Chapter II: Diffusion and Growth Processes

Size: px

Start display at page:

Download "Chapter II: Diffusion and Growth Processes"

Raymond Atkinson
6 years ago
Views:

Chapter II: Diffusion and Growth Processes Modelling and Simulation, M.Biehl, 2005 2. Diffusion Limited Aggregation See also the more detailed description in the additional hand-out, and in: W.

1 Chapter II: Diffusion and Growth Processes Modelling and Simulation, M.Biehl, Diffusion Limited Aggregation See also the more detailed description in the additional hand-out, and in: W. Kinzel and G. Reents, Physics by Computer (Springer, 1998). Further information and related Mathematica and C programs as well as Java applets are available at (square lattice DLA cluster of mass M = 10000)

2 Diffusion Limited Aggregation motivation (I): diffusion limited growth the basic algorithm (Witten-Sander) implementation and improvements fractal properties of DLA clusters self-similarity and fractal dimension variations and universality Electrostatic analogon Dielectric Breakdown Model (DBM) motivation (II): examples from a broader context

3 Diffusion Limited Aggregation (DLA) Growth of a snowflake (very much simplifying) immediate vicinity of the flake contains almost no water molecules molecules have to cross the water poor region by means of Brownian motion attachment to the growing flake is essentially irreversible Witten and Sander (1981): diffusion of single particles (low density limit) approaching the growing cluster from infinity diffusing particles stick to the cluster and becomes immobile by irreversible sticking DLA... is defined as a very simple process (algorithm) is studied almost exclusively by computer simulations displays the emergence of very complex behavior from simple rules is topic of on-going research

4 Basic algorithm: DLA on a square lattice 1. Initialize start with an immobile seed particle in the center of an otherwise empty square lattice (cluster mass M = 1, cluster radius R max = 1) 2. Launch a new particle place a single particle with equal probability on a circle with radius R start > R max about the center (as small as possible, e.g. R start = R max + 1) 3. Diffusion move the particle from its current position to a randomly chosen nearest neighbor (NN) site. Repeat 3 until a NN site of a cluster particle is reached, then go to step 4 4. Aggregation add the particle to the cluster, increase M by one and re-evaluate R max. stop if the desired mass M is reached, else go to step 2. Remark: positions on a circle have to be approximated in the square lattice

5 problems: particles can drift arbitrarily far away form the cluster particles hop around in empty regions of the lattice most of the time replace step 3 by the modified procedure 3. Diffusion (shortcuts) calculate the current distance r of the particle from the origin. If r < R jump : If R kill >r R jump : move the particle from its current position to a randomly chosen nearest neighbor site. move the particle with equal probability anywhere on a circle with radius (r R start ) around its current position. If r R kill : remove the particle from the lattice, go to step 2. repeat 3 until a nearest neighbor of a cluster site is reached, then go to 4.

6 Remarks: R jump R start should be large enough (e.g. > 10) to justify the assumption of a spherically symmetric jump Deleted particles would return to R kill or R start, eventually, the correct prob. for reaching a certain site should be used for the re-launch the simple, uncorrelated re-launch is justified for R kill (here R kill 10 R start ) Fig. 1: Illustration of the implementation of the Witten Sander algorithm. Particles start at R start, within R jump explicit diffusion hops are performed. Outside of this region shortcut jumps are implemented. Particles which reach R kill are simply removed from the lattice, dlanim(m) generates an animation.

7 DLA algorithm function cluster = dla(maxmass) % dla(m); generates a DLA cluster of mass M, % cls = dla(m) outputs the cluster, try small M first % lattice size and center site position latt = 601; cr = floor(latt/2); % use greylevel col1 for 3/4 of the cluster, then col2 col1 = -0.5; col2 = -1; rm=1; m=1; coor=zeros(latt,latt); coor(cr,cr) =col1; for ipart = 1: maxmass-1 check=0; rstart= max(rm+2,4); rkill=10*rm+10; rjump= rstart + 10; % launch particle at radius rstart pos= circle(rstart,cr,cr); ix= pos(1); iy= pos(2); check= coor(ix-1,iy)+coor(ix+1,iy)+... coor(ix,iy-1)+coor(ix,iy+1); while check == 0 % check becomes nonzero at aggregation sites ract2 =(ix-cr)*(ix-cr)+(iy-cr)*(iy-cr); % if actual radius larger rkill, re-launch particle if ract2 > rkill*rkill pos=circle(rstart,cr,cr); ix=pos(1); iy=pos(2); ract2 = rstart*rstart; end if ract2 <= rjump*rjump % explicit hop zz = floor(4*rand(1,1)); switch(zz) case 0, ix = ix+1; case 1, ix = ix-1; case 2, iy = iy+1; case 3, iy = iy-1; end else % giant jump pos = circle(sqrt(ract2)-rstart, ix,iy); ix = pos(1); iy = pos(2); end % set check<0 if a NN site of cluster is reached if (ract2 <= rstart*rstart) check= coor(ix-1,iy)+coor(ix+1,iy)+... coor(ix,iy-1)+coor(ix,iy+1); end end % end of while(check==0) % aggregate the new particle, determine mass,rm m = m+1; coor(ix,iy) = col1; if (m > floor(0.75*maxmass)) coor(ix,iy) = col2; end rm = max(rm,sqrt((ix-cr)*(ix-cr)+(iy-cr)*(iy-cr))); end % display the cluster, output mass, radius and fract. dim. cluster = coor; imshow(coor,[-1 0], notruesize ); axis([ cr-rm-3 cr+rm+3 cr-rm-3 cr+rm+3]); title([ M=,num2str(m, %3.0f ), R=,num2str(rm, %2.2f ),... D=,num2str(log(m)/log(rm), %2.2f )],... fontsize,16); function pos = circle(r,cx,cy) % function circle(rstart) returns the coordinates % (nearest lattice site) of a random position % on a circle of radius r about (cx,cy) phi = 2*pi*rand(1,1); x = cx + round(r*sin(phi)); y = cy + round(r*cos(phi)); pos = [x y];

8 Fig. 2: Growth of a square lattice DLA cluster (sizes rescaled), in each plot the 25% particles which aggregated most recently are displayed darker

9 Properties of DLA clusters branching and screening the random growth process leads to the formation of small tips tips are likely to capture diffusing particles they screen their surroundings screening self-stabilizes the tip it grows even larger..., new tips form... delicate, branched, tree like object scale invariance, lack of a typical length-scale one observes with growing M : hierarchy of arms, branches, twigs, sprouts,... fjord like empty regions of all sizes characteristic sizes cannot be defined! (apart from lattice constant and R max )

$Fig. 5: The so called Sierpinsky gasket Note: in deterministic self-similar fractals, structures repeat exactly on all possible length scales.$

10 stochastic self similarity Fig. 4: An M = cluster and three of its branches (rescaled). Substructures of the cluster look pretty much alike, i.e. statistical properties are reproduced after proper rescaling. Fig. 5: The so called Sierpinsky gasket Note: in deterministic self-similar fractals, structures repeat exactly on all possible length scales.

Fractal dimension Fig. 6: Scaling behavior of areas and lines. The mass (volume, area) of a regular D dim. object grows by a factor b D, when its linear extension R is scaled to b R.

11 Fractal dimension Fig. 6: Scaling behavior of areas and lines. The mass (volume, area) of a regular D dim. object grows by a factor b D, when its linear extension R is scaled to b R. Areas grow like A R 2, lengths grow like l R 1. generalization: assume the mass of an object grows according to M = a R D M D M is called the fractal dimension (with respect to the mass) estimate for a given object: ln M ln R D M ( + ln a ) ln R (the correction term becomes irrelevant with R ) NOTE: THE ABOVE IS NOT A PROPER MATHEMATICAL DEFINITION! (see literature, additional hand-out and assignment 2 for details)

12 Fig. 7: Six clusters with M = 1000 or 5000, the simple estimate D = ln M/ln R of the fractal dimension D M 1.75 is obtained using R = R max.

13 more systematic determination of D M : Fig. 8: Typical cluster radius R = R max as a function of M. Dots represent the result of averaging log 2 R max over 100 clusters for each of the six values of M between 125 and Error bars would be smaller than the symbols, the solid line corresponds to a least square linear fit. Here: R max 0.98 M 0.575, i.e. D M 1/ further improvement: use the mean square radius R ms = 1 M (R ms should fluctuate less then the extreme R max ) determine R max or R ms with respect to the center of mass of the cluster (compensates for random, overall offsets) radial mass density, see assignment 2 boxcounting dimension, see assignment 2 M i ri 2 1/2

Possible modifications (see the additional hand-out for more examples) Delayed aggregation A particle that reaches an aggregation site is only considered immobile if the next RW step would be onto

14 Possible modifications (see the additional hand-out for more examples) Delayed aggregation A particle that reaches an aggregation site is only considered immobile if the next RW step would be onto the cluster ( diff. boundary conditions in DBM-models) Next Nearest Neighbor (NNN) sticking Consider also NNN-sites of the cluster (diagonal neighbors) as aggregation sites Different lattices Use other regular lattice structures for diffusion and aggregation. The coordination number s denotes the # of NN each site has in the lattice. For example, s = 4 in the square and s = 6 in the triangular lattice Off lattice DLA Diffusion takes place in a continuous coordinate space (x, y). The elementary step is a random displacement by on one particle diameter. Whenever a particle touches the aggregate it is moved backward until particles do not overlap anymore Fig. 9: Off-lattice cluster with 10 7 particles source: book97/chapter2.html

15 Universality (?) Shape and other properties of DLA clusters vary with the model modifications But: up to cluster masses on the order of one finds always the same D M assumption: a universality class of two dimensional DLA exists which is characterized by the fractal dimension D M 1.71 THE SMALL PRINT: the study of larger clusters revealed some problems for instance, square lattice DLA seems to play a special role: lattice anisotropy: large clusters grow mainly along the lattice axes the fractal dimension measured in very large square lattice clusters is close to D M = 3/2 with very slowly vanishing corrections (Meakin, 1988): R max a M 2/3 (1 + b M 0.12) There are probably at least two universality classes one for small coordination number s, including s = 4 one for larger s and off lattice DLA (with D M 1.71) More subtleties concern the fractal properties and self similarity of DLA clusters...

16 Growth site probabilities diffusing particles are very unlikely to wander into one of the inner fords with high probability they attach to the protruding tips already visible from the marking of most recent particles growth occurs essentially only in a small active zone with r R max measurement of the growth site probabilities for a given cluster: Method of test particles Launch non-interacting particles, let them diffuse to the aggregation sites i, only register their frequency p i without actual growth (remove the particles) Problem: even for very many particles, most sites i will register no particles (p i 0) Electrostatic analogon Denote the probability to find a Random Walker at time t at site (x, y) of an otherwise empty lattice by p t (x, y). According to Sec. 2.1, its temporal evolution follows p t+ t (x, y) = 1 4 ( with the lattice constant a. ) p t (x + a, y) + p t (x a, y) + p t (x, y a) + p t (x, y + a) Determine the stationary p(x, y) = p t (x, y) by means of a simple iteration (Relaxation Method) under appropriate boundary conditions

Fig. 10: Boundary conditions: a) on all sites occupied by the cluster p t (x, y) = 0, the cluster acts as a trap for the RW b) very far from the cluster (in the origin), p t (x, y) must be

17 Fig. 10: Boundary conditions: a) on all sites occupied by the cluster p t (x, y) = 0, the cluster acts as a trap for the RW b) very far from the cluster (in the origin), p t (x, y) must be spherically symmetric, i.e. it depends only on (x 2 + y 2 ) realization: set p t (x, y) = const. for (x 2 + y 2 ) R out (the const. is irrelevant normalization of p t (x, y)) Mathematical structure: discretized Laplace equation 2 p(x, y) = 0, using p t (x, y) p t+ t(x, y) p t (x, y) t t 2 p x p y 2 1 ( ) a 2 p(x + a, y) + p(x a, y) + p(x, y a) + p(x, y + a) 4 p(x, y) apart from normalization, p(x, y) can be interpreted as the, e.g., electrostatic potential between two electrodes: the cluster at zero potential a circular electrode with radius R out at constant non-zero potential The growth site probability p i is proportional to p(x, y) at the NN sites of the cluster

18 Aggregation probability function ag = aggprob(m) % function ag = aggprob(m) % calls dla(m) and evaluates for the cluster the % stationary p(x,y) and normalized aggr. probability coor= sign(dla(m)); l= max(size(coor)); cr= floor(l/2); rad2 =0; % determine rad for ii=1:l, for jj=1:l rad2 = max(rad2,-coor(ii,jj)*... ((ii-cr)*(ii-cr)+(jj-cr)*(jj-cr))); end; end; rad= round(sqrt(rad2))+4; clus= -coor(cr-rad : cr+rad, cr-rad : cr+rad); p= -clus + 1; l= max(size(p)); cr = rad+1; itmax = 1000; jj=0; dev= 1; while( jj < itmax & dev > 1.e-8 ) jj = jj+1; palt = p; pwest = [p(l,:); p(1:l-1,:)]; peast = [p(2:l,:); p(1,:)]; pnorth= [p(:,l), p(:,1:l-1)]; psouth= [p(:,2:l), p(:,1)]; p = (pwest + peast + pnorth + psouth) / 4; for kx =2:l-1, for ky =2:l-1 if ((kx-cr)*(kx-cr)+(ky-cr)*(ky-cr) >= rad*rad) p(kx,ky)= 1; end if (clus(kx,ky) ~= 0), p(kx,ky)= 0; end end; end for kx=1:l p(kx,1)= 1; p(kx,l)= 1; p(1,kx)= 1; p(l,kx)= 1; end dev= sum(sum((p-palt).*(p-palt))); end % end while(...) % display the "electrostatic potential" imshow(-p-clus,[-1,0], notruesize ); axis([1 l 1 l]); % evaluate and display normalized aggregation % probability at NN sites of the cluster norm = 0; for kx =2:l-1, for ky =2:l-1 if ((clus(kx+1,ky)+clus(kx-1,ky)+... clus(kx,ky-1) +clus(kx,ky+1)) ==0) p(kx,ky) = 0; end norm = norm + p(kx,ky); end; end for kx =2:l-1, for ky =2:l-1 if ((clus(kx+1,ky)+clus(kx-1,ky)+... clus(kx,ky-1)+clus(kx,ky+1)) ==0) p(kx,ky) = p(kx,ky)/norm; end end; end figure; pagg = - p(2:l-1,2:l-1); imshow(pagg,[min(min(pagg)) 0], notruesize ); axis([1 l 1 l]);

