(591e) Transport Properties for Lennard-Jones Particles

Classical kinetic theory^1-6 provides a reliable method to obtain transport properties of multicomponent dilute gases that saves time and cost compared to experiments. In the theory, transport coefficients, including thermal conductivity, viscosity, and diffusivity, are given as algebraic functions of the collision integral Ω_ij^(l,s) arising from Chapman and Enskog's solution of the Boltzmann gas equation. For example, binary diffusion coefficients can be evaluated by D_ij = 3kT/(16nμ_ijΩ_ij^(1,1)). The particles are modeled as a group of colliding pairs, which obey the classical momentum and energy conservation laws with the following parameters: (1) the separation distance r, which is initially infinity; (2) the distance of closest approach r_m; (3) the initial relative speed g, which follows a Maxwell-Boltzmann distribution; (4) the Lennard-Jones (6-12) potential³ φ(r), which embeds the characteristics of gas molecules; and (5) the impact parameter b, which is r_m when φ(r) is zero.

For the evaluation of transport properties, the following three factors should be calculated in consecutive order: (1) the deflection angle χ(b,g); (2) the reduced collision cross section Q^(l)(g); and (3) the collision integral Ω_ij^(l,s). First, the deflection angle is given as χ(b,g) = π - 2θ(r_m), where

.               (1)

Particle trajectories during a collision event can be drawn with Eq. 1. Integration of χ(b,g) is difficult, however, because the integrand includes singularities. These can be removed analytically, allowing calculation of the deflection angle χ to high accuracy.

The trajectory map shown in Fig. 1 arises from a contour plot of the deflection angle χ(b,g) in terms of dimensionless b and g values. Four lines, demarcating deflections (χ = 0), loops (χ = π), orbits  (χ = -∞), and inflections, divide the map into six regions. Representative trajectories are also included in each region, showing how Fig. 1 helps to understand the topology of binary Lennard-Jones interactions.

The trajectory map informs the numerical procedures used when integrating Eq. 1. By comparing the grid independent χ value to the recent previously reported value⁶, we could see that our computer program is working properly as well as there are critical regions on the map. In regions IV, V and VI, to estimate χ with the same accuracy as other regions, the number of mesh points should be increased to guarantee the accuracy of the collision integral Ω_ij^(l,s).

The ongoing process is to evaluate the reduced cross section Q^(l)(g), which is necessary to obtain Ω_ij^(l,s), and ultimately the transport coefficients. The reliability will be examined by comparing the numerical results with previously reported experimental values. Once the code is successfully created, a variety of transport coefficients can be evaluated using simple algebraic relations.

References

[1] S. Chapman and T. Cowling, 2^nd Ed. Cambridge, 1960.

[2] J. Hirschfelder, R. Bird, E. Spotz, J. Chem. Phys. 16 (1948) 968-981.

[3] J. Hirschfelder, C. Curtiss, R. Bird, Wiley, New York, 1954.

[4] E. Akhmatskaya and L. Pozhar, USSR Comput. Maths. Math. Phys. 26(2) (1986) 185-190.

[5] U. Storck, ZAMM 78 8 (1998) 555-563.

[6] F. Sharipov and G. Bertoldo, J. Comp. Phys. 228 (2009) 3345-3357.