# Numerical Modelling of the Optical Properties of Plasmonic and Latex Nanoparticles to Improve the Detection Limit of Immuno-Turbidimetric Assays

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*Nanomaterials*—Recent Advances in Nanofabrication and Nanomanufacturing)

## Abstract

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## 1. Introduction

## 2. Materials and Methods

_{ext}) of NPs with different composition, size, and arrangement in aqueous solution were evaluated with the Discrete Dipole Approximation (DDA) method [34,35]. In DDA, the structure of interest, called “target”, is modelled with a number N of polarizable points (dipoles) arranged in a cubic lattice with the same geometry and permittivity of the original object [34,36,37,38]. The basis of the DDA is that the polarization P

_{j}induced on each dipole j of position r

_{j}and polarizability p

_{j}is given by [34,36,37,38].

_{Loc}is the electric field originated by the incident radiation of amplitude E

_{0}, and includes the contribution of all other dipoles [34,36,37,38]:

_{j}is expressed according to the lattice dispersion relation (LDR) developed by Draine and Goodman [34,39], i.e., as a correction of the Clausius–Mossotti polarizability by a series expansion of k

^{.}d and ε

_{m}, with d the interdipole spacing and ε

_{m}the matrix dielectric constant [34,36,37,38]:

_{j}is the dipole permittivity, b

_{1}, b

_{2}, b

_{3}, and S are coefficients of the expansions and ${p}_{j}^{CM}$ is the Clausius–Mossotti polarizability [34,36,37,38]

_{j}developed by Draine and Goodman [34,36,39]. In all cases N was comprised between 10

^{4}and 10

^{5}, as required to minimize computational errors on the absolute value of the extinction cross-section and to allow reliable comparison between calculated optical properties [34,35,36,39,40]. The targets were generated ad hoc, each with a given optical constant according to whether they are latex or metal, and disposed according to the geometry of the object under consideration. In fact, target optical constants are introduced directly from experimental data into the calculation as input numerical parameters, independently of composition or geometry, and without any need for the interpolation with analytical models of the optical constants [34,35,36,39,40]. Therefore, optical constant of gold [41], silver [42], and polystyrene (PS) latex beads [43] were adopted. The refractive index of the matrix surrounding the NPs was that of water at 25 °C [44].

_{F}the Fermi speed and A an empirical parameter usually set equal to 1 [17,45].

_{ext}resulted from the arithmetic average over 2 orthogonal polarization directions and 27 sets of Euler angles of rotation of the target with respect to the incident plane wave (i.e., a total of 54 different orientations for each extinction cross-section plotted) to simulate the dispersion of oligomers with random orientation in a liquid solution.

## 3. Results

#### 3.1. Homo-Aggregates of Latex Nanospheres

_{ext}(Figure 1A). Clustering of the NPs is associated with an increase of the C

_{ext}per single nanoparticle (C

_{ext}/NP), which is especially pronounced below 500 nm. This change is scarcely influenced by the interparticle gap when it is varied between 20 and 0 nm. Antibodies used for surface functionalization of latex spheres and most protein antigens usually have a size of 1–3 nm [49], hence at a first approximation we can assume that interparticle distance due to the antibody–antigen–antibody sandwich (Figure 1B) is of the order of 5 nm. Keeping this gap of 5 nm, the effect of nanosphere size is shown in Figure 1C. As it is well-known from the theory of electromagnetic scattering in dielectric spheres, the extinction per single nanoparticle dramatically scales with size [10,11]. This is confirmed by the change in extinction per single nanoparticle (∆C

_{ext}/NP) passing from the monomeric to the dimeric state, as shown in Figure 1D. However, it is worth stressing that the increase in aggregate size due to dimerization is responsible for a larger relative change of the extinction, compared to the monomers, as shown in the plot of ∆C

_{ext}/NP in percentage (%) of the monomer cross-section (Figure 1E). In fact, the smaller nanospheres are associated with a change of the order of +100% in the whole spectral range from 300 to 900 nm, which becomes a few ten % in spheres > 100 nm. Regarding the detection limit and sensitivity at a low analyte concentration of a turbidimetric assay, however, the absolute change in extinction is more important [1,2,4,6,7]. This change should be such that the formation of a small number of dimers is enough to be measurable, that is not the case with small (<200 nm) latex nanospheres, according to the calculations shown in Figure 1D.

_{ext}/NP, the absolute ∆C

_{ext}/NP in m

^{2}, and the relative ∆C

_{ext}/NP in percentage (%) all indicate the increase of the extinction per NPs while increasing the number of spheres in the oligomer. Thus, the calculations confirm the general principle that latex clusterization is associated with the increase of extinction and with a continuously growing signal in the turbidimetric assay [1,2,4,6].

_{ext}/NP strongly depends on the wavelength, with the largest change of the order of 10

^{−16}m

^{2}for 77 and 127 nm NP, or 10

^{−15}m

^{2}for 200 nm NP, observed below 500 nm. The relative ∆C

_{ext}/NP has an opposite trend compared to the absolute changes since the percentage systematically increases with wavelength. Besides, the relative change of ∆C

_{ext}/NP is 2–4 times larger in NPs of 77 nm compared to the 200 nm ones, suggesting the rapid loss of linearity of large nanospheres when increasing the size of the clusters.

_{ext}/NP is one order of magnitude smaller. Hence, the use of two wavelengths, one < 500 nm for low analyte concentrations and the other > 500 nm for normal assays, can be a simple strategy to increase sensitivity in the next generation of turbidimeters. It should be noted that performing the turbidimetric analysis in the blue portion of the electromagnetic spectrum is somehow risky due to the possible interference from several other biomolecules that can generate a nonspecific extinction signal [6,49,54]. Conversely, in single wavelength turbidimeters, the use of latex spheres requires the improvement of sensitivity beyond 500 nm [1,2,4,6,9].

