The response of a colloidal suspension to an alternating electric field

Advances

in Colloid and Interface Science. 16 (1982) 281-320 ElsevierScientificPublishingCompany,Amsterdam-Printed in TheNetherlands

281

THE RESPONSE OF A COLLOIDAL SUSPENSION TO AN ALTERNATING ELECTRIC FIELD R-W. O'BRIEN Dent. of Theoretical and Applied Mechanics, School of Mathematics, The University of New South !Jales,P_ O_ Box 1, Kensington, New South Wales 2033, Australia CONTENTS ABSTRACT ___.______________________-____.-_-.____-______________..___. 281 I. INTRODUCTION ________~__..._...__._._......__......_._....__........_.282 II. 283 A. Outline cf the paper ~-_..~_~.~~~.~~~...~~~~~.~~~~~~~.~~~~..~.~~~~ III_ DEFINITION OF COMPLEX CONDUCTIVITY ___.____.________.__.________._._-_ 283 A. The measurement of K* ____________________________._______________ 285 IV. THE RELATIONSHIP BETWEEN COMPLEX CONDUCTIVITY AND PARTICLE DIPOLE STRENGTH IN A DILUTE SUSPENSION ___________.__________.__..___.___.___ 285 286 V. THE ELECTROKINETIC EDUATIONS ____..______._*___.________--____.-.___-_ VI_

VII. VIII IX_ X. XIXII_ XIII XIV.

xv. XVI.

THE PERTURBATION SCHEME _____.___________________________.____________ 288 A. The dioole coefficient of a particle with small dielectric constant _________________________________.______._._______._________ 293 B. The zero frequency limit ____.____________________________________ 295 THE CALCULATION OF THE DIPOLE COEFFICIENT TO O(r,)________________.___296 THE &) FDRNULk FOR THE DIFOLE COEFFICIENT __-____-_-_-___-_-_-_____299 COMPARISON OF THE APPGOXIMATE FORMULA !.IITH THE COMPUTED RESULTS ______ 302 THE EFFECT OF SURFACE DISSOCIATION REACTIONS ON ME DIPOLE STRENGTH _. 305 APPENDIX I - THE MEASUREMENT OF COMPLEX CONDUCTIVITY _______._____.___308 APPENDIX II - DERIVING THE FORMULAE (EQS. 5.24 AND 6.3) FOR 6. and uo. 370 APPENDIX III - A METHOD FOR EVALUATING THE FORMULA (ED. 7.7) for CE __ 312 APPENDIX IV.- THE DIPOLE COEFFICIENT OF A PARTICLE IN A SYMMETRIC ELECTROLYTE WITH IONS OF APPROXIMATELY UNIFORM DIFFUSIVITIES __________-______________________.______.____ 314 ACKNOWLEDGEMENT ___________-_____-_-_____________-_______-_____-___-_317 LIST OF SYFISOLS ___._____._..____._.______________~_____~_____~_-_~_~~317

I_ ABSTRACT This paper is concerned with the calculation of the complex conductivity K* of a susoension, a quantity which may be determined exoerimentally from the measurement of the alternating current which flows between a pair of electrodes in the suspension due to an alternating voltaqe difference. A semi-analytic formula is derived for the complex conductivity of a dilute suspension of spherical particles with small dielectric constant which is reasonably accurate for c-potentials

of less than 50 mV_ For such suspensions this formula represents a very economical alternative to the exact computer calculation of K* described by DeLacey and klhite (ref. 2). Although the formula for K* is derived for particles with fixed surface charge, it

is

shown

that

the

formula

can

also be applied to a more general

class of susoensions, in which the surface charge arises from the dissociation

of a single type of surface group_ II.

INTRODUCTION East of the theoretical studies of colloidal suspensions in electric fieids have been concerned with the interpretation of measurements of the electrophoretic mobility and electrical conductivity, quantities which are measured in a steady electric field_ Such experiments are, however, limited in the amount of information they can provide; i-e_ they only yield one number for a given suspension_ This number can be used to determine only one parameter, such as the c-potential of the particles_ Furthermore, it has been shown (ref. I) that electroohoretic mobility and static conductivity are insensitive to the particle charge determining mechanism, and hence such measurements cannot provide information about such things as the dissociation constants of surface groups. As these limitations do not apply to alternating field measurements performed over a range of frequencies, there is a possibility that such measurements may yield much more information than can be obtained from the steady field measurements which are currently in use.. In this gaper we present a theoretical study of the behavior of a dilute suspension of spherical particles with low C-potential in an alternating electric field. In particular, we shall be conct:-ned with the calculation of the complex conductivity of the suspension, a quantity which is related to the complex impedance of a conductance cell containing suspension_ For a dilute suspension, the interactions between the particles are negligible, and we can determine the complex conductivity from an analysis of the behavior of an isolated particle placed in an infinite electrolyte solution with a spatially uniform alternating electric field. In general both particle and ions will be sqt in motion by the field, with the particle and counterions moving in opposite directions. As a result the double layer distorts in an alternating fashion, giving rise to an electric field, which at large distances from the particle has the same form as that of an alternating electric dipole. It can be shown (ref_ 2) that the complex conductivity of a dilute suspension may be determined from the eiectrical dipole strength of the particles_ To determine the phase and amplitude of the dipole strength of a particle, we must solve the electrokinetic equations which describe the distribution of electrical potential, ion densities and fluid flow around the particle. At present there is only one analytic solution to this problem, namely Dukhin and Shilov's formula (ref3) for a particle with a thin double layer in an electrolyte with two types of ions.

283 Although there are no analytic formulae available for particles with larger double layers or more complicated electrolytes, it is now possible to determine the dioole strength of a particle with fixed surface charge, using OeLacey and uhite's (ref. 2) recently develoned computer program to solve the electrokinetic equations numerically. Although this computer solution may be obtained for any surface potential and electrolyte, the program is very expensive to run, and the numbers which appear in the output provide little physical insight into the problem_ Outline of the Paper Our aim in this paper is to derive an approximate formula for the dipole strength of a particle with low c-potential. In the following section we define the complex conductivity K* and we briefly describe the experimental determination of this quantity. In Section IV we give the relationship between K* and the dipole strength for a dilute suspension of spherical particles. The remainder of the paper is concerned with the calculation of the dipole strength of a spherical oarticle with low zeta potential, small dielectric constant and fixed surface charge. In Section V we set out the electrokinetic equations

A.

which must be solved in order to determine the dipole strength and in Section VI we describe the perturbation scheme for solving these equations for a particle with low zeta potential_ The first step in the perturbation scheme (involving the solution ofthe electrokinetic equations for an uncharged particle) is described in Section VI. In Section VII we then solve the next set of equations to obtain the O(c) correction to the dioole strength and in Section VIII we obtain a formula for

the

dipole

strength

which

is correct

to O(s*).

Although

the

formula

contains

a number of integrals which must be evaluated numerically, the integration is straightforward and can even be carried out on a programmable hand calculator_ In Section IX we compare values of the dipole strength obtained with this approximate formula with values obtained from DeLacey and !Jhite's(ref. 2) computer program; it is found that the O(,;*)formula gives reasonably accurate results up to 5 values of about 50 mV_ For c-potentials in this range the formula represents a very economical alternative to the exact computer solution of the problem_ This comparison also provides the first independent check of DeLacey and klhite'sprogram to be reparted in the literatureIn Section X we remove the fixed surface charge condition and allow for variable dissociation of an ionogenic surface group on the particle. It is found that to D(,;*)the dipole strength is unchanged by this modificationIII.

