The role of conformational defects in solid-solid phase transitions

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The role of conformational defects in solid-solid phase

transitions

B. Bassetti, V. Benza, P. Jona

To cite this version:

The role of

conformational

defects in solid-solid

_phase

transitions

B. Bassetti

_(1,*),

V. Benza

_(2,*)

and P. Jona

₍₃₎

(1)

Dipartimento

di Fisica, Universita’

_degli

Studi di

Milano,

Via Celoria 16, 20133 _Milano,

_Italy

(2)

Facolta’ di Scienze, Universita’ della

Basilicata,

Via N. Sauro

85,

85100 Potenza,

Italy

(3)

Politecnico di

_Milano,

Istituto di Fisica, Piazza Leonardo da Vinci 32, 20133

_{Milano, Italy}

(Reçu

le 28 décembre

1988,

révisé le 28

_septembre

1989,

accepté

le 4 octobre

₁₉₈₉₎

Résumé. 2014

Le rôle du désordre de conformation dans les transitions de

phase

solide-solide de couches de chaînes linéaires de molécules

_(n-alcanes)

est étudié à l’aide d’un modèle hamiltonien. Nous définissons un nouveau _type de coordonnées de chaîne

_approprié

aux

_régimes

à faible taux

de désordre conformationnel. En effectuant une sommation exacte sur ces coordonnées, nous

obtenons un hamiltonien effectif décrivant l’interaction interchaine dans la couche. En _supposant,

d’une manière

_{phénoménologique,}

l’existence de deux structures

_{cristallographiques}

(orthorhom-bique

ou

_{monoclinique-triclinique) à}

_température

_nulle, nous montrons _{que seul le} cas

orthorhombique

conduit à une

_phase

_{rotatoire par} un

_couplage

entre

_degrés

de liberté conformationnel et _{de réseau. Nous étudions les valeurs moyennes des défauts conformationnels}

dans les différentes

_phases

du

_système.

Abstract. 2014 The role of conformational disorder in solid-solid

_phase

transitions of linear chain molecules

_(n-alkanes)

_arranged

in a

_layer

is studied in terms of a hamiltonian model. We define a new kind of chain coordinates

_appropriate

to

_regimes

with low conformational disorder.

_By

performing

an exact summation over such coordinates, we obtain the effective Hamiltonian

describing

the interchain interaction within the

_{layer. Assuming,}

on a

_{phenomenological}

_basis,

the existence of two zero _temperature

_crystal

structures

_{(orthorhombic}

or

_{monoclinic-triclinic),}

we show that

_only

for the orthorhombic one, the

_coupling

between conformational and lattice

degrees

of freedom results in a _rotary

_phase.

We

_study

the mean values of the conformational defects in the different

phases

of the _system.

Classification

Physics

Abstracts 64.60 - 64.70K

Introduction.

In recent _{years many}

_attempts

have been made to

_clarify

some

_intriguing

_aspects

of the

solid-solid

_phase

transitions in

_systems

of linear chain molecules

_[1].

At very low

temperature

such

_systems

have an

_{high degree}

of

order ;

the

_single

molecule is

a _sequence

(chain)

of identical chemical _{groups linked}

_together

without

_branchings

and

keeping

a

_trans-planar

conformation

_{[2] ;}

the chains are

_arranged

in a «

perfect

» lattice

[3].

(*)

Also INFN sez. Milano

At intermediate

_temperatures

one or more distinct solid

_phases

are observed which drive the

_system

towards

_liquid

state

_through

a

_progressive

loss of order

_[4].

_{In many} cases there is

experimental

evidence of an

_interplay

between intramolecular

_disordering

_{processes and}

structural transitions

_[4-8].

The aim of our work is to

_give

a detailed

_{representation}

of the

_system

in order to reach a

description

of intramolecular

_quantities,

such as the mean number of conformational « defects »

[2],

in the various

_phases.

From this

_point

of

_view,

the most _{economical way is} to consider a

_system

with a

_simple

molecular structure, so that one can obtain a model in terms of a

_reasonably

small number of

degrees

of freedom.

_Moreover,

one needs to

_support

his

_modelling

with

_experimental

data from well studied

_{systems ;}

this is indeed the case with some molecules of the so-called

polymethylene

chain

_class,

in

_particular

n-alkanes

_{(linear hydrocarbons) [5-12].}

In these

_systems,

the linear flexible

_part

of the molecule

_plays

the main role in

_determining

the

_equilibrium

architecture of the solid

_phases.

With n-alkanes one

_finds,

at low

temperature,

two

_possible

structures

_(«

lamellae

_»)

depending

on the

_parity

of the molecule

_(the

number of groups in the

chain) :

odd n-alkanes

crystallize

in the orthorhombic

system

[13],

even n-alkanes are monoclinic or triclinic. At

intermediate

temperatures

the observed transitional

_pattern

_depends

on both chain

length

and

_parity

_{[4] ;}

in fact all the odd

n-alkanes,

up to

_{41 length}

at normal conditions

_[9],

show a

special

solid

_{phase, just}

before

_melting,

known as « a » or « rotational »

phase.

