# Methods and tools to solve the two-temperature model (TTM)

I would like to model heat diffusion at the gold / water interface after excitation of the metal surface by an ultrafast laser pulse (ca. 80 fs).

An appropriate model to start with would be the "two temperature model" (TTM) or "parabolic two step" (PTS). This model considers the electronic and lattice systems as separate with electronic and lattice temperature $T_e$ and $T_l$, respectively. The electronic and lattice systems are coupled through an electron-phonon coupling constant $G$ and the electronic system is excited by a source term $S$ representing the incoming laser pulse. The model can be written as two coupled PDEs [from Smith (2001) Appl. Phys. Lett. 78:1240]: $$C_e(T_e) \frac{\partial T_e}{\partial t} = \frac{\partial}{\partial x} \left( k_e(T_e, T_l) \frac{\partial T_e}{\partial x} \right) - G \left[ T_e - T_l \right] + S \\ C_l \frac{\partial T_l}{\partial t} = G \left[ T_e - T_l \right],$$ where $C_e$, $C_l$ are electron and lattice heat capacities, respectively, and $k_e$ is the thermal conductivity.

I am specifically interested in determining the temperatures of the system for the first 100 ps.

The question I would like to ask is: What are the different methods and tools available to solve the problem, with an emphasis on entry-level methodologies?

• This should be relatively simple to solve with a one-dimensional finite element method. What have you tried so far? – Wolfgang Bangerth Jan 27 '18 at 20:42
• Thank you @WolfgangBangerth for you contribution. Finite element is the method I have tried already, which I implemented in Fenics to no success. I would thus like to know about the alternatives, if any. – François Jan 28 '18 at 12:26
• Well, "to no success" doesn't give much away. Was it the method that failed? Did you not get along with FEniCS? Are you looking for other methods or other software systems? – Wolfgang Bangerth Jan 29 '18 at 18:19
• @WolfgangBangerth: Indeed. I didn't want to turn this thread as a support request on FEniCS. If you are interested, though, my latest woes are documented at allanswered.com/post/wlapm/… – François Jan 29 '18 at 18:42
• I would like to use this space to enumerate and document the technics and tools people normally use to solve this type of problems. I am not an expert in the domain and this would be useful to me and, I believe, to others who might in the future be looking for an answer to that. – François Jan 29 '18 at 18:45