Multiphase Thermomechanics with Interfacial Structure 1. Heat Conduction and the Capillary Balance Law

  • Morton E. Gurtin


In [1986g, 1988g] I began the development of a nonequilibrium thermomechanics of two-phase continua, a development based on dynamical statements of the thermomechanical laws in conjunction with GIBBS’S notion of a sharp phase-interface endowed with energy and entropy. I have since come to realize that there is an additional balance law appropriate to the interface. This law, which represents balance of capillary forces, has the form1
$$\int\limits_{{\partial c}} {{\text{C}}\upsilon {\text{ + }}\int\limits_{{\text{c}}} \pi } = 0,$$
with an arbitrary subsurface of and ν the outward unit normal to the boundary curve ∂ of. Here C (x,t), the capillary stress, is a linear transformation of tangent vectors into (not necessarily tangent) vectors, while π(x,t), the interaction, is a vector field; C (x,t) represents microforces exerted across ∂c in response to the creation of new surface;π(X,t), characterizes the interaction between the interface and the bulk material. I view (1.1) as a balance law which is supplementary to the usual laws for forces and moments.


Vector Field Capillary Force Interfacial Structure Tensor Field Outward Unit 
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  1. HERRING, C., Surface tension as a motivation for sintering, The Physics of Powder Metallurgy (ed. W. E. KINGSTON) McGraw-Hill, New York. 1951.Google Scholar
  2. FRANK, F. C., On the kinematic theory of crystal growth and dissolution processes, Growth and Perfection of Crystals (eds. R. H. DOREMUS, B. W. ROBERTS, D. TURNBULL) John Wiley, New York. 1958.Google Scholar
  3. SCRIVEN, L. E., Dynamics of a fluid interface, Chem. Eng. Sci. 12, 98–108. 1960.CrossRefGoogle Scholar
  4. ERICKSEN J.L., Conservation laws for liquid crystals, Trans. Soc. Rheol. 5, 23–34. 1961.MathSciNetADSCrossRefGoogle Scholar
  5. CHERNOV, A. A., Crystal growth forms and their kinetic stability [in Russian], Kristallografiya 8, 87–93. English Transí. Sov. Phys. Crystall. 8, 63–67 (1963). 1963.Google Scholar
  6. CHERNOV, A. A., Application of the method of characteristics to the theory of the growth forms of crystals [in Russian], Kristallografiya 8, 499–505. English Transí. Spv. Phys. Crystall. 8, 401–405 (1964). 1963.Google Scholar
  7. COLEMAN, B.D.,& V.J. MIZEL, Thermodynamics and departures from Fourier’s law of heat conduction, Arch. Rational Mech. Anal. 13, 245–261. 1963.MathSciNetzbMATHGoogle Scholar
  8. COLEMAN, B. D.,& W. NOLL, The thermodynamics of elastic materials with heat conduction and viscosity, Arch. Rational Mech. Anal. 13, 167–178. 1963.MathSciNetzbMATHGoogle Scholar
  9. MULLINS, W. W.,& R. F. SEKERKA, Morphological stability of a particle growing by diffusion or héat flow, J. Appl. Phys. 34, 323–329. 1963.ADSCrossRefGoogle Scholar
  10. MULLINS, W. W.,& R. F. SEKERKA, Stability of a planar interface during solidification of a dilute binary alloy, J. Appl. Phys. 35, 444–451. 1964.ADSCrossRefGoogle Scholar
  11. VORONKOV, V. V., Conditions for formation of mosaic structure on a crystallization front [in Russian], Fizika Tverdogo Tela 6, 2984–2988. English Transí. Sov. Phys. Solid State 6, 2378–2381 (1965). 1964.Google Scholar
  12. TRUESDELL C,& W. NOLL, The non-linear field theories of mechanics, Hand-buck der Physik, vol. III/3 (ed. S. FLÜGGE), Springer-Verlag, Berlin. 1965.Google Scholar
  13. SEIDENSTICKER, R. G., Stability considerations in temperature gradient zone melting, Crystal Growth (ed. H. S. PEISER), Pergamon, Oxford (1967). 1966.Google Scholar
  14. TARSHIS, L. A., W. A. TILLER, The effect of interface-attachment kinetics on the morphological stability of a planar interface during solidification, Crystal Growth (ed. H. S. PEISER), Pergamon, Oxford (1967). 1966.Google Scholar
  15. SEKERKA, R. F., Morphological stability, J. Crystal Growth 3, 4, 71–81. 1968.ADSCrossRefGoogle Scholar
  16. CHERNOV, A. A., Theory of the stability of face forms of crystals [in Russian], Kristallografiya 16, 842–863. English Transl. Sov. Phys. Crystall. 16, 734–753 (1972). 1971.Google Scholar
