DG = DH - TDS |
[1] |
|---|
Being DH, the change in enthalpy of mixing, T the absolute temperature in the process, and DS the change in entropy of mixing. According to equation [1], and from the thermodynamic point of view, the dissolution will only take place if DG sign is negative. DS is usually positive, since in solution, the molecules display a more chaotic arrangement than in the solid state, and on the other hand, the absolute temperature must be also positive. However, DH may be either positive or negative. According to Rosen (1982), a positive DH indicates that both the polymer and the solvent are in their lower energy state, whereas a negative value suggests the solution is in its lower energy state, showing interactions such as hydrogen bonding, which are established between polymer and solvent molecules.
When a polymer solution is formed, DS is significantly small. This contrasts the found values when mixing masses or equivalent volumes of two low molecular weight liquid substances. Qualitatively, this property can be explained by means of a reticular model as shown in Figure 2.
Figure 2. Two-dimensional lattice model of solubility for a low molecular weight solute
To better understand this behavior, let�s assume that the black circles are solute molecules, and the light blue circles are solvent molecules. When we are dealing with low molecular weight solutes, their molecules can be distributed randomly within the lattice, provided that two or more molecules do not occupy the same site at the same time. Therefore, the configurational entropy of mixing, given by the Boltzmann equation, is:
DS = k ln W |
[2] |
|---|
Where W the number of possible arrangements within the lattice, and k the Boltzmann constant. With low molecular weight solutes, W is large, consequently the entropy value will be high. Nevertheless, a different situation arises when the solute molecules are part of a polymer. In such a case, these molecules must be considered as a large number of segments of identical length, bonded each other along a flexible chain.
At this point, how can we determine the possible arrangements within the lattice? To find the answer, let's consider the solution concentrated enough so that the chains are located randomly within the lattice, instead of forming isolated zones. If we began with an end of the macromolecule, we will notice that there are eight adjacent sites where the next segment can be placed.
Figure 3. Two-dimensional lattice model of solubility for a polymer solute
If the restriction that a site in the lattice cannot be occupied by two or more molecules is maintained, the location of the third segment will fall in anyone of the seven adjacent sites, and so on. The number of configurational possibilities will be lower than that of the non-polymer molecules, because a restriction concerning the distribution of the chains within the lattice has now been imposed. The term W in equation [2] is then, smaller, and therefore the change in entropy is also smaller.
This reticular model was developed separately in the early 40s, by P.J. Flory and M.L. Huggins. It explains the low entropy of mixing in polymer solutions. Nevertheless, although this model was useful for an approximate calculation of the term W, the theoretical Flory-Huggins model did not describe accurately the behavior of real polymer molecules in solution, particularly, a dilute solution. For that reason in 1950, Flory and Krigbaum developed a new model, taking into account the properties of dilute solutions, which consist in alternate regions of pure solvent and solvated polymer domains. The Flory-Krigbaum theory introduces two important concepts, which will be discussed later: the q temperature and the excluded volume.
Now let's go back to the values of change in entropy for macromolecular solutions, and the feasibility that a polymer can be dissolved in a given solvent. Since the term TDS in equation [1] is small, DH has to be small too. It must be even smaller than TDS in order to obtain a negative DG, and therefore, make the polymer be soluble. For nonpolar macromolecules that do not have specific interactions with the solvent, DH is positive and has almost the same entalphy of mixing value to that for small molecules. DH is given by a equation developed by Hildebrand:
DH = fs fp (ds - dp)2 |
[3] |
|---|
Where fs and fp are the volume fractions of solvent and polymer, respectively, whereas ds and dp represent the cohesive energy density (CED) for solvent and polymer, respectively. This magnitude is a measure of the strength of the intermolecular forces keeping the molecules together in the liquid state, and it is known commonly with the name of solubility parameter. Its units are (cal/cm3)1/2, and the equivalences to the SI units are the following:
The solubility parameters are particularly useful when studying how capable is a polymer to being dissolved in a given solvent. However, it should be pointed out that equation [3] is valid only for solutions where strong polymer-solvent interactions do not take place. Numerous tables showing solubility parameters for both solvent and polymers have been published. Some examples are detailed below.
Solvent |
ds (MPa1/2) |
Polymer |
dp (MPa1/2) |
|---|---|---|---|
Acetone |
20.3 |
Polybutadiene |
14.6-17.6 |
Benzene |
18.8 |
Polychloroprene |
15.2-19.2 |
Carbon Tetrachloride |
17.6 |
Polyethylene |
15.8-18.0 |
Chloroform |
19.0 |
Polyisobutylene |
14.5-16.5 |
Cyclohexane |
16.8 |
Polypropylene |
18.9-19.2 |
Ethanol |
26.0 |
Polyacrylonitrile |
25.3-31.5 |
n-Hexane |
14.9 |
Polymethylmethacrylate |
18.4-26.3 |
Methanol |
29.7 |
Polyvinyl acetate |
18.0-19.1 |
Methylene Chloride |
19.8 |
Polyvinyl alcohol |
25.8 |
n-Pentane |
14.3 |
Polyvinyl chloride |
19.2-22.1 |
Toluene |
18.2 |
Polystyrene |
17.4-21.1 |
Water |
47.9 |
Nylon 6.6 |
27.8 |
Table 1. Solubility parameters for solvents and polymers more commonly used. (Taken from "Polymer Handbook" / J. Brandrup and E.H. Immergut, Eds., 3rd Ed., Wiley-Interscience,
New York, 1989)
In absence of specific polymer-solvent interactions, it has been established that, for a polymer to be dissolved in a given solvent, the term (ds - dp)2 in equation [3], must be smaller than 4.0. Thus, for example, according to the data shown in Table 1, if we are trying to dissolve nylon 6.6 in water, we will see that it is not possible thermodynamically, since (dwater - dnylon6.6) = (47.9 - 27.8) MPa1/2 = 20.1 MPa1/2 >> 4.0.
However, nylon 6.6 will dissolve in toluene, since (dtoluene - dnylon6.6) = (18.2 - 27.8) MPa1/2 = -9,5 MPa1/2 << 4.0. Making similar calculations, we will see that nylon 6.6 can also be dissolved in n-hexane and carbon tetrachloride.
It should be considered that the information the solubility parameters provide is based on a thermodynamic rather than kinetic point of view. It means that if a quick dissolution is what we are looking for, kinetically good solvents must be employed. Usually, solvent mixtures of kinetically good liquids with thermodynamically good liquids, assure a quick and efficient dissolution.
In polar systems or when polymer-solvent interactions occur, for example hydrogen bonding, the calculation of the solubility parameters is carried out by means of more complicated equations.
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