Dispersity

Introduction

Two polymer samples can share the same average molar mass but have very different spreads of chain length. Dispersity, symbol Đ, gives a compact measure of that spread by dividing weight-average molar mass by number-average molar mass. It helps compare batches, but the full distribution remains important for interpreting performance.

Core explanation

Define Đ = M w/Mₙ. Because M w ≥ Mₙ, dispersity is at least one; it equals one only when every chain has the same molar mass in the ideal mathematical limit. For the sample with three 10 kg mol⁻¹ chains and one 30 kg mol⁻¹ chain, Mₙ = 15 and M w = 20 kg mol⁻¹, so Đ = 20/15 ≈ 1.33. Ordinary step-growth polymerisation under ideal balanced equal-reactivity conditions gives a most-probable chain-length distribution. In that model dispersity approaches 2 as conversion approaches one, rather than approaching 1. Controlled or living chain-growth methods can produce narrower distributions, although real initiation and side reactions prevent an absolute universal value. Radical polymerisations with termination and transfer may have broader distributions, and branching or multiple active-site populations can broaden them further. A single Đ value still does not describe distribution shape: a sample with a high-mass tail and one with two distinct peaks might have the same ratio. To analyse processing behaviour, inspect the full distribution or additional moments as well. When comparing reported dispersities, check that Mₙ and M w were measured on the same sample and calibration basis; mixing incompatible measurements makes the ratio meaningless.

Step-by-step reasoning

Calculate Mₙ and M w from the same chain-population data. Divide M w by Mₙ, retaining no units because the ratio is dimensionless. Check Đ ≥ 1. Interpret a value near one as comparatively narrow and a larger value as broader, but inspect the full curve before drawing detailed conclusions.

Visual explanation

Draw three distributions centred at the same molar mass: a narrow single peak, a broad single peak and two separated peaks. The first has Đ near one; the others may have larger values, yet a ratio alone does not show the difference between broad and bimodal shapes.

Real-world analogy

A class can have the same average test score under a tight cluster or under a mixture of very high and very low marks. One spread measure helps compare classes, but a histogram reveals more about their shapes.

Real-world example

Polymer quality control often reports Mₙ, M w and Đ together. A broad distribution can change melt flow and strength even if Mₙ remains on target, so manufacturers monitor both average size and spread.

Why?

Mass weighting favours large chains, so M w rises above Mₙ whenever masses vary. Their ratio therefore encodes some breadth. The ideal most-probable step-growth distribution remains broad because independent couplings generate many chain lengths.

Common misconception

Dispersity is not a percentage, and Đ = 1.3 does not mean 30% of chains are the wrong size. It is a ratio of two averages, not a direct probability statement or a complete picture of the distribution.

Worked example

Question: A sample has Mₙ = 50 kg mol⁻¹ and M w = 75 kg mol⁻¹. Find Đ and interpret it. Reasoning: Divide the two averages calculated on the same basis. Answer: Đ = 75/50 = 1.5; the sample has a nonzero chain-length spread, but the ratio alone does not reveal its exact shape.

Quick check

1. What is the minimum possible dispersity? Answer: One, achieved only if all chains have identical molar mass in the ideal limit.

Exam focus

Calculate both averages before dividing and include a sanity check that Đ is at least one. Mention that ideal high-conversion step growth tends toward about two only under its stated statistical assumptions.

Advanced insight

Modern nomenclature prefers the word dispersity and symbol Đ rather than the older term polydispersity index. The full molar-mass distribution remains necessary whenever rare high-mass chains or multiple populations affect the application.

Summary

Dispersity Đ = M w/Mₙ summarises the difference between mass- and number-weighted averages. It is dimensionless and at least one. Ideal most-probable step growth tends toward Đ ≈ 2 at high conversion, while controlled living methods can be narrower. No single ratio reveals the complete distribution shape.

Practice questions

1. Find Đ if Mₙ = 20 and M w = 30 kg mol⁻¹. Answer: 30/20 = 1.5, dimensionless.

2. Can a valid sample have Đ = 0.8? Answer: No. M w cannot be below Mₙ for the same nonnegative mass distribution.

3. What limiting Đ is expected for ideal high-conversion most-probable step growth? Answer: It approaches about 2 under the ideal assumptions.

4. Why inspect a distribution curve even after calculating Đ? Answer: Samples with different tails or multiple peaks can share the same average ratio.