Polymer Structure, Molecular Weight Averages, Dispersity and SEC Characterization

Unlike small synthetic molecules with uniform molecular weights (e.g. Water $18\text{ g/mol}$), synthetic polymers consist of a distribution of chain lengths. Properties such as tensile strength, melt viscosity, impact resistance, and processability depend strongly on molecular weight averages—Number-Average ($M_n$), Weight-Average ($M_w$), and Dispersity ($\text{Đ} = M_w/M_n$). Gel Permeation Chromatography / Size Exclusion Chromatography (GPC/SEC) measures these averages to verify resin grade specifications.

By completing this lesson, you will be able to:

• Calculate $M_n$, $M_w$, and dispersity ($\text{Đ}$) from molecular weight distribution data.
• Determine Degree of Polymerization ($DP_n$) from repeat unit mass.
• Interpret GPC/SEC chromatograms and hydrodynamic volume elution curves.
• Correlate molecular weight distribution width with melt processing stability.

mermaid
graph TD
    A["Polymer Sample Injection into GPC/SEC Column"] --> B["Size Exclusion Separation in Porous Gel Matrix"]
    B --> C["Large Molecules Elute First (Shorter Retention Time)"]
    C --> D["Small Molecules Elute Last (Trapped in Gel Pores)"]
    D --> E["Differential Refractometer Detection -> Mn, Mw & Dispersity Calculation"]

Molecular Weight Formulas

Worked Numerical Example:

Problem: A Polystyrene sample consists of three discrete molecular weight fractions:

• Fraction 1: $N_1 = 100\text{ moles}$, $M_1 = 10,000\text{ g/mol}$
• Fraction 2: $N_2 = 200\text{ moles}$, $M_2 = 50,000\text{ g/mol}$
• Fraction 3: $N_3 = 100\text{ moles}$, $M_3 = 100,000\text{ g/mol}$

Given styrene monomer repeat unit molar mass $M_0 = 104.15\text{ g/mol}$ ($ext{C}_8 ext{H}_8$), calculate:

1. Number-average molecular weight ($M_n$)
2. Weight-average molecular weight ($M_w$)
3. Dispersity ($\text{Đ}$)
4. Degree of Polymerization ($DP_n$)

Solution:

1. Calculate $\sum N_i$ and $\sum N_i M_i$:

1. Calculate $\sum N_i M_i^2$:

1. Calculate Dispersity ($\text{Đ}$):

1. Calculate Degree of Polymerization ($DP_n$):

• GPC Quality Control for Pipe Grade HDPE: Broad bimodal MWD ($\text{Đ} > 8.0$) for high environmental stress crack resistance (ESCR). (Illustrative Indian industry scenario based on polyolefin pipe manufacturing).

• $M_w$: Governed by high molecular weight chains; determines melt viscosity ($eta_0 propto M_w^{3.4}$).
• GPC/SEC: Gel Permeation Chromatography / Size Exclusion Chromatography.

1. ISO 16014-1:2019 — Plastics — Determination of average molecular mass and molecular mass distribution of polymers using size-exclusion chromatography, ISO.
2. Flory, P. J. (1953). Principles of Polymer Chemistry, Cornell University Press.