*3.3. Determination of Model Parameters: η*, *U, and SW*

The order of magnitude values of *<sup>η</sup>* (adiabatic cells) and *<sup>U</sup>* were 0.65 and 7 W/m2·K, respectively. These values were used to initiate the minimization of the sum of squares of differences between the experimental and predicted temperature increments. The resulting value of *η* was considered constant for all the nanoparticle concentrations. This consideration is justified by the low PPN polydispersity (monodisperse particles), the homogeneous nanoparticle morphology, and the range of particle concentrations used [21,43]. The parameter *U* changed proportionally to the nanoparticle concentration. This effect could be explained by the coefficient *U*'s dependence on the media's temperature after irradiation [27].

The resulting value *η*, from the minimization of the sum of squares, was 0.68. This value is greater than other similar systems of PPN prepared using PVP and the same oxidant [2,5] and similar to gold nanoparticles of similar sizes [44]. Other works using FeCl3 as oxidant have shown remaining Fe at the PPN surface in the form of iron oxides, improving the electrical conductivity and optical properties of the particles by the formation of a semiconductor–semiconductor heterojunction [45]. For instance, these remaining oxides could also improve the *η* value. The values of *U* for different PPN concentrations in the suspension are shown in Table 2. Furthermore, a value of 4.7 × 105 (W/m3) was fitted to *SW* by simulation of the particle-free suspension. These values agree with other values reported in the literature [46,47].


**Table 2.** Maximum temperature increments and the corresponding values of *U* as a function of the PPN concentration.

#### 3.3.1. Photothermal Effect of PPN at Different Concentrations

The photothermal effect of PPN at different concentrations under laser irradiation and the respective heat transfer model fittings are presented in Figure 6. The heating analysis as a function of PPN concentration was evaluated at a fixed pH value of 7.4 to simulate biological conditions. The concentrations of PPN evaluated were 30, 15, 7.5, 3.75, 1.875, 0 μg/mL, equivalent to the absorbances of 1.677, 0.839, 0.419, 0.210, and 0.105, respectively (Figure 6A–E). The spatiotemporal temperature distributions of irradiated PNP suspensions show the same heating profile, but different temperature increments are reached proportionally to the nanoparticle concentration increase. The profiles started with a period of rapid heating until a plateau was reached, indicating a steady state. Results show a direct dependency between the PPN concentration and the temperature increment obtained, with maximum temperature increments of 27.6, 24.8, 17.7, 11.7, and 6 ◦C for 30, 15, and 7.5, 3.75, and 1.875 μg/mL, respectively. Furthermore, a slight temperature increment for the 0 μg/mL solution is presented after irradiation, confirming that most of the temperature increments are due to the laser interactions with the PPN. Comparable results have been reported in the literature for the spatiotemporal temperature distributions of other irradiated materials, such as gold nanoparticles [48], gold nanorods [24], carbon nanotubes [49], iron oxide nanoparticles [50], etc.

**Figure 6.** Temperature profiles of PPN suspensions at pH values of 7.4 and model fitting at concentrations of (**A**) 30, (**B**) 15, (**C**) 7.5, (**D**) 3.75, (**E**) 1.875, and (**F**) 0 μg/mL. Circles represent experimental data, and the continuous solid lines represent the model fitting. Means ± SD, *n* = 3.

The mathematical analysis for the spatiotemporal temperature distributions of irradiated PNP at different concentrations shows good agreement with the experimental data. The model predicted the temperatures with an average absolute error below 1 ◦C, compared to the experimental data, as shown in Figure S4. The best agreement is found at higher PPN suspension concentrations. However, with lower nanoparticle concentrations, the model underestimates at the early stages of heating, overestimating the temperature change when reaching the thermal plateau. Furthermore, a linear correlation between the overall heat transfer coefficient and temperature increment is shown in Figure 7, contrasting with the non-linear dependency predicted by the correlation used as a starting point (Figure S2).

**Figure 7.** Estimated overall heat transfer coefficient. Circles represent the overall heat transfer coefficient, and the continuous dotted line represents a linear fitting.

#### 3.3.2. Photothermal Effect of PPN at Different pH

The photothermal effect of PPN at different pH was performed at an average nanoparticle concentration of 28.5 μg/mL. Figure 8A–E present the measured temperature profiles and their model predictions at solution pH values of 4.2, 6.0, 7.4, 8.0, and 10, respectively. The mathematical analysis at different pH values agrees well with the experimental data. The model predicted the temperatures with an average absolute error below 1 ◦C compared to the experimental data (Figure S5). The heating profiles in Figure 8 present similar behavior within them, observing a gradual temperature increase until a steady temperature is reached.

**Figure 8.** Laser irradiation effects of PPN suspensions as a function of the pH. Temperature profiles of PPN suspensions and model fitting at pH values of (**A**) 7.4, (**B**) 6.0, (**C**) 7.4, (**D**) 8.0, and (**E**) 10. Circles represent the experimental data of PPN suspensions, and the continuous solid lines represent model fitting. Means ± SD, *n* = 3. (**F**) The efficiency of transducing resonant irradiation of light to heat (*η*) for a PPN suspension of 28 μg/mL as a function of the pH. Circles represent the *η* value obtained, and the dotted line the linear adjustment.

