Materials Science. Energy and Packing. Materials and Packing. Chapter 3: Structures of Metals & Ceramics ISSUES TO ADDRESS...

Transcription

1 In the Nme of God Chpter : Structures of Metls & Cermics ISSUES TO ADDRESS... Mterils Science Dr Mohmmd Heidri-Rrni How do toms ssemble into solid structures? How does the density of mteril depend on its structure? How do the crystl structures of cermic mterils differ from those for metls? When do mteril properties vry with the smple (i.e., prt) orienttion? Chpter - 1 Chpter - 2 Non dense, rndom pcking Dense, ordered pcking Energy nd Pcking typicl neighbor bond energy typicl neighbor bond energy Energy Energy typicl neighbor bond length typicl neighbor bond length Dense, ordered pcked structures tend to hve lower energies. r r Chpter - Mterils nd Pcking Crystlline mterils... toms pck in periodic, D rrys typicl of: -metls -mny cermics -some polymers Noncrystlline mterils... toms hve no periodic pcking occurs for: -complex structures -rpid cooling "Amorphous" = Noncrystlline crystlline SiO2 Adpted from Fig..40(), Si Oxygen noncrystlline SiO2 Adpted from Fig..40(b), Chpter - 4 ١

2 Metllic Crystl Structures How cn we stck metl toms to minimie empty spce? 2-dimensions vs. Metllic Crystl Structures Tend to be densely pcked. Resons for dense pcking: - Typiclly, only one element is present, so ll tomic rdii re the sme. - Metllic bonding is not directionl. - Nerest neighbor distnces tend to be smll in order to lower bond energy. - Electron cloud shields cores from ech other Hve the simplest crystl structures. We will exmine three such structures... Now stck these 2-D lyers to mke -D structures Chpter - 5 Chpter - 6 Simple Cubic Structure (SC) Rre due to low pcking density (only Po hs this structure) Close-pcked directions re cube edges. (Courtesy P.M. Anderson) Coordintion # = 6 (# nerest neighbors) Chpter - 7 APF for simple cubic structure = 0.52 Atomic Pcking Fctor (APF) APF = Volume of toms in unit cell* Volume of unit cell *ssume hrd spheres close-pcked directions contins 8 x 1/8 = 1 tom/unit cell Adpted from Fig..42, toms unit cell R=0.5 APF = 1 volume 4 π (0.5) tom volume unit cell Chpter - 8 ٢

3 Body Centered Cubic Structure (BCC) Atoms touch ech other long cube digonls. --Note: All toms re identicl; the center tom is shded differently only for ese of viewing. ex: Cr, W, Fe (α), Tntlum, Molybdenum Coordintion # = 8 2 toms/unit cell: 1 center + 8 corners x 1/8 Chpter - 9 unit cell APF = Atomic Pcking Fctor: BCC APF for body-centered cubic structure = 0.68 toms R 2 4 π ( /4) volume tom volume unit cell 2 Close-pcked directions: length = 4R = Chpter - 10 Fce Centered Cubic Structure (FCC) Atoms touch ech other long fce digonls. --Note: All toms re identicl; the fce-centered toms re shded differently only for ese of viewing. ex: Al, Cu, Au, Pb, Ni, Pt, Ag Coordintion # = 12 4 toms/unit cell: 6 fce x 1/2 + 8 corners x 1/8 Chpter - 11 Atomic Pcking Fctor: FCC APF for fce-centered cubic structure = 0.74 mximum chievble APF 2 toms unit cell APF = Close-pcked directions: length = 4R = 2 Unit cell contins: 6 x 1/2 + 8 x 1/8 = 4 toms/unit cell 4 4 π ( 2/4) volume tom volume unit cell Chpter - 12 ٣

