Miscibility.pdf

Vol. 7

MISCIBILITY 131

MISCIBILITY

Introduction

The phase behavior of polymers, whether in solution or in mixtures with other

polymeric components, differs considerably from that of small molecules. This dif-

ference is a direct consequence of the large size of polymer molecules. For mixtures

of small molecules the driving force for miscibility is a result of the large gain in

entropy that takes place on mixing, but this is not the case for polymers. The

entropy of mixing is particularly small for polymer/polymer mixtures and, as a

result, one-phase systems are obtained only in a limited number of cases.

Here, the term miscibility is used to describe a mixture containing two or

more components that form a one-phase (solid or liquid) system. Whereas this def-

inition is unequivocal as it corresponds to a precise thermodynamic description of

the system, in practice for polymers the experimental determination of miscibil-

ity may be rather ambiguous. This is because the latter depends on the speciﬁc

experimental technique adopted and, for example, while a system may appear

one-phase if examined at sufﬁciently large length scales, it may not correspond to

true miscibility at the molecular level.

One of the primary tasks in the past few decades in polymer science has

been to control the structure and properties of multicomponent systems. Since

the properties of multicomponent systems depend on their structure, the control

and design of these structures is fundamental to produce novel properties. Phase

separation and spinodal decomposition are used to design multiphase structures.

To understand the fundamentals of these phenomena it is necessary to under-

stand thermodynamics, phase transitions, (qv) and critical phenomena in polymer

blends (qv) and be able to evaluate quantitatively the degree of miscibility between

the polymeric blend components.

132 MISCIBILITY

Vol. 7

Thermodynamics of Solutions of Small Molecules

The necessary condition for two substances to mix is that the Gibbs free energy of

the solution, G 12 , is lower than the sum of the Gibbs free energies of the separate

constituents, G 1 and G 2 . In other words, for a homogeneous solution to form,

the Gibbs free energy of mixing,

G m =

G 12 −

( G 1 +

G m , which is

related to enthalpic

H m and entropic

S m components through the relationship

G m

H m

−

S m

(1)

0 (it will be noted later that this is a

necessary though not sufﬁcient condition for thermodynamic miscibility).

Since

S m is related to the number of “distinguishable” arrangements avail-

able to a system, it is a positive quantity; mixing always increases the disorder of

the system. This is particularly evident for low molecular weight materials, where

the large entropic change occurring on mixing provides the driving force to misci-

bility. For such systems,

G m becomes more negative with increasing temperature

and this favors miscibility.

The ﬁrst term on the right-hand side of equation 1,

H m , provides a measure

H m is negative if there are

attractive interactions between the components, as this decreases

G m . Obviously,

G m increases if

H m is positive (repulsive interactions). For an ideal solution

0 and this condition is satisﬁed when the intermolecular forces between

both like and unlike molecules (ie the two components) are similar.

For an ideal mixture of small molecules, consisting of n 1 moles of component

1 and n 2 moles of component 2, the lattice model (Fig. 1a) can be used to deter-

mine the entropy of mixing. This is based on various assumptions: ( 1 ) solvent and

solute molecules are identical in size and shape and each occupies a lattice site;

( 2 ) mixing is a random process since all possible arrangements are equi-energetic

(ie

0); ( 3 ) no volume changes take place on mixing. By using this simpli-

ﬁed model, it is possible to determine the number of possible “distinguishable”

arrangements and therefore the entropy change for an ideal solution

S m

nR =−

[ x 1 ln x 1 +

x 2 ln x 2 ]

(2)

n 2 ), and R is the gas constant. Since the model assumes that

the heat of mixing is zero ( athermal mixing ), the free energy of mixing of an ideal

solution is simply

n 1 +

G m

nRT =

[ x 1 ln x 1 +

x 2 ln x 2 ]

(3)

For the ideal solution, mixing is a spontaneous process that is entropically

driven.

Real solutions may deviate from ideality, particularly since the condition

of athermal mixing is not usually encountered. The model can be modiﬁed to

G 2 ), should be less than 0.

Therefore miscibility is governed by the Gibbs free energy of mixing

It follows that the necessary condition for a mixture of two components to be

miscible at a temperature T is that ( G m <

of the extent of interactions between molecules.

