Motion, Brownian

The motion of individual particles in colloids is continuously changing direction as a result of random 

Introduction to Applied Colloid and Surface Chemistry, First Edition. Georgios M. Kontogeorgis and S0ren Kiil. 2016 John Wiley Sons, Ltd. Published 2016 by John Wiley Sons, Ltd. 

Colloidal particles-what would we like to know  

What do they look like - shape Size - key dimensions Molecular weight Molecular weight distribution Change 

Treating Brownian motion as a three-dimensional random walk , Einstein derived an important 

Brownian motion of a panicle is a result of the thermal motion of the molecular agitation of the liquid medium. Much stronger random displacement of a particle is usually observed in a less viscous liquid, smaller particle size, and higher temperature. A particle of size larger than 1 pm doesn t show a remarkable Brownian motion. There is much literature available on Brownian motion [7-9], and the Brownian motion is regarded as a diffusion process. For an isolated particle, i.e., there is no intcrparticlc action, the diffusion coefficient D , can be expressed as the Stokes-Einstein equation  

Brownian motion is a random thermal motion of a particle inside a fluid medium. The collision between the fluid molecules and suspended microparticles are responsible for the Brownian motion. The Brownian motion consists of high frequencies and is not possible to be resolved easily. Average particle displacement after many velocity fluctuations is used as a measure of Brownian motion. The mean square diffusion distance, is proportional to DAt, where D is diffusion coefficient of the particle given by Einstein relation as 

For a particle with velocity u in jc-direction, the travel distance is Ajc = uAt during the pulse interval of At. The relative error due to the Brownian motion during this time interval can be expressed as the ratio of the rms of Brownian displacement to the average motion, that is. 

The above expression indicates that the error due to Brownian motion increases as the time of measurement decreases. Larger time intervals produce flow displacements proportional to At while the Brownian motion goes as. The above expression also indicates that influence 

The other effect of Brownian motion is its influence on the depth of correlation. Depth of correlation defines the depth over which particles significantly contribute to the correlation function. Olsen and Adrian (2000) derived an expression of correlation depth as 

The last term in the right-hand side of the above expression represents the contribution of Brownian motion The depth of correlation increases due to the presence of the Brownian motion. 

1 Brownian Motion When we place a blot of ink in still water, the colored region expands with time but the color fades, eventually filling the entire water in the container. The final state is a uniform concentration of the ink. Spreading of a substance throughout accessible volume is called diffusion. The phenomena is made possible by microscopic movement of water molecules. 

The Brownian motion is stochastic. There is no knowing in advance where the particle will reach in a given time. What we can know is the transition probability P(r, r t) for the particle to move from r at f = 0 to reach r at time t. 

Polymer molecules in solution also display Brownian motion. Because the polymer molecule is not a simple sphere, each polymer conformation has its own diffusion characteristics. For rigid molecules, the shape of the molecule, spherical or rodlike, for instance, makes a difference. For a Unear flexible molecule, connectivity 

Alternatively, one can regard the motion of the particle as a random walk , consisting of a sequence of steps, the direction of each step being chosen at random and independent of the 

Now the number of steps taken in a time At is t/x, where X is the mean duration of a step, so that 

Again appealing to the equipartition principle, it can been shown that 

Note that the mass of the particle does not appear explicitly in equation (6.3) but the mean square displacement is proportional to the reciprocal of its radius and hence to the reciprocal of the cube root of its volume. This means that the smaller the particle the more extensive the Brownian motion. We also observe the important result that the root mean square displacement is proportional to the square root of the time. 

The local concentration will now be proportional to the product of the initial concentration c° and a probability function whose width increases with At 2. 

Polymers in solutions incessantly change both their shape and position randomly by thermal agitation. This Brownian motion dominates various time-dependent phenomena in polymer solutions such as viscoelasticity, diffusion, birefringence, and dynamic light scattering, which are to be discussed in subsequent chapters. In this chapter, we study the basic theory of Brownian motion. Since the general aspects of the theory of Brownian motion have already been discussed in many articles, we shall limit the discussion to topics which will be useful in the application to polymer solutions and suspensions. 

The better model, called reptation, imagines the polymer chain confined within a curved tube (deGennes, 1979). Within this tube, the Rouse model governs the chain dynamics, but the polymer diffusion is governed by the time required to escape from the tube. Because motion in the tube is one-dimensional, this escape time x is given by 

This result frequently comes close to predicting the molecular weight dependence of this case of polymer-polymer diffusion. 