4 ABCABC... Stcking Sequence 2D Projection A sites B sites C sites FCC Unit Cell FCC Stcking Sequence B B C A B B B C C B B A B C Chpter - 1 Hexgonl Close-Pcked Structure (HCP) ABAB... Stcking Sequence D Projection c Coordintion # = 12 APF = 0.74 c/ = 1.6 A sites B sites A sites 2D Projection 6 toms/unit cell Top lyer Middle lyer Bottom lyer ex: Cd, Mg, Ti, Zn Chpter - 14 Theoreticl Density, ρ Theoreticl Density, ρ where Density = ρ = ρ = Mss of Atoms in Unit Cell Totl Volume of Unit Cell n A V C N A n = number of toms/unit cell A = tomic weight V C = Volume of unit cell = for cubic N A = Avogdro s number = x 10 2 toms/mol Chpter - 15 Adpted from Fig..2(), Cllister & Rethwisch e. volume unit cell toms unit cell ρ = R x 10 2 Ex: Cr (BCC) A = g/mol R = nm n = 2 toms/unit cell = 4R/ = nm g mol ρ theoreticl ρ ctul toms mol = 7.18 g/cm = 7.19 g/cm Chpter - 16

5 Atomic Bonding in Cermics Bonding: -- Cn be ionic nd/or covlent in chrcter. -- % ionic chrcter increses with difference in electronegtivity of toms. Degree of ionic chrcter my be lrge or smll: CF 2 : lrge SiC: smll Cermic Crystl Structures Oxide structures oxygen nions lrger thn metl ctions close pcked oxygen in lttice (usully FCC) ctions fit into interstitil sites mong oxygen ions Adpted from Fig. 2.7, (Fig. 2.7 is dpted from Linus Puling, The Nture of the Chemicl Bond, rd edition, Copyright 199 nd 1940, rd edition. Copyright 1960 by Cornell University. Chpter - 17 Chpter - 18 c0tf02 Chpter - Fctors tht Determine Crystl Structure 1. Reltive sies of ions Formtion of stble structures: --mximie the # of oppositely chrged ion neighbors unstble stble stble 2. Mintennce of Chrge Neutrlity : --Net chrge in cermic CF2: C 2+ + ction should be ero. --Reflected in chemicl formul: AmXp Adpted from Fig..4, F - nions F - m, p vlues to chieve chrge neutrlity Chpter - 20

6 r ction r nion < Coordintion # nd Ionic Rdii r ction Coordintion # increses with r nion To form stble structure, how mny nions cn surround round ction? Coord # 2 liner ZnS (inc blende) Adpted from Fig..7, Computtion of Minimum Ction-Anion Rdius Rtio for Coordintion No. of Determine minimum r ction /r nion for n octhedrl site (C.N. = ) Adpted from Tble., tringulr tetrhedrl octhedrl cubic NCl (sodium chloride) Adpted from Fig..5, CsCl (cesium chloride) Adpted from Fig..6, Chpter - 21 r r nion 1 2 = 2 ction = Chpter - 22 Computtion of Minimum Ction-Anion Rdius Rtio Determine minimum r ction /r nion for n octhedrl site (C.N. = 6) 2rnion + 2rction = 2 AX Crystl Structures AX Type Crystl Structures include NCl, CsCl, nd inc blende Sme concepts cn be pplied to ionic solids in generl. Exmple: NCl (rock slt) structure r N = nm r Cl = nm = 2r nion 2r nion + 2r ction = 2 2r nion r nion + r ction = 2r nion r ction = ( 2 1)r nion rction = 2 1= rnion Chpter - 2 ctions (N + ) prefer octhedrl sites r r ction nion = = bsed on this rtio, -- coord # = 6 becuse < < 0.72 Chpter - 24

7 Exmple Problem: Predicting the Crystl Structure of FeO On the bsis of ionic rdii, wht crystl structure would you predict for FeO? Ction Al + Fe 2+ Fe + C 2+ Anion O 2- Cl - Ionic rdius (nm) Answer: rction = r nion = bsed on this rtio, -- coord # = 6 becuse < < crystl structure is NCl Adpted from Fig..5, MgO nd FeO MgO nd FeO lso hve the NCl structure Dt from Tble.4, F Chpter - 25 Chpter - 26 O 2- r Mg /r O = Mg 2+ r O = nm r Mg = nm ctions prefer octhedrl sites So ech Mg 2+ (or Fe 2+ ) hs 6 neighbor oxygen toms Cesium Chloride structure: AX 2 Crystl Structures Fluorite structure r + Cs = = 0.99 r Cl Clcium Fluorite (CF 2 ) Ctions in cubic sites Since 0.72 < 0.99 < 1.0, cubic sites preferred So ech Cs + hs 8 neighbor Cl - UO 2, ThO 2, ZrO 2, CeO 2 Antifluorite structure positions of ctions nd nions reversed Chpter - 27 Chpter - 28 ٧

