Computer simulation study of a simple cubatic mesogenic lattice model
Silvano Romano*Unità di Ricerca CNISM e Dipartimento di Fisica “A. Volta,” Università di Pavia, via A. Bassi 6, I-27100 Pavia, Italy
�Received 16 March 2006; published 11 July 2006�
Over the last 15 years, the possible existence of a cubatic mesophase, possessing cubic orientational order�i.e., along three mutually orthogonal axes� but no translational one, has been addressed theoretically, andpredicted in some cases, where the investigated interaction models involved hard-core repulsion only; on theother hand, no experimental realizations of such a phase are known at the time being. The present paperaddresses a very simple cubatic mesogenic lattice model, involving continuous interactions; we considerparticles possessing Oh symmetry, whose centers of mass are associated with a three-dimensional simple-cubiclattice; the pair potential is taken to be isotropic in orientation space, and restricted to nearest neighboring sites;let the two orthonormal triads �u j, j=1,2 ,3� and �vk, k=1,2 ,3� define orientations of a pair of interactingparticles, and let f jk=v j ·uk. The interaction model studied here is defined by the simplest nontrivial �quartic�polynomial in the scalar products f jk, consistent with the assumed symmetry and favoring orientational order;it is, so to speak, the cubatic counterpart of the Lebwohl-Lasher model for uniaxial nematics. The model wasinvestigated by mean field theory and Monte Carlo simulation, and found to produce a low-temperaturecubatically ordered phase, undergoing a first order transition to the isotropic phase at higher temperature; themean field treatment yielded results in reasonable qualitative agreement with simulation.
Over the last three decades, theoretical studies of varioussimple mesogenic models have predicted a rather rich andintriguing phase behavior, whose experimental realizationhas often proven to be a rather challenging task of its own.For example, the possible existence of biaxial nematicphases was predicted by various theoretical treatments since1970 �1�, and is still currently investigated. On the experi-mental side, stable biaxial phases have been observed in lyo-tropic systems as early as 1980 �2�; since 1986 there havebeen various claims and counterclaims of synthesizing andunambiguosly characterizing a thermotropic biaxial nematicand better experimental evidence seems to have been pro-duced over the last two years; a more detailed discussion anda more extensive bibliography can be found in Ref. �3�.
As another example, over the last 15 years, the possibleexistence of a cubatic mesophase, possessing cubic orienta-tional order �i.e., along three mutually orthogonal axes� butno translational one, has been investigated theoretically, andexplicitly predicted in some cases �4–9��: cut hard sphereswere studied in Refs. �4,5�; the possible existence of cubaticorder for hard cylinders was investigated in Ref. �6�, but noevidence of it was found in this case; Onsager crosses werestudied in Ref. �7�; arrays of hard spheres with tetragonal orcubic symmetry have been studied in Refs. �8,9�. The namedinvestigations have been carried out on hard-core models,both by simulation and by approximate analytical theories; insome cases the consituent particles were uniaxial �i.e., D�hsymmetric� �4–6�, and in other cases they possessed tetrag-onal or cubic symmetry �7–9�. As hinted above, no experi-mental realizations of a cubatic phase are known at the timebeing.
On the other hand, over the decades, mesophases possess-ing no positional order, such as the nematic one, have oftenand quite fruitfully been studied by means of lattice modelsinvolving continuous interaction potentials �10�, starting withthe Lebwohl-Lasher �LL� model and its seminal simulationpapers in the early 1970s �11,12�; this approach also yields aconvenient contact with molecular field �MF� treatments ofthe Maier-Saupe �MS� type �13–15�.
