Roughening interface transition

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. 

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. 

Anisotropy in interface roughness and in a roughening transition. Anisotropic distribution of active centers for growth, such as lattice defects, which contribute to growth. 

Since the one-dimensional roughness of the steps determines whether a spiral takes circular or polygonal form, these morphologies may be treated similarly to the roughening transition of an interface, as described in Chapter 3. It is possible to predict interface roughness either by Jackson s a factor or by Bennema-Gilmer s generalized factor (see Section 3.8). The coefficients which determine the a... 

Rig. 7. Snapshot pictures of a Monte Carlo simulation of the crystal-vacuum interface in the framework of a solid-on-solid (SOS) model, where bubbles and overhangs are forbidden. Each lattice site i is characterized by a height variable h, and the Hamiltonian then is 7i = - hf - hj[. Three temperatures are shown kT/4> — 0.545 (a), 0.600 (b) and 0.667 (c). The roughening transition temperature 7r roughly coincides with case (b). From Weeks et al. (1973). 

Fig. 36. Schematic temperature variation of intcrfacial stiffness kn I K and interfacial free energy, for an interface oriented perpendicularly to a lattice direction of a square a) or simple cubic (b) lattice, respectively. While for tl — 2 the interface is rough for all non zero temperatures, in d — 3 il is rough only for temperatures T exceeding the roughening transition temperature 7r (see sect. 3.3). For T < 7U there exists a non-zero free energy tigT.v of surface steps, which vanishes at T = 7 r with an essential singularity. While k is infinite throughout the noil-rough phase, k Tic reaches a universal value as T - T . Note that k and fml to leading order in their critical behavior become identical as T - T. ...

In sect. 2, we have summarized the general theory of phase transitions with an emphasis on low-dimensional phenomena, which are relevant in surface physics, where a surface acts as a substrate on which a two-dimensional adsorbed layer may undergo phase transitions. In the present section, we consider a different class of surface phase transitions wc assume e.g. a semi-infinite system which may undergo a phase transition in the bulk and ask how the phenomena near the transition are locally modified near the surface, sect. 3.1 considers a bulk transition of second order, while sections 3.2 and 3.4-3.6 consider bulk transitions of first order. In this context, a closer look at the roughening transitions of interfaces is necessary (sect. 3.3). Since all these phenomena have been extensively reviewed recently, we shall be very brief and only try to put the phenomena in perspective. 

Show that the equilibrium roughness of a one-dimensional interface is qualitatively larger than that of a two-dimensional interface. Show that the one-dimensional interface is always rough i.e., the roughening transition temperature is zero). 

Cho, Y. K., Interface roughening transition and grain growth in BaTiOs and NbC-Co, PhD thesis, KAIST, Daejeon, Korea, 2003. 

This roughening transition happens in the interface and thus might at first be identified with Onsager s phase transition in the two-dimensional Ising model at J/IcrT = 0.44069. However, it actually occurs [14] at J/ bTr = 0.40758 and is not described by the two-dimensional Ising model, but by the KosterUtz-Thouless transition of the two-dimensional AF-model which predicts the ratio IF /ln(L) to approach 1/a if the temperature approaches Tr from above. At Tr, no anomaly was seen in the interfadal tension. For more details we refer to Hasenbusch et al. [14]. 

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