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

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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,

Academic, New York, 1969.

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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