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Deposition of magnetic particles: A computer simulation study F. de los Santos1,2, M. Tasinkevych2, J. M. Tavares2,3 and P. I. C. Teixeira4,5 1Physics Department, Boston University 590 Commonwealth Avenue, Boston, MA 02215, USA 2Centro de F´ısica Teórica e Computacional da Universidade de Lisboa Avenida Professor Gama Pinto 2, P-1649-003 Lisbon, Portugal 3Departamento de Ciências Exactas e Tecnológicas, Universidade Aberta Rua Fernão Lopes 9, 2◦ Dto, P-1000-132 Lisbon, Portugal 4Faculdade de Engenharia, Universidade Católica Portuguesa Estrada de Tala´ıde, P-2635-631 Rio de Mouro, Portugal 5Departamento de Engenharia de Materiais and Instituto de Ciência e Engenharia de Materiais e Superf´ıcies 3 Instituto Superior Técnico 0 Avenida Rovisco Pais, P-1049-001 Lisbon, Portugal 0 (30 September2002) 2 n We report a Monte Carlo simulation of deposition of magnetic particles on a one-dimensional a substrate. Incoming particles interact with those that are already part of the deposit via a dipole- J dipole potential. The strength of the dipolar interaction is controlled by an effective temperature 0 T∗,the case of purediffusion-limited deposition being recovered in thelimit T∗→∞. Preliminary 1 resultssuggestthatthefractaldimensionofthedepositsdoesnotchangewithtemperaturebutthat there is a (temperature-dependent) cross-over from regimes of temperature-dependent to universal 1 behaviour. Furthermore,it was found that dipoles tend toalign with thelocal direction of growth. v 0 PACS numbers: 82.20.Wt, 61.43.Hv, 64.60.Cn 5 1 I. INTRODUCTION different microscopic components to organise themselves 1 0 into complex functional patterns once their interactions 3 have been appropriately tailored [1,5]. Many of these 0 Magnetic particles are a key ingredient of many mod- devices, either existing or at the design stage, have low t/ ern data recording and storage devices, from music dimensionalities (e.g., wires and films), at which simula- a tapes to computer hard disks [1]. For these applica- tions of model systems have revealed the chaining ten- m tionssmooth,regularmagneticlayersareusuallydesired. dency to be particularly strong [6,7,8,9]. - However, dipoles, both magnetic and electric, display a To our knowledge, there is detailed only one study of d fondness for arranging themselves into highly inhomo- how dipolar interactions alone affect growth. Pastor- n o geneous structures. This is a consequence of the very Satorras and Rub´ı [10] simulated an off-lattice, two- c strong anisotropy of the dipole-dipole interaction, which dimensional particle-cluster aggregation model. They : couplesthe orientationsofthe dipolemomentswiththat found a monotonic variation of the fractal dimension v i of the interparticle vector. Because the potential energy of the aggregates as a function of temperature (i.e., X at a fixed separation is lowest for a head-to-tail geome- dipole strength), the limit of diffusion-limited aggrega- r try,chainformationisparticularlyfavouredinferrofluids tion(DLA)beingrecoveredathightemperatures. Inad- a (dispersions of ferromagnetic particles) [2] or electrorhe- dition, a separate investigation by the same authors has ological fluids (dipersions of highly polarisable particles shown that highly structured layers could be obtained in solvents with low dielectric constant) [3] in magnetic at low temperatures by sequential adsorption of dipolar and electric fields, respectively. Whether such chaining particles [11]. See [12,13] for related work on other sys- can occur in zero field in the absence of any interactions tems. other than dipolar, is experimentally uncertain; it has Here we report on a simulation of dipole deposition been seen in simulations, but is not yet fully accounted on a one-dimensional (1d) substrate (i.e., a line). In the for theoretically. Likewise, what is the true equilibrium limit of zero magnetic moment this reduces to diffusion- structure of a solid of hardmagnetic particles is still un- limiteddeposition(DLD),whichshouldexhibitthesame settled (see, e.g., [4] and references therein). geometricalpropertiesasDLA[14];ourworkthus serves It is therefore of great importance, from a practical as a both a check and an extension of Pastor-Satorras as well as from a fundamental point of view, to inves- and Rub´ı’s to the case of an infinite (in one spatial di- tigate the influence of true