Conductivity and electrophoretic mobility of dilute ionic solutions

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

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Conductivity and electrophoretic mobility of dilute ionic solutions Stuart Allison ⇑, Hengfu Wu, Umar Twahir, Hongxia Pei Department of Chemistry, Georgia State University, Atlanta, GA 30302-4098, USA

a r t i c l e

i n f o

Article history: Received 24 June 2010 Accepted 2 August 2010 Available online 10 August 2010 Keywords: Electrokinetic transport Electrophoretic mobility Electrical conductance

a b s t r a c t Two complementary continuum theories of electrokinetic transport are examined with particular emphasis 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. Faraday Trans. 2 (74) (1978) 1607] are both discussed and the ‘‘large ion” model is generalized to include an ion exclusion distance and to account in a simple but approximate way for the Brownian motion of all ions present. In addition, the ‘‘large ion” model is modified to treat ‘‘slip” hydrodynamic boundary conditions in addition to the standard ‘‘stick” boundary condition. Both models are applied to the equivalent conductance of dilute KCl, MgCl2, and LaCl3 solutions and both are able to reproduce experimental conductances 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 conductances of the three electrolyte solutions. Ó 2010 Published by Elsevier Inc.

1. Introduction One of the early successes of atomic scale continuum transport modeling concerned the electrical conductance of dilute solutions of strong electrolytes [1–3]. This work, in turn, was grounded on equilibrium theory of strong electrolytes by Debye and Huckel [4]. The early theory, which was restricted to very dilute solutions of ions modeled as point charges, was subsequently extended to account for the finite size of the ions and also higher electrolyte concentrations [5–7]. For monovalent binary aqueous electrolyte solutions up to a concentration of about 0.10 mol/dm3 or M, experimental and model equivalent conductances are in excellent agreement [5–7] for reasonable choices of model parameters. Refs. [5–7] are restricted to binary electrolytes. This was subsequently extended to general electrolyte solutions made up of an arbitrary number of ions of arbitrary valence [8]. In the present work, this approach shall collectively be called the ‘‘small ion” model. Despite the successes of the ‘‘small ion” model, there have been few attempts to apply it directly to electrolyte solutions containing polyvalent ions or mixtures of electrolytes containing more complex ionic species. One of its shortcomings is that 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 linear Poisson–Boltzmann equation in representing the ionic potential of mean force [9,10].

⇑ Corresponding author. Fax: +1 404 651 1416. E-mail address: [email protected] (S. Allison). 0021-9797/$ - see front matter Ó 2010 Published by Elsevier Inc. doi:10.1016/j.jcis.2010.08.009

A transport phenomenon closely related to electrical conductivity is the free solution electrophoretic mobility. In recent years, capillary zone electrophoresis [11–15] has become a widely used separation technique for a broad array of ionic species including peptides [16–21], organic anions [22,23], proteins [24–26], and nanoparticles [27,28]. Although the conductance theories discussed in the previous paragraph [3,5–7] have been applied to mobility studies of small and weakly charged ions [22,23,29,30], they are not appropriate for large or highly charged particles including nanoparticles [27,28] or metal oxide colloidal particles [31]. For large and/or highly charged particles, there is a long established alternative that is grounded on very similar continuum electro-hydrodynamic principles, but has its origin in the electrophoresis of large colloidal particles [32–37]. In this work, it shall be called the ‘‘large ion” model. Of particular relevance to the ‘‘large ion” model is the numerical procedure of O’Brien and White [37] that has come into widespread use and can be applied to the electrophoresis of a spherical particle of arbitrary size containing a centrosymmetric charge distribution of arbitrary net charge. One factor that may limit the application of the ‘‘large ion” model to treat the mobility or conductivity of small ions is that it ignores the Brownian motion of the ion of interest. This may only be a reasonable approximation if the ion of interest is much larger than the ions comprising the surrounding electrolyte. The ‘‘small ion” model, on the other hand does account for the Brownian motion of all ions present [3,5–10]. The principle objective of the present work is to bridge the gap between the ‘‘small ion” and ‘‘large ion” models discussed above by applying both to the conductivity of a number of binary electrolytes for which experimental conductance data is available.

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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 models under conditions of larger ‘‘zeta” potential. In the course of this work, it has been necessary to modify the O’Brien and White (‘‘large particle” model) procedure in three ways. The first is to include an ‘‘ion free” layer of solvent just outside the surface of hydrodynamic shear. In addition to the distance of hydrodynamic shear from the center of ion j, aj; an ion exclusion distance, aex, is also defined. The second modification concerns hydrodynamic boundary conditions. In both ‘‘small ion” and ‘‘large ion” models, hydrodynamic boundary conditions have been handled somewhat differently and it is important to consider how this influences the results. Currently, ‘‘stick” boundary conditions are employed in the ‘‘large ion” model, and this means the particle velocity and fluid velocity match at the surface of hydrodynamic shear. In addition to the conventional ‘‘stick” hydrodynamic boundary conditions, we also consider ‘‘slip”. As shall be shown, both ‘‘stick” and ‘‘slip” models are capable of explaining the experimental conductance data equally well for about the same aex values, but different aj values must be chosen on the basis of limiting ionic conductivities. Third, although the ‘‘large ion” model does not account explicitly for the Brownian motion of the central ion of interest, we present a simple way of doing so that involves adding a correction term to the mobile ion mobilities. When this is done, the ‘‘small ion” and ‘‘large ion” model conductivities with the same or similar model parameters are comparable with each other and yield excellent agreement with experiment.

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

lj ¼ lj0  B1 zj lj0

1 X

! ðnÞ

cn r j þ B2 zj

pffiffi I

ð2Þ

n¼0 ðnÞ

where I is the ionic strength of the electrolyte, cn and r j cussed below,

are dis-

pffiffiffi 3 2:806  106 2F B1 ¼ ðe0 er RTÞ3=2 ¼ ðM1=2 Þ 12pN Av ðer TÞ3=2

ð3Þ

pffiffiffi 2   4:275  106 m2 2F ðe0 er RTÞ1=2 ¼ B2 ¼ 1=2 1=2 6pgNAv V sec M gðer TÞ

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 the remaining terms, F is the Faraday constant (=9.645  104 C/mole), NAv is Avogadros Number, e0 is the permittivity of free space, R is ðnÞ the gas constant, and other quantities (except for cn and rj ) have been described in the previous paragraph. Also, the physical basis of the B1 term in Eq. (2) is ion relaxation and the physical basis of the B2 term is the electrophoretic effect. Consider a single strong electrolyte, AaBb, or binary electrolyte, that undergoes complete dissociation according to z

