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arXiv:cond-mat/9802138v1 [cond-mat.soft] 12 Feb 1998
Preprint
Submitted to Physica A
Debye-H¨ uckel-Bjerrum theory for charged colloids
M. N. Tamashiro, Yan Levin∗and Marcia C. Barbosa
Instituto de F´ ısica, Universidade Federal do Rio Grande do Sul
Caixa Postal 15051, 91501-970 Porto Alegre (RS), Brazil
mtamash@if.ufrgs.br, levin@if.ufrgs.br, barbosa@if.ufrgs.br
We formulate an extension of the Debye-H¨ uckel-Bjerrum theory [M. E. Fisher and Y. Levin, Phys.
Rev. Lett. 71, 3826 (1993)] to the fluid state of a highly asymmetric charged colloid. Allowing for
the formation of clusters consisting of one polyion and n condensed counterions, the total Helmholtz
free energy of the colloidal suspension is constructed. The thermodynamic properties, such as the
cluster-density distribution and the pressure, are obtained by the minimization of the free energy
under the constraints of fixed number of polyions and counterions. In agreement with the current
experimental and Monte Carlo results, no evidence of any phase transition is encountered.
PACS numbers: 82.70.Dd; 36.20.−r; 64.60.Cn
I. INTRODUCTION
The technological importance of charged colloidal sus-
pensions can not be overemphasized. One comes across
these important systems in fields as diverse as the chem-
ical engineering and the environmental science. Many
water-soluble paints contain charged colloidal suspen-
sions as a main ingredient.
of great industrial importance is to stabilize the suspen-
sion against the flocculation and precipitation. On the
other extreme is the constantly growing environmental
necessity of cleaning contaminated water. For this it is
essential to find the most effective way of precipitating
the (usually) charged organic particles dissolved in the
water.
From the theoretical perspective the problem of
strongly asymmetric electrolyte solutions is extremely
difficult to study. The long-rangednature of the Coulomb
force combined with the large charge and size asymme-
try between the polyions and the counterions and coions
makes it impossible to use the traditional methods of
liquid-state theory. At high volume fractions the suspen-
sion will crystallize, that is, the polyions will become ar-
ranged in the form of a lattice. The solid state provides a
major simplification of reducing the many-polyion prob-
lem to that of one polyion inside a Wigner-Seitz (WS) cell
[1]. Unfortunately, at low densities, or in the presence of
a simple electrolyte, the suspension becomes disordered
and the WS picture is no longer valid [2]. A new strategy
must be tried.
The Debye-H¨ uckel-Bjerrum theory (DHBj) [3] was
quite successful in explaining the behavior of symmet-
ric electrolytes. The Bjerrum’s concept of association of
oppositely charged ions into dipolar pairs [4] served to
correct the Debye-H¨ uckel (DH) linearization of the non-
linear Poisson-Boltzmann equation [5]. Taking into ac-
In this case the problem
count the dipolar solvation energy made the coexistence
curve produced by the DHBj theory [3] become in excel-
lent agreement with the Monte Carlo simulations [6]. The
large surface charge of a polyion suggests that the cluster
formation should be even more important in the case of
polyelectrolytes. In the present work we shall explore to
what extent the counterion condensation influences the
thermodynamic properties of a polyelectrolyte solution.
The DHBj theory will be extended to treat the fluid state
of a charged colloidal suspension.
II. DEFINITION OF THE MODEL
We shall work with the primitive model of polyelec-
trolyte (PMP) [7]. The system will consist of Nppolyions
inside a volume V . The polyions are modeled as hard
spheres of radius a with a uniform surface charge density
σ0= −Zq
4πa2, (1)
where Z is the polyion valence (number of ionized sites)
and q > 0 is the charge of a proton. To preserve the
overall charge neutrality of the system, ZNp point-like
counterions of charge +q are present. The solvent is mod-
eled as a homogeneous medium of dielectric constant D.
Due to the strong electrostatic interaction between the
polyions and the counterions, we expect that the asym-
metric polyelectrolyte will be composed of clusters, with
density ρn, consisting of one polyion and 1 ≤ n ≤ Z asso-
ciated counterions, as well as bare polyions of density ρ0
and free (unassociated) counterions of density ρf. The
conservation of the total number of polyions and counte-
rions leads to two conservation equations,
∗Corresponding author.
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ρp=
Z
?
n=0
ρn, (2)
and
Zρp= ρf+
Z
?
n=0
nρn, (3)
where ρp= Np/V is the total density of polyions (asso-
ciated or not).
