Continuum thermodynamics of chemically reacting fluid mixtures

We consider viscous, heat conducting mixtures of molecularly miscible chemical species forming a fluid in which the constituents can undergo chemical reactions. Assuming a common temperature for all components, we derive a closed system of partial mass and partial momentum balances plus a mixture balance of internal energy. This is achieved by careful exploitation of the entropy principle and requires appropriate definitions of absolute temperature and chemical potentials, based on an adequate definition of thermal energy excluding diffusive contributions. The resulting interaction forces split into a thermo-mechanical and a chemical part, where the former turns out to be symmetric in case of binary interactions. For chemically reacting systems and as a new result, the chemical interaction force is a contribution being non-symmetric outside of chemical equilibrium. The theory also provides a rigorous derivation of the so-called generalized thermodynamic driving forces, avoiding the use of approximate solutions to the Boltzmann equations. Moreover, using an appropriately extended version of the entropy principle and introducing cross-effects already before closure as entropy invariant couplings between principal dissipative mechanisms, the Onsager symmetry relations become a strict consequence. With a classification of the factors in the binary products of the entropy production according to their parity--instead of the classical partition into so-called fluxes and driving forces--the apparent anti-symmetry of certain couplings is thereby also revealed. If the diffusion velocities are small compared to the speed of sound, the Maxwell-Stefan equations follow in the case without chemistry, thereby neglecting wave phenomena in the diffusive motion. This results in a reduced model with only mass being balanced individually. In the reactive case ..

A compressible mixture model with phase transition

We introduce a new thermodynamically consistent diffuse interface model of Allen--Cahn/Navier--Stokes type for multi-component flows with phase transitions and chemical reactions. For the introduced diffuse interface model, we investigate physically admissible sharp interface limits by matched asymptotic techniques. We consider two scaling regimes, i.e.~a non-dissipative and a dissipative regime, where we recover in the sharp interface limit a generalized Allen-Cahn/Euler system for mixtures with chemical reactions in the bulk phases equipped with admissible interfacial conditions. The interfacial conditions satify, for instance, a Young--Laplace and a Stefan type law

On jump conditions at phase boundaries for ordered and disordered phases

This is a study on jump conditions across the interface between two adjacent phases. The interface behaves as a free boundary, and in sharp interface models jump conditions are used to determine the values of thermodynamic fields at the free boundaries. In this study the jump conditions are derived from balance equations for singular surfaces that do not have singular lines, i.e. triple junctions are not considered here. At first we present the most general form of jump conditions to give a general framework, from where we consider various special cases with a focus on the influence of mechanical fields on the interfacial processes. The special cases include the Hoffmann/Cahn capillarity vector theory and jump conditions for interfaces where order/disorder transitions are involved. Furthermore we discuss interfacial chemical reaction laws, and in particular the creation and annihilation of vacancies at a liquid/solid interface

Modeling diffusional coarsening in eutectic tin/lead solders: A quantitative approach

This paper presents a quantitative simulation of the phase separation and coarsening phenomenon in eutectic tin/lead (SnPb) solders. The computer modeling is based on continuum theory and field phase models which were evaluated using the most recently available data for the free energy of the tin/lead system, diffusional and mobility coefficients, elastic constants as well as surface tensions of both phases. The model presented allows to study the influence as well as the interaction between classical diffusion of the Fickean type, surface energies according to Cahn and Hilliard, as well as stresses and strains on phase separation and coarsening. An attempt is made to compare the temporal development of a eutectic SnPb microstructure at different temperature levels and subjected to different stress levels as predicted by the model to actual experiments

A study of the coarsening in tin/lead solders

This paper presents a model, which is capable to simulate the coarsening process observed during thermo-mechanical treatment of binary tin-lead solders. Fourier transforms and spectral theory are used for the numerical treatment of the thermo-elastic as well as of the diffusion problem encountered during phase separation in these alloys. More specifically, the analysis is based exclusively on continuum theory, first, relies on the numerical computation of the local stresses and strains in a representative volume element (RVE). Second, this information is used in an extended diffusion equation to predict the local concentrations of the constituents of the solder. Besides the classical driving forces for phase separation, as introduced by Fick and Cahn-Hilliard, this equation contains an additional term which links the mechanical to the thermodynamical problem. It connects internal and external stresses, strains, temperature, as well as concentrations and allows for a comprehensive study of the coarsening and aging process