Aa Bb ! aAzþ þ bB 

ð5Þ

2. Modeling

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

2.1. Conductance and mobility of small Ion electrolytes

am0 zþ þ bm0 z ¼ 0

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

and,

pffiffiffiffiffiffiffi Kp ¼ K0  ðaK0 þ bÞ m0

ð1Þ

In Eq. (1), the ‘‘p” subscript denotes the original Onsager–Fuoss model mobility, K0 is the equivalent conductance of the solution in limit of zero ionic strength, a is the ‘‘relaxation coefficient”, b is the ‘‘electrophoresis coefficient”, and m0 would be the concentration of electrolyte in moles/dm3 or M if it did not dissociate into ions. The physical basis of a is ion relaxation, the distortion of the ion atmosphere around a particular ion from equilibrium due to the imposition of an electric and/or flow field. The physical basis of b is the additional hydrodynamic backflow produced in the vicinity of a particular ion produced by the presence of nearby ions. The coefficients a and b depend on: temperature, T, the properties of the solvent 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, however, does enter through K0 or equivalently, the electrophoretic mobility of individual ions in the limit of zero ionic strength, lj0. Underlying Eq. (1) is a model in which the ions are treated as point charges. The Onsager–Fuoss, OF, theory [2,3] starts with a general equation of continuity which specifies the concentration of ions of one species in the vicinity of ions of other species in an electrolyte solution which has reached a steady state under the influence of a weak, constant external electric field, e0 . Account is taken of the Brownian motion of the various ionic species of which an arbitrary number may be present. Electrostatics are treated at the level of the linear Poisson–Boltzmann equation. This is true not only for the ‘‘point ion” model discussed here, but the more general finite ion case considered at the end of this section. For the ‘‘point ion” model, the boundary conditions on the fluid velocity are that it remain finite at the center of an ion and vanish, on

I¼

  az m0  2 2 þ azþ þ bz ¼ ðzþ  z Þ m0 ¼ /2 m0 2 2

ð6Þ

ð7Þ

The conductivity, K, of a solution of this strong electrolyte can be written

K ¼ am0 kþ þ bm0 k ¼ am0 jzþ jKþ þ bm0 jz jK ¼ am0 Fzþ lþ þ bm0 Fz l ¼ am0 Fzþ ½lþ  l  ¼ am0 zþ K

ð8Þ

The kj terms appearing in the first equality on the right hand side of Eq. (8) are molar conductivities of specific ions. The Kj values 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 modern literature and in handbooks are Kj’s and in SI units are in S m2/mole or m2/(ohm mole). The third equality follows from the relationship between equivalent ionic conductance and ion mobility, which will be positive for + ions and negative for  ions, Kj = F |lj|. The fourth equality follows from the electroneutrality condition, Eq. (6). The fifth equality gives the equivalent conductance of the binary electrolyte, K. Dividing various equalities on the right hand side of Eq. (8) by am0z+ gives

K ¼ Kþ þ K ¼ Fðlþ  l Þ ¼ Fðjlþ j 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  z j

1 X n¼0

ðnÞ

cn r j

¼

jzþ z jq pffiffiffi 1þ q

  jlþ0 j þ jl0 j jzþ z j jzþ j þ jz j jz lþ0 j þ jzþ l0 j    jzþ z j Kþ0 þ K0 ¼ jzþ j þ jz j jz jKþ0 þ jzþ jK0

q¼

ð10Þ



ð11Þ

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

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

pffiffiffiffiffiffiffi K ¼ K0  ½B1 ðKþ0 Sþ þ K0 S Þ þ B3 ðjzþ j þ jz jÞ/ m0

ð12Þ

K0 ¼ Kþ0 þ K0

ð13Þ

B3 ¼ FB2 ¼



0:4125



m2

gðer TÞ1=2 ohm mole M1=2

ð14Þ

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

a¼

B1 ðKþ0 Sþ þ K0 S Þ/ K0

ð15Þ

b ¼ B3 ðjzþ j þ jz jÞ/

ð16Þ

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

a ¼ B1 S/

ð17Þ

In order to make contact with different relaxation theories, however, we have chosen to distinguish the relaxation terms for the different ions, Sj, defined by Eq. (10). For an electrolyte consisting of more than two ionic species, the relaxation effect is more complex than discussed in the previous paragraph, which is strictly valid only for a binary electrolyte. The OF theory can also be applied to ionic solutions containing an arbitrary number of distinct ions. Assume we have N ions present and let mj denote the concentration (in M) of species j. For this more general case, the Sj terms are given by the first equality on the right hand side of Eq. (10). We have (3)

pffiffiffi c0 ¼ ð2  2Þ=2 ¼ 0:2928932

ð18Þ

c1 ¼ 0:3535534

ð19Þ

cn ¼ cn1 r

ðnÞ



3 1 2n

¼ ð2H  IÞ  r

ð0Þ

ðr Þj ¼

ð0Þ rj



ðn > 1Þ

ð20Þ

ðn1Þ

ð21Þ PN

¼ zj 

k¼1 uk zk PN k¼1 uk =wk

!

1 wj

 ð22Þ

mj z2j uj ¼ PN 2 k¼1 mk zk

ð23Þ

wj ¼ Kj0 =jzj j

ð24Þ

Hij ¼ dij

N X uk wk w i þ wk k¼1

! þ

uj wj wi þ wj

ð25Þ

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

3

or an Excel spreadsheet, which will be shared with interested investigators upon request to the corresponding author. In terms of the dimensionless relaxation terms, Sj, generated by the above procedure, Eq. (2) can be written

pffiffi

lj ¼ lj0  ðB1 lj0 Sj þ B2 zj Þ I

ð26Þ

In the 1950s, Fuoss and Onsager generalized this theory to extend its range of validity to terms of order m10 in electrolyte concentration and also account, to lowest order, for the finite sizes of the ions [6,7]. This work was restricted to binary electrolytes and specific applications in this and subsequent work [38–40] were further restricted to monovalent (binary) electrolytes. Quint and Vaillard [8] did generalize this to an arbitrary electrolyte and in3=2 cluded terms to order m0 although some contributions at the level of m3=2 are missing [41]. In these studies [6–8,38–41] a single 0 ion exclusion distance, aex, is defined and the assumption is made that the center-to-center distance, r, between any two ions cannot be smaller than aex. The assumption is also made that the normal component of the relative fluid velocity vanishes at r = aex (see Eq. (3.4) of (7)). Closely related work was also carried out by Pitts [5] on symmetric binary electrolytes, but the assumption was made that the relative fluid velocity as a whole, and not just the normal component, vanishes at r = aex. Within the framework of the more general Fuoss–Onsager theory [6,7], the equivalent conductance of a binary electrolyte can be written

