# Molar mass distribution

The molecular weight distribution (engl. MWD, molecular weight distribution ), and molecular weight distribution , rare polydispersity describes in Science and Technology, the frequency distribution of individual molecular weights in sample polymeric substances . The breadth of this distribution is described by the polydispersity (also dispersity or polymolecularity index ). A large number of physical, mechanical and rheological properties of the sample depend on the breadth of the distribution.

Synthetic high polymer substances practically never consist of molecules of uniform size, but are in the form of polymolecular mixtures. The degrees of polymerization of the molecules (and thus also their molar masses) are distributed over a more or less broad range.

Macromolecules of biological origin, e.g. B. proteins or DNA , however, often have a completely uniform molecular weight.

## Distribution functions

In the case of certain polymerizations , a molar mass distribution ideally arises that is mathematically equivalent to a

or

is described.

In practice, due to the finite size of the individual monomers and side reactions, there are always (and intentionally) greater deviations from this. B. can be described as follows:

• narrow molar mass distribution
small number of fractions and high number of molecules per fraction or relatively few and small deviations from the mean , d. H. high uniformity;
many fractions and a small number of molecules per fraction or irregular distribution of the molecules per fraction or relatively many and high deviations from the mean value, d. H. high inconsistency (often technically desirable).
• bimodal or multimodal distribution
with two or more separate maxima in the distribution curve

## Molar mass of polymers

Various mean values ​​are defined to describe the sample statistically:

• Number average of the molar mass

The molar mass of the i-mer is weighted with the relative numerical fraction that this polymer has. The number-average molar mass therefore indicates the average molar mass of a random molecule taken from the sample. In this case, corresponds to the number of macromolecules in the sample with precisely i repeat units. ${\ displaystyle M_ {i}}$ ${\ displaystyle N_ {i}}$ ${\ displaystyle {\ overline {M}} _ {n} = {\ frac {\ sum _ {i = 1} ^ {f} N_ {i} M_ {i}} {\ sum _ {i = 1} ^ {f} N_ {i}}} = {\ frac {\ sum _ {i = 1} ^ {\ infty} n_ {i} \ cdot M_ {i}} {\ sum _ {i = 1} ^ {\ infty} n_ {i}}} = \ sum _ {i = 1} ^ {\ infty} x_ {i} \ cdot M_ {i}}$ • Viscosity average of the molar mass

The viscosity average is determined by measuring the intrinsic viscosity of the polymer solution. ${\ displaystyle {\ overline {M}} _ {\ eta}}$ ${\ displaystyle \ eta}$ ${\ displaystyle {\ overline {M}} _ {\ eta} = \ left ({\ frac {\ sum _ {i = 1} ^ {f} N_ {i} M_ {i} ^ {(1+ \ alpha )}} {\ sum _ {i = 1} ^ {f} N_ {i} M_ {i}}} \ right) ^ {\ frac {1} {\ alpha}}}$ ${\ displaystyle \ alpha}$ is a positive rational number and is usually between 0.5 and 0.9.

• Mass average of the molar mass

The molar mass of the i-mers is weighted with the relative mass fraction that this polymer has. If one were to select a random monomer unit and determine the molar mass of the associated polymer, the average would be the weight-average molar mass. ${\ displaystyle M_ {i}}$ ${\ displaystyle {\ overline {M}} _ {w} = {\ frac {\ sum _ {i = 1} ^ {f} m_ {i} M_ {i}} {\ sum _ {i = 1} ^ {f} m_ {i}}} = {\ frac {\ sum _ {i = 1} ^ {f} N_ {i} M_ {i} ^ {2}} {\ sum _ {i = 1} ^ { f} N_ {i} M_ {i}}} = {\ frac {\ sum _ {i = 1} ^ {\ infty} n_ {i} \ cdot M_ {i} ^ {2}} {\ sum _ { i = 1} ^ {\ infty} n_ {i} \ cdot M_ {i}}} = {\ frac {\ sum _ {i = 1} ^ {\ infty} x_ {i} \ cdot M_ {i} ^ {2}} {\ sum _ {i = 1} ^ {\ infty} x_ {i} \ cdot M_ {i}}} = \ sum _ {i = 1} ^ {\ infty} w_ {i} \ cdot M_ {i}}$ • Centrifuge mean of molar mass (Z mean)