All the thermodynamic properties of the system can
be determined once the free energy is known.
Helmholtz free energy can be split into two parts, the
electrostatic and entropic. The electrostatic terms are
due to the inter-particles interactions and can be at-
tributed to the polyion-counterion, the polyion-polyion,
and the counterion-counterion interactions. All the elec-
trostatic interactions will be evaluated using the DH the-
ory [5]. This is motivated by the former success of the
theory when it was applied to symmetric electrolytes. In
principle, the linear DH theory is satisfactory only for
low densities and high temperatures. However, once the
concept of clusters is introduced, the validity of the DHBj
theory is extended into the nonlinear regime [3]. The lin-
ear structure of the DHBj theory insures its internal self
consistency, a problem which is intrinsic to many of the
nonlinear theories of electrolyte solutions [8].
We shall assume that the effect of the counterion con-
densation is to renormalize the polyion charge. Thus, the
effective surface charge of a n-cluster is
The
σn= −(Z − n)q
4πa2
= σ0Z − n
Z
. (4)
All the nonlinearities related to the internal degrees of
freedom of the clusters will be included in the entropic
terms. In a previous work [7a] we have considered that
the bounded counterions condense onto the surface of the
spherical polyion. Although less realistic, this assump-
tion has allowed us to obtain closed analytical expressions
for the entropic contribution. In the present work, how-
ever, the intra-cluster interactions will be treated using
a local-density functional theory, so that the correlations
between the bound counterions are explicitly taken into
account. These correlations effects can be disregarded
only when the concentration of counterions is not too
high, a condition which may not be fulfilled in the close
vicinity of a highly charged polyion. We now proceed to
describe each one of the contributions to the Helmholtz
free-energy density f = −F/V .
III. THE POLYION-COUNTERION
INTERACTION
The polyion-counterion contribution is obtained using
the usual DH theory applied to a n-cluster of effective
surface charge σninside the ionic atmosphere [7a]. Con-
sider a n-cluster fixed at the origin, r = 0. Due to the
hard-sphere exclusion, no counterions will be found in-
side the region r < a, that is,
ρq(r < a) = 0 .(5)
Outside the spherical polyion, r ≥ a, the cluster-
counterion correlation function is approximated by a
Boltzmann factor, leading to the charge density
ρq(r ≥ a) = qρfexp[−βqψn(r)] −
Z
?
n=0
(Z − n)qρn
+σnδ(r − a) , (6)
where β−1= kBT and ψn(r) is the electrostatic poten-
tial at a distance r from the center of the polyion. Notice
that only the free counterions are assumed to get polar-
ized; the bare polyions and clusters are too massive to be
affected by the electrostatic fluctuations and contribute
only to the neutralizing background. Substituting the
charge density into the Poisson equation,
∇2ψn(r) = −4π
Dρq(r) ,(7)
one obtains the nonlinear Poisson-Boltzmann equation.
After the linearization of the exponential factor in (6),
the electrostatic potential ψn(r) satisfies the Laplace (for
r < a) and the Helmholtz (for r ≥ a) equations,
?
Dσnδ(r − a) ,
∇2ψn(r) =
0 , for r < a ,
for r ≥ a ,
κ2ψn(r) −4π
(8)
where κ =
length, and λB= βq2/D is the Bjerrum length. In prin-
ciple the linearization is valid only in the limit βqψn≪ 1,
however, since the formation of clusters is taken into ac-
count, the validity of the theory is extended into the non-
linear regime [3].
The second-order differential equation for ψn(r) can
be integrated, supplemented by the boundary conditions
of vanishing of the electrostatic potential at infinity, the
continuity of ψn(r) at r = a, and the discontinuity in
the radial component of the electric field related to the
presence of the surface charge σnat r = a. Under these
conditions, we obtain
?4πλBρf is the inverse Debye screening
ψn(r) =
−(Z − n)q
−(Z − n)qeκ(a−r)
Dr(1 + κa)
Da(1 + κa), for r < a ,
,for r ≥ a .
(9)
Using the charge density in the linearized form,
ρq(r) =
?
0 ,for r < a ,
−κ2D
4πψn(r) + σnδ(r − a) ,for r ≥ a ,
(10)
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the electrostatic energy of a n-cluster is calculated to be
Un(κ,q) =1
2
=(Z − n)2q2
D(1 + κa)
?
d3rρq(r)ψn(r)
?1
a−
κ
2(1 + κa)
?
