Temperature dependence of surface orientation of nematic liquid crystals

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Temperature dependence of surface orientation of nematic liquid crystals

A. Alexe-Ionescu, G. Barbero, Geoffroy Durand

To cite this version:

A. Alexe-Ionescu, G. Barbero, Geoffroy Durand. Temperature dependence of surface orientation of nematic liquid crystals. Journal de Physique II, EDP Sciences, 1993, 3 (8), pp.1247-1253.

�10.1051/jp2:1993195�. �jpa-00247901�

Classification Physics Abstracts

61.30 68, lo

Temperature dependence of surface orientation of nematic

liquid crystals

A. L. Alexe-Ionescu

(*),

G. Barbero

(**)

and G.

Iiyrand

Laboratoire de

Physique

des Solides, Bit. 5lo, Universit£ Paris-Sud, 91405 Orsay Cedex, France (Received 25 Febiuaiy J993, accepted ⁷ May 1993)

Abstract.-The mixed

splay

bend curvature elastic constant Kj~ ^of nematics is shown to be

proportional

to S, the modulus of the

quadrupolar

orientational parameter. ^One consequence is

presented

_on the orientation of _a nematic on a solid substrate for _an

infinitely

strong

anchoring

at an

arbitrary

angle, the surface energy contribution from Kj~

gives

_an apparent surface orientation which

depends

on the temperature.

1. Introduction.

Nematic

liquid crystals (NLC)

_are made

by organic

molecules of

elongated shape [I]. They

behave _as uniaxial

crystals,

whose

optical

axis coincides with the average orientation _n of the

major

_axes of the molecules

forming

the

phase [2],

_n is called the nematic director. In the absence of extemal

constraints,

_a solid substrate orients n~, the surface nematic director,

along

a direction _n, known _as the easy direction

[3].

When n~ _m n for every bulk orientation of _n, the

anchoring

is called

infinitely

_strong, otherwise it is finite.

Many experiments

have shown that the surface orientation of _a uniform NLC

sample

can

depend

_on the

temperature [4-9].

This

phenomenon

leads _even to

temperature

surface

« transitions _» when there exists a « critical temperature » above of below which the symmetry of the

anchoring

is

spontaneously

broken.

Sometimes,

the surface orientation

keeps changing

with the temperature. ^The temperature surface transitions _were

firstly interpreted [6]

in _terms of Parson's

[10]

or Mada's

[I ii

theories.

Phenomenological expressions [12-14]

of the surface

anchoring

_energy in term of _a power

expansion

_versus the modulus S of the orientational nematic surface order

parameter

^lead

already

to a

temperature (T) dependence

of the easy

orientation at _a solid-nematic interface. In this paper we present an altemative

interpretation

of the temperature

change

of surface orientation based

only

on the elastic

properties

of the NLC.

(*) Also _i

PhYsics Department, Polytechnical

Institute of Bucharest,

Splaiul Independentei

313, R-77216 Bucharest, Roumania.

(**) Also

Dipartimento

di Fisica, Politecnico, Corso

degli

Abruzzi 24, 10129 Torino, Italia.

1248 JOURNAL DE PHYSIQUE II N° 8

The idea is that the mixed

splay

bend volume curvature elastic constant

Kj~

^of nematics

depends linearly

_on S. Its contribution to the surface

anchoring

_energy

depends

then _on T

differently

than the

usually accepted

surface

energies

~

S~.

Our paper is

organized

_as follows. In section 2 the usual elastic

theory

of NLC is recalled.

The

generalization

of the elastic energy

density

in _terms of first and second order

spatial

derivatives of the tensorial order parameter ^is

given

in section 3. The theoretical

analysis

of the

experimental

works devoted _to detect the temperature ^{variation of the nematic surface} orientation is

reported

in section 4. There _we show that the different temperature

dependences

of the bulk and surface like elastic-constants _may be

responsible

of the

change

of the nematic orientation induced

by

the temperature.

Finally

in section 5 _we discuss the

physical acceptability

of the _use of second order derivatives in the elastic energy

density.

2. Frank-Oseen elastic

theory

for NLC.

