Fundamentals of Interface and Colloid Science, Volume V: Soft Colloids (Fundamentals of Interface and Colloid Science)

XiX ^)

interfacial potential j u m p (between phases B and a ) (V)

Xe

electric susceptibility (-)

s

X

a d s o r p t i o n energy p a r a m e t e r (-) -^crit

critical value of Xs at t h e a d s o r p t i o n / d e s o r p t i o n point (-)

i//

electric potential (V)

at

angular frequency (rad s" 1 or s"1) coi

angular frequency of incident radiation (rad s~' or s~')

a>s

angular frequency of s c a t t e r e d r a d i a t i o n (rad s^ 1 or s^ 1 )

a)

pair interaction ( J ) (ch. IV.5)

co{r)

local grand potential density (J rrr 3 ) (ch. V.4)

<w(z)

local grand potential density (J n r 3 ) (ch. V. 1)

co

d e g e n e r a c y o f s u b s y s t e m (-)

Q

d e g e n e r a c y o f a s y s t e m o r n u m b e r o f r e a l i z a t i o n s (-)

Q

a>s -coi ( r a d s" 1 o r s" 1 )

Q

solid angle (sr)

Q

grand potential (J)

xxxviii

£F 12^ Q{N,V,U)

SYMBOL LIST

interfacial (excess) grand potential (J) interfacial (excess) grand potential per unit area (J irr 2 ) n u m b e r of realizations = microcanonical partition function (-)

EFFECT OF POLYMERS ON THE INTERACTION BETWEEN COLLOIDAL PARTICLES

Gerard Fleer, Martien Cohen Stuart and Frans Leermakers 1.1

Introduction

1.1

1.2

Summary of solution properties

1.4

1.3

The interfacial Gibbs energy

1.5

1.4

1.3a

General

1.5

1.3b

Local and non-local contributions

1.6

SCF theory

1.7

1.4a

Continuum description

1.7

1.4b

Discrete description

1.4c

Ground-state approximation

1.4d 1.4e 1.5

1.6

1.7

1.8

1.9

1.9 1.11

Expressions for the local and non-local grand potential densities

1.12

Square gradient from SCF

1.13

The concentration profile

1.14

1.5a

Numerical integration

1.14

1.5b

Semidilute solutions

1.15

1.5c

Dilute solutions

1.17

Disjoining pressure and Gibbs energy of interaction

1.20

1.6a

General

1.20

1.6b

Attraction due to bridge formation

1.21

1.6c

Repulsion due to tails

1.24

1.6d

Total Gibbs energy of interaction

1.25

Numerical examples for adsorption

1.26

1.7a

Interaction curves

1.26

1.7b

Weak overlap

1.28

1.7c

Strong overlap

1.30

Depletion at flat surfaces

1.32

1.8a

Introduction

1.32

1.8b

Depletion thickness and the profile at a single surface

1.34

1.8c

Interfacial Gibbs energy for a single surface

1.37

1.8d

Gibbs energy of interaction

1.38

Depletion around a sphere

1.40

1.9a

Volume fraction profile

1.40

1.9b

Depletion thickness around a sphere

1.41

1.9c

Interfacial Gibbs energy of a sphere

1.42

1.9d

Interaction between two spheres

1.43

1.10

Adsorption versus depletion

1.45

1.11

Tethered polymers and polymer brushes

1.48

1.11a

Introduction

1.48

1.11b

Where do polymer brushes occur?

1.50

1.11c

Thermodynamics

1.52

1.1 Id Isolated grafted chains on a single surface: mushrooms

1.53

1.1 le

Brushes on a single surface

1.55

1.1 If

Interaction between surfaces covered with mushrooms

1.64

1.1 lg

Interaction between surfaces with brushes

1.66

1.1 Ih Soft depletion 1.1 li 1.12

Concluding remarks

Non-equilibrium aspects

1.73 1.75 1.76

1.12a

Introduction

1.76

1.12b

Does equilibrium apply?

1.77

1.12c

Restricted equilibrium

1.78

1.12d

Relaxation phenomena in polymer-induced surface forces

1.81

1.12e

Case study: kinetics of flocculation

1.83

1.13

Outlook

1.89

1.14

General References

1.92

1 EFFECT OF POLYMERS ON THE INTERACTION BETWEEN COLLOIDAL PARTICLES GERARD FLEER, MARTIEN COHEN STUART, AND FRANS LEERMAKERS

1.1 Introduction

In this chapter we discuss colloidal interactions induced by uncharged polymer. The interaction may be caused by adsorbed layers in which the concentration is higher than in the bulk solution or by depletion layers where the concentration is lower. In general, these layers are inhomogeneous. The surfaces may be flat or curved. For the adsorption case, where the general theory is more difficult than for depletion, we restrict ourselves to pair interactions between two (identical) flat plates, so that the results are only applicable to relatively large colloid particles at low concentrations. For depletion, we can also deal with spherical particles at various size ratios of polymer coil and colloid. At higher colloid and polymer concentrations phase separation may occur. We do not elaborate on this issue but refer to sec. IV.5.6 where such phase diagrams were discussed. When calculating the Gibbs energy of interaction, one needs to know to what extent an exchange of material is allowed during particle approach. For interacting depletion layers, the assumption of full equilibrium (constant chemical potential of both polymer and solvent in the gap) is usually justified. For adsorption layers this is only the case when, during particle approach, the adsorbed polymer desorbs sufficiently fast: if it would not desorb its chemical potential would increase. In most cases we will also assume full equilibrium for adsorption layers (although some remarks about restricted equilibrium, e.g., for a constant amount of polymer, will be made). In such a full equilibrium, the (change in) interfacial Gibbs energy is determined by the (change in) the grand potential Q= yA - pV = F -n^ -nfi, see [III.2.2.18, 19]. Here, F is the Helmholtz energy of the interfacial system containing rtj solvent molecules (with chemical potential ju^) and n polymer segments (chemical potential fi per segment). In most cases, we omit the index 2 (or p) for the polymer. On the basis of this definition, Q is an excess quantity with respect to the bulk. It can be computed from the concentration profile using a suitable model. In some cases, not the grand potential Q but the semi-grand potential for a semiopen (so) system £T°, is relevant. This is the case when the polymer cannot leave the Fundamentals of Interface and Colloid Science, Volume V J. Lyklema (Editor)

© 2005 Elsevier Ltd. All rights reserved

1.2

INTERACTION BETWEEN POLYMER LAYERS

gap, as for brushes and irreversibly adsorbed polymer. This situation is often denoted as restricted equilibrium, and resembles the case of constant surface charge for two interacting double layers (as compared with the full equilibrium case of constant chemical potential). Then // is not constant during particle approach, and i2 so is defined as £>so = F - n1/.il. When /3 S0 , rather than Q determines the interfacial Gibbs energy, the interaction between surfaces is necessarily more repulsive because any deviation from full equilibrium increases the interfacial Gibbs energy. This occurs, for example, in the case of interacting brushes, which are discussed In section 1.11. Also, in some equilibrium situations i2so is relevant, for example, in the phase behaviour of colloids in the presence of non-adsorbing polymer. We try to understand the interactions from a fundamental point of view. That implies that we start from basic principles, using theoretical models where simplifying assumptions are made. We will often rely on mean-field models where chain swelling in the bulk solution is neglected. A very versatile model, applicable to a wide range of conditions, is the lattice model of Scheutjens and Fleer (SF); the principles of this model for isolated surfaces were outlined in sec. II.5.5. In that section no expressions for state functions, like the Helmholtz energy and the grand potential, were given, but in order to describe interactions we clearly have to do that here. The SF model can be applied in a wide variety of situations, but has the disadvantage that only numerical results are obtained, which makes it difficult to analyse the trends. It may be seen as the discrete version of the Edwards continuum theory, which starts from an equation (see [1.4.1] below) that resembles the time-dependent Schrodinger equation. In general, this Edwards equation also has to be solved numerically, whereby some discretisation is necessary in the numerical scheme. The boundary condition at the surface deserves special attention, as variations in segment concentration over a length scale smaller than the segment length are unrealistic. This situation resembles that of a Stern layer which is introduced in the Poisson-Boltzmann continuum description to account for the finite size of the ions. In the discrete SF model, the finite segment size is automatically accounted for and the surface boundary condition is easier to handle. However, the Edwards equation can be used as a convenient starting point for obtaining analytical expressions by applying the so-called ground-state approximation (GSA). The basis of GSA will be explained in sec. 1.4c. We will give several examples of this GSA approach for different regimes. We also address the relation between these models and the Van der Waals-CahnHilliard theory (sees. III.2.5 and 6), which expresses the Helmholtz energy In terms of the square gradient of the concentration profile. We will see that this form of the Helmholtz energy also follows from the SF model when GSA-type approximations are made. The kinds of issues dealt with in this chapter are sketched in fig. 1.1. Analogous to [IV.3.2.3) for double layers, the Gibbs energy of interaction per unit area is defined by

INTERACTION BETWEEN POLYMER LAYERS Ga(ft) = 2[7(h)-rM] = 2[a>(/i)-.aa(oo)]

1.3 [1.1.1]

where 2y(h) is the interfacial Gibbs energy corresponding to the bottom diagrams of fig. 1.1, and 2y{°°) to those in the top diagrams. Analogously, Qa(h) = £2{h)IA is the corresponding grand potential per unit area. Note that y{h), &a(h) and A are defined per plate. Note also that in this chapter we make no distinction between y (which in principle should be measurable) and G° (which is not directly measurable for small h or solid surfaces) .

Figure 1.1. Volume fraction profiles in a slit between two surfaces at large separation (top) and at small separation (bottom), both for adsorption (left) and depletion (right). At large h , the volume fraction Pb f° r adsorption and
for the left (isolated) plate plus


' As shown in fig. 1.1, in this chapter the separation between the two surfaces is taken as 2h .

1.4

INTERACTION BETWEEN POLYMER LAYERS

bottom diagrams of fig. 1.1 as the dashed curves. The convention of putting z = 0 in the middle is convenient for narrow slits, but not for wide slits where cpm = cph . In that case we return to z = 0 at the (left) surface. In some cases it is easier to calculate the force per unit area, also known as the disjoining pressure I7(h). We write [IV.3.1.1] simply as

m)=

dGa (h)

=

dna (h)

-i£r —it1

[M 21

-

whereby we implicitly assume that the differentiation is carried out at constant temperature and pressure. When 17(h) is known, the Gibbs energy of interaction is found by integrating the disjoining pressure h

Ga(h) = -2f/7(h')dh'

[1.1.3]

In all cases we consider incompressible systems, so that there is no need to distinguish between the Helmholtz energy F and the Gibbs energy G . 1.2 Summary of solution properties In sec. II.5.2 we discussed the solution properties of polymers in some detail. Here we summarize a few important concepts needed in the present chapter. Unlike in II.5.2, where we used volume fractions 2 for polymer, we use only one volume fraction

[1.2.1]

where V is the volume of the system. From III.5.2.11, 12] we have

11

P.J. Flory, Principles of Polymer Chemistry, Cornell Univ. Press, Ithaca, New York, 1953.

INTERACTION BETWEEN POLYMER LAYERS f3 f m, ^ = -^ln
1.5

+ X
[1-2.2]

The first term in [1.2.2] is the translational entropy of the chains (small for large N ), the second is the solvent translational entropy, and the third term, in which % is the Flory-Huggins parameter, is the contact interaction term (positive when ^ > 0). In an athermal solution % = 0 , in a theta solvent % = 0.5 . By differentiating F = JV with respect to the number of solvent molecules, the solvent chemical potential fi^ (with respect to pure solvent) is found:

where 77OS is the osmotic pressure (the superscript

os

is used in order to avoid

confusion with the disjoining pressure 77). It is sometimes useful to expand the logarithm:

iZ!!^ = _ i l = ^ kT

kT

N

+

I^2+I^3+... 2

Yh

B= 1

3rb

Hence, 77OS increases monotonically with ^

_2

[L2.4]

for v > 0 . The first term is the ideal Van

't Hoff term, the quadratic term dominates in (mean-field) semidilute good solvents, and the cubic term is important in semidilute theta solvents (^ = 1/2, v = 0). Similarly, the segment chemical potential (with respect to pure amorphous polymer) is found by differentiating F = JV with respect to the number of segments:

^ =^ l n P b - [ l - l ] ( l - ^ )

+ Z(l-%)

2

[1.2.5]

We will also need the quantity d77 o s /d^, which is inversely proportional to the osmotic compressibility. It is given by ^d77^ kT Acp

=

J_ ^ b _ _ JV l-
^

b

1 N

, Vh

Vh

1.3 The interfacial Gibbs energy

1.3a General We compare a mixture of solvent and polymer in a gap between two identical surfaces (each with interfacial Gibbs energy y) with a reference state of unmixed components where each surface is in contact with pure solvent (interfacial Gibbs energy y*). According to [III.2.2.19], both yA and y*A are given by the sum of pV and the grand potential Q. When we define Q with respect to this reference state, we

1.6

INTERACTION BETWEEN POLYMER LAYERS

may write

r-r*=f = ^a

11-3.11

since the pV term cancels when there is no volume change upon mixing. For noninteracting surfaces, the difference y - y* is equal to minus the surface pressure. From the definition [III.2.2.18] we have Q = F - n^ - nfj , where F = A\ J[
h

na(h) = ja)(z)dz o

[1.3.2]

where co(z) is given by: l3co = C3J-(l-
U-3.3]

We stress that w, / , and

are separated into local and non-local contributions: / =/loc

+

j-nloc

ft,=

ft^oc+ft;nloc

[1.3.4]

For the local contribution to the free energy density the bulk expression (1.2.2] is used but with the bulk concentration
INTERACTION BETWEEN POLYMER LAYERS

^3 floe

—kT

1.7

„

= — \n) + ym(\ -
[1.3.5]

where

[ 1 3 6 ]

The non-local part, &>nloc in [1.3.6], contains two contributions: (£) a square gradient contribution xid^/dz) 2 as in the Cahn-Hilliard theory, see [III.2.6.5], where the pre-factor K is to be determined, and [li) the contact interaction between adsorbed segments and the adsorbent (which is not included in
^Ah) = -ro
l1 3 71

--

-I

We will see that for adsorbing polymer the first term determines the (negative) sign of i2a (h); the integral constitutes a positive correction. We will also see that in a GSAdescription for polymers K equals {24f.(p)~i (in units kT 113 ) and show that for h —> ~ the two terms in the integral are exactly equal (as in [III.2.6.6]), and that they are of the same order of magnitude for interacting surfaces (finite h ). 1.4 SCF theory 1.4a Continuum. Description The starting point of continuum self-consistent-field (SCF) theories is the Edwards equation1 , which is a Schrodinger-like differential equation for the end-point distribution function G[z,s). This function gives the statistical weight that walks of (contour) length s (in units of O in a field U{z) end up at position z . The Edwards equation may be written as

11

S.F. Edwards, Proc. Phys. Soc. 85 (1965) 613.

1.8

INTERACTION BETWEEN POLYMER LAYERS

— V2G = ~+uG 6 as

u^

^

[1.4.1]

kT

where U = U{z) is the local potential energy per segment in the gradient and Ub = U{°°) is the same in the bulk solution. Both U and Ub are defined with respect to infinite dilution (i.e. segments only in contact with solvent). In the bulk solution, all three terms of [1.4.1] vanish, and G(~,s) = l for any s. The combination (U -Ub)lkT occurs so often that we introduce a separate symbol u for this dimensionless field with respect to the bulk solution. The self-consistent field U(z) is determined by the local concentrations and the solvency parameter % > ^

an

d can be written as 1'

= -ln[l-0Z)]-2^(z)

^J2-= _ i n ( i _ , p b ) _ 2 m

[1.4.2a,b]

u = - l n — ' - ^ - - 2X{(p - (ph) ^ v{(p- (ph) + \{(p-
which

is

[II.5.5.21]

for

a

lattice

without

the adsorption

[1.4.2c]

term

and with

(#>(z)) = [(p[z-l) + 4(p{z) + (p(z + £)]/6 approximated as
!£ G dz

=-c

[1.4.3,

z=0

where 1/c is the so-called extrapolation length, which is related to the adsorption energy parameter ^ s as defined in [II.5.4.1]. The meaning of 1/c is illustrated in fig. 1.2. On the basis of the definition of G , G(z, N) is proportional to the volume fraction (pe(z) =
11 K.M. Hong, J. Noolandi, Macromolecules, 14 (1981) 7271 ibid 14 (1981) 1229 ; G.J. Fleer, M.A. Cohen Stuart, J.M.H.M. Scheutjens, T. Cosgrove and B. Vincent, Polymers at Interfaces, Chapman and Hall, 1993. 2) P.G. De Gennes, Scaling Concepts in Polymer Physics, Cornell Univ.Press, 1979.

INTERACTION BETWEEN POLYMER LAYERS

1.9

N

pp=


=G(z,N)

p = ^ £ l = i . f G(z,s)G(z,N-s)ds

[1.4.4]

In general, [1.4.1-4] have to be solved numerically to find the field and the volumefraction profile (which should be mutually consistent). For analytical expressions further approximations (such as GSA) are needed. Before discussing those, we summarize the discrete SCF model by Scheutjens and Fleer (which was treated in more detail in sec. II.5.5).

Figure 1.2. The extrapolation length 1/c, which is positive for adsorption (left) and negative for depletion (right), is found by extrapolating the initial slope of the end-point distribution G as a function of z. The surface is situated at z = 0 . When c = 0, the extrapolation length is infinite and G = 1 everywhere (critical point, adsorption/desorption transition). 1.4b Discrete

description

The central equation in the SF lattice model is the recurrency relation [II.5.5.7], which we write here in the form G(z,s +1) = e

u(zl

!-G(z-t.s) + -G(z,s) + -G(z + (,s)\ 6 6 6

[1.4.5]

The sum of the three terms between braces is often abbreviated as (G(z,s)}. The exact SF form of u (z) is [1.4.2c] with x{z)) (the energy is 'non-local'), but we disregard this refinement here. When this form of u (which contains no ;fsterm) is used, [1.4.5] applies to all lattice layers except z = 0 , where the train segments are located. For z = 0 , u(z) in the SF model does include xs a s a n e x tra term. It is easily shown that when the unit increments As = 1 and Az = C in [1.4.5] are allowed to approach zero (which implies that u for a unit segment has to be replaced by II As for a 'fractional segment'), [1.4.5] reduces to the continuum form [1.4.1]. We do not discuss here the more difficult question of how to relate c in [1.4.3] to xs i n the lattice model; for this we refer to the literature 1'. 11

G.J. Fleer, J. van Male, and A. Johner, Macromolecules, 32 (1999) 825; ibid, 32 (1999) 845; A.A. Gorbunov, A.M. Skvortsov, J. Van Male, and G.J. Fleer, J. Chem. Phys. 114 (2001) 5366; G.J. Fleer, F.A.M. Leermakers, in Coagulation and Flocculation: Theory and Applications, 2nd ed., Surfactant Science Series, B. Dobias and H.J. Stechemesser, Eds., M. Dekker (2004).

1.10

INTERACTION BETWEEN POLYMER LAYERS

Equations [ 1.4.4] are also used in the lattice model, obviously in a discretized form, see [II.5.5.9-11 ]. There is a minor difference in [ 1.4.4b] as compared with [II.5.5.9] (the factor Gj(z) in the latter equation, which corrects for double-counting of the joining segment, is not present in the continuum version), but that is only an unimportant detail. Much more important is the fact that the SF model gives a generalized expression for

the Helmholtz

energy density in a gradient, which includes

the non-local

contributions. This generalization of [ 1.2.2] is e3 f 0 -^- = —lncp+

(1 -
11.4.6]

Again, we have used the local energy approximation; in the exact SCF equation u and the last term of [1.4.6] contain %((p) rather than %
kr

. 1 = In

-

0 f,

I V

i

r 9

ol

—+ 1 0-0h\ +X x\

[147]

We have introduced the symbol 7b for the dimensionless grand potential density. When (ph, [1.4.7] gives a negative value for co (provided v > 0 , % < 0.5 ). We can write [ 1.4.7] in terms of the osmotic pressure profile by introducing a local osmotic pressure n°s (z), defined by [ 1.2.3] but with
n°s(z)c3


3

|j4 8 ]

This local osmotic pressure cannot be measured independently and is just an easy abbreviation.

When we denote

/7OS in [1.2.3] by

/7g s , we can now write

co(z) = /7gs - /7OS (z). Combining this with [ 1.3.2] we have h

h

na (h) = [adz = J(/7° s - /7OS (z))dz 0

[ 1.4.9]

0

This equation, with [1.4.7] for co or [1.2.3] for /7gs and [1.4.8] for /7 0 S (z), is, apart from the local energy approximation, exact. The second version of [1.4.9] immediately

INTERACTION BETWEEN POLYMER LAYERS

1.11

shows that £?a > 0 ( y > y*) for depletion, and Oa < 0 ( y < y* ) for adsorption. It also gives an interesting analogy to [III.2.3.4, 5], where the interfacial tension is written as an integral over p- p t , with p the (normal) pressure and p t the tangential pressure in the interfacial region.

1.4c Ground-state

approximation

So far all equations are general (within the mean-field approximation), but one needs to solve for the polymer concentration profile before one can apply them. Such a profile follows in principle from the Edwards equation [1.4.1]. Except for some ideal situations where the field is absent, there are no exact analytical solutions for this equation. However, an approximate solution for the Edwards equation can be found by using only the first term of an exact eigenfunction expansion: G(z,s) = g(z)e fs

[1.4.10]

where e e is the largest (= ground-state) eigenvalue and g(z) the corresponding (ground-state) eigenfunction. According to [1.4.4] with this approximation, the distribution (p(z,s)~ G(z,s)G{z,N - s) = g2eeN

for an interior segment s in the chain

becomes independent of s; all segments are assumed to have the same spatial distribution. This implies that end effects

(tails) are neglected. Tails can be

reintroduced afterwards as a perturbation of the ground state. Upon substituting [1.4.10] into the Edwards equation [1.4.1], we obtain an ordinary differential equation in g : £ 2 ^ - | = 6(e+u)g dz z \ /

[1.4.11]

This equation may be solved when e is known and when u can be expressed ing . The precise form depends on the problem at hand and differs between semldilute and dilute systems (and has different solutions for adsorption and depletion because of the different boundary conditions) as will be shown in sec. 1.5. We again need the boundary condition at the surface. Now [ 1.4.3] transforms into i^ 9 dz

=-c

[1.4.12]

z=0

Substitution of [1.4.10] into [1.4.4] gives pe ( = Ntpe I(pb ) and p { = cpl(pb ) expressed in the eigenfunction g : pe = e£Ng

p=e£Ng2

[1.4.13]

The end-point concentration is proportional to g , and the overall concentration to g2 . From [1.4.13] we have (p = (pbesNg2 . We will see later (sec. 1.5b) that in semidilute

1.12

INTERACTION BETWEEN POLYMER LAYERS

solutions e = 0 , so

a2 =

[ev b

semidilute

[l

dilute

1.4d Expressions for the local and non-local grand potential

[1.4.14]

densities

The exact expression for the grand potential density co is given In [1.4.7]. As discussed in section 1.3b, it is sometimes convenient to separate co into a local contribution coioc and a non-local part <»nloc. As in [1.4.7], we use a dimensionless form Hi (in units kT per volume C3 ) for both ftiloc and S n l o c . The local contribution Is found from the local version of [1.3.3], using [1.3.5] for / l o c and [1.2.3,5] for //, and ju:

^=$*%+*-**T^ ^

l n

N

JL

+

(pb

lz3L N

—yp -
+ l

[ -Ji)('-*,)-z('-*>?

+±

u ~ f

2yV

n>>

[1 4 15al

--

[L 4.i5b]

semidilute [1.4.15c] dilute

In the above expressions, [1.4.15a] is exact in SCF, [1.4.15b] is an expansion, and [ 1.4.15c] is the approximate form used below in the semidilute or dilute regions. The non-local contribution is obtained by subtracting the local part [1.4.15a] from the total co given in [ 1.4.7]. The result is gnloc =-m\u +— In— \^-
[1.4.16]

In the first form of [ 1.4.16] it would be possible to substitute u from [ 1.4.2], but this is unnecessary. In the second form, we neglected ln^ with respect to ln(£>b and replaced N" 1 111(1/^) by e because cpbeeN approximately equals unity (see also [1.5.11] in section 1.5c). ' A.N. Semenov, J. Bonet-Avalos, A. Johner, and J.-F. Joanny, Macromolecules, 29 (1996) 2179; G.J. Fleer, J. van Male, and A. Johner, Macromolecules, 32 (1999) 825; ibid, 32 (1999) 845.

INTERACTION BETWEEN POLYMER LAYERS

1.13

We note that, whereas the total co in [1.4.7] is negative for adsorbing polymer [
[1.4.17]

0

The integral may be written as Jgdg' where g' = dg/dz. Integration by parts gives gg'\0 -Jg'dg = -gg' z_0 - j(dg/dz) 2 dz , where we used g' = 0 at z = h and g'dg = (dg/dz)(dg/dz)dz . The term gg'| _ follows from [1.4.12] as -cg^ = -ccpo/a2 . Hence, [ 1.4.17] transforms into

^=-^0+^[f^fdz kT

[1.4.18,

6 r ° 6£ 1 {dz)

Inserting this result into i2a = ^ o c +i2° loc with C2l°c = \coXoc dz , we find for the grand potential per unit area:

MO =_^. kT

r[^i + £ir^ a L

&r° J [ kT

[L4.19]

6f {dz) \

which has the same form as [1.3.7]. Apparently, yQ in [1.3.8] may be identified with ckTI&t. With g = Jip / a, the square gradient term may be written as (24^r)" 1 (d^/dz) 2 , which shows that K in [1.3.7] is inversely proportional to

1.14

INTERACTION BETWEEN POLYMER LAYERS

middle to become equal to (ph (top part of fig. 1.1). In terms of [1.4.19], this means that a2{dg/dz)2

= 6(.wloc /kT when h -^ °o . For finite h, this has to be modified since

the gradient does not vanish at h = <*> but in the middle of the slit, where

-S1°C(^ - ^ ^

&[¥f = ^fiff 24
6

and d

IL4 201

-

\dz)

which reduces to the single-plate case for loc(loc(
a(h)

= -Yo
[2G)loc{(p) - colocl
11.4.21]

0

We now have two equations to calculate Qa from the concentration profile; [1.4.7] in combination with [1.3.2] is the general mean-field equation, and [1.4.21] is the groundstate version. Note that for adsorption the leading term in [1.4.21] is - / 0 ^ 0 ' loc

ensures Qa < 0 ; the integral over 0)

wn

icn

is a positive gradient correction. For depletion

cpQ is essentially zero so the first term in [1.4.21] vanishes, coloc is again positive, and Qa > 0, as expected. In order to make progress, we have to know more about the concentration profile cp(z). With the full expression [1.4.7] for co or [1.4.15a] for coXoc, only numerical results are possible. By making some approximations which simplify the expression for co and a>loc , such as [1.4.15c], analytical approximations become feasible. 1.5 The concentration profile In the full SCF model, the concentration profile is calculated numerically by solving the set of self-consistent equations for the profiles
2.5a Numerical integration In principle, [1.4.20] can be used to calculate the concentration profile cp{z), albeit only in an implicit form z{
C

11

or

P

loc

J24loc{(pm)j

P.G. De Genncs, Macromolecules 15 (1982) 492.

[1.5.1]

INTERACTION BETWEEN POLYMER LAYERS

1.15

Using [1.4.15] for <3loc , one may choose a certain value of
=6eg = 0, which must imply e = 0 since

9b ~ V^b~ i s non-zero. Hence, [ 1.4.13b] reduces to the simple form (p=fhg2

[1.5.2]

This result, used already in [ 1.4.14], shows that in the bulk solution g = \ (which also follows from [1.4.10] with e = 0 and Gb = 1). Inserting [1.5.2] into the expanded version [1.4.2c] of the segment potential (neglecting the quadratic term) gives the relation between II and g : u = vcpb(g2-l)

[1.5.3]

With this form of II, [1.4.11] reduces to ^ 2 d 2 g/dz 2 = 6u^ b (g 3 -g) . We may write this as

where Z is the spatial coordinate in units d;, and £ is the correlation length (or blob diameter). This is the natural length scale in semidilute solutions and scales for meanfield good solvents as #>b1/2 • Note that [ 1,5.4c] also gives the prefactor. As discussed in sec. II.5.2d, £, — ^ ~ 3 / 4 when chain swelling is taken into account. Upon multiplying [1.5.4a] with dg/dZ (dgldZ)

2

4

2

- g +2gr

we find d[(dg/dZ) 2 -g4 +2g2\/dZ = 0, so

is constant across the slit. The constant equals 9^a~'^9m since

dg/dZ = 0 in the middle of the slit. Hence,

[ft-

21

2

[1-5-51

1.16

INTERACTION BETWEEN POLYMER LAYERS

Note that this equation is a special case of [1.5.1] or [1.4.20] with a2 = (pb and (p =
1

2

2

Inserting this into [1.4.20] or [1.5.1], we obtain [1.5.5] immediately. All the equations given above are valid for adsorption and depletion in semidilute solutions. The difference is that u is positive for adsorption {g > gm > 1), and negative in depletion {g
< 1) . We discuss the depletion case separately in sec. 1.8, and

restrict ourselves here to adsorption. Then, in taking the square root of [1.5.5], we need the minus sign since g > 1 and dg/ dZ < 0 . Let us first consider wide slits (g m = 1), with z = 0 at the (left) surface. Then [1.5.5] reads dZ = dg/(l -g2).

Integration gives Z + P = arccothg, where P is an integration

constant. Hence, (p = (ph coth 2 ^±IL

g = coth^-i-£

[1.5.7]

Here, p (= P£ ) is called the proximal length. It enters through the boundary condition at the surface and is therefore closely related to the extrapolation length 1/c which, in turn, is determined by xs

an

d the volume-filling constraint in the train layer. Unless

Xs is close to the critical value ;fsc, i.e. for A%s =ZS~ZSC

sufficiently positive, the

value of p is small, on the order of the bond length t. It can be shown 1 ' that p ~ £/^3(1 - e x p - A%s) when A%s>l/JrN.

In the critical region, i.e. | A%s | VJV < 1,

GSA fails. We do not go into further detail as to the relation between p and A%s . For p«

z « J;, [1.5.7] predicts (p(z) - (ph{z/^)~2 ~ z~2 . There is thus a region in

the profile where
Z =| = f

,

dg

H= T = T

dg

U.5.8]

In [1.5.8b], the upper limit is g0 = j
11

G.J. Fleer, J. van Male, and A. Johner, Macromolecules, 32 (1999) 845. G.J. Fleer, F.A.M. Leermakcrs, in Coagulation and Flocculation: Theory and Applications, 2nd ed.. Surfactant Science Series, B. Dobias and H.J. Stechcmesser, Eds., M. Dekker (2004).

INTERACTION BETWEEN POLYMER LAYERS

1.17

Figure 1.3. Superposition approximation according to [1.5.9, 11].

For n o t t o o s t r o n g overlap, the linear superposition approximation (LSA, also used in sec. IV.3.3-3.7 for electrical double layers) is often useful. T h e principle is sketched in fig. 1.3. We p u t Z = 0 in t h e middle of the slit. It is a s s u m e d that
[1.5.14]

The parameter d may be seen as the thickness of the adsorbed layer. It is on the order of the radius of gyration, but smaller by a factor 3-5, depending on
Z =§ d

g=^-g

11.5.15] t

After multiplying [1.5.15a] with d g / d Z , we obtain d[(dg/dZ} 2 - g 4 - g 2 ]/dz = 0 . The integration constant is g^+Sm- and the dilute equivalent of [ 1.5.5] is

f^f

T-!

T

[1.5.16]

Again, this equation is a special case of [1.5.1]. In dilute solutions, [1.4.15c] may be approximated as

INTERACTION BETWEEN POLYMER LAYERS

1.19

aloc=£tp + -v
[1.5.17]

Inserting this into [1.5.1], we obtain after some rearrangement [1.5.16] First, we again consider the single-plate case (i.e. wide slits with fifm= 0 ). Now [1.5.16] reads dZ = dg/(g Jl + g 2). The solution is Z + P = arctanh[l/(l + g 2 ) 1 7 2 ] which may be rearranged to g = l/sinh(2 + P). Hence, g and

9 = -^T d

[1.5.18] d

Like in [1.5.7], p (= Pd) is the proximal length, determining the train density pQ = (fi 13vd2) sinh" 2 ip/d) - (C/3up)2 ; also in this case p = £ /^3[l-exp(-A^ s )] A plot of the volume fraction profile according to [ 1.5.18b] is given in fig. 1.4, both in a semilogarithmic and a double-logarithmic representation. This figure shows not only results for a good solvent for which [1.5.18] applies, but also for a theta solvent where a similar GSA treatment 1' leads to

Figure 1.4. Adsorbed profiles according to [ 1.5.18b] for p = 1, d = 30 , and x - ° (u = 1) on a semi-logarithmic (left) and double-logarithmic scale (right). The dotted lines in the left diagram extrapolate the exponential decay and have a slope 21 d . The dashed lines in the diagram to the right have a slope —2 {

G.J. Fleer, J. van Male, and A. Johner, loc cit.

1.20

INTERACTION BETWEEN POLYMER LAYERS

narrow, extending only over a few segment lengths, and a power-law description is hardly relevant. When we want to describe the profile in a slit, we have ~gm * 0 in [1.5.16], Again, there is no analytical solution. We can find numerical solutions analogous to [1.5.8], Alternatively, we can use LSA in the form

cp = -!—\ TT-^ + *— 1 3vd2 \sinh2{Z + H + P) sinh 2 (-Z + H + P)J where Z = z/d,

[1.5.19]

H = h/d , and P = p/d .

1.6 Disjoining pressure and Gibbs energy of interaction 1.6a General In the previous section, we discussed the volume fraction profile (p(z) in the slit. When that is known, we can calculate the grand potential i3 a per unit area (per plate), either from [1.3.2, 1.4.9], £2a - \codz , or from the GSA version [1.4.21], which reads i2a = -~/Q(PQ +]{2CO1OC -a>j£ c )dz • Here, the grand potential density co is given by [1.4.7] and its local contribution o/oc

by [1.4.15]. In this section, we try to interpret the

trends analytically, using limiting forms of co and coloc . These were given in [1.4.15c] for o>loc and in [ 1.4.7] for co = /7g s - 77OS (z). We write them as [ 1.4.8]: —v[

loc

=

co =

semidilute

2

[1.6.1]

I2 1 —22

dilute

— u(#>2 - cp^)

semidilute

2

-— JV

[1.6.2] vm2

dilute

2

In the dilute version of [1.6.1], we replaced e=-N 'ln^ b by C2 l&d2, according to [1.5.14], Note that the signs of co and « Ioc are different for adsorbing polymer (tp> = g2 (dilute), with the superposition approximation as given in [1.5.9] or [1.5.19]. With some mathematical approximations, it is now possible to find simplified expressions for the disjoining pressure FI(h) and the Gibbs energy of interaction Ga(h). It turns out that the leading term in GSA is attractive and (for weak overlap) roughly exponential in h , with a decay length that is approximately equal to d 14 (where d is

INTERACTION BETWEEN POLYMER LAYERS

1.21

the distal length) in dilute solutions, and about £/4 (with £ the correlation length) in the semidilute case. However, the presence of end points leads to a correction which is repulsive, with a decay length twice that of the attraction. As a result, Ga(h) has a (weak) repulsive barrier at relatively large separations and a (strong) attraction at short separations. 1.6b Attraction due to bridge formation For an isolated surface, the volume fraction profile in semidilute solutions is given by q>=
'

/7

mp

=

-«m-

We are interested in the sum of these two contributions to the disjoining pressure:

These discrete changes can be translated into continuous language in the following way. For the adjustment part we have h

/7 ad dh = - J (d«)dz 0

and for the midplane contribution we can write

11

P.G. De Gennes, Macromolecules 15 (1982) 492.

[1.6.4]

1.22

INTERACTION BETWEEN POLYMER LAYERS

% d h = - fflm dh

[1.6.5]

where a>m = co{
is

repulsive, because com is negative. The adjustment part /7 ad is attractive. We will see below that the net effect is still attraction, /7 ad + /7

< 0 . We note that the alternative

route to obtain the disjoining pressure, as used by De Gennes starting from a>loc (z), also gives attraction. To give a quantitative estimate of the attraction, we need the concentration profiles of the polymer within the slit. We start with the semidilute case. Upon substitution of [1.6.2a] into [1.6.4, 5] we obtain

0

L

J

For the concentration profile we use the linear superposition [1.5.9], which we rewrite as: — = coth 2 x + c o t h 2 y - l
x =Z +P

y = -Z + 2H + P

[1.6.7]

Note that in this case the left surface is situated at z = Oand as before Z = z / | , H = h/%, and P = p/£. For the evaluation of /7 a d , we need the differential quotient dip2 16H = lipdipl6H . From [1.6.7] we have d(ipl(ph)/dH = 4cothy (1 -coth 2 y) = -16e" 2 y cothy . It is sufficient to integrate over half of the slit. We take the left half where cothy = 1 and coth2 y - 1 « coth 2 x . We thus find 4

= -32e- <

H+p

2

2

)e * cosh x.

The

in first

integral

of

order 2

dip2 / &H ~ -32e~2y cosh 2 x 2

e * coth x

is

icothx(e2x-3) +

2(x + lnsinhx). Taking the leading terms at both boundaries, we obtain as a first approximation

^

/7 ad = -8v
[1.6.8]

Figure 1.5. Numerical (discrete) example of the function -OJ{Z) at separation h = 9 (closed diamonds) and at separation h + dh =10 (open circles). Areas I and II are two contributions to the disjoining pressure FI(h) . The bar diagram at the bottom gives -dm ) corresponds to area II.

INTERACTION BETWEEN POLYMER LAYERS

1.23

With «+«

[ 1 6 9 ]

We conclude that the leading contributions of 77 (repulsive) and /7 ad (attractive) exactly compensate each other! This illustrates the subtle balance between the various contributions. We also see that -77ad is larger than 7 7 ; the adjustment of the a>profile overcompensates the repulsive contribution of 77 . Integrating the disjoining pressure to obtain the Gibbs energy of interaction ([1.3]) is simple. We cast the result in the general exponential form G a (h) = -Ae~ 2/1/L

[1.6.10]

where L/2 is the decay length of the Gibbs energy of attraction. In this case L = £/2 or Ga - e~4h/%. The reason that the decay length equals £/4 is that a> is dominated by ~ [1/sinh2 x +1 /sinh 2 y], where x and y are defined as in [1.6.7], but with E, replaced by d . In this case we find Ga = Ae~ 4h/d which is the dilute version of [1.6.10] with L = d/2. Also here the prefactor is proportional to P~1=d/p. This result is qualitatively the same for the linear and quadratic terms in the dilute version of co in [1.6.2] We may conclude that GSA predicts attraction in all cases, with an exponential decay Ga — e-2h/L for ^c initial part (i.e. weak overlap). The decay length L/2 is about f/4 in semidilute solutions and around d / 4 in the dilute case. In sec. 1.7, we shall compare these predictions with the results of the numerical SCF model. The origin of the attraction. The GSA description neglects tails, so the attraction must be due to the loops. The reason for the attraction between loops is the possibility of bridge formation, making the profile more homogeneous (see fig. 1.6). When a walk starting from the left surface (black contour in fig. 1.6) reaches the midplane, the probability of returning in a loop conformation to the same surface (light gray) is

1.24

INTERACTION BETWEEN POLYMER LAYERS

Figure 1.6. Configurational entropy gain occurring when a walk starting from the left surface (black contour) reaches the midplane. It can follow some path (light gray), which returns to the same surface to make a loop, or it may follow with equal probability its mirror image (dark gray) to form a bridge.

exactly the same as that of making a bridge connecting to the opposite surface (dark gray), without any energetic penalty. The entropy is thus increased, which leads to attraction. Therefore, our statement above that there is attraction between loops has to be modified: the attraction arises because loops can transform into bridges. 1.6c Repulsion due to tails

In the previous section we considered only ground-state dominance for 'average segments', which for single surfaces describes only loops; tails are neglected. In this situation we only have attraction between two polymer layers in close proximity, both for adsorption and for depletion. For adsorbed polymer layers, end effects do play a role. This has been known since the 1980s due to the numerical work of Scheutjens et al. 1 '. Analytical theories describing tails became available only after 1995 and are largely due to Semenov et al. 2 The starting point is that, whereas the overall (loop) concentration is proportional to g2 , the end-point concentration is proportional to g (see [1.4.13]). From [1.4.13a] we have pe = N(pe / (pb = eeNg . Here, cpe = ^e(z) = (p(z,N) is the volume fraction of segments with ranking number JV . The total volume fraction of end points is then 2(pe . With £ = 0 and g = coth(Z + P) in semidilute solutions, and (f>he£N ~ 1 and g ~ l/sinh(Z + P) in dilute solutions, we may approximate the excess of end points with respect to the bulk solution as pe -1 = big -1) ~ coth 2 + P -1 pe=bg

semidilute dilute

sinh

[1.6.11a] [1.6.11b]

— d

11

J.M.H.M. Scheutjens, G.J. Fleer, J. Phys. Chem. 84 (1980) 168; G.J. Fleer, M.A. Cohen Stuart, J.M.H.M. Scheutjens, T. Cosgrove, and B. Vincent. Polymers at Interfaces. Chapman and Hall, 1993. 21 A.N. Semenov, J. Bonet Avalos, A. Johner, and J.-F Joanny, Macromolecules, 29 (1996) 2179; A.N. Semenov, J.-F. Joanny, and A. Johner. Polymer adsorption: Meanfield Theory and Ground state Dominance Approximation in: Theoretical and Mathematical Models in Polymer Research. Academic Press 1998.

INTERACTION BETWEEN POLYMER LAYERS

1.25

The factor b is a normalization constant, which ensures that the excess number of end points in the adsorbed layer equals the excess number of chains; it is defined through b\(g-l)dz

= Ug2 -l)dz for semidilute solutions and similar (without the term -1) for

dilute systems. We do not consider the precise value of b , suffice it to note that b is of order unity. In order to see the effect of chain ends on the interaction force, the GSA description as discussed in the previous sections has to be extended to include the end points. This was done by Semenov et al. 1 '. The disjoining pressure /7e(h) due to the ends was related to the end-point concentration


treatment led to attraction, the Semenov extension gives an additional contribution /7e(h) ~ +{eb) ~ [gm -1), where
.

The most important implication is the sign: end effects give rise to repulsion between adsorbed polymer layers. This repulsion may be interpreted as the repulsion originating from an ideal gas of end points. When the two surfaces are brought closer, the concentration of chain ends increases, the 'gas' pressure increases, and the plates repel each other. In order to make a quantitative estimate, we approximate gm - 1 in semidilute solutions

as

gm - 1 ~ coth(H + P ) - l = e-^h+p)/^

por

dilute

solutions,

we find

gm ~ l/sinh(H + P) ~ e~' h + P' / d for the dilute case. The result, written in the same way as [1.6.10], is Gae(h) = R e - h / L

[1.6.12]

As before, L equals £/2 in semidilute solutions and d / 2 in dilute solutions. The decay length L for the repulsion in [1.6.12] is twice the decay length L / 2 for the attraction in [1.6.10]. The reason is that the overall profile (~g2),

which determines

the attraction, decays two times faster than the end-point profile ( ~ g ).

1.6d Total Gibbs energy of interaction Upon combining [1.6.10] and [1.6.12], we obtain for the total Gibbs energy of interaction Ga =-Ae~2h/L

+ Re~h/L

where L is the decay length of the repulsion,

[1.6.13] L / 2 that of the attraction, and

A (attraction) and R (repulsion) are the amplitudes.

We use a dimensionless form

Ga = G a ( 2 IkT . A sketch of Ga[h), according to [1.6.13], is given in fig. 1.7. The curve has a maximum Gm at h = hm and passes through zero at h = hQ.

11

A.N. Semcnov, J.-F Joanny, and A. Johner,toecit.

1.26

INTERACTION BETWEEN POLYMER LAYERS

Figure 1.7. Sketch of Ga(h) according to [1.6.13]. The dashed curves are the attractive (loop/bridge) and repulsive (tail) contributions.

When L , A and R are known, these characteristic values follow from G

m

R2 = ^

2A hm = Lln—

ho=hm-Lln2

[1.6.14]

Exact numerical SCF calculations also provide a dependence Ga(h) as sketched in fig. 1.7. From such curves, the values of Gm , h m and h 0 follow directly for any bulk concentration
k

°

A = Gme2h™/L

R = 2Gmeh™/L

[1.6.15]

In the next section we will see that exact numerical results can indeed be interpreted in this way. 1.7 Numerical examples for adsorption In this section, we present some numerical results for the Gibbs energy of interaction between two surfaces with adsorbed polymer layers In full equilibrium. We first show in section 1.7a that Indeed the dominating effect is attraction, but that there is a weak repulsion at large separation. The interaction curves for weak overlap can be analyzed along the lines of [1.6.13] and fig. 1.7, using the parameters Gm (height of the maximum), h m (position of the maximum), and L (decay length of the repulsion or twice the decay length of the attraction); this will be discussed in section 1.7b. In section 1.7c, we will show how the bridging attraction found at strong overlap depends on the governing parameters. 1.7a Interaction curves

In fig. 1.8 we give the whole Interaction curve for N = 1000 and q>h = 10"2 (which is in the semidilute range). The linear plot of fig. 1.8a shows an enlarged view of the region of weak overlap around the maximum. The symbols in fig. 1.8a are the num-

INTERACTION BETWEEN POLYMER LAYERS

1.27

Figure 1.8. The Gibbs energy of interaction Ga=Ga(2/kT as a function of the plate separation 2h (in units ( ) for N = 1000, ^ = 10~2 , ^fs = 1, and % = 0 . In diagram (a) the region near the maximum is plotted with linear scales. The maximum G m , the corresponding position h m , and the position /1Q where Ga = 0 are indicated. The symbols are the SCF results, the curve is the fit with the two exponentials according to [1.6.13], using A = 1.41 10~ 2 , i? = 3.11 10" 3 , and L = 2.95 . The attractive part of the interaction curves for the range h < /IQ (solid curve) is plotted in panels (b) and (c) on semi-logarithmic and doublelogarithmic coordinates, respectively. In these graphs, the dashed curve is the extrapolation of the solid curve in diagram (a). The dotted curve was calculated from numerical integration of the superposition profile according to [ 1.5.9].

erical SCF data. The maximum is found at hm =7.0 and has a value of Gm = 1.71-10"4. The point where the interaction energy is zero corresponds to hQ = 4.96. From these data, [1.6.15] gives A = 1.4110~2, R = 3.11 10~3 , and L = 2.95 . The solid curve in fig. 1.8 is [1.6.13] with these parameters. In this region of weak overlap the fit is quite good, which shows that near the maximum the interaction curve can indeed be described as a difference between two exponentials with a single interaction length L , which is the decay length of the repulsion; that of the attraction is L / 2 . For h(z) is also overestimated so that the attraction is exaggerated. For small plate separations, this is a relatively small effect.

1.28

INTERACTION BETWEEN POLYMER LAYERS

For higher separations the discrepancy with the numerical SCF results is larger because then the end-point repulsion (not accounted for in [1.5.9]) starts to play a role. The numerical data cross over smoothly from the dotted curve at low h (which constitutes an overestimation of - G a ) to the dashed (double-exponential) curve at higher h. Even though we have no accurate analytical description over the full range, this crossover from one analytical limit to the other is gratifying. From fig. 1.8b,c we see that the attractive part of the interaction curve has a complex structure. It is neither a true exponential nor a true power-law curve. In fig. 1.8c it would be possible to force a power law through the data in the middle range with an exponent not far from -3 . For very small slit widths an exponential decay of the interaction is found in fig. 1.8b, which is not present in the GSA result. GSA is not expected to hold when the train layers start to overlap.

Figure 1.9. (a) The position hm of the maximum in the interaction curve, the position h 0 where the interaction is zero, and the interaction length L as a function of the polymer concentration tp^. For comparison, the distal length d and the correlation length £ are also plotted. All lengths are in units { . (b) The maximum G m in the interaction curve as a function of the polymer concentration cp^ . Both graphs are for JV = 1000, xs = 1 anc> X = ° •

1.7b Weak overlap In the following sets of graphs we discuss the interaction for weak overlap. One example was shown in fig. 1.8a. We now present the dependencies of Gm , its position hm, the position h0 where Ga = 0 , and of the characteristic interaction length L on is shown in fig. 1.9, which applies to JV = 1000 , Xs=l ' a n d X = ° • T r i i s figure covers the range 10"4 < cpb < 10"1 . The overlap concentration is of order 1 / JV = 10~3, so that we cross over from the dilute to the semidilute regime. The position of the maximum hm, as well as the compensation point h0 , are decreasing functions of the polymer concentration; both quantities scale with the polymer concentration with an exponent close to -0.15. The interaction length

INTERACTION BETWEEN POLYMER LAYERS

1.29

L goes through a maximum, following the distal length d in dilute solutions but the correlation length £, in the semidilute range. In fig. 1.9a, d and £, are also indicated (dotted curves). These two lengths cross at a point well into the semidilute regime. The maximum in L, however, is situated around the overlap concentration <pov . In the dilute range, L is close to d/2 and in the semidilute region to £/ 2, as expected from section 1.6b. The value of Gm is a rather strong function of the polymer concentration, roughly as Gm « Jip^ for low
Figure 1.10. (a) The position h m of the maximum in the interaction curve, the position h 0 where the interaction is zero, and the interaction length L as a function of the chain length JV. For comparison, the distal length d and the correlation length £ are also plotted, (b) The maximum G m in the interaction curve as a function of the chain length JV. Both graphs are for ^ = 1 0 - 3 , j s = 1 and x = 0 •

The interaction also depends on the chain length (fig. 1.10). In this graph the polymer concentration was fixed at h I
1.30

INTERACTION BETWEEN POLYMER LAYERS

or less logarithmic dependence with N is found. On the other side of the maximum, the drop may be explained from the fact that the adsorbed amount goes down with decreasing chain length. Thus, when one is interested to use homopolymers to stabilize particles, one should choose an intermediate degree of polymerization. Very long polymers are not effective because they do not have enough ends; short polymers do not adsorb strongly enough.

Figure 1.11. (a) The position h m of the maximum in the interaction curve, the position h 0 where the interaction is zero, and the interaction length L as a function of the (effective) adsorption energy A%s . For comparison, the distal length d is also plotted. All lengths are in units t . (b) The maximum G m in the interaction curve as a function oiAx . Both graphs are for p,j = 10" 3 , N = 1000 and x = 0 •

Figure 1.11 shows the effect of A%s = Xs~ Xsc o n the characteristics of the maximum. The adsorption energy does not dramatically influence the spatial characteristics of the maximum in the interaction curves, at least for relatively strong adsorption. Only when the adsorption energy is close to xSc ^° w e °t> serve that the position of the maximum shifts to higher h values. This signals the approach to the critical region. The height of the maximum increases monotonically with the adsorption energy. This can be understood from the fact that the adsorbed amount and, hence, the number of end points is an increasing function of the effective adsorption energy. To stabilize particles with homopolymers one should, therefore, use strongly adsorbing polymer. 1.7c Strong overlap In figs. 1.12,13 a number of plots is collected similar to fig. 1.8c under various conditions. The attractive part of the interaction energy (h < hQ ) is plotted on a doublelogarithmic scale for three values of the bulk concentration cph (fig. 1.12a), the chain length JV (fig. 1.12b), the effective adsorption energy Axs (fig- 1.13a), and the solvency X (fig. 1.13b). The first thing to notice is that for narrow gaps only the adsorption energy has an effect; for small h the curves for different

INTERACTION BETWEEN POLYMER LAYERS

1.31

Figure 1.12. Numerical SCF data for the attractive part of Ga in the range h
whereas there Is an effect of ipb , N, and %. All the curves start at 2h = 1, which corresponds to one (lattice) layer of solution between the plates. In this situation, the attraction Is linear In the effective adsorption energy. For larger plate separations, the attraction becomes weaker in all cases. For h close to h0 (where the loop/bridge attraction and the tall repulsion just compensate each other), the curves become approximately independent of %s (fig. 1.13a), but they do depend on the other parameters. The interpartlcle separation where the curves bend upwards is determined mainly by the value of h , which increases with decreasing

Figure 1.13. Numerical SCF data for the attractive part of G a , in the range h < h0 , for three values of A%s (a) and x (b)- I n diagram (a) JV = 1000, (p^ = 10" 3 , / = 0 , and in (b) JV = 1000, ( 9 b - 1 0 - 3 , Xs = 1-

1.32

INTERACTION BETWEEN POLYMER LAYERS

The solvent quality affects to a very large degree the range of the attraction, as shown in fig. 1.13b. Going from a good ( % = 0 ) to a theta solvent, the attractive component becomes stronger because the attraction between (loop) segments reduces the excluded volume effect; at x = 0.5 , the second virial coefficient vanishes. The repulsive effect of the chain ends is then relat less important. Apparently, GSA is qualitatively correct in this case and the interaction curve for ^=0.5 in the double-logarithmic coordinates of fig. 1.13b extends to very large h. Indeed, for large h the interaction is purely exponential (not shown). At shorter plate separations (2 < 2h < 15), we again find a power-law-like dependence of the interaction force. Now the exponent is to a good approximation equal to -2. 1.8 Depletion at flat surfaces 1.8a Introduction In the previous sections we have mainly considered adsorbing polymers, where the segment adsorption energy ~xskT is the driving force for accumulation of polymer near the surface. When xs i s at>ove the critical value xSc • t n i s adsorption energy overcompensates the entropy loss experienced by the chains near an impenetrable surface, so that there is positive adsorption. When such a driving force is absent, only the unfavorable entropy remains, and a depletion zone develops where the concentration is lower than in the bulk solution. A schematic picture is given in fig. 1.14. This figure is a variant of fig. II.5.9, and gives half the profile in a wide slit (fig 1.1 top right). The characteristic width 8 of the depletion zone is of the order of the radius of gyration a g in the dilute solutions, but becomes smaller as, at higher concentrations, the osmotic pressure in the solution pushes the chains closer to the surface. Beyond overlap of the coils in solution ( q>ov ), in semidilute solutions, the characteristic length scale in the solution is the blob diameter £, see [ 1.5.4c], which does not depend on chain length and in (mean field) good solvents scales as {vcph)~1/2. In such semidilute solutions, 8 turns out to be equal to £ • We will introduce a generalized form of £, which applies both in good solvents ( £ ~
Figure 1.14. Volume fraction profile for depletion at a flat wall. The continuous profile is indicated by the solid curve, the dashed lines represent a step profile with width 5, which is the depletion thickness. The hatched areas left and right of z = S are equal.

INTERACTION BETWEEN POLYMER LAYERS

1.33

The depletion thickness may be related to the adsorption F , which is the excess of material per unit area. A general definition of F was given in [II.2.1.2]. For depletion, the excess of polymer is negative, so F < 0. We will use a dimensionless form 0 ex for the excess of polymer, F = Bex/t2 (on a segment basis). In a lattice model, 0ex is found by summing (f>(z)-
'--^-Jo-'V'T-Je-*)'? rb

o

o

Hence, the depletion thickness can be computed from either the end-point concentration profile or the overall concentration profile. As in the previous sections, we can either use the Edwards equation (or some GSA equivalent) or the numerical SCF lattice model. We will show that in all cases the overall profile is well approximated as p = tanh 2 z/S . When two surfaces, each with a depletion layer, approach each other to an interparticle distance 2h , which is of order 25, the layers overlap, see fig. 1.1. We could try to solve for the profile along the lines discussed in section 1.5, and from that compute the Gibbs energy of interaction as was done in sec. 1.6 for adsorbing polymer. We would then get quantitative analytical information for weak overlap, but no easy analytical solutions for strong overlap. We shall not follow that route, as for depletion the step-function approach gives a convenient alternative. We then replace the continuous profile for each surface by the dashed lines in fig. 1.14. The idea is sketched in fig. 1.15. In this picture the interaction is zero for 2h > 25 ; the region of weak overlap is completely neglected. However, we can rather accurately describe the region of strong overlap: the attraction is roughly given by G[h) = -J7gsVov(h), where G{h) = Ga[h)A is in Joules, /7gs is the osmotic pressure in the bulk solution, and Vov(h) = 2(8-h)A . Dividing by the area A gives Ga(h) = 2/7g s (h-5), which is negative for h<8 and

Figure 1.15. Step-function approach for the overlap of two depletion layers. The overlap volume Vov is 2(5 - h) times the area A per plate.

1.34

INTERACTION BETWEEN POLYMER LAYERS

linear in h . We will see that this simple approach works quite well, but that for an accurate description of Ga the parameter S (thickness of a single depletion layer) has to be replaced by an interaction distance Si, which is slightly larger.

1.8b Depletion thickness and the profile at a single surface (i) Dilute limit. In dilute solutions, the field u in [1.4.1,2] is essentially zero, and the Edwards equation can be solved exactly. Using the boundary conditions G(O,s) = 0 and G(oo,s) = 1, Eisenriegler

derived

p e (z) = G(z,JV) = erf-^2a g

[1.8.2]

for the end points, and a slightly more complicated expression for the overall profile p(z). In the dilute limit, a g = t^N/G is the only relevant length scale. By inserting pe into [ 1.8.1 ], the depletion thickness is found as S0=~a sin 6 We use the subscript

[1.8.3]

0

to indicate that this result applies to the limit of zero

concentration. Tuinier et al.2) showed that the rather complicated exact expression for the overall profile is approximated rather accurately by p=tanh2—

[1.8.4]

fit) Semidilute limit. Above the overlap concentration, the chain-length dependence disappears and the relevant length scale is the blob diameter £,. According to [1.5.4c] it is defined as £2 I^ = 3Uh/kT

(with U b given by [ 1.4.2b]), which for good solvents

2

may be expanded to give t I ^ = 3v(pb. In GSA, the profile follows from [1.5.5] with g m = l so that dZ/dg = l-g2

. Here Z = z/£

and the minus sign was taken for the

square root of [1.5.5] because gf < 1 and dg/dZ>0.

With the boundary condition

g(0) = 0 , the solution is Z = arctanhg , or g = tanhZ . From p = g2 , /7=tanh 2 ^

[1.8.5]

We see that the coth, which followed from adsorption in the semidilute regime (see [1.5.7]), is replaced by a tanh in depletion. Note also the close analogy with [ 1.8.4]. The only difference is the length scale, as So = S0(N) in [1.8.4] is replaced by £, = ^(pb,x) in [1.8.5].

11 E. Eisenriegler, J. Chem. Phys. 79 (1983) 1052; E. Eisenriegler, Polymers near Surfaces, World Scientific (1993). 2) R. Tuinier, G.A. Vliegenthart, and H.N.W. Lekkerkerkcr, J. Chem. Phys. 113 (2000) 10768; G. J. Fleer, A.M. Skvortsov, and R. Tuinier, Macromolecules 36 (2003) 7857.

INTERACTION BETWEEN POLYMER LAYERS

1.35

Inserting [ 1.8.5] Into [ 1.8.1 ] gives the well-known De Gennes result 8=$

[1.8.6]

which expresses that in the semidilute case the depletion thickness (an interfacial property) is equal to the blob diameter (which is a solution property). It is possible to generalise the expression t2 IE,2 = 3txpb to poorer solvents by inserting in t2 IE} = 3Ub / kT the full expression [ 1.4.2b] for U b

^ = ^f

=

-3H1-^)

+ ^b]

[1-8.7]

This form reduces to the known limits in a good solvent (I2 IE,2 = 3vq>b) and in a theta solvent (t2 IE} =3cp2>/2). Moreover, it describes the transition region (£ = 0.4-0.5, relevant for many practical situations) as well. A plot of E, as a function of
Figure 1.16. The correlation length E, (in units C ) in semidilute solutions as a function of
As shown by Fleer et al.11, with this generalized form of E,, [1.8.5] describes the depletion profile in good solvents very accurately and in theta solvents quite reasonably (in the latter case [1.8.4] gets a slightly different mathematical form). Similarly, 8 = E, is accurate in good solvents and a reasonable approximation in a theta solvent. (ill) Generalization. The two limits 5 0 = <50(JV) (dilute) and £ = |(
[1.8.8]

£•£•? According to [1.8.6], the depletion thickness 8 in semidilute solutions equals the bulk solution correlation length E,, which in this case does not depend on N. Khoklov and Grossberg21 introduced a generalized (JV-dependent) correlation length <^(JV), which is 11 21

G.J. Fleer, A.M. Skvortsov, and R. Tuinier, toe. cit. A.Y. Grossberg and A.R. Khoklov, Statistical Physics of Macromolecules, AIP Press, 1994.

1.36

INTERACTION BETWEEN POLYMER LAYERS

defined through C2 /^ 2 (AT) = 3[1/ N + ((3/kT)dnos I dip]. Here, the second term is the inverse of the osmotic compressibility. Using the mean-field version of d/7 os / dip, given in [ 1.2.6], it can be shown that 8 , as defined in [ 1.8.9], is essentially the same as £(JV). This generalizes the statement "depletion thickness equals bulk solution correlation length," first formulated by De Gennes as 8 = E, (where E, does not depend on JV), to more dilute systems where 8 = §(JV), which includes the JV-dependence of 5 and £, .

Figure 1.17. The depletion layer thickness 8 (in units t ) at a flat surface, according to [1.8.9] (solid curves), as compared with exact SCF results (symbols), for two chain lengths (N = 1000, top, and N = 100, bottom) and for three solvencies. The value of xs in the lattice model was chosen as Xs = — (x + l)/6 , which is the value for which

(iv) Comparison with exact lattice results. Figure 1.17 shows <5(cpb) as calculated from 11.8.9] for three solvencies and two chain lengths, and compares the results with numerical SCF data. For % = 0 and 0.4, the agreement is quantitative, whereas for ^ = 0.5 in semidilute solutions [1.8.9] slightly overestimates the exact lattice result. The reason is a different mathematical form for the profile under theta conditions. It can be shown that the simple form 8 - % overestimates the thickness of the depletion zone by some 10%. A correction is possible11, but we do not go into details. For 0 , £j~2 —> 0 the depletion thickness equals 80 = (2l4n)a^ = 2(*JN/6K as in [1.8.3], independent of solvency. For good solvents and relatively long chains, this limit is only reached at very low q>b . We note that in a mean-field picture, a g (~ AT1^2) does not depend on solvency; chain swelling in good solvents (leading to a g ~ AT3^5 ) is not captured in such a model. As (pb increases, 5 decreases due to the increasing osmotic pressure in the solution. Therefore, this effect is stronger (and S is smaller) in better solvents. In semidilute solutions 8 = § , which follows a power law: 5 ~
G.J. Fleer, A.M. Skvortsov, and R. Tuinier, loc. cit.

INTERACTION BETWEEN POLYMER LAYERS

1.37

Figure 1.18. Depletion profiles (with z in units C ) at a flat surface for four different volume fractions in the bulk solution (indicated), at / = 0 and JV = 1OOO. The curves were computed with [1.8.8,9]; symbols are SCF results for £ s = - 1 / 6 .

lengths. Even in those cases where the depletion thickness is somewhat overestimated by [1.8.9] (as for % = °- 5 i n fig 1-17), the tanh2-profile fits the numerical data accurately, provided the width (= zeroth moment) of the profile (which may be too high by some 10% using [ 1.8.9]) is taken from the numerical data. 1.8c Interfacial Gibbs energy for a single surface It is useful to have an expression for the polymer contribution to the interfacial Gibbs energy y-y* = Qa (where Qa=Q/A is the grand potential per unit area), because this quantity is needed for the Gibbs energy of interaction between two surfaces. According to [1.1.1], Ga (h) = 2[i3 a (h)-f2 a Hl = 2[y(h)-yH] • In this section we consider only the value grand potential £2a (<x>), which we denote simply as Qa . We will find that Qa is positive, which implies a negative surface pressure (c.f. chapters III.3 and 4 for monolayers). There are two obvious ways to find Qa . The first is given in [1.4.9], Qa = I^bS ~ noslz)]dz , where the bulk osmotic pressure 77bs and the local osmotic pressure f7os(z) were defined in [1.2.3] and [1.4.8]. We denote this route to find Qa as the osmotic method. The second way is based upon integration of Gibbs' law dy = -Fdji or, in terms of 0ex and Qa , C2dQa = -9exd)i. Here F and /i are defined per segment; /i is given by [1.2.5]. A similar integration was used in [II.1.1.7] and [III.3.1.2]. Because in this method we need the (excess) adsorbed amount during addition of polymer to the system (from pure solvent to the final value
QaH=

r

?nos

J S(l-
[1.8.10]

where

1.38

INTERACTION BETWEEN POLYMER LAYERS

the system. For 5 , we can take [1.8.9] and dnos/dcp

is given by [1.2.6]. This equation

has a form which is quite different from [1.3.2], Qa = J w d z / ( . Yet, when numerically exact values for 81 C = - 0 e x / =
to give

according to [1.8.8], the three terms may be integrated

Qa = {q>h8 / t)Q. / N + 2vcpb / 3 + 23
2

[1.8.3],

1/AT =

2

(2/3n)(C15 0 ) , and [1.8.7], [C/Q = 3wpb +(3/2)
=^

|

[1.8.11]

This equation may be considered as a generalisation of the exact expression derived by Eisenriegler et al.

and Louis et al.21 for ideal chains in the dilute limit, which is

[1.8.11] with <50 instead of 5. Equation [1.8.11] extends this expression to finite concentrations.

1.8d Gibbs energy of interaction For calculating Ga[h), we need not only Qa(°°) for a single plate, but also iia{K) per plate when the plate separation is 2h . Again, we can use either the osmotic method or the adsorption method. With the osmotic route we calculate Qa = J (/7gs - J70S(z))dz , where /70S(z) depends on the concentration profile in (half) the slit, which has to be computed. In the adsorption method we can start from [1.8.10], but 0 ex = -
dn°s

y

Qa(h)= J h(l-cp)——d

0
[1.8.12]

For the Gibbs energy of interaction we subtract [1.8.10] from [1.8.12]. We write the difference 0 ex (h)-0 ex (°°) as (
where H(x) is the Heaviside step-

function, which is zero for x < 0 and unity for x > 0 . Hence, 9

r

dnos

Ga(h) = -2j [8-h](l-q>)?—H[8-h]dcp o ^

[1.8.13]

When, at a given iph, a constant value <5 = <5(ipb) would be used in [1.8.13], Ga(h) 11 21

E.E. Eisenriegler, A. Hanke, and S. Dietrich, Phys. Rev. E. 54 (1996) 1134. A.A. Louis, P.G. Bolhuis, E.J. Meijer, and J.-P. Hansen, J. Chem. Phys. 116 (2002) 10547.

INTERACTION BETWEEN POLYMER LAYERS

1.39

would be strictly linear in the range 0 < h < <5( 8{(pb). In the form of [1.8.13], 8 = 5{ 8(
interaction Ga(h) = Ga(h)C2/kT as a function of the plate separation 2h (in units t ), for IV = 1000, ^ = 0.4, and four bulk concentrations
We conclude that [1.8.13] works quite well. However, this equation still requires a numerical integration. The basic feature of fig. 1.19 is a linear Ga{h) dependence over most of the range. A simple, explicit analytical approximation for this linear part is obtained by applying the step-function approach to Ga(h) = 2[Qa{h) - Qa{oo)] with Qa (h) given by [ 1.8.12] and fla(°°) by [ 1.8.11 ]. For h -> 0 there is no polymer in the slit, so Ga(h = 0) = -2£2a(~) = -(4 / 3n)(pb 18 . For narrow slits the gap also contains only solvent, so /7os(z) = 0 and Q{h) = /7gsh according to [1.4.9]. Hence,

Ga(h) = 2ng s h-2r2 a H = -2f3aH i-Aj

[i.8.i4]

with Qa{oo) = (2/3ji)
[1.8.15]

Intuitively, one expects 5j (the zeroth moment of the osmotic pressure profile) to be of order 8 (the zeroth moment of the volume fraction profile). Indeed, from Qa = lkT/C3)
1.40

INTERACTION BETWEEN POLYMER LAYERS

Figure 1.20. The characteristic quantities Ga (0) = Ga (0)C2 / kT and 8{ as a function of b (x = °- 5 ) f° r m 'g n
G a (h) = - 2 H ° s ( 5 i - h )

[1.8.17]

is correct provided the proper interaction distance is taken. This distance is not the depletion thickness 5 at a single plate (as presupposed in fig. 1.15), but 54 as defined in [1.8.15], which is slightly higher than 8. With ng s 5j = Qa , [1.8.17b] is the same as [1.8.14] 1.9 Depletion around a sphere 1.9a Volume fraction profile In section 1.8b we discussed the zero-field solution of the Edwards equation [1.4.1] in flat geometry, which led to pe = erf(z/2a ) for the end-point distribution. For spherical geometry, with V2 in spherical coordinates, such a zero-field solution also exists11. We consider a sphere of radius a, with a radial coordinate r from the center of 11

E.E. Eisenriegler, A. Hanke, and S. Dietrich, loc. cit.

INTERACTION BETWEEN POLYMER LAYERS

1.41

the sphere. The (radial) distance from the sphere surface is then z = r - a . As for a flat plate, the surface is situated at z = 0 . The end-point distribution is now p e = [z/a + erf(z / 2a g )]/(z/a+ 1), which reduces to the flat-plate limit for a —> °°. There is also an exact expression for the overall distribution p , but we do not give that here. We restrict ourselves in this chapter to the approximation

p = [z/a + t a n h z / 5 0 ] 2 /

(z/a + 1)2, which reduces to [1.8.4] for a —> °° , with 5 0 = 2a g /V?r

according to

[1.8.3]. In a similar vein as the transition from [1.8.4] to [1.8.8] for flat plates, we may generalise this to finite concentrations by replacing 5 0 by 8 = S{N,(ph,%) as given by [1.8.9]: - + tanh — ^ - +1 \ a )

p= ^

[1.9.1]

Figure 1.21 gives an example of the profile predicted by [1.9.1] as compared with the numerical SCF results. In this example (semidilute solution, relatively good solvent) [1.9.1] works almost perfectly. In other cases (theta solvent) deviations may occur, which are related to the slight overestimation of S in a theta solvent as shown in fig. 1.17. In good solvents the predictions of [1.9.1] are very accurate for large a / 5 , but the width of the profile is slightly overestimated when a~8l}.

However, in all cases the trends are predicted very

well so that [ 1.9.1 ] is a useful approximation.

2.9b Depletion thickness around a sphere In [1.9.1], 5 is the depletion thickness at a flat plate given by [1.8.9]. This is not equal to the depletion thickness 8S around a sphere, because of geometrical reasons. Analogous to [ 1.8.1 ], 8S may be related to the negative adsorption

where p is given by [ 1.9.1 ] and 0| x is the total depleted amount of segments in a shell

Figure 1.21. Depletion profiles (with z in units I) around a sphere of radius a = 20f for four chain lengths, at
11

G.J. Fleer, A.M. Skvortsov, and R. Tuinier, loc. cit.

1.42

INTERACTION BETWEEN POLYMER LAYERS

with thickness <5S around the sphere. It is equal to Ts (the excess per unit area) times the area of the sphere 0| x =4rea 2 r s

[1.9.3]

Substitution of [1.9.1] into [1.9.2] and carrying out the integration gives the relation between 8S and 5 as (1 + 8S la)3 = l + 35/a + (7T2 /4)(5/a) 2 . This is nearly the same expression as the exact solution derived by Aarts et al. for the dilute limit (8 = So). The only difference is that the numerical factor n2 I4 is replaced by 3n 14 . In order to get agreement with the dilute limit, we use this factor 3^/4

fl + ^ f = , + 3 * 3 f * f {

a)

a

[1.9.4,

4 {a)

Figure 1.22 gives a double-logarithmic plot of <5s/5 as a function of a/8 . For large particles 8S = S as expected, but for small a/8 , Ss/a scales as (a/8) . Combination of the three equations [1.9.2-4] gives, for the excess amount per unit area, s

C3 V 4 a J

For a —> °° , this reduces to r = 0ex/f2 =-(pb8/t3 , which is the flat-plate excess as given by [ 1.8.1 ]. The second term of [ 1.9.5] is the curvature correction.

Figure 1.22. The ratio Ss/S between the depletion thickness <5S around a sphere of radius a and the depletion thickness 5 at a flat plate as a function of the ratio a/5 . The dashed line corresponds to the asymptotic behaviour for small a/8: Ss/8=

(3K/4)1/3(O/5)1/3.

1,9c Interfacial Gibbs energy of a sphere

In [ 1.9.5] we saw that the adsorption Ts per unit area around a sphere may be split up into a flat contribution, F = -s for a sphere in a solution of non-adsorbing polymer. Hence, Qs=Qa+Qc

[1.9.6]

The flat contribution Qa was discussed in section 1.8c, and we can directly use the 11 D.G.A.L. Aarts, R. Tuinier, and H.N.W. Lekkerkerker, J. Phys. Condens. Matter 14 (2002) 7551.

INTERACTION BETWEEN POLYMER LAYERS

1.43

result as given in [1.8.11]. In this section we consider only the curvature contribution Qc . In the SCF model, it is obtained as Qs - £>a , where both terms have to be evaluated numerically. In the adsorption method, we start from dQc = -[Fs - F)dn , where Fs - T = -(n / 4)((j>b / C3 )82 I a according to [1.9.5]. Analogous to the derivation of [1.8.10], this leads to <2c=^fs2(l-
[1.9.7]

The integrand depends only very weakly on cp. Let us consider two limiting cases. In dilute solutions 5 2 = 5g ={4/n)N/6 and d/7 o s / dip = 1 / JV, so Qc = < p b / 6 a . In semidilute solutions in a good solvent 82 ~ t;2 - (Suip)"1 and dnos / dq> ~ vq>, so Qc ~ [K/2)(ph/6a. Hence, Qc~
fi

--!^=iHr

[1 9 81

--

for two reasons. The first is that this form coincides with the exact solution11 Qc / kT = [
11 21

A.A. Louis, P.G. Bolhuis, E.J. Meijer, and J.-P. Hansen, J. Chem. Phys. 116 (2002) 10547. CM. Wijmans, F.A.M. Leermakers, and G.J. Fleer, Langmuir 10 (1994) 4514.

INTERACTION BETWEEN POLYMER LAYERS

1.44

Figure 1.23. The overlap volume Vov (gray area) between two spheres of radius a with a depletion layer of thickness 5S at separation 1h between the sphere surfaces.

The geometry is sketched in fig. 1.23. From elementary analytical geometry the overlap volume is easily derived: V

oV(h) = ^-{8s-h?{3a

+ 2S

s+h)

0
[1.9.9]

where 8S is given by [ 1.9.4] Now the Gibbs energy of interaction can be written as an extension of [1.8.13): G

'r' 3/7°s V 1 a( )=-] ov( -
0

[1.9.10]

^

where Vov is a function of

0, a» 8 ) with an exact result for the contact potential Ga(0) derived by Eisenriegler11. This result is Ga (0) aV

aj =-Cct

C = 4wln2 = 9.71

[1.9.11]

For h = 0 , Vov(0) in [1.9.9] reduces to 2na82 for 8S « a . In this case 8S = 8 (see fig. 1.22), which equals 8Q = (2/4n)a g for cpb -> 0 . Hence, Vov = 8 a • a | . With d/7 o s /dip = kT/NC3 , [ 1.9.10] reduces to [ 1.9.11 ] with C = 8 , which is very close. According to [1.9.11] with a | IN I2 = 1/6, in the dilute limit and for a / 5 s > l , G a (0) depends only on the product a 1, dV ov I Ah \h=0 is approximately equal to - 6 a § , which is proportional to 1 / Jvq>h . Since d/7 o s /dtp ~ v
11

E. Eisenriegler, Phys. Rev. E 55 (1997) 3116.

INTERACTION BETWEEN POLYMER LAYERS

1.45

These features show up in fig. 1.24, which gives Ga[h) for cph = 0 . 0 5 , a/£ = 10, N = 1000 and three solvencies. Indeed, the contact potentials are nearly independent of solvency, whereas the slope is higher and, hence, the range of attraction smaller for better solvents. We note that Ga(h) is roughly proportional to a so that curves for different particle sizes have the same general shape. When h but the slopes (~
Figure 1.24. The Gibbs energy of interaction Ga (in units kT per site) as a function of the slit separation 1h (in units () according to [1.9.10], for a/C = 10,
1.10

Adsorption versus depletion

In this section, we briefly discuss how adsorption or depletion phenomena influence a colloidal system and how one can experimentally recognize whether adsorption (loop/ bridge) or depletion (overlap of depletion zones) is the relevant attractive contribution in the system. This task is, as we will see, not trivial, the more so because the polymer contribution to the forces between colloids is rarely the only one; Van der Waals forces, electrostatic forces, undulation forces, etc. may mask the observations. We have seen that the depletion mechanism contributes a relatively weak interaction force to the system with an associated length scale proportional to the bulk correlation length (semidilute) or the coil size (dilute solutions). We may anticipate that, in the absence of other interactions, depletion may give rise to a phase transition in the system where there is a phase rich in polymer and dilute in particles (colloidal gas) in equilibrium with a phase dilute in polymer but rich in particles (colloidal liquid or solid), see IV.5.7. It is known11 that the ratio 8i/a between the range of interaction and the particle size determines whether the phase rich in particles remains fluid-like or is a solid (crystal). The latter occurs for 5{ la « 1, whereas a liquid phase is formed when 5j / a > . l . In sec. 1.8b we saw that 5j [~ 5) decreases with increasing polymer concentration. This complicates the theoretical description of such phase behaviour. However, the quantitative dependence 8{N,
M.G. Noro, D. Frenkel, J. Chem. Phys. 113 (2000) 2941.

1.46

INTERACTION BETWEEN POLYMER LAYERS

The relevant length scale in loop/bridge attraction is also strongly related to the bulk correlation length, but the strength of the bridging attraction is usually much stronger. We illustrate this in figure 1.25 where we present the depth Ga (0) of the attraction (at 'contact') as a function of the adsorption energy Axs with respect to the critical point Xsc for a homopolymer with chain length 1000 in good solvent (% - 0 ) and for two values of the polymer concentration; near overlap (
Figure 1.25. Numerical SCF results for the dimensionless Gibbs energy of interaction Ga (0) between two plates at 'contact' as a function of the adsorption energy Axs = xs ~ XSc ^or a w 'de range of adsorption energies (a) and for the region near the critical value for the adsorption energy (b). The dotted line in (a) is a linear extrapolation according to Ga (0) = 0.2 - Axs • Parameters N = 1000, cpb = 10~3 and 10~ 2 ,and x = °In fig. 1.25a we see that the loop/bridge attraction, which is found for positive values of Axs, is much stronger than the depletion interaction. On the scale of this figure, we can hardly see the attractive interaction caused by depletion in the range Axs < 0 . For high adsorption energy the contact interaction depends linearly on Axs • The dashed line, drawn according to Ga(0) = 0.2 - A%s, describes the data well for Axs > 0.3 . It is interesting to note that this linear dependence breaks down near the critical condition. To examine the critical region, as well as the depletion region, we also present an enlarged view (-0.1 < Axs < 0.1 ) of the same data in fig. 1.25b. Note the difference in absolute values of the interaction energy in the two figures (more than a factor of 100). Only on this scale is the attraction in the depletion range visible. In the first approximation we expect the depletion minimum for strong repulsion to follow G a (0) = 2[Qa(0)-Qa{oo)] =-2i2 a (~)~
INTERACTION BETWEEN POLYMER LAYERS

1.47

1.13a in sec. 1.7c. Obviously, for very large negative A%s (strongly repelling surface), Ga(0) becomes independent of Ax • However, the dependence of Ga(0) around Axs = 0 may be important for depletion interactions in experimental situations. A very small remaining adsorption energy 0 < xs < Xsc =0.18 may seriously influence the strength of depletion interactions. The standard depletion theories all start from the assumption of a strongly repelling surface, which may not always correspond to the real experimental conditions. Near the critical condition one would intuitively expect that the depth of the interaction curves becomes zero. However, in this region GSA is not applicable because the chains near the surface have conformations in which there is a significant ranking number dependence. (Another case where this ranking number dependence shows up so that GSA fails is for brushes, as will be discussed in the next section). In other words, tails are important in these cases. As a result, the contact Interaction becomes positive in the critical region! Note that the absolute value of the positive contact energy is of the same magnitude as the maximum contact interaction found for depletion. Figure 1.25 may be used to elaborate on the issue of the introduction to this section, i.e. how to distinguish between adsorption and depletion. For very strong adsorption the situation is obvious in most cases. All particles will be covered by polymer, and it is not difficult to verify that experimentally, e.g. by dynamic light scattering. Standard techniques will also allow the measurement of the amount of polymer adsorbed on the particles (e.g. with the classical bulk depletion technique or by reflectrometry). However, when the adsorption energy is not far from critical, one may wonder whether the attraction due to the polymers has a loop/bridge or a depletion origin. The key idea here is to use the repulsion found near the critical region to discriminate between bridging and depletion. It is well-known that the effective adsorption energy may be tuned by the addition of a so-called displacer1'. If, by adding a displacer to the system, the attraction first reduces in strength and then increases again, one must have crossed the critical region. The original system must have been in the bridging regime. This trick may become more problematic if addition of a second solvent also influences the effective solvent quality. In section 1.12 we shall discuss some kinetic effects, and at the end of that section one may find additional guidelines to distinguish adsorption from depletion in experimental situations. Such guidelines are useful since many examples can be found in the literature where one presupposes a depletion mechanism without sufficient justification. Not only depletion at a hard wall, as discussed in the present section, is important, but also 'soft depletion' is often encountered. We shall return to this issue in sec. 1.1 lh. The key idea here is to use the repulsion found near the critical region to 11 M.A. Cohen Stuart, G.J. Fleer, and J.M.H.M. Scheutjens, J. Colloid Interface Sci. 97 (1984) 515, 526.

