Bilayered smectic phase polymorphism in the dipolar Gay-Berne liquid crystal model

Bilayered smectic phase polymorphism in the dipolar Gay–Berne liquid crystal model Mohammed Houssa, Luis F. Rull, and Jose M. Romero-Enrique Citation: The Journal of Chemical Physics 130, 154504 (2009); doi: 10.1063/1.3111953 View online: http://dx.doi.org/10.1063/1.3111953 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/130/15?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Generation of frustrated liquid crystal phases by mixing an achiral nematic–smectic-C mesogen with an antiferroelectric chiral smectic liquid crystal J. Chem. Phys. 122, 144906 (2005); 10.1063/1.1872753 Landau model of the direct isotropic to smectic- C A * phase transition in antiferroelectric liquid crystals J. Chem. Phys. 121, 12038 (2004); 10.1063/1.1818672 Stability of smectic phases in the Gay–Berne model J. Chem. Phys. 121, 11183 (2004); 10.1063/1.1810472 A theory for the liquid-crystalline phase behavior of the Gay–Berne model J. Chem. Phys. 109, 2361 (1998); 10.1063/1.476804 Flow properties of liquid crystal phases of the Gay–Berne fluid J. Chem. Phys. 108, 7909 (1998); 10.1063/1.476228 Reuse of AIP Publishing content is subject to the terms: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 150.214.182.116 On: Fri, 01 Apr 2016 10:08:46 Bilayered smectic phase polymorphism in the dipolar Gay–Berne liquid crystal model Mohammed Houssa, Luis F. Rull, and Jose M. Romero-Enriquea兲 Departamento de Física Atómica, Molecular y Nuclear, Area de Física Teórica, Universidad de Sevilla, Apartado de Correos 1065, 41080 Sevilla, Spain 共Received 10 November 2008; accepted 12 March 2009; published online 17 April 2009兲 We present computer simulations of the Gay–Berne model with a strong terminal dipole. We report the existence of different stable antiferroelectric interdigitated bilayered phases in this model with diverse in-plane organization. The occurrence of these phases depends crucially on the value of the molecular elongation ␬ . For ␬ =3 we find an interdigitated bilayered smectic-A phase 共absent when there is no dipole兲and a bilayered smectic-T 共or crystal兲with positional in-plane tetragonal ordering, different from the hexatic observed in the absence of the molecular dipole. For ␬ =4, bilayered smectic-A and in-plane hexatic-ordered smectic-B 共or crystal兲phases are observed. ©2009 American Institute of Physics.关DOI: 10.1063/1.3111953兴 I. INTRODUCTION Fluids presenting liquid-crystalline behavior are frequently constituted by molecules which possess strong dipolar groups. Indeed, one of the earliest attempts to explain the occurrence of liquid crystals by Born1attributed the mesophases to the presence of dipolar interactions. Although it is known that they are not essential in the formation of the ordered phases, the dipolar forces are shown to play an important role in their stabilization.2–4The examination of the effect of polar interactions on the structure and thermodynamic properties of fluids has been the subject of numerous studies. Most of the liquid crystal computer simulations with an embedded electric dipole has been performed by using the so-called Gay–Berne 共GB兲potential.5This model is the most widely studied with uniaxial anisotropy and attractive interactions.6–13 Other models of dipolar mesogens, such as dipolar hard spherocylinders, have been also studied.14–22 A good summary of theoretical and simulation studies of dipolar mesogens can be found in a paper by Varga et al.23 The precise location and orientation of the dipole in the molecule are important in determining the structure of the liquidcrystalline phases. If we focus our attention to the case of the prolate GB molecules with the dipole located at the center of mass of the molecule and oriented along the molecular axis 共central dipole兲, the dipolar interaction has a little effect on the isotropic-nematic transition but a marked stabilization of the positionally layered liquid-crystalline phases such the smectics has been seen. For oblate dipolar GB molecules, evidences of a ferroelectric nematic fluid have been observed. A new phenomenology emerges when the dipole position shifts toward the