1. Fingering, Fronts, and Patterns in Superconductors
Alan Dorsey
University of Florida
Collaborators:
Ray Goldstein (U Arizona)
John DiBartolo (Brooklyn Poly)
Salman Ullah (Microsoft)
Wexler and Dorsey, Phys Rev B 64, (2001)
LR and ATD, PRL 88, (2002)
Support from the NSF
11 May 2005
Lorentz Center Leiden

2. Welcome to Florida!
Gainesville
11 May 2005
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3. UF Lightning Research International Center for Lightning Research
and Testing (ICLRT)
Prof. Martin Uman
Prof. Vladimir Rakov
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4. 11 May 2005
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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
11 May 2005
Lorentz Center Leiden

6. Free boundary model for the moving superconductor/normal interface
Normal regions: moving interface generates eddy currents (Ampere’s Law plus Ohm’s Law):
In the superconducting region
the magnetic field is zero.
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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7. Interfacial (Mullins-Sekerka) instability
is largest near the bump
Since the normal
velocity is largest near the
bump, so bumps grow faster!
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.
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8. Flux expulsion/dendritic growth analogy
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
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9. Modeling: time dependent Ginzburg Landau theory
Coupled nonlinear PDEs for the
order parameter and the vector
potential:
Solve numerically using “lattice
gauge theory” methods (Frahm,
Ullah, Dorsey (1991).
11 May 2005
Lorentz Center Leiden

10. 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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11. Competing interactions
Long range repulsive force: uniform phase
Short range attractive force: compact structures
Competition between forcesinhomogeneous (meso) phase
Ferromagnetic films, ferrofluids, type-I superconductors, block copolymers
Type of pattern depends upon the area fraction. When the area fraction is close to ½, one observes stripes; when it is close to zero or one, one observes bubbles (which tend to form a hexagonal lattice).
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12. 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
Ferrofluid is often mangenite (iron oxide) suspended in kerosene.
Good reference is Rosensweig, Ferrohydrodynamics.
Dimensionless number which depends the importance of surface tension is the Bond number. It is the ratio of the field energy to the surface energy.
Dynamics is highly overdamped. See Langer, Goldstein, and Jackson (1992).
Surface tension competes with dipole-dipole interaction…
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13. Results courtesy of Ken Cooper
ferromovie.mov
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14. Langmuir monolayer (phospholipid and cholesterol)
Modulated phases
Emphasize the common patterns.
Langmuir monolayer pattern is taken from M. Seul and V. S. Chen, PRL 70, 1658 (1993). Note the cybotactic groups which have local layer order.
Garnet film figure taken from Seul and Wolfe, PRA 46, 7519 (1992).
Langmuir monolayer (phospholipid and cholesterol)
Intermediate state of
type-I superconductor
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15. 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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16. 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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17. Experiments C. R. Reisen and S. G. Lipson, Phys. Rev. B (2000).
Pb-In sample, 3mm diameter, 0.14 mm thick
11 May 2005
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18. 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).
Simulations of Aranson et al.
Experiments: magnetooptics images
Of Niobium films
11 May 2005
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19. 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.
11 May 2005
Lorentz Center Leiden