Time-resolved temperature rise in a thin liquid film due to laser absorption

The temperature increase of a thin water layer is investigated, both experimentally and numerically, when the layer is heated by an infrared laser. The laser is focused to a waist of $5.3\mu \mathrm{m}$ inside a $28\mu \mathrm{m}$ gap that contains fluorescent aqueous solutions between two glass slides. Temperature fields are measured using the temperature sensitivity of rhodamine-B, while correcting for thermal diffusion using rhodamine-101, which is insensitive to temperature. In the steady state, the shape of the hot region is well fitted with a Lorentzian function whose width ranges between 15 and $30\mu \mathrm{m}$, increasing with laser power. At the same time, the maximum temperature rise ranges between 10 and $55°\mathrm{C}$ and can display a decrease at high laser powers. The total energy stored in the sample increases linearly with the laser power. The dynamics of the heating occurs with two distinct time scales: (i) a fast time (${\tau }_{\Theta }=4.2\mathrm{ms}$ in our case) which is the time taken to reach the maximum temperature at the laser position and the maximum temperature gradient, and (ii) a slow time scale for the spatial profile to reach its final width. The temperature field obtained numerically agrees quantitatively with the experiments for low laser powers but overpredicts the temperature rise while underpredicting the profile width for high powers. The total energy shows good agreement between experiments and simulations for all laser powers, suggesting that the discrepancies are due to a broadening of the laser, possibly due to a thermal lensing effect.

Figure 1

(Color online) (a) Experimental setup. (b) Lateral view of the sample (not to scale).

Figure 2

(Color online) Temperature calibration of the rho-B solution fluorescence.

Figure 3

(Color online) (a), (d) Radial fluorescence profiles after $1\mathrm{s}$ of laser heating with $P_0=76.4\mathrm{mW}$ for the rho-101 and rho-B solutions, respectively. (b), (e) Fluorescence intensity at $r=0$ as a function of time for the rho-101 and rho-B solutions, respectively. (c), (f) Total fluorescence intensity as a function of time for the rho-101 and rho-B solutions respectively. Intensity scales are in arbitrary units.

Figure 4

(Color online) Evolution of the radial temperature profile. The laser power is $P_0=76.4\mathrm{mW}$, which corresponds to an absorbed power of $P_{\mathrm{in}}=4.5\mathrm{mW}$.

Figure 5

(Color online) (a) $\Theta \left(t\right)$ for $P_{\mathrm{in}}=4.5\mathrm{mW}$. The dashed line indicates the maximum temperature increase ${\Theta }_{\infty }$. In the inset the first $40\mathrm{ms}$ are shown, and the solid line corresponds to a linear fit of the first points. (b) $\sigma \left(t\right)$ for the same data. The dashed line indicates the value of ${\sigma }_{\infty }$. The inset shows the first $100\mathrm{ms}$. (c) $\dot{\Theta }$ as a function of $P_{\mathrm{in}}$. (d) ${\tau }_{\Theta }$ as a function of $P_{\mathrm{in}}$. The solid line indicates its mean value.

Figure 6

(Color online) Evolution of the maximum thermal gradient after the laser is turned on, for an absorbed power $P_{\mathrm{in}}=4.5\mathrm{mW}$. The gradient reaches its maximum absolute value at $t=4\mathrm{ms}$.

Figure 7

(Color online) (a) Dots: Radial temperature profile after $0.5\mathrm{s}$ of laser heating at $P_{\mathrm{in}}=4.5\mathrm{mW}$. Solid line: Lorentzian fit. (b) ${\Theta }_{\infty }$ as a function of $P_{\mathrm{in}}$. (c) ${\sigma }_{\infty }$ as a function of $P_{\mathrm{in}}$. (d) $\Delta E$ as a function of $P_{\mathrm{in}}$ (dots) and linear fit (solid line). For (b)–(d), the error bars here were calculated as the standard deviation of the data for all the time steps between $0.5<t<1\mathrm{s}$.

Figure 8

(Color online) Comparison between the experimental and the numerical results. (a) ${\Theta }_{\infty }$ as a function of $P_{\mathrm{in}}$. The dots correspond to the measurements and the triangles to the numerical simulations. (b) Experimentally and numerically obtained radial temperature profiles ${\overline{T}}_{\infty }\left(r\right)$ for $P_{\mathrm{in}}=1.3\mathrm{mW}$ (circles and solid line) and $P_{\mathrm{in}}=4.1\mathrm{mW}$ (stars and dashed line). (c) $\Theta \left(t\right)$ for $P_{\mathrm{in}}=1.3\mathrm{mW}$. The dots correspond to the experiment and the solid line to the simulation. (d) $\Delta E$ as a function of $P_{\mathrm{in}}$ obtained by the experiments (dots) and by the simulations (triangles).

Figure 9

(Color online) Simulated variations of the main parameters for a laser power $P_0=20\mathrm{mW}$. (a),(b) ${\Theta }_{\infty }$ as a function of ${\omega }_0$ and $2h$, respectively.(c),(d) $\Delta E$ as a function of ${\omega }_0$ and $2h$, respectively. (e),(f) Comparison between ${\overline{T}}_{\infty }\left(r\right)$ and $\Theta \left(t\right)$ for two glass walls (dashed lines) and a combination of one glass and one PDMS wall (solid lines), as in the case of a microchannel.