end of the molecule. Experimentally different kinds of smectic phases have been observed when strong polar groups are present in the 共rodlike兲molecule: the usual monolayer SmA1phase, the bilayered SmA2phase 共with a layer spacing equal to the length of the molecule兲, and SmAd共which presents interdigitation between layers兲, among others.24–26 These new smectic phases show antiferroelectric ordering, which may be modulated across each layer as in the SmA ˜ phase.25 From a theoretical and computer simulation point of view, the GB model with a terminal dipole has been studied by Berardi et al.27–29 They carried out canonical ensemble Monte Carlo 共NVTMC兲computer simulations to study the influence of terminal dipoles in the phase diagram of the GB model, and they observed a stripped antiferroelectric bilayer structure with local ferroelectric order. More recently, they studied by NVTMC and variational theory the effect of the molecular dipole strength on the polymorphism of the smectic-A phase and a transition from noninterdigitated to interdigitated smectic-A molecular organization was observed.29 This result contrasts with the observed for hard spherocylinders with a terminal dipole, in which a dramatic stabilization of the nematic phase relative to the smectic phase is reported.17 Almost all the studies carried out up to now focus on the layer organization 共interdigitation, antiferroelectric ordering, etc.兲. However, these studies pay little attention to the way in which particles are organized on each layer. We will show by computer simulations that different layered phases are possible in the GB model with a strong terminal dipole. In addition to the SmAdphase previously reported, we find two bilayered phases with in-plane positional ordering, being tetragonal for small elongations and hexatic for large elongations. The paper is organized as follows. Section II is devoted to present the model and details of the computer simulation methodology. In Sec. III we present our results for different elongations of the GB molecule, and we end up with our conclusions. II. THE MODEL AND SIMULATION DETAILS The dipolar GB intermolecular pair potential for prolate molecules consists of a short-ranged anisotropic contribution Uij GB and a long-ranged dipolar interaction Uij dd. The terminal a兲Electronic mail: [email protected]. THE JOURNAL OF CHEMICAL PHYSICS 130, 154504 共2009兲 0021-9606/2009/130共15兲/154504/8/$25.00 © 2009 American Institute of Physics130, 154504-1 Reuse of AIP Publishing content is subject to the terms: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 150.214.182.116 On: Fri, 01 Apr 2016 10:08:46 dipole is placed at a distance dfrom the center of mass of each molecule and parallel to the main axis of the molecule. The GB potential Uij GB has the following expression:5 Uij GB共rij,ui,uj兲=4␧共r ˆij,ui,uj兲关 ␳ ij −12 − ␳ ij −6兴,共1兲 where ␳ ij =rij − ␴ 共r ˆij,ui,uj兲+ ␴ 0 ␴ 0 ,共2兲 uiis the unit vector along the symmetry axis of the molecule i,rij=兩ri−rj兩is the distance along the intermolecular vector rij joining the centers of mass of the molecules, and r ˆij =rij/rij. The anisotropic contact distance ␴ 共r ˆij,ui,uj兲and the depth of the interaction energy ␧共r ˆij,ui,uj兲depend on the orientational unit vector, the length to breathe ratio 共 ␬ = ␴ ee/ ␴ ss兲, and the energy depth anisotropy 共 ␬ ⬘= ⑀ ee/ ⑀ ss兲, which are defined as the ratio of the size and energy interaction parameters in the end-to-end 共ee兲and side-by-side 共ss兲 configurations. Their expressions are given in terms of the length scale ␴ 0and the energy unit ␧0as ␴ 共r ˆij,ui,uj兲 ␴ 0 = 冋 1− ␹ 2 冉 共r ˆij ·ui+r ˆij ·uj兲2 1+ ␹ 共ui·uj兲 +共r ˆij ·ui−r ˆij ·uj兲2 1− ␹ 共ui·uj兲 冊 册 −1/2 ,共3兲 ␧共r ˆij,ui,uj兲 ⑀ 0 =关 ⑀ 1共ui,uj兲兴 ␯ ⫻关 ⑀ 2共r ˆij,ui,uj兲兴 ␯ ⬘ ,共4兲 where ⑀ 1共ui,uj兲=关1− ␹ 2共ui·uj兲2兴−1/2,共5兲 ⑀ 2共r ˆij,ui,uj兲=1− ␹ ⬘ 2 冋 共r ˆij ·ui+r ˆij ·uj兲2 1+ ␹ ⬘共ui·uj兲 +共r ˆij ·ui−r ˆij ·uj兲2 1− ␹ ⬘共ui·uj兲 册 ,共6兲 ␹ =共 ␬ 2−1兲/共 ␬ 2+1兲and ␹ ⬘=关共 ␬ ⬘兲1/ ␯ ⬘ −1兴/关共 ␬ ⬘兲1/ ␯ ⬘ +1兴.Asin the original paper 共Ref. 5兲, we choose ␯ ⬘=2 and ␯ =1. We also set ␬ ⬘=5, allowing different values for the geometrical anisotropy between ␬ =3 and 4. The dipolar potential energy Uij dd between two molecules is given