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arXiv:cond-mat/0501523v2 [cond-mat.soft] 30 Jul 2005

Nonadditivity of Polymeric and Charged Surface Interactions:

Consequences for Doped Lamellar Phases.

O. A. Croze and M. E. Cates

School of Physics, University of Edinburgh

JCMB King’s Buildings, Mayfield Road

Edinburgh EH9 3JZ, Scotland

February 2, 2008

Abstract

We explore theoretically the modifications to the interactions between charged surfaces across

an ionic solution caused by the presence of dielectric polymers. Although the chains are neutral,

the polymer physics and the electrostatics are coupled; the intra-surface electric fields polarise

any low permittivity species (e.g., polymer) dissolved in a high permittivity solvent (e.g., water).

This coupling enhances the polymer depletion from the surfaces and increases the screening of

electrostatic interactions, with respect to a model which treats polymeric and electrostatic effects

as independent. As a result, the range of the ionic contribution to the osmotic interaction

between surfaces is decreased, while that of the polymeric contribution is increased. These

changes modify the total interaction in a nonadditive manner.

parallel surfaces, we investigate the effect of this coupling on the phase behaviour of polymer-

doped smectics.

Building on the results for

1Introduction

Many processes in soft and biological systems take place in water and involve the interaction of fatty

components, such as membranes or macromolecules. The polar nature of the aqueous environment

means these components often acquire surface charges, so that electrostatics plays a key role in

determining their physical behaviour. The subject has undergone a substantial revival recently,

especially because of its biological relevance [1].

The description of such electrostatic systems generally invokes a continuum approximation

(see, e.g., Kjellander in [1]): the electrostatic properties of the solvent (water) and of all uncharged

components are accounted for via their electrical permittivity. Membranes in water, for example,

have been usefully modelled as dielectric films of low permittivity residing in a high permittivity

medium [2, 3]. In general, any dielectric components present, even if neutral, will be involved in

modulating electrostatic interactions, because of their polarisation in the electric fields generated by

the charged components of the system. This creates nonadditivity, which is not always recognised.

Specifically, in modelling neutral polymers between charged surfaces, the electrostatic and polymeric

contributions to the intersurface forces are usually treated as independent (see, e.g., sec. 10.7 of

[4]). With charged polymers (polyelectrolytes or polyampholytes), their effects on fields is often

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attributed solely to the polymer charges, neglecting the dielectric backbone [5]. An exception is

the work of Khokhlov et al. [1], in which the polarisation of polyelectrolytes is explicitly included

in a description of their adsorption onto an oppositely charged surface. Dielectric phenomena are

also well known to be implicated in the distribution of electrolytes in the neighborhood of proteins

[6].

In this paper we directly address the interdependence of the polymer physics and electrostatics

when charged surfaces interact across solutions of neutral polymers. To gauge the extent of the

nonadditive “coupling effects”, we have constructed a simple mean field model which includes such

coupling, and calculate the force between charged surfaces across a neutral polymer solution. The

model is then adapted to predict the phase behaviour of polymer-doped lamellar phases. Recent

investigations of these composite liquid crystals have shown that a substantial amount of polymer

can be incorporated in the lamellar phase and can induce phase separation (see [7] and references

therein). The polymer has been found to reside either within the bilayers [8, 9] or within the water

layers; in the latter case it may adsorb onto the bilayers [10, 11, 12] or it may not [13, 14, 12];

this depends on the local polymer–bilayer interaction. Without charges, the problem provides an

experimental realisation of the textbook example of a polymer solution confined in a slit [13, 15].

Most lamellar phases are charged, however. Some existing descriptions correctly recognise that

polymers can affect electrostatic interactions by reducing the effective permittivity of the solution

confined between bilayers [16, 17, 18, 19], but ignore the feedback of the electrostatics on the

distribution of polymer segments themselves. A fully consistent description must either address

this, or give a good reason to neglect it. Below we explore this issue further. For simplicity our

work is restricted to the case of nonadsorbing polymers whose monomer density vanishes at either

of the confining walls.

The paper is organised as follows. Section 2 describes our mean field model and how it can be

used to investigate the properties of polymer-doped lamellar phases. The results of our model are

then presented and discussed. In Section 3 we show how, for experimentally reasonable parameters,

an account of coupling modifies the variation of both the electrostatic potential and the polymer

concentration between surfaces of fixed separation.

interaction between the plates is also modified with respect to the uncoupled results from the same

model. With the mapping described in Section 2, the osmotic pressure can be used as an equation

of state to predict the phase behaviour of doped lamellar phases. The details of this procedure

and the results of parameter variation for the phase behaviour of doped smectics are presented and

discussed in Section 4. Section 5 discusses the relation to experiment and in Section 6 we draw our

conclusions.

As a result of these changes, the osmotic

2Model

We consider a solution of neutral polymers, confined between charged surfaces (Fig. 1) and contact-

ing a reservoir with which it can exchange heat, polymer chains and electrolyte (salt). By virtue

of the surface counterions and the salt ions, the solution is electrolytic and screens the charged

surfaces. Since the polymers are nonadsorbing, their monomer density vanishes at both plates.

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σ

σ

0

D

x

D/2

Figure 1:

between charged surfaces, separated by a distance D and bearing a surface charge density σ. The

ions in solution with the polymers are not shown for clarity. Also not shown is the electrolyte and

polymer reservoir with which the system is in equilibrium.

The situation modelled: an ionic solution of nonadsorbing neutral polymers confined

2.1Variational Formulation

With these premises, the free energy characterizing the system is the grand potential Ω(Γ,T,µi,µp),

minimised at equilibrium when the solution’s volume, Γ, its temperature, T, and chemical poten-

tials, µi (i = +,−) for the ions and µp for the polymer, are fixed. Ω comprises energetic and

entropic contributions from the polymer solution, the ions and electrostatics. Adopting a mean

field approach, such contributions are conveniently expressed in terms of the monomer volume

fraction φ(r), the ion number density ni(r) and the electrostatic potential V (r). As indicated,

these variables are expected to vary with position r between the surfaces, so that the free energy

needs to be expressed as a density functional, the null variations of which yield the equations of our

model. Because the electrostatic potential and the ion densities are not independent variables, but

are constrained to obey the Maxwell equation ∇ · (ǫ∇V ) = −?

as an “action”:

FΩ=

ΓfΩ(r)dr =

The stationary value of this action is identical to Ω, as indicated by the subscript. The integrand

fΩis an “action density” which has been expressed as a sum of polymeric (fpoly), ionic (fions) and

electrostatic (fel) contributions.

To describe the polymer contribution we adopt the square gradient approximation (SGA) [21].

In this approximation the local free energy density of a polymer solution is given by the reference

free energy of the solution at position r plus a square gradient term accounting for the nonlocal

effects of concentration fluctuations on chain entropies. We choose the Flory–Huggins expression

iniqi(where qiis the ion charge),

it is convenient to adopt the variational formulation of electrostatics [20] to write the free energy

??

Γ{fpoly(r) + fions(r) + fel(r)}dr

(1)

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[22] on a cubic lattice as the reference free energy. The polymer contribution to fΩis thus:

fpoly(r)=

T

a3

+T

36a

?φ

Nlnφ

(∇φ)2

N+ (1 − φ)ln(1 − φ) + χφ(1 − φ) −µp

Tφ

?

+

φ

(2)

where T is the temperature of the solution, a is the lattice parameter of a cubic lattice, φ is the

monomer volume fraction defined on such a lattice, N is the number of monomers in a chain, χ

is the Flory interaction parameter, and µpis the monomer chemical potential, which is fixed by

the reservoir. The coefficient of the square gradient term derives from a comparison of the small

fluctuation limit of the SGA with the large wavelength limit of the free energy expansion of a

polymer solution in the random phase approximation (RPA) [15, 23, 24]. The ionic contribution,

as in Poisson–Boltzmann (PB) theory [1], comprises an entropic “ideal gas” term and a chemical

potential term added to ensure ion density conservation:

fions(r) = T

?

i=+,−

ni(lnni− 1) −

?

i=+,−

µini

(3)

where ni is the number density of ionic species i = +,−, and µi and T are respectively the

chemical potential of species i = +,− and the temperature of the system, fixed by the reservoir.

