Networks of steps

Networks of steps, seen in STM observations of vicinal surfaces on Au and Pt (110), are analyzed. A simple model is introduced for the calculation of the free energy of the networks as function of the slope parameters, valid at low step densities. It predicts that the networks are unstable, or at least metastable, against faceting and gives an equilibrium crystal shape with sharp edges either between the (110) facet and rounded regions or between two rounded regions. Experimental observations of the equilibrium shapes of Au or Pt crystals at sufficiently low temperatures, i.e. below the deconstruction temperature of the (110) facet, could check the validity of these predictions. 

The interplay between step orientation and surface reconstruction is essential for understanding the formation of the network of steps. Indeed, a step perpendicular to the missing rows cannot zig-zag forming clockwise segments parallel to the missing rows, if expensive domain boundaries between opposite reconstruction states are to be... 

When a negative free energy is associated to a crossing point the network of steps tends to condense to maximize the density of crossings. On the other side, entropic repulsions (Gruber and Mullins, 1967) favor configurations where steps are far apart and tend to stabilize the network. 

Finally in Fig. 7 the dashed lines show the truncation of the equilibrium shape of the (110) facet by ridges connecting the facet to rounded areas and Fig. 10(c) shows the expected arrangement of steps around the truncated facet. Notice that in this case there are sharp ridges between rounded regions covered by networks of steps and regions covered by non-crossing step arrays. 

We believe that the problems connected to the metastability/instability of the networks of steps deserve further experimental investigation, as well. If e is positive, as we think should be the case for gold and platinum crystals, one should be able, starting from the metastable network of steps, to observe a nucleation of arrays of parallel steps connected under a ridge. Probably this could be observed in practice, e.g. in STM experiments, only for sufficiently high temperatures and in very pure samples. 

We had several stimulating discussions with Joost Frenken, Laurens Kuipers and Misha Hoogeman, who drew our attention to the networks of steps discussed here in the first place. E.C. aknowledges financial support from grant No. CHBGCT940734 from the European Capital and Mobility Programme. 

Biochemical pathways consist of networks of individual reactions that have many feedback mechanisms. This makes their study and the elucidation of kinetics of individual reaction steps and their regulation so difficult. Nevertheless, important inroads have already been achieved. Much of this has been done by studying the metabolism of microorganisms in fermentation reactors. 

We noted above that the presence of monomer with a functionality greater than 2 results in branched polymer chains. This in turn produces a three-dimensional network of polymer under certain circumstances. The solubility and mechanical behavior of such materials depend critically on whether the extent of polymerization is above or below the threshold for the formation of this network. The threshold is described as the gel point, since the reaction mixture sets up or gels at this point. We have previously introduced the term thermosetting to describe these cross-linked polymeric materials. Because their mechanical properties are largely unaffected by temperature variations-in contrast to thermoplastic materials which become more fluid on heating-step-growth polymers that exceed the gel point are widely used as engineering materials. 

Reaction measurement studies also show that the chemistry is often not a simple one-step reaction process (37). There are usually several key intermediates, and the reaction is better thought of as a network of series and parallel steps. Kinetic parameters for each of the steps can be derived from the data. The appearance of these intermediates can add to the time required to achieve a desired level of total breakdown to the simple, thermodynamically stable products, eg, CO2, H2O, or N2. 

Liquid crystals stabilize in several ways. The lamellar stmcture leads to a strong reduction of the van der Waals forces during the coalescence step. The mathematical treatment of this problem is fairly complex (28). A diagram of the van der Waals potential (Fig. 15) illustrates the phenomenon (29). Without the Hquid crystalline phase, coalescence takes place over a thin Hquid film in a distance range, where the slope of the van der Waals potential is steep, ie, there is a large van der Waals force. With the Hquid crystal present, coalescence takes place over a thick film and the slope of the van der Waals potential is small. In addition, the Hquid crystal is highly viscous, and two droplets separated by a viscous film of Hquid crystal with only a small compressive force exhibit stabiHty against coalescence. Finally, the network of Hquid crystalline leaflets (30) hinders the free mobiHty of the emulsion droplets. 

Once this "network" of tasks has been established, your next step will be to define a work breakdown structure for each task. A work breakdown structure shows the individual steps or elements required to complete each task. Each of these work elements can then be assigned to individuals or groups for action. 

Ammonium salts of the zeolites differ from most of the compounds containing this cation discussed above, in that the anion is a stable network of A104 and Si04 tetrahedra with acid groups situated within the regular channels and pore structure. The removal of ammonia (and water) from such structures has been of interest owing to the catalytic activity of the decomposition product. It is believed [1006] that the first step in deammination is proton transfer (as in the decomposition of many other ammonium salts) from NH4 to the (Al, Si)04 network with —OH production. This reaction is 90% complete by 673 K [1007] and water is lost by condensation of the —OH groups (773—1173 K). The rate of ammonia evolution and the nature of the residual product depend to some extent on reactant disposition [1006,1008]. 

The inhibiting effect of DHQ and its NH3 product was studied on the final step in the network of Fig. 2, the alkene hydrogenation. To avoid confusion with the PCHE olefin formed from DHQ, cyclohexene (CHE) was used as the reactant, and pentylamine (PA) was used as the source of NH3. When the hydrogenation of CHE is performed in the presence of NH3, we have... 

The structure of ART networks is hard to visualize. It is in fact a theory that can better be explained by means of a sequence of steps that follow the strategy. 

Based on this physical view of the reaction dynamics, a very broad class of models can be constructed that yield qualitatively similar oscillations of the reaction probabilities. As shown in Fig. 40(b), a model based on Eckart barriers and constant non-adiabatic coupling to mimic H + D2, yields out-of-phase oscillations in Pr(0,0 — 0,j E) analogous to those observed in the full quantum scattering calculation. Note, however, that if the recoupling in the exit-channel is omitted (as shown in Fig. 40(b) with dashed lines) then oscillations disappear and Pr exhibits simple steps at the QBS energies. As the occurrence of the oscillation is quite insensitive to the details of the model, the interference of pathways through the network of QBS seems to provide a robust mechanism for the oscillating reaction probabilities. 

Traditionally, we create thermoset polymers during step growth polymerization by adding sufficient levels of a polyfunctional monomer to the reaction mixture so that an interconnected network can form. An example of a network formed from trifimctional monomers is shown in Fig. 2.12b). Each of the functional groups can react with compatible functional groups on monomers, dimers, trimers, oligomers, and polymers to create a three-dimensional network of polymer chains. 

Step 8. Spectra classified using an artificial neural network pattern recognition program. (This program is enabled on a parallel-distributed network of several personal computers [PCs] that facilitates optimization of neural network architecture). 

Pattern recognition software operating, in some cases, by distributed calculations over a network of high-performance PCs. (Step 8)... 

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