Molar mass distribution

# Molar mass distribution

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Encyclopedia
In linear polymers the individual polymer chains rarely have exactly the same degree of polymerization
Degree of polymerization
The degree of polymerization, or DP, is usually defined as the number of monomeric units in a macromolecule or polymer or oligomer molecule.For a homopolymer, there is only one type of monomeric unit andthe number-average degree of polymerization is given by...

and molar mass
Molar mass
Molar mass, symbol M, is a physical property of a given substance , namely its mass per amount of substance. The base SI unit for mass is the kilogram and that for amount of substance is the mole. Thus, the derived unit for molar mass is kg/mol...

, and there is always a distribution around an average
Average
In mathematics, an average, or central tendency of a data set is a measure of the "middle" value of the data set. Average is one form of central tendency. Not all central tendencies should be considered definitions of average....

value. The molar mass distribution (or molecular weight distribution) in a polymer
Polymer
A polymer is a large molecule composed of repeating structural units. These subunits are typically connected by covalent chemical bonds...

describes the relationship between the number of moles of each polymer species (Ni) and the molar mass (Mi) of that species. The molar mass distribution of a polymer may be modified by polymer fractionation
Polymer fractionation
Polymers are chainlike molecules that are made of the same repetition unit. Aside from few exceptions polymers consist of molecules with different chain lengths. Therefore average values are given for the molecular weight like the number average, the weight average or the viscosity average molar...

.

## Definition of molar mass averages

Different average values can be defined depending on the statistical method that is applied. The weighted mean
Weighted mean
The weighted mean is similar to an arithmetic mean , where instead of each of the data points contributing equally to the final average, some data points contribute more than others...

can be taken with the weight fraction, the mole fraction or the volume fraction:
• Number average molar mass or Mn
• Weight average molar mass or Mw
• Viscosity average molar mass or Mv
• Z average molar mass or Mz

Here a is the exponent in the Mark-Houwink equation
Mark-Houwink equation
The Mark–Houwink equation gives a relation between intrinsic viscosity [\eta] and molecular weight M:[\eta]=KM^aFrom this equation the molecular weight of a polymer can be determined from data on the intrinsic viscosity and vice versa....

that relates the intrinsic viscosity
Intrinsic viscosity
Intrinsic viscosity \left[ \eta \right] is a measure of a solute's contribution to the viscosity \eta of a solution. Intrinsic viscosity is frequently referred to as "Inherent Viscosity" in macromolecular literature...

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 small-angle laser
Laser
A laser is a device that emits light through a process of optical amplification based on the stimulated emission of photons. The term "laser" originated as an acronym for Light Amplification by Stimulated Emission of Radiation...

light scattering measures weight average molar mass. Mv is obtained from viscosimetry
Viscometer
A viscometer is an instrument used to measure the viscosity of a fluid. For liquids with viscosities which vary with flow conditions, an instrument called a rheometer is used...

and Mz by sedimentation
Sedimentation
Sedimentation is the tendency for particles in suspension to settle out of the fluid in which they are entrained, and come to rest against a barrier. This is due to their motion through the fluid in response to the forces acting on them: these forces can be due to gravity, centrifugal acceleration...

in an analytical ultracentrifuge
Ultracentrifuge
The ultracentrifuge is a centrifuge optimized for spinning a rotor at very high speeds, capable of generating acceleration as high as 2,000,000 g . There are two kinds of ultracentrifuges, the preparative and the 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.

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
Size exclusion chromatography
Size-exclusion chromatography is a chromatographic method in which molecules in solution are separated by their size, and in some cases molecular weight . It is usually applied to large molecules or macromolecular complexes such as proteins and industrial polymers...

(SEC) and gel permeation chromatography
Gel Permeation Chromatography
Gel permeation chromatography is a type of size exclusion chromatography , that separates analytes on the basis of size. The technique is often used for the analysis of polymers. As a technique, SEC was first developed in 1955 by Lathe and Ruthven. The term gel permeation chromatography can be...

(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 that 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
Static light scattering
Static light scattering is a technique in physical chemistry that measures the intensity of the scattered light to obtain the average molecular weight Mw of a macromolecule like a polymer or a protein. Measurement of the scattering intensity at many angles allows calculation of the root mean square...

. 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
Molar mass
Molar mass, symbol M, is a physical property of a given substance , namely its mass per amount of substance. The base SI unit for mass is the kilogram and that for amount of substance is the mole. Thus, the derived unit for molar mass is kg/mol...

, 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
Chemical kinetics, also known as reaction kinetics, is the study of rates of chemical processes. Chemical kinetics includes investigations of how different experimental conditions can influence the speed of a chemical reaction and yield information about the reaction's mechanism and transition...

and work-up procedure. Ideal step-growth polymerization
Step-growth polymerization
Step-growth polymerization refers to a type of polymerization mechanism in which bi-functional or multifunctional monomers react to form first dimers, then trimers, longer oligomers and eventually long chain polymers. Many naturally occurring and some synthetic polymers are produced by step-growth...

gives a polymer with polydispersity of 2. Ideal living polymerization
Living polymerization
In polymer chemistry, living polymerization is a form of addition polymerization where the ability of a growing polymer chain to terminate has been removed. This can be accomplished in a variety of ways. Chain termination and chain transfer reactions are absent and the rate of chain initiation is...

