Determining the Spheroid Geometry of Individual Metallic

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Cite This: J. Phys. Chem. C XXXX, XXX, XXX−XXX

Determining the Spheroid Geometry of Individual Metallic Nanoparticles by Two-Dimensional Single-Particle Dynamic Light Scattering Published as part of The Journal of Physical Chemistry virtual special issue “Hai-Lung Dai Festschrift”. Luis F. Guerra, Tom W. Muir, and Haw Yang*

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Department of Chemistry, Frick Laboratory, Princeton University, Princeton, New Jersey 08544, United States S Supporting Information *

ABSTRACT: Single-particle dynamic light scattering (SPDLS) is a recently developed technique that uses dark-field illumination, active real-time three-dimensional single-particle tracking, and measurements of scattered photon polarizations to nonperturbatively evaluate the shapes of single, freely diffusing particles under the assumption of the particle having either prolate or oblate spheroid geometry. As originally developed, however, SP-DLS is incapable of unambiguously assigning either of these geometries to a single particle. In this contribution, we resolve this ambiguity by introducing a second experimental observablethe scattering spectrumso that both the scattering polarization and spectrum are simultaneously recorded and analyzed. We used numerical simulations of SP-DLS to characterize the performance of this new approach as well as the effects of key experimental parameters. We anticipate that the analyses presented here will not only form a straightforward guide for researchers seeking to optimize their own SP-DLS shape measurements but also serve as the basis for future studies of time-dependent reconfiguration in single nanostructures.

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INTRODUCTION Metallic nanoparticles (MNPs) lie at the core of a wide variety of nanotechnologies, primarily as a result of their ability to support localized surface plasmon resonances (LSPRs)the coherent oscillation of conduction-band electrons in the presence of an electromagnetic field.1 The optical response of an MNP depends on factors such as the particle’s composition, size, shape, and the dielectric medium in which it is situated.2 New applications and opportunities in basic sciences continue to expand thanks to innovative chemical approaches for accessing interesting MNP morphologies.3 Solution-based syntheses of MNPs, however, inevitably result in nanoparticle populations that are heterogeneous in the nanoparticles’ makeup and appearance. Consequently, the determination of the size and shape of a sample on a particleby-particle basis is foundational to subsequent developments. For size analysis of nanoparticles, besides the well-established transmission electron microscopy (TEM), Brownian-motionbased analyses using far-field benchtop optics such as dynamic light scattering (DLS)4 and fluorescence correlation spectroscopy (FCS),5−7 which analyzes the fluctuation correlation of fluorescence signals, as well as the more modern real-time three-dimensional (3D) single-particle tracking (SPT) spectroscopy,8 provide the average size of a sample9,10 and particleby-particle sizing information,11 respectively. © XXXX American Chemical Society

Determining the 3D shape of an individual MNP, on the other hand, is less straightforward. TEM is powerful for nanoparticles with high symmetry because it is mostly based on the two-dimensional (2D) spatial projection of a sample. Since electromagnetic waves interact with nanoparticles in full 3D space, in principle, it should be possible to obtain shape information on individual nanoparticles through optical spectroscopy. Indeed, due to the shape-sensitivity of LSPR in MNPs,12 the shape class of an MNP sample (e.g., spheroids, rods and their aspect ratio, wires, cubes and polygons, etc.) can usually be deduced using an UV−vis spectrometer. Similar ideas can be extended to obtaining shape information for single nanoparticles.13 Beyond the apparent shape classes, however, extensive numerical computations such as discrete dipole approximation (DDA)14 or finite-difference time-domain (FDTD)15 methods are needed in order to discern more subtle shape differences. Moreover, common single-particle spectroscopy often requires immobilizing individual particles on a substrate surface, which unavoidably creates a dielectric interface that further complicates the computation.16 Importantly, approaches requiring substrate immobilization could be Received: May 31, 2019 Revised: July 9, 2019 Published: July 10, 2019 A

