A novel orientation-dependent potential model for prolate mesogens

A no el o ien a ion-dependen po en ial model o p ola e mesogens B. Ma ı ´nez-Haya,a) A. Cue os, and S. Lago Depa amen o de Ciencias Ambien ales, Uni e sidad Pablo de Ola ide, 41013 Se ille, Spain L. F. Rull Depa amen o de Fı ´sica A o ´mica, Molecula y Nuclea , A ea de Fı ´sica Teo ´ ica, Uni e sidad de Se illa, Apa ado Pos al 1065, 41080 Se ille, Spain 共Recei ed 1 Decembe 2003; accep ed 19 Oc obe 2004; published online 22 Decembe 2004兲 An in e molecula po en ial is in oduced o he s udy o molecula mesogenic luids. The model combines dis inc ea u es o he well-known Gay-Be ne and Kiha a po en ials by inco po a ing dispe si e in e ac ions dependen on he ela i e pai o ien a ion o a sphe ocylinde molecula co e. Resul s o a Mon e Ca lo simula ion s udy ocused on he liquid c ys al phases exhibi ed by he model luid a e p esen ed. Fo he chosen po en ial pa ame e s, molecula aspec a io L*⫽5 and empe a u es T*⫽2, 3, and 5, iso opic, nema ic, smec ic-A, and hexa ic phases a e ound. The loca ion o he phase bounda ies as well as he equa ion o s a e o he luid and u he he modynamical and s uc u al pa ame e s a e discussed and con as ed o he Kiha a luid. In compa ison o his la e luid, he model induces he o ma ion o o de ed liquid c ys alline phases a lowe packing ac ions and i a o s, in pa icula , he appea ance o laye ed hexa ic o de ing as a consequence o he g ea e a ac i e in e ac ion assigned o he pa allel side- o-side molecula pai con igu a ions. The esul s con ibu e o he e alua ion o he ole o speci ic in e ac ion ene gies in he mesogenic beha io o p ola e molecula liquids in dense en i onmen s. © 2005 Ame ican Ins i u e o Physics. 关DOI: 10.1063/1.1830429兴 I. INTRODUCTION The use o simple models o ep esen he o e all pai in e ac ions in molecula luids has p o en o be a success ul s a egy o s udy he beha io o mesogens, as well as o p edic and cha ac e ize hei liquid c ys al phases. The aim o such models is o cap u e he essen ial aspec s o he phys- ics unde laying he mesogenic beha io o he eal sys ems, such as excluded olume e ec s and dispe si e in e ac ions, while keeping easonable analy ical and compu a ional e i- ciency o heo e ical and simula ion s udies. Fluids o elonga ed o odlike molecules a e an impo - an class o mesogens wi h ele an echnological and bio- logical applica ions and, he e o e, di e en models ha e been in oduced in o de o explo e hei p ope ies. Fo in- s ance, a amily o igid molecula models o ellipsoidal symme y has been p oposed, among which he Gay-Be ne 共GB兲 luid anks as one o he mos ex ensi ely s udied.1 This model ex ended he pionee ing s udies o F enkel and co-wo ke s on he ha d ellipsoid luid,2and was speci ically in oduced as an imp o emen o he Gaussian o e lap model.3The main ea u e o he GB model is a ou - pa ame e unc ionali y ha con ols he aspec a io o he ellipsoidal co e and he aniso opy o he a ac i e in e ac- ions. The phase diag am o he GB luid has been cha ac e - ized o a b oad ange o aspec a ios and in e ac ion pa am- e e s and, in pa icula , nema ic and laye ed smec ic and hexa ic liquid c ys al phases ha e been epo ed.4–6 In spi e o he success o he Gay-Be ne model, mo e de ailed in e ac ion app oaches, such as si e-si e Lenna d- Jones chain models, indica e ha he ac ual co e o p ola e molecules is signi ican ly be e ep oduced by a sphe ocyl- inde co e 共i.e., a cylinde o heigh /diame e aspec a io L*⫽L/ ␴ , capped a bo h ends wi h a hemisphe e o he same diame e ␴ 兲.5,7 In ac , a numbe o luid models o his la e symme y ha e been in oduced in he pas decades, especially a e e icien algo i hms we e de eloped o com- pu e he minimum dis ance be ween he cen al ods o such molecules.8Examples o his amily o models a e he ha d sphe ocylinde 共HSC兲 luid and i s squa e-well 共SWSC兲o so epulsi e 共SRS兲 a ian s,9–14 and he Kiha a luid.15 The Kiha a model was in oduced as a gene aliza ion o he Lenna d-Jones luid o aniso opic molecules, and has been employed in nume ous in es iga ions o he modynamic, s uc u al, and anspo p ope ies o luids o linea molecules.16 Pe haps su p ising, i has no been un il ecen ly ha he abili y o he Kiha a luid o o m liquid c ys als phases has been sys ema ically in es iga ed.17 One d awback ha he Kiha a model sha es wi h he HSC, SWSC, and SRS models is ha i assigns he same in e ac ion ene gy o all pai o ien a ions, as long as he minimum dis ance be ween he molecules emains cons an . This is in con as wi h he in e ac ions o eal sys ems whe e he dispe si e o ces a e o ien a ion dependen and, unless speci ic in e ac ions come in o play, in he mo opic luids hey usually end o be g ea e o aligned molecula con igu- a ions 共e.g., side- o-side pa allel pai s兲 han o he mis- aligned ones 共e.g., head- o- ail o T-shaped pai s兲.18,19 This beha io o he in e ac ion ene gy is quali a i ely ep oduced by he igid chain models wi h mul iple in e ac ion si es. Howe e , al hough mul iple-si e models o his la e ype ha e been employed o s udy liquid c ys al phases,7,20 hei a兲Elec onic mail: [email p o ec ed] THE JOURNAL OF CHEMICAL PHYSICS 