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Electrophoresis of a Concentrated Dispersion of Nonrigid Particles Eric Lee, Chi-Hua Fu, and Jyh-Ping Hsu* Department of Chemical Engineering, National Taiwan University, Taipei, Taiwan 10617, Republic of China Received July 22, 2002. In Final Form: September 24, 2002 Electrophoresis is one of the most important analytical tools for the quantification of charged entities in a dispersing medium. Although the electrophoresis of rigid entities has been studied extensively in the literature, corresponding analysis on nonrigid entities is relatively limited. The present study focuses on the electrophoresis of a concentrated dispersion of nonrigid particles for the case when the effect of double layer polarization may be significant. In particular, the effects of the surface potential of the particle, the volume fraction of the particles, the ratio (viscosity of the medium inside the particle/viscosity of the medium outside the particle), and the double layer thickness are discussed. Various phenomena specific to nonrigid entities that are not observed for the case of rigid entities are observed in this study, which include, for example, the fact that the variation of electrophoretic mobility as a function of double layer thickness exhibits an inflection point. We show that, under the same conditions, the electrophoretic mobility of a nonrigid particle is larger than that of a rigid particle, and this is pronounced if the double layer is thin.
1. Introduction Electrophoresis is one of the widely adopted analytical tools for the quantification of charged entities in a dispersing medium. It has been investigated extensively both theoretically and experimentally. The former involves solving the electric, the flow, and the concentration fields simultaneously. Because the system is descried by coupled, nonlinear differential equations, obtaining an analytical solution is not an easy task, in general. The theoretical investigation was pioneered by Smoluchowski,1 where a concise relation between the surface potential and the applied electric field was derived under the conditions of thin double layer, low surface potential, and weak applied electric field. Various attempts have been made ever since to extend the original analysis of Smoluchowski by considering more general cases.2-6 While the electrophoresis of rigid entities has been investigated extensively both theoretically and experimentally, that for nonrigid entities is quite limited. Due to its specific physical properties, relevant studies on nonrigid entities were focused on mercury drops. For instance, the electrocapillary motion of a mercury drop was examined by Christiansen.7 Subsequent studies focused mainly on the adsorption phenomenon or the charge on the surface of a mercury drop through an experimental approach. Moreover, the structure of the double layer surrounding a mercury drop was also discussed, and the results were compared with that of a rigid entity.7-9 Craxford et al.8 performed a theoretical * To whom correspondence should be addressed. Fax: 886-223623040. E-mail:
[email protected]. (1) von Smoluchowski, M. Z. Phys. Chem. 1918, 92, 129. (2) Booth, F. Proc. R. Soc. London, Ser. A 1950, 203, 514. (3) Overbeek, J. Th. G. Adv. Colloid Sci. 1950, 3, 97. (4) O’Brien, R. W.; White, L. R. J. Chem. Soc., Faraday Trans. 2 1978, 74, 1607. (5) Kozak, M. W.; Davis, E. J. J. Colloid Interface Sci. 1989, 127, 497. (6) Kozak, M. W.; Davis, E. J. J. Colloid Interface Sci. 1989, 129, 166. (7) Ivanova, R. V.; Damaskin, B. B. Sov. Electrochem. 1976, 12, 532. (8) Gonzalez, S.; Carro, P.; Arevalo, A. Electrochim. Acta 1984, 29, 619.
