Equilibrium layer spacing

Figure 10.29 Response of an aligned smectic to layer dilation, (a) Initial equilibrium sample, (b) For a very small dilation Sh < Ink, the layer spacing simply increases, (c) A uniform rotation of the layers decreases the spacing toward that of equilibrium, but doesn t satisfy the boundary conditions, (d) Hence, the sample undergoes an mdulational instability, which also narrows the layer spacing while satisfying homeotropic boundary conditions, (e) For a large enough dilation, the undulation instability leads to formation of parabolic focal conic defects. (From Rosenblatt et al. 1977, with permission from EDP Sciences.)...

If a system with an artificially and permanently expanded interlayer space was to be modeled (in order to model an external surface, for example), all calculations were performed under NVT (constant mass, volume, and temperature) conditions. In certain other situations such as those described below, the artificially expanded systems were first allowed to equilibrate under NVT conditions and subsequently subjected to NPT (constant mass, pressure, and temperature) conditions, whereupon the separated layers spontaneously reannealed, thereby restoring the equilibrium interlayer spacing characteristic of that particular system. 

All surface complexes between counterions and the clay mineral surfaces were inner sphere due to our use of a monolayer of water. When present, water molecules tended to position themselves in such a way as to be in equatorial association with the counterions, i.e., they did not interpose themselves between a counterion and a clay surface. The equilibrium do0, spacings between hydrated montmorillonite layers in these simulations were reasonable [see, e.g., Brindley (1980) for experimental values], varying from 1.209 0.005 nm (K) to 1.214 0.004 nm (Rb) to 1.223 0.005 nm (Cs). 

Figure 13. Swelling behavior for a smectite clay derived from molecular dynamics simulations of montmorillonite. The equilibrium d-spacing is presented as a function of water content of the clay. The plateaus in the experimental and simulation results at 12 A and 15 A represe nt the stabiUzation of, respectively, the one-layer (insert stracture) and two-layer hydrates. No further expansion of the smectite is observed in nature beyond the two-layer hydrate. The simulations suggest that further swelling of the clay is possible although not thermodynamically favored.

A study of adsorption/desorption hysteresis loops in the swelling process previously observed experimentally by Fu and coworkers (95) was also tackled by Monte Carlo simulation (96). They conducted a series of simulations in which the interlayer water content is increased systematically from 0 to 300 mg/g of clay. Then the calculated clay layer spacing values as a function of water content were compared with experimental data. They claimed that their simulation established, for the first time, the true equilibrium clay layer spacings of the system. 

When AG = AG, there is no equilibrium Frenkel space-charge double layer instead, positive and negative adsorbed charges are of equal concentration when V =0. It is thus clear that when V 0, eVj must bg a function of (AG - AG ). Thus, the which appears in is not independent of t8e values of AG . This matter has een considered in some detail previously for the case of (AG /kT) >> 1, which allows the Langmuir/Fermi distributions of Eq. (25) to be reduced to Maxwell-BoIt mann distributions. explicit linear relation between (AG - AG ) and and n q found and used to calculate curves for a... 

Figure Bl.28.9. Energetic sitiration for an n-type semiconductor (a) before and (b) after contact with an electrolyte solution. The electrochemical potentials of the two systems reach equilibrium by electron exchange at the interface. Transfer of electrons from the semiconductor to the electrolyte leads to a positive space charge layer, W. is the potential drop in the space-charge layer.

To calculate the current-voltage characteristic of a MESFET, we assume that the conducting channel is at thermal equilibrium, and the space-chaige layer completely depleted. Eq. (14.28) becomes... 

One of the most crucial influencing factors in planar chromatography is the vapor space and the interactions involved. The fact that the gas phase is present, in addition to stationary and mobile phases, makes planar chromatography different from other chromatographic techniques. Owing to the characteristic of an open system the stationary, mobile, and vapor phases interact with each other until they all are in equihbrium. This equilibrium is much faster obtained if chamber saturation is employed. This is the reason for differences in separation quality when saturated and unsaturated chambers are used. However, the humidity of the ambient air can also influence the activity of the layer and, thus, separation. Especially during sample application, the equihbrium between layer activity and relative humidity of the... 

Figure 8.7 Tunneling through the space-charge layer at equilibrium and for an anodic overpotential. Note that the band bending is stronger after the application of the overpotential. The arrows indicate electrons tunneling through the space-charge barrier.

The investigations described in the preceding pages have been directed to one point Only the exact determination of the excess of dissolved substance in the surface layer at one particular concentration. There are, however, some further questions of great importance, the answers to which must be sought by other experimental methods. The first of these is does adsorption lead to a well-defined equilibrium in a short space of time the second is this equilibrium, assuming it to exist, a simple function of the concentration ... 

One can apply the MC technique to the same molecular model, as explored in MD. One can use the same box and the same molecules that experience exactly the same potentials, and therefore the results are equally exact for equilibrium membranes. However, MC examples of this type are very rare. One of the reasons for this is that there is no commercial package available in which an MC strategy is combined with sufficient chemistry know-how and tuned force fields. Unlike the MD approach, where the phase-space trajectory is fixed by the equations of motion of the molecules, the optimal walkthrough phase space in an MC run may depend strongly on the system characteristics. In particular, for densely packed layers, it may be very inefficient to withdraw a molecule randomly and to let it reappear somewhere else in... 

The surface Fermi level, Cp, which depends on the surface state, is not the same as the interior Fermi level, ep, which is determined by the bulk impurity and its concentration. As electron transfer equilibrium is established, the two Fermi levels are equilibrated each other (ep = ep) and the band level bends downward or upward near the surface forming a space charge layer as shown in Fig. 2-31. 

In the case in which ionic equilibrium is established between the surface and the interior, the surface ion level, qa,-, equals the interior ion level, aA.cAB>. Consequently, the unitary ion levels at the lattice and interstitial sites bend either upward or downward forming a space char ge layer in a region adjacent to the surface as shown in Fig. 3—13. When the surface A ion level, Oa+, is lower than tire interior A ion level, aA (AB A ions move from the interior to the... 

Fig. 6-48. Differential capacity of a space charge layer of an n-type semiconductor electrode as a function of electrode potential solid cunre = electronic equilibrium established in the semiconductor electrode dashed curve = electronic equilibrium prevented to be established in the semiconductor electrode AL = accumulation layer DL = depletion layer IL = inversion layer, DDL - deep depletion layer.

When electronic equilibrium is established in the space charge layer, the concentration of interfacial electrons is given by n, = n exp (- e A /k T) and the concentration of interfacial holes is given by Pt = p exp(e A lk T) n and p are the concentrations of electrons and holes, respectively, in the semiconductor interior. In general, the ionization of surface atoms (Eqn. 9-24) is in quasiequilibrium so that the concentration of surface ions depends on the overvoltage... 

In photoexcited n-type semiconductor electrodes, photoexcited electron-hole pairs recombine in the electrodes in addition to the transfer of holes or electrons across the electrode interface. The recombination of photoexcited holes with electrons in the space charge layer requires a cathodic electron flow from the electrode interior towards the electrode interface. The current associated with the recombination of cathodic holes, im, in n-type electrodes, at which the interfadal reaction is in equilibrium, has already been given by Eqn. 8-70. Assuming that Eqn. 8-70 applies not only to equilibrium but also to non-equilibrium transfer reactions involving interfadal holes, we obtain Eqn. 10-43 ... 

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