A mean-field study of a model system of grafted rods

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A mean-field study of a model system of grafted rods

Zhen-Gang Wang

To cite this version:

A

mean-field

_study

of

a

model

_system

of

_grafted

rods

Zhen-Gang

Wang

(*)

Corporate

Research Science _{Laboratories,} Exxon Research and

_{Engineering Company,}

Annan-dale, NJ 08801, U.S.A.

(Reçu

le 4 décembre 1989,

accepté

sous

_{forme définitive}

le 26

_{février 1990)}

Abstract. 2014 We

_study

the

_phase

behaviour of a model _system of

_grafted

rods on an

_impenetrable

surface,

using

a van der Waals-like mean-field

theory. By accounting

for the

_anisotropy

of the

in-plane

excluded volume effects, it is shown that the

in-plane

orientational _symmetry can be broken for

_adsorption

energy

exceeding

some critical value, in a

_density

interval, thus

_leading

to a

discontinuity

in

(the

deivative

of)

the fraction of molecules in each orientation. It is further

demonstrated,

using

a van der

_{Waals-type theory,}

that the inclusion of attractive interactions,

when

_coupled

with the

anisotropic

excluded volume

effects,

may drive the transition to first order,

in addition to

_inducing

the usual

_gas-liquid

transition.

1. Introduction.

Molecular

_{monolayers/films}

at

_gas-liquid

and

_{liquid-liquid}

interfaces are of

_great

current

interest because of their numerous

_{technological applications,}

such as molecular

electron-ics

_[1]

_]

and

_biological

membranes

_[2],

and have been the

_subject

of extensive studies

_[3],

thanks to the advent of a number of

_{newly developed}

surface

_techniques

such as

synchrotron

X-ray

diffraction

_[4]

and second harmonic

_{generation [5].}

From a

physical point

of

_view,

it is

important

to understand

the

collective

_phenomena

exhibited

by

these molecular

_monolayer

_systems

_[6-9],

which in turn are

intimately

related to

the various

applications.

Of

_particular

interest are the successive

phase

transitions that have

been observed in these

_systems

and the molecular conformation in each

_phase.

There,

a

key

issue has been the nature of the

_{liquid-expanded (LE)}

to

_{liquid-condensed (LC)}

transition. The richness in the behavior of the molecular

_monolayers

arises from the

quasi-two-dimensional nature of these

_systems,

_i.e.,

from the subtle

_coupling

between the translational

degree

of freedom and other « extra »

_degrees

of freedom of the molecules normal to the

layer

[10].

A number of statistical mechanical

theories,

including

lattice

_packing

and

_Ising

Hamiltonian

_{formulations,}

have been

_developed

to account for the orientational

_ordering

in the

_monolayers

_{[ 1 1 ] .}

Clearly

the transition from the LE

_phase

to the LC

_phase

must be associated with the

(*)

Present Address :

_Department

of

Chemistry, University

of California at Los

_Angeles,

405

_Hilgard

Avenue,

Los

_Angeles,

CA 90024, U.S.A. Classification

Physics

Abstracts 64.70M

coupling

between the translational

_degree

of freedom

_(density)

and the internal

degrees

of freedom or some

_symmetry

order

_parameter.

For

fully

flexible

_chains,

a lattice model has

been

_proposed

and

studied,

using

statistical

_{thermodynamics}

and Monte Carlo simu-lation

_[12].

More

_recently,

a similar model has been studied

_by

Cantor and

_Mcllroy[13].

Their

_theory,

which enumerates the isomeric states for a

_pair

of very short chains

(2-4 units),

better describes the

_pair

interaction out of the

_grafting

surface,

and is able to

_predict

a second

phase

transition,

apart

from the

_gas-liquid

one. The appearance of a second

liquid phase

is

also

_predicted

in a recent

_simple

mean-field calculation

_by

Shin et al.

[14].

Most

_amphiphile

molecules in the

_monolayer

are not

_long

_giant

_molecules,

and therefore cannot be viewed as

completely

flexible. A more realistic treatment would involve full consideration of the molecular

_{packing [10,}

_13-16]

and finite stiffness of the molecules.

