# Thermal Diffusivity of Aqueous Dispersions of Silicon Oxide Nanoparticles by Dual-Beam Thermal Lens Spectrometry

^{*}

## Abstract

**:**

## 1. Introduction

_{p}). Liquids, on the contrary, due to their higher C

_{p}, are able to retain heat for a long time, but due to the lower k, they provide lower heat transfers [3]. Viscosity (η) is also a problem for heat transfer fluids, which limits the use of fluids and reduces the heat-transfer efficiency. The addition of NPs to a solvent contributes to a qualitative change in the heat-conducting and viscosity properties [4]. This makes the specific heat of a nanofluid somewhat lower than that of a pure solvent but leads to a substantial increase in the thermal conductivity of NFs when even small amounts of nanophase are added to the solvent [5,6].

_{p}and k also limits the use of TLS [24].

^{−6}m

^{2}/s (although, according to reference data, it is 0.143 × 10

^{−6}m

^{2}/s [29]). Furthermore, the introduction of a correction factor for the measurement of the thermal diffusivity, as it is given in [21], in our opinion, can introduce an error in the reliability of the results.

## 2. Materials and Methods

#### 2.1. Reagents and Chemicals

_{2}nanoparticles of three different series of Ludox (SM-30, HS-40, TM-50) purchased from Merck (certificate descriptions are presented in Table 1). Aqueous dispersions were prepared using deionized water of the Milli-Q Reference system (Millipore SAS, Molsheim, France). We used the following solvents: ethanol (C

_{2}H

_{5}OH, CAS: 64-17-5, chemically pure, Khimreaktiv, Klin, Russia); chloroform (CHCl

_{3}, CAS: 67-66-3, analytical grade, Khimreaktiv, Klin, Russia); toluene (C

_{6}H

_{5}CH

_{3}, CAS: 108-88-3, analytical grade, LenReaktiv, St. Petersburg, Russia). Photostable indicator Ferroin (1,10-phenanthroline complex of iron(II), Fe(C

_{12}H

_{8}N

_{2})

_{3}SO

_{4}, M

_{r}= 596.27 g/mol, CAS: 14634-91-4, analytical grade, LenReaktiv, St. Petersburg, Russia)) and dye Sudan 1 (C

_{16}H

_{12}N

_{2}O, M

_{r}= 248.28 g/mol, CAS: 842-07-9, chemically pure, Reakhim, Moscow, Russia) were used as chromophores. The mixing of solutions was carried out using a laboratory shaker model PE-6410 (EKROSKHIM, St. Petersburg, Russia).

#### 2.2. Thermal Lens Measurements

_{e}), refractive index temperature coefficient (dn/dT):

^{2}+ V

^{2}, c = 1 + 2m + V

^{2}.

^{−1}(a/c), where $I\left(\infty \right)$ is the probe laser intensity in the stationary state. In the case of a homogeneous solution, the steady state means the average value of the last points of the transient curve (in our case, the average of the last 300 ms). The thermal diffusivity, $\tilde{D}\left(t\right)$, the so-called effective thermal diffusivity, is calculated from Equation (7) for each value, $\tilde{{t}_{c}}\left(t\right),$ at time, t. The transition from effective ${t}_{c}$ and D to true values occurs by averaging the values of the first 100 ms of the functions $\tilde{{t}_{c}}\left(t\right)$ and $\tilde{D}\left(t\right)$ using Equation (2).

#### 2.3. Thermal Lens Spectrometer

_{00}; Changchun New Industries Optoelectronics Tech. Co., Ltd., Changchun, China) passes through a shutter (model SH05, ThorLabs, Newton, NJ, USA), which is controlled by an analog-to-digital and a digital-to-analog converter (ADC–DAC) model c8051Fx-DK (Silicon Labs, Boston, MA, USA) connected to a personal computer (PC), and enters the sample in a quartz cell (l = 10.00 mm), in which a thermal lens is generated. A helium–neon laser HNL050L (wavelength of 632.8 nm, TEM

_{00}; ThorLabs, Newton, NJ, USA,) was used as the probe laser. A photodiode was used as a detector.

#### 2.4. UV Visible Spectroscopy

#### 2.5. Vibration Density Meter

#### 2.6. Differential Scanning Calorimetry

#### 2.7. Heat-Flow Method

## 3. Results and Discussion

#### 3.1. Photothermal Measurements of Reference Samples

_{2}, we measured reference samples and solutions of chromophores in water, ethanol, chloroform, and toluene. In all cases, we observed the complete development of the thermal lens and the achievement of the steady state of transient curves, which indicates the absence of the Soret effect and thermal convection. The experimental transient curves are in good agreement with the model approximation without additional fitting. As an example, Figure 2a shows transient curves for an aqueous solution of ferroin in a normalized form for the first 100 ms. A plot in logarithmic coordinates (Figure 2b) shows the behavior of the transient curve more clearly than in a linear form.

