Interface position

Figure B3.6.2. Local mterface position in a binary polymer blend. After averaging the interfacial profile over small lateral patches, the interface can be described by a single-valued function u r. (Monge representation). Thennal fluctuations of the local interface position are clearly visible. From Wemer et al [49].

On short length scales the coarse-grained description breaks down, because the fluctuations which build up the (smooth) intrinsic profile and the fluctuations of the local interface position are strongly coupled and camiot be distinguished. The effective interface Flamiltonian can describe the properties only on length scales large compared with the width w of the intrinsic profile. The absolute value of the cut-off is difficult... 

In the mean field considerations above, we have assumed a perfectly flat interface such that the first tenn in the Hamiltonian (B3.6.21) is ineffective. In fact, however, fluctuations of the local interface position are important, and its consequences have been studied extensively [, 58]. 

One can regard the Hamiltonian (B3.6.26) above as a phenomenological expansion in temis of the two invariants Aiand//of the surface. To establish the coimection to the effective interface Hamiltonian (b3.6.16) it is instnictive to consider the limit of an almost flat interface. Then, the local interface position u can be expressed as a single-valued fiinction of the two lateral parameters n(r ). In this Monge representation the interface Hamiltonian can be written as... 

A relatively simple approach suggests itself if the interfaces are known to be almost flat. In that case, the interface position can be described by a single-valued function z(x,y), where (A,y) are cartesian coordinates on a flat parallel reference plane. The functional (21) can be approximated by... 

Later we will describe both oxidation and reduction processes that are in agreement with the electrochemically stimulated conformational relaxation (ESCR) model presented at the end of the chapter. In a neutral state, most of the conducting polymers are an amorphous cross-linked network (Fig. 3). The linear chains between cross-linking points have strong van der Waals intrachain and interchain interactions, giving a compact solid [Fig. 14(a)]. By oxidation of the neutral chains, electrons are extracted from the chains. At the polymer/solution interface, positive radical cations (polarons) accumulate along the polymeric chains. The same density of counter-ions accumulates on the solution side. 

Figure 2.60 Pictorial representation of the SLIC scheme showing the updating scheme for an upwind and a downwind cell. Cells filled with fluid 1 are indicated in gray, those with fluid 2 in white. Cells containing a mixture of both fluids are represented by hatched areas. In the right column the configuration at the new time step is shown, with interface positions depicted explicitly.

The theory has been verified by voltammetric measurements using different hole diameters and by electrochemical simulations [13,15]. The plot of the half-wave potential versus log[(4d/7rr)-I-1] yielded a straight line with a slope of 60 mV (Fig. 3), but the experimental points deviated from the theory for small radii. Equations (3) to (5) show that the half-wave potential depends on the hole radius, the film thickness, the interface position within the hole, and the diffusion coefficient values. When d is rather large or the diffusion coefficient in the organic phase is very low, steady-state diffusion in the organic phase cannot be achieved because of the linear diffusion field within the microcylinder [Fig. 2(c)]. Although no analytical solution has been reported for non-steady-state IT across the microhole, the simulations reported in Ref. 13 showed that the diffusion field is asymmetrical, and concentration profiles are similar to those in micropipettes (see... 

Matching. Equations 6 and 7 demand boundary conditions. Near the constant thickness film region the interface position asymptotically approaches hQ, and the surface excess concentration limits... 

Let us consider a straight channel waveguide with the cross-section schematically shown in Fig. 4. The cross-section can be subdivided by lines parallel with the (vertical) x axis into slices s = 1, 2, S with interfaces positioned at the lateral coordinates y, y2,...,ys i. Each slice is formed by a vertical stack of layers numbered by I, 1 = 1, 2,...,L. In the vertical direction, the waveguide is bound by perfectly conducting electric or magnetic walls positioned at x = Xq and x = x. If need be, they can be substituted by PMLs as described before. 

To use Equation 3-58d or 3-58e, it is necessary to know the interface position X = 0 (i.e., p = 0) because the value of the integration depends on the exact position of X = 0. For a semi-infinite diffusion medium with fixed interface, this is easy (x = 0 is the surface). However, for a diffusion couple, the location of the... 

