Molar mass distribution
1946279
219097252
2008-06-13T15:36:05Z
193.52.208.233
/* Measurement */ mechanism of size exclusion chromatography : no interaction but limited accessibility
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 M<sub>w</sub>
*Number average molar mass or M<sub>n</sub>
*Viscosity average molar mass or M<sub><math>\nu</math></sub>
*Z average molar mass or M<sub>z</sub>
<math>
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}
</math>
<ref>R.J. Young and P.A. Lovell, Introduction to Polymers, 1991</ref>
==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]] [[scattering#electromagnetic scattering|light scattering]] measures weight average molar mass. M<sub>v</sub> is obtained from [[viscometer|viscosimetry]] and M<sub>z</sub> 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<sub>n</sub> < M<sub>v</sub> < M<sub>w</sub> < M<sub>z</sub>. [[Polydispersity]] of a sample is defined as M<sub>w</sub> divided by M<sub>n</sub> and gives an indication just how narrow a distribution is.<ref>R.J. Young and P.A. Lovell, Introduction to Polymers, 1991</ref>
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 limited accessibility of stationary phase pore volume for the polymer molecules results in shorter elution times for high 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 M<sub>w</sub> and a small increase in M<sub>n</sub> thus reducing polydispersity.
===Weight average molecular weight===
The '''weight average molecular weight''' is a way of describing the [[molecular weight]] of a [[polymer]]. Polymer [[molecule]]s, 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
<math>\bar{M}_w=\frac{\sum_i N_iM_i^2}{\sum_i N_iM_i}</math>
where <math>N_i</math> is the number of molecules of molecular weight <math>M_i</math>.
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'', ''M''<sub>w</sub>, is also related to the ''fractional monomer conversion'', ''p'', in [[step-growth polymerization]] as per [[Carothers' equation]]:
:<math>\bar{X}_w=\frac{1+p}{1-p} \quad \bar{M}_w=\frac{M_o\left(1+p\right)}{1-p}</math>, where ''M''<sub>o</sub> 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'' <ref>[http://www.polymerchemistryhypertext.com/molecular_weight_definitions.htm MOLECULAR WEIGHT DEFINITIONS - POLYMER CHEMISTRY HYPERTEXT - 2006 - Español & Português - Physical Polymer Chemistry<!-- Bot generated title -->]</ref>.
<math>\bar{M}_n=\frac{\sum_i N_iM_i}{\sum_i N_i}</math>
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==
{{reflist}}
==See also==
*[[distribution function]]
*[[equilibrium]]
*[[mass distribution]]
*[[sedimentation]]
==External Links==
*[http://openlearn.open.ac.uk/mod/resource/view.php?id=196637 Polymer Structure and Molecular Mass]
[[Category:polymer chemistry]]
[[ar:وزن جزيئي متوسط عددي]]
[[de:Molmassenverteilung]]