NANO LETTERS
One-Dimensional Capacitance Behavior of Electrolytes in a Nanopore
2003 Vol. 3, No. 2 217-221
Yuk Wai Tang,† Kwong Yu Chan,*,† and Istva´n Szalai‡ Department of Chemistry, The UniVersity of Hong Kong, Pokfulam Road, Hong Kong SAR, China, and Department of Physics, UniVersity of Veszpre´ m, H-8201 Veszpre´ m, P.O. Box 158, Hungary Received October 29, 2002; Revised Manuscript Received December 1, 2002
ABSTRACT One-dimensional capacitor behavior is reported for electrolytes confined in a nanopore. The electrical behavior of a 0.1 M electrolyte confined in a nanopore is studied by equilibrium and nonequilibrium molecular dynamics simulations. For an electrolyte in a nanopore with a radius of 4.5 Å, capacitance character is exhibited at low frequencies. A unique feature of the maximum conductivity at a resonance frequency corresponds to RLC equivalent circuit behavior.
Recently, several theoretical studies have been reported for the passage of ions through simple nanopores1-3 or modeled channels in biological membranes with specific concerns of selectivity4,5 and steady-state current response.6 For a biological channel, the unsteady-state dynamic response is probably more important than the steady-state ionic passage. The AC impedance and transient electrical or electrochemical techniques are routinely applied to the analysis of current response in a single ion channel.7,8 These analyses often yield a wealth of information about the dynamic and relaxation properties of channels. For inorganic membranes and porous electrodes, the AC impedance and transient electrical techniques have been well established. Theoretical interpretations of AC impedance or admittance of electrolytes in nanoporous structures have often been based on the concepts of equivalent electrical circuits. These analyses have not been directly related to nanoscopic or molecular contributions. It will be useful to establish a theoretical analysis of a basic model of electrolytes in a nanopore and to relate its electrical circuit analogues. The frequency dependence of the conductivity of bulk electrolytes has been studied by integral equation theory9 and nonequilibrium molecular dynamics (NEMD) simulations.10 Different behavior is expected for electrolytes confined in nanopores. By NEMD, Tang et al.2,3 have studied the steady-state conductivity of the restricted primitive model (RPM) and solvent primitive model (SPM) electrolytes in nanopores. The transient frequency-dependent conductivity of the RPM in a nanopore was studied by NEMD simulations.11 It is found that the current response of ions to the * Corresponding author. E-mail:
[email protected]. Fax: (852) 28571586. † The University of Hong Kong. ‡ University of Veszpre ´ m. 10.1021/nl025868x CCC: $25.00 Published on Web 12/20/2002
© 2003 American Chemical Society
external field is analogous to that of an LRC circuit. These simulations were performed using a restricted primitive model (RPM). Actual solvent particles are not present in the model. The packing fractions of particles of the RPM model are unrealistically low, and apparently the ion-solvent interactions, which contribute an important factor to the motion of ions, are omitted in the model. This letter reports new results with a more realistic electrolyte model with explicit molecular-solvent particles. The distinct features of electrical response are reported for the electrolytes confined in small nanopores. In our NEMD simulations, we imposed a sinusoidal electric field along the axial direction of an infinitely long nanopore containing ion and solvent particles. The current response was monitored by the changes in the positions of the ions with time. This method was described previously11 for the primitive model electrolytes. Here, the solvent molecules are treated explicitly as neutral spheres without any multipoles using the solvent primitive model (SPM) described in ref 3. The polar nature of the solvent is represented collectively and implicitly by a continuum background with a dielectric constant r. The electrostatic interaction between an ion pair is therefore given by the Coulomb pair potential damped by the dielectric constant. The short-range ion-ion, ion-solvent, and solventsolvent core interactions are described by the truncated Weeks-Chandler-Anderson-type shifted Lennard-Jones (LJ) model.12 The parameters of the cation, anion, and solvent are shown in Table 1. The electrolyte is assumed to be symmetric for easier interpretation, and the masses of the cation and anion are the same as that of sodium whereas the solvent has the mass of water. The core interaction is zero beyond the diameter of interaction. The unlike pair energy
Table 1. Parameters of Particles and the Wall Potential in the Solvent Primitive Model species
q/e
(/k)/K
m/au
σ/Å
cation anion solvent wall
+1 -1 0 0
316.35 316.35 78.2 316.35
23 23 18
3.00 3.00 3.00 3.00
