iMechanica - self assembly - Comments

thanks

Rui Huang — Mon, 29 Jan 2007 11:02:56 -0500

Yanfei:

Thanks for pointing out these two papers. I do have copies of these papers. They are indeed relevant. We probably should have cited them in addition to Lu and Suo's PRB paper. Compared to self-assembled monolayers (may be considered as a film of zero thickness), the patterns of epitaxial islands can be quite different (an effect of surface curvature as well as wetting, I think). Nevertheless, the principle of bifurcation illustrated in this paper is similar to the previous works on monolayers and surface reaction, with a root in the classical half-space problems in elasticity.

interesting wrok

Yanfei Gao — Mon, 29 Jan 2007 10:11:45 -0500

You may find out that the following two references are useful to you:

doi:10.1063/1.371999 by Yu and Suo

doi:10.1016/S1359-6454(02)00056-3 by Gao, Lu and Suo

comment on phase field model

Yanfei Gao — Mon, 20 Nov 2006 14:33:11 -0500

I just want to add some comments here. If we ignore the statistical variations (for example, thermal noise), the kinetic pathway on the energy landscape is selected by the phase-field model. Of course, the pattern can be trapped in some local energy minima, which is a well known phenomenon. (Please refer to the review paper written by Suo in Advances in Applied Mechanics in 1997.) There are some certain ways to define the patterns. For example, a correlation function can be used to examine the translational symmetry, and an angular function for rotational symmetry. I don't have reference handy, but there are references along this line. Using these statistical parameters, one can quantify the effects of initial conditions and others on the final pattern formation.

On the other hand, if there is a large thermal fluctuation, the thermal energy can substantially lead us to different patterns. When applying phase-field model for solidification problem, thermal fluctuation is necessary to be added, since both supercooling and nuclei are required to start the solidification process. In addition, the thermal energy may lead us to the global energy state. Wei and I talked about the possible transformaiton from random sripes to periodic stripes (under isotropic conditions). Energetically, it should be possible since the boundary condition governs a lower free energy for the periodic stripes, but our simulation cannot see that since the thermal fluctuations are not incorporated.

How to define the final pattern in phase field simulation

Jie Wang — Sat, 18 Nov 2006 02:25:35 -0500

Dear Wei,

Thanks for your comments. I agree with your opinion that the grid density or the length of the iteration step will not change the results much. This is true especially for the case when periodic boundary condition is employed. For many zero- dimension nanostructures such as particles, islands and quantum dots, periodic boundary condition does not hold. Under this situation, it needs to solve TDGL equation in real space by high efficient algorithm. For non periodic boundary condition, the influence of the grid density maybe becomes larger. Certainly, it also depends on the specific equation and the field variable used in the model.

How to define the final pattern is a question in phase field simulation. If defined from the total energy, as you said the pattern can rotate by any angle without changing the energy. If defined from the changes of patterns observed by the naked eye, it will not be accurate.

Jie

Some thoughts on the time-dependent TDGL equation

Wei Lu — Wed, 15 Nov 2006 01:00:02 -0500

This is an interesting question. I think the initial states and random terms that introduced to describe the nucleation process may cause different patterns. I do not think that the grid density or the length of the iteration step will change the results much. It is reasonable to expect that the grid may have a slight effect on the pattern orientation when the system is isotropic and the calculation domain is small. Simulations have shown that the pattern orientation in an isotropic system is random when the calculation domain is about 5~10 times larger than the feature size.

The system will evolve to reduce the energy toward a final state. I am not sure whether the global minimization is unique in 2D situation (for an isotropic system, the pattern can rotate by any angle without changing the energy, so it is not unique in direction. Then how about the wavelength?) For 1D situation, I think the wavelength that minimizes the energy is unique (the pattern position is not unique since one can translate it without changing the energy). Another practical question is whether the final pattern is reachable or the system will get stuck in local minimization state.