# Repulsive and attractive depletion forces mediated by nonadsorbing
polyelectrolytes in the Donnan limit

Jasper Landman  Laboratory of Physical Chemistry, Department of Chemical
Engineering and Chemistry, Eindhoven University of Technology, P.O. Box 513,
5600 MB Eindhoven, the Netherlands  Laboratory of Physics and Physical
Chemistry of Foods, Wageningen University & Research, P.O. Box 17, Bornse
Weilanden 9, 6708 WG Wageningen, the Netherlands  Max P. M. Schelling
Laboratory of Physical Chemistry, Department of Chemical Engineering and
Chemistry, Eindhoven University of Technology, P.O. Box 513, 5600 MB
Eindhoven, the Netherlands  Remco Tuinier<EMAIL_ADDRESS>Laboratory of
Physical Chemistry, Department of Chemical Engineering and Chemistry,
Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, the
Netherlands  Institute for Complex Molecular Systems, Eindhoven University of
Technology, P.O. Box 513, 5600 MB Eindhoven, the Netherlands  Mark Vis
<EMAIL_ADDRESS>Laboratory of Physical Chemistry, Department of Chemical
Engineering and Chemistry, Eindhoven University of Technology, P.O. Box 513,
5600 MB Eindhoven, the Netherlands  Institute for Complex Molecular Systems,
Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, the
Netherlands

###### Abstract

In mixtures of colloids and nonadsorbing polyelectrolytes, a Donnan potential
arises across the region between surfaces that are depleted of polyelectrolyte
and the rest of the system. This Donnan potential tends to shift the
polyelectrolyte density profile towards the colloidal surface and leads to
local accumulation of polyelectrolytes. We derive a zero-field theory for the
disjoining pressure between two parallel flat plates. Polyelectrolyte is
allowed to enter the confined interplate region at the cost of a
conformational free energy penalty. The resulting disjoining pressure shows a
crossover to a repulsive regime when the interplate separation gets smaller
than the size of the polyelectrolyte chain, followed by an attractive part. We
find a quantitative match between the model and self-consistent field
computations that take into account the full Poisson-Boltzmann electrostatics.

††preprint: AIP/123-QED

##  I Introduction

In both biological systems and industrial products, colloids or nanoparticles
are often mixed with polymers in a common solvent. In biochemistry the focus
has traditionally been on studying the properties of biologically relevant
compounds like enzymes, in dilute solution at isotonic conditions with respect
to the ionic strength; the presence of particles and macromolecules has only
came into focus more recently: there is increasing attention for the effects
of what is called ‘macromolecular crowding’ Goodsell (1991); Ellis (2001);
Zimmerman and Trach (1991); Minton (2006) on for instance enzyme activity
Norris and Malys (2011). This led to the insight that perhaps even the
organization of compartments within cells is driven by macromolecular crowding
Hancock (2004); Brangwynne (2013).

In industrial products such as paints de With (2018), cosmetics Tadros (2011a)
or food products Doublier  _et al._ (2000), polymers are also often mixed with
colloids; the phase stability of these mixtures is a key issue that needs to
be under control. In many cases there are no attractive interactions between
the polymeric compounds and the colloidal particles in colloid–polymer
mixtures. In such cases excluded volume interactions dominate the phase
stability. Also mixtures featuring charged colloids and/or charged polymers
offer the opportunity to generate complex crystal structures, including
colloidal alloy and eutectic structures. For instance Toyotama et al.Toyotama,
Okuzono, and Yamanaka (2016) reported the formation of area-filling eutectic
structures in dispersions containing charged colloidal particles and
polyelectrolyte chains.

To quantify the interactions and resulting phase behaviour in colloid–polymer
mixtures, the concept of depletion-induced attraction has been used frequently
and has been quite successful to describe well-defined systems Fleer and
Tuinier (2008). Asakura and Oosawa Asakura and Oosawa (1954, 1958) were the
first to realize that nonadsorbing polymer chains induce an indirect
attraction between colloidal particles. Near the nonadsorbing surfaces a so-
called depletion zone exists, where the polymer concentration is lower than in
the bulk of the solution. The extent of this region is approximately the
radius of gyration of the polymer in the dilute regime, whereas it equals the
correlation length in the semi-dilute regime. Once depletion zones around two
colloids overlap, an osmotic force becomes operational which pushes the
colloidal particles together. This effective attraction, mediated by excluded
volume repulsions, can lead to phase transitions, aggregation, or gelation
when exceeding a certain polymer concentration. The focus of most work related
to the depletion-mediated attraction has been on well-defined neutral systems
Poon (2002).

In practice, however, either the polymer chains and/or the colloidal particles
often carry charges, especially when dispersed in polar solvents such as
water. For instance, in food products phase stability is of key relevance.
Food dispersions often contain protein (aggregates), oil droplets and/or
polysaccharides Grinberg and Tolstoguzov (1997); Doublier  _et al._ (2000);
Syrbe, Bauer, and Klostermeyer (1998). In many cases these food colloids
contain (often negatively) charged polysaccharides and proteins. Similarly,
water-based paint formulations contain a wide variety of particles and
polymers, such as pigments and binders, which may be charged de With (2018);
Tadros (2011b). Hence it is useful to develop more understanding of the role
of polyelectrolytes on the phase stability of such colloid–polyelectrolyte
mixtures. Still, the understanding of the effects of charges in mixtures of
colloids and polymers is rather limited from a theoretical point of view, even
though Asakura and Oosawa discussed the possible effects of charges already at
an early stage Asakura and Oosawa (1958). Below we briefly mention some
earlier findings.

There is quite a number of studies on charged polymers mixed with charged
colloids. Already in the 1970s Kose and Hachisu Kose and Hachisu (1976)
studied the effect of the addition of sodium polyacrylate to polystyrene latex
particles (both negatively charged) and found that this leads to
crystallization of the colloidal spheres. This must originate from enhanced
excluded volume interactions mediated by the polyelectrolytes. Böhmer et al.
Bohmer, Evers, and Scheutjens (1990) applied numerical lattice computations
based upon the self-consistent field method of Scheutjens and Fleer Scheutjens
and Fleer (1979, 1982); Fleer  _et al._ (1993) to study the polyelectrolyte-
mediated effect on the interaction between flat walls. Below a salt
concentration of about 1 M the depletion layer thickness was found to decrease
with increasing polyelectrolyte concentration. At much lower salt
concentrations the results of the SCF computations revealed a significant
repulsive barrier in the interaction potential between two (uncharged)
parallel flat plates. This is reminiscent of the findings of Mao et al. Mao,
Cates, and Lekkerkerker (1995) who predicted such repulsive barriers for the
interaction between two hard walls (or similarly for two hard spheres) in the
presence of small hard spheres.

