<table><caption>**Table 3** Thermal parameters extracted from experimental data with genetic algorithm and confidence in this value</caption><thead><tr><th>Material</th><th>Parameter</th><th>Results obtained from fitting</th><th>Least-squares error</th><th>Most-squares error</th><th>Relative deviation from literature parameters<sup>a</sup></th></tr></thead><tbody><tr><td>Pt</td><td>α<sub>1</sub></td><td>1.14 × 10<sup>-5</sup> m<sup>2</sup> · s<sup>-1</sup></td><td>±4 %</td><td>±35 %</td><td>-53 %</td></tr><tr><td>Ti</td><td>α<sub>2</sub></td><td>6.90 × 10<sup>-6</sup> m<sup>2</sup> · s<sup>-1</sup></td><td>±33 %</td><td>±41 %</td><td>-22 %</td></tr><tr><td>SiO<sub>2</sub></td><td>α<sub>3</sub></td><td>7.38 × 10<sup>-7</sup> m<sup>2</sup> · s<sup>-1</sup></td><td>±2 %</td><td>±3.4 %</td><td>-12 %</td></tr><tr><td>Si</td><td>α<sub>4</sub></td><td>7.48 × 10<sup>-5</sup> m<sup>2</sup> · s<sup>-1</sup></td><td>±5 %</td><td>±12 %</td><td>-2 %</td></tr><tr><td>Pt</td><td>e<sub>1</sub></td><td>14 700 W · s<sup>1/2</sup> · m<sup>-2</sup> · K<sup>-1</sup></td><td>±3 %</td><td>±23 %</td><td>5 %</td></tr><tr><td>Ti</td><td>e<sub>2</sub></td><td>7920 W · s<sup>1/2</sup> · m<sup>-2</sup> · K<sup>-1</sup></td><td>±17 %</td><td>±85 %</td><td>13 %</td></tr><tr><td>SiO<sub>2</sub></td><td>e<sub>3</sub></td><td>1590 W · s<sup>1/2</sup> · m<sup>-2</sup> · K<sup>-1</sup></td><td>±1 %</td><td>±18 %</td><td>6 %</td></tr><tr><td>Si</td><td>e<sub>4</sub></td><td>13 154 W · s<sup>1/2</sup> · m<sup>-2</sup> · K<sup>-1</sup></td><td>±2 %</td><td>±3 %</td><td>-28 %</td></tr></tbody></table>

several times, and the mean and standard deviations were calculated for each parameter. The results are given in Table 3 while assuming that the layer thicknesses are known. A classical least-squares error analysis has been carried out on each obtained result by varying the parameter around its best fit value (Table 3, column 4). Interestingly, these statistical uncertainties are always smaller (≤±5 %) except for the Ti layer, for which the signal is scarcely sensitive, in comparison with the uncertainties found in simulations. This is due to a larger quantity of data exploited during experiments. Practically, these obtained results can also be affected by degeneracy (same results for different set of input values) between the fit parameters. This interdependency can be brought out by using the so-called “most squares” error analysis [28]. Moreover, in this approach, the true error values for each fitted parameter can also be found by varying all other best fitted values around its mean in the 8-D parameter space while minimizing the $\chi^2$ cost function. Table 3 also contains the true error values obtained from most squares error analysis as well as the deviations from the literature values in percentage. These results show the highest uncertainty for parameters of the thinnest layer, Ti (40 nm). Obviously, as mentioned before, this layer is thermally very thin to be effectively detected in our experimental frequency range. While, in the case of the Pt layer, the errors are higher in comparison to the SiO₂ and Si layers, because its thermal thickness is also less adequate to be scanned sufficiently through its quasi-thermally thick regime, where the determined thermal parameters are more sensitive. Furthermore, the difference between the mean diffusivity and the literature value (−53 %) can be due to reasons such as: inaccurate layer thickness value, presence of porous media [29], grain size and boundary effects [22,30], size effects [22], influence of thermal resistances between layers (not taken into account in the model) [31], or the combination of these or more effects and interfacial thermal resistance. This discrepancy is under further investigation. The result for the thermal effusivity of silicon shows a 28 % deficit in comparison with the literature value. This could be due to a possible 3-D effect, which can be even present above 50 Hz for highly diffusive materials with a spot size of approximately 8.4 mm.

<footnote><sup>a</sup> See Table 1</footnote>
## 6 Conclusions

In this work, a multilayer system having submicron and micron scale layers was investigated with PTR. The layers’ thermal parameters were extracted from the experimental signal with the use of a GA. The data analyzing procedure has been first tested and validated with noise-free simulated data. The obtained results from the experimental data were compared with literature values and some hypotheses were also put forward in order to explain the variations present in our results in comparison to literature values. The values obtained in this study are essentially useful in future investigations on samples deposited in the nanoscale on the top of such wafers for application towards thermal management of fuel cells.

**Acknowledgments** The authors would like to acknowledge the financial support for thesis work by the French Ministry of Research and to thank equally Benoit Escorne for his technical support.

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5.2 Improved methods for measuring thermal parameters of liquid samples using photothermal infrared radiometry, Meas. Sci. Technol. 24 (2013) 025603
# Improved methods for measuring thermal parameters of liquid samples using photothermal infrared radiometry

Maju Kuriakose<sup>1,2,3</sup>, Michael Depriester<sup>1,2,3</sup>, Dorin Dadarlat<sup>4</sup>
and Abdelhak Hadj Sahraoui<sup>1,2,3</sup>

E-mail: Maju.Kuriakose@univ-littoral.fr, Michael.Depriester@univ-littoral.fr and hadj@univ-littoral.fr

Received 24 September 2012, in final form 13 December 2012

Published 18 January 2013

Online at stacks.iop.org/MST/24/025603

## Abstract

High accuracy, non-contact measuring methods for finding thermal properties of liquid samples using photothermal infrared radiometry (PTR) are presented. The use of transparent windows to confine micro volume liquid samples and the implementation of front and/or back signal detection procedures helped the successful implementation of the PTR technique for measuring liquids with high proficiency. We present two configurations, the so-called back-front photothermal infrared radiometry and back photothermal infrared radiometry to find thermal diffusivity and thermal effusivity of liquid samples. Sensitivity studies and error analyses included prove the robustness of each method. As an illustration of the temperature and electric field varying studies, we have included the experimental results on a 5CB (4-cyano-4'-pentylbiphenyl) liquid crystal.

**Keywords:** photothermal radiometry, liquids, thermal diffusivity, thermal effusivity

(Some figures may appear in colour only in the online journal)

## 1. Introduction

Experimental methods for determining thermal transport properties are continuously getting modified either by new sensing methods or by changing the way it has been observed. These constant efforts ensure us high-quality, precise and accurate information about the system under investigation. Photothermal methods have been used successfully in a widespread way over the past decades to find the thermal properties of materials in a non-destructive manner. Among these, photothermal infrared radiometry (PTR), introduced by Nordal and Kanstad [1], is an efficient tool for thermal characterization of samples in a non-contact manner, which makes this method more attractive. Generally, photothermal techniques are capable of detecting small temperature variations introduced on a specimen, which allows us to find

the dynamic thermal quantities such as thermal diffusivity ($\alpha$ (m$^2$ s$^{-1}$)), thermal conductivity ($\kappa$ (W m$^{-1}$ K$^{-1}$)), thermal effusivity ($e$ (J K$^{-1}$ m$^{-2}$ s$^{-1/2}$)) and static quantities such as specific heat ($C$ (J kg$^{-1}$ K$^{-1}$)). The three dynamic thermal quantities, $\kappa$, $\alpha$ and $e$, can also be related by the equation $\kappa = e/\sqrt{\alpha}$. The general procedure for conducting PTR experiments is relatively simple. Like other photothermal techniques, the sample under investigation is heated periodically at a frequency $f = \omega/2\pi$, where $\omega$ is the angular frequency, using an intensity-modulated laser beam. As generated harmonic thermal fields diffuse into the sample layers and produce periodic temperature variations, infrared (IR) waves are emitted. These emitted IR signals are collected by the aid of proper optical arrangements and focused into an IR detector. A lock-in amplifier locked at the same modulation frequency of excitation detects the output from the IR detector. The signal

<footnote n="1"><sup>1</sup> Univ Lille Nord de France, F-59000 Lille, France</footnote>

<footnote n="2"><sup>2</sup> ULCO, UDSMM, F-59140 Dunkerque, France</footnote>

<footnote n="3"><sup>3</sup> Unité de Dynamique et Structure des Matériaux Moléculaires, EA 4496, F-59140 Dunkerque, France</footnote>

<footnote n="4"><sup>4</sup> Laboratory for Molecular and Biomolecular Physics, NIRDIMT, 71-105 Donath Street, PO Box 700, 400293 Cluj Napoca 5, Romania</footnote>
contains both the optical and thermal properties of the sample layers and also the contribution from the instrumental transfer function that comes from the underlying electronic circuitry. A proper normalization procedure allows one to filter out the instrumental contributions from the signal.

In recent decades, PTR is renowned for its non-invasive ability to find thermal parameters of solid samples. Unfortunately until now, to the best of our knowledge, only a few researchers have essayed towards its ability to examine the thermal properties of liquids [2–4]. In the article of Sreekumar and Vaidyan [2], thermal effusivity of the sample was determined by raster scanning a cell, which contains both the sample and the reference in separate cavities. Kusiak et al [3] reported a four-layer configuration (air–steel–sample–glass) for finding the thermal conductivity of liquid samples, in which the sample is sandwiched between two solid layers having known thermal parameters. Nevertheless, knowledge of sample’s density and specific heat capacity was needed to find the thermal conductivity. Dadarlat et al proposed a PPE-PTR combined detection of thermal parameters [4, 5], but in this case, the performances were limited by the presence of a thin, opaque pyroelectric sensor as the first layer. The difficulty in the simultaneous detection of any two thermal quantities (e.g., α and κ) is mainly due to the increase in number of layers and also because of the comparable thermal thickness of the light-absorbing top layer compared to the sample layer thickness [6].

Obviously in all these above-mentioned PTR methods, the procedures are not remarkably robust in comparison with the PPE methods. Moreover, the degeneracy between thermal quantities is also affecting adversely. These make the PTR method more complex and unattractive for liquid sample studies. However, since it is a non-contact method, PTR may find applications in such situations where remote sensing is a major concern. Different from the already-reported PTR techniques on liquids, which use at least four layers in the thermal model, in this paper we present a three-layer approach synonymous with a single thin sample layer [7] under investigation. The requirement to establish such a three-layer system is the thermal excitation and the signal detection must be direct from the sample surface. Here, it is achieved by using a visible-infrared transparent window (we have used a CaF₂ window) as the top layer and a visible transparent glass window as the backing to confine the liquid sample. The experiments have been performed either by using front laser excitation or by using rear laser excitation. A thin metal coating at the sample–CaF₂ and sample–backing interfaces allows the laser absorption and thereby the sample heating.

We present two configurations, the so-called back-front PTR (BF-PTR) and back PTR (B-PTR) to find thermal parameters of liquid samples. All these methods are presented theoretically and then experimentally using liquids having well-known thermal parameters. The sensitivity studies and error analyses show the robustness of each method. We have also included studies on the variations of thermal parameters versus electric field and versus temperature of a 5CB (4-cyano-4'-pentylbiphenyl) liquid crystal (LC) in order to indicate its application possibilities towards temperature and electric field varying studies.

<figure><img src="image_52.png" /><figcaption>Figure 1. A schematic diagram of the front and back IR setup. A sample is sandwiched between the semi-infinite top CaF₂ (layer 0) and backing (layer 2). (a) Front detection. (b) Back detection.</figcaption></figure>

## 2. Theory

The Fourier law of heat transfer phenomenologically models the temperature profile of a system. The response of an IR detector $V(\omega)$ due to the heating of a periodically modulated laser source ($I = \frac{1}{2}I_0(1+\exp(i\omega t))$) having peak intensity of excitation ($I_0$) and angular frequency ($\omega$) is given by

$$V(\omega) = \mathbb{Z}(\omega)\Gamma(\omega)e^{i\omega t} \quad (1)$$

Here, $\Gamma(\omega)$ is the dimensionless response factor which contains the thermal information on different layers, among which is the sample layer of interest. $\mathbb{Z}(\omega)$ represents the instrumental transfer function and can be eliminated by the normalization procedure. Considering the one-dimensional stacked configuration of layers (figure 1) and assuming no heat loss via convection or radiation, one can solve the heat diffusion equation and arrive at an expression for the temperatures at different layers [7, 8]. Assuming the light is completely absorbed at the surface of the sample layer and thus considering an optically opaque model with the sample’s optical absorption coefficient $\beta_1 \gg 1/\mu_1$, where $\mu_1$ is the thermal diffusion length of the sample layer, one can arrive at an expression for the sample temperature at $z = 0$, deriving from the equation of the three-layer model in [9, 16] as follows:

$$\Gamma_{\text{front}}(\omega, z=0) = \frac{I_f}{2\sigma_1\kappa_1} \left( \frac{e^{\sigma_1 d}(1+b_2) + e^{-\sigma_1 d}(1-b_2)}{(1+b_0)(1+b_2)e^{\sigma_1 d} - (1-b_0)(1-b_2)e^{-\sigma_1 d}} \right), \quad (2)$$

where $I_f$ is the peak intensity of the laser excitation at the front surface of the sample, $\kappa_1$ is the thermal conductivity and
d is the thickness of the sample layer. The indices 0, 1 and 2 represent the CaF₂, sample and backing layers, respectively. σ₁, b₀ and b₂ are given by

$$ \begin{aligned} \sigma_1 &= (1+i)/\mu_1, & \mu_1 &= \sqrt{\alpha_1/\pi f}, \\ b_0 &= e_0/e_1, & b_2 &= e_2/e_1 \end{aligned} \quad (3) $$

where α₁ is the thermal diffusivity of the sample and e₁ is the thermal effusivity. Index j takes the values 0, 1 and 2 according to each of these layers. Equation (2) gives the temperature at the surface of the sample (at z = 0) for a three-layer system (figure 1(a)) with light absorption at the interface between the CaF₂ window and the sample layer.

For the back excitation model (figure 1(b)), the surface temperature of the sample at the interface (z = -d) between the CaF₂ (layer 0) and the sample (layer 1) can also be derived from [9] and is given by

$$ \Gamma_{\text{back}}(\omega, z = -d) = \frac{I_b}{2\kappa_1\sigma_1} \left( \frac{2}{(1+b_0)(1+b_2)e^{\sigma_1 d} - (1-b_0)(1-b_2)e^{-\sigma_1 d}} \right), \quad (4) $$

where I_b is the peak intensity of the laser excitation. Equation (4) describes the temperature for the back detection.

## 2.1. Back-front PTR

Similar to the self-normalization procedure used to find the thermal diffusivity of solids [10], one can obtain the absolute thermal diffusivity of the liquid sample employing this configuration. Since it is a self-normalization procedure, the results have high precision and are independent of the thermal parameters of the surrounding layers. One needs to know only the thickness of the sample accurately to determine the thermal diffusivity and vice versa. The experimental configuration is depicted as in figure 1 with back-by-front normalization. In the front frequency scan, the laser heating and the signal generation are at the CaF₂-sample interface. For the back frequency scan, the laser heating is at the backing-sample interface and the signal is detected from the CaF₂-sample interface. The opaque coatings at either side of the sample provide heating by laser absorption. Since the back signal propagates through the sample, it possesses very high sensitivity to the sample's thermal diffusivity. At the same time, because of the heavily damped nature of the heat waves during propagation to the point of detection, we are bounded to some maximum range of modulation frequencies determined both by the thermal parameters of the sample and by the thickness of the sample cavity, above which the signal-to-noise ratio becomes lower.

The normalized signal for the BF-PTR configuration is obtained by dividing equation (4) with equation (2) and it can be written as follows:

$$ \Gamma_{\text{norm}}(\omega) = \frac{\Gamma_{\text{back}}(\omega, z = -d)}{\Gamma_{\text{front}}(\omega, z = 0)} = \frac{I_b}{2\kappa_1\sigma_1} \frac{2}{I_f} \left( \frac{e^{\sigma_1 d} (1+b_2) + e^{-\sigma_1 d} (1-b_2)}{e^{\sigma_1 d} (1+b_2) + e^{-\sigma_1 d} (1-b_2)} \right). \quad (5) $$

Doing further simplification, one can arrive at

$$ \Gamma_{\text{norm}}(\omega) = A_f e^{-\sigma_1 d} \left( \frac{1}{1 + R e^{-2\sigma_1 d}} \right), \quad (6) $$

with

$$ A_f = 2 \frac{I_b}{I_f (1 + b_2)}, \quad R = \frac{1 - b_2}{1 + b_2}, \quad b_2 = \frac{e_2}{e_1}. $$

The term exp(-2σ₁d) in equation (6) has a negligible contribution and it can be approximated to zero. Thus, the equation for the BF-PTR configuration can be written as follows:

$$ \Gamma_{\text{norm}}(\omega) = A_f e^{-\sigma_1 d}. \quad (7) $$

From equation (7), we can find the absolute thermal diffusivity of the sample layer. Even though the term A_f contains the effusivity of the sample and the substrate, it remains undetectable and just acts as an offset for the amplitude. We have done simulations for the full model, equation (5), and the reduced model, equation (7), and found that the reduced model is quite valid in our experimental range of frequencies.

Equation (7) is particularly useful due to its rather direct and only exponential term to find the sample's thermal diffusivity evolution as a function of time, temperature or electric field. The sensitivity to the thermal diffusivity of the sample increases with the modulation frequency. So it is advisable to realize the measurements in the high sensitivity experimental region in which the sample becomes quasi-thermally thick.

## 2.2. Back PTR

Like the back photopyroelectric technique [12, 13], the so-called B-PTR configuration possesses high sensitivity and allows simultaneous detection of thermal diffusivity and effusivity of the sample. As in the case of BF-PTR, here also we are bounded to some maximum range of modulation frequencies. As shown in figure 1(b), the heat is generated at the backing(quartz)-sample interface and the signal detection is from the CaF₂-sample interface for both the sample and reference scans. The normalized equation derived from equation (4) is given by

$$ \Gamma_{\text{norm}}(\omega) = \frac{\Gamma_{\text{back}}}{\Gamma'_{\text{back}}} = \frac{e'_1 (1 + b'_0) (1 + b'_2) e^{\sigma'_1 d}}{e_1 (1 + b_0) (1 + b_2) e^{\sigma'_1 d}} \left[ \frac{1 - R'_0 R'_2 e^{-2\sigma'_1 d}}{1 - R_0 R_2 e^{-2\sigma'_1 d}} \right], \quad (8) $$

with

$$ R_0 = \frac{1 - b_0}{1 + b_0}, \quad R_2 = \frac{1 - b_2}{1 + b_2}. \quad (9) $$

The primed variables show the thermal parameters of the reference material and those without a prime represent the sample. Figure 2 contains the simulations using equation (8) for the percentage variation of the frequency-dependent normalized amplitude and phase plotted with +1% change in thermal diffusivity and effusivity of a typical liquid sample (circles) (α₁ = 1.48 × 10⁻⁷ m² s⁻¹, e₁ = 650 J m⁻² K⁻¹ s⁻¹/²) of thickness d = 50 µm, sandwiched between the CaF₂ top layer and the bottom glass substrate (table 1). The dotted line with a cross shows a change in thermal effusivity (e₁) by +1% keeping original diffusivity (α₁, 0%), while the dashed line
<figure><img src="image_53.png" /><figcaption>**Figure 2.** Simulated plots of frequency-dependent amplitude and phase of a typical liquid sample ($\alpha_1 = 1.48 \times 10^{-7} \text{ m}^2 \text{ s}^{-1}$, $e_1 = 650 \text{ J m}^{-2} \text{ K}^{-1} \text{ s}^{-1/2}$, $d = 50 \text{ µm}$) showing the percentile change in amplitude (upper plot) and phase difference in degrees (lower plot); for +1% change in $\alpha_1$ and $e_1$ (circles); +1% change in $e_1$ keeping 0% change in $\alpha_1$ (dotted line with a cross) and +1% change in $\alpha_1$ keeping 0% change in $e_1$ (dashed line).</figcaption></figure>

<table><caption>**Table 1.** Thermal parameters used for fitting.</caption><thead><tr><th></th><th>Diffusivity<br>(m² s⁻¹)</th><th>Effusivity<br>(J m⁻² K⁻¹ s⁻¹/²)</th><th>Conductivity<br>(W m⁻¹ K⁻¹)</th></tr></thead><tbody><tr><td>CaF₂</td><td>3.58 × 10⁻⁶</td><td>5135</td><td>9.71<sup>a</sup></td></tr><tr><td>Quartz</td><td>8.4 × 10⁻⁷</td><td>1590</td><td>—</td></tr><tr><td>Water<sup>b</sup></td><td>1.43 × 10⁻⁷</td><td>—</td><td>0.598</td></tr></tbody></table>

<sup>a</sup> Data sheet, Crystran Ltd.

