MSE 561, Atomic Modeling in Material Science Assignment 2

Transcription

1 Department of Material Science and Engineering, University of Pennsylvania MSE 561, Atomic Modeling in Material Science Assignment 2 1. Construction of the Two-dimensional Block As derived in Assignment 1, rmin for the Lennard-Jones potential is, Yang Lu So the block we construct is a square that contains 400 atoms with equal spacing 3.822Å. The subroutine named SETUP is shown below, SUBROUTINE SETUP(M) INTEGER I, J, N REAL RMIN REAL M(400,2) This block is stored in M, which contains coordinates of 400 atoms OPEN (1, FILE='BLOCK') RMIN = DO I=1, 20 FORTRAN I CODES represents ARE the i th DELETED atom in y direction IN THIS COPY DO J=1, 20 CONTACT J represents ME IF the YOU j th atom NEED in x direction MORE INFO. N=(I-1)*20+J Current number of atom whose coordinates are calculated M(N,1)=(J-1)*RMIN x coordinate is calculated using j M(N,2)=(I-1)*RMIN y coordinate is calculated using i WRITE(1,*) M(N,1),M(N,2) CLOSE(1) Finally, this block is shown in Fig.1 (next page). Each atom is donated by a red circle. 1 6 r min = 2σ = 3.822Å

2 2. Generation of Neighbor List Fig.1 In order to decrease the calculation time, we need to generate a neighbor list that can be used for future calculation. The method is: 1) calculate the distance between every two atoms in this block. 2) examine whether this distance is smaller than cut-off distance (rcut) in Lennard-Jones potential. (We only consider the neighbors that interact with the central atom.) The corresponding subroutine named NEIGHBOR is shown below, SUBROUTINE NEIGHBOR(M,DIST,NEIGHBORS,NOFNEIGHBORS) INTEGER I, J, N INTEGER NEIGHBORS(400,400), NOFNEIGHBORS(400) N represents the total number of neighbors of each atom REAL RCUT FORTRAN CODES ARE DELETED IN THIS COPY REAL M(400,2), CONTACT DIST(400,400) ME Coordinates IF YOU NEED of each MORE atom is imported INFO. in array M RCUT = OPEN(1, FILE='NEIGHBOR_LIST') DO I=1,400 WRITE(1,*) 'NEIGHBORS OF',I,':' This loop goes through each atom 2

3 N=0 Reset N for each atom at the beginning of calculation DO J=1,400 DIST(I,J)=((M(I,1)-M(J,1))**2+(M(I,2)-M(J,2))**2)**(0.5) IF (DIST(I,J).LT. RCUT.AND. DIST(I,J) +.GT. 0.0) THEN N=N+1 Calculate the distance between each atom and atom I Exam whether J is within the interaction region and exclude I itself If so, count N NEIGHBORS(I,N)=J FORTRAN CODES Record atom ARE J DELETED as neighbor of IN I in THIS the array COPY NEIGHBORS WRITE(1,*) CONTACT J ME IF YOU NEED MORE INFO. IF NOFNEIGHBORS(I)=N Record N after examining all the possible J atoms WRITE(1,*) 'NUMBER OF NEIGHBORS:',N CLOSE(1) Several representative points are shown below to justify the generated neighbor list: NEIGHBORS OF 1 : NUMBER OF NEIGHBORS: 3 NEIGHBORS OF 2 :

4 NUMBER OF NEIGHBORS: 5 NEIGHBORS OF 22 : NUMBER OF NEIGHBORS: 8 3. Atomic Level Stress In the case of pair potentials, the atomic level stress can be written as (when particles have no velocities), i σ αβ = 1 Ω i dφ( ) k,k i d r α β ik Eq.(1) where Ωi is the average volume of one atom, α and β represent x axis and y axis respectively. Under Lennard-Jones potential, Eq.(1) can be written as (if rik < 7Å), i σ αβ = 1 Ω i 24ε k,k i r [ 2(σ ) 12 + ( σ ) 6 ] r α β ik Eq.(2) If rik > 7Å, i σ αβ = 1 [3A(r Ω ik r cut ) 2 + 2B( r cut )] r α β ik i k,k i Eq.(3) When doing the summation, we don t need to go through all the 400 atoms. Instead, we only consider the k atom in neighbor list of the i atom (based on the previous subroutine). So the subroutine to calculate atomic level stress named STRESS can be coded as, 4

5 SUBROUTINE STRESS(M,DIST,NEIGHBORS,NOFNEIGHBORS) INTEGER I, J, K INTEGER NEIGHBORS(400,400), NOFNEIGHBORS(400) REAL M(400,2), DIST(400,400) REAL FD, e, A, B, VI, RMIN REAL X, Y REAL SIG(400,4) Array SIG stores stress tensor (xx, xy, yx, yy) of all the 400 atoms OPEN(1,FILE='STRESS') RMIN = e = A = E-3 B = E-3 VI = 19.0*RMIN*19.0*RMIN/400.0 Calculate the average volume of one atom, Ωi DO I=1,400 SIG(I,1)=0 FORTRAN Reset the CODES stress tensor ARE for each DELETED i atom IN THIS COPY CONTACT ME IF YOU NEED MORE INFO. SIG(I,2)=0 SIG(I,3)=0 SIG(I,4)=0 DO J=1,NOFNEIGHBORS(I) For each I atom, go through J atom in I s neighbor list K=NEIGHBORS(I,J) Get the K atom in summation X=M(I,1)-M(K,1) Calculate rik(α) Y=M(I,2)-M(K,2) Calculate rik(β) IF (DIST(I,K).LT. 7.0) THEN If rik < 7.0Å FD=24.0*e/DIST(I,K)*(-2.0*(3.405/DIST(I,K))**12+ Calculate dφ/dr using Eq.(2) +(3.405/DIST(I,K))**6) ELSE FD=3.0*A*(DIST(I,K)-7.5)**2+2.0*B*(DIST(I,K)-7.5) Calculate dφ/dr using Eq.(3) IF SIG(I,1)=SIG(I,1)+FD*X*X/DIST(I,K)/VI SIG(I,2)=SIG(I,2)+FD*X*Y/DIST(I,K)/VI 5

6 SIG(I,4)=SIG(I,4)+FD*Y*Y/DIST(I,K)/VI SIG(I,3)=SIG(I,3)+FD*Y*X/DIST(I,K)/VI FORTRAN CODES ARE DELETED IN THIS COPY WRITE (1,*) 'ATOM',I,SIG(I,1),SIG(I,2),SIG(I,3),SIG(I,4) CONTACT ME IF YOU NEED MORE INFO. CLOSE(1) Due to the 4-fold symmetry: σxx=σyy and σxy=σyx =0 except for the 4 atoms at corner. They are shown in Fig.2. Fig.2 6

7 4. Radial Distribution Function Let ni(r) be number of atoms found at distances between r and r+δr from atom i. The average number of atoms found between r and r+δr from any atom of the system as, n(r) = 1 N n i (r) where N is the total number of atoms. Then RDF can be calculated using, N i=1 g(r) = 1 n(r) ρ 2πrΔr Eq.(4) Eq.(5) In two dimension, ρ = 1/Ωi as defined in Eq.(2). After Δr is given, we can simply go through every atom with a certain r, get ni(r) in Eq.(4) with distance between any two atoms calculated from previous subroutine, sum them together and calculate g(r) by Eq.(5). The corresponding subroutine named RDF is shown below, SUBROUTINE RDF(DIST) INTEGER I, J, K, N REAL RMIN, DELTAR, PI, RHO REAL R(1001), DIST(400,400) The array DIST is imported from previous subroutine REAL G G represents g(r) for each r OPEN(1,FILE='RDF') PI = RMIN = FORTRAN CODES ARE DELETED IN THIS COPY DELTAR = RMIN/100.0 CONTACT Δr is ME chosen IF YOU to be rmin/100 NEED MORE INFO. RHO = 400.0/(19.0*RMIN*19.0*RMIN) Calculate ρ DO I=1,1001 R(I)=(I-1.0)*0.015 r is increased from 0 to15å with each step as Å N=0 Reset Σni(r) for each r at the beginning of calculation DO J=1, 400 Go through each atom J DO K=1, 400 IF (DIST(J,K).GT. R(I)) THEN 7

8 IF (DIST(J,K).LT. R(I)+DELTAR) THEN N=N+1 Count Σni(r) if atom I is within r and r+δr of atom J IF IF FORTRAN CODES ARE DELETED IN THIS COPY CONTACT ME IF YOU NEED MORE INFO. G=N/400.0/RHO/2.0/PI/R(I)/DELTAR Calculate g(r) using Eq.(5) WRITE (1,*) R(I), G CLOSE(1) g(r) is shown in Fig.3, which illustrates 8 neighbors within the range from 0 to 15Å. 5. Main Program Fig.3 Finally the main program is coded as shown below, PROGRAM MAIN 8

9 REAL M(400,2), DIST(400,400) REAL NEIGHBORS(400,400), NOFNEIGHBORS(400) CALL SETUP(M) CALL NEIGHBOR(M,DIST,NEIGHBORS,NOFNEIGHBORS) FORTRAN CODES ARE DELETED IN THIS COPY CALL STRESS(M,DIST,NEIGHBORS,NOFNEIGHBORS) CONTACT ME IF YOU NEED MORE INFO. CALL RDF(DIST) 9