1st Law: Conservation of Energy Energy is conserved. Hence we can make an energy balance for a system: Sum of the inflows – sum of the outflows is equal.

1st Law: Conservation of Energy

energy flow

boundary

∑ = Δ ( Esystem

)

Energy is conserved. Hence we can make an energy balance for a system: Sum of the inflows – sum of the outflows is equal to the energy change of the system

NOTE: sign of flow is positive when into the system

UNITS: Joules, calories, electron volts …

Types of Work

Electrical

W = – dq

Magnetic

δW= μoV HdM

Surface

W = – dA

Summary so far

Variables and Parameters

System is only described by its macroscopic variables

Variables: Temperature + one variable for every work term that exchanges energy with the system + one variable for every independent component that can leave or enter the system.

Parameters: Quantities that are necessary to describe the system but which do not change as the system undergoes changes.

e.g. for a closed system: ni are parameters for a system at constant volume: V is parameter

Equations of State (Constitutive Relations): equation between the variables of the system. One for each work term or matter flow

Example of Equations of State

pV = nRT: for ideal gas

= E for uniaxial elastic deformation

M = H: paramagnetic material

Properties and State Functions

U = U(variables) e.g. U(T,p) for simple system

Heat capacity

cV =1n

δQdT

⎛ ⎝

⎞ ⎠ V

cp =1n

δQdT

⎛ ⎝

⎞ ⎠ p

Volumetric thermal expansion

Compressibility

αV =1V

dVdT

⎛ ⎝

⎞ ⎠

p

βV = −1V

dVdp

⎛ ⎝ ⎜

⎞ ⎠ ⎟

T

Some Properties Specific to ideal gasses

∂U∂V

⎛ ⎝

⎞ ⎠ T

=0

1) PV = nRT

2) cp - cv = R

3)

ONLY for IDEAL GASSES

Proof that for ideal gas, internal energy only depends on temperature dU =ncvdT

The Enthalpy (H)

H = U + PV

Gives the heat flow for any change of a simple system that occurs under constant pressure

Example: chemical reaction

Example: Raising temperature under constant pressure

dHp =δQp =ncp dT

cp =∂HdT

⎛ ⎝

⎞ ⎠ p

H

TT

m

Δ H

m

C

,p l

C

,p s

Enthalpy of Materials

is always relative

Elements: set to zero in their stable state at 298K and 1 atm pressure

Compounds: tabulated. Are obtained experimentally by measuring the heat of formation of the compounds from the elements under constant pressure.

The Enthalpy (H)

H = U + PV

Gives the heat flow for any change of a simple system that occurs under constant pressure

Example: chemical reaction

Example: Raising temperature under constant pressure

dHp =δQp =ncp dT

cp =∂HdT

⎛ ⎝

⎞ ⎠ p

H

TT

m

Δ H

m

C

,p l

C

,p s

Enthalpy of Materials

is always relative

Elements: set to zero in their stable state at 298K and 1 atm pressure

Compounds: tabulated. Are obtained experimentally by measuring the heat of formation of the compounds from the elements under constant pressure.

Entropy and the Second Law

dS ≥δQT

There exists a Property of systems, called the Entropy (S), for which holds:

How does this solve our problem ?

The Second Law leads to Evolution Laws

System

S

time

equilibrium

Isolated system

Evolution Law for constant Temperature and Pressure

TdS≥dU+pdV

⇒ d(TS) ≥dU+d(pV)

d(U +pV−TS)≤0

dG ≤ 0

G (Gibbs free energy is the relevant potential to determine stability of a material under constant pressure and temperature

Interpretation

For purely mechanical systems: Evolution towards minimal energy.

Why is this not the case for materials ?

Materials at constant pressure and temperature can exchange energy with the environment.

G is the most important quantity in Materials Science

determines structure, phase transformation between them, morphology, mixing, etc.

System

G

time

equilibrium

T,P

Phase Diagrams: One component

Describes stable phase (the one with lowest Gibbs free energy) as function of temperature and pressure.

