Light Extinction by Agglomerates of Gold Nanoparticles: A Plasmon Ruler for Sub-10 nm Interparticle Distances.

Plasmon rulers relate the shift of resonance wavelength, λl, of gold agglomerates to the average distance, s, between their constituent nanoparticles. These rulers are essential for monitoring the dynamics of biomolecules (e.g., proteins and DNA) by determining their small (<10 nm) coating thickness. However, existing rulers for dimers and chains estimate coating thicknesses smaller than 10 nm with rather large errors (more than 200%). Here, the light extinction of dimers, 7- and 15-mers of gold nanoparticles with diameter dp = 20-80 nm and s = 1-50 nm is simulated. Such agglomerates shift λl up to 680 nm due to plasmonic coupling, in excellent agreement with experimental data by microscopy, dynamic light scattering, analytical centrifugation, and UV-visible spectroscopy. Subsequently, a new plasmon ruler is derived for gold nanoagglomerates that enables the accurate determination of sub-10 nm coating thicknesses, in excellent agreement also with tedious microscopy measurements.

Introduction

Clusters (agglomerates) of Au nanoparticles (NPs) attract scientific attention for their unique optical properties that have a strong wavelength dependence due to oscillating surface electrons (i.e., plasmons). Such clusters exhibit a longitudinal surface plasmon resonance at a wavelength λl due to plasmonic coupling[1] in addition to their transversal light extinction maximum,[2] λ, at 530 nm. The Au λl shifts to longer wavelengths as the interparticle distance, s, between two Au NPs (dimers) is reduced down to 1 nm3. This shift is achieved by coating the NPs with nanothin organic[4] or inorganic[5] films. Reducing s below 1 nm does not affect λl due to quantum tunneling[6] and a quenched near-field intensity.[7]

Plasmon rulers[3] describe the dependence of Au λl on s. They are used to select the film thickness onto these particles that are extracted using tedious microscopy[8] or unstable Förster resonance energy transfer measurements.[9] Such rulers enable monitoring of the conformational dynamics of bovine serum albumin,[10] fibrinogen, γ-globulin, histone, insulin,[11] mouse matrix metalloproteinase,[12] the heat shock protein 90,[13] and single[14] and double[15] stranded DNA molecules. A plasmon ruler has been used also to detect DNA molecules using surface-enhanced Raman spectroscopy.[16] Furthermore, plasmon rulers are essential for the optimization of the Au agglomerate efficiency in the photothermal treatment of cancers.[5]

Empirical plasmon rulers have been derived based on UV–visible spectroscopy of single[17] or dimer[10] Au NPs deposited on flat surfaces. Such relations facilitate the detection of organic coatings or films as thin as 0.4 nm[18] but have been used only for in vitro biomedical applications. This can be partly attributed to the wet chemistry methods used for the Au dimer synthesis that have not been scaled up yet[19] and limit their translation to clinical applications.[20] To this end, a plasmon ruler was obtained by discrete dipole approximation (DDA) simulations[3] of Au dimers with s > 10 nm. This ruler was used to accurately estimate thicknesses of DNA coatings[15] larger than 10 nm. However, at small s (<10 nm), the Au λl redshift[3] (Δλ = λl – λs) of 90 nm is up to 40% smaller than that measured[17] from Au dimers. In addition, the use of this ruler[3] for in vivo detection of organics is not trivial, as dimers further coagulate into larger agglomerates during their administration into the blood stream[21] and their internalization by cells.[22]

The morphology of such agglomerates is typically quantified by the fractal dimension, Df, that rapidly attains its asymptotic 1.8 by coagulation within 100 ms[23] for the typical Au suspension concentrations employed in drug delivery.[21] During in vivo administration of gold NPs, hydrodynamic (e.g., shear[24]), van der Waals, or electric forces[25] could break and reassemble these agglomerates,[21] increasing their Df up to about 2 (Figure 8 in ref (24)). The extinction spectra of such agglomerates show a characteristic broad peak[26] with a λl of 650–700 nm. Khlebtsov et al.(27) simulated the light absorption of gold NP chains (Df = 1) having 4–36 NPs at s = 1, 2, 5, 10, and 20 nm. In addition, the light absorption by realistic agglomerates with Df = 2 having 2–200 NPs was investigated for s = 0.05 and 1 nm. These coarse-grid DDA[28] and T-matrix[27] simulations underestimate the measured[29] λl redshift by up to 55%. This is due to the low dipole resolution used in DDA that limits the optimization of the λl of agglomerates based on the s of their constituent gold NPs.

