Applied Energy 22 (1986) 3184
Thermal Resistances of Pressed Contacts B. Snaith, S. D. Probert and P. W. O'Callaghan Applied Energy Group, School of Mechanical Engineering, Cranfield Institute of Technology, Bedford MK43 0AL (Great Britain)
SUMMARY Contacts between two surfaces under mechanical load occur frequently in numerous structures, yet their behaviour when heat is transferred across them is usually not fully appreciated. The present review is aimed at assisting the designer in understanding the intricacy of this problem and the factors involved.
NOMENCLATURE Ac Ag
A n, a, d
A r
Macroscopic area of contact resulting from pure elastic deformations (m2). Projected gap area of the considered pressed contact (hg = A n  hr) (m2). Nominal and real areas, respectively, of the pressed contact
(mS).
Microcontact spot radius and its mean value, respectively
(m). a' b,/; b' C C'
Macroscopiccontact spot radius (m). Microscopic heatflow channel radius and its mean value, respectively (m). Macroscopic heatflow channel radius (m). Thermal conductance of the pressed contact (W K  1). Thermal conductance of unit nominal area of the contacting interface (W m  2 K  1).
31 Applied Energy 03062619/86/$0350 © Elsevier Applied Science Publishers Ltd, England, 1986. Printed in Great Britain
32
Cv
D d d' E E' F F'
f() G1, G2
g( ) H h K k km L M Mo m mo, m2, m 4
m !
N P p* P
O.
B. Snaith, S. D. Probert, P. W. O'Callaghan
Specific heat at constant volume of the interstitial gas between the contacting solids (J kg1 K1). Coefficient of the 'law' of WiedmannFranzLorenz (V 2 K  2). Specimen diameter (m). Flatness deviation (m). Young's modulus of elasticity (N m2). Hertzian elastic modulus, defined by eqn (13) (N m2). Elastic conformity modulus, defined by eqn (60). A constant characteristic of a material (see eqn (14)). A characteristic of the contact (see eqn (62)). Function of the parameter in the brackets. Dimensionless geometric factors for surfaces 1 and 2, respectively. Constriction alleviation factor. Meyer's hardness for a solid specimen (i.e. the ratio of the applied load to the projected indentation area) (N m2). Peaktovalley height (m). Constant characteristic of a pressed contact (see eqn (17)). Thermal conductivity (W m  1 K  1). Harmonic mean thermal conductivity, defined by eqn (32) (Wm1 Kl). Mean free path of a gaseous molecule (m). Microindentation hardness (N m 2). Molecular weight of the interfacial fluid. Index of the applied mechanical load in eqn (17). Surface moments (first three moments of the power spectral density of a surface profile) (m 2, dimensionless and m  2, respectively). Rootmeansquare value for the surface profile slope radians. Number of asperity bridges between the contacting surfaces. Mechanical loading, i.e. the load per unit nominal area applied to the contact ( = W/An) (N m2). Dimensionless load ( = P/E). Pressure of the ambient environment (N m2). Steadystate rate of heat flux crossing an interface in the direction of the contacting specimens' axes (W).
Thermal resistances of pressed contacts
R R' R* Rg r
S~ S~j S
T,T' T* l lt
u
z,.v, V
W X
Z
33
Steadystate rate of heat flux passing through unit area of nominal contact, crossing in the direction of the specimens' axes (W m  2). Thermal resistance of a pressed contact (K W1). Thermal contact resistance of unit nominal area of contact interface, as defined by eqn (1) (Km 2 W1). Nondimensional thermal contact resistance, as defined by eqn (26). Universal gas constant ( = 83143 x 10 3 N m kg ~K  1). Contact radial distance (m). Ultimate compressive strength (N m2). Distance between two generalised true contacts i a n d j (m). Surface parameter, defined by eqn (71) (m). Temperatures (K). Dimensionless temperature ( = T~). lnterfacial gap between dissimilar materials in contact, as defined by eqn (23) (m). Duration of the applied mechanical load (h). Mean plane separation between the surfaces in contact (m). Longitudinal and transverse velocities, respectively, of a phonon (m s 1). 1  0.8968/fl (see eqn (54)). Mechanical load applied in the direction normal to the contact plane (N). Constriction ratio; Xms a/b, Xml a'/b'. A factor (see eqn (65)). =
=
Greek symbols
~acc Ft, Ft
A( ) V
,5
Coefficient of thermal expansion (K ~). Accommodation parameter (see eqn (5)). Bandwidth parameter (see eqn (55)). Longitudinal and transverse transmission coefficients of a phonon, respectively. Ratio of the principal specific heats of the surrounding gas, i.e. that at constant pressure to that at constant volume. Difference between two values of the parameter (). Directional bias index, defined by eqn (18). Mean effective interfacial gap (m).
B. Snaith, S. D. Probert, P. W. O'Callaghan
34
tl 2 #, #' V
P p' t7 0 r T
(t3 (t3 Z or tan 0 f~ 03
Equivalent gap thickness (see eqn (70)) (m). Surface emissivity. Proportionality constant, used in eqn (59). Ratio of real contact area to macro contact area ( = Ar/Ac). Pitch between adjacent spirals on a turned surface (m). Plasticity indices for a contacting solid (see eqns (45) and (54), respectively). The Poisson ratio for a contacting solid. Radius of curvature of a contacting surface ( m  1). Density of the interstitial gas (kg m3). Electrical resistivity (ohm m  1). Rootmeansquare surface roughness parameter (m). StefanBoltzmann constant ( = 5.67 × 10  8 W m  2 K  4). Surface transmissivity. Defined by eqn (38). Normal probability function (see eqn (40)). Work function (eV). Normalised arithmetic mean surface slope ( = qJ/a). Arithmetic mean profile slope (radians). = E'x//mT/rt (N m  2). Waviness number, defined by eqn (12).
Superscripts Nondimensional version of the parameter stated.
Subscripts d e ep f fm g m max
min ml mod
Disc. Elastic. Elastoplastic. Fluid. Free molecular. Gas. Arithmetic mean value. Maximum value. Minimum value. Macroscopic scale. Modified value of the considered parameter to allow for the radiation contribution.
Thermal resistances of pressed contacts ms o
P rad s sf t or tot 1,2 12,21
35
Microscopic scale. Ambient conditions (unless stated otherwise). Plastic. Radiation. Relating to solid (unless stated otherwise). Relating to surface film. Total unless otherwise stated. Relating, respectively, to the contacting surface materials 1 and 2 (unless stated otherwise). Direction of heat flow, respectively, from material 1 to 2, and vice versa.
Abbreviations
FD RD TCR
Flatness deviation. Roughness deviation. Thermal resistance of a pressed contact.
THE T H E R M A L CONTACT PROBLEM When heat flows across the interface between two solids which are pressed together, a relatively high thermal resistance is encountered in the region of the contact. This resistance arises because real contact ensues over an area, which only need be large enough to support the total mechanical load imposed, and much of the transferred heat is constrained to flow through the zones of real contact. In many practical engineering situations, the true contact area is only a small fraction (,~ 103) of the nominal contact area. So the heatflow path lengths are considerably increased due to the presence of the pressed contact. Despite the many pertinent theoretical and experimental investigations, the heat transfer mechanisms across contacts and their dependences upon applied conditions are still not understood completely. There exists no fully comprehensive theory to predict, with hig~ accuracy, the contact resistance between even nominally flat engineering surfaces. The complexity of the problem may be realised from the schematic representation of the behaviour of a commonplace structure subjected to mechanical and thermal loads, as shown in Fig. 1.1 In this paper a review of steadystate heat transfers across pressed
•..
THERMAL APPLIEDLOADING
v
_I I
MECHANICAL CONSTRAINTS
MECHANICAL APPLED LOADING "~ .MATERIAL PROPERTIES ASSEMBLY DIMENSIONS SURFACETOPOGRAPHY (ROUGHNESS+ WAVINESS)
I I
I
coMPLEx
]
Fig. 1.
I
ISU~:ACEFI,MS OR OX,DES
Factors affecting heat transfers through a bolted assembly.
INTERACTION
CONTACTS I
CONFIGURATION:~ I BY THE DISTRIBUTIONOF MACRO I ~ METALLICCONTACTS MICRO ~ I INTERSTITIALMEDIA,
N O N  LINEAR T H E R M A L  MECHANICAL
TEMPERATUREDEFORMATION J
I
THERMAL DEFORMATION OF ASSEMBLYELEMENTS
I
THERMAL STRESSES
J MECHANICAL ]INTERACTION
!1 PRESSURE DISTRIBUTION OVER CONTACT INTERFACEDUE TO THERMALAND MECHANICAL LOADINGS
h=
Thermal resistances of pressed contacts
37
contacts is presented. In particular, the TCR which can be defined as the ratio of the average steadystate interfacial temperature drop across the contact to the corresponding mean rate of heat flow crossing unit nominal area of the interface, i.e. R' 
AT
4
(1)
is considered. Note that AT is the difference between the mean temperatures of the contacting surfaces and these are obtained by linear extrapolations of the temperature distributions in the two (well insulated laterally) contacting specimens. The thermal contact conductance, C', is the reciprocal of the TCR.
