**3. Solar absorption of plasmonic nanofluids**

The excellent optical absorption performance of plasmonic NPs make it to be a great candidate in the solar thermal conversion applications, which is critical for the solar thermal conversion applications. And the optical properties of nanofluids can be controlled by the NP size, shape, concentration and base fluid. In this section, we will discuss the optical properties of NPs or nanofluids from two aspects: theoretical design and experimental design.

#### **3.1 Theoretical design**

To achieve the optical properties of nanofluids, including transmittance, reflectance, and absorptance, the optical performance of single NP is usually determined *Solar Thermal Conversion of Plasmonic Nanofluids: Fundamentals and Applications DOI: http://dx.doi.org/10.5772/intechopen.96991*

firstly. And the dielectric function of materials are required for the optical simulation, which is taken from the experimental data of bulk materials (e.g., Johnson and Christy [44],) or a model approximating experimental results (e.g., Drude method). The Drude model is the simplest of all, but disregards radiation damping. Even today, mainly because of the simplicity, the Drude model is still used to describe the dielectric functions in many calculations. In some problems, the classical models of dielectric functions are unsatisfactory but, at the same time, full quantum theories involve a very complex treatment including non-local effects [45], polarizabilities including non-linear terms [46], electron densities calculation using mean-field theories [47] and temperature dependent effects [48]. The need for quantum treatment of the optical properties of small particles has been evidenced in recent experimental studies [49]. In large particles the resonances are influenced by retardation effects and are strongly dependent on the size of particles, but the dielectric function can be assumed as that of bulk. Based on the dielectric function, Au, Ag and Al are the three most used materials in plasmonics. Their SPR wavelengths are at visible or UV spectral bands and, therefore, of great potential in solar thermal applications.

**Mie theory:** Mie theory is a simple and theoretical method to calculate the optical properties of sphere NPs in a homogenous medium, which uses a series of coefficients *a*<sup>n</sup> and *b*<sup>n</sup> for the scattered fields and *c*<sup>n</sup> and *d*<sup>n</sup> for the internal fields to determine the scattering fields. The scattering and extinction cross sections can be calculated as: [50].

$$\mathcal{C}\_{\text{scat}} = \frac{2\pi}{k^2} \sum\_{n=1}^{\infty} (2n+1) \left( |a\_n|^2 + |b\_n|^2 \right) \tag{1}$$

$$\mathcal{C}\_{\text{ext}} = \frac{2\pi}{k^2} \sum\_{n=1}^{\infty} (2n+1) \operatorname{Re} \left( a\_n + b\_n \right) \tag{2}$$

The absorption cross section can be obtained as: *Cabs* ¼ *Cext* � *Cscat*. Despite to the less computation load, it is possible to obtain cross-sections for many wavelengths in a few seconds, using a common PC. However, a large number of terms is required for accurate cross-section calculations of spheres with very large size parameter [51]. The Mie theory has been extended to permit calculations for ellipsoidal shape, multilayer or several spheres [52].

**DDA:** To calculate the light scattering of an arbitrary shape NP, discrete dipole approximation (DDA) was first presented by Purcell and Pennypacker [53] by using a grid of dipoles. To occupy by the scattering target, DDA method discretizes the volume by an array of *N* dipoles using Clausius–Mossotti polarizability *α <sup>j</sup>* for each dipole, which interacts with the incident field and the neighbors. The polarization of dipole *j* located at *rj* can be determined by *P <sup>j</sup>* ¼ *α jE <sup>j</sup>*, and the field can be calculated as:

$$E\_j = E\_j^{inc} - \sum\_{k \neq j} A\_{jk} P\_k \tag{3}$$

where *Einc <sup>j</sup>* <sup>¼</sup> *<sup>E</sup>*0*eikr*�*iω<sup>t</sup>* . *Ajk* is the matric of dipole interaction and retardation effect.

To achieve accurate and reproduce the calculation results, two validity conditions should be verified in DDA: (a) the dipole lattice spacing *d* should be small enough, i.e., j j *m kd* ≤1, where *m* is the complex refractive index of the scattering target. (b) *d* must be small enough to refabricate accurately the NP shape. For small plasmonic NPs, or small inter-particle separations, *d* must be smaller than 1 nm.

Owing to the interaction among different NPs at the nanoscale and gravity at Earth, NPs are usually agglomerated due to Van der Waals force and then trend to be sediment at the bottom [34]. As a result, the agglomeration and sedimentation of NPs in the base fluid would affect the optical absorption and heat transfer performance, weakening the system efficiency. In addition, recent studies showed that the agglomeration or sedimentation can be worse under harsh operating conditions, such as: high temperature and pressure [35, 36]. Many methods were used to evaluate the stability of nanofluids, the simplest and direct method is the sedimentation method [18]. Interface electromotive force analysis is another common method to observe the stability of nanofluids, but this method is limited by the viscosity and concentration of the fluid [37]. Wang et al. [38] used an ultraviolet– visible spectrophotometer to study the stability of nanofluids. The NP concentration can be obtained by measuring the change in the light absorption rate of the system with the sedimentation time because the NP concentration is a linear relationship with the absorbance of nanofluid at the low concentration.

