Hostname: page-component-76fb5796d-vvkck Total loading time: 0 Render date: 2024-04-29T12:12:42.673Z Has data issue: false hasContentIssue false
  • English
  • Français

An integrable fourth-order nonlinear evolution equation applied to surface redistribution due to capillarity

Published online by Cambridge University Press:  17 February 2009

Peter Tritscher
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Members of an hierarchy of integrable nonlinear evolution equations, related to the well-known linearizable diffusion equation which has the diffusivity form as the reciprocal of the square of the concentration, are adapted to derive a new integrable nonlinear equation which models the surface evolution of an arbitrarily-oriented theoretical anisotropic material by the concomitant action of evaporation-condensation and surface diffusion. The constitutive relations are explicitly formulated and these show that the theoretical anisotropic material behaves like a liquid crystal. The integrable nonlinear equation may be used to advantage as test cases for numerical schemes. Its form has many attributes of the nonlinear governing equation for an isotropic material. Closed-form solutions are constructed for the evolution of a ramped surface by concomitant evaporation-condensation and surface diffusion.

Type
Research Article
Information
The ANZIAM Journal , Volume 38 , Issue 4 , April 1997 , pp. 518 - 541
Copyright
Copyright © Australian Mathematical Society 1997

References

[1]Bailey, G. L. J. and Watkins, H. C., “Surface tensions in the system solid copper-molten lead”, Proc. Phys. Soc. B 63 (1950) 350–358.CrossRefGoogle Scholar
[2]Blakely, J. M., Introduction to the Properties of Crystal Surfaces, Pergamon Press, Oxford (1973).Google Scholar
[3]Bluman, G. W. and Kumei, S., “On the remarkable nonlinear diffusion equation (∂/∂x)[a(u + b)−2(∂u/∂x)] − (∂u/∂t) = 0”, J. Math. Phys. 21 (1980) 1019–1023.CrossRefGoogle Scholar
[4]Bluman, G. W. and Kumei, S., Symmetries and Differential Equations Springer, New York (1989).CrossRefGoogle Scholar
[5]Bonzel, H. P. and Preuss, E., “Periodic surface profiles under the influence of anisotropic surface energy: a steady-state solution”, Surface Sci. 145 (1984) 20–32.CrossRefGoogle Scholar
[6]Bonzel, H. P., Preuss, E. and Steffen, B., “The dynamical behaviour of periodic surface profiles on metals under the influence of anisotropic surface energy”, Appl. Phys. A 35 (1984) 1–8.CrossRefGoogle Scholar
[7]Bouchiba, M., “Sur l'équation de la corrosion intergranulaire par diffusion de surface”, RAIRO Modél. Math. Anal. Numér. 21 (1987) 425–444.CrossRefGoogle Scholar
[8]Brailsford, A. B. and Gjostein, N. A., “Influence of the surface energy anisotropy on morphological changes occurring by surface diffusion”, J. Appl. Phys. 46 (1975) 2390–2397.CrossRefGoogle Scholar
[9]Broadbridge, P. and White, I., “Constant rate rainfall infiltration: a versatile nonlinear model 1. analytic solution”, Water Resour. Res. 24 (1988) 145–154.CrossRefGoogle Scholar
[10]Broadbridge, P., Knight, J. H. and Rogers, C., “Constant rate rainfall infiltration in a bounded profile: solutions of a nonlinear model”, Soil Sci. Soc. Am. J. 52 (1988) 1526–1533.CrossRefGoogle Scholar
[11]Broadbridge, P., “Exact solvability of the mullins nonlinear diffusion model of groove development”, J. Math. Phys. 30 (1989) 1648–1651.CrossRefGoogle Scholar
[12]Broadbridge, P. and Godfrey, S. E., “Exact decaying soliton solutions of nonlinear Schrödinger equations: Lie-Bäcklund symmetries”, J. Math. Phys. 32 (1991) 8–18.CrossRefGoogle Scholar
