Lecture%2003%20Structure

Lecture%2003%20Structure - Introduction To Materials...

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Unformatted text preview: Introduction To Materials Science, Chapter 1, Introduction atomic level (Chapter 2) tronic structure of individual s that defines interaction among s (interatomic bonding). MECH 340 Engineering Material Structure Structure and Order ic level (Chapters 2 & 3) ngement of atoms in materials the same atoms can have rent properties, e.g. two forms of n: graphite and diamond) roscopic structure (Ch. 4) Objectives • • • • • • • • Short-Range Order versus Long-Range Order Amorphous Materials Crystal Structures: Lattice, Unit Cells, Basis Allotropic or Polymorphic Transformations Points, Directions, and Planes in the Unit Cell Interstitial Sites Crystal Structures of Ionic Materials Covalent Structures Arrangement in Matter (a) Inert monoatomic gases have no regular ordering of atoms: (b,c) Some materials, including water vapor, nitrogen gas, amorphous silicon and silicate glass have short-range order. (d) Metals, alloys, many ceramics and some polymers have regular ordering of atoms/ions that extends through the material. Types of Solids Crystalline material: atoms self-organize in a periodic array Single crystal: atoms are in a repeating or periodic array over the entire extent of the material Polycrystalline material: comprised of many small crystals or grains Amorphous: lacks a systematic atomic arrangement Polycrystalline Material Comprised of many small crystals or grains. G Polycrystalline Materials Single Crystals and rains have different cr ystallographic orientation. Atomic mismatch within the regions where grains meet. Single crystal: atoms are in a repeating or periodic array These regions are called grain boundaries. over the entire extent of the material Introduction To Materials Science, Chapter 3, The structure of crystalline solids Polycrystalline material: comprised of many small crystals or grains. The grains have different crystallographic orientation. There exist atomic mismatch within the regions where grains meet. These regions are called grain boundaries. Introduction To Materials Science, Chapter 3, The structure of crystalline solids Polycrystalline Materials Grain Boundary University of Virginia, Dept. of Materials Science and Engineering 18 Simulation of annealing of a polycrystalline grain structure University of Virginia, Dept. of Materials Science and Engineering 20 Liquid Cr ystal Display These materials are amorphous in one state and undergo localized crystallization in response to an external electric field and are widely used in liquid crystal displays. Amorphous Materials Amorphous materials - Materials, including glasses, that have no long-range order, or crystal structure. Glasses - Solid, non-crystalline materials (typically derived from the molten state) that have only short-range atomic order. Glass-ceramics - A family of materials typically derived from molten inorganic glasses and processed into crystalline materials with very fine grain size and improved mechanical properties. Lattice , Unit Cells Lattice - A collection of points that divide space into smaller equally sized segments. Basis - A group of atoms associated with a lattice point. Unit cell - A subdivision of the lattice that still retains the overall characteristics of the entire lattice. Atomic radius - The apparent radius of an atom, typically calculated from the dimensions of the unit cell, using close-packed directions (depends upon coordination number). Packing factor - The fraction of space in a unit cell occupied by atoms. Bravais Lattice Lattice Parameters Coordination Number (a) SC and (b) BCC unit cells. Six atoms touch each atom in SC, while the eight atoms touch each atom in the BCC unit cell. Introduction To Materials Science, Chapter 3, The structure of crystalline solids Intr oduc ion To aterials Science, Chapter 3, The s , The s of crysta line solids Introductiton To MMaterials Science, Chapter 3tructuretructure lof crystalline solids Introduction To Materials Science, Chapter 3, The structure of crystalline solids Body-Centered Cubic Crystal Structure (II) Body-Centered C Cubic Crystal Structure Body-Centeredubic Crystal Structure (II) (II) Body-Centered Cubic Crystal Structure (II) Introduction To Materials Science, Chapter 3, The structure of crystalline solids Body-Centered Cubic (BCC) Crystal Structure (I) Atom at each corner and at center of cubic unit cell Cr, !