Statistics, Physics & Statistical Physics of Vulcanized Matter
UIUC Colloquium, February 2005
Statistics, Physics & Statistical
Physics of Vulcanized Matter
Paul M. Goldbart
University of Illinois at Urbana-Champaign
with Nigel Goldenfeld, Horacio Castillo, Weiqun Peng,
Kostya Shakhnovich, Swagatam Mukhopadhyay,
Xiaoming Mao, Xiangjun Xing, Tony Dinsmore,
Alan McKane & Annette Zippelius (& her Göttingen group)
Statistics, Physics & Statistical Physics of Vulcanized Matter
UIUC Colloquium, February 2005
Outline
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What is vulcanized matter
Challenges & attractions
Early views: statistics
Historical intermezzo
Modern era: statistical mechanics
Testing the picture
Prospects
Statistics, Physics & Statistical Physics of Vulcanized Matter
UIUC Colloquium, February 2005
What is molecular matter?
 Structure of matter ~ mid 1800’s
 Berzelius, Kekulé,
Pasteur, van’t Hoff,…
 Chemical reactions:
rational patterns,
optical activity
 Emerging picture
 Molecules = 3D patterns
of chemically bonded atoms
 structral- & stereo-isomers…
 Classification, but not dynamics
 until quantum theory
Statistics, Physics & Statistical Physics of Vulcanized Matter
 Eg cyclohexane
 Same atoms
& bonds
 But flexible!
UIUC Colloquium, February 2005
What is macromolecular matter?
 Much longer molecules
 Amplify flexibility?
 Random coil molecules?
 Obvious?
 Not until ~1930’s
 Staudinger: hero!
 Eg polyethylene
 Same atoms & bonds
 But very flexible!
Statistics, Physics & Statistical Physics of Vulcanized Matter
UIUC Colloquium, February 2005
Examples of macromolecular matter
 Natural rubber
 Highly polymerized isoprene
 Chewing gum, plastics, resins,…
 Many synthetic forms
Statistics, Physics & Statistical Physics of Vulcanized Matter
UIUC Colloquium, February 2005
Some scales for a typical polymer
 Monomers repeated many, many times
 ~25,000 carbon atoms
 ~75,000 atoms in all
 ~5 μm along backbone
 ~0.1 μm across coil
 bending length~1 nm
 length/dia.~6,000
 25 m mouse cable
Statistics, Physics & Statistical Physics of Vulcanized Matter
UIUC Colloquium, February 2005
What is vulcanized matter?
 Vulcanization
 Vulcan, volcano,…
 Hayward (1838)
 Goodyear (1839)
 Randomly add new bonds
 What emerges?
 Space-filling, fluctuating,
random solid network
 natural
 synthetic
 biological
 Similarities with glassy matter
Statistics, Physics & Statistical Physics of Vulcanized Matter
UIUC Colloquium, February 2005
What are random solids?
 Thermally fluctuating liquid
 add enough permanent
random constraints
 new equilib. state
 Microscopic picture
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network formation, topology
liquid destabilized
random localization & transl. SSB
structure of frozen liquid
but solid:
the random solid state
 Macroscopic picture
 diverging viscosity
 emerging static shear rigidity
 retains macro. homogeneity
Statistics, Physics & Statistical Physics of Vulcanized Matter
UIUC Colloquium, February 2005
Some scales for typical vulcanized polymers
 Elastic moduli
 Crystalline solids
 bulk or shear
 exp. values ~
 Rubber
 bulk: comparable
 shear: ~100,000 times smaller
 Reversible extensibility
 Crystal: ~1%
 Rubber: ~700%
 Eg: 1% extension of 1mm wire
 Steel ~ 350 lbs
 Rubber ~ frac. of an ounce
Statistics, Physics & Statistical Physics of Vulcanized Matter
UIUC Colloquium, February 2005
What are the challenges?
 A theory of a giant network of randomly
bonded macromolecules should explain…
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viscous liquid, rigid solid
bonding-triggered transition between them
structure, correlations, heterogeneity
remarkable nature of the elasticity
 Many levels of randomness
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architecture & topology
thermal motion
emergent structure & heterogeneity
sample-to-sample fluctuations
Statistics, Physics & Statistical Physics of Vulcanized Matter
UIUC Colloquium, February 2005
What are the attractions?
 Among Nature’s most complex systems
 many facets of randomness
 no safe ground!
 Emergent universality
 separation of length-scales
 simpler than crystals?
 Captivating mathematics
 eg polymers as…
Feynman diagrams, critical objects
 field theories on strange spaces
 And random networks matter!
 technological & biological relevance
 connections with glasses, granular media
Statistics, Physics & Statistical Physics of Vulcanized Matter
UIUC Colloquium, February 2005
Towards a theory of vulcanized matter
 Start with architectural statistics
Statistics, Physics & Statistical Physics of Vulcanized Matter
UIUC Colloquium, February 2005
Statistical views of vulcanized matter
 Random Graph Theory
 Erdős-Rényi (‘60)
 Limits? transition?
 Interpretation
 sites? bonds?
 Architecture
 infinite cluster?
 But no motion
fraction in infinite
connected component
 A statistical crit. phenom.
 Influential caricature
 Statistics,
not statistical physics
Statistics, Physics & Statistical Physics of Vulcanized Matter
measure of edge probability
UIUC Colloquium, February 2005
Statistical views of vulcanized matter
 Percolation Theory
 Broadbent-Hammersley (‘57)
 Limits? Transition?
 Statistical crit. phenom.
 Again an influential
caricature
 But still no motion
 Still statistics,
not statistical physics
Statistics, Physics & Statistical Physics of Vulcanized Matter
UIUC Colloquium, February 2005
Statistical views of vulcanized matter
 Many elaborations
& developments
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Flory
Stockmayer
Stauffer
de Gennes
Lubensky
& many other
schools
Statistics, Physics & Statistical Physics of Vulcanized Matter
UIUC Colloquium, February 2005
Vulcanized matter: Modern era
 Towards a more complete theory
 Architectural statistics
 Plus motion & its implications
 structure
 correlations
 elasticity & heterogeneity
 richer systems
 How?
 R T Deam & S F Edwards
Theory of Rubber Elasticity
Phil Trans R Soc 280A (1976) 317
Statistics, Physics & Statistical Physics of Vulcanized Matter
UIUC Colloquium, February 2005
Historical intermezzo (after H. Morawetz)
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,…
Statistics, Physics & Statistical Physics of Vulcanized Matter
UIUC Colloquium, February 2005
Historical intermezzo
Kelvin (1857): theoretical
work on thermal effects
Priestly: erasing,
coins name “rubber”
(April 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
inspiredUIUC
by Colloquium,
Kelvin February 2005
Statistics,
Physics
& Statistical
Physics of Vulcanized
Matter
Historical
intermezzo
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 World War II operation in synthetic rubber
second
in scale
only toUIUC
theColloquium,
Manhattan
project
Statistics, Physics & Statistical Physics
of Vulcanized
Matter
February
2005
Historical intermezzo
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?”
Statistics, Physics & Statistical Physics of Vulcanized Matter
UIUC Colloquium, February 2005
Historical intermezzo
Dunlop (1888):
invents the
pneumatic tyre
…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)
Conrad (1901):
Heart of Darkness
Statistics, Physics & Statistical Physics of Vulcanized Matter
UIUC Colloquium, February 2005
Towards a theory of vulcanized matter
 Start with architectural statistics
 quenched randomness
 Add motion & its implications
 annealed randomness
 Aim to understand
 random solidification transition
 properties of emergent random solid
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Microstructure: random localization, structural heterogeneity
Macroscopic: entropic elasticity
Fluctuations: role of dimensionality
Interplay of architectural & thermal disorder
Statistics, Physics & Statistical Physics of Vulcanized Matter
UIUC Colloquium, February 2005
Conventional phase transitions
 Eg: ferromagnetism
 spin freedoms on a lattice
 Energy vs entropy
 energy favors aligned spins,
→ broken spin-rotation symmetry
 entropy favors randomized spins,
→ unbroken symmetry
 temperature controls significance
of entropy
 Transition temperature
 below: energy wins, order, ferromag. state
 above: entropy wins, disorder, paramag.
 at it: strong fluctuations, critical state
Statistics, Physics & Statistical Physics of Vulcanized Matter
UIUC Colloquium, February 2005
Field theory approach: Ferromagnetism
nonlinear
coupling
temperature control parameter
space dimension


