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

**Download**(direct link)

**:**

**102**> 103 104 105 106 107 108 .. 312 >> Next

this shape are indeed observed. We shall see presently, however, that

this shape is also consistent with other mechanisms besides the one

considered until now.

In terms of spontaneous crystallization, the assumption that N

nuclei commence to grow simultaneously at t = 0 is the most unrealistic.

We can modify the model to allow for sporadic, spontaneous nucleation by

the following

The Kinetics of Crystallization: The Avrami Equation

223

argument. We draw a set of concentric rings in the plane of the disks

around point x as shown in Fig. 4.6b. If the radii are r and r + dr for

the rings, then the area enclosed between them is 2nr dr. We postulate

that spontaneous random nucleation occurs with a frequency N, having

units area-1 time-1. The rate of formation of nuclei within the ring is

therefore N27rr dr.

We continue to assume that the crystals so nucleated display a

constant rate of radial growth f. This means that it takes a crystal

originating in a ring of radius r around point x a time given by r/r to

cross x. The crystal labeled A in Fig. 4.6b has had just enough growth

time to reach x. On the other hand, a crystal nucleated in this ring

after t - r/r will not have had time to grow to x. The crystal labeled Â

in Fig. 4.6b is an example of the latter. It is only nucleation events

that occur up to t - r/r which have time to grow from the ring of radius

r and cross point x by their growth front. The increment in this number

of fronts for the ring of radial thickness dr is

dF = (N27rrdr)|t - *rj

(4.25)

The average number of fronts crossing point x at a time of

observation t is the sum of contributions from all rings which are within

reach of x in time t. The most distant ring included by this criterion is

a distance ft from x. The average number of fronts, therefore, is given

by integrating Eq. (4.25) for all rings between r = 0 and r = ft:

F = 2ttN Ä |t - v j dr

(4.26)

As far as this integration is concerned, f and t are constants, so Eq.

(4.26) is readily evaluated to give

F = - ;rNf2t3 (4.27)

3

As before, this quantity in relation to the degree of crystallinity is

given by Eq. (4.22), so equating the latter to Eq. (4.27) gives

1ï(^ò)=5^,Ý <4'28)

or

= 1 - exp (- J Nr2t3j

(4.29)

224

The Glassy and Crystalline States

Equations (4.24) and (4.29) are equivalent, except that the former

assumes instantaneous nucleation at N sites per unit area while the

latter assumes a nucleation rate of N per unit area per unit time. It is

the presence of this latter rate which requires the power of t to be

increased from 2 to 3 in this case.

In deriving these results we have focused attention on growth fronts

originating elsewhere and crossing point x. We would count the same

number if the growth originated at x and we evaluated the number of

nucleation sites swept over by the growing front. This change of

perspective is immediately applicable to a three-dimensional situation as

follows. Suppose we let N represent the number of sites per unit volume

(note that this is a different definition than given above) and assume

that a spherical growth front emanates from each. Then the average number

of fronts which cross nucleation sites in time t is

F = ~ ;r(rt)3N

(4.30)

by analogy with Eq. (4.18). Following the same argument as produced Eq.

(4.24), we obtain

â = 1 - exp (- | ÿã3Nt3J

(4.31)

for the three-dimensional case of simultaneous nucleation (concentration

N). Following the argument that produced (4.29), we obtain

â = 1 - exp ^ r3Nt4 j

(4.32)

for the three-dimensional case of sporadic nucleation (rate N).

Equations (4.24), (4.29), (4.31), and (4.32) all describe different

mechanisms for crystallization, yet all have basically the same form:

â = 1 - exp(-Ktm)

(4.33)

where Ê is a cluster-different in detail for each case-of numerical and

growth constants and m is the power dependence of time. Equation (4.33)

is known as the Avrami equation, after one of the researchers who has

made contributions in this area. Likewise, the exponent m in Eq. (4.33)

is called the Avrami exponent. The latter takes on the values 2, 3, or 4

for the four mechanisms considered above. To acquire some numerical

familiarity with the Avrami function, consider the following example.

The Kinetics of Crystallization: The Avrami Equation

225

Example 4.3

Three different crystallization systems show in values of 2, 3, and 4.

**102**> 103 104 105 106 107 108 .. 312 >> Next