1. South China University of Technology
1.高温塑性变形热模拟实验
South China University of Technology
Growth of Cluster
Xiaobao Yang
Department of Physics

2. C60 Graphene fragments C20 Clusters experimentally observed
ACS nano,6,8203(2012)
Nature 318,162(1985); 407,60(2000);

3. Diamond fragments
Science 299, 96 (2003);PRL, 103,047402(2009)

4. B20 B19 http://www.chem.brown.edu/research/LSWang/

5. The Geometry of Universe Platonic and Archimedean solids
Nature 425, October 2003

6. Wulf’s construction How Clusters are found? Dense Packing and Symmetry
in Small Clusters of Microspheres
Wulf’s construction
Science 301,403(2003)

7. Nucleation / Diffusion / Reconstruction
The growth and evolution of Clusters
Nucleation / Diffusion / Reconstruction
Substrate / bonding / Temperature

8. No diffusion vs Full diffusion
Cluster growth model
No diffusion vs Full diffusion

9. Cluster Growth Model Refer to DLA.m
1.高温塑性变形热模拟实验
Cluster Growth Model
Diffusion Limited Aggregation (DLA) Model
Discussion on programming
Choose initial position of a walker at random on a circle r0.
If the walker wanders too far from the cluster (say, >1.5r0), start a new walker.
As the cluster grows, r0 should be increased. (say, keep r0=5Rcluster)
When the walker is far from the cluster, a greater step size may be adopted.
Refer to DLA.m
Fractals? The dimensionality of clusters?

10. DLA.m 1)Place the seed 2)Atomic random walk towards the seed
clear; clf;
M = 300; N=1000; % MxM grid; N particles;
A=zeros(M,M);
for i=1:M; for j=1:M;
if(abs(i-M/2)<M/40 & j==round(M/2)) A(i,j)=-1; end;
end;
%imagesc(A); hold on
nparticle=0;
while(nparticle < N)
%generate a random walker within the belt of 3/5*M/2<R<4/5*M/2
r=(rand/5+3/5)*M/2; theta=rand*pi*2;
x=round(r*cos(theta)+M/2); y=round(r*sin(theta)+M/2);
check = 1; R=M*9/20; % checker for walkers wandering too far i.e., |walker-center|>R.
% if not meet the seed, continue wandering
while( A(x-1,y) ~= -1 & A(x+1,y) ~= -1 & A(x,y-1) ~= -1 & A(x,y+1) ~= -1)
x=x+sign(rand-0.5); y=y+sign(rand-0.5);
if( abs(x-M/2)>M*9/20 || abs(y-M/2)>M*9/20 );
check=0; break; end; %walker elimated if wandering too far!
end
if(check==1); A(x,y) = -1; nparticle = nparticle + 1; end; %meets seed; seed updated.
%if(mod(nparticle,100)==0); imagesc(A); end
colormap(winter); imagesc(A);
axis([0 M 0 M],'square','equal');
1)Place the seed
2)Atomic random walk towards the seed
3) Check if the atom is attached
4) Neglect the atom far from the seed
5) Draw the cluster

11. Matlab: illustration
colormap
image()
imagesc()
pcolor()

12. Cluster Growth Model Eden Model Eden Cluster
1.高温塑性变形热模拟实验
Cluster Growth Model
Eden Model
Consider a two dimensional lattice of points (x, y).
Placing a seed particle at the origin (x = 0, y = 0).
Growing by the addition of particles to its perimeter.
unoccupied near-neighbor sites
as the perimeter sites of the cluster.
Choose one of these perimeter sites
at random and place a particle at the chosen location.
This process is then repeated;
update perimeter and particle.
Continue this process
until a cluster of the desired size is obtained.
Eden Cluster

13. x x nearest edge tmp clear x=[0 0]; %initial nearest=[0 1 0 -1 1 0
-1 0];
edge=nearest;
%perimeter
for ii=1
growsite=ceil(length(edge)*rand);
tmp=ones(4,1)*edge(growsite,:)+nearest;
x=[x
edge(growsite,:)];
tmp=[tmp
edge];
edge=setdiff(tmp,x,'rows');
end x
edge x
tmp

14. Dimensionality of the cluster
1.高温塑性变形热模拟实验
Dimensionality of the cluster
For a straight line:
where r is small enough.

