23 Electrical Conductivity ofa Suspension of Soft Particles

23.1 INTRODUCTION

Electrokinetic equations describing the electrical conductivity of a suspension of

colloidal particles are the same as those for the electrophoretic mobility of colloidal

particles and thus conductivity measurements can provide us with essentially the

same information as that from electrophoretic mobility measurements. Several the-

oretical studies have been made on dilute suspensions of hard particles [1–3], mer-

cury drops [4], and spherical polyelectrolytes (charged porous spheres) [5], and on

concentrated suspensions of hard spherical particles [6] and mercury drops [7] on

the basis of Kuwabara’s cell model [8], which was originally applied to electropho-

resis problem [9,10]. In this chapter, we develop a theory of conductivity of a con-

centrated suspension of soft particles [11]. The results cover those for the dilute

case in the limit of very low particle volume fractions. We confine ourselves to the

case where the overlapping of the electrical double layers of adjacent particles is

negligible.

23.2 BASIC EQUATIONS

Consider spherical soft particles moving with a velocity U (electrophoretic veloc-

ity) in a liquid containing a general electrolyte in an applied electric field E. Eachsoft particle consists of the particle core of radius a covered with a polyelectrolyte

layer of thickness d (Fig. 22.1). The radius of the soft particle as a whole is thus

b= a + d. We employ a cell model [8] in which each sphere is surrounded by a con-

centric spherical shell of an electrolyte solution, having an outer radius of c such

that the particle/cell volume ratio in the unit cell is equal to the particle volume

fraction � throughout the entire suspension (Fig. 22.1), namely,

� ¼ ðb=cÞ3 ð23:1Þ

Let the electrolyte be composed of M ionic mobile species of valence zi and drag

coefficient li (i= 1, 2, . . . , M), and n1i be the concentration (number density) of

Biophysical Chemistry of Biointerfaces By Hiroyuki OhshimaCopyright# 2010 by John Wiley & Sons, Inc.

480

the ith ionic species in the electroneutral solution. We also assume that fixed

charges are distributed with a constant density of rfix in the polyelectrolyte layer.

The fundamental electrokinetic equations are the same as those for the electropho-

resis problem (Chapter 22).

23.3 ELECTRICAL CONDUCTIVITY

Consider a suspension of Np identical spherical soft particles in a general electrolyte

solution of volume V. We define the macroscopic electric field in the suspension

kEl, which differs from the applied electric field E. The field kEl may be regarded

as the average of the gradient of the electric potential c(r) (=c(0)(r) + dc(r)) in the

suspension over the volume V, namely,

Eh i ¼ � 1

V

ðV

rcðrÞdV ¼ � 1

V

ðV

rdcðrÞdV ð23:2Þ

where we have used the fact that the volume average of c(0) is zero. The electrical

conductivity K� of the suspension is defined by

ih i ¼ K� Eh i ð23:3Þ

where kil is the net electric current in the suspension given by

ih i ¼ 1

V

ðV

iðrÞdV ð23:4Þ

The electric field kEl is different from the applied field E and these two fields are

related to each other by continuity of electric current, namely,

K� Eh i ¼ K1E ð23:5Þ

where

K1 ¼XMi¼1

z2i e2n1i =li ð23:6Þ

is the conductivity of the electrolyte solution (in the absence of the soft particles). If

further we put

Ci ¼ � c2

3rd�i

dr� �i

� �r¼c

ð23:7Þ

ELECTRICAL CONDUCTIVITY 481

then, the ratio of K� to K1 is given by

K�

K1 ¼ 1þ 3�c

a3

XMi¼1

z2i n1i Ci=li

XMi¼1

z2i n1i =li

0BBBB@

1CCCCA

�1

ð23:8Þ

We obtain Ci for the low potential case and substitute the result into Eq. (23.8),

giving

K�

K1 ¼ 1� �c

1þ �c=2

�1� �c

2ð1� �cÞð1þ �c=2Þ

XMi¼1

z3i n1i =li

XMi¼1

z2i n1i =li

�ðca

1þ 2r3

a3

� �1� a3

r3

� �dy

drdr

��1

� 1� �c

1þ �c=2

�1þ �c

2ð1� �cÞð1þ �c=2Þ

XMi¼1

z3i n1i =li

XMi¼1

z2i n1i =li

�ðca

1þ 2r3

a3

� �1� a3

r3

� �dy

drdr

�

where

�c ¼ ða=cÞ3 ð23:10Þ

is the volume fraction of the particle core, which differs from the particle volume

fraction �, defined by Eq. (23.1). Equation (23.9) is the required expression for the

electrical conductivity of a concentrated suspension of spherical soft particles for

low potentials. Note that in the low potential approximation, K� does not depend on

the frictional coefficient g (or l).We consider several limiting cases of Eq. (23.9).

(i) In the limit of very low potentials, Eq. (23.9) tends to

K�

K1 ¼ 1� �c

1þ �c=2ð23:11Þ

(23.9)

482 ELECTRICAL CONDUCTIVITY OF A SUSPENSION OF SOFT PARTICLES

which is Maxwell’s relation [12, 13] respect to the volume fraction �c ofthe particle core. Note that in this case the conductivity is determined by

the volume fraction �c of the particle core (not by the particle volume frac-

tion �) and the contribution from the polyelectrolyte layer vanishes.

(ii) For a= b, soft particles become hard particles with no polyelectrolyte layer.

In this case �= �c and we find that Eq. (23.9) becomes a conductivity

expression for a concentrated suspension of spherical hard particles.

(iii) For a = 0, soft particles becomes porous spheres spherical polyelectrolytes

with no particle core. In this case �c! 0 and Eq. (23.9) reduces to

K�

K1 ¼ 1 ð23:12Þ

That is, in this limit the conductivity equals that in the absence of the parti-

cles so that spherical polyelectrolytes do not contribute to the conductivity.

Finally, we derive an approximate conductivity formula for the important case

where

la � 1; ka � 1ðand thus lb � 1; kb � 1Þ; lðb� aÞ � 1; kðb� aÞ � 1:

ð23:13Þ

In this case the potential inside the polyelectrolyte layer is essentially equal to the

Donnan potential cDON except in the region very near the boundary r= b between

the polyelectrolyte layer and the surrounding solution, where cDON is related rfix by

cDON ¼ rfixereok2

ð23:14Þ

Then Eq. (23.9) becomes

K�

K1 ¼ 1� �c

1þ �c=21� �cð1� a3=b3Þð1þ 2b3=a3Þ

2ð1� �cÞð1þ �c=2ÞecDON

kT

� �XNi¼1

z3i n1i =li

XNi¼1

z2i n1i =li

0BBBB@

1CCCCA

¼ 1� �c

1þ �c=21� �ð1� a3=b3Þð1þ a3=2b3Þ

ð1� �cÞð1þ �c=2ÞecDON

kT

� �XNi¼1

z3i n1i =li

XNi¼1

z2i n1i =li

0BBBB@

1CCCCA

ð23:15Þ

which is an approximate expression for K�/K1 for low potentials under conditions

(23.13).

ELECTRICAL CONDUCTIVITY 483

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484 ELECTRICAL CONDUCTIVITY OF A SUSPENSION OF SOFT PARTICLES