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 † 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  | i − i |  * i=1 + − X presented later in the text show that the peak height of  (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. Following existing literature, the systems in three dimensions are referred to as the “3dKA” and “3dR10” mod- t β els and those in two dimensions as the “2dmKA” and Q(c, t) exp , t τβ. (3) ∼ − τα(c, T ) “2dR10” models [37] (see Ref. [22] for complete details). " # In Fig. 1, we plot χp for the 3dR10 and 2dR10 models for different temperatures. The function shows features This is not very crucial approximation for this analysis very similar to the four-point susceptibility, χ4(t). It has though there exist huge amount of experimental and nu- a peak at time scale close to τα and the peak height merical data to support this approximation. β is the grows with decreasing temperature. Scaling arguments stretching exponent. Differentiating Eq. 3 with respect 3

max 1/d 4.0 [χ (T)] max 1/d further discussion). In both the panels of Fig. 2, we have ρ [χ (T)] ρ max 1/d ξp(T) 4 3.5 ξp(T) plotted [χp ] and ξp against the temperature for the max 1/d 3.0 four models studied. It is clear that [χp ] is indeed ξ ξ 3 proportional to the static length scale ξ . 2.5 p Motivated by this observation, we next tried to see 2.0 2 whether the concept of random pinning applies to a 1.5 broader context. Instead of actually pinning a set of ran- 0.5 1.0 1.5 2.0 2.5 3.0 0.5 1.0 1.5 2.0 2.5 3.0 T T domly chosen particles we have introduced a small num- 5.0 max 1/d ber of slightly larger sized particles (solute). If the diﬀu- [χ (T)] ξp(T) p max 1/d 5 [χ (T)] ξ (T) sion constants of these solute particles are much smaller 4.0 p p 4 than those of the solvent particles, then they should pro- ξ

ξ 3.0 duce eﬀects similar to pinned particles. Indeed it turns 3 out that small concentration of larger sized solute par-

2.0 2 ticles do produce dynamical effects similar to random pinning. This is extremely useful since the implementa- 1 1.0 0.5 1.0 1.5 2.0 2.5 3.0 0.5 1 1.5 2 2.5 3 tion of random pinning potentials in experiments in three T T dimensions can be quite challenging if not impossible. FIG. 2. Random Pinning Length Scale: Comparison of In the top left panel of Fig. 3, we plot such a pinning max 1/d [χp (T )] with the pining length-scale ξp(T ) for all the susceptibility for the 2dR10 model. This susceptibility model system studied: 3dKA (top left), 3dR10 (top right), has very similar behavior as that of the random pinning 2dmKA (bottom left) and 2dR10 (bottom right). and similar scaling arguments can be given to rationalize the observations with the static length scale, ξs being the dominant length scale. In top middle panel of Fig. 3, we have shown such a scaling collapse for the relaxation to c and setting t τα, we get, ∼ time. In different panels of Fig. 3, we have compared β ∂τα(c, T ) the temperature dependence of the peak height of χ (t) χmax(T ) (4) p p ∼ eτ (c, T ) ∂c for 2dR10, 2dmKA, 3dKA and 3dR10 model with 2% of α c=0 larger sized solute particles added to the original glass Assuming that χ attains its maximum value at t τ , p α forming liquids, with that of the static length scale ob- the above equation gives us the value of the peak∼ in tained using combination of three methods : (a) Point- χ (T, t) for a given temperature. In Refs. [35–37] it was p to-Set method, (b) Finite size scaling of relaxation time shown that for small c, the relaxation time τ (c, T ) obeys α and (c) Finite size scaling of minimum eigenvalue (see the following scaling relation. Ref. [22] for further details about these methods and τ (c, T ) related results). Notice that the static length scale com- log α = f cξd(T ) . (5) τ (0,T ) p pared here is not the pinning length scale rather it is α the static length scale of amorphous order, ξs. This is Here, the scaling function f(x) must go to zero as the because ξs is the only static length scale which governs argument goes to zero, so in the limit x 0, f(x) can the relaxation barriers in the system without any ran- be approximated as f(x) x. The validity→ of the scaling dom pinning. The comparison between peak height of ∼ max function is demonstrated in Refs. [35–37] and also found pinning susceptibility, χp and the static length scale, to describe experimental data very well [39]. Substituting ξs in Fig. 3 for all the models are indeed very good. Eq. 5 into Eq. 4, we get, We would like to emphasize that when one introduces larger sized solute particles in the system, the effective β χmax(T ) = ξd(T ). (6) packing fraction will increase and in effect the pressure p e p will also increase. To remove the effect arising from the Note that the pinning susceptibility is directly propor- change in effective packing fraction, we have maintained tional to the dth power of the pinning length scale apart the same pressure as that of the original binary mix- from a weak non-singular dependence of the stretching ture at each temperature in our simulations (see Ref. exponent β on T . In other words, [22] for further discussion). In Ref. [22], we have presented mathematical arguments for existence of the scal- d Q(T, c, t) Q(T, 0, t) ξp (T ) max lim − . (7) ing form, Eq. 5 and it is important to mention that Pin- ∝ t c 0 c → ning Susceptibility indeed obeys the mathematically re- Thus, the peak value of the χp(t) gives us a direct mea- quired constraints to have the same scaling form. On sure of the pinning length-scale ξp. We have checked that the other hand, χφ or χT may show similar increase of the results do not depend on the specific value of c chosen peak height with increasing density or decreasing tem- in the calculation as long as it is small (see Ref. [22] for perature, but they do not satisfy the required conditions 4

