Dynamics of Ion Transfer Potentials at LiquidâLiquid Interfaces - The

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Dynamics of Ion Transfer Potentials at LiquidLiquid Interfaces Konstantin Zhurov, Edmund J. F. Dickinson, and Richard G. Compton* Department of Chemistry, Physical and Theoretical Chemistry Laboratory, Oxford University, South Parks Road, Oxford, United Kingdom OX1 3QZ

bS Supporting Information ABSTRACT: The NernstPlanckPoisson finite difference method is used to simulate the dynamic evolution of a waternitrobenzene system with initially equimolar concentrations of a monovalent salt present in both liquids. The effect of single ion partition coefficients on the evolution of the liquid junction is investigated. The results from simulations reveal two separable components of the potential difference, similar to the behavior observed in recent works on the dynamic theory of membrane potentials [Ward, K. R.; et al. J. Phys. Chem. B 2010, 114, 1076310773]: a localized static component purely dependent on the ratio of single ion partition coefficients and a dynamically expanding diffuse component dependent on the mean salt partition coefficient and the diffusion coefficients of the constituent ions.

1. INTRODUCTION: THE INTERFACE BETWEEN TWO IMMISCIBLE ELECTROLYTE SOLUTIONS (ITIES) 1.1. General Understanding of ITIES. An interface between two immiscible electrolyte solutions (commonly abbreviated as ITIES) is a type of liquidliquid interface which arises from contact between two immiscible solutions in which electrolytes are dissolved. The evolving equilibrating interface may be associated with charge transfer reactions (such as electron transfer, ion transfer, or assisted ion transfer) taking place there. Ion transfer comes about as a result of differential solvation between the two solvents for each ion, such that an ion will partition into the solvent where it is more energetically stabilized by solvation (due to more preferable ionsolvent interactions, for instance). As a result, concentration gradients will arise, and hence, diffusion. ITIES have applications in numerous fields, including extraction chemistry, biochemistry, microscopy, etc.15 They have been the subject of considerable experimental and theoretical interest since the early 1970s, with various kinetic and thermodynamic approaches having been attempted to describe the system.1,614 However, we recognize that there has been a lack of study of the dynamic aspect of the evolution of potential differences at liquidliquid interfaces, which we aim to address. For an equilibrating ITIES, as different ions diffuse at different rates, charge separation will occur and that, in turn, creates an electric field. The electric field then interacts with the ions, causing migration to occur: this slows down initially faster moving ions and accelerates initially slower moving ions. The migration and the diffusion terms eventually balance out and lead to a condition of negligible flux at the liquidliquid interface; that is, equilibrium is attained. Historically, the potential difference is associated with this equilibrium condition as initially formulated by Nernst and Planck.1517 However, the dynamic r 2011 American Chemical Society

evolution of the system from the initial conditions to the steady state, including the separability of static (NernstDonnan) and dynamic components of the potential difference across the liquidliquid interface, is something that this work will address for the first time. 1.2. Theoretical Considerations. Recent theoretical work1820 has shown the feasibility of using the Nernst PlanckPoisson (NPP) equation set (eqs 1.1 and 1.2) with pseudoinfinite Dirichlet boundary conditions, effectively allowing the simulation of dynamically evolving, spatially unconstrained liquidliquid interfaces. ! D 2 Ci z i F D Dφ 1 DCi Ci ¼0 ð1.1Þ þ Dx Di Dt Dx2 RT Dx D2 φ F ¼ 2 Dx εs ε0

ð1.2Þ

∑i zi Ci

ð1.3Þ

where F¼F

where εs is the solvent-specific dielectric constant, ε0 is the permittivity of free space, Ji is flux along the x coordinate, Di is the diffusion coefficient, Ci is the concentration, and zi is the charge number of an individual species i; R is the universal gas constant, T is temperature, F is the Faraday constant, φ is the potential, and F is the local charge density. We use the NPP equation set and a similar computational setup to previous works1922 to consider the ITIES system. The Received: March 11, 2011 Revised: April 26, 2011 Published: May 06, 2011 6909

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The Journal of Physical Chemistry B

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focus of these previous works on the dynamic theory of potential difference formation was limited to single solvent interfaces (commonly “liquid junction” or “membrane” potentials), and as far as the authors are aware, no similar attempts have been made to study the dynamics of ion transfer at liquidliquid interfaces of immiscible solutions, which is the focus of this work. 1.3. Thermodynamic Considerations for ITIES. 1.3.1. Partition Coefficient. For an ITIES, a salt partitions between two phases (right and left, defined r and l) in the following manner: ½ABr ¼ KAB ½ABl

ð1.4Þ

where KAB is the partition coefficient and [AB]q is the salt concentration in each phase q. Now, it is possible to consider each ion within the salt individually such that each ion partitions as K( ¼

½ir ½il

with

i ¼ A þ , B

ð1.5Þ

where K( is the “single ion partition coefficient” and [i]q is the concentration of the ion in each phase. fr , is associated The standard Gibbs energy of transfer, ΔGQ,l trs,i with a general ion transfer process i(l) f i(r). For a binary monovalent salt partitioning between two phases, ion transfer for both Aþand B occurs such that A þ ðlÞ þ B ðlÞ f A þ ðrÞ þ B ðrÞ thus the total Gibbs energy of ion transfer is Q, l f r Q, l f r Q, l f r ΔGtrs, AB ¼ ΔGtrs, A þ þ ΔGtrs, B

ð1.6Þ ð1.7Þ

Given the standard thermodynamic relationship ΔGQ = RT ln K, eq 1.7 is extended to Q, l f r ΔGtrs, AB ¼ RT ln Ktot

The chemical potential itself can be split into chemical and electrochemical terms μi, q ¼ μoi, q þ RT ln ai, q þ zi Fφq

of species i in phase q; φq is the potential of phase q. Combining eqs 1.12 and 1.13 gives μoi, l þ RT ln ai, l þ zi Fφl ¼ μoi, r þ RT ln ai, r þ zi Fφr ð1.14Þ and rearranging μoi, r μoi, l þ RTðln ai, r ln ai, l Þ ¼ zi Fðφl φr Þ

= RT ln K(, and so Noting eq 1.8, μoi,r μoi,l = ΔGQ,lfr trs,i !! ai, r RT ln K( ln Δφ ¼ ð1.17Þ zi F ai , l For a single monovalent binary electrolyte undergoing partition, assuming full dissociation into ions in both phases, bulk electroneutrality requires Ks = ai,r/ai,l for both species and taking into consideration eq 1.11, from eq 1.17 we have RT RT 1 ðln Kþ ln Ks Þ ¼ ln Kþ lnðKþ K Þ ΔφAþ ¼ F F 2 ð1.18Þ RT ðln K ln Ks Þ F RT 1 ln K lnðKþ K Þ ¼ F 2

ΔφB ¼

ð1.8Þ and therefore, given eq 1.5, for a salt partitioning as in eq 1.6 ½A þ r ½B r ½A þ l ½B l

