Langmuir 1996, 12, 657-667
657
Particle Interactions in Diffusiophoresis: Axisymmetric Motion of Multiple Spheres in Electrolyte Gradients Huan J. Keh* and Shih C. Luo Department of Chemical Engineering, National Taiwan University, Taipei, 106-17 Taiwan, Republic of China Received July 10, 1995X
A semianalytical study of the diffusiophoretic motion of a finite string of dielectric spheres in a solution of symmetric electrolyte with a constant concentration gradient along the line through their centers is presented. The spheres may differ in radius and in zeta potential, and they are allowed to be unequally spaced. Also, the spheres can be either freely suspended in the fluid or linked by infinitesimally thin rods. The thickness of the electrical double layers surrounding the particles is assumed to be small relative to the radius of each particle and to the gap width between any two neighboring particles, but the effect of the polarization of the mobile ions in the diffuse layer is taken into account. A slip velocity of fluid and normal fluxes of solute ions at the outer edge of the thin double layer are used as the boundary conditions for the fluid domain outside the diffuse layer. Using a collocation method along with these boundary conditions, a set of electrokinetic governing equations is solved in the quasisteady state situation and the particle interaction effects are calculated for various cases. It is found that particles with the same zeta potential will interact with one another, unlike the no-interaction results obtained in previous investigations assuming that the double layer is infinitesimally thin. For most situations, the particle interaction among the spheres is a complicated function of the properties of the spheres and ions, and it no longer varies monotonically with the extent of separation for some cases. No general rule can make an adequate prediction for such complicated phenomena.
Introduction 1
Diffusiophoresis refers to the motion of colloidal particles in response to a gradient of solute concentration in a solution. If the solute is nonelectrolytic, the particle migrates toward or away from regions of higher solute concentration, depending on whether the solute is attracted to or repelled from the particle surface.2,3 However, diffusiophoresis in an electrolytic fluid is more involved and is explained as an electrokinetic motion of a charged particle resulting from two effects: (1) chemiphoresis due to the nonuniform adsorption of counterions and depletion of co-ions in the electrical double layer over the particle surface, which is similar to diffusiophoresis in nonionic media, and (2) electrophoresis due to the macroscopic electric field generated by the concentration gradient of the electrolyte and the difference in mobilities of the two ions of the electrolyte. In an unbounded solution of a symmetric electrolyte with a constant concentration gradient ∇n∞, the migration velocity of a nonconducting particle can be written as
U(0) ) A∇n∞
(1)
where the diffusiophoretic mobility4
A)
[
( )]
ζ kT 1 D2 - D1 4kT Zeζ + ln cosh 4πη Ze n∞(0) D2 + D1 Zeζ 4kT
particle center 0, D1 and D2 are the diffusion coefficients of the anion and cation, respectively, Z is the absolute value of the valences of the ions, e is the charge of a proton, /4π is the fluid permittivity, η is the fluid viscosity, and kT is the thermal energy. The first and second terms in brackets represent the electrophoretic and chemiphoretic components, respectively. Equation 2 shows that the diffusiophoretic mobility of a dielectric particle is independent of the particle size and shape. However, its validity is based on the assumptions that the double layer surrounding the particle is very thin in comparison with the particle dimension and the polarization (relaxation) effect of the ions in the diffuse layer is negligible (this would require that the magnitude of ζ be small). When the double-layer distortion from equilibrium was taken as a perturbation, Prieve and Roman5 obtained a numerical solution for the diffusiophoretic migration of a spherical particle of radius a in concentration gradients of symmetric electrolytes (KCl or NaCl) which was applicable to a broad range of ζ and κa, where κ-1 is the Debye screening length (equal to (kT/ (8πZ2e2n∞))1/2). Recently, analytical expressions for the diffusiophoretic velocity of a sphere with a thin but polarized double layer have also been derived.6,7 The result for the diffusiophoretic mobility in a symmetric electrolyte solution is7
(2)
In this expression, ζ is the zeta potential of the particle, n∞(0) is the undisturbed electrolyte concentration at the * To whom correspondence should be addressed. X Abstract published in Advance ACS Abstracts, January 1, 1996. (1) Dukhin, S. S.; Derjaguin, B. V. In Surface and Colloid Science; Matijevic, E., Ed.; Wiley: New York, 1974; Vol. 7. (2) Anderson, J. L.; Lowell, M. E.; Prieve, D. C. J. Fluid Mech. 1982, 117, 107. (3) Anderson, J. L.; Prieve, D. C. Langmuir 1991, 7, 403. (4) Prieve, D. C.; Anderson, J. L.; Ebel, J. P.; Lowell, M. E. J. Fluid Mech. 1984, 148, 247.
0743-7463/96/2412-0657$12.00/0
A)
[(
)
8kT ζ kT 1 2 D2 - D1 + b1 - b2 + (1 + 4πη Ze n∞(0) 3 D2 + D1 3Zeζ
Zeζ (4kT )] (3)
b1 + b2) ln cosh where
(5) Prieve, D. C.; Roman, R. J. Chem. Soc., Faraday Trans. 2 1987, 83, 1287. (6) Pawar, Y.; Solomentsev, Y. E.; Anderson, J. L. J. Colloid Interface Sci. 1993, 155, 488. (7) Keh, H. J.; Chen, S. B. Langmuir 1993, 9, 1142.