So far it has

been

_{proved, mainly}

_by

IR and Raman

_Spectroscopy

_[5-8],

that in the a

_phase

the conformational order decreases with

_increasing

temperature,

reaching

the

_collapse

at the

melting point.

The

_{interpretation}

of the

_{phenomenological landscape}

and its

_{comprehensive description}

within a coherent scheme has been the

_object

of research for years. Efforts have been made to

explain

the

_origin

of the a

_phase

in terms of two main mechanisms. On one hand

(intermolecular disorder),

the overall

_rigid

roto-translations of the molecules were invoked as

the dominant mechanism in

_driving

the

_system

to the a

_phase

[10].

On the other hand

(intramolecular disorder),

conformational defects localized in the flexible chains

_[11-13]

were

considered the

_origin

of the a

_phase.

In our

_previous

work we examined the

_synergism

of the above mechanisms

[14, 15].

In

paper

[14]

we introduced a Hamiltonian model where the conformational disorder was

coupled

with lattice

_elasticity.

In letter

_(15),

we extended the model

_including

the orientational disorder

(rigid

rotations of

the chains around their

_longitudinal

_axes).

We studied the existence of a solid-solid

_phase

transition marked

_by

the rotational order

_parameter.

We showed that such transition may

occur when the structure, at low

_temperature,

is

_{orthorhombic,}

while it does not occur with

monoclinic-triclinic

_systems.

In this paper we

_report

in detail some

_analytical

_aspects

of the model and extend its

applications.

We first define a set of

_spin

variables suitable to describe the

_single

chain conformational

degrees

of freedom : the choice of such variables amounts to

_{explicitly solving}

the constraints

arising

from the fact that the chains are extended

_objects.

We then

_perform

an exact summation over such

_spin

variables,

as well as over the elastic

interchain

_{couplings, obtaining}

an effective Hamiltonian in the orientational

degrees

of

freedom. Such a Hamiltonian

corresponds

to a decorated

Ising

model with bilinear and

trilinear terms

_involving

nearest as well as next-nearest

_neighbours.

The

temperature

dependent coupling

constants are

_explicitly

obtained in terms of

_single

chain and lattice

The mean values of the conformational and orientational defects are then determined

_by

a mean field treatment.

In our _{model the translations of the chains in the direction}

_orthogonal

to the « lamella »

[5, 16]

are not

_explicitly

included : we will show that these

_degrees

of freedom have no effect

on the solid-solid

_transition,

but

_only

on the distribution of disorder over the different conformational defects.

In section 1 we define the set of

_independent

« internal »

_spin

variables for the conformational

_degrees

of freedom of the

_single

chain,

consistently

with a « low disorder »

hypothesis.

In section 2 we consider the

_system

of chains

_organized

in the two-dimensional lattice. We

give

the Hamiltonian in terms of all the

_(internal

and

_external)

variables and obtain the effective interaction in the rotational

_degrees

of freedom.

In section 3 we

_apply

the mean field treatment in order to calculate the mean values of the

interesting

observables.

In section 4 we draw the conclusions and

_point

out some _open

_problems.

Fig.

1. - The backbone of n-alkanes. The site of the N

+ 2-th carbon is determined

_by

the bond vectors

VN -1--_

eN - C N -1

and

uN = C N + 1 - C N

and

_by

the _{variable SN}

representing

the

Flory angle

Q. With

sN = 0 _one has the _trans

(T)

conformation

(cp

=

03A0 ), with sn,

= ₊ 1 one has Gauche

_(G)

and Gauche’

(G’)

conformations

_(P

= _±

2 3’1r ).

The site vectors uN _ 1, i, i = 1, 2, 3 are also

_reported.

The

drawings

represent sections of chains

_containing

_{the all-trans sequence} ... TTTT..., the « dressed » kink

...TGTG’T..., the kink ... GTG’ ... and the U-like defects ... GG... and ... GG’ ... The distance _{n between}

1.

_Single

chain coordinates.

The space coordinates of the carbon atoms of an alkane chain can be

_determined,

for a

_given

sequence of

Flory angles

[2],

by

means of an iteration formula which relates each C-C bond to

the two

_preceding

ones.

More

_precisely,

we

_represent

the n-th C-C bond

by

a vector _vn

_going

from the n-th carbon

atom to the

_(n

+ 1

_)-th

_carbon,

and the

_{Flory angles}

related to _trans,

_gauche

and

_gauche

_prime

conformations

_(see

_Fig.

