# Transmittance and Reflectance Effects during Thermal Diffusivity Measurements of GNP Samples with the Flash Method

^{1}

^{2}

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^{*}

## Abstract

**:**

^{®}) on both sides to reduce reflectance of their surfaces and consequently increase the emissivity. Carrying out measurements on both samples with and without coating, a difference between the two series of measurements was found: This is attributed to a non-negligible transmittance of the uncoated samples due to the porosity of GNPs. Furthermore, assuming a spatial distribution of the light within the samples according to the Lambert-Bougert-Beer law, the extinction coefficient of GNP at different densities has been evaluated processing experimental data with a nonlinear least square regression, (NL-LSF, nonlinear least square fitting), whose model contains the extinction coefficient as unknown. The proposed method represents a further improvement of thermal diffusivity data processing, crucial to calculate the extinction coefficient when data with and without coating are available; or to correct biased thermal diffusivity data when the extinction coefficient is already known. Moreover, reflectance effects have been highlighted comparing asymptotic temperature reached during the tests on coated and uncoated samples at different densities. In fact, the decrease of asymptotic temperature of the uncoated samples gives the percentage of the light reflected and consequently an estimate of the reflectance of the GNP surface.

## 1. Introduction

- a very short energy pulse can produce a meaningful temperature increase on the irradiated surface which can even damage it; this can be avoided with a low power source (a photographic flash), but temperature increase on the opposite side results very low, and temperature detection becomes difficult;
- an error can arise when the time length of the flash pulse is comparable with the temperature transition on the opposite face: In such case the approximation of the analytical trend of the pulse with a Dirac-delta function cannot be considered any more valid, and a meaningful systematic error is introduced [6,9].

^{®}, by Agar Scientific Ltd, Stansted, Essex, UK) in order to make them opaque and good absorber in the infrared wavelength band used for measurements. The results are summarized in Figure 1. In order to avoid the error arising from the finite time length of the pulse, a special analytical solution was developed, which uses a suited regression analytical model in the nonlinear least square fitting of the experimental data.

^{®}) in order to increase the emissivity of their surface. A higher absorbance of the irradiated surface is so obtained, producing an increase of the detector output. The coating thickness is so thin (~1 μm) to negligibly influence the obtained results.

## 2. Analytical Solution

#### 2.1. Parker Solution

- homogeneous and isotropic materials;
- pulse heating (Dirac δ);
- adiabatic condition of the slab after the pulse;
- homogeneously irradiated surface;
- one-dimensional heat propagation;
- thermo-physical properties constant in the temperature range of the test;

_{∞}− T

_{0}) and the thermal diffusivity α.

#### 2.2. Double Exponential

_{1}, and R

_{2}are the inverses of the two time constants, the first linked to the lamp filament temperature increase, and the second to the time constant of the flash capacitor discharge. The way to determine R

_{1}and R

_{2}is described in [6].

#### 2.3. Solution Involving GNP Partial Transparency

_{0}is the light intensity in x = 0 and a

_{λ}is the extinction coefficient, dependent on radiation wavelength and material. In a first step, a

_{λ}can be assumed independent on wavelength, and the flash signal described by:

_{1}and R

_{2}, B is a factor proportional to the light intensity and a the extinction coefficient averaged in the wavelength range of the impinging light. Using Equation (5) as thermal input, the solution of the heat conduction equation is (details are reported in Appendix A):

## 3. Sample Preparation and Test Setup

- a quantum radiation detector (mercury cadmium telluride, MCT, active area 1 mm
^{2}, Pro-Lite Technology Ltd, Melton Mowbray, Leicestershire, UK), cooled with liquid nitrogen (77 K) in a dewar surrounding it; - a ZnSe infrared lens (focal length 50 mm, manufacturer, city, state, country), transparent to visible and IR radiation from 0.5 to 13 µm, located in front of the MCT detector (ZSL);
- a photographic flash Universal 1500 S Elinchrom (F), 200 W nominal maximum power (Elinchrom SA, Renens, Switzerland). Tests were carried out with one half of this maximum power;
- data acquisition system (NI USB-6229, National Instruments, Austin, Texas, USA) set at a sampling rate of 25 kHz, and ±10 V range.

^{®}on both sides. As already said, the coating thickness is negligible and was not taken into account in data processing. Table 1 shows the different loads and the resulting densities. Density was calculated measuring the sample weight (with an analytical balance with a 1 mg resolution) and its geometrical sizes.

^{−1}is deduced. The linearity between the detector output (in V) and the temperature increase (in °C) leads to write, as in [6]:

## 4. Results

#### 4.1. Thermal Diffusivity

- pressing samples at predefined loads;
- testing uncoated samples five times each, and evaluating the thermal diffusivity with NL-LSF analysis using Equation (3) as model;
- coating samples and testing them again five times, evaluating their thermal diffusivity again using NL-LSF with Equation (3).

^{−3}, the uncertainty bands are overlapped, that is the transmittance effects are negligible.

