [87] used QELSS to measure Ds of dilute polystyrenes in C6 H6 for 24.8 ::::; M ::::; 8870 kDa. The RMS fractional error in the fit is 18% (if all points are included) or 8.6% (if the largest point is omitted), and gives Do= 4.54 · 10- 4 M- 0 ·588 . Using FRS, Hervet, et al. [88] and Leger, et al. [89] measured Ds of monidisperse (Mw/ Mn ~ 1.06- 1.12) polystyrene:C 6 H6 for polymer molecular weights 78.3, 123, 245, 599, and 754 kDa, and measured Dp of labelled 245 and 599 kDa polystyrenes in solutions of, respectively, 599 and 1800 kDa polystyrenes. The data and their fits appear in Figs. 7 and 8. Eq. 15 works well except forDs of the 245 and 598 kDa polystyrenes, for which Ds at first increases with increasing c and then decreases at larger c. This behavior, which is visibly larger than the experimental scatter, is unique to these measurements. No other paper treated in this review reports a Ds that increases with increasing c, even over a limited concentration range. From Fig. 8, at fixed c the 245 kDa polystyrene diffuses
G. D. J. Phillies
318 10-5
10-6
10-6
10-7
?. 10-7 E
10-8
~
o"' 10-8 10-9
10-9 10-10
10-10 1
10
100
c (g/L)
Figure 3. Ds of [top to bottom] 2, 110, 233, and 350 kDa polystyrenes in CC1 4 from Callaghan and Pinder [76-78].
10
100
c (g/L)
Figure 4. Ds of [top to bottom] 110, 233, and 350 kDa polystyrenes in C 6 D6 as obtained with PFGNMR by Callaghan and Pinder [76-78], and fits to stretched exponentials.
approximately equally rapidly through 245 and 599 kDa matrices. In contrast, at fixed c diffusion of 599 kDa polystyrene is markedly slower in solutions of the 1800 kDa polymer than in solutions of the 599 kDa polymer. Nemoto, et al. [90] used FRS to study polystyrene:dibutylphthalate for 43.9 :S Mw :S 5480 kDa and two polymer concentrations: 130 and 180 g/L. Polydispersities were 1.01 :S Mw/Mn :S 1.09, except Mw/Mn ~ 1.15 for the 5480 kDa material. As seen in Fig. 9, the M-dependence of Ds changes near M ~ 800 kDa. At smaller M, a stretched exponential in M is found; at larger M, one finds Ds ,...., M- 7 with 1 ~ 2.49 at 130 g/L and 1 ~ 2.22 at 180 g/L. In a separate paper, Nemoto, et al. [91] determined Ds and the steady-state shear viscosity TJ of concentrated solutions (40 and 50 wt%) of 44 and 355 kDa polystyrene: dibutylphthalate. Under the transient lattice models, these solutions of the 44kDa polymer should be unentangled, while the 355 kDa polymer solutions should be entangled. In all four systems, Ds/T and the fluidity TJ-l have virtually the same dependence on temperature. Skirda, et al. [92] used PFGNMR to study polyethylene oxides (M = 2, 20, 40, and 3000 kDa) and polystyrenes (Mn = 240 and 1300 kDa) in chloroform, benzene, dioxane, and carbon tetrachloride over a full range of polymer volume fractions ¢. Mw/ Mn was ~ 1.1 for polyethylene oxides (except Mw/Mn ~ 2 for the 3000 kDa polymer) and~ 1.2 for polystyrenes. Figures 10 and 11 show the data. For each polymer:solvent combination, a stretched-exponential fit as shown describes the data well.
Polymer diHusion in solution
319
10-6
10-7
Cil
--E
10-8
o"'
10-9
N
~
10-10
10-11
10
100
c (g/L)
Figure 5. Ds of [top to bottom] 262, 657, and 861 kDa polystyrenes in cyclopentane near the theta temperature, as obtained with FRS by Deschamps and Leger [79], and fits to stretched exponentials.
10
100
c (g/L)
Figure 6. Ds of [top to bottom] 15, 530, and 730 kDa polydimethylsiloxane in toluene, as obtained with PFGNMR and reported by Giebel, et al. [85] based in part on work of Skirda, et al. [86], and fits to stretched exponentials.
Tao, et al. [93] measured Ds (using PFGNMR and forward recoil spectroscopy) and TJ of hydrogenated polybutadienes in alkanes. Polymer volume fractions ¢ extended from 0.2 up to the melt while M extended from 4.8 to 440kDa with Mw/Mn 1.03. Tao, et al., fit their self-diffusion data to a scaling description Ds ,..., cpa Mb. When they [93] forced a = -1.8, a one-parameter fit found that the averaged Ds¢1. 8 is ,..., M- 2 .4 1 • Figure 12 shows Tao, et al.'s data as fit to stretched-exponentials D 00 M-z exp( -acv M 7 ), the factor M-z that almost entirely determines the M-dependence of Ds appearing because the dilute-concentration diffusion coefficient scales with M. We find 1 ~ 0 and z = 2.42. Because the curvature of the stretched exponential large occurs at low c, it is difficult to determine v accurately. The fit to a scaling equation finds Ds ,..., cl. 71 M 2.4 2 , the fractional root-mean-square fit errors to the stretched exponential and power laws being equal. Tao, et al. [93] concluded that a scaling-law form fits their data. The analysis here corroborates this statement, but shows that it is incomplete, in that stretched-exponential forms describe equally accurately the measured Ds(c, M). Tinland, et al. [94] report Ds from FRAP of xanthans having M E 0.45- 9.4 MDa, Mw/ Mn E 1.2- 1.4, and concentrations 0.01-40 g/L, as shown in Fig. 13. Xanthan forms wormlike chains. This data differs from that on almost all other polymers: d ln(Ds) / d ln(c) does not decrease monotonically with increasing c. For all but the largest-M polymer, there is a concentration c** at which Ds(c) deviates from small-c behavior. The fit depends
s
G. D. J. Phillies
320 10-6
10-7
10-8 0 "'
10-9
10-10
0.1
10
100
c (g/L)
Figure 7. Ds of [88,89] [from top to bottom] 78, 123, 245, 599, and 745 kDa polystyrene in benzene and fits to eq. 15.
10-10 0.1
10
100
c (g/L)
Figure 8. Ds of [89] 245 kDa polystyrene in 245 and 599 kDa polystyrene: benzene (open, filled circles), and 599 kDa polystyrene in 599 and 1800 kDa polystyrene: benzene (open, filled squares).
marginally on the number of data points included in the analysis. For the 3800 kDa polymer, we indicate (solid, dashed lines) the fits to the first 7 or 8 data points. In Tinland, et al.'s language [94], c**, the higher-concentration boundary of the semidilute regime, decreases with increasing M. von Meerwall, et al. [95] used PFGNMR to measure Ds of linear and star polyisoprenes for polymer weight fraction 0.01 ::::; x ::::; 1, arm number f ranging from 2 to 18, and span molecular weights Ms of 5 and 18 kDa. Data were originally reported as smooth curves. From these measurements, at all c, increasing f at fixed Ms reduces D8 : Increasing f four-fold reduces Ds by two- or three-fold. At f = 8, increasing Ms reduces D 8 • Ds(c) for each f and M is described well by a stretched exponential in c. For fixed Ms and increasing f, Do and v fall while a increases, a two-fold decrease in Do via increasing f being accompanied by a 20-fold increase in a. For the smallest molecular weights (10-16 kDa) studied, v > 1 is observed. von Meerwall, et al. [96] used PFGNMR to study linear and 3-armed star polybutadienes and polystyrenes in CCl4. Mw ranged from 2.3 to 281 kDa with Mw/Mn of 1.03-1.07. For most systems the observed concentration range afforded only a one-order-of-magnitude variation in D" leading to adequate stretched-exponential fits. We concentrate on the 6.5, 8.3, 29, and 76 kDa 3-armed stars, which were studied over wider concentration ranges. As initially noted by von Meerwall, et al. [96], 'the slopes [in Fig. 14] change continuously'
Polymer diHusion in solution
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Figure 9. Ds of 130 and 180 g/L polystyrene in dibutylphthalate as a function of molecular weight, showing the lower-molecular-weight stretched-exponential and the highermolecular-weight power-law molecular-weight dependences of D., using data of Nemoto, et al. [90].
' ... deGennes' prediction of a concentration scaling regime Ds " ' c L 75 (c* < c < c**) is not borne out by our data ... at any molecular weight'. Two data sets that did not include small-c measurements were fit to stretched exponentials, after including dilute solution measurements of Hadgraft, et al. [87] as adjusted for polymer molecular weight and viscosity and temperature of the particular solvent. For brevity, Figures for these more restricted data sets are omitted. von Meerwall, et al. [97] used PFGNMR to measure Ds of 10, 37.4, 179, 498, and 1050 kDa polystyrene in tetrahydrofuran at concentrations 6-700 g/L, and also Ds of tetrahydrofuran and hexafluorobenzene in the same polymer solutions. Wesson, et al. [98] used FRS to measure Ds of polystyrenes in tetrahydrofuran and benzene. Polystyrenes had M of 32, 46, 105, 130, and 360 kDa, and were observed for 40 ::::; c ::::; 500g/L. Ds covered four orders of magnitude, but only for Ds/ Do substantially less than 1. Xuexin, et al. [99] used PFGNR to measure Ds of linear and 18-armed star polyisoprenes in CC1 4 over a wide range of c and a 100-fold range of M. Their results appear as Figs. 15, 16, and 17. In every case Ds(c) follows closely a stretched exponential. Figure 15 shows Ds for four low-molecular-weight (61, 92, 193, 216 kDa) 18-arm polyisoprenes. With increasing M and fixed f, Do and v decrease, while a increases, the increase in a being modestly more rapid with increasing M than is the decrease in D 0 • These trends continue in Fig. 16, which shows larger-M (344, 800, 6300 kDa) stars. The displayed data on 302 kDa linear polyisoprene are fit by very nearly the same D 0 , a and v as is the data on a much larger (800 kDa) 18-arm star polyisoprene. Finally, Fig. 17 shows Ds(c) for 70.8,
G. D. J. Phillies
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10-8
10-8
10-9
10-9
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10-10
0.001
10-11
0.01
0.1
c (Volume Fraction)
Figure 10. Ds of poly(ethylene oxide), [from top to bottom, at ¢ ~ 0.1] namely 2kDa in CHCl 3 and in dioxane, 20 kDa in benzene and in dioxane, 40 kDa in chloroform and in dioxane, and 3600 kDa in dioxane, after Skirda, et al. [92].
0.001
0.01
0.1
c (Volume Fraction)
Figure 11. Ds of polystyrene, namely [from top to bottom] 240kDa in benzene and 1300 kDa in benzene and carbon tetrachloride, respectively, after Skirda, et al. [92].