t ~ T  T crit T crit
order parameter:
magnetization
density
 Ferromagnetism
 T controlled, energy vs entropy
 detect/diagnose via
magnetization density
 low T: ordered, m nonzero
 high T: disordered, m zero
 Instability/resolution
 modes in Fourier space
 condensation at zero k
Statistics, Physics & Statistical Physics of Vulcanized Matter
UIUC Colloquium, February 2005
Why aim for a field theory?
 Classical
 qualitative guide
 Statistical
 impact of fluctuations
 efficient route to universal quantities
 RG philosophy
 What physicists want (or should!)
Statistics, Physics & Statistical Physics of Vulcanized Matter
UIUC Colloquium, February 2005
Random solidification vs conventional phase transitions
Conventional
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energy vs. entropy
temperature controlled
high T: entropy wins → disorder
low T: energy wins → order
detect/diagnose via order parameter
Random solidification
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constraints & repulsion vs entropy
constraint density controlled
few: entropy wins → delocalization
many: constraints win → random localization
order parameter? diagnoses what?
Statistics, Physics & Statistical Physics of Vulcanized Matter
UIUC Colloquium, February 2005
Vulcanized matter: Modern era
 Basic elements
 Gibbs’ statistical mechanics
 Polymers: fluctuating random curves
 effective repulsions
 Random constraints
 replica technique
 Develop a field theory
 symmetry structure
 Analyze classical states & fluctuations
 Meaning of the field?
 cf magnetism, superconductivity,…
Statistics, Physics & Statistical Physics of Vulcanized Matter
UIUC Colloquium, February 2005
Detecting particle localization
 Begin with one particle
 position R
 choose wave vector k
 equilibrium average
 if delocalized
 if localized
random mean position
random r.m.s. displacement
 (localization length)
Statistics, Physics & Statistical Physics of Vulcanized Matter
UIUC Colloquium, February 2005
Detecting amorphous solid order
 Collection of particles
 labelled j=1,…,J
 positions Rj in D dim’s
 equilibrium averages hi
 disorder averages []
 Randomly localized:
─ doesn’t discriminate between liquid & random solid states
─ does, via analog of Edwards-Anderson order parameter
Statistics, Physics & Statistical Physics of Vulcanized Matter
UIUC Colloquium, February 2005
Interpreting the field
 Real-space version of
“random localization detector”
 Lives on 1+n copies of space
(in the replica limit)
 Encodes statistical information
about random solid state
Statistics, Physics & Statistical Physics of Vulcanized Matter
UIUC Colloquium, February 2005
Field theory approach: Random solidification

