phys. stat. sol. (b) 242, No. 3, 626–631 (2005) / DOI 10.1002/pssb.200460381 © 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim Poisson’s ratio of degenerate crystalline phases of three-dimensional hard dimers and hard cyclic trimers M. Kowalik and K. W. Wojciechowski* Institute of Molecular Physics, Polish Academy of Sciences, Smoluchowskiego 17, 61-169 Poznań, Poland Received 31 August 2004, accepted 29 November 2004 Published online 15 February 2005 PACS 05.10.Ln, 61.44.–n, 61.50.–f, 62.20.Dc, 83.80.Ab Monte Carlo simulations of two three-dimensional hard-body models, hard dimers and hard cyclic trimers, were performed in the NpT ensemble. The Poisson’s ratio was determined for degenerate crystal- line phases in both systems. In contrast to the results of analogous simulations in two-dimensions, positive values were obtained both for the dimers and the trimers. However, the Poisson’s ratio of the trimers was lower than that for the dimers and the latter was lower than that for the spheres. This suggests that a nega- tive Poisson’s ratio may be obtained for molecules composed of more spheres. © 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 1 Introduction There are quite a few important reasons which have attracted physicists’ attention to hard-body models, i.e., systems interacting via a hard-body potential (infinite if an overlap occurs among cores of the parti- cles forming the system and zero otherwise), for over half a century. Firstly, simple models can serve as reference systems in many physical theories describing more complex systems [1]. Secondly, hard-body systems are interesting models in studying packing effects and short-range correlations which play a crucial role in both molecular fluids and solids. Finally, in spite of their simplicity, models of molecular systems interacting via a hard-body potential are able to reproduce many fundamental features of real physical systems. It is a consequence of the fact that various properties of real systems significantly de- pend on both molecular shape and size. The hard sphere system [2, 3], providing a good model for the physics underlying freezing of the rare gases [4, 5], is probably the most widely known hard-body model. Results obtained for systems of con- vex non-spherical molecules were also encouraging enough to maintain a great scientific interest in hard- body models [6]. Extensive Monte Carlo (MC) simulation studies of hard ellipsoids [7–9] and hard spherocylinders [10] showed that systems of anisotropic hard-body “molecules” are able to mimic many liquid-crystalline and solid phases. Studies for another class of anisotropic hard molecules, so called multispheres, were also per- formed [11–14]. In those papers extensive MC studies were presented concerning the simplest represen- tative of this class – the hard homonuclear dumbbell – a crude model of a diatomic molecule (and the simplest non-convex body as well) formed by two fused hard spheres, each of diameter ,s with centers separated by a distance d. It is worth adding that these systems constitute the starting point for studies of polymers [15–18]. The influence of a reduced bond length, known also as a dumbbell elongation parameter, *d d s∫ / on the phase diagram and phase stability was thoroughly investigated. It should be * Corresponding author: e-mail:
[email protected] phys. stat. sol. (b) 242, No. 3 (2005) / www.pss-b.com 627 © 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim Original Paper stressed that only periodic crystalline solid phases of the hard homonuclear dumbbell systems were stud- ied in the papers mentioned above. However, it is expected that three-dimensional hard dimers (dumbbells with elongation * 1,d = see Fig. 1) form an unusual, although thermodynamically stable solid phase, a so called aperiodic or degen- erate crystal (DC) phase. This is analogous to their two-dimensional counterpart [19–22] for which the thermodynamic properties are well established, see [19–24]. Similarly, properties of a DC phase of hard cyclic trimers (Fig. 2 shows a three-dimensional cyclic trimer) have already been thoroughly investi- gated, in two dimensions, [24, 25]. We are not aware of any investigation concerning three-dimensional counterparts of such structures and this paper is aimed to fill this gap in the aspect of the Poisson’s ratio. Other properties of the DC phase in three dimensions will be discussed elsewhere. Knowledge of the elastic properties of materials is of interest for both practical applications and fun- damental research. The Poisson’s ratio, defined as the negative ratio of the transverse strain to longitudi- nal strain when longitudinal stress is applied, is an important characteristic of materials. Everyday mate- rials have a positive Poisson’s ratio i.e., they shrink transversely while they are stretched longitudi- nally [26]. Recently, however, new group of materials exhibiting counter-intuitive behaviour has been engineered [27], so called auxetics, i.e., materials with negative Poisson’s ratio which undergo lateral expansion (contraction) upon longitiudinal tension (compression). Such materials may find many practi- cal applications and are currently the subject of intensive studies by a number of scientific groups over the world (see other papers of this volume). Various mechanisms which may lead to auxetic behaviour have been proposed, see e.g., [25, 28, 29]. One of the possibilities is to use