Primary Electroviscous Effect in a Suspension of Charged Porous Spheres

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1 Journal of Colloid and Interface Science 25, (2002) doi:0.006/jcis Primary Electroviscous Effect in a Suspension of Charged Porous Spheres Vijay Natraj and Shing Bor Chen Department of Chemical and Environmental Engineering, National University of Singapore, Singapore 7576 Received October 30, 200; accepted April 7, 2002 Primary electroviscous effect for a dilute suspension of porous spheres with fixed volumetric charge density is investigated theoretically. In the absence of flow, the electrical potential and solution charge density are assumed to satisfy the linearized Poisson Boltzmann equation. With incorporation of the electrical body force, the Brinkman equation and the Stokes equation are used to govern the fluid flow inside and outside a sphere. The theory is formulated by assuming weak deviation of the charge cloud from its equilibrium state. However, the electrical body force is not restricted to be small compared to the viscous force in the fluid momentum equation. The results show that the double layer distortion is increased with increasing particle permeability, thereby enhancing the relative importance of its stress contribution. Nonetheless, the intrinsic viscosity remains a decreasing function of permeability, similar to the case of uncharged particles. C 2002 Elsevier Science (USA) Key Words: electroviscous effect; porous particle; electric double layer; viscosity; charge density; stresslet.. INTRODUCTION Charged porous particles are often used to model crosslinked polyelectrolytes such as ficoll sulfate (), hyperbranched polyelectrolytes such as sodium salt of aromatic polyester with carboxylic acid terminal group (2) and amphiphilic polymers (3), and charged drug-delivery microspheres such as calciumbridged poly[bis(carboxylatophenoxy)phosphazene] (4). The electroviscous effect of these materials provides a means for particle characterization. The objective of this study is to investigate the charge effect on the apparent viscosity of a dilute suspension of charged porous spheres subject to a simple shear flow. A charged particle in electrolyte solution attracts oppositely charged ions and repels ions of like charge. Thus, the particle is surrounded by a charge cloud with equal but opposite charge to its own. This charge cloud is referred to as an electrical double layer. A suspension of charged particles differs in its rheological behavior from a similar uncharged system. Conway and Dobry-Duclaux (5) have proposed three mechanisms to account for this effect. These are collectively called the electroviscous To whom correspondence should be addressed. checsb@nus.edu.sg. effects. The primary effect arises from additional energy dissipation due to the distortion of the charge cloud caused by the flow. Namely, the charge effect changes the stresslet strength, as compared to that for uncharged particles. The secondary effect stems from the change in microstructure caused by the interactions between charged particles, while the tertiary effect is associated with the change in conformation of charged macromolecules in solution. We are concerned with the primary effect in this paper. Previous studies on the primary effect are all focused on a dilute suspension of charged impermeable particles. The fundamental equations are the coupled momentum equation, ion conservation equations, and Poisson s equation. Booth (6) first examined this effect for the case of charged spheres of radius a and constant surface potential ζ with the equilibrium electrical potential governed by the linearized Poisson Boltzmann equation. He assumed that the charge cloud deviates slightly from its equilibrium state due to the imposed shear flow, and the modification of the flow field around the particles caused by the electrical body force is small. The latter assumption enables one to use the flow field around an uncharged particle to deal with the ion transport instead of solving the coupled equations. This case is characterized by small Hartmann number, which is the ratio of electrical to viscous force on the fluid element. It is estimated as Ha = εζ 2 /µ 0 D for κa O() and Ha = εζ 2 /µ 0 Dκa for κa (7). Here ε is the permittivity, µ 0 is the viscosity of suspending fluid, D is the ion diffusivity, and κ is the Debye length characterizing the double layer thickness. Based on a regular perturbation method, Booth s analysis predicted the apparent viscosity as an expansion in power of ζ 2 with coefficients as functions of κa. The leading-order charge effect on the viscosity is found to be O(Ha) and increasing with increasing charge cloud thickness. For a thin double layer, Russel (8) analyzed the case the Hartmann number remains small, but the flow strength relative to the ion thermal motion can be arbitrary. Lever (9) dealt with a similar problem but for a thick double layer. He found that the deformation of the charge cloud becomes substantial and the increase in the bulk stress due to the charge effect is comparable to Einstein s result for uncharged particles. Sherwood (7) generalized the theory to allow for arbitrary Hartmann number and surface potential and conducted numerical calculations, but only for the case of weak /02 $35.00 C 2002 Elsevier Science (USA) All rights reserved. 200

2 ELECTROVISCOUS EFFECT OF POROUS SPHERES 20 double layer deformation. A similar full numerical investigation was conducted by Watterson and White (0). For high Hartmann number, Sherwood found that the apparent viscosity can be large compared with the value for a corresponding uncharged system and pointed out the appearance of eddies around each particle. Analytic results for a thin double layer were later obtained by Hinch and Sherwood (). As for nonspherical particles, the primary electroviscous effect has also been studied under various assumptions (2 4). Since all the above theories dealt with rigid impermeable particles, they may not apply to the case of charged porous particles the complexity is raised because of the flow within the particles. The only relevant work we are aware of is the theoretical study of the distortion effect on sedimentation of porous spheres (5). In this paper, we investigate the primary electroviscous effect for charged porous spheres with constant permeability and fixed charge density in simple shear flow. The problem is analyzed under the assumptions that the equilibrium electrical potential is governed by the linearized Poisson Boltzmann equation, and the distortion of the double layer is weak. However, the Hartmann number is not restricted to be small. 2. FORMULATION We consider a dilute suspension of charged porous spheres with radius a, constant permeability k, and fixed volumetric charge density ρ p subject to a simple shear flow with G being the imposed velocity gradient tensor. Electrostatic and hydrodynamic interactions between the particles are all negligible. The suspending electrolyte contains several species of ions, each with number density n m and valence z m. The particle charge attracts small ions of opposite sign to form a surrounding ion cloud which is not electrically neutral. This leads to a nonuniform electrical potential φ governed by Poisson s equation, 2 φ = (ρ + Hρ p )/ε, [] ε is the permittivity of the solution and is assumed identical inside and outside the particle, ρ is the solution charge density given by thermal energy k B T, Eq. [] can be linearized to become 2 φ 0 = κ 2 φ 0 Hρ p /ε, [2] φ 0 is the equilibrium electrical potential, and κ is the Debye screening length characterizing the thickness of the charge cloud, defined by κ 2 = e2 zm 2 εk B T n m. [3] Ohshima (6) solved Eq. [2] analytically subject to the conditions of continuous electrical potential and field at the particle surface to obtain φ 0 = φ s r φ 0 = aφ s r φ s = κar (κa + )ae κa sinh(κr) κa (κa + )e κa sinh(κa) m (r a) [4a] exp[κ(a r)] (r > a), [4b] ρ p [κa + (κa + ) exp( 2κa)] [5] 2εκ 3 a is the equilibrium potential at the particle surface and r is the radial distance from the particle center. The solution charge density is given by ρ 0 = εκ 2 φ 0. [6] We assume the particles to be sufficiently small so as to neglect the inertial effect for fluid flow. With incorporation of the electrical body force, the fluid motion is governed by the Stokes equation µ 0 2 u p ρ φ = 0 [7] outside the particle, and by the Brinkman equation (7) ρ = m n m z m e, µ 0 2 u p ρ φ = µ 0 k (u u p) [8] e is the proton charge, and H is a step function equal to unity inside the sphere and zero otherwise. Far from any charged particle, the number densities of the ions attain limiting values n m, which satisfy n m z me = 0 m for electrical neutrality of the solution. In the absence of flow, the ions obey the Boltzmann distribution. When the electrical energy eφ 0 is small compared to the inside the sphere, u and p are the fluid velocity and pressure, and u p is the particle velocity. Note that we have assumed that the effective viscosity within the porous medium is equal to the viscosity of the outside fluid. In addition, the fluid velocity must fulfill the continuity equation u = 0. [9] The imposed flow distorts the charge cloud from its equilibrium state. We assume that all ion species have an equal diffusivity D both inside and outside the particle. The ion flux can