-Fe, Mo have this crystal structure Body Centered Cube Body-Centered Cubic (BCC) Crystal Structure (I) Atom at each corner and at center of cubic unit cell Cr, !-Fe, Mo have this crystal structure aa ! The hard spheres touch one another along cube diagonal aa ! The hard spheres touch one another along cube diagonal ! the cube edge length, a= 4R/!3 ! ! theard spheres touch one ne4R/!3 long cubecdiagonal onal The h ard sphereslength, o =nother a along ube diag ! The h cube edge touch a a another ! the cube edge length, a= 4R/ R 3 ! the cube edge number, CN = ! The coordinationlength, a= 4! /!38 ! Number of atoms per unit cell, n = 2 ! The coordination number, CN = 8 ! = ! The coordination number, iCNell,8n = 1 The ber of a(t1) shared y ! Numcoordination perbunotother cells:2 x 1 = 1 Center atom oms number,cCN = 8 ! Number oftoms shared by it cell,ells: 8 x 1/8 : 1 8 corner o at oms p u ni Center a f aoms perernuight con = 2 2 other ! Numberatom t(1) sharedebytncell, n =cells= 1 x 1 = 1 Center atom (1) shared by no other cells: 1 x 1 = 1 8 corneratom 1) shared b = n.68 cells: 8 x /8 1 = ! Atomic packing(s shared by y ightother cells:11 x = 1 1 Center atom factor, APF e 0 c 8 corner atoms shared by eight o ells: 8 x 1/8 = 1 8 mic packing factor, y eight c ! Corner and atom shared APF = 0.68 ! Atocorner centersatoms arebequivalentells: 8 x 1/8 = 1 ! Atom and center factor, are equivalent ! Corneric packingatommareAPF = 0.68 ! Corner and center ato s s equivalent ! Atomic packing factor, APF = 0.68 Introduction To Materials Science, Chapter 3, The structure of crystalline solids ! Corner and center atoms are equivalent Body-Centered Cubic Crystal Structure (II) University of Virginia, Dept. of Materials Science and Engineering 10 University of Virginia, Dept. of Materials Science and Engineering 9 Universiiy f Virg g ia, D Dept. of Materials Science and Engin Universttyoof Virininia, ept. of Materials Science and Engineeringeering 10 10 University of Virginia, Dept. of Materials Science and Engineering University of Virginia, Dept. of Materials Science and Engineering 9 10 a ! The hard spheres touch one another along cube diagonal ! the cube edge length, a= 4R/!3 ! The coordination number, CN = 8 Atomic Packing Factor • APF for a body-centered cubic structure = 0.68 Close-packed directions: length = 4R ! ! = 3a R a Unit cell contains: 1 + 8 x 1/8 = 2 atoms/unit cell atoms volume 4 π ( 3a/4)3 2 unit cell atom 3 APF = volume a3 unit cell Introduc t on To Material Science, Chapter 3 T T st structure of crystal sol ss Introductiion To Materials s Science, Chapter, 3,heheructure of crystallinelineidolids Introduction To Materials Science, Chapter 3, The structure of crystalline solids Introduction To Materials Science, Chapter 3, The structure of crystalline solids Introduction To Materials Science, Chapter 3, The structure of crystalline solids Face-Centered Cubic Crystal Structure (II) Face-CenteredCubic Crystal Structure (II) Face-Centered Cubic Crystal Structure (II) Face-Centered Cubic (FCC) Crystal Structure (I) Face-Centered Cubic Crystal Structure (II) ! Atoms are located at each of the corners and on the centers oRall the faces of cubic unit cell f R ! Cu, Al, Ag, Au have this crystal structure Face Centered Cube a R R a ! The hard spheres or ion cores touch one another across a ! The hard spheres orcubecores length,one2R!2 across a face diagonal ! the ion edge touch a=aanother aa face diagonal ! the cube edge !2 ! The hard spheres or ion cer, CtN length, ua=b2Rofaclosss a ! The hard spheres or ion cores touch onehene m aer ss a est The coordination numb ores ouch another cro cro = t o n nother ! ace iagonal ! the cube edge length, a 2 a n ! fThedciagonal !hichnumbedge length, R=2 = ber of closest fneighbors to w the cubeaer, CiN bonded um!2 ber of ace d oordination an tom s ==the ! 