Boltzmann equation, molecular collision

The assumption that the probability of simultaneous occurrence of two particles, of velocities vt and v2 in a differential space volume around r, is equal to the product of the probabilities of their occurrence individually in this volume, is known as the assumption of molecular chaos. In a dense gas, there would be collisions in rapid succession among particles in any small region of the gas the velocity of any one particle would be expected to become closely related to the velocity of its neighboring particles. These effects of correlation are assumed to be absent in the derivation of the Boltzmann equation since mean free paths in a rarefied gas are of the order of 10 5 cm, particles that interact in a collision have come from quite different regions of gas, and would be expected not to interact again with each other over a time involving many collisions. 

Moreover, since the mean free path is of the order of 100 times the molecular diameter, i.e., the range of force for a collision, collisions involving three or more particles are sufficiently rare to be neglected. This binary collision assumption (as well as the molecular chaos assumption) becomes better as the number density of the gas is decreased. Since these assumptions are increasingly valid as the particles spend a larger percentage of time out of the influence of another particle, one may expect that ideal gas behavior may be closely related to the consequences of the Boltzmann equation. This will be seen to be correct in the results of the approximation schemes used to solve the equation. 

However, the intermolecular force laws play a central role in the model determining the molecular interaction terms (i.e., related to the collision term on the RHS of the Boltzmann equation). Classical kinetic theory proceeds on the assumption that this law has been separately established, either empirically or from quantum theory. The force of interaction between two molecules is related to the potential energy as expressed by... 

Preliminarily, deriving the terms on the LHS of the Boltzmann equation, we assume that the effects of collisions are negligible. The molecular motion is thus purely translational. We further assume that in an average sense each molecule of mass m is subjected to an external forces per unit mass, F(r,t), which doesn t depend on the molecular velocity c. This restriction excludes magnetic forces, while the gravity and electric fields which are more common in chemical and metallurgical reactors are retained. 

Therefore, in the limit of no molecular interactions, for which the collision term ( )coiiision vanishes, the Boltzmann equation yields... 

This assumption is difficult to justify because it introduces statistical arguments into a problem that is in principle purely mechanical [85]. Criticism against the Boltzmann equation was raised in the past related to this problem. Nowadays it is apparently accepted that the molecular chaos assumption is needed only for the molecules that are going to collide. After the collision the scattered particles are of course strongly correlated, but this is considered irrelevant for the calculation since the colliding molecules come from different regions of space and have met in their past history other particles and are therefore entirely uncorrelated. 

The conservation equations (2.202), (2.207) and (2.213) are rigorous (i.e., for mono-atomic gases) consequences of the Boltzmann equation (2.185). It is important to note that we have derived the governing conservation equations without knowing the exact form of the collision term, the only requirement is that we are considering summation invariant properties of mono-atomic gases. That is, we are considering properties that are conserved in molecular collisions. 

The starting point for the kinetic theory of dilute mono-atomic gases is the Boltzmann equation determining the evolution of the distribution function in time and space. The formulation of the collision term is restricted to gases that are sufficiently dilute so that only binary collisions need to be taken into account. It is also required that the molecular dimensions are small in comparison with the mean distance between the molecules, hence the transfer of molecular properties is solely regarded as a consequence of the free motion of molecules between collisions. 

The Cooper-Mann theory of monolayer transport was based on the model of a sharply localized interfacial region in which ellipsoidal molecules were constrained to move. The surfactant molecules were assumed to be massive compared with the solvent molecules that made up the substrate and a proportionate part of the interfacial region. It was assumed that the surfactant molecules had many collisions with solvent molecules for each collision between surfactant molecules. A Boltzmann equation for the singlet distribution function of the surfactant molecules was proposed in which the interactions between the massive surfactant molecules and the substrate molecules were included in a Fokker-Planck term that involved a friction coefficient. This two-dimensional Boltzmann equation was solved using the documented techniques of kinetic theory. Surface viscosities were then calculated as a function of the relevant molecular parameters of the surfactant and the friction coefficient. Clearly the formalism considers the effect of collisions on the momentum transport of the surfactant molecules. 

The construction of Cooper and Mann (7) for the surface viscosity includes the substrate effect by a model that represents the result of very frequent molecular collisions between the small substrate molecules and the larger molecules of the monolayer. This was done by adding a term to the Boltzmann equation for the 2D singlet distribution function that is equivalent to the friction coefficient term of the Fokker-Planck equation from which Equations 24 and 25 can be constructed. Thus a Brownian motion aspect was introduced into the kinetic theory of surface viscosity. It would be interesting to derive the collision frequency of Equation 19 using the better model (7) and observe how the T/rj variable of Equation 26 emerges. 