by Uij dd = ␮ 2 rd 3 冉 ui·uj−3共ui·rd兲共uj·rd兲 rd 2 冊 ,共7兲 where rd=rij+d共uj−ui兲is the vector joining the two point dipoles and ␮ is the dipolar moment of the molecule. The dipolar GB pair potential Uij=Uij GB+Uij dd energy contours for ␬ =3, ␮ =2冑 ⑀ 0 ␴ 0 3, and d= ␴ 0are represented in Fig. 1for parallel and antiparallel configurations. Note that they are the relevant configurations for orientationally ordered phases. For the antiparallel case, two different local minima are observed: the global minimum associated mainly with the presence of the dipole at roughly ␪ ⬇18° 共Uij⬇ −10.7 ⑀ 0兲, where ␪ is the angle between rij and the molecular orientation of the central particle ui. Additionally there is a local minimum associated with the GB part of the potential at ␪ ⬇85° 共Uij⬇−1.3 ⑀ 0兲. Finally, the effective hard-body shape corresponding to this configuration, estimated from the Uij=0 contour, has a pearlike shape, which may also lead to the formation of interdigitated bilayered smectic phases.30 For the parallel configuration, two equivalent minima are obtained for ␪ ⬇40° and ␪ ⬇140° 共Uij⬇−1.6 ⑀ 0兲. The repulsive part of the potential is swollen in the equatorial zone of the GB molecule due to the presence of the dipole, although the dipolar repulsion is softer than the induced by the GB potential 共Uij=4 ⑀ 0for ␪ =90° and rij= ␴ 0兲. Following the same strategy used to study nonpolar GB and central dipolar GB model,2–4we carry out Monte Carlo computer simulations in the isothermal-isobaric ensemble 共NPTMC兲. Our simulations are performed for N=1372 par- -4 -2 0 2 4 6 -4 -2 0 2 4 x y  -4 -2 0 2 4 6 -4 -2 0 2 4 x y ( b) (a) FIG. 1. Potential energy contours of the dipolar GB pair potential between a particle placed at the origin and its molecular axis oriented along the x ⬎0 semiaxis, and another particle placed in an arbitrary position on the z =0 plane and oriented parallel 共top兲and antiparallel 共bottom兲with respect to the particle at the origin. Lighter shadowing represents lower energies. The potential parameters are ␬ =3, ␬ ⬘=5, ␮ =2冑 ⑀ ␴ 0 3, and d= ␴ 0共see text for explanation兲. 154504-2 Houssa, Rull, and Romero-Enrique J. Chem. Phys. 130, 154504 共2009兲 Reuse of AIP Publishing content is subject to the terms: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 150.214.182.116 On: Fri, 01 Apr 2016 10:08:46 TABLE I. Thermodynamic and structural properties of the terminal dipolar GB model with ␬ =3 and ␮ ⴱ=2. Tⴱ and pⴱare the reduced temperature and pressure, ␳ ⴱ= ␳␴ 0 3is the reduced number density, uⴱ=U/N ⑀ 0is the reduced potential energy per particle, Sis the nematic order parameter, and B4is the tetragonal bondorientational order. For each state, we identify the corresponding phase 共see text for explanation兲. Tⴱpⴱ ␳ ⴱuⴱSB 4Phase 0.50 0.1 0.3352共6兲⫺15.989共14兲0.9900共4兲0.856共7兲SmTd 0.50 0.5 0.3425共5兲⫺16.085共11兲0.9905共4兲0.858共6兲SmTd 0.50 1.0 0.3503共4兲⫺16.155共15兲0.9911共4兲0.857共6兲SmTd 0.50 2.0 0.3618共4兲⫺16.199共8兲0.9909共5兲0.863共6兲SmTd 0.50 3.0 0.3748共8兲⫺16.223共12兲0.9907共3兲0.874共4兲SmTd 0.50 4.0 0.3825共4兲⫺16.187共13兲0.9909共4兲0.866共3兲SmTd 0.50 5.0 0.3891共3兲⫺16.101共13兲0.9906共3兲0.859共4兲SmTd 0.50 6.0 0.3953共3兲⫺16.042共14兲0.9916共3兲0.867共3兲SmTd 0.50 7.0 0.4070共6兲⫺16.07共3兲0.9903共9兲0.897共5兲SmTd 0.50 8.0 0.4108共3兲⫺15.998共14兲0.9927共5兲0.898共3兲SmTd 1.00 0.1 0.2897共19兲⫺13.61共7兲0.960共2兲0.512共19兲SmTd 1.00 0.5 0.3203共15兲⫺14.24共5兲0.9677共13兲0.745共16兲SmTd 1.00 4.0 0.3811共6兲⫺15.00共3兲0.9551共16兲0.825共5兲SmTd 1.00 7.0 0.3996共5兲⫺14.91共2兲0.9561共13兲0.815共6兲SmTd 1.25 0.1 0.0875共17兲⫺3.11共7兲0.029共9兲¯I 1.25 0.5 0.2845共17兲⫺12.00共9兲0.937共3兲0.073共13兲SmAd 1.25 1.0 0.3276共13兲⫺13.51共5兲0.9660共11兲0.550共13兲SmTd 1.25 7.0 0.4025共6兲⫺14.93共3兲0.9757共19兲0.905共4兲SmTd 1.50 1.0 0.21115共13兲⫺4.834共8兲0.04共4兲¯I 1.50 1.5 0.294共2兲⫺11.17共14兲0.914共6兲0.071共31兲SmAd 1.50 2.0 0.3328共17兲⫺12.89共7兲0.958共2兲0.672共19兲SmTd 1.50 3.0 0.3556共13兲⫺13.59共5兲0.968共2兲0.792共11兲SmTd 1.50 5.0 0.3794共9兲⫺14.15共4兲0.968共3兲0.857共8兲SmTd 1.50 7.0 0.3944共7兲⫺14.34共4兲0.9715共17兲0.878共5兲SmTd 2.00 4.0 0.2745共11兲⫺4.83共7兲0.06共2兲¯I 2.00 4.5 0.351共2兲⫺12.51共12兲0.965共3兲0.748共27兲SmTd 2.00 5.0 0.3555共15兲⫺12.57共8兲0.963共2兲0.740共13兲SmTd 2.00 6.0 0.3680共11兲⫺12.92共5兲0.9693共15兲0.785共10兲SmTd 2.50 7.0 0.2968共10兲⫺4.02共7兲0.057共16兲¯I 2.50 8.0 