#### 3.2. Homo-Aggregates of Au or Ag Nanospheres

_{ext}/NP has a dispersive profile, with a negative peak for wavelengths shorter than the plasmon band of isolated Au or Ag NPs, and a positive peak at longer wavelengths. A negative value of the relative and absolute ∆C

_{ext}/NP at a specific wavelength means that the extinction cross-section per single NP (that is always a quantity ≥ 0) is lower in the aggregates than in the isolated components. The resulting dispersive trend around the maximum of the plasmon resonance is reminiscent of the differential absorbance measured in solutions of optically active compounds by circular dichroism (i.e., the Cotton effect). However, the physical origin of ∆C

_{ext}/NP in the plots of Figure 3B,C,E,F is different than in the Cotton effect because the ∆C

_{ext}/NP is obtained from the difference between the C

_{ext}/NP of the aggregated and the isolated NPs, whereas in circular dichroism the spectrum is obtained from the same solution by measuring the differential absorbance of light circularly polarized in opposite directions.

_{ext}/NP is in the 550–650 nm range for Au NPs and 450–550 nm range for Ag NPs, instead of at shorter wavelengths as in latex NPs. These spectral ranges are coincident with those frequently exploited in turbidimeters [1,2,4,6,7]. When looking at the relative change of C

_{ext}/NP, it is also confirmed that the largest effect is located on the right side of the plasmon band. Overall, the red-shift and broadening increase with the number of NPs in the cluster, but there is also a dependence on the arrangement of NPs at parity of their number in the aggregate. This is especially observed between the symmetric and asymmetric octahedra, since the former is associated with the largest absolute and relative increase of the C

_{ext}/NP, similarly to what is found with the latex spheres.

#### 3.3. Hetero-Aggregates of Metal and Latex NPs

_{ext}/NP of Au and Ag homo-aggregates is comparable to that of 77 nm latex NPs. Thus, the calculations considered the optical properties of hetero-aggregates of plasmonic and latex NPs, to verify the occurrence of a synergistic effect leading to an improvement of the optical extinction upon clusterization. The calculations focused on dimers and trimers of Au and latex NPs, starting with 20 nm (Au_20) or 50 nm (Au_50) metal NPs and 77 nm latex (PS_77) NPs, subsequently considering 127 nm (PS_127) and 200 nm (PS_200) latex NPs (Figure 4, Figure 5 and Figure 6). The C

_{ext}/NP for the oligomers was compared in all cases with that of the corresponding isolated NPs. In the Au_20/PS_77 and PS_77/Au_20/PS_77 oligomers (Figure 4A), a remarkable increase of C

_{ext}/NP is observed (larger for the trimer), as well as a red-shift of the surface plasmon band. The modification of the optical properties shows little dependence on the interparticle gap, when it is varied between 5 and 1 nm (Figure 4B), which are the values typically measured by transmission electron microscopy for immuno-aggregates of metal NPs [24,55].

_{ext}/NP, although the change is less evident than in the Au_20 case. This is very appreciable in the plot of the relative ∆C

_{ext}/NP (Figure 4D), where the percent change of the Au_20 oligomers is almost twice that of the Au_50 ones. Contrary to pure latex NPs, the optical properties of the hetero-aggregates mostly change in the spectral region > 500 nm, as desirable for typical turbidimeters. Furthermore, the change on the left side of the plasmon band is weakly negative in the hetero-aggregates, showing an edge in the differential cross-section instead of the dispersive trend of pure Au NPs. In the 300–450 nm range, the interband transitions of Au NPs overlap with the scattering profile of the latex spheres. The two contributions are no more discernible in the Au_20 oligomers, while the interband transitions of gold prevail in the Au_50 oligomers.

_{ext}/NP (Figure 4E), which is one order of magnitude larger for the Au_50 oligomers compared to the Au_20 ones. This is due to the large cross-section of 50 nm gold NPs, which dominates the optical extinction spectrum. For both the Au_50 and Au_20 oligomers, the change is maximum on the right side of the plasmon band, due to its broadening and red-shift. This change is located exactly in the 500–600 nm range used in most turbidimeters. To better stress the advantage brought from hetero-aggregates, the ∆C

_{ext}/NP for PS_77 dimers and trimers are also reported (Figure 4F). One can see that ∆C

_{ext}/NP in the 500–600 nm range is of the order of 10

^{−17}m

^{2}for the Au_20 hetero-aggregates and the PS_77 homo-aggregates, while it exceeds 10

^{−16}m

^{2}for the Au_50 hetero-aggregates. This indicates a clear advantage in the sensitivity of the turbidimetric assay when a combination of Au_50 and PS_77 is used instead of pure latex NPs.

_{ext}/NP (Figure 5C) is of the same order of magnitude (ca. 50% in the 500–600 nm range) for the Au_20 and the Au_50 oligomers. This is not the case of the absolute ∆C

_{ext}/NP (Figure 5D), which in the 500–600 nm range is of the order of 10

^{−16}m

^{2}for the Au_20 hetero-aggregates, while reaching 10

^{−15}m

^{2}for the Au_50 ones. As a comparison, the PS_127 homo-aggregates have a ∆C

_{ext}/NP of 10

^{−16}m

^{2}in the same spectral range (Figure 5E). Importantly, the ∆C

_{ext}/NP of Au_50 hetero-aggregates in the 500–600 nm range is comparable to that of the PS_127 homo-aggregates in the blue region of the spectrum. Hence, at low analyte concentration, a mixture of Au_50 and PS_127 can improve the sensitivity of the turbidimetric assay of one order of magnitude compared to PS_127 alone. This is possible without resorting to the double-wavelength readout, which is more expensive and complex to be implemented in analytical routines and subjected to nonspecific signals in the blue spectral region.

_{ext}/NP (Figure 6C). Indeed, the edge in concomitance of the plasmon resonance is neat only for the Au_50 oligomers. In fact, the plots of absolute ∆C

_{ext}/NP for the Au_20 hetero-aggregate (Figure 6D) resemble that of latex homo-aggregates (Figure 6E), but with extinction values that are one order of magnitude lower. Only the Au_50 hetero-aggregates have comparable absolute ∆C

_{ext}/NP than the PS_200 homo-aggregates, in the 500–600 nm range.

_{ext}/NP always show a dispersive trend around the maximum of the plasmon resonance of the isolated gold nanoparticles. The spectral position of the plasmon resonance and the related centre of the dispersive trend in ∆C

_{ext}/NP depend on the size and shape of the isolated Au NPs, hence can be changed by acting on these parameters.

