A general gel layer model for the transport of colloids and macroions in dilute solution

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Journal of Colloid and Interface Science 263 (2003) 84–98www.elsevier.com/locate/jcis

A general gel layer model for the transport of colloids and macroioin dilute solution

Stuart Allison,a,∗ Staffan Wall,b and Mikael Rasmussonc

a Department of Chemistry, Georgia State University, Atlanta, GA 30303, USAb Department of Physical Chemistry, University of Göteborg, S-412 96 Göteborg, Sweden

c Preformulation/Experimental Formulation, AstraZeneca Research & Development, S-431 83 Mölndal, Sweden

Received 5 November 2002; accepted 11 February 2003

Abstract

A general boundary element methodology for studying the dilute solution transport of rigid macroions that contain gel layers on tsurfaces is developed and applied to several model systems. The methodology can be applied to particles of arbitrary size, shdistribution, and gel layer geometry. Account is also taken of the steady state distortion of the ion atmosphere from equilibriummakes it applicable to the transport of highly charged structures. The coupled field equations (Poisson, ion-transport, low-ReynolNavier–Stokes, and Brinkman) are solved numerically and from this, transport properties (diffusion constants, electrophoretic mexcess viscosities) can be computed. In the present work, the methodology is first applied to a gel sphere model over a wide rangecharge and the resulting transport properties are found to be in excellent agreement with independent theory under those condiindependent theory is available. It is then applied to several prolate spheroidal models of a particular silica sol sample in an attemptpossible solution structures. A single model, that is able to account simultaneously for all of the transport behavior, which does nosignificant conformational change with salt concentration, could not be found. A model with a thin (� 1-nm) gel layer at high salt contethat expands on going to low salt content is able to explain the salt dependence of the intrinsic viscosity, but not the electrophoretiHowever, a model with a fairly thick (2-nm) gel layer at high salt content, which expands slightly (2.5-nm) at low salt content, isgood agreement with experiment. In addition, the influence of particle charge and the presence of a gel layer on the Scheraga–Mparameter are examined. This parameter is proportional to the product of the translational diffusion constant and the cube root of tviscosity. It is found to be very robust with regard to net particle charge as well as properties of the gel layer. 2003 Elsevier Science (USA). All rights reserved.

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1. Introduction

Over the past few years, there has been considerabterest in the transport of model structures/particles in whthe particle “surface” is not well defined [1]. These inclubiological cells [2], branched or cross-linked polymers [3and polymer-coated latex particles [5,6], to name a few.modern theory of “soft” particles [1] appears to haveorigin in the electrically neutral “porous sphere” modelBrinkman [7] and Debye and Bueche [8]. In the 1970Felderhoff and co-workers investigated the microscofoundations of the neutral porous sphere model in contion with transport of dilute polymer solutions [9–11]. Thwork included derivations of expressions for rotational

* Corresponding author.E-mail address: [email protected] (S. Allison).

0021-9797/03/$ – see front matter 2003 Elsevier Science (USA). All rights rdoi:10.1016/S0021-9797(03)00188-7

-

and translational [10] diffusion constants of dilute porospheres as well as the intrinsic viscosity [11]. The etrophoretic mobility of a weakly charged porous sphwas first derived by Hermans and Fujita [12]. Duringpast decade, Ohshima has studied the electrophoreticbility of “soft particles” that consist of a solid inner sphesurrounded by a porous spherical shell under othergeneral conditions [1,6,13–15]. Recently, Natraj and Cnumerically investigated the primary electroviscous efof a weakly charged porous sphere [16]. In additionoverall transport, the porous medium model pioneeredBrinkman [7], Debye and Bueche [8], and Hermans andjita [12] has found applications in gel electrophoresis [17

The primary objectives of the present work are totend “soft particle” modeling to include particles of arbtrary shape both with respect to the solid inner corethe outer “gel layer,” arbitrary charge distributions with

eserved.

http://www.elsevier.com/locate/jcis

S. Allison et al. / Journal of Colloid and Interface Science 263 (2003) 84–98 85

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the particle core and gel layer, and the effects of ionlaxation of the mobile ion atmosphere. The results willapplicable to dilute solution transport, which includes tralational diffusion, electrophoretic mobility, intrinsic viscoity at low shear, and related quantities such as the primelectroviscous effect and the Scheraga–Mandelkern parter. The underlying methodology employs boundary elem(BE) techniques that have been applied to a variety of plems where the macromolecules are modeled as “hard pcles” [18–21].

The BE methodology for determining transport propties of “hard” model particles has been discussed in depreviously. The electrostatic and electrodynamic potenare obtained by numerical solution of Poisson’s equatPotential functions related to the distortion of the ionmosphere surrounding the model particle from equilibriuor “ion relaxation,” are obtained by solution of a steady-stion transport equation for each of the mobile ion specpresent. The fluid velocity in the vicinity of the particlealso needed, which requires solution of the low-Reynonumber Navier–Stokes equation for an incompressible flIn the general case under conditions where ion relaxais significant [22], these various field equations are cpled together and it is necessary to solve them by anative procedure [18]. Much of the methodology applicato “hard” particles is virtually unchanged when appliedthe more general “soft” particle model, which is the subjof interest in the present work. This includes electrostaand steady-state electrodynamics as well as steady statransport. Consequently, this part of the methodologycussed in detail previously [18] shall not be considered hFor a soft particle, calculation of the fluid velocity is qudifferent from before. However, because the technicaltails may not be of interest to many readers, they are plain Appendix A. In addition, the approach used to calculthe total forces/stresses exerted by the model particle osurrounding fluid is somewhat different from before and tis discussed in Appendix B.

In the next section, the general model is presentedthe BE methodology as applied to the transport of soft pticles is discussed. We shall focus on translational diffusfree solution electrophoretic mobility, excess viscosity, arelated quantities. In order to validate the methodologis applied to a soft model sphere and transport propeare compared with independent theory when available.the case of electrophoretic mobilities, we investigate unwhat conditions it is necessary to include ion relaxation.nally, we consider the electrophoretic mobility and primaelectroviscous effect of dilute silica sols [23–25], which aslightly aggregated and can be modeled as prolate ellipsThis example serves to illustrate the potential usefulnesthe methodology as applied to the transport of a compcolloidal system.

-

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.

2. Model and methodology

The model particle is assumed to consist of a solid cenclosed by a hydrodynamic shear surface,Sp. The solid in-ner surface in Fig. 1 representsSp. Within the solid core, thedielectric constant is taken to beεI and an arbitrary chargdistribution can be specified [18–21]. OnSp, the assumptionis made that the particle velocity and local fluid velocity aequal and this is called the “no-slip,” or “stick,” hydrodnamic boundary condition [26,27]. In addition, the solvis modeled as an electrodynamic and hydrodynamic conuum with dielectric constantε0 and viscosityη0. In the fluiddomain that lies outside ofSp, mobile ions are also treateas a continuum and their equilibrium distribution and thsteady-state nonequilibrium distributions are determineda numerical BE procedure [18–21]. As mentioned in thetroduction, it is possible that flexible segments may extfrom Sp into the fluid for a certain distance, and such stems may be characterized as soft particles. To accounthis “gel layer,” the fluid is assumed to obey the Brinkmequation [7] for an incompressible fluid. In a reference frachosen to be stationary with respect to the core of the pcle,

(1)η0∇2v − ∇p = −se + η0λ2v,

(2)∇ · v = 0,

wherev is the local fluid velocity,p is the pressure,se isthe external force/volume on the fluid, andλ is a parametewith units of length−1 that is related to the density of the gsegments [1,2,7,8]. Equation (2) follows directly from tassumption that the fluid is incompressible. FelderhofDeutch [9] derived Eq. (1) as a mean-field approximatto the equations describing the hydrodynamic interacbetween polymer segments. Ifζ andρs represent the frictionfactor of a polymer “segment” and the “segment” numdensity, respectively, then

(3)ρsζ = η0λ2.

In this mean-field model of the gel layer, the fluid behamuch like any Newtonian fluid, but with an additionexternal force term,η0λ

2v, that accounts for the presenceflexible “polymer.” Equation (3) allows one to relateλ to thephysical characteristics of the flexible strands, or polymin the gel layer. In the present work, the assumption is m

Fig. 1. Boundary element model of a prolate ellipsoidal particle. The sinner structure consists of 112 triangular plates and represents the sof hydrodynamic shear,Sp. The outer wireframe structure representssurface that encloses the gel layer,Sg. The volume of the gel layer isdenotedVg in the text and the volume contained withinSp is denotedVc.

86 S. Allison et al. / Journal of Colloid and Interface Science 263 (2003) 84–98

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thatρs and consequentlyλ are uniform within the gel layerIt then follows that one can define a secondary surfaceSg,that entirely encloses the gel layer and separates it fromouter fluid domain where theη0λ

2v-term in Eq. (1) is absenContinuity conditions onSg have been the subject of somcontroversy, as discussions on this subject by BrinkmanFelderhoff [11], and Ohshima [1,14] reveal. In the preswork, we follow the more recent (and we believe correapproach of Ohshima [14] and assume that both tangeand normal components of the hydrodynamic stress teare continuous acrossSg.

For the solvated particle, we shall use a model consisof an inner core of anhydrous material of volumeVc andan outer volume, assumed to be uniformly hydratedvolumeVg and average thicknessd . Also, letV ∗

g denote thevolume of the unhydrated gel layer. Since the total numof segments in the gel layer is constant for this model,segment density,ρs, varies inversely withVg. From Eq. (3),we thus have

(4)λ = λ∗√V ∗g /Vg.

