Origin and predictive description of critical rate tuning in muonium and hydrogen-atom addition to alkenes in supercritical carbon dioxide

Near the critical point of carbon dioxide ($T_{\mathrm{c}}=304.1$ K, $P_{\mathrm{c}}=7.38$ MPa), transverse-field muon spin rotation measurements showed that addition of muonium (a light hydrogen isotope) to ethylene in supercritical carbon dioxide exhibits critical speeding up, while addition to vinylidene fluoride and vinylidene chloride exhibits critical slowing. The origin of these opposing critical rate resonances is unresolved: the measured rate constants cannot distinguish an intrinsic change of the bimolecular rate constant from an apparent change of local reactant concentration caused by near-critical solvent clustering. The cited work also lacked a quantitative, configurationally averaged description of solute–CO$_2$ interactions, offered only empirical near-critical parametrizations of the pressure and temperature dependence, and left unvalidated whether conventional H-atom chemistry shows the same resonances. The open problem is a mechanistic and predictive account of how near-critical solvent density inhomogeneities tune elementary radical-addition rate constants in supercritical carbon dioxide.

Background

A supercritical fluid is a substance held above its critical temperature and pressure; for carbon dioxide the critical point lies at $T_{\mathrm{c}}=304.1$ K and $P_{\mathrm{c}}=7.38$ MPa. In the near-critical regime the isothermal compressibility of the fluid becomes very large, and the solvent develops strong density inhomogeneities on nanometer length scales — transient clusters and depleted regions, often called solvent density inhomogeneities or local density enhancement (Kim and Johnston, AIChE J. 1987; Tucker, Chem. Rev. 1999). Because solutes perturb this structure differently depending on their attractions to the solvent, the local environment around a dissolved molecule can differ sharply from the bulk, a phenomenon central to solvation, extraction, and reaction chemistry in supercritical fluids (Savage et al., AIChE J. 1995). Muonium (Mu) is an exotic atom consisting of a positive muon and an electron, chemically a light isotope of hydrogen with about one ninth of the proton mass. In transverse-field muon spin rotation ($\mathrm{TF}$-$\mu$SR), spin-polarized muons implanted in a solution mostly form muonium, and the time evolution of the muon polarization yields pseudo-first-order loss rates, hence bimolecular rate constants $k$ for the reaction of Mu with a dissolved substrate. The Mount Allison University muon-chemistry group used this technique to build a research program on radical kinetics in sub- and supercritical carbon dioxide (scCO$_2$). Muonium formation in scCO$_2$ was itself characterized first (Ghandi et al., J. Phys. Chem. A 2004). The muoniated ethyl radical (the adduct formed when muonium adds to ethylene, MuCH$_2$CH$_2\cdot$), the first alkyl radical observed in scCO$_2$ by magnetic resonance, revealed a gas-phase-like local environment at a density of 0.3 g cm$^{-3}$, a local density enhancement of roughly a factor of 1.5 around the radical near 0.4 g cm$^{-3}$, and an unusual density dependence of its hyperfine coupling constant (the strength of the magnetic interaction between the unpaired electron and a magnetic nucleus, which reports on the radical's local environment) (Cormier et al., J. Phys. Chem. A 2008; Cormier et al., Physica B 2009). Rate measurements for muonium addition to vinylidene fluoride in CO$_2$ (290–530 K, 40–360 bar) then showed critical slowing near the conditions of maximum isothermal compressibility, with apparent activation volumes (the pressure response of the rate, formally the pressure derivative of the activation free energy) of unprecedented magnitude on the order of $\pm 10^{6}$ cm$^{3}$ mol$^{-1}$ — the basis for the proposal of thermodynamic rate tuning of fluoralkene reactivity (Ghandi et al., PCCP 2012). The source paper (Cormier, Clarke, McFadden and Ghandi, JACS 2014) compared muonium addition to ethylene (ET), vinylidene fluoride (VDF), and vinylidene chloride (VDC) in scCO$_2$ over ranges of pressure and temperature and found that near the critical point the addition to ethylene exhibits critical speeding up while the halogenated analogues display critical slowing — a reversal suggesting that scCO$_2$ near its critical point can be used to tune alkene selectivity toward nonpolar free radicals. The paper supported a localization/segregation interpretation — selective clustering of CO$_2$ around the haloalkenes versus a locally ethylene-rich, CO$_2$-depleted environment for ethylene — with qualitative quantum-chemical calculations at the B3LYP/6-311++G(2df,p) level (a standard density-functional-theory exchange–correlation functional combined with a large Gaussian atomic-orbital basis set) including corrections for basis-set superposition error (an artificial overbinding that arises when two species described by incomplete basis sets share each other's basis functions) of up to about 10%, on a small number of relative solute–solvent configurations; accurate force fields for ET, VDF and VDC were not available, and full ab initio configurational sampling by ab initio molecular dynamics (molecular dynamics in which the interatomic forces are computed on the fly from electronic-structure theory) of the Car–Parrinello type or equivalent was out of scope. The pressure and temperature dependences were described by empirical parametrizations $k(P)$ and $k(T)$ intended only for the near-critical window. Because muonium is nine times lighter than hydrogen, kinetic isotope effects on its addition rates are large and tunneling contributes significantly, so the transfer of these muonium-based conclusions to conventional H-atom chemistry is an extrapolation, not an established result. The mechanistic crux is that a $\mu$SR-measured rate constant reports the product of the local encounter probability (governed by solvent-driven clustering and partitioning of the reactants) and the intrinsic bimolecular rate constant (governed by transition-state energetics); the measurement by itself cannot separate the two, and the required discrimination between rate-constant modification and concentration enhancement is exactly what the source work left open (Ghandi and MacLean, Hyperfine Interact. 2015).

