Journal of Colloid and Interface Science 333 (2009) 363–372

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Journal of Colloid and Interface Science

www.elsevier.com/locate/jcis

Effects of charge on osmotic reflection coefficients of macromolecules in porousmembranes

Gaurav Bhalla, William M. Deen ∗

Department of Chemical Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA

a r t i c l e i n f o a b s t r a c t

Article history:Received 10 November 2008Accepted 8 January 2009Available online 15 January 2009

Keywords:Streaming potentialOsmotic flowHindered transport theory

A computational model was developed to predict the effects of solute and pore charge on the osmoticreflection coefficients (σo) of spherical macromolecules in cylindrical pores. Results were obtained forparticles and pores of like charge and fixed surface charge densities, using a theory that combinedlow Reynolds number hydrodynamics with a continuum, point-charge description of the electricaldouble layers. In this formulation steric and/or electrostatic exclusion of macromolecules from thevicinity of the pore wall creates radial variations in osmotic pressure. These, in turn, lead to the axialpressure gradient that drives the osmotic flow. Due to the stronger exclusion that results from repulsiveelectrostatic interactions, σo with charge effects always exceeded that for an uncharged system withthe same solute and pore size. The effects of charge stemmed almost entirely from particle positionswithin a pore being energetically unfavorable. It was found that the required potential energy couldbe computed with sufficient accuracy using the linearized Poisson–Boltzmann equation, high chargedensities notwithstanding. In principle, another factor that might influence σo in charged pores is theelectrical body force due to the streaming potential. However, the streaming potential was shown tohave little effect on σo , even when it markedly reduced the apparent hydraulic permeability.

© 2009 Elsevier Inc. All rights reserved.

1. Introduction

The presence of a solvent-permeable but solute-retentive bar-rier separating two solutions of differing concentration tends togenerate an osmotic flow, depending on how “tight” or “leaky” themembrane is to the solute(s). The pertinent measure of tightnessor leakiness is the osmotic reflection coefficient (σo) introducedby Staverman [1]. For a single solute, the volume flux (v), hy-draulic permeability (κ), mechanical pressure difference (�P ), andosmotic pressure difference (�Π) are related as

v = κ(�P − σo�Π). (1)

If the solute cannot pass through the membrane, then σo = 1and the osmotic contribution is maximized; if the solute perme-ates as freely as the solvent, then σo = 0 and osmosis is absent. Weare concerned here with the prediction of σo for charged macro-molecules in porous membranes, where the pores are large enoughto permit entry of the solute but small enough to exhibit both size-and charge-selectivity.

With porous membranes and aqueous solutions, water may bemodeled as a continuum if the pore width is at least an order

* Corresponding author. Fax: +1 (617) 258 8224.E-mail address: wmdeen@mit.edu (W.M. Deen).

0021-9797/$ – see front matter © 2009 Elsevier Inc. All rights reserved.doi:10.1016/j.jcis.2009.01.019

of magnitude larger than a water molecule (i.e., if pore radii are atleast a few nm). If the solutes are also much larger than water (e.g.,globular proteins), they may be viewed as particles. These condi-tions permit one to use continuum mechanics to model osmoticflow. Consider a spherical solute of radius a and a membrane withcylindrical pores of radius b and length L. In a pioneering analysisof osmosis, applicable to completely excluded solutes (a > b), Ray[2] proposed that the end of a pore acts as an ideal, semiperme-able barrier. Thus, if the pores are long enough (L � b), osmoticnear-equilibria will exist at each end, such that �P = �Π locally,where �P and �Π are now differences between the external so-lution and that immediately inside the pore. Equivalently, P –Π isconstant near each pore end. It follows that the greater the exter-nal concentration, the more the pressure is depressed inside thepore, where Π = 0. In this way, an external concentration differ-ence is translated into an internal pressure difference between thepore ends, and it is this mechanical pressure gradient that drivesthe flow.

Anderson and Malone [3] and Anderson [4] extended theseideas to “leaky” membranes. Even if the spherical solute can en-ter the pore (a < b), its center will be excluded sterically from aregion of thickness a next to the pore wall. It was argued that thisexclusion of solute centers from the wall region creates a radialpressure difference, analogous to that for an ideal semipermeable

364 G. Bhalla, W.M. Deen / Journal of Colloid and Interface Science 333 (2009) 363–372

barrier. The radial pressure variation will be proportional to thesolute concentration at the pore centerline, which is a functionof axial position. The resulting axial dependence of the pressurewill cause flow. Thus, it is the exclusion of particle centers fromthe wall region that is responsible for the nonzero values of σo inleaky membranes. If there are also long-range solute-wall interac-tions, such as from electrical charge, the radial pressure gradientand flow rate will be altered, affecting σo .

For uncharged macromolecules, σo has been evaluated in thismanner for a variety of situations. Results for dilute solutions ofspheres in slits or cylindrical pores were obtained by Anderson andMalone [3], and their analysis was extended to capsule-shaped so-lutes by Anderson [4]. More recently, the underlying assumptionswere reexamined and calculations done for dilute solutions of pro-late or oblate spheroids in slits or cylindrical pores [5]. The effectsof solute concentration have been considered for spheres in slits orcylindrical pores [6]. The same conceptual framework was used tocalculate σo for spheres in membranes consisting of regular arraysof fibers [7]. Pore entrance and exit effects have also been exam-ined for spherical solutes [8].

There are few results in the literature for charged solutes. Sasid-har and Ruckenstein [9] used a diffuse double-layer model to ex-amine osmotic flow due to differences in electrolyte concentration.Numerical solutions of the Poisson–Boltzmann, species continuity,and momentum equations for specified differences in external saltconcentration yielded values for σo . Steric exclusion was not con-sidered (i.e., the ions were assumed to be point-sized).

