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

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2. Equation (1.34) is a normalized expression which means that,

integrated

over all values of x, it equals unity. Accordingly,

adding together the

P(x,t)dx = (2 7Ã Kt l2) 1/2 exp

630

Frictional Properties of Polymers in Solution

concentration of solute in a series of slices through the apparatus-

that is, / ñ dx = / c0P(x, t) dx-always accounts for all of the solute,

whatever its distribution.

3. The combined consideration of items (1) and (2) indicates that the

profile of ñ throughout the apparatus gets flatter as it gets broader. At

t = 0 the initial concentration is sharply peaked at c0 at x = 0; at t =

°°, ñ is uniform from one end of the apparatus to the other.

All of these points are entirely reasonable. What is confusing about

Eq. (9.83) is the step length and the undetermined constant K.

Fortunately, they can both be eliminated in a single step. If Eq. (9.83)

is a solution to Eq. (9.74), then both sides of the latter must be equal

when the indicated operations are performed on Eq. (9.83). Carrying out

these operations yields the following:

1. The first derivative of ñ with respect to t is

(Ý.

= c0 (27Ã Kt 12)~1/2 e-x2/2Kt,:

2 Kt'

1

2t

2. The second derivative of ñ with respect to x is obtained from

(S)t = ^/2Kt|2 (- ø)

followed by

Kt

Kt

3. Combining items (1) and (2) as required by Fick's second law yields

(9c/9t) x2/2 Kt2!2 - l/2t K2t2l4 Kl2

D =

(Ý2 ñ/Ýõ2 )t x2 /Ê212 Ã - 1 /Ktl2 2 Kt2 r

These manipulations show that Eq. (9.83) satisfies Fick's second law with

The Diffusion Coefficient: Experimental Aspects

631

Ê =

2D

(9.84)

so the solution becomes

c(x, t)dx - c0(47rDt) 1/2 exp

dx

(9.85)

Figure 9.11 is a plot of this function at two different times for a

solute with a diffusion coefficient arbitrarily selected to be 5 X 10-11

m2 sec-1.

We can imagine measuring experimental curves equivalent to those in

Fig. 9.11 by, say, scanning the length of the diffusion apparatus by some

optical method for analysis after a known diffusion time. Such results

are then interpreted by rewriting Eq. (9.85) in the form of the normal

distribution function, P(z) dz. This is accomplished by defining a

parameter z such that

in terms of which Eq. (9.85) becomes

The inflection point of this function-where the second derivative changes

sign-occurs at z = 1; hence the experimental analogs of Fig. 9.11 are

examined for the location of their inflection points (subscript infl).

The distance through which the material has diffused at this point is

therefore given by

from which D can be evaluated if t is known. If Fig. 9.11 had been

determined experimentally, such a procedure would be used by observing

that xinfl = 10-2 m for the curve at t = 106 sec. From this information,

D = (10~2 m)2/2(106 sec)= 5 X 10-11 m2 sec-1.

The experiment we have just described is not very satisfactory from a

practical point of view, since it is very difficult to deposit a thin

layer of solution between two bulk portions of solvent without some

mixing. An experimentally more convenient method consists of layering

equal volumes of solvent and solution so that a sharp boundary exists

between them at x = 0, with ñ = c0 for

(9.87)

(9.88)

632

Frictional Properties of Polymers in Solution

x x 102 (m)

Figure 9.11 Variation of c/c0 with x for one-dimensional diffusion

[calculated from Eq. (9.85) with D = 5 X 1ÑÃ11 m2 sec-1 ].

x < 0 and ñ = 0 for x > 0. This procedure also depends on layering

liquids, but it is easier to do this with bulk quantities of material

than to sandwich a thin layer between two bulk portions of solvent as

required above.

The Diffusion Coefficient: Experimental Aspects

633

At the outset of such an experiment, the gradient of concentration is

zero on either side of the boundary and sharply peaked near x = 0. As

diffusion occurs, the boundary becomes progressively less sharp. For x <

0, ñ < c0, and for x > 0, ñ > 0, with both the magnitude and spatial

extension of the effect increasing with time. Figure 9.12a shows how the

overall concentration profile varies with time, and Fig. 9.12b shows the

gradient dc/dx at corresponding times. Formal solution of Fick's second

law with these boundary conditions proves that the shape of the gradient

curves in this situation are identical to the curves in Fig. 9.11.

Measuring the concentration gradient poses no serious experimental

difficulty, so this method can also be analyzed by Eq. (9.87).

We are aware of the bending of a light ray when light passes from a

medium of one refractive index to another. If light passes through a

layer of optically homogeneous material and then reenters the original

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