Surface tension and ion transfer across the interface of two immiscible electrolytes

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1 Electrochemistry Communications 6 (2004) Surface tension and ion transfer across the interface of two immiscible electrolytes C.G. Verdes a, M. Urbakh b, A.A. Kornyshev a, * a Department of Chemistry, Imperial College London, London SW7 2AZ, UK b School of Chemistry, Tel Aviv University, Ramat Aviv, Tel Aviv, Israel Published online: Received 13 April 2004; received in revised form 30 April 2004; accepted 30 April 2004 Abstract We explore the surface polaron mechanism of ion transport across the interface of two immiscible electrolytes by numerical simulations of the coupled nonlinear Langevin equations of the 2d-Kramers-like theory of Kornyshev Kuznetsov Urbakh [J. Chem. Phys. 117 (2002) 6766]. A dependence of ion transfer rate on the diffusion coefficients of ion and interfacial protrusion and on interfacial tension has been investigated. In the case of strong coupling between the motions of the ion and the interface (large interfacial tension), our results reproduce the analytical limiting laws attained in this theory and also approve the suggested interpolation formula for the effective diffusion constant. The results are, however, very different in the cases of moderate and weak coupling (moderate and small interfacial tension). Possible experimental verification of the predicted effects is discussed. Ó 2004 Elsevier B.V. All rights reserved. Keywords: ITIES; Ion transfer; Stochastic transport; Surface tension 1. Introduction The interface of two immiscible electrolyte solutions (ITIES) continues to be a hot subject in electrochemistry. Considered initially relevant in relation to industrial phase transfer catalysis or biomimetic experiments performed in electrochemical cell, this system attracts more and more attention, in view of unique opportunities for self-assembling controlled by electric field. ITIES is made of two immiscible liquids (water and organic phase) and a mixture of a salt composed of hydrophilic ions and a salt composed of hydrophobic ions that distribute between the immiscible solvents: hydrophobic ions go to the organic phase, while the hydrophilic ions go the aqueous phase [1]. Under the influence of electric field two back-to-back electrical double layers are formed on the two sides of the interface, making interface ideally polarisable, up to certain * Corresponding author. Tel.: ; fax: address: a.kornyshev@imperial.ac.uk (A.A. Kornyshev). voltage drops above which the traffic of ions across the interface starts [2]. The interface is never ideally flat. It fluctuates due to capillary waves, their amplitudes increasing with electric polarization [3]. The interface has only a minor bit of molecular smearing, less due to the solvents but rather due to diversions of some of the ions into the neighbouring unfriendly medium [4,5]. In addition to the two buffer electrolytes one may add a small amount of salt that contains an ion whose chemical potential in the bulk of the unfriendly solvent is higher, but not much higher than in the friendly one. Moderately polarizing this system, one can compensate the difference in the chemical potentials and study the kinetics of across-the-interface transfer of that sort of an ion, without: (i) triggering the traffic between the two solutions of the buffer ions (the latter is indeed preserved, if the buffer ions have much larger differences between the chemical potentials in the bulk of the two solvents), or (ii) destabilizing the interface by the external field [3]. Which of the two catastrophes start first is to be experimentally detected, but dealing with moderate potential drops one can hope to measure the mobility of the given sort of ion /$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi: /j.elecom

2 694 C.G. Verdes et al. / Electrochemistry Communications 6 (2004) The transport of an ion across the interface remains to be an intriguing subject. Experiments at ITIES have shown that the effective mobility of ion passage across the interface is much smaller than the mobility of the same ion in the bulk [6 9]. The mysteriously abnormal strength of this effect has not sustained time: earlier publications claimed several orders of magnitude slowing down, whereas recent experimental reports suggest that unlikely it exceeds 50 times slowing down [10]. It is, however, a challenge to explain even such a moderate (relative to the previous claims) slowing down and to reveal the factors that may affect it. Ion transfer across ITIES is commonly considered as a first order heterogenous reaction [11 13]. Shao and Girault [14,15] were first to suggest treating it as a diffusion flux across the interface and Kakiuchi [16] obtained an analytical result for the rate constant as a function of potential drop across the interface. In order to explain experimental data he had to specially assume that diffusion coefficient in the interfacial region is much smaller than in the bulk. A number of authors tried to rationalize this presumption. Schmickler [17] has pointed out that the combination of monotonic profiles of the chemical potential and the electrical potential of the polarized interface may give rise to a well followed by a barrier in the electrochemical potential, suggesting that the ion transfer might be an activation process with potential dependent activation energy. Aoki [18] pushed forward an idea of a maximum, intrinsic to the chemical potential, whereas Indenbom [19] suggested that such a maximum will be removed by the ion-induced corrugation of the interface. Benjamin [20] has observed an ioninduced corrugation of the interface in his molecular dynamic simulations, but no maximum in the chemical potential was found and no evidence that it shows in electrochemical potential either [21]. In addition a hydrodynamic approach to ion transfer reactions have been proposed by Ferrigno and Girault [22] which have shown qualitatively how the viscosity of the two adjacent phases influences the viscous drag coefficient as an ion transfers from one phase to the another. In a comprehensive feature article, Marcus [23] discussed various experimental and theoretical aspects of the problem. His original idea was that ion transport across the interface is assisted by a favourable protrusion of one phase into the other in front of the ion. He presented estimates showing that the probability of such fluctuation can determine the ionic mobility across the interface. He has admitted that not only the ion motion is affected by the fluctuation of the interface, but the ion itself may influence the shape of the interface. However, this idea has not been explored in his article, as he did not consider in his equations the coupling between the motion of the ion and protrusion explicitly. Such coupling may however be very important because, if even there were no thermal fluctuations, the ion if it were fixed near the interface would produce such a protrusion. In other words, the protrusion is caused by the mere existence of the ion. As it will be shown in the present paper, such coupling may play a crucial role in the kinetics of ion transport across the interface. Kornyshev, Kuznetsov and Urbakh (quoted hereafter as KKU) [24] have made this coupling, called conventionally surface polaron effect, the central point of their barrier-free model (they have also studied various effects of a direct potential barrier in the electrochemical potential, having retrieved the results of Marcus [23] and Schmickler [17] as particular cases of a more general theory). The physical arguments behind their model are as follows. Consider water/oil system having a flat interface in unperturbed state, with the oil half-space at z > 0 and water half-space at z < 0. A hydrophilic ion slowly moving towards the water/oil interface will tend to keep water around, creating a water protrusion into the oil, in front of itself. Surface tension will, however, not allow the protrusion to grow unlimitedly. After a certain penetration depth, the ion will not be able to keep water anymore, and it will cross the water oil boundary. With the further motion of an ion away from the water phase, the protrusion will be moving back to the flat surface level. Similarly when the same ion moves from the oil to water, it induces water protrusion into the oil and after the ion crosses the protrusion surface, the latter relaxes to the flat state. The situation for a hydrophobic ion is reverse. Note that this is an entirely equilibrium picture, not associated with any dynamic aspects of ion and protrusion motions. Now let us turn on the dynamics and consider what will happen if the ion will try to move faster than the protrusion could follow it. This will create an additional barrier because the interface will not be in an energetically optimal configuration. If on the other hand the protrusion moves too fast this will neither correspond to the optimum path. Thus the protrusion and ion should move in a concerted way along the potential energy surface depending on the coordinate of the ions and effective coordinate of the protrusion. The KKU theory has treated the stochastic dynamics of this motion as two dimensional Kramers problem, formulated in terms of two coupled Langevin (or Fokker Planck) equations. These have been solved analytically in two opposite limiting cases, when: (i) the protrusion moves much slower than the ion, (ii) the ion moves slower than the protrusion. Not that the latter case is very realistic, but having these two cases solved, it was possible to build a simple interpolation formula which was intended to cover the whole spectrum of the system behaviour. Obviously when the protrusion is very slow, this will be the case of strongest slowing down. Indeed, here the ion should wait, as suggested by Marcus, for a favour-

3 C.G. Verdes et al. / Electrochemistry Communications 6 (2004) able protrusion-fluctuation and then inertialessly follow it. In the language of the Kramers problem, this will mean moving essentially along the protrusion coordinate and then eventually making a short move along the reaction coordinate to cross the boundary. Everything in this case will be determined by the effective diffusion coefficient of the protrusion, D h. When the ion is very slow (imagine a huge macro-ion) it will not care for the protrusion, and decisive will be the diffusion coefficient of the ion, D z. In this case, the ion moves as a tank and the protrusion inertialessly follow it. Generally the motions proceeds along the both coordinates where the system chooses its optimal trajectory, and therefore the approximation of sequential motion along these coordinates giving the effective diffusion coefficient of transfer as, D eff ½ð1=D z Þþð1=D h ÞŠ 1 will not be accurate. More accurate interpolation formula of the KKU theory (see Eq. (7) below) takes into account the complexity of this trajectory. However, a number of points remained open in this theory. First it was not clear how accurate will this interpolation work. Secondly, what happens if the coupling between the motions of the ion and of the protrusion is weak? The coupling is stronger the higher the surface tension, but it is negligible when the surface tension is low. Shall the ion wait then for a favourable protrusion-fluctuation? Pure intuition suggests that it should not, because the profits are too low. You are not waiting for an elevator on a busy day if you do not have luggage and need to get only one floor up, but you certainly do, if the 24th floor is your destination or you have a 100 pound case with you! What you will do when your target is somewhere in between and you have a reasonably loaded backpack? The KKU theory does not answer this question, because it had actually handled only the case of strong coupling. With ambition to cover the general case, we performed here numerical analysis based on the KKU model. The results appeared to be interesting and worth of reporting. They clarify the whole picture of the surface polaron mechanism and give new hints of how it can be verified experimentally. 2. Description of the model dh ov g h þ ð z; h Þ ¼ f h ðþ; t ð2þ dt oh where g z and g h are the dissipation constants for the ion and surface motions, with the thermal effects described by the d-correlated random forces f z and f h : f z ðþf t z t 0 ¼ 2kB T g z d t t 0 ; ð3þ f h ðþf t h t 0 ¼ 2kB T g h d t t 0 ; ð4þ where dðxþ is the Dirac delta function (on Langevin equations in physics and chemistry see [25]). The diffusion coefficients of the ion and protrusion are thus D z ¼ k B T =g z and D h ¼ k B T =g h, respectively. The ion dissipation constant is estimated using the Stokes equation g z ¼ 6pg 1;2 r i, with g 1;2 g 1 g 2 the viscosity, considered to have similar values for the two liquids, and r i the Stokes radius of the ion. The protrusion dissipation constant is usually considered to depend on the lateral size of the protrusion. However, since the lateral size is difficult to evaluate and the dependence itself is not very clear (there is a 2p difference between results given in [26,27]) g h is taken as a free parameter whose influence on the transfer rate at zero driving force is studied below via numerical simulations. The equations of motion are coupled through the potential V ðz; hþ. In accordance with the capillary wave theory [27] it is assumed that the excess surface energy due to the fluctuations is proportional to the square of the deviation of the interface position h from its equilibrium value in the presence of the ion h eq ðþ, z with the proportionality constant being half of the interfacial tension c. Thus the expression for the potential can be written as Vðz; hþ ¼ 1 2 c h h eq ðþ z 2 : ð5þ For simplicity, the h eq ðzþ dependence is taken of triangular shape as shown in Fig. 1, with h max ¼ z max the position at which the ion penetrates the disturbed interface, ½z i ; z f Š the interval of positions over which the ion interacts with the interface and K ¼ðz f z i Þ=2 the half-width of the triangle. The parameter that is usually experimentally measured is the transfer rate k rate defined by the equation Assuming the diffusional character of the motion of the ion and of the local corrugation of the interface, the position z of the ion and the height h of the corrugation, both relative to the flat, unperturbed interface, can be described by the system of two overdamped Langevin equations [24]: g z dz dt ov z; h þ ð Þ oz ¼ f z ðþ; t ð1þ Fig. 1. Triangular dependence of the equilibrium protuberance height on the position of the ion.

