BONDING IN SOLIDS

BONDING  IN  SOLIDS

1.1           INTRODUCTION

Materials have always been an integral part of human culture and civilization. Their properties depend on their internal structure. The properties influence the performance of a material. Closer collection of atoms result in bulk materials called as solids. Solids can be broadly classified as either crystalline or non-crystalline.

1.2 TYPES OF BONDING IN SOLIDS

     The value of the energy needed to move an atom completely away from its equilibrium position is a measure of the binding energy (also called the cohesive energy) between them. This energy varies depending on the type of bonding.

    Thus bonds are made up of attractive and repulsive forces. The attractive forces are directly associated with the valence electrons. The outer shell contains valence electrons. It can acquire more electrons from or lose all its electrons to another atom to become more stable. It can share its valence electrons with two or more atoms. This is how atomic bonds are formed.

Different Types of Bonding:-

1.      Ionic bonding

2.      Covalent bonding

3.      Metallic bonding

4.      Hydrogen bonding

5.      Van der waals bonding

1.3  IONIC BONING

    An ionic bonding is the attractive force existing between a positive ion and a negative ion when they are brought into close proximity. These ions are formed when the atoms of different elements involved lose or gain electrons in order to stabilize their outer shell electron configurations.
    As an example of ionic bonding, let us consider a molecule of NaCl . when neutral atoms of Na and Cl are brought close together, as shown in Fig.1, the outer valence electron of the sodium atom gets transferred to the chlorine atom to acquire a stable electronic configuration. There exists an electrostatic attraction between positively charged sodium cation and negatively charged chlorine anion.


    Actually, a positive charge attracts all negative  charges in the neighbourhood , and vice versa. Consequently in the crystalline solid, Na⁺  ions are surrounded by Cl^- ions and Cl^- ions by Na⁺ ions. The attraction between the neighbouring unlike charges exceeds the repulsion due to like charges. The resulting structure of sodium chloride is shown in Fig.2.

   Properties of ionic solids

1.      Ionic solids are crystalline in nature.

2.      They are hard and brittle.

3.      They have high melting and boiling points.

4.      Since all the electrons are tightly bound with the ions, ionic solids are good insulators of electricity.

5.      They are soluble in polar solvents and non-polar solvents.

6.      In an ionic crystal, a carbon is surrounded by as many anions as possible and vice-versa.

Examples of ionic solids:

NaCl , KCl , KBr , MgO , MgCl_2­ , KOH and Al₂O₃  are few examples of ionic solids.

1.3.1 BOND ENERGY OF NaCl MOLECULE

      Let us consider Na and Cl  atoms at large separation. The ionization energy of  Na atom is 5.1eV. i.e., this is the energy required to remove the outer most electron from the atom leaving  Na positively ionized.

                Na + 5.1eV                Na⁺ +e

    

     The electron affinity of chlorine is 3.6eV. I.e., when the electron removed from Na is added to chlorine atoms, 3.6eV of energy is released and Cl becomes negatively ionized.
                                 Cl + e                Cl   +  3.6eV

 Thus net energy spent in ionizing both  Na  and Cl  atoms is given by

                                      5.1    3.6 = 1.5 eV

      i.e.     Na + Cl +1.5eV                 Na⁺ +Cl


 The electrostatic attraction between Na⁺ and  Cl    ions brings them closer to the equilibrium position. At this position the spacing is r_o= 0.24nm and the potential energy is minimum. While NaCl molecule is formed energy is released. This energy called bond energy of the molecule can be calculated using the formula


                     V =           e²
                                  4πε_or_o


                        =                − (1.602 x 10^-19)²                               
                                   4π ( 8.85 x 10^-12) (2.4 x 10^-10)

                        =        9.612 x 10¹⁹
                                  1.602 x 10^-19  

                        =  − 6eV

Hence net energy released when  NaCl  molecule is formed from neutral Na and Cl atoms id given by

              (5.1 − 3.6  − 6)eV   i.e.,       −4.5eV

Thus the energy released when NaCl molecule is formed is 4.5eV.  This means that
 to dissociate NaCl  molecule into Na and Cl ions, it requires an energy of 4.5eV.

1.4 COVALENT BONDING

 In covalent bonding the stable arrangement of electrons in an outer shell is achieved by a process of valence electron sharing rather than electron transfer. Such sharing results in a lowering of the potential energy of the system . the simplest case of covalent bonding occurs in the hydrogen molecule in which two hydrogen atoms contribute their 1s¹  electrons to form an electron-pair covalent bond as shown in the Fig.3.  

