Using the Cahn–Hilliard Theory in Metastable Binary Solutions

Abstract

A solution may be in one of three states: stable, unstable, or metastable. If the solution is unstable, phase separation is spontaneous and proceeds by spinodal decomposition. If the solution is metastable, the solution must overcome an activation barrier for phase separation to proceed spontaneously. This mechanism is called nucleation and growth. Manipulating morphology using phase separation has been of great research interest because of its practical use to fabricate functional materials. The Cahn-Hilliard theory, incorporating Flory-Huggins free energy, has been used widely and successfully to model phase separation by spinodal decomposition in the unstable region. This model is used in this paper to mathematically model and numerically simulate the phase separation by nucleation and growth in the metastable state for a binary solution. Our numerical results indicate that Cahn-Hilliard theory is able to predict phase separation in the metastable region but in a region near the spinodal line.

Polymerization-Induced Phase Separation in Rubber-Toughened Amine-Cured Epoxy Resins: Tuning Morphology from the Nano- to Macro-scale

“…Indeed, nucleation has been observed in binary polymer mixtures in the metastable window between the spinodal and binodal regions, particularly close to the spinodal region. , Moderate changes to composition can rapidly push the mixture over the spinodal curve into the unstable region, yielding macro-phase-separated domains as observed here at high values of n . Interestingly, a thermal quench is typically required to freeze polymer blends in the metastable state, − yet kinetic trapping of metastable morphologies has been demonstrated by arresting evolution of the morphology with cross-links formed between domains during polymerization …”

“…We speculate this formulation is frozen at a metastable nucleation state prior to entering an unstable, macro-phase-separated state at larger values of n. 49 Indeed, nucleation has been observed in binary polymer mixtures in the metastable window between the spinodal and binodal regions, particularly close to the spinodal region. 49,50 Moderate changes to composition can rapidly push the mixture over the spinodal curve into the unstable region, yielding macro-phaseseparated domains as observed here at high values of n. Interestingly, a thermal quench is typically required to freeze polymer blends in the metastable state, 50−52 yet kinetic trapping of metastable morphologies has been demonstrated by arresting evolution of the morphology with cross-links formed between domains during polymerization. 53 To examine the effects of the curing temperature on PIPS and thus the dynamic heterogeneity of these systems, a variety of temperatures were employed to cure formulations with the same variation in composition previously employed.…”

“…62 These pronounced compositional fluctuations and length scales indicate that the thermoset at m = 0.3 and n = 0.3, including the thermosets at m = 0.1 and 0.2 at the same n = 0.3, are close to a critical point (Figure 2c). 50,63 Beyond the macro-phase separation points, the composition of phaseseparated domains approach those of the binary thermosets, and the d m approaches 5.4 nm at n = 1, close to the d x-link = 4.4 nm of the DGEBA-T3000 thermoset, that is, the thermoset phase-separates into nearly pure rubber (DGEBA-T3000) and matrix domains (Tables S2 and S3).…”

Tuneable phase behaviour and glass transition via polymerization-induced phase separation in crosslinked step-growth polymers

“…This phenomenon is commonly seen in thermally quenched systems. 40–42 Qualitatively, we found little effect of cure temperature on the ultimate phase separated structure in comparison to the effect of CA composition and rubber content. This fact is exemplified by the data shown for m = 0.2, where the metastable state persists at n = 0.4 when cured at either 60 °C or 100 °C.…”

Emergence of uniform linearly-arranged micro-droplets entrapping DNA and living cells through water/water phase-separation

“…Thus, just after mechanical mixing, for the standing solution we may expect a coarsening process of phase-separation, corresponding to spinodal decomposition. It may be reasonable to adopt the Cahn–Hilliard Equation 35 – 41 to interpret the essence of the observed phenomena: where the free energy has two different contributions: bimodality with the order parameter and the interfacial energy. Here, is a parameter of diffusivity and is time.…”