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hats:
excess constraint density:
object scale:
cubic nonlinear coupling:
HRS
can be derived semi-microscopically or argued for via
symmetries & length-scales
Statistics, Physics & Statistical Physics of Vulcanized Matter
critical freedoms:
pivotal removal of
density sector fluct’s
(stabilized by interparticle repulsion)
UIUC Colloquium, February 2005
 Localized fraction Q
 recovers Erdős-Rényi
 classical percolation
exponent
 Distribution of
localization lengths
 predicts data collapse
 universal scaling form
 Elementary derivations
Statistics, Physics & Statistical Physics of Vulcanized Matter
measure of crosslink density
probability π
 Field reports on…
localized fraction Q
Classical/Landau theory
(scaled
inverse square)
loc.2005
length
UIUC Colloquium,
February
Classical theory vs simulations
 Barsky-Plischke (’96 & ’97) MD simulations
 Continuous transition to amorphous solid state
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Q
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N chains
L segments
n crosslinks per chain
localized fraction Q grows linearly

scaling, universality
in distribution of
localization lengths
nearly log-normal
Statistics, Physics & Statistical Physics of Vulcanized Matter
UIUC Colloquium, February 2005
Classical theory vs experiments
Protein gels
 Dinsmore/Weitz
(U. Mass. Amherst/Harvard)
Colloidal gels
 roughly μm diameter particles
 confocal microscopy
 video imaging /particle tracking
 statistical analysis of motions
 μm scale thermal fluctuations
Statistics, Physics & Statistical Physics of Vulcanized Matter
UIUC Colloquium, February 2005
Classical theory vs experiments
nearly log-normal
Data (Dinsmore & Guertin, U. Mass.):
• black: gelatin with fluorescent tracer beads
• blue: particle gels by depletion attraction
• red:
particle gel by polycation adsorption
• green: colloidal crystal
Theory: heavy black curve
Statistics, Physics & Statistical Physics of Vulcanized Matter
UIUC Colloquium, February 2005
Other experimental probes
Structure and heterogeneity
 incoherent QENS?

