Study of the phase separation in polymer solutions using the pair approximation of the CVM: analytical solution of the problem

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the pair approximation of the CVM: analytical solution of the problem

F. de J. Guevara-Rodríguez, F. Aguilera-Granja, Ryoichi Kikuchi

To cite this version:

F. de J. Guevara-Rodríguez, F. Aguilera-Granja, Ryoichi Kikuchi. Study of the phase separation in polymer solutions using the pair approximation of the CVM: analytical solution of the problem.

Journal de Physique II, EDP Sciences, 1994, 4 (4), pp.589-611. �10.1051/jp2:1994150�. �jpa-00247985�

Classification Physics Abstracts

05.20 05.50 61.25H

Study of the phase separation in polymer solutions using the

pair approximation of the CVM: analytical solution of the

problem

F. de J.

Guevara-Rodriguez (~),

^F.

Aguilera-Granja

_(~) and

Ryoichi

Kikuchi (~)

(~ ^Instituto de Fisica "Manuel Sandoval Vallarta", Universidad Aut6noma de San Luis Potosi,

San Luis Potosi, S-L-P. 78000, Mexico

(~) Department of Materials Science and Engineering, University of California, Los Angeles, CA 90024-1595, U-S-A-

(Received

2 August 1993, revised 13 December 1993, accepted 14

January1994)

Abstract We study the phase diagrams of polymer solutions using the pair approximation

of the Cluster Variational Method

(CVM).

In this model the solvent

(oligomers)

and the solute

(polymers)

are chains with different lengths. In particular, _we study the dependence of the phase diagram _on the lengths of the components ^{of the} mixture, _on the lattice, and

on the relative size of the components of the mixture. We also study the critical temperature and the critical concentration. Some comparison of _our results with the Flory and Guggenheim approximations

are done for the symmetric _case

(when

oligomer chains and polymer chains have the _same

length)

and the asymmetric _case

(when

oligomer chains and polymer chains have different

length).

We present _a novel technique to solve the set of equations derived from the CVM, and _an analytical

solution of the problem is given.

1 Introduction.

Since

Meyer iii proposed

the

original

idea to work out the

polymer

solutions in _a lattice

model,

_a

large

number of works based _on this idea have been

published.

One of the

major breakthroughs

is the mean-field treatment

by Flory

_[2] and

Huggins

_[3] which is

probably

the most

widely

used

theory

of

thermodynamics

of

polymer

solutions in the last

fifty

_years.

Flory's

approximation has _some

limitations,

and the most criticized points are the

following:

I) ^the correlation in the

occupational probabilities

_are

neglected; it)

the

counting

of the number of the effective units of other chains which _are nearest

neighbors

of _a considered

(inner)

unit of _a considered chain is

neglected

_so that the two

neighboring

lattice sites must be taken

by

neighbors belonging

to the _same

chain; iii)

for the evaluation of the entropy ^{of the chains}

most of the excluded volume effects _are

neglected. Huggins' approximation develops

in _an

independent

_way and at the _same time, different from

Flory's approximation,

corrects the points _I) and

it).

After the

original

formulation

by Flory (which

is

usally

called the

Flory- Huggins approximation),

there have been _many successful attempts to

improve Flory's

entropy

exprbssion

in such _{a way} that

Flory's approximation

is recovered in _some limit. Some of them

use the

original approach

of

Flory

[4, 6] and ^others use

sophisticated techniques including

the lattice

spin

field theories with _an

expansion

in

1/z,

_z

being

the lattice coordination

number,

in order to find corrections for finite _z [7].

Nowadays,

_{many new} and

powerful analytical

as well _as numerical

techniques

have been

developed

which

depart

from the

original

lattice model

approach

of

Flory

and

Huggins

[8, 11].

However,

the lattice models still

play

_an

important

role in

understanding

and

comprehension

of _some basic

phenomena

in the

polymer

solutions. These _new

techniques

and the traditional lattice model have

complementary

functions due _to the limitations in any one of these

theories,

as consequences of

approximations,

lack of

mobility

of the chains in

simulations,

and numerical difficulties _or

large

amount of CPU-time

required

in the _case of _very

long

chains in _some methods.

