Collective Diffusion of Colloidal Hard Rods in Smectic Liquid Crystals: Effect of Particle Anisotropy
Alessandro Patti, Djamel El Masri, René van Roij, Marjolein Dijkstra
CCollective Diffusion of Colloidal Hard Rods in Smectic LiquidCrystals: Effect of Particle Anisotropy
Alessandro Patti ∗ , Djamel El Masri, and Marjolein Dijkstra † Soft Condensed Matter Group, Debye Institute for NanoMaterials Science,Utrecht University, Princetonplein 5,3584 CC, Utrecht, The Netherlands
Ren´e van Roij
Institute for Theoretical Physics, Utrecht University,Leuvenlaan 4, 3584 CE, Utrecht, The Netherlands (Dated: November 5, 2018)
Abstract
We study the layer-to-layer diffusion in smectic-A liquid crystals of colloidal hard rods with differ-ent length-to-diameter ratios using computer simulations. The layered arrangement of the smecticphase yields a hopping-type diffusion due to the presence of permanent barriers and transient cages.Remarkably, we detect stringlike clusters composed of inter-layer rods moving cooperatively alongthe nematic director. Furthermore, we find that the structural relaxation in equilibrium smecticphases shows interesting similarities with that of out-of-equilibrium supercooled liquids, althoughthere the particles are kinetically trapped in transient rather than permanent cages. Additionally,at fixed packing fraction we find that the barrier height increases with increasing particle anisotropy,and hence the dynamics is more heterogeneous and non-Gaussian for longer rods, yielding a lowerdiffusion coefficient along the nematic director and smaller clusters of inter-layer particles thatmove less cooperatively. At fixed barrier height, the dynamics becomes more non-Gaussian andheterogeneous for longer rods that move more collectively giving rise to a higher diffusion coefficientalong the nematic director.
PACS numbers: 82.70.Dd; 61.30.-v; 87.15.Vv ∗ [email protected] † [email protected] a r X i v : . [ c ond - m a t . s o f t ] A p r . INTRODUCTION Liquid crystals (LCs) are states of matter whose properties are in between those of acrystalline solid and an isotropic liquid phase [1]. They are usually classified in terms ofpositional and orientational order. Nematic LCs exhibit long-range orientational order,as the anisotropic particles are on average aligned along a preferred direction, but theylack long-range positional order. Smectic phases consist of stacks of fluid-like layers oforientationally ordered particles, where each layer is often considered to be a two-dimensionalfluid. Onsager showed in his seminal contribution the existence of a purely entropy-drivenisotropic-to-nematic (I-N) transition in a system of infinitely long hard rods [2]. Moreover,the I-N transition was confirmed by computer simulations for systems of hard rods with finite length. Additionally, Frenkel and coworkers explored the formation of smectic LCsof perfectly aligned [3] and freely rotating [4] hard rods, and found a thermodynamicallystable smectic phase of hard rods as a result of entropic effects. The equilibrium propertiesof smectic LCs are well-studied and are well-understood by now [5]. Experimental [6–8],theoretical [9–14], and computational [15, 16] studies have analyzed the phase behavior andstructure of smectic LCs of colloidal hard rods. Other investigations involve extensions tobinary mixtures with rods of different geometry [17–23], with other anisotropic [24–27] orspherical colloidal particles [28–31], or with non-adsorbing polymer as depletants [32–34].By contrast, the dynamics on a single-particle level in smectic LCs have only recentlyreceived attention, although an early study on the diffusion (or ”permeation”) of anisotropicparticles had already been reported more than forty years ago by Helfrich in order to ex-plain the capillary flow in cholesteric and smectic LCs [35]. Substantial advances in newexperimental techniques (e.g. NMR coupled to strong magnetic field gradients [36] or flu-orescent labeling [37]) disclosed the non-Gaussian nature and quasi-quantized behavior ofthe layer-to-layer diffusion. These achievements sparked off new theoretical work, basedon dynamic density functional theory, which not only confirmed the non-Gaussian layer-to-layer hopping-type diffusion and the presence of permanent barriers due to the staticsmectic background, but also showed the relevance of temporary cages due to the mutualtrapping of neighboring particles [38, 39]. We note that non-Gaussian dynamics due to arattling-and-jumping diffusive behavior is common in two-dimensional liquids [40], clustercrystals [41], and glasses [42], and has also been observed for the diffusion of a single par-2icle in a periodic external potential [43]. It is therefore not surprising to observe similarbehavior in smectic LCs. Our simulations of parallel [44] and freely rotating [45] hard rodssupported indeed these conclusions, but unveiled in addition a striking analogy with thenon-exponential structural relaxation and non-Gaussian dynamics of supercooled liquids.The non-exponential relaxation of the density fluctuations might be due to either a het-erogeneous scenario with particles relaxing exponentially at different relaxation rates, or a homogeneous scenario with particles relaxing non-exponentially at very similar rates [46].Here, we investigate the collective motion of fast-moving rods in stringlike clusters. Coop-erative diffusion, which accounts for the non-exponential decay of the correlation functions,yields an intriguing link between the dynamics observed in equilibrium smectic LCs and thatof out-of-equilibrium supercooled liquids. As far as smectic LCs are concerned, this remark-able collective motion could not be captured by the one-particle analysis of Ref. [43], andwas not observed in Ref. [47]. Two-dimensional liquids of soft disks [40] and cluster crystals[41] did not show this feature either. By contrast, several experimental and computationalstudies on glassy systems reported the existence of structural heterogeneities [48–52]. Inparticular, Glotzer and coworkers performed molecular dynamics simulations on a fragile glass-forming liquid and detected cooperative motion of stringlike clusters with an increas-ing string length of up to ∼
