Self-diffusion, definition

Molecular diffusion (or self-diffusion) is the process by which molecules show a net migration, most commonly from areas of high to low concentration, as a result of their thermal vibration, or Brownian motion. The majority of reactive transport models are designed to simulate the distribution of reactions in groundwater flows and, as such, the accounting for molecular diffusion is lumped with hydrodynamic dispersion, in the definition of the dispersivity. 

Water in skeletal carbonates bound Hp and OH , 106-107,109 liquid Hp in fluid inclusions, 106 Water molecules at day interface, modes of reorientation, 403-404 Water self-diffusion coefficient, measurement, 404-405 Weathering, definition, 4 Wintergreen, triboluminescent spectra, 255259... 

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-... 

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. 

An effusion formula for the crystal diffusivity (Dc) is obtained from the expression for in Equation 12. The average absolute flux of molecules across the face Sc is NPj /ht, with an equal flux in the opposite direction at equilibrium. For self-diffusion the equilibrium is not disturbed, and if the difference in number-fraction of labelled molecules between adjacent cells is Ax, the definition of diffusivity yields ... 

Combining Eqs. (5)-(7) leads to the definition of the corrected or self-diffusivity, the Darken equation ... 

The two-site model leads to the definition of the degree of counterion binding, p, as the ratio of counterions to surfactant ions in a surfactant self-assembly. This is a useful but incomplete characterization of the counterion distribution. The value of p can be obtained directly from self-diffusion data because the self-diffusion coefficients of free ions are easily obtainable. For free counterion diffusion a correction is made for the obstruction effect. The micellar D value is obtained as described above or estimated as Dm free. an exact Dm value is not critical. 

Obviously, if found experimentally, the exponent -2 is only a sign of reptation motion but not a sufficient condition for it. Thus, in order to prove that reptation is a really dominant mode of chain motion in polymer concentrates, we have to test it not only with self-diffusion but also with other physical properties which reflect the local motion of polymer chains. One such property is the (coherent) dynamic structure factor 5(fc, r) (see Section 3.2 of Chapter 4 for its definition). In fact, it was predicted theoretically [45-47] that the k dependence of its decay with r in the range of k defined by... 

The same water model, SPC/E at 25 °C was used by other authors too. Chowd-huri and Chandra (2001) employed 256 water molecules per ion, as well as lower ratios at increasing concentrations, and reported the average residence times of water molecules near ions in ps Na+ 18.5, K+ 7.9, and Cl 10.0. Guardia et al. (2006) also reported residence times in ps of the water molecules in the first hydration shells of ions Li+ 101, Na+ 25.0, K+ 8.2, Cs+ 6.9, F 35.5, Cl 14.0, and I 8.5, compared with 10 1 for water molecules in the bulk. These values, resulting from detailed considerations of the hydrogen bond dynamics in water and near the ions, can be compared with experimental values derived from NMR. According to Bakker (2008) these are Li+ 39, Na+ 27, K+ 15, Cl 15 (by definition the same as for K+), Br 10, and 5 ps, and for water molecules in the bulk 17 ps, calculated from the self diffusion coefficient. 

Combination of equations 18 and 19 and use of the definition of mobility leads to the following expression for the polymer self-diffusion coefficient, when... 

With microemulsions based on long-chain alcohols (e.g. decanol), the self-diffusion coefficient for water was low, indicating the presence of definite (closed) water droplets surrounded by surfactant anions in the hydrocarbon medium. Thus,... 

The multitude of transport coefficients collected can thus be divided into self-diffusion types (total or partial conductivities and mobilities obtained from equilibrium electrical measurements, ambipolar or self-diffusion data from steady state flux measurements through membranes), tracer-diffusivities, and chemical diffusivities from transient measurements. All but the last are fairly easily interrelated through definitions, the Nemst-Einstein relation, and the correlation factor. However, we need to look more closely at the chemical diffusion coefficient. We will do this next by a specific example, namely within the framework of oxygen ion and electron transport that we have restricted ourselves to at this stage. 

Though fo is normally determined by mechanical measurements, it is evident from its definition that it is related in a simple manner to translatory diffusion, and in principle it can be determined from self-diffusion measurements in a polymer with sufficiently low molecular weight to avoid entanglements. Some data of this sort have been obtained from diffusion 6f radioactively tagged molecules - and spin-echo nuclear magnetic resonance measurements, as well as from diffusion of deuterated molecules whose concentration can be followed by infrared absorption. They yield the product Pfo. since the frictional resistance for the entire molecule is measured. 

Measurements of the self-diffusion coefficients of PDMS in melts have been reported covering a wide range of molecular weights,from oligomers starting with an M = 88 g-mol up to of 10 g-mol . The results are summarized in Table 2. The third column shows how different are the values obtained for the exponent in the power law that relates Dg and M making it very difficult to reach a definitive conclusion. However, our analysis (shown below) of the literature data reveals the source of these differences. 

It is quite straightforward to evaluate the generalized GK relations for self-diffusion but it is considerably more difficult to calculate them for thermal conductivity because the Lagrangian definition of the coordinates requires an explicit time origin. ... 

First of aU, a more detailed definition of diffusion coefficients has to be made. There is a difference between the binary, chemical or salt diffusion coefficient and the tracer and self-diffusion coefficient. As can be seen, different citations use different terms for the parameters [475, 483, 484]. The mathematical relationship of these diffusion coefficients is [485]... 

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