Elastic Constants of Binary Liquid Crystalline Mixtures

Elastic Constants of Binary Liquid Crystalline Mixtures

A. Kapanowski and K. Sokalski

Institute of Physics, Jagellonian University, ul. Reymonta 4, 30-059 Cracow, Poland Z. Naturforsch. 53 a, 963–976 (1998); received September 24, 1998

Microscopic expressions for the elastic constants of binary liquid crystalline mixtures composed of short rigid uniaxial molecules are derived in the thermodynamic limit at small distorsions and a small density. Uniaxial and biaxial nematic phases are considered. The expressions involve the one-particle distribution functions and the potential energy of two-body short-range interactions. The theory is used to calculate the phase diagram of a mixture of rigid prolate and oblate molecules.

The concentration dependence of the order parameters and the elastic constants are obtained. The possibility of phase separation is not investigated.

Key words: Liquid Crystals; Nematics; Elastic Constants; Mixtures.

1. Introduction

The elastic constants of liquid crystals are the ma- terial constants that appear in the description of al- most all phenomena where the variation of the di- rector is manipulated by external fields [1]. They are of technological importance because liquid crys- tals have found wide application, e. g., in display de- vices, laser technique, holography, termography, nu- clear and microwave techniques. On the other hand, the elastic constants give information on the micro- scopic anisotropic intermolecular forces. They are also needed in the study of defects in liquid crys- tals [2].

There are microscopic theories [3 - 8] that give working expressions for the elastic constants of one- component uniaxial nematic liquid crystals. But in technical applications very often some special proper- ties are required, and chemically pure substances with the desired ones are hard to find. That is why mix- tures are widely used. It is clear that theories which allow to understand the physical properties of mix- tures are helpful in designing mixtures with the pre- scribed technical parameters. Miscibility studies are also important from a more fundamental point of view – to identify new phases. The rule that is used is the following: if two phases are continuously miscible without crossing any (first- or second-order) transi- tion line, they have the same symmetry. This method Reprint requests to Dr. A. Kapanowski;

E-mail: kapanow@izis.if.uj.edu.pl.

0932–0784 / 98 / 1200–0963 $ 06.00^c Verlag der Zeitschrift f¨ur Naturforschung, T¨ubingenwww.znaturforsch.com can often be carried out under the microscope (in a concentration gradient) and is faster than taking an X-ray picture [1].

In this paper we present a statistical theory of the elastic constants of binary liquid crystalline mix- tures. Uniaxial and biaxial nematic phases will be considered. Since a theoretical description of biax- ial phases (and mixtures) is rather complex [9, 10]

we developed our theory with some approximations:

rigid molecules, small density and the thermodynamic limit. Our aim is to express the elastic constants by means of the one-particle distribution functions and the potential energy of molecular interactions. The thermodynamic limit suggests that we neglect sur- face effects. Nevertheless we will obtain some known relations for the surface elastic constants and we will interprete those relations as consistency conditions.

The phase behaviour of liquid crystal mixtures has been studied using a number of theoretical methods.

A large variety of phase diagrams was presented in the paper by Sivardiere [11] where the Ising-like model was introduced. Brochard et al. [12] considered the less artificial Maier-Saupe model and gave a catalogue of allowed diagrams for mixtures of nematogens.

In 1973 Alben [13] considered a mean-field lattice model with discrete orientations to describe steric in- teractions in mixtures of rods and discs. He predicted the existence of a biaxial phase in the composition range between two uniaxial phases. Similar results have been obtained for van der Waals lattice mod- els in mean-field [14] and renormalization group [15]

theories. The question of thermodynamical stability

964 A. Kapanowski and K. SokalskiElastic Constants of Binary Liquid Crystalline Mixtures against phase separation was addressed by Palffy-

Muhoray et al. [16] within a mean-field theory. The geometric mean assumption about the pseudopoten- tial leads to instability of a biaxial phase, whereas a deviation from this assumption leads to a stable biaxial phase [17]. A mean-field theory was also used to describe binary mixtures of biaxial molecules [18, 19]. Mixtures of rods and spheres were studied by Agren [20], Humphries and Luckhurst [21], and Martire et al. [22]. The result is that the introduc- tion of spheres induces, via a small two-phase re- gion, a transition to an isotropic phase. Mixtures of rods of different length-to-width ratios were analysed by Peterson et al. [23], Warner and Flory [24], and Lekkerkerker et al. [25]. Recently, different demixing mechanisms in hard rod [26] and rod-plate mixtures [27] were analysed by van Roij and Mulder.

