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

nanomaterials

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

Giuliano Coletta and Vincenzo Amendola *

Department of Chemical Sciences, University of Padova, Via Marzolo 1, 35131 Padova, Italy; [email protected] * Correspondence: [email protected]

Abstract: Turbidimetric assays with latex nanoparticles are widely applied for the detection of biological analytes, because of their rapidity, low cost, reproducibility, and automatization. However, the detection limit can be lowered only at the price of a reduced dynamic range, due to the rapid saturation of the light scattering signal at high analyte concentration. Here, we use numerical calculations to investigate the possibility of increasing the performance of immuno-turbidimetric assays without compromising the measurement dynamic range, by combining plasmonic (gold, silver) and latex nanoparticles. Our modelling results show that plasmonic nanoparticles are compatible with a large signal change even when small aggregates are formed, i.e., at low analyte concentration. The working principle relies on the remarkable modiﬁcation of the surface plasmon band when noble metal nanoparticles form oligomers, and also when latex particles are included in the aggregate.

At high analyte concentration, when larger aggregates form, the latex particles can provide the required linear response of standard immuno-turbidimetric assays. Thus, the combination of the two

Citation: Coletta, G.; Amendola, V. components can be a successful strategy to improve the detection limit and the dynamic range, while Numerical Modelling of the Optical maintaining all the advantages of the homogeneous immuno-turbidimetric assays. Properties of Plasmonic and Latex Nanoparticles to Improve the Detection Keywords: gold nanoparticles; silver nanoparticles; latex nanoparticles; plasmon; optical properties; Limit of Immuno-Turbidimetric Assays. turbidimetry; DDA Nanomaterials 2021, 11, 1147. https:// doi.org/10.3390/nano11051147

Academic Editor: Mikhael Bechelany 1. Introduction In turbidimetric assays, the quantity of the analyte is correlated to optical extinction Received: 2 April 2021 through a colloid of nanoparticles (NPs) that selectively undergo aggregation and scatter- Accepted: 26 April 2021 ing of light in the presence of the target compound [1–4]. These assays are widely used in Published: 28 April 2021 medicine for measuring the concentration of various biological components but find application also in environmental, food, and agriculture analysis [3,5,6]. Immuno-turbidimetric Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in assays usually rely on surface functionalized submicrometric latex spheres because of published maps and institutional affil- their low cost, uniform tailorable diameter, ease of surface functionalization with selective iations. targeting units, colloidal stability, and linearity of the light scattering intensity in the visible range at increasing levels of aggregation [1–4,6,7]. The intensity of the light scattered by these particles exceeds the luminosity of an organic fluorescent molecule by several orders of magnitude [4,8]. The latex spheres are functionalized with antibodies or other molecular functions capable of selective binding of the analyte in a sandwich configuration, which Copyright: © 2021 by the authors. implies a growing aggregation with increasing analyte concentration, and a consequent Licensee MDPI, Basel, Switzerland. This article is an open access article growing light scattering intensity [1,2,7,9]. The resulting immuno-turbidimetric assays distributed under the terms and gather together a remarkable range of positive features: they are rapid, phase homoge- conditions of the Creative Commons neous, washing-free, single-step, low cost, selective, reproducible, automatable, and, so Attribution (CC BY) license (https:// far, are among the standard methods for detection of bioanalytes of clinical interest, in creativecommons.org/licenses/by/ particular proteins that are well recognized by antibodies [1,2,6,7]. 4.0/).

Nanomaterials 2021, 11, 1147. https://doi.org/10.3390/nano11051147 https://www.mdpi.com/journal/nanomaterials Nanomaterials 2021, 11, 1147 2 of 23

The detection limit of immuno-turbidimetric assays can be lowered by increasing the size of the latex nanospheres because the light scattering signal and its variation upon nanosphere aggregation both scale with the 6th power of particles radius [2,6,10–13]. For the same reason, however, the large latex particles introduce a high background signal in turbidimetric assays [4,6,14], which rapidly saturates beyond the instrumental photometric range at high analyte concentration, forcing sample dilution when a quantitative measurement is required [1,2]. Lowering the latex concentration is not a good strategy, because this is accompanied by the growth of the detection limit and the saturation of the available targeting units on the nanospheres with consequent saturation of the response, even if the total signal is within the spectrophotometric range of the turbidimeter (i.e., an optical absorption spectrometer working at a specific wavelength interval) [2,6]. Nonetheless, there is a continuous demand for lowering the detection limit to allow the use of small volumes of samples, such as in microfluidic circuits [7,15]. An approach to solve this problem is the combination of latex spheres of different sizes (127 and 220 nm) and functionalized with targeting molecules (for instance, antibodies) with different level of affinity for the analyte [1,2]. When the large (220 nm) latex particles have a high affinity with the analyte, these will provide high sensitivity (i.e., change of signal for a unit change of the quantity of analyte) at low analyte concentration. When analyte concentration increases, the population of 220 nm latex spheres is saturated by the analyte before reaching signal saturation in the instrument, whereas the small (127 nm) latex spheres bearing the antibodies with lower affinity are still available to react, providing a growing signal in the concentration range where sensitivity is not an issue. Although this principle is successful in most cases, tailoring the exact proportion of large and small nanospheres and selecting the appropriate combination of targeting molecules with different affinity to meet the demanded dynamic range and detection limit is not an easy task with analyte varying over several orders of magnitude of concentration. This may often result in unsatisfactory linearity in the whole measurement range. Moreover, large latex nanospheres may pose problems of reproducibility during the analysis due to aspecific agglomeration [4,6] or may affect the long-term storage of the reagent, which usually should be guaranteed for at least one year [6,14]. A more subtle problem is observed with large nanospheres, because of their large volume compared to the analyte, and consequent inertia in the formation of immuno-aggregates at very low analyte concentration. Here, we investigated a different approach that is the combination of plasmonic noble metal NPs (Au and Ag) with latex nanospheres to achieve a low detection limit, high sensitivity, and large dynamic range at the same time. Au and Ag NPs have intense extinction bands in the visible range due to the collective excitation of conduction electrons (plas- mons) [14,16–18]. These NPs have been widely exploited for spectrophotometric detection of analytes reaching a very low detection threshold also in homogeneous assays where the nanoparticles are mixed directly with the analyte in the liquid phase instead of being sup- ported on a substrate [14,15,18–20]. Although in the subwavelength scale, advanced optical structures like self-assembled plasmonic nanoarrays and metamaterials provide significant sensitivity in biosensing assays, leading to the emergence of biosensors with an ultralow limit of detection at very low analyte concentrations [21–23], the homogeneous assays still have the advantage of low-cost and high-throughput in flow continuous analysis without the need for continuous replacement of the detection substrate for each sample [24–26]. The success of Au and Ag NPs in biosensing is connected to the large extinction (i.e., absorption plus scattering [16,17]) cross-section at the plasmon resonance band and to the rapid increase of optical density at longer wavelengths upon particles aggregation [14,15]. The appearance of plasmon resonance bands in the red and near-infrared is due to the electric dipole–dipole interaction and coupling between the plasmonic modes of neighbouring particles in the cluster [14,17,18,27,28]. However, the dynamic range of the assays based on agglomeration of plasmonic NPs is also limited by the nonlinear increase of plasmon extinction versus the number of particles in the aggregate, which typically follows a sigmoidal Nanomaterials 2021, 11, 1147 3 of 23

trend versus the logarithm of the analyte, with a tight linearity interval not suited for the assay of species whose concentration changes over several orders of magnitude [18,29,30]. This is mostly due to the continuous red-shift of the plasmon band while increasing the aggregate size [14,17,18,27] that is not compatible with standard turbidimetric detection at a ﬁxed wavelength. Here, we focused on a different approach, combining the high sensitivity of the plasmon resonance in noble metal NPs with the excellent linearity of latex spheres with a size of the order of 100 nm. More speciﬁcally, we systematically modelled the change in the optical properties when NPs go from the monomeric to the oligomeric (dimer, trimer, etc.) state, as a function of particles composition (latex, Au, Ag, and their combination) and size. A variety of interesting optical effects are possible in hetero-aggregates of plasmonic and dielectric nanoparticles, including the appearance of magnetic plasmonic resonances and dark plasmonic modes or light localization phenomena [17,31,32]. The optical response of oligomers is the most relevant when pursuing high sensitivity of immunoturbidimetric methods at low analyte concentration and where small latex spheres fail [1,2,33]. At larger analyte concentration, the response of commercial latex beads is already satisfactory when the spheres have a size of ca. 100 nm, not requiring further efforts of optimization. The results of the extended modelling effort point to the use of a mixture of plasmonic and latex nanospheres as a promising strategy to obtain turbidimetric assays with a low detection limit and a wide dynamic range.

2. Materials and Methods

The extinction cross-sections (Cext) 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 [34,36–38]. The basis of the DDA is that the polarization Pj induced on each dipole j of position rj and polarizability pj is given by [34,36–38]. Pj = pjELoc rj (1)

where ELoc is the electric ﬁeld originated by the incident radiation of amplitude E0, and includes the contribution of all other dipoles [34,36–38]:

ELoc rj = E0 exp ik · rj + iωt − ∑ Ajl Pl (2) l6=j

where Ajl is the interaction matrix and Einc rj = E0 exp ik · rj + iωt is the incident monochromatic plane wave with frequency ω and wavevector k.

