Non-equilibrium Phase Separation Research Articles

On the 2D Cahn–Hilliard Equation with Inertial Term

Galenko et al. proposed a modified Cahn–Hilliard equation to model rapid spinodal decomposition in non-equilibrium phase separation processes. This equation contains an inertial term which causes the loss of any regularizing effect on the solutions. Here we consider an initial and boundary value problem for this equation in a two-dimensional bounded domain. We prove a number of results related to well-posedness and large time behavior of solutions. In particular, we analyze the existence of bounded absorbing sets in two different phase spaces and, correspondingly, we establish the existence of the global attractor. We also demonstrate the existence of an exponential attractor.

The influence of neutron irradiation in FFTF on the microstructural and microchemical development of Mo–41Re at 470–730 °C

Specimens of Mo–41 wt% Re irradiated in the fast flux test facility (FFTF) experience significant and non-monotonic changes in density that arise first from radiation-induced segregation, leading to non-equilibrium phase separation, and second by progressive transmutation of Re to Os. As a consequence the density of Mo–41Re initially decreases and then increases thereafter. Beginning as a single-phase solid solution of Re and Mo, irradiation of Mo–41 wt% Re over a range of temperatures (470–730 °C) to 28–96 dpa produces a high density of thin platelets of a hexagonal close-packed (hcp) phase identified as a solid solution of Re, Os and possibly a small amount of Mo. These hcp precipitates are thought to form in the alloy matrix as a consequence of strong radiation-induced segregation to Frank loops. Grain boundaries also segregate Re to form the hcp phase, but the precipitates are much bigger and more equiaxed in shape. Although not formed at lower dose, continued irradiation at 730 °C leads to the co-formation of late-forming chi-phase, an equilibrium phase that then competes with the preexisting hcp phase for rhenium.

Homogeneous phase separation in binary alloys under ion irradiation conditions: Role of interstitial atoms

Kinetic Monte Carlo simulations have been performed to investigate homogeneous precipitation in binary metal alloys under ion irradiation conditions. The kinetic model includes defect production, recombination, defect trapping, the formation of defect-solute complexes, and atomic mixing. While no special assumptions are made about vacancy diffusion, interstitial diffusion is assumed fast compared to the characteristic time scales of all other processes. The diffusion path of an interstitial, moreover, ends with one of the three possible outcomes: recombination, trapping in a solute-rich location, or clustering with other interstitials. Interstitials promote solute segregation in this model by the formation of mobile interstitial-solute complexes. Several unusual features of radiation-induced precipitation are revealed in this model. For ideal alloys, we find at a given temperature that for every trapping number [i.e., the minimum number of nearest-neighbor solute (type $B$) atoms required to trap a migrating interstitial, ${N}_{BT}$], there exists a corresponding defect pair production rate ${K}_{0}$, below which the alloy becomes a random solution state and above which macroscopic phase separation occurs. ${K}_{0}$ depends on the length scale of trapping sites $L$ as ${L}^{4}$. Solute-rich precipitates have the composition ${c}_{B}$ approaching ${c}_{B}={N}_{BT}∕z$, where $z$ is the lattice coordination number. This feature results in ``swelling'' of precipitates, i.e., dilution of initially pure $({c}_{B}=1)$ precipitates located in the matrix prior to ion irradiation. Ballistic mixing is observed to erode precipitates, and above some critical rate the system reverts to a random solution. For ideal solutions, and even those with a small tendency for ordering, phase separation occurs due to the interstitial interaction with solutes. At sufficiently high positive values of the heats of mixing, the usual thermal vacancy-driven precipitation prevails. Between these low and high limits, the alloy unexpectedly enters a field of solid solutions. Finally, it is shown that even in the absence of the interstitial trapping $({N}_{BT}>z)$, alloys with a small positive ordering energy can undergo nonequilibrium phase separation at a composition below its solubility limit due to an effective trapping of vacancies in solute-rich locations. The significance of these findings for real alloy systems is discussed.

Pulled fronts in the Cahn–Hilliard equation

A possibility of pulling a front by a moving inhomogeneity is considered in the context of the Cahn–Hilliard equation, which is a generic model of nonequilibrium phase-separation processes. The critical (maximum) velocity of the inhomogeneity, at which it is still able to steadily drag the front, is found in an analytical approximation, using both perturbation theory and a quasiparticle description of the front. A case of steep inhomogeneities is studied in detail by means of direct simulations, showing that the analytical prediction for the critical velocity is in very good agreement with numerical results for small and moderate values of the inhomogeneity's strength. If the driving velocity exceeds the critical value, the kink is eventually destroyed. If the perturbation is strong, the simulations show that the actual critical velocity is larger than a formally extended analytical value, i.e., the kink turns out to be more robust than it is expected from the perturbative results.

