Structure and dynamics of liquid crystals from computer simulation on simple and atomic-detail models

UNIVERSITÀ DI PISA

1 34 3

IN

S

U

P

R

EM

Æ DIG

N

IT

A

T

IS

Università Degli Studi di Pisa

S hool of graduate studies"Galileo Galilei"

PhD Thesis

Stru ture and dynami s of liquid rystals from

omputer simulation on simple and

atomi -detail models

Candidate Supervisor

LUCA DE GAETANI Prof. A. Tani

External advisor

I S ienti Ba kground 1

1 Computational Approa hes in Liquid Crystals 3

1.1 Introdu tion . . . 3 1.2 LiquidCrystals . . . 4 1.2.1 GeneralProperties . . . 4 1.2.2 Nemati Phase . . . 5 1.2.3 Sme ti Phases . . . 8 1.2.4 Some LCAppli ations . . . 9

1.3 Mole ularDynami s Simulation . . . 10

1.3.1 ComputationalApproa h inLiquid Crystal. . . 11

II Simulation of liquid rystals 13 2 Simulations: from models to atomisti simulations. 15 2.1 Model liquid rystals . . . 15

2.1.1 Introdu tion . . . 15

2.1.2 Resear h line . . . 16

2.1.3 Models . . . 16

2.2 The Adopted Model . . . 17

2.3 FRMFor e Field foratomisti simulations . . . 17

2.3.1 For e Fields . . . 17

2.3.2 Fragmentation-Re onstru tion Method . . . 20

2.3.3 Model Intermole ular Potentials . . . 21

2.3.4 Mole ularFlexibility . . . 22

2.3.5 FRM al ulations . . . 23

2.4 nCB . . . 24

3 Phase behaviour and stru ture 27 3.1 Introdu tion . . . 27

3.2 Model LC omputationaldetails . . . 27

3.4 Thermalbehaviourof the ModelLC . . . 29

3.5 5CBresults: phase behaviourand rystal . . . 34

3.6 nCBresults . . . 35

3.6.1 Equilibration . . . 35

3.6.2 8CB Sme ti Phase: analogiesand dieren es with the modelLC . 37 3.6.3 Thermodynami and stru tural properties . . . 39

3.7 Odd even ee t . . . 42

3.7.1 Chain onformations . . . 43

4 Parsons  Lee 49 4.1 Parsons Theory . . . 49

4.2 Results ofTheory . . . 53

5 Sixfold bond orientational properties 59 5.1 Introdu tion . . . 59

5.2 Computationaldetails . . . 60

5.3 Results and dis ussion . . . 62

6 Dynami s of Translational Degrees of Freedom 69 6.1 Introdu tion . . . 69

6.2 Model LC: Diusionand MAT theory. . . 70

6.2.1 Computationaldetails . . . 70

6.2.2 RelevantMAT Theory Expressions for Diusion . . . 71

6.3 Results and Dis ussion:diusion . . . 73

6.4 AnomalousDiusionin LiquidCrystals . . . 76

6.4.1 Introdu tion . . . 76

6.4.2 ComputationalDetails . . . 77

6.4.3 RelevantMCT results . . . 78

6.4.4 Results . . . 80

7 Sme ti Order Parameters from Diusion 87 7.1 Introdu tion . . . 87

7.2 MD simulations:test of the methodology . . . 89

7.3 NMR experiments: appli ationof the methodology . . . 91

8 Vis osity In Liquid Crystals 101 8.1 Vis osityRelationships in Nemati and Isotropi Phases . . . 101

8.2 RelevantMATtheory expressions for vis osity . . . 105

9 Orientational dynami s in the isotropi phase 115

9.1 Introdu tion . . . 115

9.2 ComputationalDetails . . . 118

9.3 Results and Dis ussion . . . 118

9.3.1 Singleparti leorientational relaxation . . . 121

9.3.2 Colle tive orientational relaxation . . . 127

9.4 5CBorientationalproperties . . . 130

9.4.1 Rotationaldiusion . . . 131

9.4.2 Diele tri relaxation . . . 131

10 Con lusions 139 10.1 Stru ture and Phase Behaviour . . . 139

10.1.1 Model LCphase behaviour . . . 140

10.1.2 nCB phasebehaviour . . . 141

10.1.3 Common hara teristi s for atomisti and modelsimulation . . . . 141

10.2 Diusion . . . 142

10.3 Vis osity . . . 144

10.4 OrientationalDynami s . . . 145

ABasi Orientational Properties in Liquid Crystals 147

BBasi Dynami and Reorientational Properties 149

Computational Approa hes in Liquid

Crystals

1.1 Introdu tion

The dieren e between rystals and liquids, the two most ommon ondensed matter

phases, is that mole ules in a rystal are ordered whereas in a liquid they are not. The

order in a rystal is usually both positional and orientational, in that the mole ules are

onstrainedtoo upy spe i sitesinalatti eandtopointtheirmole ularaxestospe i

dire tions: the enters of mass of their onstituting mole ules lie on the average on well

determined sites in a periodi al latti e. The mole ules in liquids, on the other hand,

dif-fuse randomly throughout the sample ontainer with the mole ular axes tumbling wildly.

The entropi fa tor isdominant: from athermodynami point ofview, the intermole ular

for esarenotsu ientto onstrainthemole ulesinalatti e,leavingthemfreetorandomly

translate and rotate. Interestingly enough, many phases with more order than present in

liquidsbutlessorderthantypi alof rystalsalsoexistinnature. Thesephasesare grouped

together and alled liquid rystals, sin e they share properties normally asso iated with

both liquidsand rystals, and their phasesare alsodened mesophases [17℄. Mesophases

maybeobtained intwodierentways. Firstthetranslationalorder anbe onned toone

ortwodimensionsratherthanthreedimensions,resultinginasme ti phase. Alternatively,

fornon-spheri almole ules, mole ularorientationsmaybe onsidered. Translationalorder

may be retained but the mole ularorientations be ome disordered, resulting in a plasti

rystal or the translational order may be lostand the orientational orderretained,

result-ing in nemati phase. A s hemati pi ture of these features is given in Fig. 1.1: it is

stressed that liquid rystals (LC) share order/disorder properties both with rystals and

liquidphases.

However, afull omprehension ofthe relationshipbetween the mole ularstru tureand

the resulting ma ros opi material properties is far from being rea hed. In parti ular, in

Figure 1.1: S hemati representation of liquid rystalline ordering properties relating to

rystal and liquid ones.

Figure1.2: Example for a rod-like mole ules: p-axozyanisole(PAA).

1.2 Liquid Crystals

1.2.1 General Properties

A number of dierent typeof mole ules form liquid rystal phases. What they allhave in

ommonisthat they are anisotropi . Either their shape issu h thatone mole ularaxis is

verydierentfromtheothertwoorinsome asesdierentpartsofthemole uleshavevery

dierentsolubilityproperties,see forexample thePAAmole uleinFig.1.2. Ineither ase,

the intera tions between theseanisotropi mole ulespromoteorientationalandsometimes

positional order in anotherwise uid phase. Shaped mole ules (i.e. one mole ularaxis is

mu h longerthat the othertwo)forms alamiti LCphases whiledisk-likemole ules form

theso alleddis oti LCphases. Itisimportantthatthemole ulebefairlyrigidforaleast

some portion of itslength, sin e it must maintain anelongated shape inorder toprodu e

intera tions that favoralignment.

Figure1.3: Generi mole ularstru ture for a liquid rystallinemoiety.

1.3 weshowhowageneri mesogen is onstituted. A ertainrigidityisrequiredtoprovide

the anisotropi mole ularstru ture, whi h is a hieved by linearly linked ring systems (A

andB),whi hmaybe onne teddire tly,orjoinedby alinkinggroup(Y)whi hmaintains

the linearityof the entral ore. This ore isnot usuallysu ient to generate aLCphase

and a ertain degree of exibility is often present, to ensure low melting points and to

help stabilizethe mole ular alignment within the mesophase stru ture. This exibility is

provided by terminal substituents (R and R'), whi h are usually alkyl or alkoxy hains;

however, one terminalunit maybeasmallpolarsubstituent. Theseterminalunitsmaybe

joineddire tlytothe entral oreorlinkedviagroupsXandZ.Thelateralsubstituents(M

andN),whilstgenerallydetrimentaltothe formationofLCphases,areused tomodifythe

mesophasemorphologyandthe physi alpropertiesof LCstogenerateenhan edproperties

for appli ations.

