From representative volume element of interacting particles to the extraction of their effective properties

Timothée Guerra; Domingos De Sousa Meneses; Olivier Rozenbaum; Cédric Blanchard

doi:10.1364/OE.438451

1. Introduction

This past decade has seen the emergence of new complex structures composed of nanoparticles embedded inside a host medium. These new types of materials have numerous applications. For example, in the field of optical fiber, nanoparticles can be integrated into the cladding of the fiber to create a sensor [1–3] or into the core of the fiber to enhance its performance [4]. Nanoparticles can also be used in solar cell applications by enhancing absorption [5,6], together with other applications [7,8].

In order to further optimize these complex materials (the composition of nanoparticles, their filling factor, their size $\cdots$), one needs to understand how light interacts with them. However, this interaction can be intricate and cumbersome to compute, in the sense that multiple scattering events between particles need to be taken into account in order to fully characterize the electromagnetic behavior of the mixture.

Nevertheless, since the employed nanoparticles are usually very small compared to the wavelength of the radiation, it is often possible to approximate this problem by a simpler one. Indeed, it is possible to homogenize the system using different effective medium approaches, such as Maxwell-Garnett (MG) [9], Bruggeman [10] or Landau, Lifshitz and Looyenga [11] models. Homogenization is the process by which a heterogeneous medium is replaced by its homogeneous equivalent, whose effective refractive index describes the electromagnetic response of the inhomogeneous material.

Unfortunately, the effective medium theories mentioned above are only valid for specific mixtures. For example, the MG model is only valid for spherical nanoparticles with only dipolar interaction. Moreover, it is known that the MG model becomes uncertain for high filling factors [12]. On the contrary, the Bruggeman model is usually thought to be valid for high filling factors, because it cannot predict the anomalous resonances of plasmonic mixtures with low filling factors [13]. Upon failure of these theories, one needs to tackle the full computation of the electromagnetic scattering problem.

To do so numerically, the notion of representative volume element (RVE) is key. In this sense, Kanit et al. proposed a quantitative definition of the RVE based on the following statistical argument: “the RVE must ensure a given accuracy of the estimated property obtained by spatial averaging” [14]. To elaborate its definition, he used the notion of integral range originally developed by Matheron [15] and Lantuéjoul [16] for stationary ergodic random functions (SERF). However, Kanit’s work did not consider cases in which strong interactions or even resonances occur in the mixture, and how they affect the RVE.

Therefore, the aim of this work is to investigate the notion of RVE for the electromagnetic properties of agglomerates composed of nanoparticles, in the case where strong interactions between particles exist, leading to potential resonances. More precisely, the behavior of the mean value and standard deviation of the scattering cross-section produced by different agglomerates is studied. Based on this knowledge, the objective is to derive criteria on the agglomerate size allowing homogenization for different particle permittivities.

Section 2. is a description of the system of interest. Using the notion of transverse and longitudinal optic frequencies, the particle permittivity is derived, leading to preliminary results about surface-plasmon condition in our agglomerates. In Sec. 3., the definition of coherent and incoherent responses alongside with their link to homogenization is reviewed. In order to extract the effective refractive index, an ensemble averaging over different agglomerates is used. However, if the corresponding standard deviation is too high, then ensemble mean is unreliable. To prevent this, adequate limitations on the particle permittivity are imposed.

In general, the scattering cross-section of an agglomerate is not a SERF. Thus, section 4. starts by the definition of the SERF of interest, which is the scattering efficiency factor when a constant ratio between agglomerate radius and radiation wavelength is imposed. Then, two phenomenological models based on power laws are derived, one for the efficiency factor mean value and another for its standard deviation. Using both models, criteria for homogenization are obtained.

In the final section, these newly formulated criteria are compared to the extraction of the refractive index. Both fail to predict the minimal surface needed in order to obtain a convergence of the extraction. However, it appears that the obtained minimal surface is independent of the particle permittivity imaginary part, except when surface-plasmon resonances are excited. In this particular case, decreasing the particle loss term results in a strong increase of resonances, implying that the agglomerate surface should increase in order to compensate this effect.

2. Description of the system

We are interested in studying the scattering of a plane wave, with infrared wavelength $\lambda$, from disordered agglomerates consisting of non-overlapping particles. These clusters of particles are 2D circles with radius $R$. Each particles are approximated to a circle with radius $r_p= {0.1}\,\mathrm{\mu}\textrm{m}$, permittivity $\varepsilon _p$ and vacuum permeability ($\mu _p = 1$), meaning that particles are non-magnetic. They are randomly located inside the agglomerate with a filling factor $f={15}{\%}$, the host medium being vacuum ($\varepsilon _m = 1$ and $\mu _m=1$). In 3D, this problem is equivalent to plane wave scattering by infinite cylinders. In our case, the magnetic field is parallel to the cylinder axis, equivalent to a p-polarization or transverse-magnetic polarization.

In addition, a minimal surface-to-surface distance between particles of $ {20}\,\textrm{nm}$ was imposed. It was demonstrated that imposing such condition reduces resonant effects inside the agglomerate, thus facilitating the homogenization process [17,18]. However the chosen distance being small compare to the particle radius, we should still have some resonant effects, or at least multiple scattering between particles.

Finally, in order to have a complete system, we need to define the particle permittivity $\varepsilon _p$. In this sense, let us recall that the driving idea of this investigation is to extract the effective refractive index of heterogeneous medium, in the presence of strong interaction between particles. For infrared wavelength and in the case of crystals, it was demonstrated that the corresponding spectral range is bounded by the transverse and longitudinal optic frequencies, $\omega _{TO}$ and $\omega _{LO}$ respectively, of a phonon and that surface-modes are excited in this range [19]. For an undamped phonon with no high-frequency contribution, the four-parameter dielectric function model gives for the particle dielectric function [20]:

(1)$$\varepsilon_p(\omega) = \frac{\omega_{LO}^{2} - \omega^{2}}{\omega_{TO}^{2} - \omega^{2}} = 1 - \frac{1}{{s_p}},$$

where $\omega$ is the pulsation of the incident wave and ${s_p}$ is a dimensionless parameter given by ${s_p} = (\omega _{TO}^{2} - \omega ^{2})/(\omega _{TO}^{2} - \omega _{LO}^{2})$. So for $\omega$ spaning from $\omega _{TO}$ to $\omega _{LO}$, ${s_p}$ ranges from $0$ to $1$. In order to modulate the interaction strength, it is important to allow non-radiative losses inside each particle. This can be achieved by adding an imaginary part $\varepsilon _p''$, called the loss parameter, to Eq. (1).

