Monte Carlo study of liquid crystal phases of hard and soft spherocylinders

Mon e Ca lo s udy o liquid c ys al phases o ha d and so sphe ocylinde s A. Cue os, B. Ma ınez-Haya, L. F. Rull, and S. Lago Ci a ion: The Jou nal o Chemical Physics 117, 2934 (2002); View online: h ps://doi.o g/10.1063/1.1491872 View Table o Con en s: h p://aip.sci a ion.o g/ oc/jcp/117/6 Published by he Ame ican Ins i u e o Physics A icles you may be in e es ed in A e-examina ion o he phase diag am o ha d sphe ocylinde s The Jou nal o Chemical Physics 104, 6755 (1996); 10.1063/1.471343 T acing he phase bounda ies o ha d sphe ocylinde s The Jou nal o Chemical Physics 106, 666 (1997); 10.1063/1.473404 O ien a ional o de ing and phase beha iou o bina y mix u es o ha d sphe es and ha d sphe ocylinde s The Jou nal o Chemical Physics 143, 044906 (2015); 10.1063/1.4923291 A new aniso opic so -co e model o he simula ion o liquid c ys al mesophases The Jou nal o Chemical Physics 128, 044906 (2008); 10.1063/1.2825292 Columna phases o disco ic sphe ocylinde s The Jou nal o Chemical Physics 129, 214706 (2008); 10.1063/1.3028539 A no el o ien a ion-dependen po en ial model o p ola e mesogens The Jou nal o Chemical Physics 122, 024908 (2004); 10.1063/1.1830429 Mon e Ca lo s udy o liquid c ys al phases o ha d and so sphe ocylinde s A. Cue os and B. Ma ı ´nez-Hayaa) Depa amen o de Ciencias Ambien ales, Uni e sidad Pablo de Ola ide, 41013 Se illa, 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, Apdo. 1065, 41080 Se illa, Spain S. Lago Depa amen o de Ciencias Ambien ales, Uni e sidad Pablo de Ola ide, 41013 Se illa, Spain 共Recei ed 2 Janua y 2002; accep ed 15 May 2002兲 We epo on a Mon e Ca lo s udy o he liquid c ys al phases o wo model luids o linea elonga ed molecules: 共a兲ha d sphe ocylinde s wi h an a ac i e squa e-well 共SWSC兲and 共b兲pu ely epulsi e so sphe ocylinde s 共SRS兲, in bo h cases o a leng h- o-b ead h a io L*⫽5. Mon e Ca lo simula ions in he iso he mal–isoba ic ensemble ha e been pe o med a a educed empe a u e T*⫽5 p obing he modynamic s a es wi hin he iso opic 共I兲, nema ic 共N兲, and smec ic A 共Sm A兲 egions exhibi ed by each o he models. In addi ion, he pe o mance o an en opy c i e ion o alloca e liquid c ys alline phase bounda ies, ecen ly p oposed o he iso opic–nema ic ansi ion o he ha d sphe ocylinde 共HSC兲 luid, is success ully es ed o he SWSC and he SRS luids and u he mo e ex ended o he s udy o he nema ic–smec ic ansi ion. Wi h espec o he mo e ex ensi ely s udied HSC luid, he in oduc ion o he a ac i e squa e well in he SWSC model b ings he I–N and N–Sm A ansi ions o highe p essu es and densi ies. Mo eo e , he so epulsi e co e o he SRS luid induces a simila bu qui e mo e signi ican shi o bo h o hese phase bounda ies owa d highe densi ies. This la e e ec is appa en ly in con as wi h e y ecen s udies o he SRS luid a lowe empe a u es, bu his disc epancy can be aced back o he di e en e ec i e size o he molecula epulsi e co e a di e en empe a u es. © 2002 Ame ican Ins i u e o Physics. 关DOI: 10.1063/1.1491872兴 I. INTRODUCTION The cu en unde s anding o he beha io o liquid c ys- alline mesogens elies la gely on de ailed s udies o simple molecula luid models. O e he pas decades, a numbe o models o a ying complexi y ha e been de eloped wi h he aim o inco po a ing he essen ial ea u es o he eal sys- ems, while minimizing he expense o hei heo e ical and nume ical ea men . Such expense is mainly imposed by he in insic o ien a ional dependence o he pai in e ac ion and he long- ange o de cha ac e is ic o he liquid c ys al phases. An impo an class o such liquid c ys al models en- compasses luids o linea molecules in e ac ing ia a ha d o so epulsi e co e o ei he ellipsoidal o sphe ocylind ical symme y, e en ually d essed by a ac i e in e ac ions o di e en na u e.1–5 In pa icula , he ha d sphe ocylinde luid 共HSC兲cons i u es one o he mos ex ensi ely s udied models and, in spi e o i s simplici y and pu ely epulsi e in e ac ion, i has been ound o display di e en ia ed iso o- pic, nema ic, smec ic, and solid phases.1,3 In he p esen wo k, we ha e concen a ed on wo o he simples modi ica ions o he HSC model which, pe haps su p isingly, ha e no dese ed as much a en ion as o he elemen al liquid c ys al models: a luid o ha d sphe ocylin- de s wi h an a ac i e squa e-well 共SWSC兲,6–8 and a luid o so epulsi e sphe ocylinde s 共SRS兲.5,9–11 We hence in end o explo e he e ec o he simple ea u es in oduced in hese wo models wi h espec o he HSC luid, namely he p esence o an a ac i e well o he so ness o he sho ange epulsi e in e ac ion, on he liquid c ys al phase dia- g