© 1996 American Chemical Society
658
b1 )
Langmuir, Vol. 12, No. 3, 1996
D2
(
Keh and Luo
D1 a2 + aβ22 - 3 aβ12 - 2aβ11 + 2 D 2a ∆(D2 + D1) 2
)
2β12β21 - 2β11β22 (4a) b2 )
D1
(
D2 a2 + aβ11 - 3 aβ21 - 2aβ22 + D 2a ∆(D2 + D1) 1 2
)
2β12β21 - 2β11β22 (4b) and
∆)
1 2 (a + aβ11 + aβ22 + β11β22 - β12β21) a2
(5)
The relaxation coefficients β11, β12, β21, and β22 in eqs 4 and 5 are defined by eqs 12. It could be found from eqs 2 and 3 that A/ζ can be either positive or negative and the polarization effect of the diffuse layer is to retard the diffusiophoretic velocity of a particle.7 The reason for this retardation is that the relaxation transport of the diffuse ions inside the double layer decreases the local electrolyte concentration gradient along the particle surface. If κa is large and Z|ζ| is small, the interaction between the diffuse ions and the particle surface is weak and the polarization of the double layer is also weak. In the limit of
f0 (Ze|ζ| 2kT )
(κa)-1 exp
(6)
β11/a ) β12/a ) β21/a ) β22/a ) 0, b1 ) D2/2(D2 + D1), b2 ) D1/2(D2 + D1), and eq 3 reduces to eq 2. In addition to the studies on a spherical particle, the diffusiophoretic mobilities of spheroidal8 and cylindrical7,9 particles have also been determined using the thin-layer polarization model. In practical applications of diffusiophoresis, collections of particles are encountered. Thus, it is important to study how the presence of its neighbors affects the movement of a particle. In the limiting situation as given by eq 6, the normalized velocity field of the immense fluid that is dragged by a particle during diffusiophoresis is the same as for electrophoresis of the particle;10 thus, the particle interaction effects in electrophoresis, which have been examined extensively in the past,10-15 can be utilized to interpret those in diffusiophoresis. One of the common conclusions from these investigations is that the diffusiophoretic or electrophoretic velocity of each particle is unaffected by the presence of the others as long as all of the particles in the suspension have the same zeta potential. When the polarization effect of diffuse ions in the double layers surrounding the particles is considered, the particle interaction behavior in diffusiophoresis can be substantially different from that in electrophoresis, due to the fact that the particle size and some other unique factors are involved in each transport mechanism.7 Recently, the electrophoretic motion of multiple spheres with thin (8) Keh, H. J.; Huang, T. Y. J. Colloid Interface Sci. 1993, 160, 354. (9) Keh, H. J.; Huang, T. Y. Colloid Polym. Sci. 1994, 272, 855. (10) Anderson, J. L. Annu. Rev. Fluid Mech. 1989, 21, 61. (11) Reed, L. D.; Morrison, F. A. J. Colloid Interface Sci. 1976, 54, 117. (12) Chen, S. B.; Keh, H. J. AIChE J. 1988, 34, 1075. (13) Keh, H. J.; Yang, F. R. J. Colloid Interface Sci. 1990, 139, 105. (14) Keh, H. J.; Yang, F. R. J. Colloid Interface Sci. 1991, 145, 362. (15) Acrivos, A.; Jeffrey, D. J.; Saville, D. A. J. Fluid Mech. 1990, 212, 95.
but polarized diffuse layers in an arbitrary configuration was examined using a boundary collocation technique.16,17 Through the use of the thin-layer polarization model, the axisymmetric diffusiophoresis of multiple spheres in nonelectrolyte gradients was also investigated.18 However, the effect of particle interactions on diffusiophoresis in electrolytic media has not been studied for the case of particles surrounded by polarized double layers. The aim of this paper is to examine the diffusiophoretic motion of a chain of spheres in a constant gradient of a symmetric electrolyte along the line of their centers. The spheres may differ in radius and in surface properties, and they are allowed to be unequally spaced. The polarized double layers are assumed to be thin compared to the radius of each particle and to the gap thickness between any two neighboring particles. The quasisteady state equations of conservation applicable to the system are solved by using the boundary collocation technique, and the particle velocities are obtained with good convergence for various cases. In the limiting case of eq 6, our results are equivalent to those obtained in previous analyses for electrophoresis.13,14 For most situations, the particle interaction among the diffusiophoretic spheres is a complicated function of the properties of the spheres and the ions. Diffusiophoresis of an Isolated Dielectric Particle in a Symmetric Electrolyte In this section, we consider the diffusiophoretic motion of a nonconducting particle of arbitrary shape in an unbounded solution of a symmetrically charged, binary electrolyte. The particle is charged uniformly on the surface, and the thickness of the electrical double layer is assumed to be small compared to the particle dimension. Hence, the fluid phase can be divided into two regions: an “inner” region defined as the double layer surrounding the particle and an “outer” region defined as the remainder of the fluid which is neutral. In the outer region, the equations of conservation of each ionic species and the fluid momentum are the Laplace equation7
∇2µm ) 0
m ) 1, 2
(7)
and the Stokes equations
η∇2v - ∇p ) 0
(8a)
∇‚v ) 0
(8b)
In eq 7, µm is the electrochemical potential energy of species m defined by
µm ) µ0m + kT ln nm + zmeΦ
(9)
where µ0m is a constant, nm and zm are the concentration and valence, respectively, of type-m ions, and Φ is the electrical potential. m equal to 1 and 2 refers to the anion and cation, respectively, so -z1 ) z2 ) Z. In eqs 8, v is the fluid velocity and p is the dynamic pressure. Note that nm and Φ also satisfy Laplace’s equation. The governing equations 7 and 8 in the outer region satisfy the boundary conditions at the particle surface (S+ p , outer edge of the thin double layer) obtained by solving for the electrochemical potentials and fluid velocity in the inner region and using a matching procedure to (16) Chen, S. B.; Keh, H. J. J. Fluid Mech. 1992, 238, 251. (17) Keh, H. J.; Chen, J. B. J. Colloid Interface Sci. 1993, 158, 199. (18) Keh, H. J.; Lin, Y. M. Langmuir 1994, 10, 3010.
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Langmuir, Vol. 12, No. 3, 1996 659
ensure a continuous solution in the whole fluid phase. Thus7,19 on S+ p: 2
∑ βmk∇2s µk k)1
n‚∇µm ) -
m ) 1, 2
(10)
v ) U + Ω×r+v(s)
(11)
with the relaxation coefficients
β11 )
β22 )
[(
)
3f1 1 4 1 + 2 exp(ζh) sinh ζh κ Z 12f1 (ζh + ln cosh ζh) (12a) Z2
[ (
]
( ) ( )
β12 ) -
1 12f1 ln cosh ζh κ Z2
(12b)
β21 ) -
1 12f2 ln cosh ζh κ Z2
(12c)
)
3f2 1 -4 1 + 2 exp(-ζh) sinh ζh + κ Z 12f2 (ζh - ln cosh ζh) (12d) Z2
]
and the apparent slip velocity
v(s) ) -
kT [(ζh + ln cosh ζh)∇sµ1 + 2πη (Ze)2 (-ζh + ln cosh ζh)∇sµ2] (13)
In eqs 10-13, ζh ) Zeζ/(4kT); fm ) (kT)2/(6πηe2Dm); n is the unit vector outwardly normal to the particle surface; r is the position vector from the particle center; ∇2s ) I:∇s∇s, where I is the unit dyadic and ∇s ) (I - nn)‚∇ represents the gradient operator along the particle surface; U and Ω are the translational and angular velocities of the particle to be determined. To obtain eqs 12 and 13, it has been assumed that the fluid is only slightly nonuniform in the undisturbed electrolyte concentration on the length scale of the particle dimension and the equilibrium concentration of each ionic species is related to the equilibrium electrical potential by the Boltzmann distribution. The electrochemical potentials far away from the particle approach the undisturbed values and the fluid is at rest there. Thus, µm and v must obey
[ (
)]
zm D2 - D1 |r| f ∞: µm f µ∞m ≡ µ0m + kT 1 Z D2 + D1
Figure 1. Geometrical sketch of the axisymmetric diffusiophoretic motion of multiple spheres.
