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Electrophoresis of a concentrated dispersion of spherical particles covered
by an ion-penetrable membrane layer
Eric Lee
a, Kuang-Ting Chou
a, Jyh-Ping Hsu
b,∗aDepartment of Chemical Engineering, National Taiwan University, Taipei, Taiwan 10617 bDepartment of Chemical and Materials Engineering, National I-Lan University, I-Lan, Taiwan 26041
Received 21 April 2004; accepted 3 August 2004 Available online 11 September 2004
Abstract
The electrophoretic behavior of a concentrated dispersion of soft spherical particles is investigated theoretically, taking the effects of double-layer overlapping and double-layer polarization into account. Here, a particle comprises a rigid core and an ion-penetrable layer containing fixed charge, which mimics biocolloids and particles covered by artificial membrane layers. A cell model is adopted to simulate the system under consideration, and a pseudo-spectral method based on Chebyshev polynomials is chosen for the resolution of the governing electrokinetic equations. The influence of the key parameters, including the thickness of the double layer, the concentration of particles, the surface potential of the rigid core of a particle, and the thickness, the amount of fixed charge, and the friction coefficient of the membrane layer of a particle on the electrophoretic behavior of the system under consideration is discussed. We show that while the result for the case of a dispersion containing rigid particles can be recovered as the limiting case of a dispersion containing soft particles, qualitative behaviors that are not present in the former are observed in the latter.
2004 Elsevier Inc. All rights reserved.
Keywords: Electrophoresis; Concentrated spherical dispersion; Soft particle; Double-layer polarization; Double-layer overlapping
1. Introduction
Electrophoresis has found wide applications in numerous areas. For the case of an isolated rigid entity in an infinite fluid under conditions of low surface potential and weak applied electric field, an analytical expression for the elec-trophoretic mobility of the entity can be derived if the dou-ble layer surrounding it is either infinitely thin or infinitely thick[1]. Many attempts have been made to consider more general and/or realistic cases. These include, for example, the electrophoresis in which the effect of double-layer polar-ization/relaxation, the level of surface potential, the level of applied electric field, the effect of double-layer overlapping, the presence of a boundary, and the concentration of parti-cles may be significant[2–7]. In particular, the development
* Corresponding author. On leave from Department of Chemical
Engi-neering, National Taiwan University. Fax: +886-3-9353731.
E-mail address:jphsu@ntu.edu.tw(J.-P. Hsu).
of computing tools and software makes the analysis of a complicated electrophoresis problem feasible. While classic electrophoresis analysis focused mainly on rigid particles, particles of a nonrigid nature such as drops and entities cov-ered by a porous layer are also not uncommon in practice. The latter is of particular importance in biological science, where electrophoresis may be conducted on microorgan-isms and cells. Apparently, representing these entities by a rigid particle is inadequate, and adopting a more realistic representation is necessary. To this end, an ion-penetrable layer model or the so-called soft-particle model [8–11] is often used, where a particle comprises a rigid core and a porous or ion-penetrable membrane layer. The peripheral zone of the human blood cell, for instance, has a glyco-protein layer about 15 nm thick, which contains ionogenic functional groups. The influence of this layer on the elec-trophoresis of the associated entity is twofold. First, because the dissociation of functional groups yields a fixed charge
[12], the mobility of the entity may either increase or
de-0021-9797/$ – see front matter 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.jcis.2004.08.005
crease, depending on the sign of the fixed charge. Second, the hydrodynamic drag arising from the flow of liquid in the porous layer leads to smaller mobility. Wunderlich[9]and Levine et al.[10]concluded that if the effect of double-layer polarization is negligible, the mobility of a particle covered by an ion-penetrable layer is larger than that of the cor-responding rigid particle. Under conditions of low surface potential and negligible double-layer polarization, Ohshima
[13,14]was able to derive an analytical expression for the electrophoretic mobility of an ion-penetrable sphere, which was justified by experimental observations. In a study of the electrophoresis of an isolated, ion-penetrable sphere in an in-finite fluid, Saville[15]and Hill et al.[16]concluded that the effect of double-layer polarization is important. Applying a network method, Lopez-Garcia et al. solved numerically the electrophoresis of a soft particle in a static electric field[17]
and that in a dynamic electric field[18].