_However,

we feel that a

study

of the

_rigid

rod

_problem,

which is at the other extreme of the

_flexibility

_spectrum

_[17],

should be

_elucidating.

Such a

_study

is also

_important

in view of the

_{corresponding}

bulk

liquid-crystal

system

which shows

_isotropic

to nematic

_phase

_{transition [18,}

_19].

The

_grafted

rods

_problem

was

_recently

studied

_{theoretically by Halperin et}

al.

_[20].

_Using

a

second virial

_{expansion (Onsager theory) [18], they numerically}

calculated the order

parameter

as a function of

_{particle density}

in the absence of

_adsorption

_{energy. It} was

concluded there that an orientational

_phase

transition does not ensue, and

_they

attributed this nonexistence of orientational transition to the

already-broken

symmetry

imposed by

the surface.

More

_recently,

Chen et al.

_[21]

consider the case where

adsorption

energy is included. Via a

simple Onsager

treatment of the excluded

_volume,

and

_using

symmetry arguments,

they

conclude that

_anisotropic

excluded volume

_alone,

even with surface

_adsorption

_{energy, is} not sufficient to

_bring

about an orientational

_phase

transition.

_They

_suggest

that attractive

interactions between the rods are

_required

for such a transiton to occur. Since the orientation

of the rods is

_{intimately coupled}

to the

_density

in the

_grafted

rod

_problem,

a

_strong

rod-rod

attractive

interaction,

which induces the usual

_gas-liquid

transition below the critical

temperature,

necessarily

leads to a discontinuous

_change

in the orientational order

parameter.

In reference

_[20],

the effect of

_anisotropy

of the excluded volume in the azimuthal

_angle

cp

is

_neglected,

thus the

_possibility

of a

_symmetry

_breaking

in that

_degree

of freedom was

precluded

from the calculation. This

_possibility

was first

pointed

out

_by

Boehm and Martire

_[22]

in the

_study

of

_rigid

rod

_adsorption

and was also discussed in the work of Chen et al.

[21]

]

but it was not included in their

_{phase diagram. Neglect}

of the

_biaxiality

of the molecules is

_equivalent

to

_performing

an _{average of the azimuth}

_angle

in the

_(biaxially)

symmetric phase,

which can be

_justified

when there is no

_adsorption

_energy.

However,

explicit incorporation

of this

_angle

becomes

_important

when the

_adsorption

energy is

sufficiently

strong

because the

_symmetry

can be broken in this

_degree

of

_freedom,

as will be

shown in this paper. This

symmetry

breaking

in turn leads to a discontinuous

_change

in the

vertical

_angle

or its derivative. This rotational

_degree

of freedom is also

_expected

to be

partially responsible

for the tilt

_{phenomena [5, 23]}

in some surfactant

_monolayer

_systems.

Thus it is desirable to _{take proper} account of the

anisotropy

of the excluded volume in the

azimuth direction. In _{this paper,} we conduct a mean-field theoretical

study

of the

grafted

rods

by incorporating

this

_degree

of freedom. For the sake of

_comparison,

we

_adopt

the

_Zwanzig

approximation [24] employed

in the work of Chen et al. The role of

_in-plane

_anisotropy,

adsorption

energy, and attractive interaction between the

rods,

are

_explicitly

taken into

consideration.

The paper is

organized

as

_{following :}

in section

2,

we

develop

a van der Waals-like

volume

part,

we use, instead of scaled

_{particle theory [27]}

which is rather

sophisticated

and

subtle

[28], especially

when

_applied

to the

_grafted

system,

a

_simpler

and more

transparent

theory, namely

the van der Waals

_equation

of state for a

_{multi-component}

fluid

_[291.

This

general theory

is then

_applied

to the

_Zwanzig

model in which

_only

orientations

along

the

principal

axes are allowed. In section

3,

we

perform

a bifurcation

_analysis

to

_study

the

possibility

of a

_symmetry

_breaking

in the

_in-plane

directions,

for the case where there is no

attractive interaction between the rods. The

importance

of the

_{one-body adsorption}

_energy is demonstrated and its critical value for the

in-plane

symmetry

breaking

is determined. Both

Onsager theory

and the mean-field

_{theory developed}

in section 2 are

_employed

to

_study

the order

parameters

as a function of the

_{monolayer density.}

In section

_4,

we

investigate

the

effects of the attractive

_pair

interaction between the

rods,

and show that in addition to the

gas-liquid

transition,

such an attractive interaction can drive the

_in-plane

symmetry

breaking

to a

first order

transition,

thus also

_resulting

in a discontinuous

change

in the other orientation

angle. Finally,

results of the

_theory

are discussed in section 5 in its relevance to

_experiments,

and some

concluding

remarks are made.

