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A model for nonreactive ceramics: Liquid metal wetting
Scripta METALLURGICA et MATERIALIA
Vol. 27, pp. 1429-1434, 1992 Printed in the U.S.A.
Pergamon Press Ltd. All rights reserved
A MODEL FOR NONREACTIVE CERAMICS: LIQUID METAL W E T T I N G Wu J i a n x i n and Li Pengxing State Key Laboratory of Metal Matrix Composite Materials, S h a n g h a i Jiao Tong University, Shanghai 200030, P.R.China
(Received July 23, 1992) (Revised September 24, 1992) l . Introduction The rapid development of metal matrix composite materials and microelectronic materials imposes an u r g e n t need for a more q u a n t i t a t i v e knowledge of ceramics: liquid metal wetting. A series of p u b l i c a t i o n s reported the experimental data on wetting (1--7), but only a few have presented theoretical considerations (8--12). W h e n considering w e t t i n g of ceramics by liquid alloys, researchers are perplexed by the problem whether the i n t e r a c t i o n between ceramic molecules and the liquid metal atoms is of a Van der Waals type or a chemical type. Some work has been done to solve this puzzle. J o h n s o n and Pepper (8), using a cluster model and molecular orbital theory, showed t h a t a direct chemical bond can be established between metal and the oxygen anions on the A1203 surface. Hichter et al. (9), using an electronic band structure mold, found t h a t metal oxide adhesion results from an electron transfer from the liquid metal band to the empty c o n d u c t i o n band of the oxide. The electron transfer is less than a u n i t electron. Both the above results are qualitative. McDonald and Eberhart (10) assumed that the chemical bonds between atoms and oxygen ions of the oxide are established at specific oxygen sites at the interface in addition to Van der Waals bonds at the remaining oxygen sites, but they did not explain why it is so. W a r r e n ' s model (11) takes the s t r u c t u r a l contribution into account in addition to the chemical c o n t r i b u t i o n , but the results are n o t satisfactory. Eustathopoulos' group (12) revised the previous models and obtained an equation for the work of adhesion W:
W =
®
2
®
c ~ [AH o~,~+-~-AH ^~.~] 2/3 No1/3V~, where VM, is the volume of a metal atom, AHi(j) is the mixing enthalpy of i in j at infinite i dilution, No is the Avgadro n u m b e r , and e is a proportional constant. This equation is held as the most satisfactory one up to date. It is semi-empirical, c o n t a i n i n g a constant c which is determined to be 0.22 for Al~03: liquid metal w e t t i n g from the regression of experimental data. Besides, calculated results of W for Fe, Co and Ni deviate considerably from the experimental data. In this work, a model is established for ceramics: liquid metal wetting, and an equation was derived for the work of adhesion. This equation automatically gives Eustathopoulos' equation with c o n s t a n t c determined directly from the atomic and ionic radii in the ceramics. In addition, considering the effect of the bond distortion as a result of the existence of the ceramics: liquid metal interface, a term is a n n e x e d to Eustathopoulos' equation. This model could give more satisfactory results than the previous ones.
1429 0956-716X/92 $5.00 + .00 Copyright (c) 1992 Pergamon Press Ltd.
