Diffusion limitation theory

A different sort of check on the diffusion-limitation theory has been performed by Northrup (1988), who reported Brownian dynamics computer simulations of ligand diffusing to cell surface receptors. His calculations were aimed at the special case in which the intrinsic binding rate constant takes on a diffusion-limited value that is, kot> = 4Ds (which... 

Unlike solid electrodes, the shape of the ITIES can be varied by application of an external pressure to the pipette. The shape of the meniscus formed at the pipette tip was studied in situ by video microscopy under controlled pressure [19]. When a negative pressure was applied, the ITIES shape was concave. As expected from the theory [25a], the diffusion current to a recessed ITIES was lower than in absence of negative external pressure. When a positive pressure was applied to the pipette, the solution meniscus became convex, and the diffusion current increased. The diffusion-limiting current increased with increasing height of the spherical segment (up to the complete sphere), as the theory predicts [25b]. Importantly, with no external pressure applied to the pipette, the micro-ITIES was found to be essentially flat. This observation was corroborated by numerous experiments performed with different concentrations of dissolved species and different pipette radii [19]. The measured diffusion current to such an interface agrees quantitatively with Eq. (6) if the outer pipette wall is silanized (see next section). The effective radius of a pipette can be calculated from Eq. (6) and compared to the value found microscopically [19]. 

The silanization of the surface of a glass pipette may be necessary for different reasons [19]. If the pipette is to be filled with an aqueous solution, its outer wall should be made hydrophobic to prevent the formation of a thin aqueous film that may cause large deviations of the experimental diffusion-limiting current from the theory (see Section II.B). Experimental voltammograms were found to quantitatively agree with the theory after the aqueous layer was eliminated by silanizing the outer pipette wall. 

It is assumed that the quantity Cc is not a function of the electrolyte concentration c, and changes only with the charge cr, while Cd depends both on o and on c, according to the diffuse layer theory (see below). The validity of this relationship is a necessary condition for the case where the adsorption of ions in the double layer is purely electrostatic in nature. Experiments have demonstrated that the concept of the electrical double layer without specific adsorption is applicable to a very limited number of systems. Specific adsorption apparently does not occur in LiF, NaF and KF solutions (except at high concentrations, where anomalous phenomena occur). At potentials that are appropriately more negative than Epzc, where adsorption of anions is absent, no specific adsorption occurs for the salts of... 

On the submicron scale, the current distribution is determined by the diffusive transport of metal ion and additives under the influence of local conditions at the interface. Transport of additives in solution may be non-locally controlled if they are consumed at a mass-transfer limited rate at the deposit surface. The diffusion of additives in solution must then be solved simultaneously with the flux of reactive ion. Diffusive transport of inhibitors forms the basis for leveling [144-147] where a diffusion-limited inhibitor reduces the current density on protrusions. West has treated the theory of filling based on leveling alone [148], In his model, the controlling dimensionless groups are equivalent to and D divided by the trench aspect ratio. They determine the ranges of concentration within which filling can be achieved. 

According to the diffusion layer theory, for which the transport process is rate-limiting, kT kR, so that k = kT. According to the interfacial barrier theory, for which the surface reaction is rate-limiting, kR kT, so that it, = R-... 

S. Jun, J. Bechhoefer, andB.-Y. Ha, Diffusion-limited loop formation of semiflexible polymers Kramers theory and the intertwined time scales of chain relaxation and closing. Europhys. Lett. 64, 420-426 (2003). 

HamF.S. (1958) Theory of diffusion-limited precipitation./.Phys. Chem. Solids 6, 335-351. 

This study employed conventional diffusion-reaction theory, showing that with diffusion-limited reactions the internal effectiveness factor of a heterogeneous catalyst is inversely related to the Thiele modulus. Using a standard definition of the Thiele modulus [100], the observed reaction rate of an immobilized-enzyme reaction will vary with the square root of the immobilized-enzyme concentration in a diffusion-limited system. In this case, a plot of the reaction rate versus the enzyme loading in the catalyst formulation will be nonlinear. 

Over molecular length scales, the diffusion distances become very short (< 1 nm) so that only very rapid events can be influenced by these short diffusion times. Necessarily, this limits the number of systems to only relatively few, where the rate at which the reactants can approach one another is slow or comparable with the rate at which the reactants react chemically with each other. Some typical systems which have been studied are discussed in Sect. 2. The Smoluchowski [3] theory of reactions in solution, which occur at a rate limited solely by how fast the reactants can approach each other (sufficiently closely to react chemically almost instantaneously) is discussed in Sect. 3. If the chemical reaction is not so rapid, the observed rate of reaction may be influenced by both the rate of approach and the rate of subsequent chemical reaction. Collins and Kimball [4], and later Noyes [5], have extended the Smoluchowski theory (1917) to consider this situation (Sect. 4). In light of these quantitative theoretical models of diffusion-limited rate processes, some of the more recent and careful experiments on diffusion-controlled reactions in solution are considered briefly in Sect. 5. As the Smoluchowski theory... 

The Smoluchowski theory of diffusion-limited (or controlled) reactions relies heavily on the appropriateness of the inital condition [eqn. (3)]. Though the initial condition does not determine the steady-state rate coefficient [eqn. (20)] because diffusion of B in towards the reactant A is from large separation distances (>10nm) in the steady-state, it does directly determine the magnitude of the transient component of the rate coefficient because this is due to an excess concentration of B present initially compared with that present in the steady-state. As a first approximation to the initial distribution, the random distribution is intuitively pleasing and there is little experimental evidence available to cast doubt upon its appropriateness. Section 6.6 and Chap. 8 Sect. 2.2 present further comments on this point. 

In the preceding sections of this chapter, a theory of diffusion-limited chemical reactions has been described for two cases, (a) where the reaction of the encounter pair is much faster than its formation and (b) where these rates are of comparable magnitude. Some experimental evidence for both these cases has been described. During this discussion, a number of other difficulties in the interpretation of diffusion-limited reactions were indicated. This section details the complications and when they may be expected. The following chapters serve to amplify these comments. Chapter 8 provides a resume and conclusion as well as recommendations for future areas of both experimental and theoretical study. 

The rate of reaction of methyl radicals is in excellent agreement with the predictions of the Smoluchowski theory (see Chap. 2, Sect. 2.6). Consequently, it appears that geminate radicals move towards and away from each other at a diffusion-limited rate. Once an encounter pair is formed, reaction is very rapid (primary recombination). Furthermore, the encounter pair is held together for a considerable time (< 0.1ns in mobile solvents) because the surrounding solvent molecules hinder their separation (solvent caging). There is much evidence which lends some support for this view the most important influences on the recombination probability are listed below. 

Nevertheless, the kinetic modelling of spurs is by far the most complex problem to which diffusion-limited chemical reaction theory has been applied. The radiation chemistry of water is of especial importance to both radiotherapy and nuclear engineering. 

If the rate of reaction of encounter pairs is comparably fast to the rate of formation of encounter pairs, Collins and Kimball [4] suggested that the slowness of the chemical reaction rate could be incorporated into the theory of diffusion-limited reaction rates by modifying the Smoluchowski [3] boundary condition, eqn. (5), to the partially reflecting boundary... 

One further conceptual point in the favour of the inclusion of a noninfinite rate of encounter reaction into the theory of diffusion-limited reactions follows from the behaviour of the time-dependent rate coefficient at short times. The time-dependent Smoluchowski rate coefficient depends on time as r1/2 at short times. Indeed, immediately after formation, the rate of reaction of reactants is essentially infinite. This is because all those reactants within a separation 5R of the encounter... 

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