Having considered the cohesion within atoms and molecules, and between atoms and molecules, we are in a position to consider the factors that determine the magnitudes of the elastic coefficients connecting stresses and strains. We wish to see how these coefficients are related to the electronic structures, that is, to chemical factors.

The elastic response coefficients are the most fundamental of all of the properties of solids; and the most important sub-set of them is the shear coefficients because they determine the existence of the solid state. If the shear stiffnesses were not sufficiently large, all matter would be liquid like. There would be no aeronautical, civil, or mechanical engineering; and modern micro-electronics, as well as opto-electronics would not be possible. They set limits on how strong materials can be, how slowly geological processes occur, and how natural structures respond to wind and rain. This is why the scientific study of them which began with Galileo (the founder of physical mechanics) continues today. However, the shear moduli are considerably more difficult to interpret in detail than the bulk moduli. They depend on both the shear plane, and the shear direction; and the structures of both of these depend on crystal symmetries and local atomic structures. The bulk modulus, on the other hand, is a scalar quantity relating an isotropic pressure to an average change in volume. It is the average of the three inverse linear compressibilities (change of length induced by pressure).

Stress is a generalization of the concept of pressure. The latter consists of a unit of force applied perpendicular to a vunit of area. However, in the case of stress, the force can be applied at any angle relative to the unit area. Thus, whereas pressure is a scalar that does not act in any special direction, two vectors are needed to define a general stress: one that indicates the direction of the force, and the other that gives the orientation of the area on which the force acts. Furthermore, in the case of solids, the orientation of the solid must be defined, except when the solid is isotropic. Stresses applied to surfaces (force per unit area) are called tractions. They can be resolved into two components: one parallel, and the other perpendicular to the unit surface areas on which they act. The orientation of the area is given by a unit vector lying perpendicular to it. For the force component that lies parallel to the unit surface area a shear traction is produced, and for the force component that lies perpendicular to the unit surface, a normal traction is produced.

Tractions tend to distort a solid material, either causing its volume to change, but not its shape, or causing its shape to change at constant volume (small changes), or both. That is, they create dilatational strain in the material, or shear strain, or both.

Textbooks often emphasize the philosophical aspects of Heisenberg's Principle calling it “The Uncertainty Principle”. However, as a physical principle there is nothing uncertain about it. In its “exact” form it is a useful relationship between physical parameters. It is an expression of the fact that particles of matter behave simultaneously in two modes: one that is particle like, and the other that is wave like. These two modes are intimately connected, and it is this connection that is expressed in the Heisenberg Principle, and more generally in quantum mechanics including Schrödinger's equation (Atkins and Friedman, 1997). From this viewpoint, the principle can be used to deduce some of the properties of atoms and molecules in a simple way. This method of analysis is quantitative, but it is limited to cases that involve only a single coordinate that is given in advance (Born, 1989).

The method has been extended somewhat by Simons and Bloch (1973) to cover some cases involving the angular momentum quantum number. It is a rudimentary form of density functional theory. In a somewhat different form than the one used here, it has been applied to molecules by Parr and his collaborators (Borkman and Parr, 1968).

Physical properties can never be measured exactly. Errors, however small, are always present in a measurement procedure. Therefore, measured values of variables are average values derived from a distribution of individual measurements.

The ultimate determinant of strength is fracture. Its upper limit is the cohesive strength of a material, but it is rare that the cohesive strength can fully manifest itself. Almost always, mechanisms intervene that concentrate the macroscopic stress into small regions where the local stress may be 10 000 times, or more, larger than the nominal applied stress. The most effective stress concentrators are cracks. Like levers, they concentrate the work that is done by macroscopic applied forces into small microscopic volumes.

