In this section we have shown how a thermomechanical model for coated sub-strates can be used to predict enhancement in thermal joint conductance.. It should also be noted that aluminum
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the highest contact conductance, and silver with the highest microhardness has the lowest contact conductance The thermal conductivity ofthe coating appears to play
a secondary role
The unusual shape ofthe curves is attributable to the fact that the assumed effective hardness curve shown in Fig 4.30 has three distinct zones Moreover, because the microhardness ofsilver is much closer to aluminum than are the microhardness of lead and tin, respectively, the transition from one region to the next is not abrupt
in the silver-on-aluminum effective microhardness curve, and this is reflected in the smoother contact conductance plot for the silver layer shown in Fig 4.34 It should
be noted that in the model that has been used, the load is assumed to be uniformly applied over the apparent contact area
4.17.2 Ranking Metallic Coating Performance
In his research on the effects of soft metallic foils on joint conductance, Yovanovich (1972) proposed that the performance of various foil materials may be ranked accord-ing to the parameterk/H , using the properties ofthe foil material He showed
empir-ically that the higher the value ofthis parameter, the greater was the improvement
in the joint conductance over a bare joint Following this thought, Antonetti and Yovanovich (1983) proposed (but did not prove experimentally) that the performance ofcoated joints can be ranked by the parameterk/(H)0.93 Table 4.21 shows the
variation in this parameter as the layer thickness is increased Table 4.21 also sug-gests that even ifthe effective microhardness ofthe layer–substrate combinations being considered is not known, the relative performance of coating materials can be estimated by assuming an infinitely thick coating (where the effective microhardness
is equal to the microhardness ofthe layer)
In this section we have shown how a thermomechanical model for coated sub-strates can be used to predict enhancement in thermal joint conductance For the particular case considered, an aluminum-to-aluminum joint, it was demonstrated that
up to an order ofmagnitude,
TABLE 4.21 Ranking the Effectiveness of Coatings
k/(H)0.93
Coating Thickness
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improvement in the contact conductance is possible, depending on the choice of coating material and the thickness employed It should also be noted that aluminum substrates are relatively soft and have a relatively high thermal conductivity, and if the joint in question had been, for example, steel against steel, improvement in the joint conductance would have been even more impressive
4.17.3 Elastomeric Inserts
Thin polymers and organic materials are being used to a greater extent in power-generating systems Frequently, these thin layers ofrelatively low thermal conduc-tivity are inserted between two metallic rough surfaces assumed to be nominally flat
The joint that is formed consists of a single layer whose initial, unloaded thickness is denoted ast0and has thermal conductivityk p There are two mechanical interfaces that consist ofnumerous microcontacts with associated gaps that are generally occu-pied with air Ifradiation heat transfer across the two gaps is negligible, the overall joint conductance is Yovanovich et al (1997)
1
h j =
1
h c,1 + h g,1+
k p
1
wheret is the polymer thickness under mechanical loading The contact and gap
conductances at the mechanical interfaces are denoted ash c,iandh g,i, respectively, andi = 1, 2.
Since this joint is too complex to study, Fuller (2000) and Fuller and Marotta (2002) choose to investigate the simpler joint, which consisted ofthermal grease
at interface 2, and the joint was placed in a vacuum Under these conditions, they assumed thath g,2 → ∞ and h g,1→ 0 They assumed further that the compression
ofthe polymer layer under load may be approximated by the relationship
t = t0
1−E P
p
whereE p is Young’s modulus ofthe polymer Under these assumptions the joint conductance reduces to the simpler relationship
h j = h1
c,1 + t0
k p
1−E P
p
−1
(W/m2· K) (4.310)
On further examination ofthe physical properties ofpolymers, Fuller (2000) con-cluded that the polymers will undergo elastic deformation of the contacting asperi-ties Fuller examined use ofthe elastic contact model ofMikic (1974) and found that the disagreement between data and the predictions was large To bring the model into agreement with the data, it was found that the elastic hardness of the polymers should
be defined as
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H ep= E p m
wherem is the combined mean absolute asperity slope The dimensionless contact
conductance correlation equation was expressed as
h cσ
k s m = a1
2.3P
mE p
a2
(4.312)
wherea1 anda2 are correlation coefficients Fuller and Marotta (2000) chose the coefficient valuesa1 = 1.49 and a2 = 0.935, compared with the values that Mikic
(1974) reported:a1= 1.54 and a2= 0.94 In the Mikic (1974) elastic contact model,
the elastic hardness was defined as
H e=mE√
whereE is the effective Young’s modulus of the joint For most polymer–metal
joints,E≈ E pbecauseE p Emetal Fuller (2000) conducted a series ofvacuum tests for validation ofthe joint conduc-tance model The thickness, surface roughness and the thermophysical properties of the polymers and the aluminum alloy are given in Table 4.22 The polymer thickness
in all cases is two to three orders ofmagnitude larger than the surface roughness (i.e.,
t/σ > 100).