2. Van der Waals-like mean-field

theory.

We first

_develop

the

theory

in a more

_general

form

_applicable

to _any

_anisotropic

fluids.

Only

in the end of this section do we

_specify

to the

_grafted

rods and the

Zwanzig

approximation.

Consider a _system of N molecules in a

_{(d-dimensional)}

volume at

_temperature

T. For the sake of

discussion,

we assume that these molecules can take m discrete

orientations,

with

Nj, j =

1,

..., m, molecules in each orientation. The continuum limit may be obtained

straightforwardly by letting

m - oo.

_Following

Gelbart and Baron

_[25]

and Cotter

_[26],

we

write the Helmholtz free _{energy per} molecule

_F/N

as

where

{3

=

1 /kT

and

QN

is the canonical

partition

function. In

equation (1),

_p is the total

particle

_{density, sj}

=

N j / N

is the fraction of molecules in orientation _wj,

_{u (r, wj,}

_{lù k)}

is the

(long-ranged)

attractive interaction

_potential

between two molecules in states _wj and W k, which are

_separated

in space

by

r, and

g (r,

’W j,

W k)

is the

pair

distribution function. The

first two terms

_represent

the translational and rotational

_entropy

of the

_rods,

_{respectively.}

The last term in

_{equation (1)}

contains the

_{short-ranged strongly repulsive potential}

U*(r, w j’, w k)

which we take to be hard core interaction.

Then,

the interaction

_part

of

equation (1) (i. e.,

the last two

_terms)

can be

_regarded

as the lowest order

_{approximation}

in the Barker-Henderson

_{perturbation theory}

for

_{liquids [30].}

We have included a

_one-body

potential

_{h (,wj)}

which can

_be,

_e.g., the

_adsorption

_energy, when a surface is

_present.

We now introduce the

following approximation,

following

Gelbart and Baron

_[25].

We

when two molecules with orientation _wj and _{cv k,}

_{respectively,}

do not

_overlap

_and,

otherwise. We note that the

_{equations (2a)}

and

_(2b)

are the lowest order term in the

_density

expansion

of the

_pair

correlation function for a hard-core

_fluid,

which takes account of the

two

_particles

involved while

_neglecting

the effect of the other molecules

_[30].

In the same

spirit,

we make the

approximation

for the last term in

_{equation (1)}

_that,

where

_bjk

is the van der Waals

_pair

excluded volume

_parameter

and is defined as

with the

_prescription

_(2a)

and

_(2b)

for

_{g (r,}

_{W j,}

_{’W k).}

The

_{approximations (2) through}

₍₄₎

are

nothing

but the van der Waals

_{approximations}

for a

_{multi-component}

fluid with

_compositions

si, s2, etc.

[29,

30].

With these

approximations, equation

(1)

now

becomes,

where

The continuum limit can be

_easily

obtained from

_{equation (5)}

as

Each term in

_{equation (5),}

or

_{equation (7),}

has a

_transparent

_{physical interpretation :}

the first term is the translational

_entropy,

the second term is the

entropy

loss due to the excluded

volume,

the third term

_represents

the orientational

_entropy,

and the last two terms arise from the two

_body

attractive interaction and one

_body

external

_{potential, respectively.}

Notice that in the absence of the last two _terms,

_{equation (7),}

when

_expanded

to the linear order in p, reduces to the

_{Onsager theory [18]}

for

_{isotropic-nematic}

transition in

_{liquid-crystals.}

Thus

equation (7)

is

_{quantitatively}

correct to order p, since it

yields

the correct second virial coefficient. It should be

_kept

in

mind, however,

that our estimate of

_ajk

in

_{equation (5) (or}

equivalently

of

_a(w,

_w’)

in

_{equation (7)) by using equation (2),}

is a rather crude

theory

still

_captures

some of thé

_{important qualitative}

features of the

_system,

such as its

symmetry

properties.