2R num t eighbors t m which 12 nouching atoo s, CN = an a CN = bonded = n closest ! The coordination number,tom is the number ofumber of ! The coordination number, CN = the number of closest t eighbors atoa whichn =a 12n i bionded= = .n (Fonum at o eighbors a ! nouching tofowtoms apertom it scell, bonded umber aofberomf Number ot ms, CN an u tom s n 4 = r n hich ouching atoms, CN m 2 12 ouching to toms a that i beraofmw CN = djacent cell, n = 4. nl o an a a ! tNums sharedas,ith = 1per unit unit cells, we o(Fy rcount tom fraction f atom a ) ell, u = cn that er of the with mu djacent n nit4. ells, ae o oay a ! Numbs sharedtoms ,p1/mu. cIn FCC unit 4ell (we have: tom ! Numiberoofatoms per ernit nit cell, n c=(ForwFoartnlm count a a . n tfraction of the atomdjacent .two cells, ells,xnlIntrnly3aiountoa aterials Science, Chapter 3, The structure of crystalline solids hat iiace atomithsm m, adjacent uFCC w6 o wy/2ount c on T M hared w s hared by In nit c e 1 o we t 6 fsssshared with a 1/m) unit cells:unit ce c oduchave: ell = that f 8 corner atoms s ). In y eight cell we have: = fraction ofatomato,mhared .bFCC unit cells: ellx /2 have:1 raction of the sm hared byIn FCCells: 6 8 1 1/8= 3 , 1/m 6 face the ato 1/m ) two c unit c x we 6 face atoms shared by two cells: 6 x 1/2 = 3 6 fmic atoms shared by by ecells: ells: /2x 1/8 = 3 8 ace a atoms shared wo ight 8 1/8 Face-Centered Cubic Crystal Structure (II) ! Atocornerackinghared by etightFcells: c6xx 18 of= volume1 8 corner p toms s factor, AP = fraction = 1 Two8 cornertationss ard spheresight cells: 8 xo1/8atomic represen by occupied atom h shared by e = (Sum f =1 ! tAtomicpacking factor, AP= A= Fraction ofm olume volume of AtomCC upacking of cell) F 0.74 (maximu vpossible) he F ices)/(Volume factor, P f = fraction of nit cell ! volum occupied o occupied ackingh factor,pheres Sum (Sum of fv atom ss = of ! Atomic pbybyhardardpheresAPF (== fraction atomicolume ic Univers ia, of f 0.74 v ccupied ityb umeinardept.cell) = 0 SaximuSum umf g atom7 volumes)/(Vo irghof D opheres(m cience a axim o oolumes)/(VoolfyVlumecell)s=Materials.74 (mnd Engineerinpossible) = ( m possible) ic volumes)/(Volumeept. ofcell) als Science a(maximug possible) of Materi = 0.74 nd Engineerinm University of Virginia, D 7 University of Virginia, Dept. of Materials Science and Engineering 7 University of Virginia, Dept. offMaterialssSScience and Engineering University of Virginia, Dept. o Material cience and Engineering R 67 a ! The hard spheres or ion cores touch one another across a face diagonal ! the cube edge length, a= 2R!2 Atomic Packing Factor • APF for a body-centered cubic structure = 0.74 Close-packed directions: length = 4R ! ! = 2a Unit cell contains: 6 x 1/2 + 8 x 1/8 = 4 atoms/unit cell a atoms volume 4 3 π ( 2a/4) 4 unit cell atom 3 APF = volume a3 unit cell ! Six atoms form regular hexagon, surrounding one atom in center. Another plane is situated halfway up unit cell (c-axis), with 3 additional atoms situated at interstices of hexagonal (close-packed) planes ! Cd, Mg, Zn, Ti have this crystal structure Introduction To Materials Science, Chapter 3, The structure of crystalline solids Hexagonal Close-Packed Crystal Structure (I) Hexagonal Closed Packing Introduction To Materials Science, Chapter 3, The stru Hexagonal Close-Packed Crysta ! HCP is one more common structure of metallic crystals ! Six atoms form regular hexagon, surrounding one atom in center. Another plane is situated halfway up unit cell (c-axis), with 3 additional atoms situated at interstices of hexagonal (close-packed) planes Introduction To Materials Science, Chapter 3, The structure of crystalline solids ! Introduction Zn, aterihave nce, ChaptertalThe structure of crystalline solids Cd, Mg, To M Ti als