PHYSICAL REVIEW A VOLUME 35, NUMBER 8 APRIL 15, 1987

Density-functional theory of curvature elasticity in nematic liquids.III. Numerical results for the Berne-Pechukas pair-potential model

Kalyan Singh' and Yashwant SinghDepartment of Physics, Banaras Hindu Uniuersity, Varanasi 22I 005, Uttar Pradesh, India

{Received 10 June 1986; revised manuscript received 12 January 1987)

The theory for curvature elasticity in nematics described in an earlier paper [Phys. Rev. A 34, 548(1986)] is applied to a simple model of Berne and Pechukas [J. Chem. Phys. 56, 4213 (1972)] in

which the strength parameter e(r[2 Q&, Q2) and the range parameter o(r», Q[,Q2) depend upon theratio of major to minor ellipsoidal axis and the relative position and orientation of the two mole-cules. The softness in the repulsion core is found to have a large effect on the value of the Frankelastic constants. The attractive part of the interaction which has the same angle dependence as therepulsive interaction is found to have a small effect on the value of all three constants associatedwith splay, twist, and bend deformations. Contrary to what has been reported for a single-center,r attractive potential, the attractive part (more accurately the perturbation potential) of thismodel makes negative contributions to all three elastic constants. The magnitude of this contribu-tion increases with the length-to-width ratio xo of the molecules. However, the value remains verysmall compared with that of the reference potential, which is also found to increase with xo and is

always positive. Numerical results for the elastic constants are reported for a range of xo, tempera-ture, and density. It is shown that with better knowledge of the order parameters P2 and P4 andtheir temperature and density dependence and the potential parameters appearing in the model, onecan calculate accurate values of all the Frank constants of any given system. The need to extend thetheory to include the flexibility of the alkyl chain and the deviation from the cylindrical molecularsymmetry is emphasized.

I. INTRODUCTION

In the previous papers of this series, ' we introduced atheory for the Frank elastic constants of a nematic phasebased on a density-functional approach which allowed usto write formally exact expressions for the Helmholtz freeenergy of a distorted system in terms of direct-pair-correlation function (DPCF). The DPCF which appearsin the expression is functional of the single particle densi-

ty distribution p(x) where x indicates both the positioncoordinate r of a molecule and its orientation Q,

I

described by Euler angles a, P,f. The functional Taylorexpansion has been used to express the DPCF of a nonun-iform system in terms of the DPCF of the isotropic liquidat the same number density. This led to the expressionfor the Frank elastic constants in successively higher-order direct correlation functions of the isotropic liquid.The result is

where

K;= —pokT fdr(2r )2 f dQ) f dQ2 5f(Q),0)F~(r)2, Q), Q2)c2(po),

E = —3 pokT f dr]2 r ]2 f dr3 f dQ~ f dQ2 f dQ3 5f(Qt 0)5f(Q3 0)F (rt2 Qt Q2)c3(pO)

(1.2)

(1.3)

Here

—(r&2 x)

(r~2 x)

0

(r&2.x)+ —,

' f"(Q2,0) (r~2 y)' .

(r)2 z)

F;(r», Q&, Q2) =f'(Q2, 0) .(rt2 z)

(r)2 z)

(1.4)

f (Q2)0)= +PL I YL O(Q2)PL o(1)L

(1.5)

K~, E2, and E3 are elastic constants associated with,respectively, splay, twist, and bend deformations. f(Q;, r)are the normalized singlet orientational distribution at r.c2(po) and c3(po) are the direct two- and three-body corre-lation functions. For a phase with D ~ symmetry andcomposed of cylindrically symmetric molecules, otherfunctions which appear in (1.2) and (1.3) are given as

f'(Q2, 0) = g PL [YL )(Q2) —YL, (Q2)]PL, (1),L

and 5f(Q;,0)=f(Q;,0)—1, i stands for 1,2,3 and x, y, zfor unit vectors along the specified X, F, and Z axes. +PL,2(1)[YL,2(Q2) + L, —2(Q2)] I

35 3535 1987 The American Physical Society

3536 KAI-YAN SINGH AND YASHWANT SINGH 35

where

P c 0(1)= —, L—(L+1)1 2L +14'

1/2

L —2! 2L+1(L+2)! 4m., (L ——1)L(L+1)(L+2)

Pg1(1)= PL—1(1)1/2

, L——L+1 (L —1)! 2L+1(L+1)! 4~

Pg2(1)

=P~ 2(1)

(1.6)

1/2

This paper is organized as follows. The model systemwhich we investigated here is described in Sec. II ~ Wealso discuss in this section the approximations used fornumerical enumerations. A perturbative theory for thedirect pair correlation function of isotropic phase isdeveloped. The results are presented and discussed in Sec.III. The paper is concluded in Sec. IV with a brief sum-mary and concluding remarks. In writing the paper, wehave assumed that the reader is familiar with previous ar-ticles of this series, and therefore, we also assume that thereader is familiar with the density-functional expansionand decoupling approximation.