this constraint, one can calculate the particle velocities U and Ω after solving eqs 7-15 for the electrochemical potentials and the fluid velocity. If the particle is a sphere and the electrolyte gradient is constant, the diffusiophoretic mobility can be determined through the above procedure and the result is given by eq 3. There is no rotational motion of the sphere due to the axial symmetry. Analysis for Multiple Spheres In this section, we consider the axisymmetric diffusiophoretic motion of a string of N dielectric spheres in an immense solution of a symmetric electrolyte, as shown in Figure 1. The spheres are neutrally buoyantly suspended in the solution, and the prescribed electrolyte concentration gradient is a constant |∇n∞|ez, where ez is the axial unit vector in the circular cylindrical coordinate system (F,φ,z) with the origin taken at the center of the first sphere for convenience. The particles may differ in radius and in zeta potential, and they are allowed to be unequally spaced along the line through their centers. It is assumed that the thickness of the electrical double layers is much smaller than the radii of the particles and the gap widths between two neighboring particles. However, the effect of polarization of the diffuse layers will be taken into account. The purpose is to determine the correction to eq 3 for the migration of each particle due to the presence of the other ones. Electrochemical Potential Distributions. The electrochemical potentials obey the Laplace equation (eq 7) and are subject to the boundary conditions (according to eqs 10 and 14)
∂µm
ln n∞
∂ri
(15)
where n∞ is the prescribed concentration distribution of the electrolyte in the absence of the particle. The second term in brackets represents the contribution from the macroscopic electric field induced by the difference of ion diffusion rates. Since there is no effective external field exerted beyond the outer edge of the double layer, the particle (charged interface plus diffuse ions) is force and torque free. With (19) O’Brien, R. W. J. Colloid Interface Sci. 1983, 92, 204.
at ri ) ai
µm f µ∞m as (F2 + z2)1/2 f ∞
m ) 1, 2 (14) vf0
2
βmki∇2s µk ∑ k)1
)-
(16a) (16b)
for m ) 1 or 2 and i ) 1, 2, ..., or N. Here, ai and βmki are the radius and relaxation coefficients (defined by eq 12) of particle i and (ri,θi,φ) are spherical coordinates measured from the center of particle i. A general solution to eq 7 suitable for satisfying eq 16 is16
µm ) µ∞m +
kT|∇n∞| ∞
n (z)0)
N
∞
Amjnrj-(n+1)Pn(qj) ∑ ∑ j)1 n)0 m ) 1, 2 (17)
660
Langmuir, Vol. 12, No. 3, 1996
Keh and Luo
where Pn is the Legendre polynomial of order n and qj denotes cos θj for brevity. This solution form satisfies the boundary condition of eq 16b immediately, and the unknown constants Amjn will be determined using eq 16a. To apply eq 17, rj and qj must be written in terms of a single coordinate system: 2
2 1/2
rj ) [F + (z - d1j) ]
(18a)
{
∑ ∑ A1jn j)1 n)0
β11i[2(n + 1)qi(1 - q2i )1/2Fn+1(rj,qj) +
(n + 1)qi2Gn+1(rj,qj) - (1 - qi2)Hn+1(rj,qj)] + 2β11i 1[(1 - q2i )1/2Fn(rj,qj) + qiGn(rj,qj)] + ri
(
{
)
}
A2jnβ12i 2(n + 1)qi(1 - qi2)1/2Fn+1(rj,qj) + (n + 1)qi2Gn+1(rj,qj) - (1 - qi2)Hn+1(rj,qj) -
}
2
[(1 - qi2)1/2Fn(rj,qj) + qiGn(rj,qj)] ) ri D2 2β11i 2β12i D1 2 -1 + qi (19a) D2 + D1 ri D2 + D1 ri
(
[
and ∞
N
∑ ∑ A1jnβ21i j)1 n)0
{
)
kT|∇n∞| ∞
n (z ) 0)
∑ ∑ Amjnrj-(n+1)Pn(qj) j)1 n)0
( )]
and then to enforce the boundary condition at K discrete points (values of θi) on the generating arc of each sphere. As a result, for N spheres in the chain, this will lead to a set of 2KN simultaneous linear algebraic equations in the truncated form of eqs 19 for the 2KN unknown constants Amjn of the truncated solution given by eq 21. These equations can be solved by any standard matrix reduction technique to yield the constants Amjn for the electrochemical potential distributions. In general, the larger the value of K, the more accurate the result will be by this truncation method. Naturally, the truncation error vanishes as K f ∞. Fluid Velocity Distribution. Now that we know the electrochemical potential distributions in the fluid phase, we can now take up the solution of the fluid velocity field. The velocity distribution for the fluid outside the thin double layers is governed by the Stokes equations (eqs 8) or the following fourth-order differential equation for axisymmetic slow viscous flow:21
E2(E2Ψ) ) 0
(22)
where Ψ is the Stokes stream function and E2 is the Stokes operator. In circular cylindrical coordinates, the stream function is related to the velocity components by
vF )
1 ∂Ψ F ∂z
(23a)
2 1/2
2(n + 1)qi(1 - qi ) Fn+1(rj,qj) + 2
vz ) -
2
(n + 1)qi Gn+1(rj,qj) - (1 - qi )Hn+1(rj,qj) 2
N K-1
(18b)
where dij is the distance between the centers of particles i and j. Substitution of eq 17 into the boundary condition of eq 16a results in ∞
µm ) µ∞m +
m ) 1, 2 (21)
qj ) (z - d1j)[F2 + (z - d1j)2]-1/2
N
