Light-Scattering Theory
Before we enter into further detail (in part II) about the application of light-scattering methods to the analysis of biopharmaceuticals,
let us review how light scattering works.
Figure 1: An array of collimated detector elements used to detect scattered light at discrete scattering angles.
|
Classical light-scattering measurements, in the absence of initial separations (such as SEC), are performed by first preparing
a series of samples at different, decreasing concentrations in a suitable solvent. The solvent chosen must be fully compatible
with the analytes and should not in any way change the nature of the original sample. Each sample is then illuminated sequentially
by a collimated light source, such as a monochromatic laser, and the scattered light from each sample is measured as a function
of scattering angle (1,2,23–29). This is shown schematically in Figure 1, where an array of collimated detector elements detect
scattered light at discrete scattering angles (30).
|
Historically, such measurements were first made using a single detector that rotated about the sample cell, stopping at each
angular location, where measurements were made. Using an array of detectors speeds up the measurement process significantly.
The relation between the measured intensities and the weight-average molecular weight (M
W) is given to the second order in the sample concentration (c), by the equation developed by Zimm:
Zimm showed that for such small concentrations, equation 1 may be written in its reciprocal form as
In equations 1 and 2,
where c is the concentration of the solute molecules (g/mL), n
0 is the refractive index of the solvent, and dn/dc is the refractive index increment of the solution (that is, the solution refractive index changes an amount dn for a solute concentration change of dc). Furthermore,
|
where, θ is the scattering angle, I(θ) is the intensity scattered by the solution into the collimated detector solid angle aboutθ, I
s
(θ) is the intensity scattered by the solvent into the collimated detector solid angle about θ, and I
0 the light intensity incident on the sample. f
geom is a geometrical factor depending on the structure of the scattering cell, the refractive indices of the solvent and the
cell, as well as the field of view of the corresponding detector and its acceptance angle for scattered light. N
A is Avogadro's number, λ0 is the incident wavelength in vacuum, M
W is the weight-average molecular weight, and A
2 is the second virial coefficient (a measure of the solvent-solute interaction). The form factor, P(θ), may always be written as an infinite series in sin2θ/2:
where α1 and α2 are constants and
and
the integration is taken over all mass elements, dm, of the molecule. The distance of mass element dm from the molecule's center of gravity is r.
Measurements of the scattered light intensity from different sample concentrations form the basis of Zimm's method to extract
the molecular parameters M
w, <r
g
2>, and A
2. Further details may be found in the original Zimm papers (26,27,33). Note from equation 2 above, in the limit, as c and the scattering angle, e, go to zero, K*c/R(0°) = 1/M
w. Although the molecular weight derived in this manner is the M
w of the sample, the derived mean square radius <r
g
2> is, in general, some type of average that is only easily defined for the case of random coil molecules in a theta solvent
(A
2 = 0). In that case, it may be shown quite easily that the measured mean square radius is a z-average value.
|
RSS
Bookstore