WaterCarbon

Temperature Dependence of the Entropy

dS=δQT

dS( )p =δQ( )p

T=

Cp dT( )p

T∂S∂T

⎛ ⎝

⎞ ⎠ p

=Cp

T

S

TTm

Tb

Temperature Dependence of the Entropy

dS=δQT

dS( )p =δQ( )p

T=

Cp dT( )p

T∂S∂T

⎛ ⎝

⎞ ⎠ p

=Cp

T

S

TTm

Tb

What is a solution ?

SYSTEM with multiple chemical components that is mixed homogeneously at the atomic scale

•Liquid solutions

•Vapor solutions

•Solid Solutions

Composition Variables

MOLE FRACTION:

ATOMIC PERCENT:

CONCENTRATION:

WEIGHT FRACTION:

Xi =ni

ntot

(at%)i =100% Xi

Ci =ni

Vor

Wi

V

wi =Wi

Wtot

Variables to describe Solutions

G=G(T,p,n1,n2, …, nN)

dG=∂G∂T

⎛ ⎝ ⎜

⎞ ⎠ ⎟

p,ni

dT +∂G∂p

⎛

⎝ ⎜

⎞

⎠ ⎟

T ,ni

dp+∂G∂ni

⎛

⎝ ⎜

⎞

⎠ ⎟

p,T

dni

dG=−SdT+V dp+μi dni

Partial Molar Quantity

V i =∂V∂ni

⎛

⎝ ⎜

⎞

⎠ ⎟

T ,p,nj

H i =∂H∂ni

⎛

⎝ ⎜

⎞

⎠ ⎟

T ,p,n j

S i =∂S∂ni

⎛

⎝ ⎜

⎞

⎠ ⎟

T ,p,nj

Partial Molar Quantity

μi ≡∂G∂ni

⎛

⎝ ⎜

⎞

⎠ ⎟

p,T

chemical potential

Partial molar quantities give the contribution of a component to a property of the solution

G =i

∑ μini

V =i

∑ V ini

Properties of Mixing

A,BA

B

Change in reaction:

XA A + XB B -> (XA, XB )

ΔHmix =Hmix −XA H A −XB H B

=Hmix − H A +XB H B −H A( )[ ]

H

HB

HA

XB

Cu-Pd

Ni-Pt

Intercept rule with quantity of mixing

ΔVmix

VB - VB

XBX*BVA - VA

ΔGmix

μB - GB

XBX*B

μA - GA

General Equilibrium Condition in Solutions

μiα =μi

β =μiγ =...

Chemical potential for a component has to be the same in all phases

For all components i

OPEN SYSTEM

Components have to have the same chemical potential in system as in environment

e.g. vapor pressure

Example: Solubility of Solid in Liquid

pure A liquid

B solid

Gliquid

GlBGlA

XB

GsB

Summary so far

1) Composition variables

2) Partial Molar Quantities

3) Quantities for mixing reaction

4) Relation between 2) and 3): Intercept rule

5) Equilibrium between Solution Phases

Standard State: Formalism for Chemical Potentials in Solutions

μi =μi0 +RT ln(ai )

chemical potential of i in solution

chemical potential of i in a standard state

effect of concentration

Choice of standard state is arbitrary, but often it is taken as pure state in same phase.

Choice affects value of ai

Models for Solution: Ideal Solution

An ideal solution is one in which all components behave Raoultian

ai =xi forall i

A,BA

B

ΔGmix =Gmixture−Gcomponents

xAμA +xBμB xAGA +xBGB

ΔGmix =RT xA lnxA +xB lnxB[ ]

ΔGmix =xA μA0 +RT lnaA( )+xB μB

0 +RT lnaB( ) − xAGA −xBGB

Summary Ideal Solutions

ΔGmix =RT xi lnxii

∑

ΔHmix =0

ΔSmix =−R xi lnxii∑

ΔVmix =0

XB

Solutions: Homogeneous at the atomic level

Random solutionsOrdered solutions

Summary so far

1) Composition variables

2) Partial Molar Quantities

3) Quantities for mixing reaction

4) Relation between 2) and 3): Intercept rule

5) Equilibrium between components in Solution Phases

For practical applications it is important to know relation between chemical potentials and composition

