George Kaptay Professor, corresponding member of the Hungarian Academy of Sciences

Application of the nano-Calphad method to select stable binary

nano-crystalline alloys

Outlines

Calphad

Nano-Calphad

Stabilization of nano-grains in polycrystals

Part 1. The essence of Calphad

happy customer unhappy customer

The subject of this talk

State parameters Equilibrium state

The subject of another talks

The subject of another talks

CALPHAD = CAlculation of PHAse Diagrams / equilibrium

What we define and what Nature (God) defines

Number and nature of components, their

average concentration + temperature + pressure (+ size, if below 100 nm)

Number of phases Nature of phases

Phase fraction of phases Composition of phases

(shape, if below 100 nm)

Empirical way

Number of experiments needed for 84 stable elements: 10085 = 10170

Years needed if each hs performes 1 experiment per day: 10158 Mission Impossible

Gibbs energy

Calphad step 1 Databank collection

and modelling

Calphad step 2 Calculation of equilibrium

(phase diagrams)

The Calphad way

Calphad

Materials balance System: your selection of a 3-D part of a Universe, including some matter n (mole), containing C (= 1, 2, …) components, each denoted as i = A, B, etc… (component = element)

𝑛𝑛 = 𝑛𝑛𝐴𝐴 + 𝑛𝑛𝐵𝐵 𝑥𝑥𝐵𝐵 ≡𝑛𝑛𝐵𝐵𝑛𝑛

𝑥𝑥𝐴𝐴 = 1 − 𝑥𝑥𝐵𝐵

Phase: a homogenous 3-D part of the system, formed spontaneously (not by us): Φ = α, β… (their number: P = 1, 2, …)

𝑛𝑛 = �𝑛𝑛ΦΦ

𝑦𝑦Φ ≡𝑛𝑛Φ𝑛𝑛

𝑦𝑦𝛼𝛼 = 1 − 𝑦𝑦𝛽𝛽

Each phase is composed of the same components as the system:

𝑛𝑛Φ = 𝑛𝑛𝐴𝐴(Φ) + 𝑛𝑛𝐵𝐵(Φ) 𝑥𝑥𝐵𝐵(Φ) ≡𝑛𝑛𝐵𝐵(Φ)

𝑛𝑛Φ 𝑥𝑥𝐴𝐴(Φ) = 1 − 𝑥𝑥𝐵𝐵(Φ)

Materials balance equation: 𝑥𝑥𝐵𝐵 = �𝑦𝑦Φ ∙ 𝑥𝑥𝐵𝐵(Φ)Φ

g

l

s(α)

s(β)

An example of a 4-phase system

Calphad

C, i, T, p

𝑥𝑥𝑖𝑖 P, Φ

𝑦𝑦Φ

𝑥𝑥𝑖𝑖(Φ)

𝐺𝐺𝑖𝑖(Φ)

𝐺𝐺 = �𝑦𝑦Φ ∙ 𝐺𝐺ΦΦ

𝐺𝐺Φ = �𝑥𝑥i(Φ) ∙ 𝐺𝐺i(Φ)i

𝐺𝐺𝑖𝑖(Φ) = 𝑓𝑓(𝐶𝐶, 𝑖𝑖,𝑝𝑝,𝑇𝑇, 𝑥𝑥i Φ ) 𝐺𝐺 → 𝑚𝑚𝑖𝑖𝑛𝑛

𝐺𝐺𝑖𝑖(α) = 𝐺𝐺𝑖𝑖(β)

𝑃𝑃𝑚𝑚𝑚𝑚𝑚𝑚 = 𝐶𝐶 + 2

𝑥𝑥𝑖𝑖 = �𝑦𝑦Φ ∙ 𝑥𝑥𝑖𝑖(Φ)Φ

1.Calculation 2. Selection

Does the solution exist?

Un-knowns for equilibrium state: 𝑦𝑦Φ and 𝑥𝑥𝑖𝑖(Φ)

Their number: (P-1) + P(C-1) = PC -1

Equations: 𝐺𝐺𝑖𝑖(α) = 𝐺𝐺𝑖𝑖(β) and 𝑥𝑥𝑖𝑖 = �𝑦𝑦Φ ∙ 𝑥𝑥𝑖𝑖(Φ)Φ

Their number: C(P-1) + C-1 = PC -1

As the number of un-knowns equals the number of equations,

the solution always exists

Why?? 𝐺𝐺 → 𝑚𝑚𝑖𝑖𝑛𝑛

The upper layer of the Earth

Towards equilibrium

Towards higher energy

There are at least two explanations….