Scie this crys 3, structure Introduction To M Hexagonal aterialcieSnce,nce, Chapterhe ,stThe urructcurettructureso(iII) C s S s cie Chapter 3, T 3 ruct st e of S of crystalline l ds Introduction To Materiallose-Packed Crystal rys alline solids Hexagonal Close-Packed Crystal Structure (II) Hexagonal CClose-Packed Crystal Structure (II) exagonal !H Unit cell haslose-PackedpCrystal Structure. (II) ratio two lattice arameters a and c Ideal ! HCP is one more common structure o ! Six atoms form regular hexagon, su in center. Another plane is situated (c-axis), with 3 additional atoms situ hexagonal (close-packed) planes ! Cd, Mg, Zn, Ti have this crystal struc ! Unit cell has two lattice parameters a and c. Ideal ratio ! Unit=cellhhaswo lattice parpmeters ters a c.nddeal Ideal ratio Unitc 1.633 ! c/a = ell as t two lattice aarame a and a I c. ratio c/a 1.633 c/a =1.633 1.633 c/a = coordination number, CN = 12 (same as in FCC) ! The ! The coordination number, CN = 12 (same as in FCC) ! The coordination number, CNCN =(same as in as in FCC) The coordination number, = 12 12 (same FCC) ! Number of attoms per uunt tcell, nn = .6. ! Number of a oms per ni i cell, = 6 ! Numiber oof toms psper it nit cell, n.t= 6. cells: 3 x 1 = 3 University of Virginia, Dept. of Materials Science and Engineering ! Numd-planeaatomersharedell, nn= 6 her ber f a toms un u 3 mid-plane atoms sharedcbbynoo ther cells: 3 x 1 = 3 3 mid-plane atoms shared by ny otherocells: 3 x 1 = 3 m o 3 o 3 hexagonal corner aatoms hared oby 6 cells: 32 x/ 1= 2 12mid-plane atoms sharedsby no byher ells: 12 x 116 /6 3= 2 gonal corner toms shared t 6 c cells: 1 x 12 hexa 12 hexagonal corner atoms shared by 6 cells: 12 x 1/6 = 2 = 12 op/bottom p corner aer aatom s shared6by 2ells: 2 x 1/2 = = 2 top/bogonalplane ccentomtoms s hared by cells:ells:x 1/6 /2 12 1 2 t hexa ttom lane ent ter s shared by 2 c c 1 2 2 x 1 = 2 top/bottom plane center atoms shared by 2 cells: 2 x 1/2 = 1 11 ! ! ! ! ! ! Atomic packing factor, APF .74 0.74 (sa e eFCC) Atomic packing factor, APF 0 (sa (sam in in FCC) Atomic packing factor, APF = 0== .74me asmasas in FCC) Atomic packing factor, APF = 0.74 (same as in FCC) All atoms s are equivalent All atom are equivalent are equivalent All atoms are equivalent 2 top/bottom plane center atoms shared by 2 cells: 2 x 1/2 = 1 University of Virginia, Dept. of Materials Science and Engineering 11 University of Virginia, Dept. of Materials Scienc c Close Packed Structure 3D Projection FCC ABCABC... Stacking Sequence A A sites B sites C sites A B B C B C B 2D Projection B C B B A B C HCP ABAB... Stacking Sequence A sites B sites A sites Top layer Middle layer Bottom layer of an atom, M) / (the volume of the cell, Vc) Atoms in the unit cell, n = 2 (BCC); 4 (FCC); 6 (HCP) Mass of an atom, M = Atomic weight, A, in amu (or g/mol) Introduc n is given tiion TtohMaterials Science,tChapter 3,To sttranslaterystaass sflrom amu e periodic able. The ructure of c m lline o ids to grams we have to divide the atomic weight in amu by Density Computations the Avogadro number NA = 6.023 " 1023 atoms/mol Since the entire crystal can be generated by the repetition Density Computation The volume oftthedcell, y of= a3rystalline maBCC) ! = the of the unit cell, he ensit Vc a c (FCC and terial, density ofRhe u(FCC); = aatoms /i#3 he unit cell, n ) " (mass a = 2 t #2 nit cell ( = 4R n t (BCC) of an atom, R )s/ t(theavolumerof the cell, Vc) where Mi he tomic adius Atomoduct on o Materi forcie= CBCC); T4 structure f (HCP) s Intr in formul cell, nce, hapter 3 he line Thus,stheitheTunit a als Snthe2d(ensity, is: (FCC); o6crystalnAolids Mass of an atom, M = Atomicomputations mu (or g/mol) Vc N A Density C weight, A, in a is given in the p# atoms/unit cell eriodic table. To translate mass from amu Atomic weight (g/mol) Since thewe havecrysdividanthe agenerated by tn amu by entire to tal c e be tomic weight ihe repetition to grams AtoAvogadro number ensityic fra crystallineany terial,ents tyou of m u w cell, and a A = .023 " 10 o m m mol thetheic nit eight the d Ntom6o adius 23fatoms/a elem ! = he != ρ = nk A cover. can findointheeutableell = (atoms of the unit cell, n ) " (mass density f th nit c at the back in the textboo front The volume of ) he(the volume of the and BCC)cNA cell, V = a (FCC cell, ) of an atom, M t /Volume/unit cell VV Avogadro's number a = 2R#2 (FCC); 3 = 4R/#3 (BCC) a (cmmic radius /unit cell) where R i the cell, Atoms in the sunit ato n = 2 (BCC); 4 (FCC); 6 (HCP) 3 c University of Virginia, Dept. of Materials Science ac d Engineering n (6.023 x 1023 atoms/mol) 16 Thus, theaformula, for = Adensityws: nA Mass of n atom M the tomic i eight, A, in amu (or g/mol) Atomic weight and atomic radius of many elements you can find in the table at the back of the textbook front cover. 3 is given in the periodic table. To translate mass from amu Vc N A to grams we have to divide the atomic weight in amu by the Avogadro number NA = 6.023 " 1023 atoms/mol != The volume of the cell, Vc = a (FCC and BCC) a =Universi2 ofFCgCia, Dept.=f 4R/#als Science and Engineering 2R#ty ( Vir in); a o Materi 3 (BCC) where R is the atomic radius Thus, the formula for the density is: 16 != nA VN Miller Indices • • • • • Miller indices - A shorthand notation to describe certain crystallographic directions and planes in a material. Denoted by [ ] brackets. A negative number is represented by a bar over the number. Directions of a form - Crystallographic directions that all have the same characteristics, although their ‘‘sense’’ is different. Denoted by h i brackets. Repeat distance - The distance from one lattice point to the adjacent lattice point along a direction. Linear density - The number of lattice points per unit length along a direction. Packing fraction - The fraction of a direction (linear-packing fraction) or a plane (planar-packing factor) that is actually covered by atoms or ions. Crystallographic Directions Direction A 1. Two points are 1, 0, 0, and 0, 0, 0 2. 1, 0, 0, -0, 0, 0 = 1, 0, 0 3. No fractions to clear or integers to reduce 4. [100] Direction B 1. Two points are 1, 1, 1 and 0, 0, 0 2. 1, 1, 1, -0, 0, 0 = 1, 1, 1 3. No fractions to clear or integers to reduce 4. [111] Direction C 1. Two points are 0, 0, 1 and 1/2, 1, 0 2. 0, 0, 1 -1/2, 1, 0 = -1/2, -1, 1 3. 2(-1/2, -1, 1) = -1, -2, 2 Directions and Repeat Distance Miller Indices of Planes Plane A 1. x = 1, y = 1, z = 1 2.1/x = 1, 1/y = 1,1 /z = 1 3. No fractions to clear 4. (111) Plane B 1. The plane never intercepts the z axis, so x = 1, y = 2, and z = 2.1/x = 1, 1/y =1/2, 1/z = 0 3. Clear fractions: 1/x = 2, 1/y = 1, 1/z = 0 4. (210) Plane C 1. We must move the origin, since the plane passes through 0, 0, 0. Let’s move the origin one lattice parameter in the y-direction. Then, x = , y = -1, and z = 2.1/x = 0, 1/y = 1, 1/z = 0 3. No fractions to clear. Exercise Draw (a) the cubic unit cell. direction and (b) the plane in a Exercise Selected Elements Element Aluminum Argon Barium Beryllium Boron Bromine Cadmium Calcium Carbon Cesium Chlorine Chromium Cobalt Copper Flourine Gallium Germanium Gold Helium Hydrogen Symbol Al Ar Ba Be B Br Cd Ca C Cs Cl Cr Co Cu F Ga Ge Au He H At. Weight (amu) 26.98 39.95 137.33 9.012 10.81 79.90 112.41 40.08 12.011 132.91 35.45 52.00 58.93 63.55 19.00 69.72 72.59 196.97 4.003 1.008 Density (g/cm3) 2.71 -----3.5 1.85 2.34 -----8.65 1.55 2.25 1.87 -----7.19 8.9 8.94 -----5.90 5.32 19.32 ----------- Crystal Structure FCC -----BCC HCP Rhomb -----HCP FCC Hex BCC -----BCC HCP FCC -----Ortho. Dia. cubic FCC ----------- Atomic radius (nm) 0.143 -----0.217 0.114 ----------0.149 0.197 0.071 0.265 -----0.125 0.125 0.128 -----0.122 0.122 0.144 ----------- Interstitial Sites Interstitial sites - Locations between the ‘‘normal’’ atoms or ions in a crystal into which another - usually different - atom or ion is placed. Typically, the size of this interstitial location is smaller than the atom or ion that is to be introduced. Cubic site - An interstitial position that has a coordination number of eight. An atom or ion in the cubic site touches