Accurate Modeling of the Polarizability of Dyes for Electromagnetic Calculations

Abstract

The wavelength-dependent complex linear polarizability of a dye is a crucial input for the modeling of the optical properties of dye-containing systems. We here propose and discuss methods to obtain an accurate polarizability model by combining absorption spectrum measurements, Kramers-Kronig (KK) tranformations, and density functional theory (DFT) calculations. We focus, in particular, on the real part of the polarizability and its link with static polarizability. In addition, we introduce simple KK-consistent analytic functions based on the theory of critical points as a much more accurate approach to model dye polarizabilities compared with existing models based on Lorentz oscillators. Accurate polarizability models based on critical points and DFT calculations of the static polarizability are derived for five commonly used dyes: Rhodamine 6G, Rhodamine 700, Crystal Violet, Nile Blue A, and Methylene Blue. Finally, we demonstrate explicitly, using examples of Mie Theory calculations of nanoparticle-dye interactions, how an inaccurate polarizability model can result in fundamentally different predictions, further emphasizing the importance of accurate models, such as the one proposed here.

Figure 1

(a) Imaginary part of the polarizability of Rhodamine 6G, as obtained using eq 1 from its UV–vis absorption spectrum including the near-UV peaks down to 200 nm (red) or only the peaks above 400 nm (blue). The molecular structure of Rhodamine 6G is also shown as an inset. (b) Real part of the polarizability of Rhodamine 6G calculated from the KK transformation of (a) with (red) or without (blue) the near-UV peaks (<400 nm) included. The difference between the two is shown in the inset and shows the effective contribution to the real polarizability from the UV peaks is approximately constant in the visible part of the spectrum (above ∼450 nm).

Figure 2

(a) Fits of the 526 nm peak of the measured absorption cross section of 10 μM Rhodamine 6G in water (pink) using eq 1 and a polarizability model with a single Lorentz oscillator (green) or a single Voigt fit (blue). (b) Corresponding real part of the polarizability functions where the condition α(λ → ∞) = α_static = 6. 94 × 10^–39 S.I. is enforced for both Lorentzian and Voigt fits.

Figure 3

(a) Fit of the measured visible Rhodamine 6G absorption cross section (purple) with a polarizability modeled as two Lorentz (green) or two Voigt (blue) oscillators. (b) Corresponding real parts of the polarizability, compared with that obtained from the KK transformation of the experimental data. A constant is added in each case to enforce the condition α(λ → ∞) = α_static = 6. 94 × 10^–39 S.I. (c, d) Same as (a, b) with a double critical point fit (red). The inset in (c) shows the individual contributions of each of the two critical points. The black dotted line in (d) delineates the region for which the fit is better than 10% for both the real and imaginary parts.

Figure 4

(a) Predicted extinction spectrum of the 60 nm diameter silver nanoparticles coated with a uniform dye layer of 0.2 dye molecules per nm² (∼1000 dyes per NP) with polarizabilities given by either the same Lorentz (green) or Voigt (blue) models as in Figure 2. Same as (b) with the resonance shifted to 460 nm to increase coupling. (c) Same as (a) for a low-coverage dye layer of 0.01 molecules per nm² on the colloid surface. The differential extinction spectrum (where the colloid-only spectrum has been subtracted) is shown to highlight the influence of the dye layer. The dashed line shows the result for the Lorentz oscillator where a real constant has been added to make it coincide with the Voigt model in the 450 nm region, as shown as a dashed line in Figure 2b (note that this also changes α(λ → ∞) and could not represent the same physical dye). (d) Same as (c) but showing differential absorption instead of extinction.