Hard-core volume

As a first approximation we postulate the existence of an equally sized hard core volume located in the center of each cylinder that defines the excluded volume per segment, according to the degree of interpenetration of neighboring cylinders. This simple construction includes complicated local intersegmental configurations, but it imposes a unique local orientation correlation of segments within the domains of a PE melt The shape of those impenetrable correlation cylinders is assumed also to be cylindrical. 

Here is the effective surface fraction of molecules and the tilde ( ) denotes reduced variables. All quantities, except v in the EOS, are reduced by the parameters in Equation 4. The specific volume v, is reduced by v, the molecular hard-core volume,... 

The concept of free volume varies on how it is defined and used, but is generally acknowledged to be related to the degree of thermal expansion of the molecules. When liquids with different free volumes are mixed, that difference contributes to the excess functions (Prausnitz et al., 1986). The definition of free volume used by Bondi (1968) is the difference between the hard sphere or hard core volume of the molecule (Vw= van der Waals volume) and the molar volume, V ... 

Mixtures of hydrocarbons are assumed to be athermal by UNIFAC, meaning there is no residual contribution to the activity coefficient. The free volume contribution is considered significant only for mixtures containing polymers and is equal to zero for liquid mixtures. The combinatorial activity coefficient contribution is calculated from the volume and surface area fractions of the molecule or polymer segment. The molecule structural parameters needed to do this are the van der Waals or hard core volumes and surface areas of the molecule relative to those of a standardized polyethylene methylene CH2 segment. UNIFAC for polymers (UNIFAC-FV) calculates in terms of activity (a,-) instead of the activity coefficient and uses weight fractions... 

The molecular hard-core volume or reference volume, v, is calculated from the group references volumes, Rk, using the expression ... 

Vj = the hard core volume of component i, given by Equation (3D-17), cubic meters per kilomole. 

The hard core volume of component i is interpolated between values calculated at 300 K and 400 K to give the hard core volume at the temperature of interest. 

Step 5 Calculate the hard core volume, V,, of each component i at 300 and 400 K from Equation (3E-28) and Equation (3E-29). 

As previously described for binary sterns, volume fractions may be replaced by segment fractions which deal with the hard core volume of the components. 

In the FH formula, the characteristic volume V can be the molar volume or a specific hard-core volume of the molecule such as the van der Waals volume Vw. The last of the above equations is the Entropic-FV formula, discussed earlier (Equation 16.51). As shown, in the Entropic-FV model the van der Waals volume is used, as a measure of the molecular volume with increments by Bondi. The Entropic-FV formula was shown to work satisfactorily for athermal polymer solutions and offers a substantial improvement over the FH term. The van der Waals repulsive term and the Entropic-FV form are functionally identical. 

The hard-core volume v is related to the temperature-independent parameter ... 

Eqnation (4.186) snms up the excluded volume of the constituent gronps to form the excluded volume of the molecule. The hnear sum is justified because the excluded volume of a group is defined as (4v ), where stands for the hard core volume. Equation (4.187) linearly combines the... 

It is remarkable that no empirical mixture parameters and no experimental data are required to use the equation. The only parameters in the Flory-Huggins equation are the hard core volumes V, which are a pme-component property, and the atomic or group contribution values are found in standard compilations. Since the v/s are significant in the FH equation only in terms of their ratios, pure-liquid molar volumes are often used for V in place of hard core volumes. For solutions of polymers of the same chemical formula, molecular masses are legitimate substitutes for V , for the same reason. Thus the volume fractions ( ) can be substituted by mass fractions W . Either volume fraction or mass fraction is directly related to laboratory data. To avoid mole fractions, the activity tti from Equations (4.368) and (4.369) can be used to calculate by / = aj. ... 

The best description of liquids and compressed gases requires the molecular hard-core volume to be a decreasing function of temperature. Thus Chen and Kreglewski express V° by means of... 

The expression for Q is readily generalized to 3-D as follows. The hard-core length / is related to the hard-core volume v by / =(gv ) , where g=geometrical factor. A similar relationship also holds for the available space /=(gv). It follows that (/—/ )— This leads in 3-D... 

Sanchez-Lacombe [1976,1978] The Sanchez and Lacombe [1976,1978] equation of state (S-L) is based on the Ising fluid model. The authors followed the Guggenheim [ 1966] approach, placing A -mers and No holes in an A -lattice. Hard-core volumes of the s-mer, as well as its flexibility, were assumed independent of Tand P. Furthermore, only the nearest neighbors of nonbonded mers contributed to the lattice energy ... 

It is supposed that an apparently reduced hard-core volume governs the increase in entropy between the surface region and the bulk phase. For the hard-core volume in the surface region we write... 

Interaction parameter xh based on hard-core volumes... 

With the necessary care, all thermodynamic expressions given above can be formulated with mass or volume or segment fractions as concentration variables instead of mole fractions. This is the common practice within polymer solution thermodynamics. Applying characteristic/hard-core volumes is the usual approach within most thermodynamic models for polymer solutions. Mass fraction based activity coefficients are widely used in Equations [4.4.7 and 4.4.8] which are related to activity by ... 

The segment number r is, in general, different from the degree of polymerization or from the number of repeating units of a polymer chain but proportional to it. One should note that Equations [4.4.12 and 4.4.13] can be used on any segmentation basis, i.e., also with r = on a hard-core volume segmented basis and segment fractions instead of... 

To extend the Flory-Orwoll-Vrij model to mixtures, one has to use two assumptions (i) the hard-core volumes n of the segments of all components are additive and (ii) the intermolecular energy depends in a simple way on the surface areas of contact between solvent molecules and/or polymer segments. Without any derivation, the final result for the residual solvent activity in a binary polymer solution reads ... 

Here, the term A includes the contribution of interaction entropy (Floiy 1970). Such a contribution can be miderstood from the concept of compressible free volume in the fluids. When two fluids are mixed with each other, part of molecules of one species enters the free volume of another species, and then the total volume is not a simple addition of the two individual components. Yamakawa made an approximate estimation from the expansion theory (Yamakawa 1971). Prigogine attributed this contribution to a combinatorial contribution of molecular geometry and a non-combinatorial contribution of molecular stmctures, and proposed an equation-of-state theory (Prigogine 1957b). Hory, Orwell and Vrij further considered the contribution of free volume, and employed separate parameters to describe the hard-core volume and surface contacts of chain units (Flory et al. 1964 Flory 1965 OrwaU and Rory 1967). This work makes the equation of state fit better to the e erimental results, and derives the so-called Flory-Orwell-Vrij equation of state for pure polymers, as given by... 

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