Effect of Brownian Motion

The deviation between theory and experiment has been ascribed to the effects of Brownian motion. Recall that LSW theory assumes that the droplets are fixed in space and that molecular diffusion is the only mechanism of mass transfer. In the case of moving particles, the contributions of molecular and convective diffusion are related by the Peclet nitmber, Pe, viz  

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The viscosity of a suspension of ellipsoids depends on the orientation of the particle with respect to the flow streamlines. The ellipsoidal particle causes more disruption of the flow when it is perpendicular to the streamlines than when it is aligned with them the viscosity in the former case is greater than in the latter. For small particles the randomizing effect of Brownian motion is assumed to override any tendency to assume a preferred orientation in the flow. 

From the image sequences, information on the velocities of nano-particles can be extracted. The statistical effect of Brownian motion on the flowing speed of the mixed liquid is found small enough to be ignored as shown in Fig. 37 where most of the particles trajectories in the liquid are straight lines and parallel with the wall basically. Therefore, Brownian diffusive motion is ignorable. 

Particle collision frequency due to Brownian motion was estimated to be less than 1% of the collision frequency due to shear. The effects of Brownian motion could therefore be neglected in the flocculation rate calculations. However, for the smallest molecular size, radius of gyration 14 nm (see Table I), the effect of Brownian motion on the particle-polymer collision efficiency was of the same order of magnitude as the effect of shear. These two contributions were assumed to be additive in the adsorption rate calculations. Additivity is not fundamentally justified (23) but can be used as an interpolating... 

The difference between the theoretical value of the emission anisotropy in the absence of motions (fundamental anisotropy) and the experimental value (limiting anisotropy) deserves particular attention. The limiting anisotropy can be determined either by steady-state measurements in a rigid medium (in order to avoid the effects of Brownian motion), or time-resolved measurements by taking the value of the emission anisotropy at time zero, because the instantaneous anisotropy can be written in the following form ... 

Applications of optical methods to study dilute colloidal dispersions subject to flow were pioneered by Mason and coworkers. These authors used simple turbidity measurements to follow the orientation dynamics of ellipsoidal particles during transient shear flow experiments [175,176], In addition, the superposition of shear and electric fields were studied. The goal of this work was to verify the predictions of theories predicting the orientation distributions of prolate and oblate particles, such as that discussed in section 7.2.I.2. This simple technique clearly demonstrated the phenomena of particle rotations within Jeffery orbits, as well as the effects of Brownian motion and particle size distributions. The method employed a parallel plate flow cell with the light sent down the velocity gradient axis. 

In most electroosmotic flows in microchannels, the flow rates are very small (e.g., 0.1 pL/min.) and the size of the microchannels is very small (e.g., 10 100 jm), it is extremely difficult to measure directly the flow rate or velocity of the electroosmotic flow in microchannels. To study liquid flow in microchannels, various microflow visualization methods have evolved. Micro particle image velocimetry (microPIV) is a method that was adapted from well-developed PIV techniques for flows in macro-sized systems [18-22]. In the microPIV technique, the fluid motion is inferred from the motion of sub-micron tracer particles. To eliminate the effect of Brownian motion, temporal or spatial averaging must be employed. Particle affinities for other particles, channel walls, and free surfaces must also be considered. In electrokinetic flows, the electrophoretic motion of the tracer particles (relative to the bulk flow) is an additional consideration that must be taken. These are the disadvantages of the microPIV technique. 

Figure 11 shows the relative-viscosity-concentration behavior for a variety of hard-sphere suspensions of uniform-size glass beads. Even though the particle size was varied substantially (0.1 to 440 xm), the relative viscosity is independent of the particle size. However, when the particle diameter was small ( 1 fJLm), the relative viscosity was calculated at high shear rates, so that the effect of Brownian motion was negligible. Figure 8 shows that becomes independent of the particle size at high shear stress (or shear rate). 

A mathematical model of a nanoparticles growth during evaporation of a micron size droplet in a low pressure aerosol reactor is developed. The main factor is found to be evaporating cooling of droplets which affects formation of supersaturated solution in the droplet. The rate of cooling can reach 2T0 K/s. The final radius of nanoparticles was found to be independent on the precursor radius. Manifestation of Lifshitz-Slezov instability is illustrated by experimental data. Effects of Brownian motion of nanoparticles inside the droplet are discussed. 