The Langmuir isotherm equation can also be derived from the formal adsorption and desorption rate equations derived from chemical reaction kinetics. In Section 3.2.2, we see that the mass of molecules that strikes 1 m2 in one second can be calculated using Equation (186), by applying the kinetic theory of gases as [dmldt = P2 (MJ2nRT)m], where P2 is the vapor pressure of the gas in (Pa), Mw is the molecular mass in (kg mol ), T is the absolute temperature in Kelvin, R is the gas constant 8.3144 (nT3 Pa mol-K-1). If we consider the mass of a single molecule, mw (kg molecule-1), (m = Nmw), where N is the number of molecules, by considering the fact that (R = kNA), where k is the Boltzmann constant, and (Mw = NAmw), we can calculate the molecular collision rate per unit area (lm2) from Equation (186) so that... 

The extension of the kinetic theory approach to include large values of a (and hence large deviations from equilibrium) requires higher order perturbations for the solution of the Boltzmann equation. It is probably unprofitable to proceed in this difficult and laborious direction until one understands the detailed analytical dependence of the transition probability a on the mechanism of molecular energy exchange and redistribution on collision. Currently available information on intermolecular forces is insufficient to establish this dependence. 

In a real material, the tendency of dipoles to align under the influence of the field is counteracted by molecular collisions (Brownian motion), which disrupt order. The Boltzmann equation of statistical mechanics provides a simple method by which the average dipole moment Ji in the direction of the electric field can be evaluated in this situation ... 

In the transition regime, the rarefaction effects dominate and the intermolecular collisions need to be taken into account. For the free-molecular flow, intermolecular collisions can be considered negligible when compared to the probability of the molecule colliding with the wall surface. As the flow enters the transition flow regime and continues into the free-molecular flow regime, the Kn becomes significant enough that the molecular approach has to be utilized. Thus, the Boltzmann equation... 

The Boltzmann equation must be solved with appropriate boundary conditions to obtain f and f. The full Boltzmann equation has not been solved analytically or numerically. Current approximate methods for extracting the desired information from the Boltzmann equation are covered in detail in a recent reference [2.84]. In view of this review, discussion of these methods will not be given. It is sufficient to indicate some of the principal methods which have been employed. These are moment or integral methods for specific molecular scattering laws, the use of "models" (of which the BGK model is the simplest) for the collisions term J(fgfg), and direct simulation by Monte Carlo or molecular-dynamics techniques. 

The principal difficulty in solving the Boltzmann equation lies in the analytically intractable collision term. For small disturbances from equilibrium, the collision term may be linearized. Another approach is the calculation of transfer processes about a particle using a relaxation model for the collision term. It would be expected that such models would be most successful in near-free-molecular conditions where the "free-streaming" terms are much more important than collisions between host-gas molecules. The so-called BGK model is perhaps the most widely applied of these models [2.5,6]. 

In the transition and free-molecular regimes, the difficulty in describing effective aerosol interaction forces lies ultimately in the intractability of the Boltzmann (or other appropriate) kinetic equation to exact solution. In the case of two transition-regime spheres, with absolutely no interaction potential, an effective attractive force arises because the zone of isotropic gas molecular collisions for each particle is truncated by the presence of the other particle. It is this effective interaction force which the dividing-sphere method approximates by assuming complete absorption for distances less than some distance defined for each pair of spheres regardless of their composition. 

Consider a dense gas of hard spheres, all with mass m and diameter a. Since the collisions of hard-sphere molecules are instantaneous, the probability is zero that any particle will collide with more than one particle at a time. Hence we still suppose that the dynamical events taking place in the gas are made up of binary collisions, and that to derive an equation for the single-particle distribution function /(r, v, /) we need only take binary collisions into account. However, the Stosszahlansatz used in deriving the Boltzmann equation for a dilute gas should be modified to take into account any spatial and velocity correlations that may exist between the colliding spheres. The Enskog theory continues to ignore the possibility of correlations in the velocities before collision, but attempts to take into account the spatial correlations. In addition, the Enskog theory takes into account the variation of the distribution function over distances of the order of the molecular diameter, which also leads to corrections to the Boltzmann equation. 

Otherwise the length of the collision cylinders must be taken into account. tThe variation of f over a molecular size provides corrections to the Boltzmann equation roughly of order ajl. At low densities these corrections are of order na, and at high densities a/lc 1, so these corrections are not insignificant. 

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