Brownian motion time-dependent diffusion

To work out the time-dependence requires a specific model for the movement of the paramagnet, for example, Brownian motion, or lateral diffusion in a membrane, or axial rotation on a protein, or jumping between two conformers, etc. That theory is beyond the scope of this book the math can become quite hairy and can easily fill another book or two. We limit the treatment here to a few simple approximations that are frequently used in practice. 

In fact, the validity of Eqs. (90) and (91) is not restricted to the simple (i.e., nonretarded) Langevin model as defined by Eq. (73). These formulas can be applied in other classical descriptions of Brownian motion in which a time-dependent diffusion coefficient can be defined. This is for instance, the case in the presence of non-Ohmic dissipation, in which case the motion of the Brownian particle is described by a retarded Langevin equation (see Section V). 

Next, Fig. 5 shows a typical center of mass trajectory for the case of a full chaotic internal phase space. The eyecatching new feature is that the motion is no more restricted to some bounded volume of phase space. The trajectory of the CM motion of the hydrogen atom in the plane perpendicular to the magnetic field now closely resembles the random motion of a Brownian particle. In fact, the underlying equation of motion at Eq. (35) for the CM motion is a Langevin-type equation without friction. The corresponding stochastic Lan-gevin force is replaced by our intrinsic chaotic force — e B x r). A main characteristic of random Brownian motion is the diffusion law, i.e. the linear dependence of the travelled mean-square distance on time. We have plotted in Fig. 6 for our case of a chaotic force for 500 CM trajectories the mean-square distance as a function of time. Within statistical accuracy the plot shows a linear dependence. The mean square distance

of the CM after time t, therefore, obeys the diffusion equation... 

Recent advances in percolation theory and fractal geometry have demonstrated that Dc is not a constant when diffusion occurs as a result of fractional Brownian motion, i.e., anomalous diffusion (Sahimi, 1993). The time-dependent diffusion coefficient, D(t), for anomalous diffusion in two-dimensional free space is given by (Mandelbrot Van Ness, 1968),... 

B.2, Here, we demonstrate once more how Brownian dynamics relates to diffusive behavior, by simulating spherical particles of radius 1 mm in water at room temperature. At time f = 0, a particle is released at the origin and undergoes 3-D Brownian motion. Write a program that repeats this simulation many times and plots the radial concentration profile of particles as a function of time. It is easier to do the data analysis if you do the simulations concurrently. Then, solve the corresponding time-dependent diffusion equation in spherical coordinates and compare the results to that obtained fi om Brownian dynamics. 

In order to examine the nature of the friction coefficient it is useful to consider the various time, space, and mass scales that are important for the dynamics of a B particle. Two important parameters that determine the nature of the Brownian motion are rm = (m/M) /2, that depends on the ratio of the bath and B particle masses, and rp = p/(3M/4ttct3), the ratio of the fluid mass density to the mass density of the B particle. The characteristic time scale for B particle momentum decay is xB = Af/ , from which the characteristic length lB = (kBT/M)i lxB can be defined. In derivations of Langevin descriptions, variations of length scales large compared to microscopic length but small compared to iB are considered. The simplest Markovian behavior is obtained when both rm << 1 and rp 1, while non-Markovian descriptions of the dynamics are needed when rm << 1 and rp > 1 [47]. The other important times in the problem are xv = ct2/v, the time it takes momentum to diffuse over the B particle radius ct, and Tp = cr/Df, the time it takes the B particle to diffuse over its radius. 

Photon correlation spectroscopy (PCS) has been used extensively for the sizing of submicrometer particles and is now the accepted technique in most sizing determinations. PCS is based on the Brownian motion that colloidal particles undergo, where they are in constant, random motion due to the bombardment of solvent (or gas) molecules surrounding them. The time dependence of the fluctuations in intensity of scattered light from particles undergoing Brownian motion is a function of the size of the particles. Smaller particles move more rapidly than larger ones and the amount of movement is defined by the diffusion coefficient or translational diffusion coefficient, which can be related to size by the Stokes-Einstein equation, as described by... 