The centrifuge agent is determined by measuring the sedimentation equilibrium. ${\ displaystyle {\ overline {M}} _ {z}}$ ${\ displaystyle {\ overline {M}} _ {z} = {\ frac {\ sum _ {i = 1} ^ {\ infty} n_ {i} \ cdot M_ {i} ^ {3}} {\ sum _ {i = 1} ^ {\ infty} n_ {i} \ cdot M_ {i} ^ {2}}} = {\ frac {\ sum _ {i = 1} ^ {\ infty} x_ {i} \ cdot M_ {i} ^ {3}} {\ sum _ {i = 1} ^ {\ infty} x_ {i} \ cdot M_ {i} ^ {2}}} = {\ frac {\ sum _ {i = 1} ^ {\ infty} w_ {i} \ cdot M_ {i} ^ {2}} {\ sum _ {i = 1} ^ {\ infty} w_ {i} \ cdot M_ {i}}}}$ Abbreviations:

${\ displaystyle M _ {\ mathrm {Mono}}}$ : Molar mass of the monomer
${\ displaystyle M_ {i}}$ : Molar mass of the polymers of the respective fraction i
${\ displaystyle m_ {i}}$ : Total mass of the respective fraction i
${\ displaystyle N_ {i}}$ : Number of macromolecules in fraction i
${\ displaystyle f}$ : Total number of all political groups
${\ displaystyle n_ {i} [m_ {i}]}$ = Amount of substance [mass] of the i-mer; = Sum of all${\ displaystyle n [m]}$ ${\ displaystyle n_ {i} [m_ {i}]}$ ${\ displaystyle x_ {i}}$ = Mole fraction of the i-mer
${\ displaystyle M_ {i}}$ = Molar mass of the i-mer, ${\ displaystyle M_ {i} = i \ cdot M_ {0}}$ ${\ displaystyle M_ {0} [n_ {0}]}$ = mean molar mass [amount of substance] of a monomeric unit
${\ displaystyle w_ {i}}$ = Mass fraction of the i-mer
${\ displaystyle w_ {i} = {\ frac {m_ {i}} {\ sum _ {i = 1} ^ {\ infty} m_ {i}}} = x_ {i} \ cdot {\ frac {M_ { i}} {\ overline {M_ {n}}}} = i \ cdot x_ {i} \ cdot {\ frac {n} {n_ {0}}}}$ The ratio of the number average and the average molar mass of a monomeric unit indicates the degree of polymerization (see below). ${\ displaystyle {\ overline {M}} _ {n}}$ ${\ displaystyle M_ {0}}$ ${\ displaystyle P_ {n}}$ ## Determination methods

The mean (see above) molar masses of a sample can be determined using various methods:

Conclusions about the breadth of the distribution can be drawn from the different values.

For the direct determination of the molar mass distribution, the

applied.

The GPC still needs a suitable calibration, while MS is an absolute method.

GPC and centrifugation are also used for preparative polymer fractionation .

## Polydispersity Comparison of two molar mass distributions with different dispersity and different mean molar mass.

The polydispersity Đ is a measure of the breadth of a molar mass distribution; it is calculated from the ratio of weight average to number average. The larger Đ , the broader the molar mass distribution. In addition to the symbol Đ , the dispersity is sometimes also given as Q, PDI (polydispersity index) or polymolecularity index .

${\ displaystyle D = {\ frac {{\ overline {M}} _ {w}} {{\ overline {M}} _ {n}}} \ geq 1}$ Instead of the polydispersity, the molecular inhomogeneity is often given, it is defined as ${\ displaystyle U}$ ${\ displaystyle U = D-1 = {\ frac {{\ overline {M}} _ {w}} {{\ overline {M}} _ {n}}} - 1}$ .

