Elastic anomalies at the interface between a nematic liquid crystal and its vapour : a microscopic approach

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Elastic anomalies at the interface between a nematic liquid crystal and its vapour : a microscopic approach

Sandro Faetti, Aurizio Nobili

To cite this version:

Sandro Faetti, Aurizio Nobili. Elastic anomalies at the interface between a nematic liquid crystal and its vapour : a microscopic approach. Journal de Physique II, EDP Sciences, 1994, 4 (9), pp.1617-1630.

�10.1051/jp2:1994221�. �jpa-00248065�

Classification

Physics

Abstracts

62.20D 61.30D

Elastic anomalies at the interface between

_a

nematic liquid crystal and its vapour

: a

microscopic approach

Sandro Faetti and Maurizio Nobili

Dipartimento

di Fisica dell'Universita' di Pisa and Consorzio Interuniversitario di Fisica della Materia, Piazza Torricelli 2, 56100 Pisa,

Italy

(Receii,ed19 November 1993, revised 25 March 1994, accepted18 May 1994)

Abstract. Close _to the interface between _a nematic liquid crystal (NLC) and another medium, the elastic constants _are

expected

to become functions of distance _z from the interface and of

angle

o between the director _n and the unit _vector k

onhogonal

to the interface. Furthermore, due _to symmetry

breaking,

_new elastic contributions that _are absent in the bulk may become imponant close _to the interface. In this paper we consider _a

simple microscopic

model based _{on van} der Waals induced

dipole-induced dipole

interactions under the

special assumption

of perfect orientational order IS I). By

using

this model, _we calculate the free energy

density

and the

excess of surface free energy. Some elastic _{constants are} obtained

by using

_a numerical

procedure.

In particular, _a new elastic contribution which is linear in the director gradients is obtained for the first time. Close to the interface, these elastic _constants

greatly depend

on distance _z and _are

simple

functions of the scalar product _n k.

1. Introduction.

Elastic constants of nematic

liquid crystals (NLC) play

_an

important

role in the

macroscopic

behaviour of these materials. Far from the interfaces of the NLC

sample,

the elastic _constants

are

independent

of the

position

and of the orientation of director _n. Close _{to an} interface the elastic constants are

expected

to become functions of distance _z from the interface and of

angle

o between the director and the unit _vector k

orthogonal

to the interface. Furthermore, new

elastic

contributions,

which _are forbidden

by

_symmetry in the

bulk,

become

possible

close _to the interface. The effect of subsurface elastic anomalies has been

extensively investigated

in

recent years

[1-7].

Some authors believe that these elastic anomalies _can

greatly

affect the

anchoring

of the director at the interfaces

[1-5],

whilst other authors

strongly disagree

with this

point

of view

[6-7].

In _our

opinion,

the

analysis

of these subsurface elastic effects is very

important

in order _to reach _a better

understanding

of the interfacial

properties

of NLC.

Many

authors have

investigated

the interfacial behaviour of NLC

using

^both

microscopic

theories

[8-15]

and

phenomenological

^models

[16-19].

Parsons

[8]

considered _van der Waals

interactions and calculated the surface tension at the free surface of _a

uniformly

oriented NLC

by using

the Kirkwood-Buff

approximation

in combination with the Fowler

approximation.

He found that the surface tension is minimized if the director is

parallel

to the interface (o ₌

ar/2),

whilst it takes _on its maximum value if the director is

orthogonal

to the interface

(o

=

0).

Therefore induced

dipole-induced dipole

van der Waals interactions favour _a

planar

director

alignment

at the free surface of _a NLC. This is the behaviour which has been observed at the free surface of the NLC PAA.

Analogous

theoretical results have been obtained _more

recently by using

_a different and _{more accurate} theoretical

analysis [14].

The

anchoring

is

predicted

to be strong ^{and the}

anchoring

coefficient is estimated to be of the order of _a few

erg/cm2.

Telo da Gama _et al.

[9, 10]

and

Tjipto-Margo

et al.

[I II investigated

the interfacial behaviour

numerically using

_a

special

model

potential

with both

repulsive

and attractive interactions.

They

showed

that, depending

_on the values of two coefficients

appearing

in the

intermolecular

potential,

either _a

homeotropic (o

=

0)

_{or a}

planar (o

=

gr/2)

director

alignment

can be favoured. In this _case, too, strong

anchoring

_energy is

expected

except in the very

special

case where interactions

favouring planar

or

orthogonal alignment

have almost the

same

intensity.

Other authors

[12, 13]

have considered hard

repulsive

interactions and showed that

they

can generate ^either a

homeotropic

_{or a}

planar

_{or a} tilted director

alignment.

In recent

papers,

Tjipto-Margo

and Sullivan

[14]

and Teixeira and Sluckin

[15]

derive _a Landau-de

Gennes free energy functional from _a

microscopic

Helmholtz free energy functional of _a

nonuniform NLC. The

explicit expressions

for the Landau-de Gennes

phenomenological

coefficients _are obtained in terms of the intermolecular

potentials.