Here we investigate a very simple lattice model capableof producing cubatic order: we are considering classical,identical particles, possessing Oh symmetry, whose centersof mass are associated with a three-dimensional �simple-cubic� lattice Z3; let x��Z3 denote the coordinate vectors oftheir centers of mass; the interaction potential is taken to beisotropic in orientation space, and restricted to nearest neigh-bors, involving particles or sites labeled by � and �, respec-tively. The orientation of each particle can be specified via anorthonormal triplet of three-component vectors �e.g., eigen-vectors of its inertia tensor�, say �w�,j, j=1,2 ,3�; in turnthese are controlled by an ordered triplet of Euler angles��= ��� ,�� ,���; particle orientations are defined with re-spect to a common, but otherwise arbitrary, Cartesian frame�which can, but need not, be identified with the latticeframe�. It also proves convenient to use a simpler notationfor the unit vectors defining orientations of two interactingmolecules �16�, i.e., u j for w�,j, and vk for w�,k, respectively,here, for each j, u j and v j have the same functional depen-dences on �� and ��, respectively �pairs of correspondingunit vectors in the two interacting molecules�; let �̃=���
= ��̃ , �̃ , �̃� denote the set of Euler angles defining the rotationtransforming u j into v j; Euler angles will be defined hereaccording to the convention used by Brink and Satchler�17–19�; let us finally define
f jk = �v j · uk� . �1�
An interaction potential consistent with the assumed symme-try can be written*Electronic address: [email protected]
here E�¯� denotes an even function of its argument; E isalso assumed to be analytical, so that Eq. �2� can be ex-panded as a convergent series of the form
= a0 + �l2
a2l��j=1
3
�k=1
3
�f jk�2l= b0 + �
l2b2l��
j=1
3
�k=1
3
P2l�f jk� , �3�
where P2l�¯� denotes Legendre polynomials of even order,and the missing second-order terms are just constants, i.e.,
�j=1
3
�k=1
3
f jk2 = 3, �
j=1
3
�k=1
3
P2�f jk� = 0. �4�
The simplest interaction model expected to produce cubaticorder is obtained by setting a4 or b4 to negative quantities,and all other higher-order coefficients to zero; in other words
= −1
6��5�
j=1
3
�k=1
3
�f jk4 � − 9 = −
4
21��
j=1
3
�k=1
3
P4�f jk�
= − �G4��̃� , �5�
where � denotes a positive quantity, setting energy and tem-perature scales �i.e., T*=kBT /��; here numerical factors havebeen adjusted by setting the isotropic average of the pair
potential to zero �i.e., b0=0�, and its minimum value to −�.
The explicit expression of G4��̃� reads
G4��̃� = 112�7�0,0
4 ��̃� + 35��0,44 ��̃� + �4,0
4 ��̃�� + 5�4,44 ��̃��
= 1768„140 cos�2�̃��1 + cos�4�̃�cos�4�̃��
+ 5 cos�4�̃��49 + cos�4�̃�cos�4�̃��
+ 7�9 + 25 cos�4�̃�cos�4�̃� + 40�cos�4�̃�
+ cos�4�̃���sin �̃�4� − 40�7 cos��̃�
+ cos�3�̃��sin�4�̃�sin�4�̃�… . �6�
The functions �p,q4 appearing in Eq. �6� are symmetry-
adapted combinations of Wigner D functions, as discussed inRefs. �7,20,21�.
G4��̃� is a function of three independent variables, and inpractice its visualization requires using projections �i.e., con-straining one of the three independent variables�; for ex-ample, let denote a fixed angle, and let
H� � = − �G4��̃, �̃ = ,�̃�; �7�
contour plots for H�� /2� are shown in Fig. 1.
MEAN FIELD AND SIMULATION ASPECTS
After applying a MF procedure �15�, the resulting expres-sion for the free energy has the form
FIG. 1. �Color online� Contour plot for the function H�� /2� �seeEq. �7��; the contour-to-contour separation is 0.05�; thin �red� linescorrespond to negative values, thicker �green� lines are associatedwith positive values, and the thickest �blue� one defines the zero-energy contour. The plot was produced by means of Maple.
FIG. 2. MF predictions and simulation results for the potentialenergy; the continuous line corresponds to MF predictions, discretesymbols have been used for simulation results, obtained with dif-ferent sample sizes, and have the following meanings: circles, q=10; squares, q=20; triangles, q=30; unless otherwise stated orshown, here and in the following figures, the associated statisticalerrors fall within symbol sizes.
