Diffusion coefficients, effects theory

The experiments which yielded the diffusion coefficients for acetaminophen in PNIPAAm gel in Fig. 16 also yielded the corresponding partition coefficients. While the diffusion coefficients fit theory, the partition coefficients as plotted in Fig. 18 do not at all. In fact, a trend opposite to theory is observed as the partition coefficients are seen to increase as the gel swelling decreases. In fact, above the transition temperature of the gel, at 35 °C, the partition coefficient is seven times the maximum possible size exclusion coefficient, 1. This implies the dominance of hydrophobic effects over steric effects, since acetaminophen is a relatively small, nonionic but hydrophobic solute, and while the gel mesh size shrinks with increasing temperature, its level of hydrophobicity increases with temperature. 

The theory of seaweed formation does not only apply to solidification processes but in fact to the completely different phenomenon of a wettingdewetting transition. To be precise, this applies to the so-called partial wetting scenario, where a thin liquid film may coexist with a dry surface on the same substrate. These equations are equivalent to the one-sided model of diffusional growth with an effective diffusion coefficient which depends on the viscosity and on the thermodynamical properties of the thin film. 

The objective of most of the theories of transport in porous media is to derive analytical or numerical functions for the effective diffusion coefficient to use in the preceed-ing averaged species continuity equations based on the structure of the media and, more recently, the structure of the solute. 

Perrin model and the Johansson and Elvingston model fall above the experimental data. Also shown in this figure is the prediction from the Stokes-Einstein-Smoluchowski expression, whereby the Stokes-Einstein expression is modified with the inclusion of the Ein-stein-Smoluchowski expression for the effect of solute on viscosity. Penke et al. [290] found that the Mackie-Meares equation fit the water diffusion data however, upon consideration of water interactions with the polymer gel, through measurements of longitudinal relaxation, adsorption interactions incorporated within the volume averaging theory also well described the experimental results. The volume averaging theory had the advantage that it could describe the effect of Bis on the relaxation within the same framework as the description of the diffusion coefficient. 

The tortuous-path and barrier theories consider the effects of the media on the electrophoretic mobility in a way similar to the effect of media on diffusion coefficients discussed in a previous section of this chapter. The tortuons-path theory seeks to determine the effect of increased path length on electrophoretic mobility. The barrier theory considers the effects of the barrier or media conductivity on the electrophoretic mobility. 

The EMD studies are performed without any external electric field. The applicability of the EMD results to useful situations is based on the validity of the Nemst-Planck equation, Eq. (10). From Eq. (10), the current can be computed from the diffusion coefficient obtained from EMD simulations. It is well known that Eq. (10) is valid only for a dilute concentration of ions, in the absence of significant ion-ion interactions, and a macroscopic theory can apply. Intuitively, the Nemst-Planck theory can be expected to fail when there is a significant confinement effect or ion-wall interaction and at high electric... 

The treatment of the two-phase SECM problem applicable to immiscible liquid-liquid systems, requires a consideration of mass transfer in both liquid phases, unless conditions are selected so that the phase that does not contain the tip (denoted as phase 2 throughout this chapter) can be assumed to be maintained at a constant composition. Many SECM experiments on liquid-liquid interfaces have therefore employed much higher concentrations of the reactant of interest in phase 2 compared to the phase containing the tip (phase 1), so that depletion and diffusional effects in phase 2 can be eliminated [18,47,48]. This has the advantage that simpler theoretical treatments can be used, but places obvious limitations on the range of conditions under which reactions can be studied. In this section we review SECM theory appropriate to liquid-liquid interfaces at the full level where there are no restrictions on either the concentrations or diffusion coefficients of the reactants in the two phases. Specific attention is given to SECM feedback [49] and SECMIT [9], which represent the most widely used modes of operation. The extension of the models described to other techniques, such as DPSC, is relatively straightforward. 

Diffusion of small molecular penetrants in polymers often assumes Fickian characteristics at temperatures above Tg of the system. As such, classical diffusion theory is sufficient for describing the mass transport, and a mutual diffusion coefficient can be determined unambiguously by sorption and permeation methods. For a penetrant molecule of a size comparable to that of the monomeric unit of a polymer, diffusion requires cooperative movement of several monomeric units. The mobility of the polymer chains thus controls the rate of diffusion, and factors affecting the chain mobility will also influence the diffusion coefficient. The key factors here are temperature and concentration. Increasing temperature enhances the Brownian motion of the polymer segments the effect is to weaken the interaction between chains and thus increase the interchain distance. A similar effect can be expected upon the addition of a small molecular penetrant. 

Hikosaka presented a chain sliding diffusion theory and formulated the topological nature in nucleation theory [14,15]. We will define chain sliding diffusion as self-diffusion of a polymer chain molecule along its chain axis in some anisotropic potential field as seen within a nucleus, a crystal or the interface between the crystalline and the isotropic phases . The terminology of diffusion derives from the effect of chain sliding diffusion, which could be successfully formulated as a diffusion coefficient in our kinetic theory. 

The infinite medium with one-dimensional diffusion and constant diffusion coefficient can be treated easily with the point source theory. Let us first assume that two half-spaces with uniform initial concentrations C0 for x < 0 and 0 for x > 0 are brought into contact with each other. The amount of substance distributed per unit surface between x and x + dx is just C0dx. From the previous result, at time t the effect of the point source C0 dx located at x on the concentration at x will be... 

The theory of linear differential equations indicates that long-term evolution depends on the boundary conditions and the determinant of the coefficients preceding the second spatial derivatives (which can actually be considered as effective diffusion coefficients). Such a system is likely to be highly non-linear. One extreme case, however, is particularly interesting in demonstrating how periodic patterns of precipitation can be arrived at. We assume that (i) species i diffuses very fast and dC /dp is large so that P is small and (ii) that species j is much less mobile and P is large. The... 

A reciprocal proportionality exists between the square root of the characteristic flow rate, t/A, and the thickness of the effective hydrodynamic boundary layer, <5Hl- Moreover, f)HL depends on the diffusion coefficient D, characteristic length L, and kinematic viscosity v of the fluid. Based on Levich s convective diffusion theory the combination model ( Kombi-nations-Modell ) was derived to describe the dissolution of particles and solid formulations exposed to agitated systems [(10), Chapter 5.2]. In contrast to the rotating disc method, the combination model is intended to serve as an approximation describing the dissolution in hydrodynamic systems where the solid solvendum is not necessarily fixed but is likely to move within the dissolution medium. Introducing the term... 

Figure 16. Schematic illustration of envelopes of gas species i, in this case Mg, surrounding a volatilizing molten chondrule in space. The size of the gas envelope is a function of ambient background pressure P by virtue of the effect that pressure has on the gas molecule diffusivity D,. The diffusion coefficient can be calculated from the kinetic theory of gases, as shown. The level of isotopic fractionation associated with volatilization of the molten chondrule depends upon the balance between the evaporative flux J vap and the condensation flux Tom When the fluxes are equal (i.e., when = 0), there is no mass-dependent isotope fractionation associated with volatilization. This will be the case when the local partial pressure P, approaches the saturation partial pressure P,...

The diffusion theory states that interpenetration and entanglement of polymer chains are additionally responsible for bioadhesion. The intimate contact of the two substrates is essential for diffusion to occur, that is, the driving force for the interdiffusion is the concentration gradient across the interface. The penetration of polymer chains into the mucus network, and vice versa, is dependent on concentration gradients and diffusion coefficients. It is believed that for an effective adhesion bond the interpenetration of the polymer chain should be in the range of 0.2-0.5 pm. It is possible to estimate the penetration depth (/) by Eq. (5),... 

There have been relatively few applications of the rate theory to GPC, presumably because of the apparent complexity of this approach. One of the most widely quoted interpretations of the rate theory to GPC is that of Ouano and Baker (4). These authors have attempted to take advantage of the undoubted potential of the rate theory approach in constructing a model. They identified the key parameters in their model as the flow rate of the eluant, gel particle size, diffusion coefficient in the stationary and mobile phases and the partition coefficient for solute between phases. Although there is little doubt that the important parameters have been correctly identified, it is not immediately apparent how they are inter-related and hence how their coupled effect can be interpreted. A critical account of the various attempts which have been made to model the GPC process will be given elsewhere. 

Another m3d h arises from the intuition that pressure effect is opposite to the temperature effect. This is not true in kinetics. Therefore, kinetic constants (reaction rate constants, diffusion coefficients, etc.) almost always increase with increasing temperature, but they may decrease or increase with increasing pressure. Both positive and negative pressure dependences are well accounted for by the transition-state theory and are not strange. 

The electron-ion recombination in high-mobility systems has also been analyzed in terms of the fractal theory [24,25]. It was postulated that even when the fractal dimension of particle trajectories is not equal to 2, the motion of particles is still described by diffusion but with a distance-dependent effective diffusion coefficient D r) = D(H-// where the parameter / is proportional to the mean free path X [24]. However, when the fractal dimension of trajectories is not equal to 2, the motion of particles is not described by orthodox diffusion. 

The case of transport through microporous membranes is different from that of macroporous membranes in that the pore size approaches the size of the diffusing solute. Various theories have been proposed to account for this effect. As reviewed by Peppas and Meadows [141], the earliest treatment of transport in microporous membranes was given by Faxen in 1923. In this analysis, Faxen related a normalized diffusion coefficient to a parameter, X, which was the ratio of the solute radius to the pore radius... 

The longitudinal diffusion coefficient D has been formulated by the hole theory in Sect. 6.3.2. If the similarity ratio X in this theory is chosen to be 0.025 for the rod with the axial ratio 50, Eq. (58) with Eq. (56) gives the solid curve in Fig. 16a. Though it fits closely the simulation data, the chosen X is not definitive because the change in D(l is small and the definition of the effective axial ratio is ambiguous. Though not shown here, Eq. (53) for D, by the Green function method describes the simulation data equally well if P and C, are chosen to be 1000 and 1, respectively. 

For the light molecules He and H2 at low temperatures (below about 50°C.) the classical theory of transport phenomena cannot be applied because of the importance of quantum effects. The Chapman-Enskog theory has been extended to take into account quantum effects independently by Uehling and Uhlenbeck (Ul, U2) and by Massey and Mohr (M7). The theory for mixtures was developed by Hellund and Uehling (H3). It is possible to distinguish between two kinds of quantum effects— diffraction effects and statistics effects the latter are not important until one reaches temperatures below about 1°K. Recently Cohen, Offerhaus, and de Boer (C4) made calculations of the self-diffusion, binary-diffusion, and thermal-diffusion coefficients of the isotopes of helium. As yet no experimental measurements of these properties are available. 

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