Here, a new plasmon ruler for gold agglomerates having s < 10 nm is derived, for the first time to the best of our knowledge, by determining the light extinction of agglomerates during coagulation with discrete element modeling (DEM) interfaced with DDA.[30] These simulations and the plasmon ruler are compared to microscopy,[4,31] dynamic light scattering,[32] analytical centrifugation,[29] and UV–visible spectroscopy data.[4,5,29,33,34] This set of data is the largest ever for benchmarking plasmon rulers, to the best of our knowledge.

Methods

DEM of Gold NP Agglomerates

Brownian coagulation or agglomeration of gold NPs in the absence of sintering or coalescence is described by DEM neglecting van der Waals, electric, and hydrodynamic forces.[23] Such DEM-derived coagulation dynamics have been validated in detail for organic (e.g., nascent[35] and mature soot[36]) and inorganic (e.g., zirconia,[37] silica,[38] and gold[39]) agglomerates dispersed in air and water[40] at a wide range of particle concentrations.[41] Here, 1000 monodisperse gold primary particles with a diameter dp = 20–80 nm (that cover the entire range of gold NPs used in the UV–vis spectroscopy data[4,5,29,33,34] presented in Figures , 3, and 5) are randomly distributed in a cubic cell containing water at 1 atm and 300 K, applying periodic boundary conditions.[35] These conditions are commonly used for the wet-phase synthesis of gold NP agglomerates.[29] Therefore, agglomerates consisting of 2–15 monodisperse NPs are formed. The distance s between constituent gold NPs is varied between 1 and 50 nm by adjusting their center of mass to account for organic[29] and inorganic[5] coatings that are used experimentally to optimize the optical properties of gold agglomerates. Quantum tunnelling effects are negligible for the present s range (1–50 nm).[42] The composition of such coatings may enhance λl by 5–9% depending on their refractive index.[1] The np, dh, and dg of these DEM-derived agglomerates are determined[23,35] accounting for their detailed morphology. In particular, the dh of NP agglomerates having np < 100 is[43]while their dg is[44]where mi is the mass, and xi is the distance of primary particle i to the agglomerate center of mass. The agglomerate dg is related to dp and np by a power law[45]where Df and kn are the fractal dimension and prefactor, respectively. The morphology of DEM-derived agglomerates has been validated already with helium ion microscopy,[35] light scattering,[36] and mass-mobility measurements of nascent[35] and mature[36] soot, zirconia,[37] silica,[38] and gold[39] NPs.

Figure 2

Normalized Qext as a function of λ of gold (a) spheres (dotted line, circles), dimers (dot-broken line, squares, inverse triangles), (b) 7- (broken line, diamonds) and 15-mers (solid and double dot-broken lines, triangles) with Df = 1.67 (dot-broken, broken, and solid lines) or 1.9 (double dot-broken lines) estimated here by DDA (lines) and compared to those measured by Zook et al.(29) (circles, diamonds, and triangles) and Esashika et al.(4) (squares and inverse triangles). The DDA-derived Qext of gold NPs increases as single spheres (dotted line) form dimers (dot-broken line), 7- (broken line), and 15-mers (solid line) by agglomeration shifting λl from 530 to 680 nm, in excellent agreement with data[4,29] (symbols).

Figure 3

Normalized Qext as a function of λ of gold dimers with dp = 30 (a: dotted line) or 50 nm (b) at s = 50 (solid line), 2 (broken & dotted lines, triangles, and diamonds), and 1 nm (dot-broken line, squares) estimated here by DDA (lines) and measured using UV–visible spectroscopy and microscopy[4,33] (symbols).

Figure 5

Estimated interparticle separation distance, s, as a function of normalized λl shift, Δλ/λs, using plasmon rulers for chains[27] (dotted line), dimers[3] (broken line), and 7- and 15-mers (eq , solid line) of gold NPs with dp = 34 (a), 50 (b), 60 (c), and 80 nm (d) compared to microscopy and UV–visible measurements of dimers (triangles,[34] and squares[4]) and agglomerates (circles[5]).

Gold Optical Properties by DDA

DDA is used to estimate light absorption, scattering, and extinction of DEM-derived agglomerates of gold NPs using the open-source DDSCAT 7.3 code.[46] The DEM-derived, fractal-like agglomerate morphology is represented on a lattice by an array of discrete dipoles interacting with each other through their electric fields.[46] The dipole electric properties are described by the input bulk gold refractive index,[47]RI, which is valid for the dp = 20–80 nm investigated here.[48] Maxwell’s equations are discretized on the lattice using the volume integral equation method and solved iteratively.[46]

The dipole spacing, x, must be small[46] compared to the incident light wavelength, λ, to calculate accurately the gold extinction coefficient, Qext, that accounts for both light absorption and scattering of gold NPs. Owing to the high light extinction of gold NPs in the NIR, x is varied for different λ and RI to have a constant ratio of 2π|RI|x/λ = 0.0192. This precision criterion is up to 16 times smaller than those used for other strongly absorbing materials (e.g., soot[30]) and results in 100,000 dipoles per gold primary particle.

