A molecular theory of surface tension in nematic liquid crystals

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A molecular theory of surface tension in nematic liquid crystals

J.D. Parsons

To cite this version:

J.D. Parsons. A molecular theory of surface tension in nematic liquid crystals. Journal de Physique,

1976, 37 (10), pp.1187-1195. �10.1051/jphys:0197600370100118700�. �jpa-00208515�

A MOLECULAR THEORY OF SURFACE TENSION

IN NEMATIC LIQUID ^CRYSTALS

J. D. PARSONS

Departamento

^{de Fisica} Universidade Federal de Santa Catarina

Florianópolis, Trindade,

^{Brasil 88000}

(Reçu

^le

2 fevrier 1976, accepte

^{le 10 mai}

1976)

Résumé. ^- On calcule la tension superficielle à la surface libre d’un cristal liquide nématique ^en

utilisant

l’approximation

^de Fowler, ^{Kirkwood et} Buff pour l’interface ^{et une} approximation ^de champ moyen pour la fonction de distribution de deux molécules. En prenant ^une interaction de Van der Waals entre molécules, ^{on trouve} que : a) les molécules préfèrent toujours être orientées dans le plan ^{de la} surface; b) ^il y ^{a un saut} de la tension de surface à la transition

nématique-isotrope,

mais elle est inférieure ^{dans la} phase nématique ; c) ^on peut ^{observer un} excès d’ordre en surface qui persiste ^{dans la masse de la} phase isotrope. ^On calcule les expressions générales ^du paramètre ^d’ordre

en surface à

partir

d’arguments d’énergie libre. On discute l’effet d’interactions entre moments

dipolaires permanents ^{et l’on trouve} que des molécules uniaxiales doivent être parallèles à la surface.

Abstract. ^- The surface tension at the free surface of a nematic liquid crystal ^is computed using

the Fowler-Kirkwood-Buff

approximation

^{to the} interface, ^{and a mean field} approximation ^{to the}

molecular pair distribution function. It is found for a Van der Waals interaction between molecules that : a) ^{the molecules} always

prefer

^an orientation in the

plane

^{of the} surface; b) ^{there is a} gap ^at the

nematic-isotropic phase transition, but the surface tension is less in the nematic

phase;

c) ^there may be observable excess surface order which persists into the bulk

isotropic

phase. ^General expressions

are given for the surface order parameter using ^free energy arguments. ^{The effect of a} permanent dipole interaction on the surface tension is considered, and it is found for uniaxial molecules, ^that this interaction also leads to the condition that the molecules be parallel ^{to the surface.}

Classification

Physics ^Abstracts

7.130

1. Introduction. ^- The surface

region

and surface tension in nematic

liquid crystals

^{is at} present ^rather

poorly

understood.

Experiments

^{so far}

performed

show several

interesting

^{effects :}

a)

Orientation at the free surface : The molecules of

some nematics

(e.g. PAA)

^{tend to lie in the}

plane

^{of the}

nematic-air free surface

[1-3]. However,

the orien- tation of MBBA has been measured ^to lie at a finite

angle

^{which is}

slightly temperature dependent [3].

Several nematics which have a low temperature smectic A

phase

^{seem to have a}

perpendicular

^orien-

tation in the nematic

phase just

above the transition

[4].

b)

Behaviour of surface tension at the nematic-

isotropic

transition temperature : ^The

early

^measu-

rements

[5, 6]

^{show that the} surface tension is discon- tinuous at the transition. Recent work

[7, 8]

^has

shown that in certain cases the

discontinuity

is nega-

tive,

^{and the} surface tension is less in the nematic

phase.

c)

^Free surface disclinations : In several nematics

having

^{a low} temperature smectic C

phase,

^discli-

nations in the orientation _appear

just

before the nematic-smectic C transition

[4].

Molecular theories of surface tension in nematics

are difficult because of the

anisotropic pair

^interac-

tions between molecules.

Recently,

^a

general

expres- sion for the surface tension of a

polyatomic

^fluid

has been derived and

applied

^{to calculate the}

dipole- dipole

contribution to the surface tension of _{low vapor} pressure ^water

[9].

