Molecular weight distributions exponential functions

He then defined a polydispersity index of relaxation times as (r )/(r ) and pointed out that this parameter increases as the MWD becomes broader. In an entangled melt, if we limit our attention to the plateau and terminal zones, and if the relaxation spectrum function is known over the full range of times, it can be shown that this ratio of times is equal to /f. As we have seen, the product /° G indicates the breadth ofthe molecular weight distribution of a linear polymer and can be calculated directly from rheological data. For example, if the relaxation modulus in the plateau and terminal zones is represented by a single exponential ... 

Three of the four resins yield extracts of the functionality of Equation 4. The slope of the exponential decay allows for the evaluation of x. The resin, see Table II, initiated by benzyl dimethyl amine (BDMA) at the stated cure cycle, when subjected to leaching yields an extract of low solubility and a distribution of oligomeric molecules of low number average molecular weight. 

To treat the effects of heterogeneity on 0, we take the familiar exponential function [Schulz (228)] for the distribution of molecular weights,... 

When there are more than one species present in a sample, each contributes to give a correlation function which is a sum of exponentials and a power spectrum which is a sum of Lorentzians. The intensity of each component in the composite function is proportional to the product of the molecular weight and the concentration in (w/v) units, assuming the refractive index increments of each component are identical. Two basic approaches are available to extract the particle distribution from the QLLS data. 

The exponential function. The mathematical forms chosen to this point were those considered possible for tails. We now consider the exponential distribution function that may be relevant to loopy adsorption, at least for high molecular weight polymers. The exponential distribution function must be artificially truncated at the physical limit of the steric layer. This leads to a normalized segment density distribution of the form... 

Generally, in order to fit the observed FID, a series of exponential functions (Eq. 8.3) are used because the distribution of dipole interaction is expressed by Lorentzian function. This is true for the solution, melt, and amorphous phases of the polymers. Actually, a PE melt with a low molecular weight (MW) exhibits a single exponential curve [17-20]. On the other hand, Weibulhan functions (Eq. 8.4) lit for the phase with partially restricted motion such as the interfacial phase [8,21]. 

The concept of a well-stirred segregated reactor which also has an exponential residence time distribution function was introduced by Dankwerts (16, 17) and was elaborated upon by Zweitering (18). In a totally segregated, stirred tank reactor, the feed stream is envisioned to enter the reactor in the form of macro-molecular capsules which do not exchange their contents with other capsules in the feed stream or in the reactor volume. The capsules act as batch reactors with reaction times equal to their residence time in the reactor. The reactor product is thus found by calculating the weighted sum of a series of batch reactor products with reaction times from zero to infinity. The weighting factor is determined by the residence time distribution function of the constant flow stirred tank reactor. 

The decay is generally not in the form of a simple exponential decay function, but usually deviates from it because of various complexities in the fluid, such as particle size distribution in solution and/or multimodes of molecular motions. Those non-single exponential decays can be expressed as a linear superposition of monoexponential decays, weighted with a distribution frmction G(r, ), the spectrum of decay rates, resulting in a Laplace transformation. 

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