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Effects of surface roughness and surface films on contact resistance
Wear, 44 (1977)
@ Elsevier
345
345 - 359
Sequoia
S.A., Lausanne
EFFECTS OF SURFACE CONTACT RESISTANCE
- Printed
in the Netherlands
ROUGHNESS
AND SURFACE
FILMS ON
T. HISAKADO Department of Precision Osaka (Japan)
(Received
Engineering,
Faculty
of Engineering,
July 22, 1976; in final form November
Osaka University,
Suita,
1, 1976)
Summary An analysis of the electrical contact resistance between two metals with a tarnish film was carried out assuming the asperities to be represented by randomly distributed cones with base angles which vary with the surface roughness, and assuming that the radius of the broken area of the film at the interface of each contact asperity is constant beyond a critical depth of penetration of an asperity. The validity of the proposed theory was confirmed by experimental data of the electrical contact resistance between a silver cone and a silver flat on which carbon films were deposited, and between a silver flat with an Ag,S tarnish and a palladium flat without tarnish. Comparison of theoretical and experimental data shows that the critical depth of penetration U, of an asperity varies mainly with the surface roughness, the thickness of the tarnish film and the amount of plastic deformation of the contact asperities with films, i.e. UJo = kR’,,,, where R,, and u are the maximum height and the standard deviation (r.m.s. roughness) of the profile ordinates, and j and k are constants dependent on the type of finish, the thickness of the film and the difference in the hardness of the mating surfaces.
1. Introduction Films on contacts create an electrical resistance that can cause failure in contact applications. However, it is very difficult to estimate theoretically the effects of thin films such as physisorbed and chemisorbed films. This is due to the changeable behaviour of the fracture of the films during plastic deformation of each contact asperity. Hence a notable feature of published work is the very wide discrepancies between the experimental results of different investigators using nominally similar materials. Also each of the theories advanced can generally be applied only to the experimental results of the investigator proposing that particular theory [ 1 - 71. Few studies of the fracture of the tarnish films [8 - lo] have been made.
346
The present work was an attempt to develop a theory that was applicable to surfaces with a tarnish film in static contact. A theory was derived by considering the contact and the fracture mech~ism of the tarnish films, i.e. using a critical penetration model of contact asperities assuming that the broken area of the film at each contact asperity was constant beyond the critical depth of penetration. The validity of the theory was checked by theoretical and experimental results of the electrical contact resistance between a silver cone and a silver flat surface with a deposited carbon film and between flat siIver surfaces. The effect of the tarnish film on the thermal contact resistance in a vacuum environment is discussed in conjunction with the theory.
2. Theory When two asperities on conductors with tarnish films touch, there is a constriction and a film resistance between them. The total resistance is given by R< = R,i + Rfi Pill 2ai
Z-+
POI do1
+ poadoz
(1)
na?
This equation is very widely used in the design and study of electrical contacts. However, there is evidence that if thin fiIms with higher resistivity than that of the metals of the asperities, i.e. pal > p1 and po2 3 p2, are broken in the contact process, the total resistance may be attributed to the constriction resistance. In the theoretical analysis the constriction resistance will be derived considering the number and size of the microcontacts and the thickness of the tarnish films. 2.1. Critical penetration model When a soft metal surface consisting of some conical asperities is pressed by a hard and smooth metal surface, as shown in Fig. 1, the number of contact points n(u) is given by [ll] n(u()) = -
&f(u)
&J, 2nG(-
mu)
du
I
(Cot20);;;”
(2)
U=Ug
The separation u. of the mean planes of a smooth and a rough surface in contact is determined by the flow pressure pf of the softer metal and the applied load IV: mo
LL,
g(u)
.IUO W--u)
&=
!!_ pi
It is assumed in this model that an effective
contact
radius a,i, i.e. the radius
347
Centre line (Reference
ur d. Penetration
plane)
depth
U,
Fig. 1. A schematic diagram of a rough surface pressed by an ideal flat surface. Fig. 2. The relationship between&he effective contact radius a,$ and the depth of penetration of a single asperity Vi.
of the region through which the electrical current can flow in a contact point, increases proportionally with the depth of penetration Vi in the range of Vi < U, until it becomes constant regardless of Vi in the range Uj 2 U,, where U, denotes the critical depth of penetration. Thus the effective contact radius u,~ is given by Vi cot 8 in the range Uj < U, and by U, cot B in the range Vi> U,,as shown in Fig. 2. This means that in the range Uj < U, the tarnish films are broken by the relative microslip motion at the interface of a microcontact with increasing plastic deformation but in the range Uj > Ue the broken area of a contact point does not increase, because the larger the contact point, the smaller the relative microslip at the interface becomes. Although the difference in the hardness values of the mating asperities has an effect on microslip, the effect is neglected in this model. Using eqns. (1) and (2), the constriction resistance is regarded as the sum of the parallel resistance of the microcontacts and is then given by
uo+ue + 1 UO
-1
(U-Ue)Fdu
/
(t&j m
(4)
348
2.2. Model considering the film thickness It is assumed in this model that tarnish films such as physisorbed and chemisorbed films do not create metal contacts at a depth of penetration less than the film thickness de. If a rough surface covered by tarnish films is pressed by a smooth surface, the effective contact radius a,; is zero when the depth of penetration Ui is less than the film thickness do. However, U,i is given by (Vi - do) cot @when Ui > de, as shown in Fig. 2. The constriction resistance is then
=
np,G(-
mu)
u
Assuming that tarnish films on asperities with a depth of penetration Vi greater than the film thickness de are ruptured perfectly during plastic deformation of the asperities, the constriction resistance is also
(U = TP,
G(-- mu) L&y
X
I(
!f!!!..$k+$g
dn(u)
ug) --
du 1
u
(l/tan
e),
t tar-k2e i m ’
(uo:do
j-~rn~rn-~j+~~~(rn)~-’
(6)
The theoretical results obtained in eqns. (4), (5) and (6), using m = 5 and @(t) = (27~-l’~ exp (-t2/2), are shown in Fig. 3. The slopes of the results from eqns. (5) and (6) plotted on a log-log scale increase with increasing film thickness de. In contrast, the slope of those from eqn. (4) decreases slightly with increasing dimensionless critical depth of penetration U,/a. Thus a comparison of the slopes of the theoretical and the experimental results may determine the most practical and reasonable model of the abovementioned theories.