₁₎

_by

a

_spin

variable s with values

0,

+

1, -

1

_{respectively.}

We have then :

It is convenient to associate with the ordered

_couple

of

_adjacent

C-C bonds

_{(v n - l’ vn)}

the

set of

_orthogonal

vectors

Starting

with a

_couple

of bonds

(vo, vl ),

each sequence of variables sl, S2, ..., sk,

through

equation

(1),

determines the

_displacement

d from the 0-th carbon to the

_(k

+

_2)-th

carbon

together

with a new set of

_orthogonal

vectors

_{Uk, 1, uk, 2, uk, 3 .}

We

_consistently

associate with the sequence Sl, S2, ... , sk the vector d and the

_application

from

_{(uo i)}

to

_{uk,

_i}

(i = 1, 2, 3 ).

From now on we will be

_mainly

concemed with defects

connecting parallel planar

sections of the chains. In our terms this

_implies

the constraint _{UO, 3}

_Uk,3.

_Furthermore,

due to the

geometry

on which the chain

lies,

from

UO, 3 || Uk, 3 it

follows that

_{UO, 1}

Il

_{Uk, 1 :}

hence

uo,1

1 identifies the overall

longitudinal

direction of the chain.

Let us call « allowed sequences » the

_particular

(s1,

_S2,

..., Sk}

satisfying

the former

constraint ;

we have for them :

(note

that aE = - 1

_implies

chain

_reversal).

We define the action

_{’G (S1, S2, - - -, Sk)}

associated with an allowed sequence

{sl, s2, ... , sk }

as :

where di (i

=

1, 2,

3 )

is the

_component

of the vector d with

respect

to

uo, i .

We will in

general

disregard

the

_longitudinal

_component

_dl,

thus

_{having simply :}

Let us consider two _{allowed sequences}

_{S1’

_S2, ...,

sk}

and

{s1,

s2] ,

...,

sk }

characterized

by

1) (S1’ S2, ..., Sk) =

(u, E, d2, d3,)

and

_{1) {s1, s2, ..., sk’} = (u’, E’, dÍ, d3);}

one can

_easily

verify

that for the sequence

(si ,

S2,

..., Sk’ S1, S2,

... , sk, }

holds the

following

composition

law :

We now introduce a

_system

of

_{independent spin}

variables associated with chain

Let us first observe that each

_{’G (S1, S2, - - -, Sk)}

can be written as a

_product

of various

13’s,

according

to the

_composition

_{law (5).}

One can think of

writing

S(s1,

_S2, ...,

Sk)

as a

product

such that each factor cannot be further

_{decomposed ; clearly}

_13(s1,

_S2,

_{... , sk )}

=

HI

= 1, k 13

(se)

is true

only

if Se = 0

(l

= 1, ... , k ).

In other words there is a set of minimal

_(not

further

_{decomposable)}

l3’s in terms of which all conformations

_preserving

a

_given

chain axis can be obtained.

The first of these

_{applications,}

in

_increasing

order of

_complexity,

are :

_{b(O), r (1,}

_{0, -}

_{1 ),}

-t (1, - 1, 1 ), t(1, 1, 1, 1 ), -t (1, 1, -

1, -

1 ),

together

with the

_{corresponding}

ones obtained

by exchanging

1 with - 1.

It is

_easily

shown that

_only

a finite number of minimal 13’s is

required,

if a maximal

transversal width is fixed for the chain conformations.

We define such a width as the

_largest

distance between all-trans

_portions

of the distorded

chain ;

it turns out that this

_quantity

is a

_multiple

A (a

=

1,

2,

... )

of the width ₁₁ of the

single

kink

_(a

_sequence Gauche-Trans-Gauche

_prime,

see

_Fig.

_1).

The variable a is then an

intrinsically

nonlocal function of

_{Flory angles.}

If,

in

_{particular, à}

=

1,

_one

needs (T =

b(O);

sK -

S(1, 0, - 1 ) ;

l3K=

(- 1, 0, 1 ) } .

The

_{requirement A}

= 1

gives,

furthermore,

two constraints on the

_ordering

of

_3T,

’GK

and

_{l3K, along}

each chain :

a)

13K

and

_13K,

can never occur

_{together along}

the same

_{chain ;}

b)

different

_13K

_{(or 13K,)}

_groups can never be

_{separated by}

an odd number of

T1S.

Hence,

disregarding

for the time

_being

the

_problems

related with the finite size of the

chain,

we have that an

_arbitrary

combination of

_l3K

(or 3K,)

and

(br)2

_generates

each allowed conformation with à = 1.

(We

exclude,

for

entropic

_reasons, U-like

defects.)

It is now clear that a natural choice of

independent

spin

variables ( §i )

is

simply

given

by :

§i =

1 for each

_13K

or

_13K,

_{and §1 =}

0 for each

(br)2

_group.