In this paper, we present _an

approach

for

studying polymer

solutions which is different from that used

by Flory

and

Huggins. Although

it is also based _{on a} lattice

model,

the present _paper

applies

the Cluster Variation Method

(CVM)

_[12] which _was

originally designed

for

alloys

and the

Ising

model. Our method

partially

corrects the limitations I) and

it) quoted above,

and

moreover the numerical output _can lead to _a simulation that takes into account the volume

exclusion effects [13]. ^{In this} paper we use the

pair approximation

_[12] with

neighboring

lattice

points

as the basic cluster to

study

the

equilibrium

structure of

polymer

solutions. There _are in the literature

only

_a few attempts to

apply

the CVM _to

polymer

solutions

[13, 14], probably

the _most elaborate of this kind of work is

by

Kurata et al. [14]. ^The pair

approximation

_we formulate in the present paper is somewhat different from Kurata's pair

approximation [14],

and is of the kind used

by

the authors elsewhere [13].

In this _{paper, we} present a novel

technique

to solve the set of equations ^{derived from the} CVM. This

technique

has the

advantage

that it leads to an

analytical

solution of the

problem.

We also describe the results of _a systematic calculation of

phase

separation

diagrams using

the CVM. We

study

in

particular

the

dependence

of the

phase diagrams

_on the

lengths

of the components ^of the

mixture,

_on the lattice used in this

model,

and _on the relative size of the components of the mixture. We derive the critical temperature, ^{the critical}

concentration,

and

briefly

comment _on the comparison ^of our results with others available in the literature.

In section 2 to section 4 _we present the model and the method of solution. The results and conclusions _are

presented

in section 5.

2. The model and the method.

In order to make it

possible

to

apply

the established entropy

expression

of lattice statistics,

we assume an

underlying crystal

lattice in which the

oligomer

chains and the

polymer

chains

are

placed.

The

oligomer

chains _as well _as the

polymer

chains _are made of segments, and each segment

occupies

a lattice

point.

In this

approximation

the coordination number _z is the

only

parameter needed to

specify

^the lattice. A twc-dimensional

representation

of the model is

given

in

figure

1, the white part

being

the

oligomer

chain and the black the

polymer

chain.

From here to the end of this _{paper, we} will call the shorter chain _an

oligomer

_or a solvent and the

larger

_one a

polymer, although

the difference in size may be small.

In order to define the

length

of the

oligomers,

or the

polymers,

_we

distinguish

the "end"

+ _~~ ~

~/~Q

Fig. i. Two-dimensional representation of the model. The white chains represent ^the oligomers

while the black chains represent the polymers.

segment and the "internal" segment ^for both

oligomers

and

polymers. Thus,

at each lattice site _we find either _an end segment

ii

₌

1),

_{or an} internal segment

ii

=

2)

of the

oligomers

and the

polymers.

To

distinguish

between the _two types ^of

chains,

_{we use a}

capital

letter

(A)

in the

probabilities

^and in the statistical

weight factors,

with values A

= for

oligomers

and A = 2 for

polymers.

An end segment ^{in this model is bonded} to _an

adjacent

segment in _one of the

z(a wiA) directions,

and _an internal segment is bonded to its two

neighboring

segments ⁱⁿ one of

z(z 1)/2(e

_w2 A) ways. These numbers _are called the statistical

weight

factors for the

single

site

probabilities

and _are listed in table I.

Including

the

bonding directions,

there _{are wi}

A different

"end

segments"

and _w2

A different "internal

segments". Therefore,

_our mathematical

problem

is to find the

equilibrium

distribution of _{wi1+w21+ wi}

2 +w2 ₂ different

species

over the lattice points, with the requirement that chemical

bondings

be

consistently

distributed.

Table I. Statistical

weight

factors _w,

A for the

single

^site

probabilities.

Since both _compc- nents in this mixture _are chains _{w, A} and _{w, B are} the _same.

~

l _z

2

z(z-1)/2

After the model is thus

defined,

_our

approach

is to write the Helmholtz free _energy F in terms of the state

variables,

which describe the state of the system, ^and then to minimize F to find the

equilibrium

distribution of the _state variables. The difference between the present work and _our

previous polymer

_{papers [13]} is that in this _paper both components ^{of the mixture}

are chains while in the

previous

work _a solvent molecule

occupies

a

single

lattice site. The pair

approximation

of the CVM used here considers two sets of state variables: _one is the set of

probability

variables for the

point configurations

_z,A, and the other is for _a

pair

of

nearest-neighboring

points,

y)j~~

_~. ^The energy E and the entropy ^S of _a system are written

as functions of these variables.

Then

_{the number of}

_species

can be

varied,

it is _more convenient to _use the

grand potential

Q rather than the Helmholtz free _energy F. The former is defined

~

Qllx, Al, lYli,j Bl)

₌

EllYli~,j

_B

I) Tsllx, Al, _lYli~,j Bl) N~PAPA

,

Ii)

where T is the absolute temperature, pA is the chemical

potential

of the A species and pA is the molar fraction _or concentration of the A~~

species.

