Polymers polynomials

SEC measurements were made using a Waters Alliance 2690 separation module with a 410 differential refractometer. Typical chromatographic conditions were 30°C, a 0.5-ml/min flow rate, and a detector sensitivity at 4 with a sample injection volume of 80 fil, respectively, for a sample concentration of 0.075%. All or a combination of PEO standards at 0.05% concentration each were used to generate a linear first-order polynomial fit for each run throughout this work. Polymer Laboratories Caliber GPC/SEC software version 6.0 was used for all SEC collection, analysis, and molecular weight distribution overlays. 

Palit S.R., Kar I. Polynomial expansion of log relative viscosity and its application to polymer solutions. Journal of Polymer Science Part A-1, 5,10 (1967) 2629-2636. 

We demonstrate that the physical properties of Xe adsorbed in mesoporous MCM-41 molecular sieves can be deduced from the analysis of the variable temperature l2,Xe NMR chemical shift data. For example, the interactions between the adsorbed Xe and the wall of the adsorbent, 8S. Our results indicate that the interactions arise from Xe adsorbed in mesoporous MCM-41 deviates significantly from not only the bulk Xe, but also from Xe adsorbed on microporous adsorbents or polymer surfaces. At a given temperature T, the pore size dependence of 8S can be described by the empirical relation 8,(T, d) = A(T)/(d + B(T)). The two temperature-dependence parameters were expressed by polynomial functions whose temperature coefficients were also revealed explicitly to the second order. 

Typically, these methods arrive at the same finite difference representation for a given problem. However, we feel that Taylor-series expansions are easy to illustrate and we will therefore use them here in the derivation of finite difference equations. We encourage the student of polymer processing to look up the other techniques in the literature, for instance, integral methods and polynomial fitting from Tannehill, Anderson and Pletcher [26] or from Milne [16] and finite volume approach from Patankar [18], Versteeg and Malalasekera [27] or from Roache [20]. 

The dependence of viscosity r of dilute polymer solutions on concentration c can be described by a polynomial in the form [31,38],... 

Alexander Polynomials as a Tool for Numerical Investigations of Polymers with Topological Constraints... 

The so-called Jones polynomials [38] are even more strongly invariant than the Alexander ones. However, their calculation requires far more computer capacity calculation of an Alexander polynomial takes in the order 0(/3) operations, where / is the number of selfintersections of contour projection on the plane on the other hand, the calculation of a Jones polynomial takes in the order 0(e ) operations. This is why the existing attempts to use Jones polynomials in computer experiments with ring polymers have not been successful as yet Nevertheless, the construction of algebraic polynomial invariants of knots and links seems to be of great importance in principle, and we shall consider it in the next section. 

Another peculiar property of LCPs is shown in Fig. 15.47, where the transient behaviour of the shear stress after start up of steady shear flow is shown for Vectra A900 at 290 °C at two shear rates. We will come back to this behaviour in Chap. 16 for lyotropic systems where this behaviour is quite common and in contradistinction to the transient behaviour of conventional polymers, as presented in Fig. 15.9. This damped oscillatory behaviour is also found for simple rheological models as the Jeffreys model (Te Nijenhuis 2005) and according to Burghardt and Fuller, it is explicable by the classic Leslie-Ericksen theory for the flow of liquid crystals, which tumble, rather than align, in shear flow. Moreover, it is extra complicated due to the interaction between the tumbling of the molecules and the evolving defect density (polynomial structure) of the LCP, which become finer, at start up, or coarser, after cessation of flow. 

The effects of plasticizers has also been studied by PAL [64, 65]. The addition of a plasticizer to polymers generally has the effect of lowering the Tg, however in some cases an anti-plasticization can occur. Borek et al [65] have shown that the fraction of free volume in PVC polymers could be fit with a fourth order polynomial as a function of plasticizer concentration. The decrease in th Tg with increasing amount of plasticizer is attributed to this increase in the free volume of the polymers. 

This method is to be used to calculate the specific volume of a pure polymer liquid at a given temperature and pressure. This procedure uses the empirical Tait equation along with a polynomial expression for the zero pressure isobar. The method requires only the equation constants for the polymer. 

For an ideal solution, then, a plot of n,ic versus c should be a straight line at constant temperature. But, as you might expect, there is a variation with concentration (Figure 12-7). Just as with real gases, however, the data can be fit to the polynomial we call a virial equation. But, what do we do about polymers that have a distribution of molecular weights ... 

The column set was calibrated with a series of 15-20 narrow-distribution polystyrene (PS) standards (Polymer Laboratories), and the data were fitted to a third-order polynomial... 

Fig. 13. Crazing stress versus temperature for a 1,800,000 molecular weight PS deformed at a rate of 4.1xl0" s . A second-order polynomial fit is drawn through the data. Also shown, as a dashed line, is a linearly decreasing shear yield stress (From Ref. courtesy J. Polymer Sci.-Polymer Phys. Wiley))...

As already mentioned in Sects. 1.1 and 1.2, a characteristic feature of intramolecular mobility in polymers is the existence of relaxation time spectra. In this case, the time dependence of the mean value of the Legendre polynomials of the 2nd order = (3/2)[(cos2e>- (1/3)] is given by Eqs. (1.2.9) and(1.2.10). 

Photoisomerization was studied from a purely photochemical point of view in which photo-orientation effects can be disregarded. While this feature can be true in low viscosity solutions where photo-induced molecular orientation can be overcome by molecular rotational diffusion, in polymeric environments, especially in thin solid film configurations, spontaneous molecular mobility can be strongly hindered and photo-orientation effects arc appreciable. The theory that coupled photoisomerization and photo-orientation processes was also recently developed, based on the formalism of Legendre Polynomials, and more recent further theoretical developments have helped quantify coupled photoisomerization and photo-orientation processes in films of polymer. 

Knot theoretical techniques are easily applicable to polymer chains that do form actual knots or links, such as some DNA fragments or various catenanes [59-72,204-213]. By appropriate modifications, the knot theoretical polynomials are also applicable to the analysis of chirality properties of general molecules that may not form knots by themselves, but the space around them can be represented by a knot. This approach has led to the concept of chirogenicity, and to a nonvisual, algorithmic, computer-based analysis of molecular chirality [62]. 

Hosoya, H. (1991). Factorization and Recursion of the Matching and Characteristic Polynomials of Periodic Polymer Networks. J.Math.Chem., 7,289-305. 

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