Biophysical Chemistry of Biointerfaces (Ohshima/Biophysical Chemistry of Biointerfaces) || Dynamic Electrophoretic Mobility of a Soft Particle

25 Dynamic ElectrophoreticMobility of a Soft Particle

25.1 INTRODUCTION

The dynamic electrophoretic mobility of colloidal particles in an applied oscillating

electric field plays an essential role in analyzing the results of electroacoustic mea-

surements of colloidal dispersions, that is, colloid vibration potential (CVP) and

electrokinetic sonic amplitude (ESA) measurements [1–20]. This is because CVP

and ESA are proportional to the dynamic electrophoretic mobility of colloidal parti-

cles. In this chapter, we develop a theory of the dynamic electrophoretic mobility of

soft particles in dilute suspensions [21].

25.2 BASIC EQUATIONS

Consider a spherical soft particle moving with a velocity U exp(�iot) in a liquid

containing a general electrolyte in an applied oscillating electric field E exp(�iot),where o is the angular frequency (2p times frequency) and t is time (Fig. 25.1). The

dynamic electrophoretic mobility m(o), which is a function of o, of the particle is

defined by

U ¼ m(o)E ð25:1Þ

where U¼ jUj and E¼ jEj. We assume that the particle core of radius a is coated

with an ion-penetrable layer of polyelectrolytes with a thickness d. The polyelectrolyte-coated particle has thus an inner radius a and an outer radius b� aþ d (Fig. 25.1).

The origin of the spherical polar coordinate system (r, �, j) is held fixed at the

instantaneous center of the particle and the polar axis (�¼ 0) is set parallel to E. Letthe electrolyte be composed of M ionic mobile species of valence zj and drag

coefficient lj ( j¼ 1, 2, . . . , M), and n1j be the concentration (number density) of

the jth ionic species in the electroneutral solution.As in the case of static electrophoresis, we adopt the model of Debye–Bueche

[22, 23] where the polymer segments are regarded as resistance centers distributed

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

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

497

polyelectrolyte layer. Let the polymer segments be spherical particles of radius asand distributed at a number density of Ns, then the frictional force is given by [22]

Fs ¼ �nu ð25:2Þ

Here, the frictional coefficient n is given by [24]

n ¼ 6pZasNs 1� igas � 1

9(gas)2

� �ð25:3Þ

with

g ¼ffiffiffiffiffiffiffiffiffiffiioroZ

s¼ (iþ 1)

ffiffiffiffiffiffiffiffiffioro2Z

r¼ (iþ 1)

1

dð25:4Þ

where ro is the mass density of the liquid, Z is the viscosity, and d¼ (2Z/oro)1/2

is the hydrodynamic penetration depth in the liquid. For a typical case where

as¼ 0.7 nm, Z¼ 0.89� 10�3 Pa s, ro¼ 1� 103 kg/m3, and o/2p¼ 1MHz, we have

jgjas¼ 2.6� 10�3. We may thus assume that

jgjas � 1 ð25:5Þ

so that Eq. (25.3) can be approximated by

n ¼ 6pZasNs ð25:6Þ

FIGURE 25.1 A soft particle in an oscillating electric field E e�iot. a¼ radius of the par-

ticle core and d¼ thickness of the polyelectrolyte layer coating the particle core. b¼ aþ d.

498 DYNAMIC ELECTROPHORETIC MOBILITY OF A SOFT PARTICLE

which is the same as the Stokes formula in the static case (o¼ 0). We also assume

that fixed charges are distributed with a density of rfix. If dissociated groups of

valence Z is distributed with a constant density (number density) N within the poly-

electrolyte layer, then we have rfix ¼ ZeN. The fundamental electrokinetic equa-

tions as well as the boundary conditions are essentially the same as those used in

static electrophoresis, except that the flow velocity of the liquid u(r, t) at position rand that of the jth mobile ionic species vj(r, t) are now functions of time t and may

be written as

u(r; t) ¼ u(r) exp(�iwt) ð25:7Þ

vj(r; t) ¼ vj(r) exp(�iot); ( j ¼ 1; 2; . . . ;M) ð25:8Þ

so that the Navier–Stokes equations become

ro@

@t[u(r; t)þU exp(�iot)]

¼ �Zr�r� u(r; t)�rp(r; t)� rel(r; t)rc(r; t)� nu(r; t); a < r < b

ð25:9Þ

ro@

@t[u(r; t)þU exp(�iot)]

