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Theory of self-assembly-driven nematic liquidcrystals revised

C. De Michele

To cite this article: C. De Michele (2019): Theory of self-assembly-driven nematic liquid crystalsrevised, Liquid Crystals, DOI: 10.1080/02678292.2019.1645366

To link to this article: https://doi.org/10.1080/02678292.2019.1645366

Published online: 27 Aug 2019.

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INVITED ARTICLE

Theory of self-assembly-driven nematic liquid crystals revisedC. De Michele

Dipartimento di Fisica, “Sapienza” Università di Roma, Rome, Italy

ABSTRACTConcentrated solutions of short blunt-ended DNA duplexes at room temperature can formliquid crystal phases due to stacking interactions between duplex terminals which induce theaggregation of the duplexes into semi-flexible linear chains. Mesophases observed in thesesystems include nematic, columnar and cholesteric ones. This experimental system is just oneof many examples, where liquid crystals ordering emerges as a result of molecular self-assembly into linear chains. In the attempt to go beyond a simple Onsager theoreticalapproach to understand the thermodynamic behavior of this system, we introduced someyears ago a general theoretical description, which models the isotropic-nematic transition byproperly taking into account molecular self-assembly, and we carefully verified the theoreticalpredictions against numerical simulations of patchy hard cylinders. Here, we provide a revisedversion of the theory in the attempt of understanding which assumptions are worth to beimproved. In particular, we focus on the Parsons-Lee approximation and the modeling oforientational entropy. We compare the results from the revised version of the theory againstoriginal ones, showing that the present version of the theory is able to capture moreaccurately the phase boundaries of the isotropic-nematic transition.

ARTICLE HISTORYReceived 1 March 2019

KEYWORDSTheory; lyotropic liquidcrystals; self-assembly; phasediagram; monte carlosimulations

1. Introduction

The spontaneous formation, through free energyminimization of reversible aggregates from basicbuilding blocks, is called self-assembly. Since build-ing block size can vary from few angstroms tomicrons, self-assembly is rather ubiquitous in nature.In addition, the interactions between building blockscan be conveniently tuned, so that through self-

assembly new materials with controlled physicalproperties can be designed. It is for these reasonsthat self-assembly is of interest in several fields, suchas material science, soft matter and biophysics [1–3].

A specific, but nonetheless less relevant, self-assemblyprocess is the formation of filamentous semi-flexibleaggregates induced by the anisotropy of attractive inter-actions between the constituent building blocks.

CONTACT C. De Michele cristiano.demichele@uniroma1.it

LIQUID CRYSTALShttps://doi.org/10.1080/02678292.2019.1645366

© 2019 Informa UK Limited, trading as Taylor & Francis Group

Examples are provided by micellar systems [4–6], for-mation of fibers and fibrils [7–10], solutions of short[11–18] and long B-DNA [19–22], chromonics [23–29]and inorganic nanoparticles [30–33].

In these systems liquid crystal (LC) phases build upabove a critical concentration, if the filamentous aggre-gates are sufficiently stiff. In order to grasp a physicalunderstanding of this complex behavior, building on thevenerable Onsager theory, we developed some years agoa novel theoretical approach for these self-assembly-driven LCs. Noticeably, our theory contains no adjustablenor fitting parameters. Predictions for the isotropic-nematic transition have been carefully tested throughcomputer simulations for polymerizing hard cylinders[34], patchy cylinder-like superquadrics [35], a realisticDNA coarse-grained model [36], patchy bent cylinders[37], bifunctional spheres [38] and patchy disks [39]. Thetheoretical treatment has been also extended to the calcu-lations of chiral strength and elastic constants [39,40]. Inthis theoretical treatment, the contribution of higherorder virial terms to free energy has been accounted forby the Parsons-Lee (PL) decoupling approximation,while the modeling of the orientational contribution tothe free energy depends on a constant (i.e. independent ofthermodynamic state point) parameter which discrimi-nates the regimes of rigid and fully flexible chains.

