# Through-Plane and In-Plane Thermal Diffusivity Determination of Graphene Nanoplatelets by Photothermal Beam Deflection Spectrometry

^{1}

^{2}

^{3}

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^{†}

## Abstract

**:**

## 1. Introduction

^{2}g

^{−1}), rigidity (Young’s modulus is approximately 1100 GPa) and strength (the fracture strength is about 125 GPa). Furthermore, it has a very high electrical current density (at the level of 108 Acm

^{−2}) and mobility of charge carriers (2 × 10

^{5}cm

^{2}V

^{−1}s

^{−1}) [1,2,3,4]. Its thermal conductivity depends on the direction of heat propagation and reaches a value of 3000 Wm

^{−1}K

^{−1}in the parallel direction and 5 Wm

^{−1}K

^{−1}in the perpendicular direction to the sample surface [5,6]. Moreover, the thermal diffusivity of graphene or freestanding graphene depends on the number of layers and can reach a value of 6.5 × 10

^{−4}m

^{2}s

^{−1}[7]. Graphene is also chemically stable and does not react with other substances [1,2]. However, it can be functionalized by various atoms, nanoparticle composites and chemical groups to produce different graphene-based hybrids and composites, which thus inherit further its performance, to be applied in medicine, energy conversion, as sensors, storage devices, structural materials, reinforced composites and catalysts in the process of water purification [8,9].

## 2. Photothermal Beam Deflection Spectrometry Theory

#### 2.1. Surface Scan Method

_{Ti}and K

_{Ti}are the thermal diffusivity and conductivity of a material or a fluid, respectively. Since GNP samples are strongly absorbing, it can be assumed that the EB absorbance occurs at the surface of the sample, and they do not occur any internal heat sources (q = 0).

_{RC}= z

_{R}–iL

_{0}is the complex Rayleigh’s length, z

_{R}= ka

^{2}n

_{0}is the Rayleigh’s length, L

_{0}is the focal distance, a is the radius of the probe beam in its waist and n

_{0}is the index of refraction in ambient temperature, whereas k = 2π/λ is the wave number, and λ is the wavelength of the probe beam in vacuum.

_{1}and z

_{1}to its coordinates expressed by Equation (4) must be found (see Appendix B):

_{0}is the refractive index of undisturbed fluid; s

_{T}= (1/n

_{0})(dn/dT) is the thermal sensitivity, and dn/dT is the temperature coefficient of refractive index; τ is the running complex coordinate along the PB trajectory; ξ and η are the PB’s coordinates in the input plane of the experimental setup (z = 0).

_{0}is the amplitude of the electric field of undisturbed PB in its waist.

_{D}), the amplitude of the PB electric field has the form (see Appendix D):

_{D}, y

_{D}, z

_{D}) and can be written as:

_{d}is the detector’s constant, h is the height of PB over the sample, S

_{PDn}is the normal component of the PD signal being a consequence of the difference in illumination between upper and lower photodiode’s halves and S

_{PDt}is the tangential component of the PD signal resulting from a difference in illumination between left and right photodiode’s halves. It is taken into account in Equations (13) and (14) that the detector is partly covered by the sample.

_{PDt}signal as a function of the distance between the EB and PB and performing the multiparameter fitting of curves obtained from Equation (14) to the experimental data.

#### 2.2. Slope Method

_{T-in}is the in-plane thermal diffusivity of GNP material.

_{T-in}determination was found by the use of an error propagation algorithm and had the form:

#### 2.3. Frequency Scan Method

_{PDn}signal are collected as a function of the modulation frequency of TOs. To the amplitude and phase of experimental data, the theoretical curves are fitted by the use of the least-squares fitting procedure.

_{f}is the amplitude of TOs at sample’s surface, φ

_{f}is the phase shift between the phase of sample surface’s temperature change and the phase of the pump beam, Ω is the angular modulation frequency of the EB and t is the time. Both b

_{f}and φ

_{f}are functions of the sample’s thermal (thermal diffusivity and conductivity) and geometrical properties (thickness), as well as parameters of the experimental setup (height of PB over the sample’s surface) [24].

_{f}is the thermal diffusivity of the fluid above the sample; z

_{l}and z

_{p}are the positions of the left and right edges of the medium where step change occurs; k

_{ft}is the wavenumber of TOs.

_{D}) then has the form:

_{1d}and ψ

_{1f}corrections to PB phase at the detector (z

_{D}) are:

_{0}is the phase of undisturbed PB. The ψ

_{1d}and ψ

_{1f}corrections to the PB phase are expressed by:

_{D0}and τ

_{s0}are the coordinate of the ray in the input plane of the experimental setup for the given observation point and the running coordinate along the ray over the sample for undisturbed probe beam, respectively.