Why?

A-B phase βA-B phase α

𝐺𝐺𝑖𝑖(α) = 𝐺𝐺𝑖𝑖(β)

Algorithm for 2-phase, 2-component macro-equilibria

1. Fix values for p, T, xA

βα ,, AA GG = βα ,, BB GG =

2. Solve the system of equations for xA,α and xA,β

In the two-phase α-β region the solution is not xA-dependent

3. Find the phase ratio yα:

βα

βα

,,

,

AA

AA

xxxx

y−

−=

This is a true tie-line

Calphad needs supercomputers

0

10

20

30

0 21 42 63 84

LG(c

ombi

natio

ns)

Components

Part 2. The essence of nano-Calphad

Nano-Calphad ≡ Calphad applied to nano-materials

Nano-materials ≡ materials with at least 1 phase with at least one of its dimensions below 100 nm

Nano came last

kmfm nm µm mm mpm

logLTµ-Tn-Tp-Tf-T

Why nano-materials are so special?

0

20

40

60

80

100

0 10 20 30 40 50

% o

f ato

ms a

long

surf

ace

r, nm

𝑥𝑥𝑠𝑠 = 𝐴𝐴𝑠𝑠𝑠𝑠 ∙𝑉𝑉𝑚𝑚𝜔𝜔

𝐴𝐴𝑠𝑠𝑠𝑠 ≡𝐴𝐴𝑉𝑉

More than 1 % of atoms are along the surface, and so all

properties are size-dependent

Size dependence of properties

80

100

120

140

160

180

200

220

0 20 40 60 80 100

Any

prop

erty

% of surface atoms

𝑌𝑌 = 𝑌𝑌𝑏𝑏 + 𝑥𝑥𝑠𝑠 ∙ 𝑌𝑌𝑠𝑠 − 𝑌𝑌𝑏𝑏

80

100

120

140

160

180

200

220

0 10 20 30 40 50

Any

prop

errt

y

r, nm

𝑌𝑌 = 𝑌𝑌𝑏𝑏 + 𝐴𝐴𝑠𝑠𝑠𝑠 ∙𝑉𝑉𝑚𝑚𝜔𝜔∙ 𝑌𝑌𝑠𝑠 − 𝑌𝑌𝑏𝑏

Size dependence of molar Gibbs energy 1

𝑌𝑌 = 𝑌𝑌𝑏𝑏 + 𝐴𝐴𝑠𝑠𝑠𝑠 ∙𝑉𝑉𝑚𝑚𝜔𝜔∙ 𝑌𝑌𝑠𝑠 − 𝑌𝑌𝑏𝑏

𝑌𝑌 ≡ 𝐺𝐺𝑚𝑚 𝐺𝐺𝑚𝑚 = 𝐺𝐺𝑚𝑚,𝑏𝑏 + 𝐴𝐴𝑠𝑠𝑠𝑠 ∙𝑉𝑉𝑚𝑚𝜔𝜔∙ 𝐺𝐺𝑚𝑚,𝑠𝑠 − 𝐺𝐺𝑚𝑚,𝑏𝑏

𝐺𝐺𝑚𝑚 = 𝐺𝐺𝑚𝑚,𝑏𝑏 + 𝐴𝐴𝑠𝑠𝑠𝑠 ∙ 𝑉𝑉𝑚𝑚 ∙ 𝜎𝜎

𝜎𝜎 =𝐺𝐺𝑚𝑚,𝑠𝑠 − 𝐺𝐺𝑚𝑚,𝑏𝑏

𝜔𝜔

Size dependence of molar Gibbs energy 2

Gibbs, 1878: 𝐺𝐺 = 𝐺𝐺𝑚𝑚 + 𝐴𝐴 ∙ 𝜎𝜎

Divide by n:

𝑛𝑛 =𝑉𝑉𝑉𝑉𝑚𝑚

𝐴𝐴𝑠𝑠𝑠𝑠 ≡𝐴𝐴𝑉𝑉

𝐺𝐺𝑚𝑚 = 𝐺𝐺𝑚𝑚,𝑏𝑏 + 𝐴𝐴𝑠𝑠𝑠𝑠 ∙ 𝑉𝑉𝑚𝑚 ∙ 𝜎𝜎

𝐺𝐺𝑚𝑚 ≡𝐺𝐺𝑛𝑛

𝐺𝐺𝑚𝑚,𝑏𝑏 ≡𝐺𝐺𝑏𝑏𝑛𝑛

Size dependence of chemical potential

𝜇𝜇𝑖𝑖 = 𝜇𝜇𝑖𝑖,𝑏𝑏 + 𝐴𝐴𝑠𝑠𝑠𝑠 ∙ 𝑉𝑉𝑚𝑚,𝑖𝑖 ∙ 𝜎𝜎𝑖𝑖

𝐺𝐺𝑚𝑚 = 𝐺𝐺𝑚𝑚,𝑏𝑏 + 𝐴𝐴𝑠𝑠𝑠𝑠 ∙ 𝑉𝑉𝑚𝑚 ∙ 𝜎𝜎

𝐺𝐺𝑚𝑚 = � 𝑥𝑥𝑖𝑖 ∙ 𝜇𝜇𝑖𝑖𝑖𝑖

𝐺𝐺𝑚𝑚,𝑏𝑏 = � 𝑥𝑥𝑖𝑖 ∙ 𝜇𝜇𝑖𝑖,𝑏𝑏𝑖𝑖

𝑉𝑉𝑚𝑚 = � 𝑥𝑥𝑖𝑖 ∙ 𝑉𝑉𝑚𝑚,𝑖𝑖𝑖𝑖

𝐺𝐺𝑚𝑚 = � 𝑥𝑥𝑖𝑖 ∙ 𝜇𝜇𝑖𝑖𝑖𝑖

= � 𝑥𝑥𝑖𝑖 ∙ 𝜇𝜇𝑖𝑖,𝑏𝑏 + 𝐴𝐴𝑠𝑠𝑠𝑠 ∙ 𝑉𝑉𝑚𝑚,𝑖𝑖 ∙ 𝜎𝜎𝑖𝑖

𝜎𝜎 = 𝜎𝜎𝑖𝑖

Chemical potential in multi-phase situations

𝜇𝜇𝑖𝑖 = 𝜇𝜇𝑖𝑖,𝑏𝑏 + 𝐴𝐴𝑠𝑠𝑠𝑠 ∙ 𝑉𝑉𝑚𝑚,𝑖𝑖 ∙ 𝜎𝜎 + 𝑉𝑉𝑚𝑚,𝑖𝑖 ∙𝐺𝐺𝑚𝑚𝑎𝑎𝑎𝑎 − 𝐺𝐺𝑓𝑓𝑠𝑠

𝑉𝑉

Case 1: sessile drop

𝜇𝜇𝑖𝑖(𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑖𝑖𝑠𝑠𝑠𝑠) = 𝜇𝜇𝑖𝑖,𝑏𝑏 +3𝑟𝑟∙ 𝑉𝑉𝑚𝑚,𝑖𝑖 ∙ 𝜎𝜎𝑠𝑠𝑙𝑙 ∙

