June  2015, 35(6): 2711-2739. doi: 10.3934/dcds.2015.35.2711

The Cahn--Hilliard--de Gennes and generalized Penrose--Fife models for polymer phase separation

1. 

Institute of Mathematics and Cryptology, Cybernetics Faculty, Military University of Technology, S. Kaliskiego 2, 00-908 Warsaw, Poland

Received  December 2013 Revised  March 2014 Published  December 2014

The goal of this paper is twofold. Firstly, we overview the known Flory--Huggins--de Gennes (FHdG) free energy and the associated degenerate singular Cahn--Hilliard--de Gennes (CHdG) model for isothermal phase separation in a binary polymer mixture. Secondly, motivated by the structure of the FHdG free energy, in which the gradient term is made up of energetic and entropic contributions, we set up a corresponding thermodynamically consistent model for nonisothermal phase separation in such mixture. The model is characterized by the modified both energy and entropy fluxes by suitable ``extra" terms. In this sense it generalizes the well-known Penrose--Fife model in which only entropy flux is modified by an ``extra" term.
Citation: Irena PawŁow. The Cahn--Hilliard--de Gennes and generalized Penrose--Fife models for polymer phase separation. Discrete and Continuous Dynamical Systems, 2015, 35 (6) : 2711-2739. doi: 10.3934/dcds.2015.35.2711
References:
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H. W. Alt, The entropy principle for interfaces. Fluids and solids, Adv. Math. Sci. Appl., 19 (2009), 585-663.

[2]

H. W. Alt and I. Pawłow, A mathematical model of dynamics of non-isothermal phase separation, Physica D, 59 (1992), 389-416. doi: 10.1016/0167-2789(92)90078-2.

[3]

H. W. Alt and I. Pawłow, Thermodynamical models of phase transitions with multicomponent order parameter, in Trends in Applications of Mathematics to Mechanics, (eds. M. D. P. M. Marques and J. F. Rodrigues), Pitman Monographs and Surveys in Pure and Applied Mathematics, 77, Longman, New York (1995), 87-98.

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H. W. Alt and I. Pawłow, On the entropy principle of phase transition models with a conserved order parameter, Adv. Math. Sci. Appl., 6 (1996), 291-376.

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S. H. Anastasiadis, I. Gancarz and J. T. Koberstein, Interfacial tension of immiscible polymer blends: Temperature and molecular weight dependence, Macromolecules, 21 (1988), 2980-2987. doi: 10.1021/ma00188a015.

[6]

M. V. Ariyapadi and E. B. Nauman, Free energy of an inhomogeneous polymer-polymer-solvent sytem, J. Polymer Sci., Part B: Polymer Physics, 27 (1989), 2637-2646.

[7]

M. V. Ariyapadi and E. B. Nauman, Gradient energy parameters for polymer-polymer-solvent systems and their application to spinodal decomposition in true ternary systems, J. Polymer Sci., Part B: Polymer Physics, 28 (1990), 2395-2409. doi: 10.1002/polb.1990.090281216.

[8]

M. V. Aryapadi and E. B. Nauman, Free energy of an inhomogeneous polymer-polymer-solvent system. II, J. Polymer Sci., Part B: Polymer Physics, 30 (1992), 533-538. doi: 10.1002/polb.1992.090300603.

[9]

S. Benzoni-Gavage, L. Chupin, D. Jamet and J. Vovelle, On a phase field model for solid-liquid phase transitions, Discrete Contin. Dyn. Syst., 32 (2012), 1997-2025. doi: 10.3934/dcds.2012.32.1997.

[10]

K. Binder, Collective diffusion, nucleation and spinodal decomposition in polymer mixtures, J. Chem. Phys., 79 (1983), 6387-6409. doi: 10.1063/1.445747.

[11]

G. Bonfanti, M. Frémond and F. Luterotti, Existence and uniqueness results to a phase transition model based on microscopic accelerations and movements, Nonlinear Anal., Real World Applications, 5 (2004), 123-140. doi: 10.1016/S1468-1218(03)00021-X.

