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The Donnan equilibrium: I. On the thermodynamic foundation of the Donnan equation of state
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2011 J. Phys.: Condens. Matter 23 194106
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IOP PUBLISHING
JOURNAL OF PHYSICS: CONDENSED MATTER
J. Phys.: Condens. Matter 23 (2011) 194106 (11pp)
doi:10.1088/0953-8984/23/19/194106
The Donnan equilibrium: I. On the
thermodynamic foundation of the Donnan
equation of state
A Philipse and A Vrij
Van ’t Hoff Laboratory for Physical and Colloid Chemistry, Debye Institute for Nanomaterials
Science, Utrecht University, Padualaan 8, 3584 CH Utrecht, The Netherlands
Received 18 January 2011, in final form 28 February 2011
Published 27 April 2011
Online at stacks.iop.org/JPhysCM/23/194106
Abstract
The thermodynamic equilibrium between charged colloids and an electrolyte reservoir is named
after Frederic Donnan who first published on it one century ago (Donnan 1911 Z. Electrochem.
17 572). One of the intriguing features of the Donnan equilibrium is the ensuing osmotic
equation of state which is a nonlinear one, even when both colloids and ions obey Van ’t Hoff’s
ideal osmotic pressure law. The Donnan equation of state, nevertheless, is internally consistent;
we demonstrate it to be a rigorous consequence of the phenomenological thermodynamics of a
neutral bulk suspension equilibrating with an infinite salt reservoir. Our proof is based on an
exact thermodynamic relation between osmotic pressure and salt adsorption which, when
applied to ideal ions, does indeed entail the Donnan equation of state. Our derivation also shows
that, contrary to what is often assumed, the Donnan equilibrium does not require ideality of the
colloids: the Donnan model merely evaluates the osmotic pressure of homogeneously
distributed ions, in excess of the pressure exerted by an arbitrary reference fluid of uncharged
colloids. We also conclude that results from the phenomenological Donnan model coincide with
predictions from statistical thermodynamics in the limit of weakly charged, point-like colloids.
earlier by Bayliss [12] who found that the osmotic pressure
of Congo-red solutions drops significantly by the addition
of sodium chloride. Bayliss [12] attributed the decrease in
osmotic pressure to the aggregation of Congo-red molecules.
Donnan and Harris [3], while not excluding the possibility
that ‘some such aggregation may occur’ [3], concluded that
the effect of added salt has a different origin. They showed
that the osmotic pressure decrease is related to the unequal
distribution of sodium chloride on both sides of the membrane
(a relation that is further explained in sections 2 and 3). This
distribution is surprising given that the membrane itself is
perfectly permeable to both Na+ and Cl− ; for good reason
Donnan and Harris [3] refer to the salt gradient as a ‘near and
hitherto quite unsuspected phenomenon’ that, nevertheless, is
‘thermodynamically necessary’ (this necessity is demonstrated
here in sections 3 and 4). Moreover, Donnan and Harris [3]
also observed that the equilibrium salt distribution is such that
the NaCl concentration is higher ‘on the side of the membrane,
opposite to that in which the Congo-red solution is present’.
In other words, Congo-red is the charged species that cannot
permeate the membrane, and its presence leads to the expulsion
1. Introduction
One century ago [1] Frederic Donnan (1870–1956) first
published on a thermodynamic equilibrium involving ions
and poly-electrolytes that now bears his name:
the
Donnan equilibrium. Donnan was inspired by earlier work
of Wilhelm Ostwald [2] who discussed two electrolyte
solutions separated by a porous wall, freely permeable to
most charged species but impermeable to at least one of
them. Ostwald’s ‘halbdurchlässige Wände’ [2], that is,
semi-permeable membranes, turned out to have peculiar
consequences, including the existence of an equilibrium
electrical potential difference across the membrane, and the
unequal salt concentrations at both sides of the membrane.
These consequences were investigated in a number of
papers by Donnan and various co-workers [3–7].
Salt
partitioning, for example, was carefully studied by Donnan and
Harris [3] for solutions of Congo-red (the di-sodium salt of
diphenylbisazonaphthylamine-sulphonic acid) separated by a
semi-permeable membrane of parchment paper from a sodium
chloride solution. These dye-solutions had been studied
0953-8984/11/194106+11$33.00
1
© 2011 IOP Publishing Ltd Printed in the UK & the USA
J. Phys.: Condens. Matter 23 (2011) 194106
A Philipse and A Vrij
Figure 1. Simple instance of the Donnan equilibrium in which membrane M separates two salt solutions of equal volume. Initially (left) the
salt concentration is the same in both volumes. However, addition of a sodium salt with a monovalent anion A− that cannot permeate the
membrane induces a salt imbalance, see equation (3).
textbooks usually do not address. These issues also lead us
to re-examine both the assumptions underlying the Donnan
equation of state, as well as its thermodynamic foundation,
which forms the main topic of this paper. An important
clue to this thermodynamic foundation is the insight, already
reported by Vrij [10, 11] and Stigter [23], that knowledge of
the salt adsorption by charged colloids, in principle, suffices
to calculate their osmotic pressure. To our knowledge, this
type of calculation has not been reported yet for the Donnan
equilibrium. In section 3 we rederive Vrij’s equation for
the salt dependence of osmotic pressure and demonstrate in
section 4 how it leads to the Donnan equation of state. Further
discussion on the latter’s interpretation is given in section 5,
followed by conclusions and outlook in section 6.
of salt, in later literature also referred to as ‘negative salt
adsorption’ [8–11].
Interestingly, Donnan’s main inspiration for investigating
membrane equilibria came from physiology [4, 7]. Donnan
believed that these equilibria were relevant to understand the
functioning of biological cell membranes [1]. Moreover,
the membrane potentials could, perhaps, also account for
electrical nerve impulses [1], a speculation also made earlier
by Ostwald [2]. Donnan, nevertheless, was well aware that
biological membranes might be much more complex than the
passive porous walls in his experiments [6]. Donnan [6]
noted that his membrane equilibrium in any case constitutes
a relatively simple model system, as a first step towards
explaining salt or electrical potential gradients in biological
systems. In this spirit, the Donnan equilibrium is still
employed in textbook treatments [13] of the physics of living
cells and their membranes.
