CHAPTER
2.5 Breadth of the Distribution
Table 2.2 lists various average molecular weights in terms of moments of the number and weight distributions, where the quantity of polymer species with par- ticular sizes are counted in terms of numbers of moles or weights, respectively.
Note that in general a given average is given by Mz1k5nU0k13
nU0k125wU0k12
wU0k11 (2-25) The moment orders in the weight distribution are one less than the corres- ponding orders in the number distributions. (CompareMnformulas.) This symme- try arises because molar and weight concentrations are generally related by Eqs. (2-9) and (2-10). Thus,
nU0k5 X
niMik5X
ðniMiÞMk2i 15X
ciMik215wU0k21 (2-26) The viscosity average molecular weight Mv, which will be discussed later in Section 3.3, is the only average listed in these tables that is not a simple ratio of successive moments of the molecular weight distribution.
Table 2.2 Molecular Weight Averagesa
Number distribution Weight distribution
Normalized Not normalized Normalized Not normalized
Mn5 nU01
nU00 5
PxiMi
Pxi 5 PniMi
Pni
5 wU00
wU21 5
Pwi
Pðwi=MiÞ 5 Pci
Pðci=MiÞ Mw5 nU02
nU01 5
PxiM2i
PxiMi 5 PniM2i
PniMi
5wU01
wU00 5
PPwiMi
wi 5
PPciMi
ci
Mz5 nU03
nU02 5
PxiM3i
PxiMi2 5 PniM3i PniM2i
5wwUU0021 5 PwiM2i
PwiMi 5 PciM2i
PciMi
Mz115 nU04
nU03 5
PxiM4i
PxiMi3 5 PniM4i PniM3i
5wwUU0032 5P
wiM3i
PwiM2i 5 PciM3i PciM2i Mv5 nU0a11
nU01
h i1=a
5
PxiMai11 PxiMi
1=a
5
PniMia11 PniMi
1=a 5½wU0a1=a 5½wiMai1=a
5 PciMai
Pci
1=a aSee Appendix 2-A for application of these formulas to mixtures of broad distribution polymers.
The breadth of a distribution will reflect the dispersion of the measured quan- tities about their mean. Simple summing of the deviation of each quantity from the mean will yield a total of zero, since the mean is defined such that the sums of negative and positive deviations from its value are balanced. The obvious expedient then is to square the difference between each quantity and the mean of the distribution and add the squared terms. This produces a parameter, called the variance of the distribution, which reflects the spread of the observed values about their mean and is independent of the direction of this spread. The positive square root of the variance is called the standard deviationof the distribution. Its units are the same as those of the mean.
The standard deviation is calculated from a moment about the mean rather than about zero. The difference between Mi, the molecular weight of any species i, and the mean molecular weightAisMi2A, and thejth moment of the normal- ized distribution about the mean is
Uj5 X
fiðMi2AÞj (2-27)
The absence of a prime superscript on U indicates that the moment is taken with reference to the arithmetic mean.
Since the arithmetic mean is the center of balance of the frequencies in the distribution, the first moment of these frequencies about the mean must be zero:
nU15 X
xiðMi2MnÞ5wU1
XwiðMi2MwÞ50 (2-28) The second moment about the mean is the variance of the distribution:
nU25X
xiðMi2MnÞ25ðsnÞ2 (2-29)
wU25X
wiðMi2MwÞ25ðswÞ2 (2-30) where sn andsw are the standard deviations of the number and weight distribu- tions, respectively. Thus the standard deviation of the distribution is the square root of the second moment about its arithmetic mean:
s5ðU2Þ0:5 (2-31)
It remains now to convertU2into terms ofMwandMn:FromEq. (2-29),
nU25P
xiðMi2MnÞ25P
xiðMi222MiMn1M2nÞ 5 P
xiM2i 22Mn
PxiMi1M2n
X xi
5nU0222MnMn1M2n nU25nU022M2n5MwMn2M2n
(2-32) sn5ðMwMn2M2nÞ0:5 (2-33) s2n=M2n5Mw
Mn21 (2-34)
Starting withEq. (2-30)instead ofEq. (2-29), it is easy to show that s2w=M2w5Mz
Mw21 (2-35)
IfMwandMnof a polymer sample are known, we have information about the standard deviation sn and the variance of the number distribution. There is no quantitative information about the breadth of the weight distribution of the same sample unlessMz and Mw 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 theMw=Mn 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).
Very highly branched polymers, like polyethylene made by free-radical, high- pressure processes, will haveMw=Mnratios of 20 and more. Most polymers made by free-radical or coordination polymerization of vinyl monomers have ratios of from 2 to about 10. TheMw=Mn ratios of condensation polymers like nylons and thermoplastic polyesters tend to be about 2, and this is generally about the nar- rowest distribution found in commercial thermoplastics.
A truly monodisperse polymer has Mw=Mn equal to 1.0. Such materials have not been synthesized to date. The sharpest distributions that have actually been made are those of polystyrenes from very careful anionic polymerizations. These have Mw=Mn ratios as low as 1.04. Since the polydispersity index is only 4%
higher than that of a truly monodisperse polymer, these polystyrenes are some- times assumed to be monodisperse. This assumption is not really justified, despite the small difference from the theoretical value of unity.
For example, let us consider a polymer sample for which Mn5100,000, Mw5104,000, and Mw=Mn51.04. In this case sn is 20,000 from Eq. (2-33).
It can be shown, however[1], that a sample with the given values of MwandMn
could have as much as 44% of its molecules with molecular weights less than 70,000 or greater than 130,000. Similarly, as much as 10 mol% of the sample could have molecular weights less than 38,000 or greater than 162,000. This poly- mer actually has a sharp molecular weight distribution compared to ordinary synthetic polymers, but it is obviously not monodisperse.
It should be understood that the foregoing calculations do not assess the sym- metry of the distribution. We do not know whether the mole fraction outside the last size limits mentioned is actually 0.1, but we know that it cannot be greater than this value with the quoted simultaneousMn andMwfigures. (In fact, the dis- tribution would have to be quite unusual for the proportions to approach this boundary value.) We also do not know how this mole fraction is distributed at the high- and low-molecular-weight ends and whether these two tails of the distribu- tion are equally populated. The Mn andMw data available to this point must be supplemented by higher moments to obtain this information.
We should note also that a significant mole fraction may not necessarily com- prise a very large proportion of the weight of the polymer. In our last example,
77 2.5 Breadth of the Distribution
if the 10 mol% with molecular weight deviating from Mn by at least 662,000 were all material with molecular weight 38,000 it would be only 3.8% of the weight of the sample. Conversely, however, if this were all material with molecu- lar weight 162,000, the corresponding weight fraction would be 16.2.
There are various ways of expressing the skewness of statistical distributions.
The method most directly applicable to polymers uses the third moment of the distribution about its mean. The extreme molecular weights are emphasized because their deviation from the mean is raised to the third power, and since this power is an odd number, the third moment also reflects the net direction of the deviations.
In mathematical terms,
nU35X
i
xiðMi2MnÞ3
nU35X
i
xiðM3i 23M2iMn13MiM2n2M3nÞ
nU35X
i
xiMi323Mn
PxiMi213M2nX
xiMi2M3nX xi
(2-36)
For a normalized distribution,
nU35MzMwMn23MnðMwMnÞ13M2nðMnÞ2M3n (2-37)
nU35MzMwMn23M2nMw12M3n (2-38) wherenU3is positive if the distribution is skewed toward high molecular weights, zero if it is symmetrical about the mean, and negative if it is skewed to low molecular weights.
Asymmetry of different distributions is most readily compared by relating the skewness to the breadth of the distribution. The resulting measure α3 is obtained by dividing U3by the cube of the standard deviation. For the number distribution,
nα35 nU3
s3n 5MzMwMn23M2nMw13M3n
ðMwMn2M2nÞ3=2 (2-39)