In this manuscript, we discuss further these twoapproximations used in the theory and we suggesta possible way to improve them. The PL approximationis not accurate at moderate and large packing fractions,since in the form used in the original theoretical treat-ment it has been used without taking into account theeffective volume occupied by a chain, which is largerthan the total volume occupied by its constituentmonomers. Concerning the constant parameter whichacts as threshold separating stiff and fully flexiblechains, in Ref [41]. it has been observed that it shoulddepend on the orientational ordering if one wants toexplain some experimental findings on the phase beha-vior of a suspension of short amyloid fibrils. In Ref[41]. it has been also noted that this parameter shouldbe comparable with, rather than the persistence length,the deflection length, as defined and discussed in Refs[42,43]. In the present manuscript a first attempt to gobeyond these two approximations will be provided,which should be intended as a guide for further futureimprovements of the theory.

In Sec. 2 we will illustrate briefly the patchy cylindermodel which will be used to test the theoretical predic-tions. Sec. 3 provides an account of the theoreticaltreatment where we discuss a possible way to improvethe approximations discussed so far. In Sec. 4 we showthe theoretical predictions of the novel theoretical

treatment and we compare them with the results forpatchy hard cylinder reported in Ref [34]. Finally, inSec. 5 the conclusions will be drawn.

2. Model

For testing the revised version of the theory developedin Refs [35]. we consider the same model adopted inRef [34]., which will be illustrated briefly in the follow-ing. We consider hard cylinders (HC) of length L anddiameter D, which are complemented with two attrac-tive sites on their bases (Figure 1).

These two attractive sites are placed along the symme-try axis of the HC at a distance L=2þ 0:15D=2 from itscenter of mass, so that no branching can occur in thesystem. Attractive sites which belong to distinct HCsinteract via the following square-well (SW) potential uSW :

βuSW ¼ �βu0 if r � δ0 if r > δ

�(1)

where r is the distance between the interacting sites,δ ¼ 0:25D is the interaction range, βu0 is the ratiobetween the binding energy u0 and the thermal energykBT ¼ β�1, with kB the Boltzmann constant. It is con-venient to define an adimensional temperatureT� ¼ kBT=u0, which will be used in the following.Note that u0 is assumed to be independent of aggregatesize, which is equivalent to assume that the self-

Figure 1. (Colour online) Patchy hard cylinder model used inthe theoretical calculations.D andL diameter and length ofcylinders respectively. The centers of the two small yellowspheres placed on the bases of the cylinders are the sitesinteracting via a square-well potential. The diameter of theyellow spheres coincides with the interaction range (i.e. thewell width), while the depth of the well u0 is the bindingenergy.

2 C. DE MICHELE

assembly process is isodesmic [44,45].

3. Theory

In the systems, which we are considering in the presentstudy, particles aggregate to form chains of variouslengths. Hence, the system is a polydisperse mixtureof semi-flexible polymers, which continuously changetheir lengths, by merging with other polymers orbreaking of the constituting bonds. The original theo-retical approach developed in Ref [35]. followed thework of van der Schoot and Cates [6,46] and its exten-sion to higher volume fractions with the use of theParsons-Lee approximation [47,48] as suggested byKuriabova et al. [49]. For a polydisperse mixture ofself-assembling linear aggregates, the Helmholtz freeenergy F can be naturally expressed as a functional ofthe number density of aggregates νðlÞ, where l is thelength (or number of monomers) of the aggregate towhich a monomer belongs. The function νðlÞ has toobey the normalization conditionX

l

lνðlÞ ¼ ρ (2)

where ρ ¼ N=V , with N the number of monomers andV the volume, is the number density of monomers. Inthe following we assume an exponential aggregatelength distribution, i.e.

νðlÞ ¼ ρMðl�1Þ=ðM � 1Þðlþ1Þ (3)

with M the average number of monomers in the aggre-gate, i.e.

