# Debye–Hückel–Bjerrum theory for charged colloids

**ABSTRACT** We formulate an extension of the Debye–Hückel–Bjerrum theory [M.E. Fisher, Y. Levin, Phys. Rev. Lett. 71 (1993) 3826] 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 a fixed number of polyions and counterions. In agreement with the current experimental and Monte Carlo results, no evidence of any phase transition is encountered.

**0**Bookmarks

**·**

**98**Views

- Citations (1)
- Cited In (0)

- Journal of Colloid Science. 04/1955; 10(2):224–225.

Page 1

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.

1

Page 2

Preprint

Submitted to Physica A

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)

2

Page 3

Preprint

Submitted to Physica A

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

3

Page 4

Preprint

Submitted to Physica A

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),

4

Page 5

Preprint

Submitted to Physica A

?

|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

600700 800 900

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)

5