251, and 302 kDa linear polyisoprenes. Nine of the above papers, namely Refs. [75,78,79,85,89,96,98,99] report Ds for a homologous polymer over a range of both c and M. A simultaneous fit of each data set to eq. 16 is then practicable, the polymer molecular weight being P and dependence on a second molecular weight being suppressed. Results of the fits appear in Table 1 and in Figs. 18-20. Most but not all fits were good, with RMS fractional errors of 6-18%. In general: The exponent a is consistently -0.5. The concentration exponent v is in the range 0.5-0.75, except Brown, et al.'s data [75] imply v ~ 0.93. The molecular weight exponent 1 is in the range 0.32-0.46, except Brown, et al.'s [75] data imply 1 ~ 0.6. A joint stretched exponential in c and polymer molecular weight thus fits most data sets well. However, Xuexin, et al. [99] cover an extremely broad range of M, so that v varied substantially with molecular weight. A fit of all of Xuexin, et al.'s data to eq. 16, with v held constant, therefore worked poorly for large or small P. A fit to intermediate P worked well and is displayed (solid curves). The two data sets that are fit less well by eq. 16 appear in Fig. 20. Leger, et al.'s [89]'s work on polystyrene:CC1 4 shows features - notably a non-monotonic dependence of Ds on c - that appears in data on no other polymer system, including other experiments on the same chemical system. Wesson, et al. 's [98] measurements lack low-concentration data where Ds ~ Dso· By inspection of Fig. 20b, the merged fit works reasonably well at
Polymer diffusion in solution
323
Table 1 Concentration and molecular weight dependences of Ds and DP for molecular weight P probes in solutions of molecular weight M matrix polymers (for Ds, one has P = M) at concentration c. The fits are to stretched exponentials Dop-a exp( -acv P 7 M 6 ), with the percent root-mean-square fractional fit error %R, the material, and the reference. Molecular weights are in kDa; concentrations except as noted are in g/L. Square brackets "[ · · · J" denote parameters that were fixed rather than floated. Notes: (a) Various, with M/P;::: 10; (b) Various, see text; (c) All four probes, see text; (d) Excluding styrene monomer, see text; (e) Not all data points, see text; (f) P = M, self diffusion. p M a a Ref v 5 %R c Do 'Y 5 5 6.27 X 10 0.93 0.61 (2) [75J 9 (b) (f) [0.5J 5.96. 10 [OJ 7.99 x w- 5 0.50 7.38. w- 4 0.58 0.46 17 (b) [77J (2) (f) [OJ 4 3.41 x w0.36 0.55 2.10. w- 3 0.64 (b) [77J 7 (b) (f) [OJ 2.94 x w- 4 0.50 1.53. w- 2 0.52 0.25 24 (b) [79J (b) (f) [OJ 7.46 x w- 4 0.501 6.17. w- 3 0.48 0.33 (b) [85J (b) (f) [OJ 7 7.55. w1.16 x w- 4 0.95 0.86 37 (b) (f) (b) [89J [0.5J [OJ 1.67 x w- 4 0.48 17 (b) [0.5J 5.08. w- 4 0.75 (b) (f) [96J [OJ 1.87 x w- 4 0.24 51 (b) [98J (b) (f) [0.5J 1.93. w- 3 0.91 [OJ 9.68 x w- 4 0.65 7.37. w- 4 0.68 0.42 (b) (f) (b) [99J [OJ 5.7 0.891 0.43 24 8000 (b) (b) [lOOJ [OJ 5.86. w- 4 0.43 [OJ 1.025 (b) [101J (b) (b) [OJ 1.53. w- 3 0.95 -0.01 0.236 4.8 7.46. w- 6 0.36 2.84. w- 4 1.15 0.26 34 (b) [106J (b) 60 [OJ 8.91. w- 6 0.34 6.80 ·10- 4 1.03 0.25 24 (b) [106J (e) 60 [OJ 3.34. w- 4 0.57 1.82. w- 5 0.99 0.139 0.287 6.4 (b) (b) (b) [107J 1.86. w- 4 0.52 4.45. w- 5 0.69 0.30 0.33 17 (e) (b) (b) [108J 8.5. w- 4 0.61 2.99. w- 2 0.61 0.19 12 (b) 1300 (b) [110J [OJ 3.0. w- 4 0.54 1.01. w- 3 0.66 0.42 (b) 1300 (b) [110J [OJ 6.6 4 1.84 . w0.50 1.77. w- 2 0.68 0.16 12 (b) [111J (b) 140 [OJ 2.20. w- 4 0.52 5.09. w- 4 0.61 0.16 0.32 32 (b) [112J (b) (b) 2.93. w- 4 0.53 2.22. w- 3 0.66 15 0.15 0.19 (b) [112J (b) (b) 4 6.51. w0.67 9.49. w- 3 0.86 0.115 (b) [104J (b) 132 [OJ 4.5. w- 3 0.68 6.67. w- 5 20 342 (a) (a) [115J [0.5J [0.3J 0.024 5 2 7.9. w0.95 (b) [116J (b) (b) [0.52J [OJ [0.16J 0.028 7.64. w- 6 130 [117J 0.56 7.61. w- 4 0.30 0.15 (a) 13 (b) (b) 7.89. w- 6 0.63 1.35. w- 3 12 0.28 0.11 180 [117J (a) (b) (b) 8.41 . w- 6 0.64 2.31. w- 4 0.74 0.26 0.29 31 (b) (b) (b) [117J 1.65. w- 4 0.52 1.56. w- 6 0.79 0.285 0.501 23 (a) (a) (b) [118J 3.98. w- 4 0.68 1.00. w- 3 [l.OJ 0.26 (c) (a) (b) [120J [OJ 7.9 1.31 . w- 4 0.52 7.56. w- 4 [l.OJ 0.30 (b) [120J (d) (a) [OJ 5.1 2.01. w- 4 0.51 1.03. w- 2 0.64 0.26 14 (c) 1300 (b) [124J [OJ 3 4.99. w0.39 8.33. w- 4 0.54 0.25 0.22 25 (c) (b) (b) [125J
324
G. D. J. Phillies 10-5 10-6 10-7
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1o-a
1
10-9
N-
o"'
10
-10
10-11 10-12 10-13
10-10
0
0.2 0.4 0.6 0.8 c (Volume Fraction)
Figure 12. Ds of hydrogenated polybutadienes in alkanes from Tao, et a!. [93], with polymer molecular weights [top to bottom] 4.9, 10.3, 23.3, 53.2, 111, 364, and 440 kDa, as fit by Ds = 5.54 · 10 3 M- 2 .4 2 exp( -5.026c0 ·5 M 0 ·0002 ).
0.01
0.1
1
10
100
c (g/L)
Figure 13. Ds of xanthan, molecular weights [from top to bottom] 0.45, 0.99, 1.9, 3.8, and 9.4 MDa. in water, for Tinland, et a!. [94].
small and large polymer P, but less well at intermediate P.
4. DIFFUSION OF PROBE CHAINS THROUGH MATRIX POLYMER SOLUTIONS This Section reviews probe diffusion measurements-studies on ternary solutions in which the molecular weight P of the probe polymer, and the molecular weight M of the matrix polymer, differ. In some cases, probe and matrix polymers have a common monomer but differ in molecular weight. In other cases, the probe and matrix polymers are chemically distinct. We proceed alphabetically through the literature. Figures show the measurements, and fits to a stretched exponential (eq. 16) inc, P, and M, leading to fitting parameters in Table 1. We also fit results on each P : M combination to a separate stretched exponential in c. These fits were almost always extremely good. For Figures and corresponding fitting parameters for the individual P : M fits, see Ref. [1]. Brown and Rymden [100] used QELSS to study diffusion of linear polystyrenes and silica spheres through polymethylmethacrylate:toluene. Toluene and PMMA are almost exactly index-matched, so scattering arises from the dilute probe chains. The matrix molecular weight M spanned 110 kDa-1.43 MDa. Probe polystyrenes had P of 2.95, 8, and 15 MDa, with Mw/Mn of 1.06, 1.08, and 1.30, respectively. Figure 21 shows Dp/ Dpo
Polymer diffusion in solution
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10-6 Cil
"'--~10-7
10-7
o"' 10-8
10-9
10-8
1
10
100
1000
c (g/L)
Figure 14. Ds of [from top to bottom] 6.5 8.7, 29, and 76 kDa three-armed polybutadienes in CC1 4 , and fits to stretched exponentials, using data of von Meerwall, et a!. [96], Fig. 4.
0
20
40
60
c (g/L)
Figure 15. Ds of [from top to bottom] 61, 92, 193, and 216 kDa 18-arm star polyisoprenes in CC1 4 , and fits to stretched exponentials, using data of Xuexin, et a!. [99], Fig. 3.
for 8MDa polystyrene in six matrix PMMAs. For the lowest-M matrix polymer, the fit predicts too strong a dependence of DP upon c. For Dp/ Dpo < 10- 3 or so, attained with the two largest matrix polymers, the measured Dp/ Dpo deviates markedly downward from a stretched exponential. The fitted curves in the Figure exclude these points. Deviations at small Dp/ Dpo might reflect a failure of eq. 16 at large cor experimental challenges at small Dp. Brown and Rymden [100] also examined how Dp depends on P. Figure 22 shows Dp of 3, 8, and 15 MDa probe polystyrenes in 445kDa PMMA solutions. Here P » M. As noted by Ref. [100], the data sets very nearly superpose, showing Dp/ Dpo to be independent from P for P / M » 1. Brown and Stilbs [101] used PFGNMR to measure DP of polyethylene oxide in aqueous dextran, as seen in Fig. 23. Polyethylene oxides had molecular weights of 73, 278, and 1200 kDa with Mw/Mn of 1.02-1.12; dextrans had molecular weights of 19, 110, and 510 kDa. The 1200 kDa PEO represented a technological outer limit; the authors restricted their analysis to the 73 and 278 kDa probes. Dp/ Dao depends only weakly on P but depends markedly on M. All data were simultaneously fit to exp( -acP'~ M 6 ), the pure exponential in c being forced because the range of Dp is small. Data and fitted curves agree well, except that for the 278 kDa probe in the 19 kDa matrix the fitting function significantly underpredicts Dp. Daivis, et a!. [102] used QELSS to measure Dp of relatively dilute 864 kDa dextran in solutions of 20.4 kDa dextran. Polymer polydispersities were in the range 1.24-1.3.
G. D. J. Phillies
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g 0
--.,
0
0.1
0 "'
1o-1o
0.01
0
50 c (g/L)
100
Figure 16. Ds of 302 kDa linear polyisoprene (filled circles) and [from top to bottom] 340, 800, and 6300 kDa 18-armed star polyisoprenes (all in CC1 4 ) and fits to stretched exponentials, using data of Xuexin, et a!. [99].