DEFIIJITIONOF COMPLEX CONDUCTIVITY Consider a colloidal suspension subjected to an alternating macroscopic electric field. In general, the local electric field and current density in the suspension fluctuate with position on a microscopic length scale (of the order of the particle

254 radius) about average values which themselves vary on a macroscopic length scale. it is these average values which are of interest here since these are the quantities which are accessible to measurement_ Ue denote the average values by angle brackets; for example cE, denotes the average electric f-reld,which may be defined mathematically as a volume

average,

viz.:

= E dV_ : _I-

(2-l) VHere c is the local electric field and V is a sample volume which contains a statistically significant number of particles but which is small in comparison with the macroscopic length scale. Other mean values are defined similarly. The local current density in the suspension is given by

3P _ where if is the current density due to the free charges and s 1s the polarization charge contribution, P being the dioole strength per unit volume. In general the charge density at any point in the suspension varies with time and thus the current density has a non-zero divergence_ However. by adding &g to the current density we obtain the divergence free vector f, given by (ref. 4, pp_ llO-117): i=i

-f

1 +=z

sD

-

(2.2)

where LJis the electrical displacement vector. Since it is customary to refer to &gas the displacement current, we shall refer to i_as the nett current density. It is assumed that the average electric field and nett current density vary sinusoidally with time; hence we may writer cE> = Eoeiwt , and = i,eiwt )

(2.3)

where as usual cF> and -=i_~ are piven by the real parts of the complex quantities on the right-hand side of these expressions and 2 TIWis the frequency. At 10~1field strengths, we expect that the quantities j. and E-Owill be linearly related. For a statistically isotropic suspension, this linear relation takes the simple form i -0 = K*Eo , where K* is defined as the complex conductivity of the suspension.

(2-4)

285 A.

The measurement of K* In the earlier works on this subject (ref. 2,4), the experimental determination of K* is only described for the case of z conductance cell with parallel plate electrodes

_ For completeness we note here that for other types of conductance the procedure for measuring K* 1s _ similar to the static conductivity procedure. If 2, denotes the complex impedance of the cell for a standard liquid of known comolex conductivity Kc, and 2 is the impedance for the suspension, then

cells

z Kf

=

K*

s

-?

_

2

(Z-5)

This result (which is derived in Appendix I) is only valid if the frequency large to restrict any inhomogeneities in the

of the applied field is sufficiently

suspension to thin layers around the electrodes_ IV_

THE RELATIONSHIP BETWEEN COMPLEX CONDUCTIVITY AND PARTICLE DIPOLE STRENGTH IN A DILUTE SUSPENSION In a dilute suspension, we may treat each of the particles as being alone in an infinite electrolyte with uniform ambient electric field EoeiL>t . In the following section it will be shown that the electrical potential at large distances from an isolated spherical particle has the form (-1

f

Co/r3)r

-

Foe itit,

(3.1)

where c is the position vector from the center of the sphere. As mentioned in the introduction, the disturbance field COT -7

r3

- (

!ioelLt

has the same form as the field due to an electrical dipole with dipole strength i:_;t proportional to COEOe _ We shall refer to the quantity Co as the dipole coefficient particle. of the The radius tained For simply

variables in Eq. 3.1 are assumed to be non-dimensional with the sphere a being the unit of length and F is the unit of ootential; thus ED is obby multiplying the dimensional field strength by E _ a dilute monodisperse susoension of spheres, the complex conductivity is given by (ref. 2)

K* = Kz (l+3+CO) =

(3.2)

where Q is the particle volume fraction and KE is the complex conductivity of the uniform electrolyte which lies beyond the equilibrium double layer. If the electrolye is dilute (an assumption which is implicitly made in the formulation of

286 electrokinetic equations in Section V) and if the frequency of the applied field is much smaller than the relaxation frequency of a water molecule, then the complex conductivity of the electrolyte is simply given by (3-3) where Km and E are the static conductivity and dielectric constant of the electrolyte V_

THE ELECTROKINETIC EQUATIONS In this section we set out the differential equations which govern the distribution of ions, electrical potential and velocity in the electrolyte. These equations must be solved in order to determine the particle dipole coefficient CQ. As

mentioned in the previous section, we will

obtained

by dividing

the

electrical

potentials

be using

non-dimensional

variables,

by

kT/e, dividing the position vector by the particle radius a and dividing the ion densities by twice the ionic strength of the electrolyte multiplied by Avogadro's constant. Finally, the fluid velocity will be multiplied by

where n is the viscosity of the e-lectrolyte. it is assumed that the electrical potential and ion densities are As usual, only slightly altered by the applied field_ Me use a r;prefix to denote these small perturbations; hence the perturbation in the-density of the type j ions at iiJt while i+eib:tand lipe a point r in the electrolyte is denoted by finj(r)elfst, denote the changes in electrical potential and fluid pressure. Since the fluid velocity is zero in the absence of an aoplied field, we will dispense with the
r2;z>t (,:a)’

Zj3nj =0 , -Lx j=I

T-lJ=o,

(4.1) (4.2)

and 0 ,Q, + Zvjdn_ili, , - ~-VI-I. 7'- (osnj + Z.n.?jb 3 J J

(4-4)

values of the potential and j= I,..-,N. Here 9’ and ny are the equilibrium ion density in the absence of an applied fieid. N is the number of ionic species

for

287

in the electrolyte,

Zj

and Dj are the valency and diffusivity of the type j ions

and 2 m. = E(kT) J 6ane2Dj is the usual non-dimensional

ionic drag coefficient.

Eq_ 4.1 is Poisson's equation, Eq. 4-2 is the incompressibility

constraint

and Eq. 4.3 is the force balance equation for the liquid; in deriving the latter equation, it is assumed that the inertial forces are negligible, a result which is valid for the frequency range of interest here (ref. 2). Thus the flow field is able to adjust instantaneously

to the variations in the body force term on

the right-hand side of Eq. 4.3. Eqs-

4.1

presence

of

to 4-4 the

are

the same as for

the

static

field

problem,

except

for

the

term

- 2 Sri7iJa J --D.j on the right-hand side of the ion conservation equation (Eq. 4.4).

As we shall

see, this extra term considerably complicates matters. In the static field problem, O'Brien and Glhite (ref. 1) were able to eliminate Poisson's equation tial not

functions possible

(Eq_ 4.1) from the problem by working in terms of ionic

related

to A;> and Znj_

In the alternating

to remove the electrostatic

field

problem,

potenit

is

equation (Eq. 4.1) and we will work here

in terms of the original variables 8:) and .t.nj_ For a coordinate system fixed to u_ and An. take the form the particle, the far field boundary conditions for .';:::, J

(4-5) J Here ME is the non-dimensional

electrophoretic

mobility of the particle defined

by:

(4.6) where

3 is

the

dimensional

mobility_

The boundary conditions to be applied on the particle surface depend on the nature of the particle charge mechanism.

To begin with, we will assume that the

particle has uniform surface charge density, unaffected by the applied field. In Section X we will allow for variable dissociation of an ionogenic surface group.

258

For a particle with fixed charge, the surface boundary conditions are

(4-7) and u __= 0 on the particle surface, where E is the particle dielectric constant and a$~ is P p the potential variation inside the particleTHE PERTURBATION SCHEME VI. As in the earlier studies of the conductivity (ref. 5, 6) and electrophoresis (ref_ 7) of particles with 10~~<-potential, we assume that each of the variables in Eqs_ 4-i to 4-4 can be expanded in power series in eG/kT_ Superscripts will be used to denote the order of the coefficients in these expansions; for example, the expansions for &nj and 5-3are zn_

J

=

&rigf (31

Bnjlt (2)'

6nS + ___, (5.1)

VJhilethe expansion for the particle dipole coefficient takes the form (5.2) where the quantities Ci, CA and Cg are obtained from the far-field forms of the potentials 6~>",~g1and cc', respectively. Our aim is to determine these quantities for a particle with a small dielectric constant. In order to avoid confusion with the "0" superscript in the equilibrium potential 0 0 and ion density ny, we will reolace e" at this point by the symbol Y and use the Boltzmann formula:

ny=n3

exp(-ZjY) ,

in place of ny, where ng is the uniform ion density at large distances from the particle_ On substituting the perturbation expansions for each of the variables in Eqs. 4-l to 4-4 and boundary conditions 4.5 and 4.7, and equating coefficients of like powers of equations and boundary conditions for each set of _ c.. _we obtain _ b-a’, @I’ and IA’. coefficients 6n:,

In

this

section

we will

determine

the

leading

terms

in

the

above

expansions;

these are the quantities associated with an uncharged particle in an alternating In the following section we look at the O(C) terms

field_

Section VIII we obtain the O(<*) term in the expansion

and then finally in

(Eq. 5.2) for the dipole

coefficient. The quantities 6o", Any, 6p” N 2 Z.&n! = 0. ,2&$0 + (i--a) t j=l J J

and u" satisfy the equations: (5.3)

v-u 0 =o,

(5.4) O=O _,

V2"O _-yap

(5.5)