_{c}and D point to the optimum measurement conditions. Thus, we confirmed that the systematic error is negligible and that the TLS measurement parameters were selected correctly and optimally. All deviations from the theory, which will be observed for dispersions, we refer to only as deviations for a particular sample.

#### 3.2. Photothermal Measurements of SiO_{2} Nanofluids

_{2}NPs with a medium particle diameter, 12 nm (HS series, Figure 4). With an increase in the concentration of the solid phase, the above effects become more pronounced. This is reflected in the appearance of a minimum value on the transient curve and in not reaching a stationary state. For the most concentrated dispersion (22.4 mg/mL, red line), the convection effects are significant compared to the diluted one (1.60 mg/mL, green line). The differences are most noticeable in logarithmic coordinates (Figure 4b). The thermal lens that appears upon the irradiation with an excitation laser develops faster in the most concentrated sample than in a blank reference solution, as evidenced by the larger initial slope of the experimental transient curve.

_{av}= 11 nm), observed a significant increase in the intensity after reaching a minimum. They recorded a decrease in the divergence of the probe beam because of the Soret effect for a dispersion with a mass fraction of only 2.5%. Thus, a converging lens appeared in the sample.

#### 3.3. Correctness of Thermal Lens Measurements

#### 3.4. Thermal Diffusivity of SiO_{2} Nanofluids

_{av}= 12 nm (HS series). At the same time, SiO

_{2}with d

_{av}= 7 nm (SM series) showed an intermediate value. Previously, it was found that the smaller the particle size, the more the thermal diffusivity decreases [36]. The authors also associated this behavior with the Brownian motion in the following way: if the particle size is very small, the Brownian motion is significant and the heat transfer rate is high, and if the particles are large, then the Brownian motion is slight and heat propagation is difficult.

## 4. Conclusions

_{2}nanofluids, at low concentrations (up to 5 mg/mL), a decrease in thermal diffusivity was revealed, due to the contribution of thermal conductivity. Further, with increasing concentration, the contribution of the heat capacity begins to dominate and the thermal diffusivity increases. Using thermal lens spectrometry, it was also found that with a change in the particle size, the thermal diffusivity also changes. Nevertheless, for the correct presentation of the results of the measurements of the thermal diffusivity of dispersions, it is necessary to consider the nanophase size. TLS revealed the heat-accumulating properties of dispersions and differences in thermal effects, which is also an important thermophysical problem. Thus, thermal lens spectrometry is a versatile tool in the analysis of thermal diffusivity in the range of low concentrations, which classical methods are unable to solve.

## Supplementary Materials

_{av}= 7 nm, (b) HS with d

_{av}= 12 nm, (c) TM with d

_{av}= 22 nm; Figure S2: Thermal diffusivity of various Ludox with different concentrations of the solid phase measured by the TLS (Equations (10) and (2); lines), and heat-flow method (Equation (14); crosses): (a) SM with d

_{av}= 7 nm, (b) HS with d

_{av}= 12 nm, (c) TM with d

_{av}= 22 nm.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Transient curve for an aqueous solution of ferroin (1 µmol/L): (

**a**) in linear scale form for the first 100 ms and (

**b**) in logarithmic scale for the first 500 ms (measurement parameters are presented in Table 2).

**Figure 3.**Dependence of absorbance at λ = 532 nm on the concentration of nanoparticles in Ludox dispersions.

**Figure 4.**Transient curve, Equation (11), for dispersions of one series of NPs (HS, d

_{av}= 12 nm), but at different concentrations for the first 600 s of thermal lens development, (

**a**) in linear and (

**b**) logarithmic scales.

**Figure 5.**Transient curve for thermal lens dissipation, Equation (12), for a sample of Ludox (HS, d

_{av}= 12 nm) with a concentration of 22.4 mg/mL.

**Figure 6.**Transient curve for thermal lens dissipation, Equation (12), for samples of the same series (SM, d

_{av}= 7 nm), but different concentrations for the first 900 s of dissipation (the first 200 ms in the inset).

**Figure 7.**Transient thermal lens signal, Equation (9), for various concentrations of a dispersion of silicon oxide nanoparticles (HS series, d

_{av}= 12 nm) for the first 600 ms.

**Figure 8.**Transient curve for a HS sample (d

_{av}= 12 nm, with a concentration of 22.4 mg/mL): red dots, experiment; the blue line is the theoretical transient curve constructed using the averaging of the last 300 points of the experimental transient curve as a steady state; the black line is the theoretical transient curve built by fitting the first 100 ms of the experiment.

**Figure 9.**Thermal diffusivity of dispersions of various Ludox series (SM, d

_{av}= 7 nm; HS, d

_{av}= 12 nm; TM, d

_{av}= 22 nm) where: (

**a**) in the concentration range from 0 to 22.4 mg/mL, (

**b**) up to 5 mg/mL (n = 5, p = 0.95). The crosses are measurements by the heat flow method, Equation (14), and the lines are measurements by thermal lens spectrometry, Equations (10) and (2).

