www.compphys.cn

South China University of Technology

www.compphys.cn

Growth of Cluster

Xiaobao Yang

Department of Physics

www.compphys.cn

Clusters experimentally observed

Nature 318,162(1985); 407,60(2000);

C60

C20

Graphene fragments

ACS nano,6,8203(2012)

www.compphys.cnScience 299, 96 (2003);PRL, 103,047402(2009)

Diamond fragments

www.compphys.cn

http://www.chem.brown.edu/research/LSWang/http://www.tsinghua.edu.cn/publish/chemen/2141/2011/20110413125809079217484/20110413125809079217484_.html

B20B19

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

www.compphys.cn

How Clusters are found?

Wulf’s construction  Dense Packing and Symmetry in Small Clusters of Microspheres

Science 301,403(2003)

www.compphys.cn

The growth and evolution of Clusters

Nucleation / Diffusion / ReconstructionSubstrate / bonding / Temperature

www.compphys.cn

Cluster growth model

No diffusion vs Full diffusion

www.compphys.cn

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

www.compphys.cn

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

www.compphys.cn

Matlab: illustrationcolormap

image()imagesc()pcolor()

www.compphys.cn

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.

www.compphys.cn

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

www.compphys.cn

Dimensionality of the cluster

where r is small enough.

For a straight line:

( ) 2m r r

2( )m r r

www.compphys.cn

Eden vs. DLA cluster

www.compphys.cn

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.

www.compphys.cn

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

www.compphys.cn

The role of energy

Shapes in square lattice:1) Diamond2) Square3) Triangular

www.compphys.cn

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

www.compphys.cn

Spanning clusters

www.compphys.cn

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)?

www.compphys.cn

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?

?

www.compphys.cn

Simulation of Percolation

www.compphys.cn

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

www.compphys.cn

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

www.compphys.cn

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

www.compphys.cn

Homework

For lecture notes, refer to http://www.compphys.cn/~xbyang/

Sending to 17273799@qq.com when ready

Apply the model in PRL,55,2515(1985)for triangular and hexagonal lattice.