J.W. Morris, Jr. University of California, Berkeley
MSE 200A Fall, 2008
Equilibrium Phase Diagrams
• Changing T, P or x may change the “phase” of a solid – Phase = state of aggregation – Gas, liquid, solid (in a particular crystal structure)
• Phase diagrams: – Maps showing equilibrium phases as a function of T, x (sometimes P) – We shall consider equilibrium phases in
• One component systems • Two component systems
– As a function of (T,x), not P – Consider
• How to read any binary phase diagram • How to understand a few simple phase diagrams
J.W. Morris, Jr. University of California, Berkeley
MSE 200A Fall, 2008
One-Component Phase Diagram: First-Order Phase Transformations
• Phase are distinct “states of aggregation” – Differ by symmetry – Gas, liquid, solid (particular crystal structure)
• Preferred phase minimizes free energy – F = E-TS - As T↓ down, gas→liquid→solid(low ΘD)→solid(high ΘD)
• Energy dominates at low T (Eβ < Eα) • Entropy dominates at high T (Sβ < Sα)
– Transformations at well-defined temperatures
• Metastability – Phase transformation requires structural change – Phase retained beyond Tαβ if slow kinetics
G
T
Tå∫Tå©
∫
©
å
∫ å
©
J.W. Morris, Jr. University of California, Berkeley
MSE 200A Fall, 2008
Mutations
• One phase simply become another at a particular Tc – Symmetry of phase change must permit morphing – Disordered high-temperature phase spontaneously orders
• Examples: – Ferromagnetism: alignment of magnetic moments – Crystal order: A and B separate to different lattice sites (some are first-order) – Ferroelectricity: ion is displaced to asymmetric position, creating dipole
T
g
Tc
C
TTc
åå' åå'
ferromagnetism order ferroelectricity
J.W. Morris, Jr. University of California, Berkeley
MSE 200A Fall, 2008
Phases in a Two-Component System
€
dg = −sdT + vdP + (µ2 −µ1)dx
• The Gibbs free energy per atom:
• Stability requires:
€
g =GN
= g(T,P,x)
g
xA
å
x1
¡µ( )x1
€
∂g∂x
T
= µ2 −µ1
€
∂ 2g∂x 2
T
≥ 0 ⇒ g(x) concave up
J.W. Morris, Jr. University of California, Berkeley
MSE 200A Fall, 2008
Phases in a Two-Component System: The Common Tangent Rule
• Binary System at (T,P) – Phases α and β – Free energy curves cross as shown
• Possible states: – Pure α: g(x) = gα(x) – Pure β: g(x) = gβ(x) – Two-phase mixture:
g(x) = fαgα(xα) + fβgβ(xβ)
• Common tangent rule: – Draw common tangent to gα(x) and gβ(x) – Tangent touches at xβ and xβ
– Then: • If x < xα, α preferred • If x > xβ, β preferred • If xα<x< xβ, two-phase mixture preferred
g
x A B
å ∫
x å x ∫ x
g
x A B
å
∫
x å x ∫ x
J.W. Morris, Jr. University of California, Berkeley
MSE 200A Fall, 2008
Phases in a Two-Component System
• Binary System at (T,P) – Phases α, β and γ – Free energy curves cross as shown
• Top figure: – Phase γ does not appear – Regions of α, β and α + β appear – The least free energy is along lower curve – Note within two-phase region
• α has fixed composition xα • β has fixed composition xβ
• Bottom figure: – Phase γ appears at intermediate x – α and γ separated by α + γ region – β and γ separated by β + γ region
J.W. Morris, Jr. University of California, Berkeley
MSE 200A Fall, 2008
Binary Phase Diagrams