2 6n.=O 0 itia and 7*&n! -F Z.n"Iv26~~0-J JJ Dj J

,

(5.6)

with boundary conditions:

QJOS-E “0 _

-0

-c,

0 s -MEEo

(5.7)

, asr-=.

andk!$_>$=O, 0

38n ---J f

(0 -= at* Zjnj

at-

and

u”

atr=

ar

(5.8)

0 =

= 0

I_

The

equations

by a zero

velocity

(5-&j

and

and

and boundary conditions for

(S-5)

pressure

field,

with

zero

electrophoretic

u_’ are mobility

satisfied (ME =o

Thus an uncharged particle and the surrounding solvent are not set in motion by an alternating field. The remaining equations involving 6,' and brig may be rewritten in the compact vector form V2V0_

-

(5-J)

witere

V

O=

(5-10;

I-

and 5 is the matrix defined by

(5-11)

Here Tj = Z$-I~ '

(5-12)

(5.13) In deriving Eq. 5.9, we have eliminated T2G@' from Eq. 5.6 by using Poisson's equation (Eq. 5.3)_ Eq. 5.9 is the vector analog of the familiar diffusion equation which describes (among other things) the temperature distribution around a body in a continuous medium, where the body contains a sinusoidally varying heat source. In this case the temperature field due to the body is found to alternate in sign with increasing distance from the body. with an exponentially decaying amplitudeAt small frequencies the decay length is large and so the disturbance in the temperature field due to the body penetrates a long way into the surrounding medium, while at high frequencies the disturbance is confined to a thin layer on the surface of the body. As we shall see, the disturbances in the ion densities obtained from the solution of Eq. 5.9 have a similar frequency dependence, although in this case there is not a single decay length but a number of decay lengths associated with the eigenvalues of the matrix A. = !-fe begin by assuming that the solution v0 to Eq. 5.9 can be expressed as a linear combination of the eigenvectors of the matrix 6, i.e. M v0 =

z j=O

u(j)8 J

'

(5.14)

where the u(j)'s are a set of linearly independent eigenvectors. On substituting the form (iq. 5.14) for y" in Eq. 5-9 and using the fact that the eigenvectors are linearly independent, we obtain the simple equations:

02*g -

(ra)2Xjfg = 0

(5.15)

the new unknobms fy- Here Xj is the eiqenvalue associated with the eigenvector ,(j) and the index j ranges from zero to M_ for

291 In order to determine the quantities fy we must first obtain the eigenvalues .'. =; these quantities are the roots of the characteristic equation: J

?.$I = 0 ,

IA-

where i is the (N+l)x (N+l) unit matrix. !Gth the aid of the formula (Eq. 5.11) for h, we can write the characteristic equation in the more explicit form:

*

(5.16)

Clearly one of the roots of the characteristic equation (lo' say) is zero- By solving the eigenvector equation:

with k = 0, we find that the eigenvector u (0) corresponding to the zero eigenvalue is simply

(5.17)

With the aid of this observation, we can write the outer boundary condition (Eq. 0 and nni in the simple vector form 5.7) for Sri

y” -

-(E.

(ci)as r --r 10 _ - rlu - -

From the expression (Eq. 5.14) for v(O), it follows that the outer boundary conditions for the fy's are simply * -E f0 -0-r 0

9

(S.lE?)

and fy s 0 for i = l,Z,...,M asr+-. From the symmetry and linearity of the problem, it follo~rsthat each of the fy functions has the form (ref. I, Section 6): f:(r) = gi(rjEo - F

,

where F= C/r_ Substituting the above forms for f:(r) in Eo. 5-15 and solvinq sub.jectto the outer boundary conditions (Eq. 5-181, VJe obtain

0 f,tr)

= -go - r+

0 CoEo-

r/r

3

, (3_13)

and

for j = 1,2,...M. The constants CC! are to be determiiled from the boundary condition (Eq. 5.8) on the particle surfase,and the square root in the exponent is taken to have a positive real part (to ensure Mith

the aid

of

a decaying

solution).

the expressions

{Eq.

5.19)

for

fp and the

and 5.17) for v_' and u_(O), we see that the perturbation

formuiae ;Eqs. 5.14

in the ion densities decay

exponentially with increasing distance from the particle, while the perturbation in potential at large distances from the particle is dominated by the fi component and has the form s::r mr -5

0

0 - r+COEO

-r/r zasr___

fhus the coefficient Co ' in the oirter form for -fi is the same as the dipole coefficient appearing in the outer form The exponential scale

of order' R(rra+T}-';

acteristic equation one

which

for

the potential.

term in the expression

approaches

(Eq. 5.19) for

fi

decays

on a length

as w approaches zero, it can be seen from the char-

(Eq. 5.16) that all the eigenvalues approach zero, except for unity.

At

high frequencies, on the other hand, it can be

shown that Xj is approximately equal to Y-, a quantity which increases in proportion to C.1. From Eq_ 5.14 it follows that the gerturbations

in ion density extend a

large distance into the electrolyte at low frequencies, while at high frequencies the perturbations which

are confined to a thin layer on the particle surface; a result,

(as we mentioned before) is similar to the solution for the scalar diffusion

equation. The constants Ci in Eq. 5-19 are obtained by rewriting the inner boundary conditions (Eq. 5-8) in terms of the new unknowns fi and substituting

the formulae

of Eq. 5.19; the result is a set of linear simultaneous equations for the unknowns Co.

In or.fer to set up these equations, it is necessary to obtain formulae for

tjhe eigenvalues

X. .

it

Unfortunately,

does

not

seem possible

to obtain

such

formulae for a geieral electrolyte, and in order to obtain any analytic results, it is necessary to place restrictions on the nature of the electrolyte or on the dielectric constant of the particle_

In this paper it will be assumed that the

particle dielectric constant is much smaller than that of the solvent, an assumption which is ‘As

valid for

many aqueous

suspensions_

usual, R denotes the real part oi the quantity_

293

The dipole coefficient of a particle with small dielectric constant For a particle with zero dielectric constant, the ion densities are unaltered by the application of the f-ield. The potential field a::~' has the same form as

A.

for a particle in a pure dielectric, viz.:' ,j,~

0

(5.20)

and

inside the particle; a result which may be verified by substitution in the governing equations and boundary conditions. Thus for an uncharged particle the dipole coefficient is simpiy

co=_l

(5.21)

2 -

0

For a particle with a non-zero dielectric constant the normal component of the electric field at the particle surface is non-zero and thus there i'sa tendency for the field to drive ions alternately into and away from the impermeable particle surface. The resulting disturbances in the ion densities then diffuse out into the electrolyte, leading to a redistribution of charge and consequently to an alteration in the particle dipole coefficient. To obtain a formula for the dipole coefficient for a particle with a small but 0 in - the boundary condition (Eq. 5.8) non-zero dielectric constant, we replace 61; by the formula (Eq. 5-20) for a particle w?th zero dielectric constant. Since this term is mu'ltipliedby E /E in the boundary condition, the errors in this approximation The inner boundary condition (Eq. 5.8) may now be written will be of order (cp/c)5 in the simple vector form 2V

?%

0

=?%,

-

2 E -O-r

Tl -1

at r = 1.

Assuming that the column vector on the right-hand side may be expressed as a linear combination of eigenvectors*, viz:

'It is possible to show that any vector can be written in this form if the electrolyte has only two ionic species, or if all the ions have uniform diffusivity. Although it seems likely that this result will hold for a general electrolyte, the proof is not available at present-

294

-1

I_! 11 = M 1 Sj”(j)

(5.22)

9

j=O

1

IN

and substituting the form (Eq. 5.14) for v" in the above boundary condition, we find

5r

-

--_Y_E

2 5

(5.23)

.r

J-0

at r = 1, for j = 0, l,...,N. Although it is not possible to determine all of the sj's for a general electrolyte it is shown in Appendix II that ;o=-(I+~*j,.jjjI

)

(5.24)

which is all tnat is required for the calculation of the particle dipole coefficient 0 in - the boundary condito O(C,/S). On substituting the formula (Eq. 5.19) for f. tion (Eq. 5.23) and using the above expression for so, vie find that the dipole coefficient is given by: (5.25)

where we have used the fact that N

_ Ikf'k _ 4_iiK_ 1Wf

(5.26)

From Eq. 5.25, it can be seen that for

the dipole strength is the same as for an uncharged sphere in a pure dielectric since (as mentioned earlier) the perturbations in ion density are small and are confined to a thin layer on the particle surface. At low frequencies the ions are arranged around the particle in such a way that the dipole coefficient is the same as for a sphere with zero dielectric constant; a result which is consistent with O'Brien and White's (ref- I) conclusions on the effect of particle dielectric constant in the static field problem.

