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Universal Binding Energy Relations in Metallic Adhesion
19
UNIVERSAL B I N D I N G ENERGY RELATIONS I N METALLIC ADHESION
J. FERRANTE N a t i o n a l A e r o n a u t i c s and Space A d m i n i s t r a t i o n , Lewis Research Center, Cleveland, Ohio
U.S.A.
44135
J. K. SMITH b e n e r a l IYotors Research L a b o r a t o r i e s ,
Warren, M i c h i g a n
U.S.A.
48UYU
J. H. KUSE Ames L a b o r a t o r y , Iowa S t a t e U n i v e r s ty, h e s , Iowa
SOU11
U.S.A.
AbSTKACT Kose, F e r r a n t e , and Smith ( r e f
7 ) have d i s c o v e r e d s c a l i n g r e l a t i o n s wnich map
t h e adhesive b i n d i n g energy o t F e r r a n t e and Smith ( r e f . d ) o n t o a s i n g l e u n i v e r s a l b i n d i n g energy curve.
The e n e r g i e s i n r e f . 6 a r e c a l c u l a t e d f o r a l l combinations o f
A l ( 1 1 1 ) , Ln(UUUl), Clg(UUOl), and Na(l1U) i n c o n t a c t .
The s c a l i n g i n v o l v e s
n o r m a l i z i n g t h e energy t o t h e maximum b i n d i n g energy and n o r m a l i z i n g d i s t a n c e s by a
A s i m p l e mathematical
s u i t a b l e c o m b i n a t i o n o f Thomas-Fermi s c r e e n i n g l e n g t h s .
e x p r e s s i o n i s f o u n d t o a c c u r a t e l y r e p r e s e n t t h e u n i v e r s a l curve, E * ( a * ) exp (-ea*)
E*
where
s e p a r a t i o n , and
6
i s t h e n o r m a l i z e d b i n d i n g energy, a* i s t h e f i t t i n g parameter.
t h a t the c a l c u l a t e d cohesive energies o f
=
-(1
+
ea*)
i s the normalized
Kose e t a l . ( r e f . 7 ) have a l s o found
K, Ba, Cu, No, and Sm s c a l e by s i m i l a r
silriple r e l a t i o n s s u g g e s t i n g t h e u n i v e r s a l r e l a t i o n may be more g e n e r a l t h a n f o r t h e s i m p l e f r e e e l e c t r o n m e t a l s f o r which i t was d e r i v e d .
I n t h i s paper we o u t l i n e these
r e S U l t S and d i s c u s s them i n r e l a t i o n t o t o p i c s o f i n t e r e s t i n adhesion, f r i c t i o n , and wear.
INTRODUCT10N An i m p o r t a n t aspect o f f r i c t i o n and wear o f m e t a l s i s t h e n a t u r e o f t h e adhesive f o r c e between t h e s u r f a c e s .
Bowden and Tabor ( r e f . 1 ) have used adhesion between
m e t a l s t o e x p l a i n p a r t s o f t h e f r i c t i o n and wear process.
Uf particular interest i n
t h e s e processes i s t h e p h y s i c a l n a t u r e o f t h e t o r c e s i n o i s s i n i i l a r m e t a l c o n t a c t s .
20
There have been numerous attempts to postulate the nature of these forces but few model calculations. For exaniple, adhesion between dissimilar metals was explained in terms o f solubility (ref. 2 ) but without accompanying mooel calculations. Buckley (ref. 3 ) has shown that metals with low mutual solubility can have higher adhesion than those with high solubility. This lack of model calculations has occurred because of the theoretical difficulties involve0 in treating solid surfaces. In the past decaae the electron density functional formalism of Hohenberg, Kohn and Sham (ref. 4) have been used to successfully calculate the surface energies (ref. 5) of simple metals. Ferrante and Smith (ref. b) recently applied this formalism to the problem of metallic adhesion for both similar and dissimilar metals in contact. Rose, Ferrante and Smith (ref. 7) found that scaling laws exist which map these results onto a universal adhesive energy curve. In this paper, we discuss the results o f these calculations and their implications to a more general category of materials, the transition metals, and then summarize the current state of adhesion theory.
USE OF KOHN-SHAM FORMALISM FOR ADHESION CALCULATIONS
The calculational formalism and methods used for obtaining self-consistent interface electronic structure will now be presented. A much more extensive description is given in refs. 6 and 8, particularly of numerical techniques. The adhesive interaction energy, Ead, between two metal surfaces is a function o f the distance between the two surfaces, a (see Fig. 1). Ead is defined as the negative of the amount of work necessary to increase the separation from a to divided by twice the cross-sectional area A. Thus 01
where E is the total energy. For identical metals, Ead is the negative of the surface energy when a is at the energy minimum. According to the density functional formalism of Hohenberg, Kohn, and Sham, (ref. 4) the total energy is given by (atomic units are used throughout unless otherwise specified)
E{n(?)}
=
/
v(?)n(?)dT
z.2. +
+
i+j
'J
F{n(?)},
21
(3)
v(7) is the
ionic potential, and n(7) is the electron number density. The first two terms in Eq. ( 2 ) are the electron-ion and ion-ion interaction energies, respectively. z is the ionic charge and K . . is the distance between ion core nuclei 1J is the (there is no ion core overlap in the systems considered here). Ts{n(?)l kinetic energy of a system of noninteracting electrons with the same density n(?), the next term is the classical electron-electron interaction energy, and Exc is the exchange-correlation energy. for metals like Zn, Mg, A1, and Na, the jellium model (Fig. 1) is a good zeroth-order approximation, and the difference between the total pseudopotential and the potential due to the jellium is small for the closest packed plane. Thus for a given separation a, one obtains E to a first-order perturbation approximation as
+
A
J
6v
(y,a)n(y,a)dy
(4)
where vJ is the potential produced by the jellium, y i s the direction normal to the surfaces, and 6V is the average, over planes parallel to the surface, of the difference in potential between an array of pseudopotentials and the jellium. Following Lang and Kohn (ref. 5 ) the Ashcroft pseudopotential is used:
vps(r)
0, r < rc =
-z/r, r > rc
where rc is determined empirically and is close to the ion core radius. The electron number density is obtained from a set of self-consistent equations. The Schroedinger equation:
where
(5)
22
k2 and k2
and with Poisson's equation (7)
@(y,a) is the electrostatic potential n+(y) is the jellium density (Fig. 1) is the Fermi wave vector magnitude and $:')(y) is the wave function (i = L, I?; i.e., left and right). The calculation is broken up into two regions, below the conduction band of the less dense metal, and above it where the solutions are doubly degenerate. It is useful to combine the first two terms of Eq. (4) along with the classical electron-electron interaction term of F{n(y,a)) as follows: I(F
where p(y,a) is the net charge density of the zeroth-order jellium solution. Mint is the exact ditterence Detween the ion-ion and the jell ium-jel lium interaction. It is shown in ref. 5 that W int . / A is negligible unless the facing planes are in registry, i.e., commensurate. I n this calculation, we assume incommensurate adhesion (Wint = U ) , since registry is not obtained with dissimilar metals in contact without corresponding loss of energy due to strains in the lattice which would be difficult to evaluate. The exchange-correlation energy Exc is written i n the local-density approximation (LDA),
where c xc (n(r)) is the exchange-correlation energy of a uniform electron gas ot number density n(r). We use Wigner's interpolation formula for the correlation energy and the Kohn-Sham exchange energy. It remains to specify the kinetic-energy functional of Eq. ( 3 ) :
23 where
(occ.)
The sum i s o v e r a l l o c c u p i e d s t a t e s . degenerate wave f u n c t i o n s as i n Eq. ( 6 ) . T h(y,a))
-
T,{n(y,O)}
The summation i n d e x
i again r e f e r s t o
I n r e f . 6, an e x p r e s s i o n f o r
i s p r e s e n t e d w h i c h i s based on k i n e t i c - e n e r g y
d e n s i t i e s i n t h e i n t e r f a c e region.
We f i n d t h i s approach more n a t u r a l f o r o u r prob-
lem, as i t i s t h e i n t e r f a c e r e g i o n i n which t h e l a r g e changes i n k i n e t i c energy dens i t y o c c u r upon adhesion. The r e s u l t s o f t h i s c a l c u l a t i o n f o r a l l d i s s i m i l a r c o m b i n a t i o n s of A l ( 1 1 1 ) , Zn(0001), Mg(U001) and Na(l1U) a r e shown i n F i g . 2.
AS can be seen t h e r e i s a wide
v a r i a t i o n o f shapes and b i n d i n g e n e r g i e s and t h e c u r v e s l o o k v e r y s i m i l a r t o b i n a i n g energy c u r v e s f o r d i a t o m i c molecules. S c a l i n g r u l e s f o r adhesive e n e r g i e s The c a l c u l a t i o n s needed t o o b t a i n t h e r e s u l t s o f F i g . Z a r e q u i t e d i f f i c u l t and time-consuming.
I t i s o f i n t e r e s t , therefore,
t o l o o k f o r some g e n e r a l s i m i l a r i t i e s
between t h e s e c u r v e s w h i c h may g e n e r a l i z e t h e r e s u l t s .
Kose, e t a l . ( r e f . 7 ) have
I t w i l l now be shown t h a t t h e c u r v e s o f F i g . 2, as w e l l as
found such s c a l i n g laws.
those o f t h e i d e n t i c a l metal contacts ( r e f . 6) (Al(111) s i m p l y s c a l e d i n t o a u n i v e r s a l curve.
-
Al(111), etc.),
can be
T h i s s c a l i n g i s m o t i v a t e d by t h e e x p e c t a t i o n
t h a t m e t a l s h a v i n g s h o r t e r s c r e e n i n g l e n g t h s would have adhesive energy c u r v e s which r i s e f a s t e r w i t h separation.
That i s , t h e m e t a l s would screen t h e d i s t u r b a n c e s
caused b y c r e a t i n g t h e s u r f a c e o v e r a s h o r t e r d i s t a n c e .
T h i s suggests t h a t f o r iden-
t i c a l m e t a l c o n t a c t s t h e s e p a r a t i o n i s s c a l e d b y t h e Thomas Fermi s c r e e n i n g l e n g t h
X
= ( Y 7 1 / 4 ) ~ 'r~s
3
'"/3
au, where t h e b u l k e l e c t r o n d e n s i t y n+ = 3 / 4 a r s .
When we encounter b i m e t a l l i c c o n t a c t s as r e p r e s e n t e d i n F i g . 1, a l e n g t h s c a l i n g a p p r o p r i a t e t o b o t h m e t a l s must b e considered. a r i t h m e t i c average,
(A1
e q u i l i b r i u m value,
AE
+
X2)/Z.
I n t h a t case, we chose t o s c a l e b y an
The energy a m p l i t u d e was s c a l e d b y i t s
Ead(am) where
am
i s t h e e q u i l i b r i u m separation.
Explicitly
Here a,,,)/
E i d ( a * ) i s t h e u n i v e r a l a d h e s i v e energy f u n c t i o n and
(xl
+
a* = 2 ( a
-
xZ).
F i g u r e 3 shows t h e r e s u l t s o f s c a l i n g t h e c a l c u l a t e d adhesive e n e r g i e s . analytical f i t i s given b y
Eid =
-
(1 + Ba*) e x p (-ea*)
with
8 = 0.90.
The
An
24
u n i v e r s a l i t y o f t h e s c a l e d a d h e s i v e energy c u r v e i s t r u l y remarkable.
One can see
t h a t the r e s u l t s f o r a l l ten b i m e t a l l i c contacts l i e very close t o the universal T h i s i s t r u e even though t h e b u l k m e t a l l i c d e n s i t i e s i n t h e v a r i o u s m e t a l s
curve.
vary by a f a c t o r o f e i g h t . I t i s i m p o r t a n t t o u n d e r s t a n d how t h e f o r t u n a t e r e s u l t o f a u n i v e r s a l energy
c u r v e comes about.
I n t h e following,
w i t h i n t h e j e l l i u m model.
we a t t e m p t t o p r o v i d e a p l a u s i b i l i t y argument
F i r s t , we have f o u n d t h a t solid-vacuum d e n s i t y d i s -
tributions, n(y) (ref. lo), scale rather accurately w i t h
x.
good a p p r o x i m a t i o n , a u n i v e r s a l number d e n s i t y d i s t r i b u t i o n
Here
a/2
i s t h e coordinate o f t h e j e l l i u m surface.
That i s , t h e r e i s , t o a n*(y
-
a / 2 ) where
There i s a s i m i l a r s c a l i n g f o r
t h e Kohn-Sham e f f e c t i v e one e l e c t r o n p o t e n t i a l :
We were m o t i v a t e d t o l o o k f o r t h i s s c a l i n g by t h e f a c t t h a t t h e Thomas Fermi e q u a t i o n s c a l e s ( r e f s . 11,12) e x a c t l y w i t h
y
i n units o f
x.
Secondly, we have f o u n d ( r e f .