3 202 NATRAJ AND CHEN thus be expressed by ( ) ezm n m J m = n m u D k B T φ + n m. [0] The three terms on the right-hand side of [0] represent the convection, electrical migration, and Brownian diffusion, respectively. The steady-state ion distribution must obey the conservation equation J m = 0. [] Since we have assumed the electrical energy to be small compared with the thermal energy, the ion concentration differs only by O(eφ c /k B T ) from its faraway value (7), φ c is the equilibrium potential at the particle center. Following Sherwood s analysis, we substitute [0] into [], neglect terms of O(eφ c /k B T ) 2 and higher, and sum over all ion species to obtain 2 ρ κ 2 ρ = u ρ. [2] D The corresponding flux of charge density is J = ρu D(εκ 2 φ + ρ). [3] To determine u, p, ρ, and φ, one needs to solve Eqs. [], [7] [9], and [2] subject to boundary conditions. Far from the sphere, the perturbations to the applied flow, electrical potential, and charge density vanish. At the particle surface, the fluid velocity u, stress n σ (n is the unit outward vector), electrical potential φ, electric field φ, and the concentration and flux of each ionic species must be continuous. The last three conditions lead to continuous ρ and J on the particle surface. It should be noted that subscripts 0 and denote the equilibrium value and the small perturbation, respectively. Substitution of the above expansions into [] and [2] yields 2 φ = ρ /ε [7] 2 ρ κ 2 ρ = D u ρ 0. [8] It should be noted that the equilibrium potential and charge density contribute only an isotropic osmotic pressure and have no dynamical significance. Therefore, the leading-order electrical body force term in the momentum equation is O(Pe) and is given by ρ 0 (φ + ρ /εκ 2 ) after again absorbing ρ φ 0 into the pressure. The ratio of the electrical to viscous force acting on the fluid is called the Hartmann number. To estimate the Hartmann number, one needs to use a characteristic electric potential. For impermeable spheres, the Hartmann number is calculated using the constant particle surface potential. For porous particles with specified volumetric charge density, the particle surface potential is no longer constant, and varies with κa. Figure presents the variation of dimensionless potential with the normalized radial variable for various κa. It can be seen from the figure that for a given ρ p, the electrical potential has a maximum at the particle center and is a monotonic decreasing function of κa. For convenience, we will choose the potential at the center of the porous sphere as the characteristic potential to estimate the Hartmann number in the present study. The Hartmann number depends on the dimensionless particle permeability k = k/a 2, dimensionless double layer thickness (κa), and total particle charge Q = 4πa 3 ρ p /3. As can be seen σ = σ H + σ M, [4] 0.4 σ H = pi + µ 0 ( u + u T ) [5] is the hydrodynamic stress, and ( σ M = ε φ φ ) 2 φ 2 I [6] εφ 0 /a 2 ρ p κa=0.0 is the Maxwell stress with I being the unit tensor. The extent of double layer deformation is characterized by the ion Peclet number. In order to simplify the analysis, we assume small Pe such that the electrical potential and solution charge density can be expanded as φ = φ 0 + φ ρ = ρ 0 + ρ, r/a FIG.. Variation of dimensionless equilibrium potential with dimensionless radial distance.