1.48

INTERACTION BETWEEN POLYMER LAYERS

discriminate between bridging and depletion. It is well known that the effective adsorption energy may be tuned by addition of a so-called displacer11. If by adding a displacer to the system the attraction first reduces in strength and then increases again, one must have crossed the critical region. The original system must have been in the bridging regime. This trick may become more problematic if addition of a second solvent also influences the effective solvent quality. In section 1.12 we shall discuss some kinetic effects, and at the end of that section one may find additional guidelines to distinguish adsorption from depletion in experimental situations. Such guidelines are useful since many examples can be found in the literature where one presupposes a depletion mechanism without sufficient justification. Not only depletion at a hard wall, as discussed in the present section, is important; 'soft depletion' is often also encountered. We shall return to this issue in sec. 1.1 lh. 1.11

Tethered polymers and polymer brushes

1.21a Introduction In this section we consider polymers that are terminally attached, or tethered by one end to flat surfaces. Such systems are easily made experimentally and provide an important tool to control colloidal stability. The combination of insights into the behaviour of end-grafted chains with those for unconstrained homopolymer at interfaces (sees. 1.1-9) provides the basic tools to understand the equilibrium characteristics of most, if not all, interfacial polymeric systems. Some aspects of end-tethered polymer chains were already discussed in sec. III.3.4 on Langmuir monolayers. In the previous sections we investigated what happens when homopolymers floating freely in solution are exposed to a solid surface. It was shown that the excess adsorbed amount is one of the important measurable quantities. The number of interfacial chains is determined on the one hand by the solution properties (concentration, chain Figure 1.26. The Huygens equal-time-of-flight oscillator (top) and the corresponding polymer conformations (bottom) in a brush (gray background). On top, a mass suspending from a string with four arbitrary 'starting' positions with zero velocity is indicated in various gray levels; the oscillation time is the same for all cases. At the bottom, the corresponding typical conformations are drawn. The local stretching of the chain increases towards the surface, which translates into the increase in velocity of the mass when it accelerates towards the bottom of the parabolic potential. Note that the relative frequency of each conformation in the brush is not the same for each starting point.

11 M.A. Cohen Stuart, G.J. Fleer, and J.M.H.M. Scheutjens, J. Colloid Interface Sci. 97 (1984) 515.

INTERACTION BETWEEN POLYMER LAYERS

1.49

length, solvent quality) and on the other by the surface properties (adsorption energy). The equilibration of the interfacial chains with those in solution may lead to several subtle effects. An important result is that the layer thickness of adsorbed chains never exceeds the coil size in solution, and for strong adsorption it is even considerably smaller. In this case, the ground-state approximation (GSA) can be successfully applied, because to first order all segments have the same distribution. Directly linked to this symmetric behaviour is the attractive loop/bridge contribution to the colloidal stability. We also saw that tails, i.e. chain fragments with an asymmetric segment distribution, give a repulsive contribution to the pair potential. However, for adsorbing polymer this is a small effect: the main features are captured by GSA, leading to attraction. If equilibration of the polymer chains is restricted, for example by imposing a confining constraint on one of the segments of the chains, the system properties change dramatically as compared with the unrestricted case. One important issue is that the number of chains on the surface can be varied independently, and does not depend on solution properties. Due to the constraint, the condition of equal chemical potentials of end-tethered chains and unconfined chains in solution no longer applies. The second key point is that each segment along the chain automatically gets its own spatial distribution, distinctly different from all others. This is easily seen when, e.g. the first segment s = 1 of the chain with ranking numbers s = 1, ..., N is tethered to the surface, the second segment has just a few degrees of freedom and can be distributed in a region 0 < z < f (where t is the bond length). The third and fourth segments can distribute more freely, and the free end s = N can be in the region 0< z<[N-1)1. The upper limit only occurs when the chain Is fully stretched, which is an unlikely event unless the chains are extremely densely grafted. From these ranking, number-dependent distributions, we must anticipate a complete failure of GSA. Indeed, this is the case, and as we will see GSA is replaced by an astonishingly simple theoretical approach based upon the equal-time-of-flight principle first suggested by Semenov1', the so-called strong, stretching limit. When the number of restricted chains is high, such that at the surface a polymer film with high (semi-dilute or higher) concentration exists, the layer becomes easily (much) thicker than the radius of gyration. This implies that the individual chains are typically stretched as compared with their random coil configurations. A layer of strongly stretched chains is called a polymer brush. The breakdown of GSA does not imply that the Edwards equation [1.4.1] cannot be used. This equation remains valid and the Semenov approach gives a remarkably simple functional form for the selfconsistent field u . The key element is that each chain, starting from its free end, has to reach the 11 A.N. Semenov, Zhur Eksp. Teor. Fix. 88 (1985) 1242; transl. as Sou Phys. JETP 61 (1985) 733.

1.50

INTERACTION BETWEEN POLYMER LAYERS

surface ( z = 0 ) in JV steps, where JV is chain length; the number of steps is the same, irrespective of the starting positions. There is a clear analogy with an harmonic oscillator where the time (the number of 'time steps') to reach the bottom (z = 0 ) is the same for any starting position of the oscillator (see fig. 1.26). Dating back to the days of Huygens, we know that such an equal-time-of-flight oscillation corresponds to a parabolic potential well; the oscillation time of an harmonic oscillator only depends on the length of the oscillator and the pending mass, but not on the amplitude it is given. In a polymer brush a similar phenomenon is present. For any position of the free end (equivalent to the amplitude), after JV steps (equivalent with time) the chain has to reach the surface. Consequently, the self-consistent potential energy field must have the equal-time-of-flight property and, therefore, must be parabolic. In good solvents we know that u ~ cp and thus the brush has as a parabolic concentration profile (more details to follow below). It is necessary to mention the approximation made in this equal time-of-flight argument. From the analogy employed, it is obvious that we only consider those conformations that have the property that segments with higher ranking numbers must be further from the surface than segments with a lower ranking number. 'Loops' in a conformation, where a segment with a lower ranking number happens to be at a higher coordinate than a segment with a higher ranking number, are neglected. This approximation is reasonable when the chains are strongly stretched (at high grafting density), but less so when loops ('fluctuations') become important at low grafting density. It is possible to solve numerically the Edwards equation (or, equivalently, use the numerical SCF model) and check the accuracy. From this exercise, we know that the predictions from the strong-stretching approximation are accurate in a wide range of relevant conditions (see fig. 1.30 below). When describing a polymer brush, one could say that it is composed of a set of polymer tails. The overlap of tails leads to an increase in local concentration. The high segment concentration, in turn, gives a high osmotic pressure, which is unfavourable to the system. Hence, repulsion is expected when two brushes in a good solvent are forced to overlap. Indeed, because there are so many chains on the surface, the expected forces are large and end-tethered polymer layers are an excellent means to stabilize colloidal systems. 1.11b Where do polymer brushes occur? End-tethered chains can be produced in several ways. One way is to chemically graft (existing) chains to a surface. In this approach, it is possible to make a homodisperse brush when the starting chains are all of the same length, but for obvious reasons it is not easy to make a dense brush. Another method is to grow new chains from a surface in contact with a solution. In this method it is easier to generate a dense brush, but it is more complicated to control the polydispersity. There are

INTERACTION BETWEEN POLYMER LAYERS

1.51

several examples in the literature 12 ' 3>4) (see also sec. III.3.8f) showing that one can indeed tune the control parameters, such as the chain length and the grafting density, to physically relevant values. Such systems are being used to test existing theories, and to control surfaces in nanotechnology applications. Polymer brushes, where each chain is fixed with one end at a surface, represent an extreme case. However, similar conformations are found in many situations as the constraints imposed on a chain may also originate from correlations within the macromolecules themselves. Brushes will typically occur when copolymeric materials are adsorbed (or grafted) on the surface. Although we will not treat these systems in any detail, it is worthwhile to briefly mention such systems. If we consider a copolymer composed of two types of segments (A and B), there are several scenarios. Let us first assume that the solvent is not selective and that the solvent quality is good for both segment types, i.e. for the whole chain. There are many ways to distribute the segments A and B along the chain. Let us consider the extreme case of a diblock copolymer A n B m . In this case there are n segments A chemically grafted to m segments B. When both n and m are large, and when the average adsorption energy is sufficiently above a critical value, the polymer chains will adsorb onto the surface already at very low polymer concentration in the bulk. Even the slightest disparity in surface affinity (xsA * # sB ) will cause one of the blocks to be preferentially on the surface because, typically, WxsA - m ^ s B | » 1 because n and m are large. The plateau of the adsorbed amount is now found when the adsorbing block (let us assume this block is composed of segments A) covers approximately all adsorption sites, such that the surface layer is nearly filled. Obviously, the B-block then protrudes into the solution, similar to end-grafted chains on the surface, but with the starting point of the B-block displaced somewhat from the surface and distributed through space to some extent. Assuming that the system is at full coverage, the grafting density is inversely proportional to the length of the A-block and the thickness of the brush is given by the degree of polymerization of the B-block. A much better scenario presents itself when the solvent is selective, i.e. one block is soluble whereas the other is not. An extra driving force (escape of the insoluble segments from the solvent) is now present to bring the insoluble chain parts together and form a brush composed of the soluble entity. When n and m are large, one could qualify such polymers as polymeric surfactants. In many polymeric surfactant systems we can observe brush formation. The corona of a micelle, irrespective of the 11 E.P.K. Currie, F.A.M. Leermakers, M.A. Cohen Stuart, and G.J. Fleer, Macromolecules 32 (1999) 487. 21 E.P.K. Currie, W. Norde, and M.A. Cohen Stuart, Ado. Colloid Interface Sci. 100-102 (2003) 205. 31 J.H. Maas, M.A. Cohen Stuart, A.B. Sieval, H. Suilhof, and E.J.R. Sudholter, Thin Solid Films 426 (2003) 135. 41 J.H. Maas, G.J. Fleer, F.A.M. Leermakers, and M.A. Cohen Stuart, Langmuir 18 (2002) 8871.

1.52

INTERACTION BETWEEN POLYMER LAYERS

geometry (spherical or cylindrical micelles or flat lamellae) is always brush-like. Polymeric surfactants are excellent substances to cover surfaces with brushes (adsorbed micelles, adsorbed monolayers and bi-layers). Again, the grafting density of the corona chains, as well as the length of these brushes, may be tuned by choosing the values of n and m. Some more details were discussed in chapter III.4. Polymer brushes may also form in systems that feature macromolecules with a more complex internal topology. As an example, we mention the generic properties of comb-like copolymer. The insight here is that in most cases there is some chemical disparity in the macromolecule; the teeth usually have different chemical groups as compared with the backbone. This disparity will generate a structure with some local asymmetry. The organization may in part be driven by unfavourable enthalpic interactions, but is also helped by a tendency to reduce the local osmotic pressure (chain crowding). In solution, a comb-like copolymer with a sufficient number of teeth may be seen as a 'bottle-brush' with a structure similar to that of worm-like micelles (in this case with a quenched 'aggregation number'). Like most macromolecules, they may also be surface active. When they have an adsorbing backbone with nonadsorbing teeth attached to it, they may form very robust brushes with high grafting densities. In conclusion, one can safely say that a brush is often a key structural element of polymers at interfaces and that the physical consequences are substantial and important. 1.11c Thermodynamics In most of section 1.11 we will be concerned with systems in which there are no chains in solution. For such a system, the number of chains at the surface is also automatically the excess number of chains. We then consider brushes in pure solvent. However, in section 1.1 lh we will briefly visit a system with a brush in the presence of an unrestricted polymer in solution. We first discuss the macroscopic thermodynamics of such systems, including the possibility of additional (free) components. The end-grafting of some chains prevents their equilibration with chains in the bulk. Due to this grafting, the chains are distinguishable from any (otherwise chemically identical) mobile ones. Even when the chains are chemically linked to the surface, we may still assign them a chemical potential, albeit they do not have the usual translational degrees of freedom. We will use the symbol fi for the chemical potential of grafted chains; ft does not have a translational part, unlike all other chemical potentials /i{. The Helmholtz energy of the system is F(iVp, {JV,}, A, V, T) = -pV + MpNp +X i / / i JV i + YA • where V is the volume of the system and where all mobile components are indicated with the index i; the grafted polymer (index p) is written explicitly. The differential form is dF = -SdT - pdV + MpdN + ^ //jdJVj + ydA . At least in a thought experiment one should be able to determine the change of the Helmholtz energy by changing the

INTERACTION BETWEEN POLYMER LAYERS

1.53

number of grafted chains, while keeping the remaining degrees of freedom fixed to obtain the chemical potential p of the grafted chains. The surface tension y is found by changing the surface area at a fixed number of grafted chains. This quantity is experimentally accessible when the chains are 'grafted' on a mobile (e.g. liquid-air) surface. On a homogeneous flat surface one typically expresses the extensive quantities per unit of surface area. The relevant thermodynamic potential that determines the interaction forces between two surfaces covered by a brush is then the semi-grand potential per unit area. Because of the constraint imposed on the grafted chains, their number per unit area is fixed (although their chemical potential as a function of h varies) and the number of solvent molecules (and all other mobile components) is variable (at constant chemical potential of the solvent). As the macroscopic pressure is also fixed, we prefer to work in a constant pressure, rather than a constant volume ensemble and thus N

p

QSO

fl

=

" "^- A"^

M i

"Z

V +P

A

= (7

~7*) + /iPa

[1 11 11

- -

where so again denotes semi-open, and where we introduced the number of grafted chains per unit area a = N / A , often denoted as grafting density. When the chains are permanently grafted to a solid substrate, one can defend the point of view that the chains (with conformational degrees of freedom) are an inherent part of the surface and thus is it possible to include the chemical potential term of the grafted polymer pa in the surface tension contribution. However, in a simple brush system where segments of the grafted chains do not interact with the surface, one may argue that the notation of [1.11.1], where the surface tension and the chemical potential of the grafted chains occur, is useful. This is so because the main contribution of interest is then pa and we can assume y -j* , independent of a . This Ansatz is used in most of the analyses in the following subsections. However, in section 1.1 lh the full equation [1.11.1] will be used and it must be understood that the numerical SCF model more easily gives access to Q^°, rather than to the surface tension or the chemical potential of the grafted chains. We mention again that Q^° [1.11.1] is the central quantity that must be computed as a function of the separation 2h between two surfaces. Typically, it is calculated per surface (i.e. over half of the slit). As before, the Gibbs energy of interaction is found as Ga(h) = 2(i2|0(h)-r2|°(oo)), where the factor 2 is needed to account for the two surfaces, as in [1.1.1]. 1.1 Id Isolated grafted chains on a single surface: mushrooms In general, segments of grafted chains interact with the substrate. They are either repelled by or attracted to the surface. For high-grafting density o~, a brush develops in which the majority of the segments cannot find a place on the surface so that the

1.54

INTERACTION BETWEEN POLYMER LAYERS

segment-surface interaction, as expressed by the term y — y* in [1.11.1], hardly makes a difference for the macroscopic properties, such as the brush height or the way brushes affect the colloidal stability; entropic (crowding) effects dominate In this regime we may use [1.11.1] in the form £2|° = jx a because y = y* . When o" is low and no substantial crowding occurs, the repulsion and attraction situations are very different, leading to so-called mushrooms for repulsion and pancakes for attraction (see fig. 1.27).

Figure 1.27. Illustration of a mushroom on a repulsive surface (top) and a pancake on an attractive surface (bottom). In a mushroom the conformation of the chain resembles that of a chain in solution, with length scale a«. In a pancake the thickness is 1/c (see fig. 1.2), which is of order C .

To describe mushrooms or pancakes, we can start again from the Edwards equation [1.4.1] and note that, for this dilute case u = 0, like in the Henry region of the adsorption isotherm or in the dilute depletion limit (section 1.8b(£)lConfigurations with one end on the surface are characterized by the function Gat (z, IV) (where the subscript a t denotes 'attached') for which the boundary conditions are quite different from those in the previous sections, Gat(0,0) = 1 (the grafted end) and Gat(°°,IV) = 0 . As before, we can express the segment-wall interaction by the parameter c as defined in [1.4.3] and fig. 1.2. (Note, however, that in this case G = 0 in the bulk 'solution'). The exact solution for terminally attached configurations, for arbitrary value of c and expressed in the scaled variable Z (in units of twice the radius of gyration a ? ) is available" (p=Gat(z,N)=-^— + cY(z-ca«)e-z2 & •Jna \ > _

&

J

Z= — 2a

[1.11.2]

&

Here e (z) is the distribution of the end point (free end) of the chain. The function Y(x) is defined by Y(x) = erfcfxje* , which has the limiting forms Y(x) = 1 / x4n for large positive x and Y(x) = 2 exp(x2) for large negative x. This result immediately gives the probability density distribution (pe(z) of free chain ends. For the case of a strongly repelling surface (c < < 0), the expression simplifies to

11

Y. Lepine, A. Caille, Can. J. Phys. 56 (1978) 403.

INTERACTION BETWEEN POLYMER LAYERS

e~z2

Gat{z,N) = -^

1.55

[1.11.3]

The overall distribution of segments {z) = JG at (z,s)G fr (z,JV-s)ds, where now the end-point distribution function Gfr for a free walk is also needed. For c < < 0, this function is given by Gfr(z,JV) = erf(Z)

[1.11.4]

which is the same as [ 1.8.2] With this form of the composition law, the overall concentration profile turns out to be
6VJra N

s- erf (2Z) - erf (Z)\ L

[1.11.5]

J

Figure 1.28 shows the two distributions (pe{z) and ip{z), according to [1.11.4 and 5], respectively. Both distributions are normalized to unity. Clearly, the behaviour at large z is reminiscent of a Gaussian, but near the surface both distributions are more like linear functions of z, starting at the origin. One also sees that the average distance of free ends (the first moment of > 0), the density distribution
l.lle

Brushes on a single surface

(i) General features. Tethered chains can be considered mutually non-interacting as long as their inverse surface density CT"1 is smaller than the area r02 = Nt2

Figure 1.28. End-point distribution
1.56

INTERACTION BETWEEN POLYMER LAYERS

Figure 1.29. Artist impression of a polymer brush. The height L and the grafting density a are indicated. Note that not all end points are in the periphery of the brush; the end-point concentration increases with increasing distance from the wall. The overall density decreases monotonically.

occupied by an isolated mushroom. Hence, this condition roughly amounts to (jNC2 ~ 1 (for ideal chains or theta solvent). When the grafting density is substantially higher, the chains interact strongly, and we have a polymer brush. We sketch such a brush in fig. 1.29. Since the segment concentration and, hence, the osmotic pressure in a brush is high, brushes tend to swell substantially (compared with a ), particularly in good solvents, thereby reducing their concentration. However, all chains have one end on the surface, so the segment density can only be lowered by stretching the brush in the direction normal to the surface. Stretching a chain (increasing the average distance between grafted and free end) will lower the conformational entropy (increase the Helmholtz energy) and, therefore, give rise to a restoring elastic force between the ends. In contrast, a brush in a poor solvent tends to contract to dimensions smaller than the ideal Gaussian coil. This contraction also reduces the number of available conformations and the associated entropy, and thus also produces an elastic force. It is possible to calculate density profiles for brushes on the basis of the Edwards equation [1.4.1] or with the numerical SCF scheme (see section 1.4b), but a simple scaling analysis of brushes, pioneered by Alexander11 and De Gennes21, already provides much insight. In this analysis it is assumed that all chains are stretched to the same extent, so that it is enough to consider the balance between stretching and osmotic pressure for a layer of homogeneous density
S. Alexander, J. Phys. France 38 (1977) 983. P.-G de Gennes, Macromolecules 13 (1980) 1069.

INTERACTION BETWEEN POLYMER LAYERS

1.57

is given by the well-known Gaussian formula: _3r^

G(r,JV) = Ce

2r

o

r$ = Nt2 = 6 a |

[1.11.6]

where C is a normalization constant and rQ the unperturbed, averaged end-to-end distance. The integral of G(r,JV) over all r is the single chain partition function. Hence, G(r, JV) can be looked upon as a partial partition function (namely, that part of the total partition function for which the end point is at position r). Quite often in the literature, the characteristic function of this partition function is given the letter F . In fact, it is the elastic stretching part of the chemical potential of the chain, which in [1.1.11] was denoted by jx . Here we choose a compromise and use the notation F e l . The elastic Helmholtz energy for stretching a free chain (with unperturbed end-to-end distance r0 ) to a distance r follows as:

^l^lni^L^t-XV-D kT G(ro,JV) 2{rg J 2 l

[1.11.7]

which is positive for r > r 0 . We have defined an expansion (deformation) coefficient a by: a =— r

[1.11.8]

0

which is larger than unity when the chain is stretched.

Note that the yardstick for

brushes is the end-to-end distance r0 = ^VJV (determining the degree of stretching a ), whereas in describing the solution properties of chains typically the radius of gyration a g ( = r0 / V6 ) Is appropriate. The elastic/orce corresponding to [ 1.11.7] is / e l = -3F el /dr or

^=4=-A a kT

r02

[M1.9]

r0

Chains confined in a slit of width r smaller than r0 undergo compression (a < 1). Also in this case the free energy is increased. One can show1'21 that for strong confinement the partition function G(z,iV) scales as exp[-7r2r02 / 6r 2 ] so that the elastic Helmholtz energy with respect to unperturbed chains is now

which is again positive because a < 1. In order to have a continuous description for any a , an interpolation formula is often used, which is simply the sum of [ 1.11.7] and [1.11.10]: 11

A.K. Dolan, S.F. Edwards, Proc. Roy. Soc. A337 (1974) 509. E.F. Casassa, J. Polym Sci B: Polym Lett. 5 (1967) 773, Macromolecules 9 (1976) 182; E.F. Casassa, Y. Tagami, Macromolecules 2 (1969) 14. 21

1.58

INTERACTION BETWEEN POLYMER LAYERS

— =-(a2 +cr2-2) kT 2

[1.11.11]

with the associated elastic force

l111121

w-H-st)

{Hi) Box model. Let us now consider a brush of homogeneous density tp and thickness L in which all chains have their free end at L, so that the expansion coefficient for any chain equals a =— r o

[1.11.13]

This simple model (step function for
is the number of

chains in the system and A the area. Clearly, the amount of segments per unit area equals JV- JVp / A = No . Since the area per segment is t2 , NG('2 is the number of segments per segmental area or, in a lattice model, per site. We use the symbol 9 for this quantity e =
d = at2

[1.11.14]

In the equations to follow, the dimensionless grafting density a will often occur. The volume occupied by the polymer is N • N £3 , so that the average volume fraction (p in the brush is N • N £3 / AL or (p = (7lV- = (7lV— L ar0

[1.11.15]

As shown above, there is an elastic force per chain fei. unit area, there is an elastic 'pressure' J7el = felo

Since there are a chains per

in the brush. Mechanical equil-

ibrium requires zero (internal) pressure; hence, the osmotic pressure /7 0S needed to balance this is simply J7os=-/elo-

[1.11.16]

Inserting J7OS ~ vq)2 /2 + q>3 / 3 (which is [1.2.4] but without the

where, analogous to [ 1.11.8], the equilibrium swelling coefficient is defined by aeq=Leq/r0 [1.11.18] We can now examine various limiting cases of [ 1.11.17 ]

INTERACTION BETWEEN POLYMER LAYERS In good solvents, l/o:

1.59

is small and we find

implying that the thickness of a brush in a good solvent increases as o"1/3 . This has been verified experimentally (see, e.g. fig. III.3.96 in section III.3.8f). We shall see in the next section that for a more realistic (parabolic) profile the same scaling applies as in [1.11.19], with a slightly higher numerical prefactor. In theta solvents (u = 0), and for low grafting densities (small 6), we have from [1.1.17] simply a = l, but when the grafting density is sufficiently high ( 0 » 3 ) we obtain

Mt)" V 2

¥-(!)"'»

Again, the numerical factor depends on the details of the model. Finally, in poor solvents (v < 0), layers contract; now the term a 3 can be ignored and we find a brush thickness proportional to 6 = dN : e

1

3|u|

C

3\v\

3 \v\

In this latter case, the factor \v\ plays the role of an effective density, which controls the thickness of the collapsed brush. Note that in all three cases, L ~ N according to [1.11.19-21], which is a direct consequence of the dominating lateral interactions in a brush, tending to align the chains. [iv) Parabolic potential profile. The 'box model' assumption that all chains are deformed to the same extent, and that the polymer density in the brush is uniform ignores fluctuations in the distribution of end points and oversimplifies the overall profile. A slightly more sophisticated approach assumes that the total Helmholtz energy F of a chain, i.e. its chemical potential (composed of an elastic and an osmotic contribution) is fixed, but that the way this is realized by a given chain may differ, depending on where its end point is located (see fig. 1.26). Strongly stretched chains have a small osmotic term and a large elastic contribution, whereas the reverse applies for weakly stretched ones. When the total Helmholtz energy is minimized, subject to this constraint of constant F , one finds quite generally that the field u = u(z) in a brush is not uniform but must have a parabolic shape1'21, u = B-C(z/L)2 . Referring once more to fig. 1.26 we may elaborate on the analogy between a particle moving in a parabolic well (harmonic 11

A.N. Semenov, Zhur. Eksp. Tear. Fiz. 88 (1985) 1242, transl. as Sou. Phys. JETP 61 (1985) 733. 21 S.T. Milner, T.A. Witten, and M.E. Cates, Europhys. Lett. 5 (1988) 503; Macromolecules 21 (1988) 2610.

1.60

INTERACTION BETWEEN POLYMER LAYERS

oscillator) and the chain ends in a polymer brush. From classical mechanics, the period x of an harmonic oscillator is given by x2 = 2n2m / C, where m is the mass of the oscillating particle and C/2 the spring constant. Here we are interested in a quarter of the oscillation time, as this is the time used for the particle to travel from its starting point (with zero velocity) to the bottom of the well (z = 0). We translate the particle problem to the brush by considering a unit mass and set N = x 14, exploiting the analogy between steps along the chain and time 'steps.' Consequently, C = In2 / T 2 = n2 /(8JV2). For a not too dense brush, u = vq> +
Z = z/L e q [1.11.22]

L

eq

(4vd)U\r

-r=brj

N

3(n2)1/3

«° = 21TJ

1/3.2/3

v a

The equilibrium thickness scales in exactly the same way as in the box model, compare [1.11.19], but is higher by a factor (24/7T2)173 =1.345 . In a theta solvent (u = 0, u =
Z = z/Leq [1.11.23]

t

% v3 )

Again, the scaling is the same as in [1.11.20], but the prefactor in L is higher by a factor (2/7r)121/4 =1.185 . Of special interest is the distribution of the free end points. In the box model and in the Alexander-De Gennes model, the end points are all located at z = L . Fluctuations of end points are not accounted for. The more exact analysis based on the equal time of flight argument shows that the ends cannot all be located at the edge, because for a given subset of conformations with fixed (free) end position, the concentration profile is an increasing function with the distance. The reason is that the local stretching in each chain is stronger near the surface than at larger z. This applies to all chains with fixed position for the end point. Hence, if all end points were at z = L , the overall profile would increase with increasing z. We know that the overall profile is doing just the opposite; therefore there must be a non-trivial distribution of free ends. In the strong stretching limit it is possible to obtain analytical expressions for the (free) endpoint distribution
E.B. Zhulina, O.V. Borisov, and V.A. Pryamitsin, J. Colloid Interface Sci. 137 (1990) 495.

INTERACTION BETWEEN POLYMER LAYERS

1.61

result is
^ =0

[1.11.24a]


# = 0.5

[1.11.24b]

In this parabolic potential profile, (pe (z) = 0 for z > L . Again, Z = z/ L , where Leq is defined in [ 1.11.22,23] for x = 0 and 0.5, respectively. From these equations it is seen that the end-point distributions are linear in z for X = 0.5 and approximately so for x - 0 (unless z is close to L ). These simple relations for the end-point distributions may be used to show that the fluctuations of the end-points can be decsribed as 5e = ((z e ) 2 -(z 2 ))/JV °= N , where (ze) = Jz 1, all systems are in the brush regime, the more so the longer the chains. It is clearly seen that for long chains the agreement is excellent, but that for short chains significant deviations occur. For relatively short chains the middle part of the profile is well described by the analytical theory, but there is a depletion zone near the surface and a diffuse tail part in the periphery of the brush. These effects, which are also found in Monte Carlo simulations1'21 and in experiments3'41 are not captured in the parabolic profile, which abruptly stops at z = L . Close inspection shows that the theoretical mastercurve slightly overestimates the density profile. This is due to the fact that in [1.11.22] the assumption was used that u = ln(l - ~ ~cp2 . These deviations become more noticeable when for fixed chain length the grafting density is increased. This increases the segment concentration in the brush and the expansion of the logarithm becomes less accurate. Wijmans et al. showed51 that if the logarithmic form of the potential is maintained, the analytical approach remains accurate up to very high segment concentrations. 11

A. Chakrabarti, R. Toral, Macromolecules 23 (1991) 2016. T. Cosgrove, T. Heath, B. van Lent, F. Leermakers, and J. Scheutjens, Macromolecules 10 (1987) 1692. 31 P. Auroy, L Auvray, and L. Leger, Macromolecules 24 (1991) 2523. 41 A Karim, V.V. Tsukruk, and J.F. Doublas, J. Phys. II 5 (1995) 1441. 51 CM. Wijmans, E.B. Zhulina, and G.J. Fleer, Macromolecules 27 (1994) 3238.

1.62

INTERACTION BETWEEN POLYMER LAYERS

0.2

I

1.5

Figure 1.30. Segment volume fraction profiles
In fig. 1.30c,d we present the normalized concentration profiles for the end points. The profiles approach the analytical forms given in [1.11.24] for long chains, but the profiles for the short chains deviate from this asymptotic behaviour. Again, this is not unexpected because the shorter chains are not very deep in the brush regime. In fig. 1.30 the theoretical brush thickness L c*=iVcr1/3 in a good solvent and L °c No112 were used to normalize the abscissa axis. The fact that these scaling coordinates work well and give (approximately) universal curves proves that the numerical profiles obey these scaling laws for the brush thickness very accurately. Numerical evaluation of the thermodynamic properties of the brush, such as the surface pressure, show that scaling predictions for these properties are very accurate as well. Some of these predictions, for example that the surface pressure scales as JV(75/3 , were discussed and compared with experiment in sec. III.3.4J for a liquid surface. We conclude that, especially for systems well in the brush regime and for cases where the polymer concentration in the brush is not too high (long chains), the simple theory is already very accurate. This implies that this simple theory can be

INTERACTION BETWEEN POLYMER LAYERS

1.63

used as a good starting point for the discussion of the interaction between brushes. [vi] Scaling analysis. As an alternative to the mean-field approach sketched above, it is possible to take into account the excluded volume correlations occurring in polymers in a good solvent. This approach distinguishes regions in which swelling due to excluded volume is dominant (so-called 'blobs') and considers the interactions between blobs as hard-sphere interactions. This is pictorially represented in fig 1.31a; blobs were also mentioned in 1.5b in the context of the properties of semidilute solutions. The scaling analysis is mostly relevant for very long and flexible chains in good solvents; these have a relatively wide semidilute concentration regime in which spatial fluctuations are important. Each blob in a chain contains JVb segments and, hence, there are NI Nb blobs of size £ in a chain. We can still use an elastic contribution to the Helmholtz energy of the form F e l ~{L/rQ)2 as in [1.11.7], but now rQ refers to an ideal chain of (swollen) blobs rather than segments, which implies that in [1.11.5b] r^ = Nt2 is replaced by 2 75 according to [H.5.2.8], as discussed also in r 2 = (jV/JVb)£; , with % given by % = fiV^ section 1.2. Eliminating Nb , one finds pel

T2

T7T N$1/3

kT

[1.11.25]

with an associated stretching force, / e l = -3F el / 3L fel

r T7T

[1.11.26]

kT JV£1/3 Using the well-known result £, ~ cp"3/4 , according to [II.5.2.18], where in this case

T«1/4

may

be

compared

/ e l ~ cc/rQ = L/r02 = L/N(2

with

the

mean-field

result

given

in

[1.11.9]

with

. The difference is a factor
1.64

INTERACTION BETWEEN POLYMER LAYERS

unity, when

^ 1 =^ - ^ / 4

[L11.28]

which also differs from its mean-field value nos ~ q>2 , according to [1.2.4], by a factor
[1.11.29]

which, with

[1.11.30]

The result has exactly the same scaling as that obtained from the mean-field argument [1.11.19] for a good solvent. Note, however, that both the osmotic pressure and the stretching force were different; these differences happen to cancel exactly for L . In semidilute solutions the blob size or correlation length scales as £ ~ (p~a , where in excluded volume scaling a = 3 / 4 and for the mean-field case a = 1/2 . In the brush the density is just a function of the grafting density,

= cr 2/3 .

The blob size in the brush is thus only a function of the grafting density, i.e. £ ~ ( j ~ 2 a / 3 , which is proportional to the average distance between grafting points in the excluded volume scaling (E,/i = 6~xl2),

but larger than that in the mean-field

1/3

picture (S,/ f. = <7~ ). As a consequence, the blobs in the mean-field picture overlap as shown in fig. 1.31b. More importantly, the Helmholtz energy per chain, which is found by counting the number of blobs per chain, is not the same for excluded volume chains as for mean-field chains. Therefore, we must expect that the interaction Helmholtz energy for the two approaches will not be the same. In other words, the cancellation just mentioned leading to the same brush height in both approaches is not expected for interaction. Yet, we will see that the predictions of both models are roughly the same. 1.1 If Interaction between surfaces covered with mushrooms (i) Mean field. Compression of a mushroom will lead to an increase of the Gibbs energy and, hence, to repulsion. We consider two flat plates at distance 2h with some mushrooms between them; the segments do not adsorb (fig. 1.32). As discussed in the text above [1.11.10], the elastic deformation work to obtain one strongly confined mushroom can be calculated by the continuous mean-field method and is approximately equal to kTn2 /6a2, where a is now given by a = 2h/r 0 (a < 1). Again, as for a simple brush, we can in [1.11.1] ignore the surface tension term and

11

P.-G. de Gennes, Scaling Concepts in Polymer Physics, Cornell Univ. Press (1979).

INTERACTION BETWEEN POLYMER LAYERS

1.65

Figure 1.32. Compression of mushrooms between two surfaces a distance 2h apart. In this figure, the top chain is grafted to the left surface and the bottom chain to the right surface. As the chains are so far apart that they do not feel each other, it is obvious that the same compression force is felt when the two chains are both grafted on the left surface and are compressed by a bare surface. In the scaling picture, one counts the number of blobs (spheres) per chain to find the interaction Helmholtz energy per chain.

identify jx = F. As we have two plates, each with a chains per unit area, we compress 2aA chains by a factor 1h / r0 . Hence, the total interaction equals Ga(h),^^ = 2 ^ M

2

=^r

,2h
[1.11.31]

The disjoining pressure fj(h) = -dG a (h)/d2h is

[ii) Scaling. A scaling argument can also be given. When a chain In a good solvent Is confined between two parallel surfaces, it can still behave as a self-avoiding walk (SAW) on length scales up to 2h. Since a three-dimensional SAW of size 2h accommodates (2h/^) 5/3 segments, chains longer than this number of segments must spread out sideways. One can consider these flattened chains as two-dimensional arrays of sub chains or 'blobs', each of which takes a diameter 2h (see fig. 1.32 where three such blobs are indicated). The Helmholtz energy required to deform a chain of length JV (which would have an end-to-end distance In solution of r = N3/5( ) would then be given by kT times the number of newly formed blobs, F e l = (IVINb)kT , where JVb is the number of segments per blob. Since the blob size is 2h = £N^5, we have JVb = (2h/(f/3. Similarly, for an unperturbed chain, N = (rQ/()5/S- For one mushroom, this leads to an elastic Helmholtz energy of the form

Since we have a (non-interacting) chains, we obtain the following scaling expression for the interaction Gibbs energy

1.66

INTERACTION BETWEEN POLYMER LAYERS

G a (h)-ff^J

-SJjJ

(2h
[1.11.34]

^
[1.11.35,

The disjoining pressure becomes ^ ~ ^ ) "

8 / 3

Note that these fi-dependencies are slightly weaker than those from the mean-field results [1.11.31,32], namely by a factor (h/f) 1/3 . Compression of mushrooms gives a weak repulsion. Assuming an unperturbed end-to-end distance of 10 nm, and a grafting density smaller than r,j2 (mushroom regime), e.g. 0.005 nm"2, one finds from [1.11.31] at In = 3 nm a compression energy of about 0.18 kT per nm2, which is about 0.6 mN/m at room temperature. This is much less than for brushes, as will be shown below. Nevertheless, this mushroom repulsion is much higher than the tail repulsion for adsorbing homopolymers (typically of order 10"4 fcT/f2) or the attraction due to depletion (of order 10"3 kT/C2). 1.1 lg Interaction between surfaces with brushes (i) From Helmholtz energy per chain to interaction forces. When the grafting density increases and a brush forms, the situation is more complicated. Now, chains do not just interact with the two surfaces, but also with each other. What happens is then determined largely by the extent of interpenetration of the two layers. In good solvents, we have seen that the grafted chains (on a single surface) repel each other and this leads to stretching (a > 1). The swelling is driven by the fact that by doing so the polymer concentration goes down, thereby reducing the osmotic effect. Interpenetration of stretched chains is always unfavourable because it increases the local polymer concentration without reducing the stretching. Therefore, interpenetration is unlikely to occur in good solvents. Instead, the thickness of the brush decreases upon compression. In poor solvents, on the other hand, interpenetration actually increases the conformational entropy. Interpenetration of the chains keeps the local density high and the chains can relax back to less compressed conformations. In a theta solvent, around v = 0, a transition occurs between these two cases. It is possible to calculate Ga{h) exactly in terms of mean-field models, e.g. via the Edwards equation [1.4.1] or the discrete SCF model. However, some extra insight can be obtained by considering the diagram of fig. 1.33. In this figure, we present an example of the Helmholtz energies F e l and F o s m per grafted chain, as a function of the deformation coefficient a = L/r 0 , for JV = 1000 , d = 0.01 and v = 1 (good solvent). The curve for F e l is given by [ 1.11.11 ]; note that F e l = 0 for a = 1. An expression for F o s was not yet given here, but this follows from the Helmholtz energy density [1.2.2] in a straightforward manner, namely by multiplying f[cp) by the volume available per

INTERACTION BETWEEN POLYMER LAYERS

1.67

Figure 1.33. Helmholtz energy F per chain (in units of fcT / fi ) in a brush (upper solid curve) as a function of the dimensionless chain extension a = L/r0 for a chain of 1000 segments in a good solvent {v = 1), at a = 0.01. The constituting contributions, due to deformation ('el') and mixing with solvent Cos'), are shown as dotted and dashed curves, respectively. The equilibrium extension a = 3.89 occurs where F has its minimum. A curve given by AF = F - Fmin is drawn in the range a < a e q ; this curve represents the interaction between two brushes compressed between surfaces in a good solvent.

chain, which is JVf3 /

1 1 j V ( y - l ) + -JVu

fcT

*

2

[1.11.36]

6

The curve labeled 'os' has been drawn according to this expression, but without the constant term N[% -1)

as this does not contribute to either a

equilibrium brush extension a. which is a

=L

or G a (h). The

/r 0 is found at the minimum of F = F e l + F ° s ,

= 3.89 in this example. From our approximate expression for a good

solvent [1.11.19], obtained by omitting the second term of [1.11.17], we find a

=3.75 ; clearly, our simple expression works adequately. Note that the slopes of

pel _ pos

ancj

p represent the forces that we used in section 1.1 le {ill) to find a

.

Although fig. 1.33, as discussed so far, applies to a single chain in a brush, it also gives information on the interaction between two brushes! As long as interpenetration can be neglected, which is the case in a good solvent, the response of two surfaces being pushed together is twice that of a single brush compressed by an inert wall. The repulsion in the latter case ( a chains per unit area) is oAF, where AF is equal to F-Fmin

in fig. 1.33. The curve labeled AF in fig. 1.33 therefore directly gives the

Gibbs (Helmholtz) energy of interaction as a function of the degree of compression a a

^ eq

=

^ / ^ e q '^ o r

com

P r e s s i ° n by an inert surface) or a I a

=2h/2L

(for two

brushes). For finding G a (2h) for two brushes, we have to multiply AF with

la

(= 0.02 in the example of fig. 1.33). The corresponding plate separation is found as 2h = 2 L a / a e q . It is clear from fig. 1.33 that Ga(h) h =L

is high (of order ikT/t2

for

, in this case) and it increases steeply with decreasing h. In [1.11.40,44] we

will find explicit analytical approximations for G a (h). The interaction curve AF = G a / 2d in fig. 1.33 is on a linear scale. Figure 1.34 gives G a = 26AF (upper solid curve) with a logarithmic scale for G a , and compares it

1.68

INTERACTION BETWEEN POLYMER LAYERS

Figure 1.34. The dimensionless Gibbs energy of interaction Ga = Ga (h) 12 / kT as a function of the scaled distance H = 1h 12L between the surfaces for N = 1000, a good solvent (v = 1), and a - 0.01 on a semi-logarithmic scale. The dashed curve is the numerical SCF result; the abscissa axis was scaled with Le_ = 119(, which is the theoretical result of [1.11.19]. The top solid curve is AF from fig. 1.33 multiplied by 26 ; the stretching parameter a of fig. 1.33 was converted into the scaled plate separation using H = a/aeq , with a e q = 3.89 . The curve with label 'mean field' is [ 1.11.40] and the curve with label 'scaling' is [1.11.44] multiplied with an (arbitrary) factor of 2.

with the result of full numerical SCF calculations (dashed curve). For convenience, we introduced a scaled particle separation H, defined as H = h / L , with L given by [1.1.19] (corresponding to L =l\9C ~lOa« in this case). The two curves labeled 'mean field' and 'scaling' will be discussed below. We see that the result of the box model matches rather well with numerically exact results. The box model gives (obviously) no interactions for H above unity, whereas in the numerical SCF theory there is a (small) repulsion for distances h > L . In the full SCF model there is a tail on each profile (compare fig. 1.30a,b), which gives rise to some repulsion when these tails start to overlap. From the comparison we see that the box model slightly overestimates the interaction energy, but the shape of the interaction curves is remarkably similar. This shows that the box model gives qualitatively correct predictions for the interaction curves. In the following subsection we consider analytical approximations for the interaction curves. (it) Analytical interaction curves. To get a simple expression for the disjoining pressure, we use nel =ajel, with Jel =-3a/r 0 (in units kT/ft) according to the approximate expression [1.11.9], J7OS ~ v
[1.11.37]

where fl{h) is the disjoining pressure. The osmotic contribution is repulsive because the segment concentration increases and the elastic term is attractive since the stretching decreases. The osmotic term depends only on
INTERACTION BETWEEN POLYMER LAYERS

(p(h)

=-'^_—

1.69

[1.11.38]

For a good solvent, L /£ = N(va/6)l/3 and

^Mj^fV^l kT

-H]

[H2

{2 )

[1.11.39,

\

The Gibbs energy of interaction is found from [1.1.3] (with the boundary h = °= replaced by h = L ). The result is 6 (h) =

a

(ir)

JVd5/3

[f + H 2 - 3 j

[1.11.40]

The latter result is plotted in fig. 1.34 with the label "mean field.' We see that [1.11.40] slightly underestimates the interaction energy as compared with the numerical analysis with the box model (label 2CTAF in fig. 1.34), of which [1.11.40] is an approximation. Interestingly, in this case the approximate [1.11.40] is closer to the numerical SCF prediction than the numerical solution of the box model. Anyhow, the simple box model, which neglects the interpenetration, rather accurately describes the full numerical SCF data. A similar exercise can be done for a theta solvent, where J7OS =cp 3 /3 (in units kT/e3). One therefore expects an H~3 dependence, according to [1.11.38]. With Leq/lV = (cT/3)1/2(see [1.11.20]), we obtain

which upon integration gives Ga(h) = IV<72p?2- + H 2 - 3 l

[1.11.42]

The disjoining pressure can also be calculated from [1.11.37] using the scaling expressions [1.11.25,28], instead of their mean-field counterparts [1.11.9, 1.2.4]. The result is ^-o»«f-L-H"«l kT

LH9/4

[1.11.43]

J

with the Integrated form G a (h)~ N6n/6\^(H-5'4

- l ) + | ( H 7 / 4 -1)1

[1.11.44]

INTERACTION BETWEEN POLYMER LAYERS

1.70

In fig. 1.34 we also plotted Ga (h) according to [1.11.44] (curve labeled 'scaling'). In the scaling approach the numerical prefactor is unknown. We multiplied [1.11.44] by 2 to bring the interaction energy to the same level as the mean-field prediction [1.11.40] at the maximum value of the interaction energy plotted (i.e. at Ga (h) = 15 ). Comparison of [1.11.44] with [1.11.40] shows that at weak compression the scaling approach gives slightly weaker repulsion than the mean-field model, whereas at strong compression

the opposite

is the case. However, the differences

are small.

Experimentally it is probably difficult to distinguish between the two. It is also interesting to note that the scaling approach gives a good match with the numerical SCF data. The repulsion due to dense brushes is much stronger than that exerted by dilute chains, mainly because the grafting density a can be so much higher. As seen from [1.11.39] the mean-field pressure scales as cr 4/3 , which implies that if a is increased by a factor of 10 the distance between grafting points is reduced by a factor Vfo and the pressure increases by a factor of about 20. As a rule, pressures measured between brushes can be very high indeed: brushes in good solvents are excellent stabilizers.

Figure 1.35. Numerical SCF results for the scaled Gibbs interaction energy as a function of the normalized slit separation H-2h/2L , where L is given by [1.11.19]. In diagram (a) the Gibbs energy of interaction is normalized by the chain length and results are shown for three degrees of polymerization. The grafting density is a - 0.01. In panel (b) the Gibbs energy of interaction is normalized by d 5 / 3 and curves are given for N = 500 and three grafting densities. (iii) Numerical

examples.

According to [1.11.40], G a should be proportional to

JVcr5/3 . In fig. 1.35 we check these dependencies. In fig. 1.35a we have varied the chain length at a fixed grafting density, a = 0 . 0 1 . We normalized the Gibbs energy of interaction by the length of the chains. There is excellent overlap of the interaction curves for small values of the normalized slit width H. For very weak overlap, i.e. near H = 1, differences are noticeable. The shorter the chain length, the more important the tail on top of the parabolic profile is, and the deviations at weak overlap are due to this diffuse periphery of the brush. In fig. 1.35b the interaction curves are presented for a fixed chain length of JV = 500 and for various grafting densities. Here, we normalized G

by d 5 / 3 . Again,

INTERACTION BETWEEN POLYMER LAYERS

1.71

there is quite satisfactory overlap of the interaction curves at relatively high compression (small H). The differences near H = 1 are again attributed to the fluctuations around the most likely trajectories. For low grafting densities these deviations are relatively more important. For very large overlap, i.e. for small values of H, we see an upward deviation for the highest grafting densities. Equation [1.11.40], based only upon the second virial coefficient, predicts a scaling Ga °= H~l in this region. However, when at strong overlap the segment concentration becomes very high in the brush we can no longer neglect the third virial coefficient. As a result, the interaction curves are more repulsive than the Ga °= H"1 curve. This is most noticeable for high grafting densities. In conclusion, the box model provides excellent guidelines to analyze the numerical SCF data; the main trends are well described and the deviations are easily interpreted. (iv) Experimental example. An experimental example for interaction between brushed surfaces is presented in fig. 1.36 where scaled interaction curves, measured with the Surface Force Apparatus, are given for brushes prepared by adsorbing diblock copolymers consisting of an insoluble polyvinylpyridine block (the 'anchor' block) and a soluble polystyrene block (the 'buoy' block) on a mica surface". The data apply to several block copolymer samples with varying lengths JVA of the anchor block and JVB of the buoy block. The value of Ga was normalized by JVo~11/6, according to [1.11.44], and h by L , according to [1.11.19], using a model to relate N (~ NB), 6 , and L to JVA and JVB . Figure 1.36 shows that all data fall nicely on a mastercurve. The scatter at weak overlap is due to limitations in measuring weak forces. The mastercurve should be described by the H-dependent part of [1.11.44]; as shown by the solid curve in fig. 1.36 this is approximately the case. The agreement could be improved by scaling up the experimental H [i.e. scaling down L eq ) by some 10%; we have not done that. We note that the mean-field expression Ga ~ 2/H + H"2 - 3 would work nearly equally well, as shown by the dashed curve. Hence, one cannot experimentally discriminate between the two models. Figure 1.36. Scaled Gibbs energy of interaction Ga(h) as a function H for two mica surfaces carrying brushes of polystyrene in toluene, modified from Patel and Tirrell11. The solid curve is drawn according to the Hdependent part of [1.11.44] (scaling), the dashed one according to [1.11.40] (mean field) and scaled such as to match at H = 0.1.

11

S. Patel, M. Tirrell, Colloids Surfaces 31 (1988) 157.

1.72

INTERACTION BETWEEN POLYMER LAYERS

Figure 1.37. (a) Gibbs energy of interaction between two plates bearing a brush with nonadsorbing segments (JV = 200 ) for 0 = 10 (a = 0.05). Curves are given for three solvency conditions: u = 0, - 0.02, and -0.04 ( x = 0.5, 0.51, and 0.52) on a linear scale. The theta solvent still gives repulsion (apart from a shallow minimum which is hardly visible on this scale), but for v = -0.02 an attractive minimum appears which deepens (quadratically) as v decreases further. See also Fleer and Scheutjens11. Panel (b) gives the theta curve of (a) on a semi-logarithmic scale, and compares it with the prediction of [1.11.421 (dashed curve), which is the box model without interpenetration.

[v] Poor solvent. Two brushes in poor solvents (x > 0.5) attract each other. The reason is two-fold: the solvent film at the edge of the brush can be removed and the chains will interpenetrate when they touch, due to the fact that the compression of the chains can be relaxed. This increases the chain entropy which is favourable. Upon further decrease of the gap width 2h, the Helmholtz energy can be lowered until a homogeneous polymer phase with a density of order |u| is obtained, which happens at distances of order H = 0.5 or h ~ 0<73|u|, as follows from [1.11.22]. The attraction is expected to start for v just below zero (depending somewhat on o"), which implies that, from a practical point of view, the stabilizing effect of a brush is lost as soon as the theta temperature is crossed. In fig. 1.37a we give an example of the interaction between two brushes as v is varied around zero. Three curves are shown in fig. 1.37a. For v = 0 (# = 0.5) the interaction is nearly completely repulsive, albeit relatively soft. Close inspection shows that before the repulsion sets in a very small attraction occurs. For v = -0.02 (£ = 0.51), this minimum is already much more pronounced and the minimum deepens rapidly as v decreases, as the curve for v = -0.04 (% = 0.52 ) shows. The repulsion at small h is due to crowding effects. For the theta solvent we presented the analytical prediction in [1.11.42]. We show the comparison with the numerical SCF result for % = °-5 in fig. 1.37b. As in the good solvent case we see once again that the numerical SCF predictions already show repulsion for 1h > 2L , due to the tail in the profile in the complete SCF profile, caused by fluctuations of chain conformations not included in the box model. For 11

G.J. Fleer, J.M.H.M. Scheutjens, Colloids Surfaces 51 (1990) 281.

INTERACTION BETWEEN POLYMER LAYERS

1.73

H < 1 this box model systematically overestimates the interaction energy. This is because the model does not allow for interpenetration of the chains, which is a poor approximation for a theta solvent. Indeed, a brush in a theta solvent gives a softer repulsion than predicted by the box model. As H = 1 corresponds to 2L ~ 60 , we note that the shallow minimum in the interaction curve shown in fig. 1.37a is outside the range of fig 1.37b. 1.1 lh Soft depletion In sees. 1.8-9 we discussed the effect of free polymer on a dispersion of hardspheres. Quite generally, we found that polymer induces a weak attractive interaction between the colloidal particles, the strength and range of which depend on the concentration of the polymer and the coil/sphere size ratio a la. In figs. 1.34-36 we saw that particles carrying tethered polymer repel each other. We now consider briefly the mixed case of colloidal particles with grafted polymer on their surfaces, dispersed in a solution of free polymer molecules. We suppose that the grafted chains (length JV, grafting density a) and the free polymer (chain length P) are chemically identical. In fig. 1.38 we present a measure for the thickness of a polymer brush immersed in a polymer solution in which P = N = 500 . The thickness is a strong function of both the concentration of the free chains and the grafting density of the tethered chains. Above we have seen that, upon overlap of stretched chains, they relax to less stretched conformations. We can apply the same argument to explain why the brush thickness decreases upon an increase in the concentration of the free chains. The free chains try to penetrate the brush and therefore increase the chain crowding. The response is a collapse of the swollen brush. The symbols in fig. 1.38 indicate the point where the polymer concentration in the (initial) brush (without outside polymer) equals the polymer concentration in the polymer solution. At those points, the osmotic pressure in the polymer solution equals that of the (initial) brush. The fact that the symbols are in the steepest part of the drop in brush thickness strengthens the argument. At very low grafting density, the (initial) tethered polymer layer has no stretched chains and free chains do not alter the thickness. Up to 9 = 10 a brush immersed in the melt,

Figure 1.38. The average position (first moment) of the free end of a brush with chain length JV = 500 and B = dN as indicated, as a function of the polymer volume fraction
1.74

INTERACTION BETWEEN POLYMER LAYERS

Figure 1.39. A set of interaction curves for a brush composed of JV = 500 segments with grafting density a = 0.01 in a good solvent (% = 0) with free identical chains in solution with P = 2000 (solid curves) and P = 125 (dashed curve) for various concentrations (pb of the free polymer. The open symbols are for
P and a is small, the depletion layer can be entirely incorporated into the grafted layer and the attractive depletion effect vanishes. An example of such a purely repulsive interaction for N = 500 and P = 125 is given in fig. 1.39 (dashed curve). With increasing P/N and at sufficiently high a, the depletion effect is visible, and Ga(h) develops an attractive well before the repulsion sets in at smaller h. The depth of this minimum depends strongly on the free polymer concentration as shown in fig. 1.39. In line with the results of fig. 1.38 we see that the position of the minimum, which reflects the thickness of the polymer brush in the system with free polymer, decreases considerably upon increasing the polymer concentration. It is also of interest to study the depletion effect as a function of the grafting density of the brush. In fig. 1.40 we show results for a system in which the free polymer concentration is fixed to q>h=0.1. The segments do not adsorb onto the surface ( Xs = 0 )• m e solvent quality is good (% = 0 ), and P » N . Both for an empty surface (a = 0, 9 = 0) and at full coverage, (
INTERACTION BETWEEN POLYMER LAYERS

1.75 Figure 1.40. Depth of the interaction Gibbs energy minimum (in units fcT//2 | for two surfaces immersed in a solution of nonadsorbing polymer (P = 300, ^ = 0.1, X = 0 ) as a function of the amount 9 = Na of grafted polymer [N = 50) on the surface. For high grafting density a linear decay is found, which extrapolates to approximately - 7 10~ 3 , which is the zero grafting density value, indicated by the dot.

by the dot at 0=0. For intermediate grafting densities there is a weakened depletion effect due to the presence of grafted chains. Indeed, this so-called 'soft depletion' effect is seen in fig. 1.40. Already very few grafted chains destroy most of the depletion force because the free chains interpenetrate with the dilute brush. As the brush becomes thicker and denser, the depletion effect passes through a minimum (so that G™'*1, which is negative, goes through a maximum). Upon further increase of a, the free chains are repelled from the brush to an increasing extent, and the attractive well deepens until it becomes comparable with the depletion attraction at a hard and inert wall. For the free chains to enter the brush, a necessary condition is aS2 < 1, where the depletion thickness S is given by [1.8.9] and varies between the value 2ag/%/^ in dilute solutions and £/[3vcpb)l^2in semidilute solutions, see fig. 1.17. Short chains thus have a much better chance to be taken up in a brush than long ones; this effect has been analysed in some detail11. l.lli Concluding remarks The literature on polymer brushes has grown tremendously in the last few years. It is impossible to cover all these developments. The fact that the chains are tethered by their ends makes computer simulations relatively easy. Such computer simulations appear to provide essentially the same picture as that obtained by numerical SCF. Furthermore, we have seen that the analytical route via the strong stretching approximation is a very accurate method, which is very helpful in analyzing brushes. The crudest, but nevertheless useful, analytical approach is the box model. The amazing aspect of this primitive model is that all results obtained scale in the same way as those following from more refined models. Only when the end-point distribution is not strongly peaked at the periphery of the brush is the box model less reliable. It is interesting to note that in recent years polyelectrolyte brushes have also received ample attention. In favourable cases, the electrostatic potential dominates the self-consistent potential. Then, the analogy with the Huygens oscillator directly leads

11

CM. Wijmans, E.B. Zhulina, and G.J. Fleer, Macromolecules 27 (1994) 3238-3248.

1.76

INTERACTION BETWEEN POLYMER LAYERS

to a parabolic electrostatic potential profile in a polyelectrolyte brush; accurate analytical solutions of the polyelectrolyte brush closely follow the numerical SCF predictions. The interaction between polyelectrolyte brushes has received some attention as well. Other interesting developments, which we did not deal with in this chapter, are brushes on curved surfaces (relevant for polymeric micelles and vesicles), brushes with nematic ordering, polydispersity effects, and the uptake of proteins or small micelles in brushes. This is a very active field of research where many new theoretical results and experimental applications are to be expected. 1.12 Non-equilibrium aspects 1.12a Introduction In sees. 1.3-7 we considered the effect of adsorbing polymers on interactions between two surfaces. We took the viewpoint that full equilibrium (a constant chemical potential) between the slit and an external reservoir could be maintained for all molecules in the system, i.e. for the solvent and the polymer. We found that such relaxed adsorbed layers give rise to a weak repulsion at weak overlap, and to a strong attraction related to bridge formation at smaller gap widths. In sec. 1.11 we discussed the case of end-attached polymers between two surfaces, where the conserved property was the number of polymer chains between the surfaces, rather than their chemical potential; only the solvent was allowed to equilibrate with the bulk solution. In this case, we found that polymers in a good solvent give rise to a strong and monotonic repulsion between the surfaces. In the present section, we consider the question as to what extent these considerations can explain the behaviour of various colloid/polymer mixtures as they occur in practical situations. We shall see that adsorbed polymer layers often (but not always) relax slowly and we qualitatively discuss the factors that affect relaxation rates. We then discuss what one may expect when the amount of adsorbed polymer in a gap between two approaching surfaces is supposed to remain constant (an example of restricted equilibrium). We examine experiments in which the distance between two surfaces is varied, for example by approach/retraction cycles, and attempt to draw conclusions on the extent of relaxation in such systems. Finally, we address the issue offlocculation induced by adsorbing polymer and the kinetic effects that are so crucial in this process. We illustrate our argument with a case study on shear-induced flocculation rates where many processes occurring simultaneously in a colloid/polymer mixture (bridge formation, conformational relaxation, stabilization) together determine the overall result. When judging experimental data it is important to emphasize that forces due to polymers almost never appear in their pure form, because other interactions such as electrostatic and Van der Waals interactions generally cannot be eliminated. If such interactions are also operative, it is usually assumed that the energetic contributions

INTERACTION BETWEEN POLYMER LAYERS

1.77

from various types of interaction are additive. We realize that this is a questionable assumption as adsorbed polymer layers may well perturb an electrical double layer, and dense polymer layers contribute to dispersive interactions between surfaces. Nevertheless, we accept it for the mainly qualitative discussion that follows. 1.12b Does equilibrium apply? In order for the equilibrium theory developed in the previous sections to be relevant, the relaxation of an adsorbed polymer layer under variations of the gap width has to be sufficiently rapid; the Deborah number De = r proces / tmeas ( s e e 1-2.3.1) should be much smaller than unity. In dilute solutions, polymer conformations relax at time scales determined by the Rouse time, which is the characteristic time for thermal relaxation of the end-to-end distance ; its value for a chain in water with a contour length of, say, 1000 nm is typically in the millisecond range. In concentrated solutions and melts, relaxations require snake-like movements (reptation) of the chains through the dense systems, which occur on the much slower reptation time scale21. At an interface, a number of segments is bound to the adsorbent as trains, and the mobility of these segments will depend both on the strength of the segment-substrate bond and on the mobility of the substrate itself. If the substrate is a solid and the segment-solid bonds are strong, a polymer molecule may be very effectively immobilized. There is plenty of experimental evidence to illustrate this point. Reduced mobility of polymer segments shows up in NMR experiments as a strongly enhanced relaxation rate of the magnetization, and a virtual disappearance of polymer signals from the usual narrow-line (high resolution) spectra that are obtained for rapidly moving molecules. In several experiments on the kinetics of exchange processes between adsorbed and free polymer molecules, slow surface processes have been detected with characteristic times varying from seconds to days. More often than not, the surface pressure generated by water-soluble polymers adsorbed at water-air interfaces shows substantial hysteresis during compression/dilation cycles (see also sec. III.3.8a). One may therefore wonder, particularly in the case of solid substrates, whether equilibrated (annealed) adsorbed polymer layers exist at all. Yet, we can mention several examples. The surface pressure of polyethyleneoxide (PEO) adsorbed at a water/air interface remains constant at about 10 mN/m under slow surface compressions, showing that this system can equilibrate properly. One would, therefore, expect that PEO adsorbed on the surfaces of a free water film would also equilibrate, so that if the weak repulsion due to tails is overcome, attraction sets in and the film breaks. Indeed, this seems to happen because stable films of PEO solutions have never been reported. Many other water-soluble polymers behave likewise; they are unable to stabilize free water films. Exceptions are found when the 11

P.E. Rouse, J. Chem. Phys. 21 (1953) 1272; M. Doi, S.F. Edwards, Theory of Polymer Dynamics, ch. 4, Oxford Science (1986). 21 M. Doi, S.F. Edwards, Theory of Polymer Dynamics, ch. 6, Oxford Science (1986).

1.78

INTERACTION BETWEEN POLYMER LAYERS

polymer has a more or less pronounced copolymeric character. In order to have equilibration of polymer adsorbed at S/L interfaces, it is probably crucial that the adsorption is very weak. One reported example concerns platelets of nbutylammonium-vermiculite clay". The interparticle spacing in swollen suspensions of this clay has been studied as a function of stress applied along the swelling axis. As a result of these experiments, the relation between gap width and applied pressure is precisely known in this system, so that it can be used as a colloidal 'instrument' to measure disjoining pressures. When PEO of varying molar mass is added to vermiculite dispersions, attraction is found to set in whenever the polymer coil dimensions are equal to, or larger than, the distance between the platelet surfaces. Once contraction occurs, the (negative) disjoining pressure is found to be independent of the PEO molar mass, but proportional to the amount of polymer adsorbed on the platelets. These experiments strongly suggest that this is a case of equilibrated polymer adsorbed between two surfaces. The polymer spontaneously enters the gap between platelets, and the platelet spacing measured does not depend on sample history. Moreover, the saturated adsorbed amount turns out to be low (about 0.1 mg/m2), which indicates weak adsorption. We should note that the case of very weak adsorption (near the adsorption/desorption threshold ^ s c ) was treated only briefly in sec. 1.10 where it was shown that for small /s the GSA approach breaks down. Hence, the expressions derived in sec. 1.6 do not apply to the vermiculite/PEO system. However, if the general arguments discussed in 1.6a still apply, namely (i) that attraction is due to the fact that the concentration in the gap is non-uniform and {ii) that the net effect is attraction, this can qualitatively account for the data. 1.12c Restricted equilibrium The interaction between polymer layers that cannot, or can only partially, equilibrate is likely to be different from that between equilibrated layers, but as yet we do not know to what extent. In order to get an idea of the trends to be expected, one can numerically calculate the interaction between two surfaces with adsorbed polymer layers, assuming that the amount of polymer between the surfaces is strictly constant; only solvent can enter and leave the gap. This may be called 'restricted equilibrium', with a quenched layer. The situation is somewhat analogous to the case of constant surface charge for interacting double layers, as opposed to that of equilibrated surfaces with a constant (electro)chemical potential. The structure of the layers (i.e. the profile) may of course adjust itself, and the starting situation is that of two surfaces at large separation, each carrying an equilibrated, adsorbed layer. This situation is difficult to treat by the analytical methods discussed in 1.3-7, but the numerical SCF scheme can cope quite well with it. 11 J. Swenson, M.V. Smalley, and H.L.M. Hatharasinghe, Phys. Rev. Lett. 81 (1998) 5840. M.V. Smalley, H.L.M. Hatharasinghe, I. Osborne, J. Swenson, and S.M. King, Langmuir 17 (2001)3800.

INTERACTION BETWEEN POLYMER LAYERS

1.79

A survey of this case was carried out by Scheutjens and Fleer1'. The quintessence of their findings can be summarized by the diagrams in figs. 1.40. In fig. 1.40 the Gibbs energy of interaction Ga(h) is plotted both for full equilibrium and for restricted equilibrium, for a good and a theta solvent and JV=1000. For the full equilibrium the bulk concentration was fixed to a value
Figure 1.41. Numerical SCF data for interaction curves for full equilibrium (solid curves) and restricted equilibrium (dashed curves) in a good solvent (a, b) and a theta solvent (c, d). Panels band d are zoomed-in versions with respect to the Gibbs interaction energy (not with respect to the slit separation). In all cases, JV = 1000. In full equilibrium, the polymer concentration in the bulk is q>b = 10~ 3 . In restricted equilibrium, the amount of polymer at 1h = 100 is fixed at the value given by the equilibrium case for that separation; 6 = 1.6191 (both walls) for the good solvent case and 6 = 3.9222 for the theta solvent.

II

J.M.H.M. Scheutjens, G.J. Fleer, Macromolecules I I I (1986)504).

18 (1985) 1882; J. Colloid Interface

Sci.

1.80

INTERACTION BETWEEN POLYMER LAYERS

Upon compression, the (total) amount was fixed. The relevant interaction energy is now Ga{h) = 2(i2 so (h)-i2 sc V)), where the semi-grand potential is the sum of the interfacial energy and the chemical work of the confined chains. The quite general result is that the restricted equilibrium curves are above the full equilibrium curves. In other words, any type of restricted equilibrium is always more repulsive than full equilibrium. This is necessarily so because the relevant thermodynamic potential can relax to lower values when there are more degrees of freedom. By the same token, interaction between diffuse double layers at constant charge gives rise to stronger repulsion than at constant potential (see sec. IV.3.4b). In a good solvent (fig. 1.41a,b), the restricted equilibrium curve is (in this case) fully repulsive, whereas the main trend for complete equilibrium was attraction. The repulsion already sets in at the initial separation 2h = 100, as can be seen in the enlargement of fig. 1.41b. The features of full equilibrium as discussed in sec. 1.6 are completely masked. For a theta solvent, the restricted equilibrium follows the complete equilibrium much better. A very characteristic local minimum in the interaction curve is now found at 2h ~\7. For shorter distances, the extra polymers that are confined and cannot escape lead to repulsion. A more systematic study of this type of restricted equilibrium is found in the work of Scheutjens and Fleer, cited above. These authors showed that the depth of the minimum is a function of the amount of polymer fixed in the slit. The more polymer that is restricted between the plates, the greater repulsion that builds up and the shallower the minimum is. At very high restricted amounts, the minimum even disappears and the interaction curves become completely repulsive. Semi-quantitative agreement with these trends was found in surface force measurements with the surface force apparatus (SFA) for polystyrene adsorbed from cyclopentane on mica11, which was very close to theta conditions. In fig. 1.42, we reproduce these data. In a surface force apparatus, it is very likely that the amount of polymer is restricted because of the large diffusion lengths involved in the setup. In fig. 1.42 the amount of polymer increases from left to right. For the lowest amount of polymer the characteristic minimum is found, and the experimental curve resembles in

Figure 1.42. A few interaction curves for homodisperse polystyrene (M = 2000 K), adsorbed from cyclopentane on mica, under theta conditions. From left to right the adsorbed amount increases as found with the surface force apparatus adapted from Y. Almog, J. Klein".

11

Y. Almog, J. Klein, J. Colloid Interface Sci. 106 (1985) 548.

INTERACTION BETWEEN POLYMER LAYERS

1.81

many aspects the restricted equilibrium curve shown in fig. 1.42d. The correlation between the SFA results and the numerical SCF predictions support the idea that polymer layers in confined spaces may not always be in full equilibrium, because of the difficulty of diffusion out of a very narrow slit with macroscopic lateral dimensions. 1.12d Relaxation phenomena in polymer-induced surf ace forces In a qualitative sense, the difference between quenched and equilibrated (relaxed) adsorbed layers shows up only at large 8 (saturated layers). In that case, relaxed layers have a non-monotonic interaction curve with a weak repulsive maximum and strong attraction at small plate distance (sec. 1.7), whereas quenched, adsorbed layers develop a strong monotonic repulsion (sec. 1.12c). The difference is the strong osmotic effect of the confined polymer, which is not compensated by the subtler entropic effects associated with the evening out of the density profile in the gap due to bridge formation. Hence, when two surfaces with saturated polymer layers are pushed together sufficiently strongly, such that the loop regions can overlap, and if relaxation occurs at experimental time scales, one may expect to observe repulsion at short times, which changes into attraction at longer times. We discuss two examples of such relaxation effects. (t) Indications from AFM experiments. The first example is taken from a study by Giesbers et al. on surface forces between a silica sphere and plate in the presence of PEO. In fig. 1.43 we show the force as a function of distance in a solution of NaCl at pH 8, containing 100 rag/l PEO of molar mass 246 kg/mol. The force is normalized with respect to the radius of the sphere, so that it is given in units of interfacial Gibbs energy (mN/m= mJ/m 2 ). The left diagram (a) refers to the case where the sphere and plate approach each other (ft decreases), the one to the right (b) is for retraction. Three

Figure 1.43 Force-distance curves, measured by AFM between a silica sphere and plate in the presence of polyethyleneoxide (PEO), at pH 8 and three concentrations of NaCl; (a) approach curves and (b) retraction curves. Molar mass of PEO is 246 kg/mol. (Redrawn from Giesbers et al. 1 ').

11 M. Giesbers, J.M. Kleijn, G.J. Fleer, and M.A. Cohen Stuart, Colloids Surfaces A: Physicochem. Eng. Aspects 142 (1998) 343.

1.82

INTERACTION BETWEEN POLYMER LAYERS

Figure 1.44. Approach (dotted curves) and retraction (solid curves), measured with AFM, between a silica sphere and plate in the presence of PEO in aqueous solution. Raw data are given as cantilever deflection A (a linear function of the force) versus piezo position AT. (arbitrary zero) (which is related to the separation 2h). Molar mass of PEO is 246 kg/mol, pH = 6.5, salt concentration 10~3 M NaCl. The approach and retraction curves for a low load force are almost identical, but for a high load force a deep minimum appears due to bridging attraction. Reference as in previous figure.

curves are given in each diagram for different salt concentrations. Clearly, the two sets of curves are rather different; the approach curves are all purely repulsive, whereas the only repulsive retraction curve is for 0.001 M salt. The other two show a clear attraction in the form of a so-called 'pull-off force. The salt dependence in fig. 1.43a must be due to double layer repulsion; at 1 M salt, the double layer force, indeed, has a very small range. Under these circumstances, the retraction curve shows a fairly strong attraction; retraction to about 100 nm is needed before the force comes back to zero. It seems likely that during the brief contact time under compressive load (typically about 0.2 s), the polymer layer was able to form bridges, thus giving attraction. The importance of the load force is shown by fig. 1.44 where the force-distance curves are reported for the same PEO-silica system at pH 6.5 and 1 mM salt. In this experiment, the two sets of curves refer to different maximum compressive load forces. When the maximum load force is small (the contact time is then also shorter), the approach and retraction curves are both repulsive and, moreover, very similar. However, at a larger load force, an attraction appears. This agrees with the idea that in that case the polymer layers relax under compression and then form bridges. Finally, the strength of the attractive pull-off force increases linearly with the amount of polymer between the plates. This is shown in fig. 1.45, where the strength of the pull-off force (the depth of the minimum in the force distance curve) is plotted

Figure 1.45. Normalized pull-off force / o f f / a between a silica sphere and plate due to PEO adsorbed on silica, as a function of the total adsorbed mass F of a saturated layer at pH = 4 and 1CT3 M NaCl. The molar mass of the PEO for each data point is indicated (in kg/mol). Same source.

INTERACTION BETWEEN POLYMER LAYERS

1.83

against the adsorbed mass of the saturated layer (which increases with the molar mass of the polymer). This plot shows that for low adsorbed amounts (below about 0.55 mg/m2) there Is no attraction, probably because these amounts can be mostly accommodated in the train layer so that hardly any loops are formed. The linear behaviour would seem to imply that all extra polymer goes into loops and then forms bridges, proportional to the amount in loops. Hi) Dispersions compressed in a centrifuge. Our next example concerns silica sols (particle radius 15 nm) with adsorbed polyvinyl pyrrolldone (PVP)11. These sols are stable in dilute dispersions, but when compressed by centrifugation, interparticle bonds are formed. Following the compression by centrifugation, the samples were brought into an aqueous medium and tested for their redispersibillty. The results are summarized in the pressure/coverage diagram presented in fig. 1.46. At low adsorbed amounts (0-0.25 mg/m2), particles were always unstable and gels were formed at any centrifugal pressure; these gels could not be re-dispersed in aqueous solution. At intermediate adsorbed amounts (0.25-0.5 mg/m2), small aggregates ('multiplets') were formed without any applied pressure, and at adsorbed amounts above 0.5 mg/m2 particles were stable (singlets) as long as no pressure was applied. By centrifugal pressure, however, the stable singlet particles could be converted Into small aggregates (multiplets); the higher the coverage 9, the higher the required pressure. Likewise, the multiplets could sometimes be converted into gels, particularly for 8 not too much above 0.25 mg/m2. Both the results of the direct surface force measurements and the centrifugation experiments seem to agree with the idea that during prolonged contact between two adsorbed layers (and assisted by applied pressure) some adsorbed polymer will sooner or later escape, allowing the remaining polymer to form bridges that lead to permanent attraction. The mobility of the polymer in the adsorbed state determines the time scale at which this happens. 1.12e Case study: kinetics qfjlocculation (i) Time scales. Polymers are often employed for the purpose of colloidJlocculation,

Figure 1.46. Aggregation states of silica spheres (80 nm diameter) with adsorbed polyvinyl pyrrolidone (PVP) of M w = 1 0 kg/mol as a function of applied centrifugal pressure p c and adsorbed mass F of PVP. Three regions are shown: singlet particles (diamonds), small aggregates (squares), large flocs/gels (triangles).

11

Ph. Ilekti, thesis, Univ. Paris 6, Protection des Surfaces de Silice avec un Polymere adsorbe (2000).

1.84

INTERACTION BETWEEN POLYMER LAYERS

i.e. the formation of aggregates of colloidal particles with the specific purpose of removing them from a dispersion. Obviously, this is possible because of the attraction that polymers can exert between colloidal particles. However, the fact that saturated, quenched layers tend to repel (unless rapid relaxation and sufficient compression occur) implies that it is important to control the adsorbed amount 0 because repulsion sets in and the flocculation stops when G becomes too high. Another factor is also important. At low 0, the adsorbed layer in its equilibrium configuration may be rather thin. If the particles to be flocculated cannot approach each other very closely (e.g. because of electrostatic repulsion), the adsorbed polymers cannot make contact with the other particle and flocculation does not occur. A freshly deposited polymer molecule may be much more extended, though, than an equilibrated one, so that flocculation occurs when the surfaces are brought into proximity before the reconfiguration has taken place. Altogether, we thus have to consider various time scales in a flocculation process: (i) the colloid/colloid collision time, which determines the overall rate of flocculation, [ii) the colloid/polymer collision time, which determines the rate at which the adsorbed amount increases and particles become 'reactive' or 'passive' during a particle encounter, and [Hi) the polymer reconformation time, which determines how long freshly adsorbed polymer molecules remain capable of forming bridges. An early example of these time scales was reported with silver iodide sols flocculated with polyvinylalcohol (PVA)11. With increasing concentration of flocculant the flocculation rate first increased and then decreased again; at a high concentration of PVA, the sol became stable. However, in the latter case there was some initial flocculation in the first five seconds after mixing. Apparently, in these experiments it took about five seconds to build up a stabilizing layer. (ii) Orthokinetic Jlocculation experiment. A case study in which many of these effects are readily seen has been carried out by Adachi et al.2). The chosen colloidal dispersion was a dilute, negatively charged polystyrene latex, which was flocculated with PEO (M = 5000 kg/mol), under conditions where the electrical double layer was entirely suppressed (1.17 M KC1). In order to distinguish the effect of the added polymer, the kinetics of salt-induced flocculation was first carefully characterized. The study focused on shear-induced (orthokinetic) flocculation where the effects of polymer show up most clearly, and the shear conditions were carefully standardized for appropriate comparison. Salt-induced coagulation rate. In the absence of the polymer, the latex coagulates rapidly because of the high KG concentration. The rate of coagulation follows the wellknown second order rate equation

11

G.J. Fleer, J. Lyklema, J. Colloid Interface Set 55 (1976) 228. Y. Adachi, M.A. Cohen Stuart, and R. Fokkink, J. Colloid Interface Set 165 (1994) 310; Y. Adachi, T. Wada, J. Colloid Interface Sci. 229 (2000) 148. 21

INTERACTION BETWEEN POLYMER LAYERS

1.85

^ = -kn 2 dt

[1.12.1]

where n is the (number) concentration of particles, and the rapid coagulation rate constant k is a function of shear rate, particle size and a capture efficiency factor aT , which accounts for the Van der Waals attraction and hydrodynamic interaction between two particles. This capture efficiency factor was calculated by Van de Ven and Mason1' by considering trajectories of a particle approaching another particle, and is given by aT=\

—T

[1.12.2]

In this equation, A is the Hamaker constant, r\ the viscosity, y the shear rate, and a0 the bare particle radius. Adachi et al. used an effective shear rate, which could be accurately controlled by a standardized tumbling method, and for which [1.12.2] remains valid. As follows from this equation, larger particles have smaller capture efficiencies, mainly because there is a severe hydrodynamic drag upon close approach of the particle surfaces. Since the volume fraction
= -k'n
[1.12.3]

and, hence, described by In—

=-fc>J

[1.12.4]

When the total number concentration n = £rij of particles (singlets + doublets + ) is plotted logarithmically against flocculation time, straight lines with a slope proportional to -fc'
"TPC a r e obtained. An example of such a plot is given in

Figure 1.47. Salt-induced orthokinetic coagulation of polystyrene latex in 1.17 M KC1, plotted as particle number concentration n(t) as a function of a scaled time t, for two particle radii a 0 , as indicated. Particle number concentration n0 = 1.7 10 13 m~ 3 . (Redrawn from Adachi et al., loc clt.)

11

T.G.M. van de Ven, S.G. Mason, Colloid Polym. Set 255 (1977) 794.

1.86

INTERACTION BETWEEN POLYMER LAYERS

Figure 1.48. Rate s of rapid orthokinetic coagulation of PS latex with 1.17 M KC1, as a function of the initial particle number concentration riQ (reference as in previous figure).

