# Ultrafast melting and solidification

Short-pulsed laser melting of thin film involves the following three steps (Qiu and Tien, 1993; Kuo and Qiu, 1996): (1) absorption of photon energy by free electrons, (2) energy transfer between the free electrons and the lattice, and (3) phase change of the lattice due to the propagation of energy. The electron-lattice thermalization time is on the order of picoseconds (10-12 sec), which is significantly shorter than the time scale for most heat transfer problems. It is common practice to assume that the electrons and the lattice are always in thermal equilibrium and have the same temperature. However, when picosecond or femtosecond (10-15 sec) lasers are used in the material processing, the time scale of the laser pulse is comparable to or shorter than the electron-lattice thermalization time, and so the electrons and lattice are not in thermal equilibrium during short-pulse laser materials processing. Furthermore, rapid phase-change phenomena do not experience controlled heat transfer at the solid-liquid interface as suggested by the following boundary conditions at solid-liquid interface:

${k_s}\frac{{\partial {T_s}(x,t)}}{{\partial x}} - {k_\ell }\frac{{\partial {T_\ell }(x,t)}}{{\partial x}} = {\rho _\ell }{h_{s\ell }}\frac{{ds(t)}}{{dt}}\quad \quad x = s(t)$

The solid-liquid interface can be heated well above the melting point during a rapid melting process, in which case the solid becomes superheated. Similarly, the solid-liquid interface can be cooled far below the melting point in the rapid solidification process, in which case the liquid becomes undercooled. Both superheated solid and undercooled liquid are thermodynamically unstable and cannot be described using equilibrium thermodynamics. Once phase change is triggered in a superheated solid or undercooled liquid, the solid-liquid interface can move at an extremely high velocity (on the order of 10 to 102 m/s). In this case, the solid-liquid interface velocity is dominated by nucleation dynamics, which deviates significantly from equilibrium thermodynamics.

Laser melting of thin film

Short-pulsed laser melting of a thin film involves two nonequilibrium processes: (1) deposition of laser energy, and (2) initiation of melting. Therefore, successful modeling of short-pulsed laser melting requires modeling of these two fundamental processes. When the photons from the laser beam interact with the solid, the photon is first absorbed by the free electrons and then transferred to the lattice by interaction between the free electrons and lattice. Since the time scales of the laser pulse and the electron-lattice thermalization are comparable, the free electrons and the lattice have their own temperatures.

The picoseconds laser melting of free-standing gold film (see figure) is solved by Kuo and Qiu (1996). The problem can be approximated to be one-dimensional because the radius of the laser beam is significantly larger than the film thickness of the gold. The dual-parabolic two-step model can be used to describe this process and the energy equations of the free electrons and the lattice are

${C_e}({T_e})\frac{{\partial {T_e}}}{{\partial t}} = \frac{\partial }{{\partial x}}\left[ {{k_e}({T_e},{T_l})\frac{{\partial {T_e}}}{{\partial x}}} \right] - G({T_e} - {T_l}) + {q'''_{laser}} \qquad \qquad(1)$
${C_l}\frac{{\partial {T_l}}}{{\partial t}} = \frac{\partial }{{\partial x}}\left( {{k_l}\frac{{\partial {T_l}}}{{\partial x}}} \right) + G\left( {{T_e} - {T_l}} \right) \qquad \qquad(2)$

The source terms in eq. (1) can be described by the following equation:

${q'''_{laser}} = 0.94\frac{{1 - R}}{{{t_p}\delta }}J\exp \left[ { - \frac{x}{\delta } - 2.77{{\left( {\frac{t}{{{t_p}}}} \right)}^2}} \right] \qquad \qquad(3)$

where R is reflectivity of the thin film, tp is laser pulse duration (s), δ is laser penetration depth (m), and J is laser pulse fluence (J/m2).