The repulsive barrier in the effective pair interaction which exists for
mixtures of large and small hard spheres, is further enhanced when both
species are like-charged, as predicted by Walz and Sharma Walz and Sharma
(1994). For low concentrations of the small charged spheres they predict that
the depletion interaction is attractive for any salt concentration and that
the strength of attraction at contact increases as the Debye length increases,
or if the charge density on the large colloidal spheres increases. At higher
concentrations of the small particles a repulsive barrier in the interaction
potential curve appears for sufficiently large size ratios of small and large
colloids and sufficiently large values of the Debye length. They could confirm
many of the theoretical trends, including the repulsive barriers and
oscillations in the interaction potential, by total internal reflection
microscopy (TIRM) experiments on the force between a latex sphere and a glass
wall in the presence of various nonadsorbing depletants. Sober and Walz
(1995); Sharma and Walz (1996); Sharma, Tan, and Walz (1997a, b)

These repulsive barriers have also been reported in mixtures of charged
colloids and polyelectrolytes. Biggs et al. Biggs  _et al._ (2000) made direct
atomic force microscopy measurements (AFM) of the force between two silica
surfaces in sodium polystyrene sulfonate solutions. They reported a secondary
minimum in the interaction potential, which became deeper on either increasing
the polymer concentration or molar mass. Burns et al. Burns  _et al._ (2002)
found similar trends upon studying polystyrene latex particles under the
influence of nonadsorbing polyacrylic acid using AFM. In case of neutral
polymeric depletants the repulsive barriers are (nearly) absent Louis  _et
al._ (2002). They seem to appear only if the depletants are either hard
particles or are charged, i.e., the self-interaction between the depletants
must be sufficiently repulsive.

Ferreira et al. Ferreira, Dymitrowska, and Belloni (2000) applied PRISM to
study mixtures of charged polyelectrolytes and charged colloids, compared to a
reference system of charged colloids and uncharged polymers. For charged
polymers mixed with charged colloids PRISM predicts that the pair interaction
between the colloidal spheres oscillates and is highly dependent on the
parameters chosen. For high salt concentration (i.e., charges are screened)
the potential at contact is repulsive due to double layer repulsions between
the colloidal particles. The repulsion is found to be stronger if the polymers
are neutral. At somewhat larger inter-particle distances there is an
attractive part in the interaction potential, which is stronger for charged
polymers compared to neutral polymers. In the presence of highly charged
polymers the contact potential becomes attractive. At larger interparticle
distances the potential then exhibits a repulsive barrier. DFT computations
and simulations by Jönssen et al. Jönsson  _et al._ (2003) show that these
structural oscillatory forces for charged spheres and polyelectrolytes are
qualitatively not so sensitive to the magnitude and sign of the charge of the
confining surfaces.

Odijk Odijk (1997) theoretically studied the depletion effects on the
interaction between (small) colloidal particles interaction mediated by like-
charged polyelectrolytes. He related the effective depletion thickness around
the colloidal particles to the Debye length, the effective charge density at
the surface of the colloidal spheres and to the Kuhn length. From this it
appears that at high salt concentrations like-charges on polymers and colloids
do not seem to affect the depletion induced attraction between colloids due to
polyelectrolytes. At low ionic strength however the situation becomes quite
complicated and detailed theories still have to be developed that enable a
computation of the stability of such systems. Also recent experimental and
theoretical studies Najafi  _et al._ (2020); Pryamitsyn and Ganesan (2014)
show that the forces between charged colloids in the presence of
polyelectrolyte solutions show nonmonotonic, sometimes oscillatory Ludwig and
von Klitzing (2020); Moazzami-Gudarzi  _et al._ (2016) trends.

In this article we show how a Donnan equilibrium Donnan (1911) arises when
polyelectrolytes are excluded from depletion zones. Unless high ionic strength
screens the charges, this strongly affects the distribution of ions between
the depletion zones and the rest of the system, and also provides a driving
force for a spontaneous shift of the polyelectrolyte towards the interplate
region. We show that when a polyelectrolyte is allowed to deform at the cost
of paying a conformational self-energy penalty, a competition between
electrostatic and conformational penalties ensues, and a resulting equilibrium
Donnan potential difference can be calculated from which we find disjoining
pressures. In certain limits we derive simplified analytic expressions that
quantify the pressures. Our approach can be seen as the so-called zero-field
limit Philipse, Kuipers, and Vrij (2013), valid when the Debye length is
sufficiently large compared to the distances between the nonadsorbing surfaces
and the electric field vanishes. Although this is arguably a rather simple
model, it has previously yielded tractable predictions that could be verified
experimentally Stojimirović  _et al._ (2020).

The disjoining pressures predicted by our model are verified by comparison
with numerical self-consistent field lattice computations in which
electrostatic interactions are accounted for on a Poisson–Boltzmann level.
Furthermore, we show under which conditions the match is expected to be
quantitative, and in which cases a moderate deviation is found.

##  II Theory

###  II.1 Disjoining pressure at contact

We start by calculating the disjoining pressure between two flat plates
submerged in a polyelectrolyte solution, at close contact. In this case, the
conformational self free energy of deformation of the polyelectrolytes is
prohibitively high and the polyelectrolyte is entirely excluded from the zone
between the flat plates, however, co- and counterions are small enough to
enter the interplate space. We simplify the system by neglecting any gradients
in the electric potential, i.e. the zero field or Donnan limit Donnan (1911);
Philipse, Kuipers, and Vrij (2013). In general, this assumption is valid when
the plate separation is significantly smaller than the Debye length. While
imperfect, the Donnan limit often works over a larger range of conditions than
expected, and has the additional advantage that the resulting models can be
quite insightful. In this case, the concentrations of ions inside the
interplate region are given by the Boltzmann distribution

$n_{\pm}^{\text{i}}=n_{\pm}^{\text{res}}\exp(\mp\Psi_{\text{D}}),$ (1)

where $n_{\pm}^{\text{i}}$ is the number density of positive and negative ions
respectively between the plates, and $n_{\pm}^{\text{res}}$ the reservoir
number density of positive and negative ions. Furthermore,
$\Psi_{\text{D}}=e\psi_{\text{D}}/k_{\text{B}}T=\Psi^{\text{i}}-\Psi^{\text{res}}$
is the dimensionless Donnan potential in the interplate region. We assume both
ideal ion behaviour, as well as a sufficiently large reservoir so that the
reservoir concentration does not noticably change. As such, the reservoir
concentrations of the ions are given by

$\displaystyle n_{+}^{\text{res}}$ $\displaystyle=n_{\text{s}},$ (2a)
$\displaystyle n_{-}^{\text{res}}$ $\displaystyle=n_{\text{s}}+zn_{\text{p}},$
(2b)

where $n_{\text{s}}$ is the number density of added monovalent salt,
$n_{\text{p}}$ the reservoir number density of polyelectrolyte molecules, each
carrying $z$ charges. In the remaining discussion we will tacitly assume that
the polyelectrolyte carries positive charges — when the charges on the
polyelectrolyte are negative, the resulting equations remain valid, with only
an exchange in the identities of the co- and counterions and a switch in sign
of the resulting Donnan potential. The distribution of ions over the two
regions is determined by the magnitude of the Donnan potential
$\Psi_{\text{D}}$, which can be found by applying the electroneutrality
condition, $n_{+}^{\text{i}}=n_{-}^{\text{i}}$, over the interplate region.
The electroneutrality condition leads to the following charge balance equation

$n_{\text{s}}\exp(-\Psi_{\text{D}})=(n_{\text{s}}+zn_{\text{p}})\exp(\Psi_{\text{D}}),$
(3)

which, when solved for $\Psi_{\text{D}}$ results in

$\Psi_{\text{D}}=-\frac{1}{2}\ln\left(\frac{n_{\text{s}}+zn_{\text{p}}}{n_{\text{s}}}\right).$
(4)

