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Abstract
Noble metal nanoparticle clusters show unique light absorption and catalysis properties, which have been widely used in the application of photocatalysis, optoelectronics, biomedical optics and so on. The absorption cross section of densely packed nanoparticle clusters, which can be enhanced or restricted due to the near field effects needs to be studied thoroughly. In this work, focusing on Au nanoparticle at the localized plasmon resonance wavelength, the effects of monomer diameter D, monomer number N, particle volume fraction Fv and complex refractive index m on the nondimensional absorption cross section η = Cabs,total/(N·Cabs) (normalized by N and the absorption cross section Cabs of a single particle) of densely packed nanoparticle clusters are studied by using the superposition T-matrix method. It is found that the enhancement (η > 1) and restriction (η < 1) mechanisms of the absorption cross section of nanoparticle clusters are determined by two competing factors (i.e. the multiple scattering and shielding effect), and the extent of these two mechanisms is mainly dependent on the monomer size parameter and the monomer number. The effect of the particle volume fraction on the nondimensional absorption cross section of nanoparticle clusters is totally different in different mechanisms. Specifically, the nondimensional absorption cross section peaks at the particle volume fraction of about 50% in the enhancement mechanism (in our calculation: D < 14 nm, N = 100), while in the restriction mechanism it decreases monotonously with increasing particle volume fraction. Moreover, the absorption efficiency of nanoparticle clusters with more absorptive monomer decreases more sharply with increasing particle volume fraction. The complex refractive index of particle shows significant effects on the nondimensional absorption cross section of nanoparticle clusters, and the largest nondimensional absorption cross section of nanoparticle clusters (N = 100) is larger than 8.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Noble metal nanoparticles due to their excellent light absorption ability have been widely used in the field of photocatalysis, photothermal, optoelectronics, biomedical optics, and cancer therapy, to name a few [1–4]. The aggregation of noble metal nanoparticles is always rigorously avoided, since that it leads to the decrease of surface area, the restriction of absorption efficiency, the particle deposition and so on. Therefore, noble metal nanoparticles are always well dispersed into fluids ensuring that light scattering of particle is independent, and the absorption cross section of a single particle only depends on the shape, size and component of particle [5–7]. Nowadays, noble metal nanoparticle clusters show unique properties, which have been widely used in the application of photocatalysis. For example, Cui et al. [8] recently proposed that Au nanoclusters show high catalytically active for visible light CO2 reduction by grafting metal cations to the Au nanoclusters. In addition, technologies of self-assembly and organization of Au nanoparticles into clusters with various aggregative morphologies and specified particle spacing have been developed. The self-assembly Au nanoparticles clusters show unique absorption properties and have been widely applied in optoelectronics, biosensors, and nanoelectronics [9–11].
The light absorption property of Au nanoparticle clusters plays a vital role in determining the efficiency of nanofluids in the application of photocatalysis, photothermal, biomedical optics and so on. For densely packed nanoparticle system in which particles are in close proximity, light scattering and absorption can differ from predictions using properties for independent scattering and absorption of a single particle [12]. This dependent scattering arises from two effects [12]: one is the near field effect that the internal radiation field in a translucent particle is affected by scattering from surrounding particles; the second effect is the far field effect that scattered radiation from one particle can constructively or destructively interfere with that from another particle. Alnimr et al. [13] proposed that the near field effect on the scattered radiation is negligible compared to the far field effect, while the dependent absorption efficiency of nanoparticles is affected by the near field effect only. Kumar and Tien [14] found that the near field effect increases the radiation absorbed by each particle due to the enhancement in the internal field of each particle, and the absorption efficiencies of particles increase obviously with the decrease of the inter-particle distance. Actually, the enhanced absorption of particles arises mainly from the multiple scattering in the near field for disordered nanoparticle system. Ma et al. [15] proposed that the multiple scattering between nanoparticles leads to the decrease of scattering and increase of absorption for lossy (with intrinsic absorption) nanoparticles.
It should be noted that in the near field the shielding effect also largely impacts the absorption cross section of nanoparticle clusters as well as the multiple scattering. The shielding effect which indicates that the incident light is shielded by other particle in vicinity, leads to a significant decrease of absorption efficiency of nanoparticle compared with that of independent one [16–22]. It is worth noting that the mutual shielding may manifest itself in the systems of particles smaller than the wavelength rather than only for the particles comparable to or larger than the wavelength [23,24]. The multiple scattering and the shielding effect are two competing factors resulting in an enhanced or restricted absorption cross section of nanoparticle clusters. Mulholland et al. [16,17] proposed that for small spheres the exciting field is increased from dipole-dipole coupling resulting in an increase in the absorption cross section, but for large spheres the dipole-dipole coupling leads to a shielding effect. Liu et al. [18] also proposed that electromagnetic interactions between monomers enhance the light absorption of particle aggregates, while the shielding effects reduce the absorption cross section. Liu et al. [20,21] found that the dipole-dipole coupling effects (alternatively, interactions among primary particles, or multiple scattering effects) are stronger as fractal dimensions increase for particle aggregate clusters. It is also pointed out that the effect of fractal dimensions on the nondimensional absorption cross section (normalized by the monomer number and the absorption cross section of a single particle) is weaker than the effect of primary particle size parameter.
The multiple scattering and shielding effect are the fundamental factors for enhancement (η > 1) and restriction (η < 1) mechanisms of the absorption cross section of nanoparticle clusters. Nevertheless, the extent of these two competing factors in the near field are not well investigated. The near field means: firstly, the distance between particles is very small (i.e. the densely packed nanoparticle system, rather than the sparse nanoparticle system), which is in the sub-wavelength range; secondly, the particle is located on the field of the scattered field by other particles. In the near field, the scattered wave composed of the traveling waves and evanescent waves, while in the far field the scattered wave only composed of the traveling waves. The enhancement and restriction mechanisms of the absorption cross section of nanoparticle clusters need to be investigated systematically with a simple but typical nanoparticle cluster model, taking into account the effects of monomer size, monomer number, nanoparticle volume fraction and the complex refractive index of particle. Therefore, the aim of this paper is to comprehensively study the near field effect on the nondimensional absorption cross section η = Cabs,total/(N·Cabs) (normalized by the monomer number N and the absorption cross section Cabs of a single particle) of densely packed nanoparticle clusters. The superposition T-matrix (STM) method was applied to calculate the absorption cross section of nanoparticle clusters. Focusing on the Au nanoparticle with the diameter D ranging from 1 to 100 nm at the localized plasmon resonance wavelength (520 nm), comparisons of the absorption property of nanoparticle clusters were made among nanoparticle clusters with monomer number N ranging from 100 to 1000 and particle volume fraction Fv ranging from 10% to 70%. In addition, the effect of the complex refractive index m on the absorption property of nanoparticle clusters was analyzed.
2. Cluster model and calculation method
Nanoparticle clusters often exhibit several morphologic structures including artificially periodic structure, fractal aggregate and so on. In this work, models in which nanoparticles are randomly and uniformly distributed in an imaginary spherical volume with specified volume fraction are used to investigate the absorption properties of nanoparticle clusters. The optical thickness of the cluster model is identical under different incident direction when the incident direction passes through the center of nanoparticle cluster. Therefore, the cluster model with spherical overall shape and monomers is very suitable for the investigation of dependent scattering and absorption of nanoparticle system, and has been used by many scholars [23–27]. To generate ensembles of randomly distributed spherical nanoparticles in an imaginary spherical volume, the Packmol software [28,29] which is good for creating high particle volume fraction configurations is used. Figure 1 presents a target configuration for an imaginary spherical volume populated by N = 100 spheres with different particle volume fractions Fv. Note that the spheres in the simulation volume are not allowed to be overlapped and cross the volume's outer boundary. The particle volume fraction Fv is defined as:
where R is the radius of the imaginary spherical volume, N is the monomer number and D is the monomer diameter.
Fig. 1 Spherical particle target sample used in MSTM computation: (a) Fv = 20%; (b) Fv = 40%; (c) Fv = 60%. The monomer number N is 100.
