Phase Transition Problem Research Articles

A Multifront Problem of Freezing-Thawing Moist Soil

Construction activities in the North of Western Siberia is carried out in specific conditions of the cryolithozone. The study of heat transfer processes in freezing soils near buildings is necessary to ensure their successful operation. Heat exchange with phase transition is traditionally described by Stefan problem, which is a system of differential equations of parabolic type with standard boundary conditions and a subsidiary condition on the phase transition boundary. It is possible to formally pass on to one equation of the conductivity type, but in that case the delta function appears in one of the coefficients. The delta function reflects the Joule heat release at the phase transition temperature. A widespread “pass-through counting” method reduces the Stefan problem to a boundary problem for a nonlinear heat conductivity equation. With this approach, the calculation results in a temperature field. But it is difficult to identify the position of the phase transition boundary on the temperature field. A large number of methods for solving the Stefan problem are developed, in which the required value is the phase transition front coordinate. The common disadvantage of these methods is their unsuitability for situations with several fronts, when these fronts appear and disappear, change the movement direction, merge with each other. The article presents a method for solving the Stefan problem, which allows to obtain the front coordinate as a zero isotherm. As an example this method is used to solve the problem of freezing-thawing of moist soil under the influence of seasonal surface temperature fluctuations. The method eliminates the need to control the evolution of each front. The Stefan problem is considered as a limiting case of the general phase transition problem in a certain temperature range. Standard transformations and application of Green’s function allow to write down a problem in the form of an integral equation. The approximate solution is obtained in the form of a recurrent formula.

The one-phase bifurcation for the p-Laplacian

A bifurcation about the uniqueness of a solution of a singularly perturbed free boundary problem of phase transition associated with the p-Laplacian, subject to given boundary condition is proved in this paper. We show this phenomenon by proving the existence of a third solution through the Mountain Pass Lemma when the boundary data decreases below a threshold. In the second part, we prove the convergence of an evolution to stable solutions, and show the Mountain Pass solution is unstable in this sense.

DISCRETE STEPS OF DIAMOND-GRAPHITE PHASE TRANSITION AT ANNEALING OF NANODIAMONDS IN WIDE TEMPERATURE RANGE

We present the short review of the works on a problem of phase transitions (PT) which are distributed in the temperature range ΔТ of PT. Experimental studies of Raman spectra (laser wavelength 514.5 nm) of nanodiamonds after annealing at 900-2000 °C and their detailed numerical analysis with decomposition of the allocated oscillatory D and G bands on the compound spectral components D and G, D (k) and G (k’) relating to various states of Brillouin’s zones were conducted. Detection of temperature maxima of intensities, frequencies and full width at half maximum (FWHM) of individual Raman components as well as intensity of a background defines discrete steps of PT of diamond - onion-shaped carbon in temperature ranges of 940-1200 °C and 1500-2000 °C.

Exhaustive Classification of the Invariant Solutions for a Specific Nonlinear Model Describing Near Planar and Marginally Long-Wave Unstable Interfaces for Phase Transition

Problems of thermodynamic phase transition originate inherently in solidification, combustion and various other significant fields. If the transition region among two locally stable phases is adequately narrow, the dynamics can be modeled by an interface motion. This paper is devoted to exhaustive analysis of the invariant solutions for a modified Kuramoto-Sivashinsky equation in two spatial and one temporal dimensions is presented. This nonlinear partial differential equation asymptotically characterizes near planar interfaces, which are marginally long-wave unstable. For this purpose, by applying the classical symmetry method for this model the classical symmetry operators are attained. Moreover, the structure of the Lie algebra of symmetries is discussed and the optimal system of subalgebras, which yields the preliminary classification of group invariant solutions is constructed. Mainly, the Lie invariants corresponding to the infinitesimal symmetry generators as well as associated similarity reduced equations are also pointed out. Furthermore, the nonclassical symmetries of this nonlinear PDE are also comprehensively investigated.

A numerical approach for solving the fractional Fisher equation using Chebyshev spectral collocation method

In this paper, an efficient numerical method is introduced for solving the fractional (Caputo sense) Fisher equation. This equation presents the problem of biological invasion and occurs, e.g., in ecology, physiology, and in general phase transition problems and others. We use the spectral collocation method which is based upon Chebyshev approximations. The properties of Chebyshev polynomials are utilized to reduce the proposed problem to a system of ODEs, which is solved by using finite difference method (FDM). Some theorems about the convergence analysis are stated. A numerical simulation and a comparison with the previous work are presented. We can apply the proposed method to solve other problems in engineering and physics.

