Relation between local density and density relaxation near glass transition in a glass forming binary mixture
RRelation between local density and density relaxation near glass transition in a glassforming binary mixture
D. C. Thakur, Jalim Singh, Anna Varughese, and Prasanth P. Jose ∗ School of Basic Sciences, Indian Institute of Technology Mandi, Kamand, Himachal Pradesh 175005, India
We employ molecular dynamics simulation to elucidate the microscopic origin of slow relaxationin supercooled Lennard-Jones liquids. These studies show a direct correlation between the growthof the highest peak of the radial distribution function g ( r ), radial force from this peak, and densityrelaxation. From the available free-volume of a neighboring pairs for string-like motion, we derive arelation between surface density of the spherical shell ρ loc around a reference particle at the highestpeak of g ( r ) and density relaxation τ α , i.e. τ α ∝ exp ( ρ / ( ρ − ρ loc )), where dynamics diverge at ρ that is akin to Vogel–Fulcher–Tammann (VFT) relation connecting τ α and temperature. Thisrelation works well in the supercooled regime of the model systems simulated. Fluids undergo glass transition under rapid compres-sion or cooling at a faster rate than structural relax-ation. Formation of transient molecular-cages duringglass transition results in the slowing down of density re-laxation. Recent experiments in two-dimensional colloidsshow that the molecular-cages have higher local density[1], where cage size fluctuation facilitates the density re-laxation [2]. Therefore, theories of dense or viscous liq-uids are useful to study the glass transition. The per-turbation theory of dense Lennard-Jones (LJ) liquids byWeeks, Chandler, and Andersen (WCA) shows that re-pulsive interactions govern the structure of dense liquids[3], besides, the dynamics also follows the same [4]. A testof WCA theory in simulations of Kob-Andersen (KA) bi-nary mixture [5] with and without attractive (KAWCA)interactions near the glass transition [6, 7] show that therelaxation dynamics is remarkably slower with attractiveinteractions at number density ρ =1.2, often studied forglass-transition in the KA model [8, 9]; the dynamics ofKA and KAWCA models are nearly identical at higherdensity ρ =1.8. Studies also show that the relaxation dy-namics becomes identical when the range of interactioninclude the whole first coordination shell [10]; besides,the attractive and repulsive interactions together form amean fluctuating potential of the cage, irrespective of thetype; identical cage potential leads to same density relax-ation [11]. Therefore, characterization of type, range, anddepth of interactions in viscous liquids are important inunderstanding of molecular-cages. There are many laterstudies with different approaches that address the ori-gin of this difference in dynamics of supercooled KA andKAWCA models.Such a study based on an explicit analysis of forcesshows that the difference between mean forces on par-ticles in KA and KAWCA models are different whendynamics differ [12, 13]. Another study shows that theconfigurational entropy obtained from the radial distri-bution function g ( r ) [14] leads to difference in dynam-ics, thus g ( r ) governs relaxation dynamics [15]. Besides,machine learning studies show that g ( r ) help in inter-preting the dynamics [16]. Interestingly, an exponential relation between the relaxation dynamics of the systemand order parameter derived from the relative angle be-tween atoms in the first coordination shell, which is sim-ilar to Vogel–Fulcher–Tammann relation [17]. Insightsfrom studies that show the importance of first coordina-tion shell [10, 17] and importance of g ( r ) [15, 16] suggestthat g ( r ) at the first coordination shell has a decisiverole in determining the relaxation dynamics. Moreover,the schematic mode-coupling theory use the value of thestructure factor at the first peak (corresponding to thelength scale of nearest neighbor separation) [18, 19] incalculations of relaxation time. Besides, studies showthat short-time β relaxation and α relaxation have iden-tical temperature dependence [20] connecting short timedynamics arising from local structure to the overall re-laxation dynamics, which is also