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Elastic constants of a system composed of interacting ellipsoidal molecules near an interacting wall
__ j_: M
s
25 April 1994 PHYSICS
LETTERS
A
4._
ELSEVIER
Physics Letters A 187 ( 1994) 331-336
Elastic constants of a system composed of interacting ellipsoidal molecules near an interacting wall A.V. Zakharov Saint PetersburgInstitutefor Machine Sciences, RussianAcademyof Sciences, Saint Petersburg199178, RussianFederation Received 29 December 1993; accepted for publication 7 February 1994 Communicated by V.M. Agranovich
Abstract A statistical-mechanical theory based upon the method of conditional distribution is applied to the calculation of the Frank elastic coefftcients of a system composed of interacting ellipsoidal molecules near an interacting wall. Three reduced elastic constants K,, K2 and K3 are presented as functions of the parameter ellipticity, the set of order parameters and the distance from the wall.
1. Introduction Although many attempts have been made to describe the elastic properties in the bulk of a liquid crystal (LC) [ l-51, there is as yet no microscopic treatment of such a problem near a solid wall. The purpose of this note is to present a simple molecular model based upon the method of conditional distribution [6], in which an attempt has been made to combine the advantages of integral equation theory and those of the cell model approach [ 71 together with some statistical-mechanical ideas [ 8 1, allowing calculation of the Frank elastic constants K,, K2 and K3 which describe the splay, torsion and bending deformation in the nematic phase near the interacting wall. Although the calculations have been made for a nematic liquid crystal (NLC ), the general approach can be applied to any liquid crystalline phase. 2. Model We consider a system of N particles in a volume V. The potential energy associated with N particles is a Elsevier Science B.V. SSDZ0375-9601(94)00157-K
sum of pair potentials 0( zj), each of which depends on the positions gi and qj and orientations ei ( 1ei I= 1) and ej of particles with numbers i and j. We divide the volume of the system V, in contact with the solid planar wall, into N equal cells. The volume of each cell is V= V/N and every cell is occupied by a molecule. The x-axis of the coordinate frame is chosen to be normal to the wall and nematic molecules occupy the half-space x> 0, and the space-fixed z-axis is chosen so that the nematic director lies in the yz plane. By integration of the Gibbs canonical distribution, a set of functions F’,(i), Fij(Q), etc., is introduced in order to define the probability densities for the molecules to be found about the positions m= W,,, ( W,,, = ~,,,@a,, a, is the volume associated with orientations, m = i, j). Using mean force potentials (MFPs) [ 6-81 we can present the abovementioned functions in the form
Fi(i)=Q-’
Fij(ij)=Q-2exP{-p[~(ii)+~ij(ii)l} where
(1)
ev[ -Bpz(i) I , 7
(2)
332
A. V. Zakharov /Physics Letters A 187 (1994) 331-336
I
a
L’
@(ij) is the interaction potential of two molecules, pi(i) and pij( ij) are the mean force potentials and are sums of the forms Pi(i)
=
,gj
Pi,j(i)
t
v)rj,/(ij)
=
,$?
(3)
Pij,/(ti)
Of course, the pij( i) will also depend on the distance of the cell i from the solid wall. The terms in the sums ( 3 ) are given by
V,Pi,j(i)=
I
dO’) Vi@(ij)Fij(ii)lFz(i) ,
(4)
I
Vipij,,(G)=
s I
d(l)Vi~(U)Fij~(iir)/Fij(ij)
.
(5)
The subscripts before the comma correspond to the MFP dependence on the positions of the molecules; those after the comma correspond to the average states. Using now the relations between the single and binary functions which follow from their definition, we have s
d(i) F,(i) = 1 ,
Fi(i)=
J”dG) F,(y)
(6)
i
The two-particle function is related to the three-particle function by an integral relation, etc. Below we take into account the first two functions of the in% nite hierarchy; this corresponds to taking into account only pair correlations between molecules. In order to make this a closed system of equations, we separate the mean force potentials into irreducible parts [ 6-8 1. In the case considered here we have VijJC,r(ii)
=%,di)
+Vj,/,rO’)
If the irreducible zero, Lj,,
.
+tij,l(ii)
part of the potential
(7)
which corresponds to neglecting three-particle and higher-order correlations, the expression for the binary function takes the form Fij(ij)=exP{B[yli,j(i) XK(g)Fi(i)FjG)
+Vj,iU)
,
exP] =
-&j(i)
s
1
dti) exp[Pfl,lO’) lK(ii)W_i)
.
(10)
Knowing the solution of this system one can compute the microscopic characteristics of the LC (expressed in terms of the one-particle and two-particle functions) and also the macroscopic characteristics, which can be expressed in terms of the free energy of the system. The free energy per molecule is given by f=-p-‘lnjd(i)exp[-pyl,(i)].
