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This paper presents an exploration of a specific type of a generalized multicomponent solution model, which appears to be first given by Saulov in the current explicit form. The assumptions of the underlying theory and a brief derivation of the main equation have been provided preliminarily for completeness and notational consistency. The resulting formulae for the Gibbs free energy of mixing and the chemical potentials are multivariate polynomials with physically meaningful coefficients and the mole fractions of the components as variables. With one additional assumption about the relative magnitudes of the solvent–solute and solute–solute interaction exchange energies, combining rules were obtained that express the mixed coefficients of the polynomial in terms of its pure coefficients. This was done by exploiting the mathematical structure of the asymmetric form of the solvent chemical potential equation. The combining rules allow one to calculate the thermodynamic properties of the solvent with multiple solutes from binary mixture data only (i.e., each solute with the solvent), and hence, are of practical importance. Furthermore, a connection was established between the osmotic virial coefficients derived in this work and the original osmotic virial coefficients of Hill found by employing a different procedure, illustrating the equivalency of what appears to be two different theories. A validation of the combining rules derived here has been provided in a separate paper where they were successfully used to predict the freezing points of ternary salt solutions of water. [ABSTRACT FROM AUTHOR]

GaN-based heterostructures have been used in high-power radio frequency applications for a number of years due to the wide bandgap of GaN and high values of spontaneous and piezoelectric polarization, resulting in high breakdown voltages and high 2D carrier concentrations. However, the accurate modeling of low-field electron mobility within such structures remains a topic of interest. This paper presents a comprehensive numerical model for calculating the carrier mobility within a 2D electron gas in Al x Ga 1 − x N/GaN high electron mobility transistors. The model is based on solving the Schrödinger and Poisson equations self-consistently, taking into account the polarization charges at material interfaces and performing semi-classical numerical calculations of low-field electron mobility within the momentum relaxation time approximation, taking into account all relevant scattering mechanisms. Both intra- and inter-subband transitions are considered, and the differences in intra- and inter-subband scattering rates are analyzed for some of the key scattering mechanisms. The importance of including inter-subband transitions in the calculations is demonstrated by comparing the calculated results with experimentally measured mobilities. [ABSTRACT FROM AUTHOR]

The phase-field equations have many attractive characteristics. First, phase separation can be induced by the phase-field equations. It transforms from a single homogeneous mixture to two distinct phases in a nascent state. Second, the solution of the phase-field equations is bounded by a finite value. It is beneficial to ensure numerical stability. Third, the motion of the interface can be described by geometric features. It is helpful for expressing natural phenomena in mathematical terms. Fourth, the phase-field equations possess the energy dissipation law. This law is about degeneration and decay. It tells us in thermodynamics that all occurrences are irreversible processes. In this paper, we would like to investigate the numerical implementation of the Allen–Cahn (AC) equation, which is the classical one of the phase-field equations. In phase field modeling, the binary phase system is described using a continuous variable called the order parameter. The order parameter can be categorized into two forms: conserved, which represents the physical property such as concentration or mass, and non-conserved, which does not have the conserved physical property. We consider both the non-conservative and conservative AC equations. Our interest is more precisely to scrutinize the utilization of the discrete Laplacian operator in the AC equation by considering the conservative and non-conservative order parameter ϕ. Constructing linearly implicit methods for solving the AC equation, we formulate a gradient-descent-like scheme. Therefore, reinterpreting the implicit scheme for the AC equation, we propose a novel numerical scheme in which solutions are bounded by 1 for all t > 0. Together with the conservative Allen–Cahn equation, our proposed scheme is consistent when mass is conserved as well. From a numerical point of view, a linear, unconditionally energy stable splitting scheme is transformed into a gradient-descent-like scheme. Various numerical simulations are illustrated to demonstrate the validity of the proposed scheme. We also make distinctions between the proposed one and existing numerical schemes. [ABSTRACT FROM AUTHOR]

Machine learning has proven exceptionally competent in numerous applications of studying dynamical systems. In this article, we demonstrate the effectiveness of reservoir computing, a famous machine learning architecture, in learning a high-dimensional spatiotemporal pattern. We employ an echo-state network to predict the phase ordering dynamics of 2D binary systems—Ising magnet and binary alloys. Importantly, we emphasize that a single reservoir can be competent enough to process the information from a large number of state variables involved in the specific task at minimal computational training cost. Two significant equations of phase ordering kinetics, the time-dependent Ginzburg–Landau and Cahn–Hilliard–Cook equations, are used to depict the result of numerical simulations. Consideration of systems with both conserved and non-conserved order parameters portrays the scalability of our employed scheme. [ABSTRACT FROM AUTHOR]