identified by the infor-mation theory [21]. Another study on gas-supercooledcoexistence in polymers show that there is a correlationbetween density relaxation and particle distribution inthe first coordination shell [22]. The transient cage for-mation leads to increase in local density [1] that reducesavailable free-volume for the molecules to relax, there-fore the relation between structure and dynamics can beaddressed from the free-volume theory [23]; for a hard-sphere system the density relaxation τ α ∝ exp( Bv /v f ),where v f = v − v is the difference between mean free-space and self-volume [24]. The free-volume have a vaguedefinition for continuous potentials with different repre-sentations such as Voronoi volume [25] or a detailed clas-sification of vibrationally accessible and hardcore free vol-ume [26] etc. This study deduces a relation connectingthe surface density ρ loc of a spherical shell with radius r ∗ at the peak of g ( r ) (cage density that represent v f ) andrelaxation time, from a systematic analysis of variationin g ( r ) with temperature. Methods:
Systems with KA [5] and KAWCA [7] po-tentials at number densities ρ =1.2 to 1.8 in grid of δρ =0.2 are simulated in uneven temperature grid fromhigh to low temperatures. The KA potential reads: V αβ ( r ) = 4 (cid:15) αβ (cid:104) ( σ αβ /r ) − ( σ αβ /r ) (cid:105) + K αβ , where pa-rameters are α, β ∈ { A, B } , (cid:15) AA = 1 . , σ AA = 1 . , (cid:15) AB = a r X i v : . [ c ond - m a t . s o f t ] F e b r g (r) (a) OAAAABBB r g (r) (e) OAAAABBB r F (r) (b)=1.2,T=0.450.6 0.8 1 1.2 r F (r) (f), =1.8,T=3.0 10 t -2 -1 r ( t ) (c)10 t -2 -1 r ( t ) (g) 10 t F s ( k ,t ) (h)10 t F s ( k ,t ) (d) Figure 1. Properties of two state points are compared inupper and lower panels: (a) and (e) shows over all (OA) andpartial radial distribution functions g αβ ( r ) ; (b) and (f) cor-responding variation in the radial force F αβ ( r ); (c) and (g) (cid:104) ∆ r ( t ) (cid:105) ; (d) and (f) F s ( k, t ) in typical supercooled states.Legends are the same in all panels. . , σ AB = 0 . , (cid:15) BB = 0 . , and σ BB = 0 .
88 [5] . K αβ en-sures zero potential at the cut-off. Length, temperatureand time are expressed in units of σ AA , σ AA (cid:112) m/(cid:15) AA ,and (cid:15) AA /k B . The KA model has potential cut-off of2 . σ αβ whereas KAWCA model is cut-off at minima2 / σ αβ . The simulated system has N = 4000 particleswith A to B ratio 4:1. Molecular dynamics using velocityVerlet algorithms simulate systems in the microcanonicalensemble [27]. Initial configuration is prepared at T = 8for densities ρ = 1.2, 1.4, and 1.6, and T = 12 for density ρ = 1.8 and quenched to the desired temperature. Thesystem is equilibrated till 20 τ α or more at all tempera-ture.The decrease of temperature results in the growth of g ( r ) in the first coordination shell; thus, shows an in-crease of local density. Various contribution to peaksof g ( r ) from partial g αβ [8] in Figs. 1(a) and (e) at( ρ, T ) =(1.2,0.45) and (1.8,3) (two representative super-cooled states) show that the first peak of g ( r ) has mostcontribution from g AB (peak of g BB ( r ) is weak), which issmaller than the second peak that is mostly from g AA ( r ).The first peak of g AB ( r ) is the tallest among g αβ ( r ) dueto larger (cid:15) AB ; however, due to lesser AB pairs contributeto smaller first smaller peak in g ( r ). The tallest peak of g ( r ) is from g AA ( r ), which is the correlation among majorcomponent A . At these two representative state points,at ρ =1.2, the g ( r ) of KA and KAWCA models differ,while at ρ =1.8, they match; the relaxation dynamics fol-lows the same trend - see discussion on relaxation dynam-ics later. To confine a reference particle, the cage formedby neighboring particles exert a fluctuating force, themean of this force at r give a measure of the confinement -the net force on the molecule in the radial direction is zeroin equilibrium. The mean force exerted on a particle bydensity at r in radial direction is F αβ ( r )ˆ r = f αβ ( r ) n αβ ( r ), where the force f αβ ( r ) = − ∂V αβ ( r ) ∂r ˆ r , V αβ ( r ) is any of thethree types of different potentials given above, n αβ is un-normalized pair density of the each species. The meannormalized force F ( r ) = Nρ (cid:80) αβ | F αβ ( r )ˆ r | gives averageradial force on a reference particle; correlation of normal-ized F αβ ( r )ˆ r is used in mode-coupling theory studies [28]to compute the memory kernel. The F ( r ) at the positionof the highest peak in Fig. 1 at ρ =1.2 considerably differfor KA and KAWCA mainly due AB component, whichhave large negative contribution that reduces at ρ =1.8that also correlates with variation in the relaxation dy-namics [29]. The difference in mean force on particles inthese models show difference in relaxation in earlier stud-ies [12, 13]. Next, we explore a relation between featuresof density relaxation and static channels of relaxation inthe cages of KA and KAWCA models.Owing to smaller σ AB and σ BB [see Figs. 2(a) and(b)], the mean square displacements (cid:104) ∆ r ( t ) (cid:105) = (cid:104)| r i (0) − r i ( t ) | (cid:105) ( r i ( t ) - position vector of i th particle at time t )of A is slower than B in Figs. 1(c) and (g). Relaxation ofany particle is due to motion of particles hindered by thecage, which implies higher energy barrier from A than B in a cage; therefore, B is a channel of relaxation ina cage at all densities. The overall density relaxation isgoverned by A due to relatively slow motion. In Figs. 1(c) and (g), (cid:104) ∆ r ( t ) (cid:105) of A and B for KA and KAWCAmodels differ at lower density ρ =1.2, while they nearlymatches at higher density ρ =1.8, where F ( r ) and g ( r )also coincide for KA and KAWCA models. Therefore,at ρ =1.2 B particles that act as a channel of relaxationin a cage is different in the KA and KAWCA modelsdue to attractive interactions, while they are nearly thesame for ρ =1.8 at a very high density. Besides, effectof attraction varies with location of r ∗ , with respect to r ∗ αβ = 2 / σ αβ , which is the cut-off distance of attrac-tive forces in KA model; they are: r ∗ AA ∼ r ∗ AB ∼ r ∗ BB ∼ ρ =1.2 at low T , r ∗ ∼ . > r ∗ AB , r ∗ BB , where forces in KA and KAWCAmodels differ due to attractive forces in Fig. 1(b), whilefor ρ =1.8, r ∗ BB > r ∗ ∼ . ∼ r ∗ AB both models haveidentical F ( r ) in Fig. 1(f) due to reduction of the at-tractive forces. The relaxation of incoherent interme-diate scattering function [14] F αs ( k, t ) = (cid:104) ρ α k ( t ) ρ α − k (0) (cid:105) ,where ρ α k ( t ) = e i k · r αi ( t ) at k =7.25 [6], where k is themean wavenumber at the first peak of S ( k ), show sim-ilar variation of in the relaxation time. Thus, the mi-croscopic structure of the cage in the first coordinationshell is related to the density relaxation. Therefore,we look for a relation between average relaxation time τ α ( τ α = (cid:82) ∞ F s ( k, t ) dt with F s ( k, t ) is averaged over F αs ( k, t )) and peak of g ( r ) from the microscopic mecha-nism of relaxation.Typical arrangement of atoms in Figs. 2(a) and (b) atthe first coordination shell near potential energy minimaresult in a dense spherical shell [1]. The inspiration ofmicroscopic mechanism of rearrangement in such a sys- Figure 2. (a) Typical 3d of distribution of KA/KAWCA par-ticles of a cage at ρ = 1.2 ( r A ∝ σ AA and r B ∝ ( σ AB + σ BB ) / g ( r ). (c) Variation of r ∗ (radius of dashed circle in (b)) with T . /( - loc ) l og ( / ) =1.2=1.4=1.6=1.8 Figure 3. Collapsed plot of logarithm of scaled relaxationtime versus relative variation of the local density ρ loc for KA(black) and KAWCA(red) models. -1 , l o c -1 -1 , l o c -1 Figure 4. τ α in semilog (left Y scale) versus 1 /T , by circlesconnected by VFT fit and ρ loc in linear scale(right Y scale)versus 1 /T as (squares), lines are Eq. 2, for KA (black) andKAWCA (red) models tem is from the string-like motion [30–33], where atomsmove among nearly identical environment with a slightexpansion of the shell [2] because the energy cost to cre-ate a hole is high, the moves that contribute predomi-nately to the relaxation is along the black arrows in Fig.2(b). Various energy barriers associated with a typicalstring-like motion include the energy of a hole, entropyof the statistical distribution of strings etc. [34, 35]. Inthis model, a short move marked in a black arrow in Fig.2(b), where the displacement of a member of a cage alongthe line joining cage particle and the reference particle isthe primary mode