With this definition the

_{sum X 2: ei}

i is the number of kinks in the chain. i = 1, L

Let us now examine how one can

_parametrize

the chain ends. We will refer to the case with

bK

defects in the

bulk,

the

_{complementary}

case

_l3K,

follows in a

straightforward

_way.

One can think of

_cutting

an indefinite

_length

chain

generated

by

13K’s

and

(br)2,s.

As a

result,

modulo the bulk

_(generated

_{by bK}

and

_(bT)2),

one can further have a

_13T,

a

13 (- 1 )

or a

b(O, -1)

on the left

end,

and a

_13T,

a

_b(1)

or a

_{’G (1, 0 )}

on the

_right

end. At each end we are left with four cases,

including

the «

empty »

case

(no

bond

added).

We

label the left end with the

_spin

variables

_{(vo, go)}

and the

_right

end with

_{(M1, v 1 )}

(Vi, ui

=

0, 1 ),

with the

correspondence

table I.

This choice is a convenient one, in that the sum _{U = U 0 + U1} counts the end-Gauche

defects

_{(Fig. 1)}

_present

both in the left terminal

_(U0)

and in the

_right

terminal

(M,1 ).

Furthermore the sums _{vl + Ui} i

(i

=

0, 1 )

and v = _vo + _v

_{1 equal}

the number of bonds

attached to each terminal

_and,

_{respectively,}

the total number of trans bonds in the terminals.

Any

chain conformation is then

_{represented by}

the

_{independent spins}

or « internal » coordinates

_{( v o,}

_1£0,

_U1,

_Pl).

The number L of

_{variables ei}

becomes in so

_doing

a

_dynamical

_variable,

related to the

number £ of C-C bonds

_{by :}

Whenever the case of low disorder is

_considered,

_namely

if the occurrence of

_adjacent

defects is excluded for _energy reasons, it is convenient to associate the

value e

= 1 with « dressed kinks »

_{corresponding}

to

_{03B1T 03B1K 03B1T}

_sequences

_(involving

five

_bonds,

see

_Fig.

₁₎

instead of

l3K.

In this

_special

case

_{equation (6)}

becomes :

Within this framework of

coordinates,

there is a

_degeneracy

of the all-trans conformation

coming

from the

_{complementary}

class of conformers

_{generated by}

the

_application

OK’.

We shall account for such

_degeneracy

in our calculations.

2. Partition functions for the

_single

chain and for the

system.

In the

_previous

section we _{gave the}

_{prescriptions}

to label all the conformations of a flexible

chain

_forming

defects with maximum transversal size

_equal

to one. Under such

conditions,

the

Flory

conformational _energy

_[2]

in terms of our coordinates assumes the form :

In this

_expression

_{J is the energy of the}

_single

kink

_(-

1 Kcal/mole

_[2]).

The first term in

equation

(7)

accounts _{for the presence of distortions} at chain

ends,

the second term counts the defects in the bulk of the chain and the last two terms contribute

_only

if

_adjacent

defects

occur.

The

_partition

function for the conformational statistics of a

_single

chain of fixed

_length

C is :

In these

_{equations Ao}

is the contribution of the all-trans

conformation ;

the first and the

second terms in

_A,

_represent

the contributions from conformations with one,

respectively

two, end-G in an otherwise

_transplanar

chain ;

the summation in

_A1

collects the contributions

from all the kinked conformations. We account for the

_{CK lJK’ degeneracy by doubling}

such a sum.

In the

_special

case

_excluding

_adjacent

_defects,

the evaluation of Z reduces to a

_simple

states with

_given

_{energy in Z is} a sum of binomial coefficients. With £ =

2 (m

+

_{3 ),}

_setting

A =

exp (- (3a),

one obtains :

where :

With even chains one has

analogous expressions.

From

_equations

₍₈₎

or

(9)

one

_easily

obtains the mean number of end-G defects and kinks for a

_single

_{chain upon}

_deriving

with

respect

to the

parameter a

or b

_{respectively.}

Let us now consider the

_system

of

_equivalent

chains in low disorder

regime.

We

_give

a Hamiltonian model for such a

_system

in a two-dimensional lattice of sites

1= .

The Hamiltonian is defined in terms of intemal variables

_{{L ;}

IL,

v, §} (Sect. 1)

and of two kinds of external variables : the rotations T and the translations x

(see

Fig.

2).

The variable T

(I )

_represents

the

_angle

between the

plane

of the chain at site 1 and a fixed direction in the

lattice ;

according

to intermolecular

_potential

calculations

_[17]

we assume discrete values for such rotations : T = ± 1 for

_alignement

of the chain

_{plane along}

the

_{x (+ )}

and

_{y (- )}

axis.

Fig.