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i0
II. Development of the Model for Ceramics: Liquid Metal Wetting Regarding the atomic configuration at the wetting interface, an exclusively clear image is not yet available. However, the geometric a r r a n g e m e n t for a binary alloy composed of two species of different atomic radii proposed by Miedema (13) can be used as an approximation. As is shown in FIG.l, the ceramics are composed of two species C + and A - . The symbols are used in such a way that they are the cation and the a n i o n respectively in the case of ionic ceramics; they are the polarized atoms in the case of polar covalent ceramics; and C+ and A - are the same in the case of homopolar covalent ceramics. For convenience, we shall refer to the two species as cation and a n i o n t h r o u g h o u t the context. The surface of the ions which is in contact with the liquid metal atoms on wetting accounts for one sixth of the total surface of the ions S. . . . = !S,o. "
(1)
6
Two major energy terms can be proposed to contribute to the work of adhesion: (i) I n t e r a t o m i c bond energy across the interface; (ii) Modification of the bond energy due to the interface. (i) Interracial I n t e r a t o m i c Bond Energy When the ceramicsare wetted by a liquid metal Me, the cation C + and the a n i o n A - in the ceramic surface are in i n t i m a t e contact with the metal atoms. So certain types of bonds are set up between C + a n d A - and Me. Since the ceramic anions usually have a much more powerful electron affinity t h a n the metal atoms in the case of ionic and ionocovalent ceramics, the electron transfer from the metal atoms to the a n i o n s occurs, j u s t as indicated by J o h n s o n and Pepper (8) as well as Hichter et al. (9), and the A - - M e + bond is essentially chemical. Let us first examine the anion-metal bond. Because the interfacial ceramic ions are in contact with a thick wall of the wetting liquid metal, u n l i k e McDonald and E b e r h a r t (10), we suppose that all the A - - M e + bond is in n a t u r e - t h e same as that in the b i n a r y A - Me solution of infinite dilution of A. This assumption is r e a s o n a b l e considering the localization of the A - - M e + bond orbitals. So the A - - M e + bond has a bond energy approximately equal to 1CS^-s^'-M.W^--M..=~-
(2)
where C~- is the surface area c o n c e n t r a t i o n of the anion, and e,^--M." is the total bond energy of an anion in the A - M e solution of infinite a n i o n dilution. The right side is divided by 6 because, according to eqn.(1), only one sixth of the anion surface is in contact with the liquid metal.
IMeM~ (a) ceramics
(b) dilute solution {X=A,B)
FIG.1 Atomic Configuration of the Ceramics and the Dilute Solution (Two Dimension Sketch) The total bond energy 8^ -M." of the anion is directly related to the e n t h a l p y of formation AH~o~.~
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of the A - Me solution at infinite A dilution (13), so C~- AH~c~,~
(3)
6 NoV~_~ where AH~e~ is in unit of J per mole, No is the Avgadro number, V^- is the v o l u m e of the anion. In the same way, we obtain the part of the work of adhesion from the C + - Me bond as C~. A H ~ . I
Wc.-M.-- -
(4) 6 NoVc~ F u r t h e r , if the liquid m e t a l is the cation metal, its atom has a size that fits well on the ceramic surface. B u t if the liquid metal is not the cation metal, its atom may h a v e a size t h a t does n o t fit well on the ceramic surface. In this case, each ceramic ion shares on the av er ag e a number, (Vc/Vm) v3, of the liquid m et al atoms. So combining eqn.s(3) and (4) gives the contribution from the bond energy to the work of adhesion W~-
(Vc/VM')v3 [ CsA A H ~ . ~ + C~. AHc~M.~] 6No V~/.3 V~3
(5)
(ii) Modification of the Bond Energy due to the Interface It is known from eqn.(1) t h a t 5/6 of the surface area of the anion is in co n t act with the cation, which d o m i n at es the bonding state of the anion. So the anion, with the outer atomic orbital fully occupied by electrons transferred from the cation atoms, would polarize t h e m et al ion Me + more severely t h a n in the case of pure A - - M e + bond. Oppositely, the metal atoms, w h i ch do not h a v e their v a l e n t electrons all transferred, only weakly polarize the anion, which is embedded in t h e ceramics. This could be easily understood by treating the wetting liquid metal atoms as epitaxial layers on the ceramic surface. When the A - - M e + bond is completely strained to the A - - C ÷ bond, the bond length l^ -~..=l^--c-; when completely free, 1A--M,.--IA--M..,1A--U._ o o is the bond length of pure A - - M e + bond. In an actual situation, t h e bond length of the A - - M e + bond at the ceramics: liquid m et al interface could be calculated ap p r o x i m a t e l y using Eshelby's model which states that the actual bond relaxes between the completely strained state and the completely free state. The bond length r e l a x a t i o n is ARM.. - rM.. - rc. 2
(6)
Because the anion, embedded in the ceramics, has already accepted electrons from t h e cation atoms, it s t r o n g l y resists deformation. Therefore, we further suppose t h a t the bond length relaxes in the m et al atoms; t h a t is, the ionic radius rM,. is strained ArM~
"
EM," -rMe"
A l t h o u g h the force constant varies with the atomic distance, for simplicity, we here take a c o n s t a n t force c o n s t a n t t h a t ' i s roughly equal to the elastic modulus of the solid metal. Then the corresponding elastic energy density over the anion is
e^-: ~1- E ~ rMe.-rc. (~)
2
(7)
w h ere EM, is the elastic modulus of the solid metal. EM, has the low t e m p e r a t u r e v a l u e because the Me + is r e g u l a r l y arranged.