<sup>b</sup> Water at 20 °C [21].

represents +1% change in diffusivity and original effusivity ($e_1$, 0%). The trends of these plots demonstrate that the sensitivity to thermal diffusivity of the sample is monotonically increasing with the increase in modulation frequency for both the amplitude and the phase. It is also evident that the amplitude is sensitive to the thermal effusivity in both the higher and lower modulation frequencies, while the phase is sensitive only at the lower modulation frequencies. Thus for such measurements, where the sample has to be replaced with a reference material, an additional proportionality factor ($A_f$) may be considered in the fitting domain in order to nullify the offset on the signal amplitude and hence the phase-sensitive lower frequency data are needed to determine both the thermal parameters of equation (8).

### 3. Experimental details

The BF-PTR and B-PTR experimental configurations are as depicted in figure 1. The sample was periodically heated using a laser source (from Laser Quantum, He–Ne 532 nm, Ventus HP 532, 1.4 mm spot size) modulated by an acousto-optic modulator (IntraAction, AOM-40). We have used <30 mW mm⁻² in our temperature-dependent phase transition studies.

The IR signal coming from the sample surface was collected and focused by two parabolic gold-coated mirrors (Edmund Optics, 90 DEG Off Axis Parabolic, M2 R184613-10140) and was detected using liquid-nitrogen-cooled HgCdTe IR detector (Judson Technologies, Model: J15D12-M204-S01M-60). A telescopic lens arrangement with a magnification factor of 6 has been used to make the spot size (8.4 mm) big enough to avoid three-dimensional effects at the experimental range of frequencies (consequently, it further reduced the applied laser power density). A 2 mm thick (24 mm diameter) CaF₂ window (Crystran, UK) was used as the top semi-infinite layer for all the front and back experimental configurations. While for the backing layer, we have chosen special quartz cuvettes (cells with detachable windows, type: QS 124, sample space thickness error $\le \pm 5\%$, from Hellma Analytics), which in principle served as the sample holder and gave excellent accuracy over the sample thickness. For both methods, the sample heating by laser absorption was achieved by making thin opaque coatings of gold (the absorption coefficient of gold, corresponding to the wavelength of the laser beam, is of the order of $10^6 \text{ m}^{-1}$) at the sample–CaF₂ and sample–backing interfaces. This method of sample cell preparation and relative theoretical approximations is similar to that reported in [14, 15]. These metallic coatings also served as the electrodes for the electric field varying experiments. The thickness of these coatings was typically a few hundreds of nanometres and assumed to have no effect on the normalized signal; the high thermal diffusion length of gold makes it thermally very thin in our experimental frequency range. It was also ensured that no incident laser beam transmitted through the opaque coating, rather it was completely absorbed or blocked at the interfaces. Figure 3 shows the cell windows used for experiments.

The CaF₂-sample-backing assembly was mounted inside a 25 mm holder and tightened together in order to maintain the sample thickness unchanged. This holder was arranged with a tungsten coil heater inside, as shown in figure 3, that served as the first heating stage near the sample in order to perform the temperature-varying studies. In our temperature-varying experiments, we have used only a single heating stage in order to verify that our method is acceptable for phase transition studies. A sinusoidal signal of 1 kHz frequency from a function generator (RIGOL, DG2021A) was amplified with a 20× voltage amplifier (model: F20A) and applied to the LC samples for electric field varying experiments.

### 4. Results and discussion

We have performed experiments on liquids such as water and glycerol having well-known thermal parameters, presenting our results in this section as the validation of the methods described in section 2. Measurements on the 5CB LC are also included in order to indicate its application possibilities towards temperature and electric field varying studies. All the experiments except for the LC are carried out at ambient temperatures (20–22 °C). The thermal parameters of the surrounding layers and reference materials used for the normalization procedure are given in table 1.
<figure><img src="image_54.png" /><figcaption>**Figure 3.** Photograph of the experimental setup's components.</figcaption></figure>

<figure><img src="image_55.png" /><figcaption>**Figure 4.** Normalized BF-PTR signal amplitude (upper plot) and phase (lower plot), plotted as a function of square root of the modulation frequency for the water sample ($d=100~\mu\text{m}$). The circles are the experimental data and the full line represents the best fit. The CaF₂ window is used as the top layer and the quartz cell as the backing layer.</figcaption></figure>

<table><caption>**Table 2.** Thermal diffusivity from BF-PTR.</caption><thead><tr><th rowspan="2"></th><th colspan="3">BF-PTR</th><th>Literature</th></tr><tr><th>Amplitude fitting</th><th>Phase fitting 10<sup>-7</sup> (m<sup>2</sup> s<sup>-1</sup>)</th><th>Complex fitting</th><th>10<sup>-7</sup> (m<sup>2</sup> s<sup>-1</sup>)</th></tr></thead><tbody><tr><td>Water</td><td>1.441±0.005</td><td>1.435±0.006</td><td>1.436±0.006</td><td>1.437±0.008<sup>a</sup></td></tr><tr><td>EG</td><td>0.926±0.006</td><td>0.924±0.007</td><td>0.922±0.009</td><td>0.918±0.002<sup>c</sup></td></tr><tr><td>Lanolin</td><td>0.50±0.02</td><td>0.49±0.02</td><td>0.53±0.03</td><td>—</td></tr></tbody></table>

## 4.1. Results obtained with the BF-PTR configuration

### 4.1.1. BF-PTR experiment on water.
The normalized experimental amplitude and phase for the water sample plotted as a function of square root of modulation frequency are shown in figure 4. The fitting has been done for the experimental amplitude, phase and complex (fitting amplitude and phase data together simultaneously) data using equation (7) and the obtained results for the thermal diffusivity of water are given in table 2 together with other literature values for comparison. The agreement of our results with the thermal diffusivity

values from literatures supports the validity of the method. The use of precision cells helped to minimize the error due to thickness uncertainties and in fact it gave higher accuracies in our experimental results.

The $\chi^2$ error values indicating the statistical uncertainties of the experimental results from each type of fitting (amplitude, phase and complex) are also included in table 2. Interestingly, the small values of the uncertainties comparable with the PPE measurement results represent the high sensitivity of the method and quality of the experimental signal. We have also included our experimental results on ethylene glycol (EG) and lanolin (wool wax) [22] in table 2.

### 4.1.2. BF-PTR experiment on 5CB with a varying electric field.
We have studied the variations of 5CB LC thermal diffusivity under the varying electric field. The frequency scans for different electric fields were conducted and the normalization for each back frequency scan has been done with the corresponding front scan. The results were obtained by fitting the normalized experimental signals using equation (7). In practice, a single front frequency scan serves the purpose of the normalization signal for all the back scans because the front signals are less sensitive to the variation of the sample's thermal parameters, thus resulting in similar or same amplitude and phase values for all the front scans provided that the scans are performed near the sample's quasi-thermally thick regime. The experiment has been carried out around ~28 °C (nematic phase: 25–35 °C). The LC is dropped into the sample holder and heated above 35 °C, until its clearing temperature, before closing the cell for the experiment. Our results on the electric field-dependent thermal diffusivity of 5CB are shown in figure 5. The thermal diffusivity enhancement is about 9% without doing any pre-surface treatments which is necessary to obtain the high-degree planar alignment and thereby increase the thermal parameter enhancement [18]. This behaviour of the thermal diffusivity of 5CB as a function of electric field is similar to that obtained with PPE in [15].

## 4.2. Results obtained with the B-PTR configuration

### 4.2.1. B-PTR experiment on glycerol.
The B-PTR method has the advantage of obtaining thermal diffusivity and effusivity of liquid samples simultaneously with high certainty. Figure 6 shows the frequency-dependent experimental result on glycerol (sample) normalized with water (reference). The thermal parameters used for the reference liquid are from table 1. The experimental data (amplitude, phase and complex)

<footnote><sup>a</sup>[11] at 25 °C, <sup>b</sup>[21] at 20 °C, <sup>c</sup>[19].</footnote>
<figure><img src="image_56.png" /><figcaption>**Figure 5.** Variation of thermal diffusivity of 5CB LC (at 28 °C) versus varying electric field (sinusoidal, 1 kHz) from the BF-PTR experiment.</figcaption></figure>

<figure><img src="image_57.png" /><figcaption>**Figure 6.** Experimental result on glycerol (sample) normalized with water (reference) plotted as a function of *f*. Amplitude (upper plot) and phase (lower plot) are fitted simultaneously. The symbols show the experimental curve and the full line represents the best fit ($d = 50$ µm).</figcaption></figure>

<table><caption>**Table 3.** B-PTR experimental results on glycerol.</caption><thead><tr><th></th><th>Diffusivity<br>10<sup>-8</sup> (m<sup>2</sup> s<sup>-1</sup>)</th><th>Effusivity<br>(J m<sup>-2</sup> K<sup>-1</sup> s<sup>-1/2</sup>)</th></tr></thead><tbody><tr><td>Amplitude fit</td><td>9.45 ± 0.02</td><td>958 ± 2</td></tr><tr><td>Phase fit</td><td>9.425 ± 0.007</td><td>940 ± 10</td></tr><tr><td>Complex fit</td><td>9.43 ± 0.02</td><td>945 ± 3</td></tr><tr><td>[20]</td><td>9.45</td><td>933</td></tr><tr><td>25 °C [21]</td><td>9.99</td><td>923</td></tr></tbody></table>

have been fitted with equation (8) to obtain the thermal diffusivity and effusivity. The results obtained for the best fitting values of glycerol with statistical uncertainties are shown in table 3 and these results are in good agreement with the already-reported values. To check if the obtained set of

values has any pairwise fitting dependence, we have performed a contour map for the $\chi^2$ cost function versus $\alpha$ and $e$ as in figure 7(a). It shows closed curves around the best fitting pair of values which confirms the absence of any interdependence between the two thermal parameters.

The absence of degeneracy between any two fitting parameters may not be trustworthy in this situation because of the third fitting parameter called the proportionality factor ($A_f$) which comes from the sample-reference replacement. This may introduce the chance of threefold degeneracy between the three unknown fitting parameters ($\alpha$, $e$ and $A_f$). It can be brought out by doing the so-called most-squares error analysis [11, 17]. In this, in contrast to the classical least-squares analysis, where only one parameter is varied keeping all other parameters fixed, the optimized set of parameters for each value of a given third parameter (e.g., $\alpha$) around its best fit is determined until minimizing the $\chi^2$ cost function. Figures 7(b) and (c) show the most-squares analysis for the thermal diffusivity and effusivity. The parabolic shape of the plots indicates the absence of degeneracy with all other fitting parameters for each of these thermal quantities. The sharpness of the parabola is a proof of lower statistical uncertainty and high accuracy of the experimental method.

### 4.2.2. B-PTR experiment on 5CB with varying electric field.
With the B-PTR experimental method, we have also investigated the variations of $\alpha$ and $e$ for the 5CB LC with varying electric field. The obtained results are from the scans performed at the high sensitive frequency region, where the sample was in the quasi-thermally thick regime. The frequency scan at zero electric field was taken as the reference signal for normalization for all other signals with applied electric field; thus, $A_f$ is kept fixed. The reference thermal parameters for the 5CB at zero electric field for thermal diffusivity were from the BF-PTR result on 5CB and for the effusivity were from [15]. Figure 8 shows the typical normalized experimental signals as a function of frequency for three different electric fields. The normalized signals were fitted using equation (8) and figure 9 shows the obtained thermal diffusivity and effusivity values plotted as a function of applied electric field.

### 4.2.3. B-PTR experiment on 5CB with varying temperature.
Furthermore, B-PTR gives the advantage of finding thermal diffusivity and effusivity simultaneously from a single frequency for those measurements in which $A_f$ is kept fixed. It helps to avoid time-consuming frequency scans while preserving better accuracy. A sample's thermal diffusivity can be calculated from the phase and the effusivity from the combined amplitude and phase signals. The analytical expressions for $\alpha$ and $e$ are given by equations (10) and (11), respectively. Both these equations are derived from equation (8) considering the sample is in the quasi-thermally thick ($\mu_s \approx d$) frequency regime where the negative exponential term can be approximated to zero ($e^{-2\sigma_1 d} \approx 0$):

$$ \alpha_s = \left( \frac{d \sqrt{\alpha_r \pi f}}{d \sqrt{\pi f} - \theta \sqrt{\alpha_r}} \right)^2 \quad (10) $$
<figure><img src="image_58.png" /><figcaption>Figure 7. Contour plot (a) showing the closed curves with $\chi^2$ minima around the best fitting values of $\alpha$ and $e$ confirms the absence of degeneracy. The most-squares error analysis (full line) plotted along with the least-squares (dashed line) for thermal diffusivity (b) and effusivity (c) confirms that the results are unique. The statistical uncertainty is inversely proportional to the sharpness of the parabolic minima. The analysis is for the B-PTR experiment (figure 6) with glycerol as sample and water as reference.</figcaption></figure>

<figure><img src="image_59.png" /><figcaption>Figure 8. Normalized experimental signals (symbols) and the best fits (lines) are plotted versus laser modulation frequency for the 5CB LC sample under three different electric fields. Signals are fitted to obtain $\alpha$ and $e$ keeping $A_f$ fixed. The reference signal was from the sample at zero electric field.</figcaption></figure>

$$e_s = -\frac{1}{2} \left\{ L - R + \sqrt{(L - R)^2 - 4J} \right\} \quad (11)$$

where

$$R = \frac{e^{\theta}}{e_r A} (e_r^2 + e_r L + J) \quad L = e_c + e_b \quad J = e_c e_b.$$

$A$ is the normalized amplitude and $\theta$ is the normalized phase. $e_r$, $e_s$, $e_c$ and $e_b$ are the thermal effusivities of reference, sample, CaF$_2$ window and backing, respectively. $\alpha_s$ and $\alpha_r$ are the thermal diffusivities of sample and reference, respectively. $f$ represents the modulation frequency and $d$ is the sample thickness. Figure 10 shows the variation of thermal diffusivity, effusivity and conductivity (calculated from $\alpha$ and $e$) versus temperature for the 5CB LC. The temperature-dependent experimental signal was normalized with the signal at the isotropic phase and the reference values were chosen from [18]. The experiment has been performed with a single stage

temperature controller with temperature resolution better than 10 mK.

We have also conducted experiments to find thermal parameters from the front scans alone. But in this case, the signal sensitivity to the sample's thermal parameters is extremely low as mentioned earlier. Mixing with the noise, the front normalized signal shows relatively less variation for the changes for sample's thermal parameters. This could be improved by choosing a highly effusive backing such as steel or aluminium instead of quartz, even though the uncertainties are nearly ten times higher than BF-PTR or B-PTR experimental results. Thus, finding thermal parameters from the front scans alone becomes more difficult and less precise.

Until now, the most used method for liquid thermal characterization is probably PPE calorimetry. The main advantage of this method compared to other PT techniques is its simplicity: it measures directly the temperature variation of the sample with the special feature for liquids of a perfect liquid/pyroelectric sensor thermal contact. In this paper, we propose the PTR technique as an alternative calorimetry. Concerning the accuracy of the investigations, in both techniques, when using sandwiched detection cells, it was demonstrated [23, 5] that it is connected with the values of the thermal parameters of the layers of the detection cell; more exactly, the accuracy is increased when the values of the thermal effusivity of the liquid and backing layers are close. In the PTR configurations proposed in this paper, the backing layer is glass with a thermal effusivity rather close to the typical liquids, so the experiment is optimized in order to obtain high accuracy. While considering the signal-to-noise ratio, PPE outperforms PTR, which limits the temperature resolution granted by PTR in highly resolved temperature-dependent measurements. Some recent advances in IR detection technologies promise us improved sensing possibilities [24], which in principle allows us to perform experiments at extremely low laser powers than it requires now. The requirement of IR transparency of the first layer of the proposed PTR detection cell can be easily fulfilled, a lot of commercially available materials satisfying this criterion.
B-PTR: Thermal Diffusivity and Effusivity under varying Electric Field

<figure><img src="image_60.png" /><figcaption>**Figure 9.** Variation of α (a) and e (b) of the 5CB LC with varying electric field. Results from B-PTR experiments performed at 28 °C when the sample is in the nematic phase.</figcaption></figure>

<figure><img src="image_61.png" /><figcaption>**Figure 10.** Experimental results on variations of thermal diffusivity (a), thermal effusivity (b) and thermal conductivity (c) from the temperature-varying B-PTR experiment at the nematic-isotropic transition of 5CB. The experiment has been performed at *f* = 32 Hz and *d* = 36 µm. The thermal conductivity has been calculated from *e* and *α*.</figcaption></figure>

**5. Conclusion**

We have explored the ability of photothermal infrared radiometry for studying thermal properties of liquid samples and demonstrated its capability to give results with high accuracy. The two presented configurations are described theoretically followed by experimental results. BF-PTR gives the absolute thermal diffusivity of a sample knowing only the sample's thickness and also the result is independent of the thermal parameters of surrounding layers. The B-PTR configuration has the potentiality of finding two

thermal parameters simultaneously. Both the back signal detection involved configurations, B-PTR and BF-PTR, possess high sensitivity to the thermal parameters of the sample and can yield results from a single modulation frequency. This is particularly helpful for finding thermal properties in temperature or time varying experiments. We have performed experiments with known samples and found very good agreement with already-reported values of thermal parameters. As a validation of prospective applications, we have included studies on the 5CB LC with varying electric field and temperature. All these results indicate the broad possibilities of using PTR as an alternative approach to investigate liquid samples thermally with high accuracy.

**Acknowledgement**

This work has been supported by the French Ministry of Research. The authors would like to thank Stephane Longuemart and Sylvain Delenclos for their fruitful discussions, and Benoit Escorne for his technical support.

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5.3 Maxwell-Wagner-Sillars effects on the thermal-transport
properties of polymer-dispersed liquid crystals, Meas.
Sci. Technol., Phys. Rev. E 89, 022511 (2014)
# Maxwell-Wagner-Sillars effects on the thermal-transport properties of polymer-dispersed liquid crystals

M. Kuriakose, S. Longuemart, M. Depriester, S. Delenclos, and A. Hadj Sahraoui*

Univ. Lille Nord de France, F-59000, France; ULCO, UDSMM, F-59140 Dunkerque, France; and Unité de Dynamique et Structure des Matériaux Moléculaires, EA 4476, F-59140 Dunkerque, France

(Received 2 August 2013; revised manuscript received 13 December 2013; published 24 February 2014)

We present the depolarization field effects (Maxwell-Wagner-Sillars effect) for the thermal transport properties of polymer dispersed liquid crystal composites under a frequency-dependent electric field. The experiments were conducted on polystyrene/4-Cyano-4'-pentylbiphenyl (PS/5CB) PDLCs of 73 vol.% and 85 vol.% liquid crystal (LC) concentrations. A self-consistent field approximation model is used to deduce the electrical properties of polymer and LC materials as well as the threshold electric field. Electric field-varying (at constant frequency) experiments were also conducted to calculate the interfacial thermal resistance between the LC droplets and polymer matrix as well as to find the elastic constant of LCs in droplet form.

DOI: 10.1103/PhysRevE.89.022511

PACS number(s): 42.70.Df, 44.10.+i, 44.35.+c, 66.30.hk

## I. INTRODUCTION

Polymer dispersed liquid crystals (PDLCs) are made of micron or submicron liquid crystal (LC) droplets encapsulated in a polymer matrix. Their much better performance than conventional LC displays as well as their broad application potential such as in flexible displays [1–3], window shutters, projection displays [4], optical switches [5], focus-tunable lenses [6], etc., have led the scientific world to consider this as a promising candidate in technology. The usual operation of PDLCs is dependent on changing the film transparency to the visible light from an opaque (OFF) state to a transparent (ON) state by the application of an external electric field (EF). The reason for such behavior is the mismatch in refractive indexes between LC droplets and the polymer matrix, which scatters away the incident light in an OFF state. Upon the application of an external EF, LC droplets modify their molecular alignment so that the refractive indexes of the host polymer layer and the LC droplets match each other in the direction of the EF. This makes the film transparent in that direction. A vast amount of studies on PDLCs’ electro-optic properties have been done by various research groups to quantify and improve PDLC performance. Nevertheless their inherent structural complexities and, at the same time, the possibility to control their physical properties via external means (e.g., by applying an electric field) have led researchers to consider them as a research interest.

In composite materials, PDLCs are different from other types of composites (e.g., metallic particles in polymers) because they give the opportunity to change their transport properties via applying an EF without changing the amount, shape, or size of inclusions (LC) in a material sample under investigation. This behavior is particularly useful while studying thermal transport through composites.

Our previous studies [7] of PS/5CB PDLC samples show that the heat transport through PDLCs are affected both by the presence of an external EF and because of the variation in LC volume fraction. Here the sample’s thermal conductivity increased monotonically and reached a saturation value with

respect to an increase in the applied EF. This change in thermal conductivity is due to the reorientation of LC droplet directors from their random orientations to the direction of the applied field, which is in parallel to the direction of heat flow through the material sample. In order to prevent the effects of depolarization fields, an EF of 1 kHz sinusoidal wave was used in such experiments.