0.3069共10兲⫺4.05共7兲0.065共16兲¯I 2.50 9.0 0.3691共15兲⫺11.33共14兲0.947共5兲0.690共17兲SmTd 2.50 10.0 0.3865共10兲⫺12.48共7兲0.9777共10兲0.845共7兲SmTd 3.00 10.0 0.3101共9兲⫺3.09共6兲0.05共2兲¯I 3.00 11.0 0.3184共11兲⫺3.10共8兲0.11共5兲¯I 3.00 12.0 0.3266共11兲⫺3.16共10兲0.19共7兲¯I 3.00 13.0 0.03432共10兲⫺3.94共8兲0.656共18兲¯N 3.00 14.0 0.3507共12兲⫺4.00共10兲0.69共2兲¯N 3.00 15.0 0.3898共12兲⫺10.16共16兲0.949共4兲0.686共15兲SmTd 3.00 16.0 0.3977共11兲⫺10.73共14兲0.961共4兲0.744共11兲SmTd 3.50 13.0 0.3200共8兲⫺2.17共6兲0.061共17兲¯I 3.50 14.0 0.3262共9兲⫺2.10共6兲0.060共11兲¯I 3.50 15.0 0.3345共10兲⫺2.13共7兲0.22共2兲¯I 3.50 16.0 0.345共2兲⫺2.55共9兲0.51共3兲¯N 3.50 17.0 0.353共3兲⫺2.65共9兲0.61共4兲¯N 3.50 18.0 0.361共3兲⫺2.75共8兲0.72共4兲¯N 3.50 19.0 0.367共4兲⫺2.73共7兲0.72共4兲¯N 3.50 20.0 0.3722共11兲⫺2.77共8兲0.77共4兲¯N 3.50 21.0 0.3758共9兲⫺2.72共9兲0.766共10兲¯N 3.50 22.0 0.3801共8兲⫺2.65共9兲0.781共11兲¯N 3.50 23.0 0.4121共9兲⫺9.09共13兲0.959共2兲0.708共10兲SmTd 3.50 24.0 0.4166共9兲⫺9.26共15兲0.966共3兲0.734共13兲SmTd 3.50 25.0 0.4201共8兲⫺9.32共12兲0.9685共18兲0.745共10兲SmTd 4.00 18.0 0.3374共9兲⫺1.04共8兲0.09共6兲¯I 4.00 19.0 0.3447共11兲⫺1.10共10兲0.25共5兲¯I 4.00 20.0 0.3563共12兲⫺1.60共10兲0.60共3兲¯N 4.00 21.0 0.3637共15兲⫺1.74共12兲0.67共3兲¯N 4.00 22.0 0.3678共10兲⫺1.62共9兲0.692共18兲¯N 154504-3 Smectic polymorphism in the dipolar Gay–Berne J. Chem. Phys. 130, 154504 共2009兲 Reuse of AIP Publishing content is subject to the terms: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 150.214.182.116 On: Fri, 01 Apr 2016 10:08:46 ticles. The reaction field method with a dielectric constant ⑀ RF=⬁and a cutoff Rc RF=4 ␴ 0共 ␬ =3兲and Rc RF=6 ␴ 0共 ␬ =4兲 has been used to deal with the long-range character of the dipolar interactions 共for details of the simulations, see the Refs. 2,19,28,29,31, and 32兲. Each MC cycle consisted of Nattempted particle translations and rotations and one volume change, where the three dimensions of the box were allowed to change independently. This procedure is particularly helpful when smectic phases appear in the simulations. In addition, an attempted dipole flip was randomly done with a 25% frequency compared to the 75% of the conventional translation-rotational moves to prevent locking in metastable states. The simulation procedure is the following. The simulation for the lowest pressure of each isotherm starts from a configuration at a very low density of particles 共 ␳ ⴱ⬅ ␳␴ 0 3 =0.01兲and a random molecular orientation distribution. Each simulation takes as initial configuration the final output of the previous simulation at lower pressure. The number of cycles for each simulation is of order of 2⫻105for disordered phases and up to 3⫻106cycles long for spacially ordered phases. After a period of equilibration 共longer for orientationally ordered systems than for isotropic systems兲a set of physical quantities, such as the average energy Uand density ␳ =N/具V典, as well as their statistical errors, are evaluated from the remainder of the run. The orientational order is characterized by the nematic order parameter S, defined as the largest eigenvalue of the order tensor Q, Q= 冓 1 N兺 i=1 N3ui丢ui−I 2 冔 .共8兲 Additionally the orientational ordering of the molecular dipoles is also monitorized by the global polarity P1, P1= 冓 1 N冏兺 i=1 N ui·n冏 冔 ,共9兲 where nis the nematic director defined as the eigenvector of Qassociated with its largest eigenvalue S. In order to characterize the microscopic structure, we evaluate a series of correlation functions. The first and second orientational correlation functions g1共r兲and g2共r兲are defined as gi共r兲= 具 兺i兺j⫽i ␦ 共r−rij兲Pi共ui·uj兲 典 具 兺i兺j⫽i ␦ 共r−rij兲 典 ⬅具V典 4 ␲ r2N2g共r兲 冓 兺 i兺 j⫽i ␦ 共r−rij兲Pi共ui·uj兲 冔 ,共10兲 where Pi共x兲is the ith-order Legendre polynomial 共i=1,2兲 and g共r兲is the usual pair correlation function. In order to observe the formation of layers in the smectic 共or crystalline兲 phases, we can split the pair distribution function into the pair correlation functions g 储 共r 储 兲and g⬜共r⬜兲, which account for the correlations between the positions of two particles along a direction parallel or perpendicular to the nematic director, respectively. Here the distance r 储 is the projection of rij along the nematic director nand r⬜is the projection onto the plane perpendicular to n. As we anticipate that dipolar correlations will be helpful to understand