Dependence of C C stretching wavenumber on polyene length in degraded polyvinyl chloride: a comparative empirical, classical mechanics, and DFT study

Mathematically describing the length-dependence of vibrational fingerprints of polyenes is challenging, yet crucial in understanding and predicting polyene-associated molecular properties of industrially-important and vital substances. To this end, we develop an analytical relationship between the wavenumbers ν∼C=C of the Raman-active CC stretching mode in polyene sequences (CHCH)n and the polyene length (n) using classical mechanics laws. Noteworthy, this relationship is derived from Newton's equations instead of regression approximations and validated against experimental data for degraded polyvinyl chloride (PVC), t-butyl end-capped all-trans polyenes, β-carotenes, and carotenoids. Furthermore, given this fundamental tool, we carefully re-examined or validated the up-to-now applied empirical tools; we find that: (i) A phenomenological exponential regression function ν~C=C=1461+151.2×exp-0.07808n proves fairly suitable for describing polyenes with lengths below 24 in degraded PVC. (ii) The derived analytical relationship agrees more closely with a long-established reciprocal-length regression function ν~C=C=1459+720/n+1 for describing carotenoids. Moreover, extensive DFT calculation results on all-trans polyenes H(CHCH)nH (n = 3-30) and polyenes end-capped with terminal vinyl chloride oligomers agree with experiment for shorter polyenes and are similar, showing that complicated calculations of ν∼C=C for infinite degraded PVC chains reduce to the calculations on finite polyene sequences. Noteworthy, unlike other polyene length-determination tools, the proposed analytical polyene length-determination based on intrinsic physical properties could well prove to be an even more versatile tool, as it comes with the added potential for determining or correcting the elasticity constants of carbon bonds in polyene chains.