NLC _are characterized

by

_a

quadrupolar

tensor order parameter

Q

defined

by [15]

Q<j ~

(3/21S in,

_n~

(1/313,~1 (li

In

ii

_{n, are} the Cartesian components ^{of the NLC director} n, 3,~ ^{are the} components of the

identity

tensor, and S is the nematic scalar order parameter

[16].

In the _case in which

Q

is

position independent,

the free energy

density

of the uniform nematic

phase

_can be written in the Landau-de Gennes form

[16],

_as follows :

fu

"

(i12 IA<

_{jti Q>j}

Qtf

+

(i13 )

B>jtfmn Q<j

Qtf

_{Qmn +}

(il4

C>jt.fmnpq Q<j

Qkf

_Qm<l

Qpq

,

(2)

where _tensors A, B and C have to be

decomposed

in _terms of the unit _tensor of components 3,~

only. Simple

calculations

[16]

show that

f~

can be put ^{in the form} :

f~

=

(1/21a (T Ti i Q,j Q,j

+

(1/31BQ,j Qji Qi;

_Qmn +

+

(

IN

tic

j(Q<j

Qj,

i~ +

C2

_Q<j

Qji ^{Qif Qf,1}

,

(31

where Tm ^is a

temperature slightly

below the

clearing

temperature T~, a

~0,

B,

Cj

and

C~

_are temperature

independent. By substituting (I)

into

(3)

_one obtains

fu

=

(3/4

_a

(T Tm )

^S~ +

(I/4 ) BS~

+

(9/16 CS~

,

(4)

where C

=

C ₊

(1/2 C~. Expression (4)

has been

widely

used

by

different authors to

analyse

the

nematic-isotropic phase

transition

[16].

Let _{us now} suppose that

Q

is

position dependent,

I.e. that the NLC is described

by

the tensorial field

Q,j

~

Q,j ^{("1 (51}

In this situation

Q<j,1=

3Q<j/%x~ ^{are different from zero. To the non}

uniformity

of

Q

it is

possible

to associate _a distortion energy, whose

meaning

is similar to the elastic

density

introduced in the classical

theory

of

elasticity [17].

In the first

approximation

the distortion energy can be

expanded

in _terms of

Q,j

~, as

[16]

fd (Q,j, ⁱⁱ

=

fd (01

₊ ' /2

K,jiimn Q,j,

_i

Qtm,

n ^,

(6)

where

f~(0)

=

f~

and K

plays

the role of elastic tensor. The elastic _term of

(6),

^I.e.

(1/2)

K,~~i~,,j Q>j i

Qfm

,>,

contains

already

a second order term in S. In this

approximation, assuming

also S _to be uniform, the elastic free energy writes _:

fe

"

(1/2 )

fi~<jt.imn Q>j,k

Qfm,

<, ^"

(1/2

_)[fi~l

(dlv

_{" )~} + fi~22

("

'i~~ " )~ +

'~33 (~

i°~ "

)~l

,

(~)

where, in this

approximation, Kjj

= K~~ =

aS~

_{and K~~}

=

pS~,

in which _a and

p

_are

temperature

independent. Expression (7)

has been

proposed long

ago

by

Frank and Oseen

[18],

and it is the fundamental

expression

of the continuum

theory

for NLC. The relation

Kjj

₌ K~~ ^holds

only

near T~. ^{In the nematic}

phase usually Kjj

# K~~.

3.

Nehring-Saupe

elastic

theory

for NLC.

More

recently Nehring

and

Saupe [19]

have shown that linear terms in the second order

spatial

derivatives of n, can

give

_a contribution to

f~

of the _same order as the

quadratic

terms in the first order

spatial

derivatives. This suggests ^that

f~

has _to be considered as a function of Q,~,~ and of Q,~, a. In order to have _a well defined variational

problem [20],

at the second order

in

S, expression (6)

has _to be written

[21]

_as :

fd (Q,j,

_i,

Q,j, iii

₌

fd (0)

₊ '/2

) K,Jim,n Q,j,

k

Qi»,,

n

+

N,jii Q;j,

_ii + +

(1/2)

M<jtfn<,>pq Q>j, if Qn<n,_pq ⁺ l~<jkfm<>/J

Qij,

_t

Qfm,

»p

(8)

At the second order in

S, K,

M and R have to be considered

temperature independent,

I.e.

they

can be

decomposed only

in _terms of the unit tensor.