  17. CHERNOV, A. A., Theory of the stability of face forms of crystals, Sov. Phys. Crystallog. 16, 734–753. 1972.Google Scholar
  18. HOFFMAN, D. W.,& J. W. CAHN, A vector thermodynamics for anisotropic surfaces —1. Fundamentals and applications to plane surface junctions, Surface Sci. 31, 368–388. 1972.ADSCrossRefGoogle Scholar
  19. SEKERKA, R. F., Morphological stability, Crystal Growth: an Introduction, North-Holland, Amsterdam. 1973.Google Scholar
  20. CAHN, J. W.,& D. W. HOFFMAN, A vector thermodynamics for anisotropic surfaces—2. curved and faceted surfaces, Act. Metall. 22, 1205–1214. 1974.Google Scholar
  21. CHERNOV, A. A., Stability of faceted shapes, J. Crystal Growth 24/25, 11–31. 1974.ADSCrossRefGoogle Scholar
  22. DELVES, R. T., Theory of interface instability, Crystal Growth (ed. B. R. PAMPLIN), Pergamon, Oxford. 1974.Google Scholar
  23. GURTIN, M. E.,& A. I. MURDOCH, A continuum theory of elastic material surfaces, Arch. Rational Mech. Anal. 57, 291–323. 1975.MathSciNetzbMATHGoogle Scholar
  24. MOECKEL, G. P., Thermodynamics of an interface, Arch. Rational Mech. Anal. 57, 255–280. 1975.MathSciNetzbMATHGoogle Scholar
  25. MURDOCH, A. I., A thermodynamic theory of elastic material interfaces, Q. J. Mech. AppL Math. 29, 245–275. 1976.MathSciNetzbMATHCrossRefGoogle Scholar
  26. BRAKKE, K. A., The Motion of a Surface by its Mean Curvature, Princeton University Press. 1978.Google Scholar
  27. MURDOCH, A. I., Direct notation for surfaces with application to the thermodynamics of elastic material surfaces of second grade, Res. Rept. ES 78–134, Dept. Eng. Sci., U. Cincinnati. 1978.Google Scholar
  28. ALLEN, S. M.,&. W. CAHN, A macroscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Act. Metall. 27, 1085–1095. 1979.Google Scholar
  29. FERNANDEZ-DIAZ &., W.O.WILLIAMS, A generalized Stefan condition, Zeit. Angew. Math. Phys. 30, 749–755.1979.zbMATHCrossRefGoogle Scholar
  30. LANGER, J. S., Instabilities and pattern formation in crystal growth, Rev. Mod. Phys. 52, 1–27. 1980.ADSCrossRefGoogle Scholar
  31. SEKERKA, R. F., Morphological instabilities during phase transformations, Phase Transformations and Material Instabilities in Solids (ed. M. E. GURTIN), Academic Press, New York. 1984.Google Scholar
  32. ALEXANDER, J. I. D., & ; W. C. JOHNSON, Thermomechanical equilibrium in solid-fluid systems with curved interfaces, J. Appl. Phys. 58, 816–824. 1984.ADSCrossRefGoogle Scholar
  33. GAGE, M., &R. S. HAMILTON, The heat equation shrinking convex plane curves, J. Diff. Geom. 23, 69–96. 1986.MathSciNetzbMATHGoogle Scholar
  34. GURTIN, M. E., On the two-phase Stefan problem with interfacial energy and entropy, Arch. Rational Mech. Anal. 96, 199–241. 1986.MathSciNetzbMATHGoogle Scholar
  35. JOHNSON, W. C., & J. I. D. ALEXANDER, Interfacial conditions for thermomechanical equilibrium in two-phase crystals, J. Appl. Phys. 58, 816–824. 1986.Google Scholar
  36. GRAYSON, M. A., The heat equation shrinks embedded plane curves to round points, J. Diff. Geom. 26, 285–314. 1987.MathSciNetzbMATHGoogle Scholar
  37. LEO, P. H., The effect of elastic fields on the morphological stability of a precipitate grown from solid solution, Ph. D. Thesis, Dept. Metall. Eng. Mat. Sci., Carnegie-Mellon U., Pittsburgh, PA. 1987.Google Scholar
  38. GURTIN, M. E., Toward a nonequilibrium thermodynamics of two phase materials, Arch. Rational Mech. Anal., 100, 275–312. 1988.MathSciNetzbMATHGoogle Scholar
  39. ANGENENT, S., & M. E. GURTIN, Multiphase thermomechanics with interfacial structure. 2. Evolution of an isothermal interface. Forthcoming. 1988.Google Scholar
  40. GURTIN, M. E.,& A. STRUTHERS. Forthcoming. 1988.Google Scholar
  41. FONSECA I., Interfacial energy and the Maxwell rule, Arch. Rational Mech. Anal. Forthcoming. 1988Google Scholar

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© Springer-Verlag Berlin Heidelberg 1999

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  • Morton E. Gurtin

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