We analyze a two-body non-Hermitian two-site Sachdev–Ye–Kitaev (SYK) model with the couplings of one site complex conjugated to the other site. This model, with no explicit coupling between the sites, shows an infinite number of second-order phase transitions, which is a consequence of the factorization of the partition function into a product over Matsubara frequencies. We calculate the quenched free energy in two different ways: first in terms of the single-particle energies and second by solving the Schwinger–Dyson equations of the two-site model. The first calculation can be done entirely in terms of a one-site model. The conjugate replica enters due to non-analyticities when Matsubara frequencies enter the spectral support of the coupling matrix. The second calculation is based on the replica trick of the two-site partition function. Both methods give the same result. The free-fermion partition function can also be rephrased as a matrix model for the coupling matrix. Up to minor details, this model is the random matrix model that describes the chiral phase transition of QCD, and the order parameter of the two-body model corresponds to the chiral condensate of QCD. Comparing to the corresponding four-body model, we are able to determine which features of the free energy are due to the chaotic nature of the four-body model. The high-temperature phase of both models is entropy dominated, and in both cases, the free energy is determined by the spectral density. The chaotic four-body SYK model has a low-temperature phase whose free energy is almost temperature-independent, signaling an effective gap of the theory even though the actual spectrum does not exhibit a gap. On the other hand, the low-temperature free energy of the two-body SYK model is not flat; in fact, it oscillates to arbitrarily low temperature. This indicates a less desirable feature that the entropy of the two-body model is not always positive in the low-temperature phase, which most likely is a consequence of the non-hermiticity. [ABSTRACT FROM AUTHOR]

The dynamics of ensembles of phase oscillators are usually described considering their infinite-size limit. In practice, however, this limit is fully accessible only if the Ott–Antonsen theory can be applied, and the heterogeneity is distributed following a rational function. In this work, we demonstrate the usefulness of a moment-based scheme to reproduce the dynamics of infinitely many oscillators. Our analysis is particularized for Gaussian heterogeneities, leading to a Fourier–Hermite decomposition of the oscillator density. The Fourier–Hermite moments obey a set of hierarchical ordinary differential equations. As a preliminary experiment, the effects of truncating the moment system and implementing different closures are tested in the analytically solvable Kuramoto model. The moment-based approach proves to be much more efficient than the direct simulation of a large oscillator ensemble. The convenience of the moment-based approach is exploited in two illustrative examples: (i) the Kuramoto model with bimodal frequency distribution, and (ii) the "enlarged Kuramoto model" (endowed with nonpairwise interactions). In both systems, we obtain new results inaccessible through direct numerical integration of populations. [ABSTRACT FROM AUTHOR]

Water management is one of the key issues in proton exchange membrane fuel cells. Fundamentally, it is related to dynamic behaviors of droplets on a gas diffusion layer (GDL) surface, and consequently they are investigated in this work. A two-dimensional hybrid method is employed to implement numerical simulations, in which the flow field is solved by using the lattice Boltzmann method and the interface between droplet and gas is captured by solving the Cahn-Hilliard equation directly. One or two liquid droplets are initially placed on the GDL surface of a gas channel, which is driven by the fully developed Poiseuille flow. At a fixed channel size, the effects of viscosity ratio of droplet to gas (μ*), Capillary number (Ca, ratio of gas viscosity to surface tension), and droplet interaction on the dynamic behaviors of droplets are systematically studied. By decreasing viscosity ratio or increasing Capillary number, the single droplet can detach from the GDL surface easily. On the other hand, when two identical droplets stay close to each other or a larger droplet is placed in front of a smaller droplet, the removal of two droplets is promoted. [ABSTRACT FROM AUTHOR]

Recently, we have derived an effective Cahn-Hilliard equation for the phase separation dynamics of active Brownian particles by performing a weakly non-linear analysis of the effective hydrodynamic equations for density and polarization [Speck et al., Phys. Rev. Lett. 112, 218304 (2014)]. Here, we develop and explore this strategy in more detail and show explicitly how to get to such a large-scale, mean-field description starting from the microscopic dynamics. The effective free energy emerging from this approach has the form of a conventional Ginzburg-Landau function. On the coarsest scale, our results thus agree with the mapping of active phase separation onto that of passive fluids with attractive interactions through a global effective free energy (motility-induced phase transition). Particular attention is paid to the square-gradient term necessary for the phase separation kinetics. We finally discuss results from numerical simulations corroborating the analytical results. [ABSTRACT FROM AUTHOR]

