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The Density Functional Theory for Electrolyte Solutions

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PyDFTele

The Density Functional Theory for Electrolyte Solutions

For an electrolyte solution close to a charged surface with temperature , total volume , and chemical potential of each species specified, the grand potential, , is written as

$$\Omega[{\rho_i(\boldsymbol{r})},\psi(\boldsymbol{r})] = F^\text{id}[{\rho_i(\boldsymbol{r})}] + F^\text{exc}[{\rho_i(\boldsymbol{r})}]+ F^\text{coul}[{\rho_i(\boldsymbol{r})},\psi(\boldsymbol{r})]+ \sum_i \int_{V} d \boldsymbol{r} [V_i^{(\text{ext})}(\boldsymbol{r})-\mu_i] \rho_i(\boldsymbol{r})+ \int_{\partial V}d \boldsymbol{r} \sigma(\boldsymbol{r}) \psi(\boldsymbol{r})$$

where $\rho_i(\boldsymbol{r})$ is the local density of the component i, $\psi(\boldsymbol{r})$ is the local electrostatic potential, and $V^\text{ext}_{i}$ is the external potential. The volume of the system is V and $\partial V$ is the boundary of the system.

The ideal-gas contribution $F^\text{id}$ is given by the exact expression

$$F^\text{id}[{\rho_i (\boldsymbol{r})}] = k_B T\sum_i \int_{V} d\boldsymbol{r}\ \rho_i(\boldsymbol{r})[\ln(\rho_i (\boldsymbol{r})\Lambda_i^3)-1]$$

where is the Boltzmann constant, is the absolute temperature, and is the well-known thermal de Broglie wavelength of each ion.

The Coulomb's free-energy is obtained by the addition of the electric field energy density and the minimal-coupling of the interaction between the electrostatic potential and the charge density , and it can be written as

$$F^\text{coul}[{\rho_{i}(\boldsymbol{r})},\psi(\boldsymbol{r})] = -\int_V d\boldsymbol{r}\ \frac{\epsilon_0 \epsilon_r}{2} |\nabla{\psi(\boldsymbol{r})}|^2 + \int_{V} d\boldsymbol{r}\ \sum_i Z_i e \rho_{i}(\boldsymbol{r}) \psi(\boldsymbol{r})$$

where is the valence of the ion i, is the elementary charge, is the vacuum permittivity, and is the relative permittivity.

The excess Helmholtz free-energy, , is the free-energy functional due to particle-particle interactions splitted in the form

$$F^\text{exc}[{\rho_i(\boldsymbol{r})}] = F^\text{hs}[{\rho_i(\boldsymbol{r})}] + F^\text{ec}[{\rho_i(\boldsymbol{r})}]$$

where $F^{\textrm{hs}}$ is the hard-sphere excess contribution and $F^{\textrm{ec}}$ is the electrostatic correlation excess contribution.

The hard-sphere contribution, $F^{\textrm{hs}}$, represents the hard-sphere exclusion volume correlation and it can be described using different formulations of the fundamental measure theory (FMT) as

The electrostatic correlation can be described using different approximations as

Finally, The chemical potential for each ionic species is defined as $\mu_i = \mu_i^\text{id} + \mu_i^\text{exc}$, where superscripts id and exc refer to ideal and excess contributions, respectively.

The thermodynamic equilibrium is obtained by the minimum of the grand-potential, $\Omega$, which can be obtained by the functional derivatives, such that, the equilibrium condition for each charged component is given by

$$\left. \frac{\delta \Omega}{\delta \rho_i(\boldsymbol{r})} \right|_{{\mu_k},V,T} = k_B T \ln[\Lambda_i^3 \rho_i(\boldsymbol{r})] + \frac{\delta F^\text{exc}}{\delta \rho_i(\boldsymbol{r})} + Z_i e \psi(\boldsymbol{r}) + V^{\text{ext}}_i(\boldsymbol{r}) - \mu_i =0$$

and for the electrostatic potential it is

$$\left. \frac{\delta \Omega}{\delta \psi(\boldsymbol{r})} \right|_{{\mu_k},V,T} = \epsilon_0 \epsilon_r\nabla^2{\psi(\boldsymbol{r})} + \sum_i Z_i e \rho_i(\boldsymbol{r}) =0$$

valid in the whole volume V, this is the well-known Poisson's equation of the electrostatic potential with the boundary conditions

$$\left. \frac{\delta \Omega}{\delta \psi(\boldsymbol{r})} \right|{{\mu_k},V,T} = \left. \epsilon_0 \epsilon_r \boldsymbol{\hat{n}}(\boldsymbol{r}) \cdot \boldsymbol{\nabla}{\psi(\boldsymbol{r})} \right|{\partial V} + \sigma(\boldsymbol{r}) = 0$$

valid on the boundary surface $\partial V$, where $\boldsymbol{\hat{n}}(\boldsymbol{r})$ is denoting the vector normal to the surface pointing inward to the system.

Examples

Voukadinova

Figure1 Figure2
Fig.1 - The ionic density profiles of an 1:1 electrolyte solution with c_+= 0.01 M and σ = -0.5C/m². Fig.2 - The electrostatic potential profile of an 1:1 electrolyte solution with c_+= 0.01 M and σ = -0.5C/m².
Figure3 Figure4
Fig.3 - The ionic density profiles of an 1:1 electrolyte solution with c_+= 1.0 M and σ = -0.5C/m². Fig.4 - The electrostatic potential profile of an 1:1 electrolyte solution with c_+= 1.0 M and σ = -0.5C/m².