LC phasesallshow atleast orientationalorder asso iatedeventuallytosome extentof

positionalorder. As aresultof orientationalorder,most physi alproperties ofliquid

rys-talsare anisotropi andmust bedes ribed by se ondranktensors. Examplesare the heat

diusion, the magneti sus eptibility, the diele tri permittivity or opti al birefringen e.

Additionally, there are new physi al qualities, whi h do not appear in simple liquids as

e.g. elasti orfri tional torques (rotational vis osity) a tingon stati ordynami dire tor

deformations, respe tively.

For example, whenviewed under ami ros ope using apolarizedlightsour e, dierent

liquid rystal phases will appear to have a distin t texture, Fig.1.4. Ea h "pat h" in

the texture orresponds to a domain where the LC mole ules are oriented in a dierent

dire tion. Within a domain,however, the mole ules are wellordered.

1.2.2 Nemati Phase

The nemati phase of alamiti liquid rystals is the simplest liquid rystal phase. In

liquid phases mole ules don't show long range translational order while they exhibit long

range orientational order, as in rystals. In this phase mole ules maintain a preferred

orientational dire tion of their axes, Fig. 1.5: this dire tion is alled dire tor and is

indi ated by

n

b

. An orientational distribution fun tion

f (θ)dθ

an be dened by either taking a snapshot of the mole ules at any one time and noting how many axes make an

angle between

θ

and

θ + dθ

with the dire tor or by following a single mole ule, noting at spe iedtimestheangle

θ

between themole ularaxisandthedire tor. Sin ealldire tions

Figure1.4: S hlieren texture of LiquidCrystal nemati phase.

Figure 1.6: Nemati and ChiralNemati Phase.

perpendi ular to the dire tor are equivalent in the most simple liquid rystal phase, the

orientationaldistributionfun tiondoesnot dependonthe azimuthalangle

φ

. The amount of orientational order in su h a liquid rystal phase is measured by an order parameter.

This an be dened inmany ways, but the most useful formulation is to nd the average

of the se ond Legendre polynomial, whi h is insensitive to head-tail inversion (see also

Appendix A):

P

2

= hP

2

(cosθ)i = h

3

2

cos

2

θ − 0.5i

(1.1)

P

2

= 1

forperfe tlyorientedmole ules whileitis0inthe isotropi phase. Ifthemole ules that forma a liquid rystal phase are hiral (la k inversion symmetry), then hiral phases

existinpla eof ertainnon- hiralphases. In alamiti liquid rystals,thenemati phaseis

repla ed by the hiral nemati phase, inwhi hthe dire torrotatesinheli alfashionabout

anaxis perpendi ular to the dire tor. Su ha phase is illustrated inFig.1.6.

The pit hof a hiralnemati phase isthe distan e alongthe helix over whi hthe dire tor

rotates by

360

◦

. It should be noted, however, that the stru ture repeats itself every half

pit h due to the equivalen y of

n

b

and

−

n

b

. Interesting opti al ee ts o ur when the wavelengthoflightinthe liquid rystal isequaltothe pit h. Thepit hofa hiralnemati

phase an be as short as 100nm. Mixing the two opti al isomers in various proportions

allows the pit h to be in reased from the pit h of either of the two pure opti al isomers.

A ra emi mixture possesses aninnite pit hand istherefore nemati . Finally, the hiral

(a) (b)

Figure1.7: On the left, panela), sme ti layer modulation. Inpanel b), representation of

sme ti A and C.

1.2.3 Sme ti Phases

In sme ti A (SmA) LC, mole ules groups together in weak planes. There is a mono

dimensional order due to the presen e of aligned planes and a long range orientational

order with the presen e of the nemati dire tor

n

b

; within the planes, however, mole ules have a liquid-like disorder and they an be onsider as a two-dimensional uid. Let us

denote the normal to the layers as the

z

-axis. The density of the enters of mass

ρ(z)

is sinusoidally modulated, as an be seen in Fig.1.7, around a quasi- onstant interlayer

distan e:

ρ(z) = ρ

0

+

X

n

ρ

n

cos(qz − φ

n

)

(1.2)

where

ρ

0

is a onstant,

q = 2π/d

,

d

is the interlayerdistan e and

φ

n

are amplitudes of density harmoni s.

InaSmAphasethemole ularlong-axispointsinadire tionnormaltotheplaneswhile

in a sme ti C (SmC) it is tilted with respe t to the planes. Other sme ti mesophases

[10℄ are mu h less ommonly en ountered that the SmA and SmC phases whi h is, to

some extent, aree tion onthe relativeemphasis ofresear h onSmAand SmCmaterials,

espe ially for ferroele tri devi es. The SmB liquid rystal phase an exhibit quasi-long

rangesixfoldbondorientationalorder,(seeChap.5): whiletranslationalorderislostwithin

afewintermole ulardistan es,orientationalorderde aysasapowerlaw. Thisfeature an

be learly seen inFig.1.8;in ontrast tothe rystal B phase where thereis true positional

order,inSmBmole ulesarepositionallymispla ed. AnalogouslytoSmAandSmCphases,

tiltedphasesofsmBare alledSmFandSmIandmole ules aretiltedtowards thesideand

apex of the hexagonal latti e respe tively (dire tion of tilt is depi ted by the triangular

mole ules in Fig.1.8). The stru ture of this pseudo-hexagonal lose pa ked unit ell for

Figure1.8: S hemati representation of liquid rystalline most diuse phases.

no interlayer orrelations, short-range in-layer orrelations, and quasi long-range bond

orientational order, with a orrelated dire tion of tilt. The plasti rystal mesophases,

G

and

J

, shown in Fig.1.8, are highly orrelated analogues of the

S

B

,

S

F

and

S

I

phases respe tively, with the repeat positionalorder being predi table over a long range inthree

dimensions [12℄.

1.2.4 Some LC Appli ations

The most remarkable features of LC with respe t to appli ations are due mainly totheir

anisotropi opti al properties [8,13℄. Sin e these properties are arried by a uid, soft

material, they therefore are extremely sensitive to external perturbations. This latter

hara teristi has allowed LCte hnology to have a major ee t onmany areas of s ien e,

engineering, and devi e te hnology.

As new properties and types of liquid rystals are investigated and resear hed, these

ma-terialsare sure togain in reasingimportan einindustrialand s ienti appli ations. The

main appli ationsof LC an be summarized insome examples:

i) LiquidCrystal Displays (LCD)

DuetoLCsensitivitytoexternalperturbations,orientationalorder anbeeasily

manip-ulated leadingto impressive magneto-opti al, ele tro-opti alopto-opti al (asthose shown

in Fig.1.4)ee ts. The most su essful LC appli ation are liquid rystals displays [14,15℄

ii) Opti alImaging

A rather new and promising appli ation of LC is opti al imaging and re ording [16℄.

In this te hnology, the LC material ondu tivity is in reased by the light that intera ts

with aphoto ondu torin whi hLC ellispla ed. This ondu tivity augmentation auses

an ele tri eld to develop in the LC ell with patterns dependent on the light intensity.

These an be transmittedby anele trode, enablingthe image re ording.

iii)Liquid Crystal Thermometers

Sin e in the hiral phase light is ree ted with a wavelength equal to the pit h of the

dire tor's helix, and the latter is dependent upon temperature, holesteri LC make it

possible to gauge temperature by looking at the olor of the thermometer. Moreover, by

mixing dierent mesogeni hiral substan es, a devi e for almost any temperature range

an be built. In this way, LC temperature sensors an be used in dierent elds, going

fromthe medi alappli ations(resear h oftumoraltissues)[8℄tothe ele troni ones

(indi-viduationof bad onne tionson a ir uit board).

iv) Other Appli ations

LCexhibitamultitudeofotherusesin ludingthevisualizationofradiofrequen ywaves

in waveguides, medi al appli ation in orthopedy, erasable opti al disks [17℄, or photoni

swit hing appli ations[18℄.

1.3 Mole ular Dynami s Simulation

Computer simulationsin ondensed matter physi s aim at al ulating stru tural and

dy-nami properties from atomisti input [1922℄. The theoreti al basis of this approa h is

statisti al me hani s. The on eptually simplest approa h is the lassi al mole ular

dy-nami s(MD)method[2326℄: onesimplysolvesnumeri allyNewton'sequationsofmotion

for the intera ting many-parti le system (atoms or mole ules intera ting, e.g., with pair

potentials).