*Advances in Microfluidics and Nanofluids*

The stability of plasmonic nanofluids is also one of the major issues limiting the applications of nanofluids. Many researchers have made much efforts to improve the

The excellent optical absorption performance of plasmonic NPs make it to be a great candidate in the solar thermal conversion applications, which is critical for the solar thermal conversion applications. And the optical properties of nanofluids can be controlled by the NP size, shape, concentration and base fluid. In this section, we will discuss the optical properties of NPs or nanofluids from two aspects: theoretical

To achieve the optical properties of nanofluids, including transmittance, reflectance, and absorptance, the optical performance of single NP is usually determined

stability of the plasmonic nanofluids from the aspect of long-time and hightemperature dispersion [39]. For example, Au@SiO2 and Ag@SiO2 core-shell NPs were synthesized using a low-temperature two-step solution process. Results showed that the synthesized metal@SiO2 nanofluids exhibited excellent dispersion stability of 93.7% for Au@SiO2 and 100% for Ag@SiO2 in 6 months without using any surfactants, and they also showed a good thermal stability after thermal exposure at 150° C for an hour [40]. An ultrastable nanofluids with the broadband photothermal absorption was achieved using citrate and polyethylene glycol-coated Au NPs, circumventing the need for free surfactants. Electrostatic stabilization provided superior colloidal stability and more consistent optical properties; chemical and colloidal stability was verified for 16 months, the longest demonstration of stable nanofluids under ambient storage in the solar literature [41]. Besides the base fluid water used above, the base fluid oil was also studied to improve the stability. A facile and effective strategy, including controlled high-temperature synthesis of nanoparticles, surface modification of particles, and post-modification particle size partition, was designed to prepare stably dispersed silicone-oil-based nanofluids that enable high-temperature operation [42]. A low cost, and scalable method was reported to synthesize solar selective nanofluids from 'used engine oil' with the excellent long-term stability and photothermal conversion efficiency. Results showed that their stability and functional characteristics can retain even after extended periods (72hours) of high temperature (300°C) heating, ultra violet light exposure and thermal cyclic loading [43].

**3. Solar absorption of plasmonic nanofluids**

design and experimental design.

**3.1 Theoretical design**

**114**

**BEM:** Boundary element method (BEM) is another method to calculate the optical properties of plasmonic nanostructures, which was introduced by García de Abajo and Howie [54] using the following equations:

$$\phi(r) = \phi\_j^{\text{ext}}(r) + \int\_{S\_j} G\_j(|r - s|) \sigma\_j(s) ds \tag{4}$$

number of publications on plasmonic nanostructures done with COMSOL

*Solar Thermal Conversion of Plasmonic Nanofluids: Fundamentals and Applications*

pendent scattering can be applied in the calculation of nanofluids.

A short comparison of these above calculation methods for NPs are listed in **Table 1**. The choice of the calculation method depends on many factors, such as: NP size, shape and dielectric environment. But the general method, such as: FEM, and FDTD, can be used in most situations by applying the periodic boundary conditions

The optical properties of nanofluids also can be calculated based on the above method, such as: FDTD or FEM, which are the direct way to achieve the absorption performance of nanofluids without the strict assumptions. But the computation load is large for nanofluids since the geometry size (� mm) is much larger than NP size or mesh size (� nm). Therefore, the optical properties of nanofluids can be obtained from the optical properties of single NP due to the low NP concentration of plasmonic nanofluids in the solar thermal conversion applications and the inde-

One method is to avoid the scattering effect of NPs due to the small size, resulting in the negligible scattering effect in the nanofluids. Therefore, the absorption efficiency of the nanofluids can be obtained by the independent scattering

*<sup>k</sup>*<sup>a</sup>λ,nf <sup>¼</sup> *<sup>k</sup>*<sup>a</sup>λ,bf <sup>þ</sup> *<sup>k</sup>*<sup>a</sup>λ,np <sup>¼</sup> <sup>4</sup>πκ

where *k*a*<sup>λ</sup>*,nf, *k*a*<sup>λ</sup>*,bf, and *k*a*λ*,np are the absorption coefficients of the nanofluid, the base fluid water and the NPs respectively. *κ* is the absorption index of water. *f* <sup>v</sup> is the NP volume fraction. *V*np is the single NP volume. *C*abs is the absorption cross section of the NP. Based on the Beer–Lambert law [61], the radiation intensity decays exponentially along the transmission direction. Therefore, the solar

Ð <sup>2</sup>*:*5μ<sup>m</sup>

where *I*sð Þ*λ* is the solar spectra at AM 1.5. *I*absð Þ*λ* is the absorbed spectra. *H* is the

Mie theory Theoretical result Accurate solution Valid for simple shapes (such as:

Fast calculation than volume methods

λ þ

<sup>0</sup>*:*3μ<sup>m</sup>*I*sð Þ<sup>λ</sup> <sup>1</sup> � <sup>e</sup>�*Hk*a*λ*,nf � �d*<sup>λ</sup>*

sphere)

large memory space for large NPs

dielectric functions; Time-consuming

Complex parameterization of boundary elements

Arbitrary shape Time-consuming;

General application Time-consuming; large

memory space for complex structures

General application Computational stability depends on

for spectral calculations

Ð <sup>2</sup>*:*5μ<sup>m</sup> <sup>0</sup>*:*3μ<sup>m</sup>*I*sð Þ*λ* d*λ*

fv*C*abs *V*np

(7)

(8)

Multiphysics has appeared in the literature.

