DILUTE SOLUTION THERMODYNAMICS,
3.2 THE SOLUBILITY PARAMETER
One of the simplest notions in chemistry is that “like dissolves like.” Qualita-tively, “like” may be defined variously in terms of similar chemical groups or similar polarities.
Quantitatively, solubility of one component in another is governed by the familiar equation of the free energy of mixing,
(3.1) where DGMis the change in Gibbs’ free energy on mixing, T is the absolute temperature, and DSMis the entropy of mixing. A negative value of DGM
indi-DGM =DHM -T SD M
3.2 THE SOLUBILITY PARAMETER 73
cates that the solution process will occur spontaneously. The term T DSM is always positive because there is an increase in the entropy on mixing. (But, note the negative sign!) Therefore the sign of DGMdepends on the magnitude of DHM, the enthalpy of mixing.
Surprisingly, the heat of mixing is usually positive, opposing mixing. This is true for big and little molecules alike. Exceptions occur most frequently when the two species in question attract one another in some way, perhaps by having opposite polarities, being acid and base relative to one another, or through hydrogen bonding. However, positive heats of mixing are the more usual case for relatively nonpolar organic compounds. On a quantitative basis, Hilde-brand and Scott (5) proposed that, for regular solutions,
(3.2)
where VMrepresents the total volume of the mixture, DE represents the energy of vaporization to a gas at zero pressure (i.e., at infinite separation of the molecules), and V is the molar volume of the components, for both species 1 and 2. The quantity v represents the volume fraction of component 1 or 2 in the mixture. The quantity DE/V represents the energy of vaporization per unit volume. This term is sometimes called the cohesive energy density. By convention, component 1 is the solvent, and component 2 is the polymer.
The reader should note that according to equation (3.2), “like dissolves like” means that the two terms DE1/V1 and DE2/V2 have nearly the same numerical values. Equation (3.2) also yields only positive values of DHM, a serious fault in the theory. However, since the majority of polymer solutions do have positive heats of mixing, the theory has found very considerable application.
The square root of the cohesive energy density is widely known as the solubility parameter,
(3.3) Thus the heat of mixing of two substances is dependent on (d1- d2)2.
These relationships are meaningful only for positive heats of mixing; that is, when the heat of mixing term opposes solution. Since (d1- d2)2cannot be negative, equations (3.2) and (3.3) break down for negative heats of mixing.
3.2.1 Solubility Parameter Tables
Tables 3.1 (6) and 3.2 (6) present the solubility parameters of common sol-vents and polymers, respectively. These tables provide a quantitative basis for understanding why methanol or water does not dissolve polybutadiene or polystyrene. However, benzene and toluene are predicted to be good solvents
d =Ê Ë
ˆ
¯ DE
V
1 2
D D D
H V E
V
E
V v v
M = M Ê
Ë ˆ
¯ -Ê Ë
ˆ
¯ È
ÎÍ ˘
˚˙
1
1 1 2
2
2 1 2 2
1 2
3.2 THE SOLUBILITY PARAMETER 75
Table 3.1 Solubility parameters of some common solvents
d H-bondinga Specific Gravityb Solvent (cal/cm3)1/2 MPa1/2 Group 20°C (g/cm3)
Acetone 9.9 20.3 m 0.7899
Benzene 9.2 18.8 p 0.87865
n-Butyl acetate 8.5 17.4 m 0.8825
Carbon tetrachloride 8.6 17.6 p 1.5940
Cyclohexane 8.2 16.8 p 0.7785
n-Decane 6.6 13.5 p —
Dibutyl amine 8.1 16.6 s —
Difluorodichloromethane 5.5 11.3 p —
1,4-Dioxane 7.9 16.2 m 1.0337
Low odor mineral spirits 6.9 14.1 p —
Methanol 14.5 29.7 s 0.7914
Toluene 8.9 18.2 p 0.8669
Turpentine 8.1 16.6 p —
Water 23.4 47.9 s 0.99823
Xylene 8.8 18.0 p 0.8611
Source: J. Brandrup, E. H. Immergut, and E. A. Grulke, eds., Polymer Handbook, 4th ed., Wiley-Interscience, New York, 1999.
aHydrogen bonding is an important secondary parameter in predicting solubility. p, Poorly H-bonded; m, moderately H-bonded; and s, strongly H-bonded.
bJ. Brandrup and E. H. Immergut, Polymer Handbook, 3th ed., Wiley-Interscience, New York, 1989, sec. III, p. 29.
Note: 1 (cal/cm3)1/2= 2.046 ¥ 103(J/m3)1/2.
Table 3.2 Solubility parameters and densities of common polymers (6)
Polymer d (cal/cm3)1/2 d (MPa)1/2 Density (g/cm3)
Polybutadiene 8.4 17.2 1.01
Polyethylene 7.9 16.2 0.85 (amorphous)
Poly(methyl methacrylate) 9.45 19.4 1.188
Polytetrafluorethylene 6.2 12.7 2.00 amorphous, estimated
Polyisobutene 7.85 16.5 0.917
Polystyrene 9.10 18.6 1.06
Cellulose triacetate 13.60 27.8 1.28a
(56% ac groups)
Cellulose tributyrate — — 1.16a
Polyamide 66 13.6 22.9 1.24
Poly(ethylene oxide) 9.9 20.0 1.20
Poly(ethylene terephalate), 10.7 21.9 1.38 partly crystalline
Poly(ethylene terephalate), 10.7 21.9 1.34 amorphous
Poly(vinyl alcohol) 12.6 25.8 1.26
Poly(vinyl chloride) 9.6 19.8 1.39
Note: 1 (cal/cm3)1/2= 2.046 ¥ 103(J/m3)1/2.
aC. J. Malm, C. R. Fordyce, and H. A. Tanner, Ind. Eng. Chem., 34, 430 (1942).
for these polymers, which they are. While solubility of a polymer also depends on its molecular weight, the temperature, and so on, it is frequently found that polymers will dissolve in solvents having solubility parameters within about one unit of their own, in (cal/cm3)1/2.
3.2.2 Experimental Determination
The solubility parameter of a new polymer may be determined by any of several means. If the polymer is cross-linked, the solubility parameter may be determined by swelling experiments (7). The best solvent is defined for the purposes of the experiment as the one with the closest solubility parameter.
This solvent also swells the polymer the most. Several solvents of varying sol-ubility parameters are selected, and the cross-linked polymer is swelled to equilibrium in each of them. The swelling coefficient, Q, is plotted against the various solvent’s solubility parameter, the maximum defining the solubility parameter of the polymer. The theoretical extent of swelling is predicted by the Flory–Rehner theory on the basis of the cross-link density and the attrac-tive forces between the solvent and the polymer (see Section 9.12).
The swelling coefficient, Q, is defined by,
(3.4)
where m is the weight of the swollen sample, m0is the dry weight, and rsis the density of the swelling agent (8,9). Typical results are shown in Figure 3.1 (9). Here, the swelling behavior of a linked polyurethane and a cross-linked polystyrene are shown, together with the 50/50 interpenetrating polymer network made from these two polymers. Both the homopolymers and the interpenetrating polymer network exhibit single peaks, albeit that the IPN
Q m m
m s
=
-0 ¥
0
1 r
Figure 3.1 The swelling coefficient, Q, reaches a maximum when the solubility parameter of the solvent nearly matches that of the polymer, for several cross-linked systems: polyurethane (), polystyrene (), and a polyurethane–polystyrene interpenetrating polymer networks (•) (9).
Solvents having solubility parameters near 2 ¥ 104(J/m3)1/2will swell the IPN best.
peak is somewhat broader and appears in-between its two homopolymers.
Polymer networks swollen to equilibrium are discussed in Section 9.12.
Alternatively, the solubility parameter may be determined by measuring the intrinsic viscosity of the polymer in these solvents, if the polymer is soluble in them. Then the intrinsic viscosity is plotted against the solubility parameter of the several solvents. Since the chain conformation is most expanded in the best solvent [see equation (3.82)], the intrinsic viscosity will be highest for the best match in solubility parameter. Such an experiment is illustrated in Figure 3.2 (10) for polyisobutene and polystyrene. The results of such experi-ments are collected in Table 3.2.
3.2.3 Theoretical Calculation: An Example
Values of the solubility parameter may be calculated from a knowledge of the chemical structure of any compound, polymer or otherwise. Use is made of the group molar attraction constants, G, for each group,
(3.5) where r represents the density and M is the molecular weight. For a polymer, M is the mer molecular weight.
Group molar attraction constants have been calculated by Small (11) and Hoy (12). Table 3.3 (11) presents a wide range of values of G for chemical groups.
d r
=
Â
GM
3.2 THE SOLUBILITY PARAMETER 77
Figure 3.2 Determination of the solubility parameter, using the intrinsic viscosity method (8), for polyisobutene (A) and polystyrene (B). The intrinsic viscosity, [h], is a measure of the indi-vidual chain size. See Section 3.8.
Table 3.3Group molar attraction constants at 25°C (according to Small; derived from measurement of heat of evaporation)† GroupGa GroupGGroup 214Ring5-membered105–115Brsingle single-bonded133Ring6-membered95–105Isingle 28Conjugation20–30CF2 n-fluorocarbons only -93H(variable)80–100CF3 190Oethers70Ssulfides double-bonded111COketones275SHthiols 19COOesters310ONOnitrates~ 285CN410NO2(aliphatic nitro-compounds)~ 222Cl(mean)260PO4(organic phosphates)~ Phenyl735Clsingle270Si(in silicones) Phenylene(o,m,p)658Cltwinned as in >CCl2260 Naphthyl1146Cltriple as in —CCl3250 Source:P.A.Small,J.Appl.Chem.,3,71 (1953). aUnits ofG=(cal-cm3)1/2/mol. †The solubility parameter can be calculated viad=rSG/M,where Mis the mer molecular weight.
CH3 CH2 CH2 CHCH CHC CCC C
}
78
For example, the solubility parameter of polystyrene may be estimated from Table 3.3. The structure is
which contains —CH2— with a G value of 133, a with G equal to 28, and a phenyl group with G equal to 735. The density of polystyrene is 1.05 g/cm3, and the mer molecular weight is 104 g/mol. Then equation (3.5) gives
(3.6) (3.7) Table 3.2 gives a value of 9.1 (cal/cm3)1/2for polystyrene.
3.3 THERMODYNAMICS OF MIXING