122, 024908 共2005兲 122, 024908-10021-9606/2005/122(2)/024908/8/$22.50 © 2005 Ame ican Ins i u e o Physics use is limi ed by he compu a ional cos associa ed o he la ge numbe o si es equi ed in o de o mimic ealis ic mesogenic molecules. The main idea behind he p esen wo k is o co ec o he de iciency o he Kiha a model commen ed abo e by inco po a ing one he mos dis inc ea u es o he Gay-Be ne luid, namely, i s pa ame ic modula ion o he pai o ien a- ion dependence o dispe si e o ces. The model is p esen ed in Sec. II and he emaining o he pape is hen de o ed o explo e, by means o Mon e Ca lo simula ions, he quali a- i e e ec s ha his ea u e in oduces in he liquid c ys al phase diag am o he luid. II. INTERACTION MODEL In his wo k, we in oduce an in e ac ion model ha in- co po a es he pai o ien a ion dependence o he dispe si e in e ac ions o he Gay-Be ne po en ial o he sphe ocylinde molecula co e o he Kiha a luid. In o de o achie e his in a s aigh o wa d and easily ecognizable way, we ha e buil a ‘‘hyb id’’ in e ac ion ene gy unc ional by mul iplying he Kiha a po en ial by he same o ien a ional p e ac o o he Gay-Be ne po en ial. These la e ac o depends explici ly on he h ee- ec o co ela ions be ween he di ec o s o he gi en pai o pa icles (uˆi,uˆj) and a uni ec o in he di ec- ion o he cen e -o -mass in e molecula dis ance ec o ( ˆij), in con as o he Kiha a in e ac ion which depends on he ela i e o ien a ion o he molecula pai s only h ough i s in luence on he minimum dis ance be ween he molecula co es dm( ij,uˆi,uˆj). We will e e o his model as he Gay- Be ne-Kiha a po en ial o , in sho , GB-K po en ial. The model is hus de ined by he ollowing exp essions: UGB-K共 ij,uˆi,uˆj兲⫽ ⑀ GB共 ˆij,uˆi,uˆj兲UK共dm兲,共1兲 UK共dm兲⫽4 ⑀ 关共 ␴ /dm兲12⫺共 ␴ /dm兲6兴,共2兲 ⑀ GB共 ˆij,uˆi,uˆj兲⫽ ⑀ GO ␯ 共uˆi,uˆj兲 ⑀ ⬘ ␮ 共 ˆij,uˆi,uˆj兲,共3兲 ⑀ GO共uˆi,uˆj兲⫽关1⫺ ␹ 2共uˆi•uˆj兲2兴⫺1/2,共4兲 ⑀ ⬘共 ˆij,uˆi,uˆj兲⫽1⫺ ␹ ⬘ 2 冋 共 ˆij•uˆi⫹ ˆij•uˆj兲2 1⫹ ␹ ⬘共uˆi•uˆj兲 ⫹共 ˆij•uˆi⫺ ˆij•uˆj兲2 1⫺ ␹ ⬘共uˆi•uˆj兲 册 .共5兲 The p e ac o ⑀ GB is cha ac e ized by he usual Gay- Be ne ou -pa ame e se 共 ␬ , ␬ ⬘, ␮ , and ␯ 兲, wi h he no a ion ␹ ⫽( ␬ 2⫺1)/( ␬ 2⫹1) and ␹ ⬘⫽( ␬ ⬘1/ ␮ ⫺1)/( ␬ ⬘1/ ␮ ⫹1). Simple geome ical a gumen s show ha , in ou model, he pa ame e associa ed o he ellipsoidal aspec a io ␬ is e- la ed o he sphe ocylinde aspec a io h ough ␬ ⫽L*⫹1 共no e ha L*is he heigh o he cen al cylinde , whe eas he o al leng h o he sphe ocylinde , including he end caps, is L*⫹1, always in uni s o he diame e ␴ 兲. On he o he hand, he aniso opy o he dispe si e in e ac ion is con- olled by ␬ ⬘; o ins ance, he a ac i e ene gy well o a pa allel pai o molecules is ␬ ⬘ imes deepe o a side- o- side con igu a ion han o a head- o- ail one. Fo he p esen s udy, we ha e aken he se o pa ame e s L*⫽5, ␬ ⫽6, ␬ ⬘⫽5, ␮ ⫽2, and ␯ ⫽1, so ha he model he e employed will be hence o h deno ed GB-K共6,5,2,1兲acco ding o he no a ion GB-K( ␬ , ␬ ⬘, ␮ , ␯ ) which is simila o he one used p e iously o he Gay-Be ne luid.14 The sphe ocylinde as- pec a io L*⫽5共and, hence, ␬ ⫽6) was chosen in o de o compa e he p esen esul s o ou ecen s udy o he Kiha a luid,17 whe eas ␬ ⬘⫽5, ␮ ⫽2, ␯ ⫽1, a e he alues o iginally sugges ed by Gay and Be ne1also employed in se e al p e- ious s udies o he liquid phase diag am o luids o sho e aspec a ios ␬ ⭐4.4–6 I mus be no ed ha he choice o ␬ ⬘, ␮ , ␯ is ac ually no a i ial ask, especially when compa ing wi h eal sys ems, and o he se s o alues ha e been p o- posed o speci ic he mo opes.14,21 Figu e 1 illus a es some o he main ea u es o he GB-K po en ial model in oduced in he p e ious pa ag aph. The op panel 关con ou plo 共A兲兴 shows he equipo en ial con ou s o he in e ac ion o wo pe ec ly pa allel GB- K共6,5,2,1兲pa icles 共i.e., uˆi⫽uˆj) whose ela i e posi ion is de ined by he in e molecula dis ance ec o in pola coo - dina es ( ij, ␪ ) in he plane o he igu e. As can be eadily seen in he igu e, he a ac i e well o he pa allel side- o- side con igu a ion ( ␪ ⫽0°) is signi ican ly enhanced wi h e- spec o he head- o- ail one ( ␪ ⫽90°). The same ep esen a- FIG. 1. Top panel: Equipo en ial ene gy su aces o he in e ac ion o wo pa allel pa icles in e ac ing h ough 共a兲 he GB-K共6,5,2,1兲po en ial in o- duced in his wo k 共wi h pa ame e s L*⫽5, ␬ ⫽6, ␬ ⬘⫽5, ␮ ⫽2, and ␯ ⫽1, see ex o de ails兲,共b兲 he Kiha a po en ial 共wi h L*⫽5), and 共c兲 he Gay-Be ne po en ial GB共6,5,2,1兲. The posi ion o he pai o pa icles is desc ibed by he pola coo dina es ( ij , ␪ ), wi h ␪ ⫽0° o he side- o-side con igu a ion and ␪ ⫽90° o he head- o- ail one. Bo om panel: GB- K共6,5,2,1兲pai in e ac ion ene gy as a unc ion o he minimum dis ance be ween he molecula co es dm o pai s o pa allel molecules in side- o- side, c ossed, head- o- ail, and T-shaped con igu a ions. 024908-2 Ma ı ´nez-Haya e al. J. Chem. Phys. 122, 024908 (2005) ion o he Kiha a luid 关plo 共B兲兴 ob iously yields a uni o m well a ound he molecula co e. On he o he hand, he simi- la diag am o he Gay-Be ne po en ial wi h he same pa- ame e s, GB共6,5,2,1兲关plo 共C兲兴, illus a es i s cha ac e is ic aniso opic well and ellipsoidal co e. A close quan i a i e ep esen a ion o he aniso opy o he dispe si e