Rovnovážné diagramy, fázové pravidlo,
Fe-C (Fe3C) diagram
… v souvislosti s mechanickými vlastnostmi
A. Kroupa, T. Kruml, Ústav fyziky materiálů, AV ČR
kroupa@ipm.cz, kruml@ipm.cz
Institute of Physics of Materials, AS CR, Brno, Czech Republic
Contents
• Laws of Thermodynamics
• Essentially a reflection of what we see around us
• Thermodynamic quantities
• Heat capacity
• Enthalpy
• Gibbs energy
• Entropy
• Properties pure substances e.g. elements
• Properties of mixtures
• Types of phase diagrams
• Fe-Fe3C (Fe-C) phase diagrams
• TTT and CCT diagrams
• Software – thermodynamics made easy
Institute of Physics of Materials, AS CR, Brno, Czech Republic
2
What is a Phase Diagram?
A phase diagram is the graphical representation of the state of a
system in thermodynamic equilibrium as a function of selected
state variables
• Phase diagrams are strictly connected with the rules of
thermodynamics
• Phase diagrams differ from “normal” (property) diagrams.
They carry a different type of information
• Phase diagrams are powerful tools to help materials
scientists to solve specific problems related to the
equilibrium state of their material system
3
What is a Phase Diagram?
A phase diagram is the graphical representation of the state of a
system in thermodynamic equilibrium as a function of selected
state variables
• Phase diagrams are strictly connected with the rules of
thermodynamics
• Phase diagrams differ from “normal” (property) diagrams.
They carry a different type of information
• Phase diagrams are powerful tools to help materials
scientists to solve specific problems related to the
equilibrium state of their material system
4
What is a Phase Diagram?
A phase diagram is the graphical representation of the state of a
system in thermodynamic equilibrium as a function of selected
state variables
• Phase diagrams are strictly connected with the rules of
thermodynamics
• Phase diagrams differ from “normal” (property) diagrams.
They carry a different type of information
• Phase diagrams are powerful tools to help materials
scientists to solve specific problems related to the
equilibrium state of their material system
5
Some Terms and Symbols
A system consists of components (element or specie – ion, molecule, ...)
A phase is of homogeneous part of the system in equilibrium. It is
determined by its crystal structure. It can also be liquid or gaseous.
A constituent is a species that can be treated as an independent entity to
describe the constitution of a phase. E.g.: element, stoichiometric
compound (e.g. oxide), a molecule in the gas phase, or a defect in a
crystalline phase
G total Gibbs energy of a system
Gm molar Gibbs energy of a system
Gm
α molar Gibbs energy of the phase α
Gi
α partial Gibbs energy of component i
in the phase α
0Gi
α Gibbs energy of the pure component i
in the phase α
6
S
y
s
t
e
m
P
h
a
s
e
Thermodynamics: State Variables
The state of a system is described by a set of state variables
• Intensive state variables: p, T, µi, ….
• do not depend on the size of the system. In equilibrium they
are the same everywhere in the system
• Extensive state variables: N, Ni, V, S, G, H, …
• are proportional to the size of the system
• Molar variables: Gm = G/N, Sm = S/N….
• derived from extensive variables by dividing them by N.
They are not dependent on the size of the system, but
have different values for different phases in equilibrium
7
Thermodynamic Equilibrium Conditions
The intensive state variables are constant in the entire system:
T=const., p = const., µi = const. for all components i in all phases
The state function G (Gibbs energy) reaches a minimum at
T, p, Ni = const.
The number of state variables necessary to describe the system is
therefore the number of independent components (c) + 2
The Gibbs phase rule relates this to the number of phases present in
the system (p) and the degree of freedom (f)
p + f = c + 2
8
• Interpretation of what we see around us
• 1st - In any process or event, energy can only be converted
from one form to another; the total energy must remain
constant
• perpetual motion machines of the first kind are impossible
• 2nd - It is not possible to convert all energy into useful work
• perpetual motion machines of the second kind are impossible
• 3rd - It is not possible to reach a temperature of absolute zero
• The entropy of a perfect crystal at absolute zero is exactly
equal to zero.