Figure 1.49. Reduced rate s of rapid orthokinetic coagulation of PS latex with 1.17 M KC1 as a function of the particle diameter a 0 (reference as in previous figure).

As can be seen from [1.12.2], the capture efficiency scales with the bare particle radius as a T ~ a^ 0 5 4 . Moreover,
INTERACTION BETWEEN POLYMER LAYERS

1.87

Figure 1.50. Polymer-induced orthokinetic flocculation plots, presented as lnn(t) as a function of time ( (arbitrary units), for various polymer concentrations c , as indicated. The straight line represents the results for salt-induced coagulation (see fig. 1.46). Polystyrene latex particles (678 nm radius), n 0 = 7.7 • 10 13 m~ 3 , salt concentration 10~4 M KC1. Flocculant: PRO (M = 5000 kg/mol)11.

increases again (fig. 1.50b). We note that several of these effects were also seen in the experiments by Fleer and Lyklema, referred to in the introduction to this section. Let us first consider the initial slope in fig. 1.50. This is higher than that for saltinduced coagulation, indicating a higher collision rate. This cannot be due to a higher sticking probability, because all electrostatic repulsion was eliminated so that the sticking probability is equal to unity at the salt concentration used. Moreover, n 0 was exactly the same. The only explanation is that the effective collision radius of the particles is larger during the polymer-induced flocculation experiment. Apparently the polymer layer on the particle contributes substantially to its size by adding a layer of effective thickness <5e . If we now suppose that polymer layers can form bridges before Van der Waals forces and hydrodynamic effects come into play, the capture efficiency no longer follows [1.12.2]. Instead, the shear-induced flocculation rate is simply proportional to a 3 = (a0 + <5e )3 , whereas that of the bare particles was proportional to C2Q-46 . Hence, by taking the ratio of the two rates we can calculate where the initial rate has its highest value <5e. It turns out that for high polymer concentrations (1 mg/1 and higher), Se is very nearly equal to the diameter of the PEO coil in solution (about 210 nm). This indicates that in these experiments the adsorbed layer contains (some) adsorbed coils, which have very nearly the same size as they had before adsorption; during a particle/particle collision there are always enough freshly adsorbed molecules on the surface that have not had the time to change their conformation. At lower polymer concentrations, the rate gradually decreases, which can be interpreted in terms of a smaller effective collision radius and a smaller <5e. Apparently, the time between successive polymer/colloid encounters is longer, and the particles cannot maintain the 'freshness' of their adsorbed layer. The molecules now have some time to

11

Y. Adachi, T. Wada, J. Colloid Interface Set 229 (2000) 148-154.

1.88

INTERACTION BETWEEN POLYMER LAYERS

Figure 1.51. 12 Effective thickness, Se , of adsorbed polymer during a flocculation experiment, as a function of the characteristic adsorption time r (see text). Flocculant: PEO M = 5000 kg/mol); PS latex particle (678 nm); salt concentration: 10~4 M KC1 (solid points) and 10~2 M KC1 (open points). Reference as in previous figure.

undergo conformation changes leading to a thinner layer, and in these dilute solutions the supply of fresh molecules is too slow. The data thus emphasize the crucial influence of relaxation of adsorbed polymer on the kinetics of flocculation. This is shown in more detail in fig. 1.51 where the value of 5e , as obtained from a series of flocculation experiments, is plotted as a function of the characteristic adsorption time T of the polymer. This adsorption time is found by considering the adsorption flux (number of polymer molecules hitting the particle per unit time) J : J p = 4K(Do + Dp)(ao+ap)np+(27t)l/2Y(ao+ap)3np

[1.12.5]

In [1.12.5], D and a are diffusion coefficients and radii, respectively, and the subscripts 0 and refer to bare particle and polymer coil, respectively; n is the number concentration of polymer molecules in the solution. Note that we consider here the adsorption rate, i.e. particle/coil collisions, whereas above (in the flocculation rate) we used particle/particle collisions. The first term in [1.12.5] is due to Brownian motion and the second one comes from collision due to shear. Taking simply the inverse of the polymer/particle collision rate, we obtain the average time T between two successive collisions between a particle and a polymer molecule. Since J decreases when the polymer concentration is lowered, T increases, and the effective thickness is clearly seen to decrease to very low values, signaling a flattening of the polymer adsorbing from dilute solutions. We finally turn to the leveling-off of n(t) in fig. 1.50, which is due to the appearance of repulsion between particles. As the data show, the final level changes nonmonotonically, being high at low concentration, low around 1 mg/1, and again high at 8 mg/1. Two trends are responsible for this effect: (i) the rate of flocculation itself goes up, allowing larger aggregates to form in a given time, and {ii) the adsorption rate increases, thus shortening the time needed to saturate the surfaces enough to get repulsion. In a subtle way, these two trends conspire to produce optimum flocculation at about 1 mg/1 in this experiment.

INTERACTION BETWEEN POLYMER LAYERS

1.89

1.13 Outlook The equilibrium characteristics of polymers at interfaces are rather accurately described by the Edwards equation, which may be solved routinely by numerical methods up to high accuracy. Due to the large number of identical repeating units along the chain there are many universal properties. These generic effects become clear from accurate analytical descriptions of the interfacial characteristics for various regimes and scenarios. The physics of polymer near an isolated surface is now accurately known, at least on the mean-field level. The extrapolation of these results to colloidal stability issues has its intricacies and we are still far from a complete picture. Nevertheless, there is a leading principle to judge the effect of polymers on the colloidal stability; the key is found in the ranking-number dependent concentration profiles. When all segments along the chain have more or less the same distribution, as for symmetric homopolymers adsorbing from good or even theta solvents with sufficiently high adsorption affinity, the ground-state approximation (GSA) is accurate, and attraction is predicted (caused by depletion when the polymers do not adsorb onto the surfaces, and by bridge formation in the adsorption regime). However, when there is a clear ranking-number dependence, as for segments in a tail, GSA breaks down and repulsion is found. This means that adsorbed polymer layers are repulsive at weak overlap (when tails dominate the interaction) and attractive at strong overlap (when loops can transform into bridges). For end-grafted polymer chains, GSA is replaced by the strong-stretching approximation, which is accurate for densely grafted polymer layers, so-called brushes. Polymer brushes are excellent tools to stabilize colloidal particles because the overlap of the brushed surfaces (each carrying a large number of tails) is very costly due to the compression of each chain in the brush upon approach of the two surfaces. Polymers relax with large characteristic time scales in response to external perturbation. This is one of the most important reasons to select polymers (rather than, e.g. short surfactants) to influence the colloidal stability. Typically, off-equilibrium effects should be superimposed on the equilibrium effects. Very generally, when not all degrees of freedom are relaxed, i.e. when the Deborah number is high because the experimental time scale is short with respect to the chain relaxation time, the interfacial layers are off equilibrium and interfacial forces are necessarily more repulsive than in full equilibrium. For example, in a dilute solution of small particles the time between collisions may be too short to allow bridging. Then there are many collisions necessary before a floe can be formed. In other words, the capture efficiency is low. This effect is probably more important for loop-to-bridge attraction than for depletion. Hence, kinetic aggregation studies could be useful to distinguish adsorption from the depletion mechanism, also when the adsorption is weak. Most of what we have reviewed in this chapter is inspired by numerically solving the Edwards equation or its lattice equivalent in the Scheutjens-Fleer model. In analytical

1.90

INTERACTION BETWEEN POLYMER LAYERS

approximations, such as GSA, the non-local contributions to the self-consistent potential are typically ignored. When the non-local contributions are kept in the analysis one obtains more complicated differential equations with higher order derivatives (up to 3 4 G/3z 4 ). The most important consequence is that one may anticipate non-monotonic density distributions. Indeed, numerical solutions that systematically include these higher order derivatives show these (very weak) effects around the periphery of the density profile; there are concentration oscillations around the bulk concentration. There must, therefore, also be non-monotonic, force distance effects. The magnitude of these effects are, however, typically (at least) an order of magnitude smaller than discussed in this review. In passing, we note that higher order derivatives play an essential role in describing wetting phenomena and liquid-liquid interfaces, which were not addressed in this chapter. A key assumption used throughout has to do with the range of the interactions. In our analysis, we have assumed in all cases that the interactions with the surface are strictly short-range, of the nearest neighbour type. This is reasonable if there are specific (key-lock) interactions between polymer segments and the surface, but not so when the interactions have a dispersive nature. We may consider, for example, the case that the interactions have a power-law decay, such as for Van der Waals forces where the adsorption energy decays as z~3. Such a scenario can be investigated by numerically solving the SCF equations. The main result is that the proximal part of the density profile is affected; this effect could be mimicked by modifying the short-range interaction term, using an effective ^ s • m ^ ne central part of the profile, the selfconsistent potential dominates the (power-law) surface contribution so that the selfsimilar density profile is not affected. Also, the distal part of the density profile decays, as usual, exponentially to the bulk density. The power-law decay of the surface interaction is most fundamentally reflected in the way in which the density profile decays to the bulk density at the very edge of the adsorption profile (at the periphery of the distal part of the profile). The additional contribution has a power-law behaviour with a decay length given by the (long-range) surface energy characteristic. This powerlaw tail may interfere with the oscillatory behavior due to the non-local, self-consistent potential discussed above. Another example of non-trivial long(er) range (surface) interactions is found in systems in which the polymer and the surface are charged. We then should solve the Edwards equation in combination with the Poisson equation, leading to polyelectrolyte theory. Such extensions are feasible and much is known about polyelectrolytes at interfaces. We refer to chapter 2 for more details. In addition to the universal features of interfacial polymer layers, there must also be non-universal aspects. We have assumed throughout our analysis that there are no specific effects on the monomeric length scale; only non-specific effects as expressed in the Flory-Huggins solvency parameter were taken into account. However, when there are complex side groups in a monomer, possibly with mesogenic features, it is

INTERACTION BETWEEN POLYMER LAYERS

1.91

expected that our approximations seriously break down. To treat such systems accurately, one could try to use computer simulations. A semi-atomistic analysis, using a numerical SCF scheme, remains an option as well. However, to account for, e.g. the stiffness of the chain, the structure of the side groups (chain branching), and the possibility of nematic ordering, a much larger set of interaction parameters is needed to describe the various chemical entities and specific interactions. Polymer mixtures and the effects of a molecular weight distribution on the adsorption characteristics have not been discussed in this chapter either. Most of these effects may be deduced from the known behaviour of ideal systems. For example, we know that the critical adsorption energy is a very weak function of the chain length, because the number of bonds is one less than the number of segments. This is one reason for preferential adsorption effects in polydisperse mixtures; it favours the adsorption of short chains. Another feature is that one long polymer chain has less translational entropy than several smaller chains containing the same total number of segments. This implies a tendency towards adsorption preference of long chains. When the first effect dominates, which is the case in a polymer melt, the shorter chains populate the interfacial region. However, in dilute solutions the opposite is true; the long chains displace the short ones from the surface. Very interesting complications arise in solvent mixtures. With a binary solvent one can either modify the solvent quality, tune the effective adsorption energy (for super critical solvent mixtures without a solubility gap), or induce capillary condensation phenomena in slits (when the solvent mixture has a solubility gap). Clearly, a combination of these effects may also occur. Last but not least, we mention that we have assumed that the surface is Ideally flat and homogeneous. We know, however, that surfaces often are chemically heterogeneous (random, patchy), rough (height fluctuations) or flexible (liquid surfaces, membranes). The classical mean-field approximation that the concentrations may be averaged laterally now breaks down. When there Is sufficient symmetry in the surface properties (stripes, spherical patches, etc.), one may use a two-gradient approach of the Edwards equation or the numerical SF equivalent of it. Surface roughness may be implemented in a lattice model using the Lego model. For surfaces with less symmetry, one needs to turn to computer simulations because it is nearly impossible to numerically solve the Edwards equation on a 3D grid. At present, computer simulations are very accurate to model the properties of short chains and relatively small systems. However, we anticipate that one day we will use computer simulations to go beyond the mean-field approximations and give more accurate answers to the many questions encountered in the behaviour of polymers at interfaces and their effects on colloidal stability.

1.92

INTERACTION BETWEEN POLYMER LAYERS

1.14 General References 1.14a Polymers in solution This field is well developed. Numerous textbooks in this area have appeared, but there are a few classical texts which cover most of the field. We list a few books in chronological order. P.J. Flory, Principles of Polymer Chemistry, Cornell University Press, Ithaca NY (1953). (The 'bible' for polymer chemistry and physics, including polymer statistics and polymers in solution. Old, but still valuable.) C. Tanford, Physical Chemistry oj Macromolecules, Wiley, New York (1961). (Classical, like Flory's book, but with more emphasis on proteins.) P.J. Flory, Statistical Mechanics of Chain Molecules, Interscience, NY (1969). (A more refined and rather mathematical treatment of chain statistics, starting from an atomic level.) P.G. de Gennes, Scaling Concepts in Polymer Physics, Cornell Univ. Press, Ithaca, NY (1979). (Scaling description for polymers in good solvents, emphasizing the 'blob' structure as compared to homogeneous 'mean-field' solutions.) H. Yamakawa, Modern Theory of Polymer Solutions, Harper & Row, NY (1971). (Thorough statistical thermodynamic treatment of polymer solution properties.) H.-G. Elias, An Introduction to Polymer Science, VCH (1997). (An elementary text, updated from classical books by the same author.) Water-soluble Polymers: Solution Properties and Applications, A. Amjad, Ed., in: Proceedings Symposium Colloids Surf. Divs. Of the American Chemical Society 1997, Kluwer (1999). (A collection of conference papers.) L. Schafer, Excluded Volume Effects in Polymer Solutions, Springer (1999). (Rather advanced text based on scaling analysis.) K. Kamide, T. Dobashi, Physical Chemistry of Polymer Solutions: Theoretical Background, Elsevier (2000). C. Wohlfarth, CRC Handbook of Thermodynamic Data of Aqueous Polymer Solutions, CRC Press (2004). (Data collection.)

INTERACTION BETWEEN POLYMER LAYERS

1.93

1.14b Polymers at Interfaces This area is newer and still developing. Only a few textbooks are available. P.G. de Gennes, Scaling Concepts in Polymer Physics, Cornell Univ. Press, Ithaca, NY (1979). (This text, mentioned above, also pays some attention to polymers adsorbing or depleting from good solvents.) E. Eisenriegler, Polymers near Surfaces, World Scientific (1993). (A valuable text for analytical solutions of the Edwards equation for ideal chains in zero field.) G.J. Fleer, M.A. Cohen Stuart, J.M.H.M. Scheutjens, T. Cosgrove and B. Vincent, Polymers at Interfaces, Chapman & Hall (1993). (The only comprehensive text so far on polymer adsorption, with emphasis on both theoretical developments and comparison with experiment.) Polymer Surfaces and Interfaces: Characterization, Modification and Application, K.L. Mittal, K.-W. Aizawa, Eds., VSP (1996). (Application-oriented, technological text.) Polymeric Systems. I. Prigogine, S.A. Rice, Eds., Adv. Chem. Phys. 94, Wiley (1996). Polymer Interfaces and Emulsions, K. Esumi, Ed., Marcel Dekker (1999). (Relation between polymer adsorption and emulsion stability.) R.A.L. Jones, R.W. Richards, Polymers at Surfaces and Interfaces, Cambridge University Press (1999). (Emphasizes the surfaces of polymer melts, wetting, and adhesion, rather than adsorption from solution.) Apart from these textbooks, several reviews are worth mentioning. The most relevant ones are (in chronological order): E. Dickinson and M. Lai, Adv. Mol. Relax. Interact. Proc. 17 (1980) 1. (A review of simulation methods applied to polymers.) A. Silberberg, in Encyclopedia of Polymer Science and Engineering Vol. I, Wiley (1985)577. I.D. Robb, Comprehensive Polymer Sci. 2 (1989) 733. H. J. Ploehn and W.B. Russel, Adv. Chem. Eng. 15 (1990) 137. M. Kawaguchi and A. Takahashi, Adv. Colloid Interface Sci. 37 (1992) 219.

1.94

INTERACTION BETWEEN POLYMER LAYERS

G.J. Fleer and F.A.M. Leermakers, in Coagulation and Flocculation: Theory and Applications, 2nd. ed., B. Dobias, H.J. Stechemesser, Eds., Surfactant Science Series, Marcel Dekker (2004). (This extensive treatment considers both adsorption and interaction.) 1.14c Effect of polymers on colloidal interactions Even though this is an important field in the context of colloids, not many books are available. We list them chronologically. The Scientific Basis of Flocculation, K.J. Ives, Ed., Sijthoff & Noordhoff (1978). (This collection of papers for a NATO Advanced Study Institute contains several contributions several contributions highlighting practical aspects of flocculation of dispersions by polymers.) D.H. Napper, Polymeric Stabilisation of Colloidal Dispersions, Academic Press (1983). (Comprehensive discussion of steric repulsion between polymer layers, but adsorption as such is not considered in detail.) W.B. Russel, D.A. Saville, W.R. Schowalter, Colloidal Dispersions, Cambridge University Press (1989). (One chapter in this advanced text (chapter 6), deals with forces due to soluble polymer.) Colloid-polymer Interactions: Partlculate, Amphiphilic, and Biological Surfaces, P. Dubin, P. Tong, Eds., ACS Symp. Series 532 (1993). (A collection of conference papers.) Colloid-polymer Interactions: from Fundamentals to Practice, R. Farinato, P. Dubin, Eds., Wiley-VCH, (1999). (A collection of conference papers.) V.A. Hackley, Polymers in Particulate Systems, in: Surfactant Science Series 104, Marcel Dekker (2001). Two recent reviews are worth mentioning: O. Spalla, Nanoparticle Interactions with Polymers and Polyelectrolytes in Current Opinion in Colloid & Interface Sci. 7 (2002) 179. (Emphasizes mixtures of polymers and nanoparticles.) G.J. Fleer and F.A.M. Leermakers, in Coagulation and Flocculation: Theory and Applications, 2nd. ed., B. Dobias, H.J. Stechemesser, Eds., Surfactant Science Series, Marcel Dekker (2004). (This extensive review considers both adsorption and interaction).

2

POLYELECTROLYTES

Martien Cohen Stuart, Renko de Vries and Hans Lyklema 2.1

2.2

Introduction

2.1

2.1a

Physicochemical properties

2.1

2.1b

Chemistry

2.1

Polyelectrolyte electric double layers 2.2a

charge distributed?

2.3

2.4

2.5

2.6

2.7

2.8

2.3

Polyelectrolytes: why are they charged and how is the 2.3

2.2b

Cylindrical electric double layers without added electrolyte

2.2c

Cylindrical electric double layers with added electrolyte

2.13

2.6

2.2d

Proton exchange equilibria of polyelectrolytes

2.16

Polyelectrolyte configurations

2.25

2.3a

Dilute solutions

2.25

2.3b

Semidilute solutions

2.32

2.3c

Grafted polyelectrolytes

2.37

2.3d

Polyelectrolyte gels

2.41

Polyelectrolyte viscosity

2.45

2.4a

General features

2.45

2.4b

Lateral information

2.47

2.4c

The intrinsic viscosity

2.50

2.4d

The dilute and semidilute range

2.53

Polyelectrolytes in electric fields

2.57

2.5a

The issue of electrokinetic binding

2.58

2.5b

Conductivity, backgrounds

2.60

2.5c

Conduction: illustrations

2.64

2.5d

Dielectric properties

2.69

Electrostatically driven complexation and phase separation

2.69

2.6a

Solubility of polyelectrolytes

2.69

2.6b

Polyelectrolyte complexes

2.71

2.6c

Compex coacervation

2.71

2.6d

Polyampholytes

2.77

2.6e

Polyelectrolyte multilayers

2.78

2.6f

Complex coacervate micelles

2.79

Applications of polyelectrolytes

2.81

2.7a

Flocculation

2.81

2.7b

Stabilization enhancement

2.82

General references

2.82

This Page is Intentionally Left Blank

2 POLYELECTROLYTES MARTIEN COHEN STUART, RENKO DE VRIES AND HANS LYKLEMA

2.1 Introduction

2.1a Physicochemical properties Polyelectrolytes are macromolecules that, when dissolved in a polar solvent like water, have a (large) number of charged groups covalently linked to them. In general, polyelectrolytes may have various kinds of such groups. Homogeneous polyelectrolytes have only one kind of charged group, e.g. only carboxylate groups. If both negative (anionic) and positive (cationic) groups occur, we call such a molecule a polyampholyte. These polyelectrolytes will only be briefly discussed at the end of this chapter. Self-assembled structures, such as linear micelles or linear protein assemblies, also often have many charged groups; these structures may have properties very similar to those of polyelectrolytes, but we shall not deal with them in this chapter. Special properties of polyelectrolytes, as compared with uncharged polymers, are their generally excellent water solubility, their propensity to swell and bind large amounts of water, and their ability to interact strongly with oppositely charged surfaces and macromolecules. Because of these features, they are widely used as rheology and surface modifiers. These typical polyelectrolyte properties are intimately related to the strong electrostatic interactions in polyelectrolyte solutions and, hence, are sensitive to the solution pH and the amount and type of electrolytes present in the solution. 2.1b Chemistry As will be explained in more detail in sec. 2.2, polyelectrolytes in solution acquire charges by the adsorption of ions or by the dissociation of surface groups. Especially important are proton exchange equilibria. For macromolecules, there are a limited number of different functional groups that are typically involved in proton exchange equilibria; we list them in table 2.1 below, along with their chemical structure. We also give in this table an indication of their tendency to acquire a charge in terms of a range of values of their deprotonation constant (expressed as its negative logarithm, i.e. pK), which characterizes each group when isolated from other charged groups.

Fundamentals of Interface and Colloid Science, Volume V J. Lyklema (Editor)

© 2005 Elsevier Ltd. All rights reserved

2.2

POLYELECTROLYTES

Table 2.1. Ionizable groups occurring in common polyelectrolytes and their proton exchange properties

Type of group

reaction

intrinsic

examples

10.5-11

poly(allylamine)

weak cationic -NH2 {aliphatic)

-NH3+

-NR2 (aliphatic)

-NR 2 H + ?^ - N R 2 + H +

10.6

poly(dimethylaminoethyl methacrylate); poly(ethylene imine)

-0-NH2 (aromatic)

(*-NH3+ ^ ^-NH2 + H +

4.5-5

poly( vinylaniline)

-Pyridine (= N: )

= N: = NH+ ^ = 1^: + H +

5.3

poly (vinylpyridine)

-

-

poly(diallyl dimethyl ammonium chloride)

-COOH

-COOH ^=i-COO" + H +

4-5

poly(acrylic acid)

-PO 4 H 2

- P O 4 H 2 , =^-PO 4 H~ + H+

0-1

phosphate esters of poly( saccharides)

-PO 4 H 2

=PO 4 H ^- - P O 4 - + H+

0-1

DNA

-

poly(styrene sulphonic acid); carrageenan

- -NH 2 4 -H+

strong cationic -NR+

weak anionic

strong anionic -SO3H

-SO3H

- - S O 3 ~ + H+

As can be seen, the groups are classified into 'weak' or 'strong,' depending on whether or not they have a pK in the range 0-14 so that they can undergo proton exchange reactions at experimentally relevant pH. Polymers with weak groups (0 < pK < 14) can adjust their average degree of dissociation to the physicochemieal conditions, and the charge on individual groups can fluctuate; molecules with such groups are also referred to as 'annealed' polyelectrolytes. Polymers with strong groups (pK < 0; pK > 14) have fixed charges and charge distributions along the chain, whatever the pH, and are therefore sometimes called 'quenched' polyelectrolytes. Many different kinds of backbones occur: all-carbon (vinyl, allyl) chains, polytesters), poly (ethers), poly (saccharides), poly(peptides), as well as poly(imines) where the charged group is itself part of the backbone. Some backbones are hydrophilic, such as those in poly(saccharides), but most are hydrophobic and the corresponding molecules would not dissolve if they would not carry a sufficient number of charges.

POLYELECTROLYTES

2.3

Copolymers, which have more than one kind of monomer (in particular, monomers with and without dissociating groups), should also be mentioned. For this category, the maximum charge that a molecule can acquire is, of course, determined by the chemical composition. Many of the polyelectrolytes discussed here owe their very existence to synthetic polymer chemistry. This is not to say, though, that nature does not produce polyelectrolytes. Many naturally occurring polysaccharides produced by plants, bacteria or animal cells are polyelectrolytes. One of the best-known natural polyelectrolytes Is DNA, which contains in-chaln phosphate groups as chargeable sites, and its close relative RNA is, of course, also a polyelectrolyte. Finally, polypeptides and proteins often carry charged groups and can thus be defined as (very complicated) polyelectrolytes. Since the structure and physicochemical behaviour of many proteins has rather special features, these molecules will receive attention in a chapter especially devoted to them (chapter 3). In this chapter, we shall first discuss how charge and countercharge In a polyelectrolyte in solution are distributed (the electrical double layer), and we consider the proton exchange equilibria from which the charges originate (2.2). We then take a look at molecular shapes (conformations) and the way these are influenced by the charge (2.3) for free chains in solution, end-grafted chains, and gels. Section 2.4 deals with the viscosity of polyelectrolyte solutions and sec. 2.5 with polyelectrolytes in external electrical fields (conductivity, electrophoresis and dielectric properties). Finally, we discuss in sec. 2.6 the solubility of polyelectrolytes and the formation of complexes in mixtures of polyelectrolytes of opposite charge. 2.2 Polyelectrolyte electric double layers 2.2a Polyelectrolytes: why are they charged and how is the charge distributed? In solution, polyelectrolytes can acquire a charge on the chain by two mechanisms: (i) ions from the solution can bind to them and/or (ii) surface groups can dissociate. For instance, as discussed in sec. 2.1, polyamines can pick up protons to form positive -NHg groups and from polyacids, protons can desorb to form negative, say, -COO ~ or -OSOg groups (we ignore other than simple ions; in particular, adsorption of ionic surfactants will not be considered here). For all the charge formation processes, the driving force is of a chemical or entropical origin, determined by the Gibbs energy difference between the bound and free state, just as with the dissociation of low-M molecules. In processes of type (i) the process is entropically unfavourable, whereas for those of type (Ii) it is entropically driven. Electric forces always oppose the charge formation; in fact, all processes come to an end when the counteracting electric forces cancel any further charging. Then, equilibrium is attained. At equilibrium the Gibbs energy is at a minimum and negative. Electric double layers around polyelectrolytes form spontaneously. It is

2.4

POLYELECTROLYTES

typical for polyelectrolytes that the equilibrium state is also determined by the degree of swelling. The spontaneous charge separation leads to the creation of a chain charge and an equal but opposite countercharge. When the solution contains low-M electrolyte, the latter consists of an excess of counterions and a deficit of co-ions. This negative adsorption of co-ions leads to the expulsion of electroneutral electrolyte, the so-called Donnan exclusion. In salt-free solutions the countercharge only contains counterions; in that case, there can be no Donnan exclusion. All of this is basically not different from the charging principles for hydrophobic colloids, see sec. II.3.2. Quantitative differences with double layers on flat plates and macroscopic spheres may come forward because of the way the chain charge is distributed. The issue is connected to the question as to whether it is allowed to treat the double layer charge as smeared out, which would allow for mean-field interpretations, such as the PoissonBoltzmann distribution for the diffuse part and Bragg-Williams type analyses for the non-diffuse part. As long as a polyelectrolyte chain may be considered as a long charged cylinder with macroscopic radius a, one can assign a surface charge density (7° to it and treat the countercharge by one of the models discussed in chapter II.3 for macrobodies. Anticipating the arguments below, it is at least likely that the outer side of the countercharge obeys PB statistics, i.e. it can be described as a diffuse double layer according to Gouy and Chapman. In figs. II.3.14 and 15, we showed that for the diffuse parts cylindrical double layers assume an intermediate position between flat and spherical ones. As intuitively expected, the larger xa, where a is now the radius of the cylinder, the more similar spherical, cylindrical and flat double layers are. However, a is often not so large that this picture is allowed over the entire electric double layer. For further discussion, it makes sense to divide the countercharge into two parts, an outer or diffuse part and an inner, non-diffuse part. The diffuse part is readily defined as that part where ion size effects are negligible, where charges may be considered smeared out, and where binding is solely caused by electrostatic attraction (zey/(r) per ion of valency z at distance r from the axis of the cylinder). In the nondiffuse part these premises no longer hold. Because of the proximity to the chain, specific ( = non-electrical) binding can usually not be ignored. The specific phenomena depend on the nature of the charged groups and on their positions along the chain. The nature effect is nothing else than the specific binding, which gives rise to the familiar lyotropic sequences (sec. IV.3.9i). However, the position effect is new. For instance, for charged groups residing on sidechains (as in polymethacrylates), it is possible for ions to accommodate between the charges whereas polyelectrolytes with all charges on the backbone (as in poly-imines) do not have this option. As positioning of ions between and outside charges can follow different patterns, interesting lyotropic sequences may be expected, beyond those generally observed for macroscopic solid surfaces. According to general experimental experience with macroscopic surfaces, plots of the diffuse (~ electrokinetically or stability-wise active) charge density ad ~ (7ek as a

POLYELECTROLYTES

2.5

function of a° start as a straight line under 45° where a^ = -o° , to level off to a plateau where further increase of a° does not lead to further enhancement of a^ . See, for instance, figs. II.3.63 and 4.13, for f and cf* , respectively. The diffuse charge density rarely exceeds a few |iC cm"2 . The meaning of the plateau is that a situation has arisen in which any further added charge on the surface is compensated by a countercharge in the non-diffuse part. This non-diffuse part was generally called the Stern part of the double layer, charge densityCT1, with a° + a1 + a^ = 0 because of electroneutrality. Obviously, in this respect double layers around polyelectrolytes resemble Gouy-Stern double layers. At this stage it is necessary to introduce the term ion condensation. This term is typical for polyelectrolyte science, and stands for the binding of (counter)ions to the charged chain. In the literature the term is used in various ways, for instance in the interpretation of experiments to account for the fraction of counterions that do not contribute to osmotic pressure, conductivity etc., or, in theory, in which models for the process of condensation are developed. Theories and the various experimental observations may, or may not, agree. The term is well established in the domain of polyelectrolytes and has the proper resonance. It is not appropriate to speak indiscriminately of 'bound' and 'free' ions, because diffuse ions are also bound in a thermodynamic sense. Hence, we shall continue to use the term condensation in this chapter as a general term for ions that are so strongly bound to (or captured by) the chain that they are osmotically, electrokinetically or conductometrically inactive, irrespective of the reason for the condensation (electric and/or specific). According to this definition, condensed ions and Stern ions are identical. History has led to differentiation between the terms, because the founding fathers of polyelectrolyte theory (Fuoss, Katchalsky, Oosawa, Manning...) were unfamiliar with the much older Stern model; on top of this came the independent development of theories. Typically, in polyelectrolyte theory (Oosawa, Manning...) two-state models have been put forward on purely electrostatic principles. We shall return to these in sees, b and c below. Description in terms of Bragg-Williams type approaches, as in Stern theories including specific binding, appears to be virtually absent in polyelectrolyte literature. As a result, in most polyelectrolyte elaborations, there is little room for ion-specificity interpretations. On the other hand, Stern and condensation theories both predict the formation of a plateau in the cfi ( a° ) relationship, as mentioned before. In order to finally arrive at theories for polyelectrolyte configurational statistics, it appears unavoidable to use a coarse-grained description of the polyelectrolyte chain and the electric double layer surrounding it. Following most of polyelectrolyte theory, the complicated polyelectrolyte molecular structure is modelled as a charged cylinder characterized by just two parameters: its radius a and its surface charge density a°. All details at length scales below the radius a are lumped into the effective parameters a and a°. Sometimes an even further simplification is used and the polyelectrolyte chain is modelled as a line charge with a linear charge density of v = 2naa° . Following

2.6

POLYELECTROLYTES

most of polyelectrolyte literature, a Stern layer to account for ion-specific binding is not taken into account. Instead, the double layers are described by the non-linear PB equation only, assuming all counterions are outside of the cylinder of radius a that models the polyelectrolyte chain. As mentioned, even in such purely electrostatic models, there is a plateau in the 0^(0°) relationship. This is the effect that is usually referred to in the polyelectrolyte literature as (counter)ion condensation. The basic idea, stemming from Imai and Ohinshi11, but elaborated by Oosawa21 and Manning31, is that upon charging the chain above a critical chain charge, the resulting potential near the chain becomes so high that any additional charge is compensated for by an electrostatically captured counterion. The simplified two-state models of Oosawa and Manning (based on the PB equation) predict that, in the limit of vanishing concentrations of both electrolyte and polyelectrolyte, o^fcr0) plots (or equivalents thereof, such as plots of the polyelectrolyte osmotic pressure versus the chain charge v) should consist of an initially linear part, which discretely breaks off into a plateau at the critical chain charge. This discrete limiting behaviour is sometimes referred to as the Manning limiting law. Indeed, for this special limiting case, the solution of the full PB equation does exhibit a mathematical discontinuity at a well-defined critical chain charge. In contrast to the simplified models however, it does not predict a distinct 'break' in, for example, the osmotic pressure as a function of the chain charge. Nevertheless, the existence of a universal critical chain charge for polyelectrolytes and the concept of counterion condensation have been very important in polyelectrolyte modelling to help develop approximate theories for polyelectrolyte properties. In the next subsection, we therefore discuss in some detail both the Oosawa two-state model and the full PB approximation for cylindrical double layers of strong polyelectrolytes without added electrolyte. Next, the case of added electrolyte is treated, again assuming strong polyelectrolytes. For weak polyelectrolytes, charges are determined by adsorption/dissociation equilibria. A final paragraph considers in detail the important case of proton exchange equilibria for linear polyelectrolytes. 2.2b Cylindrical electric double layers without added electrolyte To begin, we assume that the overall polyelectrolyte configuration is at least locally rod-like and study the cylindrical double layer in the PB approximation. While in practice there will always be some amount of electrolyte present, its presence may be neglected if the total number of co-ions is negligible as compared with the total number of counterions. Roughly speaking, this is the case when the concentration of added electrolyte is small as compared with the total polyelectrolyte monomer concentration,

11

N. Imai, T. Ohinshi, J. Chem. Phys. 30 (1959) 1115. F. Oosawa, Polyelectrolytes, Marcel Dekker (1971). 31 G. S. Manning, J. Chem. Phys. 51 (1969) 924. 21

POLYELECTROLYTES

2.7

a situation that is certainly not uncommon in practice. Polyelectrolyte electric double layers in the absence of added electrolyte are also important from an historical and theoretical point of view, since they have played an important role in the development of the theory of counterion condensation, one of the central concepts in polyelectrolyte modelling. A distinct difference with the case of excess added electrolyte is the absence of a well-defined bulk concentration of ions that can be used as a reference point (defining the zero of the electrostatic potential). Instead, models for electric double layers in the absence of added electrolyte depend on the definition of a certain volume to which the counterions are confined. That is, they are so-called cell models. In practice, the volume of the cell is related to the concentration of charged objects (colloids, polyelectrolytes). This reflects the physical effect that diluting the charged objects dilutes the counterions, and hence affects the electric double layers. Oosawa two-state model As mentioned, in the limit of vanishing concentration of both electrolyte and polyelectrolyte, the solution of the cylindrical PB equation exhibits a mathematical discontinuity at a critical chain charge. Simplified two-phase models for this limiting case, based on the PB equation, have been developed by Oosawa (loc. cit.) and Manning (loc. cit.). Their main virtue is that they clearly demonstrate the physical mechanism that leads to (electrostatic) counterion condensation. Below, we first discuss the Oosawa two-phase model. Following, we introduce the exact solution of the cylindrical PB equation in the absence of added electrolyte. Finally, we compare with experimental data for polyelectrolyte osmotic pressures. The Oosawa two-phase model provides a simple way to estimate the magnitude of the fraction /

of free counterions. The remaining fraction of counterions, ( 1 - / ) , is

assumed to be electrostatically captured or condensed into a cylindrical region close to the chain, of radius a (the polyelectrolyte chain itself is viewed as a line charge). The free counterions are assumed to move around within a cylindrical region a < r < R. The value of the radius R is set by the volume fraction (p of chains, which is approximated by (p = a2/R2

[2.2.1]

The geometry of the Oosawa two-phase model (which is a cell model) is illustrated in fig. 2.1. Without loss of generality, the strong polyelectrolyte is assumed to be negatively charged and to have a constant charge density of -v elementary charges e per unit length ( v is positive). Of course, both the counterion concentration and the electrostatic potential y/ are continuous functions of the distance r to the line charge at the centre of the cylinder. In the two-state model, the continuous distributions are approximated by bimodal distributions. The condensed counterions inside the cylinder

POLYELECTROLYTES

2.8

Figure 2.1. a) Geometry for the Oosawa two-phase model. Condensed counterions move in the inner cylinder of radius a. Free counterions move in the volume bounded by the inner and outer cylinders of radius R. b) Relation between the potential t//[r) arising from a line charge of strength -Jv and the potential difference Ayr in the Oosawa two-phase model. Note that the point of zero potential has not been defined in this graph. of radius a are assumed to have a constant concentration c cond , and the free counterions in the cylindrical region between a and R are assumed to have a constant concentration c free , given by the average values: C

free-TT-^T JVAv 7taz _

' 2 - 2 -21

1 (l-J)v

Like the counterion distribution, the electrostatic potential y/ is assumed to be bimodal: the electrostatic potential inside the cylinder of radius a is y/a and the potential in the cylindrical region between a and R is y/R . The distribution of the counterions over the two regions is then governed by a Boltzmann factor involving the difference in the electrostatic energy of the counterions: c

free cond

c

=

_J__ 1 ^ e xp(-eAiy/kT) ! " /
[2.2.4]

where Ay/ is the (average) potential difference between inner and outer cylinder (see [2.2.5]). To estimate this electrostatic potential difference, the inner cylinder is viewed as a line charge with a charge density of - Jv elementary charges e per unit length. This line charge gives rise to a continuous potential y/(r). From this potential, we estimate (as illustrated in fig. 2.1) Ayr= y/{R)-y/{a)

[2.2.5]

POLYELECTROLYTES

2.9

Standard electrostatics then gives ^ - = 2Jxln(R/a)

[2.2.6]

K1

where we have also introduced the (dimensionless) charge parameter x x = lBv

[2.2.7]

The Bjerrum length 1B used here, and throughout this chapter, is the same one that was introduced in the chapter on the adsorption of polymers and polyelectrolytes ([5.2.23]) I

B

e " 4ne0ekT

It is noted that (B is twice as large as rB , the corresponding length in low-M electrolyte solutions originally introduced by Bjerrum himself, see [1.5.2.30a]. For aqueous solutions at room temperature, ZB =0.7nm. Combining [2.2.4] and [2.2.6] gives an equation that can be solved for the fraction of free counterions J : -^— = (pfx-1

[2.2.8]

In the limit of infinite dilution,

0, the solution shows a discontinuity at a charge parameter x = 1:

r i x < \ f =\

[2.2.9] [l/x

x > l

As already alluded to, there appears to be a critical value for the charge parameter x c = 1, corresponding to a linear charge density of vc = 1 / lB. The above simple model predicts that, below this critical value (at infinite dilution), all counterions are completely free, and above it, any additional charge on the chain is compensated for by an electrostatically captured counterion. This special behaviour is the Manning limiting law mentioned previously. Solution to the full cylindrical PB equation Obviously, the above two-phase model is no more than a crude approximation. We next discuss the exact solution to the PB equation for the electrostatic potential y/{r) at a distance r from a negatively charged cylinder. As before, the cylinder radius is a, its charge density is -ve per unit length. Counterions are confined to the cylindrical volume for which a < r < R. For a negatively charged polyelectrolyte in the absence of added electrolyte, there are only positively charged counterions. These have a concentration profile c+ (r) . The Poisson equation reads

2.10

POLYELECTROLYTES

U - + ~ W ) = -(4;re/£0£)c+(r) u + ydr rdr)

[2.2.10]

and the Boltzmann distribution for the counterions is c+(r) = coexp(-e^(r)/fcT)

[2.2.11]

where c 0 is the counterion concentration at the point of zero potential {r = R) . The Poisson-Boltzmann equation for the dimensionless electrostatic potential y = ey//kT can now be written as ^ - + - ^ - y(r) = -A2exp[-y(r)] \dr rdr)

[2.2.12]

The screening parameter X is analogous to the Debye screening parameter K for the case of excess added electrolyte; A"1 is often called the Gouy-Chapman length. It is given by „ 4;rfRcn A2 = §_o. N

[2.2.13]

Av

The surface charge density at r = a is given by v/2na . This leads to a first boundary condition of d

l =2£ 3r r = a a

[2.2.14]

where * = vlB is again the dimensionless charge parameter. At the outer boundary of the cell, the electric field should vanish, which gives the second boundary condition: — 3r

=0

[2.2.15]

r=R

With respect to this equation, the point of zero potential, or equivalently, the reference concentration of counterions c 0 , can be chosen freely since adding a constant to the electrostatic potential does not affect its derivative, the electric field. A common choice in cell models is to set the potential to zero at the outer boundary of the cell. Then, c 0 equals the counterion concentration at this outer boundary. In the present case, it is more convenient to instead choose c 0 to be the average concentration in the cylindrical cell volume. For a section of length L, the number of counterions released by the charged cylinder is vL . In the limit a << R considered here, the volume in which the counterions are dispersed may be approximated by nR2L . This leads to: Av

A2=~ R

[2.2.16]

POLYELECTROLYTES

2.11

The analytical solution to the above PB equation and boundary conditions was first applied to polyelectrolytes by Fuoss, Katchalsky and Lifson1' and by Alfrey, Berg and Morawetz21. The solutions are, for a << R, \ix r2 y{r) = In — — s i n h 2 (Bin br)

[B2 R2

[2.2.17]

J

when the integration constant B is real (at low charge densities), and

[

n

2

1

.£__!_. sin 2 (|B| In br)

\B\ R

J

[2.2.18]

when the integration constant B is imaginary (at high charge density). The values of the integration constants B and b are determined by the boundary conditions at r = a and at r = R. WhenB is real, x = ( l - B 2 ) / [ l + Bcoth(Bln(R/a))] [2.2.19] 1

Blnb = - B l n R - t a n h " B and when B is imaginary, x = (l + |B| 2 ) / /[l + |B|cot(|B|ln(R/a))] [2.2.20] 1

|B|lnb = -|B|lnK-tan- |B| In the limit of vanishing polyelectrolyte concentrations l-x

x <1

\B\ -> 0

x >1

[2.2.21]

What does this imply for experimental observables, such as the counterion osmotic pressure? The counterion osmotic pressure is usually given in terms of the osmotic coefficient , n/RT

=
[2.2.22]

According to Marcus31, in the PB approximation, the osmotic pressure equals the ideal pressure resulting from the small ions at the positions where the electric field vanishes. In our case, it therefore equals the ideal ion pressure due to the counterions 11 21 31

R. Fuoss, A. Katchalsky, and S. Lifson, Proc. Natl. Acad. Sci. U.S.A. 3 7 (1951) 579. T. Alfrey, P. Berg, and H. Morawetz, J. Polym. Sci. 7 (1951) 543. R.A. Marcus, J . Chem. Phys. 2 3 (1955) 1057.

2.12

POLYELECTROLYTES

at r = R, whence 0 = exp(-y(R))

[2.2.23]

Using the limiting behaviour of the integration constant B, the osmotic coefficient is found to be (l-x/2 (p = \

x<\

[Il2x

x>\

[2.2.24]

This replaces the prediction for the fraction a of 'free' counterions of the much more approximate Oosawa two-phase model discussed earlier. The osmotic pressure itself is: [2x(l - x)

x<\

[l

x>l

n/nUm=\

[2.2.25]

The limiting value for the osmotic pressure is —M- = RT

1 NAv2xlBR2

[2.2.26]

Plots of the limiting behaviour of the counterion osmotic pressure and the osmotic coefficient are given in fig. 2.2. Note that despite the 'break' in the value of the integration constant B, the osmotic pressures does not exhibit the break predicted by the simpler Oosawa two-phase model and the Manning limiting law.

Figure 2.2. Counterion osmotic coefficient

Experimental values for osmotic coefficients of highly charged polyelectrolytes without added electrolyte are usually of the order q>= 0.1...0.3 for monovalent counterions, and agree reasonably well with the predictions of the full cell model. A general trend appears to be that the cell model overestimates the osmotic coefficients. Also, of course, it cannot explain the significant variation with counterion

POLYELECTROLYTES

2.13

type. For an example, see fig. 2.3, which shows recent data of Deserno et al. 1 ' for a rodlike polyelectrolyte, poly-(p-phenylene) with positively charged side chains. Presumably, the chlorine counterions have a stronger affinity for the polyelectrolyte chain than the iodine counterions, an effect that could be modeled in terms of a Stern model with specific affinities of the ions for the polyelectrolyte chain. Figure 2.3. Osmotic coefficient 0 as a function of monomer concentration c m for the rod-like polyelectrolyte poly-(p-phenylene) with positively charged side chains. Full symbols: iodine counterions, open symbols: chorine counterions. Full curve: PB cell model, dashed horizontal line: Manning limiting law. (Redrawn from Deserno et al. (loc. cit.)) 2.2c Cylindrical electric double layers with added electrolyte For the case of a charged cylinder with added monovalent electrolyte, the PB equation is written as [II.3.5.58] ^ - + - ^ - y ( r ) = x-2sinh y(r) ydr rdr)

[2.2.27]

First, consider the special case of a line charge of linear charge density v in the DebyeHiickel approximation. In terms of the charge parameter x = vlB , y(r) = 2XK0{KT)

[2.2.28]

More generally, in the PB approximation, the outer field ( KT » 1) of any cylinder is of the form y(r) = 2xeffK0(xr)

[2.2.29]

The parameter xeff is a function of both x and KCL . The interpretation of this parameter is that it is the charge parameter of an equivalent line charge that generates the same far field as the charged cylinder. The concept of the equivalent line charge will turn out to be very convenient when we discuss interactions of charged cylinders with weakly overlapping electric double layers. We should emphasize that the equivalent line charge parameter xeff plays an identical role as the diffuse double layer potential y d for colloids. We prefer to use xeff here since we use constant charge boundary conditions rather than constant potential or charge-regulating boundary conditions.

11

M. Deserno, C. Holm, J. Blaul, M. Ballauff, and M. Rehahn, Eur. Phys. J. E5 (2001) 97.

2.14

POLYELECTROLYTES

For a charged cylinder of radius a, the solution of the Debye-Hiickel equation is well known, and was discussed in II.3.5f. For this case, the charge parameter of the equivalent line charge is: x „= eff

[2.2.30] KaKx(Ka)

For highly charged cylinders, there is no known analytical solution, although an accurate analytical approximation has been developed by Philip and Wooding11, based on the idea that in the non-linear part of the cylindrical double layer, where y(r) » 1, the concentration of co-ions is negligible. Values for xeff (or equivalents thereof) obtained from numerical solutions of the cylindrical Poisson-Boltzmann equation have been tabulated by a number of authors, see for example Fixman and Skolnick21. Alternatively, accurate values for xeff can be obtained from the analytical approximations of Philip and Wooding. For the limit of infinite dilution and no added salt, the (approximate) Manning limiting law predicts a break at x = 1, \x

x <1

[1

x>\

*eff=

[2.2.31]

As Fixman and Skolnick observe, the limiting behaviour is only reached logarithmically upon decreasing the electrolyte concentration. Indeed, a comparison with the numerical values of Fixman (fig. 2.4) indicates that only at very small values of tea is Manning's limiting law a reasonable approximation. Figure 2.4. Effective charge parameter X = sraKj (/ra)xegas a function of bare charge parameter x. Symbols are values tabulated by Fixman (loc. cit), computed from numerical solutions of the full PB equation for various values of jra as indicated. Full curves are guide to the eyes. The lower dotted line is the Manning limiting law for tea —» 0 , the upper dotted line is the Debye-Huckel approximation.

For larger values of KCL , we slowly approach the regime where the cylindrical double layer becomes more and more similar to a flat Gouy-Chapman diffuse double layer, as mentioned in the beginning of this section, and also in II.3.5f. 11 21

J. R. Philip, R. A. Wooding, J. Chem. Phys. 52 (1970) 953. M. Fixman, J. Skolnick, Macromolecules 11 (1978) 863.

POLYELECTROLYTES

2.15

Next we turn to interactions between charged cylinders. The discussion closely follows Brenner and Parsegian11 and Fixman and Skolnick. For cylinders separated by more than a few Debye lengths, the electric double layers are only weakly overlapping. Far away from both cylinders, the electrostatic potential may then be approximated by the sum of the potentials of the isolated cylinders; this is nothing else than the linear superposition approximation (LSA) discussed, e.g., in chapter IV.3. The interaction of the cylinders depends only on the outer field of an isolated cylinder, and is conveniently calculated using the concept of the equivalent line charge.

Figure 2.5. Geometry of skewed line charges for evaluating the electrostatic interaction energy.

First consider parallel cylinders at a distance h. The interaction energy V{h) (per unit length) equals the electrostatic potential of one of the equivalent line charges, times the effective charge density of the other one: V(h)/kT = 2(B1x2ffK0(x-h)

[2.2.32]

To obtain the interaction energy V{h, y) of cylinders at a distance h and skewed at an angle y, we compute the electrostatic potential at some position Sj along one of the line charges by adding up the contributions from all positions s 2 along the other line charge, as suggested by Fixman and Skolnick, and illustrated in fig. 2.5; V(h,y) f r - j - ^ - = Jds 1 jds 2 u(s 1 ,s 2 ) u( S l , s 2 ) = I g l x ^

exof-x-ls - s i ) ^ ~ ^ ri

[2.2.33]

S

2

This gives the simple final result V(h,y) _27txlii fcT rig

11

exp(-rh) sin y

S.L. Brenner, V.A. Parsegian, Biophys. J. 14 (1974) 327.

|2 2 34]

2.16

POLYELECTROLYTES

which can be used down to angles of | y\ ~ a/L , where L is the length of the rods. For even smaller angles, it is more accurate to use the expression for parallel rods.

2.2d Proton exchange equilibria of polyelectrolytes So far it was assumed that some fixed number of polyelectrolyte groups was charged. In reality, of course, the charge is determined by adsorption/desorption equilibria, as explained in sec. II.3.6e. Here, the special, but important case, of proton exchange equilibria is treated in somewhat more detail. Proton exchange reactions can be symbolically represented by X H ^ X " + H+

[2.2.35a]

X H + ^ X + H+

[2.2.35b]

or

For simple monovalent acids or bases, the proton exchange reactions obey classical laws of chemical equilibrium, with a deprotonation constant K defined by: K=

x C

H

[2.2.36]

HX

where we employ the usual definitions pH = -loga H and pK = -logK . In these commonly accepted expressions, activities are defined with respect to an ideal solution of 1 M. To avoid taking logarithms of dimension having quantities, an alternative definition in terms of mole fractions was introduced in II.3.6e. [II.3.6.42] This latter definition differs by - log V m (- log 55.5 for aqueous solutions), where Vm is the molar volume of the solvent. Note that c x denotes both the species X" that occurs in reaction [2.2.35a] and the species X in reaction [2.2.35b]; similarly, c ^ stands both for HX and for HX+ . Also, note that we represent all proton transfer equilibria by their deprotonation constants pK, rather than using profanation

constants, which

would be just given by pKw - pK , where pKw is the dissociation constant of water; at T = 298 K, pKw =14 . We shall denote the fraction of groups (sites) that become deprotonated as a , and the fraction of charged sites as 9. If the charge is formed by deprotonation (as in case (a) above), a= 9; in the other case (b), 9=1-a.

Using this

definition, we have K=

ac a H V(1 - a)c

[2.2.37]

Hence, pH = pK + l o g ( - ^ J

[2.2.38]

As long as each macromolecule in a solution has no more than one single charge, similar proton exchange equilibrium will apply with an intrinsic equilibrium constant pK. However, polyelectrolytes usually carry many charges and these interact by

POLYELECTROLYTES

2.17

Coulomb forces. If a charge z.e on a chain in a medium of dielectric constant e is surrounded by JV other charges z^e at distances r^ , it has an extra electrostatic energy Uel given by Coulomb's law: Uel = Y - J - i —

[2.2.39]

Dissociation will cost this extra energy above the intrinsic (often termed 'chemical') work of dissociation for an isolated group. For a homogeneous polyelectrolyte, which has acquired many charges, L/ej is positive, implying that further charging becomes more difficult. In other words, each charged group must overcome the total potential energy due to all other charges, many of which are on the chain. As a result, a larger change in pH is needed to reach a certain a, i.e. the titration curve extends over a larger pH range and has a shape different from that of monovalent acids or bases. The proton binding behaviour of polyprotic systems has been extensively discussed in a review by Borkovec, Jonsson and Koper11. A general approach to a theory of titration (proton binding) isotherms requires calculation of the partition function for the macromolecule as a function of its degree of protonation, including the coupling between charge (distribution) and shape of the molecule. This is a very demanding task, often only tractable by means of molecular simulations. Here, we first neglect the coupling between the charge distribution and the shape of the macromolecule. We return to this issue at the end of this section when we discuss computer simulations of polyelectrolyte titrations. Two limiting cases are of special interest, and can be dealt with using simple approximations. If the interaction between the charged groups on the polyelectrolyte is weak, we can use a mean-field approximation. This is the limit of weak coupling. It usually applies when distances between charged groups on the polyelectrolyte are not too small. For polyelectrolytes with very small distances between charged groups, we are in the limit of strong coupling. In this limit, a charged group mainly interacts with its nearest neighbours. The problem of describing the titration of polyelectrolytes with only nearest-neighbour interactions is equivalent to the so-called Ising model, for which analytical solutions are available. First consider the limit of weak coupling. The Bragg-Williams approximation consists of assuming that the way the charged groups are distributed along the chain is not affected by the interactions. In other words, the configurational entropy is the same as that for a system of non-interacting sites, and given by s

conf =fc{alna+(l-a)ln(l-a)}

[2.2.40]

11 M. Borkovec, B. Jonsson, and G.J.M Koper, in Surface and Colloid Science 16, E. Matijevic, Ed., 99-339, Kluwcr Academic/Plenum Press (2001).

2.18

POLYELECTROLYTES

The deprotonation isotherm for this case now satisfies pH =P K - e ^ ( e ) + l O g p U V P fcTlnlO S U - « J

[2.2.41]

where if/>{0) is the potential on the chain's backbone at a particular degree of charging 0 (9= a° /o^ax )• It is quite customary to express experimental data on proton binding in terms of pKeff as a function of a or (1 - a). pK eff =pK + ApK

[2.2.42]

In the above mean-field approximation, ApK =

_ J ^ L JcTlnlO

[2.2.43,

In the Debye-Hiickel limit of low charge densities, the potential t//° is proportional to the charge density <j°, hence to 8. The shift is therefore linear in the dissociation. Hence, ApK = ~e6, with ~e defined in terms of the charge parameter x, as follows:

Figure 2 . 6 . pK e f f a s a function of 8 a t v a r i o u s c o n c e n t r a t i o n s of NaNO 3 , a s indicated in t h e figure; (a) for t h e p o l y b a s e poly (dimethylaminoethyl methacrylate) (0=1- a) a n d (b) for t h e polyacid poly (acrylic acid) [6= a).

POLYELECTROLYTES

2.19

x Ko0ra) ~ rfglnlO KjOra)

L

where Ko and K{ are the zeroth- and first-order Bessel functions of the second kind, respectively. An example is shown in fig. 2.6 for the protonation of polydimethylaminoethyl methacrylate at various concentrations of added salt. Since the above expression is based on various assumptions, such as the absence of correlation effects (mean-field assumption) and a constant double layer capacitance (Debye-Hiickel limit), deviations from linearity are likely to occur as soon as these assumptions do not hold any longer. Indeed, non-linear ApK plots are often observed in experiments, and can be represented in the form of a series expansion: ApK = W+e'02 + ...

[2.2.44]

Examples of the latter are given in fig. 2.7a and b, where the full (non-linear) PB equation was used to fit the data. At this point, we should also mention the Henderson-Hasselbalch (HH) equation [1.5.2.34], an empirical equation introduced as a generalization of [2.2.37]:

pH = pK + nlogf-^-J

[2.2.45]

Figure 2.7. a. Degree of protonation as a function of pH for hyaluronic acid at various concentrations of monovalent salt, as indicated. Experimental results are denoted by symbols. Solid curves are fits to the PB equation for a cylinder (radius 1 nm, charge spacing 1 nm and intrinsic pK = 2.75). b. Titration curves plotted as versus 1 — 0 (degree of protonation) for polylDL-glutamic acid) at two concentrations of monovalent salt. Experimental results arc denoted by symbols. The dashed and solid curves are for a PB model with cylinder radius 0.5 nm; dashed curve: flexible chain model (charge spacing 0.7 nm): solid curve: rigid rod model (charge spacing 0.5 nm). (Redrawn from (a) R.L. Cleland, J.L. Wang and D.M. Detiler, Macromolecules 15 (1982) 386; (b) D.S. Olander, A. Holtzer, J. Am. Chem. Soc. 90 (1968) 4549, with fits from M. Ullner, B. Jonsson, Macromolecules 29 (1996) 6645.

2.20

POLYELECTROLYTES

where pKgff and n are an effective deprotonation constant and a parameter accounting for the polyelectrolyte effect, respectively. This equation appears to work quite well for many cases as is best checked by making a 'Henderson-Hasselbalch plot' (pH versus log(cr/l-«)), which should be linear. Comparison with [2.2.35] suggests that if the HH equation applies, the change in pK must scale as (rt -l)log(a/l- a) , which is not justified by theoretical arguments. An illustration follows in fig. 2.12. An alternative method sometimes introduced in the literature is to use the Donnan capacitance. The idea is that the space in the neighbourhood of the chain has a welldefined 'inner' potential given by the Donnan model, and that it is this potential that modifies the proton binding and conversely. This approach may be adequate for strongly branched molecules, gel-like polymer particles or brushes, which have an inner region that is large compared with the Debye length. We shall use that approach in sec. 2.3e. For linear chains such a region does not really exist, and the Donnan approach is less appropriate; indeed, the linear dependence of the capacitance on ionic strength predicted by the Donnan model is usually not found in experiments with linear chains. When the distance between proton binding sites on the chain becomes smaller, the interaction between them becomes stronger, and at some point (when the interaction energy becomes of order kT or more) we can no longer assume weak coupling. That is, the distribution of charges along the chain is no longer entirely random. For this case, rather than separating the Helmholtz energy in an ideal entropy term and an averaged energy term, we must weigh all possible configurations of charges along the chain with their respective interaction energies. A useful approximation is to rewrite the Helmholtz energy F({Sj}) of a molecule with a specified distribution of protonated and deprotonated sites {Sj} in terms of a cluster expansion: F

(<si>) = - Z / ' i s 1 + X V i s J i=l

i<j

+

X Lijksisjsk + -

[2.2.46]

i<j
where the terms represent individual sites, pairs of neighbouring sites (doublets), triplets, etc., respectively. If the dominant contribution to the interaction energy comes from the nearest neighbours, it is reasonable to attempt a truncation after the second term:

^^H-X^i+Iv^ i-l

l2 2 471

--

i<j

This is called a (linear) Ising model11. The parameter fit = 2.303/cT(pK-pH) specifies the chemical potential of the proton, with respect to the reference state of a solution of non-interacting sites with a =0.5. The quantity u{. is the interaction energy between two neighbouring sites: it is zero unless both sites are charged, in which case it is given "After E. Ising, Z. Physlk31 (1925) 253.

POLYELECTROLYTES

2.21

as a first approximation by the screened Coulomb energy of two elementary charges at a distance C: u

= 1J

^^expHrn Ajt££Qt

(1 + ra)

The Ising model was treated in sec 1.3.8a in the context of adsorption of gas molecules (at pressure p) on a line of adsorption sites; there, the adsorption constant was taken to be KL and the interaction energy between two filled neighbouring sites was w^ j . Here, we use it for the desorption of protons from a linear polyacid, but otherwise it is entirely equivalent. Note that in sec. 1.3.8a the treatment was based on the quasichemical (Bethe-Guggenheim) approximation. It is possible to compute the partition function exactly using the method of transfer matrices, but the result is the same as that of the quasi-chemical approach for this one-dimensional case. We can therefore use the solution [1.3.8.4] and just make the substitutions: KL —> K^1

p -> a H+

6 -» a

wx j -> u

This immediately leads to the adsorption isotherm:

pH = pK-0.4343u + log l £ ± i z ^ £ ]

[2.2.49]

with p=[l- Aa(\ - a)(\ - e-ulkT)}112

.

[2.2.50]

Note that when u becomes very small, exp(-u/kT) and fi approach unity, and we recover [2.2.38] At large u, however, the charged sites tend to avoid each other. The result is that the effective pK changes strongly in a narrow window around a = 0.5 , but is changing relatively little outside that window. In other words, the titration occurs in two 'waves' as if the polyelectrolyte behaves like a divalent acid or base. Creating charges in an alternating arrangement on the chain requires no extra energy and this, therefore, happens first; in order to make all sites participate in the second 'wave,' the equilibrium constant has to change substantially. Examples of polymers with the latter kind of behaviour are poly(ethylene imine) (PEI), where the sites are nitrogen atoms (secondary amines) in the backbone chain, separated by two carbon atoms, i.e. only about 0.3 nm, and poly(maleic acid) and poly(fumaric acid), which have carboxyl groups attached to neighbouring carbon atoms. Branched variants of (PEI), like the regularly branched imine dendrimers, show these effects even more dramatically. Figure 2.8 shows data for linear PEI, both as the degree of protonation versus pH, and as effective pK as a function of degree of protonation. The pattern of two waves can be clearly discerned.

2.22

POLYELECTROLYTES

Figure 2.8. Titrations of linear poly (ethylene imine) at two different ionic strengths (indicated) plotted as degree of charging versus pH (a) and as effective pK versus 6 (b). Experimental data are indicated by points. Solid curves are calculated according to the Ising model. The parameters are: for 0.1 M, pK = 8.44, u = 0.0107, and for 1 M, pK = 8.63, u = 0.0178. (Redrawn from R.G. Smits, G.J.M. Koper and M. Mandel, J. Phys. Chem. 97 (1993) 5745.)

Figure 2.9. Titration of polymaleic acid, plotted as degree of protonation ( 1 — 0 ) versus pH, in different background electrolytes: LiCl, NaCl, and TMAC1. Solid curves are fits to a version of the linear Ising model in which chain tacticity is explicitly accounted for. (Redrawn from J. de Groot, G.J.M. Koper, M. Borkovec and J. De Bleijser, Macromolecules 31 (1998) 4182.)

Another example, for poly(maleic)acid, is given in fig. 2.9. Again, two waves can be seen in accordance with the explanations above. In this example, titrations were carried out with different monovalent counterions (lithium, sodium, and tetramethylammonium, TMA). Clearly, the differences are small in the low pH range (where less than 50% of the groups are charged, but they become very pronounced for the second 'wave': the midpoint of that wave for Li+ is almost 2.5 pH units lower than that for TMA+ . The interaction energy between charged sites is thus much larger for TMA+ , which suggests that this voluminous ion is much less strongly bound on charged sites than either Li + or Na + . The series corresponds with the specific binding sequence for alkali ions to carboxyl groups, see for instance sec. III.3.8b. As we saw above, for weak polyelectrolytes the charge density on the chain responds to the local environment, leading to a salt-dependent average charge density. As the ionic strength decreases, the charge density decreases too. This is not the only effect, though. When the size of the coil becomes comparable with the Debye length, its ends experience less interaction from neighbouring charges than middle segments. As

POLYELECTROLYTES

2.23

a result, the charges tend to accumulate at the ends, particularly when the fraction of charged monomers is on the order of a few percent. The charge density at the ends may then become 2-3 times that in the middle. This implies that when a polyelectrolyte solution is titrated, the charge 'grows in' from the ends1 .

Figure 2.10. Monte Carlo simulations of titration on flexible chains. The effective pK is plotted versus degree of protonation (1 - 0) . The chains consisted of 320 subunits connected by fully flexible bonds of 0.6 nm length. The dashed curves are calculated for a rigid chain; the points (connected by solid lines) are taken from Monte Carlo simulations.

So far, it has been assumed that coil swelling effects could be neglected, because the distance between groups separated by a long stretch of chain would be too large for any significant interaction. This turns out to be not entirely true. Monte Carlo simulations have shown beyond any doubt 21 that chain swelling does affect the effective pK. Comparison with experiments, e.g., on poly (D,L glutamic acid), confirmed that taking swelling into account improves the agreement between experiment and theory. Swelling effects are particularly dramatic for polyelectrolytes with a modestly or poorly soluble backbone. A well-known example is poly(methacrylic acid) (PMA). In its neutral state, this polymer has a certain internal cohesion, which prohibits it from forming a dilute coil. As a consequence, the distances between sites are initially almost fixed so that the pK must increase. Quite likely, the distribution of charges is very inhomogeneous at this stage with many more charges sitting at the periphery of the coil than inside. At a sufficiently high charge density ( ai.0.2 ), the electrostatic forces become strong enough to start breaking the intramolecular cohesion, and the molecule reaches a state where its pK is determined by the balance of the electrostatic and cohesive forces. Between a = 0.2 and a = 0.4, the molecule then swells substantially at a virtually constant pK. As soon as the entire molecule has reached the dilute coil state, pK may increase further because of nearest neighbour interactions. The typical behaviour of polymethacrylic acid can be shown either by means of a pKeff(a) plot (fig. 2.11) or as an HH plot (fig. 2.12). The latter plot is not simply linear as for a polyelectrolyte in a good solvent, but it has two linear sections of different slopes, and a non-linear section around a = 0.5 , which connects the two linear parts.

11 21

M. Castclnovo, Eur. Phys. J. El (2000) 115-125. See, for instance, M. Ullner, B. Jonsson, Macromolecules 29 (1996) 6645.

2.24

POLYELECTROLYTES

Figure 2.11. Effective pK as a function of degree of protonation (1 — 0) for polylmethacrylic acid) at two ionic strengths. Experimental data are indicated by points. The solid curve was calculated with a PB model for a charged cylinder (a = 0.55 nm, charge spacing 0.55 nm), but clearly docs not fit the experimental data. (Redrawn from M. Nagasawa, T. Murase, and K. Kondo, J. Phys. Chem. 69 (1965) 4005.)

Figure 2.12 illustrates how HH plots can give a quick view of conformational transitions in the chain. PAA does not exhibit such a transition; the plot is linear over the entire pH range studied. However, with the inclusion of a methyl group in the main chain (PMA) or by partial esterification of carboxyl groups (pe), the transition does occur as evidenced by the breaks. The slopes are measures of the polyelectrolyte effect in terms of n; the higher c s a j t , the lower n is. The compact state of PMA (at a < 0.2 ) also leads to an elevated n. At a = 0.5 , log(a/l - a) = 0 , pH = pK e f f , which can therefore be read from the figure.

Figure 2.12. Hcnderson-Hasselbalch (HH) plots of various polyfacrylic acid) (PAA) and polylmethacrylic acid) (PMA) either, or not, partially esterified (pe). Panel (a), •, PAA-pe + 0.01 M NaCl; x, ibid., without NaCl; A PMA, without NaCl. Panel (b), o,» A PMA-pe + 0.02 M NaCl; • , • , ibid., 0.2 M NaCl. Polyelectrolyte concentration 1000 ppm (courtesy J.T.C. Bohm).

POLYELECTROLYTES

2.25

The most pronounced swelling effect Is, of course, observed whenever the neutral chain is insoluble, so that the charging process becomes coupled to a phase transition. 2.3 Polyelectrolyte configurations

2.3a Dilute solutions No added electrolyte Kuhn et al.'' were the first to show that in the absence of screening by added electrolyte, polyelectrolytes adopt an essentially linear configuration. Consider a long, freely jointed polymer chain of contour length L, which consists of JVK Kuhn segments of length lK . In the Gaussian approximation, the Helmholtz energy of extending the chain to an end-to-end distance r is: fj^sL^J— kT LlK

[2.3.1]

Now, suppose that the chain has a linear charge density of v elementary charges e per unit length, corresponding to a charge parameter of x = iB v. Neglecting the influence of the counterions, the repulsive electrostatic interaction energy of the chains is estimated as Coulomb _ 2 ^ kT (Br

[2 3 21

where ! B is the Bjerrum length. Minimizing the total Helmholtz energy gives the scaling relation r =U2'3(!K/lB)'/3

[2.3.3]

For highly charged polyelectrolytes, it is no longer reasonable to assume that all counterions are dispersed to such an extent that their influence can be neglected. The scaling argument still applies, however, if e is interpreted as the number of 'free' counterions released into the solution per unit length of polyelectrolyte. The concentration range, where the rod-like polyelectrolytes do not overlap, is called the dilute regime. The boundary of the dilute regime is the overlap concentration c* of the rod-like polyelectrolytes, 1 c* =

NK „ ^ " ^ K

[2.3.4]

Note that in this context 'dilute' strictly means non-overlapping, which for salt-free polyelectrolytes does not necessarily mean that they do not interact. For sufficiently high polyelectrolyte molecular weights, the Afj^2 dependence implies that only solutions

11

W. Kuhn, O. Kiinzle, and A. Katchalsky, Helv. Chlm. Ada 31 (1948) 1994.

2.26

POLYELECTROLYTES

of extremely low concentration are dilute if no electrolyte is added. A further practical limit is, of course, that ionic strengths below, say, 10~5 M are difficult to obtain. Nevertheless, careful experiments have been performed in the dilute regime. A general feature of polyelectrolyte solutions without added electrolyte is that they exhibit a peak in their structure factor as measured by light-, X-ray and neutron scattering. The peak position q max is related to a characteristic distance l / q m a x • The fact that a peak is visible is thanks to two effects: 1) the strong electrostatic repulsion gives rise to rather well-defined distances between interacting polyelectrolytes and 2) the strong electrostatic repulsion gives rise to a very low compressibility and, hence, very low scattering at low q. For the dilute regime, the characteristic distance is simply the typical distance between the rod-like polyelectrolytes, q^ax ~ c • Indeed, in the dilute regime the position q max of the peak scales as q max <*= c 1 / 3 . Added electrolyte Eventually, at very high concentrations of added electrolyte, all electrostatic interactions are completely screened. Then, if the polyelectrolyte backbone is both flexible and water-soluble, its behaviour should revert to that of an uncharged, flexible, watersoluble polymer. For dilute solutions, we therefore expect a gradual transition from rod-like configurations in the absence of added electrolyte, to coiled configurations at high concentrations of added electrolyte. Quantifying this transition has been a major aim of many polyelectrolyte theories. Solving this problem has been achieved in part by mapping the problem onto the analogous problem of an uncharged polymer chain. Recall that standard theory for uncharged flexible polymer chains characterizes polymer segments in terms of two parameters: their length IK ("Kuhn length') and their second virial coefficient, or excluded volume [5 (introduced in volume II, sec. 5.2 as u = /?/fj| ). Chain expansion due to excluded volume is often characterized in terms of a linear expansion coefficient a , a or = —=a g,o where a

[2.3.5]

is the radius of gyration of the expanded chain, and a g 0 is the radius of

gyration of the equivalent chain without excluded volume interactions, a g 2 0 =ljV K !2

[2.3.6]

Chain expansion is determined solely by the dimensionless excluded volume parameter z z=

( 3 V / 2 Nl'2p —

- \ ! -

[2.3.7]

The expression for the linear expansion coefficient a that will be used here is the

POLYELECTROLYTES

2.27

Yamakawa-Tanaka11 expression: a2 =0.541 +0.459(1+ 6.04Z)046

[2.3.8]

which correctly goes to 1, as z —> 0 , and does not deviate too much from the correct asymptotic behaviour a °= z 2 / 5 at large z (see also chapter II.5). For polyelectrolytes, both the segment length (K and the excluded volume j3 can be considered as effective parameters that depend on the electrostatic interactions and, hence, on the electrolyte concentration. First consider the effective segment length. At issue is the 'stiffening' of polyelectrolyte chains due to electrostatic interactions. As we have seen, in the absence of added electrolyte, electrostatic stiffening is so strong that the polyelectrolytes have essentially rod-like conformations. Initially it was assumed that in electrolyte solutions, pieces of polyelectrolyte chains up to a Debye length long would be essentially rod-like. At larger length scales, the chain was assumed to be flexible. In other words, the 'effective' segment length was assumed to be proportional to K'X . Then, in 1977, Odijk21, and Skolnick and Fixman31 showed that, in fact, the electrostatic stiffening effect is much stronger than this, and gives rise to an effective segment length proportional to K~2 . Odijk41 (together with Houwaart) and Fixman and Skolnick51 were also the first to work out the theory for the electrostatic contribution to the segment excluded volume P, thus completing the mapping onto the uncharged polymer problem. Below, we first derive the expression for the effective segment length, and then that for the effective segment excluded volume. The theory of electrostatic stiffening was briefly introduced in II.5.14. It relies on describing the polyelectrolyte chain as 'wormlike'61 with persistence length q . Note that the Kuhn segment length equals twice the persistence length, lK = 2q . Consider a wormlike chain bent into a circle of radius R. The elastic energy of this circular wormlike chain is, per unit length :dG3asL= I J L [2.3.9] kT 2R2 For a circular polyelectrolyte, the persistence length q has both intrinsic contributions and contributions due to electrostatic interactions of the charges along the chain: q = qo+qe

11

[2.3.10]

H. Yamakawa, Modern Theory of Polymer Solutions, Harper & Row (1971). T. Odijk. J. Polym. Set 15 (1977) 477. 31 J. Skolnick, M. Fixman. Macromolecules 10 (1977) 944. 41 T. Odijk, A.C. Houwaart. J. Polym. Set Polym. Phys. 16 (1978) 627. 51 M. Fixman, J. Skolnick. Macromolecules 11 (1978) 863. 61 Such a model pictures the chain as an elastic cylinder. The persistence length q is the ratio of the bending elastic constant of the cylinder over the thermal energy kT (see also II.5.2f). 21

2.28

POLYELECTROLYTES

If the electrostatic interaction energy (per unit length) of the circular polyelectrolyte is / e l (R), the corresponding electrostatic contribution to the persistence length can be calculated from

q =_^llk He

R ^

[2.3.11]

kT dR

A serious problem in computing qe is how to deal with the electric double layers for curved polyelectrolytes of high linear charge density. This problem has not been solved satisfactorily. The pragmatic approach of Odijk, Skolnick and Fixman (and most authors after them) has been to model the polyelectrolyte as a line charge in the DebyeHuckel approximation. High linear charge densities can then only be accounted for in a very approximate way, by using an effective charge density. In the Debye-Hiickel approximation, the electrostatic interaction energy of a circular line charge, of charge density v (elementary charges e per unit length) is simply the local electrostatic potential times the local charge density (divided by two to avoid double counting of interactions). Per unit length of polyelectrolyte chain: ^ - =l v \ f d s e X p ( - y r ( S ) ) kT 2 B J r(s)

[2.3.12]

Due to the curvature, the distance r(s) between charges is smaller than the contour length s separating them (see fig. 2.13).

Figure 2.13. Circular line charge. The spatial distance r between charges is shorter than the contour distance s, which gives rise to additional electrostatic repulsion in curved configurations and, hence, to electrostatic stiffening.

This is the physical origin of the stiffening effect: curvature brings the charges closer to each other, which increases the electrostatic repulsion. The derivative of the electrostatic energy with respect to curvature, is

The derivative of the distance r(s) with respect to curvature R can be evaluated using r(s) = s ( l - ^ ( s / R ) 2 + ...)

[2.3.14]

POLYELECTROLYTES

2.29

In the other terms we may now set r{s) = s . Also, in the limit R —> °=, the integration limits may be extended to infinity. This finally gives v2in r n qe= 2_jdsexp(-x-s)(s + x-s2)

(2.3.15]

o In terms of the dimensionless charge density x = iB v, the result is q e

= ^ -

12.3.16,

As mentioned, an approximate expression valid for arbitrary charge densities is obtained by substituting an effective charge density xeff for x > 1,

%-^kr

[2.3.17]

A crude, but commonly used approximation, is to set xeff = 1 for x > 1 . The treatments of Odijk and Skolnick and Fixman (OSF for short) are much more complete than the derivation presented here, and properly take into account the fluctuating geometry of the polyelectrolyte chain. Various authors have tested the validity of [2.3.17] against numerical solutions of the full Poisson-Boltzmann equation for curved, highly charged cylinders11. For a highly charged cylinder of radius a, it turns out that [2.3.17], with xeff = 1, as compared with numerical solutions of the exact PB equation, is reasonably accurate provided KCL «1. At higher concentrations of added electrolyte, for tea = O(l) or larger, there are substantial deviations. Strictly speaking, the derivation of the expression for the electrostatic persistence length requires that the radius of curvature of the polyelectrolyte chain must always be large with respect to the Debye length K~1 . Provided the intrinsic persistence length is large enough, q 0 > x""1, i.e. for intrinsically stiff polyelectrolytes, this will certainly be the case. Furthermore, for these polyelectrolytes, excluded volume interactions are often negligible, except at extremely high molecular weights. Therefore, the above theory for electrostatic stiffening can immediately be applied to experimental data on intrinsically stiff polyelectrolytes. Indeed, shortly after the publication of the papers of Odijk and Skolnick and Fixman, it was demonstrated21 using magnetic birefringence that [2.3.17] with xeff = 1 almost quantitatively accounts for the observed electrostatic stiffening of long DNA molecules at low concentrations of added electrolyte (DNA has q0 ~ 50 nm). Its validity for intrinsically stiff polyelectrolytes, such as DNA, has since then been confirmed in many other experiments (see also fig. II.5.5) For intrinsically flexible polyelectrolytes, there are two further problems; first, the

11 21

M. Le Bret, J. Chem. Phys. 76 (1982) 6243, M. Fixman, J. Chem. Phys. 76 (1982) 6346. G. Maret, G. Wcill, Biopolymers 22 (1983) 2727.

2.30

POLYELECTROLYTES

validity of the K~2 scaling for intrinsically flexible polyelectrolytes ( q0 < K^1 ) and second, the problem of the electrostatic contribution to the excluded volume. Regarding the former problem, while some authors have reasoned that the tc~2 scaling is altogether invalid for intrinsically flexible polyelectrolytes, most authors argued that for flexible polyelectrolytes it remains valid, but only at length scales larger than a Debye length. Indeed, more recent detailed calculations have shown11 that at length scales larger than a Debye length, the K~2 scaling remains valid, even for flexible polyelectrolytes. The calculations also show that electrostatic stiffening at length scales below a Debye length is much less pronounced, which implies increased fluctuations at these short length scales. These can be corrected for by introducing an effective contour length, which is somewhat smaller than the full contour length of the polyelectrolyte chain due to fluctuations at short length scales. Next, consider the electrostatic contribution to the segment excluded volume2'31. Under conditions of strong electrostatic swelling of polyelectrolyte coils, the electrostatic contribution to the excluded volume completely overwhelms the non-electrostatic contributions; hence we can concentrate on the former. The effective segment length is
[2-3.18]

with qe given by [2.3.17] The simplest approach is that of Odijk who approximates the segments by rigid cylinders of length lK and effective diameter dgff . The effective diameter must account for the influence of the electrostatic repulsion of the segments. Assuming that a typical distance of closest approach is the distance at which the repulsive interaction energy equals the thermal energy kT, it was estimated that d e f f =2x- 1

[2.3.19]

The segment excluded volume can then be estimated using the well-known expression for the excluded volume of long slender cylinders,

£ = j£ d eff

[2.3.20]

Skolnick and Fixman instead use the expression [2.2.33] for the interaction energy V(h,y) of charged cylinders, and evaluate the full cluster integral for the excluded volume of two line charges of length lK :

P = jdr(l-exp{-V[h,y))

[2.3.21]

where r is the difference between the centre-of-mass coordinates of the rods, and the

11

H. Li, T.A. Witten. Macromolecules 28 (1995) 5921. Odijk and Houwaart, (1978) (loc. cit). 31 Fixman and Skolnick, (1978) (loc. cit.) 21

POLYELECTROLYTES

2.31

angle denotes an average over the internal coordinates of the rods (i.e. their orientations). It is convenient to define an effective diameter such that the excluded volume equals that of long slender cylinders, according to [2.3.20] An asymptotic expansion of the cluster integral results in the simple approximation deff =K- 1 (-lnvI B +21n£ eff +2.61)

[2.3.22]

This expression will be used below in the comparison with experimental data. Briefly, the procedure to calculate the polyelectrolyte radius of gyration is to first calculate the effective segment length from [2.3.17] and [2.3.18] For a polyelectrolyte of contour length L, the number of effective segments is then given by JVK = L/lK . The radius of gyration a 0 (unperturbed by excluded volume interactions) can now be calculated from [2.3.6] After calculating the electrostatic excluded volume from [2.3.20] and [2.3.22], the expansion factor a is calculated from [2.3.7] and [2.3.8] Finally, the radius of gyration a is calculated from [2.3.5] To illustrate the application of this theory, consider the data of De Nooy et al.1' on the flexible polysaccharide pullulan. By itself this polysaccharide is uncharged, but it can be given a variable linear charge density (up to about le/nm) by oxidation. As shown in fig. 2.14, the radius of gyration varies by only a factor of two when varying the salt concentration over two orders of magnitude. Adjustable parameters in comparing the experimental data with the theory are the intrinsic persistence length q0 of the polysaccharide, and a small contribution d0 to the effective diameter to account for steric repulsion at high salt concentration. As shown in fig. 2.14, the theory consistently fits the data for very reasonable values of the parameters. A scaling argument2 explains why the dependence of the coil size on the ionic strength is, in fact, rather weak. The dependencies of the electrostatic contributions to the Kuhn length and the excluded volume are:

Figure 2.14. Radius of gyration of oxidized pullulan (contour length L = 1100 nm) as a function of electrolyte (NaCl) concentration. Linear charge densities: triangles, x = 0.94 : circles, x = 0.73 ; squares, x = 0.49 . Lines: theory as described in the text, with q0 = 3 nm and d 0 = 0.3 nm. (Redrawn from de Nooy et al., loc. cit.)