6 ) t h a t t h e t o t a l number d e n s i t y i n t h e b i m e t a l l i c i n t e r f a c e i s g i v e n t o a f a i r accuracy b y a s i m p l e o v e r l a p o f t h e c o r r e s p o n d i n g solid-vacuum d i s t r i b u t i o n s . This, and t h e s t a t i o n a r y p r o p e r t y o f
E[n] i n d i c a t e s t h a t i t would b e a good a p p r o x i m a t i o n
t o use o v e r l a p p i n g solid-vacuum number d e n s i t y and p o t e n t i a l d i s t r i b u t i o n s . Thus,
i n a f i r s t o r d e r p e r t u r b a t i o n a p p r o x i m a t i o n , we have f o r i d e n t i c a l m e t a l
contacts :
Combining Eqs. ( 3 ) t o ( 5 ) , we have: Ead(a)
(l/A)(n+vg) $ X * ( y 2
-
a*/2)v;ff(y
The i n t e g r a n d i n Eq. ( 1 6 ) i s .independent o f i n t e g r a l a r e independent o f
+
a*/2)dy
rs.
The c o n s t a n t s i n f r o n t o f t h e
a, and t h u s Eq. (16) s c a l e s e x a c t l y as we s c a l e d t h e
adhesive energy c u r v e t o g i v e F i g . 3.
A l t h o u g h o u r p l a u s i b i l i t y argument i s
r e s t r i c t e d t o j e l l i u m i n t e r f a c e s , we n o t e t h a t t h e c a l c u l a t e d adhesive e n e r g i e s i n c l u d e t h e i o n - i o n t e r m e x a c t l y f o r a r i g i d l a t t i c e model and t h e e l e c t r o n - i o n t e r m i n f i r s t order p e r t u r b a t i o n theory.
DISCUSSION There a r e s e v e r a l t o p i c s o f i m p o r t a n c e i n m e t a l l i c adhesion:
the nature o f the
a t t r a c t i v e f o r c e s , t h e range o f such f o r c e s , and f i n a l l y , a r e t h e r e any
generalizations that can be found? The work of Ferrante and Smith (ref. 6 ) indicates that electron sharing similar to covalent bonding in molecules is sufficient to give strong bonding. They also determine the magnitudes of such forces for several simple metals (table 1). These results also establish that an+..approximaterange for the forces is an interplanar spacing. Finally, Rose, Ferrante and Smith (ref. 7) have found scaling laws which generalize these results. I n metallic adhesion of real surfaces further considerations come to mind. First, real surfaces are not flat and consequently, how important are the strong forces compared to the weaker, long-range van der Waals forces? Next, the results of these theoretical calculations represent ideal strengths. Real contacts fail at conditions lower than at ideal strengths. Finally, metals used in most engineering applications are either transition metals or alloys. lnglesfield (ref. 1 3 ) examined the problem of van der Waals vs. strong interaction forces using the calculations of Ferrante and Smith (ref. 9). He concluded that in spite of the fact that the strong forces only dominate at the positions of asperity contact, these strong interaction forces dominate the strength of the contact. The fact that the strength of real contacts is dominated by defect structures and the mechanical properties of the solids is an issue that ultimately must be addresseo. Also, these results apply for brittle fracture; however, metals will undergo ductile extention before fracture. Since adhesion and other contact processes are complicated combinations of physical and mechanical properties, each part of the process must be understood independently. This study has been carried out in this spirit. The results of table I, however, contain some aspects of these considerations. A question o f interest is whether a dissimilar contact is weaker., stronger or in-between contacts between the same metal. The first column in table 1 (labeled Wint = 0) gives the binding energy of a contact when perfect registry is not obtained. Perfect registry can only be obtained when the same crystallographic planes of the same metal are in COhtaCt. A more complete discussion of this issue is given in ref. 9. The second column gives the binding energy in the ideal case of perfect matching of planes which doesn't occur in a real contact. Examining table I, it can be seen that the interfacial energy is slightly lower than or comparable to the bulk energy except for the case of sodium, where the interface between sodium and any of the other metals is clearly stronger than the bulk. I n a real contact, the two surfaces would probably distort t o attempt to come into registry with the formation of defect chains to accomodate the distortions, thus modifying the interfacial energy. I n general, the interfacial binding energy is comparable to the energy of the weaker partner in the different metal contacts examined. A final concern with these results is the limitations of the calculations. The quasi-three-dimensionality of the adhesion calculations limits the reliability of 'the calculations to the densest packed planes of the simple metals.
26
Most engineering materials are transition metals. To properly handle transition metals theoretically would require extensive three-dimensional calculations which would be quite difficult at this stage. Yaniv (ref. 14) and Allan, Lannoo and Dobrzhinski (ref. 15) have applied the tight-binding approximation to the transition metal problem. Although these calculations probably give accurate trends, the approximations used make quantitative results questionable. The question arises concerning whether the results are more general than for the simple metals examined. Recently, theoretical binding energy curves have become available for several bulk metals [Carlsson et al. (ref. 16) (Mo, K, and Cu) and Herbst (ref. 17) (Sm+2, S I ~ +and ~ , Ba)]. These total cohesive energy curves were calculated as a function of the separation between at,oms for a uniformly dilated lattice. We characterize the density of the lattice in terms o f the Wigner-Seitz radius, rws = ( 3 / 4 ~ n ~ ) l 'where ~ n A is the atom density. AS shown in Fig. 4, these quite disparate cohesive energy curves can be approximately scaled into a universal function, ,:E which is also defined in Eq. (12) if we replace Ead by Ec everywhere. AE is the cohesive energy at the equilibrium spacing rwsm and a* = (rWs - rwsm)/x where x is again the Thomas-Fermi screening length. The value of rs used to determine x was determined by the equilibrium interstitial electron density (refs. 17,18). The binding energy of Mo, K, Ba, Sm [rlf (5d,6s)'] and Sm'3[4fZ(5d,6s)3] fall closely on a single curve with 6 = 1.16 where we have used the same analytic form as for the adhesive energies. The value of 6 differs from that appropriate for adhesive energies presumably in part because all atoms change their positions in the bulk cohesive energy calculations while the adhesive energy curves assume that atomic planes are moved rigidly. The results for du has the same shape, but a somewhat different 6 than the other metals. We do not understand this variation. The cohesive energy calculations of Carlsson et al. (ref. 16) (ASW density functional theory) and Herbst (ref. 18) (relativistic Hartree-Fock) are quite different from each other and from the perturbative-density functional results of ref. 6 for Ead. The nature o f cohesive bonding in these metals is quite varied. Ba is a divalent band overlap metal; Sm is an f-electron metal; Mo and Cu have important d-band interactions; while K is a simple metal. That such different metals calculated in quite different ways fall on a single curve indicates the generality of the scaling relations. We note that the "tail" region of the screening charge distribution around metal ion cores can be represented by electron gas parameters. It is known that the screening charge density and potential in this region scale as rl'' to a fair approximation (ref. 19). Then the argument of Eqs. (13 to 15) S applies to the long-range interaction of screened bulk ions with f a12 referring to the relative lattice positions and y replaced by 7.
