Kinetic-to-transport-control crossover for elementary bimolecular reactions in supercritical fluids: quantitative boundary and extension to nonideal multistep processes

Elementary bimolecular reactions in supercritical fluid (SCF) media are commonly described by ideal mass-action kinetics, which assumes that the chemical step, not the encounter of the reactants, is rate-limiting. Rabinovich and Tovbin (2011) developed a molecular theory of nonideal reaction rates for supercritical conditions — a lattice-gas model treated in the quasi-chemical approximation with activated-complex interactions — and computed, within one framework, both the elementary reaction rates and the self-diffusion and viscosity coefficients. Their analysis, however, assumes kinetic control throughout and leaves open the quantitative boundary at which, as the inert-SCF density is raised toward close packing ( $\theta \to 1$ ) and reactant concentrations are low, the reaction passes from kinetic control to reactant-transport (diffusion) control; they further identify as unresolved the treatment of full multistep reaction mechanisms with nonideality and the search for operating conditions that balance reaction rates against reactant mobility. The citing works restate the crossover only qualitatively; adjacent recent work (2025 metadynamics studies — metadynamics being an enhanced-sampling molecular-simulation method — of reactions in supercritical microaggregates) demonstrates an onset of diffusion limitation near the Widom line (the locus emanating from the critical point that separates gas-like from liquid-like supercritical states) for specific probe reactions but does not determine the crossover boundary. The problem — locating the kinetic-to-transport crossover quantitatively within a self-consistent treatment coupling nonideal elementary kinetics with molecular transport, and extending it to realistic overall supercritical processes — remains open.

Background

A supercritical fluid (SCF) is a substance held above its critical temperature and pressure, where liquid and gas phases cease to be distinct; SCFs combine gas-like mobility with liquid-like dissolving power, and their density can be tuned continuously from gas-like to liquid-like values by pressure. This tunability underlies supercritical process technology (e.g., supercritical water oxidation and reactions in supercritical carbon dioxide), but it also means that the two defining operating parameters act oppositely on chemistry: raising temperature accelerates elementary reactions, while raising pressure by adding dense SCF hinders the meeting of reactants and increases viscosity.

The conventional kinetic description is the law of mass action: for a bimolecular step $i+j \to$ products the rate is $w_{ij}=k_{ij}n_i n_j$ with an Arrhenius rate constant $k_{ij}=k_{ij}^0\exp(-E_{ij}/k_{\mathrm{B}}T)$ . This assumes randomly mixed, noninteracting molecules and that the chemical transformation, not the transport of reactants to one another, is the slow event — the condition called kinetic (activation) control. In ordinary solution chemistry the complementary regime is textbook material: when the intrinsic chemical step is fast compared with diffusive encounter, the observed rate saturates at the Smoluchowski diffusion-limited value $4\pi R(D_A+D_B)$ and the reaction is diffusion (transport) controlled; the standard criterion compares the intrinsic rate constant with this diffusion-limited value (see, e.g., the treatment of diffusion- and activation-controlled reactions in solution by Bravo-Díaz and coauthors, 2025).

For supercritical media this picture is incomplete in a specific way: the medium is dense and strongly correlated, so both the chemical rates and the transport coefficients are modified by intermolecular interactions, and the two modifications must be described consistently. Rabinovich and Tovbin (2011) addressed this with the theory of nonideal reaction rates in condensed phases. The reaction mixture is represented as a lattice gas (each cell of a notional lattice holds one molecule or a vacancy), nearest-neighbor pair interactions $\varepsilon_{ij}$ are treated in the quasi-chemical approximation (a pair closure that retains short-range-order correlations), and the elementary step is described by activated-complex (transition-state) theory, with the activated complex carrying its own interactions with the medium. The rate then acquires factors $\theta_{fg}$ and $\Lambda$ — the probability of finding reactants $A$ and $B$ on neighboring cells and the effect of their local environments — beyond the ideal product $\theta_A\theta_B$ , where $\theta_i$ is the site occupancy (the lattice analog of concentration) and $\theta=\sum\theta_i$ the total occupancy, the analog of reduced density, with $\theta\to 1$ meaning the medium approaches close packing and free volume vanishes. The observable Arrhenius slope becomes an effective activation energy $E_{AB}^{\mathrm{app}}=E_{AB}+\varepsilon_{AB}-\beta^{-1}\ln(\theta_{AB}/\theta_A\theta_B)-(z-1)\ln(S_AS_B)$ (with $z$ the lattice coordination number, $\beta=1/k_{\mathrm{B}}T$ , and $S_i$ the local-environment factors), which in the SCF-dominated limit reduces to $E_{AB}+(z-1)(1-\alpha)(\varepsilon_{AC}+\varepsilon_{BC})$ , where $\alpha$ is the ratio of activated-complex to ground-state interactions with the supercritical component $C$ : an inert SCF slows the reaction when $\alpha<1$ . The same formalism yields the self-diffusion coefficient $D_i$ (molecular motion mapped to vacancy-exchange hops $i+\mathrm{V}\to\mathrm{V}+i$ treated as bimolecular events) and a shear viscosity. The 2011 calculations showed that adding inert SCF suppresses bimolecular rates more strongly than it suppresses diffusion or viscosity, and that transport slows near the critical region (a phenomenon documented independently for near-critical diffusion, e.g., De, Shapir and Chimowitz, 2003; diffusion-controlled reactions in SCFs have been studied experimentally since the 1990s, reviewed by Okamoto, 2004).