M ¼P1

1 lρðlÞP11 ρðlÞ : (4)

The Helmholtz free energy comprises the followingcontributions [34,35,40]:

F ¼ F id þ F st þ F or þ F excl (5)

In this equation Fid is the free energy of an ideal gascomposed of a polydisperse mixture of chains withchain length distribution νðlÞ, where vd ¼ LD2π=4 isthe volume of a monomer. This ideal gas contributioncan be written in terms of the chain length distributionνðlÞ as follows:

F id ¼ V β�1X1l¼1

νðlÞ ln vdνðlÞ½ � � 1f g: (6)

The term, Fst, is the stacking free energy, whichaccounts for monomer aggregation. Expressed interms of the chain length distribution νðlÞ it takes thefollowing form:

Fst ¼ V β�1ΔðTÞX1l¼1

ðl� 1ÞνðlÞ (7)

where ΔðTÞ is the bonding free energy [35,36], i.e.

ΔðTÞ ¼ �ðβu0 þ σbÞ (8)

with u0 the (positive) stacking energy and σb the entro-pic free energy penalty for bonding, i.e. the contribu-tion to free energy due to the entropy which is lost byforming a single bond. The quantity σb can beexpressed in terms of the bonding volume Vb, whichhas been introduced within the Wertheim theory [50],as follows:

σb ¼ ln2Vb

vd

� �(9)

The bonding volume Vb can be conveniently calculatedvia Monte Carlo (MC) integration as described indetail in Ref [35]. For the present model Vb has beenalready evaluated in Ref [34].

In passing, we note that if the term Fst is missing inthe free energy, i.e. there is no aggregation betweenmolecules, other theories have been also proposed todeal with aggregate flexibility, which rely on otherapproaches than that of Khokhlov and Semenov[51,52].

For accounts for the orientational entropy lost due tothe alignment of the chains in the nematic phase(including the contribution due to their flexibility).An analytic expression for this contribution will beproposed and discussed in Sec. 3.2

The fourth term Fexcl arises from excluded volumeinteractions between chains and thus it comprisesa sum over vexclðl; l0Þ, which is the excluded volume oftwo chains with lengths l and l0, i.e.

F excl ¼ Vβ�1 ηðζϕÞ2

Xl

Xl0

νðlÞνðl 0 Þvexclðl; l 0 Þ (10)

where ηðζϕÞ is a modified Parsons-Lee factor [47,48]introduced to account for higher order terms in thevirial expansion, with

ηðxÞ ¼ 144� 3x

ð1� xÞ2 (11)

and the system volume fraction ϕ ¼ ρvd is scaled by thefactor ζ , as suggested in Ref [38]. The motivation forintroducing the factor ζ is that the so-called effectivemolecular volume of non-convex chains is expected tobe larger than the sum of the volume of the constituentmonomers, as suggested and discussed in Ref [53],where a system of linear fused hard sphere chains wasstudied. In the present approach ζ has to be regarded

LIQUID CRYSTALS 3

as an adjustable parameter which will assume differentvalues in the isotropic (ζI) and nematic (ζN) phases.This modification of the Parsons-Lee factor is the firstimprovement of the original theory which we proposein this work.

The final expression for the free energy F in terms ofthe chain length distribution νðlÞ is:

βFV

¼X1l¼1

νðlÞ ln vdνðlÞ½ � � 1f g þX1l¼1

νðlÞσoðlÞ

þ ηðζϕÞ2

X1l¼1l 0¼1

νðlÞνðl0Þvexclðl; l0Þ þ ΔðTÞX1l¼1

ðl� 1ÞνðlÞ

(12)