The number of lattice

points

in _a system is written _as N

throughout

^this _paper. Note that the definition of F does not contain the last term in

equation ii).

The

equilibrium

state is derived

by minimizing

Q with respect to

(y)j~~ ~) keeping (pA)

fixed,

rather than the

composition (pA)

fixed. ^'

3. Variables.

3.1 DEFINITIONS. We let _{xi A} denote the

probability

of

finding,

_{on a} certain lattice

point,

an end segment ^of the A chain

(A

₌ 1

being

for

oligomer

chains and A

= 2 for

polymer chains)

with its

bonding pointing

toward _one of the _wiA

Possible

directions. This is

equivalent

to

saying

that the number of such end segments ^{of the}

oligomer (A

₌ 1) or the

polymer (A

₌

2)

in _a system is Nwi _{Axi A.} For _an internal segment ^{of the}

oligomer

_or the

polymer,

_we

similarly

define _z2A with the

weight

factor _w2

A. It is

important

to

point

out that due to the fact that both components ⁱⁿ the mixture _are chains the statistical

weigth

factors

satisfy

_{w, A}

" w, B for

A

#

B. This _means that the statistical

weight

factors for the solvent

(oligomers)

_monomers

are same as those for the solute

(polymers).

We next consider _a pair of

nearest-neighboring

lattice

points,

that is, the

probabilities

of

finding

the

configuration ii j)

inside the system. ^There are two types ^of

pairs.

_I) When the segments on these _two

points

_are

adjoining

segments ^of an

oligomer

_or of _a

polymer,

_we

call them the "connected"

pair

and the

probabilities

_are denoted

by _y)j~~

~ where the Greek

superscript

_o

(in

the

probabilities

and in the statistical

weight factori)

takes the value 1.

There _are

only

connected

pairs

between segments of the _same

kind,

that is

oligomer-oligomer

and

polymer-polymer.

This _means that

w)j

~ ~ ⁼ o for A

#

B. The

weight

factors

w)j

~ ~

(A

= 1,

2)

^{for these}

configurations

_are

countid

_{from the allowed directions of chemical}

boids

and _are listed in table _II.

ii)

When the segments on these two

points

are not

directly

bonded within the _same

oligomer

_or

polymer,

_we call them "non-connected"

pair

and denote the

probability by y)j~~

_~ ^where a takes the value 2. Their

weight

factors

w)j

~ ~ are listed in table III. In

figure'2

_we illustrate the different types ^of variables used in this

~nodel

as well _as the nomenclature.

Table II. Statistical

weight

factors

w)j,~~ (A

₌ 1,

2)

for the

connecting pairs.

The _con- nected

pairs

_are

only

for

oligomer-oligomer

and

polymer-polymer

_cases, and

w)j,~

_~ = o for A

#

B.

~(i)

~A,jA

I

( j

1 2

1 o _z 1

2

(z-1) (z-1)~

Table III. Statistical

weight

factors

w)I

~ ~ for

non-connecting pairs. They satisfy

_w)(~~

~(2) ~(2) ~(2)

^{' '}

12,j2~ il,j2~ 12,jl'

~iA,jB

(2)

I 2

lz

_1)~

lz i)~lz _2)/2

2

lz i)~lz 2)/2 lz i)~lz 2)~/4

2 2,2 2,1

~ ~ M _W-

W + p© _p©

a b

,

2,2 1,2 2,1 1,1

1,1

Q' _Q. Q0 _QQ

~~~ ,~'~ ^{0' 0~ 00}

~'~

" "

Fig. 2. Illustation of different types ^{of variables. There} are four types of basic _monomers; terminal

monomers and internal _monomers for oligomers and polymers _as is shown in

a).

In the _case of _a pair

there _are four basic types of connected pairs

(oligomer-oligomer

and polymer-polymer) _as illustrated in 16), and ten different types ^of nonconnected pairs _as is shown in (c).

Inspection

of tables II and III suggests ^{that the}

weight

factors in them _cal~ be written in

a condensed form when _we define the

"semi-weights", w)j

_with

o = 1 for sites associated with connected

pairs

and _o

= 2 for sites related with non-connected pairs, I

= 1 for _an

end segment ^{and I} = 2 for _an internal segment, and A

= 1 for

oligomers

and A

= 2 for

Table IV.

Semi-weight

^factors

w)(~.

^The

w)(~

^factors are

independent

of the A index.

I

~(l) ~(2)

IA IA

1

(z-1)

2

(z

₁₎

(z 1)(z 2) /2

polymers.

Due to the fact that both components ^{in the mixture} are chains, the

semi-weight

factors

satisfy w)j~

=

w)#

for A

#

B. This _means that the

semi-weight

^factors

depend only

on the

geometrical properties

of the lattice and _on the type of segment. The

w)j~

_are listed in table IV. Then

using

the

semi-weight

^factors we can write:

w)j~~

_~ =

w)j~ wjj

_except

_w)~j

~ ⁼ o and w)(~~_~ ⁼ w)(~~_~ ^{= o} _,

(2)

We also _see that _{we can} write _w,A in table I _as

~A

W)I

~