II. APPROXIMATIONS AND DERIVATIONOF WORKINCJ EQUATIONS

and Y' (Q) are spherical harmonics.In a previous paper (hereafter referred to as I), we have

calculated K; for a system consisting of rigid moleculeswhich interact via a pair potential having short-rangerepulsion represented by interaction between two hard el-lipsoids of revolution and long-range dispersional interac-tion given by

(1.7)Ci C~

u, (r12, Q1,Q2) = —6

—6 P2(cose]2),

„12 r12

where c; and c, are constants related to the isotropic andanisotropic dispersion interaction and 012 is the angle be-tween the symmetry axis of the two molecules.

Numerical results for a range of hard-core sizes andshapes and dispersional strengths and anisotropies havebeen reported in I. Though the qualitative features exhib-ited by our calculations for both the ordinary and thediscotic nematics have been found in agreement with theexperiment and with the results of other workers, thepair potential model used in these works only crudelysimulate a real system. In this article we consider theBerne-Pechukas Gaussian overlap model" which be-longs to a general class of potentials of the form

A. Pair potential model

N N

U(x1, x2, . . . , x~)= g g u(x(. ,x~)i =1 j=1

(i&J)

(2.1)

where u(x;, xj) is the pair potential between molecules iand j; for ellipsoidal molecules, the general form of thepair potential u(x;, xj) is a function of variable r~, Q;,and Qi. A realistic model for u(x1, x2) should have a softrepulsive core and "long-range" weak attractive tail. Foraxially symmetric rigid molecules the Gaussian overlapmodel of Berne and Pechukas appears to be most satis-factory. Thus,

u(r, 2, Q1, Q2)

=4~(r12, Q„Q2)12

o(r12, Q1,Q2)

12

We consider rigid molecules which are permanently intheir ground electronic and vibrational states and have nointernal rotation. The total energy of interaction is as-sumed to be pairwise additive, i.e.,

u(r, Q) =e(r, Q)f(r/o(r, Q)), (1.8) 6o(r12, Q1, Q2)

where e(r, Q) is the energy scaling parameter and cr(r, Q)the distance scaling parameter for the interaction betweentwo molecules. Potential functions of the general form(1.8) are known as Corner potentials. Walnsley hasgiven a recent account of such potentials, and it seemslikely that a Corner description of the short-range forces,together with a multipole expansion of the long-rangeterms, will provide a suitable framework for the develop-rnent of reasonably realistic potential functions for largernolecules.

(2.2)

E(1 12, Q1, Q2) ='0[1—X (e1'e2) ]12Q1 Q2) oogr12Q1 Q2)

with

(2.3)

(2.4)

""e'e &(r12)Q1,Q2) and cr(r12, Q„Q2) are angle dependentstrength and range parameters, respectively. They aregiven by

(r12 e, )'+(r12.e2)' —2X(r12 e1)(r12 e2)g'(r12, Q1,Q2) = 1 —X

1 —X (e1.e2)

—1/2

where e1 and e2 are the unit vectors along the symmetryaxes of two interacting molecules and t p and o.

p are theconstants with units of energy and length, respectively,and

2xp —1X=2xp+1 (2.5)

is an anisotropy parameter. 0'p=2b is the length of minor

35 DENSITY-FUNCTIONAL THEORY OF. . . . III. 3537

axis of an ellipsoid. For x0 ——1.0 the Berne-Pechukasmodel reduces to that of a Lennard-Jones (12-6) modelwhich has been used extensively to study the properties ofsimple liquids and crystalline solids.

The model potential of (2.2) represents correctly at leastqualitatively, the shape of the molecular rigid core and theattractive dispersional potential. It is valid for oblate aswell as prolate shapes of arbitrary anisotropy and can beapplied to study the properties of different phases (solid,plastic, nematic, smectic, etc. ) and phase transitions. '

B. The direct pair correlation function (DPCF)of an isotropic fluid and the expression

for the elastic constants in decoupling approximation

In order to obtain the DPCF for an isotropic system in-

teracting via a pair potential of Eq. (2.2), we generalizethe perturbative method described in I to account for thesoftness in repulsive core. For this we divide the pair po-tential into reference and perturbation parts as below:"

u(r]2, Q],Q2)+e(r]2, Q]2),u„(r]2,Q],Q2)= . r]2 (r (Q]2) (2.6)

4h. = exp[ —pi]h (r]2 Q] Q2)], (2.9)

where uh, represents the potential energy of interactionbetween two hard ellipsoids of revolution. For the dis-tance of closest approach D(r]2, Q],Q2) between two hardellipsoids, we use an expression similar to (2.2), i.e.,

where r (Q]2)=2' o(r]2, Q]2) is the separation distanceat the minimum of pair potential for fixed Q] and Q2.

In terms of these potentials the excess Helmholtz freeenergy correct to first-order perturbation is found to bei

HIp, u j =HIp, u, j

+ —,p fdx]p(x])

X fdx2p(x2)u, (x»x2)gr(x»x2),

(2.8)

where KIp, u„j is the reduced excess free energy of thereference systein as a functional of p(x) and g„(x„x2) ]sthe pair correlation function of the reference system.

