Letter pubs.acs.org/JPCL
Kinetic Charging Inversion in Ionic Liquid Electric Double Layers Jian Jiang,†,§ Dapeng Cao,† De-en Jiang,‡ and Jianzhong Wu*,§ †
Department of Chemical Engineering, Beijing University of Chemical Technology, Beijing 100029, People’s Republic of China Chemical Sciences Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831-6201, United States § Departments of Chemical and Environmental Engineering and Mathematics, University of California, Riverside, California 92521, United States ‡
S Supporting Information *
ABSTRACT: The charging kinetics of electric double layers (EDLs) has a pivotal role in the performance of a wide variety of nanostructured devices. Despite the prevalent use of ionic liquids as the electrolyte, relatively little is known on the charging behavior from a microscopic perspective. Here, we study the charging kinetics of ionic liquid EDLs using a classical time-dependent density functional theory that captures the molecular excluded volume effects and electrostatic correlations. By examining variations of the ionic density profiles and the charging density in response to an electrode voltage, we find that at certain conditions, the electrode charge shows a rapid surge in its initial response, rises quickly to the maximum, and then slowly decays toward equilibrium. The electrode charge and voltage may have opposite signs when the cell width is commensurate with the layer-by-layer ionic distributions. This unusual charging behavior can be explained in terms of the oscillatory structure of ionic liquids near the electrodes. SECTION: Energy Conversion and Storage; Energy and Charge Transport charging behavior of a laminated structure filled with an ionic liquid.21 A nonmonotonic charging behavior in the charging process was observed when the humidity was fixed at certain values.21 Such unusual properties contradict the predictions of conventional EDL theories.22,23 Compared with molecular solutions, classical DFT is computationally efficient, thereby allowing for a systematic investigation of the parameters for relatively large systems.24−26 We have shown that DFT predicts the equilibrium properties of ionic liquids next to a surface or inside of a porous electrode, in good agreement with experimental data and simulation results.16,27−29 In this work, we examine the charging behavior of ionic liquid EDLs by using time-dependent density functional theory (TDDFT).30,31 As its equilibrium counterpart, TDDFT accounts for the ionic steric effects and electrostatic correlations that are neglected in the Poisson− Nernst−Planck (PNP) equations, a conventional microscopic theory of EDL charging.22 The non-mean-field effects are important for strongly correlated electrolyte systems and are often responsible for novel electrokinetic phenomena unique for ionic liquids.8 To capture the essential features of EDL charging at a constant voltage, consider a simple model electrochemical cell shown schematically in Figure 1. The system consists of an ionic liquid confined between two parallel planar electrodes at
T
he charging kinetics of electric double layers (EDLs) is closely related to the performance of a wide variety of nanostructured devices including supercapacitors,1−3 electroactuators,4,5 and electrolyte-gated transistors.6,7 Room-temperature ionic liquids (RTILs) are often used in these new applications because they have the advantages of larger electrochemical windows (and hence higher energy density), greater chemical selectivity, and higher thermal and chemical stabilities.8,9 Previous experimental and theoretical investigations show that the EDL structure of ionic liquids exhibits alternating layers of cations and anions distributed near a charged surface with the peak densities decaying to the corresponding bulk values at a length scale up to 10 times the ionic diameters.10−13 These alternating layers are responsible for a number of unexpected electrochemical phenomena at small scales. For example, Largeot et al. observed the “anomalous” increase of capacitance when the pore size of an electrode approaches the ionic dimensions.14 A similar conclusion was reached with molecular dynamics (MD) simulations and classical density functional theory (DFT) calculations.15−18 The strong electrostatic interactions and confinement effect make ionic liquid EDLs unique in terms of not only the equilibrium properties but also the charging kinetics.19 By using impedance and cyclic voltammetry, Kurig et al. analyzed EDL formation and adsorption kinetics of various ionic liquids in porous electrodes.20 They found that both the adsorption capacity and the charging kinetics depend strongly on the electrode potential as well as the chemical composition and polarity of the ionic liquids.20 Must et al. studied the © 2014 American Chemical Society
Received: May 13, 2014 Accepted: June 10, 2014 Published: June 10, 2014 2195
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In eq 2, Λi denotes the thermal wavelength, which is immaterial to the TDDFT calculations; e is the unit charge; ψ(r,t) represents the local mean electrical potential; and μex i (r,t) is the local excess chemical potential arising from ionic excluded volume effects and electrostatic correlations. The details of ψ(r,t) and μex i (r,t) are presented in the Supporting Information. The last term on the right side of eq 2 is ignored in the PNP equation. For the charging process considered in this work, the density profiles vary only in the direction perpendicular to the electrode surface, that is, ρi(r,t) = ρi(r,t). The mathematical details for numerical integration of the TDDFT equation can be found in a previous publication.30 Unlike molecular simulations, the DFT calculations are concerned only with the density profiles of ionic species as a continuous function of the position. In other words, there is no need to specify the number of particles or periodic boundary conditions. For equilibrium systems, it has been well documented that the nonmean-field contributions can be accurately described with the classical DFT.27,29 Figure 2 shows typical results from the TDDFT calculations for the evolution of the ionic density profiles during constant-
Figure 1. A model electrochemical cell for studying the charging behavior of ionic liquid EDLs. The ionic liquid, EMI-TFSI, is represented by a primitive model where both the cation and the anion are approximated by monovalent charged hard spheres. For all cartoons shown in this work, the particles illustrate the layering distributions of cations and anions near the electrodes. To avoid overcrowding, we display only a few representative particles; the majority of ions in the ionic liquid are shown as a dark background.
