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2009, 113, 17585–17589 Published on Web 09/21/2009
Diffuse Double Layer at Nanoelectrodes 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 ReceiVed: July 7, 2009; ReVised Manuscript ReceiVed: September 9, 2009
Numerical solution of the equilibrium Poisson-Boltzmann equation for hemispherical electrodes of vanishing size reveals that the effects of curvature on the diffuse double layer become significant for electrodes with radii less than 50 nm. These effects include dramatically enhanced capacitance and hence more a rapid potential drop from the outer Helmholtz plane as far as the characteristic tunnelling length for electron transfer. An enhanced driving force is therefore expected for nanoelectrodes as compared to electrodes larger than 50-100 nm, especially at low concentrations of supporting electrolyte. 1. Introduction The Poisson-Boltzmann equation describes the electric field arising at thermodynamic equilibrium for an electrolytic solution adjacent to a charged surface
∇2φ +
F s0
(
zF
i - φ ∑ zic*exp i RT i
)
)0
(1)
where φ is potential (V), F is the Faraday constant, s is the dielectric constant of the solvent medium, 0 is the permittivity of free space, R is the gas constant, and T is temperature (K); for species i, zi is its charge, and c*i is its concentration in bulk (mol m-3). The strong nonlinearity of the equation renders its exact solution difficult in most circumstances, but it has been successfully solved analytically for a planar electrode geometry in the classical work of Gouy and Chapman1-3 and by Debye and Hu¨ckel, approximately, in the spherical case for the timeaveraged distribution surrounding a single ionic charge.4 Additionally, a variety of analytical and numerical solutions, involving varying degrees of approximation, have been presented for planar, cylindrical, and spherical geometries in the past.5-12 Further, steady-state solutions of the combined Nernst-Planck-Poisson equations in a spherical geometry have yielded important information on the nature of the electric field when continuous current flow is taking place at a nanoscale electrode.13 Solving the equation itself is therefore far from novel, but to our knowledge, the practical electrochemical interpretation of size or ionic strength effects on the double layer determined by these solutions, in the spherical case, has been neglected by all past workers; this interpretation applies specifically to electrochemistry at nanoelectrodes, which is currently an area of intense study.14-18 Here, we apply a numerical solution scheme to consider the electrochemical properties of the diffuse double layer for a * To whom correspondence should be addressed. Fax: +44 (0) 1865 275410. Tel: +44 (0) 1865 275413. E-mail:
[email protected].
10.1021/jp906404h CCC: $40.75
charged nanoelectrode, including surface charge density, the electric field in the vicinity of the electrode, and the capacitance of the double layer, as predicted by the Poisson-Boltzmann equation. Substantial deviations are observed from classical Gouy-Chapman theory, derived for the same equation at a planar interface. We shall consider nanoelectrodes with sufficient excess charge for classical theory to remain applicable; we consequently ignore quantized charging and Coulomb staircase effects19-21 in order to best address the effects of curvature of the solutionphase double layer. Finite ion volume and noncontinuum effects are also ignored here; our conclusions represent the modification of the continuum electrostatic treatment of the diffuse double layer to a hemispherical geometry such that the effects of curvature on the diffuse double layer may be assessed. This curvature arises both for very small electrodes (re < 50 nm) and for larger electrodes when the ionic strength of the solution is low, that is, under weakly supported conditions. 2. Theory We consider a solution of a monovalent electrolyte A+ X-, which has a bulk concentration c*, that is completely dissociated in a solvent with uniform dielectric constant s. The solution surrounds a hemispherical electrode of radius re, which may be assumed to have a controlled surface potential E (V). The pertinent form of the Poisson-Boltzmann equation in this case is
( (
)
( ))