The full expression of Ajl Pl is: [34,36–38] ( ) exp ikrjl 1 − ikr h i = 2 × × + ij 2 − · Ajl Pl 3 k rjl rjl Pl 2 rjl Pl 3rjl rjl Pl (3) rjl rjl

where rjl = rj − rl, rjl = rj − rl . The extinction cross-section of the target is then given by [34,36–38]

N 4πk ∗ C = E · P (4) ext 2 ∑ inc j E0 j=1

∗ where Einc is the complex conjugate of the incident electric ﬁeld. In this work, calculations were performed with the DDSCAT software [39], where pj 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 Nanomaterials 2021, 11, 1147 4 of 23

. series expansion of k d and εm, with d the interdipole spacing and εm the matrix dielectric constant [34,36–38]: pCM LDR j pj = (5) CMh i k2 1 + pj b1 + b2εj + b3Sεj d

where εj is the dipole permittivity, b1, b2, b3, and S are coefﬁcients of the expansions and CM pj is the Clausius–Mossotti polarizability [34,36–38]

3 − CM d εj 1 pj = . (6) 3 εj + 2

For each calculation, we used an interdipole spacing d << λ, as required for the validity of the expression for pj developed by Draine and Goodman [34,36,39]. In all cases N was comprised between 104 and 105, as required to minimize computational errors on the absolute value of the extinction cross-section and to allow reliable comparison between calculated optical properties [34–36,39,40]. 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 [34–36,39,40]. 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]. Besides, the optical constants of Ag and Au NPs were corrected for the intrinsic size effects according to what described by Kreibig [16,17,40,45]. The intrinsic size effect is due to the conduction electrons mean free path being comparable to particles size l [17,45]. In the reasonable assumption that only the free electron behaviour is affected by the size l of nanoparticles, the metal optical constant can be expressed in the following way [17,45]:

" 2 # 1 1 ωp Γ(l) Γ ( ) = ( ) + 2 − + − ∞ ε ω, l ε∞ ω ωp 2 2 2 2 i 2 2 2 2 (7) ω + Γ∞ ω + Γ (l) ω ω + Γ∞ ω + Γ∞

where ε∞(ω) is the bulk value of the optical constant at frequency ω, Γ∞ is the electrons relaxation frequency of the bulk metal, and Γ(l) is the l-dependent value given by the following “size equation” [17,45,46]: v Γ(l) = Γ + A F (8) ∞ l

with vF the Fermi speed and A an empirical parameter usually set equal to 1 [17,45]. All the calculated Cext 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. The same spectral range was set in all calculations and related graphs between 300 and 900 nm to allow the comparison of the spectral behaviours of the different nanosys- tems modelled.

3. Results 3.1. Homo-Aggregates of Latex Nanospheres In order to model the optical extinction properties of oligomers of nanospheres, we started our calculations with the case of two latex spheres with a diameter of 127 nm in water. The size is intermediate between “large” (>200 nm) and “small” (<100 nm) latex Nanomaterials 2021, 11, 1147 5 of 23

spheres used in most research investigations or bioanalytical kits [1,2,24,47,48]. The gap between the two spheres is changed from infinite to 0, to show the effect of interparticle distance on Cext (Figure1A). Clustering of the NPs is associated with an increase of the Cext per single nanoparticle (Cext/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 (Figure1B) is of the order of 5 nm. Keeping this gap of 5 nm, the effect of nanosphere size is shown in Figure1C. 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 (∆Cext/NP) passing from the monomeric to the dimeric state, as shown in Figure1D. 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 ∆Cext/NP in percentage (%) of the monomer cross-section (Figure1E). 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 Nanomaterials 2021, 11, x 6 of 24 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 Figure1D.

Figure 1. Homo-dimers ofof PSPS NPs:NPs: EffectEffect ofof gapgap andand size.size. ((AA)) DimerDimer of of 127 127 nm nm PS PS NPs NPs in in water water at at variable variable gap. gap. ( B(B)) Sketch Sketch of dimer formationformation byby antibody–antigen–antibodyantibody–antigen–antibody sandwich sandwich immunoagglutination. immunoagglutination. (C (C) C) extCext/NP/NPfor for dimers dimers of of PS PS NPs NPs with with Δ variable size.size. AbsoluteAbsolute ((DD)) andand relativerelative ((EE))∆ CCextext/NP for dimers ofof PSPS NPsNPs withwith variablevariable sizesize areare alsoalso reported. reported.

DimersDimers areare just just the the first first type type of of oligomers oligomers that that form form at low at low antigen antigenconcentration concentration [50– 53[50–]. Hence,53]. Hence, we modelled we modelled the extinction the extinction of oligomers of oligomers formed formed by a larger by a numberlarger number of NPs (upof NPs to 11).(up Givento 11). allGiven the possibleall the possible arrangements arrangements for nanospheres for nanospheres in an oligomer, in an oligomer, the calculations the calculations have been focused on some geometric arrangements that can be associated with the formation of linear, fractal, or compact clusters [50–52], considering three nanosphere sizes corresponding to commercially available latex beads of 77 (PS_77, Figure 2A–C), 127 (PS_127, Figure 2D–F), and 200 nm (PS_200, Figure 2G–I) [1–3,7]. For the three sizes considered, the trends of Cext/NP, the absolute ΔCext/NP in m2, and the relative ΔCext/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].

Nanomaterials 2021, 11, 1147 6 of 23

have been focused on some geometric arrangements that can be associated with the formation of linear, fractal, or compact clusters [50–52], considering three nanosphere sizes corresponding to commercially available latex beads of 77 (PS_77, Figure2A–C), 127 (PS_127, Figure2D–F), and 200 nm (PS_200, Figure2G–I) [ 1–3,7]. For the three sizes 2 considered, the trends of Cext/NP, the absolute ∆Cext/NP in m , and the relative ∆Cext/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 Nanomaterials 2021, 11, x 7 of 24 latex clusterization is associated with the increase of extinction and with a continuously growing signal in the turbidimetric assay [1,2,4,6].

Δ Figure 2.2. Homo-aggregates of of PS PS NPs. NPs. ( A(A) )C Cextext/NP/NPfor for aggregates aggregates of of PS_77. PS_77. Absolute Absolute (B ()B and) and relative relative (C )(C∆)C extC/NPext/NPare are also also Δ reported. ((D)) CCextext/NP forfor aggregatesaggregates ofof PS_127. PS_127. Absolute Absolute ( E(E)) and and relative relative (F )(F∆) CextCext/NP/NPare are also also reported. reported. (G ()GC)ext C/NPext/NPfor for aggregates of PS_200. Absolute (H) and relative (I) ΔCext/NP are also reported. aggregates of PS_200. Absolute (H) and relative (I) ∆Cext/NP are also reported.

AnotherAnother generalgeneral trendtrend observedobserved from from the the calculations calculations is is that that the the more more compact compact the the oligomersoligomers are, are, the the bigger bigger the the increase increase of extinctionof extinction per per NP, NP, and and this isthis especially is especially evident evident for thefor octahedron.the octahedron. Nonetheless, Nonetheless, the formationthe formation of fractal of fractal aggregates, aggregates, such such as the as asymmetricthe asymmetric octahedron or the two asymmetric octahedra, is more representative of immunoturbidimetric assays [50,51,53]. These types of oligomers have a larger extinction per NP compared to linear chains of spheres. The absolute ΔCext/NP strongly depends on the wavelength, with the largest change of the order of 10−16 m2 for 77 and 127 nm NP, or 10−15 m2 for 200 nm NP, observed below 500 nm. The relative ΔCext/NP has an opposite trend compared to the absolute changes since the percentage systematically increases with wavelength. Besides, the relative change of ΔCext/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. The calculations suggest that the region < 500 nm is associated with the highest sensitivity, although saturation of the turbidimeters is reached earlier than for wavelength > 500 nm, where the ΔCext/NP is one order of magnitude smaller. Hence, the use of two

Nanomaterials 2021, 11, 1147 7 of 23

octahedron or the two asymmetric octahedra, is more representative of immunoturbidimetric assays [50,51,53]. These types of oligomers have a larger extinction per NP compared to linear chains of spheres. The absolute ∆Cext/NP strongly depends on the wavelength, with the largest change of the order of 10−16 m2 for 77 and 127 nm NP, or 10−15 m2 for 200 nm NP, observed below 500 nm. The relative ∆Cext/NP has an opposite trend compared to the absolute changes since the percentage systematically increases with wavelength. Besides, the relative change of ∆Cext/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. The calculations suggest that the region < 500 nm is associated with the highest sensitivity, although saturation of the turbidimeters is reached earlier than for wavelength > 500 nm, where the ∆Cext/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 The behaviour of benchmark plasmonic NPs like 20 nm Au (Au_20, Figure3A–C) and Ag (Ag_20, Figure3D–F) nanospheres has been investigated as a possible way to increase the sensitivity beyond 500 nm. For the sake of comparison with the previous calculations, the same geometric arrangements of latex spheres and the interparticle gap fixed at 5 nm are used. The extinction band due to the surface plasmon resonance of Au (Figure3A) and Ag (Figure3D) undergo a much different behaviour compared to the latex spheres. The main effect of clustering is on the plasmon band, which is red-shifted and broadened. Consequently, the ∆Cext/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 ∆Cext/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 ∆Cext/NP in the plots of Figure3B,C,E,F is different than in the Cotton effect because the ∆Cext/NP is obtained from the difference between the Cext/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. The main absolute change of Cext/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 Cext/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 Cext/NP, similarly to what is found with the latex spheres. NanomaterialsNanomaterials 2021, 11, 1147x 89 of of 23 24

Δ FigureFigure 3. Homo-aggregates ofof AuAu oror AgAg NPs. NPs. ( A(A) )C Cextext/NP/NPfor for aggregates aggregates of of Au_20. Au_20. Absolute Absolute (B ()B and) and relative relative (C )(C∆)C extC/NPext/NP Δ areare also reported. (D)) Cextext/NP/NP forfor aggregates ofof Ag_20.Ag_20. AbsoluteAbsolute ((EE)) andand relativerelative ( F(F))∆ CCextext/NP/NP areare alsoalso reported.reported.