separation L. We use the restricted primitive model of electrolytes to represent the ionic liquid, namely, cations and anions are charged hard spheres of the same size and equal but opposite valence. The model parameters are selected such that the ion size and charge approximately match those corresponding to 1-ethyl-3-methylimidazolium bis(trifluoromethanesulfonyl) imide (EMI-TFSI), a RTIL commonly used in electrochemical devices. The same model was used before to study the effect of pore size on the capacitance of ionic liquid supercapacitors.16 Specifically, the ionic diameter, σi = 05 nm, is close to the average size of EMI and TFSI ions,14 and valence Zi = ±1 corresponds to the ion charges. At 298 K and 1 bar, the molar volume and the bulk diffusion coefficient of EMI-TFSI are v = 259 cm3/mol and D = 4.29 × 10−11 m2/S, respectively.14 The interaction between ions is described in terms of the Coulomb potential with the free-space dielectric permittivity. According to our coarse-grained model, the contact energy between an anion and cation pair is about 10kBT. Because of the strong attraction between cations and anions (in comparison with 1−2kBT in an aqueous solution), an ionic liquid may be considered as not fully dissociated, similar to a weak electrolyte.32−34 EDL formation in response to an instantaneous increase of the electrode voltage is described by TDDFT, which can be intuitively understood as a modified diffusion equation for the ionic motion in the presence of an electric field31 ∂ρi (r, t ) ∂t
=
D ∇·[ρi (r, t )∇φi(r, t )] kBT
Figure 2. Evolution of the local cation density inside of the electrochemical cell during constant-voltage charging. Here, the chemical potentials of ionic species are the same as those corresponding to a bulk ionic liquid of reduced average ionic concentration ρ*bulk = ρ−σ3 = ρ+σ3 = 0.29, and the separation between two electrodes is L/σ = 12. The electrodes are neutral at t = 0 and have a constant voltage ψ0 = ±0.5 V during the charging process. The relaxation time is given in units of τD = σ2/D, where D represents ion diffusivity. The inset shows the ionic density profile at t = 0.