The basis of the method therefore is nothing but lassi al me hani s, and one reates a

deterministi traje toryinthephasespa eofthe system. The ideaistotaketimeaverages

ofthe observables ofinterestalong thistraje tory,andrelyonthe ergodi ityhypothesisof

statisti al me hani s, whi h assumes that time averages are equivalent to ensemble

aver-ages of the appropriate mi ro anoni al (

NV E

) ensemble. Of ourse, Newton's equations of motion onserve the total energy

E

, and hen e the onjugate intensive thermodynami variables su h as temperature

T

and density and pressure an only be inferred indire tly andexhibitu tuations(sin ethe parti lenumber

N

isniteand fairlysmallforasample thatshouldmimi ama ros opi sample,su hu tuationsmustnotbenegle tedandneed

areful onsideration).

Sometimes it is advantageous to dire tly realize other ensembles of statisti al me hani s,

en-semble, and this is indeed feasible by introdu ing a ouplingto appropriate 'thermostats'

or 'barostats'. An alternative way of arrying out an MD simulation at onstant

tem-perature is possible by introdu ingan arti ial weak fri tion for e, together with random

for es whose strengths are ontrolled by the u tuation-dissipation theorem. Su h

te h-niquesare very ommone.g. forthe simulationofpolymermelts [27℄. This methodisvery

losely related in spirit to sto hasti simulation methods su h as 'Brownian dynami s'

where one simulates a Langevin equation(the inertialterm in the equation of motion

be-ingomitted). Whiledynami al orrelationsforsu hmethodsdiersomewhatfromstri tly

mi ro anoni alMD methods, forthe omputationof stati propertiesfromaveragesalong

the sto hasti traje toryin phase spa e su h methods an be advantageous.

Thisstatementisalsotruefortheimportan esamplingMonteCarlo(MC)method[28℄.

As iswellknown [22℄,MC samplingmeansthat one reatesarandom-walk-liketraje tory

in onguration spa e, ontrolled by transition probabilities that ensure the approa h to

thermalequilibriumvia a detailedbalan e ondition.

Many of the pra ti al aspe ts of omputer simulations, su h as 'statisti alerrors' and

systemati errors due tothe nite size of the simulatedsystem orthe nite 'length'of the

simulatedtraje tory (orobservationtime, respe tively), areshared by allthese simulation

methods.

It mightbe observed that it is quantum me hani s that des ribes the basi physi s of

ondensed matter, and not lassi al me hani s. Attempting a numeri al solution of the

S hr

o

¨

edingerequationforasystem ofmany nu leiand ele trons isstillpremature andnot at all feasible even on the fastest omputers. Thus, one has to resort to approximations.

One very popular approa h is the 'ab initio MD' or Car-Parrinello method [29℄, where

one takes some ele troni degrees of freedom via density fun tional theory (DFT) [30,31℄.

The huge advantage of this te hnique is that one no longer relies on ee tive interatomi

potentials, whi h often are only phenomenologi ally hosen ad ho assumptions, la king

any rm quantum hemi al foundation. However, the disadvantage of this te hnique is

that itis several orders of magnitudeslower than lassi alMD, and hen e onlyveryshort

times alesandverysmallsystemsarea essible. Furthermore,themethodisunsuitableto

treatsystems withvander Waals-likefor es, su hasinraregases, whereoneis stillbetter

owith the simple Lennard-Jones(LJ)potential(perhapsamended by three-bodyfor es)

[19℄. Also, normally ioni degrees of freedom are still treated lassi ally. Alternatively,

one an still use ee tive potentials between ions and/or neutral atoms as in lassi al

MD or MC, but rely on quantum statisti al me hani s for the ioni degrees of freedom:

this is a hieved by path integral Monte Carlo (PIMC) [32℄ or path integral mole ular

dynami s (PIMD) [3335℄. Su h te hniques are indeed ru ial fora study of solids at low

temperatures,toensure thattheir thermalproperties are ompatiblewith the thirdlawof

thermodynami s.

1.3.1 Computational Approa h in Liquid Crystal

izes their dynami s[5,6℄. The latterfeature, ombined withthe omplex natureof typi al

liquid rystal forming mole ules, has suggested to adopt rather simplied des riptions of

the intermole ular intera tions. After the Lebwohl-Lasher latti e model [37℄, where even

translational freedom was missing, anisotropi intera tion models have been onsidered,

either withhard [38℄ or ontinuous potentialfun tions [39℄,the most widely employed

be-ing the Gay-Berne model [4042℄. In all these ases, mole ules are onsidered single-site

intera tion enters and no mole ular exibility is taken into a ount. Despite their

sim-pli ity, these models have proven valuable to study both the general stru ture-property

relationships and the basi features responsible of the liquid rystal behavior. However,

their simpli ity be omes a drawba k when the interest fo uses ona spe i liquid rystal

mole ule, with a well dened hemi al stru ture. A tually, along with the impressive

in- reaseof omputational apabilities,therehasbeen agrowinginterest,inthelasttenyears

or so [4354℄, in omputer simulations of liquid rystals with models of higher, physi al,

realism.

Thus, the problembe omestwofold: rst the study of simpliedmodels an help to

indi-viduate the general behavior of LC and, se ond, atomisti simulations turn out tobe

in-dispensableto at h hemi alspe i ity. In the latter ases,parti ular attention is pla ed

in the onstru tion of potentialfun tions able to a urately des ribe the intera tions

be-tween large mole ules together with the pe uliar features of mole ular exibility. This is

a parti ularlyhard task onsidering that the omplex phase behavior of mesogeni

mate-rials is the result of a deli ate balan e between subtle energeti and entropi ee ts [52℄.

In fa t, seemingly modest hanges of hemi al stru ture an lead tosigni antly dierent

positional and orientational organization in a given phase and hen e to largely dierent

ma ros opi properties. This means that, in order to keep hemi al spe i ity, it is not

possible toseparateweak attra tiveintera tionsfromex ludedvolumerepulsivefor es, as

Simulations: from models to atomisti

simulations.

2.1 Model liquid rystals

2.1.1 Introdu tion

Studying the ondensed matter eld involves dealing with a huge number of degrees of

freedomandwitha omplexbehaviourduetotheinterplaybetweenfa torsas,forexample,

ex ludedvolume, ele tostati intera tions, steri hindran e et . In this polyhedri alworld

of entangledee ts the adoptionof models an help tounderstand the physi sunderlying

phenomena exhibited by liquid rystals. Infa t it is possible to evaluate the role of ea h

spe i ontribution: for example, oulombi for es an easily be "swit hed o" so that

short-rangefor es(steri ee ts)be omethemainfeaturedrivingthedynami s;moreover,

itispossibletofollowthepathfromrigidstru tureto ompletelyexibleonesonly hanging

the for e eld. The main advantages of simulating simple models of mole ules an be

summarized asfollows

(i) basi physi s - On e deprived of spe i mole ular details, the nature of

funda-mentalaspe ts be omes leares,although quantitativepredi tions for real mesogens

propertiesbe omeimpossible

(ii) testing theories - Mole ular uids are too omplex to be studied theoreti ally

in full detail and some approximations are needed. On e simpli ations are made,

theoreti al predi tions an be testedby onvenient simulations

(iii) omputational spread - Models require less omputational eorts and make it

possible tostudy larger systems forlonger times

(iv) loss of spe i ity - Loosingthe hemi alspe i ity has as a main drawba k the

Itisworth notingthatthedeepinsightgiven bymodels anbe onsideredasastarting

point for the study of systems in their full omplexity: even with in reasingly powerful

omputational resour es, modelization still maintains its pre ious role in apturing the

main features of systems under study.

2.1.2 Resear h line

Ourrstmodelmesogenwasmadeofbeadne kla e"mole ules"whosephasediagramwas

rst studied. Then olle tiveandsingle mole uletranslationalandorientationaldynami s

was investigated.

From thetheoreti al pointof view we tested Parsons Lee DFT approa h for the phase

behavior and modied ane theory (MAT) for the dynami s. The latter was exploited

to develop a semiquantitative methodology for extra ting sme ti parameters

τ

K

from diusion data.