We developed an energetic model by integrating the generalized Langevin equation with the Cahn-Hilliard equation to simulate the diffusive behaviors of receptor proteins in the plasma membrane of a living cell. Simulation results are presented to elaborate the confinement effects from actin corrals and protein-induced lipid domains. Single-molecule tracking data of epidermal growth factor receptors (EGFR) acquired on live HeLa cells agree with the simulation results and the mechanism that controls the diffusion of single-molecule receptors is clarified. We discovered that after ligand binding, EGFR molecules move into lipid nanodomains. The transition rates between different diffusion states of liganded EGFR molecules are regulated by the lipid domains. Our method successfully captures dynamic interactions of receptors at the single-molecule level and provides insight into the functional architecture of both the diffusing EGFR molecules and their local cellular environment [ABSTRACT FROM AUTHOR]

Spatially non-uniform electric fields can phase separate initially homogeneous mixtures of liquids. Here, we investigate the dynamics of phase separation using a modified Cahn-Hilliard equation and find three kinetically distinct regimes in the phase diagram: (1) discontinuous and (2) continuous interface formation kinetics and (3) a metastable state. By considering all possible solutions of the free energy density, we are able to map the time behavior in the vicinity of the interface as a series of equilibrium interfaces "moving" in the parameter space of the equilibrium phase diagram. The kinetic phase diagram, consequently, contains an "emergence line" that delineates the experimental conditions where a non-equilibrium interface can be forbidden from forming close to a charged surface. When the interface can form on the charged surface, an abrupt transition occurs that produces electrical signatures which distinguish the discontinuous from the continuous transition region. The third kinetic regime describes non-spontaneous phase separation and potential metastable states, and is bounded by the "electrostatic spinodal" line. The equivalent kinetic regimes exist in closed systems and can be found by considering an effective concentration in an open system. [ABSTRACT FROM AUTHOR]

We calculate the alloy-disorder-limited electron mobility of AlxGa1−xN from first principles. AlxGa1−xN is a technologically important ultra-wide-bandgap alloy with promise in light emitting diodes and high-power transistors. Alloying introduces statistical disorder, which causes electrons to scatter between different crystal-momentum states, leading to a reduction in mobility for intermediate alloy compositions. The corresponding lifetime, which appears as an energy broadening in the band structure, can be evaluated by unfolding the band structure from the supercell basis to the primitive-cell basis. We fit the first-principles band broadening with a model scattering potential and evaluate the low-field electron mobility using the semiclassical Boltzmann transport equation in the relaxation-time approximation. Our calculated mobility is in agreement with experimental values. We also find the lowest alloy-scattering electron mobility (total electron mobility) across the entire composition range to be 186 c m 2 / V s (136 c m 2 / V s), which is comparable to the highest electron mobility predicted in the competitor system, β- (Al x Ga 1 − x) 2 O 3 . Our results elucidate the intrinsic limits imposed by alloy disorder on electron transport in AlxGa1−xN. [ABSTRACT FROM AUTHOR]

A binary interface defect is any interface between two (not necessarily invertible) domain walls. We compute all possible binary interface defects in Kitaev's Z / p Z model and all possible fusions between them. Our methods can be applied to any Levin-Wen model. We also give physical interpretations for each of the defects in the Z / p Z model. These physical interpretations provide a new graphical calculus, which can be used to compute defect fusion. [ABSTRACT FROM AUTHOR]