2. - Two-dimensional lattice and lattice variables used in this work. The

drawing

represents the section of a

_layer

of linear chains

(n-alkanes)

orthogonal

to the

_{(x, y )}

plane ;

the variable x represents

the chain axis

_position

and the rotational

(spin)

variable T

_gives

the orientation of the

plane

of the chain with _respect to x axis : T = _± 1 for

alignements

along

x (+ )

and

y (- )

axis. Notice the _two sublattices A

(full circles)

and B

_{(open circles)}

and the 2 x 2 cell 9 used in mean field treatment : the site numbers

The mean value

_(r)

is associated with the occurrence of either the

_{orthorhombic,}

or the

monoclinic-triclinic or the

rotatory

phase.

The variables

_x(I)

_represent

the chain axis

_position

in the two-dimensional

lattice ;

the mean

values (

_{xI -}

XI + q | >

_(I

+ _q

_labelling

the nearest

neighbours

of 1 with q-

(qx’

qy)

qx, qy

= -

1, 0,

₊

1)

_are associated with lattice

spacings.

The total energy is :

In this

_expression

the first term is the

_Flory

conformational _{energy of the}

_single

chain

(Eq. (7)),

the second and the third terms

_represent

the rotational and the elastic interaction

respectively.

The last term accounts for the interaction between the defect width and the

nearest

_neighbours

orientations modelled

_by

the

_{potential v (I, q ).}

The conformational defects are

_coupled

both with lattice

_elasticity

and with nearest

neighbour

rotations ;

the

_couplings

are

_{performed through}

the collective variables

_{A (I).}

Such

variables,

in

fact,

determine the local value of the

_equilibrium

lattice

_spacing

D (I, q )

=

5q

+ A (I) nq, d

being

the _zero

_temperature

interchain

spacing.

We define

_[15]

the

_potential

v in terms of two

_coupling

constants

_MR

and

_{MD representing}

the intermolecular

_energies

of a distorted chain inserted in a

_locally

orthorhombic or,

respectively,

monoclinic-triclinic environment.

Since in the orthorhombic

_{configuration}

there is a closer

_packing

[3],

we assume

MR > MD

[18].

In

_particular,

when

_{MR -+}

00 the

_potential

v acts as a constraint

_against

defect

formation on the I-th chain. More

_precisely,

_{for qx} = 0 _no defect _can be formed if the

neighbouring

up and down chains are

_orthogonal

to the I-th

_{chain ;}

_similarly,

for

qy

= 0 _no defect _can be formed if the

neighbouring

left and

right

chains _are

orthogonal

to the

I-th chain.

_Hence,

this constraint is active in the orthorhombic local

_ordering.

We assume that

MD

is

_proportional

to chain

_length

and

_independent

of the number of

defects ;

in

fact,

any

distortion,

if à =

1,

produces

_a

displacement

of the backbone groups which is

independent

of

such number.

_Hence,

_{the energy contribution}

_{corresponding}

to

_MD

_(to

be added to _{the pure} conformational

_term)

will be indicated with w£.

We shall now consider the

_partition

function

_Zs

of the

_system.

In order to obtain

Zs

we have to sum over the internal variables at each site 1 and over the external variables x

and T.

Since the internal

_degrees

of freedom are not

_directly

_coupled,

but

_{only through}

the

collective

variable à,

for fixed values of

_{T (I)}

and

_A(1),

we sum over the internal variables.

Moreover,

since for a

_given

distribution

A (I)

the calculation reduces to a Gaussian

_integral,

we can

_{exactly integrate}

over the x variables.

We note that the contribution to

_Zs

from the site 1 has a

_{weight proportional}

to the term

A(I)

of the

_single

chain

_partition

function

_(see

_Eq.

_(8)).

The interchain

_potential

v

_{(I, q)}

does not contain a direct interaction between the widths of

_adjacent

chains : this allows us to sum over d variables. As a result we

_get

the effective Hamiltonian as a _{power series in}

terms of the orientational variables T : one

_readily

verified

_that,

due to the discreteness of such

variables,

the Hamiltonian

_simply

reduces to a third order

_polynomial

in the form :

The

_{coupling constants Ji directly depend on 110}

and

_{A1 ;}

in the limit case

_MR

= oo one has :

We note that the nearest

_{neighbour coupling}

constant

_J,

at low

_temperature

has the

_sign

of

lJl ,

while at

_high

_temperatures

it is

_{always ferromagnetic ;}

the next nearest

_{neighbour coupling}

constant

_J2

is

_{ferromagnetic}

at all

_{temperatures.}

The trilinear term introduces an

_anisotropy

in the

_coupling

between the next nearest

neighbours

1 + _{q and I - q. In}

_fact,

in the x direction one

_gets

la n.n.n.

_coupling

J2

+

_{T (I ) J3,}

while in

_{the y}

direction one

_{has J2 -}

7 (I)J3.