^For molecules with

pair

interactions

depending only

^{on the center of mass vector R and}

on the Euler

angles [10] (Q, 0, 03C8) describing rigid body

rotations between the molecules relative ^to their centers of _mass, the result for the surface tension is :

where R ⁼ r2 - ri is the ^center of mass vector, and where the

pair potential U12 depends

^{on R and the}

internal states

e1, 62

which denote the Euler

angles :

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0197600370100118700

1188

d3ei

⁼

sin oi doi dcpi dgii. p(2)

^{is the}

pair,

^{or doublet}

distribution function defined so that

gives

^the average number of

pairs

^of

molecules,

^one

of internal state

61

^{and within}

d3r,

^of rl, the other of internal state

e2

and within

d3r2

^of r2’ We define the distribution function

g (2) by

^{the relation}

where _p is the

singlet

distribution

function,

^{or number}

density

^{of the fluid. In}

(1),

^{the z} axis is chosen to be normal to the free

surface,

^{which we shall}

conveniently

fix at z=0.

From

(1)

^and

(2)

^{we see that at the}

nematic-isotropic transition,

y will

change abruptly

because of the two factors in the

pair

distribution function :

a)

^The

nematic-isotropic

transition is

always

^first

order with a discontinuous

change

^{in the}

density

p.

Since the

density

^is

larger

in the nematic

phase,

^this

will lead to a gap in y with _y also

larger

^{in the nematic}

phase. However,

^the

density discontinuity

^{at the}

transition is _very

small,

^0.3

% being

^a

typical

^value.

Since _y oc

p2,

^this

implies

^{that the} gap

Any

in the surface tension at the transition will be no more than about 0.6

%

^{due to the}

density discontinuity.

b)

^{There is also a}

sharp discontinuity

in the distri- bution function

g (2)

^{at the}

transition,

^due

mainly

^{to the}

discontinuity

ⁱⁿ the orientational order parameter.

Using

^{a Van} der Waals interaction for

U12,

^{and a}

mean field

approximation,

^{we will show that the} gap

in y

^{due to this}

discontinuity

should be

larger

^than

that due to the

density discontinuity.

The surface tension will also

depend

^{on the orien-}

tation of the molecules at the free

surface,

^{since the} interaction

potential

^is

anisotropic,

^and

depends

^on

the direction of the ^center of mass vector R as well

as its

magnitude.

^{In the bulk}

phase

^we know that the nematic state is characterized

by long

range orienta-

tional order

along

^a certain direction

given by

^{a unit}

vector n. In the absence of boundaries and external volume torques, the direction of n is

completely arbitrary :

^{the free} energy of the nematic does not

depend

^{on n but}

only

^{on the}

degree

of order S

along

^n.

However,

^the free surface breaks the translational symmetry, ^so that the surface free _energy will

depend

on

n . k,

^{where k} is the normal to the surface. It is

important

^to realise that this is

essentially

^a geome- trical

effect,

^{and not due to} any

special

interaction between molecules at the surface. Therefore we

expect

an

anisotropic

^surface energy ^even in the case of an

isotropic

^surface

(e.g. glass,

^an

isotropic liquid, etc.) and,

ⁱⁿ

particular,

^{even in the case of a}

nematic-vapor

surface. Since the bulk free energy is still

independent

of n, the total free _energy

(bulk plus surface)

^will

be minimized for the value of

n.k

which minimizes _y.

Below we show that for a Van der Waals interaction between

anisotropic molecules,

y is minimized when

n . k

⁼ 0 i.e. when the molecules lie in the

plane

^of

the free surface.

2. Interaction

potential.

^{- In} this section we consi- der the form of

U12 in (1),

^{when the} molecules interact

through

the induced

dipole-dipole

^{term of the Cou-}

lomb interaction

(Van

^{der Waals}

forces).

This inte- raction was used

by

^{Maier and}

Saupe [11]

^{in their}

treatment of the

nematic-isotropic phase

transition.

In this section we assume that the molecules have no

permanent

dipole

^moments. This interaction will be discussed in section 6.

The

starting point

is the interaction _energy between two molecules as

given

in the second order pertur- bation

theory :

where 1 ⁰⁽¹⁾ >, 1 0(2) >

^{are the}

ground

^{states of mole-}

cules 1 and

2, and ,u(1) >, v(2) >

^{are excited states of}

energy

E,

^and

^Ev. Euv = Ell ^{+ Ev}

^{and we assume all}

molecules are in their electronic

ground

^states.

Finally,

^{H’ is}

given by

^a

dipole-dipole

interaction :

ql, qk ^are

charges

^at

positions

xi, xk in the

molecule,

and R is the center of mass

separation

between molecules.

