Characteristic time of the diffusion

FAD is able to measure precisely the characteristic time of the diffusive process, Xj, and, in some cases, the characteristic time Xj of the loss process (with a lower precision). The ratio X2/Xj seems to depend on the solvent, but not on the tempera-... 

The contribution of the subsurface-surface transfer has also been considered by Ravera et al. [69], however this will not be discussed here further in detail. The complete adsorption kinetics problem consists now of the transport by diffusion and the boundary condition (4.30). In order to estimate the influence of the three main processes going on simultaneously, a comparison of the characteristic times is helpful. The characteristic time of the diffusion process is given by the diflusion relaxation time as defined above in Eq. (4.26), which depends on the diffusion coefficient D and the surface properties of the surfactant expressed by the ratio Fo/co- The characteristic time of the orientation process is found by assuming the other processes to be at equilibrium [69]... 

Ti relaxation time of the fast step of the micellisation process T2 relaxation time of the slow step of the micellisation process Td characteristic time of the diffusion to the surface CO angular frequency... 

A convenient method of predicting whether the transport of a solvent in an amorphous polymer is FicMan or non-Fickian is to examine the diffusional Deborah number. Deg. This number is defined as the ratio of a characteristic relaxation time for the polymer-solvent system to the characteristic time of the diffusion process (Vrentas et al., 1975). Fickian transport is observed when either Dee < 0.1 or Dee >10, whereas non-Fickian transport is observed when Dee = 1. 

The characteristic time of the tliree-pulse echo decay as a fimction of the waiting time T is much longer than the phase memory time T- (which governs the decay of a two-pulse echo as a function of x), since tlie phase infomiation is stored along the z-axis where it can only decay via spin-lattice relaxation processes or via spin diffusion. 

This simplified description of molecular transfer of hydrogen from the gas phase into the bulk of the liquid phase will be used extensively to describe the coupling of mass transfer with the catalytic reaction. Beside the Henry coefficient (which will be described in Section 45.2.2.2 and is a thermodynamic constant independent of the reactor used), the key parameters governing the mass transfer process are the mass transfer coefficient kL and the specific contact area a. Correlations used for the estimation of these parameters or their product (i.e., the volumetric mass transfer coefficient kLo) will be presented in Section 45.3 on industrial reactors and scale-up issues. Note that the reciprocal of the latter coefficient has a dimension of time and is the characteristic time for the diffusion mass transfer process tdifl-GL=l/kLa (s). 

The criterion of homochronity tk/(P, where d is the characteristic dimension of the system, k is the thermal diffusivity and r is the characteristic time of the chemical reaction. Due to the strong dependence of the chemical reaction rate on temperature (see the next two criteria), it is necessary to define the temperature to which the quantity r relates. We will relate it to the theoretical temperature of combustion. 

Discussion. We can now propose a coarse description of the paraffinic medium in a lamellar lyotropic mesophase (potassium laurate-water). Fast translational diffusion, with D 10"6 at 90 °C, occurs while the chain conformation changes. The characteristic times of the chain deformations are distributed up to 3.10"6 sec at 90 °C. Presence of the soap-water interface and of neighboring molecules limits the number of conformations accessible to the chains. These findings confirm the concept of the paraffinic medium as an anisotropic liquid. One must also compare the frequencies of the slowest deformation mode (106 Hz) and of the local diffusive jump (109 Hz). When one molecule wants to slip by the side of another, the way has to be free. If the swinging motions of the molecules, or their slowest deformation modes, were uncorrelated, the molecules would have to wait about 10"6 sec between two diffusive jumps. The rapid diffusion could then be understood if the slow motions were collective motions in the lamellae. In this respect, the slow motions could depend on the macroscopic structure (lamellar or cylindrical, for example)... 

The characteristic time of this diffusion was estimated by carrying out the molecular dynamic relaxation of the film surface within the limits of the above model at 500°C. In MD calculations, the pair interaction energy between atoms is approximated by the Buckingham pair potential (Zr O, O-O) (see Table 9.4). To describe covalent bonds more correctly, a three-body O-Zr-O term in the Stillinger-Weber form was introduced in addition to the Coulomb term. 

Here, L is the length of the electrode, a the conductivity, D the diffusion coefficient of species c and Co the specific capacity of the electrode, d is a measure of the ratio of the characteristic rates of the diffusion of species c and migration, d > 1 means that the characteristic time of migration is shorter than that of diffusion, or, in other words, that the transport process associated with the inhibitor is faster than the one associated with the activator. 

Due to the relatively long lifetime of the sensitiser triplet state and the possibility of integrating data on the stilbene photoisomerisation, the apparent characteristic time of the method can reach hundreds of seconds. This unique property of the cascade system and, therefor triplet-photochrome technique, allows the investigation of slow diffusion processes, including encounters of proteins in membranes using very low concentrations of both the triplet and photochrome probes. 

Measurements of the dynamic properties of the surface water, particularly NMR measurements, have shown that the characteristic time of the water motion is slower than the bulk water value by a factor of less than 100. The motion is anisotropic. There is litde or no irrotadonally bound water. Study of a protein labeled covalently with a nitroxide spin probe (Polnaszek and Bryant, 1984a,b) has shown that the diffusion constant of the surface water is about 5-fold below the bulk water value. The NMR results are in agreement with measurements of dielectric relaxation of water in protein powders (Harvey and Hoekstra, 1972). 