require the solution of the entire infinite array of unknown constants Amjn. However, the boundary collocation method13,20 allows one to truncate the infinite series in eq 17 into a finite one with K terms,
} {
[(1 - q2i )1/2Fn(rj,qj) + qiGn(rj,qj)] + A2jn β22i[2(n +
ri
2 1/2
1)qi(1 - qi ) Fn+1(rj,qj) + (n + 1)qi2Gn+1(rj,qj) 2β22i [(1 - qi2)1/2Fn(rj,qj) + (1 - qi2)Hn+1(rj,qj)] + 1 ri D2 2β21i qiGn(rj,qj)] ) 2 + D2 + D1 ri 2β22i D1 - 1 qi at ri ) ai (19b) D2 + D1 ri
( ) } [ ( ) ( )]
for i ) 1, 2, ..., or N. In eqs 19,
Fn(rj,qj) ) -r-(n+2) (1 - qj2)1/2P′n+1(qj) j
(20a)
Gn(rj,qj) ) -r-(n+2) (n + 1)Pn+1(qj) j
(20b) (20c)
where the prime on Pn(qi) means differentiation with respect to qj. To satisfy the boundary condition of eq 16a exactly along the entire semicircular generating arc of each sphere would
(23b)
The boundary conditions for the velocity field, resulting from eqs 11 and 15, are
v ) Uiez -
kT [(ζh + ln cosh ζhi)∇sµ1 + (-ζhi + 2πη (Ze)2 i ln cosh ζhi)∇sµ2] at ri ) ai (24a) v f 0 as (F2 + z2)1/2 f ∞
(24b)
for i ) 1, 2, ..., or N. Here, ζhi denotes the value of ζh associated with particle i, and Ui is the instantaneous diffusiophoretic velocity of particle i to be determined. In eq 24a, ∇sµm can be obtained from the electrochemical potential distributions given by eq 21 with coefficients Amjn determined from the truncated form of eqs 19. The general solution to eq 22, which immediately satisfies boundary condition of eq 24b, is N
Hn(rj,qj) ) -r-(n+2) [n(n + 1)Pn+1(qj) - P′n(qj)] j
1 ∂Ψ F ∂F
Ψ)
∞
[Bjnrj-n+1 + Cjnrj-n+3]Gn-1/2(qj) ∑ ∑ j)1 n)2
(25)
where Gn-1/2(qj) is the Gegenbauer polynomial of order n (20) Gluckman, M. J.; Pfeffer, R.; Weinbaum, S. J. Fluid Mech. 1971, 50, 705. (21) Happel, J.; Brenner, H. Low Reynolds Number Hydrodynamics; Martinus Nijhoff: The Netherlands, 1983.
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Langmuir, Vol. 12, No. 3, 1996 661
and degree -1/2. The unknown constants Bjn and Cjn are to be determined from eq 24a. Substituting eqs 17 and 25 into eq 24a and using eqs 23 as well as the recurrence relations of the Legendre and Gegenbauer polynomials, it is found that the unknown constants must satisfy the following set of algebraic equations for i ) 1, 2, ..., or N: N
∞
∑ ∑ [B
jnSn(rj,qj)
+ CjnTn(rj,qj)] )
j)1 n)2
kT
Ui -
|∇n∞|
ηκ2
{∑∑ N
∞
[A1jn(4ζhi + 4 ln cosh ζhi) +
j)1 n)0
A2jn(-4ζhi + 4 ln cosh ζhi)][(1 - qi2)Gn(rj,qj) -
2
[(
Fi ) 4πηCi2
1-
)
D2 - D1 D2 + D1
)
D2 - D1 D2 + D1
(4ζhi + 4 ln cosh ζhi) +
]}
(-4ζhi + 4 ln cosh ζhi)
Ci2 ) 0 for i ) 1, 2, ..., N (26a)
∑ ∑ [BjnS*n(rj,qj) + CjnT*n(rj,qj)] ) j)1 n)2 -
∞
{∑ ∑
|∇n | ηκ2
[A1jn(4ζhi + 4 ln cosh ζhi) +
j)1 n)0
A2jn(-4ζhi + 4 ln cosh ζhi)][qi2Fn(rj,qj) qi(1 - qi2)1/2Gn(rj,qj)] -
[(
qi(1 - qi2)1/2 1 +
(
1-
)
D2 - D1 D2 + D1
)
D2 - D1 D2 + D1
(4ζhi + 4 ln cosh ζhi) +
(-4ζhi + 4 ln cosh ζhi)
]}
at ri ) ai (26b)
where
Sn(rj,qj) ) -rj-n-1Pn(qj)
(27a)
Tn(rj,qj) ) 2 2n - 3 P (q ) + P [2n -1 2n - 1
-rj-n+1
n
n-2(qj)
j
] (27b)
1 S*n(rj,qj) ) -rj-n-1(1 - qj2)1/2 P′n(qj) n
[
T*n(rj,qj) ) -rj-n+1(1 - qj2)1/2
(30)
Ui ) MiUi0
(31)
Ui0 ) Ai|∇n∞|
(32)
with
∞
n
(29)
The diffusiophoretic velocities Ui of the N particles are to be obtained by solving the above N equations simultaneously. The result can be expressed as
∞
kT
(28)
Since the particles are freely suspended in the fluid, the net force exerted by the fluid on the surface of each particle must vanish. From eq 29, we have
and N
[Bjnrj-n+1 + Cjnrj-n+3]Gn1/2(qj) ∑ ∑ j)1 n)2
If the truncated solutions given by eqs 21 and 28 are used for the electrochemical potential and velocity fields and the boundary conditions of eq 24a are satisfied at M discrete points along the generating arc of each of the N particles, then a set of 2MN linear algebraic equations in the truncated form of eqs 26 is obtained. Simultaneous solution of these equations yields the 2MN unknown constants Bjn and Cjn, in terms of the particle velocities Ui, required in eq 28 for the stream function. Velocities of Free Spheres. The drag force exerted by the fluid on each particle surface ri ) ai can be determined from the formula21
qi(1 - qi2)1/2Fn(rj,qj)] +
(1 - qi ) 1 +
(
N M+1
Ψ)
(27c)
n-3 P′ (q ) + n(n - 1) n j 2 P (q ) (27d) n - 1 n-1 j
]
Similar to the solution of electrochemical potential distributions, we can use the truncation technique to approximate the infinite-series solution of eq 25 for the stream function by one with finite terms,