Elements of cracking

Two centuries ago young men could make a living by splitting trees lengthwise into “rails” to be used for fence construction. Cracking could also be used to split large rocks in quarries. Figure 19.1 illustrates a splitting crack. Here a crack has traversed about half the length of a rod of material (perhaps a wooden log). The crack bisects the thickness, 2t, of the rod (whose width is w). The length of the crack is L, and it forms two cantilever beams each of thickness t and width w. Suppose that a wedge applies forces, F, pushing the ends of the cantilevers apart, thereby displacing each of them a distance, h, from the center-line of the rod. These displacements increase by small amounts, dh, if the tip of the crack advances incrementally by an amount dL.

Introduction

Since both crystals and polymers are constructed of repetitive units (atoms or molecules), it is difficult to differentiate them. Roughly, crystals may be taken to be arrays of repetitive units in which repetition occurs in three dimensions. Polymers may be taken to be arrays that are repetitive in one dimension. However, there are two-dimensional as well as one-dimensional crystals. And there are two-dimensional, and in some cases, three-dimensional polymers. Thus, the distinction is not clean-cut.

Another classification approach is to say that crystals tend to be inorganic in character, whereas polymers tend to be organic. In a majority of cases this is true, but by no means all cases.

Covalent bonding within one-dimensional chains of atoms (or molecules) has already been discussed in Chapter 9. Most real polymers are two-dimensional, however, at least to the extent that the bonds form “zig-zag” patterns rather than straight lines. Thus, it is not possible to discuss the extension or contraction of polymer chains in terms of radially directed forces alone. Distortion of planar angles must be considered, and in most cases changes of rotational angles. This latter degree of freedom allows the contracted configuration to be that of a helical coil. Purely planar molecules are rare.

Intramolecular bonds are usually covalent, whereas intermolecular ones are usually due to London forces (in most cases Casimir forces can be neglected), but they can also be due to hydrogen exchange, or if the polymers contain polar groups they can be due to the interactions of pairs of permanent dipoles, or permanent/induced dipole pairs.

Introduction

Since dislocations “multiply” as they move (i.e., they increase their length), and since they are almost always present initially in structural materials, their most important property is their mobility. If this is very small, as it is in covalently bonded crystals like silicon at temperatures below the Debye temperature (about 920 K in Si), the material is brittle, tending to fracture before it deforms plastically. If the mobility is large, as it is in nearly perfect gold, or copper, crystals, then the material is very ductile, or malleable. Such crystals may be hammered violently without fracturing them.

The range of dislocation mobilities is very large as measured by the stress needed to move a dislocation at an observable rate in a laboratory experiment. This range starts at zero in a nearly perfect metal, and ends at about G/4π in a covalent crystal, where G is the appropriate shear modulus (G of the (111) plane is about 5 Mbar for diamond). At the high end of this range, the velocity is proportional to the applied stress so the mobility is liquid like, and the material has no intrinsic barrier to dislocation motion (if finite barriers are observed, i.e., finite yield stresses, they result from extrinsic factors). At the low end of the range, the mobility is intrinsic, determined by the chemical structure of the crystal.

The extrinsic factors that create finite yield stresses are numerous.

Some solids are very hard and brittle, while others are soft and ductile. These differing behaviors are related to differences in resistance to plastic flow. Continuum mechanics cannot account for the differences. A quantum theory is required. In the case of plastic flow there are two levels of quantization that must be considered. The first is that, unlike elasticity, plastic flow is not a homogeneous process. It requires the inhomogeneous creation and propagation of dislocation lines. The displacements at these lines are quantized. The second is that dislocation lines themselves do not move homogeneously, except in simple metals. Kinks form on the lines. The displacements at these kinks localize and quantize the mobility process. Therefore, the wave mechanics of the bonding electrons determines the kink mobilities which in turn determine hardness and softness. Only by combining classical and quantum mechanics can a complete solid mechanics theory be developed.

