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.
J. Bardeen, L. N. Cooper, and J. R. Schreiffer, Phys. Rev. 108, 1175 (1957)
G. Caginalp, Stefan and Hele-Shaw type models as asymptotic limits of the phase field equations, Phys. Rev. A39, 5887 (1989)
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)
S. J. Chapman, Thesis, Oxford University, 1991
S. J. Chapman, Nucleation of superconductivity in decreasing fields, I & II, Europ. J. Appl. Math. 5, 449–494 (1994)
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)
S. J. Chapman, S. D. Howison, and J. R. Ockendon, Macroscopic models of superconductivity, SIAM Review 34, No. 4, 529–560 (1992)
J. C. Crank, Free and Moving Boundary Problems, Oxford, 1984
A. B. Crowley and J. R. Ockendon, Modelling mushy regions, Appl. Sci. Res. 44, 1–7 (1987)
Q. Du, M. D. Gunzburger, and S. Peterson, Analysis and approximation of the Ginzburg-Landau model of superconductivity, SIAM Review 34, 54–81 (1992)
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)
T. E. Faber, The intermediate state in superconducting plates, Proc. Roy. Soc. A248, 461–481 (1958)
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)
U. Essmann and H. Träuble, The direct observation of individual flux lines in type II superconductors, Phys. Lett. A24, 526 (1967)
V. L. Ginzburg and L. D. Landau, On the theory of superconductivity, Zh. Èksper. Teoret. Fiz. 20, 1064 (1950)
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)
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)
J. B. Keller, Propagation of a magnetic field into a superconductor, Phys. Rev. 111, 1497 (1958)
C. G. Kuper, Philos. Mag. 42, 961 (1951)
F. Liu, M. Mondello, and N. Goldenfeld, Kinetics of the superconducting transition, Phys. Rev. Lett. 66, 23, 3071–3074 (1991)
A. C. Rose-Innes and E. H. Rhoderick, Introduction to superconductivity, Pergamon, 1978
H. Träuble and U. Essmann, Ein hochauflösendes Verfahren zur Untersuchung magnetischer Strukturen von Supraleitern, Phys. Stat. Solids 18, 813–828 (1966)
H. Träuble and U. Essmann, Die Beobachtung magnetischer Strukturen von Supraleitern zweiter Art, Phys. Stat. Solids 20, 95–111 (1967)
M. Van Dyke, Perturbation Methods in Fluid Mechanics, The Parabolic Press, 1975
J. Bardeen, L. N. Cooper, and J. R. Schreiffer, Phys. Rev. 108, 1175 (1957)
G. Caginalp, Stefan and Hele-Shaw type models as asymptotic limits of the phase field equations, Phys. Rev. A39, 5887 (1989)
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)
S. J. Chapman, Thesis, Oxford University, 1991
S. J. Chapman, Nucleation of superconductivity in decreasing fields, I & II, Europ. J. Appl. Math. 5, 449–494 (1994)
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)
S. J. Chapman, S. D. Howison, and J. R. Ockendon, Macroscopic models of superconductivity, SIAM Review 34, No. 4, 529–560 (1992)
J. C. Crank, Free and Moving Boundary Problems, Oxford, 1984
A. B. Crowley and J. R. Ockendon, Modelling mushy regions, Appl. Sci. Res. 44, 1–7 (1987)
Q. Du, M. D. Gunzburger, and S. Peterson, Analysis and approximation of the Ginzburg-Landau model of superconductivity, SIAM Review 34, 54–81 (1992)
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)
T. E. Faber, The intermediate state in superconducting plates, Proc. Roy. Soc. A248, 461–481 (1958)
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)
U. Essmann and H. Träuble, The direct observation of individual flux lines in type II superconductors, Phys. Lett. A24, 526 (1967)
V. L. Ginzburg and L. D. Landau, On the theory of superconductivity, Zh. Èksper. Teoret. Fiz. 20, 1064 (1950)
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)
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)
J. B. Keller, Propagation of a magnetic field into a superconductor, Phys. Rev. 111, 1497 (1958)
C. G. Kuper, Philos. Mag. 42, 961 (1951)
F. Liu, M. Mondello, and N. Goldenfeld, Kinetics of the superconducting transition, Phys. Rev. Lett. 66, 23, 3071–3074 (1991)
A. C. Rose-Innes and E. H. Rhoderick, Introduction to superconductivity, Pergamon, 1978
H. Träuble and U. Essmann, Ein hochauflösendes Verfahren zur Untersuchung magnetischer Strukturen von Supraleitern, Phys. Stat. Solids 18, 813–828 (1966)
H. Träuble and U. Essmann, Die Beobachtung magnetischer Strukturen von Supraleitern zweiter Art, Phys. Stat. Solids 20, 95–111 (1967)
M. Van Dyke, Perturbation Methods in Fluid Mechanics, The Parabolic Press, 1975
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© Copyright 1995
American Mathematical Society
