# Optical properties of a graphene nanofluid for direct absorption solar collectors

Miguel Sainz-Mañas, Cyril Caliot, Christophe Escape, Françoise Bataille, Gilles Flamant

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Miguel Sainz-Mañas, Cyril Caliot, Christophe Escape, Françoise Bataille, Gilles Flamant. Optical properties of a graphene nanofluid for direct absorption solar collectors. Next Materials, 2025, 8, pp.100798. 10.1016/j.nxmate.2025.100798. hal-05308311

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Submitted on 13 Oct 2025

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Research article

# Optical properties of a graphene nanofluid for direct absorption solar collectors

<figure><img src="image_2.png" /></figure>

Miguel Sainz-Mañas<sup>a, c, *</sup>, Cyril Caliot<sup>b, ◐</sup>, Christophe Escape<sup>a, c, ◐</sup>, Françoise Bataille<sup>a, c, ◐</sup>,
Gilles Flamant<sup>a, c, ◐</sup>

ARTICLE INFO

Keywords:

Graphene nanoplatelets

Nanofluid

Discrete dipole approximation (DDA)

Light extinction

Solar thermal collector

A B S T R A C T

Graphene-based materials exhibit unique physical properties that are valuable for a wide range of applications. In the energy sector, this 2D carbon nanomaterial is primarily valued for its mechanical strength and electrical conductivity. However, the optical properties of graphene could enhance the performance of new-generation solar thermal collectors. This study investigates experimentally and numerically the optical properties of graphene nanoparticles dispersed in water. Spectrophotometric measurements, combined with Monte-Carlo ray-tracing simulations, permitted to determine the optical properties of the dispersion and the specific contribution of graphene particles. The influence of the shape and size of these carbonaceous particles is analyzed by comparing results from two different modeling approaches: namely, the Mie theory and the discrete dipole approximation. The results indicate that high extinction coefficients at low graphene concentrations (below 1 g/L) can be attributed to the high absorption properties of the particles. Additionally, the size and shape of the particles significantly affect both the absolute value of extinction and the spectral optical behavior of graphene. Graphene nanoplatelets with thicknesses below 10–20 nm and high length-to-thickness ratios optimize the absorption capacity of the nanofluid in the UV-Visible solar spectrum. Further, a bimodal particle size distribution is found to accurately reproduce the optical properties of the graphene dispersion obtained experimentally. These findings suggest that optimizing particle size and shape distribution can enhance the performance of graphene-based solar collectors for future applications.

1. Introduction

Nanoparticles have captured worldwide attention since last century, primarily due to their size, shape, thermo-physical properties and ability to tune the physicochemical properties of media. Carbon-based nano-particles have proven enormous potential, among which graphene has received most attention due to its excellent thermal, mechanical and electrical properties [1,2]. In addition, graphene nanoparticles offer exceptional optical absorption properties due to their 2D nature, which provides a high surface-to-volume ratio. The honeycomb lattice of gra-phene owing to the σ bond between carbon atoms is responsible for its exceptional mechanical properties, while the π bonds control the inter-action between different graphene layers [3]. These bonds are also responsible of the π-π* electron transitions exhibiting an absorption peak between 250 and 270 nm for pristine graphene [1]. Nair et al. [4]

studied the relation between the graphene transmittance (τ) decrease with the carbon-layer number (N_L) and the fine structure constant (α = 1/137), resulting in the relation τ = 1 – N_L πα, valid for wavelengths over 500 nm. These original optical properties may be very useful for their applications in new generation solar thermal receivers [5–7].

Among new generation solar thermal collectors, direct absorption solar collectors (DASC) have arisen the attention of the solar community in last decade [8]. DASCs absorb incoming solar radiation volumetri-cally throughout the heat transfer fluid (HTF), dispensing with the se-lective absorbing coating used in conventional metallic collectors. However, conventional HTF, as water or thermal oils, have poor optical absorption properties in the visible and near-infrared spectrum. Contrarily, nanoparticles, mainly metallic and carbon-based, are extremely efficient at tuning the optical properties of HTF with small added quantities. Nanoparticles dispersions, known as nanofluids, have been largely studied in the last decades for their applications in

<footnote><sup>a</sup> Processes, Materials and Solar Energy laboratory, PROMES - CNRS, 7 rue du Four Solaire, Odeillo 66120, France</footnote>

<footnote><sup>b</sup> LMAP, UPPA, CNRS, E2S, 1 allée du parc Montaury, Anglet 64600, France</footnote>

<footnote><sup>c</sup> UPVD, Rambla de la Thermodynamique, Tecnosud, Perpignan 66100, France</footnote>

<footnote>* Corresponding authors at: Processes, Materials and Solar Energy laboratory, PROMES - CNRS, 7 rue du Four Solaire, Odeillo 66120, France
E-mail addresses: Miguel.Sainz-Mana@promes.cnrs.fr (M. Sainz-Mañas), Gilles.Flamant@promes.cnrs.fr (G. Flamant).</footnote>
<table><thead><tr><th colspan="2">Nomenclature</th></tr></thead><tbody><tr><td>Variable</td><td>Definition</td></tr><tr><td>C</td><td>Cross-section</td></tr><tr><td>C<sub>p</sub></td><td>Particle concentration</td></tr><tr><td>d</td><td>Distance</td></tr><tr><td>f<sub>v</sub></td><td>Volume fraction</td></tr><tr><td>g</td><td>Asymmetry parameter</td></tr><tr><td>I<sub>AM1.5</sub></td><td>Global AM1.5 incident radiation</td></tr><tr><td>k</td><td>Absorption index</td></tr><tr><td>l<sub>p</sub></td><td>Particle length</td></tr><tr><td>L</td><td>Light path length</td></tr><tr><td>m</td><td>Complex refractive index</td></tr><tr><td>n</td><td>Refractive index</td></tr><tr><td>N<sub>T</sub></td><td>Number of particles per unit volume</td></tr><tr><td>r</td><td>Radius</td></tr><tr><td>t<sub>p</sub></td><td>Particle thickness</td></tr><tr><td>V<sub>p</sub></td><td>Particle volume</td></tr><tr><td>x</td><td>Particle size parameter</td></tr><tr><td>Acronyms</td><td>Definition</td></tr><tr><td>ADDA</td><td>Amsterdam discrete dipole approximation</td></tr><tr><td>DASC</td><td>Direct absorption solar collector</td></tr><tr><td>DDA</td><td>Discrete dipole approximation</td></tr><tr><td>HTF</td><td>Heat transfer fluid</td></tr><tr><td>NDASC</td><td>Nanofluid direct absorption solar collector</td></tr><tr><td>SEM</td><td>Scanning electron microscope</td></tr><tr><td>Greek letter</td><td>Definition</td></tr></tbody></table><table><tbody><tr><td>α</td><td>Absorption</td></tr><tr><td>ᾱ</td><td>Mean absorption</td></tr><tr><td>β</td><td>Extinction coefficient</td></tr><tr><td>β'</td><td>Normalized extinction coefficient</td></tr><tr><td>β̄</td><td>Mean extinction coefficient</td></tr><tr><td>κ</td><td>Absorption coefficient</td></tr><tr><td>κ'</td><td>Normalized absorption coefficient</td></tr><tr><td>κ̄</td><td>Mean absorption coefficient</td></tr><tr><td>λ</td><td>Wavelength</td></tr><tr><td>ξ</td><td>Aspect ratio</td></tr><tr><td>ρ</td><td>Reflectance / Density</td></tr><tr><td>σ</td><td>Scattering coefficient</td></tr><tr><td>σ'</td><td>Normalized scattering coefficient</td></tr><tr><td>τ</td><td>Transmittance</td></tr><tr><td>Φ</td><td>Scattering phase function</td></tr><tr><td>χ</td><td>Percentage of total volume fraction</td></tr><tr><td>Subscripts</td><td>Definition</td></tr><tr><td>∩</td><td>Hemispherical</td></tr><tr><td>abs</td><td>Absorption</td></tr><tr><td>cloud</td><td>Particle cloud</td></tr><tr><td>ext</td><td>Extinction</td></tr><tr><td>f</td><td>Front hemisphere</td></tr><tr><td>med</td><td>Medium</td></tr><tr><td>nf</td><td>Nanofluid</td></tr><tr><td>p</td><td>Particle</td></tr><tr><td>sca</td><td>Scattering</td></tr><tr><td>τ</td><td>Transmittance</td></tr></tbody></table>

nanofluid-based DASC (NDASC) [8–15]. Depending on the nanoparticle volume fractions and their intrinsic optical properties, the optical properties of the nanofluid can be tuned offering high versatility to NDASC and the capacity of hybridization with photovoltaic cells [16–19]. However, some key limitations must be overcome previous to their commercialization such as nanoparticle toxicity, stability, high production cost and possible erosion of components [8,20–23].

Among nanofluids for NDASC, carbon-based nanoparticles as single- or multi-wall carbon nanotubes show particularly high absorption properties in the UV and visible range [5]. Both polar and non-polar base fluids have been tested for the dispersion of graphene-based particles, either pristine graphene or graphene-oxide [24–30]. Low graphene particle concentrations, between 0.0005 % and 0.01 % weighted fractions, result in a strong increase in the optical absorption of the base fluid in the visible and UV range, up to extinction coefficients of 500 m⁻¹ as reported by Wang et al. [29] and Morozova et al. [31]. The maximum absorption peak obtained with these nanofluids is observed around 300 nm, intrinsic to the pristine-graphene structure. Base fluids used until today, as water or heat transfer oil, have minor influence on the optical properties in the UV-Vis range [24,27,29] and contribute dominantly to absorption in the infrared region. However, the choice of the based fluid influences the stability of the nanoparticles and thus the nanofluid optical properties. As reported by Hadadian et al. [32], Liu et al. [24] and Chen et al. [27], pristine graphene proved long-term stability in ionic liquids while graphene-oxide was preferred for water dispersions owing to the hydrophilic groups created on their surface. However, graphene-oxide dispersions have lower optical absorptions than pristine graphene, as noted by Chen et al. [27].

Stabilizing agents are commonly used to increase the dispersion stability, but their efficiency drops dramatically when the nanofluids are subjected to high temperatures [5,33,34]. Surface treatment of the dispersed particles can be used to create electrostatic repulsions between particles and avoid agglomeration, main cause of particle sedimentation. Several studies have analyzed the stability of crumpled graphene oxides in silicon oils and molten salts for medium-temperature

applications, highlighting the importance of the particle surface structure to reduce particle sedimentation [35–37]. In the case of graphene, hydroxides at the surface of the graphene nanoplatelets create repulsion forces between the particles, thus benefitting the dispersion stability [38]. With this technique, Bepete et al. [39] developed a surfactant-free graphene-water nanofluid from a KC8 graphite intercalation compound with long-term stability at ambient temperature.

The light extinction properties of nanofluid are highly dependent on the particle size distribution, which can be severely altered if agglomeration occurs during the material lifecycle. In many cases, owing to the complexity of experimental approaches, numerical studies are conducted to address the effect of the particle size distribution on the nanofluid optical properties. However, conventional numerical solutions, as the Mie theory, do not accurately reproduce the optical response of 2D particles. A comprehensive method for the characterization of graphene nanoplates dispersed in a fluid was proposed by Wojtkiewicz et al. [40]. They proved the influence of the orientation of a square-section nanoplatelet on the angular-dependent scattering efficiencies using the discrete dipole approximation (DDA). Inspired by this work, Yurkin et al. [41], developers of the Amsterdam DDA software package (ADDA), demonstrated the interest of using rectangular dipoles for computing particles with high length to thickness ratio with less computing cost. A graphene dispersion in ethanol was numerically and experimentally studied by Musikhin et al. [42]. The DDA could reproduce the normalized experimental extinction cross-section profiles by reconstructing the volume of crumpled graphene particles. Complex nanoparticle structures using Au with a multilayer graphene shell were modelized with the DDA method by Wu et al. [43], proving the capabilities of this method.

The previous overview of the scientific literature indicates a lack of research combining experimental and modeling approaches focusing on the effect of particle size distribution on the optical properties of graphene water-based dispersions. In this paper, a combined experimental and numerical analysis is proposed to provide deeper insights into the absorption and scattering behavior of graphene water dispersions. The
graphene morphology and the particle size distribution are key parameters to quantify the impact of agglomeration on the nanofluid light absorption properties. Retrieving these data from experimental measurements enables the identification of potential agglomeration, either at the manufacturing process or during the product lifecycle; and its impact on the nanofluid optical properties. Our approach consists in developing a radiative properties model for the graphene-water nanofluid with parameters inferred from optical measurements.

To this aim, the optical properties of hydroxylated nanoplatelets of graphene dispersed in water were experimentally analyzed in Section 2. Then, a Monte-Carlo ray-tracing software was used to calculate the normalized extinction coefficient of the graphene particles to support the experimental results. Subsequently, the influence of the size and shape of graphene particles on the optical response of the nanofluid was studied numerically using the Mie theory and the DDA in Section 3. The optical properties of a bimodal particle size distribution, calculated through an iterative process, have been found to accurately reproduce the properties of the graphene dispersion obtained experimentally. Finally, the nanofluid mean absorption variation with the fluid depth is studied to identify its potential for NDASC applications.

## 2. Measurement of the optical properties and validation

The aqueous dispersion analyzed in this study, ref. CW-GLB02A1 (supplied by Carbon-Waters®), is elaborated following several steps. First, fully exfoliated graphene sheets dispersed in tetrahydrofuran (THF) are obtained from a KC8 graphite intercalation compound [38]. After exposure to air within the solvent, graphene sheets are oxidized and turned into neutral graphene. The solution is immediately transferred to degassed water to avoid absorption of gases in water on the graphene surface. Consequently, OH⁻ ions are absorbed on the graphene surface providing a repulsion electrostatic interaction between graphene sheets. As a result, long-term stability graphene aqueous dispersion is obtained with negative zeta potentials at neutral pH [44]. Raman spectra analysis present a narrow 2D band, characteristic of single layer graphene sheets [39]. Data on the graphene dispersion are available in Table 1.

In the next section, the optical properties of this aqueous dispersion are experimentally studied.

<caption>**Table 1**</caption>

Technical data of the graphene aqueous dispersion supplied by Carbon Waters.

<table><caption>**Table 2**</caption><thead><tr><th></th><th>CW-GLB02A1</th></tr></thead><tbody><tr><td>Nanoparticle concentration</td><td>1 g/L</td></tr><tr><td>Lateral flake size</td><td>0.1–6 µm</td></tr><tr><td>Flake thickness</td><td>1–3 nm</td></tr><tr><td>Solution viscosity</td><td>0.05–0.2 Pa.s</td></tr><tr><td>pH</td><td>7.5–8.5</td></tr></tbody></table>

Technical information on the spectrophotometer and cuvettes used in the study.