Ktot ¼

2

½A r ½B r ½ABr ¼ ¼ Ks 2 þ ½A l ½B l ½ABl 2

ð1.10Þ

Hence, from eqs 1.9 and 1.10, the following relationship between the salt partition coefficient and single ion partition coefficients is established: pffiffiffiffiffiffiffiffiffiffiffiffi ð1.11Þ Ks ¼ Kþ K which holds for a binary monovalent salt (or any binary salt where both ions share the same charge magnitude). 1.3.2. Two Phase System in Equilibrium. Now, let us consider a two phase system in equilibrium. At equilibrium, the chemical potential μq in both phases should be equal, hence μl ¼ μ r

ð1.19Þ

which simplifies in both cases to

RT RT Kþ Δφ ¼ ðln Kþ ln K Þ ¼ ln 2F 2F K

ð1.9Þ

Now, assuming electroneutrality in bulk solution (i.e., [Aþ]q = [B ]q) and taking into account eq 1.4, we determine þ

ð1.15Þ

then combining the logarithms and dividing through by ziF gives the Nernst equation for the potential difference between phases r and l (Δφ) !! μoi, r μoi, l RT ai , r þ ln Δφ ¼ φr φl ¼ ð1.16Þ zi F zi F ai, l

¼ RTðln Kþ þ ln K Þ ¼ RT lnðKþ K Þ

Ktot ¼ Kþ K ¼

ð1.13Þ

where μoi,q is the standard chemical potential and ai,q is the activity

ð1.12Þ

ð1.20Þ

Thus, at equilibrium, the difference in potential between the phases r and l is directly related to the ratio of single ion partition coefficients.11 Single Ion Partition Coefficient. In the literature, K( has many commonly used names,23 especially in earlier works,7,24 but in this work it will always be referred to as the “single ion partition coefficient”. Although this thermodynamic property of the system cannot be directly experimentally determined, attempts have been made to determine it indirectly using certain extrathermodynamic assumptions.6,7,2527 One common assumption since the early 1970s is often cited as Parker’s assumption25 and was popularized by Popovych28 and states that ΔGtrs,Ph4AsþQ,lfr = ΔGtrs,Ph4BQ,lfr. This relation is then used to construct a scale of values, based initially on data derived from solubility products of the salt in question.28 In the present work, in an attempt to provide further insight into the effect on ITIES of single ion partition coefficients, the relationship between potential difference, solubility products and 6910

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single ion partition coefficients has been extensively studied. To that end a polarized ITIES, specifically the nitrobenzene system, was chosen for investigation. This has been a standard system for experimental investigations of ITIES, especially in the pioneering works in the field.1,8,29,30 1.4. Scope of this Work. Simulations were run for equilibration over the interface of tetra-n-butylammonium halides (TBAþX with X = Cl, Br, I), as well as the hypothetical case of a salt with partition coefficient (Ks) of 1. Analysis of theoretical models of an ITIES system based on data collected from the simulations was performed, confirming certain past results11 and theoretically linking bulk salt concentrations in two different phases to the potential differences. The present work directly links salt concentrations and single ion partition coefficients to potential values, but also introduces an additional term of transient nature related to partition coefficients analogous to the diffuse component of a membrane potential21 (often called a “Henderson potential”). This extra term can be clearly demonstrated in the simulation data and the component collapses in simulations in which the system reaches steady state. The potential difference is therefore composed of a static and transient component, reflecting the modern understanding1922 of the relationship between diffusional processes and potential differences.

2. THEORETICAL MODEL 2.1. LiquidLiquid Interface Model. Consider a planar interface, perpendicular to the linear coordinate x of the simulation space, between two solutions of equimolar binary monovalent electrolyte in immiscible solvents with different dielectric constants (either side of the junction). We assume the solutions are homogeneous across all planes parallel to the plane of the junction and, hence, there is negligible mass transport in all directions, except in the direction along the x coordinate. The ions are allowed to equilibrate locally at the junction, starting at time t = 0. Next, consider flux of species Aþand B along the x coordinate, with the assumption of zero convection, such that the flux can be modeled by the NernstPlanck equation, consisting of a diffusion and a migration term as follows: DCi zi F Dφ Ci ð2.1Þ þ Ji ¼ Di Dx RT Dx þ

thereby giving the following relationship: ! DCi D2 Ci zi F D Dφ Ci ¼ Di þ Dx Dt Dx2 RT Dx

ð2.2Þ

ð2.3Þ

Additionally, the potential of the system should be described at any point by the Poisson equation D2 φ F ¼ Dx2 εs ε0

∑i zi Ci

as well as D2 φ F þ Dx2 εs ε0

ð2.4Þ

∑i zi Ci ¼ 0

ð2.6Þ

Finally, eq 2.6 can be substituted into eq 2.5 to give D2 Ci F 2 zi Ci Dx2 RTεs ε0

z F DCi Dφ 1 DCi ¼0 Dx Dx Di Dt

∑k zkCk þ RTi

ð2.7Þ

2.2. Normalization. In order to deal more efficiently with the model at hand, the number of independent variables is reduced via the following conventional normalizations, to produce a dimensionless equations set:

ci ¼

Ci Ci, r

θ¼

F φ RT

0

Di ¼

Di Di , r

ð2.8Þ

* and Di,r are the bulk concentration and diffusion coefficient where Ci,r of a standard species i taken on the right-hand side. Given the initial conditions of the liquidliquid interface under consideration, * = CAþ,l * = CB,r * = CB,l * . Next, the following normalizations CAþ,r for space and time produce dimensionless coordinates X ¼ kx τ ¼ k2 Di, r t

ð2.9Þ

where

k2 ¼

F 2 Ci RTεs ε0

ð2.10Þ

Thus D2 ci zi ci DX 2

Dc Dθ

1 Dc

∑k zkck þ zi DXi DX Di Dτi ¼ 0 0

ð2.11Þ

for all species i and D2 θ þ DX 2

Further, we assume the fluxes of species A and B to be independent of each other and D not to be a function of C.31 In order for mass to be conserved, the following relation must hold: DCi DJi ¼ Dt Dx

Combining the equations above we obtain the Nernst Planck-Poisson equation set ! D2 Ci zi F D Dφ 1 DCi Ci ¼0 ð2.5Þ þ Dx Di Dt Dx2 RT Dx

∑i zi ci ¼ 0

ð2.12Þ

The physical significance of k can be understood in terms of its relation to the conventional definition of the Debye length, xD32 rffiffiffiffiffiffiffiffiffiffiffiffiffiffi RTεs ε0 k1 ¼ pffiffiffi xD ¼ ð2.13Þ 2 2F C 2 For a typical diffusion coefficient of 109 m2 s1 and a typical concentration of 1 mM, one unit X is approximately 15 nm and one unit τ is approximately 200 ns in water. Were we to normalize with respect to permittivity and viscosity in nitrobenzene, these characteristic scales would be approximately 10 nm and 150 ns respectively. 2.3. Accounting for εs,r 6¼ εs,l. Now, given that the ITIES system requires εs,r 6¼ εs,l and Di,r 6¼ Di,l, these inequalities are taken into account via relationships indicated below. For the εs,r 6¼ εs,l inequality, the following relationship is used during normalization: Rεs, l ¼ εs, r 6911