DOI: 10.1021/acs.jpcc.9b05217 J. Phys. Chem. C XXXX, XXX, XXX−XXX

Article

The Journal of Physical Chemistry C

detectors in the laboratory-frame. The fluctuations in these polarization and spectral components are coupled to rotational diffusion precisely because of the aspherical MNP’s anisotropic polarizability tensor α and its wavelength-dependence via the particle’s complex dielectric function ϵ1(λ); that is, the orientation of the particle dictates which polarization and spectral components of the electric field give rise to the induced dipole and thus dictate the far-field scattering. In the first experimental realization of SP-DLS shape analysis,19 only information from polarization measurements was utilized, where the polarization contrast fluctuation function to be used in the autocorrelation calculation is defined as

limited in their potential extension to studying dynamical structural changes in individual nanoparticles and nanostructures. Direct imaging of MNPs using liquid-cell transmission electron microscopy,17 while capable of acquiring in situ morphological evolution of single freely diffusing nanoparticles with atom-level definition,18 appears to be technically challenging and could be incompatible with the sample of interest, e.g., dynamical conformational changes of nanostructures linked by biological macromolecules. Therefore, it would be helpful to develop a shape-determining capability amenable to routine benchtop characterizations in a typical laboratory. Single-particle dynamic light scattering (SP-DLS) has been experimentally realized as a new way to study the shape of individual MNPs.19 Similar to the widely used, ensemble technique from which it partly derives its name, SP-DLS analyzes the temporal fluctuations in scattered light arising from particles’ stochastic motions via their autocorrelation. More specifically, however, it utilizes the aforementioned realtime 3D-SPT to keep a freely diffusing MNP at the center of microscope objective focus via an active feedback mechanism,20 which in turn allows one to measure the timedependent polarization of the scattered light from an MNP illuminated by a dark-field, white-light source. The technique was capable of discerning whether a sample is predominantly prolate or oblate with an asphericity sensitivity of ≈2.5% at a 10% error rate (e.g., a 2.5 nm difference between the semiaxes of a 100 nm nanoparticle). This initial proof-of-principle experiment that used scattering light polarization as the main experimental observable, however, could not unambiguously discriminate between a prolate and an oblate spheroid on a particle-by-particle basis, with assignment of particle populations as mostly prolate or oblate only possible via statistical comparisons with separate TEM measurements. The motivation of the present contribution, therefore, is to investigate whether it is theoretically possible to ascribe a spheroidal shape (prolate or oblate) to a single MNP based on the SP-DLS concept. The approach is to include an additional experimental observable, the scattering spectrum from the MNP under investigation,21 so that both the polarization and spectral distribution of the scattered light are concurrently acquired in a time dependent manneran idea that was first demonstrated in the DLS setting by Cummins, Knable, and Yeh.22

δχ =

Ix − Iy Ix + Iy

−

Ix − Iy Ix + Iy

(1)

where Ix and Iy are the X- and Y-polarized scattered photon fluxes in the laboratory frame. With the aforementioned assumptions and an experimental setup based on a typical dark-field microscope as sketched in Figure 1, δχ is a function

Figure 1. Schematic of single-particle dynamic light scattering with a typical dark-field microscopy setup.

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THEORY The theoretical basis for the current SP-DLS shape analysis is the framework put forth by Yang and references therein.23 The main idea is as follows: An MNP in solution exhibits rotational Brownian dynamics, which is characterized by a rotational diffusion tensor with three rotational axes. The rotational diffusion tensor, in turn, depends on the shape of the MNP and the hydrodynamic interaction between the MNP and the medium. It follows that if one can measure the full rotational diffusion tensor, then with reasonable assumptions for the hydrodynamic interaction under most experimental conditions, one should be able to experimentally determine the MNP’s shape. That initial work further assumes that the MNPs are nearly spherical and the aspherical nanoparticles are modeled either as prolate or as oblate. Then, by using the Gans theory under the Rayleigh scattering regime,24 the rotational diffusion can be experimentally assessed through the autocorrelation functions of either the polarization or the spectral component of the light scattered from the single MNP and collected at

of only a few key variables: the numerical apertures (NAs) of the dark-field condenser and light-collecting objective, as well as the wavelength-dependent, complex dielectric function ϵm(λ)/refractive index nm(λ) of the medium; the orientation of the particle as given by the polar and azimuthal angles, θ(t) and ϕ(t), respectively, at time t; a discrete sequence of Nλ wavelengths {λi} ≡ {λ1, λ2, ..., λNλ} approximating the continuous illumination spectrum from a white-light source; the relative intensity weights {wi} ≡ {w1, w2, ..., wNλ} of these discrete wavelengths reproducing the light-source spectrum; and the shape parameter ξ = rL/rS − 1 of the particle where rL and rS are the semimajor and semiminor axes, respectively. For the task of statistically distinguishing between prolate and oblate for an entire sample, it is sufficient to consider only the amplitude of the autocorrelation function, B


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The Journal of Physical Chemistry C Aδχ ≡ Cδχ (τ = 0, {λi}, {wi}, ξ) Ä m=l ∞ Å ÅÅ 2π 1 = dϕ ∑ ÅÅÅÅÅ ∑ 4π l = 0 ÅÅ m =−l 0 Ç

