Timescale of the diffusion

This is by orders of magnitude less than the timescale of the diffusive processes, Td — ... 

The overlaps between SPs in semidilute concentrations can be thought of in very similar terms to the entanglements defined above. Supramolecular interactions create large stmctures that physically interact to determine the mechanical response (in this case, viscous flow). The primary relaxation is the diffusion of an SP that is effectively intact on the timescale of the diffusion process. Thus, at a fixed concentration, the SP properties in dilute solution are therefore quite similar to those of covalent polymers of the same molecular weight and molecular weight distribution. 

Reverse osmosis, pervaporation and polymeric gas separation membranes have a dense polymer layer with no visible pores, in which the separation occurs. These membranes show different transport rates for molecules as small as 2-5 A in diameter. The fluxes of permeants through these membranes are also much lower than through the microporous membranes. Transport is best described by the solution-diffusion model. The spaces between the polymer chains in these membranes are less than 5 A in diameter and so are within the normal range of thermal motion of the polymer chains that make up the membrane matrix. Molecules permeate the membrane through free volume elements between the polymer chains that are transient on the timescale of the diffusion processes occurring. 

The important point here in relation to surface reactions is the timescale of the diffusion. If the surface atoms diffuse once in an hour, the surface can effectively be considered as a checkerboard (and since good images of surface atoms are often observed in STM at ambient temperature, diffusion must be slow), whereas if it occurs once every microsecond it may not be considered rigid with respect to surface reactions taking place on that kind of timescale. 

For a semi-infinite medium where the diffusion takes place only in the direction x>0 into a medium initially of zero diffusant concentration and with the concentration at the borderline x=0 instantaneously established on a constant concentration Q for the whole timescale of the diffusion experiment, the concentration at any position x in the medium and at any time t could be given by Eq. (22) [36]. 

Figure 3.3(a) shows the predicted moisture absorption through the thickness of a carbon fibre laminate [5] after 3 days in 96% RH at 50 C. This illustrates the timescale of the diffusion process. Thus it can take 13 years in a composite laminate of thickness 12 mm before saturation is reached. In service, materials are subjected to variations in temperature and kinetic processes such as diffusion follow the Arrhenius law ... 

The timescale of the diffusion process is determined by the use of the diffusion Deborah, number De, given by the following equation ... 

That volume bounded by the distance away from the electrode over which a redox-active species can diffuse to the electrode surface, within the timescale of the experiment being undertaken. From considerations of random walk in one dimension it can be shown that the distance / which a species moves in a time / is given by ... 

Mass transport can be by migration, convection or diffusion. As discussed in chapter 1, in the presence of strong electrolyte migration can be neglected, as can convection if the solution is unstirred, at a uniform temperature and the timescale of the experiment is short (i.e. a few seconds). Thus, we can make the first distinction between electrode reactions that are dominated by step 1, diffusion-controlled, and those for which steps 1 and 2 contribute to the overall observed rate. 

The thin-layer thickness calculated in this manner may well have some serious errors associated with it due to sample purity, errors in the weighing out of the solution, the diffusion of species near the thin layer into it within the timescale of the experiment, etc. 

It is useful to point out here that we frequently encounter partial steady-states. An important example is the case where the diffusion process is much faster than a surface process, and thus a quasi-steady-state is reached for the diffusion concentration profile at each changing concentration of the surface. This distinction between different timescales of the processes can lead to a significant simplification of complex problems, see end of Section 4.3 or Chapter 4 in this volume. 

The phenomenological approach does not preclude a consideration of the molecular origins of the characteristic timescales within the material. It is these timescales that determine whether the observation you make is one which sees the material as elastic, viscous or viscoelastic. There are great differences between timescales and length scales for atomic, molecular and macromolecular materials. When an instantaneous deformation is applied to a body the particles forming the body are displaced from their normal positions. They diffuse from these positions with time and gradually dissipate the stress. The diffusion coefficient relates the distance diffused to the timescale characteristic of this motion. The form of the diffusion coefficient depends on the extent of ordering within the material. 

