•   ARTICLES   • May 2018   Vol.61   No.5: 613–618https://doi.org/10.1007/s11426-017-9151-9

Structural relaxation and glass transition behavior of binaryhard-ellipse mixtures

Liang Wang1,2, Baicheng Mei1,2, Jianhui Song3, Yuyuan Lu1* & Lijia An1*

1State Key Laboratory of Polymer Physics and Chemistry, Changchun Institute of Applied Chemistry, Chinese Academy of Sciences,Changchun 130022, China

2University of Chinese Academy of Sciences, Beijing 100049, China3School of Polymer Science and Engineering, Qingdao University of Science and Technology, Qingdao 266042, China

Received September 25, 2017; accepted September 27, 2017; published online January 10, 2018

Structural relaxation and glass transition in binary hard-spherical particle mixtures have been reported to exhibit unusual featuresdepending on the size disparity and composition. However, the mechanism by which the mixing effects lead to these features andwhether these features are universal for particles with anisotropic geometries remains unclear. Here, we employ event-drivenmolecular dynamics simulation to investigate the dynamical and structural properties of binary two-dimensional hard-ellipsemixtures. We find that the relaxation dynamics for translational degrees of freedom exhibit equivalent trends as those observedin binary hard-spherical mixtures. However, the glass transition densities for translational and rotational degrees of freedompresent different dependencies on size disparity and composition. Furthermore, we propose a mechanism based on structuralproperties that explain the observed mixing effects and decoupling behavior between translational and rotational motions inbinary hard-ellipse systems.

binary ellipse mixture, molecular dynamics simulation, glass transition, translational relaxation time,rotational relaxation time

Citation: Wang L, Mei B, Song J, Lu Y, An L. Structural relaxation and glass transition behavior of binary hard-ellipse mixtures. Sci China Chem, 2018, 61:613–618, https://doi.org/10.1007/s11426-017-9151-9

1    Introduction

The physicalmechanism of glass transition in binarymixturesremains one of mysteries in condensed matter [1–7]. Signif-icant advances in our understanding of the glass state havebeen gained by the study of binarymixture models employingparticles of varying size ratios (i.e., the ratio of the character-istic dimension of the large particle to that of the small parti-cle) and component percentages, which are not only an idealprecursor of the glass state but also exhibit a wealth of dynam-ical features. A seminalmolecular dynamics simulation study

*Corresponding authors (email: yylu@ciac.ac.cn; ljan@ciac.ac.cn)

[8] showed that the characteristics of binary mixtures exertedan obvious influence on the dynamical and structural proper-ties of hard spherical colloidal suspensions when approachingglass transition. Soon after, calculations based on mode cou-pling theory (MCT) [9–11] confirmed that increasing themix-ing percentage of the smaller minority particles can increase(decrease) the rate of long-time relaxation for a large (small)size ratio, respectively. Evidence of this conclusion was fur-ther provided by molecular dynamics simulations of binaryhard spherical mixtures [12,13]. Additionally, MCT calcu-lations for both two and three-dimensional systems revealedthat mixing effects play distinct roles in relaxation dynam-ics that depend on the size ratio [14]. Moreover, it should bementioned that systems with very large size ratios exhibit rich

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dynamical features such as multiple glass transitions [15,16],sublinear diffusion of small particles [16,17], and the log-arithmic decay of the intermediate scattering function [18].However, explanations for a number of behaviors exhibitedby binary isotropic systems remain unclear. For example, thelong-time relaxation dynamics of systems with small size ra-tios tend to slow down as the percentage of small particlesincreases, which cannot be explained simply by plasticizingeffects [10].The emergence of anisotropic particle models in recent

years [19–23], such as ellipsoid and dumbbell particle sys-tems, has facilitated a renewed interest in the dynamicsof the glass transition because of the orientational degreesof freedom associated with these systems, which providesnew insight regarding the mechanism of glass transition.However, we note that most researches focused on, suchas the glass transition behavior conducted on the basis ofmonodispersed anisotropic systems [24–28]. Therefore, afull understanding of glass transition in binary anisotropicsystems is still lacking. Recently, our group [29] exploredsome phenomena induced by mixing effects in binarytwo-dimensional (2D) hard-ellipse systems; however, themechanism responsible for these newly discovered phenom-ena remains to be elucidated.In this work, we systematically investigate the structural