1!

A.E.J. de Nooy, A.C. Bcscmcr, H. van Bekkum, J.A.P.P. van Dijk, and J.A.M. Smit. Macromolecules 29 (1996) 6541. 21 T. Odijk, Macromolecules 12 (1979) 688.

2.32

POLYELECTROLYTES

lK~K~2

[3.2.23]

deff-x- 1

13.2.24]

P~lldeK~K~5

[3.2.25]

In the limit of large excluded volume (excluded volume parameter z > > 1), a g - L3>5p'%l/S

~ x- 3 / 5 ~ ns-°-3

[3.2.26]

The terms involving the excluded volume and the Kuhn length exhibit opposite dependencies on ionic strength, causing a weak overall dependence on ionic strength. Whereas for chains without excluded volume the radius of gyration increases with increasing segment length (K (at constant contour length L), the dominant effect for chains with strong excluded volume is a decrease of excluded volume interactions, with increasing lK (see [2.3.7] and [2.3.8]). The scaling dependence ai ~ n~0& is close to the typical experimental observation that a2 ~ nj° 5 . Finally, note that Monte Carlo computer simulations11 have only recently been able to reach system sizes, required to critically test the OSF theory of polyelectrolyte coil sizes using simple bead-string polymer models, having effective Debye-Hiickel interactions between beads. These simulations confirm that the mapping to the uncharged polymer problem gives a good first approximation for the coil size, and that fluctuations at length scales below a Debye length do not significantly change the K~2 scaling of the electrostatic contribution to the persistence length. 2.3b Semidilute solutions No added electrolyte Salt-free polyelectrolytes remain one of the least understood systems in polymer physics, and this holds especially for semidilute (salt-free) polyelectrolytes. From a theoretical point of view, the main problems are (i) the long-range character of the electrostatic interactions in the absence of screening electrolyte and (ii) the existence of multiple, concentration-dependent length scales. This leads to a large number of distinct regimes, as we explain below. Here we only sketch some of the features that are more or less accepted, and do not attempt to compare theory with experimental data. Note that in the present context 'salt-free' is always defined with respect to the concentration of ions released by the polyelectrolytes. This is especially important for the more dense semidilute solutions: in a polyelectrolyte solution with a monomer concentration of, say, 0.1 M, the effects of adding small amounts of electrolyte (appreciably less than the monomer concentration) are negligible. The sequence of events upon increasing the polyelectrolyte concentration in a salt11

R. Everaers, A. Milchev, V. Yamakaov. Eur. Phys. J. E8 (2002) 3; T.T. Nguyen, B.I. Shklovskii, Phys. Rev. E 66 (2002) 021801.

POLYELECTROLYTES

2.33

free solution is believed to be the following. In the dilute regime, the polyelectrolyte configurations are essentially rod-like. As will be argued below, they remain rod-like as the rods start overlapping. If they would remain globally rod-like even at still higher concentrations, we would expect, at some critical concentration, a transition to an orientationally ordered, nematic phase. No experimental evidence has been found, however, for orientational order in salt-free, semidilute, polyelectrolyte solutions (with flexible backbones). Apparently, at concentrations where anisotropic phases would be expected, the increased concentration of counterions has already screened the electrostatic interactions to such an extent that the molecules have become too flexible to form an anisotropic phase. For concentrations above the overlap concentration, scaling arguments indicate that there are multiple semidilute regimes. These are illustrated in fig. 2.15. (b, c and d). For the purpose of the scaling arguments below, consider a highly charged polyelectrolyte of contour length L, with 'monomers' of length (B and volume (g , each of which carrying one elementary charge e. We work with the volume fraction of monomers rather than with the molar concentration c. By definition, the density of counterions equals the density of monomers. Assuming that all counterions contribute to the screening of the electrostatic interactions, the relevant screening length A"1 follows from X2 = (pr2

[2.3.27]

For a rough estimate of the effective segment length lK , we use the OSF expression [2.3.17], with xeff = 1, and with the Debye length /r"1 replaced by the screening length

Z K =-S
[2.3.28]

A s s u m i n g rod-like configurations, t h e semidilute regime s t a r t s at a volume fraction
At the overlap volume fraction

2.34

POLYELECTROLYTES

Figure 2.15. Solution regimes for salt-free polyelectrolytes. a) dilute rods, c < c 1 ; b) semidilute rods, c < c ' < c " ; c) semidilute and semiflexible: c" c"] and £ < ! K .


[2.3.29]

Beyond this volume fraction, global configurations become coil-like, but the local configuration is still rod-like, as illustrated in fig. 2.15c Further increasing the monomer volume fraction decreases both the segment length and the correlation length, but the former decreases faster. At a further critical monomer volume fraction
Beyond this volume fraction, the screening by counterions is nearly complete and the behaviour of the semidilute solution becomes increasingly similar to that of uncharged flexible polymers (fig. 2.15d). Finally, consider the scaling of the position of the scattering maximum, as observed in light-, X-ray and neutron scattering experiments. As mentioned, in the dilute regime

1)

This is the characteristic length scale for density fluctuations (see also II.5.2d).

POLYELECTROLYTES

qmax~
2.35

[2.3.30]

For semidllute solutions with q>* < q><
[2.3.31]

as is indeed observed experimentally. For cp> q>** , the electrostatic interactions are screened to such an extent that the scattering maximum is no longer visible. Ultimately, in this regime, the scaling of the correlation length should tend to that for an uncharged flexible polymer (see eq. II.5.2.18), i.e. £~
[2.3.32]

Added electrolyte For the case of added electrolyte, the mapping to the uncharged polymer problem can again be used to develop approximations for the semidilute regime. In the discussion of polyelectrolytes below, following Odijk ' ', we use scaling theory for semidilute solutions of uncharged flexible polymers 21 . Some aspects of the scaling predictions are compared with the quasi-elastic light scattering data of Koene and Mandel for the highly charged, flexible polyelectrolyte sodium poly(styrene sulphonate) (NaPSS). As mentioned, for a highly charged, intrinsically flexible polyelectrolyte, the effective segment length and the segment excluded volume have the following dependencies on the Debye length: l

K~K~2

P=

K-h\~K-

[2.3.33] 5

For flexible, uncharged polymers in a good solvent, the monomer volume fraction at which coils start overlapping is (p*

^ ^ - 4 / 5 ^ - 3 / 5 ,-2/5 a

[2.3.34]

g

Substituting the expressions for the effective quantities for polyelectrolytes, it is found that
11 21

[2.3.35]

T. Odijk, Macromolecules 12 (1979) 688. P.G. de Gennes, Scaling Concepts in Polymer Physics, Cornell University Press, (1971).

POLYELECTROLYTES

2.36

Figure 2.16. Overlap concentration c* for NaPSS with M w = 6.5xl0 5 g/mol, estimated from the quasi-elastic light scattering data of Koene and Mandel, versus the concentration of added electrolyte (NaCl). The full line is the scaling prediction c* ~ c ~ 0 9 .

As shown in fig. 2.16, this compares favourably with the salt-dependence of the overlap concentration as deduced from the data of Koene and Mandel for the highly charged and intrinsically flexible polyelectrolyte NaPSS: the overlap concentration varies by about one order of magnitude when varying the salt concentration by one order of magnitude. The scaling prediction for the correlation length £, (the typical mesh size of semidilute solutions, see chapter II.5) of uncharged flexible polymers is: ^ag{

[2.3.36]

Again, substituting the expressions of the effective quantities for highly charged, intrinsically flexible polyelectrolytes, one finds,

£~x- 3 / 4 ~c°- 3 7 5

[2.3.37]

Note that the radius of gyration a g (in the dilute regime) and the correlation length
and the concentration factor {(pi (p*T^^ . Although

the radius of gyration decreases with increasing salt concentration, this decrease is overwhelmed by a stronger increase of the concentration factor (which essentially measures the distance to the dilute regime).

Figure 2.17. Effective diffusion constant of dilute (open symbols, less than 0.25 gf ) and semidilute (10g/( , closed symbols) NaPSS as a function of concentration of added electrolyte. Full lines are only guides to the eye.

2.37

POLYELECTROLYTES

The opposite dependencies of the radius of gyration and the correlation length on the salt concentration are nicely illustrated by the quasi-elastic light scattering data of Koene and Mandel for NaPSS, shown in fig. 2.17. This technique measures an effective diffusion constant Deff that is essentially inversely proportion to the radius of gyration in the dilute regime, and inversely proportional to the correlation length in the semidilute regime. 2.3c Grafted polyelectrolytes A brush is a dense layer of end-attached polymers on a surface. Brushes have rather special properties due to the fact that they can only swell in one direction, namely normal to the surface. A sketch of a polyelectrolyte brush is presented in fig. 2.18.

Figure 2.18. A polyelec-

trolyte brush.

Brushes consisting of uncharged chains were already discussed in chapter 1; here, we briefly recall the main results. The average extension of a grafted chain results from a compromise between the osmotic swelling pressure and the elastic force associated with the chain entropy. This can be cast into a simple scaling argument. Consider a flexible (Gaussian) polymer chain consisting of AT segments in a brush having a chains per unit area. If that chain is stretched to an extension L with L> rQ = NC2 (where r0 is the unperturbed end-to-end distance), the force /

between its ends equals

J = kT-^j and the associated pressure to keep the ends at this distance is Jo.

[2.3.38] This must be

balanced with the osmotic pressure 17 in the brush given by 77 = kT(fl/ 2)c2 , where c is the average segment (number) concentration equal to c ={<jN)/L and /? is the excluded volume of a segment: /3 = (1 - 2^)f3 = vC3 . Balancing these pressures immediately shows that the brush extension L scales as L/C = N{af:2v)i/3

[2.3.39]

2.38

POLYELECTROLYTES

The same scaling was found in chapter 1 (see [1.11.19]; (there is a prefactor 6~' /3 which has been omitted here). For the charged brush (with a fraction a of the segments carrying a charge) the osmotic pressure in the brush is different due to the charges. As is well known, a Donnan equilibrium exists between a solution of macro-ions and a pure salt solution. Since the brush is rather dense and since its thickness exceeds the Debye length, it can be well described as a macroscopic phase with an internal potential. For a strong polyelectrolyte, this suggests that there are several relevant cases. (1) At low salt concentrations, typically well below the concentration ac of charged monomers in the brush, the salt is almost entirely excluded from the interior of the brush, and the osmotic pressure difference is approximately given by the concentration of the counterions in the brush: 77 = kTac . This situation is called the osmotic brush. Balancing the osmotic pressure with the stretching force as given in [2.3.37], one gets the simple result: L/( = Na1/2

[2.3.40]

Note that there is no dependence on the grafting density a in this case! (2) At higher salt concentrations, the Donnan effect is that of a solution of screened cylinders, the radius of which is given by the Debye length tc~l (pertaining to the salt concentration outside the brush). This case, referred to as the salted brush, has an osmotic pressure n = kTlac\

IK-2 = kTc2 — —

[2.3.41]

for monovalent salt. This gives a brush height very similar to that of the neutral brush: L/£ = N(veo-e2f/3

[2.3.42]

but with an effective electrostatic excluded volume

V =

[2 3 431

* WT 1

l

--

BCs

(3) Finally, when the charges in the brush are completely screened (at ve « 1), the brush is expected to behave as a neutral brush with a constant 'bare' excluded volume

P=v(3 . All of this suggests a more general expression for the osmotic pressure covering all cases (1) - (3): n = kTclv.fAc) = kTcl\—

P

P[c s +orc p

+ v\

[2.3.44]

J

For a weak polyelectrolyte brush (annealed charge) the behaviour is more complicated

POLYELECTROLYTES

2.39

as the Donnan effect also acts on the protons, thereby affecting the degree of charging a . Qualitatively, the Donnan potential y/D is high at low salt concentrations; counterions are then strongly attracted and co-ions repelled, which implies that the effective

pK of the groups on the polyelectrolyte is shifted

by an amount

~ y D = {eif/D)/kT in the direction away from neutrality. At fixed pH, this implies that the fraction of charged sites decreases with increasing \yD\. Since y D increases with ac , but is also fixed by the pH, this means that dense brushes at low ionic strength have lower charge densities than dilute brushes or free chains. Added salt now has two effects: (i) it lowers y D and, hence, increases the charge density and the osmotic pressure inside the brush and (ii) it screens the charges. The net effect is that the brush height varies non-monotonically with ionic strength I: at low I (in the osmotic regime), it first shows an anomalous swelling with increasing I, and at higher I (in the salted regime) the brush shrinks. The anomalous swelling at low ionic strength can be understood more quantitatively as follows. The osmotic pressure at low I is dominated by the counterions for which we can write, for monovalent salt: 1/9

77 = kTc s exp(y D ) = A:T(e2yDj

[2.3.45]

The Donnan potential can be estimated from the electroneutrality condition: acp = 2c s sinhy D = cseyD

[2.3.46]

where a is now the fraction of dissociated monomers in the brush. This fraction responds to the external pH as follows: fy — u. —

i + gey

D

~ ~

gey

ro o A 71 [z..o.ti j

D

In this latter expression, the factor Q parametrizes the degree of dissociation in the bulk: g> = 1O(PH-PK| =

§-

[2 3 48]

«B

Eliminating a , one finds that e 2 y /7 =

fc c V / 2

= c / Qc& , and thus an osmotic pressure fcrU-^

[2.3.49]

Note that it is the geometric mean of the salt and monomer concentrations that gives the osmotic pressure here. Balancing again with the stretching pressure finally yields

2.40

POLYELECTROLYTES

K0 I / 3 L/f = JVM-

[2.3.50]

Thus, the brush height increases with increasing ionic strength and decreases with increasing grafting density. This latter effect is just the opposite of what happens with the neutral brush. Should 'bare' excluded volume be non-negligible, one might use: 77 = kTPcl

°KS „ + v\

[2.3.51]

Figure 2.19. (a) Thickness (in nm) of a layer of grafted polylacrylic acid) chains as a function of ionic strength / (on a logarithmic scale) for three pH values. The grafting density is 0.26 nm" 2 and the length of the chains is 368 monomer units, (b) Sketches of the structure of the polyelectrolyte brush in the various regimes of ionic strength, (c) Thickness of a weak polyacid brush as a function of ionic strength, calculated numerically for various grafting densities, as indicated; parameters: pH = pK + 2, JV = 20, v = 0 (courtesy of P.M. Bicsheuvel).

POLYELECTROLYTES

2.41

Figure 2.19a shows an experimental example of a weak polyelectrolyte brush, namely PAA grafted on a polystyrene film and immersed in a sodium chloride solution". Figure 19c gives a theoretical example. The height of the brush was measured as a function of salt concentration. The anomalous swelling at low salt concentration followed by a collapse at high concentration is clearly seen. 2.3d Polyelectrolyte gels By cross-linking polyelectrolyte solutions, one obtains polyelectrolyte gels or networks. The osmotic pressure of the small ions confined to the gel volume to preserve charge neutrality and give rise to the distinguishing feature of polyelectrolyte gels, their enormous swelling capacity. Not surprisingly, therefore, one of their most important applications is as a superabsorbent material. A further unique feature of polyelectrolyte gels is that they frequently exhibit first-order swelling transitions, in which the swelling of the networks changes in a jump-like fashion upon changing the solvent quality or the degree of ionization of the polyelectrolyte. The interest in polyelectrolyte networks dates back to Flory21 and Katchalsky31. Firstorder swelling transitions for polymer gels were first predicted theoretically by Dusek and Patterson41. Experimentally, they were first investigated for polyelectrolyte gels by Tanaka51 who also later discovered systems with multiple swelling transitions61 Below we give the simplest theory that captures the essential ingredients, and gives a qualitatively correct description of the swelling behaviour of polyelectrolyte gels, including the swelling transitions. After that we briefly discuss the current status of the theory of polyelectrolyte gels. We consider a polyelectrolyte gel of volume V' immersed in solvent bath of volume V » V'. The solvent is a monovalent electrolyte solution of concentration cs . Inside the polyelectrolyte gel, the electrolyte concentration is c's . The number of salt ions inside the gel volume is denoted by N's = c'sNAvV , the number of salt ions outside the gel is Ns = c s iV Av (V-V) . The polyelectrolyte gel consists of JV segments, each of which releases v counterions into the gel volume. The monomer concentration in the gel volume is c m = N / NAvV . The branch points of the gel have functionality / (number of polymer 'arms' extending from each branch point), and the number of polyelectrolyte segments between branch points is m . The so-called Flory-Rehner approximation (loc. cit.) consists in assuming that the canonical partition function can be factorized into three independent contributions. 11

E.P.K Currie, A.B. Sieval, G.J. Fleer, and M.A. Cohen Stuart, Langmuir 16 (2000) 8324. P.J. Flory, J. Rehner, J. Chem. Phys. 11 (1943) 521; P.J. Flory, Principles of Polymer Chemistry, Cornell University Press, (1953). 31 A. Katchalsky, J. Polym. Sci. 7 (1950) 393; A. Katchalsky, I. Michaeli, J. Polym. Sci. 15 (1955) 69. 41 K. Dusek, D. Patterson, J. Polym. Sci. A 2 6 (1968) 1209. 51 T. Tanaka, Phys. Rev. Lett. 40 (1978) 820. 61 M. Annaka, T. Tanaka, Nature 355 (1992) 430. 21

2.42

POLYELECTROLYTES

Then, the Helmholtz energy of the gel sample can be written as F

= F mix + F elast + Fions

I2-3-52!

where F mix is the contribution from polymer/solvent mixing, F elast is the elastic contribution of deforming the network with respect to some reference state, and F l o n s is the contribution due to the presence of mobile and bound ions. This factorization is the basis of the classic theory of polyelectrolyte gels ' ' . One of the issues of current theoretical interest is the extent to which this decoupling is a good approximation, and how to include corrections in case of its failure. This work has not yet led to a definitive conclusion, and here we proceed assuming that the factorization is valid. Historically, the mixing contribution is approximated using Flory-Huggins theory. For polyelectrolyte gels, which are usually quite dilute, we can also use a virial expansion. The advantage of the virial expansion over the Flory-Huggins theory is that the virial coefficients have a direct molecular interpretation in terms of configurational integrals (see sec. 1.3.9c), whereas the Flory % parameters do not. Following Khokhlov et al. , we therefore use a third order virial expansion: F

mix kT

= JV(Bc

+Cc2)

[2 3 53]

mm

where B and C are the second and third virial coefficients of the polyelectrolyte segments, respectively. The inclusion of terms up to the third order is neccesary to be able to account for the collapse of the network that occurs when the second virial coefficient turns negative. The elastic energy is approximated using the classic expression from the theory of rubber elasticity (see e.g. Flory and Rehner, loc. cit.):

(

9

9

9

\

at: + afr +af o where ax , a and az are deformation ratios of the network along the x, y and z directions with respect to a reference state. We consider free swelling and use ax= a = az= a. The use of this expression, valid for Gaussian chains, is, of course, questionable for highly extended polyelectrolyte chains, but it turns out that the qualitative results on gel swelling and volume phase transitions are insensitive to the specific expression used for the elastic energy. The choice of the reference state has been an issue of debate, but again, it turns out that the qualitative results are rather insensitive to this choice. Following Khokhlov et al. and de Gennes31, we assume here

11 J. Ricka, T. Tanaka, Macromolecules 17 (1984) 2916: H.H. Hooper, J.P. Baker, H.W. Blanch, and J.M. Prausnitz, Macromolecules 23 (1990) 1096. 21 A.R. Khokhlov, S.G. Starodubzev, and V. Vasilevskaya, Adv. Polym. Sci. 109 (1993) 123. 31 P.G. de Gennes, Scaling Concepts in Polymer Physics, Cornell University Press (1979).

POLYELECTROLYTES

2.43

that the reference state is close to the state of network preparation. The latter is approximated by a collection of Gaussian coils of length m that just start to overlap (the so-called c * hypothesis). This gives a reference monomer concentration of C

m.O

rn

_

1

[2 3 551

where a is the segment length. Next consider the polyelectrolyte contributions that are absent for ordinary polymer gels. The dominant contribution, which is the only one that we consider here, is the entropy of the small ions: -!5Si. = (N's + vN)ln(c's +cm) + N's Ync's

[2.3.56]

All other electrostatic contributions are neglected. This also implies that the virial coefficients in the mixing contribution only include non-electrostatic contributions. For the outside solution, the only relevant contribution to the Helmholtz energy is the entropy of the salt ions, pout

= 2i\Llnc, s

kT

[2.3.57]

s

The equation describing the Donnan equilibrium of the salt ions is the equality of the chemical potential of the salt ions in- and outside the gel volume: dF 3F out _Zf_ = ££

[2.3.58]

This gives 2 \l/2

fL= 1 +

fas-l

UcJJ

--^

[2.3.59]

2cs

At low external ionic strength, c s « vcm , almost all of the salt ions are expelled from the gel volume, whereas at high external ionic strength, c s » vcm , the concentrations of salt ions in- and outside the gel volume are nearly equal. In the latter limit, the ionic contribution to the swelling pressure becomes very small, and the polyelectrolyte network behaves similar to a neutral polymer network. At the swelling equilibrium, the total swelling pressure vanishes: '

- =0

[2.3.60]

da Even in this simple theory, this equation cannot be solved analytically. Numerical solutions are shown in fig. 2.20 for a polyelectrolyte with a backbone that is only

2.44

POLYELECTROLYTES

marginally soluble in the aqueous solvent (slightly negative value of B). First consider the limiting case of no salt in the external solution, c s = 0. Upon increasing the number of counterions v released into the gel volume, a discontinuous volume transition is found, in qualitative agreement with experiments. Then, increasing the salt concentration, the transition becomes less and less sharp, and finally disappears, again in qualitative agreement with experiments. In most experiments, the control parameter is taken to be the temperature, which basically tunes the second virial coefficient, B. At a certain temperature, corresponding Figure 2.20. Volume phase transition of polyelectrolyte gels. Deformation parameter a as a function of the number of counterions per polyelectrolyte segment v released into the gel volume. Parameter values: / = 4 , m = 100, Bcm0 m = 3 , Cc^Q m = l . Salt concentrations increase from top to bottom. For a polyelectrolyte segment length of a = 1 nm , the estimated reference monomer concentration is c m 0 = 0.17 M. For that case, from top to bottom: c s = 0 M , 1.7xlO"5M and 6.8xlO" 5 M.

Figure 2.21. Volume phase transition of mixed sodium acrylate/N-isopropylacrylamidc gels in the absence of added electrolyte. Total monomer concentration is 700mM; the concentration of ionic monomer sodium acrylate is indicated. The gels have been cross-linked with 8.6 mM JV.iVmethylenebisacrylamide. Plotted is the temperature versus the degree of swelling VI Vo , where V is the actual volume of the gel and VQ is its original volume. The full curves arc fits to an extended version of the classic theory of polymer gels, as discussed in this section. (Taken from Hirotsu et al. (loc. clt.).)

POLYELECTROLYTES

2.45

to a slightly negative value of B, there is a volume transition that may be either continuous or discontinuous. An example due to Hirotsu et al.11 is given in fig. 2.21, which shows volume phase transitions for polyelectrolyte gels of varying v in the absence of added electrolyte as a function of the temperature. The experimental data were compared with an extension of the theory as discussed here, which correctly predicted that upon increasing the number of counterions released by the polyelectrolyte segments, the transition changes from continuous to discontinuous. Clearly, a number of very drastic approximations have been made in deriving the theoretical results. Recent analytical work21 and computer simulations31 show that there can indeed be no strict decoupling of the elastic and ionic contributions in the partition function since the electrostatic interactions affect the polyelectrolyte configurations and conversely. However, it is not yet clear what consequences the breakdown of this coupling has for the dependence of measurable quantities, such as gel swelling and elastic moduli, on the degree of ionization and the ionic strength41. A further issue is that cross-links or branch points tend to be distributed rather inhomogeneously throughout the network volume in a manner that is highly sensitive to the conditions of network preparation. The consequences of this quenched disorder are being investigated theoretically, especially for neutral polymer gels51. A consequence of quenched disorder, specific to polyelectrolyte gels is that it may affect the degree of counterion binding61. 2.4 Polyelectrolyte viscosity 2.4a General features The viscosity of polyelectrolyte solutions contains a number of interesting trends as a function of the molecular mass M , the polyelectrolyte concentration c , the concentration of added low M electrolyte cs , if any, the charge on the chain, the extent of its screening and the nature of the polyelectrolyte and counterion. Generally speaking, there is no theory covering all these phenomena, but models have been elaborated for limiting situations. Lacking such general theory, it is expedient to consider well-defined experiments to assist in establishing trends. By 'well-defined' it is meant that the polyelectrolyte should be homodisperse if the effects of M , c , cs , line charge, etc. are to be investigated. These parameters should be varied over a sufficiently long range because polyelectrolyte and salt both contribute to the screening. The influence of c can most appropriately be studied by diluting the polyelectrolyte with increasing con 11

S. Hirotsu, Y. Hirokawa, T. Tanaka, J. Chem. Phys. 87 (1987) 1392. T.A. Vilgis, J. Wilder, Comp. Theor. Polym. Set 8 (1998) 61; J. Wilder, T.A. Vilgis, Phys. Rev. E57, (1998) 6865; H. Furusawa, R. Hayakawa, Phys. Rev. E58 (1998) 6145. 31 S. Schneider, P. Linse, Eur. Phys. J. 8 (2002) 457. 41 F. Schosseler, F. Ilmain, and S.J. Candau, Macromolecules 24 (1991) 225; R. Skouri, F. Schosseler, J.P. Munch, and S.J. Candau, Macromolecules 28 (1995) 197. 51 S. Panyukov, Y. Rabin, Y. Phys. Rep. 269 (1996) 1. 61 K.B. Zeldovich, A.R. Khokhlov, Macromolecules 32 (1999) 3488. 2)

2.46

POLYELECTROLYTES

centrations c s in such a way as to keep the ionic strength constant11. The greatest theoretical problem is accounting for the combined influences of electrostatic and hydrodynamic interactions. These effects are not additive since external mechanical forces can change the conformation of polyelectrolytes (sec. 2.3.) and the conformation determines the hydrodynamics. Obviously, electrolytes play a crucial additional role. Only under limiting conditions does the issue become simpler. For instance, weakly charged, dilute, polyelectrolytes behave as swollen coils (Einstein region with high volume fraction (p), whereas strongly charged polyelectrolytes in the absence of added electrolyte are extended, also if subjected to moderate shear forces. An ensuing problem is that of identifying the overlap concentration c* separating the dilute and semidilute regimes. Unlike the situation for uncharged polymers where this parameter can be simply related to M and the radius of gyration a and unlike &

polyelectrolytes at rest (fig. 2.15, [2.3.29 and 34]) 'dilute' now means that the polyelectrolyte molecules do not interact hydrodynamically. Generally, hydrodynamic interactions between particles have a much longer range than the electrostatic ones. So, for viscosity interpretations, c* may turn out lower than its static counterpart, the difference again being dependent on the line charge ve and c s . All told, rheological measurements do contain much information, but it is not yet possible to offer a general theory to rationalize the date. Accordingly, we shall in this section approach the issue empirically. To keep the scope within reasonable bounds, we shall limit the discussion to Newton fluids in the dilute and semidilute ranges. Under these conditions the measurements are straightforward (sees. IV 6.6 and 6.7), provided the system is well-defined. To describe the concentration dependence of the viscosity r]{c ) in table IV.6.1, some viscosity-quantities were defined, of which we shall mainly use the reduced viscosity2) niac=!LJ!sL

[2-4.1]

and the intrinsic viscosity li] =clim >W

[2-4.2]

In the mentioned table impenetrable particles were also considered, in which case the concentration can be well represented by the volume fraction q>. However, for polyelectrolytes, which can swell or compress depending on pH, cs and other variables, the (hydrodynamic) (p is not easy to define. Therefore, c is the preferred variable in [2.4.1 and 2]. If c is given in mol dm" 3 , rjiac and [rj] have the dimensions of a hydrodynamic molar volume, dm 3 mol"1 .

11 D.T.F. Pals, J.J. Hermans, J. Polym. Set 5 (1950) 773; Rec. Trav. Chim. Pays Bas 71 (1952) 458. These are IUPAC recommendations. Other authors sometimes use different names.

2.47

POLYELECTROLYTES

Figure 2.22 gives a sketch of often observed dependencies of the viscosity increment caused by polyelectrolytes. To the left side, the dilute regime prevails, c < c* and the relationship is linear. In this range, extrapolation to c = 0 is possible to yield [77]. At larger concentrations the polyelectrolyte itself starts to screen the charges on the other molecules, which gives rise to a reduction. The initial slope, the position of the maximum and the rate of the decrease beyond the maximum will depend strongly on cs and M (and, for heterodisperse polyelectrolytes, on its M- distribution), on the line charge ve and the countercharge distribution, in particular on the mobile fraction of the countercharge / , see [2.2.2-4]. Recall that highly charged polyelectrolytes [a = 1) at low c s are rod-like, whereas those at low a and/or high cs are coil-like; the viscosity differs substantially between these conformations. At very high c an increase is sometimes observed, probably resulting from alternative rheological behaviour (reptation ...), which is not necessarily entirely viscous. These phenomena will not be discussed here.

Figure 2.22 Schematic representation of the concentration dependence of the viscosity of polyelectrolyte solutions. Discussion in the text.

2.4b Lateral information Accepting that polyelectrolytes have their own idiosyncrasies, one can in some cases compare observations with those for more simple systems to which the behaviour has to reduce under limiting conditions. We call this 'lateral information;' it includes the behaviour of - uncharged polymers (limit for low v and/of high cs ) - low M electrolytes (limit for degree of polymerization—> 1) - particles (limit for impenetrable spheres of rods) Much of this information can be found in previous volumes of FICS, from which we shall now repeat some aspects that may turn out useful. For the range just above c* , particle sols and uncharged polymers, the Huggins equation11 sometimes works well. From [IV.6.11.9], we may write ' W = ['7] + *H['/] 2 c p +o(c p f 11

+

...

[2.4.3]

After M.L. Huggins, J. Phys. Chem. 46 (1942) 151; Ann. NY. Acad. Sci. 43 (1942) 1; J. Am. Chem. Soc. 64 (1942) 1712.

2.48

POLYELECTROLYTES

where fcH is the Huggins constant. Comparison with fig. 2.22 shows that such a simple series development is unlikely to apply for polyelectrolytes; beyond the maximum, kH should be negative; it is not a constant but depends strongly on cs , and it is questionable whether [T]]2 remains as a significant parameter. When screening causes the descending part in fig. 2.22, one should in the absence of electrolyte expect a c p 1 / 2 dependence. In fact, Fuoss en Strauss1'21 proposed a semi-empirical relationship of the type

^ = 17^2

^

where A and B are constants. In swamping electrolyte, c s » c (with c expressed as monomer concentrations), Bcp1/2 » 1, ^C=|SI/2

I2'4-5'

In passing we note that the viscosity of low M electrolyte solutions has a c p / 2 term (recall the Jones-Dole equation [1.5.3.5]), but this has a different origin. For intermediate c s I c ratios, more complicated equations are appropriate, of which illustrations will be given below. For uncharged polymers, a value for the overlap concentrations c* can be obtained from space filling (scaling) arguments (sec. IV.6.11)3).

As argued before, for polyelectrolytes c* is much lower, not only because the radius of gyration a is larger, but also because the range of hydrodynamic interaction is longer. From the particle sol domain, rather concrete information is available on the influence of the surface charge on the sol viscosity in terms of the three electroviscous effects that we shall now briefly review from chapter IV. 6. (i) The primary electroviscous effect is the influence of the electric double layer on the viscosity in the Einstein region. The ionic distribution around the polyelectrolyte chains will be polarized by the flow and this polarization will, in turn, affect the viscous energy dissipation. As interpolyelectrolyte interaction does not play a role in this phenomenon, it finds its way in [2.4.3] via modification of [77] at fixed kH . The phenomenon being determined by the mobile part of the double layer makes [77] dependent on Jve. Generally, [/;1 depends on cs , z, pH (for weak polyelectrolytes) and M . Recall

11 21 31

R.M. Fuoss, U.P. Strauss, J. Polym. Sci. 3 (1948) 602, 603. R.M. Fuoss, Dicuss. Faraday Soc. 11 (1951) 125. P.G. de Gennes, Scaling Concepts in Polymer Physics, Cornell University Press (1979) ch. 3.

POLYELECTROLYTES

2.49

that for uncharged polymers, see [IV.6.11.7], /,2\3

['] = * o ^ - = ^

/ a

a3.

*o^

I2-4-7"

where @0 is the Flory-Fox constant, under 0-conditions equal to = 2.5xlO 2 3 for [rj\ in m 3 kg~ 1 , r in m and M in kg . This equation applies to coils. The constant in it depends slightly on the solvent quality and, therefore, will be different for polyelectrolytes. (ii) The secondary electrouiscous effect accounts for the influence of pair interactions. This phenomenon controls fcH; on top of the value it has for uncharged coils there comes a second one, say kHel2- We refer to [IV.6.9.18], in which for solid spheres,

rin(«/lna)] 4/5 fcHei2~

, ,5 {icaf

J

e^eCa3 {Ka)e2lca a = -^-± -—2kT

[2.4.S]

[2.4.9]

indicating that for hard spheres the dependence on size (a) and electrokinetic potential C, is already not simple. When C, is not known, it can be converted to the mobile (diffuse) charge density o 4 usinga variant of [II.3.5.62] 11 ^ ^ K g O r o ) EQEK

[ 2 4 1 0 ]

KJ ( M I )

where KQ {Ka) and Kj {Ka) are the zeroth and first order Bessel functions of the second kind, respectively. Graphical illustrations of [2.4.10] can be found in figures II.3.14 and 15. However, this conversion is a minor problem in comparison to the issues of: (a) The rather narrow c -window where pair interactions suffice; as soon as these become measurable, multipair interactions also occur, which signals our departure from the dilute regime; (b) The objects under discussion being not hard impenetrable spheres, but more or less flexible charged chains; (iii) The tertiary electroviscous effect, which is a collective noun for the influences of conformational changes on the (reduced) viscosity. Here, we can think of increasing the hydrodynamic permeability of coils by swelling, alignment of chains in the flow field and consequences the variations of the persistence length may have on the flow behaviour. This tertiary effect is at the same time in the very heart of polyelectrolyte science but most difficult to describe, let alone generalize. It is even problematic to establish which

In this equation in Volume II, K in the denominator of the r.h.s. has to be deleted.

POLYELECTROLYTES

2.50

of these phenomena enter [2.4.3] through kH , and which through [77]; in fact, as many of these phenomena are coupled the distinction becomes academic. We shall therefore approach the issue from the empirical side by examining some literature illustrations.

Figure 2.23. Intrinsic viscosities of potassium poly (styrene sulphonate). Sample parameter as in the key. — rod limit. Drawn: Yamakawa model. (Redrawn from Davis and Russel, loc.cit.) Sample Mxicr3 % sulfonation A

1170.0

86

2.44

B

109.5

92

2.61

C

41.5

91

2.58

2.4c The intrinsic viscosity Figure 2.23 gives intrinsic viscosities for three aqueous samples of potassium poly(styrene sulphonate) solutions as a function of the reciprocal Debye length. The data are taken from a systematic and experimental study by Davis and Russel11; the Yamakawa model can be found in ref. 2) . The chain charge is expressed as the ratio lB/lc of the Bjerrum length over the distance between adjacent charges. As this ratio > 1 , no condensation is expected. Note the logarithmic scale: [//] varies by a factor of 10 3 . The [TJ] values are easily a factor of 10 2 higher than those for the corresponding uncharged polymers because of the chain expansion. As expected, [rj] increases with M and with K~1 : the lower the electrolyte concentration, the more extended the chain is. In the low salt limit, the values for extended rods are approached. To the left, at high c s a l t , the polyelectrolyte effect is suppressed and (9 conditions are approached. Here, the polyelectrolyte behaves as a non-draining Gauss-type coil. The drawn curves are based

11 21

R.M. Davis, W.B. Russel, Macromolecules 20 (1987) 518. H. Yamakawa, M. Fujii, Macromolecules 7 (1974) 128.

POLYELECTROLYTES

2.51

on the hydrodynamic theory by Yamakawa and Fuji11, combined with earlier work for such chains by Odijk21 and Fixman and Skolnick31. For theoretical background, see sec. 2.3c. We note that this reasonably accounts for the experimental trends. Key factors are chain stiffness (or for that matter, persistence length) and the excluded volume. As a trend, thermodynamic quantities are more easily accounted for than hydrodynamic ones. More to the quantitative side, it appears that [77] is, as a first approximation, proportional to M 1/2 . For a variety of systems this has been observed, for instance for poly( a -L-glutamic acid) in NaCl solution and various values of cNaC14), for poly(mono methyl itaconate) in organic solvents at various degrees of neutralization51, for sodium poly(acrylate) in aqueous NaBr solutions and at different degrees of dissociation6 and for oligo- and poly(methylmethacrylates) in acetonitrile, n-butylchloride and benzene71, where a lower slope than 1/2 was found for the low M samples. This proportionality is theoretically predicted by the familiar Stockmayer-Fixman equation81 [ti]/M1/2 = Ko+O.5l0oBM1/2

[2.4.11]

Here, B is a second virial coefficient, &0 is the Flory (or Flory-Fox) constant introduced in [2.4.7] and Ko is also a constant (independent of M ), which equals (r2\3/2

K =0

° °^r~

l2 4 121

--

where (r^ is the r.m.s. end-to-end distance under 0 conditions. Equation [2.4.11] works well if the chain is not too expanded. A plot of \rj\lM1^2 as a function of M 1 / 2 (a Stockmayer-Fixman plot) helps to assess the (hydrodynamic) virial coefficient. Intrinsic viscosities of polyelectrolytes typically depend on the electrolyte concentration c s , on the line charge and on the way this charge is distributed. As a first approximation for c s » c , one may expect [rf\(c ) to scale as c~ 1 / 2 . This scaling will hold when the polyelectrolyte behaves as a coil, expanded by intramolecular repulsion. A quick scan of the formulas for electric interaction Gibbs energies shows these to scale with c~1/2 provided / or ff11 are constants. Deviations from the c~ law are A expected for electrostatic ( \j/ and/or cfi are not constant but will regulate) and H. Yamakawa, M. Fujii, loc. cit. T. Odijk, J. Polym. Sci. Polymer Phys. Ed. 15 (1977) 477; Polymers 19 (1978) 989. 31 J. Skolnick, M. Fixman, Macromolecules 10 (1977) 944; M. Fixman, J. Skolnick, ibid. 11 (1978) 863; M. Fixman, J.Chem.Phys. 76 (1982) 6346. 41 M. Satoh, J. Komiyama, and T.Iima, Coll. Polym. Sci. 258 (1980) 136. 51 L. Gargallo, D. Radic, M. Yazdani-Pedram, and A. Horta, Eur. Polym.J. 25 (1989) 1059: also see Eur. Polym. J. 29 (1993) 609 for other solvents. 61 1. Noda, T. Tsuge, and M. Nagasawa, J. Phys.Chem. 74 (1970) 710. 71 Y. Fujii, Y. Tamai, T. Konishi, and H. Yamakawa, Macromolecules 24 (1991) 1608. 81 W.H. Stockmayer, M. Fixman, J. Polym.Sci part Cl (1963) 137. 21

2.52

POLYELECTROLYTES

conformational reasons (for high chain charges, the molecule assumes a rather rod-type conformation, see sec. 2.3). By way of illustration we give a few illustrations, emphasizing the c j 1 / 2 part and accompanying deviations.

Figure 2.24 Electrolyte concentration dependence of the intrinsic viscosity for NaPAA in NaBr solutions of concentration cs (moldm"3) . Given is the parameter B in [2.4.11]. The degree of dissociation is indicated. (Redrawn from Noda et al. loc. cit.) Figure 2.24 illustrates the expected trends for a weak polyelectrolyte. The data are plotted in terms of the parameter B in the Stockmayer-Fixman equation. This parameter is independent of M and has the dimensions of intrinsic viscosity, i.e. of a reciprocal weight concentration (in these experiments, decilitres per gram). One may expect B to consist of an electric and a non-electric part, which according to the figure, are linearly additive: B

= B non-el +B el

B e l = const •f{a)cl1/2

I2-4-13! [2.4.14]

where /(a) is a function of the degree of ionization. It appears that Bnon.ei is independent of c s and a ; this must be an intrinsic macromolecular parameter. The function J(a) is essentially a measure of the relation c^ia0) . As is always found, upon increasing a° further increases of a become gradually less active in increasing cfi (or the 'active' fraction). The trends of fig. 2.24 appear to be rather representative; they have also been reported (in less detail) by Satoh et al. (toe. cit.) for poly(a-L-glutamic acid) in NaCl.

POLYELECTROLYTES

2.53

Figure 2.25. Ion specificity in the intrinsic viscosity of Na poly(styrene sulphonatc) as a function of the electrolyte concentration. Measurements at non-zero shear (/ = 1000 s"1) . Redrawn from Jiang etal. 11 . Figures 2.25a and b are meant to illustrate the lyotropic effects for a strong polyelectrolyte solution. These measurements have been carried out as a function of the shear rate y . The intrinsic viscosity increases somewhat with decreasing y but maintains the same ionic specificity, of which the trends are typical. For instance, these have also been observed by Cohen and Priel 2 ', though at lower c s . The differences between the different counterions reflect their specific binding to the chain; increased binding leaves less charge in the diffuse {i.e. interaction-active) part. So, it is seen that the binding increases from z + = 1 to z + = 2 and for fixed z + , according to Li + < Na + < K+ and Mg 2+ < Ca 2+ < Ba 2 + . This is the familiar sequence for sulphate groups; it is a.o. reflected in the c.m.c. of dodecylsulphate micelles31 and in the surface pressure of Gibbs monolayers of the corresponding surfactants 41 . 2Ad The dilute and semidilute range Now we are considering intermolecular hydrodynamic interactions. For a variety of polyelectrolytes, the reduced viscosity as a function of c passes through a maximum, as sketched in fig. 2.22. The higher the c s , the more suppressed this maximum is. Examples include Na-pectinate in NaCl51, poly(n-butyl-4-vinylpyridinium) bromide in

11

L. Jiang, D.H. Yang, and S.B. Chen, Macromolecules 34 (2001) 3730. J. Cohen, Z. Priel, Macromolecules 22 (1989) 2356. P. Mukerjce, K.J. Mysels, Critical Micelle Concentrations of Aqueous Surfactant Systems, U.S. Natl. Bureau Standards, 36 (1971). 41 See the references in sec. III.3.10b. 51 D.T.F. Pals, J.J. Hermans, Rec. Trau. Chim. 71 (1952) 433. 21

2.54

POLYELECTROLYTES

NaCl, poly(2-vlnyl pyridine) in HCl-solutions in ethyleneglycol and triethylamine11, Napoly(vinyl sulphate) probably in NaCl or NaBr2), and Na-poly(styrenesulphonate) in saltfree31 and salt-containing media41. Figure 2.26 gives an illustration for the last-mentioned system by Cohen and Priel. In fig. 2.26a the position of the maximum is independent of M~5xlO~ 6 g ml"1 , which more or less corresponds to the overlap concentration c* . Added electrolytes lower the maximum, make it less distinct and move it to higher c . This last trend is about linear, with c (max)/ c s =4±0.5 and 77inc (max) also linear with M , and the steeper the lower c s is. The theoretical interpretation, offered by the authors, centres around this equation n J2r2 ^ i n c — ^

12-4.15]

Figure 2.26. Reduced viscosity as a function of polymer concentration for Na-neutralized poly (styrene sulphonate). (a) Influence of M at fixed, low electrolyte concentration (4xlO~ 6 M); (b) Influence of c s at fixed M = 16,000 . In fig. (a) the drawn curves are meant to guide the eye; in fig. (b) the dashed curves are theoretical. Redrawn from Cohen and Priel, loc. cit.

11 21 31 41

H. Eisenbcrg, J . Pouyct, J Polym. Sci. 13 (1954) 85. D.F. Hodgson, E.J. Amis, J. Chem. Phys. 9 1 (1989) 2635. H. Vink, Polym. 3 3 (1992) 3711. N.Imai, K. Gekko, Biophys. Chem. 4 1 (1991) 31.

POLYELECTROLYTES

2.55

with the screening now determined by K2 =4;rt R (c n + 2c.) 0

\ P

s

[2.4.16]

/

because the polyelectrolyte and the indifferent electrolyte both contribute; a h is the hydrodynamic radius. This equation was derived for extended chains, but Hess and Klein" obtained a very similar result for spherical coils. Hence, it is apparently not discriminative. In fact, apart from a numerical factor this equation contained the effective counterion charge z 4 , where z is defined as the number of counterions compensating the line charge, i.e. it is a measure of the diffuse, or mobile part of the countercharge. The factor z 4 also occurs in a paper by Antonietti et al.2), to be discussed below. That this factor does not occur in [2.4.16] stems from the formalism, according to which the structure factor S(q) is related to an 'exclusion radius' determined by K . It is also noted that the dependence o n e is a function of the cs I c ratio. Equation [2.4.15] can at least semi-quantitatively account for the trends in fig. 2.26: 77inc as a function of c passes through a maximum; this maximum increases with M via a h (for an extended chain it is linear) and gets lower with increasing c s . Even the slope cs(max)/ c s = 4 agrees with [2.4.15 and 16]. Equation [2.4.15] is, of course, more specific than the virial expansion [2.4.3]; in fact, this contribution has to be added to [2.4.12], hence the A in the l.h.s. of [2.4.15]. It is, on the one hand, gratifying that such a simple equation represents the data well over the given c range. On the other hand, one cannot infer too much from it about specific polyelectrolyte properties, such as the persistence length and the (counter) charge distribution.

Figure 2.27. Reduced viscosity increment for poly(Na-styrene sulphonatc) microgels in salt-free solutions. The molecular weight of the parental solution increases in the direction of the arrow. Degree of sulphonation 70%. Drawn curves, theory. (Redrawn from Antonietti c.s., loc.cit.}

11 21

W. Hess, R. Klein, Adv. Phys. 32 (1983) 173. M. Antonietti, A. Briel and S. Forster, Macromolecules 30 (1997) 2700.

2.56

POLYELECTROLYTES

Antonietti et a l . ' ' have presented another illustration of the same nature. These authors studied the rheology of branched polyelectrolytes, viz. cross-linked poly(styrene sulphonates) in electrolyte and salt-free solutions. So, the polyelectrolyte molecules behave as microgel particles. Below c*p there is no M -effect anymore, as would be the case for impenetrable spheres; but above c* , r][ac decreases with c just as in the r.h.s. of fig. 2.26. For this part of the curve, the authors derived an equation, which just as with the previous example consisted of the series expansion [2.4.3] plus an additional term, somewhat more elaborate than [2.4.15], but with the factor z 4 made explicit. Figure 2.27 gives an illustration. This double logarithmic plot shows that the theory can account satisfactorily for the results. Data below the predicted maximum are not available, but in analogy to fig. 2.26 a decline with decreasing c must be anticipated. For a gel the molecular mass is not so easily definable, the more so as the parental polymers (from which the gel particles were synthesized) were polydisperse. However, the trend is indicated by the arrow. This trend is reflected in the intrinsic viscosity, which in this direction increases from 0.044 via 0.052 to 0.090 dm 3 g"1 for the parental microgels, whereas the fitted values for the intrinsic viscosity and Huggins constant ku are, in this order, 0.139, 0.143 and 0.182 d m 3 g - 1 and 0.2, 0.6 and 0.9 , respectively. So, one accounts semiquantitatively for the trends.

Figure 2.28. Huggins constant for the same solution of K poly(styrene sulphonate) as in fig. 2.23.

Finally, fig. 2.28 gives an illustration of the salt effect on the Huggins constant, which displays a striking minimum between high values at low and high ionic strengths. To the left, 0 -conditions are approached. These minima result from an interplay of direct intermolecular contributions to the stress, the ensuing coil compression and the attractions appearing near 0-conditions. The authors discuss this in more detail, but 11

M. Antonictti, A. Briel and S. Forster, loc. cit.

POLYELECTROLYTES

2.57

the final quantitative answer still has to come. For further interpretations, see". Viscometry has also been invoked to monitor conformational changes and counterion specificity features. By way of illustration, we refer to the frequently discussed21 phase transition, which occurs as a function of pH in solutions of poly(methacrylic acid), but not with poly(acrylic acid). Figure 2.12 already gave an illustration. Sugai et al.3> confirmed these transitions for poly(ethacrylic acid) in NaCl and at various temperatures, and Klooster et al. 4) studied these for poly(acrylic acid) in methanol (MeOH) and found the occurrence of the transition to be ion-specific: for or > 0.1, it does not occur in LiOCH3 (as in aqueous solution of the Na-salt), whereas it is found in NaOCH3 . There appears to be more room for viscometric studies of ion specificities. In particular, the effects of higher valence cations like La3+ and Th 4+ deserve attention since it is known that, at the proper pHs these ionic species hydrolyze and give rise to superequivalent adsorption (sec. IV.3.9J) For polyelectrolytes, the phenomenon has been reported51. 2.5 Polyelectrolytes in electric fields Just as with low M electrolytes and particulate colloids, polyelectrolyte solutions can be subjected to an electric field, the response of the system acting as a tool to extract information. Three familiar techniques are electrokinetics (electrophoresis in particular), dielectric studies and conductometry. Experiments can be carried out in DC or AC. Electrophoresis is usually done in DC, dielectric investigations mostly in AC, with the aim of scanning the various relaxation ranges (dielectric spectroscopy). Conductometry is preferentially carried out in DC, but AC data are automatically produced as a side effect of dielectric spectroscopy: the (measurable) complex impedance consists of a real (resistive) part yielding the conductivity spectrum K(co) and an imaginary (capacitive) part giving a>£0£{a>), see [II.4.5.13] and sec. II.4.8a. These two quantities are related to each other through the Kramers-Kronig relations. The interpretation of the obtained electrometric data is cumbersome because a number of physical phenomena participate and interfere; for many of these no sufficient theory is available. For instance, the conduction mechanism differs substantially among the dilute, semi-dilute and concentrated regimes and may depend in a complex way on the presence of added electrolyte. In this section, some attention will be paid to the issue

11

J-L Barrat, J.F. Joanny, Adv. Chem. Phys. XCIV(1996) 1, I. Prigogine, S.A. Rice. Eds. Already in 1974 H. Okamoto, and Y. Wada cited 30 references, sec J. Polym. Sci. Polym. Phys. 12 (1974) 2413. 31 S. Sugai, K. Nitta, N. Ohno, and H. Nakano, Colloid Polym. Sci. 261 (1983) 159. 41 N.T.M. Klooster, F. van der Touw. and M. Mandel, Macromolecules 17 (1984) 2070. See for instance M. Drifford, J.P. Dalbiez, M. Delsanti, and L. Belloni, Ber. Bunsengesellsch. Phys. Chem. 100 (1996) 829. 21

2.58

POLYELECTROLYTES

of electrometrically bound counterions {= counterions that in electrometry appear not to move freely). Apart from the question as to how these are bound (electrostatically condensed, chemically or still otherwise associated), one of the issues is whether different electrometric techniques yield identical answers (are electrophoretically and conductometrically bound counterions identical?). Accepting that there is no established overall theoretical picture, it is sometimes helpful to compare electrometric properties of polyelectrolytes with those of low M electrolytes, on the one hand, and with particulate colloids on the other. One typical illustration is that the conductivity of a polyelectrolyte of JV charged monomers and its compensating counterions is always lower than that of the JV monomers free in solution. Apparently this is the result of counterion immobilization by the chain. Another line is the interpretation of the slip plane for solid particles and the application to polyelectrolytes. We shall address this issue first. 2.5a The issue of electrokinetic binding Although polyelectrolytes, together with their countercharge, are electroneutral, they move in an electric field, i.e. they exhibit electrophoresis. The occurrence of this phenomenon implies charge separation: upon moving, the polyelectrolyte may entrain part of the countercharge, but not all of it. Which part? The issue has a micromechanical origin. Upon tangential movement of a liquid (especially water) with respect to a solid, a thin, adjacent layer of this water is stagnant, meaning that it does not move with respect to the solid; in electrophoresis the stagnant water moves with the solid. In streaming potential experiments, it remains stationary with the particles constituting the porous plug. Countercharges captured in stagnant layers behave as if they are electrokinetically bound. The phenomenon is widespread; it is observed for inorganic solids, polymer latices, hydrophobic and hydrophilic surfaces, surfactant micelles, ... Hence, it presumably also occurs with polyelectrolytes, either with coils as a whole or with individual chains. Questions coming to mind include: (i) what is the origin? (ii) how can the phenomenon be experimentally detected? and (iii) can we state something about its magnitude? (i) According to present day insight, the origin has to be sought in the water structure adjacent to hard walls1'21. Mostly as a result of the repulsive part of the intermolecular interactions, this leads to the familiar stacking: oscillations of the molecular density pN{z) that peter out over a very few molecular diameters. This picture is in line with the generality of the phenomenon; it should apply to liquids adjacent to any surface that is hard (impenetrable) on the molecular scale. It is even observed for some L-L interfaces31, should not occur for clean water-air phase boundaries, but must be expected for fluids

11

J. Lyklema, S. Rovillard, and J. de Coninck, Langmuir 14 (1998) 5659. J. Lyklema, Oil Gas Set Technol. 56 (2001) 41. 31 A.M. Djerdjev, J.K. Bcatty and R.J. Hunter J. Colloid Inter/act Sci. 265 (2003) 56. 21

POLYELECTROLYTES

2.59

adjacent to polyelectrolytes. The structure formation leads to an increase of the tangential viscosity in the adjacent liquid that, because of the strongly cooperative nature of viscous flow, becomes so strong that phenomenologically speaking the layer behaves as if it were stagnant.

Figure 2.29. Relation between the electrokinetic and surface (- or line) charge, both expressed in the same units Cm" 2 or Cm- 1 . Tan a = / e k .

(ii) For particulate colloids, micelles, etc., the electrokinetically bound charge can be determined by simultaneously measuring the surface charge a° and the electrokinetically free charge crek . The latter can be inferred from £ potentials, using GouyChapman theory, of which the application is safe because the electrokinetically free part of the countercharge more or less coincides with the diffuse part of the double layer. Measurement and interpretation of f -potentials have been described in detail in chapter II.4. Figure 2.29 sketches the relationship between the two types of charges as it is usually found. At very low surface or line charge this charge equals the electrokinetic one (negligible fraction of countercharge in the stagnant layer), but upon increasing the surface or line charge, <7ek lays behind, to eventually level off in a plateau when any additional charge on the surface or the polyelectrolyte is compensated by a counterion in the stagnant layer. Phenomenologically, the trend is identical to that in fig. 2.4, but the interpretations are very different. (iii) Magnitudes can be expressed in terms of the ratio k

_ electrokinetically active charge surface- or line charge

which may, or may not, be identical to / in [2.2.2]. This ratio starts with unity, to decrease with increasing line charge, see fig. 2.29. For hydrophobic (Agl) or hydrophilic ( Fe 2 O 3 , TiO 2 ) macrosurfaces, / e k can be read from figures like II.4.13 and II.3.63; at high a° , / e k can be as low as 0.2. Such low values are also observed for ionic micelles and it will not come as a surprise if such values are also observed for polyelectrolytes. Application of these insights to polyelectrolytes remains a challenge for further study. Little is known of the micro-mechanics of water flow tangential to chains with internal degrees of freedom. The experience with macrosurfaces can perhaps best be translated to polyelectrolytes in the limiting cases of isolated chains (~ cylinders) and coil-like,

2.60

POLYELECTROLYTES

weakly charged polyelectrolytes, to which the more general and more intriguing, intermediate situations reduce under extreme conditions. In view of the developments in terms of two-state models (Oosawa, Manning, ..., see sees. 2.2b and c), there are arguments also to simplify electrokinetic binding in terms of only one parameter ( / e k ) demarcating electrokinetically, fully stagnant and fully free (as in the bulk solvent) double layer parts. This is an abstraction from reality because the tangential viscosity does, with increasing z, not drop discretely from infinite to its bulk value, but rather decreases gradually, though very steeply. However, in the case of macrosurfaces such idealizations are common and appear to serve well in practice; In electrokinetics it Is the familiar slip plane, which separates the fully stagnant layer from the fully mobile bulk, and in electrostatics this role is played by the outer Helmholtz plane, separating the Stern from the diffuse layer (see sec. II.3.6c). Actually, experience has shown that in many interpretations of data, one can get away with the identification of the potential at the slip plane (£) with that at the outer Helmholtz plane {y/1). We shall not repeat this discussion here. 2.5b Conductivity, backgrounds In principle, conductivity studies provide information on the numbers of ionic species in a solution and their mobilities. As association of ions lowers the number of charge carriers, the conductivity of a solution also depends on the degree of association. In the polyelectrolyte case it depends on the extent of counterlon association, but only if the association leads to electroneutral pairs that do not move in an electric field. On the basis of conductivities alone, one cannot obtain simultaneous information on numbers and mobilities. To that end, additional information is required, for example via measurement of transference numbers. Alternatively, one could analyze the conductivity as a function of adjustable variables, such as the degree of association of the polyelectrolyte or the concentration of added low M electrolyte. In sec. 1.6.6 the basic principles for the conductivity of simple electrolyte solutions have been discussed. Conductivity measurements are not particularly difficult provided the usual precautions are taken (avoidance of electrode polarization, appropriate thermos tatting, ...). Transference numbers can be extracted from electrolyte excesses near electrodes. The primary piece of information is the conductivity K (in Sm" 1 or Cy~ 1 s~ 1 m~ 1 J11, which follows simply from Ohm's law from the current density j (in A = Cs ) generated by an applied field E (Vm"1] j = KE

[2.5.2]

For an ideal, low M electrolyte solution, K consists of additive contributions of the ionic species i, so that it is possible to define specific ionic conductivities K{ with j^KjB

Previously called specific conductivity.

[2.5.3]

POLYELECTROLYTES

K = ZiKi

2.61

[2.5.4]

For non-ideal solutions, electric and hydrodynamic interactions have to be accounted for, but the additivity rule can be maintained (sec. 1.6.6b). It is a logical step to separate in K{ the number of charge carriers, determined by the ionic concentration c{ and the ionic velocity expressed as its mobility u; (in m 2 V*1 s^1) K^jZilFCjU;

[2.5.5]

and K = FIi\zi\ciui

[2.5.6]

Introducing for each charged species the ionic or molar conductivity Xi (in Sm~2mol~') as AJ^ZJIFUJ

[2.5.7]

[2.5.6] can be written as K = ZiciAi

[2.5.8]

For low M electrolyte solutions additivity persists, but Xi now becomes dependent on ci. The first term of the development A^cJ scales as c\12 (sec. 1.6.6b). Now, consider the application to polyelectrolytes, counting ionic species, distinguishing a polyelectrolyte, (monomer-) concentration c , a cation (c+) and anion (c ). Without loss of generality, we assume the polyelectrolyte to be negatively charged; hence cations are the counterions and anions the co-ions. K = F[| Z + |c + U + +|z_|c_u_+| Z p |c p u p ]

[2.5.9]

If c is counted per monomer, z is the number of charges per monomer. For polyelectrolytes in salt-free solution, the co-ion contribution drops out and [2.5.9] reduces to K = F[| Z + |c + u + | Z p |c p u p ]

[2.5.10]

Equations [2.5.9 and 10] are only formal in that they assume only one averaged concentration and one averaged mobility for each species. In reality, these parameters vary from place to place. Moreover, they depend on c p and cs (if electrolyte is added). Counterions have a very high concentration close to the chain, beyond which region c+ decays as in a diffuse double layer till its bulk value. For co-ions, if present, it is the other way around; their concentration is low near the chain (even zero if the line charge is high enough) and increases till their bulk value.

2.62

POLYELECTROLYTES

Here, we have to digress into the meaning of 'bulk electrolyte concentration.' For saltfree polyelectrolyte solutions, there is no bulk except for c —» 0 . Theoretically, in that case, c s = 0 , but in reality there is always some spurious electrolyte in the system (carbonate from CO 2 in the air, ions released from the vessel wall...), in addition to the H + and OH" ions stemming from the spontaneous dissociation of water. Actually, this reference state is not well-defined. In semidilute solutions the reference is the half-way cation concentration, but this reference obviously depends on c and a model is needed to establish it (compare sec. 2.2b). These ambiguities can be avoided by adding low M electrolyte to the system. However, a price has to be paid because in this situation negative adsorption of electrolyte takes place, leading to Donnan exclusion (sees. I.5.5f and 5.6a). Again, a model is needed to establish this effect because measuring c s in the dialysate would yield an overestimation. Only in swamping electrolyte (cs » c ) is the Donnan effect negligible. Intermediate situations cannot be dealt with without theory. Returning to the values of c and u in [2.5.9 and 10], there are arguments to use (again) a two-state approximation for u + , by distinguishing only bound and free counterions. The fraction n-_f c o n c M that is conductometrically bound has zero mobility. Counterions finding themselves close to the chain, but hopping along the chain, also belong to fraction (l - / c o n d J when for these ions c+A+ « c + (bulk)A + (bulk). For the free ions it is safe to presume u+ =u® , the mobility at infinite dilution, assuming c + (r) to be given by diffuse double layer theory, see [2.2.10 or 2.2.21]. The equivalent for flat plates has been developed by Bikerman 11 and gives rise to [II.4.3.60] for the excess surface conductivity in a diffuse double layer. The equivalent for cylindrical geometry has still to be elaborated. However, general experience with the interpretation of experimental surface excess conductivities in electrokinetics gives no indication that the Bikerman equation would not apply to a purely diffuse double layer. If problems arise, it is with the non-diffuse part (stagnant or Stern part), see chapter II.4. Translated into polyelectrolyte terms, it is an acceptable first approximation to work with immobile bound counterions and free counterions having bulk mobilities. With this picture in mind, setting in [2.5.9] for a symmetrical (z-z) electrolyte c+ = cs+ Jcond | z p | c p , c _ = cs and replacing the polyelectrolyte term by (1 - / c o n d ) | z p | C p U p to account for the conduction-active part, K = F[z(cs

+

/co«d|zp|cp)u0+Zcsu0+(l-/<=°»d)|Zp|Cplipj

[2.5.1!,

By the same token, for salt-free polyelectrolytes, K = F[z(cs+/c°»d|zp|cp)u0

+

(l-/»»d)|zp|cpUp]

[2.5.12]

These equations have a fair generality; variants can be found in the literature. Using 11

J.J. Bikerman, Z. Physik. Chem. A163 (1933) 378; Kolloid-Z 72 (1935) 100.

POLYELECTROLYTES

2.63

them at various c s and c , or as a function of the degree of dissociation, sometimes allows establishing parameters. However, this generality also implies the absence of any theoretical foundation in terms of polyelectrolyte configurations. In this respect, more research is wanting. For an attempt in this direction for semidilute polyelectrolytes in salt-free solutions, see11.

At this point it may be noted that so far a variety of bound (1 - / ) and free (/) fractions have been met. Different origins have been proposed (condensation, ion association, electrokinetic trapping...) and a variety of techniques have been used for measuring them (titration, osmotic pressure, electrokinetics, conductometry ...). We shall not discuss here the intriguing issue as to whether all these techniques measure the same / and, hence, whether all these fractions have the same origin. Given measurements of K and those of transference numbers, one can write equations combining the two. Generally, the transference number t of an ionic species in a mixture of species, is defined as the fraction of the current that species i transports. In terms of conductivities, K, K. 1 t=— — = —L 1 IiKi K

[2.5.13]

with 2^=1

[2.5.14]

Because of [2.5.14], one needs to measure the transport number of only one component in a mixture of two; for salt-free systems t

can be deduced if t+ is measured.

Finally, conductivities can also be represented in terms of molar conductivities A , which are the contributions per mole of electroneutral electrolyte. They are obtained by dividing K by the (molar) concentration of the electrolyte. Generally, for one electrolyte A = K/cs

(Sm 2 mor 1 )

[2.5.15]

In [1.6.6.13] it was shown that [2.5.15] also applies to asymmetric electrolytes, hence this equation may also be used for salt-free polyelectrolytes: Ap=K/cp

[2.5.16]

The problem is that in the more general situation of polyelectrolyte in low M electrolytes there are two electroneutral components, for which the molar conductivities are not additive, because of the Donnan exclusion. The usual procedure for obtaining a molar conductivity that is characteristic for the polyelectrolyte is by writing

11 R.H. Colby, D.C. Boris, W.E. Krause, and J.S. Tan, J. Polym. Sci B, Polym. Phys. 35 (1997) 2951.

2.64

POLYELECTROLYTES

/1_=^-^ P C P

[2.5.17]

This equation is simple but application is not obvious because Ks cannot be easily measured. Mostly for Ks the conductivity for c —> 0 is substituted, but this is only approximately true because of the Donnan effect. This issue comes back in establishing limiting values. For simple electrolytes, A{cs —> 0) can be found by extrapolation. If we do that for polyelectrolytes in the dilute range we obtain A (c -> 0), which is still a function of cs . In the semidilute range we can write11 A

P = AV(CP=O)

12 5 181

+ 0 C

' -

( P)

where A (c = 0) and <£> depend on cs . The overlap concentration c* also depends on c

s-

2.5c Conduction: illustrations We shall now present an anthology of experimental studies from the literature, showing the kind of results that can be obtained. The first illustration (fig. 2.30) shows the transference number of Na+ ions as the counterions of the weak poly(acrylic acid), PAA. The polyelectrolyte concentration is rather high and low M electrolytes are absent.

Figure 2.30. Transference number of Na+ (counterions) on PAA as a function of the degree of neutralization O, c — 1.5x10^^ monomol dm" 3 ; • c = 3.78 xlO~ 2 monomol dm" 3 . (Redrawn from Mandel, see ref. in sec. 2.8, who has based his figure on data by Wall et al. and Schmitt et al, respectively) Under these conditions, no c

d e p e n d e n c e is observed. For a—>0 , t P

r

. =0.5, Na+

meaning that 50% of the charge transport is accounted for by the counterion and 50% by the macro-ion. When a increases, t decreases because a fraction of the Na+ ions Na +

is bound to the chain, and therefore moves to the anode, together with the 11

H. Vink, Macromolec. Chem. 183 (1982) 2273.

POLYELECTROLYTES

2.65

polyelectrolyte. For larger a , t + even becomes negative, i.e. t > 1. This means that the polyelectrolyte does more than its share in the conduction; it even carries its counterparts in the wrong direction. The phenomenon is not unique; it is also observed for micelles and charged colloids in dilute electrolyte.

Figure 2.31 Concentration dependence of K - K{cp = 0) for poly(diallyl dimethyl ammonium chloride) (PDADMAC) over three c p ranges and for different molecular masses: O, Mn = 12,000; • 22,000; • 72,000; • 170,000 gmol" 1 . (Redrawn from Wandrey and Hunkclcr, toe. cit.)

Figures 2.31 and 32 stem from a review by Wandrey and Hunkeler11, containing much experimental information. In fig. 2.31 the average distance between the charges on the chain is 0.5nm , so the line charge ve = 3.2xlO~6 juC cm~' . As these charges reside at the periphery of the ally! groups (a five-member ring with an N(CH3);j; at its end), the backbone can be interpreted as well as a cylinder. If the radius a is ~ 0.6 nm C. Wandrey, D. Hunkeler, Study of Counterion Interactions by electrochemical Methods in Handbook of Polyelectrolytes and their Applications, S.K. Tripathy, J. Kumar, and H.S. Nalwa, Eds. Vol. 2 chapter 5. American Scientific Publ. (2002).

2.66

POLYELECTROLYTES

Figure 2.32. Molar conductivity A of Na poly(styrene sulphonate). Influences of molar mass and NaCl concentration: T = 20°C. Key: O, no added salt: V, 1CT6 (5xlCT 6 in (c)); • , 2xlO~ 6 ; D, 4xlO~ 6 ; • , 10~ 5 ; A, 2xlO~ 5 ; A, 5 x l 0 ~ 5 ; O, 10~ 4 M. The (number-average) molecular mass (in g/monomol ) is indicated. (Redrawn from C. Wandrey, loc.cit.; data points below 10 6 monomol"1 deleted.)

(depending on the radial positions of the allyl groups), this cylinder has a surface charge
C. Wandrey, Langmuir 15 (1999) 4069.

POLYELECTROLYTES

2.67

anionic polyelectrolyte Na poly(styrene sulphonate), (NaPSS); charge distance 0.5 nm ve = 4.8xl(r 6 uCcm""1; CT° = 17|iCcm-2 , also rather heterodisperse (Mw / MD <1.1). Results are collected in fig. 2.32. Curves are found with maxima that are suppressed by the addition of salt. The c range covers several decades and includes more than one concentration regime. To the right we are probably in the region (c) of fig. 2.15, where the characteristic length E, is independent of M . This regime is attained at lower c for higher M . Working our way to the left an increase of A is observed, which is more pronounced the lower c is. So, this is a typical polyelectrolyte effect. This upward trend has been observed more often. For example, Mandel in his review11 cites a number of illustrations. The occurence of a maximum is more rare; it was, for instance, reported for Na (carboxymethylhydroxyethylcellulose)21. Strictly, when in very dilute solutions the dilute regime would have been reached, A ought to be independent of c , see [2.5.18], but this is not observed. However, the caveat must be made that the measurement of A in such dilute solutions is tricky because the difference K-Ks in [2.5.17] is small; it requires a very accurate establishment of Ks and correction for salt exclusion. The positions of the maxima shift to the right with increasing cs . We note that this trend is also expected for the overlap concentration c*p ; this quantity increases because at high c the polyelectrolytes are screened better, reducing their excluded volumes.

Figure 2.33. Conductivity in dilute solutions of three carboxymcthyl-celluloses (from O via • to + with an increasing degree of substitution) and for poly(methacrylic acid) (A). (Redrawn from Vink. loc.cit.)

Another illustration was given by Vink and is reproduced in fig. 2.33. Here, K is plotted without attempting to subtract Ks . In this experiment extrapolation to c = 0 is linear and easy. The extrapolated conductivities are independent of the charge and nature of the

11

See sec. 2.8.

21

H. Vink, J. Chem. Soc. Faraday Trans. I 7 7 (1981) 2439.

31

H Vink, MacTomolec. Chem. 183 (1982) 2273.

2.68

POLYELECTROLYTES

polyelectrolyte and may, therefore, represent Ks . On the other hand, the slopes representing A are charge and nature specific, as expected. We note that the c range of this study (10~5-10~4 monomol dm" 3 ) is comparable with that in the previous study (fig. 2.32), so there are specific differences between polyelectrolytes of different natures. In the CMC series, the most highly charged samples have the lowest A values, apparently because the protons are more strongly bound. Ion specificities in ^ p (c p = 0) for polyfstyrene sulphonates) (PSS) and poly(vinylsulphates) (PVS) are collected in table 2.2; similar series were reported by Szymczak et al.1J and Kwak and Hayes21. Table 2.2. Limiting equivalent conductivities for PSS and PVS. Given is 104 A (c ->0) in S m 2 m o r 1 (fromVink, loc. cit.) Counterion

PSS

PVS

K+

46.1

64.5

38.1

55.3

112

162

Na H+

+

Analysis of such data, in terms of ionic concentrations and mobilities using [2.5.11 or 12], is ambiguous because there are two unknowns, Jcond and the mobilities of the bound counterions. For particulate colloids, there are safe ways for establishing _f c o n d , which incidentally correlates well with the free fraction inferred from particle interaction studies31. Hence, for such colloids the mobilities of bound ions can be obtained. All of this deserves to be elaborated for polyelectrolytes. Awaiting such further studies, it can be observed that A (c —> 0) in table 2.2 and in similar studies increases in the direction of Na+ < K+ < H+ , which agrees qualitatively with the increase of the ionic mobilities at infinite dilution (table I.6.5)4'. However, the increase is less quantitatively. The inference is that specific binding increases in the same direction, which is in line with the Gurney principle (sec. 1.5.4). The sequence depends on the nature of the polyelectrolyte5'. Regarding counterions of higher valencies, bivalent ions are more strongly (conductometrically) bound than monovalent ones. For example, Szymczak et al.6) report on this phenomenon for Mg2+ , Ca 2+ and Sr 2+ binding to the same polyelectrolyte. Counterions of still higher valencies like La3+ and Th 4+ can bind super11 J. Szymczak, P. Holyk, and P. Ander, J. Phys. Chem. 79 (1975) 269. Also sec P.R. Holyk, J. Szymczak and P. Andcr, J. Phys. Chem. 80 (1976) 1626. 21 J.C.T. Kwak, R.C. Hayes, J. Phys. Chem. 79 (1975) 265. 3 J. Lyklema, in [nterfacial Electrokinetics and Electrophoresis, A.V. Delgado, Ed. Marcel Dckkcr (2002) 87. See in this connection the IUPAC recommendation by van Lecuwen et al. in sec. 2.8a. 51 See for example H. Vink, J. Chem. Soc. Faraday Trans. (I) 85 (1989) 699. 61 J. Szymczak et. al, loc. cit.

POLYELECTROLYTES

2.69

equivalently, leading to overcharging11. Sometimes this phenomenon is attributed to ion correlations leading to a charge distribution, which differs from that obtained by a mean-field theory (like PB theory). However, in sec. IV.3.9j it was established that in all systems where multivalent ion binding was studied the origin was the formation and binding of hydrolyzed species. Only at rather low and rather high pH does no such charge reversal occur. To make sure that ion correlations are responsible for the overcharging, studies have to be carried out at such extreme pHs. As with viscometry, there remains much to be done in the field of polyelectrolyte conductometry, both experimentally and theoretically. 2.5d Dielectric properties We mention this theme for the sake of completeness, but refrain from a systematic discussion because the field is too complex and unsettled to review its fundamentals. For particulate colloids satisfactory techniques and elaborations are available, see sees. II.4.5e and 4.8. However, the numerous additional degrees of freedom, with the inherent number of conformational interactions for 'soft' particles, leads to almost unsurveyable complications. For an authoritative discussion of the state of affairs, a review by Mandel and Odijk is recommended21. 2.6 Electrostatically driven complexation and phase separation 2.6a Solubility of polyelectrolytes In this section we discuss the solubility of a polyelectrolyte, which carries a fixed number of charges (i.e. it has strong dissociating groups or the pH is fixed at such a value that a is unity, see [2.2.37]) and has a soluble backbone. In other words, we pose the question as to whether charges can reduce solubility. One may wonder whether this is a relevant discussion, because polyelectrolytes in water have been widely studied from the viewpoint of true solutions (i.e. one-phase systems) so that it is not immediately obvious why phase separation is an issue. Yet there is a general argument as to the reason we cannot ignore this point. This stems from the fact that in electrolyte solutions the ions are not distributed randomly. Due to electrostatic interaction, the ions tend to arrange in such a way that the distances between like (repelling) ions are larger and those between unlike (attracting) ions are smaller than the average distances. In other words, there are, even in dilute solutions, important ion-ion correlations. One of the simpler attempts to take such correlations into account in the theory of electrolyte solutions is the well-known Debye-Hiickel theory Sec, for example, M. Drifford, J-P. Dalbiez, M. Delsanti, and L. Belloni, Ber. Bunsenges. Phys. Chem. 100 (1996) 829. M. Mandel, T. Odijk, Dielectric Properties of Polyelectrolyte Solutions. Ann. Rev. Phys. Chem. 35 (1984) 75-108.

2.70

POLYELECTROLYTES

leading to the familiar limiting law for ionic activity. There is a delicate balance since both the electrostatic energy and the configurational entropy are lower than for the random case, but it is the entropy that must prevent the ions from clustering into a dense phase (see the discussion in chapter 1.5.2). An increase in the strength of the interaction as compared with the thermal energy may thus be expected to perturb that balance whereupon a dense phase appears. As is borne out by [2.2.39], such an increase can be brought about by an increase in the valencies zvz. of the ions or by a decrease in either the dielectric constant, the temperature, or the interionic distances. Experimentally, T, r, and e are not entirely independent; in fact, both r and e depend on T, but e may be controlled to some extent (e.g. in mixed solvents), and so are the valencies. As was shown by detailed experimental studies (see fig. 1.5.2) and later by Monte Carlo calculations, the correlation effects are relatively unimportant for monovalent ions in water at room temperature and moderate concentrations, so that it is justified for this case to use the Poisson-Boltzmann equation, which ignores such effects. However, correlations already become quite noticeable with divalent ions, to the extent that the Poisson-Boltzmann equation gives predictions qualitatively different from simulations in which non-smeared out charges occur. The energy/entropy balance mentioned above would, of course, shift in favour of insolubility when the valency of the ions increases or when they are tied to a macromolecule: charged groups, which are covalently linked to a polymer chain, have much less entropy than free ones. Qualitatively, this is one way of explaining why solutions containing polyions of opposite sign {polysalts) are often insoluble, whereas simple salts are soluble, and why DNA forms swollen coils when it has monovalent counterions, but collapses into a globular or toroidal state upon addition of the trivalent ion spermine or the tetravalent ion spermidine, or upon lowering the dielectric constant by, for example, adding some ethylene glycol. One theoretical approach sometimes used is to distinguish between free and bound counterions. The counterions are supposed to bind to charges on the chain, thus forming ion pairs; a polymer, which primarily carries such ion pairs, is called an ionomer. These ion pairs are strong dipoles and can again cluster with other ion pairs (into socalled multlplets). This effect, sometimes called the 'ionomer effect' produces an effective cohesion between monomers, which is opposed by the entropy associated with mixing polymer segments and solvent, and by the entropy due to the translation of the unbound ions. A feature of the theory is that counterion binding is described as a simple chemical equilibrium with a binding constant, which depends on the local dielectric constant. Kramarenko et al.11 have described models of this kind, which can account for phase transitions that are actually observed. An example is the titration of poly(acrylic acid) in

E.Yu. Kramarenko, I.Ya. Eruchimovich, and A.R. Khokhlov, Macromolecular Theory Simulation 11 (2002) 462

POLYELECTROLYTES

2.71

water/methanol mixtures with base. In their acid form, this macromolecule is soluble, but above a critical degree of neutralization the polysalts become insoluble. Another supporting observation is that polyacids in dimethyl sulfoxide (DMSO, relative dielectric constant 48.75) form very stable solutions, but as soon as a little neutral salt is added, they precipitate due to the formation of ion pairs. Hence, minor changes in the strength of Coulomb forces or in the effective entropy of the ions can tip the balance and render polyelectrolytes insoluble. 2.6b Polyelectrolyte complexes What is the physical reason for phase separation? As was said in sec. 2.6a, electrostatic forces always oppose the separation of opposite charges. The electrostatic energy of a system consisting of ions is lowest when the ions arrange in a dense phase, more or less in alternating order, as they would do in a salt crystal, with positive charges surrounded by negative ones and vice versa. To dissolve the complex, entropy is needed to overcome the attractive electrostatic energy; when the electrostatic forces are very strong, monophasic systems can only exist at extreme dilution. It should, therefore, be no surprise that stoichiometric mixtures of oppositely charged macro-ions will also phase-separate. The driving force comes from two effects: (i) the monovalent ions, which originally surrounded the macro-ion in the diffuse double layer, had a reduced entropy; when the complex forms, the diffuse double layers are stripped off the chains leading to an entropy increase; {ii) in the dense complex that forms, the average distance between positive and negative ions is smaller than that between the groups on the free macro-ion and their counterions, so that the electrostatic energy is lower. It also follows from this picture that the net change in Gibbs energy associated with the formation of the dense complex phase decreases with increasing ionic strength as the effects of (i) and {ii) both become smaller at higher ionic strength. Indeed, many complexes are redissolved at high ionic strength. A very similar case is that of colloidal spheres of opposite charge. Also here, phase separation may occur (heterocoagulation) around stoichiometric compositions. A difference is, however, that Van der Waals forces may be more important in lyophobic colloids. 2.6c Complex coacervation It has been known for a long time that mixtures of anionic and cationic polyelectrolytes phase-separate into a dense phase, containing most of the polymer, and a dilute phase essentially containing only small ions. The dense phase may sometimes have solid-like properties, in other cases it appears as a liquid. In the latter situation, the term complex coacervation is employed, and it seems likely that the phase-separated system is in thermodynamic equilibrium, so that one can meaningfully determine a phase diagram. When doing so, it is mandatory to consider the number of independent

2.72

POLYELECTROLYTES

components of such a system. The most general description calls for at least five components: solvent (water), polycationic salt, polyanionic salt, polysalt, and the simple salt consisting of the relevant counterions. Because of the Gibbs-Duhem relation, only four of these are independent. A simplified case is obtained when the simple salt is omitted and the polymers are stoichiometrically mixed in their pure acid or base form, so that only a polysalt is obtained. One could thus also consider simpler 3-component mixtures of polysalt, simple salt and water; off-stoichiometric mixtures will then require the two extra components, namely polyanionic or polycationic salt.

