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Dynamic electrophoretic mobility of spherical colloidal particles in concentrated suspensions: approximation of nonoverlapping double layers Hiroyuki Ohshima * Faculty of Pharmaceutical Sciences, Institute of Colloid and Interface Science, Science Uni6ersity of Tokyo, 12 Ichigaya, Funagawara-machi, Shinjuku-ku, Tokyo 162 -0826, Japan

Abstract An approximate expression for the dynamic electrophoretic mobility m of spherical colloidal particles in concentrated suspensions in an oscillating electric field is derived on the basis of Kuwabara’s cell model. The analysis is constrained to low zeta potentials and nonoverlapping electrical double layers of adjacent particles. The dynamic mobility depends on the frequency v of the applied electric field and the particle volume fraction f as well as on the reduced particle radius ka and the zeta potential z. The approximation of nonoverlapping double layers is found to be valid for ka\50 (where k is the Debye–Huckel parameter and a is the particle radius). Even for ka\ 10 the approximation works well for fB0.15. © 1999 Elsevier Science B.V. All rights reserved. Keywords: Dynamic electrophoretic mobility; Spherical particle; Concentrated suspension

1. Introduction In recent papers [1 – 3], we have derived the general expression for the dynamic electrophoretic mobility of spherical colloidal particles in concentrated suspensions on the basis of Kuwabara’s cell model [4]. This model assumes that each particle is surrounded by a virtual cell such that the particle/solution volume ratio in a unit cell is equal to the particle volume fraction throughout the entire system and the fluid vorticity is zero at * Tel.: +81-3-32604272 ext. 5060; fax: +81-3-32683045. E-mail address: [email protected] (H. Ohshima)

the outer surface of the unit cell. The obtained dynamic mobility, m, depends on the frequency of the applied electric field and the particle volume fraction, f as well as on the reduced particle radius ka (where k is the Debye–Huckel parameter and a is the particle radius) and the zeta potential, z [1–3]. In the limit of f 0, the obtained mobility expression reduces to those obtained for dilute suspensions [5–12], while in the limit v 0, it reduces to the static mobility of spherical particles in concentrated suspensions [13–16]. In this paper we derive a simple dynamic mobility expression for the dynamic mobility in concentrated suspensions by neglecting the over-

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H. Ohshima / Colloids and Surfaces A: Physicochem. Eng. Aspects 159 (1999) 293–297

lapping of the electrical double layers of adjacent particles. This approximation has been employed for the static problem by Kozak and Davis [15]. They derived a simple approximate expression for the static mobility of spherical particles in concentrated suspensions by ignoring overlapping of the electrical double layers of adjacent particles and showed that this approximation is quite good for practical ceases.

2. General mobility expression Consider a swarm of identical spherical colloidal particles of radius a and mass density rp in a liquid containing a general electrolyte. All the particles move with the same velocity U exp(− ivt) in an applied oscillating electric field E exp(− ivt), where v is the frequency and t is time. The dynamic electrophoretic mobility m(v) of the particle is defined by U= mE,

(1)

where U= U and E= E . We employ a cell model [4] in which each sphere is surrounded by a concentric spherical shell of an electrolyte solution, having an outer radius of b such that the particle/solution volume ratio in this unit cell is equal to the particle volume fraction f throughout the entire suspension, viz., f = (a/b)3.

(2)

The mobility m depends not only on the zeta potential z of the particle and ka (k, Debye– Huckel parameter) but also on the frequency v and the particle volume fraction f. The origin of the spherical polar coordinate system (r, u, 8) is held fixed at the center of the particle and the polar axis (u = 0) is set parallel to E. Let the electrolyte be composed of N ionic mobile species of valence zj, and n j be the concentration (number density) of the jth ionic species in the electroneutral solution. The main assumptions in our analysis are as follows: (i) The applied field E is weak so that U is proportional to E and terms of higher order in E may be neglected. (ii) The slipping plane (at which the liquid velocity relative to the particle

becomes zero) is located on the particle surface (at r= a). (iii) No electrolyte ions can penetrate the particle surface. (iv) The relative permittivity op of the particle is much smaller than that of the liquid or(op or) So that op is practically equal to zero. (v) In the absence of the applied electric field the particle surface is uniformly charged with a surface charge density s and has a corresponding zeta potential z. (vi) The fluid vorticity is zero at the outer surface of the unit cell [4]. On the basis of these assumptions we have solved the electrokinetic equations for this system, that is, the Navier–Stokes equation and Poisson–Boltzmann equation and finally arrived at the following general expressions for the dynamic mobility of spherical particles in concentrated suspensions: m=

2 3g [M(a) −G] 2

−

&

b

a

[M(a)− M(r)]G(r)dr

n

g 2b 2 (0) r el (b)Y(b) 3ha

(3)

with M(r)=

! !

