1 day 3. interfacial energies in high temperature systems george kaptay kaptay / day 3 / 1 a 4-day...
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1
Day 3.
Interfacial energies in high temperature systems
George Kaptay
Kaptay / Day 3 / 1
A 4-day short course
See J94
2
Interfacial energies Interfacial forces
Interfacial phenomena
Complex phenomena
Modeling algorithmKaptay / Day 3 / 2
3
Types of phases and interfaces to be modeledgasW(s)
Al(l)NaCl(l)
MgO(s)Si(s)
AlN(s) TiC(s)
Kaptay / Day 3 / 3
4
Modeling interfacial energies
BA
BABA
G
/
//
BBA
ABABA GGG ///
The excess interfacial Gibbs energy:
BABABA STHG ///
A B
Kaptay / Day 3 / 4
5
The excess interfacial enthalpy for the liquid/gas interface
olAA
olA
bA
iAbAgBlA UU
z
zzH
/
TRHU oAv
olA
For liquid metals (structures surface – see Day 1 / 18): (11-9)/11 = 0.182
Where the cohesion energy of the liquid metal (classic):
Kaptay / Day 3 / 5 See J48
6
The excess interfacial entropy for the liquid/gas interface
b
igsA V
VRS ln/
bb
gsA Ru
uRS
lnln
2/1
2
2
/
From the LEED measurements [Somorjai et al]:
molKJSS gBsAgBlA /14//
Kaptay / Day 3 / 6 See J48
7
The molar surface area of the liquid/gas interface
3/13/2
Avm NVf ff
fb
i
3
4
2 3 1/3/
b
Avm f
NrV 3
3
4 i
Av
f
Nr 2
For the {111} plane of the fcc crystal: fi = 0.906.
For fcc crystals fb = 0.740 → f = 1.09.
When an fcc crystal is melted, ΔmV = 1.06, → f = 1.06.
Kaptay / Day 3 / 7 See J48
8
Surface tension of pure liquid metals
3/13/2/06,1
)15,5(182,0
Avo
A
oAvo
glANV
TH
0
1
2
3
0 1 2 3elm, J/m2
kis
, J/m
2
Kaptay / Day 3 / 8
Experimental points
Calculated values
See J48
9
Surface tension of liquid metals / new age (1)
oicAgBlA HH ,/ (11-9)/11 =
0.182
)( ,,,,,o
imo
ipo
mico
ic TTCHH 0,,, o
Tic icrH
)( ,,,,,o
imo
icro
ipo
mic TTCH
2,2,1,,o
imo
imo
mic TRqTRqH
To correlate the calculated values:
Kaptay / Day 3 / 9
10
0
40000
80000
120000
160000
0 1000 2000 3000 4000
R.Tom,i J/mol
-Ho
c,i,m
J
/mo
l
Cs
Rb
K
Na
Li
Hg
q1 = 25.4 ± 1.2, q2 = 0.
2,2,1,,o
imo
imo
mic TRqTRqH
Kaptay / Day 3 / 10 From vaporization enthalpy
From critical points
11
Surface tension of liquid metals / new age (2)
i
b
f
ff
3/13/2
4
3
For the {111} plane of the fcc crystal: fi = 0.906.
For fcc crystals fb = 0.740 → f = 1.09.
For melted fcc crystals fb,fcc = 0.74/(1.12) = 0.66 → f =1.01
For melted bcc crystals fb,bcc = 0.68/(1.096) = 0.62 → f=0.97
For melted hcp crystals fb,hcp = 0.74/(1.086) = 0.68 → f=1.03
Average for melted fcc, bcc and hcp crystals: f = 1.00 ± 0.02
Kaptay / Day 3 / 11
12
Surface tension of liquid metals / new age (3)
ogiord
ogivib
ogi SSS ///
molKJSS oim
ogiord /26/
molKJS ogivib /15/
31/// ogiord
ogivib
ogi SSS
)( ,,,,,o
imo
ipo
mico
ic TTCHH
Kaptay / Day 3 / 12
13
Surface tension of liquid metals / new age (4)
3/13/2
,
,,,,
)02.000.1(
)31()02.0182.0(
Avomi
oim
omico
miNV
TH
3/13/2
,
,, )1038(
Avomi
oimo
miNV
T
Kaptay / Day 3 / 13
2,2,1,,o
imo
imo
mic TRqTRqH
q1 = 25.4 ± 1.2, q2 = 0.