• Equilibrium phase diagrams are maps – Show phases present at given (T,x) – Binary phase diagrams also give compositions, phase fractions
• We will learn – The “solid solution” phase diagram (above left) – The “eutectic” phase diagram (above right) – How to read any binary phase diagram
L L + å
å
A B
T
x
T A
T B T
A Bx
å ∫
L
å+L ∫+L
å + ∫
J.W. Morris, Jr. University of California, Berkeley
MSE 200A Fall, 2008
Binary Phase Diagrams; Three Pieces of Information for Given (T,x)
T
A Bx
xx xå ∫
å ∫
L
å+L ∫+L
å + ∫
1. Phases present • From phase field
2. Compositions of phases • Only needed for two-phase fields • xα and xβ from intersection of
isothermal line with boundaries
3. Phase fractions • From the “lever rule”
€
f α =x β − xx β − xα
f β =1− f α =x − xα
x β − xα
J.W. Morris, Jr. University of California, Berkeley
MSE 200A Fall, 2008
Complex Diagrams: Read in the Same Way
1. Phases from phase field
2. Compositions from isotherm
3. Fractions from lever rule
Al-Cu Phase Diagram
J.W. Morris, Jr. University of California, Berkeley
MSE 200A Fall, 2008
The Solid Solution Diagram
• To form a solid solution at all compositions – Pure A and pure B must have the same crystal structure – A and B must have chemical affinity
• The (α+L) region appears near the melting points – Generated as the L and α curves pass through one another – Often (but not always) between TA and TB
L
L + å
å
A B
T
x
TA
T B
g
xA B
åL
Bx
A
å
L
xA B
å
L
xåxL
L åå+L
T > TB TB > T > TA TA > T
J.W. Morris, Jr. University of California, Berkeley
MSE 200A Fall, 2008
Solidifying a Solid Solution
• Let solution have the composition shown • First solid to form is very rich in B • As T decreases:
– Fraction of α increases – Composition of α (xα) evolves toward x
• Final α has the average composition of the solution (xα = x)
L
å
A B
T
x
TA
TB
x
L
å
L + å
xåxL
x
T
x
J.W. Morris, Jr. University of California, Berkeley
MSE 200A Fall, 2008
Application: Purification (Zone Melting)
• To purify an impure (AB) liquid to pure B – Cool until first solid forms (rich in B) – Extract, re-melt and repeat (richer in B) – Repeat as often as needed to create as pure B as desired
• “Zone melting”: A continuous process that accomplishes this – Was the “enabling technology” for the transistor – Very pure starting materials are necessary to create doped semiconductors
L
å
A B
T
x
TA
TB
x
L
å
L + å
xåxL
x
T
x
J.W. Morris, Jr. University of California, Berkeley
MSE 200A Fall, 2008
The Simple Eutectic Phase Diagram
• Eutectic diagram: – Simplest diagram when α and β have different structures – Named for “eutectic reaction”: L → α + β at TE
• Source: – As T rises, L free energy curve cuts through α,β common tangent – Note TE is the lowest melting point
T
A Bx
å ∫
L
å+L ∫+L
å + ∫
g
å∫
xA B
å ∫
L
L
x1 x2 x3 x4
å∫
xA B
å ∫
L
x1 x2
å + L ∫ + L å + ∫
J.W. Morris, Jr. University of California, Berkeley
MSE 200A Fall, 2008
Eutectic Diagram: Characteristic Equilibrium Microstructure at x1
• Let L at composition x1 be cooled slowly
• L freezes gradually as T drops into α region – Likely into polygranular microstructure shown
• On further cooling β precipitates from α – Likely into microstructure shown – β precipitates on grain boundaries and/or in grain interiors
T