Figure 2.34. (a) Schematic phase diagram for complex coacervation in a mixture of a polyanion salt Pn"Cn+ and a polycation salt P n + C n ~, showing tie lines, (b) Experimental phase diagram for a mixture of gelatine and gum arabic at fixed pH (no added salt). Coexisting complex coacervate phases ( Cj , ... Cg ) and dilute electrolyte ( ej,... eg), obtained from the mixtures ( nij , ... m 5 ) are indicated. Critical points are indicated as 'Cr.p'. (Redrawn from H.G. Bungenberg dc Jong, in Colloid Science II, H.R. Kruyt, Ed., Elsevier (1949) ch. X, sec. 2g. In many cases, added salt narrows the two-phase region of the phase diagram and this often leads to complete dissolution. A schematic phase diagram is given in fig 2.34a. As can be seen, the tie-lines, corresponding to coexisting phases, are perpendicular to the polymer-polymer axis, which just means that the polymer-rich phase and the dilute phase have compositions that hardly differ when the overall composition is stoichiometric; further away from stoichiometry, the dilute phase becomes progressively rich in the excess component. For very asymmetric polyelectrolyte mixtures the system is monophasic. An example of an experimental phase diagram is shown in fig. 2.34b for the mixture of gum arabic and gelatine at pH = 5. Note that this kind of phase diagram, corresponding to associative phase separation (due to attraction between the polymers), is entirely different from the phase diagram of a mixture of two neutral, incompatible polymers in a common solvent, which usually splits into two separate polymer solutions due to repulsion {segregative phase separation). Compositions inside the closed two-

POLYELECTROLYTES

2.73

phase region will phase-separate into a dense phase (with approximately the overall composition) and a very dilute phase containing minute amounts of polymer. As is clear from this diagram, it is also possible that a concentrated phase appears with a composition, which is significantly different from the stoichiometric one (in terms of polymeric charges). Such a phase contains not only polyions of both kinds, but also necessarily a certain amount of small cations or anions to ensure overall electroneutrality. As the ionic strength increases, the composition range of the complex coacervate phase narrows, and above a critical ionic strength c sc complexation is entirely suppressed. We thus conclude that the insoluble complexes that form generally incorporate a certain amount of excess polymeric charge, which is zero in the strictly stoichiometric case and increases as the mixture becomes more asymmetric. For neutrality, a non-stoichiometric complex therefore has to take up small ions, which involves an entropy loss. Figure 2.35. (left) (a) Solubility diagrams (degree of coacervation as a function of composition) for a mixture of polyanion and polycation with added salt. Compositions within the closed area lead to phaseseparation, (below) (b) Solubility diagrams for mixtures of gum arabic and gelatine with added KC1 for various pH values as indicated. (Same reference as fig. 2.34, but sec. 2f.)

2.74

POLYELECTROLYTES

A useful representation of the phase behaviour is obtained at constant total polymer concentration, but varying polymer composition and ionic strength. Such a representation is schematically sketched in fig. 2.35a; an experimental example is presented in fig. 2.35b. One may wonder why asymmetric complexes are still insoluble. As in any electrolyte solution, the electrostatic energy is always lowered upon increasing the concentration because of increasing ion-ion correlations. For ordinary electrolytes consisting of small ions only, this attractive energy is counteracted by a significant (and salt concentrationdependent) entropy reduction, so that dissolution is possible although at the expense of reduced activity coefficients. For ions bound to a macromolecule, however, such an entropic effect is obviously much smaller. As a result, a stoichiometric mixture of macroions of opposite sign will easily become insoluble. A non-stoichiometric mixture will have to incorporate a certain number of small ions, which contribute an extra entropic term to the Gibbs energy. Hence, there is a balance between an attractive effect from the macromolecular ions and a repulsive effect from the small ions. This balance determines the shape of the phase diagram. One may also wonder what happens outside the two-phase region, but at salt concentrations below c sc . Under these conditions, the polyions still attract each other and form complexes, but there is a large mismatch in the number of available molecules, and the composition of the formed complexes will reflect that mismatch. Each molecule of the minority component will thus bind an excess of the minority component, so that complexes are formed with a high charge density. The small ions neutralizing that highly charged complex still have enough entropy to inhibit it from phase separating. Theoretical calculations of the shape of the phase diagram are complicated because the Gibbs energy of a mixture of polyions and small ions obviously depends strongly on the way in which the various ions arrange themselves. Not only will this affect the configurational entropy of the small ions, but it will also have a contribution from the entropy (conformational and translational) of the chains. To simplify matters, one could treat the electrostatics by the same model as that underlying the Debye-Hiickel theory, provided one takes into account the reduced entropy of ions on a chain. Needless to say, short range polymer-solvent and polymer-polymer interactions play a role as well; separating complexes consisting of polyelectrolytes with very hydrophobic backbones may be much more difficult than complexes, which are inherently hydrophilic. On top of this, specific ion effects will almost certainly lead to important differences in the binding energy of various kinds of ion pairs. A complete theory of complex coacervation is therefore a long way ahead. An early attempt to construct a theory for complex coacervation is due to Voorn and Overbeek11. These authors simply proposed two additive contributions to the Helmholtz 11 M.J. Voorn, Fortschr. Hochpolym. Forsch., 1 (1959) 192-233; J.T.G. Overbeek, M.J. Voorn, J. Cellular and Comparative Physiology 49 (1957) 7-26.

POLYELECTROLYTES

2.75

energy density: a polymer/solvent mixing entropy, given by the Flory-Huggins theory (see sec. V. 1.2.2) and an ionic contribution / i o n according to the Debye-Hiickel theory, given by 1 1 LJ™kT

= -K3(

3

/\2n:

[2.6.1]

Here, t is the size of a polymer segment and AT"1 is the Debye length given by the familiar expression [II.3.5.7] 2

e2 eeokT^

v

>'

In [II.3.5.7], Cj and zi are the number concentrations and number of charge per molecule of species i, e is the charge of the electron, and EQE is the dielectric permittivity of the medium. The number concentrations can be expressed in volume fractions ( CjJVj^/V =
where (B is the Bjerrum length. Note that at any combination of the chemical potential of the solvent //j and the osmotic pressure 77 turns out to be

A=^=i+vU^)+i kT

kT

N

2V \

3

C3

3+...

12.6.3]

J 2,

where a is the average number of charges per monomer unit (0 < a < 1) . This result is very similar to the osmotic pressure of an ordinary polymer solution [ 1.2.4], but with an effective excluded volume given by the term in braces with an electrostatic contribution to the excluded volume, which is negative and scales as f.|r"' . As expected, for low ionic

11

T.L. Hill, An Introduction to Statistical Thermodynamics, Addison Wesley (1960) ch. 18.

2.76

POLYELECTROLYTES

strength and vof order unity, this term is large and the excluded volume is negative. Hence, this polysalt will be insoluble. If simple salt is present, the osmotic pressure will have additional contributions from the salt. That due to the polymer can still be described by [2.6.3], but now the effective excluded volume is salt-dependent. With increasing salt concentration, it becomes less negative and eventually it may even switch from positive to negative; the polysalt becomes soluble in excess simple salt. Hence, the result is a phase diagram like that for a neutral polymer with an electrostatic term included in an effective solvency parameter. This result can be generalized to include polymer-solvent interaction by adding the usual Flory-Huggins interaction parameter x • as follows: ZeS=^-^°2+Z

= Zion+Z

[2.6.4]

where we defined the parameter j i o n = (2^/3)(^gX""1 /i3)a2 . The theory is, of course, approximate as it is based on the Debye-Hiickel theory for dilute electrolytes and low potentials, whereas complex coacervate phases may be rather concentrated. Moreover, it assumes that ions bound to a chain can distribute around a given ion in precisely the same way as free ions, which is probably only acceptable for ions separated by many Kuhn lengths, i.e. chains with low charge densities. Nevertheless, it seems to give qualitatively reasonable results. It is regrettable that the full set of equations has never been solved (which can only be done numerically), so that we do not know what the theory predicts for, for example, off-stoichiometry compositions. Finally, we should realize that the mean-field character of the treatment, which excludes fluctuations, cannot deal with a feature like the formation of soluble complexes. The next step was taken only much later by Castelnovo and Joanny11. These authors again considered mixtures of two polymers (A and B), both of which were considered to have a given linear charge density somewhere between zero and a high value. In addition, they took into account possible incompatibility effects (which lead to segregative phase separation for neutral polymers) by means of an adjustable Flory-Huggins parameter XAB for the interaction between different monomers. A contribution of the polymer's conformational entropy was included as well. The phase behaviour was analyzed in terms of solution stability, i.e. looking for spinodal lines. The theory predicts not only a region of associative phase separation due to electrostatic attraction, but also a region of miscibility and a region of segregative phase separation. As expected, associative phase separation occurs at high charge density and low ionic strength, whereas segregative phase separation occurs at high ionic strength and zero or low charge density. These two regions are separated by an intermediate one-phase region of complete miscibility. Unfortunately, results were again only given for stoichiometric compositions. We reproduce the diagram of states derived by Castelnovo and Joanny in fig. 2.36. 11

M. Castelnovo. J.F. Joanny, Eur. Phys.J. E 6 (2001) 377-386.

POLYELECTROLYTES

2.77

Figure 2.36. Theoretical phase diagram for stoichiometric mixtures of oppositely charged polyelectrolytes with added electrolyte. The parameter t characterizes the incompatibility strength as compared with the polyionic attraction, and s characterizes the salt-induced screening as compared with the polyionic attraction. Attraction is strongest for small values of s and t (lower left corner of the diagram) and decreases for points further away from this corner. (Redrawn from Castelnovo and Joanny, loc. cit.)

Complex coacervation is not restricted to pairs of flexible polyelectrolytes. Mixtures of one flexible polyelectrolyte and a globular protein also separate into dense, complex coacervate and dilute phases with qualitatively similar features. At the molecular level, the structure of such complexes has been investigated by means of various simulations11. 2.6d Polyampholytes Chain molecules with both negative and positive charges along the chain are called polyampholytes. Their behaviour in solution is strongly dependent on the ratio of positive to negative charges and on their distributions along the chain, much in the same way as the behaviour of polyelectrolyte mixtures depend on the stoichiometry. At about equal numbers of positive and negative charges, the attraction between these is the dominant feature, and the polymer collapses and forms an insoluble phase. When there is an excess of either positive or negative charges, chains may partially collapse, but the excess charge prohibits phase separation. Added salt will always counteract phase separation. The distribution of charges along the chain is also important. Strictly alternating polyampholytes (which are difficult to prepare!) are expected to dissolve more easily and 1

R. de Vries, F. Weinbreck, and C.G. de Kruif, J. Chem. Phys. 118 (2003) 4649; R. de Vries, J. Chem. Phys. 120 (2003) 3475.

2.78

POLYELECTROLYTES

be less prone to phase separation. Diblock polyampholytes, with one negative and one positive block, will show a strong tendency to form complexes, which are insoluble around the 1:1 charge ratio, but may form interesting micellar objects (mesophases) when one block is much longer than the other. Random copolymers are an intermediate case. If the dissociating groups are weak acids and bases, the various cases (excess positive, balanced, excess negative) will appear upon variation of the pH; the solubility minimum will be situated around the isoelectric point. 2.6e Polyelectrolyte multilayers The tendency of oppositely charged polyelectrolytes to form insoluble complexes also provides the mechanism underlying the fabrication of so-called polyelectrolyte multilayers. These layers are prepared by exposing a suitable (usually solid) substrate to solutions of polyanions and polycations in an alternating fashion. At each exposure, a finite amount of polyelectrolyte is deposited, and by repeating the alternating treatment one can build films up to tens or hundreds of nanometers in thickness. Many examples can be found in the literature 11 . That the alternating exposure method described above does often produces these multilayers may be surprising considering solubility diagrams like the one given in fig. 2.35. This diagram implies that, provided equilibrium can establish itself, a given quantity of complex coacervate (which forms at compositions within the two-phase region) must redissolve when a composition outside the two-phase region is imposed. Equilibration thus precludes multilayer formation. Indeed, cases have been observed where equilibration and the expected redissolution occur when a weak polyacid (polyacrylic acid, PAA) and a weak polybase polyfdimethylaminoethyl methacrylate), (PAMA) were used, and a sufficient amount of added phosphate buffer was present 21 . In this case, a rapid initial deposition was observed followed by a slow dissolution of material from the surface. An example is shown in fig. 2.37. When the ionic strength due to the buffer was lowered, the redissolution process was suppressed, and it was possible to build up a multilayer. These findings imply that polyelectrolyte complexes are either reversible, i.e. they can either equilibrate because the components are sufficiently mobile, or they are solid-like and, once formed, cannot redissolve upon changing the polyelectrolyte composition in the surrounding medium. Only in the latter situation is multilayering possible. Whether or not reversibility prevails depends on the chemical nature of the polyelectrolytes (e.g. the hydrophobicity of their backbone, the linear charge density, the nature of charged groups, etc.; typically, sulphonates form stronger complexes than carboxylates) and on the concentration and the type of added salt. Unfortunately, data underpinning such dependencies are rather scarce. A combination 11

G. Decher, Science 277 (1997) 1232; P.T. Hammond, Curr. Opin.Colloid Interface Sci. 4 (1999) 430; T.W. Graul, J.B. Schlenoff. Anal. Chem. 71 (1999) 4007. 21 D. Kovacevic, S. van der Burgh, A. de Keizer, and M.A. Cohen Stuart, Langmuir 18 (2002) 5607.

POLYELECTROLYTES

2.79

often used is poly(styrenesulphonate)(PSS)/poly(allylamine)(PAAm), which forms stable multilayers at concentrations of NaCl as high as 1 M. For the multilayering process to proceed successfully, an irreversible system is clearly required. The term 'multilayer' suggests that films prepared by the multilayering process have a well-defined periodic structure. However, neutron reflectivity studies 11 have shown that each deposited layer mixes largely with adjacent layers, so that no periodicity can be discerned at this level. In the cases studied, mixing did not extend to more than about two layers, as could be shown by reflectivity measurements on layers in which the contrast varied with a periodicity of several layers. Liquid-like complexes, in which individual polymers can move freely, would of course be expected to loose all spatial correlation over time.

Figure 2.37. Deposited mass of polymer as a function of time in an experiment where polycation (PAMA) and polyanion (PAA) are supplied at pH 6.8 to a silica substrate in alternating fashion. Curve A is for a low concentration of phosphate buffer (1 mM) and shows buildup of a stable multilayer. Curve B is for 5 mM added phosphate buffer; it is clearly seen that each deposition of PAMA is followed by a spontaneous redissolution process (the adsorbed mass decreases) and no stable multilayer is formed. (Redrawn from Kovacevic et al. toe. cit.)

2.6f Complex coacervate micelles Complex coacervation in mixtures of charged homopolymers leads to the formation of a macroscopic phase. However, if one of the homopolymers is replaced by a diblock copolymer, which has an extra neutral, water-soluble block, a macroscopic phase can be circumvented. Because incorporating the neutral block in the complex would substantially increase the Helmholtz energy, a complex coacervate particle cannot grow in three dimensions. Instead, objects of colloidal size are formed. These can be called complex coacervate core micelles (c.c.c.m.), as they are analogous to the association colloids formed by ordinary surfactants or by diblock copolymers in a selective solvent. J. Schmitt, T. Griinwald, G. Decher, P.S. Pershan, K. Kjaer. and M. Losche, Macromolecules 26 (1993) 7058.

2.80

POLYELECTROLYTES

Figure 2.38 Speciation diagram of particles occurring in mixtures of a polyelectrolyte and a charged/neutral diblock copolymer

Various systems have been reported in which stable micelles of this nature occur11. Evidence shows they are spherical, with a dense core consisting of the two kinds of charged chains in the form of a nearly neutral complex coacervate, surrounded by a more dilute corona of swollen neutral chains. The fact that two kinds of polymers are needed to form such objects implies that the sample composition plays an important role. Indeed, a pure micellar solution only forms at the 'preferred micellar composition' (p.m.c), which is somewhere close to a stoichiometric mixture in terms of charged polymer groups. Apparently, only very few small ions are incorporated. Away from the p.m.c, the number concentration of micelles decreases rapidly as they disintegrate into small soluble complexes carrying an excess positive or negative charge. This can be summarized in the following schematic speciation diagram (fig. 2.38), which presents the weight concentration of a given species as a function of the mole fraction x of either positive or negative monomers, based on the total concentration of charged monomers (i.e. we disregard the neutral monomers). We distinguish five kinds of species: free

11

A. Harada, K. Kataoka, J. Am. Chem. Soc. 121 (1999) 9241; A.V. Kabanov, T.K. Bronich, V.A. Kabanov, K.Yu, and A. Eisenberg, Macromolecules 29 (1996) 6797; S. van der Burgh, A. de Keizer, and M.A. Cohen Stuart, Langmuir 20 (2004) 1073.

POLYELECTROLYTES

2.81

polycations (p+), free polyanions (p-), positive soluble complexes (sc + ), negative soluble complexes (sc-), and complex coacervate core micelles (c.c.c.m). The composition axis can be split into four regions. In the two outer regions, 1 and IV, we have a p+ /sc+ mixture (I) or a p-/sc- mixture (IV), respectively. In the two inner regions, II and III, we have an sc+/ccm mixture (II) or an sc-/ccm mixture (III), respectively. The boundary between region II and III is at the p.m.c, where all polymer is just taken up in micelles. At the boundary between regions I and II, called 'critical excess cationic change' (c.e.c.c), we have a solution containing only sc+, and at the boundary between regions III and IV, denoted 'critical excess anionic charge'f c.e.a.c), we have just sc- in solution. Light scattering titrations have shown that the boundaries between the various regions are fairly sharp and marked by sharp changes in slope of the intensity vs /plots. An example is given in fig. 2.39. Figure 2.39. Light scattering intensity (arbitrary units) of a mixture of poly(methacrylic acid) (negatively charged, M = 113 kg/mol) and poly(dimethylaminoethyl methacrylate-co-poly glyceryl methacrylate) (positive PAMA35 and neutral PGMA100 block), as a function of mole fraction x of methacrylic acid monomers in the mixture, at starting pH 6.8 and 100 mM NaCl. The maximum intensity corresponds to the preferred micellar composition (p.m.c). (Redrawn from S. van der Burgh et al. (loc. cit.)

2.7 Applications of polyelectrolytes Polyelectrolytes are widely used to control the stability of colloidal dispersions. We discuss two specific applications, flocculation and enhanced stabilization. 2.7a Flocculation Many commercial flocculants are often cationic polyelectrolytes. In particular, high molar mass copolymers with a relatively low fraction of charged monomer units and the remaining units water-soluble but uncharged are very effective flocculants. The majority of practical systems where flocculation is required have negative surface charges (e.g. bacteria in active sludge of waste water treatment plants or anionic 'trash' in paper mill waste water); that is why cationic flocculants are preferred. Polyelectrolytes of low charge density are preferred because these do not reverse the surface charge and are thus less

2.82

POLYELECTROLYTES

prone to induce restabilization. Finally, a high molar mass is favourable because the number of bridges a given chain can form between two particles increases with increasing chain length. Some successful flocculants based on polyacrylamide copolymers can have molar masses of several thousand kg/mol. One example is the flocculation of montmorrilonite clay particles by copolymers of (uncharged) acrylamide and (cationic) N,N,N-trimethylaminoethyl chloride acrylate studied by Durand-Piana et al." The optimum flocculation occurred typically for polymers with 1% cationic units and molar mass M > 1000 kg/mol. The effect of molar mass in orthokinetic flocculation can be related to the time-dependent thickness of the adsorbed polyelectrolyte layer, in the same way as discussed in sec. 1.12.e; it is more complicated, though, because salt also influences the relaxation processes in the polymer layer21. 2.7b Stabilization enhancement Highly charged polyelectrolytes have the ability to adsorb strongly to charged surfaces such that the effective charge of the covered particle reverses. Moreover, the adsorption of weak polyelectrolytes (which is of course sensitive to pH) often shows considerable hysteresis upon cycling the ionic strength. For example, when 90% hydrolyzed polyacrylamide (effectively a copolymer of 90% acrylic acid and 10% acrylamide) adsorbs onto cationic latex from 0.5 M NaCl, the adsorbed amount is much higher than when adsorption takes place from 0.002 M NaCI. However, if adsorption is allowed to take place at 0.5 M salt, after which the ionic strength is lowered to 0.002 M, an intermediate adsorbed amount results, which is definitely higher than the one obtained upon adsorption at 0.002 M 31 . Meadows et al. called this 'enhanced adsorption,' and noticed that enhanced adsorption makes colloidal particles much more stable than particles prepared by the usual procedure (adsorption at low ionic strength). Similar enhancement effects occur upon cycling the pH41. The stabilizing effect of dense and strongly adsorbed polyelectrolyte layers is very effective indeed. Meadows found a tenfold increase in critical flocculation concentration for particles with enhanced adsorption. 2.8 General references 2.8a IUPAC recommendation Conductometric Analysis of Polyelectrolyte Solutions. Prepared for publication by H. van Leeuwen, R.F.M.J. Cleven and P. Valenta, Pure Appl. Chem. 63 (1991) 1251. 11

G. Durand-Piana, F. Lafuma, and R. Audcbert, J. Colloid Interface Sci. 119 (1987) 474; L. Eriksson, B. Aim, and P. Stenius, Colloids Surf. A70 (1993) 47. 21 Y. Adachi, T. Matsumoto, and M.A. Cohen Stuart, Colloids Surf. A207 (2002) 253. 31 J. Meadows, P.A. Williams, M.J. Garvey, and R. Harrop, J. Colloid Interface Sci. 139 (1990) 260. 4) J.G. Gobel, N.A.M. Besseling, M.A. Cohen Stuart, and C. Poncet, J. Colloid Interface Sci. 209 (1999) 129-135; C.W. Hoogendam, A. de Kcizer, M.A. Cohen Stuart, and B.H. Bijsterbosch, Langmuir 14 (1998) 3825.

POLYELECTROLYTES

2.83

2.8b Other references Aut. Div. International Symposium on Macromolecules, under auspices of IUPAC, Butterworth (1970). (Contains contributions on polyelectrolytes; old, but not dated, containing papers giving the then state of affairs.) J.-L. Barrat, J.-F. Joanny, Theory of Polyelectrolyte Solutions, in Adv. Chem. Phys. XCIV (1996) 1-66. (Review, 125 refs., emphasizing theory from a physical angle of incidence.) Polyelectrolytes in Solution and at Interfaces, J. Barthel, H. Dautzenberg, D. Horn and W. Oppermann, Eds., Ber. Bunsengesell. 100 Nr. 6 (1996). (Proceedings of the Symposium Polyelectrolytes Potsdam '93'. Containing 62 papers on polyelectrolytes in solution and at interfaces.) M. Borkovec, B. Jonsson and G.J.M. Koper, Ionization Processes and proton Binding in Polyprotic Systems: Small Molecules, Porteins, Interfaces and Polyelectrolytes, in Colloid and Surface Set 16 (2001), E. Matijevic, Ed., Kluwer/Plenum, 99-339. (Very detailed review on electric double layers in various systems, 390 refs.) H. Dautzenberg, W. Jaeger, J. Kotz, B. Philipp, C. Seidel and D. Shtcherbina, Polyelectrolytes; Formation, Characterization, Application, Carl Hanser, Miinchen (1994). S. Forster, M. Schmitz, Adv. Polymer Sci. 120 (1955) 51. M. Mandel, Polyelectrolytes, in Encyclopaedia of Polymer Science and Engineering, 2nd ed., H.F. Mark, N.M. Bikales, C.G. Overberger, and G. Mendes, Eds., Wiley, Vol. 11 (1988) 739. (Extensive review, 314 refs, covering the entire field. Although some features are nowadays better understood, this review still serves as an excellent introduction into the field.) M. Mandel, T. Odijk, Dielectric Properties of Polyelectrolyte Solutions, in Ann. Rev. Phys. Chem. 35 (1984) 75-108. (Review of the state of affairs indicating the many problems that have to be addressed.) T. Radeva, Ed., Physical Chemistry of Polyelectrolytes, Detcher (2001). K.S. Schmitz, Macroions in Solution and Colloidal Suspensions, VCH, New York (1993). Handbook of Polyelectrolytes and their Applications, S.K. Tripathy, J. Kumar and H.S. Nalwa, Eds., Am. Scientific Publishers 2002. (Three volumes: 1. Polyelectrolytebased Multilayers, Self-assemblies and Nanostructures; 2. Polyelectrolytes, their Characterization and Polyelectrolyte solutions; 3. Application of Polyelectrolytes and Theoretical Models.)

2.84

POLYELECTROLYTES

C. Wandrey, D. Hunkeler, Study of Polyion Counterion Interaction by electrochemical Methods, (2002).

3

ADSORPTION OF GLOBULAR PROTEINS

Willem Norde, Jos Buijs and Hans Lyklema 3.1

Introduction

3.2

Structure of globular proteins

3.3

3.2a

Conformational entropy

3.4

3.2b

Interactions that determine the 3D structure of proteins in aqueous solution

3.3

3.4

3.1

3.4

Adsorption of globular proteins from aqueous solution onto (solid) surfaces

3.10

3.3a

3.11

Adsorption kinetics

3.3b

Relaxation at interfaces

3.17

3.3c

Driving forces for protein adsorption

3.19

Adsorption-related structural changes in proteins

3.23

3.4a

How to measure structural properties of adsorbed proteins

3.24

3.4b

General trends

3.30

3.5

Adsorbed amount and adsorption reversibility

3.37

3.6

Influence of some system-variables on protein adsorption

3.40

3.7

3.6a

Protein and sorbent charge

3.40

3.6b

Hydrophobicity

3.41

3.6c

Protein structure stability

Adsorption at fluid interfaces

3.42 3.42

3.7a

Review of some general trends and techniques

3.43

3.7b

Some illustrations

3.45

3.8

Competitive protein adsorption and exchange between the adsorbed and dissolved states

3.52

3.9

Tuning protein adsorption for practical applications

3.55

3.10

General references

3.58

This Page is Intentionally Left Blank

3 ADSORPTION OF GLOBULAR PROTEINS WILLEM NORDE, JOS BUMS AND HANS LYKLEMA

3.1 Introduction This chapter may be considered as a sequel to chapter II.5. Chapter II.5 deals with the adsorption of relatively simple polymers and polyelectrolytes, whose molecules are built up of identical units. In solution, such molecules are rather featureless, having a flexible coily structure, and their adsorption behaviour has been well-modeled. In the present chapter we will discuss the more complicated biopolymers, which are often made up from a variety of monomeric units. For example, proteins are polymers of some twenty-two different amino acids. Because of the variation in physical-chemical properties-mainly in polarity and electrical charge, between the constituting amino acids, protein molecules are ampholytic, are more-or-less amphiphilic, and assume complex three-dimensional structures. As a result, the adsorption of biopolymers is more intricate than that of the simpler polymers treated In chapter II.5. Knowledge of the adsorption behaviour of biopolymers has progressed over recent decades but a unified predictive theory is still far away. However, the discussion of the principles of biopolymer adsorption may start from general trends observed for the adsorption of the simpler polymers, and, in particular, polyelectrolytes. Some main features of polyelectrolyte adsorption may be summarized as follows: (a) When a flexible polymer molecule adsorbs, its conformatlonal entropy is reduced. Hence, for adsorption to occur spontaneously (parts of) the polymer molecule should be attracted by the sorbent surface. Even if the attraction per adsorbing segment of the polymer is only weak the whole polymer molecule may adsorb tenaciously, because many segments adsorb. (b) A flexible, highly solvated polymer molecule typically adopts a train-loop-tall conformation in the adsorbed state, as depicted in fig. II.5.1. The extension and the density of the loops in the adsorbed layer are determined by the solubility of the polymer. A high loop-density is reached only with poor solvents. (c) Because of their polar ionic groups polyelectrolytes are usually well soluble in water. Also, mainly owing to intramolecular electrostatic repulsion, they adsorb with relatively little loop formation in a rather flat conformation.

Fundamentals of Interface and Colloid Science, Volume V J. Lyklema (Editor)

© 2005 Elsevier Ltd. All rights reserved

3.2

PROTEIN ADSORPTION

(d) More often than not, sorbent surfaces are electrically charged and they may electrostatically attract or repel polyelectrolytes. Even in the case of electrostatic repulsion the polyelectrolyte may still adsorb, namely if non-electrostatic attraction of segments outweighs the electrostatic repulsion. For example, polyelectrolytes containing hydrophobic apolar groups may adsorb readily on a (hydrophobic) surface under electrostatically unfavourable conditions. (e) Polyelectrolyte adsorption is strongly affected by the ionic strength. At elevated ionic strengths, charge-charge interaction may be effectively screened and the polyelectrolytes' behaviour approaches that of an uncharged polymer. The pH, which often controls the charge on the polyelectrolyte, and sometimes, that on the sorbent surface, influences polyelectrolytic adsorption in a similar way, at the pH where the polyelectrolyte attains a low charge density it adsorbs in a relatively thick, loopy layer, which is not sensitive to ionic strength variation. (f) In line with paragraphs c, d and e, the variation of the adsorbed amount of polyampholytes (macromolecules containing both anionic and cationic groups) as a function of pH, passes through a maximum at the isoelectric point. The pH-dependent adsorption profile flattens with increasing ionic strength. Polysaccharides, polynucleotides, and unfolded protein molecules that all attain extended structures in aqueous solution, adsorb according to the patterns mentioned above. However, globular protein molecules may deviate strongly from (some of) these principles. The polypeptide chains of globular proteins, to which enzymes, immunoproteins and most transport proteins belong, are folded into a denser structure and, as a rule, contain different structural elements such as a-helices and (3sheets as well as more unordered parts. The architecture of a globular protein is complex and heterogeneous, and highly specific for each type of protein. When interacting with an interface the protein structure may change. The kind of rearrangements depends in a complex way on the interaction with the sorbent surface. Because of the structure-specificity, the adsorption behaviour will differ between different protein species. Consequently, a general theory predicting globular protein adsorption is not to be expected but, based on the advances made during the last few decades, some general principles have been unraveled. Studies of the interaction between proteins and interfaces may help understanding of the mechanism that determines the 3D structure of protein molecules. In addition to this academic interest, protein adsorption has a high practical impact. Examples can be found in biomedical engineering, enzyme immobilization in bioreactors, drug targeting and -delivery, and its use in stabilizers in dispersions in foodstuffs, pharmaceuticals and cosmetics, etc. Conformational changes in protein molecules, which have consequences for their biological functioning, are the most intriguing and challenging aspects of protein adsorption, both from a theoretical and an applied point of view. This matter will be

PROTEIN ADSORPTION

3.3

discussed extensively in sec. 3.4. To understand the behaviour of proteins at interfaces we first have to discuss the main types of interaction that determine protein structure in (aqueous) solution. 3.2 Structure of globular proteins. Proteins are co-polymers built from some 20-22 L-a-amino acids linked together into a linear polypeptide chain. The polypeptide backbone consists of repeating identical peptide units. The structure of a peptide unit is presented schematically in fig. 3.1. Peptide units may interact through H- bonds (C=O ... H-N-) or with other adjoining hydrogenbonding entities, e.g., water molecules. Two of the three bonds in the peptide unit are free to rotate, whereas the C-N bond, shaded in fig. 3.1, is fixed because of its partial double-bond character represented by mesomerism. The side-groups, R, R', ... are specific for the type of amino acid. They vary in size, polarity, and charge. Depending on the distribution of the polar and apolar residues, the protein molecule is more or less amphiphilic and this is one of the reasons why they can be surface active. Some side groups contain cationic groups, others anionic groups. This makes the protein belong to the polyampholytes. Just as an essentially infinite number of words can be written by using the twenty six letters of the alphabet, an endless number of different polypeptides may be formed, characterized by a given sequence of amino acids. This sequence, the so-called primary structure of the protein molecule, rules the interactions in the molecule as well as those between the molecule and its environment. These, in turn, determine the spatial organization, i.e., the three-dimensional (3D) structure adopted by a protein molecule in a given environment. Within the 3D structure, different levels of organization may be distinguished, referred to as the secondary, tertiary, and quaternary structures. The secondary structure is the spatial arrangement of the polypeptide backbone, ignoring the side groups. The tertiary structure refers to the overall topology of the polypeptide chain, and the term, 'quarternary structure' is used for the non-covalent association of independent tertiary structures.

Figure 3.1. Structure of a peptide unit in a polypeptide chain. Two of the three bonds are free to rotate whereas the shaded bond is fixed.

3.4

PROTEIN ADSORPTION

In globular proteins the polypeptide chain folds into a dense structure in which the packing reaches volume fractions of 0.70-0.80. In such compactly folded conformations, the rotational freedom of the bonds in the polypeptide backbone (and side chains) is highly restricted. Such low-conformational-entropy structures are only thermodynamically stable if they are supported by interactions whose the net effect is sufficiently favourable to compensate for the low conformational entropy. Globular protein molecules in an aqueous environment have a few characteristic features: a. They are more or less spherical or ellipsoidal, with dimensions of the order of a few, to a few tens, of nanometers. b. Apolar amino acid residues tend to be accommodated in the interior of the molecule where they are shielded from contact with water. c. Almost all charged groups reside at the exterior of the protein molecule. Charges buried in the low dielectric interior occur as ion pairs. 3.2a Conformational entropy When apolar side groups are rejected from water to form the hydrophobic core of a protein molecule, part of the polar polypeptide backbone is pulled along into the interior. There, because of the absence of water molecules, the peptide units form hydrogen bonds among each other, which stabilizes the ordered structures of the polypeptide backbone. The best known secondary structures occurring in globular proteins include the a-helix and the fS-sheet. Schematic representations of these structures are shown in fig. 3.2. In distinction to a random polypeptide conformation, where two of the three bonds of a peptide unit are free to rotate, all three bonds in cc-helices and (3-sheets are blocked from rotation. This involves a reduction in conformational entropy of 2Rln2 = 11.53 J K"1 per mol of peptide units. Thus, if a polypeptide consisting of 100 amino acids (corresponding to a molar mass of ca. 10,000 Da) folds from an unordered (coily) conformation into a globular protein containing 50% of ordered structure, this goes at the expense of 580 J K^mol" 1 , which, at 300 K, yields an increase in Gibbs energy of 174kJmol~ 1 . Taking into account the restriction in the degrees of freedom of the side groups, the total increase in Gibbs energy upon folding of such a polypeptide may be in the range of a few hundreds of k j per mol. 3.2b Interactions that determine the 3D structure of proteins in aqueous solution Hydrophobic interaction Dehydration of apolar amino acid residues is the main driving force for polypeptide chains to fold into a compact globular conformation. To establish the contribution to the stabilization of a compact structure (taking the completely unfolded, hydrated conformation as the reference state) the distribution of all the amino acid residues and their individual hydrophobicities must be known. The distribution may be obtained

PROTEIN ADSORPTION

3.5

Figure 3.2. Ordered secondary structure elements in polypeptdide chains, (a) a- helix; (b) antiparallel |3- sheet; (c) parallel |3- sheet. from solving the 3D structure (by, e.g., X-ray diffraction and/or NMR) and the individual hydrophobicities can be expressed as the Gibbs energy of partitioning the amino acid between a non-polar medium and water. In this way, it has been inferred that the Gibbs energy of dehydration is about -9.2 kJnm~ 2 per mol 1 '. Based on an empirical relationship between the change in the water-accessible area upon folding and the molar mass of the polypeptide, and assuming that 60% of the interior of the folded protein molecules is made up of apolar amino acid residues21, the compact globular structure of a protein of molar mass 10,000 Da, is stabilized by a Gibbs energy of about SOOkJmor 1 at 25°C. Electrostatic interactions Proteins acquire their charge by (de)protonation of ionic groups in the side groups of the amino acid residues and the a -carboxyl group and a -amino group of both

11 21

F.M. Richards, Ann. Rev. Biophys. Bioeng. 6 (1977) 151. B. Lee, F.M. Richards, J. Mol. Biol. 5 5 (1971) 379.

3.6

PROTEIN ADSORPTION

terminal amino acids. Most of these charged groups are located at the aqueous periphery of the globular protein molecule, whereas in the unfolded conformation, depending on the primary structure, the charged groups are more or less homogeneously distributed over the fully hydrated expanded coil. The Gibbs energy of dissociation, AdissG , of a proton from any particular group can be split into a chemical ('chem') and an electrical ('el') term A

diss G = A diss G chem + A diss G el

I3-2- * 1

The chemical term contains the intrinsic contribution to the dissociation, and the electrical term the additional electrical work to remove the proton from the charged site to infinity (where the electric field strength is zero). In the simple case where all titratable groups belong to one and the same class Q0 = azFN

[3.2.2]

where Qo is the total charge of the sample, N the number of titratable groups, and a the degree of dissociation11. For the dissociation constant K diss the following expressions apply

p K diss=p H + i °g : 4r

l3-2 31

'

and AG° = -RTlnK d i s s

[3.2.4]

Combining [3.2.1], [3.2.3] and [3.2.4] yields pH = 0.434 AdissGc°hem + AdissGe°l _ j F RT

j B

^

[3 2 5 ]

a

It should be realized that A diss G° is a function of a ; at a high degree of dissociation, when the surface has attained a negative charge density and, consequently, a high negative potential, it requires more electrical work to remove a proton from the surface. The quantity AdissG°[ is often expressed as diss el =_2WE0_ RT F

=

_2WazN

[3.2.6]

in which W is the so-called electrostatic interaction factor. It depends primarily on the dielectric constant and the ionic strength of the medium. In differential form, [3.2.6] may be written as

The titration behaviour of polyelectrolytcs has been discussed in sec. 2.2d.

3.7

PROTEIN ADSORPTION

^ ^ - = -0.868 WzN + 0.434 da all-a)

[3.2.7]

M L °434 dg>0 zNFa(l-a)

[3.2.8]

or 0.868^ F

The quantity d p H / d Q 0 reaches a minimum value for a = 0.5 which is reflected by an inflection point in the titration curve for Qo versus pH. Proteins contain more than one class of titratable groups. Different classes j of groups are titrated in distinct pH regions. By way of example, the number of titratable groups, together with their intrinsic pK°-values for the protein ribonuclease (RNase), are given in table 3.1. For systems containing more than one class of titratable groups [3.2.8] has to be modified into *& = ^ dg>0 FljJVjZjajd-aj)

0.868 ?L F

[3.2.9]

and the differential titration curve, d p H / d g 0 versus QQ (or versus pH) displays more than one minimum, as is shown for RNase in fig. 3.3. Adsorption of ions other than protons can also lead to charging of the surface, provided that the adsorption is specific. The term, 'specific' implies that the adsorption

Table 3.1. Titratable groups and their intrinsic dissociation constants (pK°'s) in ribonuclease. Group

Number

pK°

Group

Number

pK°

a -carboxyl

1

3.75

e -amino

10

10.2

(5-, y -carboxyl

10

4.0-4.7

phenolic OH

3

10.0

imidazole

4

6.5

3

inaccessible

a -amino

1

7.8

4

> 12

guanidyl

Figure 3.3. Differential proton titration curve for bovine pancreas ribonuclease in 0.05 M aqueous solution. (Redrawn from W. Norde, J. Lyklema, J. Colloid Interface Sci. 66(1978) 266.)

3.8

PROTEIN ADSORPTION

Figure 3.4. Schematic picture of a globular protein molecule in solution. Shaded areas indicate hydrophobic regions. Charged groups originate from (de)protonation of amino acid residues (+/-) and from specific ion adsorption ( ©/© )• The dashed envelope iindicates the slip plane. Within this envelope the charge is Q e k . The compensating diffuse charge is not drawn.

Figure 3.5. Proton charge Qo and electrokinetic charge S e k of the same RNase as in fig. 3.3. The point of zero charge (p.z.c.) and isoelectric point (i.e.p.) are indicated. (Redrawn from W. Norde, J. Lyklema, J. Colloid Interface Sci. 66 (1978) 277.)).

forces are partly non-electric so that these ions can overcome the repelling electric potential at the surface and, by their adsorption, even increase the surface electric potential. In practice, often the electrokinetic charge @ek is measured, which results from both (de)protonation of amino acid residues and specific ion adsorption. This situation is depicted schematically in fig. 3.4. Figure 3.5 shows Q0(pH) and gek(pH) for RNase. The charge associated with specific adsorption of ions other than proteins is given by ( 9 e k - Q 0 ) . The difference between the point of zero charge (p.z.c.) and isoelectric point (i.e.p.) is a measure of the extent of specific adsorption. The protein charge is neutralized by countercharge. The electrical part of the Gibbs energy of a given charge-distribution can be evaluated as the reversible, isothermal, and isobaric work of charging the system. This is given by

Gel= J 9=0

y/'dQ'

[3.2.10]

PROTEIN ADSORPTION

3.9

where y/' is the electrostatic potential at the surface and Q' is the charge of the protein molecule during the charging process. To solve [3.2.10] a model for the charge distribution is required that relates y/ to Q . Some of such models are presented in chapter II.3. Away from the isoelectric point, where the protein surface contains a significant excess of negatively or positively charged groups, the charge may be thought of as being smeared out and the Gouy-Stern model may be applied. At, and near the isoelectric point, a model accounting for discrete charge effects is required to yield realistic results. Assuming a smeared-out distribution, and complete penetration without restraint of counterions in the expanded, unfolded polypeptide, the Donnan model may be a reasonable choice. The difference between the Gibbs energies for the compact and unfolded conformations, determines the stability of the one conformation relative to the other. It may be clear that, in the isoelectric region, a homogeneous charge distribution resists unfolding, whereas the opposite is true away from the isoelectric point. Values for the difference in Gibbs energies between the globular and the unfolded states are typically in the range of a few tens of k j per mole of protein, the effect becoming smaller at higher ionic strength. If ions reside in the interior of the protein they do so in pairs, and do not contribute significantly to the potential. In the unfolded state, such ion pairs are disrupted. Disruption of ion pairs is electrostatically unfavourable, but this effect is more or less compensated by hydration of the isolated ionic groups. For this reason, together with the fact that protein molecules usually contain no more than a few ion pairs, ion pair formation does not contribute much to the (de)stabilization of a compact protein structure. Lifshits-Van der Waals interaction Lifshits-Van der Waals interactions arise from forces between fixed and/or induced dipoles. For isolated molecules they decay steeply with the separation distance between the interacting dipoles, scaling as r"6 . Folding of the polypeptide chain into a compact protein involves the loss of Lifshits-Van der Waals interactions between units of the polypeptide and water molecules but, at the same time, such interactions are formed within the protein molecule and between water molecules. This is the Archimedes principle, discussed in sec. 1.4.6b. Quantitatively, the overall effect is not well understood for proteins. Because of the relatively high packing density in globular proteins, Lifshits-Van der Waals interactions are likely to promote a compact structure. The difference between the Hamaker constant of a protein and pure water points in the same direction. However, because of the opposing contributions of disruption and the creation of contacts, the net contributions of Lifshits-Van der Waals interaction to the stabilization of the protein structure is relatively small. Hydrogen bonds In the fully hydrated unfolded state, peptide units and some other moieties in the

3.10

PROTEIN ADSORPTION

amino acid side groups interact with water molecules through hydrogen bonds. When the protein folds into a compact structure, many of these bonds are broken and, instead, intramolecular hydrogen bonds are created in the protein molecule, and hydrogen bonds between water molecules. Peptide units dominate intramolecular hydrogen bonding, and determine the unique and fixed structures of helices and (3 sheets. Although hydrogen bonding is essential for maintaining ordered structures in the protein's Interior, its net contribution to the stabilization of the protein conformation is not particularly clear. Model studies1 2) suggest that peptide-water hydrogen bonds are more favourable than peptide-peptide and water-water hydrogen bonds. This would imply that hydrogen bonding (weakly) favours the unfolded conformation. However, hydrogen bonding between peptide units that, by the action of other factors (e.g., hydrophobic interaction) are forced into the non-aqueous inner parts of the protein, strongly stabilize ordered structures. Bond lengths and angles When the polypeptide chain folds into a tightly packed structure the bond lengths and angles may be more or less distorted. This could oppose stabilization of the globular protein structure by several kJ per mole31. In summary, the protein structure and structure stability is determined by various interactions inside the protein molecule, between the protein and water, and between the water molecules. These interactions compete with each other. As a result, the one structure is usually only some tens of kJ per mole more stable than the other. Hydrophobic interactions and the conformational freedom of the polypeptide chain play the leading parts. Both are mainly of entropic nature. Hence, the resulting protein structure is the outcome of an entropy battle between the polypeptide chain and water. Because of this marginal thermodynamic stability, none of the other factors influencing protein folding is unimportant. Consequently, even subtle changes in the environment of the protein, e.g. pH, ionic strength, ionic specificity, temperature, or additives, may cause structural transitions in the protein. In view of this, it is not surprising that most proteins, under most conditions, change their structure when adsorbing from solutions onto an interface. 3.3 Adsorption of globular proteins from aqueous solution onto (solid) surfaces The overall protein adsorption process is schematically depicted in fig. 3.6. Several steps or stages, indicated by the numbers in fig. 3.6, may be distinguished:

11

G.C. Kresheck, I.M. Klotz, Biochemistry 8 (1969) 8. W. Norde, Adv. Colloid Interface Sci. 25 (1986) 267. 31 M. Levitt, Biochemistry 17 (1978) 4277. 21

PROTEIN ADSORPTION

3.11

Figure 3.6. Schematic presentation of the protein adsorption process. Further explanation is given in the text. (1) transport from the bulk solution Into the subsurface region from where it is (2) attached at the surface. After initial adsorption the protein may (3 and 3*, etc.) relax at the sorbent surface. Relaxation may involve a fast surface-induced conformational change, but a slow spreading to optimize protein-sorbent interactions is also a very general phenomenon. Protein molecules may desorb from the sorbent surface (4, 4*, 4**). Obviously, the rate-constant of desorption is the smaller when the adsorbed protein molecule has reached a greater extent of relaxation. After desorption, the protein molecules (back in solution), may or may not (completely) regain their original native conformation. If not, the solution will eventually contain structurally perturbed protein molecules that may re-adsorb with an affinity different from the native ones. In this chapter we shall discuss all these aspects of the overall adsorption process in some detail, with special attention to adsorption-related conformational changes. In sec. 3.9 the chapter will conclude with some remarks on sorbent surface modification in order to adapt protein adsorption for practical applications. 3.3a Adsorption kinetics The rate of adsorption comprises two consecutive steps, (a) transport of the molecules to the interface and, (b), attachment at the sorbent surface. These steps will be considered separately and combined thereafter to derive an equation for the overall rate of adsorption. Transport towards the sorbent surface. The basic mechanisms of transport are diffusion and convection. The latter may be by laminar or turbulent flow. When the protein adsorbs, the subsurface region of the

3.12

PROTEIN ADSORPTION

solution becomes depleted and a protein concentration gradient is built up. This causes a flux J of protein molecules from the bulk solution towards the subsurface region. Under steady state conditions the flux is given as J = ktr(cb-ca)

[3.3.1]

where k^. is a transport rate constant, which depends on the transport mechanism and the dimensions and orientation of the protein molecules, and c b and cs are the protein concentrations in the bulk solution and the subsurface zone, respectively. In static systems, where the transport is by diffusion only, k^ may be approximated by [D/nt)l/2 in which D is the diffusion coefficient of the protein in bulk solution and t is the time1'. Clearly, the flux reaches a maximum value when cs = 0, that is, when attachment at the interface is much faster than transport to the subsurface region. This is the case when there is no activation energy for attachment, and in the initial stage of the adsorption process when there is no limitation of surface area available for adsorption.

Figure 3.7. Identification of ka and fcd .

Exchange at the interface. A simple situation is depicted in fig. 3.7. The protein molecules attach at, and detach from, the sorbent surfaces. This gives two fluxes from the subsurface region to the surface: one forward, dr/dt\+ , and one backward, dr/dt\_ , where the adsorbed amount r may be expressed in mass per unit of sorbent area. The net adsorption rate is given by, d/7dt = d/7dt| + -d/7dt|_

[3.3.21

The forward flux scales as cs and is proportional to the fraction (1 - 0] of sorbent surface area that is unoccupied with adsorbed protein. Hence, d/7dt| + =k a (l-0)c s

[3.3.3]

in which fca is the attachment rate constant. For the simple case that the adsorbing

11

For a general discussion and derivation, see sec. I.6.5e.

PROTEIN ADSORPTION

3.13

molecules do not change their sizes and shapes, as in fig. 3.7, the degree of surface occupancy 9 is defined as 77 .T 6 ^, where T"681 is the adsorbed amount when the sorbent surface is saturated with protein. For adsorption-induced conformational changes, as happens with most proteins and other (bio)polymers, the relationship between F and 6 is more complicated, and will be discussed later. The value of fca is lowered by any barrier for attachment, e.g., electrostatic repulsion, or hydration effects. A repulsive barrier may be created or reduced by the pre-adsorbed protein molecules and, hence, ka may vary with 0 and, therefore, with time. For the backward flux we can write

^j

= *d*

[3.3.4]

where kd is the detachment rate constant. In the course of the adsorption process, 6 increases and eventually a dynamic equilibrium (steady state) is reached, for which cb = cs = c

, the equilibrium concentration in solution. The equilibrium is character-

ized by fca(l-0)ceq =

fcd0

[3.3.5]

Assuming that d 77 d t_ is exclusively determined by kd and 6, the rate of adsorption off-equilibrium is still given by ka (1 - 0)c , so that ^

= ka(l-0)(cs-ceq)

With dT/dt

[3.3.6]

= J . Combining [3.3.1] and [3.3.6] gives

^ -

;b'Cef

dt

1

|

[3.3.7,

1

Polymers, including proteins, usually adsorb with a high affinity for the sorbent surface, characterized by an extremely high value of the adsorption equilibrium constant K (= ka/kd). For physical reasons, ka cannot attain extremely high values, and hence fcd must be very small. The reason for the almost infinitely low polymer desorption rate has been discussed in chapter II.5. High affinity adsorption isotherms (see fig. II.2.24) have the property that, below adsorption saturation, c is extremely low (usually below detectability). When reaching saturation, which is reflected by the (semi-)plateau in the isotherm, c (6) increases steeply and approaches c b . It follows that, upon approaching saturation, dF/dt drops strongly. This is reflected by a sharp transition in the curve for F(t) often observed for (bio)polymers. A typical example is given in fig. 3.8. Away from saturation, where c

= 0 , [3.3.7] may be rewritten as

3.14

PROTEIN ADSORPTION

Figure 3.8. Typical example of polymer adsorption kinetics.

c

b/rmax_

dO/dt

1 ,

ku

1

[3.3.8]

fca(l-0)

If fcjj. « ka(l-ff), dO/dt or, for that matter, dF/dt are determined by transport from the bulk to the subsurface region, independent of 0. This permits direct derivation of k^ from the initial linear part of the isotherm. If kb » ka (1 - 9), a condition that is probable when there is a barrier for attachment at the surface or when the surface is covered to some extent, dF/dt is determined by the rate of attachment, and hence, depends on 8. The value of ka(6) may be derived from F(t), using approximation [3.3.8]. So far, the discussion has been based on the oversimplified model shown in fig. 3.7. However, it is a rule rather than an exception that biopolymers, including proteins, adapt their conformations as a result of relaxation at the interface. This phenomenon is illustrated in fig. 3.9, where the native conformation is denoted, the 'N state' and the adapted, perturbed conformation the 'P state'. The rate-constant for relaxation is kT and those for desorption from the N state and P state, kd N and kd p . It follows that

d#M ^ f =

fc

a,Ncsa-8N-0P)-

kdN0N - kT9N

[3.3.9]

and d(9p

[3.3.10]

Figure 3.9. Adsorption and desorption of protein molecules followed by transition from the native (N) state to the perturbed (P) state. After desorption of a molecule in the P state the molecule may, or may not, return to the N state.

PROTEIN ADSORPTION

3.15

In the steady state, where dF/dt = 0 , k a V ^ N - ^ e q =fcd,N^N+

fc

d,P^P

I3-3-11'

In general, kdN *kdp. Now, contrary to the simple situation of fig. 3.7, the rate of desorption is not a unique function of the desorption rate constants but it also depends on 6P / 0^ . This ratio changes in the course of the adsorption process. As a result of surface relaxation it is expected that fcd p < kd N . The smaller the kd p / fcdN ratio, the larger the fraction of perturbed molecules in the steady state is. If kd p approaches zero, the adsorbed layer eventually consists of molecules that are all perturbed. It goes without saying, that when adsorption induces conformational changes in the protein, the expression for dF/dt is much more complicated than the one given in [3.3.7]. The adsorption kinetics are even more complicated when the increase in area per protein molecule ('footprint') due to spreading at the sorbent surface is taken into account. Let the molecular area of N state molecules be a N , and of P state molecules a p , with a p / aN = a r > 1. The maximum number of molecules in the N state per unit sorbent surface area is No , and in the P state it is JV. It follows that JV0 = aTN, and N o a N = 1. Further, 0N = nN / JV0 and 0V =np/N0, where n is the number of protein molecules in the adsorbed state. Hence, dft, -jf =

fc

a,Ncs(1-^N-aA)-'cd,N%x/(ar-%.ep)

I 3 - 3 - 12 !

in which /(a r ,# N ,i9 p ) takes into account the fact that the area-enlargement involved in the N —» P transition reduces the probability for other molecules to undergo the transition later. When a r = 1, _f (ar, #N, £?p) = 1 and it decreases for larger values of a r , the more so, the higher is the surface occupancy, {0N + 0p). Similarly, ^

= kr0Nx/(ar,0N,0p)-kdp0p

[3.3.13]

The steady state evolving from this model is given by k 0 a,NV- N

-aA)ceq

=k

d,fA

+fc

d,P0P

[3.3.14]

which is equivalent to [3.3.11] because (1-# N -ar8p) and (1-# N -&p) in the respective equations are the covered fractions of the sorbent surface. Both 0N and 6>p are expressed in number of adsorbed molecules per fixed number of sites per unit surface area. It means that (#N + 0p) is proportional to the adsorbed mass per unit surface area, F. This model explains the remarkable phenomenon of transient adsorption of polymers which has sometimes been reported. If fcdp<)c(iN, a r > 1 and the rates of attachment and spreading are of comparable magnitude, (#N + 0p) and, hence, F would pass through a maximum in the course of the adsorption process.

3.16

PROTEIN ADSORPTION

Figure 3.10. Sketch of excluded area for the deposition of molecules according to the random sequential mode.

Most theories for (blo)polymer adsorption, including proteins, start from the models presented above. Other models proposed to describe the kinetics of (bio)polymer adsorption include the random sequential adsorption (RSA) model11. In this model it is assumed that the molecules arrive randomly at the sorbent surface and that they stick where they hit. A subsequently arriving (spherical) molecule cannot be accommodated within the dashed areas around pre-adsorbed molecules, as illustrated in fig. 3.10. Clearly, the fraction 0 of the surface which is available for adsorption is a function of the degree of coverage, 0, of the surface by the adsorbate. For hard sphere molecules, 0=rmR2 (where n is the number of molecules per unit surface area, and R is the radius of the sphere. The RSA theory produces the following functionality

^=«p{ 2 -^ + i^ + i I n a - f l ) + -}

[3 3 151

--

which can be developed into 0(0) = l-46> + ^ 3 -6^+2.4243 ft + ....

[3.3.16]

K

Figure 3.11. Surface area available for the adsorption of spherical molecules in a random sequential mode. The drawn curve obeys [3.3.15], the other ones [3.3.16] with two, three or four terms of the expansion. 11

P. Schaaf, J.C. Voegel, and B. Senger, Ann. Phys. 23 (1998) 1.

PROTEIN ADSORPTION

3.17

The third term of the r.h.s. of [3.3.16] accounts for the overlap of the dashed areas in fig. 3.10, the fourth term for the double overlap, etc. Figure 3.11 shows curves for 0(9). In the kinetics according to the RSA model, the uncovered surface area (1 - 9) is replaced by

[3.3.17]

Curves for 9{t) are presented in fig. 3.12 for spheres and ellipses, and show that the saturated surface coverage decreases with increasing aspect ratio. Even with spheres it does not exceed fifty percent.

Figure 3.12. Adsorption kinetics according to the random sequential mode for spheres and ellipsoids of varying aspect radios, alb. (Redrawn from S.M. Ricci, J. Talbot, G. Tarjus, and P. Viot, J. Chem. Phys. 97 (1992) 5219.)

3.3b Relaxation at interfaces After attachment, the protein molecule will optimize its interaction with the sorbent surface, i.e., it relaxes. In the relaxation process the protein molecules expose an increasing number of segments to the surface, which usually leads to some degree of spreading. This involves rearrangements in the protein structure. The extent of spreading depends on the rate of spreading relative to the rate of attachment at the surface. More precisely, the extent of spreading is primarily determined by the ratio of the characteristic time of relaxation, rr , (which, in turn, is mainly governed by the internal cohesion in the protein molecule) and the characteristic time of filling the sorbent surface, rf (which is determined by the protein flux J towards the surface). If relaxation occurs much faster than filling the surface i.e., (rt /r f « 1) all molecules are allowed to relax completely after being attached, and the adsorbed amount under saturation conditions, P*81, is independent of J. Under most conditions, such behaviour is expected for polymers that relax quickly. However, in globular proteins the internal coherence is strong, and adsorption-induced relaxation relatively slow.

3.18

PROTEIN ADSORPTION

Then, the time of filling the surface, may approach the relaxation time (rt / rf > 1), and the degree of spreading will be affected by the protein flux, because a neighbouring site may already be occupied by a newly arriving molecule before the previously adsorbed molecule is given the time to relax. This results in less spreading, which is reflected in a higher value of r 6 ^ with increasing flux. The adsorbed layer may become heterogeneous with respect to the conformational states of the protein population. Molecules arriving at an early stage find sufficient area available for spreading, whereas this is much less the case for the molecules that arrive when the surface is already crowded.

Figure 3.13. Relaxation of protein molecules, adsorbed on silica. Surface area occupied per adsorbed IgG molecule as a function of the time required to fill the surface. (Based on data of M.G.E.G. Bremer, Immu.noglobu.lin Adsorption on Modified Surfaces, PhD thesis, Wageningen University, The Netherlands (2001).)

When the surface area occupied per adsorbed molecule (derived from / ^ a t ) is plotted vs. Tj (calculated from the flux), a curve is given, as shown in fig. 3.13. For Tt = 0, spreading is completely inhibited and the corresponding area may be compared with the dimensions of the native molecule. Extrapolation to constant r 8 ^ , where full relaxation is reached, allows estimation of the relaxation time. Thus, relaxation times in the range of tens- to thousands- of seconds are inferred for globular proteins. There is strong experimental evidence11 that at interfaces, as in solution, the conformational change of globular protein molecules is a distinctly co-operative, rather than a gradual process. It implies that the intermediate part of the curve in fig. 3.13 reflects the co-existence of native (N) and spread (P) molecules at the interface. Assuming a two state transition N —> P , the rate of spreading can be expressed as,

11

T. Zoungrana, G.H. Findenegg, and W. Norde, J. Colloid Interface Sci., 190 (1997) 437.

PROTEIN ADSORPTION ^

= -(cscN

3.19 [3.3.18]

where ks , the spreading rate constant, is determined by the activation Gibbs energy AG^ of the N —> P transition as

in which h is Planck's constant. Fitting the experimental curve to [3.3.18] yields estimates for ks and, hence, AG^ . For globular proteins, typical values for ks are in the range of lO^-lO" 1 s" 1 , and for AG+ some tens of kT . 3.3c Driving forces for protein adsorption In this section, the main contributions to protein adsorption on a smooth, rigid surface will be discussed. These contributions originate from (a) redistribution of charged groups (ions) when the electrical double layers around the protein molecules and the sorbent surface overlap, (b) dispersion forces between the protein and the sorbent material, (c) changes in the hydration of the sorbent surface and the protein, and (d) structural rearrangements in the protein molecules. Although these interactions are discussed separately, they are by no means independent of each other. Their actions could be synergistic or antagonistic. Redistribution of charged groups. As a rule the surfaces of the protein molecules and the sorbent are electrically charged, and surrounded by counterions and co-ions, together constituting the countercharge that neutralizes the surface charge. The surface charge and the countercharge together form an electrical double layer. Models for the electrical double layer are discussed in chapter II.3 and their interaction in chapter IV.3. There, it has been explained that the Gibbs energy Gcd required to invoke a charge distribution can be calculated as the isothermal, isobaric reversible work o° G c d = J y/°<&o°'

[3.3.20]

0

where y/°' and cf" are the variable surface potential and surface charge density, respectively, during the charging process. Equation [3.3.20] is the 2D equivalent of [3.2.10]; Gel in [3.2.10] is in J, whereas Gcd in [3.3.20] is in J m" 2 . As before, [3.3.20] can be integrated if a model for the double layer is available. To calculate the contribution from charge redistribution to the Gibbs energy of the adsorption process, AadsGcd , equation [3.3.20] has to be applied three times, i.e., Gcd for the bare surface and the dissolved protein has to be subtracted from Gcd for the protein-covered sorbent surface. For the sorbent surface, the Gouy-Stern model may be taken,

3.20

PROTEIN ADSORPTION

although for the protein molecule, a discrete charge model seems to be more appropriate. For the protein-covered sorbent charge-distribution models have been proposed by Norde and Lyklema11 and by Stahlberg et al.2). The protein molecules in the adsorbed layer retain a compact conformation (sec. 3.4). Under most ambient conditions of ionic strength the thickness of the adsorbed layer exceeds the Debye length, i.e., the distance over which electrostatic forces are effective. Byway of example, adsorbed protein layers usually reach a thickness of a few to a few tens of nm, whereas the Debye length in 0.01 M ionic strength is 3 nm, and in 0.1 M ionic strength it is only 1 nm. Therefore, the protein layer shields the protein-sorbent contact region from electrostatic interaction with the solution. To prevent an excessively high electric potential, the charge density in the non-aqueous, low dielectric, protein-sorbent contact region must be regulated to be essentially zero. Cross-differentiation allows calculation of the charge regulation from the dependency of the protein adsorption on the electrolyte concentration in solution. Charge regulation may occur through changes in the ionization of the protein and/or the sorbent surface, and also by the incorporation of indifferent ions from solution in the protein-sorbent contact region. As a consequence of the charge regulation, AadsGcd does not exceed a few tens of RT per mole of protein, and it is not very sensitive to the charge on the protein and the sorbent surface before adsorption. In addition to contributing to the charge regulation, ion transfer between the solution and the adsorbed layer includes a chemical effect. Compared to water, the low dielectric proteinaceous environment is a poorer 'solvent' for individual ions and, hence, the chemical effect of ion incorporation opposes protein adsorption. This explains why protein adsorption often reaches its maximum affinity when the charge density on the protein just matches that on the sorbent surface, so that no additional ions have to be incorporated to neutralize the protein-sorbent contact region. Dispersion interaction Dispersion interaction between macroscopic bodies may be computed using the Lifshits theory explained in sec. 1.4.7. There, the less accurate, but simpler microscopic treatment, known as the Hamaker-de Boer approximation, is also presented. The latter approach has the advantage of leading to relatively simple analytical expressions at the expense of less rigour. As is often the case, (adsorbed) protein molecules, and sometimes the sorbent material, are not precisely defined with respect to geometry and density, so application of the Hamaker-de Boer theory is acceptable. Then, for a sphere 1 (the protein molecule) interacting with a body 2 having a planar surface (the sorbent surface), across a medium 3, the contribution to the Gibbs energy of adsorption is given by, see [1.4.6.19 and 29] 11 21

W. Norde, J. Lyklema, J. Colloid Interface Set, 55 (1978) 285. J. Stahlberg, B. Jonsson, Anal. Chem. 68 (1996) 1536.

PROTEIN ADSORPTION

A

adsGdisp-

l 3)

6 [h

3.21

+

h + 2a + lnh + 2a\

l3 3

' -211

where A12(3) is the Hamaker constant for the system, a the radius of the sphere, and h the shortest separation distance between the sphere and the planar surface. The quantity A ads G di off

steeply with

increases with increasing dimensions of the sphere, and it drops increasing separation

distance:

A ads G di

reacnes

significant

magnitudes only for small values of h. For h « a , [3.3.21 ] reduces to I3322'

AadsGdiSp = ~ ~ ^ The Hamaker constant

A12(3) for the system can be derived from those of the

individual components, according to [1.4.6.29] A

12(3) = A2 ~ A 13 ~ A 23 + A33

[3.3.23]

When the Berthelot principle is valid, this may also be written as A -tAl/2 Al/2\jAl/2 A l/2\ 12(3)-i A ll ~ A 33 )yh.2 ~ A 33 j

A

,0 o 041 [J.J.^4]

In water, usually A,x > A 33 and A22 > A33 , and hence -A12(3) > 0 , so that AadsGdi < 0 , which implies attraction between 1 and 2. The Hamaker constant for interaction across water is ca. 6.5xlO~ 21 J for globular proteins, (l-3)xlO~ 19 J for metals and (4-12) xlCT21 J for synthetic polymers1'21. Based on these data, and applying [3.3.24] and [3.3.22], for a spherical protein molecule of radius 3 nm at a distance of 0.1 nm from the sorbent surface, AadsGdi at room temperature amounts to 6-1IRT per mol at a metal surface, and to 1-4 RT at a polymeric surface. Clearly, these estimates are semi-quantitative. Obtaining more accurate estimates requires more detailed knowledge of the system's parameters, such as Hamaker constants, sizes and shapes of the protein molecule and the sorbent material. Hydration changes Polar molecules attract water molecules, mainly through hydrogen bonding. They compete successfully with hydrogen bonds between the water molecules, so they are readily soluble in water. Apolar groups do not offer the possibility of a favourable interaction with water and therefore they are expelled from an aqueous environment. This is the hydrophobic effect. The water molecules at an interface of apolar material are strongly oriented so as to form as many hydrogen bonds as possible to other water molecules, as none can be formed with the apolar material. This reduces the entropy of the water adjacent to apolar compounds. 11

S. Nir, Progr. Surface Set 8 (1977) 1.

21

J . Visser, Adv. Colloid Interface Sci. 3 ( 1 9 7 2 ) 3 3 1 .

3.22

PROTEIN ADSORPTION

Protein molecules contain both polar and apolar groups. As discussed in sec. 3.2, for globular proteins in aqueous solution the apolar parts tend to be hidden inside the molecule, whereas the exterior is mainly polar. Other interactions and/or geometrical constraints interfere with these tendencies. Hence, although it is mainly apolar, the heart of the protein contains polar groups as well, and apolar groups occupy parts of the aqueous periphery of the molecule. However, water-soluble non-aggregating protein molecules do not have pronounced apolar patches at their surfaces. The surfaces of (model) synthetic sorbent materials are usually less complex than protein surfaces, but natural surfaces may also be rather heterogeneous. When the surfaces of the protein molecules and the sorbent are predominantly polar it is likely that some hydration water is retained between the sorbent surface and the adsorbed protein layer. However, if (one of) the surfaces are (is) apolar, dehydration would stimulate protein adsorption. The contribution from changes in hydration to the Gibbs energy of adsorption may be estimated from the partitioning of model compounds between water and a nonaqueous medium. It is thus estimated that dehydration of an apolar surface lowers the Gibbs energy by 10-20 mJ m~2 . In the case of a protein of 30,000 Da and an adsorbed mass of 1 mg m~2, this corresponds, at room temperature, to AadsGh d r ranging between -120 RT and -240 RT per mole of protein. It evidently indicates that apolar dehydration outweighs the contributions from overlapping electrical double layers and Van der Waals forces. Rearrangements of the protein structure When a protein molecule makes intimate contact with the sorbent surface, on one side of the molecule the aqueous environment is replaced by the sorbent material. As a consequence, intramolecular hydrophobic interaction becomes less important as a structure-stabilizing factor; i.e., apolar parts that were buried in the interior of the dissolved molecule may become exposed to the sorbent surface without making contact with water. Because hydrophobic interaction between amino acid side groups in the protein's interior support the formation of secondary structures, such as a-helices and |3 -sheets, a reduction of this interaction destabilizes such structures. A reduction in the a-helix and/or |J-sheet- content is, indeed, expected to occur only if peptide units released from these structures can form hydrogen-bonds with the sorbent surface, as is the case for oxides (e.g., glass, silica and metal oxides), or with surfaces retaining residual water molecules at the sorbent surface. Then, the decrease in secondary structure may lead to an increased conformational entropy of the protein. This may favour the adsorption process by tens of RT per mole of protein. If, in the non-aqueous protein-sorbent contact region, it is not possible for the peptide units to form hydrogen bonds with the sorbent surface, as is the case for hydrophobic surfaces, adsorption may induce the formation of extra intramolecular peptide-peptide hydrogen bonds, thereby promoting the formation of a-helices and/or p -sheets. Thus, whether adsorp-

PROTEIN ADSORPTION

3.23

tion on a hydrophobic surface results in an increased or decreased order in protein structure, depends on the subtle balance between energetic and entropic interactions. To summarize, it is concluded that protein-sorbent interaction is a complex phenomenon, comprising various subprocesses operating simultaneously. For example, hydrophobic interaction between the protein and the surface requires close contact between the two, which may be optimized by structural rearrangements in the protein. This may involve increased or decreased flexibility along the polypeptide chain. At hydrophilic surfaces, increased conformational freedom is expected in adsorbed protein molecules. Based on these considerations we arrive at the following 'rules of thumb'. Because of favourable dehydration, essentially all proteins adsorb on hydrophobic surfaces, even under electrostatically adverse conditions. With respect to hydrophilic surfaces, distinction may be made between proteins having a strong internal coherence ('hard' proteins) and those with a weak internal coherence ('soft' proteins). The hard proteins adsorb only if they are electrically attracted, whereas in the soft proteins the surfaceinduced conformational entropy gain is large enough to cause adsorption on a hydrophilic, electrostatically repelling surface. 3.4 Adsorption-related structural changes in proteins While searching for explanations of the clotting of blood, in the 1950s Leo Vroman discovered that water-repellent surfaces became water-wettable after contact with plasma. Later he discovered that some, but not all, of the proteins involved in blood clotting were adsorbed to hydrophobic surfaces. Together, these observations allowed him to propose that, 'globular proteins which can open easily will do so when they see a hydrophobic surface, and will turn themselves inside out to paste themselves with their fatty hearts onto that surface'11. About twenty years later Norde performed an extensive thermodynamic analysis of the protein adsorption process. This study confirmed that human plasma albumin changes its conformation when it adsorbs at polystyrene surfaces2'. This adsorption process was entropically driven, and part of the entropy gain originated from the conformational changes in the proteins31. Indications that proteins might change their conformation when they adsorb to air/ water and oil/water interfaces have been reported since the early part of last century. Information on the conformational state of proteins at the air/water interface was obtained by studying interfacial pressures and tensions using, for example, the Langmuir trough (III.3.3) and the Wilhelmy plate (III. 1.8a) techniques. Reviews cover-

11

L. Vroman, Blood, The Natural History Press (1968) 63. W. Norde, J. Lyklema, J. Colloid Interface Sci. 66 (1978) 257, 266. 31 W. Norde, J. Lyklema, J. Colloid Interface Sci. 71 (1979) 350. 21

3.24

PROTEIN ADSORPTION

ing this early work have been written by Chessman and Davies11, and Cumper and Alexander2'. For proteins adsorbed at the solid/liquid interface, information on the surface area per molecule can be derived from adsorption isotherms. However, without insight into the surface pressure or the geometry of the adsorbed protein layer, interpretation of plateau values in terms of conformational changes of adsorbed proteins becomes rather presumptuous. In one study, using infrared spectroscopy to analyze the bound fraction, the observed conformational changes of several proteins were so small that the authors felt it necessary to report that, 'caution must be exercised in interpreting the available surface area per adsorbed molecule as an indication of changes in conformation upon adsorption' 3). Generally, structural properties of proteins are either described by the energy that is required to unfold their natural state or by the relative location of all atoms in the folded structure. The folded structure is discussed in sec. 3.2. In the 1970s some experiments were performed to measure directly one of the structural properties of proteins in the adsorbed state. Structural elements of adsorbed proteins were investigated by using spectroscopic techniques, such as circular dichroism (CD), nuclear magnetic resonance (NMR), electron spin resonance (ESR), and fluorescence spectroscopy, while microcalorimetry was used to obtain thermodynamic information on the adsorption process. Reviews of this early work have often reported moderate changes in the protein structure, although the measurements were not sensitive enough to establish the type and extent of the adsorption-induced conformational changes4 5). From the late 1970's sophisticated experimental evidence became available that proved the extent of adsorption-induced structural changes to range from none to substantial denaturation of the proteins upon adsorption. A few decades of gathering experimental evidence has led to a number of general trends that relate the various physical and chemical properties of the protein/sorbent system with adsorption-induced structural changes. Nevertheless, a comprehensive theory with predictive value that describes the extent and rate of adsorption-induced structural changes in the proteins remains a challenge for the future. 3.4a How to measure structural properties of adsorbed proteins The development in experimental techniques has made it feasible to probe directly the conformational features of proteins in the adsorbed state. Not only have new and/or more sensitive instruments been developed, but more sorbent surfaces have also become available that interfere less with the measurement techniques. 11

D.F. Chessman, J.T. Davies, Adv. Protein Chem., 9 (1954) 439. C.W.N. Cumper, A.E. Alexander, Rev. Pure Appi Chem.. 1 (1951) 121. 31 B.W. Morrissey, R.T. Stromberg, J. Colloid Interface Set 46 (1974) 152. 41 W. Norde, Adhesion and Adsorption of Polymers, part B (1980) Plenum Publishing Corp. New York. 51 M.E. Soderquist, A.G. Walton, J. Colloid Interface Set. 75 (1980) 386. 21

PROTEIN ADSORPTION

3.25

Most of the techniques for measuring structural properties of proteins at interfaces are based on experimental methods that provide information on the structural properties of proteins in solution. For example, CD is a successful technique for measuring the secondary structure of proteins in aqueous surroundings. In 1974, McMillin and Walton constructed a cell that contained a stack of quartz disks to provide an optically transparent cell with enough protein layers to generate a reasonable signal11. However, difficulties with scattering and the alignment of the quartz plates made the researchers return to solution studies of desorbed proteins21. Finally, the technique regained much interest after it was shown that a suspension of small, nanometer sized colloidal particles covered with proteins generated high quality CD spectra3 . The most important approaches for determining structural properties of adsorbed proteins are based on spectroscopic methods that rely on the interaction of biopolymers with electromagnetic radiation (1.7.3), using techniques such as fluorescence-, NMR, infrared, and CD spectroscopy. Infrared and CD spectroscopy are mainly used to quantify the average amount of a -helical and (3 -sheet structures. As carbonyl groups on the protein backbone are involved In specific H-bonds in secondary structure elements, the wavelength at which infrared radiation is absorbed is slightly shifted. Circular dichroism occurs because secondary structure elements in proteins absorb left- and right- hand circularly polarized light slightly differently in the UV and far UV region. Infrared spectroscopy has a higher sensitivity for measuring changes In P -sheet structures, while CD has a higher sensitivity for quantifying a-helical structures. A major advantage of CD over infrared spectroscopy is that the spectroscopic signal is not affected by the presence of water. Fluorescence spectroscopy can generate information on the local surroundings and the solvent accessibility of fluorescent groups such as tryptophan residues or haem groups in proteins. Furthermore, fluorescence anisotropy and time-resolved fluorescence decay measurements provide information on the rotational mobility of adsorbed proteins, whereas fluorescence recovery after photo bleaching (FRAP) can be used to measure the lateral mobility. To date, applications of NMR in protein adsorption studies are rare51. Its use is limited to relatively small proteins for which complete 'H, 13C, and 15N assignments of the native structure are known. NMR spectroscopy can also be used in combination with hydrogen-deuterium exchange (HDX) experiments to probe the solvent accessibility of exchangeable hydrogens in the protein structure, because the NMR signal disappears when protein hydrogens are exchanged with deuterium atoms from solution. This type of HDX can be used to probe parts of the protein structure that are in contact with the sorbent

11

C.R. McMillin, A.G. Walton, J. Colloid Interface Set 48 (1974) 345. A.G. Walton, M.E. Soderquist, Croat. Chem. Acta 53 (1980) 363. 31 A. Kondo, S. Oku, and K. Higashitani, J. Colloid Interface Set 143 (1991) 214. 41 W. Norde, J.P. Favier, Colloids Surf. 64 (1992) 87. 51 K.Wuthrich, Acta Crystallogr. D51 (1995) 249. 21

3.26

PROTEIN ADSORPTION

surface and are thus less accessible for the deuterated solvent1'21. To characterize adsorbed proteins in terms of the energy or enthalpy involved in the stabilization of the protein structure, one can employ differential scanning calorimetry (DSC) and hydrogen-deuterium exchange in combination with mass spectrometry (HDX-MS). In DSC one measures the difference in specific heat as a function of the temperature between a sample with and without proteins. With such experiments one can derive the enthalpy and temperature of the transition between a folded and unfolded, or otherwise perturbed, conformation of proteins. In HDX-MS, one monitors the exchange between amide hydrogens on the protein backbone and protons in solution. When amide hydrogens are involved in hydrogen bonding in the secondary structure elements, their exchange rates decrease considerably. The time scale of such an exchange process is usually much larger than that used to study the solvent accessibility of exchangeable hydrogens located on the outside of the protein structure. By measuring the deuterium incorporation as a function of time, the protein is exposed to a deuterated solvent. With mass spectrometry, information is revealed on the folding/ unfolding equilibrium. This folding/unfolding equilibrium, in turn, is related to the Gibbs free energy that stabilizes the folded conformation of a protein. Compared to the spectroscopic techniques, DSC and HDX-MS monitor more general aspects of the conformational state of (adsorbed) biomolecules. Additionally, both techniques also provide information on the heterogeneity in the structural populations. Besides these techniques, most of the methods that can be used to characterize surfaces and adsorbed monolayers (III.3.7) also yield indirect information on the conformational state of proteins. For example, neutron reflectivity, ellipsometry, and

Figure 3.14. Atomic Force Microscopy images (0.75 um x 0.7 um ) of human plasma fibronectin measured in the intermittent-contact mode (Courtesy of M. Bergkvist). On the lefthand side, fibronectin is adsorbed at a hydrophilic mica surface, and on the right-hand side on a hydrophobic surface created by silanisation of silica with dichlorodimethylsilane11. 11 21

D.G. Gorcnstein, Z. Santago-Rivcra, and D.A. Keire, Polym. Mater. Sci. Eng. 71 (1994) 261. H. Nagadome, K. Kawano, and Y. Terada, FEES Letters 317 (1993) 128.