2.1.3 Models

The MC and Mole ular Dynami s (MD) te hniques, though usually omputationally

ex-pensive, oera unique tooltoperform numeri alexperimentsthat provideexa t results

for an ensemble of parti les intera ting through the input potential, against whi h the

predi tions of related theories an be tested. Hard ellipsoids and sphero ylinders proved

ee tivein onrmingtheoldOnsager's onje turethatliquid rystallinityismainlydueto

anex ludedvolumeee t[38℄,whileanisotropi softpairpotentialmodelsoftheGay-Berne

typehave shed light onthe ombined role of repulsive and attra tiveintera tions [41℄.

Among the numerous models of rodlike liquid rystal mole ules proposed and studied

during the last de ades, a parti ular attention has been re ently pointed towards those

omposed of spheri al beads, linked together to give idealized elongated parti les. These

models an be further dierentiated on the basis of the nature of the spheri al units, of

their onne tivity,and of the degreeof exibility of the overall hain.

For example, a number of hard spheres, either tangent or fused with a bond length

smaller than the diameter, an be held rigidly along a linear array. The onstraint of

rigidity an bethenremoved stepbystep, allowingasystemati investigationoftheroleof

exibility. Alotofworkhas beendoneonsu hmodelsbyboththeoryandsimulation[56℄

[57℄. Mostsimulationshavebeen arriedout viathe MonteCarlo(MC) [19℄ method,with

theinterestdire tedbasi allytothephasebehaviorandphasetransitionsinvolvingordered

uids.

Besides hard spheres, also soft spheri al beads have been adopted, either repulsive

or possessing an additionalattra tive tail. In Ref. [58℄ the phase behavior of a system of

N=600rigidrodlikeparti les,ea h omposedof11sitesandsite-siteintera tionsdes ribed

byapotentialofthe

r

−12

form,wassket hedbyMDsimulation. Thisstudywasextendedin

Figure2.1: The site-site liquid rystal model.

attra tiveintera tionswas onsidered. Elongatedliquid rystalmodels madeup oflinearly

onne tedfourLennard-Jonessiteshavebeen investigatedinRef.[61℄and inRef.[62℄;the

author of the latter omments on the possibility of using these models to study liquid

rystalsinpolymermatri es. Stiandsemiexiblemodelsofthis typewere alsostudiedin

Ref.[63℄. Theadvantageofusingsoft orebeadsisthatMDsimulationsarenotte hni ally

demanding as in the ase of hard body parti les. In these studies, the interest was again

almost ex lusively fo used onstati properties, notwithstandingthe possibility oered by

the MD te hnique toaddress dynami albehavior.

2.2 The Adopted Model

The adopted model parti leis very simple: it is omposed of 9 fused soft spheri al beads

(see Fig. 2.1). The site-site pair intera tion is equalto the repulsivepart of the

Lennard-Jones potentialin the Weeks-Chandler-Andersen (WCA) [64℄ separation, that is:

U(r

ij

) =















4ε



σ

rij



12

−



σ

rij



6

+

1

4



r

ij

≤2

1

6

_σ

0 r

ij

≥2

1

6

_σ

.

(2.1)

The site-sitebondlengthhas been kept equalto0.6

σ

sothatthe lengthtowidthratio,

κ

, ofea hmole uleis

≈

6. Wehaveassumed

ε

=6.0

·10

−22

Jand

σ = 3.9

Å .Ea h site has a mass m= 15

·

1.67

·10

−24

g. We have investigated these models in the NVE,NPT ensemble;

the number of parti les adopted and the onditions will be spe ied for ea h ase. The

number of mole ules ranged from 600 to 21600, the latter huge systems are eemploied to

study the soelusive hexati phase.

2.3 FRM For e Field for atomisti simulations

2.3.1 For e Fields

In omputer simulationstudies the stru turaland dynami ma ros opi properties of

sys-temsare derived fromtheadoptionofamodelpotential,whi h ontainsthe des riptionof

the mole ular framework and intera tions. The strong dependen e of LC phase stability

fromthemole ularfeatures[5,8,9℄makestheunderstandingofthelinkbetweenmi ros opi

a omplex interplaybetween energeti ee ts (as the mole ularintera tions: ele trostati ,

dispersive and indu tive for es) and entropi ones (like positional, orientational and

on-formationaldistributions). Fromathermodynami pointofview, thebalan eamongthese

freeenergytermsis riti alforthe(meso-)phasestability. Smallvariationsinthemole ular

framework an alter this deli ateequilibrium,ae ting signi antly the phase diagramof

the system. Thus atomisti simulations an yield valuable information about the role of

the variousmi ros opi ontributions; they are hara terizedby the following features:

(i)spe i ity -Unlikesimulationsonsimpliedmodels, anatomisti simulationaims

toreprodu e the behavior of aparti ular mole uleor phase

(ii)predi tivity -Ifasuitablefor eeld (eventuallyab-initio developed )isadopted,

simulations aninsome extentgivepredi tions ontransitiontemperature, transport

properties,phase behavior et .

(iii)experimental ounterpart -dierentfor e elds(FF) maybeemployed in

atom-isti simulationsto reprodu e real experimentalndings

(iv) physi al insights - The knowledge of position, velo ities and for e at ea h time

stepmakespossibletoevaluate hemi alandphysi alpropertiesotherwise

unobtain-able

(v) expensive simulation - The prin ipal drawba k of atomisti simulations is the

huge amount of omputer time needed to treat thousands of sites thus restraining

the time to span some tens of nanose onds

As a matter of fa t, it is in the FF spe i ation that the mi ros opi intera tions are

introdu ed in the simulation method, and the hemi al identity of the mole ule nds a

orresponden e in the bulk observables. In parti ular, in LC eld the abovementioned

deli ate interplay between the for es governing the mesogeni properties alls for very

spe i mole ularmodels. This poses some doubts on the reliability of a straightforward

adoption of the most widely employed FF's [6568℄, sin e their extension to large LC

forming mole ules, an be done only invoking a high degree of transferability. Indeed,

it has been pointed out that small dieren es in the mole ular stru ture may produ e

impressive variations in the ma ros opi properties, so that transferability must be used

with aution.

In prin iple,theCar-Parrinellomethod[29,69℄hasthe potentialityofsolvingthe

prob-lem. However, itsa tualimplementationappearsproblemati forsystemswheredispersion

for esare dominantanditsappli ationtolargemole ularsystems andlong-timedynami s

remainsa future obje tive. The same reasonsrule out a dire tappli ation of the ab initio

Born-Oppenheimer Mole ular Dynami s to mesogeni mole ules.

In this ontext, it has been proposed [7073℄ a s heme whi h ouples rst prin iple

al ulations with bulksimulations. The idea of onstru ting new FF's fromrst prin iple

properties in experimental onditions. This ab initio approa h is based on modeling the

FFonthebaseofanumberofQuantumMe hani al(QM) al ulations,performedonaset

of representative intermole ular geometries. Owing to the high omputational request of

a urate QM al ulations, this alternative s heme has been rarely used [7480℄, although

itpresentsseveral advantages. First, these ab initio derived (ABD) FF'sare suited tothe

hosen mole ule and take into a ount its hemi al identity: no degree of transferability

must be invoked forsu h intera tion potentials. Moreover, the ab initio omputed

poten-tial energy surfa e (PES) onstitutes a referen e database for parameterization of model

potentialsof dierent omplexity. This possibility appears to be of parti ular importan e

for LC's, as it allows one totune the "realism"of the modelon the set of properties that

should be reprodu ed. In some ases, e.g. where a full atomi (FA) representation is

omputationallytooexpensive,simpliedmodelsare tobeused[38,42,72,8184℄. Finally,

sin eABDFF'sarenotdrivenbyexperimentaldata,simulationmodels anbe onstru ted

even for mole ules whose properties are di ult tomeasure, givinginprin iplepredi tive

apabilities tothe whole approa h.

The major drawba k of this route toparameterization isthat the ab initio al ulation

of a urate intermole ular energies of dimers, through standard QM te hniques, be omes

qui kly unfeasible when the mole ular dimensions in rease. To ir umvent the problem,

the Fragmentation-Re onstru tion Method (FRM) was proposed [70,71℄ whi h allows us

toa urately ompute the intera tion PES of dimers of large mole ules, making use of ab

initio al ulations. Thebasi ideabehindFRMisthatmanylargemole ules anbethought

of as omposed of a rather small number of moieties, e.g. phenyl rings and hydro arbon

hains, whi h ontribute separately to the two body mole ular intera tion energy. The

smallersize (and possibly highersymmetry)of the latterfragmentsmakesthemamenable

toa urate ab initio al ulationsat areasonable omputational ost.

