Abstract
The phase separation kinetics of a two-dimensional binary mixture at critical composition confined between (one-dimensional) straight walls which preferentially attract one component of the mixture is studied for a wide range of distancesD between the walls. Following earlier related work on semiinfinite systems, two choices of surface forces at the walls are considered, one corresponding to an incompletely wet state of the walls, the other to a completely wet state (forD→∞). The nonlinear Cahn-Hilliard-type equation, supplemented with appropriate boundary conditions which account for the presence of surfaces, is replaced by a discrete equivalent and integrated numerically. Starting from a random initial distribution of the two species (say,A andB), an oscillatory concentration profile rapidly forms across the film. This is characterized by two thin enrichment layers of the preferred component at the walls, followed by adjacent depletion layers. While in these layers phase separation is essentially complete, the further oscillations of the average composition at distanceZ from a wall get rapidly damped asZ increases toward the center of the film. This structure is relatively stable for an intermediate time scale, while the inhomogeneous structure in the center of the film coarsens. The concentration correlation function in directions parallel to the walls (integrated over allZ) and the associated structure factor (describing small-angle scattering from the film) exhibit a scaling behavior, similar to bulk spinodal decomposition, and the characteristic length scale grows with time asl ‖, wherea is close to the Lifshitz-Slyozov value 1/3, and the coefficients α, β depend on film thickness only weakly. Only when one considers the local correlation function at distances close to the walls are deviations from scaling observed due to the competing effects of the grwing surface enrichment layers. However, at very late times [whenl ‖ (t) becomes comparable toD] this bulklike description breaks down, and a concentration distribution is expected to be established which is a superposition of domains separated by interfaces perpendicular to the walls, the one type of domain being rich inA and nearly homogeneous, and the other poor inA except for two thin enrichment layers at the walls.
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References
J. D. Gunton, M. San Miguel, and P. S. Salmi, InPhase Transitions and Critical Phenomena, Vol. 8, C. Domb and J. L. Lebowitz, eds. (Academic Press, London 1983), p. 267.
S. Komura and H. Furukawa, eds.,Dynamics of Ordering Processes in Condensed Matter (Plenum Press, New York 1988).
K. Binder, InMaterials Science and Technology, Vol. 5: Transformations in Materials, R. W. Cahn, P. Haasen, and E. J. Kramer, eds. (VCH, Weinheim, 1991), p. 405.
G. Kostorz, preprint.
J. W. Cahn and J. E. Hilliard,J. Chem. Phys. 28:258 (1958); J. W. Cahn,Acta Metall. 9:795 (1951).
I. M. Lifshitz and V. V. Slyozov,J. Phys. Chem. Solids 19:35 (1961).
K. Binder and D. Stauffer,Phys. Rev. Lett. 33:1006 (1974);Z. Phys. B 24:406 (1976).
A. B. Bortz, M. H. Kalos, J. L. Lebowitz, and M. A. Zendejas,Phys. Rev. B 10: 535 (1974).
J. Marro, A. B. Bortz, M. H. Kalos, and J. L. Lebowitz,Phys. Rev. B 12:2000 (1975); M. H. Kalos, J. L. Lebowitz, O. Penrose, and A. Sur,J. Stat. Phys. 18:39 (1978).
D. Huse,Phys. Rev. B 34:7845 (1986).
Y. Oono and S. Puri,Phys. Rev. Lett. 58:836 (1987); Y. Oono and S. Puri,Phys. Rev. A 38:434 (1988); S. Puri and Y. Oono,Phys. Rev. A 38:1542 (1988); A. Shinozaki and Y. Oono,Phys. Rev. Lett. 66:173 (1991).
T. M. Rogers, K. K. Elder, and R. C. Desai,Phys. Rev. B 37:196 (1988); A. Chakrabarti and J. D. Gunton,Phys. Rev. B 37:3798 (1988).
J. G. Amar, F. E. Sullivan, and R. D. Mountain,Phys. Rev. B 37:196 (1988).
K. Binder and H. L. Frisch,Z. Phys. B 84:403 (1991).
R. A. L. Jones, L. J. Norton, E. J. Kramer, F. S. Bates, and P. Wiltzius,Phys. Rev. Lett. 66:1326 (1991).
P. Wiltzius and A. Cumming,Phys. Rev. Lett. 66:3000 (1991).
U. Steiner, E. Eiser, J. Klein, A. Budkowski, and L. J. Fetters,Science 258:1126 (1992).
F. Bruder and R. Brenn,Phys. Rev. Lett. 69:624 (1992).
H. Tanaka,Phys. Rev. Lett. 70:53 (1992).
S. Puri and K. Binder,Phys. Rev. A 46:R4487 (1992); S. Puri and H. L. Frisch,J. Chem. Phys., in press.
H. W. Diehl and H.-K. Janssen,Phys. Rev. A 45:7145 (1992).
B. O. Shi, C. Harrison, and A. Cumming,Phys. Rev. Lett. 70:206 (1993).
J. F. Marko,Phys. Rev. E 48:2861 (1993).
G. Krausch, C.-A. Dai, E. J. Kramer, J. F. Marko, and F. S. Bates,Macromolecules 26:5566 (1993); G. Krausch, E. J. Kramer, F. S. Bates, J. F. Marko, G. Brown, and A. Chakrabarti, preprint.
