Macroscopic self-diffusion coefficient

Nowadays, self-diffusion coefficients are almost exclusively measured by NMR methods, through the use of methods such as the 90-8-180-8-echo technique (Stejs-kal and Tanner sequence) [10-12]. The pulse-echo sequence, illustrated in Figure 4.4-2, can be divided into two periods of time r. After a 90° radio-frequency (RF) pulse the macroscopic magnetization is rotated from the z-axis into the x-y-plane. A gradient pulse of duration 8 and magnitude g is appHed, so that the spins dephase. 

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-... 

Collecting the above results, the following final equation can be established for the self-diffusion coefficient in a macroscopic system in the gaseous state ... 

To what extent does macroscopic friction , as indicated by solvent viscosity or by inverse self-diffusion coefficient, really reflects the microscopic friction experienced by the reaction coordinate ... 

The self-diffusion approach relies on the fact that molecular displacements over macroscopic distances are very sensitive to confinement and thus to microstructure. For example, we found that at the same composition (water, oil, surfactant), the ratio between water and oil self-diffusion coefficients could differ by a factor of 100 000. This also illustrates that the microstructure is primarily determined by the spontaneous curvature of the surfactant film and not by the oil-to-water ratio. Contributions to a better understanding of microemulsion structures with FT spin-echo NMR self-diffusion starting with Stilbs, included also Nilsson, Olsson, Soderman, Khan, Guering, Monduzzi, Ceglie, Das and many others in Lund. In this work [49-63], the access to suitable systems was very important. Here, the contacts with Friberg, Shinoda, Strey and Langevin played a central role. 

For translational long-range jump diffusion of a lattice gas the stochastic theory (random walk, Markov process and master equation) [30] eventually yields the result that Gg(r,t) can be identified with the solution (for a point-like source) of the macroscopic diffusion equation, which is identical to Pick s second law of diffusion but with the tracer (self diffusion) coefficient D instead of the chemical or Fick s diffusion coefficient. 

Body force of component i Polymer mass concentration Macroscopic diffusion coefficient Cooperative diffusion coefficient Stokes-Einstein diffusion coefficient Rouse diffusion coefficient Self diffusion coefficient... 

For the purpose of macroscopic transport of water, self-diffusion coefficients of water are converted to Fickian diffusion or the chemical diffusion... 

D, Dm, Ds Diffusion, inter diffusion and self diffusion coefficient, respectively Dp Particle diffusion coefficient E Threshold energy of coagulation Edk Macroscopic bulk breaking energy... 

Diffusion. The translational diffusion coefficient D is the most commonly measured transport property of polymer solutions, but as there are several distinct types of diffusion, care must be taken to interpret D properly. For c < c, Brownian motion of isolated chains in a homogeneous solvent defines the dilute solution diffusion coefficient Dq. As c increases toward c and above, chain-chain interactions modify the friction felt during chain motion. Under these conditions, the tracer- or self-diffusion coefficient Dtr is measured by tracking the path of a single chain in a macroscopically imiform mixture of chains and solvent. To distinguish the test chain from neighbors so that its path can be identified, the chain... 

It should be noted that the radiotracer and pulsed field gradient NMR techniques measure the self-diffusion coefficient of water, Ds, rather than the Fickian or interdiffusion coefficient of water through the polymer membrane, D, and some correction is required, because it is the Fickian water diffusion coefficient that is the proper transport property to use in macroscopic studies of water diffusion [17]. The relationship is ... 

The "laminar" macroscopic flow equations contain phenomenological terms which represent averages over the macroscopic dynamics to include the effects of turbulence. Examples of these terms are eddy viscosity and diffusivity coefficients and average chemical heat release terms which appear as sources in the macroscopic flow equations. Besides providing these phenomenological terms, the turbulence model must use the information provided by the large scale flow dynamics self-consistently to determine the energy which drives the turbulence. The model must be able to follow reactive interfaces on the macroscopic scale. 

Diffusion is a physical process that involves the random motion of molecules as they collide with other molecules (Brownian motion) and, on a macroscopic scale, move from one part of a system to another. The average distance that molecules move per unit time is described by a physical constant called the diffusion coefficient, D (in units of mm2/s). In pure water, molecules diffuse at a rate of approximately 3xl0"3 mm2 s 1 at 37°C. The factors influencing diffusion in a solution (or self-diffusion in a pure liquid) are molecular weight, intermolecular... 

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. 

On a molecular scale there is no sharp boundary between hydrodynamically stagnant and movable solvent molecules. As discussed In sec. 2.2, the, say tangential, diffusion coefficient of water near many surfaces may be somewhat lower than in bulk, but it is not zero. The very existence of ionic conduction In the layer(s) adjacent to surfaces also points to non-zero mobility. Yet, phenomenologically such layers behave as immobilized. This looks like a paradox, but the phenomenon is encountered in other places as well. For Instance, a few percent of gelatin added to water may hydrodynamically immobilize the liquid completely, without markedly impairing ionic conduction or self-diffusion of dissolved ions. Macroscopic immobilization of a fluid is not in conflict with mobility on a molecular sceile. 

Neutron Scattering (QENS) [13], and Temporal-Analysis of Products (TAP) [14] (Figure 6, solid symbols). The isobutane diffusivities determined by the macroscopic methods, such as MEMBRANE, TAP, and FR show reasonable agreement. The self-difiusivity coefficient derived from QENS is about one order of magnitude lower. The E, values of 34 kJ mof and 25 kJ mol obtained by the MEMBRANE and the TAP methods, respectively, are higher than those determined by methods where the conditions of the mesurements correspond to sorption equilibrium or quasi-equilibrium, such as QENS (17 kJ mol )orFR(21 kJ mol ). 

Compared to Nafion , a stronger confinement of water in the narrow channels of the sulfonated aromatic polymers leads to a significantly lower dielectric constant of the waters of hydration (20 compared to 64 in fully hydrated Nafion [185,186]). Of particular relevance to macroscopic models are the diffusion coefficients of water. As the amount of water sorbed by the membrane increases and molecular-scale effects are reduced, the properties approach those of bulk water on the molecular scale. Figme 26 shows the trend in proton mobility. Da, and water self-diffusion, Dh20. for Nafion and the sulfonated polyetherketone membrane [134]. 

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