## Abstract

Currently, there are available a few simple analytical approximations to the complex effective refractive index that may be used for nanofluids. Namely, the Maxwell Garnett mixing formula with scattering corrections, the Maxell Garnett Mie approximation, the Foldy-Lax approximation and the small particle limit of the quasi-crystalline approximation. These approximations are valid either for very small nanoparticles (below a few nanometers in radius) or for very dilute nanofluids (below about 1% in particles’ volume fractions) and therefore, do not cover the whole domain of particle suspensions referred to as nanofluids. Here we propose a new simple analytical approximation based on local field corrections to the Foldy-Lax approximation. The new mixing formula coincides with the mentioned approximations when they are expected to be valid and provides physically sound predictions when the mentioned approximations are no longer valid, within the realm of nanofluids. We compare predictions of the analytical approximations considered in this work with experimental data published earlier for nanofluids of polystyrene in water.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

A nanofluid is a homogenous liquid with nanoparticles of sizes smaller than 100 nm in suspension. Nanofluids have become of interest in recent years because some of the physical properties of a fluid can be affected dramatically by a small amount of nanoparticles in suspension [1,2], opening a new way for designing composite fluids with properties not found in homogeneous fluids. Different applications of nanofluids have been investigated in recent years, ranging from heat transfer and solar absorbers to medical applications [1–4].

In order to understand, verify and control the properties of a nanofluid it is desirable to have simple means to characterize the nanoparticles in suspension. A simple tool could be measuring at optical frequencies, and correctly interpreting, the complex refractive index of a nanofluid. Nanofluids are in general, light scattering media. The amount of scattering losses on the propagation of a collimated beam of light will depend on the size of the nanoparticles and their refractive index contrast with the host liquid. In light scattering media it is usual to define an average (or coherent) electromagnetic wave and assign an effective propagation constant to it [5–9]. From the effective propagation constant one can define an effective Refractive Index (RI). Such RI has a non-local nature and some precautions may be needed to use it in continuous electrodynamics formulae [10,11]. Nevertheless, since it is a more common concept when dealing with fluids, in this work we will be referring to the effective refractive index of nanofluids.

The effective RI of nanofluids depends on the amount, size and state of dispersion of NanoParticles (NPs) in suspension, as well as on their RI and that of the medium surrounding the nanoparticles, commonly called the host, base fluid or matrix. The effective RI of the nanofluid is in general a complex quantity, with both its real and imaginary parts, being a function of the wavelength of light. To a very good approximation light refracts into the nanofluid according to Snell’s Law with the real part of its effective RI (see Fig. 1). Light also attenuates as it travels through the nanofluid exponentially with an extinction coefficient ${\alpha _{ext}}$ proportional to the imaginary part of the effective RI: ${n_{eff}}$. That is, the intensity of light *I* decays as a function of the path length $\ell $ travelled by light through the nanofluid as, $I(\ell )= {I_0}\exp ({ - {\alpha_{ext}}\ell } )$, where

In the dilute limit, the extinction coefficient can be expressed in terms of the extinction cross section of an isolated particle, ${\sigma _{ext}}$ [12]. That is, ${\alpha _{ext}} = \rho {\sigma _{ext}}$, where $\rho $ is the number density of particles. ${\sigma _{ext}}$ is actually the sum of the absorption cross section and the scattering cross section of an isolated particle, ${\sigma _{ext}} = {\sigma _{abs}} + {\sigma _{sca}}$. Note that in this limit the extinction coefficient ${\alpha _{ext}}$ depends linearly with the density of particles. In this case one commonly refers to the “independent-scattering” regime [8,13,14] and for this case we will denote the extinction coefficient as ${\alpha _{ind}}$. Thus, we can write,

When the density of particles is not too low, and depending on the size of the particles, the extinction coefficient may include terms proportional to the second and higher order powers of the density of particles. It is common to refer to this regime as “dependent-scattering” regime [7,13,15–22].The dependence of the real part of the effective refractive index, $\textrm{Re}({{n_{eff}}} )$, with the particles’ volume concentration, *f*, is in general closer to a linear relationship. However, some deviations from the linear dependence of the independent-scattering approximation, within the realm of nanofluids, are predicted by the different analytical approximations considered here.

