Logarithmic normal molecular weight

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.

If the observed peaks are assigned the normal sequence - dimer, trimer, tetramer etc., the plot of the retention volume versus the logarithm of the molecular weights does not produce a smooth line. The peaks representing the normal loigomeric sequence can be selected by trial and error and then a different series of peaks is discovered where the oligomerization follows a different route. The characterization of this second series of peaks has been achieved by mass spectroscopy and reported by Lattimer et. al. (14). 

Warmuth [34] when they adopted NPC for the analysis of supermolecules, such as hemicarcerplexes. Hemicarcerplexes are complexes formed with hemi-carcerand host and guest molecules. As shown in Figure 5-6, hemicarcerands possess a very hydrophobic structure with molecular weight over 2000 and is insoluble in protic solvents. A normal-phase HPLC method was developed using a silica column with dichloromethane and diethylether as the mobile-phase system. The authors demonstrate that the chromatographic retention of hemicarcerplexes is mainly dominated by its size. Furthermore, a linear relationship between the logarithmic retention factor and the size of the hemicarcerplexes was observed for linear guest molecules independent of their polarity. 

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... 

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... 

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... 

FIG. 10-9. Normalized logarithmic recoverable creep compliance plot for Rouse theory (I) and for polystyrene of high molecular weight (II) (Example III of Fig. 2-1). 

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 ... 

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