The surface of hydrodynamic shear,Sp, enclosesVc andconsequently solvent withinSg is not immobile in thismodel. For sufficiently largeλ, however, much of the fluidin the gel layer behaves as though it were immobile tgood approximation. At a depth of 1/λ beneathSg, it canbe shown that the fluid velocity drops to approximately 1/e

of its value atSg in a reference frame where the particle isrest.

A representative example of a model “soft” particleshown in Fig. 1. The inner solid ellipsoid representssurface of hydrodynamic shear,Sp, which encloses the soliinner core. The outer wireframe structure denotes thesurface,Sg. The gel layer, whereλ in Eq. (1) is not zerois the volume,Vg, contained betweenSp and Sg. Outsideof Sg lies “free fluid” which obeys Eqs. (1) and (2) witλ = 0. In the BE methodology, the surfaces are represebyN interconnected triangular plates and the approximais made that field quantities such as electrostatic potenand fluid velocities are constant over a particular plate,may vary from plate to plate. The surfaces in Fig. 1 hN = 112. In addition, the fluid in the gel layer and outer fluregions is subdivided into shells that resemble onion swhich increase in thickness moving outward fromSg. Eachshell, in turn, is subdivided into a large number of voluelements (112 in the example shown in Fig. 1), and the sdiscrete approximation is applied to the volume elemeSystematic errors that result as a consequence of this disapproximation have been accounted for in the pastcarrying out a series of calculations on structures madof a variable number of plates and extrapolating the resto a large number of plates [21,28]. Provided the starstructures consist of 100 plates or more and providedare designed to reproduce translational diffusion constof (uncharged) model structures in the limit of largeN , thesecorrection factors amount to a few percent at most.

lr

e

Appendix A explains in detail the BE procedure fdetermining the velocity field,v, around a model particlwhen a gel layer is present. This is substantially differfrom the procedure used for “hard” particles [18,21]. Tsolution of the remaining field equations (Poisson andtransport) is the same as before [18]. Once the varfield equations are solved, it is possible to computetotal force exerted by the particle on the fluid,zT, or theexcess stress for various transport cases. For a generparticle, this is discussed in Appendix B. From these forand stresses, transport properties can be determined [18In the present work, we shall consider the translatiodiffusion constant,Dt, free solution electrophoretic mobilitµ, intrinsic viscosity,[η], Scheraga–Mandelkern parame[29,30],β (discussed below), and viscosity shape factorξ ,defined by

(5)ξ = limc→0

1

cVp

(η

η0− 1

),

wherec is the number concentration of particles,Vp = Vc +Vg is the particle volume (volume of solid inner core plusgel layer),η0 is the solvent viscosity, andη is the viscosityof the solution. The intrinsic viscosity is related toξ by [19]

(6)[η] = NAvVp

Mξ,

whereNAv is Avogadro’s number andM is the anhydroumolecular weight of the particle. In addition, particuinterest will be paid to the primary electroviscous effwhich is the excess viscosity due to charge effects ofmodel particle [16,19,31–33]. Letξ represent the shapfactor of the charged model particle andξ0 the shape factoof an identical but uncharged model particle. Followcurrent convention [33,34], it is convenient to defineprimary electroviscous coefficient,

(7)p = [η][η]0 − 1 = ξ

ξ0− 1,

where[η]0 is the intrinsic viscosity in the absence of electviscous effects. The second equality in Eq. (7) is only valithe particles are monodisperse and conformationally invant. In a BE calculation,ξ0 is readily calculated in the firsiterative cycle to include the effect of ion relaxation [19].

From a well-known relation between sedimentations(in s), and diffusion constants valid in the dilute solutiregime [29],

(8)s

Dt= M(1− v′ρsol)

NAvkBT,

wherekB is Boltzmann’s constant,T is absolute temperature,v′ is the partial specific volume of the particle in tsolvent, andρsol is the mass density of solvent. The dimesionless Scheraga–Mandelkern parameter [29,30],β , is de-fined as

(9)β = NAvs[η]1/3η01/3 2/3 ′ .

(100) M (1− v ρsol)


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Combined with Eq. (6), Eq. (9) can be written in alternatforms:

(10)β =( [η]M

100

)1/3η0Dt

kBT=

(NAvVpξ

100

)1/3η0Dt

kBT.

For an uncharged solid sphere, it is straightforward to sthat Eq. (10) reduces to a constant,β0 = 2.11× 106. Thebehavior of this parameter has been examined for unchasolid prolate and oblate ellipsoid models [29,30]. Foroblate ellipsoid,β ≈ β0 regardless of the axial ratio. Foprolate ellipsoids,β increases gradually with axial rat(major axis/minor axis). For axial ratios of 5 and 18,example,β/β0 = 1.057 and 1.23, respectively. In the presework, we shall examine how charge and the presence oflayer influenceβ .

Silica surfaces repel each other at short range [and this feature also applies to silica suspensions wexhibit behavior that cannot be explained by classical DLtheory [36]. A model involving protruding surface groupor a gel layer, was proposed to explain this behavioryears ago [37] and evidence in support of this interpretahas grown steadily over the years [38–41]. In the preswork, gel layer models of silica that contain a thin adense gel layer shall be considered and reasonable vof the parametersλ and gel layer thickness,d , are neededAnhydrous SiO2 has a density of 2.2 gm cm−3 [42] and if weconsider each SiO2 as a “segment,” the segment densityanhydrous SiO2, ρs, is 2.21×1022 cm−3, which correspondto a segment volume,vseg, of 4.52 × 10−23 cm3. If we

simply equateζ to 6πη0v1/3seg, thenλ for anhydrous SiO2,

λ∗, is estimated to be 12.19 nm−1 from Eq. (3). This can betaken as an absolute upper bound onλ for the gel layer. Itis readily verified that an uncharged porous sphere min the nanometer size range with aλ this large behavehydrodynamically like a solid sphere of equal radius togood approximation. With regards to the solution behavof small silica sols, it is well known that they swell to somdegree on hydration [38–43]. An early viscosity studynonaggregated silica sols could be explained, in terma solid model sphere, by an increase in radius of ab0.35 nm, or equivalently a monomolecular layer of bouimmobile water [43]. In this particular study, the sols weprepared and stored at high pH, but the pH was adjuto 2.0 just prior to the viscosity experiments in orderminimize the absolute charge on the silica particleshence the primary electroviscous effect. Also, the partradii were estimated to be in the range 1.8 to 3.5 nAssumingλ∗ = 12 nm−1 andλ = 6.00 nm−1 (75% of thegel layer consists of solvent), BE modeling shows thatuncharged sphere with an inner core radius,r1, of 3 nmand a gel layer thickness,d , equal to 0.51 nm has thsame viscosity as a suspension of uniform solid sphwith a radius of 3.35 nm. Thus, a gel layer thicknessapproximately 0.5 nm appears appropriate on the basthe work of Iler and Dalton [43]. From the work of Ramsand co-workers on silica sols in the 7-nm size range, it

l

s

concluded that the colloidal particles coexist with oligomespecies with the latter comprising 20% or more of the tosilica present [38]. For a spherical particle with 20%the silica located in the gel layer, it is straightforwardshow thatd = 0.0833r1(λ∗/λ)2. Assumingλ∗ = 12 nm−1,λ = 6 nm−1, and r1 = 6 nm (since the 7-nm value wicorrespond approximately tor1 + d), a value ofd = 2.0 nmis estimated. Recent atomic force microscopy measuremof the net force between a glass sphere (r1 ≈ 20 µm) and aglass plane as a function of surface separation are interpto imply a gel layer thickness of 2.0 nm [41]. It is uncertaif the discrepancy between the figures 0.5 and 2.0 nmdue to differences in particle size and/or other factors. This evidence that the short-range repulsive force betwsilica particles is reduced with increasing concentrationsimple salts, NaCl [36] and KCl [44]. Despite uncertaintin relevant quantities from a variety of sources, these resdo give us a reasonable range ofd andλ values for modelingpurposes.

3. Results

3.1. Test cases of a gel sphere

The transport theory of a model gel sphere with a ctrosymmetric charge distribution has been the subject oftensive past study and therefore provides a benchmarkfor the BE methodology. Consider a sphere with an incore radius,r1, equal to 3 nm, and an outer gel layerdius,r2, equal to 4 nm in an aqueous KCl solution at 20◦C(η0 = 1.002 cp). The thickness of the gel layer,d , in thiscase is 1 nm. The ionic strength is chosen to be 10.33 mwhich givesκr1 = 1.00 (κ = Debye–Hückel screening parameter), which is in the particle size range of greainterest in the present work. The small ion diffusion costant,Dα , or equivalently the small ion mobility,mα (mα =Dα/kBT ), or hydrodynamic radius,xα (xα = 1/6πη0mα),is estimated from limiting molar ionic conductivities,λ∞

α ,and the Nernst–Einstein relation. The limiting molar ionconductivities (in 10−4 S m2 mol−1) are usually reported a25◦C. If zα is the valence of ionα, then the hydrodynamiion radius (in nm) is given by [20]

(11)xα = 9.201z2α

λ∞α

.

