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Atomic Transport

&

Phase Transformations

Lecture 2

PD Dr. Nikolay Zotov

zotov@imw.uni-stuttgart.de

2

Part I Alloy Thermodynamics

Lecture Short Description

1 Introduction; Review of classical thermodynamics

2

3 Phase equilibria, Classification of phase transitions

4 Thermodynamics of solutions I

5 Thermodynamics of solutions II

6 Binary Phase Diagrams I

7 Binary Phase Diagrams II

8 Binary Phase Diagrams III

9 Order –Disorder Phase Transitions

Atomic Transport & Phase Transformations

3

Lecture I-2 Outline

Second Law of Thermodynamics

Entropy

Third Law of Thermodynamics

Entropy – Statistical Thermodynamic Treatement

Gibbs Energy

Maxwell Relations

Temperature Dependences

Chemical potential

Partial molar properties

Atomic Transport & Phase Transformations

4

State Variables

Temperature T

Pressure p

Volume V

Adiabatic expansion of gasses

Heat transfer from hotter to colder parts of a body

Question – Are these all thermodynamic state variables?

5

Second Law of Thermodynamics

Definition:

There exist a state variale, the Entropy (S), such that for all types of processes

and all systems,

DS ≥ Q/T; DS ≥ Q/T; (1)

Implications:

# For isolated systems (δQ = 0) the entropy can only increase;

dS ≥ 0

dS/dt ≥ 0

# Once (dS/dt) becomes zero, this means that S has stopped increasing and

Equilibrium is reached.

# Reversible Process: DS = δQ/T = (DU – δW)/T (1‘)

Equality in Eq. (1) is reached for reversible processes.

‘Clausius 1865-67: ἐντροπῐᾱ́ (turning to)’

6

Equivalent Statements:

# “The entropy of the universe tends to a maximum” (Clausius)

# “Measure of how much energy is spread out during a process’’

# ‘’Entropy is a measure of the amount of energy that

cannot be transformed into work (1’)’’

Second Law of Thermodynamics

Entropy of open system:

DS = DSinter + DS*; DS * - Entropy transfered from the surrounding to the system

DS* > 0 or DS* < 0

7

Second Law of Thermodynamics

Implications:

# since S and V are state variables: U = U(S,V)

dU = (U/S)VdS + (U/V)SdV

# if work is done only by change of volume W = -pdV;

From the 1st Law: dU = Q - W = Q + pdV;

From the 2nd Law: Q = TdS (reversible, quasi-static

processes)

dU = TdS + pdV

# T = (U/S)V; p = (U/V)S

8

Relation between Heat Capacity and Entropy

V = const. (Reversible isochoric process)

dS = Q/T, but CV = (Q/dT)V ;

dS = (CV/T) dT DS = CV/T’dT’ + K

p = const. (Reversible isobaric process)

dS = Q/T, but Cp = (Q/dT)p;

dS = (Cp/T) dT DS = Cp/T’ dT’+ K

9

Third Law of Thermodynamics

Definition (1):

The entropy change of a condensed-matter system,

undergoing a reversible process, approaches zero

as the temperature approaches 0 K (DS → 0 as T → 0).

Definition (2):

The entropy of a perfect crystal at T = 0 is zero.

Implications:

# All perfect crystals will have at T = 0 the same entropy (S = 0);

# Disordered crystals and amorphous materials have a residual

entropy at T = 0;

# The 3rd law provides an absolute scale for entropy:

S = S(To) + To

T(C/T) dT = S(0) + (C/T) dT =

o

T (C/T) dT

10

EntropyEntropy values at T =298 oC

Species So (J/mol.K)

Diamond 2.38

Graphite 5.74

Sodium 51.2

Potassium 65.2

Sulfur (S) 31.8

Silver (Ag) 42.6

He (g) 126.0

Xe (g) 169.6

H2O (l) 69.9

H2O (g) 188.7

11

Third Law of Thermodynamics

Residual entropy at 0 K

N2O ~5.8 J/K.mol

H2O ~3.37 J/K.mol

CO ~ 5.8 J/K.mol

Am-Se ~ 3.95 J/mol.K (P. Richet 2001)

12

Entropy

in Statistical Thermodynamics

Implications:

# The entropy is considered as a measure of Disorder.

Perfect solid Glass or Liquid Gas

Structural Disorder

13

Entropy

in Statistical Thermodynamics

Types of states:

# Thermodynamic (Macrostate): State which can be described by only

a few state variables (e.g. p, T, V, S)

# Statistical (Microstate): State, which is described by large

number of quantities for all the individual species, constituting the system.

