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 MicrospheresWulf’s constructionScience 301,403(2003)
7 Nucleation / Diffusion / Reconstruction The growth and evolution of ClustersNucleation / Diffusion / ReconstructionSubstrate / bonding / Temperature
8 No diffusion vs Full diffusion Cluster growth modelNo 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 ModelDiffusion Limited Aggregation (DLA) ModelDiscussion on programmingChoose 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.mFractals? 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 onnparticle=0;while(nparticle < N)%generate a random walker within the belt of 3/5*M/2<R<4/5*M/2r=(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 wanderingwhile( 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!endif(check==1); A(x,y) = -1; nparticle = nparticle + 1; end; %meets seed; seed updated.%if(mod(nparticle,100)==0); imagesc(A); endcolormap(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
12 Cluster Growth Model Eden Model Eden Cluster 1.高温塑性变形热模拟实验Cluster Growth ModelEden ModelConsider 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 sitesas the perimeter sites of the cluster.Choose one of these perimeter sitesat random and place a particle at the chosen location.This process is then repeated;update perimeter and particle.Continue this processuntil 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;%perimeterfor ii=1growsite=ceil(length(edge)*rand);tmp=ones(4,1)*edge(growsite,:)+nearest;x=[xedge(growsite,:)];tmp=[tmpedge];edge=setdiff(tmp,x,'rows');endxedgextmp
14 Dimensionality of the cluster 1.高温塑性变形热模拟实验Dimensionality of the clusterFor a straight line:where r is small enough.
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 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.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 clearx=[0 0]; %initialedge=[0 10 -11 0-1 0];nearest=edge;p=0.6 % propobalityunactive=;for ii=1:200%generationgrowsite=rand(size(edge,1),1)<p;if max(mod(growsite+1,2))>0unactive=[unactiveedge((growsite==0),:)];endx=[xedge(growsite,:)];tmp=edge(growsite,:);for jj=1:size(tmp,1)edge=[edgerepmat(tmp(jj,:),size(nearest,1),1)+nearest];edge=setdiff(edge,x,'rows');if length(unactive)>0edge=setdiff(edge,unactive,'rows');1) Select the active sites with p2) Record the unactive sites3) Add atoms and update edge4) Delete edge from cluster andunactive sitesplot(x(:,1),x(:,2),'*')hold onplot(edge(:,1),edge(:,2),'o')axis equalplot(unactive(:,1),unactive(:,2),'ro')
18 Shapes in square lattice: Diamond Square Triangular The role of energyShapes in square lattice:DiamondSquareTriangular
19 Simulation of cluster1) Adding atoms 2) Atom diffusion allowed Adding and deleting to conserve the number of atoms 3) Energy estimation 4) Accept or reject the configuration
21 Percolation Problems Porous rock 1.高温塑性变形热模拟实验Percolation ProblemsPorous rock(Original percolation problem, Broadbent and Hammersley, 1957)2. Forest fires, etcHow 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 percolation2-dimension percolationPercolated systemif a spanning cluster exist(connects top and bottom exists)2x2 latticeWhat 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