In order to evaluate H I p, u„j we generalize the analysisof Andersen, Weeks and Chandler' and expand Htp, u„jabout the hard-core Boltzmann factor

u, (r]2,Q],Q2) =

0, r]2 & r (Q]2)

—E(r]2,Q]2), r]2 ( r (Q]2)

u(r]2, Q],Q2), r]2 ) r (Q]2) ' (2.7)

D(r]2 Q] Q2) —d0$(r]2 (2.10)

where do is the length of the minor axis of a ellipsoid.The result of the expansion is

HIp +r j H[p ']t'hcj 2 fdx]p(x]) fdx2p(x2)phc(r]2 Q] Q2)~4'rhc+, (2.1 1)

where—Pu& (r12,QI, Q2)

Qh (r]2 Q] Q2)e gh (r]2 Q] Q2)

and

with

P„=exp[ —Pu, (r]2,Q],Q2)] .

For a reference potential of (2.6), P„ is very similar tothe function ph, which is a step function rising froin zeroto one at r]2 ——D(r]2, Q],Q2). We, therefore, expect the

I

series (2.11) to converge fast. For a reasonable choice ofis effectively nonzero for a small range of

values of r. If that range is gd0 (it may easily be realizedthat the value of this quantity is independent of the rela-tive orientation of the two inolecules), then g is a dimen-sionless parameter which is a measure of the softness ofthe potential. For harshly repulsive potential, we mightexpect that the thermodynamic properties and correlationfunction [particularly the function y„(r]2,Q„Q2)] to bequite similar to the corresponding quantities for hard el-lipsoids of revolution, provided that the distance d0 ischosen properly.

Substituting (2.11}into (2.8) one obtains

HI p ~ j =HIp, ]ihc j ——, fdx]p(x]) fdx2p(x2)[yhc(x], x2)bpr „, pg, (x„—x2)] . (2.12)

For a uniform fluid this leads to

c2( 12 QI Q2) c2 ( 12 Q] Q2)+~4, h $ h ( 12 Q] Q2)

I

and ~h, (r]2,Q»Q2) obey simple scaling rule (decouplingapproximation )

c2'(r]2, Q],Q2) =c2'(r]2/D(r]2, Q]2) )

where

—pu, (r]2, Q],Q2)g, (r]2, Q„Q2}, (2.13}

and

C2 (r]2/d0 )) (2.14)

pQy ( r i 2, Q i, Q2 )

g, (r]2 Q] Q2)= ~h (r]2 Q] Q2) .

As in I, we assume that the functions c2 (r]2, Q],Q2)hc

Qhc(r]2 Q] Q2) /ah ( 12/ (r]2 Q]2) }

=~h.(r]2/do, n) . (2.15)

3538 KALYAN SINGH AND YASHWANT SINGH 35

Here r lz ——rlz/o(r, z, Ql, Qz), cz' and ~h, are values of theDPCF and g function of hard spheres and packing frac-tion

liPsoids is reduced in terms of o(r, z, Ql, Qz), i.e.,

D(rlz, Ql, Qz)/tr(rlz, Ql, Qz) =dp/crp=dp (2.17)1 y 4 3'g=pUhc =6 7tp (dp ) Xp (2.16)

where p' =plTp and Uh,—(m. /6)xpdp is volume of hard el-

lipsoids. The distance of closest approach of two hard el-I

We have not yet described how to calculate the value ofdp. In order to do this we start with (2.11) which, for auniform system reduces to

H(pp &r)=H(pp "hc) 2pN f drlz f dQl fdQz~hc(rlz, Ql, Qz)hg(rlz, Ql, Qz) . (2. 18)

(2.19)

We choose D(Ql, Qz) so as to make the second term in (2.18) zero. Thus D(Q„Qz) is chosen to satisfy the criterion

f drlz f dQl fdQzgh, (rlz, Ql, Qz)hp(rlz, Ql, Qz)=0 .

In the decoupling approximation (2.19) reduces to

3dr, z „3~h, (r, z/d p g)b4(r lz, Qlz) dr, zD (Ql, Qz)=0 .(dp )

Equation (2.20) is solved to give'

(ds(Qlz) )do =

bp —(o.l, /2crpp) (5(Q,z) )

where ( ) represents the unweighted averaging over Ql and Qz.

da(Qlz)=(1 —g cos 6'lz)' [1—exp[ —f3u„(rlz, Qlz)]Idrlz,0

5(Q, z) =(1—X cos Hlz)' [r lz/ds(Q, z) —1] exp[ —pu, (rlz, Qlz)] drlz,0 I I2

(2.20)

(2.21)

crpp (1 ————, z) )(1—z) )

~ll —(2 7.59+0.5~2 —5.786593—1.518')(I —~)-',and

X+.-.1 2 1 4 1

6 40 112

The relation between molecular packing fraction g and hard-sphere packing g is obvious once we note the relationbetween molecular packing and molecular density p*

1

'9m =T~P xo

dp ——dp (p*,T',xp) is found to decrease with increase of density p*, temperature T*, and length to width ratio xp. ForT ( =kT/cp) =0.75 and 1.0, dp is plotted in Fig. l. as a function of xp. The Frank elastic constant can now be written

as

(2.22)

