[Paper Review] Thermodynamic consistency and fast dynamics in phase field crystal modeling

Mowei Cheng, Stefaan Cottenier+1 more|arXiv (Cornell University)|Jul 23, 2008

Solidification and crystal growth phenomena3 citations

TL;DR

This paper presents a unified entropy functional formalism to derive equations of motion and ensure thermodynamic consistency in phase field crystal (PFC) models with both conserved order parameters and fast dynamics. By extending prior work on uniform phase field models, it establishes that PFC models with fast variables maintain thermodynamic consistency, enabling robust modeling of periodic microstructures with dynamic accuracy.

ABSTRACT

A general formulation is presented to derive the equation of motion and to demonstrate thermodynamic consistency for several classes of phase field models at once. It applies to models with a conserved phase field, describing either uniform or periodic stable states, and containing slow as well as fast thermodynamic variables. The approach is based on an entropy functional formalism previously developed in the context of phase field models for uniform states [P. Galenko and D. Jou, Phys. Rev. E {\bf 71}, 046125 (2005)] and thus allows to extend several properties of the latter to phase field models for periodic states (phase field crystal models). In particular, it allows to demonstrate the concept of thermodynamic consistency for phase field crystal models with fast dynamics.

Motivation & Objective

• To extend the entropy functional formalism from uniform phase field models to phase field crystal (PFC) models with periodic structures.
• To ensure thermodynamic consistency in PFC models that include fast thermodynamic variables.
• To derive consistent equations of motion for PFC models with conserved order parameters and both uniform and periodic stable states.
• To unify the treatment of slow and fast dynamics within a single thermodynamically consistent framework for PFC modeling.

Proposed method

• Adapts the entropy functional formalism previously used for uniform phase field models to the case of periodic structures in phase field crystal models.
• Derives the equation of motion from the entropy functional using variational principles, ensuring consistency with thermodynamic laws.
• Incorporates both conserved and non-conserved order parameters, allowing modeling of systems with fast relaxation of thermodynamic variables.
• Applies the formalism to models with either uniform or periodic stable states, demonstrating its broad applicability.
• Ensures that the derived dynamics satisfy the second law of thermodynamics by construction through the entropy functional.
• Validates the approach by showing that the resulting equations preserve thermodynamic consistency even when fast dynamics are present.

Experimental results

Research questions

• RQ1 Can the entropy functional formalism be extended to phase field crystal models with periodic structures?
• RQ2 How can thermodynamic consistency be rigorously demonstrated in PFC models that include fast-dynamic variables?
• RQ3 What is the general form of the equation of motion for PFC models with conserved order parameters and both slow and fast thermodynamic variables?
• RQ4 Does the formalism preserve thermodynamic consistency when fast dynamics are included in the model?
• RQ5 Can the same framework be applied uniformly to both uniform and periodic stable states in phase field models?

Key findings

• The entropy functional formalism successfully extends to phase field crystal models with periodic structures, enabling consistent modeling of microstructures with atomic-scale periodicity.
• The derived equations of motion are thermodynamically consistent, ensuring that entropy production remains non-negative by construction.
• The approach unifies the treatment of systems with both slow and fast thermodynamic variables, maintaining consistency across different time scales.
• Thermodynamic consistency is preserved even when fast dynamics are included, which is critical for accurate modeling of rapid relaxation processes.
• The formalism provides a general framework applicable to both conserved and non-conserved order parameters in PFC models with periodic stable states.
• The method ensures that the resulting dynamics satisfy the second law of thermodynamics, validating the physical relevance of the derived equations.