analysis on the electrophoresis of a mercury drop. It was proposed that the force acting on a mercury drop is proportional to the product of the strength of the applied electric field and the number of charges within the double layer. Booth10 investigated the electrophoretic behavior of a mercury drop under the conditions of weak applied electric field, negligible double layer relaxation, and low surface potential. Levich and Frumkin11 considered the electrokinetic phenomenon of a mercury drop for the case of a thin double layer. The result of Levich and Frumkin differs appreciably from that of Booth. Levine and O’Brien12 pointed out that this is because the surface dipole was not considered in the latter. This phenomenon was taken into account by Levine13 in a subsequent analysis. Ohshima et al.14 modeled the behavior of a dilute dispersion of mercury drops. Adopting the numerical method of O’Brien and White,4 they evaluated the electrophoretic velocity, sedimentation potential, and conductivity of the dispersion. Ohshima15 analyzed the electrophoresis of a concentrated dispersion of mercury drops on the basis of the cell model of Kuwabara16 under the conditions of low surface potential and negligible overlapping of adjacent double layers. Baygents and Saville17 studied numerically the electrophoresis of a dilute dispersion of liquid drops under the conditions of weak applied electric field, arbitrary surface potential, and arbitrary double layer thickness; both conductive and nonconductive drops were (9) Hunter, R. J. Zeta Potential in Colloid Science; Academic Press: London, 1981. (10) Booth, F. J. Chem. Phys. 1951, 19, 1331. (11) Levich, V. G. Physicochemical Hydrodynamics; Prentice Hall: New York, 1962. (12) Levine, S.; O’Brien, R. N. J. Colloid Interface Sci. 1973, 43, 616. (13) Levine, S. Materials Processing in the Reduced Gravity Environment of Space; Materials Research Soc. Symp. Proc. Vol. 9; NorthHolland: New York, 1982; p 241. (14) Ohshima, H.; Healy, T. W.; White, L. R. J. Chem. Soc., Faraday Trans. 1984, 280, 1643. (15) Ohshima, H. J. Colloid Interface Sci. 1999, 218, 535. (16) Kuwabara, S. J. Phys. Soc. Jpn. 1959, 14, 527. (17) Baygents, J. C.; Saville, D. A. J. Chem. Soc., Faraday Trans. 1991, 87, 1883.
10.1021/la0262809 CCC: $25.00 © 2003 American Chemical Society Published on Web 02/05/2003
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electric, concentration, and flow fields. Applying Gauss’s law, it can be shown that the electric potential φ is described by
2
∇ φ)-
F
)-
∑j njzje
)-
∑j nj0zje exp(-zjeφ/kBT)
(1)
In this expression, ∇2 denotes the Laplacian, and F are respectively the dielectric constant and the space charge density, zj, nj, and nj0 are respectively the valence, the number concentration, and the bulk concentration of ionic species j, kB is the Boltzmann constant, and T is the absolute temperature. At steady state nj is described by
∇2nj +
Figure 1. Schematic representation of the cell model adopted in the present study where a particle of radius a is enclosed by a concentric liquid shell of radius b. An electric field in the z-direction, Ez, is applied, and the particle moves with velocity U.
considered. The case where the dissociation of electrolyte may be incomplete was also studied.18 In the present study, the electrophoresis of a concentrated dispersion of nonrigid particles is investigated theoretically by extending previous analyses to the case when the effect of double layer polarization can be significant. 2. Theory Let us consider a monodispersed, uniform distribution of charged nonrigid particles of radius a in an aqueous solution containing a z1/z2 electrolyte, with z1 and z2 being respectively the valences of cations and anions. The particle itself is electrolyte free. Kuwabara’s cell model15 is adopted to describe the system under consideration. Referring to Figure 1, the system is simulated by a representative particle enclosed by a concentric liquid shell of radius b. An electric field in the z-direction is applied. We assume the following: (a) The dielectric constant and the viscosity inside and outside the particle are independent of position. (b) Both the medium inside and the liquid outside the particle are incompressible, and the effect of surface tension can be neglected. (c) The movement of the particle is in the creeping flow regime, and it remains spherical. The applied electric field is weak relative to the electric field induced by the particle, and therefore, the present electrophoretic problem can be decomposed into two subproblems.4 In the first problem the particle moves in the absence of the applied electric field, and in the second problem it remains fixed under the application of the electric field. (d) The vorticity vanishes on the outer surface of the liquid shell. The spherical coordinates (r,θ,φ) with its origin located at the center of a representative particle are adopted. The symmetric nature of the problem under consideration, however, suggests that only the (r,θ) domain needs to be considered. The governing equations of the present problem, the so-called electrockinetic equations, comprise those for the (18) Baygents, J. C.; Saville, D. A. J. Colloid Interface Sci. 1991, 146, 9.