Equation

(5),

or

₍₇₎

_gives

the free _energy as a functional of the distribution

_sj(s(w)).

The

equilibrium

values for

_{these sj ’s}

are obtained

_{by minimizing f}

with

_respect

_{to sj ’s}

at

_given

density

p and

temperature T,

for the set of

_parameters

_{characterizing}

the

_system.

This is

achieved

_{by setting}

These

_equilibrium

_Si*’s,

when substituted back to

_equation

_(5),

_yield

the

_equilibrium

Helmholtz free energy

f*.

Hereafter we will ommit the *

symbol

for the sake of notational

simplicity,

with the

_{understanding}

that the

_quantities

to be discussed are

_already

their

equilibrium

values.

_Using

standard

_{thermodynamic}

_relations,

the

_(osmotic)

pressure lI and chemical

_potential

» can be derived as

and

If there is

_two-phase

_coexistence,

_{the pressure and chemical}

_potential

of

_phase

a, must

equal

the pressure and chemical

potential

of the

coexisting phase

8,

i.e.,

Equations (5), (8), (9)

and

_{(10) completely}

determine the

_equilibrium

distribution

sj’s

and the

_{thermodynamics}

of the

_{system. Next,}

we

_specify

to the discrete model to be studied in what follows.

We use the

_{Zwanzig approximation [24],}

where the molecules are assumed to take

orientations

_along

the

_principal

axes, which has been

_employed

in the

_study

of bulk

isotropic-nematic transition of

_{liquid-crystals.}

Such a model is

_clearly

the

_simplest

of

all,

and at the

same time retains most of the

_{phenomenologies. Comparisons}

with

_experimental

data even

yields quantitative

agreement

to certain extent

_{[31]. Although}

when

_applied

to the

_grafted

rod

[31]

problem

some subtleties remain

(see

discussion

_below),

the model is

expected

to catch many of the

important

features of the

_problem,

_especially

the

anisotropy

in the excluded volume between the various

_{orientations,}

the

_breaking

of the

symmetry

in the direction

perpendicular

to the surface and the role of

_adsorption

_energy. The

_theory

_developed

_above,

however,

does not limit us to this

_particular

_model,

and a

_{study by using}

a continuum model is

straightforward.

We assume the molecules to be

_rectangular

_{parallelipipeds [21]}

_]

of

_{length f}

and width

d,

with one end

_always

anchored on the

_impenetrable

surface,

which can take five

orientations : the directions of z

(perpendicular

to the

_surface),

x, - x, y

and - y (in-plane).

The distinction between the orientations x and - x, and

likewise y

and - y, does not

play

a

crucial role in this

_particular

_model,

where the

_in-plane

conformations have the molecules

lying

down

_completely

on the surface. It is

_essential,

however,

in real

_systems

and when focus

is in the

_in-plane

orientation itself. Our main concern in this paper is to

_study

the

phenomenology

when

_symmetry

breaking

is allowed in the

_in-plane

direction and the

coupling

between the

_in-plane

and vertical orientations as a function of the

_density.

The

of the

_grafted

rod

_problem

and is absent in the bulk

counterpart.

For this purpose, we

ignore

the subtle differences

afore-mentioned,

and our crude model suffices.

_(We

treat x and

- _x orientations as the same,

except

for an

_entropy

contribution of In 2 associated with two

orientations as

_opposed

to

_one.)

The excluded volume

(area)

matrix

_bjk

between various orientations can be

easily

calculated to be

In order to

_{explicitly distinguish}

between the

_{in-plane (horizontal)}

and

_out-of-plane

(vertical)

orientations and to allow the

_possibility

of a

_symmetry

_breaking

in the horizontal

directions,

we define the order

_{parameters .0}

and 6 such

_that,

_{Sz = 1 - l/J, Sx = l/J (1 + u) /2,}

sy = 0

(1 - u) /2.