15 particles by cooling the system towards the glass transition[48]. Similar results were observed more recently in silica, a strong glass-former, suggestingthat stringlike motion is a universal property of supercooled liquids [52].In this paper, we investigate the effect of anisotropy of the rods on the non-Gaussianlayer-to-layer diffusion and cooperative motion of stringlike clusters in bulk smectic LCs offreely rotating hard rods. In supercooled liquids, it is generally accepted that in the case ofheterogeneous dynamics, the cooperative motion of particle clusters plays a crucial role inthe structural relaxation [53]. We argue that a similar behavior can be observed in smecticLCs, where cooperative layer-to-layer motion of strings with various sizes contributes to thelong-time relaxation behavior of the smectic phase.This paper is organized as follows. In section II, we introduce the model, the simulationdetails, and the computational tools to describe the layer-to-layer diffusion. The results onthe non-Gaussian and heterogeneous dynamics, as well as evidence of cooperative motionare discussed in section III. In the last section, we present our conclusions.3
I. MODEL AND SIMULATION METHODOLOGY
We perform simulations of a system with N = 1530 − D and length L + D , distributed over 5-10 smectic layersof approximately 300 rods each. Three different aspect ratios, L ∗ = L/D , are considered: L ∗ = 3 . , . , and 5 .
0. The region of stability of the smectic phase decreases with L ∗ , anddisappears at L ∗ ≤ .
1, where only a stable isotropic-crystal phase transition is found [14].For L ∗ = 3 . .
0, the smectic phase is stable for 2 . ≤ P ∗ ≤ . . ≤ P ∗ ≤ . P ∗ = βP D is the reduced pressure. β is 1 /k B T , with k B the Boltz-mann’s constant and T the temperature. For lower P ∗ , the smectic phase transforms into anematic phase, while for higher P ∗ the smectic freezes into a crystal phase. For L ∗ = 3 . P ∗ ≤ . P ∗ ≥ .
0. We studied the dynamics of the bulksmectic phase at the pressures and packing fractions indicated in Table I. For convenience,we label the systems with S − S (see Table I).We performed Monte Carlo (MC) simulations in a rectangular box of volume V withperiodic boundary conditions. Firstly, we performed equilibration runs in the isobaric-isothermal ( NPT ) ensemble to expand the system from an ordered crystalline phase to anequilibrated smectic phase. Each MC cycle consisted of N attempts to displace and/or rotatethe randomly selected particles, plus an attempt to change the box volume by modifyingthe three box lengths independently. Translational and rotational moves were accepted ifno overlap was detected. The systems were considered to be equilibrated when the packingfraction reached a constant value within the statistical fluctuations. A typical equilibrationrun took roughly 3 × MC cycles and was followed by a production run in the isochoric-isothermal (
NVT ) ensemble to analyze the relaxation dynamics. At this stage, we keptthe volume constant to avoid unphysical collective moves which do not mimic the Browniandynamics of the rods properly. Standard MC simulations with small displacements were usedto mimic Brownian motion. This computational approach was shown to be very efficient tostudy the slow relaxation of glasses at low temperatures [54] or at high concentrations [55].We fixed the maximum displacement according to ( i ) a reasonable time of simulation, ( ii )a satisfactory acceptance rate, and ( iii ) a suitable description of the Brownian motion ofcolloidal particles suspended in a fluid (see Fig. 1). To this end, we monitored the mean-4 ∗ . ≤ P ∗ ≤ . . ≤ P ∗ ≤ . . ≤ P ∗ ≤ . P ∗ η h/ ( L + D ) 1.018 1.014 1.023 1.015 1.048 1.030 σ/ ( L + D ) 0.050 0.043 0.075 0.052 0.093 0.043 D Lz τ /D D Lxy τ /D ¯t /τ t ∗ J /τ t maxJ /τ ¯f c f ∗ c f maxc U /k B T S S S S S S TABLE I: Details of the systems that we studied in this paper, consisting of hard spherocylinderswith varying length-to-diameter ratio L ∗ = L/D and reduced pressures P ∗ = βP D , and corre-sponding packing fractions η . For comparison, we give the pressure range of the stable smecticphase for the corresponding systems. Additionally, we give the layer spacing ( h ); the standarddeviation of the displacement from the equilibrium smectic phase ( σ ) in units of (L+D) ; the long-time in-layer D Lxy and inter-layer D Lz diffusion coefficients in units of τ /D ; the most probable ( ¯t ),median ( t ∗ J ), and maximal ( t maxJ ) jump times; the fraction of collective jumps ¯f c , f ∗ c , and f maxc ,calculated with a temporal interval ∆ t equal to ¯t , t ∗ J , and t maxJ , respectively; and finally, the heightof the energy barriers ( U ) in units of k B T . The systems are labeled by S − S . square displacement in the z and xy directions for several values of the maximum step size δ max , with δ max,z = 2 δ max,xy to take into account the anisotropy of the self-diffusion of therods [56]. We found δ max,xy = D/
10 and δ max,z = D/ IG. 1: Trajectories in a plane perpendicular to the nematic director of approximately 300 rodsin a smectic layer of hard spherocylinders with a length-to-diameter ratio L ∗ = 5 . P ∗ = 1 .