and dynamical properties of binary 2D hard-ellipse systemsusing event-driven molecular dynamics simulation. Our re-sults indicate that, for translational degrees of freedom, thestructural relaxation and glass transition of binary hard-el-lipse mixtures display similar dependences on the size ra-tio and percentage of small particles as those observed inhard-spherical mixtures. However, for the rotational degreeof freedom, the system dynamics exhibit a number of uniquefeatures, and the glass transition density shows unusual de-pendences on the size ratio and percentage of small particles.Furthermore, structural analysis reveals that the dynamicalproperties induced by mixing effects are correlated with theformed local structure of the system, which shows a depen-dence on the size ratio. Finally, we propose a novel mecha-nism responsible for the mixing effects in anisotropic particlesystems.

2    Simulation model and method

All event-driven molecular dynamics simulations of binary2D hard-ellipse mixtures employed an NVT ensemble (con-stant number of particles N, volume V, and temperatureT), and were conducted in a square box of dimension Lunder periodic boundary conditions [30,31]. The binarymixture consisted of large and small particles denotedby A and B, respectively, and N=900. The control pa-rameters that specify our systems include the size ratio

a a b b= / = /A B A B, where a and b (with α representing

particles A or B) denote the semi-major and semi-minoraxes of the particles, respectively, and the aspect ratiosof all particles are fixed at 2 (i.e., a b= 2 ), the relativeconcentration of small particles x N N N= / ( + )s B B A , whereNα denotes the number of particles A or B, and the packingfraction N a b N a b L= ( + ) /A A A B B B

2. All particles haveequivalent masses m and moments of inertia I, which areboth equal to 1 [29,32]. Moreover, we set kBT=1, wherekB is Boltzmann’s constant, because the temperature isirrelevant for a hard-particle system. Lengths and timescales are reported in units of a b4 A A and a b m k T4 /A A B ,respectively. The initial configuration was created usingan extended Lubachevsky-Stillinger compression algorithm[33,34]. Prior to data collection, the initial configuration wasequilibrated for a period of at least 3 times greater than thelongest of the characteristic translational relaxation time T,which is defined in detail later.

3    Results and discussion

3.1    Relaxation dynamics

The relaxation dynamics were mainly explored ac-cording to the self-intermediate scattering function

{ }F q t iq r t r N( , ) = exp [ ( ) (0)] /j

N

j js, = 1 , , andthe n-th order orientational correlation function

{ }L t in t N( ) = exp [ ( ) (0)] /n j

N

j j, = 1 , , , where q isthe wave number, n is a positive integer, rα,j and θα,j are theposition and orientation of particle j belonging to species α,respectively, and t is time. Here, we set q=6.0, correspondingto the first peak of the static structure factor for largeparticles, and n=3. It should be noted that the selected valuesof q and n do not affect the qualitative results and conclusionsregarding the relaxation dynamics [29,32]. The relaxationdynamics of large particles associated with Fs,A and L3,A areshown in Figure 1 with two different values of δ and threedifferent values of xs. We observe that both Fs,A and L3,Apresent two-step relaxation characteristics, indicating thatthe system is approaching glass transition. With increasingxs, the short-time translational relaxation dynamics slowdown for δ=3.33; however, the long-time translationalrelaxation dynamics speed up. Meanwhile, for δ=1.25, thelong-time translational relaxation dynamics continue toexhibit a general trend of slowing down with increasing xs.Similar trends were observed in other studies [10,12,29].However, there is no clear theoretical argument from pastliterature as to which mixing effects lead to these features.The observed speeding up of the long-time translationalrelaxation dynamics for δ=3.33 may be attributable to thestabilization of the liquid arising from a plasticizing effectinduced by  mixing  under  constant   ,  where the effectivearea fraction and excluded area decrease with  increasing  xs,

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Figure 1         (a, b) Self-intermediate scattering function Fs,A at q=6.0 and (c, d) 3rd order orientational correlation function L3,A for large 2D ellipse-shaped particlesat various relative concentrations of small particles xs for two different size ratios δ=3.33 and δ=1.25. The insets in (a) and (c) present the behaviors of Fs,A andL3,A at short-time scales. The packing fraction is fixed at 0.86 (color online).

which consequently decreases the viscosity, and thus facil-itates the relaxation dynamics [10,35]. Nonetheless, thisplasticizing effect provides no benefit for interpreting theshort-time and long-time translational relaxation dynamicsfor δ=1.25. For L3,A, we find that the rotational relaxationdynamics exhibit an equivalent xs dependence as that of thetranslational relaxation dynamics. Additionally, for bothtranslational and rotational degrees of freedom, the long-timerelaxation dynamics of the system with δ=1.25 are slowerthan those with δ=3.33 (more details see Figure S1 in theSupporting Information online).

3.2    Structural features

To elucidate the mixing effects, we analyzed the structuralfeatures of this binary system by calculating the pair corre-lation function gαβ(r), as shown in Figure 2, where r is thedistance between the centers of mass of the particles, α andβ are particle species. We note that each gαβ(r) is generallydivided into 3 peaks with respect to r. Numerical analysisshows that this is clearly related to the fact that only threetypes of packing between particle pairs are possible in 2Dspace, which are represented by contiguous packing alongthe long axes (denoted as head-head packing), short axes(denoted as shoulder-shoulder packing), and between thelong and short axes (denoted as head-shoulder packing), asillustrated in Figure 2. Obviously, head-head and shoulder-shoulder packing are prone to denser and more organizedlocal structures compared with head-shoulder packing. Thus,head-head and shoulder-shoulder packing types betweenlarge particles (i.e., gAA(r)) are preferred at xs=0.7, particu-larly for δ=1.25 (see Figure S2). In comparison, when xs=0.5,

the prominence of head-shoulder packing between largeparticles increases for both δ=3.33 and δ=1.25. Therefore,in binary 2D hard-ellipse systems, the particles are morelikely to be arranged in a more locally dense manner withincreasing xs, suggesting that the cage effect is much strongerwhen glass transition occurs. As a result, the short-timerelaxation dynamics would slow down with increasing xs.Furthermore, as shown in Figure 2, gBB(r) exhibits only a

single peak at the position corresponding to head-shoulderpacking (see Figure S3) and gAB(r) shows no peaks at the posi-tion corresponding to head-head packing (see Figure S4) forδ=3.33, which implies that the small particles prefer to as-semble together (the total structure factor S(q) for systemswith different values of δ are shown in Figure S5). To provethis analysis, we present the configurations of the binary 2Dhard-ellipse systems. As shown in Figure S6, the local struc-tures formed in the δ=3.33 and δ=1.25 systems are distinct.In the case of δ=3.33, the small particles reside within cavi-ties formed between the large particles, which can trigger theplasticizing effect, leading to a speeding-up in the long-timerelaxation dynamics with increasing xs. However, for theδ=1.25 system, the difference between large and small par-ticles is negligible, and the local structure of such a binarymixture is more dense and ordered. Thus, the plasticizing ef-fect can be ignored and the long-time relaxation dynamics areobserved to slow down with increasing xs in such a system.

3.3    Coupling and decoupling behavior of glass transition

To further examine the properties of binary anisotropic sys-tems,  we  computed  the  glass  transition  density   c  under

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Figure 2         Pair correlation functions in binary 2D hard-ellipse systems for (a, b) large particles gAA(r), (c, d) small particles gBB(r), and (e, f) between large andsmall particles gAB(r) at different values of xs for δ=3.33 and δ=1.25 (color online).

different values of δ. The values of c for both transla-tional and rotational degrees of freedom were extractedfrom MCT-based -dependence power-law fittings, i.e.,

( )c , where c and γ are both fitting parameters,and τ represents the translational and rotational relaxationtimes τT and τθ, which are defined as F q t( , = ) = 0.1Ts, andL t( = ) = 0.13, [24], respectively. The values of T

