The solubility parameter

How can AH be estimated Well, for the formation of regular solutions (those in which the solute and the solvent do not form specific interactions), the change in internal energy per unit volume of solution is given by 

AE = the change in internal energy per unit volume of solution j)i = component volume fractions 8i = solubility parameters 

The subscripts 1 and 2 usually (but not always) refer to solvent and solute (polymer), respectively. The solubility parameter is defined as follows  

CED = cohesive energy density, a measure of the intermolecular forces holding the molecules together in the liquid state AE = molar change in internal energy on vaporization V = molar volume of liquid (cmVmol) 

Traditionally, solubility parameters have been given in (cal/cm ) = hildebrands (in honor of the originator of regular solution theory), but they are now more commonly listed in (MPa) where 1 hildebrand = 0.4889 (MPa).  

This assumption is reasonable when there are no specific interactions among molecules, as is true for nonpolar molecules [26], and it allows us to replace a mixture property in terms of pure component properties. Introducing Eq. (9.5.3) into Eq. (9.5.2) yields 

If one goes back to the lattice model, N z/2 represents the total number of interactions among 1 mol of molecules. Multiplication with e yields the molar internal energy change, A17 , for vaporizing species i. Thus, 

We can extend the concept of the solubility parameter to macromolecules by defining the solubility parameter of a polymer as 

If we equate the right-hand side of Eq. (9.3.19) to the right-hand side of Eq. (9.5.5), we can relate the interaction parameter to the solubility parameter as follows  

Small values of X promote polymer solubility. Because (5 is a smrogate for Xi, it can be used in much the same way as the interaction parameter. Naturally, it suffers from the same drawbacks, as well as those resulting fi om the assumption embodied in Eq. (9.5.3). Values of the solubility parameter for selected polymer and nonpolar solvents are listed in Table 9.2. These may also be estimated using the method of group contributions [27]. For a mixture of solvents, the solubiUty parameter may be taken to be a weighted average of the solubility parameters of the constituents weighting is done with respect to the volume fraction of the components. 

Once placed in a solvent, polymers dissolve in several steps. First, the solvent must wet the polymer. Second, the solvent diffuses into the polymer, swelling it. For polymers of high molecular weight, this process may take several hours or longer, depending on sample size, temperature, and so on. Contrary to many low molecular weight substances, the polymer does not initially diffuse into the solvent. 

There are two distinguishable modes of solvent diffusion into a polymer. If the polymer is amorphous and above its glass transition temperature (i.e., a polymer melt), the diffusion of the solvent into the polymer forms a smooth composition curve, with the most highly swollen material at the outer edge. This is referred to as Fickian diffusion, following Tick s laws see Section 4.4. 

Finally the polymer diffuses out of the swollen mass into the solvent, completing the solution process. Very dilute solutions are usually required for molecular weight determination. 

One of the simplest notions in chemistry is that like dissolves like. Qualitatively, 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. 

It is intuitively obvious that a hquid has a certain amount of cohesion that holds the liquid together. It should also be apparent that the energy with which the liquid is held together is related to the heat necessary to vaporize it, which separates the molecules. In fact, the cohesion energy, E, is given by 

The work done as the vapor expands against the external (atmospheric) pressure is RT. The quantity Ec/V, where V is the molar volume, is called the cohesion energy density because it is the cohesion energy per unit volume. A basic thermodynamic relationship, 

However, we can also express the change in pressure with temperature at constant volume by making use of a well-known thermodynamic relationship. 

The numerator on the right-hand side of this equation represents the coefficient of thermal expansion, a. The denominator of the equation represents the change in volume with pressure at constant temperature, which is the coefficient of compressibility, /3. Therefore, the internal pressure is given by 

For most liquids, the internal pressure ranges from 2000 to 8000 atm. As we will see, this has important ramifications with regard to the formation of transition states in which there is a volume change. From the foregoing development, we can now write the cohesion energy as 

Regular solution theory characterises nonpolar solvents in terms of solubility parameter, 6y which is defined as 

the solubility of the undissociated species (hase), is determined at high pH, and S is determined at several different lower pH values. A plot of log[So/(S - Sq)] versus pH will have the pK as the intercept on the pH axis. 