60 collected over 5 × MC cycles.
As unit of time, we have chosen τ ≡ D /D str , where D str is the short-time translationaldiffusion coefficient at L ∗ = 5 .
0, which is the isotropic average of the diffusion coefficients inthe three space dimensions: D str ≡ ( D sz + 2 D sxy ) /
3. At short times, when the single particle israttling around its original position without feeling the presence of its surrounding neighbors,the dependence of D str on the pressure can be safely neglected. We checked that our results(measured in units of τ ) were independent of δ max .In order to characterize the layer-to-layer hopping-type diffusion and the structural relax-ation of our systems, we calculated ( i ) the energy barrier, ( ii ) the self part of the van Hovecorrelation function, ( iii ) the non-Gaussian parameter, ( iv ) the mean square displacement,( v ) the intermediate scattering function, ( vi ) the probability distribution of the size of thestringlike clusters, and ( vii ) their dynamic cooperativity. Energy barrier . We computed the energy barriers from the (relative) probability π ( z ) offinding a rod at a given position z along the nematic director ˆ n . As reported in Ref. [37],this probability is proportional to the Boltzmann factor π ( z ) ∝ exp [ − U ( z ) /k B T ] , (1)6here U ( z ) denotes the effective potential for the layer-to-layer diffusion. Self-part of the van Hove correlation function.
To quantify the heterogeneous dynamicsdue to the rattling and hopping type z -diffusion of the rods, we calculate the self-part of thevan Hove correlation function (VHF), which measures the probability distribution for thedisplacements of the rods along the nematic director ˆ n at time t + t , given their z positionsat t . It is defined as [57] G s ( z, t ) = 1 N (cid:42) N (cid:88) i =1 δ [ z − ( z i ( t + t ) − z i ( t ))] (cid:43) (2)with (cid:104) ... (cid:105) the ensemble average over all particles and initial time t , and δ is the Dirac-delta.Note that G s ( z, t ) would be a Gaussian distribution of z for freely diffusive particles. Non-Gaussian parameter.
A quantitative description of the non-Gaussian behavior of thelayer-to-layer diffusion can be obtained in terms of the non-Gaussian parameter (NGP) [58]: α ,z ( t ) = (cid:104) ∆ z ( t ) (cid:105) (1 + 2 /d ) (cid:104) ∆ z ( t ) (cid:105) − , (3)where ∆ z ( t ) = z ( t + t ) − z ( t ) denotes the z -displacement of a rod in the time interval t starting at t , and d = 1. Heterogeneous dynamics occurs on a time scale t , if the NGP isnon-vanishing. For the in-layer diffusion, a similar NGP, α ,xy ( t ), with d = 2 can be defined. Intermediate scattering function.
The structural relaxation and, in particular, the decayof the density fluctuations can be quantified by the self-part of the intermediate scatteringfunction (ISF) F s ( t ) = (cid:104) exp [ i q · ∆ r ( t )] (cid:105) (4)at the wave vectors q = (0 , , q z ) and ( q x , q y ,
0) corresponding to the main peaks of the staticstructure factor in the z and xy directions, respectively. Cluster size distribution . In order to investigate whether or not there is cooperativemotion of stringlike clusters of particles, which might be responsible for the heterogeneousdynamics and stretched-exponential decay of the ISF, we first determine the clusters offast-moving particles. The fast-moving particles can be identified as those particles thathave traveled a substantially longer distance than the average in a certain time interval andare intimately related to the particles that reside in between the center-of-masses of twosmectic layers. Hence, particles that reside more than some rattling distance δ rat from the7enter-of-mass of the nearest smectic layer in a static configuration are defined as interlayerrods. To define δ rat , we first calculate the variance of one period of π ( z ), defined as σ ≡ (cid:90) h/ − h/ z π ( z ) dz, (5)where h is the layer spacing obtained from fits to the density profiles π ( z ), and (cid:82) h/ − h/ π ( z ) dz =1. We assume that two interlayer rods belong to the same string if their z and xy distancesare smaller than h and D , respectively. We calculated the probability distribution P ( n ) ofthe number of rods n in a string for δ rat = 2 σ . In Table I, we give the values of the layerspacing h and the standard deviation (square root of the variance) σ for the systems thatwe studied. We find that denser states reveal a slightly smaller layer spacing for all aspectratios, as expected. Dynamic cooperativity.
