Restricted Dimensional Assemblies
Solid Solutions
Solid Solutions
• Many pure elements dissolve large quantities of other elements to form “solid solutions” or “alloys” (Latin alligare, “bind”)
• Substitutional (Cu-Ni, Au-Ag, Ti-Zr, K-Rb, Si-Ge, Se-Te, NaCl-KCl)
• Interstitial (H, C, B, O, N)
• Solid solutions may be disordered or ordered
Systematics
• William Hume-Rothery (1899 —1968)
• Oxford University, first chair of Metallurgy, 1958
• the “Hume-Rothery Rules”
Hume-Rothery Rule 1
• SIZE EFFECT
• If the difference between atomic radii of the elements forming an alloy exceeds 15%, solid solubility is restricted (a “negative” rule)
• Hume-Rothery used the closest distance of approach of atoms in the structure of each pure element as measure of atomic size
Complete Solubility
• Cu + Ni
• rCu = 0.128 nm, rNi = 0.125nm
• Δr/r = 2.3% (<15%), H-R #1 applies
• Cu and Ni comprise a “binary isomorphous” system; complete miscibility for any composition
wt. % Ni
at. % Ni
10Cu Ni20 4030 70 806050 90
10 20 4030 70 806050 90
°C
1600
1500
700
800
900
1000
1100
1200
1300
1400
1700
1800
1455°
1084.87°
!
L + !
L
Restricted Solubility
• Cu + Sn
• rCu = 0.128 nm, rNi = 0.158nm
• Δr/r ≈ 23% (>15%), H-R #1 does NOT applies
• Cu and Sn exhibit limited solubility
10 20 30 40 50 60 70 80 90
10 20 30 40 50 60 70 80 901100
1000
900
800
700
600
500
400
300
200
100
0
wt % Sn
at % Sn
SnCu
T (
°C)
L
!
"
#
$
%
%’
415°
189°
232°
186°
350°
520°586°
640°
756°
799°
1085°
&'
!(Sn
Hume-Rothery Rule 2
• ELECTRONEGATIVE VALENCE EFFECT
• The likelihood of formation of a stable phase is increased as one of the elements becomes more electronegative and the other more electropositive
PlatinumGold
GraphiteTitanium
Stainless SteelCopperBrassNickel
TinSteel
AluminumZinc
Magnesium
Galvanic Series
Electronegative(gains electrons)
Electropositive(loses electrons)
Noble(cathodic)
Active(anodic)
Sn 10 20 30 40 50 60 70 80 90 Au
500
600
700
800
900
1000
1100
400
300
200
100
0
°C
wt % Au
! Sn " # $ %
L
(Au)
217°252°
309°
418°
231.9681°
280°
490°86.2 95.8
80
100.3
1064.43°
rAu = 0.144 nm
rSn = 0.158 nm
Δr/r = 9.7%
Hume-Rothery Rule 3
• RELATIVE VALENCE EFFECT
• Elements with lower valence dissolve more readily in elements of higher valence
• Observation: monovalent Cu, Ag and Au dissolve in B-subgroup elements having valences >1
• For polyvalent elements, relative valence rule is less general
Extended Packing
• When size difference exceeds 15% (notably 1.225:1), Laves phases, composition AB2, may be favored
• Known examples ≈ 360
• Basic layer of smaller atoms stacked in either cubic or hexagonal arrangement
• Larger atoms have CN = 16 C14
Laves Phases
• Three basic structures
• C15 (cubic, MgCu2)
• C14 (hexagonal, MgZn2)
• C36 ( hexagonal MgNi2)
• Smaller atoms sometimes in icosahedral coordination with larger atoms
C15
Strukturbericht
• A Elements (uniary systems)
• B 1:1 stoichiometry
• C 1:2 stoichiometry
• D 1:3 stoichiometry
• E Perovskites
• H Spinels
• L long-period ordering
Ref: Strukturbericht, Akademische Verlagsgesellschaft M.B.H., Leipsig, Germany (1913 - 1939); continued as Structure Reports, International Union of Crystallography, (1940 - present)
DO3 Structure
rNi
= 0.125 nm rSn
= 0.158 nm
Sn at 0,1 Sn at !
Ni at !
Ni at !
Ni at !
Ni at !Ni at !
Sn at ! Sn at !
Sn at !
Ni at ",# Ni at ",#
Ni at ",# Ni at ",#
Sn at 0,1
Ni at 0,1
Ni at 0,1 Ni at 0,1
Ni at 0,1
Sn at 0,1
Sn at 0,1
Sn at 0,1
Ni3Sn
10Ni Sn20 30 40 50 60 70 80 90
10 20 30 40 50 60 70 80 901500
1400
1300
1200
1100
1000
900
800
700
600
500
400
wt % Sn
at % Sn
T
(°C)
19.01132°
922°
851°
978°
1176°
796°
1266°
1455°
38.0
32.5
43.189.3
L
!’
"!
