can you guess what systems that produced these patterns? · 2014-04-23 · can you guess what...
Transcript of can you guess what systems that produced these patterns? · 2014-04-23 · can you guess what...
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can you guess what systems that produced these patterns?
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can you guess what systems that produced these patterns?
Langmuir monolayer – dipolar forces (simulation)
Turing pattern in chemical system (experiment)
labyrinthine pattern in chemical system (experiment)
Ising model (simulation)
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self-assembled systems show similar patterns on macroscopic scales
Langmuir monolayers
more complex three-dimensional patterns
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time evolution of spatial patterns
what is the main difference between the top and bottom set of pictures?
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time evolution of spatial patterns
what is the main difference between the top and bottom set of pictures?
non-conserved order parameter
conserved order parameter
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Dynamics of Self-Organized and Self-Assembled Structures most material can be found in: R. C. Desai and R. Kapral, (Cambridge, 2009)
-- Introduction; binary mixtures and phase separation; order parameter; free energy and Landau expansion ; free energy functional; interfacial tension -- phase separation kinetics; Langevin Model A; Langevin Model B -- interface dynamics at late times -- propagating fronts; transverse front instabilities -- competing interactions: systems with long range repulsive interactions; front repulsion -- active materials -- self-propelled particles
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order parameter and free energy example: binary mixture phase separation
Fe-Al alloy
A and B species, concentration
critical quench off-critical quench
off-critical quenches
consolute point
critical quench
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mean field theory
configurational (entropic) term
interaction term coordination
number
thermodynamic variable conjugate to c is the chemical potential
thermodynamic space:
at the critical point
equation of state:
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Landau expansion
expand the free energy in a power series in
0
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Landau expansion of free energy
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Landau expansion of free energy for general order parameter
general form: order parameter c concentration
m magentization Q liquid crystal order parameter
density
thermodynamic space
equation of state
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Ginzburg-Landau free energy functional
f¤(Á;rÁ;r2Á; :::) = f(Á) +X
i
Li@Á
@ri
+1
2
X
ij
·2·
(1)ij
@2Á
@ri@rj+ ·
(2)ij
@Á
@ri
@Á
@rj
¸+ : : : ;
homogeneous system: free energy density free energy
inhomogeneous system:
expand f in a power series
Li = [@f¤=@(@Á=@ri)]o, ·(1)ij = [@f¤=@(@2Á=@ri@rj)]o,
·(2)ij = [@2f¤=@(@Á=@ri)@(@Á=@rj)]o
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Li vanishes, ·(1)ij = ·1±ij , and ·
(2)ij = ·2±ij
for an isotropic system f is invariant under inversion and rotation
free energy functional
Integrate by parts
Ginzburg-Landau free energy functional
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free energy functional (continued)
expansion about the critical point
free energy takes the form
correlation length
as critical point is approached
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interfacial tension
black
white
planar interface
where is the interface?
Gibbs equimolar dividing surface
equal areas
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interfacial tension (continued)
surface tension: excess surface free energy per unit area
¢f =
½(f(Á)¡ f+); for ¡1 < z < 0,(f(Á)¡ f¡); for 0 < z <1
recall
surface tension
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surface free energy is a minimum for the equilibrium profile
or
is the equilibrium profile
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multiply by
integrate
substitute into surface tension equation
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nonconserved and conserved order parameter dynamics
focus on simple cases where systems can be described by a single order parameter field; accurate treatments of some real systems require several order parameters for their descriptions
Langevin Model A – nonconserved order parameter (antiferromagnets, some chemical systems, etc.)
equation of motion for
Langevin equation
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Model A – time-dependent Ginzburg-Landau equation
early times – noise important
late times – noise can be neglected
TDGL equation
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a chemical example: bistable system:
fixed reaction-diffusion equation
has same form as the TDGL equation
let
then RD equation can be written free energy functional form
SchlÄogl model
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critical quench – Model A SchlÄogl model
early stage of evolution: formation of interfaces
late stage of evolution: well-defined interfaces
fluctuations important at early stages if interface growth
stochastic dynamics