Exploring transition pathways in the Landau-Brazovskii model

TL;DR

This work addresses how modulated phases in the three-dimensional Landau--Brazovskii model transform via nucleation-driven transitions. It develops and applies a LB saddle dynamics framework to extract minimum-energy pathways and transition states, aided by CAM-based discretization and adaptive domains. The results map a phase diagram with eight equilibria and reveal the structure of critical nuclei, Hessian spectra, and energy barriers across seven representative transitions, demonstrating systematic parameter dependencies. The approach provides a general, computationally tractable route to understanding nucleation mechanisms in modulated-phase systems and could guide analyses of block copolymers and related materials in mean-field regimes.

Abstract

The Landau-Brazovskii model provides a theoretical framework for describing various phases arising from competing short- and long-range interactions in many physical systems. In this work, we investigate phase transitions among various ordered phases within the three-dimensional Landau-Brazovskii model. We construct the phase diagram of this model, which encompasses eight distinct phases, and systematically compute the transition pathways connecting various metastable and stable states using the Landau-Brazovskii saddle dynamics. Along each transition pathway, the critical nucleus is identified with some detailed analyses of its shape, energy barrier, and Hessian eigenvalues. Furthermore, we explore how the transition state is influenced by model parameters, revealing systematic trends in critical nucleus sizes and energy barrier heights. Our results provide a comprehensive characterization of the nucleation mechanisms within the Landau-Brazovskii model and offer valuable insights into the structural transformations of modulated-phase systems.

Figures (5)

• Figure 1: Phase diagram of the LB model in the $\gamma$-$\tau$ parameter space. Colored regions separated by solid lines indicate different stable phases (disorder, FCC, DG, LAM, HEX, BCC, A15 and $\sigma$), and representative configurations of the order parameters of ordered phases are illustrated in the right panel. Dashed lines are the metastability limit of BCC and DG. Black dots mark parameter sets used in the following cases, with numbers labeling each one.
• Figure 2: (a) Transition pathway from BCC to HEX at $\tau = -0.3$, $\gamma = 0.8$. In all cases, the ordered phases and saddle points are three-dimensional structures within $\Omega=[0,L]^3$, while we present local two-dimensional slice planes of the domain for clarity of visualization, specifically the $[10\bar{1}]$–$[111]$ plane in this figure. For all figures, TS denotes the transition state, the critical nucleus is enclosed by the white line, and scale bars represent $4\pi$. White dashed lines indicate nuclei other than the critical nucleus. To the right of TS, the nuclear boundary defined as the $\Phi_{\mathrm{s}}$ level set of $\Phi_{\mathrm{s}}(\mathbf{r}) = \Phi_{\mathrm{s}}^*$ is shown as a red surface. (b) Energy barrier and (c) critical nucleus size $l_x$ in $\tau\in[-0.3,-0.26]$, $\gamma\in [0.8,0.84]$.
• Figure 3: (a) Transition pathway from HEX to BCC at $\tau = -0.008$, $\gamma = 0.8$. The slice plane is $[10\bar{1}]$-[111]. (b) Transition pathway from HEX to LAM at $\tau = -0.4$, $\gamma = 0.22$. The slice plane is $[10\bar{1}]$-$[1\bar{2}1]$.
• Figure 4: (a) Transition pathway from disorder to BCC at $\tau = 0.01$, $\gamma = 0.4$. The slice plane is $[10\bar{1}]$-$[111]$. (b) Transition pathway from BCC to disorder at $\tau = 0.05$, $\gamma = 0.6$. The slice plane is $[10\bar{1}]$-$[111]$.
• Figure 5: (a) Transition pathway from DG to HEX at $\tau = -0.14$, $\gamma = 0.7$. The slice plane is $[10\bar{1}]$-$[1\bar{2}1]$. (b) Transition pathway from DG to LAM at $\tau = -0.32$ and $\gamma = 0.08$. The slice plane is $[10\bar{1}]$-$[111]$.