Atomic Transport Phase Transformations - uni-stuttgart.de · Third Law of Thermodynamics ... From the 2nd Law: Q = TdS (reversible, quasi-static processes) ... Boltzman‘s Equation:

Atomic Transport

&

Phase Transformations

Lecture 2

PD Dr. Nikolay Zotov

[email protected]

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


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’)’’


Entropy of open system:

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

DS* > 0 or DS* < 0

7


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


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


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


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


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


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


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


Ti3+ : d1 ConfigurationCrystal-field theory

18

Entropy


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


Nature fo the microstates fi:

Electonic energy levels

Vibrational energy levels

Rotational energy levels

Spin two-level systems

20

Entropy


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


22

Entropy


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


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


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


Powders

Re gaskets

Ne/He gas

as pressure

medium

Synchrotron

Radiation

XRD

Dorfman et al (2012)

34


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


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


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


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


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


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

Documents

Atomic Transport Phase Transformations - uni-stuttgart.de · Third Law of Thermodynamics ... From the 2nd Law: Q = TdS (reversible, quasi-static processes) ... Boltzman‘s Equation: