Differential molecular weight distribution

Fig. 24a and b. Differential molecular weight distribution curves of polyisoprene prepared, a) with oligoisoprenyl-Li b) with oligoisoprenyl-Li in the presence of NNN N tetramethyl ethylenediamine [TMDA]/[OLi] = 0.01) (A. Davidjan, N. N. Lolaev, V. Sgonnik, B. Belenkii, V. Nesterov, B. Erus-salimsky, Ref. 1531... 

Also, the zeroth moment of the differential molecular weight distribution, DMWD, may be obtained by integration of the simplified equation ... 

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. 

Note the difference between K, the Mark-Houwink constant, and k defined in Equation 12. The differential sedimentation distribution can be transformed to the differential molecular weight distribution by taking the derivative of Equation 11 with respect to s, to get... 

Figure 8. A plot of the differential molecular weight distribution for the poly(l-amidoethylene) standard.

A relationship has been developed by means of which more valid values are obtained by GPC for the average degrees of polymerization for cellulose. This can be extended to other polymers. With automation of data acquisition and computer processing of data, a differential molecular weight distribution and complete information on DP of the sample are available in very short order with a low degree of error. The ready availability of narrow, well characterized polystyrene fractions for calibration makes this procedure highly attractive in view of the lack of similar standards of cellulose. 

FJg. 2. Cumulative and differential molecular weight distribution for polystyrene from mutual radiation... 

When the relation between D and M is established, we can easily convert G(D) obtained by dynamic LLS into a differential molecular weight distribution, such as fw(M). We have successfully applied the above methods to various kinds of polymeric and colloidal systems, such as for Kevlar [15, 23], fluoropolymers (Tefzel Teflon) [12,30-35,52], epoxy [53-55],polyethylene [56,57], water-soluble polymers [18,50-51,58,59], copolymers [60-62], thermoplastics [63-65] and colloids [66-72]. Three of those applications are illustrated below. 

Fig. 4 Left Fluorescence, MALLS (90°) and RI signals from a CCOA-labeled cellulose sample. Right Differential molecular weight distribution and degree of substitution of carbonyl groups (DSco) as calculated from the detector outputs...

The instantaneous differential molecular weight distribution is given by ... 

Shortt, D.W. Differential molecular weight distributions in high performance size exclusion. J. Liq. Chromatogr. Relat. Techna. 1993,16, 3371 3391. 

To obtain the molecular weight distribution, the fractions of known mass must be characterized by a molecular weight technique. The integral and differential molecular weight distribution may be evaluated using a method (154) that assumes no particular molecular weight distribution function for the fraction or treatments that assume a particular model for the molecular weight distribution of the fractions (155). Fractionation may also be achieved by crystallization from dilute solution (156). 

Fig. 10.29. Typical differential molecular weight distribution curves.

Reliable determination of the differential molecular weight distribution by means of the dasscal precipitation or extraction procedures is not possible. In the e duation of the fractionation data problems are encountered that can, in fact, only be solved satisfactorily if the distribution is known beforehand. 

The function w(M) is called the differential molecular weight distribution. We can also define a cumulative distribution, F M), which is the fraction of all the molecules in the sample that have molecular weights equal to or less than M. This can be represented in terms of w(M) as shown by Eq. 2.47. 

Using Leibniz rule and differentiating twice, we arrive at the following differential equation for the differential molecular weight distribution. 

The scales on the [G co)/Glif versus (1/ ) plot can now be shifted to yield the final estimate of the cumulative or differential molecular weight distribution. Tuminello et al. [20, 21 ] used this technique to determine the MWD of several fluoropolymers that could not be analyzed using GPC due to their insolubility. 

Fig. 23. Differential molecular-weight distribution curves of ultrasOTically degra d dM-tran (My,= 39700, =...

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