Undulation growth

Lu Taijing and I. Sunagawa, Origin of undulated growth steps on hematite crystals from Sasazawa, Japan, Mineral/., 13,1987,409-23... 

Closely related substrate-fluid-soid films can also exhibit undulation growth of the top solid layer. This is thought to occur due to the buckling of the top layer in response to a thermal expansion mismatch between solid and fluid layers [7] and van der Waals forces are not thought to play a significant role [8, 9]. This may seem curious to the reader, because we have just explained how van der Waals forces can play such a significant role in the trilayer films we are interested in. It should first be noted that in the trilayer films in which we are interested, the van der Waals forces are always attractive, due to the symmetry of the problem, while this is not necessarily the case for the substrate-fluid-solid films. However,... 

Once such method of controlling the surface undulations is through stress-guided self-assembly. Stress-driven self-assembly is a promising route to either induce ripples in a desired pattern [12] or guide undulation growth such that a specific corrugation is produced [4]. 

We have described the model used in this study to describe the evolution of the ultrathin solid-fluid-solid films. In particular, this model can capture large-amplitude height variations and simulate the late stages of undulation growth. We now describe the computational implementation of this model. 

Figure 5. Wavelength of undulation growth as a function of Young s modulus, or, more precisely, Young s modulus to the power of one-fourth. Simulations results from the model described above are directly compared with theory of Dalnoki-Veress et al.. The agreement is found to be satisfactory and the wavelength increases with Young s modulus as bending the solid layers becomes increasingly energetically unfavorable.

Figure 8. Relative wavelength of surface corrugations as a function of the magnitude of the imposed strain field. Both compressed and stretched systems are contrasted, and the wavelength of undulation growth is seen to increase or decrease with stretching or compression, respectively.

The region of stiffer material is clearly evident in Fig. 9a, which shows the system at a time of 5 minutes. The stiffer material opposes undulation growth ... 

Initially the van der Waals forces across the film drive undulation growth in these systems and cause the surface instability to occur. The growth is resisted by the elastic deformation of the solid capping layers, which also maintains the films structural integrity. Eventually, the solid capping layers locally meet and the fluid is ejected from these regions of the film to form isolated pockets of fluid. 

An alternative, and interesting, possibility is to introduce a phase-separating blend as the fluid component (a schematic of this system is shown in Fig. 11). The phase-separating A-B polymer blend will evolve, and phase separate, at its own length- and timescales. However, to minimize the interface between the A and B domains of the polymer blend it may be desirable for the length-scale of phase separation to conform to the wavelength of undulation growth found in... 

Figure 11. Schematic of potential phase separation in these films. The interaction between the length-scale of phase separation and the length-scale of undulation growth could result in the emergence of novel structures.

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