12 16.0 0.520 0.520 0.540 10 0.540 0.560 4.0 max 1/2 [χ (T)] 0.560 8.0 0.580 p 0.580 0.600 ξ (T) 0.700 s 8 0.600 0.800 0.700 (0,T) α 1.000 3.0 0.800 τ 4.0 1.250 (t) s

p 1.500 6 T)/

1.000 ξ χ

c, 2.000

1.250 ( Ax α y = e 1.500 τ 2.0 4 2.000 2.0

2 1.0 1.0 -2 -1 0 0 10 10 10 -1 0 1 2 3 4 5 d 10 10 10 10 10 10 10 cξ 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 t s T

3.0 1.6 ξ (T) ξ (T) max 1/2 s s [χ (T)] max 1/3 4.0 p [χ (T)] max 1/3 p 1.4 [χ ] ξ (T) p s 2.5 1.2 3.0 (T)

(T) 2.0 s (T) s s

ξ 1 ξ ξ

0.8 2.0 1.5

0.6 1.0 1.0 0.4 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 T T T

FIG. 3. Length-scale of Static Amorphous Order: (top left) The variation of pinning susceptibility, χp(t) with time for various temperatures for the 2dR10 model using heavy “solute” particles instead of strictly pinned particles. (top middle) Validation of the proposed scaling ansatz. Comparison of the peak height of pinning susceptibility with the static length scale of amorphous order for diﬀerent models studied: 2dR10 (top right), 2dmKA (bottom left), 3dKA (bottom middle) and 3dR10 (bottom right). Note that the static length-scale, ξs is being calculated independently using multiple conventional methods (see text for details). Very good agreement of data with conventional method clearly establishes the generic nature of the proposed method for studying growth of amorphous order in supercooled liquids approaching glass transition.

for the existence of similar scaling function (see Ref. [22] length scale associated with amorphous order in super- for an in-depth discussion on this issue). We therefore cooled liquids. The main virtue of this method is its believe that conventional susceptibilities are not equiva- simplicity and robust applicability to a wide variety of lent to our proposed pinning susceptibility. Although the systems. We believe that this is the first proposal of scaling arguments we have proposed rationalize our ob- the much sought after experimentally realizable corre- servations of a possible connection between the pinning lation function which directly picks up the size of the susceptibility and the static length scale of amorphous correlated region in glass forming liquids. All previous order, a detailed microscopic theory is essential to un- propositions involve estimating the correlation length of derstand the the observed results. amorphous order in an indirect way. This susceptibil- The Inhomogeneous Mode Coupling Theory (IMCT) ity can be measured in molecular glasses without much [6], an extension of the standard Mode Coupling Theory difficulty by introducing a small concentration of solute (MCT) [40, 41] to partially include relevant fluctuations, molecules which are somewhat larger than the the solvent suggests application of a spatially localized field which molecules. Extensions to colloidal glass formers will also couples to density fluctuations to estimate the increase be fairly simple. This new method carries the potential in cooperativity in a supercooled liquid. This theory pre- to establish the possible connection between the growth dicts a power law growth of the correlation volume while of amorphous order with the rapid increase of viscosity approaching the glass transition, the correlation volume in experimentally relevant glass forming liquids and may being dynamic in nature. Similar extension of MCT for help us understand the puzzles of glass formation and the liquids with small concentration of solute particles whose glass transition in near future. diffusion coefficients are much smaller than the solvent particles may reveal the observed microscopic connection between the pinning susceptibility and amorphous order. We would like to thank Chandan Dasgupta, Giulio In conclusion, we have proposed a new susceptibility, Biroli, Srikanth Sastry, Sriram Ramaswamy and Surajit the pinning susceptibility, which can directly measure the Sengupta for many useful discussions. 5

