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# Chemistry in Popular Novels CHEM 100

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Date Created: 10/29/15

PRL 102 015702 2009 PHYSICAL REVIEW LETTERS week ending 9 JANUARY 2009 Percolation Model for Slow Dynamics in GlassForming Materials Gregg Lois Jerzy Blawzdziewicz and Corey S O Hem Department of Physics Department of Mechanical Engineering Yale University New Haven Connecticut 06520 8284 USA Received 5 September 2008 published 6 January 2009 We identify a link between the glass transition and percolation of regions of mobility in con guration space We nd that many hallmarks of glassy dynamics for example stretched exponential response functions and a diverging structural relaxation time are consequences of the critical properties of mean eld percolation Speci c predictions of the percolation model include the range of possible stretching exponents 13 S B S l and the functional dependence of the structural relaxation time Ta and exponent B on temperature density and wave number DOI 101103PhysRevLett102015702 As temperature is decreased near the glass transition the structural relaxation time in glassy materials increases by many orders of magnitude with only subtle changes in static correlations 1 In addition structural correlations display an anomalous stretchediexponential time decay exptra where B is the stretching exponent and Ta the airelaxation time Understanding the origin of this behavior is one of the most important outstanding prob lems in statistical physics Although stretchediexponential relaxation is common to many glassiforming materials the dependence of Ta and B on temperature and density is not universal For molecular colloidal and polymer glasses where structural relaxation is measured using density autocorrelation functions the temperature dependence of Ta is affected by the fragility In magnetic glasses where structural relaxation is measured using spin autocorrelation functions Ta depends on details of the microscopic interactions 3 In all glassy systems the stretching exponent B varies between 13 and 1 depending on the scattering wave vector density and temperature and its dependence on these variables is not universal How do we understand structural relaxation in glass forming materials where correlation functions display stretchediexponential relaxation but the temperature and density dependence of Ta and B vary from one material to the next Contrary to approaches that focus on heterogei neous dynamics 5 and percolation 6 in real space we study how properties of energy landscapes 7 affect dy namics in con guration space In this picture activation from energy minima is rare at low temperatures and only infrequent hopping between minima allows structural cor relations to decay 8 We focus on the connection between anomalously slow dymamics in glassiforming materials and percolation of regions of mobility in con guration space The decay of structural correlations over a time t is related to the average distance that the system moves in con guration space during that time Thus complete relaxationidecay of structural correlations to zeroionly occurs after the sys7 tem can diffuse over a path that percolates con guration 00319007091021o157024 01570271 PACS numbers 6470Q 61 43Fs 64602211 space We demonstrate that a percolation transition in con guration space is responsible for several hallmarks of glassy dymamics i stretchediexponential relaxation of structural correlations and the experimentally observed range of values and waveinumber dependence of the stretching exponent B ii the form of the divergence of Ta and iii a diverging length scale near the glass 39amming transition for hard spheres Hard spheresiWe rst consider collections of hard spheres that interact at contact with an in nite repulsion At moderate density hard spheres behave as simple uids As density increases structural relaxation becomes anomi alously slow Upon further compression if crystallization is avoided the system becomes con ned to an amorphous collectively jammed 9 CI state at packing fraction 4 In CI states any single or collective particle displacement causes particle overlap thus because of the hardisphere contraints no motion is possible at 4 For large systems C states occur at a single 4 with to a 064 for monoi disperse systems in d 3 dimensions and 4 m 084 for bidisperse mixtures with d 2 1011 The transition from glass to liquid in disordered hard sphere systems can be understood as the percolation of allowed regions in con guration space that do not V107 late hardisphere constraints at d to only C states points in con guration space are allowed for d S to allowed regions do not percolate and relaxation is limited for d lt cm a percolating network of allowed regions spans con guration space and the system can fully relax Critical properties of the percolation transition do not depend on how allowed regions of con guration space are partitioned However to use the predictions of contini uum percolation it is useful to partition allowed regions into mobility domains each of which is associated