349
t
Fig. 3. The theoretical variation - - - - eqn. (4), - - - eqn. (5), TABLE
of the electrical eqn. (6).
contact
resistance
R, with the load W:
1
Specimen film
details
for contact
resistance
between
a cone and a flat with a deposited
carbon
Specimen
Cone angle 20 (“)
Specific resistance P (52 cm)
Finish
Knoop hardnessa
Surface roughness R, (Pm)
Cone (Ag)
140 160 140b
1.59 x lo+
Turning
118
3.7
1.59 x lo+
Turning
35.4
1.6
Film thickness do = 382 A
1.59 X 10e6 (Ag) 1.4 x 1o-3 (carbon film)
Sandpaper finishing
97.6
0.3
Flat (Ag)
aThe values of the Knoop hardness were measured under a load of 20 g. bAfter taper turning, a silver cone was annealed for 1 h at 300 “C in vacuum. The flow pressures pi obtained from the projected area of the indentations on the silver flat formed by the work-hardened cone were 48.0 kg mm-’ for 20 = 140” and 39.6 kg mmw2 for 20 = 160’. In contrast, the value of pr obtained from the area of real contact on the annealed cone was 30.4 kg mmV2.
3. Experimental procedure 3.1. Contact resistance between a silver cone and a silver flat with a deposited carbon film In order to check the validity of the theoretical analyses, electrical contact resistances were measured.
350 TABLE 2 Specimen details for the contact resistance between two flats Specimen
Material
Specific resistance P (a cm)
Finish
Vickers microhardnesz? Hv (0.01)
Surface roughnessb R-
Upper specimen
Ag
1.59 x lo+
Sandblasting
108
24.8 15.0 5.1
Lower specimen
Ag
1.59
Grinding
108
2.2 2.2 1.1
Upper specimen
Ag
1.59 x lo+
Sandblasting
108
23.5 14.9 4.8
Lower specimen
Ag
1.59 x lo@
Sandblasting
108
23.5 14.9 4.8
X
10e6
(Pm)
aThe values of H, were measured under a load of 10 g. bThe values of R m8x were measured according to the old JIS B 0601-1955 (Japanese Industrial Standard) and the values of u for the calculation of the theoretical values were estimated from the experimental relationship [ 111 of u * R-/4.
The asperity models in the shape of right circular cones with cone angles 20 = 140” and 160” were accurately produced by taper turning (Table 1). The sizes of the asperity models were then checked by a profile projector and a microscope. Silver in the form of cold drawn rods of diameter 6 mm was selected for the asperity models and some of these rods were annealed in vacuum. The end surfaces of silver rods of diameter 30 mm were prepared by sandpaper and were covered with an evaporated carbon film of thickness da = 382 A. The asperity model and the end surface were cleaned with benzene and ethyl alcohol and were then contacted using a microhardness tester. In all the experiments, after a lapse of 10 min and 2 min in increment and decrement loads, respectively, the potential difference between the asperity model and the silver flat for an electric current of 0.47 - 0.68 mA was measured by means of a d.c. potentiometer. 3.2. Contact resistance between silver flats The end surfaces of silver rods of diameter 30 mm and length 55 mm were ground and some were sandblasted isotropically. The specimens were cleaned with benzene, tri~hloroethylene and ethyl alcohol and were contacted under static pressures. After a lapse of 10 min at each pressure, the potential difference between the two flat surfaces was measured by means of a potentiometer. Specimen details are summarized in Table 2. The standard deviations (I of the profile ordinates (r.m.s. roughness) were estimated using the experimental relationship 0 = R,,,/5. from R,,,
(a)
w (g)
(b)
w (g)
Fig. 4. Variation of the electrical contact resistance R, between a silver cone and a silver flat with a deposited carbon film with the load W: (a) contact between a work-hardened silver cone and a silver flat; (b) contact between an annealed silver cone and a silver flat.
4. Results and discussion 4.1. Con tat t resistance between a silver cone and a silver flat with a deposited carbon film The measured values of the electrical contact resistance at various loads for cone angles 20 = 140” and 160” are shown in Fig. 4 as the average values of four to five test results. The broken lines were obtained by multiplying the calculated values for the contact without the carbon film by (Y,, the values of which were chosen to make the calculated values fit the experimental data, where the calculated values were obtained by (7) For contacts without a carbon film [ 121 the values of (Y, were about 2.2 regardless of the cone angle but for those with a carbon film in Fig. 4(a) the values increased with increasing cone angle. This suggests that the microslip at the interface during penetration of a work-hardened cone into a flat with a carbon film varies with the cone angle, and that the electrical contact resistance R, is inversely proportional to IV”’ because the variation of the constriction resistance with load has a greater‘effect on R, than that of film resistance.
Fig. 5. The indentations on a flat surface with the carbon film viewed with the scanning microscope (300 X ): (a) the indentation formed by a work-hardened cone with 2P = 140” ; (b) the indentation formed by a work-hardened cone with 20 = 160” ; (c) the indentation formed by an annealed cone with 20 = 160”.
In contrast, the value of a, for annealed cones pressed against a flat with a carbon film is greater than that for work-hardened cones, as shown in Fig. 4(b). The marked spread in experimental results arises from the variation of the area of the carbon film broken at the beginning of contact. The experimental results are inversely proportional to W1” under loads below 100 g but are almost constant under loads over 100 g. This suggests that the effective contact radius under loads over 100 g is independent of the load, as assumed in the critical penetration model. Figure 5 shows the indentations on a flat surface viewed with the scanning microscope. A localized fracture of the carbon film on the surface pressed by the harder cone is indicated by the circular and white lines in Figs. 5(a) and 5(b) but fracture by the softer cone was not observed (Fig. 5(c)). These results suggest that the degree of plastic deformation of the surface with the tarnish film has a significant effect on microslip at the contact interface and on the fracture of the tarnish film. Thus to estimate accurately the electrical contact resistance the values of (Y, and U,, which vary with the cone angle and hardness, are necessary but their estimation for each microcontact between two flat surfaces is difficult because of scatter, Therefore for multiple contacts the values of UJu, which may contain the effect of lye, can be determined experimentally. 4.2. Contact resistance between silver flats The measured values of the electrical contact resistance between a ground and a sandblasted silver surface at various loads for the range of roughnesses are shown in Fig. 6. They are the average values of five test results. The broken lines were obtained from eqns. (3) and (4) assuming = (tan 0),/1.5, that m = 5, a = R,,, /5 and (l/tan e), {(l/tan2 0),)-l using the experimental relationship [13] (tan e), = (?r/Z)R,,,/(3.45R,,, + ll.l), and using pm = 1.59 X low6 Sz cm and the values of U,lo which were determined by making the calculated values fit the experimental data. The variation of the slopes of the experimental results on a log-log scale is in good agreement with that for the theoretical results. However, the slopes of the calculated values obtained from eqns. (5) and (6) for de = 0 do not change with roughness and do not agree with the experimental data.