8 ABX Crystl Structures Perovskite structure Ex: complex oxide BTiO Density Computtions for Cermics Number of formul units/unit cell n ( ΣAC + ΣAA ) ρ = VC NA Avogdro s number Volume of unit cell ΣA C = sum of tomic weights of ll ctions in formul unit ΣA A = sum of tomic weights of ll nions in formul unit Chpter - 29 Chpter - 0 Exp: Density of Ncl N+ r = nm A N = g/mol Cl- r = nm A Cl = 5.45 g/mol Chpter - 1 Densities of Mteril Clsses In generl ρ metls > ρ cermics > ρ polymers Why? Metls hve... close-pcking (metllic bonding) often lrge tomic msses Cermics hve... less dense pcking often lighter elements Polymers hve... low pcking density (often morphous) lighter elements (C,H,O) Composites hve... intermedite vlues ρ(g/cm ) Metls/ Alloys Pltinum Gold, W Tntlum Silver, Mo Cu,Ni Steels Tin, Zinc Titnium Aluminum Mgnesium Grphite/ Cermics/ Semicond Polymers Dt from Tble B.1, Cllister & Rethwisch, e. Composites/ fibers Bsed on dt in Tble B1, Cllister *GFRE, CFRE, & AFRE re Glss, Crbon, & Armid Fiber-Reinforced Epoxy composites (vlues bsed on 60% volume frction of ligned fibers in n epoxy mtrix). Zirconi Al oxide Dimond Si nitride Glss -sod Concrete Silicon Grphite PTFE Silicone PVC PET PC HDPE, PS PP, LDPE Glss fibers GFRE* Crbon fibers CFRE* Armid fibers AFRE* Wood Chpter - 2 ٨

9 Silicte Cermics Most common elements on erth re Si & O Silictes Bonding of djcent SiO 4 4- ccomplished by the shring of common corners, edges, or fces Si 4+ O 2- Adpted from Figs , Cllister & Rethwisch e crystoblite SiO 2 (silic) polymorphic forms re qurt, crystoblite, & tridymite The strong Si-O bonds led to high melting temperture (1710ºC) for this mteril Chpter - Mg 2 SiO 4 C 2 MgSi 2 O 7 Adpted from Fig..12, Cllister & Rethwisch e. Presence of ctions such s C 2+, Mg 2+, & Al + 1. mintin chrge neutrlity, nd 2. ioniclly bond SiO 4 4- to one nother Chpter - 4 Bsic Unit: 4- Si04 tetrhedron Si 4+ O 2- Qurt is crystlline SiO2: Glss Structure Glss is noncrystlline (morphous) Fused silic is SiO 2 to which no impurities hve been dded Other common glsses contin impurity ions such s N +, C 2+, Al +, nd B + N + Si 4+ O 2- (sod glss) Adpted from Fig..41, Chpter - 5 Lyered Silictes Lyered silictes (e.g., clys, mic, tlc) SiO 4 tetrhedr connected together to form 2-D plne A net negtive chrge is ssocited with ech (Si 2 O 5 ) 2- unit Negtive chrge blnced by djcent plne rich in positively chrged ctions Adpted from Fig..1, Cllister & Rethwisch e. Chpter - 6 ٩