As in standard PB theory [1], we consider a dilute solution of pointlike ions. We thus ignore all

steric effects associated with finite ion size. This is a good assumption since we are interested in

the qualitative behaviour of small ions residing in a polymer solution which may be reasonably

concentrated (see Section 3), but is sufficiently far from a melt that any steric reduction of the ion

entropy by the polymer can be safely ignored.

The electrostatic contribution to the free energy action is:

fel(r) =

?

i=+,−

niqiV −1

2ǫe(φ)(∇V )2

(4)

where qi = zie is the charge of species i with valence zi, and V is the electrostatic potential.

Equation (4) represents the variational formulation of electrostatics [20]: Poisson’s equation follows

from the variation δFel=?

energy of the system follows [20]. Throughout these equations we have set kB= 1 = ǫ0.

The dielectric nature of the polymer solution is accounted for via the effective permittivity ǫe,

which is a function of the monomer concentration φ(r) at r. The polymer solution is thus treated

as a dielectric mixture of solvent with permittivity ǫ1and “dielectric monomers” with permittivity

ǫ2. Since the mean field treatment of mixtures with spherical inclusions is particularly simple, we

assign a spherical dielectric volume with characteristic radius adto each monomer, as shown in

Fig. 2. In general, we expect adto differ from the cubic lattice length a by an unknown factor γ

that depends on the local chain geometry, so that ad= a/γ. We can now use the Maxwell–Garnett

equation for the effective permittivity of a mixture with spherical inclusions [25]:

Γfeldr = 0 and if its solutions are substituted in (4), the electrostatic

ǫe(φ) = ǫ1

?

1 −

3Kαφ

1 + Kαφ

?

(5)

Here K ≡ (ǫ1− ǫ2)/(2ǫ1+ ǫ2) is the Mossotti factor. The coefficient α ≡ (4/3)π/γ3accounts for

the difference between the volume of a cubic cell of the lattice and that of the effective dielectric

sphere each of which models, in its own fashion, a single monomer.

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(i)

(ii)

ε1

a

ad

ε2

Figure 2: (i) The Flory–Huggins model on a cubic lattice, here represented by a two dimensional

square lattice. (ii) Closeup of two cells of the lattice. A dielectric sphere of permittivity ǫ2and radius

ad?= a is associated with each occupied site. The surrounding solvent has the same permittivity ǫ1

of a pure electrolytic solution. The relation between adand a is discussed in the text.

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Effective medium approaches, such as that leading to Equation (5), involve averaging the electric

and displacement fields over a region containing discrete inclusions. When such averaged fields

are used in place of the real “microscopic” fields, the medium can be properly characterised by

an effective permittivity. This will be a smooth function provided the averaging volume is large

enough to contain several inclusions [26, 27]. Thus Equation (5) is strictly valid only if R ≫ aφ−1/3,

where R is the radius of a spherical averaging volume, containing inclusions of size a and number

concentration φ/a3. If the electric field itself is not mesoscopically uniform but varying on some

length scale λ (in our case the Debye length), we also require λ ≫ R, so that the field which is

being averaged does not change in magnitude over the averaging volume itself.

Aside from this, our model is subject to the same approximations as the Poisson–Boltzmann

and Flory–Huggins mean field theories. In particular, the mean field approximation implies that all

fluctuation-induced effects are ignored and that the theory is strictly inapplicable to dilute or semi-

dilute polymer solutions in good solvents, where Flory–Huggins fails; however we expect it to give

a reasonable account of the global picture. On top of this, use of the square gradient approximation

requires externally induced concentration variations (and therefore the inducing external fields) to

be slowly varying. Another tacit assumption is that the direct effect of ions on the solvent–monomer

interactions expressed by the Flory χ parameter is negligible; this might not be true in the presence

of complexation between chain and ion but should be adequate otherwise.

