Ternary interaction

Once the results have been collected they may be analysed to show the effects of each variable separately, T, S, M, the interaction of two variables, T-S, T-M, S-M, and tlrree variable interaction, T-S-M. These interaction terms measure the effect of the change in one variable on die results for the others, and T-S-M, tire ternary interaction indicates the dependence of the results for one variable of tire simultaneous change of the others. 

Vecchio A, Finoli C, Simine DD, Andreoni V (1998) Heavy metal biosorption by bacterial cells. Fresenius J Anal Chem 361 338-342 Walker SG, Flemming CA, Ferris FG, Beveridge TJ, Bailey GW (1989) Physicochemical interaction of Escherichia coli cell envelopes and Bacillus subtilis cell walls with two clays and ability of the composite to immobililze heavy metals from solution. Appl Environ Microbiol 55 2976-2984 Wightman PG, Fein JB (2001) Ternary interactions in a humic acid-Cd-bacteria system. ChemGeol 180 55-65... 

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... 

Binary interaction parameters H23 2 ternary interaction term JV123... 

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... 

Let us take the example of a simple commercial alloy such as Ti-6AI-4V. This is the most popular structural Ti alloy used worldwide. Essentially one would need to consider Ti-AI, Ti-V and Al-V binary interactions and Ti-Al-V ternary interactions. Unfortunately, although called Ti-6AI-4V, this alloy also contains small amounts of O, C, N and Fe and it therefore exists in the multi-component space within the Ti-Al-V-O-C-N-Fe system. There are then 21 potential binary... 

Louey MD, Stewart PJ. Particle interactions involved in aerosol dispersion of ternary interactive mixtures. Pharma Res 2002 19(10) 1524—1531. 

Induced spectra actually consist of contributions arising from free-to-free, free-to-bound, bound-to-bound, and bound-to-free transitions. At temperatures much greater than the well depth of the intermolecular potential, kT e, the observed induced absorption is nearly fully due to free-to-free transitions as Welsh and associates suggest, but individual dimer lines or bands may still be quite prominent unless pressure broadening and perhaps other processes (like ternary interactions) have obliterated such structures. However, at lower temperatures, kT collision-induced is a poor choice. 

Fig. 3.29. Onset of ternary (H2-Ar-Ar) interactions the variation of fitted profiles of the H2 So lines with density argon with 3% hydrogen admixture. Solid line zero density limit (profile of H2-Ar) dashed line at 185 amagat of argon where ternary interactions have affected the profile. The functions are scaled so that the areas under the curves are equal after [152],...

It is instructive to start this survey of measured, induced rotovibrational spectra again with hydrogen because, in this case, well-resolved rotational lines may be expected, certainly at the lower temperatures. The pressure-induced fundamental band of hydrogen has been studied at pressures up to thousands of atmospheres and temperatures from 18 to over 400 K. We will be interested here in the spectra obtained at the lowest gas densities ( pair spectra ) but will also consider the modifications of the rotovibrational features observed in dense fluids when ternary interactions become discernible. 

This can be viewed as a triptych of the three possible binary T-x diagrams on the facets, with ternary interactions represented by the prism interior. Many aspects of modem alloying technology can be described in terms of such 3D graphical representations, but these and other aspects of ternary behavior lie beyond the scope of the present treatment. 

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... 

One of the inherent limitations of the Pitzer approach is the necessity that all significant interactions among ions and neutral species must be quantified. In appendix Tables B.4 to B.6, we quantify 291 binary and ternary interaction... 

Eqs. 2.76, 2.80, and 2.81 ignore volumetric ternary interaction terms (6>E, Yjk) that could P y important roles in mixed salt solutions (Monnin 1989 ... 

Ternary interactions are considered in the case of the deformation of the triplet of the connected micro-elements A-V-B. If the translation of microelements leads to the change of the angle a — ZAVB, then the force with the magnitude Fa acts on micro-elements as the resistance against this change. The magnitude of the force Fa is calculated in the limit of elastic interactions as... 

All ternary interactions involving micro-elements of type C, C, P, and P are considered to be elastic and are calculated according to Eq. (68). 

Simple binary interactions were not enough to achieve a long-term stability [24, 25]. Table 6 presents relative stabilities of binary and quaternary complex structures. This table qualitatively illustrates the concept that multiple interactions are necessary in order to stabilize the final product. Both binary and some ternary interactions in Table 7 showed instability throughout the matrix. This behavior was very common. Matrix stability was normally evaluated in 0.9 % NaCl or PBS (phosphate-buffered saline), fluids of physiologic significance. The introduction of more ingredients into the nanoparticulate formulation also increased... 

Sandri, G., Bonferoni, M. C., Chetoni, P., Rossi, S., Ferrari, F., Ronchi, C., et al. (2006), Ophthalmic delivery systems based on drug-polymer-polymer ionic ternary interaction In vitro and in vivo characterization, Eur. I. Pharm. Biopharm., 62(1), 59-69. 

The Pitzer model can be used to obtain activity coefficients for solutes in low (<0.1 mol L ), intermediate (0.1-3.5 mol L ) and high (>3.5 mol L ) ionic strength solutions. The Pitzer equations include terms for binary and ternary interactions between solute species as well as a modified DH expression. The general formula is... 

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. 

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