295

6.

The

zero

Before particle, for

frequency

have

the

shown,

order

rived

5.19)

here lrnj

(Eqs.

for

for

4.1

the

an

uncharged

4.4)

at

for

As we mentioned approach (Eq.

zero

with

5.19)for

the

more

earlier

in

formula

the

past,

some

formula,

a charged (Eq.

3.2)

researchers

taking

This

we turn

0 Sn J-;

they

a few

this

section,

give

since are

than

(ref

CO to

procedure

be

has

Thus

with

the

same

of at

small

expressions formulae

been

far-field

as

forms

those

from

the

for

the

for

an

$5:;

uncharged

particle

surfaces.

except

frequencies

eigenvalues

de-

equations

eigenvalues

low

(Eqs. were

electrokinetic

lengths

all

the

these

the

the

the

Oebve

frequency.

to

al though

also

charge,

associated

of

of

field.

error,

particle

decreasing fj's

the

arbitrary

of

the

this

a static

Si~!~”and

a charged

distances

coefficient

5).

of

oarticle,

of

from

in

(ref.

source

dipole

application In

conductivity

quantities

a particle to

particle

for

the

the

the

problem.

a particle

incorrect

locate

of

discuss

field

static of

be

to

static

the

to

to

calculation

here

the

coefficient

and

and

to

however,

In

the

pause

calculated

dioole

5.14

to

we shall

conductivity

8)

limit

proceeding

the

take

one

expresssions

the approximate

form

in

the

region

(5.27)

where of

we nave

arbitrary

tial in

619 has this

on

fj

the

the

the

f.

the

3.3)

(Eq.

3.1) the

the

have

5.27

3.1).

for

surrounding

particle (Eqs..

for

in

static

this

region;

value

exponentially and

the

now considering

v_, it and

follows in

fact

as (Ai vanishes_ decaying

potential

terms

ceases

to

a particle

that the bI< th

in

the

have

the

poten-

coefficient increasing

equation

a dipole

(Eq. form_

region

(Eq.

from

we are

5.14)

formula

significant

form

charge

since

(Eq.

particle,

except

zero

“O”,

the

form

fj’s

(Eq.

superscript

approaches

dipole

expression

regions

dipole

become

in

the

the From

form

from for

Finally,

all

the

dipole

distance 5.19)

drooped charge.

the the

and

as

and

5.28)

decayed It

to is

zero,

.&LA

conductivity particle

a result can

(ref_

extends the

and

dipole

dipole

be quite

the

potential

2) ; in

out

to

which

general,

distances

coefficients

different.

once

coefficient

the of

again

takes

appears

in

region

order

associated

the

of non1 jxj -!z,;a

with

the

_

296

THE CALCULATION OF THE DIPOLE COEFFICIENT TO O(c) VII. To determine the O(c) terms in the perturbation expansions for 69, anj, U and Bp for a particle with zero dielectric constant, we must solve the equations

v2y1 _ (c-)24 - y1 = WY1 - V&z> Ow 1

v-g

(6.11

,

=o,

(6.2)

and .2IjI- r&p I3=y

(,,)2Y&!O,

(6.3)

L

where v

1 =

s+1

9

and VJ=

1 Zldlll

0

IlZl I222 (6-4)

1 ZN6nN . INZN

In keeping with the superscript convention introduced in the previous section, Y' is the coefficient=of e&kT in the perturbation series for the equilibrium potential Y, given by Yl

= exprKa(l-r);/rl

(6-5)

for a spherical particle_ In deriving Eqs. 6.1 to 6.3, we have used the fact that the order one terms any, CD and 6p" are zero for a particle with zero dielectric constant, as shown in the previous section. From the boundary conditions 4.5 and 4.7, we find that

9'~

-M:Eo

as r-t -

(6.6)

,

and

av’ --=

ar

0

-

and u1 _ = 0 _

(6.7)

on the particle surface. Eqs.

6.2

and 6.3

are

the

same as for

the

static

field case. Thus to O(5) the

particle moves with the same mobility as in the static instantaneously

to

variations

in the

applied

field.

field

problem,

responding

29-i

The remaining equation (Eq. 6.1) is simply an inhomogeneous form of the vector diffusion equation (Eq. 5.9) discussed in the previous section; the same general conclusions concerning the frequency dependence of the solution apply here, except that in this case, since there are sources'distributed throughout the double layer, the disturbances in the ion densities at high frequencies will no longer be confined to a thin layer on the particle surface. This source term arises from the action of the field vOg" on the equilibrium charge cloud. This action results in an alternating build-up and depletion of ions on opposite sides of the particle; for example, the positive ions in the double layer around a negatively charged particle will be swept past the particle, resulting in a build-up of these ions on the downstream side of the particle. At the same time, the field will move the negative co-ions in the opposite direction, pushing them into the downstream portion of the double layer. Thus there is a build-up of both types of ions on the downstream side of the double layer and a corresponding depletion on the upstream side; the sign of the dipole coefficient CA is determined by the competition between these ions. To calculate CA, we begin by assuming (as in the previous section) that the unknown vector v1 in Eq. 6.1 can be expanded in terms of the eigenvectors of e, viz. v

1

!A_

L

=

y(j),! .I -

.j=O Substituting this form in Eq. 6.1 and using the fact that the eigenvectors are independent, we get $ff

- (,a)*X_f! =

J

3j7Yl

- vsp,

JJ

(6.9)

for j = O,l,...M, where the aj's are the coefficients in the expansion M

y=

z u(% ’_

j=O-

for the vector fi 'or 0

w.

Similarly, the boundary conditions 6.6 and 6.7 qive

as r -rm

and 1 s=Oatr=l for j = O,l,..., M.

,

,

(6.10)

298

As we mentioned in the previous section, the quantities 8~ and &nj have the same far-field form for a charged particle as for an uncharged particle. Thus at distances of more than a few Debye lengths from the particle, the quantities fI have the same exponentially decaying form (Eq. 5.19) as fy for j non-zero (vritn -3 replaced by Cj). while the iar fieid form of fi is given by Cj J.

1

co-F

fcls Co

r2

asr--0:

_

(6-H)

To determine the quantity Ci, we need only solve the equation ,2fI = +I 0

- .&.O

(6.12)

for f:, subject to the homogeneous boundary conditions 6.10. is sho>;in that the quantity ZD is given by ei: 'f I-2-D. jE1 J 3 3 $;<= - ' 0 (ias+ 4yK-)

In Appendix II it

(6.13)

The solution fi to Eq. 6.12 may be written as an integral involving the righthand side of the equation, using the procedure described on pp_ 242-243 of Refo5. On substituting the form of Eq_ 5-20 for 3-3 ln this integral and computing the far-field form of fg, we find that the required dipole coefficient is given

by CA =

-uoL(ra)

(6.14)

,

where L(x) = * + I

+

eXE (x) 25

and E5 is an exponential integral (See Chapter 5

X2 of Ref. 10). On substituting the result (Eq. 6.14) and the formula (Eq. 5.21) for CA and CE in the perturbation expansion (Eq. 5.2) for the dipole coefficient. we find that

Co = -l/Z - uoL(ita)ec/kT f O[(e
(6.15)

where no is given by the formula of Eq. 6.13. In Section IX we shall be comparing the approximate formula for Co to O(c2) with computed values for a KC1 electrolyte; in this case the ions have approximately equal diffusivities and the the O(s) correction to Co in Eq_ 6.15 is very small because the build-up in negative ions on the downstream side of the particles is approximately cancelled by the corresponding build-up in the positive ions. For other types of electrolytes (such as acids), in which the ions have very different diffusivities, the O(c) correction to Co can be quite significant; a point which is discussed in Section VI of Ref. 5 for the static field case.

299

VIII.