in e ac ions in he GB-K共6,5,2,1兲model is p o ided in he bo om panel o Fig. 1, whe e he po en ial ene gy as a unc ion o he minimum dis ance dmis ep e- sen ed o pai s o molecules in pa allel, c ossed, head- o- ail, and T-shaped con igu a ions. The i e old deepe well o he pa allel con igu a ion wi h espec o he head- o- ail, as co esponds o ␬ ⬘⫽5, is appa en in he igu e. I mus be no iced ha ou o mula ion o he GB-K model assigns a well dep h o uni y 共in educed uni s兲 o he c ossed con igu- a ion, which mus be kep in mind when compa ing, a a gi en empe a u e, he phase diag am o his luid o ha o he Kiha a luid 共 o which a well dep h o uni y applies o all molecula o ien a ions兲. III. SIMULATION DETAILS We ha e ca ied ou iso he mal-isoba ic (N-P-T) en- semble Mon e Ca lo 共MC兲simula ions o s udy he liquid c ys al phase diag am a di e en empe a u es o he GB- K共6,5,2,1兲 luid 共i.e., wi h pa ame e s ␬ ⫽6, and hence L* ⫽5, ␬ ⬘⫽5, ␮ ⫽2, and ␯ ⫽1). Fo compu ing e iciency, in he p esen simula ions he in e ac ion was unca ed a a dis ance o dC⫽3 ␴ , which co esponds o cen e -o -mass dis ances anging om ij⫽3 ␴ o ij⫽8 ␴ , depending on he ela i e pai o ien a ion, and shi ed so ha he po en ial en- e gy UGB-K anishes a he unca ion bounda y. Hence, he po en ial ac ually employed is gi en by UGB-K共 ij,uˆi,uˆj兲 ⫽ 再 ⑀ GB共 ˆij,uˆi,uˆj兲关UK共dm兲⫺UK共dC兲兴,dm⭐3 ␴ 0, dm⬎3 ␴ .共6兲 The simula ions we e un o a sys em o Np⫽1080 mol- ecules a h ee educed empe a u es, T*⫽kT/ ⑀ ⫽2, 3, and 5, whe e kdeno es he Bol zmann cons an and ⑀ he well dep h o he c ossed con igu a ion 共see Fig. 1兲. As discussed h oughou he ollowing sec ions o he pape , a hese em- pe a u es he GB-K共6,5,2,1兲 luid p esen s s able iso opic (I), nema ic (N), smec ic-A(Sm-A), and hexa ic 共Hex兲 phases. The calcula ion was s a ed a each empe a u e by comp essing he luid o a s a e o high densi y well inside he hexa ic egion o he phase diag am (P*⫽P ␴ 3/kT ⫽2.4, 2.6, and 3.1, espec i ely, o T*⫽2, 3, and 5兲. Such s a e was p ope ly equilib a ed o e some 106MC cycles be o e a sys ema ic iso he mal expansion o he luid was pe o med down o he iso opic phase. Such app oxima e p ocedu e o he es ima ion o liquid c ys al phase bound- a ies is eliable o ou pu poses and has been ex ensi ely employed in he pas ,9,11,22 in pa icula , o simila luids o p ola e molecules. I mus be s essed, howe e , ha a p ope s udy o he phase diag am should include he accu a e cal- cula ion o he ee ene gy in each o he phases, which is beyond he scope o he p esen s udy. Each s a e was ypically equilib a ed o e 106MC cycles and ensemble a e ages o he he modynamic and s uc u al p ope ies o he sys em we e compu ed o e 3 ⫻105cycles. The las molecula a angemen was hen used as ini ial con igu a ion o he subsequen un wi h a smalle sys em p essu e. Each MC cycle consis s o Npa emp s o andom displacemen s and/o eo ien a ions o he pa icles plus a ial change o he box olume. The usual pe iodic bounda y condi ions a e employed and he accep ance a ios we e kep wi hin 30%–40% o he il and displacemen o he pa icles, and wi hin 20%–30% o he box olume change. Box olume changes we e a emp ed by andomly changing he leng h o each side o he box independen ly, wi h he es ic ion ha none o hem could become sho e han wice he ange o he in e ac ion po en ial, o 8 ␴ in he p esen case. In e es ingly, we ound a endency o he box o spon aneously adop an aniso opic geome y wi h side leng hs ul illing Lx⬇Ly⬍Lz共 he subsc ip x deno es he sho es side兲, which is consis en wi h he geome y chosen o he simula ions wi h ixed box geome y (Lx⬇Ly ⬇Lz/2) o ou ea lie s udies o he SWSC, SRS, and Kiha a luids.11,17 In spi e o his la e inding, in o de o pe o m a p ope compa ison be ween he GB-K共6,5,2,1兲and he Kiha a lu- ids, we ha e also ecompu ed ull simula ions o he ou iso he ms T*⫽kT/ ⑀ ⫽1.5, 2, 3, and 5 o he Kiha a luid s udied in a p e ious wo k17 employing he same me hodol- ogy desc ibed abo e. The esul o hese new simula ions, which essen ially ep oduce he da a o he ea lie wo k,17 a e p esen ed below and include an ex ended egion a low and high densi y wi h espec o ou p e ious wo k in o de o o e lap wi h he p esen s udy o he GB-K model. The liquid c ys alline ansi ions obse ed in he expan- sion o he luid a e cha ac e ized by discon inui ies in he densi y and he nema ic o de pa ame e , as well as by sud- den quali a i e changes obse ed in he di e en co ela ion unc ions g 储 ( 储 ), g⬜( ⬜), g⬜ (n)( ⬜) de ined in p e ious wo ks.6,9,11,12,23 Fo ins ance, he unc ion g 储 ( 储 ) ep esen s he p ojec ion o he pai dis ibu ion unc ion along he nem- a ic di ec o o he luid and de elops a cha ac e is ic oscil- la o y s uc u e when laye ed phases a e o med. On he o he hand, g⬜ (0)( ⬜) accoun s o he co ela ion be ween pa icles wi hin he same laye and p esen s long- ange s uc- u e when solidlike o de is p esen . An illus a i e insigh in o hese co ela ion unc ions is p o ided below wi hin he amewo k o he luidlike smec ic-Ao hexagonally packed hexa ic phases ound in his wo k o he GB-K共6,5,2,1兲 luid. The o ma ion o hese la e hexa ic phases was in- spec ed, in addi ion, h ough he calcula ion