353
Sandblastedon ground d
Load Unl
\ \\
10
0
D
5 ;
(Agf
tAg) 24.8pm&,
2.2vmR,,.
15 Opm%ax
22ymR,..
5 Ipm%,,
1lp-h,.
1
14'
1
103
10'
10
wflxLy
Wcm2)
Fig. 6. Variation of the electrical contact resistance R, between a ground silver surface and a sandblasted silver surface with the load W.
Load Unl 0
0
T1
Sandblastedc"Sandbl~tPr (Ag) (Ag) 23.5pmA,,
,62/U 10-l
1
IO
IO' WILXLY
10' (kg/cm')
Fig. ‘7. Variation of the electrical contact resistance R, between two sandblasted silver surfaces with the load W.
The measured values of the electrical contact resistance between two sandblasted silver surfaces at various loads are shown in Fig. 7. The calculated values were obtained from eqns. (3) and (4) by the same method as the theoretical values in Fig. 6 assuming that the values of (tan e), are determined by the greater value in the roughnesses of the mating surfaces. The values of U.Jcr and the slope on a log-log scale increase with the surface roughness. These trends conform to the assumption that the greater the depth of penetration due to plastic deformation of each microcontact, the greater is the broken area in the film.
1o-51’ 10-l
’ ’
1111”
’
’
“IIll’
1
’ ’ ““‘1 lo2
10 W
(kg)
Fig. 8. Variation of the electrical contact resistance R, between a palladium surface and a silver surface with an AgzS film of thickness d 0 = 3482 A with the load W. The apparent contact area L,L, was 19.6 mm2.
4.3. Contact resistance between a palladium surface and a silver surface with an Ag2S tarnish The experimental results of electrical contact resistance between a palladium surface and a silver surface with silver sulphide films of do = 3482 A and do = 1400 A obtained by Harada and Mano [8,9] are shown in Figs. 8 and 9, respectively. The sulphide films were produced by exposure of the finished silver surfaces for 10 - 24 h to hydrogen sulphide gas at 100 Ton. The thicknesses of the sulphide films were determined from the weight of a sample silver wire of diameter 1 mm and length 40 mm before and after exposure [S] . The silver surfaces of 5 mm diameter rods with I& = 71 kg mmV2 and p1 = 1.59 X lO-‘j CI cm were isotropically prepared by sandpaper finishing before exposure, and the palladium surfaces with Hv = 140 kg mme2 and p2 = 10.8 X 10m6 L2cm were isotropically prepared by polishing or sandpaper finishing. The calculated values obtained with eqns. (3) and (4) using (tan e), = (n/2)&,,/(4.44&,, + 18.8) for a sandpaper finished surface and R,,, = 50 are shown by broken lines in Figs. 8 and 9. The slopes of the calculated values on log-log scales show good agreement with experimental results. The values of U,/o increase with surface
355
,0-4
_
0
de.MOOi (Ag,f)
’ ’ ’ ““”
lo+ . I ’ “““’ 16'
1
10 W
’ ’ ““’
L 102
(kg)
Fig. 9. Variation of the electrical contact resistance R, between a palladium surface and a silver surface with an Ag$ film of thickness do = 1400 A with the load W. The apparent contact area I.+,&, was 19.6 mm2.
roughness, so that during plastic deformation of the contact asperities with the film the broken area of the film on the rough surface may be greater than that for the smooth surface. The solid lines in Figs. 8 and 9 show the theoretical film resistance R, in eqn. (1) multiplied by 10m2, i.e. the values of R, were obtained from R, = p~d~p~~~
(9)
using p. =102 S2 cmandpf=71kgmm -2. The calculated values of the film resistance and their slopes on log-log scales are greater than those of the experimental results. This is considered to be due to the effects of film &acture by penetration of each contact asperity. In a contact between two metal surfaces covered by thin tarnish films, the films can fail so readily that it is difficult to estimate the film resistance allowing for film thickness and to separate this from the constriction resistance as in eqn. (1). It is therefore reasonable to consider the effect of fracture of tarnish films on the constriction resistance rather than that of thickness as the critical penetration model. 4.4. Critical penetration model and t~e~al con tact resistance Agreement between theory and experiment in electrical contact resistance experiments is difficult owing to the presence of non-conducting tarnish films. This difficulty is minimized, however, with thermal contact
356
stainless
iF
a 3.33pmo,2 0 1.43pma.
AllOOBE
steel 74pm 0 1 4Opma
\
on AllOOBE
1 15p~R,,~,+mR,, 0 1.5pR,,,
, 16pmRma,
0 15pmR,,,,
30pR,,x
. lOpnR,,,,
lOpR,nax
v 15pmR,ax,15p~R~ax 7 32pR,,,, -
32pRwax
Theoretfcai
I
J 1o-3
1o-2
10-l
(WILxL,)IR Fig. 10. Variation of the thermal contact metals in a vacuum with the load W.
resistance
Rh between
various
combinations
of
resistance because the difference between the thermal conductivities of the bulk metals and the tarnish films is generally small. The experimental results of thermal contact resistance measured in a vacuum [ 14 - 161 are shown in Fig. 10. The theoretical values obtained from eqn. (4), using (l/tan (I),/ ( l/tan2 0 ), = 0.6667(tan e), and values of U,/o that were chosen to make the calculated values fit the experimental results, are shown by the solid lines. The variation of the theoretical values with load agrees with the experimental results. The variation of the measured values between two aluminium surfaces (AllOOBE) with the surface roughness is due to fracture of the oxide film which varies with microslip and plastic deformation at the contact interface of the asperities. This suggests that for mating surfaces covered by a film with a lower thermal conductivity than that of the bulk metal the film has a marked effect on the thermal contact resistance and especially on the directional effect (which refers to a curious property of certain contacts by which they have a greater thermal contact resistance in one direction across the contact than in the reverse direction). 4.5. Critical depth of penetration and surface roughness The dimensionless critical depth of penetration UJa at various surface 1’2) for combinations of some metals roughnesses R,,, (= (R,,,: + R,,,;)
357
Sandblasted
on sandblasted
Sandblasted
on ground
S P ‘,n,shed
on p&shed
0’ Ag 0nAg
/
. v
AllOOBE
on AllOOBE
v
/
0 /
/
.’
/
,A
d.=3482i
( AgzS) d,=1400j MgzS)
/
on %
1
A
SP‘,n,shed
on SPfmshed
A
SPfmshed
on pollshed
ll
1O-4;
16'
4-v
’
9'
Ag
d
.P
1
lo*
10 I%,
Fig. 11. Variation of UJO with R,
(pm)
(= (R-f
+ R,,,&)1’2).