Furthermore,

for

NLC,

_tensor R is

zero because

only

tensors of _even number of indices _can be built with the unit _tensor. On the contrary, at the _same level of

approximation

for

f~,

the _tensor N has _to be

decomposed

in _terms

of the unit _tensor and of the _tensor order

parameter Q. By taking

into _account that

~<jtf

_"

_~j<kf

_~

~<jft

_"

~j>ft

,

(9)

due _to the symmetry ^of

Q

and _to Schwartz's theorem _on the inversion of the

spatial derivatives, simple

calculations

give

~>jti

" ~ ~>j

~ti

+

(1/~) fi~2(~>t ~ji

+ ~<i

~jk)

+ fi~3 ~<j

Qti

+

+

N~ 3~t

_Q,~ +

(1/2 ) N~ (3,~ _Q~t

₊

3,y

_Q~~

(10)

As

Q

is _a traceless tensor

(see Eq. (I Ill, Nj

^and

N~

do _not contribute to the elastic energy.

The elastic term

N,~~t

Q>j, if is then

equivalent

to

~>jki

_{Q>j,ki "}

~2 Qij,

>j +

N4

_{Q>j Q>j,}

kk +

~5 Qji

_Q>j.<I ^(~

^~)

It is

important

_{to stress} that in

(

II the first term is linear in

S,

whereas the second and the third _ones

depend

_on

S~,

_as the usual bulk elastic constants. The total energy of _a NLC

sample,

characterized

by

_{a strong}

anchoring

on a solid

substrate,

is obtained

by integrating (8)

_over the volume V

occupied by

the NLC.

By observing

that the r.h.s. of II

)

_can be rewritten _as

~2 _(Q<j,

_),

j +

N4 (Qij

_Q<j,

k), t +

N5 (Qji

_Q<j,I),

^IN

4 Q<j,t Q<

j, k +

N5 Qji,

_Q<j,

I1

,

(12)

we can conclude that

N~ gives only

a surface contribution _to the total energy. On the contrary

N~

and

N~

renormalize the bulk energy,

quadratic

in the first order derivatives of

Q,

and

furthermore

they give

also _a surface contribution. Both _terms connected to

N~

and

N~

_are

quadratic

^{in the scalar order} parameter ^{S. In conclusion the total elastic} energy of the distorted NLC is

given by

F

=

ljj fd (Q,j,

_i

Q,j, al

dv

,

(13 j

v

1250 JOURNAL DE PHYSIQUE II N° 8

which,

from the above

discussion,

_can be put ^{in the} form

(13)

F

=

(1/2) ljj [k,jkimn

_Q<j,k

Qim,

n

+

~ijkimnpq

Q<j,ki

Qmnpql

^{~~ +}

V

+

§[~2

_Q>m,

j

+

~4

_{Q<j Q<j,}

m

+

~5 Qji Qmj,

I "n<

~~, (14)

I

where 3 is the surface of V and _v is the unit _vector normal _to dJi. In

(14)

¹¹ is the _new elastic

tensor connected _to the first order

spatial derivatives,

renormalized

by N~

^and

N~.

By taking

into account

(ii

and that _{n n}

=

I, giving

_{n, n,}

=

0, simple

calculations

give N2 _Q,»i,

, Mm =

N~ S(n

div _n nX rot n

)

_v

1i4

Q<j Q<j,

m "m ~

°

N5 Qjf

_{Qmj f} vn< =

(1/3 IN

~ S~

[2 (nX

rot _n + n div _{n v}

It follows that

N~

does _not

give

_any contribution _to the surface energy. The surface _term

appearing

in

(14)

can be rewritten in the form

§[Kj~

^{n div n}

^K~~(n

^{div n} + nX _{rot n}

)]

_v dJi

z

where

Kj~

= 2 N

~ S ₊

(1/3 N~

S~ and K~~ =

N~

^S +

(2/3 ) N~

S~

(14')

Kj~

and K~~ are the surface like elastic constants introduced

by Nehring

and

Saupe [19].

In what

follows,

_we make the

simplifying assumption

that _{we are}

only dealing

with _a

planar problem

in which the K~~ contribution is

identically

_zero, and _we

keep only

the usual constant

~13.