am o he luid. In addi ion, unlike he HSC luid, hese models allow o he s udy o sys ems a ini e empe a u e in a na u al way. One u he undamen al aspec o his ype o s udy will necessa ily be ela ed o he assessmen , a an elemen a y le el, o he oles played by wo o he main ex ensi e magni udes, in e nal ene gy and en opy, in d i ing he liquid c ys alline ansi ions obse ed in he luids unde s udy. We will ocus, in pa icula , on he ex ension o ea lie wo ks ha ha e explo ed he ela ionship be ween s a is ical en opy and o de ing ansi ions in a omic and molecula lu- ids by eso ing o he mul ipa icle co ela ion expansion o en opy es ablished by G een and Ne le on12 and la e gen- e alized by Laza idis and co-wo ke s o nonsphe ical sys ems.13 In his con ex , many au ho s ha e s udied he esponse o o de ing in he luid o he esidual mul ipa icle en opy, de ined as he o al con ibu ions o he excess en- opy o all co ela ions in ol ing mo e han wo pa icles.14–17 Cos a e al.17 we e he i s o employ his me hodology o o ien a ional ansi ions in liquid c ys als and ound a sudden g ow h o ⌬sa he iso opic–nema ic ansi ion in a HSC luid. The au ho s o his la e wo k imposed a educed dimensionali y in he o ien a ional depen- a兲Au ho o whom co espondence should be add essed. Elec onic mail: [email p o ec ed] JOURNAL OF CHEMICAL PHYSICS VOLUME 117, NUMBER 6 8 AUGUST 2002 29340021-9606/2002/117(6)/2934/13/$19.00 © 2002 Ame ican Ins i u e o Physics dence o he adial dis ibu ion unc ion and p oposed he c osso e o ⌬s om nega i e o posi i e alues as an em- pi ical indica ion o he incipien ansi ion o he nema ic phase, a c i e ion ha had p e iously been applied o he eezing ansi ion in he ha d sphe e luid,14 bu ha showed only mode a e success in a omic luids wi h a ac i e in e ac ions.15 Thus, in he p esen s udy we ha e add essed he ques ion o whe he he en opy c i e ion, al hough con- as ed o he HSC luid, does keep i s consis ency o he liquid c ys alline ansi ions in he SWSC and SRS luids. The ele an de ails o he SWSC and SRS models and o he Mon e Ca lo simula ion me hodology employed in his wo k, including he cha ac e iza ion o he di e en liquid c ys alline phases and he compu a ion o he di e en s uc- u al and he modynamic magni udes, a e desc ibed in Sec. II. The simula ion esul s a e p esen ed and discussed ho - oughly in Sec. III. Finally, a summa y o he main conclu- sions is d awn in Sec. IV. II. SIMULATION METHOD A. Model luids and simula ion de ails We ha e applied he iso he mal–isoba ic 共NPT兲 e sion o he Mon e Ca lo echnique o e alua e he modynamic and s uc u al p ope ies o he SWSC and SRS luids. In he SWSC luid, he ha d co e o he molecules comp ises a cylinde o diame e ␴ and elonga ion L*⬅L/ ␴ ⫽5, wi h hemisphe ical caps on bo h ends o he same diame e ␴ . The a ac i e squa e well in e ac ion is cha ac e ized by a dep h ␧and a ange ␭⫽1.5 ␴ , and has he same aniso opy o he ha d-co e. Thus, he pai in e ac ion po en ial o wo such sphe ocylinde s is gi en by V共 ,⍀兲⫽ 再 ⬁dm⭐ ␴ ⫺␧ ␴ ⬍dm⭐␭ 0dm⬎␭ 共1兲 as ep esen ed in Fig. 1. The dis ance be ween a pai o mol- ecules, dm⫽dm( ,⍀), is de ined as he minimum dis ance be ween he segmen s desc ibed by he axis o he cen al cylinde . The alue o dmis a unc ion o he ela i e o ien- a ion o he molecules, ⍀, de e mined by h ee independen angles, and o he dis ance be ween hei cen e s-o -mass, , as well as o he molecula pa ame e s: diame e and elonga- ion.In he second model, he SRS luid,5,9–11 he same ype o linea molecules in e ac h ough a so and pu ely epul- si e po en ial o sphe ocylind ical shape, buil om a un- ca ion o a shi ed Kiha a po en ial 共see Fig. 1兲which assu es con inui y o he po en ial and i s i s de i a i e, V共 ,⍀兲⫽ 再 4␧关共 ␴ /dm兲12⫺共 ␴ /dm兲6⫹1/4兴dm⭐ 冑 62• ␴ 0dm⬎ 冑 62• ␴ . 共2兲 The MC-NPT simula ions we e un a educed empe a- u e T*⬅kT/␧⫽5共kdeno es he Bol zmann cons an 兲and s a ed a low p essu e (P*⬅P• ␴ 3/kT⫽0.1) deep in he iso opic egion o he phase diag am, o e a sys em o NP ⫽768 molecules ini ially a anged on a hexagonal close- packed 共hcp兲la ice. Following in pa he p ocedu e o McG o he e al.,3we buil he hcp la ice wi h he 关111兴 di ec ion along he zaxis and a dis ance be ween he close packed 关111兴planes scaled wi h (1⫹L*). The hcp uni cell con ains ou molecules and we chose a box wi h 12, 8, and 2 uni cells in he x,y, and zdi ec ions, espec i ely. In his way, he dimensions o he simula ion box along each axis ul ill he a ios Ly/Lx⫽1.15 and Lz/Lx⫽1.94. The longe dimension along he zdi ec ion is mean o inc ease he num- be o possible smec ic laye s wi hin he simula ion box 共up o 4 laye s in he close-packed con igu