Interfacial energy for nematic liquid crystals : beyond the spherical approximation

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Interfacial energy for nematic liquid crystals : beyond the spherical approximation

G. Barbero, L. Evangelista, M. Giocondo, S. Ponti

To cite this version:

G. Barbero, L. Evangelista, M. Giocondo, S. Ponti. Interfacial energy for nematic liquid crystals :

beyond the spherical approximation. Journal de Physique II, EDP Sciences, 1994, 4 (9), pp.1519-

1540. �10.1051/jp2:1994215�. �jpa-00248059�

Classification Physjcs Abstracts 61.30

Interfacial energy for nematic liquid crystals

_:

beyond the

spherical approximation (*)

G. Barbero ("

~),

L. R.

Evangelista ('.~),

_{M. Giocondo (~) and S. Ponti}

(')

(') Dipartimento

di Fisica del Politecnico, Corso Duca

Degli

Abruzzi 24, 10129 Torino, Italy (~)

Dipartimento

di Fisica, Universith della Calabria, 87036 Arcavacata di Rende (CS),

Italy

(3) Universidade Estadual de

Maringh,

Avenida Colombo 3690, 87020-900 Maringh Paranh,

Brazil

(Receii>ed 3 Maich 1994, iei,ised 25 Apiil 1994, accepted 9 June 1994)

Abstract. By means of _a simple

pseudomolecular

model the surface elastic properties of

nematic

liquid

crystals _are

analysed.

We show that in the

Maier-Saupe approximation

for the intermolecular interaction and

supposing

the interaction volume of

ellipsoidal shape,

the bulk elastic constants of splay and twist _are

expected

to be equal and different from the bend _one. The

dependence

of the bulk elastic constants on the eccentricity of the

ellipsoid

is studied. In the _same framework it is shown that the number of

phenomenological

parameters necessary to describe the surface elastic behaviour of nematic

liquid

crystals is different from the _one in the bulk. The

position dependence

of the surface elastic _constants is evaluated and the anisotropic part ^{of the} surface tension is determined. The anchoring energy strength, in the

Rapini-Papoular

sense, i,ersus

the eccentricity of the ellipsoid representing the interaction volume is also estimated.

1. Introduction.

The bulk

macroscopic

behaviour of Nematic

Liquid Crystals (NLC)

is well-described

by

the

continuum

theory proposed long

_ago

by

Oseen

[I]

and Zocher

[2].

As all the

macroscopic

theories,

the

physical

characteristics of the medium _are taken into _account

by

_means of

phenomenological

coefficients, like elastic constants, dielectric

permittivity,

surface tension

and _{so on.} In _{some cases} it is

possible

to connect the

macroscopic

parameters

entering

in the

continuum

theory

to the molecular

properties.

This has been done, in

particular,

for the elastic constants of NLC

[3].

The first model of intermolecular forces used _to evaluate the Frank

[4]

elastic _constants has been the _one

proposed by

Maier and

Saupe [5].

This model is very

simple

and it

gives

^the

possibility

to describe the nematic

-

isotropic phase

transition. For what concerns the elastic

constants this model

gives Kjj

= K~~ ₌ K~~, ^{I.e, the three bulk elastic} constants have _a

(*) Partially

supported by

Bilateral Scientific

Cooperation

^between Politecnico di Torino and Universidade Estadual de Maringfi.

common value K

[6].

The temperature

dependence

of the elastic constants near the

clearing

temperature ^is

predicted

to be

proportional

to the square of the scalar order parameter, ⁱⁿ agreement ^{with the}

experimental

data obtained

by

different groups

[7].

After the model of

Maier-Saupe

_many models have been

proposed,

in order _to

justify

the

experimental

data

Kj

_{m K~~} # K~~

[6].

These models _are based _on

multipolar expansion

of the electrostatic intermolecular interaction

[8],

_or on the induced

dipole-induced dipole

inter- action

[9],

_{or on} the steric interaction

[10].

All these models _are very

complicated

with respect

to the

Maier-Saupe

model.

Furthermore,

their

predictions,

in the

spherical approximation

for the interaction volume, are

only partially

in agreement ^{with the}

experimental

data

[3].

This fact

simply

underlines that different kinds of interactions _are

responsible

for the NLC

phase

and that it is difficult to take all of them into _account.

This conclusion is _{true not}

only

for what _concerns the bulk elastic constants, but also for the

anisotropic

part of the surface tension

[I I].

In

fact,

_as is well-known, the surface tension of NLC contains, besides the

isotropic

_part as the usual

liquids,

the surface tension of NLC

contains, besides the

isotropic

part as the usual

liquids

also _an

anisotropic

part

depending

on

the average orientation of the NLC. The latter contribution has _two different

origins.

One is connected to the NLC-NLC interaction and the other _{one to} the

anisotropic

_pan of the NLC substrate interaction. The first pan ^{is intrinsic and it} may be evaluated

starting

from the

intermolecular interaction energy

responsible

for the NLC

phase, following

the _same

procedure

as that used for the calculation of the elastic _constants. This part ^{has been} evaluated, in the

spherical approximation, by

means of the

Maier-Saupe

model

[12].

