The Muppet’s Guide to:The Structure and Dynamics of Solids

7. Phase Diagrams

Solid Solutions

Solute atoms create strain fields which can inhibit dislocations propagating in a material changing its properties

Hume-Rothery Rules – Substitutional Solutions

1. The solute and solvent should be of a similar size. (<15% difference)

2. The crystal structures must match.

3. Both solute and solvent should have similar electronegativity

4. The valence of the solvent and solute metals should be similar.

Rules to describe how an element might dissolve in a metal. Stable composition in equilibrium (thermodynamics)

Metals – Ni/Cu, Pd/Sn, Ag/Au, Mo/W

Phase Equilibria – Example

CrystalStructure

electroneg r (nm)

K BCC 0.93 0.235

Na BCC 1.00 0.191

• Both have the same crystal structure (BCC) and have similar electronegativities but different atomic radii.

• Rules suggest that NO solid solution will form.

K-Na

• K and Na sodium are not miscible.

Phase Equilibria – Example

CrystalStructure

electroneg r (nm)

Ni FCC 1.9 0.1246

Cu FCC 1.8 0.1278

• Both have the same crystal structure (FCC) and have similar electronegativities and atomic radii (W. Hume – Rothery rules) suggesting high mutual solubility.

Simple solution system (e.g., Ni-Cu solution)

• Ni and Cu are totally miscible in all proportions.

Hume-Rothery Rules – Interstitial Solution

1. The solute must be smaller than the interstitial sites in the solvent lattice

2. Solute and Solvent should have similar electro-negativities

Rules to describe how an element might dissolve in a metal. Stable composition in equilibrium (thermodynamics)

Light elements – H,C, N and O.

• Impurity atoms distort the lattice & generates stress.• Stress can produce a barrier to dislocation motion.

Modify Material Properties

• Smaller substitutional impurity

Impurity generates local stress at A and B that opposes dislocation motion to the right.

A

B

• Larger substitutional impurity

Impurity generates local stress at C and D that opposes dislocation motion to the right.

C

D

Increase material strength through substitution

(Callister: Materials Science and Engineering)

Material PropertiesDislocations & plastic deformation• Cubic & hexagonal metals - plastic deformation by

plastic shear or slip where one plane of atoms slides over adjacent plane by defect motion (dislocations).

• If dislocations don't move, deformation doesn't occur!

Adapted from Fig. 7.1, Callister 7e.

Edge Defect Motion

Dislocations & Materials Classes

• Covalent Ceramics

(Si, diamond): Motion hard. -directional (angular) bonding

• Ionic Ceramics (NaCl):

Motion hard. -need to avoid ++ and - - neighbours.

+ + + +

+++

+ + + +

- - -

----

- - -

• Metals: Disl. motion easier.

-non-directional bonding -close-packed directions for slip.

electron cloud ion cores

++

++

++++++++ + + + + +

+++++++

(Callister: Materials Science and Engineering)

Dislocation Motion• Dislocation moves along slip plane in slip direction

perpendicular to dislocation line• Slip direction same direction as Burgers vector

Edge dislocation

Screw dislocation

Adapted from Fig. 7.2, Callister 7e.

(Callister: Materials Science and Engineering)

Pinning dislocations

• dislocations make metals easier to deform• to improve strength of metals, need to stop dislocation motion

trap with:- impurity atoms;- other dislocations (work hardening;

- grain boundaries.

atom trap

(Callister: Materials Science and Engineering)

Modify Material Properties

• Grain boundaries are barriers to slip.• Barrier "strength" increases with Increasing angle of miss-orientation.• Smaller grain size: more barriers to slip.

Increase material strength through reducing Grain size

(Callister: Materials Science and Engineering)

Solid Solutions

Solid state mixture of one or more solutes in a solvent

Crystal structure remains unchanged on addition of the solute to the solvent

Mixture remains in a homogenous phase

Generally composed on metals close in the periodic tableNi/Cu, Pb/Sn etc.

Otherwise compounds tend to formNaCl, Fe2O3 etc.

• Components: The elements or compounds which are present in the mixture (e.g., Al and Cu)

• Phases: The physically and chemically distinct material regions that result (e.g., a and b).

Aluminum-CopperAlloy

Components and Phases

a (darker phase)

b (lighter phase)

Figure adapted from Callister, Materials science and engineering, 7 th Ed.

