Journal of Colloid and Interface Science 352 (2010) 1–10

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Journal of Colloid and Interface Science

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Feature Article

Conductivity and electrophoretic mobility of dilute ionic solutions

Stuart Allison ⇑, Hengfu Wu, Umar Twahir, Hongxia PeiDepartment of Chemistry, Georgia State University, Atlanta, GA 30302-4098, USA

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

Article history:Received 24 June 2010Accepted 2 August 2010Available online 10 August 2010

Keywords:Electrokinetic transportElectrophoretic mobilityElectrical conductance

0021-9797/$ - see front matter � 2010 Published bydoi:10.1016/j.jcis.2010.08.009

⇑ Corresponding author. Fax: +1 404 651 1416.E-mail address: sallison@gsu.edu (S. Allison).

Two complementary continuum theories of electrokinetic transport are examined with particularemphasis on the equivalent conductance of binary electrolytes. The ‘‘small ion” model [R.M. Fuoss, L.Onsager, J. Phys. Chem. 61 (1957) 668] and ‘‘large ion” model [R.W. O’Brien, L.R. White, J. Chem. Soc. Far-aday Trans. 2 (74) (1978) 1607] are both discussed and the ‘‘large ion” model is generalized to include anion exclusion distance and to account in a simple but approximate way for the Brownian motion of allions present. In addition, the ‘‘large ion” model is modified to treat ‘‘slip” hydrodynamic boundary con-ditions in addition to the standard ‘‘stick” boundary condition. Both models are applied to the equivalentconductance of dilute KCl, MgCl2, and LaCl3 solutions and both are able to reproduce experimental con-ductances to within an accuracy of several tenths of a percent. Despite fundamental differences in the‘‘small ion” and ‘‘large ion” theories, they both work equally well in this application. In addition, both‘‘stick-large ion” and ‘‘slip-large ion” models are equally capable of accounting for the equivalent conduc-tances of the three electrolyte solutions.

� 2010 Published by Elsevier Inc.

1. Introduction

One of the early successes of atomic scale continuum transportmodeling concerned the electrical conductance of dilute solutionsof strong electrolytes [1–3]. This work, in turn, was grounded onequilibrium theory of strong electrolytes by Debye and Huckel [4].The early theory, which was restricted to very dilute solutions of ionsmodeled as point charges, was subsequently extended to account forthe finite size of the ions and also higher electrolyte concentrations[5–7]. For monovalent binary aqueous electrolyte solutions up to aconcentration of about 0.10 mol/dm3 or M, experimental and modelequivalent conductances are in excellent agreement [5–7] for rea-sonable choices of model parameters. Refs. [5–7] are restricted tobinary electrolytes. This was subsequently extended to general elec-trolyte solutions made up of an arbitrary number of ions of arbitraryvalence [8]. In the present work, this approach shall collectively becalled the ‘‘small ion” model. Despite the successes of the ‘‘small ion”model, there have been few attempts to apply it directly to electro-lyte solutions containing polyvalent ions or mixtures of electrolytescontaining more complex ionic species. One of its shortcomings isthat electrostatics are treated at the level of the linear Poisson–Boltzmann equation which limits it to weakly charged particles.The theory has been generalized to go beyond the use of the linearPoisson–Boltzmann equation in representing the ionic potential ofmean force [9,10].

Elsevier Inc.

A transport phenomenon closely related to electrical conductiv-ity is the free solution electrophoretic mobility. In recent years, cap-illary zone electrophoresis [11–15] has become a widely usedseparation technique for a broad array of ionic species includingpeptides [16–21], organic anions [22,23], proteins [24–26], andnanoparticles [27,28]. Although the conductance theories discussedin the previous paragraph [3,5–7] have been applied to mobilitystudies of small and weakly charged ions [22,23,29,30], they arenot appropriate for large or highly charged particles including nano-particles [27,28] or metal oxide colloidal particles [31]. For largeand/or highly charged particles, there is a long established alterna-tive that is grounded on very similar continuum electro-hydrody-namic principles, but has its origin in the electrophoresis of largecolloidal particles [32–37]. In this work, it shall be called the ‘‘largeion” model. Of particular relevance to the ‘‘large ion” model is thenumerical procedure of O’Brien and White [37] that has come intowidespread use and can be applied to the electrophoresis of a spher-ical particle of arbitrary size containing a centrosymmetric chargedistribution of arbitrary net charge. One factor that may limit theapplication of the ‘‘large ion” model to treat the mobility or conduc-tivity of small ions is that it ignores the Brownian motion of the ion ofinterest. This may only be a reasonable approximation if the ion ofinterest is much larger than the ions comprising the surroundingelectrolyte. The ‘‘small ion” model, on the other hand does accountfor the Brownian motion of all ions present [3,5–10].

The principle objective of the present work is to bridge the gapbetween the ‘‘small ion” and ‘‘large ion” models discussed above byapplying both to the conductivity of a number of binary electro-lytes for which experimental conductance data is available.

2 S. Allison et al. / Journal of Colloid and Interface Science 352 (2010) 1–10

Polyvalent salts shall also be considered in order to test the modelsunder conditions of larger ‘‘zeta” potential. In the course of thiswork, it has been necessary to modify the O’Brien and White(‘‘large particle” model) procedure in three ways. The first is to in-clude an ‘‘ion free” layer of solvent just outside the surface ofhydrodynamic shear. In addition to the distance of hydrodynamicshear from the center of ion j, aj; an ion exclusion distance, aex, isalso defined. The second modification concerns hydrodynamicboundary conditions. In both ‘‘small ion” and ‘‘large ion” models,hydrodynamic boundary conditions have been handled somewhatdifferently and it is important to consider how this influences theresults. Currently, ‘‘stick” boundary conditions are employed inthe ‘‘large ion” model, and this means the particle velocity andfluid velocity match at the surface of hydrodynamic shear. In addi-tion to the conventional ‘‘stick” hydrodynamic boundary condi-tions, we also consider ‘‘slip”. As shall be shown, both ‘‘stick” and‘‘slip” models are capable of explaining the experimental conduc-tance data equally well for about the same aex values, but differentaj values must be chosen on the basis of limiting ionic conductivi-ties. Third, although the ‘‘large ion” model does not account explic-itly for the Brownian motion of the central ion of interest, wepresent a simple way of doing so that involves adding a correctionterm to the mobile ion mobilities. When this is done, the ‘‘smallion” and ‘‘large ion” model conductivities with the same or similarmodel parameters are comparable with each other and yield excel-lent agreement with experiment.

2. Modeling

2.1. Conductance and mobility of small Ion electrolytes

The original Onsager [2] and Onsager–Fuoss [3] theory treatsthe equivalent conductivity, K, or electrophoretic mobility of ionicspecies j, lj, of dilute strong electrolyte;

Kp ¼ K0 � ðaK0 þ bÞffiffiffiffiffiffiffim0p

ð1Þ

In Eq. (1), the ‘‘p” subscript denotes the original Onsager–Fuossmodel mobility, K0 is the equivalent conductance of the solutionin limit of zero ionic strength, a is the ‘‘relaxation coefficient”, b isthe ‘‘electrophoresis coefficient”, and m0 would be the concentra-tion of electrolyte in moles/dm3 or M if it did not dissociate intoions. The physical basis of a is ion relaxation, the distortion of theion atmosphere around a particular ion from equilibrium due tothe imposition of an electric and/or flow field. The physical basisof b is the additional hydrodynamic backflow produced in the vicin-ity of a particular ion produced by the presence of nearby ions. Thecoefficients a and b depend on: temperature, T, the properties of thesolvent including relative dielectric constant, er, and viscosity, g,and the valence charges of the ionic species present in solution,{zj}. They are, however, independent of ionic size. Ionic size, how-ever, does enter through K0 or equivalently, the electrophoreticmobility of individual ions in the limit of zero ionic strength, lj0.