The

analysis

shows

that the surface tension, in this

framework,

has _{not an}

anisotropic contribution,

_{contrary to} the

experimental

data

[13].

The

anisotropic

part ^{of the surface tension is found} to be different from

zero, in the same

hypothesis,

when the induced

dipole-induced dipole

interaction is

considered

[14].

In this paper we want to show that it is

possible

to recover the main elastic characteristics of the NLC

phase by considering

the

simple Maier-Saupe interaction,

in the

ellipsoidal

approximation

for the interaction volume. In this framework _we will show that the elastic

constants have _not the _same value and that the surface tension has _an

anisotropic

_pan. Our

paper is

organized

_as follows. In section 2 the

pseudomolecular approach

to evaluate the

elastic _constants is recalled and the main

hypotheses usually performed

_are stressed. The

generalization

of the

pseudomolecular

model

beyond

the

spherical approximation

is discussed

in section 3. In section 4 the bulk elastic

properties following

from the

Maier-Saupe

interaction,

in the

ellipsoidal approximation,

_are obtained. In section 5 the surface elastic

properties

and the

anisotropic

part ^{of the surface tension} are deduced. The surface elastic constants are

analysed

in section 6. The main results of _our paper are stressed in section 7.

2. Molecular

approach

to the NLC elastic constants.

In this section well-known results

concerning

the molecular

approach

to the NLC elastic

constants are stressed. Several papers are devoted to this

subject [3].

We recall the main

points

of this

theory

to the reader who is _not familiar with it.

Let be g(n, n', r) the _two

body

interaction law, between two small volumes dr and

dT' located in R and R', whose directors _{are n}

= n(R and n'

=

n(R'),

and whose relative

position

is _r

= R' R.

We suppose that

g(n,

n',

r)=0

for _r>p, where _p is the interaction range of the

intermolecular forces

giving

rise to the NLC

phase.

The NLC scalar order parameter ^{S is assumed to be}

equal

to one, I.e. the NLC order

perfect.

Consequently

the molecular direction coincides with _n.

Since _n

changes

_over distances I _{» p, An}

=

[n'- n[

« I.