Hence

_if,

_e.g.,

T (I )

=

1,

only

the next nearest

_{spins aligned along}

the x direction are

_coupled

_(J2

=

J3)

and vice _versa.

This

_anisotropy

is related to the

_anisotropic

_{response of the} environment to the defect formation at the site

_I,

as described

_by

the

_potential

_{v (I, q).}

One

_immediately

verifies that such contribution

_disappears

if

_MR

=

MD.

The variable

T (I )

describes the orientation of the

plane

of the chain with

_respect

to a fixed direction : e.g.

T(I)

= 1 _means that such

plane

is

oriented

_along

the x direction. Hence the transformation T

_(I)

- - T

(I)

for all 1 is

equivalent

to

_exchanging

the x and

_{the y}

axes. The Hamiltonian is in fact invariant under the

transformation

_{{ T (I )}

--> - T

(I ) ; (x, y )

-.

(y,

x ) } .

It is easy to see that the

_partition

function can be written in the form :

Zs

function with

_respect

to the

_{corresponding}

source w

_occurring

in the

_single

chain Hamiltonian

_Hc:

where

_{(P) S}

is the mean value of the

_polynomial

(16)

and

_{Zc = Ao}

+

_WA1

is the

_partition

Ai;*

function of a chain in the external field w£. The terms

Zc

c are the contributions to the mean

value of the observable

_0,

_coming

from the conformations with à = i and

_being

evaluated for the

_single

chain in the external field.

3. Calculation of order

_parameters.

In reference

₍₁₅₎

we

_performed

a mean field

_analysis

in terms of the rotational order

parameters

by distinguishing

two sublattices A and B

_{(Fig. 2)}

in order to describe both the monoclinic-triclinic and the orthorhombic

_phases.

In such a _way we _{determined the range of}

existence of the rotational

_phase,

which is to be identified with the

_{paramagnetic phase}

of the

spin

Hamiltonian

_Heff.

In _{this paper} we are interested in the calculation of the conformational

_parameters

(eng -

G) sand (K) s ; as

previously

shown,

these

_quantities

involve

_through

the

_polynomial

P(I) (see

Eqs.

(16)

and

_(17)),

second and third order orientational correlation functions. It is then natural to

_generalize

the mean field

_{by introducing}

new variational

_parameters

to

be associated with

_higher

order

_{correlations,}

_e.g.

_along

the lines

_presented

in reference

_[19].

We fix the size of the cell to the maximal order of the correlations to be included in the

generalized

m.f. treatment ; once the cell size is

fixed,

the construction is

_{totally general}

and

uniquely

determined

_{by assuming}

that the

_{configuration}

outside the cell is

_completely

described

_by

the variational

_parameters.

In the

_present

case we truncate to 2-nd order so that we consider a 2 x 2 cell 9

_{(Fig. 2),}

the order

_parameters

involved

_being

_T;,

_{ti, j,}

_{i, j}

=

1,

..., 4. We

perform

the exact sum of the cell

partition

function :

_{evidently, only}

linear and

_quadratic

terms occur in the cell Hamiltonian. We have :

The

_coupling

constants

_Hi

and

_{Hi , j}

are determined

_{by rearranging}

the different contributions from the

_original

hamiltonian

_Heff;

_{e. g. ,} see

_figure

_2,

the cubic terms in

In

_{particular :}

and

_{analogous expressions by cycling}

the indices. From

₍₁₈₎

we have the self

_consistency

_equations

by identifying

the variational

_parameters

with the correlations functions.

We now want to recover these

_equations

from a variational

_principle,

i.e. to determine the free

_{energy ’-;-}

of the

system.

1 In order to do

that,

we add a counterterm to

_{Hg : Hg - Hg}

+ 1 A ,

where A is an a

_priori

arbitrary

function of the variational

_parameters.

The variational

_equations

for Y are then :

and

_analogous

with

_respect

to

_rij.

From these

_equations

A can be determined as a differentiable function if and

_only

if the

couplings Hi

and

_Hij

are the

_components

of an

_{irrotational vector field.}

In such a case there exists a

_{potential 0}

such that

Four our _{purposes it is sufficient} to look for solutions such that

_T

₌

_T3

= _if,

_T2

=

_T4

=

T’ ,

F12

=

T 34

=

r,

T 13

=

F23

= T ’ . Such solutions include both the

ferromagnetic

and

antifer-romagnetic

configurations.

We are then led to the

_{self-consistency}

_{equations :}

where

The

_expressions

for A’ and B’ are obtained

_{interchanging}

T and ;F’ in A and B. We have the

analogous

s.c.

equations

for T’and r’

by changing,

in

equations (22),

the

quantities

As

_previously

_said,

we

_identify

the rotational

_phase

with the

_{paramâgnetic}

solution for T ; furthermore we

_identify

the orthorhombic and the monoclinic-triclinic structures with

antiferromagnetic

(T

= -

T’ )

and

ferromagnetic ordering

(if

=

if’)

respectively.

