Physics Reports vol.351

crosses zero. At (0, where the super#uid velocity is everywhere subcritical, the energy of quasiparticle is zero only at p"0 (in super#uid He-A at p"$p lK ). This is the topologically $ stable Fermi point discussed in Sections 4.3 and 5. At '0 the pair of horizons appears at x"$x , where x "x ( ;x in the region between the horizons, i.e. at x(x , the Fermi      point expands and becomes the Fermi surface. The Fermi surface } the 2D manifold in 3D momentum space where the quasiparticle energy in Eq. (345) is zero } is determined by equation






p#p p x !x W X# V"  , !x (x(x . (346)   p x p .  V Thus the appearance of the pair of horizons is accompanied by the quantum (Lifshitz) phase transition at which the topology of the quasiparticle spectrum drastically changes. In the extended space, which includes the momentum p and the coordinate x, the manifold of zeroes of the quasiparticle energy is the 3D Fermi hypersurface in the 4D space ( p, x). This hypersurface bounds the ergoregion in the 4D space ( p, x), where EI ( p, x)(0.

G.E. Volovik / Physics Reports 351 (2001) 195}348

319

At '0 the super#uid-comoving vacuum dissipates, say, by Hawking radiation. The Hawking process leads to the "lling of the empty negative energy states within the ergoregion, and the radiation stops, when all of them become occupied. The state with the Fermi hypersurface is thus the "nal zero-temperature state of the Hawking radiation in the systems with stable horizons. The states of the global thermodynamic equilibrium with nonzero ¹ are also possible in such systems. The thermodynamic properties of the horizons, according to the conventional rules for the degenerate Fermi system with Fermi surface, are determined by the density of states at zero energy. If is small, the momenta and energies of the occupied states are still small compared to the Planck scale: p 4p ;p , p 4p ;p , E&cp ;E . (347) V . . W . . V . The density of states is thus N(0)&p p p /E&(p/c) ;(p/c). This is one of the examples when V W X . . the e!ect of the Planck physics does not require the knowledge of the details of trans-Planckian physics, since only the "rst nonlinear correction to the energy spectrum determines the phenomenon. Since the density of states is "nite the entropy of the system with stable horizons is proportional to ¹ and thus is zero at ¹"0. If is small, the total number of the occupied states, &Ax p , where A is the area of the  . horizon, is also small. That is why one can expect that at ;1 the reconstruction of the vacuum occurs without essential modi"cation of the velocity pro"le, i.e. without the considerable back reaction of the `gravitational "elda, and thus the stable horizons in the Fermi systems is a typical phenomenon. While for essentially nonrelativistic systems this is really a typical situation to have a stable ergoregion, this is not as simple for the systems exhibiting at small energy the relativistic behavior and the event horizon. The detailed inspection of the back reaction of the order parameter "eld to the "lling of the negative energy levels behind the horizon shows that the back reaction can be signi"cant. Of course, there is no theorem which prevents the existence of the stable horizons in the pair-correlated systems. However, in a typical case discussed, e.g. in Ref. [196], the supercritical state with the Fermi surface appears to be unstable towards the collapse of the super#uidity. In such case, there are no true equilibrium states in the presence of the horizon. So that the "nal result will be the merging and mutual annihilation of the two horizons. This occurs either by the collapse of super#uidity; or, if the collapse is smoothened, by thermal relaxations of the local equilibrium states, which we shall discuss in Section 12.7; and "nally at very low temperature again by the Hawking radiation. 12.5. Hawking radiation Irrespective of the "nal destiny of the horizons, the initial stage of the dissipation of the super#uid-comoving vacuum at ¹"0 in the presence of horizons starts with the Hawking-like radiation. Let us discuss this radiation using the semiclassical description, which is valid when the quasiparticle energy is much larger than the crossover temperature of order of the Hawking temperature ¹ . In this case the Hawking radiation can be described as the quantum tunneling & between the classical trajectories. Fig. 25 shows the relevant classical trajectories of quasiparticles in the presence of the horizons in simplest 1#1 case. We consider the positive energy states, EI '0, as viewed in the laboratory (texture) frame, where the velocity "eld is time independent. These trajectories are given by the equation EI (p, x)"EI '0

320

G.E. Volovik / Physics Reports 351 (2001) 195}348

Fig. 25. Trajectories of quasiparticles in the presence of the black-hole/white-hole pair in 1#1 case. The nonlinear dispersion of the energy spectrum at high momentum p is taken into account. Only those positive-energy trajectories are considered, which are responsible for the Hawking-like radiation. They are represented by curves of constant energy in the laboratory (Killing) frame: EI (p, x)"EI '0. The trajectory between the horizons has negative energy, E(0, in the frame comoving with the super#uid vacuum. That is why the corresponding quantum states are initially occupied. But in the laboratory frame these quasiparticles have positive energy, EI '0, and thus can tunnel to the mode with the same energy EI '0, which is out-going from the black hole. The radiation from the black hole also occurs by tunneling to the mode in-going to the white hole, which due to `superluminala dispersion crosses both horizons.

with EI (p, x) given by Eq. (345). Between the horizons the positive energy states in the laboratory frame have the negative energy as viewed by the inner observer moving with the super#uid velocity * : in the `relativistica domain this energy is E(p)"!cp(0. Since the initial state of the liquid is  the super#uid-comoving vacuum, these states between the horizons are initially occupied. This super#uid-comoving vacuum is seen in the absolute laboratory frame as highly excited state with nonequilibrium distribution of quasiparticles. The equilibrization occurs via tunneling of the quasiparticles from the occupied excited states, which have EI '0 and E(0, to the modes with the same positive energy EI (p, x)"EI '0 but with E'0. The latter are the modes out-going from the black-hole horizon to in"nity. The tunneling exponent is determined by the usual quasiclassical

G.E. Volovik / Physics Reports 351 (2001) 195}348

321

action 2 Im p(x) dx. At low energy, when the nonrelativistic corrections are neglected, the momentum as a function of x on the classical trajectory has a pole at the horizon p(x)+EI /v (x!x ).   Here v is the derivative of the super#uid velocity at the horizon, which is equivalent to the surface  gravity. Using the standard prescription for the treating of poles, one obtains for the tunneling action 2 Im p(x) dx"2EI /v . The probability of tunneling, exp(!2EI /v ), reproduces the thermal   radiation with the Hawking temperature:




¹ "  , & 2 1

dv   " 1 dx

.

(348)



This can be compared with Eq. (338) for the Hawking temperature in the case of moving vierbein wall in terms of the `surface gravitya  at the horizon. The only di!erence is that in the case of the 1 wall the `surface gravitya is simulated by the gradient of the `speed of lighta instead of the gradient of the super#uid velocity in Eq. (348). There is another channel of dissipation. According to Fig. 25 the radiation from the region between the horizons also occurs by tunneling to the mode in-going to the white hole. This mode, due to the `superluminala dispersion, crosses both horizons and is transformed to the mode out-going from the black-hole horizon. The tunneling exponent is given by Eq. (348) where it is now the velocity gradient at the white-hole horizon, which determines the Hawking temperature of this tunneling. 12.6. Extremal black hole Let us consider the transition point, "0, at which the super#uid velocity "rst approaches the `speed of lighta at x"0 (Fig. 24). After rescaling of Eq. (343) at "0 one obtains the following interval: ds"!x dt#2dx dt#dx#dy#dz






dx  dx # #dy#dz . "!x dt! x x

(349)

The plane x"0, where the black and white horizons merge, marks the bridge between the two spaces, at x(0 and at x'0. The metric in Eq. (349) coincides with the metric in the vicinity of the horizon of an extremal black hole, whose mass equal to the electric charge:











r  r \ dr#r d . ds"! 1!  dt# 1!  r r

(350)

There is a strong divergence of the e!ective temperature at the horizon, ¹ J¹/x, according to  the Tolman's law in Eq. (49), where ¹ is the constant Tolman's temperature which is a real temperature in super#uids. Because of this divergence all the thermodynamic quantities have singularity at the horizon where the Planck scale of energy must intervene to cut o! the divergence. It appears again that the "rst nonlinear correction to the `relativistica energy spectrum provides the necessary cuto! already at low energy, which is much less than the Planck energy scale.

322

G.E. Volovik / Physics Reports 351 (2001) 195}348

The energy spectrum in the vicinity of the horizon of our extremal black hole is according to Eq. (345)






x p#p p 1 X # V , p '0, p#p;p . EI " cp # W V W X V 2 V x p p  V .

(351)

Thermodynamical quantities at the horizon, which follow from this spectrum can be estimated using scaling relations. Since the characteristic energy EI is of order the temperature ¹, the characteristic values of the coordinate x and momentum p are




¹  p &p , V . cp .







¹  ¹  p & p &p , x &x . W X . cp  cp . .

(352)

As a result the thermodynamic energy concentrated at the horizon is E JA¹ xp p p  V W X &(1/c)Ap x ¹. We can compare this with the thermodynamic energy in the bulk liquid far from .  the horizons. It is of order E J(1/c)Ax ¹, where x is the dimension of the container.
    Assuming that x is of order x , one obtains the ratio E /E Jp c/¹<1. Thus most of the   
  . thermodynamic energy is concentrated in the vicinity of the horizon in the region of thickness x (¹/cp ). This region is very small, but still is much larger than the Planck length scale. That is  . why all the characteristic energies and momenta are much less than the Planck scales. Thus what we need from the Planck physics here is only the small nonlinear correction to the energy spectrum; this is enough for the complete determination of thermodynamic properties of the extremal horizon. The quantum gravity is not necessary to be introduced. The most of the entropy of the system is also concentrated at the horizon, S (¹)&(1/c)Ap x ¹. The entropy of our extremal black hole tends to zero, when the temper .  ature is reduced. We considered the region of temperatures ¹ ;¹;cp , where ¹ is   .   the temperature at which the quantization of the quasiparticle levels becomes important. Let us estimate the entropy in the quantum limit. The quantum corrections become important, when p x&1. From Eq. (352) it follows that the quantum crossover temperature is V ¹ &E (p x )\. (We recall, that for the conventional black hole the quantum crossover   . .  temperature is of order the Hawking temperature, ¹ &E (p x )\.) The horizon entropy at   . .  this crossover temperature is S (¹ )&A/x &1. This means that the condensed matter     analog of the extremal black hole supports the point of view that the entropy of the extremal black hole is zero, or at least is not proportional to the area of the horizon. 12.7. Thermal states in the presence of horizons. Modixed Tolman's law In typical super#uid/superconducting systems with relativistic-like quasiparticles the pair of black and white horizons does not allow the states with global thermal equilibrium to exist because of the back reaction of the super#uid vacuum. In this case the "nal state of the pair of horizons is when the two horizons merge, i.e. it is the extremal horizon considered in the previous section. If the walls of container are properly isolated the initial stage of the evolution of the system towards this state is governed at low ¹ by Hawking radiation. Let us consider how this evolution can occur if the temperature of the liquid is higher than the quantum crossover temperature ¹ . &

G.E. Volovik / Physics Reports 351 (2001) 195}348

323

Here we discuss the scenario, in which the equilibrization occurs through the sequence of the local equilibrium states with inhomogeneous ¹ and * . In a local equilibrium the counter#ow  velocity w in Eq. (44) must be less than the `speed of lighta. That is why in the presence of horizons, when the super#uid velocity * exceeds c, the normal component velocity * must be adjusted to   keep the counter#ow velocity w"* !* below c everywhere. Thus the normal component   velocity is necessarily inhomogeneous in the presence of horizons. This violates the condition for the global thermodynamic equilibrium, * "0, and gives rise to the dissipation via viscosity of the  normal component, which plays the role of the Hawking radiation at ¹ above ¹ . Such local & equilibrium states were constructed in Ref. [27] in the simplest case of the 1#1 space}time with the #ow pro"le in the right bottom part of Fig. 24. The `speed of lighta c is kept constant, while the super#uid velocity v (x), depends on the coordinate x and exceeds c in the region between the  horizons. Though the super#uid velocity is `superluminala between the horizon, v 'c, the  counter#ow velocity w appears to be everywhere `subluminala reaching the maximum value w"c at the horizon, with w(c both outside and inside the horizons. The local equilibrium with the e!ective temperature ¹ in Eq. (44) is thus determined on both sides of the horizon.  These thermal states are obtained using the two-#uid hydrodynamics discussed in Section 2. We neglect quantum e!ects related to gravity, including the Hawking radiation process. This is permitted if all the relevant energies are much higher than temperature of the crossover to the quantum regime, which is of order of the Hawking temperature, ¹ "( /2) , where & 1  "(R v ) &c/x is the `surface gravitya at the horizon. Thus we assume that ¹<¹ , and 1 V    &

/<¹ , where  is the relaxation time due to collision of thermal quasiparticles. The latter & relation also shows that the mean free path l"c is small compared with the characteristic length, within which the velocity (or the gravitational potential) changes: l(Rv /Rx);v . This is just the   condition for the applicability of the two-#uid hydrodynamic equations, where the variables are the super#uid velocity v which, when squared, plays the part of the gravitational potential, as well as  temperature ¹(x) and velocity v (x) of the normal component, which characterize the local  equilibrium states of `mattera. The dissipative terms in the two-#uid equations can then be neglected in zeroth-order approximation, since they are small compared to the reversible hydrodynamic terms by the above parameter l(Rv /Rx);v .   If the back reaction is neglected, and thus the super#uid velocity (`gravitya) "eld is "xed, the other hydrodynamic variables, temperature ¹(x) and velocity v (x) of `mattera, are determined by  the conservation of energy and momentum. From Eq. (42) for the "0 component of the energy}momentum tensor, which corresponds to the energy conservation for the `mattera, it follows that the energy #ux Q carried by the quasiparticles is constant (note that c is constant here). In the `relativistica approximation one then has (c"1) v (1#wv ) (2s#1)  "const, " ¹ . Q"!(!g¹V "2    1!w 12

(353)

From the same Eq. (42) but for "1, which is the momentum conservation equation, there results the "rst-order di!erential equation






v w w # "2 R v . !R 2  V 1!w 1!w V  If the energy #ux is zero, Eq. (353) gives two possible states.

(354)

324

G.E. Volovik / Physics Reports 351 (2001) 195}348

(i) Eq. (353) is satis"ed by the trivial solution v "0. Then from Eq. (354) it follows that  ¹"const. This corresponds to a true equilibrium state, or global thermodynamic equilibrium discussed in Section 3.6. Such equilibrium state, however, can exist only in the absence of or outside the horizon. The e!ective temperature, which satis"es the Tolman's law, ¹ (x)"  ¹/(!g (x),¹/(1!v(x), cannot be continued across the horizon: The e!ective temperature   ¹ diverges when the horizon is approached and becomes imaginary inside the horizon, where  w '1. (ii) There is, however, another solution of Eq. (353): 1#wv "0. Since w(1, this solution can  be valid only inside the horizon, where v'1. From Eq. (354) it follows that  (x)"const/(v(x)!1), and thus the temperature behaves as ¹(x)J(x)(1!w(x))"  const/v(x), or ¹(x)"¹ / v (x) , where ¹ is the temperature at the horizon, when ap      proached from inside. Thus inside the horizon one has a quasiequilibrium state with inhomogeneous temperature. The e!ective temperature behind the horizon follows a modi"ed Tolman law: ¹   . (355) ¹ (x)"  (v(x)!1  Solution (i) outside the horizon and solution (ii) inside the horizon can be connected at the horizon. As in the case of the connection of two sister Universes across the vierbein wall, the main principle is the possibility of the superluminal exchange of energy between the matter (quasiparticles) inside and outside the horizon. This implies that the real temperature ¹ must be continuous across the horizon: ¹ "¹ . Assuming that far from the horizon the super#uid velocity     vanishes, one has ¹( x 'x )"¹ , ¹( x (x )"¹ / v (x) . Thus one has the modi"ed form of      the Tolman's law, which is valid on both sides of the black hole horizon: ¹ (x)" 

¹   " . v(x) ( g (x)  1!  c ¹





(356)

The e!ective temperature ¹ , which determines the local `relativistica thermodynamics, becomes  in"nite at the horizon. The cuto! is provided by the nonlinear dispersion of the quasiparticle spectrum with '0. The real temperature ¹ of the liquid is continuous across the horizon: c at v(x)'c . (357)    v (x)  In the presence of superluminal dispersion, all physical quantities are continuous at the horizon. On the other hand, in the limit of vanishing dispersion they experience kinks. For example, the jump in the derivative of the temperature is ¹"¹

at v(x)(c, 

¹(x)"¹


¹  !
¹  "2¹ ¹ . (358) V > V \  & This jump does not depend on details of the high-energy dispersion. This means that in the limit of a purely relativistic system, the presence of a nonzero temperature at in"nity implies a singularity at the horizon. This coordinate singularity at the horizon cannot be removed, since in the presence of

G.E. Volovik / Physics Reports 351 (2001) 195}348

325

`mattera with nonzero temperature the system is not invariant under coordinate transformations, and this produces a kink in temperature at the horizon. 12.7.1. Entropy related to horizon Let us consider the entropy of quasiequilibrium thermal state across the horizon,



R S" d"r S, S" . R¹

(359)

The `relativistica entropy, which is measured by a local observer living in the quasiparticle world is the e!ective entropy R R R¹ S " " "S(1!w . (360)  R¹ R¹ R¹   For our thermal state, in the presence of a horizon in 1#1 dimension, the total real entropy can be divided into three contributions: S"S #S #S . (361)    At ¹ <¹ , the exterior entropy, which comes from the bulk liquid, is proportional to the size  & ¸ of the external region: S J¹ ¸ . A similar estimate holds for the entropy of the interior     region, S J¹ ¸ . The entropy related to the horizon is well separated from the bulk terms,    since it contains the `Plancka energy cuto! due to the in"nite red shift at the horizon. In the 1#1 case such entropy comes from the logarithmically divergent contribution at the horizon:



 ¹  . S " dx (362)  3 1!v  The infrared cuto! is determined by x , which characterizes the gradient of the velocity "eld at  horizon or the `surface gravitya: 1!v+2 x&x/x where x is the distance from the  1  horizon. The ultraviolet cuto! x is provided by the nonlinear dispersion of the quasiparticle spectrum.  At the cuto! scale the nonlinear term becomes comparable with the linear one. We use the same energy spectrum with nonlinear dispersion as before, EI (p)"c p (1#pc/E)#pv + .   p x#c p /E, where the Planck energy scale is E "cp . Then one has for the ultraviolet 1 . . . cuto! parameters, x and p , the equation  px &c p /E&¹ and thus the following   1   .  estimation:







¹  E  . , E &cp &¹ . (363) x "\     ¹  1 E .  Again the cuto! energy E "cp appeared to be much smaller than the Planck energy scale. This   means that one does not need in the physics at Planck scale to discuss the horizon problem in the considered 1#1 space}time: only the "rst nonlinear correction to the linear spectrum well within the relativistic domain is important. From Eq. (363) it follows that the entropy related to the horizon is




1¹ E  ln . S "  9 ¹ ¹ & 

.

(364)

326

G.E. Volovik / Physics Reports 351 (2001) 195}348

This relation also means that the density of quasiparticle states diverges logarithmically at the horizon






dx dp E 1 (E!EI (p, x))" ln . , (365) 2 E 3¹ & where ¹ ;E;E . & . We discussed the entropy related to the horizon using the quasiequilibrium thermodynamic states near the horizon with temperature much larger than ¹ . In principle, it has nothing to do & with the Bekenstein entropy obtained in quantum limit. Nevertheless, one can extrapolate Eq. (364) to the crossover region to obtain S (¹ "¹ )" ln(E /¹ ). It coincides with the estimation of . &   &  the Bekenstein entropy of 1#1 black hole (see [27]). N (E)"2 

12.8. PainleveH }Gullstrand vs. Schwarzschild metric in ewective gravity. Incompleteness of space}time in ewective gravity As we have already discussed in Section 12.1, in the e!ective theory of gravity, which occurs in condensed matter systems, the primary quantity is the contravariant metric tensor gIJ describing the energy spectrum. Due to this the two seemingly equivalent representations of the black hole metric, in terms of the PainleveH }Gullstrand line element in Eq. (341) and the Schwarzschild line element






v dr 1 # r d (366) ds"! 1!  dt! # c c!v c  are not equivalent. Eqs. (366) and (341) are related by the coordinate transformation. Let us for simplicity consider the abstract #ow with the velocity exactly simulating the Schwarzschild metric, i.e. v(r)"r /r and   we put c"1 and r "1. Then the coordinate transformation is  2 1!v (r) v  #ln , dtI "dt#  dr . (367) tI (r, t)"t# v (r) 1#v (r) 1!v    What is the di!erence between the Schwarzschild and PainleveH }Gullstrand space}times in the e!ective gravity? The PainleveH }Gullstrand line elements directly follows from the contravariant metric tensor gIJ and thus is valid for the whole `absolutea Newton's space}time (t, r) of the laboratory frame, i.e. as is measured by the external experimentalist, who lives in the nonrelativistic world of the laboratory and investigates the dynamics of quasiparticles using the physical laws obeying the Galilean invariance. The time tI in the Schwarzschild line element is the time as measured by the `innera observer at `in"nitya (i.e. far from the black hole). The `innera means that this observer `livesa in the super#uid background and uses `relativistica massless quasiparticles (phonons in He or `relativistica fermionic quasiparticles in He-A) as a light for communication and for synchronization of clocks. The inner observer at some point R<1 sends quasiparticles pulse at the moment t , which arrives  at point r at t"t #0 dr/ v of the absolute (laboratory) time, where v and v are absolute  \ > \ P






G.E. Volovik / Physics Reports 351 (2001) 195}348

327

(laboratory) velocities of radially propagating quasiparticles, moving outward and inward respectively dr dE (368) v " " "$1#v .  ! dt dp P Since from the point of view of the inner observer the speed of light (i.e. the speed of quasiparticles) is invariant quantity and does not depend on direction of propagation, for him the moment of arrival of pulse to r is not t but tI "(t #t )/2, where t is the time when the pulse re#ected from    r returns to observer at R. Since t !t "0 dr/ v #0 dr/ v , one obtains for the time   \ > P P measured by inner observer as




  


0 dr 0 dr # v v P > P \ 2 2 1!v (r) 1!v (R)   " t# #ln ! #ln , (369) v (r) v (R) 1#v (r) 1#v (R)     which is just Eq. (367) up to a constant shift. In the Newtonean physical space}time of the laboratory the external observer can detect quasiparticles radially propagating into (but not out of) the black hole or out of (but not into) the white hole. For him the energy spectrum of the quasiparticles is well determined both outside and inside the horizon. Quasiparticles cross the black-hole horizon with the absolute velocity v "!1!v "!2, i.e. with the double speed of light: r(t)"1!2(t!t ). In case of a white\   hole horizon one has r(t)"1#2(t!t ). On the contrary, from the point of view of the inner  observer the horizon cannot be reached and crossed: the horizon can be approached only asymptotically for in"nite time: r(tI )"1#(r !1) exp(!tI ). Such incompetence of the local ob server, who `livesa in the curved world of super#uid vacuum, happens because he is limited in his observations by the `speed of lighta, so that the coordinate frame he uses is seriously crippled in the presence of the horizon and becomes incomplete. The Schwarzschild metric naturally arises for the inner observer outside the horizon, if the PainleveH }Gullstrand metric is an e!ective metric for quasiparticles in super#uids, but not vice versa. The Schwarzschild metric Eq. (366) can in principle arise as an e!ective metric; however, in the presence of a horizon such metric indicates an instability of the underlying medium. To obtain a line element of Schwarzschild metric as an e!ective metric for quasiparticles, the quasiparticle energy spectrum in the laboratory frame has to be 1 t #t  "t# tI (r, t)"  2 2














r  r E"c 1!  p#c 1!  p . P r r ,



(370)

Behind a horizon such spectrum has E(0 in some region of transverse momentum p . The , imaginary frequency of excitations signals the instability of the super#uid vacuum: Quasiparticle perturbations may grow exponentially without bound in laboratory (Killing) time, as eR ' #, destroying the super#uid vacuum. Nothing of this kind happens in the case of the PainleveH } Gullstrand line element, for which the quasiparticle energy is real even behind the horizon. Thus the main di!erence between PainleveH }Gullstrand and Schwarzschild metrics as e!ective metrics is: The "rst metric leads to the slow process of the quasiparticle radiation from the vacuum at the

328

G.E. Volovik / Physics Reports 351 (2001) 195}348

horizon (Hawking radiation), while the second one indicates a crucial instability of the vacuum behind the horizon. In general relativity it is assumed that the two metrics can be converted to each other by the coordinate transformation in Eq. (367). In condensed matter the coordinate transformation leading from one metric to another is not that innocent if an event horizon is present. The reason why the physical behavior implied by the choice of metric representation changes drastically is that the transformation between the two line elements, tPt#P dr v /(c!v), is singular on the horizon,   and thus it can be applied only to a part of the absolute space}time. In condensed matter, only such e!ective metrics are physical which are determined everywhere in the real physical space}time. The two representations of the `samea metric cannot be strictly equivalent metrics, and we have di!erent classes of equivalence, which cannot be transformed to each other everywhere by regular coordinate transformation. PainleveH }Gullstrand metrics for black and white holes are determined everywhere, but belong to two di!erent classes. The transition between these two metrics occurs via the singular transformation tPt#2P dr v /(c!v) or via the Schwarzschild line element, which   is prohibited in condensed matter physics, as explained above, since it is pathological in the presence of a horizon: it is not determined in the whole space}time and it is singular at horizon. It is also important that in the e!ective theory there is no need for the additional extension of space}time to make it geodesically complete. The e!ective space}time is always incomplete (open) in the presence of horizon, since this e!ective space}time exists only in the low energy `relativistica corner and quasiparticles escape this space}time to a nonrelativistic domain when their energy increase beyond the relativistic linear approximation regime [191]. Another example of the incomplete space}time in e!ective gravity is provided by the vierbein domain walls } the walls with the degenerate metric } discussed in Section 11.1. For the `innera observer, who lives in one of the domains and measures the time and distances using the quasiparticles, his e!ective space}time is #at and complete. But this is only half of the real (absolute) space}time: the other domains, which do really exist in the absolute space}time, remain unknown to this `innera observer. These examples show also the importance of the superluminal dispersion at high energy: though many results do not depend on the details of this dispersion, merely the possibility of the information exchange by `superluminala quasiparticles establishes the correct continuation across the horizon or the other classically forbidden regions. The high energy dispersion of the relativistic particles was exploited in recent works on black holes [198}202]. 12.9. Vacuum under rotation 12.9.1. Unruh and Zel'dovich}Starobinsky ewects An example of the quantum friction caused by the ergoregion without the horizon is provided by a body rotating in super#uid liquid at ¹"0 [115]. In this case the #ow of the super#uid component is tangential to the ergosurface. The e!ect we discuss is analogous to the ampli"cation of electromagnetic radiation and spontaneous emission by the body or black hole rotating in quantum vacuum, "rst discussed by Zel'dovich and Starobinsky. The friction is caused by the interaction of the part of the liquid, which is rigidly connected with the rotating body and thus represents the corotating detector, with the `Minkowskia vacuum outside the body. The emission process is the quantum tunneling of quasiparticles from the detector to the ergoregion, where the

G.E. Volovik / Physics Reports 351 (2001) 195}348

329

energy of quasiparticles is negative in the rotating frame. The emission of quasiparticles, phonons and rotons in super#uid He and Bogoliubov fermions in super#uid He, leads to the quantum rotational friction experienced by the body. The motion with constant angular velocity is another realization of the Unruh e!ect [203]. In Unruh e!ect a body moving in the vacuum with linear acceleration a radiates the thermal spectrum of excitations with the Unruh temperature ¹ " a/2c. On the other hand the observer comoving 3 with the body sees the vacuum as a thermal bath with ¹"¹ , so that the matter of the body gets 3 heated to ¹ (see references in [204]). It is di$cult to simulate in condensed matter the motion at 3 constant proper acceleration (hyperbolic motion). On the other hand the body rotating in super#uid vacuum simulates the uniform circular motion of the body with the constant centripetal acceleration. Such motion in the quantum vacuum was also heavily discussed in the literature (see the latest references in [205}207]). The latter motion is stationary in the rotating (texture) frame, which is thus a convenient frame for study of the radiation and thermalization e!ects for uniformly rotating body. Zel'dovich [208] was the "rst who predicted that the rotating body (say, the dielectric cylinder) ampli"es those electromagnetic modes which satisfy the condition !¸(0 .

(371)

Here  is the frequency of the mode, ¸ is its azimuthal quantum number, and  is the angular velocity of the rotating cylinder. This ampli"cation of the incoming radiation is referred to as superradiance [209]. The other aspect of this phenomenon is that due to quantum e!ects, the cylinder rotating in quantum vacuum spontaneously emits the electromagnetic modes satisfying Eq. (371) [208]. The same occurs for any rotating body, including the rotating black hole [210], if the above condition is satis"ed. Distinct from the linearly accelerated body, the radiation by a rotating body does not look thermal. Also, the rotating observer does not see the Minkowski vacuum as a thermal bath. This means that the matter of the body, though excited by interaction with the quantum #uctuations of the Minkowski vacuum, does not necessarily acquire an intrinsic temperature depending only on the angular velocity of rotation. Moreover the vacuum of the rotating frame is not well de"ned because of the ergoregion, which exists at the distance r "c/ from the axis of rotation.  Let us consider a cylinder of radius R rotating with angular velocity  in the (in"nite) super#uid liquid (Fig. 26). When the body rotates, the energy of quasiparticles is not well determined in the stationary frame due to the time dependence of the perturbations, caused by the rotation of the body, whose surface is never perfect. The quasiparticle energy is well de"ned in the rotating frame, where all the perturbations of the texture of the order parameter caused by the surface roughness of the cylinder are stationary. Hence it is simpler to work in the rotating (texture) frame. If the rotating body is surrounded by the stationary super#uid, i.e. * "0 far from the body, then in the rotating  frame one has * "!;r. Substituting this * in Eq. (38) one obtains that the line element, which   determines the propagation of phonons in the frame of the body, corresponds to the conventional metric of #at space in the rotating frame: ds"!(c!) dt!2 d dt#dz# d#d .

(372)

The azimuthal motion of quasiparticles in the rotating (texture) frame can be quantized in terms of the angular momentum ¸, while the radial motion can be treated in the quasiclassical

330

G.E. Volovik / Physics Reports 351 (2001) 195}348

Fig. 26. Possible simulation of Zel'dovich}Starobinsky e!ect in super#uids. The inner cylinder rotates forming the preferred rotating reference frame. In this frame the e!ective metric has an ergoregion, where the negative energy levels are empty.

approximation. Then the energy spectrum of phonons in the rotating frame is EI "E(p)#p ) * "c 



¸ #p#p!¸ . X M 

(373)

For rotons in He and Bogoliubov fermions in He-B the energy spectrum in the rotating frame is (p!p )  !¸ , EI (p)"# 2m  EI (p)"( #v (p!p )!¸ ,  $ $ where p marks the roton minimum in super#uid He, while  is a roton gap. 

(374) (375)

12.9.2. Ergoregion in superyuids For the `relativistica phonons the radius  "c/, where g "0, marks the position of the   ergosurface. In the ergoregion, i.e. at ' "c/, the energy of phonons in Eq. (373) can become  negative for any rotation velocity and ¸'0 (Fig. 27). However, the real ergosurface in super#uid He occurs at  "v /(c, where the Landau velocity in Eq. (340) is determined by the  *   roton minimum, v &/p . Let us assume that the angular velocity of rotation  is small *    enough, so that the linear velocity on the surface of the rotating cylinder is less than the Landau critical velocity: R(v . Thus excitations can never be nucleated at the surface of cylinder. *   However, at the ergosurface the velocity v " in the rotating frame reaches v , so that   *   quasiparticle can be created in the ergoregion ' . 

G.E. Volovik / Physics Reports 351 (2001) 195}348

331

Fig. 27. Vacuum seen in the frame corotating with inner cylinder is di!erent from that viewed in the laboratory frame. The states which are occupied in the vacuum viewed in the laboratory frame are shaded. If the ergoplane is close to the rotating inner cylinder and is far from the outer cylinder, which is at rest in the laboratory frame, the in#uence of the rotating cylinder on the quasiparticle behavior is dominating. Thus the relevant frame for quasiparticles is the rotating frame. In the ergoregion, some states, which are occupied in the laboratory vacuum, have positive energy in the corotating frame. The quasiparticles occupying these levels must be radiated away. At ¹"0 the radiation occurs via quantum tunneling from or to the region in the vicinity of the surface of inner cylinder, where the interaction with the rotating cylinder occurs. The rate of tunneling reproduces the Zel'dovich}Starobinsky e!ect of radiation from the rotating black hole.

The process of creation of quasiparticles in the ergoregion is determined by the interaction with the rotating body: there is no radiation in the absence of the body. If R;v one has *   
332

G.E. Volovik / Physics Reports 351 (2001) 195}348

12.9.3. Radiation to the ergoregion as a source of rotational quantum friction Let us describe one of the possible scenaria of the quantum friction, in which the interaction of quasiparticles with the container is provided by the surface roughness. Due to the surface roughness there are regions near the surface of rotating cylinder where the Bose condensate is moving together with the cylinder, i.e. the super#uid velocity in these regions is zero in the rotating frame. These regions serve as reservoir of quasiparticles with energy EI "E"0 (and also as the rotating detector). The emission of quasiparticles by rotating cylinder can be described as tunneling from these trapped states to the scattering state at the ergosurface, where the energy of quasiparticles is also EI "0 (Fig. 27). In the quasiclassical approximation the tunneling probability is e\1, where



S"Im d p (EI "0) . M For phonons with p "0 according to Eq. (373) one has X  M 1 1  ! +¸ ln  . d S"¸  ( ) R 0 

 

(376)

(377)

Thus all the phonons with ¸'0 are radiated, but the radiation probability decreases at higher ¸. If the linear velocity at the surface is much less than c, i.e. R;c, the probability of radiation of phonons with the energy (frequency) "¸ becomes






wJe\1"

R * R * R * " " , R;c ,  c c¸ 

(378)

where "¸ is the frequency of excited phonon mode. For the cylinder rotating in relativistic quantum vacuum the speed of sound c must be substituted by the speed of light, and Eq. (378) appears to be proportional to the superradiant ampli"cation of the electromagnetic waves by rotating dielectric cylinder derived by Zel'dovich [209,211]. The number of phonons with the frequency "¸ emitted per unit time can be estimated as NQ "=e\1, where = is the attempt frequency multiplied by the number of modes localized near the surface of cylinder. Since each phonon carries the angular momentum ¸, the cylinder rotating in super#uid vacuum (at ¹"0) is loosing its angular momentum, which means the quantum rotational friction. Let us consider now the emission of rotons, whose spectrum is given by Eq. (374). The minimal ¸ value of the radiated rotons, which have the gap , is determined by this gap: ¸ "

 /+v p /, where v +/p is the Landau critical velocity for the emission of rotons. *    *    Since the tunneling rate exponentially decreases with ¸, only the lowest possible ¸ must be considered. In this case the tunneling trajectory with EI "0 is determined by the equation p"p  both for rotons and Bogoliubov quasiparticles. For p "0 the classical tunneling trajectory is thus X given by p "i( p !¸/ . This gives for the tunneling exponent e\1 the equation M  M 1 1  dr ! +¸ ln  . S"Im d p "¸ M (379)   R 0 



 

G.E. Volovik / Physics Reports 351 (2001) 195}348

333

Here the position of the ergosurface is  "v /+¸/p . Since the rotation velocity  is  *    always much smaller than the gap, the momentum ¸ of the radiating roton is big. That is why the radiation of rotons (and also Bogoliubov quasiparticles with the gap) is exponentially suppressed, compared with the emission of phonons. The quantum rotational friction due to phonon emission becomes dominating mechanism of dissipation at low enough temperature where the conventional viscous friction between the rotating cylinder and the normal component of the liquid is small.

13. How to improve helium-3 13.1. Gradient expansion Though He-A and Standard Model belong to the same universality class and thus they have similar properties of the fermionic spectrum, the He-A cannot serve as a perfect model for quantum vacuum. While the properties of the chiral fermions are well reproduced in He-A, the e!ective action for bosonic gauge and gravity "elds obtained by integration over fermionic degrees of freedom is contaminated by the bad terms } the terms which are absent in a fully relativistic system. This is because the integration over the vacuum fermions is not always concentrated in the region where their spectrum is `relativistica. Thus the question arises whether we can `correcta the He-A in such a way that these uncomfortable bad terms are suppressed. If we neglect the spin degrees of freedom, the massless bosonic "elds which appear in He-A due to the breaking of symmetry ;(1) ;SO(3) is the triad "eld, e( , e( and lK . For the slow hydro, *   dynamic motion only the soft Goldstone modes, related to rotation of the triad "eld, are excited. The energetics of the Goldstone "eld is given by the so called London energy, which is the quadratic form of the gradients of the soft variables. The gradients of triad can be expressed in terms of the super#uid velocity (the torsion in Eq. (102)) and the gradients of the lK -"eld. In the reference frame where the heat bath is at rest, * "0, this energy can be written in the following  general form, satisfying Galilean invariance and the global symmetry ;(1) ;SO(3) of the , * system: [16]: m m F " n (lK ) * )# n (lK ;* )   * 2 , 2 ,

(380)

#(C(lK ;* ) ) (lK ;(
;lK ))!(C !C)(lK ) * )(lK ) (
;lK ))) (381)     #K (
) lK )#K (lK ) (
;lK ))#K (lK ;(
;lK )) . (382)  
Here Eq. (380) is the anisotropic kinetic energy of super#ow; Eq. (382) is the same as in nematic liquid crystals, it is the energy of the lK texture with coe$cients K , K and K describing the  
response of the system to splay, twist and bent, respectively; Eq. (381) is the interaction of the chiral mass current carried by texture with the super#uid velocity "eld. The temperature dependent coe$cients, n , n , C, C , K , K and K , can be considered as , ,   
phenomenological, in the same way as the parameters of the Standard Model. But they can also be

334

G.E. Volovik / Physics Reports 351 (2001) 195}348

obtained within the framework of the microscopic BCS theory if one applies the gradient expansion [212]. In the canonical form of BCS theory all the coe$cients in the gradient energy are determined by only four parameters:  (¹), p , v "p /mH, and m.  $ $ $ These four parameters play di!erent roles in e!ective theory. The "rst three parameters determine the quasiparticle spectrum in the `relativistica low-energy corner below the "rst `Planck scalea, E; /v p "mHc . They determine the `speeds of lighta, c "v and c , and the Planck  $ $ , , $ , energy scale. This is, in principle, all what is needed from the microscopic physics to construct the e!ective relativistic quantum "eld theory. All other numbers must be determined by the symmetry and by the number of the chiral fermionic species. However there is another parameter } the bare mass m of He atom. On one hand it describes the most fundamental property of the underlying microscopic physics of interacting `indivisiblea particles } the He atoms. On the other hand this parameter does not enter the relativistic spectrum of quasiparticles in the low-energy corner and thus cannot determine the fully relativistic behavior of the e!ective theory. This means that the terms in the e!ective action, which contain this parameters m, are bad. The reason, why the mass m enters the BCS theory (which is also phenomenological, but on a higher energy level) is the Galilean invariance of the underlying system of He atoms. The Galilean invariance requires that the kinetic energy of super#ow at ¹"0 must be (1/2)mn*, i.e. the  bare mass m must necessarily be incorporated into the BCS scheme to maintain the Galilean invariance. It is incorporated within the Landau theory of Fermi liquid, where the dressing of the bare particle occurs due to quasiparticles interaction. In the simpli"ed approach one can consider only that part of interaction which is responsible for the renormalization of the quasiparticle mass and which restores the Galilean invariance of the Fermi system. It is the current}current interaction with the Landau parameter F "3(mH/m!1), which relates the initial mass m and the  renormalized mass mH of quasiparticles. This is how the bare mass m enters the Fermi-liquid phenomenology and then the BCS theory. In the real He liquid the ratio mH/m varies between about 3 and 6 depending on pressure. However, in the modi"ed BCS theory this ratio can be considered as a free parameter, which one can adjust to make the system more close to the relativistic theories in the low-energy corner. As we discussed in Section 2.8, the super#uidity is bad for simulation of the relativistic e!ective theory. That is why it must be suppressed for the Einstein curvature term to prevail in the e!ective action for gravity. This must happen if mPR. In this limit the super#uid properties of the liquid are really suppressed, since the super#uid velocity is inversely proportional to m according to Eq. (102), * J1/m. In this limit of heavy mass of atoms comprising the vacuum, the vacuum becomes inert,  and the bad kinetic energy of super#ow (1/2)mn*, which is dominating over the Einstein action in  real He-A, vanishes as 1/m. Thus we can expect that in case of inert vacuum, the in#uence of the microscopic level on the e!ective theory of gauge "eld and gravity is suppressed and the e!ective action for the collective modes approaches the covariant and gauge invariant limit of the Einstein}Maxwell action. For illustration that this does really work in the right direction, let us consider how all these four parameters,  (0), p , v "p /mH, and m, enter the gradient energy in Eqs. (380)}(382)  $ $ $ in the low-temperature limit ¹; , and what happens when mPR. According to Cross  [212] one has the following expression for the coe$cients in the gradient expansion at low ¹. The normal (and thus super#uid) component densities are given by Eq. (11), which for the

G.E. Volovik / Physics Reports 351 (2001) 195}348

335

He-A quasiparticles are mH ¹ n + n , n "n , n "n!n , n "n!n , , , , , , , m ,  

(383)

mH 7 ¹ n " n , n " n , n "n!n , n "n!n . (384) , , , , , , , m 15   Here we introduced (with index 0) the bare (nonrenormalized) values of the normal and super#uid component densities, which correspond to the limit of noninteracting Fermi gas with F "0 and  thus to mH"m. In the derivation the small terms of order of the anisotropy parameter c /c &10\ have been neglected, so that the particle density is the same in normal and super#uid , , states: n"p /3. The other parameters are according to Cross [212] $ 1 n 1 , , C !C" n , C" n (385)  2m , n 2m , , 1 n , (386) K"  32mH ,








 
    

mH n n 1 n #4n #3 !1 , , K" ,  96mH , m n

,

(387)

mH n n 1 2n # !1 , , #Log , K " ,
32mH m n

(388)

d (lK ) p( )  1 RfT Log" n 1#2 . (389) dM 4 (lK ;p( ) 4mH RE  Here Log is the term which contains ln( /¹). The integral in Eq. (389) is over the solid angle in  momentum space; the energy spectrum which enters the equilibrium quasiparticle distribution function fT "(1#exp E/¹)\ is E"M# ( p( ;lK ) as is given by Eq. (97). This coe$cient  (Log) does not depend on the microscopic parameter m and represents the logarithmically diverging coupling constant in the Maxwell e!ective action for the magnetic "eld in curved space in Eq. (126). 13.2. Ewective action in inert vacuum In the limit mPR all the bad terms related to super#uidity vanish since v J1/m. Taking into  account that in this limit n "n one obtains for the remaining lK -terms: , 1 1 n (
) lK )# (n #n )(lK ) (
;lK )) F (mPR)" , , * 96mH , 32mH




#



1 n #Log (lK ;(
;lK )) . 32mH ,

(390)

336

G.E. Volovik / Physics Reports 351 (2001) 195}348

All three terms have the correspondence in QED and Einstein gravity. We have already seen that the bend term, i.e. with (lK ;(
;lK )), is exactly the energy of the magnetic "eld in curved space with the logarithmically diverging coupling constant coming from the polarization of the vacuum of massless `electricallya charged fermions. Let us now consider the twist term, i.e. (lK ) (
;lK )) and show that it corresponds to the Einstein action. 13.2.1. Einstein action in He-A The bad part of the e!ective gravity, which is simulated by the super#uid velocity "eld, vanishes in the limit of inert vacuum. The remaining part of gravitational "eld is simulated by the inhomogeneity of the lK "eld, which plays the part of the `Kasner axisa in the metric 1 , (391) gGH"c lK GlK H#c (GH!lK GlK H), g"!1, gG"0, (!g" , , c c , , 1 1 (392) g " lK GlK H# (GH!lK GlK H), g "!1, g "0 .  G GH c c , , The curvature of the space with this metric is caused by spatial rotation of the `Kasner axisa lK . For the stationary metric, R lK "0, one obtains that in terms of the lK -"eld the Einstein action is R 1 1 c p $ ((lK ) (
;lK )) . ! (!gR" 1! , (393) 16G 32G c mH  , It has the structure of the twist term in the gradient energy (387) obtained in gradient expansion. Moreover in the inert vacuum limit the twist term in Eq. (390) has the same dependence on p and $ mH as the Einstein action:









 

 

¹ p 1 2 $ ((lK ) (
;lK )) . ! (394) F "   288   mH  We thus can identify this twist term with the Einstein action and extract the gravitational constant G. Neglecting the small anisotropy factor c /c one obtains that the Newton constant in the , , e!ective gravity of the `improveda He-A is  2 G\"  ! ¹ . 9  9

(395)

While the temperature-independent part certainly depends on the details of the trans-Planckian physics (since it contains the second `Plancka energy scale  ), the temperature dependence of the  Newton constant pretends to be universal, since it does not depend on the parameters of the system. If one applies the regularization scheme provided by the trans-Planckian physics of He to the relativistic system one would suggest the following temperature dependence of Newton constant in the vacuum with N Weyl fermions (N "2 for He-A): $ $  (396) [G\]"! N ¹ . 18 $ In conclusion, in the limit of the inert vacuum the action for the metric in Eq. (393) has the general-relativistic curvature form. The example of improved He-A also shows what must be the

G.E. Volovik / Physics Reports 351 (2001) 195}348

337

variation procedure for the theories of gravity, where not all components of metric are independent. The He-A teaches that "rst the metric which enters the Einstein action must be expressed in terms of the independent observables (in He-A it is the lK vector "eld); and after that the action must be varied with regard to these observables. 13.2.2. Violation of gauge invariance Let us "nally consider the splay term (
) lK ) in Eq. (390). It has only the fourth-order temperature corrections, ¹/ . This term has similar coe$cient as the curvature term, but it is not contained in  Einstein action, since it cannot be written in the covariant form. The structure of this term can be, however, obtained using the gauge "eld presentation of the lK vector, where A"p lK . It is known that $ similar term can be obtained in the renormalization of QED in the leading order of 1/N theory if the regularization is made by introducing the momentum cuto! [213]: 1 (R A ) . ¸ " /#" 96 I I

(397)

This term violates the gauge invariance and does not appear in the dimensional regularization scheme [214], but it appears in the momentum cuto! procedure, which violates the gauge invariance. Being written in covariant form Eq. (397) can be applied to He-A, where A"p lK and $ (!g"Const: 1 p c c 1 p 1 $ , (
) lK )" , $ (
) lK ) . (!g(R (gIJA ))" (398) ¸ " I J /#" 96 96 c c 96 mH , , This term corresponds to the splay term in Eq. (390), but it contains an extra big factor of the vacuum anisotropy c /c : F "(c /c )¸ . However, in the isotropic case, where , , *   , , /#" c "c , they exactly coincide. This suggests that the regularization provided by the `trans, , Planckian physicsa of He-A represents the anisotropic version of the momentum cuto! regularization of quantum electrodynamics.

14. Discussion Let us summarize the parallels between the quantum vacuum and super#uids, which were touched upon in the review, and their possible in#uence on the quantum "eld theory. Super#uid He-A and other possible representatives of its universality class provide an example of how the chirality, Weyl fermions, gauge "elds and gravity can emergently appear in the low-energy corner, together with the corresponding symmetries which include the Lorentz symmetry and local S;(N) symmetry. This supports the `anti-grand-uni"cationa idea that the quantum "eld theory, such as Standard Model or maybe GUT, is an e!ective theory, which is applicable only in the infrared limit. Most of the symmetries of this e!ective theory are the attributes of the theory: the symmetries gradually appear in the low-energy corner together with the e!ective theory itself. The momentum-space topology of the fermionic vacuum (Section 4) is instrumental in determination of the universality class of the system. It provides the topological stability of the

338

G.E. Volovik / Physics Reports 351 (2001) 195}348

low-energy properties of the systems of given class: the character of the fermionic spectrum, collective modes and leading symmetries. The universality class, which contains topologically stable Fermi points, is common for super#uid He-A and Standard Model. This allowed us to provide analogies between many phenomena in the two systems, which have the same physics but in many cases are expressed in di!erent languages and can be visualized in terms of di!erent observables. However, in the low-energy corner they are described by the same equations if they are written in the covariant and gauge invariant form. On this topological ground it appears that some of the uni"cation schemes of the strong and electroweak interactions is more preferable than the others: this is the S;(4) ;S;(2) ;S;(2) group of the Pati}Salam model (Section 5.2). ! * 0 The advantage of He-A is that this system is complete being described by the BCS model. This scheme incorporates not only the `relativistica infrared regime, but also several successive scales of the short-distance physics, which correspond to di!erent ultraviolet `trans-Planckiana ranges of high energy. Since in BCS scheme there is no need for a cuto! imposed by hand, all subtle issues of the cuto! in quantum "eld theory can be resolved on physical grounds. It appears, however, the He-A is not a perfect object for the realization of the completely covariant e!ective gauge and gravity "elds at low energy, because the e!ect of the `transPlanckiana physics shows up even in the low-energy corner. This in particular leads to many noncovariant terms in the e!ective action, including the mass of the graviton. So far there is no condensed matter system in nature where all the symmetries of high energy physics and general relativity are reproduced exactly. The natural question is: are there any guiding principles to localize a `perfecta condensed matter? He-A has a hierarchy of `Plancka energy scales. While the Lorentz invariance is violated already at the "rst (lowest) `Plancka scale, the cuto! for the divergent integrals over the fermions in BCS scheme is mostly provided by the higher Planck scales. As a result, for the most terms in the e!ective action the main contribution comes from the energy region where the fermions are `nonrelativistica, that is why these terms are nonrelativistic. It is clear how to `correcta the He-A: one must somehow interchange the Planck scales. The Lorentz invariance must be extended far above that Planck scale, which provides the e!ective cuto!. This will kill the noncovariant nonrenormalizable terms. The killing is however never complete, the `nonrenormalizablea terms will always remain in e!ective theory. But these remaining terms become small in the low-energy limit, since they contain the Planck energy cuto! E in denominator. One example of such `nonrenormalizablea . term in He-A, which is the remnant of the `trans-Planckiana physics, corresponds to the mass term for the ;(1) gauge "eld of the hyperphoton which violates the gauge invariance 7 (Section 8.2.4). The condensed matter of the Fermi-point universality class shows one of possibly many routes from the low-energy `relativistica to high energy `trans-Planckiana physics. Of course, one might expect the many routes to high energy, since the systems of the same universality class become similar only in the vicinity of the "xed point: they can diverge far from each other at higher energies. Nevertheless, the "rst nonrenormalizable corrections could be also universal except for the magnitude of the Planck energy scale. Practically in all condensed matter systems (even of di!erent universality classes), the e!ective action for some bosonic or even fermionic modes acquires an e!ective Lorentzian metric. That is why the gravity is the "eld, which can be simulated most easily in condensed matter. The gravity can be simulated by #owing normal #uids, super#uids, and Bose}Einstein condensates; by elastic

G.E. Volovik / Physics Reports 351 (2001) 195}348

339

strains, dislocations and disclinations in crystals, etc. Though the full dynamical realization of gravity takes place only in the fermionic condensed matter with Fermi points, the acoustic type of gravity is also useful for simulation of di!erent phenomena related to marriage of gravity and quantum theory. The analog of gravity in super#uids shows the possible way how to solve the cosmological constant problem without having to invoke supersymmetry or any "ne tuning. The naive calculations of the energy density of super#uid ground state in the framework of the e!ective theory suggests that it is of the order of E, if one translates it into the language of the relativistic theories. . The standard "eld theoretical estimation gives the same huge value for the vacuum energy and thus for the cosmological constant, which is in severe contradiction with the real life. However, an exact treatment of the trans-Planckian physics in super#uids gives an exact nulli"cation of the appropriate vacuum energy density of the liquid, which enters the Lagrangian at ¹"0 (Section 2.7). This follows from the stability analysis of the ground state of the isolated super#uid liquid, which is certainly beyond the e!ective theory. Applying this to the relativistic quantum "eld theory one may conclude that the equilibrium vacuum does not gravitate. Such conclusion cannot be obtained within the e!ective theory, while in the underlying microscopic physics this result of the complete nulli"cation does not depend on the microscopic details. It is universal in terms of the transPlanckian physics, but not in terms of the e!ective theory. But what happens if the phase transition occurs in which the symmetry of the vacuum is broken, as is supposed to happen in early Universe when, say, the electroweak symmetry was broken? In the e!ective theory, such transition must be accompanied by the change of the vacuum energy, which means that the vacuum has a huge energy either above or below the transition. However, in exact microscopic theory of liquid the phase transition does not disturb the zero value of the vacuum energy (Section 10.3), if the liquid remains in equilibrium. The energy change is completely compensated by the change of the chemical potential of the underlying atoms of the liquid which comprise the vacuum state } the quantity which is not known in e!ective theory. Moreover, the analogy with the super#uids also shows that in nonequilibrium case or at nonzero ¹ (Sections 3.6 and 10.3) the e!ective vacuum energy must be of order of the energy density of matter. This is in agreement with the modern astrophysical observations. Another object, where as in the problem of gravitating vacuum the marriage of gravity and quantum theory is important, is the black hole. Having many objects for simulation of gravity, we can expect in the nearest future that the analogs of event horizon could be constructed in the laboratory. Most probably this will "rst happen in the laser trapped Bose condensates [215]. The condensed matter analogs of horizons may exhibit Hawking radiation, but in addition the other, unexpected, e!ects related to quantum vacuum could arise, such as instability experienced by vacuum in the acoustic model of gravity. Because the short-distance physics is explicitly known in condensed matter this helps to clarify the problem related to the vacuum in the presence of the horizon, or in the other exotic e!ective metric, such as the degenerate metric (Section 11.1). At the moment only one of the exotic metrics has been experimentally simulated. This is the metric induced by spinning cosmic string, which produces the analog of the gravitational Aharonov}Bohm e!ect, experienced by particles in the presence of such string. This type of Aharonov}Bohm e!ect has been experimentally con"rmed in super#uids by measurement of the Iordanskii force acting on quantized vortices (Section 11.4).