a ion兲wi h espec o he cubic box o a gi en numbe o molecules. A e equili- b a ion and a e aging, he las molecula a angemen is s o ed and used as ini ial con igu a ion o he subsequen un wi h an inc eased sys em p essu e. Thus, in his way he luid is comp essed h ough he iso opic–nema ic and nema ic–smec ic ansi ions. In o de o check o possible me as abili y and hys e ic e ec s, an expansion un, all he way back o he iso opic egion, is subsequen ly pe o med beginning wi h he smec ic s a e o highes densi y in he calcula ion. As will be discussed below, he expansion un indeed esul ed in a be e equilib a ion o he calcula ions a high densi y. The 共 i s -o de 兲liquid c ys alline ansi ions a e loca ed by he discon inui ies in he densi y and he o de pa ame e , aided by he analysis o he di e en co ela ion unc ions. The p ocedu e is simila o ha employed by McG o he e al. in hei s udy o he HSC luid3and e y ecen ly also by Ea l e al. o he SRS luid.5The nema ic o de pa am- e e S⬅ 具 P2(ui"n) 典 is calcula ed as he ensemble a e age o he mean alue o he second Legend e polynomial, i s a gu- men being he do p oduc be ween he uni a y ec o along he molecula axis o each pa icle, ui, and he 共also uni a y兲 nema ic di ec o n. Fo he p esen s udy, oge he wi h he usual pai dis ibu ion unc ion g( ), a ele an o ien a ional dis ibu ion unc ion is g2( )⬅ 具 P2(ui"uj) 典 , de ined as he a e age, o each in e molecula dis ance, o he second Leg- end e polynomial wi h a gumen he cosine o he ela i e angle be ween he axis o pa icles iand j,ui"uj⫽cos ␪ .In addi ion, he ansla ional o de o he molecules is u he examined by means o he unc ions g 储 ( 储 ) and g⬜( ⬜), FIG. 1. Pai in e ac ion po en ial ene gies as a unc ion o he minimum dis ance, dm, be ween he molecules o he model luids s udied in his wo k: squa e-well sphe ocylinde luid 关SWSC, see Eq. 共1兲兴 and so epul- si e sphe ocylinde 关SRS, see Eq. 共2兲兴. The alue o dmdepends on he dis ance be ween he cen e s-o -mass o he molecules, , and on hei o i- en a ions as de ined by he ec o s u1and u2. 2935J. Chem. Phys., Vol. 117, No. 6, 8 Augus 2002 Liquid c ys al phases o ha d and so sphe ocylinde s which a e he p ojec ions o he pai dis ibu ion unc ion a dis ance 储 pa allel and a dis ance ⬜pe pendicula o he nema ic di ec o , espec i ely. As shown by McG o he e al.,3smec ic o de ing induces well de ined oscilla ions in g 储 ( 储 ), whe eas he s uc u e wi hin he smec ic planes is e lec ed in g⬜( ⬜). Fu he mo e, we checked o smec ic B hexa ic o de ing wi hin he smec ic laye s. We al eady ad- ance, howe e , ha only smec ic A phases we e ound in he p esen s udy. The he modynamic 共densi y, ene gy兲and s uc u al p ope ies o he sys em, as well as he di e en dis ibu ion unc ions a e ob ained as ensemble a e ages in he MC simu- la ion. Typically, o each s a e an ini ial simula ion o up o 2⫻106cycles 共depending on con e gence兲is pe o med o equilib a e he sys em be o e a e aging o some 1–2⫻105 cycles. Each cycle consis s o NPa emp s o andom dis- placemen s and/o eo ien a ions o he pa icles 共whe he ei- he o bo h ypes o mo es a e pe o med is chosen an- domly兲, plus an a emp o change he olume 共ac ually done by escaling he molecula diame e , ␴ , in uni s o box leng h兲. The maximum il angle and displacemen o he pa icles and he maximum olume change a e adjus ed as o gi e an accep ance a io o be ween 30% and 40%. The usual pe iodic bounda y condi ions and minimum image con en- ions a e employed.18 B. En opy c i e ion o liquid c ys alline ansi ions As commen ed in he In oduc ion, we ha e explo ed he co ela ion be ween he con igu a ional en opy and he phase diag am o ou liquid c ys al models. Following, among o he s, he wo ks o Laza idis e al.13 and Cos a e al.,17 we ha e ocused on he beha io o he di e en componen s o he pai en opy, s2, and o he esidual en- opy, ⌬s⫽sex⫺s2, de ined as he mul ipa icle con ibu ion o he excess en opy, sex , i.e., all co ela ions in ol ing mo e han wo pa icles. As shown in hese p e ious wo ks,13 he o mal ac o iza ion o he co ela ion unc ion, g( , ␻ ) ⫽g( )•g( ␻ 兩 ), in e ms o g( ), he adial dis ibu ion unc ion, and g( ␻ 兩 ), he condi ional dis ibu ion unc ion o wo molecules wi h o ien a ions desc ibed by ␻ 共which de- no es all he ele an o ien a ion angles兲a an in e molecula dis ance , leads o a decomposi ion o he pai en opy in o ien a ional and posi ional con ibu ions. Fo he compu a ion o ⌬sin a luid o linea molecules, Cos a e al. p oposed a educed dimensionali y o he co e- la ion unc ion, g( , ␪ ), in which he ela i e o ien a ion o each pai o pa icles is desc ibed by he angle ␪ be ween he wo molecula axis, al eady in oduced in Sec. IIA. Wi h his assump ion, he o mal ac o iza ion o he co ela ion unc- ion, g( , ␪ )⫽g( )•g( ␪ 兩 ), leads o he ollowing exp es- sion o he pai con ibu ion o he excess en opy, s2共pe pa icle and in uni s o he Bol zmann cons an 兲: s2⫽s2 ⫹s2 o ,共3兲 s2 ⫽⫺2 ␲␳ 冕 关g共 兲lng共 兲⫺g共 兲⫹1兴 2d ,共4兲 s2 o ⫽4 ␲␳ 冕 g共 兲So 共 兲 2d ,共5兲 So 共 兲⫽⫺ 1 4 冕 0 ␲ g共 ␪ 兩 兲lng共 ␪ 兩 兲sin ␪ d ␪ .