Half-space diffusion

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... 

One example of half-space diffusion is the cooling of an oceanic plate. The oceanic plate when created at the mid-ocean ridge is hot, with a roughly uniform temperature of about 1600 K. It is cooled at the surface (quenched by ocean water) as it moves away from the ocean ridge. For simplicity, ignore complexities... 

Another example of half-space diffusion problem is as follows. A thin-crystal wafer initially contains some " Ar (e.g., due to decay of " K). When the crystal is heated up (metamorphism), " Ar will diffuse out and the surface concentration of " Ar is zero. If the temperature is constant and, hence, the diffusivity is constant, the problem is also a half-space diffusion problem. 

A third example is as follows. Initially a crystal has a uniform 8 0. Then the crystal is in contact with a fluid with a higher 8 0. Ignore the dissolution of the crystal in the fluid (e.g., the fluid is already saturated with the crystal). Then would diffuse into the crystal. Because fluid is a large reservoir and mass transport in the fluid is rapid, 8 0 at the crystal surface would be maintained constant. Hence, this is again a half-space diffusion problem with uniform initial concentration and constant surface concentration. The evolution of 8 0 with time is shown in Figure l-8b. 

For a diffusion couple, the definition of Amid requires some thinking because the mid-concentration of the whole diffusion couple is right at the interface, which does not move with time. This is because for a diffusion couple every side is diffusing to the other side. On the other hand, if a diffusion couple is viewed as two half-space diffusion problems with the interface concentration viewed as the fixed surface concentration, then. Amid equals 0.95387(Df), the same as the half-space diffusion problem. 

Example 1.4 Knowing D = 10 m /s, for half-space diffusion, estimate the... 

If diffusion starts from one end (surface) and has not reached the other end yet in one-dimensional diffusion, the diffusion medium is called a semi-infinite medium (also called half-space). There is, hence, only one boundary, which is often defined to be at x = 0. This boundary condition usually takes the form of CU=o = g(t), (dC/dx) x=o=g f), or (dC/dx) x=o + aC x=o=g(t), where u is a constant. Similar to the case of infinite diffusion medium, one often also writes the condition C x=x, as a constraint. 

Another experimental method to investigate diffusion is the so-called half-space method, in which the sample (e.g., rhyolitic glass with normal oxygen isotopes) is initially uniform with concentration C, but one surface (or all surfaces, as explained below) is brought into contact with a large reservoir (e.g., water vapor in which oxygen is all 0). The surface concentration of the sample is fixed to be constant, referred to as Cq. The duration is short so that some distance away from the surface, the concentration is unaffected by diffusion. Define the surface to be X = 0 and the sample to be at x > 0. This diffusion problem is the so-called halfspace or semi-infinite diffusion problem. 

For one-dimensional half-space diffusion with constant D and an initial distribution of C t o = f(x) as well as other conditions, the solutions can be found in Appendix 3. 

Figure 3-29 A half-space diffusion profile of Ar. Ca, = 0.00147 wt% is obtained by averaging 45 points at 346 to 766 /an. Points are data, and the solid curve is a fit of (a) all data by the error function with D = 0.207 /im /s and Co = 0.272 wt%, and (b) data at v < 230 fim (solid dots) by the inverse error function. In (b), for larger x, evaluation of erfc [(C —Cot)/(Co —Coa)] becomes increasingly unreliable and even impossible as (C - Cot)/(Cq — Coo) becomes negative. Data are adapted from Behrens and Zhang (2001), sample AbDArl.

Figure 1-8 Half-space diffusion profiles (cooling of oceanic plate) 43...

