Logarithmic normal molecular weight distribution

Figure 8-6. Calculated W/A , ratios as a function of the exponents t, for a Schuiz-Flory (—) or a generalized logarithmic normal (--) molecular-weight distribution for. in cadi case. w/M, 2. SI may be A , or (After H.-G. Elias, R. Bareiss and J. G.

Log normal distribution (logarithmic normal distribution). A statistical probabiUty-density function, characterized by two parameters, that can sometimes provide a faithful representation of a polymer s molecular-weight distribution or the distribution of particle sizes in ground, brittle materials. It is a variant of the familiar normal or Gaussian distribution in which the logarithm of the measured quantity replaces the quantity itself. Its mathematical for is... 

Log Normal Distribution The log normal distribution is described by the following equations. It is a two-parameter distribution derived by using the natural logarithm of n rather than n in the normal distribution. Although the log normal distribution is not the exact solution to any of the simple polymerization kinetic schemes, it is often used to fit experimental molecular weight distribution data ... 

H(x) is a continuous distribution of the logarithms of relaxation times and is called the "relaxation spectnun" by rheologists, whilst the "true" distribution of relaxation times is xH(x). We have reported on Figure 1 the normalized distribution of relaxation times for 4 polystyrene samples with polydispersity indices ranging from 1.05 to 4.2 [2]. It is clear that the distribution of relaxation times broadens with the distribution of molecular weights these features will be anal3 d in terms of molecular models in sections 3 to 6. 

This is a specific case of logarithmic distribution. It is distribution which allows one to range of molecular weights. The of the logarithm of the molecula to be normally distributed. The defined by two parameters and is... 

The normal distribution function extends to both the positive and negative sides. To avoid the negative molecular weights (which do not exist), an assumption is made that the logarithm of the molecular weight is normally distributed. Thus, we replace x by In X and m by In m. Then the weight distribution becomes... 

Under the assumption that the weight distribution of the logarithm of molecular size is normally distributed, we can replace r in the normal distribution function by In r and r ly InFn,... 

The generalized logarithmic normal distribution is skewed to large values of r, but Kotliar (237, 238) shows that this distribution is not a good representation of a polymer after either low- or high-molecular weight material is removed or degradation has occurred. 

CULAR WEIGHT FOR POLYMERS WITH A LOGARITHMIC NORMAL (LN) DISTRIBUTION OF THE MOLECULAR WEIGHT... 

WITH A logarithmic normal (LN) DISTRIBUTION OF the MOLECULAR weight ... 

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