Though Donnan’s membrane equilibrium is conceptually
quite simple (it neglects all ion correlations) it nevertheless
entails a non-trivial, nonlinear expression for the osmotic
pressure (see section 4). This Donnan equation of state
accounts for a many-body system of colloids and ions that
are all coupled to the Donnan potential, that is, the constant
electrical background potential. This makes the Donnan
model an interesting reference for charged colloids dispersed
in organic solvents with very low ionic strength [14–17]. The
usefulness of the Donnan approach to analyze such systems, is
illustrated by the recent discovery [18–21] of a macroscopic
electric field in the sedimentation–diffusion equilibria of
charged colloidal spheres at low ion concentrations; the
existence of this field can be explained in a straight forward
manner via the Donnan equation of state [22].
Such findings confirm that the Donnan equilibrium
deserves more attention in the physics of charged colloids then
it has received over the last decades. To call attention to
this point is one motivation for this paper and, consequently,
the paper will be self-contained, for readers encountering
the Donnan equilibrium for the first time. We therefore
start in section 2 with an informal treatment of the Donnan
equilibrium, in which we also point to a few puzzling issues
(such as the interpretation of the nonlinearity in Donnan’s
equation of state and the presumed ideality of colloids) that
2. Nut-shell Donnan model
The essence of the Donnan equilibrium can be explained in
a nut-shell, with a simple example that, after its introduction
by Donnan [1, 6], was later incorporated in various classical
textbooks [24–27]. Consider a sodium chloride solution with
salt concentration ρs , divided by a membrane M into two
regions i and R with equal volume (figure 1). The membrane
is permeable to Na+ and Cl− such that regions i and R can
exchange salt molecules, that is, pairs of Na+ and Cl− ions.
The probability for such an exchange is proportional to the
product of ion concentrations such that at equilibrium:
i
i
R
R
ρNa
+ρ − = ρ
Na+ ρCl− .
Cl
(1)
Here ρ i and ρ R are ion number density in, respectively, region i
and R. When in both regions only NaCl is present, of course, all
ion densities in (1) equal ρs . We now add an electrolyte NaA
to region i that fully dissociates into Na+ and A− , each with
concentration ρs . The anion A− cannot pass the membrane
because, for example, it is simply too big or because it is
irreversibly adsorbed on a surface or any other large carrier.
The added NaA produces an initial surplus of Na+ ions in
region i so part of these excess ions will diffuse to region R,
each accompanied by a Cl− ion to maintain electro-neutrality,
leading to an imbalance in salt concentration between i and
R. Since only neutral salt is displaced, the electro-neutrality
2
J. Phys.: Condens. Matter 23 (2011) 194106
A Philipse and A Vrij
(see section 4) the Donnan equation of state is a nonlinear
function of the concentration of colloidal species that cannot
permeate the membrane; the linearity in (5) is a fortuitous
result for the choice of concentrations and volumes in figure 1.
The question is then whether this nonlinearity is consistent
with the assumption of colloids obeying Van ’t Hoff’s law, the
latter stating that pressure depends linearly on colloid number
density ρc .
Apart from this issue of consistency, one can question
why Van ’t Hoff’s law has to be invoked anyhow in the
Donnan equilibrium. For the equilibrating salt molecules
the assumption of ideality is convenient to obtain the simple
expression (1) for their equilibrium concentrations. However,
the A− solutes in figure 1 are not involved in any equilibrium:
their only role in the Donnan model is to be non-diffusive and
to produce counter-ions. Given this role, it is not evident why
one should assume (either out of convenience or necessity) the
validity of Van ’t Hoff’s law (4) for these solutes. To address
these and other issues we next investigate thermodynamic
foundation of the Donnan equilibrium in sections 3 and 4.
condition itself allows any salt distribution between i and R; the
additional constraint, of course, is that the salt distribution is
the equilibrium one. Suppose that in equilibrium (see figure 1)
the salt concentrations in i and R have changed, respectively,
with an amount L s and −L s , then the equilibrium condition (1)
yields:
(2ρs + L s )(ρs + L s ) = (ρs − L s )2 ,
(2)
with the solution
L s = − 15 ρs .
(3)
Thus, due to the presence of the ‘non-diffusing’ ions A− ,
salt is expelled to the reservoir R that is devoid of A− ions.
This expulsion, also referred to as ‘negative salt adsorption’,
corresponds, in the particular setup of figure 1, to a substantial
20% decrease in region i of the initial salt concentration.
This negative salt adsorption also changes significantly the
osmotic pressure difference across the membrane. If we
would totally ignore the presence of salt and assume that is only determined by the dissociated electrolyte NaA, then
Van ’t Hoff’s law yields [28]:
= 2ρs ,
kT
(4)
3. The relation between salt adsorption and osmotic
pressure
where k is the Boltzmann constant and T the absolute
temperature (note that ρs is a number density). However,
taking the salt partitioning between i and R in figure 1 into
account we have:
Any net amount of salt transferred from an osmotic cell, such
as a red blood cell, to the surrounding salt reservoir changes the
balance in concentrations of thermal ions and, consequently,
changes the osmotic pressure difference between cell and
reservoir. In section 2 we already found in equation (5) a
simple instance of the relation between osmotic pressure and
salt adsorption. We will now derive a general differential
equation for this relation, reported earlier by one of us
elsewhere [11, 29], to show in section 4 how it entails the
Donnan equation of state. Instead of the finite reservoir in
figure 1 we consider a large electrolyte reservoir that fixes
the chemical potential ρs of the salt (figure 2). To find
the relation between negative salt adsorption and the osmotic
pressure difference across the membrane, we start from the
fundamental thermodynamic equation to clearly identify all
required assumptions. This fundamental equation for the
change in internal energy U of the cell with entropy S is given
by [30]:
dU = T d S − P dV +
μ j dn j .