In this paper we report on the effects of depolarization fields (Maxwell-Wagner-Sillars effect) on thermal transport through PDLC samples. We selected PS/5CB composites for our studies because of their well-known thermo-physical properties. Recently the photothermal radiometry technique (PTR) has been demonstrated as a simple and capable technique for measuring dynamic thermal parameters of liquid samples with high precision and resolution [8]. In particular, the noncontact capability of the PTR technique helps to extract the thermal parameters from EF-varying experiments with fewer precautions, whereas many efforts are required with contact techniques, such as photopyroelectric (PPE), to facilitate experiments free from EF-induced errors.

## II. MAXWELL-WAGNER-SILLARS EFFECT

Depolarization field effects in heterogeneous systems were first highlighted by Maxwell and later modified successively by Wagner and Sillars. So the interfacial polarization effects are often termed the Maxwell-Wagner-Sillars (MWS) effect [9,10]. In a heterogeneous mixture of LC droplets and a polymer matrix, due to the differences in their dielectric characteristics (ability to hold charges) as well as electrical conductivities, an additional charge accumulation occurs at each LC-polymer boundaries when the material is under an external EF, as shown in Fig. 1.

At low enough frequencies of the applied alternating field, the charges, near each LC-polymer boundary, start to build up (conductivity effect) and oppose the external EF. As a consequence, the effective field acting on the LC droplets reduces its strength. This in turn conversely affects the reorientation of the droplets and causes a decrease of the heat flow through it, in other words through the PDLC film, in the direction of the EF. The depolarization field effects in such multiphase systems vanish at sufficiently high frequencies of

<footnote>*hadj@univ-littoral.fr</footnote>
<figure><img src="image_62.png" /><figcaption>FIG. 1. (Color online) Maxwell-Wagner-Sillars effect in an heterogeneous medium. The simplest case with a single LC droplet surrounded by a host polymer matrix.</figcaption></figure>

an applied EF because the movement of charges will be frozen out at high frequencies and the development of depolarization fields gets interrupted.

Several electro-optic experimental studies are reported on the interfacial polarization effects in PDLCs [11–13]. Boussoalem et al. [14,15] conducted studies on polystyrene/4,4'-octylcyanobiphenyl (PS/8CB) and PS/5CB dispersed systems and reported on their frequency-dependent behavior attributed to the MWS effect. Their studies, on the light transmission vs frequency of the applied EF, were modeled using a self-consistent field approximations for calculating the average electric field ($E_{drop}$) acting on spherical LC droplets in a PDLC sample. The equation, taking into account the MWS effect and relating the average droplet electric field, $E_{drop}$, to the applied EF, $E_a$, is given by [16,17]

$$E_{\text{drop}}(\omega_e) = \frac{3\epsilon_M^*}{\epsilon_{\text{LC}}^* + 2\epsilon_M^* - \phi_{\text{LC}}(\epsilon_{\text{LC}}^* - \epsilon_M^*)} E_a \quad (1)$$

and

$$\epsilon_M^* = \epsilon_M' - i \frac{\sigma_M}{\omega_e \epsilon_0}, \quad (2)$$

$$\epsilon_{\text{LC}}^* = \epsilon_{\text{LC}\parallel}^* - i \frac{\sigma_{\text{LC}\parallel}}{\omega_e \epsilon_0}, \quad (3)$$

where $\omega_e = 2\pi f_e$ and $f_e$ is the frequency of the applied EF; $\phi_{LC}$ is the phase-separated LC fraction; $E_a$ is the magnitude of the applied electric field; $\epsilon_M'$ and $\sigma_M$ are the dielectric permittivity and electrical conductivity of the polymer matrix, respectively; and $\epsilon_{LC\parallel}$ and $\sigma_{LC\parallel}$ are dielectric permittivity and electrical conductivity of the LC in the direction of $E_a$, respectively.

Inside PDLC, a part of the LC dissolves into the polymer. A maximum amount of the LC dissolved in a polymer matrix is defined as the solubility limit ($\beta$), and it can be found by performing differential scanning calorimetry measurements. The undissolved portion of the LC forms droplets. Knowing $\beta$ and the amount of LC added to the polymer ($x$), we can calculate the phase-separated LC fraction ($\phi_{LC}$) using [18–20]

$$\phi_{\text{LC}} = \frac{100}{x} \frac{x - \beta}{100 - \beta}, \quad x \ge \beta, \quad (4)$$

where all the quantities are in a volume percentage. The value of $\beta = 55\%$ was used to calculate $\phi_{LC}$ in our experiments [14].

### III. SAMPLE PREPARATION

Polymer dispersed liquid crystal (PDLC) samples were prepared using polystyrene (Sigma-Aldrich, molecular weight of 44 000 g/mol) as the polymer matrix and 5CB (Merck) as the inclusion with two different volume fractions, LC:PS (85:15 and 73:27). Some of the advantages of PS compared to other thermoplastic compounds are a very low ac electrical conductivity ($\sigma_{ac} \approx 10^{-9}$ S/m at 1 kHz) and uniform dielectric permittivity ($\epsilon_M \sim 2.5$) in both temperature (25–80 °C) and frequency [21,22]. The density ($\rho$) of PS and 5CB are $\approx 1$ ($\rho_{PS} = 1.047$ kg/m$^3$ and $\rho_{5CB} = 1.008$ kg/m$^3$). Dielectric permittivity and ac electrical conductivity of 5CB are $\epsilon_{LC\parallel}^* = 10$ and $\sigma_{LC\parallel} = 10^{-8}$ S/m [23,24]. The temperature range of the nematic phase of 5CB is between 23 and 35 °C [25]. Thermal conductivity values for the PS matrix, $\kappa_M = 0.11$ W/mK, as well as for 5CB, aligned in parallel (long axis of LC molecules aligned in parallel to the heat flow direction), $\kappa_{LC\parallel} = 0.24$ W/mK, and perpendicular (long axis of LC molecules aligned perpendicular to the heat flow direction), $\kappa_{LC\perp} = 0.12$ W/mK, are taken from Refs. [26,27].

PDLC samples were prepared by a solvent-induced phase separation [28] procedure. At first, the required amount of both materials (5CB and PS) completely dissolved with an equal amount of chloroform. The mixture was placed in an oven for more than 24 h for a complete evaporation of solvent to obtain the final PDLC sample. Upon solvent evaporation, LC-PS phase separation happens, and 5CB droplets on the order of micron or submicron sizes develop throughout the entire polymer matrix. Micrographs of as prepared PDLCs for both LC:PS concentrations are shown in Fig. 2. The pictures were taken using a polarizing microscope with crossed polarizers.

### IV. EXPERIMENTAL

We have used an improved photothermal radiometry (PTR) technique [8,29] for measuring thermal parameters of PDLC samples. The experiments were done using the back-PTR (B-PTR) configuration because of its high sensitivity and simultaneous detectability to both the thermal diffusivity and effusivity of the sample under investigation. The experimental arrangement for the sample-cell assembly is shown in Fig. 3. The sample was kept between a CaF$_2$ window on the top and a quartz precision cell (backing) on the bottom. An internally modulated laser ($\lambda = 532$ nm, He-Ne, Ventus HP 532, 1.4 mm spot size) source was used to generate heat at the backing-sample interface with the help of a thin ($\approx 500$ nm thickness), opaque, laser-absorbing coating layer. The thermal waves diffuse into the sample and reach the top layer where a second coating, at the sample-CaF$_2$ interface, emits the infrared signal. The generated signal from the top CaF$_2$-sample interface contains the thermal information of the sample-cell assembly. It has been detected by the aid of a liquid nitrogen-cooled HgCdTe infrared detector (Judson Technologies, model J15D12-M204-S01M-60, bandwidth, dc, 250 kHz). Since the CaF$_2$ window is transparent to both the visible and infrared wavelengths of light, it acts as a semi-infinite top layer in the
<figure><img src="image_63.png" /><figcaption>FIG. 2. (Color online) PDLC micrographs under microscope with crossed polarizers. Images of PS:5CB = 27:73 (73% LC) (left) and PS:5CB = 15:85 (85% LC) (right).</figcaption></figure>

thermal configuration and passes the generated infrared signals directly to the detector.

Generally, in PTR experiments, two separate signals as a function of the laser modulation frequency (a sample scan and a reference) are used to obtain a normalized signal (by the complex division of two signals). This normalized signal, which is free from the instrumental transfer function, is analyzed to find the thermal properties of the sample under investigation while knowing the thermal properties of surrounding layers. Since B-PTR technique has the ability to find thermal parameters from a single laser modulation frequency, we have done experiments by fixing the laser modulation frequency and scanning the sample under varying EF or frequency-dependent EF ($f_e$). Thus, a sample's thermal diffusivity, as a function of EF or $f_e$, can be calculated from the phase data and the effusivity from the combined amplitude and phase signals, provided that the sample is at a quasithermally thick ($\mu_s \approx d$) regime. The analytical expressions for a sample's thermal diffusivity ($\alpha_s$) and thermal effusivity ($e_s$) are given by [8]

$$ \alpha_s = \left( \frac{d \sqrt{\alpha_r \pi f}}{d \sqrt{\pi f} - \theta \sqrt{\alpha_r}} \right)^2, \quad (5) $$

$$ e_s = -\frac{1}{2} \{L - R + \sqrt{(L - R)^2 - 4J}\}, \quad (6) $$

where

$$ R = \frac{e^{\theta}}{e_{r}A}(e_{r}^{2} + e_{r}L + J), \quad L = e_{c} + e_{b}, \quad J = e_{c}e_{b}; $$

<figure><img src="image_64.png" /><figcaption>FIG. 3. Back-PTR configuration.</figcaption></figure>

here A is the normalized amplitude and $\theta$ is the normalized phase, and $e_r$, $e_s$, $e_c$, and $e_b$ are the thermal effusivities of the reference, sample, CaF$_2$ window, and backing, respectively. $\alpha_s$ and $\alpha_r$ are the thermal diffusivities of the sample and reference, respectively. f represents the modulation frequency of the laser source, and d is the sample thickness.

In order to facilitate the experiments, the required amount of the PDLC sample was cast into the quartz precision cell using a spatula, prearranged with electric contacts for the electro-thermal measurements, and heated to the matrix-softening temperature, and we closed the cell with a CaF$_2$ window. The measurements were carried out by keeping the

<figure><img src="image_65.png" /><figcaption>FIG. 4. (Color online) Variation of thermal diffusivity (a), thermal effusivity (b), and thermal conductivity (c) as a function of the applied electric field (at constant frequency, $f_e$, 1 kHz) for PDLC samples with 85% LC (blue circles) and 73% LC (red squares).</figcaption></figure>
<figure><img src="image_66.png" /><figcaption>FIG. 5. (Color online) ρC plotted as a function of EF for PDLC samples with 85% LC (a) and 73% LC (b). Experimental data points for 85% LC (blue circles) and for 73% LC (red squares) along with the mean values of ρC (solid lines).</figcaption></figure>

sample temperature constant at ~28°C in order to avoid temperature-dependent thermal property changes. The sample cell thickness used was 50 ± 2μm.

## V. RESULTS AND DISCUSSION

The first part of the experiments on PDLCs were done with varying amplitude of the applied EF at fixed frequency ($f_e$ = 1 kHz). These results are used to find the role of interfacial thermal resistance between the polymer matrix and LC droplets. The second part of the experiments deals with the frequency dependence on the thermal properties of PDLCs.

### A. PDLC under a varying electric field

We have investigated the variation of thermal parameters of PDLC samples (85% LC and 73% LC) under varying electric field at constant frequency ($f_e$ = 1 kHz), and the results are depicted as shown in Fig. 4. The PTR experiments were conducted by fixing the laser modulation frequency, f = 18 Hz,

Equation (8) allows us to compute $r_k$, once we know the mean radius of the droplets. Micrographs obtained from a polarized microscope, for both samples (Fig. 2), are used for the estimation of droplet sizes. The majority of the droplets lie within diameters of 1–5 μm for the 85% LC sample and 0.5–2 μm for the 73% LC sample.

### 2. Order parameters of PDLC film

Considering the thermal anisotropy of the nematic LC phase, the effective thermal conductivity of the PDLC depends on the average orientation of the LC molecule with respect to

at which the sample is in a quasithermally thick regime, and varying the EF strength successively. The displayed results (Fig. 4) are from the mean of three consecutive EF scans. Additional filtering has also been done by plotting the volumetric heat capacity ($\rho C = e/\sqrt{\alpha}$) and by taking data points lying within ±1% or ±0.55% (depending on the population of experimental data points) from the linear fit value of $\rho C$ as shown in Fig. 5. Here the behavior of $\rho C$ is independent of the applied electric field as expected since the molecular reorientation in LCs does not make any contributions towards $\rho C$, and hence it stays unchanged. Nevertheless, a change in concentration of the liquid crystals in PDLC produces a shift in the volumetric heat capacity value.

#### 1. Effective medium theory and interfacial thermal resistance

In heterogeneous mixtures, the presence of interfaces and thereby an interfacial thermal resistance hinder heat transport through boundaries. Nan et al. [30] gave an estimation of effective thermal conductivity in heterogeneous mixtures with spherical inclusions on the basis of an effective medium approach, which has taken into account interfacial thermal resistance, and it is given by

$$ \kappa_{\text{PDLC}} = \kappa_m \frac{\kappa_{\text{LC}}(1 + \gamma) + 2\kappa_m + 2\phi_{\text{LC}}[\kappa_{\text{LC}}(1 - \gamma) - \kappa_m]}{\kappa_{\text{LC}}(1 + \gamma) + 2\kappa_m - \phi_{\text{LC}}[\kappa_{\text{LC}}(1 - \gamma) - \kappa_m]}, \quad (7) $$

where $\gamma = r_k/r_d$ is the ratio of the Kapitza radius ($r_k$) to droplet radius ($r_d$). $\kappa_m$ and $\kappa_{\text{LC}}$ are thermal conductivities of the polymer matrix and LC, respectively. The quantity $\gamma$ is a dimensionless parameter which compares the Kapitza radius, $r_k$, to the droplet radius, $r_d$. The Kapitza radius is related to thermal interfacial resistance, and it can be considered as a quantity which indicates whether the interfaces play a significant role in the heat transport through the material composite. If $\gamma \ll 1$, the influence of the interfaces may be ignored, but if $\gamma > 1$ the contribution of the interfaces to the effective thermal conductivity becomes more important than that of the volume of the inclusions.

By knowing the thermal conductivities of the polymer matrix and LC as well as by experimentally finding $\kappa_{\text{PDLC}}$ and $\phi_{\text{LC}}$ [Eq. (4)], one can calculate $\gamma$ by rearranging Eq. (7), which is given by

$$ \gamma = \frac{(\kappa_m / \kappa_{\text{PDLC}})(\kappa_{\text{LC}} + 2\kappa_m + 2\phi_{\text{LC}} - 2\phi_{\kappa_m}) - \kappa_{\text{LC}} - 2\kappa_m + \phi_{\kappa_{\text{LC}}} - \phi_{\kappa_m}}{2\kappa_{\text{LC}} + \phi_{\kappa_{\text{LC}}} + 2(\kappa_m \kappa_{\text{LC}} / \kappa_{\text{PDLC}})(\phi_{\text{LC}} - 1)}. \quad (8) $$

the heat flow direction. For PDLC systems, two order parameters must be considered. Zumer and Doane [31] introduced the notion of droplet director $\hat{n}_D$. The order can be quantified for each droplet as the deviation of the molecule from this director. The droplet’s order parameter, $S_D$, can be defined as

$$ S_D = \langle \frac{1}{2} \{ 3[\hat{n}_D \cdot \hat{n}(\hat{r})]^2 - 1 \} \rangle, $$

where $\hat{n}(\hat{r})$ denotes the local nematic director and $\langle \rangle$ is the averaging over the ensemble of molecules of the droplet.

Usually, in PDLCs zero field droplet directors are randomly distributed over all the sample. The effective thermal
conductivity will thus depend on these droplets' director distribution, and one may define as well a sample order parameter $S_S$:

$$S_S = \left\langle \frac{1}{2} [3(\hat{n}_D \cdot \hat{z})^2 - 1] \right\rangle = \left\langle \frac{1}{2} (3 \cos^2 \theta - 1) \right\rangle,$$

where $\hat{z}$ denotes the heat flow direction or the applied EF direction and $\theta$ the angle between the droplets' director and the $\hat{z}$ axis, the average being done over all the droplets of the sample.

Finally, the effective thermal conductivity of the LC phase in response to the applied electric field may be expressed as

$$\kappa_{LC} = \kappa_{\perp} + \frac{1}{3}(1 + 2 S_S S_D)\Delta\kappa \quad (9)$$

with $\Delta\kappa = \kappa_{\parallel} - \kappa_{\perp}$ the thermal anisotropy of the liquid crystal, and $\kappa_{\parallel}$ and $\kappa_{\perp}$ the thermal conductivity along or perpendicular to the molecular axis. Interestingly, Eq. (9), which we obtained for the effective thermal conductivity of LC droplets, is similar to the equation for the average dielectric constant of LC droplets in PDLC [32,33].

The behavior of the thermal conductivity (Fig. 4) is intimately linked with the reorientation of molecules in LC droplets. The application of an electric field will facilitate the orientation of the droplet directors along this direction and thus will increase the order parameter $S_S$ leading to a higher effective thermal conductivity as $\Delta\kappa$ is positive. For sufficiently high electric fields, all droplet directors align with the external field ($S_S = 1$) and the thermal conductivity reaches a maximum value.

For PS/5CB PDLC, nematic droplets possess the bipolar configuration; at a zero field the nematic is aligned tangentially at the droplet wall [Fig. 6(a)]. Upon application of the field, the nematic within the center of each droplet aligns quickly with the field. This process leads to a droplet, where the bulk of the nematic is aligned with the applied field, but near the wall of the droplets the nematic molecules are still aligned tangentially to the droplet surface [Fig. 6(b)]. This bipolar configuration corresponds to the calculated value of $S_D = 0.7$ for spherical droplets, which remains unchanged upon application of an electric field [32]. The orientation of the molecules along the surface of the droplets remains unchanged in the presence and in the absence of the electric field; this led us to consider that the thermal interfacial resistance is constant and independent of the electric field. For calculating $\gamma$, we have used thermal

<figure><img src="image_67.png" /><figcaption>FIG. 6. Schematic of the bipolar droplet alignment in a nematic phase (a) at a zero field (droplet director can align in any direction) and (b) at a high field (director aligns parallel to the EF).</figcaption></figure>

<figure><img src="image_68.png" /><figcaption>FIG. 7. Orientation of LC droplets inside a PS matrix. (a) Droplet directors are randomly oriented in the absence of an external EF; (b) droplet directors are aligned in the direction of an applied EF.</figcaption></figure>

conductivities when the LC molecules aligned in the direction parallel to the EF, at which $\kappa_{PDLC} = \kappa_{PDLC_{\parallel}}$ and $\kappa_{LC} = \kappa_{\perp} + \frac{1}{3}\Delta\kappa(1 + 2 S_D)$ with $S_D = 0.7$ (Fig. 7).

Table I contains the thermal conductivity values used for calculations and the results obtained for $\gamma$ values. These values indicate that the apparent thermal resistances due to the Kapitza resistance are lower for both of the samples, but not negligible. Moreover it is less prominent in the 85% LC sample than the 73% LC sample. The latter could be due to the reduction in droplet size at low LC concentrations, which in principle increases the ratio, $\gamma$, for a same Kapitza radius. Interestingly, these values are comparable to the one found previously for PS/5CB composites [7].

## B. PDLC under a frequency-varying electric field

Subsequently we have investigated the depolarization field effects in PDLC samples by varying the frequency of the applied EF, $f_e$, while fixing the amplitude at a minimum value in order to attain a maximum $\kappa_{PDLC}$ at $f_e = 1$ kHz. Figures 8(a) and 8(b), respectively, show the thermal diffusivity and thermal conductivity of the PDLC sample with 85% LC as a function of $f_e$ at constant EF of 0.127 V/µm (determined from the EF-varying experiments). The experiments (as a function of $f_e$) were conducted successively at 18, 20, 22, and 30 Hz of laser modulation frequencies ($f$) and filtered the data by plotting $\rho C$ (Fig. 9) while taking the points lying within $\pm 0.65\%$ from the mean value.

Results on thermal diffusivity and conductivity as a function of $f_e$ for PDLC with 73% LC are also displayed in Figs. 8(c) and 8(d), respectively. The results are obtained by taking the mean of three $f_e$ scans, at the same $f = 18$ Hz. Here also the data were filtered by plotting $\rho C$ (Fig. 9) and taking the points

<table><caption>TABLE I. Parameters of PDLC samples.</caption><thead><tr><th>LC vol. % (x)</th><th>κ<sub>PDLC</sub><sub>||</sub></th><th>κ<sub>LC</sub><sub>||</sub></th><th>φ<sub>LC</sub></th><th>γ</th><th>r<sub>d</sub> (μm)</th></tr></thead><tbody><tr><td>85%</td><td>0.179</td><td>0.24<sup>a</sup></td><td>0.78</td><td>0.03</td><td>0.5–2.5</td></tr><tr><td>73%</td><td>0.138</td><td>0.24<sup>a</sup></td><td>0.55</td><td>0.1</td><td>0.25–1</td></tr></tbody></table>

<footnote><sup>a</sup>Reference [27].</footnote>
<figure><img src="image_69.png" /><figcaption>FIG. 8. (Color online) Experimentally observed thermal diffusivities (a and c) and conductivities (b and d) for PDLC 85% (a and b) and for PDLC 73% (c and d) as a function of $f_e$. Applied EF for PDLC 85% and for PDLC 73% were 0.127 and 0.18 V/µm, respectively.</figcaption></figure>

lying within ±0.9% (the filtering range was chosen in a way which gives enough data points) of the ρC mean value.