Phase Diagrams

• A phase diagram is a graphical representation of the different phases present in a material.

• Commonly presented as a function of composition and temperature or pressure and temperature

Applies to elements, molecules etc. and can also be used to show magnetic, and ferroelectric behaviour (field vs. temperature) as well as structural information.

Unary Phase DiagramsA pressure-temperature plot showing the different phases present in H2O.

Phase Boundaries

Upon crossing one of these boundaries the phase abruptly changes from one state to another. Latent heat not shown

Crossing any line

results in a structural

phase transition

Reading Unary Phase Diagrams

Melting Point (solid → liquid)

Boiling Point(liquid→ gas)

Sublimation (solid → gas)

As the pressure falls, the boiling point reduces, but the melting/freezing point remains reasonably constant.

Triple Point (solid + liquid + gas)

Reading Unary Phase Diagrams

Melting Point: 0°C Boiling Point: 100°C

Melting Point: 2°C Boiling Point: 68°C

P=1atm

P=0.1atm

Water Ice

http://images.jupiterimages.com/common/detail/13/41/23044113.jpg, http://www.homepages.ucl.ac.uk/~ucfbanf/ice_phase_diagram.jpg

• When we combine two elements... what equilibrium state do we get?• In particular, if we specify... --a composition (e.g., wt.% Cu – wt.% Ni), and --a temperature (T )

then... How many phases do we get? What is the composition of each phase? How much of each phase do we get?

Binary Phase DiagramsPhase BPhase A

Nickel atomCopper atom

Phase Equilibria: Solubility Limit– Solutions – solid solutions, single phase– Mixtures – more than one phase

• Solubility Limit: Max concentration for which only a single phase solution occurs.

Question: What is the solubility limit at 20°C?

Answer: 65 wt% sugar.

If Co < 65 wt% sugar: syrup

If Co > 65 wt% sugar: syrup + sugar.65

Sucrose/Water Phase Diagram

Pu

re

Su

gar

Tem

per

atu

re (

°C)

0 20 40 60 80 100Co =Composition (wt% sugar)

L (liquid solution

i.e., syrup)

Solubility Limit L

(liquid)

+ S

(solid sugar)20

40

60

80

100

Pu

re

Wat

er

Salt-Water(ice)

http://webserver.dmt.upm.es/~isidoro/bk3/c07sol/Solution%20properties_archivos/image001.gif

Ferroelectric Materials

Na½Ba½TiO3 T. Takenaka, K. Maruyama, and K. Sakata, Jpn. J. Appl. Phys. 30, 2236 (1991)

B. Jaffe, W. R. Cook, and H. Jaffe, Piezoelectric Ceramics (Academic Press, London, 1971).

J. Rodel, W. Jo, K. T. P. Seifert, E. M. Anton, T. Granzow, and D. Damjanovic, J. Am. Ceram. Soc. 92, 1153 (2009).

Phase Diagrams• Indicate phases as function of T, Co, and P. • For this course: -binary systems: just 2 components.

-independent variables: T and Co (P = 1 atm is almost always used).

• PhaseDiagramfor Cu-Niat P=1 atm.

• 2 phases: L (liquid)

a (FCC solid solution)

• 3 phase fields: LL + aa

wt% Ni20 40 60 80 10001000

1100

1200

1300

1400

1500

1600T(°C)

L (liquid)

a (FCC solid solution)

L + aliquidus

solid

us

Figure adapted from Callister, Materials science and engineering, 7 th Ed.

Phase Diagrams• Indicate phases as function of T, Co, and P. • For this course: -binary systems: just 2 components.

-independent variables: T and Co (P = 1 atm is almost always used).

• PhaseDiagramfor Cu-Niat P=1 atm.

wt% Ni20 40 60 80 10001000

1100

1200

1300

1400

1500

1600T(°C)

L (liquid)

a (FCC solid solution)

L + aliquidus

solid

us

Figure adapted from Callister, Materials science and engineering, 7 th Ed.

Liquidus:Separates the liquid from the mixed L+ aphase

Solidus:Separates the mixed L+ a phase from the solid solution

wt% Ni20 40 60 80 10001000

1100

1200

1300

1400

1500

1600T(°C)

L (liquid)

a (FCC solid solution)

L + a

liquidus

solid

us

Cu-Niphase

diagram

Number and types of phases• Rule 1: If we know T and Co, then we know: - the number and types of phases present.

