This page calculates the temperature difference and the thermal contact conductance (thermal resistance) for the interface formed by two conforming, rectangular, rough surfaces as a function of contact pressure for the low pressure range (between 0.035 and 0.35 MPa). This is commonly occurs in microelectronic applications. This calculator focuses on bare joints filled with air, helium, as well as interfaces filled with thermal greases of varying thermal conductivities. It can be shown that radiation heat transfer is negligible and does not need to be considered for this analysis.

Also note that the interface resistance calculation is a theoretical best, and that bound line thickness, oxide scale, and failure of the interface to wet the surfaces will significantly increase interface resistance.

The temperature difference *T _{1-2}* is
calculated as:

T_{1-2} = q / (h_{j} x L x W)

Where *q* is the heat transfer, *h _{j}* the
thermal joint conductance,

r_{j} = 1 / (L x W x h_{j})

The thermal joint conductance is calculated as:

h_{j} = h_{c} + h_{g}

Where *h _{c}* is the thermal contact conductance
and

h_{g} = k_{g} / (Y + M)

The thermal conductivity of the gap substance is given by *k _{g}*,
the effective gap thickness is represented by

Y = 1.53 sig (P / H_{c})^{-0.097}

M = M_{0} (T / T_{0}) (P_{g,0} / P_{g})

Where *sig* is the effective RMS surface roughness of the
contacting asperities, *P* is the contact pressure, and *H _{c}*
is the surface microhardness of the softer of the two contacting solids. For the
calculation of the gas parameter,

The effective RMS surface roughness is calculated as:

sig = (sig_{1}^{2} + sig_{2}^{2})^{1/2}

Where *sig _{1}* is the RMS surface roughness of
surface 1 and

The contact conductance *h _{c}* is given by:

h_{c} = 1.25 k_{s} (m / sig) x (P / H_{c})^{0.95}

Where * k*_{s} is the harmonic mean thermal conductivity
of the interface, *m* is the effective mean asperity slope of the
interface, *sig* is the effective RMS surface roughness, *P* the
contact pressure, and *H _{c}* the surface microhardness. The
harmonic mean thermal conductivity can calculated as:

k_{s} = 2 k_{1}k_{2} / (k_{1}
+ k_{2})

Where *k _{1}* and

The effective mean absolute asperity slope of the interface is given by (The with surface roughnesses must be between 0.216 and 9.6 micrometers):

m = 0.125 (sig)^{0.406}

*m _{1}* and

Although the calculator is mainly for bare surfaces, it can be applied to surfaces with thermal grease. It can be assumed that the grease behaves like a liquid and fills all gaps (wets the interface surfaces completely) between the asperities, then this model is used by substituting M = 0 and the grease thermal conductivity into the gap conductance relationship. This calculator can not be used when solid interstitial materials (i.e. thermal compounds, elastomers, or adhesive tapes) are used.

**References**

M. Yovanovich, J. R. Culham and P. Teertstra, "Calculating Interface Resistance", Microelectronics Heat Transfer Laboratory, Department of Mechanical Engineering University of Waterloo

**Thanks**

Brent Webb, Intel Corporation

Copyright 1999 (c) MAYA Simulation.