2 − 3 ∙ 𝑐𝑐𝑐𝑐𝑐𝑐Θ + 𝑐𝑐𝑐𝑐𝑐𝑐3Θ4

1/3

𝜇𝜇𝑖𝑖 = 𝜇𝜇𝑖𝑖,𝑏𝑏 + 𝐴𝐴𝑠𝑠𝑠𝑠 ∙ 𝑉𝑉𝑚𝑚,𝑖𝑖 ∙ 𝜎𝜎𝑖𝑖

0

0,2

0,4

0,6

0,8

1

0 30 60 90 120 150 180

, degrees

Case 2: Liquid confined in capillaries

𝜇𝜇𝑖𝑖(𝑐𝑐𝑚𝑚𝑠𝑠) = 𝜇𝜇𝑖𝑖,𝑏𝑏 −2𝑟𝑟𝑐𝑐𝑚𝑚𝑠𝑠

∙ 𝑉𝑉𝑚𝑚,𝑖𝑖 ∙ 𝜎𝜎𝑠𝑠𝑙𝑙 ∙ 𝑐𝑐𝑐𝑐𝑐𝑐Θ

𝜇𝜇𝑖𝑖(𝑐𝑐𝑐𝑐𝑠𝑠) = 𝜇𝜇𝑖𝑖,𝑏𝑏 +2𝑟𝑟𝑐𝑐𝑚𝑚𝑠𝑠

∙ 𝑉𝑉𝑚𝑚,𝑖𝑖 ∙ 𝜎𝜎𝑠𝑠𝑙𝑙

-1

-0,5

0

0,5

1

0 30 60 90 120 150 180

, degrees

Josiah Willard Gibbs 1839 - 1903

William Thomson (Lord Kelvin) 1824 - 1907

(1869) (1878)

The historical accident: nano-Calphad came before Calphad

The Kelvin equation and the reasons of its incorrectness

𝐺𝐺𝑚𝑚 = 𝑈𝑈𝑚𝑚 + 𝑝𝑝 ∙ 𝑉𝑉𝑚𝑚 − 𝑇𝑇 ∙ 𝑆𝑆𝑚𝑚 𝑝𝑝𝑖𝑖𝑖𝑖 = 𝑝𝑝𝑜𝑜𝑜𝑜𝑎𝑎 + 𝜎𝜎 ∙1𝑟𝑟1

+1𝑟𝑟2

𝐺𝐺𝑚𝑚 = 𝐺𝐺𝑚𝑚,𝑏𝑏 + 𝐴𝐴𝑠𝑠𝑠𝑠 ∙ 𝑉𝑉𝑚𝑚 ∙ 𝜎𝜎

𝐺𝐺𝑚𝑚 = 𝐺𝐺𝑚𝑚,𝑏𝑏 +1𝑟𝑟1

+1𝑟𝑟2

∙ 𝑉𝑉𝑚𝑚 ∙ 𝜎𝜎

Reasons of incorrectness: - p is a state parameter, not an inside pressure p(in) - No nano-effect for not-curved phases ? (cubic nano-L) - The surface term of Gibbs is forgotten, - The Laplace pressure is obtained from G…. - Contradiction with nucleation theory

The nucleation contradiction

[J Nanosci Nanotechnol 12 (2012) 2625-2633]

Gibbs: critical size

Gibbs: equilibrium size

Kelvin (Gibbs-Thomson): equilibrium size

nucleus size, nm

Gibbs

energy

change,

J

Kelvin (Gibbs-Thomson) or Gibbs?

Gibbs (specific surface area)

msSb VAGG ,, ΦΦΦΦΦ ⋅⋅=− σ

Kelvin (curvature)

+⋅⋅=− ΦΦΦΦ

21,

11rr

VGG mgb σ

2r

rVm⋅⋅σ2

rVm⋅⋅σ3

0δ

δσσ moutin V⋅+ )(

CALORIMETRIC INVESTIGATION OF THE LIQUID Sn-3.8Ag-0.7Cu ALLOY WITH MINOR Co ADDITIONS

Andriy Yakymovych University of Vienna, Austria

George Kaptay University of Miskolc, Hungary

Ali Roshanghias, Hans Flandorfer, Herbert Ipser University of Vienna, Austria

. J. Phys. Chem. C,120 (2016) 1881-1890

High-Temperature Calorimeter

•SETARAM © •High temperature calorimeter (twin calorimeter) •ambient to 1000°C Equipped with an

automatic dropping device

H. Flandorfer, F. Gehringer, E. Hayer, Thermochim. Acta 382 (2002), 77-87

Materials Sn-3.8Ag-0.7Cu foil High purity metals in bulk form

Alfa Aesar (99.99%)

