Ternary interaction parameter

Mattioli and Bishop (1984), the mixture is of regular type but has a positive bulk interaction parameter between 0.6 and 3 kcal/mole. Extension to multicomponent mixtures was attempted by Berman (1990), Ganguly (personal communication), and Ottonello et al. (in prep.). Berman (1990) applied the model of Berman and Brown (1984), originally conceived for silicate melts, to aluminiferous garnets and deduced the magnitude of ternary interaction parameters by applying... 

Equation (5.21) assumes ternary interactions are small in comparison to those which arise from the binary terms. This may not always be the case and where evidence for higher-order interactions is evident these can be taken into account by a further term of the type Gijit = x< xj Xk Lijk, where Lijk is an excess ternary interaction parameter. There is little evidence for the need for interaction terms of any higher order than this and prediction of the thermodynamic properties of substitutional solution phases in multi-component alloys is usually based on an assessment of binary and ternary terms. Various other polynomial expressions for the excess term have been considered, see for example the reviews by Ansara (1979) and Hillert (1980). All are, however, based on predicting the properties of... 

Monnin 1989), which is analogous to Eq. 2.38. Equation 2.76 does not, however, include ternary interaction parameters (6 and S ) because the database for calculating these terms is inadequate at present (Monnin 1989 Pitzer 1991, 1995 Krumgalz et al. 1999). In Eq. 2.76, the Debye-Huckel term is given by... 

Equation 11.32 is used to model a single-phase liquid in a ternary system, as well as a ternary substitutional-solid solution formed by the addition of a soluble third component to a binary solid solution. The solubility of a third component might be predicted, for example, if there is mutual solid solubility in all three binary subsets (AB, BC, AC). Note that Eq. 11.32 does not contain ternary interaction terms, which ate usually small in comparison to binary terms. When this assumption cannot, or should not, be made, ternary interaction terms of the form xaXbXcLabc where Labc is an excess ternary interaction parameter, can be included. There has been httle evidence for the need of terms of any higher-order. Phase equilibria calculations are normally based on the assessment of only binary and ternary terms. 

Since the concentrations of impurities in solar cell grade silicon are in the range from ppb to a few percent, it is not necessary to take ternary interaction parameters into account. The activity coefficient of impurity, i , in a n-tli multicomponent system is given by differentiating (13.3) ... 

YOU Young, T.-H. and Chuang, W.-Y., Thermodynamic analysis on the cononsolvency of poly(vinyl alcohol) in water-DMSO mixtures through the ternary interaction parameter (exp. data by T.-H. Young), J. Membrane Sci., 210, 349, 2002. 

From this quantity w, the ternary interaction parameter y (see Equation 299) has a value w 10 . ... 

Pitzer ternary interaction parameter sane values tabulated in Appendix 5.1... 

The double-indexed g parameters are for binary interaction parameters. The first line of the above relation represents the combinatorial part, and the second line the residual part of the reduced segment molar Gibbs energy of mixing. This relation also contains a ternary interaction parameter tij that accounts for the expectation that the interaction between two components of the ternary mixture may change in the presence of a third component. 

Equation (61) makes allowance for differences in the genesis of ternary contacts but it does not yet consider that the number of segments of the third component in the coordination sphere of a certain binary contact might deviate from that expected from the average composition due to very favorable or unfavorable interactiOTis (quasi chemical equilibria). One way to model such effects consists of the introduction of composition-dependent ternary interaction parameters, as formulated in the following equation ... 

The relations presented for ternary mixtures open the possibility for investigation of the extent to which their thermodynamic behavior can be forecast (neglecting possible contributions of ternary interaction parameters) if the binary interaction parameters of the three subsystems are known as a function of composition from independent experiments. For such calculations, it is important to make sure that the size of a segment is identical for all subsystems. The fact that most of the... 

Fig. 35 Phase diagram of the PS/PVME/THP system at 20°C. Circles measured cloud points, closed stars calculated critical points, open stars calculated tie lines. The values of the specific ternary interaction parameters (61) are indicated on the edges, the binary interaction parameters are the same as in Fig. 34 [74]...

The agreement between the actually measured demixing behavior and that modeled on the basis of binary interaction parameters plus composition-independent ternary interaction parameters is surprisingly good. However, the results also demonstrate how sensitive the calculated phase diagrams can be with respect to the details of some interaction parameters. For instance, the analogous experiments... 

Furthermore, it has helped a better understanding of ternary mixtures with respect to the conditions that the subsystems must fulfill for the occurrence of cosolvency or cononsolvency, as well as concerning the necessity for the use of ternary interaction parameters. Suggested further investigations concern mixtures containing charged macromolecules and a more detailed analysis of the predictive power of the present approach. 

Ternary interaction parameter between groups i, L and k Poly(barium styrene sulfonate)... 

The following diagrams model the complete phenomenology of phase diagrams (49,55) for polymer solutions in mixed solvents. To this end all ternary interaction parameters were set zero and the number of characteristic parameters was limited to a maximum of two for each binary system. Nevertheless all experimental observations made with ternary systems are reproduced realistically. 

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