JOURNAL OF COLLOID AND INTERFACE SCIENCE 196, 316–320 (1997)ARTICLE NO. CS975220

NOTEElectrophoretic Mobility of a Spherical Particle in a Spherical Cavity

where Ç2 is the Laplace operator, r the space charge density, e thepermittivity of the liquid phase, N the number of ionic species, zj andThe electrophoretic mobility U of a spherical particle in an 0

j the valence and bulk concentration of ion species j , respectively, espherical cavity for the case of low electric field is estimatedthe elementary charge, and k and T the Boltzmann constant and absolutefor an arbitrary thick electrical double layer. We show that iftemperature, respectively. Following Zydney’s treatment (1 ) , F is de-the particle is uncharged and the cavity negatively charged,composed into F1 and F2 . The former denotes the electrical potentialthe deviation in the mobility based on the linearized Poisson–distribution in the absence of the applied electric field, and the latter,

Boltzmann equation, UL , can be serious even at a low electricalthe potential distribution outside the particle due to the presence of the

potential. In this case, the variation of the absolute deviation, electric field. Suppose that the surface potentials of particle and cavityÉU 0 ULÉ as a function of ka , k and a being, respectively, the are fixed at Fa and Fb , respectively. Then, we have (1 )reciprocal Debye length and particle radius, has a maximum.The ka at which the maximal absolute deviation occurs in-creases with l (particle size / cavity size ) , and the maximal ab- Ç

2F1 Å 0 ∑N

jÅ1

zjen 0j exp(0zjeF1 /kT )

e, [2]

solute deviation decreases with l. The latter increases with theabsolute surface potential of cavity. If the particle is positively Ç

2F2 Å 0, [3]charged and the cavity uncharged, UL is sufficiently accurate

F Å F1 / F2 , [4]if the electrical potential is low. q 1997 Academic Press

Key Words: electrophoresis; spherical particle; spherical cavity; F1 Å Fa at r Å a and F1 Å Fb at r Å b [5]boundary effects.

ÌF1

ÌuÅ ÌF2

ÌuÅ 0 at u Å 0 and u Å p. [6]

The last expression implies that F1 and F2 are symmetric about the z axis.INTRODUCTIONThe flow field is governed by the Navier–Stokes equation (1)

In a recent study, Zydney (1) analyzed the electrophoretic behavior ofa spherical particle in a spherical cavity under the condition that Henry’s

Çr£ Å 0, [7]law (2) is applicable; i.e., the applied electric field is weak so that the

hÇ2v Å Çp / rÇF, [8]distortion of the ion cloud surrounding the particle is negligible, and thedisturbance in electrical potential distribution due to fluid flow is insignifi-cant. The model proposed is applicable to an arbitrary thick electrical double

where v is the fluid velocity, Ç the gradient operator, p the pressure, andlayer at low electrical potentials. Although the system under consideration

h the fluid viscosity. Here, the particle moves with velocity U in the zis essentially imaginary, it provides valuable information about the effect

direction as a response to the applied electric field. For convenience, weof the presence of a rigid boundary on the electrophoretic behavior of a

assume that the particle is stationary, but the fluid moves in the reverseparticle. One of the interesting observations is that if an uncharged particle

direction. The boundary conditions associated with Eqs. [7] and [8] areis placed in a negatively charged cavity, the direction of motion of the

assumed to beparticle may reverse as the ratio ka (reciprocal Debye length 1 particleradius) varies. The Poisson–Boltzmann equation was linearized in the theo-retical development. This is realistic if the electrical potential is low, in £r Å 0 and £u Å 0, r Å a , [9]general. Questions arise as to what is the criterion for a low electrical

£r Å 0U cos u and £u Å U sin u, r Å b , [10]potential and whether this criterion is affected by the charged conditionson the interaction surfaces. These questions are discussed in this note.

where £r and £u are the components of v in the r and u directions, respec-tively.THEORY

The z component of the electrostatic force acting on the particle, FEz ,can be calculated byWe consider a nonconducting, spherical particle of radius a located

at the center of a spherical cavity of radius b . A uniform electric fieldE in the z direction is applied. Suppose that the movement of the particle