Obtaining activity information: Experimental

e.g. vapor pressure measurement

Pure substance i Mixture with component i in it

vapor pressure pi* vapor pressure pi

Δμi =RT lnpi

pi*

⎛

⎝ ⎜

⎞

⎠ ⎟

Δμi =RT ln ai( )

pi

pi* = ai

Obtaining activity information: Simple Models

Raoultian behavior

Henry’s behavior

ai =xi

ai =kxi

Usually Raoultian holds for solvent, Henry’s for solute at small enough concentrations.

Intercept rule

General Equilibrium Condition in Solutions

μiα =μi

β =μiγ =...

Chemical potential for a component has to be the same in all phases

For all components i

OPEN SYSTEM

Components have to have the same chemical potential in system as in environment

e.g. vapor pressure

Raoultian case and Real case

Review•At constant T and P, a closed system strives to minimize its Gibbs free energy: G = H - TS

•Mixing quantities are defined as the difference between the quantity of the mixture and that of the constituents. All graphical constructions derived for the quantity of a mixture can be used for a mixing quantity with appropriate adjustment of standard (reference) states.

G AG B

X B0 1

G

X B0 1

Δ Gmix

μA

- GA

μB

- GB

Review (continued)

Some simple models for solutions can be me made: Ideal solution, regular solution.Many real solution are much more complex than these !

ΔGmix= ΔHmix – TΔSmix

= ω xB(1−xB) + RT xA lnxA +xB lnxB[ ]

Regular Solution

The Chord Rule

What is the free energy of an inhomogeneous systems ( a system that contains multiple distinct phases) ?

CHORD RULE: The molar free energy of a two-phase system as function of composition of the total system, is given by the chord connecting the molar free energy points of the two constituent phases

The Chord Rule graphically

A B

With pure components Components are solutions

overall composition is XB*

XB

overall composition is XB*

G

0 1XB

XB*

G

0 1XB

XB*

XB

Regular Solution Model

ΔGmix= ωxB(1−xB) + RT (1−xB)ln(1−xB) +xB lnxB[ ]

0 XB 1< 0

Regular Solution Model

ΔGmix= ωxB(1−xB) + RT (1−xB)ln(1−xB) +xB lnxB[ ]

0 XB 1

> 0

Effect of concave portions of G

0 XB* 1

Single-Phase and Two-phase regions

0 XB* 1XB

XB

Single-phase

Single-phaseTwo-phase

Two-phase coexistence

When ΔGmix has concave parts (i.e. when the second derivative is negative) the coexistence of two solid solutions with different compositions will have lower energy.

The composition of the coexisting phases is given by the Common Tangent construction

The chemical potential for a species is identical in both phases when they coexist.

Common tangent does not need to be horizontal

0 XB* 1

Temperature Dependence of Two-phase region

0XB

1

ΔHmix

0XB

1

ΔHmix

0XB

1

ΔHmix

Low temperature Intermediate temperature High temperature

Phase Diagram of Regular Solution Model with > 0

0 XB 1

T

Example: Cr-W

Miscibility gap does not have to be symmetric

Lens-Type Diagram

Liquid

Solid

L+S

XB

0 1

Free energy curves for liquid and solid

0

XB

1

G

0

XB

1

G

0

XB

1

G

T > TBM > TA

M TBM > T > TA

M

TBM > TA

M > T

How much of each phase ? - The Lever Rule

Since the composition of each phase is fixed by the common tangent, the fraction of each phase can be determined from requiring that the system has the given overall composition

XB* = fαXB

α +fβ XBβ

1= fα +fβ-> solve for f and f

fα =XB

β −XB*

XBβ −XB

α fβ =XB

* −XBα

XBβ −XB

α

The result is known as the Lever Rule

Lever Rule

x x

x

x x

x

f

f

In a two-phase region chemical potential is constant

0 XB 1

0 XB 1

G

μ

Comparison between Eutectic and Peritectic

L

L

Eutectic Peritectic

Cooling: L -> Heating: -> L +

Nucleation 

(1) The structure of the liquid Many small closed packed solid clusters are present in the liquid. These clusters would form and disperse very quickly. The number of spherical clusters of radius r is given by 

 nr : average number of spherical clusters with radius r.

n0: total number of atoms in the system.

