Roughening transition

The terrace refers to a flat surface, the energy of which has a local minimum value that is, a cusp in y-plot. This means that all surfaces existing in the equilibrium shape are terraces. Due to energetics, however, terraces mostly have low Miller indices, and the energetics and structural features of terraces conform to those of the surfaces. 

A step is an entity which compensates an angular deviation of a non-equilibrium surface from a nearby terrace. If the angular deviation of a vicinal surface from a terrace is 0, then this surface can be described by the terrace that is separated by periodically spaced steps. When the average mutual spacing of steps and the step height are I and h, respectively, the step density is given as 1/1, equivalently tan d/h [53]. The free energy of this vicinal surface is then written as 

7) Readers who wish to check the derivations and detailed explanation of each equation introduced, are encouraged to check Ref. [134]. 

Interaction increases quadratically with the step density (b) Relationship between the projected surface free energy and the crystal shape as a function of temperature. 

This is called the projected surface free energy, because it is the surface free energy projected along the surface normal direction, z. One of the most important meanings of the projected surface free energy is that it is equivalent to the crystal shape through the Legendre transformation. The conversion of a polar coordinates (fr 9) into a 

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As remarked above, surface science has come to be partitioned between chemists, physicists and materials scientists. Physicists have played a substantial role, and an excellent early overview of surface science from a physicist s perspective is by Tabor (1981). An example of a surface parepisteme that has been entirely driven by physicists is the study of the roughening transition. Above a critical temperature but... 

By the end of the 1940s, however, there existed the presumption that a roughening transition could possibly occur at a crystal surface with increasing temperature [20]. Although the original idea pursued in this still... 

T. Emig, T. Nattermann. Disorder-driven roughening transition of charge-density waves and flux-Unes. Phys Rev Lett 79 5090, 1997 T. Emig, Th. 

Natterniann. Roughening transition of interfaces in disordered systems. Phys Rev Lett S7 1469 (1998). 

When the bulk transition is of first order, the above mentioned arguments based on dimensionality do not apply and the would be roughening transition temperature T j may be larger than the bulk transition temperature T, in which case there is simply no roughening transition. The situation is further complicated by the wetting phenomena. When we approach T from below, the disordered phase becomes metastable and may wet the interface a large layer of disordered phase develops in between the two ordered domains. 

We display in Fig. 2 some 2-d order-parameter and spin maps at different temperatures. These pictures already gives us a qualitative informations on the thermodynamical behavior of the APB. The APB appears to be fiat for T=1000 K and 1400 K, while it seems to be rough at T=1550 K. It becomes even more rough at higher temperature T=1675 K, as shown in Fig. 3. This visual analysis shows us, without any ambiguity, that the APB does indeed undergo a roughening transition as the temperature increases. 

The roughening transition itself is believed to be of the Kosterlitz-Thouless type , according to which one should expect the following properties ... 

The above results show that the APB undergoes a roughening transition at T j 1490 K, which is below the bulk first-order transition, and this transition is in agreement with the Kosterlitz-Thouless theory. In brief, the APB is flat below T j and, above T j, it develops transverse fluctuations and wanders through the lattice. 

For a review on the roughening transition see J. D. Weeks in Ordering in Strongly Fluctuating Condensed Matter Systems, ed. T. Riste, 293 (Plenum New-York, 1980)... 

If there is a roughening transition, then instead of occurring at a single temperature TK (obtained for the theoretical infinite surface consisting of independent units) it must be spread over a range of temperatures, AT, because of both... 

Unfortunately, even for low molecular weight material it is difficult to obtain clear experimental evidence for a roughening transition [71]. This is mainly due to the fact that during growth the interface generally assumes a metastable shape and relaxation times are long and increase with crystal size. Therefore we certainly cannot expect a definitive answer for macromolecules. We shall therefore just make several comments which hopefully will be of use when reading the literature. 

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 indication for the existence of a roughening transition was obtained by Jackson and Miller who studied the crystal shape of chloroethane and ammonium chloride. Above 370 K and 430 K, respectively, they observed a drastic change in crystal morphology, which might be interpreted as roughening. Similar observations for adamantane have been reported by Pavlovska . ... 

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. 

More recently, the question of thermal roughening has also been addressed in the study of metal surfaces. Detailed He difiraction studies from the high Miller index (113) surface of Cu and Ni proved the existence of a roughening transition on these surface. These studies were performed by means of He scattering. Let us make first two short comments. 

The relaxation of isolated, pairs of and ensembles of steps on crystal surfaces towards equilibrium is reviewed, for systems both above and below the roughening transition temperature. Results of Monte Carlo simulations are discussed, together with analytic theories and experimental findings. Elementary dynaniical processes are, below roughening, step fluctuations, step-step repulsion and annihilation of steps. Evaporation kinetics arid surface diffusion are considered. 

A profile imprinted on a crystal surface will undergo morphological changes when relaxing towards equilibrium. This morphological evolution has been foiind, in experiments and theoretically, to be significant different above and below the roughening transition of the relevant surface. - ... 

Figure 1. Monte Carlo configuration of an isolated step below the roughening transition temperature TR 1.24J ofthe standard SOS model, at t= 2000 MCS, using evaporation kinetics.

So far, we tacitly assumed that the upper and lower terraces next to the step are below their roughening transition temperature. By fixing the boundary heights of the terraces, away from the step, at, say, level 0 for the lower and level 1 (in units of the lattice spacing) for the upper terrace, one can study the time evolution of the step width w, defined, for instance, as the second moment of the gradient of the step profile also above roughening. Then one obtains s =1/4 for terrace diffusion and 1/2 for evaporation kinetics, as predicted by the continuum description of Mullins and confirmed by our Monte Carlo simulations. 

Figure 5. Profiles z(x, f) of a grating below the roughening transition temperature, at increasing time, of the standard SOS model with 80 x 320 sites and a miscut of a few lattice spacings, using evaportion kinetics in Monte Carlo simulations (full symbols). For comparison, a sinusoid is shown (open symbols). The initial amplitude of the grating is five lattice spacings.

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