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Dynamic mobility of spherical colloidal particles in concentrated suspensions: moderately thick double layers
Colloids and Surfaces A: Physiochemical and Engineering Aspects 153 Ž1999. 477]480
Dynamic mobility of spherical colloidal particles in concentrated suspensions: moderately thick double layers Hiroyuki OhshimaU Institute of Colloid and Interface Science, Faculty of Pharmaceutical Sciences, Science Uni¨ ersity of Tokyo, 12 Ichigaya Funagawara-machi, Shinjuku-ku, Tokyo 162, Japan
Abstract A simple approximate expression is derived for the dynamic electrophoretic mobility m of spherical colloidal particles in concentrated suspensions in an applied oscillating electric field. This expression is applicable when the particles have low zeta potentials, very low relative permittivity and moderately thick double layers. It is found that the present approximation produces the correct dependence of the mobility magnitude on the particle volume fraction f. That is, the mobility magnitude exhibits a maximum when plotted as a function of f, in contrast to the case of infinitesimally thin double layers, in which case the mobility magnitude increases as f increases, showing no maximum. Q 1999 Elsevier Science B.V. All rights reserved. Keywords: Dynamic electrophoretic mobility; Spherical particle; Oscillating electric field; Concentrated suspension
1. Introduction Recently we have proposed a theory for the dynamic electrophoretic mobility m of spherical colloidal particles in concentrated suspensions and derived an accurate approximate mobility expression for the case where the particles have low zeta potentials and low relative permittivity w1x. This theory combines theories of the dynamic
U
Tel: q81 3 32606725 ext. 5060; fax: q81 3 32683045. E-mail address: [email protected] ŽH. Ohshima.
electrophoretic mobility of single spherical colloidal particles w2]9x and those of the static electrophoretic mobility in concentrated suspensions w10]12x. This theory is based on Kuwabara’s cell model w13x. This model assumes that each sphere of radius a is surrounded by a concentric spherical shell of an electrolyte solution, having an outer radius of b such that the particlersolution volume ratio in the unit cell is equal to the particle volume fraction f throughout the entire suspension, viz., f s Ž arb. 3 and that the fluid vorticity is zero at the outer surface of the cell. It has been shown that for very small k a Žwhere k is
0927-7757r99r$ - see front matter Q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 7 - 7 7 5 7 Ž 9 8 . 0 0 4 7 0 - 1
478
H. Ohshima r Colloids Surfaces A: Physiochem. Eng. Aspects 153 (1999) 477]480
mass density r o and viscosity h. All the particles move with the same velocity U expŽyiv t . in an applied oscillating electric field E expŽyiv t ., where v is the frequency and t is time. The dynamic electrophoretic mobility m of the particle is defined by, Us m E where Us < U < and Es < E <. We start with the following accurate analytic expression for the dynamic mobility m w1x: e r e 0z 1 h M Ž a. y G
ms
2 1 1q S1 q S2 3 3 2 Ž 1 q ark a.
½
=
5
Ž1.
with
Fig. 1. Magnitude of the reduced dynamic electrophoretic mobility < m
the Debye-Huckel parameter. the mobility mag¨ nitude < m < decreases as f increases but for infinitely large k a it increases with f w1x. For moderate and large Žbut finite. values of k a, the mobility magnitude exhibits a maximum when plotted as a function of f w1x. Fig. 1 shows < m < as a function of f for k as 50 Žexhibiting a maximum. and k as ` Žno maximum.. The limiting mobility formula for k as ` derived in a previous paper w1x Žgiven later by Eq. Ž12.., however, cannot exhibit the correct f dependence, since it shows no maximum so that it is not very useful for practical purposes. Even at k as 50, for example, this limiting formula cannot be applied. In the present paper, we derive a simple mobility formula which is applicable for moderately thick double layers. 2. Theory Consider a swarm of identical spherical colloidal particles of radius a having zeta potential z and mass density r p in an electrolyte solution of
M Ž a. s
H Ž a. q ig aŽ 1 y R . 3f Ž 1 y ig aR . 1yfq Ž g a. 2
H Ž a. s 1 y ig ay
Ž2.
Ž g a. 2 3
Ž3.
Ž 1 q igb . eyi gŽ bya. q Ž 1 y igb . e igŽ bya. Ž 1 q igb . eyi gŽ bya. y Ž 1 y igb . e igŽ bya.
