Download - Atomic Transport Phase Transformations - uni … Transport & Phase Transformations Lecture 9 PD Dr. Nikolay Zotov [email protected]

Atomic Transport

&

Phase Transformations

Lecture 9

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

Lecture I-9 Outline

Degree of order

Order parameters (General)

Examples of order-disorder structures

Short-range order (SRO) parameters

Determination of SRO parameters

Long-range order (LRO) parameters

Determination of the LRO parameters

Degree of Order in the Solid State

Interaction parameter WS < 0

2eAB – eAA – eBB = eAB – ½ (eAA + eBB )< 0 Tendency for formation of A-B bonds

For random distribution:

Number of A-B bonds: PABRan = NozXAXB;

Degree of ordering:

No (random solution) PAB = PABRan = NozXAXB;

Partial ordering

Short-range order (SRO) PABPar > PAB

Ran

0,0 0,2 0,4 0,6 0,8 1,0

0,00

0,05

0,10

0,15

0,20

0,25

XB

XAXB;

Degree of Order in the Solid StateThermodynamic description

DGmix = DHmix – TDSmix;

DHmix = WSXAXB < 0

Non-random configurational entropy

DSmix = DSmixid + DSmix

Ex = - R[XAln(XA) + XBln(XB)] + DSmixEx ;

Quasi-Chemical model (Gugenheim 1952) for weak SRO:

DSmixEx = -R[yAAln(yAA/XA

2) + yBBln(yBB/XB2) + 2yABln(yAB/XAXB)]

yAA - fraction of A-A bonds (for random distribution XA2)

yBB - fraction of B-B bonds (for random distribution XB2)

yAB - fraction of A-B bonds (for random distribution XAXB)

In the presence of SRO yAA < XA2, yBB < XB

2 but yAB > XAXB ; → SRO reduces the configurational entropy

Order Parameters

Order Parameter (h)

G = G(T,p,ni,h) h = 1 Ordered phase with LRO

h > 0 SRO

h = 0 random distribution

Positional order-disorder

Orientational (magnetic) order-disorder

Phase Diagrams with Order- Disorder Phase Transitions

●

●

A

B

A – Homogeneous solid

solution

Disordered (Cu,Au) phase

B - Ordered CuAuI phase

Phase Diagrams with Order- Disorder Phase Transitions

ß – High-temperature

disordered structure

ß’ – Low-temperature

ordered structure

Order – Disorder Structures

T > Tc = 410oC T < Tc= 410oC

S.G F m -3 m

Strukturbericht symbol A1

Crystall class: cubic

Point Group: m -3 m

S.G P 4/mmm

Strukturbericht symbol L10

Crystall class: tetragonal

Point Group: 4/mmm

Cu

Au

CuAuI


Cu

Zn

S.G I m -3 m

Strukturbericht symbol A2


Point Group: m -3 m

S.G P m -3 m

Strukturbericht symbol B2


Point Group: m -3 m

T > TC ~ 727 KT < TC ~ 727 K

ß CuZnß’ CuZn


S.G P m -3 m

Strukturbericht symbol L12


Point Group: m -3 m

Cu3Au

S.G F m -3 m

Strukturbericht symbol A1 (fcc)


Point Group: m -3 m

T < Tc = 390oC T > Tc= 390oC

Disordered phase – the different types of atoms occupy all crystallographic sites in the

unit cell randomly with probabilities corresponding to the overall composition

All sites have the same site symmetry

Fcc (F m -3 m) 0 0 0½ ½ 00 ½ ½ ½ 0 ½

Stoichiometric CuAuI

pCu = pAu = ½

Site Symmetry 4a m -3 m


SRO, AmBn Clusters

Atom Site Site Symmetry_Coordinates______No of Atoms

Au 2e mmm 0 ½ ½ 4x1/2 = 2

Cu 1a 4/mmm 0 0 0 8x1/8 = 1

Cu 1c 4/mmm ½ ½ 0 2x1/2 = 1

Ordered structure (LRO) – Different atoms occupy different set of

equivalent atomic positions (different sites)

Each site generates a ‚sub-lattice‘.