Having this definition in hand, we can look at some preliminary results about surface-plasmon condition. In the case of a single infinitely long cylinder with permittivity $\varepsilon _p$, embedded in a medium with dielectric function $\varepsilon _m$, the condition for long-wavelength and non-retarded surface-plasmon, when electric field is normal to the cylinder axis, is given by [21]:

(2)$$\varepsilon_p + \varepsilon_m = 0.$$

This means that for a single particle of our agglomerate, the surface-plasmon condition is given by ${s_p}=0.5$. However, an agglomerate is not composed of a single cylinder, but a forest of them with filling factor $f= {15}{\%}$. A better determination of the plasmon condition for our system is given when computing the pole of the Maxwell-Garnett formula for the effective permittivity given by [9,22,23]:

(3)$$\varepsilon_{eff}^{MG} = \varepsilon_m \frac{\varepsilon_p + g\,\varepsilon_m + g\,f(\varepsilon_p - \varepsilon_m)}{\varepsilon_p + g\,\varepsilon_m - f(\varepsilon_p - \varepsilon_m)},$$

where $g$ is the dimensional factor ($g=1$ in 2D and $g=2$ in 3D). Doing so holds ${s_p}^{r}=0.425$ as plasmon condition. Finally, we can deduce that for ${s_p}$ around ${s_p}^{r}$, surface-plasmon resonances should be observed.

3. Averaging behavior of many-particle agglomerates

Now that the system of interest is defined, the objective is to extract its effective refractive index. The intuitive approach would be to simulate the electromagnetic response of macroscopic agglomerates, typically the size of real samples, in order to compare it to the response of homogeneous agglomerates and thus deduce the effective index. However, due to computation toll, it is impossible to simulate with standard means the collective response of billions of particles. Therefore, a more realistic method is to compute the response of many smaller agglomerates, and then use their average response as a comparison quantity.

More precisely, in our case, the quantity used for comparison is the coherent far-field flux $\bar {\boldsymbol {F}}_{\boldsymbol {coh}}$ computed for a given observation angle $\theta$. This quantity is equivalent to a Poynting vector and given by:

(4)$$\bar{\boldsymbol{F}}_{\boldsymbol{coh}}(\theta) = \frac{1}{2} Re\left( \langle{\boldsymbol{E_{sc}}}\rangle \times\langle{\boldsymbol{H^{*}_{sc}}}\rangle \right),$$

where $\langle {.}\rangle$ represents an ensemble averaging, $\boldsymbol {E_{sc}}$ and $\boldsymbol {H_{sc}}$ are the scattered electric and magnetic fields, respectively, symbols $\boldsymbol {{}^{*}}$ and $\times$ correspond to the complex conjugate and vector product, respectively. Ultimately, this quantity arises from an ensemble averaging (symbolized by the bar), meaning that only a standard error of the mean can be extracted but no standard deviation. The standard error of the mean is a measure of how far the computed ensemble mean is from the exact value. Moreover, it should be noted that $\bar {\boldsymbol {F}}_{\boldsymbol {coh}}$ can be very different from the ensemble total far-field flux defined as follow:

(5)$$ \overline{\boldsymbol{F}}_{\boldsymbol{t o t}}(\theta)=\left\langle\boldsymbol{F}_{\boldsymbol{t o t}}\right\rangle=\frac{1}{2} \operatorname{Re}\left(\left\langle\boldsymbol{E}_{\boldsymbol{s c}} \times \boldsymbol{H}_{\boldsymbol{s c}}^{*}\right\rangle\right). $$

The distinction between $\bar {\boldsymbol {F}}_{\boldsymbol {tot}}$ and $\bar {\boldsymbol {F}}_{\boldsymbol {coh}}$ comes from the decomposition of the electric and magnetic fields into a coherent $\langle {\boldsymbol {A_{sc}}}\rangle$ and incoherent $\delta \boldsymbol {A_{sc}}$ parts as follow: $\boldsymbol {A_{sc}} = \langle {\boldsymbol {A_{sc}}}\rangle + \delta \boldsymbol {A_{sc}}$, where $\boldsymbol {A_{sc}}$ can be either $\boldsymbol {E_{sc}}$ or $\boldsymbol {H_{sc}}$. Using this decomposition alongside with Eq. (4) and (5), we can define an incoherent far-field flux corresponding to the scattered flux produced by the statistical fluctuation of $\boldsymbol {A_{sc}}$ about $\langle {\boldsymbol {A_{sc}}}\rangle$ [24,25]:

(6)$$ \overline{\boldsymbol{F}}_{\boldsymbol{i n c o h}}=\overline{\boldsymbol{F}}_{\boldsymbol{t o t}}-\overline{\boldsymbol{F}}_{\boldsymbol{c o h}}=\frac{1}{2} R e\left(\left\langle\delta \boldsymbol{E}_{\boldsymbol{s c}} \times \delta \boldsymbol{H}_{\boldsymbol{s c}}^{*}\right\rangle\right). $$

The incoherent part is usually thought to originate from the random positions of the scatterers in the agglomerate [17]. However, incoherence also arises from electromagnetic interaction between particles [18,26,27].

Moreover, taking into account the fact that the coherent field follows Helmholtz equation:

(7)$$\nabla\times\nabla\times \langle{\boldsymbol{A_{sc}}}\rangle - \left(\frac{\omega}{c}n_{eff}\right)^{2}\langle{\boldsymbol{A_{sc}}}\rangle = 0,$$

it should always be possible to associate an effective refractive index $n_{eff}$ with the coherent part of the electromagnetic response of an inhomogeneous medium, even in the presence of strong incoherence [17,28].

Despite that fact, homogenization procedure faces challenges. One of them is to ensure that the computation of the coherent part, which is an ensemble averaging, is reliable. Imagine a case where strong resonances occur in some agglomerates of the ensemble of realizations (see Fig. 8 of [18] for example). Due to these resonances, the electromagnetic response of these particular agglomerates will be several orders of magnitudes higher than the median of the set, leading to abnormal ensemble averaging with no physical meaning. This is a case where the coherent part cannot be extracted safely, because ensemble averaging is not consistent.

