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 & Continuous Dynamical Systems, 2015, 35 (6) : 2711-2739. doi: 10.3934/dcds.2015.35.2711
References:
[1]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

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

[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.  Google Scholar

[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.  Google Scholar

[13]

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

[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.  Google Scholar

[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.  Google Scholar

[16]

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

[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.  Google Scholar

[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.  Google Scholar

[19]

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

[20]

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

[21]

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

[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.  Google Scholar

[23]

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

[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.  Google Scholar

[25]

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

[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.  Google Scholar

[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.  Google Scholar

[28]

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

[29]

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

[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.  Google Scholar

[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.  Google Scholar

[32]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[36]

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

[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). Google Scholar

[38]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[44]

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

[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.  Google Scholar

[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.  Google Scholar

[47]

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

[48]

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

[49]

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

[50]

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

[51]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[57]

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

[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.  Google Scholar

[59]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

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

[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.  Google Scholar

[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.  Google Scholar

[13]

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

[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.  Google Scholar

[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.  Google Scholar

[16]

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

[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.  Google Scholar

[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.  Google Scholar

[19]

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

[20]

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

[21]

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

[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.  Google Scholar

[23]

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

[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.  Google Scholar

[25]

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

[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.  Google Scholar

[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.  Google Scholar

[28]

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

[29]

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

[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.  Google Scholar

[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.  Google Scholar

[32]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[36]

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

[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). Google Scholar

[38]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[44]

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

[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.  Google Scholar

[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.  Google Scholar

[47]

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

[48]

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

[49]

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

[50]

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

[51]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[57]

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

[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.  Google Scholar

[59]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[1]

Kota Kumazaki, Akio Ito, Masahiro Kubo. Generalized solutions of a non-isothermal phase separation model. Conference Publications, 2009, 2009 (Special) : 476-485. doi: 10.3934/proc.2009.2009.476

[2]

Alain Miranville, Giulio Schimperna. Nonisothermal phase separation based on a microforce balance. Discrete & Continuous Dynamical Systems - B, 2005, 5 (3) : 753-768. doi: 10.3934/dcdsb.2005.5.753

[3]

Mauro Fabrizio, Claudio Giorgi, Angelo Morro. Phase transition and separation in compressible Cahn-Hilliard fluids. Discrete & Continuous Dynamical Systems - B, 2014, 19 (1) : 73-88. doi: 10.3934/dcdsb.2014.19.73

[4]

Alain Miranville, Elisabetta Rocca, Giulio Schimperna, Antonio Segatti. The Penrose-Fife phase-field model with coupled dynamic boundary conditions. Discrete & Continuous Dynamical Systems, 2014, 34 (10) : 4259-4290. doi: 10.3934/dcds.2014.34.4259

[5]

Ken Shirakawa. Solvability for phase field systems of Penrose-Fife type associated with $p$-laplacian diffusions. Conference Publications, 2007, 2007 (Special) : 927-937. doi: 10.3934/proc.2007.2007.927

[6]

Elisabetta Rocca, Giulio Schimperna. Global attractor for a parabolic-hyperbolic Penrose-Fife phase field system. Discrete & Continuous Dynamical Systems, 2006, 15 (4) : 1193-1214. doi: 10.3934/dcds.2006.15.1193

[7]

Laurence Cherfils, Alain Miranville, Sergey Zelik. On a generalized Cahn-Hilliard equation with biological applications. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 2013-2026. doi: 10.3934/dcdsb.2014.19.2013

[8]

Pavel Krejčí, Songmu Zheng. Pointwise asymptotic convergence of solutions for a phase separation model. Discrete & Continuous Dynamical Systems, 2006, 16 (1) : 1-18. doi: 10.3934/dcds.2006.16.1

[9]

Ciprian G. Gal, Alain Miranville. Robust exponential attractors and convergence to equilibria for non-isothermal Cahn-Hilliard equations with dynamic boundary conditions. Discrete & Continuous Dynamical Systems - S, 2009, 2 (1) : 113-147. doi: 10.3934/dcdss.2009.2.113

[10]

Jan Prüss, Vicente Vergara, Rico Zacher. Well-posedness and long-time behaviour for the non-isothermal Cahn-Hilliard equation with memory. Discrete & Continuous Dynamical Systems, 2010, 26 (2) : 625-647. doi: 10.3934/dcds.2010.26.625

[11]

Cecilia Cavaterra, Maurizio Grasselli, Hao Wu. Non-isothermal viscous Cahn-Hilliard equation with inertial term and dynamic boundary conditions. Communications on Pure & Applied Analysis, 2014, 13 (5) : 1855-1890. doi: 10.3934/cpaa.2014.13.1855

[12]

Erica Ipocoana, Andrea Zafferi. Further regularity and uniqueness results for a non-isothermal Cahn-Hilliard equation. Communications on Pure & Applied Analysis, 2021, 20 (2) : 763-782. doi: 10.3934/cpaa.2020289

[13]

Kota Kumazaki. Periodic solutions for non-isothermal phase transition models. Conference Publications, 2011, 2011 (Special) : 891-902. doi: 10.3934/proc.2011.2011.891

[14]

T. Tachim Medjo. A Cahn-Hilliard-Navier-Stokes model with delays. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2663-2685. doi: 10.3934/dcdsb.2016067

[15]

Pierluigi Colli, Gianni Gilardi, Danielle Hilhorst. On a Cahn-Hilliard type phase field system related to tumor growth. Discrete & Continuous Dynamical Systems, 2015, 35 (6) : 2423-2442. doi: 10.3934/dcds.2015.35.2423

[16]

Tian Ma, Shouhong Wang. Cahn-Hilliard equations and phase transition dynamics for binary systems. Discrete & Continuous Dynamical Systems - B, 2009, 11 (3) : 741-784. doi: 10.3934/dcdsb.2009.11.741

[17]

Dimitra Antonopoulou, Georgia Karali, Georgios T. Kossioris. Asymptotics for a generalized Cahn-Hilliard equation with forcing terms. Discrete & Continuous Dynamical Systems, 2011, 30 (4) : 1037-1054. doi: 10.3934/dcds.2011.30.1037

[18]

Dimitra Antonopoulou, Georgia Karali. Existence of solution for a generalized stochastic Cahn-Hilliard equation on convex domains. Discrete & Continuous Dynamical Systems - B, 2011, 16 (1) : 31-55. doi: 10.3934/dcdsb.2011.16.31

[19]

Riccarda Rossi. On two classes of generalized viscous Cahn-Hilliard equations. Communications on Pure & Applied Analysis, 2005, 4 (2) : 405-430. doi: 10.3934/cpaa.2005.4.405

[20]

Kota Kumazaki, Masahiro Kubo. Variational inequalities for a non-isothermal phase field model. Discrete & Continuous Dynamical Systems - S, 2011, 4 (2) : 409-421. doi: 10.3934/dcdss.2011.4.409

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (85)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]