parameter Rβ can be obtained by using the LorentzBerthelot combining rules. The /k parameter for the ions is chosen with reference to previous works of the restricted primitive model.10,11,13,14 These values of /k are much higher than that of the water solvent and are unrealistic especially for common cations. Since the Lennard-Jones interactions are truncated and shifted, only the “softness” of the ion cores is affected and is unlikely to have any effect on the results of this study. Large values of /k also implied polarizability, which is not considered in the present model. Since the main interionic interaction is electrostatic, the induced dipole interactions are assumed to be negligible. The univalent electrolyte we simulated has a concentration of 0.1 M and a temperature of T ) 298.15 K, and the dielectric constant of the solvent, r, is 78.3. The ion and solvent particles are confined in a cylindrical cell of radius R and length H. The ion-wall and solventwall interactions are described by a soft-core shifted and truncated LJ-type potential expressed analytically.15 The simulations were carried out with three different nanopores with radii of 4.5, 9, and 15 Å. These pore radii correspond to the location of the centers of the atoms on the pore wall if they are represented individually whereas the wall potential representation assumes that the wall atoms are smeared out. The physically accessible radii of the pores are about 0.9 Å smaller than the actual radii as determined from the density profiles and discussed in ref 3. Quirke and co-workers16,17 made a distinction between the physical radius of a pore and the chemical radius, which varies with the molecular probe. Though we have three species in the model, we assume that the physically accessible pore radius is the same for the three species and that it is 0.9 Å smaller than the pore radius. The radius of the narrowest pore is 4.5 Å, and two particles can just pass each other along this pore. The length of the pore varies such that each solution contains 24 ions and the pores are filled with uncharged spheres acting as solvent particles at a packing fraction of 0.3. This is equivalent to a density of 0.64 g cm-3 for the solution. Calculated with the above definition of a physically accessible radius, this density of solvent particles is kept the same for all pore sizes. Since a truncated LJ potential of interaction was used between the wall and the solvent particles, the wall is hydrophobic or nonwetting. Lynden-Bell and Rasaiah1 also assumed a hydrophobic wall. They argued that because of the symmetric force field inside a cylinder there is no difference in the MD simulation except for the chemical potential required to fill the pore to high density. In our case, the choice of a 0.64 g cm-3 water density value may require an enormous external chemical potential or pressure. But the motion of the ions and solvent within the pore will not be affected. A hydro218
Table 2. Setup of the Solvent Primitive Model radius of pore (Å)
length of pore (Å)
no. of ions
no. of solvent molecules
concentration (M)
ηs
4.5 9.0 15.0
4893 966 318
24 24 24
4228 4225 4213
0.1 0.1 0.1
0.3 0.3 0.3
philic wall may be more realistic but may complicate the issue with the possibility of the specific adsorption of ions. Table 2 shows the details of setups for different runs. As discussed earlier,2,3 for a sufficiently long pore where the length is 10 times the radius, the simple cutoff in the axial direction is much larger than 5 times the radius of ions, and the errors in the long-range electrostatic forces are negligible. In the NEMD simulation, a frequency-dependent alternating electric field E(ω) is applied along the axial direction of the pore. The current density of ions, J(ω), is calculated in the simulations from the velocities of all the ions in the volume. The boldface symbols denote the complex number representations. The electric field applied along the axial direction has an instantaneous value at time t, given by Ez ) Ez0 cos(ωt) where Ez0 is the amplitude of the electric field. The corresponding current density, Jz, along the same direction has the form Jz ) Jz0 cos(ωt + φ) where Jz0 is the amplitude of the current density and φ is the phase shift between the current density response and the applied electric field. The amplitudes of the sinusoidal reduced electric-field strength, E/z0, range from 0.1 to 0.6. The electric field in real units is related to the reduced electric field by E/z0 ) Ez0(de/ ) with corresponding values from 9.1 × 106 to 54.5 × 106 V m-1. The reduced frequency of the electric field, ω*, is defined as ω* ) ω(md2/)1/2 and varies from 0.1 to 28. The corresponding real-unit values range from 0.11 × 1012 to 31.6 × 1012 Hz. The period of each cycle of the electric field ranges from 0.2 to 50 ps. Figure 1 shows a typical current response in the NEMD simulations with an AC field of E/z0 ) 0.4 (in real units 3.64 × 107 V m-1) applied to a nanopore with a radius of 9 Å at ω* ) 1.12. To obtain a statistical representation of conductivity, the NEMD runs were at least 4 ns long for each frequency, each applied field strength, and each pore