<table><thead><tr><th>Perkin Elmer Lambda950</th><th></th><th>Cuvettes Perkin Elmer (ref. B0631015)</th><th></th></tr></thead><tbody><tr><td>Configuration</td><td>Transmittance or Reflectance</td><td>Cuvette</td><td>NIR SUPRASIL 300</td></tr><tr><td>Measurement range</td><td>250–2500 nm</td><td>Shape</td><td>Rectangular</td></tr><tr><td>Measurement step</td><td>10 nm</td><td>Material</td><td>Quartz</td></tr><tr><td>Uncertainty</td><td>± 1.4 %</td><td>Dimensions</td><td>45 × 12.5 × 12.5 mm</td></tr><tr><td>Reference</td><td>Spectralon 99 %</td><td>Light path</td><td>10 mm</td></tr><tr><td>Incident angle</td><td>Normal to cuvette</td><td>Volume</td><td>3.5 mL</td></tr><tr><td>Beam dimensions</td><td>4 × 12 mm</td><td>Mean transmittance</td><td>87.5 %</td></tr><tr><td>Accessories</td><td></td><td></td><td></td></tr><tr><td colspan="4">Perkin Elmer 150 mm diffuse reflectance and transmittance integrating sphere (ref. L6050011)</td></tr><tr><td colspan="4">25 mm diameter reflectance window – 16 × 25 mm oblong transmittance window</td></tr><tr><td colspan="4">Cuvette Sample Holder for the 150 mm Integrating Spheres (ref. C6951019)</td></tr><tr><td colspan="4">8.8 × 25 mm oblong transmittance window</td></tr></tbody></table>

### 2.1. Methodology

A Perkin Elmer Lambda 950 spectrophotometer was used to measure the optical properties of several nanofluid dilutions. The reflectance and transmittance of the cuvette with and without the suspension were measured. Quartz cuvettes were used with a light path of 10 mm and transmittances above 84 % in the total spectrum of the spectrometer, 250–2500 nm (Table 2). The Perkin Elmer diffuse reflectance and transmittance integrating sphere (150 mm I.D.) was incorporated to determine the hemispherical reflectance and transmittance of the samples. Depending on the sample positioning, three measurements were conducted for all samples to determine the following properties: the hemispherical transmittance ($\tau^{\uparrow}$), the directional transmittance ($\tau_{d,}$) and the hemispherical reflectance ($\rho^{\uparrow}$) (Fig. 1). $\tau_{d,}$ was obtained for a distance between the back wall of the cuvette and the sphere aperture of $d_r = 56$ cm.

To calculate the total uncertainty of the Lambda950 spectrophotometer, two uncertainty evaluations were performed. First, a 10-measurement series statistical analysis of the reflectance of the Spectralon 99 % standard was conducted (uncertainty of type A), resulting in 0.14 % standard deviation. Second, the measurement error of the machine, which is highly dependent on the specific measurement conditions, was assessed (uncertainty of type B). This was achieved by comparing the measured hemispherical reflectance ($\rho^{\uparrow}$) of the Spectralon 99 % with the expected calibration values from its hemispherical reflectance calibration certificate, resulting in a standard deviation of 0.694 %. The total uncertainty was calculated by combining the type A and B uncertainties in quadrature including a 95 % confidence level (Student's coefficient $k = 2$). The resulting uncertainty is presented in Table 2.

The hemispherical transmittance ($\tau^{\uparrow}$) evaluates the integral over the front hemisphere of the fraction of the incident beam transmitted in the forward direction. It can be decomposed as the sum of the directional transmittance, $\tau_{d,}$, and the diffuse transmittance (the scattered fraction towards the front hemisphere), $\sigma_{f}^{\uparrow}$, that can be deduced by the measurements and Eq. 1.

$$ \tau^{\uparrow} = \tau_{d,} + \sigma_{f}^{\uparrow} \quad (1) $$

The hemispherical reflectance ($\rho^{\uparrow}$) is the integral over the rear hemisphere of the fraction of the incident beam reflected and scattered in the backward directions. The energy balance of the sample results, where $\alpha$ is the absorption, is given by:

$$ 1 = \tau^{\uparrow} + \rho^{\uparrow} + \alpha \quad (2) $$

The Beer-Lambert's law relates the transmittance, $\tau_{d,}$, of a medium with an extinction coefficient ($\beta$) for the light path (L) (Eq. 3). The extinction coefficient is defined as the sum of the absorption ($\kappa$) and the scattering ($\sigma$) coefficients.

$$ \tau_{d,} = \exp(-\beta \cdot L) \quad (3) $$
<figure><img src="image_3.png" /><figcaption>Fig. 1. Schematic of the three spectrophotometer configurations using the integrating sphere.</figcaption></figure>

The extinction coefficient of a nanofluid, $\beta_{nf}$, can be defined as the sum of the extinction coefficient due to the dispersed particles, $\beta_p$, and the extinction coefficient of the base fluid medium, $\beta_{med}$ (Eq. 4).

$$ \beta_{nf} = \beta_p + \beta_{med} \quad (4) $$

In this section, results on the determination of the extinction coefficient, $\beta_{nf}$, of multiple graphene-water dispersions from Carbon Waters® are presented. Both $\tau^\circ$ and $\rho^\circ$ measurements quantify the fraction of scattering involved in the light attenuation and check if the hypothesis of a non-scattering medium is satisfied. The influence of the multiple-reflections of the quartz cuvette is quantified numerically with Monte Carlo ray-tracing simulations to support the experimental measurements. These simulations aim to analyze accurately the extinction and absorption properties of water-based graphene nanofluids.

Samples with concentration from 0.1 g/L to 0.75 g/L were obtained by diluting Carbon Waters 1 g/L solution with demineralized water (Fig. 2), which uncertainties are listed in Table 3. These samples have been analyzed with Scanning Electron Microscope (SEM) to determine the graphene particle size and shape (see Figure S1 in the supplementary material). SEM images show lateral sizes of graphene particles between 200 nm and 10 µm. However, the expected low thickness of graphene particles according to the supplier datasheet is not observed (see Table 1). This discrepancy suggests that particles may have suffered from agglomeration, forming graphite nanoplates instead of single-layer graphene.

Despite this observation, the optical properties of the nanofluid samples remained constant for several months during the experimental campaign, at ambient temperature. Table 4 presents the evolution of the mean directional transmittance, $\overline{\tau_d}$, with time for two graphene

<table><caption>**Table 3**

Samples used in the experimental characterization.</caption><thead><tr><th>Samples</th><th>Graphene concentration</th><th>Uncertainty</th></tr></thead><tbody><tr><td>DW</td><td>Demineralized water</td><td>-</td></tr><tr><td>CW-0.1</td><td>0.1 g/L</td><td>± 0.021 g/L</td></tr><tr><td>CW-0.2</td><td>0.2 g/L</td><td>± 0.021 g/L</td></tr><tr><td>CW-0.5</td><td>0.5 g/L</td><td>± 0.023 g/L</td></tr><tr><td>CW-0.75</td><td>0.75 g/L</td><td>± 0.026 g/L</td></tr><tr><td>CW-1.0</td><td>1 g/L</td><td>from supplier</td></tr></tbody></table>

<table><caption>**Table 4**

Evolution of the optical properties of two graphene dispersions, 0.1 and 0.2 g/L, with time at (i) ambient temperature, and (ii) under 80 °C water bath.</caption><thead><tr><th rowspan="2">Graphene dispersion</th><th colspan="2">Ambient temperature (20 °C)</th><th colspan="2">80 °C water bath</th></tr><tr><th>Time (h)</th><th>τ<sub>d</sub></th><th>Time (h)</th><th>τ<sub>d</sub></th></tr></thead><tbody><tr><td rowspan="5">0.1 g/L</td><td>0 h</td><td>81.1 %</td><td>0 h</td><td>80.2 %</td></tr><tr><td>144 h</td><td>80.2 %</td><td>96 h</td><td>78.9 %</td></tr><tr><td>360 h</td><td>79.9 %</td><td>768 h</td><td>79.4 %</td></tr><tr><td>1848 h</td><td>79.3 %</td><td>1440 h</td><td>80.2 %</td></tr><tr><td>0 h</td><td>61.2 %</td><td>0 h</td><td>60.8 %</td></tr><tr><td rowspan="5">0.2 g/L</td><td>144 h</td><td>60.8 %</td><td>96 h</td><td>59.2 %</td></tr><tr><td>360 h</td><td>61.5 %</td><td>768 h</td><td>58.9 %</td></tr><tr><td>1848 h</td><td>60.2 %</td><td>1440 h</td><td>59.9 %</td></tr></tbody></table>

dispersions, 0.1 and 0.2 g/ and for two scenarios: (i) dispersions maintained at ambient temperature (20 °C), and (ii) dispersions maintained at 80 °C. The $\overline{\tau_d}$ is averaged between 300 – 900 nm (Eq. 16), which is the spectral range where graphene absorbs solar radiation. The results indicate that the optical properties of the dispersion remain nearly constant for several months at ambient temperature, indicating long-

<figure><img src="image_4.png" /><figcaption>Fig. 2. Dilutions from Carbon-Waters base fluid (CW1.0) for different particle concentrations. A water sample (DW) is also shown for comparison.</figcaption></figure>
<figure><img src="image_5.png" /><figcaption>Fig. 3. Transmittance spectra $\tau_d$, obtained from an empty quartz cuvette, demineralized water (DW), and the 5 CWs samples.</figcaption></figure>

term stability of the particles. Moreover, increasing the fluid temperature close to its boiling point does not influence the particles stability. Therefore, the agglomeration of the graphene sheets appreciated in the SEM images may have occurred either during the manufacturing process or in the initial stages of the material lifecycle.