∫

∫0

π

É ÑÑ dθ δχYlm(θ , ϕ) sinθ ÑÑÑ ÑÑ ÑÖ 2Ñ Ñ

(2)

where Ylm’s are the spherical harmonics, τ is the time lapse in correlation function, and the time variable can be omitted from θ(t) and ϕ(t) by assuming δχ is ergodic. Given the setupdependent parameters described above and the experimentally measured Aδχ, the shape parameter of the individual MNP can be found by numerically solving eq 2 after judicious truncation of the infinite series. Using only the polarization correlation amplitude instead of the full rotational diffusion tensor alleviates the experimental requirements because quantifying the latter would require very long trajectories with excellent signal-to-background ratio, which could be practically challenging. The downside of the amplitude-only approach is that a specific value of Aδχ can be compatible with two distinct ξ values arising from the a priori assumption of either prolate or oblate spheroid geometry. Such a polarization-only SP-DLS shape analysis is thus limited to the asphericity of the particle via ξ and the assignment of a particular spheroid geometry to the entire sample. To resolve such an ambiguity in SP-DLS shape analysis while retaining the practicality, we introduce another experimental observablethe spectral contrast fluctuation function,23 δχsso that together with δχ one can quantitatively assess the spheroid geometry of a single, freely diffusing MNP. See Figure 2 for a possible experimental configuration, which also serves as the experimental reference for the specific theoretical derivation below. We point out that simultaneously detecting the polarization and spectral components from a single diffusing nanoparticle using this configuration has been experimentally realized previously.21 The photon fluxes Ix and Iy in eq 1 are given as summations over {λi} and {wi},

Figure 2. Experimental setups in previous and current iterations of SP-DLS shape analysis. PBS, polarizing beamsplitter. LPDC, long-pass dichroic filter. Ix and Iy, the X- and Y-polarized scattered photon fluxes, respectively; Ir and Ib, the red and blue components, respectively, of Ix. See also the experimental configuration in ref 21.

been defined in previous publications (reproduced in the Supporting Information for completeness), and α1, α2, and α3 are the components of α with α1 = α2 ≠ α3 (additional dependencies on λi throughout eqs 3 and 4 are denoted by the subscript i). It is through these polarizability components that the dependencies on V, the particle volume, and ξ arise, with the functional form of each jth component given by αj(λ) =

for prolate particles, and Ä ÑÉÑ 1 − L3 g (e) ÅÅÅ π Å − tan−1 g (e)ÑÑÑ = L 2 = L1(ξ) = 2 Å Å Ñ Å Ñ 2 2e Ç 2 Ö

Nλ

Nλ

∑ [W33,xiwi |α3,i|2 ]} i=1

(3) N l o o λ y 2 [W11, I y(t , {λi}, {wi}, V , ξ) = I0m ∑ iwi |α1, i| ] o o n i=1

and

Ir(t , {λi}, {wi}, V , ξ) = I x(t , {λi ∀ i|λi > λe}, {wi ∀ i |λi > λe}V , ξ)

Ib(t , {λi}, {wi}, V , ξ) = I x(t , {λi ∀ i|λi ≤ λe}, {wi ∀ i

∑ [W13,yiwi(α1, iα3,̅ i + α1,̅ iα3, i)]

|λi ≤ λe}, V , ξ)

i=1

(9)

with Ix reproduced from their sum. Using eqs 8 and 9, we now introduce the spectral contrast fluctuation function δχs given by

Nλ

+

(8)

and

Nλ

+

(7)

for oblate particles, and g (e) = (1 − e 2)/e 2 with e2 = 1 − (1 + ξ)−2 in both cases. Note that δχ is independent of both I0 and V. By Figure 2, the two photon fluxes for spectral contrast are straightforwardly defined in terms of Ix and the edge wavelength λe of the long-pass dichroic filter (LPDC) as

∑ [W13,xiwi(α1,iα3,̅ i + α1,̅ iα3,i)] i=1

+

(5)

Finally, the assumption of prolate or oblate geometry is made via the shape parameter-dependent functions Lj, with Ä ÉÑ ÑÑ 1 − e 2 ÅÅÅÅ 1 ji 1 + e zy ÑÑ = 1 − 2L1 j z L3(ξ) = ln − 1 Å ÅÅ 2e j 1 − e z ÑÑ 2 e ÅÇ ÑÖ k { = 1 − 2L 2 (6)