(6)
6
= 2ρs + 2(ρs + L s )− 2(ρs − L s ) = 2ρs + 4 L s = ρs , (5)
kT
5
which implies a 40% reduction of the osmotic pressure in
comparison to (4). The origin of this reduction is clear: if
in figure 1 L s salt molecules are expelled then 2 L s thermal
ions migrate from i to R such that the difference in ion number
density—and hence the difference in osmotic pressure—across
the membrane decreases with 4 L s .
The results (3) and (5) for the simple setup of figure 1
seem quite straightforward but, nevertheless, raise a few issues
that require the more extensive treatment of the Donnan
equilibrium in later sections. The choice of the non-diffusing
ion A− in figure 1, for example, is rather special: it is
monovalent and happens to have the same number density as
the salt ions. The obvious question is how the osmotic pressure
from (5) will look like for non-permeating anions with higher
valences and other concentrations. In addition, the volumes of
i and R in figure 1 are equal and so we could inquire what
happens to the osmotic pressure if solution R is actually a
very large salt reservoir that fixes the salt’s chemical potential
everywhere in i and R. Any effect of reservoir size is also of
practical importance: in a concentrated dispersion of red blood
cells the extra-cellular solution is certainly a finite reservoir in
comparison to the total volume of the blood cells. However, for
a very dilute suspension of charged colloids, the background
electrolyte solution will constitute an almost infinite reservoir.
A third issue relates to the linearity of the equation of
state in equation (5). Since equation (5) assumes Van ’t Hoff’s
law for ideal, non-interacting solute molecules, this linearity
seems at first sight quite plausible. However, in most cases
j
Here V is the cell’s volume, P is the cell’s pressure, μ j is the
chemical potential per particle of species j in the cell and n j
is the number of particles of species j . We restrict ourselves
to the following species: colloids (label c), solvent (label o),
monovalent cations (label +) and monovalent anions (label −).
Thus:
dU = T d S− P dV +μc dn c +μo dn o +μ+ dn + +μ− dn − . (7)
This equation is formally correct but, nevertheless, should be
modified because only salt molecules, i.e. pairs of cations
and anions can leave or enter the cell to maintain its electroneutrality, that is, dn + = dn − = dn s , where dn s is the number
3
J. Phys.: Condens. Matter 23 (2011) 194106
A Philipse and A Vrij
obtain a relation between osmotic pressure and salt adsorption
from (13), we first need to eliminate the solvent chemical
potential μ0 . The required second relation for μ0 follows
from the Gibbs–Duhem relation for the reservoir which relates
variations in chemical potentials of solvent and salt:
d P R = ρoR dμRo + ρsR dμRs = 0,
(14)
taking into account that the reservoir pressure P R is kept
constant. For the pure solvent, osmotic equilibrium between
cell and reservoir requires that:
μRo = μo ,
(15)
whereas the equilibrium condition for the salt is:
Figure 2. An osmotic cell with pressure P and volume V is in
equilibrium with a very large reservoir with pressure P R and volume
V R V . Colloid c cannot pass the membrane which is only
permeable to solvent (subscript 0) and small solute molecules
(subscript i); in the reservoir solute molecules carry the superscript
R. Cell and reservoir are at the same constant temperature T .
μRs = μs .
(16)
Using conditions (15) and (16) the reservoir’s Gibbs–Duhem
relation (14) becomes:
ρoR dμo + ρsR dμs = 0,
of transferred salt molecules, for example the neutral salt NaCl.
The chemical potential of the salt is a linear combination
of individual ion potentials weighted with their valency [26].
Thus for a 1:1 electrolyte μs = μ+ + μ− and equation (7)
becomes:
dU = T d S − P dV + μc dn c + μo dn o + μs dn s .
which can now be combined with the Gibbs–Duhem
relation (13) of the osmotic cell to eliminate the solvent
chemical potential:
d P = ρc dμc + L s dμs .
(8)
KNaz → K
+
+ z Na .
L s = ρo
(9)
(10)
Thus in (7) the term μs dn s guards the cell’s electro-neutrality
with regard to ion exchange with the reservoir, whereas any
dissociation inside the cell (be it from colloids or simple
electrolyte) must obey (10). We now return to (7) and integrate
it for given values of the intensive parameters T, P and μ to
obtain:
U = T S − PV + μc n c + μo n o + μs n s .
= −μc dρc + L s dμs ,
≈ ρs − ρsR ,
(19)
(20)
where we have substituted in (18). Taking the derivative of (20)
with respect to the salt chemical potential μs , for given colloid
number density ρc , we find:
∂P
∂μc
− ρc
= Ls.
(21)
∂μs ρc
∂μs ρc
(11)
Cross-differentiation in (20) yields the Maxwell relation:
∂μc
∂ Ls
−
=
,
(22)
∂μs ρc
∂ρc μs
(12)
For constant temperature we obtain from (12) the Gibbs–
Duhem relation for the osmotic cell in the form:
d P = ρc dμc + ρo dμo + ρs dμs ,
d(P − ρc μc ) = d P − ρc dμc − μc dρc
Taking the differential dU and equating it to (7) yields the
Gibbs–Duhem relation:
0 = S d T − V d P + n c d μc + n o d μo + n s d μs .
ρs
ρR
− sR
ρo
ρo
quantifies salt adsorption, relative to the imbalance in
solvent concentration between cell and reservoir. For dilute,
incompressible solutions ρ0 ≈ ρ0R , allowing the simplification
indicated in (19). We are interested in the osmotic pressure
for a given colloid concentration, so in (18) ρc should be the
variable instead of the colloid chemical potential μc . Therefore
we consider the variation in P − ρc μc rather than the pressure
differential itself:
The dissociation can neither violate the cell’s electro-neutrality
so inside the cell it must be the case that:
n + = zn c + n − .
(18)
Here the parameter
The requirement that only neutral entities can enter or leave the
cell, of course, also holds for the colloids. Suppose a neutral
colloid K is added to the cell which dissociates to produce z
sodium cations:
−z
(17)
which on substitution in (21) yields the equation that was first
reported in [11] (see also [29, 31]):
∂P
∂ Ls
= L s − ρc
.
