Roughening temperature

The result is valid below the roughening temperature, but above the roughening temperature this mean-field approximation is not sufficient... 

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. 

This profile of the phase boundary determined here looks very similar to that obtained by the mean-field approximation (19), but the result here only applies to the profile above the roughening temperature. Since this is a mean-field theory, fluctuations are also not considered correctly. 

Fig. 3.3. Growth rate versus supercooling for two different face orientations. T is above its roughening temperature and is approximately linear. 2 is below its roughening temperature and is nucleation controlled at low supercooling but the growth rapidly increases after kinetic roughening...

The roughening transition has also been studied by computer simulation methods . Figure 42 shows characteristic configurations of a f.c.c. (100) surface in the simple solid-on-solid (SOS) model, calculated by Gilmer . The roughening temperature in this model corresponds to a parameter k T/ = 0.6. 

The first first direct experimental evidence for a roughening transition was reported in 1979. Several groups have studied the thermal behavior of the basal plane of a hexagonal close-packed He crystal. In a beautiful experiment Balibar and Casting obtained for this surface a roughening temperature of Tk 1.2K. 

The initial answer to this question was provided by Mullins [4], building on the work of Herring [5]. For surfaces orientations at temperatures above their thermodynamic roughening temperature Tr (the free energy for step formation becomes zero at T = Tr), the relaxation is driven by the stiffness [6] of the surface E = E + dE ldQ, where E is the surface energy and 9 is the orientation of the surface. For mass transport by surface diffusion, the dynamics of the surface at T > Tr are described by... 

For liquid-vapor interfaces, the correlation length in the bulk is of t he order of atomic distance unless one is close to the critical point Hence the concept of local equilibrium is well justified in most practical circumstances For. solid surfaces above the roughening temperature, the concept also makes sense. Since the surface is rough adding (or removing) an atom to a particular part of the surface docs not disturb the local equilibrium state very much, and this sampling procedure can be used to determine the local chemical potential. This is the essence of the Gibbs-Thomson relation (1). 

By eomparing the numerieally ealeulated height-height eorrelation function to the expected universal behavior at the roughening transition, we have estimated the roughening temperature of the model to be given approximatey by... 

The simulation results reported below are for b = 1/2 or T - 0.687. Well-defined steps arid terraces are indeed seen during the relaxation of the groove, indicating that the surface is well below its roughening temperature. 

In this paper we review some of our recent work on the dynamics of step bunching and faceting on vicinal surfaces below the roughening temperature, concentrating on several cases where interesting two dimensional (2D) step patterns form as a result of kinetic processes. We show that they can be understood from a unified point of view based on an approximate but physically motivated extension to 2D of the kind of ID step models studied by a number of workers. For some early examples, see refs. [1-5]. We have tried to make the conceptual and physical foundations of our own approach clear, but have made no attempt to provide a comprehensive review of work in this active area. More general discussions from a similar perspective and a guide to the literature can be found in recent reviews by Williams and Williams and BartelF. 

The quantity under the square root in (5) becomes negative at small values of P5x, and P5, this happens at temperatures above the roughening temperature of the (110) surface where the solid-on-solid approximation for the step free energy f ( ) is not positive definite and the simple theory considered here breaks down. 

Heyraud and Metois (1980) studied shapes of small gold crystals in thermal equilibrium with their vapor in their samples only the (HI) and (100) facets were observed, since the range of temperatures investigated T 1000 C) is above the roughening temperature of the (110) facet. To observe some of the shapes described in this article, temperatures below the roughening and the deconstruction temperatures of the (110) facet should be considered. 

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