K ¼ Knr ð1  nÞ

ð27Þ

pffiffiffiffiffiffiffi Knr ¼ K0  b m0 =ð1 þ jaex Þ

ð28Þ

pffiffiffiffiffiffiffi

j ¼ B/ m0

ð29Þ

pffiffiffi 2F 5:028  1011 1 pffiffiffiffiffiffiffi B ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ e0 er RT er T M1=2 m

ð30Þ

pffiffiffiffiffiffiffi bD0 pffiffiffiffiffiffiffi n ¼ a m0 ð1  D1 þ D2 Þ þ 3 m0 K0

ð31Þ

The term Knr is the equivalent conductance in the absence of ion relaxation, n denotes the relaxation correction, and j is the Debye–Huckel screening parameter. In Eq. (31), the terms D1, D2, and D03 represent higher order correction terms and depend on concentration to leading order m1=2 0 . Explicit expressions are given in Section 7 of Ref. [7] specific to symmetric binary electrolytes. More general expressions (making minor corrections for sign errors) can be deduced from equations in Section 6 of Ref. [7] for general binary electrolytes. In subsequent work by Fuoss and coworkers, additional corrections were made to D2, and D03 [38–40,42]. However, these changes were minor and do not alter the relaxation corrections significantly. In addition, expressions not limited to binary electrolytes can be found in references [8,41] and include terms to order m3=2 0 . Since the 1957 paper by Fuoss and Onsager [7] carries out the most thorough comparison between theoretical and experimental conductancies and subsequent applications are mostly restricted to monovalent binary electrolytes, we shall use the equations from the 1957 paper [7] in the present work when considering the ‘‘small ion” model. The ‘‘small ion” theory pioneered by Fuoss and Onsager [2,3] remains in widespread use to this day not only for fully dissociated electrolytes, but undissociated electrolytes as well [43]. With few exceptions [41], the overwhelming majority of applications involve symmetric binary electrolytes [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

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

equation, (3) a single ion exclusion distance, aex, is included in modeling, (4) jaex is small (small ion sizes and low electrolyte concentration). In addition to Eq. (27), we shall also consider a simpler model that includes finite ion size effects in Knr as given by Eq. (28), but restricts the relaxation term to m1=2 and this allows us to 0 ignore D1, D2, and D03 altogether. Define

pffiffiffiffiffiffiffi K1 ¼ Knr ð1  a m0 Þ

ð33Þ pffiffiffiffiffiffiffi

lj;nr ¼ lj0  B2 zj / m0 =ð1 þ jaex Þ

ð34Þ

where nj is the same for both ions and the j subscript is omitted in Eq. (27) and (31). As Eq. (31) shows, higher order concentration effects (via the D1, D2, and D03 terms) can be accounted for in binary electrolytes. However, it would be useful if we could also consider ternary and higher order electrolyte solutions. This would be relevant if we were interested in the mobility of a ‘‘guest” ion in the presence of a binary electrolyte, for example. Eq. (26) makes it possible to consider such cases, where the relaxation term depends on the particular ion and we can write

pffiffi nj ffi B1 Sj I I¼

1X mk z2k 2 k

v 0 ðrÞ ¼ Tða; rÞ  u

ð35Þ 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. For ternary and higher order electrolyte solutions, we can approximate nj appearing in Eq. (33) with Eq. (35). This is equivalent to ignoring terms higher than order I1/2 in the relaxation correction. The same approximation is made in arriving at Eq. (32) for the conductivity of a binary electrolyte and Eq. (32) therefore gives us a way of determining how accurate this approximation is in specific cases. 2.2. Mobility of large spherical ions

gr v  rp ¼ r  rH ¼ s rv ¼0

can be

ð39Þ

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

Tða; rÞ ¼

3a a3 ðI þ nnÞ þ 3 ðI  3nnÞ 4r 4r

nn ¼ rr=r 2

ð40Þ ð41Þ

and for ‘‘slip” boundary conditions,

Tða; rÞ ¼

a ðI þ nnÞ 2r

ð42Þ

In Eqs. (40) and (42), I denotes the 3 by 3 identity matrix. The zero superscript on v 0 is a reminder that this refers to the special case of an uncharged spherical particle. Extending from r = a to r = aex, where aex is the ion exclusion distance, it is assumed that no ions are present. In this region of the fluid adjacent to the particle, s ¼ 0. The fluid, however, obeys Eqs. (37) and (38). For r > aex, the ion atmosphere is treated as a continuum. Let nj ðrÞ denote the local concentration of mobile ion species j in M and let zj denote the valence charge of a single ion. The charge distribution, qðrÞ, obeys the Poisson equation in general,

r  ðeðrÞrWðrÞÞ ¼ qðrÞ=e0

qðrÞ ¼ qf ðrÞ þ F

X

zj nj ðrÞ

ð37Þ ð38Þ

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

ð43Þ ð44Þ

j

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

ei



   @W @W ¼ er @r r¼a @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 external electric, e0 , or flow field, the steady state electrodynamic potential is written

WðrÞ ¼ W0 ðrÞ þ W1 ðrÞ  e0  r It shall be assumed that our model particle is spherical and contains a centrosymmetric charge distribution within a surface of hydrodynamic shear located at a distance r = a from the center of the particle. At the shear surface, ‘‘stick” or ‘‘slip” boundary conditions may prevail. In the case of ‘‘stick”, it is assumed that fluid and particle velocities match at r = a. In the case of ‘‘slip” boundary conditions, it is assumed that only the outward normal component, n, of particle and fluid velocities match at r = a. Also, if rH denotes the hydrodynamic stress tensor of the fluid, then rH  n is parallel to n at the shear surface [46,47]. Outside of the shear surface, the fluid is treated as a hydrodynamic continuum that obeys the linearized Navier–Stokes and solvent incompressibility equations. 2

v 0 ðrÞ,

ð32Þ

As demonstrated in the main body of this work, Eq. (32) works almost as well as Eq. (27) for binary electrolytes for m0 6 0.005 M. Finally, consider the ion electrophoretic mobilities within the framework of the Onsager–Fuoss theory restricted to a binary electrolyte. We can write

lj ¼ lj;nr ð1  nj Þ

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

ð46Þ

where W0 is the local equilibrium electrostatic potential and W1 is a perturbation potential that vanishes far from the particle. The local ion densities are also perturbed from their equilibrium values, nj0 ðrÞ, which are related to a new potential, Uj ðrÞ, defined by