2.1.1Model Equations

The model’s equations follow from the requirement that Equation (1) be stationary with respect to

variations in φ, niand V (Appendix A). With the additional simplifying assumption that all the

ions in solution are monovalent and with the substitution φ = ψ2for the polymer concentration

variable, the equations read:

∇

?

˜ ǫe(ψ)∇

?eV

a2

9∇2ψ

T

??

=κ2sinh

?eV

T

?

(6)

=

3

2

Kǫ1αa3

T

(∇V )2

ψ

(1 + Kαψ2)2+

?

1 − ψr 2

(7)

+ψ

Nln

?

ψ2

ψr 2

?

− ψln

1 − ψ2

?

− 2χ(ψ3− ψr 2ψ)

where ˜ ǫe≡ ǫe/ǫ1is given by the Maxwell–Garnett Equation (5), κ ≡

Debye length for a monovalent salt of number density nr

values of salt or polymer concentrations.

Equation (6) is the Poisson–Boltzmann equation with a permittivity that depends on polymer

concentration and hence on position. Equation (7) describes the concentration variations of a

dielectric polymer solution. Similar mean field descriptions of polyelectrolyte adsorption [5, 28, 1]

and its effect on surface interactions [5, 28] have been carried out. Like the latter, Equations (6) and

(7) describe the concentration variations of a polymer solution next to charged walls. However, since

we are not modelling charged polymers, the polymer profile described by (7) is only electrostatically

affected because of the dielectric nature of the polymers (described by the first term on the right

hand side of (7), which derives from the Maxwell–Garnett relation). Conversely, in Equation (7)

the polymer enters only through the modified permittivity.

?2nrse2/ǫ1T is the inverse

s, and the superscript r indicates reservoir

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The influence of confining charged surfaces is incorporated into boundary conditions on Equa-

tions (6) and (7). Assuming that the surfaces are parallel, flat, homogeneous and infinite reduces

our problem to one dimension. We shall denote the position variable by x, with origin on the “left-

most” surface (x = 0, as shown in Fig. 1). D indicates the separation between the surfaces (the

“rightmost” surface is located at x = D). For simplicity we have chosen nonadsorbing polymers,

and also now choose identical surfaces with fixed surface charge; these choices translate into the

following boundary conditions:

V′(0)=−σ

ǫ1

= −V′(D) (8)

ψ(0)=0 = ψ(D)(9)

where the primes mean d/dx. The fixed surface charge condition was chosen in view of describing

charged lamellar phases, whose unknown surface charge density can be estimated from the area

available to charged surfactant groups. (It would also be possible to impose fixed surface potential,

but this is more complicated numerically and we do not attempt it here.)

The solution of Equations (6) and (7), subject to Equations (8) and (9), allows evaluation of

the force per unit area acting between surfaces in the presence of polymer. This is the ‘net osmotic

pressure’ Πnet= Π−Πr, defined as the osmotic pressure of ions and polymers on the midplane, Π,

less that in the reservoir, Πr. Note that on the midplane, and also in the reservoir, the local osmotic

pressure equates to the normal (xx) component of a stress tensor Σαβthat elsewhere includes the

Maxwell stress arising from electric fields. The Maxwell stress vanishes in the reservoir, and on the

midplane by symmetry, which is why the normal force can be equated to the net osmotic pressure

there. To find Π, we first show in Appendix A that the stress component Σxxis independent of x.

Evaluating this (on the midplane for convenience) as a function of plate separation D gives Π(D)

and hence Πnet(D). From Equation (27) of Appendix A, evaluated at the midplane x = D/2, we

finally obtain:

Πnet(D)=4nr

sT sinh2

?eV |D/2

2T

?

−T

a3

?φ|D/2

N

?

lnφ|D/2

φr

+(φr− φ|D/2)

N

+(10)

+(1 − φ|D/2)ln

?1 − φ|D/2

1 − φr

− χ(φr− φ|D/2)2− (φr− φ|D/2)

?