# $
Ni3Sn
cubic DO3
hexagonal DO19
Perovskite Unit Cell
x
z
y
• Lattice = simple cubic (lattice points at corners of cube)
• Motif = 1 Ca2+ at 0,0,0; 1 Ti4+ at ½,½,½; and 3 O2- at ½,½,0, 0,½,½, and ½,0,½
Perovskites
NaNbO3 CaTiO3 CaSnO3 BaPrO3 YAlO3 KMgF3
KNbO3 SrTiO3 SrSnO3 SrHfO3 LaAlO3 PbMgF3
NaWO3 BaTiO3 BaSnO3 BaHfO3 LaCrO3 KNiF3
CdTiO3 CaCeO3 BaThO3 LaMnO3 KZnF3
PbTiO3 SrCeO3 LaFeO3
CaZrO3 BaCeO3
SrZrO3 CdCeO3
BaZrO3 PbCeO3
PbZrO3
Spinel• Formula A2+B3+2O4
• Eight (8) formula units per cubic unit cell (eight divalent cations, 16 trivalent cations, 32 oxygen anions), with oxygen anions in close-packed configuration
• “Normal” spinel: divalent cations on tetrahedral sites; trivalent cations on octahedral sites
• “Inverse” spinel: trivalent cations on tetrahedral sites and 1/2 of the octahedral sites; divalent cations on remaining 1/2 of octahedral sites.
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View
octahedral sitestetrahedral sites
Spinel
• Lattice = face-centered cubic
• Basis (motif) = 14 ions
• two divalent cations
• four trivalent cations
• eight oxygen anions
SpinelsNormal Inverse
SnMg2O4 MgAl2O4 CoFe2O4 ZnK2(CN)4 CuCr2S4 MgFe2O4
FeV2O4 SrAl2O4 NiFe2O4 CdK2(CN)4 CoCo2S4 TiMg2O4
WNa2O4 CrAl2O4 AlFe2O4 HgK2(CN)4 CuRh2S4 VMg2O4
MoAl2O4 PbFe2O4 FeCr2S4 SnZn2O4
FeAl2O4 MgCo2O4 MnCr2S4 MgGa2O4
CoAl2O4 TiCo2O4 MgIn2O4
NiAl2O4 CuCo2O4 FeIn2O4
CuAl2O4 CoCo2O4 CoIn2O4
ZnAl2O4 NiIn2O4
Ordering
• CuAuI is tetragonal, c/a ≈ 0.93
• Orders below Tc = 385°C
• Ordering produces change in Bravais lattice (cubic to tetragonal), P4/mmm
• Strukturbericht L10
x
z
y
Cu3AuCuAu
I
II400
L
α
Cu Au
CuAu3
Long Period Order
• CuAuII is a “long-period superlattice”
• Ordering occurs 410°C - 385°C
• Orthorhombic; b/a = 10.02; c/a ≈ 0.92
• Large unit cell comprised of 5 each CuAuI cells side-by-side, shifted by c/2
a
c
bCu3Au
CuAu
I
II400
L
α
Cu Au
CuAu3
Clustering
A B%B
ΔGm α β
L
CLUSTERING
A B%B
ΔGm
L
SISII
“spinodes”∂2(∆Gm)
∂x2= 0
100 nm 100 nm
Spinodal Decomposition
Cu-Ni-Fe
Phase Separation
• Decomposition of a solid solution can occur by
• nucleation & growth, either
• homogenous or
• heterogeneous, or by
• spinodal decomposition
• Control of thermodynamics and kinetics essential
Nano Assembly
• Formation of solid solutions governed by Hume-Rothery rules
• Solid solutions are susceptible to decomposition by phase separation (clustering) or disordering
• Ordered solid solutions generally more resistant to decomposition
Development of Interfacial Structure
Epitaxy
Greek: epi “upon” + taxis “arrangement”
Epitaxial Growth
• Occurs on a substrate (“substratum”) to induce a specific crystalline arrangement within the growing crystal
• Homoepitaxy (crystal A on substrate A)
• Heteroepitaxy (crystal B on substrate A)
• Deposition from vapor phase (VPE, MBE); deposition from liquid phase (LPE); reaction in solid phase (SPE)
Epitaxial Growth
• E. Bauer, Z. Kristallogr., 110, 423 (1958)
• W.A. Jesser and J.H. van der Merwe, in Dislocations in Solids, F.R.N. Nabarro, Ed., Elsevier, Amsterdam (1989), p. 41
• G.H. Gilmer, in Handbook of Crystal Growth, Vol. 1, D.T.J. Hurle, Ed., Elsevier, Amsterdam (1993), p. 584
Frank-Van der Merwe
Substrate
Film
Stranski-Krastanov
Substrate
Film
Volmer-Weber
Substrate
Film
Columnar
Substrate
Film
Interfacial Structure
• Exerts a profound influence on the morphology of thin films during epitaxial growth
• Control of original substrate/film interface
• Structural (“lattice matching”)
• Compositional (interphase formation)
EndLecture 05
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