Combinatory interaction parameter

If we consider the seven components in tables 5.12 and 5.13 as representative of the chemistry of natural olivines, it is clear that 21 regular binary interaction parameters (disregarding ternary and higher-order terms) are necessary to describe their mixing properties, through a combinatory approach of the Wohl or Kohler type (cf section 3.10). In reality, the binary joins for which interactions have been sufficiently well characterized are much fewer. They are briefly described below. 

Statistical thermodynamic mean-field theory of polymer solutions, first formulated independently by Flory, Huggins, and Staverman, in which the thermodynamic quantities of the solution are derived from a simple concept of combinatorial entropy of mixing and a reduced Gibbs-energy parameter, the X interaction parameter. 

Here, n is amount of substance and u the volume fraction, this last magnitude being defined by u = WiVsPji/(wivsPji + W2vSPj2), where Wi is the weight fraction and vsp>j the specific volume (i = 1, 2). Index 1 refers to solvent and index 2 to polymer, g is a phenomenological interaction parameter that takes into account deviations of AGm from its combinatorial value. Subscript u in g denotes that g is defined on a volume fraction basis. 

V being molar volume and x a phenomenological interaction parameter taking into account the deviations of Ajxi from its purely combinatorial value. Subscript v in... 

It should be pointed out that since I.G.C. measures the total free energy of the interaction, any value of the Flory-Huggins interaction parameter which is derived will be a total value including combinatorial and residual interaction parameters as well as any residual entropy contributions. Similarly when using Equation-of-state theory one will obtain Xj2 rather than Xj. The interactions are measured at high polymer concentration and are therefore of more direct relevance to interactions in the bulk state but this does not remove problems associated with the disruption of intereactions in a blend by a third component. 

Fig. 26. Hypothetical simulated spinodal curves for the phase separation of two polymers on heating illustrating the effect of the interaction parameter (Xjj) the non-combinatorial entropy parameter (Qjj) and die ratio of surface areas per unit volume Sj/Sj. The curves are all simulated using values of y, = Yz = 1 (Jcm" K" ) tj/r, - 1 VJ = 100,000 cm" mole" a, = 5x10", 04 = 4x I0" K" . If X,j = —0.6 J cm", Qjj = 0, and Sj/S, = 1, the curve is much flatter than those in the previous f ure. If there is a larger (favourable) X,j, say —1.2 and this is balanced by an unfavourable —0.0023 J cm K., then...

It is possible to simulate the spinodal curves of the phase diagram of polymer pairs using the Equation-of-state theory developed by Flory and co-workers. It is only, however, possible to do this using the adjustable non-combinatorial entropy parameter, Qjj. Another problem arises in the choice of a value for the interaction parameter Xjj. This is introduced into the theory as a temperature independent constant whereas we know that in many cases the heat of mixing, and hence is strongly temperature dependent. The problem arises because Xj was never intended to describe the interaction between two polymers which are dominated by a temperature dependent specific interaction. 

It should be noted that Eq. (13) does not apply to the interaction parameter based on volume fractions (x), due to the inclusion of a temperature dependent combinatorial entropy. In computing partial molar heats of mixing at infinite dflution, it is essential that the correction for gas pha% nonideality ( n) be included, owing to the magnitude of (< 100—300 cal/mol). 

Lichtenthaler et al. (55) determined interaction parameters for 22 solutes in poly(dimethyl siloxane) to test several expressions of the combinatorial entropy of mixing [Eq. (7)]. The magnitude of the interaction parameter is indeed directly dependent on the evaluation of the combinatorial contribution. The combinatorial contribution was computed following both the Flory-Huggins approximation and the multiple-connected-site model recently developed by Lichtenthaler, Abrams and Prausnitz (56). This model, which retains the Flory-Huggins term, also corrects for the bulkiness of the components of the mixture. Interaction parameters were computed through both approximations, showing the sensitivity of the results to the model chosen. 

The group interaction parameter anm is found from the large sets of VLE and LLE data in the literature, which are tabulated for many subgroups. It is worth noting that a m a. There are some modifications to the original UNIFAC equation in order to make the model robust for some complex systems. In the UNIFAC-DM method, the modification is made on the combinatorial part ... 

The combinatorial part In yp is calculated from pure-component properties. The residual part In yf is calculated by using binary interaction parameters for solute-solvent group pairs determined by fitting phase equilibrium data. Both parts are based on the UNIQUAC set... 

The first two terms correspond to the combinatorial entropy terms of Eq. (1) and form the non-interacting part of the structure factor which is just a weighted average of the single-chain structure factors SA(q) and SB(q) of both blend components. SA(q) and SB(q) are characterized by the radius of gyration RgA= aA(NA/6)1/2 and RgB=aB(NB/6)1/2, where aA and aB are the statistical segment lengths of polymer A and polymer B, respectively. The last term of Eq. (4) yields the SANS determined interaction parameter %SANS ... 

The first term of Equation 16.36 is due to combinatorial effects and is derived from lattice theory. The second energetic term is of a rather empirical nature and includes only the adjustable parameter of the model, the so-called FH interaction parameter... 

The combinatorial and residual terms are obtained from the original UNIFAC. An additional term is added for the FV effects. An approximation but at the same time an interesting feature of UNIFAC-FV and the other models of this type is that the same UNIFAC group-interaction parameters, i.e., those of original UNIFAC, are used. No parameter estimation is performed. The FV term used in UNIFAC-FV has a theoretical origin and is based on the Flory equation of state ... 

Parameter r measures the number of segments of a molecule for the term v in the Flory-Huggins equation. Parameters q, and are surface areas that are interchangeable for all except strongly hydrogen-bonded water and alcohols. Parameters r, q, and are pure-component molecular structure parameters. The combinatorial is dependent only on pure-component parameters. The residual depends additionally on binary interaction parameters and x ,... 

The link between GC quantities and the interaction parameters of solution theories is readily established (39). In statistical theories of solution thermodynamics, the solute activity is expressed as the sum of two terms, a combinatorial entropy and a noncombinatorial free energy of mixing. In the Flory-Huggins approximation one has,... 

The gas-lattice model considers liquids to be a mixture of randomly distributed occupied and vacant sites. P and T can change the concentration of holes, but not their size. A molecule may occupy m sites. Binary liquid mixtures are treated as ternary systems of two liquids (subscripts 1 and 2 ) with holes (subscript 0 ). The derived equations were used to describe file vapor-Uquid equilibrium of n-alkanes they also predicted well the phase behavior of -alkanes/PE systems. The gas-lattice model gives the non-combinatorial Helmotz free energy of mixing expressed in terms of composition and binary interaction parameters, quantified through interaction energies per unit contact area (Kleintjens 1983 Nies et aL 1983) ... 

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