It has been proposed that both size and charge effects canbe captured by employing the potential energy change E associ-ated with placing a solute molecule at a given radial position ina pore. In dilute solutions, and in pores long enough to allow themolecule to sample all radial positions, the relative probability ofthe molecule being at a given position is given by a Boltzmannfactor, exp(−E/kT ). For a spherical macromolecule in a cylindricalpore, the expression for the osmotic reflection coefficient that hasbeen proposed is [4,10]

σo = 1 − 4

1−λ∫0

e−E/kT (1 − β2)β dβ, (2)

where λ (=a/b) is relative solute radius and β (=r/b) is rela-tive radial position. Although results for E for charged spheres incharged cylindrical pores have been available for some time [11],corresponding results for σo have not been reported.

The objective of the present work was to provide numericalresults for σo for charged spheres in charged pores. As will beexplained, one limitation of Eq. (2) is that it neglects stream-ing potentials, which retard the flow and may alter the shape ofthe velocity profile in small, charged pores. To examine whetheror not this electrokinetic phenomenon influences σo , we devel-oped a more general formulation. Another potential source of erroris the Debye–Huckel approximation (linearization of the Poisson–Boltzmann equation), which was needed to obtain analytical re-sults for E [11]. Numerical results for E were obtained for situa-tions where that approximation should be inaccurate (high surfacecharge densities and/or large Debye lengths) and compared withthe previous analytical results. Surprisingly, neither source of errorproved to be significant in the calculation of σo , and Eq. (2) wasfound to be quite accurate.

2. Theory

2.1. System modeled

Dilute solutions of spherical macromolecules were assumed tobe separated by a membrane having long cylindrical pores, the

solutes and pores each having specified linear dimensions andsurface charge densities. Entrance and exit effects were neglectedin the hydrodynamic and electrostatic calculations, which will bevalid if the pore length greatly exceeds the pore radius or De-bye length, whichever is larger. Also present, in addition to themacromolecular solute, were univalent anions and cations, eachof negligible size relative to the macromolecule or pore. The bulkelectrolyte concentrations on the two sides were assumed to beequal, so that osmosis resulted only from an imbalance in macro-molecule concentrations. Because filtration processes typically op-erate under open-circuit conditions, zero current was imposed asthe macroscopic electrical constraint. No specific restrictions wereplaced on the Debye length or surface charge densities. However,to avoid situations where electrostatic interactions would promotemacromolecule adsorption, results were obtained only for particlesand pores of like charge.

2.2. Momentum equation

Although osmotic flow in a pore is not precisely unidirectional,the lubrication approximation is applicable if the pore is long, asdetailed in Anderson and Malone [3]. Accordingly, it was assumedthat the axial (z) momentum balance is given by

μ

r

∂

∂r

(r∂vz

∂r

)= ∂ P

∂z+ ρe

∂ψ

∂z, (3)

where vz(r, z) is the axial velocity component, μ is the viscosity,ρe is the volumetric charge density, and ψ is the electrical po-tential. At a fixed location in a pore, momentum transfer will betime-dependent, according to whether or not a particle (macro-molecule) happens to be in the vicinity. Thus, each of the fieldvariables (P , v , ψ ) is interpreted here as a time-averaged quantity.

The last term on the right side of Eq. (3) is the body force re-sulting from the streaming potential. Convection of counterions ina charged pore tends to generate a current, but in the absenceof working electrodes, the net current must be zero. Accordingly,a potential gradient (streaming potential) must develop, such thatthe total current due to ion migration balances that due to convec-tion. In small, highly charged pores, the electrical body force mightbe significant, both in retarding the flow and in altering the shapeof the velocity profile. Implicit in Eq. (2) is that the velocity profileis parabolic, as in an uncharged system.

In Eq. (3) it is assumed that the concentration of the macro-molecule is small enough so as not to influence the viscosity orotherwise affect the form of the time-averaged axial momentumbalance. In this model the macromolecule influences the flow onlyby contributing to ∂ P/∂z, as will be described shortly. In calcu-lating ρe we assumed that the contribution of the macromoleculeis negligible. In other words, the electrical body force was basedon the concentrations only of the small ions. Likewise, it was as-sumed that the macromolecules have a negligible effect on thetime-averaged concentrations of the small ions and the associatedelectric field. For a long pore, the electrical potential will then beof the form

ψ(r, z) = ψ1(r) + ψ2(z), (4)

where ψ1(r) is the equilibrium double-layer potential and ψ2(z)arises from the streaming potential. These potentials were evalu-ated in two separate calculations, as will be discussed.

2.3. Pressure distribution

Assuming osmotic equilibrium of the solvent in the radial di-rection, P –Π will be invariant in that direction. Letting P0 and Π0denote values at the pore centerline, it follows that

P (β, z) = P0(z) + Π(β, z) − Π0(z). (5)

G. Bhalla, W.M. Deen / Journal of Colloid and Interface Science 333 (2009) 363–372 365

The osmotic pressure includes contributions both from the smallions and the macromolecule, and is given by

Π(β, z) = RT[c+(β) + c−(β) + C(β, z)

], (6)

where c+ , c− , and C are the molar concentrations of the smallcation, small anion, and macromolecule, respectively. For macro-molecules, the concentration at a particular point is that of thesphere centers. Evaluating osmotic pressures using concentrationsof particle centers is not precisely correct, although it greatly sim-plifies the calculations and for neutral particles was shown to re-sult in only small errors in σo [5]. Implicit in Eq. (6) is that thesolution is ideal.