In this study, we extend previous analyses of the elec-trophoresis of particles covered by ion-penetrable layers to a general case, which takes account of the effects of particle concentration, double-layer polarization, and the overlap-ping of neighboring double layers. The unit cell model of Kuwabara[19]is adopted for the description of a concen-trated dispersion, and the pseudo-spectral method based on Chebyshev polynomials is used to solve the governing elec-trokinetic equations subject to the associated boundary con-ditions. A brief introduction to the pseudo-spectral method can be found in Lee et al.[20]. The influence of the key parameters, including the thickness of the double layer, the concentration of particles, the surface potential of the rigid core of a particle, and the thickness, the amount of fixed charge, and the friction coefficient of the membrane layer of a particle, on the electrophoretic behavior of the system under consideration is discussed.
2. Theory
Referring to Fig. 1, we consider a concentrated disper-sion of spherical particles each of which comprises a rigid core of radius a and a uniform ion-penetrable membrane layer of thickness d . The surface of the former is positively charged and the latter carries a fixed charge, which may arise from, for example, the dissociation of the functional groups. The unit cell model of Kuwabara[19]is adopted, where a dispersion is simulated by a representative cell of radius c composed of a particle of radius b (= a + d) and a liquid shell of thickness c− b. The spherical coordinates (r, θ, ϕ) are adopted with their origin located at the center of the parti-cle. An electric field E is applied, which is parallel to θ= 0, and U is the electrophoretic velocity of the particle. The liquid phase contains a z1: z2electrolyte, z1 and z2 being
respectively the valence of cations and that of anions. Let
z2= −αz1 and H = (b/c)3; the latter is a measure for the
volume fraction of particles.
Fig. 1. An electric field E is applied to a concentrated spherical disper-sion of particles, each of them comprises a rigid core of radius a and an ion-penetrable membrane layer of thickness d. The surface of the rigid core of a particle is positively charged and the membrane layer carries fixed charge. The system is simulated by a represented cell of radius c, which is composed of a particle of radius b (= a + d) and a liquid shell of thick-ness c− b. U is the electrophoretic velocity of the particle, and r and θ are the radial and the azimuthal angle of the spherical coordinates chosen.
The equations governing the present phenomenon com-prise those for the electric, concentration, and flow fields. The so-called electrokinetic equations are coupled nonlinear differential equations. On the basis of Gauss’s law, it can be shown that the electric potential φ is described by the Pois-son equation, (1) ∇2φ= −ρ+ ρfix ε , a < r < b, (2) ∇2φ= −ρ ε, b < r < c,
where ρ=jnjzje is the density of mobile ions,∇ is the
gradient operator, e is the elementary charge, nj and zj are
respectively the number concentration and the valence of the ionic species j , and ε and ρfixare respectively the
permittiv-ity of the liquid phase and the fixed charge denspermittiv-ity in the membrane layer. We assume that both ε and ρfix are
con-stant. The conservation of ionic species yields[20]
(3) ∇2n j+ zje kT(∇nj· ∇φ + nj∇ 2φ)− 1 Dj v· ∇nj= 0,
where Djis the diffusivity of the ionic species j , k and T are
respectively the Boltzmann constant and the absolute tem-perature, and v is the liquid velocity.
Suppose that the liquid phase is an incompressible New-tonian fluid with constant physical properties. For the present problem, the flow of liquid is in the creeping-flow regime, and, therefore, the flow field can be described by the continuity equation and the Navier–Stokes equations,
(4) ∇ · v = 0, (5) −∇p + η∇2v− ρ∇φ − γ v = 0, a < r < b, (6) −∇p + η∇2v− ρ∇φ = 0, b < r < c,
where p is the pressure, η is the viscosity of the liquid, and
γ is the frictional coefficient of the membrane layer.
Following the treatment of O’Brien and White[4], the electrical potential φ is expressed as the sum of the electri-cal potential in the absence of the applied electric field (or the equilibrium electrical potential), φ1, and that outside a
particle arising from the applied electric field, φ2; that is,
φ= φ1+ φ2. Also, the effect of double-layer polarization is
taken into account by expressing nj as
(7) nj= nj 0exp −zje(φ1+ φ2+ gj) kT ,
where gjis a perturbed potential and nj 0is the bulk number
concentration of ionic species j . It should be pointed out that the decomposition of φ into φ1 and φ2 is for convenience
only; it may not have actual physical meaning. Substituting Eq.(7)into Eqs.(1) and (2)and applying the relation φ=
φ1+ φ2, we obtain ∇2φ 1= − 2 j=1 zjenj 0 ε exp −zjeφ1 kT −ρfix ε , (8) a < r < b, (9) ∇2φ 1= − 2 j=1 zjenj 0 ε exp −zjeφ1 kT , b < r < c, (10) ∇2φ 2= ∇2φ− ∇2φ1 = − 2 j=1 zjenj 0 ε exp −zje(φ1+ φ2+ gj) kT − exp −zjeφ1 kT .