nj ðrÞ ¼ nj0 ðrÞeezj ðUj ðrÞþW1 ðrÞÞ=kB T ffi nj0 ðrÞ½1  ezj ðUj ðrÞ þ W1 ðrÞÞ=kB T ð47Þ nj0 ðrÞ ¼ mj ezj y0 ðrÞ y0 ðrÞ ¼

eW0 ðrÞ kB T

ð48Þ ð49Þ

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

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

r  Jj ¼ 0

ð50Þ

J j ¼ nj0 v þ Fzj nj0 Dj ðrUj þ e0 Þ=RT

ð51Þ

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

  @ Uj ðrÞ ¼ e0  n @r r¼aex

ð52Þ

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

  1 d Fe X 2 dy0 ðrÞ m z xðrÞezj y0 ðrÞ r ¼ 2 r dr dr e0 er kB T j j j

ð53Þ

  @y0 a1 ¼ @r r¼aþ a

ð54Þ

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

yLPB 0 ðrÞ

¼

8   < a1 a  ja r 1þjaex :

a1 a ejðraex Þ rð1þjaex Þ

9 a < r < aex = aex < r

;

ð55Þ

The ‘‘LPB” superscript on y0 denotes the linear Poisson Boltzmann reduced potential. In the present work, however, the general non-linear for of Eq. (53) is solved numerically subject to the boundary condition imposed by Eq. (54). For the nonequilibrium problem, we follow the strategy of carrying out two separate transport cases (37). In Case 1, the particle is translated with constant velocity, u0 , in a fluid that is otherwise at rest. In Case 2, the particle is held stationary, but it is subjected to a constant external electric field, e0 . Although the potentials, UðiÞ j ðrÞ, are not spherically symmetric (the (i) superscript has been added to distinguish the two transport cases), they can be written in terms of related functions that are [37],

1 r

ðiÞ UjðiÞ ðrÞ ¼ /ðiÞ j ðrÞb  r ð1Þ

ð56Þ

ð2Þ

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

/j ðrÞ ¼

Nþ2 X

ðiÞ

ðikÞ

dk /j

ðrÞ

ð57Þ

ðiÞ

the kth ‘‘distant” solution for case i and ion j, and the dk are the linear coefficients that are determined from boundary conditions on or near the particle as discussed later. For k = 0 or N + 1 or N + 2, the ðikÞ outer boundary condition on Uj (r) is set to 0. For k = 1 to N, ðikÞ

/j

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 ðiÞ ðrÞ, is derived (37). It is defined by

v ðiÞ ðrÞ ¼ curl½curl½gðiÞ ðrÞbðiÞ  þ uðiÞ 1

ð59Þ

ðiÞ

ð2Þ where b is defined following Eq. (55) and uð1Þ 1 ¼ u0 , and u1 ¼ 0. (ij) The ‘‘distant” solutions of g are set to zero except for j = N + 1 and N + 2. For jr 1,

g ðiNþ1Þ ðrÞ ¼ r

ð60Þ

1 r

ð61Þ

g ðiNþ2Þ ðrÞ ¼

The final expression for g(i)(r) can be written that is identical to Eq. ðikÞ (57) above with g(ik) replacing /j . ðiÞ The dk coefficients appearing in Eq. (57) and an analogous relation involving g(i)(r) and g(ik)(r) are determined from ‘‘inner” boundary conditions. These are discussed in reference [37] for the special case of ‘‘stick” boundary conditions when a = aex, but the more general conditions of interest here must be handled difðiÞ ferently. The boundary conditions on /j (r) follow from Eq. (52) ð1Þ0 ð2Þ0 and are simply /j (aex) = 0 and /j (aex) = 1, where the prime superscript denotes first derivative with respect to r. These are the same for both ‘‘stick” and ‘‘slip” hydrodynamic boundary conditions. The boundary conditions on g(i) are evaluated at r = a and 0 are different for ‘‘stick” and ‘‘slip”. For ‘‘stick”; g(1) (a) = a/2, (1)0 0 (2)0 (2)0 0 g (a) = 1/2, g (a) = 0, and g (a) = 0 where the double prime denotes second derivative with respect to r. For ‘‘slip”; careful anal0 000 0 ysis leads to the conditions: g(1) (a) = a/2, g(1) (a) = 0, g(2) (a) = 0, (2)0 0 0 and g (a) = 0 where the triple prime denotes third derivative with respect to r. With minor modifications in Eq. (7.7) of reference ðiÞ [37] that incorporate these modified boundary conditions, the dk coefficients can be uniquely determined for Cases 1 and 2. The overall solution is then taken to be the linear superposition of both Cases 1 and 2 fields that gives a net force exerted by the particle on the fluid of zero [37]. At this point, it is appropriate to discuss how the results of Case 1 and Case 2 transport studies can be used to obtain a general expression for the electrophoretic mobility, l, and then reduce that to more recognizable forms in special cases. It is possible to obtain a general expression of the electrophoretic mobility starting from the differential form of the Lorentz reciprocal theorem [48,50], ðiÞ

s0  v ðiÞ þ r  ðv ðiÞ  r0H Þ ¼ sðiÞ  v 0 þ r  ðv 0  rH Þ

ð62Þ

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

k¼0

where the sum over k extends over the inhomogeneous (k = 0) and ðikÞ different homogeneous (j = 1 to N + 2) ‘‘distant” solutions, Uj (x) is

ðxÞ ¼



Z Sp

v ðiÞ  r0H  n dS ¼ 

Z Sp

v 0  rHðiÞ  n dS þ u 

Z X

T  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 ðiÞ FH

¼

Z

ðiÞ H

r  n dS

Sp

ð64Þ

and an entirely analogous expression can be written for F 0H . For ‘‘stick” boundary conditions, the fluid velocities inside the surface integrals are constant and can be moved outside the integral directly. For the ‘‘slip” case, first recognize that we can write rH  n ¼ ðnnÞ  ðrH  nÞ. Since the normal component of the fluid velocity matches that of the particle on Sp in the slip case and since only the normal component contributes to the surface integrals, we can move them out of the integrals also. The total force exerted by the particle on the fluid is the sum of hydrodynamic and external (electrical) forces [37,49,51] ðiÞ