Thermodynamics
Institute of Physics of Materials, AS CR, Brno, Czech Republic
9
0th - Materials in thermal equilibrium have the same
temperature
•4th ? – Materials at infinite high temperature behave like ideal
systems
Leads to definitions of all thermodynamic
quantities
Thermodynamics enables us to understand
why things happen in the way they do and to
make predictions about what is likely to happen
under conditions yet to be studied
Thermodynamics
Institute of Physics of Materials, AS CR, Brno, Czech Republic
10
• For most materials we consider constant pressure
• Heat capacity is amount of heat required to raise
temperature of the unit amount of material by 1°
• Measure of capacity of material to absorb heat
• Differs from one phase to another
• Also a measure of change within the material
• Monotonous change in the heat capacity indicates stability
• Big changes with temperature indicates bond breaking
• Magnetic transition, melting of alloy
• Gas speciation
• Electronic effects in gas species
Institute of Physics of Materials, AS CR, Brno, Czech Republic
Heat capacity
11
Tfus =
505.078
Ttrs
Institute of Physics of Materials, AS CR, Brno, Czech Republic
Heat capacity of Sn for different phases
12
• Amount of heat stored in a material H = U +pV
• Has no absolute value – possible to talk only about
changes in enthalpy or differences between two states
• Need standard reference state (SER – stable phase at room temp.
and standard pressure)
• Integrated heat capacity between temperatures
• Different phases have different enthalpies
• Melting requires energy to be absorbed
• Substances reacting together give off or absorb heat
At constant volume, the heat absorbed by the system = ΔU
At constant pressure, the heat absorbed by the system = ΔH
Institute of Physics of Materials, AS CR, Brno, Czech Republic
13
Enthalpy
ΔfusH
ΔtrsH
Institute of Physics of Materials, AS CR, Brno, Czech Republic
14
Enthalpy of Sn relative to T=298K
• Any spontaneous change takes place to increase the stability of the
system
• For problems in materials science we tend to express this in terms of
Gibbs energy
• Different phases have different Gibbs energies
• At constant temperature and pressure the state with lowest Gibbs
energy is the equilibrium state
• Like enthalpy, Gibbs energy has no absolute value i.e. it is possible
to talk only about changes in the Gibbs energy
• Depends on temperature
• Material melts when you raise the temperature, i.e. difference in Gibbs
energy between liquid and solid changes with temperature
• Depends on pressure
• Reducing the pressure causes a liquid to boil
• G = U – T S + P V Play off between lowest energy state (enthalpy) and
increased disorder (entropy)
Institute of Physics of Materials, AS CR, Brno, Czech Republic
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Gibbs energy (G, state function with respect to P, T, ni)
Tfus
Ttrs
Institute of Physics of Materials, AS CR, Brno, Czech Republic
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Gibbs energy of phases of Sn relative to BCT
Derivatives of the Gibbs Energy
17
• Entropy associated with disorder
• Effect more significant at higher temperatures
• Entropy has definite value
• At low temperatures the entropy should tend towards
zero
• The effect of entropy usually offsets enthalpy
• Pure substance at transition temperature, the enthalpy change
equals entropy change * T
Institute of Physics of Materials, AS CR, Brno, Czech Republic
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Entropy
TD-19
Přehled stavových funkcí:
• Some elements mix together (eg in the liquid phase)
without giving out or absorbing heat – said to mix
ideally eg Co and Ni
• Normally there is a net heat effect which could be
either negative (giving out heat) or positive (heat is
absorbed) … or even more complex
Institute of Physics of Materials, AS CR, Brno, Czech Republic
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What happens when things mix?
regular solution negative
enthalpy of
mixing
Positive enthalpy of
mixing
Institute of Physics of Materials, AS CR, Brno, Czech Republic
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Generally two contributions
First from random mixing of the elements (ideal contribution)
“So called” excess entropy of mixing - really to account for deviations
from ideality
Institute of Physics of Materials, AS CR, Brno, Czech Republic
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Entropy of mixing
TD-23
Konfigurační entropie systému:
opět máme krystal s N atomy: n atomů A a (N-n) atomů B
[ ]
[ ]
!
ln ,
!( )!
ln ln ( )ln( )
ln (1 )ln(1 )A A A A
N
S k w w
n N n
S k N N n n N n N n
S kN x x x x
= =
−
= − − − −
=− + − −
Institute of Physics of Materials, AS CR, Brno, Czech Republic
24
Gibbs energy of mixing G = H – T S
Institute of Physics of Materials, AS CR, Brno, Czech Republic
25
Gibbs energies of two phases
Institute of Physics of Materials, AS CR, Brno, Czech Republic
26
Calculation of binary phase equilibria
.... over a range of
temperatures
Institute of Physics of Materials, AS CR, Brno, Czech Republic
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Calculation of binary phase equilibria
Institute of Physics of Materials, AS CR, Brno, Czech Republic
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Calculation of binary phase equilibria
Phase diagram and property diagram
29
Types of Phase Diagrams
A system in equilibrium is defined by a set of c+2 state variables. By
varying two of them and keeping the others constant, a twodimensional
phase diagram can be drawn.
