Formation of the Smectic-B Crystal from a Simple Monatomic Liquid
A. Metere, T. Oppelstrup, S. Sarman, A. Laaksonen, M. Dzugutov
FFormation of the Smectic-B Crystal from a Simple Monatomic Liquid
A. Metere T. Oppelstrup S. Sarman , A. Laaksonen and M. Dzugutov Dept. of Materials and Environmental Chemistry,Stockholm University, Arrhenius V¨ag. 16C S-106 91 Stockholm, Sweden Lawrence Livermore National Laboratory - 7000 East Avenue, Livermore, California 94551, USA Dept. of Mathematics and Centre for Parallel Computers,Royal Institute of Technology, S-100 44 Stockholm, Sweden (Dated: October 30, 2018)We report a molecular dynamics simulation demonstrating that the Smectic B crystalline phase(Cr- B ), commonly observed in mesogenic systems of anisotropic molecules, can be formed by asystem of identical particles interacting via a spherically symmetric potential. The Cr- B phaseforms as a result of a first order transition from an isotropic liquid phase upon isochoric coolingat appropriate number density. Its structure, determined by the design of the pair potential corre-sponds to Cr- B structure formed by elongated particles with the aspect ratio 1 .
8. The diffractionpattern, and the real-space structure inspection demonstrate dominance of the ABC-type of axiallayer stacking. This result opens a general possibility of producing smectic phases using isotropicinterparticle interaction both in simulations and in colloidal systems.
PACS numbers: 61.20.Ja, 61.30.Cz, 64.70.mf
Computer simulations using particles are now well-established tools for investigating different aspects of liq-uid crystals [1, 2]. The most interesting of these are smec-tic phases where the molecules, besides uniaxial direc-tional order, form layered structures [3, 4]. In smectic- A phase positional order within layers is entirely absent,whereas hexatic smectic- B phase is characterised by ashort-range hexagonal intralayer order [5]. This excludesany long-range periodicity and keeps the system fluid dueto Landau-Peierls instability [6]. Another modification ofsmectic- B phase is a true 3D crystal (Cr- B ). The natureof this phase was a controversial issue for some time, un-til its global 3D positional order was established in 1979[7].The simulation studies of models forming liquid-crystalphases provide a unique way to establish a relation be-tween the molecular-level properties and macroscopic be-haviour. These simulations demonstrated that the re-markable polymorphism of the mesogens forming liq-uid crystal phases can be reproduced using quite simpleparticle models. Systems of elongated molecules inter-acting via Gay-Berne (GB) potential [8–11] have beenfound successful in reproducing the liquid-crystal phasebehaviour in a variety of simulation experiments. Thesemodels were also exploited for calculating transport prop-erties of the liquid crystals [12]. Moreover, it was provedpossible to reduce the models of anisotropically interact-ing GB particles to hard spherocylinders [13, 14], whichappears to indicate the entropic origin of the LC phasesdue to geometry of excluded volume.This experience poses a question of conceptual interestfor the statistical mechanics of condensed matter: howfar further can the particle models successfully repro-ducing the basic features of smectic phases be simpli-fied? In particular, is the anisotropy of the constituentmolecules a prerequisite to producing anisotropic struc- tures like those of smectic phases, and could the entropiceffects of the particle geometry be compensated by anappropriately designed pair interaction?This question is addressed in a molecular-dynamicssimulation that we report in this Letter. We demon-strate that the Cr- B phase, a characteristic freezing formof mesogenic systems of anisotropic molecules, can beformed in a system composed of a single sort of par-ticles interacting via a spherically-symmetric potential.The crystal occurs upon cooling as a result of a first-order phase transition from isotropic liquid. It repre-sents a uniaxial structure composed of stacked layers withhexagonal close-packed intralayer structure. The struc-ture is consistent with the experimentally observed Cr- B structures [7, 15], demonstrating predominantly ABCAsequence in layers stacking. This result opens a perspec-tive of producing other (non-crystalline) types of smecticphases, like Smectic-A and hexatic phase. It also sug-gests that this class of layered mesomorphs can possiblybe produced in systems of spherically-shaped colloidalparticles.The results we report here have been produced ina molecular-dynamics simulation of a single-componentsystem comprising 16384 particles. The interparticle in-teraction was assumed to be spherically symmetric, de-scribed by the pair potential presented in Figure 1. Thefunctional form of the potential energy for two particlesseparated by the distance r is: V ( r ) = a (cid:0) r − m − a (cid:1) H ( r, b , c ) + H ( r, b , c ) , (1)where H ( r, b, c ) = (cid:40) exp (cid:16) br − c (cid:17) r < c r ≥ c. (2)The values of the parameters are presented in Table I.The first term describes the short-range repulsion branch a r X i v : . [ c ond - m a t . s o f t ] J u l m a a b c a b c
12 265.85 0.8 1.5 1.45 2.5 0.19 1.89TABLE I. Values of the parameters for the pair potential.FIG. 1. Pair potential of the potential, and its minimum, whereas the secondterm expresses the long-range repulsion. All the ther-modynamic quantities we report are expressed in termsof the reduced units that were used in the definition ofthe potential. Note that the steepness of the short-rangerepulsion, and the position pf the minimum, are consis-tent with those in the Lennard-Jones (LJ) potential [16],which makes it possible to compare the reduced numberdensities of the two systems.At the beginning, the system was equilibrated in itsstable isotropic liquid state at sufficiently high temper-ature at the density ρ = 0 .
