Cohesive-energy density

We now introduce two approaches of AH of mixing for macromolecules cohesive energy density and contact energy. Both are related to van der Waals force. 

Scatchard (1931) and Hildebrand (1933) independently suggested that this equation, with some modification, can equally be valid to describe the energy AE involved in the mixing of solute and solvent. They derived a similar equation  

The term E jV (hence AE /V) is also called the cohesive energy density (E being cohesive energy). 

Hildebrand further derived an equation to relate the heat of mixing A//mix with the solubility parameter 8, which is now called the Hildebrand equation  

The solubility parameters 81 of some solvents are given in Table 4.1 the solubility parameters of some polymers are given in Table 4.2. 

Hildebrand also relied on van Laar s expressions using the equations derived by van der Waals in 1890 for the a and b constants in the van der Waals equation of state for binary fluid mixtures, when the interactions of three kinds of molecular pairs, 1-1, 2-2 and 1-2 are present  

The bj parameter represents the space occupied by the molecules in the densely packed fluids, and not very suitable for liquids then van Laar and Lorentz modified Equation (384) by substituting the actual molar volumes, v for the b terms, in 1925  

Inserting the volume fraction term, (j iy in Equation (388) gives 

similarly to Equation (381), the total cohesive internal energy of mixing, which is the change of the cohesive internal energy before and after the mixing, can be 

A(/Tx is also the excess energy of mixing and by combining Equations (391) and (392), Scatchard obtained 

---

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. 

We shall devote a considerable portion of this chapter to discussing the thermodynamics of mixing according to the Flory-Huggins theory. Other important concepts we discuss in less detail include the cohesive energy density, the Flory-Krigbaum theory, and a brief look at charged polymers. 

The quantity AU JV° is the internal energy of vaporization per unit volume and is called the cohesive energy density (CED) of component i. The square root of the CED is generally given the symbol 6j for component i. 

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...

For benzene at 25°C this becomes AU = 33,900 - 8.314 (298) = 31,400 J mol". The molar volume of a compound is given by V° = (molecular weight)/ (density). For benzene at 25°C, this becomes V° = 78.0/0.879 = 88.7 cm mol". Tlie cohesive energy density is simply the ratio AUy/V°, but in evaluating this numerically, the question of units arises. By convention, these are usually expressed in calories per cubic centimeter, so we write... 

A linear relationship exists between the cohesive energy density of an abrasive (10) and the WoodeU wear resistance values occurring between comndum H = 9) and diamond H = 42.5). The cohesive energy density is a measure of the lattice energy per unit volume. 

The polarity of the polymer is important only ia mixtures having specific polar aprotic solvents. Many solvents of this general class solvate PVDC strongly enough to depress the melting temperature by more than 100°C. SolubiUty is normally correlated with cohesive energy densities or solubiUty parameters. For PVDC, a value of 20 0.6 (J/cm (10 0.3 (cal/cm ) has been estimated from solubiUty studies ia nonpolar solvents. The value... 

A more polar comonomer, eg, an AN comonomer, increases the water-vapor transmission more than VC when other factors are constant. For the same reason, AN copolymers are more resistant to penetrants of low cohesive energy density. AH VDC copolymers, however, are very impermeable to ahphatic hydrocarbons. Comonomers that lower T and increase the free volume in the amorphous phase increase permeability more than the polar comonomers higher acrylates are an example. Plasticizers increase permeabiUty for similar reasons. 

The permachor method is an empirical method for predicting the permeabiUties of oxygen, nitrogen, and carbon dioxide in polymers (29). In this method a numerical value is assigned to each constituent part of the polymer. An average number is derived for the polymer, and a simple equation converts the value into a permeabiUty. This method has been shown to be related to the cohesive energy density and the free volume of the polymer (2). The model has been modified to liquid permeation with some success. 

Thermodynamic Properties The variation in solvent strength of a supercritical fluid From gaslike to hquidlike values may oe described qualitatively in terms of the density, p, or the solubihty parameter, 6 (square root of the cohesive energy density). It is shown For gaseous, hquid, and SCF CO9 as a function of pressure in Fig. 22-17 according to the rigorous thermodynamic definition ... 

CRANK, J., and PARK, J, s., Dijfusion in Polymers, Academic Press, London and New York (1968) GARDON, J. L., Article enlilled Cohesive Energy Density in Encyclopaedia of Polymer Science and Technology, Vol. 3, p. 833, Interscience, New York (1969)... 

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... 

Because of the high cohesive energy density and their crystalline state the polymers are soluble only in a few liquids of similar high solubility parameter and which are capable of specific interaction with the polymers. 

Scatchard and Hildebrand defined the cohesive energy density coh-ii of a liquid 1 as the negative molar configurational energy divided by its liquid molar volume, V, ... 

The mixture cohesive energy density, coh-m> was not to be obtained from some mixture equation of state but rather from the pure-component cohesive energy densities via appropriate mixing rules. Scatchard and Hildebrand chose a quadratic expression in volume fractions (rather than the usual mole fractions) for coh-m arid used the traditional geometric mean mixing rule for the cross constant ... 

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. 

The internal pressure is a differential quantity that measures some of the forces of interaction between solvent molecules. A related quantity, the cohesive energy density (ced), defined by Eq. (8-35), is an integral quantity that measures the total molecular cohesion per unit volume. - p... 

Table 8-6. Internal Pressure and Cohesive Energy Density (ced) of Solvents...

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. ... 

---