FEz Å **S

s(0ÇF)dS , [11]is slow so that the system is at a quasi-steady state and the distributionof ions follows the Boltzmann distribution. Then the electrical potential,F, is described by the Poisson–Boltzmann equation (3 )

where s denotes the surface charge density and dS the surface element ofthe particle. The z component of the hydrodynamic drag force, FHz , can be

Ç2F Å 0 r

eÅ 0 ∑

N

jÅ1

zjen 0j exp(0zjeF /kT )

e, [1]

evaluated by (4)

3160021-9797/97 $25.00Copyright q 1997 by Academic PressAll rights of reproduction in any form reserved.

AID JCIS 5220 / 6g38$$$101 12-22-97 07:45:43 coidal

317NOTE

FIG. 1. (a) Variation of the scaled mobility as a function of ka . Solid line: present result, U* Å Uh /ejbE ; dashed line: result based on the linearizedPoisson–Boltzmann equation, U*

L Å ULh /ejbE . (b) Variations of the relative percentage deviation É(U* 0 U*L )/U*É 1 100% and the absolute deviation

ÉU* 0 U*L É as functions of ka . Key: fa Å 0, fb (ÅFbz1e /kT ) Å 01, z1 being the valence of the cation, and l Å 0.833.

In summary, Eqs. [2] , [3] , [7] , and [8] need to be solved subject toFHz Å 2pa 2 F*p

0

(tru)rÅasin2uduEqs. [5] , [6] , [9] , and [10]. Here, a collocation method based on thepseudospectral method proposed by Canuto et al. (5) is adopted. Note thatEq. [13] is an implicit relation, and, therefore, a trial-and-error procedure/ *

p

0S0p / 2m

ÌVr

Ìr DrÅa

sin u cos uduG , [12]is necessary.

where (tru)a is the shear stress on the particle surface. The mobility of theRESULTSparticle can be determined by the condition that the net force exerted on it

vanishes at the steady state. We have

Figure 1a shows the variation of the scaled mobility U* Å Uh /eFb Eas a function of ka , k being the reciprocal Debye length. For the caseFEz / FHz Å 0. [13]

AID JCIS 5220 / 6g38$$$102 12-22-97 07:45:43 coidal

318 NOTE

FIG. 2. (a) : Variation of (ka)max as a function of l (Åparticle size/cavity size) . Solid line: (ka)max is the value of ka at which É(U* 0U*

L )/U*É diverges (or U* vanishes); dotted line: (ka)max is the value of ka at which ÉU* 0 U*L É is at the maximum. (b) Variation of ÉU* 0 U*

L É

as a function of l. Key: Same as for Fig. 1 except that l is allowed to vary.

in which the particle is uncharged and the cavity negatively charged, it from this figure, ÉU*L É will underestimate ÉU*É, in general. The varia-

is defined as tions of the percentage deviation, É(U* 0 U *L ) /U*É 1 100%, and the

absolute deviation, ÉU* 0 U *L É, as functions of ka are illustrated in

Fig. 1b. Note that although the former diverges at the point U* vanishes,k Å (ekT /e 2 /∑ n0j z 2

j )01 . [14]it still provides valuable information. First, it suggests that the differ-ence between U* and U*

L is negligible if ka is small (ß2) or largeThe scaled mobility based on the linearized Poisson–Boltzmann (®20) . Second, it reveals that the deviation in the electrophoretic mo-

bility can be significant even if U* is appreciable. As an example, theequation, U*L Å ULh /eFb E , is also presented in Fig. 1a. As can be seen

FIG. 3. Variation of ÉU* 0 U*L Émax as a function of the absolute scaled surface potential of cavity ÉfbÉ. Key: Same as for Fig. 1.

