Equilibrium Phase Diagrams - mse.berkeley.edu · J.W. Morris, Jr . University of California,...

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


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

©


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


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


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


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


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 + ∫


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


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


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


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


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


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 + ∫


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


MSE 200A Fall, 2008



•  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


MSE 200A Fall, 2008


T

A Bx

å ∫

L

å+L ∫+L

å + ∫

1x 2x 3x


•  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


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


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)


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


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


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


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


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


MSE 200A Fall, 2008


•  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


MSE 200A Fall, 2008


•  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 )


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


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


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


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

Equilibrium Phase Diagrams - mse.berkeley.edu · J.W. Morris, Jr . University of California,...

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