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.

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

$$\text{Number-Average Molecular Weight: } M_n = \frac{\sum N_i M_i}{\sum N_i}$$ $$\text{Weight-Average Molecular Weight: } M_w = \frac{\sum N_i M_i^2}{\sum N_i M_i}$$ $$\text{Dispersity (IUPAC Standard Symbol): } \text{Đ} = \frac{M_w}{M_n} \ge 1.0$$ $$\text{Number-Average Degree of Polymerization: } DP_n = \frac{M_n}{M_0}$$

Worked Numerical Example:

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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 ($DP_n$)

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Solution:

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

$$\sum N_i = 100 + 200 + 100 = 400\text{ moles}$$ $$\sum N_i M_i = (100 \times 10,000) + (200 \times 50,000) + (100 \times 100,000) = 21,000,000\text{ g}$$ $$M_n = \frac{21,000,000}{400} = 52,500\text{ g/mol}$$

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

$$\sum N_i M_i^2 = (100 \times 10^8) + (200 \times 2.5 \times 10^9) + (100 \times 10^{10}) = 1.51 \times 10^{12}$$ $$M_w = \frac{1.51 \times 10^{12}}{21,000,000} = 71,905\text{ g/mol}$$

1. Calculate Dispersity ($\text{Đ}$):

$$\text{Đ} = \frac{M_w}{M_n} = \frac{71,905}{52,500} = 1.370$$

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

$$DP_n = \frac{52,500\text{ g/mol}}{104.15\text{ g/mol}} = 504.1 \text{ repeat units}$$

• 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.