THE O(,;*) FORWLA

FOR THE OlPOLE COEFFICiENT

The c* coefficient Cz 7n _ the perturbation expansion for the dipole coefficient CO is obtained from the equations for the quantities .?:>* and 6,:.

As before, we

write these equations in the vector form:

~~*,* - (ra) 24 - v 2 = -n

f VY l

1

.yl

I 0

Z26n1 1-l

!!

(7-l)

-2 1 ZN91 i: ! and we look

for

a solution

of the

form

11

.* -

=

-x

f?"(j) J-

J=

_

(7.2)

As in the previous sections, we will compute the dipole coefficient from the asymptotic formula

(7.3) for fz at larae r.

- Eq. 7-l by the expansion On replacing "2 In

(Eq. 7_2), and

expressing

the column vectors on the right-hand side of Eq. 7-l in terms of the eiaenvectors ,li), we obtain the following equation for f;: IN N

N

Z;,?

-I-

- (6nlVY'j + ,/Y'j

j=l where we have used the result of Eq. A.7 in obtaining the u (0) component of the vectors on the right-hand side of Eq. 7.1, together with the fact that 6~3' satisfies Laplace's equation.

In a similar manner, we find that the boundary condition for

fg on the particle surface is

(7.5)

at r = 1. It is possible to represent the required quantity I$ as an integral involving the terms on the right-hand side of Eq. 7.4 by using the relation 0 -_ ,y* 3n

f0 -M:$O 2 an

(7.6)

61>'GdV, V

which follows from Green's identity (ref. 9). Here G is the quantity on the right-hand side of Eq. 7.4; A denotes the surface of the particle together with a large spherical surface of radius R (centered on the origin); and V is the volume lying between these two surfaces_ If the radius R is sufficiently large, we may 0 aoproximate 6:> and fi in the integral in Eq. 7.6 over the outer surface by their asymptotic forms (Eqs. 5.7 and 7.3)_ On evaluating the resulting integral and substituting the expression from the right-hand side of Eq_ 7-4 for G, we obtain (after an application of the divergence theorem)

c; =

N 1 EG2 I .Z?D. f WI) 335~ Eg(1WE+4iiKm) - ' j=l V {

c

i-t

IjZjDj J(oY'. 06+')6$

j=l

2 - v(S:b")2dV

dV f ;

.Y2 - r(~.')~ dV I

V

I

V

N

+x

Z$DjI

j=l

(,YI - ~6:2')&nidV -f

'jmjDj

j=I

V

(uyI _

>

(7.7)

V

where E. = IEOL Unfortunately, it is only possible to evaluate two of the integrals in this 0 formula analytically. !*lith the aid of the formulae of Eqs.5.20 and 6.5 for 6$ and YI , we find

f V

-

v(~v”)2d,,

=

-8a E2 o 3

$ exp(2ra)E6(2
(7.8)

On comparing the form of Eq- 7-7 in the limit of zero frequency with a formula for Co obtained from Eq_ 5.35 of O'Brien's static conductivity paper (ref. 5), we obtain

301

J

(OYl - gl),+'dV = -4nE;N(~a)

,

(7.9)

V

where the function N(za) is defined by the formula (Eq. 5.34) in Ref. 5. The remaining integrals in the expression numerically.

(Eq. 7.7) for CE must be evaluated

Of these, the easiest to evaluate is the integral involving Y2.

We let

vY2

-

N

o(t;l>')'dV= -4;iEg ~n~Z~F(~ra).

(7.10)

j=I Values of F(t:a) (obtained by the Romberg Integration technique (Ref. 11, Section 8.3) are set out.? in Table 1 and the function is plotted in Figure 1.

Unfor-

tunately, the remaining integrals involving &ni and 61:;~ depend on so many parameters that it is impossible to provide an adquate representation a single table.

of these functions in

The best that can be done is to reduce these integrals to simpler

TAeLE 1 Computed values of the function F(,:a) which appears in the formula F(ra)


0.0219 0.0249 0.0306 0.0314 0.0422 0.0528 0.0712 0.110 O-124 0.141 0.164 0.196

of Eq. 7-10.

-ra

F(r:a)

1.1 I.0 0.9 0.8 0.7 0.6 O-5 0.4 0.3 0.2 0.1 0.01

0.216 0.242 0.274 0.315 0.369 0.445 0.554 0.726 l-028 1.673 3.778 46.95

Note: For large ka values, F = l/d ::a - 7/lr(ka)'; while at small -:a values, F = I/(2
'ldith the aid of this table. it is possible to extend O'Brien's (ref_ 5) formula for the static conductivity to the case of an unsymmetrical electrolyte; the only modification required is the addition of the term

to O'Brien's expression

(Eq_ 5-35‘1 for conductivity-

Fig. 1. The function F(s:a),related to the integrals (Eq. 7.10), which appears the formula (Eq. 7.7) for the 52 coefficient of the particle dipole strength.

in

terms which can be easily evaluated numerically. To this end, it is shown in Apoendix III that the remaining integrals in Eq. 7.7 can be calculated once the integrals ‘.,e-XE1(2b:a) - eXE1(2rra+x) ,I 0

x

3

exp(-x/q/+:a)dx

have been computed for each distinct eiqenvalue Ai. With the aid of rational approximation for El (qiven on op. 231 of Ref. lo), it is possible to evaluate the above integral on a programmable calculator. The details of the numerical procedure for evaluating the formula of Eq. 7.7 for Cg are described in Appendix III. This procedure for calculating Cg to O(c2) requires much less computer time tnan the exact numerical computation described by Delacey and White (see ref. 2), but the results are, of course, limited to low zeta potentials. In the following section we will compare our results With DeLacey and White's computed values to obtain some idea of the range of validity of our formula for dipole strength. IX.

COMPARISOh OF THE APPROXIMATE FORMULA HITH THE COMPUTED RESULTS In this section we compare values for the dipole coefficient obtained from the

O($) formula with values obtained by the DeLacey and White (ref. 2) computer program for the case of KC1 electrolyte with a rcavalue of 10. For an electrolyte such as KC1 (in which the ions have approximately equal diffusivities), it is shown in Appendix IV that the formula (Eq. 7.7) for the c2

303

term, Ci in the expansion for the dipole coefficient reduces to the much simpler form (Eq. A-30). On combining this formula with the O(c) expression (Eq. 6.15) for CO, we get N E’:*b er, L(ka) CD=-+ IjZjDj c (iGc+4nK"T - kT D j=l (8.11

1

J=I IjZ5 [G(ra, $)

* i(-&

+ e2"aE6 (*,:a)+ m~(,:a)]]}

where 6 is a weighted mean diffusivity. given by N

7

Z;IjDj.

ij = Li

2 x 'k'k k=l and m _

,(kT)* 6:ne26

L and il are given is As mentioned earlier, formulae for the quantities The only term in Eq. 8.1 which requires numerical evaluation is G(-a, L&defined by

!?ef.

5.

,

(8.3)

where g(r) is the decaying solution to the equation , satisfying the boundary condition dg/dr = 0 at r = 1. A method for simplifying _:_ and evaluating integrals in the form of Eq. 8.3 is described in Appendix III'. In the derivation of the formula of Eq. 8.1, it is assumed that the particle dielectric constant c is zero. DeLacey and White use the more realistic value P we can expect discrepancies of order cp/c between the DeLacey and White values of Ci and the values obtained from the approximate formula (Eq. 8.1). At low zeta potentials and high ka values, the O(cp/c) error in the leading term for CO may be comparable with O(r,)and 0(5*) terms for Co. In order to of

2

for

E

and

thus

P’

.z.

-2 _ 'In this case kk = iw/D

304 2 make the G formula accurate in this region, of Eq.3.1 by the quantity:

we rep lace the -1/Z in the formula

"'P -l/2 -i-i3 4 (iLlei-4i;Km)' obtained from the formula of Eq. 5.25 for the dipole strength of an uncharged particle with small dielectric constant. With this modification, the formula of Eq. 8.1 provides a vet-yaccurate representation of DeLacey and White's data at c=25 mV,with errors of one or two percent in the imaginary part of Co and smaller errors in the real part. As expected, the errors in the approximate formula for Co increase with c_ In Figure 2 we show a comparison of the computed and approximate values

-0.30 0

-o-31-

F( [C,)

. x - 0.35 0

f 0.1

I

1

1 0. .5

w/&c2

ii. 5 Fig. 2. A comparison of the computed values of the real part (a) and the imaghnary part (b) of the particle dipole coefficient with -valuesobtained from the O(r ) formula for Ka=lO, with a KC1 electrolyte. The crosses represent the computed values.