2.4. Concentrations of small ions

The concentrations of the monovalent cation and anion are re-lated to the equilibrium double-layer potential by

c±(β) = c∞ exp[∓Ψ (β)

], (7)

where Ψ = ψ1 F/RT is the dimensionless potential and c∞ is thesalt concentration in either bulk solution. Thus, the volumetriccharge density is

ρe = F (c+ − c−) = −2F c∞ sinh Ψ, (8)

where F is Faraday’s constant. Combining this with Poisson’s equa-tion yields the cylindrical Poisson–Boltzmann equation,

1

β

d

dβ

(β

dΨ

dβ

)= τ 2 sinh Ψ. (9)

The parameter τ , which is the pore radius divided by the Debyelength, is given by

τ = b

(2F 2c∞εRT

)1/2

(10)

for monovalent ions, where ε is the dielectric permittivity. Theboundary conditions, corresponding to cylindrical symmetry at thepore centerline and constant charge density at the pore surface,were

∂Ψ

∂β(0) = 0, (11)

∂Ψ

∂β(1) = qc, (12)

where qc is the dimensionless surface charge density at the cylin-drical pore wall. It is related to the dimensional charge density Q c(C/m2) as

qc = Q cbF

εRT. (13)

The nonlinear boundary value problem for Ψ was solved nu-merically using Comsol MultiphysicsTM (Comsol, Stockholm, Swe-den), a finite element package.

2.5. Macromolecule concentration

As discussed in Deen [10], for long pores the macromoleculeconcentration is a separable function, such that

C(β, z) = f (z)g(β) = f (z)exp[−E(β)/kT

]. (14)

Steric exclusion from the vicinity of the pore wall was modeled bysetting E = ∞ there, so that C = 0 for β > 1 − λ. To calculate theosmotic reflection coefficient it is unnecessary to evaluate f (z). Aswill be seen, it is sufficient to require that f (0)− f (L) = C1 − C2 =�C , where C1 and C2 are the external concentrations at the twosides of the membrane.

2.6. Velocity profile

Because of the discontinuity in C at β = 1 − λ, Eq. (3) was in-tegrated separately for 0 � β < 1 − λ (the “core” region, wherevz ≡ u) and 1 − λ < β � 1 (the “periphery,” where vz ≡ w). Withpressures and concentrations evaluated as just described, the dif-ferential equation for the core was

1

β

∂

∂β

(β

∂u

∂β

)= b2

μ

[dP0

dz+ RT

df

dz

(g(β) − g(0)

)]

− 2b2 RT c∞Λ

μLsinh Ψ, (15)

where Λ = (L F/RT )dψ2/dz is the dimensionless streaming po-tential. Integrating once, and applying the symmetry condition atβ = 0, gave

β∂u

∂β= b2

2μ

[dP0

dz− RT

df

dzg(0)

]β2

+ b2 RT

μ

df

dz

β∫0

xg(x)dx − 2b2 RT c∞Λ

μL

β∫0

x sinh Ψ dx. (16)

The differential equation for the periphery was the same as Eq.(15), except without the g(β) term. A first integration there yielded

β∂ w

∂β= b2

2μ

[dP0

dz− RT

df

dzg(0)

]β2

+ b2 RT

μ

df

dz

1−λ∫0

xg(x)dx − 2b2 RT c∞Λ

μL

β∫0

x sinh Ψ dx (17)

where Eq. (16) was used to ensure that the shear stress was con-tinuous at β = 1 − λ.

A second integration for the periphery, and application of theno-slip condition at the wall, gave the final expression for the ve-locity in that region:

w(β, z) = − b2

4μ

[dP0

dz− RT

df

dzg(0)

](1 − β2)

+ b2 RT

μ

df

dzln β

1−λ∫0

xg(x)dx

+ 2b2 RT c∞Λ

μL

1∫β

dy

y

y∫0

x sinh Ψ dx. (18)

Another integration for the core, together with matching of the ve-locities at β = 1−λ, completed the solution for the velocity profile:

u(β, z) = − b2

4μ

[dP0

dz− RT

df

dzg(0)

](1 − β2)

+ b2 RT

μ

df

dzln(1 − λ)

1−λ∫0

xg(x)dx

− b2 RT

μ

df

dz

1−λ∫β

dy

y

y∫0

xg(x)dx

+ 2b2 RT c∞Λ

μL

1∫β

dy

y

y∫0

x sinh Ψ dx. (19)

The first integral in both Eqs. (18) and (19) is proportional tothe equilibrium partition coefficient (Φ) for the macromolecule,

366 G. Bhalla, W.M. Deen / Journal of Colloid and Interface Science 333 (2009) 363–372

which is the average intrapore concentration divided by that inbulk solution (at equilibrium). That is,

Φ = 2

1−λ∫0

xg(x)dx. (20)

This and the other integrals were evaluated numerically, usinga shape-preserving spline interpolation to approximate the inte-grands. After the integrands were tabulated, the Matlab function“Fit” was employed, using the “spline interpolant” option. The in-tegration was done then using Simpson’s Rule, typically with 100intervals. The method was tested using polynomials which hadfunctional shapes similar to those in either the single or double in-tegrals of interest, and which could be integrated analytically. Thenumerical and analytical results agreed to within 1% for these testcases.

To calculate the osmotic reflection coefficient, the velocity wasintegrated piecewise over β to find the mean velocity (U ), whichis independent of z. This integration was done numerically, as justdescribed. The result obtained for U was a linear function of thegradients dP0/dz and df /dz. Integration over the pore length thenproduced a relationship of the same form as Eq. (1), permittingidentification of σo . The changes in P0 and f over the pore lengthwere related to the external pressure differences (�P and �Π ) by

P0(0) − P0(L) = �P − �Π[1 − g(0)

], (21)

RT[

f (0) − f (L)] = �Π. (22)

Remaining to be discussed are the one unknown constant thatappears in Eqs. (18) and (19) (namely, Λ) and the one unknownfunction [g(β)].