For the conventional melting process, the velocity of the solid-liquid interface is obtained by energy balance as specified by

${k_s}\frac{{\partial {T_s}(x,y,t)}}{{\partial n}} - {k_\ell }\frac{{\partial {T_\ell }(x,y,t)}}{{\partial n}} = {\rho _\ell }{h_{s\ell }}{v_n}\quad \quad x = s(y,t)$

However, this is not the case for rapid melting/solidification processes, because the velocity of the interface is dominated by nucleation dynamics. For short-pulsed laser melting of gold, Kuo and Qiu (1996) recommend that the velocity of the solid-liquid interface be obtained by

$\frac{{ds}}{{dt}} = {V_s}\exp \left( { - \frac{{{h_{s\ell ,a}}}}{{{k_b}{T_m}}}} \right)\left[ {1 - \exp \left( { - \frac{{{h_{s\ell ,a}}}}{{{k_b}{T_m}}}\frac{{{T_m} - {T_{l,I}}}}{{{T_{l,I}}}}} \right)} \right] \qquad \qquad(4)$

where Vs is the maximum interface velocity that can be approximated as the speed of sound in the liquid phase. ${h_{s\ell ,a}}$ is the latent heat of fusion per atom, kb is the Boltzmann constant, and Tl,I is the lattice temperature at the interface.

The time t = 0 is defined as the time when the peak of a laser pulse reaches the film surface. Therefore, the initial conditions of the problem are

${T_e}(x, - 2{t_p}) = {T_l}(x, - 2{t_p}) = {T_i} \qquad \qquad(5)$

The boundary conditions of the problem can be specified by assuming that the heat loss from the film surface can be neglected, i.e.,

${\left. {\frac{{\partial {T_e}}}{{\partial x}}} \right|_{x = 0}} = {\left. {\frac{{\partial {T_e}}}{{\partial x}}} \right|_{x = L}} = {\left. {\frac{{\partial {T_l}}}{{\partial x}}} \right|_{x = 0}} = {\left. {\frac{{\partial {T_l}}}{{\partial x}}} \right|_{x = L}} = 0 \qquad \qquad(6)$

Equations (1) and (2) apply to short-pulsed laser melting problems, cases in which the electrons and lattice are not in thermal equilibrium. On the other hand, when the electrons and lattice are in thermal equilibrium, Te = Tl = T, one can add eqs. (1) and (2) to obtain the following one-step model:

$C\frac{{\partial T}}{{\partial t}} = \frac{\partial }{{\partial x}}\left( {k\frac{{\partial T}}{{\partial x}}} \right) + {q'''_{laser}} \qquad \qquad(7)$

where C = Ce + Cl and k = ke + kl are the heat capacity and thermal conductivity of the thin film, respectively. In arriving at eq. (7), it is assumed that the electron-lattice thermalization time is significantly smaller than the laser pulse; this is the case for long-pulse laser melting. The corresponding initial and boundary conditions for eq. (7) are

$T(x, - 2{t_p}) = {T_i} \qquad \qquad(8)$
${\left. {\frac{{\partial T}}{{\partial x}}} \right|_{x = 0}} = {\left. {\frac{{\partial T}}{{\partial x}}} \right|_{x = L}} = 0 \qquad \qquad(9)$

The one-step model represented by eqs. (7) – (9) is presented here in comparison with the two-step model.

Numerical solutions for both two-step and one-step models were obtained by Kuo and Qiu (1996). The two-step model predicts that the melting process is initiated near or after the end of laser irradiation, which is consistent with experimental observation. The one-step model, on the other hand, predicts that the film surface always starts melting during laser pulse irradiation, which is not true for picosecond laser melting. The superheating and undercooling predicted by the two-step model are significantly smaller than those predicted by the one-step model. Consequently, the maximum melting and resolidification velocities based on the two-step model are much smaller than those predicted by the one-step model. These results show that microscopic energy transfer and phase change are very important for picosecond laser processing of materials.

## References

Faghri, A., and Zhang, Y., 2006, Transport Phenomena in Multiphase Systems, Elsevier, Burlington, MA.

Faghri, A., Zhang, Y., and Howell, J. R., 2010, Advanced Heat and Mass Transfer, Global Digital Press, Columbia, MO.

Kuo, L.S., and Qiu, T.Q., 1996, “Microscale Energy Transfer During Picosecond Laser Melting of Metal Films,” ASME HTD-Vol. 323, Vol. 1, pp. 149-157.

Qiu, T.Q., and Tien, C.L., 1993, “Heat Transfer Mechanism During Short-Pulsed Laser Heating of Metals,” ASME Journal of Heat Transfer, Vol. 115, pp. 835-841.