Strong structures fail either because elastic deflections become too large in them, or because plastic flow occurs leading to large plastic deflections, or worse to fracture. Thus an understanding of the nature of plastic resistance is important for the design of strong structures. The subject is very complex because plastic flow is so very heterogeneous, at virtually every one of the 12 levels of aggregation from atoms to large engineering structures. At each level the heterogeneity manifests itself in different ways, all of which tend to be complicated. Therefore, no attempt will be made here to discuss the subject as a whole.

General comments

Many orders of magnitude lie between the elastic response of a soft rubber (elastomer) and that of rigid diamond, more than six orders, in fact. The question for physical chemistry is: what determines the shear stiffness of a given substance? This has some sub-divisions. The two main ones are: enthalpic and entropic elasticity. This chapter discusses the first of these, and Chapter 14 discusses the latter.

The distinction between shear and dilatation is illustrated in Figure 13.1 which shows an undeformed circle at 13.1(a), the effect of shear on the circle at 13.1(b), and the effect of dilatation at 13.1(c). Note that shear does not change the area enclosed by the circle (for small deformations).

Shear moduli measure resistance to shape changes. For small shears, the volume is constant, so these moduli are conjugate to the bulk moduli which measure volume changes at constant shape.

As mentioned in Chapter 4, it was not clear until about 1887 whether one or two independent constants are needed to describe the shear responses of isotropic elastic materials (and 15 or 21 to describe trigonal crystals) (Timoshenko, 1983). In 1821, Claude Navier had proposed an atomic theory in which only central forces acted between the atoms. This related the shear and bulk moduli to one another, leaving just one independent constant for the isotropic case. Others, e.g., Stokes and Poncelet, thought that the resistance to shear was fundamentally different from the resistance to compression, so two constants were needed.

Introduction

Most fracture is athermal, either because it occurs at low temperatures, or because it occurs too fast for thermal activation to be effective. Thus it must be directly activated by applied stresses. This can occur via quantum tunneling when the chemical bonding resides in localized (covalent) bonds. Then applied stresses can cause the bonding electrons to become delocalized (anti-bonded) through quantum tunneling. That is, the bonds become broken. The process is related to the Zener tunneling process that accounts for dielectric breakdown in semiconductors. Under a driving force, bonding electrons tunnel at constant energy from their bonding states into anti-bonding states. They pass through the forbidden gap in the bonding energy spectrum.

Thermal activation

At elevated temperatures, the process known as stress-rupture occurs. It is a result of thermally activated vacancy motion. It will not be discussed in detail here. In order for atoms to move from one atomic site in a crystal to another, the temperature must be relatively high, usually above the Debye temperature where the vibrations of individual atoms are excited, and vacancies can be thermally generated. The relatively large masses of atoms prevent much activity at lower temperatures.

The mass of an electron is 1/2000 times smaller than the mass of the smallest atomic nucleus, the proton. It is about 1/20 000 times lighter than a carbon atom, and 1/200 000 times lighter than a molybdenum atom.

Introduction

In the last Chapter the intimate connection between fracture and surface energy was discussed for both the ideal and non-ideal cases. The discussion indicated the primary role played by the intrinsic surface energy, S. Now the connection of S to the electronic properties of materials will be emphasized, including the energies of interfaces. Recognizing that all such theories are approximate, only the most simple analytic theories will be presented with brief mention of detailed computer-based but approximate theories. These are sometimes said to be “first principles theories”, but they are no more based on “first principles” than other approximate theories based on quantum mechanics. The Heisenberg Theorem ensures this, by limiting the basis of any theory. Also, as this book has mentioned previously, few experimental measurements are accurate enough to justify the application of a high precision theory.

Of general interest is the pattern of surface energies throughout the Periodic Table of the Elements (Figure 20.1). Most of the data are for liquids and are from Murr (1975) and Allen (1972). An obvious exception is the value for diamond which comes from cleavage experiments. Also note that, in the solid state, anisotropic substances like Se have more than one surface energy; one of them is much larger than the value given in the figure, and the other is about the same as the given value. The gaps (e.g., atomic numbers 7–10) are simply places where there are no measured surface energies.