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AMF* = �s4
2 − T* ln��/�8�2��, � = �Eul
exp��W̃�d� ,
�8�
W̃ = 2�s4G4���, � = 1/T*, �9�
where �Eul denotes integration over Euler angles, i.e., for anyintegrable function F���
�Eul
F���d� �0
2�
d��0
�
sin �d��0
2�
F���d�; �10�
here 2�=6 denotes the lattice coordination number, and s4 isthe variational parameter �i.e., the order parameter�. More-over
�AMF*
�s4= �2���; � = s4 − �1/���
EulG4���exp��W̃�d� ,
�11�
and the consistency equation is �=0.The free energy was minimized numerically for each tem-
perature over a fine grid, by means of numerical routinesusing both the function �Eq. �8�� and its derivative �Eq. �11��;the obtained variational parameters were used to calculatethe potential energy per particle UMF
*
UMF* =
���AMF* �
��= − �s4
2, �12�
where the consistency equation has been allowed for on theright-hand expression; the configurational specific heat CMF
*
was then calculated from UMF* by numerical differentiation;
here and in the following formulas, asterisks mean scaling by� for energy quantities, and scaling by kB for the specificheat. We found a low-temperature ordered phase and a first-order transition to the disordered one, taking place at thetemperature �MF=0.7878.
Simulations were carried out on a periodically repeatedcubic sample, consisting of V=q3 particles, q=10,20,30;calculations were run in cascade, in order of increasing tem-perature, and starting from a perfectly ordered configurationat the lowest investigated temperature; each cycle �or sweep�consisted of 2V MC steps, including a sublattice sweep �22�;the finest temperature step used was �T*=0.0005, in thetransition region.
Notice that we shall find here a pronounced first-ordertransition, and that simulations carried out in order of de-creasing temperature and started from the disordered high-temperature régime may show hysteresis.
Different random-number generators were used, as dis-cussed in Ref. �22�. Equilibration runs took between 25 000and 100 000 cycles, and production runs took between200 000 and 800 000; macrostep averages for evaluating sta-tistical errors were taken over 1000 cycles. Calculated ther-modynamic quantities include mean potential energy per siteU* and configurational specific heat per particle C*.
As for the the frame-independent �rotationally invariant�order parameter, let
M =� 4
21�
�=1
V
��=1
V ��j=1
3
�k=1
3
P4�w�,j · w�,k� : �13�
then the simulation estimate for the order parameter is
FIG. 3. MF predictions and simulation results for the configu-rational heat capacity; same meaning of symbols as in Fig. 2; herethe associated statistical errors, not shown, range between 1% and5%.
FIG. 4. MF predictions and simulation results for the order pa-rameters s4, obtained with different sample sizes; same meaning ofsymbols as in Fig. 2.
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s4 =1
V�M� , �14�
and its associated susceptibility reads
�4 =1
V���M2� − �M�2� . �15�
As for computational aspects of Eq. �13�, let us remark that,by the addition theorem for spherical harmonics �17–19�, thedouble sums appearing in it can be constructed via singlesums �7�, i.e., as linear combinations of the squares of thesimpler quantities
� j,m = ��=1
V
Re�C4,m�w�,j��, � j,m = ��=1
V
Im�C4,m�w�,j��;
�16�
here m=0,1 ,2 ,3 ,4 ,C4,m�¯� are modified spherical harmon-ics, and Re and Im denote real and imaginary parts, respec-tively; in turn, each spherical harmonics is a suitable poly-nomial constructed in terms of Cartesian components of thecorresponding unit vector �see, e.g., Ref. �23��; notice alsothat in this case all second-rank order parameters are zero bysymmetry �15�.
One can also evaluate the so-called short-range order pa-rameter �24,25�
�4 =4
21��j=1
3
�k=1
3
P4�f jk�� �17�
measuring correlations between pairs molecules associatedwith nearest-neighboring sites; in the present case, the func-tional form of the interaction potential entails that the poten-tial energy is proportional to �4, i.e., U*=−��4.
Long- and short-range orientational order can be com-pared via the correlation excess
r4 = �4 − s42. �18�
RESULTS AND COMPARISONS
MF predictions and MC results for a few observables areplotted and compared in Figs. 2–6.