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 reduction in Mn thus reducing polydispersity.

### Number average molecular weight

The number average molecular weight is a way of determining the molecular weight of a polymer
Polymer
A polymer is a large molecule composed of repeating structural units. These subunits are typically connected by covalent chemical bonds...

. 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
Mean
In statistics, mean has two related meanings:* the arithmetic mean .* the expected value of a random variable, which is also called the population mean....

or average
Average
In mathematics, an average, or central tendency of a data set is a measure of the "middle" value of the data set. Average is one form of central tendency. Not all central tendencies should be considered definitions of 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
Gel Permeation Chromatography
Gel permeation chromatography is a type of size exclusion chromatography , that separates analytes on the basis of size. The technique is often used for the analysis of polymers. As a technique, SEC was first developed in 1955 by Lathe and Ruthven. The term gel permeation chromatography can be...

, viscometry via the (Mark-Houwink equation
Mark-Houwink equation
The Mark–Houwink equation gives a relation between intrinsic viscosity [\eta] and molecular weight M:[\eta]=KM^aFrom this equation the molecular weight of a polymer can be determined from data on the intrinsic viscosity and vice versa....

), colligative methods
Colligative properties
Colligative properties are properties of solutions that depend on the number of molecules in a given volume of solvent and not on the properties/identity of the molecules. Colligative properties include: relative lowering of vapor pressure; elevation of boiling point; depression of freezing point...

such as vapor pressure osmometry
Vapor pressure osmometry
Vapor pressure osmometry is a technique to measure the number average molecular weight of a polymer. It is based upon Raoult's law that governs change in vapor pressure of a solution based on the mole fraction of the solute....

, end-group
End-group
An end-group in polymer chemistry is a constitutional unit that is an extremity of a macromolecule or oligomer molecule. For example the end-group of a PET polyester may be an alcohol group or a carboxylic acid group...

determination or proton NMR
Proton NMR
Proton NMR is the application of nuclear magnetic resonance in NMR spectroscopy with respect to hydrogen-1 nuclei within the molecules of a substance, in order to determine the structure of its molecules. In samples where natural hydrogen is used, practically all of the hydrogen consists of the...

.

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
Polydispersity index
In physical and organic chemistry, the polydispersity index , is a measure of the distribution of molecular mass in a given polymer sample. The PDI calculated is the weight average molecular weight divided by the number average molecular weight. It indicates the distribution of individual...

.

High Number-Average Molecular Weight Polymers may be obtained only with a high fractional monomer conversion in the case of step-growth polymerization
Step-growth polymerization
Step-growth polymerization refers to a type of polymerization mechanism in which bi-functional or multifunctional monomers react to form first dimers, then trimers, longer oligomers and eventually long chain polymers. Many naturally occurring and some synthetic polymers are produced by step-growth...

, 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
A polymer is a large molecule composed of repeating structural units. These subunits are typically connected by covalent chemical bonds...

. Polymer molecule
Molecule
A molecule is an electrically neutral group of at least two atoms held together by covalent chemical bonds. Molecules are distinguished from ions by their electrical charge...

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

where is the number of molecules of molecular weight .

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
Small angle neutron scattering
Small angle neutron scattering is a laboratory technique, similar to the often complementary techniques of small angle X-ray scattering and light scattering, used for investigations of structure of various substances, with spatial sensitivity of about 1 - 1000 nm...

(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
Step-growth polymerization
Step-growth polymerization refers to a type of polymerization mechanism in which bi-functional or multifunctional monomers react to form first dimers, then trimers, longer oligomers and eventually long chain polymers. Many naturally occurring and some synthetic polymers are produced by step-growth...

as per Carothers' equation:
, where Mo is the molecular weight of the repeating unit.

• Distribution function
Distribution function
In molecular kinetic theory in physics, a particle's distribution function is a function of seven variables, f, which gives the number of particles per unit volume in phase space. It is the number of particles per unit volume having approximately the velocity near the place and time...

• Chemical equilibrium
Chemical equilibrium
In a chemical reaction, chemical equilibrium is the state in which the concentrations of the reactants and products have not yet changed with time. It occurs only in reversible reactions, and not in irreversible reactions. Usually, this state results when the forward reaction proceeds at the same...

• Mass distribution
Mass distribution
Mass distribution is a term used in physics and mechanics and describes the spatial distribution of mass within a solid body. In principle, it is relevant also for gases or liquids, but on earth their mass distribution is almost homogeneous.-Astronomy:...

• Sedimentation
Sedimentation
Sedimentation is the tendency for particles in suspension to settle out of the fluid in which they are entrained, and come to rest against a barrier. This is due to their motion through the fluid in response to the forces acting on them: these forces can be due to gravity, centrifugal acceleration...