3.3. Hetero-Aggregates ofof MetalMetal andand Latex Latex NPs NPs Δ It should be noted that, in in the the region region > > 550 550 nm nm the the absolute absolute ∆CCextext/NP/NP ofof Au Au and and Ag Aghomo-aggregates homo-aggregates is comparable is comparable to that to thatof 77 of nm 77 latex nm latexNPs. NPs.Thus, Thus,the calculations the calculations consid- consideredered the optical the opticalproperties properties of hetero-aggregate of hetero-aggregatess of plasmonic of plasmonic and latex and NPs, latex to verify NPs, to the verifyoccurrence the occurrence of a synergistic of a effect synergistic leading effect to an leading improvement to an improvement of the optical extinction of the optical upon extinctionclusterization. upon The clusterization. calculations focused The calculations on dimers focused and trimers on dimers of Au and latex trimers NPs, of start- Au anding with latex 20 NPs, nm starting (Au_20) with or 50 20 nm nm (Au_50) (Au_20) me ortal 50 NPs nm(Au_50) and 77 nm metal latex NPs (PS_77) and 77 NPs, nm latexsubse- (PS_77)quently NPs,considering subsequently 127 nm considering (PS_127) and 127 200 nm nm (PS_127) (PS_200) and latex 200 nm NPs (PS_200) (Figures latex 4–6). NPs The (FiguresCext/NP for4–6 the). The oligomersCext/NP wasfor compared the oligomers in all wascases compared with that inof the all casescorresponding with that isolated of the correspondingNPs. In the Au_20/PS_77 isolated NPs. and In PS_77/Au_20/P the Au_20/PS_77S_77 and oligomers PS_77/Au_20/PS_77 (Figure 4A), a oligomersremarkable (Figure4A), a remarkable increase of C /NP is observed (larger for the trimer), as well increase of Cext/NP is observed (larger forext the trimer), as well as a red-shift of the surface asplasmon a red-shift band. of The the modification surface plasmon of the band. optical The properties modiﬁcation shows of little the opticaldependence properties on the showsinterparticle littledependence gap, when it on is thevaried interparticle between 5 gap, and when 1 nm it(Figure is varied 4B), between which are 5 and the 1values nm (Figure4B), which are the values typically measured by transmission electron microscopy typically measured by transmission electron microscopy for immuno-aggregates of metal for immuno-aggregates of metal NPs [24,55]. NPs [24,55]. In the Au_50/PS_77 and PS_77/Au_50/PS_77 oligomers (Figure 4C), we still observe the red-shift and increase of Cext/NP, although the change is less evident than in the Au_20 case. This is very appreciable in the plot of the relative ΔCext/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.

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Nanomaterials 2021, 11, x 11 of 24

FigureFigure 4. 4.Hetero-aggregates Hetero-aggregatesturbidimetric ofof AuAu andand PS PSassay NPs. NPs. of ( A(oneA)) CC extorderext/NP/NP offorfor magnitude aggregatesaggregates compared ofof Au_20Au_20 andandto PS_127 PS_77.PS_77. alone. ( B)) C extThis/NP/NP isforfor possible a a PS_77/Au_20/PS_77 trimer at various gap. (C) Cext/NP for aggregates of Au_50 and PS_77. The relative (D) and absolute PS_77/Au_20/PS_77 trimer atwithout various resorting gap. (C) C toext /NPthe double-wavelengthfor aggregates of Au_50 readout, and PS_77. which The is relative more expensive (D) and absolute and com- (E) ΔCext/NP for Au_20 or Au_50 oligomers with PS_77. (F) The absolute ΔCext/NP for PS_77 homo-dimers and trimers are (E) ∆Cext/NP for Au_20 or Au_50plex oligomers to be implemented with PS_77. in (F )analytical The absolute routines∆Cext/NP andfor subjected PS_77 homo-dimers to nonspecific and trimers signals are in the reported for comparison. reported for comparison. blue spectral region. As stated before, what is more important for turbidimetric assays is the absolute change of extinction, ΔCext/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 ΔCext/NP for PS_77 dimers and trimers are also reported (Figure 4F). One can see that ΔCext/NP in the 500–600 nm range is of the order of 10−17 m2 for the Au_20 hetero-aggregates and the PS_77 homo-aggregates, while it exceeds 10−16 m2 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. Then, the calculations considered Au_20 and Au_50 hetero-aggregates with latex NPs of 127 nm (PS_127). In the Au_20 hetero-aggregates (Figure 5A), the optical extinction is dominated by their scattering profile, due to the large size of the latex NPs. Nonetheless, the red-shift and broadening of the surface plasmon band in the hetero-dimers and trimers are still appreciable. In the Au_50 case, the extinction profiles are dominated by the surface plasmon band (Figure 5B), which makes noticable the red-shift and band broadening in the oligomers. The increased scattering contribution from the PS_127 explains why the relative ΔCext/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 ΔCext/NP (Figure 5D), which in the 500–600 nm range is of the order of 10−16 m2 for the Au_20 hetero- aggregates, while reaching 10−15 m2 for the Au_50 ones. As a comparison, the PS_127 homo-aggregates have a ΔCext/NP of 10−16 m2 in the same spectral range (Figure 5E). Im- portantly, the ΔCext/NP of Au_50 hetero-aggregates in the 500–600 nm range is comparable FigureFigure 5. 5.Hetero-aggregates Hetero-aggregates ofto ofAu that Au and andof PSthe PS NPs. NPs.PS_127 (A (A) C) homo-aggregatesCextext/NP/NPfor for aggregatesaggregates in of ofthe Au_20Au_20 blue andand region PS_127.PS_127. of the ((BB)) spectrum.Cextext/NP/NP forfor aggregates aggregatesHence, at low Δ ofof Au_50 Au_50 and and PS_127. PS_127. The The relative analyterelative ( Cconcentration,(C)) and and absolute absolute (a D (mixtureD) )∆ CCextext/NP/NP of forAu_50for Au_20Au_20 and oror PS_127 Au_50Au_50 oligomerscanoligomers improve withwith the PS_127. sensitivity ( (EE)) The The of the Δ absoluteabsolute∆ CextCext/NP/NPfor for PS_127 PS_127 homo-dimers homo-dimers andand trimerstrimers areare reportedreported for comparison.

When Au_20 and Au_50 hetero-aggregates with latex NPs of 200 nm (PS_200) are considered, the scenario is different from the previous cases due to the prevalence of scattering contribution from the dielectric spheres. The surface plasmon resonance in the Au_20 oligomers is barely detectable (Figure 6A), while the Au_50 oligomers still have a well-defined plasmon band (Figure 6B). Even in the 300–450 nm range, the interband transitions of Au NPs overlap with the scattering profile of the latex spheres and the two contributions are no more discernible from each other. The effect of clustering remains that of producing a red-shift and broadening of the plasmon band, which is better appreciated from the plot of the relative ΔCext/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 ΔCext/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 ΔCext/NP than the PS_200 homo-aggregates, in the 500–600 nm range. Although the above-mentioned differences due to the size of the latex spheres, the relative and absolute ΔCext/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 ΔCext/NP depend on the size and shape of the isolated Au NPs, hence can be changed by acting on these parameters.

Nanomaterials 2021,, 1111,, 1147x 1012 ofof 23 24

Figure 6. Hetero-aggregates ofof AuAu andand PSPS NPs. NPs. ( A(A))C Cextext/NP/NP forfor aggregatesaggregates of of Au_20 Au_20 and and PS_200. PS_200. ( B(B) )C Cextext/NP/NPfor for aggregates aggregates Δ of Au_50 andand PS_200.PS_200. TheThe relativerelative ((CC)) andand absoluteabsolute ( D(D))∆ CCextext/NP/NP forfor Au_20Au_20 oror Au_50 Au_50 oligomers oligomers with with PS_200. PS_200. ( E(E)) The The Δ ext absolute ∆Cext/NP/NP forfor PS_200 PS_200 homo-dimers homo-dimers and trimer trimerss are reported forfor comparison.comparison.