The osmotic pressure in the interplate region at close contact follows from
the internal concentrations in eq. 1 and the resulting Donnan potential in eq.
4. Assuming Van ’t Hoff’s law,

$\frac{\Pi}{k_{\text{B}}T}=\sum_{i}n_{i},$ (5)

with $n_{i}$ the number density of species $i$, the osmotic pressure in the
interplate region can be written as

$\frac{\Pi^{\text{i}}}{k_{\text{B}}T}=2n_{\text{s}}\sqrt{1+\frac{zn_{\text{p}}}{n_{\text{s}}}}.$
(6)

The disjoining pressure equals the osmotic pressure difference of the
interplate region with respect to the reservoir osmotic pressure,
$\Delta\Pi=\Pi^{\text{i}}-\Pi^{\text{res}}$, where we assume that the
reservoir osmotic pressure $\Pi^{\text{res}}$ due to both ions and polymer
segments also follows from Van ’t Hoff’s law:

$\displaystyle\frac{\Pi^{\text{res}}}{k_{\text{B}}T}=(z+1)n_{\text{p}}+2n_{\text{s}}.$
(7)

For polymers, the Van ’t Hoff osmotic pressure is an appropriate assumption in
case of theta solvent conditions at low concentrations, which we adopt
throughout for the sake of simplicity, but it should be noted that more
complex models taking activity coefficients into account can and should be
used in more general situations. This applies also to the ions, where
deviations from ideality are to be expected at elevated concentrations.

###  II.2 Free energy of deformation in interplate region

The Donnan potential between the plates bears a sign opposite to the sign of
the polyelectrolyte valence, as follows from Eq. (4). As a consequence, the
Donnan potential provides a driving force that shifts the polyelectrolyte
density profile towards the interplate region, even when the interplate
separation is smaller than the typical size of the polyelectrolyte chain. To
account for this effect, we allow the polyelectrolyte to deform partially to
enter the interplate space, at the cost of a free energy penalty incurred by
the decreased conformational entropy associated with this deformation. This
free energy penalty depends on the interplate separation, and is assumed here
to be independent of the electrical potential between the plates. We expect
this deformation to occur spontaneously as the system responds to the Donnan
potential, provided the conformational free energy cost is not prohibitively
high. In this case, there is now a finite concentration of polyelectrolyte in
the interplate region that contributes to the electroneutrality condition.
This internal concentration $n_{\text{p}}^{\text{i}}$ is given by

$n_{\text{p}}^{\text{i}}(h)=n_{\text{p}}\exp\left\\{-[z\Psi_{\text{D}}(h)+\beta
F_{\text{p}}(h)]\right\\}.$ (8)

Here, $\beta=(k_{\text{B}}T)^{-1}$ is the inverse thermal energy and
$F_{\text{p}}(h)$ the (entropic) self-energy associated with a deformation to
fit between two flat plates separated by a distance $h$. In the limit of $h\gg
R_{\text{g}}$, when the interplate distance is far larger than the typical
size of the polyelectrolytes, this conformational free energy penalty is zero
and there is no concentration difference between the interplate and reservoir
regions. In the limit of $h\rightarrow 0$ the free energy penalty diverges and
the polyelectrolytes are fully excluded from the interplate region, as was
assumed in section II.1.

The canonical partition sum of a flexible polymer chain confined by two
parallel flat plates was calculated by Casassa Casassa (1967), the
conformational part of which is given by

$\displaystyle\chi(h)$
$\displaystyle=\frac{8}{\pi^{2}}\sum_{n=1,3,5,\dots}^{\infty}\frac{1}{n^{2}}\exp\left(-n^{2}\pi^{2}\frac{R_{\mathrm{g}}^{2}}{h^{2}}\right),$
(9)

where $R_{\mathrm{g}}=b\sqrt{M/6}$ is the polymer radius of gyration, with $b$
the polymer segment length and $M$ the number of segments per chain. The
conformational free energy penalty associated with this deformation is then
calculated in a straightforward manner using
$F_{\text{p}}(h)=-k_{\text{B}}T\ln\chi(h)$.

The resulting Donnan potential is a function of the interplate separation $h$,
and can be calculated from the condition of electroneutrality leading to the
charge balance equation

$(n_{\text{s}}+zn_{\text{p}})\exp\big{[}+\Psi_{\text{D}}(h)\big{]}=n_{\text{s}}\exp\big{[}-\Psi_{\text{D}}(h)\big{]}+zn_{\text{p}}\chi(h)\exp\big{[}-z\Psi_{\text{D}}(h)\big{]}.$
(10)

As mentioned before, this equation assumes that the charges on the
polyelectrolyte are positive. When the charges on the polyelectrolyte are
negative, only the sign of the Donnan potential switches. For brevity, in the
following we will drop the explicit dependencies of $F_{\text{p}}$ and
$\Psi_{\text{D}}$ on $h$.

In fig. 1a we show the Donnan potential, calculated numerically from eq. 10 as
a function of the polyelectrolyte valence $z$ at a number of different
polyelectrolyte volume fractions at fixed inter-plate distance $h$, together
with a schematic of the system under consideration in fig. 1b. At low $z$ the
Donnan potential quickly becomes more negative but soon reaches a minimum.
Surprisingly, the Donnan potential tends towards a single curve independent of
polymer volume fraction for higher $z$. This can be seen by taking the limit
of $zn_{\text{p}}\gg n_{\text{s}}$ of the charge balance eq. 10, which has the
analytical solution

$\Psi_{\text{D}}\simeq-\frac{\beta F_{\text{p}}}{1+z}.\qquad(zn_{\text{p}}\gg
n_{\text{s}})$ (11)

In the limit of $|z\Psi_{\mathrm{D}}|\ll 1$, essentially the opposite limit as
before, we can expand the exponentials in the charge balance eq. 10 up to
first order. In this limit we find the analytical solution

$\Psi_{\text{D}}\simeq-\frac{zn_{\text{p}}}{2n_{\text{s}}}\frac{1-\exp(-\beta
F_{\text{p}})}{1+\frac{zn_{\text{p}}}{2n_{\text{s}}}+\frac{z^{2}n_{\text{p}}}{2n_{\text{s}}}\exp(-\beta
F_{\text{p}})}.\qquad(|z\Psi_{\mathrm{D}}|\ll 1)$ (12)

Another possible limit can be found by taking $|\Psi_{\text{D}}|\ll 1$ and
expanding the two exponential terms independent of $z$ to first order, but
leaving the third exponential term intact. An analytical solution can then be
found that is accurate for both low $z$ and high $z$, provided the Donnan
potential is low:

$\Psi_{\text{D}}\simeq\frac{1}{z}W_{0}\left[\frac{z^{2}n_{\text{p}}}{2n_{\text{s}}+zn_{\text{p}}}\exp\left(\frac{z^{2}n_{\text{p}}}{2n_{\text{s}}+zn_{\text{p}}}-\beta
F_{\text{p}}\right)\right]-\frac{zn_{\text{p}}}{2n_{\text{s}}+zn_{\text{p}}},\qquad(|\Psi_{\mathrm{D}}|\ll
1)$ (13)

where $W_{0}$ is the principal branch of the Lambert $W$ function. These
limiting equations are useful in the appropriate conditions as shown in fig.
1, but since the effect of the Donnan potential leads to exponential changes
in the concentrations, in the general case it will be more accurate to use the
numerical result.