¼ �Zr�r� u(r; t)�rp(r; t)� rel(r; t)rc(r; t); r > b

ð25:10Þ

and the continuity equation for the ionic flow becomes

@nj(r; t)

@tþr � (nj(r; t)vj(r; t)) ¼ 0 ð25:11Þ

where rel(r, t) is the charge density resulting from the mobile charged ionic species,

c(r, t) is the electric potential, and nj(r, t) is the concentration (number density) of

the jth ionic species at r and t.In Eqs. (25.9) and (25.10), the term ro(u�r)u has been omitted, since we assume

that the liquid flow is slow as in the case of static electrophoresis. The term involv-

ing the particle velocity U exp(�iot) in Eqs. (25.9) and (25.10) arises from the fact

that the particle has been chosen as the frame of reference for the coordinate system.

25.3 LINEARIZED EQUATIONS

As in the case of static electrophoresis, for a weak field E, the electrical double

layer around the particle is only slightly distorted. Then, we may write

nj(r; t) ¼ n(0)j (r)þ dnj(r)exp(�iot) ð25:12Þ

LINEARIZED EQUATIONS 499

c(r; t) ¼ c(0)(r)þ dc(r)exp(�iot) ð25:13Þ

mj(r; t) ¼ m(0) þ dmj(r)exp(�iot) ð25:14Þ

rel(r; t) ¼ r(0)el (r)þ drel(r)exp(�iot) ð25:15Þ

where the quantities with superscript (0) refer to those at equilibrium, that is, in the

absence of E, and m(0)j is a constant independent of r.Further, symmetry considerations permit us to write

u(r) ¼ (ur; u�; uj) ¼ � 2

rh(r)E cos �;

1

r

d

dr(rh(r))E sin �; 0

� �ð25:16Þ

dmj(r) ¼ �zje�j(r)E cos � ð25:17Þ

dc(r) ¼ �Y(r)E cos � ð25:18Þ

Here, �j(r), Y(r), and h(r) are functions of r only. In terms of �j(r), Y(r), and h(r),the fundamental electrokinetic equations including Eqs. (25.9)–(25.11) can be

rewritten as

L(Lh� l2hþ g2h) ¼ G(r); a < r < b ð25:19Þ

L(Lhþ g2h) ¼ G(r); r > b ð25:20Þ

L�j � k2gj(�j � Y) ¼ dy

drzjd�j

dr� 2lj

e

h

r

� �ð25:21Þ

drel(r) ¼XMj¼1

zjednj(r) ¼ �ereoDdc(r) ¼ ereoE cos � � LY ð25:22Þ

with

l ¼ (n=Z)1=2 ð25:23Þ

gj ¼ � ioljk2kT

ð25:24Þ

where k is the Debye–Huckel parameter, L is a differential operator (Eq. (21.24)),

and G(r) is defined by Eq. (21.25).


25.4 EQUATION OF MOTION OF A SOFT PARTICLE

Consider the forces acting on an arbitrary sphere S enclosing the soft sphere at its

center. We take the radius of S tends to infinity. Since the net electric charge within

S is zero, there is no net electric force acting on S and one needs consider only the

hydrodynamic force Fh. The equation of motion for S is thus given by

ro

Z p

0

Z r

a

d

dtur cos �� u� cos �þ Uð Þe�iot� �

2pr2 sin � dr d�

þ rc4

3pa3 þ rsV s

� �d

dtUe�iot� � ¼ Fh

ð25:25Þ

where rc and rs are, respectively, the mass densities of the particle core and the

polymer segments, and Vs is the total volume of the polymer segments per particle.

The hydrodynamic force Fh is given by

Fh ¼Z p

0

srr cos �� sr� sin �ð Þ2pr2 sin � d� ð25:26Þ

where srr and srs are, respectively, the normal and tangential components of the

hydrodynamic stress.

