Article pubs.acs.org/Langmuir
Electric Double Layer at the Interface of Ionic Liquid−Dielectric Liquid under Electric Field D. W. Lee, D. J. Im, and I. S. Kang* Department of Chemical Engineering, Pohang University of Science and Technology, San 31, Hyoja-dong, Pohang 790-784, South Korea ABSTRACT: The structure of the electric double layer (EDL) is analyzed in order to understand the electromechanical behavior of the interface of ionic liquiddielectric liquid. The modified Poisson−Boltzmann equation proposed by Bazant et al. is solved to see the crowding and the overscreening effects that are the characteristics of an ionic liquid (Bazant, M. Z.; Storey, B. D.; Kornyshev, A. A. Double layer in ionic liquids: Overscreening versus crowding. Phys. Rev. Lett. 2011, 106, 046102.). From the simple one-dimensional (1-D) analysis, it is found that the changes of the composition and the material properties in the EDL are negligible except under some extreme conditions such as strong electric field overO(108) V/m. From the electromechanical view points, an ionic liquid behaves like a pure conductor at the interface with a dielectric liquid. Based on these findings, three specific application problems are considered. In the first, a new method is suggested for measuring the interfacial tension of an ionic liquid-dielectric liquid system. The deformation of a charged ionic liquid droplet translating between two electrodes is used for this measurement. The second is for the Taylor cone problem, which includes an extreme electric field condition near the tip. The size of the critical region, where the EDL effect should be considered, is estimated by using the 1-D analysis result. Numerical computation is also performed to see the profiles of electric potential and the electric stress along the interface of the Taylor cone. Lastly, the electrowetting problem of the ionic liquid is considered. The discrepancies in the results of previous workers are interpreted by using the results of the present work. It is shown that all the results might be consistent if the leaking of the dielectric layer and/or the adsorption of ions is considered.
1. INTRODUCTION Ionic liquids are salts of liquid state at room temperature. Ionic liquids have unique chemical, electrochemical, and physical properties.2−4 They have good thermal stability, negligible vapor pressure, nonflammability, and a large electrochemical window. Ionic liquids are applicable as solvents in fine chemical industry such as pharmaceutical synthesis,5 and they can also be used as a nonsolvent electrolyte in batteries, solar cells, or fuel cells.3,6 Because of their unique material properties, ionic liquids have been applied also in microfluidic systems.7−11 In many microfluidic systems or conventional processes, ionic liquid phase contacts with dielectric liquid phase (or solid) and the interface is subject to an applied electric field. Examples include electrophoretic actuation of a charged ionic liquid droplet in a dielectric liquid,12,13 jet formation from a Taylor cone of ionic liquid,14,15 and electrowetting of a ionic liquid droplet on dielectric layer.8,11 The ionic liquid−dielectric liquid interface deforms according to the distributions of the electrical stress and the osmotic pressure. In order to understand the electromechanical behaviors, it is essential to analyze the characteristics of the electric double layer (EDL) formed by the ions attracted toward the interface. It is known that an ionic liquid exhibits the crowding effects (steric effects) and the overscreening effect.1 Kornyshev derived a model equation with the crowding effect in a statistical mechanics way based on the concept of Fermi distribution.16 Almost at the same moment, Kilic et al. applied the same equation for the analysis of the concentrated electrolytes.17,18 © 2013 American Chemical Society
However, Kornyshev’s equation turns out to be the same as the modified Poisson−Boltzmann equation (MPB equation) proposed by Bickerman19 earlier for the study of steric effects of ions in electrolyte. (Thus, Kornyshev’s model is called Bickerman’s MPB equation in this work.) The full history of the modified Poisson−Boltzmann equations considering the steric effects has been reviewed by Bazant et al.20 Recently, Bazant et al.1 extended Bickerman’s MPB equation to include the electrostatic correlation effect and the overscreening effect. They proposed a modified Poisson−Boltzmann equation with some analyses. (This equation is called Bazant’s MPB equation in this work. See Bazant et al.1 and Storey and Bazant21) Both Bickerman’s MPB equation and Bazant’s MPB equation have been applied successfully for the studies of the structure and the capacitance of the EDL near the electrode. Furthermore, the validity of Bazant’s MPB equation has been confirmed through molecular dynamic simulations.22 Many other kinds of MPB equations have been proposed to describe different physical effects.20 Outhwaite et al. developed a MPB equation based on self-consistent correlation functions.23−25 Liu et al. have considered correlations and crowding effects in a nanochannel based on more complex statistical models.26 The density functional theory has also been applied to describe the EDL for high density electrolyte and Received: October 14, 2012 Revised: January 12, 2013 Published: January 18, 2013 1875
dx.doi.org/10.1021/la3040775 | Langmuir 2013, 29, 1875−1884
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shows very good agreement with a molecular dynamic simulation or a Monte Carlo simulation.27−31 However, the density functional theory requires solving nonlinear integral equations even for a flat 1-D problem. In the present work, we want to adopt Bazant’s MPB equation for the study of the EDL structure at the interface of an ionic liquid−dielectric liquid system under applied electric field. After studying the general characteristics of the EDL at the ionic liquid−dielectric liquid interface, we want to apply the results to three specific examples. The first is about the measurement of interfacial tension of the ionic liquid−dielectric liquid system. The deformation of a charged ionic liquid droplet moving between two parallel electrodes is used for estimation of the interfacial tension. This experimental setup has already been used in previous works for estimation of the charge acquired at the electrode.12,13 The second problem is about the Taylor cone. The size of the critical region near the tip, where the EDL effect should be considered, is determined. The detailed EDL structure in the region is also studied numerically. The third problem is about electrowetting of an ionic liquid droplet. The discrepancies in the results of previous workers are interpreted by using the results of the present work. It is shown that all the results might be consistent if the leaking of dielectric layer and/or the adsorbed ions are considered.
parameter γ in eq 1 represents the steric effect, and lc is the correlation length that represents the overscreening effect. In this work, for simpler analytical treatment, we pay major attention to the 1-D problems with only the crowding effect. In this case, eq 1 is reduced to the model proposed by Kornyshev.16 Since the linearized version of eq 1 allows analytical solution, we consider also the overscreening effect for some cases of very low potential on the interface. When we consider only the crowding effect, the governing equations for the 1-D problem are given as
2. THEORETICAL DEVELOPMENTS
The solution is easily obtained as follows. For the region of x > 0, by integrating eq 4 with ϕ → 0, ϕ′ → 0 as x → ∞, we have
ϕ″ = 0,
−∞