ðiÞ

FT ¼ FH þ

Z

sðiÞ dV

ð65Þ

X

Eqs. (73) and (74) are general expressions for the mobility of a sphere and simplify in limiting special cases. Consider the special case where ion relaxation is neglected. This is a reasonable approximation when the particle is weakly charged and the solution of the linear Poisson Boltzmann equation, Eq. (55), ð1Þ ð1Þ ð2Þ is appropriate. Under these conditions, /j ¼ 0; dj ¼ 0; /j ¼ ð2Þ 3 2 3 ca =r ; dj ¼ ca , where c = ðer  ei Þ=ð2er þ ei Þ. All of the terms ð2Þ with superscript (1) vanish in Eq. (73) and the dj term also drops out when we sum over j and impose the condition of charge neutrality. Under these conditions, Eq. (73) reduces to

lnr ¼

eZ 4p þ f0 f0

u 

ðiÞ FT

ðiÞ

¼u 

F 0H



þu 

Z

ðiÞ

ðI  TÞ  s dV

ð66Þ

sðiÞ ¼ F

X

ðiÞ

zj nj0 ðrUj þ e0 di2 Þ þ RT

X

j

rnðiÞ j

ð67Þ

j

Using Eqs. (40), (42), (56)–(58), and (67) in (66), it is straightforward to carry out the angular integrations. Also, the divergence theorem 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 concern ourselves with the x components of the overall forces. Eq. (66) can be written, Z ðiÞ

F T ¼ uðiÞ f0 

X X 4 pFbðiÞ zj mj dðiÞ zj j þ 4pF 3 j j

1

a

ðiÞ

r2 dr nj0 ðrÞg j ðrÞ

ð68Þ   ðiÞ /j ðiÞ 1 4 2 ðiÞ ðiÞ0 ðiÞ b þ ð1  2h1 þ 2h2 Þ/j b g j ¼ 1  h1 eðiÞ þ ð1  h1  h2 Þ 3 3 3 r ð69Þ f0 ¼ 6pgaðstickÞ;

4pgaðslipÞ

h1 ¼

3a ðstickÞ; 4r

h2 ¼

1 a3 ðstickÞ; 4 r

ð70Þ

a ðslipÞ 2r

ð71Þ

0ðslipÞ

ð72Þ

and u(1) = u0, u(2) = 0, e(1) = 0, e(2) = e0. The total force exerted by the particle on the fluid in Case 1 and Case 2 transport is not zero. We can view the steady state electrophoretic migration of our particle, where the total force exerted on the fluid is indeed zero, as a superposition of the two. If we set u0 = l e0, where l is the electrophoð1Þ ð2Þ retic mobility of our particle add F T and F T and set the sum to zero, we obtain

4pF

P j

l¼

f0  4pF

ð2Þ

j

ðiÞ

pj ¼

Z a

1

ð2Þ

zj ðmj dj =3  pj Þ

P

lstick ¼ nr

X

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

! ð1Þ zj ðmj dj =3

"

r 2 dr nj0 ðrÞ



ð73Þ

ð1Þ pj Þ

 ðiÞ /j 4 2 1  h1 di2 þ ð1  h1  h2 Þ 3 3 r #

1 ðiÞ0 þ ð1  2h1 þ 2h2 Þ/j 3

ð74Þ

1

a

r2 dr q0 ðrÞ

  4 2ca3 h1  3 ðh1  3h2 Þ 3 3r

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

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

Z

lslip nr ¼

eZ 6pga eZ

4pga

þ

þ

2 3g

2 3g

Z

1 a

Z a

1

   c a3 a5 r dr q0 ðrÞ 1   2 r r

  c a3 r dr q0 ðrÞ 1  2 r

ð76Þ

ð77Þ

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

q0 ðrÞ ¼ F

X

zj m0j ezj y0 ðrÞ ffi 2FI0 yLPB 0 ðrÞ

ð78Þ

j

P  3 0 2 0 where I 0 = j mj zj =2 is the ionic strength in moles/m , mj is the ambient concentration of species j in moles/m3, the exponential has been expanded and only the linear term in y0 has been retained. Furthermore, y0 has been approximated by its linear form given by Eq. (55). Making use of Eqs. (55) and (78) in (76) and (77),

lstick nr ¼

eZ 6pga



4aFI0 a1 3gð1 þ jaex Þ

Z

1

dre

jðraex Þ



1

aex

  c a3 a5  2 r r ð79Þ

lslip nr ¼

eZ 4pga



4aFI0 a1 3gð1 þ jaex Þ

Z

1

aex

dre

jðraex Þ

 1

 c a3 2 r

ð80Þ

The first term in brackets within the integrand on the right hand sides of Eqs. (79) and (80) can be integrated directly and the cterms can be reduced to exponential integrals. This shall not be done here. Eqs. (79) and (80) provide a straightforward way of estimating the effect of hydrodynamic boundary conditions and ion exclusion on electrophoretic mobility of weakly charged spherical particles. 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 the previous two sections offer advantages and disadvantages and the principle objective of this work is to bridge these two approaches and show how they complement each other. The greatest advantage of the ‘‘small ion” approach is that it accounts for the Brownian motion of the principle ion of interest whereas the ‘‘large ion” approach does not. The greatest advantage of the ‘‘large ion” approach is that it accounts more accurately for the finite size of the ion and is not restricted to weakly charged ions. (The ‘‘small ion” theories always employ the linearized form of the Poisson– Boltzmann equation.) From Eq. (9), we have the relationship between equivalent ionic conductance, Kj, and absolute mobility, |lj|. In the limit of zero electrolyte, relaxation and electrophoretic

7

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

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.

Fejzj j

Kj0 ¼ Fjlj0 j ¼

a2 pgaj

ð81Þ

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 are used as input data and aj will depend on the hydrodynamic boundary condition assumed. In water at 25 °C with Kj0 given in 104 m2/ (ohm mole) and aj in nm,

aj ðnmÞ ¼

55:28jzj j 4

a2 Kj0 ð10 m2 =ðohm moleÞÞ

ð82Þ

When the ‘‘slip” boundary condition is used, the aj values must be scaled 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 through the Kj0 terms and explicitly through the ion exclusion distance, aex. The effect of aex on conductance becomes significant at higher electrolyte concentration. Also, hydrodynamic boundary conditions are usually dealt with by assuming ‘‘slip” boundary conditions hold on the ion exclusion surface at r = aex, and not at r = aj. (In the theory of Pitts [5], ‘‘stick” boundary conditions are assumed to hold at r = aex.) In the ‘‘large ion” theories, the ion hydration radius does enter directly. With regard to the ion on interest, it enters through the hydrodynamic boundary condition on g(r) at r = a as discussed following Eq. (59). Ionic hydration radii also enter indirectly for all ions making up the background electrolyte in the solution of the ion transport equation, Eq. (51). The ion diffusion constant, Dj, appearing in Eq. (51) are related to aj and Kj0 by the Einstein relation [52]