A rational s hemeto implementthe FRM for the study of ondensed phase properties

an be summarized asfollows.

1) FRM: al ulation of intermole ular (and intramole ular) potentials with quantum

me hani almethods

2) Fitting: parameterization of the omputed energies through model potential

fun -tions suitablefor omputer simulations

3) Simulation: MD or MC simulations and omparison of the resulting ma ros opi

propertieswith the relevant experimental data

This s heme was tested for a sele ted group of mole ules of in reasing omplexity.

Sat-isfa torily results have been obtained for benzene [85,86℄, the series of p-n-oligophenyls

(nPh) [72,82,84℄ and 4-n-alkyl, 4' yano-biphenyl (nCB) [71,73,8790℄. The benzene

mole ule was hosen sin e its dimer onstitutes a demanding test for ab initio

al ula-tionsof theintera tion energyandhas been extensively studiedfromatheoreti alpointof

view[68,7476,9198℄. Moreover, arealisti modelwith fullatomi detail an behandled

repro-two homologues series nPh and nCB. nPh's oer unique advantages as their stru ture is

very simple, with just one type of fragment (the phenyl ring) and basi ally a single kind

of internal oordinate, the torsionalinter ring dihedral. The higher members of the series

are able to form mesophases, su h as a nemati phase for the p-quinquephenyl and even

a sme ti phase for the six ring member. The nCB series is a lassi alben hmark for LC

theoreti al studies [43,46,52,53,99105℄, both for itssimpli ity and for the abundan e of

experimental data [100,106116℄. Longer hain members present a fairly ri h

polymor-phism at room temperature, with sme ti (for

n ≥

8) as well as nemati phases (for

n ≥

5). For these properties nCB's nd dire t appli ations in material s ien e like thin lms

and liquid rystal displays [117,118℄.

2.3.2 Fragmentation-Re onstru tion Method

For the al ulations of the intera tion potential of large mole ules, through the use of

standard QMprograms, the FRM [70,71℄ has been developed by resorting to an old

sug-gestion of Claverie [119℄, who proposed approximating the intera tion energy as a sum of

atom-atom intera tions. This approa h relies on the assumption that the intera tion

en-ergyof adimer an beapproximated toagooda ura yas asum of energy ontributions

between ea h pairs of fragments into whi h the two mole ules an be de omposed. The

basi riterionbehindthisfragmentations hemeisthatthe groundstateele troni density

aroundthe atoms of ea h fragmenthas tobe as lose as possible tothat aroundthe same

atoms in the whole mole ule. The main advantage of this approa h liesin the possibility

of performing al ulationsbetween moietiesmu hsmallerthan the wholemole ules under

study. Thisallowsustoin ludeele troni orrelationee ts andtoobtainagoodestimate

of the dispersion energy whi h isexpe ted to a ountfor a largefra tion of the attra tive

intermole ular energy for most of the mole ules that an form mesophases. The

informa-tionsgainedabout theintera tionenergybetweenthe fragmentspairs arethen usedtoset

up the fullintermole ularenergy.

The rst step of the FRM approa h is a de omposition of the whole mole ule into

fragmentsby utting properly hosen singlebonds. The valen eof theresultingfragments

is then saturated by suitable "intruder"atoms or smallgroups. This allows usto express

the intermole ular energy as a sum of ontributions of all resulting pair of fragments.

Obviously, the intruder groups have to be subsequently an eled from the mole ule and

their energy ontributions properly subtra ted in order to re over the total intera tion

energy between the two original mole ules.

Bywayof example,let's onsiderthe yano-biphenyl (0CB)mole uleCNC

6

H

4

C

6

H

5

. The 0CB mole ule may be split into yano-phenyl and benzene fragments through a ut

alongthering-ringbondandthensaturatedwithhydrogenatoms. Thusthewholemole ule

an be formallywritten as

CNC

6

H

4

-C

6

H

5

=CNC

6

H

4

-H

a

+ H

b

-C

6

H

5

 H

b

-H

a

where the two intruder atoms H

a

and H

b

are rst in luded to saturate the resulting fragmentsandthenremovedasaH

a

H

b

mole ule. Itisworthnoti ing,asshowninFig. 2.2,

thatthe spatialpositionofthe fragmentsis un hangedwith respe t tothe wholemole ule

and that the lo ationof the "intruder"atoms H

a

and H

b

is unambiguously determinedby theinternalgeometryofthesaturatedfragments. Inthe 0CB ase,thisresultsinaslightly

altered bond distan e of the "intruder" H

2

mole ule (0.68 Å, instead of the equilibrium value of 0.74 Å). This fragmentationpath an besummarized by

0CB= CB +B H

2

,

where CB= yano-phenyl and B=benzene, and leads to the following expression for the

total FRMintera tion energy of the dimer

E

_0CB−0CB

F RM

= E

CB−CB

+ E

B−B

+ E

CB−B

+ E

B−CB

+ E

H

2

−H

2

−

−E

CB−H

2

− E

H

2

−CB

− E

B−H

2

− E

H

2

−B

.

Here,

E

X−Y

(X,Y=CB,B,H

2

)isthe omputedintera tionenergybetweenfragment

X

of the rst 0CBmole uleand fragment

Y

ofthe se ond. It an be easilyveried that,if N is thenumberofele tronsofthe0CB,theaboveexpression orre tlyin ludesN

2

intera tions

for intermole ularenergy. Thismethodwas su essively reproposed by Zhanget al.[120℄,

who appliedit to study intera tions between mole ules of biologi alinterest.

2.3.3 Model Intermole ular Potentials

The FRM PES,

E

F RM

(R,

Ω

), is sampled for a large number of dimer onformations, identied by the distan e R between the mole ular enters of mass and the three Euler

angles

Ω

. Theresultingenergydatabaseisthenttedontoamodelpotentialfun tion

U

(R,

=

+

+

+

+

−

+

+

+

Figure 2.2: Appli ation of the FRM s heme to the 0CB dimer. The intera tion energy of the

0CB dimerin a hosen onguration (left side) is omputed as a sum of the fragment-fragment

Ω

) suitablefor omputer simulations. The parameters P of the intermole ular fun tions, that hara terize the hosen model, are obtained from aleast square tting pro edure, by

the minimization of the fun tional

I =

P

N

geom

k=1

w

k

(E

k

F RM

− U

k

[

P

])

2

P

N

geom

k=1

w

k

(2.2)

where

N

geom

is the number of geometries onsidered,

w

k

an appropriate weight at the geometry

k ≡

[R,

Ω

℄.

The hoi eofthefun tionalformof

U

(R,

Ω

)isdriven bya ompromisebetweenmodel a ura y and omputational onvenien e.

TheFAmodel[71,73,86,88℄(FullAtomi )representsana uratepotentialfor omputer

simulationsandisobtainedby onsideringea hatomofthemole uleasanintera tingsite.

This atomisti model anbeoftensimpliedby assemblingtogether groups ofatoms (e.g.

methyl or methylene) into single intera tion sites. The atomisti FA model potential,

adopted tot the FRM energies of a dimer A

· · ·

B, an be expressed as a sum of site-site ontributions:

U

FA

(R, Ω) =

N

A

X

i

N

B

X

j

u

ij

(R)

(2.3)

where

N

A

and

N

B

are the number of intera tion sites of mole ule A and B respe tively, while

u

ij

istheintermole ularenergy ontributionofsites

i

and

j

. Thesite-sitefun tionis a modied form of the 12-6 Lennard-Jones (LJ) potential,plus the standard ele trostati

intera tion, namely

u

ij

= 4ǫ

ij



_ξ

ij

σ

ij

r

ij

+ σ

ij

(ξ

ij

− 1)



12

−



_ξ

ij

σ

ij

r

ij

+ σ

ij

(ξ

ij

− 1)



6



+

q

i

q

j

r

ij

(2.4)

The parameter

ξ

is introdu ed in the LJ ontribution, to allow the well width to vary independently from its depth and position, thus improving the model exibility and, in

parti ular,itsabilityofrepresenting orre tlytheshapeofthelowenergyrepulsivebran h.