305 for the real and imaginary parts of C0 for a zeta value of 50 mV over the frequency range for which computer results are available-

From Figure 2 it can be seen that

the errors in the imaginary part are of the order of ten percent, while the errors in the real part of CO are only a couple of percent.

The errors in the imaginary

part of CO may not be quite as serious as they sound, for since this quantity varies approximately like 5 2 , the formula for the imaginary part of CO could be used

to

compute

5 from measured

values

of

CO with

an error

of

only

five

percent.

The results displayed in Figure 2 are somewhat surprising; the real part of CO at first increases with frequency and over the entire range for which computed results are available, the computed value of this quantity is greater than the value obtained for a static field_

Furthermore, the imaginary part of CO changes sign

at low frequencies and thereafter increases monotonically

in magnitude as frequency

is increased. A more complete picture may be obtained from the curves in Figure 3, which cover a much greater frequency range.

The curves were obtained from the approximate

formula of Eq. 8.1 for CO, with ;.= 25 mV and a .ra value of unity-

As expected,

the imaginary part of CO decays to zero at large frequencies and the real part tends to the value appropriate to an uncharged sphere in a pure the

particle

ica value,

to those

charge the

has a much greater

curves

in Figure

have a similar

2,

except

that

effect

on the

dipole

dielectric.

strength

Although

at this

lower

form in the frequency range

there

is

no small

positive

maximum in the imag-

inary part of Co in this case. X.

THE EFFECT OF SURFACE DISSOCIATION

REACTIONS ON THE DIPOLE STRENGTH

The preceding analysis is based on the assumption that the surface charge density

is

unaffected

by the

application

of

the

external

field.

For those suspensions

in which the surface charge density is determined by the dissociation of ionizable surface groups, this assumption is unlikely to be true in general because the external field alters the concentration of counterions at the surface, leading to a change in the number of ionized groups. In this section, we consider a particle whose charge is determined by the dissociation of a single type of surface group-

Upon dissociating,

the

groups

yield ions of type j which go into solution; for example, if the surface reaction is SOH Z SO- + H+, where S denotes

surface

group,

the

hydrogen

ions

are the type j ions.

306

5=25mv I

b IO

Fig_ 3. The variation of the real part (a) and the imaginary part (b) of the particle dipole coefficient with frequency for a KC1 electrolyte with ::a=l. The curveswre obtained using the O(c2) formula of Eq. 8.1 for the dipole coefficient. The local charge density o is determined from the equation (ref. 12)

where Ks is the dissociation constant and Ns is the number of ionizable sites per The quantity n, in the expression here denotes the density unit area of surface. J of type j ions just beyond the surface. If the frequency of the applied field is such that the equilibrium equation (Eq. 9.1) is instantaneously satisfied, then the variation in charge density 60 due to the variation in local ion density 6nj wil 1 be given by

307

OO6” _ (9.2)

’

where the superscript "0" denotes the equilibrium value of the quantity, in the absence of the applied field. In Section VI we showed that the ion densities around an uncharged particle with

zero

dielectric

constant

are

unaffected

by

the

applied

field.

Thus

,zn. and

J

I-O

are both 0(&z)quantities and, therefore, the perturbation Lc;in charge density given by the formula of Eq. 9.2 is an O(.Q~)quantity. To O(r,)the problem of determining the ion densities and electrical potential is the same as for a particle of

fixed

charge.

The O(c*) problem is, however, different; in this case the boundary conditions associated with Eq. 7.1 for :7.>2 are no longer homogeneous. Rather surprisingly, the variation in the boundary conditions has no effect on the quantity Cg, and thus to 0(c2) the particle has the same dipole strength as a fixed charge particle. This result can in fact be proved for quite general boundary conditions. If finsdenotes the change in surface density of jth ions adsorbed onto the surface (assumed to be an O(s)' quantity),then the boundary conditions for the unknown vector _v2becomes 2 2V _-= at-

2A-v s --dY1 (>:a) dr = -

(9.3)

at

r= 1, where

form of Eq. 7.2, as a linear As usual, vie assume that v2 may be written in combination of the eigenvectors of the matrix A; the dipole coefficient = to determine .-I if to O(sL), we need only compute the leading coefficient f; in this expansion. the quantity vs has a similar expansion, that is, if M S v

= z j=O

zjp

,

(9.4)

.. 308 then the quantity e- y ’ component,for M

in the boundary condition (Eq. 9.3) will have no 5 (0)

since the eigenvalue ?.gis zero. Thtisif the eigenvector expansion (Eq. 9.4) for vs 7.5 - valid, the boundary condition for fz (obtained from Eq. 9.3) is the sameas for a fixed charge particle; and the dipole coefficient C2 (obtained from the 0 7. far-field form of fi) is unaltered by the change in boundary conditions. This result is rather disappointing because, as we mentioned at the outset, it was hoped that measurements of complex conductivity could provide information about surface dissociation reactions. kieshould emphasize, however, that the resuit derived here is not general; it only applies to suspensions of particles with low r:potential, and with only one type of SUrfaCe reaction. The result is not valid for amphoteric or zwitterionic latices in which the surface charge is determined by the competition between different surface reactions, for in this case the charge perturbation 6~ may not be an O(c,')quantity. Although complex conductivity measurements cannot be used in the study of surface reactions for the low r, suspensions considered here, these measurements can still provide more information about the suspension than can be conveniently obtained from electrophoresis and static conductivity measurements. By adjusting the parameters such as r potential, particle radius, ionic strengths and particle volume fraction so that the theoretical variation of dipole strength with frequency agrees with the measured variation, it should be possible to determine these quantities simultaneously. XI.

APPENOIX I - THE MEASUREMENT OF COMPLEX CONDUCTIVITY Consider the problem of determining the current flowing between a pair of electrodes in a suspension due to the application of an alternating potential difference. Since the net current density j is divergence free,

(A. 1) for any closed surface A, where n denotes the unit normal directed outwards from A. If A encloses one of the electrodes, then it must cut across the lead which brings the current to that electrode. In this case we may write the identity (Eq. A-1) in the form (A-2)

I(t)

where

that the

is

portion fact

by the If

the

current

which

cuts

the

integral

that free

the

charge radii

of

mean value

since

(Eq.

Eq. A.2

2.4)

for

where

of

the

this

cross-section

the

surface

A are

we may replace

A lies
the

surface

result, of

A,

minus

we have used

the

outside

in the

flowing

everywhere

j in the

fluctuations

a homogeneous

current

IO = K*

lead

is

dominated

the

much greater

intergral

are

(Eq-

cancelled

out

inhomogeneous

region

by K*$,eiWt,

using

integral suspension.

Thus in this the electrode becomes

onto

than

A-2)

by its the

in

inte-

surrounding the

case

the

constitutive the

formula

suspension

is

= -v+

I,,

given

of

range

fact

these

that

2.4)

potential

follows

interest

here,

and

suspension.

homogeneous

rj satisfies

portion

of

the

Laplace’s

equation:

the

follows

formula

the

to

the

(Eq.

E. has zero

result

is

curl

for

obtained

the constitutive A-4)

relation

we then

for

Eo,

are

thin,

the

by using obtain

Laplace’s

ratio

(Eq.

2.5)

electrodes take

Laplace's

the

boundary

with

difference

potential

in $9 between

if

and electric

difference

field

and are

we have two suspensions

from Eq. A.3 the

the

we may

and

if the

condition the

drop

(Eq.

A.5)

that

.; is

equation

in

to uniform

electrodes

is

equal

suspension

depend

difference.

potential

Hence,

suspension. it

case

applied

around

suspension,

and that

potential

Thus in this

proportional

that

substituting

expression

is small,

in the

electrode

applied

layers

layers

everywhere

on the

the

fact

the second

A.5).

over

each

from the while

divergence;

using

inhomogeneous

potential

be valid

results

of

-:i>

(Eq.

If the

the

electrical

has zero

for

equation

ities,

in the

(A-5)

frequency

the

F.

go in the

.

