Particles suspended in a fluid are continuously bombarded by the surrounding fluid molecules. This constant bombardment results in a random motion of the particles known as Brownian motion. A satisfactory description of this irregular motion ("random walk") can be obtained ignoring the detailed structure of the particle-fluid molecule interaction if we assume that what happens to the aerosol fluid system at a given time t depends only on the system state at time t. Stochastic processes with this property are known as Markov processes.
In an effort to understand quantitatively Brownian motion, let us consider a particle that is settling in air owing to the action of gravity. As we have seen, the particle eventually reaches a terminal velocity that depends on the size of the particle and the viscosity of the air. A drag force is generated, depending on the velocity of the particle, that acts in a direction opposite to the direction of motion of the particle. If our particle is sufficiently large, say, 1 pm or larger, then the individual bombardment by microscopic gas molecules will have little effect on its motion that will be determined more or less solely by the continuum fluid drag force and gravity. However, if we consider a particle that is only a few nanometers, a size comparable to that of the gas molecules, then its motion will exhibit fluctuations from the random collisions that it experiences with gas molecules.
Let us consider a particle that is initially at the origin of our coordinate system. Assuming that the only force acting on the particles is that resulting from molecular bombardment by fluid molecules, the particle will start moving randomly from its original position and after time t will be at location ri = (.t|, Vi.z.\). If we repeat the same experiment with a second particle, we will find it at r2 = (x2, >'2 ■ z2) after the same period. Let us continue this experiment with an entire population or an ensemble of particles. If we average the displacements (r) of all these particles, we expect the average (r) to be zero since there is no preferred direction in Brownian motion. Can we then say anything quantitative about Brownian motion? We know that the average mean displacements (x), (>'), (z) of a particle ensemble will be zero, but this is not enough. We need a measure of the intensity of Brownian motion, something that will allow us to distinguish between particles moving slowly and randomly and particles moving rapidly and randomly. The traditional measure of such intensity is the mean square displacement of all particles (r2), or for the three directions (x2), (y2), and (z2). Note that these means cannot be zero, as averages of positive quantities. We expect that the higher the intensity of the motion, the larger the mean square displacements. Since the mean square displacement is an important descriptor of the Brownian motion process, let us see what we can learn about this quantity.
Equations (9.35) and (9.47) provide a convenient framework for the analysis of forces acting on particles. These equations simply state that the acceleration experienced by the particle is proportional to the sum of forces acting on the particle. We have used this equation so far for "deterministic" forces, namely, the gravity, drag, and electrical forces. We now need to use the stochastic Brownian force, which is simply the product of the particle mass mp and the random acceleration a caused by the bombardment by the fluid molecules. Then the equation of motion is mp^i = —cfv + m"a (9'52)
Dividing by mp, (9.52) becomes dx 1
= —-v + a (9.53) dt x where 1 is the relaxation time of the particle. The random acceleration a is a discontinuous term, since it represents the random force exerted by the suspending fluid molecules that imparts an irregular, jerky motion to the particle. The equation of motion written to include the Brownian motion has its roots in two worlds: the macroscopic world represented by the drag force and the microscopic world presented by the Brownian force. The decomposition of the equation of motion into continuous and discontinuous pieces in (9.53) is an ad hoc assumption that is intuitively appealing and more important leads to successful predictions of particle behavior. Equation (9.53) was first formulated by the French physicist, Paul Langevin, in 1908 and is referred to as the Langevin equation. This equation will be the starting point in our effort to calculate the mean square displacement (r2).
Let us begin by taking the dot product of r and (9.53):
dx 1
dt x
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Then ensemble averaging this equation (over many particles) gives r-g) = ->.v) + (r.a) (9.55)
Since we assume that there is no preferred direction in a (directional isotropy of collisions), (r • a) will be equal to zero, giving
Now since dt x or, equivalently
The term ^mp{v2} is the kinetic energy of the system and as energy is partitioned equally in all three directions, each with an energy of \kT for a total of \kT, we obtain that (v2) = 3kT/mp. Thus (9.59) becomes d , , 1 , . 3 kT ___.
Integrating this ordinary differential equation for (r • v) we find
3kTi
Now we note that so that (9.61) becomes
I dt mD
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We saw in Section 9.3 that for t» x, the particle velocity relaxes to a pseudo-steady-state value. We assume that to be the case here, namely, that the Brownian motion of the particle is sufficiently slow, that the particle has time to "relax" after each fluctuating impulse. Under this assumption, we drop the exponential in (9.63) to obtain
L at mp which, on integration, becomes
The Brownian motion can be assumed to be isotropic so (x2) = (y2) = (z2) = 5 (r2). Thus
This result, first derived by Einstein by a different route, has been confirmed experimentally. It indicates that the mean square distance traversed by a Brownian particle is proportional to the length of time it has experienced such motion.