Stability and errors analysis of two iterative schemes of fractional steps type associated to a nonlinear reaction-diffusion equation

We present the error analysis of two time-stepping schemes of fractional steps type, used in the discretization of a nonlinear reaction-diffusion equation with Neumann boundary conditions, relevant in phase transition and interface problems. We start by investigating the solvability of a such boundary value problems in the class \begin{document}$ W^{1,2}_p(Q) $\end{document} . One proves the existence, the regularity and the uniqueness of solutions, in the presence of the cubic nonlinearity type. The convergence and error estimate results (using energy methods) for the iterative schemes of fractional steps type, associated to the nonlinear parabolic equation, are also established. The advantage of such method consists in simplifying the numerical computation. On the basis of this approach, a conceptual algorithm is formulated in the end. Numerical experiments are presented in order to validates the theoretical results (conditions of numerical stability) and to compare the accuracy of the methods.

Об одном методе решения задачи кристаллизации многокомпонентного раствора в цилиндрической ампуле

Об одном методе решения задачи кристаллизации многокомпонентного раствора в цилиндрической ампуле

Multifractality and Network Analysis of Phase Transition.

Many models and real complex systems possess critical thresholds at which the systems shift dramatically from one sate to another. The discovery of early-warnings in the vicinity of critical points are of great importance to estimate how far the systems are away from the critical states. Multifractal Detrended Fluctuation analysis (MF-DFA) and visibility graph method have been employed to investigate the multifractal and geometrical properties of the magnetization time series of the two-dimensional Ising model. Multifractality of the time series near the critical point has been uncovered from the generalized Hurst exponents and singularity spectrum. Both long-term correlation and broad probability density function are identified to be the sources of multifractality. Heterogeneous nature of the networks constructed from magnetization time series have validated the fractal properties. Evolution of the topological quantities of the visibility graph, along with the variation of multifractality, serve as new early-warnings of phase transition. Those methods and results may provide new insights about the analysis of phase transition problems and can be used as early-warnings for a variety of complex systems.

Problem of phase transitions and thermodynamic stability in complex (dusty, colloid etc) plasmas

Features of the first-order phase transitions in complex (dusty, colloid etc) plasma are under discussion. The basis for consideration is the well-known phase diagram of dusty plasma as a Debye system from Hamaguchi et al (1997 Phys. Rev. E 92 4671) in Γ-κ plane (Γ is a Coulomb non-ideality parameter, κ is a screening parameter). The initial Γ -κ phase diagram from Hamaguchi et al (1997 Phys. Rev. E 92 4671) is converted in standard thermodynamic variables in temperature-density planes. Here 2-component electroneutral systems of macro- and microions (+Z, -1) and (-Z, +1) are considered as thermodynamically equilibrium ensembles of classical Coulomb particles. An extensive area for negative compressibility of the system was revealed at the phase diagram in a fluid state of the initial Debye system when one considers the system as equilibrium two-component electroneutral mixture of macro- and microions (+Z, -1) (or (-Z, +1)) under equations of state from Hamaguchi et al (1997 Phys. Rev. E 92 4671) and Khrapak et al (2014 Phys. Rev. E 89 023102). This means thermodynamic instability of the simplified Debye system in this domain. Non-linear screening and an unavoidable existence of additional phase transitions of gas-liquid and gas-crystal type are proposed as hypothetical resolution of discussed thermodynamic instability problem.

The limit as $p \to \infty$ in a two-phase free boundary problem for the $p$-Laplacian

In this paper, we study the limit as p goes to infinity of a minimizer of a variational problem that is a two-phase free boundary problem of phase transition for the p -Laplacian. Under a geometric compatibility condition, we prove that this limit is a solution of a free boundary problem for the \infty -Laplacian. When the compatibility condition does not hold, we prove that there still exists a uniform limit that is a solution of a minimization problem for the Lipschitz constant. Moreover, we provide, in the latter case, an example that shows that the free boundary condition can be lost in the limit.

Ising model with competing “uncle–nephew” interactions

ABSTRACTWe consider the Ising model on the semi-infinite Cayley tree of second order with competing interactions up to the third-nearest-neighbors with spins belonging to the different branches of the tree and for this model investigate the problem of phase transition.

