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Simulation of Copolymer Phase Separation in One-Dimensional Thin Liquid Films

Published online by Cambridge University Press:  28 May 2015

Hidenori Yasuda
Hidenori Yasuda*
Affiliation:
Department of Mathematics, Josai University, Sakado, Saitama 350-0295, Japan
*
Corresponding author. Email: yasuda@math.josai.ac.jp
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Abstract

This paper discusses the development of an invariant finite difference scheme to simulate the microphase separation of copolymers in one-dimensional thin liquid films. The film phenomena are modelled using two-phase shallow water equations and the Ohta-Kawasaki potential, which governs the phase separation of the copolymer. Non-positive volume fractions and spurious oscillations are eventually eliminated, in simulating the one-dimensional phase separation lamellar pattern.

Keywords

76A2078M2082B26Phase separationshallow water equationscopolymerlamellar pattern
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 2 , Issue 1 , February 2012 , pp. 33 - 46
Copyright
Copyright © Global-Science Press 2012

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References

[1]Segalman, R.A., Patterning with block copolymer thin films, Materials Science and Engineering R48 (2005) pp. 191–226.Google Scholar
[2]Doi, M., Onuki, A., Dynamic coupling between stress and composition in polymer solutions and blends, J. Phys. II France 2 (1992) pp. 1631–1656.Google Scholar
[3]Tanaka, H., Viscoelastic model of phase separation, Phys. Rev. E56 (1997) 4451–4462.Google Scholar
[4]Yasuda, H., Two-phase shallow water equations and phase separation in thin immiscible liquid films, J. Sci. Comput. 43 (2010) pp. 471–487.CrossRefGoogle Scholar
[5]Ohta, T., Kawasaki, K., Equilibrium morphology of block copolymer melts, Macromolecules 19 (1986) pp. 2621–2632.CrossRefGoogle Scholar
[6]Ishii, M., Thermo-Fluid Dynamics Theory of Two-Phase Flow, Eyrolles, Paris, 1975.Google Scholar
[7]Gidaspow, D., Multiphase Flow and Fluidization, Academic Press, San Diego, 1994.Google Scholar
[8]Israelachvili, J.N., Intermolecular and Surface Forces, Academic Press, London, 1992.Google Scholar
[9]Pismen, L.M., Nanoscale effects in mesoscopic films, In: Golovin, A.A., Nepomnyashchy, A.A. (Eds.), Self-Assembly, Pattern Formation and Growth Phenomena in Nano-Systems, Springer, Amsterdam, 2006, pp. 167–193.Google Scholar
[10]Kawasaki, K., Nonequilibrium and Phase Transition, Asakura, Tokyo, 2000. (in Japanese)Google Scholar
[11]Hashimoto, T., Shibayama, M., Kawai, H., Domain-boundary structure of styrene-isoprene block copolymer films cast from solution. 4. Molecular-weight dependence of lamellar microdomains, Macromolecules 13 (1980) pp. 1237–1247.CrossRefGoogle Scholar
[12]Ohnishi, I., Nishiura, Y., Imai, M., Matsushita, Y., Analytical solutions describing the phase separation driven by a free energy functional containing a long-range interaction term, Chaos 9 (1999) pp. 329–341.Google Scholar
[13]Vladimirov, V.S., Methods of the Theory of Generalized Functions, Taylor & Francis, London, 2002.CrossRefGoogle Scholar
[14]Choksi, R., Peletier, M.A., Williams, J.F., On the phase diagram for microphase separation of diblock copolymers: An approach via a nonlocal Cahn-Hilliard functional, SIAM J. Appl. Math. 69 (2009) pp. 1712–1738.Google Scholar
[15]Yanenko, N.N., Shokin, Y.L., On the group classification of the difference scheme for systems of equations in gas dynamics, In: Holt, M. (Ed.), Lecture Notes in Physics 8, Springer, Berlin, 1971, pp. 3–17.Google Scholar
[16]Yanenko, N.N., Shokin, Y.L., Schemas numeriques invariant de groupe pour les equations de la gas, In: Cabannes, H., Temam, R. (Eds.), Lecture Notes in Physics 18, Springer, Berlin, 1973, pp. 174–186.Google Scholar
[17]LeVeque, R.J., Finite Volume Method for Hyperbolic Problems, Cambridge University Press, Cambridge, 2002.Google Scholar
[18]Dukowics, J.K., Ramshaw, J.D., Tensor viscosity method for convection in numerical fluid dynamics, J. Comput. Phys. 32 (1979) pp. 71–79.CrossRefGoogle Scholar
[19]Blossey, P.N., Durran, D.R., Selective monotonicity preservation in scalar advection, J. Comput. Phys. 227(2008) pp. 5160–5183.CrossRefGoogle Scholar
[20]Chaikin, P.M., Lubensky, T.C., Principles of Condensed Matter Physic, Cambridge University Press, Cambridge, 1995.CrossRefGoogle Scholar
[21]Gottlieb, S., Shu, C.-W., Total variation diminishing Runge-Kutta schemes, Math. Comp. 67 (1998) pp. 73–85.CrossRefGoogle Scholar