Plotted ρC data, for both PDLC samples, are unaffected by the frequency of applied EF as expected. These results clearly show the frequency dependence of the dynamic thermal properties of PDLC samples. At relatively high frequencies, depending on the LC concentration, the charge accumulation at the LC-polymer interfaces is interrupted, and the effects of the depolarization field vanish. This makes the molecules better aligned in the direction of the EF, and hence the heat transport also increases and reaches its maximum. On the other hand, the dominant depolarization field distorts the heat flow at low frequencies.

### 1. Effective thermal conductivity of LC droplets

Originally the change in thermal conductivity comes from the reorientation of the LC droplet directors in the direction

<figure><img src="image_70.png" /><figcaption>FIG. 9. (Color online) Volumetric heat capacity of PDLC samples with 85% LC (blue circles) and 73% LC (red squares) as a function of $f_e$. The solid lines represent the mean values of ρC for each sample.</figcaption></figure>

<figure><img src="image_71.png" /><figcaption>FIG. 10. (Color online) Average thermal conductivity of liquid crystal droplets as a function of $f_e$. Blue circles show the result for the 85% LC and red squares for the 73% LC sample.</figcaption></figure>

of the heat flow, inside the polymer matrix. Therefore, using Eq. (7), one can estimate the mean thermal conductivity of LC droplets, given by

$$ \kappa_{LC} = \frac{1}{\kappa_{PDLC}} \times \frac{\kappa_m[2\kappa_m(1-\phi) - \kappa_{PDLC}(2+\phi)]}{[(1+2\gamma) - \phi(1-\gamma)] - \kappa_m[1+2\gamma+2\phi(1-\gamma)]} \quad (10) $$

Evolution of the mean thermal conductivity of LC droplets, $\kappa_{LC}$, for PDLC samples of 73% LC and 85% LC is shown in Fig. 10. Calculations are done using the values of $\gamma$ and $\phi_{LC}$ in Table I.

### 2. Frequency-dependent thermal conductivity and MWS effect

The relation between droplet electric field, $E_{drop}$, and the sample order parameter, $S_s$, is given by [32]

$$ S_s = \frac{1}{4} + \frac{3(e^2 + 1)}{16e^2} + \frac{3(3e^2 + 1)(e^2 - 1)}{32e^3} \ln \left| \frac{e+1}{e-1} \right| \quad (11) $$

with $e$ being the reduced field parameter which can be expressed as $e = E_{drop}/E_0$. $E_0$ is the threshold EF (the field necessary to reorient LC molecules in the preferred direction), and its value depends on parameters such as the LC droplet's radius ($r_d$), shape, elastic free energy, surface anchoring and dielectric properties of the LC material, and polymer matrix. For nematic LC droplets, confined to a small spherical volume, the surface-to-volume ratio is relatively high. Thus, the droplet director configuration strongly depends on the interplay between bulk elastic forces and surface interactions. Considering constant and strong anchoring conditions, an approximated relation for $E_0$ is given by [17]

$$ E_0 \approx \frac{1}{r_d} \sqrt{\frac{K}{\epsilon_0 \Delta \epsilon}}, \quad (12) $$

where $\epsilon_0$ is the permittivity of free space ($8.85 \times 10^{-12}$ F/m), $\Delta \epsilon$ is the dielectric anisotropy, and $K$ is the effective elastic constant of LC droplets.
<figure><img src="image_72.png" /><figcaption>FIG. 11. (Color online) PDLC order parameter vs $f_e$. (a) PDLC 85%; EF = $1.27 \times 10^5$ V/m; circles (blue) show the experimentally found $S_S$, and the solid line is the best fit. (b) PDLC 73%; EF = $1.8 \times 10^5$ V/m; squares (red) show the experimentally found $S_S$, and the solid line represents the best fit.</figcaption></figure>

In order to establish the relation for the effective thermal conductivity variations of PDLCs as a function of frequency of the applied electric field, one can insert the equation for $E_{\text{drop}}(f_e)$ [Eq. (1)] into the equation for the sample order parameter [$S_S$, Eq. (11)], and the resultant then put into the expression for LC droplet thermal conductivity [$\kappa_{\text{LC}}$, Eq. (9)] and finally to the expression for $\kappa_{\text{PDLC}}(f_e)$ [Eq. (7)], as shown by

$$E_{\text{drop}}(f_e) \xrightarrow{\text{Eq. (1)}} S_S \xrightarrow{\text{Eq. (11)}} \kappa_{\text{LC}} \xrightarrow{\text{Eq. (9)}} \kappa_{\text{PDLC}}(f_e).$$

The outcome of the above relation for $\kappa_{\text{PDLC}}(f_e)$ can be used to model the frequency-dependent effective thermal conductivity results of PDLC samples (Fig. 4) to find electrical LC parameters and polymer matrix as well as the threshold electric field $E_0$.

Figure 11 shows the experimentally found order parameter, $S_S$ (circles), along with the numerical fit (full line) for PDLC 85% LC and 73% LC samples as a function of frequency of the applied EF. The order parameter is calculated from the experimental thermal conductivity of PDLC, $\kappa_{\text{PDLC}}$, by rearranging Eq. (9) for $S_S$ and using Eq. (10). During the fitting procedure, $E_0$ as well as the electrical conductivities of both the polymer matrix and LC are kept as free fitting parameters. Values of the materials' dielectric properties were taken from literatures. The results obtained from the fitting are shown in Table II.

<table><caption>TABLE III. Elastic constants for the PDLC samples.</caption><thead><tr><th>x</th><th>Δε</th><th>r<sub>d</sub>(μm)</th><th>E<sub>0</sub> (V/m)</th><th>K × 10<sup>-12</sup> (N)</th></tr></thead><tbody><tr><td>85%</td><td>10</td><td>0.5–2.5</td><td>(5.59 ± 0.01) × 10<sup>4</sup></td><td>0.07–1.7</td></tr><tr><td>73%</td><td>10</td><td>0.25–1</td><td>(7.93 ± 0.02) × 10<sup>4</sup></td><td>0.03–0.5</td></tr></tbody></table>

The threshold fields ($E_0$) or the threshold frequencies ($f_0$) found from the experiments show decreasing behavior with respect to the increase in LC concentration. This is expected, and it can be explained in relation to the increase in droplet size [see Eq. (12)] due to the increase in LC concentration. Knowing $E_0$ and the average droplet size, we can deduce the elastic constant, $K$, using Eq. (12). The range of values found for $K$, depending on their droplet radius, for both of the PDLC samples is shown in Table III. These values are lower than the reported values for bulk nematic 5CB ($4.1 \times 10^{-12}$) [28], even though the $K$ values are much closer to the bulk values for the biggest droplets. The electrical conductivity values for PS and LC found from the 85% LC experimental data are in good agreement with the values found in the literature (see Sec. III), while for the 73% LC sample, the results show relatively higher electrical conductivity values than reported. The differences in electrical conductivity values between the two samples can be mainly due to the change in LC solubility in PS, which can significantly affect the ionic concentration inside the material. For an increased ionic concentration in LCs, the conductive shielding also increases. This will reduce the effective EF acting on LC droplets. The variations in the elastic and electric parameters can also be due to a number of other factors that are not taken into account in our studies due to the complexity. Among these, anchoring forces and droplet shape anisotropy have a significant role in defining the threshold EF.

## VI. CONCLUSION

The Maxwell-Wagner-Sillars polarization effects on thermal transport through PDLCs have been investigated with two samples (5CB/PS) containing 73% and 85% volume fractions of LCs in PS using an improved PTR technique. Experiments are done by (i) varying the amplitude of the electric field at fixed frequency as well as (ii) varying the frequency of the electric field at constant amplitude. The results on effective PDLC thermal conductivities versus amplitude of the applied EF are used to calculate $\gamma$, a parameter which describes the contribution of LC droplet volume in heat transport through it. The determined $\gamma$ values show that the volume of the LC droplet is involved in the thermal transport.

Results on the thermal conductivity of PDLCs as a function of a frequency-dependent electric field (at constant amplitude) clearly show the thermal transport through PDLC materials is

<table><caption>TABLE II. Electrical parameters of PDLC samples from MWS fit.</caption><thead><tr><th>x</th><th>σ<sub>M</sub> (S/m)</th><th>σ<sub>LC</sub> (S/m)</th><th>E<sub>0</sub> (V/m)</th><th>f<sub>0</sub> (Hz)</th><th>E<sub>a</sub> (V/m)</th></tr></thead><tbody><tr><td>85%</td><td>(6.45 ± 0.03) × 10<sup>-9</sup></td><td>(1.217 ± 0.005) × 10<sup>-7</sup></td><td>(5.59 ± 0.01) × 10<sup>4</sup></td><td>7.0 ± 0.9</td><td>1.27 × 10<sup>5</sup></td></tr><tr><td>73%</td><td>(1.195 ± 0.006) × 10<sup>-7</sup></td><td>(1.203 ± 0.006) × 10<sup>-6</sup></td><td>(7.93 ± 0.02) × 10<sup>4</sup></td><td>377 ± 12</td><td>1.80 × 10<sup>5</sup></td></tr></tbody></table>
affected by interfacial polarization field effects. The thermal conductivity data is used to plot PDLC order parameter, $S_s$. Existing theories are used to model the frequency-dependent PDLC order parameter and extracted the threshold electric field, $E_0$, as well as the electrical conductivities of both the LC and polymer matrix. The results for the electrical conductivities from 85% LC sample are closer to the already reported values, while for the 73% LC sample they show an order of magnitude higher values. We concluded that the difference in values are mainly coming from the increase in ionic concentrations as well as due to the variations in anchoring forces and droplet shape. Recent work has investigated the

effect of anchoring forces on the electro-optical behavior of PDLC with bipolar and radial droplet configurations [34]. It shows that these configurations influence the MWS effect in PDLCs. In the future, we will focus on thermal properties of PDLCs with a radial droplet configuration.

## ACKNOWLEDGMENTS

The authors thank Prof. A. Daoudi and Prof. F. Roussel for fruitful discussions. This work was financially supported by the M.E.S.R.

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5.4 Photothermoelectric effect as a means for thermal characterization of nanocomposites based on intrinsically conducting polymers and carbon nanotubes, Journal of Applied Physics 113, 044502 (2013)
# Photothermoelectric effect as a means for thermal characterization of nanocomposites based on intrinsically conducting polymers and carbon nanotubes

Maju Kuriakose,<fnref n="1" />,<fnref n="2" /> Michael Depriester,<fnref n="1" />,<fnref n="2" /> Roch Chan Yu King,<fnref n="3" /> Frédérick Roussel,<fnref n="1" />,<fnref n="4" />
and Abdelhak Hadj Sahraoui<fnref n="1" />,<fnref n="2" />

(Received 31 October 2012; accepted 26 December 2012; published online 22 January 2013)

Recent advances in nanostructured materials have spurred the development of improved thermoelectric (TE) devices. Performances of these systems are strongly linked to their thermal conductivities. Accurate measurement techniques of thermal parameters are therefore required to assess the efficiency of new thermoelectric materials. In this work, thermal parameters (thermal diffusivity, thermal conductivity, and thermal effusivity) of nanostructured thermoelectric composites based on oxidised or unoxidised carbon nanotubes at 6.6 wt. % mixed with doped nanofibrous polyaniline have been measured with a novel photothermal method called photothermoelectricity. The thermal parameters are obtained from analyses of the phase and amplitude of the thermoelectric signal generated by the sample when it is periodically heated by a laser beam. Figure of merits are calculated from these results which indicate that the best nanocomposite exhibits almost four order of magnitude improvement compared with that of the pure doped matrix at room temperature. © 2013 American Institute of Physics. [http://dx.doi.org/10.1063/1.4788674]

## I. INTRODUCTION

Thermoelectric (TE) generators are solid-state materials that allow direct generation of electricity when a temperature gradient is created within the materials.<fnref n="1" /> Renewed interests in thermoelectrics started when theoretical predictions suggested that thermoelectric efficiency could be strongly improved with nanostructural engineering.<fnref n="2" />⁻⁵ Engineering such nanostructures can be achieved in two different ways, i.e., either through the development of superlattices⁶ or nanocomposites.⁷ For example, improved values of TE-figure of merit (ZT) have been recently recorded with the use of superlattice structures and, to the best of our knowledge, the highest value (ZT = 2.4, at room temperature) is obtained with newly designed TE devices based on Bi₂Te₃/Sb₂Te₃.⁸ However, fabrication processes of these low dimensional semi-conducting-based materials by atomic layer deposition are expensive and slow.⁹

In contrast, the integration of nanocomposites containing intrinsically conducting polymers into TE devices have generated much interests in recent years, as they are light and generally require relatively simple manufacturing processes. The use of these novel nanocomposites should lead to the reduction of the thermal conductivity via an increase in the number of contact interfaces and scattering defects¹⁰⁻¹² while keeping a good power factor (S<fnref n="2" />κₑ).¹⁰,¹³ Therefore, investigations of the thermal properties of TE generators that integrate nanostructured organic composite materials are warranted and essential in order to understand the improved transport properties.

In this work, we report on the thermal studies and thermoelectric characterisation of polyaniline (PANI)/carbon

nanotube (CNT) nanohybrids prepared previously from a novel approach of CNT-templated polymerization process. The carbon nanotubes used are either oxidized or unoxidized, single- or multiple-walled, and the as-prepared hybrids exhibit cable-like morphology with the CNT as cores wrapped by nanofibrous PANI.¹⁴ In addition, we present herein a new photothermal technique (PT) to measure the thermal conductivity of TE materials in harmonic regime. This technique is based on the photothermoelectric effect (PTE). Access to the thermal parameters is feasible through analyses of the phase or amplitude of the thermoelectric signal generated by the sample when it is periodically heated by a laser beam. Results obtained from PTE measurements are then compared with those retrieved from conventional photothermal radiometry (PTR).¹⁵

In the past, transient thermoelectric effect (TTE) has been reported for the study of thermal diffusion of photoinduced carriers.¹⁶ Some authors have also suggested to use the thermoelectric response of a sample in time domain to determine the thermal diffusivity and the thermal conductivity.¹⁷,¹⁸

The first section of this report presents the theoretical model developed to determine the thermal conductivity from the photothermoelectric effect in a modulated regime. The second part describes the experimental setup and sample processing. The Sec. IV deals with the results ended by a discussion.

## II. THEORETICAL APPROACH

The voltage difference (ΔV) between two points of a thermoelectric material of thickness L₁, when the temperature fluctuations are weak, is described by the following relation:

<footnote n="1">¹Univ Lille Nord de France, 59044 Lille, France</footnote>

<footnote n="2">²ULCO, UDSMM (EAC CNRS 4476), MREI-1, 59140 Dunkerque, France</footnote>

<footnote n="3">³University of Science and Arts of Oklahoma, Chickasha, Oklahoma 73018, USA</footnote>

<footnote n="4">⁴Univ. Lille 1, UDSMM (EAC CNRS 4476), UFR de Physique P5, 59655 Villeneuve d'Ascq, France</footnote>
$$ \Delta V = S \int_{L_1}^{z=0} \left( \frac{dT}{dz} \right) dz, \qquad (1) $$

where S is the Seebeck coefficient (considered as constant for small temperature fluctuations), z is the depth; $T_{z=0}$ and $T_{z=L}$ are the temperature at z=0 and at the depth z = L₁, respectively. Figure 1 depicts the investigated configuration.

When an optically opaque sample of thickness L₁ deposited on a semi-infinite substrate is excited by a modulated laser beam, the expression of the temperature related to the depth z is given by the formula¹⁹

$$ T(z,f) = \frac{F_0(1+R_1)}{4k_1\sigma_1} \left[ \frac{1+\gamma_{12}\exp(-2\sigma_1(L_1-z))}{1-R_1\gamma_{12}\exp(-2\sigma_1L_1)} \right] \exp(-\sigma_1 z) \qquad (2) $$

with $F_o$, the incident light power density, $k_1$ is the sample thermal conductivity, and $\sigma_1$ is the complex thermal wave number given by

$$ \sigma_1 = (1 + i)/\mu_1. \qquad (3) $$

with $\mu_1$, the thermal diffusion length defined by

$$ \mu_1 = \sqrt{\frac{\alpha_1}{\pi f}}. \qquad (4) $$

Here, f is the modulation frequency and $\alpha_1$ is the sample thermal diffusivity. The thermal wave reflection coefficient $\gamma_{12}$ is given by

$$ \gamma_{12} = \frac{b_{12} - 1}{b_{12} + 1}, \qquad (5) $$

and $b_{12} = e_1/e_2$ with $e_1$ and $e_2$ are the thermal effusivities of the sample and the substrate, respectively. $R_1$ depends on the convection coefficient $h_1$ at the air-sample interface as follows:

$$ R_1 = \frac{k_1 \sigma_1 - h_1}{k_1 \sigma_1 + h_1} \qquad (6) $$

It should be pointed out that $h_1$ cannot be neglected when the sample is thicker and/or exhibits a low thermal

<figure><img src="image_73.png" /><figcaption>FIG. 1. Schematic configuration of the PTE setup; the SEM picture shows the morphology of the TE nanohybrid material.</figcaption></figure>

diffusivity. In that case, frequency scans at very low modulation frequencies are required. Combination of Eqs. (2) and (1) and after some rearrangement, the voltage difference is expressed as

$$ \Delta V = \frac{SF_o(1+R_1)}{4k_1\sigma_1} \times \left[ \frac{1+\gamma_{12}\exp(-2\sigma_1L_1) - (1+\gamma_{12})\exp(-\sigma_1L_1)}{1-R_1\gamma_{12}\exp(-2\sigma_1L_1)} \right] \qquad (7) $$

This signal is normalized by the signal obtained with air as the backing. When the substrate is replaced by air, $\gamma_{12} = \gamma_{10}$, the voltage difference becomes

$$ \Delta V_0 = \frac{SF_o(1+R_1)}{4k_1\sigma_1} \times \left[ \frac{1+\gamma_{10}\exp(-2\sigma_1L_1) - (1+\gamma_{10})\exp(-\sigma_1L_1)}{1-R_1\gamma_{10}\exp(-2\sigma_1L_1)} \right] \qquad (8) $$

The normalized signal $V_n$ is given by

$$ V_n = \Delta V / \Delta V_0 \qquad (9) $$

If we consider (i) $\gamma_{10} \approx 1$ and (ii) that the thermal transport by conduction largely outweighs the thermal transport by convection, then the real part of $k_1\sigma_1 \gg h_1$ (i.e., $R_1 \approx 1$), and the normalized signal becomes

$$ V_n = 1 + \frac{(1 - \gamma_{12}) \exp(-\sigma_1 L_1)}{1 - \gamma_{12} \exp(-2\sigma_1 L_1)}. \qquad (10) $$

Analysis of this last expression shows that the normalized phase decreases with frequency and vanishes a first time for a value $L_1/\mu_1 = \pi$. Thus, the thermal diffusivity of the sample $\alpha_1$ can be extracted from

$$ \alpha_1 = \frac{L_1^2 f_0}{\pi}, \qquad (11) $$

where $f_0$ is the zero crossing frequency. Knowing $\alpha_1$, the thermal effusivity can be obtained after some rearrangements of Eq. (10)

$$ \gamma_{12} = \frac{(V_n - 1) + \exp(-\sigma_1 L_1)}{(V_n - 1) \exp(-2\sigma_1 L_1) - \exp(-\sigma_1 L_1)}. \qquad (12) $$

Once $\gamma_{12}$ is known, the effusivity $e_1$ can be obtained from the following expression:

$$ e_1 = e_2 \frac{1 + \gamma_{12}}{1 - \gamma_{12}}. \qquad (13) $$

Thus, each thermal parameter can be either determined by a direct least square fitting of Eq. (10) to experimental data or extracted analytically from successive use of Eqs. (11)–(13).