~~~

_~~~

3.2 REDUCTION RELATIONS. As is the standard

procedure

in the CVM

formulation,

_we

use

geometrical

relations to write cluster

probability

variables _as linear combinations of _a

larger

cluster variables. Thus z's _are written in terms of

y's

_as

GA

~ _Wj~

_Y~~,j

B' ~~~

j B

or

GA

fi _~

W)~,j

_B Y~~,j _B'

(5)

1 ~ ~~

3.3 CONSTRAINT _RELATIONS. When _we choose the

pair

variables _as the basis of formu-

lation,

these variables _are

subject

to several constraints.

3.3.1 Normalization. Since the x's and

y's

_are

probability variables, they

_are normalized to

unity

~ ~j _~~i,

j

BY~i,j

_B

(6)

iA jBa

When _we write the

grand potential

in the next section, _we have to add the normalization constraint

using

_a

Lagrange multiplier

I in the terms

3.3.2

Consistency

constraint. A

single

site

probability

_can be written in two different ways,

depending

_on whether _a bond is connected

(a

= 1) or not

(a

=

2)

_as we see in

equation (4).

Since the system is

isotropic,

a = and 2 expressions ^of _{x~ A} in

(4)

_are

equivalent:

~iA

~ _{~~~i Y~i,j}

B

~ _{~~~i Y~i,j}

B'

(8)

J B _j B

or

~ _~~~

Y~~,jB

(d~ d~)

°

,

(9)

j B _a

where

b[

is Kronecker's delta. In the minimization

procedure,

the

consistency

constraint that expresses

(8)

_can be written _as

~A

"

~ _Al

A

~ _Wj~

Y~~,j _B

(d~ d~)

,

(1°)

1A _j B _a

where

A,A Ii

₌ 1,2 and A =1,

2)

are the

Lagrange multipliers.

3.3.3 Chain

lengths.

In

minimizing

the

grand potential,

_we

specify

the average

length

of the

oligomer

chains in the system and that of the

polymers by

_means of

Lagrange multipliers.

The average

length

LA of the chains of the species ^{A is} defined _as the total number of segments,

N(wi

_{Axi A +w2 Ax2 A}

),

^divided

by

half the number of end segments, Nwi _Axi

A/2. Defining

the

following

parameter RA

la

_{w2 Ax2}

A/wi

_Axi

A),

^the _average

lengths

_can be written _as follows

LA ₌

2(RA

+

1). (11)

By controlling

the ratio

RA,

_{we can} control the chain

lengths.

In terms of the

single

^site

probabilities

and

using

the definiton of the parameter

RA,

_{we can} write _an equation

equivalent

to the _one for the

length LA

_as follows

~j

w, Ax, A

RAb) b)

₌ o.

(12)

1

Using Lagrange multipliers

rA and

equation is),

the

length

constraints _can be introduced in the

following

_way in terms of the

pair probabilities

Cr

₊

~j _rA l~j ^~j w)j~~ ~y)(~~ ~)(RAb) ^{b)) (13)}

, ,

A i jB~

Note that in

equation (13)

the

subscript

A

= 1 is for

oligomers

and A

= 2 for

polymers

and that I

= 1 is for _an end segment and I

= 2 for _an internal segment.

3.3.4

Composition

constraint. The concentration of

oligomers

^and

^polymers

are fixed

using

the chemical

potential

terms:

C~

=

~

_PAPA,

_(~~)

A

where _pA is the molar fraction of the

species

A and pA is the chemical

potential

of the species A. When there _{are no} vacancies in the system, the two

p's

_are not

independent,

but their

linear combination is _a constant. Thus _{we may} choose without loss of

generality

~j

_pA

"

0, (IS)

A

and hence _we write _{p e p2}

= -pi and call _p

simply

the "chemical

potential"

in the rest of the paper. In

analogy

with 3.3.1, 3.3.2 and 3.3.3, the chemical

potential

_can be

interpreted

_as

a

Lagrange multiplier

for the

composition.

4. Grand

potential

and its minimum.

4.I GRAND _POTENTIAL

IQ).