_{ext}/NP (Figure 7D) is of comparable entity for the Ag_20 and the Ag_50 hetero-aggregates and exceeds 40% for wavelengths > 450 nm. The absolute ∆C

_{ext}/NP (Figure 7E) shows the maxima in the 450–500 nm range, which is not the preferred one for ordinary turbidimetry. The change in C

_{ext}/NP is of the order of 10

^{−17}m

^{2}for Ag_20 and 10

^{−16}m

^{2}for Ag_50, i.e., comparable or one order of magnitude larger than in PS_77 homo-aggregates (Figure 7F).

_{ext}/NP (Figure 8C). This increment is accompanied by a negative dip at 400 nm, where the plasmon resonance of isolated Ag NPs is peaked. The absolute change of C

_{ext}/NP (Figure 8D) is twice that of the hetero-aggregates with PS_77, but it is still located prevalently in the 450–550 nm range. Compared to homo-aggregates of PS_127 (Figure 8E), the ∆C

_{ext}/NP of hetero-aggregates with Ag_50 is more than one order of magnitude larger in the 450–550 nm range.

_{ext}/NP (Figure 9C) lower than in the oligomers with PS_77 and PS_127. This is the expected consequence of the larger contribution from the 200 nm latex nanospheres, that undergo a small relative change of optical density passing from the isolated to the hetero-aggregate configuration. The absolute change of C

_{ext}/NP (Figure 9D) is comparable to that of the oligomers with PS_77 and PS_127, as well as to the homo-aggregates of PS_200 (Figure 9E) in the 450–550 nm range.

#### 3.4. Optimising Hetero-Aggregates of Metal and Latex NPs

_{ext}/NP in hetero-aggregates composed of 7 or 12 particles arranged, respectively, linearly or in asymmetric octahedra (Figure 10).

_{ext}/NP continuously grows with the number of particles in the aggregate, going from the dimer to the trimer (Figure 10A), the heptamer and the asymmetric octahedron (Figure 10B). According to the plot of the relative ∆C

_{ext}/NP (Figure 10C), the increment is always at wavelengths > 550 nm, although it is not associated with a dramatic change of the shape of the plasmon band. This result agrees with the previous observations of red agglomerates between 3 μm latex and 13 nm Au NPs [56].

_{ext}/NP is of the order of 10

^{−16}m

^{2}(Figure 10D), which is comparable to the absolute ∆C

_{ext}/NP of homo-aggregates (Figure 10E). This means that there is no advantage in combining Au_20 with PS_127.

_{ext}/NP also has a continuously growing trend passing from the dimer to the trimer (Figure 11A), the heptamer and the asymmetric octahedron (Figure 11B). The relative increment of ∆C

_{ext}/NP is located at wavelengths > 550 nm as well (Figure 11C). However, this change is of the order of 10

^{−15}m

^{2}in the absolute ∆C

_{ext}/NP (Figure 11D), which is more than one order of magnitude larger than in homo-aggregates (Figure 11E). Hence, the combination of Au_50 and PS_77 NPs provides an advantage of sensitivity compared to the separated components.

_{ext}/NP weighted on all the possible permutations still shows a remarkable broadening of the main extinction band (Figure 12A). This is reasonable considering the additional contribution due to the modification of the surface plasmon band in the 25% of Au_50/Au_50 homodimers. The permutation-weighted C

_{ext}/NP was calculated also for a 1:1 mixture of Ag_40 and PS_77 NPs. Thus, the average is performed on the Ag_40/PS_77 (50%), PS_77/PS_77 (25%) and Ag_40/Ag_40 (25%) permutations. The resulting C

_{ext}/NP retains the red-shift and broadening of the plasmon band, especially in the 450–550 nm range as in the previous calculations with silver particles.

_{ext}/NP or Au_50 and Ag_40 NPs with PS_77 also show a remarkable broadening of the plasmon band. The relative ∆C

_{ext}/NP (Figure 12C) is located again in the 550–650 nm range for the Au trimers. For Ag trimers, ∆C

_{ext}/NP extends from 450 to 600 nm, which is a larger interval than in dimers. The absolute ∆C

_{ext}/NP (Figure 12D) is of the order of 10

^{−15}m

^{2}in both cases, a higher value than in PS_77 homo-aggregates.

_{ext}/NP is lower than that of the PS_127 dimer for wavelength < 550 nm (Figure 13B), due to the larger separation between the latex spheres in the hybrid dimer. Beyond 550 nm, the ∆C

_{ext}/NP is slightly higher than in the PS_127 dimer, thanks to the plasmonic coupling of the Au_20 NPs. However, a change of less than one order of magnitude is not enough to justify the synthetic complexity of the PS_127-Au_20 hybrid.

_{ext}/NP of the dimer is homogeneously lower in the whole spectral range (Figure 13C). The [email protected] with dielectric core and gold shell are renowned for their broad plasmonic band in the red portion of the visible spectrum, as happens in the [email protected] case. This extinction band is less intense in the dimer because of the red-shift of the plasmon resonance of the dimer in the near-infrared, due to the strong plasmonic coupling between the two nanospheres. The change of extinction due to this red-shift is so intense that overwhelms the increase of the scattering contribution due to the formation of the dimer. The resulting absolute ∆C

_{ext}/NP is more than one order of magnitude lower than the PS_200 NPs (Figure 13D).

_{ext}/NP curve. Nevertheless, the resulting maxima of ∆C

_{ext}/NP are in a different spectral region depending on the orientation (Figure 13F), which makes difficult a prediction on the behaviour in real systems, where the orientation of the nanorod usually cannot be predetermined. Nonetheless, the ∆C

_{ext}/NP can be more than one order of magnitude larger than in the PS_127 dimer in the 600–650 or 700–750 nm range. Although this range is not the preferred one for turbidimetry, the resonance of nanorods can be shifted by changing their aspect ratio, to seek an optimal spectral position.

## 4. Discussion

_{ext}/NP in homo- and hetero-dimers are summarized in Figure 14A. The largest values (50–60% at 550, 575, or 600 nm) are reached when metal and latex NPs are coupled together. Exceptions are the PS_77 homo-dimer, which undergoes a similar change, and hetero-dimers with PS_200, which have a change of ca. 10% due to the dominant contribution of the dielectric particle over the plasmonic one.

_{ext}/NP. However, the behaviour of AuNR in larger hetero-oligomers deserves a detailed and specific study to verify their compatibility with a monotonic increase of the optical extinction at a fixed wavelength, as required in ordinary turbidimetry.

