Mme+521+(1)

/ National Institute of Technology, Durgapur Department of Metallurgical and Materials Engineering MME 521: Materials Science Version vO.O Mukesh Bhardwaj August 4, 2009

1

Introduction

shape and size. Virtually all important properties of solid materials may be grouped into six different categories: mechanical, thermal, electrical, magnetic, optic and deteriorative. For each there is a characteristic type of stimulus capable of provoking different responses. Categories, stimulus and their response and corresponding properties are shown in Table 1. Materials can be further classified as metals, ceramics and polymers. This scheme is based on chemical makeup and atomic structure. In addition, there are composites, which are combination of two or more of the above three basic material classes. Another classification is advanced materials-those used in high technology applications-viz. semiconductors, biomaterials, smart materials and nanoengineered materials. Metals are composed of one or more metallic elements (such as Fe, AI, Cu, Ti, Au and Ni) and often also nonmetallic elements (like, C, Nand 0) in relatively small amounts. Metals are characterized by high thermal and electrical conductivity, relatively high stiffness and ductility. Metals are resistant to fracture, opaque to light and have lusturous appearance over polished surface. These properties are attributed to nonlocalised valence electrons. In addition, some of the metals (Fe, Co, Ni) have desirable magnetic p,roperties. Pure metals are not good enough for many applications, especially structural applications. Thus metals are used in alloy form i.e. a metal mixed with another metal to improve the desired qualities. For example, aluminum, steel, brass and gold. Ceramics are compounds between metallic and nonmetallic elements and are most frequently oxides, nitrides and carbides. For example, Al203, Si02, SiC, Si3N4, porcelain, cement and glass. Ceramic materials are relatively stiff, strong and hard. Unlike metals, they are extremely brittle and susceptible to fracture. They are poor conductor of heat and electricity and are more resistant to high temperatures and harsh environments than metals and polymers. Ceramics may be transparent, translucent or opaque. Some oxide ceramics (Fe304) exhibit magnetic behavior.

Materials play key role in our life. The most important issue of security has been decided by the availability of materials. Indian Wootz steel swords is one good example. Availability of gun powder and forged steel canons further provided the edge in warfare. In modern age, it is decided by the availability of nuclear material, advanced materials for missiles and satellite and last but not least semiconductor materials, which has totally transformed our present existence. Materials are so important that early civilizations have been designated by the level of their materials development like stone age, bronze age, iron age and now it may be called as silicon age. The earliest humans had access only to a limited naturally occuring materials like stone, wood, clay, skin etc. With time they discovered techniques for producing superior materials like pottery and various metals. Further, it was realized that the properties can be altered through alloying, heat treatment and mechanical processing. In any engineering application, a set of performances are expected. Performance depends on properties of materials. Properties are governed by the structure. The structure depends on processing. Structure can be described at various dimensions. Subatomic structure deals with electrons within the individual atoms and interactions with their nuclei. On an atomic level, structure encompasses the organization of atoms or molecules relative to one another. The order of the magnitude of dimensions is Angstrom (10-10 m). The next larger structural realm, which contains large group of atoms is termed microscopic and can be observed using a microscope. The structural elements that may be viewed with the naked eye are termed macroscopic. Property is defined as the response of material when exposed to external stimuli. For example, a specimen subjected to forces will experience deformation, or a polished metal surface will reffect light. A property is a material trait in terms of the kind and magnitude of response to a specific imposed stimulus. Generally, definitions of properties are made independent of material

1

"

Category Mechanical Thermal Electrical Magnetic Optical Deteriorative

Table 1: Categories, stimulus and their response and corresponding properties. Stimulus Response Properties load deformation elastic modulus, ultimate tensile strength heat temperature thermal conductivity, heat capacity Electric field current electrical conductivity, dielectric constant magnetic field domain orientation coercivity, magnetic saturation light angle index of refraction, reflectivity environment loss of material or property reactivity

Polymers include plastic and rubber materials. Many of them are organic compounds that are chemically based on carbon, hydrogen and other nonmetallic elements (0, N, Si). They have very large molecular structures, often chain-like in nature and have a backbone of carbon atoms. For example, polyethylene, nylon, polyvinyl chloride, polycarbonate, polystyrene and silicone rubber. These materials typically have low densities. On a per mass basis, their strength and stiffness are comparable to metals and ceramics. One major drawback of polymers is their tendency to soften and/or decompose at modest temperatures. They have low electrical conductivity and are nonmagnetic. A composite is composed of two or more individual materials which come from the categories: metal, ceramics and polymers. The design goal of a composite is to achieve a combination of properties that is not displayed by any single material, and also to incorporate the best characteristics of each of the component materials. For example, fibreglass consists of small glass fibres embedded within epoxy or polyester. The glass fibres are relatively strong and stiff, whereas the polymer is ductile but also weak and flexible. Thus the resulting fibreglass is relatively stiff, strong, flexible and ductile. In addition, it has a low density. Advanced materials are utilized in high technology application. By high technology, we mean a device or product that operates or functions using relatively intricate and sophisticated principles. For example, computers, fiber-optic systems, spacecraft, aircraft and military rocketry. These advanced materials are typically traditional materials, whose properties have been enhanced. They may also be newly developed materials. Advanced materials include semiconductors, biomaterials, smart materials and nanoengineered materials. Semiconductors have electrical properties that are intermediate between electrical conductors and insulators. Furthermore, the electrical characteristics of these materials are extremely sensitive to the minute concentrations of impurity atoms. Biomaterials are employed in components implanted into the human body for replacement of diseased or damaged body parts. These materials must not produce toxic substances and must be compatible with body tissues. Metals, ceramics, polymers, composites and semi-

2

conductors may be used as biomaterials. Smart (or intelligent) materials are a group of new and state of the art materials. These materials are able to sense changes in their environments and then respond to these changes in predetermined manners. They consist of sensors and actuators. Actuators may be called upon to change shape, position, natural frequency or mechanical characteristics in response to changes in temperature, electric field or magnetic field. Four types of materials are commonly used for actuators: shape memory alloys, piezoelectric ceramics, magnetostrictive materials and electrorheological/magnetorheological fluids. Shape memory alloys, after deformation revert back to their original shape when temperature is changed. Piezoelectric ceramic expand and contract in response to electric field and vice versa. Similarly magnetostrictive materials are responsive to magnetic fields. Electrorheological and magnetorheological fluids are liquids that experience dramatic changes in viscosity upon the application of electric and mangetic fields, respectively. Materials employed as sensors include optical fibres, piezoelectric materials and microelectromechanical devices. Nanoengineered materials are made by manipulating and moving atoms and molecules to form new structures of a few atomic dimensions. This ability to carefully arrange atoms provide opportunities to develop mechanical, electrical, magnetic and other properties that are not otherwise possible. The examples are carbon nanotube and fulerene. There are technological challenges for the development of even more sophisticated and specialized materials. For nuclear energy, fuel, containment structure and facilities for the disposal of radioactive waste are to be developed. In case of automobile sector, new high strength, low density structural materials are required to be developed. For engine components, materials with high temperature performance are needed. New materials are required for economical sources of energy production. Solar cells are costly. Hydrogen fuel cell materials development is very attractive and feasible nonpolluting energy conversion technology. Recycling of these materials is also another major issue.

Category Mechanical Thermal Electrical Magnetic Optical Deteriorative

Table 1: Categories, stimulus and their response and corresponding properties. Stimulus Response Properties load deformation elastic modulus, ultimate tensile strength heat temperature thermal conductivity, heat capacity Electric field current electrical conductivity, dielectric constant magnetic field domain orientation coercivity, magnetic saturation light angle index of refraction, reflectivity environment loss of material or property reactivity

Polymers include plastic and rubber materials. Many of them are organic compounds that are chemically based on carbon, hydrogen and other nonmetallic elements (0, N, Si). They have very large molecular structures, often chain-like in nature and have a backbone of carbon atoms. For example, polyethylene, nylon, polyvinyl chloride, polycarbonate, polystyrene and silicone rubber. These materials typically have low densities. On a per mass basis, their strength and stiffness are comparable to metals and ceramics. One major drawback of polymers is their tendency to soften and/or decompose at modest temperatures. They have low electrical conductivity and are nonmagnetic. A composite is composed of two or more individual materials which come from the categories: metal, ceramics and polymers. The design goal of a composite is to achieve a combination of properties that is not displayed by any single material, and also to incorporate the best characteristics of each of the component materials. For example, fibreglass consists of small glass fibres embedded within epoxy or polyester. The glass fibres are relatively strong and stiff, whereas the polymer is ductile but also weak and flexible. Thus the resulting fibreglass is relatively stiff, strong, flexible and ductile. In addition, it has a low density. Advanced materials are utilized in high technology application. By high technology, we mean a device or product that operates or functions using relatively intricate and sophisticated principles. For example, computers, fiber-optic systems, spacecraft, aircraft and military rocketry. These advanced materials are typically traditional materials, whose properties have been enhanced. They may also be newly developed materials. Advanced materials include semiconductors, biomaterials, smart materials and nanoengineered materials. Semiconductors have electrical properties that are intermediate between electrical conductors and insulators. Furthermore, the electrical characteristics of these materials are extremely sensitive to the minute concentrations of impurity atoms. Biomaterials are employed in components implanted into the human body for replacement of diseased or damaged body parts. These materials must not produce toxic substances and must be compatible with body tissues. Metals, ceramics, polymers, composites and semi-

