Abstract
We determine the stability of a nonequilibrium interface between two coexisting solid phases in the presence of a weak external field. Starting at the coarsegrained (Cahn-Hilliard) level, we use the method of matched asymptotics to derive the macroscopic interfacial dynamics. We then show that the external field leads to an instability due to flux along the interface, in contrast with the more common Mullins-Sekerka type instability, which involves fluxes normal to the interface. We also find that the external field produces an important modification of the Gibbs-Thomson relation. With these results, we perform the linear stability analysis for an approximately flat interface. If the field is tangent to the interface, the modification of the Gibbs-Thomson relation is important and the interface is stabilized. If the field is normal to the interface, the surface flux is important, and the effect can be stabilizing or destabilizing, but the orientational dependence is opposite what would be obtained if the Mullins-Sekerka instability dominates. Numerical simulations are performed to study the effect of the surface current and are in agreement with our analytical results.
Similar content being viewed by others
References
J. S. Langer, inProceedings of Les Houches Summer School: Chance and Matter, J. Souletteet al., eds. (Elsevier, New York, 1987).
D. A. Kessler, J. Koplik, and H. Levine,Adv. Phys. 35:255 (1988).
P. Pelcé, ed.,Dynamics of Curved Fronts (Academic Press, London, 1988).
D. Bensimon, L. P. Kadanoff, S. Liang, B. I. Schraiman, and Ch. Tang,Rev. Mod. Phys. 58:977 (1986).
J. V. Maher,Phys. Rev. Lett. 54:1498 (1985); M. W. DeFrancesco and J. V. Maher,Phys. Rev. A 39:4709 (1989).
G. Tryggvason and H. Aref,J. Fluid Mech. 136:1 (1983);154:287 (1985).
J. Casademunt and D. Jasnow,Phys. Rev. Lett. 67:3677 (1991); J. Casademunt, D. Jasnow, and A. Hernàndez-Machado,Int. J. Mod. Phys. B 6:1647 (1992).
D. Jasnow, inPhase Transitions and Critical Phenomena, Vol. 10, C. Domb and J. L. Lebowitz, eds. (Academic Press, New York, 1986), p. 269, and references cited therein.
W. W. Mullins and R. F. Sekerka,J. Appl. Phys. 35:444 (1964).
P. G. Saffman and G. I. Taylor,Proc. R. Soc. A 245:312 (1958).
D. Jasnow and R. K. P. Zia,Phys. Rev. A 36:2243 (1987); Y. Oono and A. Shinozaki, University of Illinois, preprint (1992).
B. Derrida, J. L. Lebowitz, E. R. Speer, and H. Spohn,Phys. Rev. Lett. 67:165 (1991).
J. S. Langer and L. A. Turski,Acta. Metall. 25:1113 (1977); D. Jasnow, D. A. Nicole, and T. Ohta,Phys. Rev. A 23:3192 (1981).
K. Kawasaki and T. Ohta,Prog. Theor. Phys. 68:129 (1982); K. Kawasaki and T. Ohta,Physica A 118:175 (1983); T.Ohta,Ann. Phys. 158:31 (1984).
G. Caginalp,Ann. Phys. (N.Y.)172:136 (1986); G. Caginalp and P. C. Fife,SIAM J. Appl. Math. 48:506 (1988); G. Caginalp,Phys. Rev. A 39:5887 (1989).
R. L. Pego,Proc. R. Soc. Lond. A 422:261 (1989).
L. Bronsard and R. V. Kohn,J. Differential Equations 90:211 (1991).
H. K. Janssen and B. Schmittmann,Z. Phys. B 63:517 (1986).
H. K. Janssen and B. Schmittmann,Z. Phys. B 64:503 (1986).
K.-t. Leung and J. L. Cardy,J. Stat. Phys. 44:567 (1986).
K.-t. Leung, B. Schmittmann, and R. K. P. Zia,Phys. Rev. Lett. 62:1772 (1989).
A. Hernàndez-Machado and D. Jasnow,Phys. Rev. A 37:656 (1988).
K.-t. Leung,J. Stat. Phys. 50:405 (1988).
K.-t. Leung,J. Stat. Phys. 61:345 (1990).
R. K. P. Zia and K.-t. Leung,J. Phys. A 24:1399 (1991).
A. Hernàndez-Machado, H. Guo, J. L. Mozos, and D. Jasnow,Phys. Rev. A 39:4783 (1989).
S. Katz, J. L. Lebowitz, and H. Spohn,Phys. Rev. B 28:1655 (1983);J. Stat. Phys. 34:497 (1984).
J. Krug, J. L. Lebowitz, H. Spohn, and M. Q. Zang,J. Stat. Phys. 44:535 (1986); R. Dickman,Phys. Rev. A 38:2588 (1988); J. Krug,Phys. Rev. Lett. 67:1882 (1991).
L. Vallés and J. Marro,J. Stat. Phys. 43:441 (1986);49:89, 121 (1987); J. Marro, J. L. Vallés, and J. M. Gonzalez-Miranda,Phys. Rev. B 35:3372 (1987).
K. Kitahara, Y. Oono, and D. Jasnow,Mod. Phys. Lett. B 2:765 (1988).
C. Yeung, T. Rogers, A. Hernàndez-Machado, and D. Jasnow,J. Stat. Phys. 66:1071 (1992).
S. Puri, K. Binder, and S. Dattagupta,Phys. Rev. B 46:98 (1992).
D. H. Boal, B. Schmittman, and R. P. K. Zia,Phys. Rev. A 43:5214 (1991).
K.-t. Leung, K. K. Mon, J. L. Vallés, and R. K. P. Zia,Phys. Rev. Lett. 61:1744 (1988);Phys. Rev. B 39:9312 (1989).
J. W. Cahn and H. E. Hilliard,J. Chem. Phys. 28:258 (1958); H. E. Cook,Acta Metall. 18:297 (1970).
J. D. Gunton, M. san Miguel, and P. S. Sahni, inPhase Transitions and Critical Phenomena, Vol. 8, C. Domb and J. L. Lebowitz, eds. (Academic Press, New York, 1983), p. 267; H. Furukawa,Adv. Phys. 34:703 (1985); K. Binder, inPhase Transformations of Materials (Materials Science and Technology), Vol. 5, P. Haasen, ed., p. 405 (Springer-Verlag, Berlin, 1991).
T. Ohta,J. Phys. C 21:L361 (1988).
J. L. Mozos and A. Hernàndez-Machado, in preparation (1992).
T. Rogers, Ph.D. Thesis, University of Toronto (1989).
P. Pelce and A. Pumir,J. Cryst. Growth 73:337 (1985); P. Pelce,Europhys. Lett. 7:453 (1988).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Yeung, C., Mozos, J.L., Hernánez-Machado, A. et al. Surface-driven instability and enhanced relaxation in the dynamics of a nonequilibrium interface. J Stat Phys 70, 1149–1174 (1993). https://doi.org/10.1007/BF01049426
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01049426