Noble metal nanoparticles—especially shape anisotropic particles—have pronounced resonances in the optical spectrum. These sensitive absorption modes attract great interest in various fields of application. For nonspherical particles, no analytic description of the absorption spectra according to the commonly used Mie theory is possible. In this work, we present a semi-empirical approach for the explanation of the optical spectra of shape anisotropic particles such as silver nanoprisms and gold nanorods. We found an interpretation of the optical absorption spectra which is based on a single-photon-single-electron transition. This model is in a better agreement with the basic assumptions of quantum mechanics than the electrodynamic model of a localized surface plasmon excitation. Based on microfluidically obtained Ag nanoprisms and Au nanorods with very high ensemble homogeneities, dependencies between the geometrical properties of the shape anisotropic noble metal nanoparticles and the spectral position of the longitudinal absorption mode could be derived, which show that the assumption of a composed relative permittivity and the inclusion of the Rydberg constant is sufficient to describe the optical properties of the shape anisotropic particles. Within the scope of the measuring accuracy, the calculations furthermore lead to the value of the refractive index of the particle-surrounding medium.
The optical spectra of metal nanoparticles are mostly interpreted by the assumption of the excitation of plasmons [
The plasmon model is based on the idea of a field-induced collective oscillation of electrons. Analogue elongation and relaxation of all electrons always take place if photons pass matter, and this simple response on the electromagnetic field is the general cause for the effect of light refraction by elastic interaction between photons and matter. Nonelastic interactions are marked by a nonreversible energy transfer between the electromagnetic field and the energy-absorbing material. The electromagnetic resonances of atoms and molecules are well understood by quantum mechanics using the model of atom or molecule orbitals. They can be explained by one-photon-one-electron processes, typically [
Nonspherical particles have to be regarded as dimension-reduced systems, if their geometries are marked by higher aspect ratios. This means that these particles could be regarded in analogy to small compact solids in one or two directions, but they have to be regarded in analogy to clusters or molecules in the other directions. Nanorods are “one-dimensional” objects, which means they are size-reduced in two dimensions. Flat nanodisks or prisms are two-dimensional objects with a size reduction in one dimension. Therefore, their resonance behaviour can hardly be understood by models presuming a spherical symmetry. In particular, it has to be taken in mind that the reduction in dimension can cause special functions of state densities for the possible energy levels as it is known from molecular objects. Thus, the transfer of energy from the electromagnetic field into electronic states of the shape-anisotropic metal nanoparticles should be related to the electronic state distribution and the transfer of a single electron from a lower to higher energetic level.
Lower yields in the many syntheses of special nanoparticle types and larger distributions in shape and size of these particles prevented a detailed interpretation of the measured electromagnetic resonances of nonspherical particles by an excitation model. A better precondition for understanding of the optical properties came from single-particle investigations [
Next to metallic particles that show pronounced absorption in the visible range and the near UV, also charged and uncharged dielectric particles such as water, ice, and cosmic dust have already been investigated regarding the interaction between electrical charges and electromagnetic radiation. Here, it was found by Kocifaj et al. that the long-wavelength resonances in the optical spectra are an effect of excess surface charges. The physical behaviour of the net charges is related to a surface current density, which shows a linear dependence to a phenomenological surface conductivity [
For especially noble metal nanoparticles, new synthesis methods, in particular microfluidic syntheses, allow the generation of dispersed metal nanoparticles in colloidal solution with high yield and very small distribution in size and geometry [
For the microfluidic synthesis of Au nanorods as well as for Ag nanoprisms, modular reactors were designed. The reactor channels are PTFE tubes with an inner diameter of 0.5 mm. The tubing is connected to glass syringes (ILS, Ilmenau, Germany) in syringe pumps (Cetoni neMESYS, Cetoni GmbH) with standard fluid connectors, which are commercially available as HPLC equipment (IDEX Health & Science, LCC). A schematic illustration of an exemplary reactor for the synthesis of Au nanorods is shown in Figure
Exemplary sketch of a modular microflow-through reactor which is operated in a segmented flow mode. The carrier stream of an organic, immiscible liquid phase (e.g., longer alkanes or perfluorinated alkanes) is presented first in a continuous stream. Into the carrier stream, aqueous reactant solutions are dosed under simultaneous formation of aqueous droplets. These aqueous droplets can be considered as closed-up microbatch reactors with efficient droplet internal mixing conditions.
Au nanorod samples were prepared by two different synthesis pathways. Using the synthesis developed by Ye et al. [
Both synthesis strategies base on seeded growth approaches via chloroauric acid reduction in aqueous solution. Sodium borohydride reduces tetrachloroauric acid in the seed solution to elementary gold and induces nucleation whereas tetrachloroauric acid in the growth solution is reduced with ascorbic acid only to an oxidation state of +I [
The colloidal solutions of the obtained nobel metal nanoparticles were analyzed using UV-vis spectrophotometry (SPECORD 200 Analytik Jena/Cary 5000 UV-Vis-NIR spectrophotometer), SEM (Hitachi S-4800, FE-SEM), TEM (Philips CM 300), and DCS measurement for the knowledge of the differential size distribution spectra of the Stokes equivalent sedimentation diameter (DCS, DC 20000, CPS Instruments, Inc.).