340

G.E. Volovik / Physics Reports 351 (2001) 195}348

As for the other (nongravitational) analogies, the most interesting are related to the interplay between the vacuum and the matter, and which can be fully investigated in condensed matter, because of the absence of the cuto! problem. These are the anomalies, which are at the origin of the exchange of the fermionic charges between the vacuum and the matter. Such anomalies are the attributes of the Fermi systems of universality class of Fermi points: Standard Model and He-A. The spectral #ow from the vacuum to the matter, which carries the fermionic charge from the vacuum to matter, occurs just through the Fermi point. Since in the vicinity of the Fermi point the equations are the same for the two Fermi systems, the spectral #ow in both systems is described by the same Adler, and Bell and Jackiw equation for axial anomaly [85,86] (Section 7). The He-A provided the "rst experimental prove for the anomalous nucleation of the fermionic charge from the vacuum. The Adler}Bell}Jackiw equation was con"rmed up to the numerical value of the factor 1/4 with the precision of few percent. The anomalous nucleation of the baryonic or leptonic charge is in the basis of the modern theories of the baryogenesis. The modi"ed equation is obtained for the nucleation of the fermionic charge by the moving string } the quantized vortex. The transfer of the fermionic charge from the `vacuuma to `mattera is mediated by the fermion zero modes living on vortices. This condensed matter illustration of the cancellation of anomalies in 1#1 and 3#1 systems (the Callan}Harvey e!ect) has been experimentally veri"ed in He-B. The relativistic counterpart of this phenomenon is the baryogenesis by cosmic strings, which thus has been experimentally probed. The other e!ect related to axial anomaly } the helical instability of the super#uid/normal counter#ow in He-A } is also described by the same physics and by the same equations as the formation of the (hyper) magnetic "eld due to the helical instability experienced by the vacuum in the presence of the heat bath of the righthanded electrons (Section 8.3). That is why its experimental observation in He-A provided an experimental support for the Joyce}Shaposhnikov scenario of the genesis of the primordial magnetic "eld. In a future the macroscopic parity violating e!ect suggested by Vilenkin [110] must be simulated in He-A (Section 8.4). In both systems it is described by the same mixed axialgravitational Chern}Simons action. One may expect that the further theoretical and experimental exploration of the vacuum/ condensed matter analogies will clarify the properties of the quantum vacuum. References [1] C.D. Frogatt, H.B. Nielsen, Origin of Symmetry, World Scienti"c, Singapore, New Jersey, London, Hong Kong, 1991. [2] S. Chadha, H.B. Nielsen, Lorentz invariance as a low-energy phenomenon, Nucl. Phys. B 217 (1983) 125}144. [3] S. Weinberg, What is quantum "eld theory, and what did we think it is?, in: T.Y. Cao (Ed.), Conceptual Foundations of Quantum Field Theory, Cambridge University Press, Cambridge, 1999, pp. 241}251, hepth/9702027. [4] F. Jegerlehner, The `ether-worlda and elementary particles, in: H. Dorn, G. Weigt (Eds.), 31st International Ahrenshoop Symposium on the Theory of Elementary Particles, Buckow, Wiley-VCH, Berlin, 1998, p. 496, hep-th/9803021. [5] G.E. Volovik, Field theory in super#uid He: what are the lessons for particle physics, gravity and hightemperature superconductivity? Proc. Natl. Acad. Sci. USA 96 (1999) 6042}6047, cond-mat/9812381; G.E. Volovik, He and Universe parallelism, in: Yu.M. Bunkov, H. Godfrin (Eds.), Topological Defects and the Non-Equilibrium Dynamics of Symmetry Breaking Phase Transitions, Kluwer, Academic Publishers, Dordrecht, 2000, pp. 353}387, cond-mat/9902171.

G.E. Volovik / Physics Reports 351 (2001) 195}348

341

[6] C. Rovelli, Notes for a brief history of quantum gravity, gr-qc/0006061. [7] R.B. Laughlin, D. Pines, The theory of everything, Proc. Natl. Acad. Sci. USA 97 (2000) 28}31. [8] B.L. Hu, General relativity as geometro-hydrodynamics, Expanded version of an invited talk at the Second International Sakharov Conference on Physics, Moscow, 20}23 May 1996, gr-qc/9607070. [9] T. Padmanabhan, in: N. Dadhich, A. Kembhavi (Eds.), Conceptual Issues in Combining General Relativity and Quantum Theory, Festschrift Volume to be Brought out in Honour of Professor J.V. Narlikar, Kluwer, Dordrecht, 1999, hep-th/9812018. [10] G.E. Volovik, Axial anomaly in He-A: simulation of baryogenesis and generation of primordial magnetic "eld in Manchester and Helsinki, Physica B 255 (1998) 86}107. [11] I.M. Khalatnikov, An Introduction to the Theory of Super#uidity, Benjamin, New York, 1965. [12] S.P. Novikov, The Hamiltonian formalism and a multivalued analog of Morse theory, Usp. Mat. Nauk 37 (1982) 3}49 (in Russian). [13] G.E. Volovik, Poisson brackets scheme for vortex dynamics in super#uids and superconductors and e!ect of band structure of crystal, JETP Lett. 64 (1996) 845}852. [14] I.E. Dzyaloshinskii, G.E. Volovick, Poisson brackets in condensed matter, Ann. Phys. 125 (1980) 67}97. [15] G.E. Volovik, Linear momentum in ferromagnets, J. Phys. C 20 (1987) L83}L87; Wess-Zumino action for the orbital dynamics of He-A, JETP Lett. 44 (1986) 185}189. [16] D. Vollhardt, P. WoK l#e, The Super#uid Phases of Helium 3, Taylor & Francis, London, New York, Philadelphia, 1990. [17] W.G. Unruh, Experimental black-hole evaporation?, Phys. Rev. Lett. 46 (1981) 1351}1354; Sonic analogue of black holes and the e!ects of high frequencies on black hole evaporation, Phys. Rev. D 51 (1995) 2827}2838. [18] M. Visser, Acoustic black holes: horizons, ergospheres, and Hawking radiation, Class. Quantum Grav. 15 (1998) 1767}1791. [19] M. Stone, Iordanskii force and the gravitational Aharonov}Bohm e!ect for a moving vortex, Phys. Rev. B 61 (2000) 11780}11786. [20] M. Visser, Lorentzian Wormholes. From Einstein to Hawking, AIP Press, Woodbury, New York, 1995. [21] C.W. Woo, Microscopic calculations for condensed phases of helium, in: K.H. Bennemann, J.B. Ketterson (Eds.), The Physics of Liquid and Solid Helium, Part I, Wiley, New York, 1976. [22] S. Weinberg, The cosmological constant problem, Rev. Mod. Phys. 61 (1989) 1}23. [23] A.G. Riess, A.V. Filippenko, M.C. Liu, P. Challis, A. Clocchiatti, A. Diercks, P.M. Garnavich, C.J. Hogan, S. Jha, R.P. Kirshner, B. Leibundgut, M.M. Phillips, D. Reiss, B.P. Schmidt, R.A. Schommer, R.C. Smith, J. Spyromilio, C. Stubbs, N.B. Suntze!, J. Tonry, P. Woudt, R.J. Brunner, A. Dey, R. Gal, J. Graham, J. Larkin, S.C. Odewahn, B. Oppenheimer, Tests of the accelerating universe with near-infrared observations of a high-redshift type Ia supernova, astro-ph/0001384. [24] A.D. Sakharov, Vacuum quantum #uctuations in curved space and the theory of gravitation, Dokl. Akad. Nauk 177 (1967) 70}71 [Sov. Phys. Dokl. 12 (1968) 1040}1041]. [25] V. Frolov, D. Fursaev, Thermal "elds, entropy, and black holes, Class. Quant. Grav. 15 (1998) 2041}2074. [26] G.E. Volovik, Energy-momentum tensor of quasiparticles in the e!ective gravity in super#uids, gr-qc/9809081. [27] U.R. Fischer, G.E. Volovik, Thermal quasi-equilibrium states across Landau horizons in the e!ective gravity of super#uids, Int. J. Mod. Phys. D 10 (2001) 57}88, gr-qc/0003017. [28] M. Stone, Acoustic energy and momentum in a moving medium, Phys. Rev. B 62 (2000) 1341}1350. [29] L.D. Landau, E.M. Lifshitz, Classical Fields, Pergamon Press, Oxford, 1975. [30] R.C. Tolman, Relativity, Thermodynamics and Cosmology, Clarendon Press, Oxford, 1934. [31] G.E. Volovik, V.P. Mineev, Current in super#uid Fermi liquids and the vortex core structure, Sov. Phys. JETP 56 (1982) 579}586. [32] P.G. Grinevich, G.E. Volovik, Topology of gap nodes in super#uid He, J. Low Temp. Phys. 72 (1988) 371}380. [33] J.M. Luttinger, Fermi surface and some simple equilibrium properties of a system of interacting fermions, Phys. Rev. 119 (1960) 1153}1163. [34] M. Oshikawa, Topological approach to Luttinger's theorem and the Fermi surface of a Kondo lattice, Phys. Rev. Lett. 84 (2000) 3370}3373. [35] G.E. Volovik, A new class of normal Fermi liquids, JETP Lett. 53 (1991) 222}225.

342

G.E. Volovik / Physics Reports 351 (2001) 195}348

[36] K.B. Blagoev, K.S. Bedell, Luttinger theorem in one dimensional metals, Phys. Rev. Lett. 79 (1997) 1106}1109. [37] X.G. Wen, Metallic non-Fermi-liquid "xed point in two and higher dimensions, Phys. Rev. B 42 (1990) 6623}6630. [38] H.J. Schulz, G. Cuniberti, P. Pieri, Fermi liquids and Luttinger liquids, in: G. Morandi et al. (Eds.), Field Theories for Low-Dimensional Condensed Matter Systems, Springer, Solid State Sciences Series, 2000, cond-mat/9807366. [39] V.M. Yakovenko, Metals in a high magnetic "eld: a universality class of marginal Fermi liquid, Phys. Rev. B 47 (1993) 8851}8857. [40] Y. Nambu, G. Jona-Lasinio, Dynamical model of elementary particles based on an analogy with superconductivity. I, Phys. Rev. 122 (1961) 345}358; Dynamical model of elementary particles based on an analogy with superconductivity. II, Phys. Rev. 124 (1961) 246}254. [41] M. Alford, K. Rajagopal, F. Wilczek, QCD at "nite baryon density: nucleon droplets and color superconductivity, Phys. Lett. B 422 (1998) 247}256; F. Wilczek, From notes to chords in QCD, Nucl. Phys. A 642 (1998) 1}13. [42] V.M.H. Ruutu, V.B. Eltsov, A.J. Gill, T.W.B. Kibble, M. Krusius, Yu.G. Makhlin, B. Placais, G.E. Volovik, Wen Xu, Vortex formation in neutron-irradiated super#uid He as an analogue of cosmological defect formation, Nature 382 (1996) 334}336. [43] V.B. Eltsov, T.W.B. Kibble, M. Krusius, V.M.H. Ruutu, G.E. Volovik, Composite defect extends analogy between cosmology and He, Phys. Rev. Lett. 85 (2000) 4739}4742. [44] T.W.B. Kibble, Topology of cosmic domains and strings, J. Phys. A 9 (1976) 1387}1398. [45] T.D.C. Bevan, A.J. Manninen, J.B. Cook, J.R. Hook, H.E. Hall, T. Vachaspati, G.E. Volovik, Momentogenesis by He vortices: an experimental analogue of primordial baryogenesis, Nature 386 (1997) 689}692. [46] J.R. Hook, A.J. Manninen, J.B. Cook, H.E. Hall, Vortex mutual friction in rotating super#uid He, Czech. J. Phys. 46 (Suppl. S6) (1996) 2930}2936. [47] H.B. Nielsen, M. Ninomiya, Absence of neutrinos on a lattice. I } Proof by homotopy theory, Nucl. Phys. B 185 (1981) 20 [Erratum } Nucl. Phys. B 195 (1982) 541]; Absence of neutrinos on a lattice. 2 } Intuitive homotopy proof, Nucl. Phys. B 193 (1981) 173. [48] G.E. Volovik, Exotic Properties of Super#uid He, World Scienti"c, Singapore, New Jersey, London, Hong Kong, 1992. [49] G.E. Volovik, V.M. Yakovenko, Fractional charge, spin and statistics of solitons in super#uid He "lm, J. Phys.: Condens. Matter 1 (1989) 5263}5274. [50] G.E. Volovik, On edge states in superconductor with time inversion symmetry breaking, JETP Lett. 66 (1997) 522}527. [51] K. Ishikawa, T. Matsuyama, Magnetic "eld induced multi component QED in three-dimensions and quantum Hall e!ect, Z. Phys. C 33 (1986) 41}45; A microscopic theory of the quantum Hall e!ect, Nucl. Phys. B 280 (1987) 523}548. [52] M. Rice, Superconductivity: an analogue of super#uid He, Nature 396 (1998) 627}629. [53] K. Ishida, H. Mukuda, Y. Kitaoka et al., Spin-triplet superconductivity in Sr RuO identi"ed by O Knight shift,   Nature 396 (1998) 658}660. [54] T. Senthil, J.B. Marston, M.P.A. Fisher, Spin quantum Hall e!ect in unconventional superconductors, Phys. Rev. B 60 (1999) 4245}4254. [55] N. Read, D. Green, Paired states of fermions in two dimensions with breaking of parity and time-reversal symmetries, and the fractional quantum Hall e!ect, Phys. Rev. B 61 (2000) 10267}10297. [56] V.M. Yakovenko, Spin, statistics and charge of solitons in (2#1)-dimensional theories, Fizika (Zagreb) 21 (Suppl. 3) (1989) 231 [cond-mat/9703195]. [57] G.E. Volovik, Momentum-space topology of Standard Model, J. Low Temp. Phys. 119 (2000) 241}247. [58] G.E. Volovik, T. Vachaspati, Aspects of He and the standard electroweak model, Int. J. Mod. Phys. B 10 (1996) 471}521. [59] S. Ying, The quantum aspects of relativistic fermion systems with particle condensation, Ann. Phys. 266 (1998) 295}350; On the local "nite density relativistic quantum "eld theories, hep-th/9802044. [60] N.D. Mermin, T.L. Ho, Circulation and angular momentum in the A phase of super#uid He, Phys. Rev. Lett. 36 (1976) 594}597. [61] A.A. Abrikosov, Quantum magnetoresistance, Phys. Rev. B 58 (1998) 2788}2794.

G.E. Volovik / Physics Reports 351 (2001) 195}348

343

[62] J.C. Pati, A. Salam, Is baryon number conserved?, Phys. Rev. Lett. 31 (1973) 661}664; Lepton number as the fourth color, Phys. Rev. D 10 (1974) 275}289. [63] R. Foot, H. Lew, R.R. Volkas, Models of extended Pati}Salam gauge symmetry, Phys. Rev. D 44 (1991) 859}864. [64] J.C. Pati, Discovery of proton decay: a must for theory, a challenge for experiment, hep-ph/0005095. [65] H. Terazawa, High energy physics in the 21-st century, KEK Preprint 99-46, July 1999, Proceedings of 22nd International Workshop on the Fundamental Problems of High Energy Physics and Field Theory, Protvino, Moscow Region, Russia, 23}25 June 1999, to be published. [66] P.A. Marchetti, Zhao-Bin Su, Lu Yu, Dimensional reduction of ;(1);S;(2) Chern}Simons bosonization: application to the t!J model, Nucl. Phys. B 482 (1996) 731}757 and references therein. [67] C.D. Froggatt, H.B. Nielsen, Why do we have parity violation? in: N.M. Borstnik, H.B. Nielsen, C. Froggatt (Eds.), Proceedings, What comes beyond the Standard Model, Ljubljana, Slovenia, DMFA, 1999, hep-ph/9906466. [68] T.M.P. Tait, Signals for the electroweak symmetry breaking associated with the top quark, Ph.D. Thesis, hep-ph/9907462. [69] Ya.B. Zel'dovich, Interpretation of electrodynamics as a consequence of quantum theory, JETP Lett. 6 (1967) 345}347. [70] P.D. Mannheim, Implications of cosmic repulsion for gravitational theory, Phys. Rev. D 58 (1998) 1}12, 103511. [71] A. Edery, M.B. Paranjape, Classical tests for Weyl gravity: de#ection of light and time delay, Phys. Rev. D 58 (1998) 1}8, 024011. [72] P.D. Mannheim, Cosmic acceleration and a natural solution to the cosmological constant problem, gr-qc/9903005. [73] A.J. Leggett, S. Takagi, Orientational dynamics of super#uid He: a `Two-Fluida model. II. Orbital dynamics, Ann. Phys. 110 (1978) 353}406. [74] G.E. Volovik, Analog of gravitation in super#uid He-A, JETP Lett. 44 (1986) 498}501. [75] I am indebted to V.N. Gribov, who explained to me this point. [76] C.P. Martin, J.M. Gracia-Bondia, J.S. Varilly, The Standard Model as a noncommutative geometry: the low-energy regime, Phys. Rep. 294 (1998) 363}406. [77] I.S. Sogami, Generalized covariant derivative with gauge and Higgs "elds in the Standard Model, Prog. Theor. Phys. 94 (1995) 117}123; Minimal S;(5) Grand uni"ed theory based on generalized covariant derivative with gauge and Higgs "elds, Prog. Theor. Phys. 95 (1996) 637}655. [78] H. Harari, N. Seiberg, Generation labels in composite models for quarks and leptons, Phys. Lett. B 102 (1981) 263}266. [79] S.L. Adler, Fermion-sector frustrated S;(4) as a preonic precursors of the Standard Model, Int. J. Mod. Phys. A 14 (1999) 1911}1934. [80] R.D. Peccei, Discrete and global symmetries in particle physics, in: L. Mathelitsch, W. Plessas (Eds.), Proceedings Broken Symmetries, Lecture Notes in Physics, Vol. 521, Springer, Berlin, Germany, 1999, hep-ph/9807516. [81] J. Iliopoulus, D.V. Nanopoulus, T.N. Tomaras, Infrared stability or anti-granduni"cation, Phys. Lett. B 55 (1980) 141}144. [82] G.E. Volovik, M.V. Khazan, Dynamics of the A-phase of He at low pressure, Sov. Phys. JETP 55 (1982) 867}871. [83] Y. Nambu, Fermion-boson relations in the BCS-type theories, Physica D 15 (1985) 147}151. [84] S.M. Troshin, N.E. Tyurin, Hyperon polarization in the constituent quark model, Phys. Rev. D 55 (1997) 1265}1272. [85] S. Adler, Axial-vector vertex in spinor electrodynamics, Phys. Rev. 177 (1969) 2426}2438. [86] J.S. Bell, R. Jackiw, A PCAC puzzle:  P  in the  model, Nuovo Cimento A 60 (1969) 47}61.  [87] M. Trodden, Electroweak baryogenesis, Rev. Mod. Phys. 71 (1999) 1463}1500. [88] T. Vachaspati, G.B. Field, Electroweak string con"gurations with baron number, Phys. Rev. Lett. 73 (1994) 373}376; 74 (1995) 1258(E). [89] J. Garriga, T. Vachaspati, Zero modes on linked strings, Nucl. Phys. B 438 (1995) 161}181. [90] M. Barriola, Electroweak strings produce baryons, Phys. Rev. D 51 (1995) 300}304. [91] G.E. Volovik, V.P. Mineev, He-A vs Bose liquid: orbital angular momentum and orbital dynamics, Sov. Phys. JETP 54 (1981) 524}530.

344

G.E. Volovik / Physics Reports 351 (2001) 195}348

[92] R. Combescot, T. Dombre, Twisting in super#uid He-A and consequences for hydrodynamics at ¹"0, Phys. Rev. B 33 (1986) 79}90. [93] G.E. Volovik, Chiral anomaly and the law of conservation of momentum in He-A, JETP Lett. 43 (1986) 551}554. [94] A. Achucarro, T. Vachaspati, Semilocal and electroweak strings, Phys. Rep. 327 (2000) 347}426. [95] V.R. Chechetkin, Types of vortex solutions in super#uid He, Sov. Phys. JETP 44 (1976) 766}772. [96] P.W. Anderson, G. Toulouse, Phase slippage without vortex cores: vortex textures in super#uid He, Phys. Rev. Lett. 38 (1977) 508}511. [97] N.B. Kopnin, Mutual friction in super#uid He. II. Continuous vortices in He-A at low temperatures, Phys. Rev. B 47 (1993) 14354}14363. [98] G.E. Volovik, Hydrodynamic action for orbital and super#uid dynamics of He-A at ¹"0, JETP 75 (1992) 990}997. [99] G.E. Volovik, Action for anomaly in fermi super#uids: quantized vortices and gap nodes, JETP 77 (1993) 435}441. [100] T.D.C. Bevan, A.J. Manninen, J.B. Cook, H. Alles, J.R. Hook, H.E. Hall, Vortex mutual friction in super#uid He vortices, J. Low Temp. Phys. 109 (1997) 423}459. [101] C.G. Callan Jr., J.A. Harvey, Anomalies and fermion zero modes on strings and domain walls, Nucl. Phys. B 250 (1985) 427. [102] G.E. Volovik, Vortex motion in fermi super#uids and Callan}Harvey e!ect, JETP Lett. 57 (1993) 244}248. [103] M. Stone, Spectral #ow, Magnus force, and mutual friction via the geometric optics limit of Andreev re#ection, Phys. Rev. B 54 (1996) 13222}13229. [104] A.J. Leggett, Macroscopic parity nonconservation due to neutral current?, Phys. Rev. Lett. 39 (1977) 587}590. [105] M. Joyce, M. Shaposhnikov, Primordial magnetic "elds, right electrons, and the abelian anomaly, Phys. Rev. Lett. 79 (1997) 1193}1196. [106] M. Giovannini, E.M. Shaposhnikov, Primordial hypermagnetic "elds and triangle anomaly, Phys. Rev. D 57 (1998) 2186}2206. [107] O. Tornkvist, Cosmic magnetic "elds from particle physics, astro-ph/0004098. [108] V.M.H. Ruutu, J. Kopu, M. Krusius, U. Parts, B. Placais, E.V. Thuneberg, W. Xu, Critical velocity of vortex nucleation in rotating super#uid He-A, Phys. Rev. Lett. 79 (1997) 5058}5061. [109] V.M.H. Ruutu, J. Kopu, M. Krusius, U. Parts, B. Placais, E.V. Thuneberg, W. Xu, Critical velocity of continuous vortex formation in rotating He-A, Czech. J. Phys. 46 (Suppl. S1) (1996) 7}8. [110] A. Vilenkin, Macroscopic parity violating e!ects: neutrino #uxes from rotating black holes and in rotating thermal radiation, Phys. Rev. D 20 (1979) 1807}1812; Quantum "eld theory at "nite temperature in a rotating system D 21 (1980) 2260}2269. [111] G.E. Volovik, A. Vilenkin, Macroscopic parity violating e!ects and He-A, Phys. Rev. D 62 (2000) 025014. [112] T. Kita, Angular momentum of anisotropic super#uids at "nite temperatures, J. Phys. Soc. Japan 67 (1998) 216. [113] UG . Parts, V.M.H. Ruutu, J.H. Koivuniemi, Yu.N. Bunkov, V.V. Dmitriev, M. FogelstroK m, M. Huenber, Y. Kondo, N.B. Kopnin, J.S. Korhonen, M. Krusius, O.V. Lounasmaa, P.I. Soininen, G.E. Volovik, Single-vortex nucleation in rotating super#uid He-B, Europhys. Lett. 31 (1995) 449}454. [114] D. Lynden-Bell, M. Nouri-Zonoz, Classical monopoles: Newton, NUT space, gravomagnetic lensing, and atomic spectra, Rev. Mod. Phys. 70 (1998) 427}446. [115] A. Calogeracos, G.E. Volovik, Rotational quantum friction in super#uids: radiation from object rotating in super#uid vacuum, JETP Lett. 69 (1999) 281}287. [116] P. Muzikar, D. Rainer, Phys. Rev. B 27 (1983) 4243; K. Nagai, J. Low Temp. Phys. 55 (1984) 233; G.E. Volovik, Symmetry in Super#uid He, in: W.P. Halperin, L.P. Pitaevskii (Eds.), Helium Three, Elsevier Science Publishers B.V., Amsterdam, 1990, pp. 27}134. [117] B. Revaz, J.-Y. Genoud, A. Junod, K. Neumaier, A. Erb et al., d-wave scaling relations in the mixed-state speci"c heat of YBa Cu O , Phys. Rev. Lett. 80 (1998) 2267}3364.    [118] A. Vilenkin, Equilibrium parity violating current in a magnetic "eld, Phys. Rev. D 22 (1980) 3080}3084. [119] A.N. Redlich, L.C.R. Wijewardhana, Induced Chern}Simons terms at high temperatures and "nite densities, Phys. Rev. Lett. 54 (1985) 970}973. [120] R. Jackiw, V. Alan KosteleckyH , Radiatively induced Lorentz and CPT violation in electrodynamics, Phys. Rev. Lett. 82 (1999) 3572}3575.

G.E. Volovik / Physics Reports 351 (2001) 195}348

345

[121] A.A. Andrianov, R. Soldati, L. Sorbo, Dynamical Lorentz symmetry breaking from (3#1) axion-Wess}Zumino model, Phys. Rev. D 59 (1999) 025002. [122] A. Vilenkin, D.A. Leahy, Parity nonconservation and the origin of cosmic magnetic "elds, Astrophys. J. 254 (1982) 77}81. [123] J. Goryo, K. Ishikawa, Phys. Lett. A 260 (1999) 294; G.E. Volovik, Analog of quantum Hall e!ect in super#uid He "lm, Sov. Phys. JETP 67 (1988) 1804}1811. [124] D.A. Ivanov, Non-abelian statistics of half-quantum vortices in p-wave superconductors, Phys. Rev. Lett. 86 (2001) 268}271. [125] E. Witten, Superconducting strings, Nucl. Phys. B 249 (1985) 557}592. [126] G.D. Starkman, T. Vachaspati, Galactic cosmic strings as sources of primary antiprotons, Phys. Rev. D 53 (1996) 6711}6714. [127] C. Caroli, P.G. de Gennes, J. Matricon, Bound fermion states on a vortex line in a type II superconductor, Phys. Lett. 9 (1964) 307}309. [128] S.C. Davis, A.C. Davis, W.B. Perkins, Cosmic string zero modes and multiple phase transitions, Phys. Lett. B 408 (1997) 81}90. [129] N.B. Kopnin, M.M. Salomaa, Mutual friction in super#uid 3He: e!ects of bound states in the vortex core, Phys. Rev. B 44 (1991) 9667}9677. [130] M.M. Salomaa, G.E. Volovik, Quantized vortices in super#uid He, Rev. Mod. Phys. 59 (1987) 533}613. [131] G.E. Volovik, Fermion zero modes on vortices in chiral superconductors, JETP Lett. 70 (1999) 609}614. [132] N.B. Kopnin, G.E. Volovik, Flux-#ow in d-wave superconductors: low temperature universality and scaling, Phys. Rev. Lett. 79 (1997) 1377}1380; N.B. Kopnin, G.E. Volovik, Rotating vortex core: an instrument for detecting the core excitations, Phys. Rev. B 57 (1998) 8526}8531. [133] G.E. Volovik, V.P. Mineev, Linear and point singularities in super#uid He, JETP Lett. 24 (1976) 561}563. [134] V. Geshkenbein, A. Larkin, A. Barone, Vortices with half magnetic #ux quanta in `heavy-fermiona superconductors, Phys. Rev. B 36 (1987) 235}238. [135] J.R. Kirtley, C.C. Tsuei, M. Rupp, Z. Sun, Lock See Yu-Jahnes, A. Gupta, M.B. Ketchen, K.A. Moler, M. Bhushan, Direct imaging of integer and half-integer Josephson vortices in high-T grain boundaries, Phys. Rev. Lett. 76 A (1996) 1336}1339. [136] N. Read, D. Green, Paired states of fermions in two dimensions with breaking of parity and time-reversal symmetries, and the fractional quantum Hall e!ect, Phys. Rev. B 61 (2000) 10267}10297. [137] S. Bravyi, A. Kitaev, Fermionic quantum computation, quant-ph/0003137. [138] Y. Kondo, J.S. Korhonen, M. Krusius, V.V. Dmitriev, Yu.M. Mukharskiy, E.B. Sonin, G.E. Volovik, Observation of the nonaxisymmetric vortex in He-B, Phys. Rev. Lett. 67 (1991) 81}84. [139] N.B. Kopnin, V.E. Kravtsov, Conductivity and Hall e!ect of pure type-II superconductors at low temperatures, JETP Lett. 23 (1976) 578}581. [140] A.F. Andreev, The thermal conductivity of the intermediate state in superconductors, Sov. Phys. JETP 19 (1964) 1228}1231. [141] P.A. Maia Neto, S. Reynaud, Dissipative force on a sphere moving in vacuum, Phys. Rev. A 47 (1993) 1639}1646. [142] C.K. Law, Resonance response of the quantum vacuum to an oscillating boundary, Phys. Rev. Lett. 73 (1994) 1931}1934. [143] M. Kardar, R. Golestanian, The `frictiona of vacuum, and other #uctuation-induced forces, Rev. Mod. Phys. 71 (1999) 1233}1245. [144] S. Yip, A.J. Leggett, Dynamics of the He A-B phase boundary, Phys. Rev. Lett. 57 (1986) 345}348. [145] N.B. Kopnin, Movement of the interface between A and B phases in super#uid helium-3: linear theory, Sov. Phys. JETP 65 (1987) 1187}1192. [146] A.J. Leggett, S. Yip, Nucleation and growth of He-B in the supercooled A-phase, in: W.P. Halperin, L.P. Pitaevskii, (Eds.), Helium Three, Elsevier Science Publishers B.V., Amsterdam, (1990) p. 523. [147] J. Palmeri, Super#uid kinetic equation approach to the dynamics of the He A-B phase boundary, Phys. Rev. B 42 (1990) 4010}4035. [148] M. Bartkowiak, S.W.J. Daley, S.N. Fisher et al., Thermodynamics of the A-B phase transition and the geometry of the A-phase gap nodes, Phys. Rev. Lett. 83 (1999) 3462}3465.

346

G.E. Volovik / Physics Reports 351 (2001) 195}348

[149] D.I. Khomskii, A. Freimuth, Charged vortices in high temperature superconductors, Phys. Rev. Lett. 75 (1995) 1384}1387; G. Blatter, M.V. Feigel'man, V.B. Geshkenbein, A.I. Larkin, A. van Otterlo, Electrostatics of vortices in type-II superconductors, Phys. Rev. Lett. 77 (1996) 566}569; K. Kumagi, K. Nozaki, Y. Matsuda, Charged vortices in high temperature superconductors probed by NMR, Phys. Rev. B, to be published, cond-mat/0012492. [150] A.A. Logunov, Teor. Mat. Fiz. 80 (1989) 165. [151] A.J. Hanson, T. Regge, Torsion and quantum gravity, Proceedings of the Integrative Conference on Group Theory and Mathematical Physics, University of Texas at Austin, 1978; R. d'Auria, T. Regge, Gravity theories with asymptotically #at instantons, Nucl. Phys. B 195 (1982) 308. [152] I. Bengtsson, Degenerate metrics and an empty black hole, Class. Quant. Grav. 8 (1991) 1847}1858. [153] I. Bengtsson, T. Jacobson, Degenerate metric phase boundaries, Class. Quant. Grav. 14 (1997) 3109}3121; Erratum-ibid. 15 (1998) 3941}3942. [154] G.T. Horowitz, Topology change in classical and quantum gravity, Class. Quant. Grav. 8 (1991) 587}602. [155] A. Starobinsky, Plenary talk at Cosmion-99, Moscow, 17}24 October, 1999. [156] M.M. Salomaa, G.E. Volovik, Cosmiclike domain walls in super#uid He-B: instantons and diabolical points in (k, r) space, Phys. Rev. B 37 (1988) 9298}9311; Half-solitons in super#uid He-A: novel /2-quanta of phase slippage, J. Low Temp. Phys. 74 (1989) 319}346. [157] G.E. Volovik, Super#uid He-B and gravity, Physica B 162 (1990) 222}230. [158] M. Matsumoto, M. Sigrist, Quasiparticle states near the surface and the domain wall in a p $ ip -wave V W superconductor, J. Phys. Soc. Japan 68 (1999) 994, cond-mat/9902265. [159] M. Sigrist, D.F. Agterberg, The role of domain walls on the vortex creep dynamics in unconventional superconductors, Prog. Theor. Phys. 102 (1999) 965}981, cond-mat/9910526. [160] G.E. Volovik, On edge states in superconductor with time inversion symmetry breaking, JETP Lett. 66 (1997) 522}527. [161] T.A. Jacobson, G.E. Volovik, E!ective spacetime and Hawking radiation from a moving domain wall in a thin "lm of He-A, JETP Lett. 68 (1998) 874}880. [162] G.E. Volovik, Vierbein walls in condensed matter, JETP Lett. 70 (1999) 711}716. [163] Z.K. Silagadze, TEV scale gravity, mirror universe, and 2dinosaurs, hep-ph/0002255. [164] G.E. Volovik, Monopoles and fractional vortices in chiral superconductors, Proc. Natl. Acad. Sci. USA 97 (2000) 2431}2436. [165] A.S. Schwarz, Field theories with no local conservation of the electric charge, Nucl. Phys. B 208 (1982) 141}158. [166] D.D. Sokolov, A.A. Starobinsky, On the structure of curvature tensor on conical singularities, Dokl. AN SSSR 234 (1977) 1043}1046 [Sov. Phys.-Dokl. 22 (1977) 312]. [167] M. Banados, C. Teitelboim, J. Zanelli, Black hole in three-dimensional space-time, Phys. Rev. Lett. 69 (1992) 1849}1852. [168] J.D. Barrow, Varying G and other constants, Talk given at International School of Astrophysics, D. Chalonge: 6th Course: Current Topics in Astrofundamental Physics: Primordial Cosmology, Erice, Italy, 4}15 September 1997, gr-qc/9711084. [169] P.O. Mazur, Spinning cosmic strings and quantization of energy, Phys. Rev. Lett. 57 (1986) 929}932. [170] P.O. Mazur, Reply to comment on `Spinning Cosmic Strings and Quantization of Energya, hep-th/9611206. [171] A. Staruszkievicz, Gravitation theory in three-dimensional space, Acta Phys. Pol. 24 (1963) 735}740. [172] S. Deser, R. Jackiw, G. t'Hooft, Three-dimensional Einstein gravity: dynamics of #at space, Ann. Phys. 152 (1984) 220}235. [173] R.L. Davis, E.P.S. Shellard, Global string lifetime: never say forever!, Phys. Rev. Lett. 63 (1989) 2021}2024. [174] D. Harari, A.P. Polychronakos, Gravitational time delay due to a spinning string, Phys. Rev. D 38 (1988) 3320}3322. [175] B. Jensen, J. Kuc\ era, On a gravitational Aharonov}Bohm e!ect, J. Math. Phys. 34 (1993) 4975}4985. [176] G.E. Volovik, Three nondissipative forces on a moving vortex line in super#uids and superconductors, JETP Lett. 62 (1995) 65}71. [177] K.-M. Lee, Vortex dynamics in self-dual Maxwell}Higgs systems with a uniform background electric charge density, Phys. Rev. D 49 (1994) 4265}4276.

G.E. Volovik / Physics Reports 351 (2001) 195}348

347

[178] G.E. Volovik, Vortex vs spinning string: Iordanskii force and gravitational Aharonov}Bohm e!ect, JETP Lett. 67 (1998) 881}887. [179] S.V. Iordanskii, Ann. Phys. 29 (1964) 335; Zh. Eksp. Teor. Fis. 49 (1965) 225 [Sov. Phys. JETP 22 (1966) 160]. [180] E.B. Sonin, Friction between the normal component and vortices in rotating super#uid helium, Zh. Eksp. Teor. Fis. 69 (1975) 921}935 [Sov. Phys. JETP 42 (1976) 469}475]. [181] E.B. Sonin, Magnus force in super#uids and superconductors, Phys. Rev. B 55 (1997) 485}501. [182] A.L. Shelankov, Magnetic force exerted by the Aharonov}Bohm line, Europhys. Lett. 43 (1998) 623}628. [183] Y. Aharonov, D. Bohm, Signi"cance of electromagnetic potentials in the quantum theory, Phys. Rev. 115 (1959) 485}491. [184] P.O. Mazur, Mazur replies to comment on `Spinning Cosmic Strings and Quantization of Energya, Phys. Rev. Lett. 59 (1987) 2380. [185] D.V. Gal'tsov, P.S. Letelier, Spinning strings and cosmic dislocations, Phys. Rev. D 47 (1993) 4273}4276. [186] A.L. Fetter, Scattering of sound by a classical vortex, Phys. Rev. 136 (1964) A1488}A1493. [187] E. Demircan, P. Ao, Q. Niu, Interactions of collective excitations with vortices in super#uid systems, Phys. Rev. B 52 (1995) 476}482. [188] R.M. Cleary, Scattering of single-particle excitations by a vortex in a clean type-II superconductor, Phys. Rev. 175 (1968) 587}596. [189] T.D.C. Bevan, A.J. Manninen, J.B. Cook, A.J. Armstrong, J.R. Hook, H.E. Hall, Vortexmutual friction in rotating super#uid He-B, Phys. Rev. Lett. 74 (1995) 750}753. [190] S.W. Hawking, Black hole explosions?, Nature 248 (1974) 30}31. [191] T.A. Jacobson, G.E. Volovik, Event horizons and ergoregions in He, Phys. Rev. D 58 (1998) 064021. [192] K. Martel, E. Poisson, Regular coordinate systems for Schwarzschild and other spherical spacetimes, gr-qc/0001069. [193] S. Liberati, Quantum vacuum e!ects in gravitational "elds: theory and detectability, gr-qc/0009050. [194] S. Liberati, S. Sonego, M. Visser, Unexpectedly large surface gravities for acoustic horizons?, Class. Quant. Grav. 17 (2000) 2903}2923. [195] L.D. Landau, E.M. Lifshitz, Fluid Mechanics, Pergamon Press, Oxford, 1989, p. 317. [196] N.B. Kopnin, G.E. Volovik, Critical velocity and event horizon in pair-correlated systems with `relativistica fermionic quasiparticles, JETP Lett. 67 (1998) 140}145. [197] G.E. Volovik, Simulation of PainleveH }Gullstrand black hole in thin He-A "lm, JETP Lett. 69 (1999) 705}713. [198] S. Corley, T. Jacobson, Hawking spectrum and high frequency dispersion, Phys. Rev. D 54 (1996) 1568}1586. [199] S. Corley, Computing the spectrum of black hole radiation in the presence of high frequency dispersion: an analytical approach, Phys. Rev. D 57 (1998) 6280}6291. [200] S. Corley, T. Jacobson, Black hole lasers, Phys. Rev. D 59 (1999) 124011. [201] T. Jacobson, On the origin of the outgoing black hole modes, Phys. Rev. D 53 (1996) 7082}7088. [202] T.A. Jacobson, Trans-Planckian redshifts and the substance of the space-time river, hep-th/0001085. [203] W.G. Unruh, Notes on black hole evaporation, Phys. Rev. D 14 (1976) 870}892. [204] J. Audretsch, R. MuK ller, Spontaneous excitation of an accelerated atom: the contributions of vacuum #uctuations and radiation reaction, Phys. Rev. A 50 (1994) 1755}1763. [205] P.C.W. Davies, T. Dray, C.A. Manogue, Detecting the rotating quantum vacuum, Phys. Rev. D 53 (1996) 4382}4387. [206] J.M. Leinaas, Accelerated electrons and the Unruh e!ect, Talk given at 15th Advanced ICFA Beam Dynamics Workshop on Quantum Aspects of Beam Physics, Monterey, CA, 4}9 January 1998, hep-th/9804179. [207] W.G. Unruh, Acceleration radiation for orbiting electrons, Phys. Rep. 307 (1998) 163}171. [208] Ya.B. Zel'dovich, Generation of waves by a rotating body, Pis'ma Zh. Eksp. Teor. Fis. 14 (1971) 270}272 [JETP Lett. 14 (1971) 180}181]. [209] J.D. Bekenstein, M. Schi!er, The many faces of superradiance, Phys. Rev. D 58 (1998) 064014. [210] A.A. Starobinskii, Ampli"cation of waves during re#ection from a rotating `black holea, Zh. Eksp. Teor. Fis. 64 (1973) 48}57 [Sov. Phys. JETP 37 (1973) 28}32].

348

G.E. Volovik / Physics Reports 351 (2001) 195}348

[211] Ya.B. Zel'dovich, Ampli"cation of cylindrical electromagnetic waves from a rotating body, Zh. Eksp. Teor. Fis. 62 (1971) 2076}2081 [Sov. Phys. JETP 35 (1971) 1085}1087]. [212] M.C. Cross, A generalized Ginzburg}Landau approach to the super#uidity of He, J. Low Temp. Phys. 21 (1975) 525}534. [213] H. Sonoda, Chiral QED out of matter, hep-th/0005188; QED out of matter, hep-th/0002203. [214] S. Weinberg, The Quantum Theory of Fields, Cambridge University Press, Cambridge, 1995. [215] L.J. Garay, J.R. Anglin, J.I. Cirac, P. Zoller, Sonic analog of gravitational black holes in Bose}Einstein condensates, Phys. Rev. Lett. 85 (2000) 4643}4647; Sonic black holes in dilute Bose}Einstein condensates, gr-qc/0005131.

ION IMPLANTATION INTO GALLIUM NITRIDE

C. RONNING, E.P. CARLSON, R.F. DAVIS Low Temperature Laboratory, Helsinki University of Technology Box 2200, FIN-02015 HUT, Finland L.D. Landau Institute for Theoretical Physics, 117334 Moscow, Russia

AMSTERDAM } LONDON } NEW YORK } OXFORD } PARIS } SHANNON } TOKYO

Physics Reports 351 (2001) 349}385

Ion implantation into gallium nitride C. Ronning *, E.P. Carlson
, R.F. Davis
II. Physikalisches Institut, Universita( t Go( ttingen, Bunsenstr. 7-9, D-37073 Go( ttingen, Germany
Department of Materials Science and Engineering, North Carolina State University, Box 7909, Raleigh, NC 27695, USA Received November 2000; editor: D.L. Mills Contents 1. Introduction 2. Microstructural properties 2.1. Damage accumulation 2.2. Annealing procedures 2.3. Redistribution 2.4. Damage recovery 2.5. Lattice sites 3. Optical properties 3.1. Defects 3.2. Donor doping 3.3. Acceptor doping

351 352 352 356 357 358 360 363 363 365 366

3.4. Miscellaneous elements 3.5. Rare earth elements 4. Electrical properties 4.1. Implantation isolation 4.2. Donor doping 4.3. Acceptor doping 4.4. Device applications 5. Summary, conclusions, and future work Acknowledgements References

369 370 371 371 373 375 377 377 379 379

Abstract This comprehensive review is concerned with studies regarding ion implanted gallium nitride (GaN) and focuses on the improvements made in recent years. It is divided into three sections: (i) structural properties, (ii) optical properties and (iii) electrical properties. The "rst section includes X-ray di!raction (XRD), transmission electron microscopy (TEM), secondary ion mass spectroscopy (SIMS), Rutherford Backscattering (RBS), emission channeling (EC) and perturbed

-angular correlation (PAC) measurements on GaN implanted with di!erent ions and doses at di!erent temperatures as a function of annealing temperature. The structural changes upon implantation and the respective recovery upon annealing will be discussed. Several standard and new annealing procedures will be presented and discussed. The second section describes mainly photoluminescence (PL) studies, however, the results will be discussed with respect to Raman and ellipsometry studies performed by other groups. We will show that the PL-signal is very sensitive to the processes

* Corresponding author. Tel.: #49-551-397646; fax: #49-551-394493. E-mail address: [email protected] (C. Ronning). 0370-1573/01/$ - see front matter  2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 0 0 ) 0 0 1 4 2 - 3

350

C. Ronning et al. / Physics Reports 351 (2001) 349}385

occurring during implantation and annealing. The results of Hall and C}< measurements on implanted GaN are presented in Section 3. We show and discuss the di$culties in achieving electrical activation. However, optical and electrical properties are both a result of the structural changes upon implantation and annealing. Each section will be critically discussed with respect to the existing literature, and the main conclusions are drawn from the interplay of the results obtained from the di!erent techniques used/reviewed.  2001 Elsevier Science B.V. All rights reserved. PACS: 61.72.Vv; 61.80.Jh; 68.55.Ln; 85.40.Ry Keywords: Ion implantation; Gallium nitride; Structural properties; Optical properties; Dopants

C. Ronning et al. / Physics Reports 351 (2001) 349}385

351

1. Introduction In the past decade, signi"cant e!ort has been applied to the emerging wide band gap semiconductors; including diamond, silicon carbide, the III}V nitrides, and their alloys. These materials are being investigated to extend the limits of device application into regimes of higher power, higher frequency, higher temperature and optical wavelength that can be achieved via the use of the mature semiconductor materials systems namely silicon and gallium arsenide. The success of the III-nitrides among the wide band gap semiconductors is due to the continuous solid solubility between gallium nitride (GaN), aluminium nitride (AlN), and the limited solubility with indium nitride (InN) and the consequent opportunity to engineer the direct band gap between 2 (red) and 6 eV (ultraviolet). Blue and ultraviolet light-emitting diodes (LEDs) as well as laser diodes have been realized [1,2] and commercialized [3] by growing structures containing multiple, high-quality, GaV AlW InX N layers with sharp interfaces and di!erent stoichiometries (x, y, z"0,2, 1) by metal organic chemical vapor deposition (MOCVD) onto suitable substrates. Electrons are injected during operation over a highly doped back or front layer into the active layers with lower band gap, where the electrons are "nally trapped. Spontaneous or stimulated photon emission can occur by recombination with holes. The demonstration of continuous wave operation of GaN based-laser diodes at room temperature having projected lifetimes of 10,000 h has been reported [4]. In addition, GaN-based structures o!er the prospect of superior microelectronic device performance, even in comparison to those based on other wide band gap materials. Advantages include a direct band gap, strong chemical bonding, high thermal conductivity and the realization of superior electrical properties [5]. The afore-mentioned optoelectronic devices have been achieved via MOCVD, because such a fabrication procedure is limited to relative large areas and vertical, layered device structures [6,7]. Therefore, the processing techniques used to produce new devices have to be investigated and understood, as the III}V nitride devices become more complicated [8]. Advances in microelectronics would require the possibility of adequate lateral structuring and doping techniques. Di!usion of dopants into selected areas for lateral p- and n-type doping of GaN can be excluded due to the fact that surface decomposition of GaN starts at temperatures of 8003C [9]. No signi"cant di!usion of any impurity has been observed in GaN to this temperature [10]. However, a processing technique that is widely used in semiconductor industry for lateral doping is ion implantation. Ion implantation is a convenient method to incorporate electrically and optically active dopants into the host crystal using a highly energetic beam of ions that strike and penetrate into the crystal. This method allows the introduction of a precisely controlled amount of impurity into the crystal independent of the solubility of the impurity. Ion implantation doping and isolation has played a critical role in the realization of high-performance electronic and photonic devices in all mature semiconductor material systems [11]. This is also expected to be the case for GaN and its alloys as the epitaxial material quality improves and more advanced device structures are requested. However, the main disadvantage of ion implantation is that this process is compromised by the introduction of radiation damage, that may control the optical and electrical properties and which has to be removed via annealing procedures. Even complete amorphization of the host material may occur for high implantation doses [12,13].

352

C. Ronning et al. / Physics Reports 351 (2001) 349}385

The following review discusses the recent advances in the success of ion implantation into GaN obtained by di!erent groups. The microstructural changes introduced by the implantation process and its annealing behavior under di!erent annealing techniques will be treated in Section 1. The resulting optical and electrical properties of ion implanted GaN with di!erent impurities obtained by a variety of characterization tools will be subsequently presented and discussed. These results will be compared with the properties of GaN doped during "lm growth. A discussion regarding necessary future research will be presented at the end of the review.

2. Microstructural properties The microstructural changes in the host substrate which occur during ion implantation are due to the loss of kinetic energy by electronic and nuclear interactions of the impinging ions with the atoms on the host lattice. In these processes, su$cient energy can be transferred to the host atoms resulting in displacements from their lattice sites. Such atoms may displace other atoms thus creating a cascade of atomic collisions. The damage caused by the impinging ions can be in the form of vacancies, interstitials, anti-site defects and extended defects, e.g. dislocations or stacking faults. The concentrations of such defects are mainly dependent on the bonding type and structure of the host material, on the mass of the impinging ion and on the implantation temperature. They can easily reach 100}1000 for each implanted impurity atom. The precise control of the impurity and damage distribution can be calculated using Monte Carlo simulations [14]; however, the exact amount and types of defects cannot be determined at this time. 2.1. Damage accumulation Tan et al. [15,16] demonstrated by ion beam channeling spectroscopy (RBS/C) that GaN has a high threshold for damage accumulation. A high dose of 8;10 cm\ Si ions implanted with an energy of 90 keV at 77 K had a negligible e!ect on the ion channeling spectrum in that it was indistinguishable from an unimplanted sample. A dose of 2.4;10 cm\ Si ions was required to reach the random disorder level [15,16]. Other groups [13,17,18] have reported similar high levels for the dose threshold of damage accumulation in Si implanted GaN. Cross-sectional transmission electron microscopy (XTEM) has been used to con"rm the amorphous nature of GaN after high-dose implantation [15,16]. Damage accumulation in GaN has also been investigated using other implanted species including Ca [18}21], Ar [19,20], Mg [13,22,23] and Er [24]. Table 1 summarizes all measured critical doses for the amorphization for various species implanted in GaN [12,15,16,19,22,24}30]. The heavier ions have a lower critical dose, which is to be expected. The damage accumulation is lower for implanted Ca and Ar compared to implanted Si; however, the three ions have approximately the same mass. This e!ect may be due to dynamic annealing driven by the di!erent chemistries of the implanted impurities in GaN or simply due to the di!erent structural qualities of the GaN material used in the studies. Fig. 1 shows the relative lattice disorder (measured by RBS) as a function of the displacements per atom (DPA), that was calculated using TRIM [14], for various ions implanted into GaN. Full amorphization corresponds to 100 on the vertical scale. There is good agreement in the relative disorder caused by the di!erent implanted

C. Ronning et al. / Physics Reports 351 (2001) 349}385

353

Table 1 Critical dose for the amorphization of GaN for various implanted species Species

Critical dose in GaN (ions/cm)

Implantation temperature (K)

Refs.

H O C

'1;10 '5;10 2;10 5;10 '5;10 2.4;10

298 210 77 300 298 77 298 298 77 298 77 298 77 180 300

25,26,27 28 30 * 22 12,16 15 19 20 19 20 24 30 29 30,29

Mg Si Ar

6;10

Ca

6;10

Er Au

5;10 4;10 6;10 1;10; 3.6;10

The amorphization dose has not been reached for these species at these doses and temperatures.

ions compared to the calculated DPA for each ion. As seen in Fig. 1, GaN has a high threshold for damage accumulation with no signi"cant disorder occurring until a DPA of &1 is reached. Therefore, GaN has a approximately two-to-three orders of magnitude higher amorphous threshold compared to GaAs (&4;10 cm\) [31]. Conversely, the threshold for amorphization is similar to that of AlAs, where substantial dynamic annealing during implantation occurs. The addition of Al to the GaN matrix has been reported to increase its damage threshold, which is similar to the results reported for the AlGaAs system [32]. Mensching et al. [19] reported that computer simulations of the implantation induced damage yielded a factor of two higher concentration of displaced atoms than observed in RBS/C measurements. This is similar to the results of Tan et al. [15,16], who observed that GaN has a high threshold for damage accumulation and a potentially strong dynamic annealing even at liquid nitrogen temperatures. Wenzel et al. attempted to increase the dynamic annealing by performing implantation at elevated temperatures as high as 5503C [22]. However, the amount of damage increased rather than decreased with increasing temperature. A similar result was observed by Suvkhanov et al. [23], who noted that implantations performed at 7003C increased the
minimum yield in RBS/C compared to RT or 77 K implantations. However, this is in contradiction with the results of Parikh et al. [13] and Tan et al. [33], who reported that increasing the implantation temperature decreases the damage level. The nature of the implantation damage has been investigated by several groups. We found a distinctly darker area in cross-sectional TEM images near the top surface, as shown in Fig. 2a. This darker area is attributed to contrast caused by the disorder. It is highest in the center of the damage region. High-resolution TEM images of the implanted and unimplanted areas are shown

354

C. Ronning et al. / Physics Reports 351 (2001) 349}385

Fig. 1. Relative lattice disorder (percentage of the aligned to random yield in the RBS-spectrum) at the damage peak as a function of displacements per atoms occurring during implantation for various ions implanted in GaN. The displacements per atom were calculated using TRIM and assuming a threshold displacement energy of 20 eV for both Ga and N sublattices. The dashed line is a sigmoidal "t to the data to help guide the eye.

Fig. 2. Cross-sectional transmission electron picture of GaN implanted with 1;10 cm\, 200 keV Si ions at 6503C showing the implantation region in high magni"cation (a) and high resolution (b). The high resolution image (c) shows an unimplanted region for comparison.

in Fig. 2b and c, respectively. The implanted region is still crystalline with a well-de"ned atomic ordering due to the moderate implantation dose of 1;10 cm\ of Si ions used, but the arrows indicate small potentially amorphous pockets, which contain a high density of defects. These "ndings are in agreement with other studies, where low-dose Si-implanted GaN showed

C. Ronning et al. / Physics Reports 351 (2001) 349}385

355

Fig. 3. Normalized GaN (0002) XRD spectra after 180 keV Ca implantation with di!erent doses at 77 K. (From Refs. [20,21])

a near-surface region with a high density of defects including clusters, loops, and planar defects [15,16]. Similar XTEM results were obtained for low-dose Zn [34], Au [29], and C [29] implanted GaN. Examination of the energy dependence of dechanneled He ions indicate that the implantationinduced defects are mostly point defects in nature [22]. Both Hf and In implanted into GaN have heavily disturbed surroundings in the next or further nearest neighborhood after implantation, as observed by perturbed-

-angular-correlation spectroscopy (PAC) [35,36]. Liu et al. [20,37] found that a new peak appears at smaller angles in X-ray di!raction (XRD) spectra. Fig. 3 shows such peaks for Ca implanted GaN at 77 K (from Ref. [20,21]). The main (0002) peak of the virgin GaN-material decreases, and the new peak shifts to lower angles with increasing implantation dose. Other groups have also noticed this extra peak in GaN implanted with Mg [38,39] and Be [39,40]. This new peak is postulated to be an expansion of the GaN hexagonal lattice driven by the introduced impurities and displaced host atoms onto interstitials sites or by the incorporation of larger atoms on substitutional sites, and not a phase change in the GaN [20,41]. This explanation is supported by TEM, electron di!raction, and X-ray pole "gure measurement results [20,41]. The deconvolution of the XRD spectra reveals a better "t if three peaks are used instead of one. One peak for the virgin (0002) GaN, one peak for the expanded lattice (0002) GaN, and a third peak for an amorphous content in the GaN [20,41]. The amorphous peak arises in Ca implanted GaN at a dose of 3;10 cm\, begins to dominate at a dose of 1;10 cm\, then totally dominates at a dose of 6;10 cm\. The onset of the

356

C. Ronning et al. / Physics Reports 351 (2001) 349}385

Table 2 Comparison of melting temperature and activation temperature for various compound semiconductors

GaSb InP GaAs SiC GaN Sublimes at ¹(¹

Melting temperature (3C)

Activation temperature (3C)

¹ /¹  

707 1057 1237 2797 2791

500}600 700}750 750}900 1300}1600 ?

&0.75 &0.65 &0.65 0.46}0.57 ?