4.

Temperature dependence

surface orientation.

The

analysis presented

above is valid in

general.

Let _{us now} consider _a

particular

_case in which the NLC

sample

is _a slab of thickness d. The Cartesian reference frame has the z-axis normal _to the

bounding plates, placed

at z = ± d/2. The _x and y-axes are then

parallel

to the _two surfaces.

Let _us suppose furthermore that all

physical quantities,

like _n,

depend only

_on the _z-

coordinate,

and that S is

position independent.

All temperature ^effects are

supposed

_{to come}

from the uniform S

dependent

ⁱⁿ

temperature,

^from

equation (3).

Let _w be the

angle

formed

by

n with the z-axis, whose value for _z

= ± d/2 is ~fi,

imposed by

the strong

anchoring.

In the

hypothesis

of

w ^w I

(implying

~fi w I

too) expression (14),

at the second order in _{w, can} be rewritten _as

G=F/2=jl~~ _{(K((* )~+M~~~(} ~~jdz-jKj~~P(((* ((*

d/2 ^Z dz z d/2 z d/2

(15)

obtained _some years ago in _a different context

[22].

It is

important

to underline that in

(15)

K and M _are

proportional

to

S~,

whereas

Kj~

₌ c-S, as follows from

(14').

From _{now on} the

analysis

is standard.

By minimizing (15) simple

calculations

give

~4_~ ~2_~

b~ j j ⁼ 0

,

Vz _e

(- d/2,

d/2

) (16)

dz dz

and

~2_~

b~ j ^{+ R~fi =}

0,

_z

= ± d/2

(17)

dz

In

(16) b~

_=M/K is

expected

to be temperature

independent,

^{whereas in}

(17)

R

=

Kj~/K

is temperature

dependent.

As shown in reference

[22],

^the

symmetric

solution of

(16), satisfying

the

boundary

conditions

(17),

is _:

p

(z

=

ii

+ R R ch

(z/b

ch

(d/2 b)

^~

(18)

As discussed in

[21, 22]

b is _a

quasi microscopic length. Consequently

d/2 b _»

I,

and hence in the bulk _w

(z)

is

nearly

constant and

equal

to w~

given by

:

w~ =

(I

+

R)

~fi.

(19)

The surface variation of _w is of the order of

Aw=~fi-p~~-R~fi.

Equation (19)

_can be useful _to

interprete

the temperature

dependence

of surface orientation.

To do

this,

let _{us assume} that the surface

energies

involved in the

alignment

_{process are very} strong ^and characterized

by

short range interactions. In this frame ~fi is fixed

by

the surface

interactions,

and _can be considered

temperature independent.

^However the

physical

detectable

quantity

is _{not ~P,} but p~, ^at least in the

experimental

arrangement ^{of references}

[4-9].

This

quantity

for the above

discussion,

is

expected

to be

temperature dependent.

In

fact, by taking

into account the

temperature dependence

of

Kj~

₌ cS and of K

=

aS~, _{equation (19)}

can be

rewritten _as

w~(si

₌

ii

₊

(c/«i(i/sir

m

(201

Equation (20)

shows that p~ is temperature

dependent.

The variation of

p~(T) depends

on the

sign

of _c, which is _not yet ^known

[23].

We underline that the temperature

dependence

of p~ follows from the different temperature

dependence

of N

=

Kj~

with respect to K. In _our

analysis

this result follows from the tensorial

decomposition

of N. In _a recent paper Teixeira et al.

[24], by using

_a

theory

^of Poniewerski _et al.

[251,

have evaluated the ratio between surface and bulk elastic constants.

They

found that _near T~, ^{where S is}

small,

R

l/S,

in

agreement

with _our result.

5. On the

physical meaning

^of second order derivatives in

f~.

The _use of second order derivatives in the elastic energy

density

is well known for

crystals,

_as

discussed in

[21].

Its _use in _our continuum model leaves open an

important question

is it reasonable _{to use a} continuum

theory

_{on a} scale shorter than

f,

^the coherence

length [16]

associated with the usual first order derivative elastic _term ? One _{can assume} that such _a

continuous model is

acceptable

if the characteristic

length

associated with the second

derivative remains

larger

than _a molecular dimension

fo.

^This

length

is not

exactly b,

since the two

(first

and

second)

derivatives _must be used

together.

Let us estimate this

length

and its temperature

dependence.

Assume _a situation where

only

the order parameter ^{modulus S varies} in space. This situation is described

by

the additional bulk free energy

density gradient

terms

~, dS ^~

~ ~,

d~S

~

(2 Ii

I

_dz

_dz~

1252 JOURNAL DE

PHYSIQUE

II N° 8

where K' and M' _are temperature

independent. Equation (21)

is obtained

by (8), by supposing

n

position independent

and furthermore that S

=

S

(z only,

where _z indicates the distance of

the considered

point

from the wall

limiting

a semi-infinite

sample

^of NLC. For _a small

departure

_s from the

equilibrium

value _s

=

S(z) S~, by minimizing

the total free energy we

obtain the

Euler-Lagrange equation

M'

~(

K'~~(

+

~~~

=

0

(22)

dz dz dS

where

f~

is

given by equation (4). By taking

into _account that

equation (22)

writes

b~f~~ ^~ (- f~~ ^~ (+s

_=0,

₍₂₃₎

where

f~

=

K'/3

a(T~ T)

=

f(T/AT.

It exists _{now two} characteristic

lengths,

solution of

(23), given by

_:

fj

^'

=

*

~~

~~~~~~ ^~~~

(24)

For the

f expansion

to be

valid,

the second derivative term has _to be smaller than the first derivative _one. This _means

qb

_~ l. This condition is

always

verified

by

^the two roots

(24).

For the continuum model _to be

valid,

b _must be

larger

than the molecular

length fo.

The second

order elastic _constant M' _must be

large enough.

Then there exists

always

_a temperature T~ ^defined

by

2 b

= f

(T~

^{above which the two} roots

ii

^' _{present a}

_complex

_{part. However the}

s(z)

solution remains stable. If M' is smaller, I-e- when b compares with (or ^is lower

than) fo,

the introduction of the second order

elasticity

is

completely

useless. Note

finally

that, close to T~, I-e- in ^{the limit} of _a small

b/f, equation (24) predicts f~

of the order of

b, independent

of T, and

f_

of the order of

to.

It is

interesting

to compare ^this

prediction

to the

experimental

finding

of the Kent group

[26],

who could not

explain

the S

profiles

in

micropores

with _a

simple exponential decay

_~exp

z/f. They

introduced _a surface

layer

^of constant order parameter, ^{and of molecular thickness. This could be related} to _an

adsorption _property

of the

surface,

_or to the existence of _a weak second order derivative elastic constant, with

b

to.

6. Conclusion.

In this paper, ^we have shown from symmetry considerations that the surface elastic _constant

Kj~ ~cS,

to second order in

S,

whereas the bulk elastic constant K _are

proportional

to S~, to the _same order. In this frame

by assuming

_{a strong}

anchoring

at the NLC-substrate

interface,

_we have shown that the bulk director orientation is

expected

to be temperature

dependent,

in agreement with recent

published experimental

data. This result

depend mainly

on the elastic

properties

of the NLC under

consideration, although

it _can be

interpreted

in _terms

of _an effective surface energy

[27].

Of _course other

mechanisms,

due _to the temperature

variation of the

anchoring

_{energy are}

possible [28].

Each time _a

simple exponential

relaxation

in the coherence

length f

is not able _to

explain

the

spatial

variations of the order

parameter

modulus

S,

_one should think of _a

possible implication

of the second order

elasticity.

Acknowledgment.

One of _us

(A.L.A.I.

thanks I.

Dozov,

N. Kirov and A. G. Petrov for useful discussions and for the

hospitality

at the Institute of Solid State

Physics

of Sofia.

Note added in

proofs

_: after _our paper was

accepted,

_one of _us

(A.L.A.I.)

^{has obtained} the

same result _on the

Kj~-temperature dependence

in _{a mean} field

approximation (Phys.

Lett.

A175

(1993) 345).

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Physics

of

Liquid Crystals

(Clarendon Press, Oxford, 1974).

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E. B., Introduction to

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E. B. Priestley, P. J.

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Eds.

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Langevin-Crouchon,

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