×

Mesoscopic simulation of Ostwald ripening. (English) Zbl 1158.82312

Summary: The self-organization of particles in a two phase system in the coexistence region through a diffusive mechanism is known as Ostwald ripening. This phenomenon is an example of a multiscale problem in that the microscopic level interaction of the particles can greatly impact the macroscale or observable morphology of the system. Ostwald ripening is studied here through the use of a mesoscopic model which is a stochastic partial integrodifferential equation that is derived from a spin exchange Ising model. This model is studied through the use of recently developed and benchmarked spectral schemes for the simulation of solutions to stochastic partial differential equations. The typical cluster size is observed to grow like \(t^{\frac {1}{3}}\) over range of times with faster growth at later times. The results included here also demonstrate the effect of adjusting the interparticle interaction on the morphological evolution of the system at the macroscopic level.

MSC:

82C44 Dynamics of disordered systems (random Ising systems, etc.) in time-dependent statistical mechanics
65C30 Numerical solutions to stochastic differential and integral equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Lifshitz, I.; Slyozov, V., The kinetics of precipitation from supersaturated solid solutions, J. Phys. Chem. Solids, 19, 35-50 (1961)
[2] Wagner, C., Theorie der alterung von niederschlägen durch unlösen, Z. Elektrochem., 65, 581-594 (1961)
[3] Niethammer, B.; Pego, R., The LSW model for domain coarsening: asymptotic behavior for conserved total mass, J. Stat. Phys., 104, 1113-1144 (2001) · Zbl 1143.82314
[4] Newman, M.; Barkema, G., Monte Carlo methods in statistical physics (1999), Clarendon Press: Clarendon Press Oxford · Zbl 1012.82019
[5] Penrose, O., Statistical mechanics and the kinetics of phase separation, (Ball, J., Material Instabilities in Continuum Mechanics and Related Mathematical Problems (1988), Clarendon Press: Clarendon Press Oxford), 373-394
[6] Voorhees, P., The theory of Ostwald ripening, J. Stat. Phys., 38, 231-252 (1985)
[7] Voorhees, P., Ostwald ripening in two-phase mixtures, Annu. Rev. Mater. Sci., 22, 197-215 (1992)
[8] Yao, J.; Elder, K.; Guo, H.; Grant, M., Late stage droplet growth, Physica A, 204, 770-788 (1994)
[9] Voorhees, P.; Schaefer, R., In situ observation of particle motion and diffusion interactions during coarsening, Acta Metall., 35, 327-339 (1987)
[10] Niethammer, B.; Pego, R., Non-self-similar behavior in the LSW theory of Ostwald ripening, J. Stat. Phys., 95, 867-902 (1999) · Zbl 1005.82510
[11] McFadden, G.; Voorhees, P.; Boisvert, R.; Meiron, D., A boundary integral method for the simulation of two-dimensional particle coarsening, J. Sci. Comput., 1, 117-144 (1986) · Zbl 0649.65066
[12] Voorhees, P.; McFadden, G.; Boisvert, R.; Meiron, D., Numerical simulation of morphological development during Ostwald ripening, Acta Metall., 36, 207-222 (1988)
[13] Akaiwa, N.; Meiron, D., Two-dimensional late-stage coarsening for nucleation and growth at high-area fractions, Phys. Rev. E, 54, R13-R16 (1996)
[14] Imaeda, T.; Kawasaki, K., Theory of morphological evolution in Ostwald ripening, Physica A, 186, 359-387 (1992)
[15] Hou, T.; Lowengrub, J.; Shelley, M., Boundary integral methods for multicomponent fluids and multiphase materials, J. Comput. Phys., 169, 302-362 (2001) · Zbl 1046.76029
[16] Filbet, F.; Laurençot, P., Numerical approximation of the Lifshitz-Slyozov-Wagner equation, SIAM J. Num. Anal., 41, 563-588 (2003) · Zbl 1053.35085
[17] Carrillo, J.; Goudon, T., A numerical study on large-time asymptotics of the Lifschitz-Slyozov system, J. Sci. Comput., 20, 69-113 (2004) · Zbl 1097.35102
[18] Elliott, C.; French, D., A non-conforming finite element method for the two-dimensional Cahn-Hilliard equation, SIAM J. Num. Anal., 26, 884-903 (1989) · Zbl 0686.65086
[19] Elliott, C.; French, D., Numerical studies of the Cahn-Hilliard equation for phase separation, IMA J. Appl. Math., 38, 97-128 (1987) · Zbl 0632.65113
[20] Barrett, J.; Blowey, J., Finite element approximation of the Cahn-Hilliard equation with concentration dependent mobility, Math. Comput., 68, 487-517 (1999) · Zbl 1126.65321
[21] Chakrabarti, A.; Toral, R.; Gunton, J., Late-stage coarsening for off-critical quenches: scaling functions and the growth law, Phys. Rev. E, 47, 3025-3038 (1993)
[22] Gunton, J.; Toral, R.; Chakrabarti, A., Numerical studies of phase separation in models of binary alloys and polymer blends, Physica Scripta, T33, 12-19 (1990)
[23] Rogers, T.; Elder, K.; Desai, R., Numerical study of the late stages of spinodal decomposition, Phys. Rev. B, 37, 9638-9649 (1988)