Here, we will provide an explicit expression for calcu-lating vexclðl; l0 Þ and later on we will suggest simpleanalytic formulas for the isotropic and nematic phases,which are more tractable in the theoretical calculations.If we define R1 ¼ fr1;1 . . . r1;lg, R2 ¼ fr2;1 . . . r2;l0 g,U1 ¼ fu1;1 . . . u1;lg and U2 ¼ fu2;1 . . . u2;l0 g, where rγ;iand uγ;i are the position and the orientation (unitvector) of monomer i belonging to chain γ ¼ 1; 2, theaverage excluded volume is:

vexclðl; l 0 Þ ¼ � 1Vlþl0�1

ð 0

dR1dR2dΩ1dΩ2ell0

12

ðR1;U1;R2;U2ÞFcðU1ÞFcðU2Þ(13)

where dRγ ¼Ql

i¼1 drγ;i, dΩγ ¼Ql

i¼1 d!γ;i with d!γ;i

the infinitesimal solid angle around the orientation

uγ;i, ell0

12 is the Mayer function [54], i.e.:

ell0

12ðR1;U1;R2;U2Þ ¼ exp �UhðR1;U1;R2;U2Þ=kBTf g � 1;

(14)

with UhðR1;U1;R1;U2Þ the hard-core pair potential:

UhðR1;U1;R2;U2Þ ¼ 1 if 1; 2 overlap0 if 1; 2 do not overlap:

�

(15)

and FcðUÞ is the orientational distribution function ofthe chain. The meaning of the prime in the integral ofEquation (13) is that it has to be evaluated over allpositions and orientations of monomers such that (i)within each chain only two monomers are singlebonded and all the remaining monomers (if any) aredouble bonded1 and that (ii) chains do not self-overlap.The orientational distribution function of the chain canbe expressed in terms of the orientational distributionfunction of the monomers as follows:

FcðUαÞ ¼Yli¼1

f ðuα;iÞ (16)

with α ¼ 1; 2.

3.1. Isotropic phase

In the isotropic phase the orientational distributionfunction of a monomer f ðuÞ (where u is the orientationof the monomer) in Equation (16) is uniform, i.e.:

f ðuÞ ¼ 14π

(17)

The excluded volume in Equation (13) for chains ofhard cylinders can be calculated by MC integration forl; l

0 � 10 and then the results can be fit to the followingfunction:

vexclðl; l 0Þ ¼ 2 kIvdlþ l 0

2þ BIX

20 ll

0� �

(18)

where the BI and kI are fitting parameters.Under equilibrium conditions, the value of M has to

minimize the free energy, i.e.

@ðβF=VÞ@M

¼ 0 (19)

Substituting equations (3) and (18) into (12) and cal-culating the sums over chain lengths, one obtains:

βFIV

¼ ρ

Mln

vdρM

� �� 1

h iþ ρ

M� 1M

lnðM� 1Þ � ρ ln Mþ

þ ηðζIϕÞ BIX20 þ

vdkIM

� �ρ2 � ρðβu0 þ σbÞð1�M�1Þ

(20)

By plugging the isotropic free energy of Equation (20)into Equation (19), one obtains the following formula:

MI ¼ 12

1þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 4ϕeKIϕηðϕÞ=vdþβu0þσb

q� �(21)

which provides the isotropic average chain length MI

as a function of temperature and volume fraction.

3.2. Nematic phase

In the nematic phase the For contribution is differentfrom zero and the average excluded volume takesa different form, since the orientational distributionfunction of monomers f ðuÞ changes due to HCsalignment. The total free energy in Equation (5) isexpressed as a function of the average aggregatelength M and of the orientational parameter α,whose equilibrium values at a given volume fractionand temperature are obtained by minimizing the free

4 C. DE MICHELE

energy. It is also assumed, as in the isotropic phase,that the chain length distribution is exponential (seeEquation (3)).