TYPICAL OCCURRENCES OF THE PRESSED THERMALCONTACT PROBLEM
TCR and exergy degradation The economic importance of controlling the TCR may be appreciated by considering energy degradation according to temperature level. Thermal exergy is the availability of heat to perform useful work, and this decreases the closer one approaches the temperature of the ambient environment. 2 To extract the maximum utility from a degrading energy chain, energy losses to the environment and unnecessary exergetic potential decays must be minimised. Temperature reductions will occur in a system, which is above ambient temperature, as the heat progresses through the utilisation systems. In structural systems, bolted or some form of mechanical joints often occur and these can have high thermal resistances, so inhibiting the required movements of heat. An example of the importance of minimising temperature drops along desirable heat paths occurs in a solardriven power generator. 3 The system utilises evacuated heatpipes to boil a refrigerant in a manifold, the vapour from which then drives a multivane expander turbinesee Fig. 2. The minimisation of the TCRs for the ballandsocket joints connecting the solar collectors to the boiler manifold is a crucial design requirement. Figure 3 shows how increasing the temperature drop across the joint, i.e. as the contact deteriorates, reduces the overall system efficiency.
38
B. Snaith, S. D. Probert, P. W. O'Callaghan WATERCOOLED CONDENSER
/ MULTI V EXPAN( TURB
Fig. 2.
>_ 16 t~
t
Solarenergy driven power generator.
AT = TEMPERATUREDROPACROSS A BALL AND SOCKET JOINT ( C )
tt: ~
OPER ING 12
II
12
13
14
15
16
17
18
19
HEAT PIPE TEMPERATURE ('I') AMBIENT TEMPERATURE To
Fig. 3.
Effect of temperature d r o p across a ballandsocket joint of a solardriven power generator system.
Thermal resistances of pressed contacts
39
JUNCTIONTEMPERATURE
THERMAL RESISTANCE BETWEEN CHIPAND ITS PAEKABE
PACKAGE CASE TEMPERATURE
THERMAL RESISTANCE BETWEEN PACKAGEAND BOARD
BOARD TEMPERATURE
THERMAL RESISTANCE BETWEEN BOARD AND HEAT SINK (m AMBIENT ENVIRONMENT} AMBIENT T EMPER#~TUIqE
Fig. 4.
Schematic thermal resistances for a simple electronic system.
Microelectronics Recently the miniaturisation of electronic components has led to higher heat flux densities becoming more common. Simultaneously, specified maximum junction temperatures and the permitted differences in junction temperatures between electricallyconnected devices have decreased. 4 These are desirable trends because equipment reliability is reduced as its temperature rises. Figure 4 illustrates a series of thermal resistances in a simple system comprising of a dualinline package on a printedcircuit board in free air. Cooling lowers the heights of the temperature steps in the thermal ladder. Wherever interfaces are present in the system, each contact will contribute to the overall thermal resistance. Therefore reducing the interfacial resistances leads to a decrease in the cooling capacity needed.
Space vehicles In the electronics and power systems of space vehicles and communication satellites, high power densities are used in order to reduce vehicle sizes and weights. 5 Here, due to the vacuum environment, internal heat
B. Snaith, S. D. Probert, P. W.
40
EVAPORATIONSECTION (HEAT PIPE}
O'Callaghan
FLUID HEAT EXCHANGER
i"
RADIATOR
"i"
.:
BOOM SHELL ~ 
~
689 kNm 2
.. RADIATORFIN
I
•
ii
ill
:.
't
t
t
t
I,
t
MECHANICALPRESSURECLAMPINGCONCEPT THERHAL ACTUATIONEXPANSION BOLT PRESSUREIRELEASE PIECHANISM
BOOM
REA LOADING = 689 kNm2
'DRY' ATTACHHENT INTERFACE . i MULTILEGHEAT/ PIPE EVAPORATOR RADIATORFIN
Fig. 5.
Loadedcontact heat exchanger. 6
transfers occur mainly by conduction through solids. To keep the internal temperatures of such vehicles within acceptable limits, low resistance heatflow paths from the heat sources within the vehicle to the vehicle's skin must be provided. To achieve this, a knowledge of the TCRs for the various types of similar and dissimilar material junctions involved is required. One of the major challenges in the design of a heatmanagement system for future space stations is the requirement of having to transfer up to 100kW out of the heattransport circuit and into the radiator system through a 'contact' heat exchanger.6 Figure 5 illustrates a prototype fiat contact heat exchanger for this purpose; it incorporates a mechanically loaded dry interface where intimate thermal contact across the joint is a necessity.
Thermal resistances of pressed contacts
41
Aircraft structures At the design stage of a supersonic aircraft, the necessary estimates of temperature distributions and the related thermal stresses developed require a detailed knowledge of the contact resistances between the structural components. 7'8
Nuclear fuel cans Large temperature drops, across the interfaces between a nuclearfuel element and its cladding, will have an adverse effect on the system's performance. 9 So these contact resistances must be kept small.
Storage and transportation of cryogenic liquids Multiple thermal contacts in series, e.g. in the form of multilayered stacks of very hard, thin metallic discs in compression, can be used as mechanically strong, thermally insulating supports. ~°'11
Measurement applications When measuring the thermaltransport properties of a material, the temperature drops between the location being investigated and the sensing probe should be minimised. For thermalconductivity measurements, the thermal comparator method ~2,2 3 utilises the fact that the heat is constricted to flow through the asperity bridges between the surfaces of the probe and the material with which it has been pressed into contact. Also the probe technique, for the measurement of the effective thermal conductivities of constructional materials, 14 requires the thermal contact characteristics of the probemedia interface to be known accurately. A technique for the measurement of surface hardness, through the observation of thermal contact resistance, has been developed. 15,~6 MECHANISMS FOR HEAT TRANSFERS ACROSS SOLID METAL/SOLID METAL INTERFACES The modes of heat transfer across the interface are (i) solid conduction through the true contact areas; (ii) convection and conduction through the interstitial fluid; and (iii) direct thermal radiation exchanges between the nominally contacting surfaces, which are not in true contact.