Underlying Eq. (1) is a model in which the ions are treated aspoint charges. The Onsager–Fuoss, OF, theory [2,3] starts with ageneral equation of continuity which specifies the concentrationof ions of one species in the vicinity of ions of other species in anelectrolyte solution which has reached a steady state under theinfluence of a weak, constant external electric field, e0. Account istaken of the Brownian motion of the various ionic species of whichan arbitrary number may be present. Electrostatics are treated atthe level of the linear Poisson–Boltzmann equation. This is truenot only for the ‘‘point ion” model discussed here, but the moregeneral finite ion case considered at the end of this section. Forthe ‘‘point ion” model, the boundary conditions on the fluid veloc-ity are that it remain finite at the center of an ion and vanish, on

average, at a large distance away from it. The general OF mobilityexpression of an ionic species can be written [3]

lj ¼ lj0 � B1zjlj0

X1n¼0

cnrðnÞj þ B2zj

! ffiffiIp

ð2Þ

where I is the ionic strength of the electrolyte, cn and rðnÞj are dis-cussed below,

B1 ¼ffiffiffi2p

F3

12pNAvðe0erRTÞ�3=2 ¼ 2:806� 106

ðerTÞ3=2 ðM�1=2Þ ð3Þ

B2 ¼ffiffiffi2p

F2

6pgNAvðe0erRTÞ�1=2 ¼ 4:275� 10�6

gðerTÞ1=2

m2

V sec M1=2

� �ð4Þ

In the present work, SI units shall be followed for the most part,but g in Eq. (4) is in centipoise and I is in moles/dm3 = M. For theremaining terms, F is the Faraday constant (=9.645 � 104 C/mole),NAv is Avogadros Number, e0 is the permittivity of free space, R isthe gas constant, and other quantities (except for cn and rðnÞj ) havebeen described in the previous paragraph. Also, the physical basisof the B1 term in Eq. (2) is ion relaxation and the physical basis ofthe B2 term is the electrophoretic effect.

Consider a single strong electrolyte, AaBb, or binary electrolyte,that undergoes complete dissociation according to

AaBb ! aAzþ þ bBz� ð5Þ

where a and b, and z+ and z� are stoichiometries and valencies ofthe two ions. If m0 is the initial concentration of undissociated elec-trolyte, then the condition of electrical neutrality requires

am0zþ þ bm0z� ¼ 0 ð6Þ

and,

I ¼ m0

2az2þ þ bz2

�

� �¼ azþ

2ðzþ � z�Þ

� �m0 ¼ /2m0 ð7Þ

The conductivity, K, of a solution of this strong electrolyte can bewritten

K ¼ am0kþ þ bm0k� ¼ am0jzþjKþ þ bm0jz�jK�¼ am0Fzþlþ þ bm0Fz�l� ¼ am0Fzþ½lþ � l�� ¼ am0zþK ð8Þ

The kj terms appearing in the first equality on the right handside of Eq. (8) are molar conductivities of specific ions. The Kj val-ues appearing in the second equality on the right hand side of Eq.(8) correspond to equivalent conductivities of specific ions. Most,but not all, of the specific ion conductivities reported in the mod-ern literature and in handbooks are Kj’s and in SI units are inS m2/mole or m2/(ohm mole). The third equality follows from therelationship between equivalent ionic conductance and ion mobil-ity, which will be positive for + ions and negative for � ions, Kj = F|lj|. The fourth equality follows from the electroneutrality condi-tion, Eq. (6). The fifth equality gives the equivalent conductanceof the binary electrolyte, K. Dividing various equalities on the righthand side of Eq. (8) by am0z+ gives

K ¼ Kþ þK� ¼ Fðlþ � l�Þ ¼ Fðjlþj þ jl�jÞ ð9Þ

For a strong binary electrolyte of the form AaBb, the ‘‘relaxation”term in Eq. (2) according to the OF theory can be written (3)

Sj � zj

X1n¼0

cnrðnÞj ¼jzþz�jq1þ ffiffiffi

qp ð10Þ

q ¼ jzþz�jjzþj þ jz�j

� � jlþ0j þ jl�0jjz�lþ0j þ jzþl�0j

� �

¼ jzþz�jjzþj þ jz�j

� �Kþ0 þK�0

jz�jKþ0 þ jzþjK�0

� �ð11Þ

S. Allison et al. / Journal of Colloid and Interface Science 352 (2010) 1–10 3

If the binary electrolyte is also a symmetric electrolyte(|z+| = |z�|), then q = 1/2. Using Eqs. (2), (7), and (10) in (9), we havefor a general binary electrolyte

K ¼ K0 � ½B1ðKþ0Sþ þK�0S�Þ þ B3ðjzþj þ jz�jÞ�/ffiffiffiffiffiffiffim0p

ð12Þ

K0 ¼ Kþ0 þK�0 ð13Þ

B3 ¼ FB2 ¼0:4125

gðerTÞ1=2

m2

ohm mole M1=2

� �ð14Þ

It should be emphasized that g in Eq. (14) is in centipoise. Com-paring Eqs. (12) with (1), and equating K with Kp, we can nowidentify the relaxation and electrophoresis coefficients,

a ¼ B1ðKþ0Sþ þK�0S�Þ/K0

ð15Þ

b ¼ B3ðjzþj þ jz�jÞ/ ð16Þ

The expression for a simplifies further within the framework ofthe OF theory. From Eq. (10) we have S+ = S� = S and Eq. (15) thenreduces to

a ¼ B1S/ ð17Þ

In order to make contact with different relaxation theories,however, we have chosen to distinguish the relaxation terms forthe different ions, Sj, defined by Eq. (10).

For an electrolyte consisting of more than two ionic species, therelaxation effect is more complex than discussed in the previousparagraph, which is strictly valid only for a binary electrolyte.The OF theory can also be applied to ionic solutions containingan arbitrary number of distinct ions. Assume we have N ions pres-ent and let mj denote the concentration (in M) of species j. For thismore general case, the Sj terms are given by the first equality onthe right hand side of Eq. (10). We have (3)

c0 ¼ ð2�ffiffiffi2pÞ=2 ¼ 0:2928932 ð18Þ

c1 ¼ �0:3535534 ð19Þ

cn ¼ cn�13

2n� 1

� �ðn > 1Þ ð20Þ

rðnÞ ¼ ð2H � IÞ � rðn�1Þ ð21Þ

ðrð0ÞÞj ¼ rð0Þj ¼ zj �PN

k¼1ukzkPNk¼1uk=wk

!1wj

� �ð22Þ

uj ¼mjz2

jPNk¼1mkz2

k

ð23Þ

wj ¼ Kj0=jzjj ð24Þ

Hij ¼ dij

XN

k¼1

ukwk

wi þwk

!þ ujwj

wi þwjð25Þ

In Eq. (21), rðnÞ is a N by 1 column vector and H and I are N by Nmatrices. Also, I is the identity matrix and dij is the Kroneker delta.In general, it is necessary to solve for Sj iteratively. One begins bydetermining rð0Þ from Eq. (22) and known input parameters. Thesame input parameters are used to determine the components ofH. Then Eq. (21) is used to generate rðnÞ for higher order terms inn. These along with cn defined by Eqs. (18)–(20) are used in Eq.(10) to compute Sj. In most cases, the series converges rapidly withn. Despite its apparent complexity, this procedure is actually quitesimple and straightforward to implement in a computer program

or an Excel spreadsheet, which will be shared with interestedinvestigators upon request to the corresponding author. In termsof the dimensionless relaxation terms, Sj, generated by the aboveprocedure, Eq. (2) can be written

lj ¼ lj0 � ðB1lj0Sj þ B2zjÞffiffiIp

ð26Þ

In the 1950s, Fuoss and Onsager generalized this theory to ex-tend its range of validity to terms of order m1

0 in electrolyte concen-tration and also account, to lowest order, for the finite sizes of theions [6,7]. This work was restricted to binary electrolytes and spe-cific applications in this and subsequent work [38–40] were fur-ther restricted to monovalent (binary) electrolytes. Quint andVaillard [8] did generalize this to an arbitrary electrolyte and in-cluded terms to order m3=2