In view of

experiments

that show that

phase

_separation

occurs in the

oligomer-polymer

solution, _{we assume} that

oligomer

segments ^and

polymer

_seg-

ments

repel

each other when

they

sit next to each other

(as

the nearest

neighbors)

in the lattice. When segments ^{of A and} B

species

sit _on

adjacent

lattice

points,

_we ^define the _energy

parameters as

~~~

#

^~~~~

where J _> o. When _{we assume} there _{are no vacant} sites in the

lattice,

_{eA B are} the

only

_energy parameters we need. The energy for the total system _per lattice

point

is written _{as a sum} of the

nearest-neighbor energies

for the entire system as

Et ^~

~ _EABW)ij

~

Y)ij

B, (~~~

~

~AjBa ^{~ ~}

where the

1/2

factor is to avoid double

counting.

Two

things

_are to be noted in

equation (17).

(a)

the energy expression ^is exact based _on the

given

model and the

variables;

₁₆₎

only y)j

~ ~

appears because _we do not consider intra-chain

bonding

interactions. ^'

Different from the _energy, the entropy _per lattice point is written

only approximately

in terms of the

pair

variables. The CVM formula is [12, 13]

S = kB

I)

^{i +}

^iz

¹⁾

~ _W,A£ixiA) ~ _Witj ~£iyli~,j ~)

,

i18)

iA iAjBw

where kB is the Boltzmann constant and

£(v)

_represents the function _v In _{v v.} A reader who is familiar with the

quasi-chemical approximation

_{[4] or} the CVM will

recognize

the coefficient

in this expression. The

advantage

of the CVM is that the entropy expression _can be

improved systematically

when _we choose _a

larger

cluster _as the basis of the

formulation,

and the CVM expression is the most efficient

(for practical purposes)

for the chosen cluster.

When the

particle density

is _a variable parameter and is not

required

_to be

fixed,

_{as was}

pointed

out in section 2, it is convenient to mininize the

grand potential

^Q defined in

equation Ii)

while

keeping

the chemical

potential

_p

fixed,

rather than

minimizing

the Helmholtz free

energy. Q is written

explicitly

_as

Q=E-TS-C~+Cr+CA+C,. (19)

Note that the constraint terms C of

(7), (lo), (13),

and

(14)

_are included.

4.2 MINIMIZATION OF fl. The

equilibrium

state of the system ^{is found} as a minimum of

the

grand potential

Q for

given

values of the interaction

energies (eAB),

the temperature

(T),

the

length

parameters

(RA),

and _a

particularly

value of the chemical

potential (p)

_as follows

l~(

= o

(20)

dy~(~~

_~~~,T,R~,~

The differentiations with respect to

y)j~~

_~ ^{lead to the}

following

set of basic

equations:

y)j~~

_~ ⁼

y)j~exp(-eAB/kBT)yjj, (21)

where

y)j~

are to be called the

semi-pair probabilities

and _are defined _as follows:

y)j~

a (x~A)~~~~~~~exp

Iii

^{+ pA}

^{rA(RAb) b))}

⁺

^{(b( b()} _~fA ^{~~~ /zkBTj, (22)}

It is to be noted that

g)(~

are the

key

variables in _our formulation since

they

allow _us to formulate the

analytical

solution of the

problem.

The _{way to} write

equation (21) coming

from the minimization of Q is different from the _{common way to} write the CVM

equilibrium equations [12, 13],

except ^for a few

applications

[15].

Since z's _on the

right-hand

side of

(22)

_are functions of the

pair

variables y)j~,~ ~ in

(21),

the _next step is to solve

y)j~~

_~ ^from

(21), (22)

combined with the reduction

equations

in

(8)

and other constraint

relatiois. _Although

_{the set of}

_equations

_{is not}

_simple,

_it

cal~ be

proved

that the

semi-pair probabilities y)j~

can be solved

analytically

in _terms of

T,

_p,

J,

L, and the

statistical

weights.

The solution for the semi-pairs is

given

_as follows

Y)I

_" ^~~~

,

(23.a)

/~(l)~~

_~

^~(l)~

_~

2A IA ^I

Yji /w(1)

^~~~

~(l)

^{2A~2A W~~Xi A}

~~

'

(23,b)

Y~i

"

~fi,

j~~_~)

PA where

p(~

₌

~j w)jx,

A

(24)

~

and

pi(

~~ ~~~~~ ~~

~~~

jp(~

₊

pi~)H Q

PAB " ~~~~

2[1

HI

^'

where H _a

exp(-2J/kBT)

and

Q

_e

/4p(~p(~H

+

(p(~ pi~)~H2.