with a single C state For any 4 the mobility domains can be identi ed by compressing each allowable con guration speci ed by its particle positions ri to to using the LubachevskyiStillinger algorithm 12 in the large compression rate limit The point in con guration space ri belongs to the mobility domain of the resulting C state 2009 The American Physical Society PRL 102 015702 2009 PHYSICAL REVIEW LETTERS week ending 9 JANUARY 2009 o C 39 OQQ a b o 3 6 C d FIG 1 color online Schematic of allowed regions in con guration space for hard spheres a At 1 only CI states points are allowed b At 1 lt 1 motion occurs in closed mobility domains surrounding CI states c At 2 lt 1 tran sitions between contacting mobility domains shaded occur d At Plt 2 at least one network of mobility domains percolates shaded yellow and the system transitions from glass to metastable liquid The glass transition can now be described in terms of percolation of mobility domains At 4 the system is con ned to one of many possible CI states and no motion occurs Fig 1a For gt S 4 a closed mobility domain of allowed hardisphere con gurations surrounds each C state 9 Fig 1b For smaller gt mobility domains of different C states can contact which enables the system to transition from one mobility domain to another Fig 1c The system will diffuse on a network of mo bility domains if many are in contact At lower 4 a percolating cluster of mobility domains forms Fig 1d Structural relaxation in dense hardisphere systems oc7 curs via dwamical heterogeneities 13 and thus shapes of mobility domains are complex Near C states mobility domains are roughly hyperspherical and quickly explored Further from C1 states mobility domains become lameni tary and the time needed to explore these regions via cooperative motion is large However in our calculations we assume there exists an upper critical dimension Dquot of con guration space above which meani eld theory accui rately describes critical exponents of the percolation tran7 sition In this limit the complex geometry and correlations of mobility domains can be ignored We focus on large system sizes N where the dimension of con guration space dN gt Dquot and construct a meani eld theory in terms of the packing fraction of mobility domains II in con gu7 ration space Percolation occurs at a critical value HP and is controlled by meani eld exponents 14 To derive stretchediexponential relaxation our meani eld approach does not require detailed information about the lamentary features of mobility domains We quantify structural relaxation using the incoherent part of the intermediate scattering function ISF CIgt t N712 1expi Ajt where A7Jt is the displace ment of particle j over time t and 1 is the scattering wave vector 15 The ISF is well characterized by its Gaussian approximation 16 PW 7 exp 7 qur2t2d 1 where Ar2t N711IJt7 JOI2 is the mean square displacement Note that N Ar2t is identical to the meanisquare distance traveled in con guration space after time t The in niteitime value of the ISF f E ltIgt oo is an order parameter for the glass transition that is zero for a liquid and positive for a glass The percolation model predicts a glass for II lt II p and a metastable liquid for II gt II F For II lt Hp no percolati ing cluster of mobility domains exists f q gt O and the system is a glass since the maximum distance it can diffuse in con guration space is nite set by the percolation correlation length 5 0C HP 7 H 12 In the glass state Eq 1 predicts logfq 0lt 711252 0C 7q2lIP 7 l I 1 For II gt II P at least one cluster percolates and f q 0 Since percolating clusters are fractal and cover only a small fraction of con guration space the state with fl O is not necessarily ergodic The time dependence of the ISF can be determined from dynamics in con guration space using Eq 1 The model contains two important time scales short times where the system is con ned to a single mobility domain and long times where the system diffuses on a connected cluster of mobility domains Shortitime dwamics are characterized by the average transition time To between mobility do mains while longtime relaxation of the ISF is determined by diffusion on a network of mobility domains The mean square distance Ar2t traveled on networks near the mean eld percolation transition has been studied extensively 1718 and obeys the scaling form Ar2t 0C WWI3GIE 2tTo13 2 where f 0lt H 7 TIME 2 and the scaling function Gz 0lt z2 forz gtgt 1 and Gz1forz ltlt 1 The structural relaxation time Ta and stretching expo nent B can be determined when a percolating network exists If we de ne 15001 e71 Eqs 1 and 2 give 70117201 7 Hp 2 for qf ltlt 1 739 01 0 75 7011 for qf gtgt 1 3 Note that there are two contributions to Ta 1 the average transition time between mobility domains To and 2 the time scale for diffusion on percolating networks that is proportional to H 7 II p The 1 dependence in Eq 3 is consistent with experiments on dense colloidal suspeni sions 16 and vibrated granular materials 19 Since Ar2t from Eq 2 crosses over from anomalous diffusion 0C t13 at small times to normal diffusion 0lt t at large