Figure 1: Donnan potential between flat plates. a For a fixed interplate
distance of $h=$5\text{\,}\mathrm{nm}$$ the self-energy for polyelectrolytes
to enter the interplate region is
$F_{\text{p}}(h)=$6.1\text{\,}\mathrm{\mathnormal{k_{\text{B}}T}}$$. For
higher polyelectrolyte valences, the self-energy is quickly overcome and the
resulting Donnan potential collapses to a universal curve, independent of
polyelectrolyte volume fraction. The solid curves denote the numerical
solution to the charge balance eq. 10, whereas the dotted grey curve denotes
the analytical approximation in the limit of $zn_{\text{p}}\gg n_{\text{s}}$
(eq. 11), the dotted coloured curves denote eq. 13, valid in the limit
$|\Psi_{\text{D}}|\ll 1$, and the dotdashed lighter coloured curves denote eq.
12 which is valid in the limit $|z\Psi_{\text{D}}|\ll 1$. The number density
of the polyelectrolyte is related to the volume fraction through
$n_{\text{p}}=\phi_{\text{p}}/(Mb^{3})$ where $M=1000$ is the number of
polymer segments and $b=$0.3\text{\,}\mathrm{nm}$$ is the segment length. b
Schematic of the double plate geometry, illustrating how deformation of
polyelectrolyte allows a nonzero concentration in the depletion zones to yield
lower overall free energy despite the entropic penalty of deformation.

The osmotic pressures due to the ions and polyelectrolytes in the interplate
and reservoir can now be calculated from the concentrations. In the reservoir,
they follow directly from Van ’t Hoff’s law, eq. 5, however in the interplate
region, the polyelectrolyte chains give rise to an additional entropic chain
contribution. In appendix B we formally derive this contribution from the
canonical partition sum for the interplate region, which leads to the
expression

$\displaystyle\frac{\Pi^{\text{i}}}{k_{\text{B}}T}$
$\displaystyle=n_{\text{p}}^{\text{i}}\left(1+\frac{h}{\chi(h)}\frac{\partial\chi}{\partial
h}\right)+n_{+}^{\text{i}}+n_{-}^{\text{i}}$
$\displaystyle\begin{split}&=n_{\text{p}}\exp(-z\Psi_{\text{D}}-\beta
F_{\text{p}})\left(1+z+\frac{h}{\chi(h)}\frac{\partial\chi(h)}{\partial
h}\right)\\\ &\qquad{}+2n_{\text{s}}\exp(-\Psi_{\text{D}}),\end{split}\ $
(14a) $\displaystyle\frac{\Pi^{\text{res}}}{k_{\text{B}}T}$
$\displaystyle=(z+1)n_{\text{p}}+2n_{\text{s}}.$ (14b)

The disjoining pressure is then simply the difference between the internal and
reservoir osmotic pressures, $\Delta\Pi=\Pi^{\text{i}}-\Pi^{\text{res}}$. Note
that, as before, in the equations presented here, we assume that the
polyelectrolyte and ions show ideal behaviour. In appendices B and C we
present different derivations of the theory that allow one to include the
effects of a finite ion size and a different interaction parameter.

###  II.3 Self-consistent field lattice computations

To evaluate the accuracy of the model derived in the previous section, we have
performed Scheutjens–Fleer self-consistent field computations on a solution
confined between two hard walls containing solvent, polyelectrolyte/polymer
and salt ions. These computations take into account full Poisson–Boltzmann
electrostatics (on a mean-field level) and assume a finite size for all
components. In the following, we will focus on a brief description of the
parameters employed for the computations, as self-consistent field theory
already has been described in detail elsewhereFleer  _et al._ (1993);
Scheutjens and Fleer (1979); Bohmer, Evers, and Scheutjens (1990); Leermakers,
Eriksson, and Lyklema (2005).

The self-consistent field computations were carried out using the
Scheutjens–Fleer method, as implemented in the sfbox software package. The
method considers polymer chains placed on a lattice on a mean-field level,
thus fluctuations are not taken into account. The computations were carried
out on a lattice with a single gradient, terminated on either end with a hard
wall. As such, these SCF computations can be thought of as being on the level
of Flory–Huggins theory augmented with Poisson–Boltzmann electrostatics, with
a gradient in concentration occurring only in the direction normal to the hard
walls. The number of lattice layers was varied and the disjoining pressure was
computed by taking numerically the derivative of the resulting grand potential
to the distance $h$ between the walls.

A cubic lattice was used (corresponding with a lattice parameter of 1/6),
where the size of each lattice site is $b=$0.3\text{\,}\mathrm{nm}$$ with
corresponding volume $b^{3}$. The polyelectrolyte chains were composed of
$M=1000$ segments, where each segment exactly occupies a single lattice site,
thus its volume fraction is defined as
$\phi_{\mathrm{p}}=n_{\mathrm{p}}b^{3}M$. The solvent and ions all were
composed of one segment. The dielectric constant was set to $\epsilon=80$ to
model water. The Flory–Huggins interaction parameter between polyelectrolyte
and solvent was set to $\chi_{\text{polymer--solvent}}=0.5$, between
polyelectrolyte and wall to $\chi_{\text{polymer--wall}}=+1$, whereas all
other interaction parameters were set to zero ($\chi_{\text{polymer--
ions}}=0$, $\chi_{\text{solvent--ions}}=0$, $\chi_{\text{solvent--wall}}=0$,
and $\chi_{\text{ions--wall}}=0$). The charge of the polyelectrolyte was
smeared uniformly over all monomers unless otherwise stated.

##  III Results & Discussion

Figure 2: The effect of ionic strength and polyelectrolyte valence on the
disjoining pressure. a Donnan potential and b disjoining pressure between flat
plates as a function of interplate distance, for a number of different salt
concentrations. c Donnan potential and d disjoining pressure between flat
plates as a function of interplate distance, for a number of different
polyelectrolyte valences. At small interplate distances, the self-energy for
polyelectrolytes to enter the interplate region is prohibitively high, leaving
a constant Donnan potential and disjoining pressure, given by eq. 4
(horizontal dashed line, a). At intermediate distances the driving force for
polyelectrolytes towards the interplate region is sufficiently high to
overcome the self-energy, and a stabilising regime is found. Finally, at high
interplate distances both the Donnan potential and the disjoining pressure
tend towards their bulk values. The dotted grey curve in a denotes the
analytical approximation in eq. 11.

In fig. 2 we plot the Donnan potential and disjoining pressure as a function
of the inter-plate distance $h$ for a polyelectrolyte solution between two
flat plates using eqs. 10, 14a and 14b for varying ionic strength (a,b) and
varying polyelectrolyte valence (c,d). From fig. 2a,c we see that as the
separation $h$ between the plates decreases, the Donnan potential becomes more
negative, a consequence of the increasing conformational free energy penalty
for the polyelectrolyte to enter the interplate region. The universal
behavior, as predicted by eq. 11, is clearly visible in a. When the driving
force for polyelectrolyte chains to move towards this region is insufficient
to overcome the entropic penalty, the Donnan potential becomes constant,
regaining the result at close contact in eq. 4 (dashed horizontal lines in
fig. 2a). Here the presence of polyelectrolyte in the interplate region
becomes negligible. The Donnan potential and interplate separation $h$ where
this transition occurs depends on the polyelectrolyte valence and the salt
concentration in the solution. In general, both lower ionic strengths and
higher polyelectrolyte valences lead to stronger (more negative) Donnan
potentials at close contact and a concurrent smaller interplate distance where
this transition occurs.

Figure 3: Comparison of the Donnan theory with numerical SCF calculations.
Disjoining pressures are shown as a function of interplate separation $h$ for
a number of salt concentrations (a, c, d) and valences (b). For low
polyelectrolyte valence (a, $z=10$) the Donnan theory provides an excellent
quantitative match with SCF calculations of the same geometry, over a wide
range of salt concentrations. b As the polyelectrolyte valence increases, the
Donnan theory starts to overestimate the stabilisation regime slightly,
although the match is still quantitative. Moreover, the transition between
depletion and enrichment of polyelectrolyte is described remarkably well. For
a far higher polyelectrolyte valence (c, $z=100$) the overestimation of the
stabilisation regime becomes worse. When the disjoining pressure is normalised
by the disjoining pressure at contact in d, it becomes apparent that Donnan
theory shows relatively the largest errors at high salt concentrations, where
the effects electrostatic screening become increasingly important.

In fig. 2b,d we show the disjoining pressure caused by the partitioning of
(poly)ions corresponding to the Donnan potentials shown in panels a,c. The
disjoining pressure at close plate contact follows the trends of the Donnan
potentials, with a transition to constant behaviour where the concentration of
polyelectrolyte in the interplate region becomes negligible. Strikingly, we
observe a maximum in the disjoining pressure due to the presence of
polyelectrolyte between the plates — at these interplate separations, the
interaction between the plates is actually repulsive. The height of the
maximum follows the same trends as the depth of the disjoining pressure at
close contact, where lower ionic strength and higher polyelectrolyte valences
lead to a higher maximum in the disjoining pressure.

The repulsive forces between the plates are not a consequence of an electric
double layer repulsion — after all, the electric double layer is not taken
into account in this zero-field model. Instead, the repulsion arises directly
from the effect of confining the polyelectrolytes in a narrow space, and is
due to the reduced conformational entropy concurrent with this confinement.
While it would be entropically favourable, the enthalpic cost of a
polyelectrolyte leaving the interplate space is prohibitively high, scaling
with the exponent of $z\Psi_{\text{D}}$ as can be seen from eq. 8. As a
result, the polyelectrolyte concentration remains essentially constant until
small separations, and the polyelectrolyte chains effectively act as entropic
springs between the plates.

In the limit of $z$ approaching zero, the model still allows neutral polymers
to enter the depletion region. However the conformational free energy cost is
prohibitively high in the absence of a driving force provided by the Donnan
potential. As such, the concentration of neutral polymer in the depletion
regions becomes negligible and the classical behaviour of depletion
interactions with neutral polymers is recovered.

In fig. 3 we show comparisons between Donnan theory and SCF calculations. In
general, the theory is able to quantitatively predict the disjoining pressure
at close contact of the plates, as well as the sharp transition described
above. Furthermore, the maximum in the disjoining pressure is accurately
represented by the theory, and depending on the conditions the match is
quantitative. For relatively low polyelectrolyte valence ($z=10$) the match
between SCF and Donnan theory is excellent. With increasing $z$ the theory
starts to overestimate the maximum. As can be seen from fig. 3c this
overestimation is larger in absolute sense at low ionic strength. However,
upon normalising the disjoining pressure with respect to its value at close
contact, we see that the relative deviation is actually larger and extends
over a further range in the case of high ionic strength. In this salt
concentration regime, the plate separation $h$ exceeds the Debye length, and
consequently the assumption that the Donnan limit is valid — i.e., there are
no gradients in the potential — essentially breaks down. However, since the
absolute effect of charge on the disjoining pressure in this region is quite
small, this does not detract from the value of the theoretical predictions.

###  III.1 The range of the repulsive regime

Figure 4: The range of the repulsive regime. a The interplate separation at
maximum repulsion $h_{\text{max}}$, plotted as a function of the
polyelectrolyte valence, tends towards a universal curve as the valence grows.
The effect is stronger when the salt concentration is low, or the
polyelectrolyte concentration is high. The black dot-dashed line shows the
analytical result in the limit $zn_{\text{p}}\gg n_{\text{s}}$ presented in
eq. 17. b The repulsive part of the disjoining pressure decays almost
independently of the polyelectrolyte valence, except when the valence becomes
vanishingly small. The main difference is that for higher polyelectrolyte
valence, the transition to an interplate region depleted of polyelectrolyte
occurs at narrower interplate separations.

In fig. 4 we plot the interplate separation at maximum repulsion
$h_{\text{max}}$ as a function of polyelectrolyte valence $z$ for a number of
different ionic strengths and polyelectrolyte concentrations. Qualitatively,
an increase in the polyelectrolyte valence causes the maximum in the repulsion
to shift closer to contact — a direct effect of the increased affinity for
polyelectrolytes to remain between the plates even as the plates significantly
confine any present polyelectrolyte.

Surprisingly, as the valence increases, the curves tend to collapse to a
universal behaviour, an effect that is more pronounced when the ionic strength
is low or the polyelectrolyte volume fraction is high. In fact, this is the
same limit $zn_{\text{p}}\gg n_{\text{s}}$ where we earlier observed a similar
collapse in the resulting Donnan potential. To see this, we take the
derivative of the disjoining pressure with respect to the interplate
separation $h$ and set it to zero. Assuming that in this limit, counterions
dominate the osmotic pressure behaviour over salt ions, we neglect the terms
linear in $n_{\text{s}}$. As such, we get

$\frac{\partial\Delta\Pi/k_{\text{b}}T}{\partial h}\simeq
n_{\text{p}}\frac{\partial}{\partial h}\left[\exp(-z\Psi_{\text{D}}-\beta
F_{\text{p}})\left(1+z+\frac{h}{\chi(h)}\frac{\partial\chi(h)}{\partial
h}\right)-(z-1)\right]=0.$ (15)

The final term, coming from the reservoir contribution, is independent of $h$
and does not survive the differentiation. Taking the limit $zn_{\text{p}}\gg
n_{\text{s}}$ and substituting eq. 11 for $\Psi_{\text{D}}$, we can evaluate
the derivative, resulting in

$n_{\text{p}}\exp(-z\Psi_{\text{D}})\left[\frac{\partial^{2}\chi(h)}{\partial
h^{2}}h+\frac{\partial\chi(h)}{\partial
h}\left(2-\frac{z}{1+z}\frac{h}{\chi(h)}\frac{\partial\chi(h)}{\partial
h}\right)\right]=0.$ (16)

Since the factor outside the brackets will be nonzero for physical values of
$\Psi_{\text{D}}$, we can divide the factor out. The factor within brackets
can be solved for $h$ numerically, however, we can proceed analytically by
truncating Casassa’s function eq. 9 to only a single term. This is acceptable,
because the maximum typically occurs for $h\lesssim R_{\mathrm{g}}$ for highly
charged polyelectrolytes. In that case, eq. 16 has two analytical solutions,
of which one is positive and physical:

$\displaystyle h_{\text{max}}$
$\displaystyle=R_{\mathrm{g}}\sqrt{\frac{2\pi^{2}}{1+z}}.$ (17)

Even after truncating Casassa’s function to a single term, the resulting
solution is surprisingly accurate, and matches quantitatively with the
numerical solution to eq. 16, as we show in appendix D. In fig. 4 we plot the
analytical solution in eq. 17 (black dashdotted line). It is clear that the
interplate separation at maximum repulsion $h_{\text{max}}$ tends towards the
analytical solution in the limit where counterions dominate the behaviour in
the interplate region. Equation 17 can also be interpreted as a typical
electrical length scale, related to the radius of gyration of the
polyelectrolyte, modified by its valence.

