# 7. Influence of stress on rotation and reptation time

As discussed before polymer molecules have two ways of moving: the chain segments (Kuhn segments) can rotate and the entire molecule can slide along its own axis (reptate). Due to the rotation the stresses of a polymer in the glass phase reduce with time. Due to the reptation the stresses of a polymer in the rubber phase reduce with time. This reduction of glass stress or rubber stress is called relaxation.

We have learned that both the glass stress relaxation and the rubber stress relaxation are strongly temperature dependant. Temperature is in fact the same as motion of the atoms and molecules. Increasing temperatures thus give increasing mobility of atoms and molecules and thus shorter relaxation times.

A very important phenomenon that we have so far not discussed is the fact that both the glass stress relaxation and the rubber stress relaxation are also strongly dependant on the stress present.

## Influence of stress on glass stress relaxation

The relaxation of stresses in the glass phase is caused by rotation of the Kuhn segments. Their rotations will normally be forwards and backwards in a random way in the absence of a stress. When a stress is present however, rotation in one direction may increase the stress and rotation in the other direction may reduce the stress.

Rotations that reduce the stress are favoured: the stress helps the Kuhn segment to rotate. The rotations that reduce the stress will therefor speed up. The time needed for rotations that reduce the stress will strongly decrease. This causes the glass stress relaxation time to decrease strongly too.

For rotations that increase the stress the situation is opposite: the stress will counteract these rotations. The rotations that increase the stress will therefor slow down. The time needed for such rotations will increase strongly. Their contribution to the glass stress relaxation time will diminish. On average, any stress will cause the glass stress relaxation time to reduce strongly.As an example a sketch of its influence is shown below.

The horizontal axis shows the stress and the vertical axis shows the relaxation time. When the stress is zero the relaxation time is maximal at a level (in this example) of about 1000 seconds. Positive tensile stresses or negative compression stresses will strongly reduce the relaxation time. At a stress level of about 20 MPa the relaxation time is already 1000 times less.

## Influence of stress on rubber stress relaxation

In the rubber phase the Kuhn segments can freely rotate thus giving the macromolecules a large flexibility. They are easy to deform. Any deformation imposed to the polymer induces a rubber stress in the material.

In the rubber phase reptation of the macromolecules is possible. Without stress they will move along their own axis randomly forwards and backwards as long as the stress is zero. This changes when stress is present.

The deformation of the polymer also has deformed the individual macromolecules. Their normal spherical shape has been changed in, for example, an ellipsoid. If now the macromolecules reptate a step into a new position then the average deformation of the molecule changes:

1. The change may result in a reduction of the deformation of the macromolecule. Due to this the stress locally in the macromolecule will reduce.
2. The change may result in an increase of the deformation of the macromolecule. Due to this the stress locally in the macromolecule will increase.

Reptation in a direction that reduces the stress is favoured because then the stress “helps” the molecule to move into that position. It is as if it moves with the wind in the back. Therefore, the time for the macromolecule to reptate into this direction will reduce considerably.

In the same way the time to reptate into a position that increases the stress will be counteracted. Now it is like the molecule is moving against the wind. The time needed to move into those positions will strongly increase. These reptations will not contribute anymore to the total relaxation because they stop to happen at higher stresses.

Thus, the average relaxation time of all deformed polymer molecules will strongly decrease while under stress. The stress simply helps the molecules to move into new positions where their deformation becomes zero. The nett result is a rubber relaxation time that is maximal at zero stress and which reduces strongly with either compression or tensile stresses. Just like the relaxation time of the glass stress does.

## Implications

We have seen that the time with which stress disappears in time (relaxation) is strongly dependant on the actual level of the stress. This has certain implications for the behaviour of the polymers when stress is applied. We will shortly discuss three examples:

1. Multiple relaxation times. A polymer bar is suddenly deformed and this deformation is kept constant. Initially the stress is high. Due to the high stress the relaxation time will be short. In time the stress reduces and this causes the relaxation time to increase. This continues until the stress is close to zero after a long time and the relaxation time will be down to the level dictated by the temperature. This means that initially the stress will reduce rather quickly in time but this speed of reduction reduces after longer times. In fact, a spectrum of relaxation times rule this process.
2. Yield stress. A polymer in the glass phase has a certain yield stress. Below the yield stress the deformation of the polymer is relatively small and recoverable. At the yield stress the polymer will suddenly start to deform plastically until it breaks. The reached deformation at break can be up to 500 % or more. This behaviour is due to the influence of the stress on the relaxation time. At low stress the relaxation time for Kuhn segment rotation in the glass phase will be very high: in the order of a couple of years or more. When the stress is increased the relaxation time for rotation will reduce strongly. Eventually a stress will be reached where the relaxation time for Kuhn segment rotation is in the order of a few seconds. At that moment the Kuhn segments will start to rotate quickly and the polymer will deform enormously with the applied stress. This is the yield stress.
3. Viscosity. It is a well-known phenomenon that the viscosity of a polymer reduces with the shear rate. This is caused by the higher shear stress at higher shear rates. The increased stress reduces the rubber relaxation time (reptation time) which will reduce the viscosity.

## Summary:

• The relaxation time of the glass stress strongly reduces with the glass stress present.
• The relaxation time of the rubber stress strongly reduces with the rubber stress present.
• The stress in a polymer bar that is kept at a constant deformation will relax with multiple relaxation times: initially fast when the stress is high, later slow when the stress is low.
• The yield stress of a polymer is the stress at which the time for Kuhn segment rotation (glass relaxation time) has reduced to 1 second.