共6兲 In he abo e exp essions 共3兲–共6兲, ␳ ⫽N/Vdeno es he num- be densi y and s2 and s2 o e e o a o mal dis inc ion be- ween ansla ional and o ien a ional con ibu ions o s2; whe eas s2 is expec ed o moni o any changes in he ans- la ional o de o he molecules o he luid, s2 o should e- spond o hei o ien a ional o de . I mus be ema ked ha he o mal decomposi ion o he pai en opy s2⫽s2 ⫹s2 o ,as well as Eq. 共4兲 o s2 , a e independen o he app oxima ion in oduced by he use o he co ela ion unc ion g( , ␪ ), whose jus i ica ion elies upon he compa a i ely simpli ied compu a ion o s2 o , as de ined in Eq. 共6兲, and i s app op ia e beha io when applied o linea nema ogens. In ac , Cos a e al. ound a apid dec ease o s2 o owa ds la ge nega i e alues in he icini y o he iso opic–nema ic o he HSC luid and sugges ed he ze o o he esidual en opy, ⌬s ⫽sex⫺s2⫽0, as an indica o o he incipien nema ic o de ing.17 The o al excess en opy o he iso opic luid may be e alua ed om he MC equa ion o s a e, by means o he exac exp ession, sex共 ␳ 兲⫽uex T⫺ 冕 0 ␳ 冋 P kT ␳ ⬘⫺1 册 d ␳ ⬘ ␳ ⬘,共7兲 whe e uex deno es he excess ene gy 共i.e., he po en ial en- e gy兲pe pa icle in uni s o he Bol zmann cons an . Hence, he excess en opy is de e mined by he in e nal ene gy and by he in eg al o (Z⫺1)/ ␳ , whe e Z⫽P/(kT ␳ ) is he com- p essibili y ac o o he luid. In he p esen wo k, we ha e ex ended he compu a ion o he excess en opy o he nema ic and smec ic phases by including he en opy change a he co esponding phase ansi ions. Since we a e dealing wi h ansi ions a cons an N, P, T, he en opy o ansi ion, ⌬s ans , can be eadily ob ained om he en halpy o ansi ion, ⌬h ans , as gi en by he changes in he in e nal ene gy, ⌬u ans , and olume ⌬ ans , ⌬s ans⫽⌬h ans T⫽⌬u ans T⫹P*⌬ ans ,共8兲 whe e we ecall ha ⌬s ans ,⌬h ans ,⌬u ans , and P*⌬ ans(⫽P•⌬ ans /kT) a e pe pa icle and in kuni s. Thus, o ins ance, he ex ension o Eq. 共7兲 o he calcula ion o he excess en opy o a pa icula s a e a e he I–N phase ansi ion is gi en by sex共 ␳ 兲⫽uex T⫺ 冕 0 ␳ 1 冋 P kT ␳ ⬘⫺1 册 d ␳ ⬘ ␳ ⬘⫹PI–N *⌬ I–N ⫺ 冕 ␳ 2 ␳ 冋 P kT ␳ ⬘⫺1 册 d ␳ ⬘ ␳ ⬘,共9兲 2936 J. Chem. Phys., Vol. 117, No. 6, 8 Augus 2002 Cue os e al. whe e ␳ 1and ␳ 2, deno e he coexis ence densi ies o he wo phases, ep esen ed in ou NPT simula ions by he densi ies o he s a es closes on ei he side o he phase bounda y. The ansi ion p essu e is aken as he a e age o he simula ion p essu e o bo h bounda y s a es. No e ha in Eq. 共9兲 he ene ge ic pa o he en opy o ansi ion, ⌬uI–N/T, is e ec- i ely included in uex /T. The compu a ion o he i s in e- g al in Eq. 共9兲was pe o med om he Mon e Ca lo alues o (Z⫺1)/ ␳ and hei linea ex apola ion a low densi y. This la e ex apola ion should p o ide, a ␳ ⫽0, he second i ial coe icien , B2, o he luid a he i le empe a u e. In ac , as shown below 共Fig. 7兲, he ex apola ion o ␳ ⫽0o he compu ed (Z⫺1)/ ␳ cu e o he SWSC sys em is con- sis en wi h he known analy ical exp ession o B2 o his model luid.6 We ha e employed Eqs. 共3兲–共9兲 o ex end he s udy o he beha io o sex ,⌬sand he di e en componen s o s2, h ough bo h he iso opic–nema ic and he nema ic–smec ic ansi ions o he SWSC and SRS luids. III. RESULTS AND DISCUSSION We ha e pe o med MC-NPT simula ions o he SWSC and SRS luids o elonga ion L*⫽5 a a educed empe a u e T*⫽5. The he modynamic da a esul ing in he comp es- sion and expansion simula ion uns o his iso he m o bo h model luids a e lis ed in Tables I–IV. In addi ion, he equa- ions o s a e 共P* s ␳ *兲, o de pa ame e s and in e nal en- e gies ob ained o bo h models a e ep esen ed in Figs. 2, 4, and 6. Wi hin he ange o p essu es co e ed in ou s udy, P*⬇0.1–2.2, bo h he SWSC and he SRS luids exhibi di e en ia ed iso opic, nema ic, and smec ic A phases. The TABLE I. Iso he mal–isoba ic Mon e Ca lo 共MC-NPT兲simula ion esul s o he equa ion o s a e o he SWSC luid o molecula elonga ion L*⫽L/ ␴ ⫽5, squa e well dep h ␧, and ange ␭*⫽␭/ ␴ ⫽1.5 共see Fig. 1兲 a empe a u e T*⫽kT/␧⫽5. The p essu e, ixed in he simula ions, is exp essed in educed uni s ei he wi h espec o he molecula diame e , ␴ 3共 i s column兲, o o he olume o he molecula ha d co e, HSC 共second column兲, o allow o a di ec compa ison wi h ea lie wo ks. U*and Sdeno e, espec i ely, he a e ages o po en ial ene