4 ELECTROVISCOUS EFFECT OF POROUS SPHERES 203 from Eq. [8], only the radial component of the fluid velocity leads to the ion cloud distortion because ρ 0 depends merely on r. To estimate the charge cloud distortion and the Hartmann number, we can use the flow field around an uncharged sphere in simple shear flow. For thick double layer κa O(), the radial and tangential fluid velocities outside the particle are comparable. From Eq. [8], ρ is estimated to be O(uρ 0 /κ D). Therefore, ρ φ O(uρ 0 φ 0 /D) and µ 0 2 u O(µ 0 uκ 2 ) in Eq. [7]. The ratio between the two gives an estimate of the Hartmann number. Introducing the electrical potential at the particle center as the characteristic potential, we arrive at Ha εφ2 c = W η(κa), [9] µ 0 D W = 9Q 2 64π 2 εµ 0 a 2 D [20] η(κa) = 4e 2κa (κa) 4 (eκa κa ) 2. [2] For thin double layer κa, the equilibrium electrical potential and charge density exhibit significant variation only near the particle surface, as shown in Fig.. Therefore, the double layer deformation will take place mainly in this region. When the particle permeability is very high (k ), Eq. [9] remains valid because of comparable radial and tangential fluid velocities. For low permeability (k ), however, the radial fluid velocity is much smaller than the tangential velocity near the sphere surface. When (κa) 2 k, the ratio of radial to tangential velocity is O(k /2 ) (8), and thereby ρ φ O(uk /2 ρ 0 φ 0 /D). The Hartmann number becomes Ha εφ2 c k/2 µ 0 D = Wk/2 η(κa). [22] For very low permeability k κ 2 a 2 O(), the fluid velocity can be approximated by that for an impermeable particle. The ratio of radial to tangential velocity becomes O(κa) (7). As a result, Ha εφ2 c µ 0 Dκa = W η(κa). [23] κa Since the vorticity does not give rise to double layer deformation (8), only the straining part of the shear flow, E = (G + G T )/2, needs to be taken into account. Based on the symmetry nature of the elongational flow and the linear behavior at low Reynolds number, we extend Sherwood s approach (7) to propose the following solution forms u = E xg(r) + E : xxx r 2 f (r) [24] ρ = εφ s(κa) 2 D φ = φ sa 2 D E:xx r 2 n(r) [25] E:xx r 2 q(r) [26] p = µ 0 E:xx r 2 α(r) [27] outside the sphere, x is the position vector from the particle center. Within the particle, g(r), f (r), n(r), q(r), and α(r) in the above equations are respectively replaced by j(r), h(r), s(r), t(r), and β(r). We substitute the above expressions into the governing equations and nondimensionalize all lengths by the particle radius to obtain g + f + 3 fr = 0 j + h + 3hr = 0 from the continuity equation [9], [28a] [28b] f + 7 f r + 5 fr 2 = 2αr 2 + 2M(q + n)e aκ( r) r 3 α + 3 f + 5 fr = M(q + n )e aκ( r) r from Eq. [7], h + 7h r + 5hr 2 + jk [28c] [28d] = 2βr 2 + 2M aκr (aκ + )e aκ sinh(aκr) 3 (s + t) r aκ (aκ + )e aκ sinh(aκ) β + 3h + ( k r + 5r ) h + jk r = M aκr (aκ + )e aκ sinh(aκr) aκ (aκ + )e aκ sinh(aκ) (s + t ) r from Eq. [8], from Eq. [7], and q + 2q r 6qr 2 = (aκ) 2 n t + 2t r 6tr 2 = (aκ) 2 s s + 2s r [(aκ) 2 + 6r 2 ] s [28e] [28f] [28g] [28h] ( aκr) sinh(aκr) aκre aκr = A ( j + h) [28i] r n + 2n r [(aκ) 2 + 6r 2 ] n = e aκ( r) (aκ + r )(g + f ) [28j] from Eq. [8], M = εφ 2 s (κa)2 /Dµ 0 is a dimensionless

5 204 NATRAJ AND CHEN parameter, and A = (κa + )e κa /[κa (κa + )e κa sinh(κa)]. Note that in [28] and thereafter, r is the radial coordinate normalized by a. Far from the particle, the perturbations to the imposed fluid flow, electrical potential, and charge density decay to zero; i.e., Russel s expression for the particle stress S p = c [x σ H n µ 0 (un + nu)] da A 0 + V V 0 xρ 0 (φ + ρ /εκ 2 ) dv [32] f, (g ), n, q 0 as r. [29] Since u, p, ρ, and φ should each be of unique value at the particle center, one has h = s = t = β = 0 at r = 0. [30] At the particle surface (r = ), the conditions of continuous properties as mentioned earlier lead to g = j f = h f = h α = β q = t n = s q = t n = s. [3a] [3b] [3c] [3d] [3e] [3f] [3g] [3h] The governing equations [28] form a 6th-order set of ordinary differential equations. Since we have the boundary conditions at three points (r = 0,, and ), the standard techniques of solving a D boundary value problem (BVP) are not applicable. To circumvent this difficulty, we convert the problem into a two-point BVP by introducing a new independent variable δ = /r for the outside of the sphere, while δ remains r inside. All the derivatives in the differential equations and boundary conditions are then converted to those with respect to δ. In this way, the original problem can be converted to a two-point BVP (δ = 0, ). It is noted that the transformation of the boundary conditions at r = only results in sign change for f, q, and n in [3]. The new set of ordinary differential equations is solved using the routines in the IMSL package for two-point BVP, which adopt a variable order, variable step finite difference method with deferred corrections. After solving for the flow field, electrical potential, and charge density numerically, we proceed to calculate the suspension