CRYSTAL STRUCTURE, MECHANICAL BEHAVIOUR & FAILURE OF MATERIALS

MODULE ONE CRYSTAL STRUCTURE, MECHANICAL BEHAVIOUR & FAILURE OF MATERIALS CRYSTAL STRUCTURE Metallic crystal structures; BCC, FCC and HCP Coordination number and Atomic Packing Factor (APF) Crystal imperfections: point, line and surface imperfections Atomic Diffusion: Phenomenon, Fick s laws, factors affecting diffusion. 1.1 INTRODUCTION The engineering materials (either metallic or non-metallic) can be identified as crystalline or amorphous structured. But most of the metals assume crystalline form with a systematic and regular arrangement of atoms as compared to amorphous structure that lacks regular atomic arrangement. A crystalline material is thus comprised of group of atoms with a specific atomic arrangement which repeats in a 3D pattern; the small group of atoms that repeats over a 3D array is termed as a unit cell. The geometry and the atomic positions of a unit cell define the crystal structure. Figure 1.1: Unit cell lattice parameters Based on unit cell geometry for different possible combinations of a, b, c and,, seven crystal systems were identified. Further, considering atomic arrangement within a crystal system A. Bravais showed that the seven crystal systems could be arranged in 14 independent ways to obtain the 14 Bravais lattices. Pruthvi Loy, Chiranth B.P. 1 SJEC, Mangalore

Table 1.1: Crystal systems and Bravais lattices Pruthvi Loy, Chiranth B.P. SJEC, Mangalore

1. METALLIC CRYSTAL STRUCTURES Inspite of the different possible crystal structures three relatively simple structures are found for most of the common metals. Body Centered Cubic (BCC) Face Centered Cubic (FCC) Hexagonal Close Packed (HCP) 1..1 Body Centered Cubic (BCC) A cubic unit cell with atoms located at all eight corners and a single atom at the cubic center. No of atoms per unit cell = i.e., 8x (8 corner atoms each shared by 8 neighboring unit cells) + 1 (1atom at the cubic center) = Example: Chromium, Tungsten, Iron, etc. 1.. Face Centered Cubic (FCC) It also has a cubic geometry with atoms located at each of the corners and the centers of all the cube faces. No of atoms per unit cell = 4 i.e., 8x (8 corner atoms each shared by 8neighboring unit cells) + 6 x (6 atoms at the cube faces shared by unit cells) = Example: Copper, Aluminium, Silver, Gold, etc. Pruthvi Loy, Chiranth B.P. 3 SJEC, Mangalore

1..3 Hexagonal Close Packed (HCP) No of atoms per unit cell = 6 i.e., 1 x (1 corner atoms each shared by 6 neighboring unit cells) + x ( atoms at the hexagonal faces each shared by unit cells) + 3 whole atoms = 6 Example: Titanium, Zinc, Cobalt, Magnesium, etc. 1.3 CHARACTERISTICS OF CRYSTAL STRUCTURE The type of structure and its characteristics has a profound influence on the material properties. Two of the important characteristics of a crystal structure are the coordination number and the atomic packing factor (APF); apart from these the other characteristic features of interest is the stacking of planes. 1.3.1 Coordination number For metals, each atom has the same number of nearest-neighbor or touching atoms, which is the coordination number. Unit Cell Coordination number 1. Simple Cubic 06. Body Centered Cubic 08 3. Face Centered Cubic 1 4. Hexagonal Close Packed 1 1.3. Atomic Packing Factor The Atomic Packing Factor (APF) is the fraction of volume in a crystal structure that is occupied by the atoms. i.e., APF = = = Pruthvi Loy, Chiranth B.P. 4 SJEC, Mangalore

a) Simple Cubic: Number of atoms per unit cell, = 1 Volume of each atom, = Volume of a unit cell, = Therefore, APF = = (because, = r) APF = 0.5 i.e., only 5 % of the space available inside a unit cell of simple cubic structure is occupied by atoms. b) BCC: From ADC, AC = AD + DC AC = + AC = From ABC, AB =AC + BC = ( ) + = 3 = Number of atoms per unit cell, = Volume of each atom, = Volume of a unit cell, = Pruthvi Loy, Chiranth B.P. 5 SJEC, Mangalore

