Polymer Physics

2. Each polymer coil connecting two neighboring cross-links follows the Gaussian
   distribution regarding its end-to-end distances;


3. The total entropy change is a linear integration of conformational-entropy
   changes of all the network chains;


4. The deformation ratio of the network is equal to that of each network chain.

For a sample system containingNnetwork chains, according to the second and
the third assumptions, the total entropy change is then

DS 0 ¼kb^2 l^21  1

XN

1

x^2 iþ l^22  1

XN

1

y^2 iþ l^23  1

XN

1

z^2 i

"

(3.19)

According to the first assumption, we further have

Sx^2 i

N

¼

Sy^2 i

N

¼

Sz^2 i

N

¼

<R^20 >

3

(3.20)

In (3.20), the mean-square end-to-end distance of polymers<R 02 >
corresponds to a bulk polymer phase. Thus

DS 0 ¼

1

3

Nkb^2 <R^20 >ðl^21 þl^22 þl^23  3 Þ (3.21)

Fromb^2 ¼ 2 nb^32 and the characteristic ratioC¼<R

(^20) >
nb^2 , we can simplify the above
equation into

DS 0 ¼

1

2

CNkðl^21 þl^22 þl^23  3 Þ (3.22)

In the uniaxial stretching, l 1 ¼l. Since DV¼0, we obtainl 1 l 2 l 3 ¼1.
According to the fourth assumption, we have

l 2 ¼l 3 ¼

1

ffiffiffi

l

p (3.23)

Equation (3.22) can be further simplified as

DS 0 ¼

1

2

CNkðl^2 þ

2

l

 3 Þ (3.24)

Therefore, the entropic elasticity contributed by (3.24)is

fS¼T

@S

@l

¼T

@S

@l



@l

@l

¼

T

l 0



@S

@l

¼

CNkT

l 0

ðl

1

l^2

Þ (3.25)

38 3 Conformation Statistics and Entropic Elasticity