Wesslau distribution

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

Figure 8-4. Integral mass distribution for a Wesslau distribution plotted on logarithmic-cumulative frequency graph paper. The mass fraction w, and viscosity average, ( (instead of, more correctly, < degree of polymerization of the fractions (O) were measured. The number...

Figure 8-5. Integral mass distribution for a Wesslau distribution. The viscosity-average degree of polymerization (X,), and the mass fraction were measured. The viscosity-average degree of polymerization of the original material with the already known exponent the median value Xjvf, and equation (8-23) and (8-24) were used to calculate the number- and weight-average molecular weights.

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

In the extended method of moments, a CLD is reconstructed based on calculated moments, as introduced above. The simplest way is a predefined mass CLD, such as the two-parameter Wesslau distribution (Pladis and Kiparissides, 1988) ... 

MC reaction probability for reaction channel p right-hand side of population balance for impeller compartment conditional probability for branching density (Tobita-based MC) parameter in the Wesslau distribution... 

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... 

Heidemann et al also presented a discontinuous method to calculate spinodal curves and critical points using two different versions of the Sanchez-Lacombe equation of state and PC-SAFT. Moreover, Krenz and Heidemann applied the modified Sanchez-Lacombe equation of state to calculate the phase behaviour of polydisperse polymer blends in hydrocarbons. In this analysis the polymer samples were represented by 100 pseudo-components. Taimoori and Panayiotou developed a lattice-fluid model incorporating the classical quasi-chemical approach and applied the model in the framework of continuous thermodynamics to polydisperse polymer solutions and mixtures. The polydispersity of the polymers was expressed by the Wesslau distribution. 

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... 

Modifications of this equation for other initial MWD s, which can be represented by Schulz-Zimm or Wesslau distributions, have been derived by Inokuti(12) and by Saito et al.(13). 

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