## 3 Temperature Distribution \(T(t,r)\) in Glasses under Local Thermal Perturbations
### Dependence of \(T(t,r)\) on the Size of the Heating Zone
Indeed, the maximum cooling rate is approximately 2.5 times greater at \(\tau_{0}\
eq 0\) than at \(\tau_{0}=0\) both at the periphery and in the center of the heating zone; see Figure 11.  
Figure 9: \(T(t_{1},r)\) depending on the distance \(r\) at \(t_{1}=300\) ns (**a**) and time dependence of \(T(t,0)\) in the center of the hot zone (**b**) for borosilicate glass at \(r_{0}=2\) μm, \(\tau_{p}=300\) ps, \(E_{p}=120\) nJ, \(T_{in}=300\) K (\(\tau_{0}=0\), 0.3 \(\upmu\)s, 1 \(\upmu\)s, and 100 \(\upmu\)s—squares, circles, triangles facing up, triangles facing down, and diamonds). Let us consider the cooling rate \(-\mathcal{R}(t,r)\) in a thin shell in the region from \(0.95r_{0}\) to \(0.99r_{0}\). As we have established for borosilicate and sodium-line-silicate glasses, ultrafast cooling of the material occurs in this region at a rate of more than \(10^{11}\) K/s; see Figures 10-13. It is noteworthy that at the beginning of the cooling process (during the first few nanoseconds), the rate of temperature change \(\mathcal{R}(t,r)\) is the same for small and large relaxation times \(\tau_{0}\); see Figures 10a and 12a. This means that at the beginning of the cooling process at the periphery of the hot zone, the shape of the distribution function \(H(\tau_{0})\) does not affect the cooling rate \(-\mathcal{R}(t,r)\). Therefore, in this case, even at very small relaxation times \(\tau_{0}\), the influence of the time dispersion of the dynamic heat capacity is significant. Figure 10: Rate of temperature change \(\mathcal{R}(t_{1},r)\) depending on the distance \(r\) at the periphery of the hot zone at \(t_{1}=0.5\) ns (**a**) and time dependence of \(T(t,r_{1})\) at \(r_{1}=1.98\)\(\mu\)m (**b**) at the same parameters as in Figure 9 (\(\tau_{0}=0\), \(0.3\)\(\mu\)s, \(1\)\(\mu\)s, \(10\)\(\mu\)s, and \(100\)\(\mu\)s—squares, circles, triangles up, triangles down, and diamonds). Inset (**a**) shows \(\mathcal{R}(t_{1},r)\) as a function of distance \(r\) at \(t_{1}=100\) ns for different \(\tau_{0}\). The cooling rate \(-\mathcal{R}(t_{1},r)\) reaches a maximum at \(r_{max}\) for a given \(t_{1}\); see Figures 10 and 12. The rate \(-\mathcal{R}(t,r_{max})\) decreases over time, and the distance \(r_{max}\) at which the rate \(-\mathcal{R}(t,r_{max})\) is maximum shifts slightly toward the center of the heating zone. However, \(-\mathcal{R}(t,r_{max})\) still exceeds 2-3 GK/s at \(t_{1}=100\) ns; see Figures 10a and 12b. The change in cooling rate \(-\mathcal{R}(t,r_{max})\) over time for sodium-lime-silicate glass at \(r_{0}=1\) um, \(\tau_{0}=10\) us and the same conditions as in Figure 3 is presented in Table 3. The rate \(\mathcal{R}(t_{1},r)\) depending on the distance \(r\) at the periphery of the hot zone for different \(t_{1}\) is shown in Figure 13.  
Figure 12: \(\mathcal{R}(t_{1},r)\) depending on the distance \(r\) at the periphery of the hot zone at \(t_{1}=0.5\) ns (**a**) and \(t_{1}=100\) ns (**b**) for sodium-lime-silicate glass at the same parameters as in Figure 3 (\(\tau_{0}=0\) us, \(0.1\) us, \(1\) us, and \(10\) us—squares, circles, triangles up, and triangles down). Figure 13: \(\mathcal{R}(t_{1},r)\) depending on the distance \(r\) at the periphery of the hot zone at \(\tau_{0}=0.1\) μs (**a**) and \(\tau_{0}=10\) μs (**b**) at the same parameters as in Figure 3 (\(t_{1}=0.5\) ns, \(1\) ns, \(5\) ns, and \(10\) ns—squares, circles, triangles up, and triangles down). Thus, since the local structure of the material strongly depends on the local cooling rate \(-\mathcal{R}(t,r)\), the local structure of the material at the periphery of the hot zone changes significantly relative to the original glass matrix. This modification of glass practically stops as soon as the local region is cooled below the glass transition temperature \(T_{g}\). Thus, the modification process occurs until the temperature \(T(t,r_{1})\) in the local region at \(r=r_{1}\) drops below \(T_{g}\).