Chemistry Reference
In-Depth Information
Z
0
ð
o
Þ¼
Z
þ
N
t
1
þð
ot
Þ
2
dlnt
H
ð
4
:
51
Þ
N
Z
00
ð
o
Þ¼
Z
þ
N
ot
2
1
þð
ot
Þ
2
dlnt
H
ð
4
:
52
Þ
N
In the limit of high frequencies the integral for the loss modulus tends to zero
as the denominator in eqn (4.50) tends to infinity. The storage modulus tends to
G(N), which is just the integral under the relaxation spectrum:
G
ð
N
Þ¼
Z
þ
N
Hdlnt
ð
4
:
53
Þ
N
In the limit of low frequencies the integral for the imaginary part of the complex
viscosity tends to zero. The real part, Z
0
(o), tends to Z(0), which is the integral
under the relaxation spectrum after it has been multiplied by the appropriate t
value at each point,
Z
ð
0
Þ¼
Z
þ
N
Htdlnt
ð
4
:
54
Þ
N
We can illustrate the effect of the relaxation spectrum by assuming a form for
the distribution. A log normal distribution will illustrate the effect on the
storage and loss moduli,
!
ð
ln
ð
t
=
t
o
ÞÞ
2
8
:
405h
2
G
ð
N
Þ
8
:
405ph
2
p
H
¼
exp
ð
4
:
55
Þ
The spectrum has been selected such that the integral under the distribution
gives G(N). The distribution is centred on the relaxation time t
o
and the width
of the peak is given by h. The term h is the half-width of the peak at half of the
maximum height. A range of distributions with different half-width half-
heights is shown in Figure 4.11. Now we can calculate the storage and loss
moduli for a material that is described by these distributions. This is shown in
Figure 4.12.
The range of frequencies used to calculate the moduli is typically available on
many instruments. The important feature that these calculations illustrate is
that as the breadth of the distributions is increased the original sigmoidal and
bell-shaped curves of the Maxwell model are progressively lost. A distribution
of Maxwell models can produce a wide range of experimental behaviour
depending upon the relaxation times and the elastic responses present in the
material. The relaxation spectrum can be composed of more than one peak or
could contain a simple Maxwell process represented by a spike in the distribu-
tion, and this results in complex forms for all the elastic moduli.