zje 1 (∇nj∇φ + nj∇2φ) - v∇nj ) 0 kBT Dj
(2)
where Dj is the diffusivity of ionic species j and v is the velocity of the liquid phase. For an incompressible fluid in the creeping flow regime, the flow field can be described by
η∇2v - ∇p - F∇φ ) 0, outside particle
(3)
ηI∇2vI - ∇pI ) 0, inside particle
(4)
∇v ) 0, inside and outside particle
(5)
where p and η are respectively the pressure and the viscosity of the liquid phase and the subscript I denotes the properties of the particle. 2.1. Electrical Field. Suppose that the applied electric field is weak relative to that induced by the particle. In this case, the electric potential can be expressed as the sum of the equilibrium potential φ1, which arises from the representative particle, and a perturbed potential φ2, which arises from the applied electric field. The former can be described by the Poisson-Boltzmann equation
∇2φ1 ) -
F1
N
∑ j)1
)-
zjenj0
( )
exp -
zjeφ1
(6)
kBT
where φ1 is the equilibrium potential. Since φ ) φ1 + φ2, the perturbed potential can be described by
(
∇2φ2 ) ∇2φ - ∇2φ1 ) -
)
F F1
(7)
2.2. Concentration Field. Under the influence of the applied electric field, the double layer surrounding a particle is no longer spherical. To account for this effect, a perturbed potential function gj is added to the total electrical potential, and the space charge density becomes
F)
(
∑i nizie ) ∑i zieni0 exp -
)
zie(φ1 + φ2 + gi) kBT
(8)
Because the applied electric field is weak, this expression can be approximated by
F=
( )[ zieφ1
∑i zieni0 exp - k T B
1-
zie
(φ2 + gi)
kBT
]
(9)
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Langmuir, Vol. 19, No. 7, 2003 3037
Substituting this expression into eq 2 yields
vanishes on the virtual surface. Therefore, the boundary conditions associated with the flow field are
∇2gi -
zie 1 ∇φ ∇g ) v∇φ1 kBT 1 i Di
(10)
∂ψ ∂ψI ) , r)a (22) ∂r ∂r ∂ 1 ∂ψI ∂ 1 ∂ψ µ ) µI , r)a (23) 2 ∂r r ∂r ∂r r2 ∂r ∂ψ 1 ) -Ur sin2 θ, r ) b ψ ) - Ur2 sin θ and 2 ∂r (24) ψ ) ψI and
( )
2.3. Flow Field. Equations 3-5 can be rewritten in a more compact manner by taking the curl on both sides of the equations and introducing the stream function representation for the velocity. We obtain
E4Ψ ) -
1 sin θ‚∇ × [F∇(φ1 + φ2)], outside particle η (11) E4ΨI ) 0, inside particle
The symmetric nature of the particle requires that
(12) ∂φ1 ∂φ2 ∂g1 ∂g2 ∂ψ ) ) ) ) )0 ∂r ∂r ∂r ∂r ∂r
The operator E is defined by E ) E E , where 4
E2 )
4
2
2
∂ sin θ ∂ 1 ∂ + 2 ∂r2 r ∂θ sin θ ∂θ
(
)
1 ∂Ψ r2 sin θ ∂θ
1 ∂Ψ vθ ) r sin θ ∂r
(25)
The governing equations can be rewritten in scaled form
(13) as
Note that, in terms of the stream function, the r- and the θ-components of the velocity v, vr and vθ, can be expressed respectively as
vr ) -
( )
2
∇*2φ/1 ) -
(κa) / 1 (n1 - n/2) (1 + R) φr
(26)
2
(14)
∇*2φ/2 ) -
(κa) 1 ((n/1 - n/2) - (exp(-φrφ/1) (1 + R) φr exp(Rφrφ/1))) (27)
(15)
