The Thermodynamics of Phase Transitions_Perry

8/18/2019 The Thermodynamics of Phase Transitions_Perry

1/30


2/30

1. Gibbs free energy, equilibrium and chemical potential, Gibbs phase rule

2. Single component systems1. dG(T)

2. ClausiusClapeyron equation and the phase diagram of titanium

!. "inary (t#o component) systems

1. $deal solutions

2. %egular solutions

!. &cti'ity

. %eal solutions, ordered phases and $ntermediate phases

. "inary phase diagrams

1. iscibility gap

2. *rdered alloys

!. +utectics and peritectics

. &dditional useful relationships

. Ternary diagrams

. -inetics of hase transformations

Contents


3/30


4/30

Equilibrium 0 the most stable state defined by lo#est possible G

dG " #

equilibriummetastable

+.g. etastable 0 /iamond

+quilibrium 0 Graphite

Solid 0 9o# atomic inetic energy or E

⇒ lo# T and small $

9iquid 0 9arge +

⇒ high T and large $

Chemical potential or partial molar free energy µ go'erns ho# the free

energy changes #ith respect to the addition3subtraction of atoms.

This is particularly important in alloy or binary systems.

(particle numbers #ill change)

2. Gibbs free energy, equilibrium and chemical

potential, Gibbs phase rule


5/30

2. Gibbs free energy, equilibrium and chemical

potential, Gibbs phase rule

Gibbs phase rule for equilibrium phase 0

+:amples 0

Single component system ⇒ %41 and & 4 ! − '$f 1 phases in equilibrium (e.g. solid) ⇒ 2 degrees of freedom i.e. can change T and P #ithout changing the phase

$f 2 phases in equilibrium (e.g. solid and liquid) ⇒ 1 degree of freedom i.e. T is

dependent on P (or vice-versa)$f ! phases in equilibrium (e.g. solid, liquid and ) ⇒ ; degrees of freedom. !phases e:ist only at one fi:ed T and P.

7umber of degrees of freedom < 4 C = - 52

C, number of components

-, number of phases in equilibrium


6/30

(. $ingle %omponent $ystems


7/30

Clausius Clapeyron +quation

)ess

dense

more

dense

)ess

dense

more

dense

*intermediate+


8/30

. "inary (t#o component) systems 0 $deal solutions

T#o species in the mi:ture0 consider mole fractions X & and X " X & 5 X " 4 1

G1 4 X &G & 5 X "G"

T#o contributions to G from mi:ing t#o

components together0

! G1 = #eighted molar a'erage of the t#o

components

2.


9/30

. "inary (t#o component) systems 0 $deal solutions

Simplest case 0 $deal solution 0 ∆" $? 4 ;

Some assumptions 0

1.


10/30

. "inary (t#o component) systems 0 The chemical potential

Chemical potential 0 go'erns the response of the system to adding component

T#o component system need to consider partial molar µ & and µ ".

Total molar Gibbs free energy 4 −#dT 5 µ & X & 5 µ " X " (5' dP )

Simplified equations for an ideal liquid0

µ & X & 4 G & 5&T ln X &

µ " X " 4 G" 5&T ln X "

(!e! µ & is the free

energy of

component & in

the mi:ture


11/30

. "inary (t#o component) systems 0 %egular solutions and atomic bonding

Generally0 ∆" $?≠; i.e. internal energy of the system must be considered

$n a binary, ! types of bonds0 &&, "", &" of energies ε &&, ε"", ε &"

/efine0 ∆" $?4 C &"ε #here C &" is the number of &" bonds and ε4 ε &"− E(ε && 5ε"")

∆" $?4 Γ X & X " @here Γ 4$ a) ε , )*bonds per atom

$f Γ F; ⇒ &" bonding preferred

$f Γ >; ⇒ &&, "" bonding preferred

∆G$? 4 ∆" +(X &T ( X &ln X & 5 X "ln X ")

oint of note0

∆G$? al#ays decreases on addition of solute


12/30

Γ Γ

Γ Γ i:ing al#ays

occurs at high

Temp. despite

bonding

i:ing if & and

" atoms bond

& and " atoms

repel

hase separation

in to 2 phases.


13/30

. "inary (t#o component) systems 0 ctivity, a of a component

&T ln a &

; 1?"

&T ln a"

µ"

µ &

∆G$?

µ & 4 G & 5 &T ln a & G &

G" &cti'ity is simply related to chemical potential by0

µ" 4 G" 5 &T ln a"

$t is another means of describing the state

of the system. 9o# acti'ity means that the

atoms are reluctant to lea'e the solution(#hich implies, for e:ample, a lo# 'apour

pressure).

i.e.


