Self-Assembly of Synthetic and Biological Polymeric Systems of ...

elasticity or viscosity of the sample in order to characterize its material properties. In order to
obtain this information some mathematical manipulation is required. Two key features can be
utilized. The first feature is the maximum stress, , divided by the maximum strain, , which is
constant for a given frequency . This ratio is called the complex modulus :
is the radial frequency, which is , where is the applied frequency measured in Hz. The
other feature constant with time at any given frequency is , the phase difference (expressed
in radians). These two values, and , are characteristic of the material. It is straightforward
to visualise the situation where an elastic solid is placed in a cone and plate geometry. When a
tangential displacement is applied to the lower plate a strain in the sample is produced. That
displacement is transmitted directly through the sample. The upper cone will react
proportionally to the applied strain to give a stress response. An oscillating strain will give an
oscillating stress response that is in phase with the strain, so will be zero. However, if we
have a Newtonian liquid, the peak stress will be out of phase by
2.39
rad as the peak stress is
proportional to the strain rate. So, it follows that if we have a viscoelastic material some
energy is stored and some dissipated, and the stored contribution will be in phase whilst the
dissipated or loss contribution will be out of phase respect to the applied strain.
In order to describe the material properties as a function of frequency we need to use the
constitutive equation 2.30. This equation describes the relation between the stress and the
strain. However, it is most convenient to express the applied sinusoidal wave in the
exponential form of the complex number notation:
Now, the stress response lags by the phase angle :
So, substituting the complex stress and strain into the constitutive equation for a Maxwell fluid
the resulting relation is given by:
2.40
2.41
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