Plotting data as in Fig. 5.4(b) yields Sq when the line crosses the x-axis as [H+] = 0 (and S = Sq). The intercept on the y-axis gives and the slope of the line is KJSq. 

AU/V is the liquid s cohesive energy density, a measure of the attraction of a molecule from its own liquid, which is the energy required to remove it from the liquid and is equal to the energy of vaporisation per unit volume. Because cavities have to be formed in a solvent, by separating other solvent molecules, to accommodate solute molecules (as discussed earlier) the solubility parameter dj 

By itself the solubility parameter can explain the behaviour of only a relatively small group of solvents - those with little or no polarity and those unable to participate in hydrogenbonding interactions. The difference between the solubility parameters expressed as (dj - 2) will give an indication of solubility relationships. 

The solubility of a solute in a supercritical fluid can be quantitatively estimated using Giddings theory, which relies on differences between the Hildebrand solubility parameters for the SF and solute concerned. Solubility in a supercritical fluid can be understood by examining the Gibbs-Helmholtz equation  

According to Hildebrand [11], the heat of mixing can be defined in mathematical terms as 

The solubility of a solute can be defined in terms of its solubility parameters  

On the other hand, the solvent power of a supercritical fluid is defined by the equation of Guiddings as applied by Li and Lee [12], where the solubility parameters of the fluid are given by 

The solubility parameter for a supercritical fluid, S, increases with increasing density, so it is markedly dependent on pressure. This parameter also depends strongly on temperature, particularly at high pressures. Consequently, the solubility parameter for a fluid can be adjusted through both pressure and temperature. 

Attempts to account for the mole fraction solubility of a gas A in a liquid S have been dominated by Hildebrand s idea of a solubility parameter S, defined as the 

For nonpolar gases and nonpolar liquids, Hildebrand connected with N, 8a, and 5s by the following expression  

Based on his results for carbon dioxide and nitrous oxide in a number of organic liquids, Kunerth (1922) ° concluded that there was no correlation between solubility and the internal pressures of solvent and solute he expressed disagreement with Hildebrand s views. In his reply, Hildebrand (1923) emphasized that he required the condition of nonpolarity. Of the 21 liquids S used by Kunerth, only 8 had dielectric constants as low as 5. In the many references to the solubility parameter and the parameter equation since that time, there is the inherent difficulty over the terms polar and nonpolar and what constitutes a chemical reaction. 

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The solubility parameters calculated at 25°C have been tabulated for numerous hydrocarbons in the API technical data book. They can also by calculated by the relationship ... 

The values of common hydrocarbon solubility parameters vary between 300 and 600 (kJ/m3) /2 Several tables are available where the solubility parameters are shown as (cal/cm ) / Jq convert these values, it is necessary to multiply by 64.69. Thus a solubility parameter value of 10 (cal/cm ) / jg equal to 646.9 (kJ/m ) / ... 

The solubility parameter is not calculated directly. It is calculated as the square root of the cohesive energy density. There are a number of group additivity techniques for computing cohesive energy. None of these techniques is best for all polymers. 

Table 8.2 Values of the Cohesive Energy Density (CED) for Some Common Solvents and the Solubility Parameter 6 for These Solvents and Some Common Polymers...

The solubihty coefficients are more difficult to predict. Although advances are being made, the best method is probably to use a few known solubility coefficients in the polymer to predict others with a simple plot of S vs ( poiy perm Y where and are the solubility parameters of the polymer and permeant respectively. When insufficient data are available, S at 25°C can be estimated with equation 19 where k = 1 and the resulting units of cal/cm are converted to kj /mol by dividing by the polymer density and multiplying by the molecular mass of the permeant and by 4.184 (16). 

