Physical Chemistry Third Edition

28.9 Rubber Elasticity 1203

A Molecular Theory of Rubber Elasticity

We represent an ideal rubber by a model system that has the following properties:^43

(1) The equilibrium system consists of a set of polymer molecules with an equilib-

rium distribution of end-to-end lengths given by the freely jointed chain formula of

Eq. (28.9-37). (2) A certain number,N, of randomly selected polymer chains are cross-

linked. For simplicity, we assume that they are cross-linked only at their ends and that

all of the cross-linked molecules have the same number of links,n. (3) When the rubber

is stretched in thexdirection, theyandzdimensions change so that the volume remains

constant, and thex,y, andzcomponents of all end-to-end vectors change in the same

ratio as thex,y, andzdimensions of the rubber.

At equilibrium, there will beNimolecules with an end-to-end vectorri(xi,yi,

zi). If an elongation in thexdirection preserves the original volume, the end-to-end

vector of these molecules will be (x′i,yi′,z′i):

x′iαxi; yi′

yi

α^1 /^2

; z′i

zi

α^1 /^2

(28.9-13)

whereαis the degree of elongation, equal toL/L 0 , the elongatedxdimension divided

by the originalxdimension. The number of molecules with this new end-to-end vector

is still equal toNi.

To calculate the entropy change on elongation, we use the definition of the statistical

entropy of Eq. (26.1-1):

SstkBln(Ω)+const (28.9-14)

whereΩis the number of system mechanical states that are compatible with the

thermodynamic state of the system, assuming all of these states to have equal energy.

To use this formula, we calculate the probability,P, that the elongation would occur

spontaneously, equal to the probability thatN 1 chains will have end-to-end vectorr′ 1 ,

thatN 2 chains will have end-to-end vectorr′ 2 , etc. We assume that the chains act inde-

pendently so that this probability is the product of the probabilities of the individual

chains. This probability is then multiplied by the number of ways to divide the set of

polymer molecules into the specified subsets:

P′N!

∏

i

1

Ni!

p(n,x′i,y′i,z′i)NiN!

∏

i

1

Ni!

(p′i)Ni (28.9-15)

where the factorN!/

∏

iNi! is the number of ways to divide theNchains into the

required subsets and where we have abbreviatedp(n,x′i,y′i,z′i)byp′i. Using Stirling’s

approximation for ln(N!) and ln(Ni!) from Eq. (25.2-21)

ln(P′)Nln(N)−N+

∑

i

[Niln(p′i)−Niln(Ni)+Ni]



∑

i

Niln(p′iN/Ni) (28.9-16)

We now write this equation for the equilibrium distribution:

ln(P)

∑

i

Niln(piN/Ni) (28.9-17)

(^43) F. T. Wall,op. cit., Ch. 16 (note 37).