Transcript Vulcanized Matter Talk for Brown U

Seeking simplicity in complex media:
a physicist's view of vulcanized matter,
glasses, and other random solids
Paul M. Goldbart
University of Illinois at Urbana-Champaign
[email protected]
w3.physics.uiuc.edu/~goldbart
Thanks to many collaborators, including:
Nigel Goldenfeld, Annette Zippelius,
Horacio Castillo, Weiqun Peng,
Kostya Shakhnovich, Alan McKane
A little history…
Columbus (Haiti, 1492):
reports locals playing
games with elastic resin
from trees
de la Condamine (Ecuador,
~1740): latex from incisions in
Hevea tree, rebounding balls;
suggests waterproof fabric,
shoes, bottles, cement,…
a little more history…
Kelvin (1857): theoretical
work on thermal effects
Priestly erasing;
coins the name
rubber (4.15.1770)
Faraday (1826): analyzed chemistry of
rubber – “… much interest attaches to
this substance in consequence of its
many peculiar and useful properties…”
Joule (1859): experimental work inspired by Kelvin
and some
more…
F. D. Roosevelt
(1942, Special
Committee)
•“… of all critical and strategic materials…rubber presents
the greatest threat to… the success of the Allied cause”
•US WWII operation in synthetic rubber second in scale
only to the Manhattan project
yet more …
Goodyear (in Gum-Elastic and its Varieties, with a Detailed Account
of its Uses, and of the Discovery of Vulcanization; New Haven, 1855):
“… there is probably no other inert substance the properties of which
excite in the human mind an equal amount of curiosity, surprise and
admiration. Who can reflect upon the properties of gum-elastic without adoring the wisdom of the Creator?”
but…
…the invention of which led to
“frantic efforts to increase the
supply of natural rubber in the
Belgian Congo…” which led to
“some of the worst crimes of man
against man…” (Morawetz, 1985)
Dunlop (1888):
invents the
pneumatic tyre
Conrad (1901):
Heart of Darkness
Outline
• A little history
• What is vulcanized matter?
• Central themes
• What is amorphous solidification? Why study it?
• How to detect amorphous solids?
• Landau-type mean-field approach;
physical consequences
• Simulations
• Experimental probes
• Beyond mean-field theory;
connections; low dimensions
• Structural glasses
• Some open issues
What is vulcanized matter?
•Vulcanized macromolecular networks

permanently crosslinked at random
 or endlinked
•Chemical gels (atoms,
small molecules,…)

permanently covalently
bonded at random
•Form giant random
network
Central themes
• Fluid system


macromolecules, molecules, atoms,…
solution or melt, flexible or stiff macromolecules
• Introduce permanent random constraints



covalent chemical bonds (e.g. vulcanization)
do not break translational symmetry explicitly
form giant random network
• Transition to a new state: amorphous solid




structure: random localization?
static response: elastic?
correlations: liquid and solid states?
dynamic signatures?
• What can be said about?


the transition
the emergent solid near the transition & beyond
What is amorphous solidification?
• Emergence of new state of matter via sufficient
vulcanization: amorphous solid
• Microscopic picture




network formation, topology
liquid state destabilized
random localization of (fraction of) constituent particles
(e.g. random means & r.m.s. displacements)
translational symmetry broken
spontaneously, but randomly
• Macroscopic picture


emerging static shear rigidity
(& diverging viscosity)
retains homogeneity
macroscopically
Interlude: Why vulcanized matter?
• Least complicated setting for


random solid state
phase transition from liquid to it
• Why the simplicity?


