Direct Estimate of the Static Length-Scale Accompanying the Glass Transition
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] A p r Direct Estimate of the Static Length-Scale Accompanying the Glass Transition
Smarajit Karmakar, Edan Lerner, and Itamar Procaccia
Dept. of Chemical Physics, The Weizmann Institute of Sceince, Rehovot 76100, Israel
Characterizing the glass state remains elusive since its distinction from a liquid state is not obvious.Glasses are liquids whose viscosity has increased so much that they cannot flow. Accordingly therehave been many attempts to define a static length-scale associated with the dramatic slowing downof supercooled liquid with decreasing temperature. Here we present a simple method to extract thedesired length-scale which is highly accessible both for experiments and for numerical simulations.The fundamental idea is that low lying vibrational frequencies come in two types, those related toelastic response and those determined by plastic instabilities. The minimal observed frequency isdetermined by one or the other, crossing at a typical length-scale which is growing with the approachof the glass transition. This length-scale characterizes the correlated disorder in the system, whereon longer length-scales the details of the disorder become irrelevant, dominated by the Debye modelof elastic modes.
The phenomenon of the glass transition in which super-cooled liquids exhibit a dramatic slowdown in their dy-namics upon cooling is attracting a tremendous amountof effort. One of the thorny issues has to do with at-tempts to define and measure a static length-scale thatwill grow upon the approach to the glass transition. Sev-eral such attempts were published in the recent litera-ture, using delicate measures of higher-order correlationfunctions[1, 2], of the effects of boundary conditions, [3],point to set correlations [4], the scaling of the non affinedisplacement field [5] and patch correlation scale [6]. Inthis Letter we propose a different approach which yieldsa very natural static length that appears to fit the bill.An advantage of this approach is that it can be imme-diately and easily applied to both experimental [10] andnumerical data (see below).Our starting point is the fact that at lower frequencythe density of state (DOS) for a disorder system reflectsthe access of plastic modes which are not exhibited bythe density of states of the corresponding crystalline ma-terials [11, 12]. This excess of modes is sometime referredto as the ‘Boson Peak’ [13]. It is most natural to discussthe ‘modes’ in terms of the eigenfunctions of the Hessian H ij ≡ ∂ U ( r , r , . . . r N ) ∂ r i ∂ r j , (1)where U ( r , r , . . . r N ) is the total energy of the systemconsisting of N particles, and { r k } Nk =1 are their coordi-nates. Being real and symmetric, the Hessian is diagonal-izable, and in all discussions below we refer to this matrixwith the Goldstone modes (zero modes due to symme-tries) being pruned out. Recently we had progress inunderstanding the analytic form of the density of statesof generic glassy systems [14]. The eigenvalues appearin two distinct families, one corresponding to the Debyemodel of an elastic body, while the density of access plas-tic modes could be approximated as f pl ( λ ) = B ( T ) λ θ , for λ → , (2)where the pre-factor B ( T ) being strongly dependent on temperature and the exponent θ being weakly dependenton temperature. This dependence is a partial measureof the degree of disorder which grows with temperature,or for quenched systems, is partially controlled by thequench protocol from the melt to the glass. Togetherwith the standard Debye contribution one can approxi-mate the low-frequency tail of the density of states withthe following approximation P ( λ ) ≃ A (cid:18) λλ D (cid:19) d − + B ( T ) f pl (cid:18) λλ D (cid:19) . (3)The first term , which is the Debye approximation ofthe DOS for a continuum elastic medium, contains λ D ≃ µρ /d − , which is the Debye frequency and µ is the shearmodulus. For the sake of generality we do not assumehere any form for the distribution f pl , but we will seebellow that the form (2) will reappear naturally in theanalysis below.The main idea which will flash out the static typicalscale is that the minimal eigenvalue λ min observed in asystem of N particles will be determined by the first or the second term in Eq. 3. There will be a value of N where a cross-over will occur, such that for larger systemthe Debye form determines λ min whereas for small N theplastic density of states determines λ min . This cross-overwill be interpreted below in terms of a typical length-scaleseparating correlated disorder from asymptotic elasticity.To see the cross over integrate Eq. (3) from zero to h λ min i . Since every quenched system is random, everyrealization will have a somewhat different value of λ min ,and we therefore consider an ensemble average over manyrealization of system of the same number of particles N quenched with the same protocol. N Z D λ min λ D E P (cid:18) λλ D (cid:19) d (cid:18) λλ D (cid:19) = 1 . (4) −1 N < λ m i n > −3 −2 N < λ m i n > / < λ D > T = 0.800T = 0.650T = 0.600T = 0.540T = 0.520T = 0.500 ~1/N
FIG. 1: Left Panel : Minimal eigenvalue calculated for a model glass former interacting via a pure repulsive pair-wise potentialplotted as a function of system size. Right Panel: Minimal eigenvalue rescaled by the high frequency shear modulus ( which isproportional to the square of the Debye frequency) plotted as a function of system size. Notice that at larger system size therescaled eigenvalue becomes independent of temperature indicating the crossover to the continuum elasticity limit.