Also, substituting Eq.(7)into Eq.(3)yields
∇2g j− zje kT∇φ1· ∇gj= 1 Dj v· ∇φ + 1 Dj v· ∇gj (11) +zje kT∇φ2· ∇gj+ zje kT∇gj· ∇gj.
Taking the curl on both sides of Eqs.(5) and (6), apply-ing Eq.(4), and introducing the stream functionψ , it can be shown that Eqs.(5) and (6)become, respectively,
E4ψ− γ E2ψ= −sin θ η ∇ × ρ∇(φ1+ φ2) , (12) a < r < b, and (13) E4ψ= −sin θ η ∇ × ρ∇(φ1+ φ2) , b < r < c, where E4= E2E2, with (14) E2= ∂ ∂r2+ sin θ r2 ∂ ∂θ 1 sin θ ∂ ∂θ .
Note that in terms of ψ , the r-component of the liquid ve-locity, vr, and the θ -component of the liquid velocity, vθ,
can be expressed respectively as vr= −(1/r2sin θ )(∂ψ/∂θ )
and vθ= (1/r sin θ)(∂ψ/∂r).
3. Boundary conditions
Suppose that the surface of the rigid core of a parti-cle (r = a) is kept at a constant potential, both the elec-trical potential and the electric field are continuous on the membrane–liquid interface (r= b), and there is no net cur-rent across the surface of a cell (r = c). Therefore, the boundary conditions for the equilibrium potential φ1are
(15) φ1= ζa, r= a, (16) φ1|r=b−= φ1|r=b+, r= b, (17) ∂φ1 ∂r r=b− =∂φ1 ∂r r=b+ , r= b, (18) ∂φ1 ∂r = 0, r = c.
The boundary conditions for the potential φ2are assumed to
be (19) ∂φ2 ∂r = 0, r = a, (20) φ2|r=b−= φ2|r=b+, r= b, (21) ∂φ2 ∂r r=b− =∂φ2 ∂r r=b+ , r= b, (22) ∂φ2 ∂r = −Ezcos θ, r= c.
The first condition arises from the rigid core of a particle being nonconductive and impermeable to ionic species. The last condition implies that a Neumann-type condition is as-sumed for φ2[21].
We assume that the surface of the rigid core of a parti-cle is impermeable to ions, both the concentration of ionic species and its flux are continuous on the membrane–liquid interface, and the concentration of ionic species reaches the bulk value on the cell surface. Therefore, the boundary con-ditions for the concentration field are
(23) fj· n = fj· δr= 0, r = a, (24) nj|r=b−= nj|r=b+, r= b, (25) fj|r=b−= fj|r=b+, r= b, (26) nj= nj 0, r= c,
where fj is the concentration flux of ionic species j , n is the unit normal vector, and δr is the unit vector in the
r-direction.
We assume that the surface of the rigid core of a particle is no-slip. Also, the velocity of the liquid and its normal and tangential stresses are continuous on the membrane–liquid interface. On the outer boundary of a cell (the virtual sur-face), the vorticity of the fluid velocity vanishes, and there is no net fluid flow across it. Therefore, the boundary condi-tions for the flow field are
(27)
vr= U cosθ, vθ= −U sin θ, r = a,
(28)
σrθHr=b−= σrθHr=b+, σrrTr=b−= σrrTr=b+,
(29)
σrrEr=b−= σrrEr=b+, r= b,
(30)
∇ × v = 0, vr = 0, r = c.
In these expressions, σrθH is the rθ -component of the hydro-dynamic stress σH, σE
rr is the rr-component of the Maxwell
stress σE, and σrrT is the rr-component of the total stress σT= σH+ σE.
For a simpler treatment, subsequent discussions are based on scaled symbols. To this end, the governing equations and the associated boundary conditions are rewritten in di-mensionless form. The following symbols are chosen for the characteristic variables: the radius of the rigid core of a particle, a, the surface potential on the core, ζa, the
bulk number concentration of the ionic species j , nj 0, and
the electrophoretic velocity based on Smoluchowski’s the-ory when an electric field ζa/a is applied, UE= εζa2/ηa.