42
B. Snaith, S. D. Probert, P. W. O'Callaghan
Heat transfers by conduction through a contact spot These involve metaltometal conduction and also conduction via any surface contaminant films if present. For a single circular contact spot, of radius a, the thermal conductance is given 17 by C s = 2ak
(2)
If true metalmetal contact of two peaks, on the two surfaces which are being pressed together, is prevented by the intervening presence of an interfacial film, a contact spot will still occur. The contact conductance will, in general, be reduced because of the lower thermal conductivity, k, of the tarnish film compared with that of the underlying metal. Heat transfers through the interfacial fluid For configurations involving a fluid flow with a Grashof number (based on the mean clearance between the nominally contacting surfaces) below 2000, natural convection is alleged to be entirely suppressed.XS'19 This will occur for most practical situations where the nominal gap between the contact surfaces is sufficiently small ( ~ 0.1 mm). Then the heat transfers, due to the presence of the fluid, can be considered as occurring only by conduction. 20  22 The proportion of heat transferred by conduction across the fluid space, compared with that by solid conduction, depends upon the ratio of the thermal conductivities of the fluid and solid. For a gaseous interfacial fluid with a thermal conductivity considerably lower than that of the metal specimens in contact, the heat flows will tend to converge to the solid contacting bridges. Thus the gain by employing a larger nominal contact area (i.e. that involving both gasmetal plus metalmetal interfaces) will be offset by the low thermal conductivity of the gas itself, as well as by the increased constrictions of the heat flows which probably then ensue towards and from the metal bridges. However, if the interfacial fluid is a liquid with a thermal conductivity higher than that of the contacting solid specimens, it can be expected that most of the heat flow will be across the fluid region. Owing to the surface irregularities of the contacting specimens, the thickness of the gap between them which is filled by the fluid will vary across the nominal (i.e. apparent) contact area. In order to be able to
Thermal resistances of pressed contacts
43
handle this problem theoretically, an equivalent mean effective gap, iS, for the fluid is usually deduced. If it can be assumed that the heat transfer through the fluid space is essentially onedimensional, a value for the effective fluid conductance can be obtained 20.2 3 25 via 
kf
C'f = ~
(3)
If the interstitial fluid is a gas, this equation holds true at pressures above the free molecular region, where the thermal conductivity of a gas is independent of pressure. However, when the gas pressure at the interface is sufficiently low, such that the mean free path of the gaseous molecules, L, is much larger than (5 (i.e. the Knudsen number, L/6 >>1), then the heat will be transferred by free molecular conduction. 5,2s For large Knudsen numbers, the kinetic theory of gases yields the following expression for the steadystate rate of heat transfer in the free molecular regime between two parallel surfaces: 25,26 (3'+ 1"~ ~ ~ p 0fro = 0tacc~iJ48~
hgl(Z2 Zl)
(4)
where the accommodation parameter 1
(5)
+

1
and the free molecular conductance per unit nominal contact area of the contacting interface is
qfm C'fm AT
(6)
Heat transfers by direct radiation
The effective conductance per unit nominal contact area of the contacting interface due to steadystate radiation exchanges between the two parallel surfaces at absolute temperatures T 1 and T 2 is 2°
Ctrad= OJ(_
~1~2 +

.~(Z4Z4~
1 2/\T1
T2 ]
(7)
44
B. Snaith, S. D. Probert, P. W. O'Callaghan
Radiation exchanges become relatively greater (compared with other contributions for the same AT) at high absolute temperatures. This effect may be included by modifying the magnitude of kf used in eqn (3) to eat__,
)T 3
kfm°d=kf +4°"~ ~1 +E~ ~1~2
(8)
where Tm is the arithmetic mean of.T 1 and T2.24'27 Total contact conductance
If we assume that each mode of heat transfer acts in parallel, then the total contact conductance, Ct, can be expressed 5,20,24 as C t ~ C s ~ C f + Cra d
(9)
This expression is not strictly correct because all three modes of heat transfer are interdependent, but it is an acceptable approximation that introduces only a small error for all but extremely high temperature conditions. The total contact resistance is given by 1
1
1
1
Relative importance of each transport mode
The magnitude of the convective heat transfer current between pressed contacts can usually be considered insignificant because of the narrowness of the interfacial gap. Also the radiation contribution is negligible at near room temperatures, provided that the nominal area of contact is small. 21,22,27,28 It has been shown that for metallic contacts the radiation contribution seldom exceeds 2~o of the overall conductance at temperatures less than 900 K . 21'29 Fluid conductance may be important if the contact is subjected to only low mechanical loadings in an air environment, i.e. when nearly all the heat transfers occur across the air gaps. The thermal conductivity of the interfacial fluid will control the proportion of heat transferred through the fluid compared with that conducted by the solid contact. In general,
Thermal resistances of pressed contacts
45
the solid conduction contribution will become more important as the loading increases. In high vacuum conditions, solid conduction becomes the dominant mode of heat transfer because the gaseous free molecular conduction is then negligible. 29"3°
MECHANISMS OF SOLID CONTACT C O N D U C T A N C E In the absence of heat transmissions by radiation, fluid conduction and convection, the heat flux would be restricted to that occurring by solid conduction through the actual contacts (i.e. metallic asperity or metaloxide bridges between the surfaces). The heat transfer resistance of a contacting system may be considered to be due to three components: (i) The normal resistance encountered by the heat flux in passing through those regions of the contacting specimens outside the constriction zones. (ii) Macroscopic constriction towards and away from (according to the heat flux direction) the larger areas of true contact: see Fig. 6(a). This is dependent upon the longer wavelengths of the surface undulations and the resulting distribution of contact spot agglomerations. The magnitude of the macroscopic contact area is governed by the elastic deformation of the contacting members. (iii) Microscopic constrictions of the heat flow paths (within each macroscopic zone) towards or away from the microcontact areas: see Fig. 6(b). These depend upon the surface roughness. Most 'asmade' surface asperities will deform plastically initially at the lightest loadings. 31 The magnitude of the true area of contact is dictated by the hardnesses of the materials which are pressed together and the machining processes to which either (or both) of the contacting surfaces have been exposedsee Fig. 7. The contributions of the macroscopic modes of constriction to the overall resistance also depend upon the topographies of the surfaces involved and upon the loading history of the contact. Thomas and Sayles 32 considered the surface waviness and roughness in terms of the bandwidths of a continuous spectrum of surface wavelengths. Their Hertzian analysis showed that the ratio of the elastic contact radius,
46
B. Snaith, S. D. Probert, P. W. O'Callaghan
  SOLID 1 VACUUM SOLID 2
(O)
i
\
!
MACROSCOPIC CONSTRICTIONS {fulLsize view )
i
X
VACUUe VACUUM SDLID 2
(b)
Fig. 6.
MICROSCOPICCONSTRICTIONS (highly mognified view)
Heat fluxes across an interface.
1.6ram
Fig. 7.
Macro and microcontact impressions produced on a soft, optically flat copper surface by pressing a hard invarsteel surface against it.
Thermal resistances of pressed contacts
47
r, to the flow channel radius, b, is given by /
 0"44(o)) 1/3
(11)
where to is defined as the waviness number W (12)
to = E ' a d
W being the applied load, a the rms roughness of the surface of diameter d, and E' the effective Hertzian elastic modulus, where
1
1v~
E'
E1
+
lv~ E2
(13)
This analysis showed that the effect of waviness, and thus of the macroscopic resistance, predominates when ¢0 < 1, i.e. corresponding to the case where r ,~ b. The authors concluded that (i) this condition applied to all practical situations, and so (ii) the waviness must be taken into account when calculating the TCR. Clausing and Chao 33 concluded, from experiments, that the macroconstriction resistance is the dominant contribution, but Mikic et al. s4 found that the consideration of the macroscopic resistance alone is insufficient to explain the experimental results observed for rough surfaces. Mikic et al. proposed that allowances for b o t h macroscopic and microscopic constrictions of the heat flow paths should be made. In general, the former is predominant under low loadings whereas the latter becomes more important at high loadings, i.e. after elastic flattening has occurred. The loading history of a pressed contact affects its TCR. The initial contact deformation is predominantly plastic, but subsequently the contact behaves elastoplastically. 31'35 Unloading and all subsequent contact pressure variations at loads lower than the maximum previously encountered will produce only elastic deformations. 24"36'37 For a freshly assembled contact, under a relatively small load, the resistance behaviour will be determined by the microcontacts formed between the few high peaks present. They will be expected to yield plastically during the first loading, thereby bringing lower level asperities into contact until the loading stress is accommodated elastically by the substrate. The elastic strains so produced recover elastically during unloading. However, if plastic deformation and work hardening persists deep into the material, the elastic recovery can be constrained by the plastically deformed 'shell' at the surface.
48
B. Snaith, S. D. Probert, P. W. O'Callaghan
I N F L U E N T I A L P A R A M E T E R S A F F E C T I N G T H E TCR The more important of these numerous factors will now be considered.