[13]Broadbridge, P. and Rogers, C., “On a nonlinear reaction-diffusion boundary value problem: application of a Lie-Bäcklund symmetry”, J. Aust. Math. Soc. B 34 (1993) 318–332.CrossRefGoogle Scholar
[14]Broadbridge, P. and Tritscher, P., “An integrable fourth order nonlinear evolution equation applied to thermal grooving of metal surfaces”, I.M.A. J. Appl. Math. 53 (1994) 249–265.Google Scholar
[15]Cahn, J. W. and Taylor, J. E., “Surface motion by surface diffusion”, Acta Metall. Mater. 42 (1994) 1045–1063.CrossRefGoogle Scholar
[16]Christian, J., The theory of transformations in metals and alloys, Pergamon, Oxford (1965).Google Scholar
[17]Clarkson, P. A., Fokas, A. S. and Ablowitz, M. J., “Hodograph transformations of linearizable partial differential equations”, SIAMJ. Appl. Math. 49 (1989) 1188–1209.CrossRefGoogle Scholar
[18]Coleman, B. D., Falk, R. S. and Moakher, M., Rutgers Univ., unpublished research cited in Cahn, J. W. and Taylor, J. E., “Surface motion by surface diffusion”, Acta Metall. Mater. 42 (1994) 1045–1063 (p. 1047).CrossRefGoogle Scholar
[19]Davi, F. and Gurtin, M. E., “On the motion of a phase interface by surface diffusion”, J. Appl. Math. Phys. (ZAMP) 41 (1990) 782–811.CrossRefGoogle Scholar
[20]Erdelyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F. G., Higher transcendental functions, McGraw-Hill, New York (1953).Google Scholar
[21]Exner, H. E. and Artz, E., “Sintering processes”, in Physical Metallurgy (eds. Cahn, R. W. and Haasen, P.), Elsevier Science, New York (1983).Google Scholar
[22]Fokas, A. S., “A symmetry approach to exactly solvable evolution-equations”, J. Math. Phys. 21 (1980) 1318–1325.CrossRefGoogle Scholar
[23]Fujita, H., “The exact pattern of a concentration-dependent diffusion in a semi-infinite medium, part II”, Text. Res. J. 22 (1952) 823–827.CrossRefGoogle Scholar
[24]Galaktionov, V. A., Dorodnitsyn, V. A., Elenin, G. G., Kurdyumov, S. P. and Samarskii, A. A., “A quasi linear heat equation with a source: peaking, localization, symmetry exact solutions, asymptotics, structures”, J. Soviet Math. 41 (1988) 1222–1292.CrossRefGoogle Scholar
[25]Génin, F. E., Mullins, W. W. and Wynblatt, P., “The effect of stress on grain boundary grooving”, Acta Metall. Mater. 41 (1993) 3541–3547.CrossRefGoogle Scholar
[26]Ghez, R., “A generalized Gibbsian surface”, Surface Sci. 4 (1966) 125–140.CrossRefGoogle Scholar
[27]Gjostein, N. A., “Adsorption and surface energy (II): thermal faceting from minimization of surface energy”, Acta. Met. 11 (1963) 969–978.CrossRefGoogle Scholar
[28]Gjostein, N. A., “Surface self diffusion of gold (I): Analysis of the scratch-flattening process”, Trans. Metall. Soc. AIME 236 (1966) 1967–1977.Google Scholar
[29]Gruber, E. E. and Mullins, W. W., “Extended analysis of surface scratch smoothing”, Acta. Met. 14 (1966) 397–403.CrossRefGoogle Scholar
[30]Gurtin, M. E., “On thermomechanical laws for the motion of a phase interface”, J. Appl. Math. Phys. (ZAMP) 42 (1991) 370–388.CrossRefGoogle Scholar
[31]Hackney, S. A. and Ojard, G. C., “Grain boundary grooving at finite grain size”, Scripta Met. 22 (1988) 1731–1735.CrossRefGoogle Scholar
[32]Herring, C., “Effect of change of scale on sintering phenomena”, J. Appl. Phys. 21 (1950) 301–303.CrossRefGoogle Scholar
[33]Herring, C., “Surface tension as a motivation for sintering”, in The Physics of Powder Metallurgy (ed. Kingston, W. E.), McGraw-Hill, New York (1951).Google Scholar
[34]Herring, C., “The use of classical macroscopic concepts in surface-energy problems”, in Structure and Properties of Solid Surfaces (eds. Gomer, R. and Smith, C. S.), Univ. of Chicago Press, Chicago (1952).Google Scholar
