Brownian particle approximation

The first section gives a brief summary of some of the significant parameters in the study of the dynamics of ufp. The second section presents a survey of methods for description of the evolution of a ufp aerosol in size/composition, space, and time. It begins with consideration of the Brownian-particle approximation and its description in statistical physics. Thi.s is followed by a survey of the nucleation and coagulation processes. The third and final section deals with the description of transfer processes to single particles. Problems arising in specifying boundary conditions (accommodation coefficients) are covered. Two examples are discussed. 

For the Brownian-particle approximation, it is assumed that the aerosol system is stable over some experimental time. Particles are stable and their motions are uncorrelated. A special case in which the particles are treated as rigid bodies is frequently termed the Rayleigh gas. Of course, a characteristic feature of an aerosol is its inherent instability due to coagulation and deposition. Therefore, this approximation is limited to experimental times collision... 

In the Brownian-particle approximation, collisions between particles do not... 

One may then ask what applicable theory exists for the nonequilibrium evolution of a ufp aerosol system in the Brownian-particle approximation. One has still the restriction T t.. so that particle-particle collisions can be neglected. 

If the aerosol particles can be regarded as large molecular components in a dilute gas systan, then the evolution of the system can be studied with the well-developed kinetic theory of dilute gases. This requires in the Brownian-particle approximation that the particles doe not alter the state of the host gas and that changes in the state of the host gas over distances are negligible over distances of the order of the particle size. If one introduces the following restrictions ... 

In the Brownian-particle approximation, therefore, one has an exact description of the nonequilibrium evolution of the aerosol system within the limitations imposed by restrictions (2.24-29). Such systems are partially described by (2.6) with... 

In nonequilibrium systems one can postulate the separation between stochastic and mean values, although it may be very difficult to arrive at explicit descriptions. The situation is simplified somewhat by the introduction of several restrictions. In the discussion which follows, it is assumed that the Brownian-particle approximation holds. Also the additional restriction of quasistationarity is introduced for the transfer processes. A single spherical particle in this regime is characterized by the four parameters, Kn, Ma -, Sc., and Br. In addition, other parameters arise such as the accommodation coefficients specifying the transfer efficiencies between host gas and particle, and particle properties, including the thermal conductivity k. and viscosity... 

Even for the simplest problems in aerosol science, such as the approach of a particle to equilibrium in the Brownian-particle approximation, detailed analysis has required treating the particles as rigid spheres or some other axisymmetric shape. Internal states of the particle have been largely ignored. Indeed, for ultrafine particles such states cannot even be inferred from knowledge of the bulk material. 

If the Brownian particles were macroscopic in size, the solvent could be treated as a viscous continuum, and the particles would couple to the continuum solvent through appropriate boundary conditions. Then the two-particle friction may be calculated by solving the Navier-Stokes equations in the presence of the two fixed particles. The simplest approximation for hydrodynamic interactions is through the Oseen tensor [54],... 

The first paper that was devoted to the escape problem in the context of the kinetics of chemical reactions and that presented approximate, but complete, analytic results was the paper by Kramers [11]. Kramers considered the mechanism of the transition process as noise-assisted reaction and used the Fokker-Planck equation for the probability density of Brownian particles to obtain several approximate expressions for the desired transition rates. The main approach of the Kramers method is the assumption that the probability current over a potential barrier is small and thus constant. This condition is valid only if a potential barrier is sufficiently high in comparison with the noise intensity. For obtaining exact timescales and probability densities, it is necessary to solve the Fokker-Planck equation, which is the main difficulty of the problem of investigating diffusion transition processes. 

Let us consider a Brownian particle (B-particle) characterized by a mass Ma and coordinates Ra and ua. If this B-particle were so large that the fluid could be described by a hydrodynamic approximation (without fluctuations), its motion would be described by ... 

A serious difficulty now appears. The quantum master equation (3.14), obtained by eliminating the bath, does not have the required form (5.6) and therefore results in a violation of the positivity of ps(/). Only by the additional approximation rc Tm was it possible to arrive at (3.19), which does have that form (see the Exercise). The origin of the difficulty is that (3.14) is based on our assumed initial state (3.4), which expresses that system and bath are initially uncorrelated. This cannot be true at later times because the interaction inevitably builds up correlations between them. Hence it is unjustified to use the same derivation for arriving at a differential equation in time without invoking a repeated randomness assumption, such as embodied in tc rm. ) At any rate it is physically absurd to think that the study of the behavior of a Brownian particle requires the knowledge of an initial state. 

Before discussing other results it is informative to first consider some correlation and memory functions obtained from a few simple models of rotational and translational motion in liquids. One might expect a fluid molecule to behave in some respects like a Brownian particle. That is, its actual motion is very erratic due to the rapidly varying forces and torques that other molecules exert on it. To a first approximation its motion might then be governed by the Langevin equations for a Brownian particle 61... 

