Dispersity | MATSE 202: Introduction to Polymer Materials

The dispersity is an indication of the breadth of the molar mass distribution. Consider the two mass distributions shown below in Figure 1.11. The orange curve is broader than the blue, hence we would say that the orange polymer sample has greater dispersity.

Note:

The textbook Introduction to Polymers sometimes uses the term polydispersity index. Recently, the term has been changed to dispersity. IUPAC has deprecated the use of the term polydispersity index, having replaced it with the term dispersity, represented by the symbol Đ (pronounced D-stroke) which can refer to either molecular mass or degree of polymerization. Source Wikipedia: Dispersity

Diagram showing a broad distribution curve (orange) and a narrow distribution curve (blue).

Figure 1.11 - Sample Molar Mass Distributions

Source: Lauren Zarzar

We define the dispersity as the ratio of $M_{w}$ and $M_{n}$ :

$dispersity= Đ = \frac{{\bar{M}}_{w}}{{\bar{M}}_{n}}$

If your polymer is completely uniform, and every polymer molecule is exactly the same size, your dispersity would be 1. If there is any distribution in molar mass, then dispersity will be greater than 1 because $M_{w}$ is always greater than $M_{n}$ .

PROBLEM

What is the dispersity of the polymer mixture described by the data below?


N_i (mol)	M_i (g/mol)	m_i (g)	m_i * M_i (g²/mol) (g)
0.003	10,000	30	300,000
0.008	12,000	96	1,152,000
0.011	14,000	154	2,156,000
0.017	16,000	272	4,352,000
0.009	18,000	162	2,916,000
0.001	20,000	20	400,000
----	----	----	----
0.049	90,000	734	11,276,000

Click here for the answer.

ANSWER

${\bar{M}}_{n} = \frac{\sum^{} N_{i} M_{i}}{\sum^{} N_{i}} = \frac{734 g}{0.049 m o l} = 14,980 g / m o l$

${\bar{M}}_{w} = \frac{\sum^{} N_{i} M_{i}^{2}}{\sum^{} N_{i} M_{i}} = \frac{11,276,000 g^{2} / m o l}{734 g} = 15,362 g / m o l$

$dispersity= Đ = \frac{{\bar{M}}_{w}}{{\bar{M}}_{n}} = \frac{15,362 g / m o l}{14,980 g / m o l} = 1.03$

Earlier in the lesson, we learned about degree of polymerization. Well, if there is a distribution in polymer molar mass, then there must also be a distribution of degree of polymerization. So to describe the degree of polymerization for a polydisperse polymer we use degree of polymerization averages, and similarly to molar mass distributions, we have both a number average and a weight average for degree of polymerization.

${\bar{x}}_{n} = \frac{{\bar{M}}_{n}}{{\bar{M}}_{0}}$

${\bar{x}}_{w} = \frac{{\bar{M}}_{w}}{{\bar{M}}_{0}}$