*DOI: http://dx.doi.org/10.5772/intechopen.96991*

approximation, which can be described as [9]:

absorption efficiency *η*abs can be calculated as:

Ð <sup>2</sup>*:*5μ<sup>m</sup>

Ð <sup>2</sup>*:*5μ<sup>m</sup>

<sup>0</sup>*:*3μ<sup>m</sup>*I*absð Þ*λ* d*λ*

**Method Description Advantages Limitations**

<sup>0</sup>*:*3μ<sup>m</sup>*I*sð Þ*<sup>λ</sup>* <sup>d</sup>*<sup>λ</sup>* <sup>¼</sup>

*η*abs ¼

DDA Approximate result

BEM Numeric solution with

FEM Differential equations

FDTD Discretization of

Yee cell

based on discrete dipoles

surface discretization

solved over a domain

Maxwell equation with

*Comparison of calculation methods for plasmonic NPs [60].*

depth.

**Table 1.**

**117**

at the sides of the unit cell.

$$A(r) = A\_j^{ext}(r) + \int\_{S\_j} G\_j(|r - s|) h\_j(s) ds \tag{5}$$

where *ϕ*ð Þ*r* is the electric potential, *A r*ð Þ is the vector potential, *σ <sup>j</sup> r*<sup>0</sup> ð Þ is the surface charge density and *h <sup>j</sup> r*<sup>0</sup> ð Þ is the surface current density and *G <sup>j</sup>* is the Green's function of Helmholtz equation inside each homogeneous medium of dielectric function. *S <sup>j</sup>* is the boundary of the medium *j*. *s* is the point of the boundary between medias. *<sup>r</sup>* is the point inside the medium. *<sup>ϕ</sup>ext*ð Þ*<sup>r</sup>* and *Aext*ð Þ*<sup>r</sup>* are the potentials at the interface caused by external sources and the full space is filled by a homogeneous medium *j*. Therefore, a much smaller number of elements is required to evaluate the fields than volume integral based methods, but involving a complex parameterization of the boundary elements.

**FDTD:** Finite-difference time-domain (FDTD) is one of the most popular optical calculation methods in plasmonic nanostructures, which is first developed by Yee in 1966 [55]. The basis of the model is from Maxwell equations in electrodynamics. The second-order precision central difference is used to approximate the discretization of the differential form of Maxwell equations, thereby a set of timedomain propulsion formulas can be used to deal with electromagnetic wave propagation problems. Since the FDTD method directly discretizes the time-domain wave equation, it will not limit its application range due to mathematical models, and can effectively simulate various complex structures.

The popularity of this method has strongly increased in the last two decades, mainly due to the simplicity of implementation, support of arbitrary NP shape, allowing to investigate linear and non-linear properties of NPs, using Maxwell's equations directly without approximations. There are, however, some undesired effects, like the staircase of fields in non-rectangular boundaries, mainly in code implementations without adaptive meshing. To avoid this, very fine discretization or sub-pixel smoothing of the dielectric function must be applied [56]. The dielectric function of the materials requires analytical expressions (e.g., Drude–Lorentz) [57].

**FEM**: Finite element method (FEM) was developed to solve differential equations of boundary-value problems [58]. Physical problems described by differential equations over a domain, like for example the Helmholtz equation in real threedimensional space. Hence, electromagnetic (EM) field propagation around the single NP can be described by the Helmholtz Equation [59]:

$$\nabla \times \left( \mu\_\mathbf{r}^{-1} \nabla \times \mathbf{E} \right) - k\_0^2 \varepsilon\_\mathbf{r} \mathbf{E} = \mathbf{0} \tag{6}$$

where *E* is the electric field of the medium, *j* is the current density, *k*<sup>0</sup> is the wavenumber, *<sup>ε</sup>*<sup>r</sup> is the dielectric function, which is calculated as *<sup>ε</sup>*<sup>r</sup> <sup>¼</sup> ð Þ *<sup>n</sup>* � *ik* <sup>2</sup> , *n* and *k* are the complex refractive indices. Within each element *E* is approximated using a basis function expansion *<sup>E</sup>* <sup>¼</sup> <sup>P</sup>*<sup>n</sup> <sup>j</sup>*¼<sup>1</sup>*Njξj*, where the sum is over *<sup>n</sup>* interpolation point. *N<sup>j</sup>* is chosen basis function and *ξ<sup>j</sup>* is the unknown coefficient. A solution can be obtained by using the variational principle to determine *ξj*. To obtain a meaningful solution, *N<sup>j</sup>* is required to satisfy Gauss's law and appropriate boundary conditions on the surface of all elements. During the last decade, an increasing
