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 T_e with a metal lattice temperature T_l 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 T_e and T_l as well as two different heat-fluxes q _e and q _l related to the gradients of T_e and T_l, respectively, through classical Fourier law. As a result, for a homogeneous and isotropic model the positive definiteness of the heat conductivity coefficients k_e and k_l corresponding to T_e and T_l, respectively, implies that the second law of thermodynamics is satisfied for each of the pairs (T_e, q _e) and (T_l, q _l), separately. Also, the positive definiteness of k_e and k_l, and of the corresponding heat capacities c_e and c_l 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 k_l = 0 and q _l =  0 is obtained from which it follows that in a one-dimensional case the electron heat-flux q_e(x, t) has direction that is opposite not only to that of ?T_e(x, t)/?x but also to that of ?T_l(x, t + τ_T)/?x, where τ_T is an intrinsic small time of the model. Also, for a general two-temperature rigid heat conductor in which k_e, k_l, c_e, c_l, and G are positive, an inequality of the second law of thermodynamics type involving a pair (T_e ? T_l, 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.