The parallelized Multiple Sphere T-Matrix (MSTM) implementation developed by Mackowski and Mishchenko [30,31], which is based on the Superposition T-Matrix (STM) method is used in this work. The STM method is a numerically exact and highly efficient method in dealing with the electromagnetic scattering of spherical particle clusters, and the particles can be exactly defined by the positions and sizes of the spheres in the particle clusters. The MSTM implementation has been widely used and validated for applications related to clusters of spheres. The absorption cross section Cabs,i of ith sphere can be obtained:
where k is the wavenumber of incident light, bnp,i is a positive (or zero) real valued property solely of ith sphere, the amnp coefficients correspond to either the parallel or perpendicular polarization values, and Li is the maximum expansion order of vector spherical wave function for each sphere. Finally, the absorption cross section of the entire ensemble Cabs,total is:In order to well compare the absorption ability of the particle cluster and individual particle, the nondimensional absorption cross section η is defined as the ratio of the total absorption cross section of the densely packed particle clusters and a single spherical particle in the same environment:
where Cabs is the absorption cross section of a single spherical particle which is calculated by the Mie theory [5]. The refractive index of the surrounding media is supposed to be 1.33 (i.e. water). The nondimensional absorption cross section η shows the comparison between dependent and independent light absorption of particles. In addition, the absorption efficiency of densely packed nanoparticle cluster is defined as:The vector spherical wave function expansions for the fields within and scattered by the individual spheres need to contain sufficient harmonic orders to obtain the converged solution. The MSTM will automatically set the maximum order for each sphere based on the single sphere Mie criteria. The default maximum order of vector wave function expansions in MSTM may be insufficient for clustered Rayleigh spheres having large real and imaginary parts of refractive index. Figure 2(a) shows the nondimensional absorption cross section of nanoparticle clusters with different expansion order of vector spherical wave function for each sphere, the volume fraction is 50% and the complex refractive index of particle is m = 0.58 + 2.18i. Figure 2(b) shows the spectral absorption efficiency of Au nanoparticle clusters in the spectral range from 300 to 800 nm with different expansion order of vector spherical wave function for each sphere, the volume fraction is 50% and the monomer diameter is 10 nm. Figure 2(c) and 2(d) shows the nondimensional absorption cross section of nanoparticle clusters with different complex refractive index as a function of expansion order. The default expansion order of vector wave function for small monomer (D < 20 nm) is 2. As shown in Fig. 2, the converged solution can be obtained when the expansion order of vector wave function is larger than 5. In this work, all results are calculated with the expansion order of vector wave function equal to 5. The complex refractive index of Au [32] in the spectral range from 300 to 800 nm is shown in Fig. 3.
Fig. 2 (a) The nondimensional absorption cross section of nanoparticle clusters with different expansion order of vector spherical wave function for each sphere, the volume fraction is 50% and the complex refractive index of particle is m = 0.58 + 2.18i. (b) The spectral absorption efficiency of Au nanoparticle clusters in the spectral range from 300 to 800 nm with different expansion order of vector spherical wave function for each sphere, the volume fraction is 50% and the monomer diameter is 10 nm. (c-d) The nondimensional absorption cross section of nanoparticle clusters with different complex refractive index as a function of expansion order.
The absorption cross sections of nanoparticle clusters calculated in this work are orientation averaged. In MSTM, the random orientation cross sections can be obtained by using the matrix relationships for the scattered and incident field, and integrating the incident field over all propagation and polarization directions. In addition, the results are realization averaged over at least five realizations of nanoparticle clusters, which is large enough to obtain convergent results for realizations average.
3. Results and analysis
In this work, the effects of monomer diameter D, monomer number N and particle volume fraction Fv on the nondimensional absorption cross section η = Cabs,total/(N·Cabs) of densely packed nanoparticle clusters were studied firstly with nanoparticle complex refractive index m = 0.58 + 2.18i, corresponding to that of Au nanoparticle at the localized plasmon resonance wavelength of 520 nm [32]. The diameter of Au nanoparticle ranges from 1 to 100 nm. It should be noted that the use of the bulk permittivity is usually sufficient and justified to describe the physical properties of Au nanoparticle [33–35]. Thereafter, the spectral absorption efficiency of Au nanoparticle clusters with different monomer diameters and particle volume fractions are investigated in the spectral range from 300 to 800 nm. Meanwhile, the effect of the complex refractive index on the nondimensional absorption cross section η = Cabs,total/(N·Cabs) of densely packed nanoparticle clusters is studied.