Also, we define the following symbols: the scaled radius,
r∗= r/a, the scaled number concentration of ionic species j , n∗j = nj/nj 0, the scaled electric field, EZ∗ = EZ/(ζa/a),
the scaled velocity, v∗= v/UE, the scaled terminal
veloc-ity, U∗ = U/UE, the scaled equilibrium potential, φ1∗ =
φ1/ζa, the scaled perturbed potentials, φ2∗ = φ2/ζa and
gj∗= gj/ζa, and the scaled stream function, ψ∗= ψ/UEa.
Applying the method of separation of variables, the two-dimensional dependent variables, φ2∗, g1∗, g∗2, and ψ∗, can make one-dimensional. For the present problem, the fol-lowing expressions are applicable[19]: φ∗2= Φ2(r) cos θ ,
g1∗= G1(r) cos θ , g∗2= G2(r) cos θ , ψ∗= Ψ (r) sin2θ . On
the basis of these expressions and the assumption that the applied electric field is weak relative to that induced by a particle, it can be shown that the resultant governing equa-tions for φ∗2, g1∗, g2∗, and ψ∗become
(31) ∇∗2φ∗ 1= − (κa)2 (1+ α)φr exp−φrφ1∗ − expαφrφ∗1 − Qfix, a < r < b, ∇∗2φ1∗= − (κa) 2 (1+ α)φr exp−φrφ1∗ − expαφrφ∗1 , (32) b < r < c, L2Φ2− (κa)2 (1+ α) exp−φrφ1∗ + α expαφrφ1∗ Φ2 (33) = (κa)2 (1+ α) exp−φrφ∗1 G1+ α exp αφrφ1∗ G2 , (34) L2G1− φr2 dφ1∗ dr∗ = Pe1φ 2 rvr∗ dφ∗1 dr∗, (35) L2G2+ αφr2 dφ1∗ dr∗ = Pe2φ 2 rvr∗ dφ∗1 dr∗, D4Ψ− (λa)2D2Ψ = −(κa) 2 1+ α n∗1G1+ n∗2G2 dφ∗ 1 dr∗ , (36) a < r < b, (37) D4Ψ = −(κa) 2 1+ α n∗1G1+ n∗2G2 dφ∗ 1 dr∗ , b < r < c.
In these expressions, Qfix= ρfixa2/εζa is the scaled total
amount of fixed charge, φr = z1eζa/kT is the scaled
sur-face potential of particle, κ−1= [εkT /nj 0(ezj)2]1/2 is
the Debye length, Pej = UEa/Dj , j = 1, 2, is the electric
Peclet number of ionic species j , and (λa)2= (γ a2/η) is
the dimensionless group for the friction coefficient of the membrane layer. The linear operators L2and D4are defined respectively by (38) L2≡ d 2 dr∗2+ 2 r∗ d dr∗ − 2 r∗2 and (39) D4= D2D2= d2 dr∗2− 2 r∗2 2 .
The associated boundary conditions become
(40) dΦ2 dr∗ = 0, r ∗= 1, (41) Φ2|r∗=b−/a= Φ2|r∗=b+/a, r∗= b a, (42) dΦ2 dr∗ r∗=b−/a =dΦ2 dr∗ r∗=b+/a , r∗=b a, (43) dΦ2 dr∗ = −E ∗ z, r∗= c a, (44) dGj dr∗ = 0, r ∗= 1, j = 1, 2, (45) Gj|r∗=b−/a= Gj|r∗=b+/a, r∗= b a, j= 1, 2, (46) dGj dr∗ r∗=b−/a =dGj dr∗ r=b+/a , r∗=b a, j= 1, 2, (47) G1= −Φ2 and G2= −Φ2, r∗= c a, (48) Ψ∗= −1 2r ∗2U∗ and dΨ∗ dr∗ = −r ∗U∗, r∗= 1, Ψ∗|r∗=b−/a= Ψ∗|r∗=b+/a and (49) dΨ∗ dr∗ r∗=b−/a =dΨ∗ dr∗ r∗=b+/a , r∗=b a, (50) d2Ψ∗ dr∗2 r∗=b−/a =d2Ψ∗ dr∗2|r∗=b+/a, r ∗=b a, d3Ψ∗ dr∗3 − (λa) 2dΨ∗ dr∗ r∗=b−/a = d3Ψ∗ dr∗3 r∗=b+/a , (51) r∗=b a, (52) Ψ∗= 0 and d2 dr∗2− 2 r∗2 Ψ∗= 0, r∗=c a.