Both

anisotropic repulsive

and attractive intermolecular forces _are considered. At the free surface of _a NLC

[14],

repulsive

and

short-range

attractive intermolecular forces _are found _to favour _a

homeotropic

director

alignment,

whilst

long-range dispersion-like

attractive forces _can favour _a

planar

or

oblique

director

alignment.

In

particular,

induced

dipole-induced dipole

interactions favour _a

planar

^director _easy orientation _at the free

surface,

whilst

quadrupolar

interactions favour _a tilted director orientation. The

important

role of

quadrupolar

interactions has also been

emphasized

in reference

[19]

where _an ordoelectric model of

quadrupolar

interactions _was

proposed.

To the best of _our

knowledge

all theoretical

microscopic

calculations of the _excess of surface free energy and of

anchoring

_energy have been

performed by making

the

assumption

that director orientation is uniform close _to the interfaces. This

assumption completely

disregards

_any

possible

contribution due _to the subsurface elastic anomalies

[1-4]. Experimen-

tal values of the

anchoring

_energy coefficients _at different kinds of interfaces have

usually

been found _to lie in the

10~~ erg/cm2-10-' erg/cm2

_range, whilst theoretical values

predicted by microscopic

models _are of the order of _a few

erg/cm2.

In our

opinion

elastic subsurface

anomalies

might

be

responsible

for this

large discrepancy.

Different kinds of elastic anomalies _are

expected

to occur close to the interfaces of _a NLC.

The first _one is related _to the fact that the order parameter ^{S in} a thin interfacial

layer

close to the interface _can differ

greatly

from the bulk

equilibrium

value S~

[18].

The elastic _constants of

a NLC _are

increasing

functions of the scalar order parameter ^S and, thus _one expects their value to

change appreciably

in the interfacial

layer. Yokoyama

et al.

[2]

and Faetti _et al.

[1, 4]

analyzed

in detail this kind of elastic

anomaly

and showed that it _can

greatly

affect the

anchoring

energy and its temperature

dependence.

The second kind of elastic subsurface

anomaly

is related _to the presence of surface-like elastic constant

Kj~

^{in the}

expression

of the free energy

density

of NLC

[27].

Due _to the

presence ^{of this elastic constant, Barbero et al.}

[3]

showed that _a strong ^{subsurface director} distortion should _occur in _a very thin interfacial

layer

close to the surfaces of the NLC.

Hinov

[6]

and

Pergamenshchik [7] strongly

criticize these theoretical results.

According

to

Pergamenshchik,

the strong subsurface director distortion is _an artifact of the elastic

theory

due

to

disregarding higher

order elastic contributions. He thinks

that,

if all these

higher

order

contributions would be taken into account,

they

would bound from below the free energy

functional in such _a way as to make the _occurrence of any strong ^{subsurface distortion}

impossible.

Therefore he proposes that the correct director-field _must be

sought

in the class of continuous functions that solve the bulk

Euler-Lagrange equations.

In _a recent paper

[5]

Faetti

proposed

a

simple

but

rigorous

theoretical _test

(surface

torque

test)

_{to assess} the

validity

of these different models. This _test is based _{on two}

general principles

of mechanics the

principle

of virtual works and the

general

laws for the

equilibrium

of _a mechanical system. ^{The Hinov-}

Pergamenshchik

model _was found to not

satisfy

this test. Therefore _we infer that the

large

subsurface distortion _must

actually

exist if

Kj~

#0.

By introducing

second order elastic contributions into the elastic

theory

of

NLC,

Barbero et al.

[3, 20]

^{showed that the main}

macroscopic

effect of this subsurface distortion is _an

important

reduction in

anchoring

energy.

The _same main conclusions _were reached

by

Faetti

using

_a different

geometry [2 II

and

making

a

systematic expansion

of the free energy

density

at

higher

orders

[third, fourth, [5].

The third kind of elastic

anomaly

is related _to the fact that the interface breaks the translation symmetry ^{of the NLC}

and, thus,

the elastic constants are

expected

to

depend

_on the z-distance from the surface in _a

region

of thickness

comparable

with the characteristic range of molecular interactions. Furthermore,

according

to

general

_symmetry

principles,

the elastic constants are

expected

to become

dependent

_on the scalar

product

_n k.

Finally,

a great ^{number of} new

elastic contributions is

expected

to exist close _to the interfaces. In

particular,

there _{are new} elastic terms that

depend linearly

on the director

gradients.

The latter contributions _can favour the presence of _a spontaneous ^{director subsurface distortion within} a thin interfacial

layer

and

can

greatly

affect the

anchoring properties

of NLCS [3].