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Rajsekhar Das1, Saurish Chakrabarty2, and Smarajit Karmakar1 1 Centre for Interdisciplinary Sciences, Tata Institute of Fundamental Research, 21 Brundavan Colony, Narisingi, Hyderabad, 500075, India, 2 International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Shivakote, Hesaraghatta, Hubli, Bangalore, 560089, India.

This supplementary information is arranged as follows. Here, ǫαβ =1.0, σAA =1.0, σAB =1.22 and σBB =1.40. In Sec. , we introduce the systems we studied and the The potential is cut off at 1.38σαβ using a quadratic poly- simulation details. In Sec. , we give some standard defi- nomial to make the potential and its first two derivatives nitions of quantities referred to in the main text related continuous at the cutoff. The number density of particles to the physics of dynamical heterogeneity (DH) in super- is 0.81 and the temperature range is T 0.52, 3.0 . ∈{ } cooled liquids. In Sec. , we describe the methods we used 2dR10 – The system as above in two dimensions but to obtain the static length-scale ξs. In Sec. we discuss with a number density of 0.85. details about the scaling arguments presented in main ar- We have performed NVT molecular dynamics simula- ticle. In Sec. , we present a new argument, based on our tions for all the model systems where modified leap-frog obtained values of the pinning susceptibility, about the algorithm was used for the integration and Berendsen possibility of an ideal glass state at some critical value thermostat to keep the temperature constant. of the pinning concentration. We also outline a new ex- Random Pinning Protocol: A fraction of particles planation of the scaling form of τα(c,T ) on pinning con- are chosen randomly from an equilibrated configuration centration c used in Refs. [1] and [2]. The arguments are of the the studied model system and then their dynam- based on the non-singular dependence of the peak height ics are frozen. To study the dynamics of the system at of the pinning susceptibility on the pinning concentra- varying pinning concentration we have changed the con- tion. Finally in Sec. , we discuss the dependence of χp(t) centration in the range c 0.005, 0.200 [1, 2]. Subse- on the choice of δc. quently scaling analysis was∈ { done to obtain} the random pinning length scale, ξp for all the model system studied. ξp data presented in this work are taken from Refs. [1, 2]. MODELS AND THE SIMULATION DETAILS Please refer to Refs. [1, 2] for complete details on the dynamics of supercooled liquids with random pinning. We study the following four model glass-forming liq- Ternary Mixture: In order to generate dynamical uids: effects which are similar to random pinning, we have introduced a small fraction of a third type of particle with a 3dKA – The Kob-Andersen mixture [3]. A 80 : 20 bi- bigger diameter for all the four systems studied. We will nary mixture of particles interacting via Lennard-Jones refer to these systems as ternary version of the respective potentials in three spatial dimensions. models. Interaction potential for 3dKA and 2dMKA sys- 12 6 tems is same as given in Eqn.1, but here, α, β A, B, C σαβ σαβ ∈{ } Vαβ(r)=4ǫαβ . (1) and ǫ = 1.0, ǫ = 1.5, ǫ = 1.0, ǫ = 0.5, r − r AA AB AC BB ǫBC = 1.0, ǫCC = 1.0, σAA = 1.0, σAB = 0.80, σAC = 1.10, σBB = 0.88, σBC = 1.04, σCC = 1.20. Here, α, β A, B and ǫAA = 1.0, ǫAB = 1.5, ǫBB = ∈ { } For 3dR10 and 2dR10 systems the interaction potential 0.5, σAA = 1.0, σAB = 0.80, σBB = 0.88. The number density of particles is 1.2 and the temperature range is is given by Eqn. 2, and ǫαβ =1.0,σAA =1.0, σAB =1.18, T 0.45, 3.0 . The interaction potential is cut off at σAC =1.50, σBB =1.40, σBC = 1.70, σCC = 2.0. Frac- ∈ { } tion of the particle chosen as third type of particles is 2.50σαβ using a quadratic polynomial such that the po- c 0.005, 0.080 for all the four systems. NPT molec- tential and its first two derivatives are continuous at the ∈ { } cutoff distance. ular dynamics simulations are performed for the ternary models at the pressures obtained from the corresponding 2dMKA – The same system in two dimensions with the binary model simulations for respective temperatures. two types of particles having proportion 65 : 35 instead of 80 : 20. 3dR10 – A 50 : 50 binary mixture [11] interacting in DYNAMIC CORRELATION FUNCTIONS three dimensions via the potential,

