3.1 Molecular weight averages and distributions

Molecular Weight Averages and Distributions

Polymers aren't like small molecules where every molecule has the same molecular weight. A polymer sample contains chains of many different lengths, so you need statistical averages to describe the molecular weight. The two most common averages, $M_n$ and $M_w$, each emphasize different parts of the distribution, and comparing them tells you how broad or narrow that distribution is. These details directly control mechanical properties, viscosity, and processability.

Molecular Weight Averages

Number-average molecular weight ($M_n$)

$M_n$ is the simple arithmetic mean: add up the total mass of the sample and divide by the total number of molecules.

$M_n = \frac{\sum N_i M_i}{\sum N_i}$

where $N_i$ is the number of molecules with molecular weight $M_i$.

Because every molecule counts equally regardless of size, $M_n$ is sensitive to low molecular weight species like oligomers and residual monomer. Even a small population of short chains pulls $M_n$ down noticeably. This average is what you'd measure with colligative-property techniques (osmometry, for example), since those count the number of dissolved species.

$M_n$ also tells you the average chain length. The degree of polymerization is just $M_n$ divided by the repeat-unit molecular weight.

Weight-average molecular weight ($M_w$)

$M_w$ weights each chain by its mass, so heavier chains contribute more:

$M_w = \frac{\sum N_i M_i^2}{\sum N_i M_i}$

This makes $M_w$ sensitive to high molecular weight species. A few very long chains can raise $M_w$ significantly even if most chains are short. Light-scattering techniques measure $M_w$ directly.

Because the longest chains dominate bulk behavior like entanglement and load bearing, $M_w$ correlates more strongly with mechanical properties (tensile strength, toughness) and melt viscosity than $M_n$ does.

Quick comparison

Polydispersity and Molecular Weight Distribution

Polydispersity index (PDI)

The polydispersity index (also called dispersity, Đ) is the ratio:

$PDI = \frac{M_w}{M_n}$

• PDI = 1 means every chain is the same length (monodisperse). This is the theoretical lower limit.
• PDI > 1 means there's a spread of chain lengths. The larger the PDI, the broader the distribution.

Typical values to keep in mind: living/controlled polymerizations can achieve PDI ≈ 1.01–1.2, while conventional free-radical polymerization often gives PDI ≈ 1.5–2.0, and step-growth polymerization approaches PDI ≈ 2.0 at high conversion.

Shape of the distribution

The molecular weight distribution (MWD) is usually plotted as weight fraction versus $\log M$. The curve can be:

• Unimodal (single peak): most common, one dominant population of chain lengths
• Bimodal (two peaks): can result from blending two batches or from a change in reaction conditions during polymerization
• Multimodal: multiple distinct populations

The width and shape of the curve carry real information about how the polymer was made and how it will behave.

How distribution affects properties

• Mechanical strength and toughness: Higher $M_w$ and a narrower distribution generally improve tensile strength and impact resistance, because most chains are long enough to form effective entanglements.
• Melt viscosity: Higher $M_w$ raises viscosity. A broader distribution can also increase viscosity at low shear rates, making processing (extrusion, injection molding) more challenging.
• Crystallinity: A narrower distribution promotes more uniform crystallization, which affects transparency and barrier properties (gas permeability).

Interpretation of Molecular Weight Curves

Connection to synthesis method

Different polymerization mechanisms produce characteristically different distributions:

• Step-growth polymerization (e.g., polyesters, polyamides): All molecules react with each other throughout the process, producing a broad distribution. At high conversion, PDI approaches 2.0.
• Chain-growth polymerization (e.g., polyethylene, polystyrene via free-radical): Distribution width depends on reaction conditions like temperature and initiator concentration. Controlled/living variants can yield much narrower distributions.

Connection to processing

• A broader distribution can actually improve processability. The low molecular weight fraction acts somewhat like an internal plasticizer, reducing viscosity and making the melt flow more easily.
• A narrower distribution may need higher processing temperatures and pressures, but the final product tends to have more consistent and predictable properties (uniform strength, better optical clarity).

How to read a GPC/SEC curve

Molecular weight distribution data typically comes from gel permeation chromatography (GPC), also called size exclusion chromatography (SEC). The instrument separates chains by size and produces the distribution curve. From that curve, you can calculate both $M_n$ and $M_w$, determine the PDI, and spot features like shoulders or multiple peaks that indicate blending or unusual polymerization behavior.