_{ext}/NP, summarized in Figure 14B, is the relevant parameter to predict the sensitivity of a real immunoturbidimetric assay. In this case, the size of the NPs dominates the histogram, i.e., the particles with a size of 200 nm have the largest ∆C

_{ext}/NP. Remarkably, hetero-dimers of Au_50 with PS_77 and PS_127 have comparable performances with dimers containing PS_200 NPs. Ag NPs perform slightly less than Au analogues in the 550–600 nm range because the plasmon resonance of silver nanospheres is blue-shifted more than 100 nm from that of gold ones [16,17,63]. Consequently, a larger ∆C

_{ext}/NP is expected in the 450–500 nm range for Ag hetero-aggregates. More complex plasmonic NPs have a heterogeneous response, suggesting that the cost in the realization of complex architectures differing from simple metal nanospheres is justified only after a specific effort of optimization of the ∆C

_{ext}/NP, which is not unequivocally satisfactory for the morphologies considered in this study.

_{ext}/NP from larger particles. However, this is generally accompanied by a rapid loss of linearity while increasing the size of the oligomers. This is shown, for instance, in Figure 15, where the C

_{ext}/NP at 550, 575, and 600 nm are reported for the oligomers of PS_77 (Figure 15A), PS_127 (Figure 15B), and PS_200 (Figure 15C) as a function of the number of NPs. One can see how the increase of C

_{ext}/NP has a higher slope for smaller particles, while it reaches almost saturation beyond 8–10 NPs in the PS_200 spheres. Besides, the morphology of the cluster is more influential on the C

_{ext}/NP of the smallest latex spheres (PS_77), while being not so crucial for the PS_200.

_{ext}/NP on the number of particles also shows a saturation after 8–10 NPs at 550 nm, whereas at 600 nm there is continuous growth. The extinction cross-section per single NP is of the order of 10

^{−16}m

^{2}, not far from that of PS_127 homo-aggregates. The different trends at 550 and 600 nm are a consequence of the progressive red-shift of the plasmon resonance in Au NPs homo-aggregates, which in oligomers is featured by an increase of the optical density in the proximity of the extinction peak of isolated nanospheres. However, for larger aggregates the plasmon band undergoes a further red-shift and broadening, instead of continuous growth in the same spectral range as it would be required for turbidimetry. This behaviour also explains the optical extinction of Ag NPs oligomers (Figure 15E), which show a more continuous growth of the optical extinction between 550 and 600 nm but on a scale that is only of the order of 10

^{−17}m

^{2}(same as the PS_77 homo-aggregates). In fact, the plasmon resonance of silver nanospheres is blue-shifted compared to that of Au NPs analogues [16,63,64]. Hence, the saturation of the extinction in the 550–600 nm range is reached only for aggregates containing tens of particles [65].

_{ext}/NP (Figure 15F) for small oligomers is steep, suggesting that the two types of particles can be successfully combined to improve the performance of turbidimetric assays. Unfortunately, the steep portion of the curve is limited to 2–3 NPs in the aggregate. For more than just 3 NPs, the trend of C

_{ext}/NP undergoes a reduction of its slope and, most relevant, the overall value of the optical extinction remains in the 10

^{−16}m

^{2}, i.e., of the same order of magnitude of PS_127 homo-aggregates.

_{ext}/NP enters the 10

^{−15}m

^{2}range, i.e., the same as the PS_200 homo-aggregates, but with a continuous growth from 2 to 12 NPs, instead of showing a plateau as in the 200 nm latex spheres. Hence, the combination of Au_50 plasmonic NPs and PS_77 latex particles promises to improve sensitivity and dynamic range in immuno-turbidimetric assays (Figure 16).