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Similarly, with the C + - M e bond strained and the cation C ÷ assumed to be rigid, the change of the atomic radius of the metal at the interface is r . . -- rA-
Ar.. =
(8)
2 1
r.,--
so the elastic energy density over the cation is Ec. = - ~ - E . . ( ~ )
r^-
2
(9)
Finally, the modification of the bond energy per u n i t interracial area is W. = E. = 2CS-rM..E^-sign(r... - rc.) + 2C~.r..Ec.sign(r.. - ro,-)
(10)
where (2Cl-r...) and (2C~.r..) give the strained volume of the metal atoms over the anion and the cation, respectively. The sign function sign ( r . . . - r e . ) implies that the A - - M e + bond energy is larger due to the interface when r...>rc., and smaller when r...
E"
s
r M , -- r^-
..{CS._rM..( rM.. - rc- )'sign(r... - rc.)+ C c . r M . ( rMe*
)'sign(rM. -- ro,-)}
r.e
(Vc/V.,)'n.r C~- AH ® + C~. AH~<..,} 6No t V~-3 *a~.~ V~q
(11)
This is the e q u a t i o n for the work of adhesion. Because the surface area c o n c e n t r a t i o n s C[- and Cg + are strongly o r i e n t a t i o n dependent, W is also orientation dependent. The elastic moduli of nonferrous metals are relatively small, so the terms in the first bracket are i m p o r t a n t only for ferrous metals with high elastic moduli. In the above process, we totally ignored the polarization of the ceramic i o n s - - t h e y are supposedly hard to polarize. Besides, the surface relaxation and reconstruction of the ceramics are also ignored.
m. Application to Al20,/Metal systems If the ceramics is AhO3, we approximately take the surface area concentrations of the oxygen anions and aluminum cations to be the average on the stoichiametric number, so the concentrations are orientation independent
3v~::
C~.- -
C~,,.
2vZ~. -
3 v ~ ~-+ 2v~,.
(12)
3 v ~ ~-+ 2vZ~,.
where V is the ionic volume. And eqn.(ll) becomes
W
=
W.
(V^'/V"')va 213
2
[AH~.)+ 2---AHZ~.)] S/3
6No(Vo,- +-~V^l,.) The bond energy related term Wb can be converted into the form m
2
oo
[AHoy., + -~-AH^,,..,] C¢^L/V..)
2/3
Wb2
2,~
VOW"-+ ~ - v ^ , '(6-
~¢^, ~, )No(-~o )
vt~?,.
where V^, is the molar volume of aluminum. The following data (14,15) are used
(13)
Vol. 27, No. I0
LIQUID METAL WETTING
r ^ r . = 53pm,
r^,=143.1pm,
1433
ro,- = 121pm
SO
W~-
0"20~7 [AH~,s.~+ 3 A H~cM.~]
(14)
1/3 ~ 2/3 No VM. Eqn.(14) is just the Eustathopoulos' formula (12) but with the coefficient directly determined from the ionic and atomic radii.
T h e m o d i f i c a t i o n of t h e w o r k of a d h e s i o n d u e to t h e b o n d l e n g t h d e c r e a s e is c a l c u l a t e d , as is s h o w n i n T A B L E 1. TABLE 1 C a l c u l a t e d V a l u e s for M o d i f i c a t i o n of B o n d L e n g t h (rs~,. = 53pm, ro,- = 121pm (14))
Me
Fe Co Ni A1 Ag Au Cu Ga In Pb Pd Si Sn •~ D a t a
rM. (pm)
rM.. (pro)
elasticmodulus
141 139 138 143.1 144 144 127 122 162 174 117 117 141
77 72 68 53 81 82 71 61 76 79 78 40 69
1.9 1.9 1.9 -0.4 -0.4 - 0.4 -0.4 - 0.4 -0.4 - 0.4 - 0.4 -0.4 - 0.4
( x 10-"
N/m')
W. (J/m') 0.327 0.220 0.146 - 0 0.09 0.09 0.03 0.01 0.07 0.09 0.07 - 0.02 0.03
o f a t o m i c a n d i o n i c sizes a r e c i t e d from Ref.s(14) a n d (15).