In this paper we are interested in the third kind of elastic

anomaly

and _we wish _to calculate elastic _constants

using

a

microscopic

model of molecular interactions. This is

certainly

a

complex problem and,

thus, we make _use of _a very

simplified

model here. We _{assume van} der Waals induced

dipole-induced dipole

interactions between molecules and _a

perfect

orientatio- nal order of the

anisotropic

fluid. The latter

assumption

_means that the scalar order parameter ^S is assumed to be S

=

and,

thus, the local orientation of molecules coincides with the director orientation. Under these

assumptions,

the orientational part ^{of the free} energy coincides with the orientational

part

^{of the} energy, since the orientational entropy ^is zero. Furthermore the molecules _are assumed to be

spherical

in form. These _are rather strong

approximations

and,

thus,

_{we can}

expect

our theoretical results to

provide only

_a

qualitative

view of the actual interfacial behaviour of _a real NLC. In section 2 _we discuss _our model and _we calculate the

analytical expression

of the free energy

density

and of the surface _excess of free energy per unit surface _area for _a uniform director

alignment.

^{We show that the}

anchoring

_energy function is different from the

simple Rapini-Papoular expression [22].

In section 3 _{we use} a numerical

procedure

to obtain some elastic _{constants as a} function of distance _z from the interface and the scalar

product

_n k. Section 4 contains _our conclusions.

2. Free _energy

density

and surface _excess of free energy.

We consider here a semi-infinite NLC which fills the

semispace

_{z ~} 0 and has _a unit surface

area. We _use k to denote the unit vector which is oriented

along

the z-axis

orthogonal

to the

nematic-vapour

interface _z

=

0 from the nematic side toward the vapour side. The interaction energy induced

dipole-induced dipole

between _two

geometrically spherical

molecules is

[6]

_:

u =

~

(kj #

_{6~ )2 for rj~}

~ ~r

,

II

~~2

and

u =

0 for rj~ ~ «

,

(2)

where

ii

and k~ are two unit _vectors

parallel

to the

long

_axes of molecules I and

2,

t

is the tensor defined _as

t= f

_-3

ii :k~, ^f

is the

unity

tensor, v~0 is _a

positive

characteristic coefficient and _« is the characteristic molecular

length.

The interaction energy in

equation (I)

has been obtained

by assuming

the

following expression

for the

polarizability

tensor :

D,~

=

Dkj

_:d~. This _means that the

dipole

moments are assumed to be induced

only

along

the

long

_axes of each molecule. This

assumption is, obviously, conflicting

with the

assumption

of

spherical shape

of the molecules. A _more accurate

theory

of interfacial interactions should also account for the

geometrical anisotropy

of molecules. However, in this

case, theoretical calculations become much _more

complex

and,

thus,

_we consider

only

the

orientational

coupling

due _to the

long-range

part of the intermolecular

potential.

We make the

following simplifying assumptions

I)

the scalar order parameter ^{is S} = I

everywhere.

This _means that the orientational entropy is _zero and, thus, the orientational free energy coincides with the orientational energy.

Furthermore the director orientation at a

given point

coincides with the orientation of the

long

axes of molecules at the _same

point

_;

it) we use the Fowler

approximation [3

Ii which consists in

assuming

that the molecular

density

_n is _constant

everywhere

in the NLC and

abruptly

becomes _zero in the vapour

region

(z _~ 0 ). This

approximation replaces

the

liquid-vapour

interface

by

a structureless

repulsive

wall. The _same method has been used

extensively

in references

[14]

and

[15]

to make _a

systematic expansion

of the surface free energy in _terms of

spherical

harmonic contributions to the intermolecular

potential.

These papers do _not confine themselves to the

approximation

S

=

I _;

iii)

_we restrict _our attention to a

planar

case where the director lies in the _x-z

plane

of the Cartesian coordinate frame and forms _an

angle

o

(z

with the z-axis. Under this

assumption

the director _{at a}

given point

^{in the} NLC is _:

n _m

(sin (z), 0,

_cos

(z)). (3)

The total energy of the NLC per unit surface _area is then

given by

_:

=

~

1_ ~

~~~ ~' ~2)~ dZ dV

~ ~

~~~

where nj and n~ are the directors in _two different

points

and A is the

positive

coefficient :

A

=

n~ _v,

(5)

dvj

and

dV~

_are infinitesimal volume elements and the

integrals

are

performed

over the

semispace

_z ~0. Since the director-field

only depends

on the z-coordinate _{we can}

easily perform

the

integration

_over the other coordinates. After _some

straightforward

calculations, the

integral

of

equation (4)

becomes _:

1«

« «

U

=

dz u* dz'