(11)
It should be mentioned that in the nematic phase the MFPs pi,,(i) are equal for cells in the same layer, parallel to the planar wall and differ for cells in the adjacent layers, perpendicular to the wall, and, theoretically, one should take into account an infinite number of adjacent layers. But in practice, a finite number of adjacent layers shall be taken into account and it will be shown that MFPs decay to their bulk values. The decay length or the number of layers in which MFPs are different from the ones in the bulk depends strongly on an external field due to the solid wall, the pairwise interaction potential and intermolecular correlations.
3. Method of solving the non-linear integral equation
is set equal to
(8)
=O >
where p- ’ GO= kT is temperature and K(ij) = exp [ -/IlO 1. The exponential factor in this expression reflects the correlation between molecules and distinguishes the approach used here from the mean field approximation. Substitution of (9) into (6) leads to a closed system of non-linear integral equations (NIEs) for the MFPs,
I)
(9)
For a system which is non-uniform only in the xdirection the solution of Eq. ( 10) can be written in the form Vi,,(i)
=d,,F’
+hi,j(x,)
,
(12)
where qj,y)(i) is the nematic equilibrium MFP and 1h,,j( Xi) 1<< 1qi,‘j’ (i) 1. NOW our purpose is to find the rp$)(i) which satisfies Eq. (10) and the hi,l(Xi), the equation for which will be derived in the next sec-
A. V. Zakharov /Physics Letters A 187 (1994) 331-336
tion. The solution of Eq. ( 10) for the nematic phase is based upon the fact that the one-particle function F,(i) is a function of cos &, where 6i is the angle between the z,-axis and the long molecular axis of the ith molecule. The’most general form of Fi( i) is Vi(COS
Fi(i)=Ji d(i) v=exp(
ei)
~i(cOs8i)
-j@(O))
’
.
(13)
The method of solution of the five-dimensional problem ( 10) is most complicated. It is convenient to rewrite ( 10) in the form
u/i,j(Qi) =
Jrq dQjK(Qi, Qj> v/j(Qj)lWi,i(Qj) IqdQj WjCQj> ’
(14)
where Qj is a vector in the live-dimensional space Wj= Vj@,s2i.The solution can be found using successive approximations calculated from the formula
333
bit structure with six nearest neighbours, which were also taken into account in the calculations. This restriction was dictated by the computer available for the study. The implementation of the algorithm ( 15) for Eq. ( 14) was done as follows. The initial approximation was chosen to be ws’ (Qi ) = vvi[” (Qi ) = 1. Then the integral on the right-hand side of ( 15) was calculated according to ( 16) with help of an LP, sequence. This procedure was repeated for all six neighbours of the particle in cell i. The coordinate Qi in ( 15) was chosen such that vi;’ (Qi ) would be mlculated at the points forming the same stationary LP, sequences as used in the evaluation of the integrals with the initial approximation ~~,~’(Qi ). Then ” ( Qi ) was calculated by a simple multiplication of wj,yl ( Qi). The procedure was then iterated until a given accuracy was achieved, where the translational invariance of the total potential was taken into account.
wf
[Nl
LN’ll(Qi)=(lU; 1, i,g!(Qj)
WiJ
4. Derivation of the linearized integral equation for a small4 expansion
l/2
X
1
dQjK(Qi,
Qj)WJ"'(Qj)/WJ?'(Qj)
W
> (15)
and the solution does not depend on the initial approximation. The basic difficulty in solving the equation is that the algorithm ( 15) requires the successive evaluation of live-dimensional integrals. In the present paper we apply the method of Sobol [9] based on the Haar functions, calculating the multi-dimensional integrals 1 dx 1 ... 1 0
I 0
exp]
-Phi,j(Xi)1z 1-BhiJCxi)T
***%I)=
z1 ~@w. =
- ~~;“I
I/ ) l&“’
(16)
1,
for Qic Wi is about 0.1%. The cells form a simple
CU-
(17)
and on using the notation vi,j( i) = exp [ - j@i$‘)( i) 1, the linearized integral equation is l-&(xi)=
dx,_f(&
where Qa are points uniformly distributed in the ndimensional unit cube. These points belong to an LP, sequence, N is the number of points. The precision of our calculations is about R = 0 (N - ’ ln”N), where n is the dimensionality of space (our calculations were executed using N= 100). For the case with N=200, the difference between the potentials vi>“’ and t&O01 in the nematic phase 6= 1I&“]
The natural way to study the “small” solution of hij(xi) is by linearization of the non-linear functional, exp[ -&oJi) 1. Accordingly, we make the approximation
X
(
J dO’) CvjO‘){l-BVi,j(i)K(ij) j
thjCxj> -hj,i(Xj)l)lWj,iti) > (18)
where hj( Xj) = Cl+ i h,( Xi) sThe function vij( i) is the uniform nematic MFP and satisfies Eq. ( 10). Knowing the solution nj( i), one can compute h,Jx,), satisfying Eq. (18). We split the volume V into molecular layers parallel to the solid plane, where n numbers the layers. In each of these layers there are
334
A. V. Zawlarov /Physics Letters A 187 (1994) 331-336
N, molecules which occupy single cells. All MFPs belonging to a layer are equal and change with variable xi and number n. In principle, of course, this system of linear integral equations is infinite, but in the bulk fluid hi,j(Xi) =O, and in any application this system must be truncated after a finite number of disturbed layers. If we suppose further that h,j( Xi) = (Y+i, we can simplify Eq. ( 18 ) and extend the “analytical theory”. Using this concept, Eq. ( 18) can be written in the form (y.