S.P. denotes sandpaper.
is shown in Fig. 11. The values of Ue/u, which were determined from making the values calculated by eqn. (4) fit the experimental results [17] for the electrical and thermal contact reastances, increase with surface roughness and their slopes on a log-log scale are almost constant for various mating surfaces. The smaller values of U,/cr for do = 1400 8, compared with those for do = 3482 A for a given surface roughness are thought to be due to the effects of the fracture of the sulphide film by the smaller plastic deformation and smaller microslip at the interface between rough surfaces than those between a rough and a smooth surface. Thus it is necessary for the accurate prediction of the contact resistance to determine the values of UJo experimentally. The values of lJ,/a for silver surfaces prepared by sandblasting and sandpaper are given by the relationship of UJu = 2.1 X 10e2 Rrnaxl.12 (Fig. 11). The values for surfaces with tarnish films are also given approximately by the relationship UJo = kR’,,, , where j and k are constants depending on the mechanical properties of the mating surfaces and their tarnish films.
5. Conclusions A method is proposed for estimating the electrical and the thermal contact resistance between two metals with a tarnish film. Comparison of
358
theory
and experimental evidence allows the following conclusions. (1) The thin tarnish film at the interface between two contact asperities is readily broken by plastic deformation and microslip, the amounts of which have a significant effect on the constriction resistance. (2) The broken areas of the tarnish film of a given thickness at the interface vary with the geometrical shape and the relative hardness of each contact asperity as these change the amounts of plastic deformation and microslip. The less the plastic deformation of the surface with the tarnish film, the greater the contact resistance becomes, and for a large contact point the contact resistance is independent of the load because the broken area of the film does not increase with increasing load. (3) The values of UJo are given by the relationship U,/a = 2.1 X 1O-2 R maxl.lZ for sandblasted and sandpaper finished silver surfaces in air and by l_JJo = kR’,,,, where j and k are constants, for the other metals. (4) The electrical contact resistance between a palladium surface and a silver surface with a silver sulphide film increases with decreasing surface roughness. This depends on the extent of film fracture resulting in relative microslip at the interface between the contact asperities. (5) The same effect of UJo for electrical contact resistance occurs for thermal contact resistance of silver and aluminium surfaces.
Nomenclature A ai %i
-wJ, m
n
Pf
RC R, Rf % Rmax t U
ue ui
total area of real contact radii of contact points (i = 1, 2, 3, . . . . n) effective contact radius, i.e. radius of the region through which the electrical current can flow in a contact point film thickness (27r_l’2 u-l exp l--_(W) (dO)21 f(u) - f(mo) probability density of surface heights S;” g(u)dt Vickers hardness number constants depending on type of finish, thickness of film and difference in hardness of mating surfaces apparent contact area and type of machining operation: constant dependent on L,Ly ,R, m a 5 for a sandblasted surface number of contact points mean flow pressure constriction resistance electrical contact resistance film resistance thermal contact resistance per unit area of apparent contact maximum height of irregularities u/a, dimensionless separation separation of mean planes critical depth of penetration depth of penetration of a single asperity
359
0” Al>hZ hII Pl, P2
PII3 PO ; ( An
total load base angle of a single asperity thermal conductivities of solids 1 and 2 hXZ/(hl
+ ha)
specific resistances of solids 1 and 2, respectively (Pl
+ P2Y2
specific resistance of surface film r.m.s. roughness (2@‘2 exp (- t2/2) mean value of ( )
References 1 J. F. Archard, Contact and rubbing of flat surfaces, J. Appl. Phys., 24(8) (1953) 981. 2 J. A. Greenwood and J. B. P. Williamson, The contact of nominally flat surfaces, Proc. R. Sot. London, Ser. A, 295 (1966) 300. 3 S. Harada and K. Mano, Some problems of the contact resistance. In Fundamental studies on electrical contact resistance, Part 1, The Record of Electrical and Communication Engineering Conversazione, TGhoku University, Vol. 33, part 4, 1964, p. 111. 4 S. Hoshina, On the electrical contact resistance under light load (I), J. Appl. Phys. Jpn, 30 (4) (1961) 221. 5 T. Matuyama and H. Suzuki, A study of the polishing of electric contacts, Trans. JSME, 21 (106) (1955) 450. 6 T. Matuyama and H. Suzuki, A study of the polishing of electric contacts (succeeding report), Trans. JSME, 24 (146) (1958) 692. 7 T. Tsukizoe and T. Hisakado, On the mechanism of heat transfer between metal surfaces in contact (Part 1: Theoretical analysis and experimental justification of theory), Heat Transfer - Jpn. Res., 1 (1) (1972) 104. 8 S. Harada and K. Mano, The effects of surface roughness on contact resistance. In Fundamental studies on electrical contact resistance, Part 8, The Record of Electrical and Communication Engineering Conversazione, TGhoku University, Vol. 36, part 3, 1967, p. 344. 9 S. Harada and K. Mano, The effects of the surface roughness on the contact resistance of the flat contact. In Fundamental studies on electrical contact resistance, Part 9, The Record of Electrical and Communication Engineering Conversazione, Tohoku University, Vol. 36, part 4, 1967, p. 376. 10 J. R. Osias and J. H. Tripp, Mechanical disruption of surface films on metals, Wear, 9 (1966) 388. 11 T. Hisakado and T. Tsukizoe, Effects of distribution of surface slopes and flow pressures of contact asperities on contact between solid surfaces, Wear, 30 (1974) 213. 12 T. Tsukizoe, T. Hisakado and Y. Nakajima, On the mechanism of contact between sliding metal surfaces, Preprint of the 44th General Conf. (Kansai), JSME, no. 44 - 11, 1969, p. 81. 13 T. Hisakado, On the mechanism of contact between solid surfaces (1st Rep., The initial separation and the distributions of slopes of facets on surface and on the profile curve), Bull. JSME, 12 (54) (1969) 1519. 14 T. R. Thomas and S. D. Probert, Thermal contact resistance: the directional effect and other problems, Int. J. Heat Mass Transfer, 13 (1970) 789. 15 J. J. Henry, Thermal contact resistance, A. E. C. Rep. no. MIT-2079-2, 1964, p. 69. 16 T. Tsukizoe and T. Hisakado, On the mechanism of heat transfer between metal surfaces in contact (2nd Rep., Thermal contact resistance between metal surfaces in vacuum), Heat Transfer - Jpn. Res., 1 (2) (1972) 23. 17 T. Hisakado, On the mechanism of contact between solid surfaces (5th Rep., Analysis taking elastic deformations of asperities into account), Trans. JSME, 38 (314) (1972) 2657.