= F dz,

(6)

o

o o

where u* is

given by

_:

u*

=

fi (3 ^sin~

^Hi

^cos~

^{o~ + 3}

^sin~

^o~

^cos~

^{Hi +}

~

sin~

_Hi

sin~ _o~

_{+ 6}

cos~ oj cos~ ~j

8 zj~

for

(zj~(

_~ «

(7)

and

u*

=

) ^cos~

_Hi

_cos~

_o~

⁴

₁₆

^~(

+ 18

~(

+

sin~

_Hi

sin~ ~(

~/

^lo

~(

+

~~

~)

₊

« « « « 4 _«

~ ~ ~ ~

z(~

z(~

+

[sin

o _cos

o~

+ sin o~ cos o 12

~ 9

j

« «

~2 ~4

+ sin o cos

Hi

^sin

o~

cos

o~

4 ₊ 40

'(

^{36 ( for}

(zj~

_~ «

,

(8)

" "

with _z

j~ ^{= z} z' and Hi ₌ ^o (z

)

and o~ ₌ ^o

(z'),

2 _u *

(z, z')

dz dz'

corresponds

to the interaction energy between _two thin nematic slabs of thicknesses dz and dz' at

points

_z and z',

respectively.

F (z

)

_represents the free energy

density

at

depth

_z in the NLC and it is defined

as

«

F

(z,

o

=

o

u ^* (z, z' dz'

(9)

In the very

special

_case where the director orientation is uniform

everywhere

in the NLC

(Hi

₌

o~

₌ ^o ₌ ^{Cte in}

Eqs. (7)

and

(8)),

_we find the local free energy

density

F

(z

= a

(?

₊ b(z

cos~

_o

+

c(z cos~

_o

,

lo)

where coefficients

a(z), b(z)

