Heterogeneous parameters

In the case of copolymers the heterogeneity parameters need to be considered in the analysis. If we assume an AB-copolymer the heterogeneity parameters are given by... 

The stretched exponential function, A = Ao exp(—tfxf, has been applied to the fluorescence of unstained tissue [82-84], In particular, researchers at Paul French s group at Imperial college [82], show that the use of the stretched exponential, the parameters of mean, and the heterogeneity parameter (the inverse of the degree of stretch, ft) gives better tissue contrast and better fit than the mono- or multiexponential models. 

Unfortunately there is no expression corresponding to Eq. (101) for copolymers in which the measured apparent molecular weight M appears solely as a function of Ma, Mb, Mc and the true molecular weight M. The only available and tractable expression in the following one162) [Eq. (108)] in principle it is capable of yielding M and the values of the five heterogeneity parameters. 

A moving coordinate system was used to account for the velocity term of the ADE. A problem similar to the one presented in Skaggs and Kabala [58] is used for the QR study. The results are less accurate than that of the TR approach, but it is computationally less expensive. The authors claimed that it is much easier to incorporate heterogeneous parameters in the QR method. However, to date, heterogeneous parameters have not been incorporated either in the QR method or in the Tikhonov method by Skaggs and Kabala. 

The basic method yields good measurements of M but the heterogeneity parameters are generally found not to be credible for statistical copolymers. This may be due to a dependence of dn/dc on polymer molecular weight, at low molecular weights [2]. 

Representative results for a blend and a diblock copolymer are shown in Table I. It is evident for both samples that the on-line infrared and RI concentration detectors provide excellent estimates of overall composition, component molar masses, and the heterogeneity parameter H, Data for blends in Table I were obtained to assess the proposed methods. Further evidence for the capability of the SEC-LALLS procedure can be seen in Figure 2, which shows plots of data for Ws and H computed across the MMD for blend sample 4. In this figure the expected results... 

The results demonstrate that coupled chromatographic techniques with multiple detectors permit the determination of average composition data, heterogeneity parameters, and separations of homopolymers and copolymers. The methodology reviewed here enables a distinction to be made between copolymers and polymer blends. 

In the above, the heterogeneity parameters nt and m characterize the shape (width and asymmetry) of adsorption energy distribution function, and the equilibrium constant, ATi , describes the position of distribution function on energy axis... 

For the special values of heterogeneity parameters m = m s (0,1) the isotherm (7) reduces to the known Langmuir-Freundlich (LF) equation. 

P and Q are known as the "heterogeneity parameters" and are a measure of the polydispersity of chain composition. Using equation 24 together with equation 23 in the form ... 

Now we solve first the non-linear Eq. (81a) using, for example, the iteration method to obtain the value of for the assumed values of the parameters K "/ and K as well as of the heterogeneity parameters. Next, we solve the second non-linear equation (81b) for the same parameters K , Kg , kT/ci, (i=A,C), and for the new calculated value of In the same way as before, we reduce the number of the intrinsic equilibrium constants from four to two. 

The numerical procedure as described above gives credible results with respect to the values of the surface phase capacity the systems studied. On the other hand, knowledge the heterogeneity parameters Bj and X° gives possibility for evaluating the energy distribution functions by means of the suitable equation corresponding with the expression (8) [1]. 

The total mole fraction Xj depends on the bulk concentration x only and we can state that lnfj2i, vs. Xj is the function of the heterogeneity parameters n and m. For this reason this function is helpful for characterizing the liquid - solid experimental systems. The general form of the dependence lnfi2h vs. x obtained by means of Eqs. (13) and (14) has the following form (for the NBP-na model of adsorption system) ... 

Figure 12. Temperature-dependence of the fraction of liquid crystalline lipid (() and the heterogeneity parameter ("Aj, proportional to the mean square deviation of the order parameter) for the H NMR spectra of Acholeplasma laidlawii membranes shown in Figure 11 (C-14 0-m-d3 90%). The calorimetric phase transition temperature is indicated by Tc. (Unpublished data of H. C. Jarrell, R. Deslauriers, and...

Deviations, which can be expressed by an average heterogeneity parameter Hi = only indicate that the model is not fulfilled, but do not answer why... 

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