A simple but loose criteria to ensure a first approximation trustworthy ensemble averaging would be to put a condition on the relative standard deviation (rSTD) of the quantity of interest. If it is larger than a certain threshold, then the average is deemed unreliable, and in order to ensure a lower rSTD, the simulation parameters need to be changed. In our case, the quantity of interest for homogenization is $\bar {\boldsymbol {F}}_{\boldsymbol {coh}}$, from which no standard deviation can be computed. So instead, the chosen quantity is the integral of the total flux $\boldsymbol {F_{tot}}$ over all possible angle of observation $\theta$, which is nothing else than the scattering cross-section per unit length $C_{sc}^{tot}$.

In order to be as general as possible and include long range interactions, the loss parameter of the particle permittivity should be set as low as possible. However, when $\varepsilon _p''$ tends towards zero, resonances are exacerbated resulting in larger rSTD of $C_{sc}^{tot}$. This phenomenon can be observed in Fig. 1 where the relative standard deviation of $C_{sc}^{tot}$ is represented as a function of $\varepsilon _p''$ for different values of ${s_p}$. Computations were carried out by employing a T-matrix approach to the resolution of Maxwell equations [29,30]. Decreasing the loss term results in a huge increase of the rSTD, which at some point is not compatible with our simple criteria for reliable extraction of the coherent flux.

Fig. 1. Relative standard deviation of the scattering cross section as a function of the loss term, for different ${s_p}$ values and for agglomerates with radius (a) $R= {1.0}\,\mathrm{\mu}\textrm{m}$ - (b) $R= {3.0}\,\mathrm{\mu}\textrm{m}$. As the loss term is decreased the rSTD is increasing and eventually saturating. By increasing the agglomerate radius, the saturation value is decreased. Lines are guides for the eye.

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Usually, increasing the agglomerate radius results in a decrease of the rSTD of $C_{sc}^{tot}$. Going from Fig. 1(a) to (b), the agglomerate radius increased from ${1}\,\mathrm{\mu}\textrm{m}$ to $ {3}\,\mathrm{\mu}\textrm{m}$ resulting in a decrease of the rSTD by more than a factor of $3$. This reduction can be understood by considering that agglomerates with few particles are usually very different from one another regarding their spatial repartition (number and locations of particles), resulting in very large fluctuations in the electromagnetic response. On the other hand, agglomerates with large amount of particles are more similar, resulting in a set of cross-section more concentrated around the mean.

In addition, as the loss parameter drops, the number of multipolar degrees included in the computation should increase accordingly, because more and more resonant phenomena appear. For example, using an agglomerate radius of ${3}\,\mathrm{\mu}\textrm{m}$, for a particle refractive index of $n_p = 4.4.10^{-2} + 1.129i$, corresponding to ${s_p}=0.44$ and $\varepsilon _p''=10^{-1}$, around 5 degrees are needed. Whereas for $n_p = 4.4.10^{-5} + 1.128i$, corresponding to ${s_p}=0.44$ and $\varepsilon _p''=10^{-4}$, it becomes necessary to include 20 degrees.

In conclusion, in order to satisfy our simple criteria for homogenization, as $\varepsilon _p''$ decreases, both the agglomerate radius and the number of multipolar degrees used in the computation need to increase accordingly. However as both parameters rise, the computation duration also grows, up to a point where the simulation is no longer possible. This explains why for the rest of the article we had to restrict ourselves to loss parameters of $\varepsilon _p''=10^{-1}$ and $\varepsilon _p''=10^{-2}$, where computation durations are handleable. Nevertheless, it should be pointed out that the chosen loss parameters are still very small, especially when compared to losses of nanoparticles made of vitreous $SiO_2$ [31].

Moreover, four different cases for ${s_p}$ have been considered: $[0.01, 0.25, 0.44, 0.60]$. Since no strong resonances nor incoherence are expected at ${s_p}=0.01$, this point is used as a reference. As already pointed out, incoherence is related to interaction between particles which is a feature of interest for this work. Preliminary results, not published yet, pointed out that the maximum of incoherence in the range $[0, 1]$ is attained for ${s_p}=0.25$. As discussed in Section II, surface-plasmon resonances should occur around ${s_p}^{r}$, hence the choice ${s_p}=0.44$. Finally, ${s_p}=0.60$ is chosen as it is lying around the middle of the interval $[0, 1]$, implying that weak interactions between particles exist but no resonances, nor strong incoherence.

4. Characterization of the efficiency factor mean value and standard deviation

4.1 Definition of a stationary ergodic random function

The extraction of the effective refractive index is obtained by comparing the coherent flux of the mixture to the flux obtained by a homogeneous agglomerate, see [18] for a detailed explanation of the technique. As a first step towards this direction, we are interested in finding the optimal computation conditions leading to the least total flux standard deviation. In other words, given an agglomerate surface, we want the most precise value of the flux by adjusting the number of iterations. In the following, we propose to characterize the surface and the number of iterations that are required to achieve a certain degree of precision on the flux.

The determination of the representative volume element is the relevant quantity to evaluate this. It is usually understood as the minimal volume in 3D, or surface in 2D, that the simulated system should have in order to fully grasp the complexity of the entire structure [14,16,32]. The RVE depends on the physical property of interest but also on the desired precision and maximum number of iterations possible.

Using geostatistic results excavated from [15,16], Kanit prescribed that the standard deviation $D_Z$ of a stationary ergodic random function $Z$ over the surface should follow a power law. Here stationary means that two large samples should provide approximately the same result. Whereas ergodicity implies that average over time can be replaced by ensemble averaging [33]. Then using sample theory results [34], the relative error $\epsilon ^{rel}$, resulting from $n$ independent agglomerate simulations of surface $S$, is obtained using the notion of confidence interval and link to absolute error $\epsilon ^{abs}$:

(8)$$\epsilon^{rel} = \frac{\epsilon^{abs}}{\bar{Z}}=\frac{2D_Z(S)}{\bar{Z}\sqrt{n}}.$$

According to Kanit, two different visions are possible from Eq. (8). On the one hand, the surface of the agglomerate is fixed and, in order to obtain a certain relative error, the number of iterations is set according to:

(9)$$n = \left(\frac{2D_Z(S)}{\epsilon^{rel}\bar{Z}}\right)^{2}.$$

On the other hand, the number of iterations is fixed and the surface of the agglomerate is set such that a certain relative error is achieved.