14
0
0.5
1
1.5
2
2.5
3
3.5
4
0 0.5 1 1.5 2 2.5 3 3.5 4
i,m, calc, J/m2
i,m
, exp
, J/m
2
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1 1.2
i,m, calc, J/m2
i,m, e
xp
, J/m
2
NaK
Cs
CaSr
PbTl
Al
Ag
Au Cu
ZrHf
Pt
CoNi
V
Re
IrW
Ga
Hg
Bi
SnIn
Sb
Ge
Fe
Ta
q1 = 26.3 and q2 = -2.62.10-4 molK/J
Kaptay / Day 3 / 14
15
T-coefficient of surface tension / new age (5)
3/13/2
,
3/2
,
/,,
2
,2,1
1 Avo
imoi
omi
ogi
oim
oip
oim
oimo
iNTTVf
TSTTCTRqTRq
)( ,,
,o
imTo
io
mioi TT
o
mioi
Avomi
omipTo
iNV
C,3/13/2
,
,,,
3
2)31()026.0182.0(
Kaptay / Day 3 / 15
16
0
0.1
0.2
0.3
0.4
0.5
0 0.1 0.2 0.3 0.4 0.5T
i,m, calc, mJ/Km2
T
i,m, e
xp, m
J/K
m2
Na
Ag
Tl
Al
Sr
Ca
Pb
Cs
K
0
0.05
0.1
0.15
0.2
0.25
0 0.03 0.06 0.09 0.12 0.15T
i,m, calc, mJ/Km2
Ti,m
, ex
p, m
J/K
m2
Au Cu
Co
Ni
Hg
Sb
Ge
Ga
InSn
Bi
Kaptay / Day 3 / 16
17
Kaptay / Day 3 / 17
The classic model for surface tension was improved by changing 2 things at the same time:
i. the cohesion energy is made T-dependent through heat capacity (known for 130 years)
ii. a new, ordering term is taken into account in the excess surface entropy (known for 30 years)
General lesson: Models exist on almost everything. Majority of them can be improved. If you change only one thing in the model, usually you spoil it. You must be brave enough to change at least two (sometimes three), different things in the model to get it work again – in a better way.
18
Surface structure of MX type molten salts
AvXMo
MX NrrV 3)(2 AvXMoMX Nrr 2)(2 26.12 3/1 MXf
1.010
910
MX molKJSS gBlMgBlMX /23//
Kaptay / Day 3 / 18
- +
+ -
+ -
- +
- +
+ -
+ -
- +
Bulk liquid of MX associates
Vapour
See P75
19
Surface tension of MX molten salts
3/13/2/26,1
)28,3(10,0
Avo
MX
oMXvo
gMXNV
TH
0
100
200
300
400
- 100 200 300 400elm, mJ/m2
kis
, mJ/
m2
NaCl
CsCl
Kaptay / Day 3 / 19
Experimental points
Calculated values
20
Surface tension of molten monoxides (MO)
3/13/2/26,1
)28,3(10,0
Avo
MX
oMXvo
gMXNV
TH
MO TK
vH
kJ/mol
Vl
cm3/mol
MO/g,
mJ/m2
model
MO/g, J/m2
measured
FeO 1.650 413.2 15.8 532 70 550 50
MgO 2523 574.9 16.5 695 100 700 100
CaO 1823 578.3 21.1 623 90 650 100
SrO 1823 477 27.8 425 60 -
BaO 1823 336.4 34.1 239 30 250 50
Kaptay / Day 3 / 20
21
oTmglA
oTmgsA ,/,/ 15.1
Surface energy of solid metals
0
1
2
3
0 1 2 3lg, J/m2
sg, J
