Abstract
An essential parameter for crystal growth is the kinetic coefficient given by the proportionality between supercooling and average growth velocity. Here, we show that this coefficient can be computed in a single equilibrium simulation using the interface pinning method where two-phase configurations are stabilized by adding a spring-like bias field coupling to an order-parameter that discriminates between the two phases. Crystal growth is a Smoluchowski process and the crystal growth rate can, therefore, be computed from the terminal exponential relaxation of the order parameter. The approach is investigated in detail for the Lennard-Jones model. We find that the kinetic coefficient scales as the inverse square-root of temperature along the high temperature part of the melting line. The practical usability of the method is demonstrated by computing the kinetic coefficient of the elements Na and Si from first principles. A generalized version of the method may be used for computing the rates of crystal nucleation or other rare events.
Highlights
Crystal growth is of paramount importance in many branches of condensed matter physics.1,2 An important parameter in phase field equations3 describing crystal growth is the kinetic coefficient defined as the proportionality constant between supercooling and the average interface growth velocity ⟨xs⟩.4,5 We define the kinetic coefficient M using the difference in chemical potential between the two phases μsl = μs − μl as a measure of supercooling, ⟨xs⟩ = −M μsl. (1)In the spirit of the fluctuation-dissipation theorem,6 we suggest to learn about the interface dynamics by investigating spontaneous fluctuations when a bias potential is added to the Hamiltonian
We devise a stochastic model of fluctuations that assume Smoluchowski dynamics for crystal growth and show that the growth rate can be inferred from the terminal exponential relaxation of the order-parameter
The chemical potential difference between the solid and the liquid μsl is known from the average force exerted by the bias field on the system, and the proportionality constant M can be computed in a single equilibrium simulation
Summary
An important parameter in phase field equations describing crystal growth is the kinetic coefficient defined as the proportionality constant between supercooling (or superheating) and the average interface growth (or melting) velocity ⟨xs⟩.4,5. We define the kinetic coefficient M using the difference in chemical potential between the two phases μsl = μs − μl as a measure of supercooling (or superheating),. We devise a stochastic model of fluctuations that assume Smoluchowski dynamics for crystal growth and show that the growth rate can be inferred from the terminal exponential relaxation of the order-parameter. The chemical potential difference between the solid and the liquid μsl is known from the average force exerted by the bias field on the system, and the proportionality constant M can be computed in a single equilibrium simulation. The Appendix gives a detailed analysis of our stochastic model
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