assembly of non-convex objects of which hard dimers and trimers are the simplest examples. Indeed, results obtained for two-dimensional systems of such molecules were encouraging [24, 25]. The DC phase of two-dimensional hard dimers shows a positive but small Pois- son’s ratio in the close packing limit ( cp 0 051(12)n = . , see [24]). Results for the same phase in the case of two-dimensional hard trimers in the close packing limit are even more interesting as they exhibit a nega- tive Poisson’s ratio [24, 25, 30]. Thus, a question arises: what is the Poisson’s ratio for the three- dimensional counterparts of these structures. This question is the subject of the present paper. The paper is organised as follows. In Section 2 DC phases of hard dimers and trimers are described. Section 3 is devoted to the results concerning the Poisson’s ratio of the hard dimer and trimer DC phases. Finally, in Section 4 conclusions are drawn. 2 Aperiodic molecular structures at close packing Quasi-crystalline structures, characterized by long-range orientational order but lacking any periodicity at the same time, have been known for over twenty years. It is still usually assumed, however, that solid phase structures, at least at low temperatures, must coincide with translationally invariant crystalline lattices. The systems described in this paper constitute striking examples showing that the above assump- tion is not always valid. Fig. 1 Geometry of a hard three-dimensional dimer. The internal continuous line and the points at sphere centres have no physical mean- ing; they are used in schematic plots in Fig. 3. Fig. 2 Geometry of a hard three-dimensional trimer. The internal continuous lines and the points at sphere centres have no physical meaning; they are used in schematic plots in Fig. 4. 628 M. Kowalik and K. W. Wojciechowski: Poisson’s ratio of degenerate crystalline phases © 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim Two-dimensional dimer “molecules” can be arranged in such a way that, at close packing, the discs (i.e., their “atoms”) form a triangular close packed lattice. It is easy to see that the triangular lattice can be decorated with dimers in infinitely many different ways. Among them there are countably many peri- odic arrangements of the dimers. Examples of the simplest periodic dimer structures, containing one or two dimers per unit cell, can be found e.g., in [19, 20, 22]. In the thermodynamic limit, there exist also uncountably many close packed arrangements in which the dimer orientations and positions of the dimer centers of mass are non-periodic. We should stress that they are not completely random, however. Since “atoms” form a triangular lattice, the arrangements of the dimers are related to certain decorations of a kagomé lattice, see [21] for further details. Since all such configurations are macroscopically equivalent they constitute a phase already mentioned above – the degenerate crystal. In fact, there are so many such configurations in the thermodynamic limit that it leads to a positive entropy per particle [31]. This con- tribution is large enough to render the DC phase thermodynamically stable for the whole density range below melting [20]. Analogous considerations may be applied for two-dimensional hard trimers. But in this case the con- tribution of the degeneracy entropy is not yet known and whether the DC phase of hard trimers is also thermodynamically stable remains an open question. Similarly, in three dimensions, “atoms” of the hard dimers (trimers) can be arranged in a close packed structure, e.g. a fcc or hcp lattice. As in the 2D case, there are countably many periodic dimer (trimer) decorations of the lattices. Phase diagrams for three of the simplest periodic dimer arrangements were calculated in [12]. Yet again, there are uncountably many configurations in which centers of masses of orientationally disordered dimers (trimers) form an aperiodic structure while their atoms remain arranged in one of the mentioned crystalline lattices. Also in this case there is a substantial positive contribution to the entropy which comes from the degeneracy of the DC phase. Fig. 3 Stereographic projection of a hard dimer system forming the DC phase, at close packing. “Atoms” of the dimers are arranged in the fcc lattice. For sake of clarity only the atomic centres and lines connecting them (see Fig. 1) are plotted. Fig. 4 Stereographic projection of hard a trimer system forming the DC phase, at close packing. “Atoms” of the trimers are arranged in the fcc lattice. For sake of clarity only the atomic centres and lines connecting them (see Fig. 2) are plotted. phys. stat. sol. (b) 242, No. 3 (2005) / www.pss-b.com 629 © 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim Original Paper Stereographic projections of hard dimers and trimers forming a DC phase at the close packing limit are presented in Figs. 3 and 4 respectively. In both cases “atoms” of the “molecules” form the fcc lattice. For sake of clarity only molecular schemes defined in Figs. 1 and 2 are presented. The choice of these systems for further investigation was not accidental. Their two dimensional counterparts exhibit interest- ing elastic properties [23, 24, 30] and the question if they were still valid in three dimensions remained open. The “atoms” of the molecules were (at close packing) arranged in the fcc lattice because it shows the highest symmetry in three dimensions. (It is well known that, in contrast to two dimensions where the triangular lattice is elastically isotropic, there is no three-dimensional lattice for which the symmetry alone implies elastic isotropy [26]). In particular, the fcc lattice offers higher effective elastic symmetry compared to the hcp lattice. In fact, such a system is effectively cubic in the thermodynamic limit and only three elastic constants are needed to describe its elastic properties. 