which is the diffusiophoretic velocity of particle i in the absence of all the other ones. Ai is used to represent the value of A defined by eq 3 for the diffusiophoretic mobility of particle i. The mobility parameters Mi, indicating the extent of particle interactions, depend on the relative positions, sizes, and zeta potentials of the particles as well as the bulk concentrations, valences, and diffusion coefficients of the ions. When the neighboring particles are separated from particle i by an infinite distance, Mi becomes unity. Velocity of a Rigid Cluster of Spheres. We now consider the diffusiophoretic motion of a rigid cluster of N spheres. The spheres are all located along the z-axis and are connected through their centers with rigid rods of arbitrary lengths. The connecting rods are assumed to be infinitesimally thin compared to the sphere sizes; hence they make neither electrostatic nor hydrodynamic contributions but only serve to ensure the rigid-body motion of the cluster. Here, our object is to explore the diffusiophoresis of aggregates formed by flocculation or bridging of colloidal particles in a suspension. The difference between the case here and that of free spheres in the previous subsection is that the N spheres in the chain translate at the same speed U0, the diffusiophoretic velocity of the cluster, and the drag force on each individual particle no longer vanishes but the total force on the entire cluster disappears, that is, N
F)
Fi ) 0 ∑ i)1
(33)
Utilizing eq 29, the requirement of Ui ) U0 and the relation above, U0 can be determined straightforwardly. Results and Discussion In this section, we present our results for the axisymmetric diffusiophoretic motions of two free spheres, of three
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Langmuir, Vol. 12, No. 3, 1996
Keh and Luo
Table 1. The Mobility Parameters for the Axisymmetric Diffusiophoresis of Two Identical Free Spheres for the Case f1 ) 0.2 and Ka ) 100
a
M1 ()M2) D2 - D1 ) 0
(D2 - D1)/(D2 + D1) ) -0.2
Z)1
Z)2
Z)3
Z)1
Z)2
Z)3
2
0.2 0.4 0.6 0.8 0.9 0.95 0.99 1.0
1.0004 1.0027 1.0073 1.0105 1.0109 1.0109 1.0109 1.0109
1.0007 1.0052 1.0142 1.0220 1.0240 1.0245 1.0248 1.0249
1.0014 1.0107 1.0307 1.0537 1.0630 1.0669 1.0697 1.0704
1.0039 1.0284 1.0751 1.1079 1.1118 1.1118 1.1118 1.1118
1.0012 1.0091 1.0248 1.0384 1.0417 1.0426 1.0432 1.0433
1.0021 1.0159 1.0454 1.0792 1.0928 1.0985 1.1026 1.1035
5
0.2 0.4 0.6 0.8 0.9 0.95 0.99 1.0
1.0012 1.0092 1.0259 1.0440 1.0506 1.0532 1.0550 1.0554
1.0077 1.0601 1.1881 1.4048 1.5660 1.6791 1.8198 1.984
0.9990 0.9920 0.9747 0.9430 0.9155 0.8924 0.8534 0.740
1.0019 1.0142 1.0402 1.0680 1.0782 1.0821 1.0848 1.0854
0.9498 0.6107 -0.2167 -1.6120 -2.6479 -3.3743 -4.2785 -5.332
0.9994 0.9950 0.9843 0.9651 0.9485 0.9347 0.9112 0.843
8
0.2 0.4 0.6 0.8 0.9 0.95 0.99 1.0
1.0038 1.0295 1.0902 1.1841 1.2449 1.2821 1.3200 1.332
0.9994 0.9955 0.9856 0.9679 0.9525 0.9396 0.9172 0.841
0.9999 0.9995 0.9985 0.9977 0.9972 0.9969 0.9963 0.9935
1.0081 1.0624 1.1908 1.3888 1.5168 1.5950 1.6748 1.701
0.9996 0.9970 0.9906 0.9792 0.9696 0.9614 0.9473 0.899
0.9999 0.9995 0.9987 0.9979 0.9977 0.9974 0.9970 0.9952
ζe/(kT) 2a/d12
free spheres, and of a rigid cluster of two spheres in a constant gradient of a symmetric electrolyte. The details of the collocation scheme used for this work were given by Keh and Yang.13 The numerical calculations were preformed by using a DEC 3000/600 workstation. In general cases, the series in eqs 21 and 28 converge quite rapidly, and very good accuracy can be achieved with only small numbers of collocation points (K and M) on each sphere. The most important discovery is that, unlike a conclusion of the previous studies of diffusiophoresis (or electrophoresis) in which the effect of polarization of diffuse layers was ignored,11-15 interactions exist among the particles having the same zeta potential. Generally speaking, the interaction effects can be quite significant under appropriate conditions. Two Free Spheres. The diffusiophoretic velocities of two spheres along the line of their centers have been calculated for various values of (D2 - D1)/(D2 + D1), f1, Z, ζ1e/(kT), ζ2e/(kT), κa1, a2/a1, and (a1 + a2)/d12 using the collocation technique. Part of the numerical results of the mobility parameters for the case of two identical spheres (a1 ) a2 ) a, ζ1 ) ζ2 ) ζ) as a function of the separation parameter 2a/d12 are presented in Table 1 (with f1 ) 0.2, κa ) 100, Z ) 1, 2, and 3). Three constant values of 2, 5, and 8 are chosen for the parameter ζe/(kT). All of the results obtained (including the cases of 2a/d12 ) 1.0) are at least convergent to the digits as shown. It can be seen that the effect of particle interaction is stronger when the two spheres get closer. However, each sphere can be speeded up or slowed down by the other with the decrease of the separation distance, depending on the values of the relevant factors. Note that the situations associated with (D2 - D1)/(D2 + D1) ) 0 and -0.2 in Table 1 are very close to the diffusiophoresis in the aqueous solutions of KCl and NaCl, respectively. The mobility parameters of two identical spheres are plotted as a function of their dimensionless zeta potential at various values of κa and Z in Figure 2 for a case in which the cation and anion mobilities are equal (D2 - D1 ) 0 with f1 ) f2 ) 0.2 and 2a/d12 ) 0.6). Only the results at positive zeta potentials are displayed in this figure since, for D1 ) D2, the induced macroscopic electric field vanishes
b
c
Figure 2. Plots of the mobility parameters of two identical free spheres versus the dimensionless zeta potential with 2a/ d12 ) 0.6, D2 - D1 ) 0, and f1 ) f2 ) 0.2: (a) Z ) 1, (b) Z ) 2, and (c) Z ) 3.
