Multiple-site model

The wall-PRISM theory has been extended to multiple site models [95], A simple example of a multiple-site model is a vinyl polymer (e.g., polypropylene) where there are three types of united-atom sites corresponding to CH2, CH, and CH3 groups. Ignoring end effects as before the PRISM equations take the form... 

The presence of at least two fluorophores, and possibly a third, associated with metal ion binding in fulvic acid strongly suggests the need for multiple binding site models. Existing linear and nonlinear models will be reviewed for both fluorescence quenching and enhancement. A new modified 1 1 Stem - Volmer model will be introduced as well as two site and multiple site models. Application of the models to Cu binding by fulvic acid and certain well defined model systems are discussed. 

Modeling of Multisite Fluorescence Quenching. A traditional multisite fluorescence quenching model using a Stem-Volmer approach has been developed and applied to quenching curves involving residual protein fluorescence (32). More recently, however, the multiple site model has been used to describe structural characteristics in diverse polymer environments (33-3f). The multisite Stem-Volmer model shown in equation 13 may be used to define multiple fluorescent binding sites present under one emission peak... 

This approach has been used by several researchers to model the microstructures of polymers made with the Ziegler-Natta and the Phillips catalysts using the method of instantaneous distributions, method of moments, or the Monte Carlo simulation. Several techniques have also been developed to help estimate model parameters for these multiple-site models, but it still remains a difficult problem... 

Figure 2. The agreement is good at low densities but both models tend to overestimate the pressure as the density increases. The single-site model accurately reproduces the density at coexistence but overestimates the pressure while the multiple-site model accurately represents the pressure but underestimated the density.

The results of the calibration procedure are tabulated in Table 3. Note the surface-surface interaction strengths are listed rather than the surface-fluid strengths to allow comparison between single-site and multiple site models. 

Figure 4. The PSDs obtained for Vulcan using nitrogen databases. Full line single-site model, dashed line multiple-site model.

When carbon dioxide databases were calibrated on Vulcan the surface fluid interaction was optimised to yield a best fit PSD with a surface area of 70 m. g". For single-site model the optimal Ess value was 25 K while for the multiple-site model the optimal Eg was 19 K. Both databases were able to accurately fit the adsorption isotherms with an RMS error of only 0.12 and 0.016 for the single-site and multiple-site models, respectively. These fits are shown in Figure 7. The PSDs for both molecular models are shown in Figure 8. The single-site PSD comprises four peaks dominated by a large peak at 8 nm and smaller peaks a 3, 2 and 0.8 nm. The main peak in the multiple-site PSD is shifted to 7.7 nm with smaller peaks aroimd 2 nm and 1 nm. 

The above evaluation of the mixed and the true reactivity ratios in dependence of the reaction parameters, i.e., Ti loading, temperature and conversion, is a first approach to identify different active centres in copolymerization. The two centres model is a first and simplest estimation. The reality aims at a multiple site model. 

It is clear that and values in the mono-site model have great signiflcance each is the ratio of two propagation constants and their product determines aU sequence distribution functions (see Section II, where all equations presented are for the mono-site model). Is this also true of the multiple-site model, or are the Tj and parameters here simply a useful mathematical device for correlating two experimental values, F and / (/58) ... 

The case of the multiple-site model has been discussed 129) N types of sites) and a generalized expression was given for the copolymer composition. The two main conclusions of this paper are that the (rj r2)eff value taken from the mean composition measurements falls between fj y2 values of the individual species, and that when (rj r2)etf and average... 

The real blockiness of the mealed copolymers is always higher than that calculated with (rir2)e f- Henc we underestimate tl% blockiness of the copolymers obtained with a multiple-site model if we use (rj r2)eff as the measure of copolymer randomness [the same conclusion as in (129) ]. 

It is evident from the previous section that in the multiple-site model the copolymer is a mixture of copolymers of different composition this fact alone accounts to a some extent for the experimentally determined copolymer blockiness. However, a study of the copolymer fractions demonstrates that it cannot explain all the results. The blockiness of the copolymer fractions remains high therefore the tendency to block formation is an intrinsic property of active sites rather than merely the result of copolymer mixing. For example, when the pro-pylene-butene-1 copolymer was Ihictionated (156), the resulting narrow fractions of composition Cp, 30-50% exhibit X-ray diffraction curves containing crystallinity peaks of both polypropylene and polybutene-1 type. An analogous result was demonstrated us (66) the blockiness of the fractions of propylene-4-methylpentene-l copolymers insoluble... 

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