times 01570272 PRL 102 015702 2009 PHYSICAL REVIEW LETTERS week ending 9 JANUARY 2009 Eq 1 predicts that B varies with time and satis es 13 S B S 1 In experiments B is typically measured by tting the ISF to expt7390 near Ta and thus its time depen7 dence has not been observed Near Ta Eqs 1 and 2 predict that B only depends on qf with B 13 for 15 gtgt 1 and B 1 for qf ltlt 1 These limiting values have been observed in experiments 19 Thus far we have used results from meani eld percolai tion to obtain stretchediexponential relaxation To make further connections with experiment we will determine Ta versus 4 This requires that we rst express II and To in terms of gt I and ii below and calculate the packing fraction gtP at which the glass transition occurs iii i II is equal to the number of C states NCJ multiplied by the average volume V of the mobility domains normal ized by the total volume LNd NC 0C Nle zN where a gt O is a constant 2021 V is determined by the total free volume vf accessible to the system in real space and assuming that all allowed regions including laments in con guration space are explored VlN vfLd rt 7 Thus for large N H lt NN17 JN 4 ii To 0C N51 where NC is the average number of mobility domains in contact with a single domain 22 Assuming a random distribution of hyperspherical domains 751 cc N 2mm 5 For large N To 0C 2 leI 1 0C expA gtJ gtJ 7 23 iii p can be determined from Eq 4 by solving HP 0C NN1 7 P JN Using meani eld percolation results 14 percolation in con guration space occurs when NC Z Z 7 1 where z is the maximum number of mobility domains that can contact a single domain Since z increases with N we predict Hp 27M and thus rt 7 gbp 0C N71 Note that gtP lt 4 for nite N whereas gtP 4 as N gt 00 This systemisize dependence suggests a divergi ing length scale in real space since relaxation only occurs in subsystems of size N7 W where gtP7V gt 4 then 0C rm 7 NUd A diverging length with the predicted exponent has been reported in experiments on granular media 24 Using Eq 345 the asymptotic form of Ta near gtP is 127 0C Expu fgkf p 472 for P ltlt J P a PMle for P gtgt J AP 6 for qf ltlt 1 where A and B are positive constants For N gt 00 when gtP 4 the model predicts a Vogeli Fulcher divergence at 4 In this limit the functional form and location of the divergence have been veri ed in experiments of hard spheres 1925 For nite N when gtP lt 4 there is a powerilaw divergence near gtP and VogeliFulcher behavior far from p Finite energy barriersiIn contrast to hard spheres activation is important in systems with nite energy barf riers We now extend the percolation model to include activated processes in systems at constant temperature T with k3 1 We again assume that only disordered states exist as in frustrated geometries such as the pyrochlore lattice polydisperse colloidal suspensions and metallic glasses above the critical quench rate For systems with nite energy barriers the transition from glass to metastable liquid is described by the percoi lation of bonds between local energy minima Con guration space can be decomposed into basins of attraci tion surrounding each local minimum and every point in con guration space can be mapped uniquely to a single basin 26 At short times the system is con ned to a basin whereas at long times it hops from one basin to another Complete structural relaxation occurs once the system s trajectory percolates con guration space To calculate Ta we specify the ensemble of bond percolating networks on which the system can relax and select the subset that minimizes Ta To build the ensemble we prescribe a maximum energy barrier height nT and draw bonds between minima with barriers below nT If one of the networks formed in this manner percolates it is included in the ensemble For a given value of n e fraction of bonds bn and the average time 70n to make transitions between two basins are given by W KTPAEME lt7 VLT Too cc WW I PbltEgtexpltETgtdE 8 where PbE is the distribution of energy barriers For suf ciently large N we can describe the properties of the percolating networks using a meani eld description For any n a percolating network exists if bn is larger than a critical value b P and the airelaxation time is Tonq 2bn 7 bpl z for 15 ltlt 1 Tao 0C 70001175 for qf gtgt 1 9 using arguments similar to those given for Eq 3 Note that there are a range of relaxation times 7010 depending on n For large N a saddleipoint approximation holds and the system selects the rtquot that minimizes Ta Minimizing Eq 9 for qf ltlt 1 gives bn 7 b P 7 2 bn Ce Ton 7 1 where C is the proportionality constant from Eq 8 with units of time Equation 10 can be solved to determine rtquot 27 Since Ce gt 7001 for all T gt 0 Eq 10 predicts that bn gt bp and thus no glass transition occurs for T gt O The T dependence of 7101quot depends sensitively on Pb If we assume a Gaussianishaped PbE we recover experimentally observed glassy behavior including fragile strong behavior for large small standard deviation rela tive to the average energy barrier height As for hard spheres the percolation model for nite barriers predicts that structural