2

conductors may be used as biomaterials. Smart (or intelligent) materials are a group of new and state of the art materials. These materials are able to sense changes in their environments and then respond to these changes in predetermined manners. They consist of sensors and actuators. Actuators may be called upon to change shape, position, natural frequency or mechanical characteristics in response to changes in temperature, electric field or magnetic field. Four types of materials are commonly used for actuators: shape memory alloys, piezoelectric ceramics, magnetostrictive materials and electrorheological/magnetorheological fluids. Shape memory alloys, after deformation revert back to their original shape when temperature is changed. Piezoelectric ceramic expand and contract in response to electric field and vice versa. Similarly magnetostrictive materials are responsive to magnetic fields. Electrorheological and magnetorheological fluids are liquids that experience dramatic changes in viscosity upon the application of electric and mangetic fields, respectively. Materials employed as sensors include optical fibres, piezoelectric materials and microelectromechanical devices. Nanoengineered materials are made by manipulating and moving atoms and molecules to form new structures of a few atomic dimensions. This ability to carefully arrange atoms provide opportunities to develop mechanical, electrical, magnetic and other properties that are not otherwise possible. The examples are carbon nanotube and fulerene. There are technological challenges for the development of even more sophisticated and specialized materials. For nuclear energy, fuel, containment structure and facilities for the disposal of radioactive waste are to be developed. In case of automobile sector, new high strength, low density structural materials are required to be developed. For engine components, materials with high temperature performance are needed. New materials are required for economical sources of energy production. Solar cells are costly. Hydrogen fuel cell materials development is very attractive and feasible nonpolluting energy conversion technology. Recycling of these materials is also another major issue.

[ ~--

Category Mechanical Thermal Electrical Magnetic Optical Deteriorative

Table 1: Categories, stimulus and their response and corresponding properties. Stimulus Response Properties load deformation elastic modulus, ultimate tensile strength heat temperature thermal conductivity, heat capacity Electric field current electrical conductivity, dielectric constant magnetic field domain orientation coercivity, magnetic saturation light angle index of refraction, reflectivity environment loss of material or property reactivity

Polymers include plastic and rubber materials. Many of them are organic compounds that are chemically based on carbon, hydrogen and other nonmetallic elements (0, N, Si). They have very large molecular structures, often chain-like in nature and have a backbone of carbon atoms. For example, polyethylene, nylon, polyvinyl chloride, polycarbonate, polystyrene and silicone rubber. These materials typically have low densities. On a per mass basis, their strength and stiffness are comparable to metals and ceramics. One major drawback of polymers is their tendency to soften and/or decompose at modest temperatures. They have low electrical conductivity and are nonmagnetic. A composite is composed of two or more individual materials which come from the categories: metal, ceramics and polymers. The design goal of a composite is to achieve a combination of properties that is not displayed by any single material, and also to incorporate the best characteristics of each of the component materials. For example, fibreglass consists of small glass fibres embedded within epoxy or polyester. The glass fibres are relatively strong and stiff, whereas the polymer is ductile but also weak and flexible. Thus the resulting fibreglass is relatively stiff, strong, flexible and ductile. In addition, it has a low density. Advanced materials are utilized in high technology application. By high technology, we mean a device or product that operates or functions using relatively intricate and sophisticated principles. For example, computers, fiber-optic systems, spacecraft, aircraft and military rocketry. These advanced materials are typically traditional materials, whose properties have been enhanced. They may also be newly developed materials. Advanced materials include semiconductors, biomaterials, smart materials and nanoengineered materials. Semiconductors have electrical properties that are intermediate between electrical conductors and insulators. Furthermore, the electrical characteristics of these materials are extremely sensitive to the minute concentrations of impurity atoms. Biomaterials are employed in components implanted into the human body for replacement of diseased or damaged body parts. These materials must not produce toxic substances and must be compatible with body tissues. Metals, ceramics, polymers, composites and semi-

conductors may be used as biomaterials. Smart (or intelligent) materials are a group of new and state of the art materials. These materials are able to sense changes in their environments and then respond to these changes in predetermined manners. They consist of sensors and actuators. Actuators may be called upon to change shape, position, natural frequency or mechanical characteristics in response to changes in temperature, electric field or magnetic field. Four types of materials are commonly used for actuators: shape memory alloys, piezoelectric ceramics, magnetostrictive materials and electrorheological/magnetorheological fluids. Shape memory alloys, after deformation revert back to their original shape when temperature is changed. Piezoelectric ceramic expand and contract in response to electric field and vice versa. Similarly magnetostrictive materials are responsive to magnetic fields. Electrorheological and magnetorheological fluids are liquids that experience dramatic changes in viscosity upon the application of electric and mangetic fields, respectively. Materials employed as sensors include optical fibres, piezoelectric materials and microelectromechanical devices. Nanoengineered materials are made by manipulating and moving atoms and molecules to form new structures of a few atomic dimensions. This ability to carefully arrange atoms provide opportunities to develop mechanical, electrical, magnetic and other properties that are not otherwise possible. The examples are carbon nanotube and fulerene. There are technological challenges for the development of even more sophisticated and specialized materials. For nuclear energy, fuel, containment structure and facilities for the disposal of radioactive waste are to be developed. In case of automobile sector, new high strength, low density structural materials are required to be developed. For engine components, materials with high temperature performance are needed. New materials are required for economical sources of energy production. Solar cells are costly. Hydrogen fuel cell materials development is very attractive and feasible nonpolluting energy conversion technology. Recycling of these materials is also another major issue.

