Stress in ordered systems: Ginzburg-Landau-type density field theory

Abstract

We present a theoretical method for deriving the stress tensor and elastic response of ordered systems within a Ginzburg-Landau-type density field theory in the linear regime. This is based on spatially coarse graining the microscopic stress which is determined by the variation of a free energy with respect to mass displacements. We find simple expressions for the stress tensor for phase field crystal models for different crystal symmetries in two and three dimensions. Using tetradic product sums of reciprocal lattice vectors, we calculate elastic constants and show that they are directly related to the symmetries of the reciprocal lattices. We also show that except for bcc lattices there are regions of model parameters for which the elastic response is isotropic. The predicted elastic stress-strain curves are verified by numerical strain-controlled bulk and shear deformations. Since the method is independent of a reference state, it extends also to defected crystals. We exemplify this by considering an edge and screw dislocation in the simple cubic lattice.

Coarse-grained modeling of crystals by the amplitude expansion of the phase-field crystal model: an overview

“…where P = f − n(δF/δn) is a pressure term summing up to the mechanical stress, with f the integrand in Eq. ( 1), the second term arising when considering mass-conserving deformations [88], and L ≡ 1 + ∇ 2 . In terms of amplitudes, integrating over the a unit cell with n expressed via Eq.…”

Explicit temperature coupling in phase-field crystal models of solidification

“…Elasticity in PFC models can be characterized by small perturbations of the equilibrium density ψ. Therefore, it features variations of ψ over a length scale significantly larger than the lattice spacing, and a coarse-graining of ψ is usually considered in this context [35,38]. In particular, one may consider an expansion of the periodic density as in equation ( 7) with complex amplitudes encoding lattice distortion over a displacement field u.…”

“…This ansatz can be also exploited to derive a coarse-grained PFC model, namely the amplitude expansion of the PFC (APFC), where η n are the variables to solve for [39][40][41]. Within the PFC model, η n as well as ψ can be computed from a demodulation of ψ [38,42]:…”

“…The demodulation ( 9) includes a coarse-graining procedure over one unit cell UC = [0, a 1 ] × [0, a 2 ] with a 1 = p, a 2 = √ 3p/2 and p = 4π/ √ 3. Following recent works [33,36,38,43], one may obtain the mechanical stress tensor σ ij ≡ σ ij (r, t) without isotropic pressure terms from the density field:…”

“…with • a spatial average over the unit cell, which may be explicitly performed in reciprocal space through the smoothing kernel similarly to equation ( 9) for ψ. Equation ( 10) generally defines σ ij up to a phase-dependent divergence-free contribution, which can be removed by considering ψ as expressed through an onemode approximation as in equation ( 7) [38]. A coarse-grained description of the crystal is also achieved through amplitudes from equation (9).…”