z iF ziF d2φ 2 dφ Fc* + exp - φ - exp + φ 2 r dr s0 RT RT dr
)0
(2) The following conventional normalizations are applied
θ)
F φ RT
2009 American Chemical Society
(3)
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r re
R)
Letters
(4)
to yield a dimensionless Poisson-Boltzmann equation
d2θ 2 dθ + + R2e (exp(-θ) - exp(θ)) ) 0 2 R dR dR
Re ) re
RTs0 2F2c*
dθ dR
(6)
(7)
Therefore, Re represents the size of the electrode relative to the thickness of the double layer
Re )
re κ√2
(8)
The following boundary condition is applied as R f ∞
dθ dR
|
Rf∞
)0
(9)
implying that the charge on the electrode is entirely compensated by the diffuse double layer contained within the solution space, and therefore, from Gauss’s law, the electric field at the boundary of this space must be zero. The outer Helmholtz plane derived in Stern’s modification to the Gouy-Chapman model is also not considered, as this merely complicates the study by introducing an extra parameter without altering the observable effects of curvature of the diffuse double layer.22 A fixed potential (Dirichlet) boundary condition is used at the electrode surface
θ(R ) 1) ) θ0
(10)
where θ0 is defined as
θ0 )
F (E - Epzc) RT
r)re
(5)
The Debye length κ, a characteristic measure of double layer thickness, is given as
κ)
|
4πr2e qj qj q )))2 2 4πs0re 4πs0re s0
(12)
where qj is the surface charge density (C m-2). Hence, by substitution of the normalized variables introduced above
where Re is a dimensionless variable proportional to re with the form
F2c* RTs0
dφ dr
(11)
where E (V) is the applied potential versus some reference electrode and Epzc (V) is the potential of zero charge versus the same reference. By Gauss’s law, the enclosed charge, q, can be related to the electric field at the electrode surface
|
j ) -Q R)1
(13)
j is a dimensionless surface charge density defined as where Q
j ) Q
Freqj RTs0
(14)
The condition in eq 13 could equally well be used as a (Neumann) boundary condition, but in this work, we assume a potentiostatically controlled electrode with the charge density at its surface resulting as an observable. The Poisson-Boltzmann equation is solved numerically using a fully implicit finite difference method to generate a set of coupled nonlinear simultaneous equations, which are readily solved using the iterative Newton-Raphson algorithm. A space grid expanding in R away from the electrode to some characteristic Rmax was employed, where Rmax was chosen to be large enough to have no effect on the solution. Converged parameters for the expanding space grid were established by a detailed convergence study at varying Re, θ0, and so forth. The simulations were programmed in C++ and run on a standard desktop computer (Intel Core2 Quad 2.83 GHz, 3.2GB RAM). 3. Results and Discussion 3.1. Solutions of the Poisson-Boltzmann Equation Occur between Two Classical Limits. In the normalized coordinate system above, Coulomb’s law for an isolated point charge in a spherical geometry gives
j ) θ0 Q
(15)
and the Gouy-Chapman equation for a charged electrode in a planar geometry gives
()
j ) √8Re sinh Q
θ0 2
(16)
Figure 1 shows simulated results for the dimensionless surface j , as a function of Re (and hence re) for a typical charge density, Q potential step to E - Epzc ) 5F/RT (V). A size-dependent transition is observed, which corresponds to the transition between the two limits above, as the double layer becomes relatively spherical for small electrode sizes and relatively planar for large electrode sizes. This region of transition is detailed at Figure 2. 3.2. Effect of Electrode Radius. By substituting into Re typical values of s ) 78.54 (H2O) and T ) 298 K, the variation in surface charge density may be approached as a function of re; this is shown for c* ) 1 and 100 mM for a charging potential of 5RT/F (V) from Epzc at Figure 3. At re > 50 nm, a constant charge density is observed irrespective of electrode size, as predicted by the Gouy-Chapman model, whereas substantial excess charge density is observed for lower radii. The more
Letters
Figure 1. Plot of numerical simulation and the ideal Coulombic and j versus log Re. Gouy-Chapman values for log Q
Figure 2. Plot of numerical simulation and the ideal Coulombic and j versus log Re; detail of the transition Gouy-Chapman values for log Q region.
