Kroll

Gompper G and Kroll D M 1995 Phase diagram and sealing behavior of fluid vesieles Phys. Rev. E 51 514... 

Theissen O, Gompper G and Kroll D M 1998 Lattice-Boltzmann model of amphiphilic systems Euro. Phys. Lett. 42 419... 

Weaver J H, Chai Y, Kroll G H, Ohno T R, Haufler R E, Guo T, Alford J M, Conceicao J, Chibante L P F, Jain A, Palmer G and Smalley R E 1992 XPS probes of carbon caged metals Chem. Phys. Lett. 190 460... 

The metal was a laboratory curiosity until Kroll, in 1946, showed that titanium could be produced commercially by reducing titanium tetrachloride with magnesium. This method is largely used for producing the metal today. The metal can be purified by decomposing the iodide. 

It was originally separated from zirconium by repeated recrystallization of the double ammonium or potassium fluorides by von Hevesey and Jantzen. Metallic hafnium was first prepared by van Arkel and deBoer by passing the vapor of the tetraiodide over a heated tungsten filament. Almost all hafnium metal now produced is made by reducing the tetrachloride with magnesium or with sodium (Kroll Process). 

Table 3. Analysis of Kroll Process, Electrowon, and Refined Hafnium, ppm...

Magnesium-Reduction (Kroll) Process. In the 1990s, nearly all sponge is produced by the magnesium reduction process (Fig. 4). 

The tlrree impurities, iron, silicon and aluminium are present in the metal produced by the Kroll reduction of zirconium tetrachloride by magnesium to the extent of about 1100 ppm. After dre iodide refining process tire levels of these impurities are 350, 130 aird 700ppm respectively. The relative stabilities of the iodides of these metals compared to that of zirconium can be calculated from the exchange reactions... 

The Kroll process for tire reduction of tire halides of refractory metals by magnesium is exemplified by the reduction of zirconium tetrachloride to produce an impure metal which is subsequently refined with the van Arkel process to produce metal of nuclear reactor grade. After the chlorination of the impure oxide in the presence of carbon... 

Kroll, E. 1993 Modeling and Reasoning for Computer-Based Assembly Planning. In Kusiak, A. (ed.). Concurrent Engineering Automation, Tools and Techniques. NY Wiley. 

Refractory metals Zirconium Hafnium Titanium Kroll process, chlorination, and magnesium reduction Chlorine, chlorides, SiCli Wet scrubbers... 

The theory presented above has been based on the Evans-Tarazona density funetional approaeh. Therefore its generalization to multieomponent systems is not instantaneous. However, a modified Meister-Kroll theory, introdueed by Riekayzen et al. [143,144], does not suffer from the above-mentioned drawbaek and provides an aeeurate deseription of nonuniform simple (nonassoeiating) fluids. 

The prescription proposed in the original Meister-Kroll-Groot [138,139] theory for hard spheres requires the determination of the local density and the averaged density as two independent variational variables by minimizing the grand potential with respect to these variables. The modification introduced by Rickayzen et al. [143,144] arises from another definition of the average density... 

The latter relation is the final equation for the density profile, resulting from the modified Meister-Kroll-Groot theory if applied to associating fluids [145]. 

Fig. 10(a) presents a comparison of computer simulation data with the predictions of both density functional theories presented above [144]. The computations have been carried out for e /k T = 7 and for a bulk fluid density equal to pi, = 0.2098. One can see that the contact profiles, p(z = 0), obtained by different methods are quite similar and approximately equal to 0.5. We realize that the surface effects extend over a wide region, despite the very simple and purely repulsive character of the particle-wall potential. However, the theory of Segura et al. [38,39] underestimates slightly the range of the surface zone. On the other hand, the modified Meister-Kroll-Groot theory [145] leads to a more correct picture. 

In Fig. 10(b) one can see the density profiles calculated for the system with /kgT = 5 and at a high bulk density, p = 0.9038. The relevant computer simulation data can be found in Fig. 5(c) of Ref. 38. It is evident that the theory of Segura et al, shghtly underestimates the multilayer structure of the film. The results of the modified Meister-Kroll-Groot theory [145] are more consistent with the Monte Carlo data (not shown in our... 

FiG. 10 Normalized density profiles p z)/for the associating fluid at a hard wall. The association energy is jk T — 7 and the bulk density is p = 0.2098 (a), e ykgT = 5 and the bulk density equals 0.9038 (b). The solid and dashed lines denote the results of the modified Meister-Kroll theory and the theory of Segura et al., respectively. The Monte Carlo data in (a) are marked as points. (From Ref. 145.)... 

Fig. 10(b)). One of the reasons for the differences between both theories is a different form of a hard sphere part of the free energy functional. Segura et al. have used the expression resulting from the Carnahan-Starhng equation of state, whereas the Meister-Kroll-Groot approach requires the application of the PY compressibility equation of state, which produces higher oscillations. 

A = 2k/ /3 for the case of cyhnders. In order to avoid this problem, Gompper and Kroll [241] have recently argued that a more appropriate discretization of the bending free energy should be based directly on the square of the local mean curvature ... 

As yet, models for fluid membranes have mostly been used to investigate the conformations and shapes of single, isolated membranes, or vesicles [237,239-244], In vesicles, a pressure increment p between the vesicle s interior and exterior is often introduced as an additional relevant variable. An impressive variety of different shapes has been found, including branched polymer-like conformations, inflated vesicles, dumbbell-shaped vesicles, and even stomatocytes. Fig. 15 shows some typical configuration snapshots, and Fig. 16 the phase diagram for vesicles of size N = 247, as calculated by Gompper and Kroll [243]. 

FIG. 15 Conformations of fluid vesicles for different values of the bending rigidity and pressure increment (a) branched polymer (b) inflated vesicle (c) prolate vesicle (d) stomatocyte. (From Gompper and Kroll [243]. Copyright 1995 APS.)... 

G. Gompper, D. M. Kroll. Curr Opin Coll Interf Sci 2 373-381, 1997 G. Gompper, D. M. Kroll. J Phys Cond Matt (5 8795-8834, 1997. 

B. Laird, D. Kroll. Freezing of soft spheres a critical test for weighted density functional techniques. Phys Rev A 42 4810, 1990 D. Kroll, B. Laird. Comparison of weighted density functional theories for inhomogeneous liquids. Phys Rev A 42 4806, 1990. 

T. Burkhardt, H. Muller-Krumbhaar, D. Kroll. A generalized kinetic equation of crystal growth. J Cryst Growth 5S 13, 1973. 

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