Seeking simplicity in complex media: a physicist's view of vulcanized matter, glasses, and other random solids

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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
• goldbart_at_uiuc.edu
• w3.physics.uiuc.edu/goldbart

Thanks to many collaborators, including Nigel
Goldenfeld, Annette Zippelius, Horacio Castillo,
Weiqun Peng, Kostya Shakhnovich, Alan McKane
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A little history
Columbus (Haiti, 1492)reports locals playing
games with elastic resinfrom trees
de la Condamine (Ecuador,1740) latex from
incisions inHevea tree, rebounding
ballssuggests waterproof fabric,shoes,
bottles, cement,
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a little more history
Kelvin (1857) theoreticalwork on thermal effects
Priestly erasingcoins the namerubber
(4.15.1770)
Faraday (1826) analyzed chemistry ofrubber
much interest attaches tothis substance in
consequence of itsmany peculiar and useful
properties
Joule (1859) experimental work inspired by Kelvin
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and some more
F. D. Roosevelt(1942, SpecialCommittee)

• of all critical and strategic materialsrubber
  presentsthe greatest threat to the success of
  the Allied cause
• US WWII operation in synthetic rubber second in
  scale only to the Manhattan project

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yet more

• Goodyear (in Gum-Elastic and its Varieties, with
  a Detailed Accountof its Uses, and of the
  Discovery of Vulcanization New Haven, 1855)
• there is probably no other inert substance the
  properties of whichexcite in the human mind an
  equal amount of curiosity, surprise
  andadmiration. Who can reflect upon the
  properties of gum-elastic with-out adoring the
  wisdom of the Creator?

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but
the invention of which led tofrantic efforts
to increase thesupply of natural rubber in
theBelgian Congo which led tosome of the
worst crimes of managainst man (Morawetz,
1985)
Dunlop (1888)invents the pneumatic tyre
Conrad (1901)Heart of Darkness
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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

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What is vulcanized matter?

• Vulcanized macromolecular networks
• permanently crosslinked at random

• or endlinked

• Chemical gels (atoms,small molecules,)
• permanently covalentlybonded at random
• Form giant randomnetwork

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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

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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 brokenspontaneously, but
  randomly
• Macroscopic picture
• emerging static shear rigidity( diverging
  viscosity)
• retains homogeneitymacroscopically

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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

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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

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Order parameter for random localization

• One particle, position
• choose a wave vector
• equilibrium average
• delocalized
• localized
• particles, with positions
• in both liquid amorphous solid states
• doesnt distinguish between these states

random mean position
random r.m.s.displacement(localizationlength)
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Order parameter for random localization

• Edwards-Andersontype order parameter
• choose wave vectors and study
• delocalized ?
• localized

?
macroscopichomogeneity(cf. crystals)
statistical distributionof localization lengths
fraction of loc. particles

• Distinguishes liquid amorphous solid states

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Landau theory ingredients

• built from order parameter
• meaning of
• lives on (n1)-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
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Landau free energy
crosslink density control parameter
nonlinear coupling

• built from (Fourier transform of) order parameter
• in replicated real space
• subject to physical (HRS) constraints

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Instability and resolution

• What modes of 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

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Results of mean-field theory

• Order parameter
  takes the form

fraction of loc. particles
distrib. of loc. lengths

• Localized fraction
• control param. e excess x-link density

(linear neartransition)

• Universal scaling form for the loc. length
  distrib.

(plus normalization)
universal scaling function obeys
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Results of mean-field theory

• localized fraction
• linear near the transition
• Erdos-Rényi RGT form
• localization length distribution
• data-collapse for all near-criticalcrosslink
  densities
• specific universal form forscaling function

• Specific predictions

localized fraction Q
measure of crosslink density
probability p
(scaled inverse square) loc. length
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Mean-field theory vs. simulations

• Barsky-Plischke (96 97) MD simulations
• Continuous transition to amorphous solid state
• N chains
• L segments
• N crosslinks per chain
• localized fraction grows linearly
• scaling, universalityin distribution
  oflocalization lengths

nearly log-normal
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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 Hessianby
  exploiting high residual symmetry
• broken translational symmetry ? Goldstone mode

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Emergent shear elasticity

• Simple principleFree energy cost ofshear
  deformations?
• two contributions
• deformed free energy
• deformed saddle point
• Emergent elastic free energy
• Shear modulus exponent?

deformation hypothesis
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Experimental probes

• Structure and heterogeneity
• incoherent QENS?
• momentum-transfer dependencemeasures order
  parameter
• direct video imaging?
• fluorescently labeled polymers,colloidal
  particles
• probes loc. length distrib.
• Elasticity
• range of exponents?

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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

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Beyond mean-field theory

• Approach presents order-parameter field
• Correlations of order-parameter fluctuations
• meaning (in fluid state)
• localize by hand at
• will whats at be localized?
• how strongly?
• probes cluster formation
• meaning (in solid state)
• e.g. localization-length correlations

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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)
• Momentum-shell RG to order
• find percolative critical exponents for percol.
  phys. quants
• relation to percolation via the Potts model
• could it be otherwise?
• All-orders connection (see also Janssen Stenull
  01)

segments per chain
volume fraction
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Beyond mean-field theory
HRW percolation field theory
vulcanization field theory
x
x
x

• HRS constraint
• momentum conservation
• replica combinatorics
• replica limit
• ?

• ghost field sign
• by-hand elimination
• ?

works to all orders (Peng et al,. Janssen
Stenull)
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Two dimensions?

• Percolation and amorphous solidification
• several common features but
• broken symmetries?
• Goldstone modes and lower critical dimensions?
• random quasi-solidification?
• rigidity without localization?

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Structural glass?

• Covalently-bondedrandom network mediae.g.
• regard frozen-in liquid-statecorrelations as
  quenchedrandom constraints
• examine propertiesbetween two time-scalesstruct
  ure-relaxation bond-breaking
• Is there a separation of time-scales?

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Some open issues

• Elementary origin of universal distrib. of loc.
  lengths (found elsewhere? connection with
  log-normal?)
• Ordered-state structure elasticity beyond mean-
  field theory?
• Further connections with random resistor
  networks?
• Multifractality?
• Dynamics, especially of the ordered state?
• Connections with glasses?
• Experiments (Q/E INS video imaging,)?

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Acknowledgments

• CollaboratorsH. E. Castillo, N. D. Goldenfeld,
  A. J. McKane,W. Peng, K. Shakhnovich, A.
  Zippelius,,
• Simulations S. J. Barsky M. Plischke
• FoundationsS. F. Edwards, R. T. Deam, R. C.
  Ball coworkers
• Related studies of networksS. Panyukov
  coworkers
• All-orders connection with percolation see also
  H.- K. Janssen O. Stenull (via random
  resistor networks)

goldbart_at_uiuc.edu w3.physics.uiuc.edu/goldbart