Interatomic bonding in materials

1.4 Interatomic bonding in materials

Matter can exist in three states and, as atoms change directly from either the gaseous state (desublima- tion) or the liquid state (solidification) to the usually denser solid state, the atoms form aggregates in three-dimensional space. Bonding forces develop as atoms are brought into proximity to each other. Generally, there is an attractive force between the atoms but at close range a repulsive force exists.

2 Atomic mass is now expressed relative to the datum value for carbon (12.01). Thus, a copper atom has 63.55/12.01 or 5.29 times more mass than a carbon atom.

10 Physical Metallurgy and Advanced Materials

⫹ Repulsion

Increasing interatomic distance r

Potential energy

Crystal spacing

Attraction ⫺

Figure 1.2 Variation in potential energy with interatomic distance. The equilibrium spacing is given when these two forces balance. The energy of interaction decreases

as the atoms approach and has its lowest value at the equilibrium spacing, as shown in Figure 1.2. The potential energy U of a pair of atoms can be written as:

A B U =− r m + r n

(1.1) where r is the atom separation with m < n, and the first term is attractive, the second repulsive. At

r<r o , the equilibrium value, the repulsive force dominates and U rises. The force F is given by the rate of change of energy with distance dU/dr and is zero at r =r o .

The nature of the bonding forces has a direct effect upon the type of solid structure which develops and therefore upon the physical properties of the material. Melting point provides a useful indication of the amount of thermal energy needed to sever these interatomic (or interionic) bonds. Thus, some solids melt at relatively low temperatures (m.p. of tin = 232 ◦ C), whereas many ceramics melt at extremely high temperatures (m.p. of alumina exceeds 2000 ◦ C). It is immediately apparent that bond strength has far-reaching implications in all fields of engineering.

Customarily we identify four principal types of bonding in materials, namely metallic bonding, ionic bonding, covalent bonding and the comparatively much weaker van der Waals bonding. However, in many solid materials it is possible for bonding to be mixed, or even intermediate, in character. We will first consider the general chemical features of each type of bonding; in later sections we will examine the resultant disposition of the assembled atoms (ions) in three-dimensional space.

As we have seen, the elements with the most pronounced metallic characteristics are grouped on the left-hand side of the Periodic Table (Table 1.2). In general, they have a few valence electrons, outside the outermost closed shell, which are relatively easy to detach. In a metal, each ‘free’ valency electron is shared among all atoms, rather than associated with an individual atom, and forms part of the so-called ‘electron gas’, which circulates at random among the regular array of positively charged electron cores, or cations (Figure 1.3a). Application of an electric potential gradient will cause the ‘gas’ to drift through the structure with little hindrance, thus explaining the outstanding electrical conductivity of the metallic state. The metallic bond derives from the attraction between the cations and the free electrons and, as would be expected, repulsive components of force develop when cations are brought into close proximity. However, the bonding forces in metallic structures are spatially non-directed and we can readily simulate the packing and space-filling characteristics of the atoms with modeling systems based on equal-sized spheres (polystyrene balls, even soap bubbles). Other properties such as ductility, thermal conductivity and the transmittance of electromagnetic radiation are also directly influenced by the non-directionality and high electron mobility of the metallic bond.

The ionic bond develops when electron(s) are transferred from atoms of active metallic elements to atoms of active non-metallic elements, thereby enabling each of the resultant ions to attain a stable

Atoms and atomic arrangements 11

(a) Sodium (Z ⫽ 11) (b) Magnesium (Z ⫽ 12) and oxygen (Z ⫽ 8)

(c) Carbon (Z ⫽ 6) (d) Polarized atoms Figure 1.3 Schematic representation of: (a) metallic bonding, (b) ionic bonding, (c) covalent

bonding and (d) van der Waals bonding.

closed shell. For example, the ionic structure of magnesia (MgO), a ceramic oxide, forms when each magnesium atom (Z = 12) loses two electrons from its M-shell (n = 3) and these electrons are acquired by an oxygen atom (Z = 8), producing a stable octet configuration in its L-shell (Table 1.3). Overall, the ionic charges balance and the structure is electrically neutral (Figure 1.3b). Anions are usually larger than cations. Ionic bonding is omnidirectional, essentially electrostatic in character and can be extremely strong; for instance, magnesia is a very useful refractory oxide (m.p. = 2930 ◦ C). At low to moderate temperatures, such structures are electrical insulators but, typically, become conductive at high temperatures when thermal agitation of the ions increases their mobility.