and the particle velocities, which are due to the chemiphoretic effect only, will be an even function of the zeta potential. When Z ) 1, as shown in Figure 2a, the mobility
Particle Interactions in Diffusiophoresis
parameter of each sphere is a monotonic increasing function of the value of ζe/(kT) ranging from 1 to 8. Also, the mobility parameter is greater with smaller κa. However, when Z ) 2 or 3 as illustrated in Figure 2b,c, a maximum and a minimum of the particle velocity exist for some cases. When κa increases, the extremes occur at larger zeta potentials. Note that these extremes for the cases with Z ) 3 take place at smaller zeta potentials than those with Z ) 2. The mobility parameters of two identical spheres as a function of ζe/(kT) at various values of κa and Z for a case in which the cation and anion have different diffusion coefficients ((D2 - D1)/(D2 + D1) ) -0.2 with f1 ) 0.2 and 2a/d12 ) 0.6) are depicted in Figure 3. In this case, both the chemiphoretic and the electrophoretic effects contribute to the particles’ movement and the net diffusiophoretic velocities are neither an even nor an odd function of ζ. It can be seen that the mobility parameter of each particle is not a monotonic function of ζe/kT. For the representative case of Z ) 2, κa ) 100, and ζe/(kT) ) 5, as indicated in Table 1, the particles will reverse the direction of diffusiophoresis and the magnitude of their velocities can be dramatically increased when the distance between them is decreased. In general, no theoretical rule could appropriately predict the particle interactions. Whether the particle velocities are increased or decreased depends on the combination of ζ, Z, f1, f2, and 2a/d12. In Figures 4 and 5, the mobility parameters for two identical spheres are plotted versus κa in the range from 102 to 105 for cases of (D2 - D1)/(D2 + D1) ) 0 and -0.2, respectively. It is shown that there will be no particle interaction in diffusiophoresis for each case as long as the value of κa approaches infinity (the diffusiophoretic mobilities of the particles will equal the value calculated by eq 2 ignoring the polarization effect of the double layer). This behavior is in accordance with the situation of electrophoresis in the limit of eq 6. From Figures 4a and 5a for the case Z ) 1, the particle interaction is weakened steadily as κa becomes large gradually. However, as shown in Figures 4b,c and 5b,c for Z ) 2 and Z ) 3, there can be a minimum and a maximum of the particle interaction occurring at some κa for the representative cases of ζe/(kT) ) 5 and 8. If the particles are charged more highly (with greater magnitude in zeta potential) or the counterions have a larger absolute value of valence, the locations of these maximal particle interactions will shift toward large κa; that means larger values of κa are required to make the assumption of κa f ∞ valid. Because many suspensions in practical applications are composed of particles of the same material, it might be of interest to examine the interactions between two identically charged spheres (ζ1 ) ζ2 ) ζ) with unequal radii. The results of the interaction parameters for spheres with a2/ a1 ) 2 and (a1 + a2)/d12 ) 0.6 are plotted versus κa1 in Figures 6 and 7 for cases of (D2 - D1)/(D2 + D1) ) 0 and -0.2, respectively. It is understood that, although the two spheres possess the same zeta potential, they will translate at different speeds even if situated very far apart. In general, the influence of the particle interaction is more significant on the smaller one than on the larger one. It can be seen that the shift of the locations of the maximal particle interactions is also like that for two identical spheres. Again, whether the particle interaction enhances or retards the diffusiophoretic velocity of each sphere depends on the combination of relevant factors for each case. Three Free Spheres. The number of parameters involved in the problem of axisymmetric diffusiophoresis of three spheres is quite great. Also, the utilization of the boundary collocation technique for solving the three-
Langmuir, Vol. 12, No. 3, 1996 663
a
b
c
Figure 3. Plots of the mobility parameters of two identical free spheres versus the dimensionless zeta potential with 2a/ d12 ) 0.6, (D2 - D1)/(D2 + D1) ) -0.2, and f1 ) 0.2: (a) Z ) 1, (b) Z ) 2, and (c) Z ) 3.
sphere problem becomes more difficult than for the case of two spheres. Therefore, we only consider the simplest case in this subsection: three identical spheres separated
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a
a
b
b
c
c
Figure 4. Plots of the mobility parameters of two identical free spheres versus the ratio of the sphere radius to the Debye length with 2a/d12 ) 0.6, D2 - D1 ) 0, and f1 ) f2 ) 0.2: (a) Z ) 1, (b) Z ) 2, and (c) Z ) 3.
at the same spacing (d12 ) d23). The values of the mobility parameters M1 ()M3) and M2 for the case of f1 ) 0.2 and
Figure 5. Plots of the mobility parameters of two identical free spheres versus the ratio of the sphere radius to the Debye length with 2a/d12 ) 0.6, (D2 - D1)/(D2 + D1) ) -0.2, and f1 ) 0.2: (a) Z ) 1, (b) Z ) 2, and (c) Z ) 3.
κa ) 100 as a function of ζe/(kT) and 2a/d12 are listed in Table 2. For the cases of ζe/(kT) ) 5 and 8, we could not
Particle Interactions in Diffusiophoresis
Langmuir, Vol. 12, No. 3, 1996 665
a
a
b
b
c
c
Figure 6. Plots of the mobility parameters M1 (solid curves) and M2 (dashed curves) of two unequal-sized free spheres with the same zeta potential versus the ratio of the radius of sphere 1 to the Debye length for the case of a2/a1 ) 2, (a1 + a2)/d12 ) 0.6, D2 - D1 ) 0, and f1 ) f2 ) 0.2: (a) Z ) 1, (b) Z ) 2, and (c) Z ) 3.
Figure 7. Plots of the mobility parameters M1 (solid curves) and M2 (dashed curves) of two unequal-sized free spheres with the same zeta potential versus the ratio of the radius of sphere 1 to the Debye length for the case of a2/a1 ) 2, (a1 + a2)/d12 ) 0.6, (D2 - D1)/(D2 + D1) ) -0.2, and f1 ) 0.2: (a) Z ) 1, (b) Z ) 2, and (c) Z ) 3.