## 2.2. Experimental results

Fig. 3 plots the transmittance spectra, $\tau_d$, for all samples, together with that of the empty quartz cuvette and the cuvette filled with demineralized water. In the UV-Visible, water shows higher transmittances than the empty quartz cuvette. The difference between the air refractive index ($n_{air} \approx 1$) and that of the quartz cuvette ($n_{quartz} \approx 1.5$) is higher than the difference of the quartz cuvette refractive index with water ($n_{water} \approx 1.33$). This results in a higher reflectance of the empty quartz cuvette, which is supported by the forthcoming hemispherical reflectance results, $\rho^{\perp}$. The near-infrared region (NIR), 800–2500 nm, is dominated by the absorption of demineralized water, while the graphene particles mainly affect the extinction in the UV-Visible region, below 800 nm. An extinction peak at 270 nm is found, intrinsic to graphene particles as a result of the $\pi-\pi^*$ electron transition of the carbon

<figure><img src="image_6.png" /><figcaption>Fig. 4. Forward scattering ($\sigma_f^{\perp}$) obtained from Eq. 1 measured for all graphene nanofluids dilutions studied. The spectrophotometer uncertainty is drawn as a red region between –1.4 % and 1.4 %.</figcaption></figure>

bonds. The spectral transmittance decreases significantly with the nanoparticle concentration with values below 10 % for 1.0 g/L graphene concentrations. To quantify the weight of each attenuation mechanism, the hemispherical transmittance, $\tau^{\perp}$, and reflectance, $\rho^{\perp}$, must be considered.

The difference between $\tau_d$ and $\tau^{\perp}$ enables evaluating $\sigma_f^{\perp}$ for each sample (Fig. 4). In this figure, the spectrophotometer uncertainty is emphasized as a red region between –1.4 % and + 1.4 %. All samples including demineralized water show $\sigma_f^{\perp}$ values within the spectrophotometer uncertainty above 750 nm. CW-0.1 and CW-0.2 dispersions however present an increase of $\sigma_f^{\perp}$ at low wavelengths. At low wavelengths, $\sigma_f^{\perp}$ decreases with graphene concentrations, which is attributed to higher absorption by the nanofluid. These results indicate that diffuse forward scattering is negligible compared to collimated forward scattering.

Fig. 5 presents the obtained $\rho^{\perp}$ for all samples and the empty quartz cuvette. The difference between the empty cuvette and the water sample identified in the $\tau_d$ measurements is also observed in the reflectance measurements. All samples show lower values of $\rho^{\perp}$ than the empty quartz cuvette, and these values decrease with increasing particle concentration. As is well known, water absorbs nearly all incoming radiation in the NIR region. Therefore, the 4 % reflectance found in this spectral region can be attributed to the first quartz cuvette interfaces. Hence, although $\rho^{\perp}$ values of the sample range between 4 % and 8 %, these are attributed to multiple reflections at the quartz cuvette interfaces and not to the nanofluids themselves. The $\rho^{\perp}$ data related only to the nanofluid cannot be obtained experimentally with the presented protocol, and therefore it was determined numerically using Monte Carlo ray-tracing simulations. However, the results in Fig. 5 suggest that CW samples have low backscattering compared to the total light attenuation.

The cuvette extinction contribution must be subtracted from the $\tau_d$ measurements in order to determine the actual value of $\beta$ corresponding to the graphene suspension only. A ray-tracing model of the cuvette with and without nanofluid has been developed to perform this evaluation.

## 2.3. Modeling of the experimental set-up

The Monte Carlo ray-tracing software Solstice [45] has been used for modeling the spectrophotometric measurements assuming a non-scattering media and ideal smooth surfaces. The quartz cuvette has been modeled with four parallel planes acting as interfaces between different media: air, quartz, water and each graphene nanofluid. Two

<figure><img src="image_7.png" /><figcaption>Fig. 5. Hemispherical reflectance ($\rho^{\perp}$) measured for all samples studied and that of an empty quartz cuvette.</figcaption></figure>
imaginary detectors, represented as fully transparent planes, are integrated to calculate the reflectance and transmittance of each sample. Each medium is fully characterized by its complex refractive index (m) and its extinction coefficient (β), which for quartz and water are considered equal to their respective absorption coefficient. The complex refractive index, m, of the quartz cuvette was obtained from Lemarchand et al. [46,47], while values for water from Hale and Querry [48] and Kedenburg et al. [49]. The absorption coefficient of the medium (κ<sub>med</sub>) for both quartz and water is calculated from the absorption index (k) of the complex refractive index as follows:

$$ \kappa_{med} = \frac{4\pi k_{med}}{\lambda} \qquad (5) $$

The refractive index of water is considered for the graphene suspensions. A first estimate of the extinction coefficient is obtained from the raw τ<sub>d</sub> measurements in Fig. 3. Then, an iterative process has been conducted to obtain the effective β<sub>p</sub> for each graphene particle concentration. All obtained β<sub>p</sub> can be normalized to a unique variable, β'<sub>p</sub>, dividing it by the particle concentration, C<sub>p</sub>, in g/L:

$$ \beta'_{p} = \frac{\beta_{p}}{C_{p}} \qquad (6) $$

Fig. 6 presents the obtained average β'<sub>p</sub> (L·m<sup>-1</sup>·g<sup>-1</sup>) profile for the graphene contribution, independently of the particle concentration. As water fully absorbs the incoming radiation in the IR region, the particle β<sub>p</sub> is only calculated from 250 nm to 1350 nm. Beyond the extinction peak at 270 nm, graphene particles show a gray body behavior between 700 and 1350 nm. All samples show identical β'<sub>p</sub> meaning a linear increase of the extinction coefficient with the particle concentration. The obtained β'<sub>p</sub> is used for the ray-tracing model calculations and compared with the experimental results above described.

The extinction coefficient of each nanofluid, β<sub>nf</sub>, can be obtained from the addition of the graphene β'<sub>p</sub> and the water absorption coefficient, as water is assumed a non-scattering medium in the considered spectral range:

$$ \beta_{nf} = \beta'_{p} C_{p} + \frac{4\pi k_{water}}{\lambda} \qquad (7) $$

Ray-tracing results are then compared to the spectrophotometric measurements. The deviation of the numerical results with respect to the experimental data is plotted in Figure S2 in the supplementary information. Deviations observed for τ<sub>d</sub> and ρ<sup>+</sup> are within the uncertainty of the spectrophotometer measurements. Therefore, the identified

<figure><img src="image_8.png" /><figcaption>Fig. 6. Average normalized extinction coefficient (β'<sub>p</sub>) obtained from all graphene-dispersion samples.</figcaption></figure>

graphene particle β'<sub>p</sub> will be considered for the rest of the analysis.

The experimental results show low diffuse scattering by the graphene dispersions but high extinction properties in the entire solar spectrum. The limits of the experimental procedure were overcome with a Monte Carlo ray-tracing model, which enabled the determination of the extinction coefficient of the graphene particles. Nonetheless, the implemented numerical approach did not account for the scattering behavior of the media, justifying the need for a more detailed analysis of the particle directional scattering. Furthermore, the effect of the particle size and shape has not been assessed experimentally due to nanofluid manufacturing limitations. The numerical study in Section 3 calculates the particles κ, σ and β coefficients for a deeper understanding of the influence of particles size and shape on the absorption and directional scattering behavior of graphene-water dispersions.

