Why there are several averages
Imagine a class where each pupil has their own string of beads. You can count the average number of beads per pupil. Or you can ask how long the string is that holds an average bead: the answer is larger, because long strings hold more beads.
The number-average molar mass Mn is the total mass of all macromolecules divided by their number: every chain counts equally. The mass-average Mw weights each chain by its mass, so long chains count more, and always Mw ≥ Mn. There are also the z-average Mz and the viscosity average Mv, which lies between Mn and Mw.
Dispersity
If all chains are the same, the two averages coincide. The more the chains differ, the further the averages drift apart. Their ratio shows how mixed the sample is.
The ratio Đ = Mw/Mn (dispersity, formerly polydispersity index) equals 1 for a uniform sample and grows with distribution width. For the most probable distribution, typical of linear polycondensation at high conversion, Đ approaches 2. Living anionic polymerisation can give samples with Đ close to 1.
How it is measured
Some methods count molecules, others weigh their contribution. The first give Mn, the second Mw. The most common modern method separates chains by size in a column and builds the whole distribution at once.
Mn is measured by colligative properties (membrane osmometry) and end-group analysis; Mw by light scattering. Size exclusion (gel permeation) chromatography separates macromolecules by hydrodynamic volume and gives the whole distribution; with calibration standards the result is relative, with a light scattering detector absolute. Viscometry gives the viscosity average.
Viscosity and the Mark-Houwink equation
The longer the chains, the thicker the solution. This is used to estimate chain length quickly, for example for cellulose in the paper industry.
Intrinsic viscosity is related to molar mass by the Mark-Houwink (Mark-Kuhn-Houwink) equation: [η] = K·M^a, where K and a are constants for a given polymer-solvent pair at a given temperature. For flexible coils a is usually 0.5-0.8, for stiff chains higher. The degree of polymerisation of cellulose is routinely determined from the viscosity of its solution in cupriethylenediamine (ISO 5351).
Why it matters for wood
The strength of paper and fibres depends strongly on cellulose chain length. If pulping or bleaching cuts the chains too much, the paper becomes weak.
The average degree of polymerisation and distribution of cellulose govern fibre strength, solution viscosity in viscose production and the suitability of pulp for chemical conversion. The drop in pulp viscosity is used to monitor degradation during pulping and bleaching.
Key facts
- Mn
- number average
- Mw
- mass average
- always
- Mw ≥ Mn
- Đ of a uniform sample
- 1
- most probable
- Đ → 2
- Mark-Houwink
- [η] = K·M^a
Sources for this chapter
- Азаров В.И., Буров А.В., Оболенская А.В. Химия древесины и синтетических полимеров: учебник. 2-е изд. СПб.: Лань, 2010.
- Евстигнеев Э.И. и др. Химия древесины и синтетических полимеров. Ч. 1. Строение, свойства, химические реакции и производные целлюлозы: учебно-методическое пособие. СПб.: СПбГТУРП, 2010.
- Odian G. Principles of Polymerization. 4th ed. Hoboken, NJ: Wiley-Interscience, 2004.
- IUPAC. Compendium of Polymer Terminology and Nomenclature (IUPAC Recommendations 2008). Cambridge: RSC Publishing, 2009.