Results and Discussion

Validation of Au Agglomerate Morphology and Optical Properties

Agglomerates of gold NPs of diameter dp = 30 nm with well-defined morphology characteristics were created by DEM. Figure a shows the evolution of their Df (line and insets) during growth by coagulation as a function of their average diameter of gyration, dg (bottom abscissa), or the number of constituent NPs per agglomerate, np (top abscissa). The shaded area indicates the statistical variation of DEM. These Df dynamics are compared to microscopy data of gold NP agglomerates with similar dg (symbols)[31] from microscopy images based on the measured bounding length.[49] The DEM-derived Df evolution shown in Figure a was obtained using dp = 30 nm to be consistent with the dp range that is relevant for UV–vis spectroscopy measurements.[4,5,29,33,34] The Df was measured by Grogan et al.(31) for NPs with dp = 5 nm. Decreasing dp from 30 to 5 nm increases Df by 3–9% (as shown in Figure a of ref (23)). This is on par with the statistical variation of the DEM simulations (shaded area) and experiments[31] shown in Figure a.

Figure 1

(a) Evolution of Df during agglomeration of gold NPs as a function of their dg (bottom abscissa) and np (top abscissa) derived here by DEM (line and inset agglomerate images) along with microscopy data[31] (symbols). (b) Evolution of dh (solid line, symbols) and dg (broken line) of gold NP agglomerates as a function of their np derived by DEM (lines and inset images) and measured using dynamic light scattering[32] (DLS; symbols).

As single spherical Au NPs coagulate and form agglomerates, their Df decreases from 3 to about 1.6–1.7 for dg > 40 nm or np > 3. The Df is slightly smaller than the classic limit of Df = 1.78 ± 0.05 obtained by diffusion-limited cluster agglomeration[50] as their np is rather small (np < 20) for this asymptotic Df.[23] Nevertheless, the evolution of Df (Figure a, line) is in excellent agreement with microscopy data[31] of gold agglomerates of similarly small np (Figure a, symbols) pointing out the validity of DEM.

Figure b shows the hydrodynamic diameter, dh (solid line, symbols), and dg (broken line) of Au agglomerates as a function of np derived by DEM (lines) and measured using dynamic light scattering[32] (DLS; symbols). The dh and dg are the diameters of drag- and inertia-equivalent spheres, respectively.[51] For example, single spheres of dh = 30 nm have dg = dh/1.29.[51] During coagulation, dh and dg increase with increasing np based on eqs and 3, respectively (see Methods). This results in 7- and 15-mers having dh = dg and dh = 0.9dg, respectively. The latter is consistent with Stokesian dynamics of agglomerates[52] having similar np. The dh of all agglomerates is smaller than 120 nm, the desirable dh range for drug delivery.[53] The good agreement of DEM-derived Df and dh with microscopy (Figure a) and light scattering measurements (Figure b), respectively, further validates the present simulations.

Next, these realistic agglomerates were used for determining their light extinction by DDA. Figure shows the normalized extinction spectra of gold (a) spheres (dotted line, circles[29]) and dimers (dot-broken line, squares,[4] inverse triangles[29]), as well as of (b) the above 7- (broken line, diamonds[29]) and 15-mers (solid & double dot-broken lines, triangles[29]) with Df = 1.67 (dot-broken, broken and solid lines) or 1.9 (double dot-broken lines). The Df = 1.9 is selected as an average between the Df = 1.78 and 2 of agglomerates obtained by coagulation[23] and break up in the presence of van der Waals, hydrodynamic (e.g., shear)[24] or electric forces,[25] respectively, at long residence times during drug delivery. Lines correspond to present simulations and symbols to data by Zook et al.(29) (circles, diamonds, and triangles) and Esashika et al.(4) (squares and inverse triangles). All extinction spectra were obtained from Au agglomerates with the NP dp = 20–30 nm and s = 1.4 nm. The shaded areas indicate the DDA variation between dp = 20 and 30 nm for four dimers (Figure a) and 7-mers (Figure b). The measured and simulated light extinction spectra are normalized with the maximum extinction efficiency, Qext.

