Hydrodynamic phase field crystal approach to interfaces, dislocations, and multi-grain networks

Abstract

We derive a phase field crystal model that couples the diffusive evolution of a microscopic structure with the fast dynamics of a macroscopic velocity field, explicitly accounting for the relaxation of elastic excitations. This model captures better than previous formulations the dynamics of complex interfaces and dislocations in single crystals as well as grain boundary migration in poly-crystals where the long-range elastic field is properly relaxed. The proposed model features a diffusivity that depends non-linearly on the local phase. It induces more localized interfaces between a disordered phase (liquid-like) and an ordered phase, e.g., stripes or crystal lattices. For stripes, the interface dynamics is shown to be strongly anisotropic. We also show that the model is able to evolve the classical PFC at mechanical equilibrium. However, in contrast to previous approaches, it is not restricted to a single-crystal configuration or small distortions from a fixed reference lattice. To showcase the capabilities of the model, we consider a few examples, from the annihilation of dislocation loops in a single crystal at mechanical equilibrium to the relaxation of a microstructure including crystalline domains with different orientations and grain boundaries. During the self-annihilation of a mixed type dislocation loop (i.e. not shear or prismatic), long-range elastic effects cause the loop to move out of plane before the annihilation event.

A phase field crystal theory of the kinematics of dislocation lines

“…We initiate a perfect square lattice of 101 × 101 unit cells and use the sHPFC model of Ref. [46] to apply a local stress in the central region which causes the nucleation of a dislocation dipole. The PFC deforms gradually, trying to account for the externally imposed stress, increasing from linear to non-linear strains until nucleation of a pure ±a 0 e x dislocation dipole.…”

“…As detailed in Ref. [46], we solve the system of coupled equations with a Fourier pseudo-spectral method. The spatial grid of the simulation is set to ∆x = ∆y = a 0 /7.…”

Amplitude expansion of the phase-field crystal model for complex crystal structures

“…This approach allows for describing crystalline systems over diffusive timescales [3]. Different dynamics and extensions have been proposed to provide advanced modeling of elastic relaxation [36][37][38][39].…”

Evaluation of the elastic field in phase‐field crystal simulations

“…However, in its basic formulation, lattice distortions evolve at diffusive dynamics too, which is often an unphysical limit. Several extensions have been proposed to tackle more accurate modeling of elastic relaxation [6][7][8][9][10][11][12]. We consider the versatile hydrodynamic phase-field crystal approach (hPFC), recently proposed to account for the relaxation of elastic excitations through the coupling of the classical governing equation with the dynamics of a macroscopic velocity field 𝒗 [11].…”

“…Several extensions have been proposed to tackle more accurate modeling of elastic relaxation [6][7][8][9][10][11][12]. We consider the versatile hydrodynamic phase-field crystal approach (hPFC), recently proposed to account for the relaxation of elastic excitations through the coupling of the classical governing equation with the dynamics of a macroscopic velocity field 𝒗 [11]. In brief, this model builds on the linear response theory for the relaxation to equilibrium with a free energy…”