### 3. Modeling of the optical properties of the graphene-water dispersion

Several theories are employed to study the interaction between an electromagnetic wave and particles dispersed in a medium. Most of them focus on spherical particles owing to their complete spatial symmetry. Non-spherical particles such as nanoplatelets or nanotubes are more complex to model. However, as discussed by Howell et al. [50] and Van de Hulst [51], the approaches and assumptions for spheres can be applied to arbitrary shaped particles for characteristic lengths below the considered wavelength. For particle lengths equal or larger than the wavelength, several approaches have been proposed to account for the non-symmetry of non-spherical particles, for example the T-matrix method [52]. The three models mainly used for the calculation of the optical properties of carbonaceous particles are: the Mie theory, the Rayleigh-Debye-Gans approximation (RDG), and the DDA [6]. However, the RDG approximation is limited to particle sizes significantly smaller than the wavelength, and thus cannot be used for the presented graphene particles.

The ability of both the Mie theory and the DDA to reproduce the experimental optical properties of a graphene dispersion was investigated for particles with characteristics lengths comparable to the wavelength. Further, the above presented radiative transport properties of the graphene-water dispersions were compared to model results obtained with a bimodal particle size distribution. The scattering effects on light attenuation were simultaneously evaluated in depth.

#### 3.1. Methodology

The scope of application of each theory is related to the ratio between the effective particle radius (r<sub>p</sub>) and the interacting wavelength (λ). The particle size parameter, x in Eq. 8, determines the possibility of using either the Rayleigh approximation (x << 1), the Mie theory, x ≈ 1, or geometric optics (x >> 1) [53]. The SEM images previously presented show that the graphene particles have a characteristic size close to the wavelength, therefore guiding towards numerical approaches valid at x ≈ 1.

$$ x = \frac{2 \pi r_p}{\lambda} \qquad (8) $$

The complex refractive index of a particle, m<sub>p</sub> = n<sub>p</sub> + i·k<sub>p</sub>, is intrinsic to its chemical composition and wavelength dependent. However, the medium in which the particle is immersed must be taken into consideration by introducing the medium refractive index (n<sub>med,λ</sub>). Consequently, the relative complex refractive index of the solution (m<sub>λ</sub>) is obtained with:

$$ m_{\lambda} = \frac{n_{p,\lambda} + i k_{p,\lambda}}{n_{med,\lambda}} \qquad (9) $$

The complex refractive index (m<sub>p</sub>) of single-layer graphene has been
measured by several authors using spectroscopic ellipsometry [54–56]. However, fully single-layer graphene (between 0.3 and 0.4 nm) is rarely found in experimental nanofluids, as generally their thicknesses vary between 1 and 8 nm in the best cases [7,25,27,57]. As the number of layers increases, particles behave like graphite nanoplatelets rather than single-layer graphene [58]. In this study, graphene optical properties are considered for particles with thicknesses below 100 nm, while graphite properties are assumed when the thickness exceeds this threshold. The data from Kravets et al. [55] and Querry [59] (Dixon KS-2) are used for the values of $m_p$ of graphene and graphite particles respectively (Figure S3, supplementary information).

For spherical particles with dimensions close to the wavelength, the Mie theory allows calculating the scattering and extinction cross sections for a given wavelength. For further details, please refer to the methods in Van de Hulst [51], Modest [53], or Bohren and Huffman [60] radiative heat transfer books. Bohren and Huffman [60] developed the BHMIE code in Fortran to compute the extinction and scattering efficiencies from the Mie theory, which implementation to MATLAB® has been used in this study.

The DDA was first introduced by Purcell and Pennypecker [61] to evaluate the scattering and absorption of non-spherical dielectric grains. The concept consists in determining the electromagnetic field inside the particle by decomposing it into a finite number of discrete dipoles. A particle of arbitrary shape consists of an assembly of millions of atoms, an extremely large and unmanageable problem. Consequently, discretization into a significantly coarser array of many N elements, each containing several atoms, is necessary. These elements, referred to as dipoles, must remain sufficiently small to be accurately represented as dipole oscillators [53]. Each dipole is fully characterized by its complex polarizability, $\alpha_\lambda$, reproducing the bulk relative permittivity (or dielectric constant) of the material, $\epsilon_r$ [61]. Both Yurkin and Hoekstra [62] and Draine and Flatau [63] reviews offer a clear description of the methodology and general equations used in the DDA. The Amsterdam DDA (ADDA) code developed by Yurkin and Hoekstra [64] was used in this work.

Once the particle cross sections are calculated, either with the Mie theory or the DDA, the extinction coefficient ($\beta_{pcloud,\lambda}$) of a homogeneous cloud of uniform size particles, considering independent scattering (no interactions between particles) [65], can be obtained with the following equation [53]:

$$\beta_{pcloud,\lambda} = N_T C_{ext,p,\lambda} \qquad (10)$$

where $C_{ext,p}$ is the extinction cross-section of the particle. $N_T$ is the number of particles per unit volume, which is calculated as follows:

$$N_T = \frac{f_v}{V_p} \qquad (11)$$

with $f_v$ the particle volume fraction and $V_p$ the particle volume. The particle volume fraction is obtained from the particle concentration, $C_p$ (g/L), divided by the particle density, $\rho_p$, which is considered 2270 g/L for graphene and graphite particles. Eq. 10 can be used to calculate the scattering, $\sigma_{p,\lambda}$, and absorption, $\kappa_{p,\lambda}$, coefficients with $C_{sca,p,\lambda}$ and $C_{abs,p,\lambda}$ respectively. Obtained $\beta_{pcloud,\lambda}$ for each graphene concentration can be normalized to a unique variable, $\beta_{pcloud,\lambda}$, using the ratio defined in Eq. 6.

Simultaneously, the extinction coefficient of the base fluid, $\beta_{med,\lambda}$, can be calculated from the index of absorption ($k_{med}$) of its complex refractive index ($m_{med} = n_{med} + ik_{med}$) with Eq. 5. Finally, the nanofluid extinction coefficient, $\beta_{nf,\lambda}$, for a given particle concentration is obtained from the addition of both particle and medium extinction coefficients as in Eq. 7.

## 3.2. Comparison of radiative property models

A numerical comparison between the two models, Mie and DDA, was

conducted to assess the ability of the DDA to replicate the extinction cross-sections of spherical particles (see Annex 4 in the supplementary information). Nonetheless, graphene particles are 2D sheets with high length-to-thickness aspect ratios. Here, graphene sheets are modeled as square-section prisms, for which the particle length ($l_{p,prism}$) and thickness ($t_{p,prism}$) are the two characteristic dimensions. The aspect ratio, noted $\xi$, is defined as the ratio between $l_{p,prism}$ and $t_{p,prism}$:

$$\xi = \frac{l_{p,prism}}{t_{p,prism}} \qquad (12)$$

The high $\xi$ ratio of graphene sheets leads to important differences compared to the assumptions made in sphere-based calculation models. For a deep understanding of the difference between the particle shape assumption and its influence on the optical behavior, the $C_{ext}$ of a prism particle modeled with the DDA and a volume-equivalent sphere has been compared. For the spherical particles, the volume-equivalent spherical radius ($r_{p,sphere}$) is obtained from the relation in Eq. 13.

$$V_{p,prism} = V_{p,sphere} \Leftrightarrow l_{p,prism}^2 t_{p,prism} = \frac{4}{3} \pi r_{p,sphere}^3 \qquad (13)$$

Prism particles $C_{ext}$ are averaged through all possible particle orientations with respect to the incident beam direction, as they do not present complete symmetry as spherical particles. In Fig. 7, the $C_{ext}$ of a 100 nm $r_{p,sphere}$ sphere is compared to that of volume-equivalent prism particles of 4 nm thickness. The calculation set-up is summarized in Table S1 (supplementary information).

Fig. 7 underlines that the sphere-based approach underestimates $C_{ext}$ of a prism-shaped graphene particle, with a constant difference of 0.02 µm². In addition, the number of dipoles constituting the prism has a noticeable influence on the optical response of the particle. If one dipole along the particle thickness is used (90 dipoles in total), the DDA validity criterion which requires the distance between dipoles to be smaller than any structural dimension of the particle is not satisfied. Conversely, if 4 dipoles along thickness are modeled (361 dipoles in total), the dipole count remains too close to the number of atoms, given that single-layer graphene is only 1 atom thick (0.35 nm particle thickness). Therefore, a balance between the number of dipoles along thickness must be found to satisfy both main validity criteria of the DDA. In this study, dipoles were chosen to have a minimum size of 3 nm, representing at least 10 carbon atoms layers. Nevertheless, the dipole number along x- and z-axes has a smaller influence on $C_{ext}$ than the overall particle shape, as highlighted by the difference between the Mie and DDA models in Fig. 7.

Fig. 7 highlights the deviation between the Mie theory and DDA in

<figure><img src="image_9.png" /><figcaption>Fig. 7. $C_{ext}$ of a 100 nm $r_{p,sphere}$ spherical particle compared to a volume equivalent square-section prism of 4 nm thickness. The prism is modeled using the DDA with three different degrees of dipole discretization.</figcaption></figure>
<figure><img src="image_10.png" /><figcaption>**Fig. 8.** *C*<sub>ext</sub> obtained with the DDA normalized with Mie volume-equivalent results as a function of the prism particle shape ratio.</figcaption></figure>

modeling a prism particle with a fixed high aspect ratio, ξ. However, conventional sphere-based models might reproduce accurately the extinction behavior of prism particles for low values of ξ, as the prism shape approaches that of a sphere. Fig. 8 compares C<sub>ext</sub> obtained from Mie and DDA calculations for 2 µm square-section prisms of varying thicknesses as a function of the aspect ratio ξ. As ξ approaches unity, the optical properties of non-spherical particles can be accurately modeled using conventional sphere-based approaches. In contrast, the Mie theory leads to significant underestimations of the extinction capacity of prism nanoparticles with high aspect ratios. Sphere-based models can provide an accurate estimation of the optical properties of prism-shaped nano-particles for aspect ratios below 2.

In Fig. 9, the extinction cross section, C<sub>ext</sub>, calculated using the Mie theory and DDA is compared for three different cubes, considering the graphite properties of Dixon KS-2 from Querry et al. [59] in water. Both models result in similar C<sub>ext</sub> values across the entire spectrum, with a slight increase in the absolute difference at longer wavelengths (insert in Fig. 9). Nevertheless, the relative difference remains negligible, with DDA/Mie ratios ranging between 1.01 and 1.09. The computational cost

<figure><img src="image_11.png" /><figcaption>**Fig. 9.** Extinction cross section, C<sub>ext</sub>, for 3 cubical particles modelized with the DDA compared to volume-equivalent spherical particles obtained from the Mie theory. **Insert:** DDA normalized results with the Mie theory results.</figcaption></figure>

of the DDA model increases exponentially with cube size, whereas C<sub>ext</sub> is computed instantaneously using the Mie theory. Therefore, the Mie theory approach will be privileged for modeling the optical properties of prism particles with ξ ratios below 2.

### 3.3. Comparison between numerical and experimental results

The experimental results demonstrate that graphene nanofluids exhibit high absorption properties and low scattering in directions different of the incident beam. However, due to experimental limitations, scattering in directions close to the incidence beam (either forward or backward) cannot be directly measured, affecting the accurate determination of the sample extinction coefficient. To address this issue, a model of the absorption and scattering properties of graphene nanoplatelets has been developed. The following parametric study aims to guide the optimization of particle concentrations by considering particle size and shape, which is crucial for enhancing the performance of these nanofluids in future applications.

The technical data provided by Carbon Waters® describes graphene nanoplatelets close to single-layer graphene (1–3 nm thickness) with lateral sizes close to the micrometer (0.1–6 µm). However, SEM images reveal the presence of thicker particles. Therefore, DDA simulations have been performed representing graphene particles as square-section prisms of 2 × 2 µm² with various thicknesses: 3, 50, 100, 1000, and 2000 nm (Fig. 10). All calculations have been conducted using particle orientation averaging, following the methodology detailed in Table S1. In addition, graphene properties has been assigned to particles with thicknesses of 100 nm or less (from Kravets et al. [55]), while graphite properties have been attributed to particles over 100 nm thicknesses (from Querry et al. [59]). The $\beta'_{pcloud-\lambda}$ for each particle cloud is illustrated in Fig. 10. Additionally, the normalized values of these parameters, scaled relative to the maximum value within each respective particle cloud, are shown in the insert of Fig. 10.

Higher extinction coefficients are obtained for thin particles compared to thick ones. Indeed, bigger particles result in a significantly lower number of scatterers, thereby reducing the particle cloud extinction coefficient. However, as particle thickness increases, the absorption peak at 270 nm intrinsic to graphene particles, is lost, as shown in the inset of Fig. 10. Compared to the experimental data profile, the presence of thin particles in the dispersion is necessary to explain the extinction peak at low wavelengths. Nonetheless, the calculated values of $\beta'_{pcloud-\lambda}$ for thin particles (3 nm thick) are significantly higher than the

<figure><img src="image_12.png" /><figcaption>**Fig. 10.** Normalized extinction of a 0.1 g/L mono-dispersion with square-section prism particles obtained from DDA. **Insert:** Normalized results with maximum value as a function of wavelength.</figcaption></figure>
<figure><img src="image_13.png" /><figcaption>**Fig. 11.** Normalized absorption and scattering coefficients calculated using DDA for a cloud of thin prism-shaped particles with: (a) a thickness $t_{p,prism}$ of 3 nm and (b) a length $l_{p,prism}$ of 50 nm.</figcaption></figure>

<figure><img src="image_14.png" /></figure>

experimental $\beta'_p$ data. The optical properties of both thin and thick particles suggest that a unimodal particle distribution may fail to accurately replicate the experimental measurements. A deeper understanding of the scattering behavior for both thick and thin particles may provide further insights into the particle present in the studied dispersion and improve the alignment of the model with the experimental $\beta'_p$ results.

The absorption and scattering coefficients, $\kappa'_{p,cloud}$ and $\sigma'_{p,cloud}$, of various square-section prism particle clouds, considering thin graphene nano-platelets, have been calculated with the DDA and are plotted in Fig. 11. For a fixed particle thickness of 3 nm (Fig. 11.a), increasing the particle length, $l_{p,prism}$, leads to an increase in $\sigma'_{p,cloud}$ at short wavelengths. Simultaneously, the $\kappa'_{p,cloud}$ peak below 300 nm decreases significantly as $l_{p,prism}$ increases, with a slight shift in the absorption peak from 260 nm to 280 nm for particle lengths of 25 nm and 100 nm, respectively. At longer wavelengths, $\sigma'_{p,cloud}$ remains constant with increasing $l_{p,prism}$, while $\kappa'_{p,cloud}$ increases slightly.

For a fixed particle length of 50 nm (Fig. 11.b), an increase in particle thickness, $t_{p,prism}$, leads to similar trends in $\sigma'_{p,cloud}$ as observed in previous figure, at both short and long wavelengths. Regarding the absorption behavior, as $t_{p,prism}$ increases, $\kappa'_{p,cloud}$ decreases across the entire spectral range, with a shift in the absorption peak position from 270 nm to 260 nm for thicknesses of 3 nm and 9 nm respectively. This reduction in the extinction peak is attributed to the increase in particle volume, which results in a lower number of scatterers in the dispersion.

Both figures indicate that absorption dominates for small particles, whereas scattering becomes more significant as either $t_{p,prism}$ or $l_{p,prism}$ increases. Fig. 11.a highlights the influence of $l_{p,prism}$ on the shape of the $\kappa'_{p,cloud}$ curve, which may guide in the identification of particle composing the bimodal particle distribution for experimental comparison.