σ 10 The dynamic susceptibility is a measure of the fluctua- V (r)= ǫ αβ . (2) αβ αβ r tions in the two point overlap correlation function defined 2 as at a given temperature and then define a smaller spherical region of radius, R. The particles outside this spher- (1 c)N 1 − ical cavity are pinned at their respective positions and Q(T,c,t)= w ( ~r (t) ~r (0) ) . (1 c)N  | i − i |  only the particles inside the cavity are allowed to evolve * i=1 + − X according to Newtonian dynamics. The static overlap  (3) correlation is calculated for the particles inside the cav- The summation is over the (1 c)N unpinned particles. ity and in order to remove boundary effects the static The window function w(x) =Θ(−a x) where Θ(x) is the − overlap is calculated only for those particles which are Heaviside step function. The window function is chosen in the center of the cavity. This procedure is repeated to remove from the correlation function, the effects of for many different sizes of the cavity and the size depen- short time vibrational motion of a given particle inside dence of the static overlap is obtained to finally extract the “cage” formed by its neighboring particles. The value the point-to-set length scale. of the parameter a is chosen to be 0.30 for this study The external pinned boundary makes the dynamics of but qualitative results are not affected by any particular the internal particles very slow especially for small cav- choice of this parameter [2, 4]. The square brackets in- ities and one needs to employ a sophisticated sampling dicate averaging over different realizations of the pinned method (Swap Monte Carlo Method [14]) to partly over- particles and angular brackets stand for ensemble aver- come this difficulty. As this enhanced sampling method is age. not generically applicable to any model system, the sam- Here we will denote overlap correlation function for pling problems are not always easy to overcome [10] es- the system with no random pinning simply as Q (t) 0 ≡ pecially at lower temperature. In this study we have used Q(T,c = 0,t). Note that explicit temperature depen- PTS methods only at those temperature regime where it dence T is not mentioned for simplicity. With these defi- can be done within accessible computation time. Some nition the four-point dynamic susceptibility will be given of the PTS length-scale reported in this work are taken as from Ref. [2]. 1 χ (t)= Q2(t) Q (t) 2 (4) 4 N h 0 i − h 0 i Finite Size Scaling of Minimum Eigenvalue: The four-point structure factor is given by, The second method is the finite-size-scaling of the min- 1 i~k ~r (0) 2 S (k,t)= e · i w( ~r (t) ~r (0) ) (5) 4 N h| | i − i | | i imum eigenvalue of the Hessian matrix obtained for the i X inherent structure of the supercooled liquid at a given Please see Ref. [8] for a comprehensive review on the dy- temperature [11]. This method is expected to work in the low temperature regime where the supercooled liq- namical behavior of χ4(t) and S4(q,t) and their relations uid spends a considerable amount of time around a given to dynamic heterogeneity length scale, ξd. minimum of its free energy landscape and a harmonic approximation works for a finite (but short) time. We will METHODS OF CALCULATING THE STATIC briefly describe the method here. Please refer to Ref. [11] LENGTH-SCALE, ξs for the details of this method. Karmakar et. al. in [11] argued that the minimum Although there is a consensus on the existence of a eigenvalue λmin(T ) of the Hessian matrix obtained at static length-scale in supercooled liquids that grows as the inherent structure closest to the configuration of a su- the temperature is lowered, obtaining it may be quite percooled liquid at given temperature T takes the scaling challenging. In this work, we have used the following form, three methods to obtain the static length-scale. d/2 λmin(T ) d 1 Ad λmin(T ) h i = ξs (T ) h i .(6) λD(T ) F " N − 2 λD(T ) !# Point-To-Set Method: h i h i