2

2

Interatomic

bonding

2.1

Primary

interatomic

At large distances, the interatomic interations are neg- 2.1.1 Ionic bonding ligible. However, when atoms are close, the equilibrium distance between two atoms is decided by the sum of two kind of forces: attractive and repulsive. The magnitude of each of these forces is a function of interatomic distance. The origin of an attractive force FA depends on the particular type of bonding that exists between the two atoms. When the two atoms are brought closer, the // cation outer electron shells of the two atoms begin to overlap /o ,: and a strong repulsive force FR comes into play. The ! net force FN between the two atoms is given by: "Bond " ~

. - -=-~.~ ",,,....-., =stt<' =-

he= I

'"

(1)

E~y"

Eo

'" /'

bonds

E"'>'4

':rE,;:f

/

~ ; -''''-

--

E!Q\Jitibriumdistance of separation (roi H 0, I The distance at which FN becomes zero, that is the disI at E/"I'ergy minimum{Bondenergy} I tance between centre of the two atoms 1'0, is the equiI librium interatomic distance. Once in this position, the two atoms will counteract any attempt to separate them by attractive force, or to push them together by a repulsive action. Figure 1: Energetics of ionic bonding. Sometimes, it is more convenient to work with the potential energies between two atoms instead of forces. Ionic bonding is found in compounds that are comMathematically, energy, E and force F are related as posed of both metallic and nonmetallic elements. Metallic elements exist at extreme left in the periodic table (2) whereas nonmetallic one are located at extreme right EN = J F.'Vdr of the periodic table. Atoms of metallic elements easily give up their valence electrons to the nonmetallic atoms. (3) J FA d1'+ J FR dr In the process all the atoms acquire stable or inert gas EA + ER (4) configurations and in addition, an electrical charge; that is, they become ions. Sodium chloride (Na+Cl-) is the FN = 0 corresponds to minimum in EN vs l' curve Classic ionic material. A sodium atom can assume the and is denoted as Eo at interatomic distance 1'0. Eo is electron structure of neon by a transfer of its one valence called as bonding energy. 3s electron to chlorine atom. After such a transfer, the Although the preceding. treatment has dealt with an chlorine ion has a net negative charge and an electron ideal situation involving only two atoms, a similar yet configuration identical to that of argon. more complex condition exists for solid materials beThe attractive bonding forces are coulombic by virtue cause force and energy interactions among many atoms of their net electrical charge. For two isolated ions, the must be considered. The magnitude of the bonding enattractive energy EA and repulsive energy ER are reergy and the shape of the energy versus interatomic sepspectively given by following equations. aration curve vary from material to material and they both depend on the type of atomic bonding. A num(5) ber of material properties depend on Eo curve shape and bonding type. For example, materials having large (6) bonding energies have high melting and boiling points. Materials with steep energy curve with interatomic disIn these equations, A, Band n are constants whose value tances have high stiffness and low coefficient of thermal depend on the particular ionic system. The value of n expansion. is approximately 8. Three different types of primary or chemical bond are Ionic bond is termed nondirectional as the magnitude found in solids: ionic, covalent and metallic. For each of the bond is equal in all directions around an ion. For type, the bonding necessarily involves the valence elecionic materials to be stable, all positive ions must have trons. The nature of bonds depends on the electron as nearest neighbors negatively charged ions in a three structures of the constituent atoms. Secondary or pqysdimensional scheme and vice versa. Bonding energies ical forces and energies are also found in all materials. are relatively large and lies in the range between 600 and They are weaker than the primary ones. They influence 1500 KJjmol (6 and 16 eVjatom). This is reflected in the physical properties of some materials.

3

their high melting points. Ionic materials are characteristically hard and brittle and electrically and thermally insulative.

free electrons. Furthermore, at room temperature, metals and their alloys fail in a ductile manner.

2.2 2.1.2

Covalent

bonding

In covalent bonding, stable electron configurations are assumed by the sharing of electrons between adjacent atoms. Two atoms that are covalently bonded will each contribute atleast one electron to the bond and the shared electrons may be considered to belong to both atoms. The covalent bond is directional. Many nonmetallic elemental molecules (H2, C12, F2 etc) as well as molecules containing dissimilar atoms, such as CR1, H20, HN03, and HF, are covalently bonded. Furthermore, this type of bonding is found in elemental solids such as diamond, silicon and germanium and other solid compounds composed of elements that are located on the right hand side of the periodic table, such as GaAs, InSb and SiC. For N' valence electrons, an atom can covalently bond with at most 8 - N' other atoms. For example, chlorine atom can bond to only one chlorine atom, whereas in diamond, each carbon atom covalently bonds with four other carbon atoms, thus forming three-dimensional interconnecting structure. Covalent bond may be vey strong, as in diamond, which is very hard and has very high melting temperature greater than 35500C, or they may be very weak as with bismuth which melts at about 270°C. Polymeric materials are a long chain of carbon atoms that are covalently bonded together. It is possible to have interatomic bonds that are partially ionic and partially covalent. For a compound, the degree of either bond type depends on the relative positions of the constituent atoms in the periodic table or the difference in their electronegativities. The percentage ionic character of a bond between elements A and B (A being the most electronegative) may be approximated by the expression %ionic character 2.1.3

Metallic

= {1-exp[-(0.25)(XA

Secondary bonding or van der Waals bonding or physical bonding

Secondary bonds are weak in comparison to the primary chemical ones. Bonding energies are typically of the order of 0.1 eV / atom.

