Simple Diffusion Equation

As a first example of the diffusion equation approach, we consider a very simple problem the absorption of small (point) particles A, which are di-lutely dispersed in a viscous continuum, by a collection of large stationary 

The density field of the A species at point r in the fluid, (r), is measured relative to the center of a sink and is assumed to obey a simple diffusion equation  

The absorption of molecules at the surface of each sink is taken into account by a boundary condition. Perhaps one of the most useful boundary conditions is the radiation boundary condition introduced by Collins and Kimball,  

When describing transient effects in these systems, it often proves convenient to deal with the Fourier transform of the local density field. 

The half-sided Fourier transform of the rate kernel for the absorption of A, 

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Permeability of an FML is evaluated using the Water Vapor Transmission test.28 A sample of the membrane is placed on top of a small aluminum cup containing a small amount of water. The cup is then placed in a controlled humidity and temperature chamber. The humidity in the chamber is typically 20% relative humidity, while the humidity in the cup is 100%. Thus, a concentration gradient is set up across the membrane. Moisture diffuses through the membrane, and with time the liquid level in the cup is reduced. The rate at which moisture is moving through the membrane is measured. From that rate, the permeability of the membrane is calculated with the simple diffusion equation (Fick s first law). It is important to remember that even if a liner is installed correctly with no holes, penetrations, punctures, or defects, liquid will still diffuse through the membrane. 

Fig. 5. The complementary-error-function solution of the simple diffusion equation.

Then, equations (51) and (53) can be summed to cancel out the kinetic terms and provide a simple diffusion equation for the total metal ... 

The mathematical difficulty increases from homogeneous reactions, to mass transfer, and to heterogeneous reactions. To quantify the kinetics of homogeneous reactions, ordinary differential equations must be solved. To quantify diffusion, the diffusion equation (a partial differential equation) must be solved. To quantify mass transport including both convection and diffusion, the combined equation of flow and diffusion (a more complicated partial differential equation than the simple diffusion equation) must be solved. To understand kinetics of heterogeneous reactions, the equations for mass or heat transfer must be solved under other constraints (such as interface equilibrium or reaction), often with very complicated boundary conditions because of many particles. 

In the previous chapter, several factors which complicate the simple diffusion equation analysis of chemical reactions in solution were discussed rather qualitatively. However, the magnitude of these effects can only be gauged satisfactorily by a detailed physical and mathematical analysis. In particular, the hydrodynamic repulsion and competitive effects have been studied recently by a number of workers. Reactions between ionic species in solutions containing a high concentration of ionic species is a similarly involved subject. These three instances of complications to the diffusion equation all involve aspects of many-body effects. 

In this case the pressure is eliminated altogether, since by vector identity, the curl of the gradient of a scalar field vanishes. From the definition of vorticity, Eq. 2.103, a simple diffusion equation emerges for the vorticity... 

In this subsection we illustrate the attempt made in Ref. 59 to derive a generalized diffusion equation from the Liouville approach described in Section III. We are addressing the apparently simple problem of establishing a density equation corresponding to the simple diffusion equation... 

The parameter F is a measure of the relative importance of the pore and crystal diffusion processes F = 0 corresponds to an infinite value of Dc, and it is easy to see from the above equations that the crystals are in equilibrium with the local pore concentration and the exchange process is described by the simple diffusion equation... 

In most studies, the intracellular space has been assumed to be well mixed. Several investigators have shown that a simple diffusion equation coupled with appropriate reaction terms is adequate to describe intracellular transport (Kushmerick and Podolsky, 1969 Lauffenburger and Linderman, 1993). 

The validity of using a pulse model is based upon the simple diffusion equation for a liquid, which implies a Poisson process of fluctuations with time, with an average jump rate, x For the case of the scalar interaction Xp may be identified as the mean time between successive encounters of a given I spin and the S spins. 

In 1966, Hoinkis and Levi reported that Sr2+ self-diffusion in SrX appeared to occur by two processes—one fast and the other slow (33). In 1967, they (34) reported that Cs+ and Rb+ isotope exchange in zeolite A does not follow the rate law for a simple diffusion process and appears to take place with 2 steps. They plotted their isotope exchange data as 1 — U vs. time, where U is the fractional attainment of equilibrium at any instant in time. In order to ascertain whether or not the rate data fit the simple diffusion equation, they replotted their data as... 

