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can you guess what systems that produced these patterns?
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)
self-assembled systems show similar patterns on macroscopic scales
Langmuir monolayers
more complex three-dimensional patterns
time evolution of spatial patterns
what is the main difference between the top and bottom set of pictures?
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
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
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
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:
Landau expansion
expand the free energy in a power series in
0
Landau expansion of free energy
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
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
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
free energy functional (continued)
expansion about the critical point
free energy takes the form
correlation length
as critical point is approached
interfacial tension
black
white
planar interface
where is the interface?
Gibbs equimolar dividing surface
equal areas
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
surface free energy is a minimum for the equilibrium profile
or
is the equilibrium profile
multiply by
integrate
substitute into surface tension equation
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
Model A – time-dependent Ginzburg-Landau equation
early times – noise important
late times – noise can be neglected
TDGL equation
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
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
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