ð2.14Þ



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ci, l K( ¼ ci, r with 0 1 Q, l f r ΔGtrs, i A K( ¼ exp@ RT

X ¼0

where R is the constant of proportionality between the dielectric constants (εs,q) of the two solvents. Hence, from eqs 2.9 and 2.10 we have sffiffiffiffiffiffiffiffiffisffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffisffiffiffiffiffiffiffiffiffiffi 1 F 2 Ci 1 F 2 Ci x¼ x ¼ kr x ð2.15Þ kl x ¼ Rεs, l RTε0 εs, r RTε0 and applying eq 2.14, the following relationship is established: 1 pffiffiffiffi Xl ¼ Xr R

ð2.16Þ

For the Di,r 6¼ Di,l inequality, the following relationship is used during normalization: 0

0

βDi, l ¼ Di, r

ð2.17Þ

where β is a scaling factor. From the assumption that ηi, l β¼ ηi, r

ð2.18Þ

then from the StokesEinstein equation we have Di,q 1/ηi,q, where ηi,q is the viscosity of the solvent q, thus establishing a fixed expression for β. Next, applying eqs 2.16 and 2.17 to eq 2.11 we get RHS :

D2 ci zi ci DX 2

Dc Dθ

1 Dc

∑k zkck þ zi DXi DX ¼ Di, r Dτi

D2 ci LHS : R zi ci DX 2

0

∑k

Dci Dθ zk ck þ zi DX DX

! ¼

ð2.19Þ

β Dci ð2.20Þ D0i, r Dτ

2.4. Outer Boundary Conditions. The NPP equations must be solved subject to suitable boundary conditions. The outer boundary conditions are defined such that the simulation space is pseudoinfinite, so the diffusion layer does not expand outside the simulation space during the simulation time. This is achieved by defining the maximum value for simulation space, Xm, as pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi ð2.21Þ Xm, r ¼ 6 Dτm and Xm, l ¼ 6 DRτm

where D is the largest normalized diffusion coefficient and τm is the maximum τ value for the simulation, given as an input parameter. The factor of R1/2 arises from eq 2.16; that is, the simulation space is proportional to the Debye length of the solution either side of the junction. Dirichlet boundary conditions are imposed on the concentration of ions, such that for all values of τ X f þ¥

ci ¼ ci, r

X f ¥

ci ¼ ci, l

ð2.22Þ

Given the conditions imposed above, there is no material exchange at the boundaries, and therefore, the simulation space is a Gaussian box of zero enclosed charge, leading to the electric field being zero at the boundaries for all values of τ, hence Xf (¥

Dθ ¼0 DX

ð2.23Þ

2.5. Central Boundary Conditions. The concentrations are assumed to be rapidly equilibrated at the interface, so a Nernstian boundary condition is applied

ð2.24Þ

Note that in applying this condition, we approximate the liquidliquid interface as a flat plane. Therefore, the “potential of mean force” which describes the solvation of the ion as it moves it from one liquid to the other is confined to an infinitesimal distance and is simply given by an equilibrium relationship. We can expect that the detailed structure of the interface will qualitatively alter ion profiles in the range e(15 Å either side of the interface, as has been explored experimentally by Luo et al.33 This range is small with respect to a typical Debye length at low concentration and therefore we take this to be a helpful and valid approximation for the purpose of exploring the dynamic interaction of the double layer with the remainder of solution. 2.6. Numerical Methods. The simulation space is discretized along the X coordinate and 2n þ 1 points are generated such that X = 0 is the central point and X(m = X(n. The space grid is linearly expanding up to a user-defined space point Rs, after which the grid expands exponentially, thus providing a significant fraction of data points close to the liquidliquid interface, where the most significant changes occur, and at the same time ensuring simulation run-time remains reasonable for large values of τm (i.e., those with a large value of Xm and, thus, large n) by having a sparser grid near the edges. Potential difference (Δθ) is taken to be θn θn. The simulation uses a fully implicit centrally differenced finite difference discretization method. The NewtonRaphson method is then employed to iteratively solve the set of nonlinear equations for species Aþand B, as well as the potential for the entire simulation space. Appropriate parameters were established with convergence studies. All simulations were programmed in Cþþ and run on a desktop computer with 4 Intel Core2 Quad 2.85 GHz processors with 2.00 GB of RAM, with a typical runtime of 6 h.

3. THEORETICAL RESULTS AND DISCUSSION Simulations for the dynamic development of ITIES potential differences were run until some value of τ where the potential difference was approximately constant and where the fluxes of species Aþ and B across the liquidliquid interface (jAþ and jB respectively) were deemed negligible (log ji < 6). Three systems for similar TBAþX (where X = Cl, Br, I) salts were simulated (R = 2.303,1 β = 0.5,1 Ks,TBACl = 3.3389, Ks,TBABr = 2.2046, Ks,TBAI = 0.3055, DTBAþ,H2O = 5.1 106 cm2 s1,34 DBr,H2O = 2.09 105 cm2 s1, DI,H2O = 2.05 105 cm2 s1, and DCl,H2O = 2.03 105 cm2 s1)35 with values for Ks taken from experimental data for salt partitioning in nitrobenzenewater systems.36 3.1. Empirical Analysis of the Kþ = K Case. The special case of a salt with equivalent cation and anion single ion partition coefficients across an ITIES was initially considered, as predictions from eq 1.20 indicate an equilibrated potential difference of zero because both cation and anion partition to an equal extent, and so no charge separation arises. 6912



Figure 1. Dynamic evolution of the potential difference, Δθ, and maximum electric field, ξm, as a function of time for the partition of TBACl, assuming Kþ = K, on a logarithmic scale.

Figure 2. Comparison between the dynamic evolution of the maximum electric field, ξm, and its location, Xξm, as functions of time for the partition of TBACl and TBAI salts, assuming Kþ = K, on a logarithmic scale.

