26 Colloid Vibration Potential ina Suspension of Soft Particles

26.1 INTRODUCTION

When a sound wave is propagated in an electrolyte solution, the motion of cations

and that of anions may differ from each other because of their different masses so

that periodic excesses of either cations or anions should be produced at a given

point in the solution, generating vibration potentials (Fig. 26.1). This potential is

called ion vibration potential (IVP) [1–6]. A similar electroacoustic phenomenon

occurs in a suspension of colloidal particles. Since colloidal particles are much

larger and carry a much greater charge than electrolyte ions, the potential difference

in the suspension is caused by the asymmetry of the electrical double layer around

each particle rather than the relative motion of cations and anions (Fig. 26.2) [7–15].

It has been shown that the colloid vibration potential (CVP) of a suspension of col-

loidal particles is proportional to the dynamic electrophoretic mobility of the parti-

cles. Approximate expressions for the dynamic electrophoretic mobility of soft

particles are given in Chapter 25. It must also be noted that in a colloidal suspension

in an electrolyte solution, IVP and CVP are both generated simultaneously. Re-

cently, we have developed a general acoustic theory for a suspension of spherical

rigid particles, which accounts for both of CVP and IVP [16–18]. In the present

paper, we apply this theory to the suspension of spherical soft particles with the

help of an approximate expression for the dynamic electrophoretic mobility of soft

particles [19].

26.2 COLLOID VIBRATION POTENTIAL AND IONVIBRATION POTENTIAL

Consider a dilute suspension of Np spherical soft particles moving with a velocity

U exp(�iot) in a symmetrical electrolyte solution of viscosity Z and relative permit-

tivity er in an applied oscillating pressure gradient fieldrp exp(�iot) due to a soundwave propagating in the suspension, where o is the angular frequency (2p times

frequency) and t is time. We treat the case in which o is low such that the disper-

sion of er can be neglected. We assume that the particle core of radius a is coated

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

508

with an ion-penetrable layer of polyelectrolytes with a thickness d. The polyelectro-lyte-coated particle has thus an inner radius a and an outer radius b¼ aþ d (Fig. 26.3).Let the valence and bulk concentration (number density) of the electrolyte be z andn, respectively. We also denote the mass, valence, and drag coefficient of cations

by mþ, Vþ, and lþ, respectively, and those for anions by m�, V�, and l�. The dragcoefficients lþ and l� are related to the corresponding limiting conductance of cat-

ions, L0þ, and that of anions, L

0�, by

l� ¼ NAe2z

L0�

ð26:1Þ

FIGURE 26.2 Colloid vibration potential caused by the asymmetry of the electrical dou-

ble layer around the particles.

FIGURE 26.1 Ion vibration potential or periodic excesses of either cations or anions

caused by the relative motion of cations and anions.

COLLOID VIBRATION POTENTIAL AND ION VIBRATION POTENTIAL 509

where NA is Avogadro’s number. We adopt the model of Debye–Bueche (Chapter 21)

that the polymer segments are regarded as resistance centers distributed in the

polyelectrolyte layer, exerting a frictional force on the liquid flowing in the poly-

electrolyte layer, where the frictional coefficient is n. We also assume that fixed-

-charge groups of valence Z are distributed at a uniform density of N in the

polyelectrolyte layer.

We have recently proposed a general acoustic theory for a dilute suspension of

particles, which accounts for both of CVP and IVP [16–18]. Experimentally, the

total vibration potential (TVP) between two points in the suspension, which is given

by the sum of IVP and CVP, is observed. That is,

TVP ¼ IVPþ CVP ð26:2Þ

where IVP and CVP are given by [19]

IVP ¼ zen

roK�

mþ � roVþlþ

� m� � roV�l�

� �DP ð26:3Þ

CVP ¼ ½�cðrc � roÞ þ �sðrs � roÞ�roK

� mðoÞDP ð26:4Þ

with

K� ¼ K1 � ioereo ð26:5Þ

FIGURE 26.3 A spherical soft particle in an applied pressure gradient field. a¼ radius of

the particle core. d¼ thickness of the polyelectrolyte layer covering the particle core.

b¼ aþ d.

510 COLLOID VIBRATION POTENTIAL IN A SUSPENSION OF SOFT PARTICLES

K1 ¼ z2e2n1

lþþ 1

l�

� �ð26:6Þ

� ¼ ð4p=3Þa3Np

Vð26:7Þ

�c ¼VcNp

Vð26:8Þ

Vc ¼ 4

3pa3 ð26:9Þ

�s ¼V sNp

Vð26:10Þ

Here rp has been replaced with the pressure difference between the two points is

DP, K1, and K� are, respectively, the usual conductivity and the complex conduc-

tivity of the electrolyte solution in the absence of the particles, � is the particle

volume fraction, �c is the volume fraction of the particle core, Vc is the volume of

the particle core, �s is the volume fraction of the polyelectrolyte segments, Vs is the

total volume of the polyelectrolyte segments coating one particle, rc and ro, arerespectively, the mass density of the particle core and that of the electrolyte solu-

tion, and rs is the mass density of the polyelectrolyte segment, V is the suspension

volume, and m(o) is the dynamic electrophoretic mobility of the particles. Equation

(26.4) is an Onsager relation between CVP and m(o), which takes a similar form for

an Onsager relation between sedimentation potential and static electrophoretic mo-

bility (Chapter 24).