According to O’Brien and White[4], the present problem can be divided into two subproblems. In the first problem, the particle moves in the absence of E, and in the second problem E is applied, but the particle is kept fixed. In the first problem the total force acting on a particle, F1, is
the total force acting on the particle, F2, is proportional to
the applied electric field. Therefore have F1= U∗f1 and
F2= E∗zf2, where f1is independent of U∗, and f2is
inde-pendent of EZ∗. The scaled electrophoretic mobility can be defined as µ∗m= U∗/E∗z= ηU/εςaEz. Because F1+ F2= 0
at steady state, µ∗mcan be expressed as
(53) µ∗m=U ∗ Ez∗ = − f2 f1 .
Since f1is independent of U∗and f2is independent of E∗Z,
we choose|U∗| = 1 and |Ez∗| = 1, for simplicity, and, there-fore, f1= F1 and f2= F2. The forces acting on a particle
F comprises the electrical force Fe and the hydrodynamic
force Fd. Let Fzibe the magnitude of F in problem i in the
z-direction, and Fezand Fdzbe respectively the components
of Feand Fdin the z-direction. We have
(54)
Fzi= (Fez+ Fdz)i= fi, i= 1, 2.
The electric force acting on a particle can be evaluated by (55)
Fe=
r=b
σE· n ds,
where σE≡ ε(EE −12E2I ) and E= −∇φ. F
ezcan be ex-pressed as (56) Fez= Fe· δz= r=b (σE· n) · δzds,
where δz is the unit vector in the z-direction. In terms of
scaled symbols, it can be shown that
Fez= 4 3π εζ 2 a r∗2 dφ1∗ dr∗ dΦ2 dr∗ (57) + 2r∗dφ∗1 dr∗ Φ2 r∗=b/a . Fdcan be evaluated by (58) Fd= r=b σH· n ds,
where σH≡ −pI + η[∇v + (∇v)T]. Fdzcan be expressed
as (59) Fdz= Fd· δz= r=b (σH· n) · δzds.
In terms of scaled symbols, it can be shown that
Fdz= 4 3π εζ 2 a r∗2 ∂ ∂r∗(D 2Ψ )− 2r∗(D2Ψ ) r∗=b/a (60) +4 3π εζ 2 a (κa)2 (1+ α)φr r∗2Φ2 exp−φrφ∗1 − expαφrφ1∗ r∗=b/a.
µ∗m can be evaluated by substituting the results obtained from Eqs.(57) and (60)into Eqs.(53) and (54)with
fi= r∗2∂φ∗1 ∂r∗ ∂Φ2 ∂r∗ + 2r∗∂φ1∗ ∂r∗ Φ2 r∗=b/a +r∗2 ∂∂r∗(D2Ψ )− 2r∗(D2Ψ ) r∗=b/a + (κa)2 (1+α)φr r∗2Φ2[exp(−φrφ∗1) − exp(αφrφ1∗)] r∗=b/a problem i , (61) i= 1, 2.
4. Results and discussions
The applicability of the numerical scheme adopted in this study is justified by reproducing the result of Lee et al.[21], where the electrophoresis of a concentrated rigid spherical dispersion is analyzed.Fig. 2illustrates the variation of the scaled mobility µ∗mas a function of (λa)2for the case when the membrane layer is uncharged. In this case, if (λa)2→ 0, the membrane layer is frictionless to the flow of liquid, and the present case should reduce to that of Lee et al. [21], which is justified inFig. 2. On the other hand, if (λa)2→ ∞, the membrane layer is impermeable to the liquid phase, and the present problem is equivalent to that for the case of rigid spherical dispersions of radius b. Note that µ∗m approaches a constant value when (λa)2→ ∞, as it should be. The qualitative behavior of µ∗mshown inFig. 2is similar to that observed by Saville[15].
The influences of the key parameters of the present prob-lem, including the thickness of double layer κa, the relative thickness of membrane layer d/a, the scaled friction coef-ficient of the membrane layer (λa)2, the concentration of particles estimated by H , and the scaled surface potential on
Fig. 2. Variation of scaled mobility µ∗mas a function of (λa)2for the case when Qfix= 0, κa = 1.0, φr= 4.0, H = 0.421875, and d/a = 0.5. Dashed
the rigid core of a particle φr, on the mobility of a particle
are examined through numerical simulation. For illustration, the following values are chosen: εr = 78.5, η = 8.904 ×
10−3poise, ρf = 0.99704 g/cm3, ρp= 1.05 g/cm3, z1= 1,
z2 = −1, T = 298.15 K, D1 = 1.962297 × 10−5cm2/s,
D2= 2.037051 × 10−5cm2/s. These values are based on
an aqueous KCl solution at T = 298.15 K. The nature and the amount of the fixed charge carried by a particle depends largely on the nature of its membrane layer, and we consider two general cases, Qfix= 0 and Qfix= 0.
