Materials for Biomedical Engineering. Mohamed N. Rahaman

to illustrate the formation of hybrid orbitals. The 2s orbital and the three 2p orbitals in the carbon atom, for example, can combine to form four sp³ orbitals, each containing one electron (Figure 2.6). Formation of these hybrid orbitals can lead to an overall reduction in energy and, thus, to a more stable compound when the atom with the hybridized orbitals forms a bond with other atoms. Nitrogen and oxygen atoms can also form sp³ hybrid orbitals but one orbital in nitrogen and two orbitals in oxygen are filled with two electrons. As they are not available for bonding with other atoms, the two electrons in each of the filled orbitals are referred to as lone‐pair electrons.

Figure 2.6 Schematic illustration of the formation of four sp³ orbitals in (a) the carbon atom, (b) the nitrogen atom, and (c) the oxygen atom. Each hybrid orbital in carbon can form a covalent bond by sharing an electron from another atom. In comparison, one orbital in nitrogen and two orbitals in oxygen are filled with two electrons, called lone‐pair electrons, which are not available for covalent bonding but available for hydrogen bonding.

Hybrid orbitals are formed at specific angles to one another and, as bonds are formed in the directions of greatest orbital overlap, these orbitals give directionality to the bond that, in turn, determine the arrangement of the atoms in three dimensions and how the atoms pack to form solids. Hybridization of s and p orbitals, for example, can occur in the following ways (Figure 2.7):

       One s orbital and one p orbital to form two sp orbits that are diametrically opposed to each other. The angle between the orbitals is 180°, giving a linear arrangement of the orbitals

       One s orbital and two p orbitals to form three sp2 orbitals at 120° to one another. The orbitals form a triangular arrangement in a single plane and they are said to form a trigonal planar arrangement

       One s orbital and three p orbitals to form four sp3 orbitals at 109.5° to one another. The orbitals point to the corners of a tetrahedron.

Figure 2.7 Schematic comparison of the directionality of sp, sp², and sp³ hybrid orbitals.

When a carbon atom bonds with four hydrogen atoms to form a methane molecule (CH₄), for example, the shared electrons point to the corners of a tetrahedron (Figure 2.8). On the other hand, when oxygen or nitrogen atoms with sp³ hybridized orbitals bond with other atoms such as hydrogen atoms to form water (H₂O) or ammonia (NH₃) molecules, respectively, the angle between the OH bonds in H₂O or between the NH bonds in NH₃ differs slightly from 109.5° due to the presence of the lone‐pair electron orbitals. The bonds and lone‐pair electron orbitals form a slightly distorted tetrahedral arrangement, with the angle between the OH bonds in H₂O and between the NH bonds in NH₃ having a value of 104.5° and 107°, respectively.

Figure 2.8 Schematic illustration of covalent bonding in methane, ammonia, and water molecules. The polarity of the bonds and lone pair electrons in ammonia and water are also shown.

The hybridized electron orbitals in atoms such as carbon, nitrogen, and oxygen can also lead to the formation of double and triple bonds when these atoms bond to the same or other atoms. The oxygen molecule (O₂), for example, has a double bond between the atoms while the nitrogen molecule (N₂) has a triple bond between the atoms. To visualize the formation of single, double, and triple bonds between atoms, we can compare the structures of the hydrocarbon molecules ethane (C₂H₆), ethylene (C₂H₄), and acetylene (C₂H₂) (Figure 2.9). The formation of the ethane molecule, for example, can be viewed in terms of bonding among two carbon atoms, each having four sp³ hybrid orbitals, and hydrogen atoms. Each bond in the molecule shares two electrons and points directly at each other along the internuclear axis. This type of bond is called a sigma (σ) bond and it is the most common type of covalent bond.

Figure 2.9 Illustration of the formation and geometry of single, double, and triple bonds in (a) ethane, (b) ethylene, and (c) acetylene.

      In the ethylene molecule, each carbon atom has three sp² hybrid orbitals and one unhybridized p orbital. Each sp² orbital forms a σ bond with another carbon atom and with a hydrogen atom. In contrast, the two p orbitals, one from each carbon atom, can share their electrons in a side‐by‐side manner, perpendicular to the internuclear axis, to complete the valence shell of the carbon atom. This type of bond is called a pi (π) bond. The combination of the σ and π bonds results in a double bond between the carbon atoms. In the acetylene molecule, we have one σ bond between the two carbon atoms, due to sp hybrid orbitals in each carbon atom, plus two π bonds from the two unhybridized p orbitals, making a triple bond between the carbon atoms.

      In the case of the hydrocarbons, such as ethane, ethylene, and acetylene, for example, a triple bond between two carbon atoms has a higher bonding energy and a shorter interatomic distance than a double that, in turn, has a higher bonding energy and shorter interatomic distance than a single bond. Another important distinction between the single, double, and triple bonds is the freedom of one carbon atom to rotate about the internuclear axis with the other carbon atom. As the energy barrier for rotation is low, molecules can rotate freely around a single carbon–carbon bond. On the other hand, rotational motion is highly restricted around a double or a triple carbon–carbon bond. The ethylene molecule has a planar shape whereas the acetylene molecule is linear. Similar trends hold for bonds between other pairs of atoms, such as between carbon and nitrogen atoms. Overall, then, the presence of double and triple bonds has important consequences for the structure and, thus, the properties and function of covalent‐bonded molecules.

      Covalent Bonding in Ceramics

      Covalent‐bonded ceramics are characterized by intrinsic properties similar in nature to those described for ionic‐bonded ceramics (Section 2.6) but as the covalent bond can be stronger than the ionic bond, many covalent‐bonded ceramics have better mechanical properties, such as strength, elastic modulus and hardness, and a higher melting point.

      Covalent bonding appears in diamond, a crystalline form of carbon, often regarded as a prototypical example of covalent‐bonded ceramics. The atoms bonded to a given atom point to the corners of tetrahedron due to sp³ hybridization of the electron orbitals of the carbon atoms, giving a directional form of bonding (Chapter 3). Covalent bonding is the dominant type of bonding in many other ceramics, including carbides such as silicon