Since, in a nanofluid the suspended particles are several times smaller than the wavelength of visible light, its complex effective RI can be measured with standard instrumentation and with relative ease (see, for instance, Refs. [23,24]). We will consider only nanofluids consisting of monodisperse suspensions of homogeneous spherical particles of radius $a\; ({ \le 50\textrm{nm}} )$ and complex refractive index ${n_p}$, suspended in a host fluid of real refractive index ${n_m}$ and assume the particles’ volume content is less than 5%.

## 2. Established analytical approximations for ${n_{eff}}$ of nanofluids

Calculating the optical properties of suspensions of particles of arbitrary size has caught the attention of many researchers for more than 100 years [12]. Analytical, as well as numerical approaches to predict the optical properties of particle suspensions have been proposed over the years. In this section we summarize, to our knowledge, the main analytical approximations to the effective refractive index of a particle suspension proposed up to date.

#### 2.1 Very small particles and moderate volume concentrations

When particles are very small compared to the wavelength of radiation we can expect to find simple analytical approximations. In such cases, for which scattering losses may be ignored, the well-known Maxwell Garnett (MG) effective medium approximation to the effective dielectric function has been available for many years [25,26].

This approximation can only be used to predict the effective refractive index where the size of the particles is very small compared to the wavelength and scattering losses are neglected. Nevertheless, it is currently used without much restriction to calculate the optical properties of metallic nanoparticle suspensions. In most cases it produces reasonable predictions when losses are only due to absorption. But, when the particles are not sufficiently small compared to the wavelength, the MG mixing formula may not be good enough, even for very small particles’ volume concentrations,.

Simple analytical extensions to the MG approximation, to include finite size effects, have been proposed. One extended MG approximation consists of adding terms responsible of scattering losses to first order in the nanoparticles’ volume concentration [27–30]. We will refer to this approximation as the Maxwell Garnett mixing formula with scattering corrections $({\textrm{MGSC}} )$. The RI under the MGSC approximation can be written in the following way:

Another extended MG approximation was originally proposed by W. T. Doyle few decades ago [31]. It was extended further for polydisperse nanofluids recently in [32] and found that it reproduced well experimental extinction spectra of very dilute gold nanofluids of sizes around 30 nm and below. It consists of calculating the particle polarizability with the first Mie coefficient which corresponds to the induced electric dipole, but considering the actual size of the nanoparticles. This amounts to taking dynamic corrections to the particle polarizability. The approximation is referred to as the Maxwell Garnett Mie (MGM) approximation.

The MGM approximation to the effective RI can be written as,

Another known analytical approximation that goes beyond the MG approximation is the Small Particle Limit of the Quasi-Crystalline Approximation (QCA-SPL). The quasi-crystalline approximation was developed for suspensions of spherical particles of arbitrary size by Tsang and Kong [7]. It provides a simple analytical approximation only in the limit of small particles. The QCA-SPL includes scattering corrections to the MG mixing formula to second order in the particles’ volume fraction [7,22,27,28]. This approximation integrates the pair distribution function, $g(r )$, which is proportional to the density of probability of finding a particle a distance *r* from another particle. For suspensions with particles’ volume-fraction, *f*, below 10%, we can approximate $g(r )$, using a “hole” correlation function. This gives,

*b*may be referred to as the “exclusion-volume radius” (see Fig. 1) and must be equal or larger than the particles diameter ($b \ge a)$. Thus, for dilute particle suspensions the QCA-SPL reduces to the following analytical approximation [7,27,28],

#### 2.2 Particles of arbitrary size and small volume concentrations

One other approximation for the effective RI of particle suspension is the well-known Foldy-Lax Approximation (FLA) [5–7,9]. This is derived from multiple-scattering theory assuming the density of particles is small, but without restricting the particle size. It is given by,

where $S(0 )\; $ is the forward scattering amplitude of an isolated particle surrounded by the host medium. For a spherical particle $S(0 )$ is readily calculated using Mie theory [12]. The FLA assumes the field that excites any given particle is the average electromagnetic wave. It does not take into account local field corrections. The Foldy-Lax approximation is generally considered valid for dilute particles suspensions, but it can be used for small or large particles.The FLA can be expanded in powers of the particles’ volume fraction, *f*. If we keep only the first two terms we get the van de Hulst’s approximation to the effective RI:

## 3. Local field corrections to the Foldy-Lax approximation

In this work we introduce a new formula for the effective refractive index of a nanofluid, that is obtained from making local field corrections to the well-known Foldy-Lax approximation to the effective refractive index of particle suspensions. As already said, the Foldy-Lax approximation is derived from multiple scattering theory assuming the particles are excited by the average electric wave, ${{\boldsymbol E}_{avg}}$ [5–7,9]. However, in general, the field exciting any given particle may differ from the average wave due to Local Field Corrections (LFC). In this case the exciting field is called the local field and here we will denote it as ${{\boldsymbol E}_{loc}}$, [12,33,34]. Thus, the FLA is accurate whenever ${{\boldsymbol E}_{loc}} \cong {{\boldsymbol E}_{avg}}$. The latter approximation is also referred to as the Effective Field Approximation (EFA). This is in general a good approximation, regardless of the particle size, when the particle density is very low.

For small particles compared to the wavelength and for moderate particles’ density, the local field differs appreciably from the average field and the FLA will be inaccurate. We propose a way to extend the validity of the FLA approximation to within the realm of nanofluids simply by assuming the particle exciting any given particle is the local field instead of the average field. This amounts to dividing the forward scattering amplitude, *S*(0), appearing in FLA’s formula by ${E_{avg}}$ and then multiplying it by ${E_{loc}}$. We then have,

For a collection of point-like particles, embedded randomly in a homogenous host medium of dielectric permittivity ${\epsilon _m}$, the relationship between the average field and the local field for an incident (or applied) monochromatic field oscillating at frequency $\omega $ is well known, and given by [12,28,33,34],

When particles cannot be considered point-like at a given wavelength, as is the case for many nanofluids under optical radiation, particle-size corrections to the relationship between the local and average fields must be introduced. To this end we must go beyond the quasi-electrostatic approximation and consider the electrodynamic interaction among the particles. Assuming particles are spherical of radius *a*, a uniform pair distribution function with a “hole correction” of radius *b*, given by Eq. (5) above, calculating the average local field in the usual way [34], but with the electrodynamic Green tensor instead of the electrostatic one, expanding the expressions involved in a power series of the particle radius and keeping only the leading terms, yields,

Note that the FLA-LFC in Eq. (14) as well as the QCA-SPL given in Eq. (6) above, depend on the exclusion-volume radius, *b*, whereas the other analytical approximations do not. The effect of *b* being larger than the particles’ radius *a* is to increase the deviation from a linear relationship between ${n_{eff}}$ and the volume concentration of particles. Some insights into the effects of the exclusion-volume radius are discussed in [35]. Also, the FLA and FLA-LFC depend on the forward-scattering amplitude of an isolated particle, $S(0 )$. For spherical particles, $S(0 )$ can be readily expanded in a power series of the particles’ radius [12], and a few terms could be kept for nanofluids. However, it is preferable to keep the formula in terms of $S(0 )$ for future reference, since it allows using the formula for polydisperse nanofluids and for non-spherical particles. However, such cases are out of the scope of this paper.

There are two other approximations that provide analytical expressions to the extinction coefficient of light in suspensions of small particles: One was derived by J. B. Keller in 1962 for electromagnetic waves [8,13,15] and the other one was derived by V. Twersky for acoustic waves in 1978 [13,16]. Both approximations predict dependent-scattering effects in the behavior of the extinction coefficient as a function of the particles’ volume concentration. However, the former overestimates the dependent-scattering for particles of sizes in the range of nanofluids, and the latter is not derived from multiple-scattering theory. Therefore, neither is of interest in the present work. We may also note that there are alternative routes to seek dynamic corrections to the quasi-static approximations for very small particle suspensions following the Bruggeman approach [36].