1/ and1/ are plotted in Figure 3 for the systems with δ=3.33 and

δ=1.25, where = = 0T1/ 1 / when = c, such that the

extracted value of is indicated by the intercept of the fittedfunction with the abscissa axis. As shown in Figure 3(a, b),the values of c for the translational degrees of freedom ofthe large and small particles decouple at large δ, while the

c values tend to remain coupled at small δ, which is equiv-alent to that observed in binary hard-spherical systems [15].This phenomenon associated with the δ=3.33 system can beexplained by the preference for small particles to distributewithin the cavities formed by the large particles, as discussedregarding Figure S5(a). Therefore, small particles remainfree to move even though the large particles are frozen, andthe values of c for large and small particles decouple. Incomparison, the local structures of the δ=1.25 system arewell organized, resulting in a coupling of c. As shown inFigure 3(c, d), the values of c for large and small particlesin terms of the rotational degree of freedom exhibit the samedependence on size ratio as that observed in translationaldegree of freedom. However, the  extent  of  decoupling  for

the rotational degree of freedom is stronger than that for thetranslational degree of freedom at size ratio δ=3.33. Accord-ing to Stokes-Einstein and Stokes-Einstein-Debye relations,we can obtain D dT T

1 for translational degrees of free-dom, whereas D d 31 for the rotational degree of free-dom, where d denotes the particle size, and DT and Dθ are thediffusivities for translational and rotational motion, respec-tively, assuming equivalent diffusivities for large and smallparticles [26]. Therefore, rotational motion has a stronger de-pendence on the particle size than does translational motion.Moreover, the translational and rotational glass transition ofsmall and large particles becomes coupled at small δ becausethe translational and rotational motions for large and smallparticles are more cooperative due to the well-organized lo-cal structures. Meanwhile, at large δ, local structures are lessdense and rotational motion exhibits a stronger dependenceon large particles than does translational motion. Thus, thedecoupling behavior for the rotational degree of freedom isstronger than that for the translational degree of freedom atlarge δ (see Figures S7 and S8).

3.4    The dependence of glass transition behavior on sizeratio

Next, we compared the values of c obtained for the largeparticles of binary 2D hard-ellipse systems at different δ withthat of a monodispersed counterpart, denoted as 0. Relative

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Figure 3         (a, b) T1/ and (c, d) 1 / as a function of for δ=3.33 and δ=1.25 systems with xs=0.5. The lines are fitting based on MCT (i.e., ( )c ,

where γ and c are fitting parameters) (color online).

variations in the values of c (i.e., = ( ) /c-0 c 0 0) for bothtranslational and rotational degrees of freedom as a functionof xs are presented in Figure 4. We note that the values of

c-0 for the translational degrees of freedom of large particlesat relatively large δ present a single maximum pattern, where

c-0 increases with increasing xs before decreasing. Moreover,head-shoulder packing between small particles tends to be-come increasingly prevalent with increasing xs, which causesthe particles to become increasingly arranged in a loose man-ner and triggers the plasticizing effect. In spite of this, most ofthe small particles at relatively large δ are distributed withinthe cavities formed by large particles, which facilitates smallparticle motion affecting the long-time relaxation dynamics.Nevertheless, at small δ, the values of c-0 for the translationaldegrees of freedom present a single minimum pattern, whichindicates that the glassy state is stabilized with increasing xsup to about xs=0.7. This is due to the fact that more head-headand shoulder-shoulder packing between particles is obtainedin a systemwith small δ. Thus, the particles are arranged withgreater local density, and the plasticizing effect does not dom-inate in the system. In this regard, we note that the systemswestudied here are surpercooled liquids, whose temperature ap-proaches the dynamic glass transition temperature predictedby MCT theory, which is much greater than the thermody-namic glass transition temperature. As for the intermediateδ=1.67 system, the variation in the values of c-0 for the trans-lational degrees of freedom of large particles evolves froma single maximum or minimum pattern to an S-shape pat-tern, suggesting that the two mixing effects discussed abovecompete. In addition, it is worth noting that the aforemen-tioned results are not entirely equivalent with those of binaryhard-sphere and disk systems based on MCT [14]. The dif-ference is that the values of c-0