27
The use o f t h e e l e c t r o n d e n s i t y a t Miedema e t a l . e n e r g i e s (e.g.,
r w s i s r e m i n i s c e n t o f t h e approach
( r e f . 2b) whose work complements o u r own.
of
They c a l c u l a t e e q u i l i b r i u m
We, on t h e o t h e r hand, u s e s i m i l a r e q u i l i b r i u i i i
heats o f formation).
q u a n t i t i e s t o d e t e r m i n e t h e f o r m o f t n e b i n d i n g energy-distance
relation.
The p l o t t e d e n e r g i e s o f F i g . 4 were computed f o r t h e same e l e c t r o n i c conf i g u r a t i o n a t a l l a t o m i c s e p a r a t i o n s as were t h e i n t e r f a c e c a l c u l a t i o n s . a d h e s i v e energy s c a l i n g was i l l u s t r a t e d f o r s i m p l e m e t a l s .
The
However, t h e appearance
o f an analogous s c a l i n g r e l a t i o n f o r t h e b u l k energy o f t r a n s i t i o n m e t a l s i n d i c a t e s t h a t t h e a d h e s i v e energy s c a l i n g may w e l l e x t e n d beyond s i m p l e m e t a l s .
I n c o n c l u s i o n we have d i s c o v e r e d s c a l i n g r e l a t i o n s which map adhesive e n e r g i e s o n t o a u n i v e r s a l b i n d i n g energy curve.
S i m i l a r scalings f o r cohesive energies f o r a
wide range o f m e t a l s suggest t h a t t h i s r e l a t i o n may a p p l y t o more t h a n j u s t t h e simple metals.
F i n a l l y , a s i m p l e a n a l y t i c e n e r g y - s e p a r a t i o n s c a l i n g r e l a t i o n i s pro-
v i d e d which may be o f use i n p r a c t i c a l a n a l y s i s o t c o n t a c t s .
REFERENCES
1.
U. Tabor,
i n J.M. B l a k e l y (Ed.),
S u r f a c e P h y s i c s o f M a t e r i a l s , Vol.. 2, Academic,
New York, 1975, ch. l U .
2.
E. Kabinowicz, F r i c t i o n and Wear o f M a t e r i a l s , Wiley, New York, 1Y65.
3.
U.H.
4.
W.
Kohn and L.J.
Sham, Phys. Rev.,
140 (1965) A 1133-1138.
5.
[U.U.
Lang and W.
Kohn, Phys. Rev.,
ti 1 (197U) 4555-4568.
J.P.
Perdew, Phys. Rev.,
6.
Buckley, J. C o l l o i d l n t e r f a c e Sci.,
J. F e r r a n t e and J.K.
58 (1977) 36-53.
B 1 7 (1978) 2595-2611.
Smith, Phys. Kev. B 1 Y (197Y) 3Yll-392U.
J.R. Smith, B u l l . Am. Phys.
7.
J.H.
8.
J. F e r r a n t e , NASA TM-78890,
9.
Smith, Phys. Kev. L e t t . 47 (1981) 075-078.
1Y78.
10.
J. F e r r a n t e and J. K. Smith, S u r f . Sci., 38 (1973) 77-92. J.K. Smith, J. F e r r a n t e , and J.H. Rose, t o be p u b l i s h e d .
11.
6 . Clrowka and H. Recknagel, Phys. Z.
12.
S.C.
Ying, J.R.
13.
J.E.
I n g l e s f i e l d , J. Phys. F, 6 (1Y7ti) 687-701.
14.
A. Yaniv, Phys. Rev. B, 1 7 (1978) 3904-3918.
15.
6. A l l a n , PI. Lannoo and L. Uobrzynski, P h i l o s . hag.,
16.
A.E.
241-25U. 17.
J. F e r r a n t e and
SOC., 26 (1981) 428.
Rose, J. F e r r a n t e and J.K.
38 (1937) 758-765.
Smith and W. Kohn, J. Vac. S c i . Technol.,
C a r l s s o n , C.D.
K. Nonnier and
G e l a t t , Jr.,
9 (1972) 575-578.
3U (1974) 33-45.
and H. E h r e n r e i c h , P h i l . Mag. A, 41 (1980)
(An ASW c a l c u l a t i o n ) .
We would l i k e t o thank J.F.
H e r b s t f o r g i v i n g us access t o h i s u n p u b l i s h e d
c a l c u l a t i o n s o f Sin and Ba.
I n t h i s case, r s was determined f r o m t h e
e l e c t r o n d e n s i t y a t t h e c e l l boundary.
18.
V.L.
Moruzzi, J.F.
Janak and H.R.
Williams, Calculated E l e c t r o n i c Properties
o f ivletals, Pergamon, hew York, 1978.
28
19. 20.
See e.g., S. Kaimes, Wave mechanics o f electrons in metals, Interscience, New York, 1961, esp. Eq. 10.85, p. 305. A.K. Miedema, R. Boom and F.K. UeBoer, J. Less Common Met., 41 (1975) 283-298. H.K. Miedema, Phillips Tech. Hev., 36 (1976) 217-231.
-.8
-.4 0 .4 POSITION y, nm
Figure 1. - Electron number density n and jellium ion charge density n+ for an AI-Mg contact.
.8
0
r
0)
I
l
c
0
c
I
I
oo
I
0.
I
I
I
a
SCALED ADHESIVE ENERGY II
I
N
II
0
(D
s
ADHESIVE ENERGY, erg/cm*
30
TABLE I. Binding energy comparison. All energy values taken from the minimum in the adhesive energy plots (see figs. 3 and 4) Metal combination
Binding energy Wint = ’$, Perfect ergs/cm registry
A1 (111)-A1 (111) Zn ( 0001) -Zn (0001 ) Mg(OOl)-Mg(0001) Na(llO)-Na(llO) A 1 (lll)-Zn(OOOl) A1 (lll)-Mg(0001 A1 (111)-Na( 110) Zn ( 0001) -Mg ( 0001 ) Zn( 0001)-Na( 110) Mg (0001) -Na( 110)
490 505 460 195 520 505 345 490 325 310
715 545 550 230
-
41 + p a 4 exp(j3a3
SCALED CHANGE IN WIGNER SEITZ RADIUS, a*
Figure 4. - Bulk energy of various metals scaled and described in the text a* = (rws- rws )/A A value of p = 1.16 was obtained by a least squares t! to the data.
