Make two schematic plots of the logarithm of relaxation modulus versus temperature for an amorphous polymer (curve \(C\) in Figure 15.8). (a) On one of these plots demonstrate how the behavior changes with increasing molecular weight. (b) On the other plot, indicate the change in behavior with increasing crosslinking.

When learning about the fascinating world of polymers, one must grapple with the concept of viscoelastic behavior. Viscoelasticity is the property of materials that exhibit both viscous and elastic characteristics when deforming. This means they have the ability to both flow, like honey, and to stretch and recover, like a rubber band. For an amorphous polymer, this behavior is especially interesting because it doesn't have a crystalline structure, giving it unique flexibility.

In terms of practical impacts, viscoelasticity in polymers means they can dissipate energy - think of a car's shock absorbers, which are designed to absorb and dull the impact from bumps in the road. Understanding this behavior is key in material science, as it allows engineers and designers to predict how a material will react under different conditions of stress, strain, and temperature.

Molecular weight and the degree of crosslinking are two factors that significantly influence the viscoelastic behavior of polymers. Exploring how these factors change the relaxation modulus of amorphous polymers can shed light on their potential applications in various industries - from automotive to biomedical.

Molecular weight, which essentially refers to the 'size' of a polymer's molecules, strongly influences a polymer's properties. As the molecular weight of an amorphous polymer increases, so does its relaxation time - think of it as the time the polymer takes to return to its original shape after being deformed. This longer relaxation time reflects a higher resistance to flow, which in turn translates into a higher relaxation modulus.

A higher molecular weight means that individual polymer chains are longer, leading to more entanglements between chains. These entanglements act like tiny knots that hinder the movement of the chains, thereby making the material stiffer and slowing down its relaxation process. As a result, a higher molecular weight polymer would exhibit a steeper slope on a plot of the logarithm of relaxation modulus over temperature, as it takes longer for the effects of temperature to facilitate the chains' movement.

The relationship between molecular weight and viscoelastic properties is paramount when custom-tailoring materials for specific applications. For instance, thicker, higher molecular weight plastics may be sought for products that require greater toughness and durability.

Crosslinking in polymers is the process of chemically joining two or more polymer chains. This can be visualized as bridges connecting the chains, which greatly affects the mechanical properties of the material. Increased crosslinking typically makes a polymer harder, stronger, and more resistant to deformation - in other words, more rigid. This increased rigidity manifests as an increase in relaxation modulus, meaning the material resists relaxing back to its undeformed state.

In the context of the exercise, increasing crosslinking would shift the relaxation modulus curve upwards on a plot against temperature. This upward shift indicates the polymer's heightened rigidity; it is literally 'tightening up' due to the crosslinks that restrict chain movement. The outcome? A material that can withstand higher stresses without deforming, making it ideal for products that must endure rigorous operating environments.

Keep in mind, though, that excessive crosslinking can make a polymer too brittle, precisely because it restricts movement too much. This is why understanding the degree of crosslinking is crucial when designing materials - it's all about finding the optimal balance to achieve the desired mechanical behavior in the finished product.