eight other atoms or ions. Octahedral site - An interstitial position that has a coordination number of six. An atom or ion in the octahedral site touches six other atoms or ions. Tetrahedral site - An interstitial position that has a coordination number of four. An atom or ion in the tetrahedral site touches four other atoms or ions. Interstitial Sites Example: Sites for Carbon in Iron In FCC iron, carbon atoms are located at octahedral sites at the center of each edge of the unit cell (1/2, 0, 0) and at the center of the unit cell (1/2, 1/2, 1/2). In BCC iron, carbon atoms enter tetrahedral sites, such as 1/4, 1/2, 0. The lattice parameter is 0.3571 nm for FCC iron and 0.2866 nm for BCC iron. Assume that carbon atoms have a radius of 0.071 nm. (1) Would we expect a greater distortion of the crystal by an interstitial carbon atom in FCC or BCC iron? (2) What would be the atomic percentage of carbon in each type of iron if all the interstitial sites were filled? (a) The location of the ¼, ½, 0 interstitial site in BCC metals, showing the arrangement of the normal atoms and the interstitial atom (b) ½, 0, 0 site in FCC metals. (c) Edge centers and cube centers are some of the interstitial sites in the FCC structure . Solution 1. We could calculate the size of the interstitial site at the 1/4, 1/2, 0 location with the help of Figure (a). The radius RBCC of the iron atom is: From Figure (a), we find that: For FCC iron, the interstitial site such as the 1/2, 0, 0 lies along directions. Thus, the radius of the iron atom and the radius of the interstitial site are [Figure(b)]: 2. We can find a total of 24 interstitial sites of the type 1/4, 1/2, 0; however, since each site is located at a face of the unit cell, only half of each site belongs uniquely to a single cell. Thus: (24 sites)(1/2) = 12 interstitial sites per unit cell Atomic percentage of carbon in BCC iron would be: In FCC iron, the number of octahedral interstitial sites is: (12 edges) (1/4) + 1 center = 4 interstitial sites per unit cell Atomic percentage of carbon in BCC iron would be: Crystal Structure of Ionic Materials Factors need to be considered in order to understand crystal structures of ionically bonded solids: Ionic Radii Electrical Neutrality Connection between Anion Polyhedra Visualization of Crystal Structures Using Computers Charge Neutrality/Ionic Radii • Charge Neutrality: Net charge in the structure should be zero CaF2: Ca2+ + cation Fanions F- AmXp m, p determined by charge neutrality • Stable structures: maximize the # of nearest oppositely charged neighbors. - + - - + - - + - unstable stable stable Coordination Number • Coordination # increases with Issue: How many anions can you arrange around a cation? rcation ranion < .155 .155-.225 .225-.414 .414-.732 .732-1.0 Coord # 2 3 4 6 8 NaCl (sodium chloride) CsCl (cesium chloride) rcation ranion ZnS (zincblende) Example Calculation C.N. 2 rC/rA The critical ratio can be determined by simple geometrical 3 analysis <0.155 0.155-0225 Introduction to Materials Science, Chapter 13, Structure and Properties of Ceramics Geometry 4 30° 0.225-0.414 6 0.414-0.732 Cos 30°= 0.866 = R/(r+R) ! r/R = 0.155 8 0.732-1.0 University of Virginia, Dept. of Materials Science and Engineering 5 FeO example • On the basis of ionic radii, what crystal structure would you predict for FeO? Cation Ionic radius (nm) Coord # 0.053 2+ 0.077 < Fe .155 2 Fe3+ 0.069 .155-.225 3 Ca2+ 0.100 rcation Al3+ ranion .225-.414 4 6 8 (zincblende) rcation 0 .077 = ranion 0.140 NaCl = 0 .550 (sodium chloride) • AZnSer : nsw Anion .414-.732 O2Cl.732-1.0 F- 0.140 0.181 0.133 based on this ratio, --coord # = 6 CsCl --structure = NaCl (cesium chloride) AmXp Structures • Consider CaF2 : rcation 0 .100 = ≈ 0.8 ranion 0 .133 • Based on this ratio, coord # = 8 and structure = CsCl. rcation • Result: CsCl structure w/only half the cation sites ZnS Coord # (zincblende) ranion < .155 occupied. 