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Two-Scale Analysis of the Electrostatics of Dielectric Crystals: Emergence of Polarization Density and Boundary Charges

Shoham Sen, Yang Wang, Timothy Breitzman, Kaushik Dayal

TL;DR

This work develops a rigorous two-scale homogenization framework for electrostatics in dielectric crystals with locally periodic charge distributions. By employing $2$-scale convergence and a dipole scaling $\rho_l=\frac{1}{l}\tilde{\rho}_l$, it derives a homogenized potential $\Phi_0$ governed by $\Delta_x\Phi_0 = \mathbb{1}_{\Omega}\operatorname{div}(\boldsymbol{p}_0)$ together with a boundary condition that includes a bulk polarization $\boldsymbol{p}_0$ and a surface density $\sigma$ arising from partial unit cells; the analysis demonstrates that unit-cell choices alter $\boldsymbol{p}_0$ and $\sigma$ but leave the energy and the homogenized potential invariant. The paper also shows the necessity of a surface charge contribution on $\partial\Omega\setminus\Gamma_d$ and clarifies the non-uniqueness of the dipole moment while preserving a unique macroscopic potential, thereby bridging microscopic charge distributions to macroscopic electrostatic behavior in crystals. These results provide a mathematically rigorous basis for incorporating polarization and boundary effects into continuum models of dielectric crystals and solid electrolytes, with implications for energy storage and electromechanics.

Abstract

Ionic crystals, such as solid electrolytes and complex oxides, are central to modern technologies for energy storage, sensing, actuation, and other functional applications. An important fundamental issue in the atomic and quantum-scale modeling of these materials is defining the macroscopic polarization. In a periodic crystal, the usual definition of the polarization as the first moment of the charge density in a unit cell is found to depend qualitatively - allowing even a change in the sign - and quantitatively on the choice of unit cell. We examine this issue using a rigorous approach based on the framework of 2-scale convergence. By examining the continuum limit of when the lattice spacing is much smaller than the characteristic dimensions of the body, we show that the 2-scale limit provides both a bulk polarization as well as a surface charge density supported on the boundary of the body. Further, different choices of the periodic unit cell of the body lead to correspondingly different partial unit cells at the boundary; these choices give to different bulk polarization and surface charges but compensate such that the electric field and energy are independent of the choice of unit cell.

Two-Scale Analysis of the Electrostatics of Dielectric Crystals: Emergence of Polarization Density and Boundary Charges

TL;DR

This work develops a rigorous two-scale homogenization framework for electrostatics in dielectric crystals with locally periodic charge distributions. By employing -scale convergence and a dipole scaling , it derives a homogenized potential governed by together with a boundary condition that includes a bulk polarization and a surface density arising from partial unit cells; the analysis demonstrates that unit-cell choices alter and but leave the energy and the homogenized potential invariant. The paper also shows the necessity of a surface charge contribution on and clarifies the non-uniqueness of the dipole moment while preserving a unique macroscopic potential, thereby bridging microscopic charge distributions to macroscopic electrostatic behavior in crystals. These results provide a mathematically rigorous basis for incorporating polarization and boundary effects into continuum models of dielectric crystals and solid electrolytes, with implications for energy storage and electromechanics.

Abstract

Ionic crystals, such as solid electrolytes and complex oxides, are central to modern technologies for energy storage, sensing, actuation, and other functional applications. An important fundamental issue in the atomic and quantum-scale modeling of these materials is defining the macroscopic polarization. In a periodic crystal, the usual definition of the polarization as the first moment of the charge density in a unit cell is found to depend qualitatively - allowing even a change in the sign - and quantitatively on the choice of unit cell. We examine this issue using a rigorous approach based on the framework of 2-scale convergence. By examining the continuum limit of when the lattice spacing is much smaller than the characteristic dimensions of the body, we show that the 2-scale limit provides both a bulk polarization as well as a surface charge density supported on the boundary of the body. Further, different choices of the periodic unit cell of the body lead to correspondingly different partial unit cells at the boundary; these choices give to different bulk polarization and surface charges but compensate such that the electric field and energy are independent of the choice of unit cell.
Paper Structure (13 sections, 5 theorems, 86 equations)

This paper contains 13 sections, 5 theorems, 86 equations.

Table of Contents

  1. Introduction
  2. Contributions of this Work
  3. Organization
  4. Heuristics
  5. Notation
  6. Problem Formulation
  7. Weak Formulation
  8. Polarization Theorem
  9. Volume term
  10. Surface Charge Density
  11. Non-uniqueness of the dipole moment
  12. Two-scale Convergence
  13. Bounded in Sobolev Space

Key Result

Theorem 3.1

If $||\tilde{\rho}_l||_{L^2(\hat{\boldsymbol{x}}_l\oplus l\Box)}<C$ for all $\boldsymbol{x}\in\Omega$ and for all $l\in\mathcal{I}$, then $||\tilde{\rho}_l||_{L^2(\Omega)}<C$ for all $l\in\mathcal{I}$ and $C$ independent of $l$.

Theorems & Definitions (19)

  • Remark 2.1
  • Example 2.1
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Remark 4.1
  • Remark 4.2
  • Remark 4.3
  • Theorem 5.1
  • ...and 9 more