Molar mass ultracentrifugation

In order to understand polymer solution behaviour, the samples have to be characterised with respect to their molecular configuration, their molar mass and polydispersity, the polymer concentration and the shear rate. Classical techniques of polymer characterisation (light scattering, viscometry, ultracentrifugation, etc.) yield information on the solution structure and conformation of single macromolecules, as well as on the thermodynamic interactions with the solvent. In technical concentrations the behaviour of the dissolved polymer is more complicated because additional intramolecular and intermolecular interactions between polymer segments appear. 

In addition to the determination of molar mass distributions and various molar mass averages there are many experiments, requiring sometimes sophisticated data evaluation, that can be carried out with an analytical ultracentrifuge. Examples are the analysis of association, the analysis of heterogeneity, the observation of chemical reactions, and protein characterization, to mention only a few. A detailed discussion is beyond the scope of this article, but there is excellent literature available [77-79,81,87-89]... 

For the analysis of light scattering experiments the refractive indices of the DADMAC/AAM solutions at dialysis equilibrium were determined, showing the validity of the additivity principle [67]. The additivity could also be proven for the partial specific volumes which are necessary to calculate molar masses from ultracentrifugation experiments [131]. These dependencies are summarized in Fig. 24. 

Higher averages can be obtained by measuring the rate of sedimentation in solvents with the aid of an ultracentrifuge. This is based on the fact that the rate of sedimentation depends on the molar mass. The measurements supply Mw and Mz sometimes M/+. ... 

The sedimentation process gives rise to a solvent phase and a concentrated polymer solution phase which are separated by a boundary layer in which the polymer concentration varies. There is, therefore, a natural tendency for backward diffusion of the molecules in order to equalise the chemical potentials of the components in the different regions of the cell, and this causes broadening of the boundary layer. The breadth of the boundary layer also increases with the degree of polydispersity because molecules of higher molar mass sediment at faster rates. The windows in the cell enable the radial variation in polymer concentration to be measured during ultracentrifugation typically... 

Z-average molecular weight (Af ) This is also determined by ultracentrifugation technique. The molecular weight depends both on size and mass of the molecules. However, the contribution of the mass of the particle are weighed further in this type of molar mass. Mathematically, is given as... 

Weight average molecular weight may be experimentally determined by any method in which molecular size or molar mass is the parameter being measured (e.g., light scattering or ultracentrifugation). 

The molar mass distribution can also be determined by ultracentrifugation, which we will describe later. 

This is the Svedberg equation for the molar mass. If an independent value of D is available, the measurement of s suffices to determine M. Later we will show that the value oi D/s can be obtained from an ultracentrifugation experiment. 

Equation (9-114) corresponds to a Gaussian distribution function (see also Section 8.3.2.1). The molar mass can be calculated from the position of the inflection point of the function c = /(r — r ). For proteins in CsCl/ H2O, the lower molar mass limit giving a meaningful measurement is 10 000-50 000 g/mol molecule. The limit is essentially governed by the length of the ultracentrifuge cell ( 1.2 cm) and the optimal values of r — for this length. 

Figure 9-15. Kinds of preparative ultracentrifugation methods. (S) Normal ultracentrifugation, (B) band ultracentrifugation in a stabilized gradient, (I) isopycnic zone centrifugation p is the density of the gradient-forming substance (O) high molar mass, low density ( ) low molar mass, high density.

Blood serum (separated from fibrinogen and/or fibrin by ultracentrifugation) contains serum albumins. These possess a molar mass of 67 500 and an isoelectric point at pH 4.8-5.0. The serum albumins of higher mammals differ particularly in the amino acid residues at the carboxyl end of the protein chain, while the N-terminal amino acid residue is always asparagine ... 

This completes the discussion of light scattering. These data were the source of much of the information discussed in Sect. 1.3. Next, a series of other, somewhat simpler characterization techniques is discussed which can be used to determine average molar masses. With size-exclusion chromatography and ultracentrifugation, distributions can also be assessed. 

The z-average molar mass can be measured using an ultracentrifuge. Gel permeation chromatography allows the measurement of the complete distribution function and hence the estimation of all average molar masses. 

Equation 9.8.22 is needed for sedimentation equilibrium, a method of determining the molar mass of a macromolecule. A dilute solution of the macromoleeule is placed in the cell of an analytical ultracentrifuge, and the angular velocity is selected to produce a measurable solute concentration gradient at equilibrium. The solute concentration is measured optically as a function of r. The equation predicts that a plot of In (cb/c°) versus will be linear, with a slope equal to Mb (l — E p) jlRT. The partial specific volume is found from measurements of solution density as a function of solute mass fraction (page 234). By this means, the molar mass Mb of the macromolecule is evaluated. 

Example 11.1 J The molar mass of a protein from ultracentrifugation experiments... 

The data from an equilibrium ultracentrifugation experiment performed at 300 K on an aqueous solution of a protein show that a graph of In c agcunst (r/cmy is a straight line with a slope of 0.729. The rotationcd rate of the centrifuge was 50 000 rotations per minute and b = 0.70. Calculate the molar mass of the protein. 

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