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Cölfen, H.; Völkel, A.; Eda, S.; Kobold, U.; Kaufmann, J.; Puhlmann, A.; Göltner, C.; Wachernig, H. Mechanism of Nanoparticle-Enhanced Turbidimetric Assays Applying Nanoparticles of Different Size and Immunoreactivity. Langmuir
**2002**, 18, 7623–7628. [Google Scholar] [CrossRef] - Eda, S.; Kaufmann, J.; Roos, W.; Pohl, S. Development of a New Microparticle-Enhanced Turbidimetric Assay for C- Reactive Protein with Superior Features in Analytical Sensitivity and Dynamic Range. J. Clin. Lab. Anal.
**1998**, 12, 137–144. [Google Scholar] [CrossRef] - Uehara, N.; Numanami, Y.; Oba, T.; Onishi, N.; Xie, X. Thermal-Induced Immuno-Nephelometry Using Gold Nanoparticles Conjugated with a Thermoresponsive Polymer for the Detection of Avidin. Anal. Sci.
**2015**, 31, 495–501. [Google Scholar] [CrossRef] [PubMed][Green Version] - Newman, D.J.; Henneberry, H.; Price, C.P. Particle Enhanced Light Scattering Immunoassay. Ann. Clin. Biochem.
**1992**, 22–42. [Google Scholar] [CrossRef] [PubMed][Green Version] - Dzantiev, B.B.; Urusov, A.E.; Zherdev, A.V. Modern Techniques of Immunochemical Analysis: Integration of Sensitivity and Rapidity. Biotechnol. Acta
**2013**, 6, 94–104. [Google Scholar] [CrossRef][Green Version] - Ortega-Vinuesa, J.L.; Bastos-González, D. A Review of Factors Affecting the Performances of Latex Agglutination Tests. J. Biomater. Sci. Polym. Ed.
**2001**, 12, 379–408. [Google Scholar] [CrossRef] [PubMed] - Goryacheva, I.Y. Contemporary Trends in the Development of Immunochemical Methods for Medical Analysis. J. Anal. Chem.
**2015**, 70, 903–914. [Google Scholar] [CrossRef] - Bogatyrev, V.A.; Dykman, L.A.; Khlebtsov, B.N.; Khlebtsov, N.G. Measurement of Mean Size and Evaluation of Polydispersity of Gold Nanoparticles from Spectra of Optical Absorption and Scattering. Opt. Spectrosc.
**2004**, 96, 128–135. [Google Scholar] [CrossRef] - Quesada, M.; Puig, J.; Delgado, J.M.; Hidalgo-Álvarez, R. Modelling the Kinetics of Antigen-Antibody Reactions at Particle Enhanced Optical Immunoassays. J. Biomater. Sci. Polym. Ed.
**1998**, 9, 961–971. [Google Scholar] [CrossRef] - Bohren, C.F.; Huffman, D.R. Absorption and Scattering of Light by Small Particles; Wiley-Interscience: New York, NY, USA, 1983. [Google Scholar]
- Saija, R.; Iatì, M.A.; Borghese, F.; Denti, P.; Aiello, S.; Cecchi-Pestellini, C. Beyond Mie Theory: The Transition Matrix Approach in Interstellar Dust Modeling. Astrophys. J.
**2001**, 559, 993. [Google Scholar] [CrossRef] - Kato, H.; Nakamura, A.; Kinugasa, S. Effects of Angular Dependency of Particulate Light Scattering Intensity on Determination of Samples with Bimodal Size Distributions Using Dynamic Light Scattering Methods. Nanomaterials
**2018**, 8, 708. [Google Scholar] [CrossRef][Green Version] - Kato, H.; Nakamura, A.; Takahashi, K.; Kinugasa, S. Accurate Size and Size-Distribution Determination of Polystyrene Latex Nanoparticles in Aqueous Medium Using Dynamic Light Scattering and Asymmetrical Flow Field Flow Fractionation with Multi-Angle Light Scattering. Nanomaterials
**2012**, 2, 15–30. [Google Scholar] [CrossRef][Green Version] - Kim Thanh, N.T.; Rosenzweig, Z. Development of an Aggregation-Based Immunoassay for Anti-Protein A Using Gold Nanoparticles. Anal. Chem.
**2002**, 74, 1624–1628. [Google Scholar] [CrossRef] - António, M.; Nogueira, J.; Vitorino, R.; Daniel-da-Silva, A.L. Functionalized Gold Nanoparticles for the Detection of C-Reactive Protein. Nanomaterials
**2018**, 8, 200. [Google Scholar] [CrossRef][Green Version] - Amendola, V.; Bakr, O.M.M.; Stellacci, F. A Study of the Surface Plasmon Resonance of Silver Nanoparticles by the Discrete Dipole Approximation Method: Effect of Shape, Size, Structure, and Assembly. Plasmonics
**2010**, 5, 85–97. [Google Scholar] [CrossRef] - Amendola, V.; Pilot, R.; Frasconi, M.; Maragò, O.M.; Iatì, M.A. Surface Plasmon Resonance in Gold Nanoparticles: A Review. J. Phys. Condens. Matter
**2017**, 29, 203002. [Google Scholar] [CrossRef] - Byun, J.-Y.; Shin, Y.-B.; Kim, D.-M.; Kim, M.-G. A Colorimetric Homogeneous Immunoassay System for the C-Reactive Protein. Analyst
**2013**, 138, 1538–1543. [Google Scholar] [CrossRef] [PubMed] - António, M.; Ferreira, R.; Vitorino, R.; Daniel-da-Silva, A.L. A Simple Aptamer-Based Colorimetric Assay for Rapid Detection of C-Reactive Protein Using Gold Nanoparticles. Talanta
**2020**, 214, 120868. [Google Scholar] [CrossRef] [PubMed] - Venditti, I. Nanostructured Materials Based on Noble Metals for Advanced Biological Applications. Nanomaterials
**2019**, 9, 1593. [Google Scholar] [CrossRef] [PubMed][Green Version] - Sreekanth, K.V.; Alapan, Y.; Elkabbash, M.; Ilker, E.; Hinczewski, M.; Gurkan, U.A.; De Luca, A.; Strangi, G. Extreme Sensitivity Biosensing Platform Based on Hyperbolic Metamaterials. Nat. Mater.
**2016**, 15, 621–627. [Google Scholar] [CrossRef][Green Version] - Ahmadivand, A.; Gerislioglu, B.; Ahuja, R.; Kumar Mishra, Y. Terahertz Plasmonics: The Rise of Toroidal Metadevices towards Immunobiosensings. Mater. Today
**2020**, 32, 108–130. [Google Scholar] [CrossRef] - Gerislioglu, B.; Dong, L.; Ahmadivand, A.; Hu, H.; Nordlander, P.; Halas, N.J. Monolithic Metal Dimer-on-Film Structure: New Plasmonic Properties Introduced by the Underlying Metal. Nano Lett.
**2020**, 20, 2087–2093. [Google Scholar] [CrossRef] [PubMed] - Bravin, C.; Amendola, V. Wide Range Detection of C-Reactive Protein with a Homogeneous Immunofluorimetric Assay Based on Cooperative Fluorescence Quenching Assisted by Gold Nanoparticles. Biosens. Bioelectron.
**2020**, 169, 112591. [Google Scholar] [CrossRef] - Kolanowski, J.L.; Liu, F.; New, E.J. Fluorescent Probes for the Simultaneous Detection of Multiple Analytes in Biology. Chem. Soc. Rev.
**2018**, 47, 195–208. [Google Scholar] [CrossRef] [PubMed] - Wang, C.; Yu, C. Detection of Chemical Pollutants in Water Using Gold Nanoparticles as Sensors: A Review. Rev. Anal. Chem.
**2013**, 32, 1–14. [Google Scholar] [CrossRef][Green Version] - Peceros, K.E.; Xu, X.; Bulcock, S.R.; Cortie, M.B. Dipole—Dipole Plasmon Interactions in Gold-on-Polystyrene Composites. J. Phys. Chem. B
**2005**, 109, 21516–21520. [Google Scholar] [CrossRef] [PubMed][Green Version] - Iatì, M.A.; Lidorikis, E.; Saija, R. Modeling of Enhanced Electromagnetic Fields in Plasmonic Nanostructures. In Handbook of Enhanced Spectroscopies; Pan Stanford Publishing: New York, NY, USA, 2016. [Google Scholar]
- Zhao, W.; Brook, M.A.; Li, Y. Design of Gold Nanoparticle-based Colorimetric Biosensing Assays. ChemBioChem
**2008**, 9, 2363–2371. [Google Scholar] [CrossRef] [PubMed] - Li, H.; Rothberg, L. Colorimetric Detection of DNA Sequences Based on Electrostatic Interactions with Unmodified Gold Nanoparticles. Proc. Natl. Acad. Sci. USA
**2004**, 101, 14036–14039. [Google Scholar] [CrossRef] [PubMed][Green Version] - Rahmani, M.; Luk’yanchuk, B.; Hong, M. Fano Resonance in Novel Plasmonic Nanostructures. Laser Photon. Rev.
**2013**, 7, 329–349. [Google Scholar] [CrossRef] - Morandi, V.; Marabelli, F.; Amendola, V.; Meneghetti, M.; Comoretto, D. Light Localization Effect on the Optical Properties of Opals Doped with Gold Nanoparticles. J. Phys. Chem. C
**2008**, 112, 6293–6298. [Google Scholar] [CrossRef][Green Version] - Fuller, K.A.; Gonzalez, R.J.; Kochar, M.S. Light Scattering from Dimers: Latex-Latex and Gold-Latex. In Advances in Optical Biophysics; Lakowicz, J.R., Ross, J.B.A., Eds.; SPIE: San Jose, CA, SUA, 1998; Volume 3256, p. 186. [Google Scholar] [CrossRef]
- Draine, B.T.; Flatau, P.J. Discrete-Dipole Approximation for Scattering Calculations. J. Opt. Soc. Am. A