N l o o λ x 2 [W11, I x(t , {λi}, {wi}, V , ξ) = I0m ∑ iwi |α1, i| ] o o n i=1

+

V [ϵ1(λ) − ϵm(λ)] ϵm(λ) + Lj(ξ)[ϵ1(λ) − ϵm(λ)]

y 2 ∑ [W33, iwi |α3, i| ]} i=1

(4)

δχs (t , {λi}, {wi}, ξ) =

Here, I0 is an intensity scaling factor, the W’s are functions of θ(t), ϕ(t), nm(λ), and the condenser/objective NAs and have C

Ir − Ib − Ir + Ib

Ir − Ib Ir + Ib

(10)


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The Journal of Physical Chemistry C Substituting δχs for δχ in eq 2 yields Aδχs, the amplitude of the autocorrelation of δχs. The two autocorrelation amplitudes Aδχ and Aδχs form a unique 2D mapping by which both ξ, via the former, and the spheroid geometry, via the latter, can be quantified. The key here is the simultaneous recording of both the polarization and spectral contrast information. By either observable alone, each gives the same information for a rigidbody MNP because each observable follows exactly the same rotational Brownian dynamics as the MNP body axes.21 That is, at the current level of theory, the polarization and spectral contrast functions follow each other exactly for the same single MNP with a fixed correlation, but the correlation is different for spheroids of different shapes. A practical strategy to utilize the two amplitudes for shape analyses is as follows (Figure 3). The observed value of Aδχ

simulations incorporating these theoretical extensions. Specifically, P was mapped as a function of three key variablesthe elemental composition of the tracked MNP, λe, and the number of scattered photons.

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COMPUTATIONAL DETAILS A single, rotational random walk was simulated using a previously described procedure.25 Briefly, an orientation vector fixed to the particle (here, we use the unit vector aligned with α3) was rotated by three random angles about the laboratoryframe X, Y, and Z axes at each jth step. These angles (which were subsequently converted to the spherical coordinates θj and ϕj for use in the equations described earlier) were each sampled from a normal distribution with μ = 0 rad and σ = (2D̅ R )1/2 = 0.1 rad , where D̅ R = DRτs represents a dimensionless rotational diffusion coefficient defined in relation to the time duration of a single step. Finally, the random walk consisted of Ns = 106 steps, ensuring that the ratio Ns/τ̅R = 6D̅ RNs = 3 × 104 was sufficiently large to fulfill the assumption of ergodicity required for the integration in eq 2.26 The parameters D̅ R and Ns were chosen to be experimentally reasonable, with the specific values used reflecting (for example) those defining the rotational random walk of a particle in water with radius R = 40 nm, T = 293.15 K, τs = 2 μs, and a total trajectory duration of 2 s. Various key parameters were then chosen for each SP-DLS simulation. First, the particle’s experimental composition was assigned by choosing the appropriate constants in a Lorentz− Drude model for ϵ1(λ),27 and the surrounding medium was assigned the wavelength-dependent dielectric function of water at T = 293.15 K.28,29 Second, the discretized illumination spectrum {λi} was chosenhere with {wi} = {1} and Δλ = λj+1 − λj = constant to enforce uniform sampling of a homogeneous spectrumand the relevant NAs set to NAcond = 1.3 and NAobj = 0.7 (for dark-field condenser and collecting objective, respectively) as per the previously established experimental setup. Third, we set values for λe and the signal-to-noise ratio (SNR), the latter given by SNR = Np̅ 1/2 where N̅ p is the mean number of scattered photons-per-step throughout an entire trajectory. Finally, a pair of prolate and oblate shape parameters related via Aδχ(ξp) = Aδχ(ξo) were chosen for comparison. These linked shape parameters were found via numerically solving eq 2 up to l = 6 (Figure S1). At each jth step of the random walk, a number of scattered photons Np was randomly sampled from a Poisson distribution with a mean of N̅ p. Given {λi}, each kth photon pk in the jth step was stochastically assigned a wavelength λ(pk) → λi. The wavelength index i is determined by numerically finding the bracketing values qi−1 and qi such that qi−1 ≤ rk < qi, where rk is randomly sampled from a standard normal distribution. The q’s are defined as follows: q0 = −qNλ = −∞ and

Figure 3. Strategy for unambiguous shape determination. The observed polarized autocorrelation amplitude Aδχ yields two possible shape parameters for prolate, ξp, and oblate, ξo that can be differentiated via comparison of the observed spectral autocorrelation amplitude Aδχs with the predicted prolate and oblate values, Apδχs and Aoδχs, respectively. The figure-of-merit for successful shape determination is given by P, which represents the complement of the overlap between the normal distributions defined by (μp = Apδχs, σp = σApδχs) and (μo = Aoδχs, σo = σAoδχs).

