Phase Field Modeling of Electrodeposition: from Stable Thin Films to Nanorods to Dendrites
Authors: Enrique Raul , DeWitt Stephen , Thornton Katsuyo 
Published: February 2020

Electrodeposition often results in the accumulation of material in unstable morphologies such as dendrites.  In many applications, including batteries, this common yet undesirable microstructure must be avoided.  Experimentally, different types of non-planar deposits are observed: from moss- to tree- to needle-like morphologies.  The characteristic length scale of these features range from the tens of nanometers to the tens of microns.  Several hypotheses have been put forward to explain the growth of these structures.1 Broadly speaking, these hypotheses consider two different aspects of electrodeposition.  In the first, the main cause of unstable growth is attributed to a localized breakdown of the properties of the solid-liquid interface, which may play a role in some of these experimental observations.  In this presentation, however, we focus on the second category, in which the non-uniform behavior of the electrostatic potential and the ion concentration are considered.  At the mesoscopic (micron) scale, the effect of the boundary layers surrounding the electrode is approximated by a suitable set of boundary conditions for the ion concentration and electrostatic field, φ, in the electrolyte.  The resulting boundary conditions for φ are unusual compared to those of typical electrostatic problems, as they are mixed expressions that relate the normal derivative to the value of the potential.  For instance, Butler-Volmer kinetics leads to an equation expressing the normal derivative of φ in terms of an exponential function of φ.  One can describe two limiting behaviors for the solution of the electrostatic field.  In the primary potential distribution approximation,2 it is assumed that the potential drop at the deposit surface is constant.  In this approximation, the electric field at the surface of the deposit depends on the local curvature of the deposit surface.  This leads to an unstable growth, where protuberances receive more material and grow faster than flatter regions.  In the limiting potential distribution approximation,2 it is assumed that the current density is constant.  Under this approximation, growth is stable and uniform; any existing surface roughness is diminished as the deposition front evolves.  A more accurate description of the problem will fall within these two limiting cases.  The goal in this work is to determine, by phase field modeling, the conditions that lead to stable or unstable electrodeposition.  We consider a binary electrolyte where ions migrate following a linear response to the driving forces.3  The electrostatic field is determined by the continuity equation of the current.  The deposition rate is governed by Butler-Volmer kinetics.  The geometry of the deposit is described by order parameters that track the evolution of morphologies.  Finally, boundary conditions are imposed by using the Smoothed Boundary Method.4  We conduct simulations for a range of material and kinetic parameters, which allow us to identify the regions of stable growth within the parameter space.  Our model suggests that high ionic conductivities and low exchange currents promote stable electrodeposition while low ionic conductivities and high exchange currents will lead to unstable growth.   At intermediate values, we observe the growth of columnar structures similar to those recently reported by Zhang et al.4  (This work was supported as part of the Joint Center for Energy Storage Research (JCESR), an Energy Innovation Hub funded by the U.S. Department of Energy, Office of Science, Basic Energy Sciences.)

References

1-C. Monroe and J. Newman, J. Electrochem. Soc. 150, A1377 (2003)

2-Theory and Practice of Metal Electrodeposition, Yuliy D. Gamburg and Giovanni Zangari, Springer (2011)

3-B. Ornavanos, T.R. Ferguson, H.-C. Yu, M.Z. Bazant and K. Thornton, J. Electrochem. Soc. 161, A535 (2014)

4-H.-C. Yu, H.-Y. Chen, and K. Thornton, Model. Simul. Mater. Sci. Eng., 20, 0750085 (2012)

4-Y. Zhang et al, Nanolett. 14, 6889 (2014)