The solubility parameter is thus an experimentally determinable property although special methods are necessary with polymers, which cannot normally be vaporised without decomposition. Such methods are discussed in Section 5.3.3. 

It is the aim of this part of the chapter to show how certain predictions may be made about the solubility of a given material such as a polymer in any given solvent. We have seen that the solubility parameter has given us a measure of and Fbb but the magnitude of F b will have to be considered separately for the following systems ... 

Tables 5.4 and 5.5 predict that unvulcanised natural rubber (8 = 16.5) will be dissolved in toluene (8 = 18.2) and in carbon tetrachloride (8 = 17.5) but not in ethanol (8 = 26.0), all values being in units ofMPa. This is found to be true. Similarly it is found that there is a wide range of solvents for polystyrene in the solubility parameter range 17.2-19.7 MPa. ...

As already mentioned molecules cohere because of the presence of one or more of four types of forces, namely dispersion, dipole, induction and hydrogen bonding forces. In the case of aliphatic hydrocarbons the dispersion forces predominate. Many polymers and solvents, however, are said to be polar because they contain dipoles and these can enhance the total intermolecular attraction. It is generally considered that for solubility in such cases both the solubility parameter and the degree of polarity should match. This latter quality is usually expressed in terms of partial polarity which expresses the fraction of total forces due to the dipole bonds. Some figures for partial polarities of solvents are given in Table 5.5 but there is a serious lack of quantitative data on polymer partial polarities. At the present time a comparison of polarities has to be made on a commonsense rather than a quantitative approach. 

Table S.6 Partitioned values of the solubility parameter (after Hansen )...

The solubility parameters of a number of commercial plasticisers are given in Table 5.7... 

In the formulation of PVC compounds it is not uncommon to replace some of the plasticiser with an extender, a material that is not in itself a plasticiser but which can be tolerated up to a given concentration by a polymer-true plasticiser system. These materials, such as chlorinated waxes and refinery oils, are generally of lower solubility parameter than the true plasticisers and they do not appear to interact with the polymer. However, where the solubility parameter of a mixture of plasticiser and extender is within unity of that of the polymer the mixture of three components will be compatible. It may be shown that... 

Because the solubility parameter of tritolyl phosphate is higher than that of dioctyl sebacate, PVC-tritolyl phosphate blends can tolerate more of a low solubility parameter extender than can a corresponding sebacate formulation. 

Where 8 is the solubility parameter A the energy of vaporisation V the molar volume A// the latent heat of vaporisation R the gas constant T the temperature M the moleeular weight D the density. 

From this equation a useful curve relating AE and has been compiled and from this the solubility parameter may easily be assessed Figure 5.8). 

As an example of the use of Small s table the solubility parameter of polyfmethyl methacrylate) may be computed as follows ... 

In the case of crystalline polymers better results are obtained using an amorphous density which can be extrapolated from data above the melting point, or from other sources. In the case of polyethylene the apparent amorphous density is in the range 0.84-0.86 at 25°C. This gives a calculated value of about 8.1 for the solubility parameter which is still slightly higher than observed values obtained by swelling experiments. 

Since we have defined the expression (AX/T) as the solubility parameter 8, the above equation may be written... 

It was pointed out in Chapter 5 that plasticisers were essentially non-volatile solvents. Consequently they were required to have solubility parameters close to that of the polymer and a molecular weight of at least 300. If the polymer or the plasticisers had a tendency to crystallise then there would need to be some sort of specific interaction between the polymer and the plasticiser. Tables 5.4 and 5.6 gave some figures for the solubility parameters of polymers and plasticisers. 

The polymer has a low cohesive energy density (the solubility parameter 5 is about 16.1 MPa ) and would be expected to be resistant to solvents of solubility parameter greater than 18.5 MPa. Because it is a crystalline material and does... 