14/30


15/30

. "inary phase diagrams 0 The .ever rule

hase diagrams can be used to get quantitati'e information on the relati'e

concentrations of phases using the .ever rule 0

&t temperature, T and molar fraction X ;, the solid and liquid phase #ill coe:ist in

equilibrium according the ratio0

Temperature

& "

T

Solid, S

9iquid, 9

X ;

lβlα

nαlα 4 nβlβ

i.e. I2J solid and

IKJ liquid at X ;

@here nα3nβ is ratio of liquid to solid

S lid li id h di i & d " l l


16/30

Solid to liquid phase diagram in a t#o component system 0 & and " are completely

miscible and ideal solutions

"i h di Th i ibili


17/30

. "inary phase diagrams 0 The iscibility gap

& "

T 1G

liquid

solid 9

Common tangent

& "

G

S

a b c d

T 2

S

& "

T !G

9

e f

∆" $? > ;

& " X "

liquidT 1

T 2

T !e f

Single phase, mi:ed solid

2 phase0 (&5δ") and ("5δ &)Compositions e and f 8

LThe miscibility gapM


18/30

Titanium6anadium re'isited

(bcc)

(hcp)

@hat can #e deduceN

1. Ti and 6 atoms bond #ealy

2. There are no ordered phases

!. (Ti,6) phase 0 mi:ture of Ti and 6 in a fcc structure

. Ti (hcp) phase does not dissol'e 6 #ell

"lue 0 single phase

@hite 0 t#o phase(bcc)

+ ilib i i h t t


19/30

+quilibrium in heterogenous systems

αe then minimum free energy is Ge

&nd t#o phases are present

(ratio gi'en by the .ever rule = see later)

@hen t#o phases e:ist in equilibrium, the acti'ities of

the components must be equal in the t#o phases0

Common tangent

"i (t t) t / d d h


20/30

. "inary (t#o component) systems 0 /rdered phases

re'ious model gross o'ersimplification 0 need to consider sie difference bet#een &

and " (strain effects) and type3strength of chemical bonding bet#een & and ".

*rdered substitutional

/rdered phases occur for (close to) integer ratios.

i.e. 101 or !01 mi:tures.

"ut entropy of mi:ing is 'ery small so increasing

temperature can disorder the phase. &t some critical

temperature, long range order #ill disappear.

*rdered structures can also tolerate de'iations fromstoichiometry. This gi'es the broad regions on the

phase diagram

Systems #ith strong &" bonds can form /rderedand3or intermediate phases


21/30

The CopperGold system%andom mi:ture

Single phases i:ed phases

7.". &l#ays read the legendDDD (blue is not al#ays Osinge phaseB)

(fcc)(fcc)

&n intermediate phase is a mi:ture that has different structure to that of either


22/30

&n intermediate phase is a mi:ture that has different structure to that of either

component

%ange of stability depends on structure and type of bonding ($onic, metallic, co'alentH)

(ntermetallic phases are intermediate phase of integer stoichiometry e.g. 7i! &l

7arro# stability range broad stability range

"inary phase diagrams 0 *rdered phases


23/30

. "inary phase diagrams 0 *rdered phases∆" $? F ;

i!e! and 0 attract

@ea attraction Strong attraction

*rdered β phase e:tends to liquid phase

1 phase, solid

*rdered phase α

ea in liquidus line 0 attraction bet#een atoms

"inary phase diagrams 0 Simple +utectic systems


24/30

. "inary phase diagrams 0 Simple +utectic systems

∆" $? ; 8 & and " ha'e different crystal structures8

α hase is & #ith δ" dissol'ed (crystal structure &)

β hase is " #ith δ & dissol'ed (crystal structure ")

Single phase

T#o phase

+utectic point


25/30

+:ample 0 http033###.soton.ac.u3Ipasr13inde:.htm

+utectic systems and phase diagrams

"inary phase diagrams 0 eritectics and incongruent melting


26/30

. "inary phase diagrams 0 eritectics and incongruent melting

P Sometimes ordered phases are not stable as a liquid. These compounds

ha'e peritectic phase diagrams and display incongruent melting.

P $ncongruent melting is #hen a compound melts and decomposes into its

components and does not form a liquid phase.

P These systems present a particular challenge to material scientists to mae in

a single phase. Techniques lie hot pouring must be used.


27/30

Solid

soluti o

n

-(5

δ 7a)

Solid

solution

7a(5

δ

-)

(bcc)

(hcp)

(bcc)

eritectic line

(! phase equil.)9 5 -7a2

9 5 7a(δ-)

9 5 -(δ7a)

-(δ7a) 5 -7a2-7a2 5 7a(δ-)

"inary phase diagrams 0 &dditional equations


28/30

. "inary phase diagrams 0 &dditional equations

&. +quilibrium 'acancy concentration

So far #e ha'e assumed that e'ery atomic site in the lattice is occupied. "ut this is not

al#ays so. 6acancies can e:ist in the lattice.%emo'ing atoms0 increase internal energy (broen bonds) and increases configuration

entropy (randomness).

/efine an equilibrium concentration of 'acancies X 6 (that gi'es a minimum free energy)

∆G64∆" 6 − T ∆#6

@here ∆" 6 is the increase in enthalpy per mole of 'acancies added and ∆#6 is the

change in thermal entropy on adding the 'acancies (changes in 'ibrational frequencies

etc.).

X 6 is typically 1;1;! at the melting point of the solid.

". Gibbs/uhem relationship

This relates the change in chemical potential that results from a change in alloy

composition0

"inary phase diagrams 0 Ternary phase diagrams


29/30

. "inary phase diagrams 0 Ternary phase diagrams

These are complicated.

P ! elements so triangles are at

fi:ed temperature

P 6ertical sections as a function

of T and are often gi'en.

"lue = single phase

@hite = t#o phase

Qello# = three phase


30/30

Documents

The Thermodynamics of Phase Transitions_Perry