As far as the elastic constants are concerned, to our knowlwdge there are no experimental data on biaxial phases. In 1989 Kini and Chandrasekhar [28]

studied the effects of external magnetic and electric fields applied in different geometries. They showed that it is feasible to determine some of the twelve elastic constants. Our theory could help to predict the temperature and concentration dependence of them in the case of binary mixtures.

Our paper is organized as follows: In Sect. 2 we present a phenomenological continuum theory of ne- matic liquid crystals. In Sect. 3 we describe a sta- tistical theory of nematic phases and derive general expressions for the elastic constants in the case of uniaxial and biaxial nematic phases. Exemplary cal- culations of the values of the elastic constants are presented in Sect. 4, where the Corner potential en- ergy is applied and a mixture of rods and discs is analysed. In Sect. 5 we summarize the results of this work.

2. Phenomenological Approach 2.1. Description of a Phase

In this section we describe nematic liquid crys- tals from a phenomenological point of view [29].

We assume that at every point r inside a consid- ered phase we can define three orthonormal ver- sors (L(r)

;

^M(r)

;

N(r)) which reflect properties of this phase. In case of a biaxial phase they determine di- rections of its two-fold axes of symmetry. The vec- tors (L

;

^M

;

N) create the local frame which can be

expressed by means of a space-fixed reference frame (ex

;

^ey

;

^{ez) as}

L =

R

^1e

;

^{M =}

R

^2e

;

^{N =}

R

^3e

;

⁽¹⁾

where the matrix elements

R

^{i (}

i

^{= 1}

;

²

;

^{3 and}

⁼

x;y;z

) satisfy the conditions

R

ⁱ

R

^j=

^ij

;

⁽²⁾

R

ⁱ

R

^{i =}

:

⁽³⁾

Relations (2) and (3) express the orthogonality and the completeness of the local frame. Note that repeated indices imply summation. The homogeneous phase is described by

R

ⁱ⁼

^i.

2.2. Distorsion Free Energy

Let us call

F

^dthe free energy due to the distorsion of the local frame (L

;

^M

;

N). A general expression of its density

f

^d(r) was derived in [29] in the case of small distorsions. It has the form

f

^{d(r) =}

k

^ij

D

^ij+1₂

K

^{ijk l}

D

^ij

D

^{k l+1}₂

L

^ijk

S

^ijk

;

⁽⁴⁾

where

k

^ij

;K

^{ijk l}

;L

^ijk are elastic constants,

D

^{ij =1}₂

^{jk l}

R

ⁱ

R

^{k ∂}

R

^l

;

⁽⁵⁾

S

^{ijk =}

S

^{jik =∂(}

R

ⁱ

D

^jk+

R

^j

D

^ik)

;

⁽⁶⁾

and

^ijkis an element of the antisymmetric tensor (we set up the convention

¹²³ = +1). The elastic constants satisfy the symmetry relations

K

^{ijk l=}

K

^{k lij}

; L

^{ijk =}

L

^jik

:

⁽⁷⁾

In general, the linear first order terms with

k

^{ij give}

6 bulk and 3 surface terms; the quadratic first order terms with

K

^{ijk l} give 39 bulk and 6 surface terms;

the terms with

L

^ijk give 18 surface terms. The to- tal number of bulk and surface terms is 45 and 27, respectively.

When a considered phase has a

D

^{2h symmetry}

group, the distorsion free-energy density has the form

f

^d=1₂

K

¹¹¹¹⁽

D

¹¹⁾²⁺¹₂

K

¹²¹²⁽

D

¹²⁾²⁺¹₂

K

¹³¹³⁽

D

¹³⁾²

+1

2

K

²¹²¹⁽

D

²¹⁾²⁺¹₂

K

²²²²⁽

D

²²⁾²⁺¹₂

K

²³²³⁽

D

²³⁾²

+1

2

K

³¹³¹⁽

D

³¹⁾²⁺¹₂

K

³²³²⁽

D

³²⁾²⁺¹₂

K

³³³³⁽

D

³³⁾²

+

K

¹¹²²

D

¹¹

D

²²⁺

K

¹¹³³

D

¹¹

D

³³⁺

K

²²³³

D

²²

D

³³

+

K

¹²²¹

D

¹²

D

²¹⁺

K

¹³³¹

D

¹³

D

³¹⁺

K

²³³²

D

²³

D

³²

+

L

¹²³

S

¹²³⁺

L

²³¹

S

²³¹⁺

L

³¹²

S

³¹²

:

⁽⁸⁾

The terms with

K

^{ijk l}give 12 bulk and 3 surface terms, whereas the terms with

L

^ijkgive 3 surface terms. The total numbers of bulk and surface terms are 12 and 6, respectively.

