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Most, if not all, material systems are multiscale in nature that are described by different physics at different length and time scales, ranging from density functional theory (DFT) to phase field modeling to finite element analysis.

How are these different levels of theory/model coupled across length and time scales?

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    $\begingroup$ Nice to see you here Edwin! $\endgroup$ – Nike Dattani May 4 at 4:59
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One example of multi-scale modeling in materials modeling is coarse-grain modeling

In these examples larger systems can be studied in a more efficient way by treating clusters of atoms, often monomers or residues of a polymer as a single structure or "bead".

Each bead is parameterized to capture the feature of the underlying atoms, for example polarity and steric bulk. Thus the material properties of the smaller atomic level affect the larger level.

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I would limit my answer to the situation where you have only two scales in your system. For more complicated problems, it's easy to generalize the concept provided here. Usually, the computational technique that works with the smallest scale in your system sends information to the higher-level scale back and forth. For example, let's say you want to simulate the mechanical properties of nanomaterials by using multiscale approach when you want to use molecular dynamics to extract the material behavior (i.e. the relation between stress and strain) and then use the finite element to get that information from the molecular dynamics and use it in defining the governing equation of your larger-scale simulation. Basically, here molecular dynamics calculate forces between atoms or molecules in your system for complex materials when you don't have a clear material model to describe its behavior and then finite element take this information and send back the updated stress/strain values to the molecular dynamics and you repeat this loop.

Another example is about coupling molecular dynamics and DFT. One of the challenges in molecular dynamics is finding suitable force-fields for complex materials like polymers or proteins. Here, DFT could act as a really accurate but computationally expensive tool find the forces from quantum mechanical simulation and molecular dynamics would be able to use this DFT extracted force-field and update the position of atoms (keep in mind the effect of electrons-nuclei is neglected due to the fact that electrons are much faster than nuclei and the electronic orbital reaches equilibrium so fast in comparison to atomic displacement time scale) and then you repeat the DFT to update the force-field and you repeat this loop.

Now we could couple the whole range physical model for combining these two examples and doing DFT to extract force-field and then feed it to molecular dynamics to extract the behavior of material and then feed the information from molecular dynamics to finite element to update the macroscopic stress/strain and displacement, though this approach seems really computationally demanding.

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