10 Polymorphism Two or more distinct crystl structures for the sme mteril (llotropy/polymorphism) iron system titnium liquid α, β-ti 158ºC crbon dimond, grphite BCC δ-fe 194ºC FCC γ-fe BCC 912ºC α-fe Polymorphic Forms of Crbon Dimond tetrhedrl bonding of crbon hrdest mteril known very high therml conductivity lrge single crystls gem stones smll crystls used to grind/cut other mterils dimond thin films hrd surfce cotings used for cutting tools, medicl devices, etc. Adpted from Fig..16, Chpter - 7 Chpter - 8 Polymorphic Forms of Crbon (cont) Grphite lyered structure prllel hexgonl rrys of crbon toms Polymorphic Forms of Crbon (cont) Fullerenes nd Nnotubes Fullerenes sphericl cluster of 60 crbon toms, C 60 Like soccer bll Crbon nnotubes sheet of grphite rolled into tube Ends cpped with fullerene hemispheres Adpted from Fig..17, Cllister & Rethwisch e. wek vn der Wl s forces between lyers plnes slide esily over one nother -- good lubricnt Adpted from Figs..18 &.19, Cllister & Rethwisch e. Chpter - 9 Chpter - 40 ١٠

11 Crystl Systems Unit cell: smllest repetitive volume which contins the complete lttice pttern of crystl. 7 crystl systems 14 crystl lttices, b, nd c re the lttice constnts Fig..20, Chpter - 41 Chpter - 42 Point Coordintes Exmple: Chpter - 4 Chpter - 44 ١١

12 Exmple: For the unit cell shown, locte the point hving coordintes ¼ 1 ½. x Crystllogrphic Directions Algorithm 1. Vector repositioned (if necessry) to pss through origin. 2. Red off projections in terms of unit cell dimensions, b, nd c y. Adjust to smllest integer vlues 4. Enclose in squre brckets, no comms [uvw] ex: 1, 0, ½ => 2, 0, 1 => [ 201] -1, 1, 1 => [ 111] fmilies of directions <uvw> where overbr represents negtive index Chpter - 45 Chpter - 46 Exmple: Exmple: Drw [1 1 0] direction within cubic unit cell. Chpter - 47 Chpter - 48 ١٢

13 Exmple: Liner Density Number of toms Liner Density of Atoms LD = Unit length of direction vector [110] ex: liner density of Al in [110] direction = nm # toms LD = length 2 = 2.5 nm 1 Chpter - 49 Chpter - 50 HCP Crystllogrphic Directions 2 Adpted from Fig..24(), 1 ex: ½, ½, -1, 0 => - Algorithm 1. Vector repositioned (if necessry) to pss through origin. 2. Red off projections in terms of unit cell dimensions 1, 2,, or c. Adjust to smllest integer vlues 4. Enclose in squre brckets, no comms [uvtw] [ 1120] 1 2 dshed red lines indicte projections onto 1 nd 2 xes 1 - Chpter - 51 HCP Crystllogrphic Directions Hexgonl Crystls 4 prmeter Miller-Brvis lttice coordintes re relted to the direction indices (i.e., u'v'w') s follows. 2 1 Fig..24(), - [ u'v'w'] [uvtw] 1 u = (2u'-v') 1 v = (2v'-u') t = -( u + v) w = w ' Chpter - 52 ١٣

14 Exmple: Crystllogrphic Plnes Miller Indices: Reciprocls of the (three) xil intercepts for plne, clered of frctions & common multiples. All prllel plnes hve sme Miller indices. Algorithm 1. Red off intercepts of plne with xes in terms of, b, c 2. Tke reciprocls of intercepts. Reduce to smllest integer vlues 4. Enclose in prentheses, no comms i.e., (hkl) Chpter - 5 Chpter - 54 Adpted from Fig..25, Crystllogrphic Plnes Crystllogrphic Plnes exmple b c 1. Intercepts Reciprocls 1/1 1/1 1/ Reduction Miller Indices (110) exmple b c 1. Intercepts 1/2 2. Reciprocls 1/½ 1/ 1/ Reduction Miller Indices (100) x x c c b b y y Chpter - 55 Chpter - 56 ١

15 Crystllogrphic Plnes Exmple: Determine the Miller indices for the plne shown. exmple b c c 1. Intercepts 1/2 1 /4 2. Reciprocls 1/½ 1/1 1/¾ 2 1 4/. Reduction Miller Indices (64) x b y Fmily of Plnes {hkl} Ex: {100} = (100), (010), (001), (100), (010), (001) Chpter - 57 Chpter - 58 Crystllogrphic Plnes (HCP) In hexgonl unit cells the sme ide is used exmple 1 2 c 1. Intercepts Reciprocls 1 1/ Reduction Miller-Brvis Indices (1011) 2 Adpted from Fig..24(b), 1 Crystllogrphic Plnes We wnt to exmine the tomic pcking of crystllogrphic plnes Iron foil cn be used s ctlyst. The tomic pcking of the exposed plnes is importnt. ) Drw (100) nd (111) crystllogrphic plnes for Fe. b) Clculte the plnr density for ech of these plnes. Chpter - 59 Chpter - 60 ١