0
20
40
60
c (g/L)
Figure 17. Ds of [from top to bottom] 70.8, 251, and 302 kDa linear polyisoprenes in CC1 4 , and fits to stretched exponentials, using data of Xuexin, et a!. [99], Fig. 4.
The 20.4 kDa dextran had concentrations up to 166 g/L. QELSS spectra were bimodal, the slower mode corresponding to Dp of the 864 kDa dextrans. Data followed a simple exponential. Daivis, et a!. [103] used QELSS and PFGNMR to measure Dp of 110 kDa polystyrene, Mw/Mn = 1.06 in 110 kDa polyvinylmethylether, Mw/Mn ~ 1.3, in toluene. From Fig. 24a, the two physical techniques agree, except perhaps at the largest c. Stretched exponentials describe well each data set. De Smedt, et a!. [105], using FRAP, measured Dp of 71, 148, and 487 kDa dextrans (Mw/Mn < 1.35) in hyaluronic acid (Mn = 390 kDa; Mw = 680 kDa) solutions having concentrations up to 18 g/L, as seen in Fig. 24b. As noted by the original authors, stretched-exponential curves agree well with the data. Hadgraft, et a!. [87] used QELSS to study DP of 25, 162, 410, 1110, and 4600 kDa polystyrenes in 105 kDa polymethylmethacrylate: benzene, the latter being an isorefractive pair. Polystyrene and PMMA are incompatible. The radius R 9 of the polystyrene may depend on the matrix concentration. DP of higher-molecular-weight polystyrenes shows a stretched-exponential dependence on c, while Dp of 25 kDa polystyrene is substantially independent of c. The range of variation of Dp is narrow, leading to relatively imprecise fits. Hanley, et a!. [106], using QELSS, found Dp of 50, 179, 1050, and 1800 kDa polystyrenes in polyvinylmethylether: orthofiuorotoluene, the latter being an isorefractive pair. The
Polymer diffusion in solution
327
Figure 18. Ds as measured by (a) Brown, eta!. [75] (cf. Fig. 2), (b) Callaghan, eta!. [77] (cf. Fig. 3), (c) Callaghan, eta!. [77] (cf. Fig. 4), and (d) Deschamps, eta!. [79] (cf. Fig. 5), and fits of all points in each plot to eq. 16.
328
G. D. J. Phillies
10-5
10-9
10-6
--( /)
(\J
10-10
10-7
E
~
0 "'
10-11 10-8 10-12
10-9 10-10
10-13
1
100
10
1000
1
10
c (g/L)
100
1000
c (g/L)
10-6
10-6
10-7 (j)
---E
N
_£.
10-7
10-8
o"' 10-9
10-8
10-10
0
20
40
c (g/L)
60
0
20
40 c (g/L)
60
Figure 19. Ds as measured by (a) Giebel, eta!. [85], (b) von Meerwall, eta!. [96] and (c), (d) Xuexin, et a!. [99], and fits of all points in each plot to eq. 16, leading to parameters in Table 1. Other details as in Figs. 6, 14, 15, and 16, respectively.
Polymer diffusion in solution
329 10-6
10-6
10-7
10-7 (j)
"'--E
~
10-8 10-8 10-9
o"' 10-9
a
10-10 0.1
10-10 10-11 10
c (g/L)
100
10
100
1000
c (g/L)
Figure 20. Joint fits to all data of (a) Leger, et a!. [89] and separately of (b) Wesson, et a!. [98] to eq. 16.
polyvinylmethylether had Mw ~ 60 kDa with Mw/Mn ~ 3. Hanley, et al.'s data [106] appear in Fig. 24c. Curves in the data denote fits of all data to eq. 16 yielding parameters in Table 1. The lack of measurements at small c broadens the range of acceptable parameters. The fit of all data to eq. 16 is less satisfactory than the fits at each P to a separate stretched exponential, primarily because of data on the 1050 kDa probes. If data on these probes is excluded, the rms fraction error in the fit is substantially reduced while a and v only change modestly. Kent, et a!. [107] report R 9 and Dp of 233 and 930 kDa polystyrenes in 7, 66, 70, 840, and 1300 kDa polymethylmethacrylates in ethyl benzoate (for static light scattering) and toluene (for QELSS). Except for the 66 kDa PMMA, Mw/Mn ::::; 1.10. D of the polystyrene was measured as a function of polystyrene concentration, thereby obtaining both DP and the initial linear slope of D against polystyrene concentration. Figure 24d shows R 9 of the 930 kDa polystyrene as a function of PMMA concentration. R 9 follows
R 9 = R 9 o exp( -ac")
(17)
using parameters in Table 2. Unlike a scaling-law form, e.g., R 9 rv c~ 0 · 25 , eq. 17 is wellbehaved as c-+ 0. Kent, et a!. also measured DP of 233 kDa polystyrene in solutions of 66 and 840 kDa PMMA, and 930 kDa polystyrene is solutions of 840 kDa PMMA, as shown in Fig. 25a. Simple exponentials inc describe the data well. Figure 25a also shows a fit to eq. 16 using parameters in Table 1. Experimentally, the scaling prefactor a depends strongly on M, with a 12-fold change in M leading to a two-fold change in a. Kim, et a!. [108] measured DP of labeled polystyrenes through unlabeled polystyrenes
G. D. J. Phillies
330
10°
10°
10-1
10-1
10-2
10-2
-c. 1o- 3
10-3
10-4
10-4
10-5
10-5
0
0
c.
0
10-6 0.001 0.01
0.1
1 c (g/L)
10
100
Figure 21. Data of Brown and Rymden [100] on Dpl Dpo of 8 MDa polystyrene in [top to bottom] 101, 163, 268, 445, 697, and 1426 kDa polymethylmethacrylate in toluene as fit to eq. 16.
10-6 0.001 0.01
0.1
1
10
100
c (g/L)
Figure 22. DpiDpo for 3 (o), 8 (•), and 15 (D) MDa polystyrenes in 445 kDa polymethylmethacrylates.
in toluene, to test predictions that, at elevated c, Dp is independent of matrix molecular weight if M :::0: P. The probes were methyl red and polystyrenes with 10 :::; P:::; 1800kDa; matrix chains had 51 :::; M :::; 8400kDa. Polydispersities were largely < 1.06, with a maximum of 1.17. From Fig. 25b-d, the authors found Dp is only independent of M if M I P > 3. Figs. 25b-d, using Kim, et al. 's data [108], show Dp and fits of the M I P < 3 points to eq. 16 with parameters in Table 1. From the Figure, DP is described well by a joint stretched exponential in c, P, and M, except in the regime M I P > 3. Kim, et al. also studied the concentration dependence of D" and of Dp at large M I P. Figure 26a shows Ds of 900 kDa polystyrene:toluene, and the stretched-exponential fit. Figure 26b shows Dp for methyl red and for labelled 10, 35, 100, 390, 900, and 1800 kDa polystyrenes, all through toluene solutions of a range(51 :::; M :::; 8400kDa) of matrix polystyrenes, always with M I P > 3 and usually M I P > 6, the M I P ratios being chosen by Kim, et al. based on their interpretation that Dp is independent of M for M I P > 3. Stretched-exponential fits describe well the concentration dependences of D8 and Dp, even in the large M I P range which eq. 16 does not represent well the P and M dependences of DP. Lodge and collaborators have reported an extensive series of studies of probe diffusion in polymer solutions, using QELSS to measure DP of dilute probe polystyrenes in polyvinylmethylether: orthofluorotoluene. Variables studied include the probe and matrix molecular weights, the matrix concentration, and the topology (linear and star) of
Polymer diffusion in solution
0
331
8.
b. 0
0.1
0.1 0
50 c (g/L)
100
0
50 c (g/L)
100
Figure 23. Dpj Doo of (a) 73 kDa polyethylene oxide and (b) 278 kDa polyethylene oxide in [top to bottom] 19, 110, and 510 kDa dextrans in aqueous solution, and simultaneous fit of all measurements to a stretched exponential in c, M, and P, using data of Brown and Stilbs [101].
Table 2 Concentration and molecular weight dependences of the probe radius of gyration R 9 for molecular weight P probes in solutions of molecular weight M matrix polymers as functions of matrix concentration c. The fits are to stretched exponentials R 90 exp( -ac''), with the percent root-mean-square fractional fit error %RMS, the materials, and the reference. Materials include EB-ethyl benzoate, PMMA-polymethylmethacrylate, pSpolystyrene. Refs. v %RMS P(kDa) M(kDa) System 1300 pS: pMMA: EB [107] 2.2 930 390 4.11· 10 3 0.99 1.1 930 70 pS: pMMA: EB [107] 397 8.06. w- 3 0.77 [395] 4.49. w- 3 0.64 1.9 7 pS: pMMA: EB [107] 930
G. D. J. Phillies
332
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10-6
10-7
a
ff!. 10-8 E
10-7
.2-
oa.10- 9 10-10 10-11
10-8 200
0
400
4
600
c (g/L)
8
12
16
20
c (g/L)
10-6
450
10-7 400
~ 10-8
~
C\J
E
~350
.2-
a:
00.10-9
300
10-10 10-11 10
100
c (g/L)
1000
250 0.1
10
100
c (g/L)
Figure 24. (a) Dp of 110 kDa polystyrene through 110 kDa polyvinylmethylether:toluene, based on QELSS (0) and PFGNMR (D) measurements of Daivis, et al. [103] and fits (solid, dashed lines, respectively) to stretched exponentials in c. (b) Dp of [105] [from top to bottom] 71, 148, and 487 kDa dextrans in Mw 680 kDa hyaluronic acid, and fits to stretched exponentials. (c) Dp of [from top to bottom] 50, 179, 1050, and 1800 kDa polystyrenes in 60 kDa polyvinylmethylether: orthofluorotoluene, using data of Hanley, et al. [106]. (d) R 9 of [107] 930 kDa polystyrene in [from top to bottom] 7, 70, and 1300 kDa polymethylmethacrylate: ethylbenzoate against polymethylmethacrylate concentration and fits to eq. 17.
Polymer diffusion in solution
333
~
C\J
10-7
E 10-7
.20
c.
b
10-8
10-8 1
10
100
1000
105
104
c (g/L)
106
Matrix Mw (Da)
10-6 10-7
VI
--E
10-7 10-8
C\J
.20
c.
10-8
10-9
10-1 0
105
1 o6
Matrix Mw (Da)
107
L---'--L--'-L..JL.L.LLJ.___JL.......L-.L..J...J..I..LI.J..._---'-_._-'-'-'......
104
105
106
Matrix Mw (Da)
Figure 25. (a) Dp of polystyrene through polymethylmethacrylate: toluene [107] with molecular weight combinations P : M [from top to bottom] 233:66, 233:840, and 930:840 kDa. (b)-(d) Ds of polystyrene in matrix polystyrene:toluene solutions as functions Mat various c, based on Kim, et al. [108], and a single fit to all points in all three subfigures. Probe molecular weights were (b) 51 kDa, (c) 390 kDa, and (d) 900 kDa. Matrix concentrations [top to bottom] were (b) 10, 20, 50, and 100 g/L; (c) 20, 50, and 100 g/L; and (d) 10, 20, 40, and 80 g/L. Eq. 16 (solid lines) works whenever M/P < 3.
