Article pubs.acs.org/JPCB
Polymer Optical Constants from Long-Range Corrected DFT Calculations Shintaro Maekawa and Krzysztof Moorthi* R&D Center, Mitsui Chemicals, Inc., 580-32 Nagaura, Sodegaura, 299-0265, Japan
ABSTRACT: A methodology to calculate refractive indices of plastics based on the Lorentz−Lorenz equation has been proposed. The polarizability of the nonconjugated polymer repeat units is predicted using the long-range corrected functionals. The end effect corrections in repeat unit models are essential to achieve accuracy similar to that observed for molecular liquids (ca. 1% in mean absolute deviation). The functionals with 100% Hartree−Fock (HF) exchange in the long-range limit perform best for aromatic and other hydrogen-deficient compounds; the Coulomb-attenuated hybrid exchange-correlation functional (CAM-B3LYP) performs very well for hydrogen-rich (usually, fully saturated) compounds. Exceptionally good agreement is observed for the sets of wavelength-dependent refractive index data for polystyrene, poly(methyl methacrylate), and for poly(perfluoro-4-vinyloxy-1-butene) (CYTOP), for which the root-mean-square deviations are 0.004, 0.002, and 0.004, respectively.
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fields internal to other molecules as molecules cannot penetrate each other. The local field is related to the Maxwell field by7
INTRODUCTION Plastics have become key components in a variety of optical devices, because they are lightweight, strong, easy to process, have high degree of transparency, and their refractive indices rival these of glass.1,2 The search for new optical plastics, often focused at maximizing refractive index and minimizing dispersion (wavelength dependence) of refractive index, is a highly active research area.3−6 The ability to predict polymer refractive index prior to synthesis, at an appropriate level of accuracy, is important for rational design of new optical materials. The refractive index is the ratio of the speed of light in vacuum to that in a material, and characterizes macroscopic response of material to the electromagnetic radiation. When a material is placed in an external electric field, E0, either static or optical, the mean electric field in material, E (Maxwell field), induces polarization, P, which is the mean dipole moment per unit volume, V, due to the N molecules. Both fields are related by P = χE =
ε−1 E 4π
EL = E +
(2)
The mean polarization of the molecule is αEL, where α is the polarizability tensor of a molecule. Therefore, the polarization P, is P=
N α EL V
(3)
Eliminating E and EL from eqs 1−3, and invoking Maxwell relationship for an isotropic material, ε = n2, where n is the average refractive index, leads to the Lorentz−Lorenz equation n2 − 1 4π = ρ⟨α⟩ 2 3 n +2
(4)
where ρ = N/V is the number density, and ⟨α⟩ is the trace of the polarizability tensor. Equation 4 permits one to predict the refractive index, a macroscopic property, from the knowledge of average polarizability of an isolated molecule. The Lorentz
(1)
where χ and ε are the susceptibility, and dielectric constant tensors, respectively. The Maxwell field differs from the external field E0, because the polarization is itself a field source. The mean field sensed by a particular molecule, EL, (Lorentz local field) is different from the Maxwell field, E, because EL does not contain © XXXX American Chemical Society
4π P 3
Received: October 19, 2015 Revised: January 7, 2016
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DOI: 10.1021/acs.jpcb.5b10203 J. Phys. Chem. B XXXX, XXX, XXX−XXX
Article
The Journal of Physical Chemistry B
containing termini, E, αE2, is subtracted from polarizability of an E-terminated repeat unit model, αM α1 = αM − αE2 (5)
field in eq 2 introduces some level of particle correlation, but the approach is not systematic; therefore, insights from microscopic theories of refractive index are important.7 Noteworthy, a microscopic theory of a polarizable hardsphere fluid derives Lorentz−Lorenz equation without assuming local or reaction fields and rigorously including particle correlations.8 Certain scattering events at high density, and correlations with a range comparable to the wavelength, are identified as possible sources of departures from eq 4.8 In contrast to eq 4 predictions, the experimental studies indicate that the Lorentz−Lorenz function, FLL = ρ−1(n2 − 1)/(n2 + 2), depends weakly on density. However, the typical changes in FLL over the range of reduced density 0.1 < ρr < 1.7 are small:
