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# On the solution to time-dependent Ginzburg-Laudau (TDGL) equation

Time-dependent Ginzburg-Laudau (TDGL) equation is the simplest kinetic equation for the temporal evolution of a continuum field, which assumes that the rate of evolution of the field is linearly proportional to the thermodynamical driving force. The computation model based on this equation is also called phase field model. Phase field simulation can predict quite beautiful patterns of microstructures of material. It has been widely applied to simulating the evolution of microstructure by choosing different field variables. For example, using the single conserved field (concentration field), continuum phase field models has been employed to describe the pattern formation in phase-separating alloys (Nishimori and Onuki, 1990 Phys. Rev. B, 42,980) and the nanoscale pattern formation of an epitaxial monolayer (Lu and Suo, 2001 J. Mech. Phys. Solids, 49,1937). On the other hand, using the nonconserved field (polarization field), the phase field model has been utilized to simulating the formation of domain structure in ferroelectrics (Li et al. 2002 Acta Mater, 50,395). The thermodynamical driving force is usually nonlinear with respect to the field variable. In the case of nonlinearity, the solution to TDGL equation may not be unique. Different grid density, length of iteration step, initial state and random term (introduced to describe the nucleation process) may induce different results in the simulation. Does anyone investigate the effect of these factors on the final pattern? I wonder whether we can prove the solution is unique or not.

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## Comments

## Some thoughts on the time-dependent TDGL equation

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.

## How to define the final pattern in phase field simulation

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

## comment on phase field model

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.