momentum-transfer dependence
measures order parameter
 direct video imaging


fluorescently labeled “polymers”,
colloidal particles
probes loc. length distribution
Elasticity
 range of exponents?
 range of universality classes?
Statistics, Physics & Statistical Physics of Vulcanized Matter
UIUC Colloquium, February 2005
Beyond classical theory: Role of fluctuations
 Meaning of
 Field fluctuations
?
 liquid: mutual loc., clusters
 solid: correl’s in
motion & heterogeneity
 Ginzburg criterion?
 Critical dimensions?
 Percolation?
 contained, classically & beyond
Statistics, Physics & Statistical Physics of Vulcanized Matter
UIUC Colloquium, February 2005
Beyond classical theory: Critical regime
HRW percolation field theory
2
vulcanization field theory


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

ghost field sign
by-hand elimination


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

2
x
x
HRS constraint
momentum conservation
replica combinatorics
replica limit


–Houghton, Reeve, Wallace ’78
–Works to all orders (Janssen & Stenull ’01, Peng et al. ’01)
Statistics, Physics & Statistical Physics of Vulcanized Matter
x
UIUC Colloquium, February 2005
Beyond classical theory: Critical regime
 Field theory yields
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Ginzburg criterion?
Critical dimensions?
Percolation?
Critical elasticity?
 resolve shear modulus?
 phantom chains
 RRN
 incompressible fluid  percolation
Statistics, Physics & Statistical Physics of Vulcanized Matter
UIUC Colloquium, February 2005
Beyond classical theory: Goldstone fluctuations
 Explore excitations of random solid state
 low-energies, long-wavelengths: Goldstones
 structure
 meaning
 implications (esp. low dim’s)
Statistics, Physics & Statistical Physics of Vulcanized Matter
UIUC Colloquium, February 2005
Beyond classical theory: Goldstone fluctuations
liquid/high
density
gas/low
density
 Analogy: co-existing liquid-gas
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order parameter: density kink
interface location: zero-mode
Goldstones: capillary waves
stiffness: surface tension
Statistics, Physics & Statistical Physics of Vulcanized Matter
UIUC Colloquium, February 2005
Beyond classical theory: Goldstone fluctuations
 Classical
 Hill in replicated space
 Location: arbitrary
 Goldstone excitations?
 structure: ripples
 meaning: shear deformations
 cost  entropic elasticity
 stiffness  shear modulus
 implications (low dim’s?)
Statistics, Physics & Statistical Physics of Vulcanized Matter
UIUC Colloquium, February 2005
Where next?
 Ordered state fluctuations?
 Universal intrinsic heterogeneity
 Role of other freedoms?
 Liquid crystal polymers, blends,…?
 Connections
 random resistor networks, multifractality?
 with glasses, granular media,…?
 Dynamics
 especially of the ordered state ?
 Further experiments
 Q/E INS; video imaging,…?
Statistics, Physics & Statistical Physics of Vulcanized Matter
UIUC Colloquium, February 2005
Why vulcanized matter?
 Intrinsic intellectual interest

(un)usual state of matter
 Technological/biological relevance
 Simplified version of structural glass
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structure → dynamics → structure… feed-back loop cut
tough non-equilibrium problem → easier equilib. caricature
liquid-state structure frozen in, but…
extrinsically, controllably, permanently (not self-generated)
 Result: Least complicated setting for…
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
random solid state
phase transition from liquid to it
 Why the simplicity?