All chemicals used for the synthesis of Au nanorods or Ag nanoprisms were used as received from the following suppliers (purity of chemicals in brackets): sodium citrate (Merck KGaA, 99%), poly(sodium styrenesulphonate) (PSS) (Acros Organics,
The following approach is based on the idea that the electronic excitation of so-called plasmonic particles has to be understood in analogy to the electronic excitation of molecules. From the classical view of Coulomb’s law, the energy of an electrostatic field is determined from the ratio of the square of separated charges
In contrast to the classical approach, quantum mechanics is strongly particle-related. The resonant interaction of atoms and molecules with the electromagnetic field is always a process of interaction between a small number of atoms and molecules and a small number of resonant photons, typically an excitation (absorption) or emission (fluorescence) event in form of a one-photon-one-electron process.
It is assumed that the so-called plasmonic particle excitation is also a resonant one-photon-one-electron process. The resonant transition is controlled by energy eigenstates of the particle which is strongly related to the particle geometry. The involved states can be regarded in analogy to the extended pi orbitals of a dye molecule containing a chain of conjugated double bonds.
For the approximation of the resonance energy, it is assumed that the energy of absorbed photon with the wavelength
The electrical field constant
This results into the simple expression for the resonance wavelength:
It is further assumed that
The resonance wavelength of triangles was approximated by using (
This equation corresponds to the empirically found linear equation for the dependence of the resonance wavelength of triangles from their lateral size [
The resonance wavelength for nanorods can be approximated in complete analogy to the flat nanotriangles (
The investigations of Aherne and coworkers [
UV-vis spectra of five silver nanoprism samples with different particle sizes each. The spectral position of the main absorption mode is shifted bathochromically with a larger edge length, which were obtained under conditions of decreasing seed nanoparticle density [
With the flow-through synthesis of gold nanorods, a precise tuning of the physical properties by shifting the crucial reactant ratios succeeded as well. SEM images of microflow-produced Ag nanoprisms of two different sizes are exemplarily shown in Figures
(a and b) SEM images of (a) Ag nanoprisms with about 35 nm edge length and (b) Ag prisms with about 120 nm edge length.
In experiments with different seed nanoparticle densities, Au of aspect ratios between 2.4 and 3.8 has been obtained. It was found that the seed concentration is mainly influencing the final diameters of the nanorods, whereas the effect of a varying seed particle density on the nanorod’s length is less distinct. A low seed particle density leads towards slightly longer but significantly thicker rods, while from a high amount of seed nanoparticles much thinner but little shorter rods were obtained. The shift in the aspect ratio and particle size is directly connected with the change in the optical absorption spectra. Here, a red shift of the spectral position of the long-wavelength absorption results, if the seed particle concentration is enhanced. At lower seed concentrations, the absorption peak is shifted hypsochromically. Additionally, an increase in the extinction of the shorter-wavelength resonance mode is observed (Figure
(a) Optical absorption spectra of Au nanorods. Here, an increasing seed volume leads towards a red shift of the main absorption mode. With increasing seed particle density, the aspect ratio of the rods increases, not the rod length. Rods prepared under conditions of high seed particle densities are slightly shorter but markedly thinner. (b) UV-Vis spectra of Au nanorods with aspect ratios between 2.07 and 7.79.
Au nanorods with higher aspect ratios produced in wet chemical approaches show diameters between 5 nm and 30 nm with aspect ratios between 2 and 8. The spectral positions of the longitudinal absorption modes vary in the range of 700 nm to 1200 nm.
Au nanorods with tailor-made aspect ratios and adjustable longitudinal absorption modes have been synthesized by varying the amount of seed solution, the amount of ligands in the reaction solution, and the pH value, if all other conditions remain constant. By increasing the amount of seed solution, Au nanorods became thinner on average, which results in an increased aspect ratio due to less available gold per Au nanorod and a faster growth process in the longitudinal direction. The same result can be observed by increasing the amount of ligands due to earlier saturation of the Au nanorod surfaces as well as reducing the pH value by increasing the amount of hydrochloric acid. This effect can be attributed to different micelle stabilities of the CTAB micelles at different pH values [
(a and b) Electron microscope images of (a) (SEM) Au nanorods with small aspect ratio (SEM) and (b) larger aspect ratio (TEM).
The application of (
The parameter
Correlation between the spectral position of the main resonance wavelengths of silver nanoprism solutions and the SEM-determined average edge length of the individual prism samples.
The approximation of the experimental data supplies
The experimental data for the gold nanorods suggest that the parameter
It was found that the remaining factor is nearly identical with the ratio of the inverse Rydberg constant
The experimental data show a good agreement between the expected refractive index (corresponding to (
The approximation of Figure
Dependence between the axial absorption maximum of gold nanorods and their SEM-determined aspect ratio.
The parameter
This value can be approximated by
It seems that this reference length is significantly dependent neither on the metal nor on the geometry of nanoparticles, but mainly defined by the fine structure constant
The approximation of the axial resonance wavelength of gold nanorods in dependence on the aspect ratio
New microfluidic and batch syntheses generate gold nanorods and flat triangular silver nanoprisms with very high yields and high homogeneities, whereby the particles have been obtained dispersed in colloidal aqueous solution. The narrow distribution of size and shape and the possibility of tuning the extension of particles allowed obtaining well reproducible data of the long-wavelength electromagnetic resonances.
It was found that the dependence of the resonance wavelength on the particle size can be interpreted in case of both particle types by a one-photon-one-electron excitation process, which is in better agreement with the basic assumptions of quantum mechanics than the classical model of a particle plasmon excitation. The linear approximation led to an interpretation using an
The UV-Vis, SEM, and TEM data used to support the findings of this study are available from the corresponding author upon request.
There are no conflicts of interest to declare.
The authors gratefully acknowledge the financial support from the DFG (KO 1403/39-1 and HE 3494/3, Emmy Noether Program). The work of Phillip Witthöft is funded by a scholarship from the PIER Helmholtz Graduate School.