our results, so we have also evaluated the correlation functions mentioned previously, but referring the position of the molecule not to the center of mass of the molecule, but to the position of the point dipole 关g⬜ d共r⬜兲and g 储 d共r 储 兲兴. An alternative way to describe the positional ordering in layered phases is by the evaluation of the density profiles for the molecule centers of mass along the nematic director direction, ␳ 共z兲⬅ 冓 兺 i=1 N ␦ 共zi−z兲 冔 ,共11兲 where zis the position along the nematic director nand zi =ri·n. In order to see the dipolar ordering, we calculate the analog of the density profile for the dipolar centers ␳ d共z兲 ⬅具兺i=1 N ␦ 共zi d−z兲典 and the polarity profile p共z兲 ⬅具兺i=1 N共ui·n兲 ␦ 共zi d−z兲典, where zi d=zi+d共ui·n兲. The correlation functions mentioned above allow to distinguish between spacially homogeneous and stratified phases. However, these functions alone cannot unequivocally determine the nature of the layered phases. To characterize the structure of stratified phases, a bond-orientational order parameter B ␯ within the layers has been evaluated. This function is defined as B ␯ =1 ␯ N 冐 冓 兺 j=1 N 兺 k=1 ␯ exp共i ␯ ␾ jk兲 冔 冐 ,共12兲 where ␾ jk is the angle between the bond linking particles j and kand a fixed reference axis and ␯ is the number of TABLE I. 共Continued.兲 Tⴱpⴱ ␳ ⴱuⴱSB 4Phase 4.00 23.0 0.3724共11兲⫺1.57共10兲0.713共18兲¯N 4.00 24.0 0.3764共13兲⫺1.51共10兲0.75共2兲¯N 4.00 25.0 0.3810共9兲⫺1.43共9兲0.750共12兲¯N 4.00 26.0 0.3858共11兲⫺1.37共8兲0.77共2兲¯N 4.00 27.0 0.3895共15兲⫺1.20共11兲0.78共4兲¯N 4.00 28.0 0.393共3兲⫺1.15共11兲0.79共3兲¯N 4.00 29.0 0.396共3兲⫺1.05共12兲0.80共2兲¯N 4.00 30.0 0.399共3兲⫺0.96共17兲0.81共2兲¯N 4.00 31.0 0.403共4兲⫺0.8共2兲0.826共19兲¯N 154504-4 Houssa, Rull, and Romero-Enrique J. Chem. Phys. 130, 154504 共2009兲 Reuse of AIP Publishing content is subject to the terms: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 150.214.182.116 On: Fri, 01 Apr 2016 10:08:46 nearest-neighbor bonds, defined as the number of particles to be within rb⬇1.2 ␴ 0from each one. The parameter ␯ takes into account the short-range positional order. For example, ␯ =6 corresponds to a locally hexagonal packing. The order parameter B ␯ takes values close to zero when no in-plane orientational bond-order exists and close to one in the presence of perfect in-plane orientational bond order. III. SIMULATION RESULTS We have performed computer simulation for GB molecules with a terminal dipole characterized by different geometrical anisotropies ␬ in a range between ␬ =3 and ␬ =4. In our study we considered that the dipole is located at a reduced distance dⴱ⬅d/ ␴ 0=共 ␬ −1兲/2 from the molecular center of mass along the molecular axis, with a reduced dipolar moment strength given by ␮ ⴱ⬅冑 ␮ 2/ ⑀ 0 ␴ 0 3=2. In the absence of dipole, the phase diagram of the GB model has been previously characterized in the literature.12 For ␬ =3, the only reported phases are the isotropic 共I兲, either vapor or liquid, nematic 共N兲, and smectic-B 共SmB兲with hexatic in-layer bond orientational. However, there are concerns about the true nature of the SmBphase, which may be a crystal phase.12 As ␬ increases, an island of smectic A 共SmA兲phase appears for ␬ ⱖ3.6.12 When a dipole is placed at the center of mass of the molecule,2,3no new phases appear, although it is observed that the presence of the dipolar interactions favors the formation of layered phases. Thus there is a stabilization of the smectic phases with respect to either the isotropic or nematic phase. However, the relative stability between the isotropic and nematic phases is virtually unaffected by the presence of the dipole. A. ␬ =3.0 We have studied the isotherms Tⴱ⬅kBT/ ⑀ 0=0.50, 1.00, 1.25, 1.50, 2.00, 2.50, 3.00, 3.50, and 4.00 共see Table I兲. Our results show a dramatic change in the phase diagram, confirming that the presence of the dipole stabilizes layered phases with respect to the nematic or isotropic phase. This contrasts with the reported results for hard spherocylinders with a terminal dipole, in which a dramatic stabilization of the nematic phase relative to the smectic phase is observed.17 For the lower temperatures 共Tⴱ=0.50 and 1.0兲, the only observed phase for reduced pressures pⴱ=p ␴ 0 3/ ⑀ 0larger