When a considered phase possesses a

D

^{1h sym-}

metry group, the number of elastic constants is smaller. Let the

z

axis be oriented along the axis of symmetry. Then the distorsion free-energy density has the form

f

^{d =1}₂

K

^1(rN)2+1₂

K

^2[N(rN)]2

+1

2

K

^3[N(rN)]2

+1

2

K

^{4r[(Nr)N;N(rN)]}

+1

2

K

^{5r[(Nr)N + N(rN)]}

:

(9)

Therefore, in case of an uniaxial phase we have 3 bulk (

K

^1,

K

^{2 and}

K

³) and 2 surface terms (

K

^{4 and}

K

^5).

One can calculate the distorsion free energy from its density (8) or (9) by

F

^{d=Z d r}

f

^d

:

⁽¹⁰⁾

Note that we can not reject surface terms in (8) or (9) although we assume the thermodynamic limit. This will be explained in Sect. 5.

2.3. Basic Deformations

Splay, twist, and bend are known as the three ba- sic types of deformations in the continuum theory of uniaxial nematics. They describe spatial variations of the director N(r) and extract from the distorsion free energy terms with

K

^1,

K

^{2 and}

K

³, respectively. In [10] 18 basic deformations proper for the continuum

theory of biaxial nematics were given. They were di- vided into five groups and connected with relevant elastic constants: 3 twists (for

K

ⁱⁱⁱⁱ), 6 splays and bends (for

K

^ijij), 3 modified twists (for

L

^{ijk) and two}

groups of 3 double twists (for

K

^{iijjand for}

K

^{ijji). In}

the formulas for deformations a parameter

^{was used}

(1

=

is a certain length). Small

meant a small defor- mation and a conformation close to the homogeneous one (L⁽⁰⁾

;

^M(0)

;

^N(0)). The vectors of the local frame were expanded into a power series with respect to

L = L⁽⁰⁾+

^L(1)+

^2L(2)+

:::;

M = M⁽⁰⁾+

^M(1)+

^2M(2)+

:::;

N = N⁽⁰⁾+

^N(1)+

^2N(2)+

::: :

(11)

It appeared that the most important terms in (11) were those linear in

. They were sufficient to calculate the distorsion free enery up to the second order in

^and

to calculate the elastic constants of biaxial nematic liquid crystals. For the sake of completeness we list the terms linear in

from (11) for all groups of defor- mations. The first group is, for

K

^1111,

L⁽¹⁾= (0

;

⁰

;

⁰⁾

;

^{M(1)= (0}

;

⁰

;x

⁾

;

^{N(1)= (0}

;

^;

x;

⁰⁾

;

⁽¹²⁾

for

K

^2222,

L⁽¹⁾= (0

;

⁰

;

^;

y

⁾

;

^{M(1)= (0}

;

⁰

;

⁰⁾

;

^{N(1)= (}

y;

⁰

;

⁰⁾

;

⁽¹³⁾

and for

K

^3333,

L⁽¹⁾= (0

;z;

⁰⁾

;

^{M(1)= (;}

z;

⁰

;

⁰⁾

;

^{N(1)= (0}

;

⁰

;

⁰⁾

:

⁽¹⁴⁾

The second group is, for

K

^1212,

L⁽¹⁾= (0

;

⁰

;

^;

x

⁾

;

^{M(1)= (0}

;

⁰

;

⁰⁾

;

^{N(1)= (}

x;

⁰

;

⁰⁾

;

⁽¹⁵⁾

for

K

^1313,

L⁽¹⁾= (0

;

^;

x;

⁰⁾

;

^{M(1)= (}

x;

⁰

;

⁰⁾

;

^{N(1)= (0}

;

⁰

;

⁰⁾

;

⁽¹⁶⁾

for

K

^2121,

L⁽¹⁾= (0

;

⁰

;

⁰⁾

;

^{M(1)= (0}

;

⁰

;

^;

y

⁾

;

^{N(1)= (0}

;y;

⁰⁾

;

⁽¹⁷⁾

for

K

^2323,

L⁽¹⁾= (0

;y;

⁰⁾

;

^{M(1)= (;}

y;

⁰

;

⁰⁾

;

^{N(1)= (0}

;

⁰

;

⁰⁾

;