yields two possible values of the shape parameter, ξp and ξo, for assumptions of prolate or oblate particle geometry, respectively. These two shape parameters, in turn, correspond to two unique spectral autocorrelation amplitudes, Apδχs and Aoδχs. Thus, unambiguous shape determination is achieved via comparison of the observed Aδχs with these two possible values. The confidence with which the spheroid geometry of the particle can be assigned is quantified by a parameter P ∈ [0 1] given by the complement of the overlap between two normal distributions defined by (μp = Apδχs, σp = σApδχs) and (μo = Aoδχs, σo = σAoδχs), where σApδχs and σAoδχs are the errors associated with Apδχs and Aoδχs, respectively. Quantitatively, P is calculated via ÅÄ ÑÉÑ 1 ÅÅÅÅ jij μ1 − c zyz jij μ2 − c zyzÑÑÑ P = ÅÅerfcjj z − erfcjjj zzzÑÑÑ 2 ÅÅÅÇ jk 2 σ1 zz{ (11) k 2 σ2 {ÑÑÖ

i−1 * | l * o o o o o 2 ∑n = 1 [Ix (θj , ϕj , λn , ξ) + I y (θj , ϕj , λn , ξ)] o qi − 1 = − 2 erfc−1o m N }, o o o ∑n =λ 1 [Ix*(θj , ϕj , λn , ξ) + I y*(θj , ϕj , λn , ξ)] o o n ~ i | l o 2 ∑n = 1 [Ix*(θj , ϕj , λn , ξ) + I y*(θj , ϕj , λn , ξ)] o o o o o qi = − 2 erfc−1m } o o N λ o o ∑n = 1 [Ix*(θj , ϕj , λn , ξ) + I y*(θj , ϕj , λn , ξ)] o o n ~

given μ1 < μ2 and with c representing the value of Aδχs at which the distributions intersect. To investigate the effectiveness of this 2D mapping, we probed the behavior of P throughout various SP-DLS

where the asterisks denote normalization of eqs 3 and 4 by I0 and V. Assumptions of prolate or oblate geometry were made via the choice of ξp and eq 6 or ξo and eq 7, respectively. Next, D


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The Journal of Physical Chemistry C all photons in the jth step with their respective wavelengths were designated as X-polarized if sk < Ix*(θj, ϕj, λ(pk), ξ)/ [Ix*(θj, ϕj, λ(pk), ξ) + Iy*(θj, ϕj, λ(pk), ξ)] and Y-polarized otherwise, where sk is a uniformly distributed random number in the interval (0,1). The four required photon totals at every jth step were thus given by Np

Nx, j(θj , ϕj , {λi}, ξ) =

∑ [pk

∈ X‐polarized]

k Np

Ny, j(θj , ϕj , {λi}, ξ) =

∑ [pk

∈ X‐polarized]

k Np

Nr, j(θj , ϕj , {λi}, ξ) =

∑ [pk

∈ X‐polarized][λ(pk ) > λe]

k Np

Nb, j(θj , ϕj , {λi}, ξ) =

∑ [pk

∈ X‐polarized][λ(pk ) ≤ λe]

k

(12)

After repeating these stochastic calculations for every step in the random walk, the polarization and spectral contrast fluctuation functions at each jth step were simply given by δχj = (Nx, j − Ny, j)/(Nx, j + Ny, j) and δχs, j = (Nr, j − Nb, j)/ (Nr, j + Nb, j), respectively. Note that linear interpolation was used to avoid undefined values resulting from zero-photon steps. Finally, these functions’ autocorrelation amplitudes and their respective errors were calculated using a previously established procedure based on maximum-likelihood estimation (see ref 19). These latter values provide the information required to calculate the confidence parameter P for a given simulation. To investigate the application scope of the two-dimensional SP-DLS analysis, we performed simulations on three different elemental compositionssilver (Ag), gold (Au), and copper (Cu). We assumed that the shape analyses for each metal would be most successful for illumination bandwidths {λi} centered on the LSPR wavelength since the permittivitymatching condition that defines the latter, Re(ϵ1) = − 2ϵm, also maximizes the particle’s polarizability and should lead to steeper dependencies of the polarization and spectral contrast fluctuations on the particle orientation. Specifically, each illumination bandwidth was 300 nm-wide with Δλ = 1 nm and centered on the maximum value of |αs/V|2 = |3(ϵ1 − ϵm)/ (ϵ1 + 2ϵm)|2, the operative polarizability dependence in eqs 3 and 4 in the limit of a spherical particle (Figure 4). The resulting illumination bandwidths for Ag, Au, and Cu were thus 260−560, 390−690, and 380−680 nm, respectively.

Figure 4. Wavelength-dependence of volume-normalized polarizability αs/V for (a) Ag, (b) Au, and (c) Cu. Values given for the squared modulus (left axis; solid, blue line), real component (right axis; solid, red line), and imaginary component (right axis; dashed, red line).

(or ξo) for which the spheroid geometry could be assigned with high confidence (set as P > 0.95) decreased with increasing |αs/V|2, supporting the intuition that the success of the presented shape analysis depends primarily on how strongly the MNP interacts with the incoming electromagnetic field. An interesting result was observed upon comparison of the results for Ag (Figure 5a) with the results for Au and Cu (Figure 5, parts b and c, respectively). While the latter two exhibited a single, clear boundary between regions with high and low confidence shape determination (with the exceptions of indeterminate values; see Supporting Information), the former exhibited a rougher landscape with seemingly isolated regions of low confidence. This contrast was due to λedependent intersections between the curves given by Apδχs(Aδχ) and Aoδχs(Aδχ) (Figure S2), themselves resulting from the unique polarizability response of Ag. Significantly, these simulations elucidated two guiding principles for carrying out the described shape analyses. First, researchers can easily pick a near-optimal LPDC by choosing one with an edge wavelength centered on the LSPR wavelength of the metallic particle under investigation, a value readily available from theoretical models or simple spectroscopic measurements. Second, the curves Apδχs(Aδχ) and Aoδχs(Aδχ) as functions of ξ and spheroid geometry should be visualized beforehand via straightforward

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RESULTS AND DISCUSSION We first systematically examined the performance of the twodimensional SP-DLS analyses with regards to a fundamental experimental parameter, λe, the wavelength at which the dichroic beam splitter separates the blue and the red components of the scattering light (cf. LPDC in Figure 2). For each metal, we scanned λe in steps of 5 nm and calculated the resulting values of P for (ξp, ξo) pairs. As shown in Figure 5a−c, each metal displayed prominent minima in which contiguous values of λe maximized P and thus the confidence with which spheroid geometry could be assigned. Not surprisingly, the optimal λe were found near the metals’ respective LSPR wavelengths. Furthermore, the minimum ξp E


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Figure 5. Simulation results showing values of P or relative errors σξ/ξ (color gradient) for different combinations of experimental scenarios and pairs of equivalent shape parameters ξo (left axis) and ξp (right axis). (a−c) Optimization of λe for (a) Ag, (b) Au, and (c) Cu MNPs. Simulations were performed with SNR = 30. (d−f) Shape determination as a function of SNR for (d) Ag at λe = 400 nm, (e) Au at λe = 535 nm, and (f) Cu at λe = 510 nm. (g−i) Relative error in ξ as a function of SNR for (g) Ag at λe = 400 nm, (h) Au at λe = 535 nm, and (i) Cu at λe = 510 nm. In all panels: solid, red lines indicate boundaries between regions with high-confidence/high-precision shape determination (P > 0.95 or σξ/ξ < 0.05) and regions with poor-confidence/poor-precision shape determination (P ≤ 0.95 or σξ/ξ ≥ 0.05). Indeterminate results are gray (see Supporting Information). Thick, solid, black lines separate independent graphs.

ability to discriminate between Au and Cu spheroids diminished rapidly as SNR decreased, with SNR > 10 required to assign geometries of particles with ξp, ξo < 0.1 (Figure 5e and Figure 5f). Finally, we studied the precision with which ξ could be determined after assignment of prolate or oblate geometry; specifically, by investigating the behavior of the relative error σξ/ξ (see ref 19 for calculation of σξ) as a function of the SNR (Figure 5g−i). Unlike the spheroid assignment, the relationship between σξ/ξ and the SNR was insensitive toward the metallic identity of the MNP: For each element, high-precision (i.e., σξ/ξ < 0.05) estimates of the shape parameter were possible at low SNR for almost all probed prolate and oblate shape parameters. This suggests that experimental parameters

numerical calculations of eq 2 in order to identify intersections that may complicate further analyses. With the optimal λe in hand, we next explored how the success of the shape analyses correlated with the number of scattered photons. To that end, we performed simulations now holding the LPDC edge wavelength constant at its values with strongest discriminatory powerλe= 400, 535, and 510 nm for Ag, Au, and Cu, respectivelywhile scanning the SNR of the scattered signal from SNR = 3 to SNR = 100 (Figure 5d−f). In agreement with the previous scan through values of λe, Ag MNPs exhibited the greatest shape discrimination; even at the lowest interrogated SNR, SNR = 3, particles with only a 5−6% relative difference between rL and rS could still be assigned as either prolate or oblate (Figure 5d). On the other hand, the F



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for SP-DLS shape analyses should be primarily optimized for spheroid discrimination.