Dj ¼

kB T

a2 pgaj

¼

kB T Kj0 Fejzj j

ð83Þ

However, availability of limiting ionic conductance data makes explicit reference to aj unnecessary with regard to the relaxation correction. A shortcoming of the ‘‘large ion” theories is that it ignores the Brownian motion of the ion of interest, call it ion k. Provided this ion is much larger than the other ions making up the background electrolyte, that assumption is a reasonable approximation since Dk / 1/ak. However, if ion k is comparable in size the other ions, the approximation is expected to break down and a comparison of experimental conductances with both ‘‘small ion” and ‘‘large ion” theories shall give us an opportunity to evaluate this assumption. In the theory of diffusion controlled reactions, it is well known that the mutual diffusion constant of two species is simply the sum of the individual diffusion constants of the two species [53–55]. A simple way of correcting the ‘‘large ion” theory to account for the Brownian motion of the central ion is to replace Dj or aj appearing in the ion transport equation with (Dj + Dk) or

aeff j ¼



1 1 þ aj ak

1 ð84Þ

Since the relaxation process is dominated by the counterion, a single aeff j is used for binary electrolytes, where aj and ak in Eq. (85) are coion and counterion radii. Both uncorrected and corrected applications of ‘‘large ion” theory to conductance data shall be reported below. In the analysis of experimental conductance data below, which is all in aqueous media at 25 °C, we shall use data summarized in the Handbook of Chemistry and Physics [56] covering the concentration range 0.0 6 am0z+ 6 0.02 M. We shall first examine the simple monovalent salt, KCl, since it has been extensively studied in the past using the ‘‘small ion” theory [6,7]. Using the ‘‘small ion”

theory and treating aex as an adjustable parameter, we shall attempt to obtain good agreement between theory and experiment as well as confirm the conclusions of past work [6,7]. For a monovalent salt like KCl, we expect the ‘‘small ion” theory to work as well as it possibly can since the ions are both small and weakly charged. Then using this aex, the ‘‘large ion” theory will be applied to KCl for both ‘‘stick” and ‘‘slip”, and ‘‘uncorrected” and ‘‘corrected” (according to Eq. (85)) cases. The full numerical approach outlined in the previous section shall be applied in these cases. It should be emphasized that once aex is fixed, once ‘‘stick” or ‘‘slip” hydrodynamic boundary conditions are assumed, and once uncorrected or corrected mobile ion radii are selected, there are no further adjustable parameters in the ‘‘large ion” theory. 3. Results 3.1. Application to KCl For KCl, we use [56] K0 = 149.79  104 m2/(ohm mole), T = 25 °C, g = 0.89 cp, and er = 78.53 in the small ion theory for a binary symmetric electrolyte. The relative error of the equivalent conductance of this salt falls in the range of several hundredths of one percent [57]. For this case, a = 0.22940 and b = 60.575  104 m2/ (ohm mole). It is straightforward to show that aex = 0.350 nm yields best agreement between theory and experiment which confirms the finding of Fuoss and Onsager [7]. Experimental and ‘‘small ion” model conductances are summarized in Table 1 where Kp, K1, and K are given by Eqs. (1), (32), and (27), respectively. The ‘‘point ion” model for conductance, Kp, clearly does not work well at higher concentrations, but the full model, K, and the approximate finite ion model, K1, work quite well, Shown in the last two columns of Table 1 is the relative percent error defined by

  K  Kexp D ¼ 100 Kexp

ð85Þ

For D1, K1 replaces K in Eq. (86). Both K and K1 reproduce experimental conductances to well within an accuracy of several tenths of 1% although the full model is slightly better at the highest ionic strength considered. This serves to demonstrate that the approximate finite ion model works quite well in reproducing the experimental equivalent conductance of KCl. This is useful since the approximate finite ion model is much simpler than the full finite ion model and can also be applied directly to ternary and higher order electrolytes as discussed in Section 2. The corresponding results for the ‘‘large ion” model studies are summarized in Table 2. Full numerical calculations are carried out for 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 of interest (K+ or Cl) is translated with constant velocity (Case 1) or held stationary (Case 2). Diffusion constants, Dj, used in the ion transport equation are obtained from Eq. (84). In the ‘‘corrected” cases, the ion of interest is also held constant, but effective ion radii defined by Eq. (85) are used in Eq. (84). The third and fourth columns

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

a

m0 (M)

Kexp

Kp

K1

K

D1

D

0.0005 0.001 0.005 0.010 0.020

147.74 146.88 143.48 141.20 138.27

147.67 146.79 143.08 140.30 136.36

147.71 146.87 143.46 141.05 137.80

147.68 146.82 143.41 141.07 138.07

0.02 0.01 0.01 0.11 0.34

0.04 0.04 0.05 0.09 0.14

All K values are in 104 m2/(ohm mole).

8

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

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

a b c d

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 0.001 0.005 0.010 0.020

147.74 146.88 143.48 141.20 138.27

147.26 146.22 142.09 139.36 135.96

0.32 0.45 0.97 1.30 1.67

147.82 146.99 143.61 141.28 138.33

+0.054 +0.075 +0.091 +0.057 +0.043

147.88 147.02 143.61 141.31 138.41

+0.095 +0.095 +0.091 +0.078 +0.101

All K values are in 104 m2/(ohm mole). Stick boundary conditions and not corrected for Brownian motion of the ion of interest. Stick boundary conditions and corrected for Brownian motion of the ion of interest. Slip boundary conditions and corrected for Brownian motion of the ion of interest.