. (11)
The electrostatic free-energy density for the polyion-
counterion interaction is obtained through the Debye
charging process, in which all the particles are simulta-
neously charged from 0 to their final charge [5],
βfPC(ρf,{ρn}) = −
Z
?
Z
?
n=0
ρn
?1
0
dλ2βUn(λκ,λq)
λ
= −
n=0
(Z − n)2λB
2a(1 + κa)ρn. (12)
IV. THE POLYION-POLYION INTERACTION
Due to the large asymmetry between the polyions
and the counterions, the degrees of freedom associated
with the counterions can be effectively integrated out.
The long-ranged interaction between two clusters will be
screened by the cloud of free counterions, producing an
effective short-ranged potential of a DLVO form [9,10],
Veff
nm(r) = q2(Z − n)(Z − m)exp(2κa − κr)
Dr(1 + κa)2. (13)
The polyion-polyion contribution to the Helmholtz free
energy can then be calculated in the spirit of the usual
van-der-Waals theory [3,7a],
⌋
βfPP(ρf) = −1
2
Z
?
n=0
Z
?
(1 + 2κa)
κ2(1 + κa)2
m=0
βρnρm
?
d3rVeff
nm(r)
= −2πλB
Z
?
n=0
Z
?
m=0
(Z − n)(Z − m)ρnρm= −2πλB
(1 + 2κa)
κ2(1 + κa)2ρ2
f. (14)
⌈
V. THE COUNTERION-COUNTERION
INTERACTION
The counterion-counterion contribution, originating
from the interactions between the free counterions, is
calculated using the One Component Plasma (OCP) the-
ory [11]. The electrostatic free energy is found through
a Debye charging process and a closed analytic form for
fCC(ρf), valid over a wide range of densities, is presented
in [11b],
βfCC(ρf) = −ρfFcorr(ρf) ,
Fcorr(ρf) =1
4
−2
(15)
?
√3tan−1
1 +
2π
3√3+ ln
?ω2+ ω + 1
?2ω + 1
1 + 3?4πλ3
3
?
− ω2
√3
??
,(16)
ω = ω(ρf) =
?
Bρf
?1/2?1/3
. (17)
In the bulk this contribution is very small, and is included
only for completeness.
VI. THE MIXING FREE ENERGY
The mixing free energy reduces to a sum of ideal-gas
terms,
βfmix(ρf,{ρn}) =
?
s
ρs
?1 − ln?ρsΛ3
s/ζs
??,(18)
where s ∈ {f;n = 0,...,Z}, Λs are the thermal de
Broglie wavelengths associated with free counterions,
bare polyions, and clusters; ζs are the internal parti-
tion functions for an isolated specie s. Since the bare
polyions and the free (unassociated) counterions do not
have internal structure, their internal partition functions
are simply given by ζ0 = ζf = 1. For a n-cluster the
internal partition function is
ζn=
1
n!
?
Ωn
n
?
i=1
?d3ri
Λ3
c
?
exp(−βHn) , (19)
βHn= −ZλB
n
?
i=1
1
|ri|+ λB
?
i<j
1
|ri− rj|, (20)
where the integration hypervolume Ωn ≡ {a < |ri| <
Rn,∀i = 1,···,n} depends on the cutoff Rn. To fix the
value of the cutoff, we follow an argument similar to the
one used by Bjerrum in his study of dipolar formation in
simple electrolytes [4]. Suppose that we have a (n − 1)-
cluster and we want to condense one more counterion
to form a n-cluster. Because of the spherical symme-
try, the (n−1) bound counterions contribute only to the
renormalization of the polyion charge. The probability of
finding the nthcounterion at a distance r in the interval
dr is
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P(r)dr ∝ drr2exp[−βqφ(r)]
= drr2exp[(Z − n + 1)λB/r] ,
where φ(r) is the electrostatic potential generated by the
(n − 1)-cluster. The probability distribution P(r) has a
minimum at
r = Rn= (Z − n + 1)λB
which, following Bjerrum [4], we shall interpret as the
distance of closest approach at which the nthcounterion
will become bound to the (n−1)-cluster. Since Rn/a > 1,
for a given reduced temperature, T∗= a/λB, there is a
minimum value of the valence, Zmin = 2T∗− 1, below
which no counterions can condense onto a polyion, that
is, the thermal entropic energy, 2kBT, will overcome the
gain in electrostatic potential energy, (Z−n+1)q2/(Da),
preventing the confinement from taking effect.