UNIVERSAL B I N D I N G ENERGY RELATIONS I N METALLIC ADHESION
J. FERRANTE N a t i o n a l A e r o n a u t i c s and Space A d m i n i s t r a t i o n , Lewis Research Center, Cleveland, Ohio
U.S.A.
44135
J. K. SMITH b e n e r a l IYotors Research L a b o r a t o r i e s ,
Warren, M i c h i g a n
U.S.A.
48UYU
J. H. KUSE Ames L a b o r a t o r y , Iowa S t a t e U n i v e r s ty, h e s , Iowa
SOU11
U.S.A.
AbSTKACT Kose, F e r r a n t e , and Smith ( r e f
7 ) have d i s c o v e r e d s c a l i n g r e l a t i o n s wnich map
t h e adhesive b i n d i n g energy o t F e r r a n t e and Smith ( r e f . d ) o n t o a s i n g l e u n i v e r s a l b i n d i n g energy curve.
The e n e r g i e s i n r e f . 6 a r e c a l c u l a t e d f o r a l l combinations o f
A l ( 1 1 1 ) , Ln(UUUl), Clg(UUOl), and Na(l1U) i n c o n t a c t .
The s c a l i n g i n v o l v e s
n o r m a l i z i n g t h e energy t o t h e maximum b i n d i n g energy and n o r m a l i z i n g d i s t a n c e s by a
A s i m p l e mathematical
s u i t a b l e c o m b i n a t i o n o f Thomas-Fermi s c r e e n i n g l e n g t h s .
e x p r e s s i o n i s f o u n d t o a c c u r a t e l y r e p r e s e n t t h e u n i v e r s a l curve, E * ( a * ) exp (-ea*)
E*
where
s e p a r a t i o n , and
6
i s t h e n o r m a l i z e d b i n d i n g energy, a* i s t h e f i t t i n g parameter.
t h a t the c a l c u l a t e d cohesive energies o f
=
-(1
+
ea*)
i s the normalized
Kose e t a l . ( r e f . 7 ) have a l s o found
K, Ba, Cu, No, and Sm s c a l e by s i m i l a r
silriple r e l a t i o n s s u g g e s t i n g t h e u n i v e r s a l r e l a t i o n may be more g e n e r a l t h a n f o r t h e s i m p l e f r e e e l e c t r o n m e t a l s f o r which i t was d e r i v e d .
I n t h i s paper we o u t l i n e these
r e S U l t S and d i s c u s s them i n r e l a t i o n t o t o p i c s o f i n t e r e s t i n adhesion, f r i c t i o n , and wear.
INTRODUCT10N An i m p o r t a n t aspect o f f r i c t i o n and wear o f m e t a l s i s t h e n a t u r e o f t h e adhesive f o r c e between t h e s u r f a c e s .
Bowden and Tabor ( r e f . 1 ) have used adhesion between
m e t a l s t o e x p l a i n p a r t s o f t h e f r i c t i o n and wear process.
Uf particular interest i n
t h e s e processes i s t h e p h y s i c a l n a t u r e o f t h e t o r c e s i n o i s s i n i i l a r m e t a l c o n t a c t s .
20
There have been numerous attempts to postulate the nature of these forces but few model calculations. For exaniple, adhesion between dissimilar metals was explained in terms o f solubility (ref. 2 ) but without accompanying mooel calculations. Buckley (ref. 3 ) has shown that metals with low mutual solubility can have higher adhesion than those with high solubility. This lack of model calculations has occurred because of the theoretical difficulties involve0 in treating solid surfaces. In the past decaae the electron density functional formalism of Hohenberg, Kohn and Sham (ref. 4) have been used to successfully calculate the surface energies (ref. 5) of simple metals. Ferrante and Smith (ref. b) recently applied this formalism to the problem of metallic adhesion for both similar and dissimilar metals in contact. Rose, Ferrante and Smith (ref. 7) found that scaling laws exist which map these results onto a universal adhesive energy curve. In this paper, we discuss the results o f these calculations and their implications to a more general category of materials, the transition metals, and then summarize the current state of adhesion theory.
USE OF KOHN-SHAM FORMALISM FOR ADHESION CALCULATIONS
The calculational formalism and methods used for obtaining self-consistent interface electronic structure will now be presented. A much more extensive description is given in refs. 6 and 8, particularly of numerical techniques. The adhesive interaction energy, Ead, between two metal surfaces is a function o f the distance between the two surfaces, a (see Fig. 1). Ead is defined as the negative of the amount of work necessary to increase the separation from a to divided by twice the cross-sectional area A. Thus 01
where E is the total energy. For identical metals, Ead is the negative of the surface energy when a is at the energy minimum. According to the density functional formalism of Hohenberg, Kohn, and Sham, (ref. 4) the total energy is given by (atomic units are used throughout unless otherwise specified)
E{n(?)}
=
/
v(?)n(?)dT
z.2. +
+
i+j
'J
F{n(?)},
21
(3)
v(7) is the
ionic potential, and n(7) is the electron number density. The first two terms in Eq. ( 2 ) are the electron-ion and ion-ion interaction energies, respectively. z is the ionic charge and K . . is the distance between ion core nuclei 1J is the (there is no ion core overlap in the systems considered here). Ts{n(?)l kinetic energy of a system of noninteracting electrons with the same density n(?), the next term is the classical electron-electron interaction energy, and Exc is the exchange-correlation energy. for metals like Zn, Mg, A1, and Na, the jellium model (Fig. 1) is a good zeroth-order approximation, and the difference between the total pseudopotential and the potential due to the jellium is small for the closest packed plane. Thus for a given separation a, one obtains E to a first-order perturbation approximation as
+
A
J
6v
(y,a)n(y,a)dy
(4)
where vJ is the potential produced by the jellium, y i s the direction normal to the surfaces, and 6V is the average, over planes parallel to the surface, of the difference in potential between an array of pseudopotentials and the jellium. Following Lang and Kohn (ref. 5 ) the Ashcroft pseudopotential is used:
vps(r)
0, r < rc =
-z/r, r > rc
where rc is determined empirically and is close to the ion core radius. The electron number density is obtained from a set of self-consistent equations. The Schroedinger equation:
where
(5)
22
k2 and k2
and with Poisson's equation (7)
@(y,a) is the electrostatic potential n+(y) is the jellium density (Fig. 1) is the Fermi wave vector magnitude and $:')(y) is the wave function (i = L, I?; i.e., left and right). The calculation is broken up into two regions, below the conduction band of the less dense metal, and above it where the solutions are doubly degenerate. It is useful to combine the first two terms of Eq. (4) along with the classical electron-electron interaction term of F{n(y,a)) as follows: I(F
where p(y,a) is the net charge density of the zeroth-order jellium solution. Mint is the exact ditterence Detween the ion-ion and the jell ium-jel lium interaction. It is shown in ref. 5 that W int . / A is negligible unless the facing planes are in registry, i.e., commensurate. I n this calculation, we assume incommensurate adhesion (Wint = U ) , since registry is not obtained with dissimilar metals in contact without corresponding loss of energy due to strains in the lattice which would be difficult to evaluate. The exchange-correlation energy Exc is written i n the local-density approximation (LDA),
where c xc (n(r)) is the exchange-correlation energy of a uniform electron gas ot number density n(r). We use Wigner's interpolation formula for the correlation energy and the Kohn-Sham exchange energy. It remains to specify the kinetic-energy functional of Eq. ( 3 ) :
23 where
(occ.)