@ Elsevier
345
345 - 359
Sequoia
S.A., Lausanne
EFFECTS OF SURFACE CONTACT RESISTANCE
- Printed
in the Netherlands
ROUGHNESS
AND SURFACE
FILMS ON
T. HISAKADO Department of Precision Osaka (Japan)
(Received
Engineering,
Faculty
of Engineering,
July 22, 1976; in final form November
Osaka University,
Suita,
1, 1976)
Summary An analysis of the electrical contact resistance between two metals with a tarnish film was carried out assuming the asperities to be represented by randomly distributed cones with base angles which vary with the surface roughness, and assuming that the radius of the broken area of the film at the interface of each contact asperity is constant beyond a critical depth of penetration of an asperity. The validity of the proposed theory was confirmed by experimental data of the electrical contact resistance between a silver cone and a silver flat on which carbon films were deposited, and between a silver flat with an Ag,S tarnish and a palladium flat without tarnish. Comparison of theoretical and experimental data shows that the critical depth of penetration U, of an asperity varies mainly with the surface roughness, the thickness of the tarnish film and the amount of plastic deformation of the contact asperities with films, i.e. UJo = kR’,,,, where R,, and u are the maximum height and the standard deviation (r.m.s. roughness) of the profile ordinates, and j and k are constants dependent on the type of finish, the thickness of the film and the difference in the hardness of the mating surfaces.
1. Introduction Films on contacts create an electrical resistance that can cause failure in contact applications. However, it is very difficult to estimate theoretically the effects of thin films such as physisorbed and chemisorbed films. This is due to the changeable behaviour of the fracture of the films during plastic deformation of each contact asperity. Hence a notable feature of published work is the very wide discrepancies between the experimental results of different investigators using nominally similar materials. Also each of the theories advanced can generally be applied only to the experimental results of the investigator proposing that particular theory [ 1 - 71. Few studies of the fracture of the tarnish films [8 - lo] have been made.
346
The present work was an attempt to develop a theory that was applicable to surfaces with a tarnish film in static contact. A theory was derived by considering the contact and the fracture mech~ism of the tarnish films, i.e. using a critical penetration model of contact asperities assuming that the broken area of the film at each contact asperity was constant beyond the critical depth of penetration. The validity of the theory was checked by theoretical and experimental results of the electrical contact resistance between a silver cone and a silver flat surface with a deposited carbon film and between flat siIver surfaces. The effect of the tarnish film on the thermal contact resistance in a vacuum environment is discussed in conjunction with the theory.
2. Theory When two asperities on conductors with tarnish films touch, there is a constriction and a film resistance between them. The total resistance is given by R< = R,i + Rfi Pill 2ai
Z-+
POI do1
+ poadoz
(1)
na?
This equation is very widely used in the design and study of electrical contacts. However, there is evidence that if thin fiIms with higher resistivity than that of the metals of the asperities, i.e. pal > p1 and po2 3 p2, are broken in the contact process, the total resistance may be attributed to the constriction resistance. In the theoretical analysis the constriction resistance will be derived considering the number and size of the microcontacts and the thickness of the tarnish films. 2.1. Critical penetration model When a soft metal surface consisting of some conical asperities is pressed by a hard and smooth metal surface, as shown in Fig. 1, the number of contact points n(u) is given by [ll] n(u()) = -
&f(u)
&J, 2nG(-
mu)
du
I
(Cot20);;;”
(2)
U=Ug
The separation u. of the mean planes of a smooth and a rough surface in contact is determined by the flow pressure pf of the softer metal and the applied load IV: mo
LL,
g(u)
.IUO W--u)
&=
!!_ pi
It is assumed in this model that an effective
contact
radius a,i, i.e. the radius
347
Centre line (Reference
ur d. Penetration
plane)
depth
U,
Fig. 1. A schematic diagram of a rough surface pressed by an ideal flat surface. Fig. 2. The relationship between&he effective contact radius a,$ and the depth of penetration of a single asperity Vi.
of the region through which the electrical current can flow in a contact point, increases proportionally with the depth of penetration Vi in the range of Vi < U, until it becomes constant regardless of Vi in the range Uj 2 U,, where U, denotes the critical depth of penetration. Thus the effective contact radius u,~ is given by Vi cot 8 in the range Uj < U, and by U, cot B in the range Vi> U,,as shown in Fig. 2. This means that in the range Uj < U, the tarnish films are broken by the relative microslip motion at the interface of a microcontact with increasing plastic deformation but in the range Uj > Ue the broken area of a contact point does not increase, because the larger the contact point, the smaller the relative microslip at the interface becomes. Although the difference in the hardness values of the mating asperities has an effect on microslip, the effect is neglected in this model. Using eqns. (1) and (2), the constriction resistance is regarded as the sum of the parallel resistance of the microcontacts and is then given by
uo+ue + 1 UO
-1
(U-Ue)Fdu
/
(t&j m
(4)
348
2.2. Model considering the film thickness It is assumed in this model that tarnish films such as physisorbed and chemisorbed films do not create metal contacts at a depth of penetration less than the film thickness de. If a rough surface covered by tarnish films is pressed by a smooth surface, the effective contact radius a,; is zero when the depth of penetration Ui is less than the film thickness do. However, U,i is given by (Vi - do) cot @when Ui > de, as shown in Fig. 2. The constriction resistance is then
=
np,G(-
mu)
u
Assuming that tarnish films on asperities with a depth of penetration Vi greater than the film thickness de are ruptured perfectly during plastic deformation of the asperities, the constriction resistance is also
(U = TP,
G(-- mu) L&y
X
I(
!f!!!..$k+$g
dn(u)
ug) --
du 1
u
(l/tan
e),
t tar-k2e i m ’
(uo:do
j-~rn~rn-~j+~~~(rn)~-’
(6)
The theoretical results obtained in eqns. (4), (5) and (6), using m = 5 and @(t) = (27~-l’~ exp (-t2/2), are shown in Fig. 3. The slopes of the results from eqns. (5) and (6) plotted on a log-log scale increase with increasing film thickness de. In contrast, the slope of those from eqn. (4) decreases slightly with increasing dimensionless critical depth of penetration U,/a. Thus a comparison of the slopes of the theoretical and the experimental results may determine the most practical and reasonable model of the abovementioned theories.
349
t
Fig. 3. The theoretical variation - - - - eqn. (4), - - - eqn. (5), TABLE
of the electrical eqn. (6).
contact
resistance
R, with the load W:
1
Specimen film
details
for contact
resistance
between
a cone and a flat with a deposited
carbon
Specimen
Cone angle 20 (“)
Specific resistance P (52 cm)
Finish
Knoop hardnessa
Surface roughness R, (Pm)
Cone (Ag)
140 160 140b
1.59 x lo+
Turning
118
3.7
1.59 x lo+
Turning
35.4
1.6
Film thickness do = 382 A
1.59 X 10e6 (Ag) 1.4 x 1o-3 (carbon film)
Sandpaper finishing
97.6
0.3
Flat (Ag)
aThe values of the Knoop hardness were measured under a load of 20 g. bAfter taper turning, a silver cone was annealed for 1 h at 300 “C in vacuum. The flow pressures pi obtained from the projected area of the indentations on the silver flat formed by the work-hardened cone were 48.0 kg mm-’ for 20 = 140” and 39.6 kg mmw2 for 20 = 160’. In contrast, the value of pr obtained from the area of real contact on the annealed cone was 30.4 kg mmV2.
3. Experimental procedure 3.1. Contact resistance between a silver cone and a silver flat with a deposited carbon film In order to check the validity of the theoretical analyses, electrical contact resistances were measured.
350 TABLE 2 Specimen details for the contact resistance between two flats Specimen
Material
Specific resistance P (a cm)
Finish
Vickers microhardnesz? Hv (0.01)
Surface roughnessb R-
Upper specimen
Ag
1.59 x lo+
Sandblasting
108
24.8 15.0 5.1
Lower specimen
Ag
1.59
Grinding
108
2.2 2.2 1.1
Upper specimen
Ag
1.59 x lo+
Sandblasting
108
23.5 14.9 4.8
Lower specimen
Ag
1.59 x lo@
Sandblasting
108
23.5 14.9 4.8
X
10e6
(Pm)
aThe values of H, were measured under a load of 10 g. bThe values of R m8x were measured according to the old JIS B 0601-1955 (Japanese Industrial Standard) and the values of u for the calculation of the theoretical values were estimated from the experimental relationship [ 111 of u * R-/4.
The asperity models in the shape of right circular cones with cone angles 20 = 140” and 160” were accurately produced by taper turning (Table 1). The sizes of the asperity models were then checked by a profile projector and a microscope. Silver in the form of cold drawn rods of diameter 6 mm was selected for the asperity models and some of these rods were annealed in vacuum. The end surfaces of silver rods of diameter 30 mm were prepared by sandpaper and were covered with an evaporated carbon film of thickness da = 382 A. The asperity model and the end surface were cleaned with benzene and ethyl alcohol and were then contacted using a microhardness tester. In all the experiments, after a lapse of 10 min and 2 min in increment and decrement loads, respectively, the potential difference between the asperity model and the silver flat for an electric current of 0.47 - 0.68 mA was measured by means of a d.c. potentiometer. 3.2. Contact resistance between silver flats The end surfaces of silver rods of diameter 30 mm and length 55 mm were ground and some were sandblasted isotropically. The specimens were cleaned with benzene, tri~hloroethylene and ethyl alcohol and were contacted under static pressures. After a lapse of 10 min at each pressure, the potential difference between the two flat surfaces was measured by means of a potentiometer. Specimen details are summarized in Table 2. The standard deviations (I of the profile ordinates (r.m.s. roughness) were estimated using the experimental relationship 0 = R,,,/5. from R,,,
(a)
w (g)
(b)
w (g)
Fig. 4. Variation of the electrical contact resistance R, between a silver cone and a silver flat with a deposited carbon film with the load W: (a) contact between a work-hardened silver cone and a silver flat; (b) contact between an annealed silver cone and a silver flat.
4. Results and discussion 4.1. Con tat t resistance between a silver cone and a silver flat with a deposited carbon film The measured values of the electrical contact resistance at various loads for cone angles 20 = 140” and 160” are shown in Fig. 4 as the average values of four to five test results. The broken lines were obtained by multiplying the calculated values for the contact without the carbon film by (Y,, the values of which were chosen to make the calculated values fit the experimental data, where the calculated values were obtained by (7) For contacts without a carbon film [ 121 the values of (Y, were about 2.2 regardless of the cone angle but for those with a carbon film in Fig. 4(a) the values increased with increasing cone angle. This suggests that the microslip at the interface during penetration of a work-hardened cone into a flat with a carbon film varies with the cone angle, and that the electrical contact resistance R, is inversely proportional to IV”’ because the variation of the constriction resistance with load has a greater‘effect on R, than that of film resistance.
Fig. 5. The indentations on a flat surface with the carbon film viewed with the scanning microscope (300 X ): (a) the indentation formed by a work-hardened cone with 2P = 140” ; (b) the indentation formed by a work-hardened cone with 20 = 160” ; (c) the indentation formed by an annealed cone with 20 = 160”.
In contrast, the value of a, for annealed cones pressed against a flat with a carbon film is greater than that for work-hardened cones, as shown in Fig. 4(b). The marked spread in experimental results arises from the variation of the area of the carbon film broken at the beginning of contact. The experimental results are inversely proportional to W1” under loads below 100 g but are almost constant under loads over 100 g. This suggests that the effective contact radius under loads over 100 g is independent of the load, as assumed in the critical penetration model. Figure 5 shows the indentations on a flat surface viewed with the scanning microscope. A localized fracture of the carbon film on the surface pressed by the harder cone is indicated by the circular and white lines in Figs. 5(a) and 5(b) but fracture by the softer cone was not observed (Fig. 5(c)). These results suggest that the degree of plastic deformation of the surface with the tarnish film has a significant effect on microslip at the contact interface and on the fracture of the tarnish film. Thus to estimate accurately the electrical contact resistance the values of (Y, and U,, which vary with the cone angle and hardness, are necessary but their estimation for each microcontact between two flat surfaces is difficult because of scatter, Therefore for multiple contacts the values of UJu, which may contain the effect of lye, can be determined experimentally. 4.2. Contact resistance between silver flats The measured values of the electrical contact resistance between a ground and a sandblasted silver surface at various loads for the range of roughnesses are shown in Fig. 6. They are the average values of five test results. The broken lines were obtained from eqns. (3) and (4) assuming = (tan 0),/1.5, that m = 5, a = R,,, /5 and (l/tan e), {(l/tan2 0),)-l using the experimental relationship [13] (tan e), = (?r/Z)R,,,/(3.45R,,, + ll.l), and using pm = 1.59 X low6 Sz cm and the values of U,lo which were determined by making the calculated values fit the experimental data. The variation of the slopes of the experimental results on a log-log scale is in good agreement with that for the theoretical results. However, the slopes of the calculated values obtained from eqns. (5) and (6) for de = 0 do not change with roughness and do not agree with the experimental data.
353
Sandblastedon ground d
Load Unl
\ \\
10
0
D
5 ;
(Agf
tAg) 24.8pm&,
2.2vmR,,.
15 Opm%ax
22ymR,..
5 Ipm%,,
1lp-h,.
1
14'
1
103
10'
10
wflxLy
Wcm2)
Fig. 6. Variation of the electrical contact resistance R, between a ground silver surface and a sandblasted silver surface with the load W.
Load Unl 0
0
T1
Sandblastedc"Sandbl~tPr (Ag) (Ag) 23.5pmA,,
,62/U 10-l
1
IO
IO' WILXLY
10' (kg/cm')
Fig. ‘7. Variation of the electrical contact resistance R, between two sandblasted silver surfaces with the load W.
The measured values of the electrical contact resistance between two sandblasted silver surfaces at various loads are shown in Fig. 7. The calculated values were obtained from eqns. (3) and (4) by the same method as the theoretical values in Fig. 6 assuming that the values of (tan e), are determined by the greater value in the roughnesses of the mating surfaces. The values of U.Jcr and the slope on a log-log scale increase with the surface roughness. These trends conform to the assumption that the greater the depth of penetration due to plastic deformation of each microcontact, the greater is the broken area in the film.
1o-51’ 10-l
’ ’
1111”
’
’
“IIll’
1
’ ’ ““‘1 lo2
10 W
(kg)
Fig. 8. Variation of the electrical contact resistance R, between a palladium surface and a silver surface with an AgzS film of thickness d 0 = 3482 A with the load W. The apparent contact area L,L, was 19.6 mm2.
4.3. Contact resistance between a palladium surface and a silver surface with an Ag2S tarnish The experimental results of electrical contact resistance between a palladium surface and a silver surface with silver sulphide films of do = 3482 A and do = 1400 A obtained by Harada and Mano [8,9] are shown in Figs. 8 and 9, respectively. The sulphide films were produced by exposure of the finished silver surfaces for 10 - 24 h to hydrogen sulphide gas at 100 Ton. The thicknesses of the sulphide films were determined from the weight of a sample silver wire of diameter 1 mm and length 40 mm before and after exposure [S] . The silver surfaces of 5 mm diameter rods with I& = 71 kg mmV2 and p1 = 1.59 X lO-‘j CI cm were isotropically prepared by sandpaper finishing before exposure, and the palladium surfaces with Hv = 140 kg mme2 and p2 = 10.8 X 10m6 L2cm were isotropically prepared by polishing or sandpaper finishing. The calculated values obtained with eqns. (3) and (4) using (tan e), = (n/2)&,,/(4.44&,, + 18.8) for a sandpaper finished surface and R,,, = 50 are shown by broken lines in Figs. 8 and 9. The slopes of the calculated values on log-log scales show good agreement with experimental results. The values of U,/o increase with surface
355
,0-4
_
0
de.MOOi (Ag,f)
’ ’ ’ ““”
lo+ . I ’ “““’ 16'
1
10 W
’ ’ ““’
L 102
(kg)
Fig. 9. Variation of the electrical contact resistance R, between a palladium surface and a silver surface with an Ag$ film of thickness do = 1400 A with the load W. The apparent contact area I.+,&, was 19.6 mm2.
roughness, so that during plastic deformation of the contact asperities with the film the broken area of the film on the rough surface may be greater than that for the smooth surface. The solid lines in Figs. 8 and 9 show the theoretical film resistance R, in eqn. (1) multiplied by 10m2, i.e. the values of R, were obtained from R, = p~d~p~~~
(9)
using p. =102 S2 cmandpf=71kgmm -2. The calculated values of the film resistance and their slopes on log-log scales are greater than those of the experimental results. This is considered to be due to the effects of film &acture by penetration of each contact asperity. In a contact between two metal surfaces covered by thin tarnish films, the films can fail so readily that it is difficult to estimate the film resistance allowing for film thickness and to separate this from the constriction resistance as in eqn. (1). It is therefore reasonable to consider the effect of fracture of tarnish films on the constriction resistance rather than that of thickness as the critical penetration model. 4.4. Critical penetration model and t~e~al con tact resistance Agreement between theory and experiment in electrical contact resistance experiments is difficult owing to the presence of non-conducting tarnish films. This difficulty is minimized, however, with thermal contact
356
stainless
iF
a 3.33pmo,2 0 1.43pma.
AllOOBE
steel 74pm 0 1 4Opma
\
on AllOOBE
1 15p~R,,~,+mR,, 0 1.5pR,,,
, 16pmRma,
0 15pmR,,,,
30pR,,x
. lOpnR,,,,
lOpR,nax
v 15pmR,ax,15p~R~ax 7 32pR,,,, -
32pRwax
Theoretfcai
I
J 1o-3
1o-2
10-l
(WILxL,)IR Fig. 10. Variation of the thermal contact metals in a vacuum with the load W.
resistance
Rh between
various
combinations
of
resistance because the difference between the thermal conductivities of the bulk metals and the tarnish films is generally small. The experimental results of thermal contact resistance measured in a vacuum [ 14 - 161 are shown in Fig. 10. The theoretical values obtained from eqn. (4), using (l/tan (I),/ ( l/tan2 0 ), = 0.6667(tan e), and values of U,/o that were chosen to make the calculated values fit the experimental results, are shown by the solid lines. The variation of the theoretical values with load agrees with the experimental results. The variation of the measured values between two aluminium surfaces (AllOOBE) with the surface roughness is due to fracture of the oxide film which varies with microslip and plastic deformation at the contact interface of the asperities. This suggests that for mating surfaces covered by a film with a lower thermal conductivity than that of the bulk metal the film has a marked effect on the thermal contact resistance and especially on the directional effect (which refers to a curious property of certain contacts by which they have a greater thermal contact resistance in one direction across the contact than in the reverse direction). 4.5. Critical depth of penetration and surface roughness The dimensionless critical depth of penetration UJa at various surface 1’2) for combinations of some metals roughnesses R,,, (= (R,,,: + R,,,;)
357
Sandblasted
on sandblasted
Sandblasted
on ground
S P ‘,n,shed
on p&shed
0’ Ag 0nAg
/
. v
AllOOBE
on AllOOBE
v
/
0 /
/
.’
/
,A
d.=3482i
( AgzS) d,=1400j MgzS)
/
on %
1
A
SP‘,n,shed
on SPfmshed
A
SPfmshed
on pollshed
ll
1O-4;
16'
4-v
’
9'
Ag
d
.P
1
lo*
10 I%,
Fig. 11. Variation of UJO with R,
(pm)
(= (R-f
+ R,,,&)1’2).
S.P. denotes sandpaper.
is shown in Fig. 11. The values of Ue/u, which were determined from making the values calculated by eqn. (4) fit the experimental results [17] for the electrical and thermal contact reastances, increase with surface roughness and their slopes on a log-log scale are almost constant for various mating surfaces. The smaller values of U,/cr for do = 1400 8, compared with those for do = 3482 A for a given surface roughness are thought to be due to the effects of the fracture of the sulphide film by the smaller plastic deformation and smaller microslip at the interface between rough surfaces than those between a rough and a smooth surface. Thus it is necessary for the accurate prediction of the contact resistance to determine the values of UJo experimentally. The values of lJ,/a for silver surfaces prepared by sandblasting and sandpaper are given by the relationship of UJu = 2.1 X 10e2 Rrnaxl.12 (Fig. 11). The values for surfaces with tarnish films are also given approximately by the relationship UJo = kR’,,, , where j and k are constants depending on the mechanical properties of the mating surfaces and their tarnish films.
5. Conclusions A method is proposed for estimating the electrical and the thermal contact resistance between two metals with a tarnish film. Comparison of
358
theory
and experimental evidence allows the following conclusions. (1) The thin tarnish film at the interface between two contact asperities is readily broken by plastic deformation and microslip, the amounts of which have a significant effect on the constriction resistance. (2) The broken areas of the tarnish film of a given thickness at the interface vary with the geometrical shape and the relative hardness of each contact asperity as these change the amounts of plastic deformation and microslip. The less the plastic deformation of the surface with the tarnish film, the greater the contact resistance becomes, and for a large contact point the contact resistance is independent of the load because the broken area of the film does not increase with increasing load. (3) The values of UJo are given by the relationship U,/a = 2.1 X 1O-2 R maxl.lZ for sandblasted and sandpaper finished silver surfaces in air and by l_JJo = kR’,,,, where j and k are constants, for the other metals. (4) The electrical contact resistance between a palladium surface and a silver surface with a silver sulphide film increases with decreasing surface roughness. This depends on the extent of film fracture resulting in relative microslip at the interface between the contact asperities. (5) The same effect of UJo for electrical contact resistance occurs for thermal contact resistance of silver and aluminium surfaces.
Nomenclature A ai %i
-wJ, m
n
Pf
RC R, Rf % Rmax t U
ue ui
total area of real contact radii of contact points (i = 1, 2, 3, . . . . n) effective contact radius, i.e. radius of the region through which the electrical current can flow in a contact point film thickness (27r_l’2 u-l exp l--_(W) (dO)21 f(u) - f(mo) probability density of surface heights S;” g(u)dt Vickers hardness number constants depending on type of finish, thickness of film and difference in hardness of mating surfaces apparent contact area and type of machining operation: constant dependent on L,Ly ,R, m a 5 for a sandblasted surface number of contact points mean flow pressure constriction resistance electrical contact resistance film resistance thermal contact resistance per unit area of apparent contact maximum height of irregularities u/a, dimensionless separation separation of mean planes critical depth of penetration depth of penetration of a single asperity
359
0” Al>hZ hII Pl, P2
PII3 PO ; ( An
total load base angle of a single asperity thermal conductivities of solids 1 and 2 hXZ/(hl
+ ha)
specific resistances of solids 1 and 2, respectively (Pl
+ P2Y2
specific resistance of surface film r.m.s. roughness (2@‘2 exp (- t2/2) mean value of ( )
References 1 J. F. Archard, Contact and rubbing of flat surfaces, J. Appl. Phys., 24(8) (1953) 981. 2 J. A. Greenwood and J. B. P. Williamson, The contact of nominally flat surfaces, Proc. R. Sot. London, Ser. A, 295 (1966) 300. 3 S. Harada and K. Mano, Some problems of the contact resistance. In Fundamental studies on electrical contact resistance, Part 1, The Record of Electrical and Communication Engineering Conversazione, TGhoku University, Vol. 33, part 4, 1964, p. 111. 4 S. Hoshina, On the electrical contact resistance under light load (I), J. Appl. Phys. Jpn, 30 (4) (1961) 221. 5 T. Matuyama and H. Suzuki, A study of the polishing of electric contacts, Trans. JSME, 21 (106) (1955) 450. 6 T. Matuyama and H. Suzuki, A study of the polishing of electric contacts (succeeding report), Trans. JSME, 24 (146) (1958) 692. 7 T. Tsukizoe and T. Hisakado, On the mechanism of heat transfer between metal surfaces in contact (Part 1: Theoretical analysis and experimental justification of theory), Heat Transfer - Jpn. Res., 1 (1) (1972) 104. 8 S. Harada and K. Mano, The effects of surface roughness on contact resistance. In Fundamental studies on electrical contact resistance, Part 8, The Record of Electrical and Communication Engineering Conversazione, TGhoku University, Vol. 36, part 3, 1967, p. 344. 9 S. Harada and K. Mano, The effects of the surface roughness on the contact resistance of the flat contact. In Fundamental studies on electrical contact resistance, Part 9, The Record of Electrical and Communication Engineering Conversazione, Tohoku University, Vol. 36, part 4, 1967, p. 376. 10 J. R. Osias and J. H. Tripp, Mechanical disruption of surface films on metals, Wear, 9 (1966) 388. 11 T. Hisakado and T. Tsukizoe, Effects of distribution of surface slopes and flow pressures of contact asperities on contact between solid surfaces, Wear, 30 (1974) 213. 12 T. Tsukizoe, T. Hisakado and Y. Nakajima, On the mechanism of contact between sliding metal surfaces, Preprint of the 44th General Conf. (Kansai), JSME, no. 44 - 11, 1969, p. 81. 13 T. Hisakado, On the mechanism of contact between solid surfaces (1st Rep., The initial separation and the distributions of slopes of facets on surface and on the profile curve), Bull. JSME, 12 (54) (1969) 1519. 14 T. R. Thomas and S. D. Probert, Thermal contact resistance: the directional effect and other problems, Int. J. Heat Mass Transfer, 13 (1970) 789. 15 J. J. Henry, Thermal contact resistance, A. E. C. Rep. no. MIT-2079-2, 1964, p. 69. 16 T. Tsukizoe and T. Hisakado, On the mechanism of heat transfer between metal surfaces in contact (2nd Rep., Thermal contact resistance between metal surfaces in vacuum), Heat Transfer - Jpn. Res., 1 (2) (1972) 23. 17 T. Hisakado, On the mechanism of contact between solid surfaces (5th Rep., Analysis taking elastic deformations of asperities into account), Trans. JSME, 38 (314) (1972) 2657.