of Table 2 summarize the ‘‘uncorrected” model mobilities with stick boundary conditions. These model studies underestimate the equivalent conductance by 0.3–1.7% and the discrepancy increases with increasing salt. The physical basis of this discrepancy is that the relaxation effect is overestimated here as a consequence of ignoring the Brownian motion of the ion of interest. The corresponding ‘‘corrected” conductances are shown in columns 5 and 6 and here we are clearly doing much better. As in the case of the ‘‘small ion” model studies, we are now able to reproduce experimental conductances to an accuracy that is better than 0.1%. In columns 7 and 8, the corresponding ‘‘corrected” model conductances with ‘‘slip” boundary conditions are presented. As mentioned previously, we have had to scale the ionic hydration radii by 1.5 relative to the ‘‘stick” model values in order to properly account for K+0 and K0. As in the case of the corrected ‘‘stick” model conductances, the corrected ‘‘slip” conductances reproduce experimental values to an accuracy of better than 0.1%. Provided the hydrodynamic radii of the two ions are scaled in the manner discussed previously, the resulting model conductivities are very similar for ‘‘stick” and ‘‘slip” models. 3.2. Application to MgCl2 and LaCl3 As in the case of KCl, conductivity data for MgCl2 and LaCl3 is taken from reference [56]. Specifically, K0(MgCl2/2) = 129.34 (in 104 m2/(ohm mole)), K0(LaCl3/3) = 145.9, K0(Mg+2/2) = 53.0, +3  K0(La /3) = 69.7. K0(Cl ) = 76.31. The ion radii, aj, are derived from the limiting equivalent conductances using Eq. (83). With regard to the ‘‘small ion” model, the only remaining adjustable parameter is aex. For both MgCl2 and LaCl3, this parameter is varied in an attempt to get as good agreement as possible between experimental conductivities 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  104 m2/(ohm mole). For LaCl3, / = 2.449, q = 0.3668, S = 0.6853, a = 1.3148, and b = 296.76  104 m2/(ohm mole), respectively. By simple iteration, we have found aex = 0.52 nm for MgCl2 and 0.60 nm for LaCl3 give model conductivities in best agreement with experiment. Results of the ‘‘small ion” model fits with experiment are summarized in Table 3. As in the case of KCl, fits accurate to within several tenths of one percent are possible for both salts. The corresponding fits for K1 (using aex optimized in matching Kexp and K) are not as good, but still fall below a relative error of 1%. We next consider the ‘‘large ion” model. In computing the relaxation correction, Eq. (85) is used to account for the Brownian motion of both ions. Tables 4 and 5 summarize the model results and their comparison with experiment for MgCl2 and LaCl3, respectively. The y0 values represent the reduced equilibrium electrostatic potential (Eq. (49)) at aex equal to 0.52 nm (for MgCl2) and 0.60 nm (for LaCl3). These come from numerical solution of the non-linear Poisson Boltzmann equation. For y0(Mg+2, aex) the central ion has a valence charge of +2 and for y0(Cl, aex), the central ion has a valence charge of 1, etc. For Mg+2 or La+3, |y0| ranges

Table 3 Conductance Data for MgCl2 and LaCl3 (small ion model).a z+m0 (M)

a

.0005

.0010

.0050

.0100

.0200

MgCl2 Kexp Kp K1 K D1 D

125.55 125.31 125.58 125.44 +.024 .088

124.15 123.64 124.09 123.90 .048 .201

118.25 116.59 118.25 118.04 .000 .178

114.49 111.31 114.26 114.33 .201 .140

109.99 103.85 109.09 109.94 .818 .045

LaCl3 Kexp Kp K1 K D1 D

139.6 138.90 139.88 139.45 +.201 .107

137.0 136.00 137.54 136.92 +.394 .058

127.5 123.77 128.49 127.64 +.776 +.110

121.8 114.60 122.42 122.12 +.525 +.263

– – – – – –

Conductivities are in 104 m2/(ohm mole).

Table 4 Conductance/mobility data on MgCl2 (large ion model).a m0 (M)

a

Kexp

.00025 125.55

.0005 124.15

.0025 118.25

.0050 114.49

.0100 109.99

Stick y0(Mg+2, aex) y0(Cl, aex) E+(nr) E(nr) E+(r) E(r) n+ n K(nr) K(r) D(r)

2.62 1.31 4.018 5.830 4.028 5.705 .0025 +.0214 127.23 125.74 +.151

2.58 1.28 3.972 5.803 3.979 5.629 .0018 +.0300 126.29 124.13 .016

2.40 1.17 3.794 5.696 3.784 5.346 +.0026 +.0614 122.62 117.97 .236

2.28 1.10 3.678 5.628 3.656 5.191 +.0060 +.0776 120.22 114.30 .166

2.14 1.02 3.532 5.544 3.498 5.019 +.0096 +.0947 117.27 110.04 +.045

Slip E+(nr) E(nr) E+(r) E(r) n+ n K(nr) K(r) D(r)

4.018 5.830 4.029 5.708 .0027 +.0209 127.24 125.80 +.199

3.973 5.803 3.981 5.629 .0020 +.0300 126.30 124.17 +.016

3.799 5.697 3.793 5.352 +.0016 +.0606 122.69 118.16 .076

3.686 5.628 3.670 5.193 +.0043 +.0773 120.34 114.50 +.009

3.547 5.544 3.520 5.027 +.0076 +.0932 117.47 110.42 +.391

Conductivities are in 104 m2/(ohm mole).

from 2.14 to 3.37. For monovalent Cl, it lies closer to 1.0. The large absolute electrostatic potentials near Mg+2 and La+3 illustrate the importance of going beyond the linear Poisson Boltzmann equation when polyvalent ions are present.

S. Allison et al. / Journal of Colloid and Interface Science 352 (2010) 1–10 Table 5 Conductance/mobility data on LaCl3 (large ion model).a m0 (M)

a

Kexp

.000167 139.6

.000333 137.0

.001667 127.5

.00333 121.8

Stick y0(La+3, aex) y0(Cl, aex) E+(nr) E(nr) E+(r) E(r) n+ n K(nr) K(r) D(r)

3.37 1.11 5.199 5.814 5.270 5.559 .0135 +.0443 142.28 139.86 +.186

3.31 1.07 5.127 5.779 5.195 5.429 .0132 +.0617 140.91 137.17 +.124

3.03 0.95 4.853 5.654 4.879 4.988 .0054 +.1179 135.76 127.48 +.016

2.85 0.87 4.679 5.580 4.674 4.780 +.0011 +.1435 132.55 122.14 .278

Slip E+(nr) E(nr) E+(r) E(r) n+ n K(nr) K(r) D(r)

5.205 5.828 5.279 5.557 .0141 +.0464 142.54 140.00 +.286

5.134 5.792 5.206 5.409 .0140 +.0661 141.16 137.14 .102

4.867 5.660 4.906 4.932 .0081 +.1287 136.01 127.11 .306

4.700 5.582 4.716 4.702 .0034 +.1576 132.83 121.67 .107

Conductivities are in 104 m2/(ohm mole).

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

Ej ¼

3ge l 2e0 er kB T j

ð86Þ

The relaxation correction is then obtained from

nj ¼ 1 

Ej ðrÞ Ej ðnrÞ

ð87Þ

One notable difference between the ‘‘small ion” and ‘‘large ion” model results for binary electrolytes is that the relaxation correction, nj, is the same for both positive and negative ions in the ‘‘small ion” theory, but they are different in the ‘‘large ion” theory. However, these differences tend to average out when mobilities are combined to give conductivities using Eq. (9). Using the same aex 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 agreement with experiment. The ‘‘slip” model results in Table 5 were obtained using aex set equal to 0.55 nm. Under the conditions discussed above, the ‘‘large ion” models give conductivities that are accurate to within several tenths of a percent. 4. Discussion In this work, we have examined two complementary continuum theories of electrokinetic transport of dilute electrolyte solutions that are called collectively the ‘‘small ion” [1–3,5–8,38–45] and ‘‘large ion” [31–37,51] models. Conductance data of dilute binary electrolyte solutions is readily available, fairly extensive, and accurate [56,57]. This is true not only for monovalent binary electrolytes, but electrolytes made up of polyvalent ions as well. This coupled with the relative simplicity of these systems makes them ideal for evaluating the accuracy and ‘‘goodness” of the theory and

9

modeling strategies. This approach was taken more than 50 years ago in the pioneering studies of, Pitts [5], Fuoss, and Onsager [6,7] on the conductance of monovalent binary electrolytes. Their work is the basis of the ‘‘small ion” model in the present study. These early investigators were justifiably cautious about extending their modeling to polyvalent electrolytes due to the largely unknown limitations of the linear Poisson–Boltzmann equation that they employed. The ‘‘small ion” theory has been generalized to an electrolyte consisting of more than two ions of arbitrary valence [8,41], but the linear Poisson–Boltzmann equation is employed. Independently and later, progress was made in modeling the electrokinetic transport of large highly charged particles and numerical procedures made possible by computers played a vital role in these developments. First, was the development of numerical procedures to solve the non-linear Poisson Boltzmann equation around a spherical particle with a centrosymmetric charge distribution [58]. More complicated numerical procedures to solve electrokinetic transport were developed later [35–37]. These numerical procedures are the basis of the ‘‘large ion” modeling of the present study. Given the focus of the present study, the ‘‘large ion” model was generalized to include an ion exclusion distance and also to option of considering ‘‘stick” or ‘‘slip” hydrodynamic boundary conditions. A simple corrective procedure was also developed to account for the Brownian motion of all ions in the determination of the relaxation correction. In an attempt to bridge the ‘‘small ion” and ‘‘large ion” modeling methodologies, the conductance of dilute binary electrolytes made up of both monovalent and polyvalent ions represent ideal test cases for study. The high electrostatic potential around polyvalent ions tests the limits of the ‘‘small ion” model and their small size tests the limits of the ‘‘large ion” model. Both models are applied to the binary salt solutions KCl, MgCl2, and LaCl3 and these results, in turn, compared with experiment. In both ‘‘small ion” and ‘‘large ion” models, the only remaining adjustable parameter is the ion exclusion distance, aex. For aex equal 0.35, 0.52, and 0.60 nm for KCl, MgCl2, and LaCl3 both ‘‘small ion” and ‘‘large ion” models are able to reproduce experimental conductivities to an accuracy of several tenths of one percent or better. For the ‘‘large ion” model with ‘‘slip” hydrodynamic boundary conditions, an aex of 0.55 nm was necessary to get good agreement with experiment for LaCl3. Also, it is necessary to correct the ‘‘large ion” model for Brownian motion 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 given the limitations of the electrostatic model upon which it is based. The results of the present work indicate that the ‘‘small ion” theory can be applied to polyvalent electrolytes provided |zj| 6 3. It is not possible to distinguish whether or not a ‘‘large ion – stick” or ‘‘large ion – slip” model takes better account of conductance data since 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 data for KCl, MgCl2, and LaCl3, respectively. The solid line, widely spaced dashed line, and short spaced dashed line correspond to model data for KCl, MgCl2, and LaCl3, respectively. The other model studies considered, ‘‘small ion”, and ‘‘large ion-slip” are very similar to this.

5. Summary and conclusions In both the ‘‘small ion” and ‘‘large ion” models considered in this 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

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mobilities. Third, more realistic accounting of the interionic potential of mean force may be considered. Progress in this direction has already been made with regard to the ‘‘small ion” approach [9,10]. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] 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 and triangles correspond to experimental data for KCl, MgCl2, and LaCl3, respectively. The solid line, widely spaced dashed line, and short spaced dashed line correspond to model data for KCl, MgCl2, and LaCl3, respectively.

[13] [14] [15] [16] [17] [18]

are described by the linear Poisson–Boltzmann equation (which strictly limits it to weakly charged ions) and that jaex is small (limiting it to low concentration and small size). These latter two assumptions are avoided in the ‘‘large ion” model making it more appropriate for large, highly charged particles. However, the Brownian motion of the central ion is ignored in the ‘‘large ion”, but not the ‘‘small ion” model. In the present work, however, we have proposed an approximate but simple way of correcting for this assumption in the ‘‘large ion” model. Despite these differences in the two models, both are able to reproduce experimental conductivities of dilute binary electrolytes made up of monovalent or polyvalent ions to an accuracy of several tenths of a percent. Minor modifications in the ‘‘large ion” model to include an ion exclusion layer along with the above mentioned correction for Brownian motion allows us to effectively bridge the gap between the two models. These results serve to reinforce both ‘‘small ion” and ‘‘large ion” methodologies as far as application to the electrophoretic mobility and conductivity of small (spherical) ions is concerned. Despite the large absolute electrostatic potentials present when polyvalent ions are present, the use of the linear Poisson– Boltzmann equation in the ‘‘small ion” theory [2,3,5–8,38–45] does not lead to significant errors in conductivity for ions of absolute valence less than or equal to 3. The ‘‘large ion” approach [31,33– 37,51] also works well provided account is taken of the Brownian motion of all ions present. The ‘‘large ion” model can be applied to larger, more highly charged, and also ‘‘structured” particles [21,26,49,59]. As far as small ion studies are concerned, which approach an investigator chooses to use is largely a matter of personal convenience. A more exhaustive comparison of experimental and model conductivities of binary electrolytes shall be presented in future work. The principle objective of the present work has been to present a complete outline of the two approaches and demonstrate their application to three different binary electrolytes of different (cationic) valence. This work will hopefully stimulate research in several areas. First, both ‘‘small ion” and ‘‘large ion” models have a broader range of applicability than has previously been recognized. Both can be used to study conductivities of not only binary electrolytes, but ternary and more complex solutions. Second, with the growing and widespread use of capillary electrophoresis, both ‘‘small ion” and ‘‘large ion” models can be applied to studies of electrophoretic

[19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54] [55] [56] [57] [58] [59]

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