Details of the derivation

are

given

in

Appendix.

Equations

(22)

to

(25)

represent the

analytical

solution of the

problem.

When _{we use} the

equilibrium condition,

_{we can}

simplify

Q in

(19)

and we can show that the

Lagrange multiplier

for the normalization is the

equilibrium

value of the

grand potential

1 =

Q~~. (26)

5. Results.

5. I PHASE-SEPARATION DIAGRAM. The

phase diagram

calculation _can be done

by using

the intersection of the two branches of the Q _{us. p curve} [13), or

by using

the Maxwell construction [16) ^{in the}

graph

of _{p us. p.}

Considering

that the solution for the equilibrium state has been solved

analytically,

numerical

computation

is

greatly simplified.

Since the coexistence of the two

phases

p(~)

(the gas-like)

and p(~)

(the liquid-like)

is _a standard

technique

for the

calculation of the

phase diagrams

_we consider that _no

exp1al~ation

is _necessary [13).

5. 2 EFFECT _OF THE POLYMER LENGTH IN THE PHASE SEPARATION DIAGRAM. To

StUdy

this _we fix the

oligomer length

to Li ₌ 5 and

change

the

polymer length

L2. The results _are

presented

in

figure

3 for L2 ₌

5, So,

soo in the _case of _a

simple

cubic lattice _z

= 6. Let _us call

the maximum of the

phase separation diagram

the critical

point.

The critical concentration p~ and the critical temperature T~ _are defined for this

point.

The

phase separation diagrams

are more

asymmetric

_as the

polymer length

increases. We _{can see} that _p~ shifts to the lower

polymer

concentration _as the

polymer

becomes

longer,

_p~

approaching

_zero for the infinite

length

[2, 17]. ^{It is clear that the} asymmetry ^{in the}

phase diagrams

is due to the

large

^difference

in the molecular size of the two components. ^{Our results} are in

qualitative

agreement with

Flory's

calculations [2, 17] as well _as with other calculations

using

^{the CVM} [13], as

expected;

in all

previous calculations, however,

the shorter

polymer

is treated _{as a}

single

segment

occupying

one lattice

point.

The

(kBT~/zJLi)

values

corresponding

to

figure

3 _are o.35, ^{o.79 and} 1.13 for

(L2/Li)

=1, lo, and loo

respectively.

o-o

Fig. 3. Phase separation diagrams calculated by the present CVM theory based _on the simple cubic lattice

(z

= 6) and for different polymer lengths. The short dashed line is for L2 ₌ 5, the long dashed line is for L2

" 50 and the solid line is for L2

" 500. All of them

are for the fixed oligomer length

(Li

" 5).

5. 3 EFFECT OF THE COORDINATION NUMBER IN THE PHASE SEPARATION DIAGRAM. In

this

model,

in order to make it

possible

to

apply

the established entropy

expression

of lattice statistics, _we have to assume an

underlying crystal

lattice in which

oligomer

segments and

polymer

segments are

placed.

Since the lattice is

hypothetical,

it will be desirable that the results be

independent

of the coordination number

(z),

however in _some of the _cases the lattice has _some influence _on the results [3,

4,

_{7, 13].} To

study

the influence of the coordination

number,

_we calculate the

phase diagrams

for different _z values _as shown in

figure

4, in which

(a)

is for the

simple

cubic lattice

(z

=

6), (b)

is for the

body

center cubic

(z

=

8)

^and

(c)

^{is for}

face center cubic

(z

=

12).

The results in

figure

4 are for _a mixture with

(L2/Li)

_" lo with Li ₌ 5. From

figure

4, we can say that the

shape

of the

phase diagram

in _our

ajproximation

is

practically independent

of _z when the temperature axis is scaled

by

_T~. We _{can say} that when the

phase separation diagrams

_are scaled _to T~

Flory's approximation

_{[2, 17]} which is

independent

of _z is _a

fairly good approximation.

The

(kBT~/zJLi)

values obtained with _our model for _a mixture with

(L2/Li)

= lo and

Li

_" 5 _are

0.79,

0.88 and 0.97

respectively,

for

z = 6, 8 and12.

IQ) 16)

ic)

oo lo

fi

Fig. 4. Dependence of the phase separation diagram _on the coordination number _z for _a mixture with the oligomer length Li _" 5 and polymer length L2 ₌ 50, with the T axis normalized by the

critical temperature Tc. The simple cubic is presented by

la),

the bcc

by16)

and fcc by

(c).

5.4 EFFECT OF THE SOLVENT SIZE IN THE PHASE SEPARATION DIAGRAM. Now _we fix

the coordination number _{z =} 6 and the ratio

(L2/Li

" 10 for three systems with different

solvent

(oligomer)

sizes Li ₌ 5, lo and 20. The results _are

given

in

figure

5 for these three

cases. The results for the

phase diagrams

scaled to T~ indicate that, _as the solvent

(oligomer)

size is

increased,

the

phase diagrams

become

slightly

_narrower. The scaled

phase diagrams

_are

almost

independent

of the solvent size and the

important

parameter seems to be the relative size between the

polymer

and the solvent

(L2/Li).

The

(kBT~/zJLi)

values obtained _are

0.79,

0.78 and 0.77 for Li

" 5,

10,

and 20

respectively.

^Since the

(kBT~/zJLi)

ratio is almost

constant the results suggest that

(kBT~/zJ)

is

approximately

_a linear function of

Li

when _we fix the

(L2/Li)

ratio _as in the present case.

5.5 CRITICAL TEMPERATURE BEHAVIOR. In order to examine the critical temperatUre

as a funtion of the

polymer length,

_we consider two cases:

a)

_we fix the

oligomer length

and

change

the coordination number;

b)

_we fix the coordination number and

change

the

oligomer length.

The results for these two _{cases are} shown in

figure

6. The results for

a)

indicate that _as the ratio

(L2/Li)

increases the critical temperature

(kBT~/J)

increases,

being

_very

fast for low values of

(L2/Li)

and

levelling-off

for

large

values of the ratio

(L2/Li). They

also indicate that _as the coordination number _z increases the critical temperature increases.

Since the calculation for _an infinite

polymer length

is time

consuming,

_{we can}

extrapolate

in

2

la) ib) ic)

o-o i-o

p

Fig. 5. Phase separation diagrams for three systems ^{with different} component lengths _{on a} simple

cubic lattice. The ratio

(L2/Li

has been fixed in the three _cases to 10.

la)

is for the polymer length

L2 " 50, (b) for 100, and (c) for 200.

order to calculate the saturation value. For the

exptrapolation

of the temperature we can take

advantage

of

Flory's analytical

formula in the _case of the

point

solvent for the temperature

(kT~/zJ)

=

2L/(1+ li)~

[2, 17]. ^The temperature ^for _very

long polymers

_can be written

approximately

_as

(kT~/zJ)

_m

(kT~/zJ)~ 411i,

_where

_(kT~/zJ)~

_{means the} temperature for _an infinite

length polymer.

In _our case, in

analogy

with

Flory's analytical expression,

_we

assume the

following

form for the

extrapolation

of the temperatures

(kBT~/zJLi)

_"

(kBT~/zJLi)

^~'~

(27)

" "

fi

^'

where the

subscript

_{n means} the value of the ratio

(L2/Li)

and A is _a parameter to be calculated

together

^with (kBT~

/JLI _)m

The temperature

T~(oc)

for _an

infinitely long polymer

is

usally

called the 8 temperature. ^{The results of}

(kBT~/JLI)m

^for _an

oligomer length

of Li ₌ 4 _are 4.24, 8.25, 12.25 and 20.25 for _z

= 4, 6, 8 and 12,

respectively.

For the _case

b),

in which _we fix _z and examine the temperature

dependences

for two

oligomer lengths

Li ₌ 4 and 5. The results _are shown in

figure

6b. In _a similar _{way as} in the

a)

_case,

the critical temperature

(kBT~/J)

increases, _very fast for low values of

(L2/Li)

and levels off for

large

^values. The results also indicate that _as the

oligomer length (Li

increases the critical

temperature ^{increases. It is worth}

noting

that the results scale

by

a factor of

5/4,

that is, the ratio of the two

oligomer lengths.

The critical temperatures

(kBT~/JLI) extrapolated

to the infinite

length polymer using equation (27) give

the value 8.22 for both

oligomer lengths.

In

Flory's theory

the 8 temperature

plays

_an important ^{role. The 8} temperature ^{has the}

following

two characteristics:

ii)

_at

8,

the second virial coefficient vanishes and therefore the effect of the excluded volume

disappears, making

the

polymer

behave _{as a} random

walk; iii)

8

corresponds

to the critical temperature of the

phase separation diagram

for _an

infinitely long polymer. Using

the

extrapolated

value of

(kBT~/JLI)m,

_{we can} calculate

Flory's

parameter

x~(e JzLi /kB8).