Simulation results for the potential energy �Fig. 2� areindependent of sample size for T�0.68 and then T�0.72,and show a pronounced sample-size dependence in between;actually, already for q=20, Figure 2 exhibts a pronouncedjump taking place over a temperature range of 0.0005.
Figure 4 shows a similar pattern as for the jump of s4,taking place at the same temperature as for U*; on the otherhand, in the low-temperature regime, sample-size effects ap-pear to saturate for q20, and the high-temperature regionexhibits a pronounced decrease of s4 with increasing samplesize.
Both configurational specific heat �Fig. 3� and susceptibil-ity �Fig. 5� peak around the same temperature, correspondingto the named jumps; they show a recognizable sample-sizedependence over the same temperature range 0.68�T*
TABLE I. Transitional properties for the investigated model;MC results are based on the largest investigated sample size q=30.
Method � �U* s4
MF 0.7878 0.8181 0.5222
MC 0.692±0.001 0.28±0.04 0.38±0.02
FIG. 5. Simulation results for the order parameter susceptibility�4, obtained with different sample sizes: same meaning of symbolsas in Fig. 2.
FIG. 6. Simulation results for the correlation excess r4 �see Eq.�18��, obtained with different sample sizes: same meaning of sym-bols as in Fig. 2.
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�0.72, and are again largely unaffected by sample sizes out-side it.
Simulation results for the correlation excess are plotted inFig. 6, where the transition is signaled by both a peak and arecognizable jump; in the disordered region, sample-size ef-fects appear to saturate for q20.
Thus we propose a first-order transition, and the value�MC=0.692±0.001, for the transition temperature; here theerror bar is conservatively taken to be twice the temperaturestep used in the transition region. Upon analyzing the simu-lation results for the largest sample as discussed in Refs.�26,27�, we obtained the estimates for transitional propertiescollected in Table I; actually, the same analysis was alsoapplied to simulation results obtained for q=20, and yieldedconsistent results. Table I shows a fair qualitative agreementbetween MF and MC; of course, in quantitative terms MFoverestimates the transition temperature, and, even worse, itsfirst-order character, as well known for LL; let us mention,for comparison, that the ratio �MC/�MF is �0.878, and thatthe corresponding value for LL is �0.856 �10�.
Partial snapshots of configurations extracted from asample defined by q=20 and obtained at temperatures belowand above the transition �T*=0.675 and T*=0.7� are shownin Figs. 7 and 8, respectively. More precisely, in order tomaintain readability, we decided to show only a horizontalsection of the sample, i.e., the square layer consisting of �q2�particles whose centers of mass had the same value for the
vertical z coordinate; the arbitrarily chosen value was q /2.To summarize, we have defined a very simple cubatic me-sogenic lattice model �so to speak, the cubatic counterpart ofLL�, involving continuous interactions, and investigated it byMF and MC; both approaches show a first-order transition,and MF produces a reasonable qualitative agreement withMC.
ACKNOWLEDGMENTS
The present extensive calculations were carried out, on,among other machines, workstations belonging to the Sezi-one di Pavia of Istituto Nazionale di Fisica Nucleare �INFN�;allocations of computer time by the Computer Centre of Pa-via University and CILEA �Consorzio InteruniversitarioLombardo per l’ Elaborazione Automatica, Segrate-Milan�,as well as by CINECA �Centro Interuniversitario Nord-Est diCalcolo Automatico, Casalecchio di Reno-Bologna�, aregratefully acknowledged. The author also wishes to thankProfessor L. Longa �Krakow, Poland�, Professor G. R. Luck-hurst �Southampton, England, UK�, Professor E. G. Virga�Pavia, Italy�, and Dr. R. Blaak �Cambridge, England, UK�for helpful discussions and suggestions. Finally, the authorgratefully acknowledges financial support from the ItalianMinistry for Higher Education �MIUR� through the PRINGrant No. 2004024508.
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FIG. 7. Partial configuration obtained by simulation at T*
=0.675, and from a sample with q=20; see also the text.FIG. 8. Partial configuration obtained by simulation at T*
=0.700, and from a sample with q=20.
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