2.3.4 Mole ular Flexibility

A majoradvantage a hieved by in reasing the levelof omplexity fromsinglesite tomulti

site models is the possibility of taking mole ularexibility into a ount. In MC and MD

simulations this is usually done by assuming the energy as the sum of two de oupled

ontributions, namely inter- and intra-mole ular terms,i.e.

E(r

A

, r

B

; R, Ω) = E

inter

(r

A

, r

B

; R, Ω, ) + E

intra

(r

A

) + E

intra

(r

B

)

(2.5) where

r

A

and

r

B

olle t the internal oordinates of mole ules A and B, respe tively. The adopted intramole ularpotentialhas the AMBER [67℄ form

where the terms inthe sum have the expressions:

E

stretch

=

N

_X

bonds

i

k

_i

s

(r

i

− r

0

i

)

2

; E

bend

=

N

angles

_X

i

k

b

_i

(θ

i

− θ

0

i

)

2

E

tors

=

N

dihedrals

_X

i

N cos

_X

i

j=1

k

_ij

d

[1 + cos(n

i

_j

δ

i

− γ

j

i

)] ; E

LJintra

=

N

LJ intra

_X

i

X

j=i+4,i+5,...

u

LJ

_ij

where

u

LJ

ij

has the formof the standard LJ potential.

The set of parameters des ribing the intramole ularpart of the potential an be taken

from literature FF data [6567℄ or re-parameterized with the aid of single mole ule QM

al ulations. In the following, the intramole ular parameters have been taken from the

AMBERfor e-eld,ex eptforthoseinvolvinginternal oordinatesofparti ularimportan e

in determining the overall mole ular shape as, for instan e, the ore- hain dihedral angle

in nCB.

2.3.5 FRM al ulations

In view of the very large number of geometries to be onsidered for the omputation

of the intera tion energy of allpairs arising from the fragmentation s heme, a reasonable

ompromisebetweena ura yand omputational osthastobefound. Inordertoproperly

onsiderthedispersionenergy,wehave hosentheMöller-Plessetse ondorderperturbation

theory (MP2) in the supermole ule approa h. Methods like extensive implementationsof

the onguration intera tion or oupled luster with perturbative orre tions have been

dis arded be ause onsidered too expensive for the present s ope. On the other hand, no

reliable DFT fun tional has been yet found for Van der Waals intera tions. All energies

were omputed using the ounterpoise orre tion s heme, to take are of the basis set

superposition energy error (BSSE) [121℄. Up to date, although often questioned, this is

a epted as the standard orre tion [122℄ for the in ompleteness of the basis set. The

employed basis set is the standard 6-31G* basis set modied for the low exponent of the

d Gaussian Type Orbitals entered on the Carbon and Nitrogen atoms:

α

d

=0.25 versus usual values of 0.7-0.9, asoriginally proposed by Hobza et al. [94℄.

The geometry of ea h mole ule in the dimer was taken frozen in all binding energy

al ulations. Theoptimizationofthemonomer geometrywasperformedby thewelltested

density fun tional B3LYP method [123℄ with triple-zeta basis set 6-311G(2d,p). The

ob-tainedgeometries were pra ti ally oin identwith the experimentalones andno

appre ia-ble dieren es in the binding energy of the dimer was noti ed [70,71,73,86℄ by using the

2.4 nCB

In the LC eld, we hoose the nCB family of mole ules be ause they have been widely

studied [116,125132℄ so that experimental results of several quantities are available for

omparison.

Moreoverthe nCBfamilyaxhibits anoddeven ee t in many properties, inparti ular

forthe learingtemperaturewhi h o urs inanarrowrange of T, sothatitsreprodu tion

represents a hallenge for the whole modelingapproa h.

The FRM methodology was rst implemented for the 5CB mole ule, the rst in its

family that shows liquid rystalline mesophases, Fig.2.3.

Cb2

H3

C3

Cp2

Cp3

Cp2

Cp2

Cp1

Cb4

Cn

N

C1

C2 H2

C3

C2

H3

Hb2

Cb3

Cb3

Hb3

C4

Cb2

Hb3

Hb2

H2

Cb1

Figure2.3: Modelfor the5CB mole ule. Aliphati hydrogensare onsideredtogether withea h

aliphati arbon as single intera tion sites. Extension to higher homologues has been obtained

by addinga number of Cp2sites to thealiphati hain. Thedashed line indi ates the nCB ore

main axis.

The fragmentation of the 5CB mole ule (CN-C

6

H

4

-C

6

H

4

-C

5

H

11

) was performed by a very natural hoi e of the fragments:

CN-C

6

H

4

-H

a

(BCN) ; H

b

-C

6

H

4

-H

c

(B) ; H

d

-C

5

H

11

(P)

where two C-C single bonds in the 5CB are substituted by C-H bonds in the isolated

fragments. This partition retains the losed shell nature of the aromati rings as well

as of the aliphati hain and makes possible to ompute a urate intermole ular energies

between the three saturated fragments.

TheFRMPESwastted[71℄withaFApotentialsuitablefor omputersimulations,se .

2.3.3. In this model, the methyl and methylene groups of the hain are both onsidered

as a single site, thus dropping out all the aliphati hydrogen atoms. On the ontrary,

all aromati hydrogens were expli ity onsidered. Some onstraints were imposed to the

ttingparameters inorder totakeintoa ount equivalentsites and, inparti ular, togain

transferability to longer homologues: the three inner methylene groups were treated as

equivalent sites with

q

CH

2

= 0. The resulting parameters are given in Ref. [71℄, together with some details of the ttingpro edure.

Somese tionsoftheresultingFAPES are omparedinFig. 2.4withthe orresponding

Figure2.4: FRM(solidline)andttedFAmodel(dashedline)energy urvesforthe5CBdimer.

Thefa eto fa eand side byside arrangements are onsidered.

planeofthe yano-phenylring. Thisdimerenergypresentsalo alminimumof-40kJ/mol

at about 4.2 Å, the main sour e of attra tive ontributions arising from the intera tions

between the aromati rings. When the translation ve tor lies perpendi ular to both the

normal to the yano-phenyl ring and the 5CB long axis (side by side arrangements), the

resulting urvehasashallowenergyminimumatmu hhigherdistan es. Asimilara ura y

was rea hed in allthe onsidered geometries [71℄, being the ttingstandard deviationless

than 2.5 kJ/mol.

The intramole ular part of the 5CB FF was onstru ted [73℄ adopting AMBER [67℄

parametersfor the transferableinternaldegrees offreedom (asbendinganglesoraromati

ringdihedrals) andby derivingthosemorespe i (asthe ore- haindihedralorthe

inter-ring angle) fromsingle mole uleDFT al ulations.

The model 5CB has been here transferred tothe higher homologuesthrough the

addi-tion of one, two or three Cp2 intera tion sites, for 6CB, 7CB and 8CB, respe tively: the

straightforward onstru tionof nCB homologues an be inferred with the help of Fig.2.3.

Theextensionoftheintermole ularpartoftheFF anbeeasilya hieved sin eallCp2sites

of5CBwere onsideredasequivalent,thuspossessingthe sameintermole ularparameters.

Moreover, sin e ea h Cp2 was onsidered hargeless, the addition of a Cp2 site does not

alter theele tro-neutralityof themole ule. Con erningthe intramole ularpart,the

bend-ing and torsional parameters used to des ribe the exibility of the 5CB aliphati hain,

an benowused tomimi the newdegrees of freedomresultingfromthe hain elongation.

To prevent a non physi al urling of the hain on the aromati ore, LJ intramole ular

intera tion were added[73℄ in the 5CBmole ule between the lastthree sites of the pentyl

moiety and the aromati Cbn (n=1-4) arbons. Su h intera tions have been kept for all

Phase behaviour and stru ture

3.1 Introdu tion

Sofarwereportedresults frinsimulationsonsimpleand atomi detailmodelstoelu idate

themi ros opi stru tureofLC's. ItturnedoutthatthemodelLC an at hmanygeneral

features exhibited by the nCB homologue family studied by atomisti simulations. This

nding supports the idea that the model adopted an be of help extending the natural

barriers in whi h atomisti simulations are trapped due to their omputational osts.

In-fa t, examining pair orrelation fun tions in both kind of systems, a similar stru ture in

isotropi and nemati phasesturns out,determining thatthe short rangestru ture is

sub-stantiallyunaltered by the phase hange. Moreover it ould be possible to enlargewidely

the modelLCsystem toobservethe rystallinevs hexati behaviourstudyingaverylarge

ensembleofparti lesotherwisenota essible(i.e. by atomisti simulations)byourpresent

omputationalresour es.