PROTEIN ADSORPTION

3.27

reflectometry can be used to study the average density and thickness of an adsorbed protein film, while imaging techniques such as scanning force microscopy (SFM) can be used to generate information on the dimensions that a protein assumes in the adsorbed state. By way of illustration, in fig. 3.14 an AFM image is given for the adsorption of fibronectin on two different surfaces. Fibronectin is a large flexible protein that is stabilized by intermolecular ionic interactions to form a compact structure. Upon altering the solution conditions the structure can revert to a more expanded state, thereby exposing previously hidden domains, for example cell-binding sites. Upon adsorption to hydrophilic surfaces, fibronectin often adapts an elongated structure, whereas on hydrophobic surfaces the compact structure predominates. The difference in morphology is explained by the interaction between fibronectin and the negatively charged groups on the hydrophilic surface that interferes with the stabilizing interaction between the protein domains. Finally, adsorption-induced conformational changes, as well as the influence of the sorbent surface on conformational transitions in the protein molecules, may be inferred from proton titration studies2'3'41. By way of example we present some results for the adsorption of a-lactalbumin (aLA ). To that end in fig. 3.15, proton titration curves for aLA in solution and adsorbed on negatively and positively charged polystyrene particles are presented. The curves for the dissolved and the adsorbed states differ over a wide pH-range. The shifts upon adsorption reflect the influence of the electric field caused by the charges on the polystyrene surface on the pK-values of the titratable groups of the aLA molecules. Figure 3.16 shows the titration behaviour in the pH-region where the carboxyl groups are protonated. The curve for aLA in solution comprises a region reflecting the translation from the holo- to the apo-state. The non-electric Gibbs energy, A holo ^ apo G°, associated with that transition can be

Figure 3.15. Proton titration curve for a -lactalbumin in solution ( ) and on negatively (A) and positively (o) charged polystyrene particles. T = 25°C; electrolyte, 0.05 M KC1. (Redrawn from W. Norde, F. Galisteo Gonzalez, and C.A. Haynes, Polym. Adv. Technol. 6 (1995) 518).

11

M. Bergkvist, J. Carlsson and S. Oscarsson, J. Biomed. Mater. Res. (2002). C.A. Haynes, E. Sliwinsky, and W. Norde, J. Colloid Interface Set 164(1994) 394. 31 F. Galisteo, W. Norde, Colloids Surfaces B4 (1995) 389. 4 ' W. Norde, F. Galsiteo Gonzalez, and C.A. Haynes, Polymers for Adv. Technol. 6 (1995) 518. 21

3.28

PROTEIN ADSORPTION

Figure 3.16. Titration of carboxyl groups in a -lactalbumin. The drawn curve refers to the protein in aqueous solution, the upper clashed curve to the protein, adsorbed on negatively charged polystyrene particles. T = 25°C; electrolyte, 0.05 M KC1. (Redrawn from W. Norde, F. Galisteo Gonzalez, and C.A. Haynes, loc. cit.)

derived according to l A

G

2 303

holo^apo ° = -

NmaxRT\[pK(a.po)~pKihol0))da

[3.4.1]

0

with pK = pH + log[ — - 1

[3.4.2]

where K is the dissociation constant of the carboxyl groups, iVmax the total number of carboxyl groups in the oeLA molecule, and a their degree of dissociation. Hence, Aj^^ a p o G° for aLA in solution may be evaluated from the shaded area in fig. 3.16. It is estimated to be ca. -3 RT per mole of aLA . Proton titration calorimetry data, presented in fig. 3.17, reveal an endothermic holo to apo-state transition of ca. + 18 RT per mole of protein. It follows that, at room temperature, the transition is

Figure 3.17. Isothermal proton titration enthalpy of a -lactalbumin in solution (•) and adsorbed on negatively charged polystyrene particles (A ). T= 25°C ; electrolyte, 0.05 M KC1. (Redrawn from W. Norde, F. Galisteo Gonzalez, and C.A. Haynes, loc. cit.)

PROTEIN ADSORPTION

3.29

driven by an entropy gain of about 0.17 kJ K"1 mol" 1 . Remarkably, the structural transition is virtually absent when aLA is adsorbed on the polystyrene surface. It demonstrates that the structural properties of aLA are very sensitive to the local environment. In this special case of aLA it implies that adsorption causes (a) spontaneous transition to the apo-state, (b), spontaneous transition to a new conformation from which the apo-state is no longer accessible, or, (c), stabilization of the native structure such that the apo-state is no longer favoured at low pH. All the above-mentioned techniques require specific characteristics of the type of interface that can be studied. For spectroscopic techniques, these solid surfaces should not interfere with the spectroscopic signal of interest. Furthermore, for adsorption studies a selection mechanism is required to obtain experimental signals that originate only from the adsorbed proteins. Besides this selectivity, the signal sensitivity can also be a problem, as going from proteins in solutions to proteins adsorbed at a twodimensional surface means that the possibilities for concentrating the sample are limited. Because of these requirements, studies of the conformational state of proteins are restricted to a limited number of available sorbent surfaces. In fluorescence and infrared spectroscopy one can selectively excite adsorbed proteins by using an evanescent wave that is created by total internal reflection of a lightbeam within an optically transparent material [1.7.10.a]. The evanescent wave penetrates the solution to a depth that is roughly half the wavelength of the radiation used. Two prominent techniques based on this principle are total internal reflection fluorescence (TIRF) (see sees. II.2.5c and III.3.7c, iv) and attenuated total reflection-fourier transform infrared (ATR-FTIR) spectroscopy (sec. II.2.5.C). Another advantage of using total reflection is that the polarization of the evanescent wave can be controlled easily, allowing investigation of the spatial orientation of adsorbed molecules. Techniques that utilize evanescent waves require optically inert and smooth surfaces that resemble Fresnel surfaces. For example, silica interfaces as present in oxidized silicon wafers, silicon crystals, polished glasses, and quartz in fluorescence or infrared studies. These silica surfaces can be modified easily by silanization. Self-assembled monolayers (SAM), or Langmuir-Blodgett films (sec. III.3.7a) can be used to obtain a wide range of surface characteristics. The smoothness of most of these above mentioned surfaces also makes them suitable for studies with microscopic techniques such as SFM. For infrared spectroscopy, adsorption can also be studied on polymeric material that is generally spin- or dip-coated on silicon, germanium, or zinc selenide crystals. Another way to apply common biochemical techniques which are used to characterize protein structures in solution, and are suitable for proteins in the adsorbed state is to utilize nanometer-sized colloidal particles. Because of the small size, the lightscattering is very low. At the same time, the small diameter of the particle provides a relative large surface area per sample volume, which benefits the sensitivity in measurements, using for example CD, DSC and NMR spectroscopy. Ultrafine particles of teflon, polystyrene, silica, and titanium oxide have been used successfully.

3.30

PROTEIN ADSORPTION

3.4b General trends To perform their biochemical functions, many proteins fold in specific well-defined and highly ordered structures. The folded conformations of globular proteins reach atomic packing densities, expressed in volume fractions, between 0.7 and 0.8. In such a compact structure the rotational mobility along the polypeptide chain is severely restricted, which implies that it has a low conformational entropy. This low entropy is balanced by hydrophobic and Coulomb interactions, the possibility of forming intramolecular hydrogen bonding in secondary structure elements, and interactions between fixed and induced dipoles. The net result is often a compact folded protein that is marginally stable, with stabilization Gibbs energies in the order of a few- to a few tens of kJ per molar quantity, see sec. 3.2. One should note that for a protein with a stabilization Gibbs energy of 15 kJ/mol, one out of 500 is unfolded, and that folding and unfolding processes often occur within fractions of seconds. Thus, although the structures of globular proteins are described as compact and well-defined, this structure is not static. The protein structure is readily disrupted if, for example, electrostatic interactions change by varying the pH in solution. In the same way, protein structures can be affected by adding organic solvents that affect the Van der Waals interactions and the solvation or by adding molecules that compete for the interactions, such as the chaotropics sodium dodecyl sulphate (SDS) and guanidine hydrochloride. Strength of Interaction: Hydrophobic interactions As mentioned before, the same type of interactions that drive the adsorption of proteins to interfaces are the ones determining the structure of proteins. Logically, a stronger interaction between a protein and a sorbent surface is expected to increase the probability and the extent of adsorption-induced structural changes. For example, when proteins adsorb to hydrophobic surfaces, dehydration of the hydrophobic surfaces is the main driving force and, at the same time, hydrophobic interaction is an important force that keeps the polypeptide chains tightly folded. In the adsorbed state the protein/sorbent interaction can be increased by having parts of the hydrophobic core of the protein exposed to the hydrophobic sorbent, leaving more hydrophilic parts of the protein in a more flexible, and thus entropically favourable conformation. Many studies have proved that structural changes are more substantial when globular proteins adsorb to hydrophobic surfaces compared to hydrophilic surfaces. For example, early TIRF studies demonstrated that the conformation of fibronectin is not affected when it is adsorbed onto hydrophilic silica but it is affected when the surface becomes more hydrophobic11.

11

J.D Andrade, V.L. Hlady, and R.A. van Wagencn, Pure Appl. Chem. 56 (1984) 1345.

PROTEIN ADSORPTION

3.31

Strength of interaction: Electrostatic interactions A tendency similar to that above has been observed with respect to electrostatic interactions, i.e., the stronger the attraction, the larger is the extent of structural changes1 . A similar observation was made by Larsericsdotter et al., who used DSC to show that, for the adsorption of lysozyme and ribonuclease-A at hydrophilic surfaces, strong electrostatic attraction leads to high affinities whereby the protein adsorption is accompanied by a reduction in the denaturation enthalpy. Shielding the electrostatic interaction by increasing the ionic strength reduces the affinity, and the adsorptioninduced reduction of the denaturation enthalpy is diminished . Another electrostatic contribution that can affect the protein structure in the adsorbed state is the net charge density in the interfacial layer (see also sec. 3.3c). In solution, most proteins unfold when the net charge density increases, for example at low or high pH values or at low ionic strengths. If proteins adsorb to charged surfaces the interplay between charged groups, including co-adsorbed low molecular weight ions, will affect the balance between the electrostatic interactions within the protein. It is generally observed that the maximum amount of protein is adsorbed when the solution pH is close to the isoelectric point of the protein. A study in which the amount of secondary structure of monoclonal antibodies adsorbed at silica surfaces was monitored while the electrostatic interactions involved were systematically varied, revealed a clear correlation between the reduction in adsorbed amounts when the pH is shifted away from the isoelectric point and the reduction in secondary structure3'. Structural stability of proteins Conformational changes in proteins not only result from the strong interaction between a protein and a surface but can also be seen as driving forces for adsorption41. This was inferred from experiments in which protein adsorption was monitored on hydrophilic surfaces under conditions where the proteins were electrostatically repelled by the sorbent. Under these conditions, the only driving force for adsorption is generated by a total increase of the entropy of the adsorbed proteins. It was also observed that only proteins having a relatively low structural stability adsorb under these otherwise adverse adsorption conditions. This led in sec. 3.3 to the introduction of the notions of 'soft' and 'hard' proteins. Proteins that adsorb to hydrophilic surfaces under electrostatic repulsive conditions are classified as soft proteins whereas proteins that do not adsorb under those conditions are considered hard proteins . Elegant studies on the conformation of mutants and the wild-types of T4 lysozyme'.6) 11

A. Kondo, F. Murakami, and K. Higashitani, Biotech. Bioeng. 40 (1992) 889. H. Larsericsdotter, S. Oscarsson, and J. Buijs, J. Colloid Interface Set 237 (2001) 98. 31 J. Buijs, J.W.Th Lichtenbelt and W. Norde, Langmuir 12 (1996) 1605. 4) W. Norde, J. Lyklema, J. Colloid Interface Sci. 71 (1979) 350. 51 W. Norde, J. Lyklema, J. Biomater. Sci. Polymer 2 (1991) 183. 61 P. Billsten, M. Wahlgren, T. Arnebrant, J. McGuire and H. Elwing, J. Colloid Interface Sci. 175 (1995) 77. 21

3.32

PROTEIN ADSORPTION

and carbonic anhydrase II11 adsorbed on nanometer-sized silica particles revealed that the extent of structural changes was inversely related to their stabilization Gibbs energies in solution. The nature of structural changes, however, was different between the two proteins. T4 lysozyme lost part of its secondary structure upon adsorption, while for carbonic anhydrase II the secondary structure is unaffected, but the protein adopts a more open conformation which is thermally destabilized compared to the protein in solution. The examples above prove that protein stability is an important factor that determines the extent of structural changes. At the same time, no clear relationship between the stability of a protein in solution and the extent of structural changes has been found. Apparently, the structural flexibility of a protein needs to be defined in more detail before such a relationship with predictive value can arise. It is repeated that one should distinguish between adsorption-induced changes that result from strong interactions between the protein and the sorbent surface, and structural changes that result from an adsorption process in which structural rearrangements are prerequisites for adsorption to take place. Surface coverage A number of CD studies, using nanometer-sized silica particles as the sorbent surface, demonstrated that the extent of structural changes was larger when the surface coverage was less 2 ' 3 '. Apparently, the adsorbed proteins need space for structural changes to take place. This phenomenon was also observed for protein adsorption at the air-water interface, where the biological activity of an adsorbed protein monolayer was stabilized by increasing the surface pressure 4 '. A theoretical approach to this phenomenon has been given in equation [3.3.12]. From this equation it is clear that the extent of adsorptioninduced conformational changes in proteins is a function of the adsorption kinetics, i.e., the faster a surface is covered with proteins, the lower is the probability that the proteins will undergo structural rearrangements. The strength of interaction also plays an important role, as it has been observed that the adsorbed amount F passes through a maximum in the course of adsorption, because adsorbed proteins spread at the expense of the total number of proteins attached to a surface. Time-dependency of the structure of adsorbed proteins In general protein adsorption is irreversible, implying that, as a rule, the interactions between a substrate and proteins increase with contact times. It is generally 11

P. Billsten, U. Carlsson, B.H. Jonsson, G. Olofsson, F. Hook and H. Elwing, Langmuir 15 (1999) 6395. 21 A. Kondo, S. Oku and K. Higashitani, J. Colloid Interface Set 143 (1991) 214. 31 W. Norde, J.P. Favier, Colloids Surf. 64 (1992) 87. 41 A. Tronin, T. Dubrovsky, S. Dubrovskaya, G. Radicchi and C. Nicolini, Langmuir 12 (1996) 3272.

PROTEIN ADSORPTION

3.33

accepted that when proteins adsorb they do so tenaciously, because of the large number of segment contacts that can be established between them and the surface. In turn, conformational changes in the protein structure are generally thought to be responsible for increasing the number of contact points, leading to protein adsorption process models that do not allow for conformationally changed proteins to desorb spontaneously1 2). These models are supported by the often observed reduction in protein activity at longer contact-times with the substrate, and by several studies that show that proteins slowly lose secondary structure in the adsorbed state. Unfortunately, little is known about the factors that determine the overall rate of conformational changes. Adsorption isotherms of IgG and fibronectin on hydrophilic and hydrophobic substrates, obtained either by direct addition or successive addition of proteins, clearly demonstrated that conformational changes occur more rapidly on hydrophobic surfaces31. Another observation is that structurally less stable and soft proteins not only have a higher tendency to undergo structural changes upon adsorption but the structural changes occur also more readily41. Moreover, structurally less stable proteins that adsorb with high affinity to surfaces show a higher irreversibility in adsorption and, for those proteins that do desorb, the extent of refolding is lower than observed for structurally stable proteins5'6'71. These observations indicate that rearrangements in the proteins' structure occur faster under adsorption conditions that lead to a large extent of structural changes, as intuitively expected. Besides the slow structural rearrangement, a rather rapid change in the structure is also observed when the first proteins adsorb . This effect can be explained by the lack of steric hindrance for spreading on the surface. See sec. 3.3 Recent stopped-flow fluorescence spectroscopy and anisotropy measurements9 revealed clearly distinct stages for protein structural rearrangements. Specifically, the conformational changes of a -lactalbumin adsorbed on polystyrene particles occur with rate constants of 50 s"1, 8 s' 1 and 0.001 s" 1 . Following the time-dependence of the (3 -sheet content of adsorbed antibodies to silica surfaces, using FTIR spectroscopy, clearly showed that both the relatively low amounts of structure for proteins that were adsorbed in the initial phase and the slow reduction in the [3 -sheet content after longer adsorption times (see fig. 3.18)10).

11

M.E. Soderquist, A.G. Walton, J. Colloid Interface Sci. 75 (1980) 386. I. Lundstrom, H. Elwing, J. Colloid Interface Sci. 136 (1990) 68. 31 U. Jonsson, I. Lundstrom and I. Ronnberg, J. Colloid Interface Sci. 117 (1987) 127. 41 B. Singla, V. Krisdhasima and J. McGuire, J. Colloid Interface Sci. 182 (1996) 292. 51 A. Kondo, J. Mihara, J. Colloid Interface Sci. 177 (1996) 214. 61 W. Norde, J.P Favier, Colloids Surf. 64 (1992) 87. 71 W. Norde, C.E. Giacomelli, Macromolecular Symposia 145 (1999) 125. 81 A. Tronin, T. Dubrovsky, S. Dubrovskaya, G. Radicchi and C. Nicolini, Langmuir 12 (1996) 3272. 91 M.F.M. Engel, C.P.M. van Mierlo, and A.J.W.G. Visser, J. Biol. chew... 277 (2002) 10922. 101 J. Buijs, J.W.Th. Lichtenbelt, and W. Norde, Langmuir 12 (1996) 1605. 21

3.34

PROTEIN ADSORPTION

Figure 3.18. The fraction of amino acid residues in adsorbed antibodies that adopt a (5 -sheet structure (closed circles), and the adsorbed amount (open circles), as a function of the adsorption time. Both the adsorbed amount and the percentage P -sheet structure were obtained with ATR-FTIR. The data presented were obtained when IgG molecules adsorb to a hydrophilic and negatively charged silica surface created by oxidation of a cylindrical crystal made of silicon. The crystal was surrounded by a solution containing 50 jig/ml of a monoclonal IgG with an isoelectric point of 5.8, and 5 mM acetate buffer at pH 5. In the native state, roughly 60% of the amino acids are involved in a p -sheet structure. (Redrawn from J. Buijs et ai., loc. cit.}

Structure of adsorbed globular proteins As might be apparent from the paragraphs above, the final structure of a protein in the adsorbed state is the result of the interplay between many factors. Even if all the interactions between a protein and a surface, including intramolecular interactions, that determine the protein stability, are fully characterized, the problem remains that the structure of the adsorbed protein will depend on the adsorption time and on the interactions with neighbouring proteins. Furthermore, structural alterations can appear on various structural levels. For example, in fig. 3.14 an example is shown where fibronectin is adsorbed in a more elongated structure when it is adsorbed on a negatively charged hydrophilic mica surface, whereas the structure is more compact when adsorbed on a hydrophobic surface. At the same time fig. 3.14 shows that fibronectin only changed its conformation when adsorbed to hydrophobic surfaces. Most probably, the contradiction between these data is based on the differences in structural properties that were probed using different measurement techniques. The first study utilized SFM, which can be used to monitor changes in the tertiary structure, while the second study was performed with TIRF, which is sensitive to local changes involving the relocation of a couple of amino acid residues. An interesting aspect of studies of adsorption-induced changes is that very few

PROTEIN ADSORPTION

3.35

investigators report a complete unfolding of adsorbed proteins. Cases of the absence of structural changes are reported, but they are definitely outnumbered by studies that indicate partial unfolding of globular proteins upon adsorption. At hydrophobic surfaces, some proteins even increase their content of secondary structure elements121 or one type of ordered structure may be converted into another. Both the conversion from a -helix to (3 -sheet structure31 and vice versa 451 have been observed. These results are noteworthy in view of the relatively low stabilization Gibbs energies of the protein structure in solution, where a small disruption in the structure often results in complete unfolding. One should note, however, that most techniques employed to study protein structures at interfaces only monitor average properties. This implies that the results obtained mean either that each protein is partially unfolded to some extent, or that a fraction of the adsorbed molecule is strongly unfolded while another fraction is in its native conformation. Some studies claim that the proteins adopt a well-defined structure similar to the intermediate structures created by local energy minima in the folding pathway of proteins in solution61. Nevertheless, measurements using techniques that provide information on the structural heterogeneity of the protein structure in the adsorbed state, such as DSC and HDX, generally show that structures of adsorbed protein molecules are perturbed differently. Also, with other methods, such as FRAP that is used to monitor the two-dimensional diffusion of adsorbed proteins, it has been observed that one population of BSA molecules, adsorbed to various polymeric materials, is mobile, while another population is immobile. The difference in mobility between populations was ascribed to the distribution of protein conformations, established in the early stage of adsorption71. Even for proteins that show no change in average structure upon adsorption, a more heterogeneous structure can be observed. For example, lysozyme adsorbed to silica surfaces has been subjected to a broad range of techniques to assess its structural properties, such as CD, DSC, TIRF, FTIR, and HDX-MS. These studies revealed that lysozyme does not undergo significant structural alterations when adsorbed to silica in compact monolayers. Nevertheless, the results from DSC, TIRF, and HDX studies indicate that the conformation of lysozyme is rather heterogeneous. This heterogeneity fits well into the models that allow structural rearrangements as long as space is available for the proteins to spread; whereas for proteins that adsorb at a later stage, the spreading is inhibited by spatial restrictions (sec. 3.3b). 11

E.J. Castillo, J.L. Koenig, J.M. Anderson, and J. Lo, Biomater. 5 (1984) 319. M.C.L. Maste, W. Norde, and A.J.W.G. Visser, J. Colloid Interface Set 196 (1997) 224. 31 C.E. Giacomelli, W. Norde, Biomacromolec. 4 (2003) 1719. 41 J. Buijs, W. Norde, and J.W.Th. Lichtenbclt, Langmuir 12 (1996) 1605. 51 A.W.P. Vermeer, C.E. Giacomelli and W. Norde, Biochim. Biophys. Ada 1526 (2001) 61. 61 P. Billsten, U. Carlsson, B.H. Jonsson, G. Olofsson, F. Hook, and H. Elwing, Langmuir 15 (1999) 6395. 71 R.D. Tilton, C.R. Robertson, and A.P. Gast, J. Colloid Interface Set 137 (1990) 192. 21

3.36

PROTEIN ADSORPTION

To illustrate the complexity of adsorption-induced changes in protein structures, an example will now be given of how the structural stability of various segments of myoglobin is affected by adsorption on silica particles. The results were obtained using HDX-MS, a technique that is sensitive to the stability of secondary structure elements in a protein, as the exchange-rate between amide hydrogens in the protein and deuterium atoms in solution is dramatically slowed if the amide hydrogens are involved in hydrogen bonding. In this study, the hydrogen/deuterium exchange process was followed as a function of the exchange time. After the exchange process, myoglobin was cleaved enzymatically, and the deuterium incorporation into four fragments, covering 90% of the sequence, was monitored using mass spectrometry11. Structurally, myoglobin is a 153 residue, tightly folded protein with a haem group that is bound noncovalently in a hydrophobic pocket. Myoglobin contains an extremely stable core, located at the C-terminus and linked to a part of the N terminus, while the centre part of the sequence, surrounding the haem group, is less stable. Upon adsorption to silica, the myoglobin segment located in the middle of the myoglobin sequence, and close to N terminal fragments are destabilized (see fig. 3.19). Although the structural stability of the segment around the haem group did not change upon adsorption, this structure became clearly more heterogeneous. Interestingly, for the N terminal fragment, comprising residues 1-29, two distinct and equally large conformational populations were

Figure 3.19. The structure of myoglobin in solution (left), and adsorbed at silica particles (right). The structural stability is indicated in terms of grey scales and is based on the average folding/unfolding equilibrium of four segments comprising residues 1-29, residues 30-69, residues 70-106, and residues 107-137. The structure is taken from S.R. Hubard, W.A. Hendrickson, D.G. Lambright and S.G. Boxer, J. Mol. Biol. 213 (1990) 215, and the grey scale ranges from 16 kJ/mol (light) to 32 kJ/mol (dark). Upon adsorption, only the structural stabilities of the C- and N-terminal parts are reduced whereas the a -helices close to the haem group are unaffected.

11

J. Buijs, M Ramstrom, M. Danfelter, H. Larsericsdotter, P. Hakansson and S, Oscarsson, J. Colloid Interface Set 263 (2003) 441

PROTEIN ADSORPTION

3.37

observed. One of these populations has a stability similar to that in solution (~ 23 kj/mol) whereas the other population is highly destabilized upon adsorption (—11 kj/mol). From these results it is clear that, even within one protein, the structures of parts are differentially affected by adsorption. Some parts of the protein and a part of the population are highly destabilized upon adsorption whereas other parts are only affected in their structural heterogeneity.

3.5 Adsorbed amount and adsorption reversibility Protein molecules may form numerous contacts with a sorbent surface (as synthetic polymers can). Although data are very limited, there is experimental evidence for such multi-contact adsorption1'. Multiple contacts probably lead to high-affinity adsorption of the whole protein molecule. Plateau-values of the adsorbed amount F corresponding to full coverage of the sorbent surface are reached at a very low protein concentration in solution c and the region in which F depends on c is limited to values very near the /"-axis. Dilution of the sorbate in the bulk phase creates a transient difference in the chemical potential of the sorbate at the interface and that in solution. This chemical potential difference is then eliminated by spontaneous desorption of the sorbate. For high-affinity adsorption, desorption is difficult, if discernible at all, because in the c regime where F is below its plateau value the protein concentration may well be below the detectable limit. Furthermore, when equilibrium is established at such low protein concentrations, where the protein concentration gradient between the subsurface region and the bulk solution is so small that, according to [3.3.1], the transport of protein molecules into the bulk solution is extremely slow. Occasionally, protein adsorption isotherms are reported whose initial rising part deviates from the /"-axis. However, upon dilution, F(c ) rarely (if ever), follows the same path backwards, thereby making the ascending and descending branches of the isotherm distinguishable. As a rule, it is found that dilution only leads to partial desorption, if at all, even when the observation time greatly exceeds the relaxation time of the protein at the sorbent surface2'31. Desorption upon dilution is minimal, especially at hydrophobic surfaces. Such hysteresis indicates a prohibitively high barrier for (further) desorption. The system has two (meta)stable states, one on the ascending branch and the other on the descending one, each being characterized by its local minimum in Gibbs energy. The fact that the adsorption and desorption isotherms represent different metastable states, implies that a physical change has occurred in the system between adsorption and desorption. 11

B.W. Morrissey, R.T. Strombcrg, J. Colloid Interface Set 46 (1974) 152. H.P. Jennissen, in Surface and Interfacial Aspects of Biomedical Polymers, Vol. 2 J.D. Andrade, Ed., Plenum (1985) 295. 31 W. Norde, C.A. Haynes, ACS. Symp. Series 602 (1995) 26.

3.38

PROTEIN ADSORPTION

Despite the irreversible nature generally observed for protein adsorption, many authors interpret their experimental data using theories that are based on reversible thermodynamics. The most common example is the calculation of the Gibbs energy of adsorption, AadsG , by fitting the (ascending) isotherm to the Langmuir equation or variations thereof121. It is, however, questionable whether the adsorption or desorption isotherm should be used for that purpose. Another approach sometimes encountered involves calculation of the Gibbs energy of adhesion, AadhG , using A

adh G = rsp - r™ - rpw

I3-5-1'

where y is the interfacial tension of the interface indicated by the superscripts. AadhG is the reversible work (at constant temperature and pressure) to form a protein (p)/ sorbent (s) interfaces at the expense of sorbent (s)/solution (ix>) and protein (p)/solution (u>) interfaces31. Equation [3.5.1] essentially corresponds to our [III.5.1.1] for the initial spreading tension. For further analysis, see sec. III.5.2; it is noted that the interfacial tensions may change with time. It is not clear how AadhG relates to AadsG, since the latter quantity is determined not only by creation and rupture of phase boundaries but also includes other contributions such as those resulting from electrostatic interaction and protein (and sorbent) structural rearrangements. The error involved in treating protein adsorption as a reversible process may be approximated by calculating the entropy production due to the irreversibility (reflected by the hysteresis) of the process. In a closed system, the entropy change associated with any process can be written as AS = ASe + ASj

[3.5.2]

where ASe is the reversible entropy exchange between the system and its surroundings, and ASj is the internally produced entropy. For a reversible process, AS{ = 0, but for an irreversible process ASj > 0 . According to Everett41, AadsSj can be calculated from the hysteresis loop, as the closed loop integral between the ascending and descending branches of the adsorption isotherm A

adsSi=RJ ^ f d l n c p

[3.5.3]

where r* is the adsorbed amount at the upper closure point of the hysteresis loop. The integral of [3.5.3] may be evaluated as the shaded area indicated in fig. 3.20. Regrettably F(c ) is not usually known in the very dilute region (i.e., at very negative

11

T. Mizutani, J.L. Brash, Chem. Phartn. Ball. 36 (1988) 2711. E.C. Moreno, M. Krcsak and D.I. Hay, Biofouling 4 (1991) 3. 31 C.J. van Oss, Biofouling 4 (1991) 25. 4) D.H. Everett, Trans. Faraday Soc. 50 (1954) 1077. 21

PROTEIN ADSORPTION

3.39

Figure 3.20. Adsorption ( —t) and desorption (<-) of BSA on/from silica particles. Temperature 25°C, 0.05 M phosphate buffer, pH 7. The shaded area repressents the entropy change associated with the hysteresis loop; it may be calculated using [3.5.3].

lnc values). Hence, only a minimum value of AadsSj can be estimated by letting the hysteresis loop close at the lowest experimentally detectable of c . Unfortunately, no consistent molecular picture has been put forward to explain the adsorption hysteresis. The most popular explanation is that, upon first attachment, the proteins form a relatively small number of contacts with the sorbent surface, and that they gradually relax to significantly enhance bonding. An alternative explanation for hysteresis is intermolecular aggregation between adsorbed protein molecules. Then desorption requires detachment of a cluster of protein molecules rather than single ones. As explained in chapter II.5, with flexible highly solvated polymers an increasing value for F is commonly found with increasing concentration in solution. It is explained by conformational adaptation of the adsorbed molecules, involving a decreasing number of attached segments per molecule. The adsorption isotherms of globular proteins, however, develop well-defined plateau-values. Usually, the plateau value is not too far from those corresponding to a close-packed monolayer of native molecules in a side-on or end-on orientation. Under some conditions, for example for highly charged and/or structurally very labile protein molecules, considerably lower plateau adsorptions are found. They could be explained by the formation of an incomplete monolayer and/or extensive unfolding of the protein molecule to attain a rather flat conformation of the polypeptide chain on the sorbent surface (which is typical for flexible polyelectrolytes). Sometimes, but not exceptionally, a 'step' or a 'kink' is observed in the isotherm at intermediate c . An example is shown in fig. 3.21. A reasonable explanation could be that the first plateau represents a layer with randomly oriented protein molecules. Then, at higher c , intermolecular nucleation occurs to form a, 'two-dimensional protein crystal', i.e., protein molecules align in orderly oriented arrays with an increasing amount of protein in a saturated layer. Unfortunately, the validity of this suggestion has not yet been verified experimentally. Alternative explanations for the

3.40

PROTEIN ADSORPTION

Figure 3.21. Adsorption isotherm with two plateaus, frequently observed for protein adsorption on solid surfaces.

appearance of a kink In the isotherm could involve tilting of the adsorbed molecules from a side-on to a more end-on orientation, or the formation of a second adsorbed layer on top of the first one. Of course, each of these possibilities may apply to different extents for different protein-sorbent systems. 3.6 Influence of some system variables on protein adsorption 3.6a Protein and sorbent charge A systematic approach to establish the influence of the electrical charge on protein adsorption is to vary the charge on the protein (by varying the pH In the system), while keeping the charge on the sorbent-surface constant. In fig. 3.22, plateau values for serum albumin are given as a function of the pH of adsorption at different sorbents of constant charge density. If electrostatic interactions between the protein and the sorbent surface were to dominate, the /"(pH) isotherm would be a monotonic function

Figure 3.22. Plateau adsorption of human serum albumin on negatively charged polystyrene (A), positively charged polystyrene (A), negatively charged silver iodide (x) and uncharged poly(methylene oxide) (•). T = 25°C; electrolyte, 0.01 M KNO3 . (Redrawn from W. Norde, Adv. Colloid Interface Set 25 (1986) 267.)

PROTEIN ADSORPTION

3.41

of pH. Instead, all of these curves show a maximum near the isoelectric point of the protein-sorbent complex. Such a bell-shaped /^'(pH) -functionality is a feature not only of albumin, but it is quite commonly observed for protein adsorption". Lateral electrostatic repulsion between adsorbed molecules could prevent the formation of a close-packed monolayer, and thus explain the reduction in /"P1 away from the isoelectric region. Another possible reason for the maximum in /T'(pH) is related to the degree of structural rearrangement of the adsorbed molecules. In the isoelectric region, the internal cohesion is expected to be at a maximum, and adsorption-induced conformation changes resulting in a larger molecular area at the sorbent surface would be minimal. Some proteins, in particular the hard ones, show a different .TP'tpH) pattern. Unlike the 'soft' proteins, for which conformational changes may be a major driving force for adsorption, the, 'hard' proteins adsorb only in case of hydrophobic interaction and/or electrostatic attraction. Hence, for the hard proteins /T'(pH)dependencies result like those shown for ribonuclease in fig. 3.23.

Figure 3.23. As fig. 3.22 but now for bovine ribonuclease on negative polystyrene (o), uncharged poly(methylene oxide) (•) and haematite (x). (Redrawn from W. Norde, loc. dt.) 3.6b Hydrophobicity

Protein surfaces contain both hydrophilic and hydrophobic residues. It is reasonable to assume that protein surface hydrophobicity determines, at least partly, the adsorption behaviour. Studies related to hydrophobic interaction chromatography have confirmed that the more hydrophobic proteins tend to adsorb more tenaciously to (solid) surfaces . In addition to the hydrophobic fraction of the outer surface, the over all hydrophobicity may influence the adsorption as well. The probability of structural 11 2)

C.A. Haynes, W. Norde, Colloids Surf. B2 (1994) 517. A.A. Gorbunov, J. Chromatography 365 (1986) 205.

3.42

PROTEIN ADSORPTION

rearrangements in the protein molecule, when it is transferred from an aqueous solution to a (partially) non-aqueous environment (as occurs in an adsorption process) increases with an increasing contribution from intramolecular hydrophobic interaction to the stabiliztion of globular structures in solution (see sec. 3.2b). Indeed, several studies1 2) report a strong correlation between the overall hydrophobicity of a protein and its tendency to change its structure upon adsorption. It is difficult to quantify the influence of the hydrophobicity on protein adsorption. We have to compare sorbent surfaces of different hydrophobicities, different chemical compositions, and possibly different electrical charges. Nevertheless, trends may be indicated. In general, because of the contribution of dehydration, the adsorption affinity increases with increasing hydrophobicity31. Furthermore, there are indications that conformational changes in the protein molecule are triggered more strongly with increasing hydrophobicity of the sorbent surface. 3.6c Protein structure stability Protein molecules may adopt a number of conformations that have only marginally different stabilities. Hence, interaction with the sorbent surface may trigger conformational changes. By using various experimental methods, adsorption-related conformational charges have been characterized (see sec. 3.4). For some proteins, a reduction in ordered (secondary) structure, involving an increased conformational entropy of the protein, could even be a dominating force for adsorption to occur spontaneously. Thus, unlike 'hard' proteins, 'soft' proteins do adsorb at hydrophilic, electrostatically repelling surfaces, thanks to a relatively large gain in conformational entropy. The propensity of structural alteration influences the adsorption affinity and the adsorbed amount. However, it is impossible to trace correlations when comparing different types of proteins, because of interference from other factors such as charge and hydrophobicity. Unambiguous data may be obtained from comparative studies using mutants of the same type of protein. For instance, Tian et al. have compared wild type lysozyme with different mutants4'. The role of protein structural changes on the reversibility of the adsorption process, i.e., the question whether or not desorbed molecules re-adopt their original, native structure will be discussed in sec. 3.8. 3.7 Adsorption at fluid interfaces Proteins at liquid-gas (LG) and liquid-liquid (LL) interfaces form a topic of their own. First, because of their practical interest, for example in relation to the creation and 1J

K.S. Birdi, J. Colloid Interface Set 43 (1973) 545. T. Aral, W. Norde, Colloids Surf. 51 (1990) 1. W. Norde, in Physical Chemistry of Biological Interfaces, A. Baszkin, W. Norde, Eds., Marcel Dekker (2000) ch. 4. 41 M. Tian, W. Lee, M.K. Bothwell, and J.M. McGuire, J. Colloid Interface Set 200 (1998) 146. 21

PROTEIN ADSORPTION

3.43

stabilization of protein-stabilized foams and emulsions, which are relevant for the food industry (ice cream, food emulsions), see chapters 7 and 8. respectively. Another field is that of enzymatic reactions in such interfaces, e.g., for lipases. Secondly, there is academic interest. The central questions are, "What happens to their conformations when proteins adsorb at fluid interfaces? Are these conformational changes similar to those at SL interfaces, which we have discussed extensively in preceding sections?" Insight into the properties of protein monolayers at LG and LL interfaces has made steady, but slow progress. This slowness is partly caused by experimental problems that may be of methodological nature (what techniques are available, and what pitfalls must be avoided in the interpretation of signals?), or inherent in the system (e.g., the adsorption of several globular proteins is largely irreversible, leading to history dependence). For example, it is possible that a given protein with a given surface concentration/', obtained by adsorption from solution, has a conformation which differs from that in the same final state, but is obtained from spreading at a low surfacepressure, n, followed by compression). Here, we can only review some relevant trends. For older, but not necessarily dated, reviews see those by Miller and Bach, by Wahlgren and Nylander and by Izmailova, mentioned in the General References, sec. 3.10. For older reviews see refs. 12) . MacRitchie has reviewed the adsorption-desorption reversibility31 and Dickinson, in two review papers 4 ' surveyed the relevance of protein adsorption at LL interfaces for the formation of (food) emulsions. 3.7a Review of some general trends and techniques A number of differences from adsorption at SL interfaces are readily noticed. These are partly of an academic, partly of a methodical nature; sometimes a mixture of both. l)We only consider macroscopic interfaces. The interfaces are smooth, and will usually remain so upon adsorption. Hence, there is no reason for worrying about surface roughness, surface heterogeneity, or ill-defined asperities, porosities etc. The flatness is a result of the contractile action of the interfacial tension y or the interfacial pressure, n. Problems arise only when the interface contains solid-like patches: or particles, that do not yield under the influence of n . Example: small proteinaceous gel particles. An advantage is that the interfacial area A is well known. 2) Assuming adsorbing ionic surfactants to be absent, the original interface is essentially uncharged, except for charge caused by the preferential adsorption of one of the ionic species of electrolyte that might have been added, including buffers. In sees 11

I.R. Miller, D. Bach, in Surface and Colloid Science, E. Matijevic, Ed., Vol. 6 Wiley-Interscience (1973) 185-260. 21 V.N. Izmailova, Progr. Surface Membr. Sci. 13 (1979) 141. F. MacRitchie, Proteins at Liquid Interfaces, in Physical Chemistry of Biological Interfaces. A. Baszkin, W. Norde, Eds., Marcel Dekker (2000) chapter 5. 41 E. Dickinson, J. Chem. Soc. Faraday Trans. 88 (1992) 2973; ibid., 94 (1998) 1657.

3.44

PROTEIN ADSORPTION

III.4.4b and 4c this phenomenon was analyzed and shown to be minor. Moreover, it is likely that these adsorbates are readily displaced by adsorbing proteins. The conclusion is that any influence of pH or c salt stems from the influence of these parameters on the protein in solution and in the adsorbate. 3) Air does not attract proteins (nor does it attract water). So, the driving force for adsorption at the air-water interface is expulsion from the bulk of the hydrophobic parts of the protein, which may be created by some reconformation. On the other hand, non-aqueous 'oil' phases may attract proteins, mainly through their hydrophobic parts. This difference may lead to conformational stability differences between proteins at these two types of interfaces. 4) As with SL interfaces, the adsorption is, to a large extent, irreversible. This extent depends strongly on the conformational stability. Desorption can, however, be achieved by displacers. Common surfactants can act as such, but sometimes spurious admixtures may do that as well. Purity of the protein sample is therefore an essential condition for reproducibility. 5) As the surface area-volume ratio is usually small compared with adsorption on dispersed particles, it is virtually impossible to determine accurately the surface concentration F by depletion of the solution. Of course, the adsorbing area can be greatly enlarged by foaming or emulsification. However the F[c) isotherms, obtained in this way are not representative for adsorption at quiescent interfaces because the interface has passed through a complex sequence of mechanical stretching, shearing and coalescence steps, which determine the adsorption of the proteins and emulsifiers, if any. 6) The determination of F from the surface pressure, K , is not a good alternative, either, because the 2D equation of state is only ideal under very restrictive (and not particularly interesting) conditions. In general, one may not expect even the relatively simple 2D equations that work reasonably well for uncharged polymers, such as [III.3.4.58] or [III.3.8.6], to apply to proteins. 7) When we look for viable alternatives to establish F and other properties of the adsorbate, the optical techniques discussed in sec.III.3.7b-c come to mind. The adjustment and implementation of these approaches to protein monolayers is not always straightforward, but useful developments have been made, with each technique having its advantages and limitations. For example, reflectometry is a powerful tool for studying the adsorption rate, especially in the earlier stages, but does not provide additional information on the adsorbate. Ellipsometry has now been adapted to protein adsorbates1'. Neutron reflectometry, in particular at small angles, helps to provide information on the layer thickness and averaged volume fraction profile2'31. This technique cannot pick up rapid adsorption or reconformation steps, but works both at LG

11

J.A. de Feijter, J. Benjamins and F. Veer, Biopolymers 17 (1978)1759. E. Dickinson, D.S. Home, J.S. Phipps and R.M. Richardson, Langmuir 9 (1993) 242. 31 A. Eaglesham, T.M. Herrington and J. Penfold, Colloids Surf. 65 (1992) 9. 2)

PROTEIN ADSORPTION

3.45

and LL interfaces. By TIRF (total internal reflection fluorescence) the translational and rotational diffusion of adsorbed proteins can be observed in detail11) and FRAP (fluorescence recovery after photobleaching) gives information on diffusion inside the adsorbate2'. 8) Of the non-optical methods for determining F, surface pressure and radioactivity measurements may be mentioned. The former technique, and its application to a variety of systems, not including proteins, has been discussed in detail in chapter III.3. As stated, it is not straightforward to relate n to F, but compression-expansion cycles in Langmuir troughs can be enlightening. Experiments with radio labeled proteins have the well-known advantage (high sensitivity) and proviso (that the tagging of proteins should not affect the conformation of the molecules). 9) A typical additional element for fluid interfaces is the option of studying interfacial rheology. A number of interfacial rheological characteristics are obtainable. For a review of the techniques, see sec. III.3.7e, in particular we refer to the interfacial shear (77^ ) and dilatlonal (77J) viscosities (Nm^1 s) and, for oscillatory measurements, the complex, interfacial dilational modulus: K° = K°'-iK°"

[3.7.1]

where Kg' is the dilational storage modulus and Kg" the corresponding loss modulus (all K's in Nm"1). The quantity Kg" = rf^a, where co is the frequency and ?/g the interfacial dilational viscosity; see table III.3.4. The values obtained for these characteristics usually depend on the method used, as a result of differences in the way in which forces are exerted on the monolayer. In this respect, the time-dependence and the history play an important, and sometimes decisive, role. The interpretation is not always unambiguous because the observed trends are usually caused by a non-additive combination of rates of supply from solution and rate of reconformation. Therefore, a review of a variety of experimental results for the same system can give some feeling for the basic processes involved, and for their complexity. Such information is much less accessible with SL interfaces. 3.7b Some illustrations The few examples from the literature to be discussed here mostly involve rheological measurements and/or comparisons of systems and/or conditions. For background information (for this section, and for the many studies not reported here), table 3.3 collects some relevant molecular properties of proteins in aqueous solution. As protein (re-)conformation is the complicated result of a range of subtle molecular transformations it is not feasible to characterize the conformational stability by using just one V. Hlady, R.A. van Wagenen, and J.D. Andrade, in Surface and Interfacial Aspects of Biomedical Polymers; J.D. Andrade, Ed., 2 (1995) 81. 21 M. Coke, P.J. Wilde, E.J. Russell, and D.C. Clark, J. Colloid Interface Scl. 138 (1990) 489.

3.46

PROTEIN ADSORPTION

Table 3.3. Molecular properties of some proteins . Ovalbumin

BSA

Myo-

Source

hen's eggs

bovine

sperm

serum

whale

M/kDa

45

69

17.8

i.e.p.

4.7

4.9

Molecular

compact

globular

structure

globular

Lysozyme

a-Casein

P-Casein

hen's eggs

cow's milk

cow's milk

14.6

23.5

24

10.7-11.1

globin

Shape

and spherical dimensions 2.9

7.8

5.1

5.3

compact rigid

random

random

globular globular

coil

coil

ellipsoidal

ellipse

prolate

prolate

14x3.8

4.5x3.5 4.5x3.0

ellipsoidal

ellipsoidal

ellipsoidal

(nm)

X2.5

X3.0

T.D.2) stability kJ mol" 1

54

60

% a-Helix

30

55

75

27

10

1-10

% P-Sheet

27

16

-

16

20

13-16

K -Casein

BLG

RNase

a-chymotrypsin

Cytochrome- C

Source

cow's milk

cow's milk

bovine

bovine pancreas

bovine heart

M/kDa

19

18

25.2

12.2

i.e.p.

4

5.2

9.4

8.1

10.0

Molecular

spherical

rigid

globular

globular

globular

ellipse

ellipsoidal

ellipse

pancreas

structure Shape

13.68

globular and

prolate ellipsoidal 3.6X7.0

dimensions (nm) T.D.21

3.8x2.8

2.8x3.0x3.4

X2.2 50

54

38

stability kJ mol" 1 % a-Helix

14

10

11

13

51

% P-Shect

31

50

33

25

-

Data obtained from several sources, collated by J. Benjamins and W. Norde. From differential scanning calorimetry; the Gibbs energy is given for the N —> D transition at pH of maximal stability.

PROTEIN ADSORPTION

3.47

Figure 3.24. Two-dimensional equations-of-state at the air-water interface for adsorbed monolayers of five proteins and PVA (M = 42.000 ). The concentrations, c, of the solutions from which the monolayers were made (given in g d m " ' ) are indicated. Discussion in the text (courtesy, J. Benjamins). number. The thermodynamic stability given refers to thermal denaturatlon, but conformatlonal alterations upon adsorption probably follow a different path, so the conformatlonal stability may also be different. Figure 3.24 contains n{F) curves for five different proteins, including the uncharged polymer poly(vlnyl alcohol) for the sake of comparison. In these experiments it was measured by the Wilhelmy plate technique, and r was obtained ellipsometrically on the same interface.

3.48

PROTEIN ADSORPTION

The first notable feature is that data points obtained at different values of c all collapse to one 'mastercurve'; surface pressures are fully determined by F, and not by the concentration in solution needed to obtain these surface concentrations. Stated differently, for these adsorbed monolayers the n{F) relationship is unique for each protein and so we may ask whether it makes sense to speak of a 2D equation of state. The second observation is that, for all five proteins, the first 0.7-1.1 mg m~2adsorbed do not contribute measurably to the surface pressure. This is a typical protein feature, which is not exhibited by the random and heterodisperse polymer PVA. For spread monolayers, the TI(F) curves are rather different (results not shown) but the horizontal initial parts persist. These parts coincide with the initial steep rise in the isotherm, F(c). In this part of the isotherm the adsorbing molecules have the entire surface at their disposal; they can complete any conformational adjustment during the time it takes to carry out an ellipsometry measurement (ca. 10 min.). If one is interested in the occurrence, and rates, of re-conformation on much shorter time scales, dynamic measurements have to be carried out. Indeed, such processes have been observed with relaxation times of a few seconds up to a few minutes. There is some correspondence between the length of the very low n range; for more rigid molecules, it is longer. Lysozyme being the top, lower for the albumins, still lower for the structurally relatively 'weak' caseins down to virtually zero for the fully structureless and heterodisperse PVA. Semiquantitative computations with the ideal 2D equation of state (n = RTF, with F in moles m" 2 ) show that, in these initial parts, K is indeed very low; the pressure starts to build up when this layer is completed and lateral interaction sets in. This lowpressure part was also studied optically for lysozyme by Ericson et al.1'. For proteins like the albumins the plateau adsorption passes through a maximum around the i.e.p., similar to that at SL interfaces (fig. 3.22), and for the same reason. The different rates of unfolding are also reflected in the dynamic surface tension. In fig. 3.25, this tension is given as a function of the relative expansion rate of a surface,

Figure 3.25. Dynamic surface tensions obtained by the overflowing cylinder technique for 0.25 gdm~ 3 solutions of p-casein, [3lactoglobulin, ovalbumin and lysozyme. Temperature 25°C. (Redrawn from Van Kalsbeek and Prins, loc. clt.).

11

J.S. Ericson, S. Sundaram, and K.J. Stebe, Langmuir 16 (2000) 5072.

PROTEIN ADSORPTION

3.49

Figure 3.26. Apparent interfacial shear viscosities at the water-n-tetradecane interface. A, myosin; B, lysozyme; C, K-casein; D, gelatin; E, Na-caseinate; F, a s -casein; G, (5casein; H, P-lactoglobulin and I, a-lactalbumin. (Redrawn from E. Dickinson, J. Chem. Soc. Faraday Trans. 94 (1998) 1657.)

as measured by the overflowing cylinder technique described in sec. III.3.7e and illustrated in fig. III.3.73. These data have been obtained by van Kalsbeek and Prins1'. With increasing expansion rate, all the curves tend towards the surface tension of water but the 'weaker' casein can unfold far more rapidly than lysozyme, and hence take segments to the interface at a rate that is more compatible with the rate of expansion. Hence, with this protein the dynamic tension remains below the static tension over the entire range studied. Long term interfacial viscosity changes are shown in fig. 3.26. Plotted are the apparent surface shear viscosities at the water-n-tetradecane interface. Adsorption takes place from the aqueous phase, which contains 10~3 weight % of the protein at neutral pH. The method for obtaining the apparent shear viscosity was not reported, but two conclusions can be drawn. First, 77 continues to increase over tens of hours, i.e., long after adsorption has been completed. This increase must be caused by structural changes at the interface, the lateral interaction between the adsorbed protein molecules being an important characteristic. Secondly, there are dramatic differences between the relatively rigid globular proteins and the rather random caseins, the former group displaying values that are higher by several orders of magnitude. One interpretation is that the rigid spheres can acquire a much higher packing density, perhaps with some cross-linking. An additional feature is that, in several cases, the proteins can form 2D gels, which may exhibit rupture: when that occurs, a kind of slip viscosity is measured. The corresponding interfacial dilational properties exhibit the same trend with respect to the protein specificity but the

H.K.A.I, van Kalsbeek, A. Prins, in Food Emulsions and Foams, Interfaces, Interactions and Stability. E. Dickinson and J.M Rodriquez Patino, Eds., Royal Soc. Chem (London ) (1999), 91.

3.50

PROTEIN ADSORPTION

difference between them is less1'21. The inference of these observations is that the adsorbates behave like a 2D gel, with their strength depending on the nature of the protein; several protein molecules unfold to a greater extent at oil-water interfaces than during heat-denaturation in aqueous solution3 . Our last illustration, fig. 3.27, combines at least three important features. It relates the interfacial rheology to the surface pressure, different non-aqueous phases are compared, and the experiments have been carried out conscientiously. The reported rheological characteristic is the interfacial dilational modulus, K£. This quantity can be obtained by several methods, either by an oscillatory technique, using [3.7.1] to obtain Kg', or by a monotonous expansion, yielding Kg directly. The data reported stem from experiments with the dynamic drop tensiometer, described in connection with fig. HI.3.72. They have been compared to those from the barrier-andplate method (essentially a longitudinal wave technique) and were found to be very similar at given n. (For the sake of simplicity we use the symbol Kg ). The matter of the identity of results from different techniques is a separate problem, and is not generally solved. The three different non-aqueous phases are air (as in figs. 3.24 and 3.25), n-tetradecane (as in fig. 3.26), and triacylglycerol, which is a sunflower-seed oil, an important fluid for the food industry, and has useful properties as a solvent. The purity of this oil was improved (e.g., by silica extraction) and verified (by surface tension measurements) to ensure the absence of surface-active admixtures. The monolayers are formed by adsorption. As a result, n and Kg depend on the adsorption time, but for the n{F) relationship there exists a time-independent mastercurve (fig. 3.24). As the moduli also collapse into a protein-specific mastercurve at different solution concentrations, it makes sense to plot K^ as a function of n. Thus, the adsorption time is eliminated as a variable. Another matter is the frequency a> of the measurement. The higher is co, the more elastic is the behaviour, i.e., the more the first term on the r.h.s. of [3.7.1] dominates. The figure only gives data at 0.1 Hz. One would intuitively expect Kg to increase with n, and for low n this is indeed observed. For an ideal monolayer the slope should be unity, but experimental moduli are higher than that. At higher n 's, Kg increases less than linearly, and even diminishes. (3 -Casein exhibits a more capricious behaviour than its more rigid globular counterparts. Generally, the behaviour in triacylglycerol (which may be a better solvent for hydrophobic and/or even hydrophilic parts of the protein molecules) is more unequivocal than for air and tetradecane, in that the curves consist of two linear parts, even for the unruly p-casein. A kind of scaling is suggested, with at least the solvent 11 F.J.G. Boerboom, A.E.A. de Groot-Mostcrt, A. Prins and T. van Vliet, JVeth. Milk Dairy J. 50 (1996) 183. 21 A. Williams, A. Prins, Colloids Surf.. A114 (1996) 267. 31 J. Lefebvre, P. Relkin, in Surface Activity of Proteins. S. Magdassi, Ed., Marcel Dekker (1996), p.181.

PROTEIN ADSORPTION

3.51

Figure 3.27. Modulus as a function of the interfacial pressure for the indicated proteins in three non-aqueous phases. Temperature 25°C, frequency of drop oscillation 0.1 Hz. BSA = bovine serum albumin, BLG = bovine lactoglobulin. (Same source as fig. 3.24.) quality of the protein parts for the oil as one of the parameters. The inference is that the descending branch is caused by reconformation or collapse; it is certainly not a result of desorption. Formation of a second adsorbed layer is also unlikely, because there is no reason why this would reduce Kg . Experiments like those above can suggest ideas for further theoretical and experimental studies. One could think of optical investigations in conjunction with the rheol-

3.52

PROTEIN ADSORPTION

ogical ones which, In turn, may be extended by looking in more detail at the influence of

OJ12).

For reasons other than merely academic, this type of study has enormous practical relevance. Several of the proteins discussed here are milk constituents and are relevant in the preparation of food products: See the examples in chapters 7 and 8, and the book by Dickinson and Rodriguez Patino, mentioned in the General References, sec. 3.10. Other interest stems from interfacial enzymatic reactions, whose time dependence can be followed from surface pressure and surface rheology studies. By way of illustration we refer to studies from the group of Nitsch on lipase31 and catalase41 an investigation by de Roos and Walstra51 on the loss of activity of the enzymes lysozyme and bovine chymosin owing to adsorption on emulsion droplets, and a review by Panaiotov and Verger61. There is no need to state that this is a challenging area where significant developments may be expected. 3.8 Competitive protein adsorption and exchange between the adsorbed and dissolved states Protein adsorption is generally highly irreversible with respect to variation of the concentration in solution. Nevertheless, proteins can easily be desorbed by surfaceactive molecules that compete for the interaction sites, such as surfactants or other protein species. By studying protein adsorption from plasma, Vroman and co-workers observed that the adsorbed protein layer was rather dynamic. They observed that more abundant and smaller proteins adsorb first, because the transport to the sorbent is quicker. Proteins that have a higher affinity for the sorbent surface then slowly replace these adsorbed proteins7'81. For plasma proteins adsorbing to hydrophilic surfaces, HSA adsorbs first and is then followed by IgG, fibrinogen, fibronectin, Hageman factor, and high-molecular weight kininogen. This cascade of competitive protein adsorption is generally called the Vroman effect. As surfaces with an abundance of albumin tend to be less thrombogenic than those coated with fibrinogen or IgG, controlling the protein Also see, J. Benjamins, E.H. Lucassen-Reijnders, Surface Dilational Rheology of Proteins Adsorbed at Air-Water and Oil-Water Interface in Proteins at Liquid Interfaces 7 (1998) 34184. 21 M.A. Bos, T. van Vliet, Interfacial Rheological Properties of Adsorbent Protein Layers and Surfactants: a Review,. Adv. Colloid Interface Sci 91 (2001) 43-71. 31

W. Nitsch, R. Maksymiw, and H. Erdmann, J. Colloid Interface Sci.. 141 (1991) 322 R. Maksymiw, W. Nitsch, J. Colloid Interface Sci. 147 (1991) 67. 51 A.L. de Roos, P. Walstra, Colloids Surf. B6 (1996) 201. 61 I. Panaiotov, R. Verger, Enzymatic Reactions at Interfaces; Interfacial and Temporal Organization of Enzymatic Lipolysis in Physical Chemistry of Biological Interfaces, A. Baszkin, W. Norde, Eds. (see sec 3.10) ch. 11. 71 W.G. Pitt, K. Park, and S.L. Cooper, J. Colloid Interface Sci. I l l (1986) 343. 81 L. Vroman, A.L. Adams, J. Colloid Interface Sci. I l l (1986) 3 9 1 . 41

PROTEIN ADSORPTION

3.53

adsorption cascade from blood proteins is a crucial aspect of producing nonthrombogenic biomaterials. Logically, most research dealing with competitive protein adsorption has long been focused on the above-mentioned proteins to a variety of surfaces 1 ' 2 ' 34 '. As a general trend, the displacement of proteins is strongly related to their affinity for the interface. For example, human serum albumin (HSA) adsorbed to hydrophilic silica can be partly replaced by fibrinogen, whereas HSA adsorbed at hydrophobic methylated silica surfaces prevent any displacement or further adsorption of fibrinogen or IgG. A similar result was obtained when trying to replace fi -casein and fragments thereof with (3-lactoglobulin. If (}-casein is adsorbed to a methylated silica surface, no further adsorption of, or replacement by, (5 -lactoglobulin occurs. However, when the hydrophobic C-terminal half of p -casein is cut off, (3 -lactoglobulin easily replaces the hydrophilic, N-terminal half of P -casein, with adsorption kinetics similar to those for (5-lactoglobulin adsorption to a bare methylated silica surface. Thus, strong hydrophobic interactions, originating from a hydrophobic substrate or a hydrophobic protein surface, diminish the displacement of adsorbed proteins by proteins from solution. As illustrated in fig. 3.6, a requirement of adsorption reversibility is that not only can the molecules attach to and detach from the surface, but detachment should lead to their original structure. The extent of structural perturbation of adsorbed protein molecules depends on various parameters such as the hydrophobicity of the sorbent surface, the structural stability of the protein, the charge contrast between the protein and the sorbent surface and, not least, on the degree of coverage of the sorbent by the protein (see sec. 3.4). Accordingly, the structure of the exchanged protein molecules and, hence, the reversibility of the adsorption process, may be affected by the same variables. Unfortunately, very little attention has been given to this issue so far. Only a few papers have considered protein refolding after exchange from a (solid) surface. Below, some trends emerging from the few systems studied, are presented. They deal with the hard protein, lysozyme and the soft protein, serum albumin, exchanged at hydrophobic and hydrophilic colloidal particles. Samples of exchanged protein were obtained by incubating an excess of protein in a colloidal dispersion for a period of about 16 hours. As reported for various systems, 56 ' 71 after 16 hours essentially all protein molecules have been on and off the sorbent surface. The supernatant solutions may contain different populations of protein molecules with respect to the number of 11

A. Baszkin, MM. Boissonnade, Am. Chem. Soc. Symp. Sen 602 (1995) 209. B.K. Lok, Y.L. Chang, and C.R. Robertson, J. Colloid Interface Sci. 91 (1983) 2104. 31 M. Malmsten, B. Lassen, Colloids Surf. B4 (1995) 173. 41 T. Nylander, N.M. Wahlgren, J. Colloid Interface Sci. 73 (1994) 151. 51 V. Ball, P. Schaaf, and J.C. Voegel, in Surfactant Science Series 75. M. Malmsten, Ed., Marcel Dekker (1998) p. 453. 61 J.L. Brash, Q.M. Samak, J. Colloid Interface Sci. 65 (1978) 495. 71 J.C. Voegel, N. de Baillon, and A. Schmitt, Colloids Surf. 16 (1985) 289. 21

3.54

PROTEIN ADSORPTION

Figure 3.28. CD spectra (left) and DSC thermograms (right) of native BSA (drawn curve) and BSA exchanged from polystyrene surfaces (dashed). Aqueous solution, 25°C, pH 7. (Redrawn from W. Norde, C.E. Giacommelli, J. Biotechn. 79 (2000) 259.) times they have been exchanged at the surface. As they could not be separated, the experimental data are averaged over these populations. Changes In structural properties of the exchanged protein are probed by monitoring the thermal stability, and the secondary structure. By way of example, fig. 3.28 shows CD spectra and DSC thermograms for BSA in solution, before adsorption and after release from polystyrene surfaces. The homomolecular exchange of BSA at polystyrene clearly provokes a change in the secondary structure. The shifts in the CD spectra indicate the formation of (3-sheet structure at the expense of an a-helix. The structural alteration is most pronounced when exchanged with a lower adsorbed amount, corresponding to a lower protein concentration in solution. As discussed in sec. 3.3b, adsorption from a low proteinconcentration solution allows more time for adsorbed molecules to relax at the sorbent surface before neighbouring patches are occupied by protein as well. Hence, the degree of spreading, i.e., the structural perturbation, is higher at lower protein concentration, and this is reflected in the exchanged molecules. The DSC data are in line with the CD results. The thermogram of the exchanged BSA molecules deviates from that of native BSA, the more so the smaller is the adsorbed amount. Exchanged BSA is more thermostable and the heat-induced transitions occur over a wider temperature range. A broader transition region could well be caused by a heterogeneous protein population with molecules of different thermostabilities. Similar results have been reported for BSA exchanged at other hydrophobic sur-

PROTEIN ADSORPTION

3.55

faces, i.e., those of silver iodide11, and Teflon21. Formation of (3-sheet structures is often associated with intermolecular aggregation. Aggregation between exchanged BSA molecules is indeed observed, and intermolecular hydrogen bonding invoking aggregation through pi -sheet formation3'4' may cause the increased thermostability. In contrast with the hydrophobic sorbent surfaces, mentioned above, homomolecular exchange at hydrophilic silica surfaces neither leads to a modification in the secondary structure of BSA, nor to a change in the thermal stability, at any degree of surface coverage. Still, in the adsorbed state the BSA molecules have different structural characteristics, but upon release from the hydrophilic surface the protein molecules fully regain their original native structures '. Lysozyme, a hard protein, recovers its native conformation after being exchanged, 2)

irrespective of the hydrophobicity of the sorbent surface . Thus, if from the limited number of experimental data, a trend may be inferred, it would be the following: a less stable protein conformation, a more hydrophobic sorbent surface, and adsorption from protein solutions of lower concentrations, promote conformational changes that, at least partly, persist after desorption by a homomolecular exchange process. Consequently, for a given protein, to avoid irreversibility and loss of biological activity associated with interfacial exchange, a hydrophilic sorbent and a high degree of surface coverage should be selected. 3.9 Tuning protein adsorption for practical applications In various applications it is desired that proteins should be in the adsorbed state. For example, in emulsions, foams and other dispersions, adsorbed proteins may be used to stabilize the dispersed particles against coalescence and, possibly, disproportionation. For more information, the reader is referred to chapter 8. In other applications, the adsorbed proteins have to be biologically active. This requirement holds, e.g., for immobilized enzymes in bioreactors and biosensors, immunoproteins in ELISA devices and other diagnostic test kits, and for proteinaceous farmacons in drug targeting systems. As a rule, in order to retain biological activity, the structural integrity of the protein should not be perturbed too much upon binding at the surface. In other cases, where surfaces are brought into contact with protein-containing fluids, the adsorption of proteins at these interfaces should be avoided as much as possible. Adsorption of proteins from (biological) fluids is generally considered to be the first event in the biofouling process. Subsequently, bacterial and/or other biological cells (e.g., blood platelets, erythrocytes) deposit on the adsorbed protein layer. Adsorption-induced conformational changes in the protein molecules usually enhance the 11

T. Vermonden, C.E. Giacomelli and W. Norde, Langmuir 17 (2001) 3734. W. Norde, C.E. Giacomelli, Macromol. Symp. 145 (1999) 125. 31 V.J.C. Lin, J.L. Koenig, Biopolymers 15 (1976) 203. 41 R.J. Jakobsen, F.M. Wasacz, Appl. Spectrosc. 44 (1990) 1478. 21

3.56

PROTEIN ADSORPTION

interaction with the cells. After deposition, microbial cells multiply, forming so-called biofilms. Adhesion of blood platelets at surfaces of cardiovascular implant materials may lead to thrombus formation. Biofouling causes great problems in areas as diverse as biomedicine (implants, catheters, artificial kidneys, contact lenses, teeth, and dental restoratives, etc.), food processing (heat exchangers, separation membranes, etc.) and the marine environment (ship hulls, desalination units, etc.). Knowledge of the mechanism of protein adsorption provides a few clues to control the process, for example by adapting the charge and hydrophobicity of the sorbent surface and the protein molecules, and by selecting environmental conditions such as pH, ionic strength, or temperature. However, practical applications often do not allow much freedom of choice. The composition of a natural fluid is a pre-set condition, and only the surface properties of a (synthetic) material contacting the fluid can be chosen to some extent. Mutatis mutandis, the same is true for immobilized enzymes and immunoproteins in bioreactors and bio- and immunosensors. Moreover, most natural fluids contain a mixture of proteins. Selection of a combination of surface properties with respect to, e.g., charge and hydrophobicity may involve a low adsorption affinity for one protein but a high adsorption affinity for another. A generic approach to influence the magnitude of the interaction between a protein molecule (and also other macromolecules and particles) and a sorbent surface is to manipulate both the long- and short-range interaction forces by grafting soluble polymers or oligomers onto the sorbent surface. Thus, the application of oligomers of ethylene oxide (EO) on polystyrene surfaces leads to the retention of the enzymatic activity of adsorbed a-chymotrypsin, whereas this activity is lost in the absence of such pre-adsorbed oligomers. The short EO chains prevent the enzymes' making intimate contact with the polystyrene surface, without hampering adsorption. As a result, the stress exerted by the sorbent surface on the protein molecule is less, so that the protein's structural integrity, and hence its biological functioning, is less perturbed, as shown in fig. 3.29. Another interesting example is the steering effect of pre-adsorbed polyethylene-oxide (PEO) molecules on the orientation of consecutively adsorbed IgG molecules. IgG molecules are anisotropic, and by realizing proper spacing between the adsorbed PEO molecules the IgG molecules can be forced into the desired orientation at the sorbent surface, i.e., with their antigen binding sites exposed to the solution in which antigens have to be detected. According to this sieving principle, the immunological activity per molecule of IgG adsorbed on a polystyrene surface could be doubled11. By far the greatest part of the recent research on modifying surfaces by grafting

" M . G . E . G . Bremer, Immunoglobulin Adsorption on Modified Surfaces, Wageningen University, the Netherlands, ch. 7 (2001).

PhD Thesis,

PROTEIN ADSORPTION

3.57

Figure 3.29. Temperature dependency of the specific activity of a-chymotrypsin in solution (o), adsorbed on polystyrene (A) and on polystyrene covered with octaethyloxide chains (•). pH = 7.1. (Redrawn from W. Norde and T. Zoungrana, Biotechnol. Appl. Biochem. 28 (1998) 133.)

soluble polymers aims at the prevention of protein adsorption and/or adhesion of biological cells". Because the natural habitat of proteins and biological cells is an aqueous medium the polymers used must be well-soluble in water. In most cases, PEO is used. Sometimes the use of polysaccharides, e.g., dextrans, is reported. The efficacy of the grafted polymers in reducing protein adsorption depends primarily on two characteristics of the polymer layer: (i) the grafting density and, (ii) the extension Into the solution. Despite controversy In the literature data, some trends emerge. As expected, protein adsorption decreases with increasing grafting density of the polymer. Also, as a rule, protein repellency increases with increasing length of the grafted polymer chains. At a grafting density where the separation distance between neighbouring polymer molecules is smaller than twice the radius (of gyration) of the polymer molecule, the grafted polymer chains have to stretch out into the solution. The polymer layer is said to attain a 'brush' conformation. The brush conformation determines the efficacy of protein repulsion. More specifically a few more trends may be mentioned. In the case of a not-too-thick polymer layer and relatively large protein molecules, long range dispersion forces may cause accumulation of protein molecules at the outer edge of the polymer brush. Furthermore, a higher brush density is required to resist adsorption of

11 E.P.K. Currie, W. Norde, and M.A. Cohen Stuart, Adv. Colloid Interface Sci. 100-102 (2003) 205.

3.58

PROTEIN ADSORPTION

Figure 3.30. The influence of the adsorption of BSA by the PEO grafting density and chain length: 700, 445 and - - 148 EO monomers. (Redrawn from ".)

smaller protein molecules. Currie et al.11 reported an even more complex interaction between proteins, i.e., bovine serum albumin (BSA), and PEO brushes. The results of that study are shown in fig. 3.30. At relatively low grafting densities long PEO chains in a brush stimulate BSA adsorption. However, with increasing grafting densities, the adsorption gradually decreases. This result can only be explained by assuming an attractive interaction between the PEO chains and the protein molecules which, in some unknown way, is determined by a combined effect of the length of the PEO chains and the grafting density. These results are supported by data on protein-polymer brush interaction forces, obtained by Norde and Gage21 and by Leckband's group3'41. They report an activation energy up to a few tens of kT for protein (i.e., streptavidin) molecules to penetrate into a PEO brush. However, when by applying a compressive load the activation energy barrier is surpassed, PEO-streptavidin contacts are stabilized by 1-2 kT. It seems that steric and osmotic forces are involved in rejecting proteins from polymer-brushed surfaces, but more subtle interactions such as the conformationdependent hydration of the EO units may be decisive with respect to whether the PEO's layer is protein resistant or not51. Theories describing these phenomena are currently under development. 3.10 General references Physical Chemistry of Biological Interfaces, A. Baszkin and W. Norde, Eds., Marcel Dekker (2000).

11 E.P.K. Currie, J. van der Gucht, O.V. Borisov, and M.A. Cohen Stuart, Pure Appl. Chem. 71 (1999) 1227. 21 W. Norde, D. Gage, Langmuir,20 (2004) 4162. 31 N.V. Efremova, S.R. Sheth, and D.E. Leckband, Langmuir 17 (2001) 7628. 41 S.R. Sheth, N.V. Efremova, and D.E. Leckband, J. Phys. Chem. B104 (2000) 7652. 51 M. Morra, Poly(ethylene-oxide) Coated Surfaces in Water in Biomaterials Surface Science, M. Morra, Ed., John Wiley (2001) ch. 12.

PROTEIN ADSORPTION

3.59

J.P. Brash, P.W. Wojciechowski, Interfacial Phenomena and Bioproducts, Marcel Dekker (1996). (Includes various applications of proteins at interfaces.) Colloidal Biomolecules, Biomaterials and Biomedical Applications, Surfactant Science Series 116, A. Elaissari, Ed., Marcel Dekker (2003). T.E. Creighton, Proteins: Structures and Molecular Properties, 2nd ed., W.H. Freeman (1993). (Textbook) Food Emulsions and Foams, Interfaces, Interactions and Stability, E. Dickinson and J.M. Rodriguez Patino, Eds., Roy. Soc. Chem. (London) (1999). (Role of proteins in stabilizing emulsions and foams, dynamics of proteins at fluid interfaces.) E. Dickinson, Adsorbed Protein Layers at Fluid Interfaces: Interactions, Structure and Surface Rheology, in Colloids Surf. B15 (1999) 161-176. (Review, includes rheology and competition with surfactants.) Food Colloids, E. Dickinson, Ed., Curr. Opin. Colloid Interface Sci. 8 (2003) 346421. (Update, contains various aspects of proteins at interfaces.) C.A. Haynes, W. Norde, Coilloids Surfaces B2 (1994) 517. (Review, emphasizing thermodynamics of protein adsorption.) V.N. Izmailova, G.P. Yampolskaya and Z.D. Tulovskaya, Development of Rebinder's Concept on the Structure-Mechanical Barrier in the Stability of Dispersions, Stabilized with Proteins, in Coll. Surf. A160 (1999) 89-106. (Paper with a review character; emphasis on the mechanical strength of proteinaceous adsorbates.) Y. Lvov, M. Mohwald, Protein Architecture: Interfacing, Molecular Assemblies and Immobilization Biotechnology, Marcel Dekker (1999). (State of the art on protein immobilization, emphasizing biotechnological and biomedical applications.) F. MacRitchie, Chemistry at Interfaces, Academic Press (1990). (Textbook) F. MacRitchie, Proteins at Interfaces in Adv. Protein Chem. 32, G.B. Anfinsen, J.T. Edsall and F.M. Richards, Eds., Acad. Press (1978) 283-326. (Protein adsorption and its effect on biological activity.) F. MacRitchie, Spread Monolayers of Proteins, Adv. Colloid Interface Sci. 25 (1986) 341. (Review, 122 references.) Biopolymers at Interfaces, 2nd ed., M. Malmsten, Ed., Surfactant Science Series 110 Marcel Dekker (2003). (Contains 30 chapters on principles, techniques and applications.) W. Norde, Adv. Colloid Interface Sci. 25 (1986) 267. (Review, 178 references.)

This Page is Intentionally Left Blank

4

ASSOCIATION COLLOIDS AND THEIR EQUILIBRIUM MODELLING

Frans Leermakers, Jan Christer Eriksson and Hans Lyklema 4.1

4.2

4.3

4.4

4.5

4.6

4.7

4.8

Introduction

4.2

4.1a

Aqueous solution amphiphiles

4.2

4.1b

Hydrophile-lipophile balance (HLB)

4.6

4.1c

Critical micellization concentration (cm.c.)

4. Id

Surfactant packing parameter

4. le

Phase diagrams

4.7 4.14 4.16

Classical thermodynamics

4.18

4.2a

Thermodynamics of small systems

4.20

4.2b

The mass action model

4.22

4.2c

Implication for molecular modelling

4.24

4.2d

Fluctuations in micelle sizes

4.27

Molecular modelling

4.29

4.3a

Molecular simulations

4.30

4.3b

Self-consistent field (SCF) theory

4.31

4.3c

Quasi-macroscopic models

4.42

SCF for (spherical) non-ionic micelles

4.47

4.4a

4.49

Micelles at and above the c.m.c.

4.4b

Structure of C p E 5 micelles

4.53

4.4c

Trends for various micellar characteristics

4.56

4.4d

Quasi-macroscopic approaches to non-ionic micelles

4.59

4.4e

Pluronic micelles

4.61

SCF for (spherical) ionic micelles

4.64

4.5a

Micelles at the c.m.c. in various salt solutions

4.66

4.5b

Radial profiles

4.69

4.5c

Chain length dependence

4.72

4.5d

Specific ion effects

4.73

4.5e

Quasi-macroscopic approach to ionic micelles

4.75

Linear growth of micelles

4.76

4.6a

Phenomenological model for rod-shaped micelles

4.77

4.6b

SCF theory of infinitely long linear micelles

4.80

4.6c

The endcap energy

4.83

4.6d

Persistence length of wormlike micelles

4.85

4.6e

Second c.m.c. for ionic surfactants

4.86

Biaxial growth of micelles

4.88

4.7a

Thermodynamic stability of infinite bilayers

4.89

4.7b

Finite size disks

4.91

4.7c

Homogeneously curved surfactant bilayers

4.93

4.7d

On the thermodynamic stability of vesicles

4.96

Interactions between parallel lamellar surfactant layers

4.101

4.8a

Undulation forces between bilayers

4.102

4.8b

Intrinsic interactions between surfactant bilayers

4.104

4.8c 4.9

Liquid crystalline phases

Applications of the modelling

4.107 4.107

4.9a

Binary non-ionic ionic surfactant systems

4.108

4.9b

Solubilization of apolar compounds

4.113

4.10

Kinetic aspects of surfactant solutions near the c.m.c.