⁽¹⁸⁾

966 A. Kapanowski and K. SokalskiElastic Constants of Binary Liquid Crystalline Mixtures for

K

^3131,

L⁽¹⁾= (0

;

⁰

;

⁰⁾

;

^{M(1)= (0}

;

⁰

;z

⁾

;

^{N(1)= (0}

;

^;

z;

⁰⁾

;

⁽¹⁹⁾

and for

K

^3232,

L⁽¹⁾= (0

;

⁰

;z

⁾

;

^{M(1)= (0}

;

⁰

;

⁰⁾

;

^{N(1)= (;}

z;

⁰

;

⁰⁾

:

⁽²⁰⁾

The third group is, for

L

^123,

L⁽¹⁾= (0

;x;

⁰⁾

;

^{M(1)= (;}

x;

⁰

;

⁰⁾

;

^{N(1)= (0}

;

⁰

;

⁰⁾

;

⁽²¹⁾

for

L

^231,

L⁽¹⁾= (0

;

⁰

;

⁰⁾

;

^{M(1)= (0}

;

⁰

;y

⁾

;

^{N(1)= (0}

;

^;

y;

⁰⁾

;

⁽²²⁾

and for

L

^312,

L⁽¹⁾= (0

;

⁰

;

^;

z

⁾

;

^{M(1)= (0}

;

⁰

;

⁰⁾

;

^{N(1)= (}

z;

⁰

;

⁰⁾

:

⁽²³⁾

The fourth group is, for

K

^1122,

L⁽¹⁾= (0

;

⁰

;

^–

y

⁾

;

^{M(1)= (0}

;

⁰

;x

⁾

;

^{N(1)= (}

y;

^–

x;

⁰⁾

;

⁽²⁴⁾

for

K

^1133,

L⁽¹⁾= (0

;z;

⁰⁾

;

^{M(1)= (–}

z;

⁰

;x

⁾

;

^{N(1)= (0}

;

^–

x;

⁰⁾

;

⁽²⁵⁾

and for

K

^2233,

L⁽¹⁾= (0

;z;

^–

y

⁾

;

^{M(1)= (–}

z;

⁰

;

⁰⁾

;

^{N(1)= (}

y;

⁰

;

⁰⁾

:

⁽²⁶⁾

The fifth group is, for

K

^1221,

L⁽¹⁾= (0

;

⁰

;

^–

x

⁾

;

^{M(1)= (0}

;

⁰

;y

⁾

;

^{N(1)= (}

x;

^–

y;

⁰⁾

;

⁽²⁷⁾

for

K

^1331,

L⁽¹⁾= (0

;x;

⁰⁾

;

^{M(1)= (–}

x;

⁰

;z

⁾

;

^{N(1)= (0}

;

^–

z;

⁰⁾

;

⁽²⁸⁾

and for

K

^2332,

L⁽¹⁾= (0

;y;

^–

z

⁾

;

^{M(1)= (–}

y;

⁰

;

⁰⁾

;

^{N(1)= (}

z;

⁰

;

⁰⁾

:

⁽²⁹⁾

3. Microscopic Approach 3.1. Description of a System

This section is devoted to the microscopic analy- sis of binary mixtures of uniaxial nematogens which create a homogeneous phase. Let us consider a mix- ture which consists of two types of rigid uniaxial molecules

A

^and

B

. Orientations are described by

two angles

^and

or by a unit vector. We assume a small density approximation, and we take only two- body short-range interactions into account. The po- tential energies

^IJ12 depend on a vector of the distance between molecules and orientations of molecules.

The microscopic free energy of the binary mixture has the form [30]

F

^{= X}

I=^A;B

Z

d(1)

G

^I(1)ln[

G

^I(1)

^I5]^;1

;

1 2

X

I;J=^A;B

Z

d(1)d(2)

G

^I(1)

G

^J(2)

f

12^IJ

;

⁽³⁰⁾

where

G

^{I(1) =}

G

^I(r1

;

^{1) (}

I

⁼

A;B

) are the one- particle distribution functions with the normalizations

Z

d(1)

G

^{I(1) =}

N

^I

;

⁽³¹⁾

d(1) = d r1d1 = d r1d

^1d

^1sin

^1,

N

^I denotes the number of molecules

I

in the volume

V

⁽

N

^{A +}

N

^{B =}

N

^),

f

12^IJ are the Mayer functions

f

12^IJ = exp(^;

^IJ12)^;1,

^{= 1}

=k

^B

T

^{, and}

^I5 =

h

²

2

m

^I

3⁼2

h

²

2

J

^I

:

⁽³²⁾

The subscript 5 in

^I5 denotes 5 degrees of free- dom of an infinitely thin molecule, although we use this description also for spatially extended uniaxial molecules.

