McCormack Model in Gas Mixture Kinetics

• McCormack model is a collision-term approximation for the Boltzmann equation that uses a structured linear operator to accurately model gas mixture transport properties.
• It ensures conservation of mass, momentum, and energy while reproducing correct viscosity, thermal conductivity, and diffusion coefficients via a detailed Chapman–Enskog approach.
• The model captures non-equilibrium effects and cross-coupling in spherical geometries, enabling reliable computation of Onsager coefficients and fluxes under finite rarefaction conditions.

Searching arXiv for the cited paper and closely related McCormack-model mixture-kinetics work to ground the article. The McCormack model is a collision-term approximation for the Boltzmann equation of a gas mixture in which the full binary collision integral is replaced by a linear operator that remains sufficiently structured to reproduce mixture transport physics beyond single-relaxation closures. In the treatment of sublimation and deposition at a solid sphere of argon into its vapor with helium as a background gas, the model is used as the core kinetic approximation for computing Onsager coefficients, mass and energy flow rates, and non-equilibrium flow fields over a wide range of rarefaction parameters and for both hard-spheres and ab-initio intermolecular potentials (Kalempa et al., 14 Jul 2025).

1. Definition and operator structure

In the formulation considered here, the two-species mixture is indexed by $\alpha=1$ for helium and $\alpha=2$ for argon or krypton. The true Boltzmann equation is posed in spherical-polar coordinates and contains the full collision integral $Q(f_\alpha,f_\beta)$. McCormack’s idea is to replace the generally intractable double integral by a linear operator $\hat L_{\alpha\beta}$ chosen so that the mixture’s viscosity, thermal conductivity, diffusion, and thermal diffusion come out correctly, the conservation laws are respected, an $H$-theorem holds, and cross-coupling between species is accounted for (Kalempa et al., 14 Jul 2025).

In the un-linearized formulation,

$$Q(f_\alpha,f_\beta)\approx f_{0\alpha}\,\hat L_{\alpha\beta}[h_\alpha],$$

with $f_{0\alpha}$ a local Maxwellian and $h_\alpha$ the scalar perturbation. In the linearized form reported in Appendix A, the operator contains a relaxation term in $h_\alpha$ together with terms proportional to the perturbation moments $\nu_\alpha$, $\alpha=2$0, $\alpha=2$1, $\alpha=2$2, $\alpha=2$3, and to interspecies couplings through the binary-mixture coefficients $\alpha=2$4 for $\alpha=2$5. This structure distinguishes density deviation, radial-bulk-velocity deviation, temperature deviation, heat-flux deviation, and pressure-tensor deviation at the species level rather than collapsing them into a single relaxation channel.

All quantities are written in dimensionless form. The dimensionless velocity components are defined from the molecular velocities and the reference temperature $\alpha=2$6; the mean molecular mass is

$\alpha=2$7

and the reduced mass is

$\alpha=2$8

The collision-frequency parameters satisfy

$\alpha=2$9

where $Q(f_\alpha,f_\beta)$0 is the partial viscosity of species $Q(f_\alpha,f_\beta)$1 in the mixture. The coefficients $Q(f_\alpha,f_\beta)$2 are linear combinations of collision-cross-section integrals, specifically $Q(f_\alpha,f_\beta)$3-integrals.

2. Linearization and kinetic boundary-value problem

The model is inserted into the Boltzmann equation through a linearization around equilibrium with two small thermodynamic driving forces. The distribution is written as

$Q(f_\alpha,f_\beta)$4

where

$Q(f_\alpha,f_\beta)$5

Substitution and separation of the two independent perturbations $Q(f_\alpha,f_\beta)$6 and $Q(f_\alpha,f_\beta)$7 yield linearized kinetic equations for $Q(f_\alpha,f_\beta)$8 in the radial and polar variables (Kalempa et al., 14 Jul 2025).

