Constant diffusion

Kj is the equilibrium constant for the jth reaction, is the dimensionless diffusion constant of oxygen, and Pe, the Peclet number, is defined as the ratio of flow to diffusion of oxygen. 

Data on diffusion constants of impurity atoms are important to calculate incorporation of dopant impurities in silicon crystals, which control the electronic properties of silicon. However, there are fewer data on impurity diffusion constants except for that reported by Turovskii [85], Kodera [86], Shashkov and Gurevich [87], Gnesin and Raichenko [88], and Keller and Miihlbauer [89]. In the Landholt-Bomstein data book, Miihlbauer recalculated reported data using the kinematic viscosity of molten silicon (v = 3.5 x 10 m /s) [91]. Diffusion constants reported are tabulated in Table 4.3. 

Using microgravity conditions, diffusion constants of molten tin and lead were measured successfully [13,94] diffusion constants were reported to be proportional to the square of the absolute temperature, as follows  

However, for molten semiconductors, particularly for molten silicon, the temperature dependence of diffusion constants has not been made clear. The measurement of the diffusion constant under microgravity was a driving force to improve the measurement techniques development of diffusion cells made of ceramics, appHcation of a shear cell technique, and so on [94]. Application of a strong static magnetic field could also be effective in suppressing convection on Earth [95]. The diffusion 

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An appropriate value of 7 for a system modeled by the simple Langevin equation can also be determined so as to reproduce observed experimental translation diffusion constants, Dt in the diffusive limit, Dt is related to y hy Dt = kgTmy. See [22, 36], for example. 

Ifihe Bath relaxation con start t, t, is greater than 0.1 ps. yon should be able Lo calculate dyriani ic p roperlies, like time correlation fun c-tioris and diffusion constants, from data in the SNP and/or C.SV files (sec "Collecting Averages from Simulations"... 

Tlere, y Is the friction coefficien t of the solven t. In units of ps, and Rj is th e random force im parted to th e solute atom s by the solvent. The friction coefficien t is related to the diffusion constant D oflh e solven l by Em stem T relation y = k jT/m D. Th e ran doin force is calculated as a ratulom number, taken from a Gaussian distribn-... 

Here, y is the friction coefficient of the solvent, in units of ps and Rj is the random force imparted to the solute atoms by the solvent. The friction coefficient is related to the diffusion constant D of the solvent by Einstein s relation y = kgT/mD. The random force is calculated as a random number, taken from a Gaussian distribu-... 

The spreading rate of a polymer droplet on a surface has been measured (363,364). The diffusion constant was at least an order of magnitude smaller than that of the bulk. The monomer—surface friction coefficient for polystyrene has been measured on a number of surfaces and excellent... 

The flux is related both to an equiUbrium quantity, ie, the chemical activity, through the solubiUty and to a nonequilihrium coefficient D. In contrast, for an ideal membrane emphasizes the path length and the diffusion constant. Binding and heterogeneity of the membrane may compHcate these simple relationships. 

Sluggish chain mobility and low free volume result in low diffusion constants, and when combined with low solubiUty of gases lead to very low permeabihty. The diffiisivity of several gases in butyl mbber and natural mbber are shown in Table 3 (82) (see Barrier polymers). 

In the presence of a potential U(r) the system will feel a force F(rj,) = — ViT/(r) rj,. There will also be a stochastic or random force acting on the system. The magnitude of that stochastic force is related to the temperature, the mass of the system, and the diffusion constant D. For a short time, it is possible to write the probability that the system has moved to a new position rj,+i as being proportional to the Gaussian probability [43]... 

The diffusion constant should be small enough to damp out inertial motion. In the presence of a force the diffusion is biased in the direction of the force. When the friction constant is very high, the diffusion constant is very small and the force bias is attenuated— the motion of the system is strongly overdamped. The distance that a particle moves in a short time 8t is proportional to... 

