Wilson-Frenkel

This is the Wilson-Frenkel rate. With that rate an individual kink moves along a step by adsorbing more atoms from the vapour phase than desorbing. The growth rate of the step is then simply obtained as a multiple of Zd vF and the kink density. For small A/i the exponential function can be hnearized so that the step on a crystal surface follows a linear growth law... 

This Wilson-Frenkel law is in fact valid even more generally the essential assumption for the derivation was that the surface structure of the growing interface does not change from the equilibrium one. Every rough surface should then be able to grow on average according to this Wilson-Frenkel law ... 

A linear law is valid above the roughening temperature and the Wilson-Frenkel rate is an upper hmit to the growth rate which is achieved in the case of very fast surface diffusion. This is illustrated in Fig. 2. 

FIG. 2 Growth rates as a function of the driving force A//. Comparison of theory and computer simulation for different values of the diffusion length and at temperatures above and below the roughening temperature. The spinodal value corresponds to the metastability limit A//, of the mean-field theory [49]. The Wilson-Frenkel rate WF is the upper limit of the growth rate. 

For a small driving force this growth law is indeed slower than the Wilson-Frenkel law (33) with Fwf but incomparably much larger than that of the nucleation process on faceted surfaces, (24), with V exp(—c/A/.i), where c is a positive constant. Therefore, the... 

Here the lattice positions i and j should be adjacent and the -function assures that one of the two lattice positions is occupied and the other one is free, r/j is a characteristic time scale for a diffusion jump. The time-dependence of the average si) is calculated by approximating the higher moments (siSj) [49]. In practice the analysis is rather involved, so we do not give further details here. An important result, for example, is the correction to the Wilson-Frenkel rate (33) at high temperatures ... 

The crystal is growing into the layer. Freezing is accompanied by a change in entropy and structure. Therefore we assume that the freezing rate is determined by D. For the calculation of the freezing rate we use the Wilson-Frenkel equation [15] ... 

The Gibbs-Thomson equation and the Wilson-Frenkel equation are both based on free energy considerations and both lead to D = yS/y, Probably this combination of thermodynamic quantities is a means for book keeping of the free energies which are invol-... 

This is known as Wilson-Frenkel or linear kinetics. 

The linear relationship is most simply shown by analogy with the Wilson-Frenkel theory of growth fix>m the vapor (e.g., Chernov, 1984). The total growth rate Vr is the... 

V2. Van den Berg, J. W., Van Blankenstein, M., Bosman-Jacobs, E. P., Frenkel, M., Hoerchner, P., Oost-Harwig, O. I., and Wilson, J. P., Solid phase radioimmunoassay for determination of conjugated cholic acid in serum. Clin. Chim. Acta 73, 3TI-W3 (1976). Van den Ende, A., Radecker, C. E., Mairuhu, W. M., and Van Zanten, A. P., Improved extraction procedure for determination of bile acids in foeces. Clin. Chim. Acta 121, 95-109 (1982). 

H.N.W.L. wishes to thank the staff members and his PhD students and Postdocs at the Van t Hoff laboratory with whom he had the privilege to work. He benefited from a long-term collaboration with Marc Bans, Louise Bailey, Mike Cates, Bob Evans, Seth Fraden, Daan Frenkel, Jean-Pierre Hansen, Joseph Indekeu, Geoff Maitland, Theo Odijk, Roberto Piazza, Wilson Poon, Peter Pusey, Bill Russel, Patrick Warren and Ben Widom. H.N.W.L. would like to thank the Royal Netherlands Academy of Arts and Sciences for the appointment as Academy Professor for the period 2006-2011, which made it possible to write this book. 

Chavez JC, LaManna JC. HIF-la accumulation in the rat brain in response to hypoxia and ischemia is attenuated diuing aging. ISOTT 2001 21 Proceedings. In Wilson DF, Biaglow J, Evans SM, Pastuszko A, eds. New Vbrk Kluwer Academic/Plemun. In press. Frenkel-Denkbeig G, Gershon D, Levy AP. The function of hypoxia-inducible factor 1 (HIF-1) is impaired in senescent mice. FEBS Lett 1999 462(3) 341-344. 

W. O. Van Den Berg, M. Van Blankenstein, E.P. Bosman-Jacobs, M. Frenkel, P. Horchner, O.I. Oost-Harwing and I.H.P. Wilson, Solid phase radioimmunoassay for determination of conjugated cholic acid in serum, Clin. Chim. Acta 73 277 (1976). 

In normal growth all sites on the interface are considered to be equivalent and the interface advances, by the continuous random addition of atoms. The theory was based on earlier work by Wilson and Frenkel S and it predicts that the mean growth rate, R, is proportional to the undercooling, i.e. 

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