stress. We follow Russel s approach (8), which is an adaptation of Batchelor s derivation (9) by incorporating the electrical effects for an impermeable charged sphere. Our analysis finds that is also valid for porous particles. The proof is given in the Appendix. Here c is the number of particles per unit volume, A 0 refers to the surface of a particle with volume V 0, and V is an arbitrary volume surrounding the particle, which satisfies V V 0, κ 3 V, and cv. The electrical body force alters the flow of the fluid and thus modifies the surface integral in [32], compared to that for uncharged particles. Note that the volume integral attains its form in [32] because the O(Pe) correction for the electrical body force is taken into account. For impermeable spheres, Sherwood (7) used reciprocal theorem, integration by parts, and the no-slip condition at the particle surface to simplify the particle stress to obtain the net contribution of the charge effect in terms of a D integral. This simplified expression cannot apply to porous spheres because the fluid can permeate and flow through the sphere. Using [4b], [6], [5], and [24] [27], [32] can be simplified to S p = µ {[ 0χ 3 df ] } 5 f 2α + M(2I 3I 2 ) E, [33] 5 dδ δ= I = I 2 = 0 0 ( dn dδ + dq ) e aκ( /δ) dδ [34] dδ δ 2 (n + q)eaκ( /δ) 2 dδ [35] δ 3 and χ = 4πa 3 c/3 is the volume fraction of the particles. The intrinsic viscosity of the suspension can then be determined by [µ] = µ µ [ 0 3 µ 0 χ = df 0 dδ 3 2 f α ] 5 ( I + M 5 3 ) 0 I RESULTS AND DISCUSSION δ= [36] Before presenting the results, we would like to use the reported data for ficoll sulfate (crosslinked polyelectrolyte) () to examine the ranges for various physical quantities. For ficoll sulfate particles with hydrodynamic radius of about 3 nm and

6 ELECTROVISCOUS EFFECT OF POROUS SPHERES 205 [ µ ]-[ µ ] un E-3 E-4 E W κa=0. FIG. 2. Variation of [µ] [µ] un with W for k = 0.0. total charge of about 9 e, W is estimated to be about 5 using D = 0 9 m 2 s and µ 0 = Pa s. The value of κ can vary from nm (ionic strength = 0. M) to 00 nm (ionic strength = 0 5 M). However, since the Debye Huckel approximation is used for the equilibrium electrical potential, W cannot be too large. For a charged flat surface, it has been pointed out (20) that the potential based on the Debye Huckel approximation remains accurate as long as eφ c /k B T 2. Accordingly, we estimate that the largest W is about 0 for κa O() and 5(κa) 4 for κa. Therefore, the Hartmann number can be only as large as about O(0), inferred from [9] [23]. The average permeability of a polymer coil normalized by the square of its gyration radius ranges typically from 0.0 to 0. (). We now present and discuss our numerical results for the primary electroviscous effect of charged porous spheres. In order to examine the net charge effect on the suspension viscosity, we plot [µ] [µ] un against W for k = 0.0 and various κa in Fig. 2, [µ] un = ( ) 5χ 2 3 k cosh ( ) k ( + 3k ) sinh ( ) k 30k 3/2 cosh ( k ) ( + 0k + 30k 2 ) sinh ( k ) [37] is the intrinsic viscosity of uncharged spheres with equal permeability (8). It is evident from [37] that as k tends to zero, the intrinsic viscosity approaches 5/2 for impermeable spheres. For uncharged porous particles, [µ] un decreases from 2.25 for k = 0.00, to.7 for k = 0.0, to for k = 0.. As expected, Fig. 2 shows that the intrinsic viscosity increases with the particle charge. The net charge effect becomes stronger when the double layer gets thicker. It can also be seen from Fig. 2 that 5 the difference in the intrinsic viscosity exhibits a linear relation with W (or Hartmann number). We use [9] [23] to find that for W = 0, the Hartmann number is about 8.7, 2.8, and 0.02 for κa = 0.,, and 5, respectively. Booth (6) used a regular perturbation method to find that for impermeable particles, the net charge effect is proportional to the Hartmann number. Our results in Fig. 2 imply that for porous spheres, the linear behavior remains valid at least for the Hartmann number up to about 0. For k = 0.0, the increase in the intrinsic viscosity at W = 0 and κa = 0. has become substantial as compared to the intrinsic viscosity for uncharged spheres. To examine the influence of particle permeability on the primary electroviscous effect, we plot the variation of [µ]/[µ] un with k for W = 0 and various values of κa in Fig. 3. It can be found that as the permeability increases, the relative contribution of the charge effect becomes progressively important. This behavior can be understood by the increased ease of double layer deformation as the particle permeability increases. To illustrate the double layer distortion, we examine the change in charge density by considering the straining part of an applied shear flow E = γ , γ is the shear rate, and the x and y axes are the extensional and compressional axes, respectively. On the x y plane (z = 0), the change in solution charge density can be expressed by [ µ ]/[ µ ] un 3 2 ρ = εφ s(κa) 2 γ 2D s(r) cos 2θ [38] κa= FIG. 3. Variation of [µ]/[µ] un with k for W = 0. k 5

