Reconciling ab initio and experimentally extracted association rate coefficients in barrierless water cluster formation at extreme supersaturation

At extreme supersaturations, water nucleation in Laval-nozzle flows is barrierless and controlled entirely by the kinetics of monomer–cluster association. Kalikmanov and Hagmeijer (2019) showed that steady-state nucleation rates computed from ab initio, master-equation association rate coefficients (Bourgalais et al. 2016) agree well with the Laval-nozzle experiments of Lippe et al. (2018), while the predicted small-cluster size distributions differ from measurement by 1–2 orders of magnitude — a discrepancy they could attribute to the difference between experimentally extracted and ab initio rate coefficients only hypothetically. The open problem is a quantitative, attributed reconciliation of this kinetic description of barrierless cluster formation: determine whether the discrepancy between ab initio and experimentally inverted association rate coefficients (and the resulting cluster distributions) arises from experimental uncertainties, ionization/measurement biases, or limitations of the master-equation energy-transfer models; extend the nonlinear, time-dependent coalescence predictions to larger cluster sizes and wider pressure–temperature ranges; and quantify the sensitivity of the predictions to the far-extrapolated low-temperature thermodynamics entering the theory. No resolution of any of these three linked issues is known in the literature to date.

Background

Classical nucleation theory (CNT) treats gas-to-liquid condensation as thermally activated crossing of a free-energy barrier whose height controls the nucleation rate $J$ (the number of clusters passing the critical size per unit volume per unit time), with forward association rates taken from free-molecular, ideal-gas collision kinetics. At extreme supersaturation $S$ — the ratio of the vapor partial pressure to its saturation vapor pressure $p_{\mathrm{sat}}$ — the barrier can vanish: the vapor phase is unstable, cluster formation becomes barrierless, and the association rate coefficients $k_{i,j}(T,p)$ for collisions of an $i$-mer with a $j$-mer, rather than thermodynamics, control the outcome. Kalikmanov and Hagmeijer (2019) mapped this "extreme supersaturation" (XS) domain for water using mean-field kinetic nucleation theory (MKNT), in which the domain boundary is set by a microscopic temperature parameter $\theta_{\mathrm{micro}}(T)$ — a binding-energy scale reduced into units of $k_B T$ and built from the second virial coefficient $B_2(T)$ (the leading two-molecule gas non-ideality correction to the equation of state) and $p_{\mathrm{sat}}(T)$. In this domain evaporation is negligible and the kinetics is dominated by dimer formation.

The experimental anchor is the Laval-nozzle work of Lippe et al. (2018): a supersonic expansion produces a uniform post-nozzle flow at $T = 87$ K with $\ln S = 41$ and $T = 47.5$ K with $\ln S = 104$, where cluster-size distributions, reported as relative steady-state abundances $X_n = \rho_s(n)/\rho_s(1)$ (the density of $n$-mers divided by the monomer density), were measured by mass spectrometry after soft single-photon vacuum-ultraviolet (VUV) ionization. Li et al. (2019) inverted these measured populations to extract the monomer–cluster association rate constants $k_{1,n}$. Independently, Bourgalais et al. (2016) had computed $k_{i,j}(T,p)$ for water over 20–100 K and $10^{-6}$–10 bar from ab initio calculations using a master-equation approach, which couples transition-state-theory (TST) microcanonical dissociation rate coefficients with simple models of collision-induced energy transfer to the carrier gas; the association rate coefficient emerges from the eigensolutions of the resulting energy-resolved master equation.

Kalikmanov and Hagmeijer combined the ab initio rates with a coalescence equation — a Smoluchowski-type kinetic equation for cluster–cluster collisions with temperature- and pressure-dependent kernels — for the steady state. The outcome is a mixed verdict: the theoretical nucleation rates agree excellently with experiment at 87 K ($3.2\times10^{15}$ versus $(4{-}5)\times10^{15}\,\mathrm{cm^{-3}s^{-1}}$) and within roughly a factor of four at 47.5 K, but the predicted small-cluster distributions differ from the measured ones by 1–2 orders of magnitude (a factor of 3–4 at $n = 4$) and, at 47.5 K, even the qualitative trend in $n$ disagrees. The authors attribute this only hypothetically to the difference between the experimentally extracted and the ab initio $k_{1,n}$, without establishing which input is at fault. Two further limitations are explicit in the paper, and a third is inherent in the theory's construction: the linearized coalescence model (retaining only monomer–cluster collisions) was validated only by solving the full nonlinear system for a maximum cluster size $N_{\max} = 4$ at 47.5 K, where it overestimates the exact solution by 17–27%; only the steady state was treated, although the experiments probe a few hundred microseconds around the $t_s \approx 50\,\mu\mathrm{s}$ equilibration time; and $\theta_{\mathrm{micro}}(T)$ depends on $B_2$ and $p_{\mathrm{sat}}$ far-extrapolated down to 47.5–87 K, with sensitivity never quantified in the paper, as is that of the choice of carrier-gas energy-transfer model in the master-equation rates.

Problem Statement

The problem, as left open by Kalikmanov and Hagmeijer (2019), is to turn the barrierless-kinetics description of water cluster formation at extreme supersaturation into a quantitatively attributed account of the measured cluster populations, for the Laval-nozzle conditions of Lippe et al. (2018) ($T = 87$ K, $\ln S = 41$; $T = 47.5$ K, $\ln S = 104$) and neighboring extreme-supersaturation conditions:

1. Reconcile quantitatively the monomer–cluster association rate coefficients $k_{1,n}$ from ab initio transition-state/master-equation calculations (Bourgalais et al. 2016) with those extracted from experimental cluster-population inversions (Li et al. 2019) — a discrepancy that is systematic for $n \ge 3$, with the extracted values sitting near the collision limit while the ab initio values are nonmonotonic in $n$ — and determine whether it arises from experimental uncertainties, ionization/measurement biases, or limitations of the master-equation energy-transfer models, replacing the paper's merely hypothetical attribution with a demonstrated one.
2. Extend the coalescence steady-state predictions to larger maximum cluster sizes and to the time-dependent approach to steady state beyond the linearized approximation (with the full nonlinear cluster–cluster terms) over a wider pressure–temperature range, and establish whether this removes the 1–2 order-of-magnitude (and, at 47.5 K, qualitative) disagreement between predicted and measured small-cluster abundances $X_n$.
3. Quantify systematically the sensitivity of the predicted distributions to the remaining uncertainties in $\theta_{\mathrm{micro}}(T)$ (through $B_2$ and $p_{\mathrm{sat}}$) and to the carrier-gas energy-transfer models used in the master-equation calculations, so that any residual model–experiment discrepancy can be attributed to rate-coefficient differences with confidence.

These three questions are linked parts of one common objective — an internally consistent kinetic account of barrierless water cluster formation in which every model–experiment discrepancy is attributed to an identified input (experimental, detection-related, or theoretical). A full resolution requires jointly addressing all three parts: the sources named in (1) are candidate explanations rather than an exhaustive list, and the extension in (2) is carried far enough to decide whether the stated disagreement is removed.

References

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