ΔGr: excess free energy associated with the cluster.

n nG

kTrr= −⎛

⎝⎞⎠0 xp

Δ

(2) The driving force for nucleation

 The free energies of the liquid and solid at a temperature T are given by GL = HL- TSL and GS = HS- TSS

 GL and GS are the free energies of the liquid and solid respectively.

HL and HS are the enthalpy of the liquid and solid respectively.

SL and SS are the entropy of the liquid and solid respectively.

 At temperature T, we have ΔGV = GL- GS = HL-HS-T(SL-SS) =ΔH-TΔS,

 ΔGV: volume free energy.

where ΔH and ΔS are approximately independent of temperature.

For a spherical solid of radius r, 

Δ ΔG r G rV SL= − +4

343 2π π γ

For a given ΔT, the solid reach a critical radio r*, whend G

dr

( )Δ=0

rG

SL

V

* =2γΔ

ΔΔ

GG

SL

V

∗ =16

3

3

2

πγ

( )

ΔΔ

G LT

TV mm

=

rT

L TSL m

m

* =2 1γ

Δ( )22m

3SL

T

1

L3

Tm16G

Δ

πγ=Δ ∗

)(

We get

If

and

then,

and

When r < r, the solid is not stable, and when r > r, the solid is stable.

The effect of temperature on the size of critical nucleation and the actual shape of a nucleus.

(4) Heterogeneous nucleation If a solid cap is formed on a mould, the interfacial tensions balance in the plane of the mould wall, or

γSL, γSM and γML: the surface tensions of the solid/liquid, solid/mould and mould/liquid interfaces

respectively.

γ γ γ θML SM SL= + coscosθ γ γγ=−ML SM

SL

The excess free energy for formation of a solid spherical cap on a mould is ΔGhet = -VSΔGV + ASLγSL + ASMγSM - ASMγML

 Where VS: the volume of the spherical cap.

ASL and ASM: the areas of the solid/liquid and

solid/mould interfaces. It can be easily shown that,   where

Δ ΔG r G r fhet V SL= − + ×( ) ( )43 43 2π π γ θ

f ( )( cos cos )θ θ θ

=− +2 3

43

(c) Nucleation rate and nucleation time as a function of temperature

The overall nucleation rate, I, is influenced both by the rate of cluster formation and by the rate of atom transport to the nucleus, which are both influenced in term by temperature. TTT diagram gives time required for nucleation, which is inversely proportional to the nucleation rate.

600620640660680700720740760780800820840860

100 1000 10000

Actual examples of TTT diagram

Nucleation in Solid

Interface structure

Coherency loss

24

3

μ

γ= st

critr

Homogeneous Nucleation

(a) Homogeneous nucleation

SV GVAGVG ΔΔΔ +γ+−=

The free energy change associated with the nucleation process will have the following three contributions.

1 . At temperatures where the phase is stable, the creation of a volume V of will cause a volume free energy reduction of VΔGv.  

2. Assuming for the moment that the / interfacial energy is isotropic the creation of an area A of interface will give a free energy increase of Aγ.

3. In general the transformed volume will not fit perfectly into the space originally occupied by the matrix and this gives rise to a misfit strain energy ΔGv, per unit volume of . (It was shown in Chapter 3 that, for both coherent and incoherent inclusions, ΔGv, is proportional to the volume of the inclusion.) Summing all of these gives the total free energy change as

If we assume the nucleus is spherical with a radius r, we have

γp+−p−= 23 43

4rGGrG SV )( ΔΔΔ

)(*

SV GGr

ΔΔ −γ

=2

2

3

3

16

)( SV GGG

ΔΔΔ

−

pγ=∗

Similarly we have

Nucleation at Grain Boundary

The excess free energy associated with the embryo at a grain boundary will be given by

ΔG = -VΔGv + Aγ - Aγ

Where V is the volume of the embryo, A is the area of / interface of energy created, and

A the area of / grain boundary of energy destroyed during the process.