correlations such as the 10 01570273 PRL 102 015702 2009 PHYSICAL REVIEW LETTERS week ending 9 JANUARY 2009 ISF or spin correlations exhibit stretchediexponential ref laxation with 13 S B S 1 These limits for B agree with experimental values for many glassiforming materials 4 We predict that B increases slowly with time but if it is measured near Ta we expect B 13 for 15 gtgt 1 and B 1 for 15 ltlt 1 where f 0lt 7 bp 12 Our results for the limiting values of B are consistent with measurements at different 5 values in LennardAJones 28 and magnetic 2930 glasses and different 1 values in magnetic 31 and molecular 32 glasses Equations 7410 can be applied to hardisphere sys7 tems to determine Ta gt even far from p where details of lamentary regions of mobility domains are important if we generalize the assumption used to derive Eq 4 Instead of energy barriers hard spheres overcome entropic barriers to explore lamentary regions If a fraction f of the mobility domain hypervolume V is explored before mak ing a transition to a new mobility domain there is an average transition time 700 and a corresponding percoi lating network with bond fraction bf 0lt f H For N gt 00 the system will choose the f that minimizes Ta Equation 4 was derived assuming f 1 which holds near p For to ltlt p f ltlt 1 and only the hyperspherical regions of mobility domains are explored Glassy behavior begins when f becomes suf ciently large that lamentary re gions of mobility domains are explored We introduced a percolation model that predicts stretchediexponential relaxation in glassy materials which is commonly explained by invoking underlying heterogei neity 5 Percolating networks organize into densely con nected blobs below 5 and homogeneous nodes above 5 33 and thus have a builtiin heterogeneity that leads to lt 1 Further studies of the geometry of percolating net works and dymamics on these networks will lead to new predictions for slow dynamics and cooperative motion in glassy materials Financial support from NSF grant numbers CBETA 0348175 G L IE and DMR70448838 GL C S 0 and Yale s Institute for Nanoscience and Quantum Engineering GL is acknowledged We also thank the Aspen Center for Physics where this work was performed H MD Ediger CA Angell and SR Nagel J Phys Chem 100 13200 1996 2 CA Angell I Non Cryst Solids 131 133 13 1991 M Alba I Hammann M Ocio P Refregier and H Bouchiat J Appl Phys 61 3683 1987 K Gunnarsson P Svedlindh P Nordblad L Lundgren H Aruga and A Ito Phys Rev Lett 61 754 1988 4 R Bohmer KL Ngai CA Angell and DJ Plazek J Chem Phys 99 4201 1993 5 RG Palmer DL Stein E Abrahams and PW Anderson Phys Rev Lett 53 958 1984 C Monthus and J P Bouchaud J Phys A 29 3847 1996 H Sil lescu I Non Cryst Solids 243 81 1999 IS Langer E E E S 11 12 13 14 15 16 17 18 19 20 21 22 23 25 26 27 28 29 30 31 32 33 01570274 and S Mukhopadhyay Phys Rev E 77 061505 2008 MH Cohen and GS Grest Phys Rev B 24 4091 1981 The energy landscape is the hypersurface of the energy as a function of all con gurational degrees of freedom ER Weeks and DA Weitz Chem Phys 284 361 2002 B Doliwa and A Heuer Phys Rev E 67 030501R 2003 A Donev S Torquato F H Stillinger and R Connelly J Appl Phys 95 989 2004 CS O Hern LE Silbert A Liu and SR Nagel Phys Rev E 68 011306 2003 At nite N there is a distribution of jamming packing fractions which becomes a 5 function at 1 when N gt oo The percolation model can be generalized to incor porate a Gaussian distribution of jamming packing frac tions and the predictions do not change BD Lubachevsky and EH Stillinger J Stat Phys 60 561 1990 WK Kegel and A van Blaaderen Science 287 290 2000 B Doliwa and A Heuer Phys Rev E 61 6898 2000 D Stauffer Introduction to Percolation Theory Taylor and Francis London W Kob and H C Andersen Phys Rev E 52 4134 1995 W van Megen TC Mortensen SR illiams and J Muller Phys Rev E 58 6073 1998 Y Gefen A Aharony and S Alexander Phys Rev Lett 50 77 1983 T Nakayama K Yakubo and RL Orbach Rev Mod Phys 66 381 1994 PM Reis RA Inga1e and MD Shattuck Phys Rev Lett 98 188301 2007 N Xu J Blawzdziewicz and C S O Hern Phys Rev E 71 061306 2005 Since each particle is distinguishable we include the N1 This gives the time required to nd a contacting domain which is much larger than other time scales eg the time to diffuse across a mobility domain For large N a Taylor expansion yields expA J A 1NN J gt N 2 dNH 1 O Dauchot G Marty and G Biroli Phys Rev Lett 95 265701 2005 Z Cheng J Zhu PM Chaikin SE Phan and WB Russel Phys Rev E 65 041405 2002 EH Stillinger and TA Weber Phys Rev A 25 978 2 Equation 10 also follows trivially if we assume there is a probability to form bonds between minima with energy barriers E lt nT by substituting Pl gt fEgtPtEgt L Angelani G Parisi G Ruocco and G Viliani Phys Rev Lett 81 4648 1998 AT Ogielski Phys Rev B 32 7384 1985 IMD Coey DH Ryan and R Buder Phys Rev Lett 58 385 1987 IA Campbell and L Bemardi Phys Rev B 50 12643 1994 M T Cicerone FR Blackburn and MD Ediger J Chem Phys 102 471 1995 F Sciortino P Gallo P Tartaglia and S H Chen Phys Rev E 54 6331 1996 A Coniglio J Phys A 15 3829 1982

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