Therefore, APF = = ( ) APF = 0.68 i.e., 68 % of the space available in a BCC unit cell is occupied by atoms. c) FCC: From ABC, AC +BC = AB + = = Number of atoms per unit cell, = 4 Volume of each atom, = Volume of a unit cell, = Therefore, APF = = ( ) APF = 0.74 i.e., 74 % of the space available in a FCC unit cell is occupied by atoms. Pruthvi Loy, Chiranth B.P. 6 SJEC, Mangalore

d) HCP: From ABD, = h + ( ) h = - = h = Number of atoms per unit cell, = 6 Volume of each atom, = Volume of a unit cell, = Area of hexagon x c = 6 x Area of ABC x c = 6 x ( ) x c = 6 x ( ) x c = 6 c Also, = r and c = 1.633 Where, c is lattice constant, its value can be calculated as shown below(considering the atoms to be spherical in shape), Pruthvi Loy, Chiranth B.P. 7 SJEC, Mangalore

From AMN, AM = AN + MN = ( ) + ( ) cos 30 = a AN = = + AN= a = c = = 1.633 Therefore, APF = = APF = 0.74 i.e., 74 % of the space available in a HCP unit cell is occupied by atoms. Pruthvi Loy, Chiranth B.P. 8 SJEC, Mangalore

Note: It is possible to compute the theoritical density of a metallic solids having the knowledge of its crystal ctructure. Where, = = n = number of atoms per unit cell A = atomic weight V C = volume of the unit cell N A = Avogadro s number (6.03 x 10 3 atoms/mol) Table 1.: BCC, FCC and HCP Unit cell parameters Atoms/unit cell, n Coordination number, Z edge length, lattice constant, c Unit cell volume, V c APF BCC 8 FCC 4 1-0.68-0.74 HCP 6 1 r 1.633 6 c 0.74 Example Problem: Copper has an atomic radius of 0.18 nm (1.8 Å), an FCC crystal structure, and an atomic weight of 63.5 g/mol. Compute its theoretical density and compare the answer with its measured density. SOLUTION: Given, n = 4, A= 63.5 g/mol, r = 0.18 x 10-9 m = 0.18 x 10-7 cm Volume of unit cell, = = ( ) = = 16 = 16 (0.18x10-7 ) 3 = 4.75x10-3 cm 3 Therefore, Density, = = = 8.89 From periodic table of elements, the literature value of density of copper is 8.96. Pruthvi Loy, Chiranth B.P. 9 SJEC, Mangalore

1.3.3 Stacking of Planes During solidification the atoms within the solid pack together as tightly as possible, i.e., a layer of atoms stack one above the other to make up the solid material.although layers of atoms are stacked one above the other, their sequence of stacking varies for different crystal structures.the stacking sequence of few crystal strucures are as shown below. Figure 1.: Stacking of Planes in a crystal structure Pruthvi Loy, Chiranth B.P. 10 SJEC, Mangalore

1.4 CRYSTAL IMPERFECTIONS For the study of crystal structures we have assumed a perfect or ideal crystal. However, such an idealized crystal does not exist; all contain large number of various defects or imperfections. The properties of most of the metals are profoundly influenced by the presence of imperfections. Thus specific characteristics can be obtained in crystals by introducing crystalline defects. Adding alloying elements to the metal is one way of introducing a crystal defect. According to the geometry or dimensionality the defects may be classified as; Zero dimensional or Point defect o Vacancy o Interstitial defect o Substitutional defect One dimensional or Line defect o Edge dislocation o Screw dislocation Two dimensional or Surface defect o External surface o Grain boundary o Tilt boundary o Twin boundary o Stacking fault Three dimensional or Volume defect - Pores, cracks, foreign inclusions and other phases 1.4.1 Point imperfections Vacancy: The simplest of the point defect is a vacancy, where in one or more atoms are missing from their respective location within the crystal lattice. The vacancies may occur as a result of imperfect packing during crystallization or they may also arise from thermal fluctuation of atoms at high temperature. Figure 1.3: Vacancy defect Pruthvi Loy, Chiranth B.P. 11 SJEC, Mangalore