55. Note that this densityis much below of the triple-point density for the LJ sys-tem [16]. We then isochorically cooled the system, in astepwise manner, comprehensively equilibrating it aftereach temperature step. A a discontinuous change in theparameters was detected below T = 0 .
65, see Fig. 2, ac-companied by a sharp drop in the diffusivity, an apparentsignature of a first-order phase transition to a solid phase.Accordingly, its heating produced a significant hystere-sis. A non-trivial character of the low-temperature phasewas indicated by an anomalously long time required forits equilibration which amounted to several billions oftime-steps.The structure analysis of the low-temperature solidphase has been performed by inspecting the Fourier-space pattern of its density distribution. For thatpurpose, we calculated the structure factor S ( Q ) = (cid:104) ρ ( Q ) ρ ( − Q ) (cid:105) , where ρ ( Q ) is a Fourier-component of thesystem’s number density: ρ ( Q ) = 1 N N (cid:88) i =1 exp( Qr i ) (3) FIG. 2. Isochoric phase transition. Top: energy variation;bottom: pressure variation. Dots: high-temperature phase.Triangles: low-temperature phase. r i being the positions of the system’s particles. S ( Q ) rep-resents the diffraction intensity as measured in diffractionexperiments.As a first step, we calculated the diffraction intensityon the Q -space sphere corresponding to the first peak ofthe spherically averaged S ( Q ). We observed well-defineddiffraction maxima forming a regular pattern. This madeit possible to determine the global symmetry of the con-figuration: a single hexagonal axis was detected. Theaxis orientation having been found, we calculated S ( Q )within two characteristic Q -space planes: Q z = 0 and Q y = 0, Q z being the axis coordinate, and Q y coordi-nate corresponding to a translational symmetry vectororthogonal to the axis. The two diffraction patterns areshown in Fig. 3These results compel us to make the following conclu-sions. First, the observable sharpness of the diffractionmaxima indicates that the low-temperature phase pro-duced by the phase transition is a true 3D crystal. More-over, the two diffraction patterns exhibit structural fea-tures characteristic of the experimentally produced Cr- B phase [7, 15]: the configuration is a uniaxial crystal com-prised of stacked layers with dense hexagonal packing ofparticles in each layer. Based on the diffraction resultsshown in Fig. 3, we can also estimate the ratio of theinterlayer distance to the nearest-neighbour separationwithin a layer as 1 . a b FIG. 3. The isointensity plots of the structure factor S ( Q ),in two orthogonal Q -space planes. a: Q z = 0; b: Q y = 0. Q z denotes the axial dimension, and Q y corresponds to atranslational symmetry vector, orthogonal to the axis. The diffraction data presented in Fig.3 also providecomprehensive information concerning the interlayer cor-relations in the crystal. The hexagonal arrangement ofsharp diffraction peaks in the Q z = 0 plane implies theexistence of global positional interlayer correlations. Theinformation about the type of positional correlations ofthe particles of adjacent layers can be obtained by in-specting the pattern of diffraction intensity in the axialplane, Fig.3. The hexagonal close-packed layers may bestacked with two possible ordered arrangements: AAA...,ABA... or ABCA... where A, B, and C denote the rel-ative position of the layers. A random array of ABC-type planes is also possible. These types of layer packing,or their mixtures have been experimentally observed[7].The diffraction intensity profile along the axial coordi-nate in Fig. 3 demonstrates four distinct auxiliary peaks,interposed between the main peaks which represent thegeneral layers’ periodicity. This pattern can be inter-preted as representing the predominantly ABCA-type oforder in layers’ packing with possible defects [7]. Anexample of this type of arrangement of adjacent layersdiscerned by the real-space inspection of the simulatedCr- B configuration is presented in Fig. 4.A subtle problem in simulation studies of the Cr- B x y x z FIG. 4. A fragment of the simulated Cr- B configurationcomprising three adjacent layers stacked in ABC-type se-quence. Left: axial view; right: orthogonal view along anin-layer translational symmetry direction. The layers are dis-tinguished by colors. crystallisation has always been to discriminate betweenthe Cr- B phase, possessing true 3D long-range positionalorder, and the