Equation (10) includes both ionic and polymeric contributions to the net pressure: the first term

on the right hand side of the equation and the term in square brackets, respectively. The ionic

contribution is always repulsive (as one would expect from a mean field treatment of the electro-

statics). The polymeric contribution, however, can become attractive at surface separations which

unfavourably confine the polymer (values of D such that φ|D/2< φr). We find below that the in-

terplay of such opposing forces can lead to phase separation, analogous to the liquid–gas transition

of ordinary fluids. At mean field level, this shows up as a characteristic S–shaped loop in the net

osmotic pressure (like that of the isotherms predicted by the Van der Waals equation of state [29]).

Our model can thereby be used to predict the equilibria of lamellar phases, which are known to

phase separate when polymer is added to them (see [7] and references therein).

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2.2Mapping onto Lamellar Phases

A lamellar phase consists of a periodic one dimensional crystal of repeating units. Each unit

comprises a bilayer of width δ and an adjacent solvent layer of width D, so that the crystal’s repeat

distance is D +δ. Consider a lamellar phase containing Nbbilayers in contact with a polymer and

ion reservoir via a membrane impermeable to the bilayers. The solvent layers of such a phase can

be modelled as a polymer solution confined between flat parallel surfaces; a situation described by

the model we have just built. This mapping, shown in Fig. 3, is reasonable for lamellar phases with

rigid bilayers (so that the surfaces are approximately flat and D is well defined). Further, since our

model does not account for other known forces between bilayers (dispersion, hydration, Helfrich

etc.) the mapping is strictly valid only when electrostatics and polymer-induced forces dominate.

...

...

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

...

+

+

+

+

+

+

+

...

D+δ

D

δ

Figure 3: A polymer-doped lamellar phase containing Nbbilayers, approximated by a periodic

succession of units. Each unit comprises a polymer solution confined between charged parallel

planes (as described by our model) and a rigid rectangular slab (the bilayer).

Under these assumptions our model can predict the thermodynamic behaviour of a lamellar

phase, which is entirely determined by the free energy of the solvent slab. Since pressure is an in-

tensive thermodynamic variable, the osmotic pressure of the lamellar phase is identical to Equation

(10); this is the equation of state of the lamellar phase, and determines its phase behaviour.

3Results: Parallel Surfaces

In what follows we give results from the numerical solution to our model. Table 1 displays the

“baseline parameters” used in numerical evaluations, all but one of which will be held fixed at the

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values shown while the one remaining parameter is varied. The baseline values are loosely based

on experimental systems [14, 30, 17, 12, 18]: typical charged bilayers (e.g. CPCl or SDS mixed

alcohol) in an aqueous salt solution of water–soluble polymers (PVP or PEG) at room temperature.

However, particular parameters were adjusted to have values which might enhance the effects of

dielectric coupling, creating on purpose a ‘worst case’ scenario for the additive approximation [31].

Temperature

Permittivity of ionic solution

Permittivity of polymer

Dielectric factor

Polymer lattice length

Flory Parameter

Number of lattice units per chain

Surface charge density

Reservoir salt concentration

Reservoir monomer volume fraction

T

ǫ1

ǫ2

γ

a

χ

N

σ

cr

φr

298K

78.5

2

1.5

10˚ A

0.495

2000

0.1enm−2

0.02M

0.3

s

Table 1: Values of the parameters used in the numerical evaluation of the model equations and the

subsequent determination of phase diagrams. The Debye length in the reservoir corresponding to

the salt concentration shown above is λ = 21.5˚ A.

To maximise dielectric contrast, a value of 2 (the permittivity of hydrocarbon oils) is chosen

for the polymer permittivity ǫ2. Similarly, the χ parameter is set near the theta point, enhancing

the susceptibility of the polymer to external fields. Our choice of γ, the size difference between the

lattice length and the radius of the dielectric sphere, is likewise made to enhance coupling effects.

Physically, the dielectric size of a lattice monomer depends on how many “oily” hydrocarbon groups

are contained in the backbone or side groups of every chemical monomer. How much bigger the

dielectric volume occupied by the polymer is with respect to the steric volume is hard to estimate

precisely. Our choice of γ = 1.5 entails that the Flory–Huggins (“entropic”) volume available to a

lattice monomer, VFH, occupies about 80% of the polarisable volume, Vd(VFH/Vd≃ γ3/4 ≈ 0.8).

