Abstract
We review how phase-field models contributed to the understanding of various aspects of crystal nucleation, including homogeneous and heterogeneous processes, and their role in microstructure evolution. We recall results obtained both by the conventional phase-field approaches that rely on spatially averaged (coarse grained) order parameters in capturing freezing, and by the recently developed phase-field crystal models that work on the molecular scale, while employing time averaged particle densities, and are regarded as simple dynamical density functional theories of classical particles. Besides simpler cases of homogeneous and heterogeneous nucleation, phenomena addressed by these techniques include precursor assisted nucleation, nucleation in eutectic and phase separating systems, phase selection via competing nucleation processes, growth front nucleation (a process, in which grains of new orientations form at the solidification front) yielding crystal sheaves and spherulites, and transition between the growth controlled cellular and the nucleation dominated equiaxial solidification morphologies.
Highlights
The applied approaches range from discrete atomistic simulations relying on molecular dynamics (MD) [3,4,5,6], Monte Carlo (MC) [7,8,9], Brownian dynamics (BD) [10, 11], and cluster dynamics techniques [12,13,14], to continuum models including the van der Waals/Cahn-Hilliard/Ginzburg-Landau/φ4 type models [15,16,17,18], based on the square gradient (SG) approximation, and the more complex phase-field [19, 20] and classical density functional methods [21, 22]
Phase-field methods working on the molecular scale, termed phase-field crystal (PFC) models were introduced [24, 25], which can be classified as simple dynamical density functional approaches
The work of formation of such crystallites is expressed as a sum of a volumetric and an interfacial term: Whom = (4π/3)R3∆ω + 4πR2γSL, where R is the radius of the surface on which the surface tension acts, ∆ω is the thermodynamic driving force of solidification, and γSL is the free energy of the solid-liquid interface
Summary
The classical approach views the crystal-like fluctuations appearing in the undercooled liquid as small spherical domains of the bulk crystalline phase bound by a mathematically sharp solid-liquid interface [26] (known as the droplet model or capillarity approximation), while the formation, growth, and decay of these fluctuations is assumed to happen via a series of single-molecule attachment and detachment events. The large (several orders of magnitude) deviation between experimental and theoretical (CNT) nucleation rates reported for oxide glasses and other substances [1](a), [2](a), [14](a), [34] is attributable primarily to the failure of the droplet model. This view is supported by a direct evaluation of the nucleation barrier via umbrella sampling (a biased Monte Carlo technique) that shows that the droplet model relying on a constant γSL fails when predicting the nucleation barrier [5](a). For details see Ref. [37]
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