The linearization establishes the operational scope of the model in this setting. It is designed for near-equilibrium sublimation and deposition driven by a small partial-pressure difference $Q(f_\alpha,f_\beta)$9 and a small temperature difference $\hat L_{\alpha\beta}$0. A plausible implication is that the reported Onsager matrix and flow fields are controlled linear-response quantities rather than strongly nonlinear far-from-equilibrium observables.

The problem is axisymmetric. Azimuthal symmetry removes any $\hat L_{\alpha\beta}$1-dependence, so the boundary-value problem reduces to two independent partial differential equations in $\hat L_{\alpha\beta}$2 and $\hat L_{\alpha\beta}$3, one associated with the pressure perturbation and one with the temperature perturbation.

3. Physical assumptions and relation to BGK-type closures

The McCormack model carries explicit dependence on the molecular-interaction law through the $\hat L_{\alpha\beta}$4 integrals. This permits either a hard-sphere representation, for which the $\hat L_{\alpha\beta}$5-integrals are analytic, or an ab-initio representation, for which they are taken from numerical quantum-scattering data. By construction, the operator $\hat L_{\alpha\beta}$6 conserves mass, momentum, and energy. When viscosity, thermal conductivity, and diffusion coefficients are computed through the Chapman–Enskog procedure, the same first-order values are obtained as from the full Boltzmann operator, and a positive entropy production can be shown (Kalempa et al., 14 Jul 2025).

The central distinction from BGK-type mixture models is the treatment of cross-effects. Unlike a single-relaxation-time BGK or Shakhov model, McCormack’s form distinguishes partial stresses $\hat L_{\alpha\beta}$7, partial heat fluxes $\hat L_{\alpha\beta}$8, and partial bulk velocities $\hat L_{\alpha\beta}$9, and couples them through independent $H$0. This additional tensorial and species-resolved structure is described as essential for reproducing thermal diffusion and other cross-effects in mixtures.

A recurrent misconception is to regard the McCormack model as merely another algebraic simplification of the Boltzmann collision term. In the present usage, that characterization is incomplete. The model is simplified relative to the full collision integral, but it is also constructed to preserve specific mixture transport properties and reciprocity relations that a single scalar relaxation rate cannot recover. In that sense, it is not interchangeable with standard BGK-type approximations for multicomponent transport.

4. Onsager formulation and flux reconstruction

The two perturbation problems provide dimensionless moments $H$1 and $H$2 for $H$3, from which the thermodynamic fluxes are built as

$H$4

These satisfy the Onsager relations

$H$5

The kinetic coefficients are therefore

$H$6

$H$7

The dimensional mass and energy flow rates at the sphere are then

$H$8

(Kalempa et al., 14 Jul 2025).

This construction places the McCormack model directly within linear non-equilibrium thermodynamics. The coefficients $H$9 are not auxiliary outputs; they determine the interfacial response of the sublimation/deposition system to the pressure and temperature driving forces. The symmetry $$Q(f_\alpha,f_\beta)\approx f_{0\alpha}\,\hat L_{\alpha\beta}[h_\alpha],$$0 is also used numerically as a convergence check.

5. Interatomic potentials and transport sensitivity

The model permits two interaction descriptions: hard-spheres (HS) and ab-initio (AI). For HS, the collision cross section is constant and the $$Q(f_\alpha,f_\beta)\approx f_{0\alpha}\,\hat L_{\alpha\beta}[h_\alpha],$$1-integrals are analytic functions of the molecular diameters. For AI, the $$Q(f_\alpha,f_\beta)\approx f_{0\alpha}\,\hat L_{\alpha\beta}[h_\alpha],$$2-integrals are computed numerically from quantum-scattering data; Table A0 lists $$Q(f_\alpha,f_\beta)\approx f_{0\alpha}\,\hat L_{\alpha\beta}[h_\alpha],$$3 for He–Ar and He–Kr at $$Q(f_\alpha,f_\beta)\approx f_{0\alpha}\,\hat L_{\alpha\beta}[h_\alpha],$$4. All binary-collision parameters entering $$Q(f_\alpha,f_\beta)\approx f_{0\alpha}\,\hat L_{\alpha\beta}[h_\alpha],$$5, including $$Q(f_\alpha,f_\beta)\approx f_{0\alpha}\,\hat L_{\alpha\beta}[h_\alpha],$$6 and $$Q(f_\alpha,f_\beta)\approx f_{0\alpha}\,\hat L_{\alpha\beta}[h_\alpha],$$7, are built from these $$Q(f_\alpha,f_\beta)\approx f_{0\alpha}\,\hat L_{\alpha\beta}[h_\alpha],$$8-integrals (Kalempa et al., 14 Jul 2025).