On Construction of Quantum Markov Chains on Cayley trees

The main aim of the present paper is to provide a new construction of quantum Markov chain (QMC) on arbitrary order Cayley tree. In that construction, a QMC is defined as a weak limit of finite volume states with boundary conditions, i.e. QMC depends on the boundary conditions. Note that this construction reminds statistical mechanics models with competing interactions on trees. If one considers one dimensional tree, then the provided construction reduces to well-known one, which was studied by the first author. Our construction will allow to investigate phase transition problem in a quantum setting.

Solution of the Moisture Phase Transition Problem in the Insulation and Soils near Pipelines

The analysis of various approaches to the heat conductivity modeling in soils with phase transition based on Stefan problem is provided. Herein the analytical solution for Stefan problem is received only for special cases.The approximate methods of mathematical modeling of heat exchange in soils and construction materials based on the Stefan problem without initial data are offered. The found analytical solution can be a basis for approximate solutions of a wide range of heat exchange problems at harmonious oscillation of temperature on one of the borders. The mathematical apparatus for estimating problems solution of freezing and melting in areas with plane-parallel and axial symmetry is considered. Solutions for applied problems of constructions building and operation in the permafrost conditions are provided.

Applications of path-integral molecular dynamics simulations to phase transition problems

In order to predict the properties of materials from first principles simulations, both the electronic structures and the movement of nuclei should be accurately described. Till now, the electronic structure calculation methods have been relatively well developed, like the density-functional theory, the quantum chemical methods, and the diffusion quantum monte carlo method, etc. By using them, we can get the electronic structures of materials with different levels of accuracy. But the movements of nuclei are still often treated at a clasical level. More and more experimental and theoretical works have found that, the nuclear quantum effects (NQEs) are playing an important role in some phase transition porcesses. Melting line of hydrogen at high pressures were reported through coexistance simulations based on ab initio PIMD methods. We determined the melting temperature as a function of pressure and finded an atomic solid phase from 500 to 800 GPa, which melts at less than 200 K. Beyond this and up to 1200 GPa, a metallic atomic liquid is stable at temperatures as low as 50 K. The quantum motion of the protons is the key to the low melting temperature reported, as simulations with classical nuclei lead to considerably higher melting temperatures of about 300 K across the entire pressure range considered. Then, a combination of state-of-the-art theoretical methods were used to obtain an atomic level picture of classical and quantum ordering of protons in cold high-pressure solid hydrogen. Besides, Using a self-developed combination of the thermodynamic integration and the ab initio PIMD methods, we quantitatively studied the influence of NQEs on the melting of dense lithium at 45 GPa. We find that although the NQEs significantly change the free-energies of the competing solid and liquid phases, the melting temperature is lowered by only about 15 K, with values obtained using both classical and quantum nuclei in close proximity to a new experiment. So, by combining the path-integral molecular dynamics (PIMD) method with techniques like coexistance simulation and free-energy calculation method, the NQEs in many phase transition processes can be well understood.

A bifurcation phenomenon in a singularly perturbed one-phase free boundary problem of phase transition

In this paper, we prove that a bifurcation phenomenon exists in a one-phase singularly perturbed free boundary problem of phase transition. Namely, the uniqueness of a solution of the one-phase problem breaks down as the boundary data decreases through a threshold value. The minimizer of the functional in consideration separates from the trivial harmonic solution. Moreover, we prove a third solution, a critical point of the functional being minimized, exists in this case by using the Mountain Pass Lemma. We prove convergence of the evolution with initial data near the minimizer or the trivial harmonic solution to the minimizer or to the trivial solution respectively, which means both the minimizer and the trivial harmonic solution are stable, while a saddle point solution of the free boundary problem is unstable in this sense.