o he bond hex- agonal o de pa ame e :22,24 H6⫽ 冓 冏 1 Np兺 j 1 nj兺 (kl)exp共6i ␪ kl兲 冏 冔 ,共7兲 whe e njis he numbe o pai s o neighbo s wi hin he i s in-laye coo dina ion shell o pa icle j, which is de ined as a cylind ical olume o adius 1.6 ␴ and heigh 1.0 ␴ cen e ed a he cen e o mass o he molecule. The sum o e (kl) ap- plies o all possible pai s o such i s coo dina ion shell pa - icles and ␪ kl is he angle be ween he p ojec ion o he in- 024908-3 No el po en ial model o p ola e mesogens J. Chem. Phys. 122, 024908 (2005) e molecula ec o s jk and jl on o he plane pe pendicula o he di ec o o molecule j. Hence, H6is so de ined as o app oach uni y o pe ec hexa ic o de wi hin he smec ic laye s 共wi h nj⫽6 in-laye nea es neighbo s a ound each pa icle o ming a hexagon兲and end o ze o o diso de ed luidlike laye s. We employ he gene ic denomina ion hexa ic h oughou he pape , since he limi ed numbe o pa icles o ou simula ions does no allow o ully cha ac e ize his phase as a luid smec ic-Bphase o a solid c ys al Bphase, as will be discussed below. IV. RESULTS AND DISCUSSION Figu e 2 共 op panel兲 ep esen s he liquid phase diag am o he GB-K共6,5,2,1兲 luid o he h ee iso he ms T*⫽2, 3, and 5. The p essu es and densi ies o he bounda y s a es a each o he phase ansi ions obse ed a e lis ed in Table I. A mo e de ailed in o ma ion abou he iso he ms is p o ided in Figs. 3, 4, and 5 which depic , espec i ely, he equa ions o s a e o he luid, he a e age ene gy pe pa icle, and he nema ic and hexa ic o de pa ame e s o each o he s a es sampled in he p esen simula ions. The phase diag am and equa ions o s a e o he Kiha a luid, ecalcula ed he e em- ploying he a iable box geome y simula ion me hodology FIG. 2. Summa y o he densi ies o he bounda y s a es a he iso opic- nema ic (I-N), iso opic–smec ic-A(I-Sm-A), nema ic–smec ic-A (N–Sm-A), o he smec ic-A–hexa ic (Sm-A–Hex) ansi ions ob ained in he MC–N-P-Tsimula ions o he GB-K共6,5,2,1兲 luid o he p esen wo k. The bo om panel shows he phase diag am o he Kiha a luid wi h he same molecula aspec a io. These la e da a a e e ised wi h espec o hose epo ed in a p e ious wo k 共Re . 17兲. See also Tables I and II. TABLE I. Reduced p essu es and densi ies o he bounda y s a es o he liquid c ys al phase ansi ions o he GB-K共6,5,2,1兲 luid conside ed in his wo k a empe a u es T*⫽2, 3, and 5. No e he educ ion o p essu e wi h empe a u e (P*⫽P ␴ 3/kT) when compa ing he da a. I–Sm-ASm-A–Hex pI * ␳ I *pSmA * ␳ SmA *pSm-A * ␳ Sm-A *pHex * ␳ Hex * T*⫽2.0 0.70 0.086共1兲0.75 0.104共1兲1.80 0.130共1兲1.90 0.137共1兲 I–NN–Sm-ASm-A–Hex pI * ␳ I *pN * ␳ N *pN * ␳ N *pSm-A * ␳ Sm-A *pSm-A * ␳ Sm-A *pHex * ␳ Hex * T*⫽3.0 1.10 0.100共1兲1.15 0.104共1兲1.40 0.112共1兲1.45 0.123共1兲2.25 0.140共1兲2.30 0.146共1兲 T*⫽5.0 1.30 0.107共1兲1.35 0.110共1兲2.35 0.137共1兲2.40 0.141共1兲2.95 0.151共1兲3.00 0.158共1兲 FIG. 3. Equa ions o s a e o he iso he ms T*⫽2, 3, and 5 o he GB- K共6,5,2,1兲and he Kiha a luids wi h sphe ocylinde aspec a io L*⫽5. The illed ci cles co espond o he GB-K luid whe eas he open squa es a e o he Kiha a luid. The co esponding liquid c ys al phases (I,N, Sm-A,o Hex兲a e indica ed nex o he da a. In he egions whe e he iso he ms o he wo luids o e lap bu he phases a e di e en , he phase co esponding o he Kiha a luid is indica ed in pa en heses. The s a is ical e o ba s asso- cia ed o he densi y alues a e o he size o he symbols o smalle . No e he educ ion o p essu e wi h empe a u e (P*⫽P ␴ 3/kT) when compa ing he da a. 024908-4 Ma ı ´nez-Haya e al. J. Chem. Phys. 122, 024908 (2005) desc ibed in he p eceding sec ion, a e also included in he same igu es o di ec compa ison wi h he GB-K da a. I was ound ha essen ially he same esul s a e ob ained o he Kiha a luid wi h espec o ou ea lie wo k,17 excep o a⫾0.05–0.10 di e ence in he educed p essu e o he sys- em in he smec ic s a es o same densi y. This la e e ec is likely o be ela ed o aniso opic s ess e ec s25 induced by he ixed box geome y app oach o ou p e ious wo k ha should no be a ec ing he p esen esul s. O he han ha , he liquid c ys al ansi ions ele an o he p esen wo k oc- cu in he expansion o he Kiha a luid a he same packing ac ions, wi hin s a is ical unce ain y, as ound in he ixed box geome y simula ions o ou ea lie wo k.17 We ocus now on he esul s o he GB-K model. A he h ee empe a u es in es iga ed, he GB-K共6,5,2,1兲 luid p e- sen s a s able hexa ic phase a su icien ly high densi ies in which he luid is in e nally o de ed in smec ic laye s o hex- agonally packed molecules. We lea e he cha ac e iza ion o his liquid c ys alline phase o he las pa o his Sec ion. When dec easing he sys em p essu e, and hence expanding he luid, he hexa ic phase e en ually mel s o a smec ic-A phase in which he laye s become luidlike and he wo- dimensional posi ional o de wi hin hem is los . A u he expansion o he sys em leads o a ansi ion o ei he a nem- a ic phase 共 o he iso he ms T*⫽3 and 5兲o di ec ly o an iso opic phase 共 o T*⫽2). This la e phase ansi ion shows a s onge i s -o de cha ac e han he es o ansi- ions obse ed, as he luid ea anges om a ela i ely dense laye ed s uc u e o a ully diso de ed iso opic phase wi h a much