Clearly, 0 =

S x + SY’ which is the total fraction of horizontal

molecules,

whereas a nonzero

_signifies

a

_symmetry

_breaking

in the horizontal orientations. With these

order

_parameters,

_{equation (5)}

becomes,

where

_{B(0, o-)}

is defined as

p =

_pMo/2

is the reduced

_density

with

_Mo

=

4 d2, u

=

azzlMo,

and we have used reduced

parameters

_Bjk

and,

Ajk

defined as

Using (11),

the

_Bjk’S

are found to be

with r =

l /d

being

the

_aspect

ratio. The

A jk’S

will be

_specified

in section 4.

From

_{equation (8),}

the

_{equations determining}

the

_equilibrium

values

_{of 0}

and

3. Bifurcation

_analysis

and the critical

_adsorption

energy.

Equations (16a)

and

_(16b)

determine the values

of ~

and o- as functions of the

_temperature

T and

_density

_p. Because of the

_larger

excluded volumes

_{experienced by}

the

_in-plane

molecules,

0,

which is the fraction of

_in-plane

molecules,

is

_expected

to decrease as the

density

is increased. As

p --> 1, ~2013>0.

When

_{p --> 0, ~}

_approaches

a finite value

~0

determined

_by

the

_adsorption

_energy. Thus in both of these

_limits,

the

_density

of

_in-plane

molecules

_{p ~}

vanishes. On the other

_hand,

a nonzero solution of 6 is

_{possible only}

when the

density

of

_in-plane

molecules is

_{sufficiently large.}

In

_fact,

such a transition is

_expected

to take

place

at a surface _coverage of

_in-plane

rods of the order

1/ l2 [32].

Therefore it follows that

only

when the

_adsorption

energy is

sufficiently large

do we

_expect

a

_symmetry

_breaking

in the

in-plane

orientations. In the

_following,

we use bifurcation

_analysis

to determine this critical

hc

in the absence of attractive interaction.

_Although

it is desirable to discuss the behavior of the

system

in terms of the total

density

(or

surface

_coverage)

of rods p and the

density (or

surface

_coverage)

of

_in-plane

rods Pin, the latter is related to p

and 0

simply

by

P in =

P Q .

We

prefer

_{to use p} and the fraction

of in-plane

rods 0,

based on consideration of

mathematical

_{convenience, since}

decreases

_{monotonically}

with

_{increasing density}

and is thus more amenable to the

_{subsequent analysis.}

Clearly u

= 0 is

always

a solution of

equation

_(16b).

To locate the

point

where a nonzero

solution becomes

_possible,

we

_{expand equation (16b)}

to linear order in a,

where the

order o, 2term

is

identically

zero

_by

_symmetry.

Bifurcation takes

_place

when the coefficient of a

vanishes,

which

yields,

On the other

_hand,

from

_equation

_(16a),

for

The emergence of a nonzero is thus tantamount to the existence of a solution

(p-,

0 )

to the simultaneous

_equations

₍₁₈₎

and

₍₁₉₎

in the

_physical

domain

p

E

_(0,

_{1 )}

and

l/J

e

_{(o, Q o). (In}

the absence of attractive

_{interactions, 0}

is

_{monotonically decreasing}

with

p,

so Q o

is the maximum value

_{that qb}

can

_attain.)

Substituting

p/(1 2013

p B )

from

_{equation (18)}

into

_{equation (19),}

we obtain an

_equation

for

0 only,

or,

where the definition of G

_{(0 )}

is obvious from

_equation

_(20).

_Equation

₍₂₀₎

or

_{equation (21)}

is thus the determinant condition for the _emergence

of

’nonzero 0-.

The function

_{G ( Q)}

is

Fig.

1. -

Curve 1

_gives

the maximum value

of 0 (the

fraction of

_{in-plane molecules),}

i.e.,

r6o

as a function of the

_adsorption

energy

6h.

Curve 2 is the

_{graphical representation}

of the relation

G (Q ) + Bh = 0 (Eq. (21 )).

