Polymers as Viscoelastic Materials

Viscoelasticity combines a little bit of both solid behavior and fluid behavior. While ideal elastic materials store all the energy from stress in the bonds, so that the material will immediately restore itself upon release of stress, ideal viscous fluids dissipate all the stress in flow. Viscoelastic materials are somewhere in the middle, and many polymers are viscoelastic. An important characteristic of viscoelastic materials is that timescale matters. For ideal elastic solids and ideal viscous liquids, time doesn’t matter – whether you apply a stress fast or slow, the response of the material should be identical. But this isn’t so for most polymers, which show time-dependent behavior. Two important time-dependent properties of polymers are creep and stress relaxation.

Creep is a property of viscoelastic materials in which the strain of the material changes over time while under a constant load (stress) (Figure 12.12). Creep is not something that happens in a perfectly elastic solid. For an ideal elastic material, under constant stress, it would instantaneously deform and would also instantaneously go back to original shape after removal of stress.

Figure 12.12: Creep of a viscoelastic material while under a constant stress

Source: Lauren Zarzar

A strain vs. time plot for a viscoelastic material with creep would likely look something like this:

Figure 12.13: creep and recovery in a viscoelastic material

Source: Lauren Zarzar

The creep can be non-linear, as shown, where the strain changes in a non-linear fashion with time. Upon release of the stress, the polymer can recover some, but usually undergoes some amount of permanent deformation.

In comparison to creep, which is a constant stress experiment, stress relaxation is a constant strain experiment. You deform the material to a given strain, and measure the stress required to maintain that deformation over time (Figure 12.14).

Figure 12.14: Stress relaxation under constant strain

Source: Lauren Zarzar

The data for stress relaxation experiments are usually reported as a modulus vs time plot. This time dependent modulus, called the relaxation modulus, $E\left( t \right)$, is simply the time dependent stress $\sigma \left( t \right)$, divided by the (constant) strain, ${\epsilon }_0$:

$$E\left( t \right)=\frac{\sigma \left( t \right)}{{\epsilon }_0}$$

Importantly, stress relaxation is also temperature dependent. The graph shows how PMMA relaxation modulus varies as function of both time and temperature. As expected, the relaxation modulus goes down with time – after all, if at constant strain the stress is decreasing, then the modulus must be decreasing too. We also see that at higher temperatures, the modulus is also lower. This makes sense, as we are heating the polymer, we supply more energy for those bonds to rotate, for the polymers to move past each other, to overcome intermolecular interactions – thus, the modulus will decrease.

Figure 12.15: PMMA relaxation modulus

Source: Lauren Zarzar (Redrawn from the data of J.R. McLoughlin and A.V. Tobolsky. J. Colloid Sci., 7, 555 (1952))

An important characteristic of viscoelastic materials is the time-dependent mechanical properties. At short time scales, a polymer may behave as a glassy solid, but at longer time scales, it could flow more like a liquid. This time dependence is influenced by things like molecular weight which affects entanglements, as we touched on in the “polymers as liquids” section (Figure 12.16 below). If you have a low molecular weight polymer it will form less entanglements and so “untangles” more quickly. Larger polymers take longer to relax and untangle, and we see this in the “rubbery plateau” region of the plot below. A nice example of the impact of these time-dependent properties is silly putty. Silly putty behaves like an elastic solid on short time scales – you can bounce it like a rubber ball. But at long time scales, for instance if you just leave it out on the table, it flows like a liquid and will spread.

Figure 12.16: impact of polymer molecular weight on the relaxation modulus

Source: Lauren Zarzar

The plots above are a function of time, but we also know that stress relaxation should also be affected by temperature.

PROBLEM

In the figure below, we consider the effect of temperature on the modulus for polymers at a given time and strain. We already know that going from below to above the glass transition temp will have a dramatic effect on the modulus – below $T_g$ the polymer is rigid glassy solid, but above $T_g$ it begins to flow and the modulus decreases significantly. Both plots shown are for high molar mass polymer, and hence both have a rubbery plateau, but one sample is crosslinked and one is linear. Which one is which?

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ANSWER

Plot (a) is linear and plot (b) is crosslinked. The crosslinked polymer reaches a limit of lower modulus because it is being held together by crosslinks that prevent the network from deforming indefinitely.