Mean interfacial temperature For a pressed contact between similar materials, under high vacuum conditions, the thermal resistance at a specified constant mechanical loading will decrease as the mean interfacial temperature increases. 7,24'33'38 However, for metalmetal contacts, the effect is not particularly marked in the 0 ° to 100 °C temperature range. 21'a9'4° The effect is attributed primarily to the decreasing material hardness as the temperatures of the specimens rise, and thus greater true metaltometal contact areas ensue. The softening process, which occurs by atomic diffusion, is a function of temperature and time; 24 it has been described by
I("
H=H o 1F
~loget'+35~29"8
)1
(14)
where T is the absolute temperature at which F, a constant characteristic of the material under test, is determined: H and H o are the Meyer hardnesses at temperature T', the former being after t' hours and the latter after 1/180 hours of application of the mechanical load, respectively. Temperature changes will also affect other material properties such as the thermal conductivity, elastic modulus and yield strength. The TCR reduction which occurs upon increasing the temperature of a pressed contact (continuously under a constant mechanical load) will become more pronounced at higher temperatures, i.e. as the radiation heat transfer contribution becomes more significant, 41 especially with high emissivity surfaces in contact. When the contacts are formed of dissimilar materials (especially for those with large differences between their thermal expansion coefficients), then the thermal resistance may increase upon raising the mean interfacial temperature.38 This is due to thermal strain of the contacting specimen with the greater expansion coefficient causing geometric changes in the contact area: see the subsequent section on thermal rectification. If any gaseous or solid medium is present at the interface, then the contact resistance may increase or decrease, as the interfacial temperature rises, depending upon the thermophysical properties of the medium. For
Thermal resistances of pressed contacts
49
an interfacial gas, its thermal conductivity, kg, will increase with temperature because it is proportional to the square root of its absolute temperature, 2° i.e. (97  5) , a/aTT kg  2x~. ~ c,p L_ / y2(15)
Heat flux and duration of heating If the ambient temperature remains constant, then an increase of the heat flux across a contact will usually lead to a subsequent rise in its mean temperature. However, the contact conductance depends only slightly upon the heat flux magnitude. 42'43 The effect of the heating duration on the thermal contact conductance is not susceptible to simple analysis and the relevant literature is limited. Besides surface oxide growth, surface hardness is the only other influential parameter upon which the contact conductance is believed to vary with time. Cetinkale and Fishenden 24 found experimentally that H = Ho(1  F log e 180t')
(16)
for ~ o hour < t' < 24 hours, where t' is the duration of the applied load, Ho the original Meyers hardness at zero time and F a constant characteristic of the predominant metal. Thus during a period of heating, the hardnesses of the metal contacting surfaces decrease with the duration of the test load, so leading to an enhanced contact conductance. 4247 Increases (by up to 25 ~o) of the contact conductance with time over a 1.5 hour heating period have been observed for stainlesssteel interfaces in high vacua. 29 For specimens heated in air, oxide films on the metal surfaces will grow with time; 17 these will tend to reduce (relative to what would have otherwise occurred) the thermal contact conductance of the subsequently formed contact between the specimens. 48'49 It water vapour is present during the oxidisation period, the thermal contact conductance may increase. 50
Increasing load and hysteresis upon loading The dependence of the TCR upon the applied load is the most sensitive of all the parametric effects. The conductance will rise as the applied load is increased, in accordance with the general expression
C = KW"
(17)
50
B. Snaith, S. D. Probert, P. W. O'Callaghan
where Wis the applied load and Kis a constant depending on the form of the contacting surfaces and the bulk material properties. The index m is dependent upon the finish of the contacting surfaces. In addition, the loading and temperature histories of the assembly will affect the magnitudes of K and m. Most theoretical analyses leading to a relationship of the form of eqn (17), for the initial loading sequence, are based on assuming direct proportionality between the real area of contact and the applied mechanical load, i.e. the index m equals unity. For typical, nominally flat, rough surfaces pressed together producing multiple contact spots, this will hold true whether the deformation mode is elastic orplastic. 31,36 This arises from the proportionality that exists between the total real contact area and load being due to a relatively constant size for the average contact spot; i.e. as W increases, all that changes is the number of contact spots. 31'51's2 For very smooth, wavy surfaces in contact, the effect of applying the load is less marked: the real area of contact is proportional to the 2/3 power of the applied load, and is well described by the classic smoothsurface Hertzian theory of elastic contact. 36'53'54 Accordingly, it is to be expected theoretically that for a fresh contact under increasing load the index m in eqn (17) will vary, depending upon the surface topography, from 0.67 to unity. Experimental values of m within and outside this range have been reported. Values as low as 0.3 have been recorded, but typical values range from 0.670.96. 55 57 The hysteresis effect, by which the loading of a contact produces an apparent permanent reduction in the TCR, has been observed by many investigators. 28'33'42'43'46'586° Hypotheses proposed to explain this phenomenon include the following. Elastic relaxation When two surfaces are pressed together, the bulk metal sublayers, immediately below the plastically deforming asperities, will experience elastic deformations. 21,61 During unloading elastic relaxation of the sublayers will occur, which upon subsequent reloading will cause the true contact area to increase by bringing the surfaces closer together and thus forming new contact areas. Cold weMing The formation of a cold weld may occur when clean metal surfaces are pressed together. 17,62 The generation of such contacts is the basis for the
Thermal resistances of pressed contacts
51
cold (or press) welding process that has been practised since the late 1940s. The spots where cold welding has occurred (because of the rupture of any surface films present) will tend to remain in contact, even when loading is reduced. Thus the true contact area is expected to be higher (and therefore a lower thermal contact resistance ensues) than the corresponding value at the same load during the initial loading part of the cycle.
Influence of contact duration Cordier 43 investigated the influence of the period of application of the load on the contact resistance during loading and unloading cycles. It was suggested that if sufficient time were allowed to elapse between the measurements at each particular load, the hysteresis effect would disappear. That is, there would be a unique curve for each pressed interface of the form R = f ( W ) , which in practice will be a good description of behaviour only when stationary mechanical conditions have been attained. This is supported by other investigations 42'46'4v and would seem to be explained by creepstress relaxation.
Load cycling and thermal cycling The TCR can be reduced by load cycling, which will 'bed in' the contacts 6o and also scour any contaminant films on the surfaces by lateral movements, thereby producing more metaltometal contacts. Differential thermal contractions or expansions of the two surfaces in contact can aid 'beddingin', causing the real area of contact to increase and thereby reduce the contact resistance. 63'64 The mechanism is similar to that occurring as a result of load cycling in that lateral displacements will rupture any films present on the surfaces. Tests have suggested that although surface films will remain intact under high axial loads, they will fracture under tangential loads less than that required to produce macrosliding. 6 5  6 7 Alternatively, particles may be sheared from the surface peaks, and if the testing is performed in air such 'debris' will oxidise and so, when further relative movements of the contacting surfaces ensue, may increase the contact resistance.
52
B. Snaith, S. D. Probert, P. W. O'Callaghan
Nominal area of contact
Experimental evidence is available to show that the real contact area, and thus the TCR component due to conduction through the asperity bridges, is almost independent of the nominal contact area. 31.6s It is expected that this will only hold as long as all the prospective contact spots are made and the contacting specimens are of similar materials. The same is not true, however, of dissimilar materials in contact, where differential bowing of the surfaces produces a different interfacial geometry. When trying to establish dimensionless correlations for TCR in vacuo, it was noted that the nominal contact area did not play a significant role. 55 However, if any interfacial fluid is present, and the temperatures are such that significant radiation exchanges between the contacting surfaces occur, then the nominal contact area must be considered. Surface contaminant films
Many metals suffer corrosion due to an aggressive environment, galvanic reactions or other causes. The most common form, namely oxidation, results in a thin film of oxide growing on the surface of a metal specimen exposed to, say, fresh or sea water. Such films can increase the electrical contact resistance by several orders of magnitude 17 and will, to a lesser degree, affect the TCR. Experimental research has shown that the TCR will increase if oxide films are present. 9'48'5°'69  ~2 The presence of oxide films leads to more pronounced effects for smoother s u r f a c e s 49'5°'70'72"73 and at lower loads. 4s'72 Contacts experiencing higher loads and/or having rougher surfaces tend to fracture the surface films by plastic deformation and microslip, 66 thus increasing the true contact area. As well as applied load and the surface roughnesses of the contacting surfaces, the TCR will also depend upon the thermal conductivities and thicknesses of the oxide layers. For mating surfaces covered by a film with a lower thermal conductivity than that of the bulk material, it is suggested that the TCR would be altered in a manner analogous to that occurring due to the presence of low conductivity interfacial shims (see a subsequent section). But can the thermal conductivity of an extremely thin oxide film be measured with sufficient accuracy? A rise of the TCR as a result of increasing the film thickness has been reported, 9"69'72 although the existence of environmental water vapour during the oxidation period could cause a reduction in the resistance. 50 For various temperatures and oxidation periods, the growths of oxide films may be estimated.17'74
Thermal resistances of pressed contacts
53
Surface topography Considerable difficulties are encountered when it is attempted to analyse the effects of surface finish on contact resistance. This stems from the problems of defining, and measuring accurately, truly characteristic surface topographical parameters. Although significant advances have been made in this field, no universal standards exist at present. The two most widely, and it must be stressed general, descriptors used for characterising a surface are the average roughness and the flatness deviation.
Surjace roughness The smoother the contacting surfaces, i.e. the lower their average roughnesses, the better the contact and so the lower the TCR.S7.67.73.75 78 If contamination films are present, contacts between rougher surfaces may exhibit lower TCRs due to rupturing of the films. Usually the surface roughness is one of the most important parameters dictating the contact resistance at high loadings 32'34"57 or when very rough surfaces are used. 34 Thus it would appear wise if dimensionless correlations of TCR were to include the average surface roughness as a considered parameter.
Surjace flatness No surface can be made perfectly flat! In practice, the roughness of a surface may be reduced to extremely small values by means of lapping and polishing, but the elimination of flatness deviation/waviness is much more difficult. Investigations indicate that, for practical pressed contacts, the effects on the TCR on longer surface waveforms cannot be neglected, 32"33'48"7s particularly under low loading conditions. 34'5<77 The distribution of macroareas of contact is dictated by the surface waviness and the increase in the number of contact spots with load is governed partially by elastic deformation of these macroareas. Due to difficulties in identifying surface waviness effects, most models for predicting the TCR are limited to nominally fiat surfaces in contact.