G. D. J. Phillies
334
10-4
10-6
10-5
10-7
10-6
VI
--E
10-8
10-7
o"'
10-9
10-8
C\J
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10-9
a 10-10
10-10
10-11
10-11 10 c (g/L)
0.1
VI ;;;-.
100
1
10
100
c (g/L)
10-6
10-6
10-7
10-7
10-8
10-8
10-9
10-9
E
.30
c.
d
c
10-10
10-10
10-11
10-11 1
10 c (g/L)
100
1
10
100
c (g/L)
Figure 26. Data of Kim, et al. [108], and fits to stretched exponentials, for (a) Ds of 900 kDa polystyrene in toluene, and (b) Dp of (top to bottom) methyl red, and 10, 35, 100, 390, 900, and 1800 kDa Ds polystyrenes through high-molecular-weight (M/P > 3) polystyrene:toluene matrix solutions. The left axis shows zero-matrix-concentration data. (c) Dp of polystyrenes [from top to bottom: 379 kDa J = 3 star, 422 kDa linear chain, 1050 kDa linear chain, and 1190 kDa J = 3 star] through 1300 kDa polyvinylmethylether:orthofluorotoluene [110], and fits to linear (solid lines) and 3-armed star (dashed lines) data to eq. 16. (d) Dp of [111] 12-arm star polystyrenes (from top to bottom, Mw of 55, 467, 1110, and 1690 kDa)in 140 kDa polyvinylmethylether:orthofluorotoluene and fits to eq. 16.
Polymer diffusion in solution
335
Table 3 Molecular weight dependence of the self and probe diffusion coefficients Ds and Dp for molecular weight P probes in solutions of matrix polymers at a fixed concentration c. The fits are to stretched exponentials Do exp( -o:M1 ) in matrix molecular weight M. The Table gives the best-fit parameters, the percent root-mean-square fractional fit error %RMS, the system, and the reference. Square brackets "[ · · · ]" denote parameters that were fixed rather than floated. Abbreviations as per previous Tables, and DBPdibutylphthalate. Refs. 1 %RMS P(kDa) c (g/L) System 1800 l 8.54. 10 8 pS:pS:tol 0.64 0.55 17 [108] 2492P- 0 ·5 1.37. 10- 2 0.47 p 130 pS:pS:dbp 4.0 [90] 1771P- 0 ·5 1.07. 10- 2 [0.5] p 180 pS:pS:dbp 6.9 [90]
the probe polymers. Throughout the analysis, we first fit Dp for each P : M pair to a stretched exponential in c, finding uniformly excellent fits. In the following, Figures report fits of full data sets to eq. 16, the stretched exponential inc, P, and M, with parameters in Table 1. Lodge and Wheeler [110] compared Dp of linear and 3-armed star polystyrenes through polyvinylmethylether:orthofluorotoluene. Polystyrene molecular weights were 422 and 1050 kDa for chains and 379 and 1190 kDa for stars; polystyrenes were 'relatively monodisperse'. PVME had Mw = 1.3 MDa, with Mw/MN ~ 1.3. Lodge and Wheeler's measurements [110] at each P are described well by a stretched exponential inc, even though Dp varies over nearly four orders of magnitude. Figure 26c shows their data and fits (separately for chains and stars) to eq. 16. The p-a scaling of the zero-concentration diffusion coefficient and the P 1 scaling of a account for the dependence of Dp on P. Lodge and Markland [111] used QELSS to measure Dp of 12-armed star polystyrenes through 140 kDa polyvinylmethylether, Mw/ Mn ~ 1.6, in orthofluorotoluene. Probes had Mw of 55, 467, 1110, and 1690 kDa, with Mw/Mn:::; 1.10. Lodge and Markland estimated c* ~ 20 g/L and Ce ~ 100 g/L for the matrix. Figure 26d shows Lodge and Markland's data, with solid lines representing fits to eq. 16 using parameters in Table 1. For single values of P, fits were excellent, with RMS fractional errors of 2-4%; the error for the joint fit was 12%. For the smallest probe at large concentration, eq. 16 underestimates DP. Lodge, Markland, and Wheeler [112] used QELSS to measure Dp of 3-armed and 12armed star polystyrenes in polyvinylmethylether:orthofluorotoluene. The 3-armed stars had P of 379 and 1190 kDa; the 12-armed stars had P of 55, 467, 1110, and 1690 kDa. Polyvinylmethyl ethers used as matrices had M of 140, 630, and 1300 kDa. Polystyrenes all had Mw/Mn < 1.1; the PVMEs had Mw/Mn ~ 1.6. Light scattering measurements were made of R 9 of linear, 3-arm and 12-arm stars with Mw ;::: 1MDa in solutions of 250 kDa PVME for c :::; 50 g/L. Lodge, et al. [112] numerically modified their reported Dp, as described by Wheeler and Lodge [125]; this modification factor has been removed here to recover the measured diffusion coefficients .. For every P : M combination, a stretched exponential describes Ds (c) well. Lodge and collaborators' data [111,112] on 3-armed and 12-armed stars were fit (separately for each arm number) to eq. 16, the fit including all data points except for the
G. D. J. Phillies
336
10-6
10-6 10-7
10-7
~
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.20
10-8 10-8
c.
10-9 10-9
a
b
10-10
10-10
10-11
1
10 c (g/L)
1
100
10 c (g/L)
100
Figure 27. Dp of (a) 379 and (b) 1190 kDa 3-arm star polystyrenes in (from top to bottom, 140, 630, and 1300 kDa) polyvinylmethylether:ortho-fluorotoluene as fit to eq. 16.
10-6
10-6
10-7
VI ;;:;-.
10-7
10-8
E
10-8
.20
c.
10-9 10-9
a
10-10 10-11
b
10-10
1
10 c (g/L)
100
1
10 c (g/L)
100
Figure 28. Dp of 12-arm star polystyrenes (from top to bottom, Mw of 55, 467, 1110, and 1690 kDa) in (a) 140 kDa and (b) 1.3 MDa polyvinylmethylether:ortho-fluorotoluene [112], and the fit to eq. 16.
Polymer diffusion in solution
337
10-7
VI N--
E
.20
10-8
c.
10-9
b
10-10
10
20
100 50 c (g/L)
200
Figure 29. Probe diffusion of (top to bottom) 50, 100, 420, and 900 kDa polystyrenes through 110 kDa polyvinylmethyl ether in toluene, based on data in Martin [113,104]. Solid lines are joint fits of all data to a stretched exponential in c and P.
55kDa probe in the 1.3 MDa matrix polymer for c 2 10 g/L. Figures 27 and 28 show these fits. Except for the smallest (55kDa) star in the 1.3MDa matrix polymer, eq. 16 describes Dp(c) very well. The fit for 12-armed stars is markedly better than the fit for three-armed stars. Including the 55kDa probe points in the fit raises the fractional RMS error from 14.6 to 19.5%, changes the three scaling exponents by 0.01 each, and elsewise has almost no effect on the fitted curves. From Figs. 27 and 28, eq. 16 accounts well for the dependence of Dp on c, P, and M for star polymers of given arm number in linear matrices, except in the case P « M and elevated c. All three independent variables ranged over extensive domains: more than two orders of magnitude in c, a factor of 30 in P, and an order of magnitude in M. The only failure in the fit occurred for the smallest probe in the largest matrix polymer (P / M :::::! 25) at elevated matrix concentrations. Martin [104,113] examined 50, 100, 420, and 900 kDa polystyrenes, Mw/ Mn :::; 1.1, in polyvinylmethylether: toluene. The polyvinylmethylether had from intrinsic viscosity measurements a molecular weight ca. 110 kDa and a 'fairly polydisperse' molecular weight distribution. Dp was obtained with QELSS, leading to Fig. 29. The measurements were fit to a stretched exponential inc and P, obtaining the solid lines in the Figure. With only one M, theM-dependence of Dp is inaccessible. Dp for each probe was also fit accurately by a stretched exponential in c. The stretched exponential describes Dp(c) well. Martin also measured the solution viscosity. At smaller polymer concentrations, especially for the lower-molecular-weight probes, DpT] is nearly independent of c. At larger c, especially for larger P, Stokes-Einstein behavior ceases to obtain: Dp7] increases with increasing c. Probes in concentrated matrix solutions diffuse faster than expected from the macroscopic
G. D. J. Phillies
338
10-7
10-7
VI
;;;-10-8
10-8
E
.20
c.
10-9
10-9
a
b
10-10
10-10
1
10 c (g/L)
100
1
10 c (g/L)
100
Figure 30. Dp of 342 kDa polymethylmethacrylate probes in polystyrene: thiophenol. [115]) for M of [top to bottom] 44, 186, 775, and 8420 kDa, and fits (a) for each M to exp(-acv) and (b) to D 0 P-aexp(-a~M'~).
solution viscosity. In addition to the self-diffusion studies noted in the previous Section, Nemoto, et al. [115] used ultracentrifugation and QELSS to measure sedimentation coefficients and Dp of dilute polymethylmethacrylate probes in polystyrene: thiophenol. The PMMA had a molecular weight of 342 kDa; the polystyrene molecular weights were 43.9, 186, 775, and 8420 kDa, all with Mw/ Mn in the range 1.10-1.17. Figure 30a shows Nemoto, et al.'s [115] measurements of Dp of PMMA in solutions of each of the four polystyrenes. For each molecular weight of the matrix, Dp( c) is described to good accuracy by a stretched exponential in matrix concentration. Figure 30b shows the same data with all measurements simultaneously fit to a stretched exponential in c and M, yielding parameters in Table 1. Over nearly 200-fold variations in these variables, the single stretched exponential in c and M describes reasonably well the behavior of Dp, with a 20% RMS fractional error. Nemoto, et al. [115] found that Dp/ Do depends more strongly on c and P than does s/ s 0 , so at large c and M, Dp/ Do < s/ s 0 • Nemoto, et al. concluded that, at elevated c and P, s and Dp of PMMA in polystyrene solutions are quite different in their behaviors. For two samples with the same Dp/ Do but very different matrix molecular weights (44, 8420 kDa), Nemoto, et al. [115] also measured the shear viscosity, finding that 1] differed 'by more than two orders of magnitude' between the two samples. Nemoto, et al. thus showed that Dp is not governed by the shear viscosity of the matrix solution. (The original paper did not specify which solution was the more viscous. Note that the comparison is being made at fixed Dp/ D 0 , not at fixed c, where the correspondence would be self-evident.)