gy pe pa icle and he o de pa ame e o he luid. The simula ions we e un by comp essing he luid om he leas dense s a e. The alues in b acke s deno e he s a is ical unce ain y 共one s anda d de ia ion兲 in he las digi . The iso opic 共I兲, nema ic 共N兲o sme ic A 共Sm A兲phase co esponding o each s a e is indica ed in he las column. P*⫽P• ␴ 3/kT P*• HSC / ␴ 3 ␳ *⫽ ␳ • ␴ 3 ␩ ⫽ ␳ • HSC U*⫽U/␧SPhase 0.01 0.045 0.0080共3兲0.035共1兲⫺0.36共2兲0.030共1兲I 0.02 0.089 0.0135共4兲0.060共2兲⫺0.64共3兲0.030共1兲I 0.05 0.22 0.0245共4兲0.109共2兲⫺1.25共4兲0.032共1兲I 0.10 0.45 0.0360共4兲0.160共2兲⫺2.00共4兲0.032共1兲I 0.20 0.89 0.0494共4兲0.220共2兲⫺2.98共5兲0.033共1兲I 0.30 1.34 0.0580共4兲0.258共2兲⫺3.66共5兲0.036共1兲I 0.40 1.78 0.0645共4兲0.287共2兲⫺4.20共5兲0.037共1兲I 0.50 2.23 0.0699共6兲0.311共3兲⫺4.63共6兲0.042共1兲I 0.60 2.67 0.0741共4兲0.330共2兲⫺4.97共5兲0.040共1兲I 0.70 3.12 0.0773共4兲0.344共2兲⫺5.21共5兲0.049共1兲I 0.80 3.56 0.0816共4兲0.363共2兲⫺5.52共5兲0.042共1兲I 0.90 4.01 0.0845共4兲0.376共2兲⫺5.69共5兲0.060共1兲I 1.00 4.45 0.0876共4兲0.390共2兲⫺5.89共5兲0.060共1兲I 1.10 4.90 0.0910共4兲0.405共2兲⫺6.02共5兲0.072共1兲I 1.20 5.34 0.0932共4兲0.415共2兲⫺6.13共5兲0.088共2兲I 1.25 5.56 0.0948共4兲0.422共2兲⫺6.18共5兲0.068共2兲I 1.30 5.79 0.0959共4兲0.427共2兲⫺6.20共5兲0.115共1兲I 1.35 6.01 0.0980共4兲0.436共2兲⫺6.21共4兲0.233共1兲I 1.40 6.23 0.0993共4兲0.442共2兲⫺6.18共6兲0.290共1兲I 1.45 6.45 0.1016共4兲0.452共2兲⫺6.20共5兲0.486共2兲N 1.50 6.68 0.1031共4兲0.459共2兲⫺6.13共5兲0.533共2兲N 1.55 6.90 0.1047共3兲0.466共1兲⫺6.08共4兲0.665共2兲N 1.60 7.12 0.1065共3兲0.474共1兲⫺6.05共4兲0.719共2兲N 1.65 7.34 0.1085共4兲0.483共2兲⫺5.98共4兲0.784共1兲N 1.70 7.57 0.1132共6兲0.504共3兲⫺5.79共4兲0.882共2兲Sm A 1.75 7.79 0.1146共4兲0.510共2兲⫺5.80共5兲0.890共1兲Sm A 1.80 8.01 0.1153共8兲0.513共2兲⫺5.86共5兲0.852共1兲Sm A 1.85 8.23 0.1180共4兲0.525共2兲⫺5.81共5兲0.880共2兲Sm A 1.90 8.46 0.1202共4兲0.535共2兲⫺5.77共5兲0.905共3兲Sm A 1.95 8.68 0.1225共6兲0.545共3兲⫺5.73共5兲0.914共2兲Sm A 2.00 8.90 0.1240共4兲0.552共2兲⫺5.74共5兲0.915共3兲Sm A 2.05 9.12 0.1251共4兲0.557共2兲⫺5.77共5兲0.906共3兲Sm A 2.10 9.35 0.1263共4兲0.562共2兲⫺5.81共6兲0.906共4兲Sm A 2.15 9.57 0.1272共4兲0.566共2兲⫺5.82共5兲0.917共3兲Sm A 2.20 9.79 0.1285共4兲0.572共2兲⫺5.82共5兲0.920共3兲Sm A 2937J. Chem. Phys., Vol. 117, No. 6, 8 Augus 2002 Liquid c ys al phases o ha d and so sphe ocylinde s phase ansi ions in ol e ela i ely weak discon inuous changes in densi y 共see Fig. 6兲and in e nal ene gy bu a e accompanied by app eciable jumps in he o de pa ame e and/o in he s uc u e o he di e en adial co ela ion unc- ions, g( ), g2( ), and in g 储 ( 储 ), de ined abo e and depic ed in Figs. 3 and 5. As we shall see below, he pai and esidual en opies also espond o he o ien a ional and ansla ional o de ing o he luid 共Figs. 7 and 8兲. TABLE II. Same as Table I, bu o a se o simula ions un by expanding he SWSC luid om he mos dense s a e (P*⫽2.20). P*⫽P• ␴ 3/kT P*• HSC / ␴ 3 ␳ *⫽ ␳ • ␴ 3 ␩ ⫽ ␳ • HSC U*⫽U/␧SPhase 1.00 4.45 0.0881共6兲0.392共3兲⫺5.87共7兲0.085共1兲I 1.10 4.90 0.0905共4兲0.403共2兲⫺6.03共5兲0.084共1兲I 1.20 5.34 0.0930共4兲0.414共2兲⫺6.11共5兲0.146共2兲I 1.25 5.56 0.0953共6兲0.424共3兲⫺6.15共5兲0.175共2兲I 1.30 5.79 0.0968共6兲0.431共3兲⫺6.17共5兲0.215共2兲I 1.35 6.01 0.0993共4兲0.442共2兲⫺6.02共5兲0.557共2兲N 1.40 6.23 0.1007共4兲0.448共2兲⫺6.04共5兲0.609共2兲N 1.45 6.45 0.1027共8兲0.457共4兲⫺5.98共6兲0.691共2兲N 1.50 6.68 0.1043共6兲0.464共3兲⫺5.99共5兲0.716共2兲N 1.55 6.90 0.1058共6兲0.471共3兲⫺5.97共5兲0.772共3兲N 1.60 7.12 0.1078共6兲0.480共3兲⫺6.00共5兲0.789共3兲N 1.65 7.34 0.1099共5兲0.489共3兲⫺5.92共5兲0.824共3兲N 1.70 7.57 0.1155共6兲0.514共3兲⫺5.67共3兲0.907共3兲Sm A 1.75 7.79 0.1171共5兲0.521共3兲⫺5.70共3兲0.913共3兲Sm A 1.80 8.01 0.1189共4兲0.529共2兲⫺5.70共5兲0.911共3兲Sm A 1.90 8.46 0.1218共4兲0.542共2兲⫺5.72共4兲0.915共3兲Sm A 2.00 8.90 0.1243共6兲0.553共3兲⫺5.80共5兲0.915共3兲Sm A 2.10 9.35 0.1263共4兲0.562共2兲⫺5.81共5兲0.926共3兲Sm A 2.20 9.79 0.1285共4兲0.572共2兲⫺5.82共5兲0.920共3兲Sm A TABLE III. Iso he mal–isoba ic Mon e Ca lo 共MC-NPT兲simula ion esul s o he equa ion o s a e o he SRS luid 关Eq. 共2兲,Fig.1兴o molecula elonga ion L*⫽L/ ␴ ⫽5 a empe a u e T*⫽kT/␧⫽5. The simula ions we e un by comp essing he luid om he leas dense s a e. The same no a ion is employed as in Table I. In pa icula , HSC deno es he olume o a ha d sphe ocylinde wi h he same elonga ion L*⫽5. P*⫽P• ␴ 3/kT P*• HSC / ␴ 3 ␳ *⫽ ␳ • ␴ 3 ␩ ⫽ ␳ • HSC U*⫽U/␧SPhase 0.01 0.044 0.0079共4兲0,035共2兲0.12共3兲0.031共1兲I 0.02 0.089 0.0134共4兲0,060共2兲0.23共4兲0.030共1兲I 0.05 0.22 0.0249共4兲0,111共2兲0.48共6兲0.031共1兲I 0.10 0.45 0.0368共4兲0.164共2兲0.83共7兲0.033共1兲I 0.20 0.89 0.0521共6兲0.232共3兲1.4共1兲0.035共1兲I 0.30 1.34 0.0627共6兲0.279共3兲1.9共1兲0.037共1兲I 0.40 1.78 0.0708共6兲0.315共3兲2.4共1兲0.040共1兲I 0.50 2.23 0.0775共6兲0.345共3兲2.9共1兲0.040共1兲I 0.70 3.12 0.0887共6兲0.395共3兲3.7共1兲0.048共1兲I 0.80 3.56 0.0937共6兲0.417共3兲4.1共1兲0.055共1兲I 0.90 4.01 0.0982共6兲0.437共3兲4.5共1兲0.061共1兲I 1.00 4.45 0.1025共6兲0.456共3兲4.8共1兲0.071共1兲I 1.10 4.90 0.1065共6兲0.474共3兲5.2共2兲0.134共2兲I 