F’A-- (a;-%)C(i)
_x_=p-l_
1.1
I
A-~c~~B
Here we have defined the new parameter C,-,(i) as in ( 19 ), but used the solid-LC potential @(Xi) instead of the liquid-liquid potential 0( ij). Knowing the (Yi,, one can compute the full solution
l=Wi,j(i) n exP(
exP[ -P%,j(i)
j+l
-DOG,,&)
)
and the surface free energy
(19)
)
d(i) exp[ -&l’“‘(i)]
(24)
>.
where
5. Models of molecular interactions
ffj=
1
ffj,,
.
jzr
We get an infinite set of non-linear algebraic equations. The general solution of this system cannot be given. However, an important special solution can be obtained. Let n be the number of disturbed layers. In this case the last equation in ( 19) may be rewritten as a,_,,,=/3-‘(x;‘+AIB) ,
7
= [Z(i)8-‘~,_l,n11’2
whereZ(i)= one obtains lation for
(21) and (20), a recursion re-
,,
2XipB
G2)
=C(i)
+B
’
and
C(i)=C,(i),
j=O, ,
j#O.
K
aoa(e,
ej, Gj> 40
)I. 6
>(
~ofJ(ei,
-
The potential-energy parameter parameter a(ei, ej, e,) depend orientations and are given by 1-2
O(ei,ej,e,)=
[
ej, e, qij
(25)
E(e,, ej) and the size upon the molecular
(f?i’C?Jj-ejef?ij)2
(23)
X
(e~‘f?ij+ejsf?,)2
1 +xe,
(
)I ’ --I”
.t?j
(26)
and e(f?i,
xi(A_BBaj+l,j+2)
and O&j
4Eoc(ej,e,1
02(ej,ej, eo)
1 -xe, se,
where =
1=
12
X
+
1l/2
q+,j+2
ej, e,-
(21)
[B-C(i)]//3xiB.Using after some manipulation
cUj,j+
Wei,
(20)
so long as a!,= cx,_ i,+. Now, it is clear that the next Eq. ( 19) can be expressed in terms of (Y,_ I,n as a,_,,,_,
The kernel of the integral equation ( 10) is determined by the molecular interaction potential. We chose the interaction potential of two molecules in the form [lo]
f?j)
=
[
1
-X2(f?i’ej)2]-“2,
(27)
where go= qi - qj is the distance between the molecular centers, e, ej are unit vectors giving the orientation of the two molecules i and j, eij= qrj/ 1qij 1, ~0and a0 are the energy and size parameters of the potential. The shape parameter x is x= (oi -0: ) / (0: + o: ), where g,, is the separation when the molecules are end-
335
A. V. Zakharov /Physics Letters A 187 (1994) 331-336
to-end and trI that when they are side-by-side. The molecule-wall interaction is denoted by [ 111
00
m=-
I
dyC(y)y*,
p=v-‘,
0
(28) The parameters in the potential are also orientation dependent,
(
l-
CJ,(ei)=
~=(P*(COS8i))=f(2(COS28i)-l),
-l/2
>
x& 1-X*e:,
where C( qii, ei, ej) = C( qij/a) is the direct correlation function, rs is the size parameter, which depends on molecular orientations, and is given by (26 ). The quantities determining the angular correlations between molecules are
’
(29)
where Ed is a constant which we shah identify shortly, and Xi designates the distance from the wall to the molecule i, f?i= (ei,,, ei,Y, et,,).
?l
=
(p4(cos
ei)
>
=$(35(COS4Bi)-30(COS219i)+3), rl2 = =
(~6(cos&) &
Within the framework of the theory [ 41, which is based on a density functional approach, one can write a formally exact expression for the Frank elastic constants, K,/K=l+p(5-9z))
(30)
K2/K=l-p(l++Z),
(31) 1 - 32) ,
KS/K= l -4p(
(232(C0S68i)
-315(C0S48i)
constant, ?I
z= - -Y
P= 4(y2+2)
?-)I1
Y=
02-l w2+1’
K,/K=
1+A-3A’q,/q,
(36)
K,/K=
1-2A-A’q,/q,
(37)
K,/K=
1 +A+4A’q,/rj.
