Dynamic percolation and Slow Relaxation in Glass-like Materials Alexei Va´zquez and Oscar Sotolongo-Costa Department of Theoretical Physics. Havana University. Havana 10400, Cuba. (February 7, 2008) Glass-like materials are nonequilibrium systems where the relaxation time may exceed reason- abletimescales ofobservations. Inthepresentpaperadynamicpercolation modelisintroducedin order to explain the principal properties of glass-like materials near the dynamic transition. Here, contrary to conventional percolation problems, clusters are groups of particles dynamically corre- lated. Introducing a size dependent relaxation time and the scaling hypothesis for the distribution of dynamically correlated clusters the two step relaxation predicted by the mode coupling theory and observed in experimentsis obtained. 8 9 61.20.Lc 9 1 I. INTRODUCTION otherhandthetimescaleoftheβ processisconsiderably n shorter, does not diverge at T , and the β process can a c J therefore also be observed below Tc. These predictions Incommonsecondordertransitionsthenewphasehas are directly accessible to experimental verifications and 3 1 asdistinguishpropertytheexistenceoflongrangeorder, havebeentestedformanyglassformingmaterials11,13–21. leading to macroscopic structures like ferromagnetic do- Recently we have presented a fractal model which ex- 1 mains. However, while it is believed that glass-like tran- plains the hysteresis loss effect observed in supercooled v sitions are second order transitions, the glass-like state liquids7. The determinant influence of the fractal prop- 3 does not exhibit the conventional long range order but erties of the amorphous structure in the system dynam- 2 a short range order, at a mesoscopic scale, is preferred. ics was emphasized introducing a fractal distribution of 1 The relaxation time of these systems becomes as long cluster sizes. However, in that occasion, the nature and 1 0 as, or longer than, the observation time and, therefore, origin of this fractal clusters was not analyzed. 8 the system cannot reach equilibrium within observation Inthe presentpaperacluster modelforthe relaxation 9 time. Thus, it is not clear if the glass-like transition in glass-like materials is proposed following the general t/ is a true thermodynamic phase transition or a slow re- ideas presented in7. As in previous works the relaxation a laxation phenomena, usually referred to as dynamic or of the system is obtained by superposition of indepen- m freezing transition1. dent relaxing clusters with a size dependent relaxation - A major step toward resolving the dilemma would be time. Nevertheless, in our model, differing from other d the determination of an order parameter underlying the formulations, a cluster exists through a series of relax- n o transition. Thediscoveryofadiverginglengthassociated ationprocessgivingrisetoadynamicallycorrelatedclus- c with the glass-liketransition, similar perhaps to the cor- ter(DCC).Inthiswaytheslowrelaxationdynamicsand : relation length in a para-ferromagnetic transition would other features of glass-like materials are explained. Re- v i enhance our understanding of glass-like transitions and sults are compared with experimental data reported in X oftheglassystate. Fromthestructuralpointofviewthe the literature. r glassystateischaracterizedbysomelocal(quenched)dis- Thepaperis organizedasfollows. Insection2the for- a order and frustrated bonds which inhibit the long range mationofDCCisanalyzed. Themainideasandconcepts orderandleadto the formationofclustersofmesoscopic about DCC and cluster dynamics are introduced. Then, sizes. Thus,anytheorywhichtrytoexplaintheprincipal the influence of the distribution of DCC on relaxation featuresofglass-likematerialsmusttakeintoaccountthe dynamics is investigated. It is obtained that the relax- existence of such clusters. In this direction several mod- ation follows a two step process which may be identified els have been proposed2–7. with the α and β processes. Comparison with experi- During the past few years a new theoretical ap- mentaldataandMCTpredictionsispresentedinsection proach, the mode coupling theory (MCT), has been 3. Finally the conclusions are given in section 4. developed which has motivated new experimental investigations8–10. This theory provides a generalized kinetic equation approach to the density fluctuation dy- II. DYNAMIC PERCOLATION MODEL namics in supercooled fluids. It predicts two step re- laxation processes (α and β relaxations). The α relax- A. DCC relaxation time ation process exhibits a time scale which diverges at a