Consequently

~~~'

~" ~~ _~° ~

~i ~l

+ ~<j ~~<

~~/, (~. l)

where

g~ = g(n, n,

r), A,

₌ ~~,

, A~~ ⁼

~

,

(2.2)

an~ n. _n

n~~an~

~. _~

By taking

into _account that

I»p

it is

possible

_to

expand

An~ ⁱⁿ power series of x~, the Cartesian components ^of r, in the

following

_manner

An,

(R,

r)

= n,_

~

(R

_{>.~ + n~,}

~~ (R) _>.~ .i~

,

(2.3)

where n~,

~(R)

=

(an~lax~)R

and n,_ _~~

(R)

=

(b~n~lax~ ax~ )~. By substituting (2.3)

into

(2.

we have

g(n,

n',

r)

= go + A~ x~ n~

~

(R)

_{+ [A~}

n~ ~~

(R)

+ A,~ n~

~

(R

)

_n~

~

(R)],<~.t~ ^(2.4)

The free energy

density,

in the _mean field

approximation,

is

given by

F

=

jj _g(n,

n', r) ^dr

(2.5)

2

r~

where _r~ is the interaction volume defined

by

_g (n, n', _r

),

whose center is in R.

By

means of

(2.4), (2.5)

_may be rewritten _as

~

~

~o

₊

~ia

_~>,

a

+

j [~iap

^{~i, up +}

~ilap

_~i,

«

~j. PI ^(~.6)

where

Fo

₌

lj

_{go dr}

(2.7)

2

r~

is the uniform pan, and

Li«

_"

~ ⁱⁱ

^{.t« dT}

Li«p

_"

_~ ii

_{-in -ip} dT

,

L,/«p

=

~ ^Ail

^{>« 'p dT} ,

7h 7h ~h

(2.8)

are the elastic

tepsors. By decomposing

the elastic _tensors in _terms of the elements of

symmetry ^{of the NLC}

phase

(n and

taking

into account that in the bulk the NLC _{are not}

polar (and

hence _n

w

n),

it is

possible

to write

(2.6)

in the usual Frank form. This is discussed in detail in

[15, 16],

where r~ is assumed _to be of

spherical shape.

The above

reponed analysis

is valid in the bulk. The main results _are that I)

Fo

is

n-independent

it) the term linear in the first order

derivatives,

of _n is

identically

_{zero ;} and

iii)

the last term in

(2.6)

writes in the Frank

form,

in which the elastic constants _{are n-}

independent.

An extension of the molecular

approach

of the kind discussed in this section to take into

account the presence of _a surface has been

published

in

[17, 18].

There it is shown,

by

considering

a very

general

interaction law, that

iv) F~

is

n~dependent

in _a surface

layer

whose thickness is of the order of the range of the interaction law

[17]

_;

v)

the term linear in the first order derivatives of _n may be different from _zero,

giving

rise to a spontaneous

splay [18]

and

vi)

the Frank elastic constants are

expected

to be

position

and director

dependent [17, 18].

The conclusions

iv),

v) ^and

vi)

_{are very}

general.

In order _to have _an idea about the influence of the surface in the elastic

propenies

of the NLC it is necessary to consider _a well defined

expression

_{for g}

(n, n~,

_r

).

In _a recent

investigation [14]

Faetti and Nobili consider the induced

dipole-induced dipole

interaction

by supposing

that the interaction volume has _a

spherical shape. They

show that

I) Fo

^is

n-dependent,

and it _can be

expanded

in power series of

(n k)~

where k is the normal of the surface _;

II)

the elastic _constant for the spontaneous

splay, Kj,

is

z-dependent

and odd in

(n k _;

III)

the usual Frank elastic _{constants are}

z-dependent

^and their surface value is half of the

bulk _one and

IV)

the mixed

splay-bend

elastic _constant

Kj~

is

z-dependent

and _even in _n k.

A similar

analysis

has been

performed by

Alexe-Ionescu et al, for the

Maier-Saupe

interaction

[12], by supposing again

_r~ of

spherical shape. They qecover

^{the result}

(III)

of the Faetti-Nobili

analysis.

However in the

spherical approximation

for _r~,

Fo

remains _n-

independent.

Furthermore

they

find that

Kj

=

Kj~

= 0. This result is connected with the

special

form of the

Maier-Saupe

interaction, which is

independent

of the _r direction.

In all the papers

quoted

above and relevant _to the molecular

approach

to the NLC elastic

constants the interaction volume is

supposed

to be of

spherical shape.

To be _more

precise

r~ is the volume limited

by

two

spheres

of radii r~

(the inner)

and _p

(the

outer).

i~ is

supposed

to be of the order of the _« molecular dimensions _», whereas _p of the order of the

longest

interaction range of the intermolecular forces

giving

rise to the NLC

phase.

3.

Beyond

the

spherical approximation.

As is well-known the molecules of the NLC materials _are

rod-like,

_except for discotic nematic

liquid crystals,

which _are not considered in _our

analysis. Consequently

_i-o is _not well-defined.

Hence it _seems

interesting

to consider another kind of interaction

volume,

whose inner part ^is similar to a real molecule of nematic. The

shape

of the outer part ^is not very

important

because

it does _{not enter} the

theory

ⁱⁿ a crucial _manner.

Very

often the NLC molecules _are

supposed

to be

cigar-like,

I.e. to have _an

ellipsoidal shape.

For this _{reason we want to} extend the

previous approach

to the _case in which the interaction volume has _on

ellipsoidal shape

of the kind shown in

figure

I. To be _more

precise

we suppose that

g(n,

n', r) is different from _zero in the

region

limited

by

two similar

ellipsoids,

whose inner part ^{coincides with the molecular} volume, and the outer part ^{is defined}

by

the

long

_{range part} of the intermolecular interaction. The two

ellipsoids

_are

supposed

to

similar,

I.e, _to have the _same

eccentricity.

As will be shown later, the dimensions of the outer

ellipsoid

do not

play

a critical role in the elastic

properties

we want to

analyse.

For the sake of

simplicity

the

ellipsoids

_are

supposed

of revolution around _n. The semiaxes _are indicated

by

_a

and b and the

eccentricity by

e. In _{our case}

e =

(a~/b~)~

₌ l

(a~/b~)~ (3.1)

X

t~

Y

Fig. I. interaction volume limited by two similar

ellipsoids

of revolution around the z-axis. The molecular Cartesian reference frame has the z-axis coincident with the nematic director. _To is the molecular volume. _r~ is connected with the longest interaction range.

The

subscripts

0 and b refer _to the inner

(molecular)

and _outer

(bulk) ellipsoid.

In _a reference frame in which the z-axis coincides with the NLC director _n, the Cartesian

equations

^of the

ellipsoids

are

j~2 _~

y2 z2

+ = l

(3.2)

a(.

b

hi,

b

In the _event in which the NLC is limited

by

a flat surface, it is necessary also to introduce

another reference frame. It will be useful _to choose _a

laboratory

frame whose z-axis coincides with the normal _to the surface.

In the undeformed state, in which _n is

position independent,

n is

supposed

to be

parallel

to the

(z,

_y

)-plane

of the

laboratory frame, forming

_an

angle

6 with the z-axis.

In this situation the two reference frames _are connected

by

_a

simple

rotation around the _x- axis of the kind.

X=.r,

Y=ycos6-zsin6, Z=ysin6+zcos6, (3.3)

as shown in

figure

2.

z

~ z

xmX y

Fig.

2. Molecular (OXYZ and

laboratory

(0.ryz) ^Cartesian reference frames. In the undeformed state n is everywhere parallel to the ly,z) plane, at an angle with respect to the z-axis. The OXYZ and 0.<yz ^frames are connected by _a

simple

rotation of around the _<m X axis.

In the

laboratory

frame

0(x,

_y,

z)

the Cartesian

equations

of the

ellipsoids

_are

x~

₊

(I

_e

sin~

₆

_y~

_{2 yze}

cos 6 sin 6 ₊

z~(I

e

cos~

₆

₎

=

al

~,

(3.4)

as it follows

by substituting (3.3)

into

(3.2)

and

taking (3.I)

^into account.

Let _{us now} consider the presence of the surface.

According

to the relative

position

of _a

given

molecule with respect to the

surface,

the two situations shown in

figure

3 _are

possible.

z

A

^~

A 8

y

a) b)

Fig. 3. NLC sample limited by a flat surface of equation z = A in the laboratory reference frame.

According

to the

position

of the

limiting

surface with respect to the interaction volume of _a

given

molecule, the interaction volume is

complete

(a) or not (b).

Let

A~

=

~la) sin~

6 ₊

b) ^cos~

6

= a~

l~

^{~ ~'~}

^~

^~

^(3.5)

If

A >

A~

,

(3.6a)

as shown in

figure

3a, the interaction volume is

complete,

and the considered molecule

behaves _as those in the bulk in the

opposite

_case in which

A <

A~

,

(3.6b)

as shown in

figure 3b,

the interaction volume is

incomplete,

and the considered molecule is

expected

to be

responsible

for elastic

properties

different from the bulk _ones.

In the

following

we will first

analyse

for the elastic

properties

of the molecules in the bulk, I.e. whose distance from the

limiting

surface satisfies the

inequality (3.6a).

After that the

peculiar properties

of the molecules contained in the surface

layer

will be described in detail.

4. Bulk elastic

properties

^{of NLC.}

By using

^the

Maier-Saupe

interaction and the

ellipsoidal approximation

for the interaction volume it is

possible

to evaluate the bulk elastic _constants of NLC.

According

to the Maier-

Saupe theory,

the intermolecular energy

responsible

for the NLC

phase

is

g(n, n', r)

=

~

(n n~)~

=

-J(r)(n n')~, _(4.I)

r~

where-c is _a

positive

constant.

Expression (4.I)

holds in the

hypothesis

of

perfect

nematic

order,

where the director coincides with the

long

molecular axis.

By

_means of

(4.I) simple

calculations

give

~

i

in'=

n

~

~~'~

" ~

i~in(

In

n

~