For large particles, the previous model comparisons validated the use of Mie theory, which is computationally more efficient than the DDA method. Fig. 12 plots the spectral $\kappa'_{p,cloud,\lambda}$, and $\sigma'_{p,cloud,\lambda}$ of graphite spheres with $r_p$ of 3.72, 4.96 and 6.20 µm. These sizes correspond to volume-equivalent cubes of 6, 8, and 10 µm, respectively. As previously noted in Fig. 10, the absorption peak at 270 nm, characteristic of small graphene particles, is no longer observed for particles of this size. Another notable difference compared to the previously analyzed nanoplatelets is that $\sigma'_{p,cloud}$ plays a more significant role in light attenuation for micro-metric graphite particles. Similar tendencies are found for all $r_p$ cases in Fig. 12, with both $\kappa'_{p,cloud,\lambda}$, and $\sigma'_{p,cloud,\lambda}$ decreasing as particle volume

<figure><img src="image_15.png" /><figcaption>**Fig. 12.** Normalized absorption and scattering coefficients calculated with Mie of a particle cloud of various graphite spheres.</figcaption></figure>

increases due to the reduced number of scatterers in the dispersion.

Regarding the scattering phase function, the asymmetry parameter $g$ ranges from 0.87 to 0.93 across the entire spectrum. These results indicate that most of the scattered light is directed forward along the incident light direction. Such behavior is characteristic of large particles relative to the wavelength [60].

As pointed out previously, a unimodal particle distribution cannot replicate the experimental $\beta'_p$ values. Bimodal particle distributions are frequently observed in nanoparticle dispersions due to the agglomeration of small particles during the manufacturing process. While such agglomeration does not significantly affect the stability of the dispersion, it can have a notable impact on its optical properties. Multiple particle size distributions can describe the experimental results of Carbon Waters® dilutions. Based on the parametric study presented in Fig. 11 and Fig. 12, the bimodal particle distributions investigated consists of 50x50x9 nm³ graphene sheets ($P_{small}$) and 6.2 µm $r_p$ graphite spheres ($P_{large}$). By varying the fraction of the total $f_v$ of each particle type, different $\beta'_p$ values can be obtained, as shown in Fig. 13.a. Small particles significantly contribute to light extinction in the 250–400 nm
<figure><img src="image_16.png" /><figcaption>Fig. 13. Normalized extinction coefficient, $\beta_p$, of (a) various bi-modal distributions composed of 50x50x9 nm³ nanoplatelets and 6.2 µm $r_p$ spheres compared to the experimental results measured; and (b) the selected bi-modal particle distribution together with the contribution of each type of particle.</figcaption></figure>

<figure><img src="image_17.png" /></figure>

range compared to larger particles, with values of 2 orders of magnitude higher than measured data. Consequently, to replicate the experimental results, the bimodal particle distribution must be predominantly composed of large particles.

The bimodal particle distribution that best reproduces the experimental data consists of 50x50x9 nm³ graphene sheets (1.35 % of the total $f_v$) and 6.2 µm $r_p$ graphite spheres (98.65 % of the global $f_v$), as shown in Fig. 13.b. The contribution of each particle composing the bimodal distribution is also presented with their respective $f_v$. Despite their low volume fraction, small particles significantly contribute to light extinction and replicate the experimentally observed absorption peak at 270 nm. In contrast, large particles contribute nearly equally to the extinction properties in the whole spectrum range but account for almost the entire particle volume fraction. The proposed bimodal particle distribution accurately predicts the measured $\beta_p$ data. It should be noted that the presented bimodal distribution represents one of multiple possible bimodal configurations. Further experimental imaging techniques (SEM or TEM) are required to precisely characterize the shape and size distribution of the particles in the dispersion.

Both Mie and ADDA models enable calculating the light scattered away from its original direction via the scattering phase function, $\Phi_p(\theta, \varphi)$, of the particle [53]. The calculation of $\Phi_p(\theta, \varphi)$ for each particle is well described in the literature and therefore will not be detailed here [51,53, 60]. For the calculation of the bimodal dispersion phase function, $\Phi_{nf}(\theta, \varphi)$, the following expression is used based on the corresponding phase functions of each particle forming the dispersion:

$$ \Phi_{nf}(\theta, \varphi) = \frac{\Phi_{p,1}(\theta, \varphi)C_{sca,1}N_{T,1} + \Phi_{p,2}(\theta, \varphi)C_{sca,2}N_{T,2}}{C_{sca,1}N_{T,1} + C_{sca,2}N_{T,2}} \quad (14) $$

with $C_{sca}$ the scattering cross-section, and subscripts 1 and 2 referring to the small and large particles respectively. The number of particles per volume, $N_{T,i}$, for each particle cloud (i) is calculated with Eq. 15, which is based on Eq. 11:

$$ N_{T,i} = \frac{f_v}{V_{p,i}}\chi_i \quad (15) $$

with $\chi_i$ the percentage of the total $f_v$ for each particle. In our case, $\chi_1 = 1.35\%$ and $\chi_2 = 98.65\%$ correspond to the small and large particles respectively. $\Phi_{nf}(\theta, \varphi)$ is then normalized to $\int_{4\pi} \Phi_{nf}(\theta, \varphi)d\Omega = 1$.

The calculated phase function for $\lambda = 600$ nm is represented in

<figure><img src="image_18.png" /><figcaption>Fig. 14. Scattering phase function of a bimodal particle distribution composed of 50x50x9 nm³ nanoplatelets (1.35 % of $f_v$) and 6.2 µm $r_p$ spheres (98.65 % of $f_v$) at 600 nm wavelength.</figcaption></figure>

Fig. 14 together with the corresponding asymmetry parameter, $g$. The bimodal dispersion shows a strong forward scattering behavior in the direction of the incident beam. Even if small particles have an isotropic scattering, the weight of scattering is significantly low compared to absorption and extinction, as pointed in Fig. 11. Therefore, the scattering behavior of the bimodal particle distribution is dominated by the larger particles, which manifest a strong forward scattering.

Even if forward scattering dominates, the phase function is not sufficiently peaked in the forward direction to consider that the majority of the scattered radiation is collected in the $\tau_d$ measurement. Therefore, the assumption of one-dimensional radiative transfer used to identify the extinction coefficient with Eq. 3 must be validated with spectral and directional measurements to identify more accurately the nanofluid radiative properties.

### 3.4. Absorption properties of the graphene dispersion

The absorption potential of the presented nanofluids is a key parameter for future applications. For this purpose, the weighted spec-
tral average extinction coefficient ($\overline{\beta}_{nf}$) of several graphene nanofluids with different particle concentrations is calculated with Eq. 16.

$$ \overline{\beta}_{nf} = \frac{\sum_{\lambda=900\text{nm}}^{\lambda=900\text{nm}} \beta_{nf,\lambda} \cdot I_{\text{AM1.5,\lambda}}}{\sum_{300\text{nm}}^{\lambda=900\text{nm}} I_{\text{AM1.5,\lambda}}} \quad (16) $$

with $I_{\text{AM1.5,\lambda}}$ the global incident radiation assuming a AM1.5 from ASTM G173–03 [66]. Each nanofluid extinction coefficient, $\beta_{nf,\lambda}$, is calculated with Eq. 7 using the $\beta'_{p,\lambda}$ obtained with the bimodal particle distribution numerical results. Assuming that the scattered radiation is totally absorbed in the medium, the Beer-Lambert law is used to approximate the mean absorption ($\overline{\alpha}_{nf}$) of each nanofluid as a function of the optical path length, $L$:

$$ \overline{\alpha}_{nf} = 1 - \exp(-\overline{\beta}_{nf}L) \quad (17) $$

The mean absorption ($\overline{\alpha}_{nf}$) for several graphene particle concentrations over optical paths up to 70 mm is plotted in Fig. 15, alongside experimental data for comparison. Notably, the mean absorption fraction increases non-linearly with both the optical path length and particle concentration, in agreement with the logarithmic nature of the Beer-Lambert’s law.

