 Molar mass distribution

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. The molar mass distribution (or molecular weight distribution) in a polymer describes the relationship between the number of moles of each polymer species (N_{i}) and the molar mass (M_{i}) of that species. ^{[1]} The molar mass distribution of a polymer may be modified by polymer fractionation.
Contents
Definition of molar mass averages
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:
 Number average molar mass or M_{n}
 Weight average molar mass or M_{w}
 Viscosity average molar mass or M_{v}
 Z average molar mass or M_{z}
^{[2]}
Here a is the exponent in the MarkHouwink equation that relates the intrinsic viscosity to molar mass.
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 smallangle laser light scattering measures weight average molar mass. M_{v} is obtained from viscosimetry and M_{z} 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: M_{n} < M_{v} < M_{w} < M_{z}. Polydispersity of a sample is defined as M_{w} divided by M_{n} 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 highpressure 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 limited accessibility of stationary phase pore volume for the polymer molecules results in shorter elution times for highmolecularweight 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 that measures the change in refractive index of the solvent. This detector is concentrationsensitive and very molecularweightinsensitive, so it is ideal for a singledetector GPC system, as it allows the generation of mass v's molecular weight curves. Less common but more accurate and reliable is a molecularweightsensitive detector using multiangle laserlight 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 lowangle light scattering, which uses a single low angle to determine the molar mass, or RightAngleLight 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 workup procedure. Ideal stepgrowth 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 M_{w} and a small reduction in M_{n} thus reducing polydispersity.
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.
The number average molecular weight of a polymer can be determined by gel permeation chromatography, viscometry via the (MarkHouwink equation), colligative methods such as vapor pressure osmometry, endgroup determination or proton NMR ^{[3]}.
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 NumberAverage Molecular Weight Polymers may be obtained only with a high fractional monomer conversion in the case of stepgrowth polymerization, as per the Carothers' equation.
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
where N_{i} is the number of molecules of molecular weight M_{i}.
If the weight average molecular weight is w, and one chooses a random monomer, then the polymer it belongs to will have a weight of w on average (for a homopolymer).
The weight average molecular weight can be determined by light scattering, small angle neutron scattering (SANS), Xray 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 weightaverage molecular weight, M_{w}, is also related to the fractional monomer conversion, p, in stepgrowth polymerization as per Carothers' equation:
 , where M_{o} is the molecular weight of the repeating unit.
See also
References
 ^ I. Katime "Química Física Macromolecular". Servicio Editorial de la Universidad del País Vasco. Bilbao
 ^ ^{a} ^{b} R.J. Young and P.A. Lovell, Introduction to Polymers, 1991
 ^ Polymer Molecular Weight Analysis by 1H NMR Spectroscopy Josephat U. Izunobi and Clement L. Higginbotham J. Chem. Educ., 2011, 88 (8), pp 1098–1104 doi:10.1021/ed100461v
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