Given that both species Aþand B will partition equivalently, one might expect no charge separation to occur whatsoever and the system to remain electroneutral throughout. However, the partitioning of the ions between the two solvents will create concentration gradients, which, in turn, will lead to diffusional processes, and given unequal diffusion coefficients in species Aþand B, this will lead to charge separation and the creation of an electric field, with a potential difference arising. Therefore we expect a transitional potential difference to arise in the ITIES system, comparable to the type 1 liquid junction potential described by Dickinson et al.19 The typical dynamic evolution of the potential difference (Δθ) across the liquidliquid interface and the maximum electric field (ξm) are plotted in Figure 1 on a logarithmic time scale as functions of τ for a simulation of the partition of TBACl. Additionally, the location of the electric field maximum (Xξm) is

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plotted as a function of time (see Figure 2). The possibility of the electric field maximum being mobile, as indicated in recent work19 on the dynamic evolution of the liquid junction potentials, is taken into consideration by extracting dynamic data on both the magnitude of the maximum electric field and its location, rather than just extracting dynamic data on the magnitude of the electric field at X = 0. As can be seen from Figure 1, the trends in the potential difference and the maximum electric field evolution show asymptotic behavior in the limits of short and long time. For the potential difference, the limiting behaviors observed are linear increase with τ at the τ f 0 limit (short time, “stage 1”) and approach to a constant value when τ f ¥ (long time, “stage 3”). For the maximum electric field, similar limiting behavior to that of Δθ is observed in the τ f 0 limit, as ξm initially increases in magnitude as τ1/2, while in the τ f ¥ case, when Δθ approaches a near-constant value, ξm decreases in magnitude as τ1/2. It is worth noting that the transition between short-time and longtime asymptotic behavior occurs at similar values of τ for Δθ and ξm, which we describe as “stage 2”. Such behavior and the transitions between the temporal phases can be understood by considering the evolution and interplay of diffusion and migration which influence the movement of the ions in the two solvents. Under the initial conditions, there is no charge separation, and hence the electric field and therefore migration is zero. However, the thermodynamic drive for the ions of the salt to partition between the two solvents results in nonzero concentration gradients close to the liquidliquid interface, and therefore the diffusion term is nonzero as soon as τ > 0. Hence, in the limit of τ f 0, the equilibration at the liquidliquid interface is almost entirely due to diffusion, with only a negligible contribution from migration (“stage 1”). As the system evolves, charge separation occurs due to unequal diffusion coefficients between species Aþand B, and this increases with increasing time. This, in turn, increases the strength of the electric field and the interaction of the field with the ions becomes stronger, with the initially faster ion experiencing deceleration, and the initially slower ion experiencing acceleration, thus countering the diffusional transport (“stage 2”). As τ f ¥, the migration and diffusion terms tend to balance out, leading to the long time limiting behavior (“stage 3”). Thus, the evolution of Δθ and ξm in “stage 1” is primarily diffusion driven, in “stage 2” migration becomes significant and, finally, in “stage 3” migration tends toward equal magnitude to diffusion, thus leading to the second type of limiting behavior in Δθ and ξm, with “stage 2” being the transition phase. The different dynamic behaviors observed for the three “stages” are related to the comparative magnitudes of the extent of the diffusion layer and of the Debye length, xD, which determines the spatial extent of charge separation within the system. Figures 7 and 8 indicate that the spatial extent of the diffusion layer follows the Einstein relation, since the diffusion layer evolves as (2Dt)1/2, and hence three cases arise which correlate to the three stages of the dynamic behavior displayed by the system: case 1, when t < xD2/2D and the diffusion layer is smaller than the Debye length; case 2, when t ∼ xD2/2D and the spatial extent of the diffusion layer is comparable to the Debye length; case 3, when t > xD2/2D and the spatial extent of the diffusion layer greatly exceeds the Debye length. It is during stage 1, when the spatial extent of the diffusion layer is less than the Debye length, that we see most charge 6913



Figure 3. Spatial profile of the electric field (ξ) at the liquidliquid interface at short times for the partition of TBACl, assuming Kþ = K.

Figure 4. Spatial profiles of the electric field (ξ) at the liquidliquid interface at long times for the partition of TBACl, assuming Kþ = K.

Figure 5. Potential profiles (θ) at the liquidliquid interface at short times for the partition of TBACl, assuming Kþ = K.

separation occurring, via pure diffusional processes leading to charge separation due to unequal diffusion coefficients for

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Figure 6. Potential profiles (θ) at the liquidliquid interface at long times for the partition of TBACl, assuming Kþ = K.

Figure 7. Concentration profiles close to the liquidliquid interface (X = 0) at τ = 0.01 (short time) for the partition of TBACl, assuming Kþ = K.

species Aþand B . During stage 2, the diffuse component is of the same order of magnitude as xD and we see the change from short time limiting behavior to long time limiting behavior as the system starts to recover electroneutrality. During stage 3, the spatial extent of the diffusion layer far exceeds the Debye length, such that charge separation is excluded from the majority of the diffusion layer, and the system dynamics are dominated by the drive to restore electroneutrality in the entire system as it tends toward thermal equilibrium. The significant dynamic aspect to consider is the changes in magnitudes of Δθ and ξm during stage 3: although the value of ξm decreases as τ f ¥, that of Δθ becomes effectively constant. We can understand such behavior using plots of ξm as functions of X, with data presented for each increasing order of magnitude of τ (Figures 3 and 4), where it can be observed that although the magnitude of ξm is decreasing (as τ1/2) at long time, the spatial extent of the charge separation and, hence, of the electric field increases (as τ1/2) and therefore the potential difference between the liquidliquid interface becomes approximately constant (see Figures 5 and 6). 6914



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Figure 8. Concentration profiles at the liquidliquid interface displaying long time behavior for the partition of TBACl, assuming Kþ = K. Note that in all cases the system is very close to electroneutrality, so cTBAþ and cCl are not clearly distinguished.

Figure 10. Dynamic evolution of the potential difference (Δθ) as a function of time for the partition of TBACl, assuming Kþ = K, with infinite and finite supplies of bulk solution, on a logarithmic scale.

Figure 9. Dynamic evolution of the fluxes, ji, of species TBAþand Cl, as a function of time for the partition of TBACl, assuming Kþ = K, on a logarithmic scale.

magnitude i.e. as τ f ¥, jAþ f jB f 0, thus showing the ITIES system tending toward equilibrium, but not achieving it in finite time. This dynamic behavior of an unconstrained evolving ITIES system correlates well with observations made in recent works on dynamic theory of liquid junctions19 and, specifically, reveals many similarities in key dynamic aspects, such as the time dependence of Δθ and ξm in stages 1 and 3, between a binary electrolyte partitioning at a liquidliquid interface and type 1 liquid junctions. 3.2. Transient Nature of Δθ. The results discussed above arise from simulations with a spatially unconstrained evolution of the ITIES, such that the system cannot attain equilibrium in finite time, due to a limitless supply of bulk solution. In order to demonstrate the exclusively transient (nonequilibrium) nature of the potential difference arising in the simulations above, the boundary conditions were adjusted such that the system could attain equilibrium during simulation time, by ensuring that the diffusion layer will encounter an impermeable boundary during the simulation time, i.e., a finite volume of solution. As can be seen from Figure 10, the potential difference collapses, as expected, when the system reaches equilibrium. This clearly demonstrates the nonequilibrium nature of the potential difference formed by partition of a binary electrolyte at a liquidliquid interface, under conditions of equivalent single ion partition coefficients. Note that these results show agreement with existing theory11 (eq 1.20) as the equilibrated ITIES system has zero potential difference. 3.3. Asymptotic Analysis as τ f ¥ by Solution as a Power Series. Given the similarities in several key dynamic characteristics of an equilibrating ITIES and type 1 liquid junction, along with evidence for the nonequilibrium nature of the potential difference, an asymptotic analysis of long time limiting behavior was performed incorporating techniques used previously to analyze long time behavior of type 1 liquid junctions.37,38 By taking the Boltzmann transformation of the NPP equation set and performing asymptotic analysis as τ f ¥ by expansion as a power series, using boundary conditions appropriate for an ITIES, the following expression for the potential difference in the τ f ¥ limit arises (see the Supporting Information)

