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

**Download**(direct link)

**:**

**103**> 104 105 106 107 108 109 .. 312 >> Next

Calculate the value required for Ê in each of these systems so that all

will show â = 0.5 after 103 sec. Use these m and Ê values to compare the

development of crystallinity with time for these three systems.

Solution

Solve Eq. (4.33) for Ê and evaluate at t = 103 sec for each of the m

values: Ê = [-ln(l - 0)]/tm. For m = 2, Ê = (In 0.5)/(103)2 = 6.93 X 10"7

sec"2; for m = 3, Ê = 6.93 X Þ"10 sec"3; for m = 4, Ê = 6.93 X 10"13

sec"4. Note that the units of Ê depend on the value of m. Solve Eq.

(4.33) for t and evaluate at different 0's for the m and Ê values

involved:

m = 2

â (Ê = 6.93 X Þ"7) (Ê

0.1 3.89 X 102

0.2 5.67 X 102

0.3 7.18 X 102

0.4 8.59 X 102

0.5 1 X 103

0.6 1.15 X 103

0.7 1.32 X 103

0.8 1.52 X 103

0.9 1.82 X 103

t in seconds

m = 3 m = 4

= 6.93 X Þ-10) (K = 6.93 X 10-13)

5.33 X 102 6.24 X 10:

6.85 X 102 7.53 X 10:

8.02 X 102 8.47 X 10:

9.03 X 102 9.27 X 10:

1 X 103 1 X 103

1.10X 103 1.07 X 10:

1.20 X 103 1.15 X 10:

1.32 X 103 1.23 X 10:

1.49 X 103 1.35 X 10:

These three systems describe a set of crystallization curves that cross

at â = 0.5 and t = 103 sec. For the case where m = 2, the time span over

which the change occurs is widest (1430 sec from â = 0.1-0.9) and the

maximum slope is flattest (7.8 X 10-4 sec-1 between â = 0.4 and 0.6). For

m = 4, the range is narrowest (726 sec) and the maximum slope is steepest

(1.4 X 10-3 sec).

•

Although two of the mechanisms presented above yield the same power

dependence on t, it appears possible to eliminate certain mechanisms by

experimentally testing the development of â with time. A strategy for

this is suggested by Eq. (4.28). Taking the logarithm of both sides of

that equation gives

In

In ' 1

1 -

= 3 In t + const. (4.34)

226

The Glassy and Crystalline States

or for the generalized Avrami equation

In

In

(rhr

= m In t + const. (4.35)

Thus the slope of a plot of ln[ln(l - 0)_1 ] versus In t will have a

slope equal to the Avrami exponent.

Before turning to an examination of this prediction, a few more

complications must be mentioned. Until now we have considered the case of

disk or spherical growth from simultaneous or sporadic nucleation. Are

these the only possibilities? As might be anticipated, the answer is no.

Other geometries for growth have been examined; we shall include only the

case of the cylinder, which is limited in radius but not in length (i.e.,

one-dimensional growth) to produce fibrillar structures. Another

modification which has been investigated concerns the rate-determining

step of the crystallization process. Equation (4.18) and those following

from it imply that a contact between the growing disk and the surrounding

melt for time t is sufficient for crystallization. Another possibility is

that allowance must be made for the diffusion of the molecules to (or

from) the growth site. A way of dealing with this assumes that amorphous

molecules must diffuse out of the crystal domain to allow space for the

crystallizing molecules. For a crystal of radius r, the time required for

molecules to diffuse out of this domain is, according to Eq. (2.63), r =

(2Dt//2. In Eq. (4.18) this radius is written r = rt. Thus, if the growth

rate is diffusion controlled, these two expressions for r can be equated

and solved for r:

(4.36)

If this result is substituted into the previous expressions containing r,

the effect is to replace r with (2D)1/2 and to multiply those t's which

accompany r by t~'/z.

This rather complex array of possibilities is summarized in Table

4.3. Table

4.3 lists the predicted values for the Avrami exponent for the following

cases:

1. Growth geometry: fibrillar rod, disk, and sphere.

2. Nucleation mode: simultaneous and sporadic.

3. Rate determination: contact and diffusion.

Those exponents which we have discussed explicitly are identified by

equation number in Table 4.3. Other tabulated results are readily

rationalized from these. For example, according to Eq. (4.24) for disk

(two-dimensional) growth on contact from simultaneous nucleations, the

Avrami exponent is 2. If the dimensionality of the growth is increased to

spherical (three dimensional), the exponent becomes 3. If, on top of

this, the mechanism is controlled by diffusion, the

The Kinetics of Crystallization: Experimental Aspects

227

Table 4.3 Summary of Exponents in the Avrami Equation for Different

Crystallization Mechanisms

**103**> 104 105 106 107 108 109 .. 312 >> Next