application of the nano-calphad method to select stable ... · the kelvin equation and the reasons...
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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