(23)
∂μs ρc
∂ρc μs
(13)
where ρ = n/V is a number density; ρs is the number
density of salt molecules in the electrolyte reservoir. To
4
J. Phys.: Condens. Matter 23 (2011) 194106
A Philipse and A Vrij
Since in the osmotic pressure difference = P − PR , the
pressure PR of the large reservoir is a constant we may also
write:
∂
∂ Ls
= L s − ρc
.
(24)
∂μs ρc
∂ρc μs
From (28) and (29) we obtain the differential equation ρi dμi =
kT dρi , with the solution:
μi = μi,0 + kT ln(ρi /ρi,0 ),
where μi,0 is the chemical potential at some reference number
density ρi,0 . The salt chemical potential μs = μ+ +μ− follows
from (30) as:
ρ+ ρ−
;
ρs2,0 = ρ+,0 ρ−,0 . (31)
μs = μs,0 + kT ln
ρs2,0
The practical implication from this equation is clear: take a
large salt reservoir with constant salt chemical potential μs , and
measure or calculate the salt concentration in the cell, namely
L s in equation (19), as a function of colloid concentration ρc .
Then the RHS of (24) is known, and subsequent integration
yields the osmotic pressure as a function of ρc and ρs .
It should be noted that (24) is a rigorous consequence of
the fundamental equation (6) applied to neutral bulk phase.
The question is now which additional assumptions are needed
to extract an analytical expression for from (24). It turns out
that, to find such an expression, we only have to assume that
the salt dissociates into ideal ions.
Applying expression (31) to salt molecules in the osmotic cell
(no superscript) and the reservoir (superscript R) we readily
find from the equilibrium condition (16):
ρ+ ρ− = ρ+R ρ−R ,
To obtain the osmotic pressure from (24) we must further
specify the salt adsorption L s and its dependence on the colloid
concentration ρc . A charged colloid1 in the osmotic cell expels
co-ions (i.e. ions with the same charge sign as the colloids)
to the reservoir, and since each of them is accompanied by a
counter-ion we can write for the negative salt adsorption L s by
negatively charged colloids, see equation (19):
ρ− = ρs .
ρ+ ρ− = (ρsR )2 = constant,
ρ−
= −y + 1 + y 2 ;
R
ρs
(25)
(26)
zρc
.
2ρsR
ρ− − ρ−R
L−
=
= −y + 1 + y 2 − 1,
R
R
ρs
ρs
(34)
(35)
whereas for the cations (see also equation (26)):
L+
=
y
+
1 + y 2 − 1.
ρsR
(36)
The anion adsorption (35) is the salt adsorption to be inserted
in equation (24) which we first rewrite to:
1
L−
∂ L − /ρsR
∂
= R −y
,
(37)
ρsR ∂μs ρc
ρs
∂y
μs
(27)
The differential di equals the corresponding variation in
hydrostatic pressure Pi at constant chemical potential μ0 of the
solvent:
d Pi |μ0 = kT dρi .
(28)
which then together with (35) leads to:
∂
1
1
= − 1.
R
ρs ∂μs ρc
1 + y2
On the other hand, d Pi must also satisfy the Gibbs–Duhem
relation (cf equation (13)):
d Pi = ρ0 dμ0 + ρi dμi .
y=
Here y is the ratio of ions produced by the colloids, zρc ,
to the total ion concentration, 2ρsR , in the salt reservoir.
The (negative) adsorption of anions follows from substitution
of (34) in (25):
These conditions, incidentally, are a re-iteration of earlier
neutrality conditions: no new assumption is introduced here.
To find ρ− we need, in addition to (26), a second equation
in ρ− which we obtain from the equilibrium condition for the
salt in (16). We derive here the chemical potential μs of salt
in a pure solvent from Van ’t Hoff’s law for ideal ions, a
law that was already used in section 2, equation (4). For a
number density ρi of species i the differential osmotic pressure
is according to Van ’t Hoff [28]:
di = kT dρi .
(33)
is the additional equation for ρ− we were looking for.
Equations (26) and (33) lead to a quadratic equation for the
anion concentration ρ− with the positive root:
Note that the anion density ρ− in the cell equals the salt
concentration ρs in the cell. To calculate the anion adsorption
L − , we again make use of the electro-neutrality condition,
which for the reservoir reads ρ+R = ρ−R = ρsR and for the cell:
ρ+ = zρc + ρ− .
(32)
which is precisely the equilibrium condition for NaCl stated
without proof in equation (1). For the large reservoir in figure 2
the salt concentration remains constant, in contrast to the finite
reservoir in equation (2). Therefore
4. The Donnan equation of state
L s = L − ≈ ρ− − ρ−R ;
(30)
(38)
We have already assumed that ions are ideal, so the chemical
potential of the salt, set by the reservoir, follows from
substitution of ρ+ = ρ− = ρsR in (31):
(29)
1 In fact, the entity that cannot pass the membrane can be any charged object
or surface, and is not necessarily a diffusing colloid.
μs = μs,0 + 2kT ln(ρsR /ρs,0 ).
5
(39)
J. Phys.: Condens. Matter 23 (2011) 194106
A Philipse and A Vrij
From equations (38) and (39) we obtain the differential
equation:
1
1
zρc
R
d = 2ρs − 1 d(ln ρsR ); y = R , (40)
kT
2ρs
1 + y2
root term, purely due to ions, is the term calculated via the
Donnan equilibrium; this equilibrium does not (and cannot)
specify the osmotic pressure of colloids in their uncharged
state. Thus (y = 0) relates to uncharged colloids of arbitrary
shape and, in principle, arbitrary concentrations.
(y = 0), for example, could be the Carnahan–
Starling equation of state for uncharged, hard spheres [29].
Nevertheless, in the present derivation the colloid volume
fraction is still restricted to small values because otherwise
the assumption in (19), that solvent concentrations in cell and
reservoir are equal, does not hold. A finite colloid volume
fraction also decreases the available solvent volume for ions
which will also affect the Donnan equation of state. We will
deal with these topics in more detail in a future paper, and
focus in the remainder of the discussion mainly (see, however,
appendix) on the Donnan equation of state (45) for ideal
colloids.