From tables ofλ∞α , xα = 0.1252 and 0.1206 nm for K+ and

Cl−, respectively [45]. Because these values are so siman average value of 0.1229 nm is used in the present wfor both ions. In addition, Na+ is also considered later in thpresent work and hasxα = 0.1837 nm. In the gel layer, wsetλ = 0.1 (high porosity), 1.0 (intermediate porosity), a10.0 nm−1 (low porosity). In the BE calculations, the number of plates,N , is set to 128.

Shown in Figs. 2–4 are mobilities (in 10−8 m2 V−1 s−1)versus total charge=Q|e| (where|e| is the protonic charge


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Fig. 2. Electrophoretic mobilities versus particle charge for a gel sphereλ= 0.1 nm−1. The sphere has an inner radius,r1, of 3 nm and the thicknesof the gel layer is 1 nm. The aqueous solvent is at 20◦C and contains0.01033 M KCl(κr1 = 1.0). Diamonds are from the Ohshima theory [4without ion relaxation and using the linear Poisson–Boltzmann potenSquares are from BE calculations(N = 128) without ion relaxation andusing the nonlinear Poisson–Boltzmann potential. Triangles are fromcalculations with ion relaxation included.

Fig. 3. Electrophoretic mobilities versus particle charge for a gel spwith λ = 1.0 nm−1. Except for the differentλ, conditions are identical tothose of Fig. 2.

Fig. 4. Electrophoretic mobilities versus particle charge for a gel spwith λ= 10.0 nm−1. Except for the differentλ, conditions are identical tothose of Fig. 2.

for the gel sphere described in the previous paragraphλ = 0.1, 1.0, and 10.0 nm−1, respectively. The diamondcome from the theory of Ohshima of a charged sphereered with an uncharged polymer layer [46] under conditiidentical to those employed in the BE calculations. Eqtions (16) of Ref. [46] has been numerically integratedthe equilibrium electrostatic potential approximated bysolution of the linear Poisson–Boltzmann equation (Eq. (of Ref. [46]), and ion relaxation is ignored. Filled squarepresent BE mobilities in the absence of ion relaxat

Fig. 5. Electrophoretic mobilities (ion relaxation included) versus partcharge for a gel sphere withλ= 0.1, 1.0, and 10.0 nm−1. The gel sphere har1 = 3 nm, a gel layer thickness of 1 nm, in an aqueous solvent conta0.01033 M KCl at 20◦C (κr1 = 1.0). Diamonds, squares, and trianglcorrespond toλ = 0.1, 1.0, and 10.0 nm−1, respectively.

but the solution of the full (nonlinear) Poisson–Boltzmais employed. The Ohshima and BE mobilities in thesence of ion relaxation should agree with each other inlimit of small |Q|, and that is observed in all three caswhich cover a hundred fold range in gel porosity. Filledangles represent the BE mobilities with the inclusion ofrelaxation. It is observed that ion relaxation becomesnificant at about the same|Q| where the approximation othe linear Poisson–Boltzmann equation breaks downregards to unrelaxed mobilities. For the examples conered in Figs. 2–4, ion relaxation becomes significant for|Q|greater than about 20. In these examples, the reducedface potential,y = |e|ζ/kT (ζ equals the surface potention Sp), that corresponds to|Q| = 20 is 2.20. This behaviois not surprising since it has been known for a long timeleast for “hard” particles, that the effects of ion relaxatbecome significant when the nonlinear terms in the elecstatic potential become significant [22,47,48].

The presence of a gel layer influences the mobility inumber of respects. Under otherwise identical conditioa largeλ (gel layer of low porosity) causes a significadrop in mobility. This is clearly demonstrated in Fig.where relaxed mobilities forλ = 0.1, 1.0, and 10.0 nm−1 areplotted together. The relative discrepancy is actually greaat low |Q|, but persists at high|Q| as well. The less porougel layer also tends to reduce the effect of ion relaxaon mobility. Note that unrelaxed and relaxed BE curare much closer together in Fig. 4 than in Figs. 2 andA more subtle difference is seen in the absolute mobmaxima [49], which is shifted to higher|Q| for larger λ.For theλ = 10.0 nm−1 case, no absolute mobility maximis observed for the charge range studied.

Figure 6 summarizes the corresponding primary elecviscous coefficient results for the same gel sphere moThe data for eachλ are well fit by a cubic equation of thform

(12)− ln

(p

Q2

)= c0 + c1Q+ c2Q

2 + c3Q3,

where thecj ’s are polynomial coefficients. Of particulainterest are thec0’s, which equal 8.35, 8.63, and 9.62 fλ = 0.1, 1.0, and 10.0 nm−1, respectively. For a weakl


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Fig. 6. Electroviscous coefficient,p, versus particle charge for gel sphemodels. The gel sphere hasr1 = 3 nm, a gel layer thickness of 1 nm,an aqueous solvent containing 0.01033 M KCl at 20◦C (κr1 = 1.0). Dia-monds, squares, and triangles correspond toλ = 0.1, 1.0, and 10.0 nm−1,respectively. The solid lines represent cubic fits to the data.

charged “hard” sphere, the Booth theory [31] predicts

(13)p =Ay2,

wherey is the reduced dimensionless surface potentialcussed previously, andA is a constant. Under the conditioemployed in this test case,A = 0.0166 on the basis of thBooth theory [31]. Also, for a weakly charged sphere,Q isrelated toy by (in cgs units)

(14)Q =(ε0kBT r1

q2

)y(1+ κr1)

or Q = 8.459y under present conditions withε0 = 80.36.Using this in Eq. (12) along with Eq. (13), we obtaA = 0.0169, 0.0128, and 0.00475 forλ = 0.1, 1.0, and10.0 nm−1, respectively. The value forλ = 0.1 nm−1 (veryporous gel layer exterior to a “hard” spherical core of rad3 nm) is in good agreement with the Booth theory. Forλd assmall as this, the gel layer has very little effect on the flflow near the model particle and its hydrodynamic behais essential identical to that of a solid sphere of radiusr1.

We would also like to compare the BE predictiowith the recent theory of Natraj and Chen [16] on tprimary electroviscous effect of weakly charged, compleporous spheres. Somewhat different conditions from themployed above must be used in order to make a dcomparison. The conditions arer1 = 0, r2 = 4 nm, T =25◦C, η0 = 0.89 cp, 0.0058 molar monovalent salt withsmall ion diffusion constant of 10−5 cm2 s−1, ε0 = 78.3,Q = −16.44, andλ = 2.50 nm−1. Under these conditionsthe porous sphere theory [16] predictsp = 0.075. The BEcalculations yieldp = 0.0742 (single charge placed at tcenter of the sphere) or 0.0758 (charge uniformly distributhroughout the sphere). Agreement is seen to be gThe corresponding shape factors for the uncharged,ξ0, andcharged,ξ , porous spheres are 1.72 and 1.85, respectivOther transport cases have also been considered, suchtranslational diffusion constant [10] andξ0’s of unchargedporous spheres [8,11], and good agreement is obtainethese as well.

For an uncharged solid sphere, the Scheraga–Mandeparameter [29,30] equals 2.11 × 106. For a completely

.

e

r

n

Table 1Dt , ξ , and β/β0 for gel spheres (r1 = 3 nm, r2 = 4 nm, κr1 = 1.0,T = 20◦C)

Q y λ (nm−1) Dt (10−11 m2 s−1) ξ β/β0

5 0.594 0.1 7.00 1.097 1.0015 0.594 1 6.51 1.337 0.9945 0.594 10 5.43 2.324 0.997

20 2.204 0.1 6.90 1.167 1.00720 2.204 1 6.45 1.400 1.00020 2.204 10 5.41 2.366 0.99950 4.301 0.1 6.61 1.347 1.01250 4.301 1 6.25 1.557 1.00450 4.301 10 5.36 2.46 1.003

120 6.435 0.1 6.30 1.506 1.001120 6.435 1 6.06 1.695 1.001120 6.435 10 5.32 2.532 1.005

porous sphere of radiusr2, expressions forDt [10] and[η] [8,11] can be used in Eq. (10) to determine howβdepends on the dimensionless parameterσ = λr2,

(15)β

β0= 3

√G1(σ )/(1+ 10G1(σ )/σ 2)

G0(σ )/(1+ 3G0(σ )/2σ 2),

where G0(σ ) = 1 − tanh(σ )/σ 2, G1(σ ) = 1 + 3/σ 2 −3 coth(σ )/σ , and tanh and coth denote hyperbolic tangand cotangent, respectively. Forσ � 1, β/β0 can besubstantially larger than 1. Forσ = 0.1, for example,β/β0 = 3.85. However, forσ � 1, β/β0 ≈ 1.0. With thecharged gel sphere considered at the beginning ofsection (r1 = 3 nm,r2 = 4 nm,κr1 = 1.0, T = 20◦C, etc.)we are in a position to assess the effect ofλ andQ onβ/β0.The results are summarized in Table 1. Despite substavariation inDt andξ , the Scheraga–Mandelkern parametare essentially invariant withλ andQ. The robust behavioof β with particle shape has long been recognized [29],that behavior also appears to be true for gel layer poroand particle charge as well. As the results of the next secdemonstrate, this is also observed for nonspherical proellipsoid models.