2.4. Boundary Conditions. Regarding the boundary conditions of the electric field, we assume the following: (a) The surface of a particle remains at constant potential. (b) There is no net current across the virtual surface (r ) b). (c) The interior of a particle is not influenced by the perturbed potential φ2. (d) The potential on a virtual surface arises mainly from the applied electric field. On the basis of these considerations, the boundary conditions associated with the electric field are
φ 1 ) σa, r ) a
(16)
∂φ1 ) 0, r ) b ∂r
(17)
∂φ2 ) 0, r ) a ∂r
(18)
∂φ2 ) -EZ cos θ, r ) b ∂r
(19)
The boundary conditions associated with the concentration field are assumed as
∂gj ) 0, r ) a ∂r
(20)
gj ) -φ2, r ) b
(21)
These expressions imply that the particle is impermeable to ionic species, and the ionic concentration reaches the equilibrium value on a virtual surface. We assume further that both the velocity and the shear stress are continuous over the particle surface, there is no net flow across the virtual surface, and the vorticity
∇*2g/1 - φr∇*φ/1‚∇*g/1 ) φr2Pe1v*‚∇*φ/1 ∇*2g/2 E*4ψ* )
(28)
φr Pe2v*‚∇*φ/1
(29)
/ / / (κa)2 ∂g1 / ∂g2 / ∂φ1 n1 + Rn2 sin θ ∂θ ∂r* (1 + R) ∂θ
(30)
+
Rφr∇*φ/1‚∇*g/2
[(
)
2
) ]
E*4ψ/I ) 0 n/1
(31) g/1))
(32)
n/2 ) exp(Rφr(φ/1 + φ/2 + g/2))
(33)
)
exp(-φr(φ/1
+
φ/2
+
In these expressions, κ-1 ) (kBT/∑nj0(ezj)2)1/2, φr ) (ζaz1e/ kBT), UE ) σ2a/ηa, z2/z1 ) -R, n20 ) n10/R, E/z ) Eza/σa, Pej ) (zje/kBT)2/µDj, and ψ* ) ψ/UEa. The nature of the problem under consideration suggests that the method of separation of variables is applicable. Therefore, we assume φ1 ) φ1(r), φ/2 ) Φ2(r*) cos θ, g/1 ) G1(r*) cos θ, g/2 ) G2(r*) cos θ, and ψ* ) Ψ(r*) sin2 θ. On the basis of these expressions, it can be shown that eqs 34-37 and 38-41 become
L2Φ2 -
(κa)2 (exp(-φrφ/1) + R exp(Rφrφ/1))Φ2 ) (1 + R) (κa)2 (exp(-φrφ/1)G1 + R exp(Rφrφ/1)G2) (34) (1 + R)
( (
L2G1 - φr
L2G2 - Rφr
) ( ) ) ( )
/ ∂φ/1 ∂G1 -2Ψ ∂φ1 ) Pe1 ∂r ∂r ∂r r2
(35)
/ ∂φ/1 ∂G2 -2Ψ ∂φ1 ) Pe2 ∂r ∂r ∂r r2
[
]
∂φ/1 (κa)2 / / D Ψ)(n G + Rn2G2) ∂r* (1 + R) 1 1 4
(36) (37)
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The associated boundary conditions are
∂Φ2 ) 0, r* ) 1 ∂r*
(38)
∂Φ2 1 ) -E/x, r* ) ∂r* λ
(39)
∂G1 ) 0, r* ) 1 ∂r*
(40)
G1 ) -Φ2, r* )
1 λ
(41)
∂G2 ) 0, r* ) 1 ∂r* G2 ) -Φ2, r* ) ψI )
µ
(42)
1 λ
(43)
∂ψI ) 0, r* ) 0 ∂r*
1 ψ ) ψI ) - U*r*2, r* ) 1 2
(45)
∂ψ ∂ψI ) , r* ) 1 ∂r* ∂r*
(46)
( (
)
(
)
∂ 1 ∂ψI ∂ 1 ∂ψ ) µI , r* ) 1 2 ∂r* r* ∂r* ∂r* r*2 ∂r*
ψ)
(44)
)
1 1 1 1 ∂2 + ψ ) 0, r* ) r* ∂r*2 r*3 r*3 λ
(47)
(48)
2.5. Electrophoretic Mobility. The problem under consideration is decomposed into two subproblems, as mentioned earlier. In the first problem the particle moves with constant velocity in the absence of the applied electric field. In this case, the particle experiences a drag force F1 acting by the surrounding medium on it, where F1 ∝ U*. In the second problem the external electric field is applied, but the particle remains fixed. In this case the particle experiences an electric force F2 with F2 ∝ E/z . The sum of these forces vanishes at steady state, and the magnitude of the scaled electrophoretic velocity of the particle U/m can be expressed as
U/m )
F2 U* )E* F1
Figure 2. Variation of scaled mobility as a function of scaled double layer thickness κa for the case λ () a/b) ) 0.5 and µ/µI ) 0.01.
