Quantifying Shape Transition in Anisotropic Plasmonic Nanoparticles through Geometric Inversion. Application to Gold Bipyramids

Unraveling the nuanced interplay between the morphology and the optical properties of plasmonic nanoparticles is crucial for targeted applications. Managing the relationship becomes significantly complex when dealing with anisotropic nanoparticles that defy a simple description using parameters like length, width, or aspect ratio. This complexity requires computationally intensive numerical modeling and advanced imaging techniques. To address these challenges, we propose a detailed structural parameter determination of gold nanoparticles using their two-dimensional projections (e.g., micrographs). Employing gold bipyramids (AuBPs) as a model morphology, we can determine their three-dimensional geometry and extract optical features computationally for comparison with the experimental data. To validate our inversion model’s effectiveness, we apply it to derive the structural parameters of AuBPs undergoing shape modification through oxidative etching. In summary, our findings allow for the precise characterization of structural parameters for plasmonic nanoparticles during shape transitions, potentially enhancing the comprehension of nanocrystal growth and optimizing plasmonic material design for various applications.


Synthesis of Gold Bipyramids
Gold bipyramids (AuBPs) were synthesized according to a previously reported procedure using seed-mediated growth. 1 The gold seeds were prepared by fast reduction of gold precursor with NaBH 4 in the presence of both a cationic surfactant (CTAC) and citric acid in a 20 mL scintillation vial.To an aqueous mixture containing CTAC (10 mL, 50 mM), HAuCl 4 (0.05 mL, 50 mM), and citric acid (0.05 mL, 1 M), freshly prepared NaBH 4 (0.25 mL, 25 mM) was added under vigorous stirring at 20 o C. The solution turned from yellow to brownish immediately.Two minutes later, the seed solution was heated at 80 o C for 90 min in a silicone oil bath, under gentle stirring.Its color turned gradually from brown to red, indicating an increase in nanoparticle size.Finally, the solution was removed from the oil bath, stored at room temperature, and used without further treatment.Gold bipyramids with different dimensions were prepared by adding a certain volume of gold seeds (0.9, 1.1, 2.0, and 3.7 mL, for the preparation of the samples 1, 2, 3 and 4, respectively) under vigorous stirring to a growth solution comprising CTAB (100 mL, 100 mM), HAuCl 4 (5 mL, 10 mM), AgNO 3 (1 mL, 10 mM), HCl (2 mL, 1M) and AA (0.8 mL, 100 mM), at 30 o C. Five minutes later, the mixture was left undisturbed at 30 o C for 2 h.AuBPs were washed by centrifugation (2 cy-cles at 7000 rpm, 30 min), to remove excess reagents.After the second centrifugation cycle, nanoparticles were redispersed in CTAB (100 mM) to a final gold concentration of 0.5 mM.

Instrumentation
Optical extinction spectra were recorded with an Agilent Cary 3500 UV-visible spectrophotometer.TEM images were acquired with a FEI Tecnai G2 20 TWIN transmission electron microscope (TEM) operating at an acceleration voltage of 200 kV.The dimensions of AuBPs

Volume Calculation in the Geometric Inversion Model
For deriving the biconical volume (V ), we interpret the nanoparticle as a rotational solid.
Setting the origin to the center of mass and introducing a reference system with abscissa χ, directed along L * , and with ordinate axis along W , it turns out: where f c = f c (χ) and f r = f r (χ) are the functions describing respectively the truncated cone and the curved sector in the first quadrant (χ ≥ 0, f c,r ≥ 0).It is convenient to operate in the same non-dimensional units parametrizing the model projection, thus we define X = χ/W to obtain: where x, y, are defined as in the main text.A simple inspection of the integration limits ). Replacing the former two equations in the volume expression returns in the first instance: To get the final result, as reported in the main text, it does suffice recalling the constraint − y = −x 1 + y 2 and one is left with a sum of three terms: This function can recover upon the equivalent limit conditions a pure biconical volume, V c , with height h = L * /2 and base radius R = W/2: and a pure spherical volume, V r : where it is implicitly meant y/x 3 − y 4 / (1 + y 2 ) 3 → 0 + .
The biconical limit corresponds to the ordinate axis in Figure 2 of the main text, while the spherical one is recovered by the point at infinity of the frontier demarcating the validity region of the projective model (x ≤ x M (y) = cos arctan 1/y, blue line).Observe that the expressions for V c and V r would equally stem from Eq. (S4) upon the same limits, with the additional conditions → y (V c ) or → 1 (V r ).Lastly, the negative contribution to v = v(x, y) does not affect the model validity, as it produces the further constraint:

Three-Descriptor Analysis of Bipyramid Shapes
Our geometric inversion model, as extensively detailed in the main text, is based on three key approximate assumptions.We have demonstrated that these simplifications significantly streamline the model without meaningful repercussions on the predictability of computational simulations (see Figure S4).They are: 1.) the model is projective and two-dimensional; 2.) pentagonal symmetry is neglected, a circular base is used instead; 3.) lateral curvature radii are disregarded by juxtaposing two biconical structures.For the sake of generality, in this supplementary information, we introduce a less restrictive analysis in 3D, which may prove useful for future developments.To prevent confusion with the 2D model, the symbolic notation that follows is (and should be) kept distinct from that employed in the main text.
Consider thus the scheme in Figure S2, illustrating a pentagonal base of side length L, afterwards rounded at its vertices with equal junctions of radius ρ.As in the inversion 2D model, the radii delimiting circle arcs are chosen to fall perpendicularly to the pentagon side.Let W be the maximum width, i.e. the distance between two non-nearest neighbour vertices, then: R and A being respectively the distance of any vertex from the pentagon center, and the apothem.These distances at the base and along the biconical axis reduce upon rounding as: with: perimeter and area changing accordingly as: with unrounded values: Figure S2 reports instead a scheme for the longitudinal bipyramid projection, either rounded on the tipped or lateral side with radii r and R, respectively.
It is inscribed into a rhombus, with smaller diagonal W , the larger diagonal L denoting the maximum inter-tip distance.The rhombus aspect ratio of the perfect bipyramid: is fixing the tip and side angles, α and β, as: which the two shape descriptors may be simply expressed as: They can be also rearranged in terms of circularity, C = 4πA/P 2 , experimental width (W ) and length (L), i.e.: or, in terms of circularity (C): The transverse descriptor instead obeys more simply: as it has to be, since the equation S ρ = 1: represents the circumference inscribed to the pentagon base.The other two specular bounds, S r = 1 and S R = 1, return:

C. Simulation of Optical Properties
Numerical simulations of the electromagnetic response of 3D bipyramid models, rebuilt by the geometric inversion method in 2D, are performed with the boundary element method 4 as implemented in the MNPBEM toolbox. 5Particle surfaces are discretized in a number of surface elements (typically around 10 3 -10 4 ) that ensures convergence of the computed electromagnetic quantities.Current and charge distributions over the surface are determined by solving the integral equations that result from imposing proper boundary conditions to Maxwell's equations.The electromagnetic field and the optical quantities of interest then can be computed accordingly.
In order to simulate a random particle orientation with respect to the incoming beam during measurements, calculations are performed by averaging over different polarizations of the incident light excitation.Optical constants of Au are taken from the established literature on this subject. 6The surrounding medium is assumed to be water (n m = 1.333) to mimic the as-synthesized Au bipyramids in a colloidal aqueous solution.
spectra and TEM micrographs of the as-synthesized AuBPs are shown in FigureS1.

Figure
Figure S1: a) UV-Vis-NIR absorbance spectra of the synthesized bipyramids prior to oxidative etching process.TEM micrographs of initial samples.Scale bars = 100 nm.

Figure S2 :
Figure S2: Longitudinal projection scheme of a bipyramid both rounded at the tips (radii of curvature r) and lateral sides (R).The real nanoparticle profile is depicted by ABCDD C B A , with given experimental values of width (W ) and length (L).W and L here specify the maximum dimensions (tip-to-tip, side-to-side) belonging to the reference rhombus.

Figure S3 :
Figure S3: Parametric representation of Eq. (S32).Ratio between nanoparticle and cusp-tocusp aspect ratios n /y (a) as a function of x (parameterized by y) and (b) y (parameterized by x).The two quantities only coincide in the limit of x → 0 + (zero curvature radius).

Figure S4 :
FigureS4: Comparative plot of the normalized experimental absorbance spectrum of Sample 1 (black) and numerical extinction spectra of pentagonal bipyramid particles (pBP) with variation of the radius of the pentagonal-base vertices (R v ) from 1 nm to 15 nm (red, blue, magenta, and green), and the biconical particle (yellow), for fixed values of the net length (L n ), width (W), and projected area (A) of the AuBP.Insets show the TEM image of a Sample-1 AuPB and the 3D particle models.

Figure S5 :Figure S6 :Figure S7 :
Figure S5: Oxidative etching of AuBPs, samples of Series 2. (a) TEM micrographs (scale bars = 100 nm) and the corresponding 3D models.Increasing degrees of oxidation are indicated by roman numerals from i to v, with i representing no etching.(b) Length, (c) width, (d) projected area, and (e) tip radius determined by analysing TEM micrographs.The blue circles in (e) represent the calculated tip radius obtained by implementing the geometric inversion method with the knowledge of length, width, and area data.(f) Experimental absorbance spectra (solid lines) along the computed extinction cross sections of the rebuilt 3D objects.

Figure
Figure S8: (a) Normalized experimental absorbance spectra and (b) numerical extinction spectra of oxidative-etched Series.Circle, triangle, square and star markers represent respectively the samples for Series 1, 2, 3, and 4. Spectra in black denote samples before oxidative etching, while spectra transitioning from dark to light copper colors depict increasing levels of oxidative etching.

Table
1 of the main text.Absorbance

Table S1 :
Volume of Au +3 -CTAB complex used for oxidative etching of different AuBP samples, while keeping constant metallic Au concentration in all the experiments (10 mL, [Au 0 ] = 0.5 mM).