Kinetic flux-vector splitting schemes for the hyperbolic heat conduction

A kinetic solver is developed for the initial and boundary value problems (IBVP) of the symmetric hyperbolic moment system. This nonlinear system of equations is related to the heat conduction in solids at low temperatures. The system consists of a conservation equation for the energy density e and a balance equation for the heat flux 혘푖, where 푒 and 혘푖 are the four basic fields of the theory. We use kinetic flux vector splitting (KFVS) scheme to solve these equations in one and two space dimensions. The flux vectors of the equations are splitted on the basis of the local equilibrium distribution of phonons. The resulting computational procedure is efficient and straightforward to implement. The second order accuracy of the scheme is acheived by using MUSCL-type reconstruction and min-mod nonlinear limitters. The solutions exhibit second order accuracy, and satisfactory resolution of gradients with no spurious oscillations. The secheme is extended to the two-dimensional case in a usual dimensionally split manner. In order to prescribe the boundary data we need the knowledge of the 푒 and 혘푖. However, in experiments only one of the quantities can be controlled at the boundary. This problem is removed by using a continuity condition. It turned out that after some short time energy and heat flux are related to each other according to Rankine Hugoniot jump relations. To illustrate the performance of the KFVS scheme, we perform several one- and two-dimensional test computations. For the comparison of our results we use high order central schemes. The present study demonstrates that this kinetic method is effective in handling such problems

Micro-macro transitions by interpolation, smoothing, averaging and scaling of particle trajectories

We consider a Newtonian system of many diatomicmolecules each of which consisting of two atoms of equal mass whichare separated by a fixed distance. The barycenters are allowed to movealong some fixed straight line. Moreover each molecule has an additionalrotational degree of freedom. The atoms of neighbouring moleculesinteract to each other by a generic pair potential. By means of thisexample we propose a new method for deriving macroscopic models from microscopic ones. The method is based on the definition of macroscopicobservables and the derivation of corresponding balance laws by interpolation smoothing/averaging and subsequent scaling of particle trajectories

On the modelling of semi-insulating GaAs including surface tension and bulk stresses

Necessary heat treatment of single crystal semi-insulating Gallium Arsenide (GaAs), which is deployed in micro- and opto- electronic devices, generate undesirable liquid precipitates in the solid phase. The appearance of precipitates is influenced by surface tension at the liquid/solid interface and deviatoric stresses in the solid. The central quantity for the description of the various aspects of phase transitions is the chemical potential, which can be additively decomposed into a chemical and a mechanical part. In particular the calculation of the mechanical part of the chemical potential is of crucial importance. We determine the chemical potential in the framework of the St. Venant--Kirchhoff law which gives an appropriate stress/strain relation for many solids in the small strain regime. We establish criteria, which allow the correct replacement of the St. Venant--Kirchhoff law by the simpler Hooke law. The main objectives of this study are: (i) We develop a thermo-mechanical model that describes diffusion and interface motion, which both are strongly influenced by surface tension effects and deviatoric stresses. (ii) We give an overview and outlook on problems that can be posed and solved within the framework of the model. (iii) We calculate non-standard phase diagrams, i.e. those that take into account surface tension and non-deviatoric stresses, for GaAs above 786 extdegreeC, and we compare the results with classical phase diagrams without these phenomena.</o:p

A simple but rigorous micro-macro transition

This paper is devoted to a case study of micro-macro transitions. The main objective is the mathematically rigorous description of the macroscopic behavior of highly oscillating microscopic variables. In particular, we show that the theory of Young measures provides an elegant approach to this problem. A nontrivial application of the results is given in WIAS-Preprint 724

On the approximation of periodic traveling waves for the nonlinear atomic chain

We study a scheme from \cite{FV99}, which allows to approximate periodic traveling waves in the nonlinear atomic chain with nearest neighbour interactions. We prove a compactness result for this scheme, and derive some generalizations. Moreover, we discuss the thermodynamic properties of traveling waves