.

amorphous peak in XRD occurs at doses lower the onset of amorphous behavior, as observed using RBS/C [19,20]. This implies that the process of amorphization in GaN occurs in small local regions. These regions then increase with increasing implantation dose until the crystal collapses to create an amorphous layer. This interpretation is in agreement with TEM-results shown in Ref. [29], where planar defects seems to provide a nucleation site for amorphization. A change in the lattice parameter of GaN after implantation, similar to the extra XRD peak, has also been observed [42,43] along with a broadening of the X-ray rocking curve [34,44,45]. Furthermore, it was observed that high-dose H implantation resulted in the formation of voids or bubbles in GaN [25}27]. 2.2. Annealing procedures The implantation damage created in the host material can only be removed via subsequent annealing procedures. The annealing temperature for optimal implantation activation in compound semiconductors generally follows a two-thirds relationship with respect to the melting point of the material and has been intensively investigated for the common semiconductors. This behavior can be seen in Table 2, which lists the melting point of several compound semiconductors with the associated temperatures commonly reported to achieve implantation activation. The melting temperature of GaN has been determined as 27913C and [46], therefore, annealing temperatures of about 15003C are required, assuming the two-thirds relationship is applicable. Though GaN has a high melting point, it will decompose at much lower temperatures due to the very strong triple bond of molecular nitrogen (N ) that makes less negative Gibbs free energy of the nitride constituents [47,48]. The Gibbs free energy of the nitride constituents decrease with temperature faster than the Gibbs free energy of the GaN crystal resulting in decomposition before melting. Therefore, GaN surface decomposition already starts as low as 8003C [49] resulting in the formation of N and the consequent loss of nitrogen and the formation of Ga droplets on the surface [50}52]. The speed of this process depends exponentially on the temperature making it extremely di$cult to anneal GaN at high temperatures. Therefore, one has to use an annealing technique which protects the GaN surface from decomposition. In recent years, many di!erent annealing procedures have been applied or developed for implanted GaN to reach the required

C. Ronning et al. / Physics Reports 351 (2001) 349}385

357

high temperatures for annealing times in the minutes/hour range. These procedures can be divided into four groups, summaries of which are provided below. Firstly, implanted GaN samples have been annealed in nonreactive ambients. The success of this technique is limited by the product of temperature, time and pressure. Vacuum annealing o!ers the cleanest conditions; however, only temperatures around 800}9003C can be reached for several minutes without the degradation of the sample [49,53]. A slightly higher temperature can be reached under Ar or N #ow due to the higher vapor pressure, but high-purity gases are necessary to avoid the formation of thin GaO-layers on the surface. A further increase to high N overpressures (16 kbar) resulted up to annealing temperatures of 15503C [43,54]. On the other hand, the reduction of the annealing time into the second range in rapid thermal annealing (RTA) systems resulted also into maximum annealing temperatures of 12003C under N #ow [55]. Note: the measurement of the real surface temperature in RTA systems is very di$cult especially in the case of transparent samples and gas #ow conditions; thus, inaccurate values may be published. Secondly, implanted GaN has been annealed under reactive ambients [40,56}58]. In this case an exchange of nitrogen between the surface and the vapor phase occurs, i.e. the loss of nitrogen is compensated by the formation of new Ga}N bonds during the annealing process. Similar conditions exist during the growth of GaN. Therefore, these annealing techniques have been applied in MOCVD or MBE growth systems, where the implanted GaN samples are annealed under NH , an atomic nitrogen #ux or in a nitrogen plasma. Temperatures to 11003C can be reached in this way for 1 h [40,56,58]. The third way to protect the GaN surface is to deposit a capping layer that must be removed after the annealing procedure. Sputtered polycrystalline SiN and AlN have been used; however, such "lms were only stable to temperatures of 11003C [59]. The highest temperature of &13003C was achieved with epitaxial AlN-caps and subsequent vacuum annealing for 15 min [60,61]. Finally, laser processing is an additional method of annealing that uses short processing times to take advantage of GaN decomposition kinetics to suppress surface degradation. The de"nition and measurement of the equivalent annealing temperature is impossible due to the very fast nonequilibrium process. Therefore, the introduced power density is used for comparison; maximum values of 200}350 mJ/cm have been applied to GaN without an observable decomposition of the surface [51,52,62]. However, this annealing technique has not yet been applied to implanted GaN. 2.3. Redistribution Most species implanted in GaN have the anticipated Gaussian distribution around the maximum concentration with a tail extending further into the GaN substrate most likely due to channeling e!ects [63}76]. The range and distribution of the implanted impurities in GaN is in good agreement with TRIM calculations [18,19,40,66]. All implanted impurities show a high thermal stability in GaN upon annealing and di!usion or a signi"cant redistribution was only observed in a few cases, as described below. Implanted hydrogen is stable in n-GaN to 700}8003C; signi"cant loss is initiated at the surface at 9003C. However, the implanted H is still present after annealing at 12003C [25,26,63,64,66,67,74]. The shape of the implanted H depth pro"le is preserved constant even during the H loss, which implies that H decorates implantation-induced defects [63,64,77]. Further evidence that H is bonded to defects in implanted GaN is the fact that H can be readily di!used into as-grown GaN at

358

C. Ronning et al. / Physics Reports 351 (2001) 349}385

much lower temperatures [63,64,67,77]. Hydrogen implanted in p-type GaN showed a signi"cantly di!erent behavior with out-di!usion at already 5003C and a complete loss at 10003C [74]. The faster di!usion of H in p-GaN is due to its signi"cantly lower formation energy with dopants, as compared to n-type material, which has been calculated by Neugebauer and Van de Walle [78}81]. Other lighter elements also show a high thermal stability in GaN, e.g., implanted #uorine does not undergo redistribution in GaN for annealing temperatures to 8003C [68] and 9003C [66]. This is surprising, as F is generally a rapidly di!using species in III}V compounds [82]. The column IV donors elements in GaN have a very high thermal stability with no redistribution for annealing to 14503C for Si [18,65,66,68,71}73,75,83}85] and 9003C for Ge [65,66,68,75,84]. Implanted oxygen, a column VI donor in GaN, showed no redistribution to 11253C [69,70,86]; however, this is in contrast to the results of Pearton et al. [87]. The last group found that O di!used into GaN from a SiO capping layer in the temperature range of 700}9003C. The rest of the column VI donors showed no redistribution during annealing at the maximum annealing temperature of 14503C [75,84,85]. Realized and potential implanted acceptors in GaN include column II and transition elements such as Be, Mg, Ca, Zn, Cd and Hg that would be metal}site acceptors and column IV elements such as C that would be N-site acceptors. Typically, acceptor species are more likely than donor species to redistribute at high temperatures in III}V semiconductors. However, similar to donor species, acceptor species in GaN appear to have a high thermal stability. Implanted Be, a common fast interstitial di!using species due to its small size, is thermally stable for anneals to 12003C [40,65,66,68,73,84,85] with slight initial broadening beginning at 9003C [73,84,85]. The remaining column II elements of Mg [65,66,68,73,76,84,85] and Ca [69,70,72] have also a high thermal stability with no redistribution observed for annealing to 14503C or 11253C, respectively. However, there have been some reports of implanted Mg redistribution at 11503C [18,71,72]. The transition element Zn has shown the highest redistribution rate of the acceptor elements implanted in GaN. Wilson et al. reported that Zn was only stable to 7003C [66] with some redistribution observed at 8003C [65,68]. Strite et al. observed Zn di!usion at 1100}11503C [34,43] with rapid di!usion at 12503C [43] such that the Zn di!used throughout the GaN "lm. These results were con"rmed by Suski et al. [54]. The column IV acceptor carbon is stable up to the maximum annealing temperature of 14503C [65,68,75,84,85]. Several rare earth elements have been implanted in GaN including praseodymium (Pr), neodymium (Nd), europium (Eu), holmium (Ho), erbium (Er), thulium (Tm), ytterbium (Yb), lutetium (Lu), thorium (Th), and uranium (U) [68]. These elements redistribute at lower temperatures with increasing implantation dose. No redistribution was found to annealing temperatures of 8003C for doses less than 2;10 cm\; however, the onset of redistribution drops to 7003C for doses higher than 5;10 cm\ [68]. 2.4. Damage recovery The recovery of structural implantation damage can be directly observed by RBS/C, XRD, TEM and PAC. The "rst two techniques are mainly sensitive to the crystal structure; whereas, the other two are able to detect point defects. The results of all these measurement techniques show no recovery of the accumulated damage in GaN after post-implantation annealing for low

C. Ronning et al. / Physics Reports 351 (2001) 349}385

359

Fig. 4. XRD 2-axis -2 map of the (0002) GaN peak (a) directly after Be implantation with a dose of 2.5;10 cm\ and after subsequent annealing to a temperature of 9003C (b) for 10 min. (Note: contour plots have log scale, from Ref. [40])

Fig. 5. Aligned RBS spectra of Ge-implanted GaN with a dose of 5;10 cm\ and an energy of 300 keV. The aligned spectra were taken along the [0001] direction after implantation and after annealing at 9003C for 1 h. Also included are a random spectrum and an aligned spectrum of a virgin GaN sample.

temperatures up to 6003C. The spectra are almost identical to the spectra taken from the as-implanted situation. Annealing of implanted GaN between 6003C and 11003C results in a reduction of the implantation damage. This can be clearly seen in Figs. 4}6.

360

C. Ronning et al. / Physics Reports 351 (2001) 349}385

Fig. 6. Perturbed-

-angular-correlation spectroscopy (PAC) results measured on In in GaN as a function of annealing temperature. (From Ref. [60])

Fig. 4 shows that the implantation correlated extra peak in the XRD-pattern can be completely eliminated after annealing to 9003C in GaN implanted with a low implantation dose (2.5;10 cm\) of Be [38]. This indicates that the induced structural damage can be completely removed; however, a reduction and not a complete elimination of this peak is observed for heavier elements and higher implantation doses, even for higher annealing temperatures [39,40,54,56]. Fig. 5 shows RBS/C spectra taken from a high-dose Ge-implanted GaN sample, which was annealed to a temperature of 9003C. The onset of recovery is visible after the annealing step, but the near surface implantation damage is still present. The height and reduction of this damage peak strongly depends, similar to the XRD-results, on the dose and annealing temperature and is in agreement with other published studies [17}20,22,32,33,85,88}91]. A complete elimination of the damage peak was only observed for low implantation doses and very high annealing temperatures [32]. Plan view TEM con"rmed that annealing at 11003C for 30 s did not fully recover the damage [83,85,92]. For high implantation doses beyond the amorphization threshold, it was furthermore found by XTEM that the implanted region recrystallized into a polycrystalline layer with no detectable epitaxial re-growth [33,88]. The most sensitive technique to monitor the direct surrounding of the implanted impurities is PAC [93]. Fig. 6 shows the fraction of implanted In atoms with a defect-free neighborhood as well as with disturbed surroundings as a function of annealing temperature [35,60]. For isochronal annealing treatments in vacuum a gradual recovery of the implantation damage in the surrounding volume of each indium atom occurs between 6003C and 9003C. After annealing at 12003C, approximately 70(5)% of the probe atoms occupy undisturbed sites, and the remaining fraction of indium atoms occupy sites with weakly disturbed surroundings [35]. Therefore, point defects remain in the GaN sample even after this high annealing temperature. These "ndings were con"rmed by PAC measurements after ion implantation of Hf into GaN [36]. Around 65% of the Hf occupy defect free sites after annealing at 9003C. 2.5. Lattice sites Implanted impurity atoms can occupy several site locations in the host lattice including substitutional and interstitial lattice sites. The resulting electrical and optical properties of the

C. Ronning et al. / Physics Reports 351 (2001) 349}385

361

implanted GaN sample strongly depend on the local bonding con"guration and thus the lattice site location of the impurity atoms. The lattice site of the impurity may change upon post implantation annealing procedures, also a!ecting the electrical and optical properties. Therefore, the knowledge of the lattice sites of the implanted impurities is essential for device processing by ion implantation. Several groups have used assorted channeling and spectroscopy techniques to determine the lattice site location of various impurities implanted in GaN. We used the emission channeling (EC) [94] technique for the determination of the lattice site location of implanted Li [95], Na [60], In [35], Sr [60], Tm [96], and Yb [96] in GaN. The EC-technique makes use of implanted radioactive probe atoms and measures the channeling e!ects of emitted decay particles such as conversion electrons, -particles, or -particles [94]. Fig. 7 shows a contour plot of a two-dimensional -EC pattern measured around the c-axis 0002 at room temperature after implantation of Li into GaN. The emission pattern has a six-fold symmetry showing three pronounced planar channeling e!ects and three pronounced planar blocking e!ects. Furthermore, in the c-axis direction the normalized emission yield is larger than unity, indicating channeling e!ects along the c-axis direction. Channeling of -particles can only occur if the lithium emitter atoms are located in interstitial sites with respect to the c-axis atom rows. On the other hand, blocking e!ects occur if the emitter atoms are located within an atomic row or plane. The axial and the three planar channeling e!ects are then due to emitter atoms located in between the indicated equivalent (3 120), (3 210) and (01 10) planes. The observed three planar blocking e!ects are due to emitter atoms located in the equivalent (2 110), (1 010) and (1 100) planes. The Li emitter atoms are therefore unambiguously located in the center of the hexagons. Emission channeling patterns around the c-axis were also recorded for increasing substrate temperatures to 5003C [95]. The normalized yield measured in the c-axis direction as a function of temperature is plotted in Fig. 8. The onset of Li-di!usion and a lattice site change occur in the rather narrow temperature regime of 410$253C. We observed a clear indication for a lattice site change of a signi"cant fraction of Li atoms from initially interstitial sites Li' to substitutional sites L\\. We assume that Li in GaN Li1 due to a Coulomb-force-driven reaction Li> ' #VL\NLi1 occupies substitutional Ga-sites at temperatures above 4103C. A lattice site change was also observed for deuterium in GaN. Using nuclear-reaction analysis (NRA) channeling, Wampler et al. [25,27] observed H in the central region of the c-axis after ion implantation at room temperature. Annealing to 5513C and above causes the H to move from the interstitial sites to locations that could not be resolved with this technique. No deuterium remained at interstitial sites after annealing at 8093C. This temperature range is higher compared to that for the onset of Li di!usion, but close to the H redistribution temperatures reported by Wilson et al. [63,64,66,67] and the damage recovery reported above. Therefore, the lattice site change of Li can be unambiguously attributed to the di!usion of Li; whereas, the reaction of deuterium with vacancies or defects may be not only caused by deuterium di!usion, but also by di!usion of the defects during damage recovery. Further EC-investigations showed that sodium also occupies interstitial sites with no change upon annealing to 8003C [60]; whereas, the majority of the implanted heavy ions of In, Sr, Tm and Yb ions are substitutional after implantation with no change upon annealing as high as 12003C [35,60,96]. A high substitutional fraction was also observed for ion implanted Ca [97], Te [33], Er [24] and Hf [36] using RBS/C and particle induced X-ray emission (PIXE). All of these implanted heavy ions are substitutional; however, they have damaged surroundings and are slightly displaced

362

C. Ronning et al. / Physics Reports 351 (2001) 349}385

Fig. 7. Normalized emission yield of alpha particles emitted from implanted Li probe atoms measured at room temperature in GaN along the c-axis direction during Li implantation. The gray scale represents the normalized yield with respect to the yield for an o!-axis random direction. (From Ref. [95]) Fig. 8. Normalized alpha emission yield in the c-axis direction as a function of implantation temperature for undoped GaN. The change from channeling (yield '1) to blocking (yield (1) around 410$253C shows the lattice site change of a signi"cant fraction of Li from interstitial to substitutional sites. (After Ref. [95])

from the ideal substitutional position, potentially associated with defects [33,35,36,88,96}98]. Most of the heavier atoms: Ca [97], In [35], Pr(Ce) [98], Er [24], Tm [96], Yb [96] and Hf [36] exist substitutionally on the Ga lattice sites as opposed to N lattice sites. Unfortunately, the sub-lattice site of the donor Te, was not determined [33]. Annealing does not change the apparent substitutional fraction of the heavier atoms; however, it decreases the damage surrounding the implanted atoms and increases the fraction of atoms at ideal substitutional positions [33,88,96}98]. Silicon does not appear to be substitutional or interstitial direct after implantation [99,100]. The implanted Si atoms remains randomly distributed with only &20% substitutional after annealing to 10503C. However, upon annealing at 11003C there is a drastic change in the lattice location of the implanted Si with &100% of the implanted species located at substitutional Ga lattice sites [99,100]. MoK ssbauer spectroscopy was performed on GaN implanted with radioactive Cs, which decays to Sn [101]. It was indirectly determined that the Sn existed in several states in GaN including substitutional species on both Ga sites and N sites, as well as associated with defects and with O. Upon annealing the percentages of Sn occupying the various states in GaN changed. Speci"cally, the fraction on substitutional Ga sites increased at the expense of the fraction associated with defects [101].

C. Ronning et al. / Physics Reports 351 (2001) 349}385

363

Fig. 9. Photoluminescence spectra taken at 14 K of GaN implanted with various doses of Ge. Implantations were performed at a temperature of 77 K using an implantation energy of 300 keV. The inset shows an expanded view of the near band-edge emission along with its associated LO-phonon replicas for the unimplanted GaN.

3. Optical properties The optical properties of GaN can be best determined using photoluminescence (PL) and cathodoluminescence (CL) spectroscopy. The quality of the GaN sample as well as the optical activation of dopants can be identi"ed by the existence of speci"c PL-lines or -bands, by the intensity of PL-lines and by their width and shapes [2,5,7]. The "rst reported PL-study on implanted GaN was reported in 1976 by Pankove and Hutchby [58]. In this research spectra of 35 potential donor and acceptor species were recorded after ion implantation. The conclusions drawn at that time should today be used with care, because the quality of GaN samples have been improved by several orders of magnitude since that time. However, it had been shown that ion implantation resulted in nonluminescent GaN material implying the resulting implantation damage consists of many nonradiative recombination centers. This "nding is in agreement with all present studies and can easily be checked on an implanted GaN sample, which becomes nontransparent in the visible and near UV range after ion implantation. Thermal treatments are required to restore the luminescence, as reported subsequently in this section in the topic regarding the e!orts for optical donor, acceptor and rare earth activation in implanted GaN. 3.1. Defects Fig. 9(a) and its inset shows a PL spectrum of the unimplanted GaN sample for comparison. Several features can be observed including an intense near band-edge emission at 358 nm

364

C. Ronning et al. / Physics Reports 351 (2001) 349}385

(3.464 eV), commonly labeled I , that originates from recombinations of free excitons and/or excitons bound to shallow donors [102]. Two well-resolved, associated longitudinal optical (LO) phonon replicas at 367 nm (3.379 eV) and 377 nm (3.289 eV) for the I peak can also be observed, indicating the high quality of the unimplanted GaN. The position of the LO-phonon replicas results in a phonon energy of 85$5 meV for these samples, which is in agreement with values in the literature [103}105]. A weak transition at &381 nm (3.25 eV), ascribed to donor}acceptor pair (DAP) recombination [106], is observed, along with two LO-phonon replicas at 3.16 and 3.08 eV, as a shoulder on the second LO-phonon replica of the I emission. The identity of the shallow donor(s) is unknown, although both native defects and extrinsic impurities, including oxygen and silicon, have been suggested [78,107]. The identity of the acceptor is also unknown, although carbon and/or magnesium are likely candidates [108}110]. The feature observed at &550 nm (2.25 eV) is commonly referred to as the `yellowa emission band [111]. The origin of the yellow-band emission in GaN is still unknown. However, it is most likely that a variety of defects and deep level impurities causes this band and it has already been attributed [112}114] to deep acceptor levels that arise due to point defects such as Ga vacancies allowing a transition from shallow donor states to the deep acceptor state. The exact identities of both the shallow and deep defects are unknown with both native defects and extrinsic impurities possible. The broad, Gaussian line shape of several *- half-widths is typical of deep levels in semiconductors. The deep recombination partner has a large binding energy that results in a localized wavefunction (x) and thus a large range of k values [115]. Therefore, the transition occurs over a large energy range. As mentioned above, Fig. 9b}d clearly shows that the photoluminescence of GaN is severely depressed directly after ion implantation of Ge. The near-band-edge emission (3.464 eV) intensity is reduced by three orders of magnitude compared to the unimplanted sample even for a relatively low implantation dose of 1;10 cm\. The intensity is further reduced at higher implantation doses and is totally quenched at an implantation dose of 5;10 cm\. Ion implantation at higher substrate temperatures results in a reduction in the implantationinduced damage by dynamic annealing during implantation that is visible in the PL-spectra [56]. However, the reduction is minor compared to the large concentration of defects introduced during implantation, as attested by the fact the I line intensity is still four orders of magnitude lower than that of an unimplanted sample for implantation temperatures around 9003C [56]. Fig. 10 shows the PL-results of isochronal (1 h) annealing for Ge-implanted GaN with a dose of 1;10 cm\. The "rst signi"cant change in the PL spectra can be observed after an annealing temperature of 6003C. A slight decrease of the weak defect related PL-band centered around 425 nm and a slight increase of the yellow band is visible, which indicates that some species of defects are annealed out. This temperature range is in agreement with the recovery observed with PAC (see Fig. 6) and can be attributed to the onset of defect/vacancy di!usion. The intensity of the near-band-edge I emission increases after annealing at each annealing step with the intensity increasing by about three orders of magnitude at 11003C. However, the intensity of the I emission is still one}two orders of magnitude lower than that observed in unimplanted GaN indicating that the recovery is still incomplete at this temperature, which is in agreement with the presented XRD, RBS, and PAC results. (An AlN capping layer was used to stabilize the GaN surface during the high temperature anneals.) The increase in the I intensity is also the result of the recovery of implantation damage by annihilation of point defects. This can be deduced from the fact that

C. Ronning et al. / Physics Reports 351 (2001) 349}385

365

Fig. 10. Photoluminescence spectra taken at 14 K of GaN implanted with Ge at a dose of 1;10 cm\ and annealed at various temperatures for 1 h. The implantation was conducted at room temperature with an implantation energy of 300 keV. The solid arrow is positioned at the Ge-related transition.

unimplanted GaN samples do not exhibit any signi"cant changes under the same annealing conditions. The appearance of a new emission peak centered at &372 nm (3.33 eV) is observed after annealing at 11003C for 1 h. 3.2. Donor doping The new emission line at 3.33 eV in Fig. 10 is very likely due to a Ge related transition, as this peak was not observed in unimplanted GaN samples. A similar peak has been observed for GaN doped with Ge during growth [116] and by PL using radioactive Ge-doped GaN [117]. The transitional nature of this line is not known at this time. However, it should be a donor}band transition or a DAP transition with Ge as the donor. Assuming a donor}band transition, the energy level of the Ge donor would be 170 meV below the conduction band, assuming a band gap of 3.5 eV for GaN. This is not consistent with the reported [118}120] shallow nature of Ge. However, the transition may be also a DAP transition between the Ge donor and an unknown, potentially defect related, acceptor level. In this case the Ge donor level can be shallow depending on the energy level of the unknown acceptor. The emission of the 3.33 eV line is more intense for a lower implantation dose of 1;10 cm\ compared to 1;10 cm\. This can be explained by the fact that the higher implantation dose result in a greater concentration of implantation-induced defects that are still present after the 11003C annealing step.

366

C. Ronning et al. / Physics Reports 351 (2001) 349}385

Silicon-implanted GaN annealed at 11003C for 1 h shows a strong DAP emission at 3.25 eV with the associated two phonon replicas at 3.16 and 3.08 eV. Other groups have observed similar strong DAP emission for GaN implanted with Si [17,21]. On the other hand, the shallow donor level (30 meV) of Si in GaN and its associated signature in the broadening of the I -line is well known from GaN doped with this species during growth [122,123]. Therefore, the enhancement in the DAP emission must be explained by the fact that the implanted silicon act as a shallow donor and result in increased DAP transitions with an unknown acceptor level. This behavior is also seen in GaN which is doped with Si during growth [123]. A new and sharp PL-line arising at 364.5 nm (&3.40 eV) is also observed in silicon implanted GaN after thermal annealing. However, this peak is due to defects created during the implantation procedure, as this line is observed with varying intensities after implantation of Li, Be, Ge, In, Mg, Ca, and Er (see e.g. Figs. 11 and 12). It is likely that this emission is produced by nitrogen or gallium vacancies due to acceptor or donor bound excitons, because it can appear also in unimplanted GaN samples; however, it is dependent on the growth conditions [124]. 3.3. Acceptor doping Magnesium implanted and high-temperature-annealed GaN shows a strong DAP emission at 3.25 eV with two associated LO-phonon replicas at 3.16 and 3.08 eV (see Fig. 11). This emission is similar to the intrinsic DAP emission observed in undoped GaN (Fig. 9). However, the 3.25 eV emission, along with its LO-phonon replicas, is commonly seen in GaN samples doped during growth with Mg [125}141] and exhibiting p-type activation. The emission is attributed to a donor}acceptor pair transition (DAP) between a Mg acceptor and a shallow donor [134}137,140]. Therefore, it can be concluded that optical activation of the implanted Mg atoms has been realized. The donor of the DAP transition is unknown but is assumed to be the intrinsic donor in GaN. The Mg binding energy is estimated to be 235 meV assuming a band gap of 3.5 eV, a donor binding energy of 30 meV, and a value of 15 meV for coulomb interaction. This value is in good agreement with the accepted value of 225}250 meV. Fig. 11 shows also PL spectra of GaN implanted with di!erent doses of Mg and annealed at 12503C. With increasing implantation dose the Mg-related DAP transition decreases, and "nally the GaN samples implanted with a dose of 10 cm\ show no DAP-transitions. One would not expect such a behavior for increasing acceptor concentrations. Therefore, a high amount of residual defects, which strongly depends on the implantation dose, is still present in the annealed samples. This is supported by the fact that the behavior of the intensity of the yellow band is opposite to the behavior of the DAP transitions and therefore much higher in the GaN : Mg samples implanted with higher implantation doses (see Fig. 11). The PL spectrum of a GaN sample doped during MOCVD growth with 5;10 cm\ Mg is also contained in Fig. 11 [142]. The I -line of this samples is less intense compared to the emission from samples implanted with low doses. This is due to the higher intensity of the DAP-transitions. However, the di!erences between Mg-implanted and Mg-doped GaN are minor in this region of the spectrum. By contrast, the intensity of the yellow `defecta band shows a signi"cant di!erence: it is not visible in the Mg-doped GaN; whereas, the yellow band dominates the PL-spectra of the high-dose Mg-implanted GaN. Again, this demonstrates that after this high-temperature annealing

C. Ronning et al. / Physics Reports 351 (2001) 349}385

367

Fig. 11. Photoluminescence spectra measured at low temperature of GaN as a function of Mg implantation dose. (a) 10 cm\, (b) 10 cm\, and (c) 10 cm\. The post implantation annealing was performed at 12503C for 30 min under vacuum and the implantation energy was set to 60 keV. The samples were protected with an epitaxial AlN-cap during annealing. For comparison, the PL-spectrum (d) was taken from a GaN-sample that was doped with 5;10 cm\ Mg during MOCVD-growth.

procedure defects are still present in the material even when most of the implanted Mg ions are optical activated. A new PL-line at 3.36 eV appears in GaN implanted with 1;10 cm\ calcium ions and subsequently annealed at 12003C for 1 h, as shown in Fig. 12 (top). This emission was also observed, as a shoulder on the I emission peak, for higher Ca implantation doses [56]. The Ca binding energy was estimated to be 140 meV. This is close to the value of 169 meV reported by Zolper et al. [69,70,72] for the activation energy of implanted Ca in GaN as measured by temperature-dependent sheet carrier measurements. Thus, this emission peak is due to optically activated Ca. Beryllium is a promising candidate for shallow p-type doping in GaN given its calculated ionization energy of &0.06 eV [143]. We found a Be-related transition at 3.35 eV after ion implantation and annealing, as shown in Fig. 12 [144]. The Be-related PL-line is very weak compared to the Ca and Mg cases, which may be due to the possible formation of Be-defect complexes [145] or due to the occupation of non-doping interstitial sites [146]. We attributed the new line to band}acceptor (eA) recombinations. For this case, the ionization energy of Be acceptors can be calculated to be 150$10 meV. However, this is in contradiction to the theoretical value noted above. Dewsnip and co-workers [147] observed this line at 3.376 eV in GaN samples doped with Be during growth. They calculated the ionization energy to be 90}100 meV due to the

368

C. Ronning et al. / Physics Reports 351 (2001) 349}385

Fig. 12. Photoluminescence spectra measured at low temperature of GaN implanted with a variety of species: (a) calcium, 10 cm\, 300 keV, ¹ "12003C; (b) magnesium, 10 cm\, 60 keV, ¹ "12503C; (c) beryllium,   10 cm\, 100#200 keV, ¹ "9003C; (d) lithium, 10 cm\, 30 keV, ¹ "8503C; (e) indium, 10 cm\, 30 keV,   ¹ "8503C; and (f) erbium, 10 cm\, 60 keV, ¹ "8503C.  

assumption that this line is a donor-to-acceptor transition. Temperature-dependent PL are necessary to determine the nature of this Be-related transition and thus to calculate the exact activation energy of Be which can also be determined via electrical measurements. Zinc is another acceptor that has been extensively studied in GaN due to its reported high luminescence e$ciency. Implanted Zn has been reported to have a strong emission peak at 2.87 eV (&430 nm) in GaN [34,43,44,148}150], which is consistent with GaN "lms doped with Zn during growth. However, the intensity of the Zn emission in implanted "lms is typically less than in situ doped GaN [34]. Improvement of the Zn emission intensity after ion implantation was observed by Strite et al. [43,44] by annealing the implanted GaN at high temperatures or for long periods of time using a high N overpressure to stabilize the GaN during annealing. Suski et al. [150] showed that the optical activation of implanted Zn increases with increasing annealing temperatures to 13503C under high N overpressure and to 15503C under high N #Zn overpressure. This provides more evidence that the damage induced by implantation is detrimental to the optical properties of GaN and shows that the implantation induced damage is still being annealed out at temperatures in the range of 1350}15503C.

C. Ronning et al. / Physics Reports 351 (2001) 349}385

369

3.4. Miscellaneous elements We have also included in Fig. 12 the PL-spectra of Li, In and Er implanted GaN after subsequent annealing at 8503C. The only emission line appearing is positioned at 3.41 eV. As discussed above, this PL-line is related to a native defect. No other extra PL-lines are visible in this range of the spectrum for these species; thus, they do not act as shallow dopants in GaN. We have shown above by emission channeling that Li occupies substitutional sites after implantation and annealing at high temperatures [95]. On the other hand, Li on interstitial sites should act as shallow donor, but directly after ion implantation no luminescence is observed due to the high concentration of the defects. The yellow PL-band as well as a band positioned around 3.0 eV are very pronounced for the heavy elements In and Er, which is of course due to the higher amount of implantation damage introduced by these species. However, these results prove that the new lines appearing after Ge, Ca, Mg and Be implantation are only related to the respective species. Several elements have been implanted and optically investigated in GaN by other groups, including the light elements hydrogen [151,152] and helium [151,153], the isovalent elements arsenic and phosphorus [154], and the transition metal vanadium [155]. The implantation of the light elements decreased the characteristic luminescence of GaN with some recovery after annealing [151,153]. Two new luminescence lines at 2.9 eV and 0.95 eV has been observed in He [151] and H [153] implanted GaN, respectively. The peak at 0.95 eV is similar to peaks observed in electron-irradiated [156,157] and Nd implanted [158,159] GaN implying this peak is also related to defects. Metcalfe et al. reported that the cathodoluminescence spectra of the isovalent elements As and P implanted into GaN-exhibited luminescence lines at 2.58 and 2.85 eV, respectively [154]. The transition metal vanadium implanted into GaN has a luminescence line at 0.82 eV [155]. An unambiguous identi"cation of PL-lines is possible via radioactive PL [160]. This technique makes use of the chemical transition of nuclei during the radioactive decay. After implantation of radioactive ions and subsequent annealing, the appearance and the disappearance of PL-lines can be correlated with the half-life of the respective decay. This technique was applied to GaN for implantations with radioactive Ag,  Cd, Cd, As, and Se [117,161,162]. Ag produces a deep luminescence band centered at 1.52 eV, which is superimposed by four single transitions. A blue emission band centered at 2.7 eV was assigned to deep Cd-related defects; whereas, shallow levels at 3.34, 3.32 and 3.27 could be also attributed to the acceptor Cd. PL emissions at 2.58, 3.4, and 1.49 eV were caused by As (located on N-sites), Ge, and Se, respectively. Spectroscopic ellipsometry [23], optical transmission measurements [45], optical absorption measurements [121], and Raman scattering measurements [19,25,26,152,163,164] have been performed on GaN implanted with various species. The coe$cient k, which is related to the photon adsorption of the material, increased [23] while the optical transmission [45] decreased with increasing dose of implanted species, independent of ion species. This again demonstrates that the implantation-induced damage consists of many non-radiative photon adsorption centers. Silkowski et al. reported that annealing implanted GaN to 10503C in #owing NH for 90 min recovered the absorption edge in implanted GaN [121]. Raman scattering showed four new Raman peaks after implantation independent of ion species [164]. The peaks were attributed to vacancy-related crystal defects and disorder-activated Raman scattering. The intensity dropped upon annealing showing a reduction of the implantation induced defects with annealing. Raman measurements also showed a shift and a narrowing of the A (LO)-Raman peaks after implantation

370

C. Ronning et al. / Physics Reports 351 (2001) 349}385

of Ca, Ar, and C [19]. These phenomena suggest a reduction in carrier density after implantation. Raman measurements of implanted hydrogen (H) revealed vibrational modes at 3100, 3183, and 3219 cm\ [25,26,152,163]. The vibrational mode at 3100 cm\ was attributed to V% }H complexes [152,163]; the modes at 3183 and 3219 cm\ were attributed to hydrogen bonded to nitrogen atoms [25,26]. 3.5. Rare earth elements Rare-earth-doped GaN is a candidate for optoelectronic devices operating in the visible or infrared region [165,166]. Especially the 1.54 m intra-4f shell emission of erbium in the trivalent state (Er>) is promising for the telecommunications industry due to the fact that it coincides with a minimum in attenuation in silica based optical "bers. E$cient 1.54 m emission observed by photoluminescence [96,149,167}173], cathodoluminescence [171,174], and electroluminescence [149,171,175] has been reported from Er implanted GaN. The intensity of this emission increases with increasing annealing temperature [96,158,167,169,171}173] which is most likely due to the reduction of implantation defects. However, the post-implantation annealing procedure could also change the nature of the local environment or the bonding of the Er, and therefore in#uence the emission characteristic. The local environment of Er in GaN is extremely crucial for light emission: GaN "lms implanted with Er and without a co-implantation of O emit at 1.54 m [96,121], but co-implantation of O [96,149,168,169,171}174,176] and to a lesser extent F [171] substantially increases the intensity. The characteristic of the emission line changes, too, as shown in Fig. 13. The excitation mechanism by which energy from the host material is transferred to the intra-4f shell transitions of Er> in semiconductor hosts is still not completely understood. Kim et al. [177}180] report four distinct sites for Er implanted in GaN (Note: the authors do not mean lattice sites in this case, their sites are better described as four di!erent local structural environments [181]. We have shown above that about 90}100% of the RE-elements occupy substitutional Ga-lattice sites independent of the post-implantation annealing processes). The di!erent `sitesa are each excited by di!erent below-gap optical absorption processes and each has di!erent associated photoluminescence. The excitation methods for the di!erent Er sites are attributed to (1) absorption due to implantation damage-induced defects; (2) absorption due to defects or impurities characteristic of the as-grown GaN "lm; (3) an Er-speci"c absorption band which is possibly an Er-related isoelectronic trap and (4) direct Er> 4f shell absorption [177}180,182]. Visible green emission, consisting of emission peaks at 523 and 546 nm, has also been observed in Er implanted GaN [176]. This is similar to the green emission observed in GaN doped with Er during growth [183}186]. The optical properties of other rare earth elements (such as Dy [187], Nd [121,159], Pr [188,189], and Tm [187]) in GaN have also been investigated. Both Dy and Tm implanted in GaN resulted in abundant emissions upon thermal treatment [187] that were attributed to transitions between various levels in the rare earth elements. Strong room temperature red (640 nm) and infrared (several wavelengths including 1.3 m) emissions have been observed in praseodymium (Pr)-implanted GaN [188,189] similar to GaN doped with Pr during growth [190]. Several intra-4f transition emissions also appear for neodymium (Nd) implanted GaN [121]. Kim et al. report that implanted Nd> emission originates from "ve di!erent `sitesa (local environments) for the Nd> with di!erent excitation methods of the Nd> for the di!erent sites [159].

C. Ronning et al. / Physics Reports 351 (2001) 349}385

371

Fig. 13. PL spectra of GaN implanted with Er (60 keV, RT) with co-implanted oxygen (bottom) and without (top). (From Ref. [96])

4. Electrical properties The objective of ion implantation, in the context of semiconductors, is to modify the electrical properties of the semiconducting material. The goal can be either to increase or decrease the conductivity of the semiconductor, which is accomplished by dopant implantation or implantation isolation, respectively. Dopant implantation introduces an electrically active n- or p-type dopant for the purpose of increasing the free carrier concentration. The particular dopant introduced depends on the semiconductor and the desired "nal electrical properties of the semiconductor. Implantation isolation involves the implantation of various elements to produce a high resistivity layer through the introduction of mid-gap levels, that trap both electrons and holes. These levels can be either created through the implantation process itself by defects or through the chemical nature of the implanted species. 4.1. Implantation isolation Kahn et al. were the "rst to report the use of ion implantation to modify the electrical properties of GaN [191]. They implanted Be> and N> to compensate the background n-type behavior. The GaN showed a signi"cantly reduced free carrier concentration after implantation of

372

C. Ronning et al. / Physics Reports 351 (2001) 349}385

1.2;10 cm\ of either Be> or N> and annealing at 10003C. The fact that Be> and N> had the same e!ect implies that the compensation is due to the induced defects and not to the chemical nature of the implanted impurity. As discussed above, the annealing temperature of 10003C is insu$cient to remove all defects within reasonable time. The implantation of the light elements of H [74,151,192] and He [74,151,193], and the isovalent elements of N [192,194}196] and P [193], have been investigated in GaN for the purpose of creating damage during implantation, which facilitates the isolation. The resistivity obtained after implantation scaled with the mass of the implanted ion, i.e. the larger the ion, the larger the concentration of defects, and the higher the resistivity [151,192,193]. Similarly, the higher the implanted dose the higher the observed resistivity [151,192]. However, after He implantation doses exceeding 4;10 cm\, the resistivity actually began to decrease [192]. This is due to an increase in hopping or defect conduction where the trap density is so high that carriers can hop between the defects. Levels generated by the implantation of H [197,198], He [198}200], and N [201] in GaN have been investigated using deep-level transient spectroscopy (DLTS). Three defect levels located at 0.13, 0.16, and 0.20 eV were observed after H implantation [197,198]. The level at 0.20 eV was also present after He implantation, along with two deep levels located at 0.78 and 0.95 eV [198}200]. These defect levels are electron traps. The implantation of N produced a deep level located at 0.67 eV [201], which was attributed to nitrogen interstitials. The concentration of this level increased with increasing implantation dose and decreased after annealing at 9003C. An activation energy of 0.76$0.02 eV was also determined by temperaturedependent resistivity measurements for He- and N-implanted GaN, while the activation energy for H-implanted GaN was 0.29$0.04 eV [192]. This `activationa energy is a measure of the main level controlling the electrical properties and is in good agreement with the observations made by DLTS. Cao et al. implanted Ti, O, Fe, and Cr into GaN to investigate the possibility of chemically induced isolation [202]. The activation energies of the various implanted ions were all similar, indicating that the implantation defects dominate the electrical properties as opposed to the chemical nature of the implanted ion. However, all the implantations reported in this section have resulted in device quality high-resistivity GaN. To understand the in#uence of the implantation-induced damage on the electrical properties of implanted GaN, we implanted Ar into GaN as a control dopant, since it should not introduce any energy levels due to the chemical nature of the dopant. Hall e!ect and C}< electrical measurements showed that all the Ar-implanted GaN samples exhibited a high resistivity in the as-implanted state and thus could not be measured. This is in good agreement with the results of other groups reported above. Recovery of the electrical properties of GaN was observed after annealing the Ar-implanted GaN samples at 11003C for 1 h in a tube furnace. The sheet carrier concentration after annealing was in the range of 2}5;10 cm\. Taking the thickness of 2 m of the GaN "lm into account, the sheet carrier concentration can be converted to a volumetric carrier concentration in the range of 1}2.5;10 cm\, which was the background carrier concentration of the GaN "lms before implantation. This indicates that the implantation-induced electronic traps have been partially removed allowing the bulk properties of the GaN "lm to dominate the Hall e!ect measurement. This is also con"rmed by the high mobility observed in the implanted "lms. Further annealing at 12003C did not change the electrical properties signi"cantly.

C. Ronning et al. / Physics Reports 351 (2001) 349}385

373

Table 3 Electrical properties of GaN implanted with Si and annealed in a tube furnace or RTA at various temperatures and times. Ion energy: 200 keV Implantation conditions

Sheet resistivity (/䊐)

Sheet carrier concentration (cm\)

Mobility (cm/V s)

Percent activation (%)

2.47;10 1.67;10 1.77;10  '10 

6.88;10 1.44;10 1.52;10 N/A

367.47 260.87 232.51 N/A

0.69 0.14 0.15 N/A

12003C, 15 s RTA anneal 1;10 cm\, RT 1;10 cm\, RT 1;10 cm\, 6503C 1;10 cm\, 6503C

3.61;10  '10  2.41;10 5.81;10

5.4;10 N/A 6.5;10 3.99;10

321 N/A 398.56 269.8

0.054 N/A 0.0065 0.004

11003C, 1 h furnace anneal 1;10 cm\, RT 1;10 cm\, 6503C 1;10 cm\, 6503C

1.67;10 1.77;10 2.27;10

5.33;10 1.12;10 4.88;10

70.24 31.51 56.42

53.3 112 48.8

12003C, 1 h furnace anneal 1;10 cm\, RT 1;10 cm\, RT 1;10 cm\, 6503C 1;10 cm\, 6503C

N/A 2.90;10 4.87;10 1.32;10

N/A 1.04;10 1.13;10 9.97;10

N/A 20.74 113.57 47.45

N/A 104 113 99.7

11003C, 15 s RTA anneal 1;10 cm\, RT 1;10 cm\, RT 1;10 cm\, 6503C 1;10 cm\, 6503C

4.2. Donor doping Electrical measurements showed that GaN is also too resistive to measure directly after Si implantation. Annealing at 9003C for 60 s in a RTA-system did not signi"cantly change the resistivity of the samples and, therefore, electrical measurements could not be performed, but they were possible after annealing at 11003C for 15 s. However, the sheet carrier concentrations were low resulting in a very low Si activation percentages (see Table 3). The values are much lower than the 8% reported by Zolper et al. for a similar annealing procedure [71,72,194}196]. However, further work by Zolper et al. has also resulted in similar low activation percentages in the range of &0.05% [17,88,89]. The reason for this discrepancy is unclear but may be due to di!erences in the quality of the host GaN material. The low sheet carrier concentration is similar to that observed after Ar implantation and furnace annealing for our samples. This implies that the carriers observed in the Si-implanted GaN may in fact be due to the native n-type background carriers present in the GaN substrate. This is supported by the high mobility observed in the Si implanted GaN samples, which one would not expect in an implantation damaged crystal. Annealing at 12003C for 15 s in an RTA did not improve the sheet carrier concentration. Thus, either the

374

C. Ronning et al. / Physics Reports 351 (2001) 349}385

Fig. 14. Carrier concentration versus reciprocal temperature as determined from variable temperature Hall-e!ect measurements for GaN implanted with Si at 6503C with an energy of 200 keV and a dose of (a) 1;10 cm\ and (b) 1;10 cm\. Included are the data from GaN samples doped with Si during growth with similar doping levels. The solid lines are "ts of the experimental data, and the resulting activation energy (E ) for each "t is indicated. 

annealing time or temperature are still too low for electrical activation of the implanted Si. Increasing the annealing time should improve this situation. Table 3 also shows the electrical measurements of GaN implanted with Si and annealed in a tube furnace for 1 h at 11003C. The long annealing time provided a much higher percentage of Si-activated compared to RTA annealing with &100% activation for the 1;10 cm\ dose and &50% activation for the 1;10 cm\ dose. These results were con"rmed by C}< measurements. The observed mobilities are low compared to the bulk GaN due to the remaining high concentration of defects (compare with PL-results) that act as scattering centers for the free carriers. These results indicate that the Si-implanted region in the GaN is dominating the electrical properties of the "lm and that the implanted Si ions have become electrically active in the GaN substrate. Fig. 14 shows temperature-dependent Hall measurements for GaN implanted at 6503C with (a) 1;10 cm\ and (b) 1;10 cm\ Si ions along with data from a GaN doped with similar levels of Si during the growth. All samples show very little temperature dependence at low temperatures. This behavior was also observed by other groups [203}206] and can be attributed to impurity band conduction in the GaN [207]. Thus, only data above 200 K were used for the determination

C. Ronning et al. / Physics Reports 351 (2001) 349}385

375

of the activation energy. The GaN sample doped with Si during growth showed a donor activation energy, E , of 17.3 meV. This value is in good agreement with results given in Refs. [205,206,208,209]. Analysis of the 1;10 cm\ Si-implanted GaN sample indicated that two independent donor levels with activation energies of 18.1 and 273.9 meV described the temperature-dependent behavior. The activation energy of the "rst donor level is very close to that observed in the Si-doped GaN sample and, therefore, is due to the implanted Si donors. This con"rms that the implanted Si has become electrically active in the GaN. The second donor level is most likely due to deep levels related to implantation-induced defects. This is in agreement with the results reported in Refs. [197}200], where defect levels with similar activation energies were detected by DLTS. However, Zolper et al. [71,195,196] found only one activation energy level at 62 meV, which is most likely due to a combination of the implanted Si donor level with deep levels and should not be taken as a Si donor energy level. The temperature dependence of the high-dose Si-implanted GaN shows an activation energy of 5.5 meV, which is almost identical to the results determined in the sample doped with Si at a similar level during growth, indicating the electrical activation of the implanted Si ions. The results are in agreement with Ref. [17]. The low activation energy indicates a doping level above the degeneracy limit that is consistent with the high doping level. The electron mobilities in both Si-implanted GaN samples were low and did not change signi"cantly with temperature. This is due to the large amount of remaining defects that act as scattering centers. Higher temperature anneals at 12003C for 1 h in a tube furnace further increased the activation of the implanted Si and resulted in the highest mobility observed (see Table 3). However, this mobility is still lower then that observed in GaN doped during growth. This demonstrates that the optimal annealing temperature for GaN is a bit higher than 12003C, as predicted by the two-thirds rule. High activation percents have also been observed in Si implanted GaN after annealing at 13003C [90,91] and 14003C [73] in a RTA for 10 s. Electrical conductivity was observed in the Ge-implanted GaN after a 1-h 12003C tube furnace anneal. The measured sheet carrier concentrations and mobilities were lower compared to Si, resulting in activation percentages between 15 and 45%. This could be due to the higher remaining amount of implantation induced defects by the heavier ion or due to the deeper donor level of the Ge, as seen by the PL measurements above. However, the implanted Ge dominates the electrical properties of the sample and is electrically active. Other donors, including O [18,69,70], Te [10,75,85], and S [10,75,85], have been implanted and electrically investigated. Oxygen is a shallow donor in GaN [120,210] and may cause background n-type behavior [211]. However, studies on O implanted in GaN indicate a low implantation activation [18,69,70]. This was also found for S and Te implanted into GaN even after hightemperature annealing at 14003C [10,75,85]. Both S and Te have high activation energies of 48$10 and 50$20 meV, respectively, which likely results in the low implantation activation e$ciency. 4.3. Acceptor doping The challenge in ion implantation activation is to reach p-type behavior after implantation of acceptors into a material with n-type background carriers. In the case of GaN, this challenge is even more di$cult due to the absence of shallow acceptors. Table 4 summarizes the electrical

376

C. Ronning et al. / Physics Reports 351 (2001) 349}385

Table 4 Electrical properties of GaN implanted with acceptors annealed in a tube furnace at various temperatures and times. The ion implantation was performed at room temperature Ion species and implantation conditions

Sheet resistivity (/䊐)

Sheet carrier concentration (cm\)

Mobility (cm/V s)

Percent activation

11003C, 1 h furnace annealing Mg, 160 keV, 1;10 cm\ Mg, 160 keV, 1;10 cm\ Ca, 300 keV, 1;10 cm\ Ca, 300 keV, 1;10 cm\

1.02;10 8.46;10 4.96;10 8.14;10

1.90;10 1.10;10 2.96;10 1.10;10

330.4 67.59 425.1 69.46

N/A N/A N/A N/A

1.66;10  '10  6.31;10  '10 

3.40;10 N/A 8.22;10 N/A

109.5 N/A 120.4 N/A

N/A (n-type) N/A N/A (n-type) N/A

12503C, 30 min furnace annealing Mg, 60 keV, 1;10 cm\ Mg, 60 keV, 1;10 cm\ Mg, 60 keV, 1;10 cm\ Mg, 120 keV, 1;10 cm\ Mg, 120 keV, 1;10 cm\

1.4;10 2;10 1.57;10 9.5;10 1.63;10

5.3;10 3;10 5.5;10 6.7;10 1.3;10

80 100 71 97 300

N/A (n-type) n/p-type N/A (n-type) N/A (n-type) N/A (n-type)

13003C, 10 min furnace annealing Mg, 60 keV, 1;10 cm\ Mg, 60 keV, 1;10 cm\ Mg, 120 keV, 1;10 cm\

2.06;10 4.0;10 6.3;10

8;10 2.8;10 n"1.1;10

38 2.2 63

n/p-type n/p-type N/A (n-type)

12003C, 1 h furnace annealing Mg, 160 keV, 1;10 cm\ Mg, 160 keV, 1;10 cm\ Ca, 300 keV, 1;10 cm\ Ca, 300 keV, 1;10 cm\

(n-type) (n-type) (n-type) (n-type)

properties after implantation of Mg and Ca followed by subsequent annealing at various temperatures. Annealing at high temperatures and di!erent conditions enabled electrical measurements to be performed. However, the sign of the Hall coe$cient was, in the most cases, negative indicating n-type behavior in the implanted samples as opposed to the expected p-type behavior. The results of most of other studies [17,18,45,71,72,194}196,212] are in agreement with this result and showed GaN remaining n-type after Mg implantation and annealing at high temperatures. Cao et al. [84,85] were able to achieve p-type GaN after Mg implantation and annealing at very high temperatures (1100}14003C). The large deviations in the reported electrical properties of Mgimplanted GaN is due to the large activation energy of &200 meV, which means that at room temperature only &1% of the Mg ions are ionized. Thus, for a Mg implantation dose of 1;10 cm\ and an implantation activation of 100% the sheet hole concentration is expected to be 1;10 cm\, which is approximately the order of the n-type background carriers. In the case of Ca, the same argument must be applied due to a similar high activation energy of &160 meV. Only the low mobility observed in the acceptor implanted samples compared to Ar-implanted samples is an indication of electrical activation, which is compensated by the native n-type carriers.

C. Ronning et al. / Physics Reports 351 (2001) 349}385

377

Other acceptors have been predicted to be shallow, but reports of C [10,19,75,85] and Be [40] implantation have not resulted in p-type GaN. Zolper et al. reported that GaN implanted with C had increasing n-type behavior with increasing annealing temperature. 4.4. Device applications The "rst use of ion implantation for device processing was by Kahn et al., who used N> and Be> implants to increase the performance of Schottky barriers on GaN [191]. The implantation lowered the native free carrier concentration at the surface such that deposited Schottky contacts had good rectifying properties. Implantation has been also used to improve ohmic contacts by increasing the free carrier concentration at the surface of the "lm under the contact [213,214]. In both cases, Si implantations with the associated activation anneal lowered the speci"c contact resistivity of the metal contact. Simple p}n junctions have been accomplished either by implantation of Si into Mg-doped GaN "lms [45,83,215] or by implantation of Mg into Si-doped GaN [76]. The junctions yielded rectifying behavior after annealing, but they had relatively high turn-on voltages, low breakdown voltages, low forward currents and high ideality factors due to the remaining implantation defects. Zolper et al. [70,72,216] produced a JFET using Si implantation to produce the n-type channel along with the n-type source and the drain regions. Implanted Ca was used as the p-type gate in the JFET. Following an 11503C activation anneal the JFET showed good channel modulation with a measured transconductance of 7 mS/mm and a saturation current of 33 mA/mm. The transconductance value is low compared to other devices produced in the III-nitrides and is attributed to a high access resistance or a low mobility in the channel. Similarly to the implanted p}n junctions there is still signi"cant damage remaining after annealing at this temperature that is expected to degrade the performance of the device.

5. Summary, conclusions, and future work The crystal structure of GaN is very resistant to ion bombardment due to the high ionicity of the Ga}N bond. Very high doses are required for amorphization; i.e. the amorphization threshold is much higher than in other semiconductors. The damage caused during implantation consists of vacancies, interstitials, anti-site defects, and extended defects including dislocations and stacking faults. Amorphization in GaN occurs in small local regions. These regions are subsequently enlarged with increasing implantation dose until the crystal structure collapses to create an amorphous layer. Structural defects can be readily created with medium implantation doses (10}10 cm\) and lead to lattice expansion. Computer simulations usually overestimate the damage compared to that experimentally observed. The discrepancy is due to signi"cant dynamic annealing during implantation and due to the strong bonding. Implantation at high-temperatures does not necessarily result in lower damage. The implanted impurities occupy de"ned lattice sites: the alkali elements mainly occupy interstitial sites; whereas, all other elements have been found on substitutional sites directly after ion implantation. The optical properties are greatly a!ected by implantation resulting in a complete loss of the luminescence even for low doses. The largest e!ect is the introduction of

378

C. Ronning et al. / Physics Reports 351 (2001) 349}385

Fig. 15. Di!usion, recovery, and activation processes of ion-implanted impurities in GaN as function of annealing temperature.

nonradiative recombination centers due to the damage. The introduced defects have mainly deep levels within the band gap; therefore, implanted GaN is electrically highly resistive. The damage must be annealed out to achieve optical and electrical activation of the implanted impurities. Several sophisticated annealing procedures to achieve high annealing temperatures without decomposition of the GaN surface were described herein. Fig. 15 summarizes schematically the fundamental di!usion, recovery, and activation processes that occur in ion implanted GaN as a function of annealing temperature. Implanted light elements undergo measurable di!usion via interstitial mechanisms at low temperatures. Lithium becomes mobile at 4503C until it recombines with a vacancy resulting in substitutional Li that is stable to higher temperatures. Hydrogen should be also mobile on interstitial sites at even lower temperatures, as complete out-di!usion occurs at 5003C and 8003C for p- and n-type GaN, respectively. With the exception of Zn and Mg at &12003C, no other implanted element (even Be) showed signi"cant di!usion in the investigated temperature ranges. The high thermal stability of impurities is due to the strong bonding con"gurations, i.e. the high Debye temperature of GaN. The onset of di!usion of implantation defects starts around 6003C. Each interstitial atom di!uses on interstitial sites until it "nds a vacancy for recombination. This e!ect occurs only when amorphization is not reached; it results in a reduction and elimination of the lattice expansion that is visible in a change of the RBS and XRD spectra. However, anti-site defects are likely to be created during this recovery of the structural defects. The recovery of point defects, visible by PL and PAC, starts around 8003C and is mainly associated with the onset of vacancy di!usion. Small microscopic areas in the range of several tens of nanometers are completely recovered resulting in optical activation of some implanted impurities. These recovered regions grow with increasing annealing temperature to the micrometer

C. Ronning et al. / Physics Reports 351 (2001) 349}385

379

range at 12003C, which is the size of excitons in GaN. Thus, most of the implanted impurities become optically activated, but defect complexes are still present. The latter require higher temperatures to break up due to the additional formation or/and coulomb energy. These defects cause an intense yellow band in the PL spectra and are visible by PAC in this temperature range. The concentration of defect complexes signi"cantly decreases when annealed above 13003C leading to macroscopic electrical activation that can be measured by Hall or C}<. Acceptor-type activation in GaN is much more di$cult to prove compared to n-type activation due to both the native shallow donors in the virgin material and the absence of shallow acceptors. The future realization of high-performance devices on the basis of ion implanted GaN should be a matter of process engineering. However, research must be conducted in several areas. (i) Annealing procedures in the 1400}15003C range must be developed that do not result in the decomposition of the host materials for annealing times on the order of several minutes. (ii) Laser annealing has not been reported at this time. (iii) The electrical activation of implanted acceptors must be improved and, therefore, the properties of alternative implanted acceptors should be studied in detail. (iv) New, defect-free GaN substrate materials grown by lateral epitaxial overgrowth (LEO) [217] or pendeo-epitaxy [218] must be used for implantation studies and their improvement examined. (v) The properties of prototype devices must be investigated and reported. Acknowledgements The authors thank H. HofsaK { for valuable discussions and his support. We also thank Cree Research, Inc. for supplying the SiC substrates on which the GaN "lms were grown. C.R. is grateful for funding by the DFG (Ro 1198/2-1) during his stay at NCSU. R. Davis was supported in part by the Kobe Steel, Ltd. Distinguished University Professorship. The EC-studies at the University of GoK ttingen were "nancially supported by the German Bundesminister fuK r Bildung, Wissenschaft, Forschung und Technologie. Support for the research conducted at NCSU was provided by the O$ce of Naval Research under Contracts N00014-98-1-0384 (Colin Wood, monitor) and N0001498-1-0654 (John Zolper, monitor), the Kenan Institute for Engineering, Technology & Science and by Northrop Grumman. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]

S. Nakamura, M. Senoh, N. Iwasa, S. Nagahama, Jpn. J. Appl. Phys. 34 (1995) L797. S. Nakamura, G. Fasol, The Blue Laser Diode, Springer, Berlin, 1997. Cree Research Inc., Durham NC 27713. S. Nakamura, M. Senoh, S. Nagahama, N. Iwasa, T. Yamada, T. Matsushita, H. Kiyoku, Y. Sugimoto, T. Kozaki, H. Umemoto, M. Sano, K. Chocho, Appl. Phys. Lett. 72 (1998) 211. S. Strite, H. Morkoc7 , J. Vac. Sci. Technol. B 10 (1992) 1237. S.C. Jain, M. Wilander, J. Narayan, R. van Overstraeten, J. Appl. Phys. 87 (2000) 965. O. Ambacher, J. Phys. D 31 (1998) 2653. S.J. Pearton, J.C. Zolper, R.J. Shul, F. Ren, J. Appl. Phys. 86 (1999) 1. C.B. Vartuli, S.J. Pearton, C.R. Abernathy, J.D. MacKenzie, E.S. Lambers, J.C. Zolper, J. Vac. Sci. Technol. B 14 (1996) 3523. R.G. Wilson, J.M. Zavada, X.A. Cao, R.K. Singh, S.J. Pearton, H.J. Guo, S.J. Scarvepalli, M. Fu, J.A. Sekhar, S.J. Pennycook, R.J. Shu, J. Han, D.J. Rieger, J.C. Zolper, C.R. Abernathy, J. Vac. Sci. Technol. A 17 (1999) 1226.