Rs
Ž4.
2 2 Ž g a. Ž r p y r 0 . Gs 9r 0
S1 s
Ž5.
Ž k a. 2 1 y 2r3 1yf 3f P
q
g 2 Ž 1 q k aQ . q k 2 Ž 1 y ig aR . 3f Ž g2 q k2 . 1 y f q Ž 1 y ig aR . Ž g a. 2
½
5 Ž6.
2 Ž 1 q fr2.Ž k a. S2 s 9f2r3 Ž 1 y f . P
2
Ps cosh w k Ž by a.x y
Ž7. 1 sinh w k Ž b y a.x kb
s cosh w k aŽ fy1 r3 y 1 .x y
f1r3 sinh w k aŽ fy1r3 y 1 .x ka
Ž8.
H. Ohshima r Colloids Surfaces A: Physiochem. Eng. Aspects 153 (1999) 477]480
Qs s
1 y k b? tanh w k Ž b y a.x tanh w k Ž by a.x y k b 1 y k afy1 r3 ? tanh w k aŽ fy1 r3 y 1 .x tanh w k aŽ fy1 r3 y 1 .x y k afy1r3 ivr 0 s Ž i q 1. h
gs
(
as
2.5 1 q 2 exp Ž yk a.
(
vr 0 2h
Ž9. Ž 10. Ž 11.
where e r is the relative permittivity of the electrolyte solution and e 0 is the permittivity of a vacuum. In the limit of k aª `, Eq. Ž1. tends to ms
1 y ig aR 3f M Ž a. y G 4 1 y f q Ž 1 y ig aR . Ž g a. 2 Ž 12.
As mentioned in the Introduction, Eq. Ž12. shows no maximum when plotted as a function of f so that Eq. Ž12. cannot produce the correct f dependence for moderate and large k a values. In order to obtain a simple mobility formula applicable for moderate and large k a values, we employ an approximation method devised by Kozak and Davis w11x for the corresponding static electrophoresis problem. They showed that for moderate and large k a values, the overlap of electrical double layers of neighboring particles can be neglected. This approximation, as applied to our case, implies that expŽyk Ž by a.. in P ŽEq. Ž8.. and Q ŽEq. Ž9.. can be made equal to zero so that P and Q become ` and 1, respectively. Eq. Ž1. is thus simplified to ms
It can be shown that Eq. Ž13. exhibits the correct f dependence for moderate and large k a and is an excellent approximation. For very large Žbut finite. k a, Eq. Ž13. can be further simplified by approximating Ž2r3. 1 q 1r2Ž1 q ark a. 3 4 Žwhich expresses the distortion of the applied electric field due to the particles. by 1, Ž grk . 2 Ž1 q k a. by g 2 ark and Ž1 q Ž grk . 2 4 by 1, respectively, so that Eq. Ž13. becomes ms
2e r e 0 z 1 1q 3 3hŽ 1 y f . Ž 2 1 q ark a.
½
5
1 M Ž a. y G 4
e r e 0z 1 hŽ 1 y f . M Ž a . y G 4 =
e r e 0z hŽ 1 y f . =
479
1 y ig aRq g 2 ark 3f Ž 1 y ig aR . 1yfq Ž g a. 2
Ž 14.
This approximation is found to still work well and produce the correct f dependence. It is of interest to note that Eq. Ž14. differs from Eq. Ž12. for k as ` in that there is g 2 ark in the numerator of Eq. Ž14.. This implies that g 2 ark plays an essential role to account for the presence of the maximum in the < m < ]f curve. Finally we show that Eq. Ž14. gives the correct limiting form as f ª 0. The large k a limiting form for f ª 0 has been derived in a previous paper ŽEq. Ž98. in ref. w9x. as ms
e r e 0z 1 y ig a h H Ž a. y G 4 Ž 1 y igrk .
Ž 15.
ŽNote that Eq. Ž15. differs from O’Brien’s formula for low zeta potentials w2x by a factor 1rŽ1 y igrk ... Eq. Ž15. is transformed into ms
e r e 0 z 1 y ig aŽ 1 y 1rk a. q g 2 ark h H Ž a. y G 4 1 q Ž grk . 2 4
Ž 16.
