crystals Article Nucleation and Post-Nucleation Growth in Diffusion- Controlled and Hydrodynamic Theory of Solidification Frigyes Podmaniczky 1, * and László Gránásy 1,2, *   Citation: Podmaniczky, F.; Gránásy, L. Nucleation and Post-Nucleation Growth in Diffusion-Controlled and Hydrodynamic Theory of Solidification. Crystals 2021, 11, 437. https://doi.org/10.3390/cryst11040437 Academic Editors: Wolfram Miller and Koichi Kakimoto Received: 13 March 2021 Accepted: 13 April 2021 Published: 17 April 2021 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). 1 Wigner Research Centre for Physics, P.O. Box 49, H-1525 Budapest, Hungary 2 BCAST, Brunel University, Uxbridge, Middlesex UB8 3PH, UK * Correspondence: podmaniczky.frigyes@wigner.hu (F.P.); granasy.laszlo@wigner.hu (L.G.) Abstract: Two-step nucleation and subsequent growth processes were investigated in the framework of the single mode phase-field crystal model combined with diffusive dynamics (corresponding to colloid suspensions) and hydrodynamical density relaxation (simple liquids). It is found that independently of dynamics, nucleation starts with the formation of solid precursor clusters that consist of domains with noncrystalline ordering (ringlike projections are seen from certain angles), and regions that have amorphous structure. Using the average bond order parameter q 6 , we dis- tinguished amorphous, medium range crystallike order (MRCO), and crystalline local orders. We show that crystallization to the stable body-centered cubic phase is preceded by the formation of a mixture of amorphous and MRCO structures. We have determined the time dependence of the phase composition of the forming solid state. We also investigated the time/size dependence of the growth rate for solidification. The bond order analysis indicates similar structural transitions during solidification in the case of diffusive and hydrodynamic density relaxation. Keywords: classical density functional theory; molecular modelling; two-step nucleation; growth kinetics; hydrodynamic theory of freezing 1. Introduction Colloid suspensions are considered as model systems for simple molecular liquids [1,2]. A wealth of information is available for the crystallization of colloids including nucleation, which was obtained via tracing the trajectories of colloid particles via optical methods [2–4]. For example, in colloids, two-step nucleation, assisted by amorphous/liquid precursor, appears to be a general phenomenon [5–10]. Analogous experimental information is not accessible for highly undercooled simple liquids. This raises the question, to what extent the observations made on colloids are applicable to metallic liquids. There appear to be differences: e.g., crystal growth is usually diffusion controlled for colloid suspensions, whereas it may be interface controlled for a hypercooled pure liquid. The velocity of a flat crystal–liquid interface depends on the growth mechanism. In the case of diffusionless pro- cesses a steady state velocity is established, whereas in the diffusion-controlled processes the front velocity, v, is proportional to t −1/2 , where t is time [11,12]. The growth rate also depends on the curvature of the interface, as predicted by the classic kinetic model based on monomer attachment and detachment [13,14]. For spherical particles: v = 16D λ 2  3v at 4π  1/3 sinh  v at 2k B T  Δg − 2γ r  (1) where D is the self-diffusion coefficient, λ the jump distance of atoms, v at the atomic volume, k B Boltzmann’s constant, T the temperature, Δg the volumetric free energy difference between the liquid and the solid, γ the solid-liquid interfacial free energy, and r the radius of the crystallite. Note that for the critical size, r*=2γ/Δg, the deterministic growth rate is zero, for larger particles it is positive, whereas for smaller particles it is negative. This Crystals 2021, 11, 437. https://doi.org/10.3390/cryst11040437 https://www.mdpi.com/journal/crystals