Substitutional Disorder in ordered structures –

Random substitution of a given sublattice

Space group symmetry is preserved

Ordered non-stoichiometric compounds

Au

Cu


Cu

Au Cu

Cu Au

CuAuI

4-fold axis


Disordered liquid crystal Orientationally-ordered liquid crystal


LRO ↔ Substitutional Disorder → SRO ↔ (Random) Disordered structure

G G G‘ G‘

WS << 0 WS < 0 WS ~ 0

LRO parameter(s) Large Ordered Clusters? SRO parameter(s)

Change of Symmetry in the Disorder – Order Phase Transitions:

# The ordered and the disordered phases have different space groups (G ≠ G‘);

# Loss of symmetry elements in the ordered phase (G < G‘)

Types of symmetry change:

# Change of translational symmetry, the point group symmetry remains the same

Cu,Au (F m -3m) → Cu3Au (P m -3 m)

ß‘Cu,Zn (I m-3m) → ß CuZn (P m -3 m)

# Change of the point group symmetry

Cu,Au (F m -3m) → CuAuI (P 4/mmm)

Short-range order parameter (1):

hSRO = (PABSol – PAB

Ran) / (PABMax – PAB

Ran) ; Bethe SRO parameter

Short-range order parameter (2):

a = 1 – P1/XA ; P1 – probability to find an atom A as a

nearest-neighbour of an atom B.

a – Warren – Cowley SRO parameter

= 0 random distribution (P1 = XA)

< 0 SRO (P1 > XA)

> 0 Clustering (P1 < XA)

Short-range Order Parameters

random

hSRO ~ 0.3

Easterling and Potter (2009)

AB4 clusters


Warren – Cowley parameters

an = 1 – Pn/XA, Pn – Probability to find atom A on the

n-th shell, if an atom B is at (0,0,0)

a0 = a000 = 1 R0 = ( 0 0 0)

a1 = a 0 ½ ½ R1 = ( 0 ½ ½ )

a2 = a 010 R2 = ( 010 )

a (lmn)

No SRO Pn = XA and all an ≡ 0

Clustering aj > 0 and aj → 0 at large distances

SRO aj oscilate between positive and negative values

(0,0,0)

R1

R2


Determination using Diffuse X-ray and Neutron Scattering

Basics of Diffuse Scattering:

A = FT(r) A – Amplitude of scattered X-ray/neutrons

r – Scattering density

In the presence of defects (and/or SRO) r = <r> + Dr <r> - Scattering density of the average lattice

Dr – (local) variations due to presence

of SRO and/or defects

A = FT(<r>) + FT(Dr)

Itot = |A|2 ~ IBraggAv + Idiff; Small concentration of defects or degree of SRO

IBraggAv ~ |A(<r>)|2

IDiff = ISRO + IStrain + ∙∙∙



SRO Diffuse Scattering:

(C = 2 components, each atom has the same number of nearest-neighbours)

ISRO = NXAXB(fA – fB)2 + NXAXB(fA-fB)2Sj Sk≠j ajk exp(2piQ.Rjk)

= NXAXB(fA – fB)2 Sl SmSnalmn exp[2pi(lh + mk + nl]}; l,m,n -∞ +∞, a000 = 1

fA, fB – Atomic scattering factors for the A and B atoms

Rjk = Rk – Rj ; difference of radius vectors of atoms j and k

la + mb + nc; a, b c – lattice vectors

Q = 4psin(Q)/l Scattering vector length

Q = ha* + kb* + lc*; a* b* c* lattice parameters of the

reciprocal lattice

almn = aj = IFT{ISRO/ NXAXB(fA – fB)2 }

a(-l –m –n ) = a (lmn)



Q ~ 0 fCu ~ ZCu = 29

fAu ~ ZAu = 79

(fCu – fAu)2 ~ 2500

Cowley (1950) ; Cu Ka radiation

Cu3Au cubic; P m -3 m (L12) type

Cu3Au single crystal

at each temperature 450 measurements

in reciprocal space for diffrenet h,k,l



h

k

(100) Plane 405 oC

1 2 3 4 5 6 7

-0,4

-0,2

0,0

0,2

0,4

0,6

0,8

1,0

SR

O P

ara

mete

rs

Shell

405

460

550

Perfect order

(200) (400)