This explanation is also supported by the dynamic data plot (Figure 9) for the flux of species Aþand B at the liquidliquid interface (ji,X=0): as can be seen from the plot, initially the flux of the anion is greater than that of the cation (diffusion only), with both fluxes showing ji τ1/2 relationship and thus leading to increasing charge separation around the ITIES (stage 1). As τ f ¥, the system will attempt to recover electroneutrality and reach thermal equilibrium, and given that in stage 1 jB > jAþ, the only way to start reducing charge separation is to have jB < jAþ at greater times, and this is exactly what we see: after stage 1 comes a transitional stage (stage 2) as the system starts to return to equilibrium by reducing charge separation, with the migration term becoming significant, such that the flux of the slower species Aþincreases and becomes greater than that of species B. Finally, the fluxes display asymptotic behavior as τ f ¥, where the sum of fluxes (jAþ þ jB) is proportional to τ1, with the fluxes of Aþand B tending toward equality, but with vanishingly small

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Figure 11. Dynamic evolution of the potential difference (Δθ) for partition of TBACl, assuming varying Kþ/K. The asymptotes indicate the theoretically predicted values.

Δθ ¼ δ ln Ks

where

δ¼

DAþ DB DAþ þ DB

Figure 12. Dynamic evolution of the electric field maximum (ξm) for the partition of TBACl, assuming varying Kþ/K.

ð3.1Þ

As can be seen from Figure 1, the theoretical value from this expression is in very strong agreement with the data for the potential difference from simulations, with a difference of less than 0.005% at τ = 104. Note also that eq 3.1 is very similar in nature to the simplified Henderson equation for a type 1 liquid junction19 which effectively takes the form ! cl DA þ DB ð3.2Þ where δ ¼ Δθ ¼ δ ln DA þ þ DB cr Comparing eqs 3.1 and 3.2 it can be seen that both show the same simple relationship between the diffusion coefficients of species Aþand B, while the part included by the logarithm involves, explicitly or implicitly, the bulk concentration of salt either side of the liquid junction or the liquidliquid interface. This further supports the idea that the two systems are very similar in their dynamic features, but there is an inherent and significant difference between the two: the dynamic evolution of the type 1 liquid junction is primarily driven by diffusion, i.e., entropy, while the equilibration of an ITIES arises from the ΔGQ,lfr trs,AB term which is both enthalpic and entropic in nature, encompassing the difference in the sum of the effects of all the ionsolvent interactions for the ions in the two solvents. Also note that the correctness of the Henderson equation as τ f ¥ does not necessarily imply a static boundary layer as conceived by Henderson, as has previously been dismissed.19,38

4. EMPIRICAL ANALYSIS OF THE Kþ 6¼ K CASES The case of a salt with equivalent cation and anion single ion partition coefficients for an ITIES is a special case: it is unlikely to be often encountered in real-world systems, as in most systems the single ion partition coefficients are not equivalent. Thus, the case where Kþ 6¼ K is to be considered. Existing theory1 (see eq 1.20) predicts a finite potential difference at equilibrium as the cation and the anion will partition

Figure 13. Dynamic evolution of the flux of species TBAþ(jþ) for the partition of TBACl, assuming varying Kþ/K.

differently and permanent finite charge separation is expected across the ITIES, as unequal ion partitioning at the liquidliquid interface must hold true even at equilibrium. Note that ion partitioning will create concentration gradients as well as charge separation (a double layer), meaning that diffusional and migrational processes will arise as the system will attempt simultaneously to restore electroneutrality and reach thermodynamic equilibrium. Hence we expect the potential difference across the ITIES to have two contributing components: a dynamically expanding diffuse component, similar to the transient potential difference arising in the Kþ = K case, and a localized static component, comparable to the membrane potentials described by Ward et al.21 Additionally, given eq 3.1 for the diffuse component, for the case of a hypothetical salt with Ks = 1 we expect the transient component not to arise. The typical dynamic evolution of the potential difference (Δθtot) for different ratios of Kþ and K is plotted as a function of time (Figure 11) for simulations of the partition of TBACl, 6916



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Figure 14. Electric field profiles (ξ) at short time for the partition of TBACl, assuming Kþ = 0.5.

Figure 15. Electric field profiles (ξ) at long time for the partition of TBACl, assuming Kþ = 0.5. Note that the field varies in the diffusion layer, but becomes constant close to the interface (X = 0).

with the Kþ = K case included for comparative purposes. Likewise, the dynamic evolution of the maximum electric field is plotted on a logarithmic time scale (Figure 12) for several cases of Kþ 6¼ K. Unlike the Kþ = K case, the location of the electric field maximum (Xξm) for all cases of Kþ 6¼ K is not mobile in time (i.e., Xξm = 0 for all τ), and hence this set of dynamic data is omitted from plots. The temporal evolution of flux of species Aþand B at the liquidliquid interface (ji,X=0) is plotted solely for jþ in Figure 13 as temporal evolution of j showed the same behavior as in Figure 9 for all significant Kþ/K cases. Additionally, spatial data plots for ξ and ci demonstrating short and long time behaviors from a simulation of the partition of TBACl where Kþ = 0.5 can be seen in Figures 1417. As can be seen from Figure 11, the general behavior of the evolution of Δθtot remains the same: we observe two limiting behaviors at short (stage 1) and long times (stage 3), with a transition stage in between (stage 2). Again, the reasoning is similar: the thermodynamic drive for the ions to partition between the two solvents will result in nonzero concentration gradients which will result in diffusional processes taking place (stage 1), and since species Aþ and B will diffuse at unequal rates, this will lead to charge separation occurring and an electric field arising. The electric field will then interact with the moving ions in a manner that counters the diffusion effects such that initially faster species B is decelerated and initially slower species Aþ is accelerated (“stage 2”). As τ f ¥, the diffusion and migration effects will tend toward balancing each other out and the system will tend toward equilibrium (“stage 3”). Of note is the effect of Kþ/K on the magnitude of Δθtot, with greater values of Kþ/K ratio resulting in more positive Δθtot values. The general behavior of ξm and ji,X=0 is more complicated (see Figures 12 and 13). Generally, for ξm, the short-term stage 1 diffusion-dominated behavior is the same for all cases, with a τ1/2 increase in magnitude. However, significant differences in dynamic evolution start to arise during stages 2 and 3: in the τ f ¥ limit, as Kþ < Ks, ξm has a shallower gradient of decrease in magnitude with τ, with the transition into stage 3 occurring later as the value of Kþ decreases. The Kþ = K case delimits a change in the evolution of ξm in stage 2, such that cases where Kþ > Ks swap the sign of ξm during stage 2. For those cases where Kþ > Ks2 is true, ξm has the