A Bx
å ∫
L
å+L ∫+L
å + ∫
1x 2x 3x
J.W. Morris, Jr. University of California, Berkeley
MSE 200A Fall, 2008
Eutectic Diagram: Characteristic Equilibrium Microstructure at x3
• Let L at composition x3 be cooled slowly
• Solidification occurs sharply at TE – Eutectic reaction: L → α + β
• Microstructure resembles that shown (the “eutectic” microstructure) – α and β grow as plates side-by-side to maintain composition at interface – Overall microstructure is grain-like eutectic “colonies” – This transformation mechanism is easiest kinetic path
T
A Bx
å ∫
L
å+L ∫+L
å + ∫
1x 2x 3x
∫å∫
∫
∫
å
å
L
J.W. Morris, Jr. University of California, Berkeley
MSE 200A Fall, 2008
Eutectic Diagram: Characteristic Equilibrium Microstructure at x2
T
A Bx
å ∫
L
å+L ∫+L
å + ∫
1x 2x 3x
• Let L at composition x2 be cooled slowly
• Solidification occurs in two steps: – L solidifies gradually to α as it cools through L+α region – At TE, the remaining liquid solidifies by the eutectic reaction: L → α + β
• Microstructure resembles that shown (the “off-eutectic” microstructure) – Islands of “pro-eutectic” α composition at interface – Surrounded by eutectic colonies
Proeutectic å
Eutectic colony
J.W. Morris, Jr. University of California, Berkeley
MSE 200A Fall, 2008
“Eutectoid” Reactions
• The “eutectoid” is a eutectic between solid phases:
– γ → α+β – γ is a solid
• The classic example is the Fe-C diagram shown at left
– Pure Fe: • L→δ(bcc) → γ(fcc) → α(bcc)
– Eutectic (“cast iron”) • L → γ+Fe3C
– Eutectoid: • γ → α+Fe3C • The central reaction in steel
J.W. Morris, Jr. University of California, Berkeley
MSE 200A Fall, 2008
Kinetics
• Rate of change in response to thermodynamic forces
• Deviation from local equilibrium ⇒ continuous change – ∇T ⇒ heat flow ⇒ temperature changes – ∇µ ⇒ atom flow ⇒ composition changes
• Deviation from global equilibrium ⇒ discontinuous change – ΔG (ΔF) ⇒ phase change (or other change of structure)
J.W. Morris, Jr. University of California, Berkeley
MSE 200A Fall, 2008
Flow of Heat
• Let: – T2 > T1 (one-dimensional gradient) – JQ = heat flow/unit area•unit time (J/m2s) – Ignore internal sources of heat
• From the Second Law:
T1T2JQ Box of unit length,
unit cross-section area
€
JQ = −k dTdx
Fourier’s law of heat conduction
k = thermal conductivity
J.W. Morris, Jr. University of California, Berkeley
MSE 200A Fall, 2008
Evolution of Temperature
• Let: – T3 > T2>T1 (one-dimensional gradient) – ∂Q/∂t = heat added/unit time (J/m2s)
• The net heat added to the center cell is:
€
∂Q∂t
= J23Q − J12
Q[ ]dA
T1T2T3
JQ12JQ23
€
J12Q = J23
Q +dJQ
dx
dx
€
= −dJQ
dx
dV =
ddx
k dTdx
dV
€
∂Q∂t
=∂E∂t
= CV∂T∂t
dV
€
∂T∂t
=kCV
∂ 2T∂x 2
J.W. Morris, Jr. University of California, Berkeley
MSE 200A Fall, 2008
Heat Conduction in 3-Dimensions
• A temperature gradient produces a heat flux: – ∇T = (∂T/∂x)ex + (∂T/∂y)ey + (∂T/∂z)ez – JQ = JQ
xex + JQyey + Jq
zez
• How many thermal conductivities? – In the most general case, 9 – For a cubic or isotropic material, only need 1
• For a cubic or isotropic material
€
JQ = −k∇T
€
∂T∂t
= −k∇2T
J.W. Morris, Jr. University of California, Berkeley
MSE 200A Fall, 2008
Mechanisms of Heat Conduction
