Handbook of Size Exclusion Chromatography and Related ...

addition, he developed graphical means by which such measurements could be
related directly to the weight-averaged molar mass, the mean-square radius, and
the second virial coefficient. For the light scattering measurements made at
vanishingly small solute concentrations c, the familiar result of Zimm relating the
measurements of solute concentration c and the excess Rayleigh ratio R(u, c)tothe
derived macromolecular properties is given by
K c
R(u, c) ¼
1
MwP(u) þ 2A2c (1)
where Mw is the weight-average molar mass, P(u) is the scattering form factor,
A2 is the second virial coefficient, and K ¼ 4p2 (dn=dc) 2 n2 0 =(Nal4 0 ). Following
separation by SEC, at each slice (collection interval), both the MALS
measurement and a concentration measurement [corrected for its corresponding
interdetector volume (13) displacement] are made. The excess Rayleigh ratio
R(u, c) is the ratio of the scattered intensity per unit solid angle about the direction
u with respect to the direction of the incident beam to the incident light intensity
per unit area.
From Zimm’s graphical methodology, the extrapolated values of the left
hand side (l.h.s.) of Eq. (1) as c ! 0 and u ! 0 yielded molar mass directly, since
in this limit, P(u) ¼ 1 and the term proportional to A2 vanishes. We may expand
Eq. (1) for the case of small scattering angle and vanishingly small concentrations
to yield
K c
R(u, c) ¼
1
þ 2A2c
MwP(u)
1
Mw
1 þ 16p2 n 2 0
3l 2 0
kr 2 u
gl sin2
2
O sin4 u
2
þ (2)
From Eq. (2) it is easily seen that at these limits, the variation of the l.h.s. of Eq. (2)
with respect to sin 2 u=2 is16p 2 n 2 0 kr2 g l=(3Mwl 2 0 ), where K ¼ 4p 2 (dn=dc) 2 n 2 0 =
(Nal 4 0 ), l0 is the vacuum wavelength of the incident light, Na is Avogadro’s
number, and dn is the solution refractive index increment with respect to a
concentration change dc of the solute molecules. The mean square radius of a
molecule of mass M is defined by
kr 2 1
gl ¼
M
ð
r 2 dM (3)
where the integration is over all mass elements of the molecule with respect to its
center of mass. For the case of a distribution of molecules, this result is often
referred to as the z-average mean-square radius. The misnomer radius of gyration
© 2004 by Marcel Dekker, Inc.