Impurity: A pure metal consisting of only one type of atom is highly idealistic; impurity or foreign atoms will always be present. Most familiar metals are not highly pure rather they are alloys in which impurity atoms have been intentionally added to impart specific characteristics to the material. The addition of impurity atoms to a metal will result in the formation of either a solid solution and/or a new phase. A solid solution forms when as the solute atoms (impurity) are added to the solvent (host material) and the crystal structure of the parent material is retained with no new structures being formed. Impurity point defects in crystals can be, Interstitial impurity or Substitutional impurity Figure 1.4: Impurity defects Interstitial impurity: In this an interstitial foreign atom occupies a definite position in a nonlattice site within the crystal. Example: Addition of carbon atoms (0.071 nm) to iron (0.14 nm) where the carbon atoms occupy the interstitial space between the iron atoms. Substitutional impurity: When a foreign atom substitutes the parent atom and occupies its position in the lattice site, then it is known as a substitutional defect. Example: Addition of copper to nickel; copper atoms substitute the nickel atoms. Note: Point defect in ceramics may exist as both vacancies and interstitials. The atomic bonding is predominantly ionic in ceramics; i.e., their crystal structures may be thought of as being composed of electrically charged ions instead of atoms. The metallic ions, or cations, are positively charged, because they have given up their valence electrons to the nonmetallic ions, or anions, which are negatively charged. An ionic crystal possess electronegativity, i.e., there is equal number of positive and negative charges from ions; as a consequence, defects in ceramics do not occur alone rather defect for each ion type may occur; one such defect is Frenkel & Schottky defect. Figure 1.5: Frenkel and Schottky defects Pruthvi Loy, Chiranth B.P. 1 SJEC, Mangalore

Frenkel defect involves a cation vacancy - cation interstitial pair. It might be thought of as being formed by a cation leaving its normal position and moving into an interstitial site. There is no change in charge because the cation maintains the same positive charge as an interstitial. Schottky defect is a cation vacancy - anion vacancy pair. This defect might be thought of as being created by removing one cation and one anion from the interior of the crystal. Since for every anion vacancy there exists a cation vacancy, the charge neutrality of the crystal is maintained. 1.4. Line Imperfections Linear defects in crystalline solids are due to misalignent of atoms during the dislocation of atomic planes. Dislocations are of two types; Edge Dislocation and Screw Dislocation Figure 1.6: Line imperfections Edge Dislocation It is created in a crystal by insertion of an extra plane of atoms i.e., a half plane as shown in figure. The edge of the half plane terminates within the crystal, this is termed as dislocation line. The atoms above the dislocation line are squeezed together and are in a state of compression while the atoms below are pulled apart and are in a state of tension. Edge dislocation is represented by the symbol for positive dislocation and for negetive dislocation. Screw Dislocation It is said to be formed in perfect crystal when part of the crystal displaces angularly over the remaining part under the action of shear stress. The upper front region of the crystal is shifted one atomic distance to the right relative to the bottom portion. The screw dislocation derives its name from the spiral or helical path that is traced around the dislocation line by the atomic planes of atoms. Screw dislocation is represented by the symbol for clockwise or positive dislocation and for counterclockwise or negtive dislocation. Pruthvi Loy, Chiranth B.P. 13 SJEC, Mangalore

Table 1.3: Comparison of Edge and Screw dislocation Edge Dislocation It is created when a half plane of atoms is inserted in a crystal It moves in the direction of Burger s vector Burger s vector is perpendicular to the dislocation line Edge dislocation travels faster when loaded It requires less force to form and travels faster under loads. Atomic bonds around the dislocation line experiences tension and compression Symbolic representation: for positive dislocation for negetive dislocation Screw Dislocation It is created when a part of crystal displaces angularly over the remaining part It moves in the direction perpendicular to that of Burger s vector Burger s vector is parallel to the dislocation line Screw dislocation travels comparatively slower It requires comparatively high force to form and travels slower under loads Atomic bonds around the dislocation line experiences shear force. Symbolic representation: for positive dislocation for negtive dislocation 1.4.3 Surface Imperfections External Surface: One of the most obvious surface defect is the external surface, along which the crystal structure terminates. Surface atoms are not bonded to the maximum number of nearest neighbors, and are therefore in a higher energy state than the atoms at interior positions. Grain Boundary: A grain boundary is formed when two adjoining growing crystals meet at their surface.the atoms are bonded less regularly along the grain boundary and are at a higher energy state as a result the impurity atoms preferentially segregate along these boundaries.also, grain boundary acts as a barrier for dislocation motion; the smaller the grains, larger is the grain boundary area and dislocations if any moves only a short distance and stops at the grain boundary. A polycrystalline solid contains numerous grains or crystals. Each crystal has nearly the same crystal structure but different orientations. The grain boundary is few atomic radius thick and contains crystallographic misalignment between adjacent grains; various degrees of crystallographic misalignments are possible. When this orientation mismatch is slight (of the order of few degrees), then it is termed as small angle grain boundary. A small angle of misorientation (less than 10) with the edge dislocations aligned in the manner as shown in figure 1.7(b), then it is called a tilt boundary. Pruthvi Loy, Chiranth B.P. 14 SJEC, Mangalore