On the other hand, in the limit of f ª 0, Eq. Ž14. tends to
2
=
1 y ig aRq Ž grk . Ž 1 q k a. 1 q Ž grk . 2 4 1 y f q Ž 3f. 2 Ž1 y ig aR . ga Ž 13.
ms
e r e 0 z 1 y ig aq g 2 ark h H Ž a. y G
Ž 17.
If 1rk a and Ž grk . 2 in Eq. Ž16. are neglected
480
H. Ohshima r Colloids Surfaces A: Physiochem. Eng. Aspects 153 (1999) 477]480
as compared with 1, then Eq. Ž16. becomes identical to Eq. Ž17.. References w1x H. Ohshima, J. Colloid Interf. Sci. 195 Ž1997. 137. w2x R.W. O’Brien, J. Fluid Mech. 190 Ž1988. 71. w3x A.J. Babchin, R.S. Chow, R.P. Sawatzky, Adv. Colloid Interf. Sci. 30 Ž1989. 111. w4x A.J. Babchin, R.P. Sawatzky, R.S. Chow, E. Isaacs, H. Huang, in: K. Oka ŽEd.., Int. Symp. on Surface Charge Characterization, 21st Annual Meeting Fine Particle Society, San Diego, CA, Fine Particle Society, Tulsa, OK, 1990, p. 49.
w5x R.P. Sawatzky, A.J. Babchin, J. Fluid Mech. 246 Ž1993. 321. w6x M. Fixman, J. Chem. Phys. 78 Ž1983. 1483. w7x C.S. Mangelsdorf, L.R. White, J. Chem. Soc. Faraday Trans. 88 Ž1992. 3567. w8x C.S. Mangelsdorf, L.R. White, J. Colloid Interf. Sci. 160 Ž1993. 275. w9x H. Ohshima, J. Colloid Interf. Sci. 179 Ž1996. 431. w10x S. Levine, G.H. Neale, J. Colloid Interf. Sci. 47 Ž1974. 520. w11x M.W. Kozak, E.J. Davis, J. Colloid Interf. Sci. 129 Ž1989. 166. w12x H. Ohshima, J. Colloid Interf. Sci. 188 Ž1997. 481. w13x S. Kuwabara, J. Phys. Soc. Jpn. 14 Ž1959. 527.
Dynamic mobility of spherical colloidal particles in concentrated suspensions: moderately thick double layers Hiroyuki OhshimaU Institute of Colloid and Interface Science, Faculty of Pharmaceutical Sciences, Science Uni¨ ersity of Tokyo, 12 Ichigaya Funagawara-machi, Shinjuku-ku, Tokyo 162, Japan
Abstract A simple approximate expression is derived for the dynamic electrophoretic mobility m of spherical colloidal particles in concentrated suspensions in an applied oscillating electric field. This expression is applicable when the particles have low zeta potentials, very low relative permittivity and moderately thick double layers. It is found that the present approximation produces the correct dependence of the mobility magnitude on the particle volume fraction f. That is, the mobility magnitude exhibits a maximum when plotted as a function of f, in contrast to the case of infinitesimally thin double layers, in which case the mobility magnitude increases as f increases, showing no maximum. Q 1999 Elsevier Science B.V. All rights reserved. Keywords: Dynamic electrophoretic mobility; Spherical particle; Oscillating electric field; Concentrated suspension
1. Introduction Recently we have proposed a theory for the dynamic electrophoretic mobility m of spherical colloidal particles in concentrated suspensions and derived an accurate approximate mobility expression for the case where the particles have low zeta potentials and low relative permittivity w1x. This theory combines theories of the dynamic
U
Tel: q81 3 32606725 ext. 5060; fax: q81 3 32683045. E-mail address: [email protected] ŽH. Ohshima.