2.7. Streaming potential

The streaming potential (Λ) was determined from the require-ment that there be no net current. In the absence of axial varia-tions in the concentrations of the small ions, and neglecting thecurrent carried by the macromolecule, the axial component of thecurrent density was given by

iz = F[vz(c+ − c−) − (D+c+ + D−c−)Λ

], (23)

where D± are the cation and anion diffusivities. It is seen fromEqs. (18) and (19) that the axial velocity is of the form vz =GΛ + H , where the function H differs somewhat between thecore and periphery but G is the same for both regions. Evaluat-ing the concentrations using Eq. (7), and assuming (for simplicity)that D+ = D− = D , it was found that

− iz

2F c∞=

(G sinh Ψ + D

Lcosh Ψ

)Λ + H sinh Ψ. (24)

Integrating Eq. (24) over radial position to obtain the total current,and equating that to zero, led to

Λ = −∫ 1

0 H sinh Ψβ dβ∫ 10 [G sinh Ψ + (D/L) cosh Ψ ]β dβ

. (25)

Again, the integrals were evaluated numerically.

2.8. Electrostatic potential energy

The energy E(β) was needed to compute the function g(β)

via Eq. (14). This is the electrostatic free energy associated withmoving a charged sphere from bulk solution to a specified radialposition in a pore, and its evaluation required that the equilib-rium electrical potential Ψ be determined for three situations:

(i) a sphere in an unbounded solution, (ii) a pore with no sphere,and (iii) a pore with sphere present. The desired potential energychange is the electrostatic energy for system (iii) minus the sum ofthose for (i) and (ii). In each case Ψ was assumed to be governedby the nonlinear form of the Poisson–Boltzmann equation,

∇2Ψ = τ 2 sinh Ψ (26)

where ∇2 is the dimensionless Laplacian operator, spherical-radialfor system (i), cylindrical-radial for (ii) [as in Eq. (9)], and three-dimensional for (iii). For (ii) and (iii) the parameter τ was given byEq. (10); for (i), where the sphere radius was the only geometriclength scale, τ was replaced by τλ. In (ii) and (iii) the surfacecharge density on the pore wall was specified by Eq. (12). For (i)and (iii) the surface charge density on the sphere was fixed. For thecombined system, the dimensionless charge density for the spherewas expressed as

qs = Q sbF

εRT, (27)

which is analogous to Eq. (13). For each of the three problems,Ψ was determined numerically using Comsol, for various combi-nations of τ , λ, qc , and qs .

Following Sharp and Honig [12] and Reiner and Radke [13], theelectrostatic energy for each of the three systems was computedas

Ei =∫Si

Q ψ dS −∫V i

[(ε/2)|∇ψ |2 + 2RT c∞(cosh Ψ − 1)

]dV , (28)

where i = s, c, and sc for the isolated sphere, isolated pore, andcombined system, respectively. Recall that Q and ψ are the di-mensional charge density and potential, respectively, whereas Ψ

is dimensionless. The surface integration encompasses all chargedsurfaces (sphere surface and/or pore wall) and the volume inte-gration is over the entire solution volume. The energy change wascalculated as

E = Esc − Es − Ec . (29)

Although the pore was assumed to be indefinitely long, mak-ing each of the integrals in Esc and Ec divergent, the differenceEsc − Ec was made finite by the fact that ψ is the same for bothsystems as z → ±∞. For the isolated sphere, convergence wasensured by the fact that the integrand in the volume integral van-ishes as r → ∞.

If the potentials are small enough that sinhΨ ∼= Ψ (which isaccurate to within 18% for |Ψ | = 1), then Eq. (26) takes the linearform,

∇2Ψ = τ 2Ψ, (30)

which has been solved analytically for the geometry and boundaryconditions of interest here [11]. When this linearized form of thePoisson–Boltzmann equation is applicable, the electrostatic energysimplifies to

Ei = 1

2

∫Si

Q ψ dS, (31)

as given in Verwey and Overbeek [14]. The results for E calculatedpreviously for the linear problem will be compared with those ob-tained from the nonlinear formulation. The analytical evaluation ofE is detailed in Appendix A. Also discussed there are certain sim-plifications that occur in the calculation of σo when Eq. (30) isapplicable.

G. Bhalla, W.M. Deen / Journal of Colloid and Interface Science 333 (2009) 363–372 367

2.9. Parameters

As seen in Eqs. (18) and (19), the velocity profile for osmoticflow will depend on λ (sphere radius/pore radius) and the dimen-sionless parameters that influence the equilibrium potential (Ψ ),which are τ [pore radius/Debye length, Eq. (10)] and qc [dimen-sionless pore charge, Eq. (13)]. Also involved, via the function g(β),are the parameters that determine the dimensionless potential en-ergy for the macromolecule, E/kT . In addition to λ, τ , and qc ,those are qs [dimensionless sphere charge, Eq. (27)] and

ξ = (RT /F )2εb

kT, (32)

which arises when E is made dimensionless using the thermal en-ergy. Finally, there are the parameters that determine the stream-ing potential (Λ). From Eq. (25), it is found that the determinantsof Λ are those for the potential energy (λ, τ ,qc,qs, ξ ) plus

χ = μD

b2 RT c∞(33)

and two pressure ratios, �P/(RT c∞) and �Π/(RT c∞). However,substituting the expression for the streaming potential into the ve-locity profile reveals that these pressure ratios do not influence thereflection coefficient. In summary, we conclude from dimensionalanalysis that

σo = σo(λ, τ ,qc,qs, ξ,χ). (34)

In computing reflection coefficients we viewed the sphere ra-dius, sphere and pore surface charge densities, and bulk salt con-centration as the primary variables. Unless otherwise noted, theresults correspond to aqueous KCl solutions at 25 ◦C (D+ = D− =2.0 × 10−9 m2/s and μ = 8.9 × 10−4 Pa s) and a pore radius ofb = 10 nm. The corresponding value of the energy-scale param-eter is ξ = 1.1. For these reference conditions and c∞ = 1 M,the diffusivity-viscosity parameter is χ = 7.2 × 10−3. At the de-fault pore radius of 10 nm, τ = 0.1, 1, and 10 correspond toc∞ = 9×10−6, 9×10−4, and 9×10−2 M, respectively. At that poreradius, qc = qs = 1 corresponds to Q c = Q s = 1.8 × 10−3 C/m2. ForBSA, where a = 3.6 nm, qs = −11 if b = 10 nm and the net chargeis −20. At the default pore radius, λ = 0.36 for BSA.