It is noteworthy that in most treatments of the Donnan
equilibrium (see f.e. [6, 22, 24]) equation (46) is postulated,
and then (45) is derived as its consequence, without noticing
that the assumption of ideal, uncharged colloids is actually
not required. It is also custom [1, 8, 9, 22, 24, 26], as we
have also done in section 3, to employ a semi-permeable
membrane for describing the Donnan equilibrium. However,
thermodynamically speaking, this membrane setup is merely a
chosen path to gauge a change in a state parameter, in this case
an osmotic pressure difference. In essence the Donnan model
pertains to a suspension of charged colloids in equilibrium
with an electrolyte reservoir devoid of these colloids. To
monitor this equilibrium the only requirement is that—on the
experimental time scale—colloids diffuse very much slower in
(or into) the salt reservoir than ions.
This time scale requirement can be realized in various
ways. For example, one could choose colloids large enough
to make them inherently very sluggish in comparison to
ions, such that the ‘salt reservoir’ is simply the background
electrolyte in which the charged colloids are dispersed. One
could also apply an external field that acts on the colloids
but not significantly on ions; a recently studied example is a
gravitational [17] or centrifugal field [19] that confines colloids
to the bottom of a vessel, allowing the very much lighter
salt molecules to equilibrate with a supernatant electrolyte
solution. Another option is to introduce a porous medium
which excludes colloids via a steep (hard-core) repulsion,
whereas ions and solvent can diffuse through the medium.
This porous medium, of course, is the semi-permeable wall
in the case of membrane equilibrium experiments as sketched
in figures 1 and 2. Thus, the wider significance of the Donnan
model is its resultant thermodynamic, osmotic equation of state
for charged colloids, that is independent of the mechanism that
makes colloids ‘non-diffusible’ relative to ions.
with the solution:
= const. + 2ρsR [ 1 + y 2 − 1].
kT
(41)
The integration constant follows from the boundary condition
that for z = 0 (thus y = 0), the osmotic pressure reduces
to the pressure (y = 0) exerted by uncharged colloids.
Thus our final result for the osmotic pressure from the Donnan
equilibrium reads:
(y = 0)
=
+ 2ρsR [ 1 + y 2 − 1].
kT
kT
(42)
Using the expressions (35) and (36) for the adsorption of
anions and cations we can rewrite (42) as
(y = 0)
=
kT
kT
+ ρsR [−y + 1 + y 2 − 1 + y + 1 + y 2 − 1]
L−
(y = 0)
R L+
+ ρs
+ R ,
(43)
=
kT
ρsR
ρs
which on substitution of L ± = ρ± − ρ±R yields:
(y = 0)
=
+ ρ+ + ρ− − 2ρsR .
kT
kT
(44)
In case the reference fluid of uncharged colloids obeys
Van ’t Hoff’s law for ideal colloids, (y = 0) = ρc kT , (42)
simplifies to:
= ρc + 2ρsR
1 + y2 − 1 ;
kT
y=
zρc
,
2ρsR
(45)
such that
= ρc + ρ+ + ρ− − 2ρsR .
(46)
kT
Equation (46) is the usual expression for the Donnan pressure,
namely the pressure exerted by ideal colloids and ideal ions in
an osmotic cell, in excess to the osmotic pressure = 2ρsR kT
of the large salt reservoir.
5. Discussion
5.1. Generality of the Donnan equation of state
We have identified here the thermodynamic foundation of the
Donnan equation of state, namely equation (24) that is exact for
a homogeneous, electrically neutral bulk phase, supplemented
with the assumption of ideal ions. Equation (24) not merely
entails the usual formulation (46) of the Donnan equation,
valid for ideal colloids and ions but, instead, the more general
equation of state (42). The latter comprises the pressure
(y = 0) of an uncharged colloidal fluid, supplemented with
the square root term when the colloids are charged. This square
5.2. Salt adsorption
We continue the discussion of the Donnan model by inspecting
the two limiting cases for an electrolyte reservoir with,
respectively, low and high salt concentration. First we consider
salt adsorption, to continue in section 5.3 with osmotic
6
J. Phys.: Condens. Matter 23 (2011) 194106
A Philipse and A Vrij
5.3. Osmotic pressure
pressure. When the salt concentration is much lower than
the counter-ion density ρc z , the negative salt adsorption from
equation (35) vanishes as:
ρ− − ρ−R
L−
1
;
=
∼ −1 +
ρsR
ρsR
2y
y=
zρc
1.
2ρsR
In section 2 the question was raised how the simple equation
of state in (5) for the setup in figure 1 would change for a salt
reservoir that is large enough to fix the salt chemical potential.
The latter situation modifies the equation of state into (43)
with a form that is quite different from (5). Nevertheless, the
outcome for the osmotic pressure changes little: in figure 1
we have monovalent colloids (z = 1) and ρc = ρsR such that
y = zρc /2ρsR = 1/2 and equation (45) yields:
5
= ρs 2
− 1 ≈ 1.236ρs ,
(54)
kT
4
(47)
At very low salt concentration in the reservoir, in the limit
y → ∞, (47) yields ρ− → 0: in this limit the osmotic cell
is completely salt-free. On the other hand, at sufficiently high
salt concentration such that ρsR zρc , the salt adsorption from
equation (35) can be expanded as:
ρ− − ρ−R
L−
1
=
= −y + y 2 + · · · ,
R
R
ρs
ρs
2
y=
zρc
1.
2ρsR
(48)
a pressure that is only slightly larger than the pressure /kT =
1.2ρs obtained from (5). The difference is indeed due to the
finite reservoir size in figure 1: salt expulsion by the colloids
raises the salt concentration in the reservoir which diminishes
the osmotic pressure difference relative to an infinite reservoir
with constant salt concentration.
We now consider the same limiting cases (low and high
salt concentration) for the osmotic pressure, just as for the
salt adsorption in section 5.2. First we remind that the square
root term in (45) and (42) is solely due to ions and that this
nonlinearity originates in the coupling of ion densities in the
osmotic cell via the electro-neutrality condition (26) and the
salt equilibrium with the infinite reservoir that leads to (32).
