Bond moduli

An antecedent of the bond modulus is the chemical hardness of Pearson (1997) which measures the stabilities of molecules. Also, bond moduli are proportional to the physical hardnesses of Yang, Parr, and Uytterhoeven (1987) which they proposed for minerals. 

In covalent compounds with less symmetric structures than the diamond structure factors such as ionicity, in addition to the bond moduli, need to be considered (e.g., in GaP). Surface effects (e.g., friction) also play a role in polar... 

Figure 5.2 Correlation of the hardnesses of the Group IV elements, and the associated isoelectronic III-V compounds, with their bond moduli. Room temperature data. For the elements, the molecular volumes refer to the diatoms C-C, Si-Si, Ge-Ge, and Sn-Sn.

The next two figures show that crystal structure type and ionicity also play a role in determining dislocation mobility, and therefore hardness. First, if data for the III-N compounds are plotted on Figure 5.2 they do not fall on the regression line. The reason is that they have hexagonal rather than cubic crystal structures. However, when plotted by themselves as in Figure 5.3 their hardnesses are proportional to their bond moduli. 

Figure 5.5 Hardnesses of some II-VI compounds (chalcogenides) versus their bond moduli.

A plot of them (Figure 5.6) shows that they are proportional to the bond moduli. Thus the bond moduli are fundamental physical parameters which measure shear stiffness, and vice versa. Also, it may be concluded that hardness (and dislocation mobility) depends on the octahedral shear stiffnesses of this class of crystals (see also Gilman, 1973). 

Figure 5.6 Correlation of octahedral shear stiffnesses with bond moduli for Group IV crystals. The octahedral stiffnesses measure the elastic shear resistances of the covalent bonds across the (111) planes.

Perhaps the most simple crystals in this class are the alkaline earth oxides. They are II-VI compounds and have rocksalt crystal structures. Data for their hardnesses versus their bond moduli (optical band gaps per molecular volumes) are displayed in Figure 11.4. 

Figure 11.4 Hardness of alkaline earth oxides vs. bond moduli.

Three of these compounds have cubic symmetry, while T1B2 has hexagonal symmetry. Since they are metallic, bond moduli cannot be defined for them, but valence electron densities can be. The hardnesses of the cubic titanium compounds depend linearly on their VEDs the numbers of valence electrons are (4 + 4 = 8)TiC, (4 + 3 = 7)TiN, and (4 + 2 = 6)TiO. The linear dependence is shown in Figure 11.10. A similar linear dependence on their C44s is also found (Figure 11.12). 

Here / is the fraction of stiff, covalent bonds (modulus Ej) and 1 - / is the fraction of weak, secondary bonds (modulus E2). The polymer modulus is... 

The bond modulus is a quite simple parameter. It is the ratio of an atomic (or molecular) energy, and an atomic (or molecular) volume. Essentially all of the volumes of interest are known from crystallography, and the energies are... 

The shear work done for one atomic (molecular) displacement, b is the applied force times the displacement, or xb3. This work must equal the promotion energy 2Eg. Therefore, letting b3 equal the molecular volume, Vm, the required shear stress is approximately 2Eg/Vm. The parameter [Eg/Vm] is called the bond modulus. It has the dimensions of stress (energy per unit volume). The numerator is a measure of the resistance of a crystal to kink movement, while the denominator is proportional to the work done by the applied stress when a kink moves one unit distance. Overall, the bond modulus is a measure of the shear strengths of covalent bonds. 

A good indication of the importance of chemical bond strengths in determining hardness is the correlation between the heats of formation of compounds and their hardnesses. An example for III-V compounds is shown in Figure 5.13.The heat of formation density is equivalent to the bond modulus. This provides further evidence of the importance of chemical bond strength in determining hardness. 

Figure 9.8 Proportionality between hardness and bond modulus for alkaline earth fluorite crystals.

Dislocation lines do not move concertedly, that is, all at once. They move, by forming kinks along their lengths, and when the kinks move, the lines move. The open crystal structure of quartz (crystobalite) results in a relatively large amount of volume being associated with a kink on a dislocation line. This relatively large volume lowers the value of quartz s bond modulus, making its hardness consistent with those of other covalently bonded substances. 

The hardness (480kg/mm2) of the quartz version of Ge02 is also consistent with its bond modulus. Unfortunately, Sn02 does not have a quartz like structure so there are only two members of this isoelectronic set (Si02 and Ge02). 

The intrinsic energy band-gap of YAG is about 6.6 eV., and the Burgers displacement is about half the unit cell size, or 6 A. Then, if a kink volume is taken to be 6 x 3 x 3 = 54 A3, the bond modulus is 0.11 eV/A3, or 1800 kg/mm2. Given how little is known about dislocation motion in garnet, this agreement with the room temperature hardness value is largely fortuitous. 

J. J. Gilman, Bond modulus and stability of covalent solids, Phil. Mag. Lett., 87,121... 

Waferboard, a more recent wood constmction product, competes more with plywood than particle board. Waferboard and strand board are bonded with soHd, rather than Hquid, phenoHc resins. Both pulverized and spray-dried, rapid-curing resins have been successfully appHed. Wafers are dried, dusted with powdered resin and wax, and formed on a caul plate. A top caul plate is added and the wafers are bonded in a press at ca 180°C for 5—10 min. Physical properties such as flexural strength, modulus, and internal bond are similar to those of a plywood of equivalent thickness. 

The modulus of elasticity can also influence the adhesion lifetime. Some sealants may harden with age as a result of plasticizer loss or continued cross-linking. As a sealant hardens, the modulus increases and more stress is placed on the substrate—sealant adhesive bond. If modulus forces become too high, the bond may faH adhesively or the substrate may faH cohesively, such as in concrete or asphalt. In either case the result is a faHed joint that wHl leak. 

Most recent studies (69) on elevated temperature performance of carbon fiber-based composites show that the oxidation resistance and elevated temperature mechanical properties of carbon fiber reinforced composites are complex and not always direcdy related to the oxidation resistance of the fiber. To some extent, the matrix acts as a protective barrier limiting the diffusion of oxygen to the encased fibers. It is therefore critical to maintain interfacial bonding between the fiber and the matrix, and limit any microcracking that may serve as a diffusion path for oxygen intmsion. Since interfacial performance typically deteriorates with higher modulus carbon fibers it is important to balance fiber oxidative stabiHty with interfacial performance. 

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