[24] Rogers, T.; Desai, R., Numerical study of late-stage coarsening for off-critical quenches in the Cahn-Hilliard equation of phase separation, Phys. Rev. B, 39, 11956-11964 (1989)
[25] de Mello, E.; Filho, O., Numerical study of the Cahn-Hilliard equation in one, two, and three dimensions, Physica A, 347, 429-443 (2005)
[26] Kawasaki, K., Diffusion constants near the critical point for time-dependent Ising models, Phys. Rev., 145, 224-230 (1965)
[27] Gunton, J.; Gawlinski, E.; Kaski, K., Numerical simulation studies of the kinetics of first-order phase transitions, (Komura, S.; Furukawa, H., Dynamics of ordering processes in condensed matter (1988), Plenum: Plenum New York), 101-110
[28] Landau, D.; Binder, K., A guide to Monte Carlo simulations in statistical physics (2000), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0998.82504
[29] Zheng, B., Monte Carlo simulations of short-time critical dynamics, Int. J. Mod. Phys. B, 12, 1419-1484 (1998)
[30] Katsoulakis, M.; Vlachos, D., Coarse-grained stochastic processes and kinetic Monte Carlo simulators for the diffusion of interacting particles, J. Chem. Phys., 119, 9412-9427 (2003)
[31] Katsoulakis, M.; Majda, A.; Vlachos, D., Coarse-grained stochastic processes and Monte Carlo simulations in lattice systems, J. Comput. Phys., 186, 250-278 (2003) · Zbl 1034.82053
[32] Katsoulakis, M.; Majda, A.; Vlachos, D., Coarse-grained stochastic processes for microscopic lattice systems, Proc. Natl. Acad. Sci., 100, 782-787 (2003) · Zbl 1063.82033
[33] Lebowitz, J.; Orlandi, E.; Presutti, E., A particle model for spinodal decomposition, J. Stat. Phys., 63, 933-974 (1991)
[34] Katsoulakis, M.; Souganidis, P., Stochastic Ising models and anisotropic front propagation, J. Stat. Phys., 87, 63-90 (1997) · Zbl 0937.82034
[35] Hildebrand, M.; Mikhailov, A. S., Mesoscopic modeling in the kinetic theory of adsorbates, J. Phys. Chem., 100, 19089-19101 (1996)
[36] Giacomin, G.; Lebowitz, J., Exact macroscopic description of phase segregation in model alloys with long range interactions, Phys. Rev. Lett., 76, 1094-1097 (1996)
[37] Horntrop, D.; Katsoulakis, M.; Vlachos, D., Spectral methods for mesoscopic models of pattern formation, J. Comput. Phys., 173, 364-390 (2001) · Zbl 0987.65007
[38] Gaines, J., Numerical experiments with S(P)DE’s, (Etheridge, A., Stochastic Partial Differential Equations (1995), Cambridge University Press: Cambridge University Press Cambridge), 55-71 · Zbl 0829.60048
[39] Gyöngy, I., Approximations of stochastic partial differential equations, (Da Prato, G.; Tubaro, L., Stochastic Partial Differential Equations and Applications (2002), Marcel Dekker: Marcel Dekker New York), 287-307 · Zbl 1004.60073
[40] Gyöngy, I., Lattice approximations for stochastic quasi-linear parabolic partial differential equations driven by space-time white noise I, Potential Anal., 9, 1-25 (1998) · Zbl 0915.60069
[41] Gyöngy, I., Lattice approximations for stochastic quasi-linear parabolic partial differential equations driven by space-time white noise, Potential Anal., 11, 1-37 (1999) · Zbl 0944.60074
[42] Gyöngy, I.; Krylov, N., On the splitting-up method and stochastic partial differential equations, Ann. Prob., 31, 564-591 (2003) · Zbl 1028.60058
[43] Seesselberg, M.; Petruccione, F., Numerical integration of stochastic partial differential equations, Comput. Phys. Commun., 74, 303-315 (1993) · Zbl 0854.65127
[44] Yoo, H., Semi-discretization of stochastic partial differential equations on R1, Math. Comput., 69, 653-666 (2000) · Zbl 0942.65006
[45] Ghanem, R.; Spanos, P., Stochastic finite elements: a spectral approach (1991), Springer: Springer New York · Zbl 0722.73080
[46] Allen, E.; Novosel, S.; Zheng, Z., Finite element and difference approximation of some linear stochastic partial differential equations, Stochast. Stochast. Rep., 64, 117-142 (1998) · Zbl 0907.65147
[47] Deb, M.; Babuška, I.; Oden, J., Solution of stochastic partial differential equations using Galerkin finite element techniques, Comput. Met. Appl. Mech. Eng., 190, 6359-6372 (2001) · Zbl 1075.65006
[48] Du, Q.; Zhang, T., Numerical approximation of some linear stochastic partial differential equations driven by special additive noises, SIAM J. Num. Anal., 40, 1421-1445 (2002) · Zbl 1030.65002
[49] Machiels, L.; Deville, M., Numerical simulation of randomly forced turbulent flows, J. Comput. Phys., 145, 246-279 (1998) · Zbl 0926.76086
[50] D. Horntrop, Spectral schemes for stochastic partial differential equations, submitted.; D. Horntrop, Spectral schemes for stochastic partial differential equations, submitted.