0 although some contributions at the le-vel of m3=2

0 are missing [41]. In these studies [6–8,38–41] a singleion exclusion distance, aex, is defined and the assumption is madethat the center-to-center distance, r, between any two ions cannotbe smaller than aex. The assumption is also made that the normalcomponent of the relative fluid velocity vanishes at r = aex (seeEq. (3.4) of (7)). Closely related work was also carried out by Pitts[5] on symmetric binary electrolytes, but the assumption wasmade that the relative fluid velocity as a whole, and not just thenormal component, vanishes at r = aex. Within the framework ofthe more general Fuoss–Onsager theory [6,7], the equivalent con-ductance of a binary electrolyte can be written

K ¼ Knrð1� nÞ ð27Þ

Knr ¼ K0 � bffiffiffiffiffiffiffim0p

=ð1þ jaexÞ ð28Þ

j ¼ B/ffiffiffiffiffiffiffim0p

ð29Þ

B ¼ffiffiffi2p

Fffiffiffiffiffiffiffiffiffiffiffiffiffiffie0erRTp ¼ 5:028� 1011ffiffiffiffiffiffiffi

erTp 1

M1=2mð30Þ

n ¼ affiffiffiffiffiffiffim0p

ð1� D1 þ D2Þ þbD03K0

ffiffiffiffiffiffiffim0p

ð31Þ

The term Knr is the equivalent conductance in the absence ofion relaxation, n denotes the relaxation correction, and j is theDebye–Huckel screening parameter. In Eq. (31), the terms D1, D2,and D03 represent higher order correction terms and depend on con-centration to leading order m1=2

0 . Explicit expressions are given inSection 7 of Ref. [7] specific to symmetric binary electrolytes. Moregeneral expressions (making minor corrections for sign errors) canbe deduced from equations in Section 6 of Ref. [7] for generalbinary electrolytes. In subsequent work by Fuoss and coworkers,additional corrections were made to D2, and D03 [38–40,42]. How-ever, these changes were minor and do not alter the relaxationcorrections significantly. In addition, expressions not limited tobinary electrolytes can be found in references [8,41] and includeterms to order m3=2

0 . Since the 1957 paper by Fuoss and Onsager[7] carries out the most thorough comparison between theoreticaland experimental conductancies and subsequent applications aremostly restricted to monovalent binary electrolytes, we shall usethe equations from the 1957 paper [7] in the present work whenconsidering the ‘‘small ion” model. The ‘‘small ion” theory pio-neered by Fuoss and Onsager [2,3] remains in widespread use tothis day not only for fully dissociated electrolytes, but undissoci-ated electrolytes as well [43]. With few exceptions [41], the over-whelming majority of applications involve symmetric binaryelectrolytes [43–45]. The limiting assumptions of the ‘‘small ion”theory are: (1) solvent and mobile ions are treated as a continuum,(2) electrostatics are described by the linear Poisson–Boltzmann

4 S. Allison et al. / Journal of Colloid and Interface Science 352 (2010) 1–10

equation, (3) a single ion exclusion distance, aex, is included inmodeling, (4) jaex is small (small ion sizes and low electrolyteconcentration).

In addition to Eq. (27), we shall also consider a simpler modelthat includes finite ion size effects in Knr as given by Eq. (28),but restricts the relaxation term to m1=2

0 and this allows us toignore D1, D2, and D03 altogether. Define

K1 ¼ Knrð1� affiffiffiffiffiffiffim0p

Þ ð32Þ

As demonstrated in the main body of this work, Eq. (32) worksalmost as well as Eq. (27) for binary electrolytes for m0 6 0.005 M.

Finally, consider the ion electrophoretic mobilities within theframework of the Onsager–Fuoss theory restricted to a binary elec-trolyte. We can write

lj ¼ lj;nrð1� njÞ ð33Þ

lj;nr ¼ lj0 � B2zj/ffiffiffiffiffiffiffim0p

=ð1þ jaexÞ ð34Þ

where nj is the same for both ions and the j subscript is omitted inEq. (27) and (31). As Eq. (31) shows, higher order concentration ef-fects (via the D1, D2, and D03 terms) can be accounted for in binaryelectrolytes. However, it would be useful if we could also considerternary and higher order electrolyte solutions. This would be rele-vant if we were interested in the mobility of a ‘‘guest” ion in thepresence of a binary electrolyte, for example. Eq. (26) makes it pos-sible to consider such cases, where the relaxation term depends onthe particular ion and we can write

nj ffi B1Sj

ffiffiIp

ð35Þ

I ¼ 12

Xk

mkz2k ð36Þ

where Sj is given by the first term on the right hand side of Eq. (10)and the sum in Eq. (36) extends over all ions present in solution. Forternary and higher order electrolyte solutions, we can approximatenj appearing in Eq. (33) with Eq. (35). This is equivalent to ignoringterms higher than order I1/2 in the relaxation correction. The sameapproximation is made in arriving at Eq. (32) for the conductivityof a binary electrolyte and Eq. (32) therefore gives us a way of deter-mining how accurate this approximation is in specific cases.

2.2. Mobility of large spherical ions

It shall be assumed that our model particle is spherical and con-tains a centrosymmetric charge distribution within a surface ofhydrodynamic shear located at a distance r = a from the center ofthe particle. At the shear surface, ‘‘stick” or ‘‘slip” boundary condi-tions may prevail. In the case of ‘‘stick”, it is assumed that fluid andparticle velocities match at r = a. In the case of ‘‘slip” boundary con-ditions, it is assumed that only the outward normal component, n,of particle and fluid velocities match at r = a. Also, if rH denotes thehydrodynamic stress tensor of the fluid, then rH � n is parallel to nat the shear surface [46,47]. Outside of the shear surface, the fluidis treated as a hydrodynamic continuum that obeys the linearizedNavier–Stokes and solvent incompressibility equations.

gr2v �rp ¼ r � rH ¼ �s ð37Þ

r � v ¼ 0 ð38Þ

where g is the fluid viscosity, v is the local fluid velocity, p is thelocal fluid pressure, and s is the local external force/volume onthe fluid. If we had an uncharged particle (s ¼ 0) translating withvelocity u� through a fluid that is at rest far from the particle, the

solution of Eqs. (37) and (38) for the fluid velocity, v0ðrÞ, can bewritten for r > a,

v0ðrÞ ¼ Tða; rÞ � u� ð39Þ

where for ‘‘stick” boundary conditions, the tensor, Tða; rÞ, is [48]

Tða; rÞ ¼ 3a4rðI þ nnÞ þ a3

4r3 ðI � 3nnÞ ð40Þ

nn ¼ rr=r2 ð41Þ

and for ‘‘slip” boundary conditions,

Tða; rÞ ¼ a2rðI þ nnÞ ð42Þ

In Eqs. (40) and (42), I denotes the 3 by 3 identity matrix. The zerosuperscript on v0 is a reminder that this refers to the special case ofan uncharged spherical particle.

Extending from r = a to r = aex, where aex is the ion exclusion dis-tance, it is assumed that no ions are present. In this region of thefluid adjacent to the particle, s ¼ 0. The fluid, however, obeysEqs. (37) and (38). For r > aex, the ion atmosphere is treated as acontinuum. Let njðrÞ denote the local concentration of mobile ionspecies j in M and let zj denote the valence charge of a single ion.The charge distribution, qðrÞ, obeys the Poisson equation ingeneral,

r � ðeðrÞrWðrÞÞ ¼ �qðrÞ=e0 ð43Þ

qðrÞ ¼ qf ðrÞ þ FX

j

zjnjðrÞ ð44Þ

where e0 is the permittivity of free space, e is the local relativedielectric constant, W is the electrodynamic potential, qf is the fixedcharge density (within the particle) and the sum in Eq. (44) extendsover all mobile ion species present. If it is assumed eðrÞ ¼ ei for r < aand eðrÞ ¼ er for r > a, we also have the boundary condition

ei@W@r

� �r¼a�

¼ er@W@r

� �r¼aþ

ð45Þ

where a± denotes a point just outside or inside the particle surface.To proceed, we use the notation and many of the protocols of

O’Brien and White [37]. Due to the presence of a constant externalelectric, e0, or flow field, the steady state electrodynamic potentialis written