and

c(z ),

for 0

~ z ~ ~r, are

~~~~ /~~ _~/~

^~

_~/ ~/ j~

^~

^~ )~j'

_{~~ ~~}

b(z)

=

-)[-15~+28(~)~-§~ (i)~j _(j2)

« «

and

whilst,

for _z

~ «,

they

are

a(z)

=

-) [+ ~/)

^~

) _{)~j, (14)}

b(z)

=

/

_and

_c(z)

=

~

"(.

(15)

16z 32z

Note that, in the limit _z-au, the free energy

density

reaches the bulk value

Fb

"

'6 grA/15

«

~ which is

independent

of the

o-angle. Indeed,

far from the

interface,

all

spatial

directions _are

equivalent and,

thus, the free energy cannot

depend

_on H.

Figure

I shows

F.F~

o

L°g(td)

3

Fig.

I. Excess of free energy density F F

~ i,ersus z/s for cenain values of angle 0. The unities _on the

vertical axis _are nodimensional unities.

Unity corresponds

_to ^A/~r'.

the

z-dependence

of the _excess of free energy

density

F (z, ^F _b ^{for certain values of}

angle

H. Due to the rather

long

range of _van der Waals interactions the thickness of the interfacial

layer

where the _excess of free energy

density

is

appreciably higher

than _zero is of the order of

5-10 molecular

lengths (for

_z~« the _excess of free energy

density

is

proportional

to

I/z~). According

to the Gibbs

thermodynamic

of interfaces, ^the excess of surface free energy is _:

y =

°~ iF(z) F~i

dz.

(16)

o

By substituting equations (10)

to

(15)

in

equation (16)

_we

easily

find _:

y = yo ⁺ B

cos~

+ C

cos~

= yo +

W(H ), (17)

where yo is _an

isotropic contribution,

whilst

W(H)

= y yo is the

polar anchoring

_energy

function. Coefficients yo, B and C _are

given by

yo=~"~, B=~"~

_and

C=-~"~=-~. ₍₁₈₎

16« 8~r 16~r- 2

Figure

2 Shows the

H-dependence

^of the

anchoring

_energy. ^Since coefficient A is

positive,

the

excess of surface free energy is minimum if _cos

=

0,

that is

=

gr/2,

whilst it is maximum when _cos

=

I. This agrees with

previous

theoretical results

[8].

Note that the

anchoring

energy function in

equation (17)

is _{an even} function of the scalar

product

n k

= cos H,

according

to

general

_symmetry arguments

[4].

In reference

[8]

the author did _not

investigate

the

b-dependence

of the

anchoring

_energy and stated that,

probably,

it is of the

Rapini- Papoular

^form

[22].

Our theoretical results in

equation (17) clearly

show that, for _a

perfect

orientational order

(S

=

I), ^the

anchoring

^{function is different from the}

Rapini-Papoular expression

^{since it contains} an

important

contribution of the fourth order in _{cos H.} In _our

opinion

this result should hold _even if _one accounts for the effects of

imperfect

orientational order. We note that

departures

from the

Rapini-Papoular

^{form of the} same kind as those in

0 45 90 6

(deg)

Fig.

2.

Anchoring

_{energy versus}

angle

0. The unities on the venical axis _are nodimensional unities (Aim~).

equation (17)

have been observed for the

polar anchoring

of _a NLC _on different substrates

[23- 26].

We wish _to

emphasize

here that the

anchoring

_energy function

W(H

in

equation (17)

has

been obtained

by assuming

^uniform director

alignment

^of the director-field in the whole

interfacial

layer.

Therefore we can expect

equation (17)

to

really

represent ^the actual

anchoring

energy ^at the free surface of the NLC

only

if the

uniformly aligned

state is

energetically

favoured'with respect to a deformed

configuration.

Careful theoretical calculation of the

anchoring

_energy function would

require

_{one to} be able _to find the director-field which minimizes the free energy in

equation (6)

for _a fixed value of the director

angle

out of the interfacial

layer.

This is _a very difficult

problem

which _can be

only

solved

by using

numerical

methods.

However, according

to the numerical results in section 3, we expect ^molecular

interactions to favour _a strong ^{subsurface distortion if the director}

angle

is different from 0 and gr/2.

Indeed,

_we will show that _new elastic contributions that

depend

_on the first power of the director

gradients play

_an

important

role close _to the interfaces. These _new elastic _terms favour

the _occurrence of _a director distortion _near the interfaces. Therefore the surface _excess free

energy in

equation (17)

is not

expected

to

correspond

_{to a} minimum of the free energy for _a

given

value of the

angle

Hand, thus, we expect ^the

anchoring

_energy in

equation (17)

to

correspond

to an overvaluation of the actual free energy.

3. Elastic constants in the interfacial

layer.

In this section _we shall calculate _some elastic constants of the NLC

numerically

as a function of distance _z from the interface and of director

angle

H. The elastic free energy in the bulk of _a

nematic

liquid crystal

is

given by

^{the well known}

expression Kj (div

_{n )~ K~~ (n.} curl _{n )~} K~~ (n A curl _n )~

F~

= + + +

Kj~

^div (n div _n

2 2 2

(K~~ + K~~ ^div

(n

div _{n + n}

A curl _n

), (19a)

where

Kij,

K~~ ^and K~~ are the

splay,

twist and bend elastic constants,

respectively,

whilst

Kj~

^and K~~ are surface-like elastic _constants

[3].

At _a distance _{z »}

« from the

interface,

all these elastic _{constants are}

independent

of the

position

_z and of the director orientation. Close _to the interfaces, the translation symmetry ^{of the} system ^{is broken and the elastic constants}

become

dependent

on z and _on the scalar

product

n k. Furthermore, a great ^{number of} new

elastic contributions that _are forbidden

by

symmetry in the bulk, is

expected

to be present near

the interfaces. The

general expression

of the elastic free energy

density F~(z)

at a

given

position

_z can be obtained

by looking

for _a power

expansion

of the free energy in terms of the director

gradients.

Due to the presence of the interface the free _energy will

depend

_on both the

director _n and the unit _vector k

orthogonal

to the interface. F

(z)

must

satisfy

the

following physical

constraints _:

a)

it _must be _a scalar

(not pseudoscalar) function, b)

it must be invariant

with respect to the transformation _n

- n

(the

_van der Waals

potential

is _{an even} function of the director

n). By exploiting

condition

a)

we can show that there _are three _new

independent

elastic contributions that _are linear functions of the director

gradients: K(divn, Kl'(k grad )(n k)

and

K("(n grad )(n

k).

By exploiting

condition

b)

we find that the _new

elastic _constants

K(

and

Kl'

must be odd functions of the scalar

product

n.k, whilst

K(" ^is an even function of _n k.

By repeating

the _same kind of

analysis

to find the elastic contributions that

depend

on the square powers of the director

gradients

and _on the second order director

gradients (surface-like

elastic

contributions)

we find _a great ^number of

possible

new elastic terms

(15

_new elastic

constants).

In order _to

greatly simplify

the theoretical

analysis,

we concentrate our attention _on the

special

_case of

planar

director distortions where the director-field lies in the _{x z}

plane

and

only depends

_on the z-coordinate _as in

equation (3).

In this _case, the number of

important

elastic contributions is

greatly

reduced and the local elastic free energy

density

_can be written in the form

K~(~(z, H)

~

Kj(~(z, H)

~ ~

F~(z,

= F (z, +

Kj

(z,

)

sin RR' ₊ sin _{+ cos}

(H')

2 2

~~~~~'