~~~~

_{" "~}

~~~~

Consequently

L,~

_n,

~ = n, n~

~

2 _{J (>.})_x~ dr

m

0 2 ,

7~

~~ ~~

~ilap

_~>,

a

~j,

_p " I ~j ~,,

a ~/, _p

jj

~

Jl'l'taX

p d~

"

0,

7~

because n~

= I and hence n~ n~

~

= 0. It follows that the elastic energy

density (2.6)

reduces _to F

=

F~

_{+ L~~}

~ n~ _{~~ ,} which may be rewritten in the form

F

=

Fo

+

I~

~ n~,

~

n, _~

,

(4.4)

where

Fo

"

j jj J(>')

_dr,

(4.5)

7~

I~~

₌

jj _J(I)

x~ >.~ dr

(4.6)

~~~

~~

In the bulk the

integrations appearing

ⁱⁿ

(4.5)

and

(4.6)

_are

performed

_over the

complete ellipsoid.

Hence in this _case

(4.5)

and

(4.6)

_are written

j ^{2 n n p(8J} _~,

~o

~

j

^{j df'Sin 6 de d~}

^(4.7)

(J 0 ijjlHjf' and

? _{n n} p(Hj

1,,~

=

~ u~ u~ ^{d>. sin 6 de}

d~

(4.8)

0 o _jj(oj>'

where _u

=

r/r,

>.o(6

=

a~/ ~

and _p (6

=

a~/ ~.

In

(4.7)

and

(4. 8)

the

polar

axis is chosen coincident with the director _n. From

equation (4.7)

we obtain

~° i

~

( [ _I1

_{~~ ~}

~~~~

~~~~

~

i

~~~ji'~ ^~~~~

for the free _energy

density

of the uniform distribution.

We underline that in the limit _e

- 0

(spherical approximation) Fo(e

=

0)

=

~

arc

,

(4.10)

~

a( _a~

as is well-known

[12].

The trend of

Fo

_{vs. e} is shown in

figure

4.

Equation (4.9)

shows that in the bulk

Fo

is

independent

of the

n-orientation,

_as

expected.

4

)

3.5

d

_3.0

' 0.2 0.4 0. 1.0

Q~~

^2.5 e

2.0

Fig.

4. Free

density

of the uniform NLC _{state i-s.} the

eccentricity

of the interaction

ellipsoidal

volume.

In the _e

- 0 limit the well-known spherical

approximations

_are recovered. We stress that for usual NLC

molecules a/b lm and hence _e 0.9.

Let _{us now} consider the elastic _tensor

I~~.

A

simple analysis

shows that in the reference

frame in which the z-axis coincides with the NLC

director,

the matrix

I~~

is

diagonal

Ij~

= I~~ = I~~ =

0.

By taking

into account that uj =

sin 6 _cos ~, _{u~ =} sin 6 sin ~ ^and u~ = cos 6,

simple

calculations

give

Ii

=

In

_"~

i ^b

^~

_~ i

[~[[~)[~ ^~

(4.

I

I)

i~~=2arc()-)~( ²

^~

^~ _2~~

_~

~~~~~

In the limit _e

- 0 _we obtain

lim

Ii

= lim I~~ =

~

arc

(4 12)

p-o p-o 3 ~0 ah

The results

(4.