25.5 GENERALMOBILITY EXPRESSION

The dynamic electrophoretic mobility m(o) of a soft particle can be obtained by

solving Eqs. (25.19) and (25.20) with the result that

m(o) ¼ 2b2M2

9M1R1

g2

b2

Z b

a

1� r3

b3

� �G(r)dr �

Z 1

b

1� r3

b3

� �G(r)dr

�

� 2

3b2R1

Z b

a

�R2 � 3a

2b2b3M1

R3 � 2g2b2M2

3M3

� �

�f(M3 þM4br)cosh[b(r � a)]� (M4 þM3br)sinh[b(r � a)]g

� g2bM2

bM1M3

br cosh[b(r � a)]� sinh[b(r � a)]f gG(r)dr

þ 2M3R3

3b2M1R1

Z 1

a

1þ r3

2b3

� �G(r)dr

� 2R2

3g2R1

Z 1

b

1� igr1� igb

eig(r�b) � 1

� �G(r)dr

ð25:27Þ

GENERAL MOBILITY EXPRESSION 501

where

R1¼ 1�M1

M3

þ 3a

2M3b� 3G

2g2b2

� �2g2b2M2

3M1

þ 1þ 1� 3G

2g2b2

� �g2M3

b2M1

� �R3

ð25:28Þ

R2 ¼ M2(1� igb)igbM1 þM2(1� igb)

1þ g2

b2� 3g2a

2b2bM1

� �ð25:29Þ

R3 ¼ R2 � g2b2M2

M1

1� 2M1

3M3

þ a

bM3

� �ð25:30Þ

M1 ¼ 1þ a3

2b3þ 3a

2b2b3� 3a2

2b2b4

� �cosh[b(b� a)]

� 1� 3a2

2b2þ a3

2b3þ 3a

2b2b3

� �sinh[b(b� a)]

bb

ð25:31Þ

M2 ¼ 1þ a3

2b3þ 3a

2b2b3

� �cosh[b(b� a)]þ 3a2

2b2sinh[b(b� a)]

bb� 3a

2b2b3ð25:32Þ

M3 ¼ cosh[b(b� a)]� sinh[b(b� a)]

bb� a

bð25:33Þ

M4 ¼ sinh[b(b� a)]� cosh[b(b� a)]

bbþ ba2

3bþ 2bb2

3aþ 1

bbð25:34Þ

G ¼ g2[Vc(rc � ro)þ Vs(rs � ro)]6pbro

¼ 2(gb)2ff c(rc � ro)þ f s(rs � ro)g9ro

ð25:35Þ

f c ¼a

b

�3

ð25:36Þ

f s ¼Vs

4pb3=3ð25:37Þ


Vc ¼ 4p3a3 ð25:38Þ

For the limiting case of a! 0, the particle core vanishes, so a spherical soft par-

ticle becomes a spherical polyelectrolyte. In this case, Eq. (25.27) tends to

m(o)¼ 2b2

9R01f1� tanh(bb)=bbg

g2

b2

Z b

0

1� r3

b3

� �G(r)dr�

Z 1

b

1� r3

b3

� �G(r)dr

�

� 2(1� igb)f1þ (g=b)2g3b2R0

1f1� i(g=b)tanh(bb)gZ b

0

1� cosh(br)� sinh(br)=brcosh(bb)�sinh(bb)=bb

� G(r)dr

þ 2

3b2R01

(1� igb)f1þ (g=b)2gf1� i(g=b)tanh(bb)g�

(gb)2

3f1� tanh(bb)=bbg� Z 1

0

1þ r3

2b3

� �G(r)dr

� 2(1� igb)f1þ (g=b)2g3g2R0

1f1� i(g=b)tanh(bb)gZ 1

b

1� igr1� igb

eig(r�b)�1

� �G(r)dr ð25:39Þ

with

R01 ¼ 1þ 1� 3G0

2(gb)2

� �gb

� �2" #

(1� igb)f1þ (g=b)2g1� i(g=b)tanh(bb)

� (gb)2

3f1� tanh(bb)=(bb)g�

� G0

f1� tanh(bb)=(bb)g ð25:40Þ

G0¼ g2Vs(rs�ro)6pbro

¼2(gb)2f s(rs�ro)9ro

ð25:41Þ

25.6 APPROXIMATEMOBILITY FORMULA

In this section, we treat the practically important case where potential is not very

high so that dynamic relaxation effect is negligible. In this case, we have drel(r)¼ 0

or dnj(r)¼ 0. Consider first the case where potential is low and a! 0 and where

rfix¼ constant, which corresponds a uniformly charged spherical polyelectrolyte of

radius b.We thus obtain the following expression for the dynamic mobility of a spherical

polyelectrolyte:

