Molar mass distribution
In polymer chemistry, the molar mass distribution (or molecular weight distribution) describes the relationship between the number of moles of each polymer species (Ni) and the molar mass (Mi) of that species.
[1] In linear polymers, the individual polymer chains rarely have exactly the same degree of polymerization and molar mass, and there is always a distribution around an average value.
Different average values can be defined, depending on the statistical method applied.
In practice, four averages are used, representing the weighted mean taken with the mole fraction, the weight fraction, and two other functions which can be related to measured quantities:
Here, a is the exponent in the Mark–Houwink equation that relates the intrinsic viscosity to molar mass.
Mv is obtained from viscosimetry and Mz by sedimentation in an analytical ultra-centrifuge.
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:
The dispersity (also known as the polydispersity index) of a sample is defined as Mw divided by Mn and gives an indication just how narrow a distribution is.
[2][3] The most common technique for measuring molecular mass 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).
The limited accessibility of stationary phase pore volume for the polymer molecules results in shorter elution times for high-molecular-mass species.
The use of low dispersity standards allows the user to correlate retention time with molecular mass, 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.
This detector is concentration-sensitive and very molecular-mass-insensitive, so it is ideal for a single-detector GPC system, as it allows the generation of mass v's molecular mass curves.
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 latter technique does not 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 dispersity of 2.
Ideal living polymerization results in a dispersity 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 reduction in Mn, thus reducing dispersity.
Polymer molecules, even ones of the same type, come in different sizes (chain lengths, for linear polymers), so the average molecular mass will depend on the method of averaging.
The number average molecular mass is the ordinary arithmetic mean or average of the molecular masses of the individual macromolecules.
The number average molecular mass of a polymer can be determined by gel permeation chromatography, viscometry via the (Mark–Houwink equation), colligative methods such as vapor pressure osmometry, end-group determination or proton NMR.
[4] High number-average molecular mass polymers may be obtained only with a high fractional monomer conversion in the case of step-growth polymerization, as per the Carothers' equation.
where Ni is the number of molecules of molecular mass Mi.
[3] The mass-average molecular mass, Mw, is also related to the fractional monomer conversion, p, in step-growth polymerization (for the simplest case of linear polymers formed from two monomers in equimolar quantities) as per Carothers' equation:
where Mo is the molecular mass of the repeating unit.
The z-average molar mass can be determined with ultracentrifugation.
The melt elasticity of a polymer is dependent on Mz.