I

)

show that in the bulk the elastic _tensor is

independent

of the n-orientation. In

order _{to connect} the elastic tensor to the elastic constants, it is

enough

to write

(~

in the form

~jI

"

~

II

~jI

+ (133 ~

II

3j3 ~I3, (4.13)

and _to observe that the elastic contribution _to the free _energy

density

is

~

~0

+

~I

_I,

j I,1~

~ll

_I,

j I,

j + (133

~ll)I,

_I1.3 (4.14)

In the reference frame in which _n

m z-axis from

(4.14)

_we deduce that

[9]

Kjj

= K~~ =

Ii

j, K~~ = I~~ ^and

K~4

₌

Ii (4.15)

2

It follows that the

Maier-Saupe

interaction in the

ellipsoidal approximation

introduces _an elastic

anisotropy,

which

depends

_on the

eccentricity

of the

ellipsoid (representing

the molecular

volume).

The trend of the elastic _{constants >,s.} the

eccentricity

_e is shown in

figure

5.

We underline that for usual NLC molecules _e 0.9

[20].

fi

~

~

fi

33

GL

'

~d;~

1

0~0 0.2 0.4 0.6 O-S I-O

e

Fig.

5. Splay (Kji), ^twist (K~~) and bend (Kii) elastic _{constants vs.} the

eccentricity

_e.

S. Surface elastic

properties

of NLC.

The NLC molecules in the

boundary layer

A

<A~

have an

incomplete

interaction with the

other NLC molecules. From this it follows that the elastic behaviour of the surface

layer

^is

different from the bulk _one. To

analyse

these

properties

in the

following

it will be necessary ^to evaluate

integrals

of the kind

H(A

)

=

jj _lf(x,

_y,

z)

dr

,

(5.1)

~(A)

where

f(,r,

_y,

z)

is _a continuous function of the

spatial

coordinates and

r(A)

the volume limited

by

the

plane

_{z =} ^A and the

ellipsoid defining

the interaction volume (see

Fig. 6).

In

figure 6, A~

is the

quantity

introduced before

defining

the z-coordinate of the

highest point

of the

ellipsoid.

The

quantity

~ ~°

II

e

~~'~~

has _a similar

meaning

^for the inner

ellipsoid. Finally

is the z-coordinate of the intersection of the

ellipsoid

with the z-axis. For what follows it is

z

A

Fig. 6.-NLC sample limited by _a flat surface of equation z=A in the laboratory frame.

A~ ^and A~ _are the z-coordinates of the highest points of the _outer and inner

ellipsoid,

respectively.

A* is the z-coordinate of the intersection of the outer

ellipsoid

with the z-axis.

better _to rewrite (5,I) in the form

H(A

₌

jj ^f(x,

y, ^z dr

jj If

_(x,

y, z dr

=

H~

AH (A

), (5.4)

~~ i~(Ai

where

Ar(A)

is the excluded part ^{of the} interaction volume

(r~) by

the surface _at

z = A.

In

(5.4)

the first addendum _on the r-h-s- is similar _to the _one evaluated in the

previous section,

whereas

AH(A)

takes into _account for the reduction of the interaction volume.

AH(A)

is

given by

AH(A)

=

jj f(x,

_{y, z}

)

dr

(5.5)

i~(Ai

From

(5.5)

it follows that for A

=

A~,

AH

(A

=

A~

) ₌ 0 because jr

(A

₌

A~)

₌ 0. In fact for A

=

A~

the surface is

tangent

to the

ellipsoidal

interaction volume.

We stress that the

plane

_{z =} A

<

A~

^intersects the

ellipsoid.

The Cartesian

equation

of the

ellipse resulting

from this intersection is

,i~ ₊

(I

e

sin~

_{6 y~} 2

yAe

cos 6 sin 6 ₊ A ~(l _e

cos~

₆

=

al, (5.6)

obtained

by equation (3.4), by substituting

_z

= A.

It is convenient to rewrite

(5.6)

in

polar

coordinate.

Setting

x=pcos~, y=psin~, (5.7)

equation (5.6)

becomes

p~(l e

sin~

₆

_{sin~ ~}

₂

_pAe

cos 6 sin 6 sin ~

[al A~(I

_e

cos~

₆

)]

=

0

(5.8)

The

polar equation

of the

ellipse

is

p(A, ~)=

Ae _cos 6 sin 6 sin ~ _±

~la)(I

e

sin~

6

sin~

_~ A~

[(I

e) + e

sin~

₆

cos~

_~

l _e

sin~ sin~

_~

(5.9)

A

simple analysis

of

equation (5.

shows that for A _< A ^* the,r

= y =

0,

_z

= A

point

is inside the

ellipse.