Sharing of valence electrons is the key feature of the third type of strong primary bonding. Cova- lent bonds form when valence electrons of opposite spin from adjacent atoms are able to pair within overlapping spatially directed orbitals, thereby enabling each atom to attain a stable electronic configuration (Figure 1.3c). Being oriented in three-dimensional space, these localized bonds are unlike metallic and ionic bonds. Furthermore, the electrons participating in the bonds are tightly bound so that covalent solids, in general, have low electrical conductivity and act as insulators, some- times as semiconductors (e.g. silicon). Carbon in the form of diamond is an interesting prototype for covalent bonding. Its high hardness, low coefficient of thermal expansion and very high melting point (3300 ◦

C) bear witness to the inherent strength of the covalent bond. First, using the (8 − N) Rule, in which N is the Group Number 3 in the Periodic Table, we deduce that carbon (Z = 6) is tetravalent; that

3 According to previous IUPAC notation: see top of Table 1.2.

12 Physical Metallurgy and Advanced Materials is, four bond-forming electrons are available from the L-shell (n = 2). In accordance with Hund’s Rule

(Figure 1.1), one of the two electrons in the 2s-state is promoted to a higher 2p-state to give a maximum spin condition, producing an overall configuration of 1s 2 2s 1 2p 3 in the carbon atom. The outermost second shell accordingly has four valency electrons of like spin available for pairing. Thus, each carbon atom can establish electron-sharing orbitals with four neighbors. For a given atom, these four bonds are of equal strength and are set at equal angles (109.5 ◦ ) to each other and therefore exhibit tetrahedral symmetry. (The structural consequences of this important feature will be discussed in Section 1.9.2.)

This process by which s-orbitals and p-orbitals combine to form projecting hybrid sp-orbitals is known as hybridization. It is observed in elements other than carbon. For instance, trivalent boron (Z

= 5) forms three co-planar sp 2 -orbitals. In general, a large degree of overlap of sp-orbitals and/or a high electron density within the overlap ‘cloud’ will lead to an increase in the strength of the covalent

bond. As indicated earlier, it is possible for a material to possess more than one type of bonding. For example, in calcium silicate (Ca 2 SiO 4 ), calcium cations Ca 2 + are ionically bonded to tetrahedral SiO 4 4 − clusters in which each silicon atom is covalently bonded to four oxygen neighbors. The final type of bonding is attributed to the van der Waals forces which develop when adjacent atoms, or groups of atoms, act as electric dipoles. Suppose that two atoms which differ greatly in size combine to form a molecule as a result of covalent bonding. The resultant electron ‘cloud’ for the whole molecule can be pictured as pear-shaped and will have an asymmetrical distribution of electron charge. An electric dipole has formed and it follows that weak directed forces of electrostatic attraction can exist in an aggregate of such molecules (Figure 1.3d). There are no ‘free’ electrons, hence electrical conduction is not favored. Although secondary bonding by van der Waals forces is weak in comparison to the three forms of primary bonding, it has practical significance. For instance,

in the technologically important mineral talc, which is hydrated magnesium silicate Mg 3 Si 4 O 10 (OH) 2 , the parallel, covalently bonded layers of atoms are attracted to each other by van der Waals forces. These layers can easily be slid past each other, giving the mineral its characteristically slippery feel.

In thermoplastic polymers, van der Waals forces of attraction exist between the extended covalently bonded hydrocarbon chains; a combination of heat and applied shear stress will overcome these forces and cause the molecular chains to glide past each other. To quote a more general case, molecules of water vapor in the atmosphere each have an electric dipole and will accordingly tend to be adsorbed if they strike solid surfaces possessing attractive van der Waals forces (e.g. silica gel).

Worked example

The potential energy U of a pair of atoms in a solid is given by equation (1.1). (i) Outline the physical significance of the two terms and indicate the values of the two constants

m and n. (ii) Taking the value of m = 2 and n = 10, calculate the values of A and B for a stable atomic configuration where r = 3 × 10 −10 m and U = −4 eV. Calculate the force required to break the diatomic configuration.

Solution

(i) −A/r m is an attractive potential related to the type of bonding in the crystal. B/r n is a repulsive potential when the ions get close. The value of m < n, typically n ∼ 12 for ionic solids. (ii) The energy function now reads:

U =−

r 2 + r 10 .

Atoms and atomic arrangements 13 At equilibrium at r =r o ,

force F =

4 = 0.45 eV nm = 7.2 × 10 −20 J nm 2

Maximum force is at d 2 U /dr 2 = 0, at a value for r given by:

6A 110B − r 4 + r 12 = 0.

So r

= (110B/6A) 1/8 =r × (11/3) = 0.352 nm. Thus, maximum force to break bond:

N. r 8 = 0.352 3 − 0.352 8 = 14.9 eV/nm = 2.39 × 10