Deriving phase field crystal theory from dynamical density functional theory: Consequences of the approximations

Abstract

Phase field crystal (PFC) theory is extensively used for modelling the phase behaviour, structure, thermodynamics and other related properties of solids. PFC theory can be derived from dynamical density functional theory (DDFT) via a sequence of approximations. Here, we carefully identify all of these approximations and explain the consequences of each. One approximation that is made in standard derivations is to neglect a term of form ∇ · [n∇Ln], where n is the scaled density profile and L is a linear operator. We show that this term makes a significant contribution to the stability of the crystal, and that dropping this term from the theory forces another approximation, that of replacing the logarithmic term from the ideal gas contribution to the free energy with its truncated Taylor expansion, to yield a polynomial in n. However, the consequences of doing this are: (i) the presence of an additional spinodal in the phase diagram, so the liquid is predicted first to freeze and then to melt again as the density is increased; and (ii) other periodic structures, such as stripes, are erroneously predicted to be thermodynamic equilibrium structures. In general, L consists of a nonlocal convolution involving the pair direct correlation function. A second approximation sometimes made in deriving PFC theory is to replace L by a gradient expansion involving derivatives. We show that this leads to the possibility of the density going to zero, with its logarithm going to −∞ whilst being balanced by the fourth derivative of the density going to +∞. This subtle singularity leads to solutions failing to exist above a certain value of the average density. We illustrate all of these conclusions with results for a particularly simple model two-dimensional fluid, the generalised exponential model of index 4 (GEM-4), chosen because a DDFT is known to be accurate for this model. The consequences of the subsequent PFC approximations can then be examined. These include the phase diagram being both qualitatively incorrect, in that it has a stripe phase, and quantitatively incorrect (by orders of magnitude) regarding the properties of the crystal phase. Thus, although PFC models are very successful as phenomenological models of crystallisation, we find it impossible to derive the PFC as a theory for the (scaled) density distribution when starting from an accurate DDFT, without introducing spurious artefacts. However, we find that making a simple one-mode approximation for the logarithm of the density distribution log ρ(x) (rather than for ρ(x)), is surprisingly accurate. This approach gives a tantalising hint that accurate PFC-type theories may instead be derived as theories for the field log ρ(x), rather than for the density profile itself.

First order phase transitions and the thermodynamic limit

“…Both of the mean field models we have considered can be derived as local (gradient expansion) approximations of density functional theory (DFT) [19,38,66,67]. Formally, DFT is derived by averaging over all states of the system, i.e.…”

“…This local (i.e. partial differential) equation may be derived from a truncated gradient expansion in the DDFT description of an undercooled system undergoing crystallization [19,[38][39][40]. The governing equation also takes the form of conserved gradient dynamics for a scalar order parameter field f, as in equation (1), this time with the underlying free energy functional that has higher order spatial derivatives than the Cahn-Hilliard free energy functional (2).…”

“…Note that the coefficient of | | f  2 is now negative and so favors gradients in the order parameter, while the system is regularized by the strictly positive higher order term that limits the steepness of the variations in f. The parameter q represents the dominant wave number and r is the undercooling. This model has a critical point at r c =0, f c =0, and so is expected to fail before this point is reached [38]: in experiments the freezing transition is first order. The origin of this failure lies in the approximations made in deriving the PFC, but for r=r c the PFC model provides a qualitatively correct description of the freezing transition region.…”

“…Since freezing is in reality a first order transition [59], the PFC model is not correct for r>r tri . This is a consequence of the approximations made in deriving the model [38].…”

A coarse-grained phase-field crystal model of plastic motion

“…The temporal evolution of ψ is assumed to be governed by conserved and dissipative dynamics, hence changing on effectively diffusive time scales. The formulation is akin to Classical Density Functional Theory, and it has been justified to some extent on those grounds as well (Elder et al, 2007, van Teeffelen et al, 2009, Archer et al, 2019. Therefore this modeling is intermediate between fully atomistic and classical continuum theories.…”