Results

∆𝐻𝐻𝑖𝑖𝑚𝑚𝑖𝑖𝑜𝑜= −7.5 ± 1.5 𝑘𝑘𝑘𝑘/𝑚𝑚𝑐𝑐𝑚𝑚

Theoretical Consideration

𝐴𝐴𝑠𝑠𝑠𝑠 ∙ 𝑉𝑉𝑚𝑚 = 𝐴𝐴𝐵𝐵𝐵𝐵𝐵𝐵 ∙ 𝑀𝑀

( ) -2, , 2.80 0.15 J m

Dsg H Tσ ≅ ± ⋅

( ) 3 2 150 10 10 m kgBETA −= ± ⋅ ⋅3 158.933 10 kg molM − −= ⋅ ⋅

∆𝐻𝐻𝑖𝑖,𝑎𝑎ℎ= −8.2 ± 2.1 𝑘𝑘𝑘𝑘/𝑚𝑚𝑐𝑐𝑚𝑚

∆𝐻𝐻𝑖𝑖,𝑠𝑠𝑚𝑚𝑠𝑠= −7.5 ± 1.5 𝑘𝑘𝑘𝑘/𝑚𝑚𝑐𝑐𝑚𝑚

𝐺𝐺𝑚𝑚 = 𝐺𝐺𝑚𝑚,𝑏𝑏 + 𝐴𝐴𝑠𝑠𝑠𝑠 ∙ 𝑉𝑉𝑚𝑚 ∙ 𝜎𝜎

bulk-liquid nano-liquid nano-bulk

𝐻𝐻𝑚𝑚𝑖𝑖𝑚𝑚,𝑖𝑖𝑚𝑚𝑖𝑖𝑜𝑜 = 𝐻𝐻𝑚𝑚𝑖𝑖𝑚𝑚−𝑏𝑏𝑜𝑜𝑠𝑠𝑏𝑏 + ∆𝐻𝐻𝑖𝑖𝑚𝑚𝑖𝑖𝑜𝑜

∆𝐻𝐻𝑖𝑖𝑚𝑚𝑖𝑖𝑜𝑜 = (0 − 𝐴𝐴𝑠𝑠𝑠𝑠) ∙ 𝑉𝑉𝑚𝑚 ∙ 𝜎𝜎𝑠𝑠𝑠𝑠,𝐻𝐻

A new state parameter

𝜇𝜇𝑖𝑖(𝛼𝛼) = 𝜇𝜇𝑖𝑖(𝛽𝛽) 𝜇𝜇𝑖𝑖 = 𝜇𝜇𝑖𝑖,𝑏𝑏 + 𝐴𝐴𝑠𝑠𝑠𝑠 ∙ 𝑉𝑉𝑚𝑚,𝑖𝑖 ∙ 𝜎𝜎𝑖𝑖

80

100

120

140

160

180

200

220

0 10 20 30 40 50

, J/m

ol

r, nm

A new state parameter: Asp, r, or N

The extended phase rule of Gibbs

Gibbs, 1875:

Due to a new, independent variable:

CP += 3max PCF −+= 3

P = number of phases, C = number of components, F = freedom

[J. Nanosci. Nanotechnol., 2010, vol.10, pp.8164–8170]

CP += 2max PCF −+= 2

Macro-thallium

-100

-50

0

50

100

480 500 520 540 560 580 600

T, K

Go a

- G

o HC

P, J

/mol

a = LIQα = VAP2.5E-13 bar

a = HCP

a = BCC

α = VAP4.2E-11 bar

T1

T2

-15

-12

-9

-6

-3

0

3

6

480 500 520 540 560 580 600

T, K

logp

(bar

)

VAP

BCCHCP

HCP

LIQ

T2

T1

Nano-thallium (with a quaternary point)

-15

-12

-9

-6

-3

0

3

6

480 500 520 540 560 580 600

T, K

logp

(bar

)

HCP

HCP

BCC LIQ

T1 VAPT2

N>1E12

T3

-15

-12

-9

-6

-3

0

3

6

480 500 520 540 560 580 600

T, K

logp

(bar

)

VAPVAP

LIQ

LIQ

HCP

HCPHCP

T1 T2

T3

N=2E5

BCC

-15

-12

-9

-6

-3

0

3

6

480 500 520 540 560 580 600

T, K

logp

(bar

)

HCPLIQ

HCP

BCC

VAPVAP

Q

HCP

N=1.2E5

-15

-12

-9

-6

-3

0

3

6

480 500 520 540 560 580 600

T, K

logp

(bar

)

HCP

HCP

LIQ

VAP

N=1E4

T4

[G.Kaptay: J. Nanosci. Nanotechnol., 2010, vol.10, pp.8164–8170]

The size dependence of interfacial energies

Tolman, 1949:

Buff, 1951:

R

phase αphase β

r

o

δσσ

⋅+

= 21 VL ρρδ

−Γ

≡

+

⋅−⋅= ....21

ro δσσ

0

0,2

0,4

0,6

0,8

1

0 1 2 3 4 5r, nm

σ, J

/m2

,

Tolman

Buff

δ

Rcr

Samsonov

σο

( ) SrVTT

m

sslomm ∆⋅⋅+

⋅⋅−=

δσ

23

The separation dependence of interfacial energies

α = s

β = g

γ = l z

( ) )z(f5.0)z( //// ⋅σ−σ⋅+σ=σ γαβαγαγα

[ ] )z(f5.0)z( //// ⋅σ−σ⋅+σ=σ γββαγβγβ

2

)(

+

=z

zfξ

ξ

−=

ξzzf exp)(metals:

non-metals:

−⋅∆++=

ξσσσσ zz glls exp)( // gllsgs /// σσσσ −−≡∆

Surface melting

α = s

β = g

γ = l z

α = s

β = g

surface melting

( )

−−⋅∆−∆⋅−⋅=

∆ξ

σ zSTTVz

AzG o

mo

mos

m exp1)(

-80

-60

-40

-20

0

20

40

60

0 3 6 9 12 15z, nm

∆m

G/A

, mJ/

m2 equilibrium thickness of the

surface melted layer, zeq

( )

−⋅∆⋅

⋅∆⋅=

TTSVz o

mm

os

eq ξσξ ln

0

2

4

6

8

10

900 920 940 960 980 1000 1020T, K

z eq, n

m

omT

ξ

( ) )/( SeVTT mo

so

m ∆⋅⋅⋅∆≤− ξσ

α β α β

The role of the relative arrangement of phases

Same results for N > 1012

Different results for N < 1012

α β

Equilibrium arrangement corresponds to minimum Gibbs energy

∑ ΦΦΦΦ σ⋅⋅=∆s

s/spec,s/surf AVG

6. Dependence on the substrate material

γ

βα

αβ

γ

αβ

γ

Wettable substrates stabilize nano-droplets

∑ ΦΦΦΦ σ⋅⋅=∆s

s/spec,s/surf AVG

Dependence on segregation (Butler)

)s/(i)s/(B)s/(As/ ... ΦΦΦΦ σ==σ=σ=σ

3/13/2)(

)()/(

)(

)/(3/13/2

)()/()/( ln

Avi

Ei

Esi

bi

si

Avi

osisi NVf

GGx

xNVf

TR⋅⋅

∆−∆⋅+

⋅

⋅⋅⋅

+=Φ

ΦΦ

Φ

Φ

ΦΦΦ

βσσ

1xi

)s/(i =∑ Φ1)( =∑ Φ

i

bix

0

0,2

0,4

0,6

0,8

1

1,2

0 0,2 0,4 0,6 0,8 1xB*

σA, σ

B, J/

m2

A

B

solution

Size limits of thermodynamics

Thermodynamics is a statistical science

G has an average value for a large number of atoms in the given environment during a long

enough time

At small N the fluctuations will increase.

What is the critical N below which thermodynamic is not valid is a „religious” question (at N = 13: Tm ≈ 0 K, what is OK):

only the fluctuation increases with smaller N

0

200

400

600

800

1000

0 3 6 9 12logNa

T m,i,

K

13 atoms

oimT ,

Algorithm for 2-phase, 2-component nano-phase-equilibria (1)

1. Fix values for p, T, xA, N 2. Suppose a certain phase-arrangement, e.g.

3. Suppose a value for yα (0 < yα < 1), then:

αβ

NyN ⋅= αα NyN ⋅−= )1( αβ

Algorithm for nano phase equilibria (2)

βα ,, AA GG =

4. Suppose a value for xA,α (0 < xA,α < 1), then:

α

ααβ y

xyxx AA

A −

⋅−=

1,

,

αα ,, 1 AB xx −=

ββ ,, 1 AB xx −=

5. Find σα and σβ from the Butler equation (modify xAα).

6. Check, if the equations for G-s with σ-s are satisfied

7. If not, select new values for (xA,α, yα) and go back to Step 3

8. If yes, check solution for other phase arrangements / shapes and select equilibrium arrangement / shape configuration for the phases by minimizing Gibbs energy

βα ,, BB GG =

Algorithm for nano phase equilibria (3)

The final result xA,α and xA,β will be dependent on xA. Thus, tie line has not the same sense as before.