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Table 2. The Mobility Parameters M1 ()M3) and M2 for the Axisymmetric Diffusiophoresis of Three Identical Free Spheres with Equal Spacings for the Case f1 ) 0.2 and Ka ) 100 Z)1 ζe/(kT) 2a/d12
M1
M2
Z)2 M1
Z)3 M2
M1
M2
0.2 0.4 0.6 0.8 1.0
1.0004 1.0032 1.0091 1.0142 1.0163
D2 - D 1 ) 0 1.0007 1.0008 1.0054 1.0061 1.0140 1.0176 1.0187 1.0297 1.0163 1.0370
1.0014 1.0103 1.0274 1.0397 1.0370
1.0016 1.0125 1.0381 1.0718 1.1039
1.0028 1.0214 1.0597 1.0974 1.1039
0.2 0.4 0.6 0.8 0.99
1.0014 1.0107 1.0322 1.0589 1.0806
1.0024 1.0182 1.0503 1.0797 1.0830
1.0088 1.0701 1.2339 1.5497 2.2743
1.0155 1.1198 1.3691 1.7550 2.3155
0.9988 0.9907 0.9684 0.9211 0.7514
0.9980 0.9841 0.9502 0.8920 0.7435
0.2 0.4 0.6 0.8 0.99
1.0043 1.0344 1.1118 1.2475 1.4810
1.0076 1.0587 1.1764 1.3399 1.4966
0.9993 0.9947 0.9820 0.9555 0.8587
0.9988 0.9909 0.9717 0.9392 0.8542
0.9999 0.9994 0.9982 0.9968 0.9939
0.9999 0.9989 0.9971 0.9957 0.9937
0.2 0.4 0.6 0.8 1.0
(D2 - D1)/(D2 + D1) ) -0.2 1.0044 1.0077 1.0014 1.0024 1.0334 1.0565 1.0106 1.0180 1.0941 1.1449 1.0309 1.0479 1.1467 1.1933 1.0518 1.0691 1.1677 1.1677 1.0643 1.0643
1.0024 1.0186 1.0564 1.1058 1.1529
1.0042 1.0316 1.0882 1.1436 1.1529
5
0.2 0.4 0.6 0.8 0.99
1.0021 1.0166 1.0499 1.0911 1.1244
1.0038 0.9432 0.8996 0.9993 1.0283 0.5457 0.2239 0.9942 1.0779 -0.5128 -1.3874 0.9804 1.1231 -2.5470 -3.8711 0.9517 1.1280 -7.2041 -7.4693 0.8495
0.9987 0.9901 0.9692 0.9339 0.8447
8
0.2 0.4 0.6 0.8 0.99
1.0092 1.0728 1.2366 1.5228 2.0143
1.0162 1.1243 1.3732 1.7179 2.0471
0.9999 0.9990 0.9974 0.9962 0.9950
2
5
8
2
0.9996 0.9965 0.9882 0.9712 0.9102
0.9992 0.9940 0.9814 0.9607 0.9074
0.9999 0.9994 0.9983 0.9972 0.9952
obtain a convergent solution when the three spheres all touch (2a/d12 ) 1.0); thus, only the results for 2a/d12 e 0.99 are presented. As expected, the velocity of the middle sphere is affected more by the particle interaction than are those on each side of it. For the case of three touching spheres, they migrate as a single particle and their diffusiophoretic velocities are identical. It may be of interest to see how the existence of a third sphere affects the diffusiophoretic velocities of two neighboring spheres. A comparison between Tables 1 and 2 shows that the presence of the third sphere enhances the two-particle interaction effect on both spheres 1 and 2. A Rigid Cluster of Two Spheres. The velocity of a rigid cluster of spheres undergoing axisymmetric diffusiophoresis can also be determined by the procedure described in the previous section. For conciseness, here we only consider the motion of a dumbbell, along the line connecting the centers of its two spheres. For the simplest case of the diffusiophoretic motion of a dumbbell having two identical spheres along its connecting rod, the dumbbell will move at the same speed as that of either of these two spheres suspended freely and separated by the same distance, as given in Table 1. Since flocculation by bridging of particles having identical zeta potentials but different sizes can occur in a suspension of colloidal particles of the same material, it might be important to investigate the diffusiophoresis of such a dumbbell. Results for the diffusiophoretic mobility of a dumbbell composed of two spheres with a2/a1 ) 2, κa1 ) 100, and f1 ) 0.2, expressed in terms of the dimensionless form U0/U10, are listed in Table 3 for various values of Z, ζe/(kT), and (a1 + a2)/d12. When these data are compared with those in Table 1, one can find that the
Table 3. The Dimensionless Diffusiophoretic Velocity of a Rigid Dumbbell U0/U10 along the Line Connecting Its Two Spheres for the Case a2/a1 ) 2, f1 ) 0.2 and Ka1 ) 100 D2 - D 1 ) 0 ζ1e/ ζ2e/ (a1 + a2)/ (kT) (kT) d12 Z)1 Z)2 Z)3
(D2 - D1)/(D2 + D1) ) -0.2 Z)1
Z)2
Z)3
2
2
0.2 0.4 0.6 0.8 1.0
1.0443 1.0477 1.0526 1.0565 1.0583
1.0866 1.0934 1.1031 1.1114 1.1158
1.1935 1.2088 1.2314 1.2533 1.2672
1.4585 1.4946 1.5449 1.5846 1.6035
1.1514 1.1633 1.1803 1.1948 1.2023
1.2858 1.3085 1.3418 1.3741 1.3944
5
5
0.2 0.4 0.6 0.8 0.99
1.1614 1.1742 1.1928 1.2103 1.2205
2.4588 2.5761 2.7629 3.0014 3.3226
0.7772 0.7592 0.7297 0.6880 0.5989
1.2500 1.2698 1.2987 1.3257 1.3414
-8.4229 -9.1800 -10.3860 -11.9226 -13.9878
0.8632 0.8521 0.8341 0.8088 0.7551
8
8
0.2 0.4 0.6 0.8 0.99
1.6441 1.6956 1.7759 1.8706 1.9671
0.8736 0.8634 0.8466 0.8231 0.7716
0.9899 0.9891 0.9879 0.9867 0.9850
2.3617 2.4707 2.6403 2.8400 3.0434
0.9179 0.9113 0.9005 0.8855 0.8530
0.9910 0.9903 0.9892 0.9883 0.9871
-4
1
0.2 0.4 0.6 0.8 1.0
0.3696 0.3380 0.3003 0.2610 0.2238
0.4980 0.4716 0.4376 0.3979 0.3563
-1.9485 -2.0776 -2.2066 -2.3114 -2.3980
0.2797 0.2439 0.2014 0.1577 0.1169
0.3232 0.2886 0.2458 0.1987 0.1515
1.5265 1.5268 1.4921 1.4202 1.3405
larger sphere dominates the motion of the dumbbell and its influence becomes more important when the two spheres are less separated. For the case of Z ) 2, ζe/(kT) ) 5, and (D2 - D1)/(D2 + D1) ) -0.2, the dumbbell will also reverse the direction of diffusiophoresis and the magnitude of its velocity can be quite large. An interesting case is the diffusiophoresis of a neutral dumbbell (with zero averaged zeta potential, i.e., the ratio ζ2/ζ1 equal to -a12/a22). In Table 3, we also present the diffusiophoretic mobility of a neutral dumbbell with a2/a1 ) 2, ζ1e/(kT) ) -4, and ζ2e/(kT) ) 1. It can be seen that a nonuniformly charged but neutral dumbbell will migrate in gradients of electrolyte concentration. As before, the large sphere will increasingly dominate the dumbbell’s migration as the distance between the two spheres decreases. In the limit (a1 + a2)/d12 f 0, it can be shown by the linearity of the problem that the dumbbell velocity