λD(T ) is the square of the Debye frequency, d is the di- The point-to-set (PTS) length-scale was introduced by mensionality of space, A is an adjustable constant cho- Bouchaud and Biroli in Ref. [5] and it was numerically sen to get the best scaling collapse and is an unknown calculated for the first time in Ref. [7] in a model bi- scaling function. In Ref. [11], the validityF of this method nary glass former. This method is useful at somewhat was demonstrated for a few model glass forming liquids high temperatures when the length-scale is not too large. and later in Ref. [12] it was shown that the temper- The main difficulty comes from the dynamical procedure ature dependence of the PTS and minimum eigenvalue involved in calculating this length scale. In the PTS length-scales is same for various generic glass forming method, one needs to take an equilibrated configuration liquids. Recently in Ref. [13], this method was used to 3

T = 0.430 6 6 T = 0.450 T = 0.470 T = 0.500 ) ) 5 5 T = 0.550 ,T ,T T = 0.600 ∞ ∞ ( ( T = 0.700 α α 4 4 τ τ T = 0.800 / / ) ) T = 1.000 3 3 T = 1.250 N,T N,T ( ( T = 1.500 α α τ τ T = 2.000 2 2 T = 2.500 T = 3.000 1 1 2 3 4 5 10 2 10 3 10 4 10 10 10 10 10 N N/ξd(T )

7 T = 0.480 7 T = 0.500 T = 0.520 6 T = 0.540 6 T = 0.560 ) 5 T = 0.600 ) 5 ,T T = 0.650 ,T ∞ T = 0.700 ∞ ( 4 ( 4 α T = 0.800 α τ τ / / )

T = 1.000 ) 3 T = 1.250 3 N,T

T = 1.500 N,T ( ( α T = 2.000 α

τ 2 τ 2

1 1

0 0 2 3 4 10 10 10 10 2 10 4 d N N/ξs (T)

T = 0.450 15 15 T = 0.500 T = 0.550 4 T = 0.600 ) ) 3 ,T ξ 10 T = 0.700 ,T 10 ∞ ∞ (

T = 0.900 (

α 2 α /τ

T = 1.000 /τ ) ) T = 1.250 1 N,T N,T ( 0.5 1 1.5 2 5 T = 1.500 ( 5 α α

τ T T = 2.000 τ

0 2 3 4 0 2 3 4 5 10 10 10 10 10 10 10 N N/ξd(T )

FIG. 1: Top Panel: Finite size eﬀects of τα for the 3dKA model and the corresponding data collapse to obtain static length scale ξs shown in the next panel. Similar analysis done for 2dR10 model (middle panel) and 2dmKA model (bottom panel) are also shown. In all these case scaling collapse observed to be very good. 4 obtain the static length scale for a glass forming liquids 3.0 for which equilibrium state can be relatively easily sam- 0.450 0.500 pled using the Swap Monte Carlo technique. A large 0.550 2.5 0.600 change in length-scale was reported in that work. 0.700 0.800 1.000

(0,T) 1.250 α Finite Size Scaling of α-relaxation time τ 1.500 2.0 2.000 Ax

c,T)/ y = e ( α

The third method we use is the ﬁnite-size-scaling of τ the α-relaxation time, τα. The α-relaxation time of su- 1.5 percooled liquids shows a strong system size dependence. It decreases monotonically with increasing system size before attaining its asymptotic value at the thermody- 1.0 -2 -1 0 namic limit. The dependence is however controlled by a 10 ξ d 10 10 c s single static length-scale and the following scaling form is used to obtain it. 2.5 τ (N,T ) N α = . (7) d ξ (T) : PTS + FSS τα( ,T ) G ξs (T ) s ∞ ξ s(T) : Using Eqn.7 (x) is an unknown scaling function. The inﬁnite sys- 2.0 temG size relaxation time τ ( ,T ) and the static length- α ∞ scale ξs(T ) are chosen so as to get a good data collapse s d ξ when τα(N,T )/τα( ,T ) is plotted against N/ξs (T ) for 1.5 all temperatures on∞ the same graph. In Fig. 1, we have shown such data collapse to obtain the static length scale for some of the studied systems. In Ref. [6], it is demonstrated that the length-scale obtained from the 1.0 FSS analysis of τα is indeed the same as the length-scale obtained using PTS and FSS of minimum eigenvalue 0.5 1.0 1.5 2.0 method. For an in-depth discussion on the importance T of various length-scales in the dynamics of supercooled liquids see, e.g., Refs. [6] and [8]. FIG. 2: Top panel: Data collapse using Eq. 8 to obtain static length scale for 3dKA system with third type of particle with bigger diameter. Bottom panle: Comparison of length scales. PTS refer to Point-to-set method and FSS refer to Finite size SCALING ARGUMENTS scaling (see Sec. for details)

In the main article, we have argued that we can actually calculate the static length scale following scaling arguments very similar to the random pinning situations, scaling analysis for the 3dKA ternary system and com- by introducing a small fraction of solute particles whose pared the obtained length scales with the length scales diameters are larger than the solvent particles. These obtained from finite-size scaling of the α relaxation time particles, as discussed before, will have smaller diffusion for the same system. coefficients than the diffusion coefficients of the solvent Note that while extracting the functional relationship particles. If the difference between these diffusion co- between the peak height of χp(t) and the static length efficients is large, then the solute particles behave like scale, we have assumed that the peak height of χp(t) ap- pinned particles over the time-scale of structural relax- pears at a timescale which is proportional to α-relaxation ation of the solvent particles and one would expect scal- time, τα. This is an important assumption and without ing arguments similar to random pinning to hold good as this the scaling arguments will not hold good. To check well, the validity of this assumptions, we have plotted the time at which χp(t) peaks at different temperatures and τα in τ (c,T ) log α = f cξd(T ) . (8) the top panel of Fig. 3 for 2dR10 model with random τ (0,T ) s α pinning. All other models show exactly the same behaviour. For the ternary model system, again for the In order to test this scaling ansatz we have done the scaling arguments to be valid peak of χp(t) should ap- systematic studies of relaxation time with increasing frac- pear at timescale proportional to τα. In bottom panel of tion of solute particles. In Fig. 2 we have shown such a Fig. 3 we have shown such a comparison for the 2dR10 5

pinning fraction, the overlap correlation function must

3 τ decay to zero at some ﬁnite time. Therefore, there must 10 α τ χ max exist a time-scale t0(δc) such that Q[c0(T ) δc,T,t]=0 from p (t) − for all t>t0. Thus, the pinning susceptibility χp must be inﬁnite for all t>t0. It must be noted that the 2 time-scale at which χp diverges must be shorter than the

α 10

τ α-relaxation time of the system for any ﬁnite δc > 0 with t0(0) > τα . From top panel of Fig.4, it is clear that max ∼ → ∞ χp (T ) shows no tendency to diverge even when higher 1 10 and higher pinning fraction data are considered. This gives us a compelling evidence against the possibility of

0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 5.0 T 0.020 0.040 0.080 3 4.0 10 0.120 τ α max τ from χ (t) p 3.0 2

10 t) ( p χ τ 2.0

1 10

1.0

0 10

0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 0.0 0 1 2 3 4 T 10 10 10 10 10 t

FIG. 3: The time at witch the maximum of χρ(t) appears is compared with τα for 2dR10 model with random pinning (top panel) and for the ternary model (bottom panel). 8.0 c =0.000 c =0.010 c =0.020 c =0.030 c=0.040 model. One can clearly see that maximum of χp(t) oc- 6.0 curs nearly at τα for all temperatures. Similar results

were obtained for other model systems also. These ob- (t) p servations clearly validate one of the crucial assumptions χ 4.0 made in the scaling theory.

