The molar mass distribution (or molecular weight distribution) describes the relationship between the number of
moles Moles can refer to:
* Moles de Xert, a mountain range in the Baix Maestrat comarca, Valencian Community, Spain
*The Moles (Australian band)
*The Moles, alter ego of Scottish band Simon Dupree and the Big Sound
People
* Abraham Moles, French engin ...
of each
polymer
A polymer (; Greek '' poly-'', "many" + '' -mer'', "part")
is a substance or material consisting of very large molecules called macromolecules, composed of many repeating subunits. Due to their broad spectrum of properties, both synthetic a ...
species (N
i) and the
molar mass (M
i) of that species. In linear polymers, the individual polymer chains rarely have exactly the same
degree of polymerization
The degree of polymerization, or DP, is the number of monomeric units in a macromolecule or polymer or oligomer molecule.
For a homopolymer, there is only one type of monomeric unit and the ''number-average'' degree of polymerization is given by ...
and molar mass, and there is always a distribution around an average value. The molar mass distribution of a polymer may be modified by
polymer fractionation.
Definitions of molar mass average
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:
*''Number average molar mass'' (M
n), also loosely referred to as ''number average molecular weight'' (NAMW).
*''Mass average molar mass'' (M
w), where ''w'' stands for weight; also commonly referred to as ''weight average'' or ''weight average molecular weight'' (WAMW).
*''Z-average molar mass'' (M
z), where ''z'' stands for centrifugation (from German "''Zentrifuge''").
*''Viscosity average molar mass'' (M
v).
Here,
is the exponent in the
Mark–Houwink equation that relates the
intrinsic viscosity Intrinsic viscosity \left \eta \right/math> is a measure of a solute's contribution to the viscosity \eta of a solution. It should not be confused with inherent viscosity, which is the ratio of the natural logarithm of the relative viscosity to th ...
to molar mass.
[R.J. Young and P.A. Lovell, Introduction to Polymers, 1991]
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
A laser is a device that emits light through a process of optical amplification based on the stimulated emission of electromagnetic radiation. The word "laser" is an acronym for "light amplification by stimulated emission of radiation". The ...
light scattering measures mass average molar mass. M
v is obtained from
viscosimetry and M
z 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: M
n < M
v < M
w < M
z. The
dispersity (also known as the ''polydispersity index'') of a sample is defined as M
w divided by M
n and gives an indication just how narrow a distribution is.
[Stepto, R. F. T.; Gilbert, R. G.; Hess, M.; Jenkins, A. D.; Jones, R. G.; Kratochvíl P. (2009).]
Dispersity in Polymer Science
''Pure Appl. Chem.'' 81 (2): 351–353. DOI:10.1351/PAC-REC-08-05-02.
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). These techniques involve forcing a polymer solution through a matrix of
cross-linked polymer particles at a pressure of up to several hundred
bar
Bar or BAR may refer to:
Food and drink
* Bar (establishment), selling alcoholic beverages
* Candy bar
* Chocolate bar
Science and technology
* Bar (river morphology), a deposit of sediment
* Bar (tropical cyclone), a layer of cloud
* Bar ( ...
. 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.
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 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. Less common but more accurate and reliable is a molecular-mass-sensitive detector using multi-angle laser-light scattering - see
static light scattering. These detectors directly measure the molecular mass of the polymer and are most often used in conjunction with differential 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 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
Chemical kinetics, also known as reaction kinetics, is the branch of physical chemistry that is concerned with understanding the rates of chemical reactions. It is to be contrasted with chemical thermodynamics, which deals with the direction in ...
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 M
w and a small reduction in M
n thus reducing dispersity.
Number average molar mass
The number average molar mass is a way of determining the
molecular mass
The molecular mass (''m'') is the mass of a given molecule: it is measured in daltons (Da or u). Different molecules of the same compound may have different molecular masses because they contain different isotopes of an element. The related quant ...
of a
polymer
A polymer (; Greek '' poly-'', "many" + '' -mer'', "part")
is a substance or material consisting of very large molecules called macromolecules, composed of many repeating subunits. Due to their broad spectrum of properties, both synthetic a ...
. 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
In mathematics and statistics, the arithmetic mean ( ) or arithmetic average, or just the '' mean'' or the ''average'' (when the context is clear), is the sum of a collection of numbers divided by the count of numbers in the collection. The co ...
or average of the molecular masses of the individual macromolecules. It is determined by measuring the molecular mass of ''n'' polymer molecules, summing the masses, and dividing by ''n''.
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.
''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.
Mass average molar mass
The mass average molar mass (often loosely termed ''weight average molar mass'') is another way of describing the
molar mass of a
polymer
A polymer (; Greek '' poly-'', "many" + '' -mer'', "part")
is a substance or material consisting of very large molecules called macromolecules, composed of many repeating subunits. Due to their broad spectrum of properties, both synthetic a ...
. Some properties are dependent on molecular size, so a larger molecule will have a larger contribution than a smaller molecule. The mass average molar mass is calculated by
where
is the number of molecules of molecular mass
.
The mass average molecular mass can be determined by
static light scattering,
small angle neutron scattering,
X-ray scattering, and
sedimentation velocity.
The ratio of the ''mass average'' to the ''number average'' is called the
dispersity or the ''polydispersity'' index.
[
The ''mass-average molecular mass'', ''M''w, 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 ''M''o is the molecular mass of the repeating unit.
]
Z-average molar mass
The z-average molar mass is the third moment or third power average molar mass, which is calculated by
The z-average molar mass can be determined with ultracentrifugation. The melt elasticity of a polymer is dependent on Mz.[Seymore, R.B and Caraher, C.E. Polymer Chemistry: An Introduction, 1992.]
See also
* Distribution function
* Flory–Schulz distribution
* Mass distribution
* Sedimentation
References
{{Authority control
Polymer chemistry