electrophoretic mobility of single spherical colloidal particles w2]9x and those of the static electrophoretic mobility in concentrated suspensions w10]12x. This theory is based on Kuwabara’s cell model w13x. This model assumes that each sphere of radius a is surrounded by a concentric spherical shell of an electrolyte solution, having an outer radius of b such that the particlersolution volume ratio in the unit cell is equal to the particle volume fraction f throughout the entire suspension, viz., f s Ž arb. 3 and that the fluid vorticity is zero at the outer surface of the cell. It has been shown that for very small k a Žwhere k is
0927-7757r99r$ - see front matter Q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 7 - 7 7 5 7 Ž 9 8 . 0 0 4 7 0 - 1
478
H. Ohshima r Colloids Surfaces A: Physiochem. Eng. Aspects 153 (1999) 477]480
mass density r o and viscosity h. All the particles move with the same velocity U expŽyiv t . in an applied oscillating electric field E expŽyiv t ., where v is the frequency and t is time. The dynamic electrophoretic mobility m of the particle is defined by, Us m E where Us < U < and Es < E <. We start with the following accurate analytic expression for the dynamic mobility m w1x: e r e 0z 1 h M Ž a. y G
ms
2 1 1q S1 q S2 3 3 2 Ž 1 q ark a.
½
=
5
Ž1.
with
Fig. 1. Magnitude of the reduced dynamic electrophoretic mobility < m
the Debye-Huckel parameter. the mobility mag¨ nitude < m < decreases as f increases but for infinitely large k a it increases with f w1x. For moderate and large Žbut finite. values of k a, the mobility magnitude exhibits a maximum when plotted as a function of f w1x. Fig. 1 shows < m < as a function of f for k as 50 Žexhibiting a maximum. and k as ` Žno maximum.. The limiting mobility formula for k as ` derived in a previous paper w1x Žgiven later by Eq. Ž12.., however, cannot exhibit the correct f dependence, since it shows no maximum so that it is not very useful for practical purposes. Even at k as 50, for example, this limiting formula cannot be applied. In the present paper, we derive a simple mobility formula which is applicable for moderately thick double layers. 2. Theory Consider a swarm of identical spherical colloidal particles of radius a having zeta potential z and mass density r p in an electrolyte solution of
M Ž a. s
H Ž a. q ig aŽ 1 y R . 3f Ž 1 y ig aR . 1yfq Ž g a. 2
H Ž a. s 1 y ig ay
Ž2.
Ž g a. 2 3
Ž3.
Ž 1 q igb . eyi gŽ bya. q Ž 1 y igb . e igŽ bya. Ž 1 q igb . eyi gŽ bya. y Ž 1 y igb . e igŽ bya.
Rs
Ž4.
2 2 Ž g a. Ž r p y r 0 . Gs 9r 0
S1 s
Ž5.
Ž k a. 2 1 y 2r3 1yf 3f P
q
g 2 Ž 1 q k aQ . q k 2 Ž 1 y ig aR . 3f Ž g2 q k2 . 1 y f q Ž 1 y ig aR . Ž g a. 2
½
5 Ž6.
2 Ž 1 q fr2.Ž k a. S2 s 9f2r3 Ž 1 y f . P
2
Ps cosh w k Ž by a.x y
Ž7. 1 sinh w k Ž b y a.x kb
s cosh w k aŽ fy1 r3 y 1 .x y
f1r3 sinh w k aŽ fy1r3 y 1 .x ka
Ž8.
H. Ohshima r Colloids Surfaces A: Physiochem. Eng. Aspects 153 (1999) 477]480
Qs s
1 y k b? tanh w k Ž b y a.x tanh w k Ž by a.x y k b 1 y k afy1 r3 ? tanh w k aŽ fy1 r3 y 1 .x tanh w k aŽ fy1 r3 y 1 .x y k afy1r3 ivr 0 s Ž i q 1. h
gs
(
as
2.5 1 q 2 exp Ž yk a.
(
vr 0 2h
Ž9. Ž 10. Ž 11.
where e r is the relative permittivity of the electrolyte solution and e 0 is the permittivity of a vacuum. In the limit of k aª `, Eq. Ž1. tends to ms
1 y ig aR 3f M Ž a. y G 4 1 y f q Ž 1 y ig aR . Ž g a. 2 Ž 12.