3.2. Modeling of a silica sol sample

The transport behavior of several small but highly charsilica sol samples over a range of monovalent salt contrations has been analyzed in past work [23–25]. In meling, the assumption was made that samples consistemonodispersesolid prolate ellipsoids which remained coformationally invariant at all salt concentrations studiOn the basis of the work of Iler and co-workers [42,4it was also assumed that due to hydration, the solid pcles expanded by an average thickness of 0.35 nm relto the unhydrated particle [23]. From experiments at coparatively high salt content (50 mM NaCl), which yieldintrinsic viscosities and translational diffusion constants,lipsoidal dimensions were initially estimated assuming etroviscous effects were negligible. The particle charges w


at

heelecncebe

eretedterssil-

Forand

everthanssedountha

, de-pan)

se),sIfthe

to

for-el.c thr ofnten] is

L2ex

oreple

rizesad-

at-

tof thethe

E1.

of, for

el-er.rti-

w-is-ionsy, thee

overCl)some

eed-

eltesrk

esee un-son-ro-

inoseyerenore

Table 2Experimental transport properties of dilute SOL2 in NaCl solutions25◦C, pH 9.5

[NaCl] (mM) µ (−10−8 m2 V−1 s−1) [η] pa

2 – 4.65 0.675 4.1 3.9 0.40

10 3.7 3.3 0.1820 3.2 2.9 0.0430 2.9 2.9 0.0450 2.7 – –

a From Eq. (7) with[η]0 = 2.8 cm3/g.

estimated from dynamic mobility experiments [24,25]. Tinitial model parameters were then used to estimate thetroviscous effect at high salt content by BE modeling. Sia small effect of several percent was predicted to stillpresent at 50 mM NaCl, the ellipsoidal dimensions wreestimated using “high salt” intrinsic viscosities correcfor the electroviscous effect. The revised model paramewere then used to compute transport properties of fourica sol samples over the range 2 to 50 mM NaCl [25].the smaller samples, fair agreement between modelingexperiment was achieved. For the larger samples, howa substantially larger electroviscous effect was observedwas predicted by modeling. The question to be addrehere is whether or not the presence of a gel layer can accfor the observed discrepancy. It should be emphasizedthe resulting geometry of the high-salt-content structureduced by this procedure, is sensitive to the assumed exsion or degree of hydration,sv [23]. Equations (5) and (6can be combined and written

(16)ξ = ρ∗[η](1+ sv)

,

whereξ is the shape factor,[η] is the intrinsic viscosity,ρ∗is the anhydrous density of the particle (silica in this caand 1+sv = Vp/Vp0, whereVp andVp0 are the total volumeof the hydrated and unhydrated particles, respectively.fis the fraction of silica that resides in the gel layer ofparticle and use is made of Eq. (4), it is straightforwardshow that

(17)sv = [(λ∗/λ)2 − 1

]f.

Equation (17) combined with Eq. (4) provides a straightward link betweensv and the parameters of the gel modThe larger the assumed expansion, the less eccentrimodel ellipsoids. For a relatively thick assumed gel laye2 nm, the geometry that best reproduces the high salt co[η] andDt for the larger sol samples reported in Ref. [25a sphere.

In Ref. [25], the two larger silica sols examined, SOand SOL3, exhibited the greatest discrepancy betweenperiment and solid particle modeling. Since there are mextensive experimental data available on SOL2, that samshall be examined in the present work. Table 2 summathe experimental transport properities for SOL2 [25]. In

-

,

tt

-

e

t

-

Table 3SOL2 BE models and principal parameters

Model −Q (% core, % gel) bc (nm) rc d (nm) λ (nm−1) f

A 100 (100, 0) 3.28 4.06 0.0 – –B1 190 (80, 20) 2.93 4.42 0.51 6.00 0.112B2a 190 (80, 20) 2.93 4.42 0.93 4.24 0.112B2b 250 (80, 20) 2.93 4.42 0.93 4.24 0.112C1a 170 (80, 20) 2.77 4.62 0.51 8.18 0.200C1b 170 (0, 100) 2.77 4.62 0.51 8.18 0.200C2a 170 (80, 20) 2.77 4.62 2.00 3.54 0.200C2b 170 (0, 100) 2.77 4.62 2.00 3.54 0.200D1 260 (0, 100) 3.18 3.26 1.00 6.00 0.200D2 260 (0, 100) 3.18 3.26 1.50 4.66 0.200E1 420 (0, 100) 4.16 1.00 2.00 4.16 0.200E2 420 (0, 100) 4.16 1.00 2.50 3.55 0.200

dition, the hydrodynamic radius (from dynamic light sctering) was found to be 6.2 ± 0.3 nm (orDt = 3.9 ± 0.2 ×10−7 cm2 s−1). Within experimental error, this was foundbe independent of salt concentration. The parameters odifferent models studied are summarized in Table 3. Formoment, we shall focus on Models A, B1, C1a, D1, andValues of the gel layer thickness,d , and fraction of silicain the gel layer,f , are assumed, as well as distributionscharge in the core and gel layer. In Models B1 and C1aexample, 80% of the total particle charge,Q, is placed inthe core (along the principal axis between the foci of thelipsoid), and 20% is uniformly distributed in the gel layIn Models D1 and E1, on the other hand, the entire pacle charge is uniformly distributed in the gel layer. Folloing a procedure similar to that used in Ref. [25] and dcussed in the previous paragraph, the ellipsoid dimensare determined that reproduce, with reasonable accurachigh salt content[η]’s andDt ’s. The Model Q that yields thhigh-salt-content electrophoretic mobility,µ, is determinedby successive estimation. Resulting transport propertiesa range of different salt concentrations (2 to 50 mM Naare then computed using these model parameters, andkey results are summarized in Table 4.

As discussed at the end of the Methods section, we nestimates of the gel layer thickness,d , the screening parameter,λ (or equivalently, the fraction of silica in the glayer,f ), and the details of the charge distribution. Estimaof d are in the range 0.35 [43] to 2.0 nm [41]. From the woof Ramsay and co-workers, we have an estimate off of ap-proximately 0.20. Although we cannot be certain that thparameters are appropriate for this specific SOL2 samplder the conditions of the experiment, they do provide reaable values for modeling purposes. Model A is a rigid plate ellipsoid (d = 0.0 nm andf = 0.00) with minor axisbequal to 3.28 nm, axial ratio (major axis/minor axis)r equalto 4.06, andQ = −100. Although there is no gel layerthis model, it is included in the present work for the purpof comparison. Model B1 contains a relatively thin gel lawith d = 0.51 nm andf = 0.112, which is smaller than thvalue of 0.20 of Ramsay and co-workers [38]. The “c” obandr in Table 3 indicates that we are referring to the c


Table 4Transport properties of prolate ellipsoid models of a silica sol particle (SOL2)

Model [NaCl] (mM) Vp (nm3) Dt (10−11 m2 s−1) µ (−10−8 m2 V−1 s−1) ξ ξ/ξ0 − 1 β/β0

A 2 601.8 3.65 4.23 6.21 0.309 1.057A 10 601.8 3.82 3.48 5.30 0.117 1.049A 50 601.8 3.91 2.67 4.90 0.033 1.046B1 5 703.8 3.64 3.85 5.13 0.179 1.041B1 10 703.8 3.71 3.48 4.81 0.121 1.041B1 50 703.8 3.82 2.72 4.42 0.033 1.040C1a 2 635.4 3.60 4.31 6.04 0.337 1.051C1a 10 635.4 3.79 3.52 5.06 0.120 1.042C1a 50 635.4 3.89 2.73 4.65 0.030 1.042D1 2 879.2 3.54 4.34 4.38 0.361 1.035D1 10 879.2 3.71 3.59 3.63 0.130 1.018D1 50 879.2 3.82 2.75 3.31 0.031 1.017E1 2 1129.8 3.64 4.74 3.17 0.419 1.038E1 10 1129.8 3.78 3.68 2.56 0.148 1.006E1 50 1129.8 3.88 2.84 2.31 0.036 0.998B2a 2 939.6 3.38 3.92 4.81 0.252 1.044B2a 10 939.6 3.53 2.99 4.14 0.078 1.035B2a 50 939.6 3.59 1.97 3.90 0.014 1.033B2b 2 939.6 3.37 3.87 4.90 0.276 1.045B2b 10 939.6 3.52 3.10 4.18 0.088 1.034C1b 50 635.4 3.89 2.91 4.66 0.031 1.042C2a 2 1599.4 3.03 3.09 3.79 0.127 1.030C2a 10 1599.4 3.10 1.97 3.45 0.027 1.023C2a 50 1599.4 3.12 0.895 3.37 0.003 1.022C2b 2 1599.4 2.99 3.56 3.95 0.174 1.031C2b 50 1599.4 3.12 1.39 3.38 0.006 1.022D2 2 1167.7 3.32 4.10 3.94 0.303 1.031D2 10 1167.7 3.47 3.26 3.32 0.098 1.017E2 2 1416.1 3.41 4.54 3.01 0.365 1.031E2 10 1416.1 3.54 3.54 2.48 0.121 1.004

d in. Aro-tan-ior

reithinoxi-arge

sl A,C1aick-o a

rgeinn in

l

rge

il-

o-icar, is

tode-per-esengetopos-

tentns-ideip-thepar-that

theesseory

particle. It is assumed that 80% of the charge is containethe core and 20% is uniformly distributed in the gel layernet charge ofQ = −190 must be used in this case to repduce high salt mobilities. This absolute charge is substially larger than that necessary in Model A. This behavhas a simple physical interpretation. For largeλ, the fluid inthe gel layer is largely immobile relative to the particle coand consequently any charge due to the mobile ions wVg is essentially immobile as well. To a reasonable apprmation, the particle behaves as a rigid body with a net chequal toQ plus the net charge within Vg due to ions fromsolution. In order to offset this effect and bring mobilitieof the gel layer models into good agreement with Modethe absolute net charge of B1 must be increased. Modelis similar to B1 and has the same assumed gel layer thness. However,f is increased to 0.20 and corresponds tdenser gel than Model B1. In Model D1,f andd are set to0.20 and 1.0 nm, respectively. In addition, the particle chais assumed to be uniformly distributed in the gel layerthis case. We shall discuss the issue of charge distributiomore detail later. Model E1 is similar to D1, butd is set equato 2.0 nm. For a gel layer as thick as this, a sphere (rc = 1.0)must be chosen to reproduce the high-salt-contentDt and[η]observed experimentally. Also note the high absolute cha|Q| = 420, required to reproduce the high-salt-contentµ. Itis worthwhile pointing out that it was the high charge of s

,

ica sols coupled with their relatively low electrokinetic ptentials that led to the conclusion that the “surface” of silsols was penetrable by counterions [50]. This, howevevery similar to the gel model of the present work.