We should note that we have obtained the foregoing results in a more or less formal manner without attempting to justify from a rigorous mathematical point of view the validity of the Langevin equation as the basic description of the particle motion. The theoretical results we have presented can be rigorously justified. A good starting point for the reader wishing to go deeply into its theory is the classic article by Chandrasekhar (1943), which is reprinted in Wax (1954).
The movement of particles due to Brownian motion can also be viewed as a macroscopic diffusion process. Let us discuss the connection between these two different perspectives on the same process. If N(x, y, z, t) is the number concentration of particles undergoing Brownian motion, then we can define a Brownian diffusivity D, such that
If we relate the Brownian diffusivity D to the mean square displacements given by (9.66), then (9.67) can provide a convenient framework for describing aerosol diffusion. To do so, let us repeat the experiment above, namely, let us follow the Brownian diffusion of N0 particles placed at t = 0 at the y — z plane. To simplify our discussion we assume that N does not depend on y or z. Multiplying (9.67) by x2 and integrating the resulting equation over x from — oo to oo, we get r+oo am r+oo a2
The LHS can also be written as
or after integration
We can now equate this result for (x2) with that of (9.66) to obtain an explicit relation forD
which, without the correction factor Cc, is the Stokes-Einstein-Sutherland relation. Note that for particles that are larger than the mean free path of air, Cc ~ 1 and their diffusivity varies as Dpl. As expected, larger particles diffuse more slowly. In the other extreme, when Dp < 1, Cc = 1 + 1.657(2/,/D;)) and D can be approximated by 2(1.657)XkT/ 3n\iD2p. Therefore, in the free molecule regime, D varies as Dp 2.
Diffusion coefficients for particles ranging from 0.001 to 10.0 pm diameter in air at 20°C are shown in Figure 9.8. The change from DJ2 dependence is indicated by the change of slope of the line of D versus Dp.
The importance of Brownian diffusion as compared to gravitational settling can be judged by comparing the distances a particle travels as a result of each process (Twomey 1977). Over a time of 1 s a 1-pm-radius particle diffuses a distance of about 4 pm, while it falls about 200 pm under gravity. A 0.1 pm radius particle, on the other hand, in 1 s, diffuses a distance of about 20 pm compared to a fall distance of 4 pm. Even though a 1 pm particle's motion is dominated by inertia and gravity, it still diffuses several times its own radius in 1 s. The motion of the 0.1 pm particle is dominated by Brownian
diffusion. For a 0.01-pm-radius particle, Brownian diffusion further outweighs gravity; its diffusive displacement in 1 second is almost 1000 times its displacement because of gravity.
Note that the magnitude of diffusion coefficients of gases is on the order of 0.1 cm2 s 1. Therefore a 0.1 pm particle diffuses in a quiescent gas roughly 10,000 times more slowly than a gas molecule, and Brownian diffusion is not expected to be an efficient transport mechanism for aerosols in the atmosphere.
In the development of Brownian motion up to this point, we have assumed that the only external force acting on the particle is the fluctuating Brownian force mpa. If we generalize (9.52) to include an external force Fext, we get d\ ffi m"di= Fext ~ v + mp&
As before, assuming that we are interested in times for which (>t, and taking mean values, the approximate force balance is at steady state:
The ensemble mean velocity (v) is identified as the drift velocity vdrift, where
The drift velocity is the mean velocity experienced by the particle population due to the presence of the external force Fe!lt. For example, in the case where the external force is simply gravity. FeIt — mpg. and the drift velocity (or settling velocity) will simply be Vdrift — gr [see also (9.41)]. When the external force is electrical, the drift velocity is the electrical migration velocity [see also (9.49)]. Therefore our analysis presented in the previous sections is still valid even after the introduction of Brownian motion.
It is customary to define the generalized particle mobility B by
Therefore the particle mobility is given by
The mobility can also be viewed as the drift velocity that would be attained by the particles under unit external force. Recall (9.50), which is the mobility in the special case of an electrical force. By definition, the electrical mobility is related to the particle mobility by Be = qB, where q is the particle charge. A particle with zero charge, has a mobility given by (9.78) and zero electrical mobility.
Finally, the Brownian diffusivity can be written in terms of the mobility [see also (9.73)] by
a result known as the Einstein relation.
Gravitational Settling and the Vertical Distribution of Aerosol Particles Let us consider the simultaneous Brownian diffusion and gravitational settling of particles above a surface at z = 0. At t = 0, a uniform concentration N0 = 1000 cm 3 of particles is assumed to exist for z > 0 and at all times the concentration of particles right at the surface is zero as a result of their removal at the surface.
1. What is the particle concentration as a function of height and time, N(z,t}'7
2. What is the removal rate of particles at the surface?
The concentration distribution of aerosol particles in a stagnant fluid in which the particles are subject to Brownian motion and in which there is a velocity v, in the — z direction is described by
5N dN a 2N
subject to the conditions
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