(38)
> A=-
o=J7L 9 Cl
=fL(31-1)*(1-ttl)(3-~)/2~, 1+&J* ~=3Mb2P2d~Y3B(l_Y,),
s
m
0 C(Y)Y~,
0
1+&y*
b=4na&npy2 ~ 3(1-Y)
’
2R2-2 7R2+20’
A’= 27(tR2-iL) 7R2+20
’
R=w-1.
K=f(K,+K*+K,)
M= -
where P,( cos 6,) is the Legendre polynomial of order 1, and (...) =Jd( i) . . . F’i( i). There is another functional approach [ 2 1, based on the Onsager approximation for long hard spherocylinders, which allows us to obtain the Frank elastic constants in terms of the direct correlation function,
The quantities A and A’ depend on the parameters of the molecules,
y*-1
-t/2
(35)
(32)
where K, is the splay, K2 the twist, and Ka the bend
(34)
>
+lo5(c0s*ei)-5)) 6. Results for the Frank constants
(33)
For the correlation function Nemtsov took for a better approximation, C(q,, ei, ej) = C(qJa) [ 41, than the approximation introduced by Onsager [ 2 1,
>
although the difference is not very significant numerically. The calculations were carried out with the values of the potential parameters of the corresponding values for the PAA: 0,=5.01 A, co/k=520 K, and the surface parameter cow= 5.0~~.This system is charac-
336
0.
A. V. Zakharov /Physics
1 1
Letters A 187 (1994) 331-336
cause they have larger coefficients at these order parameters. Nevertheless, the values for the elastic constants calculated in the framework of these approaches are different, although the difference is not very significant in the bulk. We find that 0.5-cK,/K,c3.0 and 0.5-cK2/K1 ~0.8 for all L. These results are in agreement with experiment [ 12 1. In view of the fact that the potential model and the parameters used in the calculation only crudely simulate a real system the agreement is satisfactory. 2
3
L
Fig. 1. Dependence of the reduced constants KJKon the number of cells L from the wall, for o=B,/u~ = 3.0, tb= 5.0to, reduced density p*=O.2 and temperature 9=0.75: splay constants K,IK ((m) calculated by means of Eq. (30), (0) Eq. (36)), twist constants KJK ((a) Eq. (31), (0) Eq. (37)) and bend constantsK,/K((V) Eq. (32), (A) Eq. (38)).
Acknowledgement Support by the Russian Fond for Fundamental Research (N 99-O 1-O10 18-9) is gratefully acknowledged.
References terized by reduced parameters: the reduced density p* =~a;, the reduced temperature /I- ’ = 8= kT/c,and the anisotropy parameter x. Calculations of the Frank elastic constants for a number of cells L from the wall which were performed using the different approaches of Nemtsov and Poniewierski and Stecki are shown in Fig. 1. The ratio KJK only slightly depends on L while K,/K and KJK depend on L strongly. We can ascribe such behaviour of the elastic constants to a tendency of the molecules to be more ordered near the surface than in the bulk. In both expressions for the Frank elastic constants, (30)-(32) and (36)(38), only the order parameters q, q1 and q2 (or their combinations) depend on the value L and we see that magnitudes of the calculated splay and bend deformations are more variable than the twist one, be-
[ I] R.G. Priest, Phys. Rev. A 7 ( 1973) 720. [ 21 A. Poniewierski and J. Stecki, Mol. Phys. 38 ( 1979) 193 I. [ 31 K. Singh and Y. Singh, Phys. Rev. A 34 (1986) 548. [ 41 V.B. Nemtsov, Theor. Appl. Mech. (Minsk) 13 ( 1987) 16 [In Russian 1. [5] S.D. Lee and R.B. Meyer, J. Chem. Phys. 56 (1986) 3443. [6] L.A. Rott, Statistical theory of molecular systems (Nauka, Moscow, 1979); L.A. Rott and V.S. Vikhrenko, Fortschr. Phys. 23 (1975) 133. [ 71 A.V. Zakharov, Chem. Phys. Lett. 170 (1990) 239. [8] A.V. Sakharov, PhysicaA 174 (1991) 327; 175 (1991) 327. [9] I.M. Sobol, SIAM J. Numer. Anal. 16 (1979) 790. [lo] G.J. Gay and J. Berne, J. Chem. Phys. 74 (1981) 3316. [ 111 A.L. Tsykalo and A.D. Bagmet, Acta Phys. Pol. A 55 ( 1979) 1 Il. [ 121 F. Leenhouts, A.D. Dekker and W.H. de Jeu, Phys. Lett. A 72 (1979) 155.
