Self diffusion process

The molecular weight (M) dependence of the steady (stationary) primary nucleation rate (I) of polymers has been an important unresolved problem. The purpose of this section is to present a power law of molecular weight of I of PE, I oc M-H, where H is a constant which depends on materials and phases [20,33,34]. It will be shown that the self-diffusion process of chain molecules controls the Mn dependence of I, while the critical nucleation process does not. It will be concluded that a topological process, such as chain sliding diffusion and entanglement, assumes the most important role in nucleation mechanisms of polymers, as was predicted in the chain sliding diffusion theory of Hikosaka [14,15]. 

The significant difference in H between FCSCs and ECSCs indicates that the M dependence of V can not be controlled by the self-diffusion process within the melt (the first stage) as proposed by Hoffman et al. [40], but it should be controlled by the surface diffusion process (the second stage) as shown in Fig. 27. 

Figure 1.49. A schematic of a tagged (red) molecule in the self-diffusion process.

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... 

In Figure 5.2a, the tracer self-diffusion process, and in Figure 5.2b, the self-diffusion process, are represented. 

For a vacancy mechanism, in a self-diffusion process, the jump frequency, T, of an atom to a given adjacent site is given by [4]... 

The only microscopic feature suU remembered by the system when described by Eq. (4.3) is that of the H-bond dynamics as simulated by the variable if. As mentioned in the introduction, the integrations in time appearing on the right-hand side of Eq. (4.3) are made legitimate by the fact that the correlation functions dielectric relaxation and self-diffusion processes. Henceforth we shall neglect the third term on the right-hand side of Eq. (4.3) concerning rototranslational phenomena. This assumption allows us to obtain two independent equations for rotation and translation, respectively. 

Rotational diffusion, which in fact is a self-diffusion process in the angular domain, leads to an exchange of magnetization contributions between different angular orientations The general operator which describes rotational diffusion for an ellipsoidal particle of rotational symmetry can be adapted to the finite element scheme which leads... 

The mechanism of proton diffusion, usually referred as the Grotthuss mechanism, differs from that of other ions. Instead of a self-diffusion process, where the mass and the charge are inseparable, the proton diffusion is the movement of the protonic charge independently from the transport of the particle. This mode of propagation leads to the diffusion of the proton 0 + = 9.3 x 10 cm s being faster than the diffusion of all other ions (D <2 x 10 cm s ), and of the selfdiffusion of the water molecules in water as a solvent (for details see Ref [40]). 

In an early test case [6], the self-diffusion process of a quantum particle in a classical solvent was studied. The solvent in this case was a Lennard-Jones fluid with parameters given in Paper IV, while the solute-solvent pair potential had the simple form [75] v(r) = [B/(C + r )-where A = 0.665, B = 89,099, C= 12,608 (in atomic units). The simulation consisted of 512 total particles at a temperature T = 309 K... 

In Paper IV, the self-diffusion process in fluid neon was also studied with CMD using the pairwise pseudopotential method. In Fig. 17 the centroid velocity time correlation function is plotted for quantum neon using the pseudopotential method and for classical neon. When the quantum mechanical nature of the Ne atoms is taken into account, the diffusion constant is reduced by a small fraction. In the gas phase and to some degree in liquids, the diffusion process can be viewed as a sequence of two-body collisions, the frequency of which depends on the collision cross section. Because the quantum centroid cross section is larger than the corresponding classical value, the quantum diffusion constant is found... 

A similar value of the activation energy for the crystallization of SiC like the second one mentioned above was foimd by Yoshi [156] investigating the crystallization behavior of amorphous Sli. Q films prepared by RF sputtering. He gives a value of 685 KJ mol . In that cormection, it may also be interesting to mention that the activation energy for self diffusion processes of Si and C in SiC was found to be still higher 8 eV = 772 KJ moP [157]. 

Evangelakis, G.A., Papanicolaou, N.l. Adatom self-diffusion processes on (001) copper surface by molecular dynamics. Surf. Sci. 1996,347,376. 

Our discussion so far has considered the impurities to be fixed in the ice lattice but this is obviously not exactly true for they can move slowly by solid-state diffusion. Similarly individual water molecules migrate in a self-diffusion process which can be followed by using isotopically labelled molecules. The structure and mass of these are very little different from those of ordinary water molecules, so that a study of their diffusion gives information about self-diffusion in the ice crystal. The only case where this is not obviously true is for the isotopes of hydrogen, where the mass ratio to the proton is considerable, but the nature of the experimental results enables us to sidestep this difficulty. 

