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Temperature-Gradient Effects on Electric Double Layer Screening in Electrolytes

Kazuhiko Seki

TL;DR

This work develops a non-isothermal Poisson–Boltzmann framework that couples electrostatics with thermodiffusion via the Eastman entropy of transfer $\hat{S}_\pm = \alpha_\pm k_{\rm B}$. By linearizing around small temperature gradients, the authors derive a generalized Debye–Hückel equation with a temperature-dependent factor, introducing an effective screening length $\lambda_{\rm eff}$ that grows with temperature for $\alpha > -1$ and governs near-electrode decay and the differential capacitance. Exact analytical solutions exist for the symmetric case $\alpha_+ = \alpha_- = \alpha$, expressed in terms of modified Bessel functions (or algebraic forms at $\alpha=1$), while the asymmetric case $\alpha_+ \neq \alpha_-$ is well described by a hypergeometric-based approximate solution. Numerical results validate the analytical forms and reveal that the potential profile remains non-exponential away from the electrode, with the PZC maintaining its role as the point of minimum differential capacitance. The findings elucidate the fundamental coupling between electrostatics and thermodiffusion in non-isothermal electrolytes and have implications for ionic thermoelectrics and related colloidal/electrokinetic phenomena, including steady-state Seebeck effects dominated by the Stern layer.

Abstract

Temperature gradients drive asymmetric ion distributions via thermodiffusion (the Soret effect), leading to deviations from the classical Debye--Hückel potential.We introduce the Eastman entropy of transfer, $\hat{S}_\pm = α_\pm k_{\rm B}$ for cations and anions, respectively, where $k_{\rm B}$ is the Boltzmann constant, and analyze non-isothermal electric double layers in terms of the dimensionless Soret coefficients $α_\pm$. Analytical solutions of the generalized Debye--Hückel equation show that, for $α_+ = α_-$, the potential is exactly described by a modified Bessel function, while the marginal case $α_\pm = 1$ exhibits algebraic decay. An effective screening length, $λ_{\rm eff}$, characterizes the near-electrode potential and increases with temperature, resulting in weaker screening on the hot side and stronger screening on the cold side for $α_\pm > -1$. The differential capacitance is controlled by $α_\pm$ via $λ_{\rm eff}$, with its minimum coinciding with the potential of zero charge (PZC) even in the presence of a temperature gradient. These findings highlight the fundamental coupling between electrostatics and thermodiffusion in non-isothermal electrolytes.

Temperature-Gradient Effects on Electric Double Layer Screening in Electrolytes

TL;DR

This work develops a non-isothermal Poisson–Boltzmann framework that couples electrostatics with thermodiffusion via the Eastman entropy of transfer . By linearizing around small temperature gradients, the authors derive a generalized Debye–Hückel equation with a temperature-dependent factor, introducing an effective screening length that grows with temperature for and governs near-electrode decay and the differential capacitance. Exact analytical solutions exist for the symmetric case , expressed in terms of modified Bessel functions (or algebraic forms at ), while the asymmetric case is well described by a hypergeometric-based approximate solution. Numerical results validate the analytical forms and reveal that the potential profile remains non-exponential away from the electrode, with the PZC maintaining its role as the point of minimum differential capacitance. The findings elucidate the fundamental coupling between electrostatics and thermodiffusion in non-isothermal electrolytes and have implications for ionic thermoelectrics and related colloidal/electrokinetic phenomena, including steady-state Seebeck effects dominated by the Stern layer.

Abstract

Temperature gradients drive asymmetric ion distributions via thermodiffusion (the Soret effect), leading to deviations from the classical Debye--Hückel potential.We introduce the Eastman entropy of transfer, for cations and anions, respectively, where is the Boltzmann constant, and analyze non-isothermal electric double layers in terms of the dimensionless Soret coefficients . Analytical solutions of the generalized Debye--Hückel equation show that, for , the potential is exactly described by a modified Bessel function, while the marginal case exhibits algebraic decay. An effective screening length, , characterizes the near-electrode potential and increases with temperature, resulting in weaker screening on the hot side and stronger screening on the cold side for . The differential capacitance is controlled by via , with its minimum coinciding with the potential of zero charge (PZC) even in the presence of a temperature gradient. These findings highlight the fundamental coupling between electrostatics and thermodiffusion in non-isothermal electrolytes.