s
25 April 1994 PHYSICS
LETTERS
A
4._
ELSEVIER
Physics Letters A 187 ( 1994) 331-336
Elastic constants of a system composed of interacting ellipsoidal molecules near an interacting wall A.V. Zakharov Saint PetersburgInstitutefor Machine Sciences, RussianAcademyof Sciences, Saint Petersburg199178, RussianFederation Received 29 December 1993; accepted for publication 7 February 1994 Communicated by V.M. Agranovich
Abstract A statistical-mechanical theory based upon the method of conditional distribution is applied to the calculation of the Frank elastic coefftcients of a system composed of interacting ellipsoidal molecules near an interacting wall. Three reduced elastic constants K,, K2 and K3 are presented as functions of the parameter ellipticity, the set of order parameters and the distance from the wall.
1. Introduction Although many attempts have been made to describe the elastic properties in the bulk of a liquid crystal (LC) [ l-51, there is as yet no microscopic treatment of such a problem near a solid wall. The purpose of this note is to present a simple molecular model based upon the method of conditional distribution [6], in which an attempt has been made to combine the advantages of integral equation theory and those of the cell model approach [ 71 together with some statistical-mechanical ideas [ 8 1, allowing calculation of the Frank elastic constants K,, K2 and K3 which describe the splay, torsion and bending deformation in the nematic phase near the interacting wall. Although the calculations have been made for a nematic liquid crystal (NLC ), the general approach can be applied to any liquid crystalline phase. 2. Model We consider a system of N particles in a volume V. The potential energy associated with N particles is a Elsevier Science B.V. SSDZ0375-9601(94)00157-K
sum of pair potentials 0( zj), each of which depends on the positions gi and qj and orientations ei ( 1ei I= 1) and ej of particles with numbers i and j. We divide the volume of the system V, in contact with the solid planar wall, into N equal cells. The volume of each cell is V= V/N and every cell is occupied by a molecule. The x-axis of the coordinate frame is chosen to be normal to the wall and nematic molecules occupy the half-space x> 0, and the space-fixed z-axis is chosen so that the nematic director lies in the yz plane. By integration of the Gibbs canonical distribution, a set of functions F’,(i), Fij(Q), etc., is introduced in order to define the probability densities for the molecules to be found about the positions m= W,,, ( W,,, = ~,,,@a,, a, is the volume associated with orientations, m = i, j). Using mean force potentials (MFPs) [ 6-81 we can present the abovementioned functions in the form
Fi(i)=Q-’
Fij(ij)=Q-2exP{-p[~(ii)+~ij(ii)l} where
(1)
ev[ -Bpz(i) I , 7
(2)
332
A. V. Zakharov /Physics Letters A 187 (1994) 331-336
I
a
L’
@(ij) is the interaction potential of two molecules, pi(i) and pij( ij) are the mean force potentials and are sums of the forms Pi(i)
=
,gj
Pi,j(i)
t
v)rj,/(ij)
=
,$?
(3)
Pij,/(ti)
Of course, the pij( i) will also depend on the distance of the cell i from the solid wall. The terms in the sums ( 3 ) are given by
V,Pi,j(i)=
I
dO’) Vi@(ij)Fij(ii)lFz(i) ,
(4)
I
Vipij,,(G)=
s I
d(l)Vi~(U)Fij~(iir)/Fij(ij)
.
(5)
The subscripts before the comma correspond to the MFP dependence on the positions of the molecules; those after the comma correspond to the average states. Using now the relations between the single and binary functions which follow from their definition, we have s
d(i) F,(i) = 1 ,
Fi(i)=
J”dG) F,(y)
(6)
i
The two-particle function is related to the three-particle function by an integral relation, etc. Below we take into account the first two functions of the in% nite hierarchy; this corresponds to taking into account only pair correlations between molecules. In order to make this a closed system of equations, we separate the mean force potentials into irreducible parts [ 6-8 1. In the case considered here we have VijJC,r(ii)
=%,di)
+Vj,/,rO’)
If the irreducible zero, Lj,,
.
+tij,l(ii)
part of the potential
(7)
which corresponds to neglecting three-particle and higher-order correlations, the expression for the binary function takes the form Fij(ij)=exP{B[yli,j(i) XK(g)Fi(i)FjG)
+Vj,iU)
,
exP] =
-&j(i)
s
1
dti) exp[Pfl,lO’) lK(ii)W_i)
.
(10)
Knowing the solution of this system one can compute the microscopic characteristics of the LC (expressed in terms of the one-particle and two-particle functions) and also the macroscopic characteristics, which can be expressed in terms of the free energy of the system. The free energy per molecule is given by f=-p-‘lnjd(i)exp[-pyl,(i)].
