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Diffused solute-solvent interface with Poisson-Boltzmann electrostatics: free-energy variation and sharp-interface limit. (English) Zbl 1352.35175

The authors consider an gradient field \(E=-\nabla\psi\) satisfying the Gauss law of electrostatics \(\nabla^{\top}\epsilon\left(\phi\right)E=\rho\) in a situation, where the material occupying a bounded, smooth domain \(\Omega\) is described by a phase field \(\phi\) modeling ionized molecules solved in water. In terms of the potential \(\psi\) we have that the total charge density \(\rho\) is given as \(\rho=\rho_{f}-\left(\phi-1\right)^{2}V^{\prime}\left(\psi\right)\), where \(\rho_{f}\) is a given charge density and \(V\in C_{2}\left(\mathbb{R}\right)\) a strictly convex function describing the ionic contribution to the electrostatic interaction. Dirichlet boundary data \(\psi=\psi_{\infty}\text{ on }\partial\Omega\) are prescribed. The phase field \(\phi\) is governed by its own relaxation dynamics of the form \(\partial_{t}\phi+2P\phi+2\rho_{w}\left(\phi-1\right)U-\gamma_{0}\xi\Delta\phi+\frac{\gamma_{0}}{\xi}W^{\prime}\left(\phi\right)=F\left(\psi\right)\) with initial data \(\phi\left(0\right)=\phi_{0}\) prescribed, where \(W\) approximates the surface energy of the phase interface, \(P,\rho_{w},\)\(\gamma_{0},\xi\) are positive parameters and \(F\left(\psi\right)=\frac{1}{2}\epsilon^{\prime}\left(\phi\right)\left|\nabla\psi\right|^{2}+2\left(\phi-1\right)V\left(\psi\right)\). Matched asymptotic analysis is used to inspect the relaxation dynamics of the diffused phase interface. The paper concludes with the study of the sharp interface limit case.

MSC:

35Q60 PDEs in connection with optics and electromagnetic theory
76Z10 Biopropulsion in water and in air
76T20 Suspensions
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