Electro- and magnetooptical phenomena in colloids and suspensions are widely used for structure and kinetics analysis of those media as well as practical applications in optoelectronics [143,144]. The basic theoretical model used to study optical anisotropy of the disperse systems is the noninteracting Brownian particle ensemble. In the frame of this general approximation, several special cases according to the actual type of particle polarization response to the applied field may be distinguished (1) particles with permanent dipole moments, (2) linearly polarizable particles, (3) nonlinearly polarizable particles, and (4) particles with hysteretic dipole moment reorientation. 

One can assume that each Brownian particle of the chain is situated in a similar environment, which is approximately correct for long chains, so that we can rewrite the memory functions in (3.6) and (3.7) as... 

The time dependence of the displacement of a macromolecule, shown in Fig. 9 as a function of the ratio t/r, is typical for diffusion of Brownian particle in viscoelastic fluid (Zanten and Rufener 2000 Zanten et al. 2004). The function (5.5) for big values of B can be approximated as... 

As was demonstrated by Pyshnograi (1994), the last term in (6.7) can be written in symmetric form, if the continuum of Brownian particles is considered incompressible. In equation (6.7), the sum is evaluated over the particles in a given macromolecule. The monomolecular approximation ensures that the stress tensor of the system is the sum of the contributions of all the macromolecules. In this form, the expression for the stresses is valid for any dynamics of the chain. One can consider the system to be a dilute polymer solution or a concentrated solution and melt of polymers. In any case the system is considered as a suspension of interacting Brownian particles. 

The assumption of fast attainment of equilibrium for the off-diagonal elements makes it possible for us to set dp12(t)/dt = 0 in Eq. (8) and dp2 (t)/dt = 0 in Eq. (9). This is the famous Smoluchowski approximation [9]. The off-diagonal density elements play the same role as the velocity of the Brownian particle [9], and their time derivatives are assumed to vanish. This allows us to express pj 2 and p2 as a function of the diagonal density matrix elements,... 

As a result of AEP, the initial system of the set of Eqs. (81) is reduced to the equation describing the diffusional motion of a Brownian particle which undergoes the action of an additive and a multiplicative noise (with intensities D and Q, respectively) in the presence of a renormalized bounding potential, Eq. (90). The Markovian l t corresponds to X 00. If we take such a limit at a ed value of y, = d, and the case studied by Htoggi is recovered. Of course, having neglected the condition X c y we have reduced the problem to a trivial diffusional (lowest-order) approximation. 

The above formulae were experimentally checked by Perrin in 1908. He measured the distances traversed by Brownian particles for equal periods of time with a microscope. Based on his experiments and formulae (483) and (484), Perrin was able to define the Boltzmann constant, k, and calculated the value of Avogadro s number N., both closely approximating their values obtained by other methods. 

In the present section, it is demonstrated how the linear response of an assembly of noninteracting polar Brownian particles to a small external field F applied parallel and perpendicular to the bias field Fo may be calculated in the context of the fractional noninertial rotational diffusion in the same manner as normal rotational diffusion [8]. In order to carry out the calculation, it is assumed that the rotational Brownian motion of a particle may be described by a fractional noninertial Fokker-Planck (Smoluchowski) equation, in which the inertial effects are neglected. Both exact and approximate solutions of this equation are presented. We shall demonstrate that the characteristic times of the normal diffusion process, namely, the integral and effective relaxation times obtained in Refs. 8, 65, and 67, allow one to evaluate the dielectric response for anomalous diffusion. Moreover, these characteristic times yield a simple analytical equation for the complex dielectric susceptibility tensor describing the anomalous relaxation of the system. The exact solution of the problem reduces to the solution of the infinite hierarchies of differential-recurrence equations for the corresponding relaxation functions. The longitudinal and transverse components of the susceptibility tensor may be calculated exactly from the Laplace transform of these relaxation functions using linear response theory [72]. 

For this model, the Brownian-particle regime (as defined by the applicability of the Fokker-Planck equation) exists to good approximation for (m /m ) < 0.001 with the limitation x T. g. ... 

The value of these approaches to the problem is that they demonstrate the possibility of evaluating the memory function through the intermolecular correlation functions and structural dynamic factor of the system of interacting Brownian particles. Theoretical evaluations of the memory function are based on some approximations which, apparently, do not allow the calculation of quantities for the limiting case c M, but they are good in the cross-over region from Rouse to entanglement dynamics. 

The greatest obstacle for developing a hydrodynamic DDFT is that, the viscosity of a liquid is determined by the deviation of the two-point correlation function g r,r t) from the equilibrium value. geq( )- Approximating the first with the latter, as in the derivation of the DDFT for Brownian particles discussed above, would therefore result in a theory almost without viscosity. 

In Poiseuille flow, rigid, spherical, non-Brownian particles are subjected to lateral forces that result in migration to an equilibrium radial region located at approximately 60% of the distance from the tube axis to the tube wall, which... 

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