G. Brown and A. Chakrabarti,Phys. Rev. A 46:4829 (1992).
S. Puri and K. Binder,Phys. Rev. E 49:5359 (1994).
S. Dietrich, InPhase Transitions and Critical Phenomena, Vol. 12, C. Domb and J. L. Lebowitz, eds. (Academic Press, New York, 1988), p. 1.
D. E. Sullivan and M. M. Telo da Gama, inFluid Interfacial Phenomena, C. A. Croxton, ed. (Wiley, New York, 1986), p. 45.
P. G. de Gennes,Rev. Mod. Phys. 57:827 (1985).
M. E. Fisher,J. Stat. Phys. 34:667 (1984);J. Chem. Soc. Faraday Trans. 282:1569 (1986).
R. Lipowsky and D. A. Huse,Phys. Rev. Lett. 52:353 (1986).
M. N. Barber, InPhase Transitions and Critical Phenomena, Vol. 8, C. Domb and J. L. Lebowitz, eds. (Academic Press, New York, 1983), p. 156.
K. Binder,Ferroelectrics 73:43 (1987).
V. Privman, ed.,Finite Size Scaling and the Numerical Simulution of Statistical Systems (World Scientific, Singapore, 1990).
K. Binder,Annu. Rev. Phys. Chem. 43:33 (1992).
P. G. de Gennes,Macromolecules 13:1069 (1980).
J.-S. Wang and K. Binder,Macromol. Chem. Theory Simul. 1:49 (1992).
M. E. Fisher and Nakanishi,J. Chem. Phys. 75:5857 (1981); H. Nakanishi and M. E. Fisher,J. Chem. Phys. 78:3279 (1983).
D. Henderson, ed.,Fundamentals of Inhomogeneous Fluids (M. Dekker, New York, 1992).
K. Binder and D. P. Landau,J. Chem. Phys. 96:1444 (1992).
E. Siggia,Phys. Rev. A 20:595 (1979).
H. Furukawa,Physica A 123:497 (1984);Adv. Phys. 34:703 (1985).
T. Koga and K. Kawasaki,Phys. Rev. A 44:R817 (1991).
S. Puri and B. Dünweg,Phys. Rev. A 455:R6977 (1992), A Shinozaki and Y. Oono,Phys. Rev. E, in press.
K. Kawasaki, InPhase Transitions and Critical Phenomena, Vol. 2, C. Domb and M. S. Green, eds. (Academic Press, London, 1972), Chapter 11.
K. Binder,Z. Phys. 267:313 (1974).
R. C. Ball and R. L. H. Essery,J. Phys.: Condensed Matter 2:10303 (1990).
J. Rowlinson and B. Widom,Molecular Theory of Capillarity (Oxford University Press, Oxford, 1982).
I. Schmidt and K. Binder,Z. Phys. B 67:369 (1987).
S. Puri and K. Binder,Z. Phys. B 86:263 (1992).
D. Nicolaides and R. Evans,Phys. Rev. B 39:9336 (1989).
B. Widom, InPhase Transitions and Critical Phenomena, Vol. 2, C. Domb and M. S. Green, eds. (Academic Press, London, 1972).
M. E. Fisher,J. Phys. Soc. Japan Suppl. 26:87 (1969); see also G. G. Gabrera, R. Jullien, E. Brezin, and J. Zinn-Justin,J. Phys. (Paris)47:1305 (1986).
E. V. Albano, K. Binder, W. Paul, and D. W. Heermann,Physica A 183:130 (1992); M. Müller and W. Paul,J. Stat. Phys. 73:209 (1993).
K. Kawasaki and T. Ohta,Physica A 118:175 (1983); T. Koga and K. Kawasaki,Physica A 196:389 (1993); M. Tokuyama and Y. Enomoto,Phys. Rev. E 47:1156 (1993).
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Puri, S., Binder, K. Surface-directed spinodal decomposition in a thin-film geometry: A computer simulation. J Stat Phys 77, 145–172 (1994). https://doi.org/10.1007/BF02186836
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DOI: https://doi.org/10.1007/BF02186836