Thus the limiting cases that follow address the purely ionic
contribution to the osmotic pressure. This contribution can
be seen most clearly at low ionic strength where the counterion density outweighs the ion concentration in the reservoir.
Then (45) asymptotes towards:
Retaining only the linear term in (48) we obtain
ρ− = ρsR − 12 ρc z,
(49)
showing that at sufficiently high ionic strength, charged
colloids—or any charged objects for that matter—expel
precisely half of their counter-ions in the form of salt molecules
to the salt reservoir. Interestingly, this result does not only
follow from the phenomenological Donnan model: (49) can
also be obtained via a statistical approach, as further discussed
in section 5.4.
Here we continue with writing down the equivalent of (48)
for cations, obtained from (36):
ρ+ − ρ+R
L+
1
=
= y + y2 + · · · ,
ρsR
ρsR
2
y 1.
(50)
∼ (z + 1)ρc ;
kT
The sum of equations (48) and (50) equals the total ion
concentration in the cell of figure 2, in excess to the salt
concentration in the reservoir:
L+ + L−
= y2 + · · ·
ρsR
y 1,
y=
zρc
1.
2ρsR
(55)
This familiar result [18, 22] shows that in the low salt (strictly
speaking: no salt) limit, the osmotic pressure is solely due to
the z -valent colloids plus their ρc z counter-ions. Equation (55)
is the pendant of the negative salt adsorption in equation (47);
the latter indeed confirms that in the limit y → ∞, no salt
is present and therefore no anions are available to accompany
counter-ions to the reservoir. Then all counter-ions contribute
to the excess osmotic pressure, making (55) the maximal
excess pressure a Donnan equilibrium can generate.
The assumption of point-like colloids in (55), incidentally,
strictly implies that z is of order unity. However, replacing
points by (even fairly large) spheres will have little effect on
the osmotic pressure at low colloid densities, if the pressure is
dominated by the much larger number of counter-ions, as has
also been experimentally found for charged silica spheres in
ethanol [19, 21].
At sufficiently high salt concentrations that ρsR zρc the
square root in (45) can be expanded with the result:
z 2 ρ
1 z 4 ρc 3
c
=1+
−
ρc kT
2 ρsR
4 2
ρsR
1 z 6 ρc 5
zρc
+
···;
y = R 1.
(56)
R
8 6
ρs
2ρs
(51)
from which we find that this total excess ion density depends
quadratically on the concentration of colloidal particles in the
cell:
z2
L + + L − = R ρc2 .
(52)
4ρs
The thermodynamics in section 3 already demonstrated the
close connection between salt adsorption and osmotic pressure.
Indeed, on substitution of (52) in the osmotic equation of
state (43) we find:
(y = 0)
(y = 0)
z2
=
+ L+ + L− =
+ R ρc2 , (53)
kT
kT
kT
4ρs
showing that, at high salt concentration, the first-order
contribution from ions to the osmotic pressure of uncharged
colloids, is a quadratic in the colloid density. This quadratic
will be further discussed in section 5.3 on osmotic pressure.
7
J. Phys.: Condens. Matter 23 (2011) 194106
A Philipse and A Vrij
The ρc2 term was also found via the excess ion density L + + L −
in (52) and (53); the further expansion in (55) shows that in the
next terms only even powers of z appear, as a consequence of
the Taylor expansion of (1 + y 2 )1/2 . We define coefficients Dn
via the series expansion:
=1+
Dn ρcn−1 .
ρc kT
n=2
adsorption by the charge on the sphere. For the limiting case
of weakly charged, point-like colloids equation (60) simplifies
to:
∞
p
∼ 4π
ur 2 dr ;
for u 1 and σ → 0. (61)
ρsR
0
Since the potential u is small, it would be obvious to
substitute in (61) the Debye–Hückel potential (equation (A.9)
in appendix). However, the value of the integral in (61) can be
found, without specifying the potential u , from the requirement
of overall electrical neutrality for the point colloid and its
surrounding ion profiles:
∞
z(−e) + 4π(−e)
[ρ− (r ) − ρ+ (r )]r 2 dr = 0.
(62)
(57)
The first two ‘Donnan coefficients’ in (56) are:
D2 =
z2
;
4ρsR
D4 = −(D2 )2
1
.
4ρs
(58)
We avoid here the terms ‘virial expansion’ for (57) and ‘virial
coefficients’ for (58). First, a pressure virial series where
odd concentration powers are missing, as is the case in (56),
seems rather unphysical. Secondly, equation (24) is completely
general and anyhow does not assume the existence of a virial
expansion for the pressure; thus neither does the equation of
state (42). There is, nevertheless, a connection between the
coefficient D2 in (58) and the osmotic second virial coefficient
B2 of charged colloidal spheres: for weakly repulsive colloids
D2 coincides with B2 when the colloid diameter σ is much
smaller than the Debye screening length κ −1 . This limiting
coincidence will be further investigated in section 5.4.
0
Here (−e)[ρ− (r ) − ρ+ (r )] is the net charge density at a
distance r from the point colloid. We now invoke the
second requirement: the ion distributions ρ∓ (r ) are Boltzmann
distributions of ideal ions. Since the reduced electrical
potential u = eφ/kT is small, this second requirement yields:
ρ∓ (r ) = ρsR exp[±u] ∼ ρsR (1 ± u + 12 u 2 · · ·),
(63)
where ρsR is the ion density in the reservoir where u = 0.
From (63) and (62) we find:
∞
ze + 8πρsR e
ur 2 dr = 0.
(64)
5.4. Small, weakly charged colloids
0
Thus to satisfy electro-neutrality, the integral must be equal to:
∞
−z
ur 2 dr =
,
(65)
R
8
πρ
0
s
The treatment of the Donnan equilibrium in sections 3 and 4
clearly identifies its two pillars: first, the requirement of
overall electro-neutrality and, second, the requirement that ion
distributions are equilibrium distributions of ideal ions. We
will now show that results from statistical thermodynamics,
subject to the same two requirements, coincide with results
from the phenomenological Donnan model, in the limit of
weakly charged, point-like colloids. The term ‘point limit’,
incidentally, does not refer to a colloid that shrinks in an
absolute sense to a point; the term is merely used here for
brevity’s sake to denote a colloid with diameter σ that is
much smaller than the Debye screening length κ −1 such that
κσ 1.
First we consider the salt adsorption by a charged sphere
with diameter σ , with a total surface charge equal to z(−e).
Let p be the number of salt molecules that the sphere expels
to the surrounding electrolyte solution with constant salt
concentration ρsR :
ρ− − ρsR
p=
.
(59)
ρc
which on substitution in (61) yields:
p = − 12 z,
for σ → 0,
(66)
or in view of the definition of p in (59):
ρ− = ρsR − 12 ρc z,
for σ → 0.
(67)
This result is identical to equation (49) obtained via the Donnan
model. A recurrent theme in this paper is the close connection
between salt adsorption and osmotic pressure: thus the limiting
result (67) for the adsorption must have its pendant for the
osmotic pressure. To find this pendant we consider the osmotic
second virial coefficient for two charged spheres [29, 32]:
∞
2 3
U
B2 = πσ + 2π
1 − exp −
(68)
r 2 dr.
3
kT
σ
Here ρc is the number density of colloidal spheres and ρ− is
the (average) salt concentration in the vicinity of an individual
sphere. Stigter [37] (see also [10]) derived that:
∞
π 3
p
eφ
. (60)
=
σ
+
4
π
(1 − e−u )r 2 dr ;
u=
R
ρs
6
kT
σ/2
Here U is the double-layer interaction free energy between two
colloids at a center-to-center distance r σ . The first RHS
term in (68) is the second virial coefficient of uncharged hard
spheres; the integral is the contribution of charge to B2 . For
the limiting case of weakly charged, point-like colloids (68)
simplifies to:
∞
U 2
U
r dr ;
1 and σ → 0.
B 2 ∼ 2π
for
kT
kT
0
(69)
Here φ is the electrical potential at a distance r σ/2 from the
sphere center. The first RHS term in (60) is the salt expelled
by an uncharged sphere; the integral is the contribution to salt
8
J. Phys.: Condens. Matter 23 (2011) 194106
A Philipse and A Vrij
Just as for the adsorption p in (61), the second virial coefficient
in (69) can be found without specifying the potential in the
integrand. In (61), this potential is the electrical potential φ
around a point-like colloid with charge z(−e); in (69) U is the
work needed to bring a second point-like colloid to a distance
r from the central colloid:
φ
U
= z(−e)
= −zu.
kT
kT
Thus the second virial coefficient becomes:
∞
−z
B2 ∼ 2π(−z)
ur 2 dr = R p,
2
ρs
0
for u 1 and σ → 0,
6. Conclusions and outlook
A neutral bulk fluid containing charged colloids in equilibrium
with a large salt reservoir, satisfies an exact differential
equation between excess osmotic pressure and salt adsorption.
Its integration, under the additional assumption of ideal ions,
yields a general equation of state containing the sum of ionic
osmotic pressures and the pressure of the uncharged colloidal
fluid of, in principle, arbitrary concentration. In the limit
of ideal, uncharged colloids this equation of state reduces
to the classical Donnan result. The conclusion is that the
Donnan osmotic pressure, peculiar as its nonlinearity at first
sight may seem, has a rigorous thermodynamic foundation.
The quadratic colloid density term in the pressure equals the
osmotic second virial coefficient in the limit of weakly charged
colloids that are much smaller than the Debye screening
length. In that limit—where also the salt adsorption calculated
from the Donnan model and Debye–Hückel theory coincide—
the colloids experience only homogeneously distributed ions
subject to overall neutrality and equilibrium with the salt
reservoir, precisely the only two pillars of the Donnan model.
The Donnan model is probably the simplest approach
to a many-body fluid of unscreened colloids and ions in
(organic) solutions of very low ionic strength—and deserves
for this reason some more attention in the literature on charged
colloidal fluids. In this respect it is clearly of interest to further
refine or generalize the Donnan equation of state; an obvious
extension is the inclusion of regulation of the colloid charge
which is usually kept constant [1, 8, 22], also in this paper.
In addition, the size of the salt reservoir, usually taken to be
infinite, should also be taken into account via a generalization
of the Donnan equation of state to arbitrary reservoir volumes,
with both equations (5) and (42) as special instances.
(70)
(71)
where we have used equation (61) for the salt adsorption.
Equation (71) is another clear illustration of the tight
connection between osmotic pressure (here in the form of B2 )
and salt adsorption. From (66) and (71) we obtain
B2 =
z2
,
4ρsR
for σ → 0,
(72)
which is identical to the second Donnan coefficient in
equation (54). As mentioned above, the point limit merely
denotes colloids that are much smaller than the Debye length;
in the appendix the second virial coefficient is derived for
weakly charge colloids of arbitrary size. We note here that
various authors (see f.e. [10, 11, 32, 33]) have derived the
limiting result (72), though not via the brief reasoning given
here via equations (68)–(72)
The following comments may further elucidate the
coincidence, in equations (67) and (72), of results from the
phenomenological Donnan model and the statistical approach.
In the Donnan model all ions are coupled in the sense that
they jointly determine the average electrical Donnan potential
experienced by all charged species in the cell of figure 2.
This coupling does not depend on the thermal motion of the
colloids: their fixation (at arbitrary positions) diminishes the
osmotic pressure in (42) with (y = 0), and in (45) with
ρc kT , but leaves the nonlinear term intact. This term manifests
a salt imbalance between cell and reservoir that is maintained
by thermal as well as static point charges.
A similar salt imbalance is generated by two adjacent
colloids that are each surrounded by an electrical doublelayer. The region of double-layer overlap contains excess ions
relative to the surrounding reservoir solution (these excess ions,
it should be noted, are the origin of the DVLO-repulsion [35]).
