7.2 Molecular Weight Distribution and Polydispersity

Number-average vs Weight-average Molecular Weights

Polymers aren't like small molecules where every molecule in a sample has the same molar mass. A real polymer sample contains chains of many different lengths, so you need statistical averages to describe the distribution. Two averages dominate the field, and they weight the chains differently.

Definitions and Calculations

Number-average molecular weight ($M_n$) is the straightforward arithmetic mean: add up the total mass of all chains and divide by the total number of chains.

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

where $N_i$ is the number of chains with molecular weight $M_i$. Every chain counts equally, regardless of its size.

Weight-average molecular weight ($M_w$) weights each chain's contribution by its own mass. Heavier chains contribute more to this average than lighter ones:

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

The $M_i^2$ in the numerator is what gives larger chains disproportionate influence. Think of it this way: $M_n$ asks "what does the average chain weigh?" while $M_w$ asks "if you pick a random gram of polymer, what molecular weight does it belong to?"

Relationship and Significance

$M_w$ is always greater than or equal to $M_n$. They're equal only when every chain has exactly the same length.

Notice that $M_w$ is shifted toward the heavier chains. The gap between $M_w$ and $M_n$ directly reflects how broad the distribution is.

Polydispersity Index: Significance and Interpretation

Definition and Calculation

The polydispersity index (PDI, also called the dispersity $Đ$ in modern IUPAC notation) quantifies how broad the molecular weight distribution is:

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

• PDI = 1 means every chain has the same molecular weight (monodisperse). This is the theoretical lower bound.
• PDI > 1 means the sample contains a spread of chain lengths (polydisperse). The further above 1, the wider the spread.

Interpreting PDI Values

Typical PDI values you'll encounter:

• Living/controlled polymerizations (anionic, RAFT, ATRP): PDI ≈ 1.0–1.2
• Condensation polymerization (theoretical limit for Flory distribution): PDI → 2.0
• Free radical polymerization: PDI ≈ 1.5–2.5 or higher
• Ziegler-Natta catalysis: can vary widely depending on catalyst site heterogeneity

PDI affects real, measurable properties:

• Mechanical strength: Lower PDI generally means more uniform chain entanglements and better tensile properties.
• Viscosity: Broader distributions (higher PDI) can lower melt viscosity because shorter chains act as internal plasticizers.
• Processability: Lower PDI gives more consistent flow behavior and fewer volatile low-molecular-weight fragments during processing.

Molecular Weight Distribution: Impact on Properties

Influence on Physical and Mechanical Properties

The shape of the full distribution matters, not just the averages. A narrow distribution (low PDI) produces more predictable, uniform bulk properties.

• High-density polyethylene (HDPE) with a narrow distribution shows higher tensile strength and stiffness than HDPE with a broad distribution at the same $M_n$.
• Low-molecular-weight tails in a broad distribution act as plasticizers. They reduce the glass transition temperature ($T_g$) and lower melt viscosity. For example, blending low-$M$ polystyrene into high-$M$ polystyrene lowers $T_g$ and improves melt flow.
• High-molecular-weight tails boost toughness and wear resistance but increase melt viscosity, making processing harder. Ultra-high-molecular-weight polyethylene (UHMWPE) is a classic case: outstanding impact strength, but notoriously difficult to melt-process.

Effect on Crystallization Behavior

Molecular weight distribution strongly influences how semi-crystalline polymers pack into ordered structures.

• Narrow distributions promote faster crystallization and higher overall crystallinity. Uniform chain lengths pack more efficiently into lamellar crystals. Isotactic polypropylene with a narrow distribution crystallizes faster and reaches a higher degree of crystallinity than a broad-distribution counterpart.
• Broad distributions hinder crystallization. Short chains can't participate fully in crystal lamellae and instead accumulate in amorphous regions, disrupting regular packing. This is one reason LDPE (broad distribution, extensive branching) is less crystalline than HDPE.

Determining Molecular Weight Distribution: Methods

Three techniques come up most often. Each has trade-offs between the completeness of information you get and experimental complexity.

Gel Permeation Chromatography (GPC)

GPC (also called size exclusion chromatography, SEC) is the workhorse method for measuring the full molecular weight distribution.

How it works:

1. Dissolve the polymer in a suitable solvent.
2. Inject the solution into a column packed with porous beads of controlled pore size.
3. Large chains can't enter the smaller pores, so they pass through the column faster. Small chains explore more pore volume and elute later.
4. A detector (refractive index, UV, or light scattering) records signal vs. elution time.
5. Convert elution time to molecular weight using a calibration curve built from standards of known $M$ (polystyrene standards are the most common).

From the resulting chromatogram, you can extract $M_n$, $M_w$, PDI, and the full distribution curve.

• Advantages: Gives the complete distribution in a single run. Works for a wide range of polymers and molecular weights.
• Limitations: The polymer must be soluble in the mobile phase. Accuracy depends on how well the calibration standards match the analyte's chain architecture. Universal calibration (using intrinsic viscosity detectors) helps address this.

Light Scattering Techniques

Light scattering provides absolute molecular weight measurements without calibration standards.

Static Light Scattering (SLS): Measures the angular dependence of scattered light intensity. A Zimm plot analysis yields $M_w$ and the radius of gyration ($R_g$) directly. SLS is often coupled with GPC (GPC-MALS) to get absolute molecular weights at each elution slice, eliminating the need for calibration standards.

Dynamic Light Scattering (DLS): Measures time-dependent fluctuations in scattered intensity caused by Brownian motion. This gives the hydrodynamic radius and diffusion coefficient, which relate to molecular weight through the Stokes-Einstein equation. DLS is widely used for polymer nanoparticles and proteins.

• Advantages: Non-destructive, requires small sample volumes, provides size and shape information.
• Limitations: Extremely sensitive to dust and aggregates. Requires careful sample filtration. Polymer-solvent interactions (especially for polyelectrolytes) can complicate interpretation.

Viscometry

Viscometry is the simplest and cheapest approach, but it gives only an average molecular weight rather than the full distribution.

The Mark-Houwink equation connects intrinsic viscosity to molecular weight:

$[\eta] = K M^a$

where $K$ and $a$ are constants specific to a given polymer-solvent-temperature system. For example, polystyrene in toluene at 30°C has $K = 1.23 \times 10^{-4}$ dL/g and $a = 0.71$.

Steps:

1. Measure the intrinsic viscosity $[\eta]$ of the polymer solution (typically using a capillary viscometer at several concentrations, then extrapolating to zero concentration).
2. Look up or determine $K$ and $a$ for your polymer-solvent system.
3. Solve for the viscosity-average molecular weight $M_v$.

$M_v$ falls between $M_n$ and $M_w$ (closer to $M_w$ for typical values of $a$). It's useful for quick quality control but won't tell you about the shape of the distribution.

• Advantages: Inexpensive, minimal equipment, straightforward sample preparation.
• Limitations: Only gives a single average, not a distribution. Accuracy depends entirely on having reliable Mark-Houwink constants for your specific system.