1300
1100
.//Co Fe
).
900
ed A~ ,/
$i
70O
Cu
//
500 3O0
o~4~
Pb
100 10O
3oo
soo W
700 9oo ml/m 2
*xp
noo
13oo
FIG.2 C o m p a r i s o n of t h e E x p e r i m e n t al a n d C a l c u l a t e d V a l u e s o f t h e w o r k of A d h e s i o n
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LIQUID METAL WETTING
Vol.
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The total work of adhesion given by the sum of Wb and W° for different metals on alumina is compared with the experimental values, as is shown in FIG.2. In comparison with Eustathopoulos' formula (12), it is seen that the addition of the term W, to the work of adhesion is reasonable and necessary. IV. Concluding Remarks The most significant assumption on which our model is based is that the ceramic anion-liquid metal atoms bond (the ceramic cation-liquid metal atoms bond) has the modified energy of the pure A - - M e + (C + - Me) bonds in an A - M e ( C - M e ) solution of infinite A(C) dilution. The modification of the bond energy is due to the polarization of the metal ions and atoms. This polarization tend to strain the A - - M e + bond length to that of A - - C + and strain C + - M e to C + - A -. Therefore, the most questionable point is how much the ceramics-liquid metal bonds differ from the pure A - M e and C - M e bonds. A satisfactory treatment should be carried out based on the measurement or calculation of the interface ionocovalent bond energies, including the relaxation of the interfacial bonding and the interface atomic configuration. Our conclusion is that an equation for ceramics: liquid metal wetting is established that automatically gives Eustathopoulos' formula. In addition, we considered the interfacial bond distortion for the first time. The elastic modification of the ionocovalent bonds between the ceramic ions and the liquid metal atoms offers a more satisfactory result. This work has made it possible to calculate the work of adhesion for nonreactive wetting using eqn.(ll) with known mixing enthalpy, ionic radii and elastic modulus of the metal. It is seen that W is orientation dependent. This work also revealed the urgent need for accurate evaluation of bond energies at the ceramics: liquid metal interface. Acknowledgement The authors are grateful to Professor Gu Mingyuan at Shanghai Jiao Tong University for his useful discussion and suggestions. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
J.G. Li and H. Hausner, J. of Mater. Sci. Letts., 10, 1277(1991). H. J o h n and H. Hausner, J. of Mater. Sci. Letts., 5, 549(1986). I. Rivollet, De Chatain and N. Eustathopoulos, Acta Metall., Vol.35, No.4, 835(1987). Ali-Reza Yavari, Z. Metaakde., Bd. 79, H.9, 591(1988). T. Chou, R. Kammel and T. Oki, Z. Metallkde., Bd. 78, H.4, 287(1987). H. Nakae, K. Yamamoto and K. Sato, Met. Trans. JIM. Vol 32, No.6, 531(1991). S.-Y. Oh. J.A. Cornie and K.C. Russell, Metall, Trans. A, Vol.20, 527(Mar. 1989). K.H. J o h n s o n and S.V. Pepper, J. Appl. Phys., Vol.53, No.10, 6635(Oct, 1982). P. Hichter, D. Chatain, A. Pasturel and N. Eustathopoulos, J. Chim. Phys., 85, 941(1988). J.E. McDonald and J.G. Eberhart, Trans. Metall. Soc. AIME, 233, 512(1969). R.W. Warren, J. of Mater. Sci., 15, 2489(1980). D. Chatain, L.Coudurier and N. Eustathopoulos, Revue de Physique Appliquee, 23, 1055(1988). A.R. Miedema and P.F. de Chatel, in Book: Theory of Alloy Phase Formation, Edited by L.H. Bennet, A Publication of the Metall. Soc. of AIME, 344(1963). [14] R.D. Shannon, Acta Metallogr., A32,751(1976). [15] Lange's Handbook of Chemistry, 13th rd. McGraw-Hill(1985).
Vol. 27, pp. 1429-1434, 1992 Printed in the U.S.A.
Pergamon Press Ltd. All rights reserved
A MODEL FOR NONREACTIVE CERAMICS: LIQUID METAL W E T T I N G Wu J i a n x i n and Li Pengxing State Key Laboratory of Metal Matrix Composite Materials, S h a n g h a i Jiao Tong University, Shanghai 200030, P.R.China
(Received July 23, 1992) (Revised September 24, 1992) l . Introduction The rapid development of metal matrix composite materials and microelectronic materials imposes an u r g e n t need for a more q u a n t i t a t i v e knowledge of ceramics: liquid metal wetting. A series of p u b l i c a t i o n s reported the experimental data on wetting (1--7), but only a few have presented theoretical considerations (8--12). W h e n considering w e t t i n g of ceramics by liquid alloys, researchers are perplexed by the problem whether the i n t e r a c t i o n between ceramic molecules and the liquid metal atoms is of a Van der Waals type or a chemical type. Some work has been done to solve this puzzle. J o h n s o n and Pepper (8), using a cluster model and molecular orbital theory, showed t h a t a direct chemical bond can be established between metal and the oxygen anions on the A1203 surface. Hichter et al. (9), using an electronic band structure mold, found t h a t metal oxide adhesion results from an electron transfer from the liquid metal band to the empty c o n d u c t i o n band of the oxide. The electron transfer is less than a u n i t electron. Both the above results are qualitative. McDonald and Eberhart (10) assumed that the chemical bonds between atoms and oxygen ions of the oxide are established at specific oxygen sites at the interface in addition to Van der Waals bonds at the remaining oxygen sites, but they did not explain why it is so. W a r r e n ' s model (11) takes the s t r u c t u r a l contribution into account in addition to the chemical c o n t r i b u t i o n , but the results are n o t satisfactory. Eustathopoulos' group (12) revised the previous models and obtained an equation for the work of adhesion W:
W =
®
2
®
c ~ [AH o~,~+-~-AH ^~.~] 2/3 No1/3V~, where VM, is the volume of a metal atom, AHi(j) is the mixing enthalpy of i in j at infinite i dilution, No is the Avgadro n u m b e r , and e is a proportional constant. This equation is held as the most satisfactory one up to date. It is semi-empirical, c o n t a i n i n g a constant c which is determined to be 0.22 for Al~03: liquid metal w e t t i n g from the regression of experimental data. Besides, calculated results of W for Fe, Co and Ni deviate considerably from the experimental data. In this work, a model is established for ceramics: liquid metal wetting, and an equation was derived for the work of adhesion. This equation automatically gives Eustathopoulos' equation with c o n s t a n t c determined directly from the atomic and ionic radii in the ceramics. In addition, considering the effect of the bond distortion as a result of the existence of the ceramics: liquid metal interface, a term is a n n e x e d to Eustathopoulos' equation. This model could give more satisfactory results than the previous ones.
1429 0956-716X/92 $5.00 + .00 Copyright (c) 1992 Pergamon Press Ltd.
1430
LIQUID METAL WETTING
Vol.
27, No.
i0
II. Development of the Model for Ceramics: Liquid Metal Wetting Regarding the atomic configuration at the wetting interface, an exclusively clear image is not yet available. However, the geometric a r r a n g e m e n t for a binary alloy composed of two species of different atomic radii proposed by Miedema (13) can be used as an approximation. As is shown in FIG.l, the ceramics are composed of two species C + and A - . The symbols are used in such a way that they are the cation and the a n i o n respectively in the case of ionic ceramics; they are the polarized atoms in the case of polar covalent ceramics; and C+ and A - are the same in the case of homopolar covalent ceramics. For convenience, we shall refer to the two species as cation and a n i o n t h r o u g h o u t the context. The surface of the ions which is in contact with the liquid metal atoms on wetting accounts for one sixth of the total surface of the ions S. . . . = !S,o. "
(1)
6
Two major energy terms can be proposed to contribute to the work of adhesion: (i) I n t e r a t o m i c bond energy across the interface; (ii) Modification of the bond energy due to the interface. (i) Interracial I n t e r a t o m i c Bond Energy When the ceramicsare wetted by a liquid metal Me, the cation C + and the a n i o n A - in the ceramic surface are in i n t i m a t e contact with the metal atoms. So certain types of bonds are set up between C + a n d A - and Me. Since the ceramic anions usually have a much more powerful electron affinity t h a n the metal atoms in the case of ionic and ionocovalent ceramics, the electron transfer from the metal atoms to the a n i o n s occurs, j u s t as indicated by J o h n s o n and Pepper (8) as well as Hichter et al. (9), and the A - - M e + bond is essentially chemical. Let us first examine the anion-metal bond. Because the interfacial ceramic ions are in contact with a thick wall of the wetting liquid metal, u n l i k e McDonald and E b e r h a r t (10), we suppose that all the A - - M e + bond is in n a t u r e - t h e same as that in the b i n a r y A - Me solution of infinite dilution of A. This assumption is r e a s o n a b l e considering the localization of the A - - M e + bond orbitals. So the A - - M e + bond has a bond energy approximately equal to 1CS^-s^'-M.W^--M..=~-
(2)
where C~- is the surface area c o n c e n t r a t i o n of the anion, and e,^--M." is the total bond energy of an anion in the A - M e solution of infinite a n i o n dilution. The right side is divided by 6 because, according to eqn.(1), only one sixth of the anion surface is in contact with the liquid metal.
IMeM~ (a) ceramics
(b) dilute solution {X=A,B)
FIG.1 Atomic Configuration of the Ceramics and the Dilute Solution (Two Dimension Sketch) The total bond energy 8^ -M." of the anion is directly related to the e n t h a l p y of formation AH~o~.~
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1431
of the A - Me solution at infinite A dilution (13), so C~- AH~c~,~
(3)
6 NoV~_~ where AH~e~ is in unit of J per mole, No is the Avgadro number, V^- is the v o l u m e of the anion. In the same way, we obtain the part of the work of adhesion from the C + - Me bond as C~. A H ~ . I
Wc.-M.-- -
(4) 6 NoVc~ F u r t h e r , if the liquid m e t a l is the cation metal, its atom has a size that fits well on the ceramic surface. B u t if the liquid metal is not the cation metal, its atom may h a v e a size t h a t does n o t fit well on the ceramic surface. In this case, each ceramic ion shares on the av er ag e a number, (Vc/Vm) v3, of the liquid m et al atoms. So combining eqn.s(3) and (4) gives the contribution from the bond energy to the work of adhesion W~-
(Vc/VM')v3 [ CsA A H ~ . ~ + C~. AHc~M.~] 6No V~/.3 V~3
(5)
(ii) Modification of the Bond Energy due to the Interface It is known from eqn.(1) t h a t 5/6 of the surface area of the anion is in co n t act with the cation, which d o m i n at es the bonding state of the anion. So the anion, with the outer atomic orbital fully occupied by electrons transferred from the cation atoms, would polarize t h e m et al ion Me + more severely t h a n in the case of pure A - - M e + bond. Oppositely, the metal atoms, w h i ch do not h a v e their v a l e n t electrons all transferred, only weakly polarize the anion, which is embedded in t h e ceramics. This could be easily understood by treating the wetting liquid metal atoms as epitaxial layers on the ceramic surface. When the A - - M e + bond is completely strained to the A - - C ÷ bond, the bond length l^ -~..=l^--c-; when completely free, 1A--M,.--IA--M..,1A--U._ o o is the bond length of pure A - - M e + bond. In an actual situation, t h e bond length of the A - - M e + bond at the ceramics: liquid m et al interface could be calculated ap p r o x i m a t e l y using Eshelby's model which states that the actual bond relaxes between the completely strained state and the completely free state. The bond length r e l a x a t i o n is ARM.. - rM.. - rc. 2
(6)
Because the anion, embedded in the ceramics, has already accepted electrons from t h e cation atoms, it s t r o n g l y resists deformation. Therefore, we further suppose t h a t the bond length relaxes in the m et al atoms; t h a t is, the ionic radius rM,. is strained ArM~
"
EM," -rMe"
A l t h o u g h the force constant varies with the atomic distance, for simplicity, we here take a c o n s t a n t force c o n s t a n t t h a t ' i s roughly equal to the elastic modulus of the solid metal. Then the corresponding elastic energy density over the anion is
e^-: ~1- E ~ rMe.-rc. (~)
2
(7)
w h ere EM, is the elastic modulus of the solid metal. EM, has the low t e m p e r a t u r e v a l u e because the Me + is r e g u l a r l y arranged.
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LIQUID METAL WETTING
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Similarly, with the C + - M e bond strained and the cation C ÷ assumed to be rigid, the change of the atomic radius of the metal at the interface is r . . -- rA-
Ar.. =
(8)
2 1
r.,--
so the elastic energy density over the cation is Ec. = - ~ - E . . ( ~ )
r^-
2
(9)
Finally, the modification of the bond energy per u n i t interracial area is W. = E. = 2CS-rM..E^-sign(r... - rc.) + 2C~.r..Ec.sign(r.. - ro,-)
(10)
where (2Cl-r...) and (2C~.r..) give the strained volume of the metal atoms over the anion and the cation, respectively. The sign function sign ( r . . . - r e . ) implies that the A - - M e + bond energy is larger due to the interface when r...>rc., and smaller when r...
E"
s
r M , -- r^-
..{CS._rM..( rM.. - rc- )'sign(r... - rc.)+ C c . r M . ( rMe*
)'sign(rM. -- ro,-)}
r.e
(Vc/V.,)'n.r C~- AH ® + C~. AH~<..,} 6No t V~-3 *a~.~ V~q
(11)
This is the e q u a t i o n for the work of adhesion. Because the surface area c o n c e n t r a t i o n s C[- and Cg + are strongly o r i e n t a t i o n dependent, W is also orientation dependent. The elastic moduli of nonferrous metals are relatively small, so the terms in the first bracket are i m p o r t a n t only for ferrous metals with high elastic moduli. In the above process, we totally ignored the polarization of the ceramic i o n s - - t h e y are supposedly hard to polarize. Besides, the surface relaxation and reconstruction of the ceramics are also ignored.
m. Application to Al20,/Metal systems If the ceramics is AhO3, we approximately take the surface area concentrations of the oxygen anions and aluminum cations to be the average on the stoichiametric number, so the concentrations are orientation independent
3v~::
C~.- -
C~,,.
2vZ~. -
3 v ~ ~-+ 2v~,.
(12)
3 v ~ ~-+ 2vZ~,.
where V is the ionic volume. And eqn.(ll) becomes
W
=
W.
(V^'/V"')va 213
2
[AH~.)+ 2---AHZ~.)] S/3
6No(Vo,- +-~V^l,.) The bond energy related term Wb can be converted into the form m
2
oo
[AHoy., + -~-AH^,,..,] C¢^L/V..)
2/3
Wb2
2,~
VOW"-+ ~ - v ^ , '(6-
~¢^, ~, )No(-~o )
vt~?,.
where V^, is the molar volume of aluminum. The following data (14,15) are used
(13)
Vol. 27, No. I0
LIQUID METAL WETTING
r ^ r . = 53pm,
r^,=143.1pm,
1433
ro,- = 121pm
SO
W~-
0"20~7 [AH~,s.~+ 3 A H~cM.~]
(14)
1/3 ~ 2/3 No VM. Eqn.(14) is just the Eustathopoulos' formula (12) but with the coefficient directly determined from the ionic and atomic radii.
T h e m o d i f i c a t i o n of t h e w o r k of a d h e s i o n d u e to t h e b o n d l e n g t h d e c r e a s e is c a l c u l a t e d , as is s h o w n i n T A B L E 1. TABLE 1 C a l c u l a t e d V a l u e s for M o d i f i c a t i o n of B o n d L e n g t h (rs~,. = 53pm, ro,- = 121pm (14))
Me
Fe Co Ni A1 Ag Au Cu Ga In Pb Pd Si Sn •~ D a t a
rM. (pm)
rM.. (pro)
elasticmodulus
141 139 138 143.1 144 144 127 122 162 174 117 117 141
77 72 68 53 81 82 71 61 76 79 78 40 69
1.9 1.9 1.9 -0.4 -0.4 - 0.4 -0.4 - 0.4 -0.4 - 0.4 - 0.4 -0.4 - 0.4
( x 10-"
N/m')
W. (J/m') 0.327 0.220 0.146 - 0 0.09 0.09 0.03 0.01 0.07 0.09 0.07 - 0.02 0.03
o f a t o m i c a n d i o n i c sizes a r e c i t e d from Ref.s(14) a n d (15).
1300
1100
.//Co Fe
).
900
ed A~ ,/
$i
70O
Cu
//
500 3O0
o~4~
Pb
100 10O
3oo
soo W
700 9oo ml/m 2
*xp
noo
13oo
FIG.2 C o m p a r i s o n of t h e E x p e r i m e n t al a n d C a l c u l a t e d V a l u e s o f t h e w o r k of A d h e s i o n
1434
LIQUID METAL WETTING
Vol.
27, No. I0
The total work of adhesion given by the sum of Wb and W° for different metals on alumina is compared with the experimental values, as is shown in FIG.2. In comparison with Eustathopoulos' formula (12), it is seen that the addition of the term W, to the work of adhesion is reasonable and necessary. IV. Concluding Remarks The most significant assumption on which our model is based is that the ceramic anion-liquid metal atoms bond (the ceramic cation-liquid metal atoms bond) has the modified energy of the pure A - - M e + (C + - Me) bonds in an A - M e ( C - M e ) solution of infinite A(C) dilution. The modification of the bond energy is due to the polarization of the metal ions and atoms. This polarization tend to strain the A - - M e + bond length to that of A - - C + and strain C + - M e to C + - A -. Therefore, the most questionable point is how much the ceramics-liquid metal bonds differ from the pure A - M e and C - M e bonds. A satisfactory treatment should be carried out based on the measurement or calculation of the interface ionocovalent bond energies, including the relaxation of the interfacial bonding and the interface atomic configuration. Our conclusion is that an equation for ceramics: liquid metal wetting is established that automatically gives Eustathopoulos' formula. In addition, we considered the interfacial bond distortion for the first time. The elastic modification of the ionocovalent bonds between the ceramic ions and the liquid metal atoms offers a more satisfactory result. This work has made it possible to calculate the work of adhesion for nonreactive wetting using eqn.(ll) with known mixing enthalpy, ionic radii and elastic modulus of the metal. It is seen that W is orientation dependent. This work also revealed the urgent need for accurate evaluation of bond energies at the ceramics: liquid metal interface. Acknowledgement The authors are grateful to Professor Gu Mingyuan at Shanghai Jiao Tong University for his useful discussion and suggestions. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
J.G. Li and H. Hausner, J. of Mater. Sci. Letts., 10, 1277(1991). H. J o h n and H. Hausner, J. of Mater. Sci. Letts., 5, 549(1986). I. Rivollet, De Chatain and N. Eustathopoulos, Acta Metall., Vol.35, No.4, 835(1987). Ali-Reza Yavari, Z. Metaakde., Bd. 79, H.9, 591(1988). T. Chou, R. Kammel and T. Oki, Z. Metallkde., Bd. 78, H.4, 287(1987). H. Nakae, K. Yamamoto and K. Sato, Met. Trans. JIM. Vol 32, No.6, 531(1991). S.-Y. Oh. J.A. Cornie and K.C. Russell, Metall, Trans. A, Vol.20, 527(Mar. 1989). K.H. J o h n s o n and S.V. Pepper, J. Appl. Phys., Vol.53, No.10, 6635(Oct, 1982). P. Hichter, D. Chatain, A. Pasturel and N. Eustathopoulos, J. Chim. Phys., 85, 941(1988). J.E. McDonald and J.G. Eberhart, Trans. Metall. Soc. AIME, 233, 512(1969). R.W. Warren, J. of Mater. Sci., 15, 2489(1980). D. Chatain, L.Coudurier and N. Eustathopoulos, Revue de Physique Appliquee, 23, 1055(1988). A.R. Miedema and P.F. de Chatel, in Book: Theory of Alloy Phase Formation, Edited by L.H. Bennet, A Publication of the Metall. Soc. of AIME, 344(1963). [14] R.D. Shannon, Acta Metallogr., A32,751(1976). [15] Lange's Handbook of Chemistry, 13th rd. McGraw-Hill(1985).