# Molar mass distribution

The Molar mass distribution (also known as the molecular weight 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.

## Definition

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 M$\nu$
• 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]

## Measurement

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 is obtained from viscosimetry and Mz by sedimentation in an analytical ultracentrifuge. 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 or (GPC). These techniques involve forcing a polymer solution through a matrix of cross-linked polymer particles at a pressure of up to several thousand psi. The interaction between the crosslinked 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 (i.e., 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.

### Weight average molecular weight

The weight average molecular weight is a way of describing the molecular weight of a polymer. Polymer molecules, even if of the same type, come in different sizes (chain lengths, for linear polymers), so we have to take an average of some kind. For the weight average molecular weight, this is calculated by

$\bar{M}_w=\frac{\sum_i N_iM_i^2}{\sum_i N_iM_i}$

where $N_i$ is the number of molecules of molecular weight $M_i$.

Intuitively, if the weight average molecular weight is w, and you pick a random monomer, then the polymer it belongs to will have a weight of w on average.

The weight average molecular weight can be determined by light scattering, small angle neutron scattering (SANS), X-ray scattering, and sedimentation velocity.

An alternative measure of molecular weight for a polymer is the number average molecular weight; the ratio of the weight average to the number average is called the polydispersity index.

The weight-average molecular weight, Mw, is also related to the fractional monomer conversion, p, in step-growth polymerization as per Carothers' equation:

$\bar{X}_w=\frac{1+p}{1-p} \quad \bar{M}_w=\frac{M_o\left(1+p\right)}{1-p}$, where Mo is the molecular weight of the repeating unit.

### Number average molecular weight

The number average molecular weight is a way of determining the molecular weight of a polymer. Polymer molecules, even ones of the same type, come in different sizes (chain lengths, for linear polymers), so the average molecular weight will depend on the method of averaging. The number average molecular weight is the ordinary arithmetic mean or average of the molecular weights of the individual macromolecules. It is determined by measuring the molecular weight of n polymer molecules, summing the weights, and dividing by n [3].

$\bar{M}_n=\frac{\sum_i N_iM_i}{\sum_i N_i}$

The number average molecular weight of a polymer can be determined by gel permeation chromatography, viscometry (Mark-Houwink equation), and all colligative methods like vapor pressure osmometry or end-group titration.

An alternative measure of the molecular weight of a polymer is the weight average molecular weight. The ratio of the weight average to the number average is called the polydispersity index.

High Number-Average Molecular Weight Polymers may be obtained only with a high fractional monomer conversion in the case of step-growth polymerization, as per the Carothers' equation.

## 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
3. MOLECULAR WEIGHT DEFINITIONS - POLYMER CHEMISTRY HYPERTEXT - 2006 - Español & Português - Physical Polymer Chemistry