**4. Discussion**

Raman spectroscopy allows for temperature measurements with two methods: (a) Since the anti-Stokes and Stokes Raman intensities are proportional to the populations of their respective initial vibrational states, described by the Boltzmann distribution, the T of the sample can be estimated from the ratio of the Stokes and anti-Stokes intensities [37–39]; (b) the frequency position, the intensity or the shape of a (Stokes) Raman band may change as a function of temperature: The increase in temperature is expected to loosen chemical bonds (and hence to decrease the frequency of the mode), and/or induce more significant structural changes in the material under investigation [40–42]. Within this work, the local temperature has been investigated considering Stokes and anti-Stokes Raman spectra.

The anti-Stokes/Stokes ratio allows to determine the local temperature *T* of the sample, through the relation:

$$\rho = \frac{I\_{\rm aS}}{I\_{\rm S}} = C \cdot \frac{\left(\nu\_0 + \nu\_{\rm m}\right)^3}{\left(\nu\_0 - \nu\_{\rm m}\right)^3} \exp\left(-\frac{h\nu\_{\rm m}}{k\_B T}\right) \tag{1}$$

where *νm* is the frequency of the vibrational mode *m* considered, *ν*0 the laser frequency, *h* the Planck's constant, *kB* the Boltzmann constant and *C* is the calibration constant. A frequency dependence to the third power of the anti-Stokes/Stokes ratio is needed as the detection system is based on photon counting (CCD), whereas if the detection is energybased, a fourth power dependence is more appropriate [40]. The calibration constant is related to the experimental setup, in particular to parameters such as the polarization of the incident laser and the CCD and grating efficiency. The determination of the calibration constant, at each working excitation wavelength, is a key point to the determination of the local temperature.

#### *4.1. Determination of the Calibration Constant*

Raw Raman spectra were analyzed to obtain the parameters, such as area, intensity, frequency position and width of the peak. In particular, the anti-Stokes area of the Eg mode was divided by that of the Stokes signal to obtain the experimental anti-Stokes/Stokes ratio, *ρ* (reported as example in Table 2—Section 3.2 for the 514.5 nm excitation wavelength).

With the Raman scattering cross-section being nearly constant over the wavelength range explored, normalized intensities still differ depending on the CCD and grating efficiency (instrumental response function), which is particular of the instrumentation set-up used.

Equation (1) was fitted to the experimental values, collected at different temperatures and defined excitation wavelength, leaving C as a free parameter.

Figure 6 reports the anti-Stokes/Stokes ratio obtained from the calculated area of the anatase Raman modes, at 143, 397, 515 and 640 cm<sup>−</sup>1, in the temperature range 283–323 K, excited at 514.5 nm, and the curve resulting from the fitting with Equation (1) (the dashed lines). The corresponding calibration constants are calculated to be 0.9605, 0.9411, 0.9461 and 0.9491, respectively.

**Figure 6.** Experimental anti-Stokes/Stokes ratios in the range 283–323 K for the Raman modes of anatase, at 143 cm<sup>−</sup><sup>1</sup> (circles), 397 cm<sup>−</sup><sup>1</sup> (down-pointing triangles), 515 cm<sup>−</sup><sup>1</sup> (squares) and 640 cm<sup>−</sup><sup>1</sup> (up-pointing triangles) collected at *λexc* = 514.5 nm.

The anti-Stokes/Stokes experimental ratios of the 143 cm<sup>−</sup><sup>1</sup> Raman mode, measured at different excitation wavelengths as a function of the temperature imposed by the thermostat are reported in Table 4 and depicted in Figure 7, together with the curve obtained from the fitting and the corresponding calculated calibration constants, reported in Table 5. The curve is well fitted to experimental data, as the R2 parameters range from a minimum of 0.84, at 647 nm, to a maximum of 0.95, at 514.5 nm.

**Table 4.** Anti-Stokes/Stokes experimental ratios at different excitation wavelengths, as function of the temperature imposed by the thermostat.


**Figure 7.** Experimental anti-Stokes/Stokes ratios of the 143 cm<sup>−</sup><sup>1</sup> Eg mode of anatase as a function of the temperature, and corresponding calibration curves; data are collected by exciting at different wavelengths: 488.0 (blue squares and dashed line), 514.5 (green diamonds and dashed line), 568.2 (yellow triangles and dashed line) and 647.1 nm (red circles and dashed line).

**Table 5.** Calibration constants for the 143 cm<sup>−</sup><sup>1</sup> Raman mode at four different wavelengths.


As we can see qualitatively in Figures 6 and 7 and quantitatively in Tables 4 and 5, the calibration constant C is different depending on the excitation wavelength and the frequency of the Raman mode, due to the difference in response of the optical components of the experimental set-up to the wavelength. For this reason, it is necessary to individuate a calibration constant for a defined Raman mode, at a specific excitation wavelength.