### III. EXPERIMENT

The TE samples are pressed pellets (diameter: 13 mm) of methanesulfonic acid (CH₃SO₃H) doped nanofibrous PANI used as sheath to wrap CNT core yielding nanostructured composite materials (Fig. 1). The CNT mass fraction
<figure><img src="image_74.png" /><figcaption>FIG. 2. Evolution of the simulated normalized (a) amplitude and (b) phase as a function of the reduced thermal parameter $L_1/\mu_1$ for $\gamma_{12} = -0.2$ and various $h_1$ values.</figcaption></figure>

is ≈6.6 wt. %. Four composites are investigated: PANI-CH$_3$SO$_3$H/unoxidized single-walled carbon nanotubes (PANI/unoxSWNTs), PANI-CH$_3$SO$_3$H/unoxidized multi-walled carbon nanotubes (PANI/unoxMWNTs), PANI-CH$_3$SO$_3$H/oxidized SWNT (PANI/oxSWNT) and PANI-CH$_3$SO$_3$H/oxidized MWNT (PANI/oxMWNT). Details of pellets processing were described elsewhere.¹⁴ Each surface of the pellets were covered with a sputtered silver electrode on which electrical leads were silver pasted and plugged into an EG&G 7260 lock-in amplifier (Fig. 1). An internally modulated laser (Ventus VIS 532 nm, Laser Quantum) is employed to irradiate the pellet in order to induce a modulated heat wave across the thermoelectric nanomaterial. Experimental data retrieved from the lock-in amplifier are analyzed and compared with results obtained from a classical photothermal radiometry setup.²⁰

## IV. RESULTS

### A. Simulation results

Due to the low thermal conductivity of polymers, photothermal measurements must be carried out at low frequencies, meaning that convection effects can not be neglected. In order to take into account this phenomenon, the influence

<figure><img src="image_75.png" /><figcaption>FIG. 3. Evolution of the thermal diffusivity sensitivity [(a) amplitude and (b) phase] as a function of the reduced parameter $L_1/\mu_1$ for different values of $\gamma_{12}$. Evolution of the thermal effusivity sensitivity [(c) amplitude and (d) phase] versus $L_1/\mu_1$ for various values of $\gamma_{12}$.</figcaption></figure>
of the convection effect on the PTE signal was first investigated numerically. Figure 2 shows the evolution of the normalized phase as a function of the reduced parameter $L_1/\mu_1$, where $\mu_1$ is the thermal diffusion length. $h_1$ was set at 0, 10, and $100 \text{ W m}^{-2} \text{ K}^{-1}$ while $\gamma_{12}$ is fixed at -0.2. It can be seen that above $L_1/\mu_1 = 1$, convection has no effect on the normalized phase, thus the PTE normalized signal can be determined from Eq. (10). For $L_1/\mu_1 < 1$, convection cannot be neglected and the Eq. (9) must be used. It is interesting to note that the normalized phase damps and vanishes for $L_1/\mu_1 = \pi$ as previously reported for photopyroelectric techniques.²¹ This means that for $L_1/\mu_1 = \pi$, the thermal diffusivity $\alpha_1$ can be calculated from Eq. (11). Knowing $\alpha_1$, the thermal effusivity $e_1$ can be extracted from Eq. (12).

Figures 3(a) and 3(b) show the evolution of the thermal diffusivity sensitivity to $L_1/\mu_1$ for various values of $\gamma_{12}$. The thermal diffusivity sensitivity increases when $\gamma_{12}$ decreases, indicating that a substrate with a high effusivity should be used for the determination of the thermal diffusivity. The best frequency range found for the determination of the thermal effusivity is $0.5 \le L_1/\mu_1 \le 2$ (Figs. 3(c) and 3(d)). However, in order to avoid convection during experiments, the frequency scan will be restricted to $L_1/\mu_1$ higher than 1, limiting the available frequency range to $1 \le L_1/\mu_1 \le 2$. As expected, sensitivity on thermal effusivity increases when the thermal effusivity contrast increases ($\gamma_{12}$ far from 0).

## B. Experimental results and discussion

Experiments have been carried out using the setup described in Fig. 1. Amplitude and phase retrieved from the raw data of a photothermoelectric (PTE) signal for a 348.5 µm thick doped PANI/unoxidized SWNT are presented in Fig. 4(a) using either air or glycerol as substrates. Even though the measured voltage of the signal amplitude is weak ($\le 10 \mu V$), it can still be exploited to afford a satisfactory signal to noise ratio. Amplitude ratios and phase differences are calculated providing those of the corresponding PTE signal (Fig. 4(b)). Then, these parameters are fitted into Eq. (9) and considering $h_1 = 0$; the thermal diffusivity of $\alpha_1 = (4.68 \pm 0.04) \cdot 10^{-7} \text{ m}^2 \text{ s}^{-1}$ and a thermal effusivity of $e_1 = (1212 \pm 25) \text{ W s}^{1/2} \text{ m}^{-2} \text{ K}^{-1}$ are obtained. And, using the relationship $k_1 = e_1/\sqrt{\alpha_1}$, a thermal conductivity value of $k_1 = (0.83 \pm 0.02) \text{ W m}^{-1} \text{ K}^{-1}$ is calculated. Based on the value of the thermal diffusivity calculated from Eq. (11), the evolution of the thermal effusivity versus frequency has been calculated from Eq. (12). A zero crossing frequency $f_0 = 11.5 \text{ Hz}$ leads to a thermal diffusivity value of $(4.45 \pm 0.10) \cdot 10^{-7} \text{ m}^2 \text{ s}^{-1}$, a thermal effusivity value of $(1212 \pm 37) \text{ W s}^{1/2} \text{ m}^{-2} \text{ K}^{-1}$ and a thermal conductivity value of $(0.81 \pm 0.04) \text{ W m}^{-1} \text{ K}^{-1}$.

In order to validate these results, a classical PTR experiment¹⁵ has also been carried out. The results obtained from both techniques are summarized in Table I.

<figure><img src="image_76.png" /><figcaption>FIG. 4. Experimental amplitudes (a) and phases (b) measured for a doped PANI/oxidised SWNT composite (348.5 µm thickness) with air or glycerol as substrate. Normalized amplitude (c) and phase (d) of the PTE signal with glycerol as substrate.</figcaption></figure>
<table><caption>TABLE I. Thermal parameters retrieved from PTE and PTR techniques for a PANI/oxidized SWNT composite; *x*: thermal diffusivity; *e*: thermal effusivity, and *k*: thermal conductivity.</caption><thead><tr><th>Parameter</th><th>PTE (fit)</th><th>PTE (analytic)</th><th>PTR (fit)</th></tr></thead><tbody><tr><td>α (×10<sup>-7</sup> m<sup>2</sup>.s<sup>-1</sup>)</td><td>4.68 ± 0.04</td><td>4.45 ± 0.10</td><td>4.42 ± 0.14</td></tr><tr><td>e (W.s<sup>1/2</sup>.m<sup>-2</sup>.K<sup>-1</sup>)</td><td>1212 ± 25</td><td>1212 ± 37</td><td>1256 ± 80</td></tr><tr><td>k (W.m<sup>-1</sup>.K<sup>-1</sup>)</td><td>0.83 ± 0.02</td><td>0.81 ± 0.04</td><td>0.83 ± 0.07</td></tr></tbody></table>

To ensure the reliability of the data the procedure has been duplicated for each sample. The results on thermal diffusivities, effusivities, and conductivities are presented in Figure 5. From which it can be seen that SWNT based composites exhibit higher thermal diffusivities and thermal conductivities than their MWNT counterparts. Of interest to point out is thermal effusivities which stay roughly constant, regardless of the nature of the sample, indicating that the thermal conductivity behavior is mainly dominated by the thermal diffusivity. This behavior can be explained by the higher thermal conductivity of individual SWNTs in comparison with individual MWNTs. Although it is acknowledged that SWNTs present a higher interfacial thermal resistance than the MWNTs,²² this property, however, is not sufficient to mitigate the effects of thermal conductivity differences between the afore mentioned tubes. Thermal conductivities of oxidized nanotubes based composites are lower than thermal conductivities of unoxidized carbon nanotube composites. This behavior can be explained by the oxidation process of CNT which induces defects on the nanotubes. These defects scatter the phonons, and, thus, lower the thermal conductivity. A similar behavior has been suggested for the lowering of the electrical conductivity in the case of oxidized CNT due to induced carrier scattering centers.

<figure><img src="image_77.png" /><figcaption>FIG. 5. Experimental thermal diffusivities (a), thermal effusivities (b), and thermal conductivities (c) for PANI and PANI/CNT nanocomposites.</figcaption></figure>

<table><caption>TABLE II. Seebeck coefficient and electrical conductivity¹⁴ and thermal conductivity data retrieved from the PTE technique.</caption><thead><tr><th></th><th>S<br>(μV/K)</th><th>κ<sub>e</sub><br>(S/m)</th><th>k<br>(W m<sup>-1</sup> K<sup>-1</sup>)</th></tr></thead><tbody><tr><td>PANI</td><td>5.8</td><td>0.82</td><td>0.39</td></tr><tr><td>PANI/oxMWNT</td><td>13.5</td><td>14</td><td>0.58</td></tr><tr><td>PANI/unoxMWNT</td><td>10.6</td><td>70.1</td><td>0.74</td></tr><tr><td>PANI/oxSWNT</td><td>12</td><td>132.23</td><td>0.87</td></tr><tr><td>PANI/unoxSWNT</td><td>33</td><td>530.98</td><td>0.92</td></tr></tbody></table>

From the reported Seebeck coefficient and electrical conductivity values,¹⁴ gathered in Table II and the thermal conductivities measured in this study, the ZT has been calculated. That is the use of the formula $ZT = \frac{S^2\kappa_e}{k}$ provided interesting results at ambient temperature (300 K) which are displayed in Figure 6(a). It is observed that ZT is considerably enhanced with the incorporation of carbon nanotubes into the composite even with a small fraction of CNT (6.6 wt. %). Overall, nanocomposites containing unoxidized and single-walled CNTs exhibit better thermophysical properties than their oxidized or multiple-walled counterparts. Clearly, a careful choice of CNT and their blending with traditional intrinsically conducting polymers has led to an enhancement of the hybrid Seebeck coefficient and electrical conductivity without increasing significantly their thermal conductivity due to interfacial thermal resistances between the nanotubes and the polymeric matrix. This improvement is accompanied by a near four orders of magnitude increase in ZT value in the unoxidized, single-walled hybrid sample. However, its ZT value of $2.10^{-4}$ remains low and is far from reaching the recommended 2 – 3 ZT range required for a technological breakthrough. Nevertheless, the present work has its advantages in that the preparation of samples and

<figure><img src="image_78.png" /><figcaption>FIG. 6. (a) ZT of PANI and PANI/CNT composites at T = 300 K. (b) Ratio between the ZT value of a given PANI/CNT nanohybrid and that of PANI.</figcaption></figure>
their processing are inexpensive and straight forward and
there is room for improvement in hybrid physical character-
istics by a judicial choice/amount of each constituent of the
hybrid and synthetic procedures.

C. Conclusion

In summary, we have reported a new photothermal
experiment for the determination of TE properties based on
PhotoThermoElectric (PTE) effect. It is demonstrated that
parameters such as thermal diffusivity, effusivity, and con-
ductivity of a nanostructured thermoelectric material can be
determined analytically and numerically. The main advan-
age of this approach permits the assessment of the thermal
properties of a TE material without the need of an external
sensor. The formalism of the PTE signal is developed and a
procedure for extracting thermal parameters is also proposed.
Thus, the thermal parameters retrieved from the PTE signal
by a fitting procedure are compared with those obtained
from the conventional PhotoThermalRadiometry experiment.
A good agreement was found between the two sets of experi-
ments demonstrating the relevance and the usefulness of the
developed methodology. In addition, an analytical procedure
is also reported to determine the thermal parameters. It is
found that the values obtained for the thermal conductivity
from the analytical procedure is similar to the results
obtained from the fitting procedure with an agreement better
than 97%. Therefore, the proposed methodology provides an
accurate and simple way to determine the thermal conductiv-
ity of TE materials without any intermediate media which
makes this technique full of prospects. Various doped PANI/
(oxidized or unoxidized, single or multiple-walled) CNT
composites were investigated. Thermal parameters were
measured. Knowing the Seebeck coefficient and electrical
conductivity values, ZT values have been calculated. Among
the investigated samples, PANI/unoxidized SWNT nano-
composite exhibits the most promising TE characteristics.
This result can be explained by (i) the electrical conductivity
improvement provided by the high SWNT electrical conduc-
tivity and (ii) the weak thermal conductivity increase result-
ing from interfacial thermal resistances between nanotubes
and PANI. The CNT oxidization process creates defects on
the nanotubes leading to a decrease of the thermal conductiv-
ity. In the future, our investigations will focus on the depend-
ency of the TE efficiency on CNT amount or kind of
nanostructures incorporated in the macromolecular matrix.
The PTE method could also find potential applications for

the thermal characterization of liquids where the TE material
would be used as photothermal sensor.

ACKNOWLEDGMENTS

This work was financially supported by the M.E.S.R.
and by the “Fond Européen de Développement Economique
Régional”. The authors thank Benoit Escorne for his techni-
cal help during setup implementation. SEM experiments
were performed at the Centre Commun de Microscopie de
Lille.

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5.5 Electrical and thermal transport properties of polyani-line/silver composites and their use as thermoelectric materials, Synthetic Metals 199 (2015) 196-204
<figure><img src="image_79.png" /></figure>

# Electrical and thermal transport properties of polyaniline/silver composites and their use as thermoelectric materials

<figure><img src="image_80.png" /></figure>

CrossMark

Frederick Roussel<sup>a,b,*</sup>, Roch Chen Yu King<sup>c</sup>, Maju Kuriakose<sup>a,d</sup>, Mickael Depriester<sup>a,d</sup>, Abdelhak Hadj-Sahraoui<sup>a,d</sup>, Carole Gors<sup>a,b</sup>, Ahmed Addad<sup>b,e</sup>, Jean-Francois Brun<sup>f,g</sup>

## ARTICLE INFO

**Article history:**

Received 22 July 2014

Received in revised form 23 October 2014

Accepted 14 November 2014

Available online 1 December 2014

**Keywords:**

Functional composites

Electrical properties

Thermal properties

Thermoelectric properties

## ABSTRACT

Electrical, thermal and thermoelectric properties of chemically synthesized polyanilines (PANI)/metallic silver composites are reported and discussed in regard to their morphological features as determined by SEM. The metal-intrinsic conducting polymer power factor (PF) and figure of merit (ZT) are deduced from measurements of the electrical conductivity (σ), Seebeck coefficient (S), thermal diffusivity (α), thermal effusivity (e) and thermal conductivity (κ). It is shown that the PANI/Ag thermoelectric (TE) characteristics are much more improved as compared to those of pure PANI. In addition, the results reveal that the metal/PANI properties are inherently dependent on the volume concentration of silver and morphology of the composites. It is observed that electrical percolation takes place for samples having 16 vol% Ag content without any significant thermal percolation. The composite with 26 vol% Ag loading exhibits the best thermoelectric performance (PF ≈ 3 μW m⁻¹ K⁻²; ZT ≈ 0.5 10⁻³). The overall enhanced TE performance of the PANI/Ag composites is attributed to an improved electrical conductivity with a concomitant limitation in the rise of thermal transport via scattering.

© 2014 Elsevier B.V. All rights reserved.

## 1. Introduction

Research aimed at harnessing waste heat derived, e.g., from industrial or commercial activities and its cost-effective conversion into electricity remains challenging despite steady progress made in the past decades. It is well known that the recovery of waste heat and its conversion to electrical power can be achieved through the establishment of an effective temperature gradient across thermoelectric (TE) materials [1,2]. While TE-based on inorganic-semiconducting materials are among the most efficient today, they could, however, be scarce, costly, tedious in their fabrication and/or toxic. For these reasons, intrinsically conducting polymers (ICPs) as TE materials are currently intensively investigated. These polymers, such as polyanilines, polypyrroles,

polythiophenes or their derivatives, possess relatively low thermal conductivity and good electrical conductivity. Moreover, they are easy to prepare, air stable, light weight and much less costly in their production [3]. In addition, recent progress made in the past decade showed that nano-objects may be used to enhance TE characteristics by improving the density of states of electrons (or holes) near the band edge of a semiconductor which in turn enhances Seebeck constants for a given carrier concentration [4]. These findings, coupled with recent methods of syntheses of nanomaterials, have led to a flurry of research on TE properties that make use of nanohybrids composed of inorganics and organic conducting polymers [5,6]. Therefore, investigations of TE generators based on nanostructured organic composite materials are warranted and essential in order to understand the improved transport properties. Other relatively new TE composites make use of the good electrical conductivity of carbon nanotubes (CNTs) and the low thermal conductivity of ICPs. This strategy may introduce junctions, between the two different types of components, which could lead to phonon scattering and hoping of charge carriers [7].

<footnote><sup>a</sup> University of Lille, 1- Science and Technologies, UFR de Physique, P5, 59655 Villeneuve d'Ascq, France</footnote>

<footnote><sup>b</sup> ComUE - Lille Nord de France, 59658 Villeneuve d'Ascq, France</footnote>

<footnote><sup>c</sup> University of Science and Arts of Oklahoma, Chickasha, OK 73018, USA</footnote>

<footnote><sup>d</sup> ULCO, UDSMM, MREID, 59140 Dunkerque, France</footnote>

<footnote><sup>e</sup> University of Lille, 1-Sciences & Technologies, UMET, CNRS UMR 8207, C6, 59655 Villeneuve d'Ascq, France</footnote>

<footnote><sup>f</sup> CNRS UPR3079CEHMTI, 45071 Orléans, France</footnote>

<footnote><sup>g</sup> UFR Collégium Sciences et Techniques, Université d'Orléans, 45067 Orléans, France</footnote>

<footnote>* Corresponding author at: University of Lille, 1-Sciences and Technologies, UFR de Physique, Bat P5, 59655 Villeneuve d'Ascq, France. Tel.: +33 320436837. E-mail address: Frederick.Roussel@univ-lille1.fr (F. Roussel).</footnote>
While most research in this area make use of composites comprising variable amounts of CNT embedded into a fixed amount of PANI used as matrix, we have recently shown that the nature of CNTs (i.e., single- vs. multiple walled, oxidized vs. un-oxidized CNTs) strongly influences the TE characteristics of CNT/PANI nanohybrids [8].

Herein, we report on the electrical, thermal and TE properties of composites based on metallic silver and PANI. The choice of these two components is guided by the excellent bulk electrical conductivity of metallic silver and the relatively poor thermal conductivity of PANI. Their potential synergistic effect in enhancing TE characteristics, to our knowledge, has not been investigated. While methods for the in-situ deposition of noble metals (Ag, Pt and Au) onto nanofibers of polypyrroles are known which are, for example, synthesized with pre-fabricated $V_2O_5$ sol-gel nanofibers as the oxidizing agent and seeds for fiber growth [9,10], chemical methods on the direct deposition of Ag solid onto polyanilines with $Ag^+$ ion as the oxidizing agent was lacking for some time. Only until recently that such methods have been described in the literature [11–17]. In this work, we have slightly modified published synthetic procedures such that the reactions are conducted essentially under stirred conditions or sonication in an effort to maximize the product yields and incorporate a volume fraction of metallic silver (ranging from ca. 2% to 34%) into the composites. The morphology of the products is investigated by optical and SE microscopy and their spectral features are obtained via UV-vis IR, EDX and Raman spectroscopy. Further analyses of samples by photothermal IR radiometry provided the thermal characteristics (diffusivity, effusivity and conductivity). These data combined with the electrical conductivity and Seebeck coefficients allow the determination of the power factor (PF) and figure of merit (ZT) of a new class of thermoelectric hybrid materials.

## 2. Experimental

### 2.1. Materials and syntheses

Silver nitrate and aniline are provided by Aldrich Chemical Co.; aniline was distilled before use. The conventional synthesis of PANI requires a strong oxidizing agent, typically, ammonium peroxydisulfate (APS) whose standard reduction potential (+2.01 V) is much higher than that of $Ag^+$ (0.80 V). Thus, oxidation of Aniline with $Ag^+$ is much slower. In order to obtain a series of PANI/Ag composites having a volume fraction of silver ranging from ca. 2% to 34%, we have slightly modified published experimental procedures (Table 1). The deposition of silver solids on PANI from aq. silver nitrate was achieved by changing the reaction time and/or via control of kinetics of oxidation of aniline. The latter process could either be (a) accelerated with the use of stronger oxidants such as APS, oxygen from air (in isopropyl alcohol) [12] or in the presence of additives (p-phenylene diamine (PDA) at low pH [11] or (b) slowed down under an initial neutral (water as solvent [14])

or weakly acidic condition[11]. A control sample ("pure PANI", 0% Ag) was also prepared under the conventional procedure with APS as oxidizing agent [18] in 1.0 M $HNO_3$. An example of synthetic procedure [16] is as follows (for preparation of a PANI composite with 26.1 vol% Ag): Into a 600 mL beaker was placed a solution of 17.0 g (100.0 mmol) $AgNO_3$ in 40.0 mL 1 M $HNO_3$ (aq.). To this stirred solution was added a solution of 3.70 g (39.73 mmol) aniline in 100.0 mL 1 M $HNO_3$ (aq.). The resulting mixture was stirred for 16 days after which time the mixture was suction filtered and the deep green product was washed with 1.0 L de-ionized water, followed by ca. 50 mL acetone. The product was then high vacuumed dried.