size. For lower-frequency simulations, longer simulations were performed to obtain equivalent statistics with sufficient cycles. From the equation J(ω) ) σ(ω) E(ω), the complex conductivity σ(ω) can be obtained, with σ′(ω) and σ′′(ω) denoting the real and imaginary parts, respectively. Six different external field strengths were used for each frequency and pore size, and the results are extrapolated to get the zerofield result. The temperature in NEMD simulations is maintained at a constant value by a Gaussian thermostat as described in refs 2 and 3. The frequency-dependent conductivities of various pore sizes are also obtained by equilibrium molecular dynamics (EMD) simulations in the absence of any electric field. The conductivity is calculated by the Fourier transform of an electrical current autocorrelation function9 as described by Nano Lett., Vol. 3, No. 2, 2003
Figure 1. Current response of ions in an SPM electrolyte to an alternating field with size R ) 9 Å. The left and right scales are reduced units of the electric field and current density used in the simulation, respectively, with the relations E ) (9.09 × 107 V m-1)E* and J ) (2.01 × 1012 A m-2)J*. The top and bottom scales are real and reduced units of time, respectively. The amplitude of the axial electric field is Ez0 ) 3.64 × 107 V m-1, and ω ) 1.27 × 1012 Hz.
Figure 2. Autocorrelation functions of current density for the three different pore sizes: (s) R ) 4.5 Å, (- -) R ) 9 Å, and (‚ ‚) R ) 15 Å. The graphs are normalized to their respective zero-frequency current values.
σ(ω) )
1 kT
∫0∞〈Jz(0) Jz(t)〉eiωt dt
(1)
where k is the Boltzmann constant and T is the temperature. The autocorrelation functions of current density are obtained in EMD simulations over a time period of 50 ns. The typical current-current autocorrelations are shown in Figure 2 for the three different pore sizes. The Fourier transform of the current-current autocorrelation functions will yield the complex conductivity as given by eq 1. The autocorrelation curve for the narrowest pore crosses the horizontal axis several times, and this is evidence of the high correlation between ions due to the high confinement. The real parts of the electrical conductivities for the three nanopores are plotted in Figure 3 as a function of frequency. The data points are from NEMD simulation results at different field strengths with extrapolation to zero field Nano Lett., Vol. 3, No. 2, 2003
Figure 3. Frequency dependence of the electric conductivity of ions in models of different pore sizes. The solid points are the results of NEMD simulations: (b) pore with R ) 4.5 Å, (2) pore with R ) 9 Å, and (9) pore with R ) 15 Å. The lines are the results of Fourier transforms of EMD simulations: (s) pore with R ) 4.5 Å, (- -) pore with R ) 9 Å, and (‚ ‚) pore with R ) 15 Å. The open symbols are the results of DC-NEMD simulations.
strength, and the error bars are the standard deviations in the linear fitting of the data points of different field strengths. In the same Figure, the EMD conductivities obtained through the autocorrelation functions of the current are shown as lines. A linear interpolation was performed for the lowfrequency EMD data that show larger scattering. The EMD and NEMD results of conductivity are in good agreement in general for all pore sizes. This trend agrees with the earlier results of RPM11 except that the conductivity decay frequency is about 2 orders of magnitude higher in this model with explicit solvent particles. Since the curves of all the pore sizes merge at high frequency, it can be argued that the highfrequency behavior is due to the relaxation of the short-range ion-solvent interactions rather than a confinement effect. This conductivity decay at high frequency is similarly observed in the bulk electrolyte. For low frequencies, the conductivity decreases significantly with smaller pore sizes, indicating a strong confinement effect. The zero-frequency conductivity can be extrapolated from the NEMD simulation data at low frequencies. The lowest-frequency simulation that is practically feasible, however, is still in the gigahertz region. The agreement between the EMD zero-frequency conductivity and the NEMD AC data is reasonable, and the general trend of conductivity variation is small at the low-frequency limit. The case of the smallest pore studied with a radius of R ) 4.5 Å shows unique behavior compared to the results of the other pores. The conductivity falls off less rapidly to zero, and the fall off occurs at a higher frequency. In addition, the conductivity shows a maximum at a frequency around 10 × 1012 Hz. The physical situation is that oppositely charged ions traveling in opposite directions have difficulty passing each other. The complete electrical behavior can be better understood by analyzing the phase behavior and the complex conductivity. The phase difference between the current density of ions 219
Figure 4. Phase lag between the current density and the applied electric field as a function of frequency.