" "

(1+ igb)e − ig(b − a)H(r)− (1− igb)e + ig(b − a)H*(r) 3f 1+ H(a) (1+igb)e − ig(b − a) − (ga)2 3f 1+ H*(a) (1−igb)e + ig(b − a) (4) (ga)2 M(a)=

R=

H(a)+ iga(1− R) 3f 1− f+ (1−igaR) (ga)2

(5)

(1+ igb)e − ig(b − a) + (1− igb)e + ig(b − a) (1+igb)e − ig(b − a) + (1− igb)e + ig(b − a)

H(r)= (1−igr)exp[ig(r− a)]−

g 2r 3 3a

H*(r)= (1+igr)exp[−ig(r−a)]− H(a)= 1− iga −

(ga)2 3

H*(a)= 1+ iga −

(ga)2 3

g 2r 3 3a

(6) (7) (8) (9) (10)

H. Ohshima / Colloids and Surfaces A: Physicochem. Eng. Aspects 159 (1999) 293–297

G= g=

2(ga)2Dr . 9r0

'

'

ivr0 = (i +1) h

(11) (12)

(13)

zj ec (0) fj (r) e 2 dc (0) N 2 % n j z j exp − hkT dr j = 1 kT r (14)

N

r (0) el (r)= % zj en j exp − j=1

sa kb×cosh[k(b−a)]−sinh[k(b−a)] oro0 k(b−a)cosh[k(b−a)]−(1−k 2ab)sinh[k(b−a)] (21)

vr0 2h

Dr =rp −r0 G(r)= −

=

zj ec (0) kT

(15)

where r0 is the mass density of the liquid, h is the viscosity, e is the elementary electric charge, k is Boltzmann’ s constant, T is the absolute temperature, c (0)(r) is the equilibrium double layer potential outside the particle, and fj (r) and Y(r) are, respectively, related to the deviation of the electrochemical potential of the jth ionic species dmj (r) and that of the electric potential dc(r) by dmj (r)= − zj efj (r)E cos u

(16)

dc(r)= − Y(r)E cos u

(17)

=

kaf−1/3 cosh[ka(f−1/3−1)]−sinh[kaf−1/3] sa oro0 ka(f−1/3−1)cosh[ka(f−1/3−1)]−{1−(ka)2f−1/3} sinh[ka(f 1/3−1)] (22)

From Eq. (18) we have further derived the following simple approximate expression applicable for low z, which does not involve numerical integration: m=

!

oro0 h(1− f) [M(a) −r]

× − +

2k 2 3g 2

&

b

a

[M(a) −M(r)] 1 +

n

f 2(ka)2 (0) c (b) 1 + 2 9f

(18)

where

S1 =

k=

N 1 % z 2j e 2n j oro0kT j = 1

="

!

g 2(1+kaQ)+ k 2(1−igaR) 3f (1−igaR) (g 2 + k 2) 1− f+ (ga)2

2(1+ f/2)(ka)2 9f 2/3(1−f)P

P= cos h[k(b− a)]−

(24)

(25) 1 sin h[k(b− a)] kb

= cos h[ka(f − 1/3 − 1)] −

Q=

1/2

f 1/3 sin h[ka(f 1/3 − 1)] ka

(26)

1−kb× tan h[k(b− a)] tan h[k(b− a)]− kb 1− kaf − 1/3 × tan h[k(f − 1/3 − 1)] , tan h[ka(f − 1/3 − 1)]− kaf − 1/3

(27)

2.5 1+ 2 exp(−ka)

(28)

(20)

=

where k is the Debye – Huckel parameter and o0 is the permittivity of a vacuum. Note that in the present approximation the zeta potential z is related to s, ka and f, as follows:

a=

z= c (0)(a)

<

1 (ka)2 − 2/3 3f P 1−f

+

with a kb ×cos h[k(b −r)] − sin h[k(b− r)] c (0)(r) =z r kb ×cos h[k(b −a)] − sin h[k(b − a)] (19)

n (23)

S2 =

a 3 dc (0) dr 2r 3 dr

"

oro0z 1 2 1 1+ S1 + S2 h M(a)− G 3 (1+ a/ka)3

For low z, Eq. (3) becomes m=

295

Eq. (23) is found to be an excellent approximation to Eq. (18) with negligible errors [1].

H. Ohshima / Colloids and Surfaces A: Physicochem. Eng. Aspects 159 (1999) 293–297

296

3. Approximation of nonoverlapping double layers

where

The second term on the right-hand side of Eqs. (3), (18) and (23), which involves the excess charge rel(b) or the potential c (0)(b) on the outer surface of the unit cell, arises from the overlapping of the double layers of adjacent particles. If the double layer overlapping can be neglected, then Eq. (3) is simplified to

M1 = S1(w=0)

m=

2 3g [M(a) −G] 2

&

b

[M(a) −M(r)]G(r)dr

(29)

a

Eq. (18) to oro0 h(1− f)[M(a)− G] 2k 2 b a 3 dc (0) − 2 {M(a) −M(r)} 1 + 3 dr 3g a 2r dr

m=

&

n (30)

and Eq. (23) to

!

m(v) H(a)− G S1 = m(v= 0) M(a)− G M1

(34)