Diffusion time (diffusion time constant) — This parameter appears in numerous problems of - diffusion, diffusion-migration, or convective diffusion (- diffusion, subentry -> convective diffusion) of an electroactive species inside solution or a solid phase and means a characteristic time interval for the process to approach an equilibrium or a steady state after a perturbation, e.g., a stepwise change of the electrode potential. For onedimensional transport across a uniform layer of thickness L the diffusion time constant, iq, is of the order of L2/D (D, -> diffusion coefficient of the rate-determining species). For spherical diffusion (inside a spherical volume or in the solution to the surface of a spherical electrode) r spherical diffusion). The same expression is valid for hemispherical diffusion in a half-space (occupied by a solution or another conducting medium) to the surface of a disk electrode, R being the disk radius (-> diffusion, subentry -> hemispherical diffusion). For the relaxation of the concentration profile after an electrical perturbation (e.g., a potential step) Tj = L /D LD being - diffusion layer thickness in steady-state conditions. All these expressions can be derived from the qualitative estimate of the thickness of the nonstationary layer... 

The presence of the interface restricts the diffusive motion to a half space. More important is the presence of particle-surface interactions, which can modify the transport properties. 

The simplest case to analyze is diffusion in half space without interactions. Using G(R. t Rq) appropriate for diffusion in half space with reflecting boundary conditions 120], the expected autocorrelation response is summarized as... 

In the case of short diffusion times (i.e., only near surface penetration), it can be useful to approximate a mineral with a planar boundary as a semi-infinite medium. For the case of diffusion from a well-stirred semi-infinite reservoir at concentration Co into a half space initially at zero concentration, the concentration distribution is given by... 

Kalpna R.C. 2000. Green s function based stress diffusion solutions in the porous elastic half space for time varying finite reservoir loads. Phys, Earth Planet. Int. 120 pp93-101. 

Fig. 13 A cartoon of a profile of a smooth electrochemical interface. The half-space z < 0 is occupied by the metal ionic skeleton that, within the jellium model, is described as a continuum of positive charge density (n+) and the dielectric constant due to bound electrons (ei,), the value of which lies typically between 1 and 2. The gap accounts for a finite distance of closest approach of solvent molecules to the skeleton the gap is determined by the balance offerees that attract the molecules to the metal and the Pauli repulsion of the closed shells of the molecules from the free electron cloud of the metal of density n(z). The regions a < z < a + d and z> a + d correspond, respectively, to the first layer of solvent molecules (which can be roughly characterized by charge-dependent effective dielectric constant) and the diffuse-layer part.

There are two particularly simple problems in connection with the energy distribution of neutrons which are present in a medium of finite temperature. In the first problem the slowing down is uniform throughout the entire space which is itself uniformly filled with the slowing down material. In this case the neutron distribution is evidently the same all over space. In the second problem the neutrons enter a half space from one side with uniform intensity and diffuse into it. The question in this case is the density distribution of neutrons at large distances from the boundary plane of the half space and the exponential relaxation length of the neutron density. We shall be interested only in the first problem. 

DIFFUSION KINETICS OF PLANE LAYER SWELLING Consider two stages of swelling process in a plane layer - the initial and final. In the initial stage, the influence of the opposite layer boundary on the swelling process is inessential and therefore diffusion in a layer of finite thickness at sufficiently small values of time can be considered as the diffusion in half-space. 

Figure 4.4 shows the diffusion of a constant planar source, where at time f = 0 a source fills the half space such that the density or concentration at boundary plane x = 0 remains constant at all later times. Figure 4.4(a) shows that with an increase in time, the concentration font diffuses to a larger x-location. Figure 4.4(b) shows the variation of concentration at a particular x-location as a function of elapsed time. For large time, the concentration reaches the initial concentration, Cq, and the time to reach the equihbrium concentration, Cq, increases with distance, x, from the source location. 

However, if the channels are too wide there will be no support for the MEA, which will deflect into the channel. Wider spacing enhances conduction of electrical current and heat however, it reduces the area directly exposed to the reactants and promotes the accumulation of water in the gas diffusion layer adjacent to these regions. For a geometry shown in Figure 6-20, and with simplification of the current path in the control area (one half channel and one half spacing between the channels) the voltage loss through the control area is [28] ... 

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