Energy Landscape, Anti-Plasticization and Polydispersity Induced Crossover of Heterogeneity in Supercooled Polydisperse Liquids
Sneha Elizabeth Abraham, Sarika Maitra Bhattacharrya, Biman Bagchi
aa r X i v : . [ c ond - m a t . s o f t ] O c t Energy Landscape, Anti-Plasticization and Polydispersity InducedCrossover of Heterogeneity in Supercooled Polydisperse Liquids
Sneha Elizabeth Abraham, Sarika Maitra Bhattacharrya, and Biman Bagchi ∗ Solid State and Structural Chemistry Unit,Indian Institute of Science, Bangalore 560 012, India
Polydispersity is found to have a significant effect on the potential energy landscape; the averageinherent structure energy decreases with polydispersity. Increasing polydispersity at fixed volumefraction decreases the glass transition temperature and the fragility of glass formation analogous tothe antiplasticization seen in some polymeric melts. An interesting temperature dependent crossoverof heterogeneity with polydispersity is observed at low temperature due to the faster build-up ofdynamic heterogeneity at lower polydispersity.PACS numbers: 64.70.Pf, 82.70.Dd, 61.20.Lc
Polydispersity is ubiquitous in nature. Itis present in clays, minerals, paint pigments,metal and ceramic powders, food preservativesand in simple homogeneous liquids. It is com-mon in synthetic colloids, which frequently ex-hibit considerable size polydispersity [1] and isalso found in industrially produced polymers,which contain macromolecules with a range ofchain length. Polydispersity has significant ef-fects on both the structure and dynamics ofthe system. Experiments [2] and simulations[3, 4] on colloidal systems show that increas-ing polydispersity, at a constant volume frac-tion, lowers structural correlations, pressure,energy and viscosity. Polydisperse colloidal sys-tems are known to be excellent glass formers.William et al [5] suggest that colloidal glass for-mation results from a small degree of particlepolydispersity. Crystal nucleation in a polydis-perse colloid is suppressed due to the increaseof the surface free energy [6]. Studies by severalgroups [7] have shown that the glass becomesthe equilibrium phase beyond a terminal valueof polydispersity.Despite being natural glass formers, relation-ships between polydispersity, fragility, energylandscape and heterogeneous dynamics havenot been adequately explored in these systems.Because these systems exist in the glassy phaseover a wide range of polydispersity, they offeropportunity to test many of the theories andideas developed in this area in recent years.We find that polydispersity introduces severalunique features to the dynamics of these sys-tems not present in the binary systems usuallyemployed to study dynamical features in super- ∗ Electronic mail: [email protected] cooled liquids and glasses.In this work we particularly investigate howpolydispersity influences the potential energylandscape, fragility and heterogeneous dynam-ics of polydisperse Lennard-Jones (LJ) systemsin supercooled regime near the glass transition[8]. The polydispersity in size is introduced byrandom sampling from a Gaussian distributionof particle diameters, σ . The standard devia-tion δ of the distribution divided by its mean σ gives a dimensionless parameter, the polydis-persity index S = δσ . The mass m i of particle i is scaled by its diameter as m i = m ( σ i σ ) . Mi-cro canonical (NVE) ensemble MD simulationsare carried out at a fixed volume fraction, φ ona system of N = 864 particles of mean diameter σ = 1 . m = 1 . S = 0 . .
15 and 0 .
20 at φ = 0 .
52 and S = 0 .
10 and0 .
20 at φ = 0 .
54. All quantities in this studyare given in reduced units (length in units of σ ,temperature in units of ǫk B and time in unitsof τ = ( mσ ǫ ) ). The LJ interaction parame-ter ǫ is assumed to have the same value for allparticle pairs.At large supercooling the system settles intoglassy phase. We first analyze the system fromthe perspective of potential energy landscape(PEL), which has emerged as an important toolin the study of glass forming liquids [9, 10, 11].Fig 1(a) and (b) show the variation of the aver-age inherent structure energy ( h e IS i ) with tem-perature ( T ) at both the volume fractions stud-ied. The value of h e IS i remains fairly insen-sitive to the variation in T at high T beforeit starts to fall with T (around T ∼ . the start offall in h e IS i coincides with the onset of non-exponential relaxation in the time correlation .4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 T -7.55-7.5-7.45-7.4 < e I S > T β T -7.48-7.44-7.4-7.36 < e I S > T β S1, k = 5.0S2, k = 5.0S1, k max
S2, k max
S1, k = 10.0S2, k =10.0 (a) (b)(c) (d)
FIG. 1: (a) and (b) Temperature dependence ofthe average inherent structure energy, h e IS i . ForFig 1 (a), (b) & (c), filled circles, stars and trian-gles are for S = 0 . S = 0 .