**1994**, 11, 1491–1499. [Google Scholar] [CrossRef] - Gonzalez, A.L.; Noguez, C.; Ortiz, G.P.; Rodriguez-Gattorno, G. Optical Absorbance of Colloidal Suspensions of Silver Polyhedral Nanoparticles. J. Phys. Chem. B
**2005**, 109, 17512–17517. [Google Scholar] [CrossRef] [PubMed] - Goodman, J.J.; Draine, B.T.; Flatau, P.J. Application of Fast-Fourier-Transform Techniques to the Discrete-Dipole Approximation. Opt. Lett.
**1991**, 16, 1198–1200. [Google Scholar] [CrossRef] - Khlebtsov, N.G. An Approximate Method for Calculating Scattering and Absorption of Light by Fractal Aggregates. Opt. Spectrosc.
**2000**, 88, 594–601. [Google Scholar] [CrossRef] - Lv, R.; Feng, M.; Parak, W. Up-Conversion Luminescence Properties of Lanthanide-Gold Hybrid Nanoparticles as Analyzed with Discrete Dipole Approximation. Nanomaterials
**2018**, 8, 989. [Google Scholar] [CrossRef][Green Version] - Draine, B.T.; Flatau, P.J. User Guide for the Discrete Dipole Approximation Code DDSCAT 7.3. arXiv
**2013**, arXiv:1305.6497. [Google Scholar] - Amendola, V. Surface Plasmon Resonance of Silver and Gold Nanoparticles in the Proximity of Graphene Studied Using the Discrete Dipole Approximation Method. Phys. Chem. Chem. Phys.
**2016**, 18, 2230–2241. [Google Scholar] [CrossRef] - Olmon, R.L.; Slovick, B.; Johnson, T.W.; Shelton, D.; Oh, S.-H.; Boreman, G.D.; Raschke, M.B. Optical Dielectric Function of Gold. Phys. Rev. B
**2012**, 86, 235147. [Google Scholar] [CrossRef][Green Version] - Johnson, P.B.; Christy, R.W. Optical Constants of the Noble Metals. Phys. Rev. B
**1972**, 6, 4370–4379. [Google Scholar] [CrossRef] - Sultanova, N.; Kasarova, S.; Nikolov, I. Dispersion Properties of Optical Polymers. Acta Phys. Pol. A
**2009**, 116, 585–587. [Google Scholar] [CrossRef] - Hale, G.M.; Querry, M.R. Optical Constants of Water in the 200-Nm to 200-Μm Wavelength Region. Appl. Opt.
**1973**, 12, 555. [Google Scholar] [CrossRef] - Kreibig, U.; Vollmer, M. Optical Properties of Metal Clusters; Springer: Berlin, Germany, 1995. [Google Scholar]
- Poletti, A.; Fracasso, G.; Conti, G.; Pilot, R.; Amendola, V. Laser Generated Gold Nanocorals with Broadband Plasmon Absorption for Photothermal Applications. Nanoscale
**2015**, 7, 13702–13714. [Google Scholar] [CrossRef] - Mendoza-Herrera, L.J.; Schinca, D.C.; Scaffardi, L.B.; Grumel, E.E.; Trivi, M. Measurement of Latex Microparticle Size by Dynamic Speckle Technique. Opt. Lasers Eng.
**2021**, 140, 106528. [Google Scholar] [CrossRef] - Kubitschko, S.; Spinke, J.; Brückner, T.; Pohl, S.; Oranth, N. Sensitivity Enhancement of Optical Immunosensors with Nanoparticles. Anal. Biochem.
**1997**, 253, 112–122. [Google Scholar] [CrossRef] [PubMed] - Greenfield, E.A. Antibodies: A Laboratory Manual, 2nd ed.; CSH Press: Woodbury, NY, SUA, 2014. [Google Scholar]
- Lambert, S.; Thill, A.; Ginestet, P.; Audic, J.M.; Bottero, J.Y. Structural Interpretations of Static Light Scattering Patterns of Fractal Aggregates: I. Introduction of a Mean Optical Index: Numerical Simulations. J. Colloid Interface Sci.
**2000**, 228, 379–385. [Google Scholar] [CrossRef] [PubMed] - Serra, T.; Casamitjana, X. Structure of the Aggregates during the Process of Aggregation and Breakup under a Shear Flow. J. Colloid Interface Sci.
**1998**, 206, 505–511. [Google Scholar] [CrossRef] - Tang, S.; McFarlane, C.M.; Paul, G.C.; Thomas, C.R. Characterising Latex Particles and Fractal Aggregates Using Image Analysis. Colloid Polym. Sci.
**1999**, 277, 325–333. [Google Scholar] [CrossRef] - Khlebtsov, N.G.; Dykman, L.A.; Krasnov, Y.M.; Mel’nikov, A.G. Light Absorption by the Clusters of Colloidal Gold and Silver Particles Formed during Slow and Fast Aggregation. Colloid J.
**2000**, 62, 765–779. [Google Scholar] [CrossRef] - Mao, S.-Y. Conjugation of Fluorochromes to Antibodies BT—Immunocytochemical Methods and Protocols; Javois, L.C., Ed.; Humana Press: Totowa, NJ, USA, 1999; pp. 35–38. [Google Scholar] [CrossRef]
- Lou, S.; Ye, J.Y.; Li, K.Q.; Wu, A. A Gold Nanoparticle-Based Immunochromatographic Assay: The Influence of Nanoparticulate Size. Analyst
**2012**, 137, 1174–1181. [Google Scholar] [CrossRef] [PubMed] - Reynolds, R.A.; Mirkin, C.A.; Letsinger, R.L. A Gold Nanoparticle/Latex Microsphere-Based Colorimetric Oligonucleotide Detection Method. Pure Appl. Chem.
**2000**, 72, 229–235. [Google Scholar] [CrossRef] - Haick, H. Chemical Sensors Based on Molecularly Modified Metallic Nanoparticles. J. Phys. D. Appl. Phys.
**2007**, 40, 7173–7186. [Google Scholar] [CrossRef] - Cacciola, A.; Iatì, M.A.; Saija, R.; Borghese, F.; Denti, P.; Maragò, O.M.; Gucciardi, P.G. Spectral Shift between the Near-Field and Far-Field Optoplasmonic Response in Gold Nanospheres, Nanoshells, Homo- and Hetero-Dimers. J. Quant. Spectrosc. Radiat. Transf.
**2017**, 195, 97–106. [Google Scholar] [CrossRef] - Yong, K.T.; Sahoo, Y.; Swihart, M.T.; Prasad, P.N. Synthesis and Plasmonic Properties of Silver and Gold Nanoshells on Polystyrene Cores of Different Size and of Gold-Silver Core-Shell Nanostructures. Colloids Surf. A Physicochem. Eng. Asp.
**2006**, 290, 89–105. [Google Scholar] [CrossRef] - Gold, Nanorods 25 nm Diameter, Absorption, 650 nm, Dispersion in H
_{2}O, Citrate Capped|Au Nanorods|Sigma-Aldrich. Available online: https://www.sigmaaldrich.com/catalog/product/aldrich/900367?lang=it®ion=IT (accessed on 31 March 2021). - Alekseeva, A.V.; Bogatyrev, V.A.; Dykman, L.A.; Khlebtsov, B.N.; Trachuk, L.A.; Melnikov, A.G.; Khlebtsov, N.G. Preparation and Optical Scattering Characterization of Gold Nanorods and Their Application to a Dot-Immunogold Assay. Appl. Opt.
**2005**, 44, 6285–6295. [Google Scholar] [CrossRef] - Dey, P.; Baumann, V.; Rodríguez-Fernández, J. Gold Nanorod Assemblies: The Roles of Hot-Spot Positioning and Anisotropy in Plasmon Coupling and SERS. Nanomaterials
**2020**, 10, 942. [Google Scholar] [CrossRef] [PubMed] - Arce, V.B.; Santillán, J.M.J.; Muñetón Arboleda, D.; Muraca, D.; Scaffardi, L.B.; Schinca, D.C. Characterization and Stability of Silver Nanoparticles in Starch Solution Obtained by Femtosecond Laser Ablation and Salt Reduction. J. Phys. Chem. C
**2017**, 121, 10501–10513. [Google Scholar] [CrossRef] - Kassavetis, S.; Kaziannis, S.; Pliatsikas, N.; Avgeropoulos, A.; Karantzalis, A.E.; Kosmidis, C.; Lidorikis, E.; Patsalas, P. Formation of Plasmonic Colloidal Silver for Flexible and Printed Electronics Using Laser Ablation. Appl. Surface Sci.
**2015**, 336, 262–266. [Google Scholar] [CrossRef] - Li, H.; Sun, Z.; Zhong, W.; Hao, N.; Xu, D.; Chen, H.Y. Ultrasensitive Electrochemical Detection for DNA Arrays Based on Silver Nanoparticle Aggregates. Anal. Chem.
**2010**, 82, 5477–5483. [Google Scholar] [CrossRef]