equilibrium states, equilibrium methods
continuous transition  universal properties
Statistics, Physics & Statistical Physics of Vulcanized Matter
UIUC Colloquium, February 2005
Concluding remarks
 What attracts physicists?
 Not always in obvious settings
 Random solid state
 “standard model”
 universality: classical & beyond
 unified approach to
 structure, rigidity, heterogeneity,…
Statistics, Physics & Statistical Physics of Vulcanized Matter
UIUC Colloquium, February 2005
Acknowledgements
 Collaborators: Castillo, Goldenfeld, Mao, McKane,
Mukhopadhyay, Peng, Shakhnovich, Xing,
Zippelius (and co-workers)
 Simulations: Barsky & Plischke
 Experiments: Dinsmore and co-workers
 Foundations: Edwards & co-workers
 Related studies: Panyukov & co-workers
goldbart@uiuc.edu
w3.physics.uiuc.edu/~goldbart
Statistics, Physics & Statistical Physics of Vulcanized Matter
UIUC Colloquium, February 2005
Statistics, Physics & Statistical Physics of Vulcanized Matter
UIUC Colloquium, February 2005
Statistics, Physics & Statistical Physics of Vulcanized Matter
UIUC Colloquium, February 2005
Beyond classical theory: Critical regime
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, low D)
(Peng & PMG ’00)
segments per chain
Momentum-shell RG to order 6-D
volume fraction

find percolative critical exponents for percol. phys. quant’s

could it be otherwise?
• All-orders connection
(Janssen & Stenull ’01; Peng et al. ’01)
Statistics, Physics & Statistical Physics of Vulcanized Matter
UIUC Colloquium, February 2005
Mean-field theory vs. experiments
Data (A D Dinsmore & C F Guertin, U. Mass.):
• black: gelatin with fluorescent tracer beads
• blue: particle gels by depletion attraction
• red:
particle gel by polycation adsorption
• green: colloidal crystal
Statistics,
Statistical Physics of Vulcanized Matter
Theory:
heavyPhysics
black&curve
UIUC Colloquium, February 2005
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
Statistics, Physics & Statistical Physics of Vulcanized Matter
UIUC Colloquium, February 2005
Emergent shear elasticity
Simple principle:
Free energy cost of
shear deformations?
 two contributions

deformed free energy

deformed saddle point
deformation hypothesis
Emergent elastic free energy
Shear modulus exponent?
shear modulus ~  t
t ?
Statistics, Physics & Statistical Physics of Vulcanized Matter
UIUC Colloquium, February 2005
Goldstone fluctuations & rigidity:
More than two dimensions
Elasticity & shear modulus

homogeneous isotropic elasticity; ρ ≈ T c ε3
Order parameter fluctuations

simple shift in distribution of (squared) localization lengths
• Order parameter correlations

new diagnostics: correlations of localization parameters
Statistics, Physics & Statistical Physics of Vulcanized Matter
UIUC Colloquium, February 2005
Random solids: Two dimensions
Percolation and amorphous solidification

several common features but…

broken symmetries?

Goldstone modes and lower critical dimensions?

random quasi-solidification?

rigidity without localization?
Statistics, Physics & Statistical Physics of Vulcanized Matter
UIUC Colloquium, February 2005
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?
Statistics, Physics & Statistical Physics of Vulcanized Matter
UIUC Colloquium, February 2005
Statistics, Physics & Statistical Physics of Vulcanized Matter
UIUC Colloquium, February 2005
To include…
 Strings, nucleons & electrons, nuclei,
atoms, molecules: suppose we understand
these & their interactions
 What new ways can matter and energy
interact in space and time given these
building blocks?
 Direct attack: infeasible
 The art of condensed matter
 Sometimes looks very specific LaSrCuO but
the aim of seeking out and understanding
what the possibilities are is everpresent, if
veiled
Statistics, Physics & Statistical Physics of Vulcanized Matter
UIUC Colloquium, February 2005
To include…
 Connections with rest of physics
 What CM physics cares about
 Common ground: Goldstones, SBS,
SUSY,collective phenomena,…
 Simple statistics: percolation
 Rubber theorists: Flory, James-Guth
 Polymers: remarkable, Staudinger & Kekule
 Estimates: bulk & shear; bag of water;
quantum pressure; thermal shear
 Peierls: strength of metals, same Peierls
 Magentism: same Heisenberg, Dirac;
Goldstones
 The matter around us: techno, bio
Statistics, Physics & Statistical Physics of Vulcanized Matter
UIUC Colloquium, February 2005
 Aims of condensed
matter physics?
Statistics, Physics & Statistical Physics of Vulcanized Matter
UIUC Colloquium, February 2005
 Maxwell, Boltzmann, Gibbs
 Using statistics to solve manyparticle problems
 My bottle of water is the same
as yours
 Energy, volume – that’s it!
Statistics, Physics & Statistical Physics of Vulcanized Matter
UIUC Colloquium, February 2005
Wiener, Feynman, Kac
Fluctuations – thermal or quantal
Polymers as Feynman diagrams
In the end we’re all doing statistical physics
Whether we know it is a different matter
Semiclassical limits
Your vacuum, my ground state: their excitations
determine the character of the state
 E.g. our broken symmetry might be **, yours
might be **; our Goldstone bosons would be
phonons, yours would be **; our Meissner effect is
your Higgs effects etc. etc.