U0 )
a1U10 + a2U20 a1 + a2
(34)
where Ui0 (i ) 1 or 2) is defined by eq 32. Examination of our numerical data indicates that the above relation is very well satisfied as (a1 + a2)/d12 f 0 for all the cases considered. Concluding Remarks In this work the axially symmetric diffusiophoresis of a finite chain of dielectric spheres with thin but polarized electrical double layers in a uniform gradient of a symmetric electrolyte is examined. The spheres may differ in size and in zeta potential and they are allowed to be unequally spaced along the line of their centers. Not only the particle interactions among free spheres but also the migration of a rigid cluster of connected spheres has been investigated. The conservative equations in the “outer” region can be solved by applying the boundary conditions provided by the solution for the “inner” region. There are four factors influencing the mobility of a particle and the interactions among particles: the ratio of the particle dimension to the Debye length, the zeta potential at each particle surface, the valences of ions in solution, and the ionic drag coefficients.
Particle Interactions in Diffusiophoresis
A semianalytical procedure with the boundary collocation technique has been used to solve the electrochemical potential distributions and the velocity field for the fluid around the diffusiophoretic spheres. It is found that particle interactions actually exist among spheres with identical zeta potentials in an unbounded fluid as long as κa is finite. In addition, the particle interaction is no longer a simple monotonic function of both the spheres’ and ions’ properties and the separation distance for typical cases. The phenomena cannot be predicted systematically by a simple general rule. This novel result, which is similar to the case of electrophoresis,16 should be noted by both theoreticians and experimentalists in relevant fields. Although the analysis throughout this work is restricted to the case of a symmetric electrolyte solution, our results can also be applied to the situation of a fluid containing a uniform gradient of an arbitrary electrolyte using O’Brien’s19 reasoning that only the most highly charged counterions play a dominant role in the ionic fluxes along the particle surface. Acknowledgment. Part of this research was supported by the National Science Council of the Republic of China under Grant NSC83-0402-E002-067. Nomenclature a ) radius of particle (m) ai ) radius of particle i (m) A ) diffusiophoretic mobility defined by eq 2 or 3 (m5/s) Ai ) diffusiophoretic mobility of particle i defined by eq 3 (m5/s) A1jn, A2jn ) coefficients in the expression of eq 17 for µ1 and µ2 b1, b2 ) coefficients defined by eq 4 Bjn, Cjn ) coefficients in the expression of eq 28 for Ψ dij ) distance between the centers of particles i and j (m) D1, D2 ) diffusion coefficients of the anion and cation (m2/s) e ) proton charge (C) ez ) unit vector in z-direction f1, f2 ) dimensionless drag coefficients of the anion and cation Fi ) drag force acting on particle i (N) Fn, Gn, Hn ) functions of rj and qj defined by eq 20 Gn-1/2 ) the Gegenbauer polynomial of order n and degree -1/2 I ) unit dyadic k ) the Boltzmann constant (J/K)
Langmuir, Vol. 12, No. 3, 1996 667 K, M ) the number of collocation points on particle surface Mi ) mobility parameter of particle i defined by eq 31 n1, n2 ) concentrations of the anion and cation (m-3) n∞ ) electrolyte concentration distribution in the absence of the particles (m-3) n ) unit vector outwardly normal to particle surface N ) the number of particles in an assemblage p ) pressure distribution in the fluid (N/m2) Pn ) the Legendre polynomial of order n qj ) cos θj r ) position vector originating at the center of particle (m) r, θ, φ ) spherical coordinates ri, θi, φ ) spherical coordinates measured from the center of particle i S+ p ) outer edge of the thin double layer surrounding the particle Sn, Tn, S*n, T*n ) functions of rj and qj defined by eq 27 T ) absolute temperature (K) U ) particle velocity (m/s) Ui ) velocity of particle i (m/s) Ui0 ) velocity of particle i in the absence of all the other particles (m/s) U0 ) velocity of a cluster of particles (m/s) v ) fluid velocity distribution (m/s) v(s) ) apparent slip velocity at particle surface (m/s) vF, vz ) components of v in cylindrical coordinates (m/s) z, F, φ ) cylindrical coordinates z1, z2 ) valences of the anion and cation Z ) charge number of a symmetric electrolyte βmk ) relaxation coefficients defined by eq 12 (m) βmki ) value of βmk associated with particle i (m) ∆ ) coefficient defined by eq 5 /4π ) fluid permittivity (C2 J-1 m-1) ζ ) zeta potential at particle surface (V) ζi ) zeta potential of particle i (V) ζh ) Zeζ/4kT (V) ζhi ) Zeζi/kT (V) η ) fluid viscosity (kg m-1 s-1) κ ) reciprocal Debye length (m-1) µ1, µ2 ) electrochemical potential energies of the anion and cation defined by eq 9 (J) µ01, µ02 ) constant potential energies (J) µ∞1 , µ∞2 ) potential energies defined by eq 14 (J) F, φ, z ) cylindrical coordinates Φ ) electrical potential (V) Ψ ) the Stokes stream function of the fluid (m3/s) Ω ) angular velocity of particle (s-1) LA950564U