[12]

M. Brocate and J. Sprekels, Hysteresis and Phase Transitions, Applied Math. Sciences, 121, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-4048-8.

[13]

F. Brochard, J. Jouffroy and P. Levinson, Polymer-polymer diffusion in melts, Macromolecules, 16 (1983), 1638-1641. doi: 10.1021/ma00244a016.

[14]

F.Brochard, J. Jouffroy and P. Levinson, Polymer diffusion in blends: Effects of mutual friction, Macromolecules, 17 (1984), 2925-2927. doi: 10.1021/ma00142a084.

[15]

G. Caginalp, An analysis of a phase field model of a free boundary, Arch. Ration. Mech. Anal., 92 (1986), 205-245. doi: 10.1007/BF00254827.

[16]

J. W. Cahn, On spinodal decomposition, Acta Metall., 9 (1961), 795-801. doi: 10.1016/0001-6160(61)90182-1.

[17]

J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. I. Interfacial free energy, J. Chem. Phys., 28 (1958), 258-267. doi: 10.1063/1.1744102.

[18]

J. W. Cahn and J. E. Hilliard, Spinodal decomposition: A reprise, Acta Metall., 19 (1971), 151-161. doi: 10.1016/0001-6160(71)90127-1.

[19]

L.-Q. Chen, Phase-field models for microstructure evolution, Annu. Rev. Mater. Res., 32 (2002), 113-140.

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P. Debye, Angular dissymmetry of the critical opalescence in liquid mixtures, J. Chem. Phys., 31 (1959), 680-687. doi: 10.1063/1.1730446.

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P. G. de Gennes, Scaling Concepts in Polymer Physics, Cornell Univ. Press, Ithaca, 1979.

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P. G. de Gennes, Dynamics of fluctuations and spinodal decomposition in polymer blends, J. Chem. Phys., 72 (1980), 4756-4763. doi: 10.1063/1.439809.

[23]

S. R. de Groot and P. Mazur, Non-Equilibrium Thermodynamics, Dover Publ., New York, 1984.

[24]

D. G. B. Edelen, On the existence of symmetry relations and dissipation potentials, Arch. Ration. Mech. Anal., 51 (1973), 218-227. doi: 10.1007/BF00276075.

[25]

H. Emmerich, Advances of and by phase-filed modelling in condensed-matter physics, Advances in Physics, 57 (2008), 1-87.

[26]

M. Fabrizio, C. Giorgi and A. Morro, A thermodynamic approach to non-isothermal phase-filed evolution in continuum physics, Physica D, 214 (2006), 144-156. doi: 10.1016/j.physd.2006.01.002.

[27]

F. Falk, Cahn-Hilliard theory and irreversible thermodynamics, J. Non-Equilib. Thermodyn., 17 (1992), 53-65. doi: 10.1515/jnet.1992.17.1.53.

[28]

P. J. Flory, Principles of Polymer Chemistry, Cornell University Press, Ithaca, New York, 1953.

[29]

M. Frémond, Non-Smooth Thermomechanics, Springer, Berlin, 2002. doi: 10.1007/978-3-662-04800-9.

[30]

E. Fried and M. E. Gurtin, Continuum theory of thermally induced phase transitions based on an order parameter, Physica D, 68 (1993), 326-343. doi: 10.1016/0167-2789(93)90128-N.

[31]

P. Galenko and D. Jou, Diffuse-interface model for rapid phase transformations in nonequilibrium systems, Phys. Rev. E, 71 (2005), 046125. doi: 10.1103/PhysRevE.71.046125.

[32]

D. Y. Gao, Duality Principles in Nonconvex Systems, Kluwer Academic Publishers, Dordrecht, 2000. doi: 10.1007/978-1-4757-3176-7.