/m2
fcc
bcc
Kaptay / Day 3 / 21
22
oMs
oMCf
oMs
oMC
oMo
gsMo
gsMC H
HH
V
V
2
7173/2
//
Surface energy of solid metallic mono-carbides
M Vo
M
cm3/mol Vo
MC
cm3/mol sH
oM
kJ/mol fH
oMC
kJ/mol sM/g J/m2
MC/g J/m2
Ti 10,55 12,15 469,9 -184,1 1,88 2,50 Zr 14,01 15,34 608,8 -202,5 2,02 2,39 Hf 13,41 15,03 619,2 -208,4 2,12 2,45 V 8,34 11,49 514,2 -101,9 2,15 2,26
Nb 10,84 13,42 725,9 -140,6 2,51 2,38 Ta 10,87 13,40 782 -144,0 2,87 2,45 W 9,53 12,42 849,4 -40 3,11 2,47
Kaptay / Day 3 / 22
23
Surface energy of solid ionic mono-oxides
3/13/2/26,1
)28,3()01,008,0(
Avo
sMO
oMOso
gsMONV
TH
MX sH, kJ/mol
Vs, cm3/mol
T, K MO/g, J/m2
MO/g, J/m2
exp
NaCl 231 27,0 0 0,20 0,02 0,212
MgO 656 11,0 0 1,00 0,12 1,04
CaO 677 16,6 298 0,78 0,10 0,82
SrO 578 21,9 0 0,56 0,07 -
BaO 425 26,7 1373 0,30 0,05 0,29
FeO 523 12,3 0 0,74 0,09 0,732
Kaptay / Day 3 / 23
24
Excess interfacial enthalpy of sA/lA interface
Amo
A
oAm
cr
lAsA
T
T
V
H
r ,
/ 12
The Kelvin equation for the critical radius of nucleation:
The molar volume can be modeled as: b
AvA
oA f
NrV 3
3
4
AmAvo
A
oAm
lAsAcr
A
b T
T
NV
H
r
r
f ,3/13/2/3/1
1224.3
At T 0 K, the solid nucleus will be stable from an atom, i.e. rcr rA:
oAm
blAsA H
fH
224.3
3/1
/
Kaptay / Day 3 / 24 See J67
25
Excess interfacial entropy of sA/lA interface
From side of the solid there is not too much change in freedom:
Liquid atoms will loose part of their freedom at the s/l interface.
The entropy of melting:
0/ sAlBsAS
strmconfmvolmm SSSS
The excess interfacial entropy: )(5,0 ,,/ confBmvolBmlB
lBsA SSS
3/13/2
,,
3/1
/986,0
06,12224,3
Avo
lA
confAmvolAmoAm
b
olAsA
NV
SST
Hf
Kaptay / Day 3 / 25
26
Interfacial energy sA/lA
3/13/2
,,
3/1
/986,0
06,12224,3
Avo
lA
confAmvolAmoAm
b
olAsA
NV
SST
Hf
0
100
200
300
400
0 100 200 300 400
theor, mJ/m2
ex
p, m
J/m
2
Kaptay / Day 3 / 26
27
Summary of interfacial energies of pure metals (example: Fe)
0
0,5
1
1,5
2
2,5
3
0 500 1000 1500 2000 2500T, K
, J
/m2
sg
sl
lg
Tm
Kaptay / Day 3 / 27
28
Interfacial energy sA/lB
There is an extra excess enthalpy term, connected with the interaction of A and B atoms across the interface:
)(2
1)(2
)()(
)()( /////
BBAAABiAbA
BBbBABiBABBiBAAbAABiABAAiA
bB
iBlBsA
bA
iAlBsA
lBlBsA
sAlBsAlBsA
UUUzz
UzUzUzUzUzUz
HHHHHHH
From the theory of regular solutions:
)(
2
1BBAAABbABA UUUz
Finally, the new excess enthalpy term: BAAlBsAH 2/
Kaptay / Day 3 / 28
29
3/23/13/1
,,
3/1,
/
06,122
224,3
Avo
lBo
sA
confBmvolBmlBlAoAm
Ab
lBsANVV
SST
Hf
Interfacial energy sA/lB
0
500
1000
1500
0 500 1000 1500elm, mJ/m2
kis
, mJ/
m2
Zn/Sn (473 K), Ag/Pb (608 K), Cu/Pb (1193 és 1093 K), Fe/Cu (1373 K), Nb/Cu (1773 K), W/Cu (1773 K), Mo/Sn (1873 K), W/Sn (2273 K),
Fe/Pb (1373 and 1193 K) and Fe/Ag (1373 K)
Kaptay / Day 3 / 29
30
Interfacial energy between two immiscible liquids26.1
//
/ 12
cglBglA
lBlAlBlAlBlA T
TTS
0
40
80
120
160
0 300 600 900 1200 1500T
lA
/lB, m
J/m
2
Ga-Pb
Al-Bi
Ga-Pb: D.Chatain, L.Martin-Garin, N.Eustathopoulos: J. chim.phys., 79 (1982) 569Al-Bi: I.Kaban, W.Hoyer, M.Merkwitz: Z.Metallkunde, 94 (2003) p.831
Kaptay / Day 3 / 30
31
Covalent ceramic / liquid metal interface
The London dispersion forces connect the atoms across the interface:
62 r
AU aa
iB
iBiB II
IIA
2
3
CB
CB
CB
CBC
AB
AB
AB
ABA
lBsAC
lBsAC II
II
rrx
II
II
rrxW
66/ )()(
1
4
3
B WSi/B WN/B WSiN4/B WSiN4/B model
mJ/m2 model mJ/m2
model mJ/m2
exp mJ/m2
Cu 22,6 60,5 83,1 81,4 Ga 16,4 38,7 55,0 56,8 Sn 9,9 20,3 30,2 36,5 Pb 5,6 10,0 15,6 24,8 Bi 5,2 8,8 14,0 23,2
Kaptay / Day 3 / 31
32
Ionic ceramic / liquid metal interface (1)
The ion A induced dipole (in atom B) interaction with each other:
4BAo
B22
)(π4 rrε
ezU A
AB
The adhesion energy:
6
,
4BAo
22,
/ )(6
41
)(π463
2
BA
ACgBA
B
AvgBBAC rrrrε
ezNW
4BAo
22,
/ )(π463
2
rrε
ezNW A
B
AvgBBA
The corrected adhesion for dipole induced dipole interaction::
Kaptay / Day 3 / 32 See J66
33
Ionic ceramic / liquid metal interface (2)
The corrected adhesion energy for ion the ionic moment of ion C:
6
,
4BA
2,2
,5
/ )(3
21
)(1054,1
BA
ACgBA
B
gBACBAC rrrr
zkW
0
0,25
0,5
0,75
1
0 2 4 6 8IC/IA
k
Points from for the liquid Cu / MgO, ZrO2, Al2O3, SiO2 systems
Kaptay / Day 3 / 33
34
Wettability of solid Fe by molten chlorides
6
,
4BA
2,2
,5
/ )(3
21
)(1054,1
BA
ACgBA
B
gBACBAC rrrr
zkW
AC/B rC
nm
IC/IA k WAC/B
mJ/m2
AC/g
mJ/m2
B/AC/g
fok
MgCl2/Fe 0,074 4,89 0,39 66,4 138 121
CaCl2/Fe 0,104 3,48 0,44 85,0 147 115
BaCl2/Fe 0,138 2,62 0,48 101,9 169 114
NaCl/Fe 0,098 1,85 0,52 118,8 114 88
KCl/Fe 0,133 1,36 0,55 133,2 99 70Calculated data are confirmed experimentally by Vetiukov et al.
