Multiscale Green's function
Mathematical modeling of nanomaterials requires special techniques and is now recognized to be an independent branch of science.
[3] A mathematical model is needed to calculate the displacements of atoms in a crystal in response to an applied static or time dependent force in order to study the mechanical and physical properties of nanomaterials.
The GF based techniques are used for modeling of various physical processes in materials such as phonons,[6] Electronic band structure[7] and elastostatics.
[5] The MSGF method is a relatively new GF technique for mathematical modeling of nanomaterials.
Mathematical models are used for calculating the response of materials to an applied force in order to simulate their mechanical properties.
Multiscale modeling is needed for such materials because their properties are determined by the discreteness of their atomistic arrangements as well as their overall dimensions.
The MSGF method is also used for analyzing behavior of crystals containing lattice defects such as vacancies, interstitials, or foreign atoms.
[8] It is an extension of the lattice statics Green’s function (LSGF) method that was originally formulated at the Atomic Energy Research Establishment Harwell in U.K. in 1973.
The LSGF method is based upon the use of the Born von Karman (BvK) model[6][19] and can be applied to different lattice structures and defects.
[11][17][20] The MSGF method is an extended version of the LSGF method and has been applied to many nanomaterials and 2D materials[2] At the atomistic scales, a crystal or a crystalline solid is represented by a collection of interacting atoms located at discrete sites on a geometric lattice.
The GF in the MSGF method is calculated by minimizing the total energy of the lattice.
The matrix G is the multiparticle GF and is referred to as the lattice statics Green’s function (LSGF),.
Inversion of such a large matrix is computationally extensive and special techniques are needed for the calculation of u’s.
In this case any atom can be chosen as the origin and G(L,L') can be expressed by a single index (L'-L)[6] defined as The asymptotic limit of G(L), that satisfies Eq.
(10), for large values of R(L) is given by[8] where x = R(L) is the position vector of the atom L, and Gc(x) is the continuum Green's function (CGF), which is defined in terms of the elastic constants and used in modeling of conventional bulk materials at macroscales.
[21] The LSGF G(0,L) in this equation reduces smoothly and automatically to the CGF for large enough x as terms O(1/x4) become gradually small and negligible.
(10) is not valid anymore and the correspondence between the LSGF and the CGF, needed for their seamless linking breaks down.
(12), then leads to the following Dyson’s equation for the defect LSGF:[15] The MSGF method consists of solving Eq.
Equation (16) expresses the atomic displacements u in terms of G, the perfect LSGF even for lattices with defects.
The effective force f* can be determined in a separate calculation by using an independent method if needed, and the lattice statics or the continuum model can be used for G. This is the basis of a hybrid model that combines MSGF and MD for simulating a germanium quantum dot in a silicon lattice.
The Green's function G can be calculated independently, which can be fully atomistic for nanomaterials or partly or fully continuum for macroscales to account for surfaces and interfaces in material systems as needed [8] Tewary, Quardokus and DelRio[23] have suggested that Green's function is not just a mathematical artefact, but a physical characteristic of the solid, which can be measured by using scanning probe microscopy.
An important application of the MSGF method is in modeling the temporal (time-dependent) processes in solids, especially in nanomaterials.
This is needed in diverse applications like testing and characterization of materials, propagation of waves and heat in nanomaterials, and modeling of radiation damage in semiconductors.
[2][18] These processes need to be simulated over a wide range of times from femtoseconds to nanoseconds or even microseconds which is a challenging multiscale problem for nanomaterials.
It has been applied[25] to simulate the propagation of ripples in graphene,[9] where it has been shown that the CGFMD can model time scales over 6 to 9 orders of magnitude at the atomistic level.
At least in some idealized cases such as propagation of ripples in graphene, the CGFMD can bridge the time scales from femto to microseconds.
