Directions

This simulation shows the molecular weight distribution of a polymer and the averages M_n, M_w, and M_z calculated from it.

Choose a distribution type with the buttons at the top left:

• Schulz–Zimm: set the number-average molecular weight and the dispersity Đ = M_w/M_n (also called the polydispersity index, PDI). Đ = 1 + 1/k, and Đ = 2 is the most-probable distribution.
• step growth: the Flory most-probable distribution for a linear step-growth polymerization; set the conversion p of functional groups and the repeat-unit molecular weight M₀. For step growth, Đ = 1 + p. A conversion of 0.99 gives only 100 repeat units per chain on average.
• blend of two: mix two Schulz–Zimm polymers; set each component and the mass fraction of component 1. M_n is controlled by the component with more chains; M_w by the component with more mass.

The blue curve is the number distribution (fraction of chains) and the red curve is the weight distribution (fraction of polymer mass), both per decade of molecular weight. In blend of two, the dashed curves show the contribution of each component. The dashed vertical lines mark M_n, M_w, and M_z. Use the checkboxes to show or hide curves.

Details

Molecular weight averages

For N_i chains of molecular weight M_i, with mass fraction w_i = N_iM_i/ΣN_jM_j:

Always ; the equalities hold only for a uniform (monodisperse) polymer. M_n is sensitive to low-molecular-weight chains, M_w and M_z to high-molecular-weight chains.

Schulz–Zimm distribution

The number fraction of chains per unit molecular weight is a gamma distribution:

and the mass fraction per unit molecular weight is . The averages are

so . For (Đ = 2) this is the most-probable distribution; as the polymer becomes uniform.

Step growth (Flory most-probable distribution)

For linear step-growth polymerization with conversion p, the number and mass fractions of x-mers (M = xM₀) are

with , , and . The dispersity approaches 2 at high conversion. The curves are drawn through the discrete x-mer values as if x were continuous.

Blend of two polymers

For mass fractions w₁ and w₂ = 1 − w₁:

The fraction of chains that come from component 1 is , so a small mass of low-molecular-weight polymer contributes many chains and lowers M_n strongly.

Plotted quantities

Because molecular weights span several decades, the distributions are plotted against log M, as in a size-exclusion chromatogram. The y axis is the fraction per decade, so the area under each curve is 1:

On a log axis, the peak of the weight distribution is not at M_w, and the peak of the number distribution is not at M_n.

Definitions of terms

= dispersity, also called polydispersity index (PDI)
= cumulative number fraction and cumulative mass fraction of polymer
= Schulz–Zimm shape parameter
= molecular weight (g/mol)
= molecular weight of chains of type i (g/mol)
= molecular weight of the repeat unit (g/mol)
= number-average, weight-average, and z-average molecular weights (g/mol)
= average molecular weights of blend component j (g/mol)
= number fraction of chains per unit molecular weight (mol/g)
= total number of chains
= number of chains of type i
= number of chains with x repeat units
= conversion of functional groups in step growth
= mass fraction of polymer per unit molecular weight (mol/g)
= mass fraction of chains of type i
= mass fraction of chains with x repeat units
= mass fractions of blend components 1 and 2
= number of repeat units in a chain (degree of polymerization)
= number-, weight-, and z-average degrees of polymerization
= gamma function

About

This simulation was created in the Department of Chemical and Biological Engineering at University of Colorado Boulder for LearnChemE.com by John L. Falconer using Claude AI. It was prepared with financial support from the National Science Foundation (DUE 2336987 and 2336988). Address any questions or comments to LearnChemE@gmail.com.