For each of the following pairs of polymers, plot and label schematic stress- strain curves on the same graph [i.e., make separate plots for parts (a) to \((\mathrm{c})]\) (a) Polyisoprene having a number-average molecular weight of \(100,000 \mathrm{~g} / \mathrm{mol}\) and \(10 \%\) of available sites crosslinked; polyisoprene having a numberaverage molecular weight of \(100,000 \mathrm{~g} / \mathrm{mol}\) and \(20 \%\) of available sites crosslinked (b) Syndiotactic polypropylene having a weightaverage molecular weight of \(100,000 \mathrm{~g} / \mathrm{mol}\); atactic polypropylene having a weight- average molecular weight of \(75,000 \mathrm{~g} / \mathrm{mol}\) (c) Branched polyethylene having a numberaverage molecular weight of \(90,000 \mathrm{~g} / \mathrm{mol}\); heavily crosslinked polyethylene having a number- average molecular weight of \(90,000 \mathrm{~g} / \mathrm{mol}\)

Understanding the effect of crosslinking on polymers such as polyisoprene is key to predicting its mechanical properties. Crosslinking, which can be thought of as chemical bridges between polymer chains, significantly enhances the material's strength and elasticity. A 10% crosslinked polyisoprene, for example, would display a stress-strain curve that moderately rises, indicating elastic behavior before yielding. By contrast, a 20% crosslinked polyisoprene would show a steeper initial slope in the stress-strain curve, indicative of a higher modulus of elasticity due to the increased number of crosslinks.

Those links make the material more resilient to deformation, but also more brittle upon reaching its failure point. In applications where high elasticity and durability are required, such as in rubber bands or tires, understanding the right degree of crosslinking is crucial to achieving the desired performance.

The molecular weight of polymers, which can be measured in terms such as number-average or weight-average molecular weight, plays a significant role in determining their physical properties. Higher molecular-weight polymers generally have better mechanical properties, such as tensile strength and impact resistance, because the longer chains entangle more effectively, providing better load transfer through the material.

Most high-performance materials are made of polymers with high molecular weights, optimizing not just for strength but also for characteristics like viscosity in melts and solutions, which are crucial in the processing of polymer products. An educational example can be seen in the comparison of stress-strain curves, where a polymer with a higher molecular weight would exhibit a more pronounced elastic region and higher tensile strength than a similar polymer with lower molecular weight.

Polypropylene, a versatile polymer, can have different arrangements of its methyl groups, leading to unique material properties. Syndiotactic polypropylene, where the methyl groups alternate sides in a regular pattern, has a more orderly structure that allows for better packing and crystallinity. This would be reflected as a steeper stress-strain curve indicating a higher stiffness compared to atactic polypropylene. In contrast, atactic polypropylene has a random placement of methyl groups, leading to a disordered, amorphous structure with reduced strength and stiffness, apparent from its gentler slope on the stress-strain curve.

Thus, syndiotactic polypropylene is more suited for applications requiring rigidity and durability, whereas atactic polypropylene may be chosen for applications where flexibility and flow are needed, such as in adhesives and sealants.

Branched polyethylene and crosslinked polyethylene each exhibit unique behaviors on a stress-strain curve due to their structural differences. Branching in polyethylene results from the formation of side chains off the main backbone during synthesis and can decrease the density and crystallinity of the polymer. Consequently, this would produce a more ductile and flexible material, as reflected by a more gradual slope on the stress-strain curve.

In contrast, crosslinking creates strong connections between polymer chains, resulting in a material that is tougher and less soluble. A heavily crosslinked polyethylene would demonstrate a much steeper initial slope, indicative of a high elastic modulus, until it reaches its yield point, after which it would show a marked increase in stress with little strain, reflecting a much lower ductility compared to its branched counterpart. These properties make crosslinked polyethylene ideal for demanding environments, like in pipes and cables, where strength and chemical resistance are paramount.