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1 South China University of Technology Growth of Cluster Xiaobao Yang Department of Physics

2 Clusters experimentally observed Nature 318,162(1985); 407,60(2000); C 60 C 20 Graphene fragments ACS nano,6,8203(2012)

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

4 B 20 B 19

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

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

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

8 Cluster growth model No diffusion vs Full diffusion

9 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=5R cluster ) 4.When the walker is far from the cluster, a greater step size may be adopted. Refer to DLA.m

10 DLA.m 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)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 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 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.

13 clear x=[0 0]; %initial nearest=[ ]; 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

14 Dimensionality of the cluster where r is small enough. For a straight line:

15 Eden vs. DLA cluster

16 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 = 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 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.

17 clear x=[0 0]; %initial edge=[ ]; nearest=edge; p=0.6 % propobality unactive=[]; for ii=1:200%generation growsite=rand(size(edge,1),1)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'); end plot(x(:,1),x(:,2),'*') hold on plot(edge(:,1),edge(:,2),'o') axis equal plot(unactive(:,1),unactive(:,2),'ro') 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

18 The role of energy Shapes in square lattice: 1)Diamond 2)Square 3)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 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)?

22 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? ?

23 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)


25 Divide the data into two groups A=[ ]; 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

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)

27 Homework For lecture notes, refer to Sending to when ready Apply the model in PRL,55,2515(1985) for triangular and hexagonal lattice.


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