4.11

Outlook

4.118 4.120

4.12

General references

4.121

4 ASSOCIATION COLLOIDS AND THEIR EQUILIBRIUM MODELLING FRANS LEERMAKERS, JAN CHRISTER ERIKSSON AND HANS LYKLEMA

In this chapter we consider amphiphilic molecules in a solvent. Amphlphiles are (usually) short chain molecules that have two distinct sides. One moiety of the molecule dislikes the solvent, the other part favours it. When water is the solvent these moieties are called hydrophobic and hydrophllic, respectively. As a result of these interactions, the molecules associate and form objects of mesoscopic size, which are classically called association colloids. Without the process of self-assembly of amphiphiles into association colloids, interface and colloid science would not nearly be as challenging and significant as it is. The association of the hydrophobic (apolar) parts of the amphiphiles gives the driving force for the assembly (hydrophobic bonding). These fragments form the cores of such aggregates. The hydrophilic (polar) sides of the amphiphilic molecules accumulate at the water-core interface, forming the corona. The formation of the corona eventually counteracts the association process, and when finite objects are formed these are known as micelles. Micelles are prominent members of the colloid family and they typically introduce a large interfacial area to a system. Micelles are thermodynamically stable and it is possible to apply the powerful machinery of statistical thermodynamics to describe their self-assembly. The goal of such analyses is to understand such features as the micelle size, shape, the (in)stability against dilution and the influence of various physicochemical parameters in relation to the molecular architecture, as well as the solvent properties. In this chapter we will review the fundamental aspects of association colloids with a focus on the understanding of the underlying physicochemical phenomena. We cannot review all aspects of association colloids because the huge volume of experimental literature, the many implications to countless applications and the detailed modelling of generic and molecular-specific issues simply defy condensation into one coherent and concise text. Here we choose for an approach that largely relies on self-consistent field modelling. We like to show that, within one set of approximations, it is possible to cover a wide range of properties of these interesting systems. In particular, we will elaborate on the way in which modern computational methods can help to gain a better understanding of association colloids. Computer modelling cannot be regarded as the Fundamentals of Interface and Colloid Science, Volume V J. Lyklema (Editor)

© 2005 Elsevier Ltd. All rights reserved

4.2

ASSOCIATION COLLOIDS

end point of investigations. We therefore also pay close attention to various aspects of analytical quasi-macroscopic approaches. 4.1 Introduction Amphiphilic self-assembly introduces (liquid) hydrophobic domains in an aqueous solution. The sizes and shapes of the micelles depend strongly on the nature and concentration of the surfactant and other physicochemical conditions (pressure, temperature, ionic strength, additives) and determine in large the many applications. The use of amphiphiles in, for example, food processing or as a lubricant probably dates back to the beginning of mankind. The link between the physical properties and the molecular structure is much more recent. The term "micelle" dates back to McBain who referred to it in a Faraday Discussion remark" in a General Discussion on colloids and their viscosity (!). Early on the shape of small micelles was disputed, partly as a result of interpretational problems with X-ray data, but nowadays spherical micelles are considered as the prototypes for surfactant aggregates. Besides McBain, Hartley21, Debye31, and Stigter41, many others have in the early stages contributed to their understanding. In this connection it may also be mentioned that monolayers of surfactants at waterair interfaces have served as models for determining various parameters important for micellization. Recall Benjamin Franklin's seminal spreading experiments of oil on Clapham Lake (introduction of chapter III.3) and assessing molecular areas from j'(logc) curves for Gibbs monolayers (c.f. sec III.4.6d). 4.1a Aqueous solutions of amphiphiles Amphiphiles can hardly be discussed without explicit reference to the solvent in which they are dispersed. In order to have stable micellar structures, the solvent should be selective in the sense that it should be poor for one part of the molecule and good for the other. Self-association of surfactants in apolar media results in so-called reverse micelles with a polar core and apolar corona. These systems are close to water-in-oil microemulsions. More specifically, it is possible to tune the solvent selectivity, and thus the micellar structures, by using supercritical solvents. Space does not allow us to visit these systems (see e.g.51 for micellization in non-aqueous and mixed solvents). In this chapter the main focus is on aqueous systems. The hydrogen bonding capabilities of water makes it a highly associative liquid. 11

J. McBain, Trans. Faraday Soc IX (1913) 99. G.S. Hartley, Aqueous Solutions of Paraffin Chain Salts, Hermann (1936). 31 P. Debye, J. Phys. Coll. Chem. 53 (1949) 1; Ann N.Y. Acad. Set 51 (1949) 573. 41 D. Stigter, Rec. Trav. Chim. 73 (1954) 593; J.Th.G. Overbeek, D. Stigter, Rec. Trau. Chim. 75 (1956) 1263. 51 J. Eastoe, A. Dupont, and D.C. Steytler, Current Opin. Coll. Interf. Set 8 (2003) 267. 21

ASSOCIATION COLLOIDS

4.3

Molecules, or parts of them that are not able to take part in this hydrogen bonding network, are likely to be rejected by water. That is why oil and water do not mix. The hydrogen atoms in hydrocarbon molecules or tails are too strongly bonded to the carbon atoms to take part in hydrogen bonding. Hydrocarbon tails are therefore excellent entities to make the hydrophobic parts of surfactants. The longer the tails, the stronger the demixing with water and the stronger the driving force for micelle formation. In other words, the Helmholtz energy of mixing hydrocarbons in water is very unfavourable. It amounts to an unfavourable exchange energy of order kT per -CH 2 unit. There are many names associated with subsets of amphiphiles. Such names reflect some feature of the molecular structure or some physical property they represent or are coupled to a typical application. The name "surfactant" refers to the ability to adsorb strongly onto many surfaces while reducing the interfacial tension. Nowadays the keyword surfactant is used as a generic term. However, there are many other molecules that are surface active but which are not amphiphiles; homopolymers adsorb strongly onto almost any interface (see chapter II.5) and ions to (charged) mercury surfaces. If amphiphiles are used as a means to solubilize apolar compounds in an aqueous medium, they may be named emulsifier or they are called foaming agent if they stabilize thin liquid films. Surfactants are wetting agents when promoting, for example, the spreading of crop protection agents on leaves. If the surfactant is used to stabilize apolar particles in water (or a polymer matrix), the term compatibilizer is sometimes used. Amphiphiles are known to increase the solubility of other organic substances in water. Certain sub-classes of surfactants also have dedicated names. Substances that co-adsorb, or co-micellize, and in this way promote the activity of the surfactant, are called linkers, or, when in aqueous solution, hydrotopes. A class of its own is the set of lipid molecules, which are the main components of biological membranes. Molecules that have a hydrocarbon tail and a hydrophilic head group are the classical examples of a micelle-forming surfactant. When the head group is charged, we have the sub-class of ionic surfactants. Ionic surfactants are among the most prominent surfactants in aqueous media. We distinguish cationic and anionic surfactants. Sodium dodecyl sulfate (NaDS) or (sodium lauryl sulfate) is the best-known anionic surfactant. This surfactant is almost always polluted by dodecylalcohol, which is its hydrolysis product. Cetyltrimethylammonium bromide (CTAB) is a typical example of a cationic surfactant. We will see below that such micelles are strongly dependent on added indifferent electrolyte and that the kind of counterion is also relevant in these systems (see table 4.1). The phase diagrams of surfactants may differ significantly depending on the hydrophilicity or size of the counterion. Zwitterionic surfactants have both a negative and a positive charge in the head group. These are less sensitive to added salt. A snapshot of the structure of a small ionic micelle, as generated by molecular

4.4

ASSOCIATION COLLOIDS

Figure 4.1. a) Computer (MD) generated snapshots of an ionic micelle composed of cesium pentadecafluorooctanoate significantly deviating from the spherical structure". The dark spheres are the C-units (the fluor atoms are omitted, that is why one can "look through" the structure; the CH2 unit is somewhat more polar than the CF2 one), the light spheres are the oxygen of the carboxylic head group, the counterions (Cs) are dark gray spheres, b) MD snapshot of a spherical non-ionic micelle composed of C 12 E 6 surfactants21. The core is made up of densely packed alkyl tails (black, space-filling spheres). The lighter gray hairs are the EO fragments forming the corona. Water molecules are not shown, c) Schematic cross-section of a Na-dodecylsulphate micelle. The large spheres with a minus sign are the OSOg groups; the compensating countercharge is indicated by + and - signs, d) Schematic cross-section of a nonionic micelle, the core of densely packed alkyl chains are thin line parts, the ethylene oxides are the fat line parts. dynamics simulations, Is given in fig. 4.1a. In principle, we are interested in the average behaviour of surfactant assemblies and therefore it is always dangerous to discuss snapshots. Apart from this, we may be amazed at this stage about the complexity of the molecular assembly and the apparent importance of fluctuations (chain conformations, micelle shape, etc.). From the above, it follows that a CH2 group is a classical (but not the only) hydrophobic building unit. On the other hand, an oxygen atom in a larger molecule can 11 21

S. Balasubramanian, S. Pal, and B. Bagchi, Current Sci. 82 (2002) 8456. F. Sterpone, C. Pierleoni, G. Briganti, and M. Marchi, Langmuir 20 (2004) 4311.

ASSOCIATION COLLOIDS

4.5

accept a H-bond from water and is a typical hydrophilic unit. It is possible to combine these two in a regular fashion. The methyleneoxide (MO) chain -(CO-) n , where the Hatoms around the C are dropped for convenience, is the first member. Ethyleneoxide (EO) chain -(CCO-)n is the second; the propylene oxide (PO) chain -(C[CH3]CO-)n , where the CH3 branches off from the main chain, is the third compound. These homopolymers are unable to form micellar-like objects on their own because the amphiphilicity on the monomer length scale is insufficiently expressed. In this series, only the EO member mixes readily with water. The other two are not miscible in all proportions with water. A detailed understanding of this is still missing. Apparently the dimensions of the water H-bonding network "match up" with ethyleneoxide, but not sufficiently well with the other two. With this information, it is possible to design non-ionic surfactants. Combining sufficiently long aliphatic (hydrocarbon) tails and EO head groups leads to surfactants, which are nowadays available in high purity and with high homodispersity. The general constitution is C n E x , where E stands for EO. Results for these systems from before the 1970s should be regarded with some caution because at that time the products were rather polydisperse, especially with respect to the EO parts. When both n and x values are very small, we have weak surfactants. A typical example for a reasonably strong surfactant is for n = 12, x = 6 . I n this case, the lengths of the apolar and polar parts are about equal. A computer-generated example of the spatial structure of a spherical micelle composed of the C 12 E 6 surfactants is given in fig. 4.1b. Again, this example may serve to sharpen our intuition about these molecular aggregates. In this chapter our attention will be focused on strong surfactants, i.e. surfactants that form micelles at low surfactant concentrations. The schematic drawing of fig. 4. Id is derived from fig. 4.1b. Of course there are many polymeric amphiphiles, e.g. Pluronics, which have PPO as the apolar and EO as the polar moiety. The diblock copolymers are the polymeric analogues of the classical surfactants, but there are many other options. For example, one can have amphiphilic side chains also known as polysoaps. There exist protein molecules, such as caseins that have emulsifying properties. They have a rather apolar domain, but also a polar one, and further contain a conformationally disordered fragment that is highly solvated. This shows that amino acids can also serve as the surfactant building blocks. Polysaccharides are yet another class of water-soluble compounds. Linking aliphatic tails to a short string of these molecules leads to environmentally friendly, biocompatible amphiphiles ["sugar surfactants"). Surfactants based on n-alkyl chains generally form micelles for which the core is densely packed, c.f. fig 4.1b. In this case, the dimension of the apolar part of the surfactant molecule is a fundamental property. For this reason, it is evident that relatively small deviations from the linearity, which leads to more compact but less regularly packed tails, may have important consequences for self-assembly. We may consider a given polar head group connected to various isomers of a given apolar

4.6

ASSOCIATION COLLOIDS

constituent. The chain can be branched, it can be split into two or more subchains connected to the head, etc. The micelles composed of these isomers will differ noticeably and systematically. For example, when two ionic surfactants are coupled by a short bridge {spacer) at the position of the head groups, we have twin surfactants also called gemini. The name was coined by Menger et al.1'. In recent years the study of gemini surfactants has seen increasing interest, because it was shown that the micellization properties can be nicely varied by, for instance, controlling the length of the spacer. With the spacer, one can insert an extra steric contribution to the packing of head groups in the corona region. For overviews see refs. 23) . This again points to the fact that molecular structure is important for self-assembly. Alkyl chain lengths on the order of C16 or longer occur frequently in nature, as in lipid molecules. One then has to be aware of the fact that densely packed layers of these molecules will not necessarily be in a liquid-like state at room temperature. If linear, they can give rise to liquid-crystalline ordering. Unsaturated and branched segments in the chain will frustrate crystallization, and thus the degree of saturation is yet another parameter that should be taken into account, especially when fatty acid or mono or di-glyceride molecules are considered. An interesting, and in biological systems relevant example, is the possibility of semirigid molecules in which connected C5 and or C6 rings are grouped together, such as in cholesterol and steroids. Such molecules may have polar and apolar substituents. When the polar ones are predominantly on one face of the molecular plane and the apolar ones on the other side, we obtain facial amphiphiles that form aggregates of finite size that are distinctly different from the micelles that are discussed below. The above set of variables that are relevant for amphiphilic molecules is by no means complete, but we trust that the examples given suffice to indicate the vast scope. 4.1b Hydrophile-lipophile balance (HLB) From the above, it is clear that one can, at least in principle, generate series of surfactants in which the apolar part (usually called the tail(s)) and the polar part (usually called the head) systematically vary in size. This is particularly possible for the nonionic polymeric amphiphiles. Within a series, it is possible to rank them according to the polar/apolar ratio or the hydrophile/lipophile balance (HLB value). The empirical HLB notion was developed by William C. Griffin41 in 1949 who proposed to compute the HLB value as the molecular weight percent of the water-loving portion of the surfactant divided by 5. Experience showed that if a surfactant has an HLB = 1, it is very oil-soluble, however a surfactant with an HLB = 15 is water-soluble. Surfactants

11

F.M. Menger, C.A. Littau, J. Am. Chem. Soc. 113(1991) 1451. R. Zana, Adv. Colloid Interface Set 97 (2001) 205. 3 R. Zana, in Novel Surfactants: Preparation. Applications and Biodegradability, K. Holmberg, Ed., Marcel Dekkcr (1998) 241. 41 W.C. Griffin J. Soc. Cosmet. Chem. 1 (1949) 311. 21

ASSOCIATION COLLOIDS

4.7

with an HLB = 1-3 may be used to mix unlike oils, that water-in-oil emulsions can form with surfactants that have an HLB = 4-6, that one can compatibilize small particles in oil using surfactants with HLB = 7-9, that surfactants with HLB = 7-10 may be used to form self-emulsifying oils, that blends of surfactants in the range HLB = 8-16 can be used to make oil-in-water emulsions, that detergent solutions require surfactants with HLB = 13-15 and that surfactant blends with HLB = 13-18 may be used to solubilize oils (and form microemulsions) in water. It is also very useful to know that for making oil-in-water emulsions, the combination of the oil and the suitable HLB value changes from HLB = 6 for vegetable oils, HLB = 8-12 for silicone oils, HLB = 10 for petroleum oils and HLB = 14-15 for fatty acids and alcohols. In conclusion, the HLB notion has been, and still is11, quite useful for formulation purposes. However, the HLB value does not directly relate to the efficiency to form micelles, and for modelling purposes the HLB value is not a prominent quantity.

4.1c Critical micellization concentration (c.m.c.) Experience has shown that micelles only form above a threshold concentration, often referred to as the critical micellization concentration (c.m.c), which is for any surfactant system the most characteristic physical quantity. The lower the c.m.c, the more efficient the surfactant is to adsorb onto various surfaces. For applications in which the presence of micelles is needed, one obviously should choose a concentration above the c.m.c. The issue of sharpness of the c.m.c. will receive detailed attention below. The appearance of a c.m.c. is easily explained borrowing the ideas of macroscopic phase separation of, for example, oil and water, which also gives a rough estimate of the c.m.c. To first order, the surfactants are at low concentrations kept in solution as freely dispersed molecules because the translational entropy can (over)compensate the unfavourable local interactions the tail has with the solvent. With increasing concentration, the translational entropy per chain diminishes and, at a (fairly) well-defined point, it becomes more favourable to form a dense phase of surfactants. Obviously, the fact that objects of finite size are formed proves that such a phase separation model is flawed and more sophisticated models have to be introduced. Anticipating these extensions we nevertheless note that one can easily show, for instance, using the Flory-Huggins (FH) theory, that the maximum solubility (binodal) of oils in water (strong segregation) decreases exponentially with the chain length of the oil IV in line with experimental findings21. Below (see table 4.1) we will see that the c.m.c. indeed has this trend, proving the fact that the demixing of tails and water is crucial for determining the c.m.c. More importantly, we may use the FH theory to find our first estimate of the c.m.c.

11

X.F. Li, H. Kunieda, Curr. Opin. Coll. Interf. Sci. 8 (2003) 327. P.J. Flory, Principles of Polymer Chemistry, Cornell University Press (1953); C. Tanford, The Hydrophobic Effect. Formation of Micelles and Biological Membranes, 2nd ed. Wiley and Sons Inc. (1980). 21

4.8

ASSOCIATION COLLOIDS

The shortcomings of the simplistic phase separation model should not only be attributed to the fact that in the FH theory the solubility characteristics of the head groups are not included. Indeed, it is possible to set up a more elaborate FH-like theory for copolymers and also introduce interaction parameters for the head groupwater contacts, as well as for the head group-tail contacts. Such extended FH theory will also predict some trend of the c.m.c. as a function of the head group properties. Although such an approach is useful because the c.m.c. is reasonably accurately predicted, the theory remains fundamentally wrong. In such a simplistic phase separation picture (oil-water demixing), the chemical potential of the oil is an increasing function of the oil concentration only in the onephase regions. In the two-phase region the chemical potentials are fixed to the binodal values. Above the c.m.c. (corresponding to the two-phase region in oil-water systems), association colloids of finite sizes are formed, which has many important consequences. One of them is that the chemical potential of the surfactant is no longer fixed, but continues to increase with the micelle concentration. The key property that fundamentally changes the above picture is that the aggregates that are formed do not grow to macroscopic size but remain mesoscopic. There are stopping mechanisms, which limit the growth to be discussed, in more detail below. As finite size effects turn the step-wise phase transition into an apparent first-order transition, it is quite obvious that the sharpness of the micelle formation depends on the number of monomers in the micelle and the distribution over micelle sizes. One other consequence is that classical phenomenological thermodynamical arguments are not enough to understand micelle formation. This means that we should carefully consider how to properly use classical thermodynamic arguments in relation to the special model notions invoked. (i) Determination of the c.m.c. As c.m.c.s are characteristic properties of surfactant solutions, which indicate when micelles start to form, it is important to measure them. Such measurements can only be carried out with satisfactory precision when the c.m.c. is sufficiently defined, i.e. provided that a well-defined surfactant concentration exists above which micelles suddenly appear when the concentration is increased. When there is such a sharp c.m.c, a variety of physical properties, measured as a function of increasing surfactant concentration, show a break at the c.m.c, these breaks often coincide within experimental error. Figure 4.2 gives an illustration. We shall now briefly discuss these methods. One of the most obvious ways to probe micelle formation is to shine light on them because the scattering increases strongly with the volume of the scatterer (chapter 1.7). The shorter the wavelength of the light, the more details one can see. That is why Xrays or neutrons are better equipped to obtain insight into the internal structure of the micelles than visual light. Dynamic light scattering may be used to find the hydrodynamic or diffusion radius of the micelles. When the form factor and the structure factor are both playing a role in scattering experiments, one would prefer to construct a

4.9

ASSOCIATION COLLOIDS

Figure 4.2. Schematic figure for the dependence of some physical measurables on the surfactant concentration near the c.m.c. On a log
Zimm-plot (see fig 1.7.12). In principle this would give a direct measure of the overall mass of the micelle, its aggregation number and the second virial coefficient; the latter is a measure of the strength of interaction between the micelles. To make a Zimm plot, however, one has to change the concentration of micelles, e.g. by diluting the micellar system. However, one should be aware of the fact that in general micelles respond by making changes in the aggregation number when they are diluted. For this reason, one should be careful and restrict the dilution to regimes where size changes may be neglected. In any case, static light scattering can be of use to detect the c.m.c. because it reacts to the volume of the particles. Neutron scattering, as well as X-ray scattering, are excellent tools to study the sizes and shapes of micelles in terms of the structure factor. Extensive studies on well-defined surfactants with partially deuterated samples (for neutron scattering) have revealed much insight into the structure of micelles11. Measuring the interfacial tension as a function of the total surfactant concentration provides another frequently used method. When the interfacial tension of an air-water interface to which the surfactants adsorb is plotted as a function of the logarithm of the total surfactant concentration, and not as a function of the chemical potential (logarithm of the monomer concentration), one finds a break in the curve. This "break" is positioned where the overall surfactant concentration starts to deviate from the monomer concentration; in practice it will coincide with the c.m.c. One may use Gibbs' law to extract from the slope of the surface tension as a function of the surfactant chemical potential the excess adsorbed amount of surfactants at the interface, which is the inverse of the area per surfactant molecule a m . This is an essential parameter to make a rough estimate for the surfactant packing parameter P as given by [4.1.4]. The method has also been used to determine salt concentration effects on the c.m.c. Moreover, it is well known that 3y/31nc surf gives am I p , where p ranges from unity (high ionic strength) to 2 (no added salt). See sec. III.4.6d. There are also some practical problems. When, for example, an anionic surfactant is used one should be 11

J.R. Lu, R.K. Thomas, and J. Pcnfold, Adv. Colloid Interface Sci. 84 (2000) 143.

4.10

ASSOCIATION COLLOIDS

aware of traces of multivalent counterions in the system. These multivalent ions may coadsorb with the surfactant onto the interface, which affects the value of a m . Another familiar problem is that some minor admixtures may give rise to minima around the c.m.c. Space does not allow us to go into all the details here. Colligative properties react on the number of kinetic units and hence are sensitive to micelle formation. If a large number of surfactants group into one micelle, only the micelle (and the mobile fraction of the counterions) has full translational entropy. If one normalizes the osmotic pressure with the overall concentration of surfactant, we expect (for ideal systems) to find a constant value. However, as soon as micelles form, the osmotic pressure does not nearly go up as fast as expected for monomers and the normalized osmotic pressure goes to small values. However, osmotic measurements are tedious, therefore making this method not particularly attractive. Micelles may be used as carriers for hydrophobic molecules. We can detect the existence of micelles by measuring the capabilities of a surfactant solution to solubilize these apolar entities. Obviously, in this case one will not measure the c.m.c. of a pure surfactant system, but the c.m.c. of a mixed system. We will study solubilization in sec. 4.9b. For charged surfactants, one can use conductivity measurements to determine the c.m.c, at least when not too much extra salt is added. At low c s u r f , the molar conductivity is ideally concentration independent (in practice activity corrections cause them to be proportional to l/^c s u r f ). However, at the c.m.c. micelles appear and these micelles contribute differently to the conductivity and thus the molar conductance differs below and above the c.m.c. There are two reasons for this. First, one may expect that a micelle in which there are g surfactants will carry the charge more efficiently than g individual surfactants simply because the drag normalized per segment is less. On the other hand, counterion binding to a surfactant in a micelle is much stronger than that to isolated surfactant ions, therefore it turns out that a substantial fraction of the counterions are rather closely bound to the micelle so that the effective charge that is carried is much reduced. The overall result is a jumplike change in the slope of the molar conductivity as a function of
ASSOCIATION COLLOIDS

4.11

the overall system. For these techniques, the extent of counterion binding is relevant. These techniques thus tend to indicate the condition where the concentration of freely dispersed surfactants is approximately equal to that of micelles. Practice has shown that for pure surfactants in aqueous solutions, c.m.c.s, obtained by different techniques are within experimental error, sufficiently close to speak of THE c.m.c, which can therefore be analyzed in terms of physical principles. Small systematic differences may have a methodical origin because the micellar size distribution has in principle a certain width: it would be entropically very unfavourable to force all micelles to adopt the same aggregation number g. Different techniques may well react differently on micelles of differing sizes and therefore give a slightly different c.m.c. When experiments yield aggregation numbers, basically an average is obtained, which is different for different techniques. Colligative experiments, such as the osmotic pressure, give a number-averaged aggregation number; light scattering, on the other hand, gives a weight average. Rheology gives yet another average (see [IV.6.11.9]). The sharper the distribution is, the closer the aggregation numbers obtained by different techniques. (il) Windows for micelle formation The temperature sensitivity of micellization, both for ionic as well as non-ionic systems, is of great relevance for many applications. The c.m.c.s of ionic surfactants depend weakly on temperature and the obvious control parameter in these systems is the ionic strength. However, non-ionic surfactants, particularly those of the C n E x type, are much less sensitive to the ionic strength, but are much more temperature sensitive. In contrast to the common trend that molecules become more soluble with increasing temperature, the EO-based surfactants show the opposite trend. As a result, here the temperature is the main control parameter. Typically, such non-ionic micellar solutions rather abruptly become very turbid at a given temperature, known as the cloud point (c.p.), indicating the formation of large aggregates in the solution. We will pay attention to this phenomenon in sec. 4.4. The cloud point depends on the nature and concentration of the surfactant. We will return to this when we discuss phase diagrams (cf. sec. 4. le). For various (typically non-ionic) surfactants, there is yet another relevant temperature known as the critical micellization

temperature

(c.m.t.). When we follow

surfactant solutions at sufficiently high concentration, we may find micelles only above some rather well-defined temperature. In particular the temperature-sensitive surfactants of the Pluronic type show this phenomenon. Upon decreasing the temperature, one may lose the presence of micelles rather abruptly. Apparently, the solubility of the surfactant as a whole becomes too high to allow the formation of stable micelles in solution. The location of the c.m.t. depends on the surfactant concentration. We will discuss the c.m.t. when we consider non-ionic surfactants in more detail in sec. 4.4.

4.12

ASSOCIATION COLLOIDS

The c.p.t. and c.m.t. usually bracket the temperature window in which the non-ionic surfactant system may be used. For charged surfactants, the ionic strength is the main control parameter. In principle one can expect that upon varying the ionic strength there exists a window in which the micelles are colloidally stable. Indeed, in combination with temperature variations there can exist special conditions where this stability is lost. In this context, Table 4.1. Selected c.m.c. values1'21. Surfactant

Formula

Temp.°C

c.m.c./M

ref.

anionic sodium octyl sulphate sodium decyl sulphate sodium dodecyl sulphate sodium tetradecyl sulphate lithium dodecyl sulphate

C8SO4Na+ C10SO4Na+ C12SO4Na+ C14SO4Na+ C^SOjLi"1"

sodium dodecyl sulphate

C^SO^Na"1"

potassium dodecyl sulphate

C12SOjK+

rubidium dodecyl sulphate

C12SO4Rb+

cesium dodecyl sulphate

C^SO^Cs*

25 25 25 25 25 33 40 25 33 40 55 33 40 33 40 33 40

1.33 xlO" 1 3.33xlO- 2 8.2-8.3xlO" 3 2.1xlO" 3 8.68 xlO" 3 4.46xlO- 3 8.25X10- 3 8.2-8.3X10" 3 8.10xl0" 3 8.05X10-3 9.85xlO- 3 6.71x10-3 6.90x10-3 5.90x10-3 6.01xl0- 3 5.90X10-3 6.10X10-3

1 1 1 1 2 2 2 1 2 2 1 2 2 2 2 2 2

C12TMA+Br-

25

1.25xlO- 2

1

C,8TMA+Br-

25

3xlO" 4

1

C12Py+Br"

30

1.18x10-

1

65

1.63x10"

1

25 25 25 25 25 25 25 25 25 25

0.33 xlO~4 0.55xl0" 4 0.39 xlO" 4 0.64 xlO" 4 0.8 xlO" 4 l.OxlO- 4 9.0xl0- 3 0.87 xlO" 4 lxlO- 5 1.25xlO-4

2 2 2 2 2 2 2 2 2 2

cationic dodecyl trimethyl ammonium bromide octadecyl trimethyl ammonium bromide dododecyl pyridinium bromide linear non-ionics

C

12E2 C 12E3 C 12E4 C 12E5 C 12E6 < -'12E8 C 10E9 C 12E9

Ci 4 E 9 C

11 21

12 E 12

Mukerjee-Myscls compilation, see sec 4.12c. Taken from compilations in FICS III, tables 4.4. 4.5 and 4.6 (sec 4.6).

ASSOCIATION COLLOIDS

4.13

the Krqfft point should be mentioned. The Krafft temperature is the temperature below which the surfactants enter a two-phase region: a concentrated surfactant phase (often a lamellar phase) coexists with an aqueous solution dilute in surfactants. (ili) Experimental trends for the c.m.c. In table 4.1, a number of c.m.c.s are collected. Notwithstanding the fact that different authors (and/or) different methods) may find (or lead to) slightly different values, the list is sufficiently established to deduce the following convincing trends. (i) The c.m.c. decreases rapidly with the increasing length of the hydrocarbon chain. This is a general feature applying to all classes of surfactants; obviously it reflects the decreasing monomer solubility in this direction. Experience has shown that for each surfactant logc.m.c. = A-Bt

[4.1.1]

if t is the length of the hydrocarbon chain. Here A is a quantity of dimensions loglCg]"1, where c 0 is the extrapolated reciprocal c.m.c. for t —> 0 . This quantity has of course no physical meaning, although it is certain that A depends on the nature of the head groups only; it is more negative for non-ionics than for ionic surfactants. The constant B is about 0.3 for ionics and 0.5-0.6 for non-ionics11. It means that adding a CH2 group to the hydrocarbon chain reduces the c.m.c. by a factor of about two or three for ionics and non-ionics, respectively. The inference is that the Gibbs energy of transporting a CH2 -group from a chain in solution to that in a micelle depends on the charge of the head group. Below we will show that this effect can be explained taking into account that the ionic strength is not kept the same for all the systems considered. (ii) (Not shown in the table). The constant B also depends slightly on the nature of the head group. Branching of the chain increases the c.m.c, apparently because branching frustrates the packing, and one CH2 is sacrificed whereas one CH is added and the number of CH3 end groups increases. (iii) For ionics and non-ionics of the same tail length t, the c.m.c. is much lower for the latter. On the one hand, the ionic double layer around the micelles opposes micellization, whereas in non-ionics the repulsion between the EO fragments plays a much weaker role. The c.m.c.s reported in table 4.1 are without added salt. This means that the surfactants themselves provide the ionic strength in solution. The higher the c.m.c, the higher the ionic strength at the c.m.c. The opposing electrostatic force depends on the ionic strength. (iv) In line with the previous item, the addition of electrolytes reduces the c.m.c. because it decreases the double layer Gibbs energy. Empirically21,

11

C. Tanford, The Hydrophobic Effect. Formation of Micelles and Biological Membranes, 2nd ed. Wiley (1980). 21 H.F. Huisman, Proc. Koninkl. Nederl. Akad. Wetenschap B67 (1964) 367.

4.14 logc.m.c. = C-Dlogc s t o t

ASSOCIATION COLLOIDS [4.1.2]

where c s tot is the ionic strength, i.e. it contains the contribution of the surfactant c surf ~ c m c - a n f i t n a f of the added salt c s logc.m.c. = C-Dlog(c.m.c. + cs)

[4.1.3]

Here, C depends on t as in [4.1.1]. (v) For non-ionics, there is little influence of the EO length on the c.m.c. (vi) The counterion effect (lyotropic or Hofmelster series) is less than that of the presence of charges, but the trend is clear. On the sulphate group, specific binding increases from Li+ to Cs+ . This sequence is the same as that found for poly(styrene sulphonate)-latices (sec. IV 3.13c) and for Gibbs monolayers of NaDS (sec. III. 4.6d) and is in line with Pearson's rule (sec. IV.3.9i). In the same vein, for cationics with dodecyl trimethyl and dodecylpyridinium groups the binding also increases with the radius of the counterion, i.e. from Cl~ to I~ . Many Illustrations can be found in the Mukerjee-Mysels (toe. clt.) compilation. (vii) The Influence of the temperature is small except for non-ionics over large temperature ranges. There are indications that the c.m.c. passes through a minimum; when this minimum is close to the ambient temperature, as for many ionics, the effect is expected to be small. A minimum is in line with hydrophobic bonding as the driving mechanism. (viii) The combined effect of the nature of the head group and the affinity of the counterion is clearly visible; from table 4.1 it follows that the c.m.c. of C12TMA+Br" exceeds that of C,2SO^Na+ by a factor of about 1.5. The latter micelles form more easily. In conclusion, the tabulated data already give a first exploration of the properties of micelles. Obviously, a satisfactory theoretical treatment should account quantitatively for these observations. 4.Id. Surfactant packing parameter We mentioned above that the hydrocarbon-water system is in the strong segregation limit. This means that in the micelle the hydrocarbon tails are necessarily densely packed. Ninham and Israelachvili11 were the first to realize that for systems in which packing effects dominate one can predict to first order the physical chemical properties from a simple geometric formula featuring the (surfactant) packing parameter P also sloppily called the "surfactant parameter". We will introduce this parameter by assuming an idealized picture of a micelle. Most of the modelling afterwards will provide a justification of this picture. The first basic ingredient is that the core is a densely packed set of surfactant tails. The characteristic size of the core is given by the length

11 J.N. Israelachvili, D.J. Mitchell, and B.W. Ninham, J. Chem. Soc Faraday Trans II, 72 (1976) 1525.

ASSOCIATION COLLOIDS

4.15

Figure 4.3. Pictorial representation of various micellar topologies: the spherical micelle (left top), cylindrical micelle (left bottom), lamellar topology (middle), the cubic phase (right top) and inverted micelles (right bottom), redrawn from ref. .

of the surfactant tall I. The second Ingredient is that the volume v occupied by the tall(s) of the surfactant in the core is identical to the volume that a corresponding alkane occupies in the neat alkane phase. To first order, this volume Is proportional to the number of CH2 and CH3 groups in the surfactant tail. The third ingredient Is that the head groups cover the core such that the area per molecule am is similar to that found by the surface tension measurements at the c.m.c. (see: tables III.4.5 and 6). The combination of these quantities defines the dimensionless surfactant packing parameter [4.1.4] When P ~ 1/3 , the preferred micelle size is spherical. Cylindrical micelles are expected when P = 1/2 . Flat bilayers will dominate for P ~ 1 and reversed micelles may be found when P > 1. Here the values of 1/3, 1/2 and 1 are found from the combination V/(RA) where V is the volume, A the area and R the characteristic length in a homogeneously curved object. The surfactant packing parameter features two reasonably well-defined quantities, viz. the molecular constant / and a value v that is expected to be constant. The third parameter a m is not a constant because it depends in general on the physical chemical parameters in the system, such as the ionic strength. Moreover, this quantity is expected to vary weakly with curvature. The weakness of this surfactant packing parameter concept is thus that formally P is not a constant. 11 D.F. Evans, H. Wennerstrom, The Colloidal Domain where Physics, Chemistry, Biology and Technology Meet (1994).

4.16

ASSOCIATION COLLOIDS

We note, however, that the surfactant packing parameter incorporates various assumptions about the properties of surfactant micelles. Most of these will be subjected to molecular modelling: (i) Fluctuations of the positions of the head groups are small. To avoid a density drop in the centre of the micelle, at least one of the dimensions of the micelle should be proportional to (not exceeding) the length of the surfactant (. (ii) The density of the core is homogeneous and similar to that for the corresponding alkane phase. This also implies that little water will disperse in the core, (iil) The area per surfactant molecule is a well-defined quantity that is not a strong function of the curvature of the core. In other words, the stopping mechanism responsible for the finite aggregation number in micelles is similar to the stopping mechanism for accumulation of surfactants at the air-water interface. The fact that in the micelle there is no air-water interface introduces only second order effects. Most of these aspects will be considered in more detail below, and we will see that the fluctuations of the head groups are usually not small, that the area per molecule varies systematically with the geometry of the micelle and that the radius of the spherical micelle is larger than the cross-section radius of the cylindrical micelle, which in turn is larger than half the bilayer thickness. In fact, this implies that the apparent success of the packing parameter is not trivial. However, this concept remains rather powerful as a tool to estimate the first-order effects. Using this simple concept, one predicts that the best way to vary the surfactant parameter, and hence the structure of micelles formed by the surfactant, is mostly not to increase the tail length at given head group properties (as v <*• I, and a m does not depend much on the tail length, we expect that P varies only marginally), but rather, one should modify the area per molecule, i.e. by changing the degree of polymerization of the EO moieties in non-ionic surfactants or the ionic strength and nature of the counterion (c.f. table 4.1) in the case of ionics. Another very effective option is to change the architecture of the hydrophobic group. Splitting up a single chain into two will keep v constant but reduces I by a factor of two, which increases P by a factor of two. Indeed, this explains why single chain surfactants tend to form spherical micelles whereas double chain surfactants are more likely to form lamellar bilayers. Chain branching is also very effective in changing the packing parameter. 4.1e Phase diagrams We have stated that micelle formation is roughly related to the first-order phase separation transition between two immiscible liquids (oil and water). However, the finite sizes of the micelles give the system very distinct and unique characteristics. In sec. 4.2b we will prove that upon increasing the overall surfactant concentration above the c.m.c. the chemical potential of the surfactant does not remain fixed but continues to be a very weakly increasing function of the surfactant concentration. As a result, it is expected that upon increasing the overall surfactant concentration there are variations in the micellar size and shape. One can probe these changes and construct a phase

ASSOCIATION COLLOIDS

4.17

diagram. A plethora of such phase diagrams can be found in the literature ". By way of illustration, in fig. 4.4 we present a schematic phase diagram representing the non-ionic surfactant C 12 E 5 . On the abscissae axis we have the temperature, which is the major control parameter for this surfactant, whereas on the ordinate axis the surfactant concentration is plotted in roughly logarithmic co-ordinates. It is important to note that the c.m.c. is at rather low concentrations and various noteworthy events in the phase diagram only occur when the surfactant concentration is very high. There are many generic effects in surfactant phase diagrams. Space does not allow us to go into all the details. We just briefly comment on some of the features. Indeed, in fig. 4.4 there is a wide two-phase region at elevated temperature (the large gray region (s)) at a fixed concentration when spherical micelles exist (label Lj) upon a sudden increase in temperature of the two-phase region leads to the formation of a concentrated surfactant phase out of the small micelles (clouding; for ionic surfactants one would find the Krafft point(s)). The critical point of this phase separation region is indicated by a dot. In the phase diagram, the c.m.t. line is the first feature when coming from dilute solutions. We also have dashed the concentration above which the micelles

Figure 4.4. Generic phase diagram inspired from data of the non-ionic C 12 E 5 surfactant. The c.m.t. line is indicated by the dashed line at low surfactant concentration. The L phases are isotropic. The liquid crystalline Hj (hexagonal) and Vj (cubic) phases are found at high concentration and low temperatures. At high temperature, a large two-phase region is found (shaded parts in the diagram). The L is the micellar phase. The dashed line with labels marks the transition from spherical to rodlike micelles (second c.m.c). The La phase has a lamellar topology. The L3 (sponge) phase and the solid phase S are also indicated. Pictorial representations of various aggregate types are given in fig. 4.3. (Redrawn from R. Strey, R. Schomacker, D. Roux, F. Nallct, and U. Olsson, J. Chem. Soc. Faraday Trans. 86 (1990) 2253.)

R.G. Laughlin, The Aqueous Phase Behavior of Surfactants, Academic Press N.Y. (1994).

4.18

ASSOCIATION COLLOIDS

change from spherical (L^) to cylindrical ( L\). In many cases, this transition is rather abrupt and in that case one refers to this point as the second c.m.c. We will discuss this transition in sec. 4.6. The La phase consists of lamellae (fig. 4.3). For this particular example, these lamellae are only found at high surfactant concentrations. At high temperature the La phase can swell enormously, which is believed to be caused by undulation forces. Some aspects related to the equilibrium separation of surfactant bilayers are discussed in sec. 4.8. The L2 phase consists of inverse micelles. In this case, the hydrated EO heads form the cores of the objects. There can be coexistence between various phases. For example, the La phase can coexist with (worm-like) micelles. The gaps in the phase diagram are filled in gray. The L3 phase, also known as the sponge phase, has a local lamellar topology, but the lamellae are interconnected by bridges or so-called handles. The membranes divide the space into an inner and an outer half-space. In the region of high surfactant concentration, the transition to liquid crystalline phases is found. At sufficiently low temperature there may be a hexagonal phase (Hx) where rod-like micelles are ordered into a hexagonal lattice. There also may exist a cubic phase V composed of a complex but very regular network with zero mean curvature. Such a phase is also called the plumber's nightmare. Finally, at very high surfactant concentration we may find a solid phase of surfactant, which is labeled S. Phase diagrams can be more complicated. For surfactants with P ~ 1/3 , there may not be a strong tendency to form elongated or lamellar objects and as a result one may find a liquid crystalline phase of ordered spherical micelles. Such an ordered phase is often called a cubic phase (not indicated in the phase diagram). At present there is at least qualitative insight into most of the phenomena determining the surfactant phase diagram. We know that in phase diagrams there are many generic features, i.e. features occurring for almost any surfactant system. Although very useful, the presentation of phase diagrams for particular surfactant systems will not be the endpoint of experimental investigations because they are merely a catalogue, leaving many fundamental questions open. The challenge for the near future is to explain the molecular specific issues. Much more work is needed to explain why the phase diagrams of Cj E 5 and C 14 E 4 differ so much, although the difference between the two surfactants is just one O atom! Indeed, this simple observation calls for molecular modelling and this will be a main topic in the remainder of this chapter. 4.2 Classical thermodynamics Classical thermodynamics specifies the number of independent variables for any macroscopically homogeneous one-phase system. For each of these independent variables, there is one contribution to the change in the internal energy dU c

dL/ = TdS-pdV + ^// i dJV i i=l

14.2.1]

ASSOCIATION COLLOIDS

4.19

where the symbols have their usual meaning. Index i refers to all (c ) molecular (i.e. uncharged) components, including the solvent. Of the c + 2 variables { p , T and the c components), only c+1 are independent because of the Gibbs-Duhem relation. The most simple surfactant system has two components, a surfactant and water. For an open system of this kind, we can write for the change of the Gibbs energy dG = -SdT + Vdp + MSUT{dNSUT[ + // water dJV water

[4.2.2]

At constant p , T there is just one degree of freedom, the surfactant concentration. Classical thermodynamics provides the rules that should be obeyed when a macroscopic system goes through a phase transition and can be used to help understand those cases in which there is a macroscopic interface between two coexisting surfactant phases, e.g. when the system is above the cloud point or when a surfactant-rich lamellar phase coexists with a dilute surfactant solution which contains micelles. At issue is the question whether thermodynamics remains applicable on the mesoscopic level. In particular, the problem arises on how to use thermodynamic arguments for micelle formation. From a macroscopic point of view, the system remains completely homogeneous even when micelles are present. Thermodynamically speaking, nothing prevents or promotes molecules to choose to interact with each other and nothing prevents them from making molecular clusters or micelles still retaining [4.2.1]. Such interactions are determined by the values of the intensive variables. The extent of (non-ideal) interactions will then be a function of the chemical potentials, the pressure and the temperature. In an open system, one fixes the chemical potentials of the molecular components by letting the system equilibrate with a large reservoir. Below the c.m.c. there is no problem doing this, but as soon as the chemical potentials are increased to values consistent with the presence of micelles, the number of kinetic units of surfactants in the system suddenly becomes a very strong function of the chemical potential of the surfactants. Therefore, the closed system is preferred to examine micelle formation in the same manner as for any chemical equilibrium reaction. Macroscopic thermodynamics can actually provide further guidance to understand micellar systems. We can only make significant progress, however, upon pre-assuming that micelles exist. Formally, we then go somewhat beyond the realm of classical macroscopic thermodynamics. There exists a powerful framework, which is an extension of the classical thermodynamics, to deal with microscopically inhomogeneous systems. This approach is known as the thermodynamics of small systems. It was developed by Hill11 in the 1960s and elaborated by Hall and Pethica21 for micelle formation.

11 T.L. Hill, Thermodynamics of Small Systems. Part 1 and Part 2. Dover Publications (1991) and (1992). 21 D.G. Hall, B.A. Pethica Non-ionic Surfactants, chapter 16. Marcel Dckker (1976).

4.20

ASSOCIATION COLLOIDS

4.2a Thermodynamics of small systems In [4.2.1 ] the full set of independent variables of a homogeneous bulk composed of c types of molecules is given. From experiments, however, we know that surfactant systems organize themselves on a mesoscopic length scale. It is quite attractive to try to introduce this information into the thermodynamic analysis. However, in general one should be cautious when ad hoc extra degrees of freedom are added. Nevertheless, this is what we are going to do but we shall give an a postiori justification. Following Hill, we postulate that there is some hidden variable, which is most generally called the number of subdivisions. Without much ado, we directly identify this number as the number of micelles In the system Af = Xj-VJ, where Mi is the number of micelles of distinguishable state i. As all micelles are in mutual equilibrium, it suffices to use just one A/" • Conjugated to this hidden extensive variable, we introduce the intensive variable e. In effect, e is the energy needed to generate one extra micelle in the distribution { A/'i} at fixed entropy, volume and number of molecules. Now [4.2.1] is modified to c

dU = TdS -pAV + ^

fiidNi + £&A/

[4.2.3]

i=l

In a closed (N,V,T ) system, the Helmholtz energy is the characteristic function. For aqueous solutions we may as well choose to fix the pressure and switch to the Gibbs energy. In incompressible systems, these two quantities differ just by a constant. For a change of the Gibbs energy, we can write c

dG = -SdT + Vdp + ]T /ZjdJVj + aLV

[4.2.4]

i=l

From general thermodynamical arguments, we know that at equilibrium

an

isothermal-isobarlc system will minimize its Gibbs energy. In the small system analysis, the hidden degree of freedom is used to optimize the Gibbs energy, and thus (!^|

=£ =0

[4.2.5]

The corresponding stability condition reads

\dAr

h.p,{Ni}

W>T,p.{Nl}

The implication of [4.2.5] is that, at equilibrium, the Gibbs energy continues to be simply given by c

G

= X'"iJVi i=l

[4 2 71

- '

ASSOCIATION COLLOIDS

4.21

There is no excess Gibbs energy associated with the formation of micelles. This result is completely in line with the macroscopic thermodynamics for homogeneous systems. The extra term introduced in [4.2.3] thus does not, and should not, influence the Gibbs energy in the system. We note that the small system approach does not presuppose the presence of micelles of a particular size or shape. It anticipates, however, and this should follow from any statistical mechanical model, that there is a distribution in size (and shape). For any of these aggregates it must be true that there is no excess Gibbs energy associated with its presence, i.e. e = 0 . It is known that the micellar size (distribution) depends on a variety of factors, such as the nature of the surfactant, the temperature and the presence of additives that may be absorbed in, or repelled by, the micelle. This means that generally e is a function of p , T and the chemical potentials. However, as at equilibrium e is always equal to zero, from the e d V term no information can be obtained regarding the dependence of micellar properties on ambient conditions. To that end, models have to be developed. We note that in general we should expect that the average micelle size will be inversely related to the number of micelles in the system. This means that the Gibbs energy should also have a minimum with respect to the average micelle size. It is possible to split the Gibbs energy into a bulk part G b , referring to the solution in which the micelles are embedded and a part that can be attributed to the micelles Ga, i.e. G = Ga + G b . The total excess number of surfactants associated with micelles may be found if the bulk concentration of surfactant (i.e. outside the micelles) can be determined because the excess is counted with respect to a reference system, which fills the entire volume by the (dilute) bulk solution containing only monomers. The same applies to the other components and material balance gives N{ = N? + JVb for each component.

Because the volume work

does not enter

in the excess

thermodynamic potentials, the difference between the Gibbs and Helmholtz cases vanishes. Starting once again from [4.2.3], we may write c

Fa = Ga = ^ /ijJVf + EAf

[4.2.8]

i=l

where we have retained deliberately the EA/~ contribution to unravel more about its properties. Here we notice that the molecules in the bulk do not contribute to e. The value of e is the Gibbs energy that is in excess due to the presence of micelles. Normalizing the excess Gibbs energy by the number of micelles gives ga =

=Y//inf +£

[4.2.9]

where n? = N?IA/~ is the average excess number of molecules associated with one micelle. We may also refer to this as the most probable size of the micelle. Now E can be identified as the (excess) grand potential

4.22

ASSOCIATION COLLOIDS c

£ = ga - ^T nffii

[4.2.10]

i=l

At constant pressure and temperature we can write c

c

c

de = dg* - £ ^dnf - £ nf d/i, = - £ nf d//t i=l

i=l

[4.2.11 ]

i=l

which is the Gibbs-Duhem equation for micellization. It resembles Gibbs' law for adsorption. When we have a two-component system where surfactant {surf) is dissolved in water, we can assume that the chemical potential of the solvent is nearly constant so that the Gibbs law becomes (9^-)

=-
[4.2.12]

In words, this equation says that the excess grand potential is a decreasing function of the chemical potential of the surfactant because necessarily the aggregation number n f u r f > 0 . Recalling the stability condition [4.2.6] we use [4.2.12] and find that 3// surf / 3yl/< 0. Again in words, this says that in a closed system the chemical potential of the surfactant must go down with an increasing number of micelles. Let us next switch our attention from the closed isobaric system at constant T , where the Gibbs energy is minimized, to the open system. The grand potential is the characteristic function of the system in which all chemical potentials are fixed. In such a system both the micelle concentration, i.e. the number of micelles Af per unit volume, and the aggregation numbers (including the distribution) are variables, which the system may use to minimize its grand potential. The minimum of this grand potential should have the value zero in order to be consistent with the analysis in the closed system. In general, the thermodynamics of small systems becomes especially useful when it is applied to the interpretation of experiments. It also turns out to be of utmost importance for analysis of results of simulations or computations on micelles. 4.2b The mass action model A customary way to describe micelle formation is to consider the equilibrium between freely dispersed surfactants, usually called monomers and referred to by Xj, with micelles composed of g surfactants, X . In this approach, one assigns a chemical potential to the g -micelle, and the total number of surfactants in the system is split up into a contribution of the monomers Nsm and a contribution present in the micelles g x JVs( ,, i.e. N surf = Ns(1) + g x JVs( j . This equation is easily generalized to a range of micelle sizes. Ignoring for the moment the size distribution, the equilibrium may be expressed by gXx^±Xg

[4.2.13]

ASSOCIATION COLLOIDS

The equilibrium constant K

4.23

relates the micelle concentration to the monomer

concentration

K = J ^ i =^ L 9

^

9

[4.2.14]


This equation is often used to illustrate the first order-like transition the system passes through near the c.m.c. For large values of g , say g ~ 100 , we see that near and above the c.m.c. an increase of the monomer concentration by 1% already leads to a threefold increase in the micelle concentration. Indeed, experiments show that this is a realistic aspect of micelle formation in aqueous solution. The monomer concentration above the c.m.c. increases only very slightly when the micelle concentration is increased, e.g. by increasing the overall concentration. The model based on just one "reaction" is, however, unrealistic; intuitively one must expect the micelle to increase in size (increase the aggregation number) when the chemical potential of the monomers is increased. We will show below that this is true, as the Gibbs energy gain of size fluctuations is small compared with kT . The way to correct for these circumstances is to consider a large set of reaction equilibria and, connected to this, a set of equilibrium constants K ,. Now micelle formation becomes more complex because each of these constants has to be known before one can proceed. We shall not elaborate this route further but mention that it was shown already by Hill that the multiple equilibrium approach to micelle formation is equivalent to small systems analysis. For the time being, we accept the limitation of the single equilibrium model and proceed by pointing to the fact that equilibrium implies that S^s(l) =Mm

[4.2.15]

where in this notation jus,^ is the chemical potential of the monomer and JUSIQ^ is the chemical potential of the g -micelle (corresponding to adding one more micelle to the solution). In words, this equation points to the fact that the chemical potential of the micelle is found from the sum of the chemical potentials of the g monomers from which it is composed. Again, in line with the small system approach where £ = 0 , there is no extra Gibbs energy on top of that stored in the micelle. Ignoring activity corrections, we can write for the chemical potential of the monomer <"s(i) = >"s(i) + ; c T m ^ s i i ) a n d f° r t ^ le micelle MS{g) = fsig) + ^ m ^s(g) • w n e r e t n e volume fraction

[4.2.16]

In passing, we note that K = expf-AG^/fcT), where G®n is the standard Gibbs energy of micelle formation, which may be related to the c.m.c. (e.g. using [4.2.14], but only

4.24

ASSOCIATION COLLOIDS

after an appropriate definition of the c.m.c], and in general it is composed of an enthalpic AHj^ and an entropic ASj^ contribution, which are measurable by calorimetric methods. In a more elaborate model with multiple equilibria, similar equations apply for each equilibrium constant. We note that thermodynamics neither gives a clue about the value of g nor the values of K . To make progress, we need a molecular model to compute //°( j for a series of g values. 4.2c Implication for molecular modelling It is obvious that molecular models are needed for further progress. Models should be consistent with thermodynamics. This means, for example, that in equilibrium the (excess) grand potential of each aggregate should be zero [4.2.5], or equivalently that the chemical potential of a micelle is found to be the sum of the chemical potentials of its constituents [4.2.15]. Both the mass action model and the thermodynamics of small systems approach already require some extra-thermodynamic assumptions. In the small system approach, the existence of micelles of limited size is assumed and in the mass action model it is taken that distinct micelles exist to which a micellar chemical potential can be assigned. More detailed models will need these assumptions as well. The intensive variable e, associated with the number of micelles, is a central quantity, which in equilibrium is zero and whose derivative with respect to the number of micelles must be positive for stability. This quantity was identified as the (excess) grand potential of the micelles. It is possible, by using a model, to unravel this grand potential into its various components. Here we will illustrate this by splitting it up into a term that contains the mixing entropy and the remainder which we will call em . Referring once more to the mass action model of [4.2.13] and the equilibrium condition [4.2.15], we notice that the micelles have translational (dispersion) entropy. Ignoring the interactions between micelles and the non-ideal part of the mixing, one typically uses (,«s(1) - jU®m)/kT = lnfZ>s(1) for the monomers and

for the micelles; these logarithmic terms may be identified as (minus) the dimensionless (partial) mixing entropy term associated to the monomers and the micelles, respectively. We see that the driving force for micellization must at least overcome a significant loss of translational entropy; per micelle g monomers cede their translational entropy, whereas only the corresponding entropy of the micelle as a whole is gained. Assuming that at the c.m.c. the monomer concentration and the micelle concentration are of the same order, we see that the loss of translational entropy amounts to about -k{g -\)ln
ASSOCIATION COLLOIDS

4.25

of surfactant molecules in the micelle. Next we may set up a regular solution model to elaborate the idea of micelles having mixing entropy. Let us consider a (spherical) micelle with micellar volume Vm . We now construct a three-dimensional lattice with L sites of volume Vm , i.e. V = LVm . On these sites we distribute Af micelles that have athermal interactions with each other. From this we obtain the total (ideal) mixing entropy in the system -TS mjx //cT =
(// - ju°) / kT = In

{fiM-fJ%/l)/kT = ln{l-
[4.2.18]

We will refer a bit loosely to em as the translationally-restricted grand potential of a micelle. Alternatively, in Hill's terminology, em is the standard state subdivision potential. In fact, it is useful to point out that em is the characteristic function coupled to the partition function cpm . Here we use the notation cpm for the total volume fraction of micelles and do not refer to the size of the micelles.

4.26

ASSOCIATION COLLOIDS

All truly molecular models that aim at computing the partition function have in common that they focus on one (average) micelle at rest in a small system. We will now argue that for these models em is of key importance to evaluate the total micelle concentration. The first step is to realize that the total volume in the overall system may be split up into Af small systems with volume v = VI AT . As in the molecular modeling calculations carried out on a small system, we will optimize the Gibbs energy (for a small system) not with respect to the number of micelles (because we consider just one average micelle), but with respect to the volume of the small system v and note that dv/dAf < 0. In this small system, the molecular composition is identical to that in the overall system, and hence ni = N{/ Af for all molecules i . We take it that there is one micelle at rest in the centre of this system. The intrinsic Gibbs energy for this small system gm = £ 4 nilui + £m a n ( i hence the grand potential is e m . The subindex m reminds us of the fact that the system is constrained because it features a micelle at rest. The goal is to estimate what the overall micelle concentration is in the corresponding (unconstrained) macroscopic system. To do this we need first to convince ourselves that the micelle is stable. It turns out that it is useful to consider the excess quantities and the corresponding bulk values for each molecule I, i.e. n( = n? + H?3 , where the excess of the surfactant may be quantified in terms of the aggregation number g . Following the same arguments as above, we also find a Gibbs relation for the constrained system

(^M

=-"sVf=-9

[4-2.191

As g > 0, we infer that em is a decreasing function of ,usurf. At the same time, g = rigurf must be an increasing function of // surf , and thus e must be a decreasing function of g = n surf , to have stable micelles. Hence, the stability condition in the constrained system is given by — <0 9
[4.2.20]

Now we fix the chemical potentials in the small system and vary its volume v until the small system corresponds to the equilibrium one. Let us estimate the volume of the micelle um by the molar volume of a surfactant u surf times the aggregation number, i.e. vm = n° nrf u s u r f . Using this, the mixing entropy that a micelle will gain when the positional constraint is released depends on the volume of the small system, i.e. -S m i x Ik = ln(Pm = lnu m / v . Using [4.2.18], we have the equilibrium condition

^ m = e x p (~Sr)

|4 2 211

'-

Knowing the volume fraction of micelles, it is possible to compute v and with that the

ASSOCIATION COLLOIDS

4.27

system is completely defined. For a micelle with given em > 0 and dem/dn°urt

<0 ,

there exists an overall volume fraction of surfactant given by Psurf=«£urf

+

I42'22!

Pm

or, in other words, there is a one-to-one relation between the overall surfactant concentration, the micelle size and the micelle concentration. One of the main advantages of molecular modelling is that, besides the most probable micelle size, information can often also be obtained on the fluctuation in micelle size, which is directly linked to the micelle size distribution. Quite generally, a measure of the size distribution is found from (see sec. 1.3.7) kTJ(9L d

= a2{g)2_{g2}

[4 2.23,

<"surf

where we may identify (g) by g , that is the aggregation number of the most frequent micelle. Using Gibbs' law or [4.2.12], we may write (de / d(g))(d{g) / d/i) = -(g) or equivalently

4.2d Fluctuations in micelle sizes In general we should expect a distribution of micelle sizes. It is necessary to understand the role of the entropy associated with size distribution. We may start with the mass action model. Let there be a set of micelles with aggregation numbers g = 9mjn>"i>Sflnax • For each of them, there exists the equilibrium of [4.2.13], gXj ^ Xg . At equilibrium, AGg = fig -g/^ = 0 . We may identify AGg as the excess grand potential for the g -micelle. To distinguish this quantity from the most likely grand potential as obtained in the SCF theory and considered so far, we will refer to it as £2(g). Similarly, as done with e, we may split Q up as £2(g) = Qra(g) + kTlnip where

m(9» = -f3m«9» +

fc 2 3(9-(9»

[4.2.25]

where i2m[{g)) is the quantity that is computed in the SCF calculations and to which we refer as em [g). The measure of the width of the distribution k

is related to the

4.28

ASSOCIATION COLLOIDS

variance <jl = kT/(2k ). The statistical weight of a micelle with grand potential &mig) follows from Boltzmann's law. The distribution is Gaussian (at least around the minimum). For sufficiently large value of k&, the Gauss distribution is so narrow that integration from the smallest g = gmiB to the largest g = £fmax micelle size is the same as integrating from

g =-°°

to g = +<*>, which leads to the micelle probability

distribution

P

[K~

'H^T-

where

eXP

(-kJg-(g))2)

[

|4 2 261

kT J

'-

Ik / nkT normalizes the probability distribution function to unity. On top of

the translational entropy of the micelles, we now also need a mixing entropy. It accounts for the number of ways one can distribute distinguishable micelles into the systems. This mixing entropy may also be calledjluctuatton entropy and is given by S

fluct/'c = - X P ( 9 ) l n P ( f i f )

[4.2.27]

9

The average grand potential per micelle now contains the entropies of the translational and mixing degrees of freedom, as well as the standard state contributions: 12 = kT^

P(g) In (P(g)
9

9

9

^

[4.2.28]

'

Again, in equilibrium Q = 0 . The sum over all micelles gives the overall micelle concentration q>m =^q(p

and the volume fraction of g -micelles is found as

P(g)
I kT). Using [4.2.25], we arrive at


\j kg

y

{

kT j -J kg

V

,4.2.29]

{ kT )

Comparison of [4.2.29] with the relation [4.2.21 ] shows that the entropy due to the size distribution leads to a fluctuation correction given by Jk / nkT = -JUncr . Below we will analyze the variance of the size distribution that results from a MC simulation, as well as predicted from the SCF theory. At micelle concentrations not too close to the c.m.c, the relative fluctuations do not depend much on the micelle concentration and we typically find a I g ~ 0.2 . This means that the fluctuation contribution V2/r
ASSOCIATION COLLOIDS

4.29

appearance of micelles, and also near the sphere-to-rod transitions as well as for worm-like micelles, the distribution is not narrow and [4.2.29] is incorrect. As a result, we should keep in mind that the conversion from em to the total surfactant concentration is typically approximative, especially when for the sake of simplicity the entropy in the size fluctuations is ignored. It should therefore be understood that the relation between micelle size and micelle concentration is only expected to be accurate to first order, i.e. one should mainly apply the predictions to understand trends. Below we will present various methods that are regularly used to model micelles and micellar solutions. In the SCF theory, it is only possible to compute so-called most probable micelles (it focuses one m ) and we will use [4.2.24] to obtain information on the micelle size distribution. In analytical models that make use of surface-thermodynamics, it often proves more convenient to fix the chemical potentials and compute the grand potential &mig) (or equivalent, the g -micelle chemical potential) as a function of the micelle size. In this case, one has access to the micelle size distribution more directly, and [4.2.29] should be used to find the overall micelle concentration (provided the distribution is sufficiently

Gaussian and narrow). In (atomistic)

simulations, the size distribution of micelles is directly available from averaging over the simulation trajectory (provided that at any time in the simulation box there is a sufficient number of micelles present). In this case, it is not necessary to apply any of the thermodynamic rules. One of the most characteristic aspects of micelle formation is the pronounced gap in the size distribution. There are freely dispersed monomers in solution. These monomers have some affinity for each other and we should thus expect a small fraction of dimers, and even smaller quantities of trimers, etc. in the system. Then, the probability of finding multimers of 4, 5, ••• monomers is about zero, until at significantly higher aggregation number (order 100), the probability of finding aggregates peaks again at the most probable micelle size. Finally, this probability drops to zero again for large aggregation numbers. Any realistic molecular model should account for this gap. In this context, it is relevant to recall that for small micelles g < gmin , when dem/dg > 0 , the micelles are unstable and the Gibbs energy is at a local maximum. This is the signature for the existence of a gap in the micelle size distribution in models that focus on the most likely micelles (i.e. the SCF model). This does not mean that micelles with aggregation number g < gmin

do not exist. They may be present in the

tail of the size distribution for systems where the most likely micelle is sufficient in size, i.e. g > gmin • Such micelles may well function as an "activated state" through which growing micelles have to pass on their way towards their most probable sizes. 4.3 Molecular modelling The machinery of statistical thermodynamics gives, for any molecular model, the recipe to come to measurable observables. The partition function is the central quantity

4.30

ASSOCIATION COLLOIDS

that should be computed. Knowing the partition function gives, via the characteristic function, access to the thermodynamic, mechanical, and structural characteristics of the system. Exact partition functions for systems in which surfactants occur with molecular detail are not available. There are two options to proceed, viz. via the simulation route or one can choose to use mean-field approximations and solve meanfield partition functions. Within the latter approach, one can try to formulate the problem in such a way that (approximate) analytical results can be obtained. However, for the more detailed models computer assistance is unavoidable. 4.3a Molecular simulations For surfactant systems, one can use molecularly realistic simulations such as molecular dynamics (MD) or Monte Carlo (MC). The simulation route is not focusing on the partition function. There are simply too many possible states to visit all of them. As a result, thermodynamic quantities such as the entropy or Helmholtz energy are very hard (but not impossible) to evaluate. Instead, the aim is to generate, with the use of a computer, a large set of relevant realizations of the system. In MD the trajectory along various realizations is along a dynamic path generated by solving Newton's second law F = ma for each degree of freedom. Both structural, some mechanical parameters as well as dynamic parameters are obtained with an accuracy that is determined by the quality of the input parameters and the computer time available. It is possible to take the same molecular system as studied in an MD simulation and subject it to an MC simulation. Instead of a dynamic path, there are random changes made in the system, but the Metropolis algorithm makes sure that only relevant states are probed. Along the MC trajectory, one can then collect information on structural and mechanical properties. There is no systematic information on the dynamics. For a particular model (i.e. the specified, usually simplified, interactions and the large set of estimated force field parameters that parameterize the interactions on the atomic level) simulations give (when sufficient CPU time is available) access to close-toexact results, that is at each point along the simulation trajectory all correlations between all molecules are exactly accounted for. The system size is usually limited to linear dimensions on the order of a few nm. In addition, in MD simulations there is accurate dynamical information, but it is hard to proceed to times longer than a few ns. Both limitations make the simulation route less applicable for (dilute) micellar solutions. There are, however, several problems for which the simulations give the most accurate results available to date. One such example that will be discussed below is the lipid bilayer membrane. In the simulation of such systems, one typically makes sure that one has a suitable initial guess so that one can start the simulation already close to some equilibrated structure. In modern packages, all parameters necessary to account for all the interactions in the system, the so-called force field parameters, have

ASSOCIATION COLLOIDS

4.31

reasonable values and after a relatively short equilibration run one can already collect relevant information on the equilibrium state and fluctuations around the equilibrium. The simulation route is not limited to the atomic level. One can invoke models in which the molecules are represented by groups of atoms (united atoms) or with strings of segments (blobs). Then much larger system sizes are accessible and, for MD, longer time scales can also be probed 1 '. Of course, the standard force field cannot be used any more and one has to come up with an alternative set of parameters that specifies how the united atoms or blobs interact with each other. Such a coarse-graining step is not unambiguous. Quantifying the zooming out from the molecular details to the mesoscopic length is seen as one of the important challenges in the near future. We will not further discuss the technicalities here.

4.3b Self-consistent field (SCF) theory The second approach is the mean-field partition function route. The mean-field method pragmatically deals with the counting of the binary (and ternary, etc.) interactions in the system, which depend on the detailed local environments (i.e. distances between the molecules in each individual realization of the system), by substituting the actual surroundings by an average one. In this way, all pair interactions, three-body interactions, etc., are replaced by those for a particle in an external field. If this external field is made adjustable such that it corresponds to the average distributions of the molecules in the system, one speaks about a self-consistent field (SCF) theory. Such SCF theory may be used to model micelles in equilibrium with freely dispersed monomers. We will see below that the SCF method can be implemented on a level, ranging from that of a statistical segment (rather coarse) to that of the atom. Typically, however, the molecules are parameterized with less detail than in the all-atom simulations. For self-assembly, the structure of the surfactant molecules should be accurately accounted for. This means that architectural elements of the molecule, (one or two tails, linear or branched chains, ionic, zwitterionic nature of the head groups, etc.) must be realistically implemented. Important for a realistic modelling is that one should keep as many fluctuations in the model as possible, i.e. to impose as few constraints as possible. One weakness of SCF modelling is that it is needed to a priori specify the geometry of the system. This means that, when a spherical co-ordinate system is used, one can only obtain information on spherical micelles or spherical vesicles. In a cylindrical geometry one can model rod-like micelles or tubular vesicles. A flat geometry is used to generate information on lamellar bilayers, etc. As for each of these geometries, the partition function can be found as a function of the system variables, such as the chemical potentials and the temperature, one can a posteriori point to the thermodynamically most favourable geometry for the system. We will show this below in sec.

1

M. Krancnburg, C. Laforgc, and B. Smit, J. Phys. Chem. Chem. Phys., 19 (2004) 4531.

4.32

ASSOCIATION COLLOIDS

4.6 when we consider the sphere-to-rod transition in micellar solutions. The caveat is that more complex geometries, e.g. donut micelles or perforated disks, which may be important in practice, may be overlooked. Computer-assisted modelling very often leads to discrete rather than continuous functions. This applies also to the SCF model, which makes use of a discrete set of co-ordinates (lattice) with layers (sets of lattice sites along which the mean field approximation is applied) that depend on the chosen geometry. In other words, when the geometry is specified, the planes of lattice sites are positioned in space accordingly. For example, for a spherical micelle one can expect that it is possible to use a spherical co-ordinate system comprising spherical shells. Within each shell of sites, the density fluctuations are so small that it is possible to ignore them. Exactly how the lattice geometry is implemented will be discussed in some more detail in 4.4 and in appendix 1. As long as one is interested in the properties over length scales that exceed the size of a lattice site, there are little or no adverse effects of, nor advantages in, using a lattice. However, when one is interested in details that are small compared with the lattice site one should be aware of methodical artifacts. Modelling of micelles in which there are strong gradients in density is on the edge of the capabilities of lattice models. The adverse effects of the mean-field approximation, however, are more difficult to judge. As a result, it is necessary to compare SCF predictions with simulation results for corresponding systems. We will show results of such comparison below. The main disadvantage of using computer simulations or solving a mean-field partition function with the aid of a computer is that the equations that are used to generate a trajectory (in simulations) or to find density profiles from the optimum partition function do not yet say much about the physical properties of the system that is modelled. In general, one needs to analyze the results and do systematic variations of parameters before one can compare the results with experiments, and hence come to deeper understanding of the system of interest. Computer modelling is therefore rarely the end-point of investigations. Ideally, one would like in addition to have a model that incorporates yet other idealizations of the system, which also captures the main physics of the system of interest and ignores the less important effects. Such an approach may give insight into the balances offerees, the importance of fluctuations, etc. Somewhat closer to the theorist's dream is the advancement of analytical models. For self-assembly one can put forward formulae that capture various aspects of the problem at hand. These equations are the result of detailed theoretical modelling of a partial problem, e.g. the grand potential of the electric double layer in and around the micelle, or result from phenomenological fits of experimental or simulation (computational) data. Typically the start of such an analysis is an idealized view of the structures at hand. For instance, a usual Ansatz is to treat aqueous micelles as apolar droplets, surrounded by an oil-water interface with an adsorbate in it, to which an interfacial tension can be assigned. This means that, similarly as in computer-aided models, significant information input is needed, such as micelle geometry, the role of various

ASSOCIATION COLLOIDS

4.33

fluctuations, etc. It is fair to say that such quasi-macroscopic (analytical) models cannot (yet) take all degrees of freedom of the surfactant molecules in and outside the micelles into consideration. One rather hopes that the most important ones are sufficiently accurately accounted for. In this context it is necessary to mention that in a successful analytical model of this kind one would like it to contain the experimentally known correct results. This is typically not yet the case. The state of the art of successful quasi-macroscopic models is that there is a balance of equations, which compensates for each other's shortcomings. This point illustrates that direct comparison with a limited amount of experiments is usually insufficient. One should also critically confront these models with computer-aided molecular modelling. In other words, successful quasimacroscopic models should be developed in close harmony with molecularly realistic computational methods on the one hand and experimental results on the other. This desirable process of integration undoubtedly deserves more attention. (i) MC and SCF of surfactant micelles There are relatively few MC studies on micelles. In this section we will discuss the results of Wijmans and Linse1' who published a comparison between MC and SCF results for a rather primitive model (which at least allows such a comparison). Unfortunately, in the original paper they overlooked the fact that for a successful comparison the two models need interaction parameters that differ systematically. In particular, the through-bond contacts are irrelevant in the MC method and therefore a segment in the chain on a cubic lattice can effectively only interact with four neighbours, whereas the x parameters used in the SCF model were rescaled to six possible contacts. They incorrectly found major differences between SCF calculations and MC simulations. We will now present an improved analysis in which the original MC results are compared with the corresponding SCF model with the correct interaction parameters. We arrive at the conclusion that the results of the two methods are in almost quantitative agreement. This particular MC model is lattice-based and features an incompressible uncharged system. There is a cubic lattice with dimensions LxLxL

(L = 44) sites with

periodic boundary conditions. Chain-like surfactants of the type AN B N

are intro-

duced, where JVA = 10 is the length of the A-block and JVB = 10 the length of the B fragment in a monomeric solvent S . One may interpret an A unit as a hydrocarbon segment. In the adopted MC procedure the chains are freely jointed and self-avoiding. This means that each lattice site is occupied just once and that two segments along the chain occupy neighbouring sites. In the Metropolis weighting, nearest-neighbour con tacts were accounted for via the contact energies u AB = u A S . Assuming six neighbours, these were translated into a Flory-Huggins parameter %AB = %AS = 2.7 and ^gg = 0 .

11

CM. Wijmans, P. Linsc, Langmuir 11 (1995) 3748.

4.34

ASSOCIATION COLLOIDS

The correct parameters are a factor 2/3 smaller (at least for the middle segments) because middle segments in the chain have not six, but just four, possible contacts. Below we specify the corresponding SCF model. The simulation length is in the order of 107 Monte Carlo Steps (MCS) (in one MCS, the system is changed as many times as there are degrees of freedom). Results were typically averaged over 104 different micelles. In total 200 surfactant molecules were included in the system. The remainder of the sites was filled by solvent molecules. In the incompressible system the reference of the interaction energies is chosen such that in SCF ^ ^ = ^ B B = / c c = 0 and MC U

AA = U B B = U CC = ° •

The corresponding SCF model uses a spherical coordinate system. The molecular structure is very similar; here we choose the end segment of the A block to differ from the middle ones, i.e. -AjAN _jBN • Again, a freely-jointed chain model is used, however the chains are not self-avoiding but modelled using a first-order Markov approximation. In a Markov chain, the segments can fold back on previously occupied sites already taken by the same chain. In the SCF method, such approximate handling of intramolecular excluded volume effects is in balance with the similar handling of the intermolecular excluded volume interactions. At the cost of significant computer time, it is possible to systematically improve on the intramolecular excluded-volume correlations. Here we choose not to do this. By doing so the intermolecular excludedvolume problems (one site can be occupied by segments of two different molecules) remain. Consistent with the simulations, we take for the apolar segment-solvent interactions XAB = X^s = 2 - 7 x ( 2 / 3 ' = 1-8 and %A,S = 2.25 . The interaction parameter for A' with B was set equal to 1.8 and ZA'A= ® • ^ o r further details about the SCF method, we refer to appendix 1. Results are presented in figs. 4.5 and 6.

Figure 4.5. a) The normalized size distribution as found in the MC simulation for a system composed of lattice chains AJQBJQ in a monomeric selective solvent (left ordinate), volume fraction of micelles with size g as computed from SCF calculations m(g)//cT (right ordinate). b) The translationally restricted (standard state) grand potential fm as a function of the aggregation number as found in SCF calculations for a corresponding system and the grand potential i2m(g) corresponding to the system with most frequent micelle size g = 30 . The details about the models and parameters used are given in the text.

ASSOCIATION COLLOIDS

4.35

Figure 4.6. The radial volume fraction profiles for tail and head group segments of AJQBJQ surfactants in the micelle as found by MC simulations (continuous lines) and corresponding SCF calculations (dashed lines).

Of course an MC simulation reveals many details. We select a few relevant aspects. One important point Is that in the simulations there are, at any point along the MC trajectory, just a few (order five) micelles in the box. It is important that this number exceeds unity. If only one micelle forms in the simulation box, one cannot know whether or not the size regulation allowed the micelle to obtain its equilibrium size. The reason for this worry is that (in contrast to SCF) the thermodynamic properties are not evaluated in an MC run and one relies on the Metropolis algorithm to do its task. It is also important that the micelles continuously change in size and shape. These shape fluctuations can be estimated from the analysis of the principal moments of inertia measurable for each micelle in the system. It was found that the ratio between the main and the minor principal moments of inertia was largest (approximately 2.2) for micelles with relatively high or very low aggregation number and was relatively small (order 1.5) for intermediate (most likely) micelles (not shown). The size distribution found in the MC simulations is given in fig. 4.5a. According to expectations, there is a gap in the size distribution. Micelles with size g ~ 10 have a low (but finite) probability. Monomers and micelles with sizes around g ~ 30 are most frequent. Closer inspection of the micelle distribution shows that the distribution is not exactly symmetric, but near the maximum it can rather accurately be represented by a Gaussian. The freely dispersed monomer appears to be the most abundant species. It was found that a fraction of 0.17 unimers (volume fraction 0.008) is left in the system, while that of dimers is 0.06 (volume fraction 0.006). It suffices to add up these two amounts to obtain a measure for the c.m.c. for which we find
4.36

ASSOCIATION COLLOIDS

the aggregation number of the most likely micelles (at least in the range 17 < g < 40 ). The c.m.c. as predicted by the SCF calculations is cp° = 0.0202 (see sec. 4.4 for information on how the c.m.c. is found exactly). This value is close to the estimate found in the MC simulations. For the same overall surfactant concentration as in the simulations, the volume fraction of surfactants that is available to form micelles ~ 0.28 . This means that the total amount of surfactants in the micelles is
ASSOCIATION COLLOIDS

4.37

fluctuations. Nevertheless, the two values for the polydlspersity are of the same order of magnitude, which gives confidence to the accuracy of the modelling methods. We note that in experiments with short-chain surfactants the fluctuations in micelle size are expected to be smaller, i.e. 0.1 < a Ig < 0.2 . The relatively large fluctuations in this case study may be attributed to the primitive parameter settings that were chosen to allow comparison between simulation and theory rather than to mimic the exact experimental situation. The important conclusion is that the SCF theory accurately reproduces the micellization characteristics of this simple model. Considering the fact that the SCF procedure takes about 1 second CPU per micelle whereas the simulations are at least four to five orders of magnitude more expensive CPU-wise, this is an important result. Hence, it is practical to continue and elaborate by SCF methods more sophisticated models, e.g. with respect to the role of the solvent (water) and other structural details. (it) MD and SCF of lipid bilayers All-atom molecular dynamics simulations are more expensive (CPU-wise) than MC. Virtually all MD simulations available for surfactant self-assembly have focused on the lipid bilayer membrane. Due to the potential impact for the life sciences, there is actually a drive to generate all-atom simulation results. MD simulations have the promise to give close-to-exact results on the ns time scale and nm length scale and can therefore serve as a reference for more approximate models. Of course, it remains true that the force field parameters used to parameterize all the interactions in the system have a phenomenological origin and are subject to uncertainty. Moreover, interactions such as bond stretching and bond bending, are usually treated in the lowest order. For example, the bond stretching is typically computed by Ust[l) = Kst(l-l0)2 where I is the length of the bond, (0 the equilibrium length and Kst the "spring" constant. In reality, the deviations of this parabolic dependence may be significant, but this is not incorporated. Relevant at this stage is also that the water molecules surrounding the bilayer in the simulation box are mostly modelled in a relatively rough way and that electrostatic interactions are typically cut off after a few water diameters in order to save CPU time. To be effective, MD simulations need extremely accurate initial "guesses" of the structure of interest. This means that the experimentally determined area per lipid molecule and the lamellar topology are often used to give the simulation a smooth ride to the equilibrium structure. Accurate MD simulations should allow the area per molecule to adjust in order to keep the bilayer tensionless. This means that the simulation box, in which the simulation is performed, is allowed to change its shape (i.e., its aspect ratio) during the trajectory. As it is extremely expensive to take many degrees of freedom into account and one likes to include as many lipid molecules as possible, the water film in between the membranes (periodic boundary conditions) is typically chosen to be very small (just a few water layers). Therefore, one should

4.38

ASSOCIATION COLLOIDS

consider MD simulations to be better representative for a lamellar phase at low water content than for isolated bilayers. In sec. 4.8 we will study the interactions between lamellar bilayers and show that this is indeed a serious limitation. Most of the MD work is done on the DMPC (di-myristoyl-phosphatidylcholine) bilayer, where the zwitterionic phosphatidylcholine head group is coupled via the glycerol backbone to two saturated C 16 chains. The SOPC (l-stearoyl-2-oleoyl-sn glycero-3-phosphatidylcholine) system is chosen here because a recent detailed comparison is made to a detailed SCF model11. The comparison between the two gives insight once more into the weak and strong points of more approximate SCF analysis. The corresponding SCF analysis of the SOPC bilayers is one of the most detailed available in the literature. We cannot go into all the details here, but a brief survey is appropriate. The lipid molecules were represented on a united atom level with segments with volume v (matching a lattice site). Two C18 tails, for which the CH3 end group was given a double volume, are connected to the backbone through an ester group (intermediate polarity). The tail coupled to the snl position was fully saturated. The one connected to the glycerol on the sn2 position (central carbon) had a double bond in between positions 9 and 10. The PC head group (on the sn3 position of the glycerol) consisted of a negatively charged phosphate with volume 5 v and a positively charged choline group with a central nitrogen carrying a positive charge. In a rotational isomeric state (RIS) scheme, the chain flexibility is given by the energy difference between a gauche and a transconfiguration. The value of 0.8 kT is used, except for the double bond halfway on the sn2 chain where the gauche configuration was strongly biased by favouring the gauche configuration by 10 kT (ad hoc choice). The interaction parameters were chosen such that the c.m.c. of this lipid is of order (p° = 3 10~12 . This means that the solvent is virtually free of lipids. The solvent phase was composed of water and a 1:1 electrolyte solution with (ps = 0.01. The effect of the ionic strength on the PC head groups has been studied by Meijer et al.2). In the SCF analysis, a special compressible water model was introduced. To this end, there was one extra monomeric component in the system, which represented vacancies V (we trust that this notation will not be confused by the macroscopic volume V of the system). The concentration of vacancies was fixed in the bulk and the local amount of free volume (local compressibility) is regulated by FH interaction parameters between V and the other segments in the system. The water molecules were considered to be composed of clusters of size y: W , where the whole set was taken, i.e. y = l,---,°° . The reaction of the type W + Wl ^± W +] with fixed reaction constant Kw was implemented at each lattice layer in the system. The cluster size distribution that follows depends on the local water density. To cut a long story short, such a model can be 11

A.L. Rabinovich, P.O. Ripatti, N.K. Balabaev, and F.A.M. Lccrmakcrs, Phys. Rev. E67 (2003) 011909; F.A.M. Lcermakers, A.L. Rabinovich, and N.K. Balabaev, Phys. Rev. E67 (2003) 01 1910. 21 L.A. Mcijcr, F.A.M. Lcermakers, and A. Nelson, Langmuir 10 (1994) 1199.

ASSOCIATION COLLOIDS

4.39

used to more accurately account for water and partitioning in the core of the bilayer. The classical way to do the chain statistics in an SCF model is such that the orientation of the chain is not affected by the orientation of its neighbours. This is a poor assumption for highly ordered systems such as lipid layers. It is much better to implement a first-order correction, as used in the SOPC results, which gives a bias to the orientation of a test chain to be oriented in the same direction as the average chains in the surroundings. This has been implemented in the so-called self-consistent anisotropic field (SCAF) approach. In such a model, it is possible to find the orderdisorder phase transition in the lipid bilayer, which in the biological literature is called the gel-liquid phase transition

1(

.

It is obvious that the MD and SCF models both have a plethora of parameters. In MD these parameters are hidden in the force field. In the SCF model the number of parameters is much less. Here we choose not to mention the set, but rather refer to the

Figure 4.7. Comparison of overall structural data obtained for hydratcd SOPC lipid bilayers. In panels a and c, the MD data are plotted where the density in g/cm3 is given as a function of the distance z from the centre of the bilayer. In panels b and d, the corresponding SCF results are presented where the volume fraction is plotted as a function of the layer number z (the centre of the bilayer has z = 0 ). Profiles of the two tails, the glycerol backbone units, the head group units, the phosphate and the nitrogen (the charged members in the PC head group) and water are given for both approaches. In the SCF results, the profiles of the 1:1 electrolyte are also given (in panel d), and the profile of the free volume is indicated by the letter V. 11

F.A.M. Lcermakcrs, J.M.H.M. Schcutjcns, J. Chem. Phys. 89(1988) 6912.