[35]Hill, J. M. and Hart, V. G., “The Stefan problem in nonlinear heat conduction”, J. Appl. Math. Phys. (ZAMP) 37 (1986) 206–229.CrossRefGoogle Scholar
[36]Hilliard, J. E., Cohen, M. and Averbach, B. L., “Grain-boundary energies in gold-copper alloys”, Acta Met. 8 (1960) 26–31.CrossRefGoogle Scholar
[37]Jones, H., Spriggs, P. H. and Leak, G. M., “Relaxation of surface scratch by evaporation and condensation”, Acta Met. 13 (1965) L444–446.Google Scholar
[38]Karciga, Y., Steep evaporating grooves, Honours thesis, Dept. of Mathematics, University of Wollongong, Australia, (1995).Google Scholar
[39]Kennard, E. H., Kinetic theory of gases, McGraw-Hill, New York (1938).Google Scholar
[40]King, R. T. and Mullins, W. W., “Theory of the decay of a surface scratch to flatness”, Acta Met. 10 (1962) 601–606.CrossRefGoogle Scholar
[41]Kitada, A., “On a property of a classical solution of the nonlinear mass-transport equation ut = ”, J. Math. Phys. 27 (1986) 1391–1392.CrossRefGoogle Scholar
[42]Kitada, A. and Umehara, H., “On a property of a classical solution of the nonlinear mass-transport equation . II”, J. Math. Phys. 28 (1987) 536–537.CrossRefGoogle Scholar
[43]Kitada, A., “On a property of a classical solution of the nonlinear mass-transport equation ut = . II: comment”, J. Math. Phys. 28 (1987) 2982–2983.CrossRefGoogle Scholar
[44]Knight, J. H., Solution of the nonlinear diffusion equation: existence, uniqueness and estimation, Ph. D. Thesis, Australian National University, Canberra (1973).Google Scholar
[45]Knight, J. H. and Philip, J. R., “Exact solutions in nonlinear diffusion”, J. Eng. Math. 8 (1974) 219–227.CrossRefGoogle Scholar
[46]Kuczynski, G. C., “Self-diffusion in sintering of metallic particles”, Trans. A.I.M.E. 185 (1949) 169–178.Google Scholar
[47]Lee, M. Z. C., A numerical model for dominant surface diffusion applied to periodic sputtering of surfaces, Honours thesis, Dept of Mathematics, University of Wollongong, Australia (1995).Google Scholar
[48]Mikhailov, A. V., Shabat, A. B. and Sokolov, V. V., “The symmetry approach to classification of integrable equations”, in What is integrability? (ed. Zakharov, V. E.), Springer, Berlin (1990).Google Scholar
[49]Moore, A. J. W., “Thermal faceting”, in Metal surfaces: structure, energetics and kinetics (eds. Robertson, W. M. and Gjostein, N. A.), ASM Metals Park, Ohio (1963).Google Scholar
[50]Mullins, W. W., “Theory of thermal grooving”, J. Appl. Phys. 28 (1957) 333–339.CrossRefGoogle Scholar
[51]Mullins, W. W., “The effect of thermal grooving on grain boundary motion”, Acta Met. 6 (1958) 414–27.CrossRefGoogle Scholar
[52]Mullins, W. W., “Flattening of a nearly plane solid surface due to capillarity”, J. Appl. Phys. 30 (1959) 77–83.CrossRefGoogle Scholar
[53]Mullins, W. W. and Shewmon, P. G., “The kinetics of grain boundary grooving in copper”, Acta Met. 7 (1959) 163–170.CrossRefGoogle Scholar
[54]Mullins, W. W., “Theory of linear facet growth during thermal etching”, Philos. Mag. 6 (1961) 1313–1341.CrossRefGoogle Scholar
[55]Mullins, W. W., “Solid surface morphologies governed by capillarity”, in Metal surfaces: structure, energetics and kinetics (eds. Robertson, W. M. and Gjostein, N. A.) ASM Metals Park, Ohio (1963).Google Scholar
[56]Nichols, F. A. and Mullins, W. W., “Morphological changes of a surface of revolution due to capillarity-induced surface diffusion”, J. Appl. Phys. 36 (1965) 1826–1835.CrossRefGoogle Scholar