The diffusion coefficients listed above are easy to accept as experimentally valuable parameters, but they are harder to understand as a consequence of molecular motion. These coefficients are most often experimental values. In some cases, they are estimated from theories which imply models for the system involved. For gases, this is the model of gas molecules colliding in space. For liquids, they most often imply a solute sphere in a solvent soup. For solids, these estimates are based on a crystal lattice. In every case, the diffusion coefficients are not very directly related to random molecular motions. 

In this short section, we want to reexamine these coefficients in terms of molecular motions. Such random Brownian motions were first observed in pollen grains by Robert Brown in June of 1827. He concluded that these motions arose neither from currents in the fluid nor from gradual evaporation but from the particles [themselves]. In our terms, diffusion comes from random molecular motions. Such random motions are now widely studied, not only in physical science but in areas like fluctuations of exchange rates of currencies. 

In this section, we describe these random motions in terms of probability theory, and so connect diffusion to this broader topic. Because we want a simple, easily understood connection, we consider only the simplest case of one-dimensional motion. This simplest case depends on three rules  

Particle behavior in colloidal suspensions is dominated by forces and mechanisms different from that relevant for macroscopic objects. Their behavior will be an interplay of sedimentation, random Brownian motion, viscous drag, and interparticle interactions. Thus, in order to use colloidal particles as building blocks for patterning in the micrometer and sub-micrometer range, the balance between these forces must be controlled to provide a method for the robust and reprodudble placement of the particles. In the following a short description of the forces will be given. 

A common characteristic of small scale colloidal systems is the constant thermal movement of the components. Kinetic energy is transferred to suspended particles by collisions of solvent molecules, causing a random motion usually called Brownian motion. This motion ofpartides causes diffusion, which is a net transport of randomly moving partides along density gradients. The mean displacement of a particle by Brownian motion over time leads to the diffusion coefHdent, D (Equation 9.1) [22]  

Brownian motion is a crucial factor in partide assembly. The mass transport from a source, the bulk colloid, to a drain, the pattern, can be accomplished by diffusion. After assembly, the particles have to be trapped to prevent further random motion. 

Another result of the small mass of colloidal particles is that they behave somewhat like molecules when dispersed in liquids. For example, they diffuse through the liquid and move randomly due to the phenomenon known as Brownian motion. 

Thus the average distance that a particle will move over a period of time can be determined. Increasing temperature increases the distance travelled over a period of time while increasing particle size and fluid viscosity reduce the distance travelled. Note that the distance scales with the square root of time rather than linearly with time. 

Note that Equations (5.3)-(5.6) have been derived for the case of the onedimensional random walk. This is because these equations will be used later in analysis of sedimentation under gravity where motion only in one direction (one dimension) is of interest. Only motion of particles in the direction of the applied gravitational force field is of interest in sedimentation lateral motion in the other two orthogonal directions is not. In the case of the three-dimensional random walk, the analogy to Equation (5.3) would be L = V6af. 

In the Ref [14], the method of reflections was applied to calculations of three-particle and four-particle interactions. It was shown that, as compared to pair interactions, three- and four-particle interactions introduce corrections of the order 0(l/r ) and 0(l/r ) to the corresponding velocity perturbations, where r is the characteristic distance between particles. A generalization for the N-particle case was made in [15]. The velocity perturbation is found to be of the order 0(l/r + ). In the same work, expressions for the mobility functions are derived up to the terms of order 0(l/r ). It should be kept in mind that the corresponding expressions are power series in 1/r, so to calculate the velocities at small clearances between particles (it is this case has presents the greatest interest), one has to take into account many terms in the series, or to repeat the procedure of reflection many times. In addition to analytical solutions, numerical solutions of a similar problem are available, for example, in [16]. At small clearances between particles, the application of numerical methods is complicated by the need to increase the number of elements into which particle surfaces are divided in order to achieve acceptable accuracy of the solution. 

We begin with consideration of particles with low volume concentration buoyant in a quiescent liquid or a liquid undergoing translational motion with constant velocity. 

Random thermal motion of small particles buoyant in the liquid is known as Brownian motion. In the absence of external forces acting on particles (which can have an arbitrary size), the particles have equal kinetic energy of thermal motion 3kT/2, where k is Boltzmann s constant, T is the absolute temperature. 

Here m is the mass of the particle, - the mean square of particle s velocity. However, the observed average velocity of the particle s motion is smaller than the velocity given by (8.53). In particular, for a particle of radius 1 pm and density equal to that of water, the value is equal to 1.7 mm/s at the room temper- 

There are two ways to define the diffusion coefficient. Consider them in succession. For simplicity, begin with the one-dimensional case, that is, with the problem of one-dimensional random walk of a particle. The probability of the particle s displacement lying in the range (x,x- -dx) after n random displacements with step I, is given by the Gaussian distribution 

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If X is put equal to a distance of, say, 100 A, then t is about 10 sec, so that, due to Brownian motion, there is a very rapid interchange of molecules between the surface and the adjacent bulk region. 