The average value of triplicate measurements of Ag mass fraction in all PANI/Ag composites was determined ex-post as metallic residue obtained after complete calcination of PANI (organic component) in a DSC oven (T = 600 °C in air). Knowing the Ag mass fraction, the Ag volume fraction was deduced using the densities of PANI ($d$ = 1.33 g/cm³ [19]) and that of silver ($d$ = 10.5 g/cm³). We note that two composites contain unexpectedly high vol% Ag (26.1% and 34.3%). We attribute this finding to the removal of oligomers or by-products via washings of composites with copious amounts of water followed by acetone [15]. Thus, as the mass fraction of the organic component in the composites drops, the mass/volume fraction of Ag increases.

### 2.2. Characterization techniques

IR spectroscopy of products was recorded on KBr pressed pellet samples using an FTIR 380 Thermo electron spectrometer. UV spectroscopy was performed with an Ocean Optics USB2000+ spectrometer. Raman microspectrometry was carried out on a Horiba Jobin Yvon Labram HR operating with a HeNe laser ($\lambda$ = 632.8 nm, $P$ = 0.1 mW). Electrical conductivity was investigated using a Suss Microtec EP6 probe station equipped with a Keithley 2635 sourcemeter. X-ray diffractograms were recorded on a curved multidetector INEL-CPS 120 set up. Samples were placed into Lindeman glass capillary tubes (diameter 0.70 mm) and irradiated with a collimated monochromic beam ($\lambda$ = 0.154056 nm) during which samples were rotated around the vertical axis of the goniometer head. A scanning electron microscope (Hitachi S4700 operating at 3 kV, 10 mA) equipped with a Bruker energy dispersive X-ray analysis (EDS) system was used for analyses of product morphologies and to confirm the presence of metallic silver. Measurements of thermoelectric power factor (PF) were conducted on press pellets (average thickness = 300 µm) with an in-house made apparatus [8].

Thermal parameters of pelletized PANI/Ag samples were determined using a photothermal radiometry setup described elsewhere [20,21]. Each surface of the pellets were covered with a sputtered silver electrode on which electrical leads were silver pasted and plugged into an EG&G 7260 lock-in amplifier. An internally modulated laser (Ventus VIS 532 nm, Laser Quantum)

<table><caption>**Table 1**
Experimental conditions of synthesis.</caption><thead><tr><th>Vol% Ag in composite</th><th>AgNO<sub>3</sub> (mmol)</th><th>Aniline (mmol)</th><th>APS (mmol)</th><th>Additive (mmol)</th><th>Condition (solvent; time)</th><th>Refs.</th></tr></thead><tbody><tr><td>0.00</td><td>0.00</td><td>12.9</td><td>4.40</td><td>N/A</td><td>1.0 MHNO<sub>3</sub>, 12 h conventional stirring Synthesis</td><td>[18]</td></tr><tr><td>2.10</td><td>1.60</td><td>5.50</td><td>9.65</td><td>N/A</td><td>Water; 12 h stirring + 2 days unstirred</td><td>[14]</td></tr><tr><td>3.97</td><td>5.50</td><td>11.0</td><td>23.6</td><td>N/A</td><td>Water; 24 h stirring + 2 days unstirred</td><td>[14]</td></tr><tr><td>10.81</td><td>11.8</td><td>10.8</td><td>N/A</td><td>Isopropyl alcohol/air</td><td>1.0 MHNO<sub>3</sub>; 12 h sonication + 3-week unstirred</td><td>[12]</td></tr><tr><td>16.6</td><td>50.0</td><td>40.0</td><td>51.0</td><td>N/A</td><td>1.0 MHNO<sub>3</sub>; 1-week stirring</td><td>[15]</td></tr><tr><td>17.1</td><td>38.0</td><td>40.0</td><td>38.0</td><td>N/A</td><td>1.0 MHNO<sub>3</sub>; 2-week stirring</td><td>[15]</td></tr><tr><td>20.6</td><td>40.0</td><td>22.0</td><td>N/A</td><td>0.76 (PDA)</td><td>1.0 M HNO<sub>3</sub>; 3-day stirring</td><td>[11]</td></tr><tr><td>26.1</td><td>100.0</td><td>40.0</td><td>N/A</td><td>N/A</td><td>1.0 M HNO<sub>3</sub>; 16-day stirring</td><td>[16]</td></tr><tr><td>34.3</td><td>55.0</td><td>22.0</td><td>N/A</td><td>0.61 (PDA)</td><td>0.20 M AcOH; 3-day stirring</td><td>[11]</td></tr></tbody></table>
was employed to irradiate the pellet in order to induce a modulated heat wave across the TE material. Experimental data retrieved from the lock-in amplifier are analysed using a one dimensional model of a three-layered system (front medium, sample, and substrate). Using a fitting procedure of the normalized phase, both thermal diffusivity and thermal effusivity of the samples are simultaneously determined allowing the calculation of the thermal conductivity.

In this work, multiple repeats were performed for each sample in all measurements and mean average values are reported for electrical conductivity, Seebeck constants, PF, thermal effusivity, diffusivity and conductivity.

# 3. Results and discussion

## 3.1. Spectroscopy

UV-vis spectroscopy was first carried out on PANI/Ag composites. Samples were dedoped with $NH_3$ (aq.). Fig. 1(a) displays the UV-vis spectra of the resulting blue solutions (in NMP) of representative dedoped composites which show the typical absorptions bands of PANI in its non-conducting-emeraldine base (EB) state: $\lambda_1$ ($\pi-\pi^*$ benzenoid transition at shorter wavelength) and $\lambda_2$ ($\pi-\pi^*$ benzenoid to quinoid transition at longer wavelength). The wavelengths of the afore-mentioned transitions and the corresponding ratio of peak intensity $\lambda_1/\lambda_2$ are gathered in Table 2. It is noteworthy that for most composites, the wavelengths of the transitions fall within the normal range of a typical EB sample, i.e., 315 nm < $\lambda_1$ < 345 nm and 610 nm < $\lambda_2$ < 645 nm [22]. In addition, the above ratio of peak intensity provides a qualitative measure of the extent of oligomers in the composites. In the present work, this ratio is close to 1.2–1.3 for most samples and pure EB (which should be equal to unity under ideal condition [17]). However, a few composites exhibit a higher ratio of absorbance (2.1 vol% and 3.90 vol%) or the $\lambda_2$ transition falls below 610 nm (2.1 vol% and 34.3 vol%). These features indicate the presence of oligomers and/or by-products as they were prepared under an initially neutral (water as solvent [14]) or weakly acidic medium [11] (Table 1). The mechanisms for the formation of

<figure><img src="image_81.png" /><figcaption>Fig. 1. UV-vis spectra of (a) dedoped and (b) doped PANI/Ag hybrid.</figcaption></figure>

<table><caption>**Table 2**

UV-vis characterisitics of composites.</caption><thead><tr><th>Vol% Ag in composite</th><th>λ<sub>1</sub></th><th>λ<sub>2</sub></th><th>Abs (λ<sub>1</sub>)/Abs(λ<sub>2</sub>)</th></tr></thead><tbody><tr><td>0</td><td>323.04</td><td>617.98</td><td>1.21</td></tr><tr><td>2.1</td><td>335.8</td><td>603.16</td><td>1.54</td></tr><tr><td>3.97</td><td>328.1</td><td>612.2</td><td>1.49</td></tr><tr><td>10.8</td><td>329.3</td><td>615.6</td><td>1.27</td></tr><tr><td>16.6</td><td>340.67</td><td>621.22</td><td>1.27</td></tr><tr><td>17.1</td><td>332.01</td><td>620.03</td><td>1.31</td></tr><tr><td>20.6</td><td>332.8</td><td>617.9</td><td>1.27</td></tr><tr><td>26.1</td><td>327.9</td><td>626.84</td><td>1.16</td></tr><tr><td>34.3</td><td>336.12</td><td>570.9</td><td>1.22</td></tr></tbody></table>

by-products and oligomers from oxidation of aniline with $AgNO_3$ have been reported in the literature [11,17]. Upon addition of 1 M $HNO_3$ (aq.), the above blue NMP solutions turned green indicating the formation of doped PANI whose spectra show the three characteristic bands: 362 nm ($\pi-\pi^*$) 446 nm (polaron $\rightarrow \pi^*$) and 795 nm $\pi \rightarrow$ polaron transition (free carrier tail [23]); a representative spectrum is shown in Fig. 1b).

FTIR spectra of a typical doped PANI/Ag composite and that of $HNO_3$ doped pure PANI are displayed in Fig. 2a while Fig. 2b shows the essentially dedoped representatives of the as-synthesized composite series. Preparation of PANI often affords oligomers to

<figure><img src="image_82.png" /><figcaption>Fig. 2. IR spectra of (a) doped and (b) dedoped PANI/Ag hybrid.</figcaption></figure>
<figure><img src="image_83.png" /><figcaption>Fig. 3. Raman spectra of doped PANI/Ag composites and PANI, respectively.</figcaption></figure>

varying degrees. IR of PANI/Ag composites have been extensively analyzed by Sedenkova et al. who suggested that doped samples are best suited for the detection of oligomers and by-products as the composites are devoid of proton dopants and their counterions [22]. The spectra of the doped composites show the presence of typical PANI major absorptions found in the following ranges: 3400-3500 cm⁻¹ (N-H stretch); 1570-1592 cm⁻¹ (C=C stretch of the Quinoid ring); 1490-1509 cm⁻¹ (C=C stretch of benzenoid ring); 1290-1308 cm⁻¹ (C-N stretch of aromatic amine [15] 1143-1170 cm⁻¹ split broad band or with shoulder which includes the Benzenoid-NH-Benzenoid and/or C-H aromatic in plane deformation [17]); 822-832 cm⁻¹ (C-H deformation stretching of 1,4-disubstituted benzene ring) and 503-506 cm⁻¹. Thus, the spectra confirm PANI as the dominant organic component in the composites. However, these are accompanied with small amounts of residual dopants (small stretch of NO₃⁻ at ~1384-1385 cm⁻¹), by-products (small stretch or shoulder at 1630-1634 cm⁻¹ (which could be N-H scissoring of aromatic amines or C=O stretch of p-benzoquinone [17]) and oligomers (1444-1445 cm⁻¹; C=C stretch of substituted aromatic rings or to aminophenyl fragment, C=N stretch or N=N in azobenzenes [17]), 1288-1290 cm⁻¹ [22] (which could be associated with C-N vibration) small bands at ~695-697 cm⁻¹ (C-H out of plane ring deformations of mono-substituted benzenoid ring [17]). For doped composites, the prominent vibrations are: 3435 cm⁻¹ (N-H stretch); 1565 cm⁻¹ (C=C stretch, quinoid ring), 1483 cm⁻¹ (C=C stretch, benzenoid ring), 1384 cm⁻¹ (NO₃⁻ stretch, counterion of dopant), 1292 cm⁻¹ (C=N stretch of quinoid ring), 1241 cm⁻¹ (C=N stretch of benzenoid ring), 1138 cm⁻¹ (N=Quinoid=N stretch) [8,11,24,25] and 795 cm⁻¹ (C-H out of plane bend. of 1,4-disubstituted aromatic ring). Of interest to mention is the strong absorbance at 1138 cm⁻¹ which indicates the presence of positively charged species formed upon doping, and thus, supports the conducting state of PANI. In addition, it should be pointed out that, compared to data recorded for pure PANI, several afore-mentioned stretching peaks associated with the PANI/Ag hybrid are shifted to higher wavenumbers (at 1565, 1483, 1292 and 1241 cm⁻¹) supporting interactions between PANI and Ag. In addition, these shifts are accompanied with a significant increase of the C=N stretch (quinoid band) intensity at 1292 cm⁻¹. This can be attributed to the presence of positively charged species formed upon doping of the conducting polymer which leads to an increase of the molecular dipole moment associated with an enhancement of charge delocalization. These results suggest that silver particles, much like carbon nanotubes when blended with PANI [8,26,27], may act as a "co-dopant" of the conducting polymer.

In order to get further insights in the bending and stretching vibrations, pure PANI and PANI/Ag hybrids were characterized by Raman spectroscopy (Fig. 3). The spectrum of pure PANI shows the typical bands of the doped polymer including C-H bending

vibration at 1163 cm⁻¹; symmetric C-N stretching at 1220 cm⁻¹; C=N stretching of the quinoid ring at 1473 cm⁻¹, and C=C stretching of the quinoid rings at 1590 cm⁻¹. The broad bands centered around 1340 and 1380 cm⁻¹ have been assigned to various vibration modes of pure PANI, i.e., C-N+* vibration of delocalized polaronic structures (1340 cm⁻¹) and to more localized polaron structure or to the C-N+ring-stretching vibrations of substituted N-phenylphenazines (1380 cm⁻¹) [17]; these bands have also been assigned to N-O stretching which might be associated with the residual counter ion (NO₃⁻) of the dopant [28]. The assignment of the peaks is in good agreement with previous reports confirming that the synthesized product contains polyani-line [8,29]. It is noteworthy that the C-N stretching (1220 cm⁻¹) and the quinoid C=C stretching (1590 cm⁻¹) are slightly shifted to lower values in the case of PANI/Ag composites (1216 cm⁻¹ and 1585 cm⁻¹) reconfirming that interactions between PANI and silver particles readily occur which is consistent with the above IR observations.

## 3.2. Morphological and structural characterizations

The morphological investigations of PANI/Ag composites performed by optical and scanning electron microscopy are presented in Fig. 4(a-d) and (e-h), respectively. For samples containing less than 10% volume fraction in Ag, the composites exhibit isolated aggregates of silver particles in the PANI matrix. With increasing Ag content up to about 30 vol% Ag the number and size of Ag aggregates increase with an average size reaching several tens of microns. It is worth noting that the as-prepared PANI with low Ag contents exhibit globular or fibrillar (nano-structured) features. In addition, metallic particles are covered with a thin layer of PANI (Fig. 4(e-h)). This could imply that during polymerization the first formed Ag particles may have acted to some extent as reducing agents for the residual Ag(+) ions resulting in larger metallic crystals and in turn serve as seeds for the growing PANI chains. In order to assess the presence of metallic silver X-ray and EDS analyses were carried out. Fig. 5a shows typical EDS patterns of PANI rich (zone A) and silver rich (zone B) domains as confirmed through electron binding energy measurements which indicate the presence of carbon (0.28 keV), nitrogen (0.40 keV), oxygen (0.52 eV), and silver (2.97 keV; 3.15 keV). Fig. 5b shows a PANI/Ag X-ray diffractogram which exhibits both amorphous and ordered structures. A broad band is observed from 2θ ca. 15° to 35° indicating mainly an amorphous PANI morphology along with small sharp peaks corresponding to the periodicity of the crystallographic planes that are parallel and perpendicular to the doped PANI chains, respectively [8,30-32]. In addition, sharper peaks at angle 2θ~39°, 45°, 64°,78°, 82° and 99° reveal the presence of ordered crystalline metallic features. These diffraction peaks can be assigned to the face-centered cubic structure of silver cell in which the Miller indices of the six first reflection planes are (111), (200), (220), (311), (222) and (400) [33-35]. The average lattice constant calculated from this pattern is 0.403 nm which is consistent with the standard value of 0.408 nm. The crystallite size (L) has been evaluated using the Scherrer's formula [36]:

$$L = \frac{0.94\lambda}{\beta\psi\sigma\theta} \qquad (1)$$

where λ is the wavelength (λ(Kα₁)Cu = 0.154056 nm) of X-rays used, β is the broadening of diffraction line measured at half of its maximum intensity (in radian), and θ is Bragg's diffraction angle (in degree). The computed average crystallite size of silver nanoparticles is within the range of 70-120 nm without taking into account the correction from strain effect.
<figure><img src="image_84.png" /><figcaption>Fig. 4. Optical (a–d) and SEM (e–h) images of PANI/Ag composites with increasing silver content; black arrows show Ag particles embedded in the PANI matrix. Optical measurements were performed on pressed pellets and SEM observations were carried out on powders.</figcaption></figure>

Thus, the above results which are based on the morphology, EDS and X-ray data, confirm that PANI/Ag hybrids have been successfully prepared via in situ polymerization.

### 3.3. Electrical properties

The electrical conductivity ($\sigma$) as a function of silver volume fraction of the composites is displayed in Fig. 6a. It can be seen that, $\sigma$ increases with increasing amounts of Ag in the samples. The dependence of the electric conductivity on filler volume content is commonly described by the critical power law defined as:

$$ \sigma = \sigma_0(\phi - \phi_c)^t, \qquad (2) $$

where $t$ is the critical exponent, $\sigma_0$ the electrical conductivity prefactor, $\phi$ the volume fraction of silver in the sample, and $\phi_c$ the percolation threshold. However, this equation does not take into account the intrinsic characteristics of the system, i.e., Ag particle shape, polymer–Ag interactions, volume distribution of metallic particles, etc. Thus Eq. (2) has been rewritten as follows [37]:

$$ \sigma = \sigma_c + (\sigma_m - \sigma_c) \left[ \frac{(\phi - \phi_c)}{(F - \phi_c)} \right]^t, \qquad (3) $$

where $\sigma_c$ is the conductivity at the percolation threshold, $\sigma_m$ is the maximal conductivity, and $F$ is the filling/packing factor which

characterizes the filler phase topology that takes into account the particle shape, fractional size and spatial distribution of particles. Fitting the various parameters, a percolation threshold of $\phi_c = 16.2$ vol% Ag is obtained which is in good agreement with the theoretical 3D model of Scher and Zallen [38]. These authors predicted that the onset of macroscopic conductivity of conducting particles, when randomly dispersed in an insulating medium, occurs at a critical density of about 15.4%. It is also noteworthy that the computed $t=2.3$ strongly supports a three-dimensional transport process. The $F$ factor found to be 0.54, is close to that of the theoretical value ($F=0.64$) for spherical particles [37]. This value is linked to the percolation threshold by the relation $X_c = \phi_c/F$. Using the experimentally determined values of $\phi_c$ and $F$, a value of $X_c = 0.3$ is extracted which indicates the occurrence of interactions between Ag particles and PANI [37]. These results correlate well with the spectroscopic investigations and observed morphology of PANI/Ag composites and could be attributed to the formation of a percolating network which allows charge transport between the electric probes.

### 3.4. Seebeck coefficient

Using the well-known equation defining the Seebeck ($S$) coefficient as:

<figure><img src="image_85.png" /><figcaption>Fig. 5. EDS (a) and XRD (b) patterns of PANI/Ag hybrids.</figcaption></figure>
<figure><img src="image_86.png" /><figcaption>**Fig. 6.** Electrical conductivity (a) and Seebeck S coefficient (b) and power factor (PF) (c) as a function of volume fraction of Ag in PANI/Ag composites; symbols are experimental data points whereas the solid line is the best fit curve to Eq. (3) (correlation coefficient R² = 0.96) and dashed lines are guides to the eyes.</figcaption></figure>

$$S(T) = \frac{\Delta V}{\Delta T} \qquad (4)$$

S values were obtained by measuring the TE voltages (ΔV) as a function of temperature gradient (ΔT) at T = 30 °C. The graphs of ΔV vs. ΔT afforded linear plots whose slopes are proportional to the S constants. The positive S values obtained indicate that the composites are of p-type materials, i.e., hole carriers are the dominant contributors to S coefficients. From these values and that of the electrical conductivity σ, the TE power factor (PF) can be deduced using the formula:

$$PF = S^2 \sigma \qquad (5)$$

It can be seen from Fig. 6b that S as a function of the volume fraction of Ag decreases with increasing concentration of Ag. This behavior usually indicates a higher doping level which is accompanied by a higher concentration of charge carriers. It is interesting to point out that in this study, the doping agent is nitric acid, however, metallic silver particles act like “co-dopants” which enhance significantly the overall electrical conductivity and in turn lower the Seebeck constant value. This inverse interdependency of the Ag content with S parameter has also been observed in semi-conductor-based TEs [39–41]. In addition, the power factor PF increases by a factor of 5 decades from 10⁻⁵ μW m⁻¹ K⁻² for pure PANI to ca. 3 μW m⁻¹ K⁻² for composites incorporating 26 vol% Ag. It is also interesting to note that the evolution of PF as a function of Ag content follows a similar trend to that of σ (Fig. 6a) evidencing that the contribution of σ to PF is dominant over that of S² for the composites.

### 3.5. Thermal properties

Using the photothermal radiometry setup described earlier, the influence of silver content on the thermal parameters is investigated. Fig. 7a displays the thermal diffusivity α and effusivity e of the composites as a function of vol% Ag. It can be seen that α and e increase with increasing Ag content. Knowing α and e, the thermal conductivity κ is calculated according to the equation:

$$\kappa = e \times \alpha^{1/2} \qquad (6)$$

The evolution of κ vs. volume fraction of silver is shown in Fig. 7b. From a physical point of view, κAg (thermal conductivity of silver) and κPANI (thermal conductivity of PANI) are the upper and lower limits of the effective thermal conductivity of the composite (κc). Then as a first approach to assess the evolution of the thermal conductivity of the composite, its thermal properties have been modeled, using a weighted geometric mean, as follows [42]:

$$\kappa_c = \kappa_{PANI} \gamma \kappa_{Ag^{(1-\gamma)}}; 0 \le \gamma \le 1, \qquad (7)$$

where γ is the volume fraction of Ag. As shown in Fig 7b a good agreement is obtained between the experimental data points and the fit (solid line) providing a value of κPANI = 0.26 W m⁻¹ K⁻¹ and κAg = 429.3 W m⁻¹ K⁻¹ which is consistent with data reported in the literature for polyaniline [43–45] and silver [46]. This result demonstrates that the behavior of the thermal conductivity (κc) of PANI/Ag hybrids is in contrast with the electrical transport measurements for which a percolation phenomenon is observed with increasing silver content. Thus, electrical percolation has no

<figure><img src="image_87.png" /><figcaption>**Fig. 7.** Thermal (a) effusivity e and diffusivity α, and (b) conductivity κ as a function of volume fraction of Ag in PANI/Ag composites; the solid lines in (b) are the best fit curve to Eq. (7) (correlation coefficient R² = 0.995) and Eq. (8) (correlation coefficient R² = 0.987), respectively.</figcaption></figure>
significant effect on the thermal conductivity enhancement, i.e., the topology of the electrical conducting network is not the dominant parameter that controls the thermal transport. Such results imply that the effective channel for electric transport is along the percolating Ag particles network whereas the dominant channels of the heat flow in the PANI/Ag composites always involve the polymer media surrounding the Ag domains. In order to get further insights in the polymer/Ag particle interface, a Maxwell–Garnett type effective medium approximation, also known as MG-EMA, has been used for modeling the thermal conductivity of PANI/Ag hybrids. Such an approach is suitable for composites with low filler volume fractions when thermal percolation effects are negligible. This allows an estimation of the interfacial resistance $R_k$, also known as the Kapitza resistance, at the filler/host interface. Considering spherical metallic particulates dispersed in a polymeric host, the effective thermal conductivity $\kappa^*$ of the composite can be expressed as [47]:

$$ \kappa^* = \frac{\kappa_{\text{PANI}}(\kappa_{\text{Ag}}(1 + 2\alpha) + 2\kappa_{\text{PANI}} + 2\gamma[\kappa_{\text{Ag}}(1 - \alpha) - \kappa_{\text{PANI}}])}{(\kappa_{\text{Ag}}(1 + 2\alpha) + 2\kappa_{\text{PANI}} - \gamma[\kappa_{\text{Ag}}(1 - \alpha) - \kappa_{\text{PANI}}])} \quad (8) $$

where $\alpha = R_k \kappa_{\text{PANI}}/d$, with $d$ being the mean diameter of the particulates estimated from SEM measurements ($d \sim 25 \ \mu\text{m}$). For PANI/Ag composites containing 0–30 vol% Ag (Fig. 7b inset), a good agreement was obtained between the experimental data and the MG-EMA model providing $\kappa_{\text{PANI}} = 0.33 \text{ W m}^{-1} \text{K}^{-1}$, $\kappa_{\text{Ag}} = 425.4 \text{ W m}^{-1} \text{K}^{-1}$ and $R_k = 1.2 \text{ 10}^{-8} \text{ m}^2 \text{K W}^{-1}$. Again the thermal conductivity data are consistent with values reported in the literature [43–46]. To the best of our knowledge, the boundary thermal resistance of silver–polynaniline composites is reported here for the first time. Its value is within the range of $10^{-8} < R_k < 10^{-9} \text{ m}^2 \text{K W}^{-1}$ which is in accord with data reported for composites materials with high $\kappa$ micron meter-size particles randomly dispersed in low $\kappa$ host media [48–50]. These results seem to indicate that at low filler fractions, the interfacial resistance limits the usefulness of Ag in enhancing thermal properties of the host material which confirms that the polymer media is the dominant channel for heat flow. It is noteworthy that similar theoretical and experimental data on lack of thermal enhancement in carbon nanotube/matrix composites have been reported [45,49,50].

### 3.6. Figures of merit (ZT)

From the electrical and thermal parameters, the ZT of composites at 300 K are obtained using the equation:

$$ ZT = \frac{(S^2 \sigma) T}{\kappa} \quad (9) $$

As can be seen in Fig. 8a, ZT increases with increasing Ag content in the composites and it reaches a maximum value of ca. $0.5 \times 10^{-3}$ with 26.1% Ag. Thus, doping PANI with Ag significantly enhances TE properties of the resulting composite as compared to those of pure PANI, the ratio of the corresponding ZT's is ca. $10^{+6}$. One of the major contributors to this much improved ZT is the rapid rise of $\sigma/k$ ratio as depicted in Fig. 8b whose trend resembles that of Fig. 6a relating $\sigma$ to Ag% loading. That is the influence of $\sigma$ in ZT enhancement is more pronounced than that of $\kappa$ which is brought about by (a) tuning the electrical conductivity via percolation and (b) maintaining the thermal conductivity as low as possible, i.e., despite a steady increase in Ag content up to 26.1%, $\kappa$ remains below $2 \text{ W m}^{-1} \text{K}^{-1}$ which is much lower than of bulk Ag ($429 \text{ W m}^{-1} \text{K}^{-1}$). It is known that thermal conductivity and diffusivity in pure metallic substances are often governed by electron conduction (via scattering process of conducting electrons) rather than phonon conduction. In the present study, the presence of defects/dislocations in Ag particles or films and PANI backbone or at the metal/PANI boundary would have accounted for the process of thermal resistance via scattering, thereby a lowering in overall $\kappa$ is observed.

In comparison with other PANI/metallic or PANI/semi conductor TE performance, our best sample exhibits a good ZT value ($0.5 \times 10^{-3}$) which is above or close to that of PANI/RGO (ZT~$10^{-6}$) [51], PANI/SWNT (ZT = $2.0 \times 10^{-4}$) [45], PANI/Bi (ZT~$0.4 \times 10^{-3}$ to $1.3 \times 10^{-3}$) [52] or SWNT/PANI nanocomposites (ZT = $4 \times 10^{-3}$) [53] but is lower than that of PANI/semiconductors $Bi_2Te_3$ films (ZT~0.18) [54], or MWNT-PANI core/shell nanocomposites (ZT~0.1) [55]. It is of interest to note that there is, so far, no available analytical methods to quantify the amount of oligomers in the PANI/Ag composites. Their influence on the TE characteristics warrants further investigations. Nevertheless, insofar as electrical conductivity is concerned, their presence in the composites might not adversely affect TE performance as it has been shown that aniline tetramers and phenyl-capped octaanilines can exhibit conductivity similar to that of regular PANI i.e. $10^{-2} \text{ S/cm}$ [56] and $1 \text{ S/cm}$ [57], respectively.

Thus, the as-prepared composite TE performance shows promising characteristics with a ZT ratio enhancement up to $10^5$ as compared to that of pure PANI, but they remain to be improved in order to rival ZT values of TE-based on inorganic-semiconducting materials. Nevertheless, the present work has its advantages in that the preparation of samples and their processing are inexpensive and straight forward and there is room for improvement of TE characteristics by a judicial choice of the amount of each constituent of composite and the fine tuning of their morphology via improved synthetic procedures.

<figure><img src="image_88.png" /><figcaption>Fig. 8. (a) Figure of Merit (ZT) and (b) $\sigma/k$ ratio vs. volume fraction of Ag in PANI/Ag composites; symbols are experimental data points whereas dashed lines are guides to the eyes.</figcaption></figure>
## 4. Conclusion

The facile fabrication of PANI/Ag from aniline and silver nitrate as oxidant under stirred condition resulted in composites of variable metal contents and morphologies. The properties of these conducting polymer/metal hybrid materials are characterized via SEM, UV-vis. FTIR, Raman, EDX and X-ray. It is shown that they possess much enhanced physical properties, over those of pure PANI, such as electrical conductivity, power factor and Seebeck constants. In addition, we report for the first time the thermoelectric properties of these composites which include the thermal effusivity, thermal diffusivity, thermal conductivity and figure of merit ZT. Analyses indicate that all properties are very much dependent on the Ag loading. It is observed that electrical percolation occurs in the absence of any significant thermal percolation. The overall enhanced thermoelectric properties of composites are mainly governed by improved electrical conductivity while thermal conductivity rise is minimized via scattering of phonons. The best as-prepared sample (26.1% Ag volume fraction) exhibits good characteristics that are better than those of previously reported PANI/SWNT or PANI/Bi composites. Owing to the ease of synthesis and processing, these materials performance can be further optimized in future works with better/systematic control of morphology/Ag content through improved synthetic methods.

## Acknowledgements

R.C.Y.K. is grateful for the invitation at the University of Lille 1-Sciences and Technologies (F-59650 Villeneuve d'Ascq) as Visiting Professor and he also thanks the Univ. of Science and Arts of Oklahoma-Gladys Anderson Emerson Fund for its partial contribution in this project. SEM & EDS measurements were performed at the Centre Commun de Microscopie (Univ. Lille). The authors thank Dr. M. Moreau for providing access to the LASIR Raman facilities.

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5.6 General approach of the photothermoelectric technique for thermal characterization of solid thermoelectric materials, J. Phys. D : Appl. Phys. 50 (2017) 265501
# General approach of the photothermoelectric technique for thermal characterization of solid thermoelectric materials

Karim Touati<sup>1,2,5</sup>, Michael Depriester<sup>1,2</sup>, Emmanuel Guilmeau<sup>3</sup>, Andrés Sotelo<sup>4</sup>, Maria A Madre<sup>4</sup>, Franck Gascoin<sup>3</sup> and Abdelhak Hadj Sahraoui<sup>1,2</sup>

E-mail: karim.touati@univ-littoral.fr

Received 22 February 2017, revised 19 April 2017

Accepted for publication 22 May 2017

Published 13 June 2017

<figure><img src="image_89.png" /></figure>

CrossMark

## Abstract

This work focuses on the photothermoelectric (PTE) technique allowing the thermal characterization of solid-state thermoelectric (TE) materials. Previously, this technique was restricted to TE materials having low electrical conductivities. Here, the PTE technique is extended and generalized to all solid-state TE materials with low or high electrical conductivities. This is achieved by taking into account the Gaussian shape of the thermal excitation source. The formalism of this new methodology is developed and the procedure for extracting thermal parameters is proposed. For illustration, two different TE materials are studied: with relatively high electrical conductivity ($Bi_2Te_{2.4}Se_{0.6}$) and relatively low electrical conductivity ($Bi_2Ca_2Co_{1.7}O_x$). The thermal properties of these two materials (thermal diffusivity, effusivity and conductivity) are found and compared to those obtained by the photothermal radiometry which is a well established technique. The good concordance between the results obtained by these two techniques demonstrates the relevance of the generalized PTE technique. One of the main advantages of this technique is its non use of an external sensor.

Keywords: Seebeck effect, thermoelectric materials, thermal conductivity, photothermoelectricity

(Some figures may appear in colour only in the online journal)

## 1. Introduction

Because of the energy challenge facing our planet, a lot of research on renewable and environment friendly resources is taking place around the world. Given that more than two-thirds of the energy produced by man in the various industrial systems (nuclear plant, automobile, etc) is lost in the form of heat, more and more research organizations are interested in the recovering and valorizing of this fatal energy. Among

the most explored avenues, thermoelectric (TE) generators are attracting a particular interest. This technology presents many advantages: the non-emission of pollutant gas, the non-existence of any audible noise as a consequence of the absence of moving part, etc. Thermoelectric generators are mostly solid-state materials allowing direct conversion of thermal energy into electricity due to Seebeck effect.

The majority of research carried out these last few years on thermoelectric materials focus on improving their performances. The conversion efficiency of heat into electricity of TE materials is a function of their figure of merit given by [1]

<footnote n="1"><sup>1</sup> UDSMM (EA 4476), MREI-1, Université du Littoral Côte d'Opale, 59140 Dunkerque, France</footnote>

<footnote n="2"><sup>2</sup> Univ Lille Nord de France, 59044 Lille, France</footnote>

<footnote n="3"><sup>3</sup> CRISMAT, UMR 6508 CNRS/ENSICAEN, 6 Bd du Maréchal Juin, F-14050 Caen, France</footnote>

<footnote n="4"><sup>4</sup> Instituto de Ciencia de Materiales de Aragón (CSIC-Universidad de Zaragoza), M de Luna, 3, 50018 Zaragoza, Spain</footnote>

<footnote n="5"><sup>5</sup> Author to whom any correspondence should be addressed.</footnote>
$ZT = (S^2\sigma_e/\kappa)T$. This dimensionless parameter is dependent on the Seebeck coefficient (S), the electrical conductivity ($\sigma_e$), the thermal conductivity ($\kappa$) and the average absolute temperature of the studied material. The improvement can be achieved by increasing their power factor ($S^2\sigma_e$) or by reducing their thermal conductivity ($\kappa$). Since the electronic contribution to the thermal conductivity ($\kappa_e$) is directly linked to the electrical conductivity by the Wiedemann-Franz law, the reduction of this electronic contribution will conduct to a similar evolution of the electrical conductivity. As the lattice contribution of the thermal conductivity ($\kappa_l$) is not correlated with $\sigma_e$, more and more researches deal with the reduction of this contribution.

So, an accurate determination of the thermal and electrical properties is necessary for the improvement of TE materials performance. Although the measurement of Seebeck coefficient and electrical conductivity are often reported, thermal conductivity of these materials is scarcely measured directly. It is commonly calculated by the combination of thermal diffusivity ($\alpha$), heat capacity ($C_p$) and density ($\rho$) of these materials, which may not be as accurate as a direct measurement. For this purpose, the photothermoelectric (PTE) technique was developed [2-4]. This original technique allows a direct and accurate thermal characterization of TE materials without using an external sensor. It is based on a self-generated voltage due to the Seebeck effect. This voltage is provided by the sample when one of its two faces is thermally excited using a modulated laser at a given frequency. Access to thermal parameters is possible through analysis of the signal (amplitude and phase) evolution as function of the exciting frequency. Thermal diffusivity and effusivity ($e$) can be simultaneously determined. The combination of these two parameters allows the calculation of the TE material thermal conductivity using the relation $\kappa = e\sqrt{\alpha}$. This technique was used to study thermally many solid TE materials having low electrical conductivities. Among these we can cite the polyaniline/carbon nanotube composites and titanium trisulfide (TiS$_3$).

Until now, The PTE$_{1D}$ model presented in reference [2] was restricted to TE materials presenting low electrical conductivities. This model showed its limits when we were interested in the thermal characterization of TE materials having a relatively high electrical conductivities such as the bismuth senenido telluride (Bi$_2$Te$_{2.4}$Se$_{0.6}$). Indeed, this model does not fit the experimental data delivered by such type of materials. To remedy this issue, a new photothermoelectric model allowing the thermal characterization of all TE materials (having low or high electrical conductivity) is developed. The followed procedure and the new developed model are presented in the present work.

## 2. Theoretical approach

### 2.1. Simulation of the electrical potential distribution in a TE material subjected to a Gaussian modulated laser beam

The PTE technique is based on the voltage appearing between the two faces of a TE material kept at different temperatures ($\Delta T$). This voltage is given by:

$$ \Delta \bar{V} = -S \Delta \bar{T}. \quad (1) $$

To determine the voltage distribution, the temperature expression at each face of the TE material is needed. This expression can be obtained from the resolution of the heat equation. Since the thermal excitation source is not homogeneous (laser beam having a Gaussian shape), highly conductive thin layers (gold) are deposited on the two sides of the studied TE sample. This allows the homogenization of charges at the sample surfaces. Then a one dimensional (1D) heat propagation can be considered. This is achieved in the case of a low electrical conductive TE materials but not in the case of high electrical conductive TE materials. To visualize the effect of these two conductive layers, the electrical potential distribution in a low and high electrical conductive TE material will be simulated.

From the Onsager relations [5], heat and electric charge continuity equations [6], we established two coupled equations (equations (2) and (3)) describing all the thermal and electrical phenomena encountered in a TE material subjected to a modulated thermal excitation.

$$ \rho c_p j \omega \bar{T} - \sigma_e S T \nabla^2 \bar{V} - (\sigma_e S^2 T + \kappa_j) \nabla^2 \bar{T} = \bar{q} \quad (2) $$

$$ \sigma_e \nabla^2 \bar{V} + \sigma_e S \nabla^2 \bar{T} + j \omega \epsilon \nabla^2 \bar{V} = 0 \quad (3) $$

with $j^2 = -1$, $\omega = 2\pi f$, $f$ is the excitation frequency. $\kappa_j$ is the material thermal conductivity under zero electrical current. $\bar{T}$ and $\bar{V}$ are respectively the modulated part of temperature and electric potential across the TE material. $\epsilon$ is the material dielectric permittivity. $\dot{q}$ is the heat generation rate per unit volume [W.m$^{-3}$].

The electrical potential distribution in a TE material subjected to a Gaussian laser beam can be simulated by solving these two coupled equations. For this purpose, the Matlab solver called PDE (partial differential equations) based on the finite element method is used. Simulations are performed for low and high electrical conductive TE material by considering the system represented in figure 1. In both simulations, the thicknesses ($L_1$) of TE samples and the substrate (water) are taken equal to 450 µm and to ~3 mm, respectively. The thickness of the substrate is taken sufficiently large to prevent that the thermal wave reaches its rear side, as a consequence the Dirichlet conditions are imposed at this location. For the lateral sides, Neumann conditions are imposed. To observe the effect of the coating on the electrical potential distribution, in both cases a thin gold layer having a thickness ($L_d$) of 10 µm is used. The physical parameters of the different layers involved in these simulations are given in table 1. In order to be closer to our experimental conditions, the laser beam radius ($R_l$) is taken equal to 6 mm and the excitation frequency ($f$) is chosen in the range where the sample is thermally thin (1 Hz). In these simulations we assumed that $\dot{q}$ is equal to zero.

Figure 2 exhibits the electric potential distribution in a TE material (Bi$_2$Ca$_2$Co$_{1.7}$O$_{x}$), presenting a relatively low electrical conductivity, and in the substrate on which this sample is placed. As it was expected, this figure shows that the electrical potential distribution in the TE material follows the
<figure><img src="image_90.png" /><figcaption>**Figure 1.** Geometry of the studied system used for the simulation and the modelization of the photothermoelectric signal.</figcaption></figure>

<table><caption>**Table 1.** Physical parameters of the two thermoelectric materials simulated in this section (Bi₂Ca₂Co₁.₇Oₓ, Bi₂Te₂.₄Se₀.₆) and those of gold used as electrodes and water used as substrate at ambient temperature (20 °C).</caption><thead><tr><th>Material</th><th>κ (W · m<sup>-1</sup> · K<sup>-1</sup>)</th><th>C<sub>p</sub> (J · kg<sup>-1</sup> · K<sup>-1</sup>)</th><th>σ<sub>e</sub> (S · m<sup>-1</sup>)</th><th>S (μV · K<sup>-1</sup>)</th><th>ε (F · m<sup>-1</sup>)</th><th>ρ (kg · m<sup>-3</sup>)</th></tr></thead><tbody><tr><td>Bi₂Ca₂Co₁.₇Oₓ</td><td>1.6</td><td>479</td><td>500</td><td>150 [11]</td><td>—</td><td>5080 [12]</td></tr><tr><td>Bi₂Te₂.₄Se₀.₆</td><td>1.3</td><td>202.2</td><td>510<sup>4</sup></td><td>-175</td><td>—</td><td>7220</td></tr><tr><td>Gold</td><td>317</td><td>129</td><td>45.610<sup>6</sup></td><td>—</td><td>—</td><td>19400</td></tr><tr><td>Water</td><td>0.6</td><td>4185</td><td>5.510<sup>-4</sup></td><td>—</td><td>78.5</td><td>1000</td></tr></tbody></table>

<figure><img src="image_91.png" /><figcaption>**Figure 2.** Simulation of the electrical potential amplitude (V) distribution in a thermoelectric material presenting low electrical conductivity (Bi₂Ca₂Co₁.₇Oₓ) subjected to a modulated Gaussian thermal excitation.</figcaption></figure>

shape of the excitation thermal source (Gaussian shape). In this case, the assumption of a 1D heat propagation is not satisfied due to this inhomogeneous distribution. Then, thin gold layers are deposited at the sample faces, see figure 3. As a consequence, a complete electrical potential homogenization can be observed at the two faces, as it is shown on figure 4.

Now, the coating effect on the electrical potential distribution in the case of a high conductive material (Bi₂Te₂.₄Se₀.₆) will be highlighted.