Secondary

bonding exists between

virtually all atoms or molecules, but its presence may be obscured if any of the three primary bonding types is present. Secondary bonding forces arise from atomic or molecular dipoles. An electric dipole exists whenever there is some separation of positive and negative portions of an atom or molecule. The bonding results from the coulombic attraction between the positive end of one dipole and the negative region of an adjacent one. Dipole interactions occur between (a) polar molecules, (b) polar molecule and induced dipole in a nonpolar molecule or an atom and (c) induced dipoles between two nonpolar molecules or atoms. Hydrogen bonding, a special type of secondary bonding (category a), is found to exist between molecules having hydrogen and higly electronegative elements such as fluorine, oxygen and nitrogen as one of the constituents. These are discussed below. In case of hydrogen bond between molecules of H-F, H-O or H-N, the electron associated with hydrogen is pulled so close to anions due to their high electronegative character that the hydrogen end of the bond is essentially a positively charged bare proton that is unscreened by any electron. This highly positively charged end of the molecule is capable of a strong attractive force with the negative end of an adjacent molecule. Among the secondary bonds, hydrogen bond is strongest and the bond energy may be as high as 0.52 eV/molecule. The permanenet dipole bond exist between molecules having permanent dipoles by virtue of the difference in electronegativity of its constituent elements. For example, in case of HCl molecule, the hydrogen end is slightly positive and Cl is slightly negative. Thus there exist permanent dipole moment and negative end of one molecule is bonded to the positive end of the other. In case of polar molecule induced dipole bonds, there exist polar molecules which induce dipoles in adjacent nonpolar molecules or atoms by distorting their electron cloud. In case of nonpolar molecules or atoms, there exist no dipole moment. However, at any instant of time, the electron cloud may be slightly distorted, thus resulting in temporary dipole moment. This induces dipole moment in the neighbouring molecule/atom. It is therefore named as fluctuating induced diploe bond and is weakest in the category of secondary bonds.

-XB)2]} x 100

bonding

Metallic materials have one, two or atmost three valence electrons. These valence electrons are not bound to any particular atom in the solid and are free to drift throughout the entire crystal. The remaining nonvalence electrons and atomic nuclei form ion cores. The free electrons shield the positively charged ion cores from mutually repulsive electrostatic forces. Consequently, the metallic bond is nondirectional. Bonding may be weak or srong; energies range from 0.7 eV/atom for mercury to 8.8 eV/atom for tungsten. Their respective melting temperatures are -39 and 34100C. Metals are good conductor of heat and electricity as a consequence of their

4

I

3

Crystalline terials

and noncrystalline

ma-

In case of crystalline solids, both short range and long range order exists. In case of noncrystalline solids, only short range order exist. The structure is close to that of liquid and are called as amorphous (or glassy) solids. Crystalline solids may exist as either single crystals or polycrystals. When the periodic arrangement of atoms extends throughout the entirety of the specimen without interruption, the result is a single crystal. In case of polycrystalline solids, which is more generally the case, a collection of many small crystals separated by grain boundaries exist. These grains have random crystallographic orientations. There exists some atomic mismatch within the region where two grains meet. This region is called grain boundary. In general, the properties of a crystal in different directions are different. In a polycrystalline material, if there exist random orientation between various grains, the result is isotropic material, Le., with uniform property in all directions. However, if there exist preferential orientation, Le., more number of grains are oriented in a praticular directions. The property of the material are no more same in all directions. The material is anisotropic and is said to have a texture.

4 4.1

Structure

of crystalline

Unit cell, crystal Lattices

systems

solids and Bravais

A crystalline material is one in which the atoms are situated in a repeating or periodic array over large atomic distances, such that upon solidification, the atoms will position themselves in a repeatitive three dimensional pattern, in which each atom is bonded to its nearest neighbor atoms. All metals, many ceramic materials and certain polymers form crystalline structures under normal solidification conditions. When describing crystalline structures, atoms are thought of as being solid spheres having well defined diameter. This is termed as atomic hard sphere model. If the atoms positions are represented by points in threedimensional space, it is called as lattice. The atomic order in crystalline solids indicates that small groups of atoms form a repeatitive pattern. The smallest repeatitive unit is called as unit cell. Thus, the unit cell is the basic building block of the crystal structure and defines the crystal structure by virtue of its geometry and the atom positions within. The unit cell is chosen such that it possess highest fold of symmetry and then is of minimum size. Based on unit cell geometry, that is, the shape of the unit cell parallelopiped without regard to the atomic po-

sitions in the cell, there can exist maximum seven crystal systems. In these crystal systems, if lattice points are defined, there can exist maximum 14 Bravais lattices and are shown in Figure 2. At a lattice point, there can exist an atom (AI, Cu, Au, Pt), a molecule (methane), ion pair (NaCI), atom pair (diamond, Si, Ge). A unit cell is defined using six parameters, Le., three axes (a, b and c) passing through the unit cell corner and are defined along the unit cell edges and three interaxial angles (a, {3and ')'). 4.2