3.1 Effect of monomer size, volume fraction and monomer number
Figure 4 illustrates the nondimensional absorption cross section η = Cabs,total/(N·Cabs) of densely packed nanoparticle clusters with different particle volume fractions Fv as a function of monomer diameter D, the monomer number N is 100 and the complex refractive index of particle is m = 0.58 + 2.18i. As shown in Fig. 4, the nondimensional absorption cross section is larger than unity for nanoparticle clusters with monomer diameter smaller than about 14 nm, and smaller than unity for nanoparticle clusters with monomer diameter larger than about 14 nm. As mentioned by Mulholland et al. [16] and Liu et al. [20,21], there are two competing effects affecting the absorption property of nanoparticle clusters: the multiple scattering which enhances the absorption cross section; the shielding effect which reduces the absorption cross section. Figure 5 presents the normalized electric field intensity |E/E0|2 of nanoparticle clusters with monomer diameter of (a) 10 nm and (b) 40 nm. As shown in Fig. 5, the normalized electric field intensity inside the nanoparticle cluster with monomer diameter of 10 nm is comparable to that of the boundary area, while for nanoparticle cluster with monomer diameter of 40 nm the normalized electric field intensity inside the cluster is almost zero, implying the strong shielding effect for nanoparticle cluster with large monomer diameter. Moreover, the normalized electric field intensity of nanoparticle cluster with monomer diameter equals to 10 nm is almost four times to that of nanoparticle cluster with monomer diameter equals to 40 nm, indicating that the intensity of multiple scattering inside the nanoparticle cluster with small monomer diameter is much stronger than that of nanoparticle cluster with larger monomer diameter.
Fig. 4 The nondimensional absorption cross section η = Cabs,total/(N·Cabs) of nanoparticle clusters with different particle volume fractions Fv as a function of monomer diameter D. The monomer number N is 100, and the complex refractive index of particle is m = 0.58 + 2.18i.
Fig. 5 The normalized electric field intensity |E/E0|2 of nanoparticle clusters with monomer diameter of (a) 10 nm and (b) 40 nm, the monomer number N is 100 and the particle volume fraction Fv is 50%. E0 is the incident electric field intensity.
Figure 4 also shows that for all the particle volume fractions Fv considered the curves show a similar trend: with increasing monomer diameter, the nondimensional absorption cross section increases in the diameter range from 1 to 8 nm, then decreases sharply after it reaches a peak in the monomer diameter range from 8 to 40 nm, and reaches a plateau for monomer diameter larger than 40 nm. This phenomenon is similar to the results of Ivezić and Mengüç [36], and can be explained as below. The intensity of multiple scattering largely depends on the scattering ability of a single monomer, which increases with increasing monomer diameter obviously. For monomer with extremely small size (D < 8 nm), the shielding effect is weak and the increased scattering cross section of a single monomer leads to the increased intensity of multiple scattering, thus the nondimensional absorption cross section increases with increasing monomer diameter. However, for large monomer (D > 8 nm), the significant shielding effect leading to the sharply decreased nondimensional absorption cross section of nanoparticle clusters. In addition, for cluster with monomer diameter larger than 40nm, the monomer in the inner part of cluster is totally shielded, with almost no incident light can arrive at the surfaces of the inner monomers as shown in Fig. 5(b). The light absorption cross section of nanoparticle clusters only depends on the light absorption of nanoparticles in the outer part, resulting a nearly invariable nondimensional absorption cross section.
It is really interesting that the effect of particle volume fraction on the nondimensional absorption cross section of nanoparticle clusters is totally different in the enhancement and restriction mechanism. Figure 6 shows the nondimensional absorption cross section η = Cabs,total/(N·Cabs) of nanoparticle clusters with monomer diameter of 10 and 40 nm as a function of particle volume fraction Fv, the monomer number N is 100 and the complex refractive index of particle is m = 0.58 + 2.18i. As shown in Fig. 6, for nanoparticle cluster with monomer diameter of 40nm, the nondimensional absorption cross section decreases monotonously with increasing particle volume fraction and varies little for volume fraction larger than 40%. It is easily to understand that for large nanoparticle the shielding effect increases with increasing particle volume fraction, i.e. decreasing inter-particle clearance. For nanoparticle cluster with monomer diameter of 10 nm, the nondimensional absorption cross section increases in the particle volume fraction range from 10% to 50%. Similar phenomenon is observed in the results of Liu et al. [21] and can be explained as below. The inter-particle clearance decreases with increasing particle volume fraction [18], therefore, the intensity of multiple scattering inside the cluster increases, leading to the increased absorption cross section. Moreover, the nondimensional absorption cross section decreases with increasing particle volume fraction when the particle volume fraction larger than 50%. This can be explained that nanoparticles are easily get attached with each other forming larger particles in the disordered nanoparticle system with extremely large particle volume fraction, which reduces the absorption cross section.