• Examples:

A(1100°C, 60): 1 phase: a

B(1250°C, 35): 2 phases: L + a

B (

1250

°C,3

5) A(1100°C,60)

Figure adapted from Callister, Materials science and engineering, 7 th Ed.

wt% Ni20

1200

1300

T(°C)

L (liquid)

a(solid)L + a

liquidus

solidus

30 40 50

L + a

Cu-Ni system

Composition of phases• Rule 2: If we know T and Co, then we know: --the composition of each phase.

• Examples:TA

A

35Co

32CL

At TA = 1320°C:

Only Liquid (L) CL = Co ( = 35 wt% Ni)

At TB = 1250°C:

Both a and L CL = C liquidus ( = 32 wt% Ni here)

Ca = C solidus ( = 43 wt% Ni here)

At TD = 1190°C:

Only Solid ( a) Ca = Co ( = 35 wt% Ni)

Co = 35 wt% Ni

BTB

DTD

tie line

4Ca3

Figure adapted from Callister, Materials science and engineering, 7 th Ed.

wt% Ni20

1200

1300

30 40 50110 0

L (liquid)

a (solid)

L + a

L + a

T(°C)

A

35Co

L: 35wt%Ni

Cu-Nisystem

• Phase diagram: Cu-Ni system.

• System is: --binary i.e., 2 components: Cu and Ni. --isomorphous i.e., complete solubility of one component in another; a phase field extends from 0 to 100 wt% Ni.

• Consider Co = 35 wt%Ni.

Cooling a Cu-Ni Binary - Composition

4635

4332

a: 43 wt% Ni

L: 32 wt% Ni

L: 24 wt% Ni

a: 36 wt% Ni

Ba: 46 wt% NiL: 35 wt% Ni

C

D

E

24 36

Figure adapted from Callister, Materials science and engineering, 7 th Ed. USE LEVER RULE

• Tie line – connects the phases in equilibrium with each other - essentially an isotherm

The Lever Rule – Weight %

How much of each phase? Think of it as a lever

ML M

R S

RMSM L

L

L

LL

LL CC

CC

SR

RW

CC

CC

SR

S

MM

MW

00

wt% Ni

20

1200

1300

T(°C)

L (liquid)

a(solid)L + a

liquidus

solidus

30 40 50

L + aB

TB

tie line

CoC L Ca

SR

Figure adapted from Callister, Materials science and engineering, 7 th Ed.

• Rule 3: If we know T and Co, then we know: --the amount of each phase (given in wt%).• Examples:

At T A : Only Liquid (L) W L = 100 wt%, W a = 0

At T D : Only Solid ( a) W L = 0, W a = 100 wt%

Co = 35 wt% Ni

Weight fractions of phases – ‘lever rule’

wt% Ni20

1200

1300

T(°C)

L (liquid)

a(solid)L + a

liquidus

solidus

30 40 50

L + a

Cu-Ni system

TA A

35Co

32C L

BTB

DTD

tie line

4Ca3

R S

= 27 wt%

43 3573 %

43 32wt

At T B : Both a and L

WL= SR + S

Wa= RR + S

Figure adapted from Callister, Materials science and engineering, 7 th Ed.

wt% Ni20

120 0

130 0

30 40 50110 0

L (liquid)

a (solid)

L + a

L + a

T(°C)

A

35Co

L: 35wt%Ni

Cu-Nisystem

• Phase diagram: Cu-Ni system.• System is: --binary i.e., 2 components: Cu and Ni. --isomorphous i.e., complete solubility of one component in another; a phase field extends from 0 to 100 wt% Ni.• Consider Co = 35 wt%Ni.

Cooling a Cu-Ni Binary – wt. %

46344332

a: 27 wt%

L: 73 wt%

L: 8 wt%

a: 92 wt%

Ba: 8 wt% L: 92 wt%

C

D

E

24 36

Figure adapted from Callister, Materials science and engineering, 7 th Ed.

Equilibrium cooling

• Multiple freezing sites– Polycrystalline materials– Not the same as a single crystal

• The compositions that freeze are a function of the temperature

• At equilibrium, the ‘first to freeze’ composition must adjust on further cooling by solid state diffusion

Diffusion is not a flow

Our models of diffusion are based on a random walk approach and not a net flow

http://mathworld.wolfram.com/images/eps-gif/RandomWalk2D_1200.gif

Concept behind mean free path in scattering phenomena - conductivity