T

denotes the temperature,

m

^Iis the mass of a molecule

I

^{, and}

J

^I is a parameter with a di- mension of a moment of inertia (for a very prolate molecule it is exactly the moment of inertia with re- spect to the axis perpendicular to the molecule). Our set of state variables consists of

T

^,

V

^,

N

^{A and}

N

^B.

The free energy (30) consists of the ideal terms (with

^I5) and the excess terms directly related to inter- molecular forces. The ideal terms are those of the ideal gas.

The expression (30) was derived systematically for binary mixtures from the Bogoliubov-Born-Green- Kirkwood-Yvon hierarchy equations in the thermody- namic limit (

N

^!1

;V

^{! 1}

;N=V

= const) [30].

Two-particle distribution functions were expressed in terms of one-particle distribution functions and the two-particle correlation functions of the simple form exp(^;

^IJ12). This assumption guarantees the proper limit of the unary system.

The equilibrium distributions

G

^I minimizing the free energy (30) satisfy

ln[

G

^I(1)

^I5]^;

X

J=^A;B

Z

d(2)

G

^J(2)

f

12^IJ

= const

:

⁽³³⁾

In the homogeneous phase the distribution function

G

^Idoes not depend on the position of a molecule and

G

^{I(1) =}

G

^0I(1). In order to obtain

G

^{0I one should}

solve (33) together with (31).

3.2. Distorsion Free Energy

In order to define the microscopic distorsion free energy

F

^d one should first identify a homogeneous free energy

F

⁰. We would like to divide the total free energy

F

into a homogeneous free energy

F

^{0and the}

distorsion free energy

F

^d. We postulate that

F

^{0= (34)}

X

I=^A;B

Z

d(1)

G

^I(1)ln[

G

^I(1)

^I5]^;1

;

1 4

X

I;J=^A;B

Z

d(1)d(2)

G

^I(r1

;

¹⁾

G

^J(r1

;

²⁾

f

^12IJ

;

1 4

X

I;J=^A;B

Z

d(1)d(2)

G

^I(r2

;

¹⁾

G

^J(r2

;

²⁾

f

12^IJ

:

The definition (34) is equivalent to that by Poniewier- ski and Stecki [6]. This is a well-founded assump- tion if we also assume slow variations of the vectors (L

;

^M

;

N). We will also restrict the one-particle distri- bution functions

G

^Ito the class of

G

^0Ifunctions. This method was succesfully used in the past [5, 31, 32].

Thus, the distorsion free energy can be written as

F

^d=

F

^;

F

⁰

;

⁽³⁵⁾

were

F

^and

F

⁰are given by (30) and (34), respectively.

As we expect, for the homogeneous phase

F

^dbecomes

zero.

In the subsequent sections we will construct the distribution functions for distorted phases and we will derive the microscopic formulas for the elastic con- stants. Biaxial and uniaxial nematic phases will be considered separately.

3.3 Elastic Constants of Biaxial Phases

It was shown in [10, 33] that in the case of a homo- geneous biaxial nematic phase composed of uniaxial molecules the one-particle distribution functions

G

^I

depend on two arguments:

G

^{0I() =}

G

^0I(ex

;

^ez)

;

⁽³⁶⁾

where it is assumed that the vectors eof the reference frame coincide with the phase symmetry axes. In the distorted phase we postulate that

G

^I(r

;

^{) =}

G

^0I(

Q

¹

;Q

²⁾

;

⁽³⁷⁾

where the relevant arguments are

Q

^1(r

;

^{) =L(r)}

;

Q

^2(r

;

^{) =N(r)}

:

⁽³⁸⁾

The local orientation of the phase is described by the vectors (L

;

^M

;

N). We can use the distorsion free- energy density (8) and apply basic deformations de- scribed in Section 2. We will follow a procedure sim- ilar to one described in [10]. Let us expand the argu- ments (38) in a power series with respect to

^,

Q

^{i =}

Q

^{(0)i +}

Q

^{(1)i +}

²

Q

^{(2)i +}

;

⁽³⁹⁾

where by means of the expansions (11) we build

Q

⁽1^p)(r

;

^{) =L(p)(r)}

;

Q

⁽2^p)(r

;

^{) =N(p)(r)}

:

⁽⁴⁰⁾

The expansion of

G

^I has the form

G

^I(r

;

