Rose, Smith, and Ferrante have discovered scaling relations which map the adhesive binding energy calculated by Ferrante and Smith onto a single universal binding energy curve. These binding energies are calculated for all combinations of Al(111), Zn(0001), Mg(0001), and Na(110) in contact. The scaling involves normalizing the energy by the maximum binding energy and normalizing distances by a suitable combination of Thomas-Fermi screening lengths. Rose et al. have also found that the calculated cohesive energies of K, Ba, Cu, Mo, and Sm scale by similar simple relations, suggesting the universal relation may be more general than for the simple free electron metals for which it was derived. In addition, the scaling length was defined more generally in order to relate it to measurable physical properties. Further this universality can be extended to chemisorption. A simple and yet quite accurate prediction of a zero temperature equation of state (volume as a function of pressure for metals and alloys) is presented. Thermal expansion coefficients and melting temperatures are predicted by simple, analytic expressions, and results compare favorably with experiment for a broad range of metals

Ferrante, J.

Rose, J. J.

Smith, J. R.

English

NASA Technical Reports Server

General Disclaimer 
One or more of the Following Statements may affect this Document 
 
 This document has been reproduced from the best copy furnished by the 
organizational source. It is being released in the interest of making available as 
much information as possible. 
 
 This document may contain data, which exceeds the sheet parameters. It was 
furnished in this condition by the organizational source and is the best copy 
available. 
 
 This document may contain tone-on-tone or color graphs, charts and/or pictures, 
which have been reproduced in black and white. 
 
 This document is paginated as submitted by the original source. 
 
 Portions of this document are not fully legible due to the historical nature of some 
of the material. However, it is the best reproduction available from the original 
submission. 
 
 
 
 
 
 
 
Produced by the NASA Center for Aerospace Information (CASI) 
https://ntrs.nasa.gov/search.jsp?R=19850003824 2020-03-20T19:58:08+00:00Z

