Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Asymptotic analysis of the Ginzburg-Landau model of superconductivity: reduction to a free boundary model

Author: S. J. Chapman
Journal: Quart. Appl. Math. 53 (1995), 601-627
MSC: Primary 82D55; Secondary 35C20, 35Q55, 80A22
DOI: https://doi.org/10.1090/qam/1359498
MathSciNet review: MR1359498
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Abstract: A detailed formal asymptotic analysis of the Ginzburg-Landau model of superconductivity is performed and it is found that the leading-order solution satisfies a vectorial version of the Stefan problem for the melting or solidification of a pure material. The first-order correction to this solution is found to contain terms analogous to those of surface tension and kinetic undercooling in the scalar Stefan model. However, the ``surface energy'' of a superconducting material is found to take both positive and negative values, defining type I and type II superconductors respectively, leading to the conclusion that the free boundary model is only appropriate for type I superconductors.

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  • [1] J. Bardeen, L. N. Cooper, and J. R. Schreiffer, Phys. Rev. 108, 1175 (1957)
  • [2] G. Caginalp, Stefan and Hele-Shaw type models as asymptotic limits of the phase field equations, Phys. Rev. A39, 5887 (1989)
  • [3] G. Caginalp, The dynamics of a conserved phase field system: Stefan-like, Hele-Shaw and Cahn-Hilliard models as asymptotic limits, IMA J. Appl. Math. 44, 77-94 (1990)
  • [4] S. J. Chapman, Thesis, Oxford University, 1991
  • [5] S. J. Chapman, Nucleation of superconductivity in decreasing fields, I & II, Europ. J. Appl. Math. 5, 449-494 (1994)
  • [6] S. J. Chapman, S. D. Howison, J. B. McLeod, and J. R. Ockendon, Normal-superconducting transitions in Ginzburg-Landau theory, Proc. Roy. Soc. Edin. 119A, 117-124 (1991)
  • [7] S. J. Chapman, S. D. Howison, and J. R. Ockendon, Macroscopic models of superconductivity, SIAM Review 34, No. 4, 529-560 (1992)
  • [8] J. C. Crank, Free and Moving Boundary Problems, Oxford, 1984
  • [9] A. B. Crowley and J. R. Ockendon, Modelling mushy regions, Appl. Sci. Res. 44, 1-7 (1987)
  • [10] Q. Du, M. D. Gunzburger, and S. Peterson, Analysis and approximation of the Ginzburg-Landau model of superconductivity, SIAM Review 34, 54-81 (1992)
  • [11] A. J. W. Duijvestijn, On the transition from superconducting to normal phase, accounting for latent heat and eddy currents, IBM J. Research & Development 3, 2, 132-139 (1959)
  • [12] T. E. Faber, The intermediate state in superconducting plates, Proc. Roy. Soc. A248, 461-481 (1958)
  • [13] H. Frahm, S. Ullah, and A. T. Dorsey, Flux dynamics and the growth of the superconducting phase, Phys. Rev. Lett. 66, 23, 3067-3070 (1991)
  • [14] U. Essmann and H. Träuble, The direct observation of individual flux lines in type II superconductors, Phys. Lett. A24, 526 (1967)
  • [15] V. L. Ginzburg and L. D. Landau, On the theory of superconductivity, Zh. Èksper. Teoret. Fiz. 20, 1064 (1950)
  • [16] L. P. Gor'kov and G. M. Éliashberg, Generalisation of the Ginzburg-Landau equations for nonstationary problems in the case of alloys with paramagnetic impurities, Soviet Phys. J.E.T.P. 27, 328 (1968)
  • [17] S. D. Howison, A. A. Lacey, and J. R. Ockendon, Singularity development in moving boundary problems, Quart. J. Mech. Appl. Math. 38, 343-360 (1985)
  • [18] J. B. Keller, Propagation of a magnetic field into a superconductor, Phys. Rev. 111, 1497 (1958)
  • [19] C. G. Kuper, Philos. Mag. 42, 961 (1951)
  • [20] F. Liu, M. Mondello, and N. Goldenfeld, Kinetics of the superconducting transition, Phys. Rev. Lett. 66, 23, 3071-3074 (1991)
  • [21] A. C. Rose-Innes and E. H. Rhoderick, Introduction to superconductivity, Pergamon, 1978
  • [22] H. Träuble and U. Essmann, Ein hochauflösendes Verfahren zur Untersuchung magnetischer Strukturen von Supraleitern, Phys. Stat. Solids 18, 813-828 (1966)
  • [23] H. Träuble and U. Essmann, Die Beobachtung magnetischer Strukturen von Supraleitern zweiter Art, Phys. Stat. Solids 20, 95-111 (1967)
  • [24] M. Van Dyke, Perturbation Methods in Fluid Mechanics, The Parabolic Press, 1975

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DOI: https://doi.org/10.1090/qam/1359498
Article copyright: © Copyright 1995 American Mathematical Society

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