The layer thickness is often less than 100µm; it is in “perfect” thermal and mechanical contact with the substrate, and its bulk resistance is negligibly small relative to the contact re
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100
102
103
104
105
102
P g(torr)
K)
Experiment
Theory He
He
N2
N2
Nickel 200
= 2.32 m
P = 0.52 MPa
( / )Y YDH= 3.6
P H/ = 1.7 10e ⫻ ⫺4
Figure 4.28 Gap conductance model and data for conforming rough Ni 200 surfaces (From Song, 1988.)
200 surfaces The plastic deformation model was used to calculateY The points for
M∗ < 0.01 correspond to the high-gas-pressure tests (near 1 atm), and the points for
M∗ > 2 correspond to the low-gas-pressure tests.
In many electronics packages the thermal joint conductance across a particular joint must be improved for the thermal design to meet its performance objectives If the joint cannot be made permanent because ofservicing or other considerations, the joint
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Figure 4.29 Dimensionless gap conductance model and data for conforming rough Ni 200 surfaces (From Song, 1988.)
conductance must be “enhanced”; that is, it must be improved above the bare joint situation utilizing one ofseveral known techniques, such as application ofthermal interface materials (TIMs): for example, thermal grease, grease filled with particles
(also called paste), oils, and phase-change materials (PCMs) Enhancement ofthe
joint conductance has also been achieved by the insertion of soft metallic foils into the joint, or by the use ofa relatively soft metallic coating on one or both surfaces More recently, soft nonmetallic materials such as polymers and rubber have been used
One may consult review articles by Fletcher (1972, 1990), Madhusudana and Fletcher (1986), Madhusudana (1996), Marotta and Fletcher (1996), Prasher (2001), Savija et al (2002a, b), and other pertinent references may be found in these reviews
This section is limited to a few examples where models and data are available
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An effective method for enhancement of joint conductance consists of vapor depo-sition ofa very thin soft metallic layer on the surface ofthe substrate The layer thickness is often less than 100µm; it is in “perfect” thermal and mechanical contact
with the substrate, and its bulk resistance is negligibly small relative to the contact resistance The thermal resistance at the layer–substrate interface is also negligible
A comprehensive treatment ofthe theoretical development and experimental ver-ification ofthe thermomechanical model can be found in Antonetti (1983) and An-tonetti and Yovanovich (1983, 1985) In the following discussion, therefore, only those portions ofthe theory needed to apply the model to a thermal design problem are presented The general expression for the contact conductance of the coated joint operating in a vacuum is
h
H S
H
0.93
k1 + k2
Ck1 + k2
(W/m2· K) (4.296)
whereh c is the uncoated contact conductance,H S the microhardness ofthe softer substrate,H the effective microhardness of the layer–substrate combination, C a
spreading–constriction parameter correction factor that accounts for the heat spread-ing in the coated substrate, andk1andk2the thermal conductivities ofthe two sub-strates, respectively
The coated contact conductance relationship consists ofthe product ofthree quan-tities: the uncoated contact conductanceh c, the mechanical modification factor(H S /
H)0.93, and the thermal modification factor The uncoated (bare) contact conductance
may be determined by means ofthe conforming, rough surface correlation equation based on plastic deformation:
h c = 1.25 m
σ
2k1k2
k1 + k2
P
H S
0.95
(W/m2· K) (4.297)
whereH Sis the flow pressure (microhardness) ofthe softer substrate,m the combined
average absolute asperity slope, andσ the combined rms surface roughness of the
joint
For a given joint, the only unknowns are the effective microhardnessHand the spreading–constriction parameter correction factorC Thus, the key to solving coated
contact problems is the determination ofthese two quantities
Mechanical Model The substrate microhardness can be obtained from the fol-lowing approximate relationship (Hegazy, 1985):
H S = (12.2 − 3.54H B ) σ
m
−0.26
(GPa) (4.298)
which requires the combined surface roughness parametersσ and m and the bulk
hardness ofthe substrateH B In the correlation equation the units ofthe joint rough-ness parameterσ/m are micrometers For Ni 200 substrates, H B = 1.67 GPa.