Interstitial materials The TCR may be increased or reduced by the insertion of a suitable material layer at the interface. The insert's thermal conductivity and hardness, relative to the values of the corresponding properties of the base
54
B. Snaith, S. D. Probert, P. W. O'Callaghan
materials forming the contact, together with the insert's thickness, will dictate the resulting change of the contact's resistance, v9 84 In general, the resistance of a contact may be reduced by the insertion of a thin layer of high conductivity material which is softer than the base materials. Alternatively, hard, low conductivity, interstitial materials will increase the contact resistance, and this effect may be amplified by employing multilayered stacks 11'85'86 which have been used, in compression, as mechanicallystrong thermal insulators. A wide range of interstitial materialsmetals (solid, porous and mesh forms), plastics, ceramics, greases and oilshave been tested with the aim of achieving the required TCR. 87'88 Direction of heat flow
Electrical rectification is a wellknown phenomenon. However, experimental evidence exists that certain joints may act as thermal rectifiers under suitable conditions. Thermal rectification, if systematically predictable, could be employed profitably in association with energy storage systems and control mechanisms. Several thermal rectifier designs have been proposed.89'9° The directional effect refers to a property of a contact by which it exhibits differing thermal resistances according to the direction of heat flow across the interface. This effect has been observed for dissimilar materials in c o n t a c t 4°'44'56'57'63'9194 and also for similar materials. 42'63 A directional bias index, V, has been defined 9° in order to correlate available evidence. It is W  Rmax  Rmin
g
(18)
max
where Rmax and Rmi. are, respectively, the greater and smaller TCRs (measured at identical mechanical loadings and temperatures) according to the direction of heat flow across the contact. In order to design an effective thermal rectifier, it is desirable for Rma x~ ~ , whereas R m i n ~ 0. However, in practice for the contact assemblies considered, the latter condition cannot be satisfied as yet. The possible mechanisms controlling the value of V are so complicated that it is difficult to deduce even qualitative conclusions as to their dependence upon imposed conditions. Even the direction of greater resistance varies from one investigator to another for what appear to be
Thermal resistances of pressed contacts
55
almost nominally similar contacts. It is suggested that the means adopted for measuring the TCR have in some experiments affected the value and nature of the directional bias. One probable cause is the reversal of the heat flow direction by inverting the actual specimen assemblies relative to its source, so disturbing the geometry of the actual contact. Thus, in such measurements, the contact under investigation should not be moved or disturbedrather only the heat transfer direction should be reversed. Also the use of dissimilar material specimens with high form (i.e. lengthtoradius) ratios, for the contact, can cause different transverse heat leaks from the longitudinal heat flow specimens. This can result in temperature distributions, which indicate an apparent directional b i a s . 91'95'96 The direction of the first application of heat flux to the contact is considered to be important, but is rarely quoted in published reports. Williams 63 concluded that directional effects do exist but will disappear after repeated thermal cycling. This suggests that thermal rectification is not a permanent feature and some of the previous evidence is based upon incomplete testing. Several mechanisms have been proposed to explain the thermal rectification phenomenon.
The electronic hypothesis Moon and Keeler 97 considered theoretically the proposal of Rogers 4° that the directional effect may be due to the potential barrier (produced by interracial oxide layers), which inhibits the electronic heat conduction in one direction across the interface while assisting it in the reverse direction. The ratio of thermal contact conductances in these direct and reverse directions can be expressed a s 40'97 C1~ _ z~2 T 1 621 z21T~_2~exp _
E(
)~1 o'
')]
] \ r 2 T1
(19)
where z =transmissivity, g~ = w o r k function of the metal surface, Zsf = w o r k function of the oxide film, a ' = Boltzmann constant and T = surface's absolute temperature. It is expected that z12  z21 and that Z~ > Zsf. Therefore if T~ > T 2, then C12 > C2~. The work function Z depends upon the type of material and its surface preparation. Hence it is feasible to have a directional bias between similar materials if their surface histories are different42a point sometimes overlooked. 63 Correlation of data via eqn (19) is inhibited by a lack of knowledge of the work functions of the contacting surfaces and their oxides.
56
B. Snaith, S. D. Probert, P. W. O'Callaghan
Surface oxide phenomena Besides the oxide effect implicit in the Moon and Keeler 97 theory, surface films can lead to a directional bias. As discussed in the previous section on surface films, the fracture or distortion of oxide films affects the TCR. Williams 98 suggested that the bias found by Rogers 4° was caused by differential expansions in the plane of contact leading to abrasion of the surface films and thereby improving the true contact area. This tended to occur when the heat flowed from the aluminium to the steel specimen, but not for the reverse direction. 98 Phonon mismatches An expression derived by Little 99 for the thermal contact conductance is 6
1 Ft
2Ft'] T 4
T 4]
where A, is the nominal contact area: Ft, F , are the transmission coefficients of the longitudinal and transverse phonons, respectively; Vt, V, are the speeds, respectively, of the longitudinal and transverse phonons. No directional bias can be deduced from this phonon transport theory if the reciprocity relationship holds, i.e. if
Macrogeometric changes oj the contact area For each direction of heat flow the contacting solids assume different geometric forms, depending upon their coefficients of thermal expansion. Clausing 91 suggested a macroscopic model, in which 'spherical caps' occur, to explain the directional bias. This considered a contact formed by two dissimilar materials 1 and 2, material 2 being the better heat conductor. If the heat flows in a direction from 1 to 2, the thermal stresses cause a larger contact area than that predicted solely as a result of mechanical stresses. Thus a decrease in the macroscopic contact resistance occurs. If the heat flow is reversed, the hot spot thermal strains cause a smaller contact area than that predicted as a result of mechanical stresses alone. Thus an increase in the macroscopic contact resistance ensues. Clausing's model and experimental verification showed that the
Thermal resistances of pressed contacts
57
resistance will always be lower when heat flows from the material with the lower thermal conductivity to that with higher thermal conductivity. This contradicted the results of earlier investigators. 4°'44 Jones et al. 93"1oo,101 presented a more general macrocontact model for rectification of initially nominally flat contacts. They proposed that the axial temperature gradients (in the absence of radial losses) through the two contacting cylindrical specimens cause both contacting surfaces to deform by different amounts into bowed configurations; see Fig. 8. The curvature, p, of each contacting surface being 1
.~
P = q~
(22)
where the hotter contact end becomes convex in form and the relatively cooler one becomes concave. Due to the mismatch between the contacting surfaces, either an annular or an axial (i.e. disc) contact zone is formed, depending upon the direction of heat flow. The constriction of the heat flows to pass through these areas of the contact is greater for an axial type of contact zone, and hence the TCR will be greater. Experimental verification of this Clausing contact model for nominally flat surfaces, 10o and also for convextoflat surface contacts, 94 was achieved. Microgeometric effects Lewis and Perkins 1o2 considered a microscopic model and proposed that changes in the microscopic contact area could be produced at interfaces between materials with differing thermal conductivities according to the direction of heat flow. The actual direction of greater conductance and the magnitude of the bias are dependent upon the contacting surface conditions, i.e. flatness deviation (or waviness) and surface roughness. For contacts involving a nonwavy but rough specimen, where the rootmeansquare roughness is high and flatness deviation low, the higher conductance occurs when the heat flows from the material of the higher conductivity to that of the lower thermal conductivity, whereas the reverse could be expected for contacts having a very low rootmeansquare roughness and high flatness deviation (i.e. equivalent to the macro spherical cap model). This was proposed as the reason for the anomalies between the findings of previous investigators Clausing, Rogers and Barzelay. It was also noted 1°3 that there were inconsistencies in their experiments in that the mean interracial temperatures were not kept constant when the heatflow direction was reversed.
Fig. 8.
(o) ISOTHERHAL
SPECIMEN
~
02 (b) ANNULARCONTACT CONFG I URATO IN
~
(c) DISCCONTACT CONFG I URATO IN
Exaggerated schematic macroscopic model of the geometric changes occurring if a heat flux is imposed, and then its direction reversed, across what was an initially flat contact between two cylindrical specimens when they were isothermal.
INTERFACE
SPECIMEN 1
021
g~
g~
~o
Thermal resistances of pressed contacts
59
Combinations of macro and microgeometric effects Veziroglu and Chandra 1°4 extended the theories of Clausing, 91 Lewis and Perkins 102 and Patel 105 to propose a doubleconstriction (i.e. macro and micro) model. They considered uneven thermal stresses causing changes in the surface geometry by warping and changing the radii of curvature. The interfacial gap, t, at the contact centreline, developed by the changing radii of curvature, is given by
1 [d'~ 2
t= t, 2)
.roq
o~,~]
/
(23)
The criteria for the directional effect are: (a) If t is greater than the order of magnitude of the surface roughness, the contact conductance will be greater when the heat flows from the solid with the smaller ct/k ratio to that with the larger ~/k ratio, i.e. microeffects dominate. (b) If t is less than the order of magnitude of the surface roughness, the reverse condition to (a) will occur, i.e. macroeffects dominate. A more recent analysis by Somerset al.1°6 also indicated the need to consider both macro and microgeometrical effects simultaneously in order to explain thermal rectification. Systematic errors The value and nature of the directional biases reported in some cases depend upon the method of measurement chosen. Differential radial heat leaks through the thermocouple leads can result in temperature measurements which indicate apparent directional biases. 95 There is an obvious need for universally acceptable experimental practices to overcome such problems when measuring the TCR. 1°7 To sum up, it is probable that thermal rectification arises because of thermal strains. However, false indications occur due to systematic errors of measurement. It should be noted that whilst the directional bias may be of advantage in some applications, the mechanisms involved can lead to reductions of the thermal conductance of joints and can complicate predictions of performance.
Summary of parameter effects As has been shown, numerous parameters influence the TCR and the interdependence of their effects greatly enhances the complexity of the
60
B. Snaith, S. D. Probert, P. W. O'Callaghan TABLE 1
Summary of Effects of Varying the Stated Parameters Parameter which is changed Effect on the thermal resistance o[ the pressed contact in." High vacuum ( < 10 3 torr pressure)
Applied mechanical load Decrease: significant increased effect Mean interfacial temperature Expect a decrease; over increased the 0100 °C range little effect
Period of heating extended Applied load (or interface temperature) is cycled Surface films thickened Surface roughness increased A soft, high thermal
conductivity, material is inserted at the interface A hard, low thermal conductivity, material is inserted at the interface.
Expect a decrease as duration increases Decrease
Air or other fluid at atmospheric pressure
Decrease: significant effect May increase or decrease depending upon thermophysical properties of the interfacial fluid. Expect decrease May increase as sheared
particles oxidise Increase with film thickness May decrease as surface films are penetrated Decrease: careful material Decrease (as for in high selection can lead to a vacuum) significant reduction Increase: careful material Increase (as for in high selection can lead to a vacuum) significant increase
Increase Increase
t h e r m a l c o n t a c t p r o b l e m . H o w e v e r , guidelines for these effects o n T C R can be q u o t e d for a pressed c o n t a c t m a d e u p o f s i m i l a r materials with n o m i n a l l y f l a t c o n t a c t i n g surfaces (i.e. n o m a c r o  c o n t a c t s )   s e e T a b l e 1.
EXPERIMENTAL
TECHNIQUES
N o v e l techniques i n c o r p o r a t i n g u l t r a s o n i c transmissions, 1°8 M a c h Z e h n d e r i n t e r f e r o m e t r y , 1°9 a n d transient heat transfers 11° have been applied to d e t e r m i n e the T C R . H o w e v e r , the n o r m a l l y a d o p t e d m e t h o d uses a longitudinal heatflow system c o n t a i n i n g two cylindrical, coaxial m a t i n g specimens. T h e m e a n interfacial t e m p e r a t u r e d r o p (i.e. A T in eqn (1)), u n d e r steadystate c o n d i t i o n s , is o b t a i n e d b y e x t r a p o l a t i n g linearly
Thermal resistances o f pressed contacts
61
to the interface the temperature distributions as indicated by thermojunctions located on the axis of each specimen. When employing this method to measure the TCR, careful considerations must be given to several influential factors, 1°7 such as (i) (ii) (iii) (iv)
reduction of transverse heat losses; the specimen form ratios (i.e. length to radius); accuracy of heat metering; and temperature measurement.
Measurement of an electrical contact resistance is easier than its thermal counterpart: the periods to attain a required steady state at the contact are very much shorter (i.e. milliseconds rather than several hours) and the assembly is simple to insulate electrically. It is known that there is a relationship, j b r metals, between the thermal conductivity, k, and the electrical resistivity, fi, namely the WiedmannFranzLorenz 'law', 17 ~k = D T
(24)
where, for temperatures above 200 K, D  24 x 10  8 V 2 oC  2. From this 'law' the TCR can be treated, with certain reservations, as analogous to electrical contact resistance. However, because of the presence of surface films, which are electrically insulating though thermally conducting (i.e. permitting a flow of phonons while free electron movements are inhibited), and modes of heat transfer other than solid conductance (namely direct radiation across the interface and conduction through interfacial gas), appreciable departures from the above 'law' are to be expected 62 and have been found even for metalmetal contacts. 68,111 The electricalthermal analogue can apply exactly only when the perfectly clean metalmetal contact is in a high vacuum ( ,~ 10 6 torr or better) and at temperatures where radiation exchange effects are negligible.
A N A L Y T I C A L P R E D I C T I O N S OF T H E T H E R M A L RESISTANCE OF PRESSED CONTACTS Theoretical studies of the steadystate heat flows between solid bodies have been based on considering the flow of heat through a single microscopic contact and the adjacent regions of the solids. The resistance resulting from a single asperity bridge is an indication of the additional temperature drop due to the presence of the constriction of the heat flow
62
B. Snaith, S. D. Probert, P. W. O'Callaghan
through the contact spot. Single contact spots of noncircular form (e.g. annular, rectangular, triangular, 112  1~7 conical ~~8 and twodimensional rectangular t19) subjected to different boundary conditions have been analysed. The simplest form, and the one from which most analyses stem, is the circular disc. The solution of the constriction resistance problem assuming a true contact disc of radius a, bounded by a semiinfinite conductor, is given 17 by 1
(25)
Rd  4 a k
This was obtained by a direct analytical solution of the Laplace equation on the basis of the electricalthermal analogy (which will only apply under the conditions stated in the previous section on experimental techniques). The total heatflow constriction, i.e. on both sides of the contact bridge, will be twice the value given by eqn (25). Where the actual area of contact is a very small proportion of the apparent contact area, this disc type of constriction would be expected. A more appropriate model is the contact disc feeding into or being fed by a semiinfinite cylinder of radius b (see Fig. 9). The constriction resistance, R c, for this model is divided by the disc constriction resistance, Rd, to obtain the nondimensional resistance R*  Rc Rd
(26)
The model involves the complex mixed boundary value condition that a
aT =0 ar f o r  o. <~ r ~ < a
aY for n
Fig. 9.
b r
~
ar= o ay
i
Schematic representation of the constriction of the heat flow to the disk region a
Thermal resistances o f pressed contacts
63
temperature is prescribed over a < Ir[ < b and the heat flux over  a < r < a. This may be overcome by replacing the temperature b o u n d a r y condition with an appropriate heatflux distribution over the contact spot. Various analytical, numerical and approximation methods have been used to solve this problem, x7'24'12° 123 The Roess 12° derivation gave the nondimensional resistance as R* = 1  1.409 25(a/b) + 0.295 91(a/b) 3 + 0.052 541 9(a/b) 5 + ' "
(27)
which is almost identical to that later obtained by Gibson, 123 namely R* = 1  1.409 1839(a/b + 0"338010(a/b) 3 + 0.067 902(a/b) 5 + ' "
(28)
Typical approximations for R* were given by Cooper et al. 122 as R*  (1  a/b) 15
(29)
R* ~ 1  l'4a/b
(30)
for all values of a/b and
for a/b < 05.