###  III.2 Distribution of charge along polyelectrolyte

Figure 5: Effect of the charge distribution on the polyelectrolyte. Disjoining
pressure as a function of interplate separation. The Donnan theory is compared
to SCF calculations where the charge was placed on a single segment at the end
of a polymer (N999A1), on one or two segments near the middle of the polymer
(N500A1N499 and N499B2N499) and smeared out over all segments (C1000). Here, N
indicates a neutral segment, A a segment bearing a charge of $50$, B bearing a
charge of $25$ and C bearing a charge of $0.05$. Consequently, the total
charge on the polymer was kept constant at $50$ for all calculations. The
Donnan theory describes more accurately the SCF calculations where charge is
concentrated on a small part of the polymer.

One explanation for the overestimation of the Donnan theory with respect to
numerical SCF calculations, apart from neglecting effects of electrostatic
screening, is that the Donnan theory essentially assumes the polyelectrolyte
to be a point charge superimposed with a neutral polymer chain. We
investigated the effect of this assumption using SCF calculations. In fig. 5
we compare the Donnan theory with SCF calculations where the total
polyelectrolyte valence was distributed over the segments in four different
ways: first, we concentrated all the charges on a single segment located at
one end of the chain (N999A1). Second, we concentrated all charge on a single
segment in the middle of the chain (N500A1N499), with a third variant where
the total charge was smeared over the two middle segments (N499B2N499).
Finally, we compared this to the reference situation where the total charge is
smeared out over the whole polymer (C1000). We see that when the charge is
concentrated locally, the maximum in disjoining pressure increases when
compared to the situation where the charge is smeared out. For this effect
there is a negligible influence whether this single segment is located at the
end of a polymer chain or in the middle. The Donnan theory still somewhat
overestimates the magnitude of the maximum, yet the difference is minute in
this case.

Figure 6: Regimes where the Donnan theory is applicable. Disjoining pressures
as a function of the interplate separation $h$ for two different
polyelectrolyte valences $z$ for low $\phi_{\text{p}}$, low $n_{\text{s}}$
(a), high $\phi_{\text{p}}$, low $n_{\text{s}}$ (b), low $\phi_{\text{p}}$,
high $n_{\text{s}}$ (c) and high $\phi_{\text{p}}$, high $n_{\text{s}}$ (d).
The Donnan theory is compared to SCF calculations where the charge is
concentrated on a single segment (light symbols) or smeared out over the whole
polymer (dark symbols). In all cases, the Donnan theory describes more
accurately the SCF calculations where charge is concentrated on a single
segment, although the differences are only visible for relatively high ionic
strength (including counterions). Furthermore, when the polyelectrolyte volume
fraction is very high, the disjoining pressure at contact also deviates.

In fig. 6 we show in which conditions quantitative agreement between Donnan
theory and SCF calculations can be expected, and whether significant
difference can be expected between the behaviour of a polyelectrolyte with its
charge concentrated on a single segment (N999A1) and the behaviour of a
polyelectrolyte with its charge fully smeared out (C1000). In general, at low
polymer and salt concentrations (panel a), agreement between Donnan theory and
SCF calculations is excellent. At higher polymer volume fractions (panel b)
there is noticable difference between the N999A1 and C1000 cases. A higher
salt concentration (panel c), in contrast, causes the Donnan theory to
overestimate the maximum in disjoining pressure, and both effects are present
when the concentrations of both polymer and salt are high (panel d). In the
latter case, quantitative agreement between Donnan theory and SCF calculations
is moderate. It should be noted here that the concentration of counterions is
by far the dominant contribution to the ionic strength in panels b,d, where
the polyelectrolyte volume fraction is high. One consequence of this is that
the $100$-fold increase in salt concentration upon going from panel b to d has
only a minor effect on the magnitude of the disjoining pressure. In contrast,
upon going from panel a to panel c, the ionic strength is significantly
affected by the addition of salt in the reservoir, and as such the disjoining
pressure is decreased by more than a factor $10$.

##  IV Conclusions

In mixtures of colloids and polyelectrolytes, a Donnan potential arises across
the region depleted of polyelectrolyte. This Donnan potential provides a
driving force for the polyelectrolytes to shrink the depletion region, and
accumulate locally. In this paper, we derive a theoretical model that enables
us to calculate the depletion interaction between two flat plates that arises
due to polyelectrolyte in the reservoir. We allow the polyelectrolyte to enter
the depletion region, at the cost of a conformational free energy based on
Casassa’s expression for the conformational partition sum of a Gaussian chain
confined between flat plates Casassa (1967). In the so-called zero-field
limit, valid when the Debye length is sufficiently large compared to the
distances between the non-adsorbing surfaces, the electric field vanishes. In
such a case the partitioning of (poly)ions is governed by the requirement of
electroneutrality across the system. The electroneutrality condition leads to
a charge balance that can be solved numerically for the resulting Donnan
potential. We also derive three analytical expressions for the Donnan
potential in different limits.

The Donnan potential enables to calculate the disjoining pressure between the
flat plates due to the presence of polyelectrolyte. We find that the
disjoining pressure is constant at small interplate separations, and follows
from the analytical limit when polyelectrolytes are forbidden from entering
the interplate region. This disjoining pressure mediated by polyelectrolytes
is significantly larger in magnitude as for neutral polymers.

When the separation between the plates becomes comparable to the size of the
polyelectrolyte chains, the disjoining pressure transitions suddenly to a
repulsive regime. In this regime, polyelectrolytes confined between the plates
exert an outward pressure due to the compressed entropic spring effect of the
chains. The repulsive regime becomes larger with increasing polyelectrolyte
valence and decreasing reservoir ionic strength — either effect provides a
stronger driving force for polyelectrolytes to enter the interplate region at
smaller separations.

For low polyelectrolyte and ion concentrations, we find a quantitative match
between our model and numerical self-consistent field lattice computations on
a polyelectrolyte solution confined between two hard walls, which take into
account full Poisson-Boltzmann electrostatics. For higher concentrations, the
model still quantitatively predicts the crossover from the regime of close
plate contact to the repulsive regime. We see that at high polyelectrolyte
valence, when counterions contribute significantly to the ionic strength, the
analytic Donnan theory overestimates the repulsive regime in comparison to the
self-consistent field computations. A similar increase in the repulsive regime
is found for self-consistent field computations where the charge on the
polymer chains is concentrated on a single segment, rather than smeared out
over the full chain.

###### Acknowledgements.

MV acknowledges the Netherlands Organisation for Scientfic Research (NWO) for
a Veni grant (no. 722.017.005) and Nerd and Et for useful discussions. The
authors acknowledge prof. Frans Leermakers for the sfbox software package.

## Data availability

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

##  Appendix A Concentration in the interplate region

In the repulsive regime, quite a significant fraction of polyelectrolyte is
already depleted from the interplate region. To see this, we show in fig. 7
the polyelectrolyte segment volume fraction between the plates, as a function
of interplate separation. The figure also shows the maximum in disjoining
pressure (symbol) for each curve, as calculated using eq. 17. The
polyelectrolyte segment volume fraction between the plates is always equal or
lower than the bulk volume fraction.

Figure 7: In spite of the repulsive maximum in the disjoinin pressure between
the plates, the polyelectrolytes are always partially depleted from the
interplate region. By plotting the concentration of polyelectrolytes in the
interplate region as a function of interplate separation, it is apparent that
the concentration always monotonically drops from its bulk value to $0$ at
close contact. The repulsive barrier around $h=h_{\mathrm{max}}$ arises purely
from the entropic spring confinement of the polyelectrolytes. This can also be
observed from the position of the maximum in the disjoining pressure,
indicated with an open symbol, which tends to occur when the interal
polyelectrolyte volume fraction is about half of the bulk volume fraction.
This maximum is calculated using eq. 17.

##  Appendix B Two extended derivations in the canonical and grand canonical
ensemble

In this section, we present two more extended derivations of our model. The
first, in the canonical ensemble, provides more detail to the shorter
derivation in the main text. The second is an alternative in the grand
canonical ensemble, which takes into account the finite size of the ions. As
we will see, both provide near-identical results.

Starting in the canonical ensemble, we write down the canonical partition sum
of the system between two flat plates at separation $h$,

$Z(h)=\frac{z_{\text{p}}^{N_{\text{p}}^{\text{i}}}}{N_{\text{p}}^{\text{i}}!}\frac{z_{+}^{N^{\text{i}}_{+}}}{N^{\text{i}}_{+}!}\frac{z_{-}^{N^{\text{i}}_{-}}}{N^{\text{i}}_{-}!},$
(18)

where $N^{\text{i}}_{\text{p}},N^{\text{i}}_{+},N^{\text{i}}_{-}$ are the
numbers of particles of polyelectrolyte, positive and negative ions
respectively inside the volume enclosed by the two plates, and
$z_{\text{p}},z_{+},z_{-}$ the individual particle canonical partition sums,
which in this case are given by

$\displaystyle z_{\text{p}}$ $\displaystyle=V\exp(-z\Psi_{\text{D}})\chi(h),$
(19a) $\displaystyle z_{+}$ $\displaystyle=V\exp(-\Psi_{\text{D}}),$ (19b)
$\displaystyle z_{-}$ $\displaystyle=V\exp(+\Psi_{\text{D}}).$ (19c)

Here, $V$ is the volume enclosed by the two plates, $z$ the polyelectrolyte
valence, $\Psi_{\text{D}}$ the dimensionless Donnan potential that arises from
the partial exclusion of polyelectrolyte from the region enclosed by the
plates, and $\chi(h)$ is Casassa’s conformational part of the canonical
partition sum of a Gaussian chain confined between two flat plates at a
distance $h$, given by

$\chi(h)=\frac{8}{\pi^{2}}\sum_{n=1,3,5,\dots}^{\infty}\frac{1}{n^{2}}\exp\left(-\frac{n^{2}\pi^{2}b^{2}M}{6h^{2}}\right),$
(20)

where $b$ is the polymer segment length and $M$ the number of segments per
chain. We use Stirling’s approximation to calculate the Helmholtz free energy
from the partition sum, resulting in

$\displaystyle\frac{F}{k_{\text{B}}T}$
$\displaystyle=N_{\text{p}}^{\text{i}}\left[\ln\frac{N_{\text{p}}^{\text{i}}}{V}-1-\ln\chi(h)+z\Psi_{\text{D}}\right]$
$\displaystyle\quad+N^{\text{i}}_{+}\left[\ln\frac{N^{\text{i}}_{+}}{V}-1+\Psi_{\text{D}}\right]$
$\displaystyle\quad+N^{\text{i}}_{-}\left[\ln\frac{N^{\text{i}}_{-}}{V}-1-\Psi_{\text{D}}\right],$
(21)

from which we can calculate the osmotic pressure between the plates as

$\displaystyle\frac{\Pi}{k_{\text{B}}T}$
$\displaystyle=-\frac{\partial(F/k_{\text{B}}T)}{\partial V}$
$\displaystyle=n^{\text{i}}_{\text{p}}\left[1+\frac{h}{\chi(h)}\frac{\partial\chi(h)}{\partial
h}-hz\frac{\partial\Psi_{\text{D}}}{\partial h}\right]$
$\displaystyle\quad+n^{\text{i}}_{+}\left[1-h\frac{\partial\Psi_{\text{D}}}{\partial
h}\right]+n^{\text{i}}_{-}\left[1+h\frac{\partial\Psi_{\text{D}}}{\partial
h}\right].$ (22)

Here we have converted the particle numbers to number densities through
$n=N/V$. By requiring the system to be electrically neutral, we see that the
terms dependent on $(\partial\Psi_{\text{D}}/\partial h)$ all cancel out, as
$n^{\text{i}}_{-}=n^{\text{i}}_{+}+zn^{\text{i}}$. As such we are left with
the simpler form

$\frac{\Pi}{k_{\text{B}}T}=n^{\text{i}}_{\text{p}}\left[1+\frac{h}{\chi(h)}\frac{\partial\chi(h)}{\partial
h}\right]+n^{\text{i}}_{+}+n^{\text{i}}_{-}.$ (23)

The internal concentrations
$n_{\text{p}}^{\text{i}},n^{\text{i}}_{+},n^{\text{i}}_{-}$ can be found from
the chemical potentials that are calculated in the usual way from eq. 21. By
imposing chemical equilibrium with the reservoir surrounding the two plates,
the internal concentrations can be expressed in terms of the reservoir
concentrations and the resulting Donnan potential.

$\displaystyle n_{\text{p}}^{\text{i}}$
$\displaystyle=n_{\text{p}}\exp\big{[}-z\Psi_{\text{D}}\chi(h)\big{]},$ (24a)
$\displaystyle n^{\text{i}}_{+}$
$\displaystyle=n^{\text{res}}_{+}\exp\big{(}-\Psi_{\text{D}}\big{)}=n_{\text{s}}\exp\big{(}-\Psi_{\text{D}}\big{)},$
(24b) $\displaystyle n^{\text{i}}_{-}$
$\displaystyle=n^{\text{res}}_{-}\exp\big{(}+\Psi_{\text{D}}\big{)}=(n_{\text{s}}+zn_{\text{p}})\exp\big{(}+\Psi_{\text{D}}\big{)},$
(24c)

The Donnan potential in turn can be found by imposing charge balance on the
interplate-system and the reservoir. The resulting charge balance equation,

$(n_{\text{s}}+zn_{\text{p}})\exp(+\Psi_{\text{D}})=n_{\text{s}}\exp(-\Psi_{\text{D}})+zn_{\text{p}}\chi(h)\exp(-z\Psi_{\text{D}}),$
(25)

needs to be solved numerically to obtain the appropriate $\Psi_{\text{D}}$.

An alternative derivation can be devised, making use of the grand canonical
ensemble. The derivation is essentially lattice based: the interspace volume
is divided into lattice sites with a cell volume $v=b^{3}$ set by the course
grained length scale in the system. In this case it makes sense to set $b$
equal to the polymer segment length. The volume of the cell is chosen such
that a cell can only be occupied by a single polymer segment, ion or solvent
molecule, but all species can freely move between cells. In that case, a
single lattice site can be seen as a grand canonical ensemble in its own
right, with a grand canonical partition function given by

$\displaystyle\Xi_{\text{site}}$
$\displaystyle=\sum_{N_{\text{p}}}\sum_{N_{+}}\sum_{N_{-}}\lambda_{\text{p}}^{N_{\text{p}}}\lambda_{+}^{N_{+}}\lambda_{-}^{N_{-}}Z(N_{\text{p}},N_{+},N_{-}).$
(26)

Here, $N_{\text{p}},N_{+},N_{-}$ are the single-site copy numbers of polymers,
co-ions and counterions respectively, which by virtue of the lattice-based
setup can either be $0$ or $1$. Furthermore, each species has a fugacity
$\lambda_{i}=\exp(\beta\mu_{i})$, with $\beta=(k_{\text{B}}T)^{-1}$ the
inverse thermal energy. $Z(N_{\text{p}},N_{+},N_{-})$ is the (configurational
part of the) canonical partition sum of the site, and since we only allow a
single species to occupy the lattice site, this canonical partition sum can be
expressed as the Boltzmann factor of the excess free energy $\epsilon_{i}$
associated with species $i$ occupying the lattice site. In case of the co- and
counterions, we assume zero field and neglect any interactions between ions.
As such, their excess free energy is entirely due to their interaction with
the current Donnan potential $\Psi_{\text{D}}$.

We have chosen to model the polyelectrolyte as a point charge with valence $z$
occupying the site, surrounded by a Gaussian chain. As such, the model takes
into account the finite volume occupied by the ions, but underestimates the
volume occupied by polymer segments and polymer-polymer excluded volume
interactions are not taken care of well in this model. We do take into account
the effect of the confinement by the two flat plates surrounding the system
volume by including Casassa’s conformational part of the canonical partition
sum shown in eq. 20. We can then write for all allowed configurations

$\left\\{\begin{aligned} Z(0,0,0)&=1,\\\
Z(1,0,0)&=\exp(-z\Psi_{\text{D}})\chi(h),\\\
Z(0,1,0)&=\exp(-\Psi_{\text{D}}),\\\
Z(0,0,1)&=\exp(+\Psi_{\text{D}}).\end{aligned}\right.$ (27)

The chemical potentials of the species are set by the (much larger) reservoir.
Assuming ideal solution behaviour, $\lambda_{+}=\phi_{\text{s}}$,
$\lambda_{-}=\phi_{\text{s}}+\phi_{\text{p}}z/M$ and
$\lambda_{\text{p}}=\phi_{\text{p}}/M$, where $\phi_{\text{s}}$ is the volume
fraction of salt in the reservoir and $\phi_{\text{p}}$ the volume fraction of
polyelectrolyte segments in the reservoir. Finally, for the entire interplate-
system we assume that all sites are independent, so that the grand partition
function of the system reads

$\Xi_{\text{sys}}=\Xi_{\text{site}}^{V/v}.$ (28)

We can calculate the relevant thermodynamic variables of the system by taking
the appropriate derivatives of the grand potential
$\Omega=-k_{\text{B}}T\ln\Xi_{\text{sys}}$. The Donnan potential can be found
by imposing the electroneutrality condition, which leads to the same charge
balance eq. 25.

Figure 8: The difference between the canonical and grand canonical ensemble. a
Disjoining pressure as a function of interplate separation for the canonical
and grand canonical ensemble for a representative set of conditions. As can be
seen from the absolute deviation between the two results in b, the two
derivations can be used interchangably, yielding quantitative results with
only very minor deviations.

##  Appendix C Modified Casassa function

Tuinier and Fleer Tuinier and Fleer (2004a, b) derived a modification to
Casassa’s expression for the partition sum of a Gaussian chain confined
between flat plates. The modification takes into account the changes in radius
of gyration due to the Flory–Huggins interaction parameter between the
polyelectrolyte chain and the solvent. Substituting the modified expression
for the original in our model, we can calculate the effect of a different
interaction parameter on the disjoining pressure (although it should be noted
that we still assume the Van ’t Hoff equation for the polymer osmotic
pressure).

In fig. 9 we show how changes to the Flory–Huggins interaction parameter
$\chi$ between the polyelectrolyte and the solvent will shift the disjoining
pressure curve. When the interaction parameter is reduced from $\chi=0.5$ to
$\chi=0$, the disjoining pressure curve is compressed to smaller values of the
interplate separation $h$. Essentially, the changed radius of gyration
rescales the typical length scale in the system. The effect is far larger when
the concentration of polyelectrolyte is higher, with barely any effect for
$\phi_{\text{p}}=10^{-3}$, but a significant rescaling for
$\phi_{\text{p}}=0.05$. This rescaling is far smaller in our SCF computations,
which are shown as the data points in fig. 9.

Figure 9: The modified Casassa function shifts the range of disjoining
pressure curves depending on polymer solvency, a for $\phi_{\text{p}}=10^{-3}$
and b for $\phi_{\text{p}}=0.05$. Taking into account a different
Flory–Huggins interaction parameter between solvent and polyelectrolyte can
shift the range of the interaction between the plates. Especially at lower
concentrations, for a solvent with $\chi=0.5$ the disjoining pressure is
virtually identical to that obtained using the original Casassa function.
However, for decreasing interaction parameter $\chi$ the curves shift to
closer interplate separations. When the polyelectrolyte concentration is
higher, a small shift can be observed, although the effect does not
qualitatively change the behaviour. SCF calculations calculated with a
solvent-polyelectrolyte Flory-Huggins parameter of $\chi=0$ and $\chi=0.5$ are
provided for comparison, although at higher polyelectrolyte volume fractions,
the quantitative match between SCF and the theory for $\chi=0$ is lost.

##  Appendix D Range of the repulsive effect

In fig. 10 we show that the analytical approximation of the maximum in
disjoining pressure in eq. 17, obtained by truncating Casassa’s function after
its first term matches quantitatively with the numerical solution to eq. 16.
The truncation is acceptable since in the region where the maximum typically
occurs, $h\lesssim R_{\mathrm{g}}$, the first term of Casassa’s function
dominates.

Figure 10: Truncating Casassa’s equation provides a good analytical
approximation. The maximum in the disjoining pressure can be calculated by
numerically solving for the interplate separation $h$ where the derivative of
the disjoining pressure is zero. Truncating Casassa’s function to only a
single term allows one to find the analytical expression of eq. 17, that
matches quantitatively with the numerical solution to eq. 16. Here we work in
the limit of $zn_{\text{p}}\gg n_{\text{s}}$, where the curve becomes
independent of salt concentration and polyelectrolyte volume fraction.

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