15 & S = 0 .
20 at φ = 0 .
52 and filled diamonds and squares are for S = 0 .
10 & S = 0 .
20 at φ = 0 .
54, respectively.(c) The stretched exponent β vs. T obtained byfitting KWW equation to self-intermediate scatter-ing function, F s ( k max , t ) where k max ∼ .
0. Thelines are guide to the eye. Comparison between(a)/(b) and (c) shows that the fall of h e IS i corre-sponds to the onset of non-exponential relaxationin F s ( k max , t ). (d) β vs. T from F s ( k, t ) for dif-ferent k values. Data shown for S = 0 . S
1) & S = 0 . S
2) at φ = 0 . S = 0 .
15 omitted forclarity. functions of the system . We show in Fig 1(c)that this correlation continues to hold in poly-disperse systems. The fall of h e IS i with T isconsistent with the Gaussian landscape model.The average inherent structure energy de-creases with polydispersity (Fig 1(a) and (b)),which indicates that the packing is more effi-cient at higher S. In Fig 2 we plot the inherentstructure (IS) and the parent liquid radial dis-tribution functions (rdf). At S = 0 .
20 there ishardly any difference between the rdf of the par-ent liquid and the IS. The coordination number, N c at S = 0 .
10 and S = 0 .
20 obtained fromthe IS rdf are 13 . .
6, respectively. Thisshows that packing is more efficient at higherS and one would expect a slowing down of dy-namics at higher S. Instead, we find that similarto colloidal hard spheres polydisperse LJ sys-tems also show a speed up of relaxation withS. The presence of smaller particles at higher Sprovides some sort of lubrication [13, 14], whichspeeds up the dynamics of the whole system. A r g ( r ) T l n ( τ ) σ i T ci (a) (b) FIG. 2: (a) Average radial distribution functions(rdf) for the parent liquid (solid line) and the inher-ent structure (dashed line) for S = 0 .
10 (black) and S = 0 .
20 (red) systems at T = 0 .
50 and φ = 0 . τ from KWW fit to F s ( k max , t )for S = 0 .
10 (filled circles) and S = 0 .
20 (filled tri-angles) at φ = 0 . T ic for particles of different sizes σ i obtained from theMCT equation, D i ∼ ( T − T ic ) γ for S = 0 .
10 (opencircles) and S = 0 .
20 (open triangles) at φ = 0 . plot of the Mode Coupling Theory (MCT)[15]critical temperature T ic for particles of differentsizes σ i (inset of Fig 2(b)) shows that the T ic for the largest-sized particles in S=0.20 systemis smaller than the smallest-sized particles inS=0.10 system. This tells us that not only thesmaller particles in S = 0 .