4 696 C.G. Verdes et al. / Electrochemistry Communications 6 (2004) j ¼ k rate cðz i Þ where j is the ionic current and cðzþ is the ion concentration at position z. In the absence of coupling the transfer rate can be expressed as [16] k 0 rate ¼ D z =ðz f z i Þ. In the presence of strong ion interface coupling, i.e., for the case of large values of the interfacial tension c, the KKU model [24] gives analytical expressions for the effective diffusion coefficient of the ion transfer across the interface for limiting cases of the D z =D h ratio. Interpolation of these limiting results has led to a unified formula for the transfer rate covering the whole range of D z =D h values: k rate ¼ D eff ; ð6þ z f z i with " # D eff ¼ þ : ð7þ D z ðk=h max Þ 2 D h However, the range of applicability and accuracy of this interpolation formula was unclear, and no expression for D eff for intermediate and low values of c has been obtained. To elucidate the validity of the interpolation and to estimate values for the transfer rate in the case of weak ion interface coupling computer simulations are used as described below. The coupled system of stochastic differential equations given by (1) and (2) is solved using the Milstein scheme [28] and the ion transfer rate is calculated as the average ion velocity on the transfer across the interface. To calculate the velocity of an ion the coupled equations of motion are integrated starting with z ¼ z i and h ¼ 0 until the ion reaches the position z f or until a maximum simulation time s max has been reached. If the ion reaches z f in a time s < s max, the ion velocity becomes v ¼ðz f z i Þ=s, otherwise it is considered that s!1 and v! 0. This velocity is then averaged over a number n of ions as k rate ¼hvi ¼ D z f z E i ¼ z f z i X n s n l¼1 1 s l : ð8þ Fig. 2. Variation of transfer rate with diffusion coefficient ratio for different values of interfacial tension. Dotted lines are guidelines for the eye. interpolation formula (6) holds true. However, for typical values of c (for example, the water-nitrobenzene interfacial tension has a value of about 30 dyne/cm at room temperature) there is a significant deviation of the simulations from the analytical values in the case when D z is of the same order of magnitude or larger than D h. This effect is blown up in Fig. 3 which shows the slowing down of the transfer rate with the increase in the interfacial tension for a large value of D z =D h. This behaviour is due to the fact that, at high values of c, the interface position should always be close to h eq ðzþ in order to minimise the coupling potential barrier. Thus, even in the favourable case when thermal fluctuations facilitate the ion motion across the interface the ion 3. Results A systematic study on the variation of the ion transfer rate with respect to the value of the interfacial tension c and to the ratio D z =D h between the ion and interface diffusion coefficients has been performed. The parameters chosen for the simulations were g 1;2 ¼ 0:014 g/ (cm s), r i ¼ 1 A, h max ¼ z max ¼ 20 A, K ¼ 40 A. The results presented in Fig. 2 show that for unrealistically large values of c the simulation results match the analytical ones predicted by the KKU model (solid line) in the whole range of D z =D h values, showing that the Fig. 3. Ion transfer slowing down becomes significant only at moderate and large values of the interfacial tension: (a) fast motion of the interface; (b) fast motion of the ion.

5 C.G. Verdes et al. / Electrochemistry Communications 6 (2004) must wait for the slow interface to approach its equilibrium position corresponding to a new location of the ion. The asymptotic value of k rate =krate 0 obtained for high values of c coincides with the predictions of KKU theory. On the other hand, at low values of c, the diffusion term in the equation of motion becomes dominant and the ion no longer needs to wait for the interface to follow its movement because the thermal energy is large enough to overcome the barrier of the ion interface coupling (as clarified by the elevator analogy in the Introduction). At low values of D z, i.e., for abnormally large ions whose motion is much slower than that of the interface, the transfer rate is not affected by the ion interface coupling and the ion behaves as if it would travel in the bulk of the solution (cf. the tank analogy in the Introduction). A favourable move of the ion is immediately followed by the corresponding movement of the interface, reproducing h eq ðzþ, for high values of c. The motion of the ion is unaffected by the presence of the interface when c is low. These effects are visible in Fig. 4 showing the trajectories of the ion and interface deformation for two extreme values of the D z =D h ratio in the case of strong coupling. For the case of faster movement of the interface (Fig. 4(a)), the trajectory of the ion is unaffected by the presence of the interface while the interface closely follows the movement of the Fig. 4. Ion and interface movement in two limiting cases of D z =D h : (a) ion movement is not influenced by interface; (b) ion waits for favourable fluctuation of the interface position. ion. The interface is effectively being pushed by the ion with the rate given by the slower of the two motions, in this case that of the ion. Ion movements generated for the same sequence of random numbers, in one case considering the interaction with the fast moving interface and in the other one not taking the interaction into account (h eq ðzþ 0), would yield identical trajectories even for very large values of c. When the ion movement is faster (Fig. 4(b)), it is the ion that dully follows the interface to minimise the coupling potential. A movement of the interface to a new position h n is immediately followed by the ion moving to a new position z n such that h eq ðz n Þffih n.in this case it is the interface movement from its equilibrium position that is not affected by the coupling potential. Thus, in order to cross the interface, the ion must wait for a favourable fluctuation of the interface i.e., a fluctuation that will move the interface to h max and then back to its flat position while dragging the ion to z max and then pushing it away to z f. 4. Discussion Whereas the effects of D h and D z become pretty clear after our study, the effect of interface tension may, however, be not as simple as described above. The chemical potential, lðzþ, of the ion moving infinitely slowly across the interface, reveals the difference between the bulk solvation energies of the ion in two solvents and special interfacial effects, such as image forces and, possibly, a re-solvation barrier. Experimentally a bias voltage is applied to compensate the difference between the values of the bulk chemical potentials. However, this does not warrant that the resulting electrochemical potential will be constant across the interface. As discussed in [17,24], there could be a barrier, which would likely to be lower for lower interfacial tension since the profile of lðzþ will be less steep due to the softness of the interface. Thus, from the point of view of this effect, increasing interfacial tension will enhance the slowing down brought in by the energy barrier. Unfortunately, it is not possible to estimate the height of the resulting barrier and its interfacial tension dependence taking into account all the molecular details. We therefore cannot evaluate the strength of this effect. On the other hand, there could be an opposite effect of interfacial tension, associated with c-dependence of h eq ðzþ. Indeed the latter should decrease with the increase of c. This diminishes the coupling between h and z coordinates. At least at large jzj, h eq ðzþ is roughly /1=c,and thus the consequences of this effect will be stronger than the effect of the coefficient in Eq. (5). Formally this leads to the decrease of slowing down: no surface deformation, no surface polaron effect, no slowing down!