3. Results

3.1. Electrostatic potential energy

A key quantity both in Eq. (2) and in the more general for-mulation is the potential energy (E) for a sphere whose center islocated at a given radial position within a pore. The correspond-ing Boltzmann factor [exp(−E/kT )] is the probability that a givensphere will occupy that position. Because it determines the extentto which a particle will be excluded from a pore, the Boltzmannfactor strongly influences the osmotic reflection coefficient. In pre-liminary calculations we addressed the issue of whether numericalsolutions of the nonlinear Poisson–Boltzmann equation were nec-essary to obtain sufficiently accurate values of E at the high chargedensities of interest, or whether previously reported analytical re-sults based on the linearized Poisson–Boltzmann equation wereadequate. As shown in Fig. 1, Boltzmann factors obtained from thenonlinear and linear formulations were found to be nearly iden-tical, even for maximum values of |Ψ | exceeding unity. Evidently,the tendency of the linear formulation to overestimate the surfacepotential for a given system, and therefore overestimate Ei , waslargely offset by the fact that E is a difference between electro-static energies [Eq. (29)]. We conclude that the analytical resultsfor E , which are much easier to use, are sufficiently accurate forcalculations of σo . All of the results to be presented for σo arebased on the linear formulation for E .

Fig. 1. Comparison of Boltzmann factors [exp(−E/kT )] calculated using Eqs. (26)and (28) (“nonlinear”) or Eqs. (30) and (31) (“linear”). The nonlinear results wereobtained from finite element calculations for spheres positioned on the pore axis,and the linear results for this axisymmetric case were computed from a previousanalytical solution [11]. A variety of combinations of λ, τ , qs , and qc were used,resulting in maximum equilibrium potentials (|Ψmax|) in one of two ranges, 0.5 <

|Ψmax| < 1 or |Ψmax| > 1. For |Ψmax| < 0.5 (not shown), there was no noticeabledifference between the linear and nonlinear Boltzmann factors.

Fig. 2. Effect of τ (pore radius/Debye length) on the streaming potential. The coeffi-cients A and B , defined in Eq. (35), describe the effects of transmembrane mechan-ical and osmotic pressure differences, respectively. Note that all values have beenmultiplied by 103. It was assumed here that λ = 0.5 and qs = qc = 4.

3.2. Streaming potential

The streaming potential, which retards transmembrane flowand which may also alter the shape of the velocity profile in apore, is a linear function of the external pressure differences. Thisrelationship was expressed as

Λ = A�P

RT c∞− B

�Π

RT c∞, (35)

where A and B are dimensionless coefficients. The sign of eachcoefficient is opposite to that of the fixed charge (e.g., both are>0 if qc < 0), but the magnitudes of A and B are the same forpositively or negatively charged pores. The dependence of |A| and|B| on τ (pore radius/Debye length) is shown in Fig. 2. Under theconditions chosen both coefficients had peak magnitudes of about1.7 × 10−3 at τ = 4. Because charge effects are suppressed whendouble layers are thin relative to the pore radius, A and B eachmust vanish as τ → ∞, consistent with the plot. Both coefficients

368 G. Bhalla, W.M. Deen / Journal of Colloid and Interface Science 333 (2009) 363–372

Fig. 3. Effect of τ (pore radius/Debye length) and pore charge density (qc) on thehydraulic permeability (κ) defined in Eq. (1). The hydraulic permeability for chargedpores is compared with that for neutral pores of the same size.

Fig. 4. Effect of relative sphere size (λ = sphere radius/pore radius) and sphere andpore charge density (q = qs = qc) on the osmotic reflection coefficient (σo). It wasassumed that τ = 5.

also must vanish for large Debye lengths, or τ → 0, although themagnitude of Λ reaches a maximum, constant value in that limit.Because c∞ appears in the denominators in Eq. (35), and becauseτ ∝ c1/2∞ , constancy of Λ requires that A and B each vary as τ 2 forτ → 0, as shown in Fig. 2.

The retardation of transmembrane flow by the streaming po-tential is illustrated in Fig. 3, in which the apparent hydraulicpermeability for charged pores, relative to that for uncharged poresof the same size, is shown as a function of τ for three values of qc .The apparent hydraulic permeability was always reduced to someextent. This effect was minimal at small Debye lengths (large τ )and/or low membrane charge densities (qc), but was quite notice-able when the Debye length and/or charge density were large. Forthe parameter values in Fig. 3, there was a 13% reduction in κ forqc = 4 and τ → 0.

3.3. Osmotic reflection coefficient

The effects of molecular size and charge on the osmotic re-flection coefficient are shown in Fig. 4, where σo is plotted as afunction of λ (sphere radius/pore radius) for several charge densi-ties. In this plot the sphere and pore were assumed to have equalcharge densities (qc = qs = q) and the relative Debye length was

Fig. 5. Effect of λ (sphere radius/pore radius) and τ (pore radius/Debye length) onthe osmotic reflection coefficient (σo). It was assumed that qs = qc = 1.

Fig. 6. Effect of τ (pore radius/Debye length) and charge density (q = qs = qc) onthe osmotic reflection coefficient (σo) for λ = 0.5.

fixed (τ = 5). As expected, higher charge densities increased σo .This is because they elevated E and therefore tended to excludeparticles from the pore.