H m =

where x 1 and x 2 are the mole fractions of the two components, n is the total number

of moles ( n

Vol. 7

MISCIBILITY 133

( a )

( c )

( b )

Fig. 1. Schematic representation of the lattice model for the determination of the combi-

natorial entropy of mixing: ( a ) mixture of small molecules; ( b ) polymer chain in a solvent;

and ( c ) polymer/polymer mixture.

account for intermolecular interactions, which, within the lattice model, are of

the ﬁrst-neighbour-type. For a binary system, three kinds of interactions need to

be considered: [1, 1], [2, 2], and [1, 2] contacts; and so the process of dissolution is

given by

2 [1

2 [2

→

(4)

A contact exchange energy,

W , can be deﬁned as

W 12 = ε 12 − ε 11

2 − ε 22

(5)

In absence of special attractive forces such as those due to hydrogen bonding

and complex formation, and assuming random mixing, the enthalpic term can

be expressed as the product between the number of [1, 2] contacts in solution

and

W 12

H m =

nzN A

W 12 x 1 x 2

(6)

where z is the lattice coordination number and N A is Avogadro’s number. Equation

6 is often written in terms of the van Laar dimensionless segment–segment

from the pair interaction energies

134 MISCIBILITY

Vol. 7

interaction parameter

χ 12 :

χ 12 =

W 12

(7)

and so

H m =

χ 12 x 1 x 2

(8)

The modiﬁcation introduced above leads to the free energy of mixing of a

regular solution

G m

nRT

[ x 1 ln x 1 +

x 2 ln x 2 ]

+ χ 12 x 1 x 2

(9)

This equation, ie the van Laar model of solvent mixtures, can be considered

the predecessor of the Flory–Huggins theory but, as explained later, it can be

applied only to mixtures of small molecules.

As the temperature is decreased the interaction parameter becomes larger.

This affects the free energy of mixing (

G m ) versus composition curves as shown

in Figure 2. For

χ 12 values less than 2,

G m is negative and the composition de-

2.0 the curve changes shape. In this case, for

compositions between the two points A and D, the system can lower the free en-

ergy by separating into two phases. As the temperature decreases, the interaction

parameter becomes increasingly positive and this leads to positive

χ 12 >

G m values,

which make the system immiscible.

0.4

0.3

0.2

0.1

− 0.1

−

0.2

− 0.3

− 0.4

0.0

0.2

0.4

0.6

0.8

1.0

X 2

Fig. 2. Composition dependence of the free energy of mixing calculated for a regular

solution. Different values of the interaction parameter

χ 12 have been used to generate the

curves a (

χ 12 =

4),b(

χ 12 =

2.3), and c (

χ 12 =

1.6).

pendence is negative, while for

Vol. 7

MISCIBILITY 135

The interaction parameter

χ 12 is often associated with Hildebrand’s solu-

δ i , which is in turn related to the cohesive energy density (CED)

through the relationship (1)

δ i

CED i

E vap

(10)

where

E i vap

is the energy of vaporization of component i . The relationship be-

tween

χ 12 and

δ i is given by

χ 12 =

V m (

δ 1 − δ 2 ) 2 RT

(11)

χ 12 is necessarily a positive quantity. In practice, negative

χ 12 can be obtained when, because of speciﬁc interaction, [1, 2] contacts are fa-

vored compared to contacts between like molecules. Thus the solubility parameter

concept is only applicable to systems involving dispersive interactions and fails

for those involving hydrogen bonds, donor–acceptor interactions, etc.

Thermodynamics of Polymer Solutions

S m to deviate from the ideal case (eq. 3).

When dealing with polymer solutions where one of the components is much larger

than the other, signiﬁcant changes need to be introduced in the treatment of binary

liquid–liquid mixtures (2).

The lattice model is still applicable and so are some of the assumptions made

for simple liquids. As shown in Figure 1b, in order to position a polymer chain in the

lattice, it can be divided into a number of segments, each equal in size to a solvent

molecule and therefore each occupying a lattice site. Positioning of these subunits

cannot be considered a random event; because of chain connectivity, two consec-

utive segments in the chain have to be placed in adjacent sites. The schematic

diagram of Figure 1b clearly indicates that, for polymer solutions, one expects a

reduction of the entropy of mixing compared to solutions of small molecules.

The calculation of the entropy of mixing,

0, chain connectivity causes

S m , of polymers in solution was

carried out by Flory (3) and Huggins (4) in 1942. The Flory–Huggins (FH) theory,

based on the rigid lattice model developed for solutions of small molecules, leads

to the following expression for the free energy of mixing:

G m

[ n 1 ln

n 2 ln

2 ]

n 1

(12)

where n i represents the number of moles of component i . The FH equation can be

written in alternative forms, and, particularly useful, is the expression in terms

bility parameter

V i

V m being the molar volume. As expressed in terms of the solubility parameter, the

interaction parameter

Polymer solutions show deviations from the ideal behavior described in the previ-

ous section. In addition to intermolecular polymer–solvent interactions that make

H m

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