CONCLUSIONS In this work, we have presented two-dimensional singleparticle dynamic light scattering that allows an unambiguous assignment of individual metallic nanoparticles to either prolate or oblate spheroid geometries. The additional observablethe fluctuations in the nanoparticle’s scattering spectrumcan be readily incorporated into a polarization-only implementation of SP-DLS. We evaluated this 2D SP-DLS idea using numerical simulations and showcased its efficacy as a function of various experimental parameters, including its application scope covering different metal compositions. Furthermore, the outlined simulation/analysis pipeline ought to serve as a general procedure for further improving the SPDLS shape analysis of specific metals and shape parameters. For example, we expect that both {λi} and {wi} could be optimized more practically by considering realistic illumination bandwidths that are nonuniform and not necessarily centered around the LSPR wavelength of the metal in question. In general, the SP-DLS concept should in principle be also applicable to other nonmetallic nanoparticles. Although the initial SP-DLS implementation utilized real-time 3D-SPTS as the means to keep a freely diffusing particle at the focal volume of the microscope objective, the concept is compatible with any technique that can achieve a similar confinement. As such, we believe that the theoretical framework presented herein along with future developments would facilitate SP-DLS becoming a useful and practical tool compatible with a broad range of nanoparticle and nanostructure compositions, poised for studying their time-dependent phenomena on the particleby-particle basis.30 ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcc.9b05217. Additional function definitions, details on indeterminate values, and supplementary figures (PDF)