The

_sign

of the constant :R, determines the T = 0 solution : 31 > 0

impliçs

the orthorhombic

ground

state

_{(odd chains) ; R}

0

_implies

the monoclinic-triclinic

_ground

state

_{(even chains)}

[15].

Let us consider the case with A > 0.

With the orthorhombic ansatz,

equations (22)

have non trivial

_{(if = 0)}

solution for

temperatures

below a critical value

_To.

More

_precisely,

T = - T’

implies

r = T’ so that the

s.c.

_equation

for T does not _{contain r. The range of existence of the} non trivial solution can

then be evaluated

_by

the

_tangent

method as in the classical

_problem

of the s.c.

_{Ising equation.}

We obtain :

The function on the

_right

hand increases

_{monotonocally}

with T and tends to the number of states of the

_single

chain for T- oo. The relation

₍₂₄₎

is then satisfied for T

_To

T

(To - 7o(C)) :

from

₍₂₄₎

one estimates that the ratio

TO

is a constant. £

The free energy calculated

with

the orthorhombic solution is :

With the monoclinic-triclinic ansatz we have in

₍₂₂₎

two

_{coupled equations}

in T and OT = T - T’. One can _{argue that the} non trivial solution exists above a critical

_temperature

Tm-T-

In

fact,

it is easy to see that _{for very low}

_temperature

the

_unique

solution is

T =

0,

OT =

0,

while in the limit of

high

_temperature

the solution T =

1,

At= 0 _occurs. A

numerical estimate shows that

_TM-T

saturates for 03C4 >_ 20.

The _{free energy in the} case of the monoclinic-triclinic solution is :

It is now clear that the

_paramagnetic

(rotational)

_phase

is accessible to the

_system

if

To

T M-T ;

in such a case, the difference

T M-T - To

measures _{the range} of existence of the

phase.

With R

_0,

the relation

₍₂₄₎

is never satisfied and the accessible states have

_{ferromagnetic}

ordering.

Equations (22)

have been solved

_numerically

for different values of the chain

_length

_{(C) ;}

we

_report

in

_figure

3 T , T and Ar as functions of

_temperature.

The

_phase

_diagram

obtained in reference

₍₁₅₎

is confirmed

_by

the

_present

treatment. In

_particular,

an intermediate rotational

phase

is found for

short,

odd

n-alkanes ;

approximately

at N = 40 this

phase disappears.

We

predict

a 1-st order transition for

_longer

chains,

while for shorter ones the rotational

_phase

is reached

_through

second order transition

_{(see Figs.}

_3a,

_3b).

In

_figure

3b the two branches of

T refer to

antiferromagnetic

_ordering

and

_{ferromagnetic}

_ordering

_{respectively,}

so that a

discontinuity

occurs in T’. About the order of the

transitions,

one could _{argue, from the}

occurrence of the trilinear term in

_Heff

_(13),

that a cubic term _{appears in the}

_{corresponding}

free energy. As a _consequence,

_by

the standard Landau

_argument,

one should exclude a 2-nd order transition.

In our

_model,

due to the non scalar nature of the coefficient of the trilinear term in

Fig.

3. -

Orientational order _parameter T

_{(full line),}

two

_point

correlation r

_{(dotted line)}

and correlation

_anisotropy

Ar

_{(dashed line)}

vs. _temperature

_(K).

Mean field solutions have been calculated with R = 0.3 Kcal/mole bond and W = 0.2 Kcal/mole bond.

A)

odd chain with backbone of 21 carbons ;

B)

41 carbons.

T, both

when T = T’and when _T = -

T’,

_so that the Landau

_argument

does

not

apply.

On the

other hand it is a known fact that trilinear terms in the Hamiltonian do not

_{necessarily imply}

1-st _{order transitions : e.g. Baxter’s} exact solution of the 3-state Potts model exhibits a 2-nd order transition

_[20].

In our case one realizes that both the 1-st order and the 2-nd order transitions are associated

with the

_competition

between different bilinear terms ;

_only

second and

_higher

order correlations are sensitive to the trilinear terms.

In the

_present

treatment we concentrate on 2-nd order correlations

T,

T’ : we obtain that r

appreciably

differs from the

_product

T. T’ in the rotational

_phase

and at the onset of the

higher

temperature

phase.

In the rotational

_phase,

the result

r #

0,

AT =

0,

together

with the very low

density

of

distorted chains

_{(see after),}

shows that the defects are

trapped

in the

_crystal

structure : the’

predominant

part

of the free energy comes from the rotational interaction.

At

_higher

_{temperatures,}

_{independently}

of chain

_length,

the number of conformational

defects

_abruptly

increases at the same time as

Ar #

0 ;

such a behavior stresses the

occurrence of a

_regime

where the conformational disorder

« propagates

»,

driving

the

thermodynamics

of the

_system.