In our case, the quantity from which we want to characterize the standard deviation is the total cross section for scattering per unit length $C_{sc}^{tot}$. However, this variable does not satisfy the definition of ergodicity that is found in [16], stating that the standard deviation of an ergodic function should tend towards zero as the surface increases:

(10)$$\lim_{S\to\infty} D_Z(S) = 0.$$

Usually, when the surface of heterogeneous agglomerates increases, so does the scattering cross section. This means that the corresponding standard deviation will also increase, and therefore never tends towards zero.

In order to have a decreasing standard deviation, we should rather consider the total scattering efficiency factor $Q_{sc}^{tot} = C_{sc}^{tot} / 2R$, and maintain the ratio $R/\lambda$ constant. In order to understand why such restrictions are necessary, let us consider the case of a homogeneous agglomerate with refractive index $n_c$ and radius $R$ standing in a medium with refractive index $n_m$. Using well known results of plane-wave scattering from a disk with in plane incidence and wavelength $\lambda$, the scattered coefficients are given for any $n\in \mathbb {Z}$ by [35,36]:

(11)$$a_n ={-}\dfrac{\eta_cJ'_n(n_cx)J_n(n_mx) - \eta_mJ_n(n_cx)J'_n(n_mx)}{\eta_cJ'_n(n_cx)H_n(n_mx) - \eta_mJ_n(n_cx)H'_n(n_mx)},$$

where $H_n$ and $J_n$ are respectively Hankel and Bessel functions of the first kind, $\eta _i=\sqrt {\mu _i/\varepsilon _i}$ is the impedance of the considered material, and $x$ is a nondimensional parameter given by $x=2\pi R/\lambda$. As a result, assuming $R/\lambda =C$, where $C$ is a constant, implies that $a_n$ is also a constant for an increasing agglomerate surface. Then using the definition of the $T$-factor found in [35], one can express the far-field scattered flux of the homogeneous agglomerate as

(12)$$F_{sc}(\theta, R) = \frac{\lambda}{\pi^{2}}\left|T(\theta)\right|^{2} = \frac{R}{\pi^{2}C}\left|T(\theta)\right|^{2},$$

where $T(\theta )$ is the $T$-factor which, in our specific case of $R/\lambda =C$, only depends on the angle $\theta$ at which an observer measures the scattered flux, and is given by:

(13)$$T(\theta) = \sum_{n={-}\infty}^{+\infty} a_ne^{in\theta}.$$

The scattering cross section of the agglomerate is obtained by integrating Eq. (12) over all possible angles, yielding:

(14)$$C_{sc}(R) = \frac{R}{\pi^{2}C}\int_0^{2\pi}\left|T(\theta)\right|^{2}d\theta = \frac{R}{\pi^{2}C}I,$$

where $I$ is a constant. So finally, the scattering efficiency factor is given by :

(15)$$Q_{sc} = \frac{C_{sc}(R)}{2R} = \frac{I}{2\pi^{2}C}.$$

In conclusion, for increasing surfaces such that the ratio $R/\lambda$ is kept constant, the efficiency factor of a homogeneous agglomerate is a constant.

Going back to heterogeneous media, if homogenization is possible and the surface large enough, the ensemble total scattering efficiency factor $\bar {Q}_{sc}^{tot}$ should also be constant. However, with increasing surface, agglomerates become statistically similar; accordingly the standard deviation should decrease. So by keeping the ratio $R/\lambda$ constant and using $Q_{sc}^{tot}$ as the variable $Z$, we have a stationary ergodic random function.

4.2 Computation results and discussion

The computation of $Q_{sc}^{tot}$ mean value and associated standard deviation $D_Q$ for two different loss parameters ($\varepsilon _p''=10^{-1}$ and $\varepsilon _p''=10^{-2}$) and four different ${s_p}$ values (${s_p}=0.01$, ${s_p}=0.25$, ${s_p}=0.44$ and ${s_p}=0.60$) was conducted with $R/\lambda =1/3$. Using this ratio, instead of a greater one such as $1/2$, enables us to decrease the radius of the agglomerate to very low value, e.g. ${1}\,\mathrm{\mu}\textrm{m}$, but still keeping a ratio $r_p/\lambda$ smaller than $0.05$. Furthermore, we tried using smaller ratio, such as $1/5$. However, such value leads to imprecision in the extraction of $n_{eff}$. Indeed, when $R \ll \lambda$, the scattered flux of the inhomogeneous medium has a dipole shape that is not discriminative enough to identify the correct value.

The standard deviation and mean value of $Q_{sc}^{tot}$ are plotted in Fig. 2 and 3, respectively. The associated computation durations and number of iterations used for each case can be found in appendix. Due to simulation limitations, we were not able to consider radii greater than ${12}\,\mathrm{\mu}\textrm{m}$ for the pair ($\varepsilon _p''=10^{-2}$, ${s_p}=0.25$), and radii greater than ${10}\,\mathrm{\mu}\textrm{m}$ for ($\varepsilon _p''=10^{-2}$, ${s_p}=0.44$). For the other cases, explored radii were sufficient for our purpose.

Fig. 2. Standard deviation of $Q_{sc}^{tot}$ as a function of agglomerate surface for different values of ${s_p}$ using a loss term of (a) $\varepsilon _p''=10^{-1}$ and (b) $\varepsilon _p''=10^{-2}$ in the case $R/\lambda =1/3$. Solid lines are the data fit using Eq. (16). Two different regimes can be observed, one for small radii the other for high ones.

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Fig. 3. Mean value of $Q_{sc}^{tot}$ as a function of agglomerate surface for different ${s_p}$ values using a loss term of (a) $\varepsilon _p''=10^{-1}$ - (b) $\varepsilon _p''=10^{-2}$ in the case $R/\lambda =1/3$. Solid lines are the data fit using Eq. (17). The efficiency factor tends toward a constant for high surfaces as for homogeneous agglomerates. However, for smaller surface, the behavior is more complex

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As observed in Fig. 2, $D_Q$ tends towards $0$ for increasing surface, as required by Eq. (10). While the standard deviation of Kanit’s observable was accuratly described by a power law, instead of a bare straight line, Fig. 2 evidences a twofold behavior: one for small surfaces and one for large surfaces. Both regimes seem to follow independent power laws. Hence, the proposed model for the standard deviation is:

(16)$$D_Q(S) = \frac{1}{d_1\,S^{\delta_1} + d_2\,S^{\delta_2}},$$

where $d_i$, $\delta _i$ are coefficients that need to be determined. The result of the data fit using the model of Eq. (16) is plotted in Fig. 2, model and data are in excellent agreement. Furthermore, the extracted functions can be found in Table 1. For the case ${s_p}=0.25$ and $\varepsilon _p''=10^{-2}$, the second regime is essentially observed at the very end of the curve, meaning the fit cannot be consistent. That is why the corresponding equation of Table 1 is highlighted in red.