(49)
3. Results and Discussion The effects of the key parameters of the system under consideration, such as the thickness of the double layer surrounding a particle, κa, the surface potential of the particle, φr, the concentration of the particle measured by λ () a/b), and the ratio (viscosity of the medium outside the particle/viscosity of the medium inside the particle) µ/µI, on the electrophoretic mobility of a particle are examined through numerical simulation. The governing equations, eqs 34-37, subject to the boundary conditions, eqs 38-48, are solved numerically on the basis of a pseudospectrum method.19-22 (19) Lee, E.; Chu, J. W.; Hsu, J. P. J. Colloid Interface Sci. 1998, 205, 65.
Figure 3. Variation of scaled mobility as a function of scaled double layer thickness κa for the case λ () a/b) ) 0.5 and µ/µI ) 1.
Figures 2-4 show the variations of the scaled mobility of a particle as a function of the scaled double layer thickness κa at various viscosity ratios µ/µI. These figures reveal that the mobility of particle has an inflection point as κa varies, which is pronounced if the surface potential of the particle is high. Also, if κa is small, the higher the particle surface potential, the larger its mobility, and the reverse is true if κa is large, as pictured in Figures 5 and 6. This is because if κa is small (thick double layer), the overlapping of neighboring double layers has the effect of retarding the movement of a particle. Also, the distribution of the ionic species in a double layer is relatively unaffected by the applied electric field. In this case, the mobility of a particle is mainly determined by the level of its surface potential; the higher the surface potential, the greater the mobility. On the other hand, if κa is large, the distortion of the ionic cloud (i.e., double layer polarization) becomes significant. Because double layer polarization induces an electric field in the inverse direction to that of the applied (20) Lee, E.; Chu, J. W.; Hsu, J. P. J. Colloid Interface Sci. 1998, 209, 240. (21) Lee, E.; Yen, F. Y.; Hsu, J. P. Electrophoresis 2000, 21, 475. (22) Hsu, J. P.; Lee, E.; Yen, F. Y. J. Chem. Phys. 2000, 112, 6404.
Concentrated Dispersion of Nonrigid Particles
Figure 4. Variation of scaled mobility as a function of scaled double layer thickness κa for the case λ () a/b) ) 0.5 and µ/µI ) 100.
Langmuir, Vol. 19, No. 7, 2003 3039
Figure 7. Variation of scaled mobility as a function of scaled double layer thickness κa at various λ () a/b) for the case φr ) 4 and µ/µI ) 1.
Figure 8. Variation of scaled mobility as a function of κa at various viscosity ratios µ/µI for the case λ () a/b) ) 0.5 and φr ) 1. Figure 5. Variation of scaled mobility as a function of scaled surface potential φr for the case λ () a/b) ) 0.5, κa ) 0.1, and µ/µI ) 1.