APPROXIMATE MOBILITY FORMULA 503

m(o) ¼ rfixZb2R0

1

(1� igb)f1þ (g=b)2g1� i(g=b)tanh(bb)

� (gb)2

3f1� tanh(bb)=(bb)g�

þ rfix3Zk2R0

1

f1þ (g=b)2g(1� e�2kb)

f1� i(g=b)tanh(bb)g�

1� igb� ig=k1� ig=k

� �(coth(kb)� 1=kb)

þ (1� igb)(1þ 1=kb)(b=k)2 � 1

� �bk

coth(kb)� 1=kbcoth(bb)� 1=bb

� �� 1

� �ð25:42Þ

When jbjb� 1, kb� 1, jbj� jgj, and k�jgj, Eq. (25.42) becomes

m(o)¼ rfix

Zb2 1� igb� (gb)2

3�G0

� � 1� igb� (gb)2

3þ 2

3(1� igb)

bk

� �21þ b=2k1þ b=k

� �" #

ð25:43Þ

which, for o! 0 (b! l, g! 0, and Go! 0), further becomes Herman and Fujita’s

formula (21.54) for the static electrophoretic mobility of a spherical polyelectrolyte.

That is, Eq. (25.43) is the generalization of Herman and Fujita’s formula to the dy-

namic case.

Consider next the case where the electrolyte is symmetrical with a valence z andbulk concentration n (that is, we set M¼ 2, z1¼�z2¼ z, and n11 ¼ n12 ¼ n), andrfix(r)¼ rfix (= constant), and where

jbjb � 1; kb � 1; jbjd ¼ jbj(b� a) � 1; kd ¼ k(b� a) � 1; jbj � jgj; k � jgjð25:44Þ

In this case, the potential inside the polyelectrolyte layer can be approximated by

Eq. (21.47).

Equation (25.39) in this case gives

m(o) ¼ 2ereo3Z

1� igb1� igb� (g2b2=3)� G

� 1þ a3

2b3

� �co=km þ cDON=b

1=km þ 1=b

�

þ 1� igb� (g2b2=3)(1� a3=b3)

1� igb� (g2b2=3)� G

� rfixZb2

ð25:45Þ

Equation (25.45) is the required expression for the dynamic mobility of a soft parti-

cle, applicable for most practical cases. When o! 0 (b! l, g! 0, and G! 0),

Eq. (25.45) tends to Eq. (21.51) for the static case. When the polyelectrolyte layer


is very thin, that is, for a� b (or d� a), Eq. (25.45) reduces to

m(o) ¼ ereoZ

1� igb1� igb� (g2b2=3)� G

� co=km þ cDON=b

1=km þ 1=bþ rfixZb2

� ð25:46Þ

In the opposite limiting case of very thick polyelectrolyte layer, that is, for b� a,Eq. (25.45) becomes

m(o) ¼ 2ereo3Z

1� igb1� igb� (g2b2=3)� G

� co=km þ cDON=b

1=km þ 1=b

�

þ 1� igb� (g2b2=3)1� igb� (g2b2=3)� G

� rfixZb2

ð25:47Þ

Some results of the calculation of the magnitude and phase of the dynamic mobility

via Eq. (25.45) are given in Figs 25.2 and 25.3. All the calculations are performed

atrfix¼ ZeN¼ 4.8� 106 C/m3, 1/l¼ 1 nm,d¼ 10 nm,ro¼ 1� 103 kg/m3,rc¼ rs¼1.1� 103 kg/m3, fs¼ 0.1� (1� fc), Z¼ 0.89mPa s, and T¼ 298K. Figures 25.2 and

25.3 show the dependence of the magnitude (Fig. 25.2) and phase (Fig. 25.3) of

the dynamic mobility on the frequency o of the applied electric field for the case

where a¼ 1 mm for a 1:1 electrolyte of concentration of n¼ 0.01 and 0.1M. It is

seen that the o-dependence is negligible o/2p< 104 Hz and becomes appreciable

for o/2p> 104 Hz. That is, jmj is essentially equal to its static value (m(o) at o¼ 0)

FIGURE 25.2 Magnitude of the dynamic electrophoretic mobility m as a function of the

frequency o/2p of the applied oscillating electric field for a 1:1 electrolyte of concentration

n¼ 0.01 and 0.1M, calculated via Eq. (25.45) for a¼ 1 mm, d¼ 10 nm, rfix¼ ZeN¼ 4.8�106 C/m3 (which corresponds to N¼ 0.05M and Z¼ 1), 1/l¼ 1 nm, ro¼ 1� 103 kg/m3,

rc¼ rs¼ 1.1� 103 kg/m3, fs¼ 0.1� (1� fc), Z¼ 0.89mPa s, and T¼ 298K.

