Page 347 - Numerical Methods for Chemical Engineering

336 7 Probability theory and stochastic simulation

=

+
22 2 i = = 2
=

22 2 2
i +

=

22 2 2 n +
i
Figure 7.10 Anionic polymerization of a block copolymer.

unit during this time interval is
p = k p [M]( t) = τ (7.103)

Taking this event to be a success, and taking the limit τ → 0, n →∞ so that the Poisson
distribution applies, we find that the probability that a particular chain has grown to a length
x, starting from a length x = 1at τ = 0is
[( τ)n] (x−1) −[( τ)n]
P(x; n, τ) = e (7.104)
(x − 1)!
The distribution of chain lengths at a scaled time τ = n( τ)is
τ x−1 −τ
P(x; τ) = e (7.105)
(x − 1)!
and the average chain lengths and polydispersity are
τ τ
DP n = 1 + τ DP w = 1 + τ + P disp = 1 + (7.106)
1 + τ (1 + τ) 2
In the limit of very long chain lengths, P disp → 1; i.e., the chains are of uniform length. This
ability to generate chains of precise, uniform lengths, combined with the ability to switch
monomers in the middle of the synthesis to produce block copolymers, makes anionic
living polymerization an important tool in polymer science, particularly in the formation of
nanoscale-ordered materials through microphase separation.

Random vectors and multivariate distributions

We now extend the concept of random variables to treat random vectors, for which we need
a number of additional definitions.
Definition Covariance and correlation of two random variables
Let X and Y be two random variables. The covariance of X and Y is

cov(X, Y) = E{[X − E(X)][Y − E(Y)]} (7.107)