InOverall, the Au_50/PS_77 there are no and apparent PS_77/Au_50/PS_77 advantages in using oligomers hetero-aggregates (Figure4C), we of still Au observe and 200 thenm red-shiftlatex NPs and but, increase as stated of C previously,ext/NP, although the PS_200 the change NPs isare less not evident the best than choice in the for Au_20 turbi- case.dimetric This assays is very due appreciable to rapid saturation in the plot of of the the signal relative at large∆Cext analyte/NP (Figure concentration.4D), where the percentWhile change Au NPs of the are Au_20 the standard oligomers nanomaterial is almost twices for optical that of or the colorimetric Au_50 ones. sensing Contrary due to puretheir latexlong-term NPs, chemical the optical stability properties [15,17, of18], the hetero-aggregatesAg NPs are renowned mostly for change the best in plas- the spectralmonic performance region > 500 in nm, the as visible desirable range for [16]. typical For turbidimeters. instance, the extinction Furthermore, cross-section the change of onAg theNPs left is side5–10 oftimes the plasmonlarger than band Au is NPs weakly with negative the same in morphology, the hetero-aggregates, which may showing suggest ana proportionally edge in the differential greater effect cross-section in hetero-aggreg insteadates of the [16,17,40]. dispersive Theref trendore, of the pure calculations Au NPs. Inwere the performed 300–450 nm also range, on dimers the interband and trimers transitions containing of Au 20 NPs nm overlap (Ag_20) with or 50 the nm scattering (Ag_50) proﬁlesilver NPs of the and latex 77 nm spheres. (PS_77), The 127 two nm contributions (PS_127), or are200 no nm more (PS_200) discernible latex NPs in the (Figure Au_20 7– oligomers,9). The surface while plasmon the interband band of transitions Ag_20 hetero-aggregates of gold prevail inis sensibly the Au_50 lower oligomers. than in isolated NPs As(Figure stated 7A), before, and the what red-shift is more and important broadening for turbidimetricare observed assaysalso in isthis the case absolute like in changethe oligomers of extinction, with Au∆C andext/NP latex(Figure NPs.4 E), which is one order of magnitude larger for the Au_50In oligomersthe PS_77/Ag_20/PS_77 compared tothe trimer, Au_20 the ones. red-sh Thisift depends is due to on the the large interparticle cross-section gap of in 50 nm gold NPs, which dominates the optical extinction spectrum. For both the Au_50 the 1–5 nm range (Figure 7B), being the maximum for the smallest gap. However, band and Au_20 oligomers, the change is maximum on the right side of the plasmon band, broadening and increase of optical density in the red spectral window is scarcely affected due to its broadening and red-shift. This change is located exactly in the 500–600 nm by the gap size. range used in most turbidimeters. To better stress the advantage brought from hetero- In Ag_50 oligomers (Figure 7C), the change in the plasmon resonance band is less aggregates, the ∆Cext/NP for PS_77Δ dimers and trimers are also reported (Figure4F). One evident. However, the relative Cext/NP (Figure 7D) is of comparable−17 entity2 for the Ag_20 can see that ∆Cext/NP in the 500–600 nm range is of the order of 10 m for the Au_20 and the Ag_50 hetero-aggregates and exceeds 40% for wavelengths >− 45016 nm.2 The absolute hetero-aggregatesΔ and the PS_77 homo-aggregates, while it exceeds 10 m for the Au_50 hetero-aggregates.Cext/NP (Figure 7E) This shows indicates the maxima a clear advantagein the 450–500 in the nm sensitivity range, which of the turbidimetricis not the pre- −17 2 assayferred when one for a combination ordinary turbidimetry. of Au_50 and The PS_77 change is used in C insteadext/NP is ofofpure the order latex NPs.of 10 m for −16 2 Ag_20Then, and the10 calculations m for Ag_50, considered i.e., comparable Au_20 and or Au_50 one order hetero-aggregates of magnitude with larger latex than NPs in ofPS_77 127 nmhomo-aggregates (PS_127). In the (Figure Au_20 7F). hetero-aggregates (Figure5A), the optical extinction is dominated by their scattering proﬁle, due to the large size of the latex NPs. Nonetheless,

Nanomaterials 2021, 11, 1147 11 of 23

the red-shift and broadening of the surface plasmon band in the hetero-dimers and trimers are still appreciable. In the Au_50 case, the extinction profiles are dominated by the surface plasmon band (Figure5B), which makes noticable the red-shift and band broadening in the oligomers. The increased scattering contribution from the PS_127 explains why the relative ∆Cext/NP (Figure5C) 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 −16 2 absolute ∆Cext/NP (Figure5D), which in the 500–600 nm range is of the order of 10 m for the Au_20 hetero-aggregates, while reaching 10−15 m2 for the Au_50 ones. As a comparison, −16 2 the PS_127 homo-aggregates have a ∆Cext/NP of 10 m in the same spectral range (Figure5E ). Importantly, the ∆Cext/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. When Au_20 and Au_50 hetero-aggregates with latex NPs of 200 nm (PS_200) are considered, the scenario is different from the previous cases due to the prevalence of scattering contribution from the dielectric spheres. The surface plasmon resonance in the Au_20 oligomers is barely detectable (Figure6A), while the Au_50 oligomers still have a well-defined plasmon band (Figure6B). Even in the 300–450 nm range, the interband transitions of Au NPs overlap with the scattering profile of the latex spheres and the two contributions are no more discernible from each other. The effect of clustering remains that of producing a red-shift and broadening of the plasmon band, which is better appreciated from the plot of the relative ∆Cext/NP (Figure6C ). Indeed, the edge in concomitance of the plasmon resonance is neat only for the Au_50 oligomers. In fact, the plots of absolute ∆Cext/NP for the Au_20 hetero-aggregate (Figure6D ) resemble that of latex homo-aggregates (Figure6E), but with extinction values that are one order of magnitude lower. Only the Au_50 hetero-aggregates have comparable absolute ∆Cext/NP than the PS_200 homo-aggregates, in the 500–600 nm range. Although the above-mentioned differences due to the size of the latex spheres, the relative and absolute ∆Cext/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 ∆Cext/NP depend on the size and shape of the isolated Au NPs, hence can be changed by acting on these parameters. Overall, there are no apparent advantages in using hetero-aggregates of Au and 200 nm latex NPs but, as stated previously, the PS_200 NPs are not the best choice for turbidimetric assays due to rapid saturation of the signal at large analyte concentration. While Au NPs are the standard nanomaterials for optical or colorimetric sensing due to their long-term chemical stability [15,17,18], Ag NPs are renowned for the best plasmonic performance in the visible range [16]. For instance, the extinction cross-section of Ag NPs is 5–10 times larger than Au NPs with the same morphology, which may suggest a proportionally greater effect in hetero-aggregates [16,17,40]. Therefore, the calculations were performed also on dimers and trimers containing 20 nm (Ag_20) or 50 nm (Ag_50) silver NPs and 77 nm (PS_77), 127 nm (PS_127), or 200 nm (PS_200) latex NPs (Figures7–9). The surface plasmon band of Ag_20 hetero-aggregates is sensibly lower than in isolated NPs (Figure7A), and the red-shift and broadening are observed also in this case like in the oligomers with Au and latex NPs. NanomaterialsNanomaterials2021 2021, 11, 11, 1147, x 1312 of of 24 23

FigureFigure 7. 7.Hetero-aggregates Hetero-aggregates ofof AgAg andand PSPS NPs. NPs. ((AA)) CCextext/NP/NP forfor aggregatesaggregates ofof Ag_20Ag_20 andand PS_77.PS_77. ( (BB)) Cextext/NP/NP forfor a a PS_77/Ag_20/PS_77PS_77/Ag_20/PS_77 trimer at various gap. (C) Cext/NP/NP forfor aggregates aggregates of of Ag_50 Ag_50 and and PS_77. PS_77. The The relative relative ( (DD)) and and absolute absolute Δ Δ NanomaterialsE(E) CCext 2021/NP/NP, 11for, x Ag_20 or Ag_50 oligomers with PS_77. (FF) The absolute Cext/NP/NP for PS_77 homo-dimers and trimers are14 of 24 ( ) ∆ ext for Ag_20 or Ag_50 oligomers with PS_77. ( ) The absolute ∆ ext for PS_77 homo-dimers and trimers are reportedreported for for comparison. comparison.

The coupling of Ag NPs with 127 nm latex nanospheres has a more evident effect on plasmon resonance. In Ag_20 (Figure 8A) and Ag_50 (Figure 8B), the red-shift and broadening of the plasmon band is very evident and corresponds to an increment of the optical density in the spectral region >450 nm, as shown by the plot of the relative ΔCext/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 Cext/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 ΔCext/NP of hetero- aggregates with Ag_50 is more than one order of magnitude larger in the 450 - 550 nm range. The extinction cross-section of oligomers of Ag_20 and 200 nm latex nanospheres is dominated by the scattering profile of the big dielectric spheres (Figure 9A), while the surface plasmon resonance is easily detectable only in the Ag_50 oligomers (Figure 9B). The red-shift and broadening of the plasmon band are found also in this case, although with a relative ΔCext/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 Cext/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.

FigureFigure 8. 8.Hetero-aggregates Hetero-aggregates of of Ag Ag and and PS PS NPs. NPs. ( A(A) )C Cextext/NP/NP forfor aggregatesaggregates ofof Ag_20Ag_20 andand PS_127.PS_127. (B) Cext/NP/NP forfor aggregates aggregates of Ag_50 and PS_127. The relative (C) and absolute (D) ΔCext/NP for Ag_20 or Ag_50 oligomers with PS_127. (E) The of Ag_50 and PS_127. The relative (C) and absolute (D) ∆Cext/NP for Ag_20 or Ag_50 oligomers with PS_127. (E) The absolute ΔCext/NP for PS_127 homo-dimers and trimers are reported for comparison. absolute ∆Cext/NP for PS_127 homo-dimers and trimers are reported for comparison.

Figure 9. Hetero-aggregates of Ag and PS NPs. (A) Cext/NP for aggregates of Ag_20 and PS_200. (B) Cext/NP for aggregates of Ag_50 and PS_200. The relative (C) and absolute (D) ΔCext/NP for Ag_20 or Ag_50 oligomers with PS_200. (E) The absolute ΔCext/NP for PS_200 homo-dimers and trimers are reported for comparison.