Table 1. Function obtained from the fit of the standard deviation $D_Q(S)$. Function highlighted in red is unreliable

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Finally, from only four values of ${s_p}$ and two sets of loss parameters, it is difficult to establish any quantitative or qualitative relations between these cases using only the standard deviation. However, by studying the behavior of $\bar {Q}_{sc}^{tot}$ , it is possible to have a qualitative understanding of why two regimes are observed for the standard deviation.

For homogeneous mediums, the scattering efficiency factor is a constant if the ratio $R/\lambda$ is kept constant. However, as observed in Fig. 3, this is not the case for heterogeneous medium. In fact, the constant regime is an asymptotic regime, attained only for very large surfaces. Since the response of the mixture is reminiscent of homogeneous mediums, we can be confident that homogenization is possible for the chosen cases, as long as the agglomerates are large enough.

For smaller agglomerates, the description of $\bar {Q}_{sc}^{tot}$ as a function of the surface is more challenging. The case ${s_p}=0.25$ has the most complex behavior as its function is not monotonic, a fast increase can be observed for small surfaces, followed by a slower decrease. So far, all that can be said with certitude is that such a function, if it exists, should be decomposable into a coherent and an incoherent parts as follow $\bar {Q}_{sc}^{tot} = \bar {Q}_{sc}^{coh} + \bar {Q}_{sc}^{incoh}$. The coherent part should include a constant $Q_{\infty }$ to describe the asymptotic regime, to which we add a power law dependence accounting for smaller surfaces. The incoherent part arises from the statistical fluctuation of the total flux about the coherent flux, see Eq. (4), meaning it can be related to a standard deviation. Therefore, it is tempting to extend the model that has been given in Eq. (16) to $\bar {Q}_{sc}^{incoh}$. To sum up, we assume that:

(17)$$\bar{Q}_{sc}^{tot}(S) = \underbrace{Q_{\infty} - \frac{a}{S^{\alpha}}}_{\bar{Q}_{sc}^{coh}} + \underbrace{\frac{1}{b_1\,S^{\beta_1} + b_2\,S^{\beta_2}}}_{\bar{Q}_{sc}^{incoh}},$$

where the unknown coefficients are achieved by independently fitting the two components of Eq. (17) to the corresponding computed data. By analysing Eq. (17), it is possible to justify why we need an inverse sum of two power laws, instead of a simple power law as Kanit proposed, to describe either $D_Q$ or $\bar {Q}_{sc}^{incoh}$. Since $\bar {Q}_{sc}^{coh}$ is not constant and depends on the agglomerate surface through a power law, essentially for small surfaces, it should implies that $D_Q$ and $\bar {Q}_{sc}^{incoh}$ have two different behaviors for small and large surfaces, hence the model. This idea is equivalent to saying that if $\bar {Q}_{sc}^{coh}$ was a constant, then $D_Q$ and $\bar {Q}_{sc}^{incoh}$ would be modeled by a simple power law function. Also, looking at Eq. (6), it is clear that the incoherent flux is a second order correction of the coherent flux, the first order being zero by definition $\langle {\delta \boldsymbol {A_{sc}}}\rangle = 0$, which is in adequacy with Eq. (17).

Obtained functions from the fit of our data using Eq. (17) are plotted in Fig. 3. The agreement between model and data is remarkable; even the complex behavior of the small surface regime at ${s_p}=0.25$ is reproduced. The extracted functions can be found in Table 2. The equation of $\bar {Q}_{sc}^{incoh}$ for ${s_p}=0.25$ and $\varepsilon _p''=10^{-2}$ is highlighted in red for the same reason as before.

Table 2. Function obtained from the fit of the ensemble scattering efficiency factor $\bar {Q}_{sc}^{tot}$. Function highlighted in red is unreliable. Exponents highlighted in orange are close to unity.

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It turns out that for the incoherent efficiency factor, the second regime exponent (highlighted in orange in Table 2) is close to unity for all cases but ${s_p}=0.25$ with $\varepsilon _p''=10^{-2}$. This difference is believed to arise from the inconsistent fit; by adding more points, the exponent should tends towards $1$. Schilder et al observed that for very far off resonance scattering: $P_{incoh}/P_{coh} \propto 1/N$, where $N$ is the number of scatterers (atoms in their case) and $P$ the scattering power [17]. In our case, $N$ is the number of particles in the agglomerate with $N\propto S$, and $P_{incoh}/P_{coh}$ can be replaced by $\bar {Q}_{sc}^{incoh}/\bar {Q}_{sc}^{coh}$. Moreover, as we increase $S$, $\bar {Q}_{sc}^{coh}$ stays mostly constant for large surfaces, see Fig. 3. This means that theoretically, we should have $\bar {Q}_{sc}^{incoh} \propto 1/S$ for large surfaces and far off resonance cases, which is valid even for the case ${s_p}=0.44$. Although ${s_p} = 0.44$ corresponds to the case where surface-plasmon resonances occur, this effect was widely reduced by adding small losses to the particles, hence the unity exponent.

Finally, on both Fig. 2 and 3, it is clear that decreasing the loss term of the particle permittivity from $\varepsilon _p''=10^{-1}$ to $\varepsilon _p''=10^{-2}$ results in higher difficulties in the computation of quantities. Indeed, higher surfaces are needed in order to obtain the second regime of $D_Q$, and to have convergence of $\bar {Q}_{sc}^{tot}$. This can be explained by the fact that setting $\varepsilon _p''$ to smaller values tends to strengthens long range interactions between particles, so that larger surfaces are needed in order to account for all the scattering events between particles.