^{) =}

G

^0I(

Q

⁽⁰⁾1

;Q

⁽⁰⁾2 ) +

^X

i=1^;2

∂ⁱ

G

^0I(

Q

⁽⁰⁾1

;Q

⁽⁰⁾2 )

Q

⁽¹⁾ⁱ

+

^{2 X}

i=1^;2

∂ⁱ

G

^0I(

Q

⁽⁰⁾1

;Q

⁽⁰⁾2 )

Q

⁽²⁾ⁱ

+

²

2

X

i;j=1^;2

∂ⁱ∂^j

G

^0I(

Q

⁽⁰⁾1

;Q

⁽⁰⁾2 )

Q

⁽¹⁾ⁱ

Q

^(1)j

+

O

⁽

³⁾

:

(41)

When we substitute the expansion (41) into the dis- torsion free energy (35) we get

968 A. Kapanowski and K. SokalskiElastic Constants of Binary Liquid Crystalline Mixtures

F

^{d= 1}₄^Z

d

⁽¹⁾

d

⁽²⁾

^{2 X}

I;J=^A;B

f

12^IJ

X

i;j=1^;2

∂ⁱ

G

^0I(1)∂j

G

^0J(2)h;2

Q

^{(1)i (r1}

;

¹⁾

Q

^{(1)j (r2}

;

²⁾

+

Q

^{(1)i (r1}

;

¹⁾

Q

^{(1)j (r1}

;

^{2) +}

Q

^{(1)i (r2}

;

¹⁾

Q

^{(1)j (r2}

;

²⁾ⁱ⁺

O

⁽

³⁾

:

⁽⁴²⁾

We substitute the basic deformations into the microscopic distorsion free energy (42) and to the phenomeno- logical distorsion free energy (8). By comparison we get the microscopic formulas for the elastic constants.

The first group is

K

^{1111= 1}₂^{Z d1d2d u}

u

^{2x X}

I;J=^A;B

f

^12IJ

W

^1Iy

W

^2Jy

;

⁽⁴³⁾

K

^{2222= 1}₂^{Z d1d2d u}

u

^{2y X}

I;J=^A;B

f

12^IJ(

U

1^Iz;

W

1^Ix)(

U

2^Jz;

W

2^Jx)

;

⁽⁴⁴⁾

K

^{3333= 1}₂^{Z d1d2d u}

u

^{2z X}

I;J=^A;B

f

12^IJ

U

1^Iy

U

2^Jy

:

⁽⁴⁵⁾

The second group is

K

^{1212= 1}₂^{Z d1d2}

d

^u

u

^{2x X}

I;J=^A;B

f

12^IJ(

U

1^Iz;

W

1^Ix)(

U

2^Jz;

W

2^Jx)

;

⁽⁴⁶⁾

K

^{1313= 1}₂^{Z d1d2d u}

u

^{2x X}

I;J=^A;B

f

12^IJ

U

1^Iy

U

2^Jy

;

⁽⁴⁷⁾

K

^{2121= 1}₂^{Z d1d2d u}

u

^{2y X}

I;J=^A;B

f

12^IJ

W

1^Iy

W

2^Jy

;

⁽⁴⁸⁾

K

^{2323= 1}₂^{Z d1d2d u}

u

^{2y X}

I;J=^A;B

f

^12IJ

U

^1Iy

U

^2Jy

;

⁽⁴⁹⁾

K

^{3131= 1}₂^{Z d1d2d u}

u

^{2z X}

I;J=^A;B

f

12^IJ

W

1^Iy

W

2^Jy

;

⁽⁵⁰⁾

K

^{3232= 1}₂^{Z d1d2d u}

u

^{2z X}

I;J=^A;B

f

12^IJ(

U

1^Iz;

W

1^Ix)(

U

2^Jz;

W

2^Jx)

:

⁽⁵¹⁾

The third group is

L

¹²³⁼

L

²³¹⁼

L

^{312 = 0}

:

⁽⁵²⁾

The fourth group is

K

^{1122= 1}₄^{Z d1d2d u}

u

^x

u

^{y X}

I;J=^A;B

f

12^IJ

(

U

1^Iz;

W

1^Ix)

W

2^Jy+

W

1^Iy(

U

2^Jz;

W

2^Jx)

;

⁽⁵³⁾

K

^{2233= 1}₄^{Z d1d2d u}

u

^y

u

^{z X}

I;J=^A;B

f

12^IJ

;

U

1^Iy(

U

2^Jz;

W

2^Jx)^;(

U

1^Iz;