than 0.05 is a layered phase 共see snapshot in Fig. 2兲. Figure 3 plots the correlation functions g 储 共r 储 兲and g 储 d共共r 储 兲兲for different pressures. The bilayered character of the phase is revealed by the splitting of the second peak of g 储 corresponding to the two neighboring layers. This is confirmed by the analysis of the density and dipolar density profiles 共see Fig. 4兲and the alternance of the dipolar orientations. The dipoles of the molecules on each layer which composes the bilayer are pointing outward. Examination of g 储 shows that the distance between the layers which compose the bilayer is nearly constant and equal to 2 ␴ 0⬍ ␬ ␴ 0. This indicates a strong interdigitation between them. It was also observed correlation between particles of neighboring layers, which may indicate a solidlike character of the phase. There is also interdigitation between the closest layers corresponding to different neighboring bilayers, but their distances decrease as the pressure is increased. The examination of the dipole densities for different pressures shows that under compression, the dipole layers corresponding to neighboring bilayers coalesce into a single FIG. 2. 共Color兲Snapshot of 1372 dipolar GB molecules 共 ␬ =3, ␮ ⴱ=2兲at Tⴱ=0.5 and pⴱ=0.1, corresponding to a SmTdphase. The dipole position along the molecular axis is represented by the color change in the molecule. 0 5 10 g||(r*) p*=0.1 p*=3.0 p*=7.0 0123456 7 r * 0 5 10 g d ||(r*) FIG. 3. Plot of the longitudinal pair distribution functions of centers of masses g 储 共rⴱ兲共upper panel兲and the dipolar centers g 储 d共rⴱ兲共lower panel兲for a dipolar GB model with ␬ =3 and ␮ ⴱ=2 at Tⴱ=0.5 and different pressures: pⴱ=0.1 共dashed line兲,pⴱ=3.0 共continuous line兲,andpⴱ=7.0 共dot-dashed line兲. 154504-5 Smectic polymorphism in the dipolar Gay–Berne J. Chem. Phys. 130, 154504 共2009兲 Reuse of AIP Publishing content is subject to the terms: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 150.214.182.116 On: Fri, 01 Apr 2016 10:08:46 layer for high enough pressure 共see Figs. 3and 4兲. We have not been able, however, to check if this structural transition is a signature of the appearance of a new phase. The phase described above is analogous to the antiferroelectric smectic phase reported by Berardi et al.27–29 for other molecular parameters. In order to fully characterize this phase, we have studied the in-layer positional correlations. The hexatic bond-orientational order B6gives negligible values. However, the values of B4show that the molecules arrange in each layer in a tetragonal fashion. This positional ordering is also put into manifest by the transversal correlation function g⬜, which shows a significant in-layer structure, and the positions of the maxima are consistent with the location of the nodes of a square lattice. We will denote this interdigitated bilayered antiferroelectric smectic phase with in-layer tetragonal-positional order as the SmTdphase. This tetragonal in-layer ordering has been also observed in GB models with a terminal dipole for ␬ =4 and dⴱ=0.5 共Ref. 33兲 and with two opposing terminal dipoles.34 Experimentally this in-layer structure has been observed in ionic liquid crystals.35,36 For higher temperatures and the same range of pressures, new phases appear. The isotropic phase is observed for all the temperatures considered above Tⴱ=1.00 and small pressures. Upon compression of the system, the isotropic phase changes to bilayered structures for Tⴱ=1.25 and Tⴱ=1.50 at pⴱ=0.5 and pⴱ=1.5, respectively. Analysis of the density and dipole profiles, as well as g 储 , shows a similar interdigitated antiferroelectric stratification to the SmTdphase. However, the in-layer structure is liquidlike as B ␯ ⬇0 for ␯ =4 and 6. Furthermore, g⬜shows a liquidlike structure, so we can identify this phase as SmAd共see Fig. 5兲. It is worthwhile to note that true SmAwas never reported for ␬ =3 in the absence or with central dipole. Further pressure increase leads to a transition to the SmTdphase. The temperature and pressure range at which the SmAdphase appears is quite limited, and no trace of it appears for Tⴱⱖ2. However, the stability of this phase has been confirmed as it was obtained from both