long-range magnetic dipolar mension) system. In particular, we want to ascertain: forcesonthe geometryofparticleaggregates,soasto be firstly, whether the fractal dimension actually changes able to exertbetter controloverthem. This is especially owingtothestronganisotropyofthedipolarinteraction: relevant to the very novel field of self-assembled nanos- and, secondly,what is the correlationbetween the orien- tructured magnetic materials, where the aim is to allow tations of dipoles in the aggregateand its direction(s) of 1 growth. For computational convenience our dipoles are are already attached to the deposit. We incorporate the restrictedto resideatthe sites ofatwo-dimensional(2d) effects of these interactions through a Metropolis algo- square lattice (although they can point in any direction rithm. If the deposit contains N particles, then the in- ofthree-dimensional(3d)space): weassumethatanyef- teraction energy of the (N +1)th incoming particle (the fects coming from this discretisation of space are much randomwalker)withthe particlesin the depositis given smaller than those of the interparticle potential. That by E(r,⌣µ) = Ni=1 nφD(i,N + 1), where r and⌣µ this should be so is suggested by results for DLA [15] arethe currentpositionandthedipole orientationofthe (but remains of course to be confirmed by full off-lattice random walkerPrespecPtively (r is a 2d vector). Then we calculations). Furthermore, our analysis in terms of the randomly choose a new position r′ (r r′ = a) and concepts of fractal geometry presumes that all our de- a new dipole orientation⌣µ′ for the ra|nd−om w| alker; this posits are self-similar over some lengthscale larger than displacement is accepted with probability the mesh spacing but smaller than the deposit size [14]; again,this neednot be true of the smallestdeposits, but E(r′,⌣µ′) E(r,⌣µ) thesecontainonlyaverysmallfractionofthetotalnum- p=min 1,exp − . (2) − T∗ ber of particles. (cid:26) (cid:20) (cid:21)(cid:27) The present paper is a natural progression to non- equilibrium processes from our earlier researches on the T∗ = kBTa3/µ2 is an effective temperature, inversely thermodynamicsandphaseequilibriaofdipolarfluids[4]. proportional to the dipolar energy scale. In the limit It is organised as follows: in section II we describe our T∗ 0 only displacements that lower the energy model and the simulation method employed. Then in E(r′,→⌣µ′) are accepted. On the other hand, in the limit sectionIIIwepresentanddiscussourresults,specifically T∗ all displacements are accepted and our model → ∞ comparing them with those of Pastor-Satorrasand Rub´ı reduces to the well-known DLD [14]. [10]. Section IV summarises our findings and outlines The long range of the dipole-dipole interaction was prospects for future research. Technical details pertain- treated by the Ewald sum method (see Appendix A for ingtothetreatmentofthelongrangeofthedipole-dipole details). Inoursimulationswesetα=10/L,forwhichit interaction are relegated to two appendices. suffices to retain terms with n=0 in the real space sum of equation (A11). The sum in reciprocal space extends over all lattice points k =2πn/L with n 16, whereas II. MODEL AND SIMULATION METHOD the sum in real space is truncated at L|/2|.≤ Theparticleeventuallyeithercontactsthedeposit(i.e., Our simulations were performed on a (1+1)- becomes a nearest neighbour of another particle that is dimensional square lattice of width L = 800a sites and alreadypartofthedeposit)ormovesawayfromthesub- any height that can accommodate N dipoles, where a strate. In the latter case, if the particle reaches a dis- is the mesh spacing and the adsorbing substrate coin- tance fromthe substrate greaterthan H +2AL,it is max cides with the bottom row (henceforth we take a = 1). removedand a new one is launched. Once a particle has Periodic boundary conditions are imposed in the direc- reached the substrate or the deposit, its dipole relaxes tion parallel to the substrate. Each particle carries a 3d along the direction of the local field created by all other dipole moment of strength µ and interact through the particles in the deposit. In all simulations reported here pair potential we took A=1; larger values of A were tested and found to give the same results, but with drastically increased φD(1,2)=−rµ32 [3(⌣µ·ˆr12)(⌣µ·ˆr12)−⌣µ·⌣µ], (1) computation times. 