Correlated noise effect on the structure formation in the phase‐field crystal model

Abstract

A phase-field crystal model (PFC model) which takes into account exponential relaxation of the atomic flux and its fluctuations is developed. The model corresponds to a system undergoing phase transformation described with a partial differential equation of hyperbolic type. Such a model covers slow and rapid regimes of interface propagation at small and large driving forces during melting and solidification. The analysis is done for the evolution of atomic crystal lattices appearing from a metastable homogeneous liquid for the chemically pure system supercooled below its critical temperature. Numerical simulation of the system "Liquid -Body Centered Cubic (BCC) crystal lattice" allows us to formulate the hypothesis about the formation of metastable periodic solutions (atomic configurations which are not in the thermodynamic equilibrium) during the relaxation of atomic configurations to the stable equilibrium. These metastable states may be destroyed (or even avoided) due to the action of colored noise. Namely, considering spatiotemporal correlations of the atomic flux fluctuations, we have found that the temporal correlations promote selecting long-range atomic lattices, whereas the spatial correlations corresponding to the periodic structure scales decelerate such processes.

Structure diagram and dynamics of formation of hexagonal boron nitride in phase-field crystal model

“…This metastable structure might be seen as a distorted pattern that has often been observed in PFC modelling of different crystal lattices [10]. However, an action of the correlated noise given by the last terms in the right-hand sides of equations (2.20) and (2.21) can overcome such locally metastable states and help to relax structure to the thermodynamic equilibrium state [24]. In these peculiarities of atomic evolution, we see theoretical meaning and practical application of the obtained structure diagram, figure 2 a : one can find the equilibrium crystalline phase consistent with given initial parameters.…”

“…Also, here we do not derive a chemical potential explicitly as soon as its determination requires the separate contribution for each component. From the minimization of the free energy equation (2.12), it is possible to formulate the following dynamical equations on number densities with the spatio-temporal correlated noise sources ζi [24]: τA∂2ρAnormal∂t2+normal∂ρAnormal∂t=bold∇⋅(MA∇δscriptFδρA+bold-italicζA(r,t)) and τB∂2ρBnormal∂t2+normal∂ρBnormal∂t=bold∇⋅(MB∇δscriptFδρB+bold-italicζB(r,t)), where MA and MB are the correspondent mobilities, τA and τB are the relaxat...…”

“…Also, here we do not derive a chemical potential explicitly as soon as its determination requires the separate contribution for each component. From the minimization of the free energy equation (2.12), it is possible to formulate the following dynamical equations on number densities with the spatio-temporal correlated noise sources ζ i [24]:…”

“…Here In the following we consider a case with zero noise contributions, indeed the model can be extended to include the separate noise fluctuations correspondent to each species (in the manner of [24]). Presented equations could also be obtained using the Smoluchowski approach from the particle motion equations [44][45][46].…”

“…These simulations and the analysed patterns have been obtained for equilibrium phase co-existence as well as for the first-and second-order phase transformations [3,[17][18][19][20][21]. The PFC model allows one to describe the binary systems [2,22,23] and incorporate thermal fluctuations with proper memory of the system [14,24].…”