Nanomaterials 2021, 11, x 14 of 24

NanomaterialsFigure 20218. Hetero-aggregates, 11, 1147 of Ag and PS NPs. (A) Cext/NP for aggregates of Ag_20 and PS_127. (B) Cext/NP for aggregates13 of 23 of Ag_50 and PS_127. The relative (C) and absolute (D) ΔCext/NP for Ag_20 or Ag_50 oligomers with PS_127. (E) The absolute ΔCext/NP for PS_127 homo-dimers and trimers are reported for comparison.

FigureFigure 9.9. Hetero-aggregatesHetero-aggregates of of Ag Ag and and PS PS NPs. NPs. ( A(A) )C Cextext/NP/NPfor for aggregates aggregates of of Ag_20 Ag_20 and and PS_200. PS_200. (B ()BC) extCext/NP/NPfor for aggregates aggregates Δ ofof Ag_50Ag_50 andand PS_200.PS_200. TheThe relativerelative (C(C)) and and absolute absolute ( D(D) )∆ CCextext/NP/NP forforAg_20 Ag_20 or or Ag_50 Ag_50 oligomers oligomers with with PS_200. PS_200. ( E()E) The The Δ ext absoluteabsolute ∆CCext/NP/NP forfor PS_200 PS_200 homo-dimers and trimerstrimers areare reportedreported forfor comparison.comparison.

In the PS_77/Ag_20/PS_77 trimer, the red-shift depends on the interparticle gap in the 1–5 nm range (Figure7B), being the maximum for the smallest gap. However, band broadening and increase of optical density in the red spectral window is scarcely affected by the gap size. In Ag_50 oligomers (Figure7C), the change in the plasmon resonance band is less evident. However, the relative ∆Cext/NP (Figure7D) is of comparable entity for the Ag_20 and the Ag_50 hetero-aggregates and exceeds 40% for wavelengths > 450 nm. The absolute ∆Cext/NP (Figure7E) shows the maxima in the 450–500 nm range, which is not the preferred −17 2 one for ordinary turbidimetry. The change in Cext/NP is of the order of 10 m for Ag_20 and 10−16 m2 for Ag_50, i.e., comparable or one order of magnitude larger than in PS_77 homo-aggregates (Figure7F). The coupling of Ag NPs with 127 nm latex nanospheres has a more evident effect on plasmon resonance. In Ag_20 (Figure8A) and Ag_50 (Figure8B), the red-shift and broadening of the plasmon band is very evident and corresponds to an increment of the optical density in the spectral region >450 nm, as shown by the plot of the relative ∆Cext/NP (Figure8C). 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 Cext/NP (Figure8D) 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 (Figure8E), the ∆Cext/NP of hetero- aggregates with Ag_50 is more than one order of magnitude larger in the 450–550 nm range. The extinction cross-section of oligomers of Ag_20 and 200 nm latex nanospheres is dominated by the scattering proﬁle of the big dielectric spheres (Figure9A), while the surface plasmon resonance is easily detectable only in the Ag_50 oligomers (Figure9B). The red-shift and broadening of the plasmon band are found also in this case, although with a relative ∆Cext/NP (Figure9C) 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, Nanomaterials 2021, 11, x 15 of 24

3.4. Optimising Hetero-Aggregates of Metal and Latex NPs The calculations on plasmonic-dielectric hetero-aggregates identified gold NPs with the latex spheres of 77 or 127 nm as the optimal combination for immuno-turbidimetric assays. These combinations are featured by a remarkable change of the optical density in Nanomaterials 2021, 11, 1147 the preferred range for turbidimetry (550–600 nm), which is larger than in the correspond-14 of 23 ing homo-aggregates of latex spheres. Particles size is lower than the 200 nm threshold, for which a limited dynamic range and linearity interval has been reported [1,2]. None- thattheless, undergo the size a small of Au relative and latex change spheres of optical has to densitybe selected passing carefully, from theto obtain isolated a real to the advantage compared to latex alone. This is further demonstrated by a series of calculations hetero-aggregate conﬁguration. The absolute change of Cext/NP (Figure9D) is comparable toof C thatext/NP of thein hetero-aggregates oligomers with PS_77 composed and PS_127,of 7 or 12 as particles well as toarranged, the homo-aggregates respectively, line- of PS_200arly or in (Figure asymmetric9E) in the octahedra 450–550 (Figure nm range. 10). Calculations considered the two opposite combinations of small gold spheres 3.4.(Au_20) Optimising with the Hetero-Aggregates PS_127 NPs or of the Metal large and gold Latex particles NPs (Au_50) with the smallest latex beadsThe (PS_77). calculations When onAu_20 plasmonic-dielectric and PS_127 are hetero-aggregatescombined, the Cext identiﬁed/NP continuously gold NPs grows with thewith latex the number spheres of 77particles or 127 in nm the as aggregate, the optimal going combination from the fordimer immuno-turbidimetric to the trimer (Figure assays.10A), the These heptamer combinations and the are asymmetric featured by octahedr a remarkableon (Figure change 10B). of the According optical density to the inplot the of preferredthe relative range ΔCext for/NP turbidimetry (Figure 10C), (550–600 the increment nm), which is always is larger at wavelengths than in the corresponding > 550 nm, alt- homo-aggregateshough it is not associated of latex spheres.with a dramatic Particles change size is of lower the shape than theof the 200 plasmon nm threshold, band. This for whichresult agrees a limited with dynamic the previous range andobserv linearityations interval of red agglomerates has been reported between [1,2]. 3 Nonetheless,μm latex and the13 nm size Au of NPs Au and[56]. latex spheres has to be selected carefully, to obtain a real advantage comparedThe absolute to latex alone.change This of Δ isCext further/NP is demonstratedof the order of by 10 a−16 series m2 (Figure of calculations 10D), which of C extis /NPcom- inparable hetero-aggregates to the absolute composed ΔCext/NP ofof 7homo-aggregates or 12 particles arranged, (Figure 10E). respectively, This means linearly that orthere in asymmetricis no advantage octahedra in combining (Figure 10Au_20). with PS_127.

Figure 10. Hetero-aggregates ofof AuAu andand PS PS NPs. NPs. ( A(A))C Cextext/NP/NP forfor aa dimerdimer and and a a trimer trimer of of Au_20 Au_20 and and PS_127. PS_127. (B ()BC) extCext/NP/NPfor for Δ a heptamer andand anan asymmetricasymmetric octahedronoctahedron ofof Au_20Au_20 andand PS_127. PS_127. The The relative relative ( C(C)) and and absolute absolute (D (D) ∆) CextCext/NP/NPfor for Au_20 Au_20 Δ ext oligomers with PS_127.PS_127. (E)) TheThe absoluteabsolute ∆CCext/NP/NP forfor PS_127 PS_127 homo-aggregates areare reportedreported forfor comparison.comparison.

Calculations considered the two opposite combinations of small gold spheres (Au_20) with the PS_127 NPs or the large gold particles (Au_50) with the smallest latex beads (PS_77). When Au_20 and PS_127 are combined, the Cext/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 ∆Cext/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]. Nanomaterials 2021, 11, x 16 of 24

When Au_50 NPs are combined with PS_77 spheres, the change in Cext/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 ΔCext/NP is located at wavelengths > 550 nm as well (Figure 11C). However, this change is of the order of 10−15 m2 in the absolute ΔCext/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. Nanomaterials 2021, 11, 1147 A more accurate prediction of the optical properties of a 1:1 mixture of Au_5015 of and 23 PS_77 NPs can be achieved by accounting for the permutations of gold and latex NPs in the aggregates. For instance, in the case of the simplest immuno-aggregate occurring at −16 2 the lowestThe absolute antigen change concentration, of ∆Cext /NPthe dimer,is of the these order permutations of 10 m are(Figure Au_50/PS_77 10D), which (50%), is comparablePS_77/PS_77 to (25%) the absoluteand Au_50/Au_50∆Cext/NP of (25%) homo-aggregates (Figure 12A). (FigureThe resulting 10E). ThisCext/NP means weighted that thereon all is the no possible advantage permutations in combining still Au_20 shows with a remarkable PS_127. broadening of the main extinction bandWhen (Figure Au_50 NPs12A). are This combined is reasonable with PS_77 considering spheres, the the additional change in contributionCext/NP also due has ato continuouslythe modification growing of the trend surface passing plasmon from band the dimerin the to25% the of trimer Au_50/Au_50 (Figure 11 homodimers.A), the hep- tamerThe permutation-weighted and the asymmetric octahedron Cext/NP was (Figure calculated 11B). Thealso relative for a 1:1 increment mixtureof of∆ Ag_40Cext/NP andis locatedPS_77 NPs. at wavelengths Thus, the average > 550 nm is performed as well (Figure on the 11C). Ag_40/PS_77 However, this (50%), change PS_77/PS_77 is of the order (25%) −15 2 ofand 10 Ag_40/Ag_40m in the absolute (25%) permutations.∆Cext/NP (Figure The 11 resultingD), which C isext more/NP retains than one the order red-shift of magni- and tudebroadening larger thanof the in plasmon homo-aggregates band, especially (Figure 11inE). the Hence, 450–550 the nm combination range as in of the Au_50 previous and PS_77calculations NPs provides with silver an advantageparticles. of sensitivity compared to the separated components.