The maximal dimension of the phase diagram is c+1, because one
extensive variable can be diminished by dividing them by N:
e.g.: T, p, N1…Nj → T, p, x1…xj-1
Types of phase diagrams:
a) two intensive variables
b) one intensive, one extensive (molar) variable
c) two extensive (molar) variables
d) Sections through higher dimensional phase diagrams defined by
special restraints on extensive variables
30
Types of Phase Diagrams
A system in equilibrium is defined by a set of c+2 state variables. By
varying two of them and keeping the others constant, a twodimensional
phase diagram can be drawn.
The maximal dimension of the phase diagram is c+1, because one
extensive variable can be diminished by dividing them by N:
e.g.: T, p, N1…Nj → T, p, x1…xj-1
Types of phase diagrams:
a) two intensive variables
b) one intensive, one extensive (molar) variable
c) two extensive (molar) variables
d) Sections through higher dimensional phase diagrams defined by
special restraints on extensive variables
31
Types of Phase Diagrams
Binary Cu-Mg, type A
p = const.
32
In type A diagrams, lines denote two phase (monovariant) equilibria
and points three phase (invariant) equilibria
Unary Fe, type A
Types of Phase Diagrams
A system in equilibrium is defined by a set of c+2 state variables. By
varying two of them and keeping the others constant, a twodimensional
phase diagram can be drawn.
The maximal dimension of the phase diagram is c+1, because one
extensive variable can be diminished by dividing them by N:
e.g.: T, p, N1…Nj → T, p, x1…xj-1
Types of phase diagrams:
a) two intensive variables
b) one intensive, one extensive (molar) variable
c) two extensive (molar) variables
d) Sections through higher dimensional phase diagrams defined by
special restraints on extensive variables
33
Binary Cu-Mg, type BBinary Cu-Mg, type B
p = const. p = const.
Types of Phase Diagrams
34
In type B diagrams, tie-lines connect two separated phases in equilibrium.
Tie-lines are parallel to the extensive axis. Three phase equilibria are
defined by special tie-lines.
Types of Phase Diagrams
A system in equilibrium is defined by a set of c+2 state variables. By
varying two of them and keeping the others constant, a twodimensional
phase diagram can be drawn.
The maximal dimension of the phase diagram is c+1, because one
extensive variable can be diminished by dividing them by N:
e.g.: T, p, N1…Nj → T, p, x1…xj-1
Types of phase diagrams:
a) two intensive variables
b) one intensive, one extensive (molar) variable
c) two extensive (molar) variables
d) Sections through higher dimensional phase diagrams defined by
special restraints on extensive variables
35
Binary Cu-Mg, type C Ternary Al-Mg-Zn, type C
p = const. p = const., T = 608K
Types of Phase Diagrams
36
In type C diagrams, tie-lines can have any direction and have to be drawn
separately. Three phase equilibria are tie-triangles.
37
Reading of binary phase diagram
Rules for phase boundaries
38
39
Rules for phase boundaries
40
Rules for phase boundaries
41
Rules for phase boundaries
Types of Phase Diagrams
A system in equilibrium is defined by a set of c+2 state variables. By
varying two of them and keeping the others constant, a twodimensional
phase diagram can be drawn.
The maximal dimension of the phase diagram is c+1, because one
extensive variable can be diminished by dividing them by N:
e.g.: T, p, N1…Nj → T, p, x1…xj-1
Types of phase diagrams:
a) two intensive variables
b) one intensive, one extensive (molar) variable
c) two extensive (molar) variables
d) Sections through higher dimensional phase diagrams defined by
special restraints on extensive variables
43
Vertical sections of ternary systems
(isopleths)
• Tie-lines generally are not in the plane of section!
• The lines do not belong to monovariant equilibria - they show „zerophase-fraction“
equilibria.
• Lines show boundaries between an n- and (n-1)-phase field
(calculated from equilibrium condition for the n-phase equilibrium with
additional condition that amount of one phase is zero, although still
present in equilibrium).