(21)
2
, (22)
With the cutoff defined, we shall now attempt to
calculate the internal partition function of a n-cluster,
Eq. (19). That, in itself, is a formidable task, since it
requires evaluation of the many-body integrals (19). In-
stead of performing the integrations explicitly, we shall
use the local-density-functional theory to find the free
energy of the condensed layer of n counterions, βFcon
−lnζn. Let us define the local density of counterions in
the condensed layer of a n-cluster as
n
≡
̺c(r) =
n
?
i=1
δ(r − ri) .(23)
Within the local-density approximation (LDA), the
Helmholtz free-energy functional βFcon
sponding to density ̺c(r) is
n [̺c(r)] corre-
⌋
βFcon
n [̺c(r)] =
?
+
Vn
?
d3r̺c(r)?ln?̺c(r)Λ3
d3r ̺c(r)Fcorr[̺c(r)] ,
c
?− 1?− ZλB
?
Vn
d3r̺c(r)
|r|
+1
2λB
?
Vn
d3rd3r′̺c(r)̺c(r′)
|r − r′|
Vn
(24)
⌈
where the integrations are over the annulus Vn≡ {a ≤
|r| ≤ Rn} and Fcorris given by Eq. (16). The first term
in (24) corresponds to the usual ideal-gas contribution,
the second and the third terms are due to the electro-
static interactions and the last term is the result of the
correlations between the bounded counterions, for which
we use the expression of the OCP theory [11]. The equi-
librium configuration, ρc(r) = ?̺c(r)?, is the one that
minimizes the free-energy functional βFcon
the constraint
?
This minimization procedure leads to the Boltzmann dis-
tribution for the density profile,
n [̺c(r)] under
Vn
d3rρc(r) = n .(25)
ρc(r) =
nexp[−µcorr(r) − βqψ(r)]
d3r′exp[−µcorr(r′) − βqψ(r′)]
?
Vn
, (26)
where the electrostatic and the excess chemical potentials
are given, respectively, by
ψ(r) = −Zq
µcorr(r) = Fcorr[ρc(r)] + ρc(r)δFcorr[ρc(r)]
D|r|+ q
?
Vn
d3r′
ρc(r′)
D|r − r′|,(27)
δρc(r)
= Fcorr[ρc(r)] +
1
12
?1 − ω2[ρc(r)]?, (28)
with ω given by Eq. (17). On the other hand, the electro-
static potential and the total charge density satisfy the
Poisson equation,
− ∇2ψ(r) = ∇ · E(r) =4π
D[σ0δ(|r| − a) + qρc(r)] ,
(29)
where E(r) = −∇ψ(r) is the electric field at the point
r. Inserting (26) into (29) we find a Poisson-Boltzmann-
like equation, which is a second-order nonlinear dif-
ferential equation for the electrostatic potential ψ(r).
It should be remarked that, neglecting the correlation
term, we regain the usual Poisson-Boltzmann equation,
∇2ψ(r) ∝ exp[−βqψ(r)].
Since the boundary conditions are given in terms of
the electric field strength,
E(|r| = a) = −Zq
Da2and E(|r| = Rn) = −(Z − n)q
DR2
n
,
(30)
to perform the numerical calculations it is convenient to
rewrite the equations in terms of this variable. We now
take advantage of the spherical symmetry of the system
to eliminate the angular dependence of the equations,
that is, we replace r by r = |r| in Eqs. (26) to (29). The
electric field also has only a spherically symmetric radial
component, so that E(r) = E(r)rr. Integrating the Pois-
son equation (29) over a sphere of radius r and using the
divergence theorem, we obtain a relation for the electric
field strength E(r),
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?
|r′|<r
d3r′∇ · E(r′) =
?
|r′|=r
dS′· E(r′) = 4πr2E(r) = −4πq
D
Z −
?
|r′|<r
d3r′ρc(r′)
.(31)
Inserting (26) into (31) we obtain an integro-differential equation for the electric field,
E(r) = −
q
Dr2
Z − n
?r
?Rn
a
dr′r′2exp
?
−µcorr(r′) + βq
?r′
?r′
a
dr′′E(r′′)
?
a
dr′r′2exp
?
−µcorr(r′) + βq
a
dr′′E(r′′)
?
, (32)
⌈
where we have chosen the gauge in which ψ(r = a) = 0,
and the density profile, ρc(r), which is necessary to eval-
uate µcorr(r), is also written in terms of the electric field,
D
4πq∇ · E =
4πqr2
ρc(r) =
Dd
dr
?r2E(r)?