Figure 4         Relative variation in the glass transition density for (a) the transla-tional degrees of freedom and (b) rotational degrees of freedom for the largeparticles of 2D hard-ellipse systems. The terms 0 and c respectively denotethe glass transition densities for a monodispersed system and binary hard-el-lipse systems with different values of δ (color online).

for the translational degrees of freedom of large particles areasymmetric with respect to xs=0.5, implying that the effects oflarge and small particles on glass transition are not identical.For the rotational degree of freedom shown in Figure 4(b),

we find that the mixing effects solely stabilize the liquid statewith increasing xs up to about xs=0.7, suggesting that the ad-dition of a greater percentage of small particles tends to hin-der a decrease in the overall rotational movement of a binarymixture relative to that obtained in a monodispersed system.This phenomenon can be investigated by calculating the an-gular correlation function g2(r) (see Figure S9), which in-dicates that the short-range orientational orders g2,BB(r) and

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g2,AB(r) decrease with increasing δ. Therefore, the particlesin binary hard-ellipse mixtures move more freely than thosein a monodispersed system, and the value of c for the rota-tional degree of freedom shifts to a higher value.

4    Conclusions

In conclusion, we have studied the relaxation dynamics, glasstransition, and structural properties of binary 2D hard-ellipsesystems with different values of δ via event-driven molecu-lar dynamics simulation, and several intriguing features havebeen obtained. Our results reveal that the dependence of re-laxation dynamics on size disparity and composition for bothtranslational and rotational degrees of freedom exhibits anequivalent trend as that observed in binary hard-spherical sys-tems. This equivalence also applies to the dependence of

c on δ for translational and rotational degrees of freedom.However, the extent of decoupling of the glass transition den-sity for the rotational degree of freedom is larger than that forthe translational degree of freedom at large δ. In contrast toa monodispersed hard-ellipse system, the mixing effects ofbinary systems appear to be more complicated due to varia-tions in the local patterns of particle packing, which stabilizeboth glass and liquid states in terms of the translational de-grees of freedom, depending on δ, but solely favor the liquidstate in terms of the rotational degrees of freedom. Further-more, our results reveal that the addition of small particlesresults in a denser and more organized local structure, whichenhances the cage effect and causes a slowing down of theshort-time relaxation dynamics in both translational and ro-tational degrees of freedom. As for the long-time relaxationdynamics, binary hard-ellipse systems prefer to adopt moreorganized local structures with decreasing δ, which weakensthe plasticizing effect. Thus, the mixing effects stabilize theglass state in systems with relatively small δ. Moreover, itwas found that the small particles distribute within the cavi-ties formed by large particles at large δ, but exhibit a denselocal structure with large particles at small δ. This behav-ior explains not only the dependence of c on δ, but also theδ-dependence of the long-time relaxation dynamics. Finally,the addition of small particles strongly disrupts the originalorientational order, leading to an increase in c in terms ofthe rotational degrees of freedom. Although our research in-vestigated the influence of mixing effects on the relaxationdynamics and glass transition of binary anisotropic systemsin two dimensions, it would be worthwhile to explore theseeffects on anisotropic systems in three-dimensional space, be-cause small particles might have more latitude to diffuse outof their cages in 3D space. We expect our work to stimulatefurther experiments, theory, and simulations in this regard.

Acknowledgments            This work was supported by the National Natural Sci-ence Foundation of China (21474109, 21674055), the International Partner-ship Program of Chinese Academy of Sciences (121522KYSB20160015),and the Youth Innovation Promotion Association of Chinese Academy ofSciences (2016204).

Conflict of interest             The authors declare that they have no conflict ofinterest.

Supporting information            The supporting information is available onlineat http://chem.scichina.com and http://link.springer.com/journal/11426.The supporting materials are published as submitted, without typesettingor editing. The responsibility for scientific accuracy and content remainsentirely with the authors.

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