The first

only

field

field

(A-4)

0

the

the

the

,

.?> =

over

problem,

by

the macroscopic

(Eq.

we must determine field

where

the

(A- 3)

CO * ndA .

As in the static

to

if

for

the

Thus to calculate

to

In deriving

is

A'

and

= Ioeiwt,

I(t)

E.

electrode,

lead.

the microscopic

we may replace

relation of

the

j - fi over

scale,

Furthermore,

electrode,

the

of

curvature

length

gration.

across

to

contribution.

the microscopic
flowing

of

that

the

relating

the

ratio

complex complex

of

in the

independent of

different

the measured

conductivities. conductivities

This

of

the

properties

complex currents result

to measured

of

conductivwill is

be

equivalent

impedances.

310 APPENDIX II - DERIVING THE FORMULAE FOR EQS. 5.24 AND 6.13for so AND a0 XII. Both these expressions are special cases of the following general result. If the column vector

can be written as a linear combination of the eigenvectors of A, viz. M x= -1

yju(j)

(A-6)

j=O then the coefficient y. in this expansion is given by N 2 L-. 5 Djxj = xo+ i= y0 (itic +4:KY‘) .

(A-7)

The result (Eq. 5.24) is obtained by replacing r by the column vector on the left-hand side of Eq. 5.22, with the yj's replaced by sj_ Similarly, replacing _xby thevector y and yj by aj, we obtain the formula (Eq.6.i3) for uo. To prove the general result (Eq- A-7), we begin with the case of an electrolyte in which the ions all have different diffusivities. Clearly, "=-ji is not a root (Eq. of the characteristic equation (Eq.5.16) in this case. From the expression 5.11) for the components of the matrix A, we find that the eigenvectors u (k) cor= responding to eigenvalue xk are scalar multiples of the vector 1 'lXk -'I-xk ??'k '2-"k

(A-8)

INXk ‘N+

Thus to determine the coefficients yj in the expansion (Eq. A-6) for 5, we must solve the simultaneous equations

311

(A-9)

INIl -fN-X

. _ _ 1

where

M is the number of distinct non-zero eigenvalues. The required result (Eq. A-7) is obtained by premultiplying both sides of the above equation by the row vector

(A-10) and using the fact that 'k ----=o, - 'j for each eigen;alue hj, a result obtained by dividing the characteristic equation (Eq- 5.1') bY~~~(Yi- 'j). Only a slight modification is required to obtain the result for an electrolyte in which some of the ions have equal diffusivities.

Suppose, for instance. that

only ions 1 and N have equal diffusivities. Then one root of the characteristic

1_ Form the form (Eq. 5.11) of the equation is y,; we let this eigenvalue be r. A it follows that

matrix

L-11 is a suitable eigenvector. The only modification to the equations ( Eq. A-9) for the coefficients yj in this case is that the second column is replaced by the above column vector u (1)_ As before, the required result (Eq. A.7) is obtained by pre-multipl.yingboth sides of the equations for the yi's by the row vector (Eq. A-10) and using the fact that yl=-fN in this case.

XIII. APPENDIX III - A METHOD FOR EVALUATING THE FORMULA (Eq. 7.7) FOR C; If the electrolyte ions have approximately uniform diffusivities, the formula (Eq. 7.7) for Cg reduces to the simpler form (Eq. 8.1); the procedure for evaluating CE in this case is described in Appendix IV. In this section we describe a method for cdmputing CE for a general electrolyte. The terms in Eq. 7.7 ebich require numerical integration may be written as a single integral, viz.

*L

IjZjDj&

f

j=l Wth the aid of the eigenvector expansion (Eq. 6.8) for vl, we can replace 691 and &ni in this integral by linear combinations of the quantities fi, which satisfy the simple differential equations (Eq. 6.9). For an e7ectrolyte in which no two ionic species have the same diffusivity. the eigenvectors in the expansion (Eq. 6.8) have the form of Eq_ A-8 and the above integral may be rewritten an

0 VY1 - 065,

r.Z.D.fI +Ls f’ 3 J J 0 k&-I k

dV_

(A-11)

. To evaluate the coefficients of the ft's in the integrand, it is necessary to determine the eigenvalues of the matrix 4; thus the first step in the evaluation of c; 7s - the evaluation of the roots Xk of the characteristic equation (Eq. 5.16). The next step involves the computation of the integrals

/ V

VY

1 - v&$'f'dV k

for k=O.l,___ .M_ Wth

(A-12) the aid of the divergence theorem, it can be shown that

OY1 - o&~Of~dV = 4;iEgM(ka)f 2

.(Y1)' - v(&b")2dV

where M(ka) is given by the formula (Eq. 5.33) in Ref. 5, and the second term on the right-hand side of this expression can be eva?uated using the formu7a of Eq_ 7-8. together with the expression (Eq. 6.13) for the coefficient aoTo calculate the integral (Eq. A-12) for non-zero k, we begin by recall. ing that the quantities fi satisfy the differential equations 1 -V&J, 0 , 021 fk- (ka)21 Xkfk = (LEVY subject to the boundary conditions

313 1

afk= 0 ar

at

r

=

1

,

where the o k'~ are the (given

coefficients

in

tile

eigenvector

expansion

of

the

vector

w

in Eq. 6.4).

From the symmetry and linearity of the problem for fi, it follows that this function has the form f' k = gk (r)E -O-r^

(A-13)

-

On substituting this form for ft in Eq. 6.9, we obtain a differential equation Sy the usual Green's function technique, we write gk in terms of for g,(r). integrals involving .Y' - v&i; 0 _ Replacing fi in the integral (Eq_ A-12) by the form of Eq. A.13 and using the Green's function formula for gk, we get

where

and

L = (ka$)-'

,

_e21L 2 - 2L-l-k 'k =

(A-15) L-2

2[2+2Cl+C2]

1 -

(A. 16)

The integrals in the formula of Eq. A.14 can be easily evaluated numerically for each eigenvalue. However, for those who do not have ready access to a computer, we shall briefly describe how the integrals may be evaluated on a programmable hand calculator. Replacing 6.;~' and Y1 in the second integral on the right-hand side of Eq. A-14 by the expressions of Eqs.5.20 and 6.5, we can write this integral in terms of exponential integrals, viz. (A. 17)

where aG = -a3=ka/L, aI=-a4=ka+L and

-1 ,

a2 = -a5 = 1.

The first exponential integral El is tabulated for complex arguments on pp- 249 to 251 of Ref. 10; the higher exponential integrals in Eq. A.17 can be evaluated from El with the aid of the recurrence relation (Eq. 5-l-14) given in that reference_

314

On replacing 9,' and Y1 ln _ the remaining integral on the right-hand side of Eq. A.14 by the forms of Eqs. 5.20 and 6.5, we obtain

where ee-(s
= n,m

-

,-(ia+L-1)x dx dr

r ‘I

_Ir

r"

_

(A-19)

Xrn

Clearly En(2+:a) J

n,O =(raiL-I)

(A-20)

'

and

Jo,l = I, E 1 (.-;a+L-')e

-(qa-L-I) _ EI(2$:a)/(::a-L-l). I

The remaining integrals Jn m can be related to the Jn I 's by repeated use of the > > recurrence relation Jn,m+I =

En+m(2xa) m

(ka+L -1 ) J m n,m

(A-21)

-

Finally, the Jm I's can be computed from JI I by using the recurrence relation 3 , Jn+I,I = 11

- (?:a-L-')Jn,I+El("aCL-l)e-(~a-L-')

-

_

Thus the only integral that need be calculated numerically in the evaluation of the integral (Eq. 14) is JI I, which can be written in the form of a single , integral, namely -caXEI(2~a)-e“aXEI(2~aIx+-Il) JI,I = XIV-

e-x/Ldx

-

X

APPENDIX IV - THE DIPOLE COEFFICIENT OF A PARTICLE IN A SYMNETRIC ELECTROLYTE .WITH IONS OF APPROXIMATELY UNIFORM DIFFUSIVITIES

k!e let

Dj = D + "Df ,

(A-23)

where, as usual, Dj denotes the diffusivity of type j ions and fiis a weighted mean diffusivity. The quantity Dj is a constant and A is a small parameter. Gle seek an expression for the dipole coefficient Co which is correct to O(A)_