equilibrium states
continuous transition
 universal properties
• Simplified version of real glass



equilibrium setting
frozen-in constraints
but external, not spontaneous
• Broad technological/biological relevance
• Intrinsic intellectual interest

an (un)usual state of matter
Foundations
• S. F. Edwards and P. W. Anderson
Theory of Spin Glasses
J. Phys. F5 (1975) 965
• R. T. Deam and S. F. Edwards
Theory of Rubber Elasticity
Phil. Trans. R. Soc. 280A (1976) 280
Order parameter for random localization
•One particle, position R

choose a wave vector k

equilibrium average

delocalized: exp i k  R   k ,0

localized:

exp i k  R  exp( i k  R )  exp(  k 2 2 2)
• N particles, with positions
R j ( j  1,2,..., N )

in both liquid & amorphous solid states
N

random
mean position
1

N
j 1
exp i k  R j   k ,0
doesn’t distinguish between these states
random r.m.s.
displacement
(localization
length) 
Order parameter for random localization
• Edwards-Anderson—type order parameter

N
choose wave vectors
1

N
j 1
g
study
k1 , k 2 ,..., kand
exp i k1  R j exp i k 2  R j    exp i k g  R j

delocalized 

localized
 k , 0   k , 0      k
1
2
g
,0

Q   k1 k 2 k g ,0   d p( ) exp {  2 (|k1|2  |k 2|2      |k g|2 ) 2}
fraction of
loc. particles
macroscopic
homogeneity
(cf. crystals)
statistical distribution
of localization lengths
• Distinguishes liquid & amorphous solid states
Landau theory ingredients

built from order parameter

meaning of
[N 
1
(k , k ,..., k )
0
1
n
:
N
j 1
exp i k 0  R j    exp i k n  R j

lives on (n+1)-fold replicated space (as n → 0)

free energy: cubic theory in

pivotal removal of density sector

(stabilized by particle repulsions)

can be derived semi-microscopically

or argued for on symmetry & length-scale grounds
disorder
averaging
]
Landau free energy
 d xˆ {
1
2
1
2
3
ˆ
 ( xˆ )  | |  3! g ( xˆ ) }
2
1
2
crosslink density control parameter

 ~ N xl  N xlcrit  N xlcrit
built from (Fourier transform of) order parameter
( xˆ ) 
1
V
nonlinear coupling
ˆ) exp( ikˆ  xˆ )

(
k
n 1 kˆHRS

in replicated real space xˆ  {x 0 , x1 ,..., x n }

subject to physical (HRS) constraints
 d xˆ ( xˆ )  0

ˆ
ˆ
d
x

(
x
)
exp(
i
k

x
)0

Instability and resolution
• What modes of (k 0 , k 1 ,..., k n )
feature as critical modes?

all but 0 and 1 replica sector modes
• Instability?

all long-wavelength modes

but not resolved via 0 mode
• Frustration?

cross-linking versus repulsions
• Resolution?

“condensation” with macroscopic translational invariance

peak height & shape  loc. frac. & distrib. of loc. lengths
Results of mean-field theory
• Order parameter (k 0 , k 1 ,..., k ntakes
the form:
)
Q   k 0 k1 k n ,0   d p( ) exp {  2 (|k 0|2  |k1|2      |k n|2 ) 2}
fraction of
loc. particles
distrib. of loc. lengths
• Localized fraction:

control param. ε ~
excess x-link density
1  Q  exp{(1  ( /3))Q}
 Q  2 /3
(linear near
transition)
• Universal scaling form for the loc. length distrib.:
p( ) (4 / 3 )  (2 / 2 )
universal scaling
function; obeys
(plus normalization)
( 2 /2)(d /d )  (1   ) ( )  (   )( )
Results of mean-field theory
localized fraction Q
• Specific predictions
– localized fraction Q
measure of crosslink density

linear near the transition

Erdős-Rényi RGT form
probability π
– localization length distribution
(scaled inverse square) loc. length

data-collapse for all near-critical
crosslink densities

specific universal form for
scaling function
Mean-field theory vs. simulations
• Barsky-Plischke (’96 & ’97) MD simulations
• Continuous transition to amorphous solid state
Q




N chains
L segments
N crosslinks per chain
localized fraction Q grows linearly

scaling, universality
in distribution of
localization lengths
nearly log-normal
Symmetry and stability
•Proposed amorphous solid state
– translational & rotational symmetry broken
– replica permutation symmetry?

