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Polymer Chemistry. The Basic Concepts - Himenz P.C.

Himenz P.C. Polymer Chemistry. The Basic Concepts - Copyright, 1984. - 736 p.
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(10.45)
where normalization requires that the constant A be given by
/exp I" d5P
integrated over all fluctuations, that is, from bp = 0 to Using the
definition of an average provided by Eq. (1.9), we write
bp
2 =
f bp2 e"6G/kT d bp î
f e~6G/kT d5p 0
f° bp2 exp [- (l/2)(d2G/Ýð2)0 5p2/kT] d5p î
/ exp [- (l/2)(92G/9p2)0 5p2/kT] d5p î
(10.46)
682 Light Scattering by Polymer
Solutions
By letting ó = 5p and a = (d2G/dp2)0/2kT, these rather formidable-
looking integrals are recognized as gamma functions: / yme~ay dy. Using
tabulated values for these integrals for m = 0 and m = 2, we obtain
____ VT
bp2 = - - -
(10.47)
(d Gjbp )0
Substitution of this result into Eq. (10.42) yields
32 7Ã3 1 /_ dfl \2
kT nn.0,
r = -----r- - (n - I ----------------^-
(10.48)
3X" N* \ dp j (3 G/Ýð )o
As it stands, Eq. (10.48) is not an encouraging-looking result, but it
is actually very close to a highly useful form.
Rather than continuing to discuss the scattering of pure liquids at the
theoretical level, let us consider an example to illustrate the
application of these ideas.
Example 10.3
By an assortment of thermodynamic manipulations, the quantities dn/dp
and [N*(d2G/dp2)0] _1 can be eliminated from Eq. (10.48) and replaced by
the measurable quantities a, /3, and dn/dT: the coefficients of thermal
expansion, isothermal compressibility, and the temperature coefficient of
refractive index, respectively. With these substitutions, Eq. (10.48)
becomes
327Ã3 kT/3 dfl 2
r = -----.------ n
3X0 a2 dT
For benzenef, a = 1.21 X IO-3 deg-1, /3 = 9.5 X IO-10 m2 N_1, and dn/dT
= 6.38 X 10~4 deg-1. At 23°C, n = 1.503 for benzene at X0 = 546 nm. Use
these data to evaluate r for benzene under these conditions.
Solution
Direct substitution into the equation given provides the required value
for r:
Ç2òã3(1.38 X 10"23 J Ê"1 )(296 K)(9.5 X IO'10 m2 N"1)
(1.503)2(6.38 X IO'4 deg"1)2
r =
3 (546 X 10-9 m)4(l .21 X IO"3 deg'1)2 = 9.07 X
IO-3 m_1 (since J = N m)
t M. Kerker, The Scattering of Light and Other Electromagnetic Radition,
Light Scattering by Solutions
683
Experimentally the Rayleigh ratio for benzene at 90° has been
observed to equal about 1.58 X 1ÑÃ3 m-1 under the conditions described in
this example. By Eq. (10.6), r = (167Ã/3) R0 so the value of R0
corresponding to this calculated turbidity is Re calc = 5.41 X 10-4 m-1.
The ratio between the observed value of R0 and that calculated in the
example is called the Cabannes factor and equals about 2.9 in this case.
The origin of this enhanced scattering is understood to originate
from the anisotropy of the scattering centers. In terms of the analysis
we have presented, this amounts to some scrambling of horizontally and
vertically polarized components of light. This means that the light
scattered at 90° is not totally polarized as suggested by Fig. 10.6. By
using polarizing filters with the detection system, the intensities of
the horizontally and vertically polarized components of the scattered
light can be measured. The ratio of is h(0) to is v(0) is called the
depolarization ratio and is given the symbol pu(0) (u unpolarized
incident light; â, angle at which measured). The Cabannes factor which
corrects for this effect can be evaluated from the measured
depolarization ratio. Applied to turbidity, the Cabannes factor Ñ as a
function of pu(90) is CT= [6 + 3pu(90)] / [6 - 7pu(90)] and applied to
the Rayleigh ratio at 90°, CR90 = [6 + 6pu(90)] / [6 - 7pu(90)]. Note
that for the isotropic scatterers we have assumed, pu(90) =
0, so the Cabannes factor equals unity. Corrections of this sort are
also required for scattering by solutions. When the turbidity value
calculated in Example 10.3 is multiplied by the appropriate Cabannes
factor, the calculated and experimental results agree very well. We shall
not pursue this correction any further, but shall continue to assume
isotropic scatterers; additional details concerning the measurement and
use of the Cabannes factor will be found in Ref. 3.
As noted at the end of the last section, it is fluctuations in
concentration Sc2 rather than density which act as the scattering centers
of interest for solutions of small molecules. There is nothing in the
forgoing theory that prevents us from placing 5p by 5c2, the solute
concentration in mass volume-1 units. Therefore we write for a solution
of small molecules
In the next section, we consider the application of Eq. (10.49) to
scattering from fluctuations in concentration.
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