and the applied electric field is shown in Figure 4. The error bars are the standard deviations of NEMD results from field strengths, and the lines are from EMD results. For all pore sizes, the phase difference increases with frequency, but the increase is slower for the narrowest pore. A unique feature of the case of the smallest pore is the presence of a negative phase shift (phase lead) at frequencies below 6 × 1012 Hz. The magnitudes of the phase shift are generally smaller than 0.2 rad. A negative phase shift of 0.74 rad is observed at ω ) 2.5 × 1011 Hz and E/z0 ) 0.6. In the previous RPM simulation results,11 the phase shift at low frequencies can be as low as -1 rad. The presence of a negative phase shift at low frequencies suggests capacitance character for the SPM electrolyte in a nanopore of R ) 4.5 Å. However, the common approach of all pore sizes to a positive phase shift of π/2 and to zero conductivity at high frequencies suggests the behavior of an inductor. The electrical behavior of the SPM electrolyte is analogous to that of an RL circuit in larger pores but is analogous to that of an LRC circuit in the severely confined environment of the smallest nanopore. At the resonance frequency of the circuit, the capacitance effect cancels the inductance effect, and the ions attain the highest conductivity. Although the theory for double-layer capacitance in interfaces and nominal areas of membranes is well established, it is uncommon to expect capacitor behavior in a 1D structure. At the smallest nanopore, opposite-traveling ions of opposite charge cannot get past each other easily. The close proximity of ions, forced by the high degree of confinement, could also induce ion pairing. Such confined effects create possible accumulations of ions and nanoscopic capacitors and hence the electrical behavior. At the resonance frequency, the ions in the confined pore can migrate under the external field to the largest extent with minimal collisions with other ions along the pore. The equivalent electric circuit analyses are usually performed with the Cole-Cole plot where the imaginary part of the conductivity is plotted against the real part. The conductivities of the NEMD and EMD results are shown in Figure 5 in this format. The feature of a semicircle in the 220
Figure 5. Cole-Cole plots of the admittance at different frequencies in pores of various sizes. The open symbols are the corresponding DC-NEMD results.
first quadrant has been reported for the bulk electrolyte.9 This is characteristic of an RL circuit. We observe similar semicircles in the first quadrant for the larger pores of R ) 9 and 15 Å in Figure 5. The radius of the semicircle, however, decreases with pore size. For the narrowest pore of R ) 4.5 Å, an additional semicircle in the fourth quadrant appears at low frequencies. This feature, beyond the doubt of the error bars, has the character of a capacitor. The complete locus for the narrowest pore is that of an RLC circuit. The unique 1D capacitance feature of a molecular model of electrolytes is reported for the first time for a molecularsolvent model electrolyte. In addition, the combined LRC circuit with a resonance frequency gives the electrolyte in a nanopore system some electronic character. It will be necessary not to ignore this electrical character when investigating the electrical functions of biological ion channels. A characterization of the electrolyte-nanopore system forms the basis for further studies and an understanding of nanopores with finite lengths and networks of nanopores. These studies will aid in the interpretation of much of the AC impedance or conductivity data routinely applied to porous materials. In this paper, we study only an electrolyte concentration of 0.1 M and nanopores of three sizes. The multipolar nature of a realistic solvent and the case of asymmetric ions have also been ignored or simplified. We believe that these are secondary effects and may give additional relaxation time scales, but only further studies can illuminate the rich features of the behavior of electrolytes in nanostructures. Acknowledgment. This work was supported in part by the Research Grants Council of Hong Kong (HKU 7213/ 99P). References (1) Lynden-Bell, R. M.; Rasaiah, J. C. J. Chem. Phys. 1996, 105, 9266. (2) Tang, Y. W.; Szalai, I.; Chan, K. Y. Mol. Phys. 2001, 99, 309. (3) Tang, Y. W.; Szalai, I.; Chan, K. Y. J. Phys. Chem. A 2001, 105, 9616. Nano Lett., Vol. 3, No. 2, 2003
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