Figure 11. Hetero-aggregates of Au and PS NPs. ( (AA)) CCextext/NP/NP forfor aa dimer dimer and and a atrimer trimer of of Au_50 Au_50 and and PS_77. PS_77. (B ()B C) extCext/NP/NP for Δ fora heptamer a heptamer and and an anasymmetric asymmetric octahedr octahedronon of of Au_50 Au_50 and and PS_77. PS_77. The The relative ((CC)) andand absoluteabsolute ( D(D) )∆ CCextext/NP/NPfor for Au_50 Au_50 Δ oligomers withwith PS_77.PS_77. ((EE)) TheThe absoluteabsolute ∆CCextext/NP/NP for PS_77 homo-aggregates areare reportedreported forfor comparison.comparison.

AWhen more trimers accurate with prediction a linear arrangement of the optical ar propertiese considered, ofa the 1:1 number mixture of of permutations Au_50 and PS_77is 6 (Figure NPs can 12A,B). be achieved The permutation-weighted by accounting for the C permutationsext/NP or Au_50 of gold and andAg_40 latex NPs NPs with in thePS_77 aggregates. also show For a remarkable instance, in broadeni the caseng of of the the simplest plasmon immuno-aggregate band. The relative Δ occurringCext/NP (Fig- at theure lowest12C) is antigen located concentration, again in the 550–650 the dimer, nm these range permutations for the Au aretrimers. Au_50/PS_77 For Ag trimers, (50%), PS_77/PS_77 (25%) and Au_50/Au_50 (25%) (Figure 12A). The resulting Cext/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 Cext/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 Cext/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. Nanomaterials 2021, 11, x 17 of 24

ΔCext/NP extends from 450 to 600 nm, which is a larger interval than in dimers. The absolute ΔCext/NP (Figure 12D) is of the order of 10−15 m2 in both cases, a higher value than in PS_77 homo-aggregates. The Au_50, Ag_40, and PS_77 NPs are easily implementable in immunoturbidimetric assays because they are all commercially available at a relatively low cost and have well- known surface chemistry for immobilization of antibodies or other targeting groups that specifically bind analytes in sandwich configurations [1,2,18,24,57]. On the other hand, the plasmonic response dramatically changes with the shape of the metal object [16,17,27,28,58], hence additional calculations were performed on more complex metal nanostructures, such as latex beads decorated with gold NPs and PS@Au core@shell whose synthesis has been reported in the literature [27,59], or commercially available gold nanorods [60] (Figure 13). In the case of a hybrid bead made of a gold nanoparticle (20 nm) attached on top of a latex particle (127 nm), a remarkable change of the extinction cross-section is observed already in the monomer (Figure 13A), due to the tight coupling between the metal and the dielectric moieties. The extinction of the isolated hybrid bead is featured by a plasmon resonance band shifted beyond 550 nm and over imposed to the scattering profile of the latex sphere. After dimer formation, with a gap of 5 nm between the Au_20 NPs, there is an almost homogeneous increase of the optical density in the whole spectral range. This can be interpreted as the combined effect of the increase of light scattering in the blue portion of the spectrum due to the proximity of the two latex spheres, and the increase of extinction in the red portion of the spectrum due to the coupling of the two Au NPs. In fact, the ΔCext/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 ΔCext/NP is slightly higher than in the PS_127 dimer, thanks to the plasmonic Nanomaterials 2021, 11, 1147 coupling of the Au_20 NPs. However, a change of less than one order of magnitude16 of 23 is not enough to justify the synthetic complexity of the PS_127-Au_20 hybrid.

FigureFigure 12. Permutations 12. Permutations in hetero-aggregates in hetero-aggregates of of Au oror AgAg and and PS PS NPs. NPs. (A )(A Permutation-weighted) Permutation-weightedCext/NP Cextfor/NP a dimer for a anddimer and a trimera trimer of Au_50 of Au_50 and and PS_77. PS_77. (B) ( BPermutation-weighted) Permutation-weighted C Cextext/NP/NPfor for a a dimer dimer and and a trimer a trimer of Ag_40 of Ag_40 and PS_77. and PS_77. The relative The relative (C) and(C )absolute and absolute (D) (ΔDC) ext∆C/NPext/NP forfor Au_20 Au_20 oligomers oligomers with with PS_127.PS_127. (E ()E The) The absolute absolute∆C extΔC/NPext/NPfor PS_77for PS_77 homo-aggregates homo-aggregates are are reportedreported for comparison. for comparison. When trimers with a linear arrangement are considered, the number of permutations is 6 (Figure 12A,B). The permutation-weighted Cext/NP or Au_50 and Ag_40 NPs with PS_77 also show a remarkable broadening of the plasmon band. The relative ∆Cext/NP (Figure 12C) is located again in the 550–650 nm range for the Au trimers. For Ag trimers, ∆Cext/NP extends from 450 to 600 nm, which is a larger interval than in dimers. The −15 2 absolute ∆Cext/NP (Figure 12D) is of the order of 10 m in both cases, a higher value than in PS_77 homo-aggregates. The Au_50, Ag_40, and PS_77 NPs are easily implementable in immunoturbidimetric assays because they are all commercially available at a relatively low cost and have well- known surface chemistry for immobilization of antibodies or other targeting groups that specifically bind analytes in sandwich configurations [1,2,18,24,57]. On the other hand, the plasmonic response dramatically changes with the shape of the metal object [16,17,27,28,58], hence additional calculations were performed on more complex metal nanostructures, such as latex beads decorated with gold NPs and PS@Au core@shell whose synthesis has been reported in the literature [27,59], or commercially available gold nanorods [60] (Figure 13). In the case of a hybrid bead made of a gold nanoparticle (20 nm) attached on top of a latex particle (127 nm), a remarkable change of the extinction cross-section is observed already in the monomer (Figure 13A), due to the tight coupling between the metal and the dielectric moieties. The extinction of the isolated hybrid bead is featured by a plasmon resonance band shifted beyond 550 nm and over imposed to the scattering profile of the latex sphere. After dimer formation, with a gap of 5 nm between the Au_20 NPs, there is an almost homogeneous increase of the optical density in the whole spectral range. This can be interpreted as the combined effect of the increase of light scattering in the blue portion of the spectrum due to the proximity of the two latex spheres, and the increase of Nanomaterials 2021, 11, x 18 of 24

When a PS@Au core@shell with 20 nm thick gold shell and latex core with 160 nm of Nanomaterials 2021, 11, 1147 17 of 23 diameter is considered, the Cext/NP of the dimer is homogeneously lower in the whole spectral range (Figure 13C). The cores@shells with dielectric core and gold shell are renowned for their broad plasmonic band in the red portion of the visible spectrum, as hap- extinctionpens in the in PS@Au the red portioncase. This of theextinction spectrum band due is to less the intense coupling in of the the dimer two Au because NPs. In of fact, the thered-shift∆Cext /NPof theis lowerplasmon than resonance that of the of PS_127 the dime dimerr in for the wavelength near-infrared, < 550 due nm to (Figure the strong 13B), dueplasmonic to the larger coupling separation between between the two the nanospheres. latex spheres The in the change hybrid of dimer. extinction Beyond due 550 to nm,this thered-shift∆Cext /NPis sois intense slightly that higher overwhelms than in the the PS_127 increase dimer, of the thanks scattering to the plasmoniccontribution coupling due to ofthe the formation Au_20 NPs. of the However, dimer. The a change resulting of less absolute than one ΔCext order/NP is of more magnitude than one is notorder enough of mag- to justifynitude thelower synthetic than the complexity PS_200 NPs of the(Figure PS_127-Au_20 13D). hybrid.

FigureFigure 13.13. HybridHybrid dimersdimers ofof otherother Au Au nanostructures. nanostructures. ( A(A) )C Cextext/NP/NPfor for a a dimer dimer of of PS_127-Au_20. PS_127-Au_20. ( B(B)) The The absolute absolute∆ ΔCCextext/NP/NP forfor thethe PS_127-Au_20PS_127-Au_20 andand thethe PS_127PS_127 dimers.dimers. ((CC)) CCextext/NP forfor aa dimerdimer ofof PS@AuPS@Au core-shellcore-shell (core(core diameterdiameter 160160 nm,nm, shellshell Δ thicknessthickness 2020 nm).nm). ((DD)) TheThe absolute absolute∆ CCextext/NP/NP forfor thethe PS@AuPS@Au core@shellcore@shell andand thethe PS_200PS_200 dimers.dimers. ((EE))C Cextext/NP/NP forfor aa dimerdimer ofof PS_127PS_127 andand aa AuAu NRNR (cylinder(cylinder withwith hemisphericalhemispherical caps,caps, longlong 6565 nm,nm, diameterdiameter 2525 nm)nm) withwith longitudinallongitudinal (L)(L) oror transversaltransversal (T) orientation. (F) The absolute ΔCext/NP for the AuNR/PS_127 L and T dimers and the PS_127 dimers. (T) orientation. (F) The absolute ∆Cext/NP for the AuNR/PS_127 L and T dimers and the PS_127 dimers.