4.1. Case 1. Qfix= 0
In this case, the membrane layer is free of fixed charge.
Fig. 3shows that if d/a= 0, µ∗mincreases with κa, in gen-eral, and depending on the magnitude of d/a, µ∗mmay have a local maximum, a local minimum, and an inflection point. This can be explained as follows. If κa is small, the dou-ble layer surrounding a particle is thick, and the overlap between adjacent double layers is significant. Also, because the spatial distribution of ionic species is relatively uniform, the influence of the applied electric field is weak. These fac-tors are disadvantageous to the movement of a particle. As
κa increases, the retardation arising from the presence of
double layer decreases, and the mobility increases accord-ingly. The decrease of µ∗m in the range 1 κa 10 arises from the effects of double-layer polarization and the flow field in the membrane layer. In the former, the deforma-tion of double layer yields an induced electric field, which is in the reverse direction from that of the applied electric field. In the latter, because the double layer is thinner than the membrane layer, the effective flow region in the mem-brane layer becomes larger, and the friction resistance due to the presence of the membrane layer is also large. There-fore, µ∗m decreases with the increase in κa. However, if κa exceeds about 10, the increase in the absolute value of the
Fig. 3. Variation of scaled mobility µ∗mas a function of κa at various d/a for the case when Qfix= 0, λa = 5, φr= 1.0, and H = 0.421875.
gradient of electrical potential near the core of a particle be-comes more important than the above two negative effects, and µ∗m increases with the increase in κa. Note that for a rigid particle, as κa increases, the double layer surrounding it becomes thin, the absolute value of the potential gradi-ent near its surface is large, which implies that the electric force acting on it is large, and its mobility becomes large. In the limit as κa→ ∞, the mobility approaches that predicted by the classic Smoluchovski result; that is, we must have
µ∗m→ 1 as κa → ∞. For the present case, although
situ-ations become more complicated, similar reasoning is still applicable; that is, if κa is sufficient large, µ∗m should in-crease with κa and approach a constant value as κa→ ∞.
Fig. 3also suggests that if κa is small, the thicker the mem-brane layer the larger the µ∗mis, but the reverse is true if κa is large. The former is because that although the overlapping of double layers yields a decrease in the mobility, the pres-ence of a membrane layer alleviates this effect. Therefore, it is possible that the thicker the membrane layer the larger the mobility is. The latter is because the thicker the membrane layer the greater is the friction resistance due to its presence. The result corresponding to d/a= 0 is similar to that of Lee et al.[21]for the case of a dispersion containing rigid par-ticles. This is expected because if d/a= 0, the membrane layer is absent.
The influences of the friction coefficient of the membrane layer of a particle, measured by λa, and the scaled surface potential on its rigid core, φr, on the scaled mobility of a
particle, µ∗m, are presented inFig. 4.Fig. 4a reveals that for a fixed κa, µ∗mdecreases with the increase in λa, which is expected since the larger the λa the greater the frictional re-sistance due to the presence of the membrane layer. Note that if λa has a medium value, the variation of µ∗mas a function of κa has both a local maximum and a local minimum, if
λa is sufficiently large, µ∗mhas a local maximum only; and
if λa is small, µ∗m increases monotonically with κa. These behaviors can be explained by the same reasoning as that employed in the discussion ofFig. 3.Fig. 4a reveals that the variation of µ∗m becomes insensitive to λa when it exceeds about 100. This is because if λa exceeds a certain value, the fluid in the membrane layer is almost stagnant, and a fur-ther increase in λa has a negligible influence on the flow field inside. The qualitative behavior of µ∗mpresented inFig. 4b is similar to that for the case of a rigid spherical disper-sion [21]. Again, the specific behavior of µ∗m in the range 1 κa 10 arises from the effect of double-layer polariza-tion.
4.2. Case 2. Qfix= 0
In this case, the membrane layer carries a fixed charge.
Fig. 5illustrates the variation of the scaled mobility µ∗mas a function κa at various values of d/a for two levels of Qfix.