#### 4.2. Extinction Coefficient Results

_{1}(13250 ± 120 m

^{−1}) and b

_{2}(1.67⋅10

^{−3}± 0.01⋅10

^{−3}m

^{3}⋅kg

^{−1}).

#### 4.3. Reflectance

_{p}is the specific heat, m the sample mass, and ΔT the asymptotic temperature increase. As all measurements were carried out with the same flash power, all samples with equal mass and c

_{p}should reach the same asymptotic temperature. Table 4 and Figure 7 show instead a linearly decreasing asymptotic temperature of uncoated samples, while the one of coated samples remains constant. The coating makes the surface perfectly absorbing due to the high emissivity of the Aquadag

^{®}(about 0.99). So, the value of the GNP coated samples can be assumed as reference, because their reflectance is considered negligible. Reflectance can be calculated from the ratio of reflexed light over the incident one. Indicating with ${T}^{\infty}{}_{\mathrm{coat}}$ the asymptotic temperature of the coated samples and ${T}^{\infty}{}_{\mathrm{unc}}$ the one of the uncoated, the reflectance results:

## 5. Extinction Coefficient Uncertainty Analysis

#### 5.1. Uncertainty due to Thermal Diffusivity and Sample Thickness

#### 5.2. Uncertainty due to Convection/Radiation Effects

- data after the inflection point of the whole trend (including the temperature decrease) are processed with a nonlinear least square regression using the lumped parameter solution as model:$$T(\mathsf{\tau})={T}_{\infty}+\left({T}_{0}-{T}_{\infty}\right){e}^{-\chi \xb7\mathsf{\tau}}$$
- exponential decreasing data are extrapolated till to the start of the pulse heating;
- data used for thermal diffusivity calculation are corrected adding the difference between Equation (6) and its extrapolated initial value. This procedure returns data not influenced by convection/radiation, in fact their asymptote results horizontal.

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

Latin | ||

A | proportional factor on light intensity in Equation (2) | [V] |

a | extinction coefficient | [m^{−1}] |

B | proportional factor on light intensity in Equation (5) | [V] |

b | nonlinear least square regression coefficient | |

c | constant in Equation (7) | [V] |

c_{p} | specific heat capacity | [J·kg^{−1}·K^{−1}] |

d | constant in Equation (7) | [V·°C^{−1}] |

I | light intensity | [W·m^{−2}] |

k | thermal conductivity | [W·m^{−1}·K^{−1}] |

L | sample thickness | [m] |

m | mass | [kg] |

Q | thermal energy | [J] |

$\dot{q}$ | specific thermal flux | [W·m^{−1]} |

R | inverse of the time constant | [s^{−1}] |

S | signal | [V] |

s | standard uncertainty (estimated) | |

T | temperature | [°C or K] |

t | temperature | [°C] |

x | abscissa along the sample thickness | [m] |

Greek and composite symbols | ||

α | thermal diffusivity | [m^{2}·s^{−1}] |

β | reflectance | |

δ | Dirac delta function | |

ρ | density | [kg·m^{−3}] |

τ | time | [s] |

$\chi $ | constant function of Equation (11) | [s^{−1}] |

Subscript | ||

0 | start, initial value | |

∞ | infinity, asymptotic value | |

coat | coated | |

unc | uncoated | |

Acronyms | ||

D | Dewar | |

F | Flash | |

GNP | Graphene Nano-Plates | |

MCT | Mercury Cadmium Telluride detector | |

NL-LSF | Nonlinear least square fitting | |

PTFE | Polytetrafluoroethylene | |

SA | Sensor Amplifier | |

SH | Sample holder | |

SPS | Sensor Power Supply | |

ZSL | Zinc Selenide Lenses |

## Appendix A. Analytical Expression of the Propagation of a Double Exponential Pulse in a Solid Slab by Conduction

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**Figure 1.**Trend of thermal diffusivity α and thermal conductivity λ as a function of density (ρ) [6].

**Figure 3.**Overlapping of a typical signal recorded in a test by flash method on GNP coated and uncoated samples.

**Figure 4.**Signal recorded for three different samples of GNP not coated, pressed with different loads (100, 200 and 450 N).

**Figure 5.**Experimental apparatus used for thermal diffusivity measurements: Flash (F), dewar (D), sample holder (SH), sensor amplifier (SA), sensor power supply (SPS), zinc selenide lenses (ZSL).

**Figure 8.**Overlapped trends of α values of the colloidal graphite coated samples and uncoated, as a function of density.

**Figure 10.**Asymptotic temperature increases during thermal diffusivity tests, for coated and uncoated samples, as a function of density.

**Figure 11.**Relative deviation of a as a function of the relative deviation of sample thickness (at ρ = 570 kg⋅m

^{−3}).

**Figure 12.**Relative deviation of a as a function of relative deviation of the thermal diffusivity of coated samples, at two different densities.