of relaxation. Such paths connect-ing many particles lead to a typical string-like collectivemotion (green arrows in Fig. 2(b)). The free-energy bar-rier E from the mean-field theory for polymers [34–36]along the string of length n in available free-volume v m is E ∝ n /v m ; here we consider only pairs. Such con-nected pairs in three dimensions generate a string-like orring-like collective motion along the high-density path ona two-dimensional (2D) surface marked by the r ∗ that istypical in glasses [37]. All moves are along the quasi-linear path of inter-penetrating 2D surfaces marked asdotted circles in the one-dimensional (1D) projection inFig. 2(b). A study on the glass transition in polymerfilms give details of the barriers in typical string-like mo-tion [38]. Based on the mean-field energy barrier of a pairand free-volume theory suggest that free-volume avail-able in the shell of radius r ∗ controls the dynamics, asevery such path is through the shell around some otherparticle. Short strings at higher temperature supercooledliquid become longer as temperature reduces [39], whichis considered as cooperatively rearranging regions[40–42]in theories of glass transition [19].The free-volume change in the spherical shell has twocontributions δV = δρN + δN ρ in terms of density ρ and number of particles N . The reduction of free volume δV = v − v in a spherical shell as the system approach atransition is due to the increase of particles. That is ap-proximated as ( ρ − ρ loc ) N m , where N m is the mean shelloccupation in the supercooled state. δN is variation ofnumber of particles in a spherical-shell of infinitesimalthickness with r ∗ (cid:39) constant (fig. 2(c)) that is negligi-ble. Then the relaxation time from free-volume theorymodified as function of ρ loc reads τ α = τ ρ e (cid:16) Bv v − v (cid:17) (cid:39) τ ρ e (cid:16) Bρ ρ − ρloc (cid:17) . (1)A fit of this relation in KA and KAWCA models for ρ =1.2, 1.4, 1.6, and 1.8 in Fig. 3 show B ∼
1, which isconsistent with experiments [43]. The ρ for respectivedensities in the order above are 3.9,4.5,5.2, and 5.9 forKA and 4.6,4.8,5.3 and 5.9 show a systematic increaseof ρ and they both approach each other at high den-sity. The Eq. 1 is verified in glass forming linear flex-ible polymers near supercooled state [44]. As the Eq.1 is similar to Vogel–Fulcher–Tammann (VFT) relation τ α = τ T exp( A/ ( T − T )). Equating the Eq. 1 and VFTexpression yields relation between ρ loc and temperaturedifference ρ loc = ρ − ρ ( T − T ) A + ln ( τ T /τ ρ )( T − T ) . (2)The variation of ρ loc and inverse of temperature in Fig. 4show linear variation with concomitant logarithmic vari-ation of τ α . In the supercooled liquids Eq. 2 is valid; itis clear from the KAWCA model at ρ =1.2 where cagesbecome stable at only low temperatures, where the plotof Eq. 2 in Fig. 4(a) matches with variation of ρ loc .This study based on the simulation of KA andKAWCA models at different densities in temperaturegrids from high to low temperature near supercooledstate look into cage formation with and without attrac-tive interactions. The analysis shows that the variationin the local density and associated forces are related tothe relaxation dynamics, which is consistent with earlierinvestigations [10, 11, 17, 22]. This model based on ashort string like motion shows the connection betweenthe density in the first-coordination shell and the relax-ation dynamics. The VFT-like structure of the Eq. 1consistent with a similar relation for an order parame-ter derived among angle between particles in the firstcoordination shell [17] which is valid in 2D as well; there-fore, the results of this study provide a model for 2Dand 3D nonassociated liquids [45]. The enhancement ofthe first peak also found in repulsive 3D colloidal sys-tems that are reminiscence of molecular jamming similarto the repulsive colloids, where g ( r ) grows systematicallynear jamming [46]. Force transmission in attractive bio-polymers also shows jamming [47], it is worthwhile tohave more detailed investigations on this aspect. Theeffect of attractive force is the modification of cagingpotential; thus, ρ loc are different in KA and KAWCAmodels with attractive forces. 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