Diffusion coefficient self particle

The final section (Section 5.8) introduces dynamic light scattering with a particular focus on determination of diffusion coefficients (self-diffusion as well as mutual diffusion), particle size (using the Stokes-Einstein equation for the diffusion coefficient), and size distribution. 

Dj (self)diffusion coefficient of particles of component i DJ tracer diffusion coefficient of particles of component /... 

Whereas mutual diffusion characterizes a system with a single diffusion coefficient, self-diffusion gives different diffusion coefficients for all the particles in the system. Self-diffusion thereby provides a more detailed description of the single chemical species. This is the molecular point of view [7], which makes the selfdiffusion more significant than that of the mutual diffusion. In contrast, in practice, mutual diffusion, which involves the transport of matter in many physical and chemical processes, is far more important than self-diffusion. Moreover mutual diffusion is cooperative by nature, and its theoretical description is complicated by nonequilibrium statistical mechanics. Not surprisingly, the theoretical basis of mutual diffusion is more complex than that of self-diffusion [8]. In addition, by definition, the measurements of mutual diffusion require mixtures of liquids, while self-diffusion measurements are determinable in pure liquids. 

If a second particle is present in the system, as will be the case in almost any imaginable situation, the force field produced by one moving particle will affect and be affected by the second (Fig. 4.10b). Because the second particle produces a braking effect on the movement of the first, the effective diffusion coefficient of particle 1 will decrease in a manner proportional to the distance separating the two. If it is assumed for purposes of simplification that particle 2 does not move, the relationship between the braking effect and the self-diffusion coefficient can be written as... 

The shear viscosity is a tensor quantity, with components T] y, t],cz, T)yx> Vyz> Vzx> Vzy If property of the whole sample rather than of individual atoms and so cannot be calculat< with the same accuracy as the self-diffusion coefficient. For a homogeneous fluid the cor ponents of the shear viscosity should all be equal and so the statistical error can be reducf by averaging over the six components. An estimate of the precision of the calculation c then be determined by evaluating the standard deviation of these components from tl average. Unfortunately, Equation (7.89) cannot be directly used in periodic systems, evi if the positions have been unfolded, because the unfolded distance between two particl may not correspond to the distance of the minimum image that is used to calculate the fore For this reason alternative approaches are required. 

In a liquid that is in thermodynamic equilibrium and which contains only one chemical species, the particles are in translational motion due to thermal agitation. The term for this motion, which can be characterized as a random walk of the particles, is self-diffusion. It can be quantified by observing the molecular displacements of the single particles. The self-diffusion coefficient is introduced by the Einstein relationship... 

The self-diffusion coefficient parallel to the pore walls was computed form the mean square particle displacement,... 

Being formed in the gaseous phase medium, the electronically excited particles (EEPs) reach the solid surface by diffusion. The diffusion coefficients of EEPs are, as a rule, smaller than the self-diffusion coefficients of parent gas, a factor that is associated with increasing of the EEP elastic scattering cross-section at parent molecules due to the redis-... 

The diffusion coefficient D is one-third of the time integral over the velocity autocorrelation function CvJJ). The second identity is the so-called Einstein relation, which relates the self-diffusion coefficient to the particle mean square displacement (i.e., the ensemble-averaged square of the distance between the particle position at time r and at time r + f). Similar relationships exist between conductivity and the current autocorrelation function, and between viscosity and the autocorrelation function of elements of the pressure tensor. 

Figure 10 Size-dependence of the melting point and diffusion coefficient of silica-encapsulated gold particles. The dotted curve is calculated by the equation of Buffat and Borel. The bulk melting temperature of An is indicated by the double arrow as (oo). The solid curve (right-hand side axis) is a calculated An self-diffusion coefficient. (From Ref. 146.)...

Next, it will be helpful to anticipate a description of experimental procedures and consider the magnitude of measured diffusion coefficients. The self-diffusion coefficients for ordinary liquids with small molecules are of the order of magnitude 10 9 m2 s for colloidal substances, they are typically of the order 10"11 m2 s l. In the next section, we see that for spherical particles the diffusion coefficient is inversely proportional to the radius of the sphere. Therefore, every increase by a factor of 10 in size decreases the diffusion coefficient by the same factor. Qualitatively, this same inverse relationship applies to nonspherical particles as well. Once again, we see that diffusion decreases in importance with increasing particle size, precisely those conditions for which sedimentation increases in importance. For larger particles, for which D is very small, the diffusion coefficient also becomes harder to measure. For... 