disordered and SmTd-like configurations. For Tⴱⱖ3.00 the nematic phase intrudes between the I and SmTdphases. Again, the late appearance of the nematic phase 共in the absence of dipole, it appears for Tⴱⱖ0.83兲is a consequence of the stabilization of the smectic phases by the dipolar interaction. We have heated the system up to Tⴱ=4.00. Although for high enough temperatures 共or pressures兲the SmBphase is expected to appear, no sign of it was observed. B. ␬ =4.0 We consider now the geometrical anisotropy parameter ␬ =4. We have done simulations for Tⴱ=1.50, Tⴱ=1.75, and Tⴱ=2.00 共see Table II兲. For the lowest temperature, we observe only an antiferroelectric and interdigitated bilayered phase 共see Fig. 6兲. However, the in-layer structure differs to the corresponding to the SmTdphase for ␬ =3. Now, for ␯ =4, B ␯ is negligible, but for ␯ =6 takes nonvanishing values. Consequently, this phase has a hexatic in-layer positional ordering, so we will denote this phase as SmBd. This is also confirmed by the change in the positions of the maxima observed in the transversal correlation function g⬜共see Fig. 6 for Tⴱ=1.5 and pⴱ=1.0兲, now consistent with the location of the nodes on a triangular lattice. As the temperature is increased, new phases are observed: an isotropic fluid phase for low densities and a SmAdphase between the Iand the SmBdphase. The stability range for the SmAdphase is larger 0 2 4 6 -5 05 z * 0 2 4 6 0 2 4 6 ρ(z*) ρd(z*) FIG. 4. Center of mass densities ␳ 共zⴱ兲共continuous lines兲and dipolar center densities ␳ d共zⴱ兲共dashed lines兲measured along the director for Tⴱ=0.5 and different values of the pressure pⴱ:pⴱ=0.1 共upper panel兲,pⴱ=3.0 共central panel兲, and pⴱ=7.0 共lower panel兲. The geometrical anisotropy parameter is ␬ =3, and the arrows represent the dipole orientation at the central peaks. 024 6 r * 0 1 2 3 4 g|(r*) T*=1.5 , p*=1.5 T*=1.5 , p*=2.0 0246 r* 0 1 2 3 4 g||(r*) FIG. 5. Transversal pair correlation function g⬜共rⴱ兲for Tⴱ=1.5 and pⴱ =1.5 共continuous line兲and pⴱ=2.0 共dashed line兲. Inset: longitudinal pair correlation function g 储 共rⴱ兲for Tⴱ=1.5 and pⴱ=1.5. The geometrical anisotropy parameter is ␬ =3. 154504-6 Houssa, Rull, and Romero-Enrique J. Chem. Phys. 130, 154504 共2009兲 Reuse of AIP Publishing content is subject to the terms: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 150.214.182.116 On: Fri, 01 Apr 2016 10:08:46 than for ␬ =3, as expected since the larger elongation of the molecules favors the formation of the SmAphase, even in the absence of dipole. We do not find any trace of a nematic phase, which should be observed for higher temperatures. C. From ␬ =3.0 to ␬ =4.0 In order to understand the appearance of different smectic phases, we have considered the effect of ␬ on the in-layer structure. Thus we have done simulations for Tⴱ=0.5 and pⴱ=0.5, which corresponds to a SmTdphase for ␬ =3 and SmBdphase for ␬ =4, and we have studied different elongations ␬ =3.1, ␬ =3.2, and ␬ =3.3. Our results show that for ␬ smaller than 3.3, the in-layer structure is always tetragonal, so the equilibrium phase is SmTd. However, for ␬ ⱖ3.3, a SmBdis developed. We have checked the consistency of these results by beginning from different initial conditions. So, Fig. 7shows the evolution of B4and B6for ␬ =3.3 when the initial condition corresponds to a SmTd-like configuration. As it can be seen, the tetragonal in-layer structure crossovers to a hexatic one smoothly, indicating that the SmTdand the SmBdstructures can be transformed in each other by a continuous transformation. On the other hand, we anticipate that for some values of ␬ between 3 and 4 both SmTdand SmBdmay coexist. However, the nature of the transition between these two bilayered states is unknown and beyond our present study. IV. CONCLUSIONS In this paper we have presented computer simulations of GB molecules with a strong terminal dipole 共 ␮ ⴱ=2兲placed at dⴱ=共 ␬ −1兲/2. We have observed that the dipole stabilizes the formation of layered phases with respect to the fluid phases. Their interdigitation and dipolar ordering are similar to that already reported in the literature,27–29,33 which is a consequence of the interplay between the dipolar interaction TABLE