(1)
where kB stands for the Boltzmann constant, T is the absolute temperature, and φi(r,t) is the local electrochemical potential of ion i. In our DFT calculations, the diffusivity is fixed at D = 4.29 × 10−11 m2/S, which corresponds to that of the bulk ionic liquid at ambient conditions. This parameter reflects the selfdiffusivity of each ion in the bulk liquid and is independent of the pore size and the electrode voltage. Within the restricted primitive model, the electrochemical potential includes an ideal-solution contribution and additional terms due to the steric effects and electrostatic interactions
voltage charging. Throughout this work, the relaxation time is given in units of τD = σ2/D, which is 5.83 ns for the parameters corresponding to EMI-TFSI. Because of the symmetry in the model parameters, the cation and anion distributions are symmetric, ρ−(z,t) = ρ+(L − z,t), where L is the separation between the electrodes. Without the electrode voltage, the local electric potential is everywhere zero. However, the overall density exhibits a certain degree of inhomogeneity due to ionic excluded volume effects and electrostatic correlations. Such inhomogeneity is not captured by the Poisson−Boltzmann
φi(r, t ) = kBT ln[ρi (r, t )Λi3] + eZiψ (r, t ) + μiex (r, t ) (2) 2196
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showed an oscillating variation with respect to the pore size;16 Atkinet et al. studied the structure and dynamic properties of ionic liquid confined between two electrode materials (Au(111)) and found that the surface force between the electrodes exhibits an oscillatory behavior.36 Figure 4 shows the time dependence of the surface charge density at a given charging potential. In contrast to the EC
(PB) equation, the equilibrium counterpart of the PNP equation. Application of the electrode voltage leads to a nonuniform local electrical potential and accumulation of counterions at the electrode surfaces. In comparison with aqueous electrolytes, a distinctive feature of ionic liquids is the formation of alternating layers of cations and anions near a strongly charged surface. The layer-by-layer structure arises from the enhanced excluded volume effect and cannot be captured by the PB equation.29 From the time-dependent ionic density distributions, we can calculate the effective surface charge density for each electrode Q (t ) = −
∫0
L /2
ρc (z , t ) dz
(3)
where ρc(z,t) = e∑i Ziρi(z,t) stands for the local net charge density. Because of the symmetry, the two electrodes have the same effective surface charge density but with opposite signs. In eq 3, we conjecture that the electrode charge is always balanced by the net charge of the EDL. The hypothesis hinges on the half-cell electrostatic neutrality, which is exact only for macroscopic systems or those containing symmetric electrolytes, as discussed in this work. Conventionally, the EDL charging is described with the equivalent circuit (EC) model,22 which predicts a monotonic increase of the surface charge density according to Q (t ) = Q 0(1 − e−t / τ )
(4)
Figure 4. Evolution of surface charge density with time for different cell widths. All parameters are the same as those used in Figure 3. For the systems considered in this work, τD = 5.83 ns.
where τ is the response time reflecting the rate of charging and Q0 is the surface charge density at equilibrium. In our previous work,30 we found that TDDFT predicts the charging kinetics of EDL with aqueous electrolytes in reasonable agreement with the EC model. The charging behavior of ionic liquid EDL is distinctively different from that of aqueous electrolytes. Consider first the variation of the equilibrium surface charged density versus the cell width. Figure 3 shows that the effective surface charge
model, DFT predicts three scenarios of charging behavior for ionic liquid EDLs. We call the charging process normal if it agrees with the predictions of the EC model, as shown in the cases of cell width L/σ = 4 or 7. We have found previously that the aqueous electrolyte follows the normal charging behavior for various cell widths examined.30 However, Figure 4 shows that the ionic liquid electrolyte is distinctly different and displays nonmonotonic charging for certain cell widths, for example, in the case of L/σ = 6. Here, one can see that the surface charge rises quickly with time to a maximum and then decays monotonically to the equilibrium value. Most interestingly, as in the case of L/σ = 3, the nonmonotonic charging leads to an effective surface charge density with a sign opposite to the electrode voltage. For both cases, the decline of the electrode charge during the charging process is referred to as kinetic charging inversion. While charge inversion has been well-known for equilibrium systems,37 to our knowledge, the nonintuitive kinetic behavior has not been reported before. In general, the charging curve depends on both the voltage and the separation between the electrodes. While the former affects primarily the amplitude of the surface charge density, a change in the electrode separation may result in the variation of both the amplitude and the shape of the charging curve. How can we tell which type of charging curve will happen for a specific cell width? By analyzing a large amount of charging data, we find that the main factor dictating the type of charging (normal versus nonmonotonic versus charge inversion) is the asymptotic limit of the effective surface charge density at the infinite separation (shown by the blue line in Figure 3). When the equilibrium charge density for a specific cell width is above the asymptotic value, the charging process shows a normal behavior (top shaded area in Figure 3); below that value, the
Figure 3. Equilibrium surface charge density versus the cell width predicted by DFT. All parameters are the same as those used in Figure 2 except that the results are given for electrochemical systems of different electrode separations at fixed electrode voltages of 0.3 V.