Now that the surface dependencies of the mean value and the standard deviation of $Q_{sc}^{tot}$ are characterized, the next step is to establish a criterion for the surface of the agglomerate that enables the extraction of an effective refractive index. Looking at our data, there are two possible criteria. First criterion, homogenization is obtained if $D_Q$ is in its large surface regime; second criterion, it is achieved if $\bar {Q}_{sc}^{tot}$ is close to its final value $Q_{\infty }$, up to a tolerance that needs to be defined.

5. From efficiency factor to refractive index extraction

In order to investigate the relation between $Q_{sc}^{tot}$ and homogenization, the extraction of the effective refractive index using the coherent flux was conducted for all chosen radii, ${s_p}$ values and loss parameters, results can be found in Fig. 4. Homogenization is obtained for all cases, except for ${s_p}=0.44$ with $\varepsilon _p''=10^{-2}$, in the sense that both the real and imaginary part of the extracted $n_{eff}$ remain stable after a certain agglomerate radius. Therefore, it is possible to extract the minimum radius, $R_{min}^{1{\%}}$, required to obtain a converged extraction, inside a $ {1}{\%}$ relative error. Extracted $n_{eff}$ and $R_{min}^{{1}{\%}}$ are reported in Table 3 left and middle panels, respectively.

Fig. 4. (a) and (c) are the real and imaginary parts, respectively, of the effective refractive index for $\varepsilon _p''=10^{-1}$. (b) and (d) correspond to the same type of curves for $\varepsilon _p''=10^{-2}$.

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Table 3. Left panel: extracted $n_{eff}$ of the last computed radius. Middle panel: minimum radius needed in order to have a converged refractive index extraction inside a $ {1}{\%}$ relative error. Right panel: associated mean efficiency relative error, $\Delta \bar {Q}^{tot}_{sc} = |\bar {Q}^{tot}_{sc}(R_{min}^{{1}{\%}}) - Q_{\infty }| / Q_{\infty }$

View Table | View all tables in this article

It is interesting to note that for the case ${s_p}=0.25$, the extraction convergence is obtained for a radius of $ {2.5}\,\mathrm{\mu}\textrm{m}$, whatever the value of $\varepsilon _p''$. On the other hand, for this radius, the incoherence over total ratio $\bar {Q}_{sc}^{incoh} / \bar {Q}_{sc}^{tot}$ is around $ {10}{\%}$ and ${17}{\%}$ for $\varepsilon _p''=10^{-1}$ and $\varepsilon _p''=10^{-2}$, respectively. This is a clear demonstration that refractive index extraction on the coherent flux is possible even if incoherence is high, considering that coherence is well computed.

Although the mean efficiency factor at ${s_p}=0.25$ corresponds to the most complex behavior of all cases (mainly due to its high incoherence), the case leading to the most severe difficulties in the extraction of $n_{eff}$ is ${s_p}=0.44$. At $\varepsilon _p''=10^{-1}$, the required radius to get convergence is twice the value found for the other cases, while we were unable to reach convergence at $\varepsilon _p''=10^{-2}$, even for agglomerates as large as $R= {10}\,\mathrm{\mu}\textrm{m}$, containing around $1500$ particles. Different plateaus are observed from ${4.5}\,\mathrm{\mu}\textrm{m}$ to ${6}\,\mathrm{\mu}\textrm{m}$ and from ${7}\,\mathrm{\mu}\textrm{m}$ to ${8}\,\mathrm{\mu}\textrm{m}$. They give the illusion of convergence and thus imply that it is not possible to extrapolate the approximate radius which is required to get convergence.

These plateaus are a direct consequence of the surface-plasmon resonances, which make the homogenization process harder to achieve, if not impossible. By decreasing the loss parameter, resonances are exacerbated, explaining why for the same radius, convergence is attained for $\varepsilon _p''=10^{-1}$ but not for $\varepsilon _p''=10^{-2}$. Due to computational limitation, we were not able to investigate further the extraction convergence. However, we are inclined to think that homogenization should be possible for ${s_p}=0.44$ with $\varepsilon _p''=10^{-2}$ for two reasons. One, $\bar {Q}_{sc}^{tot}$ tends towards a constant which is reminiscent of the homogeneous behavior. Second, for large agglomerates, we have $\bar {Q}_{sc}^{incoh} \approx 1 / S$, implying that resonances are weak.

Here, we are going to demonstrate that both criteria previously established fail to estimate the minimal surface allowing homogenization. The first one states that homogenization is obtained if the standard deviation reached the second regime. Putting aside the difficulty to define when the second regime is attained, let us consider the case ${s_p}=0.25$ and $\varepsilon _p''=10^{-2}$. As already stated, for this case, the second regime is hardly observed for surfaces as large as ${452}\,\mathrm{\mu}\textrm{m}^{2}$, see Fig. 2(b). However, the refractive index extraction did converge for radii above ${2.5}\,\mathrm{\mu}\textrm{m}$, corresponding to surfaces above ${20}\,\mathrm{\mu}\textrm{m}^{2}$.

Now if we consider that the second regime is attained when the two terms appearing in the denominator of Eq. (16) are equals, corresponding to the middle of the breakup point between both regimes. Then using the function of Table 1, we can extract the surface for which this condition is met. Results are very inconsistent, for example, doing the computation for ${s_p}=0.01$ holds a minimal radius of ${2.5}\,\mathrm{\mu}\textrm{m}$ for $\varepsilon _p''=10^{-1}$ close to the actual minimal radius, but ${14}\,\mathrm{\mu}\textrm{m}$ for $\varepsilon _p''=10^{-2}$. The biggest mismatch corresponds to ${s_p}=0.60$ and $\varepsilon _p''=10^{-2}$ with a given minimal radius at ${53}\,\mathrm{\mu}\textrm{m}$, which is way above the ${4}\,\mathrm{\mu}\textrm{m}$ of Table 3. Therefore, the first criterion is not valid.

Let us consider the second criterion asserting that homogenization is possible if the mean efficiency factor is close to its final value $Q_{\infty }$, up to a tolerance that needs to be defined. Using the already derived minimal radius for convergence $R_{min}^{{1}{\%}}$, it is possible to compute the associated relative error of $\bar {Q}^{tot}_{sc}$ as follow $\Delta \bar {Q}^{tot}_{sc} = |\bar {Q}^{tot}_{sc}(R_{min}^{{1}{\%}}) - Q_{\infty }| / Q_{\infty }$, with the help of functions defined in Table 2. Results are reported in the right panel of Table 3.