Curve 3 shows the limit of

_using

the

_{Onsager expansion by setting}

p = 1 ; thus the

region

below Curve 3 cannot be accounted for

by

the

Onsager expansion.

where

_{G min}

is the minimum value of G in the interval

_{(0, Q 0).}

The

_{critical Oc}

and hence

G min

can be found

_{by letting}

ag/aQ =

0,

from which we

obtain,

and

The critical

_{adsorption energy 6 hc}

is thus seen to be

or

For the

_{rectangular parallelipipeds}

with

_aspect

ratio r,

equation (25)

is

_simply,

When - Bh > - BhB c’ equation (20)

has two solutions

₀

1

and ~ b corresponding

to the

emerging

and

_{vanishing, respectively,}

of the nonzero 6. In this case,

then,

an ordered

_phase

in the

_in-plane

orientation exists

_{between 0}

_{1 and 02,}

or

_Fi

and

_{p 2,}

the latter to be

_computed

from

_{equation (19).}

That

_{both .0}

₁

_{and 2}

and hence

_p

1 and

P2,

are within the

physical

domain, i.e.,

0 0

>

(~)

_1,

~

2)

>

0,

when -

3 h

> - {3h c’

is ensured

by equation (20),

since the

first term there is

_{always positive.}

Most of the above

_arguments

_{apply equally}

well to the

_{Onsager theory}

where

_only

the second virial coefficient is retained. In

_particular,

it can be shown that

_{equations (20)-(25)}

remain

_unchanged.

_However,

since the

_expansion

is

_only

up to order

_fi,

within the range

p E

(0, 1 ),

~

cannot decrease below some

_{minimum ~min}

which is obtained from

equation (19)

may be cut out of the

_{physical density}

_{range in the}

_Onsager

_{approximation,}

for

sufficiently large - 8 h (see Fig. 1).

We have carried out a numerical calculation

of 0

and a as functions of the

density

p at T = 300 K. For

_aspect

ratio r =

10, -l3hc

=

22/9

₊ In

(2/9)

from

equation (24) ;

thus

when - 8 h

=

2,

an ordered

_phase

is

_expected

to exist in a

_density

« window ».

Figure

2 shows

the

0-j5-

and _a-fl

_plots

for this set

_{of parameters.}

The full curves are the results from

using

our van der Waals

_type

mean-field

theory,

and the dash curves are results from

_using

the

_Onsager

theory.

Both theories show the existence of an ordered

phase

in an intermediate

density

range. It can be seen

_that,

associated with the transition in the order

parameter

o-,

0

has discontinuous

_changes

in its derivatives at both boundaries.

_Analysis

_{of the}

pressure-density

77-pr

relation and the chemical

_potential

_suggests

that the transition at both boundaries is second since there is no van der Waals

_loop.

Fig.

2. - Order

_{parameters r6 ,}

the fraction

_{of in-plane}

molecules, and a, the

in-plane

symmetry order parameter

(see

the lines below

_{Eq. ( 11 )}

for their

_definition)

as functions of the reduced

_density

p for r = 10 and

6h

= - 2, in absence of attractive interactions.

(-)

from _van der Waals

theory ;

(- - -)

from

Onsager approximation.

4. Effects of the attractive interactions.

It is well known that attractive interactions between

_particles

induce a

_{gas-liquid phase}

transition below the critical

_temperature.

For

_{simple liquids,}

such a

_transition,

as well as the critical

_point,

can be studied within the framework of various mean-field

theories,

such as van

der Waals

_theory

and

_{Bragg-Williams approximation [33].}

For more

_complex

_systems,

_e.g. a

monolayer

of adsorbed

_long

surfactant or

_polymer

_molecules,

the mean-field

_theory

becomes

more involved because of the extra

_degree

of freedom

_(rotation,

for

_example)

and of

_possible

inhomogeneity

in the

_density.

For flexible surfactant or

_polymer

molecules,

usually

a

self-consistent

_{procedure [6,}

_12(a)]

is

_required.

For the

_simple

model considered

here, however,

we can use the mean-field

_theory

_developed

in section 2 of the paper.

In the discussion that

follows,

we use the letter G for the _gas

_phase

and L for the

_liquid

phase.