WðrÞ ¼ W0ðrÞ þW1ðrÞ � e0 � r ð46Þ

where W0 is the local equilibrium electrostatic potential and W1 is aperturbation potential that vanishes far from the particle. The localion densities are also perturbed from their equilibrium values,nj0ðrÞ, which are related to a new potential, UjðrÞ, defined by

njðrÞ ¼ nj0ðrÞe�ezjðUjðrÞþW1ðrÞÞ=kBT ffi nj0ðrÞ½1� ezjðUjðrÞ þW1ðrÞÞ=kBT�ð47Þ

nj0ðrÞ ¼ mje�zjy0ðrÞ ð48Þ

y0ðrÞ ¼eW0ðrÞ

kBTð49Þ

where kB is the Boltzmann constant. It is assumed that the perturb-ing electric or flow field is sufficiently small that only terms to firstorder in those perturbing fields are significant. This justifies theexpansion of the exponential in Eq. (47). In addition to Eqs. (37),(38), and (43), an ion transport equation must be solved for eachionic species present. Retaining first order terms in the perturbingelectric or flow fields [37,49],

S. Allison et al. / Journal of Colloid and Interface Science 352 (2010) 1–10 5

r � Jj ¼ 0 ð50Þ

Jj ¼ nj0v þ Fzjnj0DjðrUj þ e0Þ=RT ð51Þ

Above, Jj is the local current density of species j and Dj is the trans-lational diffusion constant. Other quantities have been defined pre-viously. The boundary condition on j arises as a result of theconstraint that mobile ions cannot approach the particle closer thana distance aex. Setting Jj � n ¼ 0 in Eq. (51) then yields

@UjðrÞ@r

� �r¼aex

¼ �e0 � n ð52Þ

For a spherical particle, the solution of the equilibrium electro-static potential is a special case of Eq. (43). The reduced potentialdepends only on the distance from the center of the particle, r.For r > a,

1r2

ddr

r2 dy0ðrÞdr

� �¼ � Fe

e0erkBT

Xj

mjzjxðrÞe�zjy0ðrÞ ð53Þ

@y0

@r

� �r¼aþ

¼ �a1

að54Þ

In Eq. (53), x(r) is a step function which equals 0 for r < aex and 1 forr > aex. In Eq. (54), a1 = e2Z/(4e0erkBTa) (dimensionless) and Z is thenet valence charge of the particle. For a weakly charged particle,the exponential in Eq. (53) can be expanded and the resulting linearequation can be solved analytically. This is the linear Poisson–Boltz-mann equation and the reduced (dimensionless) potential for thepresent problem for r > a can be written

yLPB0 ðrÞ ¼

a1ar � ja

1þjaex

� �a < r < aex

a1arð1þjaexÞ e

�jðr�aexÞ aex < r

8<:

9=; ð55Þ

The ‘‘LPB” superscript on y0 denotes the linear Poisson Boltz-mann reduced potential. In the present work, however, the generalnon-linear for of Eq. (53) is solved numerically subject to theboundary condition imposed by Eq. (54).

For the nonequilibrium problem, we follow the strategy of car-rying out two separate transport cases (37). In Case 1, the particleis translated with constant velocity, u0, in a fluid that is otherwiseat rest. In Case 2, the particle is held stationary, but it is subjectedto a constant external electric field, e0. Although the potentials,UðiÞj ðrÞ, are not spherically symmetric (the (i) superscript has beenadded to distinguish the two transport cases), they can be writtenin terms of related functions that are [37],

UðiÞj ðrÞ ¼1r

/ðiÞj ðrÞbðiÞ � r ð56Þ

where bð1Þ ¼ u0 and bð2Þ ¼ e0. As discussed in detail previously [37],the coupled equations for the fluid velocity and ion transport arecast into the form of 1 dimensional differential equations in the ra-dial variable r. These are then solved numerically for the two trans-port cases. Let N denote the number of mobile ions species present(which is two for a binary electrolyte). For each transport case, N + 2homogeneous and one inhomogeneous set of differential equationsare solved subject to different distant boundary conditions. Theoverall solution for Case 1 or 2 is then taken to be a particular linearcombination of the above mentioned N + 3 ‘‘distant” solutions thatsatisfy appropriate boundary conditions at r = aex. The overall solu-tion of /j(r) for case i (i = 1 or 2), is

/ðiÞj ðrÞ ¼XNþ2

k¼0

dðiÞk /ði�kÞj ðrÞ ð57Þ

where the sum over k extends over the inhomogeneous (k = 0) anddifferent homogeneous (j = 1 to N + 2) ‘‘distant” solutions, Uði�kÞ

j (x) is

the kth ‘‘distant” solution for case i and ion j, and the dðiÞk are the lin-ear coefficients that are determined from boundary conditions on ornear the particle as discussed later. For k = 0 or N + 1 or N + 2, theouter boundary condition on Uði�kÞ

j (r) is set to 0. For k = 1 to N,

/ði�kÞj ðxÞ ¼ dj;k

r2 ðjr 1Þ ð58Þ

where dj,k is the Kroneker delta.The remaining two homogeneous ‘‘distant” solutions (k = N + 1

or N + 2) are associated with the distant behavior of a scalar field,g(i )(r), from which the fluid velocity, v ðiÞðrÞ, is derived (37). It isdefined by

v ðiÞðrÞ ¼ curl½curl½gðiÞðrÞbðiÞ�� þ uðiÞ1 ð59Þ

where bðiÞ is defined following Eq. (55) and uð1Þ1 ¼ �u0, and uð2Þ1 ¼ 0.The ‘‘distant” solutions of g(i�j) are set to zero except for j = N + 1

and N + 2. For jr 1,

gði�Nþ1ÞðrÞ ¼ r ð60Þ

gði�Nþ2ÞðrÞ ¼ 1r

ð61Þ

The final expression for g(i)(r) can be written that is identical to Eq.(57) above with g(i�k) replacing /jði�kÞ.

The dðiÞk coefficients appearing in Eq. (57) and an analogous rela-tion involving g(i)(r) and g(i�k)(r) are determined from ‘‘inner”boundary conditions. These are discussed in reference [37] forthe special case of ‘‘stick” boundary conditions when a = aex, butthe more general conditions of interest here must be handled dif-ferently. The boundary conditions on /ðiÞj (r) follow from Eq. (52)and are simply /ð1Þ0j (aex) = 0 and /ð2Þ0j (aex) = �1, where the primesuperscript denotes first derivative with respect to r. These arethe same for both ‘‘stick” and ‘‘slip” hydrodynamic boundary con-ditions. The boundary conditions on g(i) are evaluated at r = a andare different for ‘‘stick” and ‘‘slip”. For ‘‘stick”; g(1)0(a) = �a/2,g(1)0 0(a) = �1/2, g(2)0(a) = 0, and g(2)0 0(a) = 0 where the double primedenotes second derivative with respect to r. For ‘‘slip”; careful anal-ysis leads to the conditions: g(1)0(a) = �a/2, g(1)0 0 0(a) = 0, g(2)0(a) = 0,and g(2)0 0 0(a) = 0 where the triple prime denotes third derivativewith respect to r. With minor modifications in Eq. (7.7) of reference[37] that incorporate these modified boundary conditions, the dðiÞk

coefficients can be uniquely determined for Cases 1 and 2. Theoverall solution is then taken to be the linear superposition of bothCases 1 and 2 fields that gives a net force exerted by the particle onthe fluid of zero [37].