           A covalent bond may also be formed when two or more atoms of different non-metals share one or more pairs of valence electrons. As shown in Fig.3 the probability of finding the 1s¹ electrons in the atoms is maximum between the nuclei in the hydrogen molecule. In order to comply with Pauli’s exclusive principle, the shared two electrons should be of opposite direction. The interaction between  these two anti-spin electrons give rise to force of attraction between the two atoms.
   
    The directional nature of the covalent bond results from the restricted orbital motion of the electrons. Covalent  bonds are formed  not only due to the overlap of pure  s  orbital or pure  p orbitals  but also due to the overlap of  s and  p orbital's. such bonding is called hydrogen bonding. Carbon exhibits such a bonding. The electronic configuration of  normal  carbon is 1s²  2s² 2p².  The electron  spin distribution of normal carbon atom can be represented as


                    1s      2s      2p_x    2p_y­    2p_z
                          
      In this representation we find that1s and 2s orbital electrons are spin paired while 2p orbital electrons remain unpaired. Since these unpaired electrons are responsible for bond formation, two bonds are expected to form. But when carbon atoms approach each other, one electron of 2s orbital gets excited to 2p orbit resulting in the following spin distribution.           
                              1s     2s      2p_x    2p_y­    2p_z

     Thus this gives rise to four unpaired electron spin. The favourable bonding directions of these orbitals are directed towards four corners of a regular tetrahedron with the bond angles 109.5^o as shown in Fig.4. These four orbitals are called  sp³ hybrids and this arrangement of orbitals  is called  hybridization. In diamond, carbon atoms exhibit sp³ tetrahedral covalent bonding.

    Because of these strong , directional, primary valence forces, the crystal becomes strong with high melting point and low thermal expansion coefficient. Since it is extremely hard, it is used as an abrasive. Since valence electrons are strongly locked in covalent bonds, it is an electrical insulator. The next two elements in Group IV of the Periodic Table are silicon (Si) and germanium (Ge). These are also electrical insulators at 0k. As the temperature increases, the covalent bonds are broken up and valence electrons become free to carry electric current. These solids are semiconductors.

   Based on the strength of the covalent bonds, we can group the covalent crystals into three types.
  
   In one type of crystals, the molecules are small and they are bonded together y weak forces. They are soft and have low melting point.  ( Example: Sulphur and  Iodine).

   In the second type of crystals, the atoms are bound by strong covalent bonds and hence they are very hard with very high melting point.  ( Example: Diamond, Ge and Si).
 The third type materials have layer like structure as in the case of graphite. Within the layer, all
the atoms are bonded strongly with the neighbouring atoms. However there is no strong bonding between different layers. Hence these materials are soft. ( Example: graphite).

PROPERTIES  OF  COVALENT  SOLIDS

1. Covalent bonds are directional. Change in the direction of the bond results in the formation of different substance.
2. Since different covalent solids have very much different bond strengths, they
exhibit varying physical properties. For example, the diamond is the hardest
substance with very high melting point. It is a very good insulator of electricity.
Tin which is a covalent solid is a good conductor of electricity. It is very soft and has a low melting point. We know that silicon and germanium, the covalent solids are semiconductors.
3. Covalent solids are hard and brittle. They possess crystalline structure.
4. When compared with ionic solids, these solids have relatively low melting and boiling points.
5. Pure covalent solids are good insulators of electricity at low temperatures.
6. When covalent crystals are doped with certain impurities, they become semiconductors.

Table 1. Comparision between ionic and covalent solids

       Ionic solids                                                             Covalent solids

1. Ionic solids are formed when the                    1. Covalent solids are formed
  atoms of two different elements                             between the atoms of same                                                              
  transfer the electrons among them                        elements by sharing of electrons
  to become positive and negative ions.                   between them.

2. The bonds are non directional.                        2. The bonds are directional.

3. The bonds are relatively stronger.                   3. The bonds are relatively weaker.

4.possess high melting point and                         4. Comparatively lower
 high boiling point.                                                   melting point and low boiling
                                                                                         point.
5. Soluble in polar solvents.                                    5. Soluble in nonpolar solvents.

6. Not very hard.                                                        6. Very hard.
    Examples: NaCl, KBr, Al₂O_3.                                                         Examples: Diamond, Ge,Si.
1.5 METALIC BONDING
   The valence electrons from all the atoms belonging to the crystal are free to move throughout the crystal. The crystal may be considered as an array of positive metal ions embedded in a “ cloud ” or “ sea ” of free electrons as shown in Fig.6. This type of bonding is called metallic bonding.  A metallic structure is therefore determined largely by the packing of the positive ions alone; the electron cloud is just a sort of negatively charged glue.