During surfactant dissolution the two diffusion processes can be identified. On the molecular scale a molecule undergoes self diffusion where the diffusion coefficient is determined from its mean squared displacement. Various NMR techniques have been used to study quantitatively self diffusion processes (21-24), It is important to note that each component in the system will have a self diffusion coefficient. Diffusion coefficients for a number of mesophase systems have been collected where values of order 10 - 10 m s" were reported (25). The self diffusion coefficients of the solvent are typically reduced no more than an order of magnitude in the presence of mesophases which essentially act as obstacles to the solvent (25, 26). 

The self-diffusion process of the constituent species in a microemulsion can speak in favor of their degrees of freedom and hence their residential status in different domains. For highlighting information on the process by the NMR method, the articles of Zana and Lang [21], Nilsson and Lindman [22], Lindman and Wennerstrom [23], and others [24-27] may be consulted. 

Very slow diffusion and self-diffusion processes in solids can be determined using radiotracers. Two methods are used. In one, a thin surface containing tracer is applied to the sample. After some time has elapsed, the sample is sliced parallel to the surface and the radioactivity of each slice is measured. Another method is to use an a- or jS-active tracer. As the tracer diffuses into the sample, a direct measurement of the surface radioactivity gives information about the diffusion process. 

Here is the appropriate microscopic current and tlim denotes the thermodynamic limit. (For reviews of this theory, see Zwanzig and Steele. ) We will here focus our attention on the self-diffusion process, both because it is the simplest case from a pedagogical standpoint and because it is the one that has been most extensively studied by means of molecular dynamics calculations. Calculations of the coefficients of viscosity and thermal conductivity and the associated time-correlation functions have been reported by Alder et for hard spheres. 

The self-diffusion process is discussed by Dorfman, " who writes the self-diffusion current and conservation law as... 

In addition to combining self-diffusion coefficient measurements with impedance analysis to determine ionicity as describe above, a novel method has been developed to determine the fraction ofions in ILs exclusively from self-diffusion coefficients obtained using PFG-NMR [18]. EquiHbrium constants of the ionization process from measured ion self-diffusion coefficients were calculated using this method. Enthalpy and entropy changes of ionization and ion self-diffusion processes have been obtained for a series of ionic liquids using this method. 

The self-diffusion of Ba2+ in hydrated natural mordenite was studied. From the variation of the self-diffusion coefficient with temperature, the energy of activation and the pre-exponential factor for the self-diffusion process were calculated from the Arrhenius equations ... 

In chajiter.. tx/o x/e x/ill discuss tx/o general feature of the self diffusion process in liquids from triple to critical point the dependence on the peculiarities of the intermolecular potential and the meaning of the diffusion as an "activated process". 

SOME GENERAL FEATURES OF THE SELF DIFFUSION PROCESS IN LIQUIDS. 

The direct NMR method for determining translational difiFusion constants in liquid crystals was described previously. The indirect NMR methods involve measurements of spin-lattice relaxation times (Ti,Ti ),Tip) [7.45]. Prom their temperature and frequency dependences, it is hoped to gain information on the self-diflPusion. In favorable cases, where detailed theories of spin relaxation exist, difiFusion constants may be calculated. Such theories, in principle, can be developed [7.16] for translational difiFusion. However, until recently, only a relaxation theory of translational difiFusion in isotropic hquids or cubic solids was available [7.66-7.68]. This has been used to obtain the difiFusion correlation times in nematic and smectic phases [7.69-7.71]. Further, an average translational difiFusion constant may be estimated if the mean square displacement is known. However, accurate determination of the difiFusion correlation times is possible in liquid crystals provided that a proper theory of translational difiFusion is available for liquid crystals, and the contribution of this difiFusion to the overall relaxation rate is known. In practice, all of the other relaxation mechanisms must first be identified and their contributions subtracted from the observed spin relaxation rate so as to isolate the contribution from translational difiFusion. This often requires careful measurements of proton Ti over a very wide frequency range [7.72]. For spin - nuclei, dipolar interactions may be modulated by intramolecular (e.g., collective motion, reorientation) and/or intermolecular (e.g., self-diffusion) processes. Because the intramolecular (Ti ) and intermolecular... 

The dynamic equilibrium of micelles involves two characteristic times a fast one related to monomer exchange and a slow one related to micelle breakage. The first one is unlikely to affect the micelles self-diffusion process that we studied. 

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