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The effective microhardness must be obtained empirically for the particular layer (coating)–substrate combination under consideration This requires a series ofVick-ers microhardness measurements which will result in an effective microhardness plot similar to that shown in Fig 4.30 (e.g., a silver layer on a Ni 200 substrate)
The effective Vickers microhardness measurements, denoted H, are plotted against the relative indentation deptht/d, where t is the layer thickness and d is
the indentation depth The three microhardness regions were correlated as
H=
H S
1− t
d
+ 1.81H L t
d for 0≤
t
d < 1.0 (4.299)
1.81H L − 0.21H L
t
d − 1
for 1.0 ≤ t
d > 4.90 (4.301)
whereH S andH L are the substrate and layer microhardness, respectively The Ni
200 substrate microhardness is found to beH S = 2.97 GPa for the joint roughness
parameter values:σ = 4.27 µm and m = 0.236 rad The Vickers microhardness of
the silver layer is approximatelyH L= 40 kg/mm2= 0.394 GPa.
The relative indentation depth is obtained from the following approximate corre-lation equation (Antonetti and Yovanovich, 1983, 1985)
t
d = 1.04
t
d
P
H
−0.097
(4.302)
To implement the procedure (Antonetti and Yovanovich, 1983, 1985) for findingH from the three correlation equations requires an iterative method
To initiate the iterative method, the first guess is based on the arithmetic average ofthe substrate and layer microhardness values:
H
1= H S + H L
For a given value of t and P , the first value of t/d can be computed From the
three correlation equations, one can find a new value for H: say, H
2 The new microhardness value,H
2, is used to find another value fort/d, which leads to another
value,H
3 The procedure is continued until convergence occurs This usually occurs within three or four iterations (Antonetti and Yovanovich, 1983, 1985)
Thermal Model The spreading–constriction resistance parameter correction fac-torC is defined as the ratio ofthe spreading–constriction resistance parameter for a
substrate with a layer to a bare substrate, for the same value of the relative contact spot radius :
C = , φ n )
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Figure 4.30 Vickers microhardness ofa silver layer on a nickel substrate (From Antonetti and Yovanovich, 1985.)
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The dimensionless spreading–constriction resistance parameter is defined as
, φ n ) = 4k2aR
wherek2 is the thermal conductivity ofthe substrate that is coated,a is the con-tact spot radius for the layer on the substrate, andR
cis the spreading–constriction
resistance ofthe contact spot
The spreading–constriction resistance parameter with a layer on the substrate is (Antonetti and Yovanovich, 1983, 1985)
, φ n ) = 16
∞
n=1
J2
1(δ
(δ
n )φnγnρn (4.305)
The first ofthese,φn, accounts for the effect of the layer though its thickness and thermal conductivity; the second,γn, accounts for the contact temperature basis used
to determine the spreading–constriction resistance; and the third,ρn, accounts for the contact spot heat flux distribution For contacting surfaces it is usual to assume that the contact spots are isothermal The modification factors in this case areγn = 1.0 and
φn = K (1 + K) + (1 − K)e−2δ
(1 + K) − (1 − K)e−2δ
nτ (4.306) whereK is the ratio ofthe substrate-to-layer thermal conductivity, τ = t/ais the layer thickness-to-contact spot radius ratio, and
ρn= sinδn
2J1(δ
The parameterδ
nare the eigenvalues, which are roots ofJ1(δ
n ) = 0.