W

1^Ix)

U

2^Jy

;

⁽⁵⁴⁾

K

^{1133= 1}₄^{Z d1d2d u}

u

^x

u

^{z X}

I;J=^A;B

f

^12IJ;

U

^1Iy

W

^2Jy;

W

^1Iy

U

^2Jy

:

⁽⁵⁵⁾

The fifth group is,

K

¹²²¹⁼

K

¹¹²²

; K

¹³³¹⁼

K

¹¹³³

; K

²³³²⁼

K

²²³³

:

⁽⁵⁶⁾

To make the formulas for the elastic constants more compact we write

U

^{I =∂1}

G

^0I(

Q

⁽⁰⁾1

;Q

⁽⁰⁾2 )

;

W

^{I =∂2}

G

^0I(

Q

⁽⁰⁾1

;Q

⁽⁰⁾2 )

:

⁽⁵⁷⁾

3.4. Elastic Constants of Uniaxial Phases

In a homogeneous uniaxial phase we have one global symmetry axis which can be oriented along ez. The one-particle distribution functions depend on a one argument, thus

∂1

G

^0I(ex

;

^{ez) = 0}

; U

^{I = 0}

:

⁽⁵⁸⁾

The microscopic expressions for the elastic constants that result from (43) - (56) are

K

^{1= 1}₂^{Z d1d2d u}

u

^{2x X}

I;J=^A;B

f

12^IJ

W

1^Ix

W

2^Jx

;

⁽⁵⁹⁾

K

^{2= 1}₂^{Z d1d2d u}

u

^{2y X}

I;J=^A;B

f

^12IJ

W

^1Ix

W

^2Jx

;

⁽⁶⁰⁾

K

^{3= 1}₂^{Z d1d2d u}

u

^{2z X}

I;J=^A;B

f

^12IJ

W

^1Ix

W

^2Jx

;

⁽⁶¹⁾

K

^{4= 1}₂⁽

K

¹⁺

K

²⁾

;

⁽⁶²⁾

K

^{5= 0}

:

⁽⁶³⁾

Note that the results (62) - (63) are consistent with the wide discussion on surface elasticity by Yokoyama [34]. These expressions results from (52) and (56) when we change the symmetry of the phase from biaxial to uniaxial.

4. Exemplary Calculations

The aim of this section is to express the elastic constants by means of the order parameters which can be measured in experiments. We will apply the Corner potential energy of interactions because in principle

it allows detailed calculations without any additional approximations. On the other hand, it is quite realistic.

4.1. Homogeneous Phases

In order to simplify calculations for the biaxial phase we assume that the one-particle distribution functions

G

^0I depend only on the angle between the long axis of a molecule, determined by a unit vector

, and some symmetry axis, determined by a unit vector eI,

G

^{0I() =}

G

^0I(eI)

:

⁽⁶⁴⁾

For the uniaxial phase we will assume that eA = eB, whereas for the biaxial phase eA

eB= 0. It is conve- nient to define dimensionless functions

f

^I

G

^{I0(eI) =}

f

^I(eI)

N

^I

=

⁴

V;

⁽⁶⁵⁾

where the normalization condition is

Z

d

f

^I(eI)

=

⁴

^{=Z 1}

0

d

xf

^I(

x

^{) = 1}

:

⁽⁶⁶⁾

The order parameters are defined as

h

P

^{jiI =Z 1}

0

d

xP

^j(

x

⁾

f

^I(

x

⁾

:

⁽⁶⁷⁾

The functions

f

^Ican be expanded in an infinite series with respect to the Legendre polynomials

f

^I(

x

^{) = X}

j;even

(2

j

^{+ 1)h}

P

^jiI

P

^j(

x

⁾

:

⁽⁶⁸⁾

It is useful to describe the nematic ordering of molecules

I

in a mixture by a symmetric traceless second-rank tensor S^I with elements [35]

S

^{I =h}

^;1₃

^iI

= 1 4

Z

d

f

^I(eI)h

^;1₃

ⁱ

:

⁽⁶⁹⁾

We can show that

S

^{I =h}

P

^{2iIh(eeI)(eeI);1}₃^(ee)i

:

⁽⁷⁰⁾

970 A. Kapanowski and K. SokalskiElastic Constants of Binary Liquid Crystalline Mixtures The tensor S^I is diagonal only if e^Iis equal to e^x, e^y

or ez. Finally, we define an average tensor S as S =

x

^ASA+

x

^BSB

;

⁽⁷¹⁾

where we used concentrations

x

^{I =}

N

^I

=N

^{. We note}

that because of our approximation (64), the tensors S^I are uniaxial (2 different eigenvalues). But the average tensor S is in general biaxial (3 different eigenvalues) if eA

eB = 0.