## 4. Comparisons among analytical approximations and experimental data

#### 4.1 Numerical comparisons among different approximations

To compare the predictions of the new analytical approximation introduced in the previous sections with the currently established approximations, we will present a set of figures with three graphs in each one, calculated for selected hypothetical nanofluids. In the plots we include the “independent-scattering” calculation (defined above), and the predictions of the MGSC, the QCA-SPL, the FLA and the FLA-LFC. In each figure, the first and second graphs show the plots for the real and imaginary parts of the effective RI versus the volume concentration of nanoparticles, and the third graph shows the corresponding plots of the ratio of ${\alpha _{ext}}$ and the independent-scattering approximation. As in [8,13], we will denote this ratio as,

and show plots of $\gamma $ versus*f*in a semi-log scale so that the differences among the predictions of the different analytical approximations are more evident.

For all plots in Figs. 2–6 we chose a vacuum wavelength of 436 nm which is within the visible range. For the curves calculated with the QCA-SPL and the FLA-LFC we assumed that particles can touch each other. That is, we used $b = a$.

In Figs. 2 and 3 we show results for very small particles of 5 nm radius for polystyrene and silver nanoparticles, respectively. We can see in Fig. 2(a) that for polystyrene nanoparticles all approximations predict the same linear dependence of $\textrm{Re}({n_{eff}}$) with *f*. In Fig. 2(b) we can see that for $\textrm{Im}({n_{eff}})$ all approximations coincide with each other in the dilute limit ($f \to 0$). But for values above about 1% the QCA-SPL and the FLA-LFC depart notoriously from the other approximations and the independent-scattering approximation, but they coincide with each other. The behavior of the QCA-SPL and the FLA-LFC is the expected one due to dependent-scattering effects, which are strongest in the limit of small particles [8,13–15,20–22,37]. As the particles’ volume fractions increases above 5%, the values of $\gamma $ predicted by both, the FLA-LFC and the QCA-SPL, keep decreasing rapidly and become negative, which is non-physical. Thus, their applicability should be limited to *f* about 5% and below.

The curves for Ag nanoparticles graphed in Fig. 3 show that all approximations, except the FLA, closely follow the linear dependence of the real part of the effective RI with *f* for the range of particle’s volume density plotted. However, for the imaginary part of the effective RI, we can see from Fig. 3(c) of this figure, that the MGSC and the QCA-SPL do not coincide with the independent-scattering approximation in the limit $f \to 0$. This is non-physical since at some point, as the density of particles approaches zero, the independent-scattering assumptions must be valid. That is, at sufficiently low particle concentrations, the extinction of light past through a very thin slab of the particle suspension should be simply the sum of the extinction caused by individual particles. This flaw of the MGSC and the QCA-SPL in Fig. 3 is because these approximations assume point-like particles and include particle size corrections, sufficient in the dilute limit, but only for particles of real refractive index (non-absorbing). The MGM, on the other hand, does coincide with the independent-scattering approximation in the dilute limit, and thus, it in fact extends the MG approximation to larger particles in the dilute limit (as already shown in [32]). The FLA and the FLA-LFC do coincide with each other and with the independent-scattering approximation in the dilute limit. For values above about 1% the MGSC, MGM, QCA-SPL and FLA-LFC depart notoriously from the other approximations, but the way they bend away from the independent-scattering approximation coincide with each other. Note that the FLA bends the other way and we can suppose this is wrong. The reason is that the FLA does not take into account local field effects as the other approximations do.

In Figs. 4–6 we show plots for larger particles with a diameter of 80 nm ($a = 40$ nm), which is near the border of the nanofluid range of sizes. The figures are for Polymethyl-Methacrylate (PMMA), copper oxide (Cu_{2}O) and graphite, respectively. The first type of particles only scatter light, the second one has a high refractive index with some absorption, whereas the last type of particles is of a highly absorbing material at the chosen wavelength of 436 nm.

We can clearly see in Figs. 4(c), 5(c) and 6(c), that again the MGSC and the QCA-SPL do not coincide with the independent-scattering approximation in the dilute limit ($f \to 0$). In Fig. 4 the slope, $\partial \textrm{Im}({{n_{eff}}} )/\partial f$ in the dilute limit is higher for the MGSC and the QCA-SPL than for the independent-scattering approximation, whereas in Figs. 5 and 6 it is smaller. In fact, it is appreciably less in Fig. 6.