Figure 1.7: (a) Grain boundary (b) Tilt boundary Twin Boundary: A twin plane or boundary is a special type of grain boundary across which there is a specific morror lattice symmetry; i.e., the atoms on one side of the boundary are located in mirror image positions of the atoms on the other side. The region of material between these boundaries is termed as twinned region. Figure 1.8: Twinning and twin boundary Stacking Faults: A crystal structure has a specific stacking sequence; any deviations from the actual stacking sequence of the plane of atoms is termed as a stacking fault. For example: The stacking sequence of a FCC structure is A, B, C, A, B, C, A, B, C,. Sometimes it may appear as A, B, C, A, B, A, B, C, with a missing C plane which is termed as a stacking fault. Pruthvi Loy, Chiranth B.P. 15 SJEC, Mangalore

1.4.4 Volume Imperfections These are three dimensional imperfections that are formed inside the solid material. These includes voids, cracks, foreign inclusions and other phases which are normally introduced during processing and fabrication. 1.5 ATOMIC DIFFUSION From an atomic perspective diffusion may be defined as the mass flow process by which atoms or molecules migrate from lattice site to lattice site within a material resulting in the uniformity of composition as a result of thermal agitation. The importance and various applications of diffusion phenomenon are; Diffusion occurs more rapidly with increasing temperature and is the basis for most metallurgical processes. Diffusion is fundamental to phase changes and is important aspect in heat treatment of metals. It is important in the formation of metallic bonds (soldering, welding, brazing, etc.) 1.5.1 Diffusion Phenomenon Diffusion in solids can take place by the following methods; Vacancy diffusion Interstitial diffusion Vacancy diffusion involves the movement of an atom from original lattice position to an adjacent vacant lattice site. The extent to which vacancy diffusion can occur depends on the number of vacant sites present in the crystal; significant concentrations of vacancies may exist in metals at elevated temperature. Figure 1.9: Vacancy diffusion Interstitial diffusion involves the movement of interstitial atoms from an interstitial site to its neighbouring site without permanently displacing any of the parent atoms in a crystal lattice. With interstitial diffusion an activation energy is associated because to move into an adjacent interstitial site it must squeeze past the parent atoms in the crystal attice with the energy supplied by the vibrational energy of moving atoms. Pruthvi Loy, Chiranth B.P. 16 SJEC, Mangalore

Figure 1.10: Interstitial diffusion Interstitial diffusion occurs more rapidly than vacancy diffusion since interstitial atoms are smaller and as more empty interstitial positions are present than the vacancies. 1.5. Fick s Law of Diffusion Diffusion is a time dependent process; i.e., the quantity of an element that is transported within another is a function of time. Often it is necessary to know how fast diffusion occurs, or the rate of mass transfer. This rate is expressed as a diffusion flux (J), which is defined as the mass (m) diffusing through and perpendicular to a unit cross-sectional area of solid (A) per unit time (t). i.e., J = or J = kg/m -s or atoms/m -s The diffusion flux may or may not vary with time and accordingly we have two laws of diffusion: i. Fick s first law for steady state diffusion ii. Fick s second law for unsteady state diffusion Fick s first law of diffusion: It states that the flux of atoms (J), moving across a unit area in unit time is proportional to concentration gradient under steady state. Where, i.e., J or J = - D J diffusion flux, atoms/m -s concentration gradient D diffusivity or diffusion coefficient, m /s The negetive sign indicates that the direction of diffusion is down the concentration gradient, i.e., from a region of higher concenration to a region of lower concentration. Pruthvi Loy, Chiranth B.P. 17 SJEC, Mangalore

Concentration gradient is obtained as, = Figure 1.11: Steady state diffusion Fick s second law of diffusion: Most practical diffusion situations are usually of unsteady state. i.e., the diffusion flux and the concentration gradient at some particular point in solid vary with time resulting in net accumulation or depletion of diffusing species. Therefore, = * + Where, = rate of composition change = concentration gradient D = diffusivity, m /s Figure 1.1: Concentration profile for unsteady state diffusion Pruthvi Loy, Chiranth B.P. 18 SJEC, Mangalore

If diffusion coefficient is independent of concentration; = D i.e., the rate of composition change is equal to the diffusivity times the rate of concentration gradient. The solution to the above equation can be obtained by applying appropriate boundary conditions. For t = 0, C = C 0 (0 x ) For t > 0, C = C s (at x = 0) and C = C 0 (at x = ) Applying the above boundary conditions the solution can be obtained as, = 1 erf ( ) From the above equation and D are known. may be determined at any time and position if the parameters 1.5.3 Factors Affecting Diffusion The various factors affecting diffusion are:. Grain size 3. Atomic radius 4. Temperature 5. Concentration Crystal Structure: The ease with which the atoms diffuse increases with decreasing density of packing. Example: Atoms have higher diffusion coefficients in BCC iron than FCC iron because the former has low atomic packing factor. Grain size: As we know grain boundary diffusion is faster than diffusion within the grains, it is to be expected that overall diffusion rate would be higher in fine grained material due to increased grain boundary. Atomic radius: Diffusion occurs more rapidly when the size of the diffusing atom is small. Example: diffusion of carbon atoms in iron. Concentration: a higher concentration gradient results in faster diffusion rates. Temperature: It has most profound influence on the coefficient and diffusion rate. The diffusion coefficient (D) is related to temperature by Arrhenius type of equation as shown below, Pruthvi Loy, Chiranth B.P. 19 SJEC, Mangalore

D = D 0 Where, D 0 temperature independent pre-exponential, m /s Q activation energy, J/mol R gas constant ( R = 8.314 J/mol-K) T absolute temperature, K When the temperature increases, the diffusion coefficient increases and therefore the flow atoms also increase. Problems (Diffusion): 1. Calculate the diffusion coefficient for magnesium in aluminium at 570 C given that, D o = 1. x 10-4 m /s and Q = 131 kj/mole. Solution: D o = 1. x 10-4 m /s Q = 131 kj/mole = 131000 J/mole T = 570 C = 570 + 73 = 843 K R = 8.314 J/mol-K D = D 0 = 1. x 10-4 * + D = 9.075 x 10-13 m /s. It is proposed to enhance the surface wear resistance of a steel gear by carburizing treatment. The initial carbon content of steel is 0.15 wt%. After the treatment the surface concentration is to be maintained at 0.95 wt%. For the treatment to be effective a carbon content of 0.55 wt% must be established at 0.75mm below the surface. Specify appropriate heat treatment in terms of temperature and time for temperature 900 C to 1050 C. Take D o =.3x10-5 m /s, Q = 148000 J/mole. Solution: D o =.3x10-5 m /s Q = 148000 J/mole C 0 = 0.15 wt % C s = 0.95 wt % C x = 0.55 wt % X = 0.75 mm = 0.00075m Pruthvi Loy, Chiranth B.P. 0 SJEC, Mangalore

WKT, = 1 erf ( ) = 1 erf, where, z = erf = Table 1.4: Error-function values From error-function value table, the value of z can be obtained by interpolation as follows, 0.45 0.4755 z 0.5 0.5 0.505 z erf (z) 0.35 0.3794 0.4 0.484 0.45 0.4755 0.5 0.505 0.55 0.5633 i.e. = Therefore, z = 0.47 But z = = = 0.47 Dt = * + = 6.17 x 10-7 m Also, D = D 0 * + D = D 0 * + x For 900 C (1173 K) 6.17 x 10-7 =.3 x 10-5 * + x t = 9.6 hrs Similarly calculate for 950, 1000 and 1050 C Pruthvi Loy, Chiranth B.P. 1 SJEC, Mangalore

The following heat treatment parameters were calculated Temperature, C Time, hrs 900 9.6 950 15.9 1000 9.0 1050 5.3 3. Steel gear, having carbon content of 0.% is to be gas carburized to achieve carbon content of 0.9% at the surface and 0.4% at 0.5mm depth from the surface. If the process is to be carried out at 97 C, find the time required for carburization. Take diffusion coefficient of carbon in given steel = 10.8x10-11 m /s. Given data: Z erf(z) 0.75 0.711 Z 0.7143 8 0.741 Solution: D = 10.8x10-11 m /s C 0 = 0. % C s = 0.95 % C x = 0.4 % x = 0.5 mm = 0.0005m WKT, = 1 erf ( ) = 1 erf ( ) 0.857 = 1 erf ( ) 0.7143 = erf ( ) Let z = Equation (1) Therefore, erf(z) = 0.7143 Pruthvi Loy, Chiranth B.P. SJEC, Mangalore

From the given error-function value table, the value of z can be obtained by interpolation as follows, i.e. = Therefore, z = 0.755 Substituting the value on z in equation (1), z = t = t = 8566.35 sec = 14.8 min References: 1. Fundamentals of Materials Science & Engineering William D. Callister. Material Science and Metallurgy K. R. Phaneesh 3. Material Science and Metallurgy Kesthoor Praveen 4. Mechanical Metallurgy G. E. Dieter Pruthvi Loy, Chiranth B.P. 3 SJEC, Mangalore