(11)
It should be mentioned that in the nematic phase the MFPs pi,,(i) are equal for cells in the same layer, parallel to the planar wall and differ for cells in the adjacent layers, perpendicular to the wall, and, theoretically, one should take into account an infinite number of adjacent layers. But in practice, a finite number of adjacent layers shall be taken into account and it will be shown that MFPs decay to their bulk values. The decay length or the number of layers in which MFPs are different from the ones in the bulk depends strongly on an external field due to the solid wall, the pairwise interaction potential and intermolecular correlations.
3. Method of solving the non-linear integral equation
is set equal to
(8)
=O >
where p- ’ GO= kT is temperature and K(ij) = exp [ -/IlO 1. The exponential factor in this expression reflects the correlation between molecules and distinguishes the approach used here from the mean field approximation. Substitution of (9) into (6) leads to a closed system of non-linear integral equations (NIEs) for the MFPs,
I)
(9)
For a system which is non-uniform only in the xdirection the solution of Eq. ( 10) can be written in the form Vi,,(i)
=d,,F’
+hi,j(x,)
,
(12)
where qj,y)(i) is the nematic equilibrium MFP and 1h,,j( Xi) 1<< 1qi,‘j’ (i) 1. NOW our purpose is to find the rp$)(i) which satisfies Eq. (10) and the hi,l(Xi), the equation for which will be derived in the next sec-
A. V. Zakharov /Physics Letters A 187 (1994) 331-336
tion. The solution of Eq. ( 10) for the nematic phase is based upon the fact that the one-particle function F,(i) is a function of cos &, where 6i is the angle between the z,-axis and the long molecular axis of the ith molecule. The’most general form of Fi( i) is Vi(COS
Fi(i)=Ji d(i) v=exp(
ei)
~i(cOs8i)
-j@(O))
’
.
(13)
The method of solution of the five-dimensional problem ( 10) is most complicated. It is convenient to rewrite ( 10) in the form
u/i,j(Qi) =
Jrq dQjK(Qi, Qj> v/j(Qj)lWi,i(Qj) IqdQj WjCQj> ’
(14)
where Qj is a vector in the live-dimensional space Wj= Vj@,s2i.The solution can be found using successive approximations calculated from the formula
333
bit structure with six nearest neighbours, which were also taken into account in the calculations. This restriction was dictated by the computer available for the study. The implementation of the algorithm ( 15) for Eq. ( 14) was done as follows. The initial approximation was chosen to be ws’ (Qi ) = vvi[” (Qi ) = 1. Then the integral on the right-hand side of ( 15) was calculated according to ( 16) with help of an LP, sequence. This procedure was repeated for all six neighbours of the particle in cell i. The coordinate Qi in ( 15) was chosen such that vi;’ (Qi ) would be mlculated at the points forming the same stationary LP, sequences as used in the evaluation of the integrals with the initial approximation ~~,~’(Qi ). Then ” ( Qi ) was calculated by a simple multiplication of wj,yl ( Qi). The procedure was then iterated until a given accuracy was achieved, where the translational invariance of the total potential was taken into account.
wf
[Nl
LN’ll(Qi)=(lU; 1, i,g!(Qj)
WiJ
4. Derivation of the linearized integral equation for a small4 expansion
l/2
X
1
dQjK(Qi,
Qj)WJ"'(Qj)/WJ?'(Qj)
W
> (15)
and the solution does not depend on the initial approximation. The basic difficulty in solving the equation is that the algorithm ( 15) requires the successive evaluation of live-dimensional integrals. In the present paper we apply the method of Sobol [9] based on the Haar functions, calculating the multi-dimensional integrals 1 dx 1 ... 1 0
I 0
exp]
-Phi,j(Xi)1z 1-BhiJCxi)T
***%I)=
z1 ~@w. =
- ~~;“I
I/ ) l&“’
(16)
1,
for Qic Wi is about 0.1%. The cells form a simple
CU-
(17)
and on using the notation vi,j( i) = exp [ - j@i$‘)( i) 1, the linearized integral equation is l-&(xi)=
dx,_f(&
where Qa are points uniformly distributed in the ndimensional unit cube. These points belong to an LP, sequence, N is the number of points. The precision of our calculations is about R = 0 (N - ’ ln”N), where n is the dimensionality of space (our calculations were executed using N= 100). For the case with N=200, the difference between the potentials vi>“’ and t&O01 in the nematic phase 6= 1I&“]
The natural way to study the “small” solution of hij(xi) is by linearization of the non-linear functional, exp[ -&oJi) 1. Accordingly, we make the approximation
X
(
J dO’) CvjO‘){l-BVi,j(i)K(ij) j
thjCxj> -hj,i(Xj)l)lWj,iti) > (18)
where hj( Xj) = Cl+ i h,( Xi) sThe function vij( i) is the uniform nematic MFP and satisfies Eq. ( 10). Knowing the solution nj( i), one can compute h,Jx,), satisfying Eq. (18). We split the volume V into molecular layers parallel to the solid plane, where n numbers the layers. In each of these layers there are
334
A. V. Zawlarov /Physics Letters A 187 (1994) 331-336
N, molecules which occupy single cells. All MFPs belonging to a layer are equal and change with variable xi and number n. In principle, of course, this system of linear integral equations is infinite, but in the bulk fluid hi,j(Xi) =O, and in any application this system must be truncated after a finite number of disturbed layers. If we suppose further that h,j( Xi) = (Y+i, we can simplify Eq. ( 18 ) and extend the “analytical theory”. Using this concept, Eq. ( 18) can be written in the form (y.