The lattice length a is modelled on the water soluble polymer PVP. As shown in [32], a can be

related to the chemical monomer size, l, of the polymer of interest and the polymer molecular

weight, Mw, to the number N of lattice units per chain. On a cubic lattice, one finds a = 10˚ A and

N = 2000 for PVP with l ≃ 30˚ A and Mw≃ 500000. (Note that the N is not an important control

parameter in the parameter regime of interest to us.) Finally the reservoir monomer concentration

was chosen at a reasonably high value of φr= 0.3 so that the concentration of polymer would not

be too small, swamping out any dielectric coupling effects.

Given these “worst-case” choices, the uncoupled theory gives a surprisingly good approximation

to the full one in most cases. To a significant extent, this justifies the assumption of additivity,

tacitly made by some authors [4], and inconsistently justified by others [16, 17]. However, we are

not aware of any simple order–of–magnitude arguments that can explain this without pursuing the

more detailed calculations presented here.

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3.1Potential and Polymer Concentration Profiles

Fig. 4 displays the dimensionless electrostatic potential W ≡ eV/T and monomer concentration

profile Φ ≡ φ/φras a function of dimensionless position X ≡ x/λ, for surfaces separated by

D/λ ≃ 2.37 (recall λ is the Debye length defined in the reservoir). The solutions to the full model

(coupled equations) were obtained using an adaptation of the shooting method [33] to solve the

Equations (6) and (7) subject to the boundary conditions (8) and (9). For comparison, the solutions

found in the additive approximation (uncoupled equations) are also presented.

Qualitatively, the solutions of the two descriptions are similar. The “coupled” electrostatic

potential displays the characteristic symmetric shape of the solutions to the standard Poisson–

Boltzmann equation (uncoupled case); similarly, the monomer concentration profile compares well

with the predictions of a mean field description of nonadsorbing polymer chains confined in a slit

[34]. The characteristic depletion of monomers from between the surfaces can be observed in both

cases. It is well known that the pressure imbalance created by such depletion can cause the surfaces

to attract, if the polymer solution is sufficiently squashed that it escapes from between the plates,

reducing the osmotic pressure there.

The results, however, also highlight the interesting features which follow from a full account of

the polymers’ dielectric properties. First, the electrostatic potential is reduced around the midplane

(where the polymer monomers are more concentrated) with respect to the uncoupled solutions. The

reduction is due to the additional electrostatic screening provided by the monomers because of their

polarisation. Second, the monomer concentration profile displays an increased depletion from the

surfaces. This is also a consequence of polarisation and results from the electrostatic energy penalty

of placing dielectric monomers in the surface fields.

3.2Osmotic Pressure Between Parallel Surfaces

Fig. 5 displays the predicted variation (coupled and uncoupled cases) of total osmotic pressure

between surfaces of dimensionless surface separation D/λ. Both descriptions predict a van der

Waals loop. However, in the coupled picture the loop is translated to lower pressures and slightly

larger separations with respect to the uncoupled, additive case. In addition, while at very small

separations coupled and uncoupled profiles coincide, at large separations dielectric coupling lowers

the osmotic pressure curve below the additive prediction.

To understand these differences the ionic and polymeric contributions to the midplane pressure

are also shown in Fig. 5. For both models, the ionic contribution is always repulsive, whilst

the polymeric contribution is attractive for separations below those at which the polymer starts

to be expelled from between the plates. However, for distances large enough that no significant

amount of polymer has been expelled, the ionic contribution to the coupled pressure profile decays

more rapidly than the uncoupled prediction. This stems from the more efficient screening of the

electrostatics in the presence of dielectric monomers. Another effect of the dielectric coupling is

that the polymer contribution becomes attractive at larger D/λ than the uncoupled one. This

represents an “electrostatically enhanced” depletion of polymer from the surfaces as the dielectric

monomers are expelled from the high–field regions near the confining plates.