UNIVERSAL BINDING ENERGY RELATIONS IN METALLIC ADHESION
JOHN FERRANTE.* JOHN R. SMITH** AND JAMES H. ROSE***
*	 NASA Lewis Research Center, Cleveland, Ohio 44135
** General Motors Research Laboratories, Warren, Michigan 48090
*** Ames Laboratory, U.S. Department of Energy, Iowa State University,
Ames, Iowa 50011; Contract No. 2-7405- ENG-82; supported by the Director of
Energy Research, U.S. Office of Basic Energy Sciences, and by the National
Science Foundation under Grant No. PHY -77-27084.
ABSTRACT
Rose, Smith, and Ferrante have discovered scaling relations which map
the adhesive binding energy calculated by Ferrante and Smith onto a singleLn
M universal binding energy curve. These binding energies are calculated for
W all combinations of Al(111), Zn(0001), 14g(0001), and Na(110) in contact.
The scaling involves normalizing the energy by the maximum binding energy
and normalizing distances by a suitable combination of Thomas-Fermi screen-
ing lengths. Rose et al. have also found that the calculated cohesive
energies of	 Ba. Cu, No, and Sm scale by similar simple relations, sug-
gesting the universal relation may be more general than for the simple
free-electron metals for which it was derived. In addition, the scaling
length was defined more generally in order to relate it to measurable phys-
ical properties. Further this universality can be extended to chemisorp-
tion and molecular binding. The implications of this scaling have been
explored and have produced some interesting results and verifications. A
simple and yet quite accurate prediction of a zero temperature equation of
state (volume as a function of pressure for metals and alloys) is presented.
Thermal expansion coefficients and melting temperatures are predicted by
simple, analytic expressions, and results compare favorably with experiment
for a broad range of metals. All of these predictions are made possible by
the discovery of universality in binding energy relations for metals.
Finally some results of other researchers concerning universality in Van
der Waals forces are referred to.
DISCUSSION
The adhesive forces between metals in contact play an important role
in, for example, friction and wear and bonding of metal films and in inter-
face properties such as grain boundary energies or fracture strengths [1,2].
Metals in intimate contact interact strongly, leading to significant
changes in interfacial electronic structure. In many situations such P.s in
friction and wear, grain boundary energetics, and fracture. binding energy
as a function of separation between surfaces is of interest. However this
information is generally inaccessible experimentally. The ranges of the
strong forces are quite small [3], and mechanical deformations of the
solids add complexity to the interpretation of experiments. Consequently
it would be useful if theory could accurately give information concerning
the shape of such binding energy curves, which would complement the infor-
mation obtained from mechanical deformations.
Ferrante and Smith [3-5] have calculated the adhesive binding energy
versus separation for a number of simple metals in contact, A1(111),
Zn(0001), Mg(0001), Na(110), using the jellium model. Figure 1 shows an
example of Al and Mg "Jellia" in contact. Crystallinity is introduced into
the jellium model through first order perturbation theory and lattice
sums. These calculations have used the Hohenburg-Kohn-Sham theory. Details
can be found in Refs. 3, 4, and 6 to S.
The adhesive energy is defined as
E	
E(a) - E(•)	 (l)
ad `	 2A
where E is the total energy, a is the separation, and A is the inter-
facial cross-sectional area. The results of these calculations are shown
in Fig. 2. EAD at the energy minimum for the same metal with perfect
matching at the interface is the surface energy. The values for the bind-
ing energies at the minimums are given in Table I. From the binding ener-
gies we can see that the strength of the bond for dissimilar contacts is
comparable to that for "perfect registry" same-metal contacts. Perfect
registry refers to cleaving an ideal single crystal. Nonregistry refers to
separating two pieces of metal in contact with arbitrary relative orienta-
tion. These curves represent brittle fracture at absolute zero and do not
include effects such as ductile extension, which occurs in real contacts.
We further note that the derivative of these curves gives the tensile force
versus separation.
These calculations are limited, however; the results apply only to
simple metals and not to transition metals, which may include directional
effects. In addition, because they are only quasi-three-dimensional, only
the densest-packed planes for which variations in the plane are a minimum
are considered. Consequently it would be of interest to see if these
results have more generality than the materials considered.
There is a wide variety of shapes and well depths in the plots of
total energy Versus separation for the 6 bimetallic contacts shown in
Fig. 2. We have found that all of these curves could be scaled into the
single universal curve shown in Fig. 3. The scaled separation is given by
(a -a )
a* _	 °	 (2)
where a is the equilibrium separation and ! is the scaling length.
The scaled binding energy is given by
*	
Fad(a*)
Ead	 AE ( 3)
where AE is the well depth.
The scaling length ! was initially defined in terms of the Thomas-
Fermi screening length [9]. However, in extending these results we found
that universality applied not only to adhesion, but also to chemisorption,
cohesion, and chemical bonding [5]. In these cases the Thomas-Fermi
screening length is no longer a simply defined quantity. Therefore a dif-
ferent definition of scaling length was selected, given by
1/2
-1
f. = AE d22 2	 ( 4)da
.a ► 	 - -
This particular selection for scaling length was chosen since it can be
related to experimentally measurable properties such as the bulk modulus
for bulk solids or the vibration frequency for molecules. In Fig. 4 we
show the scaled binding energy for adhesion, chemisorption, cohesion, and
molecular bonding plotted on a single curve. We can see that for these
four diverse types of bonding the shape of the binding energy curve is the
same (i.e., there is a universal shape). These results indicate that there
is an underlying simplicity in nature that had not been realized previously.
At this point it is worth backtracking and reiterating what has been
demonstrated. These results are correlations between theoretical calcula-
tions of each property. These binding energy curves are not accessible
experimentally. As partial evidence that these results are not some arti-
fact of the theoretical techniques used, we point out that the calculation
of the binding energy for H2 is exact. There is another avenue of veri-
fication, however. The fact that the shapes of the binding energy curves
are the same permits interrelationships to be obtained between different
physical properties. A notable success is the correct prediction of the
vibration frequency for a chemisorbed atom on a given symmetry lattice site
in terms of a known vibration frequency as determined by experimental tech-
niques such as high-resolution energy loss spectroscopy [5]. Another is
the derivation of the empirically determined relationship between surface
and cohesive energy [5].
41r2a _ is 2C11 rws
Ecoh	
sb 2d 38	 (5)
where rws is the Wigner-Seitz radius, o is the surface energy, Ecoh
is the cohesive energy, the y Vs	 are the appropriate scaling lengths
for the surface and bulk, C 11 is the elastic constant appropriate to	 4
strain normal to the surface, d is the interplanar spacing, and B is the
bulk modulus. In Fig. 5 we show a plot of the theoretical relationship
along with experimental data. More such plots are given in Ref. 5.
These relationships test the proposed concepts at the minimums of the
binding energy curves. If an analytical expression for the universal shape
were known, further experimental implications of universality could be
derived. We have found that the gross features of the curves could be fit
reasonably well by Morse or Rydberg functions [10]. To fully examine the
experimental properties, a more detailed analytic form is needed. In the
spirit of the Rydberg function we propose a polynomial times an exponential.
In Ref. 13 we examine the further experimental implications in terms of the
equation of state of metals and alloys at zero temperature (the compressive
region of the binding energy curve), the melting temperature in terms of
the cohesive energy (the inflection point). and the coefficient of thermal
expansion (the anharmonic near-minimum region). In Fig. 6 we show the
equation of state, in Fig. 7 the melting temperature as a fvnction of cohe-
sive energy, and in table II the coefficient of thermal expansion. As can
be seen the theoretical results and experimental predictions agree quite
well.
At this point no complete theoretical justification for universality
is available. Ii, addition, it would be useful to obtain some semiempirical
predictions of well depth by merging these results with those of Miedema
[14]. Such a merger would enable a complete specification of the binding
energy curve and give more complete prediction of the physical properties
of materials. Similar observations have been made by Cole, Vidali, and
k
Klein for physical as opposed to strong bonding forces [15]. Smith, Gay,
Arlinghouse, and Richter are currently calculating surface energies and
adhesive energies by using fully three-dimensional models [16]. We hope to
have a sufficient understanding of universality to provide experimentalists
with simple theoretical relationships for predicting material properties.
REFERENCES
1. D.H. Buckley, J. Colloid and Interface Sci., 58, 36 (1977).
2. D. Tabor in "Surface Physics of Materials." Vol. 2, ed. by J.M.
Blakely, Academic Press, New York, (1975) chap. 10.
3. J. Ferrante and J.R. Smith, Surf. Sci., 38, 77 (1973).
4. (a) J. Ferrante and J.R. Smith, Phys. Rev. B 19, 3911 (1979).
(b) J. Ferrante, J.R. Smith, and J.W. Rose in "Microscopic Aspects of
Adhesion and Lubrication," ed. by J.M. Georges, Elsevier Scientific,
Amsterdam (1982).
5. J.H. Rose, J.R. Smith, and J. Ferrante, Phys. Rev. 8 28, 1835 (1983).
6. N.D. Lang and W. Kohn, Phys. Rev. B 1, 4555 (1970).
7. W. Kohn and L.J. Sham, Phys. Rev. A 140, 1133 (1965).
8. A.J. Bennett and C.B. Duke, Phys. Rev. 162, 578 (1967).
9. J.M. Ziman, Principles of the Theory of Solids. Cambridge University
Press, New York (1964).
10. J.C. Slater, Quantum Theory of Molecules and Solids. Vol. 1. McGraw-
Hill, New York (1963).
11. J.K. Norskov and N.D. Lang, Phys. Rev. B 21, 2131 (1980).
12. A.E. Carlsson, C.D. Gellatt, Jr., and H. Ehrenreich, Phil. Nag. A. 41,
241 (1980).
13. F. Guinea, J.H. Rose, J.R. Smith, and J. Ferrante, Appl. Phys. Lett.
44, 53 (1984).
14. (a) A.R. Miedema, R. Boom, and F.R. DeBoer, J. Less. Comm. Met. 41,
283, (1975).
(b) A.R. M*,edema, Phillips Tech. Rev., 36, 217 (1976).
15. G. Vidali, M.N. Cole, and J.R. Klein, Phys. Rev. B 28, 3064 (1983).
16. (a) J. Gay, J.R. Smith, R. Richter, F.J. Arlinghaus, and R.H. Wagoner,
J. Vac. Sci. Tech. A 2, 931 (1984).
(b) R. Richter, J.R. Smith, and J. Gay, Proc. First Int. Conf. on the
Structure of Solid Surfaces, U. of California, Berkeley, 1985.
:;:y
4
TABLE I. - BINDING ENERGY COMPARISON
[All energy values taken from the
minimums in the adhesive energy
plots (Figs. 3 and 4).]
Metal combination Nonregistry Perfect
registry
Binding energy,
ergs/cm2
A1(111)-A1(111) 490 715
Zn(0001)-Zn(0001) 505 545
Mg(001)-Mg(0001) 460 550
Na(110)-Na(110) 195 230
IAl(111)-Zn(OOD1) 520
Al(111)-Mg(0001) 505
Al(111)-Na(110) 345
Zn(DDO1)-Mg(0001) 490
Zn(0001)-Na(110) 325
Mg(0001)-Na(110) 310
TABLE II. - ^tjMPARISON OF THERMAL
EXPANSION COEFFICIENTS PRE-
DICTED WITH EXPERIMENTAL
VALUES AT 293 K
Metal a(10- 6	K-1)
Theory Experiment
W 6.0 4.5
Ir 6.7 6.5
Mo 7.7 5.0
j Ta 8.0 7.3
Nb 8.4 7.1
V 11.8 7.8
Pd 12.1 11.6
Fe .12.1 11.8
Au 12.2 14.2
Ni 13.3 12.8
Cu 16.7 16.7
Ag 17.5 19.0
Al 19.6 23.0
Cu0.7 Zn 0.3 20.2 19.9
Pb 23.6 28.7
U 60.9 46.6
Na 75.5 69.0
K 86.0 82.0
Cs 89.9 100.0
OF POOR
1.2
J .8
m
Q
OW
N
JQ
0Z}
F--
N
z .W 4C
0 1	 1	 1	 1	 1	 1	 1
-.8	 '► 	 0	 .4	 .8
POSITION, y, n m
Figure 1. - Electron number density n and jellium ion charge den-
sity n +
 for an AI-Mg contact. When a - 0. 0, the distance between
Mg and Al atomic planes is W AI + dM )12, where dAl and dMg are
the respective bulk interplanar spacings,
-200
N
E -400U
Q^L
ED -600
Wz0
W
W
N
W
-200
-400
-600
0
(a) AI(111)-Zn(0001). (b) AI(111)-Mg(0001).
I I
(c) Mg(0001)-Na(110).
OF PCOR ^.._.. .
0	 .2	 .4	 .6	 0	 .2	 .4	 .6	 0	 .2	 .4	 .6
SEPARATION, a, nm
(d) AI(111)-Na(110). 	 (e) Zn(0001)-Na(110).	 (f) Zn(0001)-Mg(0001).
Figure 2. - Adhesive binding energy versus the separation between the surfaces indicated.
)I'
1	 0	 1	 2	 3	 4	 5	 6	 7	 8
SCALED SEPARATION, ate
Figure 3. - Adhesive energy results scaled as described in the text (see Eqs. (2) to
(5)l.
-1.0
-.8
-.9
-.2
-.1
0
-.3
W
c'
.4
WZW
W
'— .5NW
DQ
-.6
QUN
-.7
0-.2
W
-.4
W
zW0z
E
z
m0
-.6
Ln
-.8
-1.0
-1	 0	 1	 2	 3	 4	 5	 6	 1	 8
SCALED SEPARATION, a"
Figure 4, - Binding energy as a function of interatomic separation for four systems as
noted, scaled as described in the text The H2+ results were taken from Ref. 10, the
Ai-Zn interface energies from Ref. 4, the oxygen chemisorption binding energy from
Ref. 11, and the Mo binding energy from Ref. 12
C:I..
OF Fes.
15x1012
7.5
10
E
0
co
5.0
N
^a
E N3
ep
°'
2.5
5
0	 5	 10	 154012
0.82 Ecoh, ergs/atom
I	 I	 i	 J
0	 2.5	 5.0	 7.5
eV/atom
Figure 5. - A plot of equation (5) along with experimental results relating surface and cohesive
energies.
01
THEORY
O	 EXPERIMENT
(a) Stainless steel.
6
0
NL
B
g,	 4
E
W
N
C/1
w
a
OF ^`+1J1i^•.Y4f i ^r
(b) Rubidium.
1.0x10-2
_rl
1.
NL
Y
W
NNw
CL
.8	 .6	 .4
VIVo
(c) Copper.
.8	 .6	 .4
VIVo
(d) Lithium.
s shown for four metals.
1.0
E) 41i
w Ad
NL
Y
cx , 5
NNW
.2
0
1.0
L
Q7E
41	 .5
NNW
a
0
1.0
tifi ta4 •^-.
OF P001
10
---  THEORY"
O	 EXPERIMENT
	
ow
8 OO
Nb	 Re
0/1
/O p
M
°'	 OZ / Hf
a 6	 Thoopt
}	 / OVCo-
W	 Ni-^	 OTiW
`n 4	 Au
	
Pd O Cr
Al	
61
p Au
 C, Be
A
Ag
Pb / r-Ba
2 — IiS/
LiO d 0 C
Na , ^p 07 `Mg
CS	 Cd	 `n
KL Rb
0	 1000	 2000
	 30)0	 4000
MELTING TEMPERATURE, T M, K
Figure 7. - Predicted and experimental melting temperatures as a function of
cohesive energy,


Universal binding energy relations in metallic adhesion

Abstract

Similar works

Full text

Available Versions

NASA Technical Reports Server