Polymer diffusion in solution
VI ;:;;-
339
10-11
E (.) ---;,_ 10-12 0 10-13 10-14 10-15
103
Figure 31. D.(filled points) and Dp(open points) from Nemoto, et al. [116], Tables 2 and 3. Dashed line marks systems with M « P; solid lines represent a joint fit to Ds for systems with M/P > 5.
Nemoto and collaborators [116,117] used FRS to study Dp of polystyrenes in polystyrene :dibutylphthalate. A first study [116] focused on diffusion of labeled polystyrene through 40 wt% solutions of very long (M/P > 5) and very short (M/P < 0.2) unlabelled polystyrenes. Thirteen probes having 2.8 :::::; Mw : : :; 8420 kDa and Mw/Mn < 1.07 (for chains larger than 1MDa, Mw/ Mn was in the range 1.09-1.17) were used, leading to Fig. 31. For D., and for Dtr of short probe chains in solutions of long matrix molecules, a best-fit to the molecular weight dependence of D gives parameters seen in Table 1, the ?-dependence of the prefactor being forced rather than obtained from the fit. All data is at one concentration, so a concentration dependence was not obtained. The data on DP of long probes in short matrix chains fits to Dop-a exp( -a;P'Y), the molecular weight dependence arising almost entirely asp-a. The data are fit well with a= 0.52, in which case the best-fit gives 1 ~ 0.03. Nemoto, et al. [117] also used FRS to measure Dp of polystyrenes in dibutylphthalate solutions at 13 and 18 % matrix concentration. Probes had Pin the range 6.1 :::::; Mw : : :; 2890 kDa, with polydispersities Mw/ Mn : : :; 1.17 (in most cases, :::::; 1.09). Table 1 shows the fits. Dp covers more than five orders of magnitude, so the errors are not large relative to the range of Dp. DP depends appreciably on M as well as P. With only a few points for any particular P or M, it is difficult to present this data graphically. Numasawa, et al. [118] used QELSS to study DP of polystyrene in polymethylmethacrylates: benzene, as seen in Fig. 32. Polystyrenes had P in the range 185-8420 kDa, with Mw/ Mn of 1.04-1.17. Polymethylmethacrylates had M in the range 850-4050 kDa, with Mw/Mn:::::; 1.08 (For the 850 kDa polymer, Mw/Mn ~ 1.35.) PMMA concentrations cov-
G. D. J. Phillies
340
ered 0-36 g/L. Numasawa, et al. [118] also report the zero-shear solution viscosities and (from static light scattering) the probe radii of gyration as functions of matrix concentration. Figure 32a shows Dp of the 420 and 8420 kDa polystyrenes in solutions of two polymethylmethacrylates at various matrix c. Figure 32b shows DP of five polystyrenes in each of four polymethylmethacrylates, all at c ~ 37 g/L. Dp decreases monotonically with increasing P, c, and M. The eight lines represent a joint fit (parameters, Table 1) of eq. 16 to all data in both Figures and one additional data point. From Fig. 32a, the stretched exponential captures well the c-dependence of Dp, and the variation of that dependence with P and M. From Figure 32b, eq. 16 captures reasonably well the dependence of Dp on M at fixed P, but-at fixed M and c-eq. 16 does less well at capturing the dependence of DP on P.
VI 10-8 -...
10-8
"'E
~
a.
0
10-9
10-9
10-10
10-10
0
10
20 30 c (g/L)
40
50
0
2
3
4
5
M (MDa)
Figure 32. Dp of polystyrenes in polymethylmethacrylate: benzene [118] (a) as a function of matrix concentration, with P:M of [top to bottom] 420 kDa:4.05 MDa, 8.42 MDa: 1.95 MDa, and 8.42 MDa: 4.05 MDa, and (b) as a function of molecular weight, for [top to bottom] 0.42, 1.26, 3.84, 5.48, and 8.42 MDa polystyrene probes in 36.7 g/L polymethylmethacrylate. All lines are the same best-fit to eq. 16.
Nyden, et al. [119] used PFGNMR to determine DP of monodisperse (Mw/Mn < 1.1) polyethylene oxides (molecular weights 10-963 kDa) in 1% and 6% aqueous solutions and a 1% chemically cross-linked gel of 100kDa ethylhydroxyethylcellulose. Figure 33a shows Dp/ Do in the 1% solution as a function of probe radius R. The solid line is a stretched exponential (18)
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where a and 6 are a scaling prefactor and exponent, and Dpo is the nominal Dp in pure solvent. Best-fit parameters were D 1 = 0.33, a = 0.33, and f3 = 0.57; the RMS fractional error in the fit was 7%. Nyden, et al. [119] note R = K pa with a = 0.53, so eq. 18 is effectively a stretched exponential in P'~ with 1 ~ 0.30, consistent with 1 for other systems (Table 1). In 6% matrix solutions, dDp/ dM is not monotonic in M. However, at the anomalous M, PFGNMR echo decays are no longer simple exponentials: Diffusive behavior is more complex than simple diffusion.
1 1o- 5 0
a.
0 "'c. 0
0.1
1o- 6
0.01
0
10
20 Rh (nm)
30
10- 7 L....------L----l..-.L.....J....JU..U.J....____I_---'--1.1 40 10 30 100 300 c (g/L)
Figure 33. (a) Dp/Dpo from Nyden, et al. [119] for polyethylene oxide probes in 1% aqueous ethylhydroxyethylcellulose against probe hydrodynamic radius R. (b) Dp of [top to bottom] styrene monomer and 580, 1200, and 2470 Da polystyrene polymers in CC14 solutions of a large-M polystyrene matrix polymer [120], and fits to eq. 16.
From PFGNMR, Pinder [120] report Dp of styrene and small (P <::: 2470 Da) probe polystyrenes in theta (cyclohexane) and non-theta (CC1 4 ) polystyrene (10.7 <::: M <::: 430 kDa) solutions, for matrix concentrations up to 300 g/L. Matrix polydispersities were Mw/ Mn <::: 1.14, with in most cases Mw/ Mn < 1.08. Pinder found that Dp of styrene monomer does not appear to depend on M but follows closely a simple exponential in c. Figure 33b shows Dp of styrene and three low-molecular-weight polystyrenes (P <::: 2470 Da) in high-molecular-weight polystyrenes (10.7 <::: M <::: 430kDa). For each probe, changing the matrix molecular weight three- to five-fold had with no apparent effect on DP. In all cases, M/P was in the range 15-20. Smith, et al. [121] used FRAPP to measure Dp of 33.6 kDa polypropylene oxide (PPO) chains through 32kDa PPO chains in a melt of 1 kDa PPO. The probe and solvent were relatively monodisperse (Mw/Mn = 1.1) while the matrix polymer had Mw/Mn = 1.6.
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10-11
VI ..._
"'E
~
10-9
a.
0
10-12
a
10-10 0.001
0.01
0.1
c (Volume Fraction)
0.2
0.4
0.6
0.8
c (Volume Fraction)
Figure 34. (a) Dp from Smith, et al. [121] of 33.6 kDa polypropylene oxide in solutions of 32 kDa PPO chains in a 1 kDa PPO melt, and a stretched-exponential fit. (b) Dp of 255 kDa deuteropolystyrene in solutions of [top to bottom] 93, 250, and 20 000 kDa polystyrene in molten 10 kDa polystyrene, at 150, 150, and 175 C, respectively, from Tead and Kramer [122], Figures 4 and 6. Solid lines are stretched-exponential fits. For clarity, the 150 C data has been multiplied by a factor of 100.
Smith, et al.'s data [121] appear in Fig. 34a. A stretched exponential with constant parameters describes Dp( c) all the way from dilute solution to the matrix melt. Tead and Kramer [122] studied diffusion of 255 kDa deuterated polystyrene through solutions of large-molecular-weight (93, 255, and 20 000 kDa) polystyrenes, with a low-M (10 kDa) polystyrene melt as the solvent. The matrix polymer volume fraction covered 0 :::::; ¢ :::::; 1. Dp was obtained using forward recoil spectroscopy to measure the timedependent probe concentration profiles. From Fig. 34b, Dp follows a stretched exponential in matrix c, no matter whether M < P, M ~ P, or M » P, from small c out to the large chain's melt. Tinland and Borsali [123] used FRAP and QELS to measure DP of 433 kDa dextran in aqueous solutions of 310 kDa polyvinylpyrrolidone (PVP) for 0 :::::; c :::::; 120g/L. Mw/ Mn was 1.5 for the matrix polymer but ca. 1.9-1.95 for the probes. Dp from the two techniques do not agree. We analyzed Dp from FRAP measurements, because FRAP does not require the detailed model assumptions needed to relate the QELSS spectrum to diffusion coefficients. Tinland and Borsali's data agree well with stretched exponentials in c. Wheeler, et al. [124] studied tracer diffusion of linear polystyrenes having molecular weights 65, 179, 422, and 1050 kDa (with Mw/ Mn :::::; 1.1) through a 1.3 MDa polyvinylmethylether matrix polymer, Mw/ Mn ~ 1.6, in orthofluorotoluene at concentrations 1-100 g/L. DP was determined using QELSS and an isorefractive matrix polymer. The matrix
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100
10
100
c (g/L)
c (g/L)
Figure 35. Merged data on Dp of polystyrenes (from top to bottom, 65, 179, 422, and 1050 kDa) in (a) 140 kDa and (b) 630 kDa kDa matrix polymers. The solid line is fit of all data to eq. 16, a single stretched exponential in c, P, and M.
10
100
c (g/L)
Figure 35. Merged data on Dp of polystyrenes (from top to bottom, 65, 179, 422, and 1050 kDa) in (c) 1300 kDa matrix polymers. The solid line is fit of all data to eq. 16, a single stretched exponential inc, P, and M.
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polymer had 1/[ry] = 2.2 g/L, so most of their data has c > c*. We fit stretched exponentials to data on all four probes. For the 179 and 422 kDa probes, fits to eqs. 15 and 16 are coincident, while for 1050 kDa probes the fits are very close. However, for the 65 kDa probe and large matrix concentration eq. 16 underestimates Dp. Wheeler and Lodge [125] then used QELSS to measure Dp of the same probes, this time with 140, 630, and 1300 kDa polyvinylmethylether matrices, Mw/ Mn ~ 1.6, and concentrations up to 300 g/L. The results of ref. [124] are a proper subset of these results. For the three matrix polymers, c* was estimated at 11, 5.7, and 3.3 g/L, respectively (based on c* = 1.5/[ry]), while Ce was estimated at 50, 12, and 6 g/L, respectively. Wheeler and Lodge [125] rescaled Dp relative to Dp of a solvent molecule in the same solution. This rescaling was here removed prior to reanalysis. We fit all of Wheeler, et al.'s data [124,125] to a joint stretched exponential inc, P, and M, as seen in Fig. 35. Solid lines in the figure are the best-fit Dp(c), plotted for fixed P and M, all lines arising from a single set of parameters. Over an order of magnitude in matrix molecular weight, a factor of 15 in probe molecular weight, and a factor of 300 in matrix concentration, eq. 16 represents reasonably well the joint dependence of Dp on c, P, and M. The error arises primarily for P » M and P « M: For the 65 kDa probe in the 630 and 1300 kDa matrix polymers and the 1050kDa probe in the 140 kDa matrix polymer, at large c the predicted DP is too small. When M and P are very different, Dp depends less strongly upon c than would be expected from eq. 16.
5. OTHER EXPERIMENTAL STUDIES ON POLYMERS IN SOLUTION The following papers had experimental foci different from those examined above. Several provide very interesting information on polymer self-diffusion in solution. Others are not amenable to the above analysis. A series of papers report on ternary polymer solutions in which neither polymeric component is dilute: Borsali, et al. [126] studied QELSS spectra of 970 kDa polystyrene: 950 kDa polymethylmethacrylate: toluene. Borsali, et al. [127] extended this work with measurements at large (weight fraction > 0.8) polystyrene concentration. Borsali, et al. [128] also studied polystyrene:polydimethylsiloxane in tetrahydrofuran (which is isorefractive with polydimethylsiloxane) and in toluene (a zero average contrast solvent for these polymers). Corrotto, et al. [129] performed static light scattering and QELSS measurements on bidisperse mixtures of polystyrene in toluene. Cosgrove and Griffiths [130] used PFGNMR to study the diffusion of protonated polystyrenes through solutions of deuterated polystyrenes, in particular the dependence of Dp on probe and matrix concentrations and molecular weights. Desbrieres, et al. [131] applied QELSS to aqueous dextran: polyvinylpyrrolidone mixtures at concentrations greater than the overlap concentration; a third slow spectral mode was found when the solution's phase separation was approached. Giebel, et al. [132,133] studied QELSS spectra ofpolydimethylsiloxane: polymethylmethacrylate in several solvents. Konak, et al. [134] report QELSS of polystyrene and polymethylmethacrylate in toluene. Sun and Wang [135-137] used QELSS to study polystyrene:polymethylmethacrylate mixtures in organic solvents over a limited concentration range. Several authors report on the wavevector dependence of the spectrum. Chang, et al.
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[138] report DP of polystyrene in polyvinylmethylether: toluene, using QELSS and FRS to measure Dp on very different length scales. Relaxation rates r scaled as q2 over 10 7 ::::; q2 ::::; lOll cm- 2 , confirming that QELSS measures simple translational diffusion and rejecting proposals that probe motion is more rapid inside the probe's correlation hole than it is over larger distances. Numasawa, et al. [139] report scattering spectra of dilute polystyrenes in polymethylmethacrylate in its isorefractive solvent benzene, obtaining the scattering-vector dependence of the linewidth r. Chu and collaborators [140] studied the motion of fluorescent-labelled DNA molecules in non-dilute solution. They propose that they have confirmed the reptation model, based on observations that (i) stretched DNA chains relax as though they were confined to tubes, (ii) Ds scales as M- 2 for nondilute probe chains, and (iii) a scaling law Ds ~ cl. 75 is consistent with their data. However: (i) Even within the reptation model a stretched chain relaxes from stretching primarily via its higher-order modes, revealing only tube confinement on a short time scale. (ii) The scaling result Ds ~ M- 2 is a common feature of many models, is in no sense a signature of reptation [141], and in any event is inconsistent with [142] data on synthetic polymers, (iii) Ref. [140]'s measurements of D.(c) do not unambiguously show scaling behavior. Furthermore, if these data had demonstrated concentration scaling, they would be inconsistent with most of the remainder of the literature, as reviewed here. In almost all other systems Ds and DP have stretched-exponential concentration dependences. Finally, the measurements of Chu and collaborators [140] reject the reptation model for their system. In the reptation model, polymer chains only perform free diffusion after escaping from their tubes at long times t > Td· At times t < Td, polymer chains are confined to their tubes and perform confined diffusion. For t < Td, the reptation model (and Schweizer's mode coupling calculation) [143] predict that a chain's meansquare displacement should satisfy (6x 2 ) ~ t• where s is 0.5 or so. Ref. [140] reports that their chains remain within the hypothesized tubes for times Td ~ 1.2 or Td > 2 minutes, depending on which estimator of Td is applied. However, ref. [140] measured directly (x 2 (t)); from Fig. 2 of ref. [140], (x 2 (t)) ~ t 1 for times as short as Td/7. That is, these chains were performing Brownian, non-constrained diffusion on time scales so short that the chains were supposed to be confined to their tubes (or equivalently for Ref. [143]). The experimental s ~ 1 is incompatible with the reptation model or with Schweizer's mode coupling calculation [143], which require t• behavior with s ~ 1/2 for the meansquare displacement for t < td. The data ref. [140] thus indicates that a key aspect of the reptation picture, namely tube confinement, is incorrect, at least in this system. Several authors report data on rodlike polymers. Jamil, et al. [144] report QELSS measurements on random-coil polystyrene probes in solutions of the rigid-rod polymer poly-( 1-stearyl-a-L-glutamate) in its isorefractive solvent toluene, fitting the Ds they found to a stretched-exponential form for its concentration dependence. They found v ~ 1.3, and interpret v > 1 as arising from end-to-end aggregation of the polymer polymers. Russo, et al. [145] measured Ds for the rodlike polymer poly( 1-benzyl-a,Lglutamate) in pyridine at concentrations above and below the isotropic: cholesteric phase transition. Dp changes its concentration dependence at the transition. Note also work of Aven and Cohen [146] on Dp of polystyrenes in polydimethylsiloxane: tetrahydrofuran. B. Chu, et al. [147,148] report on the diffusion of dilute polymethacrylate
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probes in polystyrene matrix using toluene: a-chloronaphthalene as the solvent to obtain a virtually virtually exact index matching solvent. Cotts [149] report early QELSS measurements on polystyrene:polyvinylmethylether: toluene, a precursor that flowered into the systematic studies of Lodge, Wheeler, and collaborators [110-112,124,125]. Daivis and Pinder [150] report QELSS studies of polystyrene: polyvinylmethylether mixtures dissolved in toluene (a good solvent for both polymers) or carbon tetrachloride (a marginal solvent for polystyrene), reporting spectrallineshapes and radii of gyration. Scalettar, et al. [15] used FRAP and FCS to study diffusion of phage .A DNA solutions, confirming the predictions of this author [14] on the effect of labelling fraction on fluorescence correlation spectroscopy measurements.
6. EXPERIMENTAL STUDIES ON POLYMERS IN POROUS MEDIA AND TRUE GELS Distinct from the above work on polymers in solution is an extensive series of studies on polymers and other probes diffusing through random and ordered pores and through true, physically-crosslinked, gels. The original reptation model referred to a polymer coil passing through a mesh of immobile polymer chains [33]. Extension of the original reptation model to polymer melts and solutions, based on the assumption that entangled polymer chains can be approximated as a transient pseudogel lattice through which the chain of interest is passing, represent an extension of the original gel model. We here note experiments on physical systems at least somewhat similar to those originally considered by deGennes. Cannell and Rondelez observed polystyrenes, 110:::::; Mw :::::; 600 kDa, dissolved in ethyl acetate and diffusing through polycarbonate membranes pierced by uniform cylindrical holes having diameters in the range (for different membranes) of 543-2584A. For chains whose diameters were somewhat under half the pore diameters, the measured diffusion coefficient was explained accurately by an equation that takes into account: (i) entropic considerations depress the chain concentration in the pore below its concentration in the surrounding medium, and (ii) hydrodynamic interactions between the chains and pore walls inhibit probe diffusion. Cannell and Rondelez observed a dramatic increase in diffusion through the pores on increasing the polymer concentration, in essence because the entropic effects tending to depress the polymer concentration within the pores were cancelled by entropic constraints on polymers in the bulk solution. In polycarbonate filters, pores are uniform cylinders; there are no substantial statistical fluctuations in pore diameter that would create obstacles to polymer motion. However, throughout Cannell and Rondelez's data, the hydrodynamic radius of the chain is less than the pore radius: this data does not enter the range in which an extended, large chain must thread its way through a much smaller (in transverse cross-section) pore. Teraoka, et al. [153] used QELSS to observe polystyrene:2-fluorotoluene within the pores of a porous silica glass. The 2-fluorotoluene index-matched the glass, so that the glass scattered almost no light. Teraoka, et al., used polystyrenes with various molecular weights. Their smaller polystyrenes had radii smaller than the pore radius, while the largest polystyrene had a radius of gyration modestly larger than the pore radius. QELSS measured the mutual diffusion coefficient Dm of the polystyrene, which increased with
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increasing polymer concentration. Zhou, et al. [154], used QELSS to measure the mutual diffusion coefficient of polystyrene:tmns-decahydronaphthalene through similar glasses, the solvent supplying 8 conditions for the polymer near 20 C. Dm increased weakly with increasing polymer concentration; as in other studies, the polymer chains were at most little larger than the representative pore radii. Polymer diffusion through meshworks of covalently-crosslinked polymer chains has been examined rarely. Antonietti and Sillescu [155] studied polystyrene chains diffusing through crosslinked polystyrene networks, while varying the crosslink density. These were solventfree melt systems. Increasing the number of covalent crosslinks reduced the diffusion coefficient (as measured with holographic bleaching of labelled chains) of the probe polymers. Several authors have reported on the diffusion of globular probes through crosslinked solvent-flooded polymer networks. Nishio, et al. [156] observed with QELSS the diffusion of polystyrene latex spheres through polyacrylamide gels, the gels being formed in situ around the spheres. By varying the scattering vector, the fraction of particles of given size capable of moving through particular distances was determined. Konak, et al. [157], went on to report the temperature and scattering vector dependence of probe scattering in the same systems. Park, et al. [158], Matsukawa and Ando [159], and Tokita, et al [160] observed the diffusion of small-molecule solutes and polyethylene glycols through various permanent lattices. Tokita et al. [160], using small-molecule probes having molecular weights of 18-342 Da in crosslinked polyacrylamide, report that Dp of the probes falls as a stretched exponential in probe molecular weight and matrix concentration. They interpret the dependence on P as a dependence on the probe radius R, namely exp( -R 1 c314 ). Matsukawa and Ando [159], studying polyethylene glycol probe polymers in crosslinked poly-(N,Ndimethylacrylamide) with P up to 20 kDa, find the very similar Dp/ Dpo = exp(- Rc0 ·71 ). Park, et al. [158], observing a benzospiran dye and optically labelled bovine serum albumin via holographic relaxation, found D of the probes depends on matrix concentration and probe size as exp(-3R0 ·6 c0 ·94 ). Park, et al., were also able to measure directly the matrix mesh size.; inferred from the matrix diffusion coefficient De via De= k 3 T/(61rry.;), TJ here being the solvent viscosity; they report that DP is not a universal function of R/.; because .; has the wrong dependence on concentration.