Problem Statement

The experiments established an empirical fact without a mechanism: near the critical point of CO$_2$, muonium addition to ethylene speeds up while muonium addition to vinylidene fluoride and vinylidene chloride slows down. The open problem arising from these results is to give a quantitative mechanistic and predictive account of this critical rate tuning, with four linked open aspects. (i) Mechanistic partition: determine quantitatively the relative contributions of intrinsic bimolecular rate-constant modification (solvent-modified transition-state energetics) and effective local reactant-concentration change (solvent-driven clustering, localization and segregation of solute from solvent) to the observed near-critical extrema of $k$ for at least the ethylene versus haloalkene contrast — a partition of microscopic encounter probabilities from transition-state properties that transverse-field muon spin rotation alone cannot make. (ii) Quantitative solvent-structure description: replace the qualitative few-configuration B3LYP/6-311++G(2df,p) picture with a quantitative, configurationally averaged description of solute–CO$_2$ interactions for ethylene, vinylidene fluoride and vinylidene chloride, characterized by quantities such as potentials of mean force (free-energy profiles along a chosen coordinate) and local composition profiles, capable of testing the proposed localization/segregation mechanism. (iii) Predictive theory: supersede the empirical $k(P)$ and $k(T)$ parametrizations, which are valid only close to $P_{\mathrm{c}}$ and $T_{\mathrm{c}}$ and must not be extrapolated, with a physically grounded theory that connects microscopic solvent structure to elementary rate constants beyond that near-critical window and can indicate where rate resonances occur. (iv) Isotope transfer: determine whether conventional H-atom addition to these alkenes shows the same critical resonances under comparable scCO$_2$ conditions, quantitatively accounting for the large muonium-versus-hydrogen kinetic isotope effects (the change in rate upon replacing hydrogen by the lighter muonium) and tunneling contributions. The central explanatory target is why ethylene and the haloalkenes respond in opposite directions near the critical point, not only how large each contribution is; the structural description of (ii) is the discriminating evidence for the partition in (i), the theory of (iii) is its generalization, and (iv) tests whether the account extends beyond the muonium probe to the hydrogen chemistry it is meant to represent. The four aspects are jointly posed: the mechanistic partition (i) together with the configurationally averaged solvent-structure description (ii) that discriminates it forms the core of the account, while the predictive theory (iii) and the isotope-transfer test (iv) extend and test that account rather than standing as alternative resolutions of it.

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