[33]

J. D. Gunton, M. San Miguel and P. S. Sahni, The dynamics of first-order phase transitions, in Phase Transitions and Critical Phenomena (eds. C. Domb and J. L. Lebowitz), Acadmeic Press, London, 8 (1983), 267-482.

[34]

M. E. Gurtin, Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance, Physica D, 92 (1996), 178-192. doi: 10.1016/0167-2789(95)00173-5.

[35]

B. I. Halperin, P. C. Hohenberg and S.-K. Ma, Renormalization-group methods for critical dynamics: I. Recursion relations and effects of energy conservation, Phys. Rev. B, 10 (1974), 139-153. doi: 10.1103/PhysRevB.10.139.

[36]

V. J. Klenin, Thermodynamics of Systems Containing Flexible-Chain Polymers, Elsevier, Amsterdam, 1999.

[37]

W. Köhler, A. Krekhov and W. Zimmermann, Thermal Diffusion in Polymer Blends: Criticality and Pattern Formation, Preprint Universität Bayreuth, 2013 (available from uni-bayreuth. de).

[38]

I. S. Liu, Method of Lagrange multipliers for exploitation of the entropy principle, Arch. Ration. Mech. Anal., 46 (1972), 131-148.

[39]

L. P. McMaster, Aspects of liquid-liquid phase transition phenomena in multicomponent polymeric systems, in Copolymers, Polyblends, and Composities (ed. N. A. J. Platzer), Adv. Chem. Ser., 142 (1975), 43-65. doi: 10.1021/ba-1975-0142.ch005.

[40]

A. Miranville and G. Schimperna, Nonisothermal phase separation based on a microforce balance, Discrete Contin. Dyn. Syst. Ser. B, 5 (2005), 753-768. doi: 10.3934/dcdsb.2005.5.753.

[41]

V. S. Mitlin and L. I. Manevitch, Kinetically stable structures in the nonlinear theory of spinodal decomposition, J. Polymer Sci., Part B: Polymer Physics, 28 (1990), 1-16. doi: 10.1002/polb.1990.090280101.

[42]

V. S. Mitlin, L. I. Manevitch and I. Ya. Erukhimovich, Formation of kinetically stable domain structure during spinadal decomposition of binary polymer mixtures, Zh. Eksper. Teoret. Fiz., 88 (1985), 495-506; Sov. Phys. JETP, 61 (1985), 290-296.

[43]

A. Morro, A phase-field approach to non-isothermal transitions, Mathematical and Computer Modelling, 48 (2008), 621-633. doi: 10.1016/j.mcm.2007.11.001.

[44]

I. Müller, Thermodynamics, Pitman, London, 1985.

[45]

Y. S. Nam and T. G. Park, Porous biodegradable polymeric scaffolds prepared by thermally induced phase separation, J. Biomedical Materials Research, 47 (1999), 8-17. doi: 10.1002/(SICI)1097-4636(199910)47:1<8::AID-JBM2>3.0.CO;2-L.

[46]

E. B. Nauman, M. V. Ariyapadi, N. P. Balsara, T. A. Grocela, J. S. Furno, S. H. Liu and R. Mallikarjun, Compositional quenching: A process for forming polymer-in-polymer microdispersions and cocontinuous networks, Chem. Eng. Comm., 66 (1988), 29-55. doi: 10.1080/00986448808940259.

[47]

A. E. Nesterov and J. S. Lipatov, Thermodynamics of Solutions and Mixtures of Polymers, Naukova Dumka, Kiev, 1984 (in Russian).

[48]

T. Nose, Theory of liquid-liquid interface for polymer systems, Polymer Journal, 8 (1976), 96-113. doi: 10.1295/polymj.8.96.

[49]

T. Nose, Kinetics of phase separation in polymer mixtures, Phase Transitions, 8 (1987), 245-260. doi: 10.1080/01411598708209379.