Fig. 5 shows the osmotic reflection coefficient as a function ofsolute size for various Debye lengths, with charge densities fixed.The effects of decreasing the Debye length (increasing τ ) werequalitatively similar to those of decreasing q. The curve labeled“τ = ∞” is that for an uncharged sphere and pore [3].

Fig. 6 provides another view of the effects of charge on σo . Herethe relative sphere size was fixed (λ = 0.5) and the Debye lengthand charge densities were varied. As with the two preceding plots,it was assumed that qs = qc = q. Once again, large τ and/or smallq yielded reflection coefficients that approached those for an un-charged system (σo = 0.56). However, even for a modest chargedensity (q = 0.5), σo was increased to unity if the Debye lengthwas large enough (τ small enough).

Of course, solute and pore charge densities generally will beunequal. Fig. 7 shows the effects of sphere charge density (qs)

on σo for several values of the pore charge density (qc). In thesecalculations the relative sphere size and Debye length were eachfixed (λ = 0.5 and τ = 5). As expected for a system with likecharges and repulsive electrostatic interactions, σo was elevatedby increases in either charge density. It is noteworthy that evenfor an uncharged sphere (qs = 0), σo increased noticeably as the

G. Bhalla, W.M. Deen / Journal of Colloid and Interface Science 333 (2009) 363–372 369

Fig. 7. Effect of sphere charge (qs) and pore charge (qc) on the osmotic reflectioncoefficient (σo) for λ = 0.5 and τ = 5.

pore charge density was increased (from 0.56 at qc = 0 to 0.64at qc = 4). Likewise, there were electrostatic effects for a chargedsphere in an uncharged pore, as shown by the bottom curve inFig. 7. This occurs because placing an uncharged sphere in a poredistorts the equilibrium double layer in the pore, and thereforegives a nonzero electrostatic contribution to the potential energy(E > 0, irrespective of the sign of the pore charge) [11]. Simi-larly, confining a charged sphere in an uncharged pore distorts thesphere double layer.

Surprisingly, given the ability of the streaming potential to in-fluence the hydraulic permeability (Fig. 3), the electrokinetic effecton σo was found to be negligible for all realistic combinations ofparameters that we examined. That is, the results computed withthe general formulation were never significantly different thanthose from Eq. (2). This finding is discussed further in the nextsection.

4. Discussion

A model was developed to predict the effects of solute and porecharge on the osmotic reflection coefficients of spherical macro-molecules, combining low Reynolds number hydrodynamics with acontinuum description of the electrical double layers. As expected,for molecules and pores of like charge, σo always exceeded thatfor an uncharged system with the same solute and pore size. Theeffects of charge were found to stem almost entirely from elec-trostatic exclusion of the macromolecules from the pores. For likecharges and constant surface charge densities, particle positionswithin a pore are energetically unfavorable, which decreases theequilibrium partition coefficient, Φ [11]. As recognized previously[3–5], decreases in Φ lead to increases in σo . In terms of theability to induce an osmotic flow, adding like charges to the macro-molecules and pore walls is qualitatively similar to increasing themacromolecular radius. That is, it leads to behavior more closelyresembling an ideal, semipermeable membrane, where σo = 1.However, one cannot account for charge by, say, simply adding aDebye length to the particle radius. This is because the strength ofthe electrostatic interactions depends on the surface charge densi-ties, and not just the geometric dimensions and Debye length.

The relationship between osmosis and partitioning is examinedmore quantitatively in Fig. 8, in which σo is plotted as a func-tion of Φ . The discrete symbols are numerical results obtained forvarious combinations of λ, τ , qs , and qc . A strong correlation isevident, σo decreasing as Φ increases. The curve corresponds to

Fig. 8. Relationship between the osmotic reflection coefficient (σo) and equilib-rium partition coefficient (Φ). The symbols represent numerical results obtained forvarious combinations of sphere size, Debye length, and sphere and pore charge den-sities. The curve, from Eq. (36), is a prediction for uncharged spheres in unchargedcylindrical pores.

σo = (1 − Φ)2, (36)

which was derived originally for uncharged spheres in unchargedcylindrical pores [3] and shown to be nearly exact also for un-charged spheroids (oblate or prolate) in such pores [5]. It is seenthat Eq. (36) was accurate for charged spheres in many instances,but significantly underestimated σo in others. Overall, it appearsthat Eq. (36) is somewhat less reliable for charged than for un-charged systems. It should be mentioned also that it is restrictedto cylindrical pores; for example, results obtained for neutralspheroids in slit pores followed a somewhat different relation-ship [5].

In principle, the effects of charge on σo might be mediated inpart by an electrokinetic effect. It is well known that pressure-driven flow through a small, charged pore, under open-circuit con-ditions, creates an axial variation in electrical potential [15]. This istermed the streaming potential, and it arises because the fluid inthe pore is not electrically neutral. Due to an excess of counteri-ons in the mobile fluid, any bulk flow tends to create a current,which (under open-circuit conditions) must be balanced by ionmigration. The resulting electrical body force tends to retard theflow and might alter the shape of the velocity profile. If the profileis distorted sufficiently from the parabolic shape characteristic ofPoiseuille flow, then the value of σo may be altered [4]. As stream-ing potentials had not been explored previously in this context,we sought to assess their importance. However, we found this ef-fect on σo to be negligible. This was true for a wide variety ofparameter combinations, including situations where the streamingpotential was large enough to cause a very significant reduction inthe mean velocity.