The Peclet number compares the effect of imposed shear (known as the convective effect) with the effect of diffusion of the particles. The imposed shear has the effect of altering the local distribution of the particles, whereas the diffusion (or Brownian motion) of the particles tries to restore the equilibrium structure. In a quiescent colloidal dispersion the particles move continuously in a random manner due to Brownian motion. The thermal motion establishes an equilibrium statistical distribution that depends on the volume fraction and interparticle potentials. Using the Einstein-Smoluchowski relation for the time scale of the motion, with the Stokes-Einstein equation for the diffusion coefficient, one can write the time taken for a particle to diffuse a distance equal to its radius R, as... 

There have been several other theoretical approaches towards defining the time dependence of quenching by both long-range energy (or electron) transfer and diffusion. Some of these are discussed in later chapters. For instance, improvements to the description of Brownian motion as diffusion are discussed in Chap. 11. 

CARS-CS experiments have been reported in the low-concentration limit ((N) <<1) on freely diffusing submicron-sized polymer spheres of different chemical compositions using both the E-CARS [162, 163] and the polarization-resolved CARS [163] detection scheme for efficient nonresonant background suppression. These experiments have unambiguously demonstrated the vibrational selectivity of CARS-CS, the dependence of its ACF amplitude on the particle concentration, (N), the dependence of lateral diffusion time, Tp, on the sphere size, and the influence of the microviscosity on its Brownian motion. 

It has to be noted that the detection of the various diffusive states as well as active transport depends on the time scale of observation. For short time scales, the short-range motion of tracked particles may seem similar and indicates the same local microenvironment for the particles as it is dominated by Brownian motion [37,41], Confined diffusion as well as active transport require a minimal duration for detection and appear at longer time scales (see MSD plot). To display confined diffusion, the particle has to experience the boundaries of confinement in its local microenvironment which restrict the free diffusion on longer time scales. Similarly, for active transport, the second part of (3) 4DAf is predominant on short time scales. The active transport component v2At2 becomes dominant at longer observation periods. 

In the pulmonary region, air velocities are too low to impact particles small enough to reach that region, and the mechanisms of deposition are sedimentation and Brownian diffusion. The efficiency of both processes depends on the length of the respiratory cycle, which determines the stay time in the lung. If the cycle is 15 breaths/min, the stay time is of the order of a second. Table 7.1 shows the distance fallen in one second and the root mean square distance travelled by Brownian diffusion in one second by unit density particles (Fuchs, 1964). Sedimentation velocity is proportional to particle density, but Brownian motion is independent of density. Table 7.1 shows that sedimentation of unit density particles is more effective in causing deposition than Brownian diffusion when dp exceeds 1 pm, whereas the reverse is true if dp is less than 0.5 pm. For this reason, it is appropriate to use the aerodynamic diameter dA equal to pj dp when this exceeds 1 pm, but the actual diameter for submicrometre particles. 

Many excellent introductions to quasi-elastic light scattering can be found in the literature describing the theory and experimental technique (e.g. 3-6). The use of QELS to determine particle size is based on the measurement, via the autocorrelation of the time dependence of the scattered light, of the diffusion coefficients of suspended particles undergoing Brownian motion. The measured autocorrelation function, G<2>(t), is given by... 

The diffusion of small particles depends upon many factors. In addition to Brownian motion, we must consider the effect of gravity and the motion of the fluid in which the particles reside. Ordinary diffusion as understood in colloid chemistry must be modified considerably when we deal with turbulence. However, we still retain the usual definition of diffusion, namely that it is the number or mass of particles passing a unit cross section of the fluid in unit-time and unit-concentration gradient. That is, if dw particles (or mass) move through an area / in time dt and dC/dx is the concentration increase in the jc-directior then... 

We can make a crude estimate of the conditions under which diffusion is likely to be important by comparing the time required for diffusion under given circumstances with the time required for a dependent or competing process. Thus from the kinetic theory (Sec. VI.7) of Brownian motion, the time required for a molecule to diffuse a distance x is given approximately by to = where Z>, the diffusion constant, is inversely pro-... 

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