Polymers 3/PH/AM Viscoelasticity

Polymers 3/PH/AM Viscoelasticity

A viscoelastic material is, as the name suggests, one which shows a combination of viscous and elastic effects.

The viscous term leads to energy dissipation.

The elastic term to energy storage.

Rate effects are very important for these materials.

Viscosity

For a viscous liquid with viscosity , the constitutive equation relating stress to strain is

There is dissipation of energy – and irreversible shape changes – associated with the flow.

The viscosity can be related to the diffusion equation.

In general then and D are inversely related, and as D increases with temperature viscosity decreases.

In contrast most solids exhibit pure elasticity.

Rubber Elasticity

Ideal elastic material = E—E is Young’s modulus

Energy is stored as elastic energy.

Material returns to original shape once stress removed.

Rubber bands are made from polymers, but the chains are crosslinked to provide a network.

The amorphous phase in PE is also said to be rubbery – it is above its Tgbut is constrained by the surrounding crystals and so cannot be said to be liquid-like.

For the rubber bands, it is the crosslinks which determine the properties.

In amorphous regions of uncrosslinked materials, the analogy is entanglements

The crosslinks provide a ‘memory’. When the network is stretched, entropic forces come into play which favour retraction, returning the network to its original unstretched / equilibrium state.

Changes to the Rubber Network upon stretching

Loss of entropy upon stretching means that there is a retractive force for recovery when external stress removed. This is why a rubber band returns to its original shape.

Using statistical mechanics to provide equations for the force on the chains (missed out!), we get Hookeian spring behaviour.

So far, we have talked about stretching experiments which give Young’s modulus, E. Shearing experiments are very common with viscoelastic materials, and give us the shear modulus G.

For incompressible materials (and rubber is, surprisingly, a very incompressible material) Young’s modulus E = 3G

!!! Always keep a look out for E versus G, namely stretching or shear experiments.

We get, in terms of the average MW between crosslinks Mx.

where G is the shear modulus and is the density

Note this means that for entropic elasticity (unlike enthalpic, as in steel) the modulus increases with temperature and the material gets stiffer rather than softer.

Note as cross link density goes up, (n, or equivalently Mx decreases), modulus goes up: a highly crosslinked rubber is stiffer than a lightly crosslinked one.

              

Polymeric liquids, and various solids, have attributes of both viscosity and elasticity and these are known as viscoelastic materials.

Creep and Stress Relaxation

In a creep experiment a constant load is applied and the resulting strain is measured as a function of time.

In a stress relaxation experiment a specimen is held at constant strain and the resulting stress is measured as a function of time.

              

Creep

/ immediate elastic deformation
delayed elastic deformation
These two are recoverable
Newtonian flow (i.e. permanent deformation)
This is non-recoverable

The diagram below makes the same points, but in terms of a shear experiment rather than a stretching one.

Determination with shear stress

/ A thin walled tube twisted by a torque : the shear stress in the tube is . The rotation of one with respect to the other is  and the shear strain is . If the tube is viscoelastic, then both quantities depend on time, (t) and (t).
The creep compliance J(t) under shear can be obtained by observing (t) for a fixed value of .

We can do two types of experiment, a creep experiment under constant stress  which gives us a creep compliance J as a function of time, or a stress relaxation under a constant shear strain  which gives us a shear modulus G as a function of time.

Creep experiment (constant stress) / Stress relaxation experiment

Notice how J(t) or G(t) are plotted as a function of log t. This kind of curve is called a sigmoid. This is a generally useful curve, and if you didn’t know about it you ought to, since it has so many applications in physics, as well as biology, economics, etc. See:



but note that in physics-type applications the function is usually sigmoid with respect to the logarithm of time or frequency.

The next two pictures show examples of creep and stress relaxation experiments, and how the behaviour is a function of time and temperature.

Creep as a function of time in polyethylene near the -relaxation (crystal related). / Stress relaxation with time in polyisobutylene near the glass transition
(–80°C) and onset of flow about 0°C. This last transition does not show a sigmoidal type behaviour, since what is seen is pure viscous flow.
/
.
/ This shows how viscoelastic properties change with temperature. The right hand curve is for the reference temperature T0, and the left hand one for (higher) T.
The main change is that the curve is shifted to the left by an amount log aT.
This is called the Shift Factor, and is used by engineers to predict properties.
At the higher temperature, I have drawn JU somewhat lower: remember that an elastic band contracts with increasing temperature.
I have also drawn JR higher, since a material will normally be able to creep a bit further as well as faster at a higher temperature. / Most polymer relaxations can be treated as an activated process, with relaxation times governed by the Arrhenius equation, so we get:

Superposition

Modelling Viscoelastic Behaviour

Zener models. Both of these are equivalent forms of the Standard Linear Solid (why?)

Spring and dashpot models

The Maxwell model is for stress relaxation, and gives:

while the Voigt model models creep, and gives:

They do not model real materials, because for creep, the Maxwell model would go on creeping indefinitely, while for stress-relaxation, the Voigt model cannot move with an instantaneous pull, but requires some flow before the spring can be stretched..…

Stress relaxation with time in polyisobutylene near the glass transition. /
Frequency dependence of the dynamic shear modulus of polyisobutylene. Here the glass-to-rubber relaxation is observed centred at
–10°C, well above the glass transition, because of the high frequency of observation.
Notice that at these frequencies the material is behaving as a rubber, because the entanglements are acting as temporary cross-links. In the figure a few pages up, further stress relaxation was occurring over hours as a result of viscous flow.

Observing viscoelastic losses in materials.

/ One method is by using a torsion pendulum.
This gives damped oscillations, and the logarithmic decrement  where
 = ln An / An+1
i.e. the log of the ratio of successive amplitudes.
The problem with this method is that the frequency of oscillation is specimen-dependent.

Dynamic Mechanical Analysis

or DMA. This method is by forced oscillation. In stress-strain experiments we do not as a rule apply a stress and measure the strain. Rather, we use a brute-force machine to apply a given strain, and a force transducer to measure the stress this produces. Here a sinusoidal strain is imposed on the material. The first few cycles do not count, but soon a steady-state condition is established where the stress  is ahead of the strain  by a phase angle .

and what we get from our experiments is tan.

For smallish values of the phase angle  = tan. Many classic results, when DMA instruments were not so widely available, are quoted with , so you can interpret those values in the same way as tan . Notice that when  = 0, tan. This is the point of no damping at all, i.e. Hookian behaviour.

Remember, that with DMA you have control of the frequency and can observe over a range of frequencies, a process which is much less direct if using a torsion pendulum.

Complex Modulus

E* = E' + iE"

complex
modulus / storage
modulus / loss
modulus

Dynamic Mechanical Analysis

/ Ths shows the DMA of polystyrene crosslinked with 2% divinyl benzene (110 Hz). (J.J. Fay, Lehigh University)

Five regions of viscoelastic behaviour

1. Glassy
2. Glass Transition Region
3. Rubbery Plateau
4. Rubbery Flow
5. Liquid Flow

Black line = linear, amorphous polymer
Dashed line = semicrystalline polymer
Dotted line – crosslinked amorphous polymer