O’Brien and Rider [17] have given the definition of the dynamic mobility in concentrated suspensions, which is consistent with the dynamic mobility that is actually measured in electroacoustic devices. According to this definition the dynamic mobility in a concentrated suspension is given by Eq. (1) under the condition that there is no macroscopic pressure gradient in the suspension. It can be shown that the macroscopic pressure gradient 9P in the suspension is expressed as

)

dY(r) E. dr r = b

"

(35)

(31)

In order to see the applicability of the assumption of the nonoverlapping double layers, we compare the results calculated from Eqs. (18) and (31). In general as f increases and/or ka decreases this approximation becomes worse. Some examples are shown in Figs. 1 and 2, in which we plot the magnitude of the dynamic mobility m as a function of the particle volume fraction f for several values of the dimensionless frequency r0a 2v/h and the dimensionless particle radius ka= 10 (Fig. 1) and 50 (Fig. 2). All the calculations are performed at Dr/r0 =0.1. Note that the maximum attainable volume fraction for a swarm of identical spheres is 0.74 (fmax =0.74) (Eq. (13)). It is seen that this approximation is quite good. For ka = 50, this approximation is valid for almost all attainable values of f(f B 0.6). Even for ka=10 it is valid for fB 0.15. In the static limit v 0, Eq. (31) becomes

!

In this approximation the ratio of the dynamic mobility to the static mobility in concentrated suspensions is thus simply given by

9P = r (0) el (b)

2o o z 1 1 m= r 0 1+ S 3h M(a) − G 2(1 + a/ka)3 1

m(v=0)=

(33)

"

2oro0 1 1 1+ M1 3h H(a) −G 2(1 + a/ka)3 (32)

Fig. 1. Magnitude of the reduced dynamic electrophoretic mobility m /ms (where ms oro0z/h is Smoluchowski’s mobility) as a function of the particle volume fraction f for various values of the reduced frequency r0a 2v/h( g 2 a 2). Solid lines: calculated form Eq. (18) at Dr/r0 =0.1 and ka=10. Dotted lines: calculated from Eq. (31)which is based on the assumption of nonoverlapping doubles. Note that the dynamic mobility at any v (calculated from Eqs. (18) and (31)) approaches the static mobility as f increases.

H. Ohshima / Colloids and Surfaces A: Physicochem. Eng. Aspects 159 (1999) 293–297

297

Kuwabara’s cell model is consistent with the definition of O’Brien and Rider [17].

References

Fig. 2. Same as Fig. 1 but for ka= 50. Again note that the dynamic mobility at any v (calculated from Eqs. (18) and (31)) approaches the static mobility as f increases.

The pressure gradient involves the flow field as well as the charge density in the solution. Eq. (35) states that in the cell model the contribution of the flow field to the macroscopic pressure gradient vanishes due to the zero vorticity condition and only the excess charge density on the outer cell surface (r (0) el (b)) contributes to the macroscopic pressure gradient. Therefore, if further r (0) el (b) is negligible, as in the present approximation, then the macroscopic pressure gradient becomes zero and the dynamic mobility calculated via

.

[1] H. Ohshima, J. Colloid Interf. Sci. 195 (1997) 137. [2] H. Ohshima, Colloids Surf. A Physicochem. Eng. Asp. 149 (1999) 5. [3] H. Ohshima, Colloids Surf. A Physicochem. Eng. Asp. (in press). [4] S. Kuwabara, J. Phys. Soc. Jpn. 14 (1959) 527. [5] R.W. O’Brien, J. Fluid Mech. 190 (1988) 71. [6] A.J. Babchin, R.S. Chow, R.P. Sawatzky, Adv. Colloid Interf. Sci. 30 (1989) 111. [7] A.J. Babchin, R.P. Sawatzky, R.S. Chow, E. Isaacs, H. Huang, International symposium on surface charge characterization, in: K. Oka (Ed.), 21st Annual Meeting of the Fine Particle Society, Fine Particle Society, San Diego, CA, Tulsa, OK, 1990, pp. 49. [8] R.P. Sawatzky, A.J. Babchin, J. Fluid Mech. 246 (1993) 321. [9] M. Fixman, J. Chem. Phys. 78 (1983) 1483. [10] C.S. Mangelsdorf, L.R. White, J. Chem. Soc. Faraday Trans. 88 (1992) 3567. [11] C.S. Mangelsdorf, L.R. White, J. Colloid Interf. Sci. 160 (1993) 275. [12] H. Ohshima, J. Colloid Interf. Sci. 179 (1996) 431. [13] S. Levine, G.H. Neale, J. Colloid Interf. Sci. 47 (1974) 520. [14] M.W. Kozak, E.J. Davis, J. Colloid Interf. Sci. 127 (1989) 497. [15] M.W. Kozak, E.J. Davis, J. Colloid Interf. Sci. 129 (1989) 166. [16] H. Ohshima, J. Colloid Interf. Sci. 188 (1997) 481. [17] P.F. Rider, R.W. O’Brien, J. Fluid Mech. 257 (1993) 607.