Introducing Eq. (3) into the integral we find Ad (cid:28) λ min λ D (cid:29) d/ + B ( T ) Z D λ min λ D E f pl ( λ ) dλ = 1 N . (5)Defining G (cid:18)(cid:28) λ min λ D (cid:29)(cid:19) ≡ Z D λ min λ D E f pl ( λ ) dλ , (6)we end up with G (cid:18)(cid:28) λ min λ D (cid:29)(cid:19) = " B ( T ) N − Ad (cid:28) λ min λ D (cid:29) d/ ! . (7)Changing slightly this equation, writing N = ρV , andrenormalizing the coefficient A accordingly, we write G (cid:18)(cid:28) λ min λ D (cid:29)(cid:19) = " ρB ( T ) V − ˜ Ad (cid:28) λ min λ D (cid:29) d/ ! . (8)Finally, since the function G is monotonically increasing, D λ min λ D E can be written as a scaling function of the form (cid:28) λ min λ D (cid:29) = F " ξ d ( T ) V − ˜ Ad (cid:28) λ min λ D (cid:29) d/ ! . (9)where F ≡ G − and ξ d ( T ) ≡ ρB ( T ) . The typical scale ξ ( T ) will be calculated by demanding that all the datacalculated for different system sizes and temperaturesshould collapse into a master curve just by appropriatelychoosing the ξ ( T ).The analysis is then done as follows. Two typicalglass formers were considered, both obtained from a bi-nary mixture, one with a purely repulsive potential and the other with repulsive and attractive contribution (theKob-Andersen model). The details of the potentials canbe found in [7, 8]. The systems were equilibrated at sometemperature T > T g and then instantly quenched by di-rect energy minimization to the nearest local minimumof the energy landscape. At this state the Hessian wascomputed and the minimal eigenvalue was obtained usingthe Lanczos algorithm [9]. For a given system size N andtemperature T this procedure was repeated to have anaverage h λ min i until convergence was achieved. At thispoint the temperature or the system size were changedand the procedure was repeated, to eventually have a ta-ble of h λ min i ( N, T ). The density was fixed for every typeof potential, (0.85 for pure repulsion in 2 D , 0.82 for purerepulsion in 3 D , and 1.20 for the Kob-Andersen model).In the left Panel of Fig 1, we have plotted the minimaleigenvalue h λ min i of the two-dimensional purely repulsivesuper-cooled system as a function of system size N for dif-ferent temperatures. In the right Panel the same minimaleigenvalue was rescaled by the characteristic Debye value λ D and plotted as a function of system size. Note that atlarger system size they become independent of temper-ature consistent with the fact that at larger system sizeone recovers trivial scaling predicted by the continuumelasticity theory.In Fig. 2, we re-plotted the same data of Fig. 1 accord-ing to the scaling ansatz Eq. 9. The left panel exhibitsthe results without rescaling by ξ ( T ). In the right panelwe extracted the disorder length scale ξ ( T ) by collapsingthe data. The resulting data collapse into a straight lineindicates that Eq. (2) is obeyed to high precision. Thecollapse itself supports the scaling ansatz Eq. 9. In theinset at the right panel we show how the typical scale in-creases when the glass transition is approached. In Fig, 3 −5 −4 −3 −2 −3 −2 −1 ξ d [1/V − (Ad/2)(< λ min /< µ >) d/2 >] < λ m i n > / < λ D > −5 −4 −3 −3 −2 λ min /< µ >) d/2 > < λ m i n > / < λ D > T = 0.800T = 0.650T = 0.600T = 0.540T = 0.520T = 0.500 T ξ / ξ T µ θ = 2/3 FIG. 2: Scaling of the minimal eigenvalue calculated for amodel glass former interacting via a pure repulsive pair-wisepotential in two dimensions. here and in all the figures below ξ was chosen as ξ = ξ = 1 at the highest available temperature. −6 −5 −4 −3 −2 −2 −1 λ min /< µ >) d/2 > < λ m i n > / < λ D > −5 −4 −3 −2 −2 −1 ξ d [1/V − (Ad/2)(< λ min >/< µ >) d/2 ] < λ m i n > / < λ D > T = 0.800T = 0.650T = 0.600T = 0.560T = 0.540T = 0.520 T ξ / ξ T µ θ = 3/2 FIG. 3: The measured minimal eigenvalue as a function ofsystem size for a binary system interacting via a pure repul-sive