                            
   In general, if an atom has fewer valence electrons that are more loosely held, the bonding is more metallic. Sodium, potassium, copper, silver, and gold have high electrical and thermal conductivities because their valence electrons are very mobile. As the number of valence electrons and the lightness with which they are held to the nucleus increases, the covalent nature of the bonding increases. The transition metals such as iron, nickel, tungsten and titanium have a significant fraction of covalent bonding.

    A metallic bond may be viewed as an unsaturated covalent bond. Since there are many vacancies in their outermost electron shells, their bonds do not exhibit directional preferences. Hence atoms can be rearranged in space without loss of strength. Since the electrons can wander from atom to atom because of so many unoccupied states in each atom of metal, the valance electrons in a metal are similar to that of molecules in a gas. This explains the high electrical and thermal conductivity of metals. If the volume of the metal is made to change, the energy of electrons is markedly affected. Hence metals have good cleavage strength. But by mere movement of  atoms without any change in volume, the energy of electrons is not affected. Hence metals are malleable and ductile.
 Properties of metallic solids
 1. Metallic bonds hold the atoms together in metals.
 2. Metallic bonds are relatively weak.
 3. Metallic solids are malleable and ductile.
 4. Metallic bond is non directional.
 5. They have high number of free electrons.
 6. They possess high electrical and thermal conductivity.
 7. Metals are opaque to light.
 8. Metallic solids are not soluble in polar and non-polar solvents.
 9. Metallic bonds are weaker than ionic polar and covalent bonds but stronger than van der waals bonds.
Examples of metallic solids:
                 Sodium, Copper, Gold, Silver, Aluminum.
1.6 Hydrogen Bonding
Covalently bonded atoms often produce an electric dipole configuration as shown in Fig.7. With hydrogen atom as the positive and of the dipole if bonds arise as a result of electrostatic attraction between atoms, it is known as hydrogen bonding. Let us consider the example of the water molecule H₂O.  Because of the greater electronegativity of the oxygen the electrons tend to stay closer to oxygen atom than the hydrogen atoms (Fig.8(a)). Hence the oxygen atom acts as the negative end of the dipole while hydrogen atoms act as the positive end. The positive end attracts the negative end of another water molecule and thus bonding the molecules together (Fig.8(b)). In the hydrogen atom the proton is not shielded by other surrounding electrons and hence it is attracted strongly by the negative end of another dipole. Thus the bonding is relatively strong as compared to other dipole-dipole interactions. This bond is also directional in nature. Since the most electronegative atoms like fluorine, oxygen and nitrogen procedure strong dipoles, hydrogen bonding is formed between them. The hydrogen bonding is important in many biological molecules such as DNA.
Properties of hydrogen bonded solids
 1. The hydrogen bonds are directional.
 2. The bonding is relatively strong as compared to other dipole-dipole interactions.
 3. Hydrogen bonded solids have low melting points.
 4. Since no valance electrons are available in such solids they are good insulators
     of electricity.
 5. They are soluble in both polar and non-polar solvents.
 6. They are transparent to light.
 7. Since elements of low atomic numbers form such solids, they have low
     densities.
8. When water is in the form of ice, hydrogen bond results in lower density,  but
     when it melts, it becomes more closely packed liquid and hence its
     density increases.
                     Examples of hydrogen bonded solids: Water molecule in the form of  ice, ammonia molecules.
1.7  VAN DER WAALS (MOLECULAR) BONDING
   Weak and temporary (fluctuating) dipole bonds between hydrogen are known as van der waals bonding and they are nondirectional.. If the symmetrical distribution of electrons around the nucleus (Fig.9(a)) is distributed, the centers of positive and negative charges may not coincide at that moment (Fig.9(b)) giving rise to weak fluctuating dipole.


Fig. 9 Distribution of electrons in a noble gas atom (a) symmetrical distribution (ideal)
      (b) unsymmetrical distribution (real)
      A weak attractive force exists between the opposite ends of the dipoles in the neighbouring atoms. This force only allows inert gas atoms to condense (liquefy and crystallize) at low temperatures. Van der waals bonding is typically an order of magnitude weaker than the hydrogen bonding.