• Energy must be transported through the solid – Electrons – Lattice vibrations - phonons – Light - photons (usually negligible)
• Conduction is by a “gas” of moving particles – Particles move both to left and right (JQ = JQ +-JQ-) – Particle energy increases with T – If T decreases with x, particles moving right have more energy
Net flow of heat to the right (JQ > 0)
v
J = (1/2)nev JQ+ JQ-
JQ = JQ+- JQ- T+ > T- ⇒ JQ > 0
J.W. Morris, Jr. University of California, Berkeley
MSE 200A Fall, 2008
Heat Conduction by Particles
• Transport of thermal energy – Particle reaches thermal equilibrium by collisions – Particle travels <l> = mean free path between collisions – Particle transfers energy across plane
Energy crossing plane reflects equilibrium <lx> upstream
J.W. Morris, Jr. University of California, Berkeley
MSE 200A Fall, 2008
Mechanisms of Heat Conduction
• Particles achieve thermal equilibrium by collision with one another
• Particles that cross at x were in equilibrium at x’=x-‹lx› – ‹lx› = mean free path
v
J = (1/2)nevJQ+ JQ- JQ = JQ+ - JQ-
€
JQ+ =12nevx =
12Ev (T)vx =
12Ev[T(x − lx )]vx
€
Ev T x − lx( )[ ] = Ev −∂Ev
∂T
dTdx
lx = Ev −Cv
dTdx
lx
€
JQ+ =12Evvx −
12Cvvx lx
dTdx
J.W. Morris, Jr. University of California, Berkeley
MSE 200A Fall, 2008
Mechanisms of Heat Conduction
• Total heat flux: v
J = (1/2)nevJQ+ JQ- JQ = JQ+ - JQ-
€
JQ+ =12Evvx −
12Cvvx lx
dTdx
€
JQ = JQ+ − JQ− = −Cvvx lxdTdx
= −k
dTdx
€
k = Cvvx lx
• In three dimensions:
€
k =13Cvv l =
13Cvv
2τ
€
( l = vτ )
J.W. Morris, Jr. University of California, Berkeley
MSE 200A Fall, 2008
Conduction of Heat
T1T2T3
JQ12JQ23
€
∂T∂t
=kCV
∂ 2T∂x 2
€
JQ = −k dTdx
€
k =13Cvv l =
13Cvv
2τ
• Thermal conductivity by particles:
- v = mean particle velocity - <l> = mean free path between collisions - τ = mean time between collisions
• Particles include - electrons - phonons
J.W. Morris, Jr. University of California, Berkeley
MSE 200A Fall, 2008
Heat Conduction by Electrons
• Motion of electrons = electrical conductivity
v
J = (1/2)nev
€
k =13Cvv l =
13Cvv
2τ
Wiedemann-Franz Law
€
k =
LTρ0LA
high temperature
low temperature €
(ρ = ρ0 + AT)
€
k = LσT =LTρ
J.W. Morris, Jr. University of California, Berkeley
MSE 200A Fall, 2008
Heat Conduction by Phonons
• Scattering of phonons in perfect crystals due to phonon-phonon collisions – Low energy phonons perform elastic collisions - energy conserved – High-energy phonons perform “inelastic” collisions with lattice
• Phonon thermal conductivity – Low at low T due to low CV – Low at high T due to inelastic collisions – Highest at T ~ ΘD/3
• Kphonon only important in materials with high ΘD – Diamond (ΘD ~ 2000K) is an insulator with high thermal conductivity
Much excitement in microelectronics
€
k =13Cvv l =
13Cvv
2τk
TŒ
Crystal
Glass
J.W. Morris, Jr. University of California, Berkeley
MSE 200A Fall, 2008
Heat Conduction by Phonons
• Scattering of phonons in imperfect crystals due to phonon-defect collisions – <l> is the mean spacing between defects
• Phonon thermal conductivity is low – Polygranular solids – Defective solids – Glasses
• High ΘD materials only have high k when they are nearly perfect – Defective diamond films are no particular good
€
k =13Cvv l =
13Cvv
2τk
TŒ
Crystal
Glass