We assume that the molecules are uniaxial and

rodlike,

and that the

dipole

^{moments are induced}

only along

^the

long

^{axes of each}

molecule,

all off-axis

dipole

^moments

cancelling

^out. This latter

assumption

^is unnecessary if

we are

only

interested in the orientational part ^{of the}

interaction,

but it effects the

dependence

^{on the center of}

mass vector R. Write

xi(1)

⁼

ni(1) il, XK(2)

⁼

nK(2) e2,

^where

ê1

^and

e2

^{are unit vectors}

along

^the

long

^{axes of the two}

molecules.

Then, putting (4)

^into

(3)

^{we have :}

Since

Eoo E/-Lv,

^{we have a sum of}

negative

^{terms in}

(5).

^{Hence the} interaction can be written as :

where T is the

symmetric

^tensor

n

with 1 a unit tensor.

We now average the interaction

(6)

^{over a} distribution of molecules. Introduce a

spherical

^coordinate

system ^with

polar

^{axis n and express}

61

^and

e2

in this coordinate system. Since the molecules are

uniaxial,

^all azimuthal

angles

Q 1, (P2 ^are

equally probable.

The average W over these

angles

^{then works out to be}

where the

Tij

^are the cartesian

components

^{of T.}

Next,

^we average

(8)

^{over the}

polar angles 0,, 02-

^{From the} symmetry of the nematic state, the distribution function

f(0)

is related to the order parameter ^S

by

In

doing

the average in

(8)

^{we assume that}

f(o1,’ o2)

⁼

f (o1) f (o2)

^with

f(O) given by (9).

^{This is a mean field}

approximation

^where local correlations in orientation are

ignored.

^{With this}

approximation,

the interaction

averaged

^over all orientations of the two

molecules,

^{becomes :}

Finally,

^{we must} average

(10)

^over all orientations of the center of mass vector R. We express R in the same

spherical

coordinate system ^{as before}

(with

^{n the}

polar vector).

^{Let 0 and 0 be the azimuthal and}

polar angles

of R. Since the nematic state is

cylindrically symmetric,

all values of 0 are

equally probable,

^{and an} average of

(10)

over this

angle gives,

^{with the}

help

^of

(7) :

In their treatment of the bulk

phase,

^{Maier and}

Saupe [11]

^{assume that all}

angles

^{0 are}

equally

pro-

bable,

^{i.e. the centers of mass of the molecules are}

completely random,

^{even at short} range. This

approxi-

mation is in the same

spirit

^{as the one} that invokes the

separability

^of

f(01, 02);

^i.e.

local,

^short range correlations in both

position

and orientation of molecules are

ignored. However,

^{note that the inte-} raction

potential (6)

^{is a} strong function of the direc- tion

of R,

^{therefore this is}

only

^a

rough approximation.

With this

assumption ^cos2 0 >

⁼

1/3,

^{so that the}

linear term in S

drops

out, ^and

(11)

^{reduces to :}

which is the well known result that the mean field is

proportional

^to

^S2.

In

(11)

^{we see}

if ^cos’ 0 )

>

1 /3 (molecules

^tend

to be lined _up end to end with

R // n)

^{then the term}

in S is

positive. If ^cos’ 0 > 1 /3 (molecules

^tend

to

align

ⁱⁿ

layers

^{with n 1}

R)

^{then this term is}

negative.

On the other

hand,

^{the term in}

S2

is

always positive

and rather insensitive ^to the direction of R. The interaction

(11)

^{has an extremum which is a local}

minimum when 0 =

n/2,

which favors the

tendency

to order in

layers.

^{If we have}

complete

translational symmetry, ^{the linear term in}

(11) dissapears,

^but

if this symmetry ^{is broken in some} way, this term will determine the translational

symmetry

^{of the struc-}

ture.

One _way to break the translational symmetry ^is

through

^an

equilibrium phase

transition. The inte- raction

(11)

suggests ^{that a}

possible

^low

temperature phase

^{of a nematic would be} characterized

by

^a

structure where the molecules tend to form

layers

with nematic

ordering

^{in each}

layer

^{and with the}

planes

^{of the}

layers perpendicular

^{to n. This is}

^exactly

1190

the symmetry ^{of the low} temperature ^{smectic A}

phase

observed in _many materials that form nematics

[12].

Of _course, one cannot say whether a

given

^nematic

will form such a smectic ^state

just by looking

^{at the}

interaction

(11). Nevertheless,

^the

tendency

^to

align

in

layers,

characteristic of the smectic

phase,

^is

already

present ^{in the nematic}

phase.

^We feel that

previous

molecular theories of the smectic A

phase [13, 14]

have not taken

enough

^{account of the}

anisotropy

^of

the interaction with respect ^{to the center of mass}

vector

R, corresponding

^to the linear term in S in

(11).

Translational symmetry is also broken at the free surface in the nematic

phase.

^{Even if all} orientations of R are

equally probable

^{in the}

^bulk,

this will cer-

tainly

^{not be true close to the free surface. We will}

see below that the linear term in S in

(11)

^{is crucial}

for

determining

both the surface

tension,

^{and the} orientation of molecules at the free surface. So the orientation at a free

surface,

^{and the}

tendency

^{of the}

fluid to form a

layered,

^{smectic structure are}

closely

related. A

study

of the free surface

gives

information

on molecular interactions which are

averaged

^{out in}

the

translationally symmetric,

bulk nematic

phase.

This is also the case in some types ^of

polar

^{and asso-}

ciated

liquids

^{which do not have a bulk}

liquid crystal phase [15] :

^{the fluid is}

isotropic

far from the

surface,

but there is an

anisotropy

^close

enough

^{to the}

surface, corresponding

^{to a}

preferential ordering

^{of the mole-}

cules.

3. The FKB

approximation

for surface tension. ^- The

general

^formula

(1)

for the surface tension _{is very} difficult to use without further

approximation

^{since the}

density profile

^and

pair

correlation function are

unknown in the surface

region

where the fluid is

nonhomogeneous. Indeed,

^{in an exact} treatment, the

density profile

^{cannot be}

specified beforehand,

but must be determined as part ^{of the solution}

[16].

At lov vapor pressure, the Fowler-Kirkwood- Buff

(FKB) approximation

^{leads to a}

fairly

^accurate

estimate of _y for

simple

^fluids

[17].

^{We will use the}

same

approximation

ⁱⁿ

treating

^the

nematic-vapor

free surface. The FKB

assumptions

^{are that :}

a)

^the

vapor

density

^is

negligible,

^and

b)

^the

density

^{of the}

liquid phase

^{has the} bulk value _up to the

(Gibbs’) dividing surface,

^{which we take at z = 0. These two}

assumptions imply :

where Z ⁼ _{zl - Z2’} In addition we assume that

where

1(8)

^is

given by (9).

^This

approximation

^involves

the

following assumptions : a)

^{the centers of mass}

vectors are

isotropic

^{in the bulk}

phase, b)

^local

correlations between the orientations of the molecules

are

neglected (a

^{mean field}

approximation),

^and

c)

^the

z,

dependence

of the distribution function

f(0)

^is

ignored.

^That

is,

^{we assume} that the order

parameter S

has the bulk value _up to the free surface. The first two

assumptions

^are

obviously

^{in the same}

spirit

^{as the}

Maier-Saupe theory

for the bulk

phase,

^{and the last}

assumption

^{is in the same}

spirit

^{as the FKB}

approxi-

mation

(13), although

it is much less

obvious,

^and in

fact,

^{as we shall} see, contradicts the

boundary

condition on S at the free surface which can be derived from free energy arguments. ^{As with the}

density profile,

^the detailed form of

S(z) through

^{the surface}

region

^must be obtained

self-consistently

^{in the}

solution,

^{and cannot be}

specified

in advance.

Recently,

the

S(z) profile

has been discussed

[18],

but since it

depends

^on the unknown

density profile p(z),

^the

discussion is

only qualitative.

^The

only

^{other case}

that can be treated within the FKB

approximation

is one where

S(z) decays

^{to the} bulk value S in a

characteristic distance

large compared

^{to the mole-}

cular interaction _range. This case is discussed in an

appendix

where it is shown that the

parameter S appearing

ⁱⁿ y must be taken to be the value of S at the

surface,

^{and not the} bulk value of S.

With these

approximations (1)

^{reduces to :}

with

d2e

⁼ sin 0 dO

d(p,

^the

superfluous angle 0 having

^been

integrated

^{out. The} interaction

U12

is taken to be

(6)

^{for Van} der Waals forces. Put

(6)

into

(16)

^and

integrate

^over

d 2e, d2e2 ^first, keeping

^R

constant. We get

where ( W >

^is

given by (10).

^The surface tension can

be broken _up into three terms

in W > :

First assume that n lies

along

^the surface normal

(z axis).

^{We can} then express R in ^terms of

spherical

coordinates with

k

⁼ n as the

polar

vector, and

integrate

^{over the two}

angles e,

^{0. The first term} yo in

(18) easily

^{works out to be :}

The second and third terms in

(18)

^{can be}

expressed

in terms of the

angular integrals

After a rather tedious

calculation,

^{one finds that}

From

(10), (17),

^and

(21)

^we get :

hence the surface tension for

perpendicular

^orien-

tation is

with Yo

given by (19).