~ _~~~ ~ ~

~'~~ ~i~i(Z,

~

~~~/

^{~ " (j}

^9b)

F(z, H)

is the free _energy

density

of the undistorted nematic

sample (H'(z)=0

and

H"(z)=0

in

Eq. (19b))

which has

already

been calculated in section 2

(Eq. (10)).

Kj(z,

is _{a new} effective elastic constant which is

given by Kj (z, )

=

K( (=,

+

Kl'(z, )

_{+ K("}

(z, )

_cos

(19c)

Due to the symmetry n - n,

Kj(z, H)

must be _{an even} function of the scalar

product

n k. The other elastic contributions _come from the those elastic terms which

depend

_on the second powers of the director

gradients

and _on the second director

gradients.

We have denoted the

corresponding

effective elastic constants

by using

the

symbols

_{K[(~, K§(~} and K~(~ since these

elastic _{constants are} reduced _to the

ordinary

elastic _constants

Kj,,

K~~ ^and

K,~

ⁱⁿ

equation (19a)

for _{z » «.} However, in order to avoid any confusion, it is

important

to

emphasize

that these

elastic constants _are

really

a linear combination of _a lot of different elastic contributions that

are not present ⁱⁿ the bulk

expression

of the elastic free energy

(Eq. (19a)).

For

instance,

the

splay-like

elastic contribution

Kj(~ sin~ RR'~

ⁱⁿ

equation (19b)

_comes from the

superposition

of

some different elastic terms as

Kj j/2 (div

_n)~

,

K( j/2 [(k grad )(n

k)]~

,

Kl'j/2 [(k grad )(n

k

)]

div _n

and

Kl'(/2 lgrad

(n

k)]~

Due to the symmetry n _- n, the effective elastic constants K[(~,

Kj(~

and K~(~ must be _even functions of the scalar

product

_n k.

To calculate the numerical values of these effective elastic constants, we consider _a director distortion of the form

#i~ )

=

#(z)

+

#'(=)(~ z)

+

#"iz)(~ z)2, ₍₂₀₎

where (z

), H'(z )

^and "(? _are the director

angle,

the first derivative of the director

angle

and the second derivative at distance _z from the interface,

respectively.

The numerical values of the

effective elastic _{constants can} be obtained

by substituting

_very small _constant values of

H'(z)

and

H"(z)

ⁱⁿ

equation (20), by calculating

the

corresponding

free energy

density

in

equation (9) numerically

and

by comparing

the numerical values of

F(z) f(z, H)

with

equation (19b).

We _use the

following procedure:

we fix

given

values for _z and

H(z),

then _we put

H'(z)=0

and

H"(z)#0

in

equation (20)

and _{we can} calculate

F~(z, H) F(z, H) by

numerical

integration

of

equation (9).

From

equation (19b)

_we find K[(~ =

2(F~(z, H) F(z, H))/(H"

sin 2

H).

The elastic _constants

Kj(z, H)

and the elastic

constant

K(z,

H

given by

K(z,

)

=

K[(~(z, ^sin~

+

K§(~(z, cos~

,

(21)

can be obtained

by calculating

the free energy

density

in

equation (9)

for _two different values of

H'(z)

and for

H"(z)

=

0 and

by comparing

the numerical results with the

predictions

of

equation (19b). By repeating

the _same

procedure

for different values of _z and H

(z

we obtain the

angular

and

spatial dependence

of the elastic _constants. To avoid any

spurious

effect due _to

higher

order elastic contributions _{we use} very small values of

H'(z )

and

"(z )( H'(z)

= 10~

~/«

and

H"(z)

= 10~ ~%~r~). In these conditions the elastic contribution _to the free energy is very

small with respect to the free energy

density F(z,

of the undistorted

sample and, thus,

numerical _{errors can} become

important.

For this _{reason we} have used _a

16-point

Gaussian

integration procedure

and

fourth-precision

numerical variables. In this way we obtain the elastic constants with _a relative accuracy that is better than 10~ ^~ The

analytical expressions

of the bulk elastic _constants

(z

» _« have been obtained

by

Barbero et a/.

[28, 29]

for _{a van} der Waals

potential,

a

complete

orientational order and

spherical

molecules. These theoretical

expressions

_are :

Kjj

= K~~ = K

=

~

Kj~

₌ ^{~ ~} _;

Kj

=

0, (22)

where

J

=

8 WA

j~ ^~

^dr =

~ "~

(23)

o r «

Note that this theoretical calculation

gives

identical values for the

splay

and bend elastic constants in the bulk of the

anisotropic

fluid. More

general analytical expressions

of the bulk elastic _constants have been obtained in the _case of

imperfect

orientational order

(S

_~ l

) by

Teixeira et al.

[30].

In this _case, too, the

splay

and bend elastic constants are found to have the

same values. The main result obtained

by

Teixeira et al, is that the elastic _constants have smaller values with respect to those in

equation (22).

In

particular,

the elastic

splay

^{and bend}

constants in

equation (22)

are now

multiplied by

_a factor

S~,

^{whilst the} surface-like elastic

constant is _now

multiplied by

a factor

S,

where S is the scalar order parameter. However, apart from these

quantitative features,

the main

qualitative

aspects ^remain

unchanged

with respect to the

special

_case of

perfect