The DDA-derived Qext of single gold NPs with dp = 30 nm (Figure a) attains its maximum at a transverse λs of about 530 nm (dotted line), consistent with measurements[29] (circles). The light extinction spectrum obtained by DDA for dimers of such gold NPs also exhibits a minor peak at λl = 600 nm due to plasmonic coupling.[1] This peak becomes more significant with 7- and 15-mers (Figure b, broken and solid lines), shifting λl up to 680 nm. The λl increases only by 10% as 7-mers grow into 15-mers, in good agreement with the data.[29] The small sensitivity of λl on the gold agglomerate np suggests that the light extinction derived here by DEM-DDA for np = 7 and 15 is also valid for larger agglomerates consistent with Khlebtsov et al.(27) and may not depend strongly on individual agglomerate np and Df. However, previous simulations using a single dipole per NP showed that the λl of gold agglomerates increases only up to 600 nm.[28] Here, the maximum λl = 680 nm obtained from 15-mers using 100′000 dipoles per NP (lines) is in excellent agreement with the measured λl range of 650–700 nm[23] (Figure b, triangles). Increasing Df from 1.67 to 1.9 broadens the light extinction spectra of 15-mers (again in agreement with data) but hardly affects their λl (double dot-broken line). Therefore, the plasmon ruler derived here based on λl is not affected by van der Waals, hydrodynamic, and electric forces.

The electric field enhancement by plasmonic coupling depends on dp and s, affecting the light extinction of Au agglomerates.[54]Figure a shows the normalized extinction spectra of gold dimers with dp = 30 (dotted line, diamonds[4]) and 50 nm (broken line, triangles[4]) at s = 2 nm derived here by DDA (lines) and compared to UV–visible spectroscopy and microscopy measurements (symbols[4]). Increasing dp from 30 to 50 nm enhances the plasmonic coupling between gold NPs and increases their λl from 580 to 620 nm, in good agreement with the data[4] (symbols).

Similarly, Figure b shows the normalized extinction spectra of gold dimers (dp = 50 nm) at s = 50 (solid line, circles[33]), 2 (broken line, triangles[4]), and 1 nm (dot-broken line, squares[4]) estimated here by DDA (lines) along with UV–visible spectroscopy and microscopy[4,33] measurements (symbols). The Qext of dimers of spheres with dp = 50 nm each at s = 50 nm that practically corresponds to single ones is in excellent agreement with that measured by UV–visible spectroscopy from single spheres[33] for all λ. As the s of these dimers decreases to 2 and 1 nm, the Qext exhibits a second maximum at the longitudinal λl from the electric field enhancement induced by plasmonic coupling.[55] The λl redshifts with decreasing s, consistent with previous DDA simulations[3] and is in excellent agreement with data[4] (symbols). This redshift of λl increases the light extinction of gold dimers in the NIR, consistent with the light absorption measurements of Au NPs separated by silica coatings of various thicknesses,[5] further validating the present DEM-DDA methodology.

Plasmon Ruler for Au Nanoagglomerates

Figure shows the λl as a function of s/dp by DEM-DDA for dimers of monodisperse (squares and inverse triangles) and bidisperse (circles) NPs, and DEM-derived 7- (diamonds) and 15-mers (triangles) of monodisperse NPs with dp = 20–50 nm (squares, circles, diamonds, triangles) or 75 and 80 nm (inverse triangles) and s = 1–50 nm. Dimers with dp = 75–80 nm have 7% larger λl than those with dp = 20–50 nm at s/dp = 0.025. Similarly, increasing the NP polydispersity decreases the λl of dimers by about 8% at s/dp = 0.02–0.04. The above small reductions are within the statistical variability of the DEM-DDA simulations (shaded area). By regressing s/dp to the ratio of Δλ = λl – λs to λs = 530 nm corresponding to a Au sphere suspended in water,[3] a new plasmon ruler is created (Figure , solid line):

Figure 4

Au agglomerate λl as a function of normalized interparticle separation, s/dp, estimated by DDA using dimers of monodisperse (squares and inverse triangles) and bidisperse (circles) NPs, and DEM-derived 7- (diamonds) and 15-mers (triangles) of monodisperse NPs with dp = 20–50 (squares, circles, diamonds, and triangles) or 75 and 80 nm (inverse triangles) and s = 1–50 nm. A new plasmon ruler (eq , solid line and shaded area) is derived by regressing the DDA-derived λl evolution.