non-crystalline hexatic phase where thehexagonal order and interlayer correlations exist only ina limited range [9, 10]. The difficulty is mainly causedby limitations in the system size and the simulation’stime-scale which can be comparable with the space andtime-scales of the positional order in the hexatic phase.Besides, only spherically-averaged interparticle correla-tions are usually considered [11, 17]. In the present sim-ulation, we were able to identify without any ambiguityall the distinctive structural features of the Cr- B phase,including the stacking order, both in the real-space pic-ture and in terms of the diffraction intensity patterns.To the best of our knowledge, this is the first reportedsimulation of Cr- B phase providing complete informationabout all the details of its structure.These results demonstrate that a uniaxial anisotropicstructure can be produced in a single-component systemby a spherically symmetric interparticle potential. Thisseemingly paradoxical result can be rationalised by con-sidering the structural effects of the potential’s design.This potential can be regarded as a modification of anearlier reported pair potential [18], judiciously designedto induce predominantly icosahedral ordering of the firstcoordination shell. It was found to produce a dodecago-nal quasicrystal [19], and a number of other tetrahedrallyordered structures [20]. The present potential, while re-taining the same short-range repulsive part, and the min-imum position, has two major distinctions from the ear-lier one. First, its minimum is much more narrow due toa more steep attraction part. This inhibits formation ofthe icosahedral ordering of the first neighbours due to itscharacteristic frustration, as well as any other conceivabledensely packed structure with full first coordination shelland energetically favours a low-density structure with areduced number of first neighbours. Moreover, the ex-tended width of the following maximum in the presentpotential shifts its second repulsive part, and thereby thesecond neighbours to a significantly longer distance.As a result of this potential design, a local structure isfavoured, at appropriate density, with only six equidis-tant first neighbours arranged in a hexagon. Thesehexagons are organised in flat densely packed layerswhich are uniaxially stacked with the interlayer distancedetermined by the potential’s long-range repulsion.The anisotropy of smectic phases is measured by theratio of the nearest-neighbour distance to the interlayerspacing. In systems of elongated molecules, this corre-sponds to the degree of anisotropy (aspect ratio) of theconstituent molecules. In our system, the same effect isinduced by the potential design: the interlayer separa-tion is controlled by the long-range repulsion, whereasthe in-layer nearest-neighbour distance is determined bythe short-range repulsion. This implies that the apparentaspect ratio of a Cr- B phase produced in a manner we re-port here can be manipulated by choosing the separationof the two repulsive branches of the potential.At sufficiently high density, where the short-range re-pulsion dominates the energy, a close-packed structurewill be formed, presumably hcp. At low densities andlow temperatures, where the structure will be determinedby the long-range repulsion, the same kind of lattice isexpected to be energetically favourable. A similar kindof isostructural polymorphism has been reported for astepwise pair potential [21].We conclude with the following remarks.1. So far, colloidal smectic phases have only been foundto appear in systems of rod-like colloidal particles [22].The spherically-symmetric nature of the interparticle in-teraction in the present model, and the similarity of itsmain features to classical DLVO theory for colloidal in-teraction [23–25] (amended with hard core repulsion orsteric repulsion at close to contact), suggests a possibil-ity that smectic-like layered structures can be producedin colloidal systems of spherically shaped particles, withappropriate tuning of the effective force field.2. The local structural isomorphism of the Cr- B andfluid hexatic phase suggests that the latter too can pos-sibly be produced using the general approach to the po-tential design we exploited here, as well as the smectic A phase.3. An intriguing anomaly of the Cr- B dynamics is thepresence of soft shear modes tentatively concluded fromspectroscopic measurements [26]. 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