380

C. Ronning et al. / Physics Reports 351 (2001) 349}385

[11] See for example: J.F. Ziegler (Ed.), Handbook of Ion Implantation Technology, Elsevier Science Publishers, The Netherlands, 1992, pp. 271}362; S.K. Ghandi, VSLI Fabrication Principles: Silicon and Gallium Arsenide, Wiley, New York, 1982 (Chapter 6). [12] H.H. Tan, J.S. Williams, J. Zou, D.J.H. Cockayne, S.J. Pearton, R.A. Stall, Appl. Phys. Lett. 69 (1996) 2364. [13] N. Parikh, A. Suvkhanov, M. Lioubtchenko, E.P. Carlson, M.D. Bremser, D. Bray, J. Hunn, R.F. Davis, Nucl. Instr. and Meth. B 127}128 (1997) 463. [14] J.F. Ziegler, J.P. Biersack, U. Littmark, The Stopping and Ranges of Ions in Solids, Pergamon, New York, 1985. [15] H.H. Tan, J.S. Williams, J. Zou, D.J.H. Cockayne, S.J. Pearton, R.A. Stall, Appl. Phys. Lett. 69 (1996) 2364. [16] H.H. Tan, J.S. Williams, C. Yuan, S.J. Pearton, Mater. Res. Soc. Symp. Proc. 395 (1996) 807. [17] J.C. Zolper, M.H. Crawford, J.S. Williams, H.H. Tan, R.A. Stall, Nucl. Instr. and Meth. B 127/128 (1997) 467. [18] A. Edwards, M.V. Rao, B. Molar, A.E. Wickenden, O.W. Holland, P.H. Chi, J. Electron. Mater. 26 (1997) 334. [19] B. Mensching, C. Liu, B. Rauschenbach, K. Kornitzer, W. Ritter, Mater. Sci. Eng. B 50 (1997) 105. [20] C. Liu, B. Mensching, M. Zeitler, K. Volz, B. Rauschenbach, Phys. Rev. B 57 (1998) 2530. [21] C. Liu, M. Schreck, A. Wenzel, B. Mensching, B. Rauschenbach, Appl. Phys. A 70 (2000) 53. [22] A. Wenzel, C. Liu, B. Rauschenbach, Mater. Sci. Eng. B 59 (1999) 191. [23] A. Suvkhanov, J. Hunn, W. Wu, D. Tompson, K. Price, N. Parikh, E. Irene, R.F. Davis, L. Krasnobaev, Conference Proceeding, 1999. [24] E. Alves, M.F. DaSilva, J.C. Soares, J. Bartels, R. Vianden, C.R. Abernathy, S.J. Pearton, MRS Internet J. Nitride Semicond. Res. 4S1 (1999) G11.2. [25] S.M. Myers, T.J. Headly, C.R. Hills, J. Han, G.A. Peterson, C.H. Seager, W.R. Wampler, MRS Internet J. Nitride Semicond. Res. 4S1 (1999) G5.8. [26] C.H. Seager, S.M. Myers, G.A. Peterson, J. Han, T. Headley, J. Appl. Phys. 85 (1999) 2568. [27] W.R. Wampler, S.M. Myers, MRS Internet J. Semicond. Res. 4S1 (1999) G3.73. [28] W. Jiang, W.L. Weber, S. Thevuthasan, G.J. Exarhos, B.J. Bozlee, MRS Internet J. Nitride Semicond. Res. 4S1 (1999) G6.15. [29] S.O. Kucheyev, J.S. Williams, C. Jagadish, J. Zou, G. Li, Phys. Rev. B 62 (2000) 7510. [30] W. Jiang, W.J. Weber, S. Thevuthasan, J. Appl. Phys. 87 (2000) 7671. [31] H.H. Tan, C. Jagadish, J.S. Williams, J. Zoa, D.J.H. Cockayne, A. Sikorski, J. Appl. Phys. 77 (1995) 87. [32] J.C. Zolper, J. Han, S.B. Van Deusen, M.H. Crawford, R.M. Biefeld, J. Jun, T. Suski, J.M. Baranowski, S.J. Pearton, Mater. Res. Soc. Symp. 482 (1998) 979. [33] H.H. Tan, J.S. Williams, J. Zoa, D.J.H. Cockayne, S.J. Pearton, J.C. Zolper, R.A. Stall, Appl. Phys. Lett. 72 (1998) 1190. [34] S. Strite, P.W. Epperlein, A. Dommann, A. Rockett, R.F. Broom, Mater. Res. Soc. Symp. Proc. 395 (1996) 795. [35] C. Ronning, M. Dalmer, M. Deicher, M. Restle, M.D. Bremser, R.F. Davis, H. HofsaK ss, Mater. Res. Soc. Symp. Proc. 468 (1997) 407. [36] E. Alves, M.F. da Silva, J.G. Marques, J.C. Soares, K. Freitag, Mater. Sci. Eng. B 59 (1999) 207. [37] C. Liu, B. Mensching, K. Volz, B. Rauschenbach, Appl. Phys. Lett. 71 (1997) 2313. [38] G.C. Chi, B.J. Pong, C.J. Pan, Y.C. Teng, Mater. Res. Soc. Symp. Proc. 482 (1998) 1027. [39] B.J. Pong, C.J. Pan, Y.C. Teng, G.C. Chi, W.H. Li, K.C. Lee, C.H. Lee, J. Appl. Phys. 83 (1998) 5992. [40] C. Ronning, K.J. Linthicum, E.P. Carlson, P.J. Hartlieb, D.B. Thomson, T. Gehrke, R.F. Davis, Mater. Res. Soc. Proc. 537 (1999) and MRS Internet J. Nitride Semicond. Res. 4S1 (1999) G3.17. [41] C. Liu, B. Mensching, K. Volz, B. Rauschenbach, Appl. Phys. Lett. 71 (1997) 2313. [42] M. Rubin, N. Newman, J.S. Chan, T.C. Fu, J.T. Ross, Appl. Phys. Lett. 64 (1994) 64. [43] S. Strite, A. Pelzmann, T. Suski, M. Leszczynski, J. Jun, A. Rockett, M. Kamp, K.J. Ebeling, MRS Internet J. Nitride Semicond. Res. 2 (1997) 15. [44] A. Pelzmann, S. Strite, A. Dommann, C. Kirchner, M. Kamp, K.J. Ebeling, A. Nazzal, MRS Internet J. Nitride Semicond. Res. 2 (1997) 4. [45] H.P. Maruska, M. Lioubtchenko, T.G. Tetreault, M. Osinski, S.J. Pearton, M. Schurman, R. Vaudo, S. Sakai, Q. Chen, R.J. Shul, Mater. Res. Soc. Symp. Proc. 483 (1998) 333. [46] J.A. Van Vechten, Phys. Rev. B 7 (1973) 1479. [47] J. Karpinski, S. Porowski, J. Crystal Growth 66 (1984) 11.

C. Ronning et al. / Physics Reports 351 (2001) 349}385

381

[48] S. Porowski, I. Grzegory, S. Krukowski, Mater. Res. Soc. Symp. Proc. 499 (1998) 349. [49] S.W. King, J.P. Barnak, M.D. Bremser, K.M. Tracy, C. Ronning, R.F. Davis, R.J. Nemanich, J. Appl. Phys. 84 (1998) 5248. [50] B. Molnar, A.E. Wickenden, M.V. Rao, Mater. Res. Soc. Symp. Proc. 423 (1996) 183. [51] M.K. Kelly, O. Ambacher, B. Dahlheimer, G. Groos, R. Dimitrov, H. Angerer, M. Stutzmann, Appl. Phys. Lett. 69 (1996) 1749. [52] M.K. Kelly, O. Ambacher, R. Dimitrov, H. Angerer, R. Handschuh, M. Stutzmann, Mater. Res. Soc. Symp. Proc. 482 (1998) 973. [53] C.B. Vartuli, S.J. Pearton, C.R. Abernathy, J.D. MacKenzie, E.S. Lambers, J.C. Zopler, J. Vac. Sci. Technol. B 14 (1996) 3523. [54] T. Suski, J. Jun, M. Leszczynski, H. Teisseyre, I. Grzegory, S. Porowski, J.M. Baranowski, A. Rocket, S. Strite, A. Stonert, A. Turos, H.H. Tan, J.S. Williams, C. Jagadish, Mater. Res. Soc. Symp. Proc. 482 (1998) 949. [55] M. Fu, V. Scarvepalli, R.K. Singh, C.R. Abernathy, X. Cao, S.J. Pearton, J.A. Sekhar, Mater. Res. Soc. Symp. 483 (1998) 345. [56] E.P. Carlson, Ph.D. Thesis, North Carolina State University. [57] A. StoK tzler, M. Deicher, University of Konstanz, private communication. [58] J.I. Pankove, J.A. Hutchby, J. Appl. Phys. 47 (1976) 5387. [59] J.C. Zopler, D.J. Rieger, A.G. Baca, S.J. Pearton, J.W. Lee, R.A. Stall, Appl. Phys. Lett. 69 (1996) 538. [60] C. Ronning, M. Dalmer, M. Uhrmacher, M. Restle, U. Vetter, L. Ziegeler, H. HofsaK ss, T. Gehrke, K. JaK rrendahl, R.F. Davis, J. Appl. Phys. 87 (2000) 2149. [61] C. Ronning, H. HofsaK ss, A. StoK tzler, M. Deicher, E.P. Carlson, P.J. Hartlieb, T. Gehrke, P. Rajagopal, R.F. Davis, MRS Internet J. Nitride Semicond. Res. 5S1 2000 W11.44. [62] W.S. Wong, L.F. Schloss, G.S. Sudhir, B.P. Linder, K.-M. Yu, E.R. Weber, T. Sands, N.W. Cheung, Mater. Res. Soc. Symp. 449 (1997) 1011. [63] J.M. Zavada, R.G. Wilson, C.R. Abernathy, S.J. Pearton, Appl. Phys. Lett. 64 (1994) 2724. [64] J.M. Zavada, R.G. Wilson, S.J. Pearton, C.R. Abernathy, Mater. Res. Soc. Symp. Proc. 339 (1994) 553. [65] R.G. Wilson, S.J. Pearton, C.R. Abernathy, J.M. Zavada, Appl. Phys. Lett. 66 (1995) 2238. [66] R.G. Wilson, C.B. Vartuli, C.R. Abernathy, S.J. Pearton, J.M. Zavada, Solid State Electron. 38 (1995) 1329. [67] R.G. Wilson, S.J. Pearton, C.R. Abernathy, J.M. Zavada, J. Vac. Sci. Technol. A 13 (1995) 719. [68] R.G. Wilson, Electrochem. Soc. Proc. 95-21 (1995) 152. [69] J.C. Zolper, R.G. Wilson, S.J. Pearton, R.A. Stall, Appl. Phys. Lett. 68 (1996) 1945. [70] J.C. Zolper, R.G. Wilson, S.J. Pearton, R.A. Stall, Mater. Res. Soc. Symp. Proc. 423 (1996) 189. [71] J.C. Zolper, M.H. Crawford, A.J. Howard, S.J. Pearton, C.R. Abernathy, C.B. Vartuli, C. Yuan, R.A. Stall, J. Ramer, S.D. Hersee, R.G. Wilson, Mater. Res. Soc. Symp. Proc. 395 (1996) 801. [72] J.C. Zolper, S.J. Pearton, R.G. Wilson, R.A. Stall, Proceedings of the Eleventh International Conference on Ion Implantation Technology, Austin, TX, June 16}21, 1996, p. 705. [73] X.A. Cao, C.R. Abernathy, R.K. Singh, S.J. Pearton, M. Fu, V. Scarvepalli, J.A. Sekhar, J.C. Zolper, D.J. Rieger, J. Han, T.J. Drummond, R.J. Shul, R.G. Wilson, Appl. Phys. Lett. 73 (1998) 229. [74] S.J. Pearton, R.G. Wilson, J.M. Zavada, J. Han, R.J. Shul, Appl. Phys. Lett. 73 (1998) 1877. [75] X.A. Cao, R.G. Wilson, J.C. Zoper, S.J. Pearton, J. Han, R.J. Shul, D.J. Rieger, R.K. Singh, M. Fu, V. Scarvepalli, J.A. Sekhar, J.M. Zavada, J. Electron. Mater. 28 (1999) 261. [76] E.V. Kalinina, V.A. Solov'ev, A.S. Zubrilov, V.A. Dmitriev, A.P. Kovarsky, MRS Internet J. Nitride Semicond. Res. 4S1 (1999) G6.53. [77] S.J. Pearton, S. Bendi, K.S. Jones, V. Krishnamoorthy, R.G. Wilson, F. Ren, R.F. Karlicek Jr., R.A. Stall, Appl. Phys. Lett. 73 (1998) 1877. [78] J. Neugebauer, C.G. Van de Walle, Phys. Rev. B 50 (1994) 8067. [79] J. Neugebauer, C.G. Van de Walle, Phys. Rev. Lett. 75 (1995) 4452. [80] J. Neugebauer, C.G. Van de Walle, Appl. Phys. Lett. 68 (1996) 1829. [81] J. Neugebauer, C.G. Van de Walle, Mater. Res. Soc. Symp. Proc. 395 (1996) 645. [82] K.T. Short, S.J. Pearton, J. Electrochem. Soc. 135 (1988) 2835.

382

C. Ronning et al. / Physics Reports 351 (2001) 349}385

[83] X.A. Cao, J.R. LaRoche, F. Ren, S.J. Pearton, J.R. Lothian, R.K. Singh, R.G. Wilson, H.J. Guo, S.J. Pennycook, Solid State Electron. 43 (1999) 1235. [84] R.G. Wilson, J.M. Zavada, X.A. Cao, R.K. Singh, S.J. Pearton, H.J. Guo, S.J. Pennycook, M. Fu, J.A. Sekhar, V. Scarvepalli, R.J. Shul, J. Han, D.J. Rieger, J.C. Zolper, C.R. Abernathy, J. Vac. Sci. Technol. A 17 (1999) 1226. [85] X.A. Cao, S.J. Pearton, R.K. Singh, C.R. Abernathy, J. Han, R.J. Shul, D.J. Rieger, J.C. Zolper, R.G. Wilson, M. Fu, J.A. Sekhar, H.J. Guo, S.J. Pennycook, MRS Internet J. Nitride Semicond. Res. 4S1 (1999) G6.33. [86] G. Dang, X.A. Cao, F. Ren, S.J. Pearton, J. Han, A.G. Baca, R.J. Shul, J. Vac. Sci. Technol. B 17 (1999) 2015. [87] S.J. Pearton, H. Cho, J.R. LaRoche, F. Ren, R.G. Wilson, J.W. Lee, Appl. Phys. Lett. 75 (1999) 2939. [88] J.C. Zolper, H.H. Tan, J.S. Williams, J. Zou, D.J.H. Cockayne, S.J. Pearton, M.H. Crawford, R.F. Karlicek Jr., Appl. Phys. Lett. 70 (1997) 2729. [89] J.C. Zolper, S.J. Pearton, J.S. Williams, H.H. Tan, R.J. Karlicek Jr., Mater. Res. Soc. Symp. 449 (1997) 981. [90] J.C. Zolper, J. Han, R.M. Biefeld, S.B. Van Deusen, W.R. Wampler, S.J. Pearton, J.S. Williams, H.H. Tan, R.J. Karlicek Jr., R.A. Stall, Mater. Res. Soc. Symp. 468 (1997) 401. [91] J.C. Zolper, J. Han, R.M. Biefeld, S.B. Van Deusen, W.R. Wampler, D.J. Reiger, S.J. Pearton, J.S. Williams, H.H. Tan, R.F. Karlicek Jr., R.A. Stall, J. Electron. Mater. 27 (1998) 179. [92] R.G. Wilson, J.M. Zavada, X.A. Cao, R.K. Singh, S.J. Pearton, H.J. Guo, S.J. Pennycook, M. Fu, J.A. Sekhar, V. Scarvepalli, R.J. Shul, J. Han, D.J. Rieger, J.C. Zolper, C.R. Abernathy, J. Vac. Sci. Technol. A 17 (1999) 1226. [93] T. Wichert et al., Appl. Phys. A 48 (1989) 59; in: G. Langouche (Ed.), Hyper"ne Interactions of Defects in Semiconductors, Elsevier, Amsterdam, 1992, p. 77!. [94] H. HofsaK ss, G. Lindner, Phys. Rep. 201 (1991) 123; Hyper"ne Interactions 97/98 (1996) 247. [95] M. Dalmer, M. Restle, M. Sebastian, U. Vetter, H. HofsaK ss, M.D. Bremser, C. Ronning, R.F. Davis, U. Wahl, K. Bharuth-Ram, ISOLDE Collaboration, J. Appl. Phys. 84 (1998) 3085. [96] M. Dalmer, M. Restle, A. StoK tzler, U. Vetter, H. HofsaK ss, M.D. Bremser, C. Ronning, R.F. Davis, Mater. Res. Soc. Symp. Proc. 482 (1998) 1021. [97] H. Kobayashi, W.M. Gibson, Appl. Phys. Lett. 74 (1999) 2355. [98] U. Wahl, A. Vantomme, G. Langouche, J.P. Araujo, L. Peralta, J.G. Correia, ISOLDE collaboration, J. Appl. Phys. 88 (2000) 1319. [99] H. Kobayashi, W.M. Gibson, Appl. Phys. Lett. 73 (1998) 1406. [100] H. Kobayashi, W.M. Gibson, J. Vac. Sci. Technol. A 17 (1999) 2132. [101] M. Fanciulli, M. Lindroos, G. Weyer, T.D. Moustakas, ISOLDE Collaboration, Mater. Sci. Forum 196}201 (1995) 61. [102] R. Dingle, D.D. Sell, S.E. Stokowski, M. Ilegems, Phys. Rev. B 4 (1971) 1211. [103] W. GoK tz, N.M. Johnson, R.A. Street, H. Amano, I. Akasaki, Appl. Phys. Lett. 66 (1995) 1340. [104] S. Fischer, C. Wetzel, E.E. Haller, B.K. Meyer, Appl. Phys. Lett. 67 (1995) 1298. [105] P. Perlin, J. Camassel, W. Knap, T. Taliercio, J.C. Chervin, T. Suski, I. Grzegory, S. Porowski, Appl. Phys. Lett. 67 (1995) 2524. [106] W. GoK tz, L.T. Romano, B.S. Krusor, N.M. Johnson, R.J. Molnar, Appl. Phys. Lett. 69 (1996) 242. [107] P. Boguslawski, E. Briggs, T.A. White, M.G. Wensell, J. Bernholc, Mater. Res. Soc. Symp. Proc. 339 (1994) 693. [108] P. Boguslawski, E.L. Briggs, J. Bernholc, Appl. Phys. Lett. 69 (1996) 233. [109] S. Fischer, C. Wetzel, E.E. Haller, B.K. Meyer, Appl. Phys. Lett. 67 (1995) 1298. [110] A. Ishibashi, H. Takeishi, M. Mannoh, Y. Yabuuchi, Y. Ban, J. Electron. Mater. 25 (1996) 779. [111] E.R. Glaser, T.A. Kennedy, S.W. Brown, J.A. Freitas, W.G. Perry, M.D. Bremser, T.W. Weeks, R.F. Davis, Mater. Res. Soc. Symp. Proc. 395 (1996) 667. [112] T. Ogino, M. Aoki, Jpn. J. Appl. Phys. 19 (1980) 2395. [113] J. Neugebauer, C.G. Van de Walle, Appl. Phys. Lett. 69 (1996) 503. [114] T. Mattila, R.M. Nieminen, Phys. Rev. B 55 (1997) 9571. [115] E.F. Schubert, Doping in III}V Semiconductors, University of Cambridge Press, New York, 1993. [116] X. Zhang, P. Kung, A. Saxler, D. Walker, T.C. Wang, M. Razeghi, Appl. Phys. Lett. 67 (1995) 1745. [117] A. StoK tzler, R. Weissenborn, M. Deicher, ISOLDE Collaboration MRS Internet J. Nitride Semicond. Res. 5S1 (2000) W12.9. [118] R.D. Cunningham, R.W. Brander, N.D. Knee, D.K. Wickenden, J. Lumin. 5 (1972) 21.

C. Ronning et al. / Physics Reports 351 (2001) 349}385

383

[119] S. Nakamura, T. Mukai, M. Senoh, Jpn. J. Appl. Phys. 31 (1992) 2883. [120] W. GoK tz, R.S. Kern, C. Chen, H. Liu, D.A. Steigerwald, R.M. Fletcher, Phys. Rev. B 59 (1999) 211. [121] E. Silkowski, Y.K. Yeo, R.L. Hengehold, M.A. Khan, T. Lei, K. Evans, C. Cerny, Mater. Res. Soc. Symp. Proc. 395 (1996) 813. [122] E.F. Schubert, I.D. Goepfert, W. Griueshaber, J.M. Redwing, Appl. Phys. Lett. 71 (1997) 921. [123] A. Cremades, L. GoK rgens, O. Ambacher, M. Stutzmann, F. Scholz, Phys. Rev. B 61 (2000) 2812. [124] S. Fischer, G. Steude, D.M. Hofmann, F. Kurth, F. Anders, M. Topf, B.K. Meyer, F. Bertram, M. Schmidt, J. Christen, L. Eckey, J. Holst, A. Ho!mann, B. Mesching, B. Rauschenbach, J. Crystal Growth 189/190 (1998) 556, and references therein. [125] H. Amano, M. Kitoh, K. Hiramatsu, I. Akasaki, J. Electrochem. Soc. 137 (1990) 1639. [126] M. Smith, G.D. Chen, J.Y. Lin, H.X. Jiang, A. Salvador, B.N. Sverdlov, A. Botchkarev, H. Morkoc7 , B. Goldenberg, Appl. Phys. Lett. 68 (1996) 1883. [127] M.R. Khan, T. Detchproh, H. Nakayama, K. Hiramatsu, N. Sawaki, Solid State Phenom. 55 (1997) 218. [128] M. Kunzer, J. Baur, U. Kaufmann, J. Schneider, H. Amano, I. Akasaki, Solid State Electron. 41 (1997) 189. [129] L. Eckey, U. Von Gfug, J. Holst, A. Ho!mann, B. Schineller, K. Heime, M. Heuken, O. SchoK n, R. Beccard, J. Crystal Growth 189/190 (1998) 523. [130] P. Kozodoy, S. Keller, S.P. DenBaars, U.K. Mishra, J. Crystal Growth 195 (1998) 265. [131] B. Schineller, A. Guttzeit, P.H. Lim, M. Schwambera, K. Heime, O. SchoK n, M. Heuken, J. Crystal Growth 195 (1998) 274. [132] E. Oh, H. Park, Y. Park, Appl. Phys. Lett. 72 (1998) 70. [133] U. Kaufmann, M. Kunzer, M. Maier, H. Obloh, A. Ramakrishnan, B. Santic, P. Schlotter, Appl. Phys. Lett. 72 (1998) 1326. [134] A.K. Viswanath, E.J. Shin, J.I. Lee, S. Yu, D. Kim, B. Kim, Y. Choi, C.H. Hong, J. Appl. Phys. 83 (1998) 2272. [135] T.W. Kang, S.H. Park, H. Song, T.W. Kim, G.S. Yoon, C.O. Kim, J. Appl. Phys. 84 (1998) 2082. [136] J.K. Sheu, Y.K. Su, G.C. Chi, B.J. Pong, C.Y. Chen, C.N. Huang, W.C. Chen, J. Appl. Phys. 84 (1998) 4590. [137] S. Hess, R.A. Taylor, J.F. Ryan, N.J. Cain, V. Roberts, J. Roberts, Phys. Stat. Sol. 210 (1998) 465. [138] D. Corlatan, J. KruK ger, C. Kisielowski, R. Klockenbrink, Y. Kim, S.Y. Peyrot, M. Rubin, E.R. Weber, Mater. Res. Soc. Symp. Proc. 482 (1998) 673. [139] P.H. Lim, B. Schineller, O. SchoK n, K. Heime, M. Heuken, J. Crystal Growth 205 (1999) 1. [140] M.A. Reshchikov, G.C. Yi, B.W. Wessels, MRS Internet J. Nitride Semicond. Res. 4S1 (1999) G118. [141] F. Shahedipour, B.W. Wessels, Appl. Phys. Lett. 76 (2000) 3011. [142] A.D. Hanser, Ph.D. Thesis, North Carolina State University. [143] F. Bernardini, V. Fiorentini, A. Bosin, Appl. Phys. Lett. 70 (1997) 2990. [144] C. Ronning, E.P. Carlson, D.B. Thomson, R.F. Davis, Appl. Phys. Lett. 73 (1998) 1622. [145] K.H. Ploog, O. Brandt, J. Vac. Sci. Technol. B 16 (1998) 1609. [146] G. Neugebauer, C.G. Van der Walle, Appl. Phys. Lett. 85 (1999) 3003. [147] D.J. Dewsnip, A.V. Andrianov, I. Harrison, J.W. Orton, D.E. Lacklison, G.B. Ren, S.E. Hooper, T.S. Cheng, C.T. Foxon, Semicond. Sci. Technol. 13 (1998) 500. [148] J.I. Pankove, J.A. Hutchby, J. Appl. Phys. 24 (1974) 281. [149] J.T. Torvik, R.J. Feuerstein, C.H. Qiu, J.I. Pankove, F. Namavar, Mater. Res. Soc. Symp. Proc. 482 (1998) 579. [150] T. Suski, J. Jun, M. Leszczynski, H. Teisseyre, S. Strite, A. Rockett, A. Pelzmann, M. Kamp, K.J. Ebeling, J. Appl. Phys. 84 (1998) 1155. [151] C. Uzan-Saguy, J. Salzman, R. Kalish, V. Richter, U. Tisch, S. Zamir, S. Prawer, Appl. Phys. Lett. 74 (1999) 2441. [152] M.G. Weinstein, M. Stavola, C.Y. Song, C. Bozdog, H. Przbylinski, G.D. Watkins, S.J. Pearton, R.G. Wilson, MRS Internet J. Nitride Semicond. Res. 4S1 (1999) G59. [153] V.A. Joshkin, C.A. Parker, S.M. Bedair, L.Y. Krasnobaev, J.J. Cuomo, R.F. Davis, A. Suvkhanov, Appl. Phys. Lett. 73 (1998) 3875. [154] R.D. Metcalfe, D. Wickenden, W.C. Clark, J. Lumin. 16 (1978) 405. [155] B. Kaufmann, A. DoK rnen, V. HaK rle, H. Bolay, F. Scholz, G. Penzl, Appl. Phys. Lett. 66 (1994) 992. [156] M. Linde, S.J. Uftring, G.D. Watkins, V. HaK rle, F. Scholz, Phys. Rev. B 55 (1997) 10,177.

384

C. Ronning et al. / Physics Reports 351 (2001) 349}385

[157] I.A. Buyanova, M.T. Wagner, W.M. Chen, B. Monemar, J.L. LindstroK m, H. Amano, I. Akasaki, Appl. Phys. Lett. 73 (1998) 2968. [158] E. Silkowski, Y.K. Yeo, R.L. Hengehold, B. Goldenberg, G.S. Pomrenke, Mater. Res. Soc. Symp. Proc. 422 (1996) 69. [159] S. Kim, S.J. Rhee, X. Li, J.J. Coleman, S.G. Bishop, Phys. Rev. B 57 (1998) 14,588. [160] R. Magerle, A. Burchard, M. Deicher, T. Kerle, W. Pfei!er, E. Recknagel, Phys. Rev. Lett. 75 (1995) 1594. [161] A. Burchhard, E.E. Haller, A. StoK tzler, R. Weissenborn, M. Deicher, ISOLDE Collaboration, Physica B 273}274 (1999) 96. [162] A. StoK tzler, R. Weissenborn, M. Deicher, ISOLDE Collaboration, Physica B 273}274 (1999) 144. [163] M.G. Weinstein, C.Y. Song, M. Stavola, S.J. Pearton, R.G. Wilson, R.J. Shul, K.P. Killeen, M.J. Ludowise, Appl. Phys. Lett. 72 (1998) 1703. [164] W. Limmer, W. Ritter, R. Sauer, B. Mensching, C. Liu, B. Rauschenbach, Appl. Phys. Lett. 72 (1998) 2589. [165] A.J. Steckl, J.M. Zavada, MRS Bull. 24 (1999) 33. [166] G.S. Pomrenke, P.B. Klein, D.W. Langer, Rare Earth Doped Semiconductors, Mater. Res. Soc. Proc. 301. S. Co!a, A. Polman, R.N. Schwartz (Eds.), Rare Earth Doped Semiconductors II, Mater. Res. Soc. Proc. 422. [167] U. HoK mmerich, M. Thaik, T. Robinson-Brown, J.D. MacKenzie, C.R. Abernathy, S.J. Pearton, R.G. Wilson, R.N. Schwartz, J.M. Zavada, Mater. Res. Soc. Symp. Proc. 482 (1998) 685. [168] R.G. Wilson, R.N. Schwartz, C.R. Abernathy, S.J. Pearton, N. Newman, M. Rubin, T. Fu, J.M Zavada, Appl. Phys. Lett. 65 (1994) 992. [169] J.T. Torvik, R.J. Feuerstein, C.H. Qiu, M.W. Leksono, J.I. Pankove, F. Namavar, Mater. Res. Soc. Symp. Proc. 422 (1996) 199. [170] M. Thaik, U. HoK mmerich, R.N. Schwartz, R.G. Wilson, J.M. Zavada, Appl. Phys. Lett. 71 (1997) 2641. [171] J.T. Torvik, C.H. Qiu, R.J. Feuerstein, J.I. Pankove, F. Namavar, J. Appl. Phys. 81 (1997) 6343. [172] D.M. Hansen, R. Zhang, N.R. Perkins, S. Safvi, L. Zhang, K.L. Bray, T.F. Kuech, Appl. Phys. Lett. 72 (1998) 1244. [173] J.M. Zavada, M. Thaik, U. HoK mmerich, J.D. MacKenzie, C.R. Abernathy, F. Ren, H. Shen, J. Pamulapati, H. Jiang, J. Lin, R.G. Wilson, MRS Internet J. Nitride Semicond. Res. 4S1 (1999) G11.1. [174] C.H. Qiu, M.W. Leksono, J.I. Pankove, J.T. Torvik, R.J. Feuerstein, F. Namavar, Appl. Phys. Lett. 66 (1995) 562. [175] J.T. Torvik, R.J. Feuerstein, J.I. Pankove, C.H. Qiu, F. Namavar, Appl. Phys. Lett. 69 (1996) 2098. [176] L.C. Chao, B.K. Lee, C.J. Chi, J. Cheng, I. Chyr, A.J. Steckl, Appl. Phys. Lett. 75 (1999) 1833. [177] S. Kim, S.J. Rhee, D.A. Turnbull, E.E. Reuter, X. Li, J.J. Coleman, S.G. Bishop, Appl. Phys. Lett. 71 (1997) 231. [178] S. Kim, S.J. Rhee, D.A. Turnbull, X. Li, J.J. Coleman, S.G. Bishop, P.B. Klein, Appl. Phys. Lett. 71 (1997) 2662. [179] S. Kim, S.J. Rhee, D.A. Turnbull, X. Li, J.J. Coleman, S.G. Bishop, Mater. Res. Soc. Symp. Proc. 468 (1997) 131. [180] S. Kim, S.J. Rhee, X. Li, J.J. Coleman, S.G. Bishop, P.B. Klein, J. Electron. Mater. 27 (1998) 246. [181] S. Kim, private communication, MRS fall meeting 1997. [182] S.J. Rhee, S. Kim, X. Li, J.J. Coleman, S.G. Bishop, Mater. Res. Soc. Symp. Proc. 482 (1998) 667. [183] A.J. Steckl, R. Birkhahn, Appl. Phys. Lett. 73 (1998) 1700. [184] R. Birkhahn, A.J. Steckl, Appl. Phys. Lett. 73 (1998) 2143. [185] A.J. Steckl, M. Garter, R. Birkhahn, J. Sco"eld, Appl. Phys. Lett. 73 (1998) 2450. [186] M. Garter, J. Sco"eld, R. Birkhahn, A.J. Steckl, Appl. Phys. Lett. 74 (1999) 182. [187] H.J. Lozykowski, W.M. Jadwisienczak, I. Brown, Appl. Phys. Lett. 74 (1999) 1129. [188] L.C. Chao, A.J. Steckl, Appl. Phys. Lett. 74 (1999) 2364. [189] J.M. Zavada, R.A. Mair, C.J. Ellis, J.Y. Lin, H.X. Jiang, R.G. Wilson, P.A. Grudowski, R.D. Dupuis, Appl. Phys. Lett. 75 (1999) 790. [190] R. Birkhahn, M. Garter, A.J. Steckl, Appl. Phys. Lett. 74 (1999) 2161. [191] M.A. Khan, R.A. Skogman, R.G. Schulze, M. Gershenzon, Appl. Phys. Lett. 42 (1983) 430. [192] S.C. Binari, H.B. Dietrich, G. Kelner, L.B. Rowland, K. Doverspike, D.K. Wickenden, J. Appl. Phys. 78 (1995) 3008. [193] G. Hanington, Y.M. Hsin, Q.Z. Liu, P.M. Asbeck, S.S. Lau, M.A. Khan, J.W. Yang, Q. Chen, Electron. Lett. 34 (1998) 193. [194] S.J. Pearton, C.B. Vartuli, J.C. Zolper, C. Yuan, R.A. Stall, Appl. Phys. Lett. 67 (1995) 1435.

C. Ronning et al. / Physics Reports 351 (2001) 349}385

385

[195] J.C. Zolper, M.H. Crawford, S.J. Pearton, C.R. Abernathy, C.B. Vartuli, C. Yuan, R.A. Stall, J. Electron. Mater. 25 (1996) 839. [196] S.J. Pearton, C.B. Vartuli, C.R. Abernathy, J.D. Mackenzie, J.C. Zolper, C. Yuan, R.A. Stall, Inst. Phys. Conf. Ser. 142 (1996) 1023. [197] F.D. Auret, S.A. Goodman, F.K. Koschnick, J.M. Spaeth, B. Beaumont, P. Gibart, Appl. Phys. Lett. 74 (1999) 407. [198] S.A. Goodman, F.D. Auret, F.K. Koschnick, J.M. Spaeth, B. Beaumont, P. Gibart, MRS Internet J. Nitride Semicond. Res. 4S1 (1999) G6.12. [199] F.D. Auret, S.A. Goodman, F.K. Koschnick, J.M. Spaeth, B. Beaumont, P. Gibart, Appl. Phys. Lett. 73 (1998) 3745. [200] S.A. Goodman, F.D. Auret, F.K. Koschnick, J.M. Spaeth, B. Beaumont, P. Gibart, Appl. Phys. Lett. 74 (1999) 809. [201] D. Haase, M. Schmid, W. KuK rner, A. DoK rnen, V. HaK rle, F. Scholz, M. Burkard, H. Schweizer, Appl. Phys. Lett. 69 (1996) 2525. [202] X.A. Cao, S.J. Pearton, G.T. Dang, A.P. Zhang, F. Ren, R.G. Wilson, J.M. Van Hove, J. Appl. Phys. 87 (2000) 1091. [203] P. Hacke, A. Maekawa, N. Koide, K. Hiramatsu, N. Sawaki, Jpn. J. Appl. Phys. 33 (1994) 6443. [204] D.K. Gaskil, A.E. Wickenden, K. Doverspike, B. Tadayan, L.B. Rowland, J. Electron. Mater. 24 (1995) 1525. [205] W. GoK tz, N.M. Johnson, C. Chen, H. Liu, C. Kuo, W. Imler, Appl. Phys. Lett. 68 (1996) 3144. [206] W. GoK tz, L.T. Romano, B.S. Krusor, N.M. Johnson, R.J. Molnar, Appl. Phys. Lett. 69 (1996) 242. [207] R.J. Molnar, T. Lei, T.D. Moustakas, Appl. Phys. Lett. 62 (1993) 72. [208] W. GoK tz, N.M. Johnson, D.P. Bour, C. Chen, H. Liu, C. Kuo, W. Imler, Mater. Res. Soc. Symp. Proc. 395 (1996) 443. [209] W. GoK tz, R.S. Kern, C.H. Chen, H. Liu, D.A. Steiferwald, R.M. Fletcher, Mater. Sci. Eng. B 59 (1998) 211. [210] B.C. Chung, M. Gerzhenzon, J. Appl. Phys. 72 (1992) 651. [211] C. Wetzel, T. Suski, J.W. Ager III, E.R. Weber, E.E. Haller, S. Fischer, B.K. Meyer, R.J. Molnar, P. Perlin, Phys. Rev. Lett. 78 (1997) 3923. [212] J.S. Chan, N.W. Cheung, L. Schloss, E. Jones, W.S. Wong, N. Newman, X. Liu, E.R. Weber, A. Gassman, M.D. Rubin, Appl. Phys. Lett. 68 (1996) 2702. [213] J. Burm, K. Chu, W.A. Dfavis, W.J. Scha!, L.F. Eastman, Appl. Phys. Lett. 70 (1997) 464. [214] J. Brown, J. Ramer, K. Zheng, L.F. Lester, S.D. Hersee, J. Zolper, Mater. Res. Soc. Symp. 395 (1996) 855. [215] W.C. Lai, M. Yokoyama, C.C. Tsai, C.S. Chang, J.D. Guo, J.S. Tsang, S.H. Chan, Phys. Stat. Sol. B 216 (1999) 561. [216] J.C. Zolper, R.J. Shul, A.G. Baca, R.G. Wilson, S.J. Pearton, R.A. Stall, Appl. Phys. Lett. 68 (1996) 2273. [217] O.H. Nam, M.D. Bremser, T.S. Zheleva, R.F. Davis, Appl. Phys. Lett. 71 (1997) 2638. [218] K. Linthicum, T. Gehrke, D. Thomson, E. Carlson, P. Rajagopal, T. Smith, D. Batchelor, R.F. Davis, Appl. Phys. Lett. 75 (1999) 196.

PHYSICS OF COLLOIDAL DISPERSIONS IN NEMATIC LIQUID CRYSTALS

Holger STARK Low Temperature Laboratory, Helsinki University of Technology Box 2200, FIN-02015 HUT, Finland L.D. Landau Institute for Theoretical Physics, 117334 Moscow, Russia

AMSTERDAM } LONDON } NEW YORK } OXFORD } PARIS } SHANNON } TOKYO

Physics Reports 351 (2001) 387–474

Physics of colloidal dispersions in nematic liquid crystals Holger Stark Fachbereich Physik, Universitat Konstanz, D-78457 Konstanz, Germany Received December 2000; editor: M:L: Klein Contents 1. Motivation and contents 2. Phenomenological description of nematic liquid crystals 2.1. Free energy 2.2. Routes towards the director 0eld 2.3. Hydrodynamic equations 2.4. Topological defects 3. Nematic colloidal dispersions 3.1. Historic account 3.2. Nematic emulsions 4. The paradigm—one particle 4.1. The three possible con0gurations 4.2. An analytical investigation of the dipole 4.3. Results and discussion of the numerical study 4.4. Conclusions 5. Two-particle interactions 5.1. Formulating a phenomenological theory 5.2. E9ective pair interactions

389 390 392 394 396 400 406 406 408 410 410 412 415 422 423 423 425

6. The Stokes drag of spherical particles 6.1. Motivation 6.2. Theoretical concepts 6.3. Summary of numerical details 6.4. Results, discussion, and open problems 7. Colloidal dispersions in complex geometries 7.1. Questions and main results 7.2. Geometry and numerical details 7.3. Results and discussion of the numerical study 7.4. Coda: twist transition in nematic drops 8. Temperature-induced >occulation above the nematic-isotropic phase transition 8.1. Theoretical background 8.2. Paranematic order in simple geometries 8.3. Two-particle interactions above the nematic-isotropic phase transition 9. Final remarks Acknowledgements References

427 428 429 432 433 436 437 438 440 444 450 451 454 457 465 466 466

Abstract This article reviews the physics of colloidal dispersions in nematic liquid crystals as a novel challenging type of soft matter. We 0rst investigate the nematic environment of one particle with a radial anchoring of the director at its surface. Three possible structures are identi0ed and discussed in detail; the dipole, the Saturn-ring and the surface-ring con0guration. Secondly, we address dipolar and quadrupolar two-particle

E-mail address: [email protected] (H. Stark). c 2001 Elsevier Science B.V. All rights reserved. 0370-1573/01/$ - see front matter
PII: S 0 3 7 0 - 1 5 7 3 ( 0 0 ) 0 0 1 4 4 - 7

388

H. Stark / Physics Reports 351 (2001) 387–474

interactions with the help of a phenomenological theory. Thirdly, we calculate the anisotropic Stokes drag of a particle in a nematic environment which determines the Brownian motion of particles via the Stokes– Einstein relation. We then turn our interest towards colloidal dispersions in complex geometries where we identify the dipolar con0guration and study its formation. Finally, we demonstrate that surface-induced nematic order above the nematic-isotropic phase transition results in a strongly attractive but short-range two-particle interaction. Its strength can be controlled by temperature and thereby induce >occulation in c 2001 Elsevier Science B.V. All rights reserved. an otherwise stabilized dispersion.
PACS: 77.84.Nh; 61.30.Jf; 61.30.Cz; 82.70.Dd Keywords: Colloidal dispersions; Nematic liquid crystals; Topological defects; Two-particle interactions; Stokes drag; Complex geometries; Flocculation

H. Stark / Physics Reports 351 (2001) 387–474

389

1. Motivation and contents Dispersions of particles in a host medium are part of our everyday life and an important state of matter for fundamental research. One distinguishes between emulsions, where surfactant-coated liquid droplets are dispersed in a >uid environment, colloidal suspensions, where the particles are solid, and aerosols, with >uid or solid particles >oating in a gaseous phase. Colloidal dispersions, whose particle size ranges from 10 nm to 10
m, appear in food, with milk being the best-known fat-in-water emulsion, in drugs, cosmetics, paints, and ink. As such, they are of considerable technological importance. In nature, one is confronted by both the bothering and appealing side of fog, and one can look at the beautiful blue-greenish color of berg lakes in the Canadian Rockies, caused by light scattering from a 0ne dispersion of stone >ower in water. The best-known example of an ordered colloid, a colloidal crystal, is the opal, formed from a uniform array of silica spheres compressed and fused over geological timescales. In fundamental research, colloidal dispersions are ideal systems to study Brownian motion and hydrodynamic interactions of suspended particles [202,110]. They provide model systems [134,89] for probing our understanding of melting and boiling, and for checking the Kosterlitz–Thouless– Halperin–Nelson–Young transition in two-dimensional systems [233,238,23]. The main interest in colloidal dispersions certainly focusses on the problem how to prevent the particles from >occulation, as stated by Russel, Saville and Schowalter [202]: Since all characteristics of colloidal systems change markedly in the transition from the dispersed to the aggregated state, the question of stability occupies a central position in colloid science. There exists a whole zoo of interactions between the particles whose delicate balance determines the stability of a colloidal dispersion. Besides the conventional van der Waals, screened Coulombic, and steric interactions [202], >uctuation-induced Casimir forces (e.g., in binary >uids close to the critical point [116,157]) and depletion forces in binary mixtures of small and large particles [202,56,55,196] have attracted a lot of interest. Entropic e9ects play a major role in the three latter types of interactions. In a subtle e9ect, they also lead to an attraction between like-charged macroions [109,45,125] by Coulomb depletion [4]. The present work focusses on the interesting question of how particles behave when they are dispersed in a nematic solvent. In a nematic liquid crystal, rodlike organic molecules align on average along a unit vector n, called director. The energetic ground state is a uniform director 0eld throughout space. Due to the anchoring of the molecules at the surface of each particle, the surrounding director 0eld is elastically distorted which induces additional long-range two-particle interactions. They are of dipolar or quadrupolar type, depending on the symmetry of the director con0guration around a single particle [22,190,200,182,140]. The forces were con0rmed by recent experiments in inverted nematic emulsions [182,179,183]. On the other hand, close to the suspended particles, topological point and line defects in the orientational order occur which strongly determine their behavior. For example, point defects give rise to a short-range repulsion [182,183]. Colloidal dispersions in a nematic environment are therefore an ideal laboratory for studying the statics and dynamics of topological defects. Before we deal with the physics of such dispersions in the following, we review the phenomenological description of the nematic phase in Section 2. We introduce the total free energy

390

H. Stark / Physics Reports 351 (2001) 387–474

and the Ericksen–Leslie equations governing, respectively, the statics and dynamics of the director 0eld. We also provide the basic knowledge of topological point and line defects in the orientational order necessary for the understanding of di9erent director con0gurations around a single spherical particle. In Section 3 we review the work performed on colloidal dispersions in nematic liquid crystals relating it to recent developments in the liquid crystal 0eld. Furthermore, with nematic emulsions, we introduce one particular model system which motivated the present investigation. In Section 4 we investigate, what I consider the paradigm for the understanding of nematic colloidal dispersions, i.e., the static properties of one particle. We concentrate on a radial anchoring of the director at its surface, for which we identify three di9erent types of nematic environment; the dipole con0guration, where the particle and a companion point defect form a tightly bound topological dipole, the Saturn-ring con0guration, where the particle is surrounded by a ring defect and a structure with an equatorial surface ring, which appears for decreasing anchoring strength of the molecules at the surface of the particle. In Section 5 we address colloidal interactions with the help of a phenomenological theory and illustrate how they depend on the overall symmetry of the director con0guration around a single particle. Then, in Section 6 we calculate the Stokes drag of one particle moving in a nematic environment. Via the Stokes–Einstein relation, one immediately has access to the di9usion constant which determines the Brownian motion of spherical objects. In Section 7 we turn our interest towards colloidal dispersions in complex geometries. In particular, we consider particles, e.g., droplets of water, in a large nematic drop. We identify the dipole con0guration and illustrate possible dynamic e9ects in connection with its formation. Finally, in Section 8 we demonstrate that surface-induced nematic order above the nematic-isotropic phase transition leads to another novel colloidal interaction which strongly in>uences the stability of dispersions. It is easily controlled by temperature and can, e.g., induce >occulation in an otherwise stabilized system. 2. Phenomenological description of nematic liquid crystals Typical liquid crystalline compounds consist of organic molecules. According to their elongated or disc-like shape one distinguishes between calamatic and discotic liquid crystals. Fig. 1a presents the molecular structure of the well-studied compound N -(p-methoxybenzylidene)-pM At suNciently high butylaniline (MBBA). Its approximate length and width are 20 and 5 A. temperatures, the liquid crystalline compound behaves like a conventional isotropic liquid; the molecules do not show any long-range positional and orientational order, as illustrated in the right box of Fig. 1b for rod-like molecules. Cooling below the clearing point Tc , the liquid becomes turbid, which indicates a phase transition to the liquid crystalline state. Finally, below the melting point Tm the system is solid. There exists a wealth of liquid crystalline phases [29,51,27]. Here we concentrate on the simplest, i.e., the nematic phase, which consists of non-chiral molecules. Their centers of mass are disordered as in the isotropic liquid, whereas their main axes align themselves on average parallel to each other, so that they exhibit a long-range orientational order. The average direction is given by a unit vector n, called Frank director. However, n merely characterizes an axis in space, e.g., the optical axis of the birefringent nematic phase. As a result, all physical quantities, which we formulate in the following, have to be invariant under the inversion of the director (n → −n). From the topological point

H. Stark / Physics Reports 351 (2001) 387–474

391

Fig. 1. (a) The compound MBBA. (b) The nematic liquid-crystalline phase below the clearing point Tc . The average direction of the molecules is indicated by the double arrow {n; −n}.

of view, the order parameter space of the nematic phase is the projective plane P 2 = S 2 =Z2 , i.e., the unit sphere S 2 in three dimensions with opposite points identi0ed [226]. Unlike the magnetization in ferromagnets, the nematic order parameter is not a vector. This statement can be understood from the following argument. Organic molecules often carry a permanent electric dipole moment along their main axis but so far no ferroelectric nematic phase with a spontaneous polarization has been found. Therefore, the same number of molecules that point into a certain direction in space also have to point into the opposite direction. In thermotropic liquid crystals, the phase transitions are controlled by temperature. On the other hand, increasing the concentration of rod- or disc-like objects in a solvent can lead to the formation of what is called lyotropic liquid crystalline phases. The objects can be either large macromolecules, like the famous tobacco mosaic virus [78], or micelles, which form when amphiphilic molecules are dissolved, e.g., in water. All directions in the isotropic >uid are equivalent. The phase transition to the nematic state breaks the continuous rotational symmetry of the isotropic liquid. As a result, domains with di9erently oriented directors appear like in a ferromagnet. These domains strongly scatter light and are one reason for the turbidity of the nematic phase. Hydrodynamic Goldstone modes appear in systems with a broken continuous symmetry [76,27]. They are “massless”, i.e., their excitation does not cost any energy for vanishing wave number. In the nematic phase, the Goldstone modes correspond to thermal >uctuations of the director about its equilibrium value. Such >uctuations of the local optical axis also scatter light very strongly. In the next four Sections 2.1–2.4 we will lay the basis for an understanding of the static and dynamical properties of the nematic phase, and we will apply it in the following sections to nematic colloidal dispersions. Sections 2.1 and 2.2 provide the necessary knowledge for determining the spatially non-uniform director 0eld in complex geometries (e.g., around particles) under the in>uence of surfaces and external 0elds and in the presence of topological defects. Furthermore, with the help of the dynamic theory in Section 2.3, we will calculate the Stokes drag of a particle in a nematic environment, and we will demonstrate that it is in>uenced by

392

H. Stark / Physics Reports 351 (2001) 387–474

the presence of topological defects close to the particle. Finally, in Section 2.4 we will review the basic knowledge of point and line defects in nematics. 2.1. Free energy Thermodynamics tells us that a complete knowledge of a system on a macroscopic level follows from the minimization of an appropriate thermodynamic potential [25]. We use the free energy, which consists of bulk and surface terms,

3 Fn = Fel + F24 + FH + FS = d r (fel + f24 + fH ) + dS fS ; (2.1) and discuss them in order. The energetic ground state of a nematic liquid crystal is a spatially uniform director 0eld; any deviation from it costs elastic energy. To describe slowly varying spatial distortions of the director 0eld n(r), one expands the free energy density into the gradient of n(r), ∇i nj , up to second order, and demands that the energy density obeys the local point symmetry D∞h of the nematic phase. The point group D∞h contains all the symmetry elements of a cylinder, i.e., all rotations about an axis parallel to n(r), a mirror plane perpendicular to n(r), and an in0nite number of two-fold axes also perpendicular to n(r). The result is the Oseen–ZUocher–Frank free energy density [167,242,79], which consists of two parts, fel = 12 [K1 (div n)2 + K2 (n · curl n)2 + K3 (n × curl n)2 ]

(2.2)

and K24 (2.3) div(n div n + n × curl n) ; 2 where K1 , K2 , K3 , and K24 denote, respectively, the splay, twist, bend, and saddle-splay elastic constants. Fig. 2 illustrates the characteristic deformations of the director 0eld. The splay and bend distortions can be viewed, respectively, as part of a source or vortex 0eld. In the twist deformation, the director rotates about an axis perpendicular to itself. In calamatic liquid crystals one usually 0nds the following relation, K3 ¿ K1 ¿ K2 . For example, in the compound pentylcyanobiphenyl (5CB), K1 = 0:42 × 10−6 dyn, K2 = 0:23 × 10−6 dyn, and K3 = 0:53 × 10−6 dyn. In discotic liquid crystals, the relationship K2 ¿ K1 ¿ K3 is predicted [216,168,221], which is in good agreement with experiments, where K2 ¿ K1 ¿ K3 is observed [232,103]. The saddle-splay term is a pure divergence. It can be transformed into integrals over all surfaces of the system,
1 F24 = − K24 d S · (n div n + n × curl n) ; (2.4) 2 where it prefers the formation of a saddle (see Fig. 2). A Cauchy relation for K24 follows from the Maier–Saupe molecular approach [163], f24 = −

K24 = (K11 + K22 )=2 :

(2.5)

Exact measurements of K24 are still missing but it is of the order of the bulk elastic constants K1 , K2 , and K3 [41,5,40]. There is also the possibility of another surface term with a free

H. Stark / Physics Reports 351 (2001) 387–474

393

Fig. 2. Illustration of the characteristic deformations in a nematic liquid crystal: splay, twist, bend, and saddle-splay.

energy density K13 div(n div n), which we will not consider in this work [163,166,9,172,173]. The controversy about it seems to be solved [174]. In the one-constant approximation, K = K1 = K2 = K3 , the Frank free energy takes the form

K K − K24 3 2 d r(∇i nj ) + d S · (n div n + n × curl n) : (2.6) Fel = 2 2 It is often used to obtain a basic understanding of a system without having to deal with e9ects due to the elastic anisotropy. The bulk term is equivalent to the non-linear sigma model in statistical 0eld theory [241,27] or the continuum description of the exchange interaction in a ferromagnet [156]. In nematic liquid crystals we can assume a linear relation between the magnetization M and an external magnetic 0eld H , M = H , where  stands for the tensor of the magnetic susceptibility. The nematic phase represents a uniaxial system, for which the second-rank tensor  always takes the following form:  = ⊥ 1 + V(n ⊗ n) ;

(2.7)

1 is the unit tensor, and ⊗ means dyadic product. We have introduced the magnetic anisotropy V =  − ⊥ . It depends on the two essential magnetic susceptibilities  and ⊥ for magnetic 0elds applied, respectively, parallel or perpendicular to the director. The general expression for the magnetic free energy density is −H ·H =2 [123]. A restriction to terms that depend on the director n yields V (2.8) fH = − [(n ·H )2 − H 2 ] : 2 In usual nematics V is positive and typically of the order of 10−7 [51]. For V ¿ 0, the free energy density fH favors an alignment of the director n parallel to H . By adding a term

394

H. Stark / Physics Reports 351 (2001) 387–474

−V H 2 =2 on the right-hand side of Eq. (2.8), we shift the reference point in order that the mag-

netic free energy of a completely aligned director 0eld is zero. This will be useful in Section 4 where we calculate the free energy of the in0nitely extended space around a single particle. The balance between elastic and magnetic torques on the director de0nes an important length scale, the magnetic coherence length  K3 H = : (2.9) V H 2