On the other hand there are features not amenable to an analysis based on simple

model data, notably the dynami s of internal degrees of freedom and their relationswith

the phasediagram ofthe system. An exampleis the hain behaviourinthe nCb series for

whi hadetailed onformationaland stru turalanalysis isne essary togive insighrsonits

role on the transitionfroman ordered toa disordered phase.

Both systems havebeen analyzedthrough orderparameters and pair orrelation

fun -tions: the more general ones are explained in Appendix A and B, while the others are

introdu ed inthe body of the text.

3.2 Model LC omputational details

We have onsidered a system of N=600 parti les des ribed in Se . 2.1.3. The system

has been simulated by MD in the isothermal  isobari ensemble with a timestep of 5fs.

The Nose'-Hoover thermostat [133135℄ has been used to onstrain average temperature

of the omputational box to u tuate, thus permitting the system to rea h the imposed

value of the average pressure. The parti le beads are maintained in the onguration

des ribed in se tion 2.1.3 by means of the method of onstraints reviewed in Ref. [137℄.

Wehave onsidered twopressures, 1.0and2.5Kbarand several temperatures inthe range

[50 − 375]

K,for the lower pressure, and

[300 − 700]

Kfor P=2.5Kbar.

At the lowest temperatures, we havestarted the simulation froma perfe t rystal in a

hexagonal lose pa ked onguration, stret hed along the x oordinate of the laboratory

frame and with all parti les parallel to this dire tion. The resulting omputational box

wasorthorhombi ,withthe following ellve torsandangles: a=

(a

x

, 0, 0)

,b=

(0, b

y

, 0)

and =

(0, c

y

, c

z

)

;

α = 90

◦

,

β = 90

◦

and

γ = 60

◦

. The z-axis is of the order of 140 Å while x

and y axes of the order of 40 Å. The initial onguration of the other temperatures has

beenthenalone ofthe equilibrationrunofthe respe tiveprevioustemperature. Toavoid

unphysi alrotationofthesimulationbox,ahasbeen onstrainedtolieonx andbtomove

in the xy plane.

For every state point onsidered, a thermalization run of 500 ps has been followed by

a produ tionrun of 1ns (atsele ted state pointswe haveextended the time interval

sim-ulatedup to 2ns, to he k the stability of ourresults, whi hhas proved very satisfa tory)

duringwhi haverages ofthermodynami alandstru tural propertiesof interest have been

a quired.

3.3 Atomisti simulation details

All MD simulations were arried out in the NPT ensemble with a parallel version of a

modied [73℄ Mos ito3.9 [138℄ pa kage. Temperature and pressure were kept onstant

using the weak oupling s heme of Berendsen et al. [139℄, allowing the aspe t ratio of the

simulation elltovaryduringsimulationruns. Duringallequilibrationandprodu tionruns

the bond lengths were kept xed at their equilibrium value using the SHAKE algorithm

[140℄ and a timestep of 1 fs was used. The short range intermole ular intera tions have

been trun ated at R

c

= 10 Å, employing standard orre tions for energy and virial [19℄. Charge- harge long range intera tions were treated with the parti lemesh Ewald (PME)

method [141,142℄, using a onvergen e parameter

α

of 5.36/2R

c

and a

4

th

order spline

interpolation.

The onstru tion and equilibration of the 5CB system diers from the other nCB

ho-mologues. Infa t, FRMwas rst deeply tested on 5CB from rystalline toisotropi phase

throughthe nemati phase[73℄;on eitsabilitytoreprodu ethe mainmi ros opi features

and stru ture of su h phases was assessed, the FRM potential was extended to the other

nCB homologues. In the latter ase, no rystallinestru ture was built, the interest being

3.3.1 nCB simulation details

Starting ongurations were prepared as follos. First, a omplete optimization of ea h

mole ulargeometrywas performed through quantum me hani al al ulations, arriedout

with the density fun tional B3LYP method [123℄ using a polarized triple

ζ

6-311G(2d,p) basisset. Afterremovingthealiphati Hydrogenatomsfromea hoptimized onformation,

an antiparalleldimer arrangement was reated. The nCB dimers were repli ated 3 and 4

times along the X and Y laboratory axis, respe tively, to obtain a layer of 48 mole ules.

Finally, a low density, orientationally ordered (

<

P

1

>

= 0;

<

P

2

>

= 1) latti e was on-stru ted by repli ating ea h layer 4 times along the Z axis. This way, we obtained 192

nCB mole ules for a total of more than 5000 intera tion sites. These low density

stru -tures were used as starting ongurations for short, high pressure (P= 1000 atm) runs,

performed atT =100 K. The nal ongurations were all hara terized by high densities

(

>

1.1 g/ m

3

),highorder parameters(

<

P

2

> >

0.65) andsome degreeofpositionalorder along the phase dire tor. These systems were then equilibrated at atmospheri pressure

and dierenttemperatures.

The internal onformation of allhomologues atdierent temperatures has been

moni-tored, withparti ular attention tothe torsionaldihedralsand tothe elongation(L)of the

aliphati hain . The latter has been evaluatedas the distan e between the Cb4 site (see

Fig. 2.3) and the proje tion of the Cp3 site on the long mole ular axis. The distan e of

the Cp3 site from the long axis, L

⊥

,allows us to ompute the ratio

R

L

= L/L

⊥

. Their mean values, averaged on all mole ules, will be indi ated as

<

L

>

,

<

L

⊥

>

and

<

R

L

>

, respe tively.

Anotherusefultooltodes ribethemole ular'shape'istheaspe tratioofthe moments

of inertia,whi h is onne ted to the mole ularaspe t ratio:

κ

mol

=

(I

1

· I

2

)

1

2

I

3

(3.1)

Here

I

1

and

I

2

are the two largest and intermediate eigenvalues of the mole ular inertia tensor and

I

3

the minimum eigenvalue asso iated tothe major axisof the mole ules, that areassumedrodlike. Finally,theinertiatensoranisotropy(

∆

I)was omputedassuggested by Ref. [54℄, that is:

∆I =

(I

1

+ I

2

)/2 − I

3

I

1

+ I

2

+ I

3

(3.2)

3.4 Thermal behaviour of the Model LC

Webeginthepresentationofour resultsbyreportinginTab. 3.1the valuesofthe

thermo-dynami alquantitiesenteringtheequationofstate,i.e. pressure,temperatureanddensity.

The last olumn ontains the symbol of the orresponding typeof phase.

P (Kbar)

∆

P (Kbar) T (K)

∆

T (K) 1000

ρ

0

(Å

−3

) 1000

∆ρ

0

(Å

−3

) phase 1.0 0.1 50 5 2.88 0.04 C 1.0 0.1 100 5 2.73 0.03 C 1.0 0.05 150 6 2.53 0.02 C 1.0 0.05 165 5 2.47 0.04 C 1.0 0.03 175 5 2.35 0.02

S

A

1.0 0.01 200 2 2.09 0.01 N 1.0 0.03 250 4 1.97 0.01 N 1.0 0.05 275 8 1.89 0.02 N/I 1.0 0.07 300 6 1.81 0.04 I 1.0 0.05 325 9 1.78 0.02 I 1.0 0.07 350 9 1.74 0.03 I 1.0 0.07 375 8 1.71 0.03 I 2.5 0.08 300 7 2.79 0.01 C 2.5 0.09 350 9 2.71 0.01 C 2.5 0.09 375 10 2.67 0.015 C 2.5 0.09 400 10 2.50 0.015

S

A

2.5 0.08 435 12 2.32 0.01 N 2.5 0.08 450 11 2.30 0.01 N 2.5 0.07 500 11 2.23 0.01 N 2.5 0.09 550 14 2.17 0.015 N 2.5 0.08 600 15 2.08 0.015 N/I 2.5 0.09 625 16 2.05 0.015 I 2.5 0.09 650 17 2.02 0.015 I 2.5 0.1 675 18 2.00 0.015 I 2.5 0.1 700 18 1.98 0.015 I

Table 3.1: Cal ulated values forthe quantities entering the equation ofstate of the model

0

0.2

0.4

0.6

0.8

1

Ψ

0

0.2

0.4

0.6

0.8

1

τ

300

350

400

450

500

550

600

650

700

T(K)

0

0.2

0.4

0.6

0.8

1

P

6

2

(a)

(b)

(c)

Figure3.1: (a): Bond Orientational(

Ψ

6

);(b) sme ti (

τ

);( ) nemati (

P

2

)order parame-ters, as afun tion of temperatureatP=2.5 Kbar.

slightanddevelop onlyinthelong rangelimit. InChap. 5we willfo usonhexati phases

and show someee tive toolstoanalyze B phases and the how todis riminate between a

rystal and a sme ti phase. Infa t, a simple analysis of

g(r)

and of

Ψ

6

(see Eq.3.3) isn't enoughto a urately assess the natureof su h phases.

