[Paper Review] Crystallization of Simple Fluids: Relative Stability of f.c.c. and b.c.c Structures

Swarn Lata Singh, Yashwant Singh|arXiv (Cornell University)|Feb 27, 2009

Material Dynamics and Properties11 citations

TL;DR

This paper develops a free-energy functional for crystallizing fluids that includes both symmetry-conserved and symmetry-broken parts of the direct pair correlation function (DPCF), using the Ornstein-Zernike equation with Roger-Young closure and a perturbative expansion for the symmetry-broken component. It correctly predicts that inverse power fluids with n=12 freeze into f.c.c., while softer potentials (n≤6) favor b.c.c., resolving a long-standing challenge in density functional theory.

ABSTRACT

A free-energy functional for a crystal that contains both the symmetry conserved and symmetry broken parts of the direct pair correlation function is developed. The free-energy functional is used to investigate the crystallization of fluids interacting via the inverse power potential ; $u(r)=ε{(σ/r)}^n$. In agreement with simulation results we find that for $n=12$ the freezing is into close packed f.c.c structure while for soft repulsions $(n\leq 6)$ b.c.c phase is more stable.

Motivation & Objective

• To develop a first-principles statistical mechanical theory that correctly captures the relative stability of f.c.c. and b.c.c. phases during fluid crystallization.
• To address the failure of conventional density functional theory in describing the fluid-solid transition due to neglect of symmetry-broken contributions to the direct pair correlation function.
• To incorporate the qualitatively new, symmetry-breaking component of the DPCF that emerges at the crystal phase, which vanishes at melting.
• To test the predictive power of this improved functional against Monte Carlo simulation data for inverse power potentials across varying softness (n=12, 6, 4).

Proposed method

• Formulate a free-energy functional A[ρ] = A_id[ρ] + A_ex[ρ] where A_ex includes both symmetry-conserved c^(0) and symmetry-broken c^(b) parts of the DPCF.
• Calculate c^(0)(r) using the Ornstein-Zernike equation with the Roger-Young closure, ensuring thermodynamic consistency across densities.
• Model c^(b)(r₁,r₂;[ρ]) via a functional Taylor expansion, with the leading-order term expressed as an integral over the three-body DPCF of the isotropic fluid and density fluctuations ρ_b(r).
• Express c^(b) in Fourier space using reciprocal lattice vectors G, with coefficients c^(G)(r) depending on the difference vector r and the crystal structure.
• Perform functional integration over density and order parameter fields using a double parameterization (λ and ξ) to compute the excess free energy A_ex^(b) of the crystal.
• Use the grand thermodynamic potential ΔW = W_l - W to locate the fluid-solid coexistence point, with ΔW/N = 0 at transition.

Experimental results

Research questions

• RQ1 Does a free-energy functional that includes both symmetry-conserved and symmetry-broken DPCF components correctly predict the relative stability of f.c.c. and b.c.c. phases in inverse power fluids?
• RQ2 How significant is the contribution of the symmetry-broken DPCF to the free energy difference between fluid and solid phases, especially for soft repulsions?
• RQ3 Can the perturbative treatment of c^(b) via the three-body DPCF of the isotropic fluid provide a reliable estimate of the solid phase stability?
• RQ4 Why do conventional DFT approaches fail to predict the correct crystallization pathway for soft repulsive potentials?
• RQ5 To what extent does the inclusion of c^(b) improve agreement with Monte Carlo simulation results for n=12, 6, and 4?

Key findings

• For n=12, the f.c.c. structure is predicted to be more stable, in agreement with Monte Carlo simulations, due to favorable symmetry-conserved and symmetry-broken contributions.
• For n=6 and n=4, the b.c.c. structure becomes more stable than f.c.c., correctly reproducing simulation results, with the symmetry-broken DPCF playing a crucial stabilizing role.
• The symmetry-broken contribution to the free energy difference ΔW/N is non-negligible—accounting for about 25% of the symmetry-conserved part for n=12 and nearly 50% for n=4.
• Without the symmetry-broken term, the theory overestimates fluid stability, especially for soft potentials, leading to incorrect predictions.
• The perturbative expression for c^(b) via the three-body DPCF overestimates its magnitude for softer potentials (n=6,4), causing the predicted transition to occur at lower γ values than in simulations.
• The functional form of c^(G)(r) differs significantly in magnitude and r-dependence between f.c.c. and b.c.c. structures, explaining their distinct stabilities.