We can thus summarize the overall differences between the net pressure profiles: at large dis-

tances the shorter range of the coupled electrostatic repulsion lowers the pressure profile with

respect to the uncoupled prediction; the larger range of the polymer expulsion then promotes the

occurrence of the van der Waals loop; finally, at small enough separations the polymer is fully

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0.8

0.6

0.4

0.2

0

0 0.5 1 1.5 2

0

0.5

1

1.5

2

2.5

3

(i)

(ii)

x / λ

eV/kBT

φ / φ

r

Figure 4: Plots of the numerical solutions of Equations (6) and (7) subject to the boundary con-

ditions (8) and (9), for a surface separation is D ≃ 2.37λ, and using the parameters of Table 1.

The results display both coupled (solid line) and uncoupled (dotted line) solutions for (i) the di-

mensionless electrostatic potential, eV/kBT, and (ii) the monomer concentration rescaled by the

reservoir value, φ/φr, as functions of the dimensionless position between surfaces, x/λ.

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-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1 1.5 2 2.5 3 3.5 4 4.5 5 5.5

D/λ

Πnet(Pa/105)

polymer

ionic

total

Figure 5: Net osmotic pressure as a function of D/λ, together with its ionic and polymeric contri-

butions. The plots were obtained from Equation (10) by substituting the midplane values of V and

φ from the solutions of (6) and (7) for various D. The coupled (solid lines) and uncoupled (dashed

lines) cases are shown. Parameters as in Table 1.

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expelled, the dielectric coupling is removed, and coupled and uncoupled predictions coincide.

4Results: Phase Diagrams

4.1Expected Phase Behaviour

As mentioned in the Introduction, the addition of polymer to charged lamellar phases has exper-

imentally been observed to induce phase separation. X–ray scattering studies on these systems

have mainly evidenced two kinds of separation [10, 14]: LαLαcoexistence between two lamellar

phases with different spacings, and LαL coexistence between a single lamellar phase and an isotropic

solution of polymer (with trace surfactant). The occurence of phase separation and its modality

depend, if all other parameters are held fixed, on the relative composition of the prepared mixture.

This has been usefully mapped on density–density phase diagrams plotting the polymer content of

the sample against its surfactant composition (e.g. [10, 12]).

4.2Predicting Phase Coexistence

We can explain the phase behaviour just described using our model’s equation of state, Equation

(10). Recall that in mean field theories like ours, phase coexistence shows up as a Van der Waals

loop in plots of the net osmotic pressure as a function of bilayer separation D. To deal with this we

can deploy the Maxwell construction [29], finding a horizontal line which cuts the van der Waals

loop into two regions of equal area (Fig. 6). The horizontality of the line represents the equality

of pressures for coexisting phases of different spacings D, whereas the equal areas represent the

equality of chemical potentials [29]. The coexistences found in this way connect two lamellar phases

of different layer spacings (LαLαcoexistence). This type of coexistence only occurs for a positive

pressure: referring to Fig. 5, the polymer-induced attraction causes phase separation, but cannot

overcome the electrostatic repulsion between bilayers.

However, when the polymer contribution is large enough, the pressure can become negative and

a special type of Van der Waals loop occurs across the zero-pressure axis (Fig. 7). This amounts

to a coexistence between a lamellar phase of finite D, found where the pressure profile crosses

the zero axis, and a lamellar phase of infinite D (identical to the reservoir) which is represented

asymptotically by the pressure tending to zero at infinite D. Hence this modified construction

allows the prediction of LαL coexistence. However, the resulting areas are not equal in general; this

is because the chemical potential of the surfactant is undefined in the bilayer-free state of infinite D.

Recalling that the Maxwell construction is equivalent to the common-tangent construction on the

Helmholtz free energy, the failure to equate chemical potentials under these conditions corresponds

to the so-called ‘virtual tangency’ condition in which a phase at finite surfactant density connects

with one at zero density [35]. A more accurate treatment would allow for the finite molecular or

micellar solubility of surfactant in water, giving a slightly more elaborate calculation (in which

the slope of the free energy is rapidly varying at very low surfactant concentration) but an almost

identical result for the coexistence properties.