The MGM has the same slope, $\partial \textrm{Im}({{n_{eff}}} )/\partial f$, than the independent-scattering approximation in Fig. 4, but a small difference can be appreciated in Fig. 5 (about 4%), and a larger difference (about 15%) is clearly seen in Fig. 6. Indicating that for larger particles but still within the nanofluid range of sizes, with larger refractive index contrast between the nanoparticles and the host liquid, the MGM fails in the dilute limit as well. The error arises from the fact that the MGSC and the QCA-SPL assume point-like particles, and even though the MGM is in fact applicable to larger particles than the MGSC, for particles as large as 40 nm it already produces non-physical predictions in the dilute limit. The FLA and the FLA-LFC do coincide with the independent-scattering approximation in the dilute limit, but for higher particle densities, however, only the FLA-LFC shows the expected dependent-scattering behavior for dense nanofluids.

Also, in Fig. 4(a) we can see that all approximations for the real part of the effective RI are close to each other but the MGM deviates slightly at larger particles’ volume fractions, *f*. However, in Fig. 5(a) we can see that it is the MGSC and the QCA-SPL that have a slightly smaller slope, $\partial \textrm{Re}({{n_{eff}}} )/\partial f$, than the other approximations, and in Fig. 6(a), the FLA-LFC deviates slightly from linear relationship and goes somewhat above the independent-scattering approximation, whereas the MGM deviates slightly below. This means that absorption by the particles affects differently the value of the real part of the effective RI predicted by each approximation.

#### 4.2 Comparison with experimental data

In the graphs presented in Fig. 7 we compare the values of the ratio $\gamma $ (see Eq. (15)) predicted by the different analytical approximations with experimental data reported in 1982 by A. Ishimaru and Y. Kuga [13] for suspensions of polystyrene particles in water at a vacuum wavelength of 633 nm. The experimental data in [13] were later tested against full numerical calculations done with the quasi-crystalline approximation [37] finding good agreement. Two sets of experimental data correspond to suspensions within the range of nanofluids, namely, those reported for particles of nominal diameters of 91 nm and 109 nm. However, after careful measurements in the dilute limit, the authors of [13] concluded that the correct diameters were 80 nm and 103 nm respectively. We use these latter diameters for the calculations presented in Fig. 7.

We can see that overall, the FLA-LFC reproduces noticeably better the experimental data than all other approximations. The QCA-SPL does take into account the dependent-scattering behavior evident in the experimental data but has the incorrect dilute limit. All other approximations fail to reproduce the dependent-scattering behavior of the experimental curve. In the graphs in Fig. 7(a) and 7(b) we also include a curve calculated with the FLA-LFC but for an exclusion-volume radius, *b*, slightly larger than the particles radius *a* that fits better the theoretical curve to the experimental data. In fact, we could let the exclusion-volume radius, *b*, to be a function of the density of particles, *f*. This would allow for a better match of the FLA-LFC with the experimental data, but this would require support by physicochemical modeling.

We are not aware of more validated experimental data we may compare the predictions of the considered analytical approximations to the effective RI of nanofluids. Although measuring the complex effective RI of nanofluids must be straight forward, obtaining accurate measurements for comparison with theory requires many precautions and very well characterized and stable nanofluids. Further comparison with experiment and a careful delineation of the regions of applicability of each approximation or numerical simulations should be of interest for future works.

## 5. Conclusions

Considering the well-known Foldy-Lax approximation to the effective propagation constant and improving the assumed exciting field with an approximation to the local field, but with additional electrodynamics corrections terms, yields a new formula for the effective refractive index, not proposed previously. We refer to the new approximation as the Foldy-Lax approximation with local field corrections, the FLA-LFC. The FLA-LFC reduces to the established approximations to the effective propagation constant or RI when these are expected to be valid and produces physically sound predictions outside the regions of applicability of the established approximations. It predicts correctly the dependent-scattering effects, which are stronger for smaller particles, and it includes the effects of the exclusion-volume radius, *b*, allowing to take into account more of the physics behind the optical properties of nanofluids. The parameter *b* is also included in the small particle limit formula of the quasi-crystalline approximation (QCA-SPL); however, the QCA-SPL is limited to point-like particles whereas the FLA-LFC works well for larger particles. Both, the QCA-SPL and FLA-LFC are limited to moderate volume fractions around 5% and below. We compared the predictions by all the analytical approximations considered plus those of the FLA-LFC with experimental data reported by A. Ishimaru and Y. Kuga in 1982 and found that the FLA-LFC reproduces appreciably better the experimental data than all other approximations considered.