opposite direction in all phases of development compared to cases where Kþ e Ks. Analogous behavior can be observed for the dynamic evolution of ji,X=0, with stage 1 behavior being similar for all systems: the flux for both species flows in the direction of favorable solvent interactions and the species B initially has greater flux since DB > DAþ. For the stage 2 and 3 behaviors of ITIES where Kþ < 1, we observe the flux of species Aþ changing direction during stage 2 and recovering the asymptotic behavior typical for Kþ = K systems in stage 3. For the Kþ > Ks2 case (with K < 1), we observe the opposite behavior: it is the flux of species B that changes direction during stage 2, but unlike in the Kþ < 1 cases, there is no crossover of flux magnitudes during stage 3, with jB tending directly toward equal magnitude with jAþ in the τ f ¥ limit. The special case of Kþ = 1 likewise displays atypical behavior, mirrored by the Kþ = Ks2 (K = 1) case, where during stage 1 the flux of species Aþ increases in magnitude before leveling out during stage 2, and demonstrating the standard crossing-over point in stage 3 so that the flux of the initially slower species overtakes that of the initially faster species as the system recovers electroneutrality. All of the above-mentioned behavior can be rationalized by considering the interplay of the factors which give rise to the two components of the potential difference across an ITIES: the diffuse transient component, which arises from the interaction of migration and diffusion, and the local static component, which arises from the different extent of partition of cations and anions at the ITIES. Keeping note of the relationship established for the Kþ = K case between the mean ion partition coefficient and the potential difference across the liquidliquid interface (eq 3.1), and noting the effect of Kþ/K on Δθ (Figure 11), for a given salt, the Kþ/K ratio is expected to have the greatest effect on the local static component. This is because it governs the degree of permanent charge separation at the ITIES by determining the magnitude and direction of partition for individual ions, and thereby the magnitude and direction of the electric field due to the double layer. The dynamics of the diffuse, expanding component will be affected to a much lesser degree, as the key dynamic factors in establishing a transient potential difference 6917



Figure 16. Concentration profile st short time for the partition of TBACl, assuming Kþ = 0.5. The different partition directions have caused a double layer to form.

will effectively remain unchanged. This arises as a result of the localization of the static component of potential difference in a boundary layer, as charge separation over distances much greater than the Debye length is energetically unfavorable. The energy from ion partition is sufficient for charge separation over a limited distance, with the spatial extent of the charge separation determined by the Debye length, xD, of the solution. As a result, charge separation within the system is limited to the locality of the liquidliquid interface and does not expand beyond the boundary layer, even at long time. Note that in real units, the region of charge separation corresponds to |x| < 50 nm either side of the interface, for typical coefficients. It will extend less far in nitrobenzene since the Debye length is less in a less polar solvent. Moreover, the local component of the system displays Nernstian behavior (see eq 1.16) as in the τ f ¥ limit, cl,iKs f cr,i at the boundary layer. This means that from the point of view of the electroneutral diffusion layer, the relation cl,iKs = cr,i holds at the interface, which is equivalent to the central boundary condition for the Kþ = K case. Therefore the expanding diffuse component does not interact significantly with the internal dynamics of the local static component within the boundary layer, and as such, at long times, the two components are separable. Furthermore, given the equivalence of the conditions on cq,i relationships at the outer 6 K edge of the boundary for layer for the Kþ = K and the Kþ ¼ cases, the same equations apply for the formation of the diffuse component of the potential difference (eq 3.1) and, hence, we expect Δθdif to be of the same magnitude, irrespective of Kþ/K. Hence, the single ion partition coefficient ratio will determine whether the two components of the electric field will act in the same or opposing directions, thus producing varying overall potential differences. This is supported by the data from Figures 16 and 17, which demonstrate concentration profiles at short and long times for the partitioning of the TBACl with Kþ = 0.5 with several key features holding true for all Kþ/K ratios investigated: in the τ f 0 limit, the ions will immediately start to partition across the ITIES, with single ion partition coefficients determining the magnitude and direction of partitioning, but with the spatial extent of concentration perturbation from initial conditions

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Figure 17. Concentration profiles at long times for the partition of TBACl, assuming Kþ = 0.5. Note that in all cases the system is very close to electroneutrality, so cTBAþ and cCl are not clearly distinguished. The inset magnifies the region where a static double layer persists.

Table 1. Seven Behavioral Cases for an Equilibrating ITIES System TBAX salt 1

Kþ < 1

K > Ks2

2

Kþ = 1

K = Ks2

3 4

1 < Kþ < Ks Kþ = Ks

Ks < K < Ks2 K = Ks

5

Ks < Kþ < Ks2

1 < K < Ks

6

Kþ = Ks2

K = 1

7

Kþ > Ks2

K < 1

always greater for the faster species B, thus giving rise to both transient and static components of the potential difference across the ITIES; in the τ f ¥ limit, it can be seen that the charge separation due to unequal single ion partition coefficients in the system stays local to the ITIES with spatial extent of |X|∼3 and the Nernst equation cr,i/cl,i f Ks applying at |X| g 10 which correlates with cr,i/cl,i = Ks holding true in the τ f ¥ limit for finite volume simulations where the system reaches equilibrium in finite time, meaning that there is a spatially well-defined profile for the static component of the overall potential difference of the system. Meanwhile the spatial extent of deviation from bulk concentration for species Aþ and B (cAþ ∼ cB 6¼ cr*) increases with time, reaching |X|∼200 at τ = 104, which is in line with the Einstein relation of the root-mean-square displacement of molecules in time due to diffusion only39 ((X2)1/2 = (2D0 τ)1/2 in dimensionless units) and holds true for all time phases, providing a smooth transition from cr,i/cl,i ∼Ks near the ITIES to cr,i/cl,i = 1 in the bulk solution, such that there is a spatially well-defined profile for the dynamic component of the overall potential difference of the system. These observations hold true for all the major ITIES systems (save certain special cases, which, nevertheless, still follow the overall dynamic behavior of an equilibrating ITIES) and indicate the possibility of separability of the two components of the potential difference. In light of this and noting identical behavior for any Kþ/K ratio observed for all three phases for the dynamic evolution of Δθ, as well as for stage 1 in ξm and ji,X=0 (Figure 11), the different 6918



Figure 18. Electric field profiles (ξ) at long times for a system in which the electric field maximum, ξm, changes sign during the partition of TBACl (assuming Kþ = 5).

