Molecular weight distribution calculation

Molecular Weight and Molecular Weight Distribution Calculations... 

In GPC-SEC with universal calibration, at each retention volume i of the chromatogram, a value of is read, and for molecular-weight and molecular-weight-distribution calculations the values of [rj]i are needed. These are obtained, whenever the equation is known, from the Mark-Houwink constants of the polymer in the solvent and at the temperature of chromatographic elution. 

Fig. 11.8 The molecular-weight distribution calculated using the Schulz function...

Fig. 4 Readings from the concentration and photometer analysis for a polyacrylamide in aqueous solution and the absolute molecular weight distribution calculated by using the Zimm-Debye equation for 1ow-angle-laser-1i ght-scatteri ng.

Figure 2.2 Molecular weight distributions calculated using the log-normal distribution function (Eq. 2.70) for = 100,000 and PI = 1.01,1.03, and 1.10. Although these samples might be described as "nearly monodisperse" in comparison to commercial polymers, there are many molecules that are significantly larger or smaller than average.

At 25°C, the Mark-Houwink exponent for poly(methyl methacrylate) has the value 0.69 in acetone and 0.83 in chloroform. Calculate (retaining more significant figures than strictly warranted) the value of that would be obtained for a sample with the following molecular weight distribution if the sample were studied by viscometry in each of these solvents ... 

The width of molecular weight distribution (MWD) is usually represented by the ratio of the weight—average and the number—average molecular weights, MJM. In iadustry, MWD is often represented by the value of the melt flow ratio (MER), which is calculated as a ratio of two melt indexes measured at two melt pressures that differ by a factor of 10. Most commodity-grade LLDPE resias have a narrow MWD, with the MJM ratios of 2.5—4.5 and MER values in the 20—35 range. However, LLDPE resias produced with chromium oxide-based catalysts have a broad MWD, with M.Jof 10—35 and MER of 80-200. 

If the initiation reaction is much faster than the propagation reaction, then all chains start to grow at the same time. Because there is no inherent termination step, the statistical distribution of chain lengths is very narrow. The average molecular weight is calculated from the mole ratio of monomer-to-initiator sites. Chain termination is usually accompHshed by adding proton donors, eg, water or alcohols, or electrophiles such as carbon dioxide. 

The significance of knowing the K and a values of fully hydrolyzed PVA is that molecular weight distribution data can be directly calculated using two methodologies. The first is the Mark-Houwink method, which requires prior knowledge of K and a values for fully hydrolyzed PVA and calibration standards such as PEG, PEO, or PSC. The second method is the intrinsic viscosity method. This method utilizes a simple ratio of the concentration signal to the specific... 

Analytical expressions have been derived for calculating dispcrsitics of polymers formed by polymerization with reversible chain transfer. The expression (eq. 17) applies in circumstances where the contributions to the molecular weight distribution by termination between propagating radicals, external initiation, and differential activity of the initial transfer agent are negligible.21384... 

Fig. 44a. Theoretical molecular weight distribution of a polymer sample degraded along the central streamline at different strain rates, calculated with a pre-exponential factor A = 1014s-1 (I) strain rate e = 75000s-1 (II) strain rate e = 88000s-1 (III) strain rate e = 190000 s- b Theoretical molecular weight distribution of a polymer sample degraded along the central streamline at different strain rates, calculated with A = 104 s-1 (I) strain rate e = 100000 s -1 (II) strain rate e = 120000 s 1 (III) strain rate e = 300000 s -1 (Solid line polymer before degradation, dotted line, degraded polymer)...

Calculated Molecular Weight Distributions. The calculated weight fraction distributions for the micro-mixed, segregated, and micro-mixed reactor with dead-polymer models for Runs 2, 5,... 

The fair degree of consistency observed in the values of <()j) for Seeds II and III and the excellent agreement between the experimental molecular weight distribution and those calculated with, lends credibility to the dead-polymer model. The... 

Table 13.4 tabulates results at the same value of as in Table 13.3. The polydispersities are lower than when the same average chain length is prepared by a binary polycondensation going to completion. The stoichiometry-limited binary polycondensations have a higher polydispersity because the monomer in stoichiometric excess (the B monomer) is included in the calculations. This broadens the molecular weight distribution.

Example 13.4 Calculate the molecular weight distribution for a self-condensing polymerization with — + Stop the calculations... 

If Mark-Houwink coefficients were supplied at setup time, the chromatogram may be converted into the differential molecular weight distribution of the specimen. Various averages characterizing this molecular weight distribution are then calculated. The molecular weight distribution may be written to a file. 

To generate the necessary distribution functions, the ratio of is used to approximate the true molecular weight distribution by a Schulz-Zimm distribution. It is also assumed that the reactive functional groups are distributed randomly on the polymer chain. The Schulz-Zimm parameters used to calculate distribution functions and probability generating functions (see below) are defined as follows ... 

These parameters are used calculate the site and mass distribution functions assuming a Schulz-Zimm molecular weight distribution. The Schulz-Zimm parameters are calculated in lines 930-950. The weight fraction of diluent (as a fraction of the amount of polymer) is then sought. If there is no diluent enter 0. If there is a diluent, the functionality and molecular weight of the diluent is requested (line 1040). The necessary expectation values are computed (lines 1060-1150). 

Advanced computational models are also developed to understand the formation of polymer microstructure and polymer morphology. Nonuniform compositional distribution in olefin copolymers can affect the chain solubility of highly crystalline polymers. When such compositional nonuniformity is present, hydrodynamic volume distribution measured by size exclusion chromatography does not match the exact copolymer molecular weight distribution. Therefore, it is necessary to calculate the hydrodynamic volume distribution from a copolymer kinetic model and to relate it to the copolymer molecular weight distribution. The finite molecular weight moment techniques that were developed for free radical homo- and co-polymerization processes can be used for such calculations [1,14,15]. 

As early as 1952, Flory [5, 6] pointed out that the polycondensation of AB -type monomers will result in soluble highly branched polymers and he calculated the molecular weight distribution (MWD) and its averages using a statistical derivation. Ill-defined branched polycondensates were reported even earlier [7,8]. In 1972, Baker et al. reported the polycondensation of polyhydrox-ymonocarboxylic acids, (OH)nR-COOH, where n is an integer from two to six [ 9]. In 1982, Kricheldorf et al. [ 10] pubhshed the cocondensation of AB and AB2 monomers to form branched polyesters. However, only after Kim and Webster published the synthesis of pure hyperbranched polyarylenes from an AB2 monomer in 1988 [11-13], this class of polymers became a topic of intensive research by many groups. A multitude of hyperbranched polymers synthesized via polycondensation of AB2 monomers have been reported, and many reviews have been published [1,2,14-16]. 

Fig. 53.—Molecular weight distribution in poly-(hexa-methylene adipamide) as obtained by fractionation (points) compared with curves calculated from Eq. (3) for two values of p, (Taylor. )...

Fig. 56.—Molecular weight distributions for multichain polymers for the several values of / indicated, as calculated from Eq. (22 ). Number averages for the different distributions are identical. (Schulz. )...

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