**Figure 1.**Homo-dimers of PS NPs: Effect of gap and size. (

**A**) Dimer of 127 nm PS NPs in water at variable gap. (

**B**) Sketch of dimer formation by antibody–antigen–antibody sandwich immunoagglutination. (

**C**) C

_{ext}/NP for dimers of PS NPs with variable size. Absolute (

**D**) and relative (

**E**) ∆C

_{ext}/NP for dimers of PS NPs with variable size are also reported.

**Figure 2.**Homo-aggregates of PS NPs. (

**A**) C

_{ext}/NP for aggregates of PS_77. Absolute (

**B**) and relative (

**C**) ∆C

_{ext}/NP are also reported. (

**D**) C

_{ext}/NP for aggregates of PS_127. Absolute (

**E**) and relative (

**F**) ∆C

_{ext}/NP are also reported. (

**G**) C

_{ext}/NP for aggregates of PS_200. Absolute (

**H**) and relative (

**I**) ∆C

_{ext}/NP are also reported.

**Figure 3.**Homo-aggregates of Au or Ag NPs. (

**A**) C

_{ext}/NP for aggregates of Au_20. Absolute (

**B**) and relative (

**C**) ∆C

_{ext}/NP are also reported. (

**D**) C

_{ext}/NP for aggregates of Ag_20. Absolute (

**E**) and relative (

**F**) ∆C

_{ext}/NP are also reported.

**Figure 4.**Hetero-aggregates of Au and PS NPs. (

**A**) C

_{ext}/NP for aggregates of Au_20 and PS_77. (

**B**) C

_{ext}/NP for a PS_77/Au_20/PS_77 trimer at various gap. (

**C**) C

_{ext}/NP for aggregates of Au_50 and PS_77. The relative (

**D**) and absolute (

**E**) ∆C

_{ext}/NP for Au_20 or Au_50 oligomers with PS_77. (

**F**) The absolute ∆C

_{ext}/NP for PS_77 homo-dimers and trimers are reported for comparison.

**Figure 5.**Hetero-aggregates of Au and PS NPs. (

**A**) C

_{ext}/NP for aggregates of Au_20 and PS_127. (

**B**) C

_{ext}/NP for aggregates of Au_50 and PS_127. The relative (

**C**) and absolute (

**D**) ∆C

_{ext}/NP for Au_20 or Au_50 oligomers with PS_127. (

**E**) The absolute ∆C

_{ext}/NP for PS_127 homo-dimers and trimers are reported for comparison.