This figure indicates that for the case where Qfixis small, if
κa is large, the thicker the membrane layer the smaller the µ∗m, as shown in Fig. 5a, but the reverse is true if Qfix is
(a)
(b)
Fig. 4. Variation of scaled mobility µ∗mas a function of κa at various λa
(a) and various φr (b) for the case when Qfix= 0, H = 0.421875, and
d/a= 0.5. φr= 1.0 in (a) and λa = 5 in (b).
large, as illustrated inFig. 5b. As the thickness of the mem-brane layer increases, both the friction of liquid flow inside and the total amount of fixed charge increase. The former is disadvantageous to the movement of a particle, but the latter is advantageous. The behavior of µ∗m shown in Fig. 5 re-flects the net result of these two competing factors. Note that inFig. 5a, although the influence of membrane friction is more important than that of total amount of fixed charge, µ∗m increases with d/a if κa is small. This is because if κa is suf-ficiently small, since the double layer extends to the outside of the membrane layer, the flow of liquid inside it becomes relatively insignificant, that is, the influence of friction is rel-atively unimportant to that of total amount of fixed charge.
The influence of the friction coefficient of the membrane layer of a particle, measured by λa, on its electrophoretic
be-(a)
(b)
Fig. 5. Variation of scaled mobility µ∗mas a function of κa at various d/a at
two values of Qfixfor the case when λa= 10, φr= 1.0, and H = 0.421875.
(a) Qfix= 0.5, (b) Qfix= 5.
havior is presented inFig. 6. Here, µ∗mis negative because the membrane layer is negatively charged.Fig. 6 indicates that for a fixed κa,|µ∗m| decreases with the increase in λa, and for a fixed λa, µ∗m exhibits a local minimum as κa varies. The former is expected since the larger the λa, the more significant the friction of the membrane layer, and the latter can be explained as follows. If κa is small, the over-lapping between neighboring double layers is significant and the spatial distribution of mobile ions is relatively uniform. Since both of these factors are disadvantageous to the move-ment of a particle,|µ∗m| is small. As κa increases, those fac-tors become less significant, and|µ∗m| increases accordingly. The rationale behind the presence of a local minimum in µ∗m for κa in the range 1 to 10 is similar to that for the case when
Fig. 6. Variation of scaled mobility µ∗mas a function of κa at various λa for
the case when Qfix= −10, φr= 1.0, H = 0.421875, and d/a = 0.5.
the membrane layer is free of fixed charge. That is, the po-larization of the double layer induces an electric field, which is in the reverse direction from that of the applied electric field. Also, when the double layer is thinner than the mem-brane layer, the friction of the latter becomes significant. It is interesting to note that after passing the local minimum, if
κa is increased further,|µ∗m| does not increase again as that
for the case the membrane layer is free of fixed charge. This is because the surface of the rigid core of a particle is pos-itively charged, and, therefore, the thinner the double layer the greater the absolute value of the potential gradient near the surface, which leads to a higher driving force for the par-ticle to move in the direction of the applied electric field. The negative fixed charge in the membrane layer of the par-ticle, however, tends to balance this driving force, and the net result is that|µ∗m| decreases with the increase in κa. 4.3. Effect of surface potential φr
The influence of the surface potential on the rigid core of a particle on its mobility is illustrated in Fig. 7, where the scaled mobility µ∗m is plotted against κa at various φr.
A comparison betweenFigs. 7 and 4b reveals that the in-fluence of φr on µ∗mfor the case when the membrane layer
is negatively charged is relatively less important than that for the case where it is free of fixed charge. This is be-cause, in the former, the induced electric field arising from double-layer polarization is in the same direction as that of the electric force acting on a particle due to the presence of the fixed charge in its membrane layer, and, therefore,|µ∗m| does not decrease significantly as φrincreases.
4.4. Effect of volume fraction H
The influence of the volume fraction of particle, esti-mated by H = (b/c)3, on its scaled mobility µ∗m is
illus-Fig. 7. Variation of scaled mobility µ∗mas a function of κa at various φrfor
the case when Qfix= −10, λa = 5, H = 0.421875, and d/a = 0.5.
Fig. 8. Variation of scaled mobility µ∗mas a function of κa at various H for
the case when Qfix= −10, λa = 5, φr= 1.0, and d/a = 0.5.
trated inFig. 8. The general behavior of µ∗m as H varies is similar to that of rigid particle[21]; that is, the larger the H the smaller the µ∗m. This is because, the higher the concen-tration of particles, the more significant the overlap between neighboring double layers and the drag acting on a particle arising from the neighboring particles, which are disadvan-tageous to the movement of a particle.Fig. 8also reveals that the κa at which µ∗mhas a local minimum increases with H . This is because the larger the H the closer the particle sur-face to cell sursur-face, and, therefore, a larger κa is necessary to achieve the condition that the boundary of double layer is within a cell, that is, to avoid double-layer overlap.