smalle packing ac ion. One no iceable aspec o he phase diag am o he GB- K共6,5,2,1兲 luid is he subs an ial educ ion in he ange o s abili y o he nema ic phase when cooling down he sys em om T*⫽5 oT*⫽3, as he ansi ion om smec ic-A o nema ic shi s o lowe p essu es and densi ies. In ac , he nema ic phase disappea s al oge he a T*⫽2, as al eady poin ed ou abo e. Hence, he esul o he MC simula ions sugges s he p esence o an iso opic–nema ic–smec ic-A iple poin a a empe a u e o oughly T*⬇2.5 and a den- si y close o ␳ *⬇0.1 共see Fig. 2兲. The compa ison o he phase diag am o he GB- K共6,5,2,1兲 luid o ha o he Kiha a luid in he same ange o empe a u es should p o ide ele an in o ma ion abou he ela i e impo ance o ene gy and en opy as ‘‘d i ing o ces’’ o he liquid c ys al ansi ions. As can be seen in Fig. 2, he o e all ea u es o he phase diag am o bo h luids p esen many simila i ies, al hough i is in e es ing o ind ha he I–N–Sm-A iple poin o he Kiha a luid is loca ed a oughly simila empe a u e bu a a highe densi y FIG. 4. A e age pai po en ial ene gy (U*⫽U/ ⑀ ) along he iso he ms s ud- ied in his wo k o he GB-K共6,5,2,1兲and he Kiha a luids. The illed ci cles co espond o he GB-K luid and he open squa es o he Kiha a luid. The e ical e o ba s co espond o one s anda d de ia ion o he compu ed ene gies. FIG. 5. Top panel: Nema ic o de pa ame e S2and bond hexagonal o de pa ame e H6as a unc ion o numbe densi y o he GB-K共6,5,2,1兲 luid a educed empe a u e T*⫽3. The e ical e o ba s co espond o one s an- da d de ia ion o he compu ed o de pa ame e s. Bo om panel: In-laye dis ibu ion unc ion in he di ec ion pe pendicula o he di ec o g⬜ (0)( ⬜ *) in he smec ic-A共sho -dash cu e兲and hexa ic 共long-dash and solid cu es兲 phases o he same luid. The a iable ⬜ *⫽ ⬜/ ␴ ep esen s he p ojec ion o he pai in e molecula dis ance ec o on o he plane pe pendicula o he di ec o . Hence, his dis ibu ion shows he spa ial pai co ela ion be ween he pa icles in he same smec ic laye . The educed p essu e o each s a e is indica ed in he legend, whe eas he empe a u e is T*⫽3 in all cases. 024908-5 No el po en ial model o p ola e mesogens J. Chem. Phys. 122, 024908 (2005) han in he GB-K共6,5,2,1兲 luid. The e a e u he signi ican di e ences be ween bo h phase diag ams which ha e o be a ibu ed o he aniso opy o he dispe si e in e ac ions in he GB-K po en ial a ound he molecula co e in compa ison o he iso opic in e ac ions o he Kiha a luid. A i s el- e an aspec is ha he smec ic phase o he Kiha a luid does no p esen any ace o in-laye hexagonal packing o de wi hin he ange o densi ies scoped in he p esen s udy. In o de o co obo a e his obse a ion, he densi y ange co - e ed in ou ea lie wo k o he Kiha a luid17 was ex ended o o e lap wi h he p esen simula ions. In ac , he hexa ic s a es ound o he GB-K共6,5,2,1兲 luid mel ed sys ema i- cally o smec ic-As a es when he GB-K po en ial was ex- changed by he Kiha a po en ial. Ano he pa icula i y o he GB-K共6,5,2,1兲 luid eadily obse ed when compa ing he wo panels o Fig. 2 is he sys ema ic dec ease o he educed p essu es and densi ies o he bounda y s a es sepa a ing he phase ansi ions ha akes place as he empe a u e o he luid cools down om high (T*⫽5) o low (T*⫽2) empe a u e. This end, which is especially p onounced o he N–Sm-A ansi ion, as com- men ed abo e, is as well appa en o he Sm-A–Hex and I-N ansi ions o he GB-K luid. In con as o his beha - io , he phase bounda ies o he Kiha a luid 共Table II兲a e conside ably less sensi i e o empe a u e 共wi h he excep- ion o he N–Sm-A ansi ion, due o he ins abili y o he nema ic phase a low empe a u e兲. A i s o e all conclusion ha can be d awn om hese indings is ha he o ien a ion dependence in oduced in he GB–K po en ial and, in pa - icula , he enhanced in e ac ion o aligned pai con igu a- ions 共see Fig. 1兲, a o s he o ma ion o he liquid c ys al phases explo ed in he p esen wo k. In ac , he iso opic phase becomes signi ican ly des abilized in compa ison o he Kiha a luid, especially a low empe a u e. A T*⫽2, o ins ance, he densi y o he bounda y iso opic s a e a he I–Sm-A ansi ion is ␳ *⫽0.086 in he GB-K共6,5,2,1兲 luid, in compa ison o he much highe alue o ␳ *⫽0.112 in he Kiha a luid 共see Table I兲. A somewha unexpec ed inding ha o some ex en seems o con adic his la e conclusion is ha he nema ic phase o he GB-K共6,5,2,1兲 luid a he highes empe a u e in es iga ed, T*⫽5, p esen s a g ea e ange o s abili y han in he Kiha a luid. In o he wo ds, he N–Sm-A ansi ion is delayed owa d highe densi ies wi hin he nema ic phase in he o me luid in compa ison o he la e one. In addi ion, i is also obse ed ha he change in densi y a he ansi ion is la gely educed in he GB-K共6,5,2,1兲 luid. A possible ex- plana ion o hese ea u es is ha he GB-K nema ic s a es become ac ually mo e comp essible han he Kiha a ones as a consequence o he slowe ise o he sho - ange epulsi e o ces o a la ge ac ion o pai o ien a ions. In Fig. 1 i can be app ecia ed ha he epulsi e pa o he in e ac ion a dis ances sho e han dm/ ␴ ⫽1共 he ze o o he po en ial o all pai o ien a ions兲becomes weake o molecula con igu- a ions wi h a p e ac o ⑀ GB smalle han uni y. A he same ime, he co e epulsion is enhanced in he GB-K共6,5,2,1兲 po en ial wi h espec o he Kiha a po en ial o he side- o- side pa allel con igu a ions ypical o he smec ic phase. As a consequence o his, he e ec i e olume excluded by he molecules in he smec ic laye s is g ea e in he GB-K luid, so ha he en opic cons ains wo k in a o o he mo e comp essible nema ic phase 共in which many pai o ien a ions a om he side- o-side one a e ele an 兲, hence delaying he N–Sm-A ansi ion. Since he sho pai dis ances a which he epulsi e pa o he po en ial domina es a e mo e e icien ly accessed by he molecules o he luid a high empe a u e, i is no su p ising ha his e ec is only ob- se ed in ou simula ions a T*⫽5, whe eas a he lowe empe a u es, mo e sensi i e o he a ac i e pa o he in- e ac ion, he smec ic-Aphase is s able down o signi ican ly lowe densi ies in he GB-K共6,5,2,1兲 luid han in he Kiha a one.Figu e 3 compa es he equa ions o s a e (P* ⫽P ␴ 3/kT e sus ␳ *⫽ ␳␴ 3) along he iso he ms T*⫽2, 3, and 5 o he GB-K共6,5,2,1兲and he Kiha a luids. We i s o all ema k ha in all cases he iso he ms o bo h models consis en ly con e ge a low densi y. Wi h inc easing em- pe a u e he beha io o he luids is expec ed o become p og essi ely less sensi i e o he de ails o he pai in e ac- ion po en ial 共especially a low densi y whe e he a ac i e o ces a e dominan 兲and, in ac , a T*⫽5 he good o e lap be ween he equa ions o s a e o he GB-K共6,5,2,1兲and he Kiha a luids ex ends o a subs an ial egion o he nema ic TABLE II. Reduced p essu es and densi ies o he bounda y s a es o he liquid c ys al phase ansi ions o he Kiha a luid conside ed in his wo k a empe a u es T*⫽1.5, 2.0, 3.0, and 5.0. No e he educ ion o p essu e wi h empe a u e (P*⫽P ␴ 3/kT) when compa ing he da a. The abula ed alues di e sligh ly wi h espec o hose gi en in a ecen wo k 共Re . 17兲due o he g ea e sys em size and he imp o ed p essu e equilib a ion me hodology o he p esen simula ions 共see ex o de ails兲. I–Sm-A pI * ␳ I *pSMA * ␳ SMA * T*⫽1.5 1.75 0.114共1兲1.80 0.132共1兲 T*⫽2.0 1.65 0.112共1兲1.70 0.125共1兲 I–NN–Sm-A pI * ␳ I *pN * ␳ N *pN * ␳ N *pSM-A * ␳ SM-A * T*⫽3.0 1.40 0.108共1兲1.45 0.111共1兲1.70 0.119共1兲1.75 0.130共1兲 T*⫽5.0 1.30 0.107共1兲1.35 0.110共1兲1.75 0.128共1兲1.80 0.135共1兲 024908-6 Ma ı ´nez-Haya e al. J. Chem. Phys. 122, 024908 (2005) phase. A he su icien ly high densi y, howe e , signi ican di e ences be ween bo h sys ems e en ually a ise. A T* ⫽5, o smec ic-As a es o simila densi y, he p essu e is signi ican ly g ea e o he GB-K luid han o he Kiha a luid. In e es ingly, his end ends o e e se a he lowe empe a u es, so ha a T*⫽2 and 3 he p essu e o he Kiha a s a es becomes compa able o ha o he GB-K s a es wi h he same densi y. A hese lowe empe a u es, howe e , he compa ison be ween he equa ions o s a e o he GB- K共6,5,2,1兲and he Kiha a luids a high densi y is no as s aigh o wa d as o T*⫽5, since he o e lap be ween he liquid c ys al phases is g ea ly educed. Fo ins ance, he acciden al coincidence o he equa ions o s a e o bo h luids is no iceable a T*⫽2 in he high densi y ange ( ␳ * ⬎0.137) whe e he GB-K共6,5,2,1兲 luid has en e ed he hexa ic phase, whe eas he Kiha a luid emains in he Sm-A phase. The eason o inding g ea e p essu es in he GB-K luid a high empe a u e is again ela ed o he la ge p e - ac o in he GB-K in e ac ion po en ial which enhances he sho - ange epulsion o he side- o-side pai con igu a ions 共a he same ime as i inc eases he dep h o he well a la ge dis ances兲. The epulsi e wall is el mo e e icien ly by he molecules a high empe a u es wi h he ne e ec o inc easing he p essu e o he sys em wi h espec o he Ki- ha a luid. A lowe empe a u es, he g ea e a e age a ac- ions educe he p essu e o he GB-K sys em, especially in he smec ic-Aphase. F om he compa ison o he phase diag ams and equa- ions o s a e o he GB-K共6,5,2,1兲and he Kiha a luids i becomes appa en ha , e en hough s e ic excluded olume e ec s cons i u e he main cons ain ha d i es he liquid c ys al ansi ions o he p ola e molecula luids o he ype conside ed he e, he ene ge ic con ibu ion o he ee ene gy has an impo an ole ha canno be neglec ed and, hence, ha an app op ia e desc ip ion o he dispe si e o ces is equi ed in o de o model he p ope ies o eal mesogens. Fu he suppo o hese conside a ions is p o ided in Fig. 4, which e eals he quali a i ely di e en e olu ion in he GB-K and Kiha a luids o he a e age in e nal ene gy pe pa icle along he h ee iso he ms conside ed in ou wo k. In he Kiha a luid, he ene gy g ows mono onously wi h den- si y a high empe a u e (T*⫽3 and 5兲, whe eas a weak dec easing end is obse ed a lowe empe a u es (T* ⫽2) in he low densi y side o he iso opic b anch and o he smec ic-As a es.17 The ac ha he in e nal ene gy g ows wi h densi y a high empe a u e can be aced back o he p og essi e ele ance o he epulsi e pa o he pai po en ial a he ela i ely high densi ies conside ed in ou s udy. In he Kiha a luid, he a ac i e well only domina es a su icien ly low empe a u es o , al e na i ely, in a dilu ed densi y egime, much close o he ideal gas limi han he e conside ed. Thus, i u ns ou ha in he ange o densi ies ele an o