What the 2011 analysis did not do is solve the coupled problem. All its rate expressions presuppose kinetic control — no diffusion limitation at the micro or macro level — while its transport coefficients are computed separately as equilibrium properties. The authors state that, as the inert-SCF density is raised toward $\theta\to 1$ , a bimolecular reaction can pass from the kinetic regime to the diffusion regime, especially at low reactant concentrations, requiring a separate treatment that includes diffusion; the boundary of this crossover is not determined quantitatively. They add that realistic overall supercritical processes require calculating all elementary steps of the full mechanism under nonideality (since temperature and pressure changes alter not only elementary rates but also equilibria, the mechanism, and the states of accompanying materials), and that process implementation requires optimum conditions balancing reaction rates against reactant mobility, because raising temperature or pressure alone can worsen some elementary steps. Twelve years later, the same author's comprehensive review of molecular modeling of supercritical processes (Tovbin, 2023) still describes this transition to the diffusion regime only qualitatively and concludes that "a comprehensive analysis of the entire system as a whole is required."

Problem Statement

Determine, quantitatively, the boundary of the crossover from kinetic control to reactant-transport (diffusion) control for an elementary bimolecular reaction $A+B\to$ products in an inert supercritical fluid medium, as the total medium density is raised toward $\theta\to 1$ (where $\theta$ is the lattice-gas total site occupancy, the analog of reduced density, so $\theta\to 1$ means the medium approaches close packing and free volume vanishes) — that is, the critical occupancy/density $\theta^*$ and its dependence on temperature, the intermolecular interaction parameters $\varepsilon_{ij}$ , the activated-complex interaction ratios $\alpha_{ij}$ (the ratios of activated-complex to ground-state interaction strengths with the medium), and the reactant concentrations (especially dilute reactants) at which the encounter/transport stage rather than the chemical step becomes rate-limiting. The determination must be made within a self-consistent molecular treatment in which the nonideal elementary rate expressions and the medium's transport coefficients (self-diffusion, viscosity) derive from the same intermolecular interactions and remain consistent with the equilibrium properties of the supercritical mixture, so as to delimit the range of validity of the kinetic-control assumption underlying the nonideal rate expressions, including how the criterion changes when the supercritical component is itself a reactant. In passing from the elementary step to realistic overall supercritical processes, the same coupled treatment is to be extended to full multistep reaction mechanisms computed with nonideality, in which changes of temperature and pressure alter not only the elementary rates but also the equilibria, the mechanism, and the states of accompanying materials, and used to identify the optimum operating conditions that balance reaction rates against reactant mobility, given that raising temperature or pressure alone can worsen individual elementary steps. The central question of this programme is the quantitative crossover boundary $\theta^*$ , determined within the self-consistent coupled treatment; the extension of the same coupled treatment to full multistep mechanisms under nonideality, and the identification of operating windows that balance reaction rates against reactant mobility, form a second part of the same programme. The objective of the programme is to predict when supercritical processes are kinetically rather than transport controlled, and to convert that prediction into usable operating windows for supercritical reaction technology.

References

A. B. Rabinovich and Yu. K. Tovbin, "Supercritical fluid effect on the rates of elementary bimolecular reactions", Kinetics and Catalysis 52(4), 471–479 (2011). DOI: 10.1134/S0023158411040148, https://doi.org/10.1134/S0023158411040148
Yu. K. Tovbin, "Molecular Modeling of Supercritical Processes and the Lattice—Gas Model", Processes 11(9), 2541 (2023). DOI: 10.3390/pr11092541, https://doi.org/10.3390/pr11092541
Yu. K. Tovbin, "Possibilities of the Molecular Modeling of Kinetic Processes under Supercritical Conditions", Russian Journal of Physical Chemistry A 95(3), 429–444 (2021). DOI: 10.1134/S0036024421030225, https://doi.org/10.1134/S0036024421030225
Yu. K. Tovbin and S. V. Titov, "Role of local environment relaxation in calculating the reaction rates for nonideal reaction systems", Russian Journal of Physical Chemistry B 5(7), 1135–1143 (2011). DOI: 10.1134/S1990793111070153, https://doi.org/10.1134/S1990793111070153
Yu. K. Tovbin, "Modeling of Processes and Physicochemical Properties of Fluids in the Near- and Supercritical Regions of State Parameters: Review", Russian Journal of Physical Chemistry B 17(8), 1569–1592 (2023). DOI: 10.1134/S1990793123080018, https://doi.org/10.1134/S1990793123080018
Junhao Guo, Yutong Wang, and Guozhu Liu, "Enhanced Reaction in Supercritical Inhomogeneous Microaggregation", JACS Au 6(1), 554–564 (2025). DOI: 10.1021/jacsau.5c01500, https://doi.org/10.1021/jacsau.5c01500
Masami Okamoto, "Diffusion-Controlled Reaction in Supercritical Fluid", The Review of High Pressure Science and Technology 14, 30–36 (2004). DOI: 10.4131/jshpreview.14.30, https://doi.org/10.4131/jshpreview.14.30
S. De, Y. Shapir, and E. H. Chimowitz, "Diffusion in dilute binary fluids confined in porous structures near the solvent critical point", The Journal of Chemical Physics 119, 1035–1044 (2003). DOI: 10.1063/1.1577532, https://doi.org/10.1063/1.1577532
Carlos Bravo-Díaz, Sonia Losada-Barreiro, José G. Santos, and Margarita E. Aliaga, "Reactions in Solution: Diffusion-Controlled and Activation-Controlled Reactions", in From Experimental Kinetic Data to Reaction Mechanisms (Physical Chemistry in Action series), Springer, Cham, 113–134 (2025). DOI: 10.1007/978-3-031-90702-9_5, https://doi.org/10.1007/978-3-031-90702-9_5