At first glance, since lessening the loss parameter enhances resonant effects, the minimal agglomerate size would be rather expected to increase. However, it can be observed that $\Delta \bar {Q}^{tot}_{sc}$ is smaller for $\varepsilon _p''=10^{-1}$ than $\varepsilon _p''=10^{-2}$, which implies that the condition for convergence is more severe in the first case than the second. This last point is in contradiction with the extraction of $n_{eff}$ which demonstrated that $R_{min}^{{1}{\%}}$ is independent of the loss value, while the convergence inside a certain tolerance of $\bar {Q}^{tot}_{sc}$ deteriorates with decreasing loss parameter, as it can be observed on Fig. 3. Therefore, the second criterion is not valid.

To ensure the validity of the existence of a single minimal representative radius independent of the loss parameter, more computations were conducted. Specifically, the minimal radius for ${s_p}=0.25$ is only ${2.5}\,\mathrm{\mu}\textrm{m}$, meaning that we do not need extensive computation with large radius and large multipolar degree. Simulation with radius up to ${7}\,\mathrm{\mu}\textrm{m}$ were achieved for ${s_p}=0.25$ and $\varepsilon _p''=10^{-5}$. The result is identical as before, the minimal radius such that the refractive index extraction converged within a ${1}{\%}$ relative error for ${s_p}=0.25$ with $\varepsilon _p''=10^{-5}$ is ${2.5}\,\mathrm{\mu}\textrm{m}$. The same conclusion is obtained for ${s_p}=0.60$ and $\varepsilon _p''=10^{-5}$

Finally, we demonstrated that the study of $Q_{sc}^{tot}$ mean value and standard deviation to predict the condition for the homogenization of a mixture has limited applications, in the sense that no simple relation can be established. However, we clearly established that the convergence of $n_{eff}$ extraction is independent of the loss parameter, except when surface-plasmon resonances are excited. In this case, adding losses inside each particle tends to weaken resonances, allowing simpler homogenization.

6. Conclusion

The object of this work was to investigate the behavior of the representative volume element for homogenization in the case where strong interactions between particles exist. The systems of interest were 2D agglomerates composed of non-overlapping circular particles whose permittivity real part was derived from the $\omega _{TO}$ - $\omega _{LO}$ spectrum. By using this specific range of frequencies, we ensured multiple scattering between particles. Small losses were added to the system in order to modulate the strength of interactions, and make homogenization simpler.

Homogenization being only possible on the coherent response of a set of agglomerates, an ensemble averaging of the quantity of interest is needed. However, if the set of response is spread over orders of magnitude, due to resonances inside some agglomerates, then the ensemble averaging can be inconsistent. Accordingly, adequate limitations for the particle permittivity were imposed to avoid such unreliable cases.

In addition, the study that was proposed here can only be applied to a stationary ergodic random function. To comply with this restriction, the scattering simulations were conducted with a constant ratio between the agglomerate radius and the wavelength of the incident light. For such condition, the quantity of interest was the total scattering efficiency factor $Q^{tot}_{sc}$.

While in Kanit work it was observed that the standard deviation of a SERF can be described by a power law function, the standard deviation of the efficiency factor $D_Q$ had a more complex behavior composed of two different regimes for small and large surfaces. Using a new phenomenological model consisting of the inverse sum of two power laws, data were perfectly matched.

In the case of homogeneous agglomerates, the scattering efficiency factor is constant for any agglomerate radius. For inhomogeneous agglomerates, $\bar {Q}_{sc}^{tot}$ tends towards a constant as the surface increases, supporting the idea that homogenization should be possible. However, for small surfaces, its behavior can be very complex. In order to model this transient regime, $\bar {Q}^{tot}_{sc}$ can be decomposed into coherent and incoherent parts. The former can be modeled by the sum of a constant and a power law. The latter being essentially equivalent to a standard deviation, it is well approximated by the same model as $D_Q$.

Using these newly known behaviors, we formulated two possible criteria on the minimal surface allowing homogenization, one for the mean value the other for the standard deviation of $Q^{tot}_{sc}$. Both criteria were unsuccessful to predict the minimal surface needed in order to have a converged refractive index extraction.

However, this work demonstrated that when no surface-plasmon resonances arise, a single minimal representative radius, independent of the loss parameter, can be defined for different particle permittivity real parts. In the case where surface-plasmon resonances exist, decreasing the loss parameter results in an increase of the resonances strength. In order to reach homogenization, higher agglomerate surfaces are needed in order to compensate these resonances. However, it is believed that at some point resonances will be so strong that homogenization becomes impossible

Appendix: Computation durations and number of iterations

In this appendix, we summarize simulation durations alongside with the rule used to select the number of iterations as a function of the agglomerate radius. Most of the simulations were carried out, in parallel computation, on two different servers. The first one has a processor Intel Xeon Silver 4114 containing 20 cores with a clock frequency of ${2.2}\,\textrm{GHz}$ each and ${512}\,\textrm{GB}$ of RAM. This means that during parallel computation, each core has at its disposal around ${25}\,\textrm{GB}$ of RAM. The second one has a processor Intel Xeon Silver 4215R composed of 16 cores with ${3.2}\,\textrm{GHz}$ each and ${1.5}\,\textrm{TB}$ of RAM, meaning that each core has around ${94}\,\textrm{GB}$ of dedicated RAM for simulation.

The largest simulation carried out in this work was composed of $1024$ agglomerates with radius ${10}\,\mathrm{\mu}\textrm{m}$ and a multipolar degree of $17$, resulting in matrices consuming around ${41}\,\textrm{GB}$ of RAM for each iterations. To achieve this simulation on the second server, more than 11 days of computation were required. Different simulation durations, carried out on the first server, are reported on Fig. 5. It is very clear that computation duration quickly escalates for high multipolar degree. The rules used to decide how many iterations are needed for each agglomerate radius are reported as annotations in Fig. 5.

Fig. 5. Simulation duration carried out on the first server as a function of agglomerate radius for different multipolar degrees. Framed annotations are the number of iterations used for each radius ranges. Lines are guides for the eye

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Funding

Agence Nationale de la Recherche (ANR-19-CE05-0021).