To check whether or not the rods in a given string move col-lectively from one layer to another layer, we require a cluster criterion involving a spatialproximity of particles and a temporal proximity of jumping rods. To this end, we assumethat two jumping rods i and j are moving cooperatively if (1) their arrival times t ( i ) and t ( j ) in their new layers (i.e. the time at which their distance to the middle of the new layerequals δ rat ) satisfy | t ( i ) − t ( j ) | < ∆ t , and (2) the center-of-mass positions of the two particles, r i ( t ( i ) ) and r j ( t ( j ) ), satisfy the above mentioned static cluster criterion. The time interval∆ t is fixed on the basis of the distribution of jump times and will be introduced in thefollowing section. III. RESULTS
The layered structure of smectic LCs yields an effective periodic potential U ( z ) for thediffusion of rods out of the middle of a smectic layer to another layer. The permanentenergy barriers for the layer-to-layer diffusion are determined from the effective potential U ( z ) introduced in Eq. (1). In order to evaluate the effect of the pressure on the inter-layerdiffusion, two separate state points are considered for each aspect ratio: one in the proximityof the nematic-smectic (N-Sm) or isotropic-smectic (I-Sm) transition, and the other in theproximity of the smectic-crystal (Sm-K) transition. In Fig. 2, we show U ( z ) for the sixsystems along with the fit U ( z ) = (cid:80) mi =1 U i [sin( πz/h )] i , where m is an integer number,8 ≡ (cid:80) mi =1 U i is the barrier height, and h the interlayer spacing given in Table I. FIG. 2: Potential energy barriers for the layer-to-layer diffusion in smectic LCs of hard spherocylin-ders with varying length-to-diameter ratios L ∗ and reduced pressures P ∗ . The open symbols referto weak smectic states L ∗ = 3 . P ∗ = 2 .
85 ( (cid:77) ), L ∗ = 3 . P ∗ = 2 .
35 ( (cid:3) ), and L ∗ = 5 . P ∗ = 1 . (cid:35) ). The solid symbols denote state points deep in the smectic phase L ∗ = 3 . P ∗ = 3 .
00 ( (cid:78) ), L ∗ = 3 . P ∗ = 2 .
50 ( (cid:4) ), and L ∗ = 5 . P ∗ = 2 .
00 ( (cid:32) ). The solid lines are fits with m = 5 harmonicmodes. As a general consideration, we observe that the energy barriers increase with increasingpacking fraction. The denser state is characterized by a much stronger confinement of theparticles to the middle of the smectic layers. This is especially evident at L ∗ = 5 . . k B T to 7 . k B T from η = 0 .
508 to 0.557. A similarbehavior was detected in experiments on smectic LCs of fd viruses, where the height ofthe energy barriers was found to increase from 0 . k B T to 1 . k B T by decreasing the ionicstrength [37]. According to the authors, a low ionic strength gives rise to stronger correlationsdue to more pronounced electrostatic interactions between the virus particles, resulting inhigher energy barriers. Older experiments on thermotropic liquid crystals estimated energybarriers of ∼ k B T [59, 60]. Our results are in good quantitative agreement with thoseobtained by computer simulations in Ref. [61], where rods with L ∗ = 3 . k B T higher than that to diffuse from layerto layer, confirming the difficulty to detect transverse particles in between smectic layers[62]. We further observe that the packing fraction is not the only parameter affecting the9ffective potentials U ( z ) as displayed in Fig. 2. It is interesting to note that the twosystems S and S , with L ∗ = 3 . P ∗ = 2 .
85 as denoted by the open triangles ( (cid:77) )and with L ∗ = 5 . P ∗ = 2 .
00 as shown by the solid circles ( (cid:32) ), corresponding tovery similar packing fractions η = 0 .
556 and 0.557, respectively, exhibit barrier heightsthat differ by approximately 1 . k B T . The barrier height at fixed η thus increases withthe particle anisotropy. The effect of the particle anisotropy on the effective potential canalso be illustrated by comparing the barriers for the systems S and S with L ∗ = 3 . P ∗ = 3 .
00 ( (cid:78) ) and L ∗ = 5 . P ∗ = 2 .