4. Surface tension for

parallel alignment.

^{- Next}

consider the surface tension when the molecules lie in the

plane

^{of the}

surface,

^{i.e. when n is}

along

^{the x}

axis. Since R is

expressed

^{in terms of}

0, 0

^{in a coordi-}

nate system ^{with n the}

polar axis,

^{we can calculate} _y

by simply interchanging x

^{and z in the} coefficients

(22) (i.e. Ixx

->

Izz, Ixy

->

Izy, etc.). ^Then, using

^the

general

results

(17)

^and

(10),

^{one finds :}

We see

that y

II V’ 1. for all values of S. Further- more, the symmetry ^{of the} interaction

(10)

^{rules out a}

smaller value

of y

^for any other orientation. If there are no external torques

applied,

and the influence of other boundaries is

ignored,

^the total free _energy

(volume plus surface)

will be minimized when the orientation is

parallel

^to the free surface. We conclude

that,

^for molecules

interacting

^{with Van der Waals}

forces,

^the orientation will tend to be

parallel

^{to the} free surface.

The

result, (25)

^also

predicts

^a gap in the surface tension vs. temperature ^{curve at the}

nematic-isotropic phase

transition

T,,

^{due to} the discontinuous

change

in S there :

where AS ⁼

S(T Tk) - S(T > Tk).

^{If we assume}

that the S in

(25)

^{is the bulk}

value,

^{then we know that}

S(T > Tk)

^{= 0 and}

S(T Tk) N 1/3.

^This

gives

^a gap of about

10 %

^{in the surface tension.}

Moreover,

^the

gap is

negative

^{- the surface} tension is less in the nematic

phase.

^{We can} relate this to molecular interactions as follows : the surface tension is essen-

tially proportional

^{to the}

pair interaction ( W >

averaged

^over all orientations. From

(11)

one can see

that W )

^{will be less in the nematic}

phase

^{when the}

linear term in S becomes

negative,

i.e. when the dis- tribution of the centers of mass vectors favors a

layered

^{like structure close to the free surface. The}

decrease

in ( W >

^{with S}

implies

^{there is less}

restoring

force on s,urface molecules in the nematic

phase,

than in the

isotropic phase.

^This

implies, ignoring

^the

small

density effect,

^that y will be less in the nematic

phase.

Note that this can

only happen

^{when the}

orientation is

parallel

^{to the free}

surface;

^{for a} perpen- dicular orientation the surface tension _{gap would} be

positive,

^{as can be seen from}

(24).

We can make some further remarks on the

y(T)

curve away from

Tk.

^{Since y oc}

pe

^we expect y to

slowly

^{increase with}

decreasing

temperature ^{due to} thermal

expansion.

^In fact this is the main temperature

dependence of y in

^normal

organic liquids [19].

^{For a}

typical

^{value of}

10-3/deg

^{for the volume}

expansion

coefficient we expect ^{a value}

Ayfy = 0.2 %

per

degree.

Now for T

Tk,

^{we know that}

S

^{increases with}

decreasing temperature,

^{with the rate} of increase

being largest

^near

Tk.

Since for

parallel orientation,

^an

increase in S leads to a decrease in _y, it is

possible

that close to

Tk

^one may ^see y

decreasing

with decreas-

ing temperature.

^{Far below}

Tk,

where the order parameter is almost constant, ^the thermal

expansion

effect will

probably

win out, and y should start to increase

again.

The situation is further

complicated by

^{the fact that near}

Tk

the thermal

expansion

^coef-

ficient shows

pretransitional

anomalies. A careful measurement of

y(T)

^above

Tk

^could

give

information

on the temperature

dependence

^and

magnitude

^of

the order parameter ^{at the surface}

(see

^section

5).

The

dependence

^of y ^{on an}

applied magnetic

^field

has received some attention in the literature

[20].

If the field is

applied parallel

^to the free surface there should be no bulk

distortions,

^{and one can see} from

(25)

^that y should decrease with H because of the increase in S with H. However this effect _{is very} small in nematics because of the small value of the

anisotropy

^{of the}

magnetic susceptibility

^{and it is not}

expected

^to be observable.

We have calculated _y for

parallel

^{and for}

perpendi-

cular orientation and shown that y ^{|| yl. For certain} distortion

problems

^near the free surface it is of interest to calculate

y(03C8 ) ; where 0

^{is the}

angle

^{that n}

makes with the surface normal

k :

:

cos 0

⁼

n.k.