    Solids with van der waals bonding are electrical insulators since the electrons are localized. They cannot respond to electrical conductivity when external electrical field is applied. These solids are characterized by low melting points, high thermal expansion coefficients. Due to the presence of weak forces they are soft and mechanically weak.
Properties of solids with van der Waals bonding
1. Van der waals bond are nondirectional.
2. Van der waals bonding is weaker than the hydrogen bonding.
3. Van der waals bonded solids have low melting point.
4. Since no valance electrons are available, such solids are good insulators of electricity.
5. They are soluble in both polar and non polar liquids.
6. They are usually transparent to light.
               Examples of van der Waals bonded solids:
                                           Solid nein, Solid argon.

1.8 VARIATION OF INTERATOMIC FORCE WITH
      INTERATOMIC SPACING
  
     The attractive forces between the atom bring them close together until a strong repulsive force arises due to overlap of electron shells. When two atoms approach each other, the negatively charged electron shells come much closer than their positive nuclei. At a certain separation, called the equilibrium separation, r_o, the attractive and repulsive forces are equal. The two atoms come to a stable condition and have minimum potential energy. At the equilibrium position, the bonding force, F, between the two atoms may be represented by the general equation
                     
                           (n>m)                                                   (1)   

where r is the interatomic distance (i.e. center-to-center spacing between the atoms) and A, B, m and n are constants that depend on the type of bond. The first term represents the attractive force and the second the repulsive force. At larger separation, the attractive force predominates. The two atoms approach further, the repulsive force predominates, tending to push them back to their equilibrium spacing.

Fig. 10 Variation of interatomic force, F with interatomic spacing, r
   Since the attractive forces in interatomic bonds are largely electrostatic, m is usually 2 following coulomb’s inverse square law of electrostatics. The value of n is not so easy to approximate, but it usually takes the values from 7 to 10 for metallic bonds and 10 to 12 for ionic and covalent bonds. Fig. 10 shows the variation of the net force between atoms (according to Eq. (1) with m = 2 and n = 7). At the equilibrium spacing r_o, the net force is zero.  r_o is of the order of 10^-10 m i.e. 1A. For different bonds r_o varies between 1 and 4A. To separate the atoms completely from the structure, a force F_m, the maximum ordinate of the curve is to be applied. This force corresponds to the cohesive strength of the material.
Equilibrium spacing, r_o
  The general expression for the bonding force between two atoms is

        F(r) =   A           B        
                     r^m           rⁿ       
At equilibrium spacing _o,  F = 0
                                                            Hence          A             B                               
                                                                                          r^m             rⁿ      
           
                                                 i.e., ( r_o­ )^m-n =

                                     
                                                     or          r_o=                                   (2)
1.9 Estimation of Cohesive Energy
         To calculate the cohesive energy let us consider the general situation of two identical atoms. As the atoms approach, the attractive forces increase. Since the atoms do the work during attraction, the energy of attraction is negative and hence the potential energy decreases (i.e. negative value increases). When the separation decreases to the order of few atomic diameters, repulsive forces begin to act. Since external work must be done to bring two such atoms close together, the repulsive force is positive and hence the potential energy increases. At the equilibrium position, the potential energy of either atom is given by
          U = Decrease in potential energy due to attraction + Increase in potential energy due
                to repulsion.

      Since the work done on the system is stored as potential energy, it can be calculated by integrating Eq. (1) as follows:

     Work done in moving through a small distance

                            du( r ) = F( r ) dr

Hence the potential energy of the atom

                  U(r) =   ∫ du(r)     =  ∫ F(r) dr
                                                          
                                                    =∫ [ A/r^M  ─  B/r^N]dr
       
                                                    =         +        +  C

                                                     = 
where a and b are new constants related to A and B as a =  , b =   , m = M−1 , and n = N−1.
      when r = ∞, U = 0
                                                 Hence C = 0
                                              Therefore     U(r) =                       (3)

         All the stable arrangements of atoms in solids are such that the potential energy U(r) is a minimum. This happens at equilibrium position ( r = r_o ).

                    Thus               = 0

Fig .11 variation of potential energy U  with interatomic  spacing  r

From Eq. (3),
                                                     =                        (4)

 i.e.     

 i.e.   

   or                                    (5)           

Eq. (3) can be written as     
  This energy corresponding to the equilibrium position (r  = r_o) is called the bonding energy or the energy of cohesion of the molecule. Since this is the energy required to dissociate the atoms, this is also called the energy of dissociation. This can be calculated as follows:

     Rearranging Eq. (5) we get 
                                                                  
Substituting this in the above equation
         
                                                     

                                                                                                      (6)
  Thus the minimum value of energy  is negative. The positive quantity  is the dissociation energy of the molecule; i.e. The energy required to separate the two atoms. Since m ≠ n the attractive and repulsive energies are not equal though the attractive and repulsive forces are equal in equilibrium. Infact, the total binding energy is essentially determined by the energy of attraction. This can be shown by employing the condition that
     
       _{r = r₀}         > 0                                     (7)

since U has a minimum at r = r_o.
 