To see if there are any conditions in which the streaming po-tential might influence σo , we mapped out combinations of pa-rameters for which the velocity profile in a pore without macro-molecules was either parabolic or nonparabolic. For Poiseuille flowin a cylindrical pore, the centerline velocity is exactly twice themean value. The electrokinetic effect (when present) reduces theratio of local to mean velocity near the pore wall and increases itnear the centerline. We defined a “nonparabolic” velocity profileas one in which the centerline ratio was altered by >5% (velocityratio >2.1). The results of that exploration are shown in Fig. 9,in which the curve corresponds to combinations of dimension-less pore charge density (|qc|) and ratio of pore radius to Debyelength (τ ) at which the velocity profile changed from parabolic to

370 G. Bhalla, W.M. Deen / Journal of Colloid and Interface Science 333 (2009) 363–372

Fig. 9. Combinations of dimensionless pore charge density (qc) and ratio of poreradius to Debye length (τ ) for which the velocity profile in a pore was predicted tobe either parabolic or nonparabolic. A nonparabolic profile is defined here as one inwhich the ratio of centerline to mean velocity is >2.1.

nonparabolic. For a given τ , nonparabolic profiles resulted for val-ues of |qc| above the curve and parabolic ones for values below it.(Without macromolecules, the shape of the velocity profile can beinfluenced by one additional parameter, χ . However, with μD andT fixed, as assumed here, χ ∝ 1/τ 2, so that χ is not independent.)

The results in Fig. 9 indicate that nonparabolic profiles mightindeed occur, especially if τ is small (Debye length exceeding poreradius) and pore charge density large. As a benchmark for sur-face charge density, if that for the pore were similar to that ofBSA under physiological conditions (Q c = Q s = −2.0×10−2 C/m2),then |qc| = 5.5, 11, and 22 for b = 5, 10, and 20 nm respec-tively. Based on Fig. 9, nonparabolic profiles will result for τ < 2if |qc| = 20. However, σo computed for such high pore charge den-sities usually did not differ significantly from unity. Although forrelatively small molecules (λ = 0.1) we sometimes found σo < 1under nonparabolic conditions, electrokinetic effects on σo werestill insignificant. Thus, under conditions where the streaming po-tential most affects the apparent hydraulic permeability (small τand large |qc|, Fig. 3), and where it would be expected to mostaffect σo , there was not a significant difference between the fullmodel and Eq. (2). This was largely because the ideal semiperme-able limit was predicted by each. In summary, we were unable toidentify any realistic conditions for which Eq. (2) was inaccurate.

Serum albumin is the predominant protein in blood plasma,and its loss from the circulation by permeation through capillarywalls — especially in the kidney — is of considerable pathophysi-ological importance [16]). In addition to being negatively charged,albumin is not precisely spherical, and it is often modeled as aprolate spheroid. Accordingly, one might ask whether the siev-ing of BSA through a porous membrane is influenced more by itsshape or its charge. A complete answer is not yet possible, butthe present results concerning the effects of charge for spheres,together with previous predictions of the effects of shape for un-charged spheroids, permit some comparisons to be made. Becausethe reflection coefficient for filtration (σ f ) is approximately equalto that for osmotic flow (see below), it is permissible to draw con-clusions about sieving from the results for σo . Fig. 10 shows σo asa function of relative molecular size for three cases: an unchargedsphere and pore; a sphere and pore each with a charge densitylike that of BSA; and an uncharged prolate spheroid and unchargedpore. The abscissa (λ) in each case is the ratio of Stokes–Einsteinradius to pore radius; with the Stokes–Einstein radius fixed at thevalue for BSA (3.6 nm), it is the pore radius (b) that was varied

Fig. 10. Predicted effects of molecular size, shape, and charge on the osmotic re-flection coefficient (σo) for BSA. Results for an uncharged prolate spheroid in anuncharged pore and a charged sphere in a charged pore are compared with thosefor an uncharged sphere in an uncharged pore. In each case λ = Stokes–Einsteinradius/pore radius. The calculations for the prolate spheroid were as described inBhalla and Deen [5]. Based on data reviewed in Vilker et al. [17], the major and mi-nor semiaxes were taken to be 7.05 and 2.08 nm, respectively, corresponding to anaxial ratio of 3.39 and a Stokes–Einstein radius of 3.56 nm. For the charged case,it was assumed that a = 3.6 nm, c∞ = 0.15 M, and Q c = Q s = −2.0 × 10−2 C/m2.The sphere charge density corresponds to 20 negative charges per molecule, whichis representative of BSA under physiological conditions [17].

along each curve. Because the other dimensionless groups in thecharge model (τ ,qc,qs, ξ,χ ) also contain b, they were varied alongwith λ in computing σo for that case. The axial ratio for the prolatespheroid was taken to be 3.4 [17]. Generally, σo for the chargedsphere and uncharged prolate spheroid each exceeded that for theneutral sphere, by roughly similar amounts. There was an excep-tion to this trend for very tightly fitting particles, where values ofσo for prolate spheroids became slightly lower than those for un-charged spheres. This reflects the fact that the minor semiaxis ofa prolate spheroid is less than its Stokes–Einstein radius [5]. Thus,if aligned properly, it will fit in a pore even if λ > 1. (For an axialratio of 3.4, the theoretical cutoff for partitioning or sieving occursat λ = 1.7.) The effects of charge depend, of course, on the chargedensity of the pore wall. Reducing Q c to half the value for BSAgave a curve that overlapped with that for the prolate spheroid(results not shown). Overall, Fig. 10 suggests that the effects ofmolecular shape and charge on the sieving of BSA are of roughlycomparable importance.

