## 3 Temperature Distribution \(T(t,r)\) in Glasses under Local Thermal Perturbations
### Dependence of \(T(t,r)\) on the Size of the Heating Zone  
For example, let us compare the temperature distributions \(T(t,r)\) at \(r_{0}=1\) and \(r_{0}~{}=~{}2\) um for borosilicate glass; see Table 1. To obtain the same thermal response value, let us set \(E_{p}=120\) nJ at \(r_{0}=2\) um, increasing the energy in proportion to the volume of the heating zone. Thus, in both cases, we obtain the same amplitudes of the temperature response to heating pulses; see Figures 8 and 9. As expected, the temperature \(T(t,0)\) in the center of the heating zone relaxes in a time period of the order of \(r_{0}^{2}/D_{0}\), which for borosilicate glass is about 15 us at \(r_{0}=2\) um and \(D_{0}=2.6\cdot 10^{-7}\) m\({}^{2}\)/s. For comparison, \(T(t,0)\) relaxes four times faster at \(r_{0}=1\) um than at \(r_{0}=2\) um; see Figures 8b and 9b. As the size of the heating zone increases, the saturation of the influence of the time dispersion of the dynamic heat capacity on \(T(t,r)\) shifts towards larger values of \(\tau_{0}\). For example, compare \(T(t,r)\) at \(\tau_{0}=1\) us for \(r_{0}=1\) and \(r_{0}=2\) um; see Figures 8 and 9. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Substance** & \(f_{a}\) & \(B\) & \(T_{a}\) \\  & \(\rm Hz\) & \(\rm K\) & \(\rm K\) \\ \hline Borosilicate glass & 5\(\cdot 10^{7}\) & 1.9\(\cdot 10^{4}\) & 392 \\ \hline Sodium-time-magnesium-silicate glass & 1\(\cdot 10^{4}\) & 1.1\(\cdot 10^{4}\) & 480 \\ \hline Sodium-silicate glass & 1.24\(\cdot 10^{4}\) & 1.29\(\cdot 10^{4}\) & 407 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Relaxation parameters of silicate glasses used for model calculations. Cooling Rate Distribution \(-\mathcal{R}(t,r)\) and Its Influence on Microstructuring Processes of Glasses  
Now consider the spatial distribution of the rate of temperature change \(\mathcal{R}(t,r)\). Due to thermal expansion, the size of the heating zone changes with temperature by about 10 nm or less. We neglect these changes with respect to \(r_{0}\). Let us consider the rate of temperature change \(\mathcal{R}(t,r)=\frac{\partial\mathcal{T}(t,r)}{\partial t}\) due to thermal diffusion. This temperature change affects the physical properties of the material. We will focus on very fast (about \(10^{9}\) K/s or more) local temperature changes that have the greatest impact on the microstructuring process and will study the effect associated with the time dispersion of the dynamic heat capacity. It is worth noting that this effect is significant even well above the glass transition temperature \(T_{g}\). For example, the rate of temperature change \(\mathcal{R}(t,r)\) for borosilicate glass at \(r_{0}=2\) um, \(\tau_{p}=300\) ps, \(E_{p}=120\) nJ and various \(\tau_{0}\) values is presented in Figures 10 and 11. Note that the rate of temperature change \(\mathcal{R}(t,r)\) is greatest near the periphery of the hot zone; see Figure 10a. The cooling rate \(-\mathcal{R}(t,r)\) is about 600 GK/s at \(r_{1}=1.98\) um and \(t_{1}=0.36\) ns; see Figure 11a. However, in the center of the heating zone, the cooling rate \(-\mathcal{R}(t,0)\) does not exceed 0.6 GK/s; i.e., \(-\mathcal{R}(t,0)\) is three orders of magnitude less than the cooling rate at the periphery--see Figure 11. Therefore, the local structure of glass in the center of the hot zone should differ significantly from the structure at the periphery of the heating zone after laser modification. This conclusion is consistent with experiments [19; 33; 35; 39; 45; 48; 49; 50; 51; 52]. However, the effect associated with dynamic heat capacity (at \(\tau_{0}\
eq 0\)) is significant both in the center and at the periphery of the heating zone.