Weight and number distributions

If A/w and of a polymer sample are known, we have information about the standard deviation and the variance of the number distribution. There is no quantitative information about the breadth of the weight distribution of the same sample unless and are known. As mentioned earlier, it is often assumed that the weight and number distributions will change in a parallel fashion and in this sense the Mw/M ratio is called the breadth of the distribution although it actually reflects the ratio of the variance to the square of the mean of the number distribution of the polymer (Eq. 2-34). 

Figure 10. Weight and number distributions of mesogens. Mesogen length is defined as the number of aromatic units per mesogen.

Since Hatch and Choate first published their equation, special graph paper has been developed and printed whereby plotting the cumulative percent of particles by number or weight, oversize or undersize, against particle size, results in a straight line. The mathematics of the distribution are such that one can readily transfer between weight and number distributions, and even area and diameter distributions [4],... 

There is a simple relationship between the weight and number distribution functions, regardless of the form of the distribution. This can be demonstrated as follows ... 

In general, resins are compatible with a large number of materials (oils, plasticizers, polyethylene waxes, rubbers). Compatibility depends on resin type, molecular weight and its distribution, resin structure and configuration, and finally on application requirements. 

N umerical simulations of reactor start-up were programmed, predicting monomer and initiator concentrations, total polymer concentration, weight and number average molecular weights, viscosity and population density distribution dynamics. The following two relationships obtained from steady state observations were utilized in the simulation. 

Number of units in a given primary molecule, and the weight and number averages for the distribution as a whole. 

We often use the ratio between the weight and number average molecular weights as a guide to summarize a polymer s overall molecular weight distribution. 

FIGURE 5 Typical lognormal distribution of particles based on weight and number. 

The width of the distribution function can be roughly characterized by the polydispersity coefficient, v, which is equal to the ratio of the weight and number averages... 

The breadth of the molecular weight distribution is described by the ratio of the weight and number average molecular weights or degrees of polymerization, and is referred to as the polydispersity index (PDI) or molecular weight distribution (MWD) [Eq. (8)]. 

For analysis of sorption data, the particles were represented as spheres with radius equal to the hydraulic radius of the particles. For cubes, this leads to an equivalent spherical radius equal to one-half the length of the cube side or a. The weight average radius may be found from the previous data and the relation among weight and number average radii and the standard deviation of the distribution given by Herdan (6). 

The equations for the weight- and number-average distribution functions can be used to calculate the different average depees of polymerization. The number-average depee of polymerization (x ) is... 

Shalitin and Katchalski [79] derived analytical expressions for the molecular weight and compositional distributions of the copolymers. Poissonian distributions were calculated for the over-all molecular weights and for the concentrations of molecules containing a specified number of A units regardless of the number of B present. 

Here n is the number of distinct fractions, w,- the fraction weights and March distributions. The refinable parameters are n — 1 weights plus the distinct strength parameters r,. 

To define more clearly the aggregation mechanism, the process may be characterized on the basis of the shape of the aggregate mass (or size) distribution, and the relative variation as a function of time of the weight and number average masses [46], This information well characterized the diffusion- or reaction-limited processes, and these characteristics were determined to be less affected by a concomitant fragmentation process. 

---