The dimensionless joint conductance data for three polymers (delrin, polycarbon-ate, and PVC) are plotted in Fig 4.35 against the dimensionless contact pressure over approximately three decades Two sets ofdata are reported for delrin The joint conductance model and the data show similar trends with respect to load At the higher loads, the data and the model approach asymptotes corresponding to the bulk resistance ofthe polymers The dimensionless joint conductance goes to different asymptotes because the bulk resistance is defined by the thickness ofthe polymer
TABLE 4.22 Thickness, Surface Roughness, and Thermophysical Properties
of Test Specimens
Material (103m) (106m) (rad) (W/m· K) (GPa)
Polycarbonate A 1.99 0.773 0.470 0.22 2.38 0.38
Source: Fuller (2000).
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10⫺4
10⫺3
10⫺2
10⫺5
2.3P
E m p
F
F
h c,2
k p
Delrin1; Data Delrin1; Model
Delrin2; Data Delrin2; Model
Polycarbonate; Data Polycarbonate; Model
PVC; Data PVC; Model
h km j
s
6
6
6
5
5
5
4
4
4
3
3
3
2
2
2
2 3 4 5 6 7 8 2 3 4 5 6 7 8 2 3 4 5 6 7 8
k1
k2
Figure 4.35 Dimensionless joint conductance versus dimensionless contact pressure for polymer layers (From Fuller and Marotta, 2000.)
layers In general, there is acceptable agreement between the vacuum data and the model predictions
4.17.4 Thermal Greases and Pastes
There is much interest today in the use ofthermal interface materials (TIMs) such as thermal greases and pastes to enhance joint conductance Prasher (2001) and Savija
et al (2002a,b) have reviewed the use ofTIMs and the models that are available
to predict joint conductance The thermal joint resistance or conductance ofa joint formed by two nominally flat rough surfaces filled with grease (Fig 4.22) depend
on several geometric, physical, and thermal parameters The resistance and conduc-tance relations are obtained from a model that is based on the following simplifying assumptions:
• Surfaces are nominally flat and rough with Gaussian height distributions
• The load is supported by the contacting asperities only
• The load is light; nominal contact pressure is small; P /H c≈ 10−3to 10−5.