Suppose the director is planarly anchored at a wall, and a magnetic 0eld is applied perpendicular to it. Then H gives the distance from the wall that is needed to orient the director along the applied 0eld [51]. The coherence length tends to in0nity for H → 0. Finally, we employ the surface free energy of Rapini–Papoular to take into account the interaction of the director with boundaries: W fS = [1 − (n · ˆ)2 ] : (2.10) 2 The unit vector ˆ denotes some preferred orientation of the director at the surface, and W is the coupling constant. It varies in the range 10−4 –1 erg=cm2 as reviewed by Blinov et al. [14]. In Section 4.3.4 we will give a lower bound of W for the interface of water and the liquid crystalline phase of 5CB in the presence of the surfactant sodium dodecyl sulfate, which was used in the experiment by Poulin et al. [182,183] on nematic emulsions. From a comparison between the Frank free energy and the surface energy one arrives at the extrapolation length [51] K3 S = : (2.11) W It signi0es the strength of the anchoring. Take a particle of radius a in a nematic environment with an uniform director 0eld at in0nity. (We will investigate this case thoroughly in Section 4.) The Frank free energy of this system is proportional to K3 a whereas the surface energy scales as Wa2 . At strong anchoring, i.e., for Wa2 K3 a or S a, the energy to turn the director away from its preferred direction ˆ at the whole surface would be much larger than the bulk energy. Therefore, it is preferable for the system that the director points along ˆ. However, n can deviate from ˆ in an area of order S a. In Section 4.3.4 we will use this argument to explain a ring con0guration around the particle. Rigid anchoring is realized for S → 0. Finally, S a means weak anchoring, where the in>uence of the surface is minor. Since in our discussion we have always referred S to the radius a, it is obvious that the strength of the anchoring is not an absolute quantity but depends on characteristic length scales of the system. 2.2. Routes towards the director 4eld The director 0eld n(r) in a given geometry follows from a minimization of the total free energy Fn = Fel + F24 + FH + FS under the constraint that n is a unit vector: Fn = 0

with n ·n = 1 :

(2.12)

H. Stark / Physics Reports 351 (2001) 387–474

395

Even in the one-constant approximation and under the assumption of rigid anchoring of the director at the boundaries, this is a diNcult problem to solve because of the additional constraint. Typically, full analytical solutions only exist for one-dimensional problems, e.g., for the description of the FrXeedericksz transition [29,51], or in two dimensions when certain symmetries are assumed [140]. To handle the constraint, one can use a Lagrange parameter or introduce an appropriate parametrization for the director, e.g., a tilt () and twist () angle, so that the director in the local coordinate basis takes the form n = (sin  cos ; sin  sin ; cos ) :

(2.13)

If an accurate analytical determination of the director 0eld is not possible, there are two strategies. First, an ansatz function is constructed that ful0ls the boundary conditions and contains free parameters. Then, the director 0eld follows from a minimization of the total free energy in the restricted space of functions with respect to the free parameters. In Section 4.2 we will see that this method is quite successful. Secondly, one can look for numerical solutions of the Euler–Lagrange equations corresponding to the variational problem formulated in Eq. (2.12). They are equivalent to functional derivatives of Fn [; ], where we use the tilt and twist angle of Eq. (2.13) to parametrize the director: Fn Fn 9ni = =0 ;  ni 9

(2.14)

Fn Fn 9ni = =0 :  ni 9

(2.15)

Einstein’s summation convention over repeated indices is used. To arrive at the equations above for (r) and (r), we have employed a chain rule for functional derivatives [219]. These chain rules are useful in numerical problems since they allow to write the Euler–Lagrange equations, which can be quite complex, in a more compact form. For example, the bulk and surface equations that are solved in Section 4.3 could only be calculated with the help of the algebraic program Maple after applying the chain rules. Typically, we take a starting con0guration for the director 0eld and relax it on a grid via the Newton–Gauss–Seidel method [187]. It is equivalent to Newton’s iterative way of determining the zeros of a function but now generalized to functionals. We illustrate it here for the tilt angle : Fn =(r) new (r) = old (r) − 2 : (2.16) “ Fn =2 (r)” There are two possibilities to implement the method numerically. If the grid for the numerical investigation is de0ned by the coordinate lines, one determines the Euler–Lagrange equations analytically. Then, they are discretized by the method of 0nite di9erences for a discrete set of grid points r [187]. Finally, “2 Fn =2 (r)” is calculated as the derivative of Fn =(r) with respect to (r) at the grid point r. We put “2 Fn =2 (r)” into quotes because it is not the discretized form of a real second-order functional derivative, which would involve a delta function. If the geometry of the system is more complex, the method of 0nite elements is appropriate (see Section 7). In two dimensions, e.g., the integration area is subdivided into 4nite elements, which in the simplest case are triangles. In doing so, the boundaries of the complex geometry

396

H. Stark / Physics Reports 351 (2001) 387–474

are well approximated by polygons. The 0nite-element technique generally starts from an already discretized version of the total free energy Fn and then applies a numerical minimization scheme, e.g., the Newton–Gauss–Seidel method. Both the 0rst and second derivatives in Eq. (2.16) are performed with respect to (r) at the grid point r. 2.3. Hydrodynamic equations In the last subsection we concentrated on the static properties of the director 0eld. In this subsection we review a set of dynamic equations coupling the >ow of the liquid crystal to the dynamics of the Frank director. The set consists of a generalization of the Navier–Stokes equations for the >uid velocity C to uniaxial media and a dynamic equation for the director n. We will not provide any detailed derivation of these equations, rather we will concentrate on the explanation of their meaning. The main problem is how to 0nd a dynamic equation for the director. An early approach dates back to Oseen [167]. Ericksen [67–71] and Leslie [128,129] considered the liquid crystal as a Cosserat continuum [73] whose constituents possess not only translational but also orientational degrees of freedom. Based on methods of rational thermodynamics [72], they derived an equation for the >uid velocity from the balance law for momentum density and an equation for the director, which they linked to the balance law for angular momentum. The full set of equations is commonly referred to as the Ericksen–Leslie equations. An alternative approach is due to the Harvard group [77,27] which formulated equations following rigorously the ideas of hydrodynamics [76,27]. It only deals with hydrodynamic variables, i.e., densities of conserved quantities, like mass, momentum, and energy, or broken-symmetry variables. Each one obeys a conservation law. As a result, hydrodynamic modes exist whose, in general, complex frequencies become zero for vanishing wave number. Excitations associated with broken-symmetry variables are called hydrodynamic Goldstone modes according to a concept introduced by Goldstone in elementary particle physics [92,93]. The director is such a variable that breaks the continuous rotational symmetry of the isotropic >uid. In a completely linearized form the Ericksen–Leslie equations and the equations of the Harvard group are identical. In the following, we review the Ericksen–Leslie equations and explain them step by step. In a symbolic notation, they take the form [29,51] 0 = div C ; %

dC = div T dt

(2.17) with T = −p1 + T 0 + T  ;

0 = n × (h0 − h ) ;

(2.18) (2.19)

where the divergence of the stress tensor is de0ned by (div T )i = ∇j Tij . The 0rst equation states that we consider an incompressible >uid. We also assume constant temperature in what follows. The third equation balances all the torques on the director. We will discuss it below. The second formula stands for the generalized Navier–Stokes equations. Note that d 9 (2.20) = + C· grad dt 9t

H. Stark / Physics Reports 351 (2001) 387–474

397

means the total or material time derivative as experienced by a moving >uid element. It includes the convective part C· grad. Besides the pressure p, the stress tensor consists of two terms: Tij0 = −

9fb ∇i nk 9∇j nk

with fb = fel + f24 + fm ;

Tij = !1 ni nj nk nl Akl + !2 nj Ni + !3 ni Nj + !4 Aij + !5 nj nk Aik + !6 ni nk Ajk :

(2.21) (2.22)

In addition to p, T 0 introduces a second, anisotropic contribution in the static stress tensor. It is due to elastic distortions in the director 0eld, where fb denotes the sum of all free energy densities introduced in Section 2.1. The quantity T  stands for the viscous part of the stress tensor. In isotropic >uids, it is simply proportional to the symmetrized gradient of the velocity 0eld, Aij = 12 (∇i vj + ∇j vi ) :

(2.23)

The conventional shear viscosity equals !4 =2 in Eq. (2.22). The uniaxial symmetry of nematic liquid crystals allows for further contributions proportional to A which contain the director n. There are also two terms that depend on the time derivative of the director n, i.e., the second dynamic variable, N=

dn −!×n dt

with ! = 12 curl C :

(2.24)

The vector N denotes the rate of change of n relative to the >uid motion, or more precisely, relative to a local >uid vortex characterized by the angular velocity ! = curl C=2. The viscosities !1 ; : : : ; !6 are referred to as the Leslie coeNcients. We will gain more insight into T  at the end of this subsection. Finally, Eq. (2.19) demands that the total torque on the director due to elastic distortions in the director 0eld (h0 ) and due to viscous processes (h ) is zero. The elastic and viscous curvature forces are h0i = ∇j

9fb 9fb − ; 9∇j ni 9ni

hi = %1 Ni + %2 Aij nj

with %1 = !3 − !2 and %2 = !2 + !3 :

(2.25) (2.26)

De Gennes calls h0 a molecular 0eld reminiscent to a similar quantity in magnetism [51]. In Eq. (2.19) the curvature force h0 −h is only de0ned within an additive expression &(r; t)n(r; t). It has the meaning of a Lagrange-multiplier term, and &(r; t) is to be determined by the condition that the director is normalized to unity. In static equilibrium, we obtain h0 (r) + &(r)n(r) = 0, i.e., the Euler–Lagrange equation in the bulk which follows from minimizing the total free energy Fn introduced in Section 2.1. One can easily show that the saddle-splay energy f24 does not contribute to h0 . The 0rst term of the viscous curvature force h describes the viscous process due to the rotation of neighboring molecules with di9erent angular velocities. The coeNcient %1 is, therefore, a typical rotational viscosity. The second term quanti0es torques on the director 0eld exerted by a shear >ow. An inertial term for the rotational motion of the molecules is not included in Eq. (2.19). One can show that it is of no relevance for the timescales of

398

H. Stark / Physics Reports 351 (2001) 387–474

micro-seconds or larger. In the approach of the Harvard group, it does not appear since it results in a non-hydrodynamic mode. The energy dissipated in viscous processes follows from the entropy production rate
dS T (2.27) = (T  ·A + h ·N ) d 3 r ; dt where T is temperature, and S is entropy. The 0rst term describes dissipation by shear >ow and the second one dissipation by rotation of the director. Each term in the entropy production rate is always written as a product of a generalized >ux and its conjugate force. The true conjugate force to the >ux A is the symmetrized viscous stress tensor Tijsym = (Tij + Tji )=2. The >ux N is conjugate to the generalized force h : Note, that h corresponds to the dual form of the antisymmetric part of T  , i.e., hi = 'ijk (Tjk − Tkj )=2. The Harvard group calls T sym and h >uxes since they appear in the currents of the respective balance laws for momentum and director [77,27]. In hydrodynamics the viscous forces are assumed to be small, and they are written as linear functions of all the >uxes:  sym    T A = : (2.28) N h The matrix must be compatible with the uniaxial symmetry of the nematic phase, and it must be invariant when n is changed into −n. Furthermore, it has to obey Onsager’s theorem [52], which demands a symmetric matrix for zero magnetic 0eld. Ful0lling all these requirements results in Eqs. (2.22) and (2.26). One additional, important Onsager relation is due to Parodi [170]: !2 + !3 = !6 − !5 :

(2.29)

It reduces the number of independent viscosities in a nematic liquid crystal to 0ve. The Leslie coeNcients of the compound 5CB are [39] !1 = −0:111 P; !2 = −0:939 P; !3 = −0:129 P ; !4 = 0:748 P; !5 = 0:906 P; !6 = −0:162 P :

(2.30)

At the end, we explain two typical situations that help to clarify the meaning of the possible viscous processes in a nematic and how they are determined by the Leslie coeNcients. In the 0rst situation we perform typical shear experiments as illustrated in Fig. 3. The director 0eld between the plates is spatially uniform, and the upper plate is moved with a velocity v0 relative to the lower one. There will be a constant velocity gradient along the vertical z direction. Three simple geometries exist with a symmetric orientation of the director; it is either parallel to the velocity 0eld C, or perpendicular to C and its gradient, or perpendicular to C and parallel to its gradient. The director can be 0rmly aligned in one direction by applying a magnetic 0eld strong enough to largely exceed the viscous torques. For all three cases, the shear forces T  per unit area are calculated from the stress tensor T  of Eq. (2.22), yielding T  = )i C0 =d, where d is the separation between the plates. The viscosities as a function of the Leslie coeNcients for all three cases are given in Fig. 3. They are known as Mie8sowicz viscosities after the scientist who 0rst measured them [150,151]. If one chooses a non-symmetric orientation for the director, the viscosity !1 is accessible in shear experiments too [84].

H. Stark / Physics Reports 351 (2001) 387–474

399

Fig. 3. De0nition of the three MiZesowicz viscosities in shear experiments.

Fig. 4. Permeation of a >uid through a helix formed by the nematic director.

The second situation describes a Gedanken experiment illustrated in Fig. 4 [102]. Suppose the nematic director forms a helical structure with wave number q0 inside a capillary. Such a con0guration is found in cholesteric liquid crystals that form when the molecules are chiral. Strictly speaking, the hydrodynamics of a cholesteric is more complicated than the one of nematics [139]. However, for what follows we can use the theory formulated above. We assume that the velocity 0eld in the capillary is spatially uniform and that the helix is not distorted by the >uid >ow. Do we need a pressure gradient to press the >uid through the capillary, although there is no shear >ow unlike a Poiseuille experiment? The answer is yes since the molecules of the >uid, when >owing through the capillary, have to rotate constantly to follow the director in the helix, which determines the average direction of the molecules. The dissipated energy follows from the second term of the entropy production rate in Eq. (2.27). The rate of change, N = C0 · grad n, is non-zero due to the convective time derivative. The energy dissipated per unit time and unit volume has to be matched by the work per unit time performed by the pressure gradient p . One 0nally arrives at p = %1 q02 v0 :

(2.31)

Obviously, the Gedanken experiment is determined by the rotational viscosity %1 . It was suggested by Helfrich [102] who calls the motion through a 0xed orientational pattern permeation. This motion is always dissipative because of the rotational viscosity of the molecules which have to follow the local director. Of course, the Gedanken experiment is not suitable for measuring %1 . A more appropriate method is dynamic light scattering from director >uctuations [99,29]. Together with the shear experiments it is in principle possible to measure all 0ve independent viscosities of a nematic liquid crystal.

400

H. Stark / Physics Reports 351 (2001) 387–474

2.4. Topological defects Topological defects [111,146,226,27], which are a necessary consequence of broken continuous symmetry, exist in systems as disparate as super>uid helium 3 [230] and 4 [235], crystalline solids [224,81,160], liquid crystals [30,121,127], and quantum-Hall >uids [204]. They play an important if not determining role in such phenomena as response to external stresses [81,160], the nature of phase transitions [27,164,222], or the approach to equilibrium after a quench into an ordered phase [21]; and they are the primary ingredient in such phases of matter as the Abrikosov >ux-lattice phase of superconductors [1,13] or the twist-grain-boundary phase of liquid crystals [193,94,95]. They even arise in certain cosmological models [34]. Topological defects are points, lines or walls in three-dimensional space where the order parameter of the system under consideration is not de0ned. The theory of homotopy groups [111,146,226,27] provides a powerful tool to classify them. To identify, e.g., line defects, homotopy theory considers closed loops in real space which are mapped into closed paths in the order parameter space. If a loop can be shrunk continuously to a single point, it does not enclose a defect. All other loops are divided into classes of paths which can be continuously transformed into each other. Then, each class stands for one type of line defect. All classes together, including the shrinkable loops, form the 4rst homotopy or fundamental group. The group product describes the combination of defects. In the case of point singularities, the loops are replaced by closed surfaces, and the defects are classi0ed via the second homotopy group. In the next two subsections we deal with line and point defects in nematic liquid crystals whose order parameter space is the projective plane P 2 = S 2 =Z2 , i.e., the unit sphere S 2 with opposite points identi0ed. They play a determining role for the behavior of colloidal dispersions in a nematic environment. There exist several good reviews on defects in liquid crystals [111,146,226,27,30,121,127]. We will therefore concentrate on facts which are necessary for the understanding of colloidal dispersions. Furthermore, rather than being very formal, we choose a descriptive path for our presentation. 2.4.1. Line defects = disclinations Line defects in nematic liquid crystals are also called disclinations. Homotopy theory tells us that the fundamental group ,1 (P 2 ) of the projective plane P 2 is the two-element group Z2 . Thus, there is only one class of stable disclinations. Fig. 5 presents two typical examples. The defect line with the core is perpendicular to the drawing plane. The disclinations carry a winding number of strength + or −1=2, indicating a respective rotation of the director by + ◦ or −360 =2 when the disclination is encircled in the anticlockwise direction (see left part of Fig. 5). Note that the sign of the winding number is not 0xed by the homotopy group. Both types of disclinations are topologically equivalent since there exists a continuous distortion of the director 0eld which transforms one type into the other. Just start from the left disclination in Fig. 5 and rotate the director about the vertical axis through an angle  when going outward from the core in any radial direction. You will end up with the right picture. The line defects in Fig. 5 are called wedge disclinations. In a Volterra process [111,27] a cut is performed so that its limit, the disclination line, is perpendicular to the spatially uniform director 0eld. Then the surfaces of the cut are rotated with respect to each other by an angle of 2S about the disclination line, and material is either 0lled in (S = +1=2) or removed (S = −1=2). In twist

H. Stark / Physics Reports 351 (2001) 387–474

401

Fig. 5. Disclinations of winding number ±1=2. For further explanations see text.

disclinations the surfaces are rotated by an angle of  about an axis perpendicular to the defect line. Disclinations of strength ±1=2 do not exist in a system with an vector order parameter since it lacks the inversion symmetry of the nematic phase with respect to the director. In addition, one 0nds ,1 (S 2 ) = 0, i.e., every disclination line of integral strength in a ferromagnet is unstable; “it escapes into the third dimension”. The same applies to nematic liquid crystal as demonstrated by Cladis et al. [35,236] and Bob Meyer [149] for S = 1. The director 0eld around a disclination follows from the minimization of the Frank free energy (2.6) [111,29,51]. In the one-constant approximation the line energy Fd of the disclination can be calculated as   , R 1 Fd = K : (2.32) + ln 4 2 rc The surface term in Eq. (2.6) is neglected. The second term on the right-hand side of Eq. (2.32) stands for the elastic free energy per unit length around the line defect where R is the radius of a circular cross-section of the disclination (see Fig. 5). Since the energy diverges logarithmically, one has to introduce a lower cut-o9 radius rc , i.e., the radius of the disclination core. Its line energy, given by the 0rst term, is derived in the following way [111]. One assumes that the core of the disclination contains the liquid in the isotropic state with a free energy density 'c necessary to melt the nematic order locally. Splitting the line energy of the disclination as in Eq. (2.32) into the sum of a core and elastic part, Fd = ,rc2 'c + K, ln(R=rc )=4, and minimizing it with respect to rc , results in 'c =

K 1 ; 8 rc2

(2.33)

so that we immediately arrive at Eq. (2.32). The right-hand side of Eq. (2.33) is equivalent to the Frank free energy density of the director 0eld at a distance rc from the center of the disclination. Thus rc is given by the reasonable demand that the nematic state starts to melt when this energy density equals 'c . With an estimate 'c = 10−7 erg=cm3 , which follows from a description of the nematic-isotropic phase transition by the Landau–de Gennes theory [126], and K = 10−6 dyn, we obtain a core radius rc of the order of 10 nm. In the general case (K1 = K2 = K3 ), an analytical expression for the elastic free energy does not exist. However,

402

H. Stark / Physics Reports 351 (2001) 387–474

Fig. 6. Radial and hyperbolic hedgehog of point charge Q = 1.

a rough approximation for the core energy per unit length, Fc , can be found by averaging over the Frank constants: , K1 + K2 + K3 Fc = : (2.34) 8 3 In Section 4.3 we will make use of this form for Fc . A more re0ned model of the disclination core is derived from Landau–de Gennes theory [48,96,212], which employs a traceless second-rank tensor Q as an order parameter (see Section 8.1). The tensor also describes biaxial liquid crystalline order. Investigations show that the core of a disclination should indeed be biaxial [141,144,207], with a core radius of the order of the biaxial correlation length b , i.e., the length on which deviations from the uniaxial order exponentially decay to zero. Outside of the disclination core, the nematic order is essentially uniaxial. Therefore, the line energy of a disclination is still given by Eq. (2.32) with rc ∼ b , and with a core energy now determined by the energy di9erence between the biaxial and uniaxial state rather than the energy di9erence between the isotropic and nematic state. 2.4.2. Point defects Fig. 6 presents typical point defects in a nematic liquid crystal known as radial and hyperbolic hedgehogs. Both director 0elds are rotationally symmetric about the vertical axis. The second homotopy group ,2 (P 2 ) of the projective plane P 2 is the set Z of all integer numbers. They label every point defect by a topological charge Q. The result is the same as for the vector order parameter space S 2 since close to the point singularity the director 0eld constitutes a unique vector 0eld. For true vectors it is possible to distinguish between a radial hedgehog of positive and negative charge depending on their vector 0eld that can either represent a source or a sink. In a nematic liquid crystal this distinction is not possible because n and −n describe the same state. Note, e.g., that the directors close to a point defect are reversed if the defect is moved around a ± 1=2 disclination line. Therefore, the sign of the charge Q has no meaning in nematics, and by convention it is chosen positive. The charge Q is determined by the number of times the unit sphere is wrapped by all the directors on a surface enclosing the defect core. An analytical expression for Q is 
 1   Q= dSi 'ijk n · (∇j n × ∇k n) ; (2.35) 8 where the integral is over any surface enclosing the defect core. Both the hedgehogs in Fig. 6 carry a topological charge Q=1. They are topologically equivalent since they can be transformed

H. Stark / Physics Reports 351 (2001) 387–474

403

Fig. 7. The hyperbolic hedgehog at the center is transformed into a radial point defect by a continuous distortion of the director 0eld. Nails indicate directors tilted relative to the drawing plane.

Fig. 8. A radial (left) and a hyperbolic (right) hedgehog combine to a con0guration with total charge 0 = |1 − 1|.

into each other by a continuous distortion of the director 0eld. Just start from the hyperbolic hedgehog and rotate the director about the vertical axis through an angle  when going outward from the core in any radial direction. By this procedure, which is illustrated in Fig. 7 with the help of a nail picture, we end up with a radial hedgehog. The length of the nail is proportional to the projection of the director on the drawing plane, and the head of the nail is below the plane. Such a transition was observed by Lavrentovich and Terentjev in nematic drops with homeotropic, i.e., perpendicular anchoring of the director at the outer surface [126]. In systems with vector symmetry, the combined topological charge of two hedgehogs with respective charges Q1 and Q2 is simply the sum Q1 + Q2 . In nematics, where the sign of the topological charge has no meaning, the combined topological charge of two hedgehogs is either |Q1 + Q2 | or |Q1 − Q2 |. It is impossible to tell with certainty which of these possible charges is the correct one by looking only at surfaces enclosing the individual hedgehogs. For example, the combined charge of two hedgehogs in the presence of a line defect depends on which path around the disclination the point defects are combined [226]. In Fig. 8 we illustrate how a radial and a hyperbolic hedgehog combine to a con0guration with total charge 0 = |1 − 1|. Since the distance

404

H. Stark / Physics Reports 351 (2001) 387–474

Fig. 9. A hyperbolic hedgehog can be opened up to a −1=2 disclination ring.

d of the defects is the only length scale in the system, dimensional arguments predict an interaction energy proportional to Kd [169]. It grows linear in d reminiscent to the interaction energy of quarks if one tries to separate them beyond distances larger than the diameter of a nucleus. The energies of the hedgehog con0gurations, shown in Fig. 6, are easily calculated from the Frank free energy Fel + F24 [see Eqs. (2.2) and (2.3)]. The director 0elds of these con0gurations are n = (x; y; z)=r for the radial and n = (−x; −y; z)=r for the hyperbolic hedgehog, where r = (x; y; z) and r = |r|. In a spherical region of radius R with free boundary conditions at the outer surface, we obtain for their respective energies: Fradial = 4(2K1 − K24 )R → 4(2K − K24 )R ; 4 4 Fhyper = (6K1 + 4K3 + 5K24 )R → (2K + K24 )R ; (2.36) 15 3 where the 0nal expressions apply to the case of equal Frank constants. When K24 = 0, these energies reduce to those calculated in Ref. [126]. Note that the Frank free energy of point defects does not diverge in contrast to the distortion energy of disclinations in the preceding subsection. The hyperbolic hedgehog has lower energy than the radial hedgehog provided K3 ¡ 6K1 − 5K24 or K ¿ K24 for the one-constant approximation. Thus, if we concentrate on the bulk energies, i.e., K24 = 0, the hyperbolic hedgehog is always energetically preferred, since K1 is always of the same order as K3 . This seems to explain the observation of Lavrentovich and Terentjev, already mentioned [126], who found the con0guration illustrated in Fig. 7 in a nematic drop with radial boundary conditions at the outer surface. However, a detailed explanation has to take into account the Frank free energy of the strongly twisted transition region between hyperbolic and radial hedgehog [126]. In Section 7.4 we will present a linear stability analysis for the radial hedgehog against twisting. In terms of the Frank constants, it provides a criterion for the twist transition to take place, and its shows that the twisting starts close to the defect core. If, in addition to the one-constant approximation, K24 also ful0ls the Cauchy relation (2.5), i.e., K = K24 , the energies of the two hedgehog con0gurations in Eqs. (2.36) are equal, as one could have predicted from Eq. (2.6), which is then invariant with respect to rigid rotations of even a spatially varying n. The twisting of a hedgehog in a nematic drop takes place at a length scale of several microns [26,126,182]. However, point defect also possess a 0ne structure at smaller length scales, which has attracted a lot of attention. Fig. 9 illustrates how a hyperbolic hedgehog opens up to a −1=2

H. Stark / Physics Reports 351 (2001) 387–474

405

disclination ring by 0lling in vertical lines of the director 0eld. The far0eld of the disclination ring is still given by the hedgehog so that the ring can be assigned the same topological point charge Q = 1. Similarly, a radial point defect is topologically equivalent to a 1=2 disclination ring. Mineev pointed out that the characterization of a ring defect requires two parameters; the index of the line and the charge of the point defect [152]. The classi0cation of ring defects within homotopy theory was developed by Garel [86] and Nakanishi et al. [161]. It can be asked whether it is energetically favorable for a hedgehog to open up to a disclination ring [155,225]. One can obtain a crude estimate of the radius R0 of the disclination ring with the help of Eqs. (2.32) and (2.36) for disclination and hedgehog energies. When R0 rc , the director con0guration of a charge 1 disclination ring is essentially that of a simple disclination line, discussed in the previous subsection. It extends up to distances of order R0 from the ring center. Beyond this radius, the director con0guration is approximately that of a hedgehog (radial or hyperbolic). Thus, we can estimate the energy of a disclination ring of radius R0 centered in a spherical region of radius R to be    1  R0 Fring ≈ 2R0 K + 8!K(R − R0 ) ; (2.37) + ln 4 2 rc where ! = 1 for a radial hedgehog and ! = 1=3 for a hyperbolic hedgehog. We also set K24 = 0. Minimizing over R0 , we 0nd    16 3 !− : (2.38) R0 = rc exp  32 Though admittedly crude, this approximation yields a result that has the same form as that calculated in Refs. [155,225] using a more sophisticated ansatz for the director 0eld. It has the virtue that it applies to both radial and hyperbolic far-0eld con0gurations. It predicts that the core M of a radial hedgehog should be a ring with radius R0 ≈ rc e3:6 , or R0 ≈ 360 nm for rc ≈ 100 A. The core of the hyperbolic hedgehog, on the other hand, will be a point rather than a ring because R0 ≈ rc e−0:2 ≈ rc . As in the case of disclinations, more re0ned models of the core of a point defect use the Landau–de Gennes free energy, which employs the second-rank tensor Q as an order parameter. Schopohl and Sluckin [208] chose a uniaxial Q but allowed the degree of orientational order, described by the Maier–Saupe parameter S [29,51], to continuously approach zero at the center of the defect. A stability analysis of the Landau–de Gennes free energy demonstrates that the radial hedgehog is either metastable or unstable against biaxial perturbations in the order parameter depending on the choice of the temperature and elastic constants [195,88]. Penzenstadler and Trebin modeled a biaxial defect core [171]. They found that the core radius is of the order of the biaxial correlation length b , which for the compound MBBA gives approximately 25 nm. This is an order of magnitude smaller than the estimate above. The reason might be that the ansatz function used by Penzenstadler and Trebin does not include a biaxial disclination ring. Such a ring encircles a region of uniaxial order, as illustrated in the right part of Fig. 9, and it possesses a biaxial disclination core. Numerical studies indicate the existence of such a ring [218,87] but a detailed analysis of the competition between a biaxial core and a biaxial disclination ring is still missing. We expect that Eq. (2.37) for a disclination ring and therefore Eq. (2.38) for its radius can be justi0ed within the Landau–de Gennes theory for R0 rc ∼ b .

406

H. Stark / Physics Reports 351 (2001) 387–474

However, the Frank elastic constant and the core energy will be replaced by combinations of the Landau parameters. Since the ring radius R0 varies exponentially with the elastic constants and the core energy, and since rc is only roughly de0ned, it is very diNcult to predict with certainty even the order of magnitude of R0 . Further investigations are needed. 3. Nematic colloidal dispersions In this section we 0rst give a historic account of the topic relating it to recent developments in the liquid crystal 0eld and reviewing the work performed on colloidal dispersions in nematic liquid crystals. Then, with nematic emulsions, we introduce one particular model system for such colloidal dispersions. 3.1. Historic account Liquid crystal emulsions, in which surfactant-coated drops, containing a liquid crystalline material, are dispersed in water, have been a particularly fruitful medium for studying topological defects for thirty years [147,61,26,126,121,60]. The liquid crystalline drops typically range from 10 to 50
m in diameter and are visible under a microscope. Changes in the Frank director n are easily studied under crossed polarizers. The isolated drops in these emulsions provide an idealized spherical con0ning geometry for the nematic phase. With the introduction of polymer-dispersed liquid crystals as electrically controllable light shutters [58,60], an extensive study of liquid crystals con0ned to complex geometries, like distorted drops in a polymer matrix or a random porous network in silica aerogel, was initiated [60,44]. Here, we are interested in the inverse problem that is posed by particles suspended in a nematic solvent. Already in 1970, Brochard and de Gennes studied a suspension of magnetic grains in a nematic phase and determined the director 0eld far away from a particle [22]. The idea was to homogeneously orient liquid crystals with a small magnetic anisotropy by a reasonable magnetic 0eld strength through the coupling between the liquid-crystal molecules and the grains. The idea was realized experimentally by two groups [31,75]. However, even in the highly dilute regime the grains cluster. Extending Brochard’s and de Gennes’ work, Burylov and Raikher studied the orientation of an elongated particle in a nematic phase [24]. Chaining of bubbles or microcrystallites was used to visualize the director 0eld close to the surface of liquid crystals [191,36]. A bistable liquid crystal display was introduced based on a dispersion of agglomerations of silica spheres in a nematic host [62,118,117,91]. The system was called 4lled nematics. Chains and clusters were observed in the dispersion of latex particles in a lyotropic liquid crystal [181,188,189]. The radii of the particles were 60 and 120 nm. Therefore, details of the director 0eld could not be resolved under the polarizing microscope. Terentjev et al. [225,119,201,213] started to investigate the director 0eld around a sphere by both analytical and numerical methods, 0rst concentrating on the Saturn-ring and surface-ring con0guration. Experiments of Philippe Poulin and coworkers on inverted nematic emulsions, which we describe in the following subsection, clearly demonstrated the existence of a dipolar structure formed by a water droplet and a companion hyperbolic hedgehog [182,179,183,184]. A similar observation at a nematic-isotropic interface was made by Bob Meyer in 1972 [148].

H. Stark / Physics Reports 351 (2001) 387–474

407

Lately, Poulin et al. [180] were able to identify the dipolar structure in suspensions of latex particles and they could observe an equatorial ring con0guration in the weak-anchoring limit of nematic emulsions [153]. In a very recent paper, Gu and Abbott reported Saturn-ring con0gurations around solid microspheres [97]. Particles in contact with the glass plates of a cell were studied in Ref. [98]. Lubensky, Stark, and coworkers presented a thorough analytical and numerical analysis of the director 0eld around a spherical particle [182,140,219]. It is discussed in Section 4. Ramaswamy et al. [190] and Ruhwandl and Terentjev [200] determined the long-range quadrupolar interaction of particles surrounded by a ring disclination, whereas Lubensky et al. addressed both dipolar and quadrupolar forces [140] (see Section 5). Recently, Lev and Tomchuk studied aggregates of particles under the assumption of weak anchoring [130]. Work on the Stokes drag of a spherical object immersed into a uniformly aligned nematic was performed by Diogo [57], Roman and Terentjev [194], and Heuer et al. [112,105]. The calculations were extended to the Saturn-ring con0guration by Ruhwandl and Terentjev [199] and to the dipolar structure by Ventzki and Stark [228], whose work is explored in detail in Section 6. The Stokes drag was also determined through molecular dynamics simulations by Billeter and Pelcovits [10]. Stark and Stelzer [220] numerically investigated multiple nematic emulsions [182] by means of 0nite elements. We discuss the results in Section 7. It is interesting to note that dipolar con0gurations also appear in two-dimensional systems including (1) free standing smectic 0lms [132,175], where a circular region with an extra layer plays the role of the spherical particle, and (2) Langmuir 0lms [74], in which a liquid-expanded inclusion in a tilted liquid-condensed region acts similarly. Pettey et al. [175] studied the dipolar structure in two dimensions theoretically. In cholesteric liquid crystals particle-stabilized defect gels were found [239], and people started to investigate dispersions of particles in a smectic phase [80,108,12]. Sequeira and Hill were the 0rst to measure the viscoelastic response of concentrated suspensions of zeolite particles in nematic MBBA [209]. Meeker et al. [143] reported a gel-like structure in nematic colloidal dispersions with a signi0cant shear modulus. Perfectly ordered chains of oil droplets in a nematic were produced from phase separation by Loudet et al. [137]. Very recent studies of the nematic order around spherical particles are based on the minimization of the Landau–de Gennes free energy using an adaptive grid scheme [83], or they employ molecular dynamics simulations of Gay–Berne particles [10,6]. The 0ndings are consistent with the presentation in Section 4. With two excellent publications [210,211], Ping Sheng initiated the interest in partially ordered nematic 0lms above the nematic-isotropic phase transition temperature Tc using the Landau–de Gennes approach. In 1981, Horn et al. [106] performed 0rst measurements of liquid crystal-mediated forces between curved mica sheets. Motivated by both works, Poniewierski \ and Sluckin re0ned Sheng’s study [177]. Bor\stnik and Zumer explicitly considered two parallel plates immersed into a liquid crystal slightly above Tc [18], and thoroughly investigated short-range interactions due to the surface-induced nematic order. An analog work was presented by de Gennes, however, assuming a surface-induced smectic order [50]. The e9ect of such a presmectic 0lm was measured by Moreau et al. [154]. Recent studies address short-range forces of spherical objects using either analytical methods [15], which we report in Section 8, or numerical calculations [85]. In Section 8 we also demonstrate that such forces can induce >occulation of colloidal particles above the nematic-isotropic phase transition [16,17]. In a more general context, they were also suggested by LUowen [135,136]. Mu\sevi\c et al. probe these

408

H. Stark / Physics Reports 351 (2001) 387–474

interactions with the help of an atomic force microscope [158,159,113], whereas BUottger et al. [19] and Poulin et al. [178] are able to suspend solid particles in a liquid crystal above Tc . Even Casimir forces arising from >uctuations in the liquid-crystalline order parameter were investigated both in the nematic [3,2,223] and isotropic phase [240] of a liquid crystal. 3.2. Nematic emulsions In 1996, Philippe Poulin succeeded in producing inverted and multiple nematic emulsions [182,183]. The notion “inverted” refers to water droplets dispersed in a nematic solvent, in contrast to direct liquid-crystal-in-water emulsions. If the solvent itself forms drops surrounded by the water phase, one has multiple emulsions. We introduce them here since they initiated the theoretical work we report in the following sections. Philippe Poulin dispersed water droplets of 1 to 5
m in diameter in a nematic liquid crystal host, pentylcyanobiphenyl (5CB), which formed larger drops (∼ 50
m diameter) surrounded by a continuous water phase. This isolated a controlled number of colloidal droplets in the nematic host which allowed to observe their structure readily. As a surfactant, a small amount of sodium dodecyl sulfate was used. It is normally ine9ective at stabilizing water droplets in oil. Nevertheless, the colloidal water droplets remained stable for several weeks, which suggested that the origin of this stability is the surrounding liquid crystal—a hypothesis that was con0rmed by the observation that droplets became unstable and coalesced in less than one hour after the liquid crystal was heated to the isotropic phase. The surfactant also guaranteed a homeotropic, i.e., normal boundary condition of the director at all the surfaces. The multiple nematic emulsions were studied by observing them between crossed polarizers in a microscope. Under such conditions, an isotropic >uid will appear black, whereas regions in which there is the birefringent nematic will be colored. Thus the large nematic drops in a multiple emulsion appear predominately red in Fig. 10a, 1 whereas the continuous water phase surrounding them is black. Dispersed within virtually all of the nematic drops are smaller colloidal water droplets, which also appear dark in the photo; the number of water droplets tends to increase with the size of the nematic drops. Remarkably, in all cases, the water droplets are constrained at or very near the center of the nematic drops. Moreover, their Brownian motion, usually observed in colloidal dispersions, has completely ceased. However, when the sample is heated to change the nematic into an isotropic >uid, the Brownian motion of the colloidal droplets is clearly visible in the microscope. Perhaps the most striking observation in Fig. 10a is the behavior of the colloidal droplets when more than one of them cohabit the same nematic drop: the colloidal droplets invariably form linear chains. This behavior is driven by the nematic liquid crystal: the chains break, and the colloidal droplets disperse immediately upon warming the sample to the isotropic phase. However, although the anisotropic liquid crystal must induce an attractive interaction to cause the chaining, it also induces a shorter range repulsive interaction. A section of a chain of droplets under higher magni0cation (see Fig. 10b) shows that the droplets are prevented from approaching each other too closely, with the separation between droplets being a signi0cant 1

Reprinted with permission from P. Poulin, H. Stark, T.C. Lubensky, D.A. Weitz, Novel colloidal interactions in anisotropic >uids, Science 275 (1997) 1770. Copyright 1997 American Association for the Advancement of Science.

H. Stark / Physics Reports 351 (2001) 387–474

409

Fig. 10. (a) Microscope image of a nematic multiple emulsion taken under crossed polarizers, (b) a chain of water droplets under high magni0cation, (c) a nematic drop containing a single water droplet.

fraction of their diameter. A careful inspection of Fig. 10b even reveals black dots between the droplets which we soon will identify as topological point defects. The distance between droplets and these host->uid defects increases with the droplet radius. To qualitatively understand the observation, we start with one water droplet placed at the center of a large nematic drop. The homeotropic boundary condition enforces a radial director 0eld between both spherical surfaces. It exhibits a distinctive four-armed star of alternating bright and dark regions under crossed polarizers that extend throughout the whole nematic drop as illustrated in Fig. 10c. Evidently, following the explanations in Section 2:4:2 about point defects, the big nematic drop carries a topological point charge Q = 1 that is matched by the small water droplet which acts like a radial hedgehog. Each water droplet beyond the 0rst added to the interior of the nematic drop must create orientational structure out of the nematic itself to satisfy the global constraint Q = 1. The simplest (though not the only [140]) way to satisfy this constraint is for each extra water droplet to create a hyperbolic hedgehog in the nematic host. Note that from Fig. 8 we already know that a radial hedgehog (represented by the water droplet) and a hyperbolic point defect carry a total charge zero. Hence, N water droplets in a large nematic drop have to be accompanied by N − 1 hyperbolic hedgehogs. Fig. 11 presents a qualitative picture of the director 0eld lines for a string of three droplets. It is rotationally symmetric about the horizontal axis. Between the droplets, hyperbolic hedgehogs appear. They prevent the water droplets from approaching each other and from 0nally coalescing since this would involve a strong distortion of the director 0eld. The defects therefore mediate a short-range repulsion between the droplets.

410

H. Stark / Physics Reports 351 (2001) 387–474

Fig. 11. The director 0eld lines of a nematic drop containing a string of three spherical particles.

In the following sections, we demonstrate the physical ideas which evolved from the experiments on multiple nematic emulsions. We will explain the chaining of droplets by introducing the topological dipole formed by one spherical particle and its companion hyperbolic defect. This leads us to the next chapter where we investigate the simplest situation, i.e., one particle placed in a nematic solvent which is uniformly aligned at in0nity. 4. The paradigm---one particle The multiple nematic emulsions that we introduced in Section 3.2 are already a complicated system. In this section we investigate thoroughly by both analytical and numerical means what I regard as the paradigm for the understanding of inverted nematic emulsions. We ask which director 0eld con0gurations do occur when one spherical particle that prefers a radial anchoring of the director at its surface is placed into a nematic solvent uniformly aligned at in0nity. This constitutes the simplest problem one can think of, and it is a guide to the understanding of more complex situations. 4.1. The three possible con4gurations If the directors are rigidly anchored at the surface, the particle carries a topological charge Q = 1. Because of the boundary conditions at in0nity, the total charge of the whole system is zero; therefore, the particle must nucleate a further defect in its nematic environment. One possibility is a dipolar structure where the particle and a hyperbolic hedgehog form a tightly bound object which we call dipole for short (see Fig. 12). As already explained in Fig. 8, the topological charges +1 of a radial hedgehog, represented by the particle, and of a hyperbolic point defect “add up” to a total charge of zero. In the Saturn-ring con0guration, a −1=2 disclination ring encircles the spherical particle at its equator (see Fig. 12). Of course, the disclination ring can be moved upward or downward, and by shrinking it to the topologically

H. Stark / Physics Reports 351 (2001) 387–474

411

Fig. 12. A spherical particle with a preferred homeotropic anchoring at its surface that is placed into a uniformly aligned nematic liquid crystal exhibits three possible structures: the dipole con0guration where the particle is accompanied by a hyperbolic hedgehog, the Saturn-ring con0guration where the particle is surrounded by a −1=2 disclination ring at the equator, and the surface-ring con0guration.

equivalent hyperbolic hedgehog, the Saturn ring is continuously transformed into the dipole con0guration. However, our calculations show that a non-symmetric position of the defect ring is never stable. When the surface anchoring strength W is lowered (see Fig. 12), the core of the disclination ring prefers to sit directly at the surface of the particle. For suNciently low W , the director 0eld becomes smooth everywhere, and a ring of tangentially oriented directors is located at the equator of the sphere. In the case of tangential boundary conditions, there exists only one structure. It possesses two surface defects, called boojums, at the north and south pole of the particle [145,26,120,231]. We will not investigate it further. It is instructive to 0rst consider the director 0eld far away from the particle, which crucially depends on the global symmetry of the system [22,140]. With its knowledge, ansatz functions for the director con0gurations around a particle can be checked. Furthermore, the far 0eld determines the long-range two-particle interaction. Let the director n0 at in0nity point along the z axis. Then, in the far 0eld, the director is approximated by n(r) ≈ (nx ; ny ; 1) with nx ; ny 1. In leading order, the normalization of the director can be neglected, and the Euler–Lagrange equations for nx and ny arising from a minimization of the Frank free energy in the one-constant

412

H. Stark / Physics Reports 351 (2001) 387–474

approximation are simply Laplace equations: ∇2 n3 = 0 :

(4.1)

The solutions are the well-known multipole expansions of electrostatics that include monopole, dipole, and quadrupole terms. They are all present if the suspended particle has a general shape or if, e.g., the dipole in Fig. 12 is tilted against n0 . In the dipole con0guration with its axial symmetry about n0 , the monopole is forbidden, and we obtain x zx y zy nx = p + 2c 5 and ny = p + 2c 5 ; (4.2) r3 r r3 r where r = (x2 + y2 + z 2 )1=2 . We use the expansion coeNcients p and c to assign both a dipole (p) and quadrupole (c ) moment to the con0guration: p = p n0

and

c = c(n0 ⊗ n0 − 1=3) :

(4.3)

The symbol ⊗ means tensor product, and 1 is the unit tensor of second rank. We adopt the convention that the dipole moment p points from the companion defect to the particle. Hence, if p ¿ 0, the far 0eld of Eqs. (4:2) belongs to a dipole con0guration with the defect sitting below the particle (see Fig. 12). Note, that by dimensional analysis, p ∼ a2 and c ∼ a3 , where a is the radius of the spherical particle. Saturn-ring and surface-ring con0gurations possess a mirror plane perpendicular to the rotational axis. Therefore, the dipole term in Eqs. (4:2) is forbidden, i.e., p = 0. We will show in Section 6 that the multipole moments p and c determine the long-range two-particle interaction. We will derive it on the basis of a phenomenological theory. In the present section we investigate the dipole by both analytical and numerical means. First, we identify a twist transition which transforms it into a chiral object. Then, we study the transition from the dipole to the Saturn ring con0guration, which is induced either by decreasing the particle radius or by applying a magnetic 0eld. The role of metastability is discussed. Finally, we consider the surface-ring con0guration and point out the importance of the saddle-splay free energy F24 . Lower bounds for the surface-anchoring strength W are given. 4.2. An analytical investigation of the dipole Even in the one-constant approximation and for 0xed homeotropic boundary conditions, analytical solutions of the Euler–Lagrange equations, arising from the minimization of the Frank free energy, cannot be found. The Euler–Lagrange equations are highly non-linear due to the normalization of the director. In this subsection we investigate the dipole con0guration with the help of ansatz functions that obey all boundary conditions and possess the correct far-0eld behavior. The free parameters in these ansatz functions are determined by minimizing the Frank free energy. We will see that this procedure already provides a good insight into our system. We arrive at appropriate ansatz functions by looking at the electrostatic analog of our problem [182,140], i.e., a conducting sphere of radius a and with a reduced charge q which is exposed to an electric 0eld of unit strength along the z axis. The electric 0eld is E (r ) = ez + qa2

r

r3

− a3

r 2 ez − 3z r : r5

(4.4)

H. Stark / Physics Reports 351 (2001) 387–474

413

Fig. 13. Frank free energy (in units of ,Ka) for the topological dipole as a function of the reduced distance rd =a from the particle center to the companion hedgehog.

In order to enforce the boundary condition that E be normal to the surface of the sphere, an electric image dipole has to be placed at the center of the sphere. The ansatz function for the director 0eld follows from a normalization: n(r) = E (r)= |E (r)|. An inspection of its far 0eld gives  x xz y yz n(r ) ≈ qa2 3 + 3a3 5 ; qa2 3 + 3a3 5 ; (4.5) r r r r in agreement with Eqs. (4:2). The electrostatic analog assigns a dipole moment qa2 and a quadrupole moment 3a3 =2 to the topological dipole. The zero of the electric 0eld determines the location −rd ez of the hyperbolic hedgehog on the z axis. Thus, q or the distance rd from the center of the particle are the variational parameters of our ansatz functions. Note that for q = 3, the hedgehog just touches the sphere, and that for q ¡ 3, a singular ring appears at the surface of the sphere. In Fig. 13 we plot the Frank free energy in the one-constant approximation and in units of ,Ka as a function of the reduced distance rd =a. The saddle-splay term is not included, since for rigid anchoring it just provides a constant energy shift. There is a pronounced minimum at rd0 = 1:17a corresponding to a dipole moment p = qa2 = 3:08a2 . The minimum shows that the hyperbolic hedgehog sits close to the spherical particle. To check the magnitude of the thermal >uctuations of its radial position, we determine the curvature of the energy curve at rd0 ; its approximate value amounts to 33,K=a. According to the equipartition theorem, the average thermal displacement rd0 follows from the expression

rd0 kB T ≈ ≈ 2 × 10−3 ; (4.6) a 33Ka where the 0nal estimate employs kB T ≈ 4 × 10−14 erg; K ≈ 10−6 dyn, and a=1
m. These >uctuations in the length of the topological dipole are unobservably small. For angular >uctuations of the dipole, we 0nd 5 ≈ 10−2 , i.e., still diNcult to observe [140]. We conclude that the spherical particle and its companion hyperbolic hedgehog form a tightly bound object. Interestingly, we note that angular >uctuations in the 2D version of this problem diverge logarithmically with the sample size [175]. They are therefore much larger and have indeed been observed in free standing smectic 0lms [132].

414

H. Stark / Physics Reports 351 (2001) 387–474

Fig. 14. (Top) Image of a single droplet with its companion defect as observed under crossed polarizers obtained by Poulin [183]. (Bottom) Simulated image of the same con0guration using the Jones matrix formalism [140]. The two pictures are very similar. From Ref. [140].

The droplet-defect dipole was observed by Philippe Poulin in inverted nematic emulsions [183]. In the top part of Fig. 14 we present how it looks like in a microscope under crossed polarizers, with one polarizer parallel to the dipole axis. In the bottom part of Fig. 14 we show a calculated image using the Jones matrix formalism [60] based on the director 0eld of the electrostatic analog. Any refraction at the droplet boundary is neglected. The similarity of the two images is obvious and clearly con0rms the occurrence of the dipole con0guration. The electric 0eld ansatz is generalized by no longer insisting that it originates in a true electric 0eld. This allows us to introduce additional variational parameters [140]. The Frank free energy at rd0 is lowered, and the equilibrium separation amounts to rd0 = 1:26a. The

H. Stark / Physics Reports 351 (2001) 387–474

415

respective dipole and quadrupole moments turn out to be p = 2:20a2 and c = −1:09a2 . We are also able to construct ansatz functions for the dipole–Saturn ring transition utilizing the method of images for the related 2D problem and correcting the far 0eld [140]. The results agree with the numerical study presented in the next subsection. 4.3. Results and discussion of the numerical study Before we present the results of our numerical study, we summarize the numerical method. Details can be found in [219]. 4.3.1. Summary of numerical details The numerical investigation is performed on a grid which is de0ned by modi0ed spherical coordinates. Since the region outside the spherical particle is in0nitely extended, we employ a radial coordinate 6 = 1=r 2 . The exponent 2 is motivated by the far 0eld of the dipole con0guration. Such a transformation has two advantages. The exterior of the particle is mapped into a 0nite region, i.e., the interior of the unit sphere (6 6 1). Furthermore, equally spaced grid points along the coordinate 6 result in a mesh size in real space which is small close to the surface of the particle. In this area the director 0eld is strongly varying, and hence a good resolution for the numerical calculation is needed. On the other hand, the mesh size is large far away from the sphere where the director 0eld is nearly homogeneous. Since our system is axially symmetric, the director 0eld only depends on 6 and the polar angle 5. The director is expressed in the local coordinate basis (er ; e5 ; e7 ) of the standard spherical coordinate system, and the director components are parametrized by a tilt [(6; 5)] and a twist [(6; 5)] angle: nr = cos ; n5 = sin  cos , and n7 = sin  sin . The total free energy Fn of Eq. (2.1) is expressed in the modi0ed spherical coordinates. Then, the Euler–Lagrange equations in the bulk and at the surface are formulated with the help of the chain rules of Eqs. (2.14) and (2.15) and by utilizing the algebraic program Maple. A starting con0guration of the director 0eld is chosen and relaxed into a local minimum via the Newton–Gauss–Seidel method [187] which was implemented in a Fortran program. So far we have described the conventional procedure of a numerical investigation. Now, we address the problem of how to describe disclination rings numerically. Fig. 15 presents such a ring whose general position is determined by a radial (rd ) and an angular (5d ) coordinate. The free energy Fn of the director 0eld follows from a numerical integration. This assigns some energy to the disclination ring which certainly is not correct since the numerical integration does not realize the large director gradients close to the defect core. To obtain a more accurate value for the total free energy F, we use the expression F = Fn − Fn |torus + Fc=d × 2rd sin 5d ;

(4.7)

where Fc and Fd are the line energies of a disclination introduced in Eqs. (2.32) and (2.34). The quantity Fn |torus denotes the numerically calculated free energy of a toroidal region of cross section R2 around the disclination ring. Its volume is R2 × 2rd sin 5d . The value Fn |torus is replaced by the last term on the right-hand side of Eq. (4.7), which provides the correct free energy with the help of the line energies Fc or Fd . We checked that the cross section R2 of the cut torus has to be equal or larger than 3V6 V5=2, where V6 and V5 are the lattice

416

H. Stark / Physics Reports 351 (2001) 387–474

Fig. 15. Coordinates (rd ; 5d ) for a −1=2 disclination ring with a general position around the spherical particle. From Ref. [219].

Fig. 16. The reduced distance rd =a of the hyperbolic hedgehog from the center of the sphere as a function of the reduced splay (K1 =K3 ) and twist (K2 =K3 ) constants.

constants of our grid. For larger cross sections, the changes in the free energy F for 0xed core radius rc were less than 1%, i.e., F became independent of R2 . What is the result of this procedure? All lengths in the free energy Fn can be rescaled by the particle radius a. This would suggest that the director con0guration does not depend on the particle size. However, with the illustrated procedure a second length scale, i.e., the core radius rc of a disclination, enters. All our results on disclination rings therefore depend on the ratio a=rc . In discussing them, we assume rc ≈ 10 nm [111] which then determines the radius a for a given a=rc . 4.3.2. Twist transition of the dipole con4guration In this subsection we present our numerical study of the topological dipole. We always assume that the directors are rigidly anchored at the surface (W → ∞) and choose a zero magnetic 0eld. In Fig. 16 we plot the reduced distance rd =a of the hedgehog from the center of the sphere as a function of the reduced splay (K1 =K3 ) and twist (K2 =K3 ) constants. In the

H. Stark / Physics Reports 351 (2001) 387–474

417

Fig. 17. (a) Nail picture of a closeup of the twisted dipole con0guration. Around the hyperbolic hedgehog the directors are tilted relative to the drawing plane. From Ref. [219]. (b) Phase diagram of the twist transition as a function of the reduced splay (K1 =K3 ) and twist (K2 =K3 ) constants. A full explanation is given in the text.

one-constant approximation, we 0nd rd =1:26 ± 0:02, where the mesh size of the grid determines the uncertainty in rd . Our result is in excellent agreement with the generalized electric-0eld ansatz we introduced in the last subsection [140]. However, Ruhwandl and Terentjev using a Monte-Carlo minimization report a somewhat smaller value for rd [200]. In front of the thick line rd is basically constant. Beyond the line, rd starts to grow which indicates a structural change in the director 0eld illustrated in the nail picture of Fig. 17a. Around the hyperbolic hedgehog the directors develop a non-zero azimuthal component n7 , i.e., they are tilted relative to the drawing plane. This introduces a twist into the dipole. It should be visible under a polarizing microscope when the dipole is viewed along its symmetry axis. In Fig. 17b we draw a phase diagram of the twist transition. As expected, it occurs when K1 =K3 increases or when K2 =K3 decreases, i.e., when a twist deformation costs less energy than a splay distortion. The open circles are numerical results for the transition line which can well be 0tted by the straight line K2 =K3 ≈ K1 =K3 − 0:04. Interestingly, the small o9set 0.04 means that K3 does not play an important role. Typical calamatic liquid crystals like MBBA, 5CB, and PAA should show the twisted dipole con0guration. Since the twist transition breaks the mirror symmetry of the dipole, which then becomes a chiral object, we describe it by a Landau expansion of the free energy: 2 max 4 F = F0 + a(K1 =K3 ; K2 =K3 )[nmax 7 ] + c[n7 ] :

(4.8)

we have introduced a simple order parameter. With the maximum azimuthal component nmax 7 Since the untwisted dipole possesses a mirror symmetry, only even powers of nmax are allowed. 7 The phase transition line is determined by a(K1 =K3 ; K2 =K3 ) = 0. According to Eq. (4.8), we expect a power-law dependence of the order parameter with the exponent 1=2 in the twist region close to the phase transition. To test this idea, we choose a constant K2 =K3 ratio and determine nmax for varying K1 . As the log–log plot in Fig. 18 illustrates, when approaching the 7 phase transition, the order parameter obeys the expected power law: nmax ∼ (K1 =K3 − 0:4372)1=2 7

with K2 =K3 = 0:4 :

(4.9)

418

H. Stark / Physics Reports 351 (2001) 387–474

Fig. 18. Log–log plot of the order parameter nmax versus K1 =K3 close to the twist transition (K2 =K3 = 0:4); 7 ◦ : : : numerical values, – : : : 0t by a straight line. Fig. 19. The free energy F in units of K3 a as a function of the angular coordinate 5d . The parameter of the curves is the particle size a. Further parameters are indicated in the inset.

