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Subsections

Phase transitions

Collaborator: W. Dreyer, F. Duderstadt, S. Qamar, B. Wagner

Cooperation with: W.H. Müller (Technische Universität Berlin), P. Colli, G. Gilardi (Università di Pavia, Italy), S. Eichler, G. Jurisch (Freiberger Compound Materials GmbH), T. Hauck (Motorola, München/Berlin), B. Niethammer (Humboldt-Universität zu Berlin)

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)

Description:

1. Improvement of the existing phase model for morphological changes in solder materials

A careful study of a sharp interface limit of the existing phase field model, that was used in the past to describe morphological changes in eutectic tin/lead alloys, exhibited that anisotropic surface energies can only be generated due to mechanical effects. This is one of the reasons that the diffusion flux was reformulated by W. Dreyer and B. Wagner. The resulting thermodynamically consistent form now reads
Jk = $\displaystyle \left.\vphantom{ -\frac{B_{ki}}{T}\frac{\partial}{\partial X_{i}}... ... c}\frac{\partial c}{\partial X_{j}}% \frac{\partial c}{\partial X_{l}}}\right.$ - $\displaystyle {\frac{{B_{ki}}}{{T}}}$$\displaystyle {\frac{{\partial}}{{\partial X_{i}}% }}$($\displaystyle {\frac{{\partial\psi(c,\varepsilon_{rs})}}{{\partial c}}}$ -2Ajl(c,$\displaystyle \varepsilon_{{rs}}^{}$)$\displaystyle {\frac{{\partial^{2}c}}{{\partial X_{j}\partial X_{l}}}}$ - $\displaystyle {\frac{{\partial A_{jl}(c,\varepsilon_{rs})}}{{\partial c}}}$$\displaystyle {\frac{{\partial c}}{{\partial X_{j}}% }}$$\displaystyle {\frac{{\partial c}}{{\partial X_{l}}}}$  
    $\displaystyle \left.\vphantom{ -2\frac{\partial A_{jl}(c,\varepsilon_{rs})}{\pa... ...mn}}% \frac{\partial^{2}\varepsilon_{mn}}{\partial X_{j}\partial X_{l}}}\right.$ -2$\displaystyle {\frac{{\partial A_{jl}(c,\varepsilon_{rs})}}{{\partial\varepsilon _{mn}}}}$$\displaystyle {\frac{{\partial c}}{{\partial X_{j}}}}$$\displaystyle {\frac{{\partial\varepsilon_{mn}% }}{{\partial X_{l}}}}$ - $\displaystyle {\frac{{\partial^{2}a_{jl}(c,\varepsilon_{rs})}}{{\partial \varepsilon_{mn}\partial\varepsilon_{op}}}}$$\displaystyle {\frac{{\partial\varepsilon_{op}% }}{{\partial X_{j}}}}$$\displaystyle {\frac{{\partial\varepsilon_{mn}}}{{\partial X_{l}}% }}$ - $\displaystyle {\frac{{\partial a_{jl}(c,\varepsilon_{rs})}}{{\partial\varepsilon_{mn}}% }}$$\displaystyle {\frac{{\partial^{2}\varepsilon_{mn}}}{{\partial X_{j}\partial X_{l}}}}$). (1)

The variables are the (tin) concentration c and the strain $ \varepsilon_{{rs}}^{}$. Bki gives the mobility tensor and T is the temperature. The functions $ \psi$(c,$ \varepsilon_{{rs}}^{}$), Ajl(c,$ \varepsilon_{{rs}}^{}$), and ajl(c,$ \varepsilon_{{rs}}^{}$) are explicitly known, see [3]. In the sharp interface limit of this model there results a surface tension that can be calculated from the gradient coefficients Ajl(c,$ \varepsilon_{{rs}}^{}$).

In the first numerical treatment of the resulting phase field equation, B. Wagner ignored the mechanical contributions and developed a non-stiff boundary integral formulation to efficiently simulate the long-time evolution of precipitates during coarsening.

2. Completion of the model for the chemistry and arsenic precipitation in semi-insulating GaAs

The description of semi-insulating GaAs, which includes thermo-mechanical coupling, diffusion, interface motion, precipitation of arsenic droplets, and the determination of various thermodynamic equilibria, has been completed. The material parameters have been tested and make it possible to calculate the phase diagram of GaAs, which is in exact agreement with the experimental data.

\begin{figure}\ProjektEPSbildNocap{10cm}{fg7phbildn.eps} \end{figure}

The Figure shows a region of the phase diagram of GaAs, where the doping substances are distributed randomly over the three sublattices. Outside this region, precipitation of arsenic droplets sets in. The upper endpoints of the isobars are triple points, where solid, liquid, and gas phase are in equilibrium.