Theoretical models of the film viscosity lead to values about 10 times smaller than those often observed [113, 114]. It may be that the experimental phenomenology is not that supposed in derivations such as those of Eqs. rV-20 and IV-22. Alternatively, it may be that virtually all of the measured surface viscosity is developed in the substrate through its interactions with the film (note Fig. IV-3). Recent hydrodynamic calculations of shape transitions in lipid domains by Stone and McConnell indicate that the transition rate depends only on the subphase viscosity [115]. Brownian motion of lipid monolayer domains also follow a fluid mechanical model wherein the mobility is independent of film viscosity but depends on the viscosity of the subphase [116]. This contrasts with the supposition that there is little coupling between the monolayer and the subphase [117] complete explanation of the film viscosity remains unresolved. 

When a system is not in equilibrium, the mathematical description of fluctuations about some time-dependent ensemble average can become much more complicated than in the equilibrium case. However, starting with the pioneering work of Einstein on Brownian motion in 1905, considerable progress has been made in understanding time-dependent fluctuation phenomena in fluids. Modem treatments of this topic may be found in the texts by Keizer [21] and by van Kampen [22]. Nevertheless, the non-equilibrium theory is not yet at the same level of rigour or development as the equilibrium theory. Here we will discuss the theory of Brownian motion since it illustrates a number of important issues that appear in more general theories. 

In the general case, (A3.2.23) caimot hold because it leads to (A3.2.24) which requires GE = (GE ) which is m general not true. Indeed, the simple example of the Brownian motion of a hannonic oscillator suffices to make the point [7,14,18]. In this case the equations of motion are [3, 7]... 

Mori H 1965 Transport, collective motion and Brownian motion Prog. Theor. Phys. 33 423... 

Smoluchowski theory [29, 30] and its modifications fonu the basis of most approaches used to interpret bimolecular rate constants obtained from chemical kinetics experiments in tenus of difhision effects [31]. The Smoluchowski model is based on Brownian motion theory underlying the phenomenological difhision equation in the absence of external forces. In the standard picture, one considers a dilute fluid solution of reactants A and B with [A] [B] and asks for the time evolution of [B] in the vicinity of A, i.e. of the density distribution p(r,t) = [B](rl)/[B] 2i ] r(t))l ] Q ([B] is assumed not to change appreciably during the reaction). The initial distribution and the outer and inner boundary conditions are chosen, respectively, as... 

In liquid solution. Brownian motion theory provides the relation between diffiision and friction coefficient... 

Kramers H A 1940 Brownian motion in a field of force and the diffusion model of chemical reactions Physica 7 284-304... 

We call the correlation time it is equal to 1/6 Dj, where Dj is the rotational diffusion coefficient. The correlation time increases with increasing molecular size and with increasing solvent viscosity, equation Bl.13.11 and equation B 1.13.12 describe the rotational Brownian motion of a rigid sphere in a continuous and isotropic medium. With the Lorentzian spectral densities of equation B 1.13.12. it is simple to calculate the relevant transition probabilities. In this way, we can use e.g. equation B 1.13.5 to obtain for a carbon-13... 

Woessner D E 1996 Brownian motion and correlation times Encyclopedia of Nuclear Magnetic Resonance ed D M Grant and R K Harris (Chichester Wiley) pp 1068-84... 

Woessner D E 1962 Nuclear spin relaxation in ellipsoids undergoing rotational Brownian motion J. Chem. Rhys. 37 647-54... 

Doi, M. and Edwards, S.F., 1978. Dynamics of concentrated polymer systems 1. Brownian motion in equilibrium state, 2. Molecular motion under flow, 3. Constitutive equation and 4. Rheological properties. J. Cheni. Soc., Faraday Trans. 2 74, 1789, 1802, 1818-18.32. 

Molecular dynamics is a simulation of the time-dependent behavior of a molecular system, such as vibrational motion or Brownian motion. It requires a way to compute the energy of the system, most often using a molecular mechanics calculation. This energy expression is used to compute the forces on the atoms for any given geometry. The steps in a molecular dynamics simulation of an equilibrium system are as follows ... 

Statistically, in a high-pressure region, an ion will be struck by neutral molecules randomly from all angles. The ion receives as many collisions from behind as in front and as many collisions from one side as from the other. Therefore, it can be expected that the overall forward motion of the ion will be maintained but that the trajectory will be chaotic and similar to Brownian motion (Figure 49.4b). Overall, the ion trajectory can be expected to be approximately along the line of its initial velocity direction, since it is still influenced by the applied potential difference V. 

