Diffusion coefficient random walk

In this regime the typical distance from the origin of motion increases as the square root of time. Thus, the dispersion in turbulent flows at long times is analogous to molecular diffusion or random walks with independent increments and comparison of Eq. (2.24) with (2.16) relates the turbulent diffusion coefficient, Dt, to the integral of the Lagrangian correlation function, Tl, as... 

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... 

In contrast to normal diffusion, Ar2n does not grow linearly but with the square root of time. This may be considered the result of superimposing two random walks. The segment executes a random walk on the random walk given by the chain conformation. For the translational diffusion coefficient DR = kBT/ is obtained DR is inversely proportional to the number of friction-performing segments. 

The simplest and most basic model for the diffusion of atoms across the bulk of a solid is to assume that they move by a series of random jumps, due to the fact that all the atoms are being continually jostled by thermal energy. The path followed is called a random (or drunkard s) walk. It is, at first sight, surprising that any diffusion will take place under these circumstances because, intuitively, the distance that an atom will move via random jumps in one direction would be balanced by jumps in the opposite direction, so that the overall displacement would be expected to average out to zero. Nevertheless, this is not so, and a diffusion coefficient for this model can be defined (see Supplementary Material Section S5). 

The vacancy will follow a random-walk diffusion route, while the diffusion of the tracer by a vacancy diffusion mechanism will be constrained. When these processes are considered over many jumps, the mean square displacement of the tracer will be less than that of the vacancy, even though both have taken the same number of jumps. Therefore, it is expected that the observed diffusion coefficient of the tracer will be less than that of the vacancy. In these circumstances, the random-walk diffusion equations need to be modified for the tracer. This is done by ascribing a different probability to each of the various jumps that the tracer may make. The result is that the random-walk diffusion expression must be multiplied by a correlation factor, / which takes the diffusion mechanism into account. 

The list below shows the last position reached, in units of the jump step a, during a random walk for 100 atoms, each of which makes 200 jumps. If the jump time is 10-3 s and the jump distance, a, is 0.3 nm, estimate the diffusion coefficient (a) in units of a2 s-1 and (b) in units of m2 s-1 ... 

If both ionic conductivity and ionic diffusion occur by the same random-walk mechanism, a relationship between the self-diffusion coefficient, D, and the ionic... 

This equation shows that it is possible to determine the diffusion coefficient from the easier measurement of ionic conductivity. However, Da is derived by assuming that the conductivity mechanism utilizes a random-walk mechanism, which may not true. 

The correlation factor,/, is defined by the ratio of the tracer diffusion coefficient to the random-walk diffusion coefficient (Section 5.6) ... 

Fick s (continuum) laws of diffusion can be related to the discrete atomic processes of the random walk, and the diffusion coefficient defined in terms of Fick s law can be equated to the random-walk displacement of the atoms. Again it is easiest to use a one-dimensional random walk in which an atom is constrained to jump from one... 

To obtain a more complete description, we need to find an analytic expression for the pre-exponential factor Dq of the diffusion coefficient by considering the microscopic mechanism of diffusion. The most straightforward approach, which neglects correlated motion between the ions, is given by the random-walk theory. In this model, an individual ion of charge q reacts to a uniform electric field along the x-axis supplied, in this case, by reversible nonblocking electrodes such that dCj(x)/dx = 0. Since two... 

Such a mechanism is not incompatible with a Haven ratio between 0.3 and 0.6 which is usually found for mineral glasses (Haven and Verkerk, 1965 Terai and Hayami, 1975 Lim and Day, 1978). The Haven ratio, that is the ratio of the tracer diffusion coefficient D determined by radioactive tracer methods to D, the diffusion coefficient obtained from conductivity via the Nernst-Einstein relationship (defined in Chapter 3) can be measured with great accuracy. The simultaneous measurement of D and D by analysis of the diffusion profile obtained under an electrical field (Kant, Kaps and Offermann, 1988) allows the Haven ratio to be determined with an accuracy better than 5%. From random walk theory of ion hopping the conductivity diffusion coefficient D = (e /isotropic medium. Hence for an indirect interstitial mechanism, the corresponding mobility is expressed by... 

