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@uiuc.eduw3.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 resinfrom trees

de la Condamine (Ecuador,~1740): latex from incisions inHevea tree, rebounding balls;suggests waterproof fabric,shoes, bottles, cement,…

a little more history…

Kelvin (1857): theoreticalwork on thermal effects

Joule (1859): experimental work inspired by Kelvin

Priestly erasing;coins the namerubber (4.15.1770)

Faraday (1826): analyzed chemistry ofrubber – “… much interest attaches tothis substance in consequence of itsmany peculiar and useful properties…”

and some more…

F. D. Roosevelt(1942, Special

Committee)

•“… of all critical and strategic materials…rubber presentsthe 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 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?”

but…

Dunlop (1888):invents the pneumatic tyre

…the invention of which led to“frantic efforts to increase thesupply of natural rubber in theBelgian Congo…” which led to“some of the worst crimes of managainst man…” (Morawetz, 1985)

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 randomnetwork

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 brokenspontaneously, 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 GlassesJ. Phys. F5 (1975) 965

• R. T. Deam and S. F. EdwardsTheory of Rubber ElasticityPhil. Trans. R. Soc. 280A (1976)

280

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

doesn’t distinguish between these

states

)2exp()exp(exp 22kii RkRk0kRk ,exp i

jRrandom

mean position

random r.m.s.displacement(localizationlength)

R

k

0kRk ,1

1 exp N

j jiN

N),...,2,1( Nj

}{ 2exp)( )( 222212

,||||||21

gpdQ g kkk0kkk

• Edwards-Anderson—type order parameter

choose wave vectors and

study

delocalized localized

Order parameter for random localization

0k0k0k ,,, 21 g

N

j jg

jj iiiN1

211 expexpexp RkRkRk

gkkk ,...,, 21

fraction ofloc. particles

statistical distributionof localization lengths

macroscopichomogeneity(cf. crystals)

• Distinguishes liquid & amorphous solid states

Landau theory ingredients

built from order parameter

meaning of :

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

),...,,( 10 nkkk

disorderaveraging

][1

01 expexp

N

j jn

j iiN RkRk

Landau free energy

built from (Fourier transform of) order parameter

in replicated real space

subject to physical (HRS) constraints

}{ 3!3

122

122

1 )ˆ(ˆ)ˆ(ˆ || xgxxd

0)exp()ˆ(ˆ

0)ˆ(ˆ

xkixxd

xxd

HRSkn

xkikV

x ˆ1)ˆˆexp()ˆ(

1)ˆ(

},...,,{ˆ 10 nx xxx

crosslink density control parameternonlinear coupling crit

xlcritxlxl~ NNN

• 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

),...,,( 10 nkkk

Instability and resolution

Results of mean-field theory

}{ 2exp)( )( 221202

,||||||10

npdQ n kkk0kkk

fraction ofloc. particles distrib. of loc. lengths

• Order parameter takes the form:),...,,( 10 nkkk

)/2()/4()( 23 p

}{ )( )3/(1exp1 QQ

))(()()1()/)(2/( 2 dd

• Localized fraction: control param. ε ~

excess x-link density

• Universal scaling form for the loc. length distrib.:

3/2 Q (linear neartransition)

universal scalingfunction; obeys

(plus normalization)

– localized fraction linear near the transition

Erdős-Rényi RGT form

– localization length distribution data-collapse for all near-

criticalcrosslink densities

specific universal form forscaling function

Results of mean-field theory

Q

measure of crosslink density

(scaled inverse square) loc. length

localize

d f

racti

on

Q

pro

bab

ilit

y π

• Specific predictions

nearly log-normal

• 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

Q

Mean-field theory vs. simulations

Q

•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

Symmetry and stability

• Simple principle:Free energy cost ofshear deformations?

– two contributions deformed free energy

deformed saddle point

• Emergent elastic free energy

• Shear modulus exponent?

Emergent shear elasticity

deformation hypothesis

t~modulusshear

?t

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?

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

will what’s at be localized?

how strongly?

probes cluster formation

– meaning (in solid state):

e.g. localization-length correlations

x

y

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.

quant’s

relation to percolation via the Potts model

could it be otherwise?

• All-orders connection (see also Janssen & Stenull ’01)

)6/(2)6/()2(/ dddlL

d6 volume fraction

segments per chain

3

!3

12

2

122

1 )ˆ(ˆ)ˆ(ˆ xgxxd

Beyond mean-field theory

3

!3

12

2

122

1

2

2

122

1

)()(

)(

gx

xdx

HRW percolation field theory vulcanization field theory

2

ghost field signby-hand elimination

2

HRS constraintmomentum conservation replica combinatorics replica limit

works to all orders (Peng et al,. Janssen & Stenull)

x x 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-bondedrandom network mediae.g.– regard frozen-in liquid-state

correlations as quenchedrandom constraints

– examine propertiesbetween two time-scales:structure-relaxation & bond-breaking

•Is there a separation of time-scales?

yxyx 12 SeAsGe,SiO,Si

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,…)?

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)

goldbart@uiuc.edu

w3.physics.uiuc.edu/~goldbart