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REFERENCES

(1) Kreibig, U.; Zacharias, P. Surface Plasma Resonances in Small Spherical Silver and Gold Particles. Z. Phys. A: Hadrons Nucl. 1970, 231, 128−143. (2) Bohren, C. F.; Huffman, D. R. Absorption and Scattering of Light by Small Particles; Wiley-VCH: Mörlenbach, 2004. (3) Xia, Y.; Xiong, Y.; Lim, B.; Skrabalak, S. E. Shape-Controlled Synthesis of Metal Nanocrystals: Simple Chemistry Meets Complex Physics? Angew. Chem., Int. Ed. 2009, 48, 60−103. (4) Berne, B. J.; Pecora, R. Dynamic Light Scattering: With Applications to Chemistry, Biology, and Physics; Dover Publications: Mineola, 2000. (5) Magde, D.; Elson, E.; Webb, W. W. Thermodynamic Fluctuations in a Reacting SystemMeasurement by Fluorescence Correlation Spectroscopy. Phys. Rev. Lett. 1972, 29, 705−708. (6) Elson, E. L.; Magde, D. Fluorescence Correlation Spectroscopy. I. Conceptual Basis and Theory. Biopolymers 1974, 13, 1−27. (7) Magde, D.; Elson, E. L.; Webb, W. W. Fluorescence Correlation Spectroscopy. II. An Experimental Realization. Biopolymers 1974, 13, 29−61. (8) Cang, H.; Wong, C. M.; Xu, C. S.; Rizvi, A. H.; Yang, H. Confocal Three Dimensional Tracking of a Single Nanoparticle with Concurrent Spectroscopic Readouts. Appl. Phys. Lett. 2006, 88, 223901. (9) Pecora, R. Doppler Shifts in Light Scattering from Pure Liquids and Polymer Solutions. J. Chem. Phys. 1964, 40, 1604−1614. (10) Dominguez-Medina, S.; Chen, S.; Blankenburg, J.; Swanglap, P.; Landes, C. F.; Link, S. Measuring the Hydrodynamic Size of Nanoparticles Using Fluctuation Correlation Spectroscopy. Annu. Rev. Phys. Chem. 2016, 67, 489−514. (11) Xu, C. S.; Cang, H.; Montiel, D.; Yang, H. Rapid and Quantitative Sizing of Nanoparticles Using Three-Dimensional Single-Particle Tracking. J. Phys. Chem. C 2007, 111, 32−35. (12) Myroshnychenko, V.; Rodríguez-Fernández, J.; PastorizaSantos, I.; Funston, A. M.; Novo, C.; Mulvaney, P.; Liz-Marzán, L. M.; García de Abajo, F. J. Modelling the Optical Response of Gold Nanoparticles. Chem. Soc. Rev. 2008, 37, 1792−1805. (13) Olson, J.; Dominguez-Medina, S.; Hoggard, A.; Wang, L.-Y.; Chang, W.-S.; Link, S. Optical Characterization of Single Plasmonic Nanoparticles. Chem. Soc. Rev. 2015, 44, 40−57. (14) Purcell, E. M.; Pennypacker, C. R. Scattering and Absorption of Light by Nonspherical Dielectric Grains. Astrophys. J. 1973, 186, 705−714. (15) Yee, K. Numerical Solution of Initial Boundary Value Problems Involving Maxwell’s Equations in Isotropic Media. IEEE Trans. Antennas Propag. 1966, 14, 302−307. (16) Noguez, C. Surface Plasmons on Metal Nanoparticles: The Influence of Shape and Physical Environment. J. Phys. Chem. C 2007, 111, 3806−3819. (17) Williamson, M. J.; Tromp, R. M.; Vereecken, P. M.; Hull, R.; Ross, F. M. Dynamic Microscopy of Nanoscale Cluster Growth at the Solid−liquid Interface. Nat. Mater. 2003, 2, 532−536. (18) Liao, H.-G.; Zheng, H. Liquid Cell Transmission Electron Microscopy. Annu. Rev. Phys. Chem. 2016, 67, 719−747. (19) Guerra, L. F.; Muir, T. W.; Yang, H. Single-particle Dynamic Light Scattering: Shapes of Individual Nanoparticles. Nano Lett. 2019, DOI: 10.1021/acs.nanolett.9b02066. (20) Welsher, K.; Yang, H. Imaging the Behavior of Molecules in Biological Systems: Breaking the 3D Speed Barrier with 3D Multiresolution Microscopy. Faraday Discuss. 2015, 184, 359−379. (21) Cang, H.; Montiel, D.; Xu, C. S.; Yang, H. Observation of Spectral Anisotropy of Gold Nanoparticles. J. Chem. Phys. 2008, 129, No. 044503. (22) Cummins, H. Z.; Knable, N.; Yeh, Y. Observation of Diffusion Broadening of Rayleigh Scattered Light. Phys. Rev. Lett. 1964, 12, 150−153. (23) Yang, H. Single-particle Light Scattering: Imaging and Dynamical Fluctuations in the Polarization and Spectral Response. J. Phys. Chem. A 2007, 111, 4987−4997.

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AUTHOR INFORMATION

Corresponding Author

*(H.Y.) E-mail: [email protected]. Telephone: +1 (609) 258-3578. ORCID

Luis F. Guerra: 0000-0002-8960-1720 Tom W. Muir: 0000-0001-9635-0344 Haw Yang: 0000-0003-0268-6352 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This research was supported by a grant from the Gordon and Betty Moore Foundation (No. 4741) and by the National Science Foundation Graduate Research Fellowship Program under Grant No. DGE-1656466 (to L.F.G.). G


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The Journal of Physical Chemistry C (24) Gans, R. Ü ber die Form Ultramikroskopischer Goldteilchen. Ann. Phys. (Berlin, Ger.) 1912, 342, 881−900. (25) Hunter, G. L.; Edmond, K. V.; Elsesser, M. T.; Weeks, E. R. Tracking Rotational Diffusion of Colloidal Clusters. Opt. Express 2011, 19, 17189−17202. (26) Lu, C. Y.; Vanden Bout, D. A. Effect of Finite Trajectory Length on the Correlation Function Analysis of Single Molecule Data. J. Chem. Phys. 2006, 125, 124701. (27) Rakić, A. D.; Djurišić, A. B.; Elazar, J. M.; Majewski, M. L. Optical Properties of Metallic Films for Vertical-cavity Optoelectronic Devices. Appl. Opt. 1998, 37, 5271−5283. (28) Jones, F.; Harris, G. ITS-90 Density of Water Formulation for Volumetric Standards Calibration. J. Res. Natl. Inst. Stand. Technol. 1992, 97, 335−340. (29) The International Association for the Properties of Water and Steam Release on the Refractive Index of Ordinary Water Substance as a Function of Wavelength, Temperature and Pressure; 1997. (30) Hartland, G. V. Optical Studies of Dynamics in Noble Metal Nanostructures. Chem. Rev. 2011, 111, 3858−3887.

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