Figure 3. Plot of numerical simulation for qj/µC cm-2 versus re/m, shown on a logarithmic scale for electrode size; s ) 78.54, T ) 298 K, c* ) 1 and 100 mM.
extensive double layer at lower concentrations allows this double layer curvature effect to manifest itself at larger electrode sizes. This is evident from the plot, which shows that in the case c* ) 1 mM, the charge density increases relatively more rapidly with reduced re than that for c* ) 100 mM. The division shown in Figure 3 is that where, to the left, the charge on the electrode is less than 10 elementary charges, such
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Figure 4. Simulated potential field in the vicinity of an electrode charged to a potential of Epzc + 5RT/F/V for re ) 5 and 50 nm; s ) 78.54, T ) 298 K, c* ) 100 mM.
Figure 5. Simulated potential field in the vicinity of an electrode charged to a potential of Epzc + 5RT/F/V for re ) 5 and 50 nm; s ) 78.54, T ) 298 K, c* ) 1 mM.
that quantum effects may be expected to dominate and hence our theory is not appropriate. It must be noted that for both concentrations, double layer curvature induces deviations from Gouy-Chapman theory at radii larger than those (re < 10 nm) where observably quantized charging is expected. 3.3. Effect of Electrolyte Concentration. If a species can undergo electrolysis at a charged electrode, the driving force for electron transfer is considered to be the potential drop between the electrode and the point in solution at which an electron is able to tunnel between the electrode and the solutionphase species. The maximum range of such tunnelling occurs at r - re ≈ 10 Å ) 1 nm. As the electric field in the vicinity of a nanoelectrode has been demonstrated to be nonclassical, we now investigate how this affects the potential drop as far as the maximum tunnelling distance. Potential profiles for the first 50 Å ()5 nm) away from the electrode surface are shown for electrode sizes 5 and 50 nm (c* ) 100 and 1 mM, E - Epzc ) 5 RT/F) in Figures 4 and 5. A size-dependent difference is perceptible in both but is distinctly more marked at lower concentrations. For constant applied potential, the surface charge density is enhanced for small electrodes. Therefore, in accordance with Gauss’s law, the electric field is stronger, and therefore, the potential drop from the outer Helmholtz plane to the maximum electron tunnelling length is increased. This corresponds to reduced absolute potentials (versus Epzc) at this maximum tunnelling
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Letters
Figure 6. Plot of numerical simulation for potential at the maximum tunnelling length for an applied potential of φ(r ) re) ) 5RT/F, φ(r re ) 10 Å ) 1 nm)/mV versus c*/mM, shown on a logarithmic scale for concentration; s ) 78.54, T ) 298 K, re ) 5, 10, 20, and 100 nm.
length, as shown in Figure 6, which demonstrates the effect of electrolyte concentration for constant typical electrode sizes of re ) 5, 10, 20, and 100 nm, with an applied potential of Epzc + 5RT/F (V). Under potentiostatic control, smaller electrodes are therefore predicted to afford more powerful driving forces to electrolysis as the potential drop to the maximum tunnelling distance is greater; the relative difference is more marked for lower supporting electrolyte concentrations where the double layer is more extensive and uncompensated ohmic drop at the tunnelling length is more significant. 3.4. Analysis in Terms of Differential Capacitance. The j d (F m-2), for a diffuse double differential capacitance density, C layer may be defined as
j d ) dqj C dE
(17)
where qj is the enclosed surface charge density (C m-2) and E is the applied potential versus Epzc. For an ideal Coulombic point charge
jd ) C
s0 re
(18)
which is independent of potential. Equivalently, for the Gouy-Chapman model for a planar electrode
jd ) C
2F2s0c* F cosh (E - Epzc) RT 2RT
(
)
(19)
Differential capacitance curves were produced by simulating qj at various values of (E - Epzc) and using a three-point numerical derivative to differentiate the resulting curves. The simulated results for c* ) 1 mM and varying electrode size are shown at Figure 7, again demonstrating the behavior of the diffuse double layer as undergoing transition between two well-defined regimes at electrode radii in the nanoscale region. Coulombic behavior, where capacitance is not a function of potential, is observed across a wider potential window on either side of Epzc with
Figure 7. Simulated differential capacitance curves for electrodes of varying sizes, shown with the theoretical predictions of Coulomb’s Law (for the two smallest radii cases) and the Gouy-Chapman model s ) 78.54, T ) 298 K, re ) 0.3162, 1, 3.162, and 10 nm.