crossover temperature T , which is somewhat above the c calorimetric glass transition temperature T , and struc- Molecular dynamics simulations of soft-sphere g tural relaxation experiences rapid slowing down. On the mixtures22 haverevealedthatinadditiontothe stochas- 1 ticjumpmotionofsingleatomscorrelatedjumpmotions, coefficient D = δ2/τ0. In this mean field approach the in which a cluster of several atoms jump at successive normalized relaxation is given by27 closed times by permuting their positions, take place. x(t) t Similarresultshavebeenobtainedinmoleculardynamics φ (t)= h i =exp , (5) s simulations of water, a frustrated system with multiple x(0) (cid:16)− τs(cid:17) h i random hydrogen bond network structures23. Besides, MonteCarlosimulations of spin glasses24,25 also revealed with the size dependent relaxation time the existence of such cooperative relaxation dynamics. kT 1 These results suggest that the cooperative motion of τs = =τ0s . (6) D c particles (atoms, spins, etc.) in glass-like materials does nottakeplaceasthemotionoffrozenclustersbutassuc- Thus, in this mean field approximation the relaxation cessive concatenations of single particle motions. Hence, time of a DCC is proportional to its size and a pertur- in this sense, a cluster is formed through the perturba- bation of the equilibrium state relaxes exponentially to- tionpropagationfromoneparticletoits neighbors. This wards equilibrium. As will be seen, the existence of a kindofclusterwillbereferredhereasadynamicallycor- broad distribution modifies the relaxation rate. relatedcluster(DCC).Forshorttimesonlysmallclusters participate in the relaxation dynamics however, with in- creasingtime,largerandlargerclustersareincorporated. B. Relaxation dynamics Let us denote a characteristic physical quantity of the i-th DCC particle by x . It may be the displacements Letn betheclusternumbers,i.e. thenumberofDCC i s around an equilibrium position in ordinary glasses, the with size s per unit lattice. We assume that, in spite of magnetic moment in a spin glass, or other. x will be the dynamic nature of the clusters, the scaling assump- i randomvariablefollowingcertaindistribution, with zero tion about the cluster number still holds. Therefore, if mean value and variance x2 = δ2, δ being a typical the systemis close to percolationǫ=(1 T/T ) 0 the valueoffluctuations. Morehovieir,letτ0 be acharacteristic scaling assumption establishes that28 − c ∼ time scale of the fluctuations in x . The different DCC configurationis will be then charac- ns =s−τf[ǫsσ] , (7) | | terized by the sum where τ and σ are scaling exponents and f(x) is a cutoff s function (f(z 1) 1, and f(z 1) 1). Thus, the x= xi , (1) averagednorm≪alized≈decay will be≫given≪by Xi=1 φ(t)= sn φ (t) sn . (8) where s is the number of particles in the cluster, 1 s s s ≥ X .X s < . In general the fluctuations x are correlated i but,i∞nafirstapproximation,they maybeconsideredin- Close to percolationthe sum is dominated by the con- dependent. In such an approximation the central limit tribution oflarge cluster andwe may substitute the sum theorem26 tells us that, for large s, P (x) the limit dis- by an integral. Thus, substituing equations (5), (6) and s tribution of the sum in equation (1) follows a Gaussian (7) in equation (8), with the variable change z =t/τs, it distribution with variance δs1/2, i.e. is obtained P (x)= 1 exp x2 . (2) φ(t)= 0t/τ0dzza−1e−zf[(t/τǫ)σz−σ] , (9) s √2δ2s (cid:18)− 2δ2s(cid:19) R t/τ0dzza−1f[(t/τ )σz−σ] 0 ǫ R conPfisg(xu)rarteipornesseanstssoctihaeterdeltaotivaevnaulumeboefrxo,ffmoricarocsltuastteesroorf where τǫ =τ0|ǫ|−1/σ and a=τ −2 Thefirstthingwenotefromthisexpressionistheexis- size s. The entropy of this ’temporally isolated’ subsys- tem is thus given by tenceoftwocharacteristictimesτ0 andτǫ. Theformeris acharacteristictimeforthemicroscopicmotionwhilethe k last one is the characteristic time of the largest clusters Ss(x)=klnPs(x)=−2δ2sx2+const. , (3) with size ǫ−1/σ. ∼ It is tentative to associate these characteristic times and the force associated to an entropy change with the β and α relaxation processes, respectively. Moreover, the distribution of cluster sizes is divided in ∂S (x) kT s f (x)=T = x= cx . (4) two regions. The initial part which follows a power law s ∂x −δ2s − decay and a second part with a sharp cutoff. The cutoff Now, from the kinetic point of view, the relaxation of size is of the order of ǫ−1/σ with characteristic time τ . ǫ the DCC may be reduced to the Brownian motion of a Hence, τ divides the time scaleintwo regionswhere the ǫ harmonic oscillator with generalized coordinate x, un- relaxation function will show different behaviors. These der the elastic force in equation (4) and with diffusion two time scales are analyzed below. 