Free energy of an inhomogeneous polymer–polymer–solvent system. II

A thin-interface phase-field model of electrochemical interfaces is developed based on Marcus kinetics for concentrated solutions, and used to simulate dendrite growth during electrodeposition of metals. The model is derived in the grand electrochemical potential to permit the interface to be widened to reach experimental length and time scales, and electroneutrality is formulated to eliminate the Debye length. Quantitative agreement is achieved with zinc Faradaic reaction kinetics, fractal growth dimension, tip velocity, and radius of curvature. Reducing the exchange current density is found to suppress the growth of dendrites, and screening electrolytes by their exchange currents is suggested as a strategy for controlling dendrite growth in batteries.Understanding the cause of dendrite growth during electrodeposition is a challenging problem with important technological relevance for advanced battery technologies [1]. Controlling the growth of dendrites would solve a decades-old problem and enable the use of metallic electrodes such as lithium or zinc in rechargeable batteries, leading to significant increases in energy density.Due to the complexity of observed deposition patterns [2-5], a complete theoretical understanding of the formation of dendrites from binary electrolytes has not been developed. Modeling of electrodeposition has largely focused on analysis of diffusion equations without consideration of morphology [6][7][8][9][10], or variations of diffusion limited aggregation [1,11] which are applicable only at the limit of very small currents, and which do not account for surface energy.In contrast, the phase-field method [12, 13] has succeeded at quantitatively modeling dendritic solidification at the limit of zero reaction kinetics [14][15][16], but has had only limited application to electrochemical systems with Faradaic reactions at the interface. The advantage of the phase-field method is that boundaries are tracked implicitly, and interfacial energy, interface kinetics, and curvature-driven phase boundary motion are incorporated rigorously.Phase-field models of electrochemical interfaces have recently been developed [17][18][19][20][21][22][23] and applied to dendritic electrodeposition [20,21,23], but these models suffer from significant limitations. Perhaps the most serious oversight in current electrodeposition models is the assumption of linearized or Butler-Volmer kinetics. It has been known for several decades that even seemingly simple metal reduction reactions are in fact multi-step and limited by electron transfer [24,25]. As a consequence, curved Tafel plots that deviate from Butler-Volmer have been reported for zinc reduction [26,27].Simulating experimental length and time scales is a second challenge. Guyer et al. [17,18] provided a diffuseinterface description of charge separation at an electrochemical interface capable of modeling double layers and Butler-Volmer kinetics, but the model is essentially too complex for practical use. The evolution equations are numerically unstable an...

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“…κ ij ∇c i ·∇c j (7) This equation, which has been previously reported in literature [6,7,33], may be simplified by combining the summation terms:…”

Section: B Free Energy Of a Multicomponent Systemmentioning

confidence: 99%