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Phase transitions

Collaborator: W. Dreyer (FG 7), F. Duderstadt (FG 7), J. Sprekels (FG 1), B. Wagner (FG 1)

Cooperation with: S. Brodie, C.M. Brown (Heriot-Watt University Edinburgh, UK), P. Colli, G. Gilardi (Università di Pavia, Italy), S. Eichler (Freiberger Compound Materials GmbH), T. Hauck (Motorola, München), W.H. Müller (Technische Universität Berlin), B. Niethammer (Universität Bonn), E. Radkevich (Lomonosov University, Moscow, Russia)

Supported by: BMBF: ``Mathematische Modellierung und Simulation der Entstehung, des Wachstums und der Auflösung von Arsenausscheidungen in einkristallinem Galliumarsenid'' (Mathematical modeling and simulation of the formation, growth and dissolution of arsenic precipitation in single crystal gallium arsenide)


There are two models for different aspects of phase transitions that were studied during the last report period.

(1) The standard model, which was designed by Dreyer/Müller, is able to describe morphological changes of binary alloys on the $\mu m$ scale. In particular, a study of the coupling of mechanical stresses, diffusion and interface motion is of special interest for practical applications. An important difference between the Dreyer/Müller model and other similar models concerns the higher gradient part, which models surface energy of the interface boundary. That part is given by second derivatives of the concentration as $a_{ij}(c)\partial
^{2}c/\partial x_{i}\partial
x_{j} $. The anisotropy and the concentration dependence of the matrix aij(c) leads to quite new sharp interface limits that have not been studied before. While the numerical exploitation of the consequences was already done in former years, B. Wagner started a serious mathematical treatment of the problem. Moreover, the microscopic model to calculate the explicit structure of the matrix aij(c) that was used up to now has been changed during the last report period. The atomic interactions were formerly described by central forces, however, their capability to describe properly atomic interaction in anisotropic crystals of, say, fcc or tetragonal lattice symmetry has become apparent. For this reason Dreyer/Müller started a rederivation of all atomic contributions to the free energy containing in particular the gradient matrix. This rederivation relies on microscopic embedded force potentials, where noncentral forces are modeled by the introduction of more than two body forces.

Regarding the applications of the model there are two aspects that were considered. The data basis for tin/lead alloys has been further extended and a collaboration with Bosch has been started to find out whether the Dreyer/Müller model can be described similarly to the formation of shear bands, the appearance of localized phase transitions that were observed in the Bosch labs and which might lead to the initiation of cracks. On the other hand, the collection of data for the intermetallic silver/tin is still of high importance, because due to environmental reasons silver/tin will probably substitute the tin/lead system as solder joint material in microelectronic devices in the very near future.

(2) The other model that was developed and extensively studied is designed to describe a process in semi-insulating gallium arsenide (GaAs), where a complex coupling between eigenstrain fields, due to dislocations and misfit, and diffusion, which is initiated by a heat treatment of GaAs, leads to the appearance of liquid arsenic droplets. Their inhomogeneous distribution on the mm scale means a serious problem regarding the use of GaAs as wafer material for optoelectronic devices.

The original mathematical model, which was developed at WIAS, contains a coupled system of partial differential equations for the variables strain, (As) concentration and a time- and space-dependent distribution indicating the size of the droplets. It has turned out during the last report period that a description of semi-insulating GaAs with only these variables is insufficient for its realistic thermodynamic description. There are further constituents which appear in semi-insulating GaAs. These are oxygen (O), silicon (Si), boron (B) and carbon (C) in very small quantities, but nevertheless these trace elements induce very important phenomena. Moreover, these elements as well as vacancies and the arsenic may carry charges and there are chemical reactions among these constituents and with electrons and holes, which must also be considered.

Several conferences with the wafer manufacturer FCM and with other experimenters have led to an extended description to simulate the evolution of arsenic droplets in GaAs. There are now 16 constituents included in the extended model that is designed as follows: Roughly speaking, there are two different time scales. Mechanical and chemical equilibrium is reached so much faster than diffusional and interfacial equilibrium that the mechanics and the chemistry are described by quasistatic equations whose time dependence is given by the evolution of diffusion and of interface motion. This leads to an elliptic second-order system to describe the strains and stresses and to 16 nonlinear algebraic equations to describe the chemistry. There is only a single diffusing constituent, which has turned out to be the uncharged arsenic interstitial, and there is thus (as before) a single diffusion equation. However, that equation is now of a quite complicated structure, because it couples to the solution of the chemical algebraic system and (as before) to the mechanical system.

Due to these new complexities, the evolution of droplets is currently described as a free boundary value problem for a single droplet. If all results of this simplification are fully exploited, the complete Becker/Döring model will be coupled again to the described model. To this end a multiscale transition is needed and will be carried out in collaboration with B. Niethammer (Bonn), who is the leading expert in Becker/Döring and related LSW models.


  1. C.M. BROWN, W. DREYER, W.H. MÜLLER, Discrete Fourier transforms and their application to stress-strain problems in composite mechanics: A convergence study, Proc. Roy. Soc. London Ser. A, 458 (2002), pp. 1-21.
  2. E. BONETTI, P. COLLI, W. DREYER, G. GILARDI, G. SCHIMPERNA, J. SPREKELS, On a model for phase separation in binary alloys driven by mechanical effects, Phys. D, 165 (2002), pp. 48-65.
  3. W. DREYER, W.H. MÜLLER, Toward quantitative modeling of morphology changes of solids with phase field theories: Atomistic arguments for the determination of higher gradient coefficients, in: Bioceramics 15. Proceedings of the 15th International Symposium on Ceramics in Medicine, Sydney, Australia, December 4-8, 2002, B.B. Nissan, D. Sher, W. Walsh, eds., vol. 240-242 of Key Engineering Materials, Trans Tech Publications Ltd, Ütikon-Zürich, 2003, pp. 901-914.
  4. W. DREYER, F. DUDERSTADT, Modeling of the appearance of arsenide droplets in gallium arsenide, to appear as WIAS Preprint.
  5. W. DREYER, On jump conditions of moving interfaces in crystals, to appear as WIAS Preprint.

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