LITERATURE REVIEW
2.1 Polymer-Polymer Miscibility
2.1.1 Thermodynamics of Polymer-Polymer Miscibility
The state of miscibility of any mixture is governed by the free energy of mixing, ∆Gmix, which is defined as:
∆Gmix =∆Hmix −T∆Smix (1.0)
Where ∆Hmix is the enthalpy of mixing, T is the absolute temperature, and
∆Smix is the entropy of mixing.
The thermodynamic driving force for mixing is minimisation of ∆Gmix. Thus if the free energy is positive, the system is immiscible. While for small molecules the entropy is high enough to ensure miscibility, for polymers the entropy is almost zero, causing enthalpy to be decisive in determining miscibility. For spontaneous mixing, ∆Gmix must be negative, and so
∆Hmix −T∆Smix <0 (1.1)
Tg shifted lower Tg shifted lower
For complete miscibility to occur, a negative free energy of mixing is necessary but not sufficient. Figure 2.7 shows that ∆Gmix for a binary mixture can vary with composition in several ways [28]:
a.) Complete immiscibility (Curve A) exists if ∆Gmix is positive b.) Complete miscibility (Curve B) exists only if
∆Gmix<0 (1.2)
and that the second derivative of ∆Gmix with respect to the volume fraction of either component must be greater than zero over the whole composition
Figure 2.7 Possible free energy of mixing diagram for binary mixtures [28]
c.) Curve C represents a system that is partially miscible as it only satisfies equation (1.2) but not the derivative criterion expressed by equation (1.3) at all points along the ∆Gmix-composition curve since it passes through points of inflexion defined by
0
at points X and X’ in Figure 2.7. Between these spinodal points the system will phase separate spontaneously into bimodal compositions Y and Y’, with a decrease in free energy.
For two-component blends it is possible to construct a phase diagram, which may exhibit upper or lower critical solution temperature (UCST or LCST) as shown in Figure 2.8.
Generally, UCST behaviour is characteristic of systems which mix endothermically while LCST behaviour is a characteristic of exothermic mixing (which could arise from specific chemical interactions) and associated with entropy effects.
For low molecular weight materials, increasing temperature generally leads to increasing miscibility as the T∆Smix term increase, thus driving ∆Gmix to more negative values. Thus liquid-liquid and polymer-solvent mixtures usually exhibit UCST.
LCST behaviour is more commonly observed in polymer blends as phase separation occurs when temperature increases because the intermolecular attractive forces responsible for the miscibility behaviour tend to disappear as the internal energy of the molecules becomes high enough to overcome them.
Figure 2.8 Schematic phase diagram for a system exhibiting both UCST and LCST behaviour [29]
Mathematical models of polymer solutions like Flory-Huggins model [30] are useful for understanding how various factors can affect polymer solubility. The Flory-Huggins model uses combinatorial analysis to estimate the increase in configurations available to the system when a flexible polymer in a disordered state is mixed with solvent.
LCST
The entropy of mixing, ∆Smix, arising from the increased number of ways of arranging the polymer and solvent molecules in the solution is given by
∆Smix=−R
(
n1lnφ1+n2lnφ2)
(1.5)where n1 and n2 are mole fractions of solvent and solute; φ1 and φ2 are their volume fractions; and R is the gas constant. In almost all polymer solutions the increase in entropy is the driving force for the mixing process.
An expression for the enthalpy of mixing, ∆Hmix, can be obtained by considering the change in adjacent neighbour (molecules or segments) interactions on the lattice upon mixing:
∆Hmix =RTχ12n1φ2 (1.6)
where
χ
12 is the Flory-Huggins interaction parameter.Thermodynamically,
χ
12 is one of the key parameters to determine the miscibility of polymer blends and it may be shown that the Flory-Huggins parameter and solubility parameters are related by
( )
RT Vr 1 2 2
12
δ
χ = δ − (1.7)
where δi are the solubility parameters of two homopolymers or copolymers, R is the gas constant, T is the absolute temperature, and Vr is a reference volume, taken as 100 cm3 for polymers.