density oscillates versus the cell width with the peak values decaying to the asymptotic limit at large cell width (almost 35 times the ionic diameter). The oscillatory behavior corroborates recent investigations on surface forces and supercapacitors for ionic liquid systems.35 For example, Jiang et al. reported that the capacitance of an ionic liquid in a charged nanopore 2197
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Figure 5a, the initial response of the electrode voltage mainly occurs in the Stern layer. Further charging leads to the formation of the layered distributions of cations and anions. The key difference is that, in this case, the L layer enriches coions (highlighted by the black arrow in Figure 5c), which has a net charge opposite to that of the Stern layer. As a result, the effective surface potential shows a maximum upon the formation of the Stern layer, and a further charging, dominated by the L layer, leads to the decrease of the effective surface charge density. At L/σ = 3, the surface charge density Q(t) shows a rapid increase in its initial response, quickly rises to the maximum, and then decays monotonically to a negative value (see Figure 4). Figure 5e and f shows the corresponding local cation and charge densities, respectively. After formation of the Stern layer, the charging process is dominated by adsorption of co-ions and depletion of counterions in the second layer. As a result, the effective surface charge density declines upon further charging. Charge inversion occurs when the Stern layer is overcompensated by the co-ions. While the nonintuitive phenomenon is often attributed to strong electrostatic correlations, here the charge inversion is also related to the geometric constraint that prohibits the formation of the multilayer structure. In other words, the separation between the two electrodes is too narrow to accommodate any more layers of counterions. In summary, we have studied the charging process of ionic liquid EDLs by using the TDDFT. With a relatively simple model for the electrochemical cell, TDDFT allows us to examine the microscopic details of EDL charging by exploring a broad range of the parameter space at large time scales. The electrokinetic behavior of ionic liquid EDL charging is distinctively different from that predicted by the EC model, and the novel phenomena cannot be captured by conventional microscopic theories. At the small scale, the equilibrium surface charge density exhibits strong oscillation versus the separation between electrodes. The confinement effect is similar to that recently revealed in the dependence of the ionic liquid capacitance on the pore size and ionic-liquid-mediated surface forces. While the EC model predicts a monotonic increase of the surface charge upon application of an electrode voltage, the results from TDDFT show three types of charging behavior depending on the cell width and voltage. The normal charging occurs only when the equilibrium surface charge density has a sign the same as that of a single surface (i.e., at the asymptotic limit when the two electrodes are far apart); kinetic charging inversion takes place when the equilibrium charge is below or opposite of the asymptotic charge density. It is worthwhile to note that the kinetic charging inversion is different from charge inversion occurring in equilibrium systems. In the latter case, it is referred to the appearance of a local charge density in EDLs with the sign opposite to that of the charged surface. By contrast, kinetic charging inversion is referred to the decline of the surface charge and electrical energy during the charging process. The counterintuitive charging kinetics in ionic liquids is closely affiliated with the local segregation of counterions and co-ions near the electrode surface, leading to formation of layer-by-layer ionic densities. For the symmetric system considered in this work, the shape of the charging curve is dictated by the commensurate distributions of cations and anions within the half cell. The novel phenomena discussed in this Letter demonstrate the rich electrokinetic behavior of ionic liquid EDLs. We believe that similar results should be observable in real materials, provided
charging process is nonmonotonic; and charge inversion happens when the equilibrium charge density is negative. We may illustrate mechanisms underlying the different charging kinetics with the variations of the local ionic and charge densities in the half cell, that is, from the anode to the center of the cell. Consider three representative cases with cell widths L/σ = 7, 6, and 3 corresponding to normal charging, nonmonotonic charging, and charge inversion, respectively. Figure 5a and b shows the results for cell width L/σ = 7. As
Figure 5. Evolutions of the local cation density (a, c, and e) and the local net charge density (b, d, and f) during the charging process. The cell width for (a) and (b) is L/σ = 7; that for (c) and (d) is L/σ = 6; and for (e) and (f), it is L/σ = 3. All parameters are the same as those used in Figure 3. The inset cartoons show ion distributions near the anode.
expected, the early stage of charging is mainly related to the formation of the Stern layer, which is characterized by a rapid increase of the surface charge density and accumulation of counterions at the electrode surface. At this stage, the charging behavior is not much different from that predicted by the traditional EDL models. However, further charging leads to the formation of oscillatory distributions of counterions and coions. Figure 5b shows that the formation of layer-by-layer ionic structure is accompanied by the oscillation of the local charge density. Interestingly, electrical charging affects mainly the amplitudes of the ionic density profiles, suggesting that the layer formation arises from the local segregation of cations and anions without significant variation of the overall charge. As a result, the integrated charge, as reflected by the effective surface charge density Q(t), is primarily determined by the last layer of ions, here designated as the L layer, near the symmetric surface (highlighted by the black arrow in Figure 5a). In this case, the L layer is dominated by counterions, similar to the Stern layer. Accordingly, the charge kinetics appears normal because it corresponds to that for the formation of the L layer. Figure 5c and d shows the evolutions of the local cation and charge densities when the cell width is L/σ = 6. As shown in 2198
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that one is able to control the separation between electrodes within a few ion layers (