Figure 5 shows that the gold layers deposited on a TE material having a $\sigma_e = 510^4$ S·m⁻¹ allow a better electrical potential distribution but not a complete charges homogenization at
<figure><img src="image_92.png" /><figcaption>**Figure 3.** Geometry of the studied system used for the simulation and the modelization of the photothermoelectric signal after depositing two thin gold layers at the sample faces.</figcaption></figure>

<figure><img src="image_93.png" /><figcaption>**Figure 4.** Simulation of the influence of the deposition of two gold layers on the electrical potential amplitude (V) distribution in a Bi₂Ca₂Co₁.₇Oₓ sample subjected to a modulated Gaussian thermal excitation.</figcaption></figure>

the sample faces. Consequently, an influence of the electrical contacts position allowing the photothermoelectric signal collection appears. The access to thermal properties of such TE materials by the PTE technique can be achieved by following two ways. The first consists of depositing a thicker gold layer allowing a complete potential homogenization on the sample faces. In this case, the PTE₁D model can be used. The second way to follow consists of developing a new PTE model taking into account the positions of the electrical contacts on the sample faces.

Because the first solution is economically unsustainable (a huge gold quantity is needed to reach a complete potential homogenization), the second solution will be retained. We propose in the following section a new theoretical model

allowing the thermal characterization of all TE materials (with high or low electrical conductivity). In contrast to the PTE₁D model, this new theory takes into account the contacts position as well as the laser beam radius.

## 2.2. Development of a photothermoelectric model taking into account the Gaussian shape of the laser beam

Considering a TE sample subjected to a modulated heating on its front face ($z = 0$), see figure 1. When the heating flux density displays a Gaussian distribution ($F(r, \omega) = \frac{F_0}{2} \exp(-r^2/R_l^2)$), the temperature distribution in the sample as function of its depth ($z$) and the exciting frequency can be written as:
<figure><img src="image_94.png" /><figcaption>Figure 5. Simulation of the influence of the deposition of two gold layers on the electrical potential amplitude (V) distribution in a Bi$_{2}$Te$_{2.4}$Se$_{0.6}$ sample subjected to a modulated Gaussian thermal excitation.</figcaption></figure>

$$ \begin{aligned} \bar{T}_{1(3D)}(r, z, \omega) = & 2R_l^2 \int_0^\infty \bar{T}_{1(1D)}(z, s_1(\lambda, \omega)) \\ & \times \exp(-\lambda^2 R_l^2/4) J_0(\lambda r) \lambda d\lambda. \end{aligned} \quad (4) $$ $$ \begin{aligned} \bar{T}_{1(3D)}(r, z, \omega) = & 2R_l^2 \int_0^\infty \frac{F_0}{2\kappa_1 s_1(1 + B_{01})} \left[ \frac{1 + \Gamma_{12} \exp(-2s_1(L_1 - z))}{1 - \Gamma_{10}\Gamma_{12} \exp(-2s_1L_1)} \exp(-s_1z) \right] \\ & \times \exp(-\lambda^2 R_l^2/4) J_0(\lambda r) \lambda d\lambda \end{aligned} \quad (5) $$

This expression is obtained by applying the Hankel transforms and the Hankel inverse transforms to the equation describing the temperature distribution in the same sample when the thermal excitation is considered homogeneous [7, 8] ($\bar{T}_{1(1D)}$). $F_0$ represents the heat flux density generated by the light source at the center of the light spot ($r = 0$ and $z = 0$). $J_0$ is the zero-order Bessel function of the first kind, $\lambda$ represents Hankel's variable which is a superposition of planar waves with different spatial frequencies [9]. $s_1$ is a complex thermal wave number given by:

Since the sample is a TE material that exhibits a Seebeck coefficient ($S_1$), the generated voltage ($\Delta\bar{V}$) between two points located on its two faces ($z = 0$ and $z = L_1$) is given by:

$$ \begin{aligned} \Delta\bar{V}(\omega) = & \bar{V}_1(r = r_0, z = 0, \omega) - \bar{V}_1(r = r_L, z = L_1, \omega) \\ = & -S_1 \times [\bar{T}_1(r = r_0, z = 0, \omega) - \bar{T}_1(r = r_L, z = L_1, \omega)]. \end{aligned} \quad (6) $$

$r_0$ and $r_L$ are respectively the points at the surfaces $z = 0$ and $z = L_1$ where the electrical potentials are measured.

By introducing equations (5) in (6), the photothermoelectric signal that the TE sample generates is expressed by:

$$ s_1^2 = \lambda^2 + \sigma_1^2, \quad \sigma_1 = \frac{1+i}{\mu_1}, \quad \mu_1 = \sqrt{\frac{\pi f}{\alpha_1}}. $$

Here, $\alpha_1$ is the sample's thermal diffusivity. As the complex thermal wave number is dependent on $\lambda$, the thermal wave reflection coefficients in the case of a three dimensional (3D) heat propagation $\Gamma_{ij}$ are given by [7, 9]:

$$ \begin{aligned} \Delta\bar{V}_{(3D)}(\omega) = & 2R_l^2 \int_0^\infty \frac{-S_1 F_0}{2\kappa_1 s_1(1 + B_{01})} \left[ \frac{1 + \Gamma_{12} \exp(-2s_1L_1)}{1 - \Gamma_{10}\Gamma_{12} \exp(-2s_1L_1)} \right] \\ & \times \exp(-\lambda^2 R_l^2/4) J_0(\lambda r_0) \lambda d\lambda \\ - & 2R_l^2 \int_0^\infty \frac{-S_1 F_0}{2\kappa_1 s_1(1 + B_{01})} \left[ \frac{(1 + \Gamma_{12}) \exp(-s_1L_1)}{1 - \Gamma_{10}\Gamma_{12} \exp(-2s_1L_1)} \right] \\ & \times \exp(-\lambda^2 R_l^2/4) J_0(\lambda r_L) \lambda d\lambda. \end{aligned} \quad (7) $$

$$ \Gamma_{ij}(\lambda) = \frac{B_{ij}(\lambda) - 1}{B_{ij}(\lambda) + 1}, \quad B_{ij}(\lambda) = \frac{\kappa_i s_i(\lambda)}{\kappa_j s_j(\lambda)}. $$

The indexes $i$ and $j$ refer to the different layers represented on figure 1. Indexes 0, 1, 2 refer to the front layer, TE sample and the substrate, respectively. $\kappa$ is the thermal conductivity. In contrary to the 1D heat conduction case where the thermal wave reflection coefficient is only dependent on the layers effusivities contrast, here $\Gamma_{ij}$ is also dependent on the exciting frequency and the Hankel's variable.

After replacing $\bar{T}_{1(1D)}$ by its expression developed in [2], the temperature distribution in the sample is given by:

This signal is dependent on the thermal properties of the TE sample, the electrical contacts positions and the laser beam radius. To remove any instrumental contribution which can affect this photothermoelectric signal, a normalization procedure should be performed. To achieve that, the substrate on which the TE sample is placed should be replaced by another substrate (e.g. air) having a known thermal effusivity. Hereinafter, the new signal generated by the TE sample when placed on air is written:
$$ \begin{aligned} \Delta \bar{V}_{s(3D)}(\omega) = & 2R_l^2 \int_0^\infty \frac{-S_1 F(r, z=0, \omega)}{2\kappa_l s_l(1+B_{01})} \left[ \frac{1+\Gamma_{10} \exp(-2s_l L_l)}{1-\Gamma_{10}^2 \exp(-2s_l L_l)} \right. \\ & \left. \times \exp(-\lambda^2 R_l^2/4) J_0(\lambda r_0) \lambda d\lambda \right. \\ & \left. -2R_l^2 \int_0^\infty \frac{-S_1 F(r, z=0, \omega)}{2\kappa_l s_l(1+B_{01})} \left[ \frac{(1+\Gamma_{10}) \exp(-s_l L_l)}{1-\Gamma_{10}^2 \exp(-2s_l L_l)} \right] \right. \\ & \left. \times \exp(-\lambda^2 R_l^2/4) J_0(\lambda r_L) \lambda d\lambda. \right] \end{aligned} \quad (8) $$

The ratio of $\bar{V}_{s(3D)}(\omega)$ to $\bar{V}_{s(3D)}(\omega)$ gives the normalized photothermoelectric signal ($\bar{V}_{n(3D)}(\omega)$) free from all instrumental contributions:

$$ \bar{V}_{n(3D)}(\omega) = \frac{\Delta \bar{V}_{(3D)}(\omega)}{\Delta \bar{V}_{s(3D)}(\omega)}. \quad (9) $$

When thermal effusivities of the front and back layers are known, the thermal diffusivity and effusivity of the TE material can be obtained by fitting equation (9) to the normalized experimental signal. This procedure will be referred in the following sections as PTE$_{3D}$ technique.

## 3. Experiment

To demonstrate the consistency of our theoretical developments, two TE materials with high and low electrical conductivities are elaborated and then thermally characterized using the PTE technique. In this part, we describe the sample preparation procedure and the experimental device of the PTE technique.

### 3.1. Material synthesis

As high electrical conductive material, a Bismuth selenidotelluride (Bi$_{2}$Te$_{2.4}$Se$_{0.6}$) sample was elaborated from pure elemental precursors [10] (Alfa Aesar) of bismuth (needles, 99.99%), tellurium (shots, 99.99%), and selenium (shots, 99.99%). Appropriate stoichiometric mixture of the elements were loaded in a 20 ml tungsten carbide ball mill jar, containing seven tungsten carbide 10 mm-diameter balls. The mixture was then subjected to mechanical alloying for 30 min divided in 15 cycles of 2 min each at 700 rpm in a Fritsch Pulverisette 7 (Fritsch GmbH) premium line device. The obtained powder was compacted using a spark plasma sintering (SPS) process in a graphite die of 15 mm diameter at a pressure of 25 MPa during 30 min at a temperature of 723 K.

As low electrical conductive material, a Bi$_{2}$Ca$_{2}$Co$_{1.7}$O$_{x}$ sample was prepared by the classical solid state method [11]. Bi$_{2}$O$_{3}$ (99.9%, Aldrich), CaCO$_{3}$ (≥99%, Aldrich), and Co$_{3}$O$_{4}$ (99.5%, Panreac) were mixed and milled in an agate ball mill for 2h at 300 rpm in acetone media. The resulting slurry was then sieved and the balls were washed with acetone to separate the grinding media and the powders mixture. The suspension was then totally dried under infrared radiation until a soft brownish powder was obtained. The dry mixture was then calcined twice at 750 and 800 °C for 12h, with an intermediate manual milling, to decompose the CaCO$_{3}$, uniaxially pressed at about 400 MPa and sintered at 800 °C for 24h with a final furnace cooling.

<figure><img src="image_95.png" /><figcaption>Figure 6. Frequency dependence of the photothermoelectric experimental signal (amplitude and phase) generated by the thermoelectric sample Bi$_{2}$Te$_{2.4}$Se$_{0.6}$ at ambient temperature.</figcaption></figure>

<figure><img src="image_96.png" /><figcaption>Figure 7. Normalized amplitude (upper plot) and phase (lower plot) of the photothermoelectric signal, plotted as a function of frequency. The circles represent the experimental data and the full lines represent the best fits of the 1D and the 3D models. The Bi$_{2}$Te$_{2.4}$Se$_{0.6}$ sample has a thickness of 450 µm.</figcaption></figure>

### 3.2. PTE technique setup

For each of the two studied materials, adapted samples were cut from the prepared rods. The Bi$_{2}$Ca$_{2}$Co$_{1.7}$O$_{x}$ sample was square-shaped with parallel opposite faces, its side and thickness are around 3 mm and 609 µm, respectively. The geometry of the Bi$_{2}$Te$_{2.4}$Se$_{0.6}$ sample was round-shaped with parallel opposite faces, its diameter and thickness are about 15 mm and 450 µm, respectively.

In order to get equipotential surfaces on the Bi$_{2}$Ca$_{2}$Co$_{1.7}$O$_{x}$ sample, a highly electrically conductive coating (gold) was deposited by sputtering (Q150R rotary pumped coating system) on its two faces. This will avoid a 3D effect at this location. In the purpose of reducing as much as possible the 3D effect, thin gold layers were also deposited on the faces of the Bi$_{2}$Te$_{2.4}$Se$_{0.6}$ sample. Then, two electrical leads connected to the electrodes were silver pasted and plugged into a
<figure><img src="image_97.png" /><figcaption>**Figure 8.** Most squares (full line) and least squares error analysis (dashed line) on thermal diffusivity (a) and effusivity (b) of the Bi₂Te₂.₄Se₀.₆ sample.</figcaption></figure>

EG&G 7260 lock-in amplifier. Afterwards, the surfaces of the two studied samples were blackened in order to increase their light absorption.

To create a temperature difference and a voltage between the two faces of the TE sample, an internally modulated laser (Ventus VIS 532 nm, Laser Quantum) was employed to illuminate the front sample's face. For the frequency range used here ($1<f<100$ Hz), the temperature difference generated between the two sides of the sample is less than few hundreds of milli-Kelvin.

## 4. Experimental results and discussion

Amplitude and phase of the signal provided by the Bi₂Te₂.₄Se₀.₆ sample are presented in figure 6. The signal amplitude decreases with the exciting frequency because of the temperature difference decrease between the two sides of the sample. Amplitude ratios and phase differences were calculated, providing the normalized amplitude and phase represented in figure 7. As depicted on this figure, the model considering a 1D heat propagation in the sample (PTE₁D model) is not able to fit the normalized experimental signal. Conversely, the model taking into account the Gaussian shape of the excitation (PTE₃D model) described by equation (9) is successfully fitted to this experimental normalized signal using a non linear least square fitting procedure. To meet the requirements of the model (thermally thick substrates and negligible convection effect) and because of sensitivity, scans were performed in the frequency range of 1–100 Hz.

From the numerical analysis of the experimental signal, a thermal diffusivity $\alpha_1 = (8.71 \pm 0.3) \times 10^{-7} \text{ m}^2 \text{ s}^{-1}$ and a thermal effusivity $e_1 = (1582 \pm 72) \text{ W s}^{1/2} \text{ m}^{-2} \text{ K}^{-1}$ were obtained. Calculus gives a thermal conductivity $k_1 = (1.47 \pm 0.1) \text{ W m}^{-1} \text{ K}^{-1}$. The error values indicate the statistical uncertainties of the experimental results arising from the fit. The most-squares as well as the least-squares

<table><caption>**Table 2.** Thermal parameters retrieved from (PTE₃D) technique compared to those obtained by the PTR technique for a Bi₂Te₂.₄Se₀.₆ sample at ambient temperature.</caption><thead><tr><th>Thermal parameters</th><th>PTE<sub>3D</sub></th><th>PTR</th></tr></thead><tbody><tr><td>α (10<sup>-7</sup> m<sup>2</sup> s<sup>-1</sup>)</td><td>8.71 ± 0.30</td><td>9.79 ± 0.02</td></tr><tr><td>e (W s<sup>1/2</sup> m<sup>-2</sup> K<sup>-1</sup>)</td><td>1582 ± 72</td><td>1432 ± 08</td></tr><tr><td>κ (W m<sup>-1</sup> K<sup>-1</sup>)</td><td>1.47 ± 0.10</td><td>1.42 ± 0.01</td></tr></tbody></table>

<figure><img src="image_98.png" /><figcaption>**Figure 9.** Normalized amplitude (upper plot) and phase (lower plot) of the photothermoelectric signal, plotted as a function of frequency. The circles represent the experimental data and the full lines represent the best fits of the 1D and the 3D models. The Bi₂Ca₂Co₁.₇O₈ sample has a thickness of 609 µm.</figcaption></figure>

error analysis [13] represented in figure 8 exhibit a single and identical minimum in the case of thermal diffusivity (figure 8(a)) and effusivity (figure 8(b)). This single minimum means that the calculated solution is unique (if the most squares error analysis shows a flat line, it indicates a multi-fold degeneracy
<table><caption>**Table 3.** Thermal parameters retrieved from PTE<sub>3D</sub> and PTE<sub>1D</sub> techniques compared to those obtained by the PTR technique for a Bi<sub>2</sub>Ca<sub>2</sub>Co<sub>1.7</sub>O<sub>8</sub> sample at ambient temperature.</caption><thead><tr><th>Thermal parameters</th><th>PTE<sub>1D</sub></th><th>PTE<sub>3D</sub></th><th>PTR</th></tr></thead><tbody><tr><td>α (10<sup>-7</sup> m<sup>2</sup>·s<sup>-1</sup>)</td><td>6.57 ± 0.04</td><td>6.67 ± 0.15</td><td>6.32 ± 0.05</td></tr><tr><td>e (W·s<sup>1/2</sup>·m<sup>-2</sup>·K<sup>-1</sup>)</td><td>1936 ± 12</td><td>1987 ± 87</td><td>2018 ± 23</td></tr><tr><td>κ (W·m<sup>-1</sup>·K<sup>-1</sup>)</td><td>1.57 ± 0.02</td><td>1.62 ± 0.10</td><td>1.60 ± 0.02</td></tr></tbody></table>

which means that the fit parameter satisfies more than one solution for the same experimental data).

In order to validate this new technique, the obtained thermal parameters were compared to those obtained by the photothermal radiometry (PTR) on the same sample. The PTR technique is well described in the article of Depriester et al [14]. As it can be seen in table 2, the PTE<sub>3D</sub> calculated thermal conductivity is in close agreement with the one obtained by the PTR technique. The deviation between the two conductivities is less than 4%, confirming a good accuracy of the PTE<sub>3D</sub> technique.

The experimental signals generated by the low conductive TE material (Bi<sub>2</sub>Ca<sub>2</sub>Co<sub>1.7</sub>O<sub>8</sub>) are processed by following the same procedure already met in the case of the high conductive material, see figure 9. As we deposited thin gold layers on the two faces of the sample, the PTE<sub>1D</sub> model was also fitted to the same experimental signal. The two models, fitting very well the normalized experimental signal, are quasi-superposed. The results obtained by using PTE<sub>3D</sub> and PTE<sub>1D</sub> are reported in table 3. These results show that both PTE models (PTE<sub>1D</sub> and PTE<sub>3D</sub>) are suitable in the case of low conductive materials. In this case also, the PTR technique confirms the PTE results.

## 5. Conclusion

In this work, we have reported a simple tool allowing an accurate thermal characterization of high efficient thermoelectric materials. After being restricted to low conductive TE materials, the PTE technique was extended to the most conductive materials.

Simulations using the finite element method have shown that it is possible to use the PTE<sub>1D</sub> model for the thermal characterization of materials with low electrical conductivities. This is possible by depositing thin conductive layers allowing the homogenization of the electrical potential distribution at the surfaces of the sample. When the TE materials electrical conductivity comes closer to the one of the deposited gold layer, which is the case for the most efficient TE materials, the deposit of a coating may be insufficient and the PTE<sub>1D</sub> model is then no longer valid.

In this work, we demonstrated that thermal parameters can then be extracted from the photothermoelectric signal by taking into account the Gaussian form of the thermal excitation source (PTE<sub>3D</sub>). The formalism of this new methodology is developed and a procedure for extracting thermal parameters is also proposed. Thermal parameters retrieved by the PTE technique are confirmed by those obtained by the conventional PTR technique. One of the main advantages of this approach is the non use of a temperature external sensor. In

this work, it was also demonstrated that thermal parameters (thermal diffusivity, effusivity and conductivity) of these materials can be simultaneously determined.

This study paves the way for further developments. Once thermal parameters of a TE material are known, the relative evolution of Seebeck coefficient can be obtained, in principle, from signal amplitude using the same configuration. Furthermore, transversal thermal conductivity could be obtained by the use of the PTE<sub>3D</sub> model. This will be the subject of our next investigations.

## Acknowledgments

This work was financially supported by 'la région Hauts de France', 'le Syndicat Mixte de la Côte d'Opale' and by the 'Fond Européen de Développement Economique Régional'. M A Madre and A Sotelo acknowledge MINECO-FEDER (project MAT2013-46505-C3-1-R) and Gobierno de Aragon-FEDER (Grupo de Investigación Consolidado T12) for financial support.

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Résumé

Ce mémoire d'Habilitation à Diriger les Recherches présente une synthèse de mes travaux de recherche au sein de l'UDSMM. Les thématiques abordées concernent de manière générale l'étude des phénomènes de transport thermique. Une partie des résultats obtenus à l'aide de techniques photothermiques est exposée. Nous avons utilisé d'une part la radiométrie infrarouge et d'autre part une nouvelle technique photothermique que nous avons développée au sein du laboratoire : la photothermoélectricité. Dans cette technique, la force électromotrice générée en régime harmonique par un matériau thermoélectrique est analysée afin de déterminer les paramètres thermiques. Elle constitue un moyen simple de déterminer la conductivité thermique des thermopiles, paramètre crucial dans leurs performances. Un développement théorique tenant compte du couplage thermique/électrique a permis de mettre en évidence une contribution des porteurs de charge aux mécanismes de transport thermique.

Différents types de matériaux ont été étudiés à l'aide de l'une ou l'autre de ces techniques, dont des matériaux thermoélectriques à l'état solide ou liquide. À la fin de ce rapport, quelques voies sont envisagées pour la suite de ce travail aussi bien dans le domaine de l'instrumentation que dans celui du développement de nouveaux matériaux thermoélectriques.

Mots clés : photothermie, thermoélectricité, transfert thermique, régime harmonique, radiométrie infrarouge, composites, problèmes inverses.