Statistics, Physics & Statistical Physics of Vulcanized Matter
UIUC Colloquium, February 2005
 Onsager, Landau, Wilson
 Symmetry and symmetry breaking
 Clear limits e.g. Bose-Einstein
condensation
 Special role played by big systems: efects
become qualitative
 Role of exactly solvable models
 Coarse-graining and the RG
Statistics, Physics & Statistical Physics of Vulcanized Matter
UIUC Colloquium, February 2005
 The very existence of polymers
Staudinger’s spurned idea
The remarkable act of staying together
Kekule’s structure of benzene
Scales in polymers: atoms, step lengths,
coil sizes
 Effect of repulsion; solvent-mediated
interactions (cf. HEP)
 Edwards: the polymers are the diagrams
 Irony: it took a student of Schwinger’s to
bring Feynman’s path-integral methods to
condensed matter




Statistics, Physics & Statistical Physics of Vulcanized Matter
UIUC Colloquium, February 2005
 Edwards and Anderson
 How to cope with qualitative effects of
disorder
 Electron localization
 Spin glasses
 Vulcanized matter
Statistics, Physics & Statistical Physics of Vulcanized Matter
UIUC Colloquium, February 2005
 Critical phenomena and universality
 Large ratios of length scales
 Critical point in water
 Long polymers
Statistics, Physics & Statistical Physics of Vulcanized Matter
UIUC Colloquium, February 2005
Statistics, Physics & Statistical Physics of Vulcanized Matter
UIUC Colloquium, February 2005
Outline





A little history
What are random solids
Ordinary phase transitions vs. random solidification
Detection and diagnosis
A Landau-type approach
 describing the emergent state
 Beyond mean-field theory
 role of critical fluctuations
 connections with percolation, elasticity
 Goldstone-type fluctuations
 nature, meaning, consequences
 Implications in two dimensions
 quasi random solid state
 contrast with percolation
 Concluding remarks, future directions
Statistics, Physics & Statistical Physics of Vulcanized Matter
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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
Statistics, Physics & Statistical Physics of Vulcanized Matter
UIUC Colloquium, February 2005
Emergent state: Random solid
Constraint-induced instability (with
“frustration”: cross-linking vs. repulsion)
Resolution

condensation with MTI

determines
and
Interpretation
–
magnitude: localized fraction
–
wave-vector dependence: heterogeneity
(a.k.a. distribution of localization lengths)
• Not a number but a function!
Statistics, Physics & Statistical Physics of Vulcanized Matter
UIUC Colloquium, February 2005
What forms random solids ?
Macromolecular networks

permanently cross-linked or end-linked at random
Chemical gels (atoms, small molecules,…)

permanently covalently bonded at random
Key point: Form giant random
structures thermally fluctuating
in equilibrium
Statistics, Physics & Statistical Physics of Vulcanized Matter
UIUC Colloquium, February 2005
Molecular bound state
Transition as condensation in replica space
order parameter
dispersion of atoms/replicas
Localization
 atoms/replicas bound
But random
with cm uniformly distributed
Statistics, Physics & Statistical Physics of Vulcanized Matter
UIUC Colloquium, February 2005
Modern approach: Basic elements
 Average over configs. (Gibbs’ stat. mech.)
 Average over random constraints
 Replica technique to handle log
 Effective pure theory of coupled replicas
Statistics, Physics & Statistical Physics of Vulcanized Matter
UIUC Colloquium, February 2005
Modern approach: Illustrating replicas
 Average over configs:
 Av. random constr’s:
 Replicas for log:
 Av. config’s of coupled replicas (no randomness):
Statistics, Physics & Statistical Physics of Vulcanized Matter
UIUC Colloquium, February 2005
Landau approach: Random solidification






