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

Himenz P.C. Polymer Chemistry. The Basic Concepts - Copyright, 1984. - 736 p.
Download (direct link): polymerchemistry1984.djvu
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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 - -'/) <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 in radians per second. Remember that there are 2n
radians in a full cycle, so = 2irv. The reciprocal of 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 is induced in a sample:
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