2 1. β relaxation with For t τǫ equation (9) is approximated by τǫ± =cσ,θ±τǫ , (17) ≪ t −a t φ(t) a γ ,a , (10) σθ± ≈ (cid:16)τ0(cid:17) (cid:16)τ0 (cid:17) β± = . (18) 1+σθ± where γ(x,a) is the incomplete Gamma Euler function. c isaconstantwhichdoesnotdependonǫand,there- Moreover,fort τ0,usingtheseriesexpansionofγ(x,a) σ,θ± for small x, we≪obtain fore, τǫ± diverges as τǫ near Tc. Equation(16)isthewellknownKolraushstretchedex- a t a t ponential often used to fit the experimental data in the φ(t) 1 exp , (11) ≈ − 1+aτ0 ≈ (cid:16)− 1+aτ0(cid:17) α relaxation process. Moreover, the characteristic time τǫ± τǫ diverges at T = Tc as it is expected for the α ∝ while for t τ0, γ(t/τ0,a) Γ(a), resulting relaxation process. ≫ ≈ t −a φ(t) Γ(1+a) . (12) ≈ (cid:16)τ0(cid:17) 3. Static viscosity Thus, the initial decay, due to single particle motions, is The static viscosity is related to the mean cluster size exponential. However, for t τ0 cooperative dynamics ≫ through the expression leads to the formation of DCC, with power law distribu- tion sizes, giving the power tail in equation (12). The fractalbehaviorinthe distributionofDCC sizesleadsto η =Ghτsi=Gτ0hsi , (19) fractal properties in the time decay. where G is an elasticity modulus and s is given by The imaginary part of the susceptibility χ′′ is related h i to the relaxation function through the equation s = s2n sn . (20) s s ′′ ′ h i X .X χ (ω)=ωFT [φ(t)] , (13) > For T T the mean cluster size defined in equation whereFT′ denotestherealpartoftheFouriertransform. (20) will∼divecrge according to s ǫ−1/σ28 and, there- h i∼| | Then, for τ0 t τǫ, using equation(12)itis obtained fore, ≪ ≪ χ′′(ω)≈asec(cid:16)aπ2(cid:17)(ωτ0)a . (14) η ∼τǫ ∼τǫ± ∼hsi∼|ǫ|−1/σ . (21) Thisfrequencydependencecorrespondswiththeβ relax- Hence, τǫ± the characteristictime for the α relaxation process shows the same temperature dependence of the ation. The characteristic time for these time scale is τ0 static viscosity. which does not depend on ǫ. Hence, if we assume that the divergences near T are given by the percolative na- c ture of the DCC, τ0 will not diverge at T = Tc as it is III. DISCUSSION expected for the β process. Thepresentmodelexhibitsthetwosteprelaxationpre- 2. α relaxation dictedbytheMCT8–10. Theexponentaoftheβ process (χ′′ ωa) is identified here with a = τ 2. More- ∼ − over, since 2 τ 5/2 as it is obtained in conventional Nowletusinvestigatetheinfluenceofthecutoffinthe percolation28≤then≤0 a 1/2, which is in the range distribution of cluster sizes in the long time (t ≫τǫ) re- predictedbythe MCT≤.On≤the otherhand,the Kolraush laxation dynamics. In this case the form of the cutoff stretched exponent of the α process, given by equation function f(x) is determinant and cannot be rulled out. (18), satisfies 0 β± < 1 which is also the range pre- We will assume that this cutoff function is of the form ≤ dicted by the MCT. Moreover,the characteristictime of f(x)=exp( xθ±) , (15) the α process has the same temperature dependence as − the static viscosity, in agreement with the MCT. where θ− and θ+ stand for ǫ<0 and ǫ>0, respectively. Thus,ourmodelseemstoarrivetothesameresultsob- Anasymptoticexpansion,fort τ ,ofφ(t)definedin tainedbytheMCT,whichhavebeenextensivelytestedin equation(9)withf(x)inequation≫(15ǫ)givetheestimate lightscatteringandneutronscatteringexperiments11–21. In order to perform a more accurate comparison let us ′′ t β± investigate the behavior of χ around the minimum be- φ(t) exp . (16) ∼ h−(cid:16)τǫ±(cid:17) i tween the α peak and the β process. This minimum 3 should be near the frequency 1/τǫ since it marks the model with β− and β+, respectively. The main contri- transition from the β to the α process in our model. In bution to this temperature dependence is given by θ± this frequency rangewe will have the contributionofthe in equation (18). For instance, in conventional percola- β process with relax according to (t/τ0)−a and of the tion theory28 σθ− = 1 (β− = 1/2) and σθ+ = 1 1/d − α process relaxing as exp[ (t/τǫ±)β±] 1 (t/τǫ±)β±. (β+ = (d 1)/(2d 1)), where d is the space dimen- − ≈ − − − Thus, a gross estimate of the relaxation function in the sionality, and therefore β− > β+. While this behavior timescalewillbegivenbyconsideringbothcontributions is in agreement with the experimental observations the which, resulting values of β±, using conventional percolation