Crystallographic planes

points,

directions

and

Six parameters of a unit cells were defined. In the unit cell, conventions have been developed to designate point locations, directions and planes. Three indices are used for such purposes. The basis for determining index values is the unit cell, with a right-handed coordinate system consisting of three axes situated at one of the corners and coinciding with the unit cell edges. For some crystal systems- namely, hexagonal, rhombohedral (trigonal), monoclinic and triclinic- the three axes are not mutually perpendicular, as in the cartesian coordinate scheme. The position of any point located within a unit cell may be specified in terms of its coordinates as fractional multiples of the unit cell edge lengths (i.e., in terms of a, b and c). For example index of a point given as is equivalent to coordinate (la, 1b, !c). A crystallographic direction is defined as a line between two points, or a vector. Any vector can be translated throughout the crystal lattice without alteration, if parallelism is maintained. Therefore, a vector of convenient length is positioned such that it passes through the origin of the coordinate system. The length of the vector projection on each of the three axes is determined. These are measured in terms of the unit cell dimensions a, band c. These three numbers are multiplied or divided by a common factor to reduce them to the smallest integer values. The three indices, not separated by commas, are enclosed in square Drackets. Any negative index is denoted by placing bar above it. For example, cube diagonals can be expressed as [111], [III], [Ill] and [111]. This entire family of directions is denoted as < 111 >. Directions in cubic crystals having the same indices without regard to order or sign, for example, [123] and [213], are equivalent. This is in general, not true for other crystal systems. For example, for crystals of tetragonal symmetry, [100] and [010] directions are equivalent, whereas [100] and [001] are not. A problem arises for crystals having hexagonal symmetry in that some crystallographic equivalent directions will not have the same set of indices. This is circumvented by utilizing a four-axis coordinate system. The three ai, a2 and a3 axes are contained within a sin-

lq

5

7 Crystal systems

14 Bravais Lattices

cubic a=b=c

a=~gOo tetragonal a=b:;t:C

a=/3=y=900

orth orho 1"'1 bie a;:tJ:;cC o:=I3='y=900

rhombohedral a=b=c a=(3=)f#9
Ijexagonal a=I);=C 0:=[3=90<) \1=1200

.

/ .1/ '

monodinic a;:t):;cC

I

/17

<X='p9G<):t=13

triclii"c

~ o:~~IOO

Figure 2: The 7 crystal systems and the 14 Bravais lattices.

6

gle plane called the basal plane and are at 1200 angle to one another. The z axis is perpendicular to this basal plane. The direction is then denoted by four indices as [uvtw]. The first three indices pertain to projections along the respective aI, a2 and a3 axes in the basal plane. The orientation of planes for a crystal structure are represented in a similar manner. Except for hexagonal system, the planes are specified by three Miller indices as (hkl). Any two planes parallel to each other are equivalent and have identical indices. Following procedure is followed to obtain the indices. If the plane passes the selected origin, another parallel plane must be constructed within the unit cell by an appropriate translation. The plane either intersects or parallels each of the three axes. The length of the planar intercept for each axis is deterrnined in terms of lattice parameters a, b and c. The reciprocal of these numbers are taken. Thus for parallel axis, the value becomes zero. For other intercepts, nonzero values are obtained. Multiply them with a suitable common factor to obtain the set of smallest integers. The integer indices, not separated by commas, are enclosed within parenthesis as (hkl). An intercept on negative side of origin is indicated by a bar. The family of planes are denoted by enclosing indices in curly braces as {hkl}. In case of cubic crystals, planes and directions having the same indices are perpendicular to one another. For other crystal systems, there are no simple geometrical relationships between planes and directions having the same indices. In case of denoting planes in hexagonal crystals, four indices are used as discussed earlier in case of directions.

4.3

by multiplying n by 4/37fR3 where R is the radius of an atom in a unit cell. The coordination number is defined as the total number of nearest neighbours or touching atoms. Atomic packing factor (AP F) is defined as the ratio of volume of atoms in a unit cell (Vs) and total unit cell volume (Ve).

p=-

. .

Metallic bonding is not directional.