Nonequilibrium Lipid Domain Growth in the Gel−Fluid Two-Phase Region of a DC16PC−DC22PC Lipid Mixture Investigated by Monte Carlo Computer Simulation, FT-IR, and Fluorescence Spectroscopy

The nonequilibrium ordering dynamics of lipid domains in the gel−fluid coexistence region of an equimolar DC16PC−DC22PC lipid mixture has been studied by means of Monte Carlo computer simulation, fluorescence, and Fourier transform infrared spectroscopy. The results reveal that the nonequilibrium phase separation process after a sudden temperature or pressure quench of the binary mixture from the one-phase fluid region into the two-phase gel−fluid coexistence region has a strong influence on the lateral membrane organization on different length scales. This is manifested as the formation of a heterogeneous lateral bilayer structure composed of long-living gel and fluid lipid domains characterized by a relaxation time on the order of hours. In the early time stage of the phase separation process a distinct local lipid structure of ordered DC16PC lipids is formed at the dynamically changing network of domain boundaries. Our combined theoretical and experimental investigations suggest that nonequilibrium eff...

Differential Scanning Calorimetry Study of Phase Transitions in Poly(lactide)−Chloroform−Methanol Systems

The phase separation processes occurring in poly(l-lactide) (PLLA)−chloroform−methanol mixtures and poly(dl-lactide) (PDLLA)−chloroform−methanol mixtures have been studied using differential scanning calorimetry, cloud point measurements, and optical microscopy. It is demonstrated that liquid−liquid demixing occurs in ternary solutions of PDLLA at sufficiently high methanol concentrations. For PLLA-containing solutions, both liquid−liquid demixing processes and solid−liquid demixing processes occur. The most important parameters for the phase separation process are the applied cooling rate and the polymer concentration. Nonequilibrium phase separation processes play an important role in the phase separation of ternary PLLA solutions. Due to the high activation energy needed for crystallization, liquid−liquid demixing will usually precede solid−liquid demixing even if solid−liquid demixing is favored thermodynamically. Only at low cooling rates and high polymer concentration does solid−liquid demixing take...

Phase transitions of polysulfone solution during coagulation

Coagulation of a polysulfone solution film was induced by water vapour sorption under atmospheric exposure, followed by water diffusion in a coagulation bath. Demixing in the skin region of the solution film can involve both equilibrium and non-equilibrium phase separation. Surface pore size of the solidified film is proportional to the duration of the liquidliquid phase separation during vapour sorption.

Microstructural evolution and thermal stability Associated with a Gas-Atomized Cu-Nb Alloy

High-pressure inert gas atomization (HPGA) has been used to produce rapidly solidified Cu-21Nb-2Mo (weight percent) powders with a range of particle sizes and microstructures. The associated microstructures depend on particle size. Specifically, fine-scale particles (\s15 μm@#@) are characterized by a predominance of multiphase spheroids and a small population of Nb-based dendrites in an almost pure matrix of Cu. In contrast, large particles (45 to 75 μm@#@) contain only Nb-based dendrites in a Cu matrix. The volume fraction of the dispersed constituent is much higher in the former instance than in the latter. The change in microstructure with particle size is analyzed in terms of both the amount of undercooling and cooling rate of the liquid droplets prior to and during solidification. In particular, the large undercoolings in the fine particles are believed to induce a nonequilibrium liquid phase separation which results in a high volume fraction of spheroidal, multiphase Nb-Cu particles within a Cu-rich matrix containing Nb-rich dendrites. High-temperature (900 °C) isothermal annealing treatments have also been performed on consolidated material to determine the inherent thermal stability of these microstructures.

Interfacial Phenomena in Change-of-Phase Heat Transfer: Low Concentration Polymer Solutions

Interferometry was used to measure in situ the thickness profiles of evaporating thin films as a function of heat flux for low concentration polystyrenel/cyclohexane solutions. Evaporative heat flow rates per unit width of contact line were calculated based on polynomial fits of interference data, combined with a fluid flow model. Temperature profiles obtained from thermocouple measurements were employed in evaluating macroscopic interline region heat sink capacities using a one-dimensional heat conduction model. Reasonable agreement between evaporative heat fluxes derived from both methods was established. Pure solvent integral heat fluxes were not significantly altered through polymer addition. Fluid flow toward the triple interline to replace that which evaporated resulted in an apparent non-equilibrium phase separation of concentrated polymer solution from the bulk phase; the separation increased with increasing polymer molecular weight and concentration. A substantial rise in the reflectivity beyond ...