hats:
excess constraint density:
object scale:
particle density:
cubic nonlinear coupling:
HRS
can be derived semi-microscopically or argued for via
symmetries & length-scales
Statistics, Physics & Statistical Physics of Vulcanized Matter
Critical freedoms:
pivotal removal of
density sector fluct’s
(stabilized by interparticle repulsion)
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Mean field theory
Localized fraction Q obeys:

control parameter 
» excess crosslink density
1  Q  exp {(1  ( /3))Q}
 Q   (linear near transition)
Universal scaling form for the loc. length distribution:
 scaling & collapse
universal scaling
function; obeys
(magic normalization)
Mean-field theory
Cavity approach
(Castillo et al. ’94; (w/ Mao & Mézard ’04)
Statistics,
Peng
et al.Physics
’98) & Statistical Physics of Vulcanized Matter
UIUC Colloquium, February 2005
Mean field theory
localized fraction Q
Specific predictions
 localized fraction Q

linear near the transition

Erdős-Rényi random graph
theory form
probability π
measure of crosslink density
 localization length distribution

data-collapse for all near-critical
crosslink densities

specific universal form for
scaling function
(scaled inverse
square)
loc. Physics
length of Vulcanized Matter
Statistics,
Physics &
Statistical
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Beyond classical theory: Critical regime
 Approach presents order-parameter field
 Correlations of order-parameter fluctuations
 meaning (in fluid state):

localize by hand at

will what’s at

how strongly?

probes cluster formation
be localized?
 meaning (in solid state):

e.g. localization-length correlations
(Peng & PMG ’00)
Statistics, Physics & Statistical Physics of Vulcanized Matter
UIUC Colloquium, February 2005
Real-space view
 Classical value of (x0,x1,…) ?
– hill in replicated space
 section-area
 form
gives Q
givs N(2)
 width
gives typical 2
ridge location is Goldstone mode
 Goldstone excitations
– (x) ! (x - u(xcm))
 ripples of the ridge-line
 Physical meaning ?
 shear deformations
 nD fields on xcm
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Energetics of Goldstone excitations
•Free energy
•Stiffness
– derives elasticity
μ : controls constraint density
c : particle density
a & g : Landau parameters
– recovers classical stiffness exponent
Statistics, Physics & Statistical Physics of Vulcanized Matter
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Implications of Goldstone excitations
Effects of ripples
on order parameter

simple shift in loc. length distribution
on correlator

distribution & correlations
of localization parameters
Statistics, Physics & Statistical Physics of Vulcanized Matter
UIUC Colloquium, February 2005
Implications: 3D
Fluctuations suppress order parameter
moderately: remains nonzero
symmetry: remains broken
loc. length distribution: finite shift
Order parameter fluctuation correlators
decay conventionally in space
Rigid state of matter
Statistics, Physics & Statistical Physics of Vulcanized Matter
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Implications: 2D
Fluctuations suppress order parameter
strongly: infinite shift of loc. len. distribution
o.p. vanishes: quasi-localized fraction
Correlators decay algebraically
power-law exponent
k is ‘probe’ scale
Translational symmetry restored
But rigidity remains
Percolation physics?
Connections with 2D melting
Statistics, Physics & Statistical Physics of Vulcanized Matter
UIUC Colloquium, February 2005
Statistics, Physics & Statistical Physics of Vulcanized Matter
UIUC Colloquium, February 2005
Elasticity beyond the classical theory
Shear modulus near transition?

Scaling approaches, simulations
– de Gennes ’79 (random resistor network analogy):
 f =
 D - g
– Daoud & Coniglio ’81 (percolation),
Del Gado et al. ’02 (simulations),…:
 f =

 D
RG approach to LW model
– phantom chains  RRN physics
– incompressible fluid  percolation physics
(Xing et al., cond-mat/0406411)
Statistics, Physics & Statistical Physics of Vulcanized Matter
UIUC Colloquium, February 2005