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The resistance offered to the flow of heat by the mismatch of the elastic constants at the interface between two materials has been calculated. It is shown that for a perfectly joined interface the heat flow is proportional to the difference of the fourth powers of the temperature on each side of the interface. Deviations from this temperature dependence are to be expected for rough surfaces and for surfaces pressed into contact with one another. The calculated contact resistance between some common solids is given, and graphs are presented from which the heat flow between any two materials may be computed. It is shown too that the spin–phonon, phonon–electron, and phonon–phonon relaxation processes give rise to additional resistive processes in some solids, some of which restrict the heat flow at the surface. The theoretical results compare well with the available experimental data. However, the problem of the contact resistance between helium and metals is still unresolved. A tentative explanation of this is presented which predicts that there should be an appreciable difference between the contact resistance of a metal in the normal and in the superconducting state.

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