Paper Structure

This paper contains 9 sections, 68 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Theoretical formulation
  3. One Dimension
  4. Approximate Solution
  5. Exact solution for $\alpha_+ = \alpha_-$
  6. Three-Dimensional Isotropic System
  7. Numerical Results
  8. Discussion
  9. Conclusion

Figures (4)

  • Figure 1: Schematic of the electrostatic potential in a linear temperature gradient. The origin is located at the edge of the diffuse layer, and charge neutrality is satisfied at the distance $L$. The parameter $\lambda_{\rm eff}$ denotes the effective screening length.
  • Figure 2: $\lambda_{\rm eff}$ calculated from Eq. (\ref{['eq:lmdeff']}). $T(0)$ and $T(L)$ denote the local temperature inside the electric double layer and the bulk temperature, respectively, as illustrated in Fig. \ref{['fig:schematicfig']}. (a) Short-dashed, long-dashed, and solid lines correspond to $\alpha_+ = \alpha_- = 1$, $\alpha_+ = \alpha_- = 4$, and $\alpha_+ = \alpha_- = 7$, respectively. (b) Long-dashed, short-dashed, and solid lines correspond to $\alpha_+ = \alpha_- = 4$, $\alpha_+ = \alpha_- = 40$, and $\alpha_+ = 4, \alpha_- = 40$, respectively.
  • Figure 3: Electrostatic potential as a function of the dimensionless distance from the electrode ($x/\lambda_D$) with $\psi(0)=1$, $T_{\rm r}=T(0)/T(L)=0.98$, and $g=0.00165$. Thick solid line: exact solution for $\alpha_+=\alpha_-$ [Eq. (\ref{['eq:sol1_1']})]; thin line: hypergeometric approximation [Eq. (\ref{['eq:hyper1']})]; dashed line: exponential decay with a power-law correction [Eq. (\ref{['eq:asym3']})] using $\lambda_{\rm eff}$ [Eq. (\ref{['eq:lmdeff']})]; circles: numerical solution of Eq. (\ref{['eq:DH']}) with the additional boundary condition $\psi(10\lambda_D)=0$. (a) Upper and lower curves/circles correspond to $\alpha_+=\alpha_-=4$ and $\alpha_+=\alpha_-=40$, respectively; (b) $\alpha_+=4$ and $\alpha_-=40$.
  • Figure 4: Electrostatic potential as a function of the dimensionless distance from the electrode ($x/\lambda_D$) with $\psi(0)=1$, $T_{\rm r}=T(0)/T(L)=0.98$, and $g=0.00165$. Thick solid line: exact solution for $\alpha_+=\alpha_-$ [Eq. (\ref{['eq:sol2']})]; dashed line: exponential decay with a power-law correction [Eq. (\ref{['eq:asym3']})] using $\lambda_{\rm eff}$ [Eq. (\ref{['eq:lmdeff']})]; circles: numerical solution of Eq. (\ref{['eq:DH']}) with the additional boundary condition $\psi(10\lambda_D)=0$. (a) Upper and lower curves/circles correspond to $\alpha_+=\alpha_-=-4$ and $\alpha_+=\alpha_-=-40$, respectively; (b) $\lambda_{\rm eff}$ calculated from Eq. (\ref{['eq:lmdeff']}). Short dashed, long dashed, and solid lines correspond to $\alpha_+=\alpha_-=-1$, $\alpha_+=\alpha_-=-4$, and $\alpha_+=\alpha_-=-7$, respectively.