Interface condition for heat exchange

I would like to compute the heat distribution of a piece of metal with some surrounding material. The heat is assumed to propagate by diffusion, so inside the metal piece and also on the outside, the governing law would famously be $$\rho c_p \frac{\partial u}{\partial t} = \nabla(\kappa \nabla u).$$ What would be the appropriate condition for the interface between the two materials?

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The following are closely related, but I'm not sure consensus was ever reached. physics.stackexchange.com/questions/52900/…, physics.stackexchange.com/questions/56992/… – joshphysics Apr 5 '13 at 15:41