4.3.3. Dipole versus Saturn ring There are two possibilities to induce a transition from the dipole to the Saturn-ring con0guration; either by reducing the particle size or by applying, e.g., a magnetic 0eld. We always assume rigid anchoring in this subsection, set K24 = 0, and start with the 0rst point. 4.3.3.1. E<ect of particle size. In Fig. 19 we plot the free energy F in units of K3 a as a function of the angular coordinate 5d of the disclination ring. For constant 5d , the free energy F was chosen as the minimum over the radial coordinate rd . The particle radius a is the parameter of the curves, and the one-constant approximation is employed. Recall that 5d = =2 and 5d =  correspond, respectively, to the Saturn-ring or the dipole con0guration. Clearly, for small particle sizes (a = 180 nm) the Saturn ring is the absolutely stable con0guration, and the dipole enjoys some metastability. However, thermal >uctuations cannot induce a transition to the dipole since the potential barriers are much higher than the thermal energy kB T . E.g., a barrier of 0:1K3 a corresponds to 1000kB T (T = 300 K; a = 1
m). At a ≈ 270 nm, the dipole assumes the global minimum of the free energy, and 0nally the Saturn ring becomes absolutely unstable at a ≈ 720 nm. The scenario agrees with the results of Ref. [140] where an ansatz function for the director 0eld was used. Furthermore, we stress that the particle sizes were calculated with the choice of 10 nm as the real core size of a line defect, and that our results depend on the line energy (2.32) of the disclination. The reduced radial coordinate rd =a of the disclination ring as a function of 5d is presented in Fig. 20. It was obtained by minimizing the free energy for 0xed 5d . As long as the ring is open, rd does not depend on 5d within an error of ±0:01. Only in the region where it closes to the hyperbolic hedgehog, does rd increase sharply. The 0gure also illustrates that the ring sits closer to larger particles. The radial position of rd =a = 1:10 for 720 nm particles agrees very well with analytical results obtained by using an ansatz function (see Refs. [140]) and with numerical calculations based on a Monte-Carlo minimization [200]. Recent observations of

H. Stark / Physics Reports 351 (2001) 387–474

419

Fig. 20. The reduced radial coordinate rd =a of a disclination ring as a function of 5d for two particle sizes. Further parameters are indicated in the inset.

the Saturn-ring con0guration around glass spheres of 40, 60, or 100
m in diameter [97] seem to contradict our theoretical 0ndings. However, we explain them by the strong con0nement in 120-
m-thick liquid crystal cells which is equivalent to a strong magnetic 0eld. 4.3.3.2. E<ect of a magnetic 4eld. A magnetic 0eld applied along the symmetry axis of the dipole can induce a transition to the Saturn-ring con0guration. This can be understood from a simple back-of-the-envelope calculation. Let us consider high magnetic 0elds, i.e., magnetic coherence lengths much smaller than the particle size a. The magnetic coherence length H was introduced in Eq. (2.9) as the ratio of elastic and magnetic torques on the director. For H a, the directors are basically aligned along the magnetic 0eld. In the dipole con0guration, the director 0eld close to the hyperbolic hedgehog cannot change its topology. The 0eld lines are “compressed” along the z direction, and high densities of the elastic and magnetic free energies occur in a region of thickness H . Since the 0eld lines have to bend around the sphere, the cross section of the region is of the order of a2 , and its volume is proportional to a2 H . The Frank free energy density is of the order of K=2H , where K is a typical Frank constant, and therefore the elastic free energy scales with Ka2 =H . The same holds for the magnetic free energy. In the case of the Saturn-ring con0guration, high free energy densities occur in a toroidal region of cross section ˙2H around the disclination ring. Hence, the volume scales with a2H , and the total free energy is of the order of Ka, i.e., a factor a=H smaller than for the dipole. Fig. 21 presents a calculation for a particle size of a=0:5
m and the liquid crystal compound 5CB. We plot the free energy in units of K3 a as a function of 5d for di9erent magnetic 0eld strengths which we indicate by the reduced inverse coherence length a=H . Without a 0eld (a=H = 0), the dipole is the energetically preferred con0guration. The Saturn ring shows metastability. A thermally induced transition between both states cannot happen because of the high potential barrier. At a 0eld strength a=H = 0:33, the Saturn ring becomes the stable con0guration. However, there will be no transition until the dipole loses its metastability at a 0eld strength a=H = 3:3, which is only indicated by an arrow in Fig. 21. Once the system has changed to the Saturn ring, it will stay there even for zero magnetic 0eld. Fig. 22a schematically illustrates how a dipole can be transformed into a Saturn ring with the help of a magnetic 0eld.

420

H. Stark / Physics Reports 351 (2001) 387–474

Fig. 21. The free energy F in units of ,K3 a as a function of the angular coordinate 5d . The parameter of the curves is the reduced inverse magnetic coherence length a=H . Further parameters are indicated in the inset. Fig. 22. (a) The Saturn ring is metastable at H =0. The dipole can be transformed into the Saturn ring by increasing the magnetic 0eld H beyond Ht2 where the dipole loses its metastability. Turning o9 the 0eld the Saturn ring remains. (b) The Saturn ring is unstable at H = 0. When the magnetic 0eld is decreased from values above Ht2 , the Saturn ring shrinks back to the dipole at Ht1 where the Saturn ring loses its metastability. A hysteresis occurs. From Ref. [219].

If the Saturn ring is unstable at zero 0eld, a hysteresis occurs (see Fig. 22b). Starting from high magnetic 0elds, the Saturn ring loses its metastability at Ht1 , and a transition back to the dipole takes place. In Fig. 19 we showed that the second situation is realized for particles larger than 720 nm. We also performed calculations for a particle size of 1
m and the liquid crystal compound 5CB and still found the Saturn ring to be metastable at zero 0eld in contrast to the result of the one-constant approximation. To be more concrete, according to Eq. (2.9), a=H = 1 corresponds to a 0eld strength of 4:6 T when 0:5
m particles and the material parameters of 5CB (K3 = 0:53 × 10−6 dyn; V = 10−7 ) are used. Hence, the transition to the Saturn ring in Fig. 21 occurs at a rather high 0eld of 15 T. Assuming that there is no dramatic change in a=H = 3:3 for larger particles, this 0eld decreases with increasing particle radius. Alternatively, the transition to the Saturn ring is also induced by an electric 0eld with the advantage that strong 0elds are much easier to apply. However, the large dielectric anisotropy V' = ' − '⊥ complicates a detailed analysis because of the di9erence between applied and local electric 0elds. Therefore, the electric coherence length E = [4K3 =(V' E 2 )]1=2 , which replaces H , only serves as a rough estimate for the applied 0eld E necessary to induce a transition to the Saturn ring. 4.3.4. In>uence of 4nite surface anchoring In the last subsection we investigate the e9ect of 0nite anchoring on the director 0eld around the spherical particle. The saddle-splay term with its elastic constant K24 is important now. We always choose a zero magnetic 0eld. In Fig. 23 we employ the one-constant approximation and plot the free energy versus the reduced surface extrapolation length S =a for di9erent reduced

H. Stark / Physics Reports 351 (2001) 387–474

421

Fig. 23. The minimum free energy F in units of ,K3 a as a function of the reduced surface extrapolation length S =a for di9erent K24 =K3 . A 0rst-order phase transition from the dipole to the surface ring occurs. Further parameters are indicated in the inset.

Fig. 24. The saddle-splay free energy F24 in units of ,K3 a as a function of S =a for the same curves as in Fig. 23. Inset: F24 in units of ,K24 a versus the angular width of the surface ring calculated from the ansatz functions in Eqs. (4.10). Fig. 25. Phase diagram of the dipole-surface ring transition as a function of S =a and K24 =K3 . Further parameters are indicated in the inset.

saddle-splay constants K24 =K3 . Recall that S is inversely proportional to the surface constant W [see Eq. (2.11)]. The straight lines belong to the dipole. Then, for decreasing surface anchoring, there is a 0rst-order transition to the surface-ring structure. We never 0nd the Saturn ring to be the stable con0guration although it enjoys some metastability. For K24 =K3 =0, the transition takes place at S =a ≈ 0:085. This value is somewhat smaller than the result obtained by Ruhwandl and Terentjev [200]. One could wonder why the surface ring already occurs at such a strong anchoring like S =a ≈ 0:085 where any deviation from the homeotropic anchoring costs a lot of energy. However, if V5 is the angular width of the surface ring where the director deviates from the homeotropic alignment (see inset of Fig. 24) then a simple energetical estimate allows V5 to be of the order of S =a. It is interesting to see that the transition point shifts to higher anchoring

422

H. Stark / Physics Reports 351 (2001) 387–474

strengths, i.e., decreasing S =a when K24 =K3 is increased. Obviously, the saddle-splay term favors the surface-ring con0guration. To check this conclusion, we plot the reduced saddle-splay free energy F24 versus S =a in Fig. 24. The horizontal lines belong to the dipole. They correspond to the saddle-splay energy 4K24 a which one expects for a rigid homeotropic anchoring at the surface of the sphere. In contrast, for the surface-ring con0guration the saddle-splay energy drops sharply. The surface ring at the equator of the sphere introduces a “saddle” in the director 0eld as illustrated in the inset of Fig. 24. Such structures are known to be favored by the saddle-splay term. We modeled the surface ring with an angular width V5 by the following radial and polar director components:      5 − =2 5 − =2 −1 nr = −tanh and n5 = − cosh ; (4.10) V5 V5 where V5=2 to ensure that nr = 1 at 5 = 0; , and calculated the saddle-splay energy versus V5 by numerical integration. The result is shown in the inset of Fig. 24. It 0ts very well to the full numerical calculations and con0rms again that a narrow “saddle” around the equator can considerably reduce the saddle-splay energy. For the liquid crystal compound 5CB we determined the stable con0guration as a function of K24 =K3 and S =a. The phase diagram is presented in Fig. 25. With its help, we can derive a lower bound for the surface constant W at the interface of water and 5CB when the surfactant sodium dodecyl sulfate is involved. As the experiments by Poulin et al. clearly demonstrate, water droplets dispersed in 5CB do assume the dipole con0guration. From the phase diagram we conclude S =a ¡ 0:09 as a necessary condition for the existence of the dipole. With a ≈ 1
m; K3 = 0:53 × 10−6 dyn, and de0nition (2.11) for S we arrive at W ¿ 0:06 erg=cm2 :

(4.11)

If we assume the validity of the Cauchy relation (2.5), which for 5CB gives K24 =K3 = 0:61, we conclude that W ¿ 0:15 erg=cm2 . Recently, Mondain-Monval et al. were able to observe an equatorial ring structure by changing the composition of a surfactant mixture containing sodium dodecyl sulfate (SDS) and a copolymer of ethylene and propylene oxide (Pluronic F 68) [153]. We conclude from our numerical investigation that they observed the surface-ring con0guration. 4.4. Conclusions In this section we presented a detailed study of the three director 0eld con0gurations around a spherical particle by both analytical and numerical means. We clearly 0nd that for large particles and suNciently strong surface anchoring, the dipole is the preferred con0guration. For conventional calamitic liquid crystals, where K2 ¡ K1 , the dipole should always exhibit a twist around the hyperbolic hedgehog. It should not occur in discotic liquid crystals where K2 ¿ K1 . According to our calculations, the bend constant K3 plays only a minor role in the twist transition. The Saturn ring appears for suNciently small particles provided that one can realize a suNciently strong surface anchoring. According to our investigation, for 200 nm particles the surface constant has to be larger than W = 0:3 erg=cm2 . However, the dipole can be transformed into the Saturn ring by means of a magnetic 0eld if the Saturn ring is metastable at H = 0. Otherwise a hysteresis is visible. For the liquid crystal compound 5CB, we 0nd the Saturn ring

H. Stark / Physics Reports 351 (2001) 387–474

423

to be metastable at a particle size a = 1
m. Increasing the radius a, this metastability will vanish in analogy with our calculations within the one-constant approximation (see Fig. 19). Lowering the surface-anchoring strength W , the surface-ring con0guration with a quadrupolar symmetry becomes absolutely stable. We never 0nd a stable structure with dipolar symmetry where the surface ring possesses a general angular position 5d or is even shrunk to a point at 5d = 0; . The surface ring is clearly favored by a large saddle-splay constant K24 . The dispersion of spherical particles in a nematic liquid crystal is always a challenge to experimentalists. The clearest results are achieved in inverted nematic emulsions [182,179,183,153,184]. However, alternative experiments with silica or latex spheres do also exist [181,188,189,153, 180,98] and produce impressive results [97]. We hope that the summary of our research stimulates further experiments which probe di9erent liquid crystals as a host >uid [180], manipulate the anchoring strength [153,97,98], and investigate the e9ect of external 0elds [97,98]. 5. Two-particle interactions To understand the properties of, e.g., multi-droplet emulsions, we need to determine the nature of particle–particle interactions. These interactions are mediated by the nematic liquid crystal in which they are embedded and are in general quite complicated. Since interactions are determined by distortions of the director 0eld, there are multi-body as well as two-body interactions. We will content ourselves with calculations of some properties of the e9ective two-particle interaction. To determine the position-dependent interaction potential between two particles, we should solve the Euler–Lagrange equations, as a function of particle separation, subject to the boundary condition that the director be normal to each spherical object. Solving completely these non-linear equations in the presence of two particles is even more complicated than solving them with one particle, and again we must resort to approximations. Fortunately, interactions at large separations are determined entirely by the far-0eld distortions and the multipole moments of an individual topological dipole or Saturn ring, which we studied in Section 4.1. The interactions can be derived from a phenomenological free energy. We will present such an approach in this section [190,182,140]. 5.1. Formulating a phenomenological theory In Section 4, we established that each spherical particle creates a hyperbolic hedgehog to which it binds tightly to create a stable topological dipole. The original spherical inclusion is described by three translational degrees of freedom. Out of the nematic it draws a hedgehog, which itself has three translational degrees of freedom. The two combine to produce a dipole with six degrees of freedom, which can be parametrized by three variables specifying the position of the particle, two angles specifying the orientation of the dipole, and one variable specifying the magnitude of the dipole. As we have seen, the magnitude of the dipole does not >uctuate much and can be regarded as a constant. The direction of the dipole is also fairly strongly constrained. It can, however, deviate from the direction of locally preferred orientation (parallel to a local director to be de0ned in more detail below) when many particles are present. The particle–defect pair is in addition characterized by its higher multipole

424

H. Stark / Physics Reports 351 (2001) 387–474

moments. The direction of the principal axes of these moments is speci0ed by the direction of the dipole as long as director con0gurations around the dipole remain uniaxial. The magnitudes of all the uniaxial moments like the strengths p and c of the dipole and quadrupole moment (see Section 4.1) are energetically 0xed, as we have shown in Section 4.2. When director con0gurations are not uniaxial, the multipole tensors will develop additional components, which we will not consider here. We can thus parametrize topological dipoles by their position and orientation and a set of multipole moments, which we regard as 0xed. Let e! be the unit vector specifying the direction of the dipole moment associated with droplet !. Its dipole and quadrupole moments are then p! = pe! and c ! = c(e! ⊗ e! − 1=3), where p and c are the respective magnitudes of the dipole and quadrupole moments calculated, e.g., by analytical means in Section 4.2. The symbol ⊗ means tensor product, and 1 is the second-rank unit tensor. Note, that this approach also applies to the Saturn-ring and surface-ring con0guration but with a vanishing dipole moment p = 0. It even applies to particles with tangential boundary conditions where two surfaces defects, called boojums [145,26,120], are located at opposite points of the sphere and where the director 0eld possesses a uniaxial symmetry, too. We now introduce dipole- and quadrupole-moment densities, P (r) and C (r), in the usual way. Let r! denote the position of droplet !, then   P (r ) = p! (r − r ! ) and C (r ) = c ! (r − r ! ) : (5.1) !

!

In the following, we construct an e9ective free energy for director and particles valid at length scales large compared to the particle radius. At these length scales, we can regard the spheres as point objects (as implied by the de0nitions of the densities given above). At each point in space, there is a local director n(r) along which the topological dipoles or, e.g., the Saturn rings wish to align. In the more microscopic picture, of course, the direction of this local director corresponds to the far-0eld director n0 . The e9ective free energy is constructed from rotationally invariant combinations of P , C , n, and the gradient operator ∇ that are also even under n → −n. It can be expressed as a sum of terms F = Fel + Fp + FC + Falign ;

(5.2)

where Fel is the Frank free energy, Fp describes interactions between P and n, FC describes interactions between C and n involving gradient operators, and
 Falign = −D d 3 r Cij (r)ni (r)nj (r) = −DQ {[e! ·n(r ! )]2 − 1=3} (5.3) !

e!

forces the alignment of the axes along the local director n(r! ). The leading contribution to Fp is identical to the treatment of the >exoelectric e9ect in a nematic [147,51]
Fp = 4K d 3 r[ − P ·n(∇·n) + :P · (n × ∇ × n)] ; (5.4) where : is a material-dependent unitless parameter. The leading contribution to FC is
FC = 4K d 3 r[(∇·n)n ·∇(ni Cij nj ) + ∇(ni Cij nj ) · (n × ∇ × n)] :

(5.5)

There should also be terms in FC like Cij ∇k ni ∇k nj . These terms can be shown to add contributions to the e9ective two-particle interaction that are higher order in separation than those arising

H. Stark / Physics Reports 351 (2001) 387–474

425

from Eq. (5.5). One coeNcient in Fp and all coeNcients in FC are 0xed by the requirement that the phenomenological theory yields the far 0eld of one particle given by Eq. (4.2) (see next subsection). Eq. (5.5) is identical to that introduced in Ref. [190] to discuss interactions between Saturn rings, provided ni Cij nj is replaced by a scalar density 6(r) = ! (r − r! ). The two energies are absolutely equivalent to leading order in the components n3 of n perpendicular to n0 provided all e! are restricted to be parallel to n0 . Since P prefers to align along the local director n, the dipole-bend coupling term in Eq. (5.4) can be neglected to leading order in deviations of the director from uniformity. The −P ·n(∇·n) term in Eq. (5.4) shows that dipoles aligned along n create local splay as is evident from the dipole con0guration depicted in Fig. 12. In addition, this term says that dipoles can lower their energy by migrating to regions of maximum splay while remaining aligned with the local director. Experiments on multiple nematic emulsions [182,183] support this conclusion. Indeed, the coupling of the dipole moment to a strong splay distortion explains the chaining of water droplets in a large nematic drop whose observation we reported in Section 3.2. We return to this observation in Section 7. 5.2. E<ective pair interactions In the following we assume that the far-0eld director n0 and all the multipole moments of the particles point along the z axis, i.e., e! = ez = n0 . Hence, we are able to write the dipole and quadrupole densities as P (r ) = P(r )n0

and

C (r ) = 32 C(r )(n0 ⊗ n0 − 1=3) ;

(5.6)

where P(r) and C(r) can be both positive and negative. We are interested in small deviations from n0 ; n = (nx ; ny ; 1), and formulate the e9ective energy of Eq. (5.2) up to harmonic order in n3 :
(5.7) F = K d 3 r[ 12 (∇n3 )2 − 4P 93 n3 + 4(9z C)93 n3 ] : The dipole-bend coupling term of Eq. (5.4) does not contribute because P is aligned along the far-0eld director. The Euler–Lagrange equations for the director components are ∇2 n3 = 493 [P(r ) − 9z C(r )] ;

which possess the solution
1 9 [P(r  ) − 9z C(r  )] : n 3 (r ) = − d 3 r  |r − r  | 3

(5.8)

(5.9)

For a single droplet at the origin, P(r) = p(r) and C(r) = 23 c(r), and the above equation yields exactly the far 0eld of Eq. (4.2). This demonstrates the validity of our phenomenological approach. Particles create far-0eld distortions of the director, which to leading order at large distances are determined by Eq. (5.8). These distortions interact with the director 0elds of other particles which leads to an e9ective particle–particle interaction that can be expressed to leading order

426

H. Stark / Physics Reports 351 (2001) 387–474

Fig. 26. A chain of three topological dipoles formed due to their dipolar interaction.

as pairwise interactions between dipole and quadrupole densities. Using Eq. (5.9) in Eq. (5.7), we obtain
F 1 d 3 r d 3 r  [P(r)VPP (r − r )P(r ) + C(r)VCC (r − r )C(r ) = 4K 2 + VPC (r − r )][C(r)P(r ) − P(r)C(r )] ; (5.10) with 1 1 VPP (r) = 93 93 = 3 (1 − 3 cos3 5) r r 1 1 VCC (r) = −92z 93 93 = 5 (9 − 90 cos2 5 + 105 cos4 5) r r 1 cos 5 VPC (r) = 9z 93 93 = 4 (15 cos2 5 − 9) ; r r

(5.11)

where 5 is the angle enclosed by the separation vector r and n0 . The interaction energy between droplets at positions r and r with respective dipole and quadrupole moments p; p ; c; and c is thus   4 2 U (R) = 4K pp VPP (R) + cc VCC (R) + (cp − cp)VPC (R) ; (5.12) 9 3 where R = r − r . The leading term in the potential U (R) is the dipole–dipole interaction which is identical to the analogous problem in electrostatics. Minimizing it over the angle 5, one 0nds that the dipoles prefer to form chains along their axes, i.e., pp ¿ 0; 5 = 0; . Such a chain of dipoles is illustrated in Fig. 26. It is similar to con0gurations seen in other dipolar systems such as magnetorheological >uids and in magnetic emulsions under the in>uence of an external 0eld [90,133]. The chaining was observed by Poulin et al. in inverted emulsions [179,183] or in a suspension of micron-size latex particles in a lyotropic discotic nematic [180]. Both systems were placed in a thin rectangular cell of approximate dimensions 20
m × 1 cm × 1 cm. The upper and lower plates were treated to produce tangential boundary conditions. Thus the total topological charge in the cell was zero. The dipolar forces were measured recently by a method introduced by Poulin et al. [179]. When small droplets are 0lled with a magnetorheological >uid instead of pure water, a small magnetic 0eld of about 100 G, applied perpendicular to the

H. Stark / Physics Reports 351 (2001) 387–474

427

chain axis, induces parallel magnetic dipoles. Since they repel each other, the droplets in the chain are forced apart. When the magnetic 0eld is switched o9, the droplets move towards each other to reach the equilibrium distance. In a chain of two moving droplets, the dipolar force on one droplet has to be balanced by the Stokes drag, pp 24K 4 = 6)e9 av ; (5.13) R where v is the velocity of one particle, and )e9 is an e9ective viscosity, which we will address in Section 6. Inertial e9ects can be neglected since the movement is overdamped. By measuring the velocity as a function of R, Poulin et al. could show that the origin of the attractive force is indeed of dipolar nature down to a separation of approximately 4a. Furthermore, they found that the prefactor of the dipolar force scales as a4 , as expected since both the dipole moments p and p scale as a2 (see Section 4.1). In Section 6 we will calculate the Stokes drag of a spherical particle. If p; p = 0, the quadrupolar interaction is dominant. A minimization over 5 predicts that ◦ the quadrupoles should chain under an angle of 5 = 49 [190]. In experiments with tangential boundary conditions at the droplet surface, where a quadrupolar structure with two opposite ◦ surface defects (boojums) forms, the chaining occurred under an angle of 5 = 30 , probably due to short-range e9ects [183]. A similar observation was made in a suspension of 50 nm latex particles in a lyotropic discotic nematic [153], where one expects a surface-ring con0guration because of the homeotropic surface anchoring (see Section 4.3.4). Finally, we discuss the coupling between dipoles and quadrupoles in Eq. (5.12). Their moments scale, respectively, as a2 or a3 . The coupling is only present when the particles have di9erent radii. Furthermore, for 0xed angle 5, the sign of the interaction depends on whether the small particle is on the right or left side of the large one. With this rather subtle e9ect, which is not yet measured, we close the section about two-particle interactions. 6. The Stokes drag of spherical particles In Section 2.3 we introduced the Ericksen–Leslie equations that govern the hydrodynamics of a nematic liquid crystal. Due to the director as a second hydrodynamic variable besides the >uid velocity, interesting new dynamical phenomena arise. With the MiZesowicz viscosities and Helfrich’s permeation, we presented two of them in Section 2.3. Here we deal with the >ow of a nematic around a spherical particle in order to calculate the Stokes drag, which is a well-known quantity for an isotropic liquid [217,202]. 2 Via the celebrated Stokes–Einstein relation [63– 65], it determines the di9usion constant of a Brownian particle, and it is, therefore, crucial for a 0rst understanding of the dynamics of colloidal suspensions [202]. In Section 6.1 the existing work on the Stokes drag, which has a long-standing tradition in liquid crystals, is reviewed. Starting from the Ericksen–Leslie equations, we introduce the theoretical concepts for its derivation in Section 6.2. We calculate the Stokes drag for three director con0gurations; a uniform director 0eld, the topological dipole, and the Saturn-ring 2

We cite here on purpose the excellent course of Sommerfeld on continuum mechanics. An English edition of his lectures on theoretical physics is available.

428

H. Stark / Physics Reports 351 (2001) 387–474

structure. Since a full analytical treatment is not possible, we have performed a numerical investigation. A summary of its details is presented in Section 6.3. Finally, we discuss the results and open problems in Section 6.4. 6.1. Motivation Due to the complexity of the Ericksen–Leslie equations, only few examples with an analytical solution exist, e.g., the >ow between two parallel plates, which de0nes the di9erent MiZesowicz viscosities [47], the Couette >ow [8,46], the Poiseuille >ow [7], which was 0rst measured by Cladis et al. [234], or the back >ow [176]. Besides the exploration of new e9ects, resulting from the coupling between the velocity and director 0eld, solutions to the Ericksen–Leslie equations are also of technological interest. They are necessary to determine the switching times of liquid crystal displays. A common way to measure viscosities of liquids is the falling-ball method, where the velocity of the falling particle is determined by a balance of the gravitational, the buoyancy, and Stokes’s friction force. Early experiments in nematic liquid crystals measured the temperature and pressure dependence of the e9ective viscosity )e9 in the Stokes drag [234,122]. Cladis et al. [234] argued that )e9 is close to the MiZesowicz shear viscosity )b , i.e., to the case where the >uid is >owing parallel to the director (see Fig. 3 in Section 2.3). Nearly twenty years later, Poulin et al. used the Stokes drag to verify the dipolar force between two topological dipoles in inverted nematic emulsions [179]. BUottger et al. [19] observed the Brownian motion of particles above the nematic-isotropic phase transition. Measuring the di9usion constant with the help of dynamic light scattering, they could show that close to the phase transition the e9ective viscosity in the Stokes drag increases due to surface-induced nematic order close to the particle. It is obvious that the hydrodynamic solution for the >ow of a nematic liquid crystal around a particle at rest, which is equivalent to the problem of a moving particle, presents a challenge to theorists. Diogo [57] assumed the velocity 0eld to be the same as the one for an isotropic >uid and calculated the drag force for simple director con0gurations. He was interested in the case where the viscous forces largely exceed the elastic forces of director distortions, i.e., Ericksen numbers much larger than one, as we shall explain in the next subsection. Roman and Terentjev, concentrating on the opposite case, obtained an analytical solution for the >ow velocity in a spatially uniform director 0eld, by an expansion in the anisotropy of the viscosity [194]. Heuer et al. presented analytical and numerical solutions for both the velocity 0eld and the Stokes drag again assuming a uniform director 0eld [112,105]. They were 0rst investigating a cylinder of in0nite length [104]. Ruhwandl and Terentjev allowed for a non-uniform but 0xed director con0guration, and they numerically calculated the velocity 0eld and the Stokes drag of a cylinder [198] or a spherical particle [199]. The particle was surrounded by the Saturn-ring con0guration (see Fig. 12 of Section 4.1), and the cylinder was accompanied by two disclination lines. Billeter and Pelcovits used molecular dynamics simulations to determine the Stokes drag of very small particles [10]. They observed that the Saturn ring is strongly deformed due to the motion of the particles. The experiments on inverted nematic emulsions [182,179] motivated us to perform analogous calculations for the topological dipole [228], which we present in the next subsections. Recently, Chono and Tsuji performed a numerical solution of the Ericksen–Leslie equations around a cylinder determining both the velocity and director 0eld [32]. They could show that

H. Stark / Physics Reports 351 (2001) 387–474

429

the director 0eld strongly depends on the Ericksen number. However, for homeotropic anchoring their director 0elds do not show any topological defects required by the boundary conditions. The Stokes drag of a particle surrounded by a disclination ring strongly depends on the presence of line defects. There exist a few studies, which determine both experimentally [37] and theoretically [107,49,203] the drag force of a moving disclination. In the multi-domain cell, a novel liquid crystal display, the occurrence of twist disclinations is forced by boundary conditions [206,205,192]. It is expected that the motion of these line defects strongly determines the switching time of the display. 6.2. Theoretical concepts We 0rst review the Stokes drag in an isotropic liquid and then introduce our approach for the nematic environment. 6.2.1. The Stokes drag in an isotropic >uid The Stokes drag in an isotropic >uid follows from a solution of the Navier–Stokes equations. Instead of considering a moving sphere, one solves the equivalent problem of the >ow around a sphere at rest [217]. An incompressible >uid (div C =0) and a stationary velocity 0eld (9C= 9t =0) are assumed, so that the 0nal set of equations reads div C = 0 and − ∇p + div T  = 0 : (6.1)  In an isotropic >uid the viscous stress tensor T is proportional to the symmetrized velocity gradient A, T  = 2)A, where ) denotes the usual shear viscosity. We have subdivided the pressure p = p0 + p in a static (p0 ) and a hydrodynamic (p ) part. The static pressure only depends on the constant mass density % and, therefore, does not appear in the momentum-balance equation of the set (6.1). The hydrodynamic contribution p is a function of the velocity. It can be chosen zero at in0nity. Furthermore, under the assumption of creeping >ow, we have neglected the non-linear velocity term in the momentum-balance equation resulting from the convective part of the total time derivative d C=dt. That means, the ratio of inertial (%v2 =a) and viscous ()v=a2 ) forces, which de0nes the Reynolds number Re = %va=), is much smaller than one. To estimate the forces, all gradients are assumed to be of the order of the inverse particle radius a−1 , the characteristic length scale of our problem. Eqs. (6.1) are solved analytically for the non-slip condition at the surface of the particle [C(r = a) = 0], and for a uniform velocity C∞ at in0nity. Once the velocity and pressure 0elds are known, the drag force FS follows from an integration of the total stress tensor −p1 + T  over the particle surface. An alternative method demands that the dissipated energy per unit time, (T  ·A) d 3 r, which we introduced in Eq. (2.27) of Section 2.3, should be FS v∞ [11]. The 0nal result is the famous Stokes formula for the drag force: FS = %v∞ with % = 6)a : (6.2) The symbol % is called the friction coeNcient. The Einstein–Stokes relation relates it to the di9usion constant D of a Brownian particle [63– 65]: kB T D= ; (6.3) 6)a where kB is the Boltzmann constant and T is temperature.

430

H. Stark / Physics Reports 351 (2001) 387–474

We can also calculate the Stokes drag for a 0nite spherical region of radius r = a=' with the particle at its center [228]. The result is FS = %' v'

with %' = 6)a

1 − 3'=2 + '3 − '5 =2 ; (1 − 3'=2 + '3 =2)2

(6.4)

where v' denotes the uniform velocity at r = a='. The correction term is a monotonically increasing function in ' on the interesting interval [0; 1]. Hence, the Stokes drag increases when the particle is con0ned to a 0nite volume. For ' = 1=32 the correction is about 5%. 6.2.2. The Stokes drag in a nematic environment To calculate the Stokes drag in a nematic environment, we have to deal with the Ericksen– Leslie equations, which couple the >ow of the >uid to the director motion. We do not attempt to solve these equations in general. Analogous to the Reynolds number, we de0ne the Ericksen number [49] as the ratio of viscous ()v∞ =a2 ) and elastic (K=a3 ) forces in the momentum balance of Eq. (2.18): )v∞ a Er = : (6.5) K The elastic forces are due to distortions in the director 0eld, where K stands for an average Frank constant. In the following, we assume Er 1, i.e., the viscous forces are too weak to distort the director 0eld, and we will always use the static director 0eld for C = 0 in our calculations. The condition Er 1 constrains the velocity v∞ . Using typical values of our parameters, i.e., K = 10−6 dyn; ) = 0:1 P, and a = 10
m, we 0nd
m v∞ 100 : (6.6) s Before we proceed, let us check for three cases if this constraint is ful0lled. First, in the measurements of the dipolar force by Poulin et al., the velocities of the topological dipole are always smaller than 10
m=s [179]. Secondly, in a falling-ball experiment the velocity v of the falling particle is determined by a balance of the gravitational, the buoyancy, and Stokes’s friction force, i.e., 6)e9 av = (4=3)a3 (% − %> )g, and we obtain v=


m 2 (% − %> )a2 g → 10 : 9 )e9 s

(6.7)

To arrive at the estimate, we choose )e9 =0:1 P and a=10
m. We take %=1 g=cm3 as the mass density of the particle and % − %> = 0:01 g=cm3 as its di9erence to the surrounding >uid [202]. Thirdly, we consider the Brownian motion of a suspended particle. With the time t = a2 =6D that the particle needs to di9use a distance equal to the particle radius a [202], we de0ne an averaged velocity v=

a 6D
m → 10−3 = : t a s

(6.8)

The estimate was calculated using the Stokes–Einstein relation of Eq. (6.3) with thermal energy kB T = 4 × 10−14 erg at room temperature and the same viscosity and particle radius as above.

H. Stark / Physics Reports 351 (2001) 387–474

431

After we have shown that Er 1 is a reasonable assumption, we proceed as follows. We 0rst calculate the static director 0eld around a sphere from the balance of the elastic torques, n × h0 = 0 [see Eqs. (2.19) and (2.25)]. It corresponds to a minimization of the free energy. For C = 0, the static director 0eld de0nes a static pressure p0 via the momentum balance, −∇p0 + div T 0 = 0, where the elastic stress tensor T 0 depends on the gradient of n [see Eqs. (2.18) and (2.11)]. If we again divide the total pressure into its static and hydrodynamic part, p=p0 +p , the velocity 0eld is determined from the same set of equations as in (6.1), provided that we employ the viscous stress tensor T  of a nematic liquid crystal [see Eq. (2.22)]. In the case of an inhomogeneous director 0eld, both the di9erent shear viscosities and the rotational viscosity %1 , discussed in Section 2.3, contribute to the Stokes drag. In general, the friction force FS does not point along C∞ , and the friction coeNcient is now a tensor . In the following, all our con0gurations are rotationally symmetric about the z axis, and the Stokes drag assumes the form FS = C∞

with  = %⊥ 1 + (% − %⊥ )ez ⊗ ez :

(6.9)

There only exist two independent components % and %⊥ for a respective >ow parallel or perpendicular to the symmetry axis. In these two cases, the Stokes drag is parallel to C∞ . Otherwise, a component perpendicular to C∞ , called lift force, appears. In analogy with the  isotropic >uid, we introduce e9ective viscosities )e9 and )⊥ e9 via 

% = 6)e9 a

and

%⊥ = 6)⊥ e9 a :

(6.10)

It is suNcient to determine the velocity and pressure 0elds for two particular geometries with C∞ either parallel or perpendicular to the z axis. Then, the friction coeNcients are calculated with the help of the dissipated energy per unit time [see Eq. (2.27)] [11,57]:
=⊥ FS v∞ = (T  ·A + h ·N ) d 3 r : (6.11) It turns out that the alternative method via an integration of the stress tensor at the surface of the particle is numerically less reliable. Note that the velocity and pressure 0elds for an arbitrary angle between C∞ and ez follow from superpositions of the solutions for the two selected geometries. This is due to the linearity of our equations. It is clear that the Brownian motion in an environment with an overall rotational symmetry is governed again by two independent di9usion constants. The generalized Stokes–Einstein formula of the di9usion tensor D takes the form kB T D = D⊥ 1 + (D − D⊥ )ez ⊗ ez with D=⊥ = : (6.12) %=⊥ At the end, we add some critical remarks about our approach which employs the static director 0eld. From the balance equation of the elastic and viscous torques [see Eqs. (2.19), (2.25) and (2.26)], we derive that the change n of the director due to the velocity C is of the order of the Ericksen number: n ∼ Er. This adds a correction T 0 to the elastic stress tensor T 0 in the momentum balance equation. In the case of a spatially uniform director 0eld, the correction T 0 is by a factor Er smaller than the viscous forces, and it can be neglected. However, for a non-uniform director 0eld, it is of the same order as the viscous term, and, strictly speaking,

432

H. Stark / Physics Reports 351 (2001) 387–474

should be taken into account. Since our problem is already very complex, even when the directors are 0xed, we keep this approximation for a 0rst approach to the Stokes drag. How the friction force changes when the director 0eld is allowed to relax, must be investigated by even more elaborate calculations. Two remarks support the validity of our approach. First, far away from the sphere, n has to decay at least linearly in 1=r, and T0 is negligible against the viscous forces. Secondly, the non-linear term in the Navier–Stokes equations usually is omitted for Re1. However, whereas the friction and the pressure force for the Stokes problem decay as 1=r 3 , the non-linear term is proportional to 1=r 2 , exceeding the 0rst two terms in the far0eld. Nevertheless, performing extensive calculations, Oseen could prove that the correction of the non-linear term to the Stokes drag is of the order of Re [217]. One might speculate that the full relaxation of the director 0eld introduces a correction of the order of Er to the Stokes drag. 6.3. Summary of numerical details In this subsection we only review the main ideas of our numerical method. A detailed account will be given in Ref. [228]. The numerical investigation is performed on a grid which is de0ned by modi0ed spherical coordinates. Since the region outside the spherical particle is in0nitely extended, we employ a reduced radial coordinate  = a=r. The velocity and director 0elds are expressed in the local spherical coordinate basis. With this choice of coordinates, the momentum balance of Eqs. (6.1) with the viscous stress tensor of a nematic becomes very complex. We, therefore, used the algebraic program Maple to formulate it. The two equations in (6.1) are treated by di9erent numerical techniques. Given an initial velocity 0eld, the momentum balance including the inertial term 9C= 9t can be viewed as a relaxation equation towards the stationary velocity 0eld, which we aim to determine. The Newton– Gauss–Seidel method, introduced in Section 2.2, provides an e9ective tool to implement this relaxation. Employing the discretized version of the momentum balance equation, the velocity at the grid point r relaxes according to vinew (r) = viold (r) −

[ − ∇p + div T  ]i : [9(−∇p + div T )]i = 9vi (r)

(6.13)

Note that the denominator can be viewed as the inverse of a variable time step for the 0ctitious temporal dynamics of C. A relaxation equation for the pressure involving div C = 0 is motivated by the method of arti0cial compressibility [33]. Let us consider the complete mass-balance equation. For small variations of the density, we obtain  9p % 9p = − 2 div C with c = : (6.14) 9t c 9% The quantity c denotes the sound velocity for constant temperature, and c2 =% is the isothermal compressibility. In discretized form we have % pnew = pold − 2 Vt div C : (6.15) c

H. Stark / Physics Reports 351 (2001) 387–474

433

Note that the reduced 0ctitious time step % Vt=c2 cannot be chosen according to the Newton– Gauss–Seidel method since div C does not contain the pressure p. Instead, it should be as large as possible to speed up the calculations. In Ref. [187] upper bounds are given beyond which the numerical scheme becomes unstable. To obtain the friction coeNcient % , an e9ective two-dimensional problem has to be solved due to the rotational symmetry of the director con0gurations about the z axis. In the case of %⊥ (C∞ ⊥ez ), the velocity 0eld possesses at least two mirror planes which are perpendicular to each other and whose line of intersection is the z axis. As a result, the necessary three-dimensional calculations can be reduced to one quadrant of the real space. A description of all the boundary conditions will be presented in Ref. [228]. The director 0elds for the topological dipole and the Saturn ring are provided by the respective ansatz functions of Eqs. (22) and (33) in Ref. [140]. The parameters of minimum free energy are chosen. In Section 4 we showed that these ansatz functions give basically the same results as the numerical investigation. We checked our programs in the isotropic case. It turned out that the three-dimensional version is not completely stable for an in0nitely extended integration area. We therefore solved Eqs. (6.1) in a 0nite region of reduced radius r=a = 1=' = 32. For ' = 1=32, our programs reproduced the isotropic Stokes drag, calculated from Eq. (6.4), with an error of 1%. 6.4. Results, discussion, and open problems We begin with an investigation of the stream line patterns, discuss the e9ective viscosities, and formulate some open problems at the end. 6.4.1. Stream line patterns In Fig. 27 we compare the stream line patterns around a spherical particle for an isotropic liquid and a spatially uniform director 0eld parallel to C∞ . A uniform n can be achieved by weak surface anchoring and application of a magnetic 0eld with a magnetic coherence length smaller than the particle radius. In the isotropic >uid the bent stream lines occupy more space around the particle, whereas for a uniform director con0guration they seem to follow the vertical director 0eld lines as much as possible. This can be understood from a minimum principle. In Section 2.3 we explained that a shear >ow along the director possesses the smallest shear viscosity, called )b . Hence, in such a geometry the smallest amount of energy is dissipated. Indeed, for a uniform director 0eld, one can derive the momentum balance from a minimization of the dissipation function stated in Eq. (2.27) [104]. A term −2p div C has to be added because of the incompressibility of the >uid. It turns out that the Lagrange multiplier −2p is determined by the pressure p. In the case of the topological dipole parallel to C∞ , we observe a clear asymmetry in the stream lines as illustrated in Fig. 28. The dot indicates the position of the point defect. It breaks the mirror symmetry of the stream line pattern, which exists, e.g., in an isotropic liquid relative to a plane perpendicular to the vertical axis. In the far0eld of the velocity, the splay deformation in the dipolar director con0guration is clearly recognizable. Since we use the linearized momentum balance in C, the velocity 0eld is the same no matter if the >uid >ows upward or downward. The stream line pattern of the Saturn ring [see Fig. 29 (right)] exhibits the mirror symmetry,

434

H. Stark / Physics Reports 351 (2001) 387–474

Fig. 27. Stream line pattern around a spherical particle for an isotropic liquid (right) and a uniform director 0eld parallel to C∞ (left). Fig. 28. Stream line pattern around a spherical particle for an isotropic liquid (right) and the topological dipole parallel to C∞ (left).

and the position of the ring disclination is visible by a dip in the stream line close to the equator of the sphere. If C∞ is perpendicular to the dipole axis, the missing mirror plane of the dipole con0guration is even more pronounced in the stream line pattern. It is illustrated in Fig. 30, where the point defect is indicated by a dip in the stream line. Although the pattern resembles the one of the Magnus e9ect [217], symmetry dictates that FS⊥ C∞ . A lift force perpendicular to C∞ does not exist. We 0nd a non-zero viscous torque acting on the particle whose direction for a >uid >ow from left to right is indicated in Fig. 30. Symmetry allows such a torque M since the cross product of the dipole moment p and C∞ gives an axial or pseudovector M ˙ p × C∞ . In the Saturn-ring con0guration a non-zero dipole moment and, therefore, a non-zero torque cannot occur. 6.4.2. E<ective viscosities In Table 1 we summarize the e9ective viscosities of the Stokes drag, de0ned in Eq. (6.10), for a uniform director 0eld, the dipole and the Saturn-ring con0guration. The values are calculated for the two compounds MBBA and 5CB. For a reference, we include the three MiZesowicz viscosities. In the case of C∞ parallel to the symmetry axis of the three con0gurations, we  might expect that )e9 is close to )b as argued by Cladis et al. [234]. For a uniform director  0eld, )e9 exceeds )b by 30% or 60%, respectively. The increase originates in the stream lines  bending around the particle. The e9ective viscosity )e9 of the dipole and the Saturn ring are larger than )b by an approximate factor of two. In addition to the bent stream lines, there exist

H. Stark / Physics Reports 351 (2001) 387–474

435

Fig. 29. Stream line pattern around a spherical particle for the Saturn ring (right) and the topological dipole (left) with their respective symmetry axis parallel to C∞ . Fig. 30. Stream line pattern around a spherical particle for the topological dipole perpendicular to C∞ .

Table 1 E9ective viscosities of the Stokes drag for the two compounds MBBA and 5CB and for three di9erent director con0gurations. As a reference, the three MiZesowicz viscosities are included

 )e9 (P) )⊥ e9 (P)  )⊥ e9 =)e9

MBBA: )a = 0:416 P; )b = 0:283 P; )c = 1:035 P

5CB: )a = 0:374 P; )b = 0:229 P; )c = 1:296 P

Uniform n

Dipole

Saturn ring

Uniform n

Dipole

Saturn ring

0.380 0.684 1.80

0.517 0.767 1.48

0.493 0.747 1.51

0.381 0.754 1.98

0.532 0.869 1.63

0.501 0.848 1.69

strong director distortions close to the particle which the >uid has to >ow through constantly changing the local direction of the moving molecules. Recalling our discussion of the permeation in Section 2.3, a contribution from the rotational viscosity %1 arises which does not exist in a  uniform director 0eld. In all three cases, we 0nd )e9 either close to or larger than )a , so that )b is  not the only determining quantity of )e9 , as argued by Cladis et al. [234]. For C∞ perpendicular to the symmetry axis, )e9 ⊥ assumes a value between )a and )c , which is understandable since the >ow velocity is mainly perpendicular to the director 0eld.  The ratio )⊥ e9 =)e9 for the uniform director 0eld is the largest since the extreme cases of a respective >ow parallel or perpendicular to the director 0eld is realized the best in this con0guration. Furthermore, both the dipole and the Saturn ring exhibit nearly the same anisotropy, and we conclude that they cannot be distinguished from each other in a falling-ball experiment.

436

H. Stark / Physics Reports 351 (2001) 387–474 

The ratios )⊥ e9 =)e9 that we determine for the Saturn ring and the uniform director 0eld in the case of the compound MBBA agree well with the results of Ruhwandl and Terentjev who 0nd   ⊥ )⊥ e9 =)e9 |uniform = 1:69 and )e9 =)e9 |Saturn = 1:5 [199]. However, they di9er from the 0ndings of Billeter and Pelcovits in their molecular dynamics simulations [10]. In the ansatz function of the dipolar con0guration, we vary the separation rd between the hedgehog and the center of the particle. Both the e9ective viscosities increase with rd since the non-uniform director 0eld with its strong distortions occupies more space. However, the ratio   )⊥ e9 =)e9 basically remains the same. For the Saturn ring, )e9 increases stronger with the radius rd than does )⊥ e9 . This seems to be reasonable since a >ow perpendicular to the plane of the  Saturn ring experiences more resistance than a >ow parallel to the plane. As a result, )⊥ e9 =)e9 decreases when the ring radius rd is enlarged. 6.4.3. Open problems One should try to perform a complete solution of the Ericksen–Leslie equations including a relaxation of the static director 0eld for C = 0. In the case of Er 1, a linearization in the small deviation n from the static director 0eld would suNce. Such a procedure helps to gain insight into several open problems. First, it veri0es or falsi0es the hypothesis that the correction to the Stokes drag is of the order of Er. Secondly, the Stokes drag of the topological dipole is the same whether the >ow is parallel or anti-parallel to the dipole moment. This is also true for an object with a dipolar shape in an isotropic >uid. If such an object is slightly turned away from its orientation parallel to C∞ , it will experience a viscous torque and either relax back or reverse its direction to 0nd its absolute stable orientation. The topological dipole will not turn around since it experiences an elastic torque towards its initial direction, as explained in Section 5.1. Nevertheless, a full solution of the Ericksen–Leslie equations would show whether and how much the dipole deviates from its preferred direction under the in>uence of a velocity 0eld. It would also clarify its orientation when C∞ is perpendicular to the dipolar axis. Furthermore, we speculate that the non-zero viscous torque, discussed in Section 6.4.1, is cancelled by elastic torques. Preliminary results [228] for the two-dimensional problem with the relaxation of the director 0eld included show that the Stokes drag of the dipolar con0guration varies indeed linearly in Er for Er ¡ 1. Furthermore, it is highly non-linear depending on C∞ being either parallel or anti-parallel to the topological dipole. The Stokes drag of particles in a nematic environment still presents a challenging problem to theorists. On the other hand, clear measurements of, e.g., the anisotropy in Stokes’s friction force are missing. 7. Colloidal dispersions in complex geometries In this section we present a numerical investigation of water droplets in a spherically con0ned nematic solvent. It is motivated by experiments on multiple nematic emulsions which we reported in Section 3.2. However, it also applies to solid spherical particles. Our main purpose is to demonstrate that the topological dipole provides a key unit for the understanding of multiple emulsions. In Sections 7.1–7.3 we 0rst state the questions and main results of our investigation.

H. Stark / Physics Reports 351 (2001) 387–474

437

Fig. 31. Scenario to explain the chaining of water droplets in a large nematic drop. The right water droplet and its companion hyperbolic hedgehog form a dipole, which is attracted by the strong splay deformation around the droplet in the center (left picture). The dipole moves towards the center until at short distances the repulsion mediated by the point defect sets in (middle picture). A third droplet moves to the region of maximum splay to form a linear chain with the two other droplets.

Then we de0ne the geometry of our problem and summarize numerical details. In particular, we employ the numerical method of 0nite elements [227] which is most suitable for non-trivial geometries. Finally we present our results in detail and discuss them. The last subsection contains an analytical treatment of the twist transition of a radial director 0eld enclosed between two concentric spheres. It usually occurs when the inner sphere is not present. We perform a linear stability analysis and thereby explain the observation that a small water droplet at the center of a large nematic drop suppresses the twisting. 7.1. Questions and main results In our numerical investigation we demonstrate that the dipolar con0guration formed by one spherical particle and its companion hyperbolic point defect also exists in more complex geometries, e.g., nematic drops. This provides an explanation for the chaining reported in Section 3.2 and in Refs. [182,183]. One water droplet 0ts perfectly into the center of a large nematic drop, which has a total topological charge +1. Any additional water droplet has to be accompanied by a hyperbolic hedgehog in order not to change the total charge. If the dipole forms (see Fig. 31, left), it is attracted by the strong splay deformation in the center, as predicted by the phenomenological theory of Section 5.1 and in Refs. [182,140], until the short-range repulsion mediated by the defect sets in (see Fig. 31, middle). Any additional droplet seeks the region of maximum splay and forms a linear chain with the two other droplets. In the following we present a detailed study of the dipole formation in spherical geometries. For example, when the two water droplets in the middle picture of Fig. 31 are moved apart symmetrically about the center of the large drop, the dipole forms via a second-order phase transition. We also identify the dipole in a bipolar con0guration which occurs for planar boundary conditions at the outer surface of the nematic drop. Two boojums, i.e., surface defects appear [145,26,120], and the dipole is attracted by the strong splay deformation in the vicinity of one of them [182,183,140]. Besides the dipole we 0nd another stable con0guration in this geometry, where the hyperbolic hedgehog sits close to one of the boojums, which leads to a hysteresis in the formation of the dipole. In the experiment it was found that the distance d of the point defect from the surface of a water droplet scales with the radius r of the droplet like d ≈ 0:3r [182,183]. In the following we will call this relation the scaling law. By our numerical investigations, we con0rm this scaling

438

H. Stark / Physics Reports 351 (2001) 387–474

Fig. 32. (a) Geometry parameters for two water droplets with respective radii r1 and r2 in a large nematic drop with radius r3 . The system is axially symmetric about the z axis, and cylindrical coordinates 6; z are used. The coordinates z1 ; z2 , and zd are the respective positions of the two droplets and the hyperbolic hedgehog. The two distances of the hedgehog from the surfaces of the droplets are d1 and d2 . From Ref. [220]. (b) Triangulation of the integration area (lattice constant: b = 0:495). Between the small spheres a re0ned net of triangles is chosen. From Ref. [220].

law within an accuracy of ca. 15%, and we discuss the in>uence of the outer boundary of the large drop. Finally, we show that water droplets can repel each other without a hyperbolic defect placed between them. 7.2. Geometry and numerical details We numerically investigate two particular geometries of axial symmetry. The 0rst problem is de0ned in Fig. 32a. We consider two spherical water droplets with respective radii r1 and r2 in a large nematic drop with radius r3 . The whole system possesses axial symmetry, so that the water droplets and the hyperbolic hedgehog, indicated by a cross, are located always on the z axis. We employ a cylindrical coordinate system. The coordinates z1 ; z2 , and zd denote, respectively, the positions of the centers of the droplets and of the hyperbolic hedgehog on the z axis. The distances of the hedgehog from the surfaces of the two water droplets are, respectively, d1 and d2 . Then, the quantity d1 + d2 means the distance of the two small spheres, and the point defect is situated in the middle between them if d1 = d2 . We, furthermore, restrict the nematic director to the (6; z) plane, which means that we do not allow for twist deformations. 3 3

In nematic droplets with homeotropic anchoring a twist in the director 0eld is usually observed (see [26] and Section 7.4). In Section 4.3.2 we demonstrated that it even appears in the dipole con0guration close to the hyperbolic hedgehog. However, for the Frank elastic constants of 5CB, the distance of the defect from the surface of the water droplet di9ers only by 10% if the director 0eld is not allowed to twist. We do not expect a di9erent behavior in the geometry under consideration in this section. Here, we want to concentrate, as a 0rst step, on the principal features of the system. Therefore, we neglect twist deformations to simplify the numerics. The same simpli0cation to catch the main behavior of nematic drops in a magnetic 0eld was used by other authors, see, e.g., [115,114].

H. Stark / Physics Reports 351 (2001) 387–474

439

The director is expressed in the local coordinate basis of the cylindrical coordinate system, n(6; z)=sin (6; z)e6 +cos (6; z)ez , where we introduced the tilt angle . It is always restricted to the range [−=2; =2] to ensure the n → −n symmetry of the nematic phase. At all the boundaries we assume a rigid homeotropic anchoring of the director, which allows us to omit any surface term in the free energy. In Ref. [140] it was shown that rigid anchoring is justi0ed in our system and that any deformation of the water droplets can be neglected. In the second problem we have only one water droplet insider a large nematic drop. We use the same coordinates and lengths as described in Fig. 32a, but omit the second droplet. The anchoring of the director at the outer surface of the large nematic sphere is rigid planar. At the surface of the small sphere we again choose a homeotropic boundary condition. Because of the non-trivial geometry of our problem, we decided to employ the method of 0nite elements [227], where the integration area is covered with triangles. We construct a net of triangles by covering our integration area with a hexagonal lattice with lattice constant b. Vertices of triangles that only partially belong to the integration area are moved onto the boundary along the radial direction of the appropriate sphere. As a result, extremely obtuse triangles occur close to the boundary. We use a relaxation mechanism to smooth out these irregularities. The 0nal triangulation is shown in Fig. 32b. In the area between the small spheres, where the hyperbolic hedgehog is situated, the grid is further subdivided to account for the strong director deformations close to the point defect. The local re0nement helps us to locate the minimum position of the defect between the spheres within a maximum error of 15% by keeping the computing time to a reasonable value [220]. In the following, we express the Frank free energy, introduced in Section 2.1, in units of K3 a ^ The quantity a is the characteristic length scale of our system, and denote it by the symbol F. typically several microns. The saddle-splay term, a pure surface term, is not taken into account. The Frank free energy is discretized on the triangular net. For details, we refer the reader to Ref. [220]. To 0nd a minimum of the free energy, we start with a con0guration that already possesses the hyperbolic point defect at a 0xed position zd and let it relax via the standard Newton–Gauss–Seidel method [187], which we illustrate in Eq. (2.16) of Section 2.2. Integrating the free energy density over one triangle yields a line energy, i.e., an energy per unit length. As a rough estimate for its upper limit we introduce the line tension Fl =(K1 +K3 )=2 of the isotropic core of a disclination [51]. Whenever the numerically calculated local line energy is larger than Fl , we replace it by Fl . Note that Fl di9ers from Eq. (2.34). However, its main purpose is to stabilize the hyperbolic point defect against opening up to a disclination ring whose radius would be unphysical, i.e., larger than the values discussed in Section 2.4.2. All our calculations are performed for the nematic liquid crystal pentylcyanobiphenyl (5CB), for which the experiments were done [182,183]. Its respective bend and splay elastic constants are K3 = 0:53 × 10−6 dyn and K1 = 0:42 × 10−6 dyn. The experimental ratio r3 =r1=2 of the radii of the large and small drops is in the range 10 –50 [182,183]. The diNculty is that we want to investigate details of the director 0eld close to the small spheres which requires a 0ne triangulation on the length scale given by r1=2 . To keep the computing time to a reasonable value we choose the following lengths: r3 = 7; r1=2 = 0:5–2, and b = 0:195 for the lattice constant of the grid. In addition, we normally use one step of grid re0nement between the small spheres (geometry 1) or between the small sphere and the south pole of the large nematic drop (geometry 2). With such parameters we obtain a lattice with 2200 –2500 vertices.