Spontaneous imbibition is well-known to be one of the most effective processes of oil recovery in fractured reservoirs. However, the detailed pore-scale mechanisms of the counter-current imbibition process and the effects of different fluid/rock parameters on this phenomenon have not yet been deeply addressed. This work presents the results of a newpore-level numerical study of counter-current spontaneous imbibition, using coupled Cahn-Hilliard phase field and Navier-Stokes equations, solved by a finite element method. A 2D fractured medium was constructed consisting of a nonhomogeneous porous matrix, in which the grains were represented by an equilateral triangular array of circles with different sizes and initially saturated with oil, and a fracture, adjacent to the matrix, initially saturated with water and supported by low rate water inflow. Through invasion of water into the matrix, oil drops were expelled one by one from the matrix to the fracture, and in the matrix, water progressed by forming capillary fingerings, with characteristics corresponding to the experimental observations. The effects of wettability, viscosity ratio, and interfacial tension were investigated. In strongly waterwet matrix, with grain contact angles of θ < π/8, different micro-scale mechanisms were successfully captured, including oil film thinning and rupture, fluids' contact linemovement,water bridging, and oil drop detachment. It was notified that there was a specific grain contact angle for this simulated model, θ = π/4, above it, matrix oil recovery was negligible by imbibition, while below it, the imbibition rate and oil recovery were significantly increased by decreasing the contact angle. In simulated mixed wet models, water, coming from the fracture, just invaded the neighboring water-wet grains; the water front was stopped moving as it met the oil-wet grains or wide pores/throats. Increasing water-oil interfacial tension, in the range of 0.005-0.05 N/m, resulted in both higher rate of imbibition and higher ultimate oil recovery. Changing the water-oil viscosity ratio (M), in the range of 0.1-10, had a negligible effect on the imbibition rate, while due to co-effects of capillary fingering and viscous mobility ratio, the model with M = 1 had relatively higher ultimate oil recovery. [ABSTRACT FROM AUTHOR]

We examine effects of hydrodynamics on phase separation morphologies developed during drying of thin films containing a volatile solvent and two dissolved polymers. Cahn-Hilliard and Flory-Huggins theories are used to describe the free energy of the phase separating systems. The thin films, considered as Newtonian fluids, flow in response to Korteweg stresses arising due to concentration non-uniformities that develop during solvent evaporation. Numerical simulations are employed to investigate the effects of a Peclet number, defined in terms of system physical properties, as well as the effects of parameters characterizing the speed of evaporation and preferential wetting of the solutes at the gas interface. For systems exhibiting preferential wetting, diffusion alone is known to favor lamellar configurations for the separated phases in the dried film. However, a mechanism of hydrodynamic instability of a short length scale is revealed, which beyond a threshold Peclet number may deform and break the lamellae. The critical Peclet number tends to decrease as the evaporation rate increases and to increase with the tendency of the polymers to selectively wet the gas interface. As the Peclet number increases, the instability moves closer to the gas interface and induces the formation of a lateral segregation template that guides the subsequent evolution of the phase separation process. On the other hand, for systems with no preferential wetting or any other property asymmetries between the two polymers, diffusion alone favors the formation of laterally separated configurations. In this case, concentration perturbation modes that lead to enhanced Korteweg stresses may be favored for sufficiently large Peclet numbers. For such modes, a second mechanism is revealed, which is similar to the solutocapillary Marangoni instability observed in evaporating solutions when interfacial tension increases with the concentration of the non-volatile component. This mechanism may lead to multiple length scales in the laterally phase separated configurations. [ABSTRACT FROM AUTHOR]

We present steady non-linear solutions of films of confined polymer blends deposited on a solid substrate at off-critical concentrations with a free deformable surface. The solutions are obtained numerically using a variational form of the Cahn-Hilliard equation in the static limit, which allows for internal diffuse interfaces between the two components of the mixture. Existence of most of the branches of non-linear solutions at off-critical concentrations can be predicted from the knowledge of the branching points obtained with a linear stability analysis plus the non-linear solutions at critical concentrations. However, some families of solutions are found not to have correspondence at critical compositions. We take a value for surface tension that allows strong deformations at the sharp free upper surface. Varying the average composition and the length and thickness of the films we find a rich morphology of static films in the form of laterally structure films, layered films, droplets on the substrate, droplets at the free surface, and checkerboard structures. We show that laterally structured solutions are energetically favorable over homogeneous and other structured solutions within the whole spinodal region and even close to the absolute stability binodal boundary. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the formation of brine channels in sea ice. The model includes a time-dependent Ginzburg-Landau equation for the solid-liquid phase change, a diffusion equation of the Cahn-Hilliard kind for the solute dynamics, and the heat equation for the temperature change. The macroscopic motion of the fluid is also considered, so the resulting differential system couples with the Navier-Stokes equation. The compatibility of this system with the thermodynamic laws and a maximum theorem is proved. [ABSTRACT FROM AUTHOR]

We simulate a droplet impacting on a solid surface using a phase field model consisting of the Cahn-Hilliard and Navier Stokes equations with the generalized Navier boundary condition. An efficient gradient stable scheme is used to solve the system in axisymmetric coordinates. Our objective is to understand numerically the mechanisms leading to different impacting phenomena such as adherence, bouncing, partial bouncing, and splashing. In particular, we study how various processes are affected by the relevant dimensionless parameters: the Reynolds number, Weber number, density ratio, viscosity ratio, and the wettability of the solid surface. We also compare our numerical results with the experimental results and observe almost quantitative agreement. [ABSTRACT FROM AUTHOR]