00 ( (cid:32) ), respectively. In this case, although thepacking fractions are significantly different ( η = 0.568 and 0.557), both barrier heights are7 . k B T . This result shows that a more pronounced particle anisotropy yields significantlyhigher energy barriers. In conclusion, the barrier height increases with increasing packingfraction and particle anisotropy. Moreover, in our recent work on perfectly aligned hard rods,we noticed that the freezing out of the rotational degrees of freedom has also a quite tangibleimpact on the height of the barriers, which were found to be higher than those observed insystems of freely rotating rods, especially at low packing fractions [44]. Such barriers fadeout gradually by approaching the continuous N-Sm transition, while they remain finite inthe case of the first order N-Sm transition of freely rotating hard rods. One might expectthat structural defects, such as screw dislocations and stacking faults, could facilitate thelayer-to-layer diffusion by creating barrier-free nematic-like pathways through the layers [63].The small size of our system and the periodic boundary conditions preclude the developmentof such defects.The periodic shape of the effective potential determines a hopping-type diffusion alongthe direction parallel to the nematic director ˆ n , with the particles rattling around in theiroriginal layer until they find the appropriate conditions to overcome the barrier and jumpto a neighboring smectic layer. An efficacious way to quantify the diffusion of rattling andjumping rods along ˆ n is provided by the computation of the self part of the VHFs defined inEq. (2). In Fig. 3, we show the self VHFs for the six systems of interest as a function of z at several equidistant times t . We detect the appearance of peaks at well-defined locationscorresponding to the center-of-mass of the smectic layers along ˆ n , in agreement with previousexperimental [37] and theoretical [38] results. For each aspect ratio, we note that the heightof the peaks is larger and the number of peaks is generally higher at the lowest packingfraction. This indicates that decreasing the pressure leads to a higher number of jumping10articles which are able to diffuse longer distances. These fast particles determine theheterogeneous dynamics of the system and affect its structural relaxation. FIG. 3: Self part of the van Hove function G s ( z, t ) for smectic LCs of hard spherocylinders withvarying length-to-diameter ratios L ∗ and reduced pressures P ∗ . Length-to-diameter ratio, pressure,and packing fraction are given in each frame as ( L ∗ , P ∗ , η ). The curves refer to the time evolution,from t = 0 . τ (dotted lines) to t = 40 τ (solid lines) with increments of ≈ τ . Deviations from the Gaussian behavior of the VHFs have been extensively analyzed inliquid [40, 64], glassy [42, 43], and liquid crystalline [38, 39] systems in terms of the non-Gaussian parameter defined in Eq. (3). The NGPs as measured in our six systems areshown in Fig. 4. The in-layer NGPs, α ,xy , are essentially negligible for the whole timerange, implying a Gaussian in-layer dynamics which is typical of a liquid-like system. Thelayer-to-layer NGPs exhibit a time-dependent behavior, which is strictly linked to the cagingeffect exerted on the rods by their nearest neighbors. More specifically, α ,z is basicallyzero at short times when the rods are rattling around their original location and do notperceive the presence of the surrounding cage. Between t/τ = 0 . α ,z starts toincrease, indicating the development of dynamical heterogeneities. During this time interval,the motion of the rods is hampered by the trapping cages and becomes subdiffusive. It isreasonable to assume that the average life-time of the cages corresponds to the time between α ,z (cid:39) α ,z = α max ,z , the maximum value of the NGP, which occurs at time t = t max with11 (cid:46) t max /τ (cid:46)
10, depending on the system. This peak increases with packing fraction andits location determines the beginning of the long-time diffusive regime, where the deviationsfrom Gaussian behavior start to decrease. Increasing the pressure affects the z -diffusion ina twofold manner: ( i ) it increases its heterogeneous behavior, and ( ii ) it delays the onset ofthe long-time diffusive behavior. Comparison of the two systems S and S , correspondingto very similar packing fractions, shows that the dynamics becomes more non-Gaussian forsystem S consisting of longer rods and higher energy barriers. On the other hand, forsystems S and S , displaying similar barrier heights, the non-Gaussian dynamics is againmore pronounced for system S with longer particles. Hence, increasing the anisotropy ofthe particles yields higher energy barriers and dynamics that is more heterogeneous andnon-Gaussian. FIG. 4: Non-Gaussian parameter α ( t ) for the layer-to layer ( (cid:32) ) and in-layer ( (cid:35) ) diffusion forsmectic LCs of hard spherocylinders with varying length-to-diameter ratios L ∗ and reduced pres-sures P ∗ . Length-to-diameter ratio, pressure, and packing fraction are given in each frame as ( L ∗ , P ∗ , η ). Most of the information obtained by the analysis of the non-Gaussian parameter can alsobe deducted by the mean square displacements (MSDs), (cid:104) ∆ z ( t ) (cid:105) and (cid:104) ∆ x ( t ) + ∆ y ( t ) (cid:105) ,shown in Fig. 5. The xy -MSD (open circles) is characterized by a relatively smooth crossover12 IG. 5: Mean square displacement (MSD) in units of D along the nematic director ˆ n (solid circles)and along the plane perpendicular to ˆ n (open