Multiple contacts: nominally fiat surfaces By applying the single circular contact spot resistance, eqn (25), to a nominally flat joint with N independent spots of mean radius t~, under vacuum conditions, we obtain R   
1
2dNk m
(31)
where k m is the harmonic mean of the thermal conductivities of the two contacting solids, i.e. 2
1
km
kl
~
1
k2
(32)
To allow for the interference between the multiple heatflow channels, a constriction alleviation factor, g(x), is incorporated. Normally the Roess formula, i.e. eqn (27), is used, which now takes the form g(x) = 1  1.409 25x + 0295 91x 3 + 0052 541 9x 5
(33)
64
B. Snaith, S. D. Probert, P. W. O'Callaghan
where x== b
(34)
and eqn (31) becomes R
g(x) 26Nk m
(35)
The use of eqn (35) requires a knowledge of the total number, N,ot, of contact spots, the mean radius of these regions, d, and the mean heatflow channel radius,/~, which are produced under a particular applied load. Values of these variables may be estimated from topographical data for the contacting surfaces and the mean plane separation, u, which depends upon the applied load and the physical specifications of the materials in contact. An analysis by Tsukizoe and Hisakado, 124'125 assuming on each contacting surface a Gaussian distribution of conical asperities, which deform in an ideal plastic manner (i.e. no interference between neighbouring asperities occurs), results in the number of contact spots
N,o,
~2f~(t)
8
(36)
and the mean contact spot radius ti =
2 rcWf
(37)
where WI( = I~l/a) is the normalised arithmetic mean surface slope and f (=u/a) is the normalised meanplane separation. For a Gaussian distribution of surface heights, the relationship between the normalised mean plane separation and the applied loading is given by
1,
~(t') = ~  ~ =
fl q~(t)dt
(38)
where M is the surface microhardness of the softer of the two materials in contact. The nominal contact pressure p= W (39) An and the normal probability function is 4~(?) = ~
1
exp (  f2/2)
(40)
Thermal resistances of pressed contacts
65
Thus, at a particular applied load, the normalised mean plane separation may be calculated and thus the number of contacts and the mean contact radius evaluated. The mean flow channel radius, /~, required for the constriction alleviation factor g(d/b) can be estimated from
/)=(
A'°t ) ° s = ( = N , ot)°5 ,,/ZNtotAtot /
(41)
where Ntot is the total number of microcontacts in the defined area Atot. Substituting expressions for this alleviation factor, the mean contact spot radius and the number of contact spots into eqn (35) yields the TCR. Mikic 126 undertook an analysis, assuming a Gaussian distribution of profile heights, and that the distribution of profile slopes is independent of the profile heights, under various deformation modes. He concluded that
c.' = ,,3 tano( )o9 (7
(42)
for ideal plastic deformations. To account for contact interactions under plastic deformations
C,v=l.13ktanO(
p ,]0.94 (7 \ ~  ~  ~ j
(43)
which is a modified version of eqn (42). To account for elastic deformations of the substrate Cep = Cp(1 + 0"6#)
(44)
where M  E' tan 0
(45)
Finally, for a completely elastic model
C, = l.55 ktanO ( pw/~ a
"~0.94
~E tan 0J
(46)
The parameter # was suggested by Mikic as a plasticity index for the deformation mode criterion. The deformation is predominantly plastic for ~ < ~ and predominantly elastic for g > 3. Bush and Gibson lz7 treated statistically an isotropic rough surface, where the geometry is specified completely in terms of moments of its power spectra.~Z8'x29 Using these surface moments, expressions for the
66
B. Snaith, S. D. Probert, P. W. O'Callaghan
nondimensional variables, namely the contact spot radius a*, the flow channel radius b* and the contact conductance C*, were deduced as a
a* x//~ (3m___qyi4 \m4/ b b*  ~_/3mo,xll,
(47)
(48)
and C* 
Cs
(49)
t mo/ where the contact conductance, Cs, is obtained from the reciprocal of Rs as given by eqn (35). Employing Gibson's own alleviation factor (see eqn (28)) leads to
2kmx Cs = nbg(x)
(50)
The analysis showed the variation of the contact conductance with the applied normal pressure, P, to be of the form / p "~0.94
Cp* = 2 " 1 3 t ~ )
(51)
for plastic deformation and //p\O.89
(52) for elastic deformation, where f~ = E ' ~ m ~
(53)
For their model, Bush and Gibson defined a plasticity index as p'
=(8)l/2(m°m4v~x/4(E'~ t,3)
t,
,~ )
t,M)
(54)
Thermal resistances of pressed contacts
67
where v = 1  08968/fl; the criterion for elastic deformation being #' < 0.4, and for ideal plastic contact #' > 2. The surface moments m 0, m 2, m 4 are, respectively, the variances of the distributions of profile heights, slopes and curvatures, and fl, Nayak's bandwidth parameter, 128 is defined by mom4
fl
mY
(55)
Multiplecontactswavysurfaces Holm ~3° considered the specialised problem of N identical circular metallic contact areas of radius a, uniformly distributed over a relatively large circular region (i.e. a macroregion) of radius b'. His result (obtained by applying the electricalthermal analogy) for the total TCR can be expressed as
1
R
2Nak m
q
1 2b'k m
(56)
where the first term represents the selfresistance of the microcontacts and the second is a macroscopic interaction term. A mathematical and physical interpretation by Greenwood T M recommended a more general expression, namely
1
3n ~ ~~S~.j.
R = 2km~ai
~ 32N2km
(57)
where S u is the distance between the two generalised true contacts i and j within a cluster, which for a wellfilled circular disc will lead to Holm's equation exactly. Theoretically it was shown that the total resistance is often close to that found by assuming the entire area covered by the cluster is a single contact spot, i.e. macroscopic effects dominate. The above expressions are for contacts between semiinfinite bodies. Clausing and Chao 33 considered two semiinfinite cylinders with contacting surfaces, and introduced the appropriate alleviation factors. Thus R = g(Xms)
g(Xml)
2Nakm
2b'k m
(58)
where the Roess constriction alleviation factor (eqn (27)) for both microand macroconstrictions is employed. For their model of spherical
68
B. Snaith, S. D. Probert, P. W. O'Callaghan
capped cylinders, of radius b', with an equivalent flatness deviation, d' (i.e. the distance from the spherical cap base to the apex at zero load), they proposed R = na g ( X m s ) M ~
2
kmP
F
nb'
g([1.285/~] 1/3)
2k m [1.285/~] 1/3
(59)
where ~ is the proportionality constant between P and M (i.e. ¢ = 1 for pure plastic flow) and /~ is an elastic conformity modulus, which represents the ratio of elastic deformation to the flatness deviation, namely
where d~ot (=d'l + d'2) is the total equivalent flatness deviation. Popov and Yanin 132 presented a unique approach for predicting the rate of heat transfer across the pressed contact between two metal surfaces with both waviness and microroughness. The basic multiplecontacts (eqn (35)) was incorporated but with the Roess alleviation factor expressed in terms of the relative area of the actual contact. For this factor they gave
g(q) = 1  l'4r/1/2 + 0"296q 3/2
(61)
where q is the ratio of the real contact area, A r, to the actual macroscopic contact area, A¢. They proposed a simplified expression I , / W \10.8 r/= LF ~A~)]
(62)
where F' is a constant dependent upon the average rms roughness of the contacting surfaces, i.e. F' = f ( a l + a~_). Using Hertzian formulae, the macroscopic contact area can be determined for various spherical and cylindrical wavy surfaces in contact. Yovanovich 133 performed an analysis for the thermal resistance of a contact formed by a hard, smooth, flat surface with a softer turnedfinish surface. The model comprises of a twodimensional heatflow channel, and the contact was based upon the plastic deformation of a ridge formed by the turning process, i.e. a continuous spiraltype finish. The macroscopic term of the total resistance is defined as 1
Rm, = 2(Gl/k 1 + G2/k2)
(63)
Thermal resistances of pressed contacts
69
where 2 is the pitch between adjacent spirals and is a characteristic of the machining process; G~ and G 2 a r e dimensionless geometric factors, which depend upon the halfwidth of the contact strips, halfwidth of the heatflow channel and the heatflow channel contact angle. The microscopic term was based on a Cooper eta/. 122 expression: 1 Rm~
1.45 km (~___)°'985 tan 0 a
(64)
For values of P/M less than 3 x 10 2, it was recommended that both macro and micro terms for the TCR were necessary, but that if P/M exceeded 5 x 10 2, the microscopic term is negligible. Yovanovich's macromodel was employed by A1Astrabadi et al. 77 to predict the thermal resistances of contacts with idealised surfaces, produced by the shaping and endmilling manufacturing processes.
Limitations of the analytical predictions The analysis leading to eqn (31) assumed (i) circular contact regions, randomly distributed and separated by large distances; (ii) that the contact plane was isothermal; and (iii) the heat flow lines were straight and parallel when remote from the contact plane. In reality, circular contact regions are rarely found, even at very low loads. The temperature distribution in the vicinity of a contact bridge is threedimensional and the distributions of heat flux in the abutting solids depend upon the shapes of the contacting members and upon the magnitudes of the transverse heat flows. Unfortunately the amount of corroboratory experimental data concerning the mean contact spot size, d, and the number, N, of microcontacts that exists is still limited. Thus the appropriateness of many of the simplifying assumptions made is difficult to ascertain. Some of the models are, of course, limited by the assumption of an ideal plastic deformation, which will not occur in practice as work hardening and material interaction ensue. The normal technique for surface parameter measurements is to sample profile heights at regular intervals using a stylustype instrument. The parameters normally incorporated to define the surface are the average roughness, o, and the arithmetic mean absolute profile slope, tan 0. The latter is strongly dependent upon the sampling interval
70
B. Snaith, S. D. Probert, P. IV. O'Callaghan
employed, 134 and the former will increase as the square root of the sampling length used.135 Careful consideration must be given to these conclusions when specifying and measuring the surface parameters. For the values used for the microhardness of each material in contact, the conventional models employ the bulk hardness. More recent analyses 136'137 have indicated that this is inadequate and proposed an iterative model based on a surface microhardness distribution. It should be noted that most analytical developments are concerned with plane pressed contacts between surfaces with micro and/or macro roughnesses. As yet only limited knowledge is available for predicting the thermal contact behaviours of bolted, welded or riveted joints and the more complex cylindrical joints. By considering such joints, with combinations of various surface textures, similar or dissimilar materials, interfacial materials or fluids, one can begin to appreciate the vast scale of the problems still to be solved.