From

(10)

^and

(17)

we can see that

only

^even powers of

cos #

^can enter y, but the

resulting expression

^{is not}

easy ^to carry ^out because the interaction

(10)

^involves

1192

the _squares of the tensor components

Tij

^{rather than}

these

quantities

themselves. For

qualitative

^discus-

sions it _may be sufficient to use the

simple expression :

From

(24)

^and

(25)

^it follows that

In the nematic

phase

^{where S} >

1/3,

^{it is}

important

to

recognize

^{that this is a} very

large

difference in surface

energies;

^{at most an order of}

magnitude

^less

than the actual interfacial surface tension of the

liquid.

This means that it will be _very difficult to tilt the surface orientation

by

^the

application

^{of bulk} torques. ^For

example,

^{if one}

applies

^a

magnetic

^field

perpendicular

to the

surface,

^the orientation at the surface will be

given by

^an

angle t/Jo (measured

^with

respect

^{to the}

normal)

^of

[21]

where 3 DS 2 ~

k,

^{where k is a} Frank orientation elastic constant

[22].

Now with k ~ 10-6

dynes,

xa N 10-’ c.g.s., H ~ 104

G,

^and _{yl -}

Y II N

¹

dyne/

cm we find that cos

t/Jo

^{3 x}

10-3,

^{which is}

hardly

observable. So our

theory

suggests ^{that the}

general

solutions of the elastic

equations

for the distortions in surface orientation

produced by

^volume torques

[21, 23]

^{will be difficult to} observe in the nematic

phase.

^The surface orientation is then

subject

^to

strong

anchoring :

ⁱⁿ

problems involving

orientation at the free

surface,

^{instead of}

minimizing

^{the bulk}

plus

^{surface free}

energies,

^{one may}

simply

^minimize

the bulk free energy and

require

^{that the}

boundary

condition

n-k =

0 be satisfied at the free surface.

Instead of

using

^bulk torques, ^one may try ^to

change

the orientation at the free surface

by using

solid bound- aries.

Suppose

^{we introduce a} thin nematic film on top of a solid that

rigidly

^{maintains a}

perpendicular alignment

^{of molecules}

[24].

^{For a thin}

enough layer,

one may be able to tilt the

alignment

^at the free surface.

This

problem

^was considered in ref.

[21]

^{where it}

was shown that the

alignment

^at the surface satisfies :

where d is the thickness of the film and

using

^{the same estimates as before.} For d >

dc

^we

have

approximately parallel alignment

^{at the surface}

(tf 0 ~ x/2),

^but for d

dc,

^the

alignment

^{at the free}

surface is forced to be

perpendicular

^to

satisfy

^the

rigid boundary

^{condition at the} solid surface. It would be

interesting

^to try ^to observe this

effect,

^as the gap

in _{y at} the transition temperature is very sensitive to the orientation

tf¡ 0’

^{and even reverses}

sign (see

eq.

(24))

^for

^{00 -+}

^0. The surface tension itself will

depend

^{on the}

layer thickness d, varying from 01,

I

when d >>

dc

^{to yl when d}

dc.

Finally,

it should be

pointed

^{out that the surface}

energy favors

parallel orientation,

^{but the direction} of n within the surface is

arbitrary.

^This

degeneracy

is inherent in the

problem,

since what we have chosen

to call the X axis is

arbitrary.

^The

degeneracy

^{will be}

removed,

ⁱⁿ

practice, by

^small

perturbing

^{forces in}

the

bulk,

^whose

origin

may ^come from other boun- daries in the

problem,

^{or from an external}

magnetic

field

applied

^{in a}

given

^direction

parallel

^to the surface.

5. Excess surface order. ^- The fact that

Any

⁰ for the

preferred parallel

orientation also

implies

that there will be excess surface

order,

^and

that S #

⁰

at the surface even above

Tk.

^It has been shown elsewhere

[21]

^{that the}

boundary

condition that S must

satisfy

^{at a}

plane

interface is

given by :

where

5(S)

^is the volume free energy. To get ^a

rough

estimate of the

magnitude

^{of this}

effect,

^we may ^use

a Landau

expansion

^for

f(S) [25] :

where A ⁼

a(T - Tc*)

^with

T * slightly

^{below the}

(first order) phase

transition temperature

Tk,

^and

a,

B, C,

^{D are}

positive

constants. Since D >

0,

^note that excess surface order can

only

^{occur when}

which is the case for

parallel

orientation. The

equi-

librium

S(z)

^must

satisfy

^{the Euler}

Lagrange

equa- tion :

which for

(32) gives

^{a nonlinear}

equation

^for

S(z) :

with the

boundary

conditions S

= Sb, dS/dz

^{= 0}

at z ^{= -} oo.