Effect Of Natural Convection On Directional Solidification Of Pure Metal

AbstractThe paper is focused on the modeling of the directional solidification process of pure metal. During the process the solidification front is sharp in the shape of the surface separating liquid from solid in three dimensional space or a curve in 2D. The position and shape of the solid-liquid interface change according to time. The local velocity of the interface depends on the values of heat fluxes on the solid and liquid sides. Sharp interface solidification belongs to the phase transition problems which occur due to temperature changes, pressure, etc. Transition from one state to another is discontinuous from the mathematical point of view. Such process can be identified during water freezing, evaporation, melting and solidification of metals and alloys, etc.The influence of natural convection on the temperature distribution and the solid-liquid interface motion during solidification of pure copper is studied. The mathematical model of the process is based on the differential equations of heat transfer with convection, Navier-Stokes equation and the motion of the interface. This system of equations is supplemented by the appropriate initial and boundary conditions. In addition the continuity conditions at the solidification interface must be properly formulated. The solution involves the determination of the temporary temperature and velocity fields and the position of the interface. Typically, it is impossible to obtain the exact solution of such problem. The numerical model of solidification of pure copper in a closed cavity is presented, the influence of the natural convection on the phase change is investigated. Mathematical formulation of the problem is based on the Stefan problem with moving internal boundaries. The equations are spatially discretized with the use of fixed grid by means of the Finite Element Method (FEM). Front advancing technique uses the Level Set Method (LSM). Chorin’s projection method is used to solve Navier-Stokes equation. Such approach makes possible to uncouple velocities and pressure. The Petrov-Galerkin formulation is employed to stabilize numerical solutions of the equations. The results of numerical simulations in the 2D region are discussed and compared to the results obtained from the simulation where movement of the liquid phase was neglected.

Quark-hadron phase transition in DGP brane gravity with bulk scalar field

A DGP brane-world framework is picked out to study quark-hadron phase transition problem. The model also includes a bulk scalar field in agreement with string theory prediction. The work is performed using two formalisms as: smooth crossover approach and first order approach, and the results are plotted for both branches of DGP model. General behavior of temperature is the same in these two approaches and it decrease with increasing time and expanding Universe. Phase transition occurs at about micro-second after the big bang. The results show that transition time depends on brane tension value in which larger brane tension comes to earlier transition time.

Stochastic rotation dynamics for nematic liquid crystals.

We introduce a new mesoscopic model for nematic liquid crystals (LCs). We extend the particle-based stochastic rotation dynamics method, which reproduces the Navier-Stokes equation, to anisotropic fluids by including a simplified Ericksen-Leslie formulation of nematodynamics. We verify the applicability of this hybrid model by studying the equilibrium isotropic-nematic phase transition and nonequilibrium problems, such as the dynamics of topological defects and the rheology of sheared LCs. Our simulation results show that this hybrid model captures many essential aspects of LC physics at the mesoscopic scale, while preserving microscopic thermal fluctuations.

Low symmetry phases of(Al, Ga)Sbunder low pressure

The interesting problem of the first phase transition which is induced by pressure in AlSb and GaSb antimonide binaries is revisited. Then, the case of the AlGaSb ternary has been investigated too. The choice of ternary AlGaSb is due to the particular attention which is given actually to it and to the quasi-absence of investigations of the phase transitions in it. We found that low pressures applied to GaSb and AlSb induce transitions from zinc-blende to Imm2 and from zinc-blende to Cmcm, respectively. Our result for AlSb is in agreement with literature, and in the case of GaSb which poses the problem, we find that the Imm2 phase presents a lower total energy than the b-β-Sn phase. For ternaries, the transition is found to be to a phase which is obtained from the Cmcm phases of binaries. Calculations were performed using the plane wave version of the FPLMTO (PLW-FPLMTO) first principle method based on the local density approximation (LDA) and the generalized gradient approximation (GGA96).

Numerical simulation of a phase transition problem with natural convection using hybrid FEM/EFGM technique

Purpose – The purpose of this paper is to study the flow and heat transfer characteristics of a phase transition, melting problem. In this problem, phase transition between solid and liquid takes place within a square enclosure in the presence of natural convection. Design/methodology/approach – The physical problem, described with non-linear partial differential equations, is simulated using a hybrid finite element and element free Galerkin method (FEM/EFGM) approach. In energy conservation equation, the fixed-domain, effective heat capacity method is used to take into account the latent heat of phase change. The governing partial differential equations are solved with a meshfree, EFGM near the phase transition front while in the region away from the front with uniform nodal distribution; problem is simulated with traditional FEM. Findings – A sensitivity analysis of characteristic dimensionless numbers Rayleigh number (Ra), Prandtl number (Pr), Stefan number (ste) is presented in order to investigate their impact on thermal and flow fields. Typically computational times of EFGM are higher than that of FEM. Therefore, by using EFGM only in that portion of physical problem where phase transition occurs, the hybrid FEM/EFGM strategy employed in present paper could reduce the computational time of EFGM while still retaining its accuracy. Also, the consistent performance of the results obtained with this hybrid approach is validated with those already available in literature for some special cases. Originality/value – The hybrid methodology adopted in this paper, is quite new for solving such type of phase transition problem.