At this point, it is appropriate to discuss how the results of Case1 and Case 2 transport studies can be used to obtain a generalexpression for the electrophoretic mobility, l, and then reduce thatto more recognizable forms in special cases. It is possible to obtaina general expression of the electrophoretic mobility starting fromthe differential form of the Lorentz reciprocal theorem [48,50],

s0 � v ðiÞ þ r � ðv ðiÞ � r0HÞ ¼ sðiÞ � v0 þr � ðv0 � rðiÞH Þ ð62Þ

where quantities with superscript (i) denote the actual fieldsaround a charged spherical particles (i = 1 or 2) and quantities withsuperscript 0 denote an arbitrary flow field that obeys Eqs. (37) and(38) subject to appropriate hydrodynamic boundary conditions. Forthe arbitrary fields, choose an uncharged sphere of radius a (same asthe radius of our charged particle) where v0 is given by Eqs. (39)–(42). Integrate Eq. (62) over the fluid domain, X, exterior to a singleisolated particle enclosed by surface Sp with outward normal (intothe fluid), n. Applying the divergence theorem yields

�Z

Sp

v ðiÞ � r0H � ndS ¼ �

ZSp

v0 � rðiÞH � ndSþ u� �Z

XT � sðiÞ dV ð63Þ

6 S. Allison et al. / Journal of Colloid and Interface Science 352 (2010) 1–10

The total hydrodynamic force exerted by the particle on the fluid is

FðiÞH ¼ �Z

Sp

rðiÞH � ndS ð64Þ

and an entirely analogous expression can be written for F0H . For

‘‘stick” boundary conditions, the fluid velocities inside the surfaceintegrals are constant and can be moved outside the integral di-rectly. For the ‘‘slip” case, first recognize that we can writerH � n ¼ ðnnÞ � ðrH � nÞ. Since the normal component of the fluidvelocity matches that of the particle on Sp in the slip case and sinceonly the normal component contributes to the surface integrals, wecan move them out of the integrals also. The total force exerted bythe particle on the fluid is the sum of hydrodynamic and external(electrical) forces [37,49,51]

FðiÞT ¼ FðiÞH þZ

XsðiÞ dV ð65Þ

Using Eqs. (64) and (65) in Eq. (63),

u� � FðiÞT ¼ uðiÞ � F0H þ u� �

ZXðI � TÞ � sðiÞ dV ð66Þ

In Eq. (66), u� is the velocity of our uncharged particle and uðiÞ thevelocity of our charged particle (Case 1 or 2). The external forceterm can be written [49,51]

sðiÞ ¼ FX

j

zjnj0ðrUðiÞj þ e0di2Þ þ RTX

j

rnðiÞj ð67Þ

Using Eqs. (40), (42), (56)–(58), and (67) in (66), it is straightfor-ward to carry out the angular integrations. Also, the divergence the-orem is applied to the second term on the right hand side of Eq.(67). Without loss of generality, we can also assume the electric/flow fields are along the x direction with ðu�Þx ¼ 1 and only concernourselves with the x components of the overall forces. Eq. (66) canbe written,

FðiÞT ¼ uðiÞf0 �43pFbðiÞ

Xj

zjmjdðiÞj þ 4pF

Xj

zj

Z 1

ar2 dr nj0ðrÞgðiÞj ðrÞ

ð68Þ

gðiÞj ¼ 1�43

h1

� �eðiÞ þ2

3ð1�h1�h2Þ

/ðiÞj

rbðiÞ þ1

3ð1�2h1þ2h2Þ/ðiÞ0j bðiÞ

ð69Þ

f0 ¼ 6pgaðstickÞ; 4pgaðslipÞ ð70Þ

h1 ¼3a4rðstickÞ; a

2rðslipÞ ð71Þ

h2 ¼14

ar

� �3ðstickÞ; 0ðslipÞ ð72Þ

and u(1) = u0, u(2) = 0, e(1) = 0, e(2) = e0. The total force exerted by theparticle on the fluid in Case 1 and Case 2 transport is not zero. Wecan view the steady state electrophoretic migration of our particle,where the total force exerted on the fluid is indeed zero, as a super-position of the two. If we set u0 = l e0, where l is the electropho-retic mobility of our particle add Fð1ÞT and Fð2ÞT and set the sum tozero, we obtain

l ¼4pF

Pj

zjðmjdð2Þj =3� pð2Þj Þ

f0 � 4pFP

jzjðmjd

ð1Þj =3� pð1Þj Þ

! ð73Þ

pðiÞj ¼Z 1

ar2 dr nj0ðrÞ 1� 4

3h1

� �di2 þ

23ð1� h1 � h2Þ

/ðiÞj

r

"

þ 13ð1� 2h1 þ 2h2Þ/ðiÞ0j

#ð74Þ

Eqs. (73) and (74) are general expressions for the mobility of asphere and simplify in limiting special cases.

Consider the special case where ion relaxation is neglected. Thisis a reasonable approximation when the particle is weakly chargedand the solution of the linear Poisson Boltzmann equation, Eq. (55),is appropriate. Under these conditions, /ð1Þj ¼ 0; dð1Þj ¼ 0; /ð2Þj ¼ca3=r2; dð2Þj ¼ ca3, where c = ðer � eiÞ=ð2er þ eiÞ. All of the termswith superscript (1) vanish in Eq. (73) and the dð2Þj term also dropsout when we sum over j and impose the condition of charge neu-trality. Under these conditions, Eq. (73) reduces to

lnr ¼eZf0þ 4p

f0

Z 1

ar2 drq0ðrÞ

43

h1 �2ca3

3r3 ðh1 � 3h2Þ� �

ð75Þ

The ‘‘nr” subscript denotes the ‘‘no relaxation” limiting case and q0

is the equilibrium charge density. For the ‘‘stick” and ‘‘slip” cases,Eq. (75) reduces to

lsticknr ¼

eZ6pga

þ 23g

Z 1

ar drq0ðrÞ 1� c

2ar

� �3� a

r

� �5� �� �

ð76Þ

lslipnr ¼

eZ4pga

þ 23g

Z 1

ar drq0ðrÞ 1� c

2ar

� �3� �

ð77Þ

It is straightforward to reduce these equations further. The chargedensity equals zero for r < aex and for r > aex,

q0ðrÞ ¼ FX

j

zjm0je�zjy0ðrÞ ffi �2FI0yLPB

0 ðrÞ ð78Þ

where I0 =P

jm0jz

2j

� �=2 is the ionic strength in moles/m3, m0j is the

ambient concentration of species j in moles/m3, the exponentialhas been expanded and only the linear term in y0 has been retained.Furthermore, y0 has been approximated by its linear form given byEq. (55). Making use of Eqs. (55) and (78) in (76) and (77),

lsticknr ¼

eZ6pga

� 4aFI0a1

3gð1þjaexÞ

Z 1

aex

dre�jðr�aexÞ 1� c2

ar

� �3� a

r

� �5� �� �

ð79Þ

lslipnr ¼

eZ4pga

� 4aFI0a1

3gð1þ jaexÞ

Z 1

aex

dre�jðr�aexÞ 1� c2

ar

� �3� �

ð80Þ

The first term in brackets within the integrand on the right handsides of Eqs. (79) and (80) can be integrated directly and the c-terms can be reduced to exponential integrals. This shall not bedone here. Eqs. (79) and (80) provide a straightforward way of esti-mating the effect of hydrodynamic boundary conditions and ionexclusion on electrophoretic mobility of weakly charged sphericalparticles. One example shall be given. Consider a = aex and c = 1/2,Eq. (79) reduces to the Henry law mobility [32,50]. The ratio of‘‘slip” to ‘‘stick” mobilities increases from 1.50, 1.56, 1.77, 2.07,and 3.30 for ja = 0, 0.12, 0.52, 1.10, and 3.39, respectively.

2.3. Analysis of conductance data

Both the ‘‘small ion” and ‘‘large ion” theories described in theprevious two sections offer advantages and disadvantages andthe principle objective of this work is to bridge these two ap-proaches and show how they complement each other. The greatestadvantage of the ‘‘small ion” approach is that it accounts for theBrownian motion of the principle ion of interest whereas the ‘‘largeion” approach does not. The greatest advantage of the ‘‘large ion”approach is that it accounts more accurately for the finite size ofthe ion and is not restricted to weakly charged ions. (The ‘‘smallion” theories always employ the linearized form of the Poisson–Boltzmann equation.) From Eq. (9), we have the relationship be-tween equivalent ionic conductance, Kj, and absolute mobility,|lj|. In the limit of zero electrolyte, relaxation and electrophoretic

Table 1Comparison of KCl equivalent conductance between experiment and small iontheory.a

m0 (M) Kexp Kp K1 K D1 D

0.0005 147.74 147.67 147.71 147.68 �0.02 �0.040.001 146.88 146.79 146.87 146.82 �0.01 �0.040.005 143.48 143.08 143.46 143.41 �0.01 �0.050.010 141.20 140.30 141.05 141.07 �0.11 �0.090.020 138.27 136.36 137.80 138.07 �0.34 �0.14

a All K values are in 10�4 m2/(ohm mole).

S. Allison et al. / Journal of Colloid and Interface Science 352 (2010) 1–10 7

effects vanish and the first terms on the right hand side of Eq. (80)can be used to related the limiting equivalent ionic conductance,Kj0, to ionic hydration radius, aj.

Kj0 ¼ Fjlj0j ¼Fejzjj

a2pgajð81Þ

where zj is the valence charge on ion j and a2 equals 6 for ‘‘stick”and 4 for ‘‘slip” boundary conditions. In the present work, Kj0 areused as input data and aj will depend on the hydrodynamic bound-ary condition assumed. In water at 25 �C with Kj0 given in 10�4 m2/(ohm mole) and aj in nm,

aj ðnmÞ ¼ 55:28jzjja2Kj0ð10�4 m2=ðohm moleÞÞ

ð82Þ

When the ‘‘slip” boundary condition is used, the aj values must bescaled by a factor of 3/2 relative to the ‘‘stick” condition. In the‘‘small ion” theory [2,3,5–8,38–45], size enters implicitly throughthe Kj0 terms and explicitly through the ion exclusion distance,aex. The effect of aex on conductance becomes significant at higherelectrolyte concentration. Also, hydrodynamic boundary conditionsare usually dealt with by assuming ‘‘slip” boundary conditions holdon the ion exclusion surface at r = aex, and not at r = aj. (In the theoryof Pitts [5], ‘‘stick” boundary conditions are assumed to hold atr = aex.) In the ‘‘large ion” theories, the ion hydration radius does en-ter directly. With regard to the ion on interest, it enters through thehydrodynamic boundary condition on g(r) at r = a as discussed fol-lowing Eq. (59). Ionic hydration radii also enter indirectly for allions making up the background electrolyte in the solution of theion transport equation, Eq. (51). The ion diffusion constant, Dj,appearing in Eq. (51) are related to aj and Kj0 by the Einstein rela-tion [52]

Dj ¼kBT

a2pgaj¼ kBTKj0

Fejzjjð83Þ

However, availability of limiting ionic conductance data makes ex-plicit reference to aj unnecessary with regard to the relaxationcorrection.

A shortcoming of the ‘‘large ion” theories is that it ignores theBrownian motion of the ion of interest, call it ion k. Provided thision is much larger than the other ions making up the backgroundelectrolyte, that assumption is a reasonable approximation sinceDk / 1/ak. However, if ion k is comparable in size the other ions,the approximation is expected to break down and a comparisonof experimental conductances with both ‘‘small ion” and ‘‘largeion” theories shall give us an opportunity to evaluate this assump-tion. In the theory of diffusion controlled reactions, it is wellknown that the mutual diffusion constant of two species is simplythe sum of the individual diffusion constants of the two species[53–55]. A simple way of correcting the ‘‘large ion” theory to ac-count for the Brownian motion of the central ion is to replace Dj

or aj appearing in the ion transport equation with (Dj + Dk) or

aeffj ¼

1ajþ 1

ak

� ��1

ð84Þ

Since the relaxation process is dominated by the counterion, a sin-gle aeff

j is used for binary electrolytes, where aj and ak in Eq. (85) arecoion and counterion radii.

Both uncorrected and corrected applications of ‘‘large ion” the-ory to conductance data shall be reported below.

In the analysis of experimental conductance data below, whichis all in aqueous media at 25 �C, we shall use data summarized inthe Handbook of Chemistry and Physics [56] covering the concen-tration range 0.0 6 am0z+ 6 0.02 M. We shall first examine the sim-ple monovalent salt, KCl, since it has been extensively studied inthe past using the ‘‘small ion” theory [6,7]. Using the ‘‘small ion”

theory and treating aex as an adjustable parameter, we shall at-tempt to obtain good agreement between theory and experimentas well as confirm the conclusions of past work [6,7]. For a mono-valent salt like KCl, we expect the ‘‘small ion” theory to work aswell as it possibly can since the ions are both small and weaklycharged. Then using this aex, the ‘‘large ion” theory will be appliedto KCl for both ‘‘stick” and ‘‘slip”, and ‘‘uncorrected” and ‘‘cor-rected” (according to Eq. (85)) cases. The full numerical approachoutlined in the previous section shall be applied in these cases. Itshould be emphasized that once aex is fixed, once ‘‘stick” or ‘‘slip”hydrodynamic boundary conditions are assumed, and once uncor-rected or corrected mobile ion radii are selected, there are no fur-ther adjustable parameters in the ‘‘large ion” theory.

3. Results

3.1. Application to KCl

For KCl, we use [56] K0 = 149.79 � 10�4 m2/(ohm mole),T = 25 �C, g = 0.89 cp, and er = 78.53 in the small ion theory for a bin-ary symmetric electrolyte. The relative error of the equivalent con-ductance of this salt falls in the range of several hundredths of onepercent [57]. For this case, a = 0.22940 and b = 60.575 � 10�4 m2/(ohm mole). It is straightforward to show that aex = 0.350 nm yieldsbest agreement between theory and experiment which confirms thefinding of Fuoss and Onsager [7]. Experimental and ‘‘small ion” mod-el conductances are summarized in Table 1 where Kp, K1, and K aregiven by Eqs. (1), (32), and (27), respectively. The ‘‘point ion” modelfor conductance, Kp, clearly does not work well at higher concentra-tions, but the full model, K, and the approximate finite ion model,K1, work quite well, Shown in the last two columns of Table 1 isthe relative percent error defined by

D ¼ 100K�Kexp

Kexp

� �ð85Þ

For D1, K1 replaces K in Eq. (86). Both K and K1 reproduce exper-imental conductances to well within an accuracy of several tenthsof 1% although the full model is slightly better at the highest ionicstrength considered. This serves to demonstrate that the approxi-mate finite ion model works quite well in reproducing the experi-mental equivalent conductance of KCl. This is useful since theapproximate finite ion model is much simpler than the full finiteion model and can also be applied directly to ternary and higherorder electrolytes as discussed in Section 2.

The corresponding results for the ‘‘large ion” model studies aresummarized in Table 2. Full numerical calculations are carried outfor each ion. Mobilities are computed for both K+ (a+ = 0.125 nm(stick), 0.1875 nm (slip)) and Cl� (a� = 0.121 nm (stick), 0.1815 nm(slip)) and aex = 0.350 nm. In the ‘‘uncorrected” cases, the ion ofinterest (K+ or Cl�) is translated with constant velocity (Case 1) orheld stationary (Case 2). Diffusion constants, Dj, used in the iontransport equation are obtained from Eq. (84). In the ‘‘corrected”cases, the ion of interest is also held constant, but effective ion radiidefined by Eq. (85) are used in Eq. (84). The third and fourth columns

Table 2Comparison of equivalent conductance between experiment and large ion theory.a

m0 (M) Kexp K (stick,u)b D (stick,u) K (stick,c)c D (stick,c) K (slip,c)d D (slip,c)

0.0005 147.74 147.26 �0.32 147.82 +0.054 147.88 +0.0950.001 146.88 146.22 �0.45 146.99 +0.075 147.02 +0.0950.005 143.48 142.09 �0.97 143.61 +0.091 143.61 +0.0910.010 141.20 139.36 �1.30 141.28 +0.057 141.31 +0.0780.020 138.27 135.96 �1.67 138.33 +0.043 138.41 +0.101

a All K values are in 10�4 m2/(ohm mole).b Stick boundary conditions and not corrected for Brownian motion of the ion of interest.c Stick boundary conditions and corrected for Brownian motion of the ion of interest.d Slip boundary conditions and corrected for Brownian motion of the ion of interest.