To evaluate the formula (Eq. 7.7) for the O(s') correction to the dipole coeffi1 1 cient, we require expressions for the quantities 1+2 and 13n.,obtained from the J solution of the vector equation (Eq. 6.1). For small A, we assume that the unknown . vector v? in _ this equation can be written in the form (A-24) On substituting this form in Eq- 6.1 and equating the O(1) and O(:.,) terms, we obtain the equations v

2

1 - (-a)2A - v 1 = ,yl _ "&)O, , VO =o -0

(A-25)

and 2A ,lzvl - (bra) ' = (ka)2e1- 2: -1 =o - v-1

=

(A-26)

where the vector y is defined in Eq. 6.4, and the matrices froand cl are defined by the formula

thus the components of /IOmay be obtained from the expression (Eq. 5.11) for e by replacing the -tjisby the uniform value * and r.D 0

0

_--

0

0;

0

__-

0

0

Di

e-e

0

0

0

By using the fact that

%

1

(A-27)

N II Ij'j = O ' j=l for a symmetric e?ectro?yte, it can be shown that !I is an eigenvector of the =. corresponding to eigenvalue 3 _ Thus the solution to the equation matrix A DK (Eq. A. 25) for y: is simply ' = g(r)(E -0 - p)w YO

'

where g(r) is the decaying solution to the differential equation

C

(~.28)

316

satisfying the boundary condition @X=0 dr at r = l_ From the formula of Eq. 6.4. it can be seen that the first component of 9 is zero. Thus from Eq_ A-28 it follows that the potential perturbation a-+ ' is zero for an electrolyte in which the ions have uniform diffusivity; the ion densities are altered by the field, but the ions are arranged in such a way that the charge density is everywhere zero. Thus the integral involving 6:) ' in the formula (Eq. 7.7) for CE is an O(a) quantity, and since it is multiplied by a term which is itself of O(n) for a symmetric electrolyte, we may neglect this term in the calculation of Cg to O(L)_ To calculate the terms involving 5ni in Eq. 7.7, we first rewrite these terms in the vector form iV

.yl .v,j;Ola-++-

+0(C2)1

dV ,

(A-29)

where 5 = (O,Z,D,,Z,,D,,--.,Z,ON), and _b= 6(0,z,,z,,._.2,) In formulating this expression we have used the approximate formula (Eq. A-24) for v_l_ Pre-multiplying both sides of Eq. A-26 by b_,and using the formula (Eq_ A-28) for yi, we find that the quantity _S- yi (which appears in the integral in Eq- A-29) satisfies the equation v

N

2

g(r)5 - r j=l

with the usual homogeneous boundary conditions. Thus if we define our mean diffusivity 6 by N

jj

=$ Zj20jEj/~Z~Ii j=l

,

i=l

the sum on the right-hand side of the equation for k - yi

I-y:=0

-

Thus the integral (Eq. A-29) reduces to OY

dV >

is zero and we find

31'7 where we have used the formula (Eq. A-28) for vl_ Substituting the form (Eq. :O 5.20) for &:3'and integrating with respect to r we obtain

2

E$f

IjZ5

j=l

J-g(r) I

$-

(I--$)r'dr _ r

Substituting this expression for the integrals involving c;niin Eq. 7.7 and using the formulae of Eqs. 7.8 and 7.9, we get

xv.

ACKNOWLEDGEMENT I would like to thank Pk. Emma DeLacey and Dr. Lee White for providing the computed complex conductivities used in Section IX. This work has been partially supported by a grant from the Australian-American Educational Foundation. XVI.

LIST OF SYMBOLS

a

particle radius

ai

coefficients defined after the formula of Eq. A.17

a

a row vector defined in Appendix

A

a

50

a square matrix, defined by Eq. 5.11

BOY!&

matrices defined in Appendix IV

b

a row vector defined in Appendix IV

cO C;

particle dipole coefficient, defined by Eq. 3.1

b 'j

IV

closed surface

coefficients in the asymptotic forms of Eq. 5.19 mean

ionic

diffusivity.

defined

by Eq_

8.2

diffusivity of type j ions

P

electric displacement vector

e

the electron charge

_E

electric field strength

CO

amplitude of the mean field strength (see Eq. 2.3)

EO

the magnitude of co

E,(x)

the nth exponential integral of x, defined

in

Section

VI

of Ref. 10.

31s fk J F(+:a)

coefficients

g(r)

a function defined after Eq_ 8.3

i

nett current density, defined by Eq. 2.2

ff

free charge current density

!*

amplitude of average nett current density, defined by Eq_ 2-3

in eigenvector expansion of vector yk

a function defined in Eq_ 7-10

the current flowing between the electrodes; see Appendix IO

the amplitude of I

‘j

a quantity related to ionic concentration,

J

an integral defined by Eq. A.19

n,m k-i

I

defined by Eq. 5.12

the absolute temperature, in energy units

K*

complex conductivity of suspension, defined by Eq_ 2-4

K" E 0, K

complex conductivity of electrolyte

KS

dissociation constant of surface group

static conductivity of electrolyte

a complex quantity defined by Eq. A-15 L(-:a)

a function defined after Eq. 6.14

iii

a mean ionic drag coefficient, defined after Eq_ 8.2

m. J M

non-dimensional

H(r:a)

a function defined in Eq. 5.33 of Ref. 5

ME

ionic drag coefficient, defined after Eq. 4.4

the number of linearly independent eigenvectors of-e, minus one

non-dimensional

electrophoretic

mobility, defined by Eq. 4.6

n. J no-' J

number density of type j ions

n

unit outward normal

N

the number of ionic species in the electrolyte

N(ka)

a function defined by Eq. 5.34 of Ref. 5

Ns

density of type j ions beyond the equilibrium double layer

number density of ionizable sites on the particle surface

P

fluid pressure

P

polarization vector

r

distance from particle center

319 r

position vector

r

unit vector in the direction of r

_U ,(j)

fluid velocity jth eigenvector of the matrix B

VU,? 2 column vectors defined in Sections VI, VII and VIII, respectively _ a!! W

a column vector, defined in Eq_ 6.4

Y

non-dimensional equilibrium potential

'j

valences of type j ions

Greek Symbols coefficients in the eigenvector expansion of the vector M coefficients in the expansion (Eq. 5.22) non-dimensional frequency, defined by Eq. 5.13 a prefix, denoting a small perturbation in the quantity concerned the deviation of the diffusivity of -Lype J_ ions from the mean value D dielectric constant of solvent particle dielectric constant electrical potential on the particle shear plane viscosity of the solvent Debye length of the bulk electrolyte jth eigenvalue of the matrix A electrophoretic mobility of the particle particle surface charge density particle volume fraction macroscopic e7ectrica7 potentia7 electrical potentials in the electrolyte and particle, respectively angular frequency of the applied field The angle brackets around a quantity denote a volume average of that quantity, as in Eq. Z-I_ Unless otherwise noted, the superscripts on a quantity refer to the coefficients in the perturbation expansion for that quantity, as in Eqs. 5-l and 5.2.

320

XVII. REFERENCES R-G:.O'Brien and L-R. White, J. Chem. Sot_ Faraday Trans. II, 74(1978)1607. E.H.B. OeLacey and L-R. blhite,J. Chem. Sot. Faraday Trans. II, to appear. V.N. Shilov and S.S. Dukhin, Kolloidny Zh., 32(1970)117S-S. Dukhin, Surface and Colloid Science, 3(1971)83. R.GJ. O'Brien, J. Colloid Interface Sci., 81(1981)234_ D.A_ Saville, J. Colloid Interface Sci., 72(1979)477. F_ Booth, Proc. Roy- Sot- London Ser. A, 203(1950)5. S-S. Dukhin and 8-V. Derjaguin, Surface and Colloid Science, 7(1974)49. 14-H.Protter and H-F. bleinberger,Maximum Principles in Oifferentiai Equations, Englewood Cliffs, N-J., Prentice-Hall, 1967. 10 M. Abromowitz and I-A_ Stegun, Handbook of Mathematical Functions, Dover, New York, 1972. II R-W_ Hornbeck, Numerical plethods,Quantum, New York, 1975. 12 R-0. James, J-A_ Davis and J-O_ Leckie, J. Colloid Interface Sci., 65(1978) 331.