In this

_regime

the mean chain

_{width à>s}

undergoes

fast

saturation and the frozen

_degrees

of freedom of our model should be « switched on » in order

With mean field

_solutions,

we evaluate

_{P > s and}

calculate the conformational order

parameters

as in

_equation

_{(17) ;}

we have :

where

_{. > c}

means

_single

chain statistics from

Zc.

The

_probability

of the all-trans conformation is a further measure of

_{disorder ;}

we can

evaluate it as :

It is

_interesting

to note

_that,

with this

calculation,

we estimate that the onset of end-G defects

_always

occurs at lower

_temperatures

than kink formation. One

has,

in fact :

Fig.

4. - Mean number of end-G defects

(dashed line),

mean number of kinks

_{(full line)}

and

_probability

of the all-trans conformation

_{(dotted line)}

vs. _temperature as

_resulting

from the solutions shown in

Fig.

The ratio

₍₂₉₎

becomes smaller than 1 at a

_temperature

_Tr

which

_depends

on chain

_length.

With

_{medium-length}

and

_long

_{chains Tr}

results lower than the

temperature

of the transition to

the rotational

_{phase To ;}

for short

chains, instead,

provided

that,

as T - oo, the

_asymptotic

value of

_{(K) c}

is

_larger

than the

_asymptotic

value of

_{(end -}

_{G ) c,}

one has

_{Tr> To.}

We show in

_figure

_{4 (end -}

_{G)s’ (K)s}

and

_{PS (t )}

as functions of

_temperature

for

medium-length

and

_long

chains. Note that the conformational

_collapse

occurs at the end of the rotational

_phase

_{(if any),}

at the transition to the

_higher

_temperature

_phase.

The behavior of the fluctuations

_(reported

in

_Fig.

_5),

as estimated in our _treatment, stresses

that both

_phase

transitions are marked

_by

conformational disorder.

_Noting

that the

_plots

of

figure

5 are to be identified with the differential intensities of defect markers

_{[8, 21],}

a

comparison

with

_experimental

results allows us to associate the well-known observed

discontinuities with the structural transitions

_predicted

on the basis of our model.

Fig.

5. - Fluctuations and. _as calculated in the model. The

drawing

are to be identified with the differential intensities of

spectroscopic

markers of conformational defects.

_A)

21

carbons ;

B)

41 carbons.

Conclusions.

Assuming

as a

_{phenomenological}

datum the existence of « lamellae » of linear chain

molecules with two

_possible

_crystal

structures at _{very low}

_temperature,

we show that

_only

one

of these structures can _go over to a rotational

_{phase through coupling}

of the internal

_degrees

of freedom with the external ones.

_According

to

_experiments

we show that :

i)

the rotational

_phase

occurs

_only

in

_systems

with orthorhombic structure at low

ii)

the range of existence of the rotational

phase

decreases with

_increasing

chain

_{length ;}

iii)

for

_long

odd chains and for even chains

(monoclinic-triclinic

cell at low

_temperature)

the rotational

_phase

does not

_{exist ;}

iv)

the conformational _{defects appear} at the onset and

_during

the rotational

_{phase ;}

v)

the conformational

_collapse

occurs at the transition from the rotational

_phase

to the

_high

temperature

regime.

These results have been obtained

_{by freezing}

the

_degrees

of freedom associated with

translations

_orthogonal

to the

_planes

of the

lamella,

thus

_describing

the

_system

as a

two-dimensional lattice of rotators.

We think that such

_degrees

of freedom do not affect the overall behavior of the

_system

as

far as the existence and the

_stability

of the rotational

_phase

are concemed. It is instead clear that

_orthogonal

translations

_strongly

affect the behavior

_{of (end -}

_{G > S/ K> s}

vs. T.

In the

_present

work we assumed that end-Gauche

_defects,

as well as

_kinks,

can form

_only

inside the bulk. One can

_try

to include within the model a

_rough

_description

of the formation

of end-G defects also on the surface of the lamella

[5, 16].

In order to do that we tried to

modify

the d = 0 _sector in the

partition

function

by adding

the contribution from the states

with one end-G in an otherwise

_trans-planar

chain. We obtained that the

_qualitative

behavior

of the rotational

_phase

does not

_change

with

_respect

to the

_previous

results ;

the ratio

(end -

G)sl (K)s’

instead,

changes dramatically,

as shown in

_{figure 6.}

Fig.

6. -

Calculation of mean numbers of defects

_(end-G

dashed line, kinks full

_line)

and all-trans

Acknowledgments.

We wish to thank

_prof.

G. Zerbi for

_suggestions

and useful discussions.

References

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Contemporary

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[2]

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[3]

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=

MD,

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