20 system but thewhole system has a faster relaxation. The rateof growth of relaxation time upon lowering of Tdecreases with S (Fig 2(b)). Hence as the sys-tem is cooled, vitrification is expected to occurat a lower T for the system at higher S. Thisshould lead to a lowering of the glass transitiontemperature with S.Fragility is a term being used to character-ize and quantify the non-Arrhenius transportbehavior in glass-forming liquids as they ap-proach glass transition[24]. To study the effectof polydispersity on fragility, we plot the diffu-sion coefficients in an Angell-like fragility plotin Fig 3. The plot clearly shows that increasingpolydispersity at fixed volume fraction reducesthe fragility of the liquid so that the system is astronger glass former at higher polydispersity.2 T r / T - l n D S = 0.10, φ = 0.52S = 0.15, φ = 0.52S = 0.20, φ = 0.52S = 0.10, φ = 0.54S = 0.20, φ = 0.540.1 0.15 0.2 S m FIG. 3: Angell-like fragility plot at different S forthe two φ studied. The thick lines are VFT fit tothe diffusivity data, D = D exp ( E D T − T ). The refer-ence temperature T r is chosen such that D ( T r ) =4 . × − . The VFT extrapolation is used to locate T r . The plot shows that fragility decreases with S and that for a given S fragility increases with in-crease in φ . [Inset: Strength parameter m (where m = E D T [24]) obtained from VFT fit as a functionof S at φ = 0 . This effect is analogous to the antiplasticiza-tion that has been observed in polymer melts[16]. PEL analysis shows that the antiplasti-cized system has smaller barriers to overcomein order to explore the configuration space [17].In the rest of the paper we explore the correla-tions between fragility and non-exponential re-laxation/heterogeneous dynamics.Fragility is usually correlated to the stretchexponent β which is found to be valid for manymaterials [18]. From PEL perspective, fragileliquids display a proliferation of well-separatedbasins which result in a broad spectrum of re-laxation times leading to stretched exponentialdynamics [10]. The correlation is also consis-tent within the framework of coupling model(CM) [19] according to which the strength ofthe intermolecular coupling is given by (1 − β ).The rate of growth of intermolecular couplingwith decrease in T is a measure of fragilitywhich according to CM would depend on therate of fall of β with T. We indeed find thatas S increases (fragility decreases) the rate offall of β with T decreases (Fig 1(c)). However, α ( t ) ln (time) α ( t ) ln (time) (a) T * = 0.45 (b) T * = 0.50(c) T * = 0.55 (d) T * = 0.60 FIG. 4: The non-Gaussian parameter, α ( t ) for S = 0 .
10 (solid line), S = 0 .
15 (dashed line) and S = 0 .
20 (dot-dashed line) at four different T de-picting the crossovers between different S . Data isshown for φ = 0 . if we look only at the β values and not its T-dependence we find that at high T, stretching isanti-correlated with fragility whereas at low T,we get the reverse scenario where the stretch-ing is correlated with fragility . This leads toa cross-over of the β values for different S atintermediate T as shown in Fig 1(c). The β values in Fig 1(c) are obtained by KWW fitto F s ( k max , t ). However, these cross-overs areindependent of k values as shown in Fig 1(d).The interplay between the T-independent in-trinsic heterogeneity (due to the particle sizeand mass distribution) and the dynamic het-erogeneity which builds up at low T seems tobe the microscopic origin of the anti-correlationbetween fragility and stretching at high T andthe observed crossover at intermediate T.To investigate this point in further details,we study the non-Gaussian parameter, α ( t )which also shows a correlation with fragilityfor most materials [22]. The non-zero values of α ( t ) in a monodisperse system is purely due tothe presence of dynamic heterogeneity whereasin polydisperse system, in addition to dynamicheterogeneity, there is an intrinsic heterogene-ity due to particle size and mass distributionwhich is present at all T . Thus for the lat-ter, α ( t ) reflects a coupled effect of both theseheterogeneities. As seen in Fig 4, for a polydis-perse system α ( t ) is nonzero both in the shorttime limit (due to the mass distribution [20])and in the long time limit (due to the spreadin diffusion coefficients with particle size and3ass). At high T , the non-zero value of α ( t )is predominantly due to the intrinsic hetero-geneity and thus increases with S (Fig 4(d)).As T is lowered, the effects of dynamic hetero-geneity starts to dominate, as was shown bythe onset of connected clusters of fast movingparticles [4, 21] whose size increases as one ap-proaches glass transition. Since the relaxationtime increases with decrease of S (Fig. 2(b)),there is a faster build-up of dynamic hetero-geneity at lower S which leads to the observedcrossovers (Fig 4(c)&(b)) in the values of α ( t )between different S (similar to that observedfor β in Fig 1(c)). Hence at low T , one gets thescenario where α ( t ) decreases with polydisper-sity (Fig 4(d)). Since fragility decreases with S,these crossovers in β and α ( t ) would mean thatfragility is correlated only to the dynamic het-erogeneity and not to the intrinsic heterogeneityin the system .When the polydispersity is increased at con-stant volume, we get results that are oppositeto that obtained from constant volume fractionstudies. We find that the dynamics slows down with increase in polydispersity [20]. This is be-cause at constant volume as polydispersity in-creases, the packing fraction increases [23] andhence we find a coupled effect of polydispersityand density.Our results show that at constant volumefraction, although the increase of polydisper-sity leads to a more efficient packing, the dy-namics become faster due to the lubricationeffect. 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