Extending the Flory-Huggins Theory for Crystalline Multicomponent Mixtures

TL;DR

This work extends the Flory–Huggins free-energy framework to crystalline multicomponent mixtures by introducing crystallinity degrees of freedom and four state-dependent pair interactions that couple amorphous demixing with crystallization. The resulting mean-field model yields a comprehensive free-energy functional $\Delta G$ (and the density form $\Delta G_V$) that accounts for amorphous-amorphous, amorphous-crystalline, crystalline-amorphous, and crystalline-crystalline interactions, along with latent-heat $\Delta h_i$ and surface-energy $\Delta \sigma_i$ contributions. Chemical potentials $\mu_i$ are derived in closed form, and the formalism recovers the classical FH limit for purely amorphous binaries, while enabling melting-point depression analyses and phase diagrams for binary and ternary blends through convex-hull constructions. The framework captures a wide array of phenomena, including spinodal and binodal behavior in both amorphous and ordered states, and co-crystallization scenarios, offering a tractable path to qualitatively understand complex crystallization-demixing interplay in multi-component systems.

Abstract

The Flory-Huggins theory is a well-established lattice model that is commonly used to study the mixing of distinct chemical species. It can successfully predict phase separation phenomena in blends of incompatible materials. However, it is limited to amorphous mixtures, excluding systems where the phase segregation is shaped by the concurrent crystallization of one or several blend components. A generalization of the Flory-Huggins formalism is thus necessary to capture the coupling and the interplay of crystallization with amorphous demixing mechanisms, such as spinodal decomposition. This work therefore revolves around the derivation of a free energy model for multicomponent mixtures that encompasses the physics of both processes. It is detailed which concepts from the original Flory-Huggins theory are required to apprehend the presented developments and how the current framework is built upon them. Furthermore, additional discussion points address chemical potential calculations and selected examples of binary and ternary phase diagrams, thereby highlighting the variety of blend behaviors that can be represented.

Figures (6)

• Figure 0: Graphical abstract
• Figure 1: Schematic illustration of the mixing of a polymer solution on a two-dimensional Flory-Huggins lattice. The size proportions are $N_2=8$ for the polymer (in yellow) to $N_1 = 1$ for the solvent (in green).
• Figure 2: Phase diagrams of binary mixtures subject to a) UCST-type and b) LCST-type amorphous demixing. In a), the blend is strongly asymmetric ($N_1 = 100$ and $N_2 = 1$), which results in a miscibility gap that leans toward compositions richer in the smaller constituent. In b), the asymmetry is not as severe ($N_1 = 1$ and $N_2 = 2$) and the immiscible region is accordingly more centered. All relevant parameters used for the calculation of the diagrams are provided in the SI (SI-D).
• Figure 3: Phase diagrams of binary mixtures containing one species that can crystallize. In the first row, only the crystalline-amorphous parameter $\Delta \chi_{12}^{(ca)}$ is active and increased from a) $100/T$ to b) $0.2 + 350/T$. In the second row, $\Delta \chi_{12}^{(ca)}$ is maintained at $0.2 + 350/T$ while the effect of amorphous-amorphous interactions ranging from c) $\chi_{12}^{(aa)} = 0.3 + 110/T$ to d) $\chi_{12}^{(aa)} = 0.4 + 250/T$ is added. All relevant parameters used for the calculation of the diagrams are provided in the SI (SI-D).
• Figure 4: Phase diagrams of binary mixtures where both components can crystallize. In the first row, the amorphous-amorphous, amorphous-crystalline and crystalline-amorphous interaction parameters are active (i.e. $\chi_{12}^{(aa)}$, $\Delta \chi_{12}^{(ac)}$, and $\Delta \chi_{12}^{(ca)}$). $\chi_{12}^{(aa)}$ is increased from a) $0.3 + 150/T$ to b) $0.4 + 480/T$, while $\Delta \chi_{12}^{(ac)}$ and $\Delta \chi_{12}^{(ca)}$ are maintained constant at $0.1 + 100/T$ and $0.05 + 50/T$, respectively. In the second row, the effect of crystalline-crystalline compatibility is added. $\Delta \chi_{12}^{(cc)}$ is accordingly decreased from c) $0.3-30/T$ to d) $-60/T$. In both c) and d), the values of $\chi_{12}^{(aa)}$, $\Delta \chi_{12}^{(ac)}$, and $\Delta \chi_{12}^{(ca)}$ are the same as in a). All relevant parameters used for the calculation of the diagrams are provided in the SI (SI-D).