Polymers as Solids

First, let’s consider polymers as solids. When we treat a polymer like a solid, we consider things like the strength, stiffness, and toughness of a material. Hooke’s law is relevant here (stress = Young’s modulus x strain):

$$\sigma =E\epsilon$$

Stress, $\sigma$, the load on an object divided by its cross-sectional area, is a measure of the force at any point inside a material (much like how we describe the pressure exerted by a gas on the walls of its container). Stress therefore equals force divided by area:

$$stress=\sigma =\frac{force}{area}$$

Strain is defined as the normalized extension, the change in length ($\Delta l$) divided by the original length of the object ($l_0$).

$$strain=\epsilon =\frac{\Delta l}{l_0}$$

Stress is proportional to the stain and independent of loading rate. If we plot stress versus strain, then the modulus of the material is the slope of the curve, toughness is the area under the curve, and strength is the stress at breaking. This relationship for an ideal elastic solid is shown below.

Figure 12.2: Modulus of an ideal elastic solid

Source: Lauren Zarzar

Think about how these properties could tie back to the molecular scale. We basically have a series of atoms linked up together by a network of bonds, which are a little like tiny springs (and thus we can more easily visualize how Hooke’s law is relevant here). Stiffness or modulus of a solid material is related to the stiffness of the chemical bonds within the system (how much does the spring stretch?). Strength of an ideal solid material is related to the cohesive strength of the bonds, which is proportional to the depth of the potential energy well of the bonds.

For an ideal elastic solid, if we apply and subsequently remove a force on the material, it should go back to its original shape without any permanent deformation. There is no perfectly elastic material. When it comes to polymers, probably the closest we can come are single crystals of polymers, which are rare.  “Real” solid materials deviate from such ideal behavior.

Figure 12.3: Mechanical propertiesof polymers

Source: Figure 19.4 and Table 19.1 from Young, Robert J., and Peter A. Lovell.
Introduction to Polymers, Third Edition, CRC Press, 2011.

Above are some examples of how varied the mechanical properties of polymers can be. The breaking points for materials on this plot is shown with the “x”. Polymer fibers which are crystalline can be high modulus, and so can glassy polymers below $T_g$. But semi-crystalline polymers and elastomers (above $T_g$) are “softer” and certainly have lower modulus. Thus, the mechanical properties are highly tunable and span orders of magnitude in scale. Comparing the modulus of some common polymers to other materials (Table 12.1 below) we see that polymers, in comparison, are not very stiff! Even a glassy polymer like atactic polystyrene has a modulus of only about 3,000 MPa  (about 1/20^th that of window glass).

Why are the mechanical properties of polymers so variable, and why so different than other solid materials? Polymers are unique in that they are made of giant molecules – these molecules interact very differently than small molecules. Many factors affect polymer mechanical properties. Some of these we already discussed in detail. $T_g$ for example, by definition, is a change in the mechanical properties. Above the glass transition, polymers can flow and deform with lower modulus, but below the glass transition, polymers are glassy solids and are more brittle. Cross-linking most definitely affects mechanical properties; thermosets are characterized by very high levels of crosslinking, and they tend to be more rigid and higher modulus than a low-crosslinking elastomer. Crystallinity is a big factor as well. Consider the difference in modulus between and high and low density PE in the table above. Recall that HDPE is linear, and the chains pack together much more easily. Whereas for LDPE, there is branching which prevents close packing, and prevents crystallization, of the PE. Because HDPE has more crystallinity, we find that its modulus is much higher.  

We have already stated that single-crystal polymers are very rare; semi-crystalline polymers, which have some amorphous and some crystalline regions, are much more common. If the polymer is semi-crystalline, then it has some regions that are crystalline and some that are amorphous and the mechanical properties of the bulk tend to be a combination or “average” of the properties associated with each. So for example, we expect crystalline polymers to have higher modulus and amorphous polymers to have lower modulus, thus as we increase the degree of crystallinity, we find that the modulus also increases:

Figure 12.4: Variations of Young's modulus (E) with the degree of crystallinity for different polymers

Source: Figure 19.9 from Young, Robert J., and Peter A. Lovell.
Introduction to Polymers, Third Edition, CRC Press, 2011.

Even within polymers that are the same “chemically”, i.e., polyisoprene, we can still have geometric isomers and/or different skeletal structure that influence the mechanical properties. For example, “cis” isomers tend to not pack together well, and therefore do not crystallize to a significant degree. Thus, polyisoprene that is predominantly in the “cis” conformation is an elastomeric material. But the “trans” form does pack better, induces more crystallization, and causes the material to be more rigid. You could thus imagine tuning the overall mechanical properties of this polymer by just varying the ratio of cis and trans bonds in the polymer.  

Figure 12.5: cis and trans bonds of 1,4-polyisoprene

Source: Lauren Zarzar

Given the unique mechanical properties of elastomers, it’s worth considering the thermodynamics associated with mechanical deformation. Try this at home! Hold a rubber band to your lips and stretch it, then release it. Do you feel a change in temperature? When you stretch the rubber band, you should feel heat (exothermic, $\Delta H$ is negative). When you release the rubber band, you feel cool (endothermic, $\Delta H$ positive). What’s going on, and can we explain it in terms of entropy and enthalpy?

Strain induced crystallization means that polymer chains become more aligned when stretched, facilitating crystallization (which contributes to the exothermic nature of the rubber band stretching!) This isn’t necessarily a good property….. We know that the degree of crystallinity of a polymer can have dramatic impact on its mechanical properties. So if the degree of crystallinity changes, then the properties, change, perhaps in undesirable ways. It also means the polymers behave differently upon application and release of stress (hysteresis) (see Figure 12.6 below).

Figure 12.6: Illustration of mechanical hysteresis for a strain-crystallizing elastomer

Source: Figure 21.9 from Young, Robert J., and Peter A. Lovell.
Introduction to Polymers, Third Edition, CRC Press, 2011.
(Data taken from Andrews, E.H., Fracture in Polymers, Oliver and Boyd Ltd. London, 1968)