Almeida-Thouless instability? RSB? Intact?
– full local stability analysis

put lower bounds on eigenvalues of Hessian
by exploiting high residual symmetry

broken translational symmetry  Goldstone mode
Emergent shear elasticity
• Simple principle:
Free energy cost of
shear deformations?
– two contributions


deformed free energy
deformed saddle point
• Emergent elastic free energy
• Shear modulus exponent?
shear modulus ~  t
t ?
deformation hypothesis
Experimental probes
•Structure and heterogeneity
– incoherent QENS?

momentum-transfer dependence
measures order parameter
– direct video imaging?


fluorescently labeled polymers,
colloidal particles
probes loc. length distrib.
•Elasticity
– range of exponents?
Interlude: 3 levels of randomness
•Quenched random constraints (e.g. crosslinks)

architecture (holonomic)

topology (anholonomic)
•Annealed random variables

Brownian motion of particle positions
•Heterogeneity of the emergent state

distribution of localization lengths

characterize state via distribution
•Contrast with percolation theory etc.

just the one ensemble
Beyond mean-field theory
• Approach presents order-parameter field

• Correlations of order-parameter fluctuations
– meaning (in fluid state):

localize by hand at x

will what’s at y be localized?

how strongly?

probes cluster formation
– meaning (in solid state):

e.g. localization-length correlations

Beyond mean-field theory
• Landau-Wilson minimal model

cubic field theory on replicated d-space

upper critical dimension?

Ginzburg criterion (cf. de Gennes ’77):
cross-link density window (favours short, dilute chains)
segments per chain
L / l ( d 2) /(6d )  2 /(6d )
• Momentum-shell RG to order 6  d
volume fraction

find percolative critical exponents for percol. phys. quant’s

relation to percolation via the Potts model

could it be otherwise?
• All-orders connection (see also Janssen & Stenull ’01)
Beyond mean-field theory
HRW percolation field theory
 dx





1
2
  ( x) 2  12 

   12   ( x) 2  12 

2
vulcanization field theory
2



2




 31! g (   ) 3 
2
1
1 ˆ
1
2
3

ˆ
ˆ
ˆ
d
x
  ( x
)




g

(
x
)


2
3!
  2




 ghost field sign
 by-hand elimination



2
x
x
 HRS constraint
 momentum conservation
 replica combinatorics
 replica limit


works to all orders (Peng et al,. Janssen & Stenull)
x
Two dimensions?
• Percolation and amorphous solidification

several common features but…

broken symmetries?

Goldstone modes and lower critical dimensions?

random quasi-solidification?

rigidity without localization?
Structural glass?
• Covalently-bonded
random network media
e.g.   Si, SiO 2 , Ge x As ySe1 x  y
– regard frozen-in liquid-state
correlations as quenched
random constraints
– examine properties
between two time-scales:
structure-relaxation & bond-breaking
•Is there a separation of time-scales?
Some open issues
• Elementary origin of universal distrib. of loc. lengths
(found elsewhere? connection with log-normal?)
• Ordered-state structure & elasticity beyond meanfield theory?
• Further connections with random resistor networks?
• Multifractality?
• Dynamics, especially of the ordered state?
• Connections with glasses?
• Experiments (Q/E INS; video imaging,…)?
Acknowledgments
• Collaborators:
H. E. Castillo, N. D. Goldenfeld, A. J. McKane,
W. Peng, K. Shakhnovich, A. Zippelius,,…
• Simulations:
S. J. Barsky & M. Plischke
• Foundations:
S. F. Edwards, R. T. Deam, R. C. Ball & coworkers
• Related studies of networks:
S. Panyukov & coworkers
• All-orders connection with percolation: see also
H.- K. Janssen & O. Stenull (via random resistor networks)
[email protected]
w3.physics.uiuc.edu/~goldbart