For an illustration of the involved field equations in non-equilibrium processes, we give, as an example, the diffusion equation, which describes the concentration field y(t, r) of the interstitial arsenic in the vicinity of a spherical arsenic droplet.

$\displaystyle {\frac{{\partial y}}{{\partial t}}}$ = D($\displaystyle {\frac{{\partial}}{{\partial r}}}$(r2$\displaystyle {\frac{{\partial y}}{{\partial r}}}$) + $\displaystyle {\frac{{y(1-y)}}{{RT}}}$$\displaystyle {\frac{{180MGb^{2}}}{{\bar{\rho }_{S}r^{8}}}}$) - $\displaystyle {\frac{{D}}{{RT}}}$$\displaystyle {\frac{{\partial y(1-y)}}{{\partial r}}}$$\displaystyle {\frac{{36MGb^{2}% }}{{\bar{\rho}_{S}r^{7}}}}$. (2)
Herein, D denotes the diffusion constant, R and T are the gas constant and the temperature, respectively, M is the molecular weight of the arsenic, G denotes the shear modulus, and $ \bar{{\rho}}_{{S}}^{}$ is the mass density of the solid. The quantity b measures the phenomenon that the liquid droplet needs more space than a solid of the same number of particles, leading to a multiaxial stress field in the vicinity of the droplet. Note that b depends implicitly on time via the evolving radius rI(t) of the droplet. More details can be found in [4].

3. Various initial- and boundary problems for interface motions

are due to the Grinfeld instability, which is a phenomenon of growing interest. The Grinfeld instability states that a plane interface between a solid and its melt may become unstable if multiaxial stress fields appear in the solid. Surface tension and gravity provoke a stabilization of a plane interface. We found a further phenomenon in the competition of these effects. It is important whether the creation/annihilation of the melt decreases or increases the volume that is occupied by the two-phase mixture. This has the consequence that the stability of a plane interface depends on further conditions, which are:

(i) Do we consider an infinite system without mass conservation or a finite system with mass conservation?

(ii) Do we control the total volume or the total pressure of the considered system?

Our study of the Grinfeld instability relies on the solution of a quasistatic elastic problem for the stresses in the solid and the pressure in the melt. Both are used to calculate the interfacial normal speed w$\scriptstyle \nu$, which is given by (see [1])

w$\scriptstyle \nu$ = MI($\displaystyle \mu_{{L}}^{}$ - $\displaystyle \mu_{{S}}^{}$ + $\displaystyle {\frac{{1}}{{\rho_{S}}}}$$\displaystyle \sigma^{{<ij>}}_{}$$\displaystyle \nu^{{i}}_{}$$\displaystyle \nu^{{j}}_{}$). (3)
Here MI > 0 denotes the interfacial mobility, $ \mu_{{L}}^{}$ and $ \mu_{{S}}^{}$ are chemical potentials of the melt and the solid, respectively, $ \rho_{{S}}^{}$ is the mass density of the solid at the interface, $ \sigma^{{<ij>}}_{}$ denote the trace-free components of the stress tensor, and $ \nu^{{i}}_{}$ are the components of the interfacial normal. This equation becomes a nonlinear and nonlocal PDE determining the geometry of the interface, if the liquid pressure and the stress fields of the mechanical boundary value problem have been inserted. The main results can be found in [2].

References:

  1. W. DREYER, On jump conditions at phase boundaries for ordered and disordered phases, WIAS Preprint no. 869, 2003 .

  2. W. DREYER, A. MÜNCH, B. WAGNER, On the Grinfeld instability, to appear as WIAS Preprint.

  3. W. DREYER, B. WAGNER, Sharp-interface model for eutectic alloys. Part I: Concentration dependent surface tension, WIAS Preprint no. 885, 2003 .

  4. W. DREYER, F. DUDERSTADT, S. QAMAR, Diffusion in the vicinity of an evolving spherical arsenic droplet, to appear as WIAS Preprint.

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LaTeX typesetting by I. Bremer
2004-08-13