(466b) A New Equation for the Mean Free Path of Air.

The mean free path of air, λ, is a fundamental quantity in aerosol science. Through kinetic theory, it enters established mathematical expressions defining the macroscopic properties of air (Jeans, 1967). The mean free path of air further dictates if particle transport occurs in the free molecular, transient, or continuum regimes (Fuchs, 1964). Distinctly different theories of particle dynamics hold in the three regimes, therefore, an accurate λ is essential for correctly assessing the relevant transport regimes in aerosol processes.

Unfortunately, measuring directly the mean free path of air is not feasible. However, λ can be indirectly estimated using kinetic theory equations that relate it to transport properties (e.g., viscosity) (Jeans, 1967). To overcome uncertainties in the calculation of λ, resulting from approximations in the corresponding theoretical treatments, one can resort to detailed atomistic simulations where the atomic structure is maintained, and accurate force fields are available. Recently, such an approach was introduced to compute λ at 300 K and 1 atm by combining explicit-atom molecular dynamics (MD) simulations with a detailed collision algorithm capable of identifying molecular collisions and computing λ (Tsalikis et al., 2023). The calculated λ was 43% smaller than the currently known and widely used value of 67.3 nm at 300 K and 1 atm. This result was independent of the employed force field, if this could reproduce the macroscopic values of air density, diffusivity and viscosity. Here, the above methodology is applied to a vast range of temperatures and pressures, namely from 0.5 to 5 atm and from 100 to 3000 K, to cover the entire regime relevant to atmospheric and industrial aerosol processes.

The MD simulations were performed with large simulation cells containing 50000 air molecules (79 mol % N₂ and 21 mol % O₂) utilizing the atomistic force field by Zambrano et al (Zambrano et al., 2014). All production runs were conducted in the microcanonical (NVE) ensemble, to avoid perturbations of the atomistic trajectories mainly by the barostat and, to a lesser extent, by the thermostat (Tsalikis et al., 2023). Initial configurations for the NVE simulations were generated after long simulation in the isothermal-isobaric (NPT) ensemble, at the T and P conditions of interest. The NVE MD runs were conducted at density and energy conditions corresponding to temperatures from 100 to 3000 K and pressures from 0.5 to 5 atm. To ensure the reproducibility of the MD results, simulations were repeated three times, utilizing initial configurations with entirely different atomic positions and velocities. In all cases, fully consistent results were obtained.

The force field was exhaustively validated by comparing the MD-extracted air density, diffusivity, and viscosity values against experimental measurements and theoretical expressions over the broad temperature range of 100 – 3000 K. In all cases, a solid agreement was found.

The MD-calculated λ were systematically smaller than those predicted by the classic kinetic theory and its variants (Bird, 1994). The temperature scaling (at constant pressure) of the MD-computed λ values was almost identical to that of the established expression of Jennings (Jennings, 1998) and agreed reasonably well with those suggested by kinetic theory variants. On the other hand, all models agreed with the MD calculations concerning the corresponding pressure scaling.

The MD-extracted λ were mapped into a simple equation (Tsalikis et al., 2024) of the form:

λ(T, P) = b T^c P^d (1)

where b, c, and d, are numerical constants. The equation describes well the T and P dependence of the MD-computed λ, as shown in Figure 1a below. The corresponding determination coefficient is R² = 99.97%, implying a very accurate description of the MD data. In particular, the mean and maximum deviations between MD-measured values and predictions from Equation (1) being equal to 1.8 and 4.3 %, respectively, in the region 200 K < T ≤ 3000 K for all pressures examined. The MD-derived λ reported in Figure 1a (symbols) span almost uniformly the (T, P) surface of Equation (1), suggesting that the new equation provides a reliable description of the raw data independent of the specific (T, P) pair in the range of phase state points covered.

The MD-data were also expressed in terms of the Jennings equation. The latter is an established expression relating the mean free path of air with experimental observables and is given by

λ = (π/8)^1/2×(μ/u) ×(ρP)^-1/2 (2)

where μ, and ρ are the air viscosity and density respectively, whereas parameter u is a numerical prefactor, equal to u = 0.4987445, derived assuming gas molecules as rigid elastic spheres (Pekeris and Alterman, 1957). To best fit the MD-extracted λ with Equation (2), the prefactor u is allowed to vary. In Figure 1b, the simulation results for λ are reported (symbols) as a function of air viscosity, μ and ρP. The latter term, ρ and P are grouped to allow the graphical representation of λ as a two-dimensional surface in 3-D space (Figure 1b). It is evident that, the updated Jennings expression with a new and higher value for u, can describe very accurately the simulation results for λ over the entire range of μ and ρP conditions examined (Tsalikis et al., 2024). This is reflected in the corresponding determination coefficient R² being equal to R² = 99.92%.

References

G.A. Bird, Molecular gas dynamics and the direct simulation of gas flows (Clarendon Press, Oxford, 1994).

N.A. Fuchs, The mechanics of aerosols, (Dover Publications Inc, New York, 1964).

J. Jeans, An introduction to the kinetic theory of gases, (Cambridge University Press, Cambridge, 1967).

S.G. Jennings, The mean free path in air, J. Aerosol Sci. 19 (1998) 159-166, doi.org/ 10.1016/0021-8502(88)90219-4.

C.L. Pekeris, Z. Alterman, Solution of the Boltzmann-Hilbert integral equation II. The coefficients of viscosity and heat conduction, Proc. Natl. Acad. Sci. U.S.A. 43 (1957) 998-1007, doi.org/ 10.1073/pnas.43.11.998.

D.G. Tsalikis, V.G. Mavrantzas, S.E. Pratsinis, Dynamics of molecular collisions in air and its mean free path, Phys. Fluids. 35, (2023) 097131, doi.org/10.1063/5.0166283.

D.G. Tsalikis, V.G. Mavrantzas, S.E. Pratsinis, A new equation for the mean free path of air, Aerosol Sci. Tech. (2024) In press.

H. Zambrano, J.H. Walther, R. Jaffe, Molecular dynamics simulations of water on a hydrophilic silica surface at high air pressures. J. Mol. Liq. 198 (2014) 107-113, doi.org/ 10.1016/j.molliq.2014.06.003.