This is not a true tie-line

T

A BxA

α βα + β

xA,βxA,α

p, N, T = const

A BxA

α

β

α + β

xA,β

xA,α

p, N = const

xA,β

xA,α

not a tie line

tie line

Part 3. Thermodynamic stabilization of nano-alloys vs grain coarsening and precipitation

The case of pure metals

𝐺𝐺𝑚𝑚,𝐴𝐴𝑜𝑜 = 𝐺𝐺𝑚𝑚,𝐴𝐴,𝑏𝑏

𝑜𝑜 + 𝐴𝐴𝑠𝑠𝑠𝑠 ∙ 𝑉𝑉𝑚𝑚,𝐴𝐴𝑜𝑜 ∙ 𝜎𝜎𝐴𝐴,𝑠𝑠𝑏𝑏

𝑜𝑜

𝐴𝐴𝑠𝑠𝑠𝑠 ≡𝐴𝐴𝑉𝑉

=𝑘𝑘𝑟𝑟

𝐺𝐺𝑚𝑚,𝐴𝐴𝑜𝑜 = 𝐺𝐺𝑚𝑚,𝐴𝐴,𝑏𝑏

𝑜𝑜 +𝑘𝑘𝑟𝑟∙ 𝑉𝑉𝑚𝑚,𝐴𝐴

𝑜𝑜 ∙ 𝜎𝜎𝐴𝐴,𝑠𝑠𝑏𝑏𝑜𝑜

G.Kaptay: Nano-Calphad: extension of the Calphad method to systems with nano-phases and complexions. J Mater Sci, 47

(2012) 8320-8335

k = 3 (sphere); k = 3.72 (cube); k = 3.36 ± 0.36 for a grain

0

2000

4000

6000

8000

10000

0 20 40 60 80 100

, J/m

ol

r, nm

coarsening, no stability

2-component alloys (GB = grain boundayr)

49

A = bulk component, B = segregated component, = average mole fraction 𝑥𝑥𝐵𝐵

y = GB-ratio, = atomic radius, = bulk and GB filling ratios, x = B mole fraction in bulk, 𝑟𝑟𝑚𝑚𝑖𝑖𝑖𝑖 = minimum possible grain size, 𝜔𝜔𝐵𝐵

𝑜𝑜 ,𝜎𝜎𝐵𝐵𝑜𝑜 = molar surface area and GB energy of component B, Ω = bulk interaction energy, 𝑉𝑉𝑚𝑚 = molar volume, 𝐺𝐺𝑚𝑚 molar Gibbs energy.