The comparison with experiment can be made at several levels. The first, and most common, is in the comparison of derived quantities that are not directly measurable, for example, a set of average crystal coordinates or a diffusion constant. A comparison at this level is convenient in that the quantities involved describe directly the structure and dynamics of the system. However, the obtainment of these quantities, from experiment and/or simulation, may require approximation and model-dependent data analysis. For example, to obtain experimentally a set of average crystallographic coordinates, a physical model to interpret an electron density map must be imposed. To avoid these problems the comparison can be made at the level of the measured quantities themselves, such as diffraction intensities or dynamic structure factors. A comparison at this level still involves some approximation. For example, background corrections have to made in the experimental data reduction. However, fewer approximations are necessary for the structure and dynamics of the sample itself, and comparison with experiment is normally more direct. This approach requires a little more work on the part of the computer simulation team, because methods for calculating experimental intensities from simulation configurations must be developed. The comparisons made here are of experimentally measurable quantities. 

The amplitude of the elastic scattering, Ao(Q), is called the elastic incoherent structure factor (EISF) and is determined experimentally as the ratio of the elastic intensity to the total integrated intensity. The EISF provides information on the geometry of the motions, and the linewidths are related to the time scales (broader lines correspond to shorter times). The Q and ft) dependences of these spectral parameters are commonly fitted to dynamic models for which analytical expressions for Sf (Q, ft)) have been derived, affording diffusion constants, jump lengths, residence times, and so on that characterize the motion described by the models [62]. 

Center-of-mass translational motion in MD simulations is often quantified in tenns of diffusion constants, D, computed from the Einstein relation. 

Analysis of neutron data in terms of models that include lipid center-of-mass diffusion in a cylinder has led to estimates of the amplitudes of the lateral and out-of-plane motion and their corresponding diffusion constants. It is important to keep in mind that these diffusion constants are not derived from a Brownian dynamics model and are therefore not comparable to diffusion constants computed from simulations via the Einstein relation. Our comparison in the previous section of the Lorentzian line widths from simulation and neutron data has provided a direct, model-independent assessment of the integrity of the time scales of the dynamic processes predicted by the simulation. We estimate the amplimdes within the cylindrical diffusion model, i.e., the length (twice the out-of-plane amplitude) L and the radius (in-plane amplitude) R of the cylinder, respectively, as follows ... 

We finish this section by comparing our results with NMR and incoherent neutron scattering experiments on water dynamics. Self-diffusion constants on the millisecond time scale have been measured by NMR with the pulsed field gradient spin echo (PFGSE) method. Applying this technique to oriented egg phosphatidylcholine bilayers, Wassail [68] demonstrated that the water motion was highly anisotropic, with diffusion in the plane of the bilayers hundreds of times greater than out of the plane. The anisotropy of... 

Table 3 Diffusion Constants and Rotational CoiTelation Tunes of Water Molecules from an MD Simulation of a Fully Flydrated Fluid Phase DPPC Bilayer ...

Chemicals have to pass through either the skin or mucous membranes lining the respiratory airways and gastrointestinal tract to enter the circulation and reach their site of action. This process is called absorption. Different mechanisms of entry into the body also greatly affect the absorption of a compound. Passive diffusion is the most important transfer mechanism. According to Pick s law, diffusion velocity v depends on the diffusion constant (D), the surface area of the membrane (A), concentration difference across the membrane (Ac), and thickness of the membrane (L)... 

Fig. 7 gives an example of such a comparison between a number of different polymer simulations and an experiment. The data contain a variety of Monte Carlo simulations employing different models, molecular dynamics simulations, as well as experimental results for polyethylene. Within the error bars this universal analysis of the diffusion constant is independent of the chemical species, be they simple computer models or real chemical materials. Thus, on this level, the simplified models are the most suitable models for investigating polymer materials. (For polymers with side branches or more complicated monomers, the situation is not that clear cut.) It also shows that the so-called entanglement length or entanglement molecular mass Mg is the universal scaling variable which allows one to compare different polymeric melts in order to interpret their viscoelastic behavior. 

Thus, in order to reproduce the effect of an experimentally existing activation barrier for the scission/recombination process, one may introduce into the MC simulation the notion of frequency , lo, with which, every so many MC steps, an attempt for scission and/or recombination is undertaken. Clearly, as uj is reduced to zero, the average lifetime of the chains, which is proportional by detailed balance to Tbreak) will grow to infinity until the limit of conventional dead polymers is reached. In a computer experiment Lo can be easily controlled and various transport properties such as mean-square displacements (MSQ) and diffusion constants, which essentially depend on Tbreak) can be studied. 

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