Although, from the results presented in this work, we cannot know yet whether the FLA-LFC gives accurate predictions for the complex effective RI for all nanofluids of interest, we can be confident it is applicable beyond the regions of applicability of the established approximations, and, at least for non-absorbing nanofluids with a refractive index contrasts between the NP an the host liquid of 1.2, it predicts accurately the extinction coefficient up to about 5% of particles’ volume concentration. It should be of interest and useful to evaluate further in the future the accuracy of the FLA-LFC, either by comparison with numerical simulations or with experimental results throughout the nanofluids domain. In this work we considered only monodisperse nanofluids of spherical particles; however, the FLA-LFC formula can be extended to polydisperse nanofluids of non-spherical particles. As a final note, we must say that although the motivation and context of this work was on nanofluids, all formulas presented in this work can also be used when the host medium is solid.

## Appendix

A rigorous method based on multiple-scattering theory that yields the Foldy-Lax approximation was presented in [38]. In this reference, the coherent component of the refraction and reflection of a plane wave incident to a half-space of a random particle suspension was derived. To obtain the Foldy-Lax approximation in a somewhat simplified way, one can follow the procedure presented in detail in [38] assuming a normally incident wave. Here, such method is summarized, but leaving open the possibility of improving the effective field approximation.

Consider a boundless medium of refractive index ${n_m}$ and assume the half-space $z > 0$ contains a random system of embedded particles. Suppose a plane wave is normally incident to the nanofluid from $z < 0$. Let us write the incident wave as ${E_0}\textrm{exp}({ikz - i\omega t} )\hat{{\boldsymbol e}}$ where *k* is the wavenumber in the medium surrounding the particles (the host or matrix), $\omega $ is the radial frequency, and $\hat{{\boldsymbol e}}$ is the electric polarization vector lying in the $xy$-plane. The electromagnetic field at any point in space is given by the incident wave plus the fields scattered by all the particles. We may write,

*p*particle. Clearly, all fields must have a time dependence given by $\textrm{exp}({ - i\omega t} )$. We will assume this factor is implicit in all fields and omit writing. For a system of identical particles, ${\boldsymbol E}_p^S({\boldsymbol r} )$ may be expressed with the transition operator formalism and we can write Eq. (16) in the following way [7,9,10,38],

^{th}*k*, since the difference between them is of order $\sigma S$, and we obtain, This is the FLA. The equivalent expression was obtained in Eq. (24) of [38] but for an oblique angle of incidence. (Note an unfortunate but evident clerical mistake in the just cited equation of [38]). Now, using ${k_{eff}} = n_{eff}^{FLA}{k_0}$ and $k = {n_m}{k_0}$ in Eq. (22) yields Eq. (7) above. Note that solving for ${k_{eff}}$ in Eq. (22), linearizing it to order $\sigma S(0 )$ (results in $n_{eff}^{vdH}$ given in Eq. (8) above) and substituting the resulting expression in (19), with ${E_{exc}} = {E_{avg}}$, gives ${E_{avg}} \cong {E_0}$.

If we now assume that the exciting field is the local field, which is also a plane wave traveling with the effective propagation constant, ${k_{eff}}$, (see Sect. 3 above), then we should only replace ${E_{exc}} \cong {E_{loc}}$ in Eq. (21), and we then obtain,

This equation readily gives Eq. (10) above. Again, solving for ${k_{eff}}$, linearizing it to order $\sigma S(0 )$ and using this result with $\sigma = \frac{3}{2}\frac{f}{{{{({ka} )}^3}}}$ in (19), gives again ${E_{avg}} \cong {E_0}$.## Funding

Dirección General de Asuntos del Personal Académico, Universidad Nacional Autónoma de México (IN102218); Consejo Nacional de Ciencia y Tecnología (708764).

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