[50]

D. R. Paul and S. Newman (eds), Polymer Blends, 1, Academic Press, New York, 1978.

[51]

I. Pawłow, Three dimensional model of thermomechanical evolution of shape memory materials, Control Cybernet., 29 (2000), 341-365.

[52]

I. Pawłow, Thermodynamically consistent Cahn-Hilliard and Allen-Cahn models in elastic solids, Discrete Contin. Dyn. Syst., 15 (2006), 1169-1191. doi: 10.3934/dcds.2006.15.1169.

[53]

O. Penrose and P. C. Fife, Themodynamically consistent models of phase-field type for the kinetics of phase transition, Physica D, 43 (1990), 44-62. doi: 10.1016/0167-2789(90)90015-H.

[54]

O. Penrose and P. C. Fife, On the relation between the standard phase-field model and a "thermodynamically consistent" phase-field model, Physica D, 69 (1993), 107-113. doi: 10.1016/0167-2789(93)90183-2.

[55]

P. Pincus, Dynamics of fluctuations and spinodal decomposition in polymer blends. II, J. Chem. Phys., 75 (1981), 1996-2000. doi: 10.1063/1.442226.

[56]

G. Schimperna and I. Pawłow, A Cahn-Hilliard equation with singular diffusion, J. Differential Equations, 254 (2013), 779-803. doi: 10.1016/j.jde.2012.09.018.

[57]

M. Šilhavý, The Mechanics and Thermodynamics of Continuous Media, Springer, Berlin, 1997. doi: 10.1007/978-3-662-03389-0.

[58]

I. Singer-Loginova and H. M. Singer, The phase field technique for modeling multiphase materials, Rep. Prog. Phys., 71 (2008), 106501. doi: 10.1088/0034-4885/71/10/106501.

[59]

H. Tanaka, Viscoelastic model of phase separation, Phys. Rev. E, 56 (1997), 4451-4462. doi: 10.1103/PhysRevE.56.4451.

[60]

A. Voit, A. Krekhov and W. Köhler, Laser-induced structures in a polymer blend in the vicinity of the phase boundary, Phys. Rev. E, 76 (2007), 011808. doi: 10.1103/PhysRevE.76.011808.

[61]

A. Vrij and G. J. Roebersen, Inhomogeneous polymer solutions: Theory of the interfacial free energy and the composition profile near the consolute point, J. Polym. Sci.: Polym. Physics, 15 (1977), 109-125. doi: 10.1002/pol.1977.180150110.

[62]

D. Zhou, P. Zhang and Weinan E, Modified models of polymer phase separation, Phys. Rev. E, 73 (2006), 061801. doi: 10.1103/PhysRevE.73.061801.

show all references

References:
[1]

H. W. Alt, The entropy principle for interfaces. Fluids and solids, Adv. Math. Sci. Appl., 19 (2009), 585-663.

[2]

H. W. Alt and I. Pawłow, A mathematical model of dynamics of non-isothermal phase separation, Physica D, 59 (1992), 389-416. doi: 10.1016/0167-2789(92)90078-2.

[3]

H. W. Alt and I. Pawłow, Thermodynamical models of phase transitions with multicomponent order parameter, in Trends in Applications of Mathematics to Mechanics, (eds. M. D. P. M. Marques and J. F. Rodrigues), Pitman Monographs and Surveys in Pure and Applied Mathematics, 77, Longman, New York (1995), 87-98.

[4]

H. W. Alt and I. Pawłow, On the entropy principle of phase transition models with a conserved order parameter, Adv. Math. Sci. Appl., 6 (1996), 291-376.

[5]

S. H. Anastasiadis, I. Gancarz and J. T. Koberstein, Interfacial tension of immiscible polymer blends: Temperature and molecular weight dependence, Macromolecules, 21 (1988), 2980-2987. doi: 10.1021/ma00188a015.

[6]

M. V. Ariyapadi and E. B. Nauman, Free energy of an inhomogeneous polymer-polymer-solvent sytem, J. Polymer Sci., Part B: Polymer Physics, 27 (1989), 2637-2646.

