Paul M. Goldbart University of Illinois at Urbana-Champaign goldbart@uiuc

1. 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.edu w3.physics.uiuc.edu/~goldbart Thanks to many collaborators, including: Nigel Goldenfeld, Annette Zippelius, Horacio Castillo, Weiqun Peng, Kostya Shakhnovich, Alan McKane

2. 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,…

3. Kelvin (1857): theoreticalwork on thermal effects a little more history… Priestlyerasing;coins 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

4. and some more… F. D. Roosevelt(1942, SpecialCommittee) “… 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

5. 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?”

6. …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) but… Dunlop (1888):invents the pneumatic tyre Conrad (1901):Heart of Darkness

7. 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

8. Vulcanized macromolecular networks • permanently crosslinked at random What is vulcanized matter? • or endlinked • Chemical gels (atoms,small molecules,…) • permanently covalentlybonded at random • Form giant randomnetwork

9. 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 Central themes

10. 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 What is amorphous solidification?

11. 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

12. 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

13. 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 random mean position random r.m.s.displacement(localizationlength)

14. Order parameter for random localization Edwards-Anderson—type 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

15. Landau theory ingredients disorder averaging 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

16. 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

17. 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

18. Order parameter takes the form: Results of mean-field theory fraction of loc. particles distrib. of loc. lengths • Localized fraction: • control param.ε~excess x-link density (linear neartransition) • Universal scaling form for the loc. length distrib.: (plus normalization) universal scaling function; obeys

19. Specific predictions Results of mean-field theory localized fraction Q measure of crosslink density 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 probability π (scaled inverse square) loc. length

20. 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

21. 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

22. Emergent shear elasticity deformation hypothesis Simple principle:Free energy cost ofshear deformations? two contributions deformed free energy deformed saddle point Emergent elastic free energy Shear modulus exponent?

23. 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?

24. 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

25. 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

26. Beyond mean-field theory segments per chain 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 exponentsfor percol. phys. quant’s relation to percolation via the Potts model could it be otherwise? All-orders connection (see also Janssen & Stenull ’01) volume fraction

27. 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)

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

29. Structural glass? Covalently-bondedrandom network mediae.g. regard frozen-in liquid-statecorrelations as quenchedrandom constraints examine propertiesbetween two time-scales:structure-relaxation & bond-breaking Is there a separation of time-scales?

30. 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,…)?

31. 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