For f ðuÞ we adopted the form suggested many yearsago by Onsager [55], i.e.:

f ðuÞ ¼ fOðuÞ ¼ α

4π sinh αcoshðα cos θÞ (22)

where θ is the angle between the monomer’s symmetryaxis and the nematic direction and α, which controlsthe width of the angular distribution, is related to thenematic order parameter S as follows:

S ¼ðπ0dθ sin θ

32cos2 θ� 1

2

� �fOðcos θÞ

¼ 1� 3coth αα

þ 3α2

�!highα1� 3

α(23)

As in the isotropic phase, the excluded volume inEquation (13) for chains of hard cylinders can be calcu-lated, for different chain lengths l and value of α, by MCintegration of Equation (13) using the Onsager orienta-tional distribution function fOðuÞ (see Equation (22)) andthen the results can be fit to the following function:

vexclðl; l0; αÞ ¼ 2 kNðαÞvd lþ l0

2þ BNðαÞX2

0 ll0

� �(24)

where the parameters kN and BN are now functions ofα. As in Ref [34]., the functional forms adopted forkNðαÞ and BNðαÞ are:

kNðαÞ ¼ kHCN ðαÞ (25)

BNðαÞ ¼ π

4D3 η1 þ

η2α1=2

þ η3α

� �(26)

where the term 2vdkHCN ðαÞ is the end-midsection contribu-tion (i.e. the termwhich scales as l) to the excluded volumeof two hard cylinders which is known analytically (see Ref[36].) and ηk with k ¼ 1; 2; 3 are fitting parameters.

The excluded volume vexcIðl; l0; αÞ is calculated numeri-cally by generating Ntrial pairs of chains of length l (formore details see Refs [35,36]) in a box of volume V , wheremonomer orientations were randomly extracted from theOnsager distribution in Equation (22) for a given α.

With such procedure, ifNoverlap is the number of timesthat the two generated chains overlap, the excludedvolume is:

vexclðl; αÞ ¼Noverlap

NtrialV: (27)

The excluded volume vexcl is calculated for several valuesof α ranging from 10 to 45, and l ranging from 1 to 5 andby fitting Equation (24) to these numerical estimates ofvexcl for different l and α, we determined ηi and thus

BNðαÞ. In Equation (24) we implicitly assume that allaggregates, independently from their length, are nematicwith the same orientational distribution function, since αis independent of l. A better approach would be to intro-duce an l-dependent α, although this way theoreticalcalculations would become rather demanding.A simpler strategy consists in assuming short aggregateswith l< liso (in the present case liso ¼ 2) isotropic and allother aggregates with l � liso nematic with the sameorientational distribution (i.e. with the same α).According to this simplifying assumption a correctionterm has to be calculated and added to the free energyas discussed in Ref [35].

Finally, the term For has to be evaluated. At presentthere is no exact analytic expression for this term, butthere exist the following two analytic expressions of σo forthe limiting cases of very stiff (RC) and very flexible (FC)chains [35,36,43,56, 7], i.e. when l � lp (where lp is thepersistence length) and l � lp respectively:

σoðlÞ ¼ l8lp

ð@f@θ

� �2

f�1dΩ � 2 ln

ðf 1=2dΩ

� �

þ lnð4πÞ ðlp � lÞ(28)

σoðlÞ ¼ðf lnð4πf ÞdΩ þ l

12lp

ð@f@θ

� �2

f�1dΩ

ðlp � lÞ (29)

By plugging into Equations (28) and (29) the exponentialchain length distribution, as in Equation (3), and theOnsager orientational distribution function, and carryingout the calculations one obtains the following limitingexpressions for σo:

σ RCo ðlÞ ¼ logðαÞ � 1þ α� 1

6lpl αl � lp (30)

σ FCo ðlÞ ¼ logðα=4Þ þ α� 1

4lpl αl � lp (31)

As discussed in detail in Ref [35], the orientational con-tribution For to free energy can be expressed as the sum ofcontributions having the form of either one or the otherof the two limiting expressions RC and FC, i.e.

For ¼ Vβ�1Xl¼l0�1

l¼1

νðlÞ logðαÞ � 1½ � þ þ α� 16lp

l

�

þXl¼1

l¼l0

νðlÞ logðα=4Þ þ α� 14lp

l

�

(32)

LIQUID CRYSTALS 5

where the contribution of chains of length l< l0 is treatedwith the RC approximation, while the contribution oflonger chains enters with the FC expression. The persis-tence length lp for the present model have been alreadyevaluated in Ref [34], while the details on the numericalprocedure adopted for its calculation can be found inRef [35].