behaviors of ξm and ji,X=0 in phases 2 and 3 can be considered as particular cases which arise from differences in the relationships between the Kþ, K, and Ks values, and can be seen as variations on the general dynamic processes taking place at an equilibrating ITIES. Classifying the different relationships between the three factors results in seven categories summarized in Table 1. Within the seven categories, two are special cases where one of the ions does not partition between the two solvents (categories 2 and 6) and the rest can be split into two major cases, where ions either partition in opposite directions (categories 1 and 7), or in the same direction (categories 35). Within the latter resides the special case of both ions partitioning with equal thermodynamic drive in the same direction (the Kþ = K case considered earlier). Thus, each category will give rise to certain particular cases in the dynamic evolution of ξm and ji (as is clearly demonstrated by Figures 12 and 13), such as the atypical increase in magnitude of jþ during stage 1 for the Kþ = 1, K 6¼ 1 case (category 2), the change in sign of ξm for the Ks < Kþ < Ks2 case (category 5) or the change in sign of jþ during stage 2 for Kþ < 1 case (category 1). In the first case (category 2), there is no initial thermodynamic drive for species Aþ to partition and, hence, no initial nonzero concentration gradients arise to start diffusional processes. However, species B does partition, resulting in increasing charge separation, which leads to increasing transport of species Aþ, and therefore increasing flux, as the system attempts to restore electroneutrality and equilibrate itself. For the second case, spatial data plots of the electric field at certain points in time (see Figure 18) indicate that in this case the interplay between the static and diffuse components of the potential difference results in an electric field with multiple field maxima, as the negative electric field arising from the diffuse component and the positive electric field from the local static component are superimposed, with the central point becoming more positive in time and changing sign as the magnitude of the electric field arising from the diffuse component decreases as the electric expands away from the interface. The final case, where jþ changes sign, arises from the two ions partitioning in opposite directions, causing large charge separation to occur (stage 1), and as the system equilibrates and tries to return to electroneutrality, the charge separation can only start to

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Figure 19. Concentration profiles and the electric field profiles (ξ) for partition of a salt with Ks = 1, assuming Kþ = 0.5, at long time. Note that in all cases the system is very close to electroneutrality, so cTBAþ and cCl are not clearly distinguished. The inset magnifies the region where a static double layer persists.

decrease if the fluxes of both species are in the same direction. This situation arises during stage 2 for the species which has smaller preference in partitioning between the two solvents. It is therefore clear that the dynamic behavior of an equilibrating ITIES is initially governed by the magnitude and direction of single ion partitioning (stage 1), while long time behavior is a result of the system attempting to regain electroneutrality and reach equilibrium (stage 3), and that the transitional dynamic behavior seen in stage 2 is simply a result of the system finding ways to recover electroneutrality from the initial conditions. Likewise, the overall behavior in terms of the total potential difference arising across the ITIES is the result of two contributing components, one diffuse and transient, one local and static, arising from mean salt partition coefficient and single ion partition coefficients. As the diffusion layer grows in time, as (2Dt)1/2, from the Einstein relation, it can be compared to the Debye length (xD). During stage 1, t < xD2/2D, and therefore the spatial extent of the diffusion layer is smaller in magnitude than xD, and we observe most of the charge separation within the system arising during this period of time. Stage 2 occurs when t ∼ xD2/2D; that is, the spatial extent of diffusion layer is comparable in magnitude to the Debye length, and indicates a time when the system starts to reduce charge separation initiated by the ion partition at the liquidliquid interface during stage 1 since this charge separation cannot be supported over the full diffusion layer. Stage 3 occurs when t > xD2/2D, such that the diffuse component has grown to exceed the Debye length, and, hence, the boundary layer. No meaningful charge separation is supported outside the boundary layer and the system attempts to restore electroneutrality outside the boundary layer to the maximum extent possible, so tending to thermal equilibrium as τ f ¥. As such, the system dynamics are dominated by the drive to re-establish electroneutrality throughout the system and compensate the charge separation encountered outside the boundary layer in stage 1. 4.1. Ks = 1 Case. Noting the strong evidence (Figures 16 and 17) for the possibility of separating the two components, the case for a salt with Ks = 1 was considered, wherein the transient component should not arise (as indicated by eq 3.1) and we 6919



Figure 20. Comparison of theoretical results, Δθtot, with simulated results for the limiting potential difference Δθ, for a range of Kþ/K ratios for the partition of TBACl, TBAI, and a hypothetical salt with Ks = 1.

expect to observe solely the local static component, as existing theory1 indicates a nonzero potential difference at equilibrium (eq 1.20). Figure 19 shows the concentration profiles of species Aþ and B and electric field for large values of τ. As can be clearly seen, as τ f ¥, the concentration profiles reveal a charge separation local to the ITIES, with a spatial extent of |X|∼3, and the electric field is also seen to remain local to the ITIES with the same spatial extent at τ = 103, with no evidence of a diffuse transient component at any stage of ITIES equilibration. Therefore, this is a clear profile of the evolution of a local static potential difference in isolation, as these data are in close agreement with the contribution from the local static potential difference in the Kþ 6¼ K cases for the TBAþ halide salts. Coupled with the diffusion only profile from the Kþ = K case, which also shows close agreement with the diffuse contribution to the data profiles for the TBAþ halide salts, it is clear that the two components simply add up to the total potential difference of an equlibrating ITIES. Equation eq 1.20 was compared to the simulation data for the Kþ = 1 case and excellent agreement is displayed between the two: at τ = 103 there was less than 0.0005% difference between values from the expression and values from the simulation for the potential difference of the system. In light of this, from eqs 3.1 and 1.20, with the latter equation having a Δθ = 1/2 ln(Kþ/K) dimensionless form, the following expression was derived 1 Kþ Δθtot ¼ Δθdyn þ Δθstat ¼ δ ln Ks þ ln ð4.1Þ 2 K The comparison of eq 4.1 with the simulation data showed good agreement with all the data, with less than 0.2% difference at τ = 104, thus showing that the potential difference of an equilibrating ITIES can be numerically separated into its constituent components. A plot of Δθtot vs ln(Kþ/K) (figure 20) with Δθtheory shows excellent correlation between the two, with the intercept giving the dynamic diffuse component contribution to Δθtot and the gradient showing the contribution of the local static component. As with the Kþ = K case, in order to demonstrate that one of the components of the potential difference is, indeed, transient in

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Figure 21. Dynamic evolution of the potential difference, Δθ, for the partition of TBACl, assuming Kþ = 0.5, with infinite and finite supplies of bulk solution.

nature and the other component is static, the outer boundaries of the simulations were changed such that the system could attain equilibrium during simulation time by providing only a finite amount of solution. Simulations were run for varying degrees of spatial constraint (by varying the value of Xm) and as can be seen from figure 21, the transient component collapses when the system attains equilibrium, while the local static component remains. If Xm is large enough (typically Xm > 10), then the full potential difference, as predicted by eq 4.1, is achieved before the Δθdyn component starts to collapse when the diffusion layer reaches the spatial impermeable boundary and the potential difference is reduced to the Δθstat component at equilibrium, in accordance with existing theory1 (eq 1.20). Simulations with a finite volume of solution for the Ks = 1 salt show a small, temporary decrease in magnitude of the potential difference when the diffuse layer encounters the outer boundary of simulation space, before the potential difference reaches the value of Δθstat.