(220) (420)

Cu3Au

Cowley (1950)

(110) (200) (211) ∙∙∙

nA(i) = NiPi = NiXA(1 – ai) Crystal

i=1 12 x 0.75 x 1.152 = 10.4 (12)

i=2 6 x 0.75 x 0.814 = 3.7 (0)



●1 2 3 4 5 6 7

-0,2

-0,1

0,0

0,1

0,2

0,3

SR

O P

ara

mete

rs

Shell

426 oC CuAuI

Long-range Order Parameters

Au Cu

Cu Au

a

b

Ideally, A atoms ordered on a sites

B atoms ordered on b sites

Definitions:

ra – fraction of a sites occupied by A atoms

rb – fraction of b sites occupied by B atoms

wa – fraction of wrong (B) atoms occupying a sites

wa = 1 – ra ;

wb – fraction of wrong (A) atoms occupying b sites

wb = 1 – rb ;

hLRO = (ra – XA)/(1 – XA)

hLRO = (rb – XB)/(1 – XB)hLRO = 1 ideal LRO

hLRO < 1 Increasing degree of substitutional disorder


Quasi-chemical thermodynamic model of LRO:

B2 and L10 type structures (Z=8), Independent disordereing on the a and b sites and XA = XB = 1/2!

Number of A atoms on a sites = [(1+h)/4]N

Number of B atoms on a sites = [(1-h)/4]N

Number of A atoms on b sites = [(1-h)/4]N

Number of B atoms on b sites = [(1+h)/4]N

Molar enthalpy of mixing: DHmix ~ NA/2{(1 – h2)(eAA + eBB) + 2(1 + h2)eAB}

Configurational entropy of mixing:

W = [(N/2)!/ (# A atoms on a sites)! (# B atoms on a sites)!] x [(N/2)!/ (# B atoms on b sites)! (# A atoms on b sites)!]

= (N/2)!/{[(1+h)/4]N}!{[(1-h)/4]N}! x (N/2)!/{[(1-h)/4]N}!{[(1+h)/4]N}! N = NA ;

Sterling formula

DS mix = kBln(W) = R{ln(2) – ½ [(1+h)ln(1+h) + (1-h)ln(1-h)]}

h = 0 Random distribution: DS mix = Rln(2) ; DSmixid = -R[XAln(XA) + XBln(XB)]; XA=XB = 0.5

h ~ 1 perfect LRO: DS mix = R{ln(2) – ½ [2ln(2) – 0ln(0)]} = R{ln(2) – ln(2)} = 0


Quasi-chemical thermodynamic model of LRO:

# Gibbs energy of mixing (excluding vibrational entropy)

DG mix (h) = N/2{(1 – h2)(eAA + eBB) + 2(1 + h2)eAB } - RT {ln(2) – ½ [(1+h)ln(1+h) + (1-h)ln(1-h)]}

DGmix will have at every T an optimum LRO parameter, which minimizes the Gibbs energy

# Condition for extremum ∂DGmix/∂h = 0

∂DGmix/∂h = NAh[2eAB – (eAA + eBB)] + (NAkBT/2) ln (1+h)/(1-h);

2hDe + kBT ln[(1+h)/(1-h)] = 0

T ln[(1+h)/(1-h)] = 2h|De|/kB ; Critical temperature TC: h = 0 ; ln(1+h)/(1-h) → 2h for h → 0

TC = |De|/kB ; T = 2hTC/ ln[(1+h)/(1-h)]



Activity Coefficients

RT ln (gA) = W (1 – XA)2 = W(1 + h – 2ra)2

XA = ra + wb = 2ra – h

RT ln (gB) = W (1 –XB)2 = W(1 + h – 2rb)2

h = 1 ra = rb = 1 g = 1

hLRO = ra – wb or hLRO = rb – wa ;


Stoichiometric CuAu

Second-order phase transition

Quasi-chemical model (1st neighbours only):

Tln[(1+h)/(1-h)] = 2hTC

Cowley model (multiple shells):

