Stress Calculation And Boundary Conditions Of Silicon Crystals In Molecular Dynamics

In this paper,the molecular dynamics simulation is combined with Cauchy-Born approximation algorithm of finite temperature and artificial neural network respectively.The PK stress and boundary conditions diffusion wave in the center of silicon crystal are obtained.Therefore,the main research is divided into the following two parts:(1)Finite temperature Cauchy-Born approximation was used to calculate PK stress of silicon crystals at different temperatures and different deformations.According to Cauchy-Born rule of finite temperature,all the atoms in the alloy system will produce consistent deformation,inner displacement and thermal vibration with Boltzmann dis-tribution in the average position at finite temperature and given deformation.The derivative of Helmholtz free energy with respect to the deformation matrix can obtain the expression of PK stress.Since Helmholtz free energy is the form of the partition function of atomic displacement u_I,the PK stress expression derived before is sim-plified,and the PK stress can be expressed as the ensemble average of virial terms.the quasi-harmonic approximation algorithm can make it further simple,the idea is to make harmonic potential energy in the average location,after the removal of high-order items,the original Boltzmann distribution can be approximate to gaussian distribu-tion.Do the same operation to virial term(the second order Taylor expansion),PK stress can be written about temperature T linear form,only the linear dependence coef-ficient G(F):H(F)is known,then add virial item at zero temperature,the PK stress at any temperature can be calculated out.However,the alloy system we generally study is very large,and it may not be easy to calculate the inverse of a high-dimensional matrix and calculate H(F).So we want to derive a much simpler and faster algorithm.When the Fourier transform is used to convert the original computation space into reciprocal space,and the point is taken in the first Brillouin region.The translation invariance of the system and various initial conditions are added to obtain the final and simpler PK stress expression,instead of needing to calculate the inverse of the high-dimensional matrix,we only need to calculate the inverse of multiple 3×3or 6×6matrices.In order to prove the effectiveness of this fast algorithm,we calculate the expressions of three types of deformation in silicon crystal system:no deformation,tensile deformation and shear deformation,and compare the PK stress calculated at 0K to 500K(every 100K)with the PK stress calculated by MD.In addition,we also show the effect of inner displacement in shear deformation,and the PK stress calculated in the case of inner displacement and no inner displacement is different.(2)Artificial neural network is used to train the boundary conditions of silicon crystals.We want to learn that the displacement of the atom on the outside of the boundary is a function of the historical summation of the displacement of the atom on the inside of the boundary by artificial neural networks.while training,we built a displacement field in the center of a larger scale silicon crystal system,and took the internal and external atomic displacement of the boundary of a small region as the research object,and input a neural network to find the relationship between them.In order to simplify the model and make use of translational invariance,we only pick up the boundary of the small-scale displacement for training.After the small neural network is trained well,we consider the direction of the wave,increase the complexity of the data,and make the neural network trained by the boundary conditions play a role in waves in all directions.We will also use the trained boundary conditions to do dynamic tests.With each running step,MD will update the position of the external atoms with the neural network,and then re-influence the position of the internal atoms.Based on this,we repeatedly observe that the boundary conditions we trained do play a role,and the velocity wave is absorbed when it reaches the boundary.