The isotropic molecular polarizabilities of single methyl-branched alkanes in the terahertz range

Additive models for molecular parameters have a long history, and remain important for the rapid prediction of molecular properties. To test the validity of an additive approach, we study the polarizabilities of methyl branched alkanes in the terahertz spectral range, and compare these to the polarizabilities of their straight chain counterparts. A single branched methyl group increases the isotropic polarizability by a fixed amount, regardless of the carbon backbone chain length or the methyl branch’s position. These results, which are also compared to DFT calculations, establish the validity of an empirical additive approach for these prototype non-polar and non-hydrogen-bonding liquids. 2013 Elsevier B.V. All rights reserved. The additive properties of molecular subunits, such as atoms, bonds, and groups, have been recognized as a useful tool to predict the properties of molecules for more than a century [1]. The additivity of bond and group polarizabilities, in particular, was first demonstrated in the 1940s. Even today, more complex variants of the early empirical and theoretical additive models still prove to be relevant in computational physical chemistry [2–8], largely due to a decreased computational cost compared to pure ab initio methods. Typically, experimental verification of a theoretical model of molecular polarizability relies on the measurement of a material’s index of refraction in the visible spectral range. Terahertz time-domain spectroscopy (THz-TDS) presents an interesting and sometimes advantageous alternative to these methods. With THz-TDS both the amplitude and phase of a broadband THz pulse are measured and, from comparison with a reference pulse, a material’s indices of refraction and absorption coefficients can be easily extracted [9,10]. In particular, the n-alkanes, a homologous series of saturated hydrocarbons with the general formula CnH2n+2, are a prototypical set of a molecules for which the THz frequency range is an ideal window on the additive behavior of their mean molecular polarizabilities. They exhibit very low absorption (<1 cm ), with no measurable resonant absorption features below a few THz. As a result, the dielectric behavior is essentially dispersionless, allowing for the accurate determination of their mean, isotropic, molecular polarizabilities [10]. Here, we describe a study of the mean molecular polarizability of both linear and branched liquid alkanes. The effects of adding a branched methyl group to a linear carbon chain are measured using THz-TDS and characterized using a simple additive model for the polarizability. This permits us to independently measure the polarizabilities of methyl, methanediyl, and methanetriyl. We compare these measured polarizabilities to density functional theory (DFT) calculations. The refractive indices and absorption coefficients of single methyl branched alkanes and their linear counterparts, for carbon chain lengths from 9 to 16 carbons, were measured using a conventional THz time domain spectroscopy setup configured in reflection geometry. The liquid samples, obtained from commercial vendors with >99% purity, were used without further purification. Liquids were injected into a custom-designed stainless steel sample cell with a single 6 mm-thick high resistivity Si window and a micrometer adjustable inner (rear) stainless steel wall. The THz pulse transmits through the window and sample, reflects off of the adjustable rear wall and then back through the sample and window; this configuration is equivalent to a double-pass transmission measurement. The reflection geometry and custom sample cell allows the sample path length to be varied without requiring a different window thickness or the removal of the sample cell from the setup between measurements. Taking the ratio of the sample’s Fourier transformed THz waveform to an empty cell’s Fourier transformed reference waveform (or, alternatively, using the first reflection off of the air–Si window interface as a reference) permits the determination of the sample’s index of refraction and absorption coefficient by using the complex transmission function [11,12]. In this analysis, corrections for the Fresnel reflections at each interface are included; however, the most important aspect is the complex exponential term which represents the propagation through the sample itself. Each sample was measured over a minimum of five different path lengths, ranging from 5 to 25 mm, and the absorption D.V. Nickel et al. / Chemical Physics Letters 592 (2014) 292–296 293 and refraction spectra were averaged to produce the final results. The sample temperatures were controlled to within 0.1 C using a water circulator andweremonitored using a probe inserted directly into the liquid sample. All the n-alkanes measured in this experiment remain liquid over the measured temperature range from 20 to 60 C. Figure 1 illustrates typical results. This shows the mean indices of refraction, n(m), and absorption coefficients, a(m), respectively for two of the linear n-alkanes (nonane and tridecane) as well as their single methyl branched counterparts (2-methylnonane and 2methyltridecane), measured at 20 C. The corresponding molecular structures are overlaid above the index spectra in Figure 1a. Each pair of molecules shares the same ‘backbone’ carbon chain structure, i.e. the number of carbon atoms in the linear carbon chain segment, Nb, are equivalent. The branched structures simply have another methyl group branching from one of their internal (nonend) carbons. Like the straight chain alkanes, the methyl branched alkanes’ indices are approximately dispersionless within the measured frequency range. We observe an increase in n(m) for the branched species relative to their straight chain counterparts. Despite the change in n(m), the absorption coefficients a(m) for all the measured alkanes and branched alkanes are indistinguishable from each other, within the repeatability of the measurement. In Figure 1. (a) Representative n(m) at T = 20 C for the linear n-alkanes and their single methyl branched counterparts with the typical errors in n plotted at 0.4 THz. Their corresponding structures are overlaid above the spectra. (b) a(m) at T = 20 C for the same representative n-alkanes. The typical errors in a are plotted near 0.4 THz. The dispersionless index and small, monotonically increasing absorption coefficient is typical behavior for all the measured samples. all cases, their absorption coefficients increase with frequency smoothly and monotonically. The increase in the index of refraction with the addition of a methyl branch can be seen more clearly in Figure 2a, which shows the indices of refraction at a representative frequency of 1 THz and at 20 C, for both the linear carbon chain structures and the branched structures, plotted vs. Nb, the number of carbons in their linear carbon chain backbones. Plotting the results in this manner facilitates the direct comparison of two related molecules with equivalent carbon chain structures. For nonane (Nb = 9), we measured multiple different species, with the methyl group branching from the 2nd, 3rd or 4th carbon atoms in the linear chain. For completeness (Figure 2b), we also show the absorption coefficients vs. Nb for both the branched and linear structured alkanes at the same representative frequency (1 THz). These have no measurable dependence on the number of carbons in the linear chain backbones. Since the absorption coefficients have no measurable chain length dependence and considering n(m) a(m)c/(4pm) for all the samples, the absorption coefficients are negligible and can be ignored for the subsequent calculations. The mean molecular polarizability of each sample is calculated using the Lorentz–Lorenz equation [13] (Eq. (1)), which is used to Figure 2. (a) Refractive index at a representative frequency of 1 THz and 20 C for each measured alkane, both branched (red, green, blue circles) and linear (black squares), plotted vs. the number of carbons in their linear chain backbone, Nb. A methyl branched structure for Nb = 12 was not commercially available. (b) The absorption coefficient at 1 THz and 20 C for each measured alkane plotted vs. Nb. The typical error in a for these measurements are plotted for Nb = 9. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) 294 D.V. Nickel et al. / Chemical Physics Letters 592 (2014) 292–296 relate the macroscopic index of refraction to the microscopic mean molecular polarizability, according to: