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

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predicts can be

taken into account by imagining that the rise consists of a series

of n smaller

(and unresolved) steps. This is equivalent to expanding the model so

that it consists of n Voigt elements as shown in Fig. 3.10b. Each of

these Voigt elements is characterized by its own value for G*, 77*, and

r.

2. With this modification of the model, Eq. (3.66) becomes

J(t) = I J,(t) = E Jj(~)(l - e"t/T0

(3.70)

i=l i=l

or

J(t) = /"j(i-) (1 - å-'/ã) <ir

(3.71)

0

if a continuous distribution of retardation times is considered

[compare Eq. (3.62)].

3. In addition to the set of Voigt elements, a Maxwell element

could also be

included in the model. The effect is to include a

contribution given by

Eq. (3.69) to the calculated compliance. This long time flow

contribution to the compliance is exactly what we observe for non-cross-

linked polymers in Fig. 3.12.

4. Therefore, in the most general case, we write for an actual polymer

J(t) = J(0) + -!;+/ J(T)(1 - e-'^)d7

(3.72)

V* 0

An advantage of having the relaxation spectrum defined by Eq. (3.63)

is that it can be adapted to expressions like this to calculate

mechanical behavior other than that initially measured.

The Maxwell and Voigt models of the last two sections have been

investigated in all sorts of combinations. For our purposes, it is

sufficient that they provide us with a way of thinking about relaxation

and creep experiments. Probably one of the reasons that the various

combinations of springs and dash-pots have been so popular as a way of

representing viscoelastic phenomena is the fact that simple and direct

comparison is possible between mechanical and electrical networks, as

shown in Table 3.3. In this parallel, the compliance of a spring is

equivalent to the capacitance of a condenser and the viscosity of a

dashpot is equivalent to the resistance of a resistor. The analogy is

complete

Dynamic Viscoelasticity

173

Table 3.3 Comparison of Mechanical and Electrical Models Consisting of

Different Arrangements of Springs and Dash-pots or Their Equivalents,

Capacitance and Resistance, Respectively

Mechanical Electrical

F/A Electromotive force

7 Charge

7 Current

J Capacitance

V Resistance

Maxwell element Series Parallel

Voigt element Parallel Series

because electrical work done on a network is stored in capacitors and

dissipated by resistors. Likewise, mechanical energy is stored and

dissipated by the elastic and viscous units of a mechanical model system.

The only word of caution about the use of this analogy is that the rules

for combination are reversed-that is, series becomes parallel and vice

versa-between the electrical and mechancial networks to produce the close

correspondence between the storage and dissipative units. Electrical

circuits can thus be designed and analyzed which embody the appropriate

features of a model mechanical system. The usefulness of this analogy

will be even more evident-although we shall not pursue it-in the next

section, in which we take up periodic stress-strain relationships, the

mechanical analog of alternating current.

3.10 Dynamic Viscoelasticity

The relaxation and creep experiments that were described in the preceding

sections are known as transient experiments. They begin, run their

course, and end. A different experimental approach, called a dynamic

experiment, involves stresses and strains that vary periodically. Our

concern will be with sinusoidal oscillations of frequency v in cycles per

second (Hz) or ñî in radians per second. Remember that there are 2n

radians in a full cycle, so ñî = 2irv. The reciprocal of ñî gives the

period of the oscillation and defines the time scale of the experiment.

In connection with the relaxation and creep experiments, we observed that

the maximum viscoelastic effect was observed when the time scale of the

experiment is close to r. At a fixed temperature and for a specific

sample, r or the spectrum of r values is fixed. If it does not correspond

to the time scale of a transient experiment, we will lose a considerable

amount of information about the viscoelastic response of the system. In a

dynamic experiment it may

174

The Elastic and Viscoelastic States

be possible to vary the frequency in such a way that the period and the

range of r values overlap optimally. This sort of dynamic mechanical test

yields the maximum amount of information about a viscoelastic substance.

Suppose an oscillating strain of frequency ñî is induced in a sample:

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