Macromolecules of biological origin, e.g. B. proteins or DNA , often have a completely uniform molar mass, so they have a non-uniformity of zero, or polydispersity of one, so the following applies:

${\ displaystyle {\ overline {M}} _ {n} = {\ overline {M}} _ {w} (= {\ overline {M}} _ {z})}$ For synthetic polymers, however, the following applies:

${\ displaystyle {\ overline {M}} _ {n} <{\ overline {M}} _ {\ eta} <{\ overline {M}} _ {w} <{\ overline {M}} _ {e.g. }}$ ## Average degree of polymerization

In general, the average degree of polymerization of a homopolymer is obtained by dividing the average molar mass by the molar mass of the repeating unit. In individual cases (e.g. in the case of polycondensation ) this can differ from that of the monomer.

### Numerical mean

${\ displaystyle {\ overline {X}} _ {n} = {\ frac {\ sum _ {i = 1} ^ {f} N_ {i} M_ {i}} {\ sum _ {i = 1} ^ {f} N_ {i}}} \ cdot {\ frac {1} {M _ {\ mathrm {Mono}}}} = {\ frac {{\ overline {M}} _ {n}} {M _ {\ mathrm {Mono}}}}}$ ### Weight average

${\ displaystyle {\ overline {X}} _ {w} = {\ frac {\ sum _ {i = 1} ^ {f} m_ {i} M_ {i}} {\ sum _ {i = 1} ^ {f} m_ {i}}} \ cdot {\ frac {1} {M _ {\ mathrm {Mono}}}} = {\ frac {\ sum _ {i = 1} ^ {f} N_ {i} M_ {i} ^ {2}} {\ sum _ {i = 1} ^ {f} N_ {i} M_ {i}}} \ cdot {\ frac {1} {M _ {\ mathrm {Mono}}}} = {\ frac {{\ overline {M}} _ {w}} {M _ {\ mathrm {Mono}}}}}$ ### Viscosity agent

${\ displaystyle {\ overline {X}} _ {\ eta} = \ left ({\ frac {\ sum _ {i = 1} ^ {f} N_ {i} M_ {i} ^ {(1+ \ alpha )}} {\ sum _ {i = 1} ^ {f} N_ {i} M_ {i}}} \ right) ^ {\ frac {1} {\ alpha}} \ cdot {\ frac {1} { M _ {\ mathrm {Mono}}}} = {\ frac {{\ overline {M}} _ {\ eta}} {M _ {\ mathrm {Mono}}}}}$ ## literature

• JMG Cowie: Chemistry and Physics of Synthetic Polymers. Vieweg, 2 ed., 1991.
• K. Matyjaszewski , TP Davis: Handbook of Radical Polymerization. Wiley, 2002.
• Bernd Tieke: Macromolecular Chemistry. An introduction. Wiley-VCH, Weinheim 2005, ISBN 3-527-31379-6 .

## Individual evidence

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3. ^ Karl August Wolf: Structure and physical behavior of plastics . Springer Berlin Heidelberg, Berlin, Heidelberg 1962, ISBN 978-3-662-25000-6 , pp. 29 .
4. Sebastian Koltzenburg, Michael Maskos, Oskar Nuyken: Polymers: Synthesis, Properties and Applications . Springer Spectrum, Berlin 2014, ISBN 3-642-34772-X , p. 302-310 .
5. a b Bernd Tieke: Makromolekulare Chemie . 2., completely revised and exp. Edition. Wiley-VCH, Weinheim 2005, ISBN 3-527-31379-6 , pp. 9 .
6. ^ MD Lechner, EH Nordmeier and K. Gehrke: Makromolekulare Chemie. Birkhäuser, 2010, ISBN 978-3-7643-8890-4 , p. 245.
7. Kang-Jen Liu: NMR studies of polymer solutions. VI. Molecular weight determination of poly (ethylene glycol) by NMR analysis of near-end groups . In: The Macromolecular Chemistry . 116, No. 1, August 1968, pp. 146-151. doi : 10.1002 / macp.1968.021160115 .
8. Sebastian Koltzenburg, Michael Maskos, Oskar Nuyken: Polymers: Synthesis, Properties and Applications . Springer Spectrum, Berlin 2014, ISBN 3-642-34772-X , p. 46 .
9. Walter Krauß: Binder for solvent-based and solvent-free systems . 2., ext. and rework. Edition volume 2.1 . Hirzel, Stuttgart a. a. 1998, ISBN 3-7776-0886-6 , pp. 133 .