For a spherical soft particle, an approximate expression for m(o) for the dynamic

electrophoretic mobility is given by Eq. (25.45), which is a good approximation

when the following conditions are satisfied:

jbjb � 1; kb � 1; jbjd ¼ jbjðb� aÞ � 1; kd ¼ kðb� aÞ � 1; jbj � jgj; k � jgjð26:11Þ

which hold for most practical cases.

Some results of the calculation of the magnitude and phase of the dynamic mo-

bility via Eqs. (26.2)–(26.4) are given in Figs 26.4 and 26.5, in which we have used

the following values: ro¼ 1� 103 kg/m3, er¼ 78.5 (water at 25�C), and rp¼ rs¼1.1� 103 kg/m3 in an aqueous KCl solution at 25�C (Lo

þ ¼ 73:5� 10�4 m2=O=mol

and mþ¼ 39.1� 10�3 kg/mol for Kþ, and Lo� ¼ 76:3� 10�4m2=O=mol and m�¼35.5� 10�3 kg/mol for Cl�). For the ionic volumes for Kþ and Cl� ions, we have

COLLOID VIBRATION POTENTIAL AND ION VIBRATION POTENTIAL 511

used the values of their partial molar volumes reported by Zana and Yeager [5],

that is, Vþ¼ 3.7� 10�6m3/mol (for Kþ) and V�¼ 22.8� 10�6m3/mol (for Cl�).The values of �s and fs have approximately been set equal to zero. It is to be noted

that the phase of CVI agrees with that of dynamic electrophoretic mobility m(o)(Eq. (25.45)) and the phase of IVI is zero (in the present approximation).

Figures 26.4 and 26.5 show the dependence of the magnitude (Fig. 26.4) and

phase (Fig. 26.5) of each of CVI, IVI, and TVI on the frequency o of the pressure

gradient field due to the applied sound wave for the case where a¼ 1 mm for an

aqueous KCl solution of concentration n¼ 0.01M. It is seen that the o-dependenceis negligibly small o/2p< 104Hz and becomes appreciable for o/2p> 104Hz. That

is, CVI is essentially equal to its static value at o¼ 0 for o/2p< 104Hz and drops

sharply to zero for o/2p> 104Hz, while the phase of CVI is zero for o/2p< 104Hz

and increases sharply for the frequency range o/2p> 104 Hz. The magnitude

of IVI, on the other hand, is constant independent of o, while the phase of IVI is

always zero, since in the present approximation IVI is a real quantity.

It can be shown that the CVI tends to a nonzero limiting value at very high

electrolyte concentrations, as in the case of other electrokinetics of soft particles

(Chapters 21 and 24). This is a characteristic of the electrokinetic behavior of soft

particles, which comes from the second term of the right-hand side of Eq. (25.45).

FIGURE 26.4 Magnitudes of CVI, IVI, and TVI divided by DP as a function of the fre-

quency o/2p of the applied pressure gradient field for a suspension of soft particles in a

KCl solution of concentration n¼ 0.01M. Calculated via Eqs. (26.2)–(26.4) as combined

with Eq. (9.45) for a¼ 1 mm, d¼ 10 nm, N¼ 0.05M, Z¼ 1, 1/l¼ 1 nm, ro¼ 1� 103 kg/m3,

rc¼ rs¼ 1.1� 103 kg/m3, �s¼ 0, fs¼ 0, Z¼ 0.89mPa s, T¼ 298K. er¼ 78.5, Loþ ¼ 73:5�

10�4 m2=O=mol, Vþ¼ 3.7�10�6m3/mol, mþ¼ 39.1� 10�3 kg/mol, Lo� ¼ 76:3� 10�4 m2=

O=mol, V�¼ 22.8� 10�6m3/mol, and m�¼ 35.5� 10�3 kg/mol. From Ref. 19.

512 COLLOID VIBRATION POTENTIAL IN A SUSPENSION OF SOFT PARTICLES

The limiting CVI value is obtained from Eq. (26.4) (as combined with Eq. (25.45))

with the result that

CVI ¼ ½�cðrp � roÞ þ �sðrs � roÞ�ro

1� igb� ðg2b2=3Þð1� a3=b3Þ1� igb� ðg2b2=3Þ � G

� �ZeN

Zb2DP

ð26:12Þ

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FIGURE 26.5 Phases of CVI, IVI, and TVI as a function of the frequency o/2p of the

applied pressure gradient field for a suspension of spherical soft particles in a KCl solution.

Calculated via Eqs. (26.2)–(26.4) as combined with Eq. (9.45). Numerical values used are the

same as in Fig. 26.4. From Ref. 19.

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514 COLLOID VIBRATION POTENTIAL IN A SUSPENSION OF SOFT PARTICLES