Interfaces, equilibrium constructions

We noted in Section VII-2B that, given the set of surface tension values for various crystal planes, the Wulff theorem allowed the construction of fhe equilibrium or minimum firee energy shape. This concept may be applied in reverse small crystals will gradually take on their equilibrium shape upon annealing near their melting point and likewise, small air pockets in a crystal will form equilibrium-shaped voids. The latter phenomenon offers the possible advantage that adventitious contamination of the solid-air interface is less likely. 

In the case of supported metalhc particles, the construction is modified by introducing the adhesion energy (Wulff-Kaishew construction) [Henry, 1998]. The equilibrium shape is a Wulff polyhedron, which is truncated at the interface by an amount Ahs, according to the relation Ahs/hj = /3/(t where /3 is the adhesion energy of the crystal on the substrate. 

The heptane water and toluene water interfaces were simulated by the use of the DREIDING force field on the software of Cerius2 Dynamics and Minimizer modules (MSI, San Diego) [6]. The two-phase systems were constructed from 62 heptane molecules and 500 water molecules or 100 toluene molecules and 500 water molecules in a quadratic prism cell. Each bulk phase was optimized for 500 ps at 300 K under NET ensemble in advance. The periodic boundary conditions were applied along all three directions. The calculations of the two-phase system were run under NVT ensemble. The dimensions of the cells in the final calculations were 23.5 A x 22.6 Ax 52.4 A for the heptane-water system and 24.5 A x 24.3 A x 55.2 A for the toluene-water system. The timestep was 1 fs in all cases and the simulation almost reached equilibrium after 50 ps. The density vs. distance profile showed a clear interface with a thickness of ca. 10 A in both systems. The result in the heptane-water system is shown in Fig. 3. Interfacial adsorption of an extractant can be simulated by a similar procedure after the introduction of the extractant molecule at the position from where the dynamics will be started. 

Figure 13.15. Mechanism, nomenclature, and constructions for absorption, stripping and distillation in packed towers, (a) Two-film mechanism with equilibrium at the interface, (b) Sketch and nomenclature for countercurrent absorption or stripping in a packed tower, (c) Equilibrium and material balance lines in absorption, showing how interfacial concentrations are found, (d) Equilibrium and material balance lines in stripping, showing how interfacial concentrations are found, (e) Equilibrium and material balance lines in distillation, showing how interfacial concentrations are found.

Let us consider the system illustrated in Figure 10-2.1 vo large crystals (a and fi) with sufficient buffer capacity are in equilibrium (juf = /if) and possess surfaces of equal size. These surfaces are lattice planes characterized by their Miller (hkl) indices (h ,) and (h ) (m = 1,2,3). Construction of an arbitrary interface can be achieved... 

In the context of the morphological evolution of non-equilibrium systems, let us then ask whether the reaction path, when constructed for a system with stable interfaces, can tell us something about the instability of moving boundaries. For this we... 

We wish to prove by means of the Wulff construction (Section C.3.1) that the equilibrium shape of the grain boundary nucleus in Fig. 19.12 is indeed composed of two spherical-cap-shaped interfaces. 

Construct the equilibrium line. The equilibrium line relates the mole fraction of ammonia in the gas phase to that in the liquid phase when the two phases are at equilibrium. Equilibrium is assumed to exist between the two phases only at the gas-hquid interface. For dilute systems, Henry s law will apply. It applies for liquid mole fractions less than 0.01 in systems in general, and, as can be seen in Example 11.5, for the ammonia-water system it applies to liquid mole fractions as high as... 

Similar in spirit is the Milestoning [90] method by Fiber and coworkers, who assume that the diffusion of interest occurs through a tube in configuration space, and translate the rare process into a non-Markovian hopping between configuration space hyperplanes, the so-called milestones (which are in fact rather similar to the TIS interfaces, except that they do not form a foliation). The kinetics is obtained from starting an equilibrium ensemble on a milestone, and measuring the time distribution needed to reach the next milestone. The distribution can subsequently be used to construct the kinetics. The assumption is that there is an equilibrium (Boltzmann) distribution on each milestone. 

Fig. 32a-c. A thin blend film bounded by antisymmetric surfaces exerting opposing fields [93] a Cahn construction with trajectories -2kV( ) (solid and dashed lines) plotted for Ap= 0 and (here) for c —The bare surface free energy derivatives (—dfsL/dcf))s and (+dfsR/d( >)s due to left (L) and right (R) surface are marked by dotted lines. Boundary conditions (Eq. 51) are met at points 1 and 2 b the profile 1-2 as determined by Cahn construction (a) for a rather thick (due to the limit (ft,-Hfi) hlm c the corresponding bilayer equilibrium morphology with the interface between phases and < >2 running parallel to both surfaces... 

Microtubules are not static constructs. They exist in an equilibrium, by which heterodimers add permanently to one end (the plus-end ) and are shed from the other (the minus-end ). In the heterodimer, the a-tubuUn bonded GTP at the interface of the dimer is enclosed by a loop of )5-tubulin and thus protected from hydrolysis. The GTP attached to )5-tubulin is however hydrolysed to GDP shortly after the addition of another heterodimer. This destabilises the microtubules, and causes a more facile depolymerisation of the microtubules from the minus-end in the direction of the plus-end. Both processes (polymerisation and depolymerisation) occur in the cell simultaneously, and... 

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