Examples: Coordinates and velocities of atoms(molecules);

Magnetic moments (spins);

Polarization vectors (dipoles), etc.

M ~ 0 M > 0

14

Entropy

in Statistical Thermodynamics

Different Microstates could lead to the same Macrostate

Boltzman‘s Equation: S = kB ln(W)

W is the number of microstates leading to a given

macrostate

Implications:

# At T = 0 (most) systems are in their ground state (W = 1) → S = 0

# Additivity of the Entropy: (S1, W1) and (S2, W2)

New system with W = W1xW2 microstates

S = kBln(W) = kBln(W1 W2) = S1 + S2;

# In condensed-matter systems: S = Sele + Svib + Sconf + ∙∙

15

Entropy

in Statistical Thermodynamics

Contributions to the Entropy

# Static (Configurational) contributions:

Species Effect_____________________________

atoms Distribution of atoms over different sites

electrons Distribution over different (degenerate)

elecronic states (levels)

electron spins Different orientations of the spins

(Orientational order/disorder)

# Dynamic contributions:

Species Effect_____________________________

atoms lattice vibrations (Phonons)

electrons Excitations across the Fermi surface

electron spins spin waves (magnons)

16

Entropy

in Statistical Thermodynamics

A/ Static electon entropy

# Transition elements (Ti,Fe,Mn)

have only d-electons in their valence shell

and unfilled d-orbitals;

# Transition metals easily oxidize M+p;

# dn Configuration;

n = group number – oxidation state

Ti3+; group 4; n = 4 – 3 = 1; d1 Configuration

17

Octahedral

coordination

Tetrahedral

coordination

Entropy

in Statistical Thermodynamics

Ti3+ : d1 ConfigurationCrystal-field theory

18

Entropy

in Statistical Thermodynamics

Definitions:

M - Number of species in the system

r Number of microstates f of a given species (f1, f2, ... fr)

A macrostate J is defined by a r-dimensional distribution function

(m1, m2, . . . , mr)J; mi – Number of species in microstate fi

belonging to macrostate J

(Occupation number of a given species state)

W = M!/ m1 !m2 ! … mr!

M = S1

rm

i;

19

Entropy

in Statistical Thermodynamics

Nature fo the microstates fi:

Electonic energy levels

Vibrational energy levels

Rotational energy levels

Spin two-level systems

20

Entropy

in Statistical Thermodynamics

S = kB ln (W) = kB ln (M!/ m1 !m2 ! … mr!)

= kB [ln M! –S ln(mi!)]

= kB {Mln(M) – M – S[miln(mi) – mi]} =

= kB { Mln(N) – M – S mi ln(mi) + M} =

= kB { S mi [ ln(M) – ln(mi)]}

Sterling’s approximation ln(x!) xln(x) – x

for large number of species

21

S = - kB S mi ln(mi/M)

S = S(J) = S(m1,m2, …mr)

# Condition for maximum of the entropy

dS = 0

dS = = -kBS d{mi ln(mi/M)} =

= -kBS d{miln(mi) – miln(M)} = ∙∙∙ = -kBS ln(mi/M)dmi;

# Additional condition M = const → dM = S dmi = 0

# Lagrange equation (method):

dS + adM = 0 →

Entropy

in Statistical Thermodynamics

22

Entropy

in Statistical Thermodynamics

In the equilibrium macrostate all

microstates are equally occupied

For a system without change of the

total number of species (M = const)

m1 = m2 = … = M/r

a = -kBln(r)

S{-kBln(mi/M) + a }dmi = 0

-kBln(mi/M) + a = 0 for every i

start

end

23

In an isolated system the internal energy U is also constant.