**Figure 13.**How raw acquired temperature data are corrected to take into account the convection heat transfer; red line: Pure convection trend evaluated with the lumped parameter model.

Sample # | Load (N) | Diameter (mm) | Thickness (mm) | Mass (g) | ρ (kg⋅m^{−3}) |
---|---|---|---|---|---|

1 | 100 | 31 | 2.36 | 0.184 | 103.1 ± 1.7 |

2 | 200 | 31 | 1.49 | 0.201 | 180.7 ± 3.1 |

3 | 450 | 34 | 0.88 | 0.200 | 248.1 ± 4.8 |

4 | 550 | 32 | 0.65 | 0.186 | 352.9 ± 9.3 |

5 | 1500 | 34 | 0.39 | 0.200 | 569.3 ± 9.7 |

6 | 3000 | 34 | 0.29 | 0.201 | 785 ± 27 |

7 | 5000 | 34 | 0.23 | 0.184 | 859 ± 38 |

8 | 7000 | 35 | 0.21 | 0.184 | 950 ± 28 |

**Table 2.**Results of thermal diffusivity measurements of graphene samples with and without coating (* Uncoated, ** Coated).

Sample # | Load (N) | ρ (kg⋅m^{−3}) | α·10^{7} (m^{2}·s^{−1}) * | s_{α}/α (%) | α·10^{7} (m^{2}·s^{−1}) ** | s_{α}/α (%) |
---|---|---|---|---|---|---|

1 | 100 | 103.1 ± 1.7 | 419 ± 2.1 | 0.5 | 355 ± 10.3 | 2.9 |

2 | 200 | 180.7 ± 3.1 | 304 ± 1.2 | 0.4 | 265 ± 2.6 | 1.0 |

3 | 450 | 248.1 ± 4.8 | 187 ± 1.6 | 0.9 | 163 ± 1.5 | 0.9 |

4 | 550 | 352.9 ± 9.3 | 158 ± 0.8 | 0.5 | 130 ± 1.4 | 1.1 |

5 | 1500 | 569.3 ± 9.7 | 85.6 ± 1.8 | 2.1 | 69.1 ± 0.8 | 1.1 |

6 | 3000 | 785 ± 27 | 68.3 ± 3.7 | 5.4 | 53.0 ± 0.2 | 0.4 |

7 | 5000 | 859 ± 38 | 51.1 ± 0.6 | 1.1 | 47.2 ± 0.4 | 0.9 |

8 | 7000 | 950 ± 28 | 37.1 ± 0.6 | 1.6 | 36.6 ± 0.5 | 1.4 |

ρ (kg⋅m^{−3}) | L (mm) | α·10^{7} (m^{2}⋅s^{−1}) * | α·10^{7} (m^{2}⋅s^{−1}) ** | a (m^{−1}) |
---|---|---|---|---|

103.1 | 2.36 | 419 | 355 | 3406 |

180.7 | 1.49 | 304 | 265 | 5607 |

248.1 | 0.88 | 187 | 163 | 8975 |

352.9 | 0.65 | 158 | 130 | 11,170 |

569.3 | 0.39 | 85.6 | 69.1 | 17,629 |

785 | 0.29 | 68.3 | 53.0 | 29,034 |

859 | 0.23 | 51.1 | 47.2 | 42,451 |

ρ (kg⋅m^{−3}) | T (°C) * | T (°C) ** | β |
---|---|---|---|

103.1 | 26.7 | 19.4 | 0 |

180.7 | 20 | 19.1 | 0.02 |

248.1 | 20.7 | 24.2 | 0.05 |

352.9 | 17.6 | 18.7 | 0.09 |

569.3 | 15.8 | 19.8 | 0.21 |

785 | 12.7 | 26.1 | 0.29 |

859 | 15.8 | 23.2 | 0.35 |

950 | 15.15 | 24.5 | 0.41 |

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**MDPI and ACS Style**

Bellucci, S.; Bovesecchi, G.; Cataldo, A.; Coppa, P.; Corasaniti, S.; Potenza, M.
Transmittance and Reflectance Effects during Thermal Diffusivity Measurements of GNP Samples with the Flash Method. *Materials* **2019**, *12*, 696.
https://doi.org/10.3390/ma12050696

**AMA Style**

Bellucci S, Bovesecchi G, Cataldo A, Coppa P, Corasaniti S, Potenza M.
Transmittance and Reflectance Effects during Thermal Diffusivity Measurements of GNP Samples with the Flash Method. *Materials*. 2019; 12(5):696.
https://doi.org/10.3390/ma12050696

**Chicago/Turabian Style**

Bellucci, Stefano, Gianluigi Bovesecchi, Antonino Cataldo, Paolo Coppa, Sandra Corasaniti, and Michele Potenza.
2019. "Transmittance and Reflectance Effects during Thermal Diffusivity Measurements of GNP Samples with the Flash Method" *Materials* 12, no. 5: 696.
https://doi.org/10.3390/ma12050696