orientational order. Similar effects of _a finite value of the order

parameter are also

expected

as far _as the interfacial values of the elastic _{constants are} concerned. In

particular,

_we expect our numerical values of the effective

splay

and bend elastic

constants to have _to be

multiplied by

S~ _to obtain the correct order of

magnitude

and the values

of the K[(~ surface-like effective elastic constant and of the

Kj

elastic constant to have _to be

multiplied by

S. Our numerical values of the elastic _constants for z/«

=

10~

_agree with the

analytical predictions

in

equations (22)

and

(23)

within _our relative numerical accuracy

(10~~). _Figures

3, 4 and 5 show the

z-dependence

of the effective elastic _constants

K

o o

i

Iwg(tt)

Fig.

3. Elastic _constant K _i>ersus z/~r for certain values of 0. The elastic _constant is expressed in nodimensional unities (A/~r).

~ elf K

iso

0 ~ 0 e

I I

lwg(tbj

²

lwg(tb)

Fig.

4.

Fig.

5.

Fig.

4. Elastic _constant A~( i,ersus z/~r for certain values of 0. The elastic _constant is expressed in nodimensional unities (A/~r).

Fig. 5. Elastic _constant K, _i,ersus z/~r for certain values of 0. The elastic _constant is expressed in nodimensional unities (Aim)~.

K, _{K~(~ and}

K~, respectively,

for certain values of

angle

H. Note that K behaves very

peculiarly

since it

rapidly

^increases

starting

from ₌

=

0 toward _z

= « and

abruptly

reaches _a constant value for

z ~ ~r which coincides with the bulk value

predicted

in

equation (22).

This

peculiar

behaviour is,

probably, closely

related _to the

special

kind of intermolecular

potential being

considered here. It is

important

to

emphasize

here that the use of the Fowler

approximation

is

known _to

produce poorly

accurate results for _z

= « and'much _{more accurate} results for

z » «. In

particular,

the

physical meaning

of theoretical results obtained for _{z ~ «}

using

the Fowler

approximation

is

questionable. Indeed,

in the present

approach,

_{z represents} the

gravity

center of molecules that have been assumed _to be

spheres

of radius _«. Furthermore _{we note}

that, within the relative numerical accuracy

(10~

^~

),

K is

completely independent

of

angle

for any value of _z. Since K is defined

by equation (21),

we infer that

equality

_K~(~

= K§(~ = K,

which holds for the bulk elastic constants with this

model,

remains satisfied at any

point

_z.

Therefore both K~(~ and K§(~ are

independent

of

angle

H. On the contrary, K~(~ and

Kj

_are found _to

decay

_to the bulk values within _a thicker interfacial

layer (10-30

characteristic

lengths «)

and _to

greatly depend

on

angle

H. An

important

feature of K[(~, _K§(~ ^and K~(~ is that their value at interface _z

=

0 is

exactly

half the bulk value. This property can be

easily

understood since the _nearest

neighbours

for _a molecule which lies _on the interface _are

exactly

half of those for _a bulk molecule.

According

to our

previous analysis,

the effective elastic _constant

Kj

is

expected

to be _an odd function of the scalar

product

_n k, ^whilst the other

effective elastic _{constants are}

expected'to

be _even functions of the _same parameter.

Figures

6a,

6b and 6c show the numerical values of K

=

K[(~

_{= K§(~,} K((~ ^and

Kj

at z ₌ « versus

angle

H.

The full

points

denote the numerical values of the elastic _constants and the full lines denote the

polynomial

best fits of the numerical results to K~~ ⁼ ao + aj x + a~ x~ _{+ a~} x~ _{+ a~}

x~,

where

ao, aj, a~, a~ and a~ are numerical coefficients and _x

= n.k

= cos H.

By repeating

the

polynomial

fit for different values of _{z, we} find that the

angular dependence

of

K[(~(z, )

and

Kj

(z,

H)

_are

given (within

the numerical

accuracy) by K()(Z,

#

=

a0(Kj3

₊

a2(Kj3)(n

_{k )~}

(24)

0 45 90 135 180

6

(deg)

a)

-

1

t f

(

_o

ar

j~

p

_m

z-i.o ~

~ ^-i

0 45 90 135 180 0 45 90 135 180

6

(deg)

⁶

(deg)

b) c)

Fig.

6. a) Elastic constant K

K() K~(

_iersus 0 at _{z = ~r.} Black

points

denote numerical values, whilst the full line denotes the

polynomial

best fit. The elastic constant is expressed ⁱⁿ nodimensional unities (Al_~r ). b) Elastic constant K((~ i,ersus 0 at _z ~r. Black

points

denote numerical values, ^{whilst the} full line denotes the polynomial best fit. The elastic _constant is

expressed

in nodimensional unities (A/~r). c) Elastic _constant Kj _versus oat _{z ~r.} Black

points

denote numerical values, whilst the full line

denotes the polynomial best fit. The elastic constant is expressed in nodimensional unities

(A/~r)~.

and

Kj (z,

₌ a

(Kj )(n

k +

a~(Kj )(n

k_)~

,

(25)

where

ao(Kj~), a~(Kj~), aj(Kj)

^and

a~(Kj)

_are functions of _z. Coefficients

ao(Kj~),

a~(Kj~ ),

_aj

(Kj

and a~

(Kj

_{versus z}

ire

_{shown in}

figures

7a,

7b,

7c and

7d, respectively.

We note that the