Starting from measured C versus x profile, roughly estimate the interface position (e.g., at midconcentration). Plot x (as vertical axis) versus C (as horizontal axis) as in Figure 3-8a. 

The interface position of the isotopic fraction profile is not necessarily the same as that of the concentration profile of the same element. 

Based on these observations, the diffusivity extracted from isotopic fraction profiles is usually regarded to be similar to intrinsic diffusivity or self-diffusivity even in the presence of major element concentration gradients. That is, the multicomponent effect does not affect the length of isotopic fraction profiles (but it affects the isotopic fractions and the interface position). On the other hand, the diffusion of a trace or minor element is dominated by multicomponent effect in the presence of major element concentration gradients. 

Denote boundary motion speed as u that may or may not depend on time. For crystal growth, the interface moves to the right with x = Xo>0. For crystal dissolution, the interface moves to the left with x = Xo<0. That is, u is positive during crystal growth and negative during crystal dissolution under our setup of the problem. The interface position can be found as... 

To solve the above problem, the usual method is to eliminate the moving boundary by adopting a reference frame that is fixed to the crystal-melt interface. That is, in the new reference frame and new coordinates y, the interface position is always at y=0. Hence, we let... 

Figure 3-28 H2O diffusion profile for a diffusion-couple experiment. Points are data, and the solid curve is fit of data by (a) error function (i.e., constant D) with 167 /irn ls, which does not fit the data well and (b) assuming D = Do(C/Cmax) with Do = 409 /im ls, which fits the data well, meaning that D ranges from 1 /rm /s at minimum H2O content (0.015 wt%) to 409 firn ls at maximum H2O content (6.2 wt%). Interface position has been adjusted to optimize the fit. Data are adapted from Behrens et al. (2004), sample DacDC3.

Crystal growth Consider the case for crystal growth along one direction (hence a one-dimensional problem). Define the initial interface to be at x = 0 and the crystal is on the side with negative x (left-hand side) and the melt is on the positive side (Section 3.4.6). Due to crystal growth, the interface advances to the positive side. Define the interface position at time t to be at x = Xq, where Xq > 0 is a function of time. Let w be the mass fraction of the main equilibriumdetermining component then the diffusion equation in the melt is... 

A long, hollow, cylindrical bowl is suspended by a flexible spindle and driven from the top as shown in Figure 13. Axial ribs in the bowl ensure full acceleration of the liquid during its short time in the bowl. Feed is jetted into the bottom of the bowl and clarified liquid overflows at the top, leaving deposited solids as compacted cake on the bowl wall. The clarifying performance of the bowl is reduced as the deposited cake decreases the effective outer radius of the bowl in accordance with equation 11. Consequently, cake capacity of the industrial model is limited to 0.1—10 L. For liquid—liquid separation, the interface position (eq. 26) is determined by selection of ring-dam diameter or by the length of a hollow nozzle-type screw dam. 

At this point, the interface position, x(f), is an unknown function of time. However, it can be determined by imposing the requirement that the net rate at which heat flows into the boundary must be equal to the rate at which heat is delivered to supply the latent heat needed for the melting. Any small difference between the densities p3 and pL may be neglected and the resulting uniform density is represented by p. Then, if the boundary advances a distance Sx, an amount of heat pHm Sx must be supplied per unit area. Also, if the time required for this advance is St, the heat that has entered the boundary from the liquid is JLSt (at x = x) and the heat that has left the interface through the solid is JsSt (at x = x)- Therefore,... 

This type of relationship, accounting for the flux into and out of the interface, is generally known as a Stefan condition [1]. The Stefan condition introduces a new variable, the interface position, and one new equation. 

A figure similar to Figure B.3 is displayed once the run is complete. The figure shows the water-ice and ice-hydrate interface positions as a function of time. This plot can be used to determine the time when the pipeline can be restarted or when methanol can be flowed based upon the annulus spacing available. 

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