Cohesive Energies and the Solubility Parameter

Interactions among atoms and molecules, as we have seen, are a result of various forces stemming from their atomic or molecular structure, including electrostatic or charge interactions, steric or entropic phenomena, and the ever-present van der Waals forces. Of these, electrostatic and steric interactions may be repulsive in that they act to force the interacting units apart or at least reduce the net attraction between units. The van der Waals forces, on the other hand, are usually (but not always) attractive. When one discusses the use of a surfactant as an emulsion stabilizer, as in the above sections, the concept of the function of the surfactant is that it have a strong tendency to 

FIGURE 11.8. The solubility parameter concept predicts that the optimum interactions among surfactant, oil phase, and aqueous phase will occur when the two portions of the surfactant molecule closely match the parameters of each phase 

The first firm steps in the quantification of material interactions in terms of their molecular cohesive (stick together) properties came in 1950 with the studies of Hildebrand on the solubility of nonelectrolytes. Hildebrand characterized the cohesive energy density of a material as an intensive property he called the solubility parameter, usually given the symbol 8, measured in (J cm ). The reference to solubility stems from the fact that the original studies were based on the solubility of the materials of interest in various solvents and the correlations between the chemical structures of the two. The square root was chosen because it was found to allow one to calculate an average value of 8 for mixtures of materials and to estimate values for materials based on their atomic and functional group composition. 

When one applies the solubility parameter idea to mixtures, the same basic concepts allow one to visualize the molecular situation in terms of the interactions among the various molecular species present. As a first approximation, one can estimate the cohesive interaction between two unlike molecules, Eo(ah), as the product of the two solubility parameters 

TABLE 11. Solubility Parameters (Cohesive Energy Densities) of Some Commonly Encountered Materials 

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Although rigorous additivity rules are not applicable in this case, a fair estimation of the cohesive energy and the solubility parameter of polymers can be made by group contribution methods. 

Correlations for the cohesive energy and the solubility parameter will be presented in Chapter 5, to allow the calculation of these properties at the same level of accuracy as can be attained by group contributions but for much wider classes of polymers. The pitfalls of using solubility parameters in miscibility calculations will also be highlighted in the context of a discussion of the various types of phase diagrams that are observed for blends and mixtures. 

Selected values of the cohesion energy and the solubility parameter 6 are listed below for definitions, see Ge(CH3)4, p. 35 [41]. 

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 polymer has a low cohesive energy density (the solubility parameter 8 is about 16.1 MPaU2) and would be expected to be resistant to solvents of solubility parameter greater than 18.5 MPa1/2 (Chapter 4). Since polyethylene is a crystalline hydrocarbon polymer incapable of specific interaction, there are no solvents at room temperature. Materials of similar solubility parameters and low molecular weight will however cause swelling, the more so in low-density polymers. LDPE has a gas permeability in the range normally expected with rubbery materials. HDPE has a permeability of about one-fifth that of LDPE. 

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 Rao function has the same form as the Sugden function or Molar Parachor (Ps = My1/4/p), derived by Sugden in 1924, which correlates the surface tension with the chemical structure. Also the Small function or Molar Attraction Function, which correlates the cohesion energy density, ecoh, and the solubility parameter, 8, with the chemical structure, has this form ... 

Small considered that the F quantity of Equation 16.9 shows better additive characteristics than the cohesive energy. The total molar attraction constant is calculated from Equation 16.1 and the solubility parameter is calculated from the equation ... 

Dioxane, ethylene glycol, water-soluble esters, and short-chain alcohols at high bulk phase concentrations may increase the CMC because they decrease the cohesive energy density, or solubility parameter, of the water, thus increasing the solubility of the monomeric form of the surfactant and hence the CMC (Schick, 1965). An alternative explanation for the action of these compounds in the case of ionic surfactants is based on the reduction of the dielectric constant of the aqueous phase that they produce (Herzfeld, 1950). This would cause increased mutual repulsion of the ionic heads in the micelle, thus opposing micellization and increasing the CMC. 

The solubility parameter will be estimated indirectly, by combining the correlations for the cohesive energy and the molar volume. 

The same types of electrical forces which determine the values of the molar polarization and the dielectric constant also determine the cohesive energy density [2], An approximate correlation, given by Equation 9.4, has thus been found [3] between at room temperature and the solubility parameter 8 for polymers. This equation can be used to provide a rough estimate of the value of the dielectric constant at room temperature. 

Besides eos, the statistical thermoc amic theories also predict variations of the reduced cohesive energy density, CED solubility parameter, and the internal pressure, p, ... 

A computational model based on molecular dynamics was developed to predict the miscibility of indomethacin in the carriers polyethylene oxide (PEO), glucose, and sucrose (Gupta et al. 2011). The cohesive energy density and the solubility parameters were determined by simulations using the condensed-phase optimized molecular potentials for atomistic simulation studies (COMPASS) force field. The simulations predicted miscibility for indomethacin with PEO (A5 < 2), borderline miscibility with sucrose (A5 < 7), and immiscibility with glucose (A5 > 10 Table 2.2). 

The remaining interactions in molecules that have permanent dipole moments, that is, the dipole-dipole and dipole-induced-dipole interactions, have the same dependence on intermolecular separation as the London potential, varying as and are of lesser magnitude at ordinary temperatures (Atkins, 1998). These two interactions and the previously mentioned dispersion interaction are collectively known as van der Waals interactions. They are related to such measurable properties as surface tension and energy of vaporization, and to concepts such as the internal pressure, the cohesive energy density (energy of vaporization per unit volume, A UIV), and the solubility parameter, 6, which is the square root of the cohesive energy density (Hildebrand and Scott, 1962). 

The cohesive energy density is defined as the energy per unit volume, and the solubility parameter is the square root of that. In the cgs system of units, the cohesive energy density was measured as cal/cm, so the solubility parameter was (cal/cm ). In the current SI system used in this text, cohesive energy density is given in MPa, so the solubility parameter is MPa. Multiply the old (cal/cm ) value by 2.046 to convert to MPa. ... 

An important point is that the percolation threshold, depends on the morphology of the percolating clustei-s, and thus on the incompatibility between the two phases. We postulate that = fgQ, so that once hard phase spheres aggregate into infinite cylinders, percolated hard phase is formed. This assumption is a crucial link that relates thermodynamic information about the hard segment (its cohesive energy density or solubility parameter) to mechanical properties of polyurethanes based on that hard segment. 

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. 

We encountered the quantity AE ap/V in Eq. (8-35) it is the cohesive energy density. The square root of this quantity plays an important role in regular solution theory, and Hildebrand named it the solubility parameter, 8. 

This expression is known as the cohesive energy density and in S.I. is expressed in units of megapascals. The square root of this expression is more commonly encountered in quantitative studies and is known as the solubility parameter and given the symbol 6, i.e. ... 

This equation is the one most often used to calculate the cohesion energy of a liquid. From the molar mass and density of the liquid, the molar volume can be determined, and by means of Eq. (6.36) the value of S can be determined. The importance of the solubility parameter for interpreting several types of interactions will now be illustrated. 

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