[7]

M. V. Ariyapadi and E. B. Nauman, Gradient energy parameters for polymer-polymer-solvent systems and their application to spinodal decomposition in true ternary systems, J. Polymer Sci., Part B: Polymer Physics, 28 (1990), 2395-2409. doi: 10.1002/polb.1990.090281216.

[8]

M. V. Aryapadi and E. B. Nauman, Free energy of an inhomogeneous polymer-polymer-solvent system. II, J. Polymer Sci., Part B: Polymer Physics, 30 (1992), 533-538. doi: 10.1002/polb.1992.090300603.

[9]

S. Benzoni-Gavage, L. Chupin, D. Jamet and J. Vovelle, On a phase field model for solid-liquid phase transitions, Discrete Contin. Dyn. Syst., 32 (2012), 1997-2025. doi: 10.3934/dcds.2012.32.1997.

[10]

K. Binder, Collective diffusion, nucleation and spinodal decomposition in polymer mixtures, J. Chem. Phys., 79 (1983), 6387-6409. doi: 10.1063/1.445747.

[11]

G. Bonfanti, M. Frémond and F. Luterotti, Existence and uniqueness results to a phase transition model based on microscopic accelerations and movements, Nonlinear Anal., Real World Applications, 5 (2004), 123-140. doi: 10.1016/S1468-1218(03)00021-X.

[12]

M. Brocate and J. Sprekels, Hysteresis and Phase Transitions, Applied Math. Sciences, 121, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-4048-8.

[13]

F. Brochard, J. Jouffroy and P. Levinson, Polymer-polymer diffusion in melts, Macromolecules, 16 (1983), 1638-1641. doi: 10.1021/ma00244a016.

[14]

F.Brochard, J. Jouffroy and P. Levinson, Polymer diffusion in blends: Effects of mutual friction, Macromolecules, 17 (1984), 2925-2927. doi: 10.1021/ma00142a084.

[15]

G. Caginalp, An analysis of a phase field model of a free boundary, Arch. Ration. Mech. Anal., 92 (1986), 205-245. doi: 10.1007/BF00254827.

[16]

J. W. Cahn, On spinodal decomposition, Acta Metall., 9 (1961), 795-801. doi: 10.1016/0001-6160(61)90182-1.

[17]

J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. I. Interfacial free energy, J. Chem. Phys., 28 (1958), 258-267. doi: 10.1063/1.1744102.

[18]

J. W. Cahn and J. E. Hilliard, Spinodal decomposition: A reprise, Acta Metall., 19 (1971), 151-161. doi: 10.1016/0001-6160(71)90127-1.

[19]

L.-Q. Chen, Phase-field models for microstructure evolution, Annu. Rev. Mater. Res., 32 (2002), 113-140.

[20]

P. Debye, Angular dissymmetry of the critical opalescence in liquid mixtures, J. Chem. Phys., 31 (1959), 680-687. doi: 10.1063/1.1730446.

[21]

P. G. de Gennes, Scaling Concepts in Polymer Physics, Cornell Univ. Press, Ithaca, 1979.

[22]

P. G. de Gennes, Dynamics of fluctuations and spinodal decomposition in polymer blends, J. Chem. Phys., 72 (1980), 4756-4763. doi: 10.1063/1.439809.

[23]

S. R. de Groot and P. Mazur, Non-Equilibrium Thermodynamics, Dover Publ., New York, 1984.

[24]

D. G. B. Edelen, On the existence of symmetry relations and dissipation potentials, Arch. Ration. Mech. Anal., 51 (1973), 218-227. doi: 10.1007/BF00276075.

[25]

H. Emmerich, Advances of and by phase-filed modelling in condensed-matter physics, Advances in Physics, 57 (2008), 1-87.

[26]

M. Fabrizio, C. Giorgi and A. Morro, A thermodynamic approach to non-isothermal phase-filed evolution in continuum physics, Physica D, 214 (2006), 144-156. doi: 10.1016/j.physd.2006.01.002.