𝑦𝑦 =𝑟𝑟∗

𝑟𝑟 + 3 ∙ 𝑟𝑟𝑚𝑚 𝑟𝑟∗ ≡

43∙ 𝑘𝑘 ∙

𝑓𝑓𝑠𝑠𝑏𝑏𝑓𝑓𝑏𝑏

∙ 𝑟𝑟𝑚𝑚

𝑟𝑟𝑚𝑚 𝑓𝑓𝑏𝑏, 𝑓𝑓𝑠𝑠𝑏𝑏

𝑥𝑥 =𝑥𝑥𝐵𝐵 − 𝑦𝑦1 − 𝑦𝑦 𝑟𝑟𝑚𝑚𝑖𝑖𝑖𝑖 ≅

𝑟𝑟∗

𝑥𝑥𝐵𝐵− 3 ∙ 𝑟𝑟𝑚𝑚

𝑟𝑟𝑚𝑚 =3 ∙ 𝑓𝑓𝑏𝑏 ∙ 𝑉𝑉𝑚𝑚4 ∙ 𝜋𝜋 ∙ 𝑁𝑁𝐴𝐴𝑙𝑙

1/3

𝜔𝜔𝐵𝐵𝑜𝑜 = 𝜋𝜋 ∙ 𝑟𝑟𝑚𝑚2 ∙

𝑁𝑁𝐴𝐴𝑙𝑙𝑓𝑓𝑠𝑠𝑏𝑏

𝐺𝐺𝑚𝑚 = 𝑅𝑅 ∙ 𝑇𝑇 ∙ 𝑥𝑥 ∙ 𝑚𝑚𝑛𝑛𝑥𝑥 + 1 − 𝑥𝑥 ∙ 𝑚𝑚𝑛𝑛 1 − 𝑥𝑥 + Ω ∙ 𝑥𝑥 ∙ 1 − 𝑥𝑥 +

+𝑘𝑘

𝑟𝑟 + 3 ∙ 𝑟𝑟𝑚𝑚∙𝑉𝑉𝑚𝑚𝜔𝜔𝐵𝐵𝑜𝑜

∙ 𝜔𝜔𝐵𝐵𝑜𝑜 ∙ 𝜎𝜎𝐵𝐵𝑜𝑜 ∙ 1 +3 ∙ 𝑟𝑟𝑚𝑚𝑟𝑟 − 𝑅𝑅 ∙ 𝑇𝑇 ∙ 𝑚𝑚𝑛𝑛𝑥𝑥 − Ω ∙ 1 − 𝑥𝑥 2

W-”Ag” alloys, 500 K, 15 mol% Ag

50 6M meeting, 29 March 2017, Miskolc, Hungary

-1000

0

1000

2000

3000

4000

0 4 8 12 16 20

Gm

, J/m

ol

r, nm

OMEGA = 20 kJ/mol

-1000

0

1000

2000

3000

4000

0 4 8 12 16 20

Gm

, J/m

ol

r, nm

OMEGA = 35 kJ/mol

-1000

0

1000

2000

3000

4000

0 4 8 12 16 20

Gm

, J/m

ol

r, nm

OMEGA = 50 kJ/mol

Stability diagram for W-based alloys

51 6M meeting, 29 March 2017, Miskolc, Hungary

-40

-20

0

20

40

60

80

100

120

140

160

0 5 10 15 20 25 30 35

, kJ/

mol

, kJ/molunstable

kinetically stable

thermodynamically stable

Ti

Cr Sc

YAg, Mn

Ce

Cu

Ta

Mo, Nb99%

75%

ZrFe - Co - Ni Al

???

W-Ag phase diagram

52 6M meeting, 29 March 2017, Miskolc, Hungary

0

500

1000

1500

2000

2500

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

T, K

xAg

bulk bcc-W + bulk liq-Ag

bulk bcc-W + bulk fcc-Ag

bulk bcc-W + bulk vap-Ag (1 bar)

W Ag

0

500

1000

1500

2000

2500

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

T, K

xAgW Ag

bulk bcc-W + bulk fcc-Ag

bulk bcc-W + bulk liq-Ag

bulk bcc-W + bulk vap-Ag (1 bar)

100

3010 3 1

bcc-W-nano-grains with

segregated Ag atoms

Maximum Hall-Petch

Maximum stabity

Theoretically selected stable nc alloys

W-based alloys: Ag, Au, Ba, Bi, Cd, Ce, Cr, Cs, Cu, Eu, Gd, Hg, Ho, In, K, La, Li, Mg, Na, Nd, Pb, Pu, Rb, Sb, Sc, Sm, Sn, Tb, Th, Tl, Tm, U, Y, Yb, Zn.

Mo-based alloys: Ag, Ba, Bi, Ca, Cd, Ce, Cs, Cu, Er, Eu, Hg, Ho, In, K, La, Li,

Mg, Na, Nd, Pb, Rb, Sm, Sr, Tl, Yb.

Nb-based alloys: Ag, Ba, Bi, Ca, Cd, Ce, Cs, Cu, Er, Eu, Gd, Hg, K, La, Li, Mg, Na, Rb, Sc, Sm, Tl, Y, Yb.

Ti-based alloys: Ba, Ca, Ce, Cs, Eu, Gd, K, La, Li, Mg, Na, Nd, Rb, Sr, Yb.

Al-based alloys: Bi, Cd, Cs, In, K, Na, Pb, Rb, Tl.

Mg-based alloys: Cs, K, Na, Rb.

Key papers

Calphad, 56 (2017) 169-184.

J. Phys. Chem. C, 120 (2016) 1881-1890.

J. Mater. Sci., 51 (2016) 1738-1755.

Langmuir 31 (2015) 5796-5804.

Acta Mater, 60 (2012) 6804-6813.

J Mater Sci, 47 (2012) 8320-8335.

Int. J. Pharmaceutics, 430 (2012) 253-257.

J Nanosci Nanotechnol, 12 (2012) 2625-2633

J. Nanosci. Nanotechnol., 10 (2010) 8164–8170

kaptay@hotmail.com