2.0 A NOTE ON THE POSSIBLE IDEAL GLASS TRANSITION WITH RANDOM PINNING

0.0 -2 -1 0 1 2 3 4 5 If there is an ideal glass transition as a function of 10 10 10 10 10 10 10 10 the pinning strength, the pinning susceptibility must di- t verge for all times larger than some ﬁnite time for that FIG. 4: Top panel: The variation of pinning susceptibility value of the pinning strength. This can be proven as fol- with time for various pinning fractions for the 3dKA model lows. Suppose, the ideal glass transition takes place at with 2000 particles at T = 0.80. Bottom panle: The variation c = c0(T ) at temperature T . In that case, at tempera- of pinning susceptibility with time for various concentrations ture T and pinning fraction c0(T ), the overlap correlation of third particles for the ternary version of 2dR10 model with function must not decay to zero at any ﬁnite time. Thus, 1000 particles at T = 0.60 with δc = 0.01. Q[c0(T ),T,t] > 0 for all t. However, for a slightly lower 6 achieving an ideal glass state by just increasing the number of pinned particles in a system. In the bottom panel 18.0 of Fig. 4, we have shown the variations of peak heights 16.0 ρ = 0.850 with increasing concentrations of larger third particles ρ = 0.860 for the ternary 2dR10 model to demonstrate the similar- 14.0 ρ = 0.870 ity with the pinning susceptibility for random pinning. ρ = 0.880 12.0 ρ = 0.890 This clearly shows that peak value of pinning suscepti- ρ = 0.900 bility does not depend on the pinning concentrations or 10.0 ρ = 0.910 (t)

amount of third particles in the case of ternary models. φ Similar results are obtained for other model systems also. χ 8.0

6.0 EXISTENCE OF SCALING FUNCTION 4.0

In this section, we discuss the criterion for the existence 2.0 of the observed scaling function. We ﬁrst ﬁnd out if there 0.0 -2 -1 0 1 2 3 4 5 is an optimum value of the pinning fraction, cP , at which 10 10 10 10 10 10 10 10 the temporal peak of the pinning susceptibility takes its t highest value for a given temperature. For this, we set the FIG. 5: The variation of χ (t) with time for various number derivative of χp[c,T,τα(c,T )] with respect to c to zero. φ Using the stretched exponential form for Q(c,t) (valid density for the 2dR10 model with 1000 particles at T = 0.80 with δρ = 0.01. The peak height increasing quite strongly at the α-relaxation regime), this gives us the following with increasing density. This is stark contrast with the pin- relation: ning susceptibility where peak height does not change with increasing pinning concentrations. ∂2τ ∂τ 2 τ α = α . (9) α ∂c2 ∂c P P where the temperature dependence of all the quantities On the other hand, instead of introducing a third type are not explicitly written. Solving this, we get, of particle particle (or random pinning) if we simply in-

τα(cP ,T )= A(T ) exp [B(T )cP ] . (10) crease the number density ρ and calculate χφ(t) at different number density we see that peak height of χφ(t) A corollary to this is that if the peak height of χp(t) increases very strongly with the increase of ρ as shown for a given temperature is independent of c, then the in Fig. 5 for the 2dR10 model. This is completely diﬀer- derivative considered to get the maximum in the above ent than the pinning susceptibility where the peak height analysis vanishes for all values of c. In that case, we get does not change with increasing pinning concentrations. after renaming A(T ) and B(T ) appropriately, Thus one can rule out similar scaling function to exist for

d χφ. We would also like to mention that the peak does not τα(c,T )= τα(0,T ) exp[cξ (T )]. (11) p appear at a time scale proportional to τα for a given den- Thus it is essential for the existence of the proposed sity as a function of temperature. This also rules out the possibility of similar scaling function to exist for χφ(t). scaling function that the peak height of χp(t) for a given temperature is independent of the pinning concentration, Thus, we feel that our proposed pinning susceptibility is unique in its own way to directly estimate the c. In top panel of Fig. 4, we have plotted χp(t) for different pinning concentrations and one observes that the growth of amorphous order in supercooled liquids and peak height does not grow with increasing pinning con- the above mentioned scaling arguments clearly suggest centration. It is rather constant within the numerical that the peak height is directly proportional to the correlation volume without any unknown exponents. statistical error in our data. Thus peak height of χp(t) can be considered to be independent of the pinning concentration. This directly suggests that the temperature max at which the ideal glass transition occurs is independent DEPENDENCE OF χp (T ) ON δc of the pinning fraction as assumed in Ref. [1]. Similarly max in the bottom panel of Fig. 4, we have shown similar data In this section we have checked the dependence of χp for the ternary model of 2dR10. Here also one can clearly for various temperature on the choice of δc while calcu- max see that with increasing concentration of larger size third lating χp via numerical diﬀerentiation. In Fig. 6, we max particles, the peak height of pinning susceptibility does have plotted χp using diﬀerent values δc for our model not change. Thus, it may be safe to expect a similar systems. One can clearly see that the peak height does scaling function to hold good for this case also. not depend explicitly on the choice of δc as long as it is 7

