Mechanistic origin of near-critical rate resonances in radical additions to alkenes in supercritical carbon dioxide: intrinsic rate-constant change versus local concentration enhancement

Transverse-field muon spin rotation measurements showed that addition of muonium (an ultra-light isotope of hydrogen) to ethylene in supercritical CO2 exhibits a local maximum in the bimolecular rate constant near CO2's critical pressure (critical speeding up), while addition to the halogenated alkenes vinylidene fluoride and vinylidene chloride exhibits local minima (critical slowing), shifting relative rates by more than two orders of magnitude. Whether these resonances arise from a genuine change in the intrinsic bimolecular rate constant (solvent-modified transition-state energetics) or from an effective change in local reactant concentration caused by near-critical solvent clustering and solute segregation remains unresolved, because the technique cannot separate the two contributions. The problem asks for a quantitative, mechanism-resolved account of these resonances, with an optional experimental extension asking whether the same critical resonances occur for conventional H-atom chemistry under comparable conditions.

Background

A supercritical fluid is a substance held above its critical temperature and pressure ($T_{\mathrm{c}}$ = 304.1 K, $P_{\mathrm{c}}$ = 7.38 MPa for CO2), where liquid and gas phases merge into a single, highly compressible medium. Near the critical point the fluid develops large-scale density fluctuations: the local density around a dissolved solute can substantially exceed the bulk density, a phenomenon called local density augmentation or solvent clustering, long recognized as a distinctive feature of supercritical solvents (Tucker, Chem. Rev. 1999). A central, still-unsettled question of supercritical-fluid chemical kinetics is whether these microscopic inhomogeneities change elementary reaction rates by altering the intrinsic rate constant — the transition-state energetics in the usual transition-state-theory picture — or mainly by changing effective local reactant concentrations and encounter probabilities.

Radical cage-effect experiments in supercritical CO2 (scCO2) gave early, partially conflicting evidence: chlorine-atom cage effects were used to probe free-radical chlorination (Fletcher, Suleman & Tanko, JACS 1998; Tanko & Pacut, JACS 2001), but no consensus emerged on how near-critical clustering propagates into elementary bimolecular rate constants. Direct, pressure-resolved rate measurements were scarce because conventional H-atom techniques (flash photolysis, pulse radiolysis) are unsuitable in neat CO2, which offers no intrinsic proton source, and additives perturb the solvent.

Muon methods bypass this limitation. Muonium (Mu, $\mu^{+}e^{-}$) is a light hydrogen isotope (mass 0.11 amu) formed when a positive muon thermalizes in a medium; it adds to unsaturated bonds like a hydrogen atom, and the reaction can be monitored by transverse-field muon spin rotation (TF-μSR), which tracks the spin precession of implanted muons and yields pseudo-first-order, hence bimolecular, rate constants without any added probe reagent. Cormier, Arseneau, Brodovitch, Lauzon, Taylor & Ghandi (J. Phys. Chem. A 2008) established Mu formation and addition to ethylene in scCO2; Ghandi, McFadden, Cormier, Satija & Smith (PCCP 2012) mapped Mu + vinylidene fluoride rate constants across 290–530 K and 40–360 bar and observed critical slowing near maximum isothermal compressibility.