44
45
Vertical sections of ternary systems
(isopleths)
Liquidus projection: eutectic, peritectic, phase transformation
Example: Au-Ge-Sn
46
Reading ternary phase diagram
Phase transformations on liquidus surface
E (eutectic) U (phase transformation) P (peritectic)
47
TD-48
Fázový diagram Fe–C:
TD-49
Fázový diagram Fe–Fe3C:
(metastabilní)
TD-50
Legury železa: feritotvorné a austenitotvorné
TD-51
Mikrostruktura oceli v závislosti na zpracování:
TD-52
Jednofázové materiály: souvislost složení tuhého roztoku
s mechanickými vlastnostmi
substituční nebo intersticiální zpevnění (dobře rozpracovaná
teorie, Fischer, Labusch, Lukáč),
tj. zvýšení zejména skluzového napětí a posunutí celé
deformační křivky k vyšším napětím, interakce dislokací s
cizími atomy (P.-L. jev)
Hume-Rotheryho empirická pravidla pro rozpustnost
příměsí (velikost atomů, elektronegativita, mocenství prvků)
Vícefázové materiály:
Vliv mikrostruktury a fázového složení na průběh plastické
deformace (geometrie rozložení fází, složení fází a jejich
objemový podíl, velikost a rozložení částic jednotlivých
fází).
TD-53
Precipitace
Eutektoidní FT
Typy fázových transformací řízených difúzí:
Uspořádání
TD-54
Stadia: nukleace, růst a hrubnutí (precipitátů, zrn)
Kulovitý precipitát:
Energie mechanických napětí může snižovat i
zvyšovat energetickou bariéru!
Nukleace homogenní nebo heterogenní na
různých poruchách (vakance, dislokace,
vrstevné chyby, hranice zrn, volný povrch)
(pozn.)
Hnací síla nukleace roste s rostoucím podchlazením pod křivku
rozpustnosti a významně ovlivňuje rychlost nukleace.
Za malého podchlazení je hnací síla nukleace malá. Za příliš
vysokého podchlazení je mobilita atomů malá. Tedy existuje
optimum pro rychlost nukleace.
55
Scheil –Gulliverova rovnice
Při běžném ochlazování nejsou obecně splněny podmínky pro
ustavení kompletní termodynamické rovnováhy
Lépe situaci popisuje přístup použitý v práci Scheila a Gulivera
Předpoklady:
1 – Žádná difuze v pevné fázi, jakmile se jednou zformuje
2 – Nekonečně rychlá difuze v tavenině
3 – Termodynamická rovnováha existuje přímo na rozhraní
(4 – Křivky liquidu a solidu mohou být extrapolovány přímkami)
CL = C0. (fL)k-1
CS = k.C0. (1-fS)k-1
Příklad: ocel Fe-9%Cr-2%Mo-0,3%V-0,3%Ti-0,1%C-0,1%N
56
Scheil –Gulliverova rovnice
57
Scheil –Gulliverova rovnice
58
Scheil –Gulliverova rovnice
TD-59
Zpracování Fe-C slitin:
TD-60
TTT křivky Fe–C
Perlit Fe-
0,8wt%C
TD-61
Souvislost mikrostruktury se způsobem tepelného zpracování:
Výrazné a rychlé podchlazení:
bezdifúzní transformace austenitu na
(nerovnovážný) martensit
TD-62
CCT diagramy:
TD-63
Další témata:
• Jak pracují programy pro výpočet fázových diagramů
(fenomenologický přístup, CALPHAD metoda, extrapolace unárních a
binárních f.d. na složitější soustavy, rozvoj dodatkových příspěvků v
mocninách molárních zlomů, optimalizace parametrů na základě exp.
dat)
• Existují i programy pro modelování kinetiky fázových
transformací (teorie difúze, aktivity složek, mobility, …)
• TD databáze mohou být spojené s diagramy
vlastností materiálů
• Tvarová paměť (NiTi, …)
64
Institute of Physics of Materials, AS CR, Brno, Czech Republic
Few comments about the software
• All are based on global minimization of total Gibbs energy of system in
dependence of temperature, pressure and overall composition,
• Thermodynamic data valid from the room temperature
• Different level of user-friendliness
• Originally different level of versatility and power, but now more closer to
each other, the differences are small, depends on personal preferences
• The thermodynamic databases in the form of black boxes, mostly not
compatible (conversion software usually not commercially available)
• Different qualities of graphical outputs
• Different reliability of global minimum search, detection of miscibility gaps,
detection of errors of the assessment (liquid MG, LT phases at HT etc.)