5. CONCLUSIONS Detailed dynamic analysis was performed on an ITIES with a monovalent binary electrolyte initially present in equimolar concentrations. The effect of single ion partition coefficients on the dynamic properties of the system was investigated. The nature of the liquidliquid potential difference formation was elucidated, with distinct qualitative and quantitative separation into two components: one local to the ITIES and static in nature, arising from the magnitude and direction of individual ion partitioning, another dynamic and diffuse, arising from the interplay between the diffusion coefficients of the ions and the mean salt partitioning. Specific cases where only one of the components is present have been investigated (Kþ = K and Ks = 1 cases), with appropriate expressions presented for both which correlate the ion partitioning effect to the potential difference. Charge separation at equilibrium is restricted to a few unit Debye lengths either side of the interface (tens of nanometres). The dynamic development of the potential differences has also been discussed in terms of three “stages”, with a transition between developing charge separation on a nanosecond time scale and the 6920


The Journal of Physical Chemistry B recovery of equilibrium by restoration of electroneutrality at longer time scales. An expression for the total potential difference arising in a spatially unconstrained ITIES has been derived with numerically separable contributions from the two components by analogy to the NernstDonnan equation for membrane potentials. The Kþ = K case was compared to type 1 liquid junction, and the analytical expression derived for the liquidliquid potential difference was compared to the Henderson equation for type 1 liquid junctions. The temporal and spatial evolutions of certain aspects of the system, such as the electric field and the flux across the ITIES have been analyzed and quantified, with different trends categorized into conditions related to the magnitude and direction of single ion and mean ion partition coefficients between the two solvents, and placed in the context of a general dynamic evolution of an equilibrating ITIES.

’ ASSOCIATED CONTENT

bS

Supporting Information. τ f ¥ asymptotic analysis. This material is available free of charge via the Internet at http://pubs.acs.org.

’ AUTHOR INFORMATION Corresponding Author

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(23) These alternative names include: single-ion solvent actiivity coefficient, medium effect of a single ion, medium activity coefficient, solvent activity coefficient, and individual extraction constant. (24) Siekierski, S. J. Radioanal. Chem. 1974, 21, 9–13. (25) Parker, A. Chem. Rev. 1969, 69, 1–32. (26) Popovych, O. Anal. Chem. 1974, 46, 2009–2013. (27) Girault, H. H. Electrochim. Acta 1987, 32, 383–5. (28) Popovych, O.; Dill, A. Anal. Chem. 1969, 41, 456–462. (29) Danil de Namor, A. F.; Hill, T.; Sigstad, E. J. Chem. Soc., Faraday Trans. 1 1983, 79, 2713–2722. (30) Koczorowski, Z.; Geblewicz, G. J. Electroanal. Chem. 1983, 152, 55–66. (31) Buck, R. J. Membr. Sci. 1984, 17, 1–62. (32) Morrison, F. A., J.; Osterle, J. F. J. Chem. Phys. 1965, 43, 2111–2115. (33) Luo, G.; Malkova, S.; Yoon, J.; Schultz, D. G.; Lin, B.; Meron, M.; Benjamin, I.; Vany sek, P.; Schlossman, M. L. Science 2006, 311, 216–218. (34) Koczorowski, Z.; Geblewicz, G. J. Electroanal. Chem. 1980, 108, 117–120. (35) Bard, A. J.; Faulkner, L. R. Electrochemical Methods: Fundamentals and Applications; John Wiley & Sons: New York, 2001. (36) Markin, V.; Volkov, A.; Volkova-Gugeshashvili, M. J. Phys. Chem. B 2005, 109, 16444–16454. (37) Hickman, H. J. Chem. Eng. Sci. 1970, 25, 381–398. (38) Jackson, J. L. J. Phys. Chem. 1974, 78, 2060–2064. (39) Compton, R. G.; Banks, C. E. Understanding Voltammetry, 2nd ed.; Imperial College Press: London, 2011.

*Fax: þ44 (0) 1865 275410. Tel: þ44 (0) 1865 275413. E-mail: [email protected].

’ REFERENCES (1) Samec, Z. Pure Appl. Chem. 2004, 76, 2147–2180. (2) Vany sek, P.; Ramírez, L. J. Chil. Chem. Soc. 2008, 53, 1455–1463. (3) Kivlehan, F.; Lanyon, Y. H.; Arrigan, D. W. M. Langmuir 2008, 24, 9876–9882. (4) Watarai, H. Solvent Extr. Liq. Membr. 2008, 21–57. (5) Ishimatsu, R.; Kim, J.; Jing, P.; Streimer, C. C.; Fang, D. Z.; Fauchet, P. M.; McGrath, J. L.; Amemiya, S. Anal. Chem. 2010, 82, 7127–7134. (6) Popovych, O. Anal. Chem. 1966, 38, 558–563. (7) Alexander, R.; Parker, A.; Sharp, J.; Waghorne, W. J. Am. Chem. Soc. 1972, 94, 1148–1158. (8) Gavach, C.; Savajols, A. Electrochim. Acta 1974, 19, 575–581. (9) Koryta, J.; Vanysek, P.; Brezina, M. J. Electroanal. Chem. 1977, 75, 211–228. (10) Samec, Z. J. Electroanal. Chem. Interfacial Electrochem. 1979, 99, 197–205. (11) Hung, L. J. Electroanal. Chem. 1980, 115, 159–174. (12) Girault, H. H. J.; Schiffrin, D. J. J. Electroanal. Chem. Interfacial Electrochem. 1985, 195, 213–27. (13) Samec, Z. Chem. Rev. 1988, 88, 617–32. (14) Kakiuchi, T. Anal. Chem. 1996, 68, 3658–3664. (15) Nernst, W. H. Z. Phys. Chem. 1889, 4, 165. (16) Planck, M. Wied. Ann. 1890, 39, 161–186. (17) Planck, M. Wied. Ann. 1890, 40, 561–576. (18) Perram, J.; Stiles, P. Phys. Chem. Chem. Phys. 2006, 8, 4200–4213. (19) Dickinson, E. J. F.; Freitag, L.; Compton, R. G. J. Phys. Chem. B 2010, 114, 187–197. (20) Ward, K. R.; Dickinson, E. J. F.; Compton, R. G. J. Phys. Chem. B 2010, 114, 4521–4528. (21) Ward, K. R.; Dickinson, E. J. F.; Compton, R. G. J. Phys. Chem. B 2010, 114, 10763–10773. (22) Ward, K. R.; Dickinson, E. J. F.; Compton, R. G. Int. J. Electrochem. Sci. 2010, 5, 1527–1534. 6921


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