Low-Frequency (Gigahertz to Terahertz) Depolarized Raman Scattering Off n-Alkanes, Cycloalkanes, and Six-Membered Rings: A Physical Interpretation

Abstract

Molecular liquids have long been known to undergo various distinct intermolecular motions, from fast librations and cage-rattling oscillations to slow orientational and translational diffusion. However, their resultant gigahertz to terahertz spectra are far from simple, appearing as broad shapeless bands that span many orders of magnitude of frequency, making meaningful interpretation troublesome. Ad hoc spectral line shape fitting has become a notoriously fine art in the field; a unified approach to handling such spectra is long overdue. Here we apply ultrafast optical Kerr-effect (OKE) spectroscopy to study the intermolecular dynamics of room-temperature n-alkanes, cycloalkanes, and six-carbon rings, as well as liquid methane and propane. This work provides stress tests and converges upon an experimentally robust model across simple molecular series and range of temperatures, providing a blueprint for the interpretation of the dynamics of van der Waals liquids. This will enable the interpretation of low-frequency spectra of more complex liquids.

Figure 1

General form of the low-frequency vibrational spectrum of a weakly interacting liquid. The contributions are generally identified as α-relaxation (orientational relaxation), β-relaxation (translational diffusion), the fast-β process (cage rattling), and librations. The inset shows the same general spectrum on a logarithmic frequency axis, which is advantageous for highlighting the very low frequency (gigahertz and lower) part of the spectrum.

Figure 2

Definition of the molecular principal axes. The x- and y-axes have been chosen to correspond to the n-alkanes’ major and minor axes and the cycloalkanes’ two major axes. The x-axis is aligned with the double bond(s) of the unsaturated rings. The z-axes are the out-of-plane axes for all molecules.

Figure 3

Anisotropic molecular polarizabilities of various hydrocarbons. Alkanes and cycloalkanes as a function of chain length are shown as well as cyclohexene, 1,4-cyclohexadiene, and benzene. The blue line shows the linear dependence of the anisotropic polarizabilities of n-alkanes longer than propane with n, while the red dashed curve is a quadratic line to guide the eye.

Figure 4

Time-domain OKE signal of various n-alkanes at 25 °C from pentane to hexadecane.

Figure 5

OKE spectra of various n-alkanes at 25 °C from pentane to hexadecane.

Figure 6

Experimental OKE spectra of liquidn-alkanes at 25 °C and fits. The fits (blue line) of the data (black line) are comprised of the α-relaxation (red), β-relaxation (yellow), fast-β mode (green), and librational mode (blue).

Figure 7

Fit parameters of alkane spectra. Shown are the fitting parameter values of the α-relaxation (red ●), β-relaxation (yellow ■), fast-β (green □), and libration (blue ○) line shapes as a function of n-alkane chain length, n. The parameters displayed are the amplitudes (top), diffusive lifetimes (middle), and vibrational frequencies (bottom). See also Table S2.

Figure 8

OKE spectrum of liquid methane at 95 K (black line) fit with the Bucaro–Litovitz function (purple dashed line).

Figure 9

OKE spectrum of liquid methane at 95 K (black line) with the fit (blue line) comprised of two modes: β-relaxation (yellow) and fast-β (green). Also shown are the best fit for Brillouin-zone edge TA and LA phonon modes (dashed lines). See the Supporting Information for fit parameters.

Figure 10

OKE spectra of liquid propane at various temperatures.

Figure 11

OKE spectra of propane at a range of temperatures and fits. The fits (blue line) of the data (black line) are comprised of the α-relaxation (red), β-relaxation (yellow), fast-β mode (green), and librational mode (blue).

Figure 12

Fit parameters of propane spectra. Shown are the fitting parameter values of the α-relaxation (red ●), β-relaxation (yellow ■), fast-β (green □), and libration (blue ○) line shapes as a function of temperature, T. The parameters displayed are the amplitudes (top), diffusive lifetimes (middle), and vibrational frequencies (bottom).

Figure 13

OKE spectra of various cycloalkanes at 25 °C.

Figure 14

Fitted OKE spectra of cycloalkanes at 25 °C. The fits (red line) of the data (black line) are comprised of the α-relaxation (red), β-relaxation (yellow), fast-β mode (green), and librational mode (blue).

Figure 15

Fitting parameters of cycloalkane spectra. Shown are the fitting parameter values of the α-relaxation (red ●), β-relaxation (yellow ■), fast-β (green □), and libration (blue ○) line shapes as a function of the number of carbons, n. The parameters displayed are the amplitudes (top), diffusive lifetimes (middle), and vibrational frequencies (bottom). See also Table S5 for parameter values.

Figure 16

OKE data of six-membered rings at 25 °C. C₆H₈ is 1,4-cyclohexadiene.

Figure 17

Fitted OKE spectra of saturated to unsaturated six-membered rings at 25 °C. The fits (green line) of the data (black line) are comprised of the α-relaxation (red), β-relaxation (yellow), fast-β mode (green), and librational mode (blue).

Figure 18

Fitting parameters of the six-membered rings spectra, and the square of the anisotropic polarizabilities (inset). Shown are the fitting parameter values of the α-relaxation (●), β-relaxation (■), fast-β (□), and libration (○) line shapes for each molecule as indicated on the x-axis. The parameters displayed are the amplitudes (top), diffusive lifetimes (middle), and vibrational frequencies (bottom). See also Table S6 for parameter values.