EM PIRICAL C ORRELATI ONS Because the analytical prediction of TCR still presents such a formidable problem, a pragmatic approach which has been adopted widely is to correlate existing experimental data. 5557,78'138143 It would be convenient for the thermal design engineer, in need of contact resistance predictions, if the parameters involved are few and readily available or easily measurable. What are considered to be the more usable correlations for thermal contact conductance, in high vacua, are presented here.
Typical dimensionless correlations Popov's 14° handling of experimental data for nominally flat surfaces (namely 80 data pointsfor a variety of materials and finishing processes) yielded
c'=2.7 × lO
k~
4/pz\o956 (65)
where k s is the harmonic mean thermal conductivity (W m  1 K  1); SB is
Thermal resistances of pressed contacts
71
the ultimate compressive strength of the weaker material in the contact (N m2); and Z is a [actor defined by 12
Z = (hmaxl + hmax2)
for 5/~m > (hmaxl + hmax2) > 1 pm
20
Z=(hmaxl +hmax2) for 10/am>(hmaxl +hmax2)>5#m 30
Z=(hmaxl +hmax2 ) for 30#m>(hmaxl +hmax2)> 10#m where hmax is the maximum peaktovalley height of the surface profile. A correlation coefficient of 0.947 was found for the equation in the dimensionless loading range PZ 3 x 10s < ~  ~  < 1 × 10 3 There is a length dimension hidden in eqn (65) due to an assumed value for the mean contact spot radius. A semiempirical correlation for assumed nominally fiat surfaces (the condition being that the surface average roughness had to be greater than onetenth of the flatness deviation for this to hold) of similar materials in contact, by Tien, 138 is C'a / p \0.85  0"55m'// (66)
k
\M/
This takes an expected similar form to the theoretical correlations given by eqns (42) and (64) because it is partly based upon Mikic's 116 theoretical study. The exponent of the pressure (namely 085) given in the above equation based on experimental data differs from the theoretical exponents, i.e. 0.94 to 0.99. It could be argued that this is due to the combined effects of nonideal plastic deformations of the asperities, elastic deformation of the smoother surfaces and the elastic displacement of the sublayers, i.e. a more practical elastoplastic effect, thus giving rise to a more realistic exponent for the pressureconductance relationship. Empirical correlations deduced by Thomas and Probert 55 from 350 published thermal resistances for contacts with a broad range of surfaces (under high vacuum environments) are (a)
In ~
= (0.743 _+0.067)1n ~
+ 2.26_+ 0.88
(67)
B. Snaith, S. D. Probert, P. W. O'Callaghan
72
for stainless steelstainless steel contacts and (b)
In ~
= (0.72_+ 0.044) 1n ~
+ 0.66 + 0.62
(68)
where aluminium alloy was used for either one or both of the contacting surfaces (the nonaluminium metal surface always being the harder). Probable errors are indicated. Correlation coefficients were 0.915 for data set (a) and 0.913 for set (b). An important question arises from these correlations in that whereas the slopes of the correlations are in good agreement the lines do not coincide. Fried raises the point, in the discussion section of reference 55, that a suitable general correlation of thermal contact conductance may not exist or cannot be developed. Considering, as we have seen in previous sections of this review, the number and complexity of parameters that may influence the TCR, this does have a strong possibility of being true. One parameter rarely used in contact conductance correlations, which may have a significant role to play, is the mean interracial temperature, T m. A semiempirical correlation by Fletcher and Gyorog 139 does bring in this temperature parameter, i.e. C'
k 1" /170P*T*\q 5 2 =~oLexp~  &*o* ) J [ " 2 × 1 0  6 6 " +0"036P'T*]°56
(69)
where T* = ~Tm; P* = P/E; and r* = rio~r,where r is the specimen radius: 6 o is an equivalent gap thickness obtained via the empirical relationship 60= 106(0.5217 +8.06 x 102s  6.22 + 2"108
x 104s 2
(70)
× 1 0  6 s 3" • ')
where s = (FD + 2a)rough surface  0"5(FD
+
20")smooth surface
(71)
FD and a being, respectively, the flatness deviation and rms surface roughness (both in microns). Introducing the mean interracial temperature and an equivalent gap thickness resulted in a more complex form of correlation, but the expression 69 is alleged to predict the thermal contact conductance with an average deviation of 24 ~o over a wide range of loadings (up to 48 MNm2), temperatures (  150 °C to 260 °C) and surface conditions (with flatness deviations from 0.38 to l 1 4 # m and rms roughness deviations from 0.076 to 3.05 pm).
120
Fig. I0.
uJ "1
8
Z
8
r'~ Z
Z
/ /~/
/
I
~f"
I
J
I
I
/ 7
~
~
." .." " " "
/ /~
/
/ / EQUATION (66) (TIEN)
EQUATION (65) (POPOV)
EQUATION
EQUATION _. .~ "  (68) ........." " " " (THOMAS& PROBERT)
//
/
/
/
/
SURFACES:TURNEDFINISH (7= 0471 ]~m =RD hmax = 2"56/Jm m' = 0024 rndians FO taken os lO/~m LOADINGSAND TEMPERATURES TAKEN FROMFRIED28 Trn= 1B33"C specimen rodius = 25/. x 102m
MATERIAL:ALUHINIUM ALLOY 6061T 6 k = 200 Wm4 Kq ~x = 24 x 106 "C1 E =6? GNm2 H = 1000 MNm2 SB=260 MNrn2
(69) _(FLETCHER E l ~  ~ ' . . . . ;    ,~ . " '~    &, GYOROG) 1000 2000 3000 z,O00 5000 6000 7000 8000 MECHANICAL LOADING(kNrn 2 ) Thermal contact conductance versus loading derived from the stated correlation for the specified aluminium alloy aluminium alloy contact.
20
#o
60
80
~o I00
)=
%
.,,.
%0
4
,i
Fig. I !.
"tI..
8
z
u
8
r,, z
z
%
.
1000
.
~
/
.
~
2000
.
/
/
/
.
~
/
3000
~ .I I
7
/
i
/
/,.000
~
/
/
7
5000
111
6000
"
/
7000
i
~ / ~   ' . ~   "
/
' .....
"
/
/
/
/
/
f
specimen radius=25/,x 10= m
(7= 0/*pm=RD hmox = 2TB/J.m m'= 0.031 rodians FD token os 10/4m LOABN6SAND TEMPERATURES TAKEN FROMFRIED28 Tm=24..29C .2
SURFACES:GROUNDFINISH
SB= 530 MNm2
o< = 1Bxl06"C 1 E = 20? GNm2 H = 1800 MNm2
MATERIAL:STAINLESSSTEEL304 k = 16 Wm I ICI
8000
!69)
g GYOROG)
EQUATION ( FLETCHER
(65) (POPOV)
EQUATION
EQUATION (66) (TIEN)
(67) (THOMASg PROBERT)
EQUATION
ME[HANICAL LOA01NG ( k N m2) Thermal contact conductance versus loading derived from the stated correlation for the specified stainless steelstainless steel contact.
0
0 '~
~
2
///
//
4
6
10
12
14.
Cb
tb
h~
Ib
4~
Thermal resistances of pressed contacts
75
Correlation comparisons The correlations given are compared for aluminium alloyaluminium alloy and stainless steelstainless steel type contacts in Figs 10 and 11, respectively. Because no published experimental data provide the complete range of parameters (especially of the surface characterisation descriptors) required to perform the comparison, a number of data sources have been solicited. The material properties used are typical, the surface parameters being taken from the Cranfield library of surface topographical data. The loading and temperature values are taken from Fried 28 for contacts of the same material and surfaces similar to the ones specified. The divergence of these correlations, shown in Figs 10 and 11, strengthens the case that a suitable general correlation cannot be achieved. Which one, or any, of these correlations can the design engineer take to give a representative value of the thermal conductance of his contact ? Only when steps are taken towards some form of international standardisation in surface topography and contact conductance measurements can any improvements be expected in the experimental correlations.
ACKNOWLEDGEMENT The authors wish to express their appreciation for the financial support received from the Science and Engineering Research Council.
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76
5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22.
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Thermal resistances of pressed contacts
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78
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48. 49.
50.
51. 52. 53. 54. 55. 56. 57.
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80
78. 79. 80. 81.
82. 83. 84.
85. 86. 87. 88.
89. 90. 91. 92.
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