potential in 3 dimension. Every data point represents anaverage over 2000 Inherent structures. and 4 we show the same the same analysis for the purelyrepulsive glass in 3D, a and for the Kob-Andersen modelin 3D. Both the data collapse and the resulting increasein the typical scale appear very encouraging and supportthe approach proposed above. Note that the approx-imate linearity in the collapsed scaling function in theright panel of Fig. 3 again indicates the relative accu- −5 −4 −3 −1 λ min /< µ >) d/2 > < λ m i n > / < λ D > −7 −6 −5 −4 −3 −2 −1 ξ d [1/V − (Ad/2)(< λ min >/< µ >) d/2 ] < λ m i n > / < λ D > T ξ / ξ T µ T = 1.000T = 0.700T = 0.600T = 0.550T = 0.500T = 0.470T = 0.450T = 0.430 θ =1.0 FIG. 4: The measured minimal eigenvalue as a function ofsystem size for a binary system interacting via Kob-Andersenpotential in 3 dimension. Every data point represents an av-erage over 2000 Inherent structures.
100 200 300 400 500 600 7000.350.40.450.50.550.60.650.7 N S c
100 200 300 400 5000.750.80.850.90.9511.05N/ ξ d S c / S c ( ∞ ) T = 0.700T = 0.600T = 0.550T = 0.500T = 0.470T = 0.450 T S c ( ∞ ) FIG. 5: The scaling of configurational entropy with the lengthscale extracted from the finite size scaling of minimal eigen-value for the Kob-Anderson Model. The data is taken from[2, 15] . racy of the functional form (2) for the density of plasticmodes. We have written the value of appropriate expo-nent θ in Eq. (2) directly into the graphics. In the rightpanel of Fig. 4 we see some curvature that indicates thathigher order terms in the density of states already playsome role. Nevertheless the data collapse is superb.Finally, it is interesting to ask how our typical scalehelps in understanding other measures of disorder thatwere proposed in the past. As an example we con-sider here the configurational entropy S c ( T ) of the Kob-Andersen model in 3D. This entropy was computed inRefs. [2, 15] and the reader is referred to these publica-tions for a full description of the method and the results.The configurational entropy is expected to characterizethe degree of disorder in the supercooled liquid. Once S c ( T ) was computed, it was discovered phenomenolog-ically that the data at different temperatures could becollapsed by rescaling the system size by a typical scale ℓ ( T ). It remains however mysterious what that length-scale might be, and how to measure it independently ofthe configurational entropy. At this point we can offera resolution of that mystery. In Fig. 5 we show, in theleft panel, the configurational entropy measured at differ-ent temperatures as a function of the system size. Sincethe number of accessible minima reduces as a functionof temperature, (high energy ’states’ are excluded whenthe temperature decreases), the configurational entropygoes down as seen in the left panel of Fig. 5. In the rightpanel we exhibit S C ( T ) /S C ( N → ∞ ) as a function of there-scaled system size N/ξ ( T ). The rescaling is achievedusing our data for ξ ( T ) as shown in the right panel ofFig. 4 without any re-fitting. The data collapse showsthat the mysterious ℓ ( T ) is precisely our typical scale aspropose in this Letter.We should stress that the estimate of our typical scaleis easily achieved also in appropriate experiments. It wasshown in Ref. [10] that the density of states can be calcu-lated in experiments on colloidal glassy systems, and in ξ ψ /K B T τ α
3d Repulsive : ψ = 12d Repulsive : ψ = 1 3d KaLJ : ψ = 2 FIG. 6: The logarithm of the relaxation time τ α ( T ) measuredfor all the three models discussed above, plotted against ξ ψ /T .The straight lines are a guide to the eye. particular λ min is available. We propose that such mea-surements should be repeated as a function of system sizeat different values of the packing fraction. Such data canbe used as explained above to determined the dependenceof our typical scale on the packing fraction, throwing newlight on the interesting physics of these complex systems.Finally, it is impossible to end this Letter without ask-ing how the newly found length-scale correlates with thedramatic slowing down that is observed in our super-cooled liquids as the temperature decreases. If we takethe point of view that the typical size of relaxation eventsalso depends on ξ , either proportional to ξ if they arestringy in nature or on ξ if they are planar, then weexpect the relaxation time τ α to be of the order of τ α ∝ exp [ Cξ ψ /k B T ] , (10)where C is some unknown scale and ψ is 1 or 2 dependingon the geometry of the relaxation events. We see that Eq.(10) is validated with ψ = 1 for the repulsive model inboth 2D and 3D, and with ψ = 2 for the Kob-Andersenmodel. It is not known at this point why indeed ψ maydiffer when the attractive part of the potential is added,and we must leave this issue for further research in thefuture, and see Ref. [16, 17] for some comments on thisissue.In summary, we have proposed here a very simplemethod to extract the typical scale that separates a dis-order dominated regime from an elastic dominated be-havior. All that is needed is the measurement of theminimal eigenvalue of the Hessian matrix (or, equiva-lently, the minimal harmonic frequency of the system),for systems of different size and temperature. A simple re-plotting procedure of the data is then used to extractthe typical scale. This scale appears to properly collapsethe data of the configurational entropy, resolving a rid-dle that existed for some time regarding the nature of thelength-scale that does it. Finally, and only in a manner ofpassing, we also considered the relation of the obtainedlength-scale to the relaxation time of the super-cooledliquids, and found a strong indication that this lengthscale also determines the observed dynamics. We trustthat the newly proposed length-scale would be computedin further numerical and laboratory experiments by othergroups to enhance the understanding of the glass tran-sition. In particular it would be useful to compare thislength-scale to other length-scales that were proposed byother groups as mentioned in the introduction.This work had been supported in part by an ERC“ideas” grant, the Israel Science Foundation and by theGerman Israeli Foundation. A discussion with Satya Ma-jumdar is acknowledged. [1] L. Berthier, G. Biroli, J.-P. Bouchaud, L. Cipilletti, D.El Masri, D. L’Hˆote, F. Ladieu and M. Pierno, Science , 1797 (2005).[2] S. Karmakar, C. Dasgupta, and S. Sastry, Proc. Nat.Acad. Sci. (USA) 106, 3675 (2009).[3] G. Biroli, J.-P. Bouchaud, A. Cavagna, T.S. Grigera andP. Verrocchio, Nature Physics, , 771 (2008).[4] S. Franz, and A. Montanari, J. Phys. A, , F251 (2007).[5] M. Mosayebi, E. Del Gado, P. Ilg and H.C, ¨Ottinger,Phys. Rev. Lett, , 205704 (2010).[6] J. Kurchan and D. Levine, ArXiv cond-matt 0904.4850.[7] S. Karmakar, E. Lerner and I. Procaccia, Phys.Rev. E, , 055103(R) (2010).[8] W. Kob and H. C. Andersen, Phys. Rev. E , 4626(1995).[9] http://en.wikipedia.org/wiki/Lanczos algorithm[10] A. Ghosh, V. K. Chikkari, P. Schall, J. Kurchan and D.Bonn Phys. Rev. Lett., , 248305 (2010).[11] A. Tanguy, J.P. Wittmer, F. Leonforte, J.L. Barrat,Phys.Rev. B , 174205 (2002).[12] A. Sokolov, http://online.kitp.ucsb.edu/online/glasses-c10/sokolov/[13] V. Ilyin, I. Procaccia, I. Regev, Y. Shokef, Phys. Rev. B , 174201 (2009).[14] H.G.E. Hentschel, Smarajit Karmakar, Edan Lerner,Itamar Procaccia, submitted to Phys. Rev. E, alsoarXiv:1101.0101.[15] S. Karmakar, PhD Thesis, Numerical Studies Of SlowDynamics And Glass Transition In Model Liquids,http://etd.ncsi.iisc.ernet.in/handle/2005/633.[16] L. Berthier and G. Tarjus, Phys. Rev. Lett. , 170601(2009).[17] U. R. Pedersen, T. B. Schrøder, and J. C. Dyre, Phys.Rev. Lett.105