Diffusion can be modeled as a random walk in three dimensions, and the value of the diffusion coefficient can be computed by the correlation formula... 

From the theoretical point of view, it is necessary to show that no microphysical difference exists between the processes of diffusion, i.e. the transfer of molecules according to a gradient of their chemical potential or concentration, and self-diffusion, i.e. the re-distribution of molecules in space due to their random walk at equilibrium. The corresponding coefficients... 

The size of a surface available for field ion microscope study of surface diffusion is very small, usually much less than 100 A in diameter. The random walk diffusion is therefore restricted by the plane boundary. For a general discussion, however, we will start from the unrestricted random walk. First, we must be aware of the difference between the chemical diffusion coefficient and the tracer diffusion coefficient. The chemical diffusion coefficient, or more precisely the diffusion tensor, is defined by a generalized Fick s law as... 

The effective polarizability of surface atoms can be determined with different methods. In Section 2.2.4(a) a method was described on a measurement of the field evaporation rate as a function of field of kink site atoms and adsorbed atoms. The polarizability is derived from the coefficient of F2 term in the rate vs. field curve. From the rate measurements, polarizabilities of kink site W atoms and W adatoms on the W (110) surface are determined to be 4.6 0.6 and 6.8 1.0 A3, respectively. The dipole moment and polarizability of an adatom can also be measured from a field dependence of random walk diffusion under the influence of a chemical potential gradient, usually referred as a directional walk, produced by the applied electric field gradient, as reported by Tsong et a/.150,198,203 This study is a good example of random walk under the influence of a chemical potential gradient and will therefore be discussed in some detail. 

An expression for the coefficient of diffusion identical with that given in Chapman and Cowling (C3) has also been obtained by considering diffusion as a random-walk process (F12). 

In this section we give a proof of the Kawabata formula (52), following a method due to Kaveh (1984) and Mott and Kaveh (1985a, b). We assume that an electron undergoes a random walk, which determines an elastic mean free path l and diffusion coefficient D. If an electron starts at time t=0 at the point r0 then the probability per unit volume of finding it at a distance r, at time U denoted by n(r, t) obeys a diffusion equation... 

Describe how the random walk statistics are used to relate the random walk to the diffusion coefficient. 

Polymer molecules in a solution undergo random thermal motions, which give rise to space and time fluctuations of the polymer concentration. If the concentration of the polymer solution is dilute enough, the interaction between individual polymer molecules is negligible. Then the random motions of the polymer can be described as a three dimensional random walk, which is characterized by the diffusion coefficient D. Light is scattered by the density fluctuations of the polymer solution. The propagation of phonons is overdamped in water and becomes a simple diffusion process. In the case of polymer networks, however, such a situation can never be attained because the interaction between chains (in... 

Now, everything falls into place We set out to study the laws of random walk by using the simple model of Fig. 18 and found the Bernoulli coefficients. We then saw that for large n (which is equivalent to large times), the Bernoulli coefficients can be approximated by a normal distribution whose standard deviation, a, grows in proportion to the square root of time, tm (Eq. 18-3). And now it turns out that the solution of the Fick s second law for unbounded diffusion is also a normal distribution. In fact, the analogy between Eqs. 18-3b and 18-17 gave the basis for the law by Einstein and Smoluchowski (Eq. 18-17) that we used earlier (Eq. 18-8). The expression (2Dt)U2 will also show up in other solutions of the diffusion equation. 

According to the model of random walk in three dimensions, the diffusion coefficient of a molecule i, can be expressed as one-third of the product of its mean free path A, and its mean three-dimensional velocity u, (Eq. 18-7a). In the framework of the molecular theory of gases, u, is (e.g., Cussler, 1984) ... 

The coefficient Ex is called the turbulent (or eddy) diffusion coefficient it has the same dimension as the molecular diffusion coefficient [L2 1]. The index x indicates the coordinate axis along which the transport occurs. Note that the turbulentjliffusion coefficient can be interpreted as the product of a mean transport distance Lx times a mean velocity v = (Aa At) l Egex, as found in the random walk model, Eq. 18-7. 

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