7. ANALYSIS The above sections summarize a detailed examination [1] of nearly the entirety of the published literature on polymer self-diffusion and probe diffusion in polymer solutions. Dependences of Ds and Dp on polymer concentration, probe molecular weight, and matrix molecular weight were determined. We now attempt to extract systematic behaviors from the above particular results, asking: What features are common to self- and probediffusion of all polymers in solution? To answer: First, the concentration and molecular weight dependences of Ds and Dp are considered. Second, having found that D. and Dp uniformly follow stretched exponentials in c, correlations of their scaling parameters and other polymer properties are examined. Third, for cases that examined a series of homologous polymers, a joint function of matrix
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concentration and matrix and probe molecular weights is found. Fourth, we consider the few cases in which stretched-exponential behavior is not seen. Particular behaviors of a few specific systems clarify the proposed systematic behavior. Other results implying a generalized phenomenology for polymer diffusion are examined. First, the above summarized the published literature on self-diffusion and probe diffusion of random-coil polymers in solution. The concentration dependences of Ds and Dp are essentially always described well by a stretched exponential (eq. 15) in the matrix concentration c. On a log-log plot of Ds against c, stretched exponentials appear as smooth curves, while scaling ( power-law) behavior leads to straight lines. Almost without exception, log-log plots of measured D.(c) give smooth curves, not straight lines. Correspondingly, the hypothesis that the concentration dependence of D.(c) shows scaling ( power-law) behavior is uniformly rejected by the published literature. The observation that experimental data follows some mathematical form does not prove that the form is physically significant. Several different equations may describe the same data to within experimental error. However, if experimental data is uniformly not described by a mathematical form, to well beyond experimental error, models that predict that form are inadequate. Straight lines are almost never observed in plots of Ds or Dp against c. Correspondingly, models that predict or assume that D. and DP of polymers in solution follow power laws inc, P, and/or Mare very definitely rejected by the literature. Second, there are systematic correlations between the scaling parameters a and v and the molecular weights of the probe and matrix polymers. In particular: The scaling prefactor a depends strongly on M. Figure 36 shows a from Ds and eq. 15, plotted against polymer molecular weight, including almost all data on linear and star polymers. Concentrations were in g/L. The best-fit line shows a power-law relation between a and M. Figure 36a also presents a of 3-, 8- and 18-armed stars. a of an 18-armed star polymer tends to be somewhat smaller than a for a representative linear chain. However, a of star polymers almost always lies within the scatter in the values for a observed for the linear chains. a for three-armed stars includes results [96] in which only a limited number of concentrations were studied for a given polymer. Figure 36a emphasizes polymer topology. Figure 36b gives a against M with points labelled by reference, including Tinland, et al.'s [94] measurements on associating polymers. The scaling exponent v depends on M at low M. Figure 37 shows the dependence of v on polymer molecular weight. The filled circles refer to linear polymers as studied in refs. [74-76,79,80,85,86,89,95,97-99]. For polymers larger than 250 kDa, v approaches very closely to 0.5. For small polymers, v increases toward v = 1 with decreasing M. In sixteen systems the fitting process led to v = 1. Most of these fits refer to polymers with M < 200 kDa, consistent with an interpretation that v increases toward 1.0 at small M. These fits appear in Figure 37 as heavily-overlapped open diamonds. Figure 37 also shows v for star polymers. These data are substantially confined to M < 250 kDa. At very small M, v ~ 1; above 200 kDa, v for stars appears from limited data to trend toward 0.5. Third, several papers [75,100,104,106-108,11Q-113,115-118,120,124,125] report Ds and Dp for multiple homologous polymers. Data from each paper were separately fit to
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0.1 0.1 (!j
0.01 0.01 0.001
0.001
0.0001
0.0001 10
100 1000 10000 M (kDa)
10
100 1000 10000 M (kda)
Figure 36. (a) Scaling pre-factor a for linear polymers (•) and 3-, 8-, and 18- armed star polymers (6, D, *, respectively) against polymer molecular weight M. Solid line is a power law a rv MI.!. (b) Data from refs. (0) [76] for polymers in CC1 4 , ( •) [79], (D) [80], (•) [85,86], (6) [89], (.&) [95,99] with linear chains, (+) [95,96] with f = 3, (x) [95] with f = 8, (IZI) [99] with f = 18, (O) [74], (+) [75], (V) [76] for polymers in C5D 6 , (~) [79], (\>9) [98], (0) [97], (*) [96] for linear polybutadiene, (ffi) [96] for f = 3 polybutadiene, (1>) (75] for PEO in water, and (not in (a)) (<1) (94] for xanthan in water.
eq. 16.
(For Ds, P and M are the same variable.) The joint stretched exponential
Dop-a exp( -acv yr M 0 ) describes well a great number of measurements of Dp in systems with 0.3:::; M/P:::; 3. Outside this range of M/P, eq 16 predicts a concentration dependence for Dp that is stronger than that found experimentally. In making these fits, for each paper v was treated as a single constant, the dependence of v on M discussed above being suppressed. From Table 1: (i) a is almost always in the range 0.5 to 0.6. (ii) When it was fit, v is generally in the range 0.5-0. 75, as seen in Figure 37. In two cases, v was larger (0.86, 0.99); in one case it was smaller (0.43). (iii) For linear chains, "'(is almost always in the range 0.25-0.3. There are outliers at 0.14, 0.19, and 0.43; each of these is associated with a system in which v was unusually small (0.43) or unusually large (0.86, 0.99). For star polymers, "'(is~ 0.15. (iv) For linear polymers and three-armed stars, 8 is usually in the range 0.25-0.30. Equation 16 is less accurate when M and P differ substantially. In particular, for M/ P > 3 or for P/M > 3, eq. 16 tends to overstate the decrease in Dp with increasing c. Specific results of Section IV pertaining to the range of validity of eq. 16 include: Brown, et al. [100]'s data for Dp/ Dpo < 10- 3 show an experimental Dp that is smaller than calculated from eq. 16. Also, for P / M ~ 80, the measured DP is larger than
G. D. J. ?billies
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1.5
:;::...
0
0.5
0
*0
00 0 0
0*)
0 0
oo
0
0
0 0
*
oo
0 250
500
750
1000
M (kDa)
Figure 37. Scaling exponent v for polymers of Table 1 plotted against the polymer molecular weight M. A large-molecular- weight asymptote v ~ 0.5, and a low-molecular-weight increase in v toward v = 1 are apparent for linear polymers (•) and for 3- ( 6, 8- (D), and 18- (*) armed star polymers. Open diamonds refer to linear polymers in which an adequate fit was obtained with v = 1 forced during the fitting process.
predicted. Kim, et al. [108] found that eq. 16 generally works well for M I P ::::; 3, but overstates the dependence of Dp on M for Ml P > 3. From this data, as cis increased the range of values of MIP, for which eq. 16 works well, appears to narrow. For MIP::::; 3, Lodge and Wheeler [110] found that eq. 16 worked well for linear chains and f = 3 stars diffusing through linear-chain matrices. For large f = 12 polystyrene stars in a short linear polyvinylmethylether, Lodge and Markland [111]'s data follow eq. 16 even for PI M ~ 12. (If the comparison is made by arm rather than total molecular weights, Lodge and Markland's data is confined to a region PalMa< 2.) Lodge, et al. [112] examined the diffusion of small f = 12 stars through solutions of a large linear polyvinylmethylether: For M I P 2: 3 and elevated concentrations, the measured Dp is larger than expected from eq. 16. Lodge, et al [111,112] also studied three-armed stars diffusing through linear chains; for M I P > 3 the measured D is too large relative to fits based on eq. 16. Equation 16 fits well Martin's [113,104] joint data, but these were limited to MIP < 2.2. Nemoto, et al.'s [115] results on PMMA:polystyrene solutions followed eq. 16 more accurately when 2 2: MIP 2: 0.5. Numasawa, et al.'s data [118] on polystyrene in polymethylmethacrylate:benzene follows reasonably well eq. 16, but the joint stretched exponential clearly gives a better description of the c-dependence of Dp than it does of the M-dependence. Fits to Wheeler, et al.'s [124,125] measurements show that eq. 16 underestimates Dp for M I P 2: 7 and for PI M 2: 3. Fourth, in two cases stretched exponentials did not describe the c or M dependences.
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Nemoto, et al. [90] studied the M-dependence of Ds at large fixed c, finding a crossover from a stretched-exponential to a power-law molecular weight dependence for M 2 800 kDa. Tao, et al. [93] studied very large polymers in highly concentrated solutions. Their data are described well by power laws, consistent with the proposal [93] that polymer transport coefficients have different phenomenologies in the melt/near-melt domain and in the solution domain treated here. Fifth, there is evidence for other correlations. Several papers report Ds or Dp and also the solution viscosity 7). With a 110 kDa matrix polymer, Martin [113,104] found for 50 kDa probe polystyrene that Dp7] increased up to 6-fold with increasing c. For larger probe polymers (100 kDa, 420 kDa), the increase in Dp7] was less dramatic; with 900 kDa probes, " ... this product is very nearly independent of concentration." Dp, 7) and probe radius of gyration R 9 were reported by Numasawa, et al. [118] for polystyrene in polymethylmethacrylate:benzene. Numasawa, et al. identified a regime in which Dp7]R9 is approximately constant. However, for small probe chains diffusing through larger matrix chains but not large probe chains diffusing through smaller matrix chains, Nemoto, et al. found that the product Dp7]R9 increases up to 100-fold with increasing c and M. Martin [104] and Numasawa, et al. [118] each proposed that the failure of the StokesEinstein equation, with Dp7] increasing markedly with increasing c, is associated with a transition to reptation dynamics. However, the diffusion ofrigid spherical probes through solutions of high-molecular-weight polymers shows nearly the same phenomenology for Dr] [114]; that is, Dr] increases dramatically with increasing c at large M, even though there is no possibility that spheres diffuse via reptation. Like effects do not prove that like causes are at work. The existence of non-reptating spherical probe-polymer systems in which D7) increases markedly with increasing c does not prove that polymer-probe systems showing the same phenomenology are not reptating. However, because nonreptating sphere-probe systems have a D7) that can increase markedly at large matrix c, the observation that Dp7] of probe polymers increases markedly at large matrix c is not evidence for reptation dynamics. Sixth, Xuexin, et al. [99] report DP for linear and f = 18 star polymers having very nearly the same D 0 • For these two polymers, a and v are also very nearly the same, consistent with an interpretation that a and v are determined by chain size and not by chain topology. 8. CONCLUSIONS AND DISCUSSION
In the above, virtually the entirety of the published literature on polymer self-diffusion and on the diffusion of chain probes in polymer solutions has been reviewed. Without exception the concentration dependences of Ds and Dp are described by stretched exponentials in polymer concentration. The measured molecular weight dependences compare favorably with the elaborated stretched exponential, eq. 16, except that, when P » M or M » P, there is a deviation from eq. 16, that deviation referring only to the molecular weight dependences. The deviation uniformly has the same form: The elaborated stretched exponential overestimates the concentration dependence of DP, so that at elevated c the predicted Dp/ Do is less than the measured Dp/ D 0 • Contrarywise, almost without exception the experimental data on solutions is inconsistent with models that
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predict power-law dependences forDs and Dp on c and M. There is no indication in the experimental data that exponential models cease to be adequate, at smaller concentrations, if the polymer molecular weight is increased. This Chapter has not considered melt systems. The extremely thorough experiments of Tao, et al. [93] do show that as one moves from polymer solutions toward the melt, one encounters in some systems a region of very large c in which scaling models are at least approximately correct, implying that power laws could be asymptotically valid in the near-melt regime.
