Pinning Susceptibility: a Novel Method to Study Growth Of

Pinning Susceptibility : A Novel Method to Study Growth of Amorphous Order in Glass-forming Liquids 1 2 1 Rajsekhar Das ,∗ Saurish Chakrabarty ,∗ and Smarajit Karmakar y 1TIFR Center for Interdisciplinary Science, Tata Institute of Fundamental Research, Narsingi, Hyderabad 500075, India, 2International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Shivakote, Bangalore, 560089, India Existence and growth of amorphous order in supercooled liquids approaching glass transition is a subject of intense research. Even after decades of work, there is still no clear consensus on the molecular mechanisms that lead to a rapid slowing down of liquid dynamics approaching this putative transition. The existence of a correlation length associated with amorphous order has recently been postulated and also been estimated using multi-point correlation functions which cannot be calculated easily in experiments. Thus the study of growing amorphous order remains mostly restricted to systems like colloidal glasses and simulations of model glass-forming liquids. In this Letter, we propose an experimentally realizable yet simple correlation function to study the growth of amorphous order. We then demonstrate the validity of this approach for a few well-studied model supercooled liquids and obtain results which are consistent with other conventional methods. Glasses are ubiquitous in nature and are of immense found to be sensitive to these correlations in glassy liq- practical importance in our day-to-day lives as well as uids. The intricate nature of the correlation functions in modern technology. In spite of knowing their pres- limit the applicability of them only to systems where the ence and usefulness from the early history of mankind, dynamical details of individual constituent molecules are the nature of the glassy state and the glass transition, available, e:g: colloidal glasses [25, 26] and in silico model where the viscosity of a liquid increases by many or- glass-forming liquids [3, 27]. The point-to-set correlation ders of magnitude within a narrow temperature or den- function[7, 18], finite size scaling of stiffness of local po- sity window, still puzzles the scientific community and tential energy landscape [16, 19, 21] and other measures is believed to be one of the major unsolved problems in of local order [20, 28] played an important role in iden- condensed matter physics [1{9]. The viscosity of a glass tifying the length scale associated with the amorphous forming liquid increases so dramatically that one is often order. These methods also require microscopic details of tempted to believe that viscosity probably diverges at a the system in order to calculate the correlation length certain critical temperature associated with a thermody- and are thus not useful for experiments of a variety of namic phase transition. Thus, not surprisingly, there are molecular glass forming liquids. two types of theories on glass transition. The first type Some recent proposals to study the growth of amor- assumes that the (ideal) glass transition is a thermody- phous order using higher order non-linear susceptibili- namic phase transition and glassy states are believed to ties have yielded encouraging results. In Refs. [29{ be a thermodynamic state of matter. This approach is 31], higher order non-linear dielectric susceptibilities were taken by the Random First Order Transition (RFOT) measured for a few molecular glasses at different tem- theory [10{12]. The other approach is to consider the peratures and it was shown that the third and fifth or- glass transition to be a purely dynamic phenomenon and der non-linear dielectric constants were sensitive to the the slowdown of dynamics is thought to be a result of growth of amorphous order. It is important to note that an ever increasing number of self-generated kinetic con- measuring the higher order non-linear dielectric suscep- straints with supercooling[13{15]. tibilities is extremely difficult as they are many orders In spite of all the differences in opinion about the ex- of magnitude smaller than the leading linear contribu- istence of a true thermodynamic glass transition, there tion. Special experimental techniques were needed to es- is a consensus about the existence and growth of corre- timate these higher order susceptibilities [29, 30]. Since lation lengths along with the rapid increase in viscosity χ4(t) cannot be estimated directly in experiments, an in- or relaxation time[2{5, 7{9]. Recently there have been a direct approach is taken. Its peak value is estimated us- P 2 P 2 lot of progress in identifying at least two different length ing an approximate