Thus, using the reservoir monomer concentration to control the polymer content of our lamellar

phase, Equation (10) allows us to determine the equilibrium spacings D. The relation between

bilayer volume fractions φband separations D follows from simple geometry:

φb=

δ

D + δ

(11)

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Π

Π∗

D

φp

D1

D2

φp

1

2

φp

Figure 6: The Maxwell construction for the osmotic pressure (10) of a polymer doped lamellar

phase, predicting LαLα coexistence at spacings D1,D2. Once these spacings are identified, the

corresponding polymer content of coexisting phases can be obtained from a graph of Equation

(12), as shown.

D

Π

Π∗=0

D1

infinity

f

φb

0

φb

1

Figure 7: The special Maxwell construction for LαL coexistence. The isotropic fluid is represented

by a putative lamellar phase with D = ∞ and hence zero osmotic pressure. In reality an exponen-

tially small amount of surfactants coexisting with the lamellar phase resides in the reservoir. The

inset shows the equivalent condition of virtual tangency, when the common tangent construction is

used to determine phase equilibria from the free energy density f as a function of volume fraction

φb; see [35].

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Where we recall δ is the bilayer thickness. Similarly, the bulk polymer volume fraction φpwithin

a lamellar phase is found by integrating the polymer profile φp(x) ≡ ψ2(x) (from the solution of

Equation (7)) for a given D, normalising by the full volume now including the bilayers themselves:

φp=

1

D + δ

?D

0

φp(x)dx (12)

4.3 Semi-Grand Ensemble for Salt

At fixed chemical potential for salt, the preceding constructions allow us to find all states of coexis-

tence and the polymer and surfactant densities in them; below we create phase diagrams essentially

by ‘joining the dots’ when these coexistences are plotted on the (φb,φp) plane. In most experiments,

however, the experimenter fixes the total amount of salt (as well as that of bilayer and polymer) in

the system. To deal with this is possible in principle, by a similar integration: the ion concentration

obeys (16), so that defining the total salt concentration as ns≡ n+, we have:

ns=

1

D + δ

?D

0

ni(x)dx =

nr

s

D + δ

?D

0

e−eV (x)/Tdx(13)

Equation (13) expresses the Donnan equilibrium for a lamellar phase [36, 37]: since V (x) is always

positive for bilayers of finite separation, the total amount of salt ns within a lamellar phase is

smaller than in the the reservoir. This salt expulsion is more efficient, the closer together the

bilayers, so that coexisting phases with different periods will contain different amounts of salt.

However, this additional calculation represents a major numerical complication. We have de-

cided to neglect this, and thus work in a semi-grand ensemble with respect to salt. Experimentally

our phase diagrams are those of a lamellar system in which a fixed volume of solvent, and fixed

amounts of bilayer and polymer, are in contact with a salt reservoir through a dialysis membrane.

This type of experiment is perfectly possible [36], but is not typical in studying polymer-doped

lamellar phases. How the use of the salt reservoir affects our results will be discussed in Section 5.

4.4Effect of Bilayer Surface Charge Density

With these considerations in mind, we now present phase diagrams from our model by mapping

pairs of coexisting phase points. The error bars on the phase points result from the uncertainties

associated with the graphical method by which coexistences were found, or, when this was very

accurate, from the intrinsic accuracy of the numerics. The phase boundaries themselves have been

drawn as guides for the eye and do not represent numerical fits to our data.

We study first the effect of changing the surface charge density, σ. Density–density phase

diagrams, with polymer and bilayer volume fractions as composition variables, are shown in Fig.

8 for doped lamellar phases with σ = 0.05,0.1 and 0.2enm−2(top, middle and bottom panels

respectively). The middle phase diagram, calculated using the same parameters as Table 1, provides

a reference for studying the influence of parameter variation on phase behaviour. All other phase

diagrams are calculated by changing the parameter of interest above and below its reference value.

Results are shown for both coupled and uncoupled equations. In both cases, an increase in σ

changes the phase diagram topology from a large area of LαL coexistence (top panel), through an

intermediate “pinch off” region (middle panel, coupled case), to two miscibility gaps, separated

by a single-phase Lα region (middle panel, uncoupled; bottom panel). The LαL coexistence in

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