II. Thermodynamic and structural properties of the terminal dipolar GB model with ␬ =4 and ␮ ⴱ=2. For each state, we identify the corresponding phase 共see text for explanation兲. Tⴱpⴱ ␳ ⴱuⴱSB 6Phase 1.50 0.1 0.2081共18兲⫺17.51共12兲0.941共5兲0.628共15兲SmBd 1.50 0.25 0.2170共14兲⫺17.16共10兲0.963共2兲0.644共17兲SmBd 1.50 0.5 0.2306共12兲⫺17.74共10兲0.9723共14兲0.611共16兲SmBd 1.50 1.0 0.2429共10兲⫺18.47共8兲0.9775共13兲0.488共16兲SmBd 1.50 2.0 0.2598共7兲⫺19.06共7兲0.9840共8兲0.522共11兲SmBd 1.50 3.0 0.2660共7兲⫺19.12共7兲0.9845共7兲0.512共10兲SmBd 1.75 0.25 0.0838共2兲⫺3.98共4兲0.0365共6兲¯I 1.75 0.5 0.2011共13兲⫺14.90共10兲0.9472共19兲0.037共18兲SmAd 1.75 0.75 0.2121共12兲⫺15.51共9兲0.9550共17兲0.058共25兲SmAd 1.75 1.0 0.2204共13兲⫺15.95共9兲0.9595共17兲0.062共32兲SmAd 1.75 1.25 0.2342共14兲⫺16.98共13兲0.9692共17兲0.512共33兲SmBd 2.00 0.5 0.1024共11兲⫺3.68共10兲0.033共10兲¯I 2.00 0.75 0.1229共12兲⫺4.34共10兲0.038共13兲¯I 2.00 1.0 0.1988共16兲⫺13.43共18兲0.925共5兲0.028共17兲SmAd 2.00 1.25 0.2140共12兲⫺14.73共10兲0.9481共19兲0.050共25兲SmAd 2.00 1.5 0.2200共13兲⫺15.06共11兲0.952共2兲0.054共30兲SmAd 2.00 1.75 0.2258共11兲⫺15.39共10兲0.956共2兲0.073共34兲SmAd 2.00 2.0 0.2315共10兲⫺15.73共9兲0.9590共16兲0.104共42兲SmAd 2.00 2.50 0.2468共11兲⫺17.11共10兲0.9732共14兲0.607共17兲SmBd 01234 5 r * 0 1 2 3 4 5 g| ( r *) T*=1.5 , p*=1.0 T*=1.75 , p*=0.75 0123456 r* 0 2 4 6 8 10 g||(r*) FIG. 6. Transversal pair correlation function g⬜共rⴱ兲for Tⴱ=1.5 and pⴱ =1.0 共continuous line兲and Tⴱ=1.75 and pⴱ=0.75 共dashed line兲. The geometrical anisotropy parameter is ␬ =4. Inset: longitudinal pair correlation function g 储 共rⴱ兲for the same conditions. 050000 100000 150000 200000 MC cycles 0 0.2 0.4 0.6 0.8 1 B4 B6 FIG. 7. 共Color兲Monte Carlo evolution of the bond-order parameters B4and B6for 1372 dipolar GB molecules with geometrical anisotropy parameter ␬ =3.3 at Tⴱ=0.5 and pⴱ=0.5. The initial condition corresponds to a SmTd state. 154504-7 Smectic polymorphism in the dipolar Gay–Berne J. Chem. Phys. 130, 154504 共2009兲 Reuse of AIP Publishing content is subject to the terms: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 150.214.182.116 On: Fri, 01 Apr 2016 10:08:46 共controlled by the dipole position and its strength兲and the GB interaction 共controlled by the geometrical anisotropy ␬ and the strength of the dispersive interactions兲. However, we observe different layered phases with diverse in-layer molecular organizations. We report the existence of three different antiferroelectric bilayered phases: SmAd,SmTd, and SmBdphases, depending on the geometrical elongation of the molecule. For all these phases, the layers which compose the basic bilayer are at a nearly constant distance smaller than ␬ ␴ 0, indicating interdigitation between them. On the other hand, the distance between bilayers increases as the pressure is lowered. For ␬ =3 we observe the appearance of a SmAdand the SmTdwith tetragonal-positional order on the layer. These phases are induced by the presence of an offcentered strong dipole, since no true SmAphase is observed in the absence or with a central dipole. On the other hand, the only layered phase observed without dipole or for a central dipole is a SmBphase, with hexatic in-layer positional order. At this point it is worthy noting that the GB parametrization used by Berardi et al.27–29 共different from ours兲 allows for the appearance of a SmAphase in the absence of dipole even for ␬ =3. As the GB molecule becomes more elongated, the tetragonal in-layer ordering in the SmTd changes to the more usual hexatic ordering of the SmBd phase, and for ␬ =4 we do not observe the SmTdphase. For intermediate values of ␬ both SmBdand SmTdphases may coexist. These results show that the molecular geometry plays an important role not only on the onset of the bilayered phases but also in the microscopic organization of the phase. ACKNOWLEDGMENTS The authors acknowledge financial support from the Junta de Andalucía 共Spain兲through Grant No. P06-FQM- 01869 and “Ayudas PAIDI FQM-205” and Ministerio de Educación y Ciencia 共Spain兲through Grant No. ENE2007- 68040-C03-02. J.M.R.-E. also acknowledges a “Ramón y Cajal” Fellowship. 1M. Born, Sist. Phys. Math. 26,614共1916兲; Ann. Phys. 55, 221 共1918兲. 2M. 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Downloaded to IP: 150.214.182.116 On: Fri, 01 Apr 2016 10:08:46