#### *4.2. Comparison between the Four Raman Modes of Titanium Dioxide*

In order to individuate the best Raman mode, to obtain the more efficient signal in the determination of the local temperature, it is necessary to compare the behavior of the four TiO2 Raman modes as a function of the temperature variation. Figure 8a shows, for example, the theoretical anti-Stokes/Stokes ratios of the Titania modes, calculated with excitation wavelength at 514.5 nm, in the temperature range of interest, 283–323 K. The 143 cm<sup>−</sup><sup>1</sup> Eg anti-Stokes/Stokes ratio shows the highest value, in comparison with the other Raman modes, it presents values in the range 0.50–0.55, while the 397 cm<sup>−</sup><sup>1</sup> mode, the 515 cm<sup>−</sup><sup>1</sup> and the 640 cm<sup>−</sup><sup>1</sup> modes present values in the ranges 0.15–0.19, 0.08–0.12 and 0.05–0.07, respectively. The total variations of the ratio, in the whole T range (Δ*ρ*), decreases from 0.0476, with the 143 cm<sup>−</sup><sup>1</sup> mode, to 0.0233, with the 640 cm<sup>−</sup><sup>1</sup> mode. At the excitation wavelengths of 488.0, 568.2 and 647.1 nm the behavior and the values obtained are comparable to that obtained at 514.5 nm.

**Figure 8.** (**a**) Theoretical behavior and (**b**) sensitivity of the anti-Stokes/Stokes ratio of the four anatase modes, at 143, 397, 515, and 640 cm<sup>−</sup>1, calculated with *λexc* = 514.5 nm, in the range 283–323 K.

To evaluate the efficiency in temperature detection, one of the most used figures of merit of thermometry is the sensitivity, calculated, for Raman measurements, through the following Equation [9,16]:

$$S = \left| \frac{\partial \rho}{\partial T} \right| = \left| -\frac{\left(\nu\_0 + \nu\_m\right)^3}{\left(\nu\_0 - \nu\_m\right)^3} \frac{h\nu\_m}{kT^2} \exp\left(-\frac{h\nu\_m}{kT}\right) \right| \tag{2}$$

where *ρ* is the anti-Stokes over Stokes ratio.

The derivative with respect to the temperature, has been evaluated at 514.5 nm, for all the Raman modes of anatase. The results, plotted in Figure 8b, show an increasing in sensitivity corresponding to the decreasing in frequency of the Raman modes and a decrease of the sensitivity of all Raman mode as temperature increases, also already observed in literature [43]. This can be attributed to the fact that at high temperature the differences between the population of the ground state and the first vibrational excited state are smaller than at room temperature.

In particular, it is possible to evaluate the sensitivity, at 300 K close to the room temperature, and to compare the obtained values for different wavelengths and different Raman modes of titanium dioxide. The outcome, shown in Table 6, is that the sensitivity at constant excitation wavelength decreases with increasing frequency of the Raman mode and for a given Raman mode it increases with increasing excitation wavelength. Corresponding to the 143 cm<sup>−</sup><sup>1</sup> Eg mode a thermal resolution in the range 1.2 (@ 514.5 nm) to 3.4 K (@ 647.1 nm) have been calculated. All these experimental data are comparable with those already reported in literature [9].

**Table 6.** Sensitivity at 300 K of the anti-Stokes/Stokes ratio to temperature, calculated for the 143, 397, 515 and 640 cm<sup>−</sup><sup>1</sup> modes of anatase.


The behavior of the theoretical ratio can be compared to the experimental anti-Stokes/Stokes ratios for the four Raman modes, reported in Figure 6. It is evident that

there is a good agreement, confirming the 143 cm<sup>−</sup><sup>1</sup> Raman mode, the lowest in Raman frequency, the more sensitive to Temperature variation. Moreover, the sensitivity calculated starting from the experimental data overlaps well the theoretical ones.

#### *4.3. Validation of the Method and Temperature Determination: Test Sample*

By using the calibration constants, it is possible to determine the local temperature. In Figure 9a, the temperature derived from repeated measurements, performed on a different region of the *test sample*, of the anti-Stokes/Stokes ratio for the 143 cm<sup>−</sup><sup>1</sup> mode of anatase at 514.5 nm, at 297 K, is reported; the figure also shows the mean temperature calculated for these measurements and the standard deviation. In this context, the standard deviation of temperatures, rather than the errors derived from each measurement, is reported, as it is preferable when repeated measures are conducted; moreover, the two values have the same order of magnitude (few kelvins). Detailed values are reported in Table 7, and it turns out that the data show an excellent overlap between the expected and measured data, with a deviation as low as 3 K, maximum. Moreover, the results obtained by changing the input power, reported in Figure 9b, confirm that the local sample temperature is not affected by the laser power, when an incident power of few mW is used, while at higher input powers the laser is heating the sample, at all the used wavelengths.

**Figure 9.** (**a**) Temperature determined from the anti-Stokes/Stokes values of the 143 cm<sup>−</sup><sup>1</sup> Eg mode at *λexc* = 514.5 nm of the *Test Sample*, with the mean value (solid line) and the standard deviation (dashed lines); (**b**) difference between the temperature derived from the anti-Stokes/Stokes ratio (corrected with the proper calibration constant) for the 143 cm<sup>−</sup><sup>1</sup> Eg mode of anatase as function of the laser power incident on the sample at 488.0 (blue squares), 514.5 (green diamonds), 568.2 (yellow triangles) and 647.1 nm (red circles). The dashed line represents the room temperature.

**Table 7.** Results of repeated measurements at four different excitation wavelengths conducted at room temperature.


These results confirm the validation of the method thus verifying that the temperature measured all over the sample is uniform.

It is possible to conclude that the Raman modes of anatase, in particular the Eg one at 143 cm<sup>−</sup>1, are excellent candidate for the local temperature detection in the visible range. However, the need remains to investigate the behavior at longer wavelengths, towards the near IR, where the biological window is located.

The performances obtained with this TiO2 based Raman thermometer are compatible with data reported in literature [9,44]; the lower thermal resolution, with respect to fluorescent thermometer, is compensated with the wider wavelength working range.