**Figure 6.**Hetero-aggregates of Au and PS NPs. (

**A**) C

_{ext}/NP for aggregates of Au_20 and PS_200. (

**B**) C

_{ext}/NP for aggregates of Au_50 and PS_200. The relative (

**C**) and absolute (

**D**) ∆C

_{ext}/NP for Au_20 or Au_50 oligomers with PS_200. (

**E**) The absolute ∆C

_{ext}/NP for PS_200 homo-dimers and trimers are reported for comparison.

**Figure 7.**Hetero-aggregates of Ag and PS NPs. (

**A**) C

_{ext}/NP for aggregates of Ag_20 and PS_77. (

**B**) C

_{ext}/NP for a PS_77/Ag_20/PS_77 trimer at various gap. (

**C**) C

_{ext}/NP for aggregates of Ag_50 and PS_77. The relative (

**D**) and absolute (

**E**) ∆C

_{ext}/NP for Ag_20 or Ag_50 oligomers with PS_77. (

**F**) The absolute ∆C

_{ext}/NP for PS_77 homo-dimers and trimers are reported for comparison.

**Figure 8.**Hetero-aggregates of Ag and PS NPs. (

**A**) C

_{ext}/NP for aggregates of Ag_20 and PS_127. (

**B**) C

_{ext}/NP for aggregates of Ag_50 and PS_127. The relative (

**C**) and absolute (

**D**) ∆C

_{ext}/NP for Ag_20 or Ag_50 oligomers with PS_127. (

**E**) The absolute ∆C

_{ext}/NP for PS_127 homo-dimers and trimers are reported for comparison.

**Figure 9.**Hetero-aggregates of Ag and PS NPs. (

**A**) C

_{ext}/NP for aggregates of Ag_20 and PS_200. (

**B**) C

_{ext}/NP for aggregates of Ag_50 and PS_200. The relative (

**C**) and absolute (

**D**) ∆C

_{ext}/NP for Ag_20 or Ag_50 oligomers with PS_200. (

**E**) The absolute ∆C

_{ext}/NP for PS_200 homo-dimers and trimers are reported for comparison.

**Figure 10.**Hetero-aggregates of Au and PS NPs. (

**A**) C

_{ext}/NP for a dimer and a trimer of Au_20 and PS_127. (

**B**) C

_{ext}/NP for a heptamer and an asymmetric octahedron of Au_20 and PS_127. The relative (

**C**) and absolute (

**D**) ∆C

_{ext}/NP for Au_20 oligomers with PS_127. (

**E**) The absolute ∆C

_{ext}/NP for PS_127 homo-aggregates are reported for comparison.

**Figure 11.**Hetero-aggregates of Au and PS NPs. (

**A**) C

_{ext}/NP for a dimer and a trimer of Au_50 and PS_77. (

**B**) C

_{ext}/NP for a heptamer and an asymmetric octahedron of Au_50 and PS_77. The relative (

**C**) and absolute (

**D**) ∆C

_{ext}/NP for Au_50 oligomers with PS_77. (

**E**) The absolute ∆C

_{ext}/NP for PS_77 homo-aggregates are reported for comparison.

**Figure 12.**Permutations in hetero-aggregates of Au or Ag and PS NPs. (

**A**) Permutation-weighted C

_{ext}/NP for a dimer and a trimer of Au_50 and PS_77. (

**B**) Permutation-weighted C

_{ext}/NP for a dimer and a trimer of Ag_40 and PS_77. The relative (

**C**) and absolute (

**D**) ∆C

_{ext}/NP for Au_20 oligomers with PS_127. (

**E**) The absolute ∆C

_{ext}/NP for PS_77 homo-aggregates are reported for comparison.

**Figure 13.**Hybrid dimers of other Au nanostructures. (

**A**) C

_{ext}/NP for a dimer of PS_127-Au_20. (

**B**) The absolute ∆C

_{ext}/NP for the PS_127-Au_20 and the PS_127 dimers. (

**C**) C

_{ext}/NP for a dimer of [email protected] core-shell (core diameter 160 nm, shell thickness 20 nm). (

**D**) The absolute ∆C

_{ext}/NP for the [email protected] [email protected] and the PS_200 dimers. (

**E**) C

_{ext}/NP for a dimer of PS_127 and a Au NR (cylinder with hemispherical caps, long 65 nm, diameter 25 nm) with longitudinal (L) or transversal (T) orientation. (

**F**) The absolute ∆C

_{ext}/NP for the AuNR/PS_127 L and T dimers and the PS_127 dimers.

**Figure 14.**Comparative results for dimers. Relative (

**A**) and absolute (

**B**) ∆C

_{ext}/NP for the homo- and hetero-dimers of this study, computed at 550, 575, and 600 nm.

**Figure 15.**Extinction versus number (N) of particles in the aggregate at 550, 575, and 600 nm. (

**A**) PS_77. (

**B**) PS_127. (

**C**) PS_200. (

**D**) Au_20. (

**E**) Ag_20. (

**F**) Au_20 with PS_127. (

**G**) Au_50 with PS_77.

**Figure 16.**The combination of Au_50 and PS_77 NPs promises to improve the sensitivity and dynamic range in immuno-turbidimetric assays, while avoiding the low signal of PS_77 NPs homo-aggregates and the high background and limited dynamic range of large (>200 nm) latex nanospheres.

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**MDPI and ACS Style**

Coletta, G.; Amendola, V. Numerical Modelling of the Optical Properties of Plasmonic and Latex Nanoparticles to Improve the Detection Limit of Immuno-Turbidimetric Assays. *Nanomaterials* **2021**, *11*, 1147.
https://doi.org/10.3390/nano11051147

**AMA Style**

Coletta G, Amendola V. Numerical Modelling of the Optical Properties of Plasmonic and Latex Nanoparticles to Improve the Detection Limit of Immuno-Turbidimetric Assays. *Nanomaterials*. 2021; 11(5):1147.
https://doi.org/10.3390/nano11051147

**Chicago/Turabian Style**

Coletta, Giuliano, and Vincenzo Amendola. 2021. "Numerical Modelling of the Optical Properties of Plasmonic and Latex Nanoparticles to Improve the Detection Limit of Immuno-Turbidimetric Assays" *Nanomaterials* 11, no. 5: 1147.
https://doi.org/10.3390/nano11051147