(2.25)

where subscript r and a indicate, respectively, the contributions arising from the reference and perturbation. In thedecoupling approximation, their expression reduces to

&+,."'=—po~pTo(do )'cz,.(g) f dQl f dQz f drlz5f«l 0)F (rlz Ql»z)~'(rlz, Qlz), (2.23)

IC = —3 ppT*o'p(d.

p ) cz, (g ) fdQl f dQz fdQ3 6f(Ql, 0)5f(Q3 0)

X

X f drz3o (rz3 Qz Q3)

dr]20. r]2,Q],Q2 F; r]2,Q],Q2 (2.24)

K;+,' '= ppT*crp f dQl f dQz—f drlz5f(Ql 0)cz, (z))F;(rlz, Ql, Qz)a (rlz, Ql, Qz),

35 DENSITY-FUNCTIONAL THEORY OF. . . . III. 3539

K;+,'= —, p—oT oo f dQ~ f dQz fdQ35f(Q] 0)5f(Q3 0)cz g(7l)

x f dr&3o (r~3 Q] Q3)

X fdrz3o'(r, Qz, Q3)

X fdr)zo (r)z, Q), Qz)F;(r)z, Q), Qz) .Here

K;apK;

&o

cz „(g)= f (r ~z ) cz'(r ~z, zl)dr fz

4g —11'+1680(1—zl )

Bcz „(zl)cz „(rI)=

8'g

(2.26)

1 21/6cz~(zl)= ~ (1—X cos g&z)

' f dr&z (r&z) ~h(r fz/do, zl) exp — (1—g cos g~z) '~z[(r*, )'z —(r~ ) 6+ —']

l

)g6dr lz[(r 12) r 12) ]$ hs(r 12/do z))21/6

Bcz,(zl )cz (zl) =

0'g

In writing above equations, we have taken forc z'(r *,z /d O, z) ) the analytic solution of Wertheim andThiele' of Percus —Yevick equation.

III. RESULTS AND DISCUSSIONS

As explained in I, our theory involve expansion to in-creasingly higher orders in the correlation functions, i.e.,

K, = g (K,'"'),n=0

(3.1)

where K ' contains the pair correlation function, K ' thethree-body correlation function, and so on. Each term ofthe series (3.1) is expressed as sum over contributionswhich are quadratic or higher order in the order parame-ters, i.e.,

K'"=K "(2,2)+2K;"'(2,4)+K,' "=E;"'(2,2,2)+ ~ ~ ~,

(3.2a)

(3.2b)

1.015—

0.995—

where K '(L,L') ~ Pl PL, and K;"'(L,L',L")~ PL PL PL-.Since Pz &P4 &P6 and P4/Pz-0. 3 for a typicalnematic, the series (3.2) is expected to converge fast. Inwriting (3.2), we used the following relations which werefound from symmetry considerations:

Kg' '(L,L') =K '(L',L),E '(L,L',L")=K '(L', L,L")=K '(L",L',L),

+O0.975—

0 955—

0.935—

I

2.0I

2. 5I

3.00«~ A, ]1

l

3 5 g.0

FICx. 1. Variation of the distance of minor axis of hard ellip-

soid (do ) as a function of xo (or 1/xo) at g =0.45.

etc.We evaluate the terms written explicitly in (3.2a) and

(3.2b) and approximate the contributions of higher-orderterms in the series (3.2a) using a [1,0] Fade approximant.Since the vlaue of K;"'(2,2, 2) is found to be very smallcompared to K '(2, 2) the contributions of higher-orderterms in the series (3.2b) are neglected. We also use a[1,0] Fade approximant to evaluate the contributions ofhigher-order terms in the series (3.1).

For numerical enumerations of the terms of the series(3.2) we, however, need to know the values of the orderparameters P2 and P4 as a function of temperature and

3540 KALYAN SINGH AND YASH%'ANT SINGH 35

TABLE I. Contribution of individual terms in the series (3.1) and (3.2) to each elastic constant forboth hard-core repulsions and attractive interactions for a system of prolate (xo ——3.0) ellipsoid of revo-lution. For the results shown here P2 ——0.5, P4 ——0. 15, g =0.45, T*=0.697, eo/k=575 K, and

0o.o ——5 A. The values of K; are given in the unit of 10 dyn. Symbols are same as in I.Ki Ki r +Ki a ~ Kl 4 90, K2 ——2.37, E3 ——9.26 ( X 10 dyn).