2 3 4 6 8 .155-.225 .225-.414 .414-.732 .732-1.0 • Only half the cation sites are occupied since NaCl #Ca2+ ions = 1/2 (sodium # F- ions. chloride) CsCl (cesium chloride) Anion-Cation Connections NaCl (a) The cesium chloride structure, a SC unit cell with two ions (Cs+ and CI-) per lattice point. (b) The sodium chloride structure, a FCC unit cell with two ions (Na+ + CI-) per lattice point. Note: Ion sizes not to scale. Example For potassium chloride (KCl), (a) verify that the compound has the cesium chloride structure and (b) calculate the packing factor for the compound. rcation Coord # SOLUTION ranion 3 rK+/ rCl- = 0.133/0.181 = 0.735 4 ZnS (zincblende) 2 a.< rom your Book, rK+ = 0.133 nm and rCl- = 0.181 nm, so: F.155 .155-.225 .225-.414 NaCl (sodium chloride) .414-.732 6 < 0.735 < 1.000, the coordination number for Since 0.732 CsCl each type of ion is eight and the CsCl structure is likely. (cesium chloride) .732-1.0 8 b. The ions touch along the body diagonal of the unit cell, so: a0 = 2rK+ + 2rCl- = 2(0.133) + 2(0.181) = 0628 nm a0 = 0.363 nm Example 2 Show that MgO has the sodium chloride crystal structure and calculate the density of MgO. rcation SOLUTION Coord # ranion ZnS (zincblende) From your Book, rMg+2 = 0.066 nm and rO-2 = 0.132 nm, so: < .155 2 .155-.225 .225-.414 .414-.732 .732-1.0 3 4 6 8 rMg+2 /rO-2 = 0.066/0.132 = 0.50 NaCl (sodium chloride) Since 0.414 < 0.50 < 0.732, the coordination number fCsCl or each (cesium ion is six, and the sodium chloride structure is possible. chloride) Density Calculation: The atomic masses are 24.312 and 16 g/mol for magnesium and oxygen, respectively. The ions touch along the edge of the cube, so: a0 = 2 rMg+2 + 2rO-2 = 2(0.066) + 2(0.132) = 0.396 nm = 3.96 × 10-8 cm Introduction to Materials Science, Chapter 13, Structure and Properties of Ceramics More examples of crystal structures in ceramics (will not be included in the test) Zinc Blende Structure: typical for compounds where covalent bonding dominates. C.N. = 4 ZnS, ZnTe, SiC have this crystal structure Zinc Blende University of Virginia, Dept. of Materials Science and Engineering 8 ZnS, ZnTe, SiC have this crystal structure Introduction to Materials Science, Chapter 13, Structure and Properties of Ceramics More examples of crystal structures in ceramics (will not be included in the test) Fluorite (CaF 2): rC = rCa = 0.100 nm, rA = rF = 0.133 nm ! rC/rA = 0.75 From the table for stable geometries we see that C.N. = 8 Fluorite FCC structure with 3 atoms per lattice point University of Virginia, Dept. of Materials Science and Engineering 9 Perovskite Corundum Corundum structure of alpha-alumina (α-AI203). Structure of Covalently Bounded Material Covalently bonded materials frequently have complex structures in order to satisfy the directional restraints imposed by the bonding. Diamond cubic (DC) - A special type of face-centered cubic crystal structure found in carbon, silicon, and other covalently bonded materials. Diamond Structure (a) Tetrahedron and (b) the diamond cubic (DC) unit cell. This open structure is produced because of the requirements of covalent bonding. Polymer Microstructure • Polymer = many mers mer HHHHHH CCCCCC HHHHHH Polyethylene (PE) mer HHHHHH CCCCCC H Cl H Cl H Cl Polyvinyl chloride (PVC) mer HHHHHH CCCCCC H CH3 H CH3 H CH3 Polypropylene (PP) Adapted from Fig. 14.2, Callister 6e. • Covalent chain configurations and strength: secondary bonding Linear Branched Cross-Linked Network Direction of increasing strength Molecular Weight & Crystallinity • Molecular weight, Mw: Mass of a mole of chains. smaller Mw larger Mw • Tensile strength (TS): --often increases with Mw. --Why? Longer chains are entangled (anchored) better. • % Crystallinity: % of material that is crystalline. --TS and E often increase with % crystallinity. crystalline region --Annealing causes crystalline regions amorphous to grow. % crystallinity region increases. ngly bonded carbon atoms is e singly bonded carbon atoms is form a zigzag pattern in a s form a zigzag pattern in a ing taining nds n bonds tate rotate nds bonds are nds are s coils e like re: Molecular Shape Introduction