Tabulated values ofC were reported by Antonetti (1983) for a wide range of the
parametersK and τ Details ofthe thermomechanical model development are given
in Antonetti (1983) and Antonetti and Yovanovich (1983, 1985)
The thermomechanical model ofAntonetti and Yovanovich (1983, 1985) has been verified by extensive tests First the bare joint was tested to validate that part ofthe model Figure 4.31 shows the dimensionless joint conductance data and theory plotted versus the relative contact pressure for three joints having three levels of surface roughness The two surfaces were flat; one was lapped and the other was glass bead blasted All tests were conducted in a vacuum The agreement between the model given by the correlation equation and all data is very good over the entire range of relative contact pressure
The bare surface tests were followed by three sets of tests for joints having three levels ofsurface roughness Figure 4.32 shows the effect ofthe vapor-deposited silver layer thickness on the measured joint conductance plotted against the contact pres-sure For these tests the average values ofthe combined surface roughness parameters wereσ = 4.27 µm and m = 0.236 rad For the contact pressure range the substrate
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100
101
Relative Pressure H P ⫻104
h
mk = 1.25( (H P 0.95
Specimens 08/09 Specimens 10/11 Specimens 26/27 Specimens 34/35
Figure 4.31 Dimensionless contact conductance versus relative contact pressure for bare Ni
200 surfaces in a vacuum (From Antonetti and Yovanovich, 1985.)
microhardness was estimated to beH S = 2.97 GPa The layer thickness was between
0.81 and 39.5µm The lowest set ofdata and the theoretical curve correspond to the
bare surface tests Agreement between data and model is very good The highest set ofdata for layer thickness oft = 39.5 µm corresponds to the infinitely thick layer
where thermal spreading occurs in the layer only and the layer microhardness con-trols the formation of the microcontacts Again, the agreement between experiment and theory is good
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100
101
102
Pressure (kN/m )2
K)
Specimens Coating 08/09 None
0.81 m
39.5 m 5.1 m
1.4 m 1.2 m 0.81 m
1.2 m 1.4 m 39.5 m 5.1 m
” 22/23
16/17
10/11
12/13 18/19 14/15
Upper bound Infinite coating
Lower bound
No coating
Figure 4.32 Effect of layer thickness and contact pressure on joint conductance: vacuum data and theory (From Antonetti and Yovanovich, 1985.)
The difference between the highest and lowest joint conductance values is ap-proximately a factor of 10 The enhancement is clearly significant The agreement between the measured values ofjoint conductance and the theoretical curves for the layer thicknesses: t = 0.81, 1.2, 1.4, and 5.1 µm is also very good, as shown in
Fig 4.32 All the test points for bare and coated surfaces are plotted in Fig 4.33
as dimensionless joint conductance versus relative contact pressure The agreement between experiment and theory is very good for all points
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100
101
102
Relative Pressure ⫻104
h
mk⬘⬘ = 1.25 H P⬘0.95
h mk
⬘ ⬘
h
mk⬘⬘ = 1.25 H P⬘0.95
h mk
⬘ ⬘
P H⬘
( (
Series A = 4.27 m Series B = 1.28 m Series C = 8.32 m
Figure 4.33 Dimensionless joint conductance for a bare and silver layer on Ni 200 substrates versus relative contact pressure (From Antonetti and Yovanovich, 1985.)
A parametric study was conducted to calculate the enhancement that can be achieved when different metal types are used The theory outlined earlier will now
be applied to a common problem in electronics packaging: heat transfer across an aluminum joint What is required is a parametric study showing the variation in joint conductance as a function of metallic coating type and thickness for fixed surface
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TABLE 4.20 Assumed Nominal Property Values of Four Coatings
roughness and contact pressure The thermophysical properties ofthe coatings and the aluminum substrate material are presented in Table 4.20
Figure 4.34 shows the effect of the metallic layers on joint conductance As shown
in this figure, except for a very thin layer (about 1µm), the performance curves are
arranged according to layer microhardness Lead with the lowest microhardness has
Figure 4.34 Effect of layer thickness for four metallic layers (From Antonetti and Yovano-vich, 1983.)
... theory outlined earlier will nowbe applied to a common problem in electronics packaging: heat transfer across an aluminum joint What is required is a parametric study showing the variation...
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The effective microhardness must be obtained empirically for the particular layer (coating)–substrate combination under consideration This requires a series ofVick-ers... data-page="5">
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