As an example, parabolic trough solar collectors for large-scale electricity production have linear tubular receivers with inner diameters of 70 mm and a 96 % mean surface coating absorption [67]. In this dimension range, graphene particle concentrations of 0.2 g/L would generate similar absorption fractions, but would require high volumes of nanofluid, which might entail important economic constraints. On the other hand, small and medium size commercial parabolic trough collectors for heat generation have receiver inner diameters of around 30 mm with similar surface absorption properties [67]. In this case, only graphene concentrations of 0.4 g/L or higher can reproduce the 96 % absorption but lower nanofluid volumes will be needed. A compromise between particle concentration and fluid volume, and thus the optical path, must be identified to find the optimized NDASC for the desired application.

## 4. Discussion

The numerical analysis has shown that the thickness of graphene nanoplatelets is the main parameter affecting the light absorption properties of graphene dispersions. The size of particles present in the

dispersion can be deduced from the results of the optical properties model. In particular, since a high absorption peak at 270 nm is observed, thin graphene particles must be present in the dispersion, as large particles do not exhibit this optical behavior. Even if thin graphene particles are used in the manufacturing process, agglomeration can lead to large carbon particles. These changes in the particle morphology can dramatically alter the optical properties of the nanofluid dispersions. In some cases, particle agglomeration does not affect the stability of the particles and may thus be difficult to detect.

Thin graphene nanoplatelets below 10–20 nm with high aspect ratios, $\xi$, should be preferred for tuning the optical absorption properties of the base fluid. A higher number of carbon layers do not only reduce the extinction capacity but also increases the scattering effects, further reducing the absorption properties of the dispersion. High aspect ratio graphene particles, however, need to be modelized with non-spherical numerical models as the DDA, since sphere-based models as the Mie theory significantly underestimate the particles absorption capacity. For NDASC applications, the absorption peak at 270 nm found for thin graphene particles (below 20 nm) has a small effect on the absorption of the incident solar spectrum, as the main solar irradiance starts at approximately 320–350 nm. The optical optimization must focus on increasing the absorption above this range, which may require increasing the particle length (see Fig. 11.a).

While optical absorption optimization is an important parameter in the design of nanofluid dispersions, particle stability is the key parameter that limits its practical use. An optimized, high absorbing nanofluid has no potential for industrial applications if particle agglomeration and sedimentation occur within hours. The manufacturing process must ensure constant optical and thermal properties during the material lifecycle, which can only be achieved with stable nanofluids. Therefore, the particles’ stability needs to be further evaluated under real-world conditions, such as high temperatures, pressures, and high solar irradiation for NDASC utilization.

The graphene morphology in the dispersion was expected to be within the nanometer scale. However, SEM images and DDA evaluation showed that micrometric particles dominated the particle distribution in the nanofluid, likely due to small agglomerations that formed early in the material lifecycle. As a result, the absorption capacity of the dispersion was significantly reduced. Nevertheless, the optical properties of the graphene dispersion were constant for several months during the experimental campaigns at ambient temperature. As shown in Fig. 15, the nanofluid is able to absorb over 95 % of the incident radiation even if the optical absorption properties did not meet initial expectations. This analysis highlights the importance of achieving stable dispersions, rather than focusing on highly absorbing dispersions. In some cases, both objectives must be pursued, as larger particles may be more prone to cause sedimentation.

## 5. Conclusions and perspectives

This study contributes to the understanding of the optical behavior of graphene nanoparticles dispersed in water, using both experimental and numerical approaches. The spectral properties of graphene dispersions in water (from Carbon Waters®) with concentrations from 0.1 g/L to 1 g/L have been measured experimentally. The measured optical properties were found constant for several months during the measuring campaign, indicating long term stability of the particles at ambient temperature. The measured extinction and scattering properties of the graphene-based nanofluid have been numerically corrected with a ray-tracing software (Solstice) to account for the effects of multiple reflections caused by the quartz cuvette containing the nanofluid. The measured diffuse transmittance and reflectance were found to be extremely low, falling within the uncertainties of the spectrophotometer. Therefore, assuming one dimensional radiative transfer, the extinction coefficient of the nanofluid was identified from the directional transmittance measurements.

<figure><img src="image_19.png" /><figcaption>Fig. 15. Mean absorption, $\overline{\alpha}_{nf}$, variation for different graphene concentrations as a function of the optical path. The mean absorption experimental results obtained from $\tau_d$ are also plotted as purple diamonds.</figcaption></figure>
A modeling approach was chosen to evaluate the optical response of graphene nanoplatelets based on the size and shape given by the manufacturer and visualized with SEM. First, the ability of the Mie theory to reproduce non-spherical particle properties has been compared with the DDA approach, a method typically adapted for non-spherical particles analysis. Results underlined the importance of considering prism shaped particles for thin nanoplatelets as spherical methods underestimated their extinction efficiency. However, the Mie theory was able to replicate the optical response of graphene cubical particles with acceptable accuracy. Consequently, the DDA was implemented for thin particles while the conventional Mie theory was used for larger cubical particles with volume-equivalent spheres.

Modeling results underlined the outstanding absorption properties of thin graphene nanoplatelets, with a significant absorption peak at 270 nm as expected. Larger cubical particles, however, presented a gray behavior in the whole spectrum with lower extinction coefficients. By comparison with the experimental data, a bimodal particle distribution was found to replicate the extinction properties of the samples. The bimodal distribution was composed of 50 × 50 × 9 nm³ nanoplatelets (1.35 % of the total $f_v$) and 6.2 µm radius spheres (98.65 % of the global $f_v$). Furthermore, the scattering phase function of the bimodal particle dispersion has been calculated resulting in a mainly forward scattering behavior of the nanofluid, with $g = 0.896$. Future dispersions must privilege thin graphene sheets, below 10–20 nm, to increase absorption and reduce scattering effects.

Numerical results indicated that the effect of scattering depends on the shape and size distribution of large particles. The distribution of large particles is either the result of the manufacturing process or aggregation mechanisms depending on the pH of the demineralized water used. Deeper numerical analysis must be done combining radiative transfer models (in absorbing and anisotropically scattering media) with spectral and directional measurements to identify more accurately the nanofluid radiative properties. Simultaneously, further experimental visualizations (SEM or TEM) must be performed to validate the presented bimodal particle distribution. However, the challenges found during SEM analysis must be overcome to precisely identify the size of the particles. Dynamic Light Scattering (DLS) measurements may elucidate the particle distribution functions for deeper analysis of the scattering behavior.

The presented graphene-based nanofluids demonstrate significant potential for future applications in engineering sectors such as solar energy or optical engineering. However, further research will be necessary, particularly experimental research that simulate real-world conditions, such as exposure to solar radiation, to collect data about their performance, limitations (such as corrosion, erosion or toxicity) and elaborate future development. On the other hand, the stability of the nanofluid must be further examined, by studying the influence of temperature, pH of the fluid, and other environmental factors.

### CRediT authorship contribution statement

**Miguel Sainz-Manas:** Methodology, Software, Validation, Formal Analysis, Investigation, Data curation, Writing - Original draft, Visualization. **Cyril Caliot:** Conceptualization, Software, Resources, Writing - Review & editing, Supervision. **Christophe Escape:** Validation, Investigation, Resources. **Françoise Bataille:** Writing - Review & editing, Supervision, Project administration, Funding acquisition. **Gilles Flamant:** Conceptualization, Resources, Writing - Review & editing, Supervision, Project administration, Funding acquisition.

### Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

### Acknowledgments

The Perkin Elmer Lambda 950 spectrophotometer belongs to the PCM platform at the PROMES laboratory in Odeillo. The SEM imaging was achieved by Yonko Gorand at the EnRMAT-C2M platform at the University of Perpignan Via Domitia (UPVD). This work was supported by the French "Investments for the future" ("Investissements d'Avenir") and "France 2030" programs managed by the National Agency for Research (ANR) under contract ANR-10-LABX-22-01 (labex SOLSTICE) and ANR-22-PESP-0005 (SHIP4D project), respectively.

### Appendix A. Supporting information

Supplementary data associated with this article can be found in the online version at doi:10.1016/j.nxmate.2025.100798.

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