Thermo-Calc Pandat
MT DATA FactSage
• Thermo–Calc (grandpa of all softwares) (http://www.thermocalc.com)
• Probably most powerful, most versatile
• Two regimes – command mode, “windows”type mode
• Keeps all results for given session, can be extracted later
• Less user friendly, the user-friendly version just for routine calculations
• Free download of limited version for academic use (ternary systems only,
limited time use only)
• The assessment modulus PARROT accessible only in Command mode
• It is possible to define all quantities derived as partial derivations of the Gibbs
energy
• DICTRA (software for modelling of diffusion processes)
• Older versions “”hate users”
• Uses mobilities instead of diffusion coeficients
• Local equilibrium at interface
65
Institute of Physics of Materials, AS CR, Brno, Czech Republic
• Thermo–Calc (grandpa of all softwares) (http://www.thermocalc.com)
• Probably most powerful, most versatile
• Two regimes – command mode, “windows”type mode
• Keeps all results for given session, can be extracted later
• Less user friendly, the user-friendly version just for routine calculations
• Free download of limited version for academic use (ternary systems only,
limited time use only)
• The assessment modulus PARROT accessible only in Command mode
• It is possible to define all quantities derived as partial derivations of the Gibbs
energy
• DICTRA (software for modelling of diffusion processes)
• Older versions “”hate users”
• Uses mobilities instead of diffusion coeficients
• Local equilibrium at interface
66
Institute of Physics of Materials, AS CR, Brno, Czech Republic
• Thermo–Calc (grandpa of all softwares) (http://www.thermocalc.com)
• Probably most powerful, most versatile
• Two regimes – command mode, “windows”type mode
• Keeps all results for given session, can be extracted later
• Less user friendly, the user-friendly version just for routine calculations
• Free download of limited version for academic use (ternary systems only,
limited time use only)
• The assessment modulus PARROT accessible only in Command mode
• It is possible to define all quantities derived as partial derivations of the Gibbs
energy
• DICTRA (software for modelling of diffusion processes)
• Older versions “”hate users”
• Uses mobilities instead of diffusion coeficients
• Local equilibrium at interface
67
Institute of Physics of Materials, AS CR, Brno, Czech Republic
• Thermo–Calc (grandpa of all softwares) (http://www.thermocalc.com)
• Probably most powerful, most versatile
• Two regimes – command mode, “windows”type mode
• Keeps all results for given session, can be extracted later
• Less user friendly, the user-friendly version just for routine calculations
• Free download of limited version for academic use (ternary systems only,
limited time use only)
• The assessment modulus PARROT accessible only in Command mode
• It is possible to define all quantities derived as partial derivations of the Gibbs
energy
• DICTRA (software for modelling of diffusion processes)
• Older versions “”hate users”
• Uses mobilities instead of diffusion coeficients
• Local equilibrium at interface
68
Institute of Physics of Materials, AS CR, Brno, Czech Republic
• Thermo–Calc (grandpa of all softwares) (http://www.thermocalc.com)
• Probably most powerful, most versatile
• Two regimes – command mode, “windows”type mode
• Keeps all results for given session, can be extracted later
• Less user friendly, the user-friendly version just for routine calculations
• Free download of limited version for academic use (ternary systems only,
limited time use only)
• The assessment modulus PARROT accessible only in Command mode
• It is possible to define all quantities derived as partial derivations of the Gibbs
energy
• DICTRA (software for modelling of diffusion processes)
• Older versions “”hate users”
• Uses mobilities instead of diffusion coeficients
• Local equilibrium at interface
69
Institute of Physics of Materials, AS CR, Brno, Czech Republic
• Thermo–Calc (grandpa of all softwares) (http://www.thermocalc.com)
• Probably most powerful, most versatile
• Two regimes – command mode, “windows”type mode
• Keeps all results for given session, can be extracted later
• Less user friendly, the user-friendly version just for routine calculations
• Free download of limited version for academic use (ternary systems only,
limited time use only)
• The assessment modulus PARROT accessible only in Command mode
• It is possible to define all quantities derived as partial derivations of the Gibbs
energy
• DICTRA (software for modelling of diffusion processes)
• Older versions “”hate users”
• Uses mobilities instead of diffusion coeficients
• Local equilibrium at interface
70
Institute of Physics of Materials, AS CR, Brno, Czech Republic
71
Institute of Physics of Materials, AS CR, Brno, Czech Republic
• Open CALPHAD (free software and databases)
http://www.opencalphad.com/
• OpenCalphad is an informal international collaboration of scientists and
researchers interested in the development of high quality software and
databases for thermodynamic calculations for all kinds of applications.
• The intention is to allow interested scientists to develop more fundamental
models to incorporate results from DFT calculations as well as to provide
scientists interested in materials simulations a flexible and reliable software
using consistent thermodynamic databases.
• Basic principles are identical with Thermo-Calc, main author is Dr. Sundman,
who was one of authors of original CALPHAD method and ThermoCalc
software
• Currently includes only basic modules and limited amount of databases
• Anybody can download the code
• Anybody can take part in the collaboration
Trying to incorporate the description of thermodynamic function from 0K