Table 3Conductance Data for MgCl2 and LaCl3 (small ion model).a

z+m0 (M)

.0005 .0010 .0050 .0100 .0200

MgCl2Kexp 125.55 124.15 118.25 114.49 109.99Kp 125.31 123.64 116.59 111.31 103.85K1 125.58 124.09 118.25 114.26 109.09K 125.44 123.90 118.04 114.33 109.94D1 +.024 �.048 .000 �.201 �.818D �.088 �.201 �.178 �.140 �.045

LaCl3Kexp 139.6 137.0 127.5 121.8 –Kp 138.90 136.00 123.77 114.60 –K1 139.88 137.54 128.49 122.42 –K 139.45 136.92 127.64 122.12 –D1 +.201 +.394 +.776 +.525 –D �.107 �.058 +.110 +.263 –

a Conductivities are in 10�4 m2/(ohm mole).

8 S. Allison et al. / Journal of Colloid and Interface Science 352 (2010) 1–10

of Table 2 summarize the ‘‘uncorrected” model mobilities with stickboundary conditions. These model studies underestimate the equiv-alent conductance by 0.3–1.7% and the discrepancy increases withincreasing salt. The physical basis of this discrepancy is that therelaxation effect is overestimated here as a consequence of ignoringthe Brownian motion of the ion of interest. The corresponding ‘‘cor-rected” conductances are shown in columns 5 and 6 and here we areclearly doing much better. As in the case of the ‘‘small ion” modelstudies, we are now able to reproduce experimental conductancesto an accuracy that is better than 0.1%. In columns 7 and 8, the cor-responding ‘‘corrected” model conductances with ‘‘slip” boundaryconditions are presented. As mentioned previously, we have hadto scale the ionic hydration radii by 1.5 relative to the ‘‘stick” modelvalues in order to properly account for K+0 and K�0. As in the case ofthe corrected ‘‘stick” model conductances, the corrected ‘‘slip” con-ductances reproduce experimental values to an accuracy of betterthan 0.1%. Provided the hydrodynamic radii of the two ions arescaled in the manner discussed previously, the resulting model con-ductivities are very similar for ‘‘stick” and ‘‘slip” models.

Table 4Conductance/mobility data on MgCl2 (large ion model).a

m0 (M)

.00025 .0005 .0025 .0050 .0100Kexp 125.55 124.15 118.25 114.49 109.99

Sticky0(Mg+2, aex) 2.62 2.58 2.40 2.28 2.14�y0(Cl�, aex) 1.31 1.28 1.17 1.10 1.02E+(nr) 4.018 3.972 3.794 3.678 3.532�E�(nr) 5.830 5.803 5.696 5.628 5.544E+(r) 4.028 3.979 3.784 3.656 3.498�E�(r) 5.705 5.629 5.346 5.191 5.019n+ �.0025 �.0018 +.0026 +.0060 +.0096n� +.0214 +.0300 +.0614 +.0776 +.0947K(nr) 127.23 126.29 122.62 120.22 117.27K(r) 125.74 124.13 117.97 114.30 110.04D(r) +.151 �.016 �.236 �.166 +.045

SlipE+(nr) 4.018 3.973 3.799 3.686 3.547�E�(nr) 5.830 5.803 5.697 5.628 5.544E+(r) 4.029 3.981 3.793 3.670 3.520�E�(r) 5.708 5.629 5.352 5.193 5.027n+ �.0027 �.0020 +.0016 +.0043 +.0076n� +.0209 +.0300 +.0606 +.0773 +.0932K(nr) 127.24 126.30 122.69 120.34 117.47K(r) 125.80 124.17 118.16 114.50 110.42D(r) +.199 +.016 �.076 +.009 +.391

a Conductivities are in 10�4 m2/(ohm mole).

3.2. Application to MgCl2 and LaCl3

As in the case of KCl, conductivity data for MgCl2 and LaCl3 is ta-ken from reference [56]. Specifically, K0(MgCl2/2) = 129.34 (in10�4 m2/(ohm mole)), K0(LaCl3/3) = 145.9, K0(Mg+2/2) = 53.0,K0(La+3/3) = 69.7. K0(Cl�) = 76.31. The ion radii, aj, are derived fromthe limiting equivalent conductances using Eq. (83). With regard tothe ‘‘small ion” model, the only remaining adjustable parameter isaex. For both MgCl2 and LaCl3, this parameter is varied in an attemptto get as good agreement as possible between experimental conduc-tivities and full model conductivities from Eqs. (27)–(31). For MgCl2,Eqs. (9), (11), (10), (15), and (16) give: / = 1.732, q = 0.4199,S = 0.5096, a = 0.6913, and b = 157.38 � 10�4 m2/(ohm mole). ForLaCl3, / = 2.449, q = 0.3668, S = 0.6853, a = 1.3148, and b =296.76 � 10�4 m2/(ohm mole), respectively. By simple iteration,we have found aex = 0.52 nm for MgCl2 and 0.60 nm for LaCl3 givemodel conductivities in best agreement with experiment. Resultsof the ‘‘small ion” model fits with experiment are summarized in Ta-ble 3. As in the case of KCl, fits accurate to within several tenths of onepercent are possible for both salts. The corresponding fits for K1

(using aex optimized in matchingKexp andK) are not as good, but stillfall below a relative error of 1%.

We next consider the ‘‘large ion” model. In computing the relax-ation correction, Eq. (85) is used to account for the Brownian mo-tion of both ions. Tables 4 and 5 summarize the model results andtheir comparison with experiment for MgCl2 and LaCl3, respec-tively. The y0 values represent the reduced equilibrium electro-static potential (Eq. (49)) at aex equal to 0.52 nm (for MgCl2) and0.60 nm (for LaCl3). These come from numerical solution of thenon-linear Poisson Boltzmann equation. For y0(Mg+2, aex) the cen-tral ion has a valence charge of +2 and for y0(Cl�, aex), the centralion has a valence charge of �1, etc. For Mg+2 or La+3, |y0| ranges

from 2.14 to 3.37. For monovalent Cl�, it lies closer to 1.0. The largeabsolute electrostatic potentials near Mg+2 and La+3 illustrate theimportance of going beyond the linear Poisson Boltzmann equationwhen polyvalent ions are present.

Table 5Conductance/mobility data on LaCl3 (large ion model).a

m0 (M)

.000167 .000333 .001667 .00333Kexp 139.6 137.0 127.5 121.8

Sticky0(La+3, aex) 3.37 3.31 3.03 2.85�y0(Cl�, aex) 1.11 1.07 0.95 0.87E+(nr) 5.199 5.127 4.853 4.679�E�(nr) 5.814 5.779 5.654 5.580E+(r) 5.270 5.195 4.879 4.674�E�(r) 5.559 5.429 4.988 4.780n+ �.0135 �.0132 �.0054 +.0011n� +.0443 +.0617 +.1179 +.1435K(nr) 142.28 140.91 135.76 132.55K(r) 139.86 137.17 127.48 122.14D(r) +.186 +.124 +.016 �.278

SlipE+(nr) 5.205 5.134 4.867 4.700�E�(nr) 5.828 5.792 5.660 5.582E+(r) 5.279 5.206 4.906 4.716�E�(r) 5.557 5.409 4.932 4.702n+ �.0141 �.0140 �.0081 �.0034n� +.0464 +.0661 +.1287 +.1576K(nr) 142.54 141.16 136.01 132.83K(r) 140.00 137.14 127.11 121.67D(r) +.286 �.102 �.306 �.107

a Conductivities are in 10�4 m2/(ohm mole).

S. Allison et al. / Journal of Colloid and Interface Science 352 (2010) 1–10 9

Mobilities, lj, of both positive and negative ions are computedin the absence, nr, and presence, r, of ion relaxation. Presented inthe table are dimensionless reduced mobilities defined by

Ej ¼3ge

2e0erkBTlj ð86Þ

The relaxation correction is then obtained from

nj ¼ 1� EjðrÞEjðnrÞ ð87Þ

One notable difference between the ‘‘small ion” and ‘‘large ion”model results for binary electrolytes is that the relaxation correc-tion, nj, is the same for both positive and negative ions in the ‘‘smallion” theory, but they are different in the ‘‘large ion” theory. How-ever, these differences tend to average out when mobilities arecombined to give conductivities using Eq. (9). Using the sameaex values we found above in fitting experiment with ‘‘small ion”model conductivities, 0.52 nm for MgCl2 and 0.60 nm for LaCl3,we also obtain very good fits for the ‘‘large ion” model with ‘‘stick”boundary conditions. The same is true for the ‘‘large ion – slip”model for MgCl2. However, for the ‘‘large ion – slip” model of LaCl3,it is necessary to reduce aex to 0.55 nm in order to get good agree-ment with experiment. The ‘‘slip” model results in Table 5 were ob-tained using aex set equal to 0.55 nm. Under the conditionsdiscussed above, the ‘‘large ion” models give conductivities thatare accurate to within several tenths of a percent.

4. Discussion

In this work, we have examined two complementary contin-uum theories of electrokinetic transport of dilute electrolyte solu-tions that are called collectively the ‘‘small ion” [1–3,5–8,38–45]and ‘‘large ion” [31–37,51] models. Conductance data of dilute bin-ary electrolyte solutions is readily available, fairly extensive, andaccurate [56,57]. This is true not only for monovalent binary elec-trolytes, but electrolytes made up of polyvalent ions as well. Thiscoupled with the relative simplicity of these systems makes themideal for evaluating the accuracy and ‘‘goodness” of the theory and

modeling strategies. This approach was taken more than 50 yearsago in the pioneering studies of, Pitts [5], Fuoss, and Onsager[6,7] on the conductance of monovalent binary electrolytes. Theirwork is the basis of the ‘‘small ion” model in the present study.These early investigators were justifiably cautious about extendingtheir modeling to polyvalent electrolytes due to the largelyunknown limitations of the linear Poisson–Boltzmann equationthat they employed. The ‘‘small ion” theory has been generalizedto an electrolyte consisting of more than two ions of arbitrary va-lence [8,41], but the linear Poisson–Boltzmann equation is em-ployed. Independently and later, progress was made in modelingthe electrokinetic transport of large highly charged particles andnumerical procedures made possible by computers played a vitalrole in these developments. First, was the development of numer-ical procedures to solve the non-linear Poisson Boltzmann equa-tion around a spherical particle with a centrosymmetric chargedistribution [58]. More complicated numerical procedures to solveelectrokinetic transport were developed later [35–37]. Thesenumerical procedures are the basis of the ‘‘large ion” modeling ofthe present study. Given the focus of the present study, the ‘‘largeion” model was generalized to include an ion exclusion distanceand also to option of considering ‘‘stick” or ‘‘slip” hydrodynamicboundary conditions. A simple corrective procedure was alsodeveloped to account for the Brownian motion of all ions in thedetermination of the relaxation correction.

In an attempt to bridge the ‘‘small ion” and ‘‘large ion” modelingmethodologies, the conductance of dilute binary electrolytes madeup of both monovalent and polyvalent ions represent ideal testcases for study. The high electrostatic potential around polyvalentions tests the limits of the ‘‘small ion” model and their small sizetests the limits of the ‘‘large ion” model. Both models are appliedto the binary salt solutions KCl, MgCl2, and LaCl3 and these results,in turn, compared with experiment. In both ‘‘small ion” and ‘‘largeion” models, the only remaining adjustable parameter is the ionexclusion distance, aex. For aex equal 0.35, 0.52, and 0.60 nm forKCl, MgCl2, and LaCl3 both ‘‘small ion” and ‘‘large ion” models areable to reproduce experimental conductivities to an accuracy ofseveral tenths of one percent or better. For the ‘‘large ion” modelwith ‘‘slip” hydrodynamic boundary conditions, an aex of 0.55 nmwas necessary to get good agreement with experiment for LaCl3.Also, it is necessary to correct the ‘‘large ion” model for Brownianmotion of both ions using Eq. (85). The fact that the ‘‘small ion”model works as well as it does for MgCl2, and LaCl3 is surprising gi-ven the limitations of the electrostatic model upon which it isbased. The results of the present work indicate that the ‘‘small ion”theory can be applied to polyvalent electrolytes provided |zj| 6 3. Itis not possible to distinguish whether or not a ‘‘large ion – stick” or‘‘large ion – slip” model takes better account of conductance datasince both are capable of comparable accuracy.

Fig. 1 summarizes the reduced conductivity data for data KCl,MgCl2, and LaCl3 for both experiment and ‘‘large ion-stick” models.Squares, diamonds and triangles correspond to experimental datafor KCl, MgCl2, and LaCl3, respectively. The solid line, widely spaceddashed line, and short spaced dashed line correspond to modeldata for KCl, MgCl2, and LaCl3, respectively. The other model stud-ies considered, ‘‘small ion”, and ‘‘large ion-slip” are very similar tothis.

5. Summary and conclusions

In both the ‘‘small ion” and ‘‘large ion” models considered inthis work, solvent and mobile ions are treated as a continuum,and a single ion exclusion distance, aex, is included in modeling.Particle transport is also being considered in an infinite domain.For the ‘‘small ion” model, it is also assumed that electrostatics

Fig. 1. Experimental and model equivalent conductances for KCl, MgCl2, and LaCl3.Symbols are from experiment (56) and lines are from ‘‘large ion” model with ‘‘stick”boundary conditions. Other model studies are very similar. Squares, diamonds andtriangles correspond to experimental data for KCl, MgCl2, and LaCl3, respectively.The solid line, widely spaced dashed line, and short spaced dashed line correspondto model data for KCl, MgCl2, and LaCl3, respectively.

10 S. Allison et al. / Journal of Colloid and Interface Science 352 (2010) 1–10

are described by the linear Poisson–Boltzmann equation (whichstrictly limits it to weakly charged ions) and that jaex is small (lim-iting it to low concentration and small size). These latter twoassumptions are avoided in the ‘‘large ion” model making it moreappropriate for large, highly charged particles. However, theBrownian motion of the central ion is ignored in the ‘‘large ion”,but not the ‘‘small ion” model. In the present work, however, wehave proposed an approximate but simple way of correcting forthis assumption in the ‘‘large ion” model. Despite these differencesin the two models, both are able to reproduce experimental con-ductivities of dilute binary electrolytes made up of monovalentor polyvalent ions to an accuracy of several tenths of a percent.Minor modifications in the ‘‘large ion” model to include an ionexclusion layer along with the above mentioned correction forBrownian motion allows us to effectively bridge the gap betweenthe two models. These results serve to reinforce both ‘‘small ion”and ‘‘large ion” methodologies as far as application to the electro-phoretic mobility and conductivity of small (spherical) ions is con-cerned. Despite the large absolute electrostatic potentials presentwhen polyvalent ions are present, the use of the linear Poisson–Boltzmann equation in the ‘‘small ion” theory [2,3,5–8,38–45] doesnot lead to significant errors in conductivity for ions of absolute va-lence less than or equal to 3. The ‘‘large ion” approach [31,33–37,51] also works well provided account is taken of the Brownianmotion of all ions present. The ‘‘large ion” model can be appliedto larger, more highly charged, and also ‘‘structured” particles[21,26,49,59]. As far as small ion studies are concerned, whichapproach an investigator chooses to use is largely a matter of per-sonal convenience. A more exhaustive comparison of experimentaland model conductivities of binary electrolytes shall be presentedin future work. The principle objective of the present work hasbeen to present a complete outline of the two approaches anddemonstrate their application to three different binary electrolytesof different (cationic) valence.

This work will hopefully stimulate research in several areas.First, both ‘‘small ion” and ‘‘large ion” models have a broader rangeof applicability than has previously been recognized. Both can beused to study conductivities of not only binary electrolytes, but ter-nary and more complex solutions. Second, with the growing andwidespread use of capillary electrophoresis, both ‘‘small ion” and‘‘large ion” models can be applied to studies of electrophoretic

mobilities. Third, more realistic accounting of the interionic poten-tial of mean force may be considered. Progress in this direction hasalready been made with regard to the ‘‘small ion” approach [9,10].

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