7 206 NATRAJ AND CHEN inside the particle and by the same equation with function s replaced by n outside the particle, θ is the angle measured from the x axis. Owing to the symmetry nature of the extensional flow, it is only necessary to investigate the behavior in the first quadrant. We find from the above expression that for a given r, ρ is symmetric in magnitude about θ = 45, but has opposite signs on the two sides of θ = 45. Figure 4 presents the variations of s and n with dimensionless r for W = 0 and κa =. It can be seen from the figure that s and n are always negative, and their magnitudes increase with increasing permeability. The results reveal that more counterions are driven by the flow to accumulate in the region θ<45, and thus the solution charge density is reduced in magnitude in the region θ>45. As a result, a quadrupole moment of the charge distribution forms, analogous to the finding by Lever (9) and Sherwood (7) for impermeable particles. The charge cloud deformation causes the circular contours of charge density at equilibrium to deform into ellipse-like contours with the long axis in the x direction. Such an anisotropic ion distribution then induces a flow in the opposite direction to counteract the applied one. It should be noted that despite the increase in the relative contribution, the intrinsic viscosity itself decreases with increasing dimensionless permeability for constant W and κa, as shown in Fig. 5. For a very thick charge cloud, the porous sphere behaves as a point charge, so the particle stress is insensitive to the permeability inasmuch as the far-field flow plays a comparatively important role in the ion cloud deformation. Despite not showing the result in Fig. 5, our numerical calculation does find that the intrinsic viscosity ( 4) becomes very weakly dependent upon the permeability for κa = 0.0. s,n r/a k =0.0 FIG. 4. Functions for the change in charge density versus normalized radial distance for W = 0 and κa =. [ µ ] κa= k FIG. 5. Variation of [µ] with k for W = 0. For very large W and thick double layer, our numerical computations for the fluid flow predict the occurrence of eddies around the particle, similar to what Sherwood (7) pointed out for impermeable charged sphere. This can be attributed to a substantial flow modification caused by the electric quadrupole moment associated with the double layer deformation. In spite of the interesting, versatile behavior of eddies as to their number and rotation directions obtained from our numerical computations, we do not intend to show the results of streamlines. The reason is that such large W (or Hartmann number) is beyond the validity range for linearization of the Poisson Boltzmann equation adopted in our analysis. Nonetheless, this finding implies the possibility of observing eddies experimentally, provided that the particles carry sufficient charge. It is also encouraged to develop a theory using the Poisson Boltzmann equation without linearization to verify the existence of eddies and study their behavior. 4. CONCLUSIONS We have presented the formulation and conducted numerical calculations for the primary electroviscous effect of charged porous spheres without restriction to a small Hartmann number. The effect of double layer deformation caused by the imposed flow is twofold. First, the electrical interaction between the distorted ion cloud and the particle charge leads to part of the particle stress. Second, the electrical body force associated with the cloud distortion modifies the fluid flow, compared to that for an uncharged sphere. The numerical results have revealed that the additional stress due to the charge effect increases with increasing particle permeability. However, the overall stress remains reduced as the permeability increases, analogous to the behavior of uncharged porous particles.