Cormier, Clarke, McFadden & Ghandi (JACS 2014) then compared Mu addition to ethylene (ET), vinylidene fluoride (VDF), and vinylidene chloride (VDC) in scCO2. Near the pressure $P_{\mathrm{c}}$ of maximum isothermal compressibility, Mu + ET shows a local maximum in $k$ (critical speeding up, the first such report for a reaction in a pure supercritical fluid), while Mu + VDF shows a local minimum (critical slowing); at fixed density the Mu + VDC rate shows a resonance near $T_{\mathrm{c}}$. The data were fitted with the empirical forms $k = A + \alpha P\left|\frac{P-P_{\mathrm{c}}}{P_{\mathrm{c}}}\right|^{\gamma}$ and $k = A\left|\frac{T-T_{\mathrm{c}}}{T_{\mathrm{c}}}\right|^{\gamma}e^{-E_{\mathrm{a}}/RT}$, and the selectivity ratio $k_{\mathrm{ET}}/k_{\mathrm{VDF}}$ changes by more than two orders of magnitude near $P_{\mathrm{c}}$. The authors proposed a microscopic picture based on solute segregation: qualitative calculations at the B3LYP/6-311++G(2df,p) level (a standard density-functional-theory method combined with a triple-zeta basis set) on a small number of relative configurations suggest VDF–CO2 attractions exceed CO2–CO2, while ET–CO2 and H–CO2 are weaker, so near criticality the halogenated alkenes preferentially partition into high-density regions while Mu and ET avoid them, raising (ET) or lowering (VDF, VDC) effective encounter rates.

Several limitations of the current evidence remain open. TF-μSR cannot by itself distinguish an intrinsic change of the bimolecular rate constant from an apparent change of local reactant concentration, and the published rate data carry no stated uncertainties, so even the statistical significance of the resonances against monotonic pressure dependence rests on qualitative grounds. The quantum-chemical survey was limited to a few configurations and is qualitative — accurate force fields for the solutes are lacking, and configurationally averaged solute–CO2 potentials of mean force (the free-energy profile of a solute pair as a function of separation, obtainable in principle from ab initio molecular dynamics — molecular dynamics with interatomic forces computed from electronic-structure theory on the fly, as in the Car–Parrinello scheme) do not exist for these systems. The empirical parametrizations were fitted to near-critical data only, and their behaviour outside the near-critical regime is unconstrained. And because Mu is nine times lighter than H — kinetic isotope effects (rate ratios between isotopologues) for Mu versus H reactions are large and tunneling-dominated — the extension of these resonances to conventional H-atom chemistry is an outstanding experimental question.

Problem Statement

For muonium addition to ethylene, vinylidene fluoride, and vinylidene chloride in supercritical CO2 near the critical point — where critical speeding up (a local maximum in the observed rate constant for Mu + ethylene) and critical slowing (local minima for the halogenated alkenes) were reported — determine the mechanistic origin of the rate resonances and establish a quantitative microscopic account that connects them to prediction, specifically:

1. Decompose each observed resonance into its contributions from (a) a change in the intrinsic bimolecular rate constant (solvent-modified transition-state energetics) and (b) an effective change in local reactant concentration or encounter probability caused by near-critical density inhomogeneity, solute clustering, and solute segregation between high- and low-density regions. The decomposition must be quantitative for the actual Mu + alkene/CO2 systems studied (the $T = 305$ K pressure scans and the fixed-density scans near $T_{\mathrm{c}}$), not only by analogy in other supercritical fluids.

2. Provide the microscopic underpinning the original study lacked: quantitative, configurationally averaged descriptions of the relevant solute–CO2 and solute–solute interactions near criticality sufficient to test the proposed localization/segregation mechanism and its predicted interaction-energy ordering (VDF–CO2 more attractive than CO2–CO2, ET–CO2 and H–CO2 weaker), replacing the qualitative few-configuration B3LYP survey.

3. As an optional experimental extension beyond the core of items 1–2, motivated by muonium's role as a probe isotope nine times lighter than hydrogen, validate experimentally whether corresponding critical resonances occur for conventional H-atom addition to these alkenes under comparable near-critical scCO2 conditions, accounting for the large muonium–hydrogen kinetic isotope effects and tunneling differences.

The single overarching objective is a mechanism-resolved, quantitative account of how near-critical CO2 microstructure controls elementary radical-addition kinetics at the measured near-critical state points, in place of purely empirical near-critical parametrizations; items 1 and 2 jointly define the core of the problem, and item 3 is an optional experimental extension of the same objective to real hydrogen chemistry.

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