However, in contrast to the homogeneous Donnan cell, the
distribution of excess ions is inhomogeneous, which is why B2
generally differs from the second Donnan coefficient D2 . Only
in the limit of a very dilute solution of very small colloids,
each colloid inhabits a large Debye cube κ −3 in which ions are
homogeneously distributed, subject only to electro-neutrality
and salt equilibrium. What equations (67) and (72) express is
that the point-like colloids could hardly distinguish such a large
Debye cube from an osmotic Donnan cell as in figure 2.
Acknowledgments
Mrs Marina Uit de Bulten-Weerensteijn is thanked for her help
in the preparation of the manuscript. Dr Ben Erné, Dr René
van Roij and Dr Mircea Rasa are acknowledged for various
discussions on the Donnan equilibrium, within a collaboration
that lead to [21].
Appendix. Second virial coefficient B2 for colloids of
finite size
To include finite colloid size in the second virial coefficient
we start from (42) and consider dilute, uncharged hard spheres
with an osmotic pressure given by:
(y = 0)
= ρc + B2HS ρc2 ;
kT
B2HS = (2/3)πσ 3 , (A.1)
where B2HS is the second virial coefficient of spheres with
diameter σ . Substitution of (A.1) in (A.2) and expansion of
the square root term up to order ρc2 yields:
= ρc + (B2HS + D2 )ρc2 ;
kT
9
D2 =
z2
.
4ρsR
(A.2)
J. Phys.: Condens. Matter 23 (2011) 194106
A Philipse and A Vrij
We remind here that only the coefficient D2 follows from
the Donnan model whereas the appearance of B2HS is solely
due to our choice for the pressure of uncharged colloids
in (A.1). For charged spheres the usual osmotic second virial
coefficient [29, 32] in the virial expansion up to order ρc2
= ρc + (B2HS + B2DL )ρc2 ,
kT
Substitution of this dimensionless DLVO-repulsion
in (A.5) yields:
2 ∞
2κ
z2
DL
B2 = R
exp[−κ(r − σ )]r dr
4ρs 2 + κσ
σ
2 z2
κσ
= R 1−
(A.12)
.
4ρs
2 + κσ
(A.3)
Thus B2DL factorizes into a term z 2 /4ρsR that merely stems from
electro-neutrality plus salt equilibrium, and a term that depends
on the Debye screening length. The latter term vanishes when
κ −1 approaches zero (κσ → ∞) where B2DL in (A.12) also
vanishes, as it should. In the other limit of the colloid size
approaching zero, (A.12) reduces to:
contains a contribution B2DL from the electrical double-layer of
charged spheres:
B2DL = 2π
U
r 2 dr,
1 − exp −
kT
∞
σ
(A.4)
that depends on the double-layer interaction free energy U
between two spheres at a center-to-center distance r . For
weakly charged spheres, or sufficiently high salt concentration,
the double-layer repulsion is weak at all distances such that:
B2DL
≈ 2π
∞
σ
U 2
r dr ;
kT
U
1.
kT
B2DL
∼
(A.5)
πε0 εr σ 2 φ02
exp[−κ(r − σ )].
r
(A.6)
Here φ0 is the surface potential of the spheres, ε0 εr is the dielectrical constant and κ −1 is the Debye screening length:
κ
−2
ε0 εr kT
=
.
2e2 ρsR
∞
σ/2
ur 2 dr =
−z
;
8πρsR
eφ
,
kT
(A.8)
where the lower integration boundary is now the colloid radius
σ/2. For u we take the Debye–Hückel potential [35]:
u = u0σ
exp[−κ(r − σ/2)]
;
2r
u0 =
eφ0
,
kT
(A.9)
for which the integral in (A.8) can be solved to yield the result,
also given in [35]:
ze = πεo εr φ0 σ (2 + κσ ).
(A.10)
Combining (A.6), (A.7) and (A.10) we obtain the double-layer
repulsion in the form:
2
z2
U
2κ
exp[−κ(r − σ )]
= R
.
kT
4ρs 2 + κσ
2πr
(A.14)
[1] Donnan F G 1911 Z. Electrochem. 17 572
[2] Ostwald W 1890 Z. Phys. Chem., Stoichiometrie
Verwantschaftslehre 6 71
Semi-permeable membranes, it should be noted, were already
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extensive diffusion studies, see: Graham T 1854 The
Bakerian lecture: on osmotic force Phil. Trans. R. Soc.
144 177
[3] Donnan F G and Harris A B 1911 J. Chem. Soc. 99 1554
[4] Donnan F G 1917 Scientia 24 282
[5] Donnan F G and Garner W E 1919 J. Chem. Soc. 115 1314
[6] Donnan F G 1924 Chem. Rev. 1 73
[7] Donnan F G 1929 Annual Report of the Smithsonian Institution
p 309
[8] Overbeek J T G 1953 J. Colloid Sci. 8 594
[9] Overbeek J T G 1956 Prog. Biophys. Biophys. Chem. 6 57
[10] Vrij A 1959 Thesis Light-scattering from charged colloidal
particles in salt solutions, Utrecht University, The
Netherlands
[11] Vrij A 1990 Colloids Surf. A 51 61
[12] Bayliss W M 1909 Proc. R. Soc. B 81 269
[13] Glaser R 2001 Biophysics (Berlin: Springer)
[14] Philipse A P and Vrij A 1988 J. Chem. Phys. 88 6459
[15] Philipse A P and Vrij A 1989 J. Colloid Interface Sci. 128 121
[16] Thies-Weesie D M E, Philipse A P, Nägele G, Mandl B and
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[17] Philipse A P and Koenderink G H 2003 Adv. Colloid Interface
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[18] van Roij R 2003 J. Phys.: Condens. Matter 15 2569
[19] Rasa M and Philipse A P 2004 Nature 429 857
[20] Warren P 2004 Nature 429 822
[21] Rasa M, Erné B H, Zoetekouw B, van Roij R and Philipse A P
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[22] Philipse A P 2004 J. Phys.: Condens. Matter 16 4051
(A.7)
u=
for κσ → 0,
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(A.13)
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10
J. Phys.: Condens. Matter 23 (2011) 194106
A Philipse and A Vrij
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