Condition for constant internal energy: dU = 0

U ~ S eimi → dU = S eidmi = 0

Lagrange equation (method):

dS + adM + ßdU = 0

S{-kBln(mi/M) + a + ßei}dmi = 0

-kBln(mi/M) + a + ßei = 0 for every i;

mi /M = exp( - ei/kBT) / Z; Z = Sexp( - ei/kBT) Partion function

Entropy

in Statistical Thermodynamics

24

Entropy

in Statistical ThermodynamicsBoltzman distribution

pi = exp( - ei/kBT) / Z; gives the probability of finding species in a given microstate

fi, characterized by energy ei.

Validity (Limits):

# species in different states are non-interacting;

# the microstates fi (ei) are not changing;

# valid for system with very large number of species (particles)

25

Entropy

in Statistical Thermodynamics

S = - kB S mi ln(mi/M) = - kB S mi ln [(exp(-ei/kBT)/Z] =

= - kB S mi [-ei/kBT – ln(Z)]

= 1/T S mi ei + kBMln(Z); S = U/T + kBMln(Z) ;

26

Gibbs EnergyThermodynamic Functions of State

System of constant composition:

Gibbs Energy (G): G = H – TS

Most convinient state function at constant p and/or T

Differential: dG = (∂G/∂p)Tdp + (∂G/∂T)pdT

Local Minimum: (∂G/∂p) = 0 ; T = const

(∂G/∂T) = 0 ; p = const (dG =0)dG =0

dG =0

dG =0

G

Path variable

Stable (Equilibrium) State G*

at constant p:

(∂G/∂T)p = 0

(∂2G/∂2T)p > 0

G ≥ G*

J. Gibbs

27

Implications:

# The Entropy of an isolated system with fixed U has a maximum value

in the equilibrium state

G = H –TS = U + pV – TS U - TS

Path

G

Path

S

Gibbs Energy

G*

Units: J (Joule; J/mol)

V = 1 cm3 = 1x10-6 m3

p = 1.01x105 Pa = 1.01x105 kg/ms2;

pV = 0.1 J

Maximum Entropy Principle

28

Helmholz Free Energy (F)

Definition: F = U – TS

Differential: dF = dU – TdS - SdT

Implications:

# for reversible processes

dF = dU – TdS – SdT = (δQ – δW) – T δQ/T – SdT

# … at constant temperature

dF = δQ – δW – δQ

= - δW; dF is equal to the total (reversible) work done on the system

H. von Helmholz

F = U – TS = U - T[U/T + kBMln(Z) ]; F = - kBMln(Z)

29

Gibbs Energy

Implications:

# if there is no other types of work (δW‘ = 0)

V = (∂G/∂p)T and S = - (∂G/∂T)p;

# at constant pressure and temperature dG = – δW‘

# Differential:

dG = dH – (SdT + TdS) = d(U + pV) – (SdT + TdS) =

= dU + Vdp + pdV – SdT – TdS =

= TdS – pdV – δW‘+ Vdp + pdV – SdT – TdS

dG = Vdp – SdT – δW‘dG = (∂G/∂p)Tdp + (∂G/∂T)pdT

30

From the internal energy:

(∂p/∂S)V = - (∂T/∂V)S

From the Helmholz free energy:

(∂S/∂p)T = - (∂p/∂T)V

From the Gibbs free energy:

(∂S/∂p)T = - (∂V/∂T)p

Maxwell Relations

Z = Z(X,Y)

∂(∂Z/∂X)/∂Y = ∂(∂Z/∂Y)∂X

31

Gibbs Energy

# Gibbs – Helmholz Equation

G = H – TS = H + T (∂G/∂T)p;

GdT = HdT + TdG

(TdG – GdT)/ T2 = - HdT/T2;

(1/T)dG – GdT/ T2 = - HdT/T2;

d(G/T)/dT = - H/T2

32

Equations-of-State (EOS)

In the Gibbs energy description the volume V is a function of p and T.