Fig. 9. Variation of scaled mobility µ∗mas a function of Qfixfor the case
when λa= 5, κa = 1.0, φr= 4.0, H = 0.125, and d/a = 0.5.
4.5. Effect of Qfix
Fig. 9shows the variation of the scaled mobility of a par-ticle µ∗m as a function of the scaled total amount of fixed charge in its membrane layer Qfix for the case where the
surface of its core is positively charged. This figure indicates that if Qfix exceeds about −5, µ∗m is positive, that is, the
particle moves in the direction of the applied electric field; otherwise, it moves in the reverse direction. This is expected because if Qfix is sufficiently small, the net charge carried
by a particle is negative.
5. Conclusions
The electrophoresis of a concentrated dispersion of soft, spherical particles is investigated theoretically under the conditions when the effects of double-layer overlap and double-layer polarization can be significant. The results of numerical simulations can be summarized as follow: (a) If the double layer surrounding a particle is either very thick or very thin, the thinner the double layer the greater the mo-bility of the particle. For a medium-thick double layer, the mobility may exhibit both a local maximum and a local min-imum as the thickness of the double layer varies. (b) The electrophoresis of a rigid spherical dispersion can be recov-ered as a special case of the present analysis by letting the friction coefficient of the membrane layer surrounding a par-ticle approach either a very large or very small value. (c) If
the membrane layer is free of fixed charge, it is thicker for smaller mobility. On the other, if the membrane layer car-ries fixed charge, increasing its thickness can have either a positive effect or a negative effect on the mobility, depend-ing on the parameters assumed. (d) If the membrane layer is free of fixed charge, the higher the surface potential of the rigid core of a particle the larger its mobility when the double layer is thick, but the reverse is true if it is thin. This phe-nomenon does not occur when the sign of the fixed charge in the membrane layer is the same as that on the surface of the rigid core. (e) If the sign of the fixed charge in the mem-brane layer is different from that on the surface of the rigid core, the direction of electrophoresis depends on the relative amount of the fixed charge on the surface of the rigid core and that in the membrane layer. (f) The more concentrate a dispersion is the smaller the mobility.
Acknowledgments
This work is supported by the National Science Council of the Republic of China.
References
[1] R.J. Hunter, Foundations in Colloid Science, vols. I and II, Clarendon, Oxford, 1989.
[2] J.Th.G. Overbeek, Adv. Colloid Sci. 3 (1950) 97. [3] F. Booth, Proc. R. Soc. London Ser. A 203 (1950) 514.
[4] R.W. O’Brien, L.R. White, J. Chem. Soc. Faraday Trans. 2 74 (1978) 1607.
[5] M.W. Kozak, E.J. Davis, J. Colloid Interface Sci. 127 (1989) 497. [6] M.W. Kozak, E.J. Davis, J. Colloid Interface Sci. 129 (1989) 166. [7] S. Levine, G.H. Neale, J. Colloid Interface Sci. 47 (1974) 520. [8] E. Donath, V. Pastushenko, Bioelectrochem. Bioenerg. 6 (1979) 543. [9] R.W. Wunderlich, J. Colloid Interface Sci. 88 (1982) 385.
[10] S. Levine, M. Levine, K.A. Sharp, D.E. Brooks, Biophys. J. 42 (1983) 127.
[11] K.A. Sharp, D.E. Brooks, Biophys. J. 47 (1985) 563.
[12] G.V. Seaman, in: D. Sergenor, N. Mac (Eds.), The Red Blood Cells, vol. 2, Academic Press, New York, 1975, pp. 1136–1229.
[13] H. Ohshima, J. Colloid Interface Sci. 163 (1994) 474. [14] H. Ohshima, J. Colloid Interface Sci. 228 (2000) 190. [15] D.A. Saville, J. Colloid Interface Sci. 222 (2000) 137.
[16] R.J. Hill, D.A. Saville, W.B. Russel, J. Colloid Interface Sci. 258 (2003) 56.
[17] J.J. Lopez-Garcia, C. Grosse, J. Horno, J. Colloid Interface Sci. 265 (2003) 327.
[18] J.J. Lopez-Garcia, C. Grosse, J. Horno, J. Colloid Interface Sci. 265 (2003) 341.
[19] S. Kuwabara, J. Phys. Soc. Jpn. 14 (1959) 527.
[20] E. Lee, J.W. Chu, J.P. Hsu, J. Colloid Interface Sci. 205 (1998) 65. [21] E. Lee, J.W. Chu, J.P. Hsu, J. Colloid Interface Sci. 209 (1999) 240.