[51] Canuto, C.; Hussaini, M.; Quarteroni, A.; Zang, T., Spectral methods in fluid dynamics (1988), Springer: Springer New York · Zbl 0658.76001
[52] Vlachos, D.; Katsoulakis, M., Derivation and validation of mesoscopic theories for diffusion-reaction of interacting molecules, Phys. Rev. Lett., 85, 3898-3901 (2000)
[53] Lam, R.; Basak, T.; Vlachos, D.; Katsoulakis, M., Validation of mesoscopic theories and their application to computing concentration dependent diffusivities, J. Chem. Phys., 115, 11278-11288 (2001)
[54] Doob, J., Stochastic processes (1953), Wiley: Wiley New York
[55] Elliott, F.; Majda, A.; Horntrop, D.; McLaughlin, R., Hierarchical Monte Carlo methods for fractal random fields, J. Stat. Phys., 81, 717-736 (1995) · Zbl 1107.82326
[56] Elliott, F.; Horntrop, D.; Majda, A., Monte Carlo methods for turbulent tracers with long range and fractal random velocity fields, Chaos, 7, 39-48 (1997) · Zbl 0953.76528
[57] Elliott, F.; Horntrop, D.; Majda, A., A Fourier-wavelet Monte Carlo method for fractal random fields, J. Comput. Phys., 132, 384-408 (1997) · Zbl 0876.65096
[58] D. Horntrop, A. Majda, An overview of Monte Carlo simulation techniques for the generation of random fields, in: P. Muller, D. Henderson (Eds.), Monte Carlo simulations in oceanography, Proceedings of the Ninth Aha Huliko Hawaiian Winter Workshop, 1997, pp. 67-79.; D. Horntrop, A. Majda, An overview of Monte Carlo simulation techniques for the generation of random fields, in: P. Muller, D. Henderson (Eds.), Monte Carlo simulations in oceanography, Proceedings of the Ninth Aha Huliko Hawaiian Winter Workshop, 1997, pp. 67-79.
[59] Prigarin, S., Spectral models of random fields in Monte Carlo methods (2001), VSP: VSP Utrecht
[60] Sabelfeld, K., Monte Carlo methods (1991), Springer: Springer Berlin
[61] P. Kramer, O. Kurbanmuradov, K. Sabelfeld, Extensions of multiscale Gaussian random field simulation algorithms, submitted.; P. Kramer, O. Kurbanmuradov, K. Sabelfeld, Extensions of multiscale Gaussian random field simulation algorithms, submitted. · Zbl 1124.65009
[62] Kloeden, P.; Platen, E., Numerical solution of stochastic differential equations (1992), Springer: Springer Berlin · Zbl 0752.60043
[63] Burrage, K.; Platen, E., Runge-Kutta methods for stochastic differential equations, Ann. Num. Math., 1, 63-78 (1994) · Zbl 0824.65148
[64] Burrage, K.; Burrage, P., High strong order explicit Runge-Kutta methods for stochastic ordinary differential equations, Appl. Num. Math., 22, 81-101 (1996) · Zbl 0868.65101
[65] Glotzer, S.; Di Marzio, E.; Muthukumar, M., Reaction-controlled morphology of phase-separating mixtures, Phys. Rev. Lett., 74, 2034-2037 (1995)
[66] Fradkov, V.; Glicksman, M.; Marsh, S., Coarsening kinetics in finite clusters, Phys. Rev. E, 53, 3925-3932 (1996)
[67] Hönig, A.; Niethammer, B.; Otto, F., On first-order corrections to the LSW theory I: infinite systems, J. Stat. Phys., 119, 61-122 (2005) · Zbl 1073.82013
[68] Hönig, A.; Niethammer, B.; Otto, F., On first-order corrections to the LSW theory II: finite systems, J. Stat. Phys., 119, 123-164 (2005) · Zbl 1100.82007
[69] Mandyam, H.; Glicksman, M.; Helsing, J.; Marsh, S., Statistical simulations of diffusional coarsening in finite clusters, Phys. Rev. E, 58, 2119-2130 (1998)
[70] D. Horntrop, Mesoscopic simulation for self-orgranization in surface processes, in: V. Sunderam, et al. (Eds.), Computational Science-ICCS 2005, Springer Lecture Notes in Computer Science, vol. 3514, 2005, pp. 852-859.; D. Horntrop, Mesoscopic simulation for self-orgranization in surface processes, in: V. Sunderam, et al. (Eds.), Computational Science-ICCS 2005, Springer Lecture Notes in Computer Science, vol. 3514, 2005, pp. 852-859. · Zbl 1129.82310
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.