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The opposite opinion was also expressed (see and references therein). This issue is addressed in Sec. 5.
-
The opposite opinion was also expressed (see and references therein). This issue is addressed in Sec. 5.
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29
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68949147089
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Improved treatments of the compact layer combine electrostatics with a statistical treatment of the lattices of absorbed ions and reorientable dipoles. Microscopic effects due to metal electrons are discussed below.
-
Improved treatments of the compact layer combine electrostatics with a statistical treatment of the lattices of absorbed ions and reorientable dipoles. Microscopic effects due to metal electrons are discussed below.
-
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34
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68949151097
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The inequality [Eq. 4] is applicable both to primitive ionic models and to mixtures of ions with (nonpolarizable) molecular multipoles (the extension of the model considered in). From the arguments of, it can be shown that Eq. 4 also accounts for classical image effects.
-
The inequality [Eq. 4] is applicable both to primitive ionic models and to mixtures of ions with (nonpolarizable) molecular multipoles (the extension of the model considered in). From the arguments of, it can be shown that Eq. 4 also accounts for classical image effects.
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A major step in developing the microscopic theory of metal electrons in contact with surface-inactive electrolytes was realizing that the solvent layer shifts in response to charging, so that both ze and zH are σ dependent (see, and references therein). Correlated relaxation of ze and zH can significantly contract the gap lH in the moderate anodic range (σ>0) resulting in a sharp decrease in CH -1 (σ). This phenomenon, supported by experiment, is completely reversed in theories with the solvent film's position frozen (see, e.g., for review). Relaxation of lH also typically leads to CH (σ) <0, in the spirit of the RGC model. In contacts of metals with solid electrolytes where solvent is absent, outward displacement of ze in the cathodic range readily accounts for the qualitative behavior and anomalously high value of C for Au/AgCl and Au/Ag4 RbI5 interfaces. Both CH and C can naturally become negative at large cathodic charges causing instability. In general, predictions of very low and negative values of CH -1 indicate that the inequality [Eq. 2] can easily fail.
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68949160199
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For example, the original GC model for a 1:1 electrolyte leads to the effective gap contracting with charging as l∼ln (σ) /σ; condition 8 is not satisfied. The contraction rate is too small for the anomaly to arise. This is clearly true for other GC-type models accounting for "steric effects," which further reduce or even reverse the compression rate (see for reviews). In general, Cd is strictly non-negative in all "local statistical" PB-type models. A recent attempt to extend this analysis beyond local models was proven incorrect.
-
For example, the original GC model for a 1:1 electrolyte leads to the effective gap contracting with charging as l∼ln (σ) /σ; condition 8 is not satisfied. The contraction rate is too small for the anomaly to arise. This is clearly true for other GC-type models accounting for "steric effects," which further reduce or even reverse the compression rate (see for reviews). In general, Cd is strictly non-negative in all "local statistical" PB-type models. A recent attempt to extend this analysis beyond local models was proven incorrect.
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The relation between the EC model and its prototype, the relaxing gap interfacial capacitor, is not literal. Thus, EDL electrocompression does not require simultaneous physical shift of the ionic positions toward the electrode resembling the relaxing plate of the EC. An efficient mechanism promoting contraction of the effective gap and, consequently, the appearance Cd (σ) <0 in diffuse layers is "overscreening" condensation of the induced charge density near the electrode at the expense of some depletion in the tail regions of the ionic distribution. This is closely related to ionic correlations absent in PB-type theories. Similarly, the "spring" is a metaphor for the entropic, electrostatic, and molecular forces defining the equilibrium charge distributions in an EDL.
-
The relation between the EC model and its prototype, the relaxing gap interfacial capacitor, is not literal. Thus, EDL electrocompression does not require simultaneous physical shift of the ionic positions toward the electrode resembling the relaxing plate of the EC. An efficient mechanism promoting contraction of the effective gap and, consequently, the appearance Cd (σ) <0 in diffuse layers is "overscreening" condensation of the induced charge density near the electrode at the expense of some depletion in the tail regions of the ionic distribution. This is closely related to ionic correlations absent in PB-type theories. Similarly, the "spring" is a metaphor for the entropic, electrostatic, and molecular forces defining the equilibrium charge distributions in an EDL.