K '(2, 2)2E '(2, 4)K(o)

lK(0)P

t

E "(2,2, 2)Ki

KPP

Ki

8.38—2.04

6.346.73

—0.795.555.956.02

K;,E2

4.06—0.68

3.383.48

—0.632.752.852.95

K3

8.382.72

11.1012.41

—1.149.96

11.2711.37

—1.400.35

—1.05—1.12

—1.05—1.12—1 ~ 12

K;,K2

—0.680.12

—0.56—0.58

—0.56—0.58—0.58

—1.40—0.47—1.87—2.11

—1.87—2.11—2.11

density and the values of the potential parameters ep/kand o.p. While accurate values of Pz are available for anumber of systems over a range of temperature and densi-ty from experiments, our knowledge of P4 is meager. Forour calculation we, however, choose P2 ——0.5 andP4/Pz ——0.3. These values correspond to a typical nemat-ic phase at about intermediate temperature of its range ofexistence. It should, however, be remembered that P2, P4and the ratio P4/P2 are temperature and density depen-dent. In this article we, therefore, prefer to give thevalues of elastic constants in reduced form, i.e.,

K;upK; =

to make it, as far as possible, independent of the value ofthe parameters used in the calculation.

In order to estimate the value of Ep/K we assume thatthe liquid (crystal)-solid transition temperature at triplepoint obeys a simple scaling law; T,*(=kT, /Eo) =c',where c' is a constant independent of xp. Guided by theresult obtained for Lennard-Jones (12-6) system (i.e., forxo ——1) from computer simulation' we choose c'=0.68.This value of c' is reduced triple point temperature for

the Lennard-Jones (12-6) system. Indeed, taking PAA(para-azoxyanisole) as an example, we find that the condi-tion T'=0.68 corresponds to ep/k =575 K. ' This is afairly reasonable value because it gives the energy of in-teraction in minimum energy configuration of two PAAmolecules equal to 2 Kcal/mole which is in agreementwith the values obtained from approximate quantummechanical calculation. ' For prolate rnolecules, we takeo.

p ——5.0 A which roughly corresponds to a PAA moleculeand oblate op ——15.0 A which corresponds to a discoticnematic phase of hexa n hexyl-ox-y benzoate ( n =4) of tri-phenylene.

In Tables I and II, we give the contribution of individu-al terms of the series (3.1) and (3.2) for xo =3.0 and 1/3. 0respectively, at q =0.45. These results correspond toT=400 K for prolate molecules and 600 K for oblatemolecules. The results obtained using the Pade, approxi-mant are also given in the tables to show the convergenceof the series and the magnitude of contribution of higher-order terms. Notations used here are the same as those ofI. The values listed in these tables can be compared withthe corresponding values of the tables of I in order to seethe effect of the softness of the repulsive core of the po-tential and the attractive interaction.

TABLE II. Contribution of individual terms in the series (3.1) and (3.2) to each elastic constant forboth hard-core repulsions and attractive interactions for a system of oblate (xo ——1/3. 0) ellipsoid of re-volution. For the results shown here P2 ——0.5, P4 ——0. 15, g =0.45; T =0.718; eo/k =830 K, ando.o ——15 A. The values of K; are given in the unit of 10 ' dyn. Symbols are same as in I.E'' =K +K Ki =4.58, K2 =5.57 K3 =2.76 ( X 10 dyn).

K,"'(2,2)2E '(2, 4)K:

lK(o)

I

KI(2, 2, 2)K;K,'E'

5.211.016.226.46

—0.935.295.535.64

K;,K2

7.350.337.687.69

—0.966.726.736.83

S.21—1.34

3.874. 14

—0.863.013.283.43

Kl

—0.3S—0.17—1.02—1.06

—1.02—1.06—1.06

K;,E2

—1.20—0.06—1.26—1.26

—1.26—1.26—1.25

—0.8S0.23

—0.62—0.67

—0.62—0.67—0.67

35 DENSITY-FUNCTIONALL THEORY OF. . . . III ~ ~ ~ 3541

In Figs. 2(a) and 2(b), we plot E*asd I/, ti l, fo

o t.ib to of fs o re erence and ertur'

ar1 h h f

h b f }1

the one reported in I H'on o t e perturbation potential K;, from

in . Here we find thatis negative for all i

at t~is contributioni and the ina nitude i

1 d 'h1/i xp or oblate moleccules. The valueys small corn ared t

f f po ential K;„which ind th xp or prolate and

For the potential model taken in I'b ' f h

elastic constant wa 1

e perturbation otenp ential to the Frankwas a ways positive an

in the system ofand decreased with x

o prolate molecules anxp

1/xp in the case of bles and increased with

expected on the ro o ate molecules. Su

e groun s that the ma'uch a behavior was

the elastic const tan s came from thee major contribution t

g Persion interaction and

h io i hi h hcreased with x f

ic t is interactionon was significant, de-

creased with 1/x fxp or prolate molecul in-u e of fixed 2b and in-

Pechukas pair-poten'

xp or oblate molececules. The Berne-- o entia model taken in thin is article h'as

identical an leg e dependence in re ulsiveof h ac ion. T e stren th o

11

h F k 1

ases wit xp, therefore it

Thc constants is positive.

