Physical Chemistry Third Edition

1196 28 The Structure of Solids, Liquids, and Polymers

this the value forn2 can be calculated, and so on. From the initial condition in

Eq. (28.7-2)

〈x^2 〉 0  0 (28.7-9)

so that

〈x^2 〉 1 

a^2

3

(28.7-10)

Each iteration of Eq. (28.7-8) adds a term equal toa^2 /3, so that

〈x^2 〉n

na^2

3

(28.7-11)

The three directions are all equivalent so that〈x^2 〉n〈y^2 〉n〈z^2 〉n. By the theorem

of Pythagoras, the mean-square end-to-end distance in three dimensions is

〈r^2 〉n〈x^2 〉n+〈y^2 〉n+〈z^2 〉nna^2 (28.7-12)

and the root-mean-square end-to-end distance is

rrms〈r^2 〉^1 /^2 n^1 /^2 a (28.7-13)

As expected, the root-mean-square distance is proportional to the length of a link. It is

proportional to the square root of the number of links in the polymer chain, not to the

number of links. This comes from the fact that a longer chain has more ways to coil

up than does a short chain, so that adding more links increases the root-mean-square

distance less rapidly than the number of links.

In order to solve for the distribution of end-to-end lengths, we approximate

Eq. (28.7-1) by a differential equation.^38 We expand the functionpin four differ-

ent Taylor series inn,x,y, andz, treatingnas though it could take on nonintegral

values:

p(n+1,x,y,z)p(n,x,y,z)+

∂p

∂n

a+ ··· (28.7-14)

p(n,x±a,y,z)p(n,x,y,z)±

∂p

∂x

a+

1

2

∂^2 p

∂x^2

a^2 ± ··· (28.7-15)

with equations like Eq. (28.7-15) foryand z. These series are substituted into

Eq. (28.7-1). The lowest-order terms that do not cancel are kept, and the higher-order

terms are neglected:

∂p

∂n



a^2

6

[

∂^2 p

∂x^2

+

∂^2 p

∂y^2

+

∂^2 p

∂z^2

]



a^2

6

∇^2 p (28.7-16)

where∇^2 is called theLaplacian operator. Equation (28.7-16) is valid in the case that

nis large compared with unity and thatais small compared with the values ofx,y,

andzthat are important.

(^38) F. T. Wall,op. cit., p. 341ff (note 37).