Kaptay / Day 3 / 34
35
Interfacial energy in liquid metal / molten salt systems
6
,
4BA
2,2
,5
/ )(3
21
)(1054,1
BA
ACgBA
B
gBACBAC rrrr
zkW
T.Utigard, J.M.Toguri, T.Nakamura, Metall.Trans. B., 17B (1986) 339
NaF/Bi (1373 K), NaCl/Bi (1373 K), NaF/Pb (1273 K), NaCl/Pb (1173 K), NaF/Ag (1273 K), NaCl/Ag (1273 K), NaF/Cu (1373 K) and NaCl/Cu (1373 K)
0
500
1000
1500
0 500 1000 1500AC/B, mJ/m
2, elm
AC
/B, m
J/m
2 , kis
Kaptay / Day 3 / 35
36
The surface excess (by definition):
BBoBB xx * B
oB
1
n
iii dd
1
The Gibbs equation:
ioii aTR ln ii adTRd ln
+Gibbs-Duhem (for binaries):
Ba
BBoA adTR
0
ln
Kaptay / Day 3 / 36
Concentration dependence of (Gibbs, 1878)
37
Langmuir (1918)
The equilibrium between bulk and surface phases:
A* + B = A + B*
RT
GK
oadsexp
BB
AB
ax
axK
*)1(
*
For the infinitely diluted solution of B in A: aA = 1, aB= B xB
B
BB xb
xbx
1
* BKb
Kaptay / Day 3 / 37
38
Belton (1976)
B
BB xb
xbx
1
* BBoBB xx *
B
oB
1
1
1 BB
BB xb
bx
Ba
BBoA adTR
0
lnIntegration at infinitely diluted solution of B: dlnaB = dxB/xB.
BBB
oA xxb
TR
)1ln(
Kaptay / Day 3 / 38
39
[??]
)1ln( BaoB
oA aK
TR
TRK
oB
oB
oA
oA
a
exp
RT
GK
oads
a exp
At infinitely diluted solution of B: xB << 1 (and if b >>1)
A* + B = A + B*
Kaptay / Day 3 / 39
Gibbs – Langmuir – Belton (1878 – 1976)
40
Theoretical Concentration dependence of Fe-O/g
)2841ln(2911960)1823( OCK
800
1200
1600
2000
0 0,05 0,1
CO , t%
Fe-O
/g, m
J/m
2 '
Kaptay / Day 3 / 40
41
Theoretical concentration dependence of Fe-S/g
)5231ln(1911960)1823( SCK
500
1000
1500
2000
0 0,5 1 1,5
CS, wt%
Fe-S
, mJ/
m2
Kaptay / Day 3 / 41
42
Additive extension to ternary Fe-O-S system
0
500
1000
1500
2000
-500 0 500 1000 1500 2000
additiv, mJ/m2
kis
, mJ
/m2
)5231ln(191)2841ln(2911960/ SOgSOFe aa
Kaptay / Day 3 / 42
Experimental points
Calculated values
43
Taking into account the competition between O and S atoms for surface sites
CCaBBaoC
CCa
oB
BBa
CCaBBa
oACBA aKaK
aKaK
aKaK
TR
,,,,
,,
1ln
0
500
1000
1500
2000
0 500 1000 1500 2000
elm, mJ/m2
kis
, mJ/
m2
Kaptay / Day 3 / 43
Calculated values
Experimental points
44
On the Butler equation (1932)
gBgAgBA ///
Compared to the Gibbs equation:
i. it is easier to teach and to apply, but still
ii. at equal assumptions it provides the same results.
B
BoB
ogB
A
AoA
ogAgBA a
aTR
a
aTR *ln
*ln ///
The surface activity coefficient as function of surface composition is to be modelled
Kaptay / Day 3 / 44
45
Modeling surface excess Gibbs energy
The model is based on the ratio of broken bonds ():
EiAAA GxTRaTR *ln*ln
EA
EiA GG )1(
For liquid metals: Hoar, Melford, 1957:
Monma, Suto, 1961: = 0.16 -0.20,
Speiser, Poirier, Yeum, 1987 – 1989: = 0.25,
Tanaka, Iida, Hara, Hack, 1994 – 2000: = 0.17,
For molten salts: Tanaka et al.: = 0.06, later -0.1 (!!??)