V = (∂G/∂p)T , also V = V(p,T) – Equation of state

Murnaghan EOS

V(p) = Vo [1 + p(B‘/B)] (1/B‘)

B‘ – first derivative of B with respect to p

B = Bo + B‘p (K = Ko + Ko‘p)

Diamond Anvil Cell (DAC)

Birch - Murnaghan EOS

P(V) = 3Bo/2[(Vo/V)7/3 – (Vo/V)5/3] {1 +

¾ (Bo‘ – 4)[(Vo/V)2/3 – 1]}

Bulk Modulus

B = - V(∂p/∂V)T = - V/ (∂V/∂p)T

33

Equations-of-State (EOS)

Powders

Re gaskets

Ne/He gas

as pressure

medium

Synchrotron

Radiation

XRD

Dorfman et al (2012)

34

Equations-of-State (EOS)

35

Temperature Dependences (1)

At p = const

Cp = Cv + a2VTB(T)

T > 298 K

DH = H(T) - Ho = ∫298

TCp(T‘)dT‘ (Kirchhof‘s law)

S(T) = So + ∫0

T(Cp/T‘)dT‘ = ∫

0

T(Cp/T‘) dT‘

G(T) = H – TS = Ho + ∫298

TCp(T‘)dT‘ - T ∫

298

T[Cp(T‘)/T‘] dT‘

Standart reference state

T = 298 oC

P = 1 atm = 101325 Pa

36

Temperature Dependences (2)

0 200 400 600 800 10000

5

10

15

20

25

CV

(J/m

ol.K

)

Temperature (K)

a-Sn

Debey Model

QD ~ 230 K

0 200 400 600 800 10000,00

0,05

0,10

0,15

0,20

0,25

En

tro

py (

J/m

ol.K

)

Temperature (K)

S(T) = ∫0

T(Cp/T‘)dT‘

37

Temperature Dependences (2)

a-Sn

Debey Model

QD ~ 230 K

G is negative, decreases with increasing

Temperature and the Slope (∂G/∂T)p = -S.

200 400 600 800 1000

-150

-100

-50

0

50

100

150

200

250

En

erg

y (

J/m

ol)

Temperature (K)

H

TS

G

38

Temperature Dependences (3)

Empirical Model ( T > 300 K)

cp (J/mol.K) = a + bT + c/T2;

Species a bx103 c x 105

Al 20.7 12.3

Cu 22.6 5.6

Fe 37.12 6.17

Ag 21.3 8.5 1.5

Si 23.9 2.5 -4.1

Temperature Dependences (3)Heat capacities

DeHoff (2008)

39

300 400 500 600 700 800 900

24

26

28

30

32

CP

(J/m

ol.K

)

T (K)

Si

Al

Temperature Dependences (3)Heat capacities

40

Gibbs Energy

System with changing composition*:

Gibbs Energy (G): G = G(p,T,n1,n2,…)

ni – Number of moles of species i

Differential:

dG = Vdp – SdT + Si (∂G/∂ni) dni; i = 1, 2 .. , C

Chemical Potential:

µi = (∂G/∂ni) p,T, n≠ni;

* Open systems (Diffusion)

Closed systems with chemical reactions

N = S ni

41

Gibbs Energy

Partial Molar Properties

System with (a possible change) of composition:

For any extensive state function A = A(p,T, n1, n2,…):

dA = (A/T)p,n dT + (A/p)T,n dp + S(A/ni)T,p,n≠ni dni;

Ai = (A/ni)T,p,n≠ni Partial molar property of A for component i.

For a system at T = const and p = const.

dA = S(A/ni)T,p,n≠ni dni = S A i dni ;

A = S A i dni = S Ā i ni ; The total property A is a weighted

sum of the partial molar properties

42

Gibbs Energy

Partial Molar Properties

Gm = S µk nk; Gk = µk;

All general thermodynamic relations can be expressed

in terms of the partial molar properties;

µk = Gk = Hk - TSk;

Vm = S Vk nk; Vk = (µk/p)T,n

Sm = S Sk nk; Sk = -(µk/T)p,n

Hm = S Hk nk; Hk = µk - T(µk/T)p,n

43

Gibbs Energy

Partial Molar Properties

Generalized Gibbs – Duhem Equation

dA = d(S Ā i ni) = S d(Ā ini) = S [Āidni + S nid Āi ]

= dA + S nid Ā i

S nidĀi = 0 Not all partial molar properties are

independent

System with C = 2 components:

n1 + n2 = N; Molar fractions: x1 = n1/N; x2 = n2/N; x1 + x2 = 1

dµ1 = - (x1/1-x1) dµ2 ;

A ≡ G

S ni dµi = 0