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South China University of Technology
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Growth of Cluster
Xiaobao Yang
Department of Physics
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Clusters experimentally observed
Nature 318,162(1985); 407,60(2000);
C60
C20
Graphene fragments
ACS nano,6,8203(2012)
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www.compphys.cnScience 299, 96 (2003);PRL, 103,047402(2009)
Diamond fragments
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http://www.chem.brown.edu/research/LSWang/http://www.tsinghua.edu.cn/publish/chemen/2141/2011/20110413125809079217484/20110413125809079217484_.html
B20B19
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www.compphys.cnNature 425, 593-595. 9 October 2003
http://blog.sciencenet.cn/home.php?mod=space&uid=279992&do=blog&id=509211
The Geometry of Universe
Platonic and Archimedean solids
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How Clusters are found?
Wulf’s construction Dense Packing and Symmetry in Small Clusters of Microspheres
Science 301,403(2003)
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The growth and evolution of Clusters
Nucleation / Diffusion / ReconstructionSubstrate / bonding / Temperature
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Cluster growth model
No diffusion vs Full diffusion
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Cluster Growth Model►Diffusion Limited
Aggregation (DLA) Model
►Discussion on programming
►Fractals? The dimensionality of clusters?
1. Choose initial position of a walker at random on a circle r0.
2. If the walker wanders too far from the cluster (say, >1.5r0), start a new walker.
3. As the cluster grows, r0 should be increased. (say, keep r0=5Rcluster)
4. When the walker is far from the cluster, a greater step size may be adopted. Refer to DLA.m
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DLA.mclear; 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; 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); endend;
colormap(winter); imagesc(A);axis([0 M 0 M],'square','equal');
1)Place the seed2)Atomic random walk towards the seed3) Check if the atom is attached4) Neglect the atom far from the seed5) Draw the cluster
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Matlab: illustrationcolormap
image()imagesc()pcolor()
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Cluster Growth Model►Eden Model
Eden Cluster
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.
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clearx=[0 0]; %initialnearest=[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
nearest
x
edge
tmp
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Dimensionality of the cluster
where r is small enough.
For a straight line:
( ) 2m r r
2( )m r r
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Eden vs. DLA cluster
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Morphology of a Class of Kinetic Growth Models
PRL,55,2515(1985)
(a) p = l, (b) p = 0.9, (c) p = 0.8, (d) p = 0.7,(e) p = 0.6, (f) p = 0.545.
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.
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 p2)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.
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clearx=[0 0]; %initialedge=[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]; end
edge=setdiff(edge,x,'rows'); if length(unactive)>0 edge=setdiff(edge,unactive,'rows'); endend
plot(x(:,1),x(:,2),'*')hold onplot(edge(:,1),edge(:,2),'o')axis equalplot(unactive(:,1),unactive(:,2),'ro')
1) Select the active sites with p2) Record the unactive sites3) Add atoms and update edge4) Delete edge from cluster and unactive sites
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The role of energy
Shapes in square lattice:1) Diamond2) Square3) Triangular
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Simulation of cluster
1) Adding atoms2) Atom diffusion allowed Adding and deleting to conserve
the number of atoms3) Energy estimation4) Accept or reject the configuration
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Spanning clusters
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Percolation Problems
1. Porous rock (Original percolation problem,
Broadbent and Hammersley, 1957)
2. Forest fires, etc
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
How far from each other should trees in a forest (orchard) be planted in order to minimize the spread of fire (blight)?
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What is percolation
2-dimension percolation
6x6
120x120
Infinite x infinite (critical coverage)
2x2 lattice
Percolated system if a spanning cluster exist (connects top and bottom exists)
What is the probability for a system to be percolated for a given coverage?
?
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Simulation of Percolation
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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 endend
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Divide the data into two groups
A=[ 0 0 1 1 0 1 1 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 0 0 1 0 0 1 1 0 1 0 1 0 0 0 1 0 0 0 0 0 0 1 0 1 1 1 0 0 0 0 1 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0];
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 ii jj]; end endend
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Find the spanning clustersfor 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 tp2=intersect(tp2,hole,'rows'); end cluster(ii,:) =[min(tp2(:,1)) max(tp2(:,1)) min(tp2(:,2)) max(tp2(:,2))];end
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Homework
For lecture notes, refer to http://www.compphys.cn/~xbyang/
Sending to [email protected] when ready
Apply the model in PRL,55,2515(1985)for triangular and hexagonal lattice.