Tln[(1+h2)/(1-h2)] = 2h2TC

0

1

200 300 400 500 600 700 800

0,0

0,2

0,4

0,6

0,8

1,0

Eta

Temperature oC

QC

Cowley


Non-Stoichiometric CuAu

CuAu

XAu < 0.5 Maximum Order: Cu site fully occupied,

some Cu on the Au site

ra = 1; wb = (XCu-0.5)/0.5 = [(1-XAu) – 0.5]/ 0.5 = 1 – 2XAu;

h = 2XAu

XAu > 0.5 Maximum order: Au site fully occupied;

some Au on the Cu site

rb = 1; wa = (XAu - 0.5)/ 0.5 = 2XAu - 1;

h = 2 - 2XAu0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0

0,0

0,2

0,4

0,6

0,8

1,0

LR

O P

ara

mete

r

XAu

Maximum LRO

hLRO = ra – wb or hLRO = rb – wa ;


AmBn and/or Non-stoichiometric – General treatment for two sub-lattices:

(m + n = 1)

hLRO = ra – wb = rb – wa ;

NAa/N = mXA + mn h fraction of A atoms on the a sub-lattice

NAb/N = nXA – mn h fraction of A atoms on the b sub-lattice

NBb/N = nXB + mn h fraction of B atoms on the b sub-lattice

NBa/N = mXB – mn h fraction of B atoms on the a sub-lattice NA

a +NAb = NA; NB

b +NBa = NB;

Configurational Entropy

S = kB ln(WaxWb); Wa = (mN)!/(NAa)! (NB

a)! ; Wb = (nN)!/(NAb)! (NB

b)!


I. Cu3Au (m = ¾, n = ¼) Stoichiometric

Quasi-Chemical model: T ln{(1/3 + h)(3 + h)/(1-h2)} = h(16/3) TC

Cowley model: T ln{(1/3 + h2)(3 + h2)/(1-h2)2} = h2(16/3) TC

TC

First-order

phase transition

II. CuAu (m = ½ , n = ½ ) Non-Stoichiometric

Tln{(XA + h/2)(XB+h/2)/(XA-h/2)(XB-h/2)} = 4hTC

Basics:

Ihkl ~ Fhkl2 ; Fhkl – Structure factor for reflexion (hkl)

Fhkl = Sj fjexp(2pi(hxj + kyj + lzj))

Determination of Long-range Order Parameters

1

2

Disordered bcc structure

Disordered bcc structure:

Fhkl = <f>{exp[i2p(h.0+k.0+l.0)] + exp[i2p(h.1/2+k1/2 + l1/2)]}

= <f>{1 + exp[ip(h+k+l)]}

h+k+l = even F = 2<f> (fundamental, allowed reflexions)

h+k+l = odd F = 0

<f> = XA fA + XBfB

1 (0 0 0 )2 ( ½ ½ ½ )

# Lowering of the symmetry of the low-tenperature ordered phase

# Appearence of additional (super-structure) reflections

# C = 2 components and ordered structure with 2 sub-lattices (a and b)

# Fhkl = Sj

a(rafA + wafB)exp(2pi(hxj + kyj + lzj)) + Sj

b(rbfB + wbfA)exp(2pi(hxj + kyj + lzj))

(Partially) ordred bcc structure

2 sub-lattices: a =(000); b( ½ ½ ½ )

Fhkl = (rafA + wafB) + (rbfB + wbfA)exp(pi(h+k+l))

h+k+l = even Fundamental reflexions

Fhkl = (rafA + wafB) + (rbfB + wbfA) = (rafA + wbfA) + (wafB + rbfB )= 2(fAXA + fBXB) = 2<f>



h+k+l = odd; Super-structure reflexions

Fhkl = (rafA + wafB) - (rbfB + wbfA) = (rafA - wbfA) - (-wafB + rbfB )= h(fA – fB)

Ihkl ~ h2(fA – fB)2

Cu3Au

Disordered

Quenched from 600 oC

Ordered

Annealed at 365 oC 20 for days

Warren (1990)


Warren (1990)

Cu3Au

1st Order


CuAuI

2nd-Order

Ordering Mechanisms

?

Vacancies

Diffusion