The interaction parameter
χ
12 is a useful measure of the solvent power [31]. It has been shown both theoretically and experimentally that aχ
12 value of about 0.55 is the dividing line between poor solvents and non-solvents [32].The region of poor solvency extends from about 0.31 to 0.55. Values of
χ
12less than 0.30 indicate good solvents. In general, the smaller the
χ
, thestronger the polymer-diluent interaction and consequently the better the solvent.
The Hildebrand solubility parameters of homopolymers can be calculated using
M
∑Fi
=ρ
δ (1.8)
where δ is the solubility parameter of the polymer, ΣFi is the sum of the molar attraction constants of all the groups in the repeat group of the polymer, M is the molecular weight of the repeat group, and ρ is the density of the polymer at the temperature of interest.
The solubility parameters of random copolymers can be calculated using
δc =∑δiφi (1.9)
where δc is the solubility parameter of the copolymer, δi is the solubility parameter of the homopolymer corresponding to repeat group i, and φi is the volume fraction of repeat group i in the copolymer.
Substitution of the expression for entropy in equation (1.5) and enthalpy in equation (1.6) into the expression for free energy of mixing in equation (1.0) yields the well-known Flory Huggins expression for the Gibbs free energy of mixing
∆Gmix =RT
(
n1lnφ1+n2lnφ2+χ12n1φ2)
(2.0)From equation (2.0), the smaller that
χ
12 is, the more stable is the solution relative to the pure components and the more likely that the system is miscible over a wide range of concentrations. For most systemsχ
12 decreases with increasing temperature and increases with increasing concentration ofpolymer. In dilute solutions, the polymer molecules are isolated from each other by regions of pure solvent, i.e., the polymer segments are not uniformly distributed in the lattice. In view of this, the Flory-Huggins theory is least satisfactory for dilute polymer solutions and only applies to concentrated solutions or mixtures.
The majority of polymer pairs exhibit an endothermic heat of mixing when blended together, which does not favour the formation of a single amorphous phase. Values of ∆Hmix which indicate whether pairs may be miscible, can be estimated for nonpolar components by using the Hildebrand approach to regular solutions which introduces the concept of the solubility parameter δ. Demixing of liquids is attributed to the tendency of molecules to attract their own species more strongly than a dissimilar species. This idea is expressed quantitatively in the equation
∆Hmix =Vmixφ1φ2
(
δ1 −δ2)
2 (2.1)where Vmix is the molar volume of the mixture.
Chemically similar molecules are in most cases found to have similar solubility parameters, and thus a reduced tendency to demix. Therefore, miscibility or solubility will be predicted if the absolute value of the (δ1 - δ2) difference is zero or small (less than 2 MPa1/2). Specific effects such as hydrogen bonding and charge transfer interactions can lead to negative ∆Hmix but these are not taken into account by equation (2.1) since the right hand side of the equation cannot be negative, the solubility parameter method does not comprehend exothermic mixing. Therefore, to improve the prediction of polymer miscibility, a three-dimensional solubility parameter which gives individual contributions for dispersive (i.e. van der Waals), polar, and hydrogen bonding interactions as proposed by Hansen is sometimes used.
The overall solubility parameter is then the sum of the various contributions miscibility) is the distance between the two components of the blend in three-dimensional space using the three components of the solubility parameter as coordinates. Mathematically, it is represented by δm and is defined as
( ) ( ) (
,1 ,2)
2Huang et al [33] applied the three-dimensional solubility parameter method to predict the miscibility trend of PVC/homopolymer and PVC/copolymer blends.
A homopolymer system was reported to be miscible when δm was 11 but for copolymers the miscibility turned semi-miscible when δm was 5.8. They concluded that the smaller distance between the solubility parameter of copolymer and that of PVC indicate better chances of miscibility due to the ability to adjust the solubility parameter through the copolymer composition.