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**PRINCIPLES OF PRODUCTION ENGINEERING**

Week 1 STRUCTURE OF MATERIALS

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**Why Study Crystal Structure of Materials?**

The properties of some materials are directly related to their crystal structures Significant property differences exist between crystalline and noncrystalline materials having the same composition For example, pure and undeformed magnesium and beryllium, having one crystal structure, are much more brittle (i.e., fracture at lower degrees of deformation) than are pure and undeformed metals such as gold and silver that have yet another crystal structure Opposite of “Crystalline” is Amorphous. It means lacking definite form; having no specific shape; formless. Diamond is crystalline while graphite is amorphous. Silica may opaque or may be transparent

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**Crystalline and Crystal Structure**

A crystalline material is one in which the atoms are situated in a repeating or periodic array over large atomic distances All metals, many ceramics, and some polymers make crystalline structure Some of the properties of crystalline solids depend on the crystal structure of the material 0. Solid materials may be classified according to the regularity with which atoms or ions are arranged with respect to one another … that is, long-range order exists, such that upon solidification, the atoms will position themselves in a repetitive 3D pattern, in which each atom is bonded to its nearest-neighbor atoms Ice, NaCl are crystalline solids; the glass for drinking water is an amorphous solid. Heat-treatment makes glass partly crystalline so that it becomes transparent. Nylon, polypropylene are crystalline polymers. 3. the manner in which atoms, ions, or molecules are spatially arranged. There is an extremely large number of different crystal structures all having long range atomic order; these vary from relatively simple structures for metals to exceedingly complex ones, as displayed by some of the ceramic and polymeric materials.

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Lattice In crystalline structures, atoms are considered as being solid spheres having well-defined diameters Atomic hard sphere model -> in which spheres representing nearest-neighbor atoms touch one another Lattice is a regularly spaced array of points that represents the structure of a crystal 3. Sometimes the term lattice is used in the context of crystal structures; in this sense “lattice” means a three-dimensional array of points coinciding with atom positions

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Unit Cells Unit Cell is the smallest group of atoms or molecules whose repetition at regular intervals in three dimensions produces the lattices of a crystal They are parallelepipeds or prisms having three sets of parallel faces A unit cell is chosen to represent the symmetry of the crystal structure 0. The atomic order in crystalline solids indicates that small groups of atoms form a repetitive pattern 2. parallelepiped -> a three-dimensional figure formed by six parallelograms; Prism -> a polyhedron with two congruent and parallel faces (the bases) and whose lateral faces are parallelograms. In first figure of previous slide it happens to be a cube. 3. … wherein all the atom positions in the crystal may be generated by translations of the unit cell integral distances along each of its edges. Thus, the unit cell is the basic structural unit or building block of the crystal structure and defines the crystal structure by virtue of its geometry and the atom positions within

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**Metallic Crystal Structures**

The Face-Centered Cubic Crystal Structure The Body-Centered Cubic Crystal Structure The Hexagonal Close-Packed Crystal Structure 0. The atomic bonding in this group of materials is metallic and thus nondirectional in nature. Consequently, there are minimal restrictions as to the number and position of nearest-neighbor atoms; this leads to relatively large numbers of nearest neighbors and dense atomic packings for most metallic crystal structures. The atoms tend to arrange themselves in a position that guarantees minimum energy of the system, thus the atoms in a unit cell of metals tend to as close as possible. This gives rise to dense crystalline structures

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**Face-Centered Cubic Structure (FCC)**

FCC -> a unit cell of cubic geometry, with atoms located at each of the corners and the centers of all the cube faces For the fcc crystal structure, each corner atom is shared among eight unit cells, whereas a face-centered atom belongs to only two Therefore, one-eighth of each of the eight corner atoms and one-half of each of the six face atoms, or a total of four whole atoms, may be assigned to a given unit cell copper, aluminum, silver, and gold have fcc The cell comprises the volume of the cube, which is generated from the centers of the corner atoms 2. Bottom figure shows a hard sphere model for the FCC unit cell. …

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**FCC - Exercise Derive: Where a = side length of the unit cell cube**

And R = Radius of the atom sphere

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**FCC – Coordination Number and APF**

For metals, each atom has the same number of touching atoms, which is the coordination number For fcc, coordination number is 12 The APF (Atomic Packing Factor) is the sum of the sphere volumes of all atoms within a unit cell divided by the unit cell volume For fcc, APF is 0.74 1. the coordination number of a central atom in a molecule or crystal is the number of its nearest neighbors 2. Take the front-most yellow atom in the figure and determine its coordination number (the front face atom has four corner nearest-neighbor atoms surrounding it, four face atoms that are in contact from behind, and four other equivalent face atoms residing in the next unit cell to the front) 3. … (assuming the atomic hard sphere model) 4. … which is the maximum packing possible for spheres all having the same diameter (calculated on a piece of paper). …. Metals typically have relatively large atomic packing factors to maximize the shielding provided by the free electron cloud (a deep solid state physics concept). This is y fcc metals are very good conductors (electrons repelled from electrons of nearest atoms and become free)

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**Body-Centered Cubic Structure (BCC)**

BCC -> a cubic unit cell with atoms located at all eight corners and a single atom at the cube center Center and corner atoms touch one another along cube diagonals

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**BCC - Exercise Derive: Where a = side length of the unit cell cube**

And R = Radius of the atom sphere First take diagonal of the cube and apply Pythagoras theorem and then take diagonal of the face of the cube and then apply Pythagoras theorem and then solve the two equations

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**BCC Chromium, iron, tungsten exhibit bcc structure**

Two atoms are associated with each BCC unit cell The coordination number for the BCC is 8 the atomic packing factor for BCC lower—0.68 versus 0.74 (FCC) 2. … :the equivalent of one atom from the eight corners, each of which is shared among eight unit cells, and the single center atom, which is wholly contained within its cell 3. … each center atom has as nearest neighbors its eight corner atoms 4. Since the coordination number is less for BCC than FCC … (Derive APF 0.68 by yourself)

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**Packing Factor – FCC vs BCC**

Left -> FCC (more packed and dense); Right -> BCC

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**Hexagonal Close-Packed Crystal (HCP)**

The top and bottom faces of the unit cell consist of 6 atoms that form regular hexagons and surround a single atom in the center Another plane that provides 3 additional atoms to the unit cell is situated between the top and bottom planes The atoms in this mid-plane have as nearest neighbors atoms in both of the adjacent two planes

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**Hexagonal Close-Packed Crystal (HCP)**

The equivalent of six atoms is contained in each unit cell If a and c represent, respectively, the short and long unit cell dimensions the c/a ratio should be 1.633 The coordination number and the APF for the HCP are the same as for FCC: 12 and 0.74, respectively The HCP metals include cadmium, magnesium, titanium, and zinc, etc … : one-sixth of each of the 12 top and bottom face corner atoms, one-half of each of the 2 center face atoms, and all 3 midplane interior atoms (c/a calculation is bit complicated) … however, for some HCP metals this ratio deviates from the ideal value.

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Density Computations Density of a material can be computed from its crystalline structure n = number of atoms associated with each unit cell A = atomic weight VC = volume of the unit cell NA = Avogadro’s number (6.023 X 1023 atoms/mol) The mole is defined as the amount of substance that contains as many elementary entities (e.g., atoms, molecules, ions, electrons) as there are atoms in 0.012 kg of the isotope carbon-12 (12C).[1] Thus, by definition, one mole of pure 12C has a mass of exactly 12 g. The amount of substance n (in mol) and the number of atoms or molecules contained in it, N, are proportional, and the proportionality constant N/n is known as Avogadro constant. By definition it has the dimension of the inverse of amount of substance (unit mol−1) and its experimentally determined value is (30)×1023 mol−1