e softness in the repulsive core is found to have pro-

its dependence on x .e va ue of the elastic cconstants and on

11 d f 1~ ~

n xp. or example, for

. , and 173 (X10 d n r1a e o I), wherea

q. . with the same parameters (ande values are 6.0, 3.0

dyn) (see Table I).7

tween the two sets ofThe difference be-

of softness in the r 1

se s o values are mainl y due to the effecte repu sive core of the airp pncrease with the increase of

p . The reason for thi'

hT, d ( Fi. l). T

is is t e de enden

which actually entig. . he hard-core ac

'packing fraction g

reference pot t' 1

en ers in the calcula'

en ia is usually differentlation of K. of thI, r

increases with th e increase of xi erent (and this difference

o1 1 ki fc ing raction g becausee of the deviation of'

y. ince for the ssystems of interest

3.2,

4.5— 1.6—

3 2 0.0—K3p a

1-6—

00——0.8—

10—

K3,a

48—

3.2—

$ C4 05—0 0—

00—

2.4—K2~ 48—

, a

32—

1 ~ 2

06—0.0—

—08—Klpa

4"0

I

2QI

25I

30l

2.0l

25I

3.01

3-5 3-5

Xp

FIG. 2. Coomparison of the c t 'b s a racontn utions'

a ra

1X

4.0

ecules as a function ofc t s of repulsive and attra o t e

0

n o xo. (b) Same as in (a'a ractive interactions to the

n a), but as a function of }/xo t e elastic constants of a s

o xo or a system of obl ate molecules.o a system of prolate mol-

3542 KALYAN SINGH AND YASHWANT SINGH 35

5.0— 6.4—Xp

l, .p 5.2—

3.0—lj.p—

2.0—2-8—

1.0—

0 ~ 0—xo= 3,0

1.6—p v

Qw"o

3.0— 3Q—

20— Kg

1-0— 1.0—0.65

I

Q.75I

0.85 0.95 1.0

0.0 I

0.65I

0.75I

Q.esI

0.95 1 0

(a) Elastic constant (E; ) as a function of reduced temperature ( T*) with xo ——3.0 and 4.0 for prolate ellipsoids. (b) Sameas in {a)but with xo ——1/3. 0 and 1/4.0 for oblate ellipsoids.

do &1, q &g and therefore the E;, of the reference po-tential of Eq. (2.7) is less than those of the hard-core po-tential having do ——1.

We plot K;* as a function of T* for systems of prolatemolecules with xo=3.0 and 4.0 and for systems of oblatemolecules with 1/xo ——3.0,4.0 in Figs. 3(a) and 3(b),respectively. These figures indicate that IC;* increasesT* almost linearly for all i and xp. In calculating thesevalues we, however, assumed that P4/P2 ——0.3 and is tem-perature and density independent. This may not be truein real systems. With little reflection one finds that thevalue of the ratio P4/Pz should increase with density andxo and decrease with temperature. As we know thatK;&IC~ is because of the contribution of a term involv-ing P4/P2. The contribution of this term to E;* is posi-tive for i = 3 and negative for i = 1 and 2 for the case ofreference potential and negative for i =3 and positive fori =1, and 2 for the case of attractive interaction. Sincethe contribution of perturbation potential is very smallcompared to that of the reference potential for all value ofi we find Kz &K~ &K2 for a system of prolate rnole-cules. If the ratio of P4/P2 decreases with the tempera-ture, the values of Kz will decrease and the values of E

~

and Kz will increase with T* from these given in Fig.3(a).

For oblate ellipsoids the contribution of the terrri in-volving P4/P2 to K;* is found to be negative for i =3 and

positive for i = 1 and 2 for the reference potential. On theother hand, for the perturbation potential this contribu-tion is positive for i =3 and negative for i =1 and 2.Since the magnitude of the contribution of the attractiveinteraction is small we find in agreement with previous re-

sult ' that for a system of oblate molecules

K2 & E ] & K3 ~ If the value of the ratio P4 /P2 decreaseswith the temperature from its value of 0.3 taken in thecalculation, the value of Kz will increase and K& and K2will decrease with T* than those given in the Fig. 3(b).Therefore, the temperature dependence of K;* depends onthe temperature dependence of P4/P2.

In Fig. 4, we plot K as a function of molecular pack-ing fraction g for both a system of prolate moleculeswith xo ——3.0 and a system of oblate molecules with

1/xo ——3.0. The values given here are based on the as-sumption that P4/Pz ——0.3 and the value of the ratio isdensity independent. Since, we believe that the value ofthe ratio P4/Pz increases with density, the actual depen-dence of E;* on density will be more pronounced than theone given in the figure.

Taking PAA as an example for comparing the theoreti-cal values with experiments, we find that at 400 K and

=0.50, K&.Kz.K& ——2. 1:1:3.9 compared to the observedratios of 2.0:1:3.2. ' The absolute magnitude (6.3, 3.0, and11.9X10 dyn) of our calculated elastic constants arealso in good agreement with the experiment (7.0, 3.5, and

35 DENSITY-FUNCTIONAL THEORY OF. . . . III. 3543

4 8 —(b)

4.Q—

5.0

4.0—

3.0—

2 4—

1.6—

o.e-

20—I

032.5—

2 0—

I

0.4pa/p

l

0.5

40 (a)1.5—

3 ~ 2 %1,0

2.4— K3 K1

1.6—

0.8—

I I

3-0 2 01/

l I

10 20I l

3.0 4.0Xp

l

0.40I

0.45 0.50I

Q.55

FIG. 4. (a) Elastic constants (K;*) as a function of molecular

packing fraction g with xo ——3.0 for prolate ellipsoids. (b)

Same as in (a) but, with xo ——1/3. 0 for oblate ellipsoids.