F’A-- (a;-%)C(i)
_x_=p-l_
1.1
I
A-~c~~B
Here we have defined the new parameter C,-,(i) as in ( 19 ), but used the solid-LC potential @(Xi) instead of the liquid-liquid potential 0( ij). Knowing the (Yi,, one can compute the full solution
l=Wi,j(i) n exP(
exP[ -P%,j(i)
j+l
-DOG,,&)
)
and the surface free energy
(19)
)
d(i) exp[ -&l’“‘(i)]
(24)
>.
where
5. Models of molecular interactions
ffj=
1
ffj,,
.
jzr
We get an infinite set of non-linear algebraic equations. The general solution of this system cannot be given. However, an important special solution can be obtained. Let n be the number of disturbed layers. In this case the last equation in ( 19) may be rewritten as a,_,,,=/3-‘(x;‘+AIB) ,
7
= [Z(i)8-‘~,_l,n11’2
whereZ(i)= one obtains lation for
(21) and (20), a recursion re-
,,
2XipB
G2)
=C(i)
+B
’
and
C(i)=C,(i),
j=O, ,
j#O.
K
aoa(e,
ej, Gj> 40
)I. 6
>(
~ofJ(ei,
-
The potential-energy parameter parameter a(ei, ej, e,) depend orientations and are given by 1-2
O(ei,ej,e,)=
[
ej, e, qij
(25)
E(e,, ej) and the size upon the molecular
(f?i’C?Jj-ejef?ij)2
(23)
X
(e~‘f?ij+ejsf?,)2
1 +xe,
(
)I ’ --I”
.t?j
(26)
and e(f?i,
xi(A_BBaj+l,j+2)
and O&j
4Eoc(ej,e,1
02(ej,ej, eo)
1 -xe, se,
where =
1=
12
X
+
1l/2
q+,j+2
ej, e,-
(21)
[B-C(i)]//3xiB.Using after some manipulation
cUj,j+
Wei,
(20)
so long as a!,= cx,_ i,+. Now, it is clear that the next Eq. ( 19) can be expressed in terms of (Y,_ I,n as a,_,,,_,
The kernel of the integral equation ( 10) is determined by the molecular interaction potential. We chose the interaction potential of two molecules in the form [lo]
f?j)
=
[
1
-X2(f?i’ej)2]-“2,
(27)
where go= qi - qj is the distance between the molecular centers, e, ej are unit vectors giving the orientation of the two molecules i and j, eij= qrj/ 1qij 1, ~0and a0 are the energy and size parameters of the potential. The shape parameter x is x= (oi -0: ) / (0: + o: ), where g,, is the separation when the molecules are end-
335
A. V. Zakharov /Physics Letters A 187 (1994) 331-336
to-end and trI that when they are side-by-side. The molecule-wall interaction is denoted by [ 111
00
m=-
I
dyC(y)y*,
p=v-‘,
0
(28) The parameters in the potential are also orientation dependent,
(
l-
CJ,(ei)=
~=(P*(COS8i))=f(2(COS28i)-l),
-l/2
>
x& 1-X*e:,
where C( qii, ei, ej) = C( qij/a) is the direct correlation function, rs is the size parameter, which depends on molecular orientations, and is given by (26 ). The quantities determining the angular correlations between molecules are
’
(29)
where Ed is a constant which we shah identify shortly, and Xi designates the distance from the wall to the molecule i, f?i= (ei,,, ei,Y, et,,).
?l
=
(p4(cos
ei)
>
=$(35(COS4Bi)-30(COS219i)+3), rl2 = =
(~6(cos&) &
Within the framework of the theory [ 41, which is based on a density functional approach, one can write a formally exact expression for the Frank elastic constants, K,/K=l+p(5-9z))
(30)
K2/K=l-p(l++Z),
(31) 1 - 32) ,
KS/K= l -4p(
(232(C0S68i)
-315(C0S48i)
constant, ?I
z= - -Y
P= 4(y2+2)
?-)I1
Y=
02-l w2+1’
K,/K=
1+A-3A’q,/q,
(36)
K,/K=
1-2A-A’q,/q,
(37)
K,/K=
1 +A+4A’q,/rj.
