perfect crystal imperfect crystal point defect vacancy self-interstitial atom substitutional impurity atom interstitial impurity atom line defect dislocation planar defect surface grain boundary interface magnetic domain bulk defect void crack mechanical electrical magnetic optical chemical microstructure semiconductor hole elecronic defect structural defect
N n n E V F E V F Z Z E remove = Zε b ε b E surface Z ε b E F V = E remove E surface Zε b sublimation E E sub = zε b
E F V E sub fcc, face centered cubic E V F E V F E sub T m E 9k V F B T m k B N av R k B = R N av = 1.41 10 J/K ev electron-volt J 1 ev =1.60 10 19 J Table 1. Comparison of calculated and experimental E F V with E sub in fcc metals. Material E sub (ev) calculated E F (ev) experimental E F V (ev) Al 1).04 0.75 Cu*.54 1.8 1. Ag*.85 0.97 1.1 Au*.9 1.0 0.9 Ni* 4.45 1.6 1.6 Pd*.91 1.44 1.4 Pt* 5.77 1.4 1.5 1) R. O. Simmons and R. W. Ballufi, Physical Review,, 570(1960) *M. S. Daw, S. M. Foiles, M. I. Baskes, The embedded atom method: a review of theory and applications, Material Science Report, 9, 59(199)
N n H = ne F n N N! W = ( N n)! n! S mix = kblnw = k B [ln N! ln(n n)! ln n!] H, T S, G S vib G = H T S mix Tn S vib = ne F kbt[ln N! ln(n n)! ln n!] Tn S vib H = ne F H n -T S mix -Tn S vib G= H T S G N, n >> 1 ln N! N ln N N ln( N n)! ( N n) ln( N n) ( N n) ln n! n ln n n G ne F k B T[N ln N ( N n)ln( N n) n ln n] Tn S vib G G n = 0 Τ S
G n E F k B T ln n N n T S vib = 0 C V = n N N >> n C V = n N n N n = exp S vib exp E F k B T k B exp( S vib / k B ) A C V = A V exp E F k B T A =1 ~ 10 σ Ω W = σω E * F = EF W = EF σω C V (σ) = Aexp E * F = Aexp E σω F = C k B T k B T V exp σω k B T σ C V C V σω = CV ( σ ) CV = CV exp 1 kbt σω << k B T σω σω exp + kbt 1 kbt C V σω C k T B V
- C S C S = A S exp E S F k B T E S F A E S F E = E + E + E F V F strain int E V F E int solute atom - Hume Rothery Rules solvent atom 15 eletronegativity valence E int
R R+ R R R + R R R + R p σ r σ θ σ φ (R + R) σ r = p σ r θ = σ φ = p (R+ R) r R+ R 1 1+ ν ε r = { σ r ν s ( σθ + σφ )} = σ r E E ε φ = ε θ m 1 ν = 1 { σφ ν ( σθ + σ r )} = + σ E E r E ν p r 1+ ν = E σ 1+ ν r + 4E σ r 4πr dr = 4π (R + R) 1+ν 6 R + R E (1+ ν) p 4π(R + R) = E dr R + R r 4 ε = R R = 1+ ν E p (1+ ν) p 4π(R + R) = E = Eε 4πR 1+ ν (1+ε) E G = (1 + ν ) ε << 1 GΩε
Al R RCu Al = 0.14 nm E Al = 70 GPa ν Al = 0.5 Cu = 0.18 nm E Cu =10 GPa ν Cu = 0.4 Cu Al Cu Al R = R Cu E = E Cu ν = ν Cu R = R Al RCu ε = R R = 0.105 = 0.050 10 19 J = R s = R Al E s = E Al ν s = ν Al R = R Cu R 19 ε = 0.117 = 0.16 10 J = 0.10 ev 0.01eV Al Al Cu Cu Al Al Cu
E int A B AA BB AB E AA E BB E AB AB 1 = E AB ( EAA + E BB ) AA BB AB = 0 AB covalent bond + + A B AB(covalent) A B A B AB AB A B HF HI E = 561 kj/mol = 95 kj/mol HF E HI H H I I 1 ( E HH E + II ) = 9 kj/mol HI H I HF H H I I H F Pauling x AB = 96.5( xa xb ) (kj/mol) AB E 1 AB = ( EAA + EBB) + 96.5( xa xb ) kj/mol ev E 1 AB = ( EAA + EBB) + 0.0104( xa xb ) ev Table Pauling AB
AB intermetallic compound Table. Electronegativity defined by Pauling. 1 4 5 6 7 8 9 10 11 1 1 14 15 16 17 18 IA IIA IIIA IVA VA VIA VIIA VIII IB IIB IIIB IVB VB VIB VIIB 0 H He.1 Li Be B C N F O Ne 1 1.5.5 4.5 Na Mg Al Si P Cl S Ar 0.9 1. 1.5 1.8.1.5 K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge As Br Se Kr 0.8 1 1. 1.5 1.6 1.6 1.5 1.8 1.8 1.8 1.9 1.6 1.6 1.8.8.4 Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb I Te Xe 0.8 1 1. 1.4 1.6 1.8 1.9... 1.9 1.7 1.7 1.8 1.9.5.1 Cs Ba La Hf Ta W Re Os Ir Pt Au Hg Tl Pb Bi At Po Rn 0.7 0.9 1.1 1. 1.5 1.7 1.9....4 1.9 1.8 1.8 1.9. electronic phase - valence e/a e/a valence electrons per atom A B Z A Z B f A f B AB e / a = Z f + Z A A B f B A Cu Cu IB Z Cu = 1 Zn IIB Z Zn = f Zn Zn Cu-Zn e / a = 1+ f Zn
Cu-Zn e / a < 1. 4 fcc Cu-Zn 1.4 < e / a < 1. 6 1.6 < e / a < 1. 75 bcc CuZn Cu 5 Zn 8 1.75 < e / a CuZn 5 Cu Mole fraction of Zn Zn Mott Jones e/a Table 1 IB Cu Ag Au electronic phase
Table. Valence electrons per atom for α, β and γ-phases. Alloy Maximum e/a Minimum e/a Range of e/a Cu-Zn 1.84 1.48 1.58-1.66 Cu-Al 1.408 1.48 1.6-1.77 Cu-Sn 1.70 1.49 1.67-1.67 Cu-Ga 1.406 Cu-Si 1.40 Cu-Ge 1.60 Ag-Cd 1.45 1.50 1.59-1.6 Ag-Zn 1.78 1.58-1.66 Ag-Hg 1.5 Ag-In 1.40 Ag-Al 1.408 Ag-Ga 1.80 Ag-Sn 1.66 Au-Zn 1.48 Au-Cd 1.49 Au-Al 1.7 N. F. Mott and H. Jones, The theory of the properties of metals and alloys, (Dover, New York, 1958), p.17 self-interstitial atom fcc fcc R a a = R R i a R R i = = ( 1) R 0. 414R = GΩ + ε (1 ε ) R Ri R ( 1) R ε = = = R ( 1) R i R i a R
= 4 GΩ (1 + ) Al = 9.1 10 19 J = 5.7eV 0.75eV C SIA C SIA E exp kbt strain C SIA 10 96 C SIA 10 1 Frenkel pair radiation damage fcc R i R 41 H C N O - Fe-C C 91 bcc body centered cubic ferritic phase 91 194 fcc austenitic phase 155 bcc δ
1600 δ 1400 155 δ+l δ+γ 19 149 L Temperature ( o C) 100 1000 91 γ γ+l.14 114 γ+fe C 4. L+Fe C 800 α 600 α+γ 0.0 0.76 77 α+fe C 400 0 1 4 5 6 6.7 Carbon (wt%) Fe-Fe C Fe Fe C bcc 77 0.0wt% C Fe Fe C cementite C 77 C fcc 77 0.76wt% C 1147.14wt% C bcc C 149 0.09wt% C bcc fcc bcc O T R a 4R a = O R oct
a R R oct = = 1 R 0. 155R Fe R = 0.14 nm R oct = 0.019 nm C R C = 0.07 nm R oct.5 T R tet R oct 1.9 R tet = 0.06 nm C O T bcc bcc C z Fe 4 Fe y <100> z c c = ( R+ R C ) a (0.14 + 0.07) 0.87 = 0.105 nm C Fe 0.05nm z x
ε z c = a = 0.66 <100> x y x y z ε x ε y ε z ε x = ε y C xy C Fe Fe a 4 a = R R = 1 R + 6 ε x = ε y = a C R C a / = 1+ 6 4 1+R C R = 0.0 z x y 1/10 -Fe C fcc C quench C martensite C c z a x y body-centered tetragonal cubic, bct C C C 0.5wt% C x y z bct c Zener
C <100> V p V V/V p( V / V ) = V K p V p = K V E V = (1 ν ) V K = E /{ (1 ν )} EV V = V 6(1 ν ) V V = a V V ε x + ε y + ε z = 0.66 0.0 = 0.0
E Fe = 10 GPa ν Fe = 0.9 = W 1. 10 0 J = 0.75 ev C C C C E exp kbt strain 7 C C = 0.0017 at% =0.006 wt% Fe-C 7 C 0.0 wt% C 0.6 ev C C x10 9 wt% C C graphite Fe Fe C cementite C C - C fcc - O T a R R oct = = ( 1) R 0. 414R R oct = 0.051 nm ε = 4 Ω = RC R R oct oct πr oct = 0.40 = GΩε (1 + ε ) = 4.1 10 0 J = 0.6 ev C C C C C E exp kbt strain 7 C C = 5 at% = 1.1 wt% 1147 C C = 1 at% =.9 wt% C C = 0.76 wt% C C =. wt%
T 1/ C O T fcc fcc Al Cu 5 A-B B f a f w f a f w A M A B M B N av -Fe -Fe -Fe 0.65nm