Kaptay / Day 3 / 45
46
The Butler equation, applied to associated liquids
Experimental points: V.N.Eremenko, V.I.Nizhenko, N.I.Levi, B.B.Bogatirenko: Ukr. Him. Zh., 1962, vol.28, No.4, pp.500-505
BA
oAaBbfBBAA
AaBb ba
Hba
Al-Ni
Kaptay / Day 3 / 46
47
A = 1 J/m2, B = 0.5 J/m2, VA = 1 10-5 m3/mol, VB = 2 10-5 m3/mol, = 20 kJ/mol. Then: Tc = 1202.79 K. At T = 875.86 K, bulk separation at xB = 0.1 and xB = 0.9
gBgA //
Kaptay / Day 3 / 47
Surface phase separation in monotectic alloys (a)
0,96
0,97
0,98
0,99
1
0 0,2 0,4 0,6 0,8 1xB*
A,
B, J
/m2
BA
Fig.1.a: XB = 0.00068
solution
Partial surface tensions as function of surface content
See J99
48
Kaptay / Day 3 / 48
Surface phase separation in monotectic alloys (b)
0,96
0,97
0,98
0,99
1
0 0,2 0,4 0,6 0,8 1xB*
A,
B, J
/m2
BA
Fig.1.b: XB = 0.00072
solution
Partial surface tensions as function of surface content
See J99
49
Kaptay / Day 3 / 49
Surface phase separation in monotectic alloys (c)
0,96
0,97
0,98
0,99
1
0 0,2 0,4 0,6 0,8 1xB*
A,
B, J
/m2
BA
Fig.1.c: XB = 0.000755
2 solutions
Partial surface tensions as function of surface content
See J99
50
Kaptay / Day 3 / 50
Surface phase separation in monotectic alloys (d)
0,96
0,97
0,98
0,99
1
0 0,2 0,4 0,6 0,8 1xB*
A,
B, J
/m2
B
A
Fig.1.d: XB = 0.00080
solution
Partial surface tensions as function of surface content
See J99
51
Kaptay / Day 3 / 51
Surface phase separation in monotectic alloys (e)
0,96
0,97
0,98
0,99
1
0 0,2 0,4 0,6 0,8 1xB*
A,
B, J
/m2
B
A
Fig.1.e: XB = 0.00085
solution
Partial surface tensions as function of surface content
See J99
52
Kaptay / Day 3 / 52
Surface phase separation in monotectic alloys (f)
-4
-3
-2
-1
0
-6 -5 -4 -3 -2 -1 0
logxB
log
xB*
Surface composition as function of bulk composition
See J99
53
Kaptay / Day 3 / 53
Surface phase separation in monotectic alloys (g)
Surface tension as function of bulk composition
0
0.2
0.4
0.6
0.8
1
1.2
-6 -5 -4 -3 -2 -1 0
logxB
, J
/m2
See J99
54
Kaptay / Day 3 / 54
Surface phase separation in monotectic alloys (h)
A phase diagram with a surface phase separation line
0
200
400
600
800
1000
1200
1400
-6 -5 -4 -3 -2 -1 0
logxB
T, K
1 bulk liquid
2 bulk liquids
1 bulk liquid+ nanolayer
SPT line
*crT
crT
Fig.2.c
See J99
55
Kaptay / Day 3 / 55
Surface phase separation in monotectic alloys (i)
T-coefficient of surface tension as function of bulk composition
-2
0
2
4
6
8
-4 -3.5 -3 -2.5 -2 -1.5 -1
logxB
d
/dT
10
4 , J/m
2 K
T=800 K
T=986 K
T=1200 K
See J99
56
Two shapes of welding poolsTwo shapes of welding pools
0dT
d0
dT
d
0
200
400
600
800
1000
1200
1400
-6 -5 -4 -3 -2 -1 0
logxB
T, K
1 bulk liquid
2 bulk liquids
1 bulk liquid+ nanolayer
SPT line
*crT
crT
Kaptay / Day 3 / 56
57
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