Weakly Nonlocal Non-Equilibrium Thermodynamics: the Cahn-Hilliard Equation

  • Péter Ván
Part of the Advanced Structured Materials book series (STRUCTMAT, volume 89)


The Cahn-Hilliard and Ginzburg-Landau (Allen-Cahn) equations are derived from the second law. The intuitive approach of separation of full divergences is supported by a more rigorous method, based on Liu procedure and a constitutive entropy flux. Thermodynamic considerations eliminate the necessity of variational techniques and explain the role of functional derivatives.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



The work was supported by the grants National Research, Development and Innovation Office - NKFIH 116197 (116375), NKFIH 124366 (124366) and 123815.


  1. Allen SM, Cahn JW (1979) A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metallurgica 27(6):1085–1095Google Scholar
  2. Alt HW, Pawlow I (1992) A mathematical model of dynamics of non-isothermal phase separation. Physica D: Nonlinear Phenomena 59(4):389–416Google Scholar
  3. Anders D, Weinberg K (2011) A variational approach to the decomposition of unstable viscous fluids and its consistent numerical approximation. ZAMM-Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik 91(8):609–629Google Scholar
  4. Anderson DM, McFadden GB, Wheeler AA (1998) Diffuse-interface methods in fluid mechanics. Annual Rev in Fluid Mechanics 30:139–65Google Scholar
  5. Antanovskii LK (1995) A phase field model of capillarity. Physics of Fluids 7(4):747–753Google Scholar
  6. Antanovskii LK (1996) Microscale theory of surface tension. Physical Review E 54(6):6285Google Scholar
  7. Bedeaux D, Johannessen E, Rojorde A (2003) A nonequilibrium Van der Waals square gradient model. (I). The model and its numerical solution. Physica A 330:329–353Google Scholar
  8. Berezovski A, Ván P (2017) Internal Variables in Thermoelasticity. SpringerGoogle Scholar
  9. Berezovski A, Engelbrecht J, Maugin GA (2011) Generalized thermomechanics with dual internal variables. Archive of Applied Mechanics 81(2):229–240Google Scholar
  10. Cahn JW (1961) On spinodal decomposition. Acta Metallica 9:795–801Google Scholar
  11. Cahn JW, Hilliard JE (1958) Free energy of a nonuniform system I. Interfacial free energy. Journal of Chemical Physics 28:258–267Google Scholar
  12. Capriz G (1985) Continua with latent microstructure. Archive for Rational Mechanics and Analysis 90(1):43–56Google Scholar
  13. Capriz G (1989) Continua with Microstructure. Springer, New YorkGoogle Scholar
  14. Cimmelli V, Oliveri F, Pace A (2016) Phase-field evolution in Cahn–Hilliard–Korteweg fluids. Acta Mechanica 227(8):2111–2124Google Scholar
  15. Cimmelli VA (2007) An extension of Liu procedure in weakly nonlocal thermodynamics. Journal of Mathematical Physics 48:113,510Google Scholar
  16. Coleman BD, Gurtin ME (1967) Thermodynamics with internal state variables. The Journal of Chemical Physics 47(2):597–613Google Scholar
  17. Coleman BD, Mizel VJ (1967) Existence of entropy as a consequence of asymptotic stability. Archive for Rational Mechanics and Analysis 25:243–270Google Scholar
  18. Coleman BD, Noll W (1963) The thermodynamics of elastic materials with heat conduction and viscosity. Archive for Rational Mechanics and Analysis 13:167–178Google Scholar
  19. Dunn JE, Serrin J (1985) On the thermomechanics of interstitial working. Archive of Rational Mechanics and Analysis 88:95–133Google Scholar
  20. Fabrizio M, Giorgi C, Morro A (2006) A thermodynamic approach to non-isothermal phase-field evolution in continuum physics. Physica D: Nonlinear Phenomena 214(2):144–156Google Scholar
  21. Frémond M (2001) Non-Smooth Thermomechanics. SpringerGoogle Scholar
  22. Germain P (1973) The method of virtual power in continuum mechanics. Part 2: Microstructure. SIAM Journal of Applied Mathematics 25:556–575Google Scholar
  23. Giorgi C (2009) Continuum thermodynamics and phase-field models. Milan Journal of Mathematics 77(1):67–100Google Scholar