In this _case 0 _w ~ _w 2

ar and in

(5.9)

the

sign

in front _to the square ^root is _a

positive

one, because _p is _a

positive quantity (see Fig. 7a).

On the contrary ^for A*<A<

A~

^the x ₌ y = 0, _{z =} A

point

is _out of the

ellipse,

as shown in

figure

7b. In this _case

~j(A)«

~ _«

~~(A)

= w

~j(A), (5.io)

where

JA

e

cos~

₆

al

_a~

j(

_{Al A *}

~[(A)#tg~

)2

~ ~

,=tg~ ~

~

(5.ii)

a~(I

-esin 6)-

(1-e)A-

, -eA

(A~/A)

-1

In this situation _p

changes

between _p~

= 0m connected with the

sign,

to pM ~ 0M

connected with the ₊

sign

in

(5.9) (see Fig. 7b).

Let us now consider AH

(A) given by (5.5). By

means of the

previous analysis

we have

AM " i(C) PM(z. 4)

AH(A

^* _< A

<

A~)

=

dz

d~ f(p

_cos ~, _p sin

~,

_{z P}

dp (5.12)

A ~j

(=)

p~(z,

~

and

A* 2 _n p(C, WI

AH

(A~

< A

< A *

)

=

dz

d~ f(p

_cos ~, _p sin ~, z p

dp

₊

A 0 0

AM " @i (zl PM(C. 41

+

A*

dz

#i(=) d~ ^m(=, f(p

_cos

~,

_p sin

~,

_z

)

_p

dp (5.13)

y y

q,

M

x

A<k

_x

A>x

a) b)

Fig. 7.-Ellip~e resulting

from the interaction between the flat surface _{at z} =A

<A~

and the interaction

ellipsoidal

volume. Ca~e (a)

corresponds

to A

< A*, (b) refers to the _case A*

< A

<

A~.

Let _us

apply

the considerations

reported

above _to the calculations of the energy

density

of the undistorted

configuration Fo(A).

In this _case

f(.r,

_y,

z)

=

~~, (5.14)

J.

as discussed in section 4.

Consequently

AM

^{" 4 (=) PM (z, 4} ~.

AF

~

(A

^*

< A

< A

~ ⁼ dz

d~

~ ~ ~ p

dp

,

(5.

15

A

#

lo)

m(=,

#

(P

_{+ Z}

and

AFO(A~<A<A*)

=

A* 2n P(z. 4

~. AM " 4 j(z) PM(z, @)

=

dz d~§

~ ~ ~ p

dp

+ dz

d~

^~

p

dp

A 0 0

(P + Z

)

A #

(z

p~(=,

j )

(p

^~ ₊ Z~)~

(5.16)

The

A-dependence

of

AF~

is shown in

figure

8 for different values of _e and 6

=

gr/6, by assuming

_a~

= 30 ao.

Figure

8 shows that

AF~

decreases very

rapidly

when A increases. The

surplus

of energy,

AFO,

^is localized near the surface in _a surface

layer

^{whose thickness} is of the order of several a~.

+

I _e

=

0, 0.25,

0.5

f

Qk

^°'~

0.

0 4 8 12 16 20

Ala~

Fig.

8.- Excess of free energy

density

AFO due _to the presence of the

limiting

surface

i,s.

A. AFO vanishes for A larger than several molecular dimensions.

By

means of

AF~

it is

possible

to introduce _a surface energy defined _as

G(e,

6 AM

=

AFO(e,

_;A dA

,

(5.17)

A~

whose trend _{vs. H,} for different values of _e, is shown in

figure

9. From this

figure

_we deduce that G

(e,

6 is different from the surface energy

proposed by Rapini-Papoular [2 Ii-

However the easy ^direction is the

homeotropic

_one, and the

anchoring

_energy

strength,

defined

by

d~G _(e,

₆

w2 "

~

(5, 18)

de _{8 0}

fi

@

~

'

~

@

~j

^0.

0.

0.0 0.4 O-S 1.2 1.6

~/(rad.)

Fig.

9. Surface energy G(@, e) _i-s. @for different values of the eccentricity. Note that G(@, _e) tends _to

a constant value for

e -

0

(spherical approximation).

In this limit the

anisotropic

part ^{of the surface} tension G(@, e) vanishes. Fore # 0, Gl@, e) ^{is minimum for 0. This} means that the easy direction is the homeotropic one.

vs. e is

reported

in

figure

^10. It its

imponant

to stress that

w~(e

₌ 0) ₌

w~(e

₌

1)

= 0, and that

W(e)

is

symmetric

around the value _e

= 0.5.

0,14

fi

~

~

e

°o

~ ~ ~~

o

~$ jaw

°°°

~ °o $~~

~

~~4

i~°°~

O~

am

~

° am am ox aw

e

O.02

O,I O.3 O.5 0.7 0.9

e

Fig.