440

H. Stark / Physics Reports 351 (2001) 387–474

Fig. 33. The free energy F^ as a function of the distance d1 + d2 between the small spheres which are placed symmetrically about z = 0 (r1 = r2 = 1). Curve 1: zd = 0, curve 2: position zd of the defect can relax along the z axis. From Ref. [220]. Fig. 34. The free energy F^ − F^min as a function of d1 =r1 = d2 =r2 . The small spheres are placed symmetrically about z = 0. Curve 1: r1 = r2 = 0:5, curve 2: r1 = r2 = 1, and curve 3: r1 = r2 = 2. From Ref. [220].

7.3. Results and discussion of the numerical study In this subsection we discuss the results from our numerical investigation. First, we con0rm the scaling law d1=2 ≈ 0:3r1=2 , which was observed in experiment, by varying the di9erent lengths in our geometry. Secondly, we demonstrate that the topological dipole is also meaningful in complex geometries. Finally, we show that the hyperbolic hedgehog is not necessary to mediate a repulsion between the water droplets. 7.3.1. Scaling law In Fig. 33 we plot the reduced free energy F^ as a function of the distance d1 + d2 between the surfaces of the small spheres, which are placed symmetrically about the center, i.e., z2 = −z1 . Their radii are r1 = r2 = 1. Curve 1 shows a clear minimum at d1 + d2 ≈ 0:7, the defect stays in the middle between the two spheres at zd = 0. In curve 2 we move the defect along the z axis and plot the minimum free energy for each 0xed distance d1 + d2 . It is obvious that beyond d1 + d2 = 2 the defect moves to one of the small spheres. We will investigate this result in more detail in the following subsection. In Fig. 34 we take three di9erent radii for the small spheres, r1 = r2 = 0:5; 1; 2, and plot the free energy versus d1 =r1 close to the minimum. Recall that d1 is the distance of the hedgehog from the surface of sphere 1. Since for such small distances d1 + d2 the defect always stays at zd = 0, i.e., in the middle between the two spheres, we have d1 =r1 = d2 =r2 . The quantity F^min refers to the minimum free energy of each curve. For each of the three radii we obtain an energetically preferred distance d1 =r1 in the range of [0:3; 0:35], which agrees well with the experimental value of 0.3. Why does a scaling law of the form d1=2 = (0:325 ± 0:025)r1=2 occur? When the small spheres are far away from the surface of the large nematic drop, its 0nite radius r3 should hardly in>uence the distances d1 and d2 . Then, the only length scale in the system is r1 = r2 , and we expect d1=2 ˙ r1=2 . However, in Fig. 34 the in>uence from the boundary of the

H. Stark / Physics Reports 351 (2001) 387–474

441

large sphere is already visible. Let us take curve 2 for spheres with radii r1=2 = 1 as a reference. It is approximately symmetric about d1 =r1 = 0:35. The slope of the right part of curve 3, which corresponds to larger spheres of radii r1=2 = 2, is steeper than in curve 2. Also, the location of the minimum clearly tends to values smaller than 0.3. We conclude that the small spheres are already so large that they are strongly repelled by the boundary of the nematic drop. On the other hand, the slope of the right part of curve 1, which was calculated for spheres of radii r1=2 = 0:5, if less steep than in curve 2. This leads to the conclusion that the boundary of the nematic drop has only a minor in>uence on such small spheres. When we move the two spheres with radii r1=2 = 1 together in the same direction along the z axis, the defect always stays in the middle between the droplets and obeys the scaling law. We have tested its validity within the range [0; 3] for the defect position zd . Of course, the absolute minimum of the free energy occurs in the symmetric position of the two droplets, z2 = −z1 . We further check the scaling law for r1 = r2 . We investigate two cases. When we choose r1 = 2 and r2 = 0:6, we obtain d1=2 ≈ 0:3r1=2 . In the second case, r1 = 2 and r2 = 1, we 0nd d1 ≈ 0:37r1 and d2 ≈ 0:3r2 . As observed in the experiment, the defect sits always closer to the smaller sphere. There is no strong deviation from the scaling law d1=2 = (0:325 ± 0:025)r1=2 , although we would allow for it, since r1 = r2 . 7.3.2. Identi4cation of the dipole In this subsection we demonstrate that the topological dipole is meaningful in our geometry. We place sphere 2 with radius r2 = 1 in the center of the nematic drop at z2 = 0. Then, we determine the energetically preferred position of the point defect for di9erent locations z1 of sphere 1 (r1 = 1). The position of the hedgehog is indicated by > = (d2 − d1 )=(d1 + d2 ). If the defect is located in the middle between the two spheres, > is zero since d1 = d2 . On the other hand, if it sits at the surface of sphere 1, d1 = 0, and > becomes one. In Fig. 35 we plot the free energy F^ versus >. In curve 1, where the small spheres are farthest apart from each other (z1 = 5), we clearly 0nd the defect close to sphere 1. This veri0es that the dipole is existing. It is stable against >uctuations since a rough estimate of the thermally induced mean displacement of the defect yields 0.01. The estimate is performed in full analogy to Eq. (4.6) of Section 4.2. When sphere 1 is approaching the center (curve 2: z1 = 4 and curve 3: z1 = 3:5), the defect moves away from the droplet until it nearly reaches the middle between both spheres (curve 4: z1 = 3). This means, the dipole vanishes gradually until the hyperbolic hedgehog is shared by both water droplets. An interesting situation occurs when sphere 1 and 2 are placed symmetrically about z = 0. Then, the defect has two equivalent positions on the positive and negative part of the z axis. In Fig. 36 we plot again the free energy F^ versus the position > of the defect. From curve 1 to 3 (z1 = z2 = 4; 3; 2:5) the minimum in F^ becomes broader and more shallow. The defect moves closer towards the center until at z1 = −z2 ≈ 2:3 (curve 4) it reaches >=0. This is reminiscent to a symmetry-breaking second-order phase transition [27,124] which occurs when, in the course of moving the water droplets apart, the dipole starts to form. We take > as an order parameter, where >=0 and > = 0 describe, respectively, the high- and the low-symmetry phase. A Landau expansion of the free energy yields ^ F(>) = F^0 (z1 ) + a0 [2:3 − z1 ]>2 + c(z1 )>4 ;

(7.1)

442

H. Stark / Physics Reports 351 (2001) 387–474

Fig. 35. The free energy F^ as a function of > = (d2 − d1 )=(d1 + d2 ). Sphere 2 is placed at z2 = 0. The position z1 of sphere 1 is the parameter. Curve 1: z1 = 5, curve 2: z1 = 4, curve 3: z1 = 3:5, and curve 4: z1 = 3. The radii are r1 = r2 = 1. From Ref. [220]. Fig. 36. The free energy F^ as a function of > = (d2 − d1 )=(d1 + d2 ). The small spheres are placed symmetrically about z = 0. Curve 1: z1 = −z2 = 4, curve 2: z1 = −z2 = 3, curve 3: z1 = −z2 = 2:5, curve 4: z1 = −z2 = 2:3, curve 5: z1 = −z2 = 2. The radii are r1 = r2 = 1. From Ref. [220].

where z1 = −z2 plays the role of the temperature. Odd powers in > are not allowed because ^ ^ −>). This free energy qualitatively describes the curves of the required symmetry, F(>) = F( in Fig. 36. It should be possible to observe such a “second-order phase transition” 4 with a method introduced recently by Poulin et al. [179] to measure dipolar forces in inverted nematic emulsion. We already explained the method in Section 5.2 after Eq. (5.12). Two small droplets 0lled with a magnetorheological >uid are forced apart when a small magnetic 0eld of about 100 G is applied perpendicular to the z axis. When the magnetic 0eld is switched o9, the two droplets move towards each other to reach the equilibrium distance. In the course of this process the phase transition for the dipole should be observable. 7.3.3. The dipole in a bipolar con4guration It is possible to change the anchoring of the director at the outer surface of the large nematic drop from homeotropic to planar by adding some amount of glycerol to the surrounding water phase [182]. Then the bipolar con0guration for the director 0eld appears [26,120], where two boojums [145], i.e., surface defects of charge 1 are situated at the north and south pole of the large nematic drop (see con0guration (1) in Fig. 37). The topological point charge of the interior of the nematic drop is zero, and every small water droplet with homeotropic boundary condition has to be accompanied by a hyperbolic hedgehog. In the experiment the hedgehog sits close to the water droplet, i.e., the dipole exists and it is attracted by the strong splay deformation 4

There is strictly speaking no true phase transition since our investigated system has 0nite size. However, we do not expect a qualitative change in Fig. 36, when the nematic drop is much larger than the enclosed water droplets (r3 r1 ; r2 ), i.e., when the system reaches the limit of in0nite size.

H. Stark / Physics Reports 351 (2001) 387–474

443

Fig. 37. Planar boundary conditions at the outer surface of the large sphere create boojums, i.e., surface defects at the north and the south pole. A water droplet with homeotropic boundary conditions nucleates a hyperbolic hedgehog. Two con0gurations exist that are either stable or metastable depending on the position of the water droplet; (1) the dipole, (2) the hyperbolic hedgehog sitting at the surface. From Ref. [220]. Fig. 38. The free energy F^ as a function of the position z1 of the water droplet for the con0gurations (1) and (2). For z1 ¿ − 3:5, (1) is stable, and (2) is metastable. The situation is reversed for −4:3 ¡ z1 ¡ − 3:5. Con0guration (1) loses its metastability at z1 = −4:3. From Ref. [220].

close to the south pole [182], as predicted by the phenomenological theory of Section 5 and Refs. [182,140]. A numerical analysis of the free energy F^ is in agreement with experimental observations but also reveals some interesting details which have to be con0rmed. In Fig. 38 we plot F^ as a function of the position z1 of the small water droplet with radius r1 =1. The diagram consists of curves (1) and (2), which correspond, respectively, to con0gurations (1) and (2) in Fig. 37. The free energy possesses a minimum at around z1 = −5:7. The director 0eld assumes con0guration (2), where the hyperbolic hedgehog is situated at the surface of the nematic drop. Moving the water droplet closer to the surface, induces a repulsion due to the strong director deformations around the point defect. When the water droplet is placed far away from the south pole, i.e., at large z1 , the dipole of con0guration (1) forms and represents the absolute stable director 0eld. At z1 = −3:5 the dipole becomes metastable but the system does not assume con0guration (2) since the energy barrier the system has to overcome by thermal activation is much too high. By numerically calculating the free energy for di9erent positions of the hedgehog, we have, e.g., at z1 = −4:0, determined an energy barrier of K3 a ≈ 1000kB T , where kB is the Boltzmann constant, T the room temperature, and a ≈ 1
m. At z1 = −4:3, the dipole even loses its metastability, the hyperbolic defect jumps to the surface at the south pole and the water droplet follows until it reaches its energetically preferred position. On the other hand, if it were possible to move the water droplet away from the south pole, the hyperbolic hedgehog would stay at the surface, since con0guration (2) is always metastable for z1 ¿ −3:5. The energy barrier for a transition to the dipole is again at least 1000kB T . We have also investigated the distance d1 of the defect from the surface of the water droplet. For z1 ∈ [ − 2; 4], d1 >uctuates between 0.3 and 0.35. For z1 ¡ − 2, it increases up to 0.5 at z1 = −4:3, where the dipole loses its metastability.

444

H. Stark / Physics Reports 351 (2001) 387–474

Fig. 39. An alternative, metastable con0guration. Both droplets are surrounded by a −1=2 disclination ring which compensates the topological charge +1 of each droplet. An additional +1=2 disclination ring close to the surface of the nematic drop satis0es the total topological charge +1. From Ref. [220]. Fig. 40. The free energy F^ as a function of the distance d of the droplets. A repulsion for d ¡ 0:6 is clearly visible. From Ref. [220].

7.3.4. Repulsion without defect We return to the 0rst geometry with two water droplets and homeotropic boundary conditions at all the surfaces. When we take either a uniform director 0eld or randomly oriented directors as a starting con0guration, our system always relaxes into the con0guration sketched in Fig. 39. Both water droplets are surrounded in their equatorial plane by a −1=2 disclination ring which compensates the point charge +1 carried by each droplet. That means, each droplet creates a Saturn-ring con0guration around it, which we introduced in Section 4.1 (see also Refs. [225,119]). To obtain the total point charge +1 of the nematic drop there has to be an additional topological defect with a point charge +1. In the numerically relaxed director 0eld, we 0nd a +1=2 disclination ring close to the outer surface. This con0guration has a higher energy than the one with the hyperbolic hedgehog. It is only metastable. Since a transition to the stable con0guration needs a complete rearrangement of the director 0eld, the energy barrier is certainly larger than K33 a ≈ 1000kB T . We, therefore, expect the con0guration of Fig. 39 to be stable against thermal >uctuations. It would be interesting to search for it in an experiment. We use the con0guration to demonstrate that even without the hyperbolic hedgehog the two water droplets experience some repulsion when they come close to each other. In Fig. 40 we plot the free energy F^ versus the separation d of the two spheres. For large d, the free energy oscillates which we attribute to numerical artifacts. For decreasing d, the free energy clearly increases, and the water droplets repel each other due to the strong deformation of the director 0eld lines connecting the two droplets. 7.4. Coda: twist transition in nematic drops Already thirty years ago, in connection with nematic emulsions, the two main director con0gurations in a nematic drop were discussed both experimentally and theoretically [147,61]: for

H. Stark / Physics Reports 351 (2001) 387–474

445

homeotropic boundary conditions, a radial hedgehog at the center of the drop appears, whereas tangential surface anchoring leads to the bipolar structure already discussed above. The simple picture had to be modi0ed when it was found that nematic drops in both cases also exhibit a twisted structure [26]. For the bipolar con0guration, a linear stability analysis of the twist transition was performed [237]. A numerical study of the twisting in the radial structure of capillaries was presented in Refs. [185,186]. Lavrentovich and Terentjev proposed that the twisted director 0eld in a nematic drop with homeotropic surface anchoring is given by a combination of a hyperbolic hedgehog at the center of the drop and a radial one at its periphery [126] as illustrated in Fig. 7 of Section 2.4.1. This con0guration was analyzed by means of an ansatz function, and a criterion for the twist transition was given [126]. In this subsection we focus on the director 0eld between two concentric spheres with perpendicular anchoring at both the surfaces and present a stability analysis for the radial con0guration against axially symmetric deformations. In particular, we will derive a criterion for the twist transition, and we will show that even small spheres inside a large one are suNcient to avoid twisted con0gurations. This has been recently observed in the experiments on multiple nematic emulsions [182,183]. Throughout the paper we assume rigid surface anchoring of the molecules. In nematic emulsions it can be achieved by a special choice of the surfactant [182,183]. For completeness we note that in a single droplet for suNciently weak anchoring strength an axial structure with an equatorial disclination ring appears [66,60]. In the following three subsections, we 0rst expand the Frank free energy into small deviations from the radial con0guration up to second order. Then, we formulate and solve the corresponding eigenvalue equation arising from a linear stability analysis. The lowest eigenvalue leads to a criterion for the twist transition. We close with a discussion of our results. 7.4.1. Expansion of the elastic energy We consider the defect-free radial director con0guration between two concentric spheres of radii rmin and rmax and assume rigid radial surface anchoring at all the surfaces. If the smaller sphere is missing, the radial director con0guration exhibits a point defect at the center. We will argue below that this situation, rmin = 0, is included in our treatment. The twist transition reduces the SO(3) symmetry of the radial director con0guration to an axial C∞ symmetry. In order to investigate the stability of the radial con0guration n0 = er against a twist transition, we write the local director in a spherical coordinate basis, allowing for small deviations along the polar (5) and the azimuthal (7) direction: n(r; 5) = (1 − 12 b2 f2 − 12 a2 g2 )er + age5 + bfe7 :

(7.2)

f(r; 5) and g(r; 5) are general functions which do not depend on 7 due to our assumption of axial symmetry. The amplitudes a and b describe the magnitude of the polar and azimuthal deviation from the radial con0guration. The second-order terms in a and b result from the normalization of the director. The radial director 0eld between the spheres only involves a splay distortion, and its Frank free energy is Fradial = 8K11 (rmax − rmin ) ;

(7.3)

446

H. Stark / Physics Reports 351 (2001) 387–474

where we did not include the saddle-splay energy. If an azimuthal (b = 0) or a polar component (a = 0) of the director is introduced, the splay energy can be reduced at costs of non-zero twist and bend contributions depending on the values of the Frank elastic constants K1 , K2 , and K3 . We expand the Frank free energy of the director 0eld in Eq. (7.2) up to second order in a and b and obtain

2 VF = 2b dr d cos 5[ − 4K1 (f2 + rfr f) + K2 (cot 5f + f5 )2 + K3 (f + rfr )2 ]

2 + 2a dr d cos 5[ − 4K1 (g2 + rgr g) + K1 (cot 5g + g5 )2 + K3 (g + rgr )2 ] (7.4) as the deviation from Fradial . The respective subscripts r and 5 denote partial derivatives with respect to the corresponding coordinates. Note that there are no linear terms in a or b, i.e., the radial director 0eld is always an extremum of the Frank free energy. Furthermore, there is no cross-coupling term ab in Eq. (7.4), and the stability analysis for polar and azimuthal perturbations can be treated separately. For example, for any function f(r; 5) leading to a negative value of the 0rst integral in Eq. (7.4), the radial con0guration (a = b = 0) is unstable with respect to a small azimuthal deformation (b = 0), which introduces a twist into the radial director 0eld. Therefore, we will call it the twist deformation in the following. An analogous statement holds for g(r; 5) which introduces a pure bend into the radial director 0eld. We are now determining the condition the elastic constants have to ful0l in order to allow for such functions f(r; 5) and g(r; 5). As we will demonstrate in the next subsection, the solution of this problem is equivalent to solving an eigenvalue problem. 7.4.2. Formulating and solving the eigenvalue problem In a 0rst step, we focus on the twist deformation (b = 0). We are facing the problem to determine for which values of K1 , K2 , and K3 the functional inequality

dr d x {K2 (1 − x2 )[xf=(1 − x2 ) − fx ]2 + (K3 − 4K1 )f2 + (2K3 − 4K1 )rfr f + K3 r 2 fr2 } ¡ 0

(7.5)

possesses solutions f(r; x). The left-hand side of the inequality is the 0rst integral of Eq. (7.4) after substituting x = cos 5. After some manipulations (see Ref. [197]), we obtain

dr d x (K2 fD(x) f + K3 fD(r) f)

¡ 2K1 ; (7.6) dr d x f2 where the second-order di9erential operators D(x) and D(r) are given by D(x) = (1 − x2 )

92 9 1 + 2x + 2 9x 9x 1 − x2

and

D(r) = −r 2

92 9 − 2r : 2 9r 9r

(7.7)

The inequality in Eq. (7.6) is ful0lled the best when the left-hand side assumes a minimum. According to the Ritz principle in quantum mechanics, this minimum is given by the lowest

H. Stark / Physics Reports 351 (2001) 387–474

447

eigenvalue of the operator K2 D(x) + K3 D(r)

(7.8)

on the space of square-integrable functions with f(rmin ; 5) = f(rmax ; 5) = 0 for 0 6 5 6  (0xed boundary condition) and f(r; 0) = f(r; ) = 0 for rmin 6 r 6 rmax . The eigenvalue equation of the operator K2 D(x) +K3 D(r) separates into a radial and an angular part. The radial part is an Eulerian di9erential equation [20] with the lowest eigenvalue  2  1 (r) &0 = + (7.9) 4 ln(rmax =rmin ) and the corresponding eigenfunction   ln(r=rmin ) 1 (r) f (r) = √ sin  : ln(rmax =rmin ) r

(7.10)

The angular part of the eigenvalue equation is solved by the associated Legendre functions Pnm=1 . The lowest eigenvalue is &0(x) = 2, and the corresponding eigenfunction is f(x) (5) = P11 (5) = sin 5. With both these results, we obtain the instability condition for a twist deformation:   2  1 K3 1  K2 + ¡1 : (7.11) + 2 K1 4 ln(rmax =rmin ) K1 This inequality is the main result of the paper. If it is ful0lled, the radial director 0eld no longer minimizes the Frank free energy. Therefore it is a suNcient condition for the radial con0guration to be unstable against a twist deformation. It is not a necessary condition since we have restricted ourselves to second-order terms in the free energy, not allowing for large deformations of the radial director 0eld. Hence, we cannot exclude the existence of further con0gurations which, besides the radial, produce local minima of the free energy. To clarify our last statement, we take another view. The stability problem can be viewed as a phase transition. Let us take K3 as the “temperature”. Then condition (7.11) tells us that for large K3 the radial state is the (linearly) stable one. If the phase transition is second-order-like, the radial state loses its stability exactly at the linear stability boundary, while for a 0rst-order-like transition the system can jump to the new state (due to non-linear >uctuations) even well inside the linear stability region. Thus, as long as the nature of the transition is not clear, linear stability analysis cannot predict for sure that the radial state will occur in the linear stability region. Furthermore, if the transition line is crossed, the linear stability analysis breaks down, and there could be a transition from the twisted to a new con0guration. However, there is no experimental indication for such a new structure. Keeping this in mind, we will discuss the instability condition (7.11) in the next subsection. We 0nish this subsection by noting that the elastic energy for a bend deformation (a = 0) has the same form as the one for the twist deformation (b = 0), however, with K2 replaced by K1 . Therefore, we immediately conclude from (7.11) that the instability condition for a polar component (a = 0) in the director 0eld (7.2) cannot be ful0lled for positive elastic constants. A director 0eld with vanishing polar component is always stable in second order.

448

H. Stark / Physics Reports 351 (2001) 387–474

Fig. 41. (a) Stability diagram for the twist transition [cf. Eq. (7.11)]. The dark grey corresponds to the ratios of Frank constants where the radial con0guration is unstable for a ratio rmax =rmin = 50. The light grey triangle is the region where the radial con0guration is unstable for rmax =rmin ¿ 50. The circles represent the elastic constants for the liquid crystal compounds MBBA, 5CB, and PAA. (b) A comparison between the regions of instability for a radial director 0eld against twisting derived in this work (full line) and by Lavrentovich and Terentjev (dashed line) for rmax =rmin → ∞. The regions di9er by the areas I and II.

7.4.3. Discussion5 The instability condition (7.11) indicates for which values of the elastic constants K1 , K2 , and K3 the radial con0guration is expected to be unstable with respect to a twist deformation. The instability domain is largest for rmax =rmin → ∞ and decreases with decreasing ratio rmax =rmin , i.e., a water droplet inside a nematic drop can stabilize the radial con0guration. In Fig. 41a, the instability condition (7.11) is shown. If the ratios of the Frank elastic constants de0ne a point in the grey triangles, the radial con0guration can be unstable depending on the ratio rmax =rmin . The dark grey area gives the range of the elastic constants where a twisted structure occurs for rmax =rmin = 50. With increasing ratio rmax =rmin the instability domain enlargens until it is limited by K3 =(8K1 )+K2 =K1 =1 for rmax =rmin → ∞. The light grey triangle is the region where the radial con0guration is unstable for rmax =rmin ¿ 50 but where it is stable for rmax =rmin ¡ 50. The circles in Fig. 41a, represent, respectively, the elastic constants for the liquid crystal compounds MBBA, 5CB, and PAA. For 5CB the elastic constants are in the light grey domain, i.e., a twisted structure is expected for rmax =rmin → ∞ (no inner sphere) but not for rmax =rmin ¡ 50. Such a behavior has been recently observed in multiple nematic emulsions [182]. It has been found that a small water droplet inside a large nematic drop prevents the radial con0guration from twisting. Two examples of nematic drops observed under the microscope between crossed polarizers can be seen in Fig. 42. In the left image the director con0guration is pure radial, in the right one it is twisted. The left drop contains a small water droplet that stabilizes the radial con0guration according to Eq. (7.11). The water droplet is not visible in this image because of the limited resolution. A better image is presented in [182]. We have calculated the polarizing microscope picture of the twisted con0guration by means of the 2 × 2 Jones matrix formalism [60]. We took the director 0eld of Eq. (7.2) and used the eigenfunction of Eq. (7.10) with an amplitude 5

Reprinted with permission from A. RUudinger, H. Stark, Twist transition in nematic droplets: A stability analysis, Liq. Cryst. 26 (1999) 753. Copyright 1999 Taylor and Francis, http:==www.tandf.co.uk.

H. Stark / Physics Reports 351 (2001) 387–474

449

Fig. 42. Radial (left) and twisted (right) con0guration of the director 0eld in a nematic drop (diameter ≈ 20
m) of 5CB observed under the microscope between crossed polarizers. In the radial con0guration there is a small isotropic liquid droplet in the center of the nematic drop (invisible in this image).

Fig. 43. Calculated transmission for the twisted con0guration of the director 0eld in a nematic drop whose diameter is 20
m. The transmission amplitude was obtained by summing over 20 wave lengths between 400 and 800 nm. The amplitude b of the twist deformation was set to 0.15. This 0gure has to be compared to the right image of Fig. 42. Fig. 44. Radial dependence of f(r) (r) [cf. Eq. (7.10)] for rmax =rmin = 50. The function is strongly peaked close to rmin .

b = 0:15. The result shown in Fig. 43 is in qualitative agreement with the experimental image on the right in Fig. 42. In Fig. 44 we plot the radial part f(r) (r) [see Eq. (7.10)] of the eigenfunction f(r; 5) = f(r) (r)f(x) (5) governing the twist deformations. For large values of rmax =rmin it is strongly

450

H. Stark / Physics Reports 351 (2001) 387–474

peaked near rmin . The maximum of f(r) (r) occurs at a radius r0 which is given by ln

r0 ln(rmax =rmin ) 2 = arctan : rmin  ln(rmax =rmin )

(7.12)

Hence, for rmax =rmin 1 the maximal azimuthal component bf(r0 ; 5) of the director 0eld is located at r0 =rmin = e2 ≈ 7:39, i.e., close to the inner sphere. From the polarizing microscope pictures it can be readily seen that the twist deformation is largest near the center of the nematic drop. In the opposite limit, rmax =rmin ≈ 1, the position of maximal twist is at the geometric mean of rmin and rmax : r0 = (rmin rmax )1=2 . In the limit rmin → 0, where the inner sphere is not existing, a point defect with a core radius rc is located at r = 0. In this case our boundary condition, f(r) (rmin ) = 0, makes no sense since the director is not de0ned for r ¡ rc . Fortunately, for rmin → 0 the lowest eigenvalue of the operator (7.8) and therefore the instability condition is insensitive to a change of the boundary condition. Furthermore, the shape of the eigenfunction is also independent of the boundary condition, in particular its maximum is always located close to rmin . A last comment concerns the work of Lavrentovich and Terentjev [126]. In Fig. 41b, we plot as a dashed line the criterion, K3 =(4K1 ) + K2 =(2K1 ) = 1, which the authors of Ref. [126] derived for the twist transition in the case rmax =rmin → 0. They constructed an ansatz function which connects a hyperbolic hedgehog at the center via a twist deformation to a radial director 0eld at the periphery of a nematic drop. Then they performed a stability analysis for an appropriately chosen order parameter. The region of instability calculated in this article and their result di9er by the areas I and II. This is due to the complementarity of the two approaches. While the authors of Ref. [126] allow for large deviations with respect to the radial con0guration at the cost of 0xing an ansatz function, we allow the system to search the optimal con0guration (i.e., eigenfunction) for small deformations. We conclude that both results together give a good approximation of the region of instability for the radial con0guration against twisting. However, we cannot exclude that a full non-linear analysis of the problem leads to a change in the stability boundaries. In conclusion, we have performed a stability analysis of the radial con0guration in nematic drops with respect to a twist deformation. Assuming strong perpendicular anchoring at all the surfaces, we have derived an instability condition in terms of the Frank constants. We could show that a small water droplet inside the nematic drop stabilizes the radial con0guration.

8. Temperature-induced +occulation above the nematic-isotropic phase transition Ping Sheng [210,211] was the 0rst to study the consequences of surface-induced liquid crystalline order above the nematic-isotropic phase transition. He introduced the notion paranematic order in analogy to the paramagnetic phase, in which a magnetic 0eld causes a non-zero magnetization. He realized that the bounding surfaces of a restricted geometry in>uence the bulk transition temperature Tc . In nematic 0lms, e.g., the phase transition even vanishes below a critical thickness [210]. Sheng’s work was extended by Poniewierski and Sluckin [177], who studied two plates immersed in a liquid crystal above Tc and who calculated an attractive force

H. Stark / Physics Reports 351 (2001) 387–474

451

between the two plates due to the surface-induced order. This force was investigated in detail \ by Bor\stnik and Zumer [18]. The work presented in this section explores the liquid crystal mediated interaction between spherical particles immersed into a liquid crystal above Tc . It has to be added to the conventional van der Waals, electrostatic, and steric interactions as a new type of interparticle potential. Its strength can be controlled by temperature, and close to the clearing temperature Tc , it can induce a >occulation transition in an otherwise stabilized colloidal dispersion. In Section 8.1 we review the Landau–de Gennes theory, which describes liquid crystalline order close to the phase transition, and we present Euler–Lagrange equations for the director and the Maier–Saupe order parameter to be de0ned below. Section 8.2 illustrates paranematic order in simple plate geometries and introduces the liquid crystal mediated interaction of two parallel plates. In Section 8.3 we extend it to spherical particles and investigate its consequences when combined with van der Waals and electrostatic interactions. 8.1. Theoretical background We start with a review of the Landau–de Gennes theory and then formulate the Euler–Lagrange equations for restricted geometries with axial symmetry. 8.1.1. Landau–de Gennes theory in a nutshell The director n, a unit vector, only indicates the average direction of the molecules. It tells nothing about how well the molecules are aligned. To quantify the degree of liquid crystalline order, we could just vary the magnitude of n, i.e., choose a polar vector as an order parameter. However, all nematic properties are invariant under inversion of the director, thus every polar quantity has to be zero. The next choice is any second-rank tensor, e.g., the magnetic susceptibility tensor . The order parameter Q is de0ned by the relation   9 1  − 1 tr  ; Q= (8.1) 2 tr  3 where tr  = ii stands for the trace of a tensor, and Einstein’s summation convention over repeated indices is always assumed in the following. We subtract the isotropic part 1 tr =3 from , in order that Q vanishes in the isotropic liquid. The prefactor is convention. The order parameter Q describes, in general, biaxial liquid crystalline ordering through its eigenvectors and eigenvalues. The uniaxial symmetry of the nematic phase demands that two eigenvalues of Q are equal, which then assumes the form   3( − ⊥ ) 1 3 Q= S n⊗n− 1 with S = : (8.2) 2 3 2⊥ +  The Maier–Saupe or scalar order parameter S indicates the degree of nematic order through the magnetic anisotropy V =  − ⊥ . It was 0rst introduced by Maier and Saupe in a microscopic treatment of the nematic phase [142]. The microscopic approach was generalized by Lubensky to describe biaxial order [138].

452

H. Stark / Physics Reports 351 (2001) 387–474

In his seminal publication (see Ref. [48]) de Gennes was interested in pretransitional e9ects above the nematic-isotropic phase transition. He constructed a free energy in Q and ∇i Qjk in the spirit of Landau and Ginzburg, commonly known as Landau–de Gennes theory:
FLG = d 3 r (fb + f∇Q ) ; (8.3) with fb = 12 a0 (T − T ∗ ) tr Q 2 − 13 b tr Q 3 + 14 c(tr Q 2 )2 ;

(8.4)

f∇Q = 12 L1 (∇i Qjk )2 + 12 L2 (∇i Qij )2 :

(8.5)

The quantity fb introduces a Landau-type free energy density which describes a 0rst-order phase transition, and f∇Q is necessary to treat, e.g., >uctuations in Q , as noticed by Ginzburg. Both free energy densities are Taylor expansions in Q and ∇i Qjk , and each term is invariant under the symmetry group O(3) of the isotropic liquid, i.e., the high-symmetry phase. The Landau parameters of the compound 5CB are a0 = 0:087 × 107 erg=cm3 K; b = 2:13 × 107 erg=cm3 ; c = 1:73 × 107 erg=cm3 , and T ∗ = 307:15 K [38]. The elastic constants L1 and L2 are typically of the order of 10−6 dyn. It can be shown unambiguously that fb is minimized by the uniaxial order parameter of Eq. (8.2), for which the free energies fb and f∇Q take the form 3 9 1 fb = a0 (T − T ∗ )S 2 − bS 3 + cS 4 ; 4 4 16

(8.6)

9 3 f∇Q = L1 (∇i S)2 + L1 S 2 (∇i nj )2 : 4 4

(8.7)

To arrive at Eq. (8.7), we set L2 = 0 in order to simplify the free energy as much as possible for our treatment in Sections 8.2 and 8.3. L2 = 0 merely introduces some anisotropy, as shown by de Gennes [48]. Assume, e.g., that S is 0xed to a non-zero value at a space point rs in the isotropic >uid, then the nematic order around rs decays exponentially on a characteristic length scale called nematic coherence length. If L2 = 0, the respective coherence lengths along and perpendicular to n are di9erent. In Fig. 45 we plot fb as a function of S using the parameters of 5CB. Above the superheating temperature T † = T ∗ + b2 =(24a0 c), there exists only one minimum at S = 0 for the thermodynamically stable isotropic phase. At T † a second minimum for the metastable nematic phase evolves, which becomes absolutely stable at the clearing temperature Tc =T ∗ +b2 =(27a0 c). A 0rst-order phase transition occurs, and the order parameter as a function of temperature assumes the form

1b 2a † S(T ) = (8.8) + (T − T ) : 6c 3c Finally, at the supercooling temperature T ∗ the curvature of fb at S = 0 changes sign, and the isotropic >uid becomes absolutely unstable. For the compound 5CB, we 0nd Tc − T ∗ = 1:12 K and T † − Tc = 0:14 K.

H. Stark / Physics Reports 351 (2001) 387–474

453

Fig. 45. The free energy density fb in units of 1000a0 T ∗ as a function of the Maier–Saupe order parameter S for various temperatures. The Landau coeNcients of the compound 5CB are employed. A 0rst-order transition occurs at Tc .

8.1.2. Euler–Lagrange equations for restricted geometries In the following, we determine the surface-induced liquid crystalline order above Tc . As usual, it follows from a minimization of the total free energy, F = FLG + Fsur ;

(8.9)

where we have added a surface term Fsur to the Landau–de Gennes free energy FLG . We restrict ourselves to uniaxial order and employ a generalization of the Rapini–Papoular potential, introduced in Section 2.1,
3 Fsur = dA (WS (S − S0 )2 + 3Wn SS0 [1 − (n · ˆ)2 ]) ; (8.10) 4 where dA is the surface element. The quantity S0 denotes the preferred Maier–Saupe parameter at the surface, and ˆ is the surface normal since we always assume homeotropic anchoring. The surface-coupling constants WS and Wn penalize a respective deviation of S from S0 and of the director n from ˆ. In recent experiments, anchoring and orientational wetting transitions of liquid crystals, con0ned to cylindrical pores of alumina membranes, were analyzed [42,43]. It was found that WS and Wn vary between 10−1 and 5, with the ratio Wn =WS not being larger than 0ve. If WS = Wn = W , the intergrand in Eq. (8.10) is equivalent to the intuitive form W tr(Q − Q0 )2 =2 with the uniaxial Q from Eq. (8.2) and Q0 = 32 S0 (ˆ ⊗ ˆ − 13 1). It was introduced by Nobili and Durand [165]. In formulating the elastic free energy density f∇Q of Eq. (8.5), one also identi0es a contribution which can be written as a total divergence, ∇i (Qij ∇k Qjk − Qjk ∇k Qij ). When transformed into a surface term and when a uniaxial Q is inserted, it results in the saddle-splay energy of Eq. (2.4). To simplify our calculations, we will neglect this term. It is not expected to change the qualitative behavior of our system for strong surface coupling. In what follows, we assume rotational symmetry about the z axis. We introduce cylindrical coordinates and write the director in the local coordinate basis, n(6; z) = sin (6; z)e6 + cos (6; z)ez , restricting it to the (6; z) plane. The same is assumed for the surface normal ˆ(6; z) = sin 0 (6; z)e6 + cos 0 (6; z)ez . Expressing and minimizing the total free energy under

454

H. Stark / Physics Reports 351 (2001) 387–474

all these premises, we obtain the Euler–Lagrange equations for S and the tilt angle  in the bulk,   1 3c 3 b 2 sin2  2 2 ∇ S− 2S+ S − S − 3S (∇) + =0 ; (8.11) 2L1 2L1 62 N sin  cos  ∇2  − =0 ; (8.12) 62 and the boundary equations are 1 3 (ˆ ·∇)S − (S − S0 ) − S0 sin2 ( − 0 ) = 0 ; %S N 2%n N 1 S0 (ˆ ·∇) − sin[2( − 0 )] = 0 : 2%n N S The meaning of the nematic coherence length  N = L1 =[a0 (T − T ∗ )]

(8.13) (8.14)

(8.15)

will be clari0ed in the next subsection. At the phase transition, NI = N (Tc ) is of the order of 10 nm, as can be checked by the parameters of 5CB. The surface-coupling strengths WS and Wn are characterized by dimensionless quantities   a0 (T − T ∗ )L1 a0 (T − T ∗ )L1 1 L1 1 L1 = and %n = = ; (8.16) %S = N WS WS N Wn Wn which compare the respective surface extrapolation lengths L1 =WS and L1 =Wn to the nematic coherence length N . For W = 1 erg=cm2 and L1 = 10−6 dyn, the extrapolation lengths are of the same order as N at Tc , i.e., 10 nm. 8.2. Paranematic order in simple geometries In the 0rst two subsections we study the paranematic order in a liquid crystal compound above Tc for simple plate geometries. It is induced by a coupling between the surfaces and the molecules. We disregard the non-harmonic terms in S in Eq. (8.11) to simplify the problem as much as possible and to obtain an overall view of the system. In Section 8.2.3 the e9ect of the non-harmonic terms is reviewed. 8.2.1. One plate We assume that an in0nitely extended plate, which induces a homeotropic anchoring of the director, is placed at z = 0. Its surface normals are ±ez , and its thickness should be negligibly small. A uniform director 0eld along the z axis obeys Eqs. (8.12) and (8.14), and the Maier–Saupe order parameter S follows from a solution of Eqs. (8.11) and (8.13), S0 S(z) = exp[ − |z |=N ] : (8.17) 1 + %S The order parameter S decays exponentially along the z axis on a characteristic length scale given by the nematic coherence length N . The value of S at z = 0 depends on the strength %S

H. Stark / Physics Reports 351 (2001) 387–474

455

of the surface coupling, i.e., on the ratio of the surface extrapolation length L1 =W and N . The plate is surrounded by a layer of liquid crystalline order whose thickness N decreases with increasing temperature since N ˙ (T − T ∗ )−1=2 . The total free energy per unit surface, F=A, consisting of the Landau–de Gennes and the surface free energy, is F 3 %S : (8.18) = WS S02 A 2 1 + %S √ Note that the energy increases with temperature since %S ˙ T − T ∗ . The whole theory certainly becomes invalid when N approaches molecular dimensions. For 10 K above Tc , we 0nd N ≈ 3 nm, i.e., the theory is valid several Kelvin above Tc . Finally, we notice that a nematic wetting layer can be probed by the evanescent wave technique [214]. 8.2.2. Two plates If two plates of the previous subsection are placed at z = ±d=2, the order parameter pro0le S(z), determined from Eqs. (8.11) and (8.13), is cosh(z=N ) S(z) = S0 : (8.19) cosh(d=2N ) + %S sinh(d=2N ) For separations d2N , the layers of liquid crystalline order around the plates do not overlap, as illustrated in the inset of Fig. 46. 6 If d 6 2N , the whole volume between the plates is occupied by nematic order, which induces an attraction between the plates. The interaction energy per unit area, VF=A, is de0ned as VF=A = [F(d) − F(d → ∞)]=A. It amounts to   VF F(d) − F(d → ∞) 3 tanh(d=2N ) 1 2 − : (8.20) = = WS S0 %S A A 2 1 + %S tanh(d=N ) 1 + %S In Fig. 46 we plot VF=A versus the reduced distance d=2NI for di9erent temperatures at Tc and above Tc . The material parameters of 5CB are chosen; WS = 1 erg=cm2 , and S0 = 0:3. The energy unit 3WS S02 =2=104 kB T is determined at room temperature. Note, that NI is the coherence length at Tc . If dN , the interaction energy decays exponentially in d; VF=A ˙ exp(d=N ). The interaction is always attractive over the whole separation range. This can be understood by a simple argument. Above Tc , the nematic order always possesses higher energy than the isotropic liquid. Therefore, the system can reduce its free energy by moving the plates together. The minimum of the interaction energy occurs at d = 0, i.e., when the liquid with nematic order between the plates is completely removed. This simple argument explains the deep potential well in Fig. 46. It extends to a separation of 2N where the nematic layers start to overlap. Since N ˙ (T − T ∗ )−1=2 , the range of the interaction decreases with increasing temperature, and the depth of the potential well becomes smaller. 8.2.3. E<ect of non-harmonic terms In this subsection we review the e9ects on the two-plate geometry when the complete Landau–de Gennes theory including its non-harmonic terms in S is employed. A wealth of \ Figs. 46, 47, 51 and 52 are reprinted with permission from A. Bor\stnik, H. Stark, S. Zumer, Temperature-induced >occulation of colloidal particles above the nematic-isotropic phase transition, Prog. Colloid Polym. Sci. 115 (2000) 353. Copyright 2000 Springer Verlag. 6

456

H. Stark / Physics Reports 351 (2001) 387–474

Fig. 46. Interaction energy per unit area, VF=A, as a function of the reduced distance d=2NI for various temperatures. For further explanation see text.

phenomena exists, which we illustrate step by step [210,211]. Their in>uence on the interaction \ of two plates was studied in detail by Bor\stnik and Zumer [18]. First, we assume rigid anchoring at the nematic-plate interfaces, i.e., S(±d=2) is 0xed to S0 [210]. For d → ∞, there is a phase transition at the bulk transition temperature Tc∞ = Tc from the nematic to the surface-induced paranematic phase, as expected. When the plates are moved together, the transition temperature Tcd increases until the 0rst-order transition line in crit ). For d ¡ d , no phase transition a d–T phase diagram ends in a critical point at (dcrit ; Tcd crit between the nematic and the paranematic phase is observed anymore. This is similar to the gas–liquid critical point in an isotropic >uid. For S0 = S(±d=2) = 0:5 − 1 and typical values of crit is situated approximately 0.2 K above T the Landau parameters, Tcd c∞ = Tc and 0.1 K above the superheating temperature T † . Secondly, we concentrate on a basically in0nite separation, dNI , and allow a 0nite surfacecoupling strength WS [211]. For suNciently small WS , both the boundary [S(±d=2)] and the bulk [S(0)] value of the scalar order parameter exhibit a jump at Tc . That means, the surface coupling is so small that S(±d=2) follows the bulk order parameter. However, in a 0nite interval WS0 ¡ WS ¡ WScrit , the discontinuity of S(±d=2), which Sheng calls a boundary-layer phase transition, occurs at temperatures Tbound above Tc . Beyond the critical strength WScrit , the boundary transition vanishes completely. Sheng just used the linear term ˙ WS S of our surface potential for his investigation. The separate boundary-layer transition occurred in the approximate interval 0:01 erg=cm2 ¡ WS ¡ 0:2 erg=cm2 . We do not expect a dramatic change of this interval for the potential of Eq. (8.10). Thirdly, we combine the 0nite separation of the plates with a 0nite surface-coupling strength WS . The boundary-layer transition temperatures Tbound and the interval WS0 ¡ WS ¡ WScrit are not e9ected by a 0nite d. In addition, a jump of S(±d) occurs at the bulk transition temperature Tcd 6 Tbound . It evolves gradually with decreasing d. When Tcd becomes larger than Tbound

H. Stark / Physics Reports 351 (2001) 387–474

457

in the course of moving the plates together, the separate boundary-layer transition disappears. Finally, at a critical thickness dcrit the nematic-paranematic transition vanishes altogether. crit − T All these details occur close to Tc∞ = Tc within a range of Tcd c∞ = 0:5 K [211]. The calculations are non-trivial. Since we do not want to render our investigation in the following subsection too complicated, we will skip the non-harmonic terms in the Landau–de Gennes theory. Furthermore, we use a relatively high anchoring strength of about WS = 1 erg=cm2 , so crit − T that Tcd c∞ is even smaller than 0.5 K. The simpli0cations are suNcient to bring out the main features of our system. 8.3. Two-particle interactions above the nematic-isotropic phase transition In this subsection we present the liquid crystal mediated interaction above Tc as a new type of two-particle potential. We combine it with the traditional van der Waals and electrostatic interaction and explore its consequences, namely the possibility of a temperature-induced >occulation. We start with a motivation, introduce all three types of interactions, and 0nally discuss their consequences. Our presentation concentrates on the main ideas and results (see also Ref. [17]). Details of the calculations can be found in Refs. [15,16]. 8.3.1. Motivation In Section 3 we already mentioned that the stability of colloidal systems presents a key issue in colloid science since their characteristics change markedly in the transition from the dispersed to the aggregated state. There are always attractive van der Waals forces, which have to be balanced by repulsive interactions to prevent a dispersion of particles from aggregating. This is achieved either by electrostatic repulsion, where the particles carry a surface charge, or by steric stabilization, where they are coated with a soluble polymer brush. Dispersed particles approach each other due to their Brownian motion. They aggregate if the interaction potential is attractive, i.e., if it possesses a potential minimum Umin ¡ 0 at 0nite separations. Two situations are possible. In the case of weak attraction, where |Umin | ≈ 1–3kB T , an equilibrium phase separation of a dilute and an aggregated state exists. The higher interaction energy of the dispersed particles is compensated by their larger entropy in comparison to the aggregated phase. Strong attraction, i.e., |Umin | ¿ 5 − 10kB T , causes a non-equilibrium phase with all the particles aggregated. They cannot escape the attractive potential in the observation time of interest of, e.g., several hours. Due to Chandrasekhar, the escape time tesc can be estimated as [28] tesc =

a2 D0 exp(−Umin =kB T )

with D0 =

kB T : 6)a

(8.21)

D0 is the di9usion constant of a non-interacting Brownian particle with radius a, and ) is the shear viscosity of the solvent. The quantity tesc approximates the time a particle needs to di9use a distance a in leaving a potential well of depth Umin . More re0ned theories suggest that the complete two-particle potential has to be taken into account when calculating tesc [131,100]. Here, we study the in>uence of liquid crystal mediated interactions on colloidal dispersions above Tc , which are stabilized by an electrostatic repulsion. We demonstrate that the main e9ect of the liquid crystal interaction ULC is an attraction at the length scale of N , whose strength can be controlled by temperature. If the electrostatic repulsion is suNciently weak, ULC induces

458

H. Stark / Physics Reports 351 (2001) 387–474

Fig. 47. Two particles at a separation d2N do not interact. At d ≈ 2N both a strong attraction and repulsion set in.

a >occulation of the particles within a few Kelvin close to the transition temperature Tc . It is completely reversible. A similar situation is found in polymer stabilized colloids. There, the abrupt change from a dispersed to a fully aggregated state within a few Kelvin is called critical >occulation [162,202]. The reversibility of >occulation has interesting technological implications. For example, in “instant” ink, the particles of dried ink redisperse rapidly when put into water [162]. So far, experiments on colloidal dispersions above the clearing temperature Tc are very rare [19,178]. They would help to explore a new class of colloidal interactions. Furthermore, they could provide insight into wetting phenomena above Tc with all its subtleties close to Tc , which we reviewed in Section 8.2.3. Also, experiments by Mu\sevi\c et al. [158,159], who probe interactions with the help of an atomic force microscope, are promising. 8.3.2. Liquid crystal mediated interaction One particle suspended in a liquid crystal above the clearing temperature Tc is surrounded by a layer of surface-induced nematic order whose thickness is of the order of the nematic coherence length N . The director 0eld points radially outward when a homeotropic anchoring at the particle surface is assumed. Two particles with a separation d2N do not interact. When the separation is reduced to d ≈ 2N , a strong attraction sets in since the total volume of nematic order is decreased as in the case of two plates (see Fig. 47). In addition, a repulsion due to the elastic distortion of the director 0eld lines connecting the two particles occur. In this subsection we quantify the two-particle interaction mediated by a liquid crystal. In principle, the director 0eld and the Maier–Saupe order parameter S follow from a solution of Eqs. (8.11) – (8.14). Since the geometry of Fig. 48a cannot be treated analytically, we employ two simpli0cations. First, we approximate each sphere by 72 conical segments, whose cross sections in a symmetry plane of our geometry are illustrated in Fig. 48a. In the following, we assume a particle radius a = 250 nm, and, therefore, each line segment has a length of

H. Stark / Physics Reports 351 (2001) 387–474

459

Fig. 48. (a) Two spheres A and B are approximated by conical segments as illustrated in the blowup. From Ref. [15]. (b) At separations d ≈ 2N , the director is chosen as a tangent vector nc of a circular segment whose radius is determined by the boundary condition (8.14).

26 nm. Secondly, we construct appropriate ansatz functions for the 0elds S(r) and n(r). To arrive at an ansatz for S(r), we approximate the bounding surfaces Ai and Bi of region i by two parallel ring-like plates and employ the order parameter pro0le of Eq. (8.19), where d is replaced by an average distance di of the bounding surfaces. Since the particle radius is an order of magnitude larger than the interesting separations, which do not exceed several coherence lengths, the analogy with two parallel plates is justi0ed. Furthermore, we expect that only a few regions close to the symmetry axis are needed to calculate the interaction energy with a suNcient accuracy. In the limit of large separations (d2N ), the director 0eld around each sphere points radially outward. In the opposite limit (d ≈ 2N ), the director 0eld lines are strongly distorted, and we approximate them by circular segments as illustrated in Fig. 48b, for the third region. The radius of the circle is determined by the boundary condition (8.14) of the director. With decreasing separation of the two particles, the director 0eld should change continuously from n∞ at d2N to the ansatz nc at small d. Hence, we choose n(r) as a weighted superposition of nc and n∞ : n(r ) ˙ 'i nc + (1 − 'i )n∞ ;

(8.22)

where the free parameter 'i follows from a minimization of the free energy in region i with respect to 'i . As in the case of two parallel plates, the interaction energy is de0ned relative to the total free energy of in0nite separation: ULC (d) = F(d) − F(d → ∞) :

(8.23)

In calculating ULC , we employ the free energy densities of Eqs. (8.6) and (8.7) and the surface potential of Eq. (8.10), neglecting the non-harmonic terms in S. The volume integrals cannot be performed analytically without further approximations which we justi0ed by a comparison with a numerical integration. The 0nal expression of ULC is very complicated, and we refer the reader to Ref. [15] for its explicit form. We checked that regions i = 1; : : : ; 9 are suNcient to

460

H. Stark / Physics Reports 351 (2001) 387–474

Fig. 49. The liquid crystal mediated interaction ULC in units of kB T as a function of the particle separation d. The interaction is shown at Tc ; Tc + 1 K; Tc + 3 K; and Tc + 11 K. It strongly depends on temperature. Large inset: ULC is composed of an attractive and repulsive part. Small inset: A weak repulsive barrier occurs at d ≈ 60 nm.

calculate ULC . The contribution of region 9 to the interaction energy is less than 5%. Hence, the orientational order outside these nine regions is not relevant for ULC . We subdivide the interaction energy in an attractive part which results from all terms in the free energy depending on the order parameter S or its gradient, only. The repulsive part is due to the elastic distortion of the director 0eld and a deviation from the homeotropic orientation at the particle surfaces. All the graphs, which we present in the following, are calculated with the Landau parameters of the compound 8CB [38], i.e., a0 = 0:12 × 10−7 erg=cm3 K; b = 3:07 × 10−7 erg=cm3 ; c=2:31×10−7 erg=cm3 , and L1 =1:8×10−6 dyn, which gives Tc −T ∗ =b2 =(27a0 c)= 1:3 K. The surface-coupling constants are WS = 1 erg=cm2 and Wn = 5 erg=cm2 . In the large inset of Fig. 49 we plot the attractive and repulsive contribution at the clearing temperature Tc in units of the thermal energy kB T . As in the case of two parallel plates, the total interaction energy exhibits a deep potential well with an approximate width of 2NI . At larger separations, it is followed by a weak repulsive barrier whose height is approximately 1:5kB T , as indicated by the small inset in Fig. 49. If d2N ; ULC decays exponentially: ULC ˙ exp(−d=N ). Fig. 49 illustrates further that the depth of the potential well, i.e., the liquid crystal mediated attraction of two particles decreases considerably when the dispersion is heated by several Kelvin. That means, the interaction can be controlled by temperature. It is turned o9 by heating the dispersion well above Tc . The same holds for the weak repulsive barrier. As expected, both the depth of the potential well and the height of the barrier decrease with the surface-coupling constants, where WS seems to be more important [15]. 8.3.3. Van der Waals and electrostatic interactions The van der Waals interaction of two thermally >uctuating electric dipoles decays with the sixth power of their inverse distance, 1=r 6 . To arrive at the interparticle potential of two macroscopic objects, a summation over all pair-wise interactions of >uctuating charge distributions is performed. In the case of two spherical particles of equal radii a; the following,

H. Stark / Physics Reports 351 (2001) 387–474

always attractive, van der Waals interaction results [202]:   d(d + 4a) 2a2 A 2a2 UW = − + ln : + 6 d(d + 4a) (d + 2a)2 (d + 2a)2

461

(8.24)

Here d is the distance between the surfaces of the particles, and A is the Hamaker constant. For equal particles made of material 1 embedded in a medium 2, it amounts to [202]   3 '1 − '2 2 3hBuv (n21 − n22 )2 A = kB T + √ ; (8.25) 4 '1 + '2 16 2 (n21 + n23 )3=2 where '1 and '2 are the static dielectric constants of the two materials, and n1 and n2 are the corresponding refractive indices of visible light. The relaxation frequency Buv belongs to the dominant ultraviolet absorption in the dielectric spectrum of the embedding medium 2. Typical values for silica particles immersed into a nematic liquid crystal are '1 = 3:8; n1 = 1:45; '2 = 11; n2 = 1:57; and Buv = 3 × 1015 s−1 [15]. As a result, the Hamaker constant equals A = 1:1 kB T . Note, that for separations da the particles are point-like, and the van der Waals interaction decays as 1=d6 . In the opposite limit, da; it diverges as a=d. We stabilize the colloidal dispersion against the attractive van der Waals forces by employing an electrostatic repulsion. We assume that each particle carries a uniformly distributed surface charge whose density qs does not change under the in>uence of other particles. Ionic impurities in the liquid crystal screen the surface charges with which they form the so-called electrostatic double layer. For particles of equal radius a embedded in a medium with dielectric constant '2 ; the electrostatic two-particle potential is described by the following expression [202]: UE = −kB T

aqs2 ln(1 − e−Cd ) : z 2 e02 np

(8.26)

Here, e0 is the fundamental charge, and z is the valence of the ions in the solvent, which have a concentration np . The range of the repulsive interaction is determined by the Debye length  (8.27) C−1 = '2 kB T=(8e02 z 2 np ) ; whereas the surface-charge density qs controls its strength. The potential UE decays exponentially at dC−1 . Expression (8.26) is derived via the Derjaguin approximation [53,202], which is only valid for d; C−1 a. In the following, we take a monovalent salt (z = 1); choose '2 = 11; and vary np between 10−4 and 10−3 mol=l. Then, at room temperature the Debye length C−1 ranges from 10 to 3.5 nm. Together with typical separations d not larger than a few coherence lengths N and a = 250 nm; the Derjaguin approximation is justi0ed. Furthermore, we adjust the surface-charge density around 104 e0 =
m2 . The ranges of np and qs are well accessible in an experiment. In Fig. 50 we plot the electrostatic and the van der Waals interactions and their sum in units of kB T . The surface-charge density qs is 0:5 × 104 e0 =
m2 and C−1 = 8:3 nm. All further parameters besides the Hamaker constant A are chosen as mentioned above. We increased A from 1.1 to 5.5. Even then it is clearly visible that the strong electrostatic repulsion determines the interaction for d ¡ 30 nm; the dispersion of particles is stabilized. At about 55 nm, UE + UW exhibits a shallow potential minimum (see inset of Fig. 50), and at dC−1 ; the algebraic decay

462

H. Stark / Physics Reports 351 (2001) 387–474

Fig. 50. The electrostatic (dashed) and van der Waals (dotted) interaction and their sum UE +UW (full line) in units of kB T as a function of particle separation d. The parameters are chosen according to the text. Inset: A shallow potential minimum appears at d ≈ 55 nm.