16 Solution: Plnr Density of (100) Iron Adpted from Fig..2(c), toms 2D repet unit Plnr Density = 2 re 2D repet unit At T < 912 C iron hs the BCC structure. 1 = (100) 2D repet unit Rdius of iron R = nm 1 2 = R toms nm 2 4 = R toms = 1.2 x m 2 Chpter - 61 Plnr Density of (111) Iron Solution (cont): (111) plne toms 2D repet unit Plnr Density = re 2D repet unit 1 tom in plne/ unit surfce cell 2 toms in plne toms bove plne toms below plne h = re = 2 h = = R = R R toms = 7.0 = nm toms 0.70 x m 2 Chpter - 62 Crystls s Building Blocks Some engineering pplictions require single crystls: -- dimond single -- turbine bldes crystls for brsives (Courtesy Mrtin Dekins, GE Superbrsives, Worthington, OH. Used with permission.) Fig. 9.40(c), Cllister & Rethwisch e. (Fig. 9.40(c) courtesy of Prtt nd Whitney). Polycrystls Properties of crystlline mterils often relted to crystl structure. -- Ex: Qurt frctures more esily long some crystl plnes thn others. (Courtesy P.M. Anderson) Chpter - 6 Chpter - 64 ١

17 Single Crystls -Properties vry with direction: nisotropic. Single vs Polycrystls -Exmple: the modulus of elsticity (E) in BCC iron: E (digonl) = 27 GP X-Ry Diffrction Polycrystls -Properties my/my not vry with direction. -If grins re rndomly oriented: isotropic. (E poly iron = 210 GP) -If grins re textured, nisotropic. E (edge) = 125 GP 200 µm Chpter - 65 Diffrction grtings must hve spcings comprble to the wvelength of diffrcted rdition. Cn t resolve spcings < λ Spcing is the distnce between prllel plnes of toms. Chpter - 66 X-Rys to Determine Crystl Structure Incoming X-rys diffrct from crystl plnes. c X-Ry Diffrction Pttern c c extr distnce trvelled by wve 2 θ θ λ reflections must be in phse for detectble signl spcing between plnes d Adpted from Fig..7, Intensity (reltive) x b y (110) x b (200) y x (211) b y Mesurement of criticl ngle, θc, llows computtion of plnr spcing, d. X-ry intensity (from detector) θ c d = nλ 2 sin θc θ Chpter - 67 Diffrction ngle 2θ Diffrction pttern for polycrystlline α-iron (BCC) Adpted from Fig..20, Cllister 5e. Chpter - 68 ١٧

18 SUMMARY Atoms my ssemble into crystlline or morphous structures. Common metllic crystl structures re FCC, BCC, nd HCP. Coordintion number nd tomic pcking fctor re the sme for both FCC nd HCP crystl structures. We cn predict the density of mteril, provided we know the tomic weight, tomic rdius, nd crystl geometry (e.g., FCC, BCC, HCP). Intertomic bonding in cermics is ionic nd/or covlent. Cermic crystl structures re bsed on: -- mintining chrge neutrlity -- ction-nion rdii rtios. Crystllogrphic points, directions nd plnes re specified in terms of indexing schemes. Crystllogrphic directions nd plnes re relted to tomic liner densities nd plnr densities. SUMMARY Mterils cn be single crystls or polycrystlline. Mteril properties generlly vry with single crystl orienttion (i.e., they re nisotropic), but re generlly non-directionl (i.e., they re isotropic) in polycrystls with rndomly oriented grins. Some mterils cn hve more thn one crystl structure. This is referred to s polymorphism (or llotropy). X-ry diffrction is used for crystl structure nd interplnr spcing determintions. Chpter - 69 Chpter - 70 ١٨