circles) for smectic LCs of hard spherocylinders withvarying length-to-diameter ratios L ∗ and reduced pressures P ∗ . Length-to-diameter ratio, pressure,and packing fraction are given in each frame as ( L ∗ , P ∗ , η ). The solid lines are a guide for the eye. from the short- to long-time diffusion, as observed in slightly dense liquids. By contrast, forthe z -MSD (solid circles) one clearly detects a more sophisticated behavior as three separatetime regimes can be identified. The short-time dynamics, with the rods still rattling intheir cages, are diffusive, that is (cid:104) ∆ z ( t ) (cid:105) ∝ t . In this regime, (cid:104) ∆ z ( t ) (cid:105) > (cid:104) ∆ x ( t ) + ∆ y ( t ) (cid:105) because of the anisotropy of the rods [56]. After an induction time, we observe the formationof a plateau which extends up to t max and quantifies the time to escape from the trappingcages. In this time window, the dynamics becomes subdiffusive. Finally, at different times,the xy -MSD and z -MSD become linear with time and the long-time diffusive regime isreached.From the MSDs in the diffusive regime, we computed the long-time diffusion coefficients inthe z - and xy -directions by applying the well-known Einstein relation [4]. The values of thelong-time diffusion coefficients, D Lxy and D Lz , are presented in Table I in units of D /τ . Thedynamics of each system is characterized by a diffusion coefficient in the xy -direction thatis larger than the one in the z -direction. This result is in agreement with the dynamics in13 IG. 6: Diffusion ratio γ (top) and dimensionless diffusion coefficients (bottom) as a function ofthe packing fraction for smectic LCs of hard spherocylinders with length-to-diameter ratio L ∗ = 5 . D ∗ xy,z = D Lxy,z τ /D denotes the reduced diffusion coefficients, as given in Table I. The solid linesare power law fits. thermotropic smectogenic LCs [65], but in contrast with recent experiments on the diffusionof fd viruses [37], most probably because of their huge aspect ratio ( L ∗ > γ ≡ D Lxy /D D Lz / ( L + D ) , (6)for L ∗ = 5 . D Lxy and D Lz as well as γ are well fitted by power law functions of the type η − ν , with ν (cid:39) D Lz decreases muchfaster than D Lxy . This is to be expected as the energy barriers (see Fig. 2) hamper thediffusion more in dense systems. For the lowest packing fractions, i.e., where the smecticphase almost coexists with the nematic phase, the two diffusion coefficients approach eachother. Similar considerations are also valid for the systems with aspect ratio L ∗ = 3 . F s,xy ( t ) and F s,z ( t ) at the wave vectors q = ( q x , q y ,
0) and (0 , , q z ), respectively, corresponding to the14ain peaks of the static structure factor. We found that D (cid:113) ( q x + q y ) (cid:39) Dq z (cid:39) .
4, 1.3, and 1.0 for the systems with L ∗ = 3 . , . , and 5.0, respectively.Regardless the aspect ratio, we can affirm that the in-layer structural relaxation is severalorders of magnitude faster than the inter-layer relaxation. If we define the relaxation time t r as the time at which F s ( t ) decays to e − , then t r,xy /τ is of the order of 10 − and t r,z /τ > .We also find that F s,xy ( t ) decays very fast to zero with slightly stretched exponential decay,as expected for dense liquid-like dynamics [66].By contrast, the inter-layer relaxation develops in two steps separated by a plateau, thebeginning of which corresponds to the development of the cage regime. During the initialdecay, which is relatively fast ( t/τ ≤ F s,z ( t ) ischaracterized by an exponential decay. After this short time lapse, we detect a plateau, whoseheight and temporal extension depend on the packing of the system, as was also found incolloidal glasses [66]. In our previous analysis, we observed that the Gaussian approximationof the self-intermediate scattering function, that is F Gs,z ( t ) = exp [ − q z (cid:104) ∆ z ( t ) (cid:105) ], does notshow any significant plateau [45]. This result indicates that the existence of a plateaumust be linked to the non-vanishing NGP α ,z , and hence to the heterogeneous inter-layerdynamics. After the plateau, a second decay, which corresponds to the escape from thetemporal cages, leads the systems towards the structural relaxation on a time scale which islong at the highest packing fractions. We were only able to estimate the long-time relaxationdecay in the z -direction for systems S and S , due to their relatively low packing fractions.In particular, we fit the long-time decay of F s,z ( t ) with a stretched exponential function ofthe form exp (cid:2) − ( t/t r ) β (cid:3) , with t r /τ ∼ = 2500 and β ∼ = 0 . S , and t r /τ ∼ = 650 and β ∼ = 0 . S . The relaxation time of the remaining systems are beyond our simulation time.The results shown so far give clear evidence of the existence of fast-moving particlesdetermining a distribution of decay rates which affects the long-time structural relaxation.We now turn our attention to the occurrence of collectively moving particles which mightplay a crucial role, or might even be responsible, for the heterogeneous inter-layer dynamicsin smectic LCs. To this end, we first label the fast-moving particles by applying the staticcluster criterion defined below Eq. (5). Fig. 8 shows the probability size distribution P ( n )of the number of interlayer rods n in a string-like cluster using rattling distance δ rat = 2 σ ,where σ is the standard deviation as specified in Eq. (5) and presented in Table I. In15 IG. 7: Self-intermediate scattering function for smectic LCs of hard spherocylinders with length-to-diameter ratio L ∗ = 3 .