[27]

F. Falk, Cahn-Hilliard theory and irreversible thermodynamics, J. Non-Equilib. Thermodyn., 17 (1992), 53-65. doi: 10.1515/jnet.1992.17.1.53.

[28]

P. J. Flory, Principles of Polymer Chemistry, Cornell University Press, Ithaca, New York, 1953.

[29]

M. Frémond, Non-Smooth Thermomechanics, Springer, Berlin, 2002. doi: 10.1007/978-3-662-04800-9.

[30]

E. Fried and M. E. Gurtin, Continuum theory of thermally induced phase transitions based on an order parameter, Physica D, 68 (1993), 326-343. doi: 10.1016/0167-2789(93)90128-N.

[31]

P. Galenko and D. Jou, Diffuse-interface model for rapid phase transformations in nonequilibrium systems, Phys. Rev. E, 71 (2005), 046125. doi: 10.1103/PhysRevE.71.046125.

[32]

D. Y. Gao, Duality Principles in Nonconvex Systems, Kluwer Academic Publishers, Dordrecht, 2000. doi: 10.1007/978-1-4757-3176-7.

[33]

J. D. Gunton, M. San Miguel and P. S. Sahni, The dynamics of first-order phase transitions, in Phase Transitions and Critical Phenomena (eds. C. Domb and J. L. Lebowitz), Acadmeic Press, London, 8 (1983), 267-482.

[34]

M. E. Gurtin, Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance, Physica D, 92 (1996), 178-192. doi: 10.1016/0167-2789(95)00173-5.

[35]

B. I. Halperin, P. C. Hohenberg and S.-K. Ma, Renormalization-group methods for critical dynamics: I. Recursion relations and effects of energy conservation, Phys. Rev. B, 10 (1974), 139-153. doi: 10.1103/PhysRevB.10.139.

[36]

V. J. Klenin, Thermodynamics of Systems Containing Flexible-Chain Polymers, Elsevier, Amsterdam, 1999.

[37]

W. Köhler, A. Krekhov and W. Zimmermann, Thermal Diffusion in Polymer Blends: Criticality and Pattern Formation, Preprint Universität Bayreuth, 2013 (available from uni-bayreuth. de).

[38]

I. S. Liu, Method of Lagrange multipliers for exploitation of the entropy principle, Arch. Ration. Mech. Anal., 46 (1972), 131-148.

[39]

L. P. McMaster, Aspects of liquid-liquid phase transition phenomena in multicomponent polymeric systems, in Copolymers, Polyblends, and Composities (ed. N. A. J. Platzer), Adv. Chem. Ser., 142 (1975), 43-65. doi: 10.1021/ba-1975-0142.ch005.

[40]

A. Miranville and G. Schimperna, Nonisothermal phase separation based on a microforce balance, Discrete Contin. Dyn. Syst. Ser. B, 5 (2005), 753-768. doi: 10.3934/dcdsb.2005.5.753.

[41]

V. S. Mitlin and L. I. Manevitch, Kinetically stable structures in the nonlinear theory of spinodal decomposition, J. Polymer Sci., Part B: Polymer Physics, 28 (1990), 1-16. doi: 10.1002/polb.1990.090280101.

[42]

V. S. Mitlin, L. I. Manevitch and I. Ya. Erukhimovich, Formation of kinetically stable domain structure during spinadal decomposition of binary polymer mixtures, Zh. Eksper. Teoret. Fiz., 88 (1985), 495-506; Sov. Phys. JETP, 61 (1985), 290-296.

[43]

A. Morro, A phase-field approach to non-isothermal transitions, Mathematical and Computer Modelling, 48 (2008), 621-633. doi: 10.1016/j.mcm.2007.11.001.