Let us consider the Corner potential energy of the form

^IJ12(

u=

^{IJ), where}

u

is the distance between molecules

I

^and

J

^{, u =}

u

^,

^IJ depends on vectors

1, ₂ and. For

^IJ one can write the general expansion proposed by Blum and Torruela [36]. It involvs the 3-j Wigner symbols and the standard rota- tion matrix elements. The same expression was used to describe interactions of biaxial molecules in [33].

In the case of uniaxial molecules, the lowest order terms of the expansion give

^IJ(1

;

²

;

^{) =}

0^IJ+

^I11^J(1)² (72) +

12^IJ(2)²+

2^IJ(12)²

:

There are a number of posibilities for the functional dependence of

^IJ12 on

u=

^IJ, and some of them were given in [10]. We do not have to specify it now, be- cause this dependence will be hidden in a function

B

^s(

T

) defined as

B

^s(

T

^{) =Z 1}

0

d

x x

^s

f

¹²⁽

x

⁾

=

Z

1

0

d

x x

^sexp(;

¹²⁽

x

^));1

;

⁽⁷³⁾

where

T

^{= 1}

=

is a dimensionless temperature and

is a depth of the potential energy (we assume for sim- plicity that it is the same for both types of molecules).

Thanks to the form of the Corner potential energy one can rewrite (30) and (33) in the form

F

^{= (74)}

X

I=^A;B

Z 1

0

d

xN

^I

f

^I(

x

^)ln[

f

^I(

x

⁾

N

^I

^I5

=

⁴

V

^];1

;

1 2

X

I;J=^A;B

N

^I

^{IJ X}

j;even

K

^jjIJh

P

^jiIh

P

^jiJ

P

^j(eIeJ)

;

ln[

f

^{I(1eI)]; X}

J=^A;B

^{IJ X}

j;even

K

^{jjIJ (75)}

P

^j(1eI)h

P

^jiJ

P

^{j(eIeJ) = const}

;

where

^{IJ = (}

^{IJ0 )3}

B

²⁽

T

⁾²

²

N

^J

=

⁴

V;

⁽⁷⁶⁾

K

^IJ(cos

¹

;

^cos

^{2) =}

¹²

Z

d

^1d(

^IJ

=

^IJ0 )³

:

⁽⁷⁷⁾

As a consequence of the definition (77) we can ex- press the kernel

K

^IJ as a sum with even Legendre polynomials

K

^IJ(

x;y

^{) = X}

j;k ;even

K

^jkIJ

P

^j(

x

⁾

P

^k(

y

⁾

;

⁽⁷⁸⁾

K

^{jkIJ = (2}

j

^{+ 1)(2}

k

^{+ 1)}

Z 1

0

d

x

^{Z 1}

0

d

yK

^IJ(

x;y

⁾

P

^j(

x

⁾

P

^k(

y

⁾

:

⁽⁷⁹⁾

Note that for

^IJgiven by (72) the kernel is diagonal.

Equations (75) imply that

f

^I should be written as ln[

f

^I(

x

^{)] = X}

j;even

C

^jI

P

^j(

x

⁾

:

⁽⁸⁰⁾

It allows as to transform (75) into a set of equations

C

^{jI = X}

J=^A;B

^IJ

K

^jjIJh

P

^jiJ

P

^{j(eI eJ)}

;

⁽⁸¹⁾

where

j

^{= 2}

;

⁴

;

^{6 and}

I

⁼

A;B

. The normalization condition (66) must be also enclosed. In the case of the uniaxial (biaxial) phase we have eI

eJ = 1 (eI eJ=

^IJ). The stable solution of (81) which has the lowest free energy will describe a homogeneous phase.

4.2. Elastic Constants for Uniaxial Phases

We insert the expansion (68) into (59) - (61). It appears that only a finite number of terms gives a nonzero contribution. Thus we get the explicit depen- dence the elastic constants on the order parameters:

K

^{s= X}

I;J=^A;B

X

j;k ;even

^IJ

F

^{s;jkIJ h}

P

^jiIh

P

^kiJ

;

⁽⁸²⁾

^{IJ =}

T

B

⁴⁽

T

⁾⁽

0^IJ)⁵

N

^I

N

^J

=

⁽⁴

V

⁾²

;

⁽⁸³⁾