A common measure of membrane selectivity in filtration pro-cesses is the sieving coefficient (Θ) for a given solute, which is itsfiltrate-to-retentate concentration ratio. If the Peclet number basedon pore length is large and concentration polarization is negligible,then Θ = 1−σ f . Because the hydrodynamic problem that must besolved to predict σo is much simpler than that needed to computeσ f (as reviewed in [18]), osmotic flow calculations provide an at-tractive way to gain insight into filtration selectivity, provided thatσo does not differ greatly from σ f . A comparison of the two reflec-tion coefficients for charged spheres and pores is shown in Fig. 11.Results are shown for two charge densities, with those for thesphere and pore assumed to be equal in both cases (qs = qc = q).The values of σ f were computed from

σ f = 1 − 4

1−λ∫0

Ge−E/kT (1 − β2)β dβ, (37)

which is the same as Eq. (2), but including now the lag coeffi-cient (G) [10]. In the absence of values of G for charged particles

G. Bhalla, W.M. Deen / Journal of Colloid and Interface Science 333 (2009) 363–372 371

Fig. 11. Reflection coefficients for osmosis (σo) and filtration (σ f ) as a function ofrelative sphere size (λ). Results are shown for two charge densities (q = qs = qc = 1or 4), with τ = 5 in each case.

in charged pores, results for uncharged systems were used [18]. Forboth charge densities in Fig. 11, σ f slightly exceeded σo , the agree-ment improving somewhat as the charge density was increasedfrom q = 1 to 4. We conclude that the approximate equality of σo

and σ f found previously for neutral spheres and pores [4,5] con-tinues to hold when electrostatic interactions are present.

The present approach could be used to analyze the effects ofcharge on σo in more complex geometries, such as a sphericalmacromolecule passing through an array of fibers. With the ef-fects of the streaming potential on σo apparently negligible, ef-forts to extend the theory to more complex geometries can focuslargely on evaluating the corresponding electrostatic energies (E).Moreover, our finding that Boltzmann factors [exp(−E/kT )] de-rived using the linear and nonlinear Poisson–Boltzmann equationswere nearly identical suggests that linearization can be invoked tosimplify the calculation of E for other systems. (This conclusionapplies only to systems of like charge, where E > 0.) Electrostaticenergies have been reported for a sphere interacting with a sin-gle cylindrical fiber [19]; a task that remains for fiber arrays is toaccount for the simultaneous electrostatic interactions of a spherewith multiple fibers.

Appendix A

A.1. Simplifications for small potentials

If the pore radius, pore surface charge density, and Debyelength are such that the equilibrium double layer potential is small,certain simplifications can be made. If |Ψ | � 1, then sinh Ψ ∼= Ψ

and Eq. (9) becomes

1

β

d

dβ

(β

dΨ

dβ

)= τ 2Ψ, (A.1)

which is a modified Bessel equation. The solution that satisfies Eqs.(11) and (12) is

Ψ (β) = qc

τ

I0(τβ)

I1(τ ). (A.2)

Thus, Eq. (A.2) may be used in place of the numerical solutionof Eq. (9), provided that |Ψ | � 1. In addition, the integral in thestreaming potential contribution to the fluid velocity [Eqs. (18) and(19)] may be evaluated analytically:

1∫β

dy

y

y∫0

sinh Ψ x dx ∼=1∫

β

dy

y

y∫0

Ψ x dx = qc

τ 3

I0(τ ) − I0(τβ)

I1(τ ). (A.3)

Aside from being able to make the substitutions sinh Ψ = Ψ andcosh Ψ = 1 in Eq. (23), there was no particular simplificationin evaluating the streaming potential. Computing Λ, local valuesof vz , and σo each still required numerical integration. Accordingly,the code written to evaluate σo did not use the approximationsembodied in Eqs. (A.1)–(A.3). If macromolecules are absent ( f = 0in the velocity expressions), the linearization for small potentialspermits the analytical evaluation of Λ and vz(β) [15]. This washelpful for checking those parts of our code, but did not simplifythe calculation of σo .

A.2. Analytical evaluation of electrostatic potential energies

A major computational saving was achieved by using previ-ous analytical results for the particle potential energy, E(β). Thoseresults were based on eigenfunction solutions of the linearizedPoisson–Boltzmann equation for the three-dimensional geometrycorresponding to a sphere positioned eccentrically within a cylin-drical pore [11]. As discussed in connection with Fig. 1, accuratevalues of the Boltzmann factor were obtained in that manner evenif |Ψ | was not small. As shown previously [11], when the sur-face charge densities are constant, E is a quadratic function of thecharge densities. Thus,

E

ξkT= αsq2

s + αscqsqc + αcq2c , (A.4)

where the coefficients αs , αsc, and αc each depend on β , λ, and τbut are independent of qs and qc . The energy scale factor ξ is thatdefined in Eq. (32). To good approximation, the coefficients may becalculated as

αs = 8πτλ4eτλ

(1 + τλ)2

Ω(τ ,β)

δ(τ ,λ), (A.5)

αsc = 4π2λ2 I0(τβ)

(1 + τλ)I1(τ )δ(τ ,λ), (A.6)

αc =[π I0(τβ)

τ I1(τ )

]2(eτλ − e−τλ)τλL(τλ)

(1 + τλ)δ(τ ,λ). (A.7)

The functions δ, L, and Ω are defined as

δ(τ ,λ) = πτe−τλ − 2(eτλ − e−τλ)τλL(τλ)Ω(τ ,β)

1 + τλ, (A.8)

L(τλ) = coth(τλ) − 1

τλ, (A.9)

Ω(τ ,β) = I0(τβ)

2

∞∑t=0

βt

2tt!It(τβ)

τ t

∞∫ς=0

ς2t[

K1([τ 2 + ς2]1/2)

I1([τ 2 + ς2]1/2)

]dς,

(A.10)

where In and Kn are modified Bessel functions of the first andsecond kinds, respectively, of order n. Truncating the series inEq. (A.10) after 20 terms is usually sufficient for 1% accuracy. Thisexpression, which is from Eq. (2.121) in Smith [20], can be simpli-fied further if the Debye length is relatively small. If τ � 3, Ω iswell-represented by Eq. (30) in Smith and Deen [11].

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