To make distinction between a

_symmetric

_(with

_respect

to the

_{in-plane orientations)}

and a

_{symmetry-broken phase,}

we attach the letter S for the former and B for thé latter. Thus

in the full parameter space, four different

phases

are

_possible,

at least

_{mathematically.}

These

S/B

boundaries and

_{p G}

and

_PL

the

_density

for the

_coexisting

gas and

liquid phase,

respectively,

then

_{mathematically,}

the

_{following possibilities}

_exist,

with their

_{corresponding}

sequence of

phase

transitions :

In

_practice,

_however,

not all of the above

_{possibilities}

are realizable for

_physically

reasonable

parameters.

The

_analysis

in the case where attractive interactions are

_included,

_proceeds

in much the

same manner as the last section. We first consider the situation where the interaction is

independent

of the orientations. In this case,

_A,,

=

A xx

=

A xy

=

0,

and

the ~ - P

relation

and the values

_{of p}

_{1 and p 2}

remain

_unchanged.

However,

if the attractive interaction is

_strong

enough,

then a first order transition from a _gas

_phase

to a

_{liquid phase}

takes

_place.

As in _any

mean-field

_theories,

a Maxwell construction is

_required

in this case to eliminate the

thermodynamically

unstable

_region

and locate the values of the densities at coexistence

_[33],

j5-G

and

_pL’

Figure

3 shows the results for T = 300

K, r

=

10,

u = - 1 000 K and

h = - 600 K. This

corresponds

to the second

possibility

mentioned above.

Figure

3 shows

that,

associated with the discontinuous

_change

in the

_density

at the

transition,

the order

parameters 0

and a also

_{change discontinuously.}

The

_jump

in

₍

at the

_gas-liquid

transition has been

_pointed

out in the work of Chen et al.

_Figure

4 shows another situation for the same

temperature

and

_aspect

ratio but with h = - 450 K and u = - 850

K ;

it

corresponds

to

possibility (1).

In both of these cases, the transition between the

_symmetric

and

symmetry-broken

_phases

remains second

order,

except

when it is overtaken

by

the

_gas-liquid

_transition,

in which case it becomes first order.

Fig.

3.

_{- Isotherms ~ -}

_{II, 6 -} II and _{p -} II for r = 10, B h = - 2, u = - 1 000 K and T = 300 K.

There is no

_anisotropy

in the attractions. The

_regions

I, II and III

_{correspond respectively,}

to the GS,

Fig.

4. -

Isotherms r6 -

II, 6 - II and p - II for r = 10, {3h = - 1.5, u = - 850 K and T = 300 K.

Again

there is no attraction

_{anisotropy. Regions}

_{I, II,} III and IV are

_respectively

the GS, GB, GS, and LS

_phases.

It is of interest to

_investigate

the effects of

_anisotropic

attractions. To this

_end,

we make the

following

ad hoc choices for the

_Ajk’S:

Figure

5 shows the results for T = 300 K and the same

_aspect

ratio as the other two. An obvious feature in these

_results,

is

that,

besides the

_gas-liquid

transition,

the transition GB to

GS becomes

_{discontinuous,}

as

_opposed

to the

_isotropic

case. In this case we have the

following

sequence : GS --> GB ==> GS =>

LS,

where we have two discontinuous

changes

(denoted by

the =>

symbol)

in the

_density

and the order

_parameters.

Fig.

5.

_{- Isotherms r6 -}

_II, o- - II and _{p -} II for r = 10,

{3h

= - 2, u = - 1 000 K, T = 300 K and

Discussions of the results of

figures

3,

4 and 5

_depend

on the

_validity

of

_using

a

density-independent

ajk

or

_a(w,

_{w’) (see}

_{Eq. (5)}

or

_Eq.

₍₇₎₎

which can be

put

into doubt. The

phenomenology pointed

out,

however,

can be

_expected

on

_physical

_{considerations,}

and

should not be affected

_by

the drawbacks of the

_{approximations}

we make.

Results in this section

_suggest

that,

attractive interactions can induce a

_gas-liquid

transition,

which in turn leads to discontinuous

_changes

in the orientational order

_parameters.

Moreover,

when

_anisotropy

in the attractive interactions are

_allowed,

it is

_possible

to have another first order transition. This latter transition is

_mainly

orientational in nature. At this

juncture,

it is

_interesting

to _compare the behavior studied here and the LE to LC

_phase

transition observed in

_typical

surfactant

_monolayer

_{systems :}

both situations can exhibit two

successive first order

_transitions,

one of which is associated

_mainly

with conformational

changes.