FIG. 5. The ratios K&/K& and K2/EC& plotted as a functionof xo {prolate) and 1/xo for oblate molecules. Experimentalpoints are due to Leenhonts et al. (Ref. 18). ~: p-azoxyanisole(PAA). )&: p-methoxybenzylidene- p '-cyanoaneline (MBCA).~: aniaylidene-p-aminophenylacetate (APAPA): D: p-methoxybenzylidene- p '- (MBAPB).

11.2&& 10 dyn). Experimentally it is, however, difficultto measure the absolute values of the elastic constants. '

It is the ratio's Kz/K& and K2/K& which are measuredmore accurately.

From experimental data' ' one finds that (i) the ratioK2/K& lies within 0.5 to 0.7 and is more or less tempera-ture independent and only slightly depends on xo, (ii)K3 /K~ increases with increasing xo for rigid molecules,and (iii) for molecules with flexible alkylene chains, theratio Kq/K~ decreases with increasing chain length. InFig. 5, we plot calculated values of the ratios K2/K~ andK3/K& as a function of xo for both prolate and oblatemolecules. We also give on the figure the experimentalvalues of de Jeu and coworkers' for Kz/K~. As has al-ready been argued the value of the ratio K&/K, is sensi-tive to the value of P4/P2 used in the calculation. In or-der to show this we plot in Fig. 5 (in box) the value ofK3 /K

&as a function of P4 /P2 for x0

——4.0. We find thatthe experimental data of de Jeu et al. can easily be fittedby giving small variation to the value of the ratio P4/P2with xo. For example, to fit the value of K3/E& forMBAPB (p-methoxybenzylidene-p-quinophenyl ben-zoate), we have to choose the value of P4/P2-0. 36 whichappears to be very reasonable.

In order to understand (iii) we note that the P2P4 at, -tractive interactions contribution tends to make K& largerthan E3, whereas the opposite is true for P2P4 repulsion

terms. If molecular polarizability increases sufficientlythrough a homologous series of compounds 2K;, (2,4)grow faster than 2K; „(2,4) and account in this way forthe decreasing value of K3/K&. The effect of flexiblealkyl chain in a molecule will also be to make the repul-sive core of the intermolecular interactions more soft.

The other quantity which one can find fairly accuratelyfrom experiments is K;/K where K= —, (K~+Kq+Kq).In Fig. 6, we plot this quantity for both systems of prolatemolecules with x0 ——3.0 and 4.0 and systems of oblatemolecules with I/xo ——3.0 and 4.0 as a function of TWe find that the values of the ratio K;/K are temperatureindependent for all values of i and xo. The values areonly slightly affected by the values of x0. While K3/Kincreases with x0 for a system of prolate molecules, K i /Kand K2/K decrease. In the case of oblate molecules, wefind just the opposite trend. While we find quantitativeagreement with the experim. ental data of K;/E for allvalues of i for a system of prolate molecules, the experi-mental values show weak temperature dependence. '

K3 /K is found to decrease with T* while K ~ /K andK2/E increase. This can be explained by varying thevalue of P4/P2 with temperature. As has already been ar-gued, the value of P4/Pz should decrease with tempera-ture which in turn will decrease the value of K3/K andincrease the value of K&/K and K2/E with temperaturefor a system of prolate molecules in accordance with theexperiment.

KALYAN SINGH AND YASHWANT SINGH 35

1.5—

1 25—

(b) lg„= 3.0————-1/ gp/ XO

Kz/ K

1.0—

0.75—K3/K

0.5—(a)

2.0—

1.5—K3/ K

I~

1.0— Kq/K

05— Kz/K

0.00 65

I

0.75I

0 85I

pe5 1 0

IV. SUMMARY AND CONCLUSIONS

Our theory of curvature elasticity in the nematics re-

quires as input the direct correlation functions of an iso-tropic system and the order parameters P2 and P4 as afunction of number density and temperature. Though thepair-potential function does not appear explicitly in ourtheory, it appears through the direct correlation function.The integral equations of liquid-state theory which relatecorrelation functions with the potential energy of interac-tion can always be solved numerically to give informationabout cz as a function of intermolecular separation andorientation for a given pair-potential model, such calcula-tions in practice are, however, often quite difficult or timeconsuming. The perturbation theory developed in Sec.II B to obtain cz is a generalization to molecular liquids ofthe WCA theory of atomic fluids. The %'CA theory hasbeen used extensively to obtain accurate values of thestructure and thermodynamics of simple fluids.