LD

= number

of atoms centred on direction vector length of direction vector

Planar density (PD) is defined as follows. PD

= number

of atoms centred on a plane area of plane

In case of cubic crystals, the interplanar spacing (dhkd can be calculated as:

_

d hkl

-

yh2

+

a k2

+

(7) [2

FCC and HCP are the closest packed system with APF=0.74. In case of HCP, basal plane is the closest packed plane. In case of FCC, {Ill} is the closest packed plane. In closest packed plane, the central atom is surrounded by 6 atoms. Thus there are six valleys, on which maximum 3 atoms can be placed. Thus there are two ways to place the second layer. Similarly third layer can also be placed in two possible ways. If the atoms in the third layer are exactly above that of first layer, we obtain ABAB... arrangement. This result in HCP lattice. The atom in second layer can not be above first layer for closest packed structure. If the atom in the third layer are also not above the atoms in first layer, we obtain ABCABC... arrangement. This result in FCC lattice. Thus FCC and HCP differ only in the sequence atoms are placed in the third layer. In closed packed lattices, there are two kind of voids: tetrahedral and octahedral. Tetrahedral void exist just between three atoms in one layer and one atom placed above the valley formed by three atoms in the first layer. A tetrahedral void is defined by joining the atoms centre. A model of the void can be obtained by joining 4 equilateral triangles. Octahedral void exist in the valley which was uncovered by the atoms in the second layer.

Nearest neighbor distances tend to be small in order

to lowerbond energy.

.

nA VeNA

where A is atomic weight and NA is Avogadro number (6.023 x 1023). In materials science, closest packed planes and closest packed directioI1s have special significance. Therefore, quite often, it is necessary to calculate linear and planar densities. Linear density (LD) is defined as follows.

The simplest crystal structure exist for metals. The metals tend to be densely packed because of the following reasons. Typically, only one element is present, so all atomic radii are the same.

Ve

Theoretical density p can be calculated as:

Metallic crystal structures

.

= Vs

AP F

Electron cloud shields cores from each other

Most of the metals either crystallize as face centred cubic (FCC), body centred cubic (BCe) or hexagonal closed packed (HCP). Metals having FCC structure are Cu, AI, Ag, Ni, Pt, Au, Pb and 'Y Fe. Metals having BCC structure are Cr, W, a and 'Y Fe, Mo and Ta. Metals having HCP structure are Cd, Co, a Ti and Zn. The number of atoms per unit cell (n) is calculated as the summation of the fraction of all atoms within a unit cell. The total volume of atoms can thus be obtained 7

An octaheral model can be obtained by joining a square with 8 equilateral triangles: 4 above and 4 below. Some metals and nonmetals may have more than one crystal structure. This phenomena is known as polymorphism. In case of elemental solids, this condition is known as allotropy. The prevailing crystal structure depends on both the temperature and the external pressure.

5

X-ray

diffraction

In order to characterize crystal structure (dimensions ~ 1 A), the wavelength of the electromagnetic radiation (signal) used for probing the crystal structure must be of same order of magnitude. X-rays are used for the purpose. X-rays are electromagnetic radiation. The wave length

lies approximately

in the range 0.01 to 100

A

and centred around 1 A. The energy of the radiation of

4.4

wavelength1 A is ~ 12400eV/photon using the formula

Ceramic crystal structures

E

= he/A.

Comparing

this value with the seventh ion-

Ceramics are composed of at least two elements. The ization energy of nitrogen equal to 668 eV / atom, it can atomic bonding may be ionic or covalent or mixed de- be concluded that x-rays have enormous power to boil pending on the electronegativities of its constituents. off even the inner shell electrons. Following are the factors that govern the crystal strucFor characterization, we model atoms as mirrors. ture of ceramics. Therefore angle of incidence is equal to angle of reflecIn case of ionic bonding, it is necessary to main- tion. Secondly, interference criteria is considered. If the tain charge electroneutrality. Thus in case of cal- waves are in phase, constructive interference occurs. If cium fluorite, there must be twice as many F- as they are out of phase, destructive interference occurs. Ca2+ ions, which is reflected in the chemical for- The signal consists of coherent (in phase) monochromatic (single wavelength) x-rays. The diffraction phemula CaF 2. nomena is shown in Fig. 3. The distance betweeen A-A' . The ionic radius of the cations (rc) is generally less and B-B' is dhkl. For constructive interference, the extra than that of anions (ra). Each cation prefer to have distance travelled by ray 2 compared to ray 1 must be as many nearest neighbor anions as possible. Simi- equal to whole number times wavelength. This result in larly, the anions also desire a maximum number of Bragg's law given as: cations nearest neighbors. 2dhkl sin () = nA (8) Stable ceramic crystal structures form when those anions surrounding a cation are all in contact with that Using Bragg's law, interference criteria is obtained. Incation. The coordination number is related to the terference criteria and crystal structure together result cation-anion radius ratio. For rc/ra < 0.155, cation is in a set of expected reflections. An example is shown in bonded to two anions in a linear manner. If 0.155 < Fig. 4. This result in finger printing of crystal structure. rc/ra < 0.225, cation is surrounded by three anions For example {001} results in reflection in case of simple in the form of a planar equilateral triangle, with the cubic but not in case of BCC or FCC, because there excation located in the center. For 0.225 < rc/ra < 0.414, ist one more plane at the position of body centre or face the cation is located at the centre of tetrahedron. For centre, which causes destructive interference. The rule 0.414 < rc/ra < 0.732, the cation is located at the centre in nutshell can be described as follows. All the planes of octahedron. For 0.732 < rc/ra < 1, the coordination causes reflection in case of simple cubic. In case of BCC, number is 8 and the cation is positioned at the centre of the sum of miller indices must be even. In case of FCC, the cube. For rc/ra > 1, the coordination number is 12. either all the miller indices must be odd or all must be Such relations are purely based on geometrical consid- even. No reflection will occur in mixed cases. The value eration. These relations are only approximate and there of (h2 + k2 + [2) of allowed reflections in case of simple are exceptions. cubic is 1, 2, 3, 4, 5, 6, 8, 9 In case of BCC, the values are 2, 4, 6, 8, 10, 12, 14, 16. . .. This is equivalent 4.5 Polymer crystal structures to the values 1, 2, 3, 4, 5, 6, 7, 8 In case of FCC, the values are 3, 4, 8, 11, 12. . . . Since molecules are involved at lattice positions, the Eliminating dhkl between equations 7 and 8, following atomic arrangement is more complex for polymers. equation is obtained. Polymer molecules are often only partially crystalline, having crystalline regions dispersed within the remainA2 sin2 () (9) ing amorphous material. Any chain disorder or misalign4a2 - h2 + k2 + [2 ment will result in an amorphous region, since twisting, kinking and coiling of the chains prevent the strict or- Therefore for determining crystal structure, first sin2 () dering of every segment of every chain. Cross-linking, values are tabulated. Then the values are normalized higher molecular weight and higher cooling rate are op- with respect to the first reflection. The values are then posing factors to crystallization of polymers. multiplied with a minimum natural number necessary