We dene the rystalline/hexati phase as C phases. At P=1.0 Kbar a C phase is

observed up to T=165K,while atP=2.5 Kbarit ispresent up to T=375K.The Cphase

then undergoes a transition to a sme ti A(

S

A

) phase, between T=165 K and T=175 K, atthe lower pressure, and between T=375K and T=400 K,atthe higher pressure.

Werst inferredthe type ofthe phasesfromthe analysisof variousorder paramateres.

The thermal behavior of the bond orientational order parameter,

Ψ

6

, and of the sme ti order parameter,

τ

, shown in Fig.3.1(a)and (b), respe tively, fora pressure of 2.5Kbar.

The former is dened as:

Ψ

6

=

1

3

N













X

i

X

j∈nn(i)

exp (

i

6ϑ

ij

)













,

(3.3)

where

j ∈ nn(i)

meansthat

j

isa nearestneighborof

i

and

ϑ

ij

is the angle formed by the proje tion of r

ij

onto a plane normal to the dire tor with an axis in this plane. By onstru tion, we have

Ψ

6

= 1

in the ase of perfe t in plane hexagonal order and

Ψ

6

= 0 when that oder is ompletely absent, as in the sme ti or nemati phases. Intermediate

values of

Ψ

6

orrespond toaredu ed inplane hexagonalorder,asit happens inareal h p rystal. Whileat T= 400K

Ψ

6

= 0, the sme ti order parameter is far from being zero.

τ

is dened as:

τ =











exp



i

2π

z

d







,

(3.4)

It is 0 for non-layered phases (i.e. nemati and isotropi phases) while it is

6= 0

where planes are present (sme ti s, rystals). It vanishes at the next temperature examined at

P=2.5 Kbar, that is T=435 K, and at T=175 K at the lower pressure investigated. This

is a lear symptom that a transition to a positionally homogeneous phase has o urred,

as the featureless behavior of the pair orrelation fun tions further testies. This liquid

phase is to be lassied as nemati (N), being non zero the orientationalorder parameter

P

2

(Fig.3.1( ), for P=2.5 Kbar). The latter is ade reasing fun tion of temperature, with a dis ontinuity at T

≃

600 K, being lose to zero beyond. We an assume 600 K as the approximate temperature at whi h a nemati to isotropi (I) phase transitiontakespla e

atP=2.5Kbar. Analogously, theNI phase transitiono urs atT

≃

275 Kfor apressure of 1 Kbar.

We note that the sme ti region is quite narrow, if ompared with that exhibited by

hardsphero ylindersoflengthtowidthratioequalto6[143℄. Inthe latter aseweobserve

the same sequen e of phases, that is: C (2.34)

S

A

(1.44) N (1.12) I. We have indi ated in parentheses the pressure in redu ed unit,

P

∗

₌

PD

3

/

k

B

T (D is the diameter of the sphero ylinder), of the orresponding phase transition. To make a omparison, we an

hoose forour parti lesavalue of D su hthat the NI phase transitiono urs atthe same

redu ed pressure as in the ase of sphero ylinders, thus obtaining: C (1.78)

S

A

(1.68) N (1.12) I. Sme ti A phase seems to be destabilized with respe t to both the rystalline

phase and the nemati phase.

The on lusions drawn fromthe analysis of the pair orrelation fun tionsare

orrobo-rated by the analysis of various orderparameters.

InFig. 3.2we anseethehighstru tured

g(r)

atT=300Kinthe rystalphase. Therst peaka ountsforthesixrstneighborswhilethese ondandthirdpeaksarerepresentative

of the h p pa king of mole ules. Note alsothat

g(r)

is very lowbetween peaks, indi ating that pra ti allythe mole ules keep their positionin the rystal.

The existen e of the C-

S

A

phase transition is indi ated by the qualitatively dierent behavior of the layer pair orrelation fun tion,

g

l

(r)

Eq.A-3, together with the qualita-tivelysimilarbehavioroftheparallelpair orrelationfun tion,

g

||

(r)

Eq.A-1,see Appendix A. They are shown in Fig.3.3 (a) and (b), respe tively,at temperatures of T=375K and

T=400K, for apressure of P=2.5Kbar.

In

g

l

(r) ℓ

isthelengthofaparti le,hereassumedtobe22.6Å.

g

l

(r)

givesinformationon thepositionalorderwithinalayer: ina rystallinephaseitpresentsalongrangestru ture,

Figure3.2:

g(r)

(bla k line) and

g

l

(r)

atT=300K in the Crystal phase.

0

1

2

3

4

5

g (r)

T=375 K

T=400 K

0

5

10

15

20

25

30

r

0

0.5

1

1.5

2

2.5

3

g (r)

(A)

°

||

l

(a)

(b)

Figure 3.3: The layer (a) and parallel (b) pair orrelation fun tions on either side of the

inFig.3.2. The rst peak orresponds toinplane nearestneighbors, whilethe positionsof

the two still resolved su essive peaks are the mark of an inplane hexagonal order.

g

l

(r)

at T= 375 K keeps os illatingat large distan es where the fun tionat T= 400 Kalready

approa hesthelimitingvalueofunity. Thisprovestheliquidlike hara terofthepositional

order insidea layerof the phase atT= 400 K.The latter isdenitely lassiedas sme ti

fromthe behaviorof the parallelpair orrelation fun tion of Fig. 3.3 (b).

Atboth T=400 Kand T= 375 K,aswellas atlowertemperatures,

g

||

(r)

,whi hgives information on positional order along the dire tor, shows an os illating stru ture with

peaks being lower and wider asthe temperatureis in reased, and separated by a distan e

of 22-23 Å.

3.5 5CB results: phase behaviour and rystal

The ellparameters(a,band ), thedensity(

ρ

)of the rystal phaseare reportedinTable 3.2together with the experimental values [108℄. and the equilibrationtime (

t

) of the run.

T (K)

t

(ns)

ρ

(g/ m

3

) a(Å) b (Å) (Å) 200 1.5 1.229

±

0.0013 8.26 15.39 10.58 250 1.5 1.207

±

0.0016 8.33 15.44 10.65 280 13.5 1.194

±

0.0021 8.38 15.45 10.69 253 [108℄ - 1.15 8.25 16.02 10.93

Table 3.2: Cal ulatedand experimental[108℄ rystaldata at severaltemperatures.

At 280 K, the system was equilibrated for 13.5 ns and the omputed pair orrelation

fun tionsare reported inFig. 3.4together with thosederived fromthe experimental[108℄

X-ray dira tion rystal data (see Ref.[73℄ for details). Nomajordieren e results,being

the simulated stru ture ingoodagreement with the experimentalfun tionat short range,

despite overestimatingof the rystal density by about 5%.

Upper panel of Fig 3.4 an be ompared with

g(r)

in Fig. 3.2: both fun tions exhibit high peaksand very lowminima and are stilldierent from1in the wholerange inwhi h

they are al ulated. Dieren es arise only in the rystal type, whi h in the h p of model

LC displays a double peak between 7.4 and 8.6 Å. Lower panel of Fig 3.4 is instead

qualitatively in ex ellent a ord with the lower one in Fig. 3.2: they both represent the

layering ofmole ules inthe rystalphase and asubstantiallyfeatureless urve fornemati

orisotropi phases.

Atransitionfromthisorderedphasetoapositionallydisordered,orientationallyordered

nemati phase o urs by heating at

∼

290K. To assess its stability and to verify the ergodi ityof the simulations, threedierentstarting stru tures were reated, namely0.75

(run A), 0.52 (run B) and 0.43 (run C), ea h with a dierent degree of orientational

order but equal values of pressure and temperature. This kind of study was previously