[57]Nolan, T P. and Sinclair, R., “Modeling of agglomeration in polycrystalline thin films: application to TiSi2 on a silicon substrate”, J. Appl. Phys. 72 (1992) 720–724.CrossRefGoogle Scholar
[58]Olver, P. J., “Evolution equations possessing infinitely many symmetries”, J. Math. Phys. 18 (1977) 1212–1215.CrossRefGoogle Scholar
[59]Preuss, E., Freyer, N. and Bonzel, H. P., “Surface self-diffusion on Pt(110): directional dependence and influence of surface-energy anisotropy”, Appl. Phys. A 41 (1986) 137–143.CrossRefGoogle Scholar
[60]Ratke, L. and Vogel, H. J., “Theory of grain boundary grooving in the convective-diffusive regime”, Acta Metall. Mater. 39 (1991) 915–923.CrossRefGoogle Scholar
[61]Riedel, H., Fracture at high temperatures Springer-Verlag, Berlin (1987).CrossRefGoogle Scholar
[62]Robertson, W. M., “Grain-boundary grooving by surface diffusion for finite slopes”, J. Appl. Phys. 42 (1971) 463–467.CrossRefGoogle Scholar
[63]Rogers, C. and Shadwick, W. M., Bäcklund transformations and their applications Academic Press, New York (1982).Google Scholar
[64]Rogers, C., Stallybrass, M. P. and Clements, D. L., “On two phase filtration under gravity and with boundary infiltration: application of a Backlund transformation”, Nonlin. Anal. Theory Methods Appl. 7 (1983) 785–799.CrossRefGoogle Scholar
[65]Rogers, C., “Application of rciprocal Backlund transformations to a class of nonlinear boundary value problems”, J. Phys. A: Math. and Gen. 16 (1983) L493–495.CrossRefGoogle Scholar
[66]Rogers, C., “Application of a reciprocal transformation to a two-phase Stefan Problem”, J. Phys. A: Math. and Gen. 18 (1985) L105–109.CrossRefGoogle Scholar
[67]Rogers, C., “On a class of moving boundary problems in non-linear heat conduction: application of a Bäcklund transformation”, Int. J. Non-Linear Mech. 21 (1986) 249–256.CrossRefGoogle Scholar
[68]Sander, G. C., Parlange, J. Y., Kühnel, V., Hogarth, W. L., Lockington, D. and O'Kane, J. P. J., “Exact nonlinear solution for constant flux infiltration”, J. Hydrol. (Neth.) 97 (1988) 341–346.CrossRefGoogle Scholar
[69]Srinivasan, S. R. and Trivedi, R., “Theory of grain boundary grooving under the combined action of the surface and volume diffusion mechanisms”, Acta Metall. 21 (1973) 611–620.CrossRefGoogle Scholar
[70]Srolovitz, D. J. and Safran, S. A., “Capillary instabilities in thin films. I. energetics”, J. Appl. Phys. 60 (1986) 247–254.CrossRefGoogle Scholar
[71]Srolovitz, D. J. and Safran, S. A., “Capillary instabilities in thin films. II. kinetics”, J. Appl. Phys. 60 (1986) 255–260.CrossRefGoogle Scholar
[72]Storm, M. L., “Heat conduction in simple metals”, J. Appl. Phys. 22 (1951) 940–951.CrossRefGoogle Scholar
[73]Sushumna, I. and Ruckenstein, E., “The role of physical and chemical interactions in the behavior of supported metal-catalysts: iron on alumina, a case study”, J. Catal. 94 (1985) 239–288.CrossRefGoogle Scholar
[74]Taylor, J. E., Cahn, J. W. and Handwerker, C. A., “I-Geometric models of crystal growth”, Acta Metall. Mater. 40 (1992) 1443–1474.CrossRefGoogle Scholar
[75]Tritscher, P., Integrable nonlinear evolution equations applied to solidification and surface redistribution, Ph. D. Thesis, University of Wollongong, Australia (1996).Google Scholar
[76]Tritscher, P. and Broadbridge, P., “Grain boundary grooving by surface diffusion: an analytic nonlinear model for a symmetric groove”, Proc. Roy. Soc. A (1995) 450, 569–587.Google Scholar
[77]Vogel, H. J. and Ratke, L., “Instability of grain boundary grooves due to equilibrium grain boundary diffusion”, Acta Metall. Mater. 39 (1991) 641–649.CrossRefGoogle Scholar