In this

_study

it results from a

_symmetry

_{breaking coupled}

with orientation

_dependent

attractive

interactions,

whereas in the LE to LC

transition,

the effect of conformational

packing

is

_suggested

to be

_{important [10, 13-16]}

in addition to the orientational

_change.

5. Discussion.

In _{this paper,} we

_study

a model

system

of

_grafted

rods

_using

a mean-field

_{theory. Emphasis}

is

placed

on the

_importance

of the orientational

_degree

of freedom in the azimuth

_angle

in

determining

the

_phase

behavior of the

_system.

This

_complements

earlier studies which have

not taken this

_degree

of freedom into consideration. It is shown that when the

_adsorption

energy is

large enough,

a

_symmetry

breaking

transition in the afore-mentioned orientation can take

place,

which

_necessarily

leads to a discontinuous

_change

_(second

_order)

in the other

orientation

_angle.

_Furthermore,

it is shown that attractive

_{interactions,}

can lead to a

_jump

discontinuity

in the order

_parameters,

which was

_pointed

out

_by

Chen et al.

_[21],

and in some

cases

_{(anisotropic interactions)}

_may also cause another first order transition _{besides the}

gas-liquid

transition.

It is

_interesting

to note that the

_possibility

of an orientational

_phase

transition for

_grafted

rods when

_adsorption

energy

(which

favors the flat

_{orientations)}

is

_included,

was

_suggested

in

the work of

_{Halperin et}

al.

_[20].

_They

first attributed this

_possibility

to the

_weakening

of the

alignment by

the

_impenetrable

surface. In a more recent

_{discussion, however,}

these authors

point

out that such surface

_potential

have the effect of

_constraining

the rods to lie

_parallel

to

the

surface,

and hence

_causing

a

_symmetry

breaking

in the

in-plane

orientations at some

appropriate

densities.

_Although

the

_{potential proposed}

in reference

[32]

is

_slightly

different in form than the

_adsorption

_energy

_employed

_here,

the effect of such a

_potential

in the two

studies is the same,

namely

to favor the

_in-plane

orientations. In the

_{preliminary study}

presented

in this paper, we find no

_{alignment phase}

transition if the

_symmetry

_breaking

in the

azimuth directions is

_neglected,

thus

_confirming

the

_symmetry

_argument

_{given by}

Chen et al. Our

_{study employs}

a discrete

_model,

where

_only

three orientations are allowed. While this

kind of

_{approximation}

is

_commonplace

in

_studying

bulk

_systems,

a few comments should be made in

_applying

it to the

_grafted

_system.

Such an

_{approximation}

makes sense

_only

if the

fraction of molecules in each direction is

_comparable

to each other. This is

_possible

in the

grafted

rod

_problem

if there is

₍₁₎

a

_strong

surface attraction that favors the

_flat-lying

orientation or

₍₂₎

other

_energetic

factors,

e.g.

bending

energy, that

strongly

confine the

molecules to orient at some

_{prefered 0-angle.}

The second

possibility

arises,

for

example,

in

systems

of

_fatty

acid molecules at water-air

interface,

where _{the -COOH group, with the}

hydrophilic

part

immersed in water, orients the

-CH2-

tails. This is

_probably

one of the factors

in the tilt transition observed in these

_systems.

Another

_experimental

_realization,

of which the

theory

presented

in this paper is

expected

to be a more faithful

_{representation,}

is to use

_rigid

observe the

breaking

and

_restoring

of the

_in-plane

_symmetry.

A more direct test of the

predictions

given

in this paper can be done

by performing

Monte Carlo simulations of the

same model

_[35].

Since the

general

mean-field

theory developed

in this paper is not limited to

the discrete

_{approximation,}

the continuum can also be studied

[36, 37].

Acknowledgments.

1 thank T. A.

Witten,

A.

_Halperin

and M. W. Kim for

_helpful

discussions,

and W. M. Gelbart for

_sending

me a _copy of their work

(Ref. [21]) prior

to

_publication.

1 am

grateful

to W. M. Gelbart for his critical

_reading

of the

_manuscript

and some valuable

_suggestions.

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