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Index Ammann-Beenker tiling, 60, 74, 75, 77, 78, 80-85 annealed, 62 annealed average, 238, 241, 247, 254, 259, 266, 268 asymptotic expansion, 71-73, 83, 84, 91, 95 averaging annealed, 14, 19, 151, 155, 159, 161, 162, 285 disorder, 10 generalized, 229 quenched, 10, 151, 155, 159, 161, 162, 192, 285 sample, 12, 14 self-, 43 thermal, 9, 12
crossover, 116, 117, 141, 283, 284, 293 crossover exponent, 63, 64 data analysis, 202 delocalization, 235, 236, 260, 263-267, 269 density matrix, 239 density of states, 252, 256 des Cloizeaux relation, 198, 223, 225 generalized, 221, 223, 225 percolation, 221, 223, 225 Sierpinski, 204, 208, 223, 225 differential approximants, 71, 83-85 diffusion, 305, 306, 308, 309, 311-314, 316, 318, 319, 323, 324, 330, 335, 337339, 341, 342, 344-348, 350, 351 in multimacrocomponent solutions, 306 inter, 307 mutual, 305, 306, 346, 347 self, 305, 307, 313, 314, 316, 319, 338, 344, 347, 348, 351 tracer, 305, 307, 308, 342 Dijkstra's algorithm, 287 dimension backbone, 115, 118 critical, 197, 226 fractal, 113--115, 118, 290 random walk, 114, 115, 128 spectral, 115 upper critical, 28, 105, 106, 119, 128, 129, 143 directed polymer, 11, 141, 275 Hamiltonian for the pure case, 13 for the random case, 13 disorder correlated, 16, 278 random, 272 weak, 285, 288 distribution, 10, 13 Boltzmann, 9 Gaussian, 13 probability, 9, 18, 31 DNA, 14, 29, 34, 37, 345, 346
backbone, 115, 118, 121, 122 Bethe ansatz, 23 binding energy, 241, 251, 260-263, 266, 267 blob picture, 32 branched polymer, 94, 95, 149, 152, 175178, 192 bridges, 93 Brownian motion, 307-309, 345 Burgers equation, 25 burning algorithm, 215, 221 Callan-Symanzik, 124, 125 cell Kadanoff, 104, 121 Wheatstone, 121, 122, 140, 142 chemical metric, 198, 215 collapse transition, 63, 64, 149, 151, 152, 170, 172-178, 180-184, 187, 192 connectivity constant, 14, 60, 68, 87 copolymers, 59, 62-64 correlation function, 66, 238, 307, 309 correlation length, 105, 107, 113-117, 121, 124, 132, 142, 242, 284 Coulomb system, 63, 64 357
358 DNA denaturation, 59, 60, 87, 88, 94 effective coordination number, 197 percolation, 216 Sierpinski, 203 energy landscapes, 202 enhancement exponent, 197 percolation, 216 Sierpinski, 203 ensemble fixed distance, 25, 31 fixed force, 32 inequivalence, 15, 37 entropy, 240, 241, 253, 256, 257, 260, 263, 266-268 enumeration, 105, 114, 116, 139 epsilon expansion, 61 exact enumeration technique, 202, 223, 227 exponent, 1-6, 9, 17, 21, 27, 29, 39, 60, 61, 63, 64, 67, 68, 73, 74, 83-89, 91-93, 96, 98, 103-105, 107, 112119, 121, 122, 124, 125, 128, 132, 134, 135, 138-143, 149-152, 164, 166-170, 176, 177, 179, 189, 192, 252, 254, 258, 273, 278 crossover, 40, 114, 117, 142, 173, 181, 183-185 exact, 27 length scale, 39 reunion, 21 size, 29 specific heat, 39 wandering, 28 extreme value statistics, 251, 252 finite lattice method, 73, 98 finite-size scaling, 284 Fisher relation, 198, 225 £-space, 200 percolation, 219 Sierpinski, 204, 208 Fisher renormalisation, 3, 4 fixed point, 112, 121, 124, 125, 128-135, 140, 142, 143 strong disorder, 129, 143
Flory exponent, 197 £-space, 200 percolation, 220 Sierpinski, 204 Flory formula, 61 Flory theory, 3-5, 235, 236, 240, 253, 259, 268, 269 fluctuation free energy, 17, 26 fluorescence spectroscopy, 306, 307, 346 flux-line, 239 forward recoil spectroscopy, 319, 342 fractal dimension, 5, 6 fractals, 5, 7, 111, 113-118, 149-155, 159, 162,167,170,171,174,176,178, 179, 184, 192, 193, 290 deterministic, 196, 203, 223 finitely ramified, 203, 223 infinitely ramified, 203, 208, 223 Given-Mandelbrot, 151, 153, 171, 203 random, 196, 213, 223 Sierpinski, 153, 171, 175, 177, 192, 203, 223 carpet, 208, 223 gasket, 203, 223 square lattice, 208, 223 triangular lattice, 203, 223 free energy, 62, 63, 65, 91, 94, 236, 240, 241, 244, 245, 254, 282, 292 fugacity, 67-69, 71, 88-90, 94 Gaussian chain, 235, 236, 239-241, 252, 259, 260, 268 glassy behavior, 236, 243, 244, 248 growth constant, 60, 61, 64, 68, 71, 72, 87, 92, 93, 98 Harris criterion, 2, 3, 11, 12, 39-41, 61, 105-107, 141 Hoshen-Kopelman algorithm, 201, 221 impurity annealed, 4 quenched, 3, 60, 62, 63 irrelevant variable, 10 Kesten's pattern theorem, 68
359 KPZ, 25, 26 £-space, 198, 215 lacunarity, 208 Langevin equation, 309, 310, 312 lattice Bethe, 151 hierarchical, 12, 37 Husimi, 151, 155 hyperbolic, 86, 87 regular, 197 simplex, 149, 151-157, 159, 162-164, 166,167,171-174,187 Leath algorithm, 201, 214, 215, 221, 228 localization, 62, 63, 235, 236, 241-243, 251, 252, 256, 260, 268, 269 localized eigenstates, 243, 249 magnetization, 64 marginal relevance disorder random medium, 22 RANI model, 35 marginal variable, 10 mass, 60, 70 mass insertion, 136, 137 McKennzie-Moore relation, 198, 208, 220 microviscosity, 309 mode-coupling model, 312 Monte Carlo, 59-61, 63, 64, 73, 98, 99, 201, 223 Morita approximation, 62 most probable paths, 273 directed, 281 non-directed, 291 multifractality, 140, 216, 226 n-vector model, 2-4, 64, 106, 108, 111, 112, 119, 133, 149, 178 Ngai model, 313 nuclear magnetic resonance, 306 optimal path, 271, 272, 294 directed, 27 4 non-directed, 285 overlap, 28
partition function, 14, 281, 282 path integral, 14 path-integral, 236, 238, 243, 246, 249 Penrose tiling, 60, 73-76, 78-83, 85 percolation, 2, 7, 213, 223, 289 backbone, 5, 214, 223 backbone dimension, 6 cluster, 5, 6, 61, 290 correlation length, 6 dangling ends, 6 fractal, 6 invasion, 290 spectral dimension, 5 phase diagram, 63, 94, 96, 294 phase transition, 283, 291, 293 power laws, 9, 311, 312, 321, 348, 351, 352 quantum particle, 235, 236, 239, 241, 256, 259, 268 quasi-crystals, 72, 73 quenched average, 238, 243, 244, 246 radius of gyration, 311, 314, 331, 346, 351 ramification finite, 196, 203, 223 infinite, 196, 208, 223 random obstacles, 235, 236, 253-256, 258, 259, 266, 268, 269 random potential, 235, 253, 254, 259-261, 263, 265-269 random walk, 273 RANI annealed case four chain, 36 three chain, 36 two chain, 34 RANI model, 34 flow equation, 35 Rayleigh scattering, 306 recursion equations, 151, 156-158, 160, 162-168, 171-178, 180, 181, 183, 184, 188-190, 192 reentrant behaviour, 96-98 regularization cut-off, 125
360 dimensional, 125 relevant variable, 10 renormalization additive, 138 multiplicative, 138-140 renormalization group, 103-147, 155 dimensional regularization, 45 fixed point, 22, 47 stable, 29 unstable, 23 flow, 21, 22, 26, 47 KPZ, 26 minimal subtraction scheme, 46 real space, 37, 39 scale transformation, 26 replica, 17, 28 interaction, 35 symmetry breaking, 28, 235, 236, 269 replica method, 62, 236, 241, 244-246, 249, 253 reptation dynamics, 308, 351 response fluctuation, 15 resummation Borel, 126 Chisolm-Borel, 127, 134, 135 Pade-Borel, 126, 127 RG see renormalization group, 21 river basin network, 31 scaling theory, 6, 65, 66, 68, 95, 310, 312, 319, 329 Schrodinger equation, 235, 236, 241-244, 251, 252, 268 self-averaging absence of, 215, 226 self-avoiding interaction, 235-237, 256, 259261, 263, 265, 266, 269 self-avoiding loops, 62, 88, 89, 92, 93, 294 slab geometry, 68 star polymers, 313, 337, 348, 349, 351 stretched exponential, 305, 314-326, 331, 332, 334, 335, 337, 338, 340-344, 347-351
strip geometry, 60, 67, 68, 96 strong disorder, 288, 290 surface adsorption, 149, 150, 178-186 surface desorption, 88, 96 susceptibility, 66 theta solvents, 316 topological metric, 198, 215 transfer matrix, 31, 53, 67, 69 for energy, 54 for overlap, 54 transfer matrix method finite temperature, 282 zero temperature, 276, 285 Travelling Salesman Problem, 294 bounds, 295, 298, 299 dilute lattice, 295 directed, 295 tube model, 311-315 universality class, 9 unzipping, 14, 36 force, 14, 30 pure case, 29 re-entrance, 30 transition, 29, 36, 37 Variational method, 246 vesicle, 59, 60, 88, 94 wedge geometry, 64, 66, 67 Wittkop procedure, 203, 229