equality χ4 kBT (χT ) =cP , where arXiv:1608.01474v2 [cond-mat.stat-mech] 11 Jan 2017 ' scales { (a) a dynamic length scale akin to the length χT ( @Q=@T ), the temperature, T being the control scale characterizing the heterogeneity present in the dy- variable≡ and Q the two-point density overlap correla- namics of supercooled liquids [4, 5, 8] and (b) a static tion function to be defined later. kB is the Boltzmann length scale of the so called \amorphous order" [7, 16{21]. constant and cP is the specific heat of the system. In Simple two-point density correlation functions have been Ref. [5], frequency dependent χT was measured for pure shown to be unsuitable for the identification of the build glycerol in a desiccated Argon environment in order to up of these correlations in the supercooled liquids. A four eliminate moisture absorption, using capacitive dielec- point susceptibility, χ4(t) and its corresponding structure tric spectroscopy close to the calorimetric glass transi- factor, S (q; t) (see [22] for definitions) [8, 23, 24] were tion temperature. Similarly χ @Q=@φ, where φ is the 4 φ ≡ 2 packing fraction is measured using dynamic light scat- 16 0.520 tering (DLS) experiments on colloidal glass formers for 0.540 14 0.560 which the packing fraction is the relevant control param- 0.580 eter. It was shown that the dynamic length scale does 12 0.600 0.650 grow as the glass transition is approached and becomes 10 0.700 (t) 0.800 (assuming exponent η to be in the range 2 to 4) 1:5nm p ∼ χ 8 1.000 at the lowest temperature studied (T = 192K) for glyc- 1.500 erol. In short, all these methods are experimentally deli- 6 2.000 2.500 cate and indirect when it comes to estimating the growth 4 3.000 of cooperativity in supercooled liquids. 2 In this Letter, we propose a very simple correlation function which directly estimates the growth of amor- 0 -1 0 1 2 3 4 5 10 10 10 10 10 10 10 phous order and the associated static length scale with- t out involving unknown fitting parameters. The idea came 25 from the recent interest in dynamics of supercooled liq- 0.450 20 0.470 0.500 uids and glass transition with quenched random pinning 0.550 0.600 sites. It was claimed in Refs. [32{34] that an ideal glass 0.700 0.800 15 0.900 state can be obtained by introducing quenched disorder 1.000 1.250 (t) P 1.500 in the form of pinned particles in supercooled liquids. Al- χ 2.000 10 2.500 though the existence of such an ideal glass state is still 3.000 debated [35, 36], random pinning has become a diagnos- tic tool to study the dynamics of the supercooled liquids 5 and to test the validity of existing theories of glass transi- 0 tion [37]. Similar to χT and χφ, in the context of random 10-1 100 101 102 103 104 105 pinning, we define the \pinning susceptibility" as t @Q(T; c; t) FIG. 1. Pinning Susceptibility: Time-Temperature de- χp(T; t) = (1) @c pendence of χp(t) for 3dR10 model (top panel) and 2dR10 c=0 model (bottom panel). Peak height of χ4(t) which directly measures the correlation volume of amorphous order, grows where c is the fraction of pinned particles and the overlap correlation function Q(T; c; t) is defined as, monotonically with decreasing temperature. (1 c)N 1 − Q(T; c; t) = w ( ~r (t) ~r (0) ) : (1 c)N 2 j i − i j 3 * i=1 + − X presented later in the text show that the peak height of 4 5(2) this newly defined \pinning susceptibility", χp is directly The window function w(x) = Θ(a x), Θ(x) being the proportional to ξd, where ξ is the static length scale ob- − p p Heaviside step function. The square brackets indicate tained using random pinning [35{37] and d is the number averaging over different realizations of the pinned par- of spatial dimensions. This pinning length scale, ξp is in ticles and angular brackets stand for ensemble average turn related to the static length scale, ξs associated with (see Ref. [22] for details). The pinning susceptibility of d=(d θ) the amorphous order as ξs ξp − , where θ is the a liquid measures the sensitivity of Q(0; t) (we drop the surface tension exponent in RFOT∼ theory. temperature argument for brevity) to small changes in The maximum of χ (t), χmax(T ) appears at time the number of pinned particles [38]. χ has a peak at a p p p t τ , so χmax(T ) χ [T; τ (T )]. At times much time comparable to τ [defined as Q(0; τ ) = 1=e] and is α p p α α α larger∼ compared to the≈ short time β-relaxation time scale zero for both short and long times. (i:e:, comparable to τ time scale), Q(c; t) can be approx- We calculate the pinning susceptibility for four differ- α imated very well by stretched exponential functions as, ent models systems in both two and three dimensions.

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