Brownian collision rate

Brownian collisions also occur during shear aggregation. The collision rate due to shear is more important than the Brownian collision rate of particles for shear aggregation. This condition occurs for particles larger than a critical size, given by [87]... 

Particle sizes larger than those given are those where the shear collision rate is dominant over Brownian collision rate. 

Brownian collision rate [m s ] shear collision rate [m s ] fast agglomeration rate [s ] slow agglomeration rate [s ] mean surface roughness [m] max surface roughness [m] surface roughness parameter maximum radius of curvature [m] minimum radius of curvature [m] substrate radius [m]... 

Assuming an infinite repulsion potential, the particles would be stable for ever however, since in reality, repulsion potentials are finite there is always the probability of particle aggregation due to thermal fluctuations. The rate of particle coagulation is a function of the frequency of particles encounters, and of the probability of coagulation at this state [65]. Without repulsion coagulation will proceed very rapidly, even in fairly dilute dispersions, with the particles aggregating at the same rate at which they become encountered, by diffusion through the continuous phase. This rate is termed the Brownian collision rate or the... 

In general, particles of radius RSy2 will also be executing random Brownian motion (i.e., diffusion). In such a case, Dx should be replaced by Dn = (Dx + D2). The collision rate Z12 (where the second subscript now reminds us that particle 2 is also executing diffusive motion) is then... 

Neutral molecules, dissolved, dispersed or suspended in a liquid medium, are in continuous random motion (Brownian motion) with a mean free path (x) and collision diameter (xe), depending on c and vex effects. At a far separation distance, is negative, increasing to 0 at xe, where repulsion counterbalances attraction and the amphiphiles are at dynamic equilibrium in a primary minimum energy state. At x High concentrations shorten x and make the collision rate nonlinear with c, (Hammett, 1952). A separation distance of x < xe is sterically forbidden without fusion. 

Even relatively weak attraction between droplets or solid particles in aerosols suffices to create an enhanced collision rate that can change particle-size distributions and overall stability. Think in kT thermal-energy units. Alone, small suspended bodies do a Brownian bop, randomly jiggling from the kT kicks of the air. Should their mutually random paths bring two particles to separations comparable with their size, their van der Waals attraction energy also approaches kT. To previous randomness, attraction adds strength of purpose and increased chance of collision, aggregation, or fusion.59... 

These trajectory methods have been used by numerous researchers to further investigate the influence of hydrodynamic forces, in combination with other colloidal forces, on collision rates and efficiencies. Han and Lawler [3] continued the work of Adler [4] by considering the role of hydrodynamics in hindering collisions between unequal-size spheres in Brownian motion and differential settling (with van der Waals attraction but without electrostatic repulsion). The results indicate the potential significance of these interactions on collision efficiencies that can be expected in experimental systems. For example, collision efficiency for Brownian motion will vary between 0.4 and 1.0, depending on particle absolute size and the size ratio of the two interacting particles. For differential... 

Howarth (HI5) assumed a mechanism analogous to Brownian motion. But his derivation for collision rates is wrong because he introduced a variable particle diffusion coefficient in the Smoluchowski equation (06, S27) which had been derived on the basis of a constant coefficient. 

Brownian motion also contributes insigniflcantly to the collision rates due to large drop or particle sizes in actual industrial operations. 

The theory of the Brownian motion, which we have described, is distinguished by a characteristic feature—namely, the concept of a collision rate—which is the inverse of the time interval between successive collision events of the Brownian particle with its surroundings we recall the words of Einstein [2],... 

Before proceeding, we remark that just as in the Einstein theory of the Brownian motion of which the Debye theory is a rotational version, the characteristic microscopic time scale is a time interval t so long that the motion of the particle at time t is independent of its motion at time t t, but small compared to the observation time intervals [17]. It is also supposed that during the time t, which is the mean of the time intervals between collision events, any external nonstochastic forces which may be applied to the system do not alter. In the fractal waiting time picture, however, the concept of collision rate does not hold and the mean of the time intervals between collision events diverges. 

In water, this ratio is 1.2x10" y for 50 nm particles and 0.12 y for 500 nm particles. This means that shear rates due to pouring a suspension are already sufficient for shear rate collisions to dominate in 500 nm particle suspensions, while in heavily stirred sols of 50 nm particles Brownian collisions between particles still dominate. 

Perikinetic, where agglomeration takes place as a result of Brownian motion. The following equation was proposed for determining the collision rate of particles (78)... 

