A non-hydrostatically strained solid in contact with its own melt or vapor can partially relieve its elastic energy by producing an undulated interface. This is the mechanism behind a morphological instability of the interface giving rise to the evaluation of grooves with a definite spacing under uniaxial stress and, possibly, island formation, if the stress is biaxial. The instability was first predicted by Asaro and Tiller [1] Experimentally, it has been observed and studied by Torii and Balibar[2]. Since the independent rediscovery of the instability by Grinfeld [3], it has often been referred to as the Grinfeld or Asaro-Tiller-Grinfeld instability (ATG). Further, exhaustive, information may be obtained in the references [4].
I want to show some pictures/movies of the timedevelopement of the 2 dimensional periodic surface structure of a solid in contact with its own melt under uniaxial stress/strain. The code is simply solving a linear system Ax=b for every step of the evolution. But building up the matrix A and the vector b(=By) is not that trivial. The solution x gives the displacement along the surface for the timestep. Out if this the normal velocity of the interface can be extracted. And with this information the surface is shifted in the next iteration step. If you have N points for your surface, the matrizes A and B are of size 2N times 2N. If N is large (>300) I use parallel algorithms for the linear algebra ( PLAPACK ( For further information about MPI applications one may look here: MPI libraries )) in order to solve the problem in appropriate time. The building of the matrizes is parallelised too and scales linear with the number of prozessors.
P_l is the pressure in the liquid phase. Sigma_0 is the uniaxial stress/strain that acts on the solid phase. E is the Young-modulus and \nu is the Poisson ratio. The last two quantities are measures for the elastical properties of a solid. If E is large, the solid is very inelestic. If you stress a solid in x-direction and therefore change its lenght, what is the change in lenght in z-direction? The answer is given by \nu, because \nu is the ratio of that changes. The liquid pressure and the stress/ strain can be easily varied in the experiments. But to leading order only the prestress is important.
First of all, as a starting point, we have our undulated surface which is uniaxial stressed. This is displayed in the top plot of the movie frame. My program computes the combination (\sigma_{tt}-sigma_{nn})^2 of the elements of the 2-D stress tensor \sigma . This combination is the change in chemical potential due to the applied stress. The interesting quantity for the evolution of the surface is the normal velocity, v_n, of the surface. In our case this velocity consists of three parts. They are seperatly displayed in the three bottom plots of the movie frame. Adding them and multipliing them with a constant gives v_n. The bottom plot is surface tension \gamma times the curvature \kappa. Above this the difference in density beween solid and liquid times gravitational acceleration g times z-component of the interface \zeta(x) is displayed. And above this we have (1-\nu^2)/ (2 E) times [ (\sigma_{tt}-sigma_{nn})^2 - const]. The constant is a shift in equilibrium position of a flat interface. If constant is (prestress)^2, then the equilibrium position of the flat interface is at z=0, like in our examples. The curves from the first frame are displayed in pink in the following frames. As titletext all the input data such as E, \nu, etc are displayed. dt is the timestep, needed to shift the surface via r_new=r_old+ dt v_n normal. Every PIC timesteps the data are written on the disc.
| A relax.mpg | A small movie, showing the relaxation to a flat surface for a surface with periodicity 1. |
| ~0.5 Mb 120 frames |
| [1] | R. J. Asaro and W. A. Tiller, Metall. Trans. {\bf 3}, 1789 (1972) |
| [2] | R.H. Torii and S. Balibar, J. Low Temp. Phys. {\bf 89}, 391 (1992) |
| [3a] | M.A. Grinfeld, Doklady Akademii Nauk SSSR {\bf 265} 836 (1982) |
| [3b] | M.A. Grinfeld, Sov. Phys. Dokl. {\bf 31}, 831 (1986) |
| [3c] | M.A. Grinfeld, Europhys. Lett. {\bf 22}, 723 (1993), and references therein. |
| [4a] | K. Kassner, C. Misbah, J. Müller, J. Kappey, P. Kohlert, J. Crystal Growth {\bf 225}, 289 (2001) |
| [4b] | K. Kassner, C. Misbah, J. Müller, J. Kappey, P. Kohlert, Phys. Rev. E {\bf 63}, 036117 (2001) |