. (33)
The integro-differential equation was solved iteratively to
obtain the electric field E(r). The charge density ρc(r)
is then calculated using Eq. (33). Finally, the internal
free energy of a n-cluster can be expressed in terms of
the charge density and the electric field to be
⌋
βFcon
n
= βFcon
n [ρc(r)] =
?
+
Vn
?
d3rρc(r)?ln?ρc(r)Λ3
d3r ρc(r)Fcorr[ρc(r)] ,
c
?− 1?+βD
8π
?
Vn
d3r E2(r) +(Z − n)2λB
2Rn
−Z2λB
2a
Vn
(34)
⌈
while the internal partition function of a n-cluster is
ζn= exp(−βFcon
n
).
VII. THERMODYNAMIC PROPERTIES
The total Helmholtz free energy of the polyelectrolyte
solution is a sum of the entropic and the electrostatic
contributions,
f(ρf,{ρn}) = fmix+ fPC+ fPP+ fCC.(35)
Minimization of the total Helmholtz free energy under
the constraints of fixed number of polyions and counte-
rions leads to the law of mass action,
µ0+ nµf= µn, (36)
where the chemical potential of a specie s is µs =
−∂f/∂ρs. This results in a set of Z coupled nonlinear
algebraic equations for the densities ρn, whose form is
suitable to the use of an iterative method. Starting from
a uniform distribution of clusters {ρn}, we were able to
solve the coupled system numerically. A sample of the
distributions obtained is presented in Fig. 1. Two fea-
tures are worth remarking. The counterion condensation
600 700800 900
n
0.00
0.05
0.10
0.15
0.20
ρn /ρp
FIG. 1. Cluster-density distribution {ρn} for Z = 1000,
volume fraction φ =4
temperature. From left to the right, the values of the reduced
temperature are T∗= 100,50, and 20.
3πa3Np/V = 0.01, and various values of
is more effective as the temperature decreases, and the
width of the distribution is not very sensitive to the vari-
ations in temperature. The pressure can be obtained as a
Legendre transform of the Helmholtz free-energy density,
p = f(ρf,{ρn}) +
?
s
µsρs.(37)
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0.00 0.050.10
φ
0.15 0.20
0.000
0.004
0.008
0.012
βpa
3/ Z
Z=5000
Z=2000
Z=1000
FIG. 2. Dependence of the dimensionless (total) pressure,
βpa3/Z, on the volume fraction φ =
values of Z(1000,2000,5000) and T∗= 100.
4
3πa3Np/V , for several
0.000.05 0.10
φ
0.150.20
-0.004
0.000
0.004
0.008
0.012
0.016
βpa
3/ Z
FIG. 3.
the dimensionless (total) pressure, βpa3/Z, on the vol-
ume fraction φ=
T∗
= 100. Solid line:
tion; long-dashed line:polyion-counterion contribution;
dot-dashed line: polyion-polyion contribution; dashed line:
counterion-counterion contribution.
Dependence of the various contributions to
4
3πa3Np/V , for Z= 1000,and
mixing (ideal-gas) contribu-
In Fig. 2 we present the total pressure inside the poly-
electrolyte solution, which is a monotonically increasing
function of the density of polyions. In agreement with the
current experimental and Monte Carlo results [12], no ev-
idence of any phase transition is encountered. To allow
for a better appreciation of the relative importance of
all the terms, in Fig. 3 we present separately the various
contributions to the total pressure. All the electrostatic
terms give a negative contribution to the total pressure,
what can be interpreted as a form of an induced effective
attraction between all the particles.
ACKNOWLEDGMENTS
This work has been supported by the Brazilian agency
CNPq (Conselho Nacional de Desenvolvimento Cient´ ıfico
e Tecnol´ ogico).
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[10] We shall leave untouched the still controversial question
of the existence of short-ranged attractive interactions
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between the two polyions [but see I. Sogami and N. Ise, J.
Chem. Phys. 81, 6320 (1984), and Yan Levin (to be pub-
lished)]. If this attraction exists, it will not affect strongly
the thermodynamic properties of the solution, which are
dominated by the counterions and their interactions with
the polyions (see Fig. 3).
[11] (a) S. Nordholm, Chem. Phys. Lett. 105, 302 (1984); (b)
R. Penfold, S. Nordholm, B. J¨ onsson, and C. E. Wood-
ward, J. Chem. Phys. 92, 1915 (1990).
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