exponents, aresmallerthanthoseobservedinexperiments. Thus,we φ(t) ǫa/σ t −a t b , (22) foundagainthatconventionalpercolationtheorycannot ∼| | h(cid:16)τǫ±(cid:17) −(cid:16)τǫ±(cid:17) i give a precise quantitative description of the properties ofglass-likematerials. However,the scalingassumptions where b = β± a. Substituting this expression in equa- are still valid, but with different scaling exponents. To − tion (13) and assuming a,b 1 it is obtained obtaina goodagreementwith the experimentalobserva- ≪ tionsσθ± shouldtakelargervalueswhichimpliesthat,in χ′′(ω) ǫa/σ a(ωτǫ±)a+b(ωτǫ±)−b , (23) glass-forming materials, the cutoff for larger DCC sizes ∼| | h i is stronger than the one expected in conventionalperco- This expression constitutes an interpolation formula of- lation. ten used to fit the experimental data. The values of a This sharp cutoff for large sizes may be attributed to and b obtained from the fit to experimental data are in frustration effects which destroy the long range order. generalsmall, thus our suppositiona,b 1 is consistent The existence of frustration, which is an inherent prop- ≪ with experiments. From this expression we obtain that erty of glass-like materials, may lead to different critical ′′ thepositionoftheminimumωmin andthevalueofχ (ω) exponents. The percolation problem including frustra- at the minimum χ′′ satisfies tioneffects (frustratedpercolation)is a subjectof recent min study, and exhibits properties which are different from ωmin ǫ1/σ , (24) the corresponding unfrustrated case32. ∼| | Finally we want to mention that other authors have used the idea of DCC, for instance the models by Domb χ′′ ǫa/σ . (25) et al2, Cohen et al3, and Chamberlin4. However, these min ∼| | models does not analyze the properties of glass-like sys- Ifa/σ =1/2thenweobtaintheMCTpredictionwhich tems around the dynamic transition, which is the main is consistent with the light scattering data13,15,18–21. contribution of the present work. However, under this assumption the static viscosity in equation(21)willdivergenearT accordingtoη ǫ−γ, c ∼| | with γ = 1/2a, while the MCT predicts γ = 1/(2a)+ IV. CONCLUSIONS 1/(2b). Thus, when a/σ = 1/2 we obtain the MCT pre- dictions, except for the exponent γ. Our result is more generalsincetheratioa/σ isnotnecessarilyrestrictedto The complex dynamics of glass-forming materials was this value. reduced to the formation of independent dynamic cor- The value of a obtained from the fit to light scatter- related clusters (DCC) with a size dependent relaxation ing data for different glass-forming materials is found time. The relaxation of the system was obtained as a temperature independent and around 0.3. Therefore, superposition of exponential relaxations over the distri- τ =2+a 2.3isslightlylargerthantheoneobtainedin bution of the DCC sizes. In this way we have obtained percolatio∼n theory in three dimensions τ 2.1531. Be- the two step relaxation process observed in supercooled ≈ sides, as was shown above, a/σ =1/2 in order to obtain liquids. a good agreemen with the MCT and experiments. In Ourmodelcontainsasa particularcase the MCT pre- the present model a/σ = (τ 2)/σ is the critical expo- dictions,itismoresimpleanditisbasedinaverygeneral nentβ ofconventionalpercola−tion28whichisalsoslightly principle, the scaling assumption for the distribution of smaller (0.4) in three dimensions31. These results sug- DCC near T . Besides, the present model gives the cor- c gest that the scaling exponent τ is larger than the one rect temperature dependence for the Kolraush exponent obtained in conventionalpercolationtheory. The scaling β, while the MCT predicts a constant value in disagree- assumption holds but the power decay is stronger. ment with the experimental bahavior. On the other hand, the MCT predicts that the expo- The comparison of our results with the experimental nentβ oftheα-processdoesnotdependontemperature. dataforglass-materialssuggestthatthepercolationcrit- However,itisfoundtobetemperaturedependent13,15,12. icalexponentτ shouldbelargerthantheoneobtainedin It takes a constant value above T and then decreases conventionalpercolationtheory. Moreover,thecutofffor c near T , taking again a constant value below T . This larger DCC sizes should also be stronger. These results c c high and low temperature limits are identified in our were attributted to the existence of frustration effects, 4 which inhibit the formationof long range order dynamic 14D. L. Sidebottom, R. bergan, L. B´’orjesson, and L. M. structures. Torell, Phys. Rev.Lett. 68, 3587 (1992). 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