10.-

Anchoring

_energy

strength

^for small deformations around the easy direction _i,.I. the

eccentricity.

Note that w>~(0 = w>~(l ) ⁰ as expected. The inset shows _{u.2 vs. e} for 0.9

< e < I which _is the actual range for _e in usual NLC molecules.

6. Surface elastic constants.

By

_means of the formalism

presented

in sections 4 and 5 _{we can} now evaluate the surface

elastic _constants of _a NLC. As shown in section 4 the elastic tensor is defined

by (4.6).

In the

event in which the interaction volume is not

complete

we have

Iji(A

)

=

Iii AI,i

(A ),

(6.1)

where

(~

is the bulk elastic _tensor and

A(~ given by

AI,~ (A

=

I)

x~ x~ dr

,

(6.2) A7(A)

>'

the variation introduced

by

the surface. AI~~ ^{can be evaluated}

by

_means of

expressions

of the kind of

(5.12)

and

(5.13) according

to the value of A.

Since _near the surface the elements of symmetry ^{of the NLC} are n

(the director)

and k

(the

normal _to the

surface),

it is

possible

to

decompose AI~~(A)

^{in the}

following

_manner

AI~~(A

⁾ = cj (A

&~~ ⁺

c~(A

_{n~ n~ + cl}

(A

k~ k~ +

c4(A )(n~

k~ + n~ _k~ ).

(6.3)

2

By using (6.3) simple

calculations

give

3 _{cj + c~ +} c~ +

Pc4

₌ p ₌ AI

cj+c~+P~c~+Pc4=p~=n,n~AI~~,

~~,

~

(6.4) cj+P c.~+c~+Pc4=p~=k~k~AI~~,

~~l

+

~C2

₊

~Cl

₊ (~ +

~~)C4

"

~4

_{" ~j}

~I ~~jI,

where P

= n k

= cos 6 and

Pi

₌

~

(

~

dx

dy

dz

~7(A

(X _{+ y + Z )~}

P2

_"

~ ~ ~ (Y ^{sin 6} + z cos 6 )~d~i

dy

dz

A7

IA1 (fi + y~ + Z

j6.5) P3

=

~ ~ i

z~ dx

dy

dz

Ar(A)

IA + y~ + Z 1'

P4

"

~ ~ i

(Y sin 6 _{+ z cos} 6 _z dA.

dy

^dz

A7jA)

(X + y + Z )~

By solving (6.4)

with respect to c~ (I

= 1, 2, 3, 4) _we obtain

c~ = T~~

p, ^(6.6)

where the matrix T is

(l ^{P~)~ (l P~)}

(I

P~)

2

P(I P~)

T

=

~ ~

~~

~)~

^~

~

~

~~

_{~ ~}

(6.7)

(1-P

(I

-P +P 2 -4P

2P(1-P~)

-4P -4P

2(1+3P~)

as it follows from

(6.4).

The trend of the surface elastic _{constants c-j}(A

),

I

= 1, 2, 3, 4, _»s. A is

shown in

figures11,

^12, 13, 14, 15 for different values of 6 and _e,

by assuming

a~ =

30a~.

^We stress that

changing

6 in _{ar +} 6, cj, c~ ^and c~ do _not

change,

whereas

c4(6)

=

-c4(ar

₊ 6), as

expected.

In fact the term

AI~~n~,,

n~ _~ has _to be invariant for

n - n. This

implies,

_as it follows from

(6.3),

that

c4(n)

₌

c4(- n).

Furthermore for _e

=

0,

_c~

= c4 w 0, whereas _c-j and _c~ remain different from _zero.

o.

e=0 e =0.75

8 = ~/6 8 ₌ z/6

) )

/£ @

' '

£

o-i

£

o 5 io 15 20 25 30 0 5 io 15 20 25 30

Alao Alar

a)

b)

o.14

e = 0.85 _{e =} 0.95

8 ₌ ~/6 8 ₌ n/6

) )

@

~ ~

/£

' '

, ,

~ o. 05

~

0 5 lo 15 20 25 30 0 5 lo 15 20 25 30

Alao Alao

c) d)

e=0.90 e=o.90

8=0 8=~/6

) )

qf @

' _o.4 '

£ $

0 5 lo 15 20 25 30 35 40 0 5 lo 15 2O 25 30 35 40

Alao Alar

e) fl

e = 0.90 _0, e = 0.90

8 ₌ n/3 8 ₌ n/2

~i ~i

1

0

3

0.4

~ ~

£

0.i

£

o-1

0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40

Alao Alao

g) h)

Fig.

I. Surface elastic constant _c v-I- A for different values of Hand _e. Note that cj vanishes for A larger than _a few (5 101molecular dimensions. Its surface value is half of the bulk elastic _constant.

0.

e = 0.75

~ ^{e =} 0.85

I 8 ₌

n/6

_I

8

= n/6

( I

~ ~

~

rq rq

~ o

~ o.

o

0 5 lo 15 20 25 30 0 5 lo 15 20 25 30

Alao Alao

ai b)

o. o.

e = 0.95 _e

= 0.90

-~ °. 8

=

n/6

-~ o.

