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 - A, 2015, 35 (6) : 2711-2739. doi: 10.3934/dcds.2015.35.2711
References:
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[2]

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

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H. W. Alt and I. Pawłow, Thermodynamical models of phase transitions with multicomponent order parameter,, in Trends in Applications of Mathematics to Mechanics, 77 (1995), 87.

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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.

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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. doi: 10.1021/ma00188a015.

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

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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., 28 (1990), 2395. doi: 10.1002/polb.1990.090281216.

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

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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. doi: 10.3934/dcds.2012.32.1997.

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

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

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

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

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

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P. J. Flory, Principles of Polymer Chemistry,, Cornell University Press, (1953).

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

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

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

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

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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), 8 (1983), 267.

[34]

M. E. Gurtin, Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance,, Physica D, 92 (1996), 178. 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. doi: 10.1103/PhysRevB.10.139.

[36]

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

[37]

W. Köhler, A. Krekhov and W. Zimmermann, Thermal Diffusion in Polymer Blends: Criticality and Pattern Formation,, Preprint Universität Bayreuth, (2013).

[38]

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

[39]

L. P. McMaster, Aspects of liquid-liquid phase transition phenomena in multicomponent polymeric systems,, in Copolymers, 142 (1975), 43. 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. 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., 28 (1990), 1. 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.

[43]

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

[44]

I. Müller, Thermodynamics,, Pitman, (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. 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. doi: 10.1080/00986448808940259.

[47]

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

[48]

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

[49]

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

[50]

D. R. Paul and S. Newman (eds), Polymer Blends,, 1, 1 (1978).

[51]

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

[52]

I. Pawłow, Thermodynamically consistent Cahn-Hilliard and Allen-Cahn models in elastic solids,, Discrete Contin. Dyn. Syst., 15 (2006), 1169. 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. 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. 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. doi: 10.1063/1.442226.

[56]

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

[57]

M. Šilhavý, The Mechanics and Thermodynamics of Continuous Media,, Springer, (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). doi: 10.1088/0034-4885/71/10/106501.

[59]

H. Tanaka, Viscoelastic model of phase separation,, Phys. Rev. E, 56 (1997), 4451. 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). 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. 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). 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.

[2]

H. W. Alt and I. Pawłow, A mathematical model of dynamics of non-isothermal phase separation,, Physica D, 59 (1992), 389. 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, 77 (1995), 87.

[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.

[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. doi: 10.1021/ma00188a015.

[6]

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

[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., 28 (1990), 2395. 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., 30 (1992), 533. 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. 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. 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., 5 (2004), 123. doi: 10.1016/S1468-1218(03)00021-X.

[12]

M. Brocate and J. Sprekels, Hysteresis and Phase Transitions,, Applied Math. Sciences, 121 (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. doi: 10.1021/ma00244a016.

[14]

F.Brochard, J. Jouffroy and P. Levinson, Polymer diffusion in blends: Effects of mutual friction,, Macromolecules, 17 (1984), 2925. 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. doi: 10.1007/BF00254827.

[16]

J. W. Cahn, On spinodal decomposition,, Acta Metall., 9 (1961), 795. 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. doi: 10.1063/1.1744102.

[18]

J. W. Cahn and J. E. Hilliard, Spinodal decomposition: A reprise,, Acta Metall., 19 (1971), 151. 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.

[20]

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

[21]

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

[22]

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

[23]

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

[24]

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

[25]

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

[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. doi: 10.1016/j.physd.2006.01.002.

[27]

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

[28]

P. J. Flory, Principles of Polymer Chemistry,, Cornell University Press, (1953).

[29]

M. Frémond, Non-Smooth Thermomechanics,, Springer, (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. 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). doi: 10.1103/PhysRevE.71.046125.

[32]

D. Y. Gao, Duality Principles in Nonconvex Systems,, Kluwer Academic Publishers, (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), 8 (1983), 267.

[34]

M. E. Gurtin, Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance,, Physica D, 92 (1996), 178. 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. doi: 10.1103/PhysRevB.10.139.

[36]

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

[37]

W. Köhler, A. Krekhov and W. Zimmermann, Thermal Diffusion in Polymer Blends: Criticality and Pattern Formation,, Preprint Universität Bayreuth, (2013).

[38]

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

[39]

L. P. McMaster, Aspects of liquid-liquid phase transition phenomena in multicomponent polymeric systems,, in Copolymers, 142 (1975), 43. 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. 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., 28 (1990), 1. 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.

[43]

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

[44]

I. Müller, Thermodynamics,, Pitman, (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. 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. doi: 10.1080/00986448808940259.

[47]

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

[48]

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

[49]

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

[50]

D. R. Paul and S. Newman (eds), Polymer Blends,, 1, 1 (1978).

[51]

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

[52]

I. Pawłow, Thermodynamically consistent Cahn-Hilliard and Allen-Cahn models in elastic solids,, Discrete Contin. Dyn. Syst., 15 (2006), 1169. 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. 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. 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. doi: 10.1063/1.442226.

[56]

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

[57]

M. Šilhavý, The Mechanics and Thermodynamics of Continuous Media,, Springer, (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). doi: 10.1088/0034-4885/71/10/106501.

[59]

H. Tanaka, Viscoelastic model of phase separation,, Phys. Rev. E, 56 (1997), 4451. 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). 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. 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). doi: 10.1103/PhysRevE.73.061801.

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