In the original formulation of the theory, the parameterl0 – of the order of the persistence length lp –was fixed andconveniently adjusted depending on the specific modelused in the calculations [35,36,39]. Anyway, in Ref [41]. ithas been observed that, according to Ref [43,57], l0 has tobe related to the deflection length λ ¼ lp=α, rather than tothe persistence length lp. The rationale behind this is thatbending fluctuations make linear aggregates effectivelymore flexible on increasing α, since they are more signifi-cant if α is large, i.e. if monomers are more aligned. Hence,following the same approach of Ref [41], we setl0ðαÞ ¼ �λ ¼ �lp=α, where � is an adjustable parameter,which, as in Ref [41], is set equal to 0:29.

In summary, the free energy in the nematic phasecan be written as follows:

βFNV

¼ ρ

Mln

vdρM

� �� 1

h iþ ρ

M � 1M

lnðM � 1Þ � ρ ln M

þ ηðζNϕÞ BNX20 þ

vdkNM

� �ρ2 þ�ρðβu0 þ σbÞ 1� 1

M

� �

þXl¼l0�1

l¼1

νðlÞ logðαÞ � 1½ � þ α� 16lp

l

�

þXl¼1

l¼l0

νðlÞ logðα=4Þ þ α� 14lp

g�

(33)

If we assume that the orientational entropy σo can beapproximated by the expression in Equation (31) validfor long chains, minimization with respect to M yields:

MN ¼ 12

1þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ αϕeKNðαÞϕηðϕÞ=vdþβu0þσb

q� �(34)

while if Equation (30) valid for short chains is assumed,one obtains:

MN ¼ 12

1þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 4αϕeKNðαÞϕηðϕÞ=vdþβu0þσb�1

q� �(35)

3.3. Phase coexistence

Coexistence lines can be calculated by imposing theconditions of equal pressure and chemical potential inthe isotropic and in the nematic phase, along with therequirements that the free energy of isotropic phase inEquation (20) is minimum with respect to isotropic

average chain length MI and that the free energy ofnematic phase in Equation (33) is minimum withrespect to both the nematic average chain length MN

and the parameter α. These conditions translate intothe following set of equations:

@FIðρI;MIÞ@MI

¼ 0 (36)

@FNðρN ;MN ; αÞ@MN

¼ 0 (37)

@FNðρN ;MN ; αÞ@α

¼ 0 (38)

PIðρI ;MIÞ ¼ PNðρN ;MN ; αÞ (39)

μIðρI ;MIÞ ¼ μNðρN ;MN ; αÞ (40)

where I stands for isotropic and N for nematic and ρI andρN are the number densities of the isotropic and nematicphase respectively. In the calculation of phase boundarieswe had to adjust the values of ζ I and ζN to 1:075 and 1:165to best match theoretical predictions with numericalresults. As already observed, the modified PL approachhas been introduced to account for the larger effectivevolume occupied by each aggregate, hence it is reassuringthat optimal values both for ζI and ζN are greater than 1.

4. Results and discussion

For the model discussed in Sec. 2, by using the revisedtheoretical approach, we calculated the phase bound-aries in the ϕ-T plane, which are shown in Figure 2(Th. Revised) together with the results obtained by theoriginal version of the theory (Theory) as reported inRef [34].

In this figure, results from MC simulations reportedin Ref [34]. are also shown. Since on increasingT chains on average are shorter (as it can be evincedfrom Equations (21) and (34)), thus reducing the driv-ing force for nematization, the volume fractions ofboth the isotropic and nematic phases at coexistenceare expected to increase as shown in Figure 2. In thisfigure it can be seen that the revised version of theorycaptures much better the isotropic branch of the phaseboundaries and the accuracy of the nematic branch isimproved as well.

If the phase diagram in the ϕ-MX0 plane is consid-ered as shown in Figure 3, similar conclusions can bedrawn, even though the improvement is much lessapparent.