Fig. 6 The nondimensional absorption cross section η = Cabs,total/(N·Cabs) of nanoparticle clusters with monomer diameter of 10 nm and 40 nm as a function of particle volume fraction Fv. The monomer number N is 100, and the complex refractive index of particle is m = 0.58 + 2.18i.
Figure 7 shows the nondimensional absorption cross section η = Cabs,total/(N·Cabs) of densely packed nanoparticle clusters with different monomer numbers N as a function of monomer diameter D, the particle volume fraction Fv is 50% and the complex refractive index of particle is m = 0.58 + 2.18i. As shown in Fig. 7, similar pattern is observed for the curves with different monomer numbers. As the monomer number increases, the nondimensional absorption cross section rises faster to reach a peak at a somewhat smaller nanoparticle diameter and then decays faster after the peak. It can be explained that for nanoparticle cluster with larger nanoparticle number, the electromagnetic interactions are occurred between more monomers and simultaneously more monomers are shielded. The largest nondimensional absorption cross section of densely packed nanoparticle clusters is almost independent of monomer number. In addition, for nanoparticle clusters with extremely small monomer, the effect of monomer number on the nondimensional absorption cross section is weak, which is in line with the analysis by Kumar et al. [14].
Fig. 7 The nondimensional absorption cross section η = Cabs,total/(N·Cabs) of nanoparticle clusters with different monomer numbers N as a function of monomer diameter D. The particle volume fraction Fv is 50%, and the complex refractive index of particle is m = 0.58 + 2.18i.
3.2 Effect of complex refractive index of particle
The radiative properties of a single nanoparticle are also the fundamental mechanism for the nanoparticle cluster absorption enhancement or restriction as well as the primary nanoparticle interactions. The radiative properties of a single spherical nanoparticle largely depend on the complex refractive index of nanoparticle as well as the size parameter. Meanwhile, previous study has revealed that the complex refractive index of nanoparticle exhibits significant effect on the dependent scattering and absorption of densely packed nanoparticle clusters [26]. Considering the widely application of the Au nanoparticles in the solar energy utilization, the spectral absorption efficiency of Au nanoparticle clusters with different particle volume fraction is investigated.
Figure 8 shows the spectral absorption efficiency Qabs of Au nanoparticle clusters with different particle volume fractions Fv in the spectral range from 300 to 800 nm, the monomer number N is 100. As shown in Fig. 8, for nanoparticle clusters with monomer diameter of 10 nm, the pattern of the spectral absorption efficiency is similar to that of a single Au nanoparticle. The absorption efficiency of Au nanoparticle cluster increases slightly with increasing nanoparticle volume fraction in the spectral range from 300 to 800 nm. In addition, the peak position of the absorption efficiency of Au nanoparticle cluster slightly red shifts with increasing particle volume fraction.
Fig. 8 The spectral absorption efficiency Qabs of Au nanoparticle clusters with different particle volume fractions Fv, the monomer number N is equal to 100.
For Au nanoparticle with diameter of 40nm, the aggregation of Au nanoparticles leads to significant decrease of the absorption in the spectral range of 300 to 600 nm, especially at the localized plasmon resonance wavelength of about 520 nm. The absorption efficiency of nanoparticle cluster with more absorptive monomer decreases more sharply with increasing volume fraction. This feature is also pointed out in the analysis of Liu et al. [18] and Okada et al. [27]. Liu et al. [18] showed that nondimensional absorption cross section of aggregates with more absorptive spherules (m = 2 + i) is smaller than that of aggregates with less absorbing monomers (m = 1.75 + 0.5i). Okada et al. [27] proposed that the absorption cross section of particle cluster decreases more rapidly with increasing particle volume fraction for particle with larger imaginary part k of complex refractive index. Interestingly, the absorption efficiency of Au nanoparticle cluster is still larger than that of a single Au nanoparticle and tends to increase with increasing nanoparticle volume fraction for the wavelength larger than 600 nm. This may due to the decreased size parameter with increasing wavelength and the weak light absorption of Au nanoparticle for wavelength larger than 600 nm.