Phase field models frequently provide insight into phase transitions and are robust numerical tools to solve free boundary problems corresponding to the motion of interfaces. A body of prior literature suggests that interface motion via surface diffusion is the long-time, sharp interface limit of microscopic phase field models such as the Cahn-Hilliard equation with a degenerate mobility function. Contrary to this conventional wisdom, we show that the long-time behaviour of degenerate Cahn-Hilliard equation with a polynomial free energy undergoes coarsening, reflecting the presence of bulk diffusion, rather than pure surface diffusion. This reveals an important limitation of phase field models that are frequently used to model surface diffusion. [ABSTRACT FROM AUTHOR]

Solutal Marangoni instability (SMI) is investigated both in 2D and 3D using a combined Cahn-Hilliard and Navier-Stokes model in a finite system. Fe-Sn is chosen as a representative alloy system since the phase diagram reveals a region with a miscibility gap, where two liquid phases, namely, the Fe-rich phase L1 and the Sn-rich phase L2, are in chemical equilibrium. In 3D, considering a perturbed liquid cylinder (L2 phase) with a length of ƛ and a radius of R0 embedded in the middle of a simulation box of ƛ x H x H (length x width x height) surrounded by the phase L1, we find that the perturbation induced Marangoni flow is either clockwise or anti-clockwise depending on the mean curvature difference between the convex and concave regions which is affected by the ratio of ƛ/R0. The critical ratio of ƛ/R0 for SMI is shown to be invariant for different Marangoni numbers as well as independent of the geometrical properties of the L1 phase. In 2D, a perturbed liquid pipe with a length of ƛ and a radius of R0 embedded in the middle of a simulation box of ƛ x H (length x height) is taken into account. Due to different curvature constitution, the critical ratio of ƛ/R0 for SMI depends on the height of the L1 phase. [ABSTRACT FROM AUTHOR]

Temporal evolution of polymerization-induced spinodal decomposition (PISD) under electric fields was investigated numerically in ethylene glycol/phenolic resin solutions with different initial composition. A model composed of the nonlinear Cahn-Hilliard-Cook equation for spinodal decomposition and a rate equation for curing reaction was utilized to describe the PISD phenomenon. As initial composition varied, deformed droplet-like and aligned bi-continuous structures were observed in the presence of an electric field. Moreover, the anisotropic parameter (D), determined from the 2D-FFT power spectrum, was employed to quantitatively characterize the degree of morphology anisotropy. The value of D increased quickly in the early stage and then decreased in the intermediate stage of spinodal decomposition, which was attributed to the resistance of coarsening process to morphology deformation and the decline of electric stress caused by polymerization reaction. The results can also provide a guidance on how to control the morphology of monolithic porous polymer and carbon materials with anisotropic structures. [ABSTRACT FROM AUTHOR]

The author comments on the paper "Degenerate Mobilities in Phase Field Models Are Insufficient to Capture Surface Diffusion" by Alpha A. Lee et al. He highlights Cahn-Hilliard equations presented by Lee as approximations for motion by surface diffusion. He also criticizes the general statement of the paper's title and discusses the association of the term "phase field models" with equations based on a quartic free energy.

The authors respond to a commentary by Axel Voigt on their paper "Degenerate Mobilities in Phase Field Models Are Insufficient to Capture Surface Diffusion." They highlight their argument about the insufficiency of degenerate mobilities in phase field models to capture surface diffusion. They also discuss the combination of a Cahn-Hilliard equation and a quadratic degenerate mobility.

We study initial boundary value problems for the unstable convective Cahn-Hilliard (CH) equation, i.e., the Cahn Hilliard equation whose energy integral is not bounded below. It is well-known that without the convective term, the solutions of the unstable CH equation ∂tu+∂x4u+∂x2(|u|pu)=0 may blow up in finite time for any p > 0. In contrast to that, we show that the presence of the convective term u∂xu in the Cahn-Hilliard equation prevents blow up at least for 049. We also show that the blowing up solutions still exist if p is large enough (p >= 2). The related equations like Kolmogorov-Sivashinsky-Spiegel equation, sixth order convective Cahn-Hilliard equation, are also considered. [ABSTRACT FROM AUTHOR]