Furthermore, a small increment in the suspensions' maximum temperature occurs when pH increases. This increment in the maximum temperature is not related to the absorbance of the suspensions but to variations in the efficiency of transducing resonant irradiation of light to heat. As a result, the model predicts that this efficiency increases linearly with the pH values of the suspension (Figure 8F). This finding is supported by the ζ value measurements presented in Figure 5B, in which a decrease in the ζ value occurs as the pH increases, indicating that the nanoparticle surface charge affects the transduction of light.

The predicted temperature contours for a concentration of PPN of 3.75 μg/mL and pH 7.4 are presented in Figure 9. At a glance, no matter which contour is selected, the laser region is the hottest spot in the contour, unlike the suspension/air interface in the suspension region, which remains almost at room temperature during the experiment. The temperature of ~51 ◦C is reached in the laser region, with an exposure time of 25 min. The contour analysis helps to understand and control the maximum temperature reached in the irradiated volume as a function of time, which is necessary for all biological applications to avoid undesirable effects, for example, necrosis, damage surrounding healthy tissue, and the possible effects of prolonged laser exposure.

**Figure 9.** Computed temperature contours at different times during the suspensions heating at PPN of 3.75 μg/mL and pH of 7.4. Plane xz at y = 6.3 mm located at the midpoint of the cuvette depth. The color scale represents temperatures in K.

#### **4. Conclusions**

Polypyrrole nanoparticles (PPN) were synthesized by oxidative chemical polymerization, resulting in an appropriate size, good stability, and strong absorption in the NIR region, suitable for applications in photothermal therapies. The zeta potential (ζ) of PPN could be tuned from positive to negative values by controlling the pH of the suspensions while maintaining similar sizes. A three-dimensional mathematical model that considers the extinct radiation of suspensions of PPN and the photothermal transduction efficiency was developed. A direct approach was proposed to determine the order of magnitude of the photothermal transduction efficiency and the overall heat transfer coefficient. The heat transfer model shows good agreement between the experimental and the predicted temperature changes. Furthermore, a linear dependency of the overall heat transfer coefficient with the temperature was found. For PPN of 60 nm dispersed at a pH of 7.4, the photothermal transduction efficiency had a value of 0.68.

Additionally, a linear dependency was found between the photothermal transduction efficiency and the pH of the suspensions. The model could predict temperature zones with potential photothermal therapy use around the laser region regarding nanoparticle concentration and power. The current approach for modeling the conversion of NIR light into heat by using nanoparticle suspensions could contribute to the analysis and design of systems/devices for photothermal ablation of cells and tissues.

**Supplementary Materials:** The following supporting information can be downloaded at: https://www. mdpi.com/article/10.3390/polym14153151/s1, Figure S1: Temperature gradients at the thermocouple position (6.25, 6.25, 16 mm) for the adiabatic cell for different photothermal transduction efficiency (*η*). The experimental data correspond to a concentration of PPN of 30 μg/mL and a pH value of 7.4; Figure S2: Calculated temperature gradients for different values of overall heat transfer coefficient (*U*). The solid dots correspond to the experimental heating data at a concentration of PPN of 30 μg/mL and a pH value of 7.4; Figure S3: PPN calibration curve at a wavelength of 808 nm. The mass extinction coefficient (ε) of PPN is 0.0559 *a*.*u*./ *μg mL* ·*cm* . Mean ±SD, *n* = 3; Figure S4: Comparison of simulated temperature gradients and experimental temperature gradients for the different concentrations of PPN at a pH value of 7.4; Figure S5: Comparison of simulated temperature gradients and experimental temperature gradients for the different pH values at a concentration of PPN of 30 μg/mL.

**Author Contributions:** Conceptualization, A.L.-A.; methodology, A.L.-A., O.P.-M. and M.O.-M.; formal analysis, M.O.-M., J.A.I.-E. and K.Y.H.-G.; investigation, O.P.-M. and R.J.R.-C.; resources, A.L-A., M.O.-M. and P.Z.-R.; data curation R.J.R.-C. and D.F.-Q.; writing—original draft preparation, A.L.-A., O.P.-M., M.O.-M. and K.Y.H.-G.; writing—review and editing, A.L.-A., D.F.-Q. and P.Z.-R.; supervision, A.L.-A.; project administration, A.L.-A.; funding acquisition, A.L.-A., P.Z.-R., M.O.-M., R.J.R.-C. and D.F.-Q. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Acknowledgments:** Omar Peñuñuri-Miranda, Karol Yesenia Hernández-Giottonini, and Rosalva Josefina Rodríguez-Córdova thank the National Council of Science and Technology of Mexico (CONA-CYT) for the graduate scholarships 638,788, 302,373, and 281,669, respectively. We are grateful for the use of the transmission electron microscopy (TEM) facilities of the University of Sonora.

**Conflicts of Interest:** The authors declare no conflict of interest.

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