-
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57
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C. W. McCombie, in Problems in Thermodynamics and Statistical Physics, edited by, P. T. Landsberg, (Pion, London, 1971), p. 459.
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McCombie, C.W.1
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68949164064
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w (σ), the energy of the isolated EC, does not include the variable l. Instead, l is replaced by its equilibrium value, l (σ). The simple interpretation is that σ is a "slow variable" and that the EC is always equilibrated at the current value of σ. However, one can use W (σ,l) instead, and optimize over both variables σ and l. The result is unaltered, reflecting the uniqueness of the equilibrium state.
-
w (σ), the energy of the isolated EC, does not include the variable l. Instead, l is replaced by its equilibrium value, l (σ). The simple interpretation is that σ is a "slow variable" and that the EC is always equilibrated at the current value of σ. However, one can use W (σ,l) instead, and optimize over both variables σ and l. The result is unaltered, reflecting the uniqueness of the equilibrium state.
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59
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68949152068
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The physical reason for the instability of the EC under control is the steep ∼ 2 / l2 increase in the attractive electrostatic force at small separations l. Increasing initially leads to a continuous shift of the relaxing plate; it also flattens the local minimum corresponding to its equilibrium position. Near the critical voltage, cr =u (σcr), the local minimum disappears completely (the inflection point). This leads to collapse when the gap and the charge on the plates change discontinuously). A small change in the controlled parameter leads to a dramatic discontinuous change in the equilibrium state of EC, similar to Zeeman's "catastrophe machine" and the Euler buckling instability. Such behavior is interpretable in terms of simple models with variable charges bound to their equilibrium positions by linear restoring forces.
-
The physical reason for the instability of the EC under control is the steep ∼ 2 / l2 increase in the attractive electrostatic force at small separations l. Increasing initially leads to a continuous shift of the relaxing plate; it also flattens the local minimum corresponding to its equilibrium position. Near the critical voltage, cr =u (σcr), the local minimum disappears completely (the inflection point). This leads to collapse when the gap and the charge on the plates change discontinuously). A small change in the controlled parameter leads to a dramatic discontinuous change in the equilibrium state of EC, similar to Zeeman's "catastrophe machine" and the Euler buckling instability. Such behavior is interpretable in terms of simple models with variable charges bound to their equilibrium positions by linear restoring forces.
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60
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The special case ξi =1 follows directly from Eq. 26 in the limit s→0.
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The special case ξi =1 follows directly from Eq. 26 in the limit s→0.
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61
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This can be justified by the fact that the nonuniform field due to the periodic perturbation of the surface charge density with the wave vector K decays as ∼exp (- qK z) where qK = (κ2 + K2) 1/2 and κ=2π/ λD is the Debye wave vector. Clearly, this can be neglected at interelectrode separations d λD.
-
This can be justified by the fact that the nonuniform field due to the periodic perturbation of the surface charge density with the wave vector K decays as ∼exp (- qK z) where qK = (κ2 + K2) 1/2 and κ=2π/ λD is the Debye wave vector. Clearly, this can be neglected at interelectrode separations d λD.
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73
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It is worth noting that the so-called "molecular models" simulated the compact layer as a two-dimensional lattice of "molecular dipoles" inserted between two conductors, the electrode and the solvent, often treated in the PC approximation (with =). In this approximation, a region of CH ≤0 under σ control would signify phase transition in the compact layer. However, in the PC model, which effectively places all the countercharge in the Helmholtz plane, Cd vanishes, and the compact layer is simply equivalent to the EDL.
-
It is worth noting that the so-called "molecular models" simulated the compact layer as a two-dimensional lattice of "molecular dipoles" inserted between two conductors, the electrode and the solvent, often treated in the PC approximation (with =). In this approximation, a region of CH ≤0 under σ control would signify phase transition in the compact layer. However, in the PC model, which effectively places all the countercharge in the Helmholtz plane, Cd vanishes, and the compact layer is simply equivalent to the EDL.
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Here, we ignore more conventional reasons for the inconsistency of two-layer EDL models, among them, specific ionic absorption on the electrode, strongly dissimilar sizes of cations and anions (which can be partially overcome by corresponding modification of zH (σ) or by considering a restricted σ range where one counterion species dominates), penetration of the electron-density distribution beyond the limits of the compact layer, etc..