1.15 5.12 0.1088共8兲0.484共4兲5.4共2兲0.110共3兲I 1.20 5.34 0.1117共6兲0.497共3兲5.5共2兲0.198共2兲I 1.25 5.56 0.1157共6兲0.515共3兲5.7共2兲0.541共2兲N 1.30 5.79 0.1200共8兲0.534共4兲5.8共2兲0.722共2兲N 1.35 6.01 0.1216共8兲0.541共4兲6.0共2兲0.715共2兲N 1.40 6.23 0.1240共8兲0.552共4兲6.2共2兲0.752共2兲N 1.45 6.45 0.1258共8兲0.560共4兲6.3共2兲0.766共2兲N 1.50 6.68 0.1281共8兲0.570共4兲6.5共2兲0.795共2兲N 1.55 6.90 0.1303共8兲0.580共4兲6.6共2兲0.820共3兲N 1.60 7.12 0.1319共8兲0.587共4兲6.8共2兲0.826共4兲N 1.65 7.34 0.1339共8兲0.596共4兲7.0共2兲0.838共3兲N 1.70 7.57 0.1360共8兲0.605共4兲7.1共2兲0.855共3兲N 1.75 7.79 0.1370共8兲0.610共4兲7.3共2兲0.862共3兲N 1.80 8.01 0.1398共8兲0.622共4兲7.4共2兲0.880共4兲N 1.85 8.23 0.145共1兲0.644共5兲7.4共2兲0.915共3兲N 1.90 8.46 0.147共1兲0.655共5兲7.5共2兲0.920共2兲Sm A 1.95 8.68 0.151共1兲0.673共6兲7.5共2兲0.937共2兲Sm A 2.00 8.90 0.153共1兲0.681共5兲7.6共2兲0.940共3兲Sm A 2.10 9.35 0.158共1兲0.702共5兲7.9共2兲0.950共3兲Sm A 2938 J. Chem. Phys., Vol. 117, No. 6, 8 Augus 2002 Cue os e al. A. Phase diag am o he SWSC luid We ocus now on he discussion o he simula ion esul s o he SWSC luid. The iso he m ob ained by comp essing he sys em om he iso opic phase 共Table I and Fig. 2兲 e ol es smoo hly wi h inc easing p essu e up o P*⬇1.30. In his low p essu e in e al, bo h he densi y and he abso- lu e alue o he po en ial ene gy become p og essi ely la ge , while he o de pa ame e emains small (S⬍0.1) and oughly cons an . A P*⫽1.35–1.40, he numbe densi y is su icien ly high and close o he incipien iso opic–nema ic ansi ion as o induce luc ua ions in he sys em ha ake he o de pa ame e abo e 0.2. A he same ime, he in e nal ene gy app oaches an ex emum and s abilizes a U* ⬇⫺6.2. In ac , a P*⫽1.45 he o de pa ame e jumps o S⫽0.486 and en e s al eady he nema ic phase o he luid; his is he i s s a e included in he comp ession un ha displays long- ange o ien a ional o de , as moni o ed by g2( ). The densi y, in e nal ene gy and o de pa ame e o his i s nema ic s a e, as well as he beha io o he he mo- dynamic a iables in he p oximi y o he ansi ion, a e con- sis en wi h he esul o simila simula ions o Williamson and del Rı ´o6pe o med o e a la ge numbe o pa icles (NP⫽1020). Wi hin he nema ic phase, he u he inc ease o p es- su e and densi y o he SWSC luid induces a g ea e o ien- a ional o de ing which akes he o de pa ame e o S⬎0.7 o P*⬎1.55. A he same ime he po en ial ene gy becomes sligh ly less nega i e 共i.e., he ene gy inc eases兲. The densi y, ene gy and o de pa ame e main ain hei inc easing end wi h g owing P*, and a P*⫽1.70 he ansla ional o de ing cha ac e is ic o a smec ic A phase de elops and pe sis s in he emaining highe p essu e s a es included in ou s udy. No e idence o smec ic B o a solid phase o de wi hin each o he smec ic laye s was de ec ed, which is no unexpec ed, since o he HSC luid o same elonga ion he smec ic FIG. 2. MC-NPT esul s o he equa ion o s a e 共 op兲, o de pa ame e 共middle兲, and in e nal ene gy pe pa icle 共bo om兲 o he SWSC luid model. Solid ci cles and open iangles deno e he simula ion se ies un by comp essing and expanding he luid, espec i ely 共see ex 兲. The e ical dashed lines indica e he coexis ence densi ies a he iso opic–nema ic 共I–N兲and a he nema ic–smec ic A 共N–Sm A兲phase ansi ions, ob ained in he simula ions when expanding he luid 共see Table V兲. TABLE IV. Same as Table III, bu o a se o simula ions un by expanding he SRS luid om he mos dense s a e (P*⫽2.10). P*⫽P• ␴ 3/kT P*• HSC / ␴ 3 ␳ *⫽ ␳ • ␴ 3 ␩ ⫽ ␳ • HSC U*⫽U/␧SPhase 1.00 4.45 0.1025共6兲0.456共3兲4.8共1兲0.058共2兲I 1.10 4.90 0.1067共6兲0.475共3兲5.2共2兲0.096共2兲I 1.20 5.34 0.1119共8兲0.498共4兲5.5共2兲0.267共2兲I 1.25 5.56 0.1157共8兲0.515共4兲5.7共2兲0.504共2兲N 1.30 5.79 0.1195共8兲0.532共4兲5.8共2兲0.698共2兲N 1.35 6.01 0.1220共8兲0.543共4兲6.0共2兲0.722共2兲N 1.40 6.23 0.1240共8兲0.552共4兲6.2共2兲0.740共2兲N 1.50 6.68 0.1285共8兲0.572共4兲6.5共2兲0.802共2兲N 1.60 7.12 0.1323共8兲0.589共4兲6.8共2兲0.830共2兲N 1.65 7.34 0.1343共8兲0.598共4兲7.0共2兲0.823共3兲N 1.70 7.57 0.1371共8兲0.610共4兲7.1共2兲0.864共3兲N 1.75 7.79 0.1387共8兲0.617共4兲7.3共2兲0.879共3兲N 1.80 8.01 0.1431共8兲0.637共5兲7.4共2兲0.904共3兲Sm A 1.85 8.23 0.1465共8兲0.652共5兲7.5共2兲0.931共3兲Sm A 1.90 8.46 0.1492共8兲0.664共6兲7.7共2兲0.938共3兲Sm A 2.00 8.90 0.1557共8兲0.693共6兲7.8共2兲0.953共3兲Sm A 2.10 9.35 0.1577共8兲0.702共5兲7.9共2兲0.950共3兲Sm A 2939J. Chem. Phys., Vol. 117, No. 6, 8 Augus 2002 Liquid c ys al phases o ha d and so sphe ocylinde s A–solid ansi ion is obse ed a signi ican ly highe p es- su e (P*⬇2.5) and densi y ( ␳ *⬇0.138).3 As men ioned abo e, in o de o assess he s abili y o he di e en phases obse ed in he simula ions, an expan- sion un was pe o med om he smec ic s a e o highes p essu e (P*⫽2.20) back o he iso opic egion (P* ⫽1.00). The ele an da a conce ning he expansion simula- ion se a