8 ELECTROVISCOUS EFFECT OF POROUS SPHERES 207 Finally, we would like to address the extensibility of the present formulation. The theoretical analysis was carried out by assuming the porous spheres to be rigid and to possess a constant volumetric charge density and porosity. Some polyelectrolytes are flexible, however, and hence their conformations change in an applied flow. To take into account chain flexibility, Jiang and Chen (2) recently used a charged dumbbell model to investigate the coupled primary and tertiary effects in extensional flow. They found that the double layer distortion enhances the average chain elongation. For crosslinked or hyperbranched polyelectrolytes with certain flexibility, it would be a difficult task to develop a theory even by simply using a constant elastic modulus for the particle since the particle deformation is coupled with the other equations. In spite of specifying the particle charge density, the present analysis can be easily modified for the case of constant surface potential or for charge-regulated condition, which will lead to changes in the equilibrium potential and the solution charge density. If the volumetric charge density of a particle varies radially, our formulation can also be applied by first numerically solving for the equilibrium electrical potential and solution charge density and then making a corresponding change in the equations for the double layer distortion. As for a radially varying porosity, Eqs. [28] remain valid with k being a function of r. APPENDIX: DERIVATION OF PARTICLE STRESS The bulk stress for the suspension can be expressed as (8) S = pi + 2µ 0 E + S p. [A.] The particle stress S p is given by S p = c [σ H + σ M + pi µ 0 ( u + u T )] dv, [A.2] V c is the number of particles per unit volume, A 0 refers to the surface of a particle with volume V 0, and V is an arbitrary volume surrounding the particle, which satisfies V V 0, κ 3 V, and cv. Outside the particles, only σ M exists in the integrand because σ H = pi + µ 0 ( u + u T ). Therefore, [A.2] can be rewritten as S p = c [σ H + σ M + pi µ 0 ( u + u T )] dv V 0 + c V V 0 σ M dv. [A.3] We use the identities σ H + σ M = [x(σ H + σ M )] σ M = (xσ M ) x σ M = (xσ M ) + xρ φ and divergence theorem to obtain S p = c [xσ H n µ 0 (nu + un)] da+ c A 0 V V 0 [A.4] [A.5] xρ φ dv. [A.6] Note that the isotropic contribution from the pressure has been excluded, and an integral over the enclosing surface of V has vanished because κ 3 V. Introducing the perturbations ρ and φ, and using integration by part for the volume integral in [A.6], yields Eq. [32], in which the contribution of ρ φ 0 has already been absorbed in σ H, as mentioned in the main text. ACKNOWLEDGMENT The authors are grateful to the National University of Singapore for supporting the work through Grant R REFERENCES. Deen, W. M., and Smith, F. G., J. Membrane Sci. 2, 27 (982). 2. Turner, S. R., Walter, F., Voit, B. I., and Mourey, T. H., Macromolecules 27, 6 (994). 3. Bayoudh, S., Laschewsky, A., and Wischerhoff, E., Colloid Polym. Sci. 277, 59 (999). 4. Cohen, S., Bano, M. C., Visscher, K. B., Chow, M., Allcock, H. R., and Langer, R., J. Am. Chem. Soc. 2, 7832 (990). 5. Conway, B. E., and Dobry-Duclaux, A., in Rheology (F. R. Eirich, Ed.), Vol. 3, Chap. 3, Academic Press, San Diego, Booth, F., Proc. Roy. Soc. A 203, 533 (950). 7. Sherwood, J. D., J. Fluid Mech. 0, 609 (980). 8. Russel, W. B., J. Fluid Mech. 85, 673 (978). 9. Lever, D. A., J. Fluid Mech. 92, 42 (979). 0. Watterson, I. G., and White, L. R., J. Chem. Soc. Far. Trans. 2 77, 5 (98).. Hinch, E. J., and Sherwood, J. D., J. Fluid Mech. 32, 337 (983). 2. Sherwood, J. D., J. Fluid Mech., 347 (98). 3. Chen, S. B., and Koch, D. L., Phys. Fluids 8, 2792 (996). 4. Allison, S. A., Macromolecules 3, 4464 (998). 5. Liu, Y. C., and Keh, H. J., Colloid Surf. A 40, 245 (998). 6. Ohshima, H., and Kondo, T., J. Colloid Interface Sci. 55, 499 (993). 7. Brinkman, H. C., Appl. Sci. Res. A, 27 (947). 8. Chen, S. B., and Ye, X., J. Colloid Interface Sci. 22, 50 (2000). 9. Batchelor, G. K., J. Fluid Mech. 4, 545 (970). 20. Hunter, R. J., Foundations of Colloid Science, Vol., Oxford Univ. Press, New York, Jiang, L., and Chen, S. B., J. Non-Newtonian Fluid Mech. 96, 445 (200).