4, and reduced pressures P ∗ = 2 .
85 and 3.00 (top), L ∗ = 3 .
8, and P ∗ = 2 .
35 and 2.50 (middle), and L ∗ = 5 .
0, and P ∗ = 1 .
60 and 2.00 (bottom). The solid andopen symbols refer to the lowest and highest pressure, respectively. Squares and circles refer,respectively, to the in-layer and inter-layer relaxation. The solid lines are fits.
Fig. 9, we give an illustrative example of static strings observed in system S . Regardlessthe particle anisotropy, two interesting conclusions can be drawn: ( i ) the observed stringsconsist mostly of 2 or 3 rods, while clusters of more than 5 rods are extremely rare, butdo exist; and ( ii ) strings containing more than 2 rods are more often formed in the denserstate, as usually observed in supercooled liquids and glassy systems, where the averagecluster size increases when the caging effect becomes stronger [48]. The semilog plot ofFig. 8 shows that the size probability distribution is roughly exponential. In particular,we observed that P ( n ) ∝ exp ( − αn ), from which the average cluster size can be estimated: (cid:104) n (cid:105) = (1 − exp( − α )) − . The largest clusters are thus found for the shortest rods and thehighest pressure.The energy barriers of Fig. 2 and the periodically peaked shape of the VHFs of Fig. 3unfold the rattling-and-jumping layer-to-layer diffusion of the rods, but only provide a global16 IG. 8: Size probability distribution P ( n ) of the number of interlayer rods n in a stringlike clusters(with δ rat = 2 σ , see text), in a smectic LCs of hard spherocylinders with length-to-diameter ratio L ∗ = 3 .
4, and reduced pressures P ∗ = 2 .
85 and 3.00 (top), L ∗ = 3 .
8, and P ∗ = 2 .
35 and 2.50(middle), and L ∗ = 5 .
0, and P ∗ = 1 .
60 and 2.00 (bottom). The solid and open symbols refer to thelowest and highest pressure, respectively. The solid lines denote the fit P ( n ) ∝ exp( − αn ), with α given in the figure. picture of the dynamics in smectic LCs. In order to gain a deeper understanding of the actualdynamics on the particle scale behind the layer-to-layer hopping-type diffusion, we followedthe trajectories of single particles. Interestingly, we observed that some rods diffuse veryfast, others move to the inter-layer spacing, where they might reside for a long time, andthen return to their original layer, and others move from one layer to another several timesor perform consecutive jumps as shown in Fig. 10b for system S . Furthermore, transverseinter-layer rods, although extremely rare due to the high energy barriers [61, 62], have alsobeen detected. These transverse inter-layer particles might diffuse either to a new layer orgo back to the old one by keeping or changing their original orientation. This variegatedbehavior suggests a rather broad distribution of layer-to-layer jump times Π( t J ), where t J is the time between the first and last contact with the new and old layer, respectively. Such17 IG. 9: (color online). Snapshot of 3000 rods with length-to-diameter ratio L ∗ = 3 . η = 0 . D/ a contact is established as soon as the particle is at a rattling distance δ rat = 2 σ from themiddle of a smectic layer. The probability distributions of jump times for systems S − S have been computed by averaging over at least N jumps, with N the number of rods in eachsystem. In Fig. 10a, we give the probability distribution of jump times Π( t J ) for system S . As expected, the distribution is not particularly narrow, but extends over two timedecades 0 . < t J /τ <
1, with the most probable jump time ¯t =0.17 τ and the median time(i.e. the time at which 50% of the jumps have been performed) t ∗ J = 0 . τ . These timesincrease at η = 0 . t ∗ J before another jump is started. Single jumps are significantly morefrequent than multiple jumps, especially at high densities where the latter are less than 1%of the total number of jumps. Most of the multiple jumps consist of double or triple jumps,18 IG. 10: (color online). (a) Probability distribution of jump times Π( t J ) based on δ rat = 2 σ forlenght-to-diameter ratio L ∗ = 3 . η = 0 . xz plane, with the dashed lines locating the centre of thesmectic layers. The arrow indicates the trajectory of a transverse interlayer particle. while quadruple ones are basically irrelevant. The largest number of multiple jumps wasobserved in the system with the lowest packing fraction, i.e., S : L ∗ = 5 . , η = 0 . ≈
5% of the total jumps is multiple, of which ≈
83% is a double and the remaining fractionis a triple jump.The formation of static stringlike clusters does not necessarily imply the occurrence ofdynamic cooperativity, as a rod that belongs to a cluster might still diffuse individually fromlayer-to-layer or might fail to jump. In order to ascertain the occurrence of collective motionof strings, we introduced a dynamic cluster criterion. More specifically, we assume that twojumping rods i and j are actually moving cooperatively, if (1) their arrival times t ( i ) and t ( j ) in their new layers satisfy the condition | t i − t j | < ∆ t and (2) their z and xy distances at t ( i ) and t ( j ) are smaller than h and D , respectively. To select a consistent value for ∆ t , we usethe distribution of jump times Π( t J ). If we assume that ∆ t is the maximal jump time t maxJ (i.e. long enough for all jumps to be performed according to Π( t J )), we find that for L ∗ = 3 . f maxc asgiven in Table I. These jumps involve mainly two or three rods, whereas collective jumps of4 or more rods are extremely rare. We further note that the vacated space of a jumping rodcan be either occupied by another rod jumping in the same direction or by a rod jumping inthe opposite direction, with roughly the same probability. A less restrictive spatial criterionwould not affect significantly these values. By contrast, the number of collective jumps israther sensitive to the temporal criterion. If we reduce ∆ t to the median jump time t ∗ J , then19he fraction of collective jumps, f ∗ c , decreases substantially as shown in Table I. Regardlessthe details of the cluster criterion, we thus find a fraction of 10 − − − of the jumps to becollective, the more so for longer rods at lower pressures.The motion is indeed strongly cooperative at low packing fractions, despite the largerstatic stringlike clusters detected in the denser systems (see Fig. 8). This result is most prob-ably due to the permanent smectic barriers which increase upon approaching the smectic-to-solid phase transition and hence hamper the attempted jumps of the rods in the strings. Bycontrast, in glass-forming systems, where no permanent barriers are observed, the clustersize increases upon approaching the glass transition [48, 67]. IV. CONCLUSIONS
In summary, we have studied the diffusion and structural relaxation in equilibrium smec-tic LC phases of hard rods with different anisotropies by computer simulations. Remarkably,these systems exhibit non-Gaussian layer-to-layer diffusion and dynamical heterogeneitieswhich are similar to those observed in out-of-equilibrium supercooled liquids. The simulta-neous presence of temporary cages due to the trapping action of neighboring rods and thepermanent barriers due to the static smectic background, provokes a rattling-and-jumpingdiffusion which influences the long-time structural relaxation decay. In analogy with glassysystems, one can clearly distinguish three separate time regimes for z -motion. The short-time diffusive regime, with the rods rattling around their original location without feeling thesurrounding neighbors, is characterized by a Gaussian distribution of the z -displacementsand an exponential temporal relaxation. The subdiffusive regime at intermediate times dis-plays a non-Gaussianity and a plateau in both mean square displacement and intermediatescattering function. At this stage, the interlayer dynamics is heterogeneous with fast-movingparticles diffusing individually or cooperatively in a stringlike fashion. Finally, at long times,the systems enter a second diffusive regime with Gaussian distributions of the displacementsand non-exponential decay of the intermediate scattering function. By contrast, at all time-scales, the in-layer diffusion is typical for two-dimensional fluids with a negligible NGP anda structural relaxation which is at least 4 time decades faster than the inter-layer one.The analysis of the self-VHFs points out the tendency for the rods to diffuse from layerto layer through quasi-discretized jumps. This hopping-type motion is significantly ham-20ered in very dense systems, where the barriers for the layer-to-layer diffusion intensify theconfinement of the rods to the middle of the smectic layers. The temporal extension of thejumps is not uniform, but characterized by a rather broad time distribution which coversapproximately two time decades. Depending on the dwelling time between two successivejumps of the same particle, single and multiple jumps have been detected, with the formersignificantly more frequent than the latter, especially at high densities. Although the inter-layer rods are usually oriented along the nematic director, some of them assume a transverseorientation. The long tails of the VHFs indicate the presence of particles that are able todiffuse much longer distances than the average, especially at low packing fractions. Inter-estingly, the dynamic behavior of such fast-moving particles supports the intriguing analogywith glassy systems even further. In particular, fast-moving particles assemble in string-like clusters whose average length increases upon approaching the smectic-to-crystal phasetransition. Likewise, fragile and strong glass-formers show a similar tendency when cooleddown towards the glass transition temperature [48, 52]. We also gave clear evidence thatthe strings detected in static configurations can promote collective diffusion of jumping par-ticles. This is especially tangible at low packing fractions where the hampering action of thepermanent energy barriers is less effective. Finally, we also investigated the effect of particleanisotropy on the non-Gaussian layer-to-layer diffusion and cooperative motion in smecticLCs. We find that at fixed packing fraction, the barrier height increases with increasing par-ticle anisotropy, and hence the dynamics is more heterogeneous and non-Gaussian for longerrods, yielding a lower diffusion coefficient along the nematic director and smaller clustersof inter-layer particles that move less cooperatively. 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