.

8

on introduction of vacancies, the entropy of the system increases. However, this can not continue endlessly as work has to be done in order to break bonds and create a vacancy. There is a trade off and equilibrium number of vacancies Nv increases with temperature according to the following relation

Figure 3: Diffraction of x-rays by planes of atoms A-A' and B-B'.

<,

[) '!. :.~

Figure 4: Diffraction

pattern

for powdered

lead.

where, N is the total number of atomic sites, Qv is the energy required for the formation of a vacancy, T is the absolute temperature in Kelvin and k is Boltzmann's constant. For specification of composition of an alloy in terms of "its constituent elements, the two most common ways are ight percent and atom percent. The basis for weight percent (wt%) is the weight of a particular element relative to the total alloy weight. For an alloy that contains two hypothetical atoms denoted by 1 and 2, the concentration of 1 in wt%, CI, is defined as

to clear the fraction. Then miller indices are guessed. (10) Finally the value of sin2 ()/ (h2+ k2 + [2) is calculated to check that the values obtained for all the reflections are same. Then miller indices pattern is studied in order to where mi and m2 represent the weight (or mass) of elunderstand the type of crystal structure. Using equation ements 1 and 2, respectively. The concentration of 2 9, the value of lattice parameter is obtained. would be computed in an analogous manner. The basis for atom percent (at%) calculations is the Q. Determine the crystal structure and lattice parame- number of moles of an element in relation to the total ter, given that >. = 1.5418 A and 2() values are 44.48, moles of the elements in the alloy. The number of moles 51.83, 76.35, 92.90, 98.40, 121.87, 144.54, 155.51. The in some specified mass of a hypothetical element 1, nml, answer is FCC lattice with lattice parameter equal to may be computed as follows: 3.53 A which is that of Ni. (11)

6

Imperfections

6.1

Point defects

in solids

where mi is the mass in grams and Al is the atomic The lattice is never perfect. There are always various weight of element 1. Concentration in terms of atom percent of element 1 kind of defects or imperfections. They are classified as in an alloy containing 1 and 2 atoms, C~ is defined as point defect (0 dimension), line defect (1 dimension), follows. surface defect (2 dimension) or volume defect (3 dimen(12) sion). Q. Knowing the definition of weight percent and atom percent, write the formulae for conversion from one unit to other. In case of an alloy, two things may happen. Either there exist complete solubility between solute and solvent and is called solid solution or there may occur precipitation of separate phase. If an atom of solute substitute for an atom of solvent, it is called as substitutional solid solution. If an atom of solute occupies the void space between solute atoms, it is called as interstitial solid solution. These are called impurity point defects. For substitutional solid solution to occur throughout the composition range (extended solid solubility), following Hume Rothery rules must satisfy.

Point defects are the result of change in local order at a regular lattice position. If the atom is missing from the regular lattice position, it is called as vacancy. If an extra atom is located between two regular lattice positions, it is called as self-intersti tial. Point defects are equilibrium defects. In a perfect lattice of say pure metal, all the atoms are alike and indistinguishable. Therefore, only one configuration is possible. However, if some vacancies are introduced then the number of lattice positions will be greater than the total number of atoms in the lattice and in such condition, more than one configuration is possible. Therefore 9

\ '1