For a concentrated system this represents the ratio of the diffusive timescale of the quiescent microstructure to the convection under an applied deforming field. Note again that we are defining this in terms of the stress which is, of course, the product of the shear rate and the apparent viscosity (i.e. this includes the multibody interactions in the concentrated system). As the Peclet number exceeds unity the convection is dominating. This is achieved by increasing our stress or strain. This is the region in which our systems behave as non-linear materials, where simple combinations of Newtonian or Hookean models will never satisfactorily describe the behaviour. Part of the reason for this is that the flow field appreciably alters the microstructure and results in many-body interactions. The coupling between all these interactions becomes both philosophically and computationally very difficult. 

As with spherical particles the Peclet number is of great importance in describing the transitions in rheological behaviour. In order for the applied flow field to overcome the diffusive motion and shear thinning to be observed a Peclet number exceeding unity is required. However, we can define both rotational and translational Peclet numbers, depending upon which of the diffusive modes we consider most important to the flow we initiate. The most rapid diffusion is the rotational component and it is this that must be overcome in order to initiate flow. We can define this in terms of a diffusive timescale relative to the applied shear rate. The characteristic Maxwell time for rotary diffusion is... 

At the merging temperature the a-relaxation time matches that of the j8-re-laxation. Around this temperature, the dynamic structure factor has to be gen-erahzed, in order to include also the segmental diffusion process underlying the a-process. The )0-process can be considered as a local intrachain relaxation process, which takes place within the fixed environment set by the other chains. When the segmental diffusion reaches the timescale of the local relaxation, given atoms and molecular groups will noticeably participate simultaneously... 

We next consider the effective force balance for all >N variables, while treating the system as an unconstrained system. For simplicity, we consider the case in which the crossover from ballistic motion to diffusion occurs at a timescale much less than any characteristic relaxation time for vibrations of the hard coordinates, so that the vibrations are overdamped, but in which the vibrational relaxation times are much less than any timescale for the diffusion of the soft coordinates. In this case, we may assume local equilibration of all 3N momenta at timescales of order the vibration time. Repeating the analysis of the Section V.A, while now treating all 3N coordinates as unconstrained, yields an effective force balance... 

The analytical solution of the Smoluchowski equation for a Coulomb potential has been found by Hong and Noolandi [13]. Their results of the pair survival probability, obtained for the boundary condition (11a) with R = 0, are presented in Fig. 2. The solid lines show W t) calculated for two different values of Yq. The horizontal axis has a unit of r /D, which characterizes the timescale of the kinetics of geminate recombination in a particular system For example, in nonpolar liquids at room temperature r /Z) 10 sec. Unfortunately, the analytical treatment presented by Hong and Noolandi [13] is rather complicated and inconvenient for practical use. Tabulated values of W t) can be found in Ref. 14. The pair survival probability of geminate ion pairs can also be calculated numerically [15]. In some cases, numerical methods may be a more convenient approach to calculate W f), especially when the reaction cannot be assumed as totally diffusion-controlled. 

It has then to be concluded that the charge transfer in the ferrocene/ferroci-nium couple in this specific solvent is a fast process at the timescale of the hole diffusion in the semiconductor space charge layer. However, at the present time, it seems that the constraints arising from the construction of a perfectly tight device will hinder the development of these electrochemical photovoltaic cells. 

When the electrochemical properties of some materials are analyzed, the timescale of the phenomena involved requires the use of ultrafast voltammetry. Microelectrodes play an essential role for recording voltammograms at scan rates of megavolts-per-seconds, reaching nanoseconds timescales for which the perturbation is short enough, so it propagates only over a very small zone close to the electrode and the diffusion field can be considered almost planar. In these conditions, the current and the interfacial capacitance are proportional to the electrode area, whereas the ohmic drop and the cell time constant decrease linearly with the electrode characteristic dimension. For Cyclic Voltammetry, these can be written in terms of the dimensionless parameters yu and 6 given by... 

The fluorescence spectra of the N-P -N were examined in order to obtain further information on the cavities experienced by the N-P -N molecules. At a loading level of 1 X 10-3 mol/g-film, the N-P -N show monomeric and weak intramolecular excimer emissions. Because the rate of translational diffusion of the naphthoate groups in the LDPE films is slow on the timescale of the decay of the excited singlet states [32,35,112-114], the excimers probably form from chro-mophores that are at or near excimerlike conformations prior to excitation. On the basis of the relative weakness of the intramolecular excimer emissions, it is pro-... 

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