A. Rudin University of Waterloo
Waterloo, Ontario, Canada
Molecular Weight Distributions of Polymers
I n most textbooks, polymer molecular weight averages are presented as definitions which the student is obliged to commit to memory. Examples are given of uses of the different molecular weight averages, hut the student is usually given no reason why these particular concepts were selected in the first place. An order of presentation which emphasizes that these averages are simply arithmetic means of different distributions eliminates this mystery. I t is somewhat limited, however, compared t o definitions in terms of moments of the molecular weight distrihution, since moments are more generally useful to characterize polymers. Both approaches are combined in this article. The appropriate mathematical relations are derived from a mechanical analogy which students trained in science may grasp more readily than definitions directly from statistics, and the use of moments to derive the average, breadth and skewness of the distribution is described.
The numher and weight average molecular weights, M , and M,, are simply the arithmetic means of the number and weight - distributions and are defined from eqn. (3). The distrihution we have assumed to define the arithmetic mean is a number distribution, since the record consists of numbers of molecules of specified sizes. The sums of these numbers comprise the integral (cumulative) number distrihution. Figure 1represents such a distribution. The scale along the abscissa is the
Arithmetic Mean Molecular Weights
The distribution of molecular sizes in a polymer sample is usually expressed as the proportions of the sample with particular molecular weights. The mass of data contained in the distrihution can be understood more readily by condensing the information into parameters descriptive of various aspects of the distrihution. One such summarizing parameter expresses the central tendency of the distribution. A numher of choices are available for this measure, including the median, mode and various averages, such as the arithmetic, geometric, and harmonic means. Each may be most appropriate for different distrihutions. I n polymer science, however, only the arithmetic mean has been used. This is mainly because direct measurement of this mean is much easier experimentally than measurement of the distrihution and subsequent computat,ion of its central tendency. The distribution must he known, to derive the other summarizing parameters. T o define the arithmetic mean, A , let us assume a sample of N polymer molecules containing nl molecules with molecular weight MI, nz molecules with molecular weight Mz,. . .njmolecules with molecular weight Mi. n l + n l + ... +n, = N (1) The arithmetic mean molecular weight, A , is given by
The ratio n , / N is the proportion of the sample with molecular weight M,. If we call this proportion f r the arithmetic mean molecular weight is given by
MOLECULAR WEIGHT, Figure 1.
Mi
-
Normalized integral number dirtnbution cune.
molecular weight while that on the ordinate could he the total number of molecules with molecular weights less t,han or equal to the corresponding value on the ahscissa. However, it is easier to compare different distributions if the cumulative figures along the ordinate are expressed as fractions of the total number of molecules in each sample and Figure 1 is drawn in this way. The units of the ordinate are therefore mole fractions and extend from 0 to 1; the integral distrihution is said to he normalized. I n mathematical terms, the cumulative numher (or mole) fraction, X ( M ) , is defined as M
X(M)=
zj
(4)
i
where rt is the fraction of molecules with molecular weight M,. The differential number function is simply the mole fraction, xi, and a plot of these values against corresponding M,'s yields a differential numher distribution curve, as in Figure 2. If the distribution is normalized the area under the xi-Mi curve in Figure 2 will he unity. To compile the numher distrihution we have expressed the proportion of species with molecular weight M ,as Volume 46, Number 9, September 1969
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beams from second moments of cross-sectional areas about particular axes. By extending the above examples we could say that a moment in mechanics is generally defined as where U , is the jth moment, about a specified line or plane, a, of a quantity F (for example, force, weight, mass, area), d is the distance from F to the reference line or plane and j is a number. The moment is named according to the power, j, to which d is raised. If F is composed of elements, F,, each located a distance d l from the same reference, the moment of F is the sum of the moments of its elements
Mathematically, there is no restriction on the choice of F or j, but use of moments to solve practical mechanics problems usually confines F to the examples listed above and j to values of 1 or 2. The reference line or plane must be specified when the value of the moment is quoted. I n polymer science the mathematical formulation for moments corresponds to that in eqn. (15). While the reference line may be located anywhere, the usefulness of choosing the ordinate (M = 0) in the graph of the differential molecular weight distribution is so great that this reference is usually not mentioned explicitly. The distance d from the reference line is measured along the abscissa in terms of the molecular weight, M, and the quantity F is replaced by f,, the proportion of the polymer with molecular weight M,. As a matter of utility, j assumes a wider range of values in polymer science than in mechanics. With these differences, which are mainly matters of emphasis, the concepts of moments correspond closely in both disciplines. This correspondence is illustrated in the next section through consideration of parallel situations in polymer science and elementary mechanics. Arithmetic Mean or Center of Bolonce of the Distribution
When we derived eqn. (3), defining the arithmetic mean, we postulated a molecular weight distribution comprising nl polymer molecules with size MI, nr with size M2, and so on. Let us picture each species represented by a weight of magnitude (in pounds) equal to the number of molecules, ni, of the particular size M,. Each weight is suspended from a horizontal bar (which is itself weightless) a t a distance (in feet) from the fulcrum equal to the corresponding molecular weight, M,. (Fig. 5). The clockwise turning effect of each weight, n, pounds, about the fulcrum is equal to niMi foot-pounds. This is the first moment of the weight about the fulcrum. The total first moment of all the weights about the fulcrum, which is a t M = 0, is En&'