[44]

I. Müller, Thermodynamics, Pitman, London, 1985.

[45]

Y. S. Nam and T. G. Park, Porous biodegradable polymeric scaffolds prepared by thermally induced phase separation, J. Biomedical Materials Research, 47 (1999), 8-17. doi: 10.1002/(SICI)1097-4636(199910)47:1<8::AID-JBM2>3.0.CO;2-L.

[46]

E. B. Nauman, M. V. Ariyapadi, N. P. Balsara, T. A. Grocela, J. S. Furno, S. H. Liu and R. Mallikarjun, Compositional quenching: A process for forming polymer-in-polymer microdispersions and cocontinuous networks, Chem. Eng. Comm., 66 (1988), 29-55. doi: 10.1080/00986448808940259.

[47]

A. E. Nesterov and J. S. Lipatov, Thermodynamics of Solutions and Mixtures of Polymers, Naukova Dumka, Kiev, 1984 (in Russian).

[48]

T. Nose, Theory of liquid-liquid interface for polymer systems, Polymer Journal, 8 (1976), 96-113. doi: 10.1295/polymj.8.96.

[49]

T. Nose, Kinetics of phase separation in polymer mixtures, Phase Transitions, 8 (1987), 245-260. doi: 10.1080/01411598708209379.

[50]

D. R. Paul and S. Newman (eds), Polymer Blends, 1, Academic Press, New York, 1978.

[51]

I. Pawłow, Three dimensional model of thermomechanical evolution of shape memory materials, Control Cybernet., 29 (2000), 341-365.

[52]

I. Pawłow, Thermodynamically consistent Cahn-Hilliard and Allen-Cahn models in elastic solids, Discrete Contin. Dyn. Syst., 15 (2006), 1169-1191. doi: 10.3934/dcds.2006.15.1169.

[53]

O. Penrose and P. C. Fife, Themodynamically consistent models of phase-field type for the kinetics of phase transition, Physica D, 43 (1990), 44-62. doi: 10.1016/0167-2789(90)90015-H.

[54]

O. Penrose and P. C. Fife, On the relation between the standard phase-field model and a "thermodynamically consistent" phase-field model, Physica D, 69 (1993), 107-113. doi: 10.1016/0167-2789(93)90183-2.

[55]

P. Pincus, Dynamics of fluctuations and spinodal decomposition in polymer blends. II, J. Chem. Phys., 75 (1981), 1996-2000. doi: 10.1063/1.442226.

[56]

G. Schimperna and I. Pawłow, A Cahn-Hilliard equation with singular diffusion, J. Differential Equations, 254 (2013), 779-803. doi: 10.1016/j.jde.2012.09.018.

[57]

M. Šilhavý, The Mechanics and Thermodynamics of Continuous Media, Springer, Berlin, 1997. doi: 10.1007/978-3-662-03389-0.

[58]

I. Singer-Loginova and H. M. Singer, The phase field technique for modeling multiphase materials, Rep. Prog. Phys., 71 (2008), 106501. doi: 10.1088/0034-4885/71/10/106501.

[59]

H. Tanaka, Viscoelastic model of phase separation, Phys. Rev. E, 56 (1997), 4451-4462. doi: 10.1103/PhysRevE.56.4451.

[60]

A. Voit, A. Krekhov and W. Köhler, Laser-induced structures in a polymer blend in the vicinity of the phase boundary, Phys. Rev. E, 76 (2007), 011808. doi: 10.1103/PhysRevE.76.011808.

[61]

A. Vrij and G. J. Roebersen, Inhomogeneous polymer solutions: Theory of the interfacial free energy and the composition profile near the consolute point, J. Polym. Sci.: Polym. Physics, 15 (1977), 109-125. doi: 10.1002/pol.1977.180150110.

[62]

D. Zhou, P. Zhang and Weinan E, Modified models of polymer phase separation, Phys. Rev. E, 73 (2006), 061801. doi: 10.1103/PhysRevE.73.061801.

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