  Differentiating Eq. (4) and applying condition given by Eq. (7)
         
 _{r = r₀}      =
                        −am(m+1) 
                            
                               am(m+1) 

                                or        am(m+1)      
Substituting for r₀ from Eq. (5), we get

                                               am(m+1)       

                                               (m+1)
                                               
                                              i.e.                              (8)

Hence from Eq. (1) we understand that the force acting between the atoms are mostly electrostatic in nature.

1.10  CALCULATION OF COHESIVE ENERGY OF IONIC SOLIDS
      The cohesive energy (U) of a crystal containing opposed charged ions with charges Z₁ and Z₂ can be written as (following Born)

                                                                (9)
   The first term is due to attraction and the other due to repulsion.  A, the Madelung constant depend on the geometrical arrangement of ions in the crystal (i.e. crystal structure); B is the repulsion constant, ρ is the repulsion exponent and r is the distance between the oppositely charged ions. The constants B and ρ are respectively a measure of the strength and the range of repulsive interaction.

1.10.1 APPLICATION TO SODIUM CHLORIDE CRYSTAL

           For a uni-valent crystal like NaCl

                                                             Z₁ = Z₂ = 1

Eq. (9) becomes
                                                                           (10)

The total energy per kmol of the crystal is

                                                               (11)
where N is the Avogadro’s number(=0.623 x 10²⁶ k mol^-1 ).
     To evaluate the repulsion constant B, let us consider the fact that at equilibrium separation , the potential energy U is minimum.

                i.e.        _{r = r₀}    

                        or                                   (12)
Substituting this Eq. (11), the potential energy at equilibrium separation can be written as
 
                                                                         (13)
   The term  is the Madelung or electrostatic energy.  This equation gives the
cohesive energy (also called lattice energy) of an ionic solid like NaCl. This is the energy released during the formation of NaCl crystal or the energy spent to separate the solid ionic crystal into its constituent ions. If the Madelung constant  A and repulsion exponent ρ are known, the cohesive energy of ionic solids can be obtained experimentally from compressibility measurement of the crystal. For NaCl structure it can be shown that

                                             
where  is the bulk modulus (which is the reciprocal of the compressibility k).

     The final equation of the equilibrium energy or the lattice energy per kmol of ionic crystal can be written as

                                                                      (14a)
where n the repulsive exponent varies from 9 to 15.

   Hence the cohesive energy of a molecule which is equal and opposite to the lattice energy is given by
                                                   (14b)

   For NaCl molecule,  and . the cohesive energy of sodium chloride molecule is about 7.95 eV.
1.10.2  EVALUATION OF MADELUNG CONSTANT FOR NaCl


          Fig. 12 equilibrium positions of ions in NaCl crystal
    The  Madelung constant, A is a function of crystal structure. It can be calculated from the geometrical arrangement of ions in the crystal. Let us consider the equilibrium positions of ions in NaCl structure as shown in Fig.12. let us choose the central   ion as the reference ion having a single positive charge on it. Six  ions are surrounding this  ion as first nearest neighbours. Let us consider them at unit distance. Twelve  ions are the second nearest neighbours at a distance . Eight  ions are the third nearest neighbours at a distance  and so on. The Madelung constant for the NaCl structure can be written as a summation series
                     

This converges to a value 1.74756        

  Madelung constants for some typical ionic crystals are given below:
            
             Sodium Chloride              1.74756
             Cesium Chloride               1.76267
             Fluorite                                2.51939
             Zinc Blende                         1.63805
Table 2 types of solids and their structure dependent properties

Type	Structure	Property	Examples	Approximate cohesive energy in kj/mol
Ionic	Positive and negative ions	Brittle,non conducting,high melting point.	NaCl LiF	184 244
Covalent	Bonded to one another	Hard, non conducting (white pure),high melting point.	Diamond SiC	170 244
Metallic	Positive ion in a cloud of electron gas	High conductive	Na Fe	26 94
Hydrogen	Molecular held together by hydrogen bonding	Insulators low melting	H2O (ice) HF	12 7
Van der  Waals	Atoms and molecules	Soft insulating, low melting, volatile	Argon	2

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