6.1 Introduction

We can now bring together the factors underlying the moduli of materials. First, look back to Figure 3.6, the bar chart showing the moduli of materials. Recall that most ceramics and metals have moduli in a relatively narrow range: 30–300 GN m^–2. Cement and concrete (45 GN m^–2) are near the bottom of that range. Aluminum (69 GN m^–2) is higher up; and steels (200 GN m^–2) are near the top. Special materials lie outside it—diamond and tungsten lie above; ice and lead lie a little below—but most crystalline materials lie in that fairly narrow range. Polymers are different: most of them have moduli that are smaller, some by several orders of magnitude. Why is this? What determines the general level of the moduli of solids? And is there the possibility of producing stiff polymers?

6.2 Moduli of Crystals

We showed in Chapter 4 that atoms in crystals are held together by bonds that behave like little springs. We defined the stiffness of these bonds as

S0=d2Udr2r=r0

Imagine a solid held together by these little springs, linking the atoms between two planes within the material, as shown in Figure 6.1. For simplicity we put atoms at the corners of cubes of side r₀.

The total force exerted across unit area, if the two planes are pulled apart a distance (r – r₀) is the stress σ with

σ=nS0(r–r0)

S₀ can be calculated from the theoretically derived U(r) curves described in Chapter 4. This is the realm of the solid-state physicist and quantum chemist, but we shall consider one example: the ionic bond, for which U(r) is given in Equation (4.3). We showed in Example 4.8 that

S0=αq24πɛ0r03

for the ionic bond. The coulombic attraction is a long-range interaction (it varies as 1/r; an example of a short-range interaction is one that varies as 1/r¹⁰). Because of this, a given Na⁺ ion not only interacts (attractively) with its shell of six neighboring Cl^– ions, it also interacts (repulsively) with the 12 slightly more distant Na⁺ ions, with the eight Cl^– ions beyond that, and with the six Na⁺ ions that form the shell beyond that. To calculate S₀ properly, we must sum over all these bonds, taking attractions and repulsions properly into account. The result is identical with Equation (6.6), but with α = 0.58.

The Table of Physical Constants on the inside front cover gives values for q and ɛ₀. r₀, the atom spacing, is close to 2.5 × 10^–10 m. Inserting these values gives:

S0=0.58(1.6×10–19)24π×8.85×10–12(2.5×10–10)3=8.54Nm–1

The stiffnesses of other bond types are calculated in a similar way (in general, the cumbersome sum just described is not needed because the interactions are of short range). The resulting hierarchy of bond stiffnesses is as shown in Table 6.1.

A comparison of these predicted values of E with the measured values plotted in the bar chart of Figure 3.6 shows that, for metals and ceramics, the values of E we calculate are about right: the bond-stretching idea explains the stiffness of these solids. We can be happy that we can explain the moduli of these classes of solid. But a paradox remains: there exists a whole range of polymers and rubbers that have moduli that are lower—by up to a factor of 100—than the lowest we have calculated. Why is this? What determines the moduli of these floppy polymers if it is not the springs between the atoms?

6.3 Rubbers and Glass Transition Temperature

This is because rubber, like many polymers, is composed of long spaghetti-like chains of carbon atoms, all tangled together, as we showed in Chapter 5. In the case of rubber, the chains are also lightly cross-linked, as shown in Figure 5.10. There are covalent bonds along the carbon chain, and where there are occasional cross-links. These are very stiff, but they contribute very little to the overall modulus because when you load the structure it is the flabby Van der Waals bonds between the chains that stretch, and it is these that determine the modulus.

Well, that is the case at the low temperature, when the rubber has a “proper” modulus of a few GPa. As the rubber warms up to room temperature, the Van der Waals bonds melt. (In fact, the stiffness of the bond is proportional to its melting point: that is why diamond, which has the highest melting point of any material, also has the highest modulus.) The rubber remains solid because of the cross-links, which form a sort of skeleton: but when you load it, the chains now slide over each other in places where there are no cross-linking bonds. This, of course, gives extra strain, and the modulus goes down (remember, E = σ/ɛ_n).

Many of the most floppy polymers have half-melted in this way at room temperature. The temperature at which this happens is called the glass temperature, T_G. Some polymers, which have no cross-links, melt completely at temperatures above T_G, becoming viscous liquids. Others, containing cross-links, become leathery (e.g., PVC) or rubbery (as polystyrene butadiene does). Some typical values for T_G are: polymethylmethacrylate (PMMA, or perspex), 100°C; polystyrene (PS), 90°C; polyethylene (low-density form), –20°C; natural rubber, –40°C.

To summarize, above T_G, the polymer is leathery, rubbery or molten; below, it is a true solid with a modulus of at least 2 GN m^–2. This behavior is shown in Figure 6.2 which also shows how the stiffness of polymers increases as the covalent cross-link density increases, towards the value for diamond (which is simply a polymer with 100 percent of its bonds cross-linked, Figure 4.7). Stiff polymers, then, are possible; the stiffest now available have moduli comparable with that of aluminum. Oriented polymer fibers can be even stiffer.