A functional Taylor expansion is used to derive a rela-

FIG. 6. (a) Temperature dependence of ratio K;/K for pro-late ellipsoids with xo ——3.0 and 4.0. (b) Same as in (a), but foroblate ellipsoids with xo ——1/3. 0 and 1/4.0.

tion between the thermodynamic and structural propertiesof the reference system and the system of hard ellipsoidsof revolution. The first (lowest-order) approximation gen-erated by the expansion equates the free energy andy(r, Q) for the reference system to the respective functionsappropriate to a system of hard ellipsoids of revaluationwith distance of closest approach D(Q, z). A prescriptionis obtained for choosing a temperature, density, and aniso-tropy dependent minor axis (do ) of the hard ellipsoids sothat the first approximation for the free energy containserrors of order P only, and the corrections to the first ap-proximation for g„(r~z, Q~, Qz) are of order g . The soft-ness parameter is essentially the range of intermoleculardistance in which b P(r, z, Q &, Qz) is nonzero. Computersimulation study is needed to test the validity of this ex-pansion.

The Berne-Pechukas model provides a suitable frame-work for the development of reasonably reliable potentialfunctions for large molecules. There is, however, a longway to go before such potential functions make the transi-tion from idealized models to representations of a real po-tential. One unsatisfactory feature of Berne-Pechukas po-tential, for example, is that the potential as a function of ris simply scaled according to o.(Q, , Qz), so that whereo(Q„Qz) is large the potential is softer. The other defectof this model is that it does not take into account the flex-ibility of alkyl chain of the molecules. One possibile wayof taking care of this is to integrate over the coordinatescharacterizing the configuration of the flexible chainswith suitable weight factor for distribution of chains tovarious accessible conformations. Such a procedure willgive an effective interaction between rigid cores of themolecules with possibly renormalized values ofo(Q&z), e(Q&z) and the dependence of interactions on theintermolecular separations.

The softness in the repulsive core of the interaction hasbeen found to have large effect on the value of the Frankelastic constants. The effect depends on the value of xo,density and temperature. This is because the effectivelength of the minor axis of hard ellipsoids in terms ofwhich the Frank elastic constant of the reference potentialis calculated, changes with xo, density and temperature.

In order to obtain the values of K; and their tempera-ture and density dependence for a given system, we needto know the values of P2 and P4 as a function of tempera-ture and density and the potential parameters eo/k and cr0.In particular, we found that the quantities K~/K& andE;/K are very sensitive to the value of P4/P2 and itstemperature and density dependence. While one can, inprinciple, obtain from quantum-mechanical calculationsthe values of the potential parameters, at present ourknowledge of them is very scanty. As our knowledge ofthese input parameters improve, more accurate values ofthe elastic constants will be obtained. Attempts should,however, be made to extend our theory to include the flex-ibility of the alkyl chain and deviation from the cylindri-cal molecular symmetry.

35 DENSITY-FUNCTIONAL THEORY OF. . . . III. 3545

'Permanent address: M.L.K. Post Graduate College, Bal-rampur 271 201, Uttar Pradesh, India.

Y. Singh, Phys. Rev. A 30, 583 (1984).Y. Singh and K. Singh„Phys. Rev. A 33, 3481 (1986).K. Singh and Y. Singh, Phys. Rev. A 34, 548 (1986).

4W. M. Gelbert and A. Ben-Shaul, J. Chem. Phys. 77, 916(1982).

5B. J. Berne and P. Pechukas, J. Chem. Phys. 56, 4213 (1972).J. Corner, Proc. R. Soc. London, Ser. A 192, 275 (1948).

7S. J. Walnsley, Chem. Phys. Lett. 49, 390 (1977).sSee, for example, J. P. Hansen and I. R. McDonald, Theory of

Simple Fluids (Academic, New York, 1976).J. Kushick and B. J. Berne, J. Chem. Phys. 59, 3732 (1973); 64,

1362 (1976).' A. L. Tsykalo and A. D. Bagmet, Mol. Cryst. Liq. Cryst. 46,

111 (1978).' J. D. Weeks, D. Chandler, and H. C. Andersen, J. Chem.

Phys. 54, 5237 (1971).' H. C. Andersen, J. D. weeks, and D. Chandler, Phys. Rev. A

4, 1597 {1971).U. P. Singh and S. K. Sinha (unpublished).M. S. Wertheim, Phys. Rev. Lett. 10, 321 (1963).J. P. Hansen and L. Verlet, Phys. Rev. 184, 151 (1969).

' This value of ep/k is in good agreement with the one reportedby Tsykalo and Bagmet (Ref. 10) which they found in theirmolecular dynamical study to give good agreement for the

temperature dependence of the order parameter between thecalculated and experimental data of PAA. The values of theforce parameters found by them are ep/k = 525 K andop ——5.36 A.J. W. Baron and A. Les, Mol. Cryst. Liq. Cryst. 54, 273(1979).

' W. H. de Jeu, W. A. P. Classen, and A. M. J. Sprujit, Mol.Cryst. Liq. Cryst. 37, 269 (1976); F. Leenhonts, A. J. Dekker,and W. H. de Jeu, ibid. 72A, 155 (1979); E. F. Gramsbergenand W. H. de Jeu, ibid. 97A, 199 (1983).

'9See, for example, W. H. de Jeu, Physica/ Properties of LiquidCrystalline Materials (Gordon and Breach, New York, 1980)~