4.40

ASSOCIATION COLLOIDS

original literature for a full discussion of this1'. The primary results of both methods are predictions of the structural characteristics of the bilayer system. In fig. 4.7 we present a selection. The first problem that has to be overcome when a detailed comparison is made is that the MD method is not lattice-based whereas the SCF model is. This means that the natural units in MD simulations are densities in mass per unit volume whereas the SCF model gives scaled densities in terms of volume fractions. The translation of true to scaled densities depends on the Van der Waals radii used in the simulations and in the simulation this parameter is different for each atom. Accepting this problem, we still can compare the results on a qualitative rather than a quantitative level. The spatial coordinate also differs between the two methods. In the MD simulations, true dimensions are found whereas in the SCF method all lengths are normalized to the characteristic length of a lattice site. In principle, the size assigned to a lattice site in the system contains some compromise. Water molecules will fit on a lattice site with size I ~ 0.3 nm. The C-C bond is significantly shorter. Therefore, an average of 0.2 nm was chosen as a reasonable compromise between these two lengths. We met this problem before (sec. 1.3.8b). For the segment potentials, both the short-range, nearest-neighbour contacts and the longer range electrostatic interactions are accounted for as discussed in sec. 4.5 as well as in appendix 1. In a way we can refer to this approach as a variant of the GouyChapman model in which the finite size of the ions and the chain statistics of the chainmolecules are accounted for. Referring to figs. 4.7a and b, we find many remarkable points of resemblance and interesting differences regarding the overall concentration profiles across the bilayer. One directly notices the fact that the MD data are noisy, which is due to the limited statistics in the sampling. As there was no averaging done over the two halves of the bilayer, one can judge the level of noise from the left-right asymmetry. The major important observation is that water apparently does not penetrate into the core. In both models the water penetrates up to and perhaps slightly beyond the glycerol units, but not much further. This is in line with general experience. The overall density in the tail region is very homogeneous throughout the core; there is no dip in density in the centre as sometimes reported in MD studies. We expect that in those investigations the membranes were not sufficiently equilibrated. The core density closely resembles the density of a hydrocarbon melt (of, for example, C lg chains). The overall profile of the head groups is decomposed in a P and N profile for the phosphate group and the choline group in panels 4.7c and d, respectively. It is seen from these profiles that in both models on average the head group is laying rather flat, i.e. parallel to the membrane surface. However, the N profile, which is positioned at the end of the head group, has a wider distribution. This has important consequences for this potential profile (not shown, but deductible from the ion A.L. Rabinovich, P.O. Ripatti, N.K. Balabaev, and F.A.M. Leermakers. loc. cit.

ASSOCIATION COLLOIDS

4.41

distributions). It turns out that the electrostatic potential is somewhat positive on the water side of the membrane and also positive at the inside of the bilayer, but negative at the positions of the phosphates. Outside the bilayer in the aqueous phase, there is the classical diffuse part of the electric double layer with a decay length given by the Debye length. Inside the bilayer, there is also a diffuse-like layer. Here the difference in the distributions of the positive and negative ions decays much slower than in the water phase. The fact that at the centre of the bilayer the densities of positively and negatively charged ions approach each other shows that at these ionic strength conditions the electric field in the core approaches zero. In the SCF calculations, there is a free-volume profile across the bilayer (c.f. panel 4.7d). The fraction of free volume in the bilayer is higher than that in the water phase. Indeed, the free volume is more expelled by water phase than by the alkyl chains. This is consistent with the fact that alkyl chains spontaneously adsorb from water onto the vapour-water interface. The free volume further tends to be slightly elevated in the interfacial regions, that is near and around the glycerol backbone, the reduced density of molecular components allows for a reduction of unfavourable interactions. As the presence of free volume is important, e.g. for the transport of gases across the bilayer (see also sec. 6.7a), there are attempts to come up with such numbers from MD simulations11. Further, we should note that without any free volume the membrane would be in the gel state (chains highly ordered in all-trans conformation). The unsaturation in the sn2

chain keeps the bilayer in the liquid state. Within the same

parameter setting, the DPPC membrane is in the gel-state (C 18 lipids). Such gel membranes have been studied both in MD21 as with the SCF model31. There is a large number of measurables in MD. The SCF computations are less detailed, but nevertheless one can find various detailed predictions of measurable quantities. The order parameter profile of both alkyl tails, the interdigitation of tails across the midplane of the bilayer, the average position of segments along the tails and their fluctuations and several other parameters of the bilayers were found to be very similar between the two approaches. Moreover, they appear to follow the experimental findings. For example, order parameters can be extracted from deuterium NMR41. Considering the efficiency with respect of the computation times needed between SCF and MD (at least an order of 105 in CPU time), it is obvious that only the SCF method can be used for a parameter study, e.g. on the effect of the interdigitation of the alkyl tails as a function of the solvation of the phosphate group. Here we refer to the original literature51. In conclusion, this comparison shows the potentialities of MD and SCF simulations. 11

S.J. Marrink, R.M. Sok, and H.J.C. Bcrcndscn, J. Chem. Phys. 104 (1996) 9090. R.M. Vcnable. B.R. Brooks, and R.W. Pastor, J. Chem. Phys. 112 (2000) 4822. 31 F.A.M. Lecrmakers, J.M.H.M. Scheutjcns, J. Chem. Phys. 89 (1988) 6912. 41 K. Gawrisch, N.V. Eldho, and I.V. Polozov, Chem. Phys. of Lipids 116 (2002) 135 5i F.A.M. Leermakers, et al.(2003) loc. cit. 21

4.42

ASSOCIATION COLLOIDS

4.3c Quasi-macroscopic models Another strategy to extend the theoretical modelling of association colloids is to construct (based on experimental evidence or insight from realistic molecular models) new theories in which some carefully selected degrees of freedom are deliberately Ignored In order to end up with a simpler, and, more importantly, more transparent description of the system. These models typically deal with micellar aggregates on the macroscopic level although it remains necessary to mix in some molecular features as well. As a result, these models typically pragmatically combine features of the macroscopic with that of the microscopic world and we may refer to the approach as being quasi-macroscopic. They may subsequently be used to consider problems that are beyond the reach of detailed simulations or numerical SCF methods. The target of this level of modelling is the Gibbs energy of a micelle (or the micellar solution), which is assumed to be composed of a set of separable contributions for which relatively simple expressions are formulated. Although the computer is often still needed to solve the equations, we may also refer to this approach as being analytical. Important contributions may be found in the literature1'. The mass action model is the starting point for many models of this kind. The important property is the difference in standard chemical potential of a g -aggregate and g monomeric surfactants in solution, //°( , - g/i0^ • This quantity is once more identified to be the average grand potential of a micelle In the micellar standard state £m identical to the translationally-restrlcted grand potential. Upon dividing by, g we obtain the difference in the standard chemical potentials between a surfactant molecule present in an aggregate of size g and a singly dispersed surfactant in water or, equlvalently, the surfactant contribution to em uslg)-9Msa) 9

[ 4 3 i ]

g

which is a function of the size of the aggregate (as well as the temperature and pressure). The target of analytical modelling (as with molecular modelling) is to come up with accurate formulas that account for the major contributions to nfl. Typically it is assumed that these are additive and that one can evaluate them individually. In general, there may be a relatively large number of contributions. We will introduce and subsequently discuss these in more detail 12° =U2°) tr+ U2O) def( +U2£) intf +U2°) s t e r i c + -

[4.3.2]

' R. Nagarajan, Theory of Micelle formation: Quantitative Appraoch to Predicting Micellar Properties Jrom Surfactant Molecular Structure in: Structure-Performance Relationships in Surfactants, K.Esumi and M. Ucno, Eds., Surf. Sci. Series 70 (1997) 1; D. Blankschtcin, A. Shiloach, N. Zocllcr, Current Opinion Coll. Int. Sci. 2 (1997) 294; J.C. Eriksson, S. Ljunggren. Langmuir. 6 (1990) 895; J.C. Eriksson, S. Ljunggren. and U. Henriksson J. Chem. Soc. Faraday Trans II. 81 (1985) 833.

ASSOCIATION COLLOIDS

4.43

In this equation, the meanings of the various contributions are the following. The term labeled tr accounts for the transfer of a hydrophobic tail from water to the micelle, deft expresses the deformation of the tail when it is densely packed and oriented (on average) in the normal direction in the micelle as compared with the bulk liquid state, intj

gives the work to create a core-water interface, and steric points to the

interactions between the (uncharged) head groups. When the surfactants are charged we need to add an extra term (•OSlei • w n l c n accounts for the electrostatic contribution. For non-ionic surfactants, it is necessary to account for the solvation (i2°)sol and stretching of the corona chains (i2° ) def . Let us illustrate the procedure by considering a primitive model for a spherical micelle composed of a linear alkyl-based surfactant with a small uncharged hydrophilic head group with area a h , postponing more detailed treatments to sees 4.4 and 4.5. Assume that the core has a density close to that of a liquid hydrocarbon. The transfer of Gibbs energy of a CH2 , ( G2 ) differs from that of a CH3 , ( G3 ) and may be estimated from solubility data for hydrocarbons in water. At room temperature one has for a tail with (t -1) CH2 s and one CH3 (the other end is connected to the surfactant head) a transfer-free energy per chain amounting to: — s - i L = G 2 (t-l) + G3 =1.47(£-l)-3.53-lnx s ( 1 )

[4.3.3]

The translational entropy of the monomer, -lnx^j, where x s ,j, is the monomer mole fraction, opposes micellization whereas the two non-logarithmic terms are the most important contribution to the driving force for self-assembly; both terms are negative. The values for G2 and G3 can be obtained from alkyl-water phase equilibria as reported by Tanford11. In the literature the temperature dependence of the transfer energy is also available21. ( &)ti

depends on the tail length t and not on the aggregation

number, and in fig. 4.8 (where the translational entropy contribution is dropped) it is just a horizontal line. One may argue, however, that in a spherical aggregate with g tails in the core, not all segments manage to find a place deep in the core and as a result some segments have to be interfacial. This would imply that there is a g dependence. This feature is incorporated in the intj term to be discussed below. The volume of the core, composed of g chains with tail length t, can be computed when the volumes occupied by a CH2 and CH3 are known31. The molecular volumes v2 for a CH2 and u3 for a CH3 are estimated from the density of aliphatic hydrocarbons. One chain occupies the volume ut

1 C. Tanford, The Hydrophobic Effect. Formation of Micelles and Biological Membranes, 2nd ed. Wiley and Sons, New York (1980). 21 R. Nagarajan, E. Ruckenstein, Langmuir 7 (1991) 2934. 31 R. Nagarajan, Theory of Micelle Formation: Quantitative Appraoch to Predicting Micellar Properties from Surfactant Molecular Structure in: Structure-Performance Relationships in Surfactants, K.Esumi and M. Ueno, Eds., Surf. Set Series 70 (1997) 1.

4.44

ASSOCIATION COLLOIDS

Figure 4.8. a) The (tr) transfer (excluding the l n x ^ , contribution), (def) tail deformation, (intf) = interfacial, and (int) head group interactions contributions to the micellar standard state grand potential per surfactant of a g -micelle, as well as the (tot) total standard chemical potential as a function of the aggregation number for a surfactant with ( = 12 and head group area of approximately a^ =; 0.5 nm^. An alkyl-water interfacial tension y = 50 raN/m is assumed, b) The micelle size distribution around the maximum on a logarithmic scale (arbitrary units). vt = u 2 ( t - l ) + u 3 = [0.0269(t-l) + 0.0546]10- 2 7

[m]3

[4.3.4]

The estimate for the tail length contributes to the length of a methylene group l2 and a methyl group I3 is It = l ^ t - 1 ) + I3 =[O.13(£-1) + O.28]1CT9

[m]3

[4.3.5]

For a spherical micelle, we can now directly compute the area per molecule a g (t) occupied by a g -micelle as a function of the tail length t

„ ( « = ( — ] (vt)213 g \ 9 )

[4.3.6]

The radius of the g -micelle with surfactants with length t is given by Kg(t) = H f - M g ^ 4/r )

[4.3.7]

where it is assumed that there is a sharp interface between core and aqueous phase. In reality the interface is somewhat diffuse; we argued above that it is a few water molecule diameters wide and for macroscopic interfaces, this is well established (chapter III.2). As a result, the radius requires an assumption as to how to define and locate the core-corona interface. In homogeneously curved assemblies, the surfactant tails are on average all oriented in the normal direction of the micelle surface and as one end of the surfactant tail is attached to the polar head, which is constrained at the core-water interface, we should expect that the chain conformation differs from that in the bulk. There are several ways to estimate this conformational contribution, which involves an

ASSOCIATION COLLOIDS

4.45

entropy loss. One way to do this is to enumerate in a dedicated mean-field theory the conformational entropy of tails that are grafted by one end on a curved interface such that the density of chains in the core has a preset value. The results for such singlechain partition function for various g and t values can be fitted to obtain interpolation formulas". Typically, for very short chains it is possible to account for all intramolecular excluded-volume effects, finite extensibility, etc. The disadvantage of such interpolation equations is that they do not give much physical insights. An alternative method was suggested by Semenov21 who launched an equation inspired by the entropy loss that is suffered by stretching an ideal or Gaussian chain. When a chain with contour length L = tl, where / is the length of a methylene unit, is forced into a micelle with radius R q (t), the entropy loss depends on the flexibility of the chain. The statistical segment size of an alkyl chain is often taken to be 3.6 methylene groups. This means that the statistical segment size (k = 0.46 nm and the cross-sectional area of an alkane molecule in the liquid state is (£ =0.21 nm 2 . The number of statistical segments in an alkyl chain of length t is thus given by rk = t/3.6 . The deformation contribution to the standard chemical potential is given by t^gW

Kit)

where for the spherical geometry Ks = 3/r2 / 80 . In cylindrical geometry with radius of the cylindrical crosssection R , the coefficient Ks is replaced by Kc = 5/3Ks. For lamellar geometry Kj = 2KC and R is replaced by the half-bilayer thickness. The deformation of the tails potentially contributes to a significant stopping force especially when the characteristic size of the micelle is on the order of the fully extended length of the tails. In the Gauss-type stretching model, the chain can extend unrealistically further than its contour length and typically the entropy loss of the fully stretched chain is underestimated. Therefore, in the Gauss-type model the chain stretching usually contributes to a minor extent to the stopping force. As can be seen in fig. 4.8a, the deformation term is positive and it only weakly depends on the aggregation number. When a micelle is formed, there is a core-water interface left because the head groups cannot cover the entire area. This means that we overestimated the transfer energy contribution. We have to implement a correction. Assuming a sharp core-water interface, a first estimate is -G 2 A(u2) 2/3 l a N'')~ a h'' w n e r e " R I ^ ' 2 3 i s *-ne a r e a e x ' posed to the water phase of a surface CH2 group and (a N (£)-a h ) is the area that is not shielded by the head group (the dimensions of the head group area is given by a h ). However, in this approach the finite width of the core-water interface is ignored. Therefore, one usually uses the interfacial tension approach 11 J.C. Eriksson, S. Ljunggren Langmuir 6 (1990) 895; J.C. Eriksson, S. Ljunggren, and U. Henriksson J. Chem. Soc. Faraday Trans 2, 81 (1985) 833. 21 A.N. Semenov, Soviet Phys. JETP, 61 (1985) 733.

4.46

^

ASSOCIATION COLLOIDS

L

=^ ( a g ( t ) -

V

[4.3.9,

where y^w is the macroscopic interfacial tension yhw = 50 niN/m of the interface between water and hydrocarbon (composed of chains with length t ). The interfacial term has the tendency to enlarge micelles as for given t the area is a decreasing function of the aggregation number and the free energy cost per molecule decreases as is also shown in fig. 4.8a. Hence, this term raises the cooperativity of the micellization process. An equation such as [4.3.9] gives a first-order approximation of a quantity that is more complex. The free energy associated with the interface between core and corona has non-trivial curvature corrections, non-trivial anisotropic chain orientational aspects, etc. We further stress that the interfacial tension as used in this equation is not thermodynamically defined because there is no way to measure unambiguously. We will see below that one can not extract this quantity from molecular modelling either. In this primitive model the most important stopping mechanism is the contribution due to repulsive interactions between the head groups. The simplest way to account for the head group repulsion is the van der Waals (regular solution) approach. Assuming head groups of size a h = f^ , the area fraction occupied by the heads is given by Ja-gah/{4:7rRg{t)2). The ideal pressure of a 2D lattice gas is given by (nah)lkT = -\n(\-(t>a)

or

tester ^ t e r =-ln(l-/a)

[4.3.10]

Of course, this ideal lattice gas pressure will not be appropriate at large values of Ja and [4.3.10] should be replaced by (i2°)ster /kT = ( l - / a ) " 2 , which follows the simulation results for hard disc pressures in 2D rather accurately even at relatively high values of Ja . We mention once more that we must include extra terms if the head group is charged (of electrostatic origin) or oligomeric (elastic-free energy of stretching of the head group). Such contributions will be discussed separately below. The micelle size distribution is, for given surfactant length t, given by

*>m(9)-exp U ^ ? -

[4.3.11]

The shallow minimum of the total i2° in fig 4.8a leads, even on a logarithmic scale, to a sharp maximum in
ASSOCIATION COLLOIDS

4.47

stretching of the tails may be significantly underestimated and we may consequently have underestimated the contribution of chain-stretching to the "stopping" force for micelle formation. Further, the absence of micelles with very small g is only qualitatively correct because the assumptions of sphericity of the structure of such small aggregates is likely to be far too restrictive. A better description should also feature curvature corrections for y hw . Evaluating the quasi-macroscopic model, we see that the interfacial tension yhc essentially increases the cooperativity for self-assembly whereas the head group repulsion, I.e. the interaction term, is the anti-cooperative term. The combination of the two gives rise to a minimum in the standard chemical potential of the micelle as a function of g and the corresponding maximum in the size distribution. State of the art quasi-macroscopic models are developed in rather large detail and are targeted to be molecularly specific. For example, such analytical models may be used to obtain insight into the selection of the geometry for a particular surfactant system. The temperature dependence of micellization may be investigated after the temperature dependencies of the parameters of the model are specified11. In the following sections, we will elaborate specific molecular models for the selfassembly of a specified set of surfactants. We base our analysis on the self-consistent field model because this method is best suited for this. Results will be complemented by those from the corresponding quasi-macroscopic models and checked against experiments where appropriate. 4.4 SCF for non-ionic (spherical) micelles In this section we will discuss two types of non-ionic surfactants, the C n E x and Pluronic surfactants, which have the EO moiety as the head(s) and the alkyl chain or PPO unit, respectively as the tails. Most of the discussion will be focused around SCF predictions. Quasi-macroscopic models for non-ionics will receive some attention at the end of this section. Water is a reasonably good solvent for polyethylene oxide at room temperature. On the E = C2O -level, the FH parameter with water (W) is at room temperature at approximately XWE ~ 0-4 • PEO has a lower critical solution temperature (LCT) meaning that there exists a temperature, higher than room temperature, such that / W E > 0.5 and, depending on the molecular weight, the system has a solubility gap. The surfactants that rely on the EO part of the molecule to have micelles that are soluble in water are correspondingly sensitive to the temperature. Likewise, the solubility of the PPO chain is temperature sensitive. As a result the most important parameter to control the micellization of EO-based surfactants is the temperature. The micellization for the

J)

R. Nagarajan, loc.cit.: D. Blankschtein, et ai. Loc. clt.: J.C. Eriksson, et at, Loc. cit.

4.48

ASSOCIATION COLLOIDS

family of non-ionic surfactants is a very weak function of the ionic strength. Only at relatively high ionic strength do electrolytes affect the solvent quality of the ethylene oxide part of the chain. In this chapter, we choose to construct an SCF model for which the united CH2 atoms determine the discretization length. The simple reason for this is that the alkyl chain may also be modelled using these entities. Using these united atoms allows one, in principle, to account accurately for the chain architecture. We use a model pioneered by Barneveld11 for EO-containing and PO-containing surfactants that has fewer parameters than the two-state model, but still captures the main temperature characteristics (c.m.t, c.p.t). The generic features of micellization can easily be illustrated by a very primitive model, cf sec. 4.3b ad (i). However, eventually one needs to be molecule specific. The challenge for the near future is, of course, to choose parameters and molecular architectures such that the results can be generalized. One of the problems for a realistic molecular model is dealing with the water phase. Water is an associative fluid that forms via relatively strong and many H-bonds small water clusters that dynamically break and form. This complexity is beyond reach for models that are targeted to describe association colloids. At present we can only advance via ad hoc water models. Below we will discuss SCF results with spatial resolution in two dimensions, and for such a coordinate system we need (for the time being) to be less ambitious. In short, we will model water to be composed of a cluster of W units that occupy five sites (one central surrounded by four others). In addition, we allow for discrete amounts of free volume. Free volume is implemented as units called V that occupy one lattice site each. For details about this model, we refer to appendix 1. The alkyl-EO non-ionics are modelled as(Cg)1[C]n_[[(O)1(C)2]x(O)j. Here we choose to introduce the united atom level of description with the C 3 pointing to the terminal methyl group and the O representing oxygen. The terminal O represents the alcohol group. We will model the methyl group to be a bit more hydrophobic than the methylene units and therefore set %C W= 1 • 1 a n d Xc w = 1 ^ • The overall solubility parameter for an ethyleneoxide unit should be such that it is soluble in water. Typically it turns out that one should choose a negative value and use the Ansatz that this parameter is inversely proportional to the temperature Xow = /£ow'300)-22fi where %ow(300) = -0.6 is the value at room temperature. In passing, we note that in the literature there exists a two-state model to deal with the temperature dependence of non-ionics2'. We will not discuss this approach here. An important parameter that drives the demixing of the heads and the tails is the repulsion between O and C for which we choose Xoc = %oc = ^ • The exact value of this parameter is uncertain; this value becomes important when the

1

' P.A. Barneveld, The Bending Elasticity of Surfactant Monolayers and Bilayers, PhD thesis Wageningcn University (1991). 21 P. Linse, M. Bjorling, Macromolecules 24 (1991) 6700.

ASSOCIATION COLLOIDS

4.49

Table 4.2. The Flory-Huggins interaction parameters used in the SCF analysis of the non-Ionic surfactant system at T = 300 K X

ca

c

O

W

V

ca c o w

0

0.5

2

1.5

1.5

0.5

0

2

1.1

2

2

2

0

-0.6

2.5

1.5

1.1

-0.6

0

2.5

2

2.5

2.5

0

V

1.5

bilayer is normally compressed. Bilayers under compression will be discussed in sec. 4.8. To make the set of FH interaction parameters complete, we mention that with respect to the Interaction with the free volume the O is treated as W , i.e. Xyo = 2.5 , and the mixing energy between methylene and methyl groups is set to j

cc

= 0.5 .

Below all lengths are made dlmenslonless by normalization to the characteristic size of the lattice cell for which a value of I = 0.2 nm Is used. The surface tension (grand potential per unit area) is given in dimenslonless units kT 112 . This implies that y = 1 corresponds to 100 mN/m. The tabulated set of FH interaction parameters is relatively large and admittedly to some extent ad hoc. We have used the comparison with the more detailed modelling to estimate most of the parameters, but they still should be regarded as adjustable. We may judge the quality of the set by their consistency with respect to the predictions that result from it and to the comparison of the results with experiments. One should realize, though, that the MD or MC simulations typically include even more parameters In their force fields. Similarly, in analytical models the number of parameters is not small either. We have to live with this. We note that the present free-volume model has approximately twice the number of input parameters as compared with the more primitive incompressible model of sec. 4.3b (i). However, as we will see, this set suffices to deal both with the whole set of non-Ionics, including the Pluronics and related compounds, as well as mixtures of these components. Finally, this parameter set is sufficient to predict the adsorption of the surfactant at the air-water interface without the need to introduce additional parameters (space does not allow us to illustrate this).

4.4a Micelles at and above the c.m.c. We will now discuss in some depth the state of the art of SCF modelling of non-ionic surfactants of the C n E x type, for which we choose three representatives (n,x) = (10,6), (12,5), and (14,4). Later we will also discuss other members of this surfactant family. This set of three surfactants has been selected because all three surfactants have the same number of carbons and as compared with the (12,5), the (10,6) has one more O

4.50

ASSOCIATION COLLOIDS

Figure 4.9. SCF results for three C n E x surfactants, (n,x ) = (10,6), (12,5) and (14,4). a) The translationally restricted (standard state) grand potential as a function of the aggregation number, b) The bulk concentration of the freely dispersed surfactants (monomers) as a function of the aggregation number of the micelles, c) The total volume fraction of surfactant in the system as a function of the monomer volume fraction. The arrows point to the theoretical c.m.c. of the surfactant system, d) The relative size distribution ag/g as a function of the average aggregation number g . The open circles in panels a and b represent the system, which coexists with infinitely long cylindrical micelles to be discussed in sec. 4.6b.

and the (14,4) has one less O unit. Hence, the overall molecular weights of the three components are similar. Nevertheless, the three surfactants are remarkably different in their association behaviour as is known experimentally11. We may refer to fig. 4.1b to a detailed MD-generated snapshot of a non-ionic micelle. Here we will be Interested not in snapshots but in the average micellar properties. Any SCF analysis of micellization starts by analyzing £m(g) as a function of the aggregation number. These functions are given in fig. 4.9a. Typically, £m(g) has a positive slope at very small values of g (dotted line), there exists a part with a negative slope (drawn line), and subsequently there may again be a part with a negative slope (again dotted). As explained, only the sections with negative slopes correspond to micelles that are macroscopically stable. The maxima in the curves correspond to the smallest micelles that are stable. We will refer to these micelles as the ones corresponding to the theoretical c.m.c. " R.G. Laughlin, The Aqueous Phase Behavior of Surfactants, Academic Press (1994).

ASSOCIATION COLLOIDS

4.51

For the surfactant with the largest head group (10,6), there is an aggregation number above which em is negative. We argued above that the physically realistic values for this quantity should be positive, but we have dotted the continuation of the curve towards the negative values. The two other curves do not reach negative values, but develop a minimum in £m{g) • This minimum signals the upper limit of the size of spherical micelles that is thermodynamically stable. This minimum also tells us that the micelle concentration cannot increase further. For these systems we need to consider alternative aggregate shapes. In fig 4.9b we present the volume fraction of the freely dispersed surfactants (monomers) that coexist with micelles with aggregation number g . We note that the chemical potential of the surfactant is (in first order) given by the logarithm of this volume fraction and hence one can interpret the figure also as the chemical potential of the surfactant versus g . It follows from [4.2.19] that there is a direct relation between em and the chemical potential. According to [4.2.19], when £m(g) decreases, ^ s u r f (g) increases and vice versa. Again, we have used a solid line for those parts that are physically significant and dotted the irrelevant parts. Consistent with the idea of a c.m.c, the smallest stable micelles are found at the minimum of the (monomer) chemical potential curve. We will see that other definitions of the c.m.c. may be given. To elaborate on an alternative measure of the c.m.c, we present in fig. 4.9c the relation between the overall surfactant concentration, computed by [4.2.22], i.e. (^ = b , as a special event and assigns the c.m.c. to this concentration. For this reason we will call this point the experimental c.m.c. The break is clearly visible in all but the (14,4) system. For the (10,6) surfactant, the exponential growth of the total surfactant concentration (above the experimental c.m.c.) continues to very high concentrations (volume fraction unity). For the (12,5) surfactant, the total surfactant concentration stops because the micelle concentration is limited. We will see below that at these high surfactant concentrations non-spherical micelles will be stable and dominate the surfactant systems. These alternative aggregated states are disregarded in the present curve. For the (14,4) system, the surfactant head is too small and the apolar tail too long to allow spherical micelles to

4.52

ASSOCIATION COLLOIDS

be sufficiently stable to allow for a high concentration of (spherical) micelles. Below we will argue that this system is close to a cloud point (at room temperature). Several alternative measures of the c.m.c. exist. A practical definition of the c.m.c. is the concentration where half of a differential surfactant addition goes into micelles, i.e. where S implying (pm ~
Figure 4.10. Overall radial volume fraction profiles for spherical micelles of C12 E 5 . a) Near the theoretical c.m.c. for an aggregation number g = 45 . b) Near the experimentally accessible c.m.c. for an aggregation number g = 102 .

ASSOCIATION COLLOIDS

4.53

4.4b Structure of C 12 E 5 micelles Once the set of macroscopically stable micelle sizes is known, it is of interest to discuss their structural characteristics. In fig. 4.10 we present radial volume fraction profiles of C 12 E 5 surfactant micelles. In panel (a) we show the smallest stable spherical micelle at the theoretical c.m.c. In this case the aggregation number is just g = 45 and aggregates only exist at extremely low micelle concentrations. In panel (b) a significantly larger micelle is presented, which is more characteristic of a spherical micelle at such concentration as might be seen by light scattering (i.e. near the experimentally expected c.m.c). The micelle with the smaller aggregation number appeals to our intuition. In the core the tail density is rather homogeneous; the core is virtually free of water and the free volume has slightly accumulated in the tail region with respect to the water phase. All these properties are in broad agreement with expectations. Less obvious is the amount of overlap between tails and head groups. The ethyleneoxide unit is not completely insoluble in a hydrocarbon phase1' because there is just one O near a pair of C's. Hence, we should anticipate that the separation between heads and tails is not as dramatic as one would expect for a charged surfactant. Indeed, the transition region between core and corona is very wide and extends over much of the micelle structure. Such a rather diffuse core/corona interface poses a major problem for accurate analysis by quasi-macroscopic models where this interface is assumed to be (fairly) sharp. The micelle size is rather modest and the core radius is significantly less than the extended tail length. From the thermodynamic analysis of this surfactant system (fig. 4.9a), it was found that £mig) has a minimum. This minimum indicates that spherical micelles above a particular size are unstable. We wish to find some molecular interpretation of this instability. The radial volume fraction profiles presented in fig. 4.9b show a micelle that is close to this instability regime. In this case the aggregation number is approximately 100 at or near the "experimental" c.m.c. As compared with the small micelles, the aggregation number has doubled and the size of the core has extended significantly and is now on the order of the extended length of the tails. From this point of view, it is not surprising that growing the micelles still further becomes incompatible with the spherical shape. There are a number of significant changes in the radial volume fraction profiles as compared with the small micelles that are consistent with the incipient instability. The tail density in the core tends to drop to some lower value in the centre. The head groups now interpenetrate significantly into the core. The water phase, however, is still kept outside the micelle. Eventually, with a still larger aggregation number, the head groups start facilitating the uptake of water in the micelle and the micelle structure becomes unstable. In passing, we note that this scenario of micelle size instability, which presents itself in the SCF calculations, will remain unnoticed for all approaches that impose 11

S. Burauer, T. Sachcrt, T. Sottmann, and R. Strey, Phys. Chem. Chem. Phys 1 (1999) 4299.

4.54

ASSOCIATION COLLOIDS

conformational constraints on the surfactants. One may wonder how sensitive these instabilities are for the parameters in the model. Without going into all the details, it is necessary to mention that all presented results are seen in all reasonable parameter settings. Of course, one can suppress the instabilities by increasing ^ c o or by making Xyjo m o r e negative. These observations are relevant in the context of the temperature sensitivity of the micellar solutions, which will be discussed in the section where we consider the Pluronic surfactants. It Is of considerable interest to also present the radial distributions of individual segments along the surfactant. To this end, we evaluate the number of segments with ranking number s at coordinate r given by ns[r) which are related to the segment volume fraction profiles <jo(r,s) by ns{r) = L{r)(p{r,s), where L{r) is the number of sites in the spherical coordinate system at coordinate r (see appendix 1). We have selected the C 12 E 5 surfactant at the theoretical c.m.c. for this exercise and have numbered the terminal CH3 as segment s = 1 and the terminal oxygen as segment s = 28. One intuitively expects that each segment will have a distribution that is centered around a given most likely position. This average position is expected to be at a lower r for smaller values of the ranking number s . In other words, the tail end should be in the micelle core and the end of the hydrophilic head should mostly point towards the water phase. Moreover, the width of the distribution is expected to be wider for terminal segments than for middle segments. Inspection of fig. 4.11a,b, which provides the segment distribution functions, gives several rather surprising results. Referring to fig. 4.1 la, where some of the tail segments are presented, we notice that most of the tail segments have virtually the same distribution! All distributions peak near the edge of the core and the width of the distribution is only a very weak function of the ranking number. In fig. 4.11b, where the distributions of the carbon groups just next to the oxygen atoms are shown, the situation is more according to intuition. The distribution shifts to larger r and widens for increasing s . The fact that radial distributions of the tail segments are rank independent must be attributed to the spherical topology; only few segments are needed to populate the

Figure 4.11. Segmental distribution functions ns(r) for segments with ranking number s as indicated: a) for segments in the tail and b) for segment next to an oxygen unit in the head group for the C 1 2 E 5 surfactant at the theoretical c.m.c.

ASSOCIATION COLLOIDS

4.55

centre. Moreover, the observation that the core-corona interface is wide makes it is possible that almost every segment in the tail can visit the centre. As most of the volume of the core is at its periphery, this explains that the most likely position of each tail segment is at the core-corona interface. Most "artists' view" representations of spherical micelles are definitely misleading in this respect. From the appendix and specifically [A.5.2], we know that it is possible to define and compute a grand potential density a{r). It is possible to show that this quantity is equivalent to the difference between the normal and the tangential pressure a>(r) = lp N (r)- pT{r)] ( a o / u o ) , where ao/vQ <x \/l is the ratio of the area and the volume of a lattice site. Close inspection of [A.5.21 also reveals that there are various options for defining this difference p N - p T (i.e., the grand potential density). Indeed, this quantity cannot be unambiguously defined. The fundamental reason for this is that there are no strict rules on how to attribute the non-local interaction terms to the various contrib.utions to the local pressure tensor components. The choice made here to compute the grand potential density is that the binary interactions are assumed to be equally distributed over the two components that take part in each pair interaction. Other options are possible; these may lead to different grand potential densities. The caveat is that one choice is not necessarily better than the other. Hence, there should not be any physical consequence of how this bookkeeping is done. Having mentioned this ambiguity, we still may be interested in the question of how the radial distribution of the grand potential density develops in the micelle. In fig. 4.12 we present these distributions, made dimensionless with I3 IkT for the two micelles with g = 45 and g = 101 for which the overall distributions were shown in fig. 4.10. The proper volume integral over the grand potential density distribution is equal to £ m , which is different in the two cases ( em ~ 25 for g = 45 and £m - 1 3 for g = 101.). The grand potential density assumes both positive and negative values. In the head group region it tends to be positive, indicating repulsion between the head groups. The grand potential density drops sharply to negative values near the core-corona interface; the negative values keep the micelle together. In the centre of the micelle the pressure tends to grow again. For the large micelle the local value for p N - p T is clearly positive in the centre, while it remains negative for the smaller one.

Figure 4.12. The radial grand potential density profiles for two micelles of C 12 E 5 with number of surfactants g = 45 and g = 101 as indicated.

4.56

ASSOCIATION COLLOIDS

4.4c Trends for various micellar characteristics It is known experimentally and all models also predict that log c.m.c. drops linearly with the tail length, see [4.1.1 ]. In fig. 4.13a this linear dependence is illustrated for the non-ionic surfactant series with eight EO segments. In the calculations this linear dependence is predicted accurately. The slope is strongly dependent on the actual value used for the j c w parameter for which a value of Zcw = * •1 w a s chosen. One should realize that the conversion from volume fraction to molar concentrations involves a division by the chain length, which is JV = 25 +1. Implementing this conversion has minor effects on the slope. The present parameter settings give logic.m.c.) <* -0.351. Most likely this is a slight underestimation of the experimental slope. Theoretically one can easily increase the slope by increasing ^ c w . In fig. 4.13a it is shown

Figure 4.13. a) The concentration of surfactant at the c.m.c. (in volume fraction units) on a logarithmic scale as a function of the length of the surfactant tail for CtEg non-ionic surfactants. The solid line is the theoretical c.m.c. (i.e. first measure of the c.m.c. where the maximum occurs for e m (g)), the dotted one is the second measure for the c.m.c. (where the volume fraction of micelles equals that of the freely dispersed surfactants, i.e.
ASSOCIATION COLLOIDS

4.57

that two definitions of the c.m.c. ((i) theoretical c.m.c. defined by the appearance of the first stable micelles and (ii) an "experimental" c.m.c. where the micelle concentration equals that of freely dispersed ones) differ systematically in line with the fact that the c.m.c. is not infinitely sharp. However, the scaling of the c.m.c. with the tail length is not affected by this ambiguity in the definition of the c.m.c. Applying arguments based on the surfactant packing parameter, we argued above that the aggregation number in spherical micelles should grow more than linearly with the tail length. As shown in fig. 4.13b, we see that the aggregation number for both measures of the c.m.c. is almost linear with the tail length, albeit that the data are better fitted by a quadratic dependence. We notice however that near the theoretical c.m.c. virtually all surfactant systems form spherical micelles, but that especially as the surfactant packing parameter is not close to 1/3, one should anticipate that shape changes may take place, which cause the aggregation numbers to grow much more quickly than predicted in fig. 4.13b. The growth in aggregation number as a function of the tail length must imply the increase of the size of the core on the one hand and the micelle as a whole on the other 1 '. As shown above, we have information on the distribution function of the individual segments in the micelle. It is rather straightforward to obtain the first moment of such distribution functions to find the average location of a particular segment. We estimate the core size by the average position of segment s = t, which is the last apolar segment of the tail connected directly to the head group. The size of the whole micelle is estimated from the first moment analysis of the terminal oxygen segment s = N . These two size measures are presented in fig. 4.13c. Completely in line with expectations, we find that the core radius increases with the length of the surfactant tail. On top of the core there is a corona, the size of which being a good approximation, independent of the surfactant tail length. This can be deduced from the constant difference between the micelle size and that of the core. From this constant corona size, we may anticipate that the area per molecule am is a weak function of the tail length. We may compute the dimensionless area per molecule am straightforwardly as am = 4/rR20re Ig . The conversion to a real area per molecule is a technical issue. We suggest using the conversion factor 2!2 = 0.08 nm 2 . The factor 2 corrects for the anisotropic dimensions of a lattice site (which may be traced back to the value of Aj =1/3, cf. appendix 1). In fig. 4.13d, it is shown that the area per molecule is a slightly increasing function of the tail length. In real dimensions we predict the area to be on the order of 0.4-0.5 nm 2 per molecule. From experiments it is known that the c.m.c. is only a weak function of the head group size (see table 4.1). In fig. 4.14a we present the volume fraction of monomers at the theoretical and "experimental" c.m.c. for a series of non-ionic surfactants for which 11

C. Tanford, Y. Nozaki, and M.F. Rhode, J. Phys. Chem. 81 (1977) 1555.

4.58

ASSOCIATION COLLOIDS

Figure 4.14. a) The concentration of surfactant at the c.m.c. (in volume fraction units) on a logarithmic scale as a function of the length of the surfactant head x for C16EX non-ionic surfactants. The solid line is the theoretical c.m.c, the dotted one is the experimentally expected c.m.c. b) The corresponding aggregation number at the theoretical c.m.c. (closed symbols) and that at the experimental c.m.c. (open symbols) as a function of the tail length. The drawn curves are power-law fits with an exponent of approximately -0.6 . c) The size of the micelle core and the overall micelle in lattice units as a function of the head group size for the C 16 E X with £m = 10 kT . d) Corresponding dimensionless area per molecule as a function of head group size.

the tail length is fixed to 16 units and the head group size varies from 8 to 32. In line with expectations, the experimental c.m.c. shifts to a higher concentration in line with the theoretical one. The chain length JV, needed for the conversion from volume fractions to molar quantities, is a strong function of the head group size. In this case AT = 16 + 3x + l. With this type of correction, the c.m.c. is an even weaker (but still growing) function of the head group size. The corresponding aggregation numbers are presented in fig 4.14b. As expected, with increasing head group size the aggregation number decreases. The SCF predictions indicate that the aggregation number follows the scaling behaviour g « x~° 6 , both for the micelles near the theoretical c.m.c, as well as near the experimental one. Referring to fig 4.14c, we note that the overall size of the micelle is a linearly increasing function of the head group size, i.e. Rm = 10 + 0.09x . Indeed, the aggregation number goes down, and directly coupled to this the dimension of the core reduces, but this does not compensate for the growth of the corona layer. The thickness of the corona is

ASSOCIATION COLLOIDS

4.59

•^corona x x ° * a n d it exceeds the unperturbed radius of gyration of the head group moiety significantly. The area per molecule is an increasing function of the head group size as can be seen in fig 4.14d. To a good approximation we find that am =*= x02 . This scaling exponent is significantly smaller than 0.5, indicating again that the head group moiety is stretched normal to the core-corona interface.

4.4d

Quasi-macroscopic approaches to non-ionic micelles

From the above it is obvious that it is not trivial to make transparent and predictive (analytical) models for non-ionic surfactant systems. Nevertheless it is of interest to review briefly some of the attempts that are present in the literature. For this, we must return to sec. 4.3c and, in particular, to [4.3.2] where various contributions to the chemical potential of exchange of a surfactant from the solution to the micelle are summed up. To use such a model for the alkyl non-ionics, we need to add terms that capture the relevant aspects of forming a corona made up of the EO head groups. Basically we must account for the solvation effects (i7i ) m i x and the deformation of the head group U^) d e f h • We have seen, e.g. in fig. 4.10, that the corona chains do not have a trivial distribution and there are several ways to approximate what happens. As a first model, it is assumed that the head group density is uniform over some region D, which may be called the effective corona thickness. In such an approach the tail is non-uniformly stretched. In a second model, we assume that there is a radial distribution of segments and propose that the stretching of the corona chains is uniform. Although these models are not completely confirmed by accurate SCF predictions, both lead to relatively simple equations that may serve as first estimates. Regarding the mixing contribution in a corona of homogeneous density, it must be realized that the surfactant in solution has an EO fragment, which is viewed as an isolated free polymer coil swollen in water. In the corona, it is viewed as forming a solution denser in polymer segments compared with the isolated polymer coil. The differences in hydration between the two states contribute to the Gibbs energy of the aggregate. To estimate this, we need the volume of the corona Vs, which may be expressed in the volume of the core Vc , the radius of the core Rc and the corona thickness D VS=VC (1 + — ) 3 - l R

I

c

[4.4.1]

)

The average volume fraction is given by (pc = g(3x +1)/3 / Vc . Introducing the excluded volume parameter v = 1 - 2 j

E W

, where / E W is the overall FH interaction parameter

for an ethylene oxide segment with water, we write for the mixing energy

(i?U

i

4.60

ASSOCIATION COLLOIDS

When the density is allowed to be non-uniform and when it is assumed that the stretching is uniform, the segment density drops with 1/r2 . In this case the mixing term is given by (^W —

1 1 = -xmv

M/1Q]

4.4.3

c

kT 2 l + D/Rc where
[4.4.4,

More recently Persson et al.2) have proposed complementary equations derived from the conventional Flory-Huggins theory for a homogeneously distributed corona region composed of ethylene oxide layers, which account for the contact energy and the mixing energies of core-corona and corona-solvent contributions. When the volume fraction in the head group region is expected to drop with 1/r2 , we may use Flory's31 rubber elasticity theory to estimate the deformation free energy of a chain

kT

2[xl2

D

\

We are not going to review the performance of these phenomenological equations here. For this we refer to an extensive review given by Nagarajan41. In general it will be clear that the SCF results are significantly more in line with experimental findings than the analytical approximations. The fundamental reason for this is rather obvious: the EO head groups are typically too short to allow polymer theories to become accurate and the profiles of the corona cannot be accounted for with sufficient accuracy. Moreover, the overlap between core and corona is, at least for the short EO surfactants, important. The latter effects may be accounted for by a renormalization of the interfacial Gibbs energy y such that the finite fraction of EO groups in the water phase is accounted for.

11

A.N. Semenov, Soviet Phys. JETP 61 (1985) 733. CM. Persson, U.R.M. Kjellin, and J.C. Eriksson, Langmuir 19 (2003) 8152. 31 P.J. Flory, Principles of Polymer Chemistry, Cornell Univesity Press (1953). 4 R. Nagarajan, Theory of Micelle Formation: Quantitative Approach to Predicting Micellar Properties from Surfactant Molecular Structure in: Structure-Performance Relationships in Surfactants, K. Esumi and M. Ucno, Eds., Surf. Sci. Series 70 (1997) 1. 21

ASSOCIATION COLLOIDS

4.61

4.4e Pluronic micelles Not all surfactant systems are based on alkyl chains as the hydrophobic fragment. It is of considerable interest to briefly discuss surfactants that have the propylene oxide PO as the hydrophobic unit. PO is composed of two CH2 s one O and a side-group CH3 . Here we refrain from the side-chain complexity and model the PO unit as (CJgtCHgljfOlj. There is an interesting family of commercial surfactants, the so-called Pluronics, which are tri-block copolymers that have the PPO as a central block and two PEO blocks as the hydrophilic heads. Here we will present the SCF predictions for Pluronic type surfactants. It is of interest to mention that the parameter set used for the alkyl-based non-ionic surfactants (table 4.1) can be used directly to model the Pluronic systems. Below we consider P84, which is (EO)19 (PO)43 (EO)19 . For this surfactant, which according to the literature has an HLB = 14, a c.m.t. (of a 1 % solution) of 28.5°C and a c.p.t. of 74°C1), we will present structural information on the micelles and investigate what is expected upon changing the temperature. Little is known in the literature about the internal structure of Pluronic surfactant micelles. As the PO is not a pure alkyl moiety, the amount of solvent accumulated in the core is unknown. All-atom MD simulations on these polymeric micelles are rather expensive and we do not know of any attempts in this direction. However, SCF predictions for the overall segment volume fraction profiles can be made, and results are shown in fig. 4.15a. In this graph, we present a micelle at the theoretical c.m.c. for which the micelle has an aggregation number of approximately 50. Not unexpectedly the overall structure of the micelle is very similar to that of the shorter alkyl chain surfactants. Because of the polymeric nature of the surfactant, all length scales have increased somewhat. The core has a homogeneous density of approximately 0.8. This includes the CH2 and CH3 and the O units of the PO moiety. Again there is a wide

Figure 4.15. a) Radial volume fraction profiles for the hydrophobic (PPO) and hydrophilic (PEO) moieties of the P84 pluronic surfactant, the water and the free volume profiles, b) The radial grand potential density for the micelle at the theoretical c.m.c. ( g = 49 ). 11

P. Alexandridis, T.A. Hatton, Colloids Surf. A 96 (1995) 1; V.G. de Bruijn, L.J.P. van den Broeke, F.A.M. Leermakers, and J.T.F. Keurentjes Langmuir 18 (2002) 10467.

4.62

ASSOCIATION COLLOIDS

overlap region of core and corona and the head group profile has a characteristic shape very similar to that found for the short-chain non-ionics. The water penetrates significantly into the core. This is also not unexpected because of the O unit in the PO, which attracts water to some extent. One may argue that as the core is significantly larger, the grand potential density has developed such that one would be able to find some signature of an isotropic (Laplace) pressure. This is not the case as shown in fig. 4.12. It is of interest to see that the grand potential density is a scaled-up version of the grand potential density of the non-ionic micelle at the c.m.c. given in fig. 4.12. There is a positive value on the outside where the hydrated head groups feel a positive pressure. The grand potential density is negative inside the micelle. Although the segment density in the core is rather homogeneous, the chain conformations in the micelle are not isotropic, they are stretched because of the "anchoring" on the EO groups in the water phase. As a result, the local pressure difference p N — p T is not homogeneous in the core.

Figure 4.16. a) Grand potential as a function of the aggregation number for P84 Pluronic surfactants for various values of the Zow a s indicated. Note that the line parts with a positive slope correspond to unstable micelles, b) The ^QW " e ^ ordinate) and c.m.t. (K) as a function of the surfactant bulk volume fraction cp° for P84.

(i) C.m.t. andc.p.t As both PO and EO solubilities in water depend on the temperature, it is no surprise that the temperature is one of the main control parameters for non-ionics. De Bruijn et al.11 showed that the SCF model can be used to predict the c.p.t. and the c.m.t. for these surfactants in a very simple way. These authors found an excellent correlation for these quantities between experimental findings and the theoretical predictions, assuming that /ow is the only FH parameter that is temperature-dependent. Of course this is a simplification because all interaction parameters depend on the temperature. However, Zcw a P P e a r s to be a very weak function of the temperature, at

" V.G. dc Bruijn, L.J.P. van den Broeke, F.A.M. Lccrmakers, and J.T.F. Kcurcntjes Langmuir 18 (2002) 10467.

ASSOCIATION COLLOIDS

4.63

least around room temperature. We may also assume that this is the case for the CH3 water contacts. As it is not known how the various parameters that control the free volume depend on the temperature, we keep these parameters fixed here as well. Hence, we propose the following relation between XQSN a n c ' t n e temperature T *OW = " 0 . 6 ^

[4.4.6]

This equation shows that with increasing temperature J O w becomes less negative. This renders both PO and EO less water-soluble. In general, the temperature dependence of the FH parameter can be written as % = A + B/T,

where the A part

refers to the entropic and B to the enthalpic part. Using [4.4.6], therefore, means that we assume here that for all interactions, the FH parameter is temperature independent, i.e. that B = 0 except for Xow f° r which the value of A is zero. This is of course far from realistic. In fig. 4.16a we present predictions for the grand potential as a function of the aggregation number for a series of ^Ow

vames m

the range -0.4 to -1.0. There are

large effects for relatively small changes of the Xow parameter. This means that the micellization is a strong function of this parameter. For %ow > - 0 . 6 , the surfactant becomes gradually less soluble. The region of £m(g) with a negative slope becomes smaller and smaller and eventually for Xow > ~ 0 - 4 vanishes. The absence of the stable micellar region effectively means that the solution cannot maintain free floating spherical micelles. The situation is very similar to the situation discussed for C 14 E 4 , where the stability region was also much narrower than for surfactants with a larger head group. Here we will formally identify the cloud point by the temperature [ Xow^ for which no stable micelles are present. In this case the cloud point is close to %ow > ~0-4 • We realize, however, that already for more negative values of ;fOw • * n e maximum value of the micelle concentration is sufficiently low so that the micelles will remain experimentally undetected. In practice, the c.p.t. is therefore not as high as predicted here. De Bruijn et a!.1' examined a large set of Pluronic molecules and proved that there is a strong correlation between the c.p.t. identified by the loss of stability of the spherical micelles and the experimental data. Experimentally it is known that for non-ionics the c.m.c. is a strong function of the temperature 2 . In an experiment where the surfactant concentration is fixed and the temperature is changed, one may find a micellization transition at a temperature known as the c.m.t. In fig. 4.16b we present the predictions of the c.m.t. as a function of the given bulk concentration of the surfactant P84 using the ad hoc relation [4.4.6] between Xow

anc

'

tne

temperature T . In line with experimental observations, it is

found that that the c.m.t. is a strong function of the bulk concentration, albeit our 11 De Bruijn ct al., loc. cit. 2 M. Tanaka. S. Kaneshina, G. Sugihara, N. Nishikino, and Y. Murata', Solution Behaviour of Surfactants. Theoretical and Applied Aspects 1 (1982) 41.

4.64

ASSOCIATION COLLOIDS

temperature scale is obviously not accurate. Again, de Bruijn et al.1' proved a very good correlation between experimental data for the c.m.t. (for a 1% solution) and the theoretical predictions for a large class of Pluronic surfactants. In conclusion, we expect that the first-order effects of the temperature are reasonably well accounted for in this simple model. 4.5 SCF for (spherical) ionic micelles An important class of surfactants is that of the alkyl-based, single-chain, ionic amphiphiles. Spherical micelles composed of these surfactants are centrally placed in this section. Basically we will take the same approach as in the previous section. We will first discuss the SCF predictions and thereafter summarize the pertaining aspects of the semi-macroscopic approaches. SCF analyses of micelles composed of charged surfactants give a unique opportunity to include electric double layers. We not only study the diffuse part of the double layer, but also account for the mechanism by which the surface charge is generated (i.e. the self-assembly process) and for specific interactions with counterions. The surfactant charge and diffuse double layer in ionic micelles have many similarities with the electric double layer of ionic surfactants densely packed at the air-water interface [see III.fig. 3.17, p.3.52]. In the SCF model, the conformational statistics and the short-range interactions are already accounted for. The extension to account for the electrostatic interactions involves the integration of the formalism with the Poisson equation, which allows the evaluation of the electrostatic potentials resulting from given charge distributions. In order to compute the distribution of the (charged) molecules, one should generalize the segment potentials by including the appropriate electrostatic terms. We again refer to appendix 1 for the details and limit ourselves to the most important issues here. The idea is to borrow as much as we can in terms of molecular characteristics from the surfactant models discussed above. In fig. 4.17 we give a schematic representation of the molecules that we shall consider in the modelling of the ionic surfactant systems. We will use the same water model as for the non-ionics, including the free volume component. In addition, there are small cations and anions where it is understood that the theory can be developed in the absence and presence of extra electrolytes. We take the ionic surfactant as the alkyl chain with a charged head group. In fig. 4.17 we give the cationic surfactant CTA, which has a C16 tail and a positively charged group surrounded by three CH3 units and the anionic surfactant DS, which has a C12 with a sulphate group modelled here as a unit that occupies five sites all having a charge of v = -0.2. These molecules mimick the paradigms of surfactant science CTAB and NaDS. 11

De Bruijn et al., toe. cit.

ASSOCIATION COLLOIDS

4.65

Figure 4.17. Molecular building bricks of the molecules involved in the SCF modelling of ionic surfactants. CTA = cetyltrimetylammonium (cation). DS = dodecylsulphate (anion), a cation composed of 5 volume units, a similar anion and a water unit with the same size.

We evaluate the electrostatic potential in and around the charged micelles from the charge distribution using the Poisson equation. In order to accomplish this, the dielectric permittivity profile in the system must be known. In the core of the micelle the dielectric permittivity is low, but in the water phase the relative dielectric constant should be close to 80. To account for the gradients in the dielectric permittivity, we use the local density average, i.e. f £

o A^A' r '

£0£{r)=

[4.5.1]

A

where the index A runs over all segment types at position r, and £A is the relative dielectric constant of the phase composed of pure component A and eQ is the permittivity of vacuum. In the following calculations, we take all relative dielectric constants to be 80 except for the V units for which we choose unity (vacuum) and except for ec = £CH = 2 . It is typical for this kind of analysis that the concentrations of dissociating substances (NaBr, NaDS, CTAB) are counted as electroneutral volume fractions. This is in line with the thermodynamic tenet that the solution and a micelle (including the counter charge) are electroneutral. The conversion from volume fractions to concenTable 4.2. The Flory-Huggins interaction parameters used in the SCF analysis of the ionic surfactant systems at T = 300 K. S is a unit of the sulphate head group, N is the central unit in the trimethyl ammonium group.

c3

C

S

N

cation

anion

W

V

0

0.5

2

2

2

2

1.5

1.5

c s

0.5

0

2

2

2

2

1.1

2

2

2

0

0

0

0

0.5

2.5

N

2

2

0

0

0

0

0.5

2.5

cation

2

2

0

0

0

0

0

2.5

anion

2

2

0

0

0

0

0

2.5

W

1.5

1.1

0.5

0.5

0

0

0

2.5

V

1.5

2

2.5

2.5

2.5

2.5

2.5

0

X

4.66

ASSOCIATION COLLOIDS

trations depends on the molar volume. A conversion factor of 10 for the small ions is rather accurate, I.e. (ps = 1 corresponds to 10 Molar. As compared with the modelling of the non-ionic systems, we now have to consider several additional molecular species. Except where mentioned otherwise, the model encompasses the following features. The ions are modelled as the water molecules, but with a central unit that carries either a positive or a negative charge (c.f. fig. 4.17). The Flory-Huggins parameters for all molecular species are collected in table 4.2. Here we specifically have chosen to model the cationic and the ionic species to be structurally identical and completely symmetric in respect to the short-range interactions. So, in this approximation specific differences between cationics and (structurally identical) anionics are ignored. So, the main difference between the two surfactants is the chain length; secondary differences, such as those mentioned in sec. 4.1c (iii), trend (viii), are ignored, but differences in hydrophilicity of the head groups are accounted for. Typically we assigned a large repulsion to the interactions between the charged species with the hydrocarbon units C and CH3 , even more repulsion with the free volume V and athermal interactions with water. We know that ion-specific effects do matter in micellization and briefly touch upon these effects in sec. 4.5d. The short-range interactions felt by the head group units (sulphate and N in the trimethylammonium) are of course much less important than the effects of their charges. We give these groups a weak repulsion with water, an athermal interaction with the small ions and a significant repulsion with the hydrocarbon units. 4.5a Micelles at the c.m.c. in various salt solutions We will start our analysis by comparing the thermodynamic data for the two mentioned surfactant systems over a large range of ionic strengths. As before, we consider em . In fig. 4.18a we present a set of curves for the anionic NaDS and in fig. 4.18b similar data for the CTAB. With respect to the excluded volume characteristics, we have modelled both surfactants very similarly (c.f. fig. 4.17). The CTAB surfactant has three methyl groups surrounding the positive charge and is therefore more hydrophobic than the NaDS. We have modelled the sulphate by five identical units, all with a reasonable solvency and a charge of -0.2 elementary units. From fig. 4.18a, one can observe a gradual transition regarding the ionic strength contribution of the added electrolyte. For relatively low electrolyte concentration, i.e. cps < 0.002 , the micellar properties are independent of the added salt. At higher ionic strength, there is a significant effect on the micellization of the amount of added salt. This reflects the fact that the ionic strength in solution is for an increasing extent determined by the added electrolyte. When this concentration becomes high, increasing screening of the double layer allows the micelle to grow in size, in the example by up to a factor of two. For the more hydrophobic CTAB surfactant, the same is found at lower ionic strength. However, as shown in fig. 4.18b, it is predicted that the stability region for

ASSOCIATION COLLOIDS

4.67

Figure 4.18. Standard state grand potential e m as a function of the aggregation number for various values of the salt concentration of a 1:1 electrolyte for a) NaDS and b) CTAB surfactant system. To convert the volume fractions of ions to a molar concentration, one needs a conversion factor of 10, i.e. the molar concentration is 10 times the volume fraction of the ions.

micelles (i.e. dem{g)/dg <0) is gradually lost with increasing ionic strength. For tps > 0.002, there appears a minimum in £m{g) and the value of this minimum shifts to higher £m(g) with increasing ionic strength. At cps > 0.05 the stability is lost and the surfactant does not dissolve anymore (Krafft point). This behaviour mirrors the clouding of non-ionic surfactants. Referring to 14.2.24] we know that the slope of e(g) is related to the fluctuations in the aggregation number. The steeper this curve, the more narrow the relative size fluctuation. The width of the size distribution has many features in common with that of the non-ionic surfactants. Near the c.m.c, there are relatively large size fluctuations at surfactant concentrations near some experimental c.m.c, i.e. where the surfactant monomer volume fraction is approximately equal to the volume fraction of micelles. The fluctuations are relatively independent of the surfactant concentration and almost independent of the average aggregation number. These fluctuations are found smaller for the ionic micelles than for the non-ionics. Inspecting fig. 4.18a, we notice that for NaDS the fluctuation in aggregation number is a weak function of the ionic strength in the solution. In line with experimental findings, we predict a I g-0.2 for low salt and a /g ~ 0.15 for the high salt concentration. The same applies for CTAB in the regime of relatively low ionic strength. However, when the ionic strength is increased to values near the Krafft point, a strong increase in the size fluctuations is observed. The volume fraction of freely dispersed surfactants that coexist with the smallest stable micelles is the (theoretical) c.m.c. In fig. 4.19a we present the c.m.c.s for NaDS and CTAB as a function of the ionic strength. In line with the results discussed above, we see that at very low ionic strength the c.m.c. is a very weak function of the ionic strength. At relatively high ionic strength, there is a power law-like dependence on the ionic strength, i.e. c.m.c. »= [
4.68

ASSOCIATION COLLOIDS

Figure 4.19. a) The theoretical c.m.c. in volume fraction units as a function of the (bulk) volume fraction of 1:1 electrolyte for NaDS and CTAB surfactant systems, b) The aggregation number at the c.m.c. as a function of the volume fraction of 1:1 salt for NaDS and CTAB.

range solvatlon; Its size etc.) starts to determine the repulsion between head groups rather than the electrostatic repulsion. Such leveling off of the electrostatic contribution is not (yet) seen in the presented systems. The dependence of the c.m.c. on the ionic strength of NaDS is similar to that of CTAB except that for the latter spherical micelles cannot be found at salt concentrations in excess of cps = 0.05; the CTAB curve stops at this salt strength value. Apparently CTAB micelles need a sufficiently strong electrostatic repulsion between the head groups to have a sufficient stopping force for micellization; the three methyl groups in the head group give it a hydrophobic character. In this case, the scaling exponent of the ionic strength dependence of the c.m.c. is slightly higher than in the NaDS case. For CTAB we find an exponent near -0.8. In experiments similar power law values are found11. The aggregation number at the theoretical c.m.c. also depends on the salt concentration. In fig. 4.19b we present the corresponding aggregation numbers in the ionic strength regime of 10~3 <
C. Tanford, J. Phys. Chem. 78 (1974) 2461.

ASSOCIATION COLLOIDS

4.69

4.5b Radial profiles In fig. 4.20a we present overall radial density profiles for the spherical micelle at the c.m.c. of NaDS surfactants at the added salt volume fraction of tps = 0.01 (equivalent to 0.1 molar). Radial distributions are presented for the apolar tails, the polar sulphate group, the water and the free volume. The apolar core is virtually free of water and the free volume enriches the core slightly more than it does bulk water. The head group profile is situated on the aqueous side of the micelle. There is a significant overlap of the tail distribution with that of the head groups. The head groups are not positioned at an infinitely sharp plane. In fact, the distribution of the head groups is bell-shaped and extends over some five lattice layers. This means that the head group distribution is at least 1 nm wide, which is comparable with the radius of the core. The distribution of the ions is given in fig. 4.20b. As expected, the co-ions, labeled Cl, are expelled from the head group region, and even more so from the core. The counterion (Na) concentration is significantly increased in the head group region. Indeed, the counterion can easily penetrate between the head groups. The head group density in the corona remains very low. The latter is of course correlated to the relative large width of the head group distribution. In passing we note that the low concentrations of the co-ion (Cl) in and around the micelle imply the general phenomenon that micelles expel salt.

Figure 4.20. a) Radial volume fraction profiles for spherical NaDS micelles at the c.m.c. at volume fraction of (salt NaCl)
4.70

ASSOCIATION COLLOIDS

As a result of the penetration of ions into the head group region, only a fraction of the countercharge resides in the diffuse part of the double layer. In addition, no very high potentials are created. The radial charge density profile is presented in fig. 4.6c in combination with the radial electrostatic potential profile. There is an excess negative charge in the head group region and an excess of positive charge both in the core and in the bulk solution just outside the micelle. The electrostatic potential is negative with respect to the bulk solution; it is most negative inside the core and decreases exponentially outside the micelle, as expected for the diffuse part of the electric double layer. In fig. 4.20d we present yet one more example of the dimensionless radial grand potential density profile. We note that the non-local contributions to the grand potential density are equally distributed over the monomers to which the interactions refer. This choice is arbitrary and in line with the fact that the local pressure tensor components p N and p T in the micelle are not experimentally observable quantities. Nevertheless, we find a complex structure of the grand potential density, which we are not going to try to explain in any detail. Note that the local grand potential density in the core is negative. This means that the volume contribution to the formation of micelles is negative (i.e. favourable); clustering of tails is the driving force for micellization.

Figure 4.21. a) Radial volume fraction profiles for spherical CTAB micelles at the c.m.c. at volume fraction of salt q>s = 0.001 . Given are the tail, head, water and V components, b) The distribution of the anion and the cation, c) The radial electrostatic potential (left ordinatc) and charge density (right ordinatc) profiles. In the inset the electrostatic potential profile on the outside of the micelle is plotted on a logarithmic scale, d) The radial grand potential density profile.

ASSOCIATION COLLOIDS

4.71

In fig. 4.21 we present the radial profiles for the CTAB surfactant system at cps = 0.001. In first order all the results are very similar to those of NaDS, but there are significant, specific differences worth discussing. The spherical micelle at the c.m.c. CTAB is slightly larger than that of NaDS: the interface between core and corona is shifted slightly to larger r values. It is significant that the overlap of tails and head groups is larger for the CTAB micelles than for the NaDS and, therefore, the average head group position is approximately the same for both micelles. The overlap between heads and tails must be attributed to the more hydrophobic nature of the CTAB head group. This has corresponding consequences for the charge-potential relations. The electrostatic potential inside the micelle is significantly higher for the CTAB micelle (about 225 mV as compared with roughly -70 mV for the NaDS micelle). The main reason for this is that the ionic strength is ten times lower in the first case. The high electrostatic potentials in the corona of the CTAB micelles result from the fact that the counterions do not readily enter the interior of the micelle; there is only a modest increase of the counterions in the head group region in the CTAB micelles as compared with that in the NaDS ones. As a result, the excess charge in the head group region is significantly larger for the CTAB micelles (about 50%) as compared with that of the NaDS. We may compare the electrostatic potential just outside the micelle, i.e. at layer r = 10, and compare this with the £ -potential (potential at the shear plane). The absolute value of this potential is i// ~ 7 mV in 0.1 M salt (NaDS micelle) and a factor of 6.5 higher ( ^ = 45 mV) in 0.01 M salt (CTAB micelle). Ion specific effects can disturb this (idealized) prediction. Such effects are discussed below (sec. 4.5d) in more detail. The fact that not all counterions are in the diffuse part of the double layer, but that a significant fraction of the counterions can penetrate the corona and locally compensate the head group charge is familiar. This effect is not easily quantified from the SCF results. Before one can do this, one needs to define the head group region and specify some rule to judge those ions considered to be diffuse and those ions directly taking part in the charge compensation. To get a rough estimate, we evaluate the direct charge compensation at r = 6 at the maximum in the head group profile. At this coordinate, a fraction 0.46 of the charges of CTAB (0.01 M salt) is masked by counterions whereas this fraction grows to 0.67 for the NaDS case (0.1 M salt). Otherwise stated, the "free" fractions of the countercharges are / = 0.54 and / = 0.33 , in 0.01 and 0.1 M electrolyte, respectively. Again, these numbers are likely to increase when counterions are specifically bound to the head groups. In the MD-generated snapshot of an ionic micelle as given in fig. 4.1a, we can also see that many of the counterions have accumulated in the head group region. Quantifying this counterion binding via MD simulations is not trivial either; it is similarly difficult in SCF modelling. In passing, we point to the inset of fig. 4.20c where we present the logarithm of the electrostatic potential as a function of r in the diffuse part of the electric double layer. The linear slope proves that in this region the electrostatic potential profile drops

4.72

ASSOCIATION COLLOIDS

exponentially in full accordance with the Debye Hiickel prediction. The Debye length is approximately 1.6 nm (here we use the lattice site with length 0.3 nm). For reasons of completeness, we also present the grand potential density profile across the CTAB micelle. Without going into further details, we mention that the pressure profile in the head group region is very similar to that for NaDS micelles; there is a positive excess pressure in the head group region and a negative one in the tail region. 4.5c Chain length dependence Experimental evidence (see [4.1.1 ]) has shown that often log c.m.c. = A-Bt where the slope B is about 0.3 for ionics and 0.5-0.6 for non-ionics. SCF theory can account for this difference. To this end, in fig. 4.22 we present computed c.m.c.s for a series of alkyl trimethylammonium bromide surfactants for three values of background electrolyte concentration. From this figure it is seen that both at very high ionic strength, i.e.
Figure 4.22. The c.m.c. as a function of the tail length for trimethylammonium bromide surfactants for three values of the bulk volume fraction of added salt as indicated.

ASSOCIATION COLLOIDS

4.73

4.5d Specific ion effects Ion specificity can also be modelled and by way of illustration we shall now demonstrate how to account for variations in ionic solvation and radii. In practice these two trends are closely related, see 1.5.3d. In a model, one can separate these issues. (i) Counterion solvation We choose the CTAB system with (ps = 10~3 as our reference system. Referring to fig. 4.17, we have modelled the anion as a negatively charged unit surrounded by four units of W (mimicking water). The key idea here is to replace the W component by some new unit named Y and give the anion the chemical composition (Br)(Y) 4 . If we choose the interaction parameters %YA = _£WA for all segment types A, we return to the CTAB system. Let %Yc a n c ' %Y CH ' 3e *-ne on ty parameters that differ with respect to this set. These two parameters will control the change in Gibbs energy upon placing a unit Y near or in the hydrophobic core. We have seen above that the corona of the CTAB micelles is not too polar (the head groups are buried more or less in the interface between core and water) and in this way we expect to modifiy the efficiency by which the counterion can enter the core as well as the corona region. This, in turn, is expected to influence the micellization. As in the starting system, we set %Y CH - %YC = 0.4 for simplicity. This means that in this model the results are fully specified by the value of %xc • The thermodynamic data of the micellar systems collected for the CTA surfactant systems with modified counterion hydrophobicities are given in fig. 4.23a. All systems contain a volume fraction of a 1:1 electrolyte (with the same counterion and the default co-ion) of
Figure 4.23. a) The standard state grand potential £nl as a function of the aggregation number for cetyltrimcthylammonium surfactants with a counterion, of which the hydrophobicity varied as indicated. The volume fraction of the 1:1 electrolyte is tps = 10~3 and all other parameters arc the same as in the CTAB system, b) The c.m.c. in volume fraction units as a function of the hydrophobicity of the counterion.

4.74

ASSOCIATION COLLOIDS

more easily come near the core and partition into the corona. Consequently, the charge is more efficiently screened. It is therefore no surprise that the grand potential curves of the micelles develop similarly as with increasing ionic strength (cf. fig. 4.18b). The curve for %YC = 0.6 , is very near the Krafft point, and it resembles the starting system with the added salt volume fraction of cps = 0.05 . Apparently the overall hydrophilicity of the head group is the sum of that of the head and the corresponding counterion. With decreasing %YC, the hydrophilicity of the head group diminishes and the surfactant packing parameter P increases. As a result, the stability of the spherical micelles diminishes. In line with the discussion of these systems at elevated ionic strength, we anticipate that the more hydrophobic head groups will more likely promote the formation of cylindrical micelles. These effects are well known in the literature and we will return to these effects below. Above we have analyzed what happens when the counterion is made more hydrophobic. Similar physics may be observed when the counterion is explicitly attracted to the head group. This is another aspect of specific adsorption. Again, we should expect that the charges in the corona are more effectively screened and the effective hydrophilicity of the head group diminishes. (ii) Counterion size The second important ion-specific effect that can be sequestered in SCF modeling is the size of the counterion. We choose the NaDS system to investigate this. As mentioned above, the counterion in the original system occupies five sites and may be expressed as A4 = (Na)1(W)4 with the four W units surrounding the centrally charged unit. The key idea to investigate the effect of the size is to reduce the number of W units around the Na but retain the lattice size. Introducing A3 = (Na)j(W)3 , A2 = (Na)1(W)2 , Al = (Na)j(W)j and A0 = (Na)j, we obviously reduce the size of the counterion. We should realize from the start that by removing W units we are also going to change the overall hydrophilicity of the counterion. The smaller the ions, the more the ions will behave as point charges and the less they are influenced by the non-electrostatic, excluded volume effects. The standard state (translationally restricted) grand potential is plotted in fig. 4.24a as a function of the aggregation number g . The micelles formed in the presence of small counterions are typically larger than those in the presence of large counterions. Indeed, the smaller counterions can more easily penetrate into the head group region making the micellization more favourable. In fig. 4.24b it is shown that the c.m.c. decreases with the decreasing size of the counterion and that the aggregation numbers go up (as mentioned above). These changes are significant but modest. The results are in line with the experimentally observed dependencies of the c.m.c. for various counterions (see table 4.1). We add the caveat that in practical situations we must expect that by changing the counterion one invariably also changes more parameters at the same time and the experimental situation is obviously more complicated.

4.75

ASSOCIATION COLLOIDS

Figure 4.24. a) The standard state grand potential e m as a function of the aggregation number g or 5 different counterions A0, Al, A2, A3 and A4 as defined in the text (only the first and last one are indicated), b) The theoretical c.m.c. (in volume fraction units) (left ordinate) and the aggregation number at the c.m.c. (right ordinate) for various counterions A0, ..., A4 as indicated. The lines are to guide the eye. 4.5e Quasi-macroscopic approach to ionic micelles When anionic or cationic surfactants form micelles, one should account for the presence of the electric double layer(s) in the system. The analytical computation of these interactions is obviously complicated by a number of factors, such as discrete charge effects (which is beyond Poisson Boltzmann and SCF), specific ion effects (as discussed in sec. 4.5d), the exact size and shape of the surfactant head group and the dielectric permittivity in the corona. One invariably has to make drastic approximations. Typically one considers the head groups to lay in a plane with a Stern layer on top. Let the radius of the core be given by Rc and within a Stern approximation, let the positions of the first counterions be located a distance S from the core. The head group area a h is evaluated at the distance 8 from the core and will be expressed as as. There is an exact solution of the grand potential of the diffuse part of the flat electric double layer, see [II.3.5.20]. Evans and Ninham11 also give an accurate curvature correction so that the results can be applied to various geometries

^ ^ = 2[lnf- + Vl + (s/2) 2 ]--(Vl + (s/2}2-l) - ^ l n f I + iVl + (s/2)2] kT

[ (2

) s\

I

KS

(2

2

j

[4.5.2]

where s/2 is the dimensionless charge in the diffuse part of the double layer which in [II.3.5.14] was written as s/2 = - p o 4 , where s=

Ane2

[4.5.3]

eoeKaskT and K the reciprocal Debye length. The first two terms of [4.5.2] are appropriate for the flat electric double layer whereas the third term comprises the curvature correction. The parameter C must be chosen according to the geometry. It is C = 2/(Rc + S) 11

D.F. Evans. B.W. Ninham, J. Phys. Chem. 87 (1983) 50025.

4.76

ASSOCIATION COLLOIDS

for spherical geometry and C = 1/(RC + S] for cylindrically shaped surfaces. Note that [4.5.2] yields an electrostatic surface (pressure) contribution of approximately 2/cT/a charge (planar case)1'. We expect that the radial density profiles of the spherical micelles of the charged surfactant systems as given in figs 4.20 and 4.21 are realistic. However, using [4.5.2] to stand for the complete electrostatic contribution to i2£ implies a number of ad hoc simplifications. We have argued above that the success of any model lies in a matching balance of approximations. Taking the full charge density of the head groups sitting on the micelle surface obviously ignores the distribution of the head group charges, the local charge compensation of small ions that penetrate the corona region and effectively reduce the charge, the fact that the head group charges are in a relatively low dielectric permittivity (especially when the charge is very close to the core-water interface), etc. Moreover, it neglects the part of the electric double layer inside the core of the micelle. A micellar model, which makes use of [4.5.2], has 8 as an adjustable parameter, which may be used to change the surface charge density of the micelle to some extent. In such a way, one can "correct" for the fact that only a fraction / of the ions is in the diffuse part of the double layer and the fraction (1-/) is in between the head groups. All of this is somewhat artificial. For an illustration of a model in which Stern layers are explicitly accounted for, see ref. . 4.6 Linear growth of micelles With increasing overall surfactant concentration, the primary effect is that the number of micelles rises. A secondary effect is that the aggregation number increases. When the number of micelles per unit volume grows, the translational entropy assigned to each micelle invariably goes down. This means that em diminishes as well. Of course, in reality systems compensate for this by increasing the aggregation number. For sufficiently low values of em, it becomes relatively inexpensive (in terms of Gibbs energy) to allow for shape fluctuations. As a result, we might anticipate a transition from spherical geometry into elongated ones. When that occurs the strategy of the system has completely altered. The primary effect of raising surfactant concentration is now the growth of micelles rather than the increase in number of micelles in solution. It is often found that a system rather suddenly changes its mode of response and in experiments one can notice this change in the rather abrupt changes in measurable properties. The transition from spherical to elongated micelles takes place at a concentration that is known as the second c.m.c. Depending on the surfactant architecture and concentration, there are also limits for further expansion of the aggregation number and linear micelles will in turn give way to lamellar aggregates. We

11

V. Srinivasan, D. Blankschtein, Langmuir 19 (2003) 9932.

ASSOCIATION COLLOIDS

4.77

shall now discuss the basic elements of these transitions starting with the appearance of linear micelles.

4.6a Phenomenological model for rod-shaped micelles Consider a linear micelle with a cylindrical middle part of length Lc , which on both sides ends with a hemispherical cap. A central assumption in the modelling of linear micelles is that the excess grand potential of a surfactant in such a micelle can be expressed as a superposition of the excess grand potentials of the surfactant felt in the various microenvironments. A surfactant in the cylindrical part is in a different environment than those positioned in the ends and already from a chain-packing point of view we then expect that there are gradients in (Gibbs) free energy densities along the micelle. For linear micelles to compete favourably with spherical micelles, it is necessary that the excess grand potential per molecule for a cylindrical part i2j? j is lower than that of the molecules in the endcap .£§phere . The latter quantity was discussed in the previous sections. The cylindrical counterpart can be evaluated similarly. For example, the conformational entropy loss for the surfactant tail packed in a cylindrical geometry differs from that in a spherical geometry, and the Gibbs energy to develop an electrostatic double layer is a function of the geometry. At this point we assume that for a given chemical potential of the surfactant both £2^x, as well as •Gl?pilere . are known. We note that at equilibrium there are no gradients in the chemical potential of the surfactants. This means that the differences between Q®, and •fi°iiere will give rise to an excess Gibbs energy associated with the endcaps. For a spherocylindrical micelle containing g-m surfactant molecules in the cylindrical part and m 12 molecules in each hemispherical end, we have for the excess chemical potential per molecule in a spherocylindrical micelle with an aggregation number g, &!c{g) <&<9> = ^ y ^ y l ^ ^ p h e r e = < 1 + J(Sphere " < l )

I4-6-1!

for g > m . Note that for such micelles the value of the aggregation number g can go to infinity without a violation of the packing constraints. From [4.6.1] we can define the excess Gibbs energy associated with the presence of the two endcaps AGec= S p h e r e - < l )

I4'6'21

We may also consider the formation of rings. For a worm-like micelle longer than a few times its persistence length, it should be possible that the two ends of the same micelle merge forming a closed loop. The system will gain the endcap energy by doing so. However, the entropy loss associated with the formation of a ring is typically larger than this gain so that rings can safely be ignored. To get information on the length distribution of linear micelles, it is quite common

4.78

ASSOCIATION COLLOIDS

to set up a mass action model in which between all micelle sizes an equilibrium occurs where the aggregation number g grows by integer numbers11. It appears to us that in the first-order model without any loss of physical reality the "discretization length" can be chosen larger, i.e. the size of the spherical micelle. In such a coarse model, the linear micelles exist with discrete lengths L = 1,2

,°° , where the unit length Rs is the

size of a spherical micelle size that coexists with the linear ones. Let there be various equilibria between micelles of different sizes. In particular, we consider the equilibrium between micelles that differ by one length unit X

L+Xl^XL+l

[4.6.3]

for each value of L . In [4.6.3] the X refers to a species and the subindex refers to the length. When we consider the reaction constant for this reaction the true reason for the coarse way of discretization becomes clear. In this "reaction," the endcap energy of the Xj component is gained when the micelle grows from length L to L +1 . The reaction constant K is assumed to be independent of L and only a function of this endcap Gibbs energy K = exp

^

[4.6.4]

Now the volume fraction of a micelle with length L can be expressed in terms of that of the volume fraction of micelles with unit length. Realizing that the micelle with length L assumes L times the volume of that with L = 1 , we find the size distribution
^e-aL

[4.6.5]

where -a = lnfK^), in line with more detailed treatments11. This size distribution in terms of the length of the worm-like micelles may be converted into the corresponding distribution as a function of the aggregation number. In first order one assumes that the aggregation number is proportional to the length L . The proportionality constant is the number of surfactants per unit length gL . The size distribution is for very small L linearly increasing with L (and thus with g ) and for large L (large g ) exponentially decaying. Such an exponential size distribution is of course much wider than the size distribution found for spherical micelles (in first approximation Gaussian). The value I / a may be interpreted as the (dimensionless) most probable length (in units of size Rs ) of the micelle. At fixed value of K, we see that the most likely micelle size increases with an increasing concentration of the spherical micelles with which they coexist. Similarly, at a fixed concentration of spherical micelles the size of the wormlike micelles increases when the endcap energy increases. Note that in this model the product Kip^ will always be smaller than unity.

11

J.C. Eriksson, S. Ljunggren, Langmuir 6 (1990) 895.

ASSOCIATION COLLOIDS

4.79

The total volume fraction of micelles is found by
Pm = P l + 2 K t f + 3 K g g f + - =

( i

_^)2

[4.6.6]

This equation may be inverted to find the volume fraction of the micelles with unit length as a function of the overall volume fraction of micelles 1 l-j4K