reducing electrode radius. The smallest electrodes here are presented for continuity of theory rather than to represent any real behavior for subnanoscale electrodes; for hypothetical electrodes of atomic radius or smaller and ignoring the quantum effects which must necessarily occur here, our solutions exhibit that excellent correlation with the Coulombic expression for capacitance is achieved across a wide potential range at c* ) 1 mM (not shown). It is therefore projected that at micromolar concentrations, and so very small Re, this perfect Coulombic behavior should be observable for small nanoscale electrodes where the absolute values of capacitance are larger, and therefore neglecting quantum effects or finite ion volume effects is not inaccurate. 4. Conclusion A rational approach to analysis of numerical solutions of the Poisson-Boltzmann equation in a spherical space demonstrates that the diffuse double layer at a charged nanoelectrode must differ significantly from the predictions made by classical Gouy-Chapman theory. These effects are caused by marked curvature of the double layer, which arises when the electrode radius becomes large compared to the Debye length. This is the case for nanoelectrodes with re < 50 nm, for typical concentrations of electrolyte; as electrolyte concentration is reduced, deviations from classical theory are observed for increasingly large electrodes (re > 50 nm). In either case, our prediction is that a curved double layer will cause nonclassical behavior in these systems, irrespective of any additional quantum effects or finite ion volume effects that may arise at the nanoscale. Provided the potential is not too distant from the potential of zero charge, the effect of double layer curvature is to cause electric fields in the diffuse double layer to tend toward the ideal situation of a Coulombic point charge. As curvature increases, the capacitive properties of the diffuse double layer obey Coulomb’s law, rather than the Gouy-Chapman equation, over an increasing potential range. We note that this research does not take account of the compact double layer, quantum effects concerning the electrostatics of the nanoelectrode, or dynamic variation of the diffuse double layer under conditions of electrolysis but is rather intended to provide an understanding of equilibrated diffuse double layer behavior as a framework for future study. Increased
Letters electrochemical driving forces at the characteristic tunnelling length and increased charging current are logical predictions of this research for electrochemical investigation at nanoelectrodes. Acknowledgment. E.J.F.D. thanks St John’s College, Oxford, for funding. References and Notes (1) Gouy, L. G. C. R. Hebd. Se´ances Acad. Sci. 1909, 149, 654–657. (2) Gouy, L. G. J. Phys. Theor. Appl. 1910, 9, 457–467. (3) Chapman, D. L. Philos. Mag. 1913, 25, 475–481. (4) Debye, P.; Hu¨ckel, E. Phys. Z. 1923, 24, 185–206. (5) Stillinger, Frank, H., J. J. Chem. Phys. 1961, 35, 1584–1589. (6) Loeb, A. J.; Overbeek, J.; Wiesema, P. The Electrical Double Layer Around a Spherical Colloid Particle; MIT Press: Cambridge, MA, 1961. (7) Delmastro, J. R.; Booman, G. L. J. Electroanal. Chem. 1971, 32, 157–163. (8) Stokes, A. N. J. Chem. Phys. 1976, 65, 261–264. (9) Frahm, J.; Diekmann, S. J. Colloid Interface Sci. 1979, 70, 440–447. (10) Tenchov, B.; Brankov, I. J. Colloid Interface Sci. 1986, 109, 172– 180.
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