(38)
> A=-
o=J7L 9 Cl
=fL(31-1)*(1-ttl)(3-~)/2~, 1+&J* ~=3Mb2P2d~Y3B(l_Y,),
s
m
0 C(Y)Y~,
0
1+&y*
b=4na&npy2 ~ 3(1-Y)
’
2R2-2 7R2+20’
A’= 27(tR2-iL) 7R2+20
’
R=w-1.
K=f(K,+K*+K,)
M= -
where P,( cos 6,) is the Legendre polynomial of order 1, and (...) =Jd( i) . . . F’i( i). There is another functional approach [ 2 1, based on the Onsager approximation for long hard spherocylinders, which allows us to obtain the Frank elastic constants in terms of the direct correlation function,
The quantities A and A’ depend on the parameters of the molecules,
y*-1
-t/2
(35)
(32)
where K, is the splay, K2 the twist, and Ka the bend
(34)
>
+lo5(c0s*ei)-5)) 6. Results for the Frank constants
(33)
For the correlation function Nemtsov took for a better approximation, C(q,, ei, ej) = C(qJa) [ 41, than the approximation introduced by Onsager [ 2 1,
>
although the difference is not very significant numerically. The calculations were carried out with the values of the potential parameters of the corresponding values for the PAA: 0,=5.01 A, co/k=520 K, and the surface parameter cow= 5.0~~.This system is charac-
336
0.
A. V. Zakharov /Physics
1 1
Letters A 187 (1994) 331-336
cause they have larger coefficients at these order parameters. Nevertheless, the values for the elastic constants calculated in the framework of these approaches are different, although the difference is not very significant in the bulk. We find that 0.5-cK,/K,c3.0 and 0.5-cK2/K1 ~0.8 for all L. These results are in agreement with experiment [ 12 1. In view of the fact that the potential model and the parameters used in the calculation only crudely simulate a real system the agreement is satisfactory. 2
3
L
Fig. 1. Dependence of the reduced constants KJKon the number of cells L from the wall, for o=B,/u~ = 3.0, tb= 5.0to, reduced density p*=O.2 and temperature 9=0.75: splay constants K,IK ((m) calculated by means of Eq. (30), (0) Eq. (36)), twist constants KJK ((a) Eq. (31), (0) Eq. (37)) and bend constantsK,/K((V) Eq. (32), (A) Eq. (38)).
Acknowledgement Support by the Russian Fond for Fundamental Research (N 99-O 1-O10 18-9) is gratefully acknowledged.
References terized by reduced parameters: the reduced density p* =~a;, the reduced temperature /I- ’ = 8= kT/c,and the anisotropy parameter x. Calculations of the Frank elastic constants for a number of cells L from the wall which were performed using the different approaches of Nemtsov and Poniewierski and Stecki are shown in Fig. 1. The ratio KJK only slightly depends on L while K,/K and KJK depend on L strongly. We can ascribe such behaviour of the elastic constants to a tendency of the molecules to be more ordered near the surface than in the bulk. In both expressions for the Frank elastic constants, (30)-(32) and (36)(38), only the order parameters q, q1 and q2 (or their combinations) depend on the value L and we see that magnitudes of the calculated splay and bend deformations are more variable than the twist one, be-
[ I] R.G. Priest, Phys. Rev. A 7 ( 1973) 720. [ 21 A. Poniewierski and J. Stecki, Mol. Phys. 38 ( 1979) 193 I. [ 31 K. Singh and Y. Singh, Phys. Rev. A 34 (1986) 548. [ 41 V.B. Nemtsov, Theor. Appl. Mech. (Minsk) 13 ( 1987) 16 [In Russian 1. [5] S.D. Lee and R.B. Meyer, J. Chem. Phys. 56 (1986) 3443. [6] L.A. Rott, Statistical theory of molecular systems (Nauka, Moscow, 1979); L.A. Rott and V.S. Vikhrenko, Fortschr. Phys. 23 (1975) 133. [ 71 A.V. Zakharov, Chem. Phys. Lett. 170 (1990) 239. [8] A.V. Sakharov, PhysicaA 174 (1991) 327; 175 (1991) 327. [9] I.M. Sobol, SIAM J. Numer. Anal. 16 (1979) 790. [lo] G.J. Gay and J. Berne, J. Chem. Phys. 74 (1981) 3316. [ 111 A.L. Tsykalo and A.D. Bagmet, Acta Phys. Pol. A 55 ( 1979) 1 Il. [ 121 F. Leenhouts, A.D. Dekker and W.H. de Jeu, Phys. Lett. A 72 (1979) 155.