The solubility parameters of these extenders are generally somewhat lower than that of PVC. They are thus tolerated in only small amounts when conventional plasticisers of low solubility parameter, e.g. the sebacates, are used but in greater amounts when phosphates such as tritolyl phosphate are employed. 

The anomalous effect of the last two rubbers in the table with their low solubility parameters is possibly explained by specific interaction of PVC with carbonyl and carboxyl groups present respectively in the ketone- and fumarate-containing rubbers to give a more than expected measure of compatibility. It is important to note that variation of the monomer ratios in the copolymers and terpolymers by causing changes in the solubility parameter and eompatibility will result in variation in their effect on impact strength. 

As may be expected of an amorphous polymer in the middle range of the solubility parameter table, poly(methyl methacrylate) is soluble in a number of solvents with similar solubility parameters. Some examples were given in the previous section. The polymer is attacked by mineral acids but is resistant to alkalis, water and most aqueous inorganic salt solutions. A number of organic materials although not solvents may cause crazing and cracking, e.g. aliphatic alcohols. 

The solubility parameter is calculated at 20 MPa and therefore the polymer is swollen by liquids of similar cohesive forces. Since crystallisation is thermodynamically favoured even in the presence of liquids of similar solubility parameter and since there is little scope of specific interaction between polymer and liquid there are no effective solvents at room temperature for the homopolymer. 

The solubility parameter is in the range 18.4-19 MPa and the polymer is predictably dissolved by halogenated and aromatic hydrocarbons of similar solubility parameter. Stress cracking can occur with some liquids. 

The solubility parameter is about 19.2MPa and being amorphous they dissolve in such solvents as tetrahydrofuran, mesityl oxide, diacetone alcohol and dioxane. Since the main chain is composed of stable C—C and C—O—C linkages the polymer is relatively stable to chemical attack, particularly from acids and alkalis. As already mentioned, the pendant hydroxyl groups are reactive and provide a site for cross-linking. 

The solubility parameter of poly(ethylene terephthalate) is about 21.8 MPa but because it is a highly crystalline material only proton donors that are capable of interaction with the ester groups are effective. A mixture of phenol and tetrachloroethane is often used when measuring molecular weights, which are about 20 000 in the case of commercial polymers. 

Regular solutions, the solubility parameter and Scatchard-Hildebrand theory... 

Large databases for the solubility parameter exist, most notably Barton s Handbook of Solubility Parameters and Other Cohesive Parameters [112], A small listing of values is given in Table 4. The units used are those of (J/cm ) / = (MPa) /, although the older literature uses units of (cal/cm ). The units are related by ... 

The solubility parameter concept has found widespread use in predicting the compatibility of components used in paints and coatings, and the patent literature contains numerous references to the solubility parameter or solubility parameter ranges in specifying formulations. Its use in predicting adhesion should apply in... 

Fig. 21. Peel strengths of various adhesives against poly(ethylene terephthalate) adherends vs. the solubility parameter difference between adhesive and adherend. Drawn using data from ref. [72].

Hansen [137-139], and later van Krevelen [114] proposed the generalization of the solubility parameter concept to attempt to include the effects of strong dipole interactions and hydrogen bonding interactions. It was proposed that the cohesive energy density be written as the sum of three terms, viz. 

Fig. 23. The compatibility sphere delined by the three-dimensional solubility parameters . +, the solubility parameter coordinates of a given polymer , coordinates of solvents showing a high degree of compatibility (e.g., full mi.scibility) with the polymer U, solvents showing a lower degree of compatibility.

As pointed out earlier, acrylics differ from the commonly used rubber precursors for PSA formulation in the fact that they often incorporate polar monomers, such as acrylic acid, A-vinyl pyrrolidone, vinyl acetate, or acrylamide. As a result, the solubility parameters of acrylic polymers are typically higher than those of rubbers, like polyisoprenes or polybutadienes. 

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