The collision rate is initially extremely fast (actually it starts at infinity) but for t 4Rp/nD, it approaches a steady-state value of /coi = 8nRp DNq. Physically, at t — 0, other particles in the vicinity of the absorbing one collide with it, immediately resulting in a mathematically infinite collision rate. However, these particles are soon absorbed by the stationary particle and the concentration profile around our particle relaxes to its steady-state profile with a steady-state collision rate. One can easily calculate, given the Brownian diffusivities in Table 9.5, that such a system reaches steady state in 10-4 s for particles of diameter 0.1 pm and in roughly 0.1 s for 1 pm particles. Therefore neglecting the transition to this steady state is a good assumption for atmospheric applications. 

The first pubUshed criticism of the binary collision model was due to Fixman he retained the approximation that the relaxation rate is the product of a collision rate and a transition probabihty, but argued that the transition probability should be density dependent due to the interactions of the colliding pair with surrounding molecules. He took the force on the relaxing molecule to be the sum of the force from the neighbor with which it is undergoing a hard binary collision, and a random force mA t). This latter force was taken to be the random force of Brownian motion theory, with a delta-function time correlation ... 

The collection mechanisms may include intercq)tion, gravity settling and Brownian difliision. If the overall collection efficiency for a single bubble is r t de ed as the ratio of the particle to bubble collision rate dh ed by the particle-to-bubble approach rate then r t can be considered to conoprise the sum of the efficiencies for the individual mechanisms so ... 

At the point of zero charge, there is no repulsive electrical force on the particles and so the full adhesion between the grains is developed. If this adhesion is strong, then each Brownian collision between singlet particles will produce a doublet. This was the case considered by Smoluchowski in 1917 and extended by Fuchs in 1934. The theory was based on the idea that colloidal particles behave like molecules which can react to form abimolecular compound. Thus the rate of appearance of doublets dAf /df is proportional to the square of singlet concentration N per unit volume by the law of mass action, and is limited by the Brownian diffusion coefficient to give... 

In his seminal work (as it is frequently called), Kramers treated the escape over a potential barrier by a particle undergoing Brownian motion, i.e. thermal noise-assisted escape [1]. Hence, his focus was on the effect of the medium - solvent or bath gas - on the solute reaction rate. While much of the physical chemistry community was at the time focused on the rate of reaction of an isolated molecule -- and would remain so occupied for many years to come, Kramers work was not completed in a vacuum. Indeed, Lindemann, Rice and Ramsperger, Kassel, Slater, Christiansen, and others had already published their collision-rate-based theories of the role of the bath gas in promoting chemical reactions in low-density gases [3, 4]. Thus, one must ask,... 

Let us now consider the contribution of diffusion to the maximum rates of homogeneous reactions involving associations as a first step in the interactions of proteins with both small and large substrates. In the present context association to form a specific complex does not include any covalent bond formation. Three events are involved in the approach to equilibrium collision, alignment to correct mutual orientation and for some processes separation after the event. These events depend on translational diffusion for approach and separation and on rotational diffusion for alignment. Earlier in this section the diffusion constant was defined using the principle of random Brownian motion and this will now be used in equations which have been derived to evaluate maximum collision rates. 

Studies on orthokinetic flocculation (shear flow dominating over Brownian motion) show a more ambiguous picture. Both rate increases (9,10) and decreases (11,12) compared with orthokinetic coagulation have been observed. Gregory (12) treated polymer adsorption as a collision process and used Smoluchowski theory to predict that the adsorption step may become rate limiting in orthokinetic flocculation. Qualitative evidence to this effect was found for flocculation of polystyrene latex, particle diameter 1.68 pm, in laminar tube flow. Furthermore, pretreatment of half of the latex with polymer resulted in collision efficiencies that were more than twice as high as for coagulation. 

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 time-dependent decrease in the concentration of particles (N = number of particles per cubic centimeter) in a monodisperse suspension due to collisions by Brownian motion can be represented by a second-order rate law... 

The rate constant kp is on the order of 5 10 12 cm3 sec1 for water at 20° C and for ctp = 1. Thus, for example, a turbid water containing 106 particles cm 3 will reduce its particle concentration by half within a period of ca. 2.5 days (2 105 sec) provided that all particles are completely destabilized and that the particles are sufficiently small (e.g., d < 1 pm) so that collisions result from Brownian motion only. 

The rate of coagulation of particles in a liquid depends on the frequency of collisions between particles due to their relative motion. When this motion is due to Brownian movement coagulation is termed perikinetic when the relative motion is caused by velocity gradients coagulation is termed orthokinetic. 

Pick s laws describe the interactions or encounters between noninteracting particles experiencing random, Brownian motion. Collisions in solution are diffusion-controlled. As is discussed in most physical chemistry texts , by applying Pick s Pirst Law and the Einstein diffusion relation, the upper limit of the bimolecular rate constant k would be equal to... 

---