The Second Law of Thermodynamics for a Two-Temperature Model of Heat Transport in Metal Films

ABSTRACT The second law of thermodynamics asserts that heat will always flow “downhill”, i.e., from an object having a higher temperature to one having a lower temperature. For a parabolic rigid heat conductor with a single temperature T and a single heat-flux q this amounts to the statement that the inner product of q and ∇T must be non-positive for every point x of the conductor and for every non-negative time t. For a homogeneous and isotropic body in which classical Fourier law with a heat conductivity coefficient k is postulated, the second law is satisfied if k is a positive parameter. For ultra-fast pulse-laser heating on metal films, a parabolic two-temperature model coupling an electron temperature Te with a metal lattice temperature Tl has been proposed by several authors. For such a model, at a given point of space x and a given time t there are two different temperatures Te and Tl as well as two different heat-fluxes q e and q l related to the gradients of Te and Tl, respectively, through classical Fourier law. As a result, for a homogeneous and isotropic model the positive definiteness of the heat conductivity coefficients ke and kl corresponding to Te and Tl, respectively, implies that the second law of thermodynamics is satisfied for each of the pairs (Te, q e) and (Tl, q l), separately. Also, the positive definiteness of ke and kl, and of the corresponding heat capacities ce and cl as well as of a coupling factor G imply that a temperature initial-boundary value problem for the two-temperature model has unique solution. In the present paper, an alternative form of the second law of thermodynamics for the two-temperature model with kl = 0 and q l = 0 is obtained from which it follows that in a one-dimensional case the electron heat-flux qe(x, t) has direction that is opposite not only to that of ∂Te(x, t)/∂x but also to that of ∂Tl(x, t + τT)/∂x, where τT is an intrinsic small time of the model. Also, for a general two-temperature rigid heat conductor in which ke, kl, ce, cl, and G are positive, an inequality of the second law of thermodynamics type involving a pair (Te − Tl, q e − q l) is postulated to prove that a two-heat-flux initial-boundary value problem of the two-temperature model has a unique solution. For a one-dimensional case, the semi-infinite sectors of the plane ( q l, q e) over which uniqueness does not hold true are also revealed.