e lis ed in Table II and also shown in Fig. 2. As he sys em p essu e is dec eased in his expansion un, he luid unde goes he e e se sequence o phase ansi ions Sm A–N–I and a mode a e hys e esis is obse ed. The iso opic–nema ic and he nema ic–smec ic A phase bound- a ies a e in his case loca ed wi hin he in e als (P* ⫽1.30–1.35, ␳ *⫽0.0968–0.0993) and (P*⫽1.65–1.70, ␳ *⫽0.1099–0.1155), espec i ely 共see Table V兲, and, hence, appea sligh ly shi ed owa d smalle p essu es and/o densi ies in compa ison o he comp ession un dis- cussed in he p eceden pa ag aphs. Simila e ec s we e ob- se ed in he simula ions o Williamson and del Rı ´o o he iso opic–nema ic ansi ion o his same luid.6Since he equilib a ion o an o de ed s a e om an ini ial diso de ed con igu a ion is mo e demanding han he e e se case when pe o ming MC simula ions, we conclude ha , in he com- p ession un, he s a es o highe densi y wi hin he iso opic and nema ic phases a e ac ually me as able, i.e., uns able wi h espec o he mo e o de ed nema ic and smec ic phases, espec i ely. The e o e, when discussing he liquid c ys al- line beha io o he SWSC luid in he emaining o he pape , we will conside solely he esul s a ising om he simula ions o he expansion un. Figu e 3 ep esen s he adial unc ions g( ), g2( ), and g 储 ( 储 )共see abo e o de ini ions兲 o ele an s a es o he expansion MC un o he SWSC luid. I is ema kable, o ins ance, how g2( ) clea ly e lec s, o P*⭓1.35, he long ange o ien a ional o de ha di e en ia es quali a i ely he nema ic and smec ic phases om he iso opic one. On he o he hand, he smec ic o de ha cha ac e izes he s a es a P*⭓1.70 in he expansion un is clea ly obse ed no only in g 储 ( 储 ) bu also in he adial dis ibu ion unc ion g( ). As can be seen in Fig. 3, wi h g owing densi y all h ough he iso opic and nema ic phases, g( ) de elops p og essi ely mo e di e en ia ed maxima associa ed o he i s and second neighbo ing molecules 共a / ␴ ⬇1.1 and 2.3, espec i ely兲. Howe e , a e he nema ic–smec ic A ansi ion 共i.e., om P*⫽1.65 o P*⫽1.70兲, he s uc u e o g( ) changes quali- a i ely, wi h a sudden inc ease o he a ea o he i s , sec- ond, and e en hi d, nex -neighbo peaks, and he appea ance o a dep ession a / ␴ ⬇3.5–6.0, whe e g( ) s ays below uni y, as a consequence o he laye ed s uc u e o he luid. I mus be no ed, howe e , ha , due o he weak i s -o de cha ac e o he liquid c ys al ansi ions p esen ly s udied, and in spi e o he quali a i e changes unde gone by g( ), g2( ), and g 储 ( 储 ) a he I–N and N–Sm A phase ansi ions, i is no necessa ily s aigh o wa d o assign he bounda y s a es o each ansi ion in MC simula ions una oidably pe - o med o e a ini e numbe o pa icles. Fo ins ance, he assignmen o he N–Sm A ansi ion, based on he obse a- ion o an app eciable jump in densi y 共see Fig. 6兲accompa- nied by a simul aneous jump in he ampli ude o he laye ing maxima in g 储 ( 储 ) and in he s uc u e o g( ), lea es a weak onse o laye ing 关⬇1.2 ampli ude in g 储 ( 储 )兴in he s a e o highes densi y conside ed nema ic in ou s udy 共P*⫽1.65 in he SWSC luid, P*⫽1.75 in he SRS luid, see bo om o Figs. 3 and 5兲. This same e ec was no ed by McG o he e al. in hei s udy o he HSC luid.3I seems also imely o commen on he appa en mo e p onounced oscilla ions o g 储 ( 储 ) a he edge o he simula ion cell, whe eas he co ec s uc u e o his unc ion expec ed in he smec ic phase would be a sequence o maxima o he same heigh . How- e e , due o he limi ed numbe o pa icles employed in ou simula ions, in he compu a ion o g 储 ( 储 ) he i s neighbo - ing smec ic laye s a e no sampled as e icien ly as he cen- al laye , and hus he compu ed maxima a 兩 储 / ␴ 兩 ⬇6.5 a e oo na ow and o e es ima ed in heigh . Simula ions wi h a la ge numbe o pa icles 共and, hus, an inc eased size o he simula ion cell兲, as hose o Re . 3 o he HSC sys em wi h L*⫽5, la gely co ec o his e ec . Summa izing, he p essu es and coexis ence densi ies o he iso opic–nema ic and nema ic–smec ic A ansi ions o FIG. 3. Co ela ion unc ions o ep esen a i e s a es o he SWSC luid in he p esen s udy 共see Sec. II A o de ini ions兲. The sys em p essu e o each s a e is indica ed nex o he co esponding cu e. 