Figure 7,5 Sliear coagulation of a monodisperse latex dispersion. Straight lines were obtained for differing shear rates in accordance with a simple theory assuming additivity of the effects of Brownian motion and shear. The points are experimental results (Swift and Friedlander, 1964).

Quemada (1978a, 1978b) examined the rheology and modelling of concentrated dispersions and described simple viscosity models that incorporate the effects of shear rate and concentration of filler and separate effects of Brownian motion (or aggregation at low shear) and particle orientation and deformation (at high shear). The ratio of structure-build-up and -breakdown rates is an important parameter that is influenced by the ratio of the shear rate to the particle diffusion. A simple form of viscosity relation is given here ... 

In practice, it is difficult to measure the DEP force due to the effects of Brownian motion and electrical field-induced fluid flow [3]. Instead, the DEP crossover frequency can be measured as a function of medium conductivity and provides sufficient information to determine the dielectric properties of the suspended particles. The DEP crossover frequency,is the transition frequency point where the DEP force switches from pDEP to nDEP or vice versa. According to Eq. (6), the crossover frequency is defined to be the frequency point where the real part of the Clausius-Mossotti factor equals zero ... 

With surface forces absent, in the limit of Pe l, the distribution of particles is only slightly altered from the Einstein limit. To order cj> which takes into account two-particle interactions, Batchelor (1977) calculated the effect of Brownian motion on the stress field in a suspension of hard spheres and determined the low shear limit relative viscosity to be given by the Einstein relation with an added term equal to 6.14>. This result is found to agree satisfactorily with experiment for shear limit with interparticle surface forces, including questions as to the existence of a uniquely defined asymptotic limit, we choose not to discuss this case further, instead referring the reader to Russel et al. (1989) and van de Ven (1989). 

BATCHELOR, G.K. 1977. The effect of Brownian motion on the bulk stress in a suspension of spherical particles. ]. Fluid Mech. 83, 97—117. 

Brownian forces are important for suspended particles smaller than 1000 run, which includes most latexes. However, because intetparticle forces also become important in this range, it is difficult to see the effects of Brownian motion... 

Jeffrey [1923] extended Einstein s analysis to flow around an impermeable, rigid ellipsoid of revolution, and Simha [1940] further incorporated the effect of Brownian motion, deriving an equation of the form... 

Figure 5 Effect of Brownian motion on the measurement of the DSD in a sediment. Small droplets take part in chaotic thermal motion in a direction normal to the sediment plane. The histogram shows that a significant part of the smaller droplets are withdrawn from the DSD as measured in the sediment 67% of the droplets measured in the 1-pm class were not foimd within the sediment.

The bombardment of suspended water droplets, by molecules in the surrounding oil phase, will impart forces on the droplets causing them to move (Brownian motion). Small droplets are more susceptible to this effect than larger ones and this may result in collisions between neighboring droplets. Friedlander and Wang (42) investigated the effect of Brownian motion on dispersions, and the droplet size distribution was found to be self-preserving. In a CEC, flic dynamic forces created by laminar and... 

Olsen and Adrian [16] performed a theoretical smdy of the effect of Brownian motion on the pPIV correlation signal peak and derived a function quantifying the broadening of the correlation function. They postulated that this widthwise broadening of the correlation function could be used to calculate the temperature of the fluid, since Brownian motion has a direct... 

The phenomena in which the effect of Brownian motion afq>ears most clearly is diffusion small particles placed at a certain point will spread out... 

In eqn (7.234) the effect of Brownian motion was not taken into account. If this is included, the final equation becomes... 

Even though there are objections to the theoretical model (hyphae treated as flexible chains forming spherical coils network interaction between branched hyphae randomizing effect of Brownian motion essential for validity of polymer rheology theory), this formal approach is quite useful (Metz et al., 1979). 

It is obvious that this Brownian error establishes a lower limit concerning the measurement time interval At. For shorter times, the measurements are dominated by uncorrelated Brownian motion. Brownian motion becomes an important factor when tracing 50-500 nm particles in flow field experiments with flow velocities of less than about 1 mm s . For a velocity of 0.5 mm s and a seed particle diameter dp = 500 nm, the lower limit is At 100 ps for an error of 20% due to Brownian motion. It is possible to reduce this error by averaging over several particles in a single interrogation spot and by ensemble averaging over several realizations [3]. On the other hand. Equation (4.15) shows that the effect of Brownian motion is relatively less important for faster flows. 

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