The five Models A, B1, C1a, D1, and E1 correspondvery different structures which have been specificallysigned to reproduce the high-salt-content transport proties of SOL2. Comparing the transport properties of thfive models given in Table 4 over the salt concentration rafrom 2 to 50 mM NaCl, they are seen to be very similareach other. Thus, each of these five models representssible SOL2 structures on the basis of their high-salt-condiffusion constants and intrinsic viscosities. High salt traport properties of all five models can be made to coincwith each other, to a good approximation, by simple manulation of the net charge. Attention shall now be turned tosalt dependence of the model transport properities. Coming these with experimental results in Table 2, it is seenalthough theµ’s are fairly well reproduced,p from experi-ment is substantially larger at low salt content thanξ/ξ0 − 1from Table 4. From Table 4,ξ/ξ0 − 1 in 2 mM salt goesfrom 0.31 to 0.42 on going from Model A to E1 whereasexperimental value is 0.67. Thus, increasing the thicknof the gel layer does improve the agreement between thand experiment, but only to a limited extent.


iclen, ispor

est aonalsera

n ofns.ccurcle1ins

esstheE2,of

thesent

e ath-

delat

ighmMbleous

tion4eis

delstent

etictha

inilitycle.

s.

d in

allengel

his

thendtheceary

1byicleodelple

tioniallyityore

hickionsield

lygetheargen of38,gelfavor

tentL2ich

iverope

tal

hisfromionhe

ded(ex-re-

tend

It appears that any static model, in which the partconformation does not change with the salt concentratiounable to account simultaneously for the observed transbehavior (Dt, µ, andp) of SOL2. Although the invariancof Dt observed experimentally with salt argues againsignificant conformational change, a modest conformatichange is possible sinceDt is fairly insensitive to changein particle size. Consequently, we shall next consider sevadditional models that mimic the effects of an expansiothe gel layer on going from high-salt to low-salt conditioThere is independent evidence that such an expansion o[36,44]. On going from B1 to B2a, the core of the partiremains the same, as doQ andf , butd increases from 0.5to 0.93 nm. Since the fraction of Si in the gel layer remaconstant,λ is reduced from 6.0 to 4.24 nm−1. Similarly,on going from Model C1a to C2a, the gel layer thicknexpands from 0.51 to 2.0 nm; on going from D1 to D2,expansion is from 1.0 to 1.5 nm; and on going from E1 tothe expansion is from 2.0 to 2.5 nm. The primary quantityinterest here in theapparent electroviscous coefficient,p′.This is similar to Eq. (7), but account must be take ofchange in conformation with salt concentration. For prepurposes, it can be written

(18)p′ = ξVp

(ξ0)hs(Vp)hs− 1,

where the “hs” subscript indicates the model appropriathigh salt. If we use Models B1, C1a, D1, or E1 for the higsalt case and B2a, C2a, D2, or E2 for 2 mM NaCl, thenp′ =0.50 (Model B1 at high salt and B2a at low salt), 1.11 (MoC1a at high salt and C2a at low salt), 0.626 (Model D1high salt and D2 at low salt), or 0.692 (Model E1 at hsalt and E2 at low salt) when the salt concentration is 2NaCl. At first glance, it might appear that an expandagel layer model readily accounts for the large electrovisceffect. There is, however, a problem with this interpretasince theµ’s for Models B2a, C2a, and D2 given in Tablefall below experimental mobilities. This is particularly trufor Model C2a where the “expansion” of the gel layerlargest of all of the cases considered. For these moalthough the expandable gel layer model is consiswith the salt dependence ofp′, it is inconsistent withthe salt dependence ofµ. The possibility that the particlcharge varies with salt is inconsistent with electrokinemeasurements on these particular sol samples [24] andpossibility shall not be considered further.

As discussed previously, “trapping” of counterions withthe gel layer reduces the absolute electrophoretic mobrelative to that expected for a solid, impermeable partiFor highly charged particles, simply increasing|Q| doesnot necessarily lead to higher|µ|, as Fig. 5 demonstrateFor solid spherical particles, the dependence ofµ onQ, ormore precisely, the “zeta” potential, has been exploredetail [49]. Depending on a number of conditions,µ mayplateau at sufficiently high|Q| or actually go through amaximum. In going from Models B2a to B2b (d = 0.93 nm),

t

l

s

,

t

|Q| increases from 190 to 250, but|µ| remains nearlyconstant at about 3.9 and 3.0 × 10−8 m2 V−1 s−1 in 2 and10 mM NaCl, respectively. These mobilities, in turn, fsubstantially below experimental values. This finding thraises a new question. Can we conclude that a thicklayer model is simply inconsistent with experiment for tparticular sample on the basis of the low predicted|µ|?

To help address this question, we shall consideradditional variable of the detailed charge distribution ahow this influences transport. For solid-particle models,details of the charge distribution do, in principle, influenµ [51], but that influence appears to be of secondimportance [18,20]. In going from Model C1a to C(d andQ remain constant),|µ| in 50 mM NaCl increases b7% when 80% of the net charge is moved from the partcore to the gel layer. Thus, a significant change in the mproduces only a modest change in mobility in this examwhere the gel layer is thin (d = 0.51 nm). ComparingModels C2a and C2b, a similar change in charge distribuis carried out, but in this case, the gel layer is substantthicker (d = 2 nm). In this case, the changes in mobilgiven in Table 4 are much greater. Thus, mobilities are msensitive to the details of the charge distribution when a tgel layer is present. Furthermore, the charge distributfor Models D1 and E1 have been designed in order to ymodels that have fairly thick gel layers (d = 1.0 or 2.0 nm)but are also able to reproduceµ measured experimentalin 50 mM NaCl. Thick gel layer models can yield larabsolute mobilities provided a substantial fraction ofparticle charge resides in the gel layer. Since the high chof silica sols arises as a consequence of deprotonatiosurface and gel layer silanol and silicilic acid residues [40,42], placing a high fraction of the net charge in thelayer is not unreasonable. The present modeling resultssuch a charge distribution on the basis ofµ’s.

The “expandable gel layer” model that is most consiswith all of the experimental transport properties of SOis the Model E1–E2 pair. In this case, the gel layer, whis quite thick at high salt content (d = 2.0 nm), expandsto d = 2.5 nm at low salt content. Because the relatexpansion of the gel layer is modest in this case, the din low salt mobility is also modest (Table 4). At the samtime, however, this expansion yields a predictedp′ of 0.69at 2 mM NaCl which is comparable to the experimenvalue of 0.67. In Fig. 7, experimentalp’s (diamonds) areplotted versus salt concentration along with apparentp′ forModels B1 (solid line) and B2–B1 (dashed line). On tbasis, it can be concluded that a modest expansion (d = 2.0 to 2.5 nm) of the gel layer at a salt concentratbelow approximately 5 mM NaCl is consistent with texperimental primary electroviscous effect of SOL2.

Finally the Scheraga–Mandelkern ratios are also incluin Table 4. Although they exceed 1 by several percentcept for the spherical Models E1 and E2), it should bemembered thatβ/β0 ≈ 1.057 for a solid, uncharged prolaellipsoid with an axial ratio of 5. Thus, particle charge a


1alt”withof

this

enurendort

lec-iteize,Thea-les

per-gthe

x-

dentionchly

s ef-lec-

en-sesthe

ime.hly

meteon-

ar-s aheev-

ilarirlyn-

rgersuf-d onto

v-he

s isthe. Al-dict

ffectctedffu-lowwithndsinlsoas a2],entgel

utionent

. Inforears[43]ple[23]into

er-te auch

crep-ent.ithousiththeysols

eterthe

nt tont,

delleson

Fig. 7. p′ versus [NaCl] for SOL2. Experiment (diamonds), Model B(solid line), and B2–B1 (dashed line). In the B2-B1 model, the “high sshape factor and particle volume of B1 are used in Eq. (18) alongthe “low salt” (2 to 10 mM NaCl) shape factors and particle volumeModel B2.

the presence of a gel layer appear to have little effect onparameter. For SOL2, it was not possible to computeβ ex-perimentally since the particle molecular weight,M, was un-known.

4. Discussion and summary

The primary objective of the present work has bethe implementation of a boundary element, BE, procedapplied to the transport of a general model polyion aits application to several problems of interest. Transpproperties considered include translational diffusion, etrophoretic mobility, and viscosity. The model itself is qugeneral and can accommodate particles of arbitrary sshape, geometry of the gel layer, and charge distribution.effect of ion relaxation is included which makes it applicble to highly charged polyions. It is assumed the particare dilute, rigid and monodisperse and have a uniformmeability,λ, within the gel layer. In addition, the resultintransport properties are derived in the linear regime ofapplied flow, electric, or shear fields.