12 where r ( a) is the distance between particles 1 and 12 2, ˆr12 is th≥e two-dimensional (2d) unit vector along the III. RESULTS AND DISCUSSION interparticle axis, and⌣µ and⌣µ are the 3d unit vectors in the directionof the dipole moments of particles 1 and 2 respectively. Finally, ‘1’ and ‘2’denote the full set Four effective temperatures were considered: T∗ = of positional and orientational coordinates of particles 1 10−1(28deposits),10−2(41deposits),10−3(42deposits) and 2. and 10−4 (54 deposits), chosen to be in the range where A particle is introduced at a lattice site (x ,H + thefractaldimensionofdipolarDLAclustersisexpected in max AL),wherex isarandomintegerinthe interval[1,L], to change [10]. Eachdeposit contains 50000dipoles. We in H is the maximum distance from the substrate to havealsogenerated30DLDdeposits onthe samelattice max any particle in the deposit, and A is a constant. The by this same method, with T∗ = ; known results for ∞ dipolemomentofthereleasedparticleisorientedatran- DLD (see, e.g., [16]) were used to check the validity of dom. The particle then undergoes a random walk by a our algorithm. On the other hand, comparison between series of jumps to nearest-neighbour lattice sites, while these and the results for finite temperatures reveal the experiencing dipolar interactions with the particles that effect of dipolar interactions on DLD. 2 L 1 ρ(h)= η(i,h) , (3) hL i i=1 X b) 103 * h ) h ( ρ FIG.1. Snapshots of two deposits obtained for T∗ =10−1 (black) and T∗=10−4 (grey). 102 Figure 1 is a snaphsot of two deposits for T∗ = 10−1 (black) and T∗ =10−4 (grey, only part of the deposit is 10-4 10-3 10-2 10-1 shown – in fact, it grows up to a height of about 8000). h/h* Bothdepositshavethesamegeneralappearance,already observed in DLD: they consist of several trees compet- FIG.2. (a)Meandensityρ(h)ofdepositsatheighth: ran- ing to grow. As the size of the deposit increases, fewer dom deposition (solid line), T∗ = 10−1 (dotted line), 10−2 and fewer trees ‘survive’ (i.e., carry on growing), as a (dashed line), 10−3 (long-dashed line) and 10−4 (dot-dashed consequenceofthe so-calledshieldingorscreeningeffect. line). (b) h∗ρ(h) vs h/h∗: random deposition (circles), From figure 1 this seems more pronouncedat lowertem- T∗ = 10−1 (squares), 10−2 (diamonds), 10−3 (triangles up) peratures,since abovea heightof1000(about1/8ofthe and10−4 (trianglesdown). Thesolidlineisalinearfittothe maximum height attained by this deposit) only one tree DLD results in therange of thegraph; it has slope −0.28. survives. where η(i,h) is 1 (0) if the site with coordinates (i,h) Inordertocomparequantitativelytheresultsobtained is occupied (unoccupied), and the average (denoted by for different temperatures we have measured the mean angular brackets) is taken over all available deposits at density of dipoles ρ(h) at a height h, each temperature. ρ(h) is plotted in figure 2a: all the 3 curves have similar shapes, with a smooth decrease at g(x)wouldalsoneedto haveanexplicittemperaturede- small h, levelling off (saturating) at intermediate h, and pendence. This dependence should be able to describe with a sharp drop at large h, when the top of the de- the trends observed in figure 2b for finite temperatures: positisreached. Itisimmediatelynoticedthatthefinite- a crossover between an approximately linear regime for temperature curves differ from that for DLD in one im- very low relative heights, characterised by an exponent portant respect: the density at a given height and the greaterthanα=0.27;andalinearregimeforintermedi- maximum height h∗ attained by the deposits vary with ate heights, characterisedby roughly the same exponent temperature. These variations are monotonic: ρ(h) is as DLD. smaller and h∗ is larger at lower temperatures and thus the increase in the strength of dipolar interactions en- In [10] it was argued that the change in the confor- hances the shielding effect. mational properties of DLA clusters introduced by the For DLD, ρ(h) was found to be of the form [14], presence of dipolar interactions could be interpreted as a change in their fractal dimension. The fractal dimen- ρ(h) h−αg(h/h∗), (4) sion for each temperature was determined by measur- ∝ ing the dependence of the radius of gyration of dipolar where α, the so-called codimensionality, is the difference DLA clusters on the number of particles in a cluster. between the dimension of space d (= 2, in the presente Between T∗ = 10−1 and T∗ = 10−4 a fractal