APPROXIMATE MOBILITY FORMULA 505

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

m(o) is zero for o/2p< 104 Hz and increases sharply for the frequency range

o/2p> 104Hz.

It is to be noted that in the limit of very high electrolyte concentrations,

Eq. (25.45) tends to a nonzero limiting value given by

m(o) ! m1(o) ¼ 1� igb� (g2b2=3)(1� a3=b3)

1� igb� (g2b2=3)� G

� rfixZb2

ð25:48Þ

which is not subject to the ionic shielding effect. This is a characteristic of electro-

phoresis of soft particles (see Eq. (21.62) for the static electrophoresis problem).

REFERENCES

1. R. W. O’Brien, J. Fluid Mech. 190 (1988) 71.

2. A. J. Babchin, R. S. Chow, and R. P. Sawatzky, Adv. Colloid Interface Sci. 30

(1989) 111.

3. A. J. Babchin, R. P. Sawatzky, R. S. Chow, E. E. Isaacs, and H. Huang, in: K. Oka (Ed.),

International Symposium on Surface Charge Characterization: 21st Annual Meet-ing of Fine Particle Society, San Diego, CA, August 1990, p. 49.

4. R. P. Sawatzky and A. J. Babchin, J. Fluid Mech. 246 (1993) 321.

5. M. Fixman, J. Chem. Phys. 78 (1983) 1483.

6. R. O. James, J. Texter, and P. J. Scales, Langmuir 7 (1991) 1993.

FIGURE 25.3 Phase of the reduced dynamic electrophoretic mobility m as a function of

the frequency o/2p of the applied oscillating electric field calculated via Eq. (25.45). Numer-

ical values used are the same as in Fig. 25.2.


7. C. S. Mangelsdorf and L. R. White, J. Chem. Soc., Faraday Trans. 88 (1992) 3567.

8. C. S. Mangelsdorf and L. R. White, J. Colloid Interface Sci. 160 (1993) 275.

9. H. Ohshima, J. Colloid Interface Sci. 179 (1996) 431.

10. M. Lowenberg and R. W. O’Brien, J. Colloid Interface Sci. 150 (1992) 158.

11. R. W. O’Brien, J. Colloid Interface Sci. 171 (1995) 495.


13. H. Ohshima, in: H. Ohshima and K. Furusawa (Eds.), Electrical Phenomena at Interfaces,2nd edition, Dekker, New York, 1998, Chapter 2.

14. H. Ohshima and A. S. Dukhin, J. Colloid Interface Sci. 212 (1999) 449.

15. A. S. Dukhin, H. Ohshima, V. N. Shilov, and P. J. Goetz, Langmuir 15 (1999) 3445.

16. A. S. Dukhin, V. N. Shilov, H. Ohshima, and P. J. Goetz, Langmuir 15 (1999) 6692.

17. A. S. Dukhin, V. N. Shilov, H. Ohshima, and P. J. Goetz, Langmuir 16 (2000) 2615.

18. J. A. Ennis, A. A. Shugai, and S. L. Carnie, J. Colloid Interface Sci. 223 (2000) 21.

19. J. A. Ennis, A. A. Shugai, and S. L. Carnie, J. Colloid Interface Sci. 223 (2000) 37.

20. H. Ohshima, Theory of Colloid and Interfacial Electric Phenomena, Elsevier/Academic

Press, 2006.


22. P. Debye and A. Bueche, J. Chem. Phys. 16 (1948) 573.

23. J. J. Hermans and H. Fujita, Koninkl. Ned. Akad. Wetenschap. Proc. Ser. B. 58

(1955) 182.

24. L. D. Landau and E. M. Lifshitz, Fluid Mechanics, Pergamon, Oxford, 1966, Sec. 24

REFERENCES 507

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Biophysical Chemistry of Biointerfaces (Ohshima/Biophysical Chemistry of Biointerfaces) || Dynamic Electrophoretic Mobility of a Soft Particle