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EXAMPLE PROBLEM 3.3 Copper has an atomic radius of nm, an FCC crystal structure, and an atomic weight of 63.5g/mol. Compute its theoretical density and compare the answer with its measured density Solution: The crystal structure is FCC, n = 4 ACu = 63.5g/mol VC = a3 = [2R(2)1/2]3 (For FCC)= 16R3(2)1/2 ; R (atomic radius) = 0.128nm Using the equation: Vc = a^3 = [2R(2)^0.5]^3

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EXAMPLE PROBLEM 3. The literature value for density for Cu is 8.94g/cm3 4. … which is in very close agreement that we have calculated

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**Crystalline and Non Crystalline Materials**

Single Crystal Polycrystalline Materials Anisotropy

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1. Single Crystal For a crystalline solid, when the repeated arrangement of atoms is perfect or extends throughout the entirety of the specimen without interruption, the result is a single crystal If the extremities of a single crystal are permitted to grow without any external constraint, the crystal will assume a regular geometric shape having flat faces Within the past few years, single crystals have become extremely important in many of our modern technologies. … All unit cells interlock in the same way and have the same orientation. Single crystals exist in nature, but they may also be produced artificially. They are ordinarily difficult to grow, because the environment must be carefully controlled … as with some of the gem stones; the shape is indicative of the crystal structure. A photograph of a garnet single crystal is shown … in particular electronic microcircuits, which employ single crystals of silicon and other semiconductors.

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**2. Polycrystalline Materials**

Most crystalline solids are composed of a collection of many small crystals or grains; such materials are termed polycrystalline Initially, small crystals or nuclei form at various positions. These have random crystallographic orientations 2. Various stages in the solidification of a polycrystalline specimen are represented schematically … … as indicated by the square grids

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**2. Polycrystalline Materials**

Growth of the crystallites; the obstruction of some grains that are adjacent to one another Upon completion of solidification, grains having irregular shapes have formed The grain structure as it would appear under the microscope; dark lines are the grain boundaries there exists some atomic mismatch within the region where two grains meet; this area, called a grain boundary 1. The small grains grow by the successive addition from the surrounding liquid of atoms to the structure of each … … The extremities of adjacent grains impinge on one another as the solidification process approaches completion 2. 3. … The crystallographic orientation varies from grain to grain.

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3. Anisotropy The physical properties of single crystals of some substances depend on the crystallographic direction in which measurements are taken This directionality of properties is termed anisotropy The extent and magnitude of anisotropic effects in crystalline materials are functions of the symmetry of the crystal structure … For example, the elastic modulus, the electrical conductivity, and the index of refraction may have different values in the [100] and [111] directions … and it is associated with the variance of atomic or ionic spacing with crystallographic direction. Substances in which measured properties are independent of the direction of measurement are isotropic the degree of anisotropy increases with decreasing structural symmetry—triclinic structures (having three unequal crystal axes intersecting at oblique angles) normally are highly anisotropic Table -> The modulus of elasticity values at [100], [110], and [111] orientations for several materials are presented in the table

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3. Anisotropy For many polycrystalline materials, the crystallographic orientations of the individual grains are totally random. Under these circumstances, even though each grain may be anisotropic, a specimen composed of the grain aggregate behaves isotropically Sometimes the grains in polycrystalline materials have a preferential crystallographic orientation, in which case the material is said to have a “texture.” … Also, the magnitude of a measured property represents some average of the directional values In materials science, texture is the distribution of crystallographic orientations of a polycrystalline sample. A sample in which these orientations are fully random is said to have no texture. If the crystallographic orientations are not random, but have some preferred orientation, then the sample has a weak, moderate or strong texture.

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3. Anisotropy The magnetic properties of some iron alloys used in transformer cores are anisotropic—that is, grains (or single crystals) magnetize in a <100>-type direction easier than any other crystallographic direction Energy losses in transformer cores are minimized by utilizing polycrystalline sheets of these alloys into which have been introduced a “magnetic texture”

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**Numerical Problems Problems 3.2 to 3.19, 3.23 to 3.25, 3.27 to 3.32,**

and 3.37 to 3.43

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