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**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

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**C60 Graphene fragments C20 Clusters experimentally observed**

ACS nano,6,8203(2012) Nature 318,162(1985); 407,60(2000);

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

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

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**The Geometry of Universe Platonic and Archimedean solids**

Nature 425, October 2003

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**Wulf’s construction How Clusters are found? Dense Packing and Symmetry**

in Small Clusters of Microspheres Wulf’s construction Science 301,403(2003)

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**Nucleation / Diffusion / Reconstruction **

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

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**No diffusion vs Full diffusion**

Cluster growth model No diffusion vs Full diffusion

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**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?

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**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

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

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**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

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**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

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**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.

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1.高温塑性变形热模拟实验 Eden vs. DLA cluster

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**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)

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**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')

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**Shapes in square lattice: Diamond Square Triangular**

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

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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

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Spanning clusters

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**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

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**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

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**Simulation of Percolation**

1.高温塑性变形热模拟实验 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 end

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**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=[ ];

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**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))];

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**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

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