Distribution function Wesslau

The broadness of the distribution is given by the parameter j, obtained from the slope (tan 0) of the log-normal plot where tan d = 7A/2. The polydispersity ratio (Af /Afp = exp(7 /2) no and x are constants in the distribution function not determinable from the log-normal line. (Special cases of the general distribution function have also been employed, e.g., the Lansing and Kraemer relationship where x = 0 [304] and the Wesslau relationship [305], where x = —1). Fractionation data can also be adequately represented by other exponential distributions such as that of Tung [306] W(n) dn = abn exp(—an ) dn. (W(n) = 1 — exp(—an )). Fig. 14a(c). 

We can also assume a theoretical form of the distribution function such as the log-normal (Wesslau) distribution. In this case... 

The polydispersity of polyethylene can often be described with the help of the Wesslau distribution function ... 

Important differential mass-distribution functions (probability density function of mass-distribution) are the most probable distribution (Schulz-Flory), the Schulz-Zimm distribution, the Poisson distribution, Tung distribution, and logarithmic normal distribution (Wesslau distribution) [08IUP2]. Methods for the determination of distribution functions of molar mass are listed in Table 4.1.4. 

For widely distributed polymers such as radically polymerized polyethene the Schulz-Flory distribution function is unable to describe the high degree of asymmetry in the distribution. In this case, the Wesslau distribution (logarithmic normal distribution) is used and given by ... 

The spinodal curve and the critical points (including multiple critical points) only depend on few moments of the molar-mass distribution of the polydisperse system while the cloud-point curve the shadow curve and the coexistence curves are strongly influenced by the whole curvature of the distribution function. The methods used that include the real molar-mass distribution or an assumed analytical distribution replaced by several hundred discrete components have been reviewed by Kamide. In the 1980s continuous thermodynamics was applied, for example, by Ratzsch and Kehlen to calculate the phase equilibrium of a solution of polyethene in supercritical ethene as a function of pressures at T= 403.15 K. The Flory s model was used with an equation of state to describe the poly-dispersity of polyethene with a a Wesslau distribution. Ratzsch and Wohlfarth applied continuous thermodynamics to the high-pressure phase equilibrium of ethene [ethylene]-I-poly(but-3-enoic acid ethene) [poly(ethylene-co-vinylace-tate)] and to the corresponding quasiternary system including ethenyl ethanoate [vinylacetate]. In addition to Flory s equation of state Ratzsch and Wohlfarth also tested the Schotte model as a suitable means to describe the phase equilibrium neglecting the polydispersity with respect to chemical composition of the... 

In order to model polymer transport phenomena of this type, where polydispersity effects are important, it is not adequate to consider the polymer as a single component of concentration, c, as has been done so far in this chapter. The polymer itself is made up of many components which are different only in their size (although the Mark-Houwink parameters that apply for the polymer will be esentially the same for each of the polymer subcomponents). Thus it is necessary to use a multicomponent representation of the polymer molecular weight distribution in order to model the polymer behaviour adequately in such experiments. Brown and Sorbie (1989) have adopted this approach in order to model the Chauveteau-Lecourtier results quantitatively. They used a multicomponent representation of the MWD based on a Wesslau distribution function (Rodriguez, 1983, p. 134) with 26 discrete fractions being used to represent the xanthan. For this case, a set of convection-dispersion equations including dispersion and surface exclusion... 

Wesslau MWD. The model based on the Wesslau MWD has been described previously (JS). The weight fraction distribution of x-mer, where X is the degree of polymerization (DP), measured as a function of the logarithm of the degree of polymerization, is given by... 

Peak Shapes. In the case of the Wesslau MWD, the shapes of the peaks from the three detectors are always the same. For the Flory-Schulz distribution, the peak shapes are slightly different and the differences increase with increasing polydispersity. As the polydispersity increases, the LS and viscosity signals become narrower relative to the concentration detector signal and they also become less skewed. Figure 3 shows the peak variance of the viscosity and LS signals relative to the concentration detector peak variance as a function of polydispersity. The concentration detector peak variance increases from 0.25 mL when the polydispersity is 1.1 to 3.65 mL when the polydispersity is 3.3. The LS peak variance increases more slowly. The viscometer variance is in between the two but closer to the LS peak behavior. Figure 4 shows the relative skew of the peaks compared with the refractometer, where the skew is defined as... 

Crosslinking and main-chain scission in Marlex-50 has been reinvestigated recently by Dole and co-workers [272] using the Charlesby— Pinner function modified for a Wesslau molecular weight distribution (Table 18). Evidence of increase of Gc L with dose has been obtained and related to vinylene decay. GCL and Gcs at zero dose are almost independent of temperature but at 27 Mrad, GCL increases with temperature. 

---