dimen- case) and the fractal dimension D of the deposit, h∗ is sion was obtained ranging from about 1.7 to about 1.2. the maximum height, and g(x) is a universal function. We have performed linear fits to the data shown in fig- g(x) 1 when x 1 and decays faster than any power ure 2b, using only those points corresponding to heights ≈ ≪ of x when x 1. In order to compare DLD and finite- below the region where the crossover referred to above → temperature results, we propose a general form for ρ(h), seems to take place. These points follow straight lines inspired by equation (4): withtemperature-dependentslopesrangingfrom0.3(for T∗ = 10−1) to 0.6 (for T∗ = 10−4). On the basis of ρ(h,T∗)=A(T∗)h−αg(h/h∗), (5) the analysis of just this part of the deposit, we obtain where A(T∗) is some unknown function of T∗ only. It is an (apparent) variation of the fractal dimension of the easily seen that A(T∗) can be found as a function of h∗ deposits with T∗, from D = 1.7 to D = 1.4. We conclude, as was already pointed out in [14] and seems to andofthenumberofparticlesinthedepositN,byusing be confirmed by the present work, that the results of the normalization condition, [10]can be interpreted in terms of a crossoverbetween a h∗ h∗ temperature-dependentfractaldimensionatshortlength N =L ρ(h,T∗) L ρ(h,T∗)dh, (6) scales, to D 1.7 at long length scales, with a crossover × ≈ ≈ h=1 Z1 height that itself depends on temperature. This conclu- X sion must, however,be tested with longer simulations at whence the lowest temperatures, where the statistics are a little Nh∗(α−1) poorer. A(T∗)= . (7) L 11/h∗x−αg(x)dx There are other routes to estimating the fractal dimension of the deposits. We shall use one other to show R Since g(x) is a universal function and L and N are the that it is not necessary to assume that the finite tem- same for all the deposits we are analysing, equation (5) peraturedepositshaveafractaldimensiondifferentfrom becomes DLD. The mean height of the upper surface, h , when m the deposit contains M particles, is defined as [16], −α h ρ(h,T∗)h∗ g(h/h∗). (8) ∝ h∗ (cid:18) (cid:19) In figure 2b we plot h∗ρ(h,T∗) as a function of h/h∗, 1 L for h/h∗ 1 on a log-log scale. The data points for hm(M)= hmax(i,M) , (9) hL i ≪ DLD follow, as expected, a straight line. A linear re- Xi=1 gression gives α 0.27 and thus D 1.73, in good ≈ ≈ agreementwith what wasobtained previously by several other methods [17,16]. If the full functional dependence of ρ(h,T∗) were captured by equation (5), two possibil- where hmax(i,M) is the maximum height of the occu- ities would arise: (i) α is T∗-independent and all curves pied sites of column i when there are M particles in the deposit. In a DLD deposit this quantity is expected to are paralell straight lines; (ii) α depends on T and all scale with M, as [16] curves are straight lines with different slopes. However, we arrive at neither of these scenarios, so the situation is a little more complex. If we were to interpret our results in terms of a function similar to equation (5), then h Mφ. (10) m ∝ 4 foundthat 1.33<φ<1.44,whichcorrespondsto a fractal dimension, 1.69 < D < 1.75. Moreover, we have found no evidence of any regular variation of φ with T∗ over a given range of M. Figure 3a shows that, in the initial stages of growth, the mean height of the upper surface grows identically at every temperature. There is then a crossover region at intermediate stages when the less dense deposits grow slightly faster. Finally, in the laterstagesallthedepositsgrowatthesamerateregard- less of temperature, as evidenced by figure 3b. However, note that, as is clear from figure 3a, if the deposits had beenallowedtogrowonlytointermediatestages(e.g.,up to M = 10000), an increasing value of φ with increasing temperature would have obtained, and thus an apparent variationofD withT∗,withthesametrendsasobserved through the calculation of ρ(h). We conclude this analysis by attempting to make a firstconnectionbetween the orientationofthe dipoles in thedepositanditsgrowth. Figure4isasnapshotofpart ofa deposit forT∗ =10−1. Dipoles whose horizontal(or lateral) component is smaller (greater) in absolute value than their vertical component are coloured black (grey). Sincewehaveverifiedthatthez-component(i.e.,out-of- plane) of the dipoles in the deposit is zero after a short time,figure4suggeststhatthedipolestendtoalignwith the direction of growth of the deposit