Calculational procedure of all the dynamic variables appearing in the above expressions—namely, the dynamic structure factor F(q,t) and its inertial part, Fo(q,t), and the self-dynamic structure factor Fs(q,t) and its inertial part, Fq (q, t) —is similar to that in three-dimensional systems, simply because the expressions for these quantities remains the same except for the terms that include the dimensionality. Cv(t) is calculated so that it is fully consistent with the frequency-dependent friction. In order to calculate either VACF or diffusion coefficient, we need the two-particle direct correlation function, c(x), and the radial distribution function, g(x). Here x denotes the separation between the centers of two LJ rods. In order to make the calculations robust, we have used the g(x) obtained from simulations. 

It is evident from the above expressions that the appropriate diffusion coefficient must also be measured in order that molecular or particle masses may be determined from sedimentation velocity data. In this respect, a separate experiment is required, since the diffusion coefficient cannot be determined accurately in situ, because there is a certain self-sharpening of the peak due to the sedimentation coefficient increasing with decreasing concentration. 

MD simulation is advantageous for obtaining dynamic properties directly, since the MD technique provides not only particle positions but also particle velocities that enable us to utilize the response theory (e.g., the Kubo formula [175,176]) to calculate the transport coefficients from time-dependent correlation functions. For example, we will examine the self-diffusion process of a tagged PFPE molecular center of mass (Fig. 1.49) from the simulation to gain insight into the excitation of translational motion, specifically, spreading and replenishment. The squared displacement of the center mass of a molecule or a bead is used as a measure of translational movement. The self-diffusion coefficient D can be represented as a velocity autocorrelation function... 

With the Einstein relationship =6DA, where is the average squared distance particles with a self diffusion coefficient D cover in a time A, the maximum average size of the domains can be estimated as =5 pm, when we use A=50 milliseconds as the smallest mixing time for which cross-peaks appear, and 10 10 m2/s for D, the value found for... 

As schematically shown in Figure 12.5 the self-diffusion coefficient D can be determined from the PFGE experiment for different values of A. For normal diffusion, for which Fick s law holds 1, D is independent of A and the average square distance the diffusing particles cover during the diffusion time A is directly proportional to A (Einstein relationship) ... 

If the tracer is composed of the same species as that of the solid host, then the diffusion coefficient is named the tracer self-diffusion coefficient, where DA is the tracer self-diffusion coefficient. It is necessary to clarify that self-diffusion is a particle transport process that takes place in the absence of a chemical potential gradient [13]. This process is described, as explained later, by following the molecular trajectories of a large number of molecules, and determining their mean square displacement (MSD). 

The aim of this Chapter is the development of an uniform model for predicting diffusion coefficients in gases and condensed phases, including plastic materials. The starting point is a macroscopic system of identical particles (molecules or atoms) in the critical state. At and above the critical temperature, Tc, the system has a single phase which is, by definition, a gas or supercritical fluid. The critical temperature is a measure of the intensity of interactions between the particles of the system and consequently is a function of the mass and structure of a particle. The derivation of equations for self-diffusion coefficients begins with the gaseous state at pressures p below the critical pressure pc. A reference state of a hypothetical gas will be defined, for which the unit value D = 1 m2/s is obtained at p = 1 Pa and a reference temperature, Tr. Only two specific parameters, Tc, and the critical molar volume, VL, of the mono-... 

The relative density qr is again considered a starting point for additional dynamic processes occurring in the system. Self-diffusion of the particles is an example of such a process. Based on the same mathematical assumption (i), the magnitude of the diffusion coefficient D = D exp(qr) is derived as an exponential function of qr with an unit amount This assumption is further supported by many empirically established equations describing dynamic properties of macroscopic systems. 

Self-diffusion under equilibrium conditions may also be monitored in multicomponent systems. Again, with both eqs 2 and 3 a self-diffusion coefficient (of a particular component) may be defined. This coefficient depends on the nature and the concentration of all molecular species involved as well as on the nature of the catalyst particle. 

Powerful methods for the determination of diffusion coefficients relate to the use of tracers, typically radioactive isotopes. A diffusion profile and/or time dependence of the isotope concentration near a gas/solid, liq-uid/solid, or solid/solid interface, can be analyzed using an appropriate solution of - Fick s laws for given boundary conditions [i-iii]. These methods require, however, complex analytic equipment. Also, the calculation of self-diffusion coefficients from the tracer diffusion coefficients makes it necessary to postulate the so-called correlation factors, accounting for nonrandom migration of isotope particles. The correlation factors are known for a limited number of lattices, whilst their calculation requires exact knowledge on the microscopic diffusion mechanisms. 

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