6 C. DE MICHELE

The theoretical isotropic branch of the revised the-ory is closer to numerical results from MC simulations,although the nematic branch at higher volume frac-tions underestimates the numerical results to the sameextent at which the original theory overestimates them.

An interesting feature of the phase diagram in theϕ-MX0 plane is the reentrant (non-monotonic) beha-vior of the nematic branch of the phase boundaries.We note that Onsager theory predicts a monotonicbehavior of both branches [34], hence this non-

monotonic behavior is a distinctive feature of self-assembly-driven nematic liquid crystals. Nevertheless,self-assembly alone is not enough to guarantee theemergence of the reentrance, since it is crucial toaccurately calculate the excluded volume betweenpolymers. Indeed, as shown in Figure 5(b) of Ref[34]. the results predicted by Lu and Kindt’s theory[58] and the ones from the work of Kuriabova et al.[49] do not exhibit the reentrant behavior of thephase diagram in the ϕ� X0M plane.

If one introduces the ratio R between the averagechain length of the nematic phase and that of the isotropicphase, i.e.

R ¼ MN=MI (41)

this quantity at coexistence strongly depends on pack-ing fraction through an entropic contribution relatedto the orientational order in the system.

At high temperatures nematization takes place atlarge ϕ and from Equations (21) and (34) one has:

R ¼ MN

MI ffiffiffi

αp

(42)

According to Equation (22) the width of the orienta-tional distribution of monomers is controlled by α,hence α is expected to increase dramatically at largeϕ, thus making R very large. At the same time, at largevolume fractions MI 1 [35], hence from Equation(42) it follows that MN is also very large. Likewise, atlow temperatures from Equations (21) and (34) one hasthat [36]:

0.1 0.15 0.2 0.25 0.3T*

0.2

0.3

0.4

0.5

0.6φ

Th. RevisedMCTheory

I

I+N

N

Figure 2. (Colour online) Isotropic-nematic phase diagram inthe packing-fraction ϕ vs temperature T� plane. Circles areresults from MC simulations from Ref [34]. Straight lines indi-cate theoretical predictions according to the present revisedversion of the theory. Dashed line are theoretical predictionsfrom Ref [34], which were obtained by the original version ofthe theory.

10 20 30 40 50X

0M

0.1

0.2

0.3

0.4

0.5

φ

TheoryTh. RevisedMC

I

I+N

N

φI

φN

T*=0.135

T*=0.229

T*=0.106

T*=0.083

Figure 3. (Colour online) Isotropic-nematic coexistence lines in the average aspect ratio X0M and volume fraction ϕ plane. Symbolsare numerical results from MC simulations of Ref [34]. Solid lines are theoretical predictions from the present revised version of thetheory, while dashed lines are original theoretical predictions as found in Ref [34]. Full circles along the isotropic (ϕI) and nematic(ϕN) phase boundaries, which are joined by dotted lines, indicate ϕ and M for isotropic and nematic phases at coexistence at thesame temperature.

LIQUID CRYSTALS 7

MN /ffiffiffiffiffiffiffiffieβu0

p; (43)

which can increase unbounded. We deduce that MN

can reasonably exhibit a minimum as a function ofϕ, i.e. a reentrant behavior as shown in Figure 2.

We note that the reentrant behavior of thenematic branch has been by now robustly con-firmed by both theoretical predictions and simula-tions but it still awaits an experimentalconfirmation. The revised version of the theory ofself-assembly-driven liquid crystals embodies twoimprovements: (i) the α dependence of l0 and (ii)the use of a modified PL factors. In order to assessthe relative role of these two improvements, inFigure 4 we show the phase boundaries whereboth improvements are used (Th. Revised) andwhere only the modified PL factors are used (Th.Revised PL) with l0 fixed and equal to 4 (i.e. thevalue used in Ref [34]). Resorting only to the mod-ified PL factors the isotropic branch of the phaseboundaries is almost unchanged while the nematicbranch is clearly a bit off from MC results. If weswitch to the phase diagram in the ϕ-MX0 plane,we have a similar situation where the nematicbranch overestimates the numerical results and theisotropic branch is even closer to MC results thanthe full revised version of the theory. We can con-clude that the modified PL factors mostly affect theisotropic branch while the nematic branch is mostlycontrolled by the α dependence of l0. In otherwords, both improvements surely matter in predict-ing correct results.

