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# Molar mass distribution

The Molar mass distribution in a polymer describes the relationship between a polymer fraction and the molar mass of that polymer fraction. In linear polymers the individual polymer chains rarely have the exact same degree of polymerization and there is always a distribution around an average value.

Different average values can be defined depending on the statistical method that is applied. The weighted mean can be taken with the weight fraction, the mole fraction or the volume fraction:

• Weight average molar mass or Mw
• Number average molar mass or Mn
• Viscosity average molar mass or Mν
• Z average molar mass or Mz

$M_n=\frac{\sum M_i N_i} {\sum N_i}, M_w=\frac{\sum M_i^2 N_i} {\sum M_i N_i}, M_z=\frac{\sum M_i^3 N_i} {\sum M_i^2 N_i}, M_\nu=\left[\frac{\sum M_i^{1+a} N_i} {\sum M_i N_i}\right]^\frac{1} {a}$ [1]

These different definitions have true physical meaning because different techniques in physical polymer chemistry often measure just one of them. For instance osmometry measures number average molar mass and small angle laser light scattering measures weight average molar mass. Mv and Mz are obtained from respectively viscosimetry and sedimentation analysis. The quantity a in the expression for the viscosity average molar mass varies from 0.5 to 0.8 and depends on the interaction between solvent and polymer in a dilute solution. In a typical distribution curve the average values are related to each other as follows Mn < Mv < Mw < Mz. Polydispersity of a sample is defined as Mw divided by Mn and gives an indication just how narrow a distribution is.[2]

The most common technique for measuring molecular weight used in modern times is a variant of high pressure liquid chromatography (HPLC) known by the interchangeable terms of size exclusion chromatography (SEC) and gel permeation chromatography (GPC). These techniques involve forcing a polymer solution through a matrix of crosslinked polymer particles at a pressure of up to several thousand psi. The interaction between the crossliked polymer stationary phase and the polymer in the mobile phase results in higher retention times for low molecular weight species. The use of low polydispersity standards allows the user to correlate retention time with molecular weight although the actual correlation is with the Hydrodynamic volume. If the relationship between molar mass and the hydrodynamic volume changes (ie the polymer is not exactly the same shape as the standard) then the calibration for mass is in error.

The most common detectors used for size exclusion chromatography include online methods similar to the bench methods used above. By far the most common is the differential refractive index detector which measures the change in refractive index of the solvent. This detector is concentration sensitive and very molecular weight insensitive so it is ideal for a single detector GPC system as it allows the generation of mass v's molecular weight curves. Less common but more accurate and reliable is a molecular weight sensitive detector using multi-angle laser light scattering - see Static Light Scattering. These detectors directly measure the molecular weight of the polymer and are most often used in conjunction with differental refractive index detectors. A further alternative is either low angle light scattering, which uses a single low angle to determine the molar mass or Right Angle Light Laser scattering in combination with a Viscometer, although this last technique does not actually give an absolute measure of molar mass but one relative to the structural model used.

The molar mass distribution of a polymer sample depends on factors such as chemical kinetics and work-up procedure. Ideal step-growth polymerization gives a polymer with polydispersity of 2. ideal living polymerization results in a polydispersity of 1. By dissolving a polymer an insoluble high molar mass fraction may be filtered off resulting in a large reduction in Mw and a small increase in Mn thus reducing polydispersity.

Molecular weight distribution is often found in the literature but this phrase is technically incorrect and actually refers to the 1st derivative of the area against molar mass plot.

## References

1. ^ R.J. Young and P.A. Lovell, Introduction to Polymers, 1991
2. ^ R.J. Young and P.A. Lovell, Introduction to Polymers, 1991