3 Simulation results In the simulations performed, we used the constant pressure MC method in which the simulation box can change both its size and shape. Employing the method described in [32, 33] we were able to determine the Poisson’s ratio from fluctuations of the box matrix elements [34]. Figure 5 presents Poisson’s ratio ( )Nn as a function of the relative volume for defect-free DC phase of hard dimer systems consisting of different numbers of molecules. As one can see, it can be approxi- mated by a linear function ( ) ( ) ( ) cp cp( 1)N N Na V Vn n= / - + , (1) where V is the volume, cpV is the volume at close packing, ( )cpNn and ( )Na denote the Poisson’s ratio in the close packing limit and the slope coefficient, respectively, for samples consisting of N molecules. The values of ( )Na and ( )cpNn for different N are collected in Table 1 along with the extrapolation to the ther- modynamic limit obtained from the linear approximation shown in Fig. 6. 0.10 0.15 0.20 0.25 0.30 1.00 1.05 1.10 1.15 1.20 1.25 ν (N ) V/Vcp N = 54 N = 432 N = 1458 Fig. 5 Poisson’s ratio vs. the relative volume for different sizes of the hard dimers system. Linear fits to the data for the large, medium and small system are represented by continuous, dashed and dotted line, respectively. Fig. 6 N-dependence of the linear approximation parameters obtained from Eq. (1). The lines represent linear fits to the points depicted. 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.000 0.005 0.010 0.015 0.020 a (N ) , ν (N ) cp 1/N a(N) ν (N) cp 630 M. Kowalik and K. W. Wojciechowski: Poisson’s ratio of degenerate crystalline phases © 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim Table 1 The coefficients of linear approximation defined by Eq. (1). Numbers in brackets denote accu- racy of the results. N a(N) ( ) cp N n 54 0.49(1) 0.159(1) 432 0.48(2) 0.154(2) 1458 0.48(2) 0.152(3) • 0.48(1) 0.153(2) It can be easily seen that the Poisson’s ratio depends rather weakly on the system size and its value for the smallest system is close to the value obtained by extrapolation to the thermodynamic limit. This fact, confirmed also in other cases [33, 35], was exploited in simulations of the hard trimers system. Figure 7 shows a comparison of the Poisson’s ratio for the hard spheres (the data are taken from [35]), the dimers, and the trimers. All the presented data were obtained for systems consisting of the same number 108N = spheres. Considering the weak N-dependence mentioned in the paragraph before we expect however, that possible discrepancy between the obtained results and those corresponding to the thermodynamic limits will not exceed a few percent. It can be seen in Fig. 7 that the investigated molecular systems in the DC phase exhibit positive Pois- son’s ratio behaviour. Although positive, the value for triatomic particles (trimers) is less than that ob- tained for diatomic particles (dimers) which in turn is less than that of mono-atomics (spheres). This suggests that the Poisson’s ratio decreases when the number of spheres in molecules is increased. Studies of some higher multisphere systems should answer this question. 4 Concluding remarks In this paper the Poisson’s ratio of the DC phase for the two simplest molecular hard-core systems, dimers and trimers, was calculated. Although in both cases its value is definitely positive, it is observed that with increasing number of spheres in the molecules the Poisson’s ratio decreases: its value for hard dimers is less than that for hard spheres, and the Poisson’s ratio of hard trimers is even smaller. By analogy to two- dimensional systems for which the same inequalities are fulfilled (but already two-dimensional trimers exhibit negative Poisson’s ratio) it is attractive to assume that by increasing the number of spheres one can further decrease the Poisson’s ratio, and may lead to negative Poisson’s ratio for some molecules com- posed of higher numbers of spheres. Work in this direction is currently in progress. 0.0 0.1 0.2 0.3 0.4 0.5 1.00 1.05 1.10 1.15 1.20 1.25 ν V/Vcp spheres dimers trimers Fig. 7 Poisson’s ratio as a function of relative volume for different hard-core systems. The data for hard spheres come from [35]. The lines represent linear fits to the data. phys. stat. sol. (b) 242, No. 3 (2005) / www.pss-b.com 631 © 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim Original Paper The present simulations show very weak dependence of the Poisson’s ratio on the system size in three dimensions. Such behaviour was earlier found for two-dimensional systems. This indicates that studies of quite small systems should be sufficient to characterize correctly the thermodynamic limit value of the Poisson’s ratio for uniform phases, i.e., when small samples are representative for the structure. The latter conclusion is of particular interest in the case of systems consisting of more complex molecules which we plan to study in the future. 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