• There is plastic deformation ofthe contacting asperities ofthe softer solid
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• Grease is homogeneous, fills the interstitial gaps completely, and wets the
bound-ing surfaces perfectly
In general, the joint conductanceh j and joint resistanceR j depend on the contact and gap components The joint conductance is modeled as
whereh crepresents the contact conductance andh grepresents the gap conductance
The joint resistance is modeled as
1
R j = 1
R c + 1
whereR cis the contact resistance andR gis the gap resistance For very light contact
pressures it is assumed thath c h g andR c R g The joint conductance and
resistance depend on the gap only; therefore,
h j = h g and R1j =
1
R g where h j =
1
A a R j (4.316)
Based on the assumptions given above, the gap conductance is modeled as an equiv-alent layer ofthicknesst = Y filled with grease having thermal conductivity k g The joint conductance is given by
h j = k g
The gap parameterY is the distance between the mean planes passing through the
two rough surfaces This geometric parameter is related to the combined surface roughnessσ = σ2
1+ σ2
2, whereσ1 andσ2 are the rms surface roughness of the two surfaces and the contact pressureP and effective microhardness of the softer
solid,H c The mean plane separationY , shown in Fig 4.22, is given approximately
by the simple power law relation (Antonetti, 1983)
Y
P
H c
−0.097
(4.318)
The power law relation shows thatY/σ is a relatively weak function of the relative
contact pressure Using this relation, the joint conductance may be expressed as
h j = k g
k g
1.53σ(P /H c ) −0.097 (W/m2· K) (4.319) which shows clearly how the geometric, physical, and thermal parameters influence the joint conductance The relation for the specific joint resistance is
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A a R j = 1
h j = 1.53
σ
k g
P
H c
−0.097
In general, ifthe metals work-harden, the relative contact pressureP /H c is ob-tained from the relationship
P
c1(1.62σ/m) c2
1/(1+0.071c2)
(4.321)
where the coefficientsc1andc2are obtained from Vickers microhardness tests The Vickers microhardness coefficients are related to the Brinell hardnessH B for a wide range ofmetals The units ofσ in the relation above must be micrometers The units
ofP and c1must be consistent
The approximation ofHegazy (1985) for microhardness is recommended:
H c = (12.2 − 3.54H B ) σ
m
−0.26
whereH c, the effective contact microhardness, andH B, the Brinell hardness, are in GPa and the effective surface parameter (σ/m) is in micrometers Ifthe softer metal
does not work-harden,H c ≈ H B SinceH B < H c, ifwe setH c = H Bin the specific joint resistance relationship, this will give a lower bound for the joint resistance or an upper bound for the joint conductance
The simple grease model for joint conductance or specific joint resistance was compared against the specific joint resistance data reported by Prasher (2001) The surface roughness parameters of the bounding copper surfaces and the grease thermal conductivities are given in Table 4.23 All tests were conducted at an apparent contact pressure of1 atm and in a vacuum Prasher (2001) reported his data as specific joint resistancer j = A a R j versus the parameterσ/k g, wherek gis the thermal conductivity
ofthe grease
TABLE 4.23 Surface Roughness and Grease Thermal Conductivity
Roughness, Conductivity,
Source: Prasher (2001).
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4.17.5 Phase-Change Materials
Phase-change materials (PCMs) are being used to reduce thermal joint resistance in microelectronic systems PCMs may consist ofa substrate such as an aluminum foil supporting a PCM such as paraffin In some applications the paraffin may be filled with solid particles to increase the effective thermal conductivity of the paraffin
At some temperature T m above room temperature, the PCM melts, then flows through the microgaps, expels the air, and then fills the voids completely After the temperature ofthe joint falls belowT m, the PCM solidifies Depending on the level
ofsurface roughness, out-of-flatness, and thickness ofthe PCM, a complex joint is formed Thermal tests reveal that the specific joint resistance is very small relative to the bare joint resistance with air occupying the microgaps (Fig 4.36) Because ofthe complex nature ofa joint with a PCM, no simple models are available for the several types ofjoints that can be formed when a PCM is used
Figure 4.36 Specific joint resistance versusσ/k for grease (From Prasher, 2001.)