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. 

There is an intimate connection at the molecular level between diffusion and random flight statistics. The diffusing particle, after all, is displaced by random collisions with the surrounding solvent molecules, travels a short distance, experiences another collision which changes its direction, and so on. Such a zigzagged path is called Brownian motion when observed microscopically, describes diffusion when considered in terms of net displacement, and defines a three-dimensional random walk in statistical language. Accordingly, we propose to describe the net displacement of the solute in, say, the x direction as the result of a r -step random walk, in which the number of steps is directly proportional to time ... 

Diffusional interception or Brownian motion, ie, the movement of particles resulting from molecular collisions, increases the probability of particles impacting the filter surface. Diffusional interception also plays a minor role in Hquid filtration. The nature of Hquid flow is to reduce lateral movement of particles away from the fluid flow lines. 

Figure 5 relates N j to collection efficiency particle diffusivity from Stokes-Einstein equation assumes Brownian motion same order of magnitude or greater than mean free path of gas molecules (0.1 pm at... 

The natural process of bringing particles and polyelectrolytes together by Brownian motion, ie, perikinetic flocculation, often is assisted by orthokinetic flocculation which increases particle coUisions through the motion of the fluid and velocity gradients in the flow. This is the idea behind the use of in-line mixers or paddle-type flocculators in front of some separation equipment like gravity clarifiers. The rate of flocculation in clarifiers is also increased by recycling the floes to increase the rate of particle—particle coUisions through the increase in soUds concentration. 

Other Factors Affecting the Viscosity of Dispersions. Factors other than concentration affect the viscosity of dispersions. A dispersion of nonspherical particles tends to be more viscous than predicted if the Brownian motion is great enough to maintain a random orientation of the particles. However, at low temperatures or high solvent viscosities, the Brownian motion is small and the particle alignment in flow (streamlining) results in unexpectedly lower viscosities. This is a form of shear thinning. 

Perikinetic flocculation is the first stage of flocculation, induced by the Brownian motion. It is a second-order process that quickly diminishes with time and therefore is largely completed in a few seconds. The higher the initial concentration of the soflds, the faster is the flocculation. 

Sols are dispersions of colloidal particles in a Hquid. Sol particles are typically small enough to remain suspended in a Hquid by Brownian motion. 

Acid mist eliminators use three aerosol collection mechanisms inertial impaction, interception, and Brownian motion. Inertial impaction works well for aerosols having particle diameters larger than 3 p.m Brownian motion and interception work well with aerosols having smaller particle diameters. 

Ions of an electrolyte are free to move about in solution by Brownian motion and, depending on the charge, have specific direction of motion under the influence of an external electric field. The movement of the ions under the influence of an electric field is responsible for the current flow through the electrolyte. The velocity of migration of an ion is given by ... 

In view of the facts that three-dimensional coUoids are common and that Brownian motion and gravity nearly always operate on them and the dispersiag medium, a comparison of the effects of particle size on the distance over which a particle translationaUy diffuses and that over which it settles elucidates the coUoidal size range. The distances traversed ia 1 h by spherical particles with specific gravity 2.0, and suspended ia a fluid with specific gravity 1.0, each at 293 K, are given ia Table 1. The dashed lines are arbitrary boundaries between which the particles are usuaUy deemed coUoidal because the... 

Behavior. Diffusion, Brownian motion, electrophoresis, osmosis, rheology, mechanics, and optical and electrical properties are among the general physical properties and phenomena that are primarily important in coUoidal systems (21,24—27). Of course, chemical reactivity and adsorption often play important, if not dominant, roles. Any physical and chemical feature may ultimately govern a specific industrial process and determine final product characteristics. 

SPACEEIL has been used to study polymer dynamics caused by Brownian motion (60). In another computer animation study, a modified ORTREPII program was used to model normal molecular vibrations (70). An energy optimization technique was coupled with graphic molecular representations to produce animations demonstrating the behavior of a system as it approaches configurational equiHbrium (71). In a similar animation study, the dynamic behavior of nonadiabatic transitions in the lithium—hydrogen system was modeled (72). 

In other words, the lower the mass of the particle, the higher its velocity, because the average energy of any particle at a given temperature is constant, kT. A dispersed particle is always in random thermal motion (Brownian motion) due to coUisions with other particles and with the walls of the container (4). If the particles coUide with enough energy and are not well dispersed, they will coagulate or flocculate. 

Factors which adversely influence the separation of veiy fine particle systems are brownian motion and London forces. However, it is possible to counter these forces by the use of dispersants, temperature control, and so on. 

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