tooMaterials Science, Chapter 15, Polymer Structures Introduction t Materials Science, Chapter 15, Polymer Structures Polymer Crystals Polymer Introduction to Materials Science, Chapter 15, Polymer Structures Thin ccrystallineplatelets grown from solution - chains ffold - chains old Thin rystalline platelets grown Polymer bbackaandfforth: chain--folded model ack nd orth: chain folded Crystallinity (I) Atomic arrangement in polymer crystals is more complex than in metals or ceramics (unit cells are tPolyethylene ypically large Polyethylene and complex). Polyethylene The average chain length is much greater than the The average chain length is much greater than the partially crystalline (semi thickness of the crystallite thickness of the crystallite crystalline crystalline), with crystalline University of Virginia, Dept. of Materials Science and Engineering University of Virginia, Dept. of Materials Science and Engineering Polymer molecules are often region 24 24 amorphous region aterials Science and Engineering t. of Materials Science and Engineering 12 12 regions dispersed amorphous material. within University of Virginia, Dept. of Materials Science and Engineering 21 Thermoplastics vs. Thermosets • Thermoplastics: T --little cross linking --ductile --soften w/heating --polyethylene (#2) polypropylene (#5) polycarbonate polystyrene (#6) mobile liquid viscous liquid Callister, rubber Fig. 16.9 tough plastic Tm Tg crystalline solid partially crystalline solid • Thermosets: --large cross linking (10 to 50% of mers) --hard and brittle --do NOT soften w/heating --vulcanized rubber, epoxies, polyester resin, phenolic resin Molecular weight Anisotropy Different directions in a crystal have a different packing. For instance, atoms along the edge of FCC unit cell are more separated than along the face diagonal. This causes anisotropy in the properties of crystals, for instance, the deformation depends on the direction in which a stress is applied. In some polycrystalline materials, grain orientations are random, so bulk material properties are isotropic Some polycrystalline materials have grains with preferred orientations (texture), so properties are dominated by those relevant to the texture orientation and the material exhibits anisotropic properties Polymorphism • Demonstrates "polymorphism" Temperature, C 1536 1391 longer heat up FCC Stable Liquid BCC Stable The same atoms can have more than one crystal structure. 914 Tc 768 BCC Stable cool down shorter! longer! magnet falls off shorter Polymorphism and Allotropy Introduction To Materials Science, Chapter 3, The structure of crystalline solids Some materials may aterials in morxist in more crystal structure, this is called Some m exist may e e than one than one crystal polymorphism.tructure, tmaterial is olymorphism. If tsolid, ital iscalled allotropy. An s If the his is called p an elemental he materi is an elemental is carbon, which can example of allotropysolid, it is called allotropy. exist as diamond, graphite, and An example of allotropy is carbon, which can exist as amorphous carbon. diamond, graphite, and amorphous carbon. Polymorphism and Allotropy Pure, solid carbon occurs in three crystalline forms – diamond, graphite; and large, hollow fullerenes. Two kinds of fullerenes Pure, solid carbon noccursuckmthreee lcrystalline formsc–diamond, are show here: b in inst rful erene (buckyball) and arbon n large graphite; andanotube., hollowfullerenes. Two kinds of fullerenes are University f Virg (buckyball) and d Engineerin 17 here: buckminsterfulleroeneinia, Dept. of Materials Science ancarbong nanotube. shown Terms Allotropy Amorphous Anisotropy Atomic packing factor (APF) Body-centered cubic (BCC) Coordination number Crystal structure Crystalline Face-centered cubic (FCC) Grain Grain boundary Hexagonal close-packed (HCP) Isotropic Lattice parameter Non-crystalline Polycrystalline Polymorphism Single crystal Unit cell Assignment #1 Reading: Chapter 2 Assignment Problems: ...
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