As mentioned in the Introduction, Eq. Ž12. shows no maximum when plotted as a function of f so that Eq. Ž12. cannot produce the correct f dependence for moderate and large k a values. In order to obtain a simple mobility formula applicable for moderate and large k a values, we employ an approximation method devised by Kozak and Davis w11x for the corresponding static electrophoresis problem. They showed that for moderate and large k a values, the overlap of electrical double layers of neighboring particles can be neglected. This approximation, as applied to our case, implies that expŽyk Ž by a.. in P ŽEq. Ž8.. and Q ŽEq. Ž9.. can be made equal to zero so that P and Q become ` and 1, respectively. Eq. Ž1. is thus simplified to ms
It can be shown that Eq. Ž13. exhibits the correct f dependence for moderate and large k a and is an excellent approximation. For very large Žbut finite. k a, Eq. Ž13. can be further simplified by approximating Ž2r3. 1 q 1r2Ž1 q ark a. 3 4 Žwhich expresses the distortion of the applied electric field due to the particles. by 1, Ž grk . 2 Ž1 q k a. by g 2 ark and Ž1 q Ž grk . 2 4 by 1, respectively, so that Eq. Ž13. becomes ms
2e r e 0 z 1 1q 3 3hŽ 1 y f . Ž 2 1 q ark a.
½
5
1 M Ž a. y G 4
e r e 0z 1 hŽ 1 y f . M Ž a . y G 4 =
e r e 0z hŽ 1 y f . =
479
1 y ig aRq g 2 ark 3f Ž 1 y ig aR . 1yfq Ž g a. 2
Ž 14.
This approximation is found to still work well and produce the correct f dependence. It is of interest to note that Eq. Ž14. differs from Eq. Ž12. for k as ` in that there is g 2 ark in the numerator of Eq. Ž14.. This implies that g 2 ark plays an essential role to account for the presence of the maximum in the < m < ]f curve. Finally we show that Eq. Ž14. gives the correct limiting form as f ª 0. The large k a limiting form for f ª 0 has been derived in a previous paper ŽEq. Ž98. in ref. w9x. as ms
e r e 0z 1 y ig a h H Ž a. y G 4 Ž 1 y igrk .
Ž 15.
ŽNote that Eq. Ž15. differs from O’Brien’s formula for low zeta potentials w2x by a factor 1rŽ1 y igrk ... Eq. Ž15. is transformed into ms
e r e 0 z 1 y ig aŽ 1 y 1rk a. q g 2 ark h H Ž a. y G 4 1 q Ž grk . 2 4
Ž 16.
On the other hand, in the limit of f ª 0, Eq. Ž14. tends to
2
=
1 y ig aRq Ž grk . Ž 1 q k a. 1 q Ž grk . 2 4 1 y f q Ž 3f. 2 Ž1 y ig aR . ga Ž 13.
ms
e r e 0 z 1 y ig aq g 2 ark h H Ž a. y G
Ž 17.
If 1rk a and Ž grk . 2 in Eq. Ž16. are neglected
480
H. Ohshima r Colloids Surfaces A: Physiochem. Eng. Aspects 153 (1999) 477]480
as compared with 1, then Eq. Ž16. becomes identical to Eq. Ž17.. References w1x H. Ohshima, J. Colloid Interf. Sci. 195 Ž1997. 137. w2x R.W. O’Brien, J. Fluid Mech. 190 Ž1988. 71. w3x A.J. Babchin, R.S. Chow, R.P. Sawatzky, Adv. Colloid Interf. Sci. 30 Ž1989. 111. w4x A.J. Babchin, R.P. Sawatzky, R.S. Chow, E. Isaacs, H. Huang, in: K. Oka ŽEd.., Int. Symp. on Surface Charge Characterization, 21st Annual Meeting Fine Particle Society, San Diego, CA, Fine Particle Society, Tulsa, OK, 1990, p. 49.
w5x R.P. Sawatzky, A.J. Babchin, J. Fluid Mech. 246 Ž1993. 321. w6x M. Fixman, J. Chem. Phys. 78 Ž1983. 1483. w7x C.S. Mangelsdorf, L.R. White, J. Chem. Soc. Faraday Trans. 88 Ž1992. 3567. w8x C.S. Mangelsdorf, L.R. White, J. Colloid Interf. Sci. 160 Ž1993. 275. w9x H. Ohshima, J. Colloid Interf. Sci. 179 Ž1996. 431. w10x S. Levine, G.H. Neale, J. Colloid Interf. Sci. 47 Ž1974. 520. w11x M.W. Kozak, E.J. Davis, J. Colloid Interf. Sci. 129 Ž1989. 166. w12x H. Ohshima, J. Colloid Interf. Sci. 188 Ž1997. 481. w13x S. Kuwabara, J. Phys. Soc. Jpn. 14 Ž1959. 527.