# 8. Stress relaxation

In chapter 6 we have learned how a deformation of the polymer causes stress in the polymer. We have also learned in chapter 4 that the polymer molecules are mobile: the Kuhn segments of the macromolecule rotate and the entire molecules reptate. Due to this mobility any stress will reduce with time. This is called stress relaxation.

## Stress relaxation in the glass phase

In the glass phase the polymer molecules are not very mobile. Rotation of the Kuhn segments is possible, but at a very low frequency. At the glass transition temperature the rotation time is 1 second. At lower temperatures one single rotation may need a time as much as a million seconds or more.

Yet it is these rotations that cause the stress to disappear. Let us visualize the polymer molecule with its Kuhn segments as a folding ruler. The rotating parts of the ruler are the Kuhn segments.

Now let us suppose that the hinges between the rotating parts of the folding ruler are rusty; it is difficult to rotate them. If we would deform the ruler a little bit then the parts of the folding ruler would first bend a little, as shown in the figure below.

However, after some time one or more of the hinges will give way and some parts of the folding ruler will rotate. This will effectively reduce the bending of the parts and will thus reduce the stress. This is shown in the picture above at the right-hand side. The stress is relaxing due to the rotation of the parts.

In the same way the stress in a polymer in the glass phase (the glass stress) will relax due to the rotation of the Kuhn segments. Once a sufficient amount of Kuhn segments has rotated all bending will have disappeared. The stress has then reduced considerably. The glass stress, due to bending of the chain segments, has been replaced by the rubber stress which is due to the rotation of the Kuhn segments. The relaxation time of stresses in the glass phase is identical with the Kuhn segment rotation time.

The typical time for this stress reduction (the glass relaxation time) is equal to the time that the Kuhn segments need for making one rotation. At the glass transition temperature, where the Kuhn segments need 1 second for a rotation, the relaxation time is always 1 second. At a temperature far below the glass transition temperature the relaxation time may increase to many millions of seconds.

The change of the glass stress with time due to relaxation can be mathematically described by means of a differential equation. If there is no other influence on the glass stress than the glass relaxation process then the change of stress with time looks as follows:
[change of glass stress per unit of time] = – [glass stress] / [glass relaxation time]

The differential equation above describes a situation where glass stress can only disappear due to relaxation. The level of the stress will gradually reduce to zero with time. This is a very special situation. More common is that a polymer is continuously deformed. In such a case the “change of the glass stress per unit time” consists of two parts:

1. An increase of the stress due to the deformation
2. A decrease of the stress due to the relaxation: “-glass stress / glass relaxation time”

The differential equation now looks like:
[change of glass stress per unit of time] = + [change of glass stress due to deformation per unit of time] – [glass stress] / [glass relaxation time]

We will show later how such differential equations can be used to calculate the yield stress of the polymer.

## Stress relaxation in the rubber phase

In the rubber phase the time that the Kuhn segments need to rotate is much less than 1 second. That implies that any glass stress will disappear almost immediately to zero. Glass stress is not relevant in the rubber phase.

The quickly rotating Kuhn segments will deform the macromolecules in a random shape. It is called the random coil configuration and that is the natural shape of the macromolecules. If we would now apply a deformation to the polymer then the random coil configuration is disturbed. The shape of the macromolecules changes from a sphere into an ellipsoid. The macromolecules will react on this deformation by creating a stress that tends to bring their shape back to that of a sphere. This stress is called the rubber stress.

We can visualize this rubber stress by means of a rotating jumping rope. Due to the rotating rope the ends of the rope are pulled together. And the faster that the rope rotates the more force on the rope ends is created. In the same way the rubber stress increases with temperature!

In the rubber phase the macro molecules can slowly change their position due to reptation. This gives the macromolecules the possibility to reshape themselves into their wished random coil configuration. Thus, the rubber stress will disappear in time. The time scale on which this happens is simply the reptation time. Therefor the rubber relaxation time is equal to the reptation time.

The change of the rubber stress with time due to relaxation can be mathematically described by means of a differential equation that is very similar with the one for the glass stress. If there is no other influence on the rubber stress than the rubber relaxation process then the change of stress with time looks as follows:

[change of rubber stress per unit of time] = – [rubber stress] / [rubber relaxation time]

The differential equation above describes a situation where rubber stress can only disappear due to relaxation. The level of the stress will gradually reduce to zero with time. This is a very special situation. More common is that a polymer is continuously deformed. In such a case the “change of the rubber stress per unit time” consists of two parts:

1. An increase of the stress due to the deformation
2. A decrease of the stress due to the relaxation: “-rubber stress / rubber relaxation time”

[change of rubber stress per unit of time] = + [change of rubber stress due to deformation per unit time] – [rubber stress] / [rubber relaxation time]

We will show later how such differential equations can be used to calculate the viscosityof the polymer.

With increasing temperatures the reptation time reduces. At the rubber – melt transition temperature the reptation time is 1 second and the rubber relaxation time is 1 second too. Above this temperature the rubber stress will disappear so fast that it is not relevant anymore. The polymer is now in the melt phase.

## Summary:

• Any stress in a polymer reduces in time due to the mobility of the molecules.
• The reduction of stress is called relaxation.
• In the glass phase the glass stress is caused by bending of the Kuhn segments.
• In the glass phase the relaxation of the stress is caused by the rotation of the Kuhn segments.
• In the rubber phase the rubber stress is caused by the fast rotation of the Kuhn segments that tend to bring the shape of the macromolecules back to spherical.
• In the rubber phase the relaxation of the rubber stress is caused by the reptation of the macromolecules.
• Both the change of the glass stress and the change of the rubber stress with time can be described with a differential equation.