Local motions which occur in macromolecular systems can be probed from the diffusion process of small molecules in concentrated polymeric solutions. The translational diffusion is detected from NMR over a time scale which may vary from about 1 to 100 ms. Such a time interval corresponds to a very large number of elementary collisions and a long random path consequently, details about mechanisms of molecular jump are not disclosed from this NMR approach. However, the dynamical behaviour of small solvent molecules, immersed in a polymer melt and observed over a long time interval, permits the determination of characteristic parameters of the diffusion process. Applying the Langevin s equation, the self-diffusion coefficient Ds is defined as... 

The technique is limited at high velocity, when the characteristic time of the data acquisition system is comparable to 1/v, leading to a decrease of the signal to noise ratio. It is also limited at low velocities, when the bleached pattern relaxes by diffusion faster than the appearance of the oscillations due to the flow. The easily available range is typically lO pm/s high velocity limit is reached, we switch to a bulk method which uses another bleaching pattern only one beam (width - 50 Xm) is shined in the sample, at an angle smaller... 

We shall show how these deviations in water are essentially related to the multiplicative nature of the stochastic forces in the model used to picture the H-bond dynamics (Chapters II and X). Starting from Eq. (4.5), if the characteristic times of the H-bond dynamics are not infinitely small, super-Bumett coefficients appear in the diffusion equation. To do tffis, rather than performing calculation on the basis of the variable i) defined by Eq. (3.2) we shall use as basis the operator defined by Eq. (12 ) ... 

The diffusion equations involved in our theoretical analysis, which are derived via the AEP from the fluctuating reduced model of Section IV, can be regarded as a five-state version of the Anderson two-state model supplemented by a quantitative description of the secondary process. We would especially stress that, in accordance with the point of view of other authors, the characteristic time of the principal dielectric relaxation band roughly coincides with the residence time in the structured part of the liquid. ... 

Now, we shall demonstrate that the characteristic times of the normal diffusion process, namely, the inverse of the smallest nonvanishing eigenvalue 1 //.], the integral and effective relaxation times xint and xef obtained in [8,62,63], also allow us to evaluate the dielectric response of the system for anomalous diffusion using the two-mode approximation just as normal diffusion (Ref. 8, Section 2.13). Here, we can use known equations for xint, x,f, and X for the normal diffusion in the potential Eq. (163) [8,62,63] these equations are... 

Thus the anomalous relaxation in a double-well potential is effectively determined by the bimodal approximation, Eq. (159) the characteristic times of the normal diffusion process—namely, the inverse of the smallest nonvanishing eigenvalue, the integral, and effective relaxation times—appear as time... 

In the present section, it is demonstrated how the linear response of an assembly of noninteracting polar Brownian particles to a small external field F applied parallel and perpendicular to the bias field Fo may be calculated in the context of the fractional noninertial rotational diffusion in the same manner as normal rotational diffusion [8]. In order to carry out the calculation, it is assumed that the rotational Brownian motion of a particle may be described by a fractional noninertial Fokker-Planck (Smoluchowski) equation, in which the inertial effects are neglected. Both exact and approximate solutions of this equation are presented. We shall demonstrate that the characteristic times of the normal diffusion process, namely, the integral and effective relaxation times obtained in Refs. 8, 65, and 67, allow one to evaluate the dielectric response for anomalous diffusion. Moreover, these characteristic times yield a simple analytical equation for the complex dielectric susceptibility tensor describing the anomalous relaxation of the system. The exact solution of the problem reduces to the solution of the infinite hierarchies of differential-recurrence equations for the corresponding relaxation functions. The longitudinal and transverse components of the susceptibility tensor may be calculated exactly from the Laplace transform of these relaxation functions using linear response theory [72]. 

Changes in microstructure of the suspension become important when the diffusion time fj becomes long compared to the characteristic time of the process, fp. This number hcis been discussed earlier as the De number. The importance of convection relative to diffusion is compared in the Peclet number Pe (in which u is the fluid velocity). The importance of convection forces relative to the dispersion force is compared in Nf just as the dispersion force compared to the Brownian force. The electrical force compared to the dispersion or Brownian force is given by N. The particle size compared to the range of the electrical force is compared in UK. 

Thermal lensing contribution to the measured nonlinear optical properties. If the pulse duration is longer than the characteristic time of the heat diffusion in the medium, or if this time is itself longer than the delay between successive pulses, material heating may lead to an observable transient thermal lens phenomenon [120, 165, 212, 218, 219], This can show itself, in experiments, with characteristics similar to those of a pure (electronic) Kerr effect. There have been some attempts to extract the respective values of the thermal and electronic contributions to y from z-scan measurements [136, 160, 165, 166, 175, 220], However, de Nalda et al. proved later that this method was not reliable enough to get quantitative results [219],... 

In some cases another kind of heterogeneous nucleation may be observed at the droplet surface. High solute concentration is created at the drop surface, and when the droplet is small enough, so that the characteristic time of the solute diffusion x dc = is smaller than both the evaporation... 

Two timescales can be distinguished in the adsorption process of ionic species. The first timescale is characterized by the diffusion relaxation time of the EDL, = 1 / (D,k /) see Equations 5.32 and 5.34 above. It accounts for the interplay of electrostatic interactions and diffusion. The second scale is provided by the characteristic time of the used experimental method, tgxp, that is, the minimum interfacial age that can be achieved with the given method typically,... 

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