In 1968, Hoinkis and Levi studied Ba2+ isotopic ion exchange in BaA using the temperature jump method (35). They found that the reaction rate could not be described by the simple diffusion equation over the temperature range 45° to 100 °C. The reaction does obey the simple diffusion equation from 100° to 120 °C. They concluded from this data that below 100 °C sited and unsited Ba2+ ions are present and that above 100 °C very few ions are sited. 

In 1968, Dyer, Gettins, and Molyneux (28) studied Ba2+ isotopic ion exchange in BaA over the temperature range 19° to 65 °C. They did not show plots of dimensionless time as a function of time but stated that these plots were linear and, therefore, obey the simple diffusion equation. There is some discrepancy between this work and the work of Hoinkis and Levi at the Hahn-Meitner Institute (35, 37) that is described above, in which they state that in their study of Ba2+ isotopic ion exchange, linear plots of dimensionless time vs. time are not obtained and fit the data by an equation for a two-step process. 

Simple diffusion equations of this type have also been applied to much wider classes of reaction, for example, to cases of sinks and particles being absorbed that are both moving and similar in size (e.g., colloid coagulation and small molecule reactions). The physical background for such applications is widely discussed in the literature." When the sink density is not small, competition effects come into play and it is no longer sufficient to consider reaction at a single sink." " These competition effects lead to a nonanalytic dependence of the rate coefficient on the sink density. Such effects are not discussed here. 

We now adopt a somewhat different point of view suppose we have an ensemble of isolated pairs of potentially reactive A and B molecules and wish to describe the time evolution of the probability that a given pair at relative separation r remains unreacted at time t, / (r, /). In contrast to the simple diffusion equation, we now also allow for the possibility that forces act between the molecules in the pair. It is customary to assume that the dynamics for this situation may be modeled by a Smoluchowski equation ... 

In the limit of no forces other than simple excluded volume effects (g(r) = 9 r-a)), (3.15) reduces to the simple diffusion equation result in Section III.A. In the w = 0 limit, k may be written in the form ... 

Fig. 6.2. Rate kernel in units of kjD /kpR as a function of time (tD /R ) for simple diffusion equation dynamics. The ratio A j//cp 1.0 for this graph. The heavy vertical axis indicates the singular contribution to the rate kernel. For diffusion equation dynamics, My(/) also diverges at / — O (cf. (3.9)). This will not be true in general. The breakdown of the simple diffusion model comes as no surprise. Apart from this, the gross features of this diagram illustrate the general situation a singular part followed by a negative relaxing part, which decays as t" for long times.

This equation can be solved numerically for simple diffusion equation dynamics analytical results can be obtained. ... 

As an illustration of the structure of 7(z) [or 7(/)], we examine the case where simple diffusion equation dynamics is used in place of sm> that is. 

Fig. ILL The unreacted pair probability P(t) versus t for several values of the initial separation of the pairs. Simple diffusion equation dynamics is assumed and X k j/k0=1.0.

Theoretical treatments often focus on either the primary or secondary processes. The traditional approaches that make use of a simple diffusion equation for the pair dynamics are necessarily restricted to a description of... 

Overall rotational tumbling is regulated by frequent collisions with light water molecules. For a nearly rigid protein, this physical model should lead to diffusive rotational behavior, where the reorientation of a unit vector attached to the molecule undergoes a random walk on the surface a sphere. If c(n, t) is the probability density for finding the vector pointing direction n at time f, a spherical molecule should follow a simple diffusion equation [31,32] ... 

Kelley and Rentzepis [297] have recently studied the recombination of iodine atoms in liquid and fluid xenon over times to 150 ps after photolysis. The iodine molecule can be biphotonically dissociated through the state to produce geminate pairs with larger initial separations. Some degree of spin relaxation of excited iodine atoms ( Pi/2) produced by biphotonic excitation may occur and reduce the probability of recombination. There is also evidence that the 11 state of I2 may be collisionally predissociated and that recombination may be more rapid than the rate of vibrational relaxation of the excited 12 state in polyatomic solvents (see also ref. 57). Despite these complications, several workers have attempted to model the time dependence of the recombination (or survival) probability of iodine atom reactions. The simple diffusion equation analysis of recombination probabilities [eqn. 

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