Figure 9 shows the nondimensional absorption cross section η = Cabs,total/(N·Cabs) of nanoparticle clusters as a function of the real part n and the imaginary part k of complex refractive index with different nanoparticle size parameters x, the monomer number N is equal to 100. The line inside is the contour lines of the nondimensional absorption cross section with value of 1. As can be seen in Fig. 9, for all the complex refractive index considered, the nondimensional absorption cross section is always larger than 1 for the size parameter x = 0.01 and 0.1, while for the size parameter x = 1 it is always smaller than 1, indicating the significant effect of the size parameter on the nondimensional absorption cross section of nanoparticle cluster. The effect of complex refractive index on the nondimensional absorption cross section of nanoparticle clusters is different for nanoparticle clusters with different monomer size parameters. Meanwhile, the effect of complex refractive index on the boundary between the enhancement (η > 1) and restriction (η < 1) mechanisms is significant only for a moderate monomer size parameter (x = 0.5 in our calculated case). For all the size parameters considered, the largest nondimensional absorption cross section is observed for particle with real part n of complex refractive index smaller than 0.5 and a relative large imaginary part k of complex refractive index. Such complex refractive index corresponds to Au in the spectral range from about 600 nm to near infrared region and TiO2 in the far infrared spectral range by example. In addition, the largest nondimensional absorption cross section of clusters (N = 100 in our calculated case) is larger than 8. Moreover, the cases for clusters with different monomer numbers are also calculated but not illustrated due to that similar phenomenon is observed with that of nanoparticle cluster with monomer number N = 100.
Fig. 9 The nondimensional absorption cross section η = Cabs,total/(N·Cabs) of nanoparticle clusters as a function of complex refractive index with the monomer size parameter x = 0.01, 0.1 0.5 and 1, respectively. The monomer number N = 100 and the particle volume fraction Fv is 50%. The line inside is the contour line of the nondimensional absorption cross section with value of 1.
4. Conclusions
In conclusion, the nondimensional absorption cross section η = Cabs,total/(N·Cabs) of densely packed nanoparticle clusters was studied by using the superposition T-matrix (STM) method. Focusing on the Au nanoparticle with diameter D ranging from 1 to 100 nm at the localized plasmon resonance wavelength (m = 0.58 + 2.18i at wavelength of 520 nm), the nondimensional absorption cross section of densely packed nanoparticle clusters was studied with monomer number ranging from 100 to 1000 and particle volume fraction ranging from 10% to 70%. In addition, the effect of the complex refractive index on the nondimensional absorption cross section of densely packed nanoparticle clusters was analyzed.
The enhancement (η > 1) and restriction (η < 1) mechanisms of the absorption cross section of clusters are determined by the two competing factors, i.e. the multiple scattering and shielding effect. The extent of these two mechanism is mainly dependent on the monomer size parameter and the monomer number. The effect of particle volume fraction on the nondimensional absorption cross section of Au nanoparticle clusters is totally different in different mechanisms. The nondimensional absorption cross section peaks at the particle volume fraction of about 50% in the enhancement mechanism (in our calculation: D < 14 nm, N = 100), while in the restriction mechanism it decreases monotonously with increasing particle volume fraction. In addition, the largest nondimensional absorption cross section of densely packed nanoparticle cluster is almost independent of monomer number N. We further reveal that the absorption efficiency of particle cluster with more absorptive monomer decreases more sharply with increasing particle volume fraction. Moreover, the effect of complex refractive index of monomer on the nondimensional absorption cross section is significant. In addition, the largest nondimensional absorption cross section of nanoparticle clusters (N = 100 in our calculated case) is larger than 8.
Funding
National Natural Science Foundation of China (51336002, 51806124, 51421063).