The sum i s o v e r a l l o c c u p i e d s t a t e s . degenerate wave f u n c t i o n s as i n Eq. ( 6 ) . T h(y,a))
-
T,{n(y,O)}
The summation i n d e x
i again r e f e r s t o
I n r e f . 6, an e x p r e s s i o n f o r
i s p r e s e n t e d w h i c h i s based on k i n e t i c - e n e r g y
d e n s i t i e s i n t h e i n t e r f a c e region.
We f i n d t h i s approach more n a t u r a l f o r o u r prob-
lem, as i t i s t h e i n t e r f a c e r e g i o n i n which t h e l a r g e changes i n k i n e t i c energy dens i t y o c c u r upon adhesion. The r e s u l t s o f t h i s c a l c u l a t i o n f o r a l l d i s s i m i l a r c o m b i n a t i o n s of A l ( 1 1 1 ) , Zn(0001), Mg(U001) and Na(l1U) a r e shown i n F i g . 2.
AS can be seen t h e r e i s a wide
v a r i a t i o n o f shapes and b i n d i n g e n e r g i e s and t h e c u r v e s l o o k v e r y s i m i l a r t o b i n a i n g energy c u r v e s f o r d i a t o m i c molecules. S c a l i n g r u l e s f o r adhesive e n e r g i e s The c a l c u l a t i o n s needed t o o b t a i n t h e r e s u l t s o f F i g . Z a r e q u i t e d i f f i c u l t and time-consuming.
I t i s o f i n t e r e s t , therefore,
t o l o o k f o r some g e n e r a l s i m i l a r i t i e s
between t h e s e c u r v e s w h i c h may g e n e r a l i z e t h e r e s u l t s .
Kose, e t a l . ( r e f . 7 ) have
I t w i l l now be shown t h a t t h e c u r v e s o f F i g . 2, as w e l l as
found such s c a l i n g laws.
those o f t h e i d e n t i c a l metal contacts ( r e f . 6) (Al(111) s i m p l y s c a l e d i n t o a u n i v e r s a l curve.
-
Al(111), etc.),
can be
T h i s s c a l i n g i s m o t i v a t e d by t h e e x p e c t a t i o n
t h a t m e t a l s h a v i n g s h o r t e r s c r e e n i n g l e n g t h s would have adhesive energy c u r v e s which r i s e f a s t e r w i t h separation.
That i s , t h e m e t a l s would screen t h e d i s t u r b a n c e s
caused b y c r e a t i n g t h e s u r f a c e o v e r a s h o r t e r d i s t a n c e .
T h i s suggests t h a t f o r iden-
t i c a l m e t a l c o n t a c t s t h e s e p a r a t i o n i s s c a l e d b y t h e Thomas Fermi s c r e e n i n g l e n g t h
X
= ( Y 7 1 / 4 ) ~ 'r~s
3
'"/3
au, where t h e b u l k e l e c t r o n d e n s i t y n+ = 3 / 4 a r s .
When we encounter b i m e t a l l i c c o n t a c t s as r e p r e s e n t e d i n F i g . 1, a l e n g t h s c a l i n g a p p r o p r i a t e t o b o t h m e t a l s must b e considered. a r i t h m e t i c average,
(A1
e q u i l i b r i u m value,
AE
+
X2)/Z.
I n t h a t case, we chose t o s c a l e b y an
The energy a m p l i t u d e was s c a l e d b y i t s
Ead(am) where
am
i s t h e e q u i l i b r i u m separation.
Explicitly
Here a,,,)/
E i d ( a * ) i s t h e u n i v e r a l a d h e s i v e energy f u n c t i o n and
(xl
+
a* = 2 ( a
-
xZ).
F i g u r e 3 shows t h e r e s u l t s o f s c a l i n g t h e c a l c u l a t e d adhesive e n e r g i e s . analytical f i t i s given b y
Eid =
-
(1 + Ba*) e x p (-ea*)
with
8 = 0.90.
The
An
24
u n i v e r s a l i t y o f t h e s c a l e d a d h e s i v e energy c u r v e i s t r u l y remarkable.
One can see
t h a t the r e s u l t s f o r a l l ten b i m e t a l l i c contacts l i e very close t o the universal T h i s i s t r u e even though t h e b u l k m e t a l l i c d e n s i t i e s i n t h e v a r i o u s m e t a l s
curve.
vary by a f a c t o r o f e i g h t . I t i s i m p o r t a n t t o u n d e r s t a n d how t h e f o r t u n a t e r e s u l t o f a u n i v e r s a l energy
c u r v e comes about.
I n t h e following,
w i t h i n t h e j e l l i u m model.
we a t t e m p t t o p r o v i d e a p l a u s i b i l i t y argument
F i r s t , we have f o u n d t h a t solid-vacuum d e n s i t y d i s -
tributions, n(y) (ref. lo), scale rather accurately w i t h
x.
good a p p r o x i m a t i o n , a u n i v e r s a l number d e n s i t y d i s t r i b u t i o n
Here
a/2
i s t h e coordinate o f t h e j e l l i u m surface.
That i s , t h e r e i s , t o a n*(y
-
a / 2 ) where
There i s a s i m i l a r s c a l i n g f o r
t h e Kohn-Sham e f f e c t i v e one e l e c t r o n p o t e n t i a l :
We were m o t i v a t e d t o l o o k f o r t h i s s c a l i n g by t h e f a c t t h a t t h e Thomas Fermi e q u a t i o n s c a l e s ( r e f s . 11,12) e x a c t l y w i t h
y
i n units o f
x.
Secondly, we have f o u n d ( r e f .
6 ) t h a t t h e t o t a l number d e n s i t y i n t h e b i m e t a l l i c i n t e r f a c e i s g i v e n t o a f a i r accuracy b y a s i m p l e o v e r l a p o f t h e c o r r e s p o n d i n g solid-vacuum d i s t r i b u t i o n s . This, and t h e s t a t i o n a r y p r o p e r t y o f
E[n] i n d i c a t e s t h a t i t would b e a good a p p r o x i m a t i o n
t o use o v e r l a p p i n g solid-vacuum number d e n s i t y and p o t e n t i a l d i s t r i b u t i o n s . Thus,
i n a f i r s t o r d e r p e r t u r b a t i o n a p p r o x i m a t i o n , we have f o r i d e n t i c a l m e t a l
contacts :
Combining Eqs. ( 3 ) t o ( 5 ) , we have: Ead(a)
(l/A)(n+vg) $ X * ( y 2
-
a*/2)v;ff(y
The i n t e g r a n d i n Eq. ( 1 6 ) i s .independent o f i n t e g r a l a r e independent o f
+
a*/2)dy
rs.
The c o n s t a n t s i n f r o n t o f t h e
a, and t h u s Eq. (16) s c a l e s e x a c t l y as we s c a l e d t h e
adhesive energy c u r v e t o g i v e F i g . 3.
A l t h o u g h o u r p l a u s i b i l i t y argument i s
r e s t r i c t e d t o j e l l i u m i n t e r f a c e s , we n o t e t h a t t h e c a l c u l a t e d adhesive e n e r g i e s i n c l u d e t h e i o n - i o n t e r m e x a c t l y f o r a r i g i d l a t t i c e model and t h e e l e c t r o n - i o n t e r m i n f i r s t order p e r t u r b a t i o n theory.
DISCUSSION There a r e s e v e r a l t o p i c s o f i m p o r t a n c e i n m e t a l l i c adhesion:
the nature o f the
a t t r a c t i v e f o r c e s , t h e range o f such f o r c e s , and f i n a l l y , a r e t h e r e any
generalizations that can be found? The work of Ferrante and Smith (ref. 6 ) indicates that electron sharing similar to covalent bonding in molecules is sufficient to give strong bonding. They also determine the magnitudes of such forces for several simple metals (table 1). These results also establish that an+..approximaterange for the forces is an interplanar spacing. Finally, Rose, Ferrante and Smith (ref. 7) have found scaling laws which generalize these results. I n metallic adhesion of real surfaces further considerations come to mind. First, real surfaces are not flat and consequently, how important are the strong forces compared to the weaker, long-range van der Waals forces? Next, the results of these theoretical calculations represent ideal strengths. Real contacts fail at conditions lower than at ideal strengths. Finally, metals used in most engineering applications are either transition metals or alloys. lnglesfield (ref. 1 3 ) examined the problem of van der Waals vs. strong interaction forces using the calculations of Ferrante and Smith (ref. 9). He concluded that in spite of the fact that the strong forces only dominate at the positions of asperity contact, these strong interaction forces dominate the strength of the contact. The fact that the strength of real contacts is dominated by defect structures and the mechanical properties of the solids is an issue that ultimately must be addresseo. Also, these results apply for brittle fracture; however, metals will undergo ductile extention before fracture. Since adhesion and other contact processes are complicated combinations of physical and mechanical properties, each part of the process must be understood independently. This study has been carried out in this spirit. The results of table I, however, contain some aspects of these considerations. A question o f interest is whether a dissimilar contact is weaker., stronger or in-between contacts between the same metal. The first column in table 1 (labeled Wint = 0) gives the binding energy of a contact when perfect registry is not obtained. Perfect registry can only be obtained when the same crystallographic planes of the same metal are in COhtaCt. A more complete discussion of this issue is given in ref. 9. The second column gives the binding energy in the ideal case of perfect matching of planes which doesn't occur in a real contact. Examining table I, it can be seen that the interfacial energy is slightly lower than or comparable to the bulk energy except for the case of sodium, where the interface between sodium and any of the other metals is clearly stronger than the bulk. I n a real contact, the two surfaces would probably distort t o attempt to come into registry with the formation of defect chains to accomodate the distortions, thus modifying the interfacial energy. I n general, the interfacial binding energy is comparable to the energy of the weaker partner in the different metal contacts examined. A final concern with these results is the limitations of the calculations. The quasi-three-dimensionality of the adhesion calculations limits the reliability of 'the calculations to the densest packed planes of the simple metals.
26
Most engineering materials are transition metals. To properly handle transition metals theoretically would require extensive three-dimensional calculations which would be quite difficult at this stage. Yaniv (ref. 14) and Allan, Lannoo and Dobrzhinski (ref. 15) have applied the tight-binding approximation to the transition metal problem. Although these calculations probably give accurate trends, the approximations used make quantitative results questionable. The question arises concerning whether the results are more general than for the simple metals examined. Recently, theoretical binding energy curves have become available for several bulk metals [Carlsson et al. (ref. 16) (Mo, K, and Cu) and Herbst (ref. 17) (Sm+2, S I ~ +and ~ , Ba)]. These total cohesive energy curves were calculated as a function of the separation between at,oms for a uniformly dilated lattice. We characterize the density of the lattice in terms o f the Wigner-Seitz radius, rws = ( 3 / 4 ~ n ~ ) l 'where ~ n A is the atom density. AS shown in Fig. 4, these quite disparate cohesive energy curves can be approximately scaled into a universal function, ,:E which is also defined in Eq. (12) if we replace Ead by Ec everywhere. AE is the cohesive energy at the equilibrium spacing rwsm and a* = (rWs - rwsm)/x where x is again the Thomas-Fermi screening length. The value of rs used to determine x was determined by the equilibrium interstitial electron density (refs. 17,18). The binding energy of Mo, K, Ba, Sm [rlf (5d,6s)'] and Sm'3[4fZ(5d,6s)3] fall closely on a single curve with 6 = 1.16 where we have used the same analytic form as for the adhesive energies. The value of 6 differs from that appropriate for adhesive energies presumably in part because all atoms change their positions in the bulk cohesive energy calculations while the adhesive energy curves assume that atomic planes are moved rigidly. The results for du has the same shape, but a somewhat different 6 than the other metals. We do not understand this variation. The cohesive energy calculations of Carlsson et al. (ref. 16) (ASW density functional theory) and Herbst (ref. 18) (relativistic Hartree-Fock) are quite different from each other and from the perturbative-density functional results of ref. 6 for Ead. The nature o f cohesive bonding in these metals is quite varied. Ba is a divalent band overlap metal; Sm is an f-electron metal; Mo and Cu have important d-band interactions; while K is a simple metal. That such different metals calculated in quite different ways fall on a single curve indicates the generality of the scaling relations. We note that the "tail" region of the screening charge distribution around metal ion cores can be represented by electron gas parameters. It is known that the screening charge density and potential in this region scale as rl'' to a fair approximation (ref. 19). Then the argument of Eqs. (13 to 15) S applies to the long-range interaction of screened bulk ions with f a12 referring to the relative lattice positions and y replaced by 7.
27
The use o f t h e e l e c t r o n d e n s i t y a t Miedema e t a l . e n e r g i e s (e.g.,
r w s i s r e m i n i s c e n t o f t h e approach
( r e f . 2b) whose work complements o u r own.
of
They c a l c u l a t e e q u i l i b r i u m
We, on t h e o t h e r hand, u s e s i m i l a r e q u i l i b r i u i i i
heats o f formation).
q u a n t i t i e s t o d e t e r m i n e t h e f o r m o f t n e b i n d i n g energy-distance
relation.
The p l o t t e d e n e r g i e s o f F i g . 4 were computed f o r t h e same e l e c t r o n i c conf i g u r a t i o n a t a l l a t o m i c s e p a r a t i o n s as were t h e i n t e r f a c e c a l c u l a t i o n s . a d h e s i v e energy s c a l i n g was i l l u s t r a t e d f o r s i m p l e m e t a l s .
The
However, t h e appearance
o f an analogous s c a l i n g r e l a t i o n f o r t h e b u l k energy o f t r a n s i t i o n m e t a l s i n d i c a t e s t h a t t h e a d h e s i v e energy s c a l i n g may w e l l e x t e n d beyond s i m p l e m e t a l s .
I n c o n c l u s i o n we have d i s c o v e r e d s c a l i n g r e l a t i o n s which map adhesive e n e r g i e s o n t o a u n i v e r s a l b i n d i n g energy curve.
S i m i l a r scalings f o r cohesive energies f o r a
wide range o f m e t a l s suggest t h a t t h i s r e l a t i o n may a p p l y t o more t h a n j u s t t h e simple metals.
F i n a l l y , a s i m p l e a n a l y t i c e n e r g y - s e p a r a t i o n s c a l i n g r e l a t i o n i s pro-
v i d e d which may be o f use i n p r a c t i c a l a n a l y s i s o t c o n t a c t s .
REFERENCES
1.
U. Tabor,
i n J.M. B l a k e l y (Ed.),
S u r f a c e P h y s i c s o f M a t e r i a l s , Vol.. 2, Academic,
New York, 1975, ch. l U .
2.
E. Kabinowicz, F r i c t i o n and Wear o f M a t e r i a l s , Wiley, New York, 1Y65.
3.
U.H.
4.
W.
Kohn and L.J.
Sham, Phys. Rev.,
140 (1965) A 1133-1138.
5.
[U.U.
Lang and W.
Kohn, Phys. Rev.,
ti 1 (197U) 4555-4568.
J.P.
Perdew, Phys. Rev.,
6.
Buckley, J. C o l l o i d l n t e r f a c e Sci.,
J. F e r r a n t e and J.K.
58 (1977) 36-53.
B 1 7 (1978) 2595-2611.
Smith, Phys. Kev. B 1 Y (197Y) 3Yll-392U.
J.R. Smith, B u l l . Am. Phys.
7.
J.H.
8.
J. F e r r a n t e , NASA TM-78890,
9.
Smith, Phys. Kev. L e t t . 47 (1981) 075-078.
1Y78.
10.
J. F e r r a n t e and J. K. Smith, S u r f . Sci., 38 (1973) 77-92. J.K. Smith, J. F e r r a n t e , and J.H. Rose, t o be p u b l i s h e d .
11.
6 . Clrowka and H. Recknagel, Phys. Z.
12.
S.C.
Ying, J.R.
13.
J.E.
I n g l e s f i e l d , J. Phys. F, 6 (1Y7ti) 687-701.
14.
A. Yaniv, Phys. Rev. B, 1 7 (1978) 3904-3918.
15.
6. A l l a n , PI. Lannoo and L. Uobrzynski, P h i l o s . hag.,
16.
A.E.
241-25U. 17.
J. F e r r a n t e and
SOC., 26 (1981) 428.
Rose, J. F e r r a n t e and J.K.
38 (1937) 758-765.
Smith and W. Kohn, J. Vac. S c i . Technol.,
C a r l s s o n , C.D.
K. Nonnier and
G e l a t t , Jr.,
9 (1972) 575-578.
3U (1974) 33-45.
and H. E h r e n r e i c h , P h i l . Mag. A, 41 (1980)
(An ASW c a l c u l a t i o n ) .
We would l i k e t o thank J.F.
H e r b s t f o r g i v i n g us access t o h i s u n p u b l i s h e d
c a l c u l a t i o n s o f Sin and Ba.
I n t h i s case, r s was determined f r o m t h e
e l e c t r o n d e n s i t y a t t h e c e l l boundary.
18.
V.L.
Moruzzi, J.F.
Janak and H.R.
Williams, Calculated E l e c t r o n i c Properties
o f ivletals, Pergamon, hew York, 1978.
28
19. 20.
See e.g., S. Kaimes, Wave mechanics o f electrons in metals, Interscience, New York, 1961, esp. Eq. 10.85, p. 305. A.K. Miedema, R. Boom and F.K. UeBoer, J. Less Common Met., 41 (1975) 283-298. H.K. Miedema, Phillips Tech. Hev., 36 (1976) 217-231.
-.8
-.4 0 .4 POSITION y, nm
Figure 1. - Electron number density n and jellium ion charge density n+ for an AI-Mg contact.
.8
0
r
0)
I
l
c
0
c
I
I
oo
I
0.
I
I
I
a
SCALED ADHESIVE ENERGY II
I
N
II
0
(D
s
ADHESIVE ENERGY, erg/cm*
30
TABLE I. Binding energy comparison. All energy values taken from the minimum in the adhesive energy plots (see figs. 3 and 4) Metal combination
Binding energy Wint = ’$, Perfect ergs/cm registry
A1 (111)-A1 (111) Zn ( 0001) -Zn (0001 ) Mg(OOl)-Mg(0001) Na(llO)-Na(llO) A 1 (lll)-Zn(OOOl) A1 (lll)-Mg(0001 A1 (111)-Na( 110) Zn ( 0001) -Mg ( 0001 ) Zn( 0001)-Na( 110) Mg (0001) -Na( 110)
490 505 460 195 520 505 345 490 325 310
715 545 550 230
-
41 + p a 4 exp(j3a3
SCALED CHANGE IN WIGNER SEITZ RADIUS, a*
Figure 4. - Bulk energy of various metals scaled and described in the text a* = (rws- rws )/A A value of p = 1.16 was obtained by a least squares t! to the data.