Equation has been derived for gold NPs with dp < 100 nm and is not affected by dynamic depolarization and structural retardation.[56] The light extinction of such NPs is given only by their dipole mode resulting in λs ∼ 530 nm.[27] Therefore, the critical diameter for this ruler is 100 nm, as the light extinction of gold NPs with dp > 100 nm is determined by the sum of both the dipole and quadrupole modes and increases their λs > 530 nm.[27] Furthermore, eq is valid for agglomerates with s ≥ 1 nm that are not affected by quantum tunneling[6] and the quenched near-field intensity.[7]

Next, eq is compared to the experimental data and previously reported plasmon rulers. Figure shows the estimated s (lines) as a function of the normalized λl shift, Δλ/λs, using plasmon rulers for chains (dotted line),[27] dimers (broken line),[3] and 7- and 15-mers (eq , solid line) of gold NPs with dp = 34 (a), 50 (b), 60 (c), and 80 nm (d) in comparison to microscopy and UV–vis measurements of dimers (triangles,[34] and squares[4]) and agglomerates (circles[5]). Jain et al.(3) derived and validated their plasmon ruler for dimers at s > 10 nm. Below 10 nm, this ruler (broken line) results in an error of 220% on average. The plasmon ruler of Khlebtsov et al.(27) was derived for chains with Df = 1 at s = 1–20 nm (dotted line) and is in better agreement with data. Nevertheless, it still overestimates the measured s by 70% (on average) due to the neglect of the realistic structure of agglomerates (Df > 1) that are commonly present in measurements. In contrast, the plasmon ruler derived here with Df = 1.67–1.9 (solid line and within its shade) is in better agreement with the data for all dp studied here.

Therefore, eq can be used to measure and accurately select sub-10 nm organic and inorganic coatings of gold nanoagglomerates and to monitor the dynamics of biomolecules. For example, during protein adsorption on the NP surface, a corona monolayer is formed.[57] The monolayer thickness may vary from 3.3[58] to 16 nm[59] depending on the adsorbed protein[57] and its concentration,[60] as well as the NP size[59] and surface charge.[61] The protein corona monolayer is formed within 10–50 min and covers up to 80% of the particle surface.[60] At such long residence times, gold NPs coagulate into agglomerates with various s.[11] In specific, Figure shows the s of gold agglomerates with dp = 30 nm coated by common blood proteins, that is, fibrinogen (red), histone (green), albumin (orange), and γ-globulin (blue) estimated by interfacing UV–vis spectroscopy data[11] with eq (filled bars). The gold agglomerate s ranges from 3.9 (histone, green filled bar) to 13.3 nm (blue filled bar). In this regard, using plasmon rulers for gold dimers[3] (open bars) and chains[27] (lined bars) overestimates by up to a factor of 2.8 and 2.3 , the gold agglomerate s formed by histone (green filled bar) and γ-globulin (blue filled bar), respectively. Therefore, the sub-10 nm ruler derived here for gold NP agglomerates is essential to accurately monitor the dynamics of protein corona formation.

Figure 6

Separation distance, s, of gold agglomerates with dp = 30 nm coated by fibrinogen (red), histone (green), albumin (orange), and γ-globulin (blue) estimated using UV–vis spectroscopy data[11] with plasmon rulers for dimers[3] (open bars), chains[27] (lines bars) and agglomerates (eq , filled bars).

Summary & Conclusions

The evolution of gold agglomerate morphology and optical properties during agglomeration is investigated here by coupling DEM with DDA. The morphology and hydrodynamic diameter of DEM-derived gold NP agglomerates are validated with microscopy[31] and light scattering[32] measurements, respectively. The evolution of gold light extinction during agglomeration reveals that the longitudinal surface plasmon resonance wavelength, λl, increases up to 680 nm as single gold NPs coagulate to 15-mers, in excellent agreement with data from UV–visible spectroscopy.[4,29]

The λl shift of gold dimers, 7-, and 15-mers increases with decreasing s and can be described by a universal power law resulting in a new plasmon ruler (eq ) that enables the estimation of Au nanoagglomerate coating thickness, s, in excellent agreement with the microscopy data.[4,5,34] In contrast, existing, widely used plasmon rulers for dimers[3] and chains[27] estimate coating thicknesses smaller than 10 nm with an average error of 220 and 70%, respectively. Therefore, the new plasmon ruler obtained here for agglomerates of Au NPs can be used instead of tedious microscopy measurements to determine the thickness of sub-10 nm organic and inorganic coatings. This can facilitate monitoring of the dynamics of biomolecules, such as proteins[10] and DNA,[12] and the optimization of gold agglomerate coating thickness for photothermal therapy of cancer.[5]