[2] Chakrabarty, S., Das, R., Karmakar, S. & Dasgupta, 12.0 C. Understanding the dynamics of glass-forming liquids δc = 0.010 with random pinning within the random ﬁrst order tran- 10.0 δc = 0.020 sition theory. J. Chem. Phys. 145, (2016). δc = 0.030 δc = 0.040 [3] Kob,W.& Andersen,H. C. Testing mode-coupling theory 8.0 for a supercooled binary Lennard-Jones mixture I: The van Hove correlation function. Phys. Rev. E 51, 4626 (T) (1995). max 6.0 ρ

χ [4] Laˇcevi´c, N., Starr, F. W., Schrøder, T., Novikov, V. & 4.0 Glotzer, S. Growing correlation length on cooling below the onset of caging in a simulated glass-forming liquid. Phys. Rev. E 66, 030101 (2002). 2.0 [5] Bouchaud,JP.& Biroli,G. On the Adam-Gibbs- Kirkpatrick-Thirumalai-Wolynes scenario for the viscos- 0.0 ity increase in glasses. 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 T J. Chem. Phys. 121, 7347 (2004). 8.0 [6] Karmakar, S., Dasgupta, C. & Sastry, S. Length scales in glass-forming liquids and related systems: a review. δc = 0.025 δ Rep. Prog. Phys. 79, 016601 (2016). c = 0.030 [7] Biroli, G., Bouchaud, J.-P., Cavagna, A., Grigera, T. S. 6.0 δc = 0.035 δc = 0.040 & Verrocchio, P. Thermodynamic signature of growing amorphous order in glass-forming liquids. Nat. Phys. 4, 771–775 (2008). (T) 4.0

max [8] Karmakar, S., Dasgupta, C. & Sastry, S. Growing length p χ scales and their relation to timescales in glass-forming liquids. Annu. Rev. Condens. Matter Phys. 5, 255–284 2.0 (2014). [9] Gutiérrez, R., Karmakar, S., Pollack, Y. & Procaccia, I. The static lengthscale characterizing the glass transition at lower temperatures. Europhys. Lett. 111, 56009 0.0 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 (2015). T [10] Hocky, G. M., Markland, T. E. & Reichman, D. R. Grow- ing point-to-set length scale correlates with growing re- max FIG. 6: Top panel: χp for different choices of δc for 2dR10 laxation times in model supercooled liquids. Phys. Rev. Ternary model. Notice that up to δc = 0.04, there is not Lett. 108, 225506 (2012). much dependence of χp on δc. Bottom Panel: Similar plot [11] Karmakar, S., Lerner, E. & Procaccia, I. Direct esti- for 3dKA Ternary model. One can clearly see that the peak mate of the static length-scale accompanying the glass max value of the χp does not depend on the choice of the δc for transition. Physica A: Statistical Mechanics and its Ap- the studied models. plications 391, 1001–1008 (2012). [12] Biroli, G., Karmakar, S. & Procaccia, I. Comparison of Static Length Scales Characterizing the Glass Transition. Phys. Rev. Lett. 111, 165701 (2013). small. This also directly establishes the importance of [13] Gutiérrez, R., Karmakar, S., Pollack, Y. & Procaccia, our proposed susceptibility along with behaviour. I. The static lengthscale characterizing the glass transition at lower temperatures. Europhys. Lett. 111, 56009 (2015). [14] T.S. Grigera and G. ParisiFast Monte Carlo algorithm for supercooled soft spheres, Phys. Rev. E. 63, 045012 [1] Chakrabarty, S., Karmakar, S. & Dasgupta, C. Dynamics (2001). of glass forming liquids with randomly pinned particles. Sci. Rep. 5, (2015).