8

= n/6

~

£

@ _@

~ ~

rq rq

0 5 lo 15 20 25 30 35 40 o 5 lo 15 20 25 30 35 40

Alao Alao

c) d)

o-lo

e = 0.90

e = 0.90

m

8

=

n/3

_m ° °°

8

=

n/2

~ ~

@ @

_0.06

~

0.04

~

0.04

~ ~

0.02 0.

o,oo

0 5 lo 15 20 25 30 35 40 0 5 lo 15 20 25 30 35 40

Alao _Alao

e) o

Fig.

12. Surface elastic constants c~ i,s. A for different values of Hand _e. It vanishes for large values of A. Its value is small with respect to c-j. In the

spherical approximation

(e 0), _c~

m 0.

e = 0.90 _e

= 0.75

8 ₌ n/6 8 ₌ n/6

) ~i

1

0.4

~

~ ~

~ ~

0.1

°'~

0 5 lo 15 20 25 30 35 40 0 5 lo 15 2o 2s 3o

Alao Alao

a) b)

e=0.85

e=095 8 ₌ K/6

8 ₌ n/6

-~ _-s

~ 5

@ _%

~ ~

~

g

0 5 lo 15 20 25 30 0 5 lo 15 20 25 30 35 40

Alao Mao

c) d)

e = 0.90 e = 0.90

8 ₌ n/6 8

= n/3

~i ~i

@

d

~

~

@

0.1

@

0 5 lo 15 20 25 30 35 40 0 5 lo 15 20 25 30 35 40

Alao Alao

e) ~

e = 0.90

o.4 8 ₌ n/2

~i

@

~

~

~ o-1

0 5 lo15 20 25 30 35 40

Alao

g)

Fig.

13. Surface elastic constants c, i-s. A for different values of and _e. Its trend _vs. A is

nearly

monotonic. The surface value of _c~ is of the order of the surface value of cj. It vanishes for

large

value of A (with respect to an).

0.2

e = 0.75 _e

= 0.85

_~ o,1

8

= n/6

_~ ~, ~ 8 ₌ n/6

~ £

~

0.0

~

0.

~ ~

~-O.l ~-O.l

~0.2

0 5 lo 15 20 25 30 35 40 45 50 0 5 lo 15 20 25 30 35 40 45 50

Alao Alao

a) b)

0.2

e = 0.95 e = 0.90

~ ~

8

= n/6 _{o 1} 8 ₌

n/6

~/ ~/

d

0.0

~

0

~ ~

~

-0,1

~

-0,1

~0.2

o 5 lo 15 20 25 30 35 40 45 50 0 5 lo 15 20 25 30 35 40 45 50

Alao Alao

c) d)

0.04 ~~~°~'

e = 0.90

e = 0.90

a

" n/3

fi 4xio-' 3

= n/2

fi ~

q @

q~

~ ⁰

~

j

^~~ -4xio~'

-8xio~'

0 lo 20 30 40 50 0 5 lo 15 20 25 30

Alao Alao

e) 0

Fig.

14. Surface elastic constants c~ vs. A for different values of _e and H. Its trend is similar _to that of c, surface elastic constant, as well _as its order of magnitude. Note that c~(6 ) c~(w + and that c~(e ₌ ⁰ ₌ 0.

e=0 _e =0.25

8=nA6

^° 8=nA6

j/ j/

_°.

@ @

o,

~

~

$ $

_C2

0.

~

0 5 lo 15 20 25 0 5 lo 15 20 25

Alao Alao

a) b)

1. 0.4

e = 0.50 _e = 0.75

°.

8

₌ nA6

8

₌ nA6

p

^0. c,

j/

0

qf

_0.4

¹

0.1

~

~

.~ c~

$

~ -O.i c,

0 5 lo 15 20 0 lo 20 30 40 5O

Alao ^Alao

c) d)

Fig.

15. c-j, r.~, r.~ r.4 surface elastic _constants fore

=

0 (a), _e

=

0.25 (b), _e

=

0.5 (c), _e 0.75 (d) and

= w/6 _1-s. A.

This is in

agreement

^{with the results}

published

in

[12].

The

surplus

of elastic energy localized in the

boundary layer

is

given by

6F

=

j _{ILi (A}

₎

n,. j n,.

j +

c2(A

)(n x Curl _{n )~ +} C.3(A

)i(k v) ni~

c~(A

[(k

V

n]

_{n x} curl

n) (6.8) Equation (6.8)

shows that the presence of _a surface introduces additional elastic _terms into the free energy. The first two terms

give

rise to an elastic contribution similar _to the Frank elastic

energy

density.

The last _{two terms are new} elastic contributions connected with the presence of the surface. The

c~(A )-term

has been introduced for the first time

by

Dubois-Violette and

Parodi

[22]

and, more

recently, by

Mada

[23].

The c~-term ^is new and it has been

justified,

in

our paper, for the first time _{on a}

pseudomolecular

basis. It is _zero in the

spherical

approximation,

as discussed in

[12].