_{d}is the detector constant, I

_{0g}is the light intensity of undisturbed PB, S

_{PDnd}and S

_{PDnf}are the components of PDS resulting from deflection of PB in the field of TOs and its phase change, and A

_{t}and φ

_{t}are the amplitude and additional phase changes in total PDS.

#### 2.4. Fitting Accuracy

_{Ds}) was estimated in both surface and frequency scan methods by calculating the square root of the covariance matrix [26]:

_{r}is the variance on residuals

_{f}denotes the Jacobian matrix; meanwhile, P′ is the matrix of fitted parameters for which the minimum error function was reached.

## 3. Materials and Methods

#### 3.1. Sample Preparation

^{3}.

#### 3.2. Experimental Setup

## 4. Results and Discussion

#### 4.1. Surface Scan Method

_{PDt}tangential component of the signal dependence on the excitation-probe beam offset y. It is observed that the phase of S

_{PDt}changes linearly with the increase in the value of y. The results of the least-squares fitting procedure of the S

_{PDt}phase of experimental data to the theoretical curves obtained by the use of Equation (14) are presented in Table 1.

#### 4.2. Slope Method

_{GS}small compared to the spatial range of the thermal disturbance L

_{TH}that is defined as [25,28,29,30]:

_{th}higher than 5 mm for the lowest obtained values of D

_{T-in}and highest frequency used. Since the examined samples are nanoplatelets, the condition of L

_{CH}≪ L

_{TH}is satisfied, and the induced TOs are not affected by the material’s structure. Therefore, the measured thermal diffusivity is a bulk effective diffusivity of the examined material, and Equation (16) can be used for its determination. The values of in-plane thermal diffusivities were measured as an average of three different measurements on different spots on the GNP samples’ surface for which SD of D

_{T-in}was determined to evaluate the determination repeatability.

#### 4.3. Frequency Scan Method

#### 4.4. Comparison of In-Plane and Through-Plane Thermal Properties

^{−1}K

^{−1}). This is the result of breaking the GNP platelets in this direction, which introduces additional interfaces and thus, limits the heat conduction within the samples and its exchange with surroundings. Since no pressing loads are applied in the direction parallel to the sample’s surface, the resulting change in their structure (folding, rolling, breaking platelets) in this direction exceeds much less extends and results in much higher in-plane values of thermal properties.

#### 4.5. Thermal Conductivity Evaluation

_{p}is the material-specific heat and ρ its density. The specific heat of GNPs was assumed to be 710 Jkg

^{−1}K

^{−1}[32], whereas their densities as described in [12,13]. The results are presented in Table 5 and Figure 6.

_{T-through}increases with the increase in D

_{T-through}for the whole range of the loading pressure used. Such behavior is caused by different heat propagation in relation to the direction of load application, as a result of different changes introduced in the GNP samples structure after their compression, as described above.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

_{0}is the average energy flux that impinges on the illuminated sample’s surface, d is the sample’s thickness, a is the radius of PB in its waist, D

_{Tm}, D

_{Tf}, k

_{Tf}, k

_{Tm}are the GNP material and fluid over its surface thermal diffusivities and conductivities, respectively.

## Appendix B

## Appendix C

_{1f}results from the PB phase change caused by the change in fluid refractive index, Φ

_{1d}is the consequence of existence of fluid refractive index gradients that lead to the change in PB geometrical path. The proper correction is given by:

_{D}, y

_{D}, z

_{D}are the coordinates of PB’s rays in the detector plane.

## Appendix D

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**Figure 1.**Schematic diagram of EB and PB configuration in the in-plane thermal diffusivity determination by the slope method.

**Figure 3.**Scheme of the experimental setup for BDS measurement. L1, L2, L3, L4, L5, L6: lenses, M: reflecting mirrors, QP: quadrant photodetector, IF: filter, LA: lock-in amplifier, EB: excitation beam, PB: probe beam, SG: signal generator.

**Figure 4.**Variation in the photothermal deflection phase with pump-probe beam offset for GNP samples pressed with different loads.

**Figure 6.**The relation between thermal diffusivity and conductivity for in-plane and through-plane configuration of the measurements.

**Table 1.**Values of the GNP samples in-plane thermal diffusivities and conductivities pressed with loads from 500 to 2000 N obtained by the use of CRT theory.

Sample | P, N | D_{T-in}, ×10^{−6} m^{2}s^{−}^{1} | κ_{T-in}, W m^{−1}K^{−1} |
---|---|---|---|

S1 | 500 | 46.0 ± 2.2 | 10.2 ± 0.4 |

S2 | 1000 | 34.4 ± 1.4 | 10.8 ± 0.4 |

S3 | 2000 | 30.2 ± 0.8 | 13.4 ± 0.6 |

S4 | 700 | 41.5 ± 1.4 | 11.2 ± 0.6 |

S5 | 700 | 45.0 ± 1.5 | 12.7 ± 0.7 |

S6 | 700 | 42.6 ± 1.1 | 11.9 ± 0.8 |

S10 | 0 | 107.0 ± 2.0 | 24.5 ± 1.8 |

Sample | a, m^{−1} | b, ° | R^{2} |
---|---|---|---|

S1 | 0.0454 | 98.13 | 0.9868 |

S2 | 0.0532 | 98.66 | 0.9943 |

S3 | 0.0586 | 103.98 | 0.9915 |

S4 | 0.0374 | 108.03 | 0.968 |

S5 | 0.0384 | 96.92 | 0.9742 |

S6 | 0.0490 | 88.33 | 0.9944 |

**Table 3.**Values of the GNP samples in-plane thermal diffusivities pressed with loads from 500 to 2000 N.

Sample | P, N | f, Hz | D_{T-in}, ×10^{−6} m^{2}s^{−1} |
---|---|---|---|

S1 | 500 | 11 | 50.0 ± 3.2 |

S2 | 1000 | 11 | 39.2 ± 1.6 |

S3 | 2000 | 11 | 33.3 ± 1.1 |

S4 | 700 | 11 | 45.0 ± 1.8 |

S5 | 700 | 11 | 50.0 ± 2.0 |

S6 | 700 | 11 | 47.0 ± 1.7 |

S10 | 0 | 11 | 110 ± 2.4 |

**Table 4.**Values of the GNP samples through-plane thermal diffusivities pressed with loads from 500 to 2000 N.

Sample | P, N | D_{T-through}, ×10^{−6} m^{2}s^{−1} | κ_{T-through}, W m^{−1}K^{−1} |
---|---|---|---|

S1 | 500 | 14.4 ± 0.2 | 3.23 ± 0.10 |

S2 | 1000 | 9.30 ± 0.11 | 3.16 ± 0.08 |

S3 | 2000 | 7.60 ± 0.08 | 3.52 ± 0.12 |

S4 | 700 | 11.0 ± 0.1 | 3.28 ± 0.14 |

S5 | 700 | 10.0 ± 0.1 | 3.11 ± 0.12 |

S6 | 700 | 9.10 ± 0.11 | 2.74 ± 0.10 |

S7 | 500 | 5.80 ± 0.06 | 1.33 ± 0.06 |

S8 | 1000 | 6.40 ± 0.08 | 2.25 ± 0.08 |

S9 | 2000 | 2.10 ± 0.02 | 1.08 ± 0.04 |

S10 | 0 | 99.5 ± 1.8 |

**Table 5.**Values of the GNP samples through-plane and in-plane thermal conductivities pressed with loads from 500 to 2000 N.

Sample | ρ, kg m^{−3} | κ_{T-in}, W m^{−1}K^{−1} | κ_{T-through}, W m^{−1}K^{−1} |
---|---|---|---|

S1 | 300 ± 7 | 11.7 ± 0.6 | 3.07 ± 0.12 |

S2 | 461 ± 9 | 12.8 ± 0.7 | 3.04 ± 0.10 |

S3 | 623 ± 18 | 14.7 ± 0.7 | 3.36 ± 0.16 |

S4 | 398 ± 8 | 12.7 ± 0.6 | 3.11 ± 0.11 |

S5 | 398 ± 8 | 14.1 ± 0.8 | 2.83 ± 0.09 |

S6 | 398 ± 8 | 13.3 ± 0.7 | 2.57 ± 0.10 |

S7 | 300 ± 7 | - | 1.24 ± 0.04 |

S8 | 461 ± 9 | - | 2.09 ± 0.06 |

S9 | 623 ± 18 | - | 0.93 ± 0.03 |

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**MDPI and ACS Style**

Cabrera, H.; Korte, D.; Budasheva, H.; Abbasgholi N. Asbaghi, B.; Bellucci, S.
Through-Plane and In-Plane Thermal Diffusivity Determination of Graphene Nanoplatelets by Photothermal Beam Deflection Spectrometry. *Materials* **2021**, *14*, 7273.
https://doi.org/10.3390/ma14237273

**AMA Style**

Cabrera H, Korte D, Budasheva H, Abbasgholi N. Asbaghi B, Bellucci S.
Through-Plane and In-Plane Thermal Diffusivity Determination of Graphene Nanoplatelets by Photothermal Beam Deflection Spectrometry. *Materials*. 2021; 14(23):7273.
https://doi.org/10.3390/ma14237273

**Chicago/Turabian Style**

Cabrera, Humberto, Dorota Korte, Hanna Budasheva, Behnaz Abbasgholi N. Asbaghi, and Stefano Bellucci.
2021. "Through-Plane and In-Plane Thermal Diffusivity Determination of Graphene Nanoplatelets by Photothermal Beam Deflection Spectrometry" *Materials* 14, no. 23: 7273.
https://doi.org/10.3390/ma14237273