As a first application, a sphere with a gel layer is eamined over a broad range of polyion charge,Q, and λ.The BE results are in excellent agreement with indepentheory where available. This includes translational diffusand intrinsic viscosity of “porous spheres” [8–11] whilack a solid inner core, electrophoretic mobility of weakcharged gel spheres [1,6,13–15,46], the electroviscoufect of weakly charged porous spheres [16], and the etroviscous effect of gel spheres in the limitλ → 0 in the gellayer [31–33]. Even for spherical polyions containing a ctrosymmetric charge distribution, however, there are cawhere independent theory is currently unavailable andtransport properties are being reported here for the first tThis includes electrophoresis and electroviscosity of higcharged gel spheres. The Scheraga–Mandelkern para[29,30] is also examined and found to be essentially cstant over a broad range ofλ andQ.

The BE method is then applied to the transport of a pticular silica sol sample, SOL2 [25], which is modeled aprolate ellipsoid with and without a gel layer. Provided tnet particle charge is adjusted, it is possible to identify s

r

eral gel layer models which have transport behavior simto that of a solid model, and these, in turn, reproduce fawell the experimental mobility data. However, they all uderestimate the primary electroviscous effect,p. Althoughincreasing the thickness of the gel layer does yield lap’s at low salt content, the effect does not appear to beficient to reconcile experimental and model values basea single static structure. A thick gel layer, in turn, leadslow mobilities, but this effect can be partially offset by moing much of the net particle charge into the gel layer. Tpossibility that a conformational change with salt occurconsidered next. In the “expandable gel layer” model,gel layer expands as the salt concentration decreasesthough this expandable gel layer model does indeed prea stronger dependence of the primary electroviscous eon salt concentration than the static models, the prediabsolute electrophoretic mobilities and translational dision constants tend to fall below experimental values atsalt content. However, an expandable gel layer modela thick (2-nm) gel layer at high salt content which expaonly slightly on going to low salt content is able to explaquite well all of the transport measurements. It should abe mentioned that a gel layer has also been suggestedpossible explanation for anomalous transport of Ludox [5which is a silica sol similar to those considered in the preswork. This analysis does not prove that the expandablelayer model discussed above describes the actual solstructure of SOL2, only that this model is most consistwith the measured transport properties (Dt, µ, and[η]) overthe monovalent salt concentration range from 2 to 50 mMfact, we are not entirely satisfied with this interpretationtwo reasons. First of all, a gel layer as thick as 2 nm appat odds with the expansion observed by Iler and Daltonon small silica sols comparable in size to the SOL2 samconsidered here. Second, electron micrographs of SOL2show particles that are heterogeneous and aggregatedrodlike or prolate ellipsoidal structures. Also, X-ray scatting studies on an aged SOL2 sample [53] clearly indicaprolate ellipsoid or rod-shaped structure. Other factors sas sample heterogeneity may also be responsible for disancies between the prolate ellipsoid models and experimAdditional experimental studies are currently underway wthe aim of isolating and characterizing more homogenesamples. BE modeling shall be carried out in parallel wexperimental measurements of transport properities asbecome available. Thus, the present analysis of the silicais preliminary.

What the analysis of the Scheraga–Mandelkernparam[29,30], β , has shown is that particle charge andpresence of a gel layer have little influence onβ for thecases considered in the present work. This is equivalesaying that the product of translational diffusion constaDt, and the cube root of the intrinsic viscosity,[η]1/3, ispredicted to be nearly constant for a single rigid mostructure. With regard to highly charged colloidal particor rigid macromolecules, this implies that electrolyte fricti


gh1ssaltbeally.

ther–

ermgelTheions

ity,

ityionsle,

ed)bol

torsseisin

ere.thenatethe

tof

the

ofrem

fyang

andrite

t

rger

e

ce.

eThein

(throughDt) and the primary electroviscous effect (throu[η]) are closely related to each other. Because of the/3power law dependence of[η], the primary electroviscoueffect is predicted to depend much more strongly onconcentration and charge than electrolyte friction. It willinteresting to see if this behavior is observed experiment

Appendix A. Velocity field

For the problem of interest in the present work,Green’s function for the low-Reynolds-number NavieStokes equation with the presence of an additional tthat accounts for the friction between “segments” in thelayer and the fluid (the Brinkman equation) is needed.momentum balance and solvent incompressibility equatcan be written

(A.1)∇ · Tλk = η0∇2uλ

k − ∇pλk = ekδ(x − y)+ η0λ

2uλk ,

(A.2)∇ · uλk = 0,

where Tλk , uλ

k , and pλk denote the stress tensor, veloc

and pressure of the singular solution,η0 is the solventviscosity, ek is a unit vector in directionk, δ representsthe delta function, andλ (which has units of length−1) isa measure of gel density. In what follows, the fluid velocat “field” position,y, is represented as an integral equatover the “source” position,x. The differential operatorappearing in Eqs. (A.1), (A.2) act on the “source” variabx, instead of the “field” variable,y. In this appendix asthroughout the entire paper, bold (or double underlinupper case symbols represent tensors or matrices, and(or single underlined) lower case symbols represent vecThe solution of Eq. (A.1) is well known for the special caλ = 0 [27,54]. The solution of the general problem [56]straightforward using Fourier transforms, as illustratedRef. [27], and only a brief outline shall be presented hEquations (A.1), (A.2) are first Fourier transformed andtransformed equations can then be combined to elimithe velocity term. The resulting algebraic equation forFourier transform ofpλ

k is again transformed to yield

(A.3)pλk = − rk

4πr3 ,

wherer = |r|, r = x − y, andrk denotes thekth componenof r. With the pressure term solved, the Fourier transformEq. (A.1) is then retransformed to yield

(A.4)uλk = ek · Uλ,

whereUλ is a 3 by 3 cartesian matrix and them, n compo-nent is

Uλmn = − 1

8πη0

{[2w1 + 2

λw2 − 2

λ2(v3 −w3)

]δmn

(A.5)−[2w1 + 6

λw2 − 6

λ2 (v3 −w3)

]rmrn

r2

},

wherevp = 1/rp, wp = e−λr/rp, andδmn is the Kronekerdelta. Equations (A.3)–(A.5) can be combined to yield

d.

singular stresses(Tλk

)mn

= −pλk δmn + η0

[∇mUλkn + ∇nU

λkm

]

(A.6)

= f1rkδmn + f2(rmδnk + rnδmk)+ f3rkrmrn

r3 ,

where

(A.7)f1 = 1

4π

{v3 + 2w3 + 6

λw4 − 6

λ2 (v5 −w5)

},

(A.8)f2 = 1

4π

{λw2 + 3w3 + 6

λw4 − 6

λ2(v5 −w5)

},

(A.9)f3 = − 1

2π

{λw2 + 6w3 + 15

λw4 − 15

λ2 (v5 −w5)

}.

In the limit λ goes to zero, Eqs. (A.5) and (A.6) reduce to

(A.10)U0mn = − 1

8πη0r

(δmn + rmrn

r2

),

(A.11)(T0k

)mn

= 3rmrnrk4πr5

,

which are well known.The starting point of any boundary element solution

the velocity/pressure field is the Lorentz reciprocal theo[26,27,54–56], which in differential form can be written

(A.12)s′ · v + ∇ · (v · S′) = s · v′ + ∇ · (v′ · S),

where v and v′ represent any velocity fields that satisthe low-Reynolds-number Navier–Stokes equation ofincompressible fluid,s and s′ represent the correspondinexternal force/unit volume at a particular point in space,S andS′ are the corresponding stress tensors. We can w

(A.13)∇ · S = η0∇2v − ∇p = −s

with an equivalent expression for{v′, s′,S′}. In the presenwork, it is convenient to expresss as

(A.14)s = se − η0λ2v,

wherese represents the external forces on the fluidexceptthose due to segment friction in the gel layer. Local chadensities in the fluid may makese significant. ConsidevolumeV bounded by outer surfaceS2 and inner surfaceS1.Integrating Eq. (A.12) overV and using the divergenctheorem yields∫V

v · s′ dVx +∫S2

v · S′ · ndSx −∫S1

v · S′ · ndSx

(A.15)

=∫V

v′ · sdVx +∫S2

v′ · S · ndSx −∫S1

v′ · S · ndSx,

wheren is a local outward normal to the respective surfa(For S1, n points into the volume,V , and forS2, n pointsoutward fromV .) For the primed field in Eq. (A.15), choosthe singular solution discussed earlier in this appendix.parameterλ is chosen to match that of the actual fluid


id,e.

ea

larste

fine)

uid

gel,ear,e

e

ningdaryare

In

ors

re

–

8)

domainV . The assumption is made that in the actual fluλ is constant over a well-defined domain of the fluid volumUsing Eqs. (A.1), (A.13), (A.14) in (A.15) yields

vk(y)Φ(y,V )= −∫V

se · uλk dVx

+∫S2

(uλk · f + v · Tλ

k · n)dSx

(A.16)−∫S1

(uλk · f + v · Tλ

k · n)dSx,

wheref = −S · n is the local hydrodynamic stress force/aron S2 or S1 and Φ(y,V ) equals 1 ify lies within V , 0if y lies outside ofV , or 1/2 if y lies on either boundingsurfaceS2 orS1. The advantage of using the general singusolution withλ chosen to match that of the fluid of interein domainV is that theη0λ

2v term drops out of the volumintegral over the external forces. Sincev is the quantity beingdetermined, it is desirable to eliminate this term. Also de(Dλ)km = (Tλ

k · n)m which makes it possible to write (A.16in compact form as

v(y)Φ(y,V )= −∫V

Uλ(x,y) · se(x) dVx

+∫S2

[Uλ(x,y) · f(x)+ Dλ(x,y) · v(x)

]dSx

(A.17)−∫S1

[Uλ(x,y) · f(x)+ Dλ(x,y) · v(x)

]dSx.

For the model of interest in the present work, the fldomain is divided into an outer volume,V0, which extendsfrom the surface that defines the outer boundary of thelayer,Sg, to a surface at infinity,S∞; and an inner volumeVg, that extends from the surface of hydrodynamic shSp, to Sg. For the model ellipsoid shown in Fig. 1, thshaded inner ellipsoid representsSp in this example, and thouter wireframe surface representsSg. The volume betweenthese two surfaces constitutes the gel layer. The screeparameter,λ, vanishes inV0, but is finite (and assumeconstant) inVg. The reference frame is taken to be stationwith respect to the particle and stick boundary conditionsassumed to hold onSp. Thus,v = 0 for points onSp. In thisreference frame, letv∞(y) denote the fluid velocity aty if theparticle were absent. This velocity field,v∞(y), is specifiedby the boundary conditions in a particular application.domainV0, setλ = 0. Also, the surface integral overS∞contributesv∞(y) to the rhs of Eq. (A.17) and

v(y)Φ(y,V )= v∞(y)−∫V0

U0(x,y) · se(x) dVx

(A.18)−∫S

[U0(x,y) · f(x)+ D0(x,y) · v(x)

]dSx.

g

In domainVg, λ �= 0 and Eq. (A.17) becomes

v(y)Φ(y,V )= −∫Vg

Uλ(x,y) · se(x) dVx

+∫Sg

[Uλ(x,y) · f(x)+ Dλ(x,y) · v(x)

]dSx

(A.19)−∫Sp

Uλ(x,y) · f(x) dSx.

Next, the surfacesSp and Sg are subdivided intoNtriangular platelets and the assumption is made thatf andv are constant over a particular platelet. For plateletj onSg,{f,v} → {fg

j ,vgj }; and for plateletj on Sp, {f,v} → {fp

j ,0}.Also define the integrals

(A.20)Uλgj (y)≡

∫Sgj

Uλ(x,y) dSx,

(A.21)Uλpj (y)≡

∫Spj

Uλ(x,y) dSx,

where thej subscript denotes a platelet index and gp denotes the surface isSg or Sp, respectively. Analogou

expressions can be written forDλgj (y) andDλp

j (y) as well ascorresponding terms withλ = 0. In the special case whethe field point is on plateleti of Sg, ygi

(A.22)Uλggij ≡ Uλg

j (ygi ),

(A.23)Uλgpij ≡ Uλp

j (ygi).

Similarly, when the field point is on plateleti of Sp, ypi

(A.24)Uλpgij ≡ Uλg

j (ypi ),

(A.25)Uλppij ≡ Uλp

j (ypi ).

Analogous expressions forDλ corresponding to Eqs. (A.22)(A.25) must also be defined. In addition let

(A.26)v(1)ei ≡ v∞(ygi)−∫V0

U0(x,ygi) · se(x) dVx,

(A.27)v(2)ei ≡ −∫Vg

Uλ(x,ygi) · se(x) dVx,

(A.28)v(3)ei ≡ −∫Vg

Uλ(x,ypi) · se(x) dVx.

Choosey = ygi and with the above definitions, Eq. (A.1becomes

(A.29)1

2vgi = v(1)ei −

N∑[U0ggij · fg

j + D0ggij · vg

j

],

j=1


s

odewn

in

(

to

t

eit

ce in

.g

for

-ell.o-

haseyerr

rti-s-rse an

orkslyuc-s to

fforn-

id,ress

tri-

ionsto

s the

nichs

while Eq. (A.19) can be written

(A.30)

1

2vgi = v(2)ei +

N∑j=1

[Uλggij · fg

j + Dλggij · vg

j

] −N∑j=1

Uλgpij · fp

j .

Alternatively, if we choosey = ypi , then Eq. (A.19) become

(A.31)

0 = v(3)ei +N∑j=1

[Uλpgij · fg

j + Dλpgij · vg

j

] −N∑j=1

Uλppij · fp

j .

Equations (A.29)–(A.31) constitute 9N equations in 9Nunknowns which are thevg

i , fgi , andfp

i . All of the U andDmatrices are numerically determined once a particular mis defined. In addition, either Eqs. (A.26)–(A.28) are knoor estimates are available.

It shall prove convenient to put Eqs. (A.29)–(A.31)“supermatrix” form. Letvg, fg, etc. denote 3N -by-1 columnvectors formed by stacking the 3-by-1 column vectorsvg

1,vg

2, etc.) on top of each other. Also letU0gg denote the 3N -

by-3N matrix formed from theN2 3-by-3 matricesU0ggij ,

(A.32)U0gg≡

U0gg11 U0gg

12 · · ·U0gg

21 U0gg22 · · ·

· · · · · · U0ggNN

.

Similar 3N -by-3N matrices are formed fromD0ggij , etc.

These 3N -by-3N matrices, in turn, can be combinedform a grand 9N -by-9N supermatrix,G. Equations (A.26)–(A.28) can be combined and written

(A.33)

[ I/2+ D0gg U0gg 0I/2− Dλgg −Uλgg Uλgp

−Dλpg −Uλpg Uλpp

][vg

fg

fp

]=

v(1)e

v(2)e

v(3)e

,

where I in Eq. (A.33) is the 3N -by-3N identity matrixand 0 is the 3N -by-3N null matrix. It is straightforwardto construct this 9N -by-9N matrix and then inverting i(to yield G−1) makes possible the determination of the 9N

unknowns by simple matrix multiplication. Although thconstruction ofG and its inversion are time-consuming,should be emphasized that this only has to be done onthe entire calculation. Once thevg

i , fgi , and fp

i are known,Eq. (A.18) or (A.19) can be used to determinev anywherein the fluid.

In the absence of a gel layer (Vg = 0), the proceduresimplifies substantially. TakeSp andSg to be coincident andfrom stick boundary conditions,vg = 0 (wherevg is a 3N -by-1 column vector). Equation (A.29) reduces to

(A.34)v(1)e = U0pp · fp,

where the components ofv(1)e are given by Eq. (A.26)The unknown surface stresses,fp, are obtained by invertinthe 3N -by-3N matrix, U0pp, and multiplying v(1)e withthis inverse matrix. This is the standard BE procedureproblems without a gel layer.

l

There is nothingformally wrong with using this apparently simpler procedure when a gel layer is present as wIn general,λ used in the singular solution for a particular dmain of solvent does not necessarily have to equal theλ inthe actual fluid domain. However, setting the two equalthe advantage that theη0λ

2v portion of the external forcterm present in the gel layer is eliminated. When a gel lais present, butλ = 0 is used in the singular solution fothe entire fluid domain, an additional term,η0λ

2v, must beadded tose in Eq. (A.26) in that domain near the model pacle whereλ �= 0 in the actual fluid. The difficulty this createis we do not knowv (which is the quantity we can only determine aftervg, fg, andfp are determined), yet it appeain v(1)e . However, an iterative approach can be used wherestimate ofv is used to determinev(1)e , which yields a bet-ter estimate ofv, and the procedure is repeated untilv con-verges. Our experience has been that this approach wreasonably well whenλ is small (less than approximate1 nm−1 for model particles comparable to the SOL2 strtures defined in Table 3), but the whole procedure failconverge for largeλ. Physically, a small uncertainty inv canresult in a large uncertainty inv(1)e (due to the presence othe η0λ

2v-term), and the calculation becomes unstablelargeλ. The whole point of this appendix is the circumvetion of this problem.

Appendix B. Forces and stresses

The total force exerted by a soft particle on the fluzT, can be written as an integral of the hydrodynamic sttensor,S, over a large, but arbitrary spherical surface,S∞,centered on the model particle [49]

(B.1)zT = −∫S∞

S · ndSx,

wheren is a local outward normal toS∞. In the fluid do-main,

(B.2)∇ · S = −se + η0λ2v,

wherese is the external force/volume term (due to eleccal interactions in the present work) andη0λ

2v representsexternal forces due to segmental hydrodynamic interactwithin the gel layer. Applying the divergence theoremEq. (B.2), and choosing the volume of integration,Vo, tobe the fluid confined between the surface that enclosegel layer,Sg, andS∞ (whereλ = 0), it is straightforward toshow

(B.3)zT =∫Vo

sedVx +∫Sg

fdSx,

wheref = −S · n. Alternatively, the volume of integratiocould be chosen to include the entire fluid domain, whincludes not onlyVo, but the domain of the gel layer a


ci-the

erltetheof

r, by

e-.)ac-

menar-ntn-

if-ng

ligi-s,

ts (o

ns.gg

ible

iderheri-ure.o bepar-tro-les

83)

317

o,

58

0.

in

ics,

n,

od

.

