11 May 2005Lorentz Center Leiden1 Fingering, Fronts, and Patterns in Superconductors Alan Dorsey...
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Transcript of 11 May 2005Lorentz Center Leiden1 Fingering, Fronts, and Patterns in Superconductors Alan Dorsey...
11 May 2005 Lorentz Center Leiden 1
Fingering, Fronts, and Patterns in Superconductors
Alan DorseyUniversity of Florida
Collaborators:Ray Goldstein (U Arizona)John DiBartolo (Brooklyn
Poly)Salman Ullah (Microsoft)
Support from the NSF
11 May 2005 Lorentz Center Leiden 2
Welcome to Florida!
Gainesville
11 May 2005 Lorentz Center Leiden 3
UF Lightning Research
Prof. Martin UmanProf. Vladimir Rakov
International Center for Lightning Research and Testing (ICLRT)
11 May 2005 Lorentz Center Leiden 4
11 May 2005 Lorentz Center Leiden 5
Outline
• Interface motion in superconductors
• Interfacial instabilities• Analogies with
dendritic growth• Propagating fronts• Modulated phases and
the intermediate state of type-I superconductors
• Nonequilibrium vortex patterns and thermal instabilities
http://www.fys.uio.no/super/dend/
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Free boundary model for the moving superconductor/normal
interface
BDB Bt2
iBni nBDvB )/(
)1( 0 nci vdHB
• In the superconducting region the magnetic field is zero.
• Normal regions: moving interface generates eddy currents (Ampere’s Law plus Ohm’s Law):
• At the interface we have the boundary condition:
• For a flat interface the field at the interface is the critical field; for a curved interface:
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Interfacial (Mullins-Sekerka) instability
• is largest near the bump
nBvn /
B
• A linear stability analysis shows that the growth rate is positive at long wavelengths. Surface tension stabilizes the growth at short wavelengths.
• A similar instability occurs in the dendritic growth of solids.
• Since the normal velocity is largest near the bump, so bumps grow faster!
11 May 2005 Lorentz Center Leiden 8
Flux expulsion/dendritic growth analogy
TDT Tt2
])/()/([ ''liquidpTsolidpTn nTcDnTcDLv
)1( 0 nmi vdTT
• A piece of solid grows into its supercooled liquid phase. This releases a latent heat L that must diffuse away from the interface for the solid to grow.
• At the interface the rate of heat production is equal to the rate at which heat flows into the solid and liquid.
• The Gibbs-Thomson condition:
11 May 2005 Lorentz Center Leiden 9
Modeling: time dependent Ginzburg Landau theory
222
||)/2(2
])/2([
baem
eit
A
i
)(4 sn JJA
• Coupled nonlinear PDEs for the order parameter and the vector potential:
• Solve numerically using “lattice gauge theory” methods (Frahm, Ullah, Dorsey (1991).
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Propagating front solutions
• DiBartolo and Dorsey (1996): special one dimensional solutions of TDGL equations for an interface.
• Exact solution for special parameter values.
• Matched asymptotics and marginal stability analysis.
• Pulled vs. pushed fronts (Ebert and van Saarloos).
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Competing interactions
•Long range repulsive force: uniform phase•Short range attractive force: compact structures•Competition between forcesinhomogeneous (meso) phase•Ferromagnetic films, ferrofluids, type-I superconductors, block copolymers
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Ferrofluid in a Hele-Shaw cell
•Ferrofluid: colloid of 1 micron spheres. Fluid becomes magnetized in an applied field.
•Hele-Shaw cell: ferrofluid between two glass plates
Surface tension competes with dipole-dipole interaction…
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Results courtesy of Ken Cooper
http://www.its.caltech.edu/~jpelab/Ken_web_page/ferrofluid.html
ferromovie.mov
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Modulated phases
Langmuir monolayer (phospholipid and
cholesterol)
Intermediate state of type-I superconductor
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The intermediate state• For thin films complete
flux explusion is energetically unfavorable.
• The sample breaks up into normal and superconducting regions that coexist.
• The domain size is set by a competition between: – Demagnetizing energy
(favors finely divided structure).
– Surface energy (favors a coarse structure).
• Laminar model developed by Landau in 1937.
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Current loop model
• Supercurrents circulate on the normal/superconductor boundries.
• There is a long range Biot-Savart interaction that causes branching.
• The instability is regulated on short scales by surface tension.
• Overdamped dynamics proposed by Dorsey and Goldstein (1998).
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Experiments
C. R. Reisen and S. G. Lipson, Phys. Rev. B (2000).Pb-In sample, 3mm diameter, 0.14 mm thick
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Nonequilibrium vortex patterns
• Vortex entry in type-II superconductors often results in “dendrites”.
• Subtle interplay of geometry, thermal effects, and nonlinear IV characteristics.
• Recent theoretical work by I. S. Aranson et al., Physical Review Letters (2005).
Experiments: magnetooptics imagesOf Niobium films
Simulations of Aranson et al.
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Summary
• Fingering: dynamical instabilities during magnetic flux entry (free boundary problem, Mullins-Sekerka instability).
• Fronts: novel propagating front (interface) solutions in time-dependent GL theory.
• Patterns:– Competing interactions: attractive short
range and repulsive long range lead to mesoscale patterns.
– Intermediate state patterns in type-I superconductors.
– Nonequilibrium vortex patterns during field entry and exit.