6 698 C.G. Verdes et al. / Electrochemistry Communications 6 (2004) However, we have to take into account that the halfwidth of the triangle in Fig. 1 may also diminish with the increase of c. According to Eq. (7) this may compensate the effect of the decrease of h max (the force that corresponds to the potential (5) is proportional to the derivative of h eq ðzþ). We know very little about it, and thus only the experiments or molecular dynamic computer simulations can truly reveal the effect of surface tension. Are the real experiments feasible? Can we vary surface tension keeping other parameters unchanged? It does not seem easy. The most efficient way to change surface tension could be via adding a surfactant, but would not the surfactant layer add some purely mechanical barrier for ion transfer? We thus see that there are still a number of gray areas in the picture of ion transfer across ITIES. 5. Conclusions The simulations performed show that the analytical formula (6) obtained in [24] for k rate is accurate in the limiting case of very large interfacial tension. However, when thermal energy becomes comparable with the ion interface coupling potential, the ion can cross the interface by overcoming the potential barrier rather than by moving simultaneously with the interface across the h eq ðzþ path; this leads to a smaller reduction of the ion transfer rate. The difference is noticeable when the ion diffusion coefficient is of the same order of magnitude or larger than that of the interface. For the case of high interface diffusion coefficient the movement of the ion is not affected by the coupling, and no reduction of the transfer rate is expected. The bottom line of these simulations is that they show that the ion interface coupling can be indeed accounted for the observed reduction in the ion transfer rate, but its effect can be dramatically different in different situations. Verification of these differences could help to distinguish the surface polaron mechanism from other suggested mechanisms of slowing down. It is not easy to perform such experiments but they are worth of trying. With reservations in mind raised in Section 4, assuming that the interfacial tension between two liquids could be experimentally varied (possibly by addition of surfactant molecules to the system without significantly affecting other properties of the interface), the ion interface coupling hypothesis could be tested by verifying the c-dependence of the transfer rate shown in Fig. 3. Another possibly verifiable behaviour predicted by the theory is the variation of the slowing down in the transfer rate with the bulk diffusion coefficient of the ion. This can be realized by using ions of different size at unchanged properties of the interface, which is easier than changing the interfacial tension. As our results show, it should be expected that the relative slowing down in the transfer rate for different ions across a given interface will decrease with the Stokes radii of the ion. Another possibility to change a diffusion coefficient is by varying a viscosity of one of the contacting solvents. Few works in this direction have been already performed [9,14,15,29] and demonstrated that the viscosity of the most viscous solvent had a direct effect on the rate constant. Our findings are in agreement with this conclusion. However, it should be noted that varying viscosity should influence both the diffusion coefficients of ions and the interface. Thus using this approach it is difficult to distinguish between the surface polaron effect discussed here and the hydrodynamic effects discussed above. In order to get a direct experimental proof of the suggested mechanism of the protrusion assisted ion transfer one has to measure the effect of ionic transfer on corrugations of ITIES. According to our model, the ion current should induce nonequilibrium amplification of the interface fluctuations, which leads to an increase of the interfacial area as compared the case of zero current. The excess area stored in the interface fluctuations can be measured quantitatively by the micropipet technique [30]. Thus a predicted correlation between the ionic current and the excess area can be verified. Acknowledgements Financial support of this work by the Leverhulme Trust (Grant NF/07 058/P) and Israel Science Foundation (Grant No. 573/00) is greatly appreciated. AAK acknowledges Royal Society Wolfson Merit Research Award. References [1] Z. Samec, T. Kakiuchi, in: E. Gersicher, C.W. Tobias (Eds.), Advances in Electrochemistry and Electrochemical Sciences, VCH, Weinheim, 1995, p [2] H.H. 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