5. Conclusions

In this paper we proposed two possible improvementsof the theory developed in Ref [35] for hard cylinder-like particles: (i) the use of a modified PL factor and (ii)the introduction of a α-dependent l0 parameter. Thepredictions from the original theory were also com-pared against MC simulations of hard cylinder in Ref[34]. In this work, we compare novel predictions fromthe revised theory against simulations and originaltheoretical results, providing evidence of a significantimprovement. Modeling of orientational entropy is stillbased on the two limiting expressions for rigid andflexible rods proposed many years ago by Khokhlovand Semenov [43,56,59], where the threshold separat-ing the two regimes l0 is now α-dependent to accountfor bending fluctuations [57]. A better description ofthe intermediate regime instead of resorting to an α-dependent l0 would be advisable to have a betterdescription of this contribution. On the other hand,a better estimate of the isotropic branch could beachieved by a better treatment of higher order termsin the virial expansion, which are at present accountedfor by the PL factor. In the present version of thetheory two adjustable parameters are introduced inthe modified PL factors of the isotropic and nematicphases and they have been adjusted to maximize theagreement between theory and simulations. A futuregoal would be to avoid the use of adjustable parametersby a suitable microscopic treatment of the higher ordercontributions to the virial expansion. Despite the lim-itations of the present approach, where these two

0.1 0.15 0.2 0.25 0.3

T*

0.2

0.3

0.4

0.5

0.6

φ

Th. RevisedTh. Revised PLMC

I

I+N

N

Figure 4. (Colour online) Isotropic-nematic phase diagram inthe packing-fraction ϕ vs temperature T� plane. Symbols andlines as in Figure 2 except that dashed lines are now theoreticalpredictions from the revised version of the theory, where onlythe modified Parsons-Lee approximation is used and l0 ¼ 4 (i.e.l0 is kept fixed and does not depend on α).

10 20 30 40 50X

0M

0.1

0.2

0.3

0.4

0.5

φ

Th. RevisedTh. Revised PLMC

I

I+N

N

φI

φN

T*=0.135

T*=0.229

T*=0.106

T*=0.083

Figure 5. (Colour online) Isotropic-nematic coexistence lines inthe average aspect ratio X0M and volume fraction ϕ plane.Symbols and lines as in Figure 3 except that the dashed linesare now theoretical predictions from the revised version of thetheory where only the modified Parsons-Lee approximation isused and l0 ¼ 4 (i.e. l0 is kept fixed and does not depend on α).

8 C. DE MICHELE

parameters have to be adjusted ad-hoc for a givenmodel, the importance of the present findings relieson the fact that they clearly point towards the need ofgoing beyond the PL decoupling approximation.

A further improvement of the theory would be toemploy a joint orientation and chain length distribution,as suggested first for non self-assembling systems by vander Schoot and Cates [6,46]. A first attempt in thisdirection has been done in Ref [49] and the presentapproach can be modified accordingly. A possible sim-pler strategy would be to introduce an l-dependent α. InRef [38], where bifunctional hard spheres are studied, ithas been observed that if the theory accounts for the l-dependence of α, the nematic phase is expected to befavored against the isotropic phase, thus resulting ina wider coexistence region, i.e. an isotropic branch closerto numerical results. Although theoretical calculationsbecome rather demanding and cumbersome by introdu-cing an l-dependent α, the residual gap found betweentheoretical predictions and numerical results, as shownin Figure 1, could possibly be filled this way.

Note

1. The only exception is when we have two chains, eachconstituted of two monomers.

Disclosure statement

No potential conflict of interest was reported by the author.

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