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4.18 THERMAL RESISTANCE AT BOLTED JOINTS
Bolted joints are frequently found in aerospace systems and less often in microelec-tronics systems The bolted joints are complex because oftheir geometric configu-rations, the materials used, and the number ofbolts and washers used The pressure distributions near the location ofthe bolts are not uniform, and the region influenced
by the bolts is difficult to predict A number of papers are available to provide infor-mation on measured thermal resistances and to provide models to predict the thermal resistance under various conditions
Madhusudana (1996) and Johnson (1985) present material on the thermal and mechanical aspects ofbolted joints For bolted joints used in satellite thermal design, the publications ofMantelli and Yovanovich (1996, 1998a, b) are recommended
For bolted joints used in microelectronics cooling, the publications ofLee et al
(1993), Madhusudana et al (1998) and Song et al (1992b, 1993a) are recommended
Mikic (1970) describes variable contact pressure effects on joint conductance
NOMENCLATURE
Roman Letter Symbols
Fourier coefficient, dimensionless geometric parameter related to radii ofcurvature, dimensionless
A a apparent contact area, m2
A g effective gap area, m2
A n coefficient in summation, dimensionless
A r real contact area, m2
mean contact spot radius, m semimajor diameter ofellipse, m correlation coefficient, dimensionless radius ofcircle, m
strip half-width, m side dimension ofplate, m radius offlat contact, m
a e elastic contact radius, m
a ep composite elastic–plastic contact radius, m
a L thick-layer limit ofcontact radius, m
a p plastic contact radius, m
a s thin-layer limit ofcontact radius, m
a∗ combination ofterms, dimensionless
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B Fourier coefficient, dimensionless
geometric parameter related to radii ofcurvature, dimensionless
Bi Biot modulus, dimensionless
Bi(x,y) beta function of argumentsx and y, dimensionless
B n coefficient in summation, dimensionless
b semiminor diameter ofellipse, m
side dimension ofplate, m radius ofcompound disk, m correlation coefficient, dimensionless channel half-width, m
b1 radius ofcylinder, m
C Fourier coefficient, dimensionless
correction factor, dimensionless
C c contact conductance, dimensionless
c correlation coefficient, dimensionless
length dimension, m flux channel half-width, m side dimension ofisoflux area, m
diameter ofcircular cylinder, m
uniform gap thickness, m side dimension ofisoflux area, m
d o reference value for average diagonal, m
E modulus ofelasticity (Young’s modulus), N/m2
modulus ofelasticity for polymer, N/m2 complete elliptic integral ofsecond kind, dimensionless
E effective modulus of elasticity, N/m2 erf(x) error function of argumentx, dimensionless
erfc(x) complementary error function of argumentx, dimensionless
F emissivity factor, dimensionless
F i factor, dimensionless,i = 1, 2, 3
total normal load on a contact, N
F (k, ψ) incomplete elliptic integral ofthe first kind ofmoduluskand
amplitudeψ, dimensionless
F∗ combination ofterms, dimensionless
F load per unit cylinder length, N
Fo Fourier modulus, dimensionless
f ep elastic–plastic parameter, dimensionless
f g combination ofterms, dimensionless
f (u) axisymmetric heat flux distribution, dimensionless
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H effective microhardness, MPa
H B Brinnell hardness, MPa
bulk microhardness ofsubstrate, MPa
H∗
B adjusted Brinnell hardness, MPa
H e effective microhardness, MPa
H ep elastic–plastic microhardness, MPa
H p microhardness ofsofter contacting asperities, MPa
substrate microhardness, MPa
H S microhardness ofsofter material, MPa
H V Vickers microhardness, MPa
H1 microhardness, layer 1, MPa
H2 microhardness, layer 2, MPa
h conductance or heat transfer coefficient, W/m2· K
h coated joint conductance in vacuum, W/m2· K
h c contact conductance, W/m2· K
h g gap conductance, W/m2· K
h j joint conductance, W/m2· K
I g gap conductance integral, dimensionless
I g," line contact elastogap integral, dimensionless
I g,p point contact elastogap integral, dimensionless
I o relative layer thickness, dimensionless
I q layer thickness conductivity parameter, dimensionless
Iγ relative layer thickness conductivity parameter, dimensionless
K thermal conductivity parameter, dimensionless
complete elliptic integral offirst kind, dimensionless
Kn Knudsen number, dimensionless
k modulus related to the ellipticity, dimensionless
thermal conductivity, W/m· K
k g effective gas thermal conductivity, W/m· K
grease thermal conductivity, W/m· K
k g,∞ gas thermal conductivity under continuum conditions,
W/m · K
k s harmonic mean thermal conductivity ofa joint, W/m· K
k1 layer 1 thermal conductivity, W/m· K
k2 layer 2 thermal conductivity, W/m· K
L relative contact size, dimensionless
" point in flux tube where flux lines are parallel, m
M gas rarefaction parameter, m
M g molecular weight ofgas, g-mol
M s molecular weight ofsolid, g-mol
Hertz elastic parameter, dimensionless absolute asperity slope, dimensionless
... MPaH2 microhardness, layer 2, MPa
h conductance or heat transfer coefficient, W/m2· K
h coated joint... foil supporting a PCM such as paraffin In some applications the paraffin may be filled with solid particles to increase the effective thermal conductivity of the paraffin
At some temperature... data-page="8">
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