liquid c ys al beha io , he iso opic Kiha a a - ac ion inco po a es li le no el quali a i e ea u es wi h e- spec o he simila beha io also ound o he pu ely epul- si e SRS luid 共basically a unca ed Kiha a luid wi hou a ac i e well兲.11 On he con a y, he speci ic opology o he a ac i e in e ac ions in he GB-K luid leads o a quali- a i ely di e en ole o ene gy in he s abili y o he liquid c ys al phases. Especially no iceable is he d op in ene gy ha akes place as he sys em en e s he smec ic-Aphase om ei he he iso opic (T*⫽2) o nema ic (T*⫽3 and 5兲 phase. Such ene gy discon inui y is absen o much weake in any o he sphe ocylinde model luids p oposed in he pas , such as he Kiha a17 o he SRS and SWSC luids,11 and is a di ec consequence o he bias o he GB-K po en ial a o ing speci ic pai con igu a ions. I ollows ha he s a- bili y o he smec ic phases is suppo ed no only by s e ic e ec s, bu also by he bene icial con ibu ion o he dispe - si e in e ac ions o he ee ene gy. In e es ingly, a signi ican u he d op in ene gy is also obse ed in he h ee iso he ms a he Sm-A–hexa ic ansi ion. This ein o ces he idea ha he appea ance o hexagonal o de in he GB-K共6,5,2,1兲 luid a densi ies whe e he Kiha a luid main ains a s able smec ic-Aphase is also closely ela ed o he ene ge ic e - ec s induced by he aniso opic na u e o he in e ac ion po- en ial. In o de o cha ac e ize he hexa ic phases obse ed in he GB-K共6,5,2,1兲 luid and clea ly disce n hem om he smec ic-Aphases, se e al diagnos ics we e applied du ing he simula ions. The ansi ion om he smec ic-A o he hexa ic phase implies a sudden change o he hexagonal bond o de pa ame e 关Eq. 共7兲兴, in ou case om oughly H6⬇0.2 o H6⬇0.5, as can be seen in he uppe panel o Fig. 5 whe e H6is ep esen ed along wi h he nema ic o de pa- ame e o he iso he m T*⫽3. The o ma ion o he hexa ic phase can also be isualized h ough he long- ange s uc u e appea ing ab up ly in he in-laye dis ibu ion unc ion g⬜ (0)( ⬜)共 he pai dis ibu ion wi hin he plane o he laye 兲. Figu e 5 illus a es he quali a i e change ha his unc ion unde goes a he Sm-A–hexa ic ansi ion a T*⫽3, om a smoo h liquidlike dis ibu ion wi h only sho - ange o de o a much mo e s uc u ed wo-dimensional hexa ic c ys al-like co ela ion unc ion wi h well-de ined nea es -neighbo posi- ions o e se e al coo dina ion shells. A con o e sial opic p esen in se e al ecen wo ks on he Gay-Be ne luid has been whe he he hexa ic phases o med in ha sys em a e ac ually liquidlike smec ic-B phases o solidlike c ys al Bphases on a mac oscopic basis.6 We ad ance ha ou p esen esul s do no se e o close his a om i ial ques ion. A su icien ly high densi y a s able solid c ys alline phase is known o o m in sphe ocylinde co e luids.9In ac , i has been a gued ha he absence o a well-de ined phase ansi ion when comp essing and/o cool- ing down he hexa ic Gay-Be ne luid indica es ha he phase mus be c ys alline om he beginning. Fu he e i- dence in a o o he c ys alline na u e o he hexa ic phases o he ype obse ed o he Gay-Be ne sys em a ise om he calcula ion o shea iscosi ies.23 Howe e , he di e ence be ween he smec ic-Band he c ys al Bphases is ac ually a sub le one o he ini e size sys ems employed in compu e simula ions. In addi ion, ca e mus be aken when dealing wi h such high densi y s a es whe e he mobili y o he pa - icles is g ea ly educed and me as able s a es o ine icien sampling o he phase space may a ec he appa en s uc- u e o he luid. One u he aspec abou he smec ic-B phases is ha he long- ange o de wi hin he laye s is un- ca ed on a mac oscopic scale by he p esence o a la ge num- 024908-7 No el po en ial model o p ola e mesogens J. Chem. Phys. 122, 024908 (2005) be o punc ual de ec s, in con as wi h he c ys al Bphase whe e he long- ange o de ex ends o e a mac oscopic dis ance.6We close by no ing o his espec ha , wi hin he ange o densi ies and empe a u es explo ed in ou simula- ion, he hexa ic o de pa ame e emains a alues below 0.6 共see Fig. 5兲, ha is much smalle han he limi ing alue o uni y, which is an indica ion o a subs an ial p esence o de ec s wi hin he laye s o he luid, in p inciple compa ible wi h a smec ic-Bphase. Howe e , he limi ed size o ou simula ion box p e en s us om making de ini e s a emen s abou he ange o he in-laye co ela ions and we canno he e o e d aw conclusions in a o o he smec ic-Bo c ys- al Bcha ac e o he obse ed hexa ic phase. V. SUMMARY AND CONCLUSIONS A igid model po en ial, e e ed o as Gay-Be ne-Kiha a o GB-K po en ial, has been in oduced which is expec ed o be eliable o he s udy o molecula mesogenic luids. The GB-K model ea u es a sphe ocylinde molecula co e d essed wi h dispe si e in e ac ions dependen on he ela i e pai o ien a ion. The ma hema ical o mula ion o he model is compac and combines he unc ionali ies o he well- known Kiha a and Gay-Be ne po en ials. The esul s o he Mon e Ca lo simula ion s udy o he model luid o a speci ic se o pa ame e s a empe a u es T*⫽2, 3, and 5 p esen ed in his pape show ha i is ca- pable o ep oduce he iso opic, nema ic, smec ic-A, and hexa ic liquid c ys al phases obse ed in eal mesogens. A he h ee empe a u es in es