γγ

=ϑ2

cos

Fick I

It would be reasonable to take the flux across a given plane o be proportional to the concentration gradient across the plane:

⎥⎦

⎤⎢⎣

⎡∂∂

−=x

cDJ

J: the flux, [quantity/m2s]

D: diffusion coefficient, [m2/s]

⎥⎦

⎤⎢⎣

⎡∂∂x

cConcentration gradient, [quantity /m-4]

Here, quantity can be atoms, moles, kg etc.

Fick II:

2

2

x

CD

t

C

∂

∂=

∂

∂

1. Thin Film (Point Source) Solution

M( )x

X

1D

X

2Dy

M( ) x per unit y

X

3D

M( ) x per unit area

c(x,t) =M

4πDtexp−

x2

4Dt

⎛

⎝ ⎜

⎞

⎠ ⎟

assume boundaries at infinity c(x,0) =Mδ(x)

c(∞,t)=c(−∞,t)=0

Solution to Fick II

c(x,t) =M

4πDtexp−

x2

4Dt

⎛

⎝ ⎜

⎞

⎠ ⎟

Measurement of diffusion coeffiecient

c(x,t) =M

4πDtexp−

x2

4Dt

⎛

⎝ ⎜

⎞

⎠ ⎟

Dt4

xtconsc

2−= tanln

If c versus x is known experimentally, a plot of ln(c) versus x2 can then be used to determine D.

x

Error functions

2π

exp(−x2)

2π

exp(−u2)0

x

∫ du =erf(x)

)()(

)()(

)(

)(

zerfzerf

xerf1xerfc

1erf

00erf

−=−−=

=∞=

The final solution should be:

⎥⎦

⎤⎢⎣

⎡⎟⎠

⎞⎜⎝

⎛+=

Dt2

xerf1

2

ctxc

'),(

( )

⎥⎥⎦

⎤

⎢⎢⎣

⎡ −−

p= ∫∞ d

Dt4

x

Dt2

ctxc 0

2exp

'),(

⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+=

Dt2

xerf1Ctxc s),(

When x=0, the composition is always kept at c=c’/2

⎥⎦

⎤⎢⎣

⎡⎟⎠

⎞⎜⎝

⎛+=

Dt2

xerf1

2

ctxc

'),(

erf(-z)=-erf(z)

⎥⎦

⎤⎢⎣

⎡⎟⎠

⎞⎜⎝

⎛−=

Dt2

xerf1Ctxc s),(

Diffusion Mechanis

Interstitial mechanism Self Diffusion with Vacancy

Ring Mechanism Interstitialcy

These are the two most common mechanisms

A Random Jump Process

1B1 n6

1J Γ=

2n

6

1J B2 Γ=

)( 21B21B nn6

1JJJ −Γ=−=

⎟⎠

⎞⎜⎝

⎛∂∂

−=−x

C2C1C B

BB )()(

x

C

6

1J B2

BB ∂∂

⎟⎠

⎞⎜⎝

⎛ Γ−=

Diffusion by Vacancy mechanism

⎟⎠

⎞⎜⎝

⎛ Δ−υ=Γ

RT

GXz m

vB exp

⎟⎠

⎞⎜⎝

⎛−=RT

GX v

v exp

2BA 6

1D Γ=

⎟⎠

⎞⎜⎝

⎛ Δ+Δ−⎟

⎠

⎞⎜⎝

⎛ Δ+Δυ=

RT

HH

R

SSz

6

1D vmvm2

A expexp

⎟⎠

⎞⎜⎝

⎛−=RT

QDD 0A exp

⎟⎠

⎞⎜⎝

⎛ Δ+Δυ=

R

SSz

6

1D vm2

0 exp

Effeect of temperature on diffusion

⎟⎠

⎞⎜⎝

⎛−=RT

QDD 0A exp

Effect of defect on diffusion