References
1. A. Kasaeian, A. T. Eshghi, and M. Sameti, “A review on the applications of nanofluids in solar energy systems,” Renewable and Sustainable Energy Reviews. 43, 584–598 (2015).
2. O. Mahian, A. Kianifar, S. A. Kalogirou, and S. W. I. Pop, “A review of the applications of nanofluids in solar energy,” Int. J. Heat Mass Transf. 57, 582–594 (2013).
3. B. W. Xie, J. Dong, J. M. Zhao, and L. H. Liu, “Radiative properties of hedgehog-like ZnO-Au composite particles with applications to photocatalysis,” J. Quant. Spectrosc. Radiat. Transf. 217, 1–12 (2018). [CrossRef]
4. B. Khlebtsov, V. Zharov, A. Melnikov, V. Tuchin, and N. Khlebtsov, “Optical amplification of photothermal therapy with gold nanoparticles and nanoclusters,” Nanotechnology 17(20), 5167–5179 (2006). [CrossRef]
5. C. F. Bohren and D. R. Huffman, Absorption and scattering of light by small particles (Wiley, 1983).
6. M. I. Mishchenko, A. A. Lacis, and L. D. Travis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge University, 2002).
7. M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Multiple Scattering of Light by Particles: Radiative Transfer and Coherent Backscattering (Cambridge University, 2006).
8. X. Cui, J. Wang, B. Liu, S. Ling, R. Long, and Y. Xiong, “Turning Au Nanoclusters Catalytically Active for Visible-Light-Driven CO2 Reduction through Bridging Ligands,” J. Am. Chem. Soc. 140(48), 16514–16520 (2018). [CrossRef] [PubMed]
9. Y. X. Zhang and H. C. Zeng, “Surfactant-Mediated Self-Assembly of Au Nanoparticles and Their Related Conversion to Complex Mesoporous Structures,” Langmuir 24(8), 3740–3746 (2008). [CrossRef] [PubMed]
10. J. Lee, H. Zhou, and J. Lee, “Small molecule induced self-assembly of Au nanoparticles,” J. Mater. Chem. 21(42), 16935–16942 (2011). [CrossRef]
11. B. L. Frankamp, A. K. Boal, and V. M. Rotello, “Controlled interparticle spacing through self-assembly of Au nanoparticles and poly(amidoamine) dendrimers,” J. Am. Chem. Soc. 124(51), 15146–15147 (2002). [CrossRef] [PubMed]
12. J. R. Howell, M. P. Mengüç, and S. D. Robert Siegel, Thermal Radiation Heat Transfer (CRC Press, 2010).
13. M. A. Al-Nimr and V. S. Arpaci, “Radiative properties of interacting particles,” J. Heat Transfer 114(4), 950–957 (1992). [CrossRef]
14. S. Kumar and C. L. Tien, “Dependent absorption and extinction of radiation by small particles,” J. Heat Transfer 112(1), 178–185 (1990). [CrossRef]
15. Y. Ma, V. K. Varadan, and V. V. Varadan, “Enhanced Absorption Due to Dependent Scattering,” J. Heat Transfer 112(2), 402–407 (1990). [CrossRef]
16. G. W. Mulholland, C. F. Bohren, and K. A. Fuller, “Light Scattering by Agglomerates: Coupled Electric and Magnetic Dipole Method,” Langmuir 10(8), 2533–2546 (1994). [CrossRef]
17. G. W. Mulholland and R. D. Mountain, “Couple dipole calculation of extinction coefficient and polarization ratio for smoke agglomerates,” Combust. Flame 119(1-2), 56–68 (1999). [CrossRef]
18. L. Liu, M. I. Mishchenko, and W. P. Arnott, “A study of radiative properties of fractal soot aggregates using the superposition T-matrix method,” J. Quant. Spectrosc. Radiat. Transf. 109(15), 2656–2663 (2008). [CrossRef]
19. J. Yon, C. Rozé, T. Girasole, A. Coppalle, and L. Méès, “Extension of RDG-FA for Scattering Prediction of Aggregates of Soot Taking into Account Interactions of Large Monomers,” Part. Part. Syst. Charact. 25(1), 54–67 (2008). [CrossRef]
20. F. Liu and G. J. Smallwood, “Effect of aggregation on the absorption cross-section of fractal soot aggregates and its impact on LII modelling,” J. Quant. Spectrosc. Radiat. Transf. 111(2), 302–308 (2010). [CrossRef]
21. F. Liu, C. Wong, D. R. Snelling, and G. J. Smallwood, “Investigation of Absorption and Scattering Properties of Soot Aggregates of Different Fractal Dimension at 532 nm Using RDG and GMM,” Aerosol Sci. Technol. 47(12), 1393–1405 (2013). [CrossRef]
22. J. Dong, J. M. Zhao, and L. H. Liu, “Morphological effects on the radiative properties of soot aerosols in different internally mixing states with sulfate,” J. Quant. Spectrosc. Radiat. Transf. 165, 43–55 (2015). [CrossRef]
23. V. P. Tishkovets, E. V. Petrova, and M. I. Mishchenko, “Scattering of electromagnetic waves by ensembles of particles and discrete random media,” J. Quant. Spectrosc. Radiat. Transf. 112(13), 2095–2127 (2011). [CrossRef]
24. V. P. Tishkovets and E. V. Petrova, Light Scattering Reviews 7: Light scattering by densely packed systems of particles: near-field effects (Springer, 2013).
