Crystal macroscopic steps

In the formation of a crystal two steps are required (1) the birth of a new particle and (2) its growth to macroscopic size. The first step is called nucleation. In a crystallizer, the CSD is determined by the interaction of the rates of nucleation and growth, and the overall process is complicated kinetically. The driving potential for both rates is supersaturation, and neither crystal growth nor formation of nuclei from the solution can occur in a saturated or unsaturated solution. Of course, very small crystals can be formed by attrition in a saturated solution, and these may act just like new nuclei as sites for further growth if the solution later becomes supersaturated. 

These slip lines are microscopic ledges produced by dislocations (Figure 7.1c) that have exited from a grain and appear as lines when viewed with a microscope. They are analogous to the macroscopic steps found on the surfaces of deformed single crystals (Figures 7.8 and 7.9). 

For the explanation of macroscopic phenomena, the thickness of the phase boundary (interface) often plays no important role. As an example, we describe the movement of a phase boundary in two dimensions or the movement of a step edge on a crystal surface. We start with a Ginzburg-Landau equation [69]... 

The maintenance of a connection to experiment is essential in that reliability is only measurable against experimental results. However, in practice, the computational cost of the most reliable conventional quantum chemical methods has tended to preclude their application to the large, low-symmetry molecules which form liquid crystals. There have however, been several recent steps forward in this area and here we will review some of these newest developments in predictive computer simulation of intramolecular properties of liquid crystals. In the next section we begin with a brief overview of important molecular properties which are the focus of much current computational effort and highlight some specific examples of cases where the molecular electronic origin of macroscopic properties is well established. 

Macroscopic growth during electrocrystallization occurs through fast movements of steps, 10-4-10 5cm high, across the crystal face. Under certain conditions, spirals also appear, formed of steps with a height of a thousand or more atomic layers, so that they can be studied optically (Fig. 5.50). 

While microscopic techniques like PFG NMR and QENS measure diffusion paths that are no longer than dimensions of individual crystallites, macroscopic measurements like zero length column (ZLC) and Fourrier Transform infrared (FTIR) cover beds of zeolite crystals [18, 23]. In the case of the popular ZLC technique, desorption rate is measured from a small sample (thin layer, placed between two porous sinter discs) of previously equilibrated adsorbent subjected to a step change in the partial pressure of the sorbate. The slope of the semi-log plot of sorbate concentration versus time under an inert carrier stream then gives D/R. Provided micropore resistance dominates all other mass transfer resistances, D becomes equal to intracrystalline diffusivity while R is the crystal radius. It has been reported that the presence of other mass transfer resistances have been the most common cause of the discrepancies among intracrystaUine diffusivities measured by various techniques [18]. 

The even more involved quantitative transformation of 410 to give 411 requires four cycUzation steps within the crystal. These cycHzations can be performed on a hot stage at 180 °C for 30 min and the macroscopic shape of the crystals does not change at least for the octaphenyl case [ 130]. Importantly, the tetraphenyl-substi-tuted benzocyclobutene bonds of 411 exhibit the extraordinary length of 1.726 A [130], which is the world record for sp -sp single bonds [131] (Scheme 65). 

Table 5.5. Miller indices, stepped surface designations and angles between the macroscopic surface and terrace planes for fee crystals...

The results in sections 2 and 3 describe the adsorption isotherms and diffusivities of Xe in A1P04-31 based on atomistic descriptions of the adsorbates and pores. The final step in our modeling effort is to combine these results with the macroscopic formulation of the steady state flux through an A1P04-31 crystal, Eq. (1). We make the standard assumption that the pore concentrations at the crystal s boundaries are in equilibrium with the bulk gas phase [2-4]. This assumption cannot be exactly correct when there is a net flux through the membrane [18], but no accurate models exist for the barriers to mass transfer at the crystal boundaries. We are currently developing techniques to account for these so-called surface barriers using atomistic simulations. 

All seven steps require time, resulting in a rate of incorporating clusters into the growing crystal surface, which is called crystal growth kinetics. The following two sections consider translation of such a rate into a macroscopic equation for correlation and prediction. It is difficult to say which of the steps control the process, or even if the conceptual picture is valid. However, the first step—species transport to the solid surface—is well established and a brief description is given in Section 3.2.1.2. 

The parameters of the JT distortions were calculated by the X -method for a series of crystals in good agreement with experimental melting temperatures [14]. The details of the theory and specific calculations seemingly require additional refinements, but the main idea of the JT origin of the liquid-crystal phase transition seems to be quite reasonable. This work thus makes an important next step toward a better understanding of the relation between the macroscopic property of SB and the microscopic electronic structure, the JT effect parameters. 

The argument can perhaps be put forward In the following way A step of radius r centered on the dislocation maintains in its neighborhood a concentration c(r) determined by the curvature of the step. Under steady state conditions the concentration in the solution decreases in all directions approaching the value c maintained at a macroscopic distance from the crystal surface. It follows then that the concentration at the dislocation c(0) must obey the condition c(r) > c(0) > c. In fact, a brief calculation shows that the relation among these three quantities is... 

Consider first the case of clean steps. The curvature of the step is then entirely determined by its macroscopic radius r. Consequently, c(r) increases continuously with r reaching rather rapidly the saturation concentration ce. If it is now assumed that for a new step to be nucleated c(0)/ce has to be equal to the critical value c0/ce given by Eq. (2), it is clear that r/a has to become macroscopically large whatever the value given to c in other words, the first step has to make room for the following one to be nucleated. In particular, if c0/ce 11 0.20 as appears to be the case for LiF in water solutions considered above, one deduces from Eq. (4) that r/a -106 even for c 0. Needless to say, such a large value of r/a corresponds to an extremely flat crystal surface and no visible pit. 

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