WhenFinally, a PS@Authe effect core@shell of dimerization with 20 on nm a cylindrical thick gold shellAuNR and with latex hemispherical core with 160 caps nm (65 of diameternm total length is considered, and diameter the Cext of/NP 25 ofnm) the with dimer PS_127 is homogeneously was investigated lower (Figure in the 13E). whole Nano- spec- tralrods range of gold (Figure are known 13C). The to exhibit cores@shells two plasmo with dielectricn bands, coreone most and gold intense shell at are longer renowned wave- forlengths, their broadthat corresponds plasmonic bandto the inpolarization the red portion of conduction of the visible electrons spectrum, along as the happens main rod in theaxis, PS@Au and a case.second This resonance, extinction less band intense, is less located intense inat thea similar dimer wavelength because of the as red-shiftin nano- ofspheres, the plasmon due to resonancepolarization of along the dimer the short in the axis near-infrared, [16,17,40,61]. due The to spectral the strong location plasmonic of the couplingmain resonance between is thetuneable two nanospheres. by changing The the changeaspect ratio of extinction of the rod, due i.e., to thisthe red-shiftratio between is so intensethe main that and overwhelms the minor axis the [16,17,40,61,62]. increase of the scattering It should contributionbe noted that due asymmetric to the formation NPs can ofbind the the dimer. latex The moiety resulting with absolutea different∆C orientext/NPationis more and, than for onethe orderAuNR, of we magnitude considered lower the thantwo limiting the PS_200 cases NPs of (Figurelongitudinal 13D). or transversal binding along the dimer axis. The orien- tationFinally, makes the a big effect difference of dimerization in the extinction on a cylindrical of the hetero-dimer, AuNR with hemisphericalwhere the plasmon caps (65resonance nm total band length is blue-shifted and diameter from of 25720 nm) to 700 with nm PS_127 for the was longitudinal investigated configuration, (Figure 13E). or Nanorods620 nm for of transversal gold are known orientation. to exhibit For transversal two plasmon orientation, bands, one the most large intense blue-shift at longer origi- wavelengths,nates a dispersive that corresponds profile of the to theΔCext polarization/NP curve. ofNevertheless, conduction electronsthe resulting along maxima the main of rodΔCext axis,/NP are and in aa different second resonance, spectral region less depending intense, located on the atorientation a similar (Figure wavelength 13F), which as in nanospheres, due to polarization along the short axis [16,17,40,61]. The spectral location of the main resonance is tuneable by changing the aspect ratio of the rod, i.e., the ratio between the main and the minor axis [16,17,40,61,62]. It should be noted that asymmetric NPs can

bind the latex moiety with a different orientation and, for the AuNR, we considered the two Nanomaterials 2021, 11, 1147 18 of 23

limiting cases of longitudinal or transversal binding along the dimer axis. The orientation Nanomaterials 2021, 11, x makes a big difference in the extinction of the hetero-dimer, where the plasmon resonance19 of 24 band is blue-shifted from 720 to 700 nm for the longitudinal configuration, or 620 nm for transversal orientation. For transversal orientation, the large blue-shift originates a dispersive profile of the ∆Cext/NP curve. Nevertheless, the resulting maxima of ∆Cext/NP aremakes in a difficult different a spectralprediction region on the depending behaviour on in the real orientation systems, (Figurewhere the 13F), orientation which makes of the Δ difficultnanorod a usually prediction cannot on the be behaviour predetermined. in real systems, Nonetheless, where the the orientationCext/NP can of thebe more nanorod than one order of magnitude larger than in the PS_127 dimer in the 600–650 or 700–750 nm usually cannot be predetermined. Nonetheless, the ∆Cext/NP can be more than one order of magnituderange. Although larger this than range in the is PS_127 not the dimer prefe inrred the one 600–650 for turbidimetry, or 700–750 nm the range. resonance Although of na- thisnorods range can is be not shifted the preferred by changing one their for turbidimetry, aspect ratio, to the seek resonance an optimal of nanorods spectral position. can be shifted by changing their aspect ratio, to seek an optimal spectral position. 4. Discussion 4. DiscussionIdeally, the extinction properties of a colloidal solution for immunoturbidimetric assays Ideally,should theundergo extinction a high properties change ofat alow colloidal analyte solution concentration, for immunoturbidimetric when oligomers as-like saysdimers, should trimers, undergo etc. are a highformed, change followed at low by analyte a continuous concentration, increase when of the oligomers signal at higher like dimers,analyte trimers,concentration, etc. are formed,over the followed whole byrange a continuous of analytical increase interest of the [1,2,4,6,7,14]. signal at higher Im- analyteportantly, concentration, this should come over without the whole saturation range of of analytical the signal interest beyond [1 the,2,4 dynamic,6,7,14]. Impor-range of tantly,the turbidimeter this should and come without without losing saturation the linear ofity the in signal the upper beyond limit the of dynamic analyte concentra- range of thetion turbidimeter [1,2,4,6,7,14]. and Although without latex losing beads the linearityhave excellent in the performances upper limit of and analyte have concentra- been used tionfor several [1,2,4,6 decades,7,14]. Although for immunoturbidimetric latex beads have excellent detection, performances their extinction and properties have been useddo not forallow several for meeting decades all for the immunoturbidimetric features of the idea detection,l reagent for their immunoturbidimetric extinction properties assays do not of allow for meeting all the features of the ideal reagent for immunoturbidimetric assays of analytes that must be measured over several orders of magnitude of concentration [6,9]. analytes that must be measured over several orders of magnitude of concentration [6,9]. On the other hand, the responsivity of the surface plasmon resonance in noble metal NPs On the other hand, the responsivity of the surface plasmon resonance in noble metal NPs (Au, Ag) upon changes of the dielectric environment, has been exploited multiple times (Au, Ag) upon changes of the dielectric environment, has been exploited multiple times for for optical sensing, even for ultrasensitive assays [14,15,18,19]. Hence, the optical proper- optical sensing, even for ultrasensitive assays [14,15,18,19]. Hence, the optical properties of ties of oligomers containing both latex and noble metal NPs have been studied to identify oligomers containing both latex and noble metal NPs have been studied to identify possible appropriatepossible appropriate combinations combinations with positive with prospectspositive prospects in immunoturbidimetry. in immunoturbidimetry. The results The results on the relative ΔCext/NP in homo- and hetero-dimers are summarized in Figure on the relative ∆Cext/NP in homo- and hetero-dimers are summarized in Figure 14A. The largest14A. The values largest (50–60% values at (50–60% 550, 575, at or 550, 600 575, nm) or are 600 reached nm) are when reached metal when and metal latex NPsand arelatex coupledNPs are together.coupled together. Exceptions Exceptions are the PS_77 are homo-dimer,the PS_77 homo-dimer, which undergoes which a undergoes similar change, a sim- andilar change, hetero-dimers and hetero-dimers with PS_200, with which PS_200, have awhich change have of ca.a change 10% due of ca. to the10% dominant due to the contributiondominant contribution of the dielectric of the particle dielectric over particle the plasmonic over the one. plasmonic one.

Figure 14. Comparative resultsresults forfor dimers.dimers. RelativeRelative ( A(A)) and and absolute absolute (B ()B∆) ΔCCextext/NP/NPfor for the the homo- homo- and and hetero-dimers hetero-dimers of of this this study,study, computed atat 550,550, 575,575, andand 600 600 nm. nm.

More complex nanostructures like the PS_127-Au_20 or the PS@Au core@shell do not provide appreciable advantages over the hetero-aggregates of simple spheres. The AuNR may be a convenient choice if the binding with latex particles happens with a transversal geometry that is associated with the largest relative increase of ΔCext/NP. However, the behaviour of AuNR in larger hetero-oligomers deserves a detailed and specific study to

Nanomaterials 2021, 11, 1147 19 of 23

More complex nanostructures like the PS_127-Au_20 or the PS@Au core@shell do not Nanomaterials 2021, 11, x 20 of 24 provide appreciable advantages over the hetero-aggregates of simple spheres. The AuNR may be a convenient choice if the binding with latex particles happens with a transversal geometry that is associated with the largest relative increase of ∆Cext/NP. However, the verifybehaviour their of compatibility AuNR in larger with hetero-oligomers a monotonic increase deserves of athe detailed optical and extinction specific at study a fixed to wavelength,verify their compatibilityas required in with ordinary a monotonic turbidimetry. increase of the optical extinction at a fixed wavelength,In any case, as required the absolute in ordinary ΔCext/NP turbidimetry., summarized in Figure 14B, is the relevant parameter toIn predict any case, the the sensitivity absolute of∆C aext real/NP immuno, summarizedturbidimetric in Figure assay. 14B, isIn the this relevant case, the parame- size of theter toNPs predict dominates the sensitivity the histogram, of a real i.e., immunoturbidimetric the particles with a size assay. of 200 In this nm case,have thethe sizelargest of ΔtheCext NPs/NP. dominates Remarkably, the hetero-dimers histogram, i.e., of the Au_50 particles with with PS_77 a size and of PS_127 200 nm have have comparable the largest performances∆Cext/NP. Remarkably, with dimers hetero-dimers containing ofPS_200 Au_50 NPs. with Ag PS_77 NPs andperform PS_127 slightly have comparableless than Au analoguesperformances in the with 550–600 dimers nm containing range because PS_200 th NPs.e plasmon Ag NPs resonance perform of slightly silver lessnanospheres than Au isanalogues blue-shifted in the more 550–600 thannm 100 range nm from because that theof gold plasmon ones resonance[16,17,63]. of Consequently, silver nanospheres a larger is blue-shifted more than 100 nm from that of gold ones [16,17,63]. Consequently, a larger ΔCext/NP is expected in the 450–500 nm range for Ag hetero-aggregates. More complex ∆C /NP is expected in the 450–500 nm range for Ag hetero-aggregates. More complex plasmonicext NPs have a heterogeneous response, suggesting that the cost in the realization 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 of complex architectures differing from simple metal nanospheres is justified only after a specific effort of optimization of the ΔCext/NP, which is not unequivocally satisfactory for specific effort of optimization of the ∆C /NP, which is not unequivocally satisfactory for the morphologies considered in this study.ext the morphologies considered in this study. In the plot of Figure 14B there is an obvious prevalence of ΔCext/NP from larger par- In the plot of Figure 14B there is an obvious prevalence of ∆Cext/NP from larger parti- ticles. However, this is generally accompanied by a rapid loss of linearity while increasing cles. 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 Cext/NP at the size of the oligomers. This is shown, for instance, in Figure 15, where the Cext/NP 550,at 550, 575, 575, and and 600 600nm are nm reported are reported for the for oligomers the oligomers of PS_77 of PS_77(Figure (Figure 15A), PS_12715A), PS_127(Figure 15B),(Figure and 15 PS_200B), and (Figure PS_200 15C) (Figure as 15a functionC) as a function of the number of the numberof NPs. ofOne NPs. can One see how can seethe increase of Cext/NP has a higher slope for smaller particles, while it reaches almost satura- how the increase of Cext/NP has a higher slope for smaller particles, while it reaches almost tionsaturation beyond beyond 8–10 NPs 8–10 in NPs the in PS_200 the PS_200 spheres. spheres. Besides, Besides, the themorphology morphology of ofthe the cluster cluster is ext moreis more influential influential on on the the CCext/NP/NP ofof the the smallest smallest latex latex spheres spheres (PS_77), (PS_77), while beingbeing notnot soso crucialcrucial for the PS_200.