Sb

is the bulk value

(the

value the order parameter ^{would have} if the fluid were

unbounded).

It satisfies :

For T

Tk

the solution which minimizes

(32)

^is

with

For T >

Tk

the solution that minimizes

(32)

^is

Sd

^{= 0.}

At T ⁼

Tk

^{there is a}

discontinuity

ⁱⁿ

Sb

^{and the}

transition is first order.

An exact solution of

(34)

^{for all cases} would involve

elliptic

functions and would be _very difficult to carry out. Here we

only

^consider

approximate

^solutions

for T >

Tk

^{and T}

Tk.

^First suppose T

Tk

^so

we are in the bulk nematic

phase.

^In

particular,

choose T ⁼

Tc *,

^then

A = 0, Sb

⁼

B jC.

^Assume

that the effect of the surface is small and write S

= Sb

⁺

S’(z)

^{and linearize}

(34)

ⁱⁿ

S’(z).

^This

gives

^a

simple exponential equation

^for

S’(z) :

where

a2

⁼

(3 CSb -

²

CSb)/D

⁼

B 2/CD

> 0. The solution for S is then

where

So

⁼

S(O)

^is the order at the surface. The interfacial

boundary

^condition

(31) along

^with

(25)

then

gives

For

yo/Da ->

oo,

So approaches

^the

limiting

^value

of

3/4.

^In

evaluating (39)

^{we use the}

typical

^values

C N 1

J/cm3,

^{D ~}

^lO-12 J/cm,

yo N 30

ergs/cm’,

Sb

^{= 0.5. Then}

(39) gives So N

0.68 which is about 30

%

^excess surface order. The

decay

distance is

a -1 N

100

A

which is

small,

^but still much

larger

than the _range of interaction of Van der Waals forces

(only

^{a few}

A).

The case T >

Tk

poses ^more difficulties since

Sb

^{= 0. A first}

integral

^of

(34)

^is

readily

^{obtained :}

Note that the r.h.s. is

always positive

^{for A} > 0 since this is

just

^{the free} energy of the bulk nematic

phase

which has an absolute minimum at S ⁼ 0 when T >

Tk.

^Thus

(40) represents

^{a nonlinear}

decay

^from

the surface : oscillations in

S(z)

^{cannot occur.}

Using (40)

^{and the}

boundary

^condition

(31)

^{we can} get ^a condition on

So

^without

knowing

^the

complicated dependence

^{on z :}

Again

^when yo -> oo,

So -> 3/4. Eq. (41)

^{can be}

solved

numerically

^for

So

^{at all}

temperatures

^{if the}

quantities A, B, C, D,

yo ^are known. For the values

A/C

⁼

1/10, B/C

⁼

1/2,

^and yo and D as estimated

before, (41)

^{has the}

approximate

solution of

So

^{= 0.5}

just

^above

Tk.

^{This rather}

large

value decreases

roughly inversely

^{to the} temperature difference T -

Tc*.

A more detailed

qualitative

discussion of the excess

order

parameter, taking

^{into account} the fact that the surface

region

^is

really

diffuse rather than

sharp,

has been

given

^{in ref.}

[18].

6. Effect of a

permanent dipole-dipole

interaction. ^- Interactions between permanent ^molecular

dipoles

are

ignored

^{in the}

theory

of bulk nematics because no

ferroelectric effects have ever been measured in

nematics;

^{the states n and - n seem to be}

completely equivalent.

^This

implies

^{that the} molecular distri- bution function

f(O)

^{has the} symmetry property

f(n - 0)

⁼

f(O),

^{so that even if the molecules have} permanent

dipole

moments, ^as many

point

up ^as

point

^{down on the} average and _any

long

range ferro- electric effect cancels out.

However,

^{since a} free surface breaks the translational

symmetry,

^{it is}

possible

^to

have cos

0 > :0

^{0 at the}

surface,

where there is an

.anisotropy

ⁱⁿ

the

distribution of center of mass vectors.

Since

^cos

0 >

^{= 0 in the}

bulk,

^{it must}

decay

to zero in a characteristic

distance

^{from the free} surface.

In this section we calculate the contribution to the surface tension

produced by

permanent

dipole- dipole

interactions. We assume that the permanent

dipoles

^are

along

^the

long

^axes of the molecules. Then the interaction is

given by :

where T is

given by (7), and p

^{is the}

dipole

^{moment of}

the molecule.

