Phase diagram and commensurate-incommensurate transitions in the phase field crystal model with an external pinning potential

Abstract

We study the phase diagram and the commensurate-incommensurate transitions in a phase field model of a two-dimensional crystal lattice in the presence of an external pinning potential. The model allows for both elastic and plastic deformations and provides a continuum description of lattice systems, such as for adsorbed atomic layers or two-dimensional vortex lattices. Analytically, a mode expansion analysis is used to determine the ground states and the commensurate-incommensurate transitions in the model as a function of the strength of the pinning potential and the lattice mismatch parameter. Numerical minimization of the corresponding free energy shows reasonable agreement with the analytical predictions and provides details on the topological defects in the transition region. We find that for small mismatch the transition is of first order, and it remains so for the largest values of mismatch studied here. Our results are consistent with results of simulations for atomistic models of adsorbed overlayers.

Marginal stability analysis of the phase field crystal model in one spatial dimension

“…Indeed, it is straightforward to show that the stability analysis of the present hyperbolic PFC-model, described by Eq. (19), gives the same boundaries for stable-metastable-unstable regions in the "ǫ-k" phase diagram as it is treated for periodic patterns in the parabolic PFC-equation 37 and the hyperbolic SwiftHohenberg equation 24 . As a result of such analysis, the equilibrium 38 value of k ≈ 1, is obviously always linearly stable.…”

“…(26) and (27) for the local equilibrium limit τ → 0. Thus, one can characterize the behavior of φ in the hyperbolic PFC-equation (19) as an oscillatory relaxation in the high-frequency (short-wave) regime and monotonic relaxation to equilibrium in the low-frequency (long-wave) regime.…”

“…(19) shows that, in addition to dissipation described by the parabolic PFC-equation (4), inertia ∝ ∂ 2 φ/∂t 2 is also taken into account due to kinetic contribution (17 …”

“…The hyperbolic PFC-equation (19) describes relaxation of the slow thermodynamic variable φ as well as of the fast variable J (in a sense of the model of fast transformations 32 consistent with a general thermodynamics of transport processes 35 ). Eq.…”

“…It can be related to other continuum fields theories such as classical density-functional theory 8,9 and the atomic density function theory 10 . The PFC-model may also be considered as a conserved version of the Swift-Hohenberg equation and provides an efficient method for simulating liquid-solid transitions 11,12 , colloidal solidification 13 , dislocation motion and plasticity 14,15 , glass formation 16 , epitaxial growth 6,17 , grain boundary premelting 18 , surface reconstructions 19 , and grain boundary energies 20 .…”