In his

theory

_x~ is

always 1/2

for _any coordination number. Our results

presented

in

figure

⁷

(square marks)

indicate that _x~ deviate from

1/2

for _any finite coordination

Z=12

<

Z=8

@

m

~ Z=6

(a)

0.0

O loo 200

L2/Li

Li = 5

~

'

f ~l

" '

r~

(b)

O.0

O 250 SOD

L2/Li

Fig. 6. The critical temperature _as a function of

(L2/Li)

for three different lattices in la) and for two different oligomer lengths in 16).

number. In the _same

figure,

_we alto present results obtained when the solvent is _a

point particle (circle marks)

_{[13] as} well _as the results obtained

by

Saleur

(triangle)

_[18] in the _case of _{a square}

lattice. When _{we compare our} present calculations in which _we consider that the solvent has

a structure with the calculation where the solvent is _a point

particle

_{[13], we} find that x~ is

larger

when the solvent molecule has _{a structure.}

5.6 CRITICAL CONCENTRATION. The behavior of the critical concentration _p~ is illus-

trated in

figure

8. In the _same

figure Flory's

calculations _are

presented

for comparison. A mean-field

theory

like

Flory's

has the

advantage

of

simplicity,

and _even the

possibility

to de-

, I

I I

',

'

O

~K

~i

0. _A ", Q

',,

'O~,

""'-4X

___

2 4 6 8 lo 12 14 16 18 20

Coordination Number

Fig. 7. The critical _xc

(+

JzLi

/kBB)

parameter ^{for different} coordination numbers and for different

approximations. The results of this work _are in squares. Circles _are the results when the solvent is _a

point particle _[13], and _a triangle is for _{a square} lattice by Saleur [18]. ^{The continuous} curves are aids for the _eye.

°.

Flory

~

cvm Ct

o-o

i io ioo

L2/Ll

Fig. 8. The critical concentration _{pc as a} function of the normalized length

(L2/Li).

The solid line presents ^{the results of this work and the dashed line} Flory's results.

rive _an

analytical

expression for the _{p~ as} is

given

ⁱⁿ references [2] and [17]

where the solvent is considered _{as a}

single point

and L is the

polymer length.

The results show that _p~

begins

at the value 0.5 for the

symmetric

_case

(Li

"

L2).

As the difference

in size between the

polymer

and the

oligomer increases,

_p~ decreases and _goes to zero in the

limit when the ratio

(L2/Li)

_goes to

infinity.

In

general

_our results for _{p~ are} in

qualitative

agreement ^with

Flory's

calculations. It may be noted that the results

presented

here show that the

phase diagrams

calculated

by Flory's approximation

_are

slightly

_more

asymmetric

than

our

phase diagrams.

5. 7 PHASE-SEPARATION DIAGRAM FOR POINT SOLVENT. Now _we compare the

phase

dia-

grams calculated here

using

the structural solvent

(CVM-II)

with the other calculation where the solvent is _a

point particle (CVM-1)

_[13]. Both calculations _were

performed using

the CVM

pair approximation

_{on a}

simple

cubic lattice. The results _are

presented

in

figure

9 for two

different

lengths

_cases

(L2/Li)

_" 12 and 102. The dashed _{curves are} for Li

= 1 and solid

curve for Li ₌ 4. The

comparison

indicates that the solvent structure increases asymmetry

of the

phase diagram

and that it is

slightly

narrower in the _case of

long polymers.

For the comparison of the critical temperature, we calculate (kBT~

/JLI ),

which _are 4.975 and 6.825 for

(L2/Li )=

12 and 102,

respectively.

The

correponding

results for the

point

solvent _are 5.4 and 7.375. The temperature ^decreases to about 92% in both _cases. This decrease is in agreement with the

generally acepted

_property that the critical temperature decreases _as the statistical

description

of the system is

improved.

.2

(a) (b)

L~/Li

₌ 12

L~/Li

= l02

o-o i-o o-o i-o

p p

Fig. 9. Comparison between the two different CVM approaches. The solid

curves represent ^the

result of this work, and the dashed _{curves are} when the solvent is considered _a point particle _[13].

5.8 SYMMETRIC OLIGOMER-POLYMER MIXTURE. This _case is the

simplest

_one to

Study

among mixtures. The

phase diagrams

_are

symmetric

around the concentration 0.5. In order to compare our results with those

by Flory

_{[2, 17]} and

Guggenheim

_{[4], we} calculate the ratio

(zJL/kBT~),

where L denotes the

polymer length,

_or the

oligomer length.

This ratio in

Flory's

approximation ^{is 2 and is}

independent

of the lattice. In the _case of the

Guggenheim approximation

this ratio is

given by

(zJL/kBT~)

= z L

In(1

^~

,

(29)

JOURNAL DE PHhS'QUE '< -T 4 N'4 APR<L 1994