FigureFigure 15. 15. ExtinctionExtinction versus versus number number (N) (N) of of particle particless in in the the aggregate aggregate at at 550, 550, 575, 575, and and 600 600 nm. nm. (A) ( APS_77.) PS_77. (B) (PS_127.B) PS_127. (C) PS_200.(C) PS_200. (D) (Au_20.D) Au_20. (E) Ag_20. (E) Ag_20. (F) Au_20 (F) Au_20 with with PS_127. PS_127. (G) (Au_50G) Au_50 with with PS_77. PS_77.

When the Au_20 homo-aggregates areare consideredconsidered (Figure(Figure 1515D),D), thethe dependencedependence ofof Cextext/NP/NP onon the the number number of of particles particles also also shows a saturation after 8–10 NPsNPs atat 550550 nm,nm, whereas at 600 nm there is continuous growth. The extinction cross-section per single NP is of the order of 10−16 m2, 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

Nanomaterials 2021, 11, 1147 20 of 23 Nanomaterials 2021, 11, x 21 of 24

whereas at 600 nm there is continuous growth. The extinction cross-section per single NP is −16 2 theof theoptical order density of 10 inm the, not proximity far from thatof the of PS_127extinction homo-aggregates. peak of isolated The nanospheres. different trends How- ever,at 550 for and larger 600 nm aggregates are a consequence the plasmon of the band progressive undergoes red-shift a further of the plasmonred-shift resonance and broaden- in Au NPs homo-aggregates, which in oligomers is featured by an increase of the optical ing, instead of continuous growth in the same spectral range as it would be required for density in the proximity of the extinction peak of isolated nanospheres. However, for larger turbidimetry. This behaviour also explains the optical extinction of Ag NPs oligomers aggregates the plasmon band undergoes a further red-shift and broadening, instead of (Figurecontinuous 15E), growth which inshow the a same more spectral continuous range growth as it would of the be optical required extinction for turbidimetry. between 550 andThis 600 behaviour nm but also on explainsa scale that the optical is only extinction of the order of Ag of NPs 10 oligomers−17 m2 (same (Figure as the 15E), PS_77 which homo- aggregates).show a more In continuous fact, the plasmon growth of resonance the optical of extinction silver nanospheres between 550 is and blue-shifted 600 nm but compared on a toscale thatthat of Au is onlyNPs ofanalogues the order [16,63,64]. of 10−17 m 2Hence,(same asthe the saturation PS_77 homo-aggregates). of the extinction In in fact, the 550– 600the nm plasmon range resonance is reached of only silver for nanospheres aggregates is containing blue-shifted tens compared of particles to that [65]. of Au NPs analoguesWhen [the16,63 Au_20,64]. Hence, NPs (that the saturation perform ofbetter the extinction than Ag in NPs) the 550–600 are combined nm range with is the PS_127reached particles only for (that aggregates represent containing the best tens compromise of particles [among65]. latex spheres), one expects to When the Au_20 NPs (that perform better than Ag NPs) are combined with the PS_127 observe high sensitivity and persistent linearity. The increase of Cext/NP (Figure 15F) for particles (that represent the best compromise among latex spheres), one expects to observe small oligomers is steep, suggesting that the two types of particles can be successfully high sensitivity and persistent linearity. The increase of Cext/NP (Figure 15F) for small combinedoligomers to is steep,improve suggesting the performance that the two of types turb ofidimetric particles canassays. be successfully Unfortunately, combined the steep portionto improve of the the curve performance is limited of turbidimetricto 2–3 NPs in assays. the aggregate. Unfortunately, For more the steep than portionjust 3 NPs, of the trendthe curve of Cext is/NP limited undergoes to 2–3 NPs a reduction in the aggregate. of its slope For and, more most than relevant, just 3 NPs, the the overall trend value of of −16 2 theCext optical/NP undergoes extinction a remains reduction in of the its 10 slope m and,, i.e., most of the relevant, same order the overall of magnitude value of of the PS_127 homo-aggregates.optical extinction remains in the 10−16 m2, i.e., of the same order of magnitude of PS_127 homo-aggregates.The evolution of the optical extinction is much improved when hetero-aggregates of largerThe gold evolution particles of (Au_50) the optical andextinction smaller latex is much spheres improved (PS_77) when are used hetero-aggregates (Figure 15G). The of larger gold particles (Au_50) and smaller latex spheres (PS_77) are used (Figure 15G). Cext/NP enters the 10−15 m2 range, i.e., the same as the PS_200 homo-aggregates, but with a −15 2 continuousThe Cext/NP growthenters from the 10 2 to m12 NPs,range, instead i.e., the of same showing as the a plateau PS_200 homo-aggregates,as in the 200 nm latex but with a continuous growth from 2 to 12 NPs, instead of showing a plateau as in the spheres. Hence, the combination of Au_50 plasmonic NPs and PS_77 latex particles prom- 200 nm latex spheres. Hence, the combination of Au_50 plasmonic NPs and PS_77 latex isesparticles to improve promises sensitivity to improve and sensitivity dynamic and range dynamic in immuno-turbidimetric range in immuno-turbidimetric assays (Figure 16).assays (Figure 16).

FigureFigure 16. 16. TheThe combination of of Au_50 Au_50 and and PS_77 PS_77 NPs NPs promises promises to improve to improv the sensitivitye the sensitivity and dynamic and dy- namicrange range in immuno-turbidimetric in immuno-turbidimetric assays, while assays, avoiding while the avoiding low signal the of low PS_77 signal NPs of homo-aggregates PS_77 NPs homo- aggregatesand the high and background the high background and limited dynamic and limited range dynamic of large (>200range nm) of large latex nanospheres.(>200 nm) latex nanospheres. 5. Conclusions 5. ConclusionsWe presented a systematic study aimed at identifying a combination of plasmonic and latex NPs that can improve the detection limit of immune-turbidimetric assays, while We presented a systematic study aimed at identifying a combination of plasmonic and latex NPs that can improve the detection limit of immune-turbidimetric assays, while keeping a wide dynamic range. Our calculations suggest that the combination of Au NPs with latex spheres permits a remarkable change of optical density in the preferred range for turbidimetry (550–600 nm), by the formation of oligomeric plasmonic-dielectric hetero-aggregates. The change of optical density is sensibly larger than in the corresponding

Nanomaterials 2021, 11, 1147 21 of 23

keeping a wide dynamic range. Our calculations suggest that the combination of Au NPs with latex spheres permits a remarkable change of optical density in the preferred range for turbidimetry (550–600 nm), by the formation of oligomeric plasmonic-dielectric hetero- aggregates. The change of optical density is sensibly larger than in the corresponding homo-aggregates of latex spheres. Therefore, a mixture of plasmonic and latex NPs can meet the ideal requirement of high increment of extinction at low analyte concentration followed by a continuous increase of the optical density at higher analyte concentration, over the whole range of analytical interest. To this end, the size of Au and latex spheres must be selected carefully, to obtain a real advantage compared to latex alone. More speciﬁcally, we found an optimal response with 50 nm Au NPs and 77 nm latex spheres. The 50 nm Au NPs provide an intense variation of the optical properties already after the formation of dimers or trimers, that are the aggregates appearing ﬁrst at low analyte concentration. Furthermore, the use of relatively small 77 nm latex particles is crucial to prevent the saturation of the signal read by the turbidimeter while keeping linearity in the upper limit of analyte concentration. Overall, the appropriate combination of plasmonic and latex particles promises to improve the detection limit and dynamic range of immuno- turbidimetric assays.

Author Contributions: Conceptualization, methodology, software, validation, formal analysis, inves- tigation, data curation, visualization: G.C. and V.A.; resources, writing—original draft preparation, writing—review and editing, supervision, project administration, funding acquisition: V.A. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by the University of Padova P-DiSC project “DYNAMO”. Data Availability Statement: The data presented in this study are available on request from the corresponding author. Conﬂicts of Interest: The authors declare no conﬂict of interest.

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