19 Examples Fig. 11: Top row: full stationary p(x, y) for clusters of mass M = 100, 500, and 1000 Bottom: corresponding normalized growth site probability at NN sites of the cluster Note: the growth site prob. distribution is a highly non-trivial multi fractal

Fig. 12: stationary p(x, y) and growth site probabilities p i for a cluster of M = 2000.

20 Fig. 12: stationary p(x, y) and growth site probabilities p i for a cluster of M = Lowest/highest probabilities correspond to white/black lattice sites. Left, the cluster is displayed in black for clarity

21 Dielectric Breakdown Model (DBM) stochastic model of a gaseous discharge pattern (lightning) assume the probability q(x, y) for growth of the ionization channel (zero pot. cluster) at (x, y) increases with the local field or potential p(x, y): q(x, y) [p(x, y)] η DBM growth algorithm 1. given the current cluster, evaluate p(x, y) and q(x, y) by means of the relaxation method under appropriate boundary conditions 2. choose one of the aggregation sites with prob. q(x, y) and occupy site (x, y) update boundary conditions and go to 1 The role of parameter η η = 0 : growth with equal probability at all NN sites of the cluster η = 1 : (a version of the so-called Eden model) q(x, y) = p(x, y) is identical with the DLA growth site probability (apart from subtle difference in the boundary conditions) η : growth occurs only at the sites of maximal p(x, y), i.e. at the tips of the cluster one-dimensional growth Note: the fractal dimension of DBM-clusters depends on η

22 Examples from a broader context many physical, chemical, biological growth or branching processes are more or less accurately described by a Laplace equation with moving boundary conditions among others: solidification in a melt or vapor (snow flakes) viscous fingering, mixing of different fluids aggregation of monolayer islands on a crystal surface electrochemical deposition crack propagation in a solid growth of bacteria colonies Shape and (fractal) properties of the clusters may depend on the degree of randomness in the progress for instance: expect more regular, compact clusters for, e.g., DLA growth with additional diffusion of particles along the cluster edge parallel occupation of all growth sites according to q(x, y) or p(x, y) in the DBM/DLA

islands resemble DLA clusters, at higher

23 example: aggregation of Pt atoms on a Pt(111) surface (hexagonal lattice) Fig. 13: Pt island formation at temperatures T = 300K (left) and T = 400K (right) c/o T. Michely, Univ. Aachen/Germany At low temperatures, islands resemble DLA clusters, at higher T diffusion along edges smoothens the shape Fig. 14: Various growth or branching processes, from left to right: electrochemical deposition, mineralization pattern, Triacontane on SiO x,bacteria, termite colonies source: and ( )

24 Selected Literature P. Meakin, The Growth of Fractal Aggregates in: C.Domb and J.L.Lebowitz (eds.), Phase Transitions and Critical Phenomena Volume 12, Academic Press (London) 1988 L.M. Sander, Growth and aggregation far from equilibrium in: C. Godrèche (ed.), Solids far from equilibrium Cambridge University Press (Cambridge) 1991 T. Vicsek, Fractal Growth Phenomena World Scientific (Singapore) 1992 A.-L. Barabási and H.E. Stanley, Fractal Concepts in Surface Growth Cambridge University Press (Cambridge) 1995 A. Bunde and S. Havlin (eds), Fractals and Disordered Systems Springer (Berlin) 2nd ed. 1996

1.1.6 Island Shapes (see Michely / Krug book, Chapter 3)

1 1.1.6 Island Shapes (see Michely / Krug book, Chapter 3) The last section was concerned with nucleation, which determines the number density of islands. This section is concerned with their shape. The