共Top兲 adial dis ibu ion unc ion g( ); 共middle兲o ien a ional dis ibu ion unc ion g2( ) ⬅ 具 P2(cos ␪ ) 典 ;共bo om兲p ojec ion o he pai dis ibu ion unc ion a dis- ance 储 pa allel o he nema ic di ec o g 储 ( 储 ). Oscilla ions in his la e unc ion a e indica i e o laye ed smec ic o de ing in he luid. 2940 J. Chem. Phys., Vol. 117, No. 6, 8 Augus 2002 Cue os e al. he SWSC luid, as ob ained by a e aging he p essu es o he wo bounda y s a es o each phase ansi ion 共in he expansion MC un兲and om hei indi idual densi ies, a e PI–N *⫽1.325; ␳ I *⫽0.0968, ␳ N *⫽0.0993, and PN–Sm A * ⫽1.675; ␳ N *⫽0.1109, ␳ Sm A *⫽0.1155, espec i ely 共see Table V兲. The speci ic ole o he squa e well in he liquid c ys al beha io o he SWSC luid may be assessed by con as ing he p esen esul s wi h he MC-NPT simula ions o McG o he e al. o he ha d sphe ocylinde HSC luid.3We begin by ecalling he ema kable s abiliza ion o he in e nal ene gy o he SWSC luid a oughly cons an alues a ound U*⫽⫺6.0 a high densi y ( ␳ *⬎0.09) 共lowe panel o Fig. 2兲. Since such s abiliza ion akes place igh be o e he I–N ansi ion, i migh be en a i ely concluded ha he ene ge ic con ibu ion o he ee ene gy o he sys em ‘‘sa u a es’’ and has a negligible e ec in he liquid c ys alline beha io o he luid. In ac , such a sa u a ion e ec may be unde s ood as a di ec consequence o he squa e-well in e ac ion imposed in he model: a high enough densi ies all nea es -neighbo s a e a dis ance close han he SW ange and hus mo e eely inside he squa e well wi hou change o ene gy.19 Unde his in e p e a ion i ollows ha he SWSC luid should i ually esemble he liquid c ys alline beha io o he HSC luid. Howe e , we ind ha bo h he I–N and he N–Sm A phase ansi ions o he SWSC luid a e delayed owa d highe densi ies and p essu es wi h espec o he HSC luid. This can be seen in Fig. 6 which depic s in mo e de ail he equa ion o s a e o he HSC and he SWSC luids in he icini y o he I–N and he N–Sm A ansi ions. The co e- sponding ansi ion p essu es and densi ies a e compa ed in Table V. Fo he HSC sys em, he I–N and N–Sm A ansi- ions we e de e mined o be wi hin (PI–N *⫽1.19, ␳ I–N *⫽0.0914–0.0932) and (PN–SmA *⫽1.540, ␳ N–Sm A * ⫽0.1061–0.1095), espec i ely.3Thus, he ene ge ic con i- bu ion o he a ac i e well o he ee ene gy does seem o d i e pa ly hese o de ing ansi ions. The p esence o he squa e well imposes con igu a ional cons ains ha s abilize he iso opic phase wi h espec o he nema ic phase, and also his la e one wi h espec o he smec ic A phase. I ollows ha he ole o he in e nal ene gy opposes in his case ha o he main d i ing o ce o he liquid c ys alline ansi ions; he con igu a ional en opy, a magni ude ha is con olled by excluded olume e ec s 共 he only ele an e - ec o he HSC luid兲, mainly a compe i ion be ween he phase space accessible o he o a ion and ansla ion o he molecules. We inally no e ha o a bi a ily high empe a- u es 共i.e., T→⬁兲 he phase diag am o he SWSC sys em should end asymp o ically o ha o he HSC luid o same elonga ion. Hence, wi h inc easing empe a u e he loca ion o he liquid c ys al ansi ions o he o me a e expec ed o shi smoo hly owa d smalle p essu es and densi ies. B. Phase diag am o he SRS luid We concen a e now on he simula ion esul s o he SRS luid. The iso he ms ob ained when comp essing he luid om he iso opic phase 共Table III and Fig. 4兲o ex- panding he sys em back om he smec ic phase 共Table IV and Fig. 4兲display a sequence o iso opic–nema ic–smec ic A phases in ol ing he same quali a i e beha io in he el- e an s uc u e pa ame e s and co ela ion unc ions 共Fig. 5兲 as ha obse ed o he SWSC sys em. The simula ions o he SRS luid showed a weake hys e esis han he SWSC luid, yielding mo e simila esul s in he comp ession and expansion uns. Howe e , some deg ee o hys e esis is s ill app eciable and we will e e o he esul s o he expansion un in he ollowing discussion o he liquid c ys alline p op- e ies o he SRS luid. The e a e qui e signi ican quan i a i e di e ences be- ween he SRS sys em wi h espec o he SWSC and HSC models. As can be eadily obse ed in Figs. 4 and 6, o a gi en p essu e, he so co e o he SRS in e ac ion b ings he luid o conside ably la ge densi ies in compa ison o he HSC 共Re . 3兲and he SWSC luids. In addi ion, wi h espec o hese la e sys ems, he SRS luid p esen s I–N and N–Sm A ansi ions unde qui e di e en p essu e/ densi y condi ions, namely wi hin he in e als (PI–N *⫽1.20–1.25, ␳ I–N *⫽0.1119–0.1157) and (PN–Sm A * FIG. 4. MC-NPT esul s o he equa ion o s a e 共 op兲, o de pa ame e 共middle兲, and in e nal ene gy pe pa icle 共bo om兲 o he SRS luid model. The same no a ion as in Fig. 2 is used. The iso opic–nema ic 共I–N兲and he nama ic–smec ic A 共N–Sm A兲coexis ence densi ies a e indica ed by e i- cal dashed lines 共see Table V兲. No e ha , in spi e o he so co e o he SRS luid 关Eq. 共2兲兴, he packing ac ion gi en in he uppe axis 共 o a mo e di ec compa ison wi h ea lie wo ks兲is de ined wi h espec o HSC , he olume o a ha d sphe ocylinde o he same elonga ion L*⫽5. 2941J. Chem. Phys., Vol. 117, No. 6, 8 Augus 2002 Liquid c ys al phases o ha d and so sphe ocylinde s