**Figure 10.**Effect of particle concentration on thermal diffusivity of dispersions for different Ludox series (SM, d

_{av}= 7 nm; HS, d

_{av}= 12 nm; TM, d

_{av}= 22 nm), where crosses are heat flow measurements, Equation (14) and lines are TLS measurements, Equations (10) and (2).

**Figure 11.**Increase in thermal conductivity for SiO

_{2}dispersions of various Ludox series in comparison with pure water. A decrease is denoted with negative values (blue color), and an increase is denoted with positive values (red color).

Ludox | Average Particle Diameter d_{av}, nm | Concentration, % w/w | Density, kg/m^{3} | Specific Surface, m^{2}/g |
---|---|---|---|---|

SM-30 | 7 | 30 | 1.220 | 350 |

HS-40 | 12 | 40 | 1.310 | 220 |

TM-50 | 22 | 50 | 1.400 | 140 |

Parameter | Value |
---|---|

Excitation laser | |

Wavelength, λ_{e} (nm) | 532 |

Focusing lens focal length, f_{e} (mm) | 200 |

Confocal distance, Z_{ce} (mm) | 10.9 |

Laser power at cell, P (mW) | 120 |

Spot size at the waist, ω_{e}_{0} (µm) | 42 |

Probe laser | |

Wavelength, λ_{p} (nm) | 632.8 |

Focusing lens focal length, f_{p} (mm) | 300 |

Confocal distance, Z_{cp} (mm) | 2.7 |

Laser power at cell (mW) | 4.5 |

Spot size at the waist, ω_{p}_{0} (µm) | 23 |

Spot size at cell, ω_{p} (µm) | 90 |

Other constants | |

Cell length (mm) | 10 |

Sample-to-detector distance, Z_{2} (cm) | 230 |

Mode mismatch factor, m | 4.59 |

Geometric parameters, V | 4.89 |

Modulator frequency (Hz) | 0.25 |

Number of transient curves to average | 300 |

Number of experiment repetitions | 5 |

Solvent | Characteristic Time, ms | Thermal Diffusivity, mm^{2}/s | ||||
---|---|---|---|---|---|---|

Calculation, Equation (2) | Experiment, Equation (7) | Δ, % | Theory [32] | Experiment | Δ, % | |

Water | 3.10 | 3.04 ± 0.04 | 2 | 0.142 | 0.145 ± 0.002 | 2 |

Ethanol | 4.95 | 4.95 ± 0.03 | <1 | 0.089 | 0.089 ± 0.001 | <1 |

Chloroform | 5.44 | 5.44 ± 0.10 | <1 | 0.081 | 0.081 ± 0.002 | <1 |

Toluene | 4.79 | 4.85 ± 0.23 | 1 | 0.092 | 0.091 ± 0.005 | 1 |

**Table 4.**Error in finding the characteristic time and thermal diffusivity, Equation (13), between using the averaging of the last 300 ms of the transient curve as a stationary state (parameters without apostrophe) and using the adjusted intensities of the probe beam in the stationary state (parameters with apostrophe, true value).

Ludox | c, mg/mL | Not Correct, Equation (7) | Correct, Equation (10) | Δ(D), % | ||
---|---|---|---|---|---|---|

t_{c}, ms | D, mm^{2}/s | t_{c}′, ms | D′, mm^{2}/s | |||

SM (d_{av} = 7 nm) | 1.60 | 2.23 | 0.198 | 3.32 | 0.133 ± 0.002 | 49 |

3.98 | 1.48 | 0.298 | 3.24 | 0.136 ± 0.001 | 120 | |

8.79 | 0.38 | 1.161 | 3.20 | 0.138 ± 0.002 | 740 | |

14.39 | 0.20 | 2.205 | 2.88 | 0.153 ± 0.002 | 1340 | |

22.40 | 0.12 | 3.675 | 2.53 | 0.174 ± 0.004 | 2010 |

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**MDPI and ACS Style**

Khabibullin, V.R.; Usoltseva, L.O.; Mikheev, I.V.; Proskurnin, M.A.
Thermal Diffusivity of Aqueous Dispersions of Silicon Oxide Nanoparticles by Dual-Beam Thermal Lens Spectrometry. *Nanomaterials* **2023**, *13*, 1006.
https://doi.org/10.3390/nano13061006

**AMA Style**

Khabibullin VR, Usoltseva LO, Mikheev IV, Proskurnin MA.
Thermal Diffusivity of Aqueous Dispersions of Silicon Oxide Nanoparticles by Dual-Beam Thermal Lens Spectrometry. *Nanomaterials*. 2023; 13(6):1006.
https://doi.org/10.3390/nano13061006

**Chicago/Turabian Style**

Khabibullin, Vladislav R., Liliya O. Usoltseva, Ivan V. Mikheev, and Mikhail A. Proskurnin.
2023. "Thermal Diffusivity of Aqueous Dispersions of Silicon Oxide Nanoparticles by Dual-Beam Thermal Lens Spectrometry" *Nanomaterials* 13, no. 6: 1006.
https://doi.org/10.3390/nano13061006