Acknowledgments

This work is part of the OUTWARDS Project (ID: ANR 19-CE05-0021), funded by the Agence Nationale de la Recherche (ANR). The authors acknowledge Florent Poupard for helpful support regarding the management of CEMHTI’s computing resources

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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$1 / D_{Q} (S)$	$ε_{p}^{″} = 10^{- 1}$	$ε_{p}^{″} = 10^{- 2}$
$s_{p} = 0.01$	$10.529 S^{0.280} + 3.913 S^{0.613}$	$13.553 S^{0.362} + 0.771 S^{0.807}$
$s_{p} = 0.25$	$2.940 S^{0.278} + 0.037 S^{0.927}$	$2.632 S^{0.189} + 10^{- 5} S^{1.892}$
$s_{p} = 0.44$	$5.774 S^{0.374} + 0.463 S^{0.750}$	$2.239 + 1.147 S^{0.520}$
$s_{p} = 0.60$	$6.102 S^{0.480} + 0.006 S^{1.235}$	$3.272 S^{0.187} + 0.282 S^{0.748}$

${\bar{Q}}_{s c}^{c o h}$	$ε_{p}^{″} = 10^{- 1}$	$ε_{p}^{″} = 10^{- 2}$	$1 / {\bar{Q}}_{s c}^{i n c o h}$	$ε_{p}^{″} = 10^{- 1}$	$ε_{p}^{″} = 10^{- 2}$
$s_{p} = 0.01$	$0.142 - 0.097 / S^{0.587}$	$0.142 - 0.098 / S^{0.579}$	$s_{p} = 0.01$	$0.322 + 14.553 S$ ^1.048	$0.789 + 14.274 S$ ^1.053
$s_{p} = 0.25$	$0.775 - 0.481 / S^{0.579}$	$0.868 - 0.569 / S^{0.539}$	$s_{p} = 0.25$	$2.283 S^{0.291} + 0.342 S$ ^1.040	$2.267 S^{0.330} + 0.003 S^{1.400}$
$s_{p} = 0.44$	$0.518 - 0.173 / S^{2.413}$	$\approx 0.567$	$s_{p} = 0.44$	$1.506 S^{0.269} + 1.516 S$ ^1.027	$1.333 S^{0.078} + 0.253 S$ ^0.936
$s_{p} = 0.60$	$0.465 - 0.145 / S^{1.720}$	$0.592 - 0.212 / S^{2.503}$	$s_{p} = 0.60$	$0.585 + 4.471 S$ ^1.048	$2.344 S^{0.084} + 0.628 S$ ^0.999

	$n_{e f f}$		$R_{m i n}^{1 %}$		$Δ {\bar{Q}}_{s c}^{t o t}$
	$ε_{p}^{″} = 10^{- 1}$	$ε_{p}^{″} = 10^{- 2}$	$ε_{p}^{″} = 10^{- 1}$	$ε_{p}^{″} = 10^{- 2}$	$ε_{p}^{″} = 10^{- 1}$	$ε_{p}^{″} = 10^{- 2}$
$s_{p} = 0.01$	$1.173 + 0.002 i$	$1 .173 + 0.002 i$	$2 μ m$	$2 μ m$	$12.1 %$	$12.5 %$
$s_{p} = 0.25$	$1.469 + 0.057 i$	$1.484 + 0.037 i$	$2.5 μ m$	$2.5 μ m$	$1.15 %$	$5.00 %$
$s_{p} = 0.44$	$1.161 + 0.961 i$	$1.174 + 1.123 i$	$7 μ m$	No convergence	$0.71 %$	No convergence
$s_{p} = 0.60$	$0.593 + 0.305 i$	$0.505 + 0.265 i$	$4 μ m$	$4 μ m$	$0.75 %$	$4.87 %$

$1 / D_{Q} (S)$	$ε_{p}^{″} = 10^{- 1}$	$ε_{p}^{″} = 10^{- 2}$
$s_{p} = 0.01$	$10.529 S^{0.280} + 3.913 S^{0.613}$	$13.553 S^{0.362} + 0.771 S^{0.807}$
$s_{p} = 0.25$	$2.940 S^{0.278} + 0.037 S^{0.927}$	$2.632 S^{0.189} + 10^{- 5} S^{1.892}$
$s_{p} = 0.44$	$5.774 S^{0.374} + 0.463 S^{0.750}$	$2.239 + 1.147 S^{0.520}$
$s_{p} = 0.60$	$6.102 S^{0.480} + 0.006 S^{1.235}$	$3.272 S^{0.187} + 0.282 S^{0.748}$

${\bar{Q}}_{s c}^{c o h}$	$ε_{p}^{″} = 10^{- 1}$	$ε_{p}^{″} = 10^{- 2}$	$1 / {\bar{Q}}_{s c}^{i n c o h}$	$ε_{p}^{″} = 10^{- 1}$	$ε_{p}^{″} = 10^{- 2}$
$s_{p} = 0.01$	$0.142 - 0.097 / S^{0.587}$	$0.142 - 0.098 / S^{0.579}$	$s_{p} = 0.01$	$0.322 + 14.553 S$ ^1.048	$0.789 + 14.274 S$ ^1.053
$s_{p} = 0.25$	$0.775 - 0.481 / S^{0.579}$	$0.868 - 0.569 / S^{0.539}$	$s_{p} = 0.25$	$2.283 S^{0.291} + 0.342 S$ ^1.040	$2.267 S^{0.330} + 0.003 S^{1.400}$
$s_{p} = 0.44$	$0.518 - 0.173 / S^{2.413}$	$\approx 0.567$	$s_{p} = 0.44$	$1.506 S^{0.269} + 1.516 S$ ^1.027	$1.333 S^{0.078} + 0.253 S$ ^0.936
$s_{p} = 0.60$	$0.465 - 0.145 / S^{1.720}$	$0.592 - 0.212 / S^{2.503}$	$s_{p} = 0.60$	$0.585 + 4.471 S$ ^1.048	$2.344 S^{0.084} + 0.628 S$ ^0.999

From representative volume element of interacting particles to the extraction of their effective properties

Abstract

1. Introduction

2. Description of the system

3. Averaging behavior of many-particle agglomerates

4. Characterization of the efficiency factor mean value and standard deviation

4.1 Definition of a stationary ergodic random function

4.2 Computation results and discussion

5. From efficiency factor to refractive index extraction

6. Conclusion

Appendix: Computation durations and number of iterations

Funding

Acknowledgments

Disclosures

Data availability

References

Data availability

Cited By

Figures (5)

Tables (3)

Equations (17)

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