-
Here, we ignore more conventional reasons for the inconsistency of two-layer EDL models, among them, specific ionic absorption on the electrode, strongly dissimilar sizes of cations and anions (which can be partially overcome by corresponding modification of zH (σ) or by considering a restricted σ range where one counterion species dominates), penetration of the electron-density distribution beyond the limits of the compact layer, etc..
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Consider an EC. "Normally" C (σ) >0 and the equilibrium state are unaffected when the isolated EC with fixed σ and corresponding equilibrium v (σ) is connected to a potential source of equal voltage =v (σ). In contrast, if C (σ) <0 the same procedure leads to dramatic changes. Upon connecting to a source of the same voltage, the equilibrium state (local minimum) becomes unstable (local maximum); there is immediate charge transfer from the source to the plates with an associated gap contraction (see also Appendix 0, Sec. 03). Charging leads to a new equilibrium state if a stable branch with C () >0 exists at larger σ.
-
Consider an EC. "Normally" C (σ) >0 and the equilibrium state are unaffected when the isolated EC with fixed σ and corresponding equilibrium v (σ) is connected to a potential source of equal voltage =v (σ). In contrast, if C (σ) <0 the same procedure leads to dramatic changes. Upon connecting to a source of the same voltage, the equilibrium state (local minimum) becomes unstable (local maximum); there is immediate charge transfer from the source to the plates with an associated gap contraction (see also Appendix 0, Sec. 03). Charging leads to a new equilibrium state if a stable branch with C >0 exists at larger σ.
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80
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68949100755
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Despite artificiality that can be introduced (e.g., unphysically small cations and neglect of ionic polarizability in), this approach, rigorously and consistently applied, still addresses the legitimacy and consequences of NC, in a spirit of exactly solved statistical models. Furthermore, in combination with the suggested parametrization of CH (σ), the results of tuning may well be physically adequate.
-
Despite artificiality that can be introduced (e.g., unphysically small cations and neglect of ionic polarizability in), this approach, rigorously and consistently applied, still addresses the legitimacy and consequences of NC, in a spirit of exactly solved statistical models. Furthermore, in combination with the suggested parametrization of CH (σ), the results of tuning may well be physically adequate.
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81
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Essentially, this is similar to a parametrized or σ dependence of the distance of closest ionic approach to the electrode. Originally designed to reflect typical differences in ionic size, it can also be modified to account for the asymmetry of the electron response.
-
Essentially, this is similar to a parametrized or σ dependence of the distance of closest ionic approach to the electrode. Originally designed to reflect typical differences in ionic size, it can also be modified to account for the asymmetry of the electron response.
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Uniform charge distributions, planar spherical or cylindrical, are often used in applications to electrochemical and colloid interfaces. In the former, the surface charge density is defined by the excess of electrons appropriate to free lateral redistribution and can be reasonably considered homogeneous under "normal" (absent criticality, see below) conditions if the electrode surface is smooth. Examples are a mercury electrode or a monocrystalline surface. For colloids, the macroionic charge is distributed over the irregularly sited ionizable surface groups; a uniformly charged shell description is just a rough approximation. In some applications, where surface modulation of σ is of special importance (see for review), it can be modeled as a fixed laterally inhomogeneous distribution σ (r). This is still σ control because the lateral distribution is imposed by the "observer" (see also). The alternative is allowing the surface charge to equilibrate self-consistently with the electrolyte charge distributions. We know of no such attempts in macroionic (colloidal macroparticles) studies, where surface charge redistribution, caused by instability, might laterally displace the charged groups and deform the particles.
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92
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A point made in the earliest discussions of NC as well as many more recent studies (see, and references therein).
-
A point made in the earliest discussions of NC as well as many more recent studies (see, and references therein).
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93
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68949140117
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In microscopic treatments of the electrode, the electrochemical potential of the electron (the Fermi level) is fixed everywhere on the electrode while its electrostatic and chemical components can vary for various reasons including fluctuation in response to ionic fields.
-
In microscopic treatments of the electrode, the electrochemical potential of the electron (the Fermi level) is fixed everywhere on the electrode while its electrostatic and chemical components can vary for various reasons including fluctuation in response to ionic fields.
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