981)

.ce

.6.

well, Vg. In this case,

(B.4)zT =∫

Vo+Vg

sedVx − η0λ2∫Vg

v dSx +∫Sp

fdSx.

Equations (B.1), (B.3), and (B.4) are equivalent in prinple, but Eqs. (B.1) and (B.2) are more practical sinceintegration overη0λ

2v in the gel layer is avoided. Considthe special case of largeλ when the velocity field in the gelayer nearSg varies rapidly. In order to accurately calculathe second integral on the right hand side of Eq. (B.4),gel layer would have to be divided into a large numberclosely spaced shells. This problem is avoided, howeveusing Eq. (B.1) or (B.3). (In the calculation ofv discussedin Appendix A, it is also possible to avoid related volumintegrations overη0λ

2v by using Green’s functions with differentλ in the gel layer and the more distant fluid domain

In calculating intrinsic viscosities or the related shape ftors, a procedure similar to that applied tozT, discussed inthe previous paragraph, is employed. The present treatwill focus on the general problem of a nonspherical soft pticle. Letη denote the viscosity of a dilute solution (solveviscosity= η0) of monodisperse particles of number cocentrationc and subjected to a shear gradient ofγ . It is as-sumed thatγ is sufficiently small compared to rotational dfusion so that no significant orientation of the particles alothe flow direction occurs. It is also assumed thatc is suffi-ciently small so that interparticle interactions have a negble effect on the solution viscosity. Under these condition

1

c

(η − η0

η0

)= − 1

γ η0

{⟨∫Sp

[f1x2 + η0(v1n2 + v2n1)

]dSx

⟩

(B.5)− η0λ2⟨ ∫Vg

v1x2dVx

⟩+

⟨ ∫Vo+Vg

se1x2dVx

⟩},

where the 1 and 2 subscripts denote spatial componenvectors) along the “1” and “2” directions and “〈〉” denotesisotropic averaging over all possible particle orientatioThe volume integral overη0λ

2v can be eliminated usinthe following relation, which is readily derivable usindivergence theorem arguments and the properties ofS,∫Vg

se1x2dVx =∫Sg

f1x2dSx −∫Sp

f1x2dSx

(B.6)+ η0

∫Sg

(v1n2 + v2n1) dSx + η0λ2∫Vg

v1x2dVx,

and substituting this into Eq. (B.5),

1

c

(η − η0

η0

)= − 1

γ η0

{⟨∫Sg

[f1x2 + η0(v1n2 + v2n1)

]dSx

⟩

(B.7)+⟨∫

se1x2dVx

⟩}.

Vo

t

f

Equation (B.7) is used in the present work. It is also possto generalize Eq. (B.7) to replaceSg with any surface,S′,which liesexterior to the gel layer providedVo is then takento be the fluid domain exterior toS′.

As discussed previously [19], it is necessary to consfive independent shear configurations for general nonspcal model particles and follow a careful averaging procedHowever, the number of shear configurations that have tconsidered is reduced to three for axisymmetric modelticles and one for spheres provided the equilibrium elecstatic potential reflects the symmetry of the model particthemselves.

References

[1] H. Ohshima, Adv. Colloid Interface Sci. 62 (1995) 189.[2] S. Levine, M. Levine, K. Sharp, D.E. Brooks, Biophys. J. 42 (19

127.[3] W.M. Deen, F.G. Smith, J. Membr. Sci. 12 (1982) 217.[4] A. Larsson, M. Rasmusson, H. Ohshima, Carbohydrate Res.

(1999) 223.[5] K. Makino, T. Taki, M. Ogura, S. Handa, M. Nakajima, T. Kond

H. Ohshima, Biophys. Chem. 47 (1993) 261.[6] H. Ohshima, Electrophoresis 16 (1995) 1360.[7] H.C. Brinkman, Appl. Sci. Res. A 1 (1947) 27.[8] P. Debye, A.M. Bueche, J. Chem. Phys. 16 (1948) 573.[9] B.U. Felderhof, J.M. Deutch, J. Chem. Phys. 62 (1975) 2391.

[10] B.U. Felderhoff, Physica A 80 (1975) 63.[11] B.U. Felderhoff, Physica A 80 (1975) 172.[12] J.J. Hermans, H. Fujita, Koninkl. Ned. Akad. Wet. Proc. Ser. B

(1955) 182.[13] H. Ohshima, J. Colloid Interface Sci. 163 (1994) 474.[14] H. Ohshima, J. Colloid Interface Sci. 228 (2000) 190.[15] H. Ohshima, Electrophoresis 23 (2002) 1993.[16] V. Natraj, S.B. Chen, J. Colloid Interface Sci. 251 (2002) 200.[17] D. Stigter, Macromolecules 33 (2000) 8878.[18] S.A. Allison, Macromolecules 29 (1996) 7391.[19] S.A. Allison, Macromolecules 31 (1998) 4464.[20] S. Mazur, C. Chen, S.A. Allison, J. Phys. Chem. B 105 (2001) 110[21] S.A. Allison, Biophys. Chem. 93 (2001) 197.[22] S.A. Allison, D.S. Stigter, Biophys. J. 78 (2000) 121.[23] D. Biddle, C. Walldal, S. Wall, Colloids Surf. A 118 (1996) 89.[24] M. Rasmusson, S. Wall, Colloids Surf. A 122 (1997) 169.[25] M. Rasmusson, S.A. Allison, S. Wall, J. Colloid Interface Sci.,

press.[26] J. Happel, H. Brenner, Low Reynolds Number Hydrodynam

Martinus Nijhoff Publishers, The Hague, 1983.[27] S. Kim, S.J. Karilla, Microhydrodynamics, Butterworth–Heineman

London, 1991.[28] S.A. Allison, Macromolecules 32 (1999) 5304.[29] K.E. Van Holde, Physical Biochemistry, Prentice–Hall, Englewo

Cliffs, NJ, 1971, Chap. 7.[30] H.A. Scheraga, L. Mandelkern, J. Am. Chem. Soc. 75 (1953) 179[31] F. Booth, Proc. R. Soc. London Ser. A 203 (1950) 533.[32] J.D. Sherwood, J. Fluid Mech. 101 (1980) 609.[33] I.G. Watterson, L.R. White, J. Chem. Soc. Faraday Trans. 2 77 (1

1115.[34] M.J. Garcia-Salinas, F.J. de las Nieves, Langmuir 16 (2000) 7150[35] G. Vigil, Z. Xu, S. Steinberg, J. Israelachvili, J. Colloid Interfa

Sci. 165 (1994) 367.[36] H. Yotsumoto, R.-H. Yoon, J. Colloid Interface Sci. 157 (1993) 434[37] G. Frens, J.Th.G. Overbeek, J. Colloid Interface Sci. 38 (1972) 37


han,inton,

in:iety,

37

ce

ed.,

78)

tro-

ace

om-

ss-ics,

6)

[38] J.D.F. Ramsay, S.W. Swanton, A. Matsumoto, G.D.C. Boberdin: Bergna (Ed.), The Colloid Chemistry of Silica, in: AdvancesChemistry Series, Vol. 234, American Chemical Society, WashingDC, 1994, Chap. 3.

[39] T. Healy, in: Bergna (Ed.), The Colloid Chemistry of Silica,Advances in Chemistry Series, Vol. 234, American Chemical SocWashington, DC, 1994, Chap. 7.

[40] J. Israelachvili, H. Wennerstrom, Nature 379 (1996) 219.[41] J.J. Adler, Y.I. Rabinovich, B.M. Moudgi, J Colloid Interface Sci. 2

(2001) 249.[42] R.K. Iler, The Chemistry of Silica, Wiley, New York, 1979.[43] R.K. Iler, R.L. Dalton, J. Am. Chem. Soc. 60 (1956) 955.[44] Y.I. Rabinovitch, B.V. Derjaguin, N.V. Churaev, Adv. Colloid Interfa

Sci. 16 (1982) 63.[45] D.R. Lide (Ed.), CRC Handbook of Chemistry and Physics, 74th

CRC Press, Boca Raton, FL, 1993.[46] H. Ohshima, Electrophoresis 23 (2002) 1993.

[47] J.Th.G. Overbeek, Kolloid-Beih. 54 (1943) 287.[48] F. Booth, Proc. R. Soc. London Ser. A 203 (1950) 514.[49] R.W. O’Brien, L.R. White, J. Chem. Soc. Faraday Trans. 2 74 (19

1607.[50] Th.F. Tadros, J. Lyklema, Electroanal. Chem. Interfacial Elec

chem. 17 (1968) 267.[51] Y.E. Solomentsev, Y. Pawar, J.L. Anderson, J. Colloid Interf

Sci. 158 (1993) 1.[52] J. Laven, H.N. Stein, J. Colloid Interface Sci. 238 (2001) 8.[53] A. Larsson, Colloid Polymer Sci. 277 (1999) 680.[54] O.A. Ladyzhenskaya, The Mathematical Theory of Viscous, Inc

pressible Flow, Gordon & Breach, New York, 1963, Chap. 3.[55] J. Richardson, H. Power, in: C.A. Brebbia, S. Kim, T.A. O

wald (Eds.), Boundary Elements XVII, Computational MechanSouthampton/Boston, 1995, p. 585.

[56] H.A. Lorentz, Versl. Koninkl. Akad. Wetensh. Amsterdam 5 (189168.

A general gel layer model for the transport of colloids and macroions in dilute solution

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Transcript of A general gel layer model for the transport of colloids and macroions in dilute solution