Potential	$$Q(f_\alpha,f_\beta)\approx f_{0\alpha}\,\hat L_{\alpha\beta}[h_\alpha],$$9-integrals	Reported consequence
Hard-sphere	Analytic	Constant collision cross section
Ab-initio	Numerical from quantum scattering	Table A0 for He–Ar and He–Kr at $f_{0\alpha}$0

In the free-molecular limit $f_{0\alpha}$1, collisions vanish and every potential gives the same analytical result. For finite $f_{0\alpha}$2, however, the choice of potential materially affects the transport predictions: HS versus AI can differ by tens of percent in the cross-coupling coefficient $f_{0\alpha}$3, and by up to approximately $f_{0\alpha}$4 in $f_{0\alpha}$5 or $f_{0\alpha}$6. A common simplifying assumption is that realistic scattering laws only weakly perturb mixture kinetics once the geometry and rarefaction are fixed. The reported results contradict that assumption for finite rarefaction, especially in the cross-coefficients.

6. Spherical boundary conditions and predicted non-equilibrium structure

The sphere is held at temperature $f_{0\alpha}$7. Helium is completely accommodated at the surface, while argon is completely absorbed and re-emitted. The linearized boundary conditions at the sphere are therefore, for helium,

$f_{0\alpha}$8

and for argon or krypton,

$f_{0\alpha}$9

At infinity, $h_\alpha$0 (Kalempa et al., 14 Jul 2025).

These boundary conditions produce a specifically spherical non-equilibrium structure. The model predicts Knudsen layers with density-jumps and temperature-jumps and even inversion of the temperature gradient. Because the operator retains the cross-coupling terms $h_\alpha$1, it also reproduces the heat-of-sublimation effect, namely a heat flow driven by a pressure or mass gradient through the quantities $h_\alpha$2. The paper states that a single-relaxation BGK would fail here, whereas the McCormack form succeeds.

The transitional regime, characterized by $h_\alpha$3, is especially significant. In that regime the cross-coefficient $h_\alpha$4 can change sign and can differ by over $h_\alpha$5 from its free-molecular limit. This indicates that the interplay between intermolecular collisions, surface boundary conditions, and curvature is not perturbative in the crossover between rarefied and more collisional transport.

7. Position within kinetic theory

Within the setting described here, the McCormack model functions as the “minimal, yet sufficiently rich collision-term approximation” for binary-mixture sublimation and deposition problems. The phrase reflects a precise collection of properties: exact conservation of mass, momentum, and energy; correct Chapman–Enskog transport coefficients for mixtures; an $h_\alpha$6-theorem; reproduction of cross-coupling effects such as diffusion and thermal diffusion; compatibility with both HS and realistic potentials; and numerical implementability in spherical geometry for computing mass and heat fluxes (Kalempa et al., 14 Jul 2025).

Its significance lies less in formal simplification than in selective fidelity. The model replaces the full collision integral, but it does so while preserving the species-resolved moments and coupling channels needed for Onsager reciprocity, thermal diffusion, and heat-of-sublimation effects. In the context of rarefied-gas mixture kinetics around curved interfaces, this makes it a transport-accurate surrogate for the full Boltzmann collision operator rather than a generic relaxation ansatz.