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

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subchain is small compared to ncl0. Without going into full mathematical

details, let us examine how this constraint can be lifted.

The first step is to write an expression for the energy stored in a

network U as a result of the elongation a. This may be written as

where the F's are the x, y, and z components of the distorting force and

x0, y0, and z0 are the initial dimensions of the sample. Assuming that

the applied force is in the z direction, as in Fig. 3.1, we note that the

contraction in the x and ó directions is much less than the elongation in

z. The x and ó components of the force are therefore assumed to obey

Hooke's law; that is, Fx = kHx, with kH given by Eq. (3.45). A similar

expression describes F , but Fz is described differently because of the

greater elongation it experiences. For the latter the the force component

may be shown to be

in which rz is the z component of the average end-to-end distance in the

subchain. Note that Eq. (3.48) gives a force which is identical to the x

and ó components if rz < ncl0. Equation (3.48) gives the first three

terms in the series expansion of the inverse Langevin function of

rz/ncl0; this describes

xo /s/"

Ó0/s/"

óî Na azo

U = /

Fx dx + / Fy dy + / Fz dz

(3-47)

Óî zo

F.

Z

kT

[o

(3.48)

154

The Elastic and Viscoelastic States

the chain-end separation without assuming stringent limitations on

r/ncl0. Equations (3.48) and (3.45) with n = nc can be substituted into

Eq. (3.47) and the result integrated and then multiplied by v to give the

elastic energy stored per unit volume of elastomer. The derivative of

this last quantity with respect to a equals at. When the indicated

operations are carried out, the following result is obtained:

(3.49)

For the case of a < nc'/2, terms higher than first order in this

expression are negligible and the result becomes identical to Eq. (3.39).

However, the derivation of Eq. (3.49) does not limit its applicability to

these small values of a. Figure 3.5 shows that experimental results are

still not exactly described by Eq. (3.49), although the general shape of

the theoretical and experimental lines is similar.

at =

ukT V

1.1 rz~

ÃÒ + 7 V nc

3 -

VST

a + 9 / a \ 297 5 / a v

fif s m ç (;/=-)

Relative length

Figure 3.5 Comparison of experiment (points) and theory [Eq. (3.49)] for

the entropy elasticity of the same sample shown in Fig. 3.3. [Reprinted

with permission from H. M. James and E. Guth, J. Chem. Phys. 11:455

(1943).]

The Shear Modulus and the Compliances

155

In connection with Eq. (3.45) we noted that the deformation of

individual chains can be studied directly from random flight statistics.

Using equivalent expressions for the x, y, and z components of force and

following the procedure outlined above gives a more rigorous derivation

of Eq. (3.39) than that presented in the last section.

Even better agreement between theory and experiment has been obtained

in other theories by abandoning the notion of affine deformation and

recognizing that shorter subchains experience a greater strain than do

longer subchains for a given stress. We shall not pursue this development

any further, however, and shall turn next to a consideration of other

types of deformation.

3.6 The Shear Modulus and the Compliances

Until now we have restricted ourselves to consideration of simple tensile

deformation of the elastomer sample. This deformation is easy to

visualize and leads to a manageable mathematical description. This is by

no means the only deformation of interest, however. We shall consider

only one additional mode of deformation, namely, shear deformation.

Figure 3.6 represents an elastomer sample subject to shearing forces.

Deformation in the shear mode is the basis

Figure 3.6 Definition of variables to define the shear deformation of an

elastic body.

156

The Elastic and Viscoelastic States

for the definition of viscosity. An examination of elasticity according

to the same mode of deformation will make it easy to combine the two for

a description of viscoelasticity. We shall begin, however, by describing

the purely elastic response to a body under conditions of shear.

By analogy with Eq. (3.1), we seek a description for the

relationship between stress and strain. The former is the shearing force

per unit area, which we symbolize as as, as in Chap. 2. For shear strain

we use the symbol ys; it is the rate of change of ys that is involved in

the definition of viscosity in Eq. (2.2). As in the analysis of tensile

deformation, we write the strain AL/L, but this time AL is in the

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