25. L. X. Ma, J. Y. Tan, J. M. Zhao, F. Q. Wang, and C. A. Wang, “Multiple and dependent scattering by densely packed discrete spheres: Comparison of radiative transfer and Maxwell theory,” J. Quant. Spectrosc. Radiat. Transf. 187, 255–266 (2017). [CrossRef]
26. L. X. Ma, J. Y. Tan, J. M. Zhao, F. Q. Wang, C. A. Wang, and Y. Y. Wang, “Dependent scattering and absorption by densely packed discrete spherical particles: Effects of complex refractive index,” J. Quant. Spectrosc. Radiat. Transf. 196, 94–102 (2017). [CrossRef]
27. Y. Okada and A. A. Kokhanovsky, “Light scattering and absorption by densely packed groups of spherical particles,” J. Quant. Spectrosc. Radiat. Transf. 110(11), 902–917 (2009). [CrossRef]
28. L. Martínez, R. Andrade, E. G. Birgin, and J. M. Martínez, “Packmol: A package for building initial configurations for molecular dynamics simulations,” J. Comput. Chem. 30(13), 2157–2164 (2009). [CrossRef] [PubMed]
29. J. M. Martínez and L. Martínez, “Packing optimization for automated generation of complex system’s initial configurations for molecular dynamics and docking,” J. Comput. Chem. 24(7), 819–825 (2003). [CrossRef] [PubMed]
30. D. W. Mackowski, “A general superposition solution for electromagnetic scattering by multiple spherical domains of optically active media,” J. Quant. Spectrosc. Radiat. Transf. 133, 264–270 (2014). [CrossRef]
31. D. W. Mackowski and M. I. Mishchenko, “A multiple sphere T -matrix Fortran code for use on parallel computer clusters,” J. Quant. Spectrosc. Radiat. Transf. 112(13), 2182–2192 (2011). [CrossRef]
32. E. D. Palik, Handbook of optical constants of solids (Academic, 1985).
33. C. Burda, T. C. Green, S. Link, and M. A. El-Sayed, “Electron Shuttling Across the Interface of CdSe Nanoparticles Monitored by Femtosecond Laser Spectroscopy,” J. Phys. Chem. B 103(11), 1783–1788 (1999). [CrossRef]
34. J. H. Hodak, A. Henglein, and G. V. Hartland, “Electron-phonon coupling dynamics in very small (between 2 and 8 nm diameter) Au nanoparticles,” J. Chem. Phys. 112(13), 5942–5947 (2000). [CrossRef]
35. G. Baffou, R. Quidant, and C. Girard, “Thermoplasmonics modeling: A Green’s function approach,” Phys. Rev. B Condens. Matter Mater. Phys. 82(16), 165424 (2010). [CrossRef]
36. Z. Ivezić and M. P. Mengüç, “An investigation of dependent/independent scattering regimes using a discrete dipole approximation,” Int. J. Heat Mass Transf. 39(4), 811–822 (1996). [CrossRef]