Fig. 51. The total two-particle interaction ULC + UE + UW as a function of particle separation d for various temperatures. A complete >occulation of the particles occurs within a temperature range of about 0.3 K. qs =0:5 × 104 e0 =
m2 ; C−1 = 8:3 nm, and further parameters are chosen according to the text.

of the van der Waals interaction takes over. In the following subsection, we investigate the combined e9ects of all three interactions for the Hamaker constant A = 1:1. 8.3.4. Flocculation versus dispersion of particles In Fig. 51 we plot the total two-particle interaction ULC + UE + UW as a function of particle separation d for various temperatures. We choose qs = 0:5 × 104 e0 =
m2 and C−1 = 8:3 nm. At 4.5 K above the transition temperature Tc ; the dispersion is stable. With decreasing temperature, a potential minimum at 0nite separation develops. At TFD = Tc + 0:54 K; the particle doublet or aggregated state becomes energetically preferred. We call TFD the temperature of >occulation

H. Stark / Physics Reports 351 (2001) 387–474

463

Fig. 52. In comparison to Fig. 51 the surface-charge density is increased to 0:63 × 104 e0 =
m2 . As a result, >occulation does not occur.

transition. Below TFD ; the probability of 0nding the particles in the aggregated state is larger than the probability that they are dispersed. Already at Tc + 0:3 K the minimum is 7 kB T deep, and all particles are condensed in aggregates. That means, within a temperature range of about 0.3 K there is an abrupt change from a completely dispersed to a fully aggregated system, reminiscent to the critical >occulation transition in colloidal dispersions employing polymeric stabilization [162,202]. Between d = 30 and 50 nm, the two-particle interaction exhibits a small repulsive barrier of about 1:5 kB T . Such barriers slow down the aggregation of particles, and one distinguishes between slow and rapid >occulation. The dynamics of rapid >occulation was 0rst studied by Smoluchowski [215]. Fuchs extended the theory to include arbitrary interaction potentials [82]. However, only after Derjaguin and Landau [54] and Verwey and Overbeek [229] incorporated van der Waals and electrostatic interactions into the theory, became a comparison with experiments possible. In our case, the repulsive barrier of 1:5 kB T slows down the doublet formation by a factor of three, i.e., it does not change very dramatically if the barrier is reduced to zero. If the surface-charge density qs is increased to 0:63 × 104 e0 =
m2 ; the dispersed state is thermodynamically stable at all temperatures above Tc ; as illustrated in Fig. 52. An increase of the Debye length C−1 ; i.e., the range of the electrostatic repulsion, has the same e9ect. In Fig. 53 we present >occulation phase diagrams as a function of temperature and surfacecharge density for various Debye lengths C−1 . The inset shows one such diagram for C−1 = 8:3 nm. The full line represents the >occulation temperature TFD as a function of qs . For temperatures above TFD ; the particles stay dispersed while for temperatures below TFD the system is >occulated. To characterize the aggregated state further, we have determined lines in the phase diagram of C−1 = 8:3 nm; where the escape time tesc of Eq. (8.21) is, respectively, ten (dash-dotted) or hundred (dotted) times larger than in the case of zero interaction. These lines are close to the transition temperature TFD ; and indicate again that the transition from the dispersed to a completely aggregated state takes place within less than one Kelvin. The large plot of Fig. 53 illustrates the >occulation temperature TFD as a function of qs for various Debye

464

H. Stark / Physics Reports 351 (2001) 387–474

Fig. 53. Flocculation phase diagrams as a function of temperature and surface-charge density for various Debye lengths C−1 . Inset: Phase diagram for C−1 = 8:3 nm. The full line represents the >occulation temperature TFD as a function of qs . The dash-dotted and dotted lines indicate escape times from the minimum of the interparticle potential which are, respectively, ten or hundred times larger than in the case of zero interaction. From Ref. [16].

lengths C−1 . TFD increases when the strength (qs ) or the range (C−1 ) of the electrostatic repulsion is reduced. The intersections of the transition lines with the T = Tc axis de0ne the “>occulation end line”. In the parameter space of the electrostatic interaction (surface charge density qs versus Debye length C−1 ), this line separates the region where we expect the >occulation to occur from the region where the system is dispersed for all temperatures above Tc (see Ref. [16]). 8.3.5. Conclusions Particles dispersed in a liquid crystal above the nematic-isotropic phase transition are surrounded by a surface-induced nematic layer whose thickness is of the order of the nematic coherence length. The particles experience a strong liquid crystal mediated attraction when their nematic layers start to overlap since then the e9ective volume of liquid crystalline ordering and therefore the free energy is reduced. A repulsive correction results from the distortion of the director 0eld lines connecting two particles. The new colloidal interaction is easily controlled by temperature. In this section we have presented how it can be probed with the help of electrostatically stabilized dispersions. For suNciently weak and short-ranged electrostatic repulsion, we observe a sudden >occulation within a few tenth of a Kelvin close to Tc . It is reminiscent to the critical >occulation transition in polymer stabilized colloidal dispersions [202]. The >occulation is due to a deep potential minimum in the total two-particle interaction followed by a weak repulsive barrier. Thermotropic liquid crystals represent polar organic solvents, and one could wonder if electrostatic repulsion is realizable in such systems. In Ref. [101] complex salt is dissolved in nematic liquid crystals and ionic concentrations of up to 10−4 mol=l are reported which give

H. Stark / Physics Reports 351 (2001) 387–474

465

rise to Debye lengths employed in this section. Furthermore, when silica spheres are coated − with silanamine, the ionogenic group [ = NH+ 2 OH ] occurs at the particle surface with a density of 3 × 106 mol=
m2 . It dissociates to a large amount in a liquid crystal compound [59]. In addition, the silan coating provides the required perpendicular boundary condition for the liquid crystal molecules. These two examples illustrate that electrostatic repulsion should be accessible in conventional thermotropic liquid crystals, and we hope to initiate experimental studies which probe the new colloidal force. Our work directly applies to lyotropic liquid crystals [51], i.e., aqueous solutions of non-spherical micelles, when the nematic-isotropic phase transition is controlled by temperature [181,180]. They are appealing systems since electrostatic stabilization is more easily achieved. When the phase transition is controlled by the micelle concentration 6m ; as it is usually done, then our diagrams are still valid but with temperature replaced by 6m . In polymer stabilized dispersions, we 0nd that the aggregation of particles sets in gradually when cooling the dispersion down towards Tc . This is in contrast to electrostatic stabilization where >occulation occurs in a very narrow temperature interval (see Ref. [16]). 9. Final remarks In this article we have demonstrated that the combination of two soft materials, nematic liquid crystals and colloidal dispersions, creates a novel challenging system for discovering and studying new physical e9ects and ideas. Colloidal dispersions in a nematic liquid crystal introduce a new class of long-range twoparticle interactions mediated by the distorted director 0eld. They are of either dipolar or quadrupolar type depending on whether the single particles exhibit the dipole, Saturn-ring or surface-ring con0guration. The dipolar forces were veri0ed in an excellent experiment by Poulin et al. [179]. Via the well-known >exoelectric e9ect [147], strong director distortions in the dipole con0guration should induce an electric dipole associated with each particle. It would be interesting to study, both theoretically and experimentally, how this electric dipole contributes to the dipolar force. On the other hand, there exists a strong short-range repulsion between particles due to the presence of a hyperbolic point defect which prevents, e.g., water droplets from coalescing. Even above the nematic-isotropic phase transition, liquid crystals mediate an attractive interaction at a length scale of 10 nm. Its strength is easily controlled by temperature, and it produces an observable e9ect since it can induce >occulation when the system is close to the phase transition. To understand colloidal dispersions in nematics in detail, we have performed an extensive study of the three possible director con0gurations around a single particle. These con0gurations are ideal objects to investigate the properties of topological point and line defects. The dipolar structure should exhibit a twist in conventional calamitic compounds. The transition from the dipole to the Saturn ring can be controlled, e.g., by a magnetic 0eld which presents a means to access the dynamics of topological defects. Furthermore, we have studied how the strength of surface anchoring in>uences the director con0guration. Surface e9ects are of considerable importance in display technology, and there is fundamental interest in understanding the coupling between liquid crystal molecules and surfaces. In addition, we have clari0ed the mechanism due to which the saddle-splay term in the Frank free energy promotes the formation of the

466

H. Stark / Physics Reports 351 (2001) 387–474

surface-ring structure. The Stokes drag and Brownian motion in nematics have hardly been studied experimentally. Especially the dipole con0guration with its vector symmetry presents an appealing object. We have calculated the Stokes drag for a 0xed director 0eld. However, we have speculated that for small Ericksen numbers (Er 1) >ow-induced distortions of the director 0eld result in corrections to the Stokes drag which are of the order of Er. Preliminary studies support this conclusion. Furthermore, for growing Er they reveal a highly non-linear Stokes drag whose consequences seemed to have not been explored in colloidal physics. Finally, we have demonstrated that the dipole, consisting of the spherical particle and its companion point defect, also exists in more complex geometries, and we have studied in detail how it forms. Acknowledgements I am grateful to Tom Lubensky, Philippe Poulin and Dave Weitz for their close and inspiring collaboration on nematic emulsions at the University of Pennsylvania which initiated the present work. I thank Anamarija Bor\stnik, Andreas RUudinger, Joachim Stelzer, Dieter Ventzki, \ and Slobodan Zumer for collaborating on di9erent aspects of nematic colloidal dispersions. The results are presented in this review article. A lot of thanks to Eugene Gartland, Thomas Gisler, Randy Kamien, Axel Kilian, R. Klein, G. Maret, R.B. Meyer, Michael Reichenstein, E. Sackmann, Thorsten Seitz, Eugene Terentjev, and H.-R. Trebin for fruitful discussions. Finally, I acknowledge 0nancial assistance from the Deutsche Forschungsgemeinschaft through grants Sta 352=2-1=2 and Tr 154=17-1=2. References [1] A.A. Abrikosov, On the magnetic properties of superconductors of the second group, Sov. Phys. JETP 5 (1957) 1174–1182. [2] A. Ajdari, B. Duplantier, D. Hone, L. Peliti, J. Prost, “Pseudo-Casimir” e9ect in liquid crystals, J. Phys. II France 2 (1992) 487–501. [3] A. Ajdari, L. Peliti, J. Prost, Fluctuation-induced long-range forces in liquid crystals, Phys. Rev. Lett. 66 (1991) 1481–1484. [4] E. Allahyarov, I. D’Amico, H. LUowen, Attraction between like-charged macroions by Coulomb depletion, Phys. Rev. Lett. 81 (1998) 1334–1337. [5] D.W. Allender, G.P. Crawford, J.W. Doane, Determination of the liquid-crystal surface elastic constant K24 , Phys. Rev. Lett. 67 (1991) 1442–1445. [6] D. Andrienko, G. Germano, M.P. Allen, Computer simulation of topological defects around a colloidal particle or droplet dispersed in a nematic host, Phys. Rev. E., to be published. [7] R.J. Atkin, Poiseuille >ow of liquid crystals of the nematic type, Arch. Rational Mech. Anal. 38 (1970) 224–240. [8] R.J. Atkin, F.M. Leslie, Couette >ow of nematic liquid crystals, Q. J. Mech. Appl. Math. 23 (1970) S3–S24. [9] G. Barbero, C. Oldano, Derivative-dependent surface-energy terms in nematic liquid crystals, Nuovo Cimento D 6 (1985) 479–493. [10] J.L. Billeter, R.A. Pelcovits, Defect con0gurations and dynamical behavior in a Gay-Berne nematic emulsion, Phys. Rev. E 62 (2000) 711–717. [11] R.B. Bird, W.E. Stewart, E.N. Lightfoot, Transport Phenomena, Wiley, New York, 1960. [12] C. Blanc, M. Kleman, The con0nement of smectics with a strong anchoring, Eur. Phys. J. E., to be published.

H. Stark / Physics Reports 351 (2001) 387–474

467

[13] G. Blatter, M.V. Feigel’mann, V.B. Geshkenbein, A.I. Larkin, V.M. Vinokur, Vortices in high-temperature superconductors, Rev. Mod. Phys. 66 (1994) 1125–1388. [14] L.M. Blinov, A.Y. Kabayenkov, A.A. Sonin, Experimental studies of the anchoring energy of nematic liquid crystals, Liq. Cryst. 5 (1989) 645–661. \ [15] A. Bor\stnik, H. Stark, S. Zumer, Interaction of spherical particles dispersed in liquid crystals above the nematic–isotropic phase transition, Phys. Rev. E 60 (4) (1999) 4210 – 4218. ∗ \ [16] A. Bor\stnik, H. Stark, S. Zumer, Temperature-induced >occulation of colloidal particles immersed into the isotropic phase of a nematic liquid crystal, Phys. Rev. E 61 (3) (2000) 2831–2839. ∗∗ \ [17] A. Bor\stnik, H. Stark, S. Zumer, Temperature-induced >occulation of colloidal particles above the nematic-isotropic phase transition, Prog. Colloid Polym. Sci. 115 (2000) 353–356. \ [18] A. Bor\stnik, S. Zumer, Forces in an inhomogeneously ordered nematic liquid crystal, Phys. Rev. E 56 (1997) 3021–3027. [19] A. BUottger, D. Frenkel, E. van de Riet, R. Zijlstra, Di9usion of Brownian particles in the isotropic phase of a nematic liquid crystal, Mol. Cryst. Liq. Cryst. 2 (1987) 539 –547. ∗ [20] W.E. Boyce, R.C. Di Prima, Elementary Di9erential Equations, Wiley, New York, 1992. [21] A. Bray, Theory of phase-ordering kinetics, Adv. Phys. 43 (1994) 357–459. [22] F. Brochard, P.G. de Gennes, Theory of magnetic suspensions in liquid crystals, J. Phys. (Paris) 31 (1970) 691–708. ∗∗ [23] R. Bubeck, C. Bechinger, S. Neser, P. Leiderer, Melting and reentrant freezing of two-dimensional colloidal crystals in con0ned geometry, Phys. Rev. Lett. 82 (1999) 3364–3367. [24] S.V. Burylov, Y.L. Raikher, Orientation of a solid particle embedded in a monodomain nematic liquid crystal, Phys. Rev. E 50 (1994) 358–367. [25] H.B. Callen, Thermodynamics and an Introduction to Thermostatistics, 2nd Edition, Wiley, New York, 1985. [26] S. Candau, P.L. Roy, F. Debeauvais, Magnetic 0eld e9ects in nematic and cholesteric droplets suspended in an isotropic liquid, Mol. Cryst. Liq. Cryst. 23 (1973) 283–297. [27] P. Chaikin, T.C. Lubensky, Principles of Condensed Matter Physics, Cambridge University Press, Cambridge, 1995. [28] S. Chandrasekhar, Stochastic problems in physics and astronomy, Rev. Mod. Phys. 15 (1943) 1–89. [29] S. Chandrasekhar, Liquid Crystals, 2nd Edition, Cambridge University Press, Cambridge, 1992. [30] S. Chandrasekhar, G. Ranganath, The structure and energetics of defects in liquid crystals, Adv. Phys. 35 (1986) 507–596. [31] S.-H. Chen, N.M. Amer, Observation of macroscopic collective behavior and new texture in magnetically doped liquid crystals, Phys. Rev. Lett. 51 (1983) 2298–2301. [32] S. Chono, T. Tsuji, Numerical simulation of nematic liquid crystalline >ows around a circular cylinder, Mol. Cryst. Liq. Cryst. 309 (1998) 217–236. [33] A.J. Chorin, A numerical method for solving incompressible viscous >ow problems, J. Comput. Phys. 2 (1967) 12–26. [34] I. Chuang, R. Durrer, N. Turok, B. Yurke, Cosmology in the laboratory: defect dynamics in liquid crystals, Science 251 (1991) 1336–1342. [35] P.E. Cladis, M. KlXeman, Non-singular disclinations of strength S =+1 in nematics, J. Phys. (Paris) 33 (1972) 591–598. [36] P.E. Cladis, M. KlXeman, P. PiXeranski, Sur une nouvelle mXethode de dXecoration de la mXesomorphe du p, n-mXethoxybenzilid_ene p-bXetylaniline (MBBA), C. R. Acad. Sci. Ser. B 273 (1971) 275–277. [37] P.E. Cladis, W. van Saarloos, P.L. Finn, A.R. Kortan, Dynamics of line defects in nematic liquid crystals, Phys. Rev. Lett. 58 (1987) 222–225. [38] H.J. Coles, Laser and electric 0eld induced birefringence studies on the cyanobiphenyl homologues, Mol. Cryst. Liq. Cryst. Lett. 49 (1978) 67–74. [39] P. Collings, Private communication, 1995. [40] G.P. Crawford, D.W. Allender, J.W. Doane, Surface elastic and molecular-anchoring properties of nematic liquid crystals con0ned to cylindrical cavities, Phys. Rev. A 45 (1992) 8693–8708. [41] G.P. Crawford, D.W. Allender, J.W. Doane, M. Vilfan, I. Vilfan, Finite molecular anchoring in the escaped-radial nematic con0guration: A 2 H-NMR study, Phys. Rev. A 44 (1991) 2570–2576.

468

H. Stark / Physics Reports 351 (2001) 387–474

\ [42] G.P. Crawford, R. Ondris-Crawford, S. Zumer, J.W. Doane, Anchoring and orientational wetting transitions of con0ned liquid crystals, Phys. Rev. Lett. 70 (1993) 1838–1841. \ [43] G.P. Crawford, R.J. Ondris-Crawford, J.W. Doane, S. Zumer, Systematic study of orientational wetting and anchoring at a liquid-crystal–surfactant interface, Phys. Rev. E 53 (1996) 3647–3661. \ [44] G.P. Crawford, S. Zumer (Eds.), Liquid Crystals in Complex Geometries, Taylor & Francis, London, 1996. [45] J.C. Crocker, D.G. Grier, When like charges attract: The e9ects of geometrical con0nement on long-range colloidal interactions, Phys. Rev. Lett. 77 (1996) 1897–1900. [46] P.K. Currie, Couette >ow of a nematic liquid crystal in the presence of a magnetic 0eld, Arch. Rational Mech. Anal. 37 (1970) 222–242. [47] P.K. Currie, Apparent viscosity during viscometric >ow of nematic liquid crystals, J. Phys. (Paris) 40 (1979) 501–505. [48] P.G. de Gennes, Short range order e9ects in the isotropic phase of nematics and cholesterics, Mol. Cryst. Liq. Cryst. 12 (1971) 193–214. [49] P.G. de Gennes, Nematodynamics, in: R. Balian, G. Weill (Eds.), Molecular Fluids, Gordon and Breach, London, 1976, pp. 373–400. [50] P.G. de Gennes, Interactions between solid surfaces in a presmectic >uid, Langmuir 6 (1990) 1448–1450. [51] P.G. de Gennes, J. Prost, The Physics of Liquid Crystals, 2nd Edition, International Series of Monographs on Physics, Vol. 83, Oxford Science, Oxford, 1993. [52] S.R. de Groot, Thermodynamics of Irreversible Processes, Selected Topics in Modern Physics, North-Holland, Amsterdam, 1951. [53] B.V. Derjaguin, Friction and adhesion. IV: The theory of adhesion of small particles, Kolloid Z. 69 (1934) 155–164. [54] B.V. Derjaguin, L. Landau, Theory of the stability of strongly charged lyophobic sols and the adhesion of strongly charged particles in solutions of electrolytes, Acta Physicochim. URSS 14 (1941) 633–662. [55] A.D. Dinsmore, D.T. Wong, P. Nelson, A.G. Yodh, Hard spheres in vesicles: Curvature-induced forces and particle-induced curvature, Phys. Rev. Lett. 80 (1998) 409–412. [56] A.D. Dinsmore, A.G. Yodh, D.J. Pine, Entropic control of particle motion using passive surface microstructures, Nature 383 (1996) 239–244. [57] A.C. Diogo, Friction drag on a sphere moving in a nematic liquid crystal, Mol. Cryst. Liq. Cryst. 100 (1983) 153–165. \ [58] J.W. Doane, N.A. Vaz, B.G. Wu, S. Zumer, Field controlled light scattering from nematic microdroplets, Appl. Phys. Lett. 48 (1986) 269–271. [59] I. Dozov, Private communication, 1999. [60] P.S. Drzaic, Liquid Crystal Dispersions, Series on Liquid Crystals, Vol. 1, World Scienti0c, Singapore, 1995. X [61] E. Dubois-Violette, O. Parodi, Emulsions nXematiques. E9ects de champ magnXetiques et e9ets piXezoXelectriques, J. Phys. (Paris) Coll. C4 30 (1969) 57–64. [62] R. Eidenschink, W.H. de Jeu, Static scattering in 0lled nematic: new liquid crystal display technique, Electron. Lett. 27 (1991) 1195. ∗ U [63] A. Einstein, Uber die von der molekularkinetischen Theorie der WUarme geforderte Bewegung von in ruhenden FlUussigkeiten suspendierten Teilchen, Ann. Phys. (Leipzig) 17 (1905) 549–560. [64] A. Einstein, Eine neue Bestimmung der MolekUuldimensionen, Ann. Phys. (Leipzig) 19 (1906) 289–306. [65] A. Einstein, Zur Theorie der Brownschen Bewegung, Ann. Phys. (Leipzig) 19 (1906) 371–381. \ [66] J.H. Erdmann, S. Zumer, J.W. Doane, Con0guration transition in a nematic liquid crystal con0ned to a small spherical cavity, Phys. Rev. Lett. 64 (1990) 1907–1910. [67] J.L. Ericksen, Anisotropic >uids, Arch. Rational Mech. Anal. 4 (1960) 231–237. [68] J.L. Ericksen, Theory of anisotropic >uids, Trans. Soc. Rheol. 4 (1960) 29–39. [69] J.L. Ericksen, Conservation laws of liquid crystals, Trans. Soc. Rheol. 5 (1961) 23–34. [70] J.L. Ericksen, Continuum theory of liquid crystals, Appl. Mech. Rev. 20 (1967) 1029–1032. [71] J.L. Ericksen, Continuum theory of liquid crystals of nematic type, Mol. Cryst. Liq. Cryst. 7 (1969) 153–164. [72] A.C. Eringen (Ed.), Continuum Physics: Vols. I–IV, Academic Press, New York, 1976. [73] A.C. Eringen, C.B. Kafadar, Part I: Polar 0eld theories, in: A.C. Eringen (Ed.), Continuum Physics: Polar and Nonlocal Field Theories, Vol. IV, Academic Press, New York, 1976, pp. 1–73.

H. Stark / Physics Reports 351 (2001) 387–474

469

[74] J. Fang, E. Teer, C.M. Knobler, K.-K. Loh, J. Rudnick, Boojums and the shapes of domains in monolayer 0lms, Phys. Rev. E 56 (1997) 1859–1868. [75] A.M. Figueiredo Neto, M.M.F. Saba, Determination of the minimum concentration of ferro>uid required to orient nematic liquid crystals, Phys. Rev. A 34 (1986) 3483–3485. [76] D. Forster, Hydrodynamic Fluctuations, Broken Symmetry, and Correlation Functions, Frontiers in Physics: A Lecture Note and Reprint Series, Vol. 47, W.A. Benjamin, Massachusetts, 1975. [77] D. Forster, T. Lubensky, P. Martin, J. Swift, P. Pershan, Hydrodynamics of liquid crystals, Phys. Rev. Lett. 26 (1971) 1016–1019. [78] S. Fraden, Phase transitions in colloidal suspensions of virus particles, in: M. Baus, L.F. Rull, J.P. Ryckaert (Eds.), Observation, Prediction, and Simulation of Phase Transitions in Complex Fluids, NATO Advanced Studies Institute Series C: Mathematical and Physical Sciences, Vol. 460, Kluwer, Dordrecht, 1995, pp. 113–164. [79] F.C. Frank, I. Liquid Crystals: On the theory of liquid crystals, Discuss. Faraday Soc. 25 (1958) 19–28. [80] G. Friedel, F. Grandjean, Observation gXeomXetriques sur les liquides aX coniques focales, Bull. Soc. Fr. Mineral 33 (1910) 409–465. [81] G. Friedel, Dislocations, Pergamon Press, Oxford, 1964. U [82] N. Fuchs, Uber die StabilitUat und Au>adung der Aerosole, Z. Phys. 89 (1934) 736–743. [83] J. Fukuda, H. Yokoyama, Director con0guration and dynamics of a nematic liquid crystal around a spherical particle: numerical analysis using adaptive grids, Eur. Phys. J. E., to be published. ∗ [84] C. GUahwiller, Direct determination of the 0ve independent viscosity coeNcients of nematic liquid crystals, Mol. Cryst. Liq. Cryst. 20 (1973) 301–318. [85] P. Galatola, J.B. Fournier, Nematic-wetted colloids in the isotropic phase, Mol. Cryst. Liq. Cryst. 330 (1999) 535–539. [86] A. Garel, Boundary conditions for textures and defects, J. Phys. (Paris) 39 (1978) 225–229. [87] E.C. Gartland, Private communication, 1998. [88] E.C. Gartland, S. Mkaddem, Instability of radial hedgehog con0gurations in nematic liquid crystals under Landau–de Gennes free-energy models, Phys. Rev. E 59 (1999) 563–567. [89] A.P. Gast, W.B. Russel, Simple ordering in complex >uids, Phys. Today 51 (1998) 24–30. [90] A.P. Gast, C.F. Zukoski, Electrorheological >uids as colloidal suspensions, Adv. Colloid Interface Sci. 30 (1989) 153–202. [91] A. Glushchenko, H. Kresse, V. Reshetnyak, Yu. Reznikov, O. Yaroshchuk, Memory e9ect in 0lled nematic liquid crystals, Liq. Cryst. 23 (1997) 241–246. [92] J. Goldstone, Field theories with “superconductor” solutions, Nuovo Cimento 19 (1) (1961) 154–164. [93] J. Goldstone, A. Salam, S. Weinberg, Broken symmetries, Phys. Rev. 127 (1962) 965–970. [94] J.W. Goodby, M.A. Waugh, S.M. Stein, E. Chin, R. Pindak, J.S. Patel, Characterization of a new helical smectic liquid crystal, Nature 337 (1989) 449–452. [95] J.W. Goodby, M.A. Waugh, S.M. Stein, E. Chin, R. Pindak, J.S. Patel, A new molecular ordering in helical liquid crystals, J. Am. Chem. Soc. 111 (1989) 8119–8125. [96] E. Gramsbergen, L. Longa, W.H. de Jeu, Landau theory of the nematic–isotropic phase transition, Phys. Rep. 135 (1986) 195–257. [97] Y. Gu, N.L. Abbott, Observation of saturn-ring defects around solid microspheres in nematic liquid crystals, Phys. Rev. Lett. 85 (2000) 4719 – 4722. ∗∗ [98] M.J. Guardalben, N. Jain, Phase-shift error as a result of molecular alignment distortions in a liquid–crystal point-di9raction interferometer, Opt. Lett. 25 (2000) 1171–1173. [99] Groupe d’Etude des Cristaux Liquides, Dynamics of >uctuations in nematic liquid crystals, J. Chem. Phys. 51 (1969) 816 –822. [100] P. HUanggi, P. Talkner, M. Borkovec, Reaction-rate theory: Fifty years after Kramers, Rev. Mod. Phys. 62 (1990) 251–341. [101] I. Haller, W.R. Young, G.L. Gladstone, D.T. Teaney, Crown ether complex salts as conductive dopants for nematic liquids, Mol. Cryst. Liq. Cryst. 24 (1973) 249–258. [102] W. Helfrich, Capillary >ow of cholesteric and smectic liquid crystals, Phys. Rev. Lett. 23 (1969) 372–374.

470

H. Stark / Physics Reports 351 (2001) 387–474

[103] G. Heppke, D. KrUuerke, M. MUuller, Surface anchoring of the discotic cholesteric phase of chiral pentayne systems, in: Abstract Book, Vol. 24, Freiburger Arbeitstagung FlUussigkristalle, Freiburg, Germany, 1995. [104] H. Heuer, H. Kneppe, F. Schneider, Flow of a nematic liquid crystal around a cylinder, Mol. Cryst. Liq. Cryst. 200 (1991) 51–70. [105] H. Heuer, H. Kneppe, F. Schneider, Flow of a nematic liquid crystal around a sphere, Mol. Cryst. Liq. Cryst. 214 (1992) 43–61. [106] R.G. Horn, J.N. Israelachvili, E. Perez, Forces due to structure in a thin liquid crystal 0lm, J. Phys. (Paris) 42 (1981) 39–52. [107] H. Imura, K. Okano, Friction coeNcient for a moving disclination in a nematic liquid crystal, Phys. Lett. A 42 (1973) 403–404. [108] A. JXakli, L. AlmXasy, S. BorbXely, L. Rosta, Memory of silica aggregates dispersed in smectic liquid crystals: E9ect of the interface properties, Eur. Phys. J. B 10 (1999) 509–513. [109] G.M. Kepler, S. Fraden, Attractive potential between con0ned colloids at low ionic strength, Phys. Rev. Lett. 73 (1994) 356–359. [110] R. Klein, Interacting brownian particles – the dynamics of colloidal suspensions, in: F. Mallamace, H.E. Stanley (Eds.), The Physics of Complex Systems, IOS Press, Amsterdam, 1997, pp. 301–345. [111] M. KlXeman, Points, Lines and Walls: in Liquid Crystals, Magnetic Systems, and Various Ordered Media, Wiley, New York, 1983. [112] H. Kneppe, F. Schneider, B. Schwesinger, Axisymmetrical >ow of a nematic liquid crystal around a sphere, Mol. Cryst. Liq. Cryst. 205 (1991) 9–28. [113] K. Ko\cevar, R. Blinc, I. Mu\sevi\c, Atomic force microscope evidence for the existence of smecticlike surface layers in the isotropic phase of a nematic liquid crystal, Phys. Rev. E 62 (2000) R3055 –R3058. ∗ [114] S. Komura, R.J. Atkin, M.S. Stern, D.A. Dunmur, Numerical analysis of the radial–axial structure transition with an applied 0eld in a nematic droplet, Liq. Cryst. 23 (1997) 193–203. \ [115] S. Kralj, S. Zumer, FrXeedericksz transitions in supra-
m nematic droplets, Phys. Rev. A 45 (1992) 2461–2470. [116] M. Krech, The Casimir E9ect in Critical Systems, World Scienti0c, Singapore, 1994. \ [117] M. Kreuzer, R. Eidenschink, Filled nematics, in: G.P. Crawford, S. Zumer (Eds.), Liquid Crystals in Complex Geometries, Taylor & Francis, London, 1996, pp. 307–324. [118] M. Kreuzer, T. Tschudi, R. Eidenschink, Erasable optical storage in bistable liquid crystal cells, Mol. Cryst. Liq. Cryst. 223 (1992) 219 –227. ∗ [119] O.V. Kuksenok, R.W. Ruhwandl, S.V. Shiyanovskii, E.M. Terentjev, Director structure around a colloid particle suspended in a nematic liquid crystal, Phys. Rev. E 54 (1996) 5198–5203. ∗∗ [120] M. Kurik, O. Lavrentovich, Negative-positive monopole transitions in cholesteric liquid crystals, JETP Lett. 35 (1982) 444–447. [121] M.V. Kurik, O.D. Lavrentovich, Defects in liquid crystals: Homotopy theory and experimental studies, Sov. Phys. Usp. 31 (1988) 196–224. [122] E. Kuss, pVT -data and viscosity-pressure behavior of MBBA and EBBA, Mol. Cryst. Liq. Cryst. 47 (1978) 71–83. [123] L.D. Landau, E.M. Lifschitz, Electrodynamics of Continuous Media, Course of Theoretical Physics, Vol. 8, Pergamon Press, Oxford, 0rst English Edition, 1960. [124] L.D. Landau, E.M. Lifschitz, Statistische Physik, Teil 1, Lehrbuch der Theoretischen Physik, Vol. 5, 6th Edition, Akademie, Berlin, 1984. [125] A.E. Larsen, D.G. Grier, Like-charge attractions in metastable colloidal crystallites, Nature 385 (1997) 230–233. [126] O. Lavrentovich, E. TerentXev, Phase transition altering the symmetry of topological point defects (hedgehogs) in a nematic liquid crystal, Sov. Phys. JETP 64 (1986) 1237–1244. [127] O.D. Lavrentovich, Topological defects in dispersed liquid crystals, or words and worlds around liquid crystal drops, Liq. Cryst. 24 (1998) 117–125. [128] F.M. Leslie, Some constitutive equations for anisotropic >uids, Quart. J. Mech. Appl. Math. 19 (1966) 357–370. [129] F.M. Leslie, Some constitutive equations for liquid crystals, Arch. Rational Mech. Anal. 28 (1968) 265–283.

H. Stark / Physics Reports 351 (2001) 387–474

471

[130] B.I. Lev, P.M. Tomchuk, Interaction of foreign macrodroplets in a nematic liquid crystal and induced supermolecular structures, Phys. Rev. E 59 (1999) 591–602. [131] S. Lifson, J.L. Jackson, On the self-di9usion of ions in a polyelectrolyte solution, J. Chem. Phys. 36 (1962) 2410–2414. [132] D. Link, N. Clark, Private communication, 1997. [133] J. Liu, E.M. Lawrence, A. Wu, M.L. Ivey, G.A. Flores, K. Javier, J. Bibette, J. Richard, Field-induced structures in ferro>uid emulsions, Phys. Rev. Lett. 74 (1995) 2828–2831. [134] H. LUowen, Kolloide – auch fUur Physiker interessant, Phys. Bl. 51 (3) (1995) 165–168. [135] H. LUowen, Solvent-induced phase separation in colloidal >uids, Phys. Rev. Lett. 74 (1995) 1028–1031. [136] H. LUowen, Phase separation in colloidal suspensions induced by a solvent phase transition, Z. Phys. B 97 (1995) 269–279. [137] J.-C. Loudet, P. Barois, P. Poulin, Colloidal ordering from phase separation in a liquid-crystalline continuous phase, Nature 407 (2000) 611– 613. ∗∗ [138] T.C. Lubensky, Molecular description of nematic liquid crystals, Phys. Rev. A 2 (1970) 2497–2514. [139] T.C. Lubensky, Hydrodynamics of cholesteric liquid crystals, Phys. Rev. A 6 (1972) 452–470. [140] T.C. Lubensky, D. Pettey, N. Currier, H. Stark, Topological defects and interactions in nematic emulsions, Phys. Rev. E 57 (1998) 610 – 625. ∗∗∗ [141] I.F. Lyuksyutov, Topological instability of singularities at small distances in nematics, Sov. Phys. JETP 48 (1978) 178–179. [142] W. Maier, A. Saupe, Eine einfache molekular-statistische Theorie der nematischen kristallin>Uussigen Phase. Teil I, Z. Naturforsch. Teil A 14 (1959) 882–889. [143] S.P. Meeker, W.C.K. Poon, J. Crain, E.M. Terentjev, Colloid-liquid-crystal composites: An unusual soft solid, Phys. Rev. E 61 (2000) R6083–R6086. ∗∗ [144] S. Meiboom, M. Sammon, W.F. Brinkman, Lattice of disclinations: The structure of the blue phases of cholesteric liquid crystals, Phys. Rev. A 27 (1983) 438–454. [145] N. Mermin, Surface singularities and super>ow in 3 He-A, in: S.B. Trickey, E.D. Adams, J.W. Dufty (Eds.), Quantum Fluids and Solids, Plenum Press, New York, 1977, pp. 3–22. [146] N.D. Mermin, The topological theory of defects in ordered media, Rev. Mod. Phys. 51 (1979) 591–648. [147] R.B. Meyer, Piezoelectric e9ects in liquid crystals, Phys. Rev. Lett. 22 (1969) 918–921. [148] R.B. Meyer, Point disclinations at a nematic-isotropic liquid interface, Mol. Cryst. Liq. Cryst. 16 (1972) 355 –369. ∗ [149] R.B. Meyer, On the existence of even indexed disclinations in nematic liquid crystals, Philos. Mag. 27 (1973) 405–424. [150] M. MiZesovicz, In>uence of a magnetic 0eld on the viscosity of para-azoxyanisol, Nature 136 (1935) 261. [151] MiZesowicz, The three coeNcients of viscosity of anisotropic liquids, Nature 158 (1946) 27. [152] V.P. Mineev, Topologically stable defects and solitons in ordered media, in: I.M. Khalatnikov (Ed.), Soviet Scienti0c Reviews, Section A: Physics Reviews, Vol. 2, Harwood, London, 1980, pp. 173–246. [153] O. Mondain-Monval, J.C. Dedieu, T. Gulik-Krzywicki, P. Poulin, Weak surface energy in nematic dispersions: saturn ring defects and quadrupolar interactions, Eur. Phys. J. B 12 (1999) 167–170. ∗∗ [154] L. Moreau, P. Richetti, P. Barois, Direct measurement of the interaction between two ordering surfaces con0ning a presmectic 0lm, Phys. Rev. Lett. 73 (1994) 3556–3559. [155] H. Mori, H. Nakanishi, On the stability of topologically non-trivial point defects, J. Phys. Soc. Japan 57 (1988) 1281–1286. [156] A.H. Morrish, The Physical Principles of Magnetism, Wiley Series on the Science and Technology of Materials, Wiley, New York, 1965. [157] V.M. Mostepanenko, N.N. Trunov, The Casimir E9ect and its Application, Clarendon Press, Oxford, 1997. [158] I. Mu\sevi\c, G. Slak, R. Blinc, Observation of critical forces in a liquid crystal by an atomic force microscope, in: Proceedings of the 16th International Liquid Crystal Conference, Abstract Book, Kent, USA, 1996, p. 91. [159] I. Mu\sevi\c, G. Slak, R. Blinc, Temperature controlled microstage for an atomic force microscope, Rev. Sci. Instrum. 67 (1996) 2554–2556. [160] F.R.N. Nabarro, Theory of Crystal Dislocations, The International Series of Monographs on Physics, Clarendon Press, Oxford, 1967.

472

H. Stark / Physics Reports 351 (2001) 387–474

[161] H. Nakanishi, K. Hayashi, H. Mori, Topological classi0cation of unknotted ring defects, Commun. Math. Phys. 117 (1988) 203–213. [162] D.H. Napper, Polymeric Stabilization of Colloidal Dispersions, Colloid Science, Vol. 3, Academic Press, London, 1983. [163] J. Nehring, A. Saupe, On the elastic theory of uniaxial liquid crystals, J. Chem. Phys. 54 (1971) 337–343. [164] D.R. Nelson, Defect-mediated Phase Transitions, in: C. Domb, J. Lebowitz (Eds.), Phase Transitions and Critical Phenomena, Vol. 7, Academic Press, New York, 1983, pp. 1–99. [165] M. Nobili, G. Durand, Disorientation-induced disordering at a nematic-liquid-crystal–solid interface, Phys. Rev. A 46 (1992) R6174–R6177. [166] C. Oldano, G. Barbero, An ab initio analysis of the second-order elasticity e9ect on nematic con0gurations, Phys. Lett. 110A (1985) 213–216. [167] C.W. Oseen, The theory of liquid crystals, Trans. Faraday Soc. 29 (1933) 883–899. [168] M.A. Osipov, S. Hess, The elastic constants of nematic discotic liquid crystals with perfect local orientational order, Mol. Phys. 78 (1993) 1191–1201. [169] S. Ostlund, Interactions between topological point singularities, Phys. Rev. E 24 (1981) 485–488. [170] O. Parodi, Stress Tensor for a nematic liquid crystal, J. Phys. (Paris) 31 (1970) 581–584. [171] E. Penzenstadler, H.-R. Trebin, Fine structure of point defects and soliton decay in nematic liquid crystals, J. Phys. France 50 (1989) 1027–1040. [172] V.M. Pergamenshchik, Surfacelike-elasticity-induced spontaneous twist deformations and long-wavelength stripe domains in a hybrid nematic layer, Phys. Rev. E 47 (1993) 1881–1892. [173] V.M. Pergamenshchik, P.I. Teixeira, T.J. Sluckin, Distortions induced by the k13 surfacelike elastic term in a thin nematic liquid-crystal 0lm, Phys. Rev. E 48 (1993) 1265–1271. [174] V.M. Pergamenshchik, K13 term and e9ective boundary condition for the nematic director, Phys. Rev. E 58 (1998) R16–R19. [175] D. Pettey, T.C. Lubensky, D. Link, Topological inclusions in 2D smectic C 0lms, Liq. Cryst. 25 (1998) 579 –587. ∗ [176] P. Pieranski, F. Brochard, E. Guyon, Static and dynamic behavior of a nematic liquid crystal in a magnetic 0eld. Part II: Dynamics, J. Phys. (Paris) 34 (1973) 35–48. [177] A. Poniewierski, T. Sluckin, Theory of the nematic-isotropic transition in a restricted geometry, Liq. Cryst. 2 (1987) 281–311. [178] P. Poulin, Private communication, 1999. [179] P. Poulin, V. Cabuil, D.A. Weitz, Direct measurement of colloidal forces in an anisotropic solvent, Phys. Rev. Lett. 79 (1997) 4862– 4865. ∗∗ [180] P. Poulin, N. Franc_es, O. Mondain-Monval, Suspension of spherical particles in nematic solutions of disks and rods, Phys. Rev. E 59 (1999) 4384 – 4387. ∗ [181] P. Poulin, V.A. Raghunathan, P. Richetti, D. Roux, On the dispersion of latex particles in a nematic solution. I. Experimental evidence and a simple model, J. Phys. II France 4 (1994) 1557–1569. ∗ [182] P. Poulin, H. Stark, T.C. Lubensky, D.A. Weitz, Novel colloidal interactions in anisotropic >uids, Science 275 (1997) 1770 –1773. ∗∗∗ [183] P. Poulin, D.A. Weitz, Inverted and multiple emulsions, Phys. Rev. E 57 (1998) 626 – 637. ∗∗∗ [184] P. Poulin, Novel phases and colloidal assemblies in liquid crystals, Curr. Opinion in Colloid & Interface Science 4 (1999) 66–71. [185] M.J. Press, A.S. Arrott, Theory and experiment of con0gurations with cylindrical symmetry in liquid-crystal droplets, Phys. Rev. Lett. 33 (1974) 403–406. [186] M.J. Press, A.S. Arrott, Elastic energies and director 0elds in liquid crystal droplets, I. Cylindrical symmetry, J. Phys. (Paris) Coll. C1 36 (1975) 177–184. [187] W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery, Numerical Recipes in Fortran: The Art of Scienti0c Computing, Cambridge University Press, Cambridge, 1992. [188] V.A. Raghunathan, P. Richetti, D. Roux, Dispersion of latex particles in a nematic solution. II. Phase diagram and elastic properties, Langmuir 12 (1996) 3789–3792. [189] V.A. Raghunathan, P. Richetti, D. Roux, F. Nallet, A.K. Sood, Colloidal dispersions in a liquid-crystalline medium, Mol. Cryst. Liq. Cryst. 288 (1996) 181–187.

H. Stark / Physics Reports 351 (2001) 387–474

473

[190] S. Ramaswamy, R. Nityananda, V.A. Raghunathan, J. Prost, Power-law forces between particles in a nematic, Mol. Cryst. Liq. Cryst. 288 (1996) 175 –180. ∗∗ [191] J. Rault, Sur une mXethode nouvelle d’Xetude de l’orientation molXeculaire aX la surface d’un cholestXerique, C. R. Acad. Sci. Ser. B 272 (1971) 1275–1276. [192] M. Reichenstein, T. Seitz, H.-R. Trebin, Numerical simulations of three dimensional liquid crystal cells, Mol. Cryst. Liq. Cryst. 330 (1999) 549–555. [193] S.R. Renn, T.C. Lubensky, Abrikosov dislocation lattice in a model of the cholesteric–to–smectic-A transition, Phys. Rev. A 38 (1988) 2132–2147. [194] V.G. Roman, E.M. Terentjev, E9ective viscosity and di9usion tensor of an anisotropic suspension or mixture, Colloid J. USSR 51 (1989) 435–442. [195] R. Rosso, E.G. Virga, Metastable nematic hedgehogs, J. Phys. A 29 (1996) 4247–4264. [196] D. Rudhardt, C. Bechinger, P. Leiderer, Direct measurement of depletion potentials in mixtures of colloids and nonionic polymers, Phys. Rev. Lett. 81 (1998) 1330–1333. [197] A. RUudinger, H. Stark, Twist transition in nematic droplets: A stability analysis, Liq. Cryst. 26 (1999) 753–758. [198] R.W. Ruhwandl, E.M. Terentjev, Friction drag on a cylinder moving in a nematic liquid crystal, Z. Naturforsch. Teil A 50 (1995) 1023–1030. [199] R.W. Ruhwandl, E.M. Terentjev, Friction drag on a particle moving in a nematic liquid crystal, Phys. Rev. E 54 (1996) 5204 –5210. ∗∗ [200] R.W. Ruhwandl, E.M. Terentjev, Long-range forces and aggregation of colloid particles in a nematic liquid crystal, Phys. Rev. E 55 (1997) 2958–2961. ∗ [201] R.W. Ruhwandl, E.M. Terentjev, Monte Carlo simulation of topological defects in the nematic liquid crystal matrix around a spherical colloid particle, Phys. Rev. E 56 (1997) 5561–5565. ∗∗ [202] W.B. Russel, D.A. Saville, W.R. Schowalter, Colloidal Dispersions, Cambridge University Press, Cambridge, 1995. [203] G. Ryskin, M. Kremenetsky, Drag force on a line defect moving through an otherwise undisturbed 0eld: Disclination line in a nematic liquid crystal, Phys. Rev. Lett. 67 (1991) 1574–1577. [204] S.D. Sarma, A. Pinczuk (Eds.), Perspectives in Quantum Hall Fluids, Wiley, New York, 1997. [205] M. Schadt, Optisch strukturierte FlUussigkristall-Anzeigen mit groaem Blickwinkelbereich, Phys. Bl. 52 (7=8) (1996) 695–698. [206] M. Schadt, H. Seiberle, A. Schuster, Optical patterning of multi-domain liquid-crystal displays with wide viewing angles, Nature 318 (1996) 212–215. [207] N. Schopohl, T.J. Sluckin, Defect core structure in nematic liquid crystals, Phys. Rev. Lett. 59 (1987) 2582–2584. [208] N. Schopohl, T.J. Sluckin, Hedgehog structure in nematic and magnetic systems, J. Phys. France 49 (1988) 1097–1101. [209] V. Sequeira, D.A. Hill, Particle suspensions in liquid crystalline media: Rheology, structure, and dynamic interactions, J. Rheol. 42 (1998) 203–213. [210] P. Sheng, Phase transition in surface-aligned nematic 0lms, Phys. Rev. Lett. 37 (1976) 1059–1062. [211] P. Sheng, Boundary-layer phase transition in nematic liquid crystals, Phys. Rev. A 26 (1982) 1610–1617. [212] P. Sheng, E.B. Priestly, The Landau–de Gennes theory of liquid crystal phase transition, in: E.B. Priestly, P.J. Wojtowicz, P. Sheng (Eds.), Introduction to Liquid Crystals, Plenum Press, New York, 1979, pp. 143–201. [213] S.V. Shiyanovskii, O.V. Kuksenok, Structural transitions in nematic 0lled with colloid particles, Mol. Cryst. Liq. Cryst. 321 (1998) 45–56. [214] R. Sigel, G. Strobl, Static and dynamic light scattering from the nematic wetting layer in an isotropic crystal, Prog. Colloid Polym. Sci. 104 (1997) 187–190. [215] M. Smoluchowski, Versuch einer mathematischen Theorie der Koagulationkinetik kolloider LUosungen, Z. Phys. Chem. (Leipzig) 92 (1917) 129–168. [216] K. Sokalski, T.W. Ruijgrok, Elastic constants for liquid crystals of disc-like molecules, Physica A 113A (1982) 126–132. [217] A. Sommerfeld, Vorlesungen uU ber Theoretische Physik II. Mechanik der deformierbaren Medien, 6th Edition, Verlag Harri Deutsch, Frankfurt, 1978.

474

H. Stark / Physics Reports 351 (2001) 387–474

[218] A. Sonnet, A. Kilian, S. Hess, Alignment tensor versus director: Description of defects in nematic liquid crystals, Phys. Rev. E 52 (1995) 718–722. [219] H. Stark, Director 0eld con0gurations around a spherical particle in a nematic liquid crystal, Eur. Phys. J. B 10 (1999) 311–321. ∗∗ [220] H. Stark, J. Stelzer, R. Bernhard, Water droplets in a spherically con0ned nematic solvent: A numerical investigation, Eur. Phys. J. B 10 (1999) 515 –523. ∗ [221] J. Stelzer, M.A. Bates, L. Longa, G.R. Luckhurst, Computer simulation studies of anisotropic systems. XXVII. The direct pair correlation function of the Gay–Berne discotic nematic and estimates of its elastic constants, J. Chem. Phys. 107 (1997) 7483–7492. [222] K.J. Strandburg, Two-dimensional melting, Rev. Mod. Phys. 60 (1988) 161–207. [223] B.D. Swanson, L.B. Sorenson, What forces bind liquid crystal, Phys. Rev. Lett. 75 (1995) 3293–3296. [224] G.I. Taylor, The mechanism of plastic deformation of crystals, Proc. R. Soc. London, Ser. A 145 (1934) 362–415. [225] E.M. Terentjev, Disclination loops, standing alone and around solid particles, in nematic liquid crystals, Phys. Rev. E 51 (1995) 1330 –1337. ∗∗ [226] H.-R. Trebin, The topology of non-uniform media in condensed matter physics, Adv. Phys. 31 (1982) 195–254. [227] E.H. Twizell, Computational Methods for Partial Di9erential Equations, Chichester, Horwood, 1984. [228] D. Ventzki, H. Stark, Stokes drag of particles suspended in a nematic liquid crystal, in preparation. [229] E.J.W. Verwey, J.T.G. Overbeek, Theory of the Stability of Lyophobic Colloids, Elsevier, Amsterdam, 1948. [230] D. Vollhardt, P. WUol>e, The Phases of Helium 3, Taylor & Francis, New York, 1990. [231] G.E. Volovik, O.D. Lavrentovich, Topological dynamics of defects: Boojums in nematic drops, Sov. Phys. JETP 58 (1983) 1159–1168. [232] T.W. Warmerdam, D. Frenkel, R.J.J. Zijlstra, Measurements of the ratio of the Frank constants for splay and bend in nematics consisting of disc-like molecules—2,3,6,7,10,11-hexakis(p-alkoxybenzoyloxy)triphenylenes, Liq. Cryst. 3 (1988) 369–380. [233] Q.-H. Wei, C. Bechinger, D. Rudhardt, P. Leiderer, Experimental study of laser-induced melting in two-dimensional colloids, Phys. Rev. Lett. 81 (1998) 2606–2609. [234] A.E. White, P.E. Cladis, S. Torza, Study of liquid crystals in >ow: I. Conventional viscometry and density measurements, Mol. Cryst. Liq. Cryst. 43 (1977) 13–31. [235] J. Wilks, D. Betts, An Introduction to Liquid Helium, Clarendon Press, Oxford, 1987. [236] C. Williams, P. PieraXnski, P.E. Cladis, Nonsingular S = +1 screw disclination lines in nematics, Phys. Rev. Lett. 29 (1972) 90–92. [237] R.D. Williams, Two transitions in tangentially anchored nematic droplets, J. Phys. A 19 (1986) 3211–3222. [238] K. Zahn, R. Lenke, G. Maret, Two-stage melting of paramagnetic colloidal crystals in two dimensions, Phys. Rev. Lett. 82 (1999) 2721–2724. [239] M. Zapotocky, L. Ramos, P. Poulin, T.C. Lubensky, D.A. Weitz, Particle-stabilized defect gel in cholesteric liquid crystals, Science 283 (1999) 209 –212. ∗∗ \ [240] P. Ziherl, R. Podgornik, S. Zumer, Casimir force in liquid crystals close to the nematic-isotropic phase transition, Chem. Phys. Lett. 295 (1998) 99–104. [241] J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, 3rd Edition, International Series of monographs on Physics, Vol. 92, Oxford Science, Oxford, 1996. [242] H. ZUocher, The e9ect of a magnetic 0eld on the nematic state, Trans. Faraday Soc. 29 (1933) 945–957.

475

CONTENTS VOLUME 351 B. Gumhalter. Single- and multiphonon atom}surface scattering in the quantum regime

1

H. Heiselberg. Event-by-event physics in relativistic heavy-ion collisions

161

G.E. Volovik. Super#uid analogies of cosmological phenomena

195

C. Ronning. E.P. Carlson, R.F. Davis. Ion implantation into gallium nitride

349

H. Stark. Physics of colloidal dispersions in nematic liquid crystals

387

PII: S0370-1573(01)00055-2

476

FORTHCOMING ISSUES D.G. Yakovlev, A.D. Kaminker, O.Y. Gnedin, P. Haensel. Neutrino emission from neutron stars B. Ananthanarayan, G. Colangelo, J. Gasse, H. Leutwyler. Roy analysis of pi-pi scattering R. Alkofer, L. von Smekal. The infrared behaviour of QCD Green's functions J.-P. Minier, E. Peirano. The PDF approach to turbulent polydispersed two-phase #ows D.F. Measday. The nuclear physics of muon capture C. Schubert. Perturbative quantum "eld theory in the string-inspired formalism M. Bordag, U. Mohideen, V.M. Mostepanenko. New developments in the Casimir e!ect G.E. Mitchell, J.D. Bowman, S.I. PenttilaK , E.I. Sharapov. Parity violation in compound nuclei: experimental methods and results G. Grynberg, C. Robilliard. Cold atoms in dissipative optical lattices I. Pollini, A. Mosser, J.C. Parlebas. Electronic, spectroscopic and elastic properties of early transition metal compounds R. Fazio, H. van der Zant. Quantum phase transitions and vortex dynamics in superconducting networks D.I. Pushkarov. Quasiparticle kinetics and dynamics in nonstationary deformed crystals in the presence of electromagnetic "elds V.M. Shabaev. Two-time Green's function method in quantum electrodynamics of high-Z few-electron atoms G. Bo!etta, M. Cencini, M. Falcioni, A. Vulpiani. Predictability: a way to characterize complexity A. Wacker. Semiconductor superlattices: a model system for nonlinear transport I.L. Shapiro. Physical aspects of the space}time torsion Ya. Kraftmakher. Modulation calorimetry and related techniques M.J. Brunger, S.J. Buckman. Electron}molecule scattering cross sections. I. Experimental techniques and data for diatomic molecules S.Y. Wu, C.S. Jayanthi. Order-N methodologies and their applications

PII: S0370-1573(01)00056-4