  24. Glavatskiy KS (2015) Lagrangian formulation of irreversible thermodynamics and the second law of thermodynamics. The Journal of Chemical Physics 142(20):204,106Google Scholar
  25. Glavatskiy KS, Bedeaux D (2008) Nonequilibrium properties of a two-dimensionally isotropic interface in a two-phase mixture as described by the square gradient model. Physical Review E 77:061,101Google Scholar
  26. Grmela M (2008) Extensions of classical hydrodynamics. Journal of Statistical Physics 132(3):581–602Google Scholar
  27. Grmela M, Öttinger HC (1997) Dynamics and thermodynamics of complex fluids. I. Development of a general formalism. Physical Review E 56(6):6620–6632Google Scholar
  28. de Groot S (1959) Thermodynamics of Irreversible Processes. North HollandGoogle Scholar
  29. de Groot SR, Mazur P (1962) Non-Equilibrium Thermodynamics. North-Holland Publishing Company, AmsterdamGoogle Scholar
  30. Gurtin ME (1965) Thermodynamics and the possibility of spatial interaction in elastic materials. Archive for Rational Mechanics and Analysis 19:339–352Google Scholar
  31. Gurtin ME (2000) Configurational Forces as Basic Concepts of Continuum Physics. Springer, New YorkGoogle Scholar
  32. Gurtin MG (1996) Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance. Physica D 92:178–192Google Scholar
  33. Gyarmati I (1970) Non-equilibrium Thermodynamics - Field Theory and Variational Principles. Springer, BerlinGoogle Scholar
  34. Heida M, Málek J, Rajagopal K (2012) On the development and generalizations of Cahn–Hilliard equations within a thermodynamic framework. Zeitschrift für Angewandte Mathematik und Physik 63(1):145–169Google Scholar
  35. Hohenberg PC, Halperin BI (1977) Theory of dynamic critical phenomena. Reviews of Modern Physics 49(3):435–479Google Scholar
  36. Hohenberg PC, Krekhov A (2015) An introduction to the Ginzburg-Landau theory of phase transitions and nonequilibrium patterns. Physics Reports 572:1–42Google Scholar
  37. Johannessen E, Bedeaux D (2003) A nonequilibrium van der Waals square gradient model. (II). Local equilibrium of the Gibbs surface. Physica A 330:354–372Google Scholar
  38. Johannessen E, Bedeaux D (2004) A nonequilibrium van der Waals square gradient model. (III). Heat and mass transfer coefficients. Physica A 336:252–270Google Scholar
  39. Landau LD, Ginzburg VL (1950) K teorii sverkhprovodimosti. Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki 20:1064, English translation: On the theory of superconductivity, in: Collected papers of L. D. Landau, ed. D. ter Haar, (Pergamon, Oxford, 1965), pp. 546–568Google Scholar
  40. Landau LD, Khalatnikov IM (1954) Ob anomal’nom pogloshchenii zvuka vblizi tochek fazovogo perekhoda vtorogo roda. Dokladu Akademii Nauk, SSSR 96:469–472, English translation: On the anomalous absorption of sound near a second order transition point. in: Collected papers of L. D. Landau, ed. D. ter Haar,(Pergamon, Oxford, 1965), pp. 626–633Google Scholar
  41. Liu IS (1972) Method of Lagrange multipliers for exploitation of the entropy principle. Archive of Rational Mechanics and Analysis 46:131–148Google Scholar
  42. Mariano PM (2002) Multifield theories in mechanics of solids. Advances in Applied Mechanics 38:1–94Google Scholar
  43. Matolcsi T, Ván P, Verhás J (2005) Fundamental problems of variational principles: objectivity, symmetries and construction. In: Sieniutycz S, H F (eds) Variational and Extremum Principles in Macroscopic Problems, Elsevier, Amsterdam-etc., pp 57–74Google Scholar
  44. Maugin G (1999) The Thermomechanics of Nonlinear Irreversible Behaviors (An Introduction). World Scientific, Singapore-New Jersey-London-Hong KongGoogle Scholar
  45. Maugin G (2013) The principle of virtual power: from eliminating metaphysical forces to providing an efficient modelling tool. Continuum Mechanics and Thermodynamics 25:127–146Google Scholar
  46. Maugin GA (1980) The principle of virtual power in continuum mechanics. Application to coupled fields. Acta Mechanica 35:1–70Google Scholar
  47. Maugin GA (2006) On the thermomechanics of continuous media with diffusion and/or weak nonlocality. Archive of Applied Mechanics 75:723–738Google Scholar