decay

of the

K~(~-elastic

constant toward the bulk constant value _occurs within _a rather

large

characteristic

length compared

with the other elastic constants. In

particular,

coefficient

a~(Kj~)

in

figure

7b remains

appreciably

^{different from the bulk value} up to

z ₌ 30 _«.

1.0

~ _~

~ _Q

~ ~

~ ~

jP m

@

.1 0 1 2 3 -1 0 1 2 3

Log(zla) Log(zla)

a) b)

-l

0

og(zla) og(z/o)

c)

d>

Fig.

_7. -a)

Coefficient ao(K,~) _versus z/~r. The unity _on the

ertical axis is A/~r.⁾ Coefficient

a~(K,~) versus z/~r.

The

unity on the vertical _axis is A/~r. c) a, (K, ) ersus

the

4. Conclusions.

In this paper we

give

the first

microscopic

calculation of _some elastic constants of _a NLC in the interfacial

layer.

To make this calculation _{we use a} very

simplified

kind of intermolecular

potential

based _{on van} der Waals

induced-dipole

interactions and _we make the

simplifying assumption

of

perfect

orientational order and _a

spherical geometric shape

of molecules.

Furthermore _we consider

only planar

director distortions and, thus, _we cannot find the twist elastic constant K~~ or the K~~ ^elastic constant. These _are rather

strong approximations and,

thus, theoretical results _are

expected

to

give only qualitative

information _on the actual elastic behaviour of real NLC materials. For instance, this

simple

model is _not able to

predict

_any

difference between the

splay

^{and bend} elastic constants. With this very

simplified

model of

molecular interactions _we calculate the

analytical expression

of the free energy

density

and the director

anchoring

_energy for _a

uniformly aligned

NLC. We find that the

anchoring

_energy is not

represented by

the

simple Rapini-Papoular expression

in close

agreement

with

experimental

results _on different substrates.

By using

_a numerical

procedure

we obtain the

dependence

^{of the}

effective elastic constants on z and

Hand,

for the first

time,

_we evaluate the symmetry

breaking

elastic constant

Kj.

We find that this _new elastic _constant is different from _zero within _a thin interfacial

layer.

It is

important

_to

emphasize that,

due to the

approximations

of the present

model,

_our theoretical results _are

expected

to

give only

_a

qualitative

view of the very

complex

interfacial behaviour of _a real NLC. However, these

simple

theoretical results

strongly

suggest ^{that the} director orientation in _a very thin interfacial

layer

can be

strongly

distorted due _to the presence of the symmetry

breaking

contribution which is

proportional

to the _new effective elastic constant

Kj

which favours the _occurrence of _a subsurface director distortion if the director is tilted with respect ^{to the interface}

(Kj

=

0 for

planar alignment).

However, in _our

opinion,

an

elastic

description

of the subsurface director distortion cannot

provide

accurate results.

Indeed,

as shown in reference

[5],

_an infinite number of

higher

order elastic contributions

play

_an

important

role when the characteristic

length

3 of the director distortion is

comparable

with the molecular

length

_«.

Therefore,

as far _as the subsurface director distortions _are concemed, more

accurate theoretical results _can be obtained

by using

a

microscopic expression

of the total free

energy as that in

equation (4).

In

particular

one has _to look for the interfacial director-field

which minimizes the total free energy

by using,

for instance, numerical minimization

procedures.

In the _case of induced

dipole-induced dipole

interactions _we have shown that the favoured surface director orientation is

parallel

to the interface and,

thus,

_we

expect

^the

equilibrium

director-field _to be

uniformly

oriented

parallel

to the free surface

since,

in this

case, both the K((~ ^and

Kj

elastic constants do not

play

_any role

[5].

Some

preliminary

numerical calculations with the free energy in

equation (4)

seem to confirm that the minimum of the total free energy is reached for _a uniform director

alignment

in the

plane

of the free

surface. A subsurface director distortion _can be

expected

to occur if the free energy contains contributions which favour _a tilted

alignment.

For instance, _a tilted surface orientation and,

thus,

_a subsurface director distortion is

expected

to _occur if _we consider _a NLC with the _same

intermolecular interactions in

equation (I )

but in the presence of _a

magnetic

^field

perpendicular

to the interface. Another

interesting

_case is

represented by

the model

potentials

of references

[14]

and

[15]

that have been shown to

favour,

in _{some cases, an} easy tilted

alignment.

In this

case we expect ^that a spontaneous ^{subsurface director distortion should also} occur in the absence of external fields.

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