iga ed, he GB-K共6,5,2,1兲 luid p esen s a s able hexa ic phase a su icien ly high densi ies, cha ac e ized by laye s o hexagonally packed molecules. When expanding he luid, he hexa ic phase e en ually mel s o a smec ic-Aphase s uc u ed in luidlike diso de ed laye s. A u he expansion o he sys em leads o a ansi ion o ei he a nema ic phase 共a T*⫽3 and 5兲o di ec ly o an iso opic phase 共a T*⫽2). In his la e phase change he luid unde goes a s ongly i s -o de ansi ion om a dense laye ed s uc u e o a ully diso de ed iso opic phase wi h a subs an ial change in densi y. In compa ison o he Kiha a luid, he GB-K共6,5,2,1兲 luid is ound o a o he o ma ion o he o de ed liquid c ys alline phases and, speci ically, he appea ance o laye ed hexa ic o de , a lowe packing ac ions. This p ope y can be in e p e ed as being a di ec consequence o he g ea e dispe si e in e ac ions assigned in he GB-K luid o speci ic pai o ien a ions, such as he pa allel side- o-side con igu a- ion. In ac , he speci ic opology o he GB-K po en ial wi h espec o he Kiha a po en ial leads o a quali a i ely di e - en beha io o he in e nal ene gy and o i s in luence on he s abili y o he liquid c ys al phases. In pa icula , i is ound ha he en ance o he luid in he smec ic phases (Sm-Ao hexa ic兲is accompanied by a subs an ial s abiliza ion o he in e nal ene gy o he luid, in con as o he beha io o o he sphe ocylinde model luids s udied p e iously, such as he Kiha a, o he SRS and SWSC luids. Hence, i becomes appa en ha he ene ge ic con ibu ion o he ee ene gy has an impo an ole in he mesogenic beha io o p ola e mo- lecula liquids in dense en i onmen s, and ha an app op i- a e desc ip ion and ea men o he dispe si e o ces is e- qui ed in o de o model accu a ely he p ope ies o eal mesogens, e en a a quali a i e le el. The in luence o he sho - ange in e ac ions on he in e nal s uc u e o he mo- lecula luids has as well been s essed ecen ly in a s udy o sys ems composed o linea dipola molecules.26 The main ad an age o he GB-K po en ial is ha i combines, wi hin he in insic limi a ions o he igid models, a quali a i ely mo e adequa e desc ip ion o bo h, he pai in e ac ions and he molecula shape o he ypical mesogens, in compa ison o p e ious models, whe eas i keeps a compa able compac - ness and nume ical e iciency in i s o mula ion. The p esen s udy has ocused on he p esen a ion o he GB-K model and has s essed he quali a i e e ec s in o- duced by he aniso opic dispe si e in e ac ions o he model in he equa ion o s a e and in he phase diag am o he luid. Fu u e wo k in ou g oup will be de o ed o compa e he beha io o he 共sphe ocylinde 兲GB-K luid o ha o he 共ellipsoidal兲Gay-Be ne luid, so ha he ele ance o he exac shape o he molecula co e a supe c i ical and sub- c i ical empe a u es will be exposed. In addi ion, he imple- men a ion o he GB-K model o obla e 共disk-like兲mesogens will be explo ed. ACKNOWLEDGMENTS The au ho s acknowledge suppo om he Spanish Di eccio ´n Gene al de In es igacio ´n Cien ı ´ ica y Te ´cnica 共G oups No. BQU2001-3615-C02兲and Plan Andaluz de In es igacio ´n共G an Nos. FQM-205 and FQM-319兲. 1J. G. Gay and B. J. Be ne, J. Chem. Phys. 74, 3316 共1981兲. 2M. P. Allen, G. T. E ans, D. F enkel, and B. M. Mulde , Ad . Chem. Phys. 86,1共1993兲, and e e ences he ein. 3B. J. Be ne and P. Pechukas, J. Chem. Phys. 56,4213共1972兲. 4C. Zannoni, J. Ma e . Chem. 11, 2637 共2001兲. 5L. F. Rull, Physica A 220,113共1995兲, and e e ences he ein. 6E. de Miguel and C. Vega, J. Chem. Phys. 117, 6313 共2002兲. 7G. V. Paolini, G. Cicco i, and M. Fe a io, Mol. Phys. 80,297共1993兲. 8C. Vega and S. Lago, Compu . Chem. 18,55 9S. C. McG o he , D. C. Williamson, and G. Jackson, J. Chem. Phys. 104, 6755 共1996兲, and e e ences he ein. 10D. J. Ea l, J. Ilny skyi, and M. Wilson, Mol. Phys. 99, 1719 共2001兲. 11 A. Cue os, B. Ma ı ´nez-Haya, L. F. Rull, and S. Lago, J. Chem. Phys. 117, 2934 共2002兲;117, 11405共E兲共2002兲. 12M. S. Al-Ba wani and M. P. Allen, Phys. Re . E 62, 6706 共2000兲. 13F. del Rı ´o, E. A ´ alos, R. Espı ´ndola, L. F. Rull, G. Jackson, and S. Lago, Mol. Phys. 100, 2531 共2002兲. 14M. A. Ba es and G. R. Luckhu s , J. Chem. Phys. 110,7087共1999兲. 15T. Kiha a, Ad . Chem. Phys. 5, 147 共1963兲. 16S. Lago, B. Ga zo ´n, S. Cale o, and C. Vega, J. Phys. Chem. 101,6763 共1997兲. 17A. Cue os, B. Ma ı ´nez-Haya, S. Lago, and L. F. Rull, Phys. Re . E 68, 011704 共2003兲. 18R. A. K omhau and B. Linde , J. Phys. Chem. 99, 16909 共1995兲. 19D. P. Ojha and V. G. K. M. Pisipa i, Z. Na u o sch., A: Phys. Sci. 57a, 645 共2002兲. 20J. Ilny skyi and M. R. Wilson, Compu . Phys. Commun. 134,23共2001兲. 21G. R. Luckhu s and P. S. J. Simmonds, Mol. Phys. 80, 233 共1993兲. 22M. Houssa, L. F. Rull, and S. C. McG o he , J. Chem. Phys. 109, 9529 共1998兲. 23J. T. B own, M. P. Allen, E. Ma ı ´n del Rı ´o, and E. de Miguel, Phys. Re . E57, 6685 共1998兲. 24H. Zewdie, Phys. Re . E 57, 1793 共1998兲. 25H. Domı ´nguez, E. Velasco, and J. Alejand e, Mol. Phys. 100,273共2002兲. 26S. Lago, S. Lo ´pez-Vidal, B. Ga zo ´n, J. A. Mejı ´as, J. A. An a, and S. Cale o, Phys. Re . E 68, 021201 共2003兲. 024908-8 Ma ı ´nez-Haya e al. J. Chem. Phys. 122, 024908 (2005)