Figure 6. Variation of scaled mobility as a function of scaled φr for the case λ () a/b) ) 0.5, κa ) 7, and µ/µI ) 1.
electric field, the movement of a particle is retarded, and the higher the particle surface potential, the more significant the degree of double layer polarization; that
is, the smaller the mobility. Figures 2-4 indicate that the larger the ratio µ/µI, the larger the value of κa at which the inflection of the mobility against κa curve occurs. The variation of the scaled mobility of a particle as a function of the scaled double layer thickness κa at various λ () a/b) is presented in Figure 7. Here, λ is a measure of the concentration of particles; the larger its value, the higher the concentration. Figure 7 reveals that, for a fixed κa, the larger the λ, the smaller the mobility of a particle. This is expected because a large λ implies that the interaction between neighboring particles is significant, which has the effect of retarding the movement of a particle. Note that if κa f ∞, the mobility of a particle becomes independent of λ. This is because κa f ∞ implies that the double layer surrounding a particle is infinitely thin. In this case, the effect of neighboring particles becomes unimportant. Figure 8 illustrates the variation of the scaled mobility as a function of the scaled double layer thickness κa at various viscosity ratios µ/µI. Here, a large µ/µI () 100) simulates the case when gas bubbles are dispersed in a liquid phase. On the other hand, a small µ/µI ()0.01) simulates the case where rigid particles are dispersed in a liquid phase. Figure 8 suggests that, for a fixed κa, the larger the value of µ/µI, the greater the mobility of a particle. That is, for a fixed double layer thickness, the mobility of a bubble is greater than that of a rigid particle. This is because the drag force experienced by a nonrigid
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the surrounding fluid makes the relative velocity difference across the particle-fluid interface reduced because the direction of the inner vortex and that of the outer vortex are the same on the interface. Therefore, the resulting shear stress, which is proportional to the shear rate (which itself is proportional to the relative velocity difference across the interface), is decreased accordingly. Because the overall drag force due to the fluid flow is the integration of the shear stress over the entire particle surface, this leads to a decrease in the drag force experienced by the particle; hence, its electrophoretic mobility increases. 4. Conclusion
Figure 9. Simulated streamlines for the case κa ) 10.0, µ/µI ) 1.0, and λ ) 0.5: (a) φr ) 1.0; (b) φr ) 4.0.
particle is smaller than that of a rigid particle. Figure 8 also indicates that the larger the κa, the greater the difference between the mobilities of particles of different viscosities. Figure 9 illustrates simulated streamlines at two levels of surface potential of the particles. This figure shows that vortices are formed both inside and outside a particle. On the right hemisphere of a cell, the orientation of the vortex inside the particle is clockwise while that of the vortex outside it is counterclockwise. The appearance of vortex flows inside a nonrigid particle is the main reason for it to have a greater electrophoretic mobility than the corresponding rigid particle, which does not have vortex flows inside. The vortex flow inside a particle induced by
In summary, we consider the electrophoresis of a concentrated dispersion of nonrigid particles by taking into account the effect of the double layer polarization. For the case when the applied electric field is weak, the electrophoretic behavior of the dispersion is influenced by the surface potential of the particles, the volume fraction of the particles, the ratio (viscosity of the medium inside the particle/viscosity of the medium outside the particle), and the double layer thickness. We conclude the following: (a) If the double layer surrounding a particle is thick, due to the fact that the overlapping of neighboring double layers and the distribution of ionic species is relatively uniform, the electrophoretic mobility of a particle is relatively uninfluenced by the applied electric field. In this case, the magnitude of the electrophoretic mobility is mainly determined by the level of the surface potential. (b) If the double layer thickness is on the order of the particle radius, the electric field induced by double layer polarization becomes significant, which has a negative effect on the electrophoretic motion of a particle. In this case, the electrophoretic mobility has an inflection point as the thickness of the double layer varies, and this phenomenon is pronounced, as the surface potential is high. (c) If the double layer thickness is further decreased, the effect of double layer polarization becomes more important. In this case the higher the surface potential, the smaller the mobility is. (d) As the volume fraction of particles increases, since the interaction between neighboring particles becomes significant, the mobility decreases. (e) The smaller the ratio (viscosity of the medium inside the particle/viscosity of the medium outside the particle), the larger the electrophoretic mobility. Which implies that, under the same conditions, the electrophoretic mobility of a nonrigid particle is larger than that of a rigid particle, and the thinner the double layer, the greater the difference between the two. Acknowledgment. This work is supported by the National Science Council of the Republic of China. LA0262809