15. 1.高温塑性变形热模拟实验
Eden vs. DLA cluster

16. Morphology of a Class of Kinetic Growth Models
Place a seed particle at a site on a two dimensional square lattice.
1)Check the four neighbors of the seed and occupy each one, independently, with a probability p
2)Sample the nearest neighbors of the second generation and fill these sites independently with a probability p.
sites which are not filled are blocked and cannot be filled at a later time.
p = l, (b) p = 0.9, (c) p = 0.8,
(d) p = 0.7,(e) p = 0.6, (f) p =
If we modify our model so that all perimeter sites are active growth sites for all time, then we approach the Eden model in the limit p->0.
PRL,55,2515(1985)

17. 1) Select the active sites with p 2) Record the unactive sites
clear
x=[0 0]; %initial
edge=[0 1
0 -1
1 0
-1 0];
nearest=edge;
p=0.6 % propobality
unactive=[];
for ii=1:200%generation
growsite=rand(size(edge,1),1)<p;
if max(mod(growsite+1,2))>0
unactive=[unactive
edge((growsite==0),:)];
end
x=[x
edge(growsite,:)];
tmp=edge(growsite,:);
for jj=1:size(tmp,1)
edge=[edge
repmat(tmp(jj,:),size(nearest,1),1)+nearest];
edge=setdiff(edge,x,'rows');
if length(unactive)>0
edge=setdiff(edge,unactive,'rows');
1) Select the active sites with p
2) Record the unactive sites
3) Add atoms and update edge
4) Delete edge from cluster and
unactive sites
plot(x(:,1),x(:,2),'*')
hold on
plot(edge(:,1),edge(:,2),'o')
axis equal
plot(unactive(:,1),unactive(:,2),'ro')

18. Shapes in square lattice: Diamond Square Triangular
The role of energy
Shapes in square lattice:
Diamond
Square
Triangular

19. Simulation of cluster
1) Adding atoms 2) Atom diffusion allowed Adding and deleting to conserve the number of atoms 3) Energy estimation 4) Accept or reject the configuration

20. Spanning clusters

21. Percolation Problems Porous rock
1.高温塑性变形热模拟实验
Percolation Problems
Porous rock
(Original percolation problem, Broadbent and Hammersley, 1957)
2. Forest fires, etc
How far from each other should trees in a forest (orchard) be planted in order to minimize the spread of fire (blight)?
Suppose a large porous rock is submerged under water for a long time, will the water reach the center of the stone?
p=0.48

22. What is percolation ? 2-dimension percolation 2x2 lattice 6x6
1.高温塑性变形热模拟实验
What is percolation
2-dimension percolation
Percolated system
if a spanning cluster exist
(connects top and bottom exists)
2x2 lattice
What is the probability for a system to be percolated for a given coverage?
6x6 ?
Infinite x infinite
(critical coverage)
120x120

23. Simulation of Percolation
1.高温塑性变形热模拟实验
Simulation of Percolation

24. Percolation clear, clf, colormap gray; M=24; p=0.7; A=rand(M,M);
for i=1:M; for j=1:M;
if(A(i,j)<p ) A(i,j)=0; else A(i,j)=1; end
end; end
imagesc(A)
for ii=1:size(A,1)
for jj=1:size(A,2)
text(jj,ii,num2str(A(ii,jj)));
hold on
end

25. Divide the data into two groups
rock=[];
hole=[];
for ii=1:size(A,1)
for jj=1:size(A,2)
if A(ii,jj)==1
rock=[rock
ii jj];
else
hole=[hole
end
A=[ ];

26. Find the spanning clusters
for ii=3%1:size(hole,1)
tp1=hole(ii,:);
tp2=[tp1
repmat(tp1,size(nearest,1),1)+nearest];
tp2=intersect(tp2,hole,'rows');
while size(tp1,1)<size(tp2,1)
tp1=tp2;
for jj=1:size(tp1,1)
tp2=[tp2
repmat(tp1(jj,:),size(nearest,1),1)+nearest];
end
cluster(ii,:) =[min(tp2(:,1)) max(tp2(:,1)) min(tp2(:,2)) max(tp2(:,2))];

27. Apply the model in PRL,55,2515(1985)
1.高温塑性变形热模拟实验
Homework
Apply the model in PRL,55,2515(1985)
for triangular and hexagonal lattice.
Sending to when ready
For lecture notes, refer to