  48. Maugin GA, Drouot R (1983) Internal variables and the thermodynamics of macromolecule solutions. International Journal of Engineering Science 21(7):705–724Google Scholar
  49. Muschik W, Ehrentraut H (1996) An amendment to the Second Law. Journal of Non-Equilibrium Thermodynamics 21:175–192Google Scholar
  50. Öttinger HC (2005) Beyond Equilibrium Thermodynamics. Wiley-InterscienceGoogle Scholar
  51. Öttinger HC, Grmela M (1997) Dynamics and thermodynamics of complex fluids. II. Illustrations of a general formalism. Physical Review E 56(6):6633–6655Google Scholar
  52. Pawłow I (2006) Thermodynamically consistent Cahn-Hilliard and Allen-Cahn models in elastic solids. Discrete and Continuous Dynamical Systems 15(4):1169–1191Google Scholar
  53. Penrose O, Fife PC (1990) Thermodynamically consistent models of phase-field type for the kinetics of phase transitions. Physica D 43:44–62Google Scholar
  54. Penrose R (2004) The Road to Reality. Jonathan CapeGoogle Scholar
  55. Prigogine I, Stengers I (1986) La nouvelle alliance. Gallimard, ParisGoogle Scholar
  56. Sieniutycz S, Farkas H (2005) Progress in variational formulations for macroscopic processes. In: Sieniutycz S, Farkas H (eds) Variational and Extremum Principles in Macroscopic Problems, Elsevier, Amsterdam, pp 3–24Google Scholar
  57. Triani V, Papenfuss C, Cimmelli VA, Muschik W (2008) Exploitation of the Second Law: Coleman- Noll and Liu procedure in comparison. Journal of Non-Equilibrium Thermodynamics 33:47–60Google Scholar
  58. Ván P (2002) Weakly nonlocal irreversible thermodynamics - the Ginzburg-Landau equation. Technische Mechanik 22(2):104–110Google Scholar
  59. Ván P (2008) Internal energy in dissipative relativistic fluids. Journal of Mechanics of Materials and Structures 3(6):1161–1169Google Scholar
  60. Ván P (2009)Weakly nonlocal non-equilibrium thermodynamics - variational principles and Second Law. In: Quak E, Soomere T (eds) AppliedWave Mathematics (Selected Topics in Solids, Fluids, and Mathematical Methods), Springer-Verlag, Berlin-Heidelberg, chap III, pp 153–186Google Scholar
  61. Ván P (2013) Thermodynamics of continua: The challenge of universality. In: Pilotelli M, Beretta GP (eds) Proceedings of the 12th Joint European Thermodynamics Conference, Cartolibreria SNOOPY, Brescia, pp 228–233Google Scholar
  62. Ván P (2017) Galilean relativistic fluid mechanics. Continuum Mechanics and Thermodynamics 29(2):585–610Google Scholar
  63. Ván P, Fülöp T (2006) Weakly nonlocal fluid mechanics - the Schrödinger equation. Proceedings of the Royal Society, London A 462(2066):541–557Google Scholar
  64. Ván P, Muschik W (1995) Structure of variational principles in nonequilibrium thermodynamics. Physical Review E 52(4):3584–3590Google Scholar
  65. Ván P, Nyíri B (1999) Hamilton formalism and variational principle construction. Annalen der Physik (Leipzig) 8:331–354Google Scholar
  66. Ván P, Papenfuss C (2010) Thermodynamic consistency of third grade finite strain elasticity. Proceedings of the Estonian Academy of Sciences 59(2):126–132Google Scholar
  67. Ván P, Berezovski A, Engelbrecht J (2008) Internal variables and dynamic degrees of freedom. Journal of Non-Equilibrium Thermodynamics 33(3):235–254Google Scholar
  68. Ván P, Papenfuss C, Berezovski A (2014) Thermodynamic approach to generalized continua. Continuum Mechanics and Thermodynamics 25(3):403–420, erratum: 421–422Google Scholar
  69. Verhás J (2014) Gyarmati’s variational principle of dissipative processes. Entropy 16:2362–2383Google Scholar
  70. Yourgrau W, Mandelstam S (1999) Variational Principles in Dynamics and Quantum Theory, 2nd edn. Pitman, New York-Toronto-LondonGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Theoretical PhysicsWigner Research Centre for PhysicsBudapestHungary
  2. 2.Department of Energy Engineering, Faculty of Mechanical EngineeringBudapest University of Technology and EconomicsBudapestHungary
  3. 3.Montavid Thermodynamic Research GroupBudapestHungary

Personalised recommendations