## 1.3 Molecular Weight

### 1.3.1 Molecular-Weight Distribution

A typical synthetic polymer sample contains chains with a wide distribution of chain lengths. This distribution is seldom symmetric and contains some molecules of very high molecular weight. A representative distribution is illustrated in Figure 1-8. The exact breadth of the molecular-weight distribution depends upon the specific conditions of polymerization, as will be described in Chapter 2. For example, the polymerization of some olefins can result in molecular-weight distributions that are extremely broad. In other polymerizations, polymers with very narrow molecular-weight distributions can be obtained. As will be shown in subsequent chapters, many polymer properties, such as melt viscosity, are dependent on molecular weight and molecular-weight distribution. Therefore, it is useful to define molecular-weight averages associated with a given molecular-weight distribution as detailed in this section.

Figure 1-8 A representation of a continuous distribution of molecular weights shown as a plot of the number of moles of chains, *N*, having molecular weight *M*, against *M*.

### 1.3.2 Molecular-Weight Averages

For a discrete distribution of molecular weights, an average molecular weight, , may be defined as

where *N _{i}* indicates the number of moles of molecules having a molecular weight of

*M*and the parameter α is a weighting factor that defines a particular average of the molecular-weight distribution. The weight,

_{i}*W*, of molecules with molecular weight

_{i}*M*is then

_{i}Molecular weights that are important in determining polymer properties are the number-average, (α = 1), the weight-average, (α = 2), and the *z*-average, (α = 3), molecular weights.

Since the molecular-weight distribution of commercial polymers is normally a continuous function, molecular-weight averages can be determined by integration if the appropriate mathematical form of the molecular-weight distribution (i.e., *N* as a function of *M* as illustrated in Figure 1-8) is known or can be approximated. Such mathematical forms include theoretical distribution functions derived on the basis of a statistical consideration of an idealized polymerization, such as the Flory, Schultz, Tung, and Pearson distributions ^{9} (see Example 1.1 and Problem 1.3) and standard probability functions, such as the Poisson and logarithmic-normal distributions.

It follows from eq. (1.1) that the *number-average molecular weight* for a discrete distribution of molecular weights is given as

where *N* is the total number of molecular-weight species in the distribution. The expression for the number-average molecular weight of a continuous distribution function is

The respective relationships for the *weight-average molecular weight* of a discrete and a continuous distribution are given by

and

In the case of high-molecular-weight polymers, the number-average molecular weight is directly determined by membrane osmometry, while the weight-average molecular weight is determined by light-scattering and other scattering techniques as described in Chapter 3. As mentioned earlier, a higher moment of the molecular-weight distribution is the *z*-average molecular weight () where α = 3. As discussed later in Chapter 3 (Section 3.3.3), a viscosity-average molecular weight, , can be obtained from dilute-solution viscometry. The viscosity-average molecular weight falls between and depending upon whether the solvent is a good or poor solvent for the polymer. In the case of a good solvent, .

A measure of the breadth of the molecular-weight distribution is given by the ratios of molecular-weight averages. For this purpose, the most commonly used ratio is , called the *polydispersity index* or PDI ^{9}. Recent IUPAC recommendations suggest the use of the term *molar-mass dispersity*, *D*_{M}, for this ratio ^{10}. The PDIs of commercial polymers vary widely. For example, commercial grades of polystyrene with a of over 100,000 have polydispersity indices between 2 and 5, while polyethylene synthesized in the presence of a stereospecific catalyst may have a PDI as high as 30.^{*} In contrast, the PDI of some vinyl polymers prepared by “living” polymerization (see Chapter 2) can be as low as 1.06. Such polymers with nearly *monodisperse* molecular-weight distributions are useful as molecular-weight standards for the determination of molecular weights and molecular-weight distributions of commercial polymers (see Section 3.3.4).

#### Example 1.1

A polydisperse sample of polystyrene is prepared by mixing three *monodisperse* samples in the following proportions:

1 g |
10,000 molecular weight |

2 g |
50,000 molecular weight |

2 g |
100,000 molecular weight |

Using this information, calculate the number-average molecular weight, weight-average molecular weight, and PDI of the mixture.

*Solution*

Using eqs. (1.3) and (1.5), we obtain the following:

#### Example 1.2

A polymer is fractionated and is found to have the continuous molecular-weight distribution shown below as a plot of the weight, *W*, of molecules having molecular weight, *M*, versus *M*. Given this molecular-weight distribution, calculate and .

*Solution*

Using eqs. (1.4) and (1.6), we obtain the following:

#### Example 1.3

The single-parameter Flory distribution is given as

*W*(*X*) = *X*(ln *p*)^{2} *p*^{x}

where *X* is the degree of polymerization and *p* is the fractional monomer conversion in a step-growth polymerization. Using this equation, obtain expressions for the number-average and weight-average *degrees of polymerization*^{*} in terms of *X* and *p.*^{†}

*Solution*

Using the following geometric series:

Since it can be shown that *B*(1 – *p*) = *A*(1 + *p*), it follows that

, and then