A theory for the stabilization of polar crystal surfaces by a liquid environment

TL;DR

This paper develops a theoretical framework for understanding how liquid environments stabilize polar crystal surfaces by modeling the system as a stack of capacitors, predicting suppressed charge fluctuations and clarifying boundary conditions in slab geometries.

Contribution

It extends existing theories by incorporating the solution environment explicitly, predicting charge fluctuation suppression and clarifying electrostatic boundary conditions for polar surfaces in solution.

Findings

Surface charge fluctuations decrease with increasing crystal thickness.

Electric displacement fields in slabs originate from geometry, not boundary conditions.

The theory explains limitations of standard slab corrections in simulations.

Abstract

Polar crystal surfaces play an important role in the functionality of many materials, and have been studied extensively over many decades. In this article, a theoretical framework is presented that extends existing theories by placing the surrounding solution environment on an equal footing with the crystal itself; this is advantageous, e.g., when considering processes such as crystal growth from solution. By considering the polar crystal as a stack of parallel plate capacitors immersed in a solution environment, the equilibrium adsorbed surface charge density is derived by minimizing the free energy of the system. In analogy to the well-known diverging surface energy of a polar crystal surface at zero temperature, for a crystal in solution it is shown that the "polar catastrophe" manifests as a diverging free energy cost to perturb the system from equilibrium. Going further than existing theories, the present formulation predicts that fluctuations in the adsorbed surface charge density become increasingly suppressed with increasing crystal thickness. We also show how, in the slab geometry often employed in both theoretical and computational studies of interfaces, an electric displacement field emerges as an electrostatic boundary condition, the origins of which are rooted in the slab geometry itself, rather than the use of periodic boundary conditions. This aspect of the work provides a firmer theoretical basis for the recent observation that standard "slab corrections" fail to correctly describe, even qualitatively, polar crystal surfaces in solution.