AID JCIS 5220 / 6g38$$$102 12-22-97 07:45:43 coidal

319NOTE

FIG. 4. (a) Variation of ka at which É(U* 0 U*L )/U*Émax 1 100% occurs, (ka)max , as a function of l (particle size/cavity size) . (b) Variation of

É(U* 0 U*L )/U*Émax 1 100% as a function of l. Key: fa Å 1, fb Å 0.

of l (particle size / cavity size ) are illustrated in Figs. 2a and 2b, respec-relative deviation of U*L from U* is on the order of 15% at ka Å 3.3

tively. The former shows that (ka )max increases with l, and the latter(or 5.2 ) , and U* is 0.15 (or 00.15) at this ka . Figure 1b also showsthat the absolute deviation, ÉU* 0 U*

L É, has a maximum, ÉU * 0 reveals that ÉU* 0 U*L Émax decreases with l. The variation of ÉU * 0

U*L Émax , at a certain ka , (ka )max . For convenience, (ka )max is also de- U*

L Émax as a function of the absolute scaled surface potential of cavityÉfbÉ (ÅÉFbÉz1e /kT , z1 being the valence of the cation ) is illustrated infined as the value of ka at which É(U* 0 U*

L ) /U*É diverges (or U*vanishes ) . The variations of (ka )max and ÉU* 0 U*

L Émax as functions Fig. 3. This figure suggests that the higher ÉfbÉ is, the greater the

AID JCIS 5220 / 6g38$$$102 12-22-97 07:45:43 coidal

320 NOTE

FIG. 5. Variation of É(U* 0 U*L )/U*Émax 1 100% as a function of the absolute scaled surface potential of particle ÉfaÉ (ÅFaz1e /kT ) , z1 being the

valence of the cation, for the case l Å 0.833. Key: fa Å 1, fb Å 0.

absolute deviation. It should be pointed out that the deviation of U*L REFERENCES

from U* can be serious even if the electrical potential is low. Forexample, if l Å 0.833, the relative percentage deviation, É(U* 0

1. Zydney, A. L., J. Colloid Interface Sci. 169, 476 (1995).U*L ) /U*É 1 100%, is on the order of 10%, even if ÉfbÉ is lower than

2. Henry, D. C., Proc. R. Soc. A 133, 106 (1931).0.5 (or ÉFbÉ ° 12.5 mV) .3. Hunter, R. J., ‘‘Foundations of Colloid Science.’’ Oxford Univ. Press,For the case in which a particle is positively charged and a cavity

Oxford, 1989.uncharged, U* never vanishes, and, therefore, É(U* 0 U*L ) /U*É is

4. Happel, J., and Brenner, H., ‘‘Low Reynolds Number Hydrodynam-finite. Figure 4 summarizes the variations of (ka )max and É(U* 0ics.’’ Martinus Nijhoff, The Hague, 1983.U*

L ) /U*Émax 1 100% as functions of l. Here, (ka )max is the value of5. Canuto, C., Hussaini, M. Y., Quarteroni, A., and Zang, T. A.,

ka at which É(U* 0 U*L ) /U *Émax 1 100% has the maximum. The

‘‘Spectral Methods in Fluid Dynamics.’’ Springer-Verlag, Newvariation of É(U* 0 U*L ) /U*Émax 1 100% as a function of the absolute

York, 1986.scaled surface potential of particle ÉfaÉ (ÅFaz1e /kT ) is shown in Fig.5. In general, the percentage deviation in the electrophoretic mobility Eric Leeis smaller than that if a particle is uncharged and a cavity negatively Jhih-Wei Chucharged, and the mobility based on the linearized Poisson–Boltzmann Jyh-Ping Hsu1

equation is satisfactory even if the surface potential of particle ismedium. Department of Chemical Engineering

We conclude that, depending on the charges on particle and cavity sur- National Taiwan Universityfaces, the deviation in the mobility based on the linearized Poisson–Boltz- Taipei, Taiwan 10617mann equation can be serious even if the electrical potential is low. In Republic of Chinaparticular, if a particle is uncharged and a cavity negatively charged, useof the linearized Poisson–Boltzmann equation can be inappropriate.

Received June 6, 1997; accepted October 9, 1997

ACKNOWLEDGMENT

1 To whom correspondence should be addressed. Fax: 886–2–3623040.This work was supported by the National Science Council of the RepublicE-mail: t8504009@ccms.ntu.edu.tw.of China.

AID JCIS 5220 / 6g38$$$102 12-22-97 07:45:43 coidal