Again

^we express the

6i

ⁱⁿ

spherical

coordinates with

polar

^{vector n and} average

(42)

over the two azimuthal

angles

wi, Q2. The

only surviving

^{term is :}

An average

over 01 and 02 gives

^a

factor cos () >2 ^{== G2}

in a mean field

approximation. Expressing TZZ

ⁱⁿ

polar

coordinates as

before,

^the average interaction turns out to be :

As

expected,

^{for a} random distribution of centers of _mass,

cos’ 0 )

⁼

1/3,

and the interaction vanishes in the bulk

phase.

^{But near the}

surface,

the interaction

(44)

^{will not} average ^out.

1194

The contribution of the

permanent dipole

^interac-

tion to the surface tension is then

where U > = R -3(1 - 3 ^cos’ 0). Doing

^the angu- lar

integrals

^{as before we find for}

perpendicular alignment

and for the

parallel alignment

So we have yd ^{11 yal and hence the} permanent

dipole forces,

^{like the Van der Waals’}

forces,

^{lead to a}

parallel alignment

of molecules at the surface. This appears ^to rule out the ferroelectric interaction between permanent

dipoles

^{in the}

interpretation

^of

the

perpendicular

orientation of MBBA in a thin film with two free surfaces

[2].

We have assumed the

dipole

^{moment of a molecule}

is

along

^the

long

^{axis. If we have}

permanent

^{off axis}

dipole

^moments which make an

angle

^{a to the direction}

of _easy

polarizability

^{then the molecules are no}

longer

uniaxial but biaxial

[26].

^{In this} case, if the

dipole-dipole

interaction is the

only interaction,

^the above arguments go

through exactly

^{as before and}

the surface tension is minimized when the

dipole

moments are

parallel

^{to the surface which means that}

the director makes an

angle

^{a to the surface. When}

there are Van der Waals forces present, ^{these will tend}

to line _{up the} director

parallel

^{to the}

surface,

^{so that} the actual

angle tf

^{that n} makes with the surface will be less than ^a. The

angle tf

is thus determined

by

^two

competing

forces and will in

general

^be

temperature dependent

since the two order parameters ^{S and e} may be different functions of temperature. ^{In MBBA} the molecules are tilted at an

angle r

^{750 from the}

surface which

depends

upon temperature

[3].

^{The case}

of MBBA would be an

interesting

^{one to}

study

ⁱⁿ

detail since we have

experimental

information on y at the transition

[27],

^{the tilt}

angle,

^and the electric

dipole

^moment

[28].

Other mechanisms that can lead to a tilt in the molecular orientation at the surface are :

a)

Interactions of surface molecules with

impu-

rities

selectively

^{adsorbed to} the surface which tend to

orient the

polar

^{heads of the} molecules and

thereby

cause a term in _y which favors

perpendicular align-

ment. In this _case, the tilt

angle

^would vary with the concentration of the surface

impurities.

b)

^The formation of smectic-like

layers

^{near the}

surface, resulting

^{in a} breakdown of the translational symmetry ^at finite distances from the surface. If the

layers

^{tend to} lie in the

plane

of the free

surface,

^as

they

do in

smectics,

^{this will lead to a}

perpendicular

^orien-

tation at the surface. This kind of

ordering,

^where

one has a non-zero smectic order parameter ^{close to} the surface which

decays

^{to zero in the}

bulk,

^{is ana-}

logous

^{to the case} of orientational surface order above the bulk

clearing

temperature ^{where the bulk order} parameter ^{vanishes. A better}

understanding

^{of the}

relationship

between order and molecular interactions in smectics will

probably

^be necessary, before such a

model could be worked out in detail.

Appendix. Spatial dependence

^{of S. - If there is}

excess surface

order,

^{the order}

parameter

^will

decay

to the bulk value within a characteristic distance from the surface. In other

words,

the order _para- meter, and therefore the

pair

distribution

function, depend

^on the coordinate _{z, in}

(1).

We assume the FKB

approximation

^and put

(13), (14), (15)

^into

(1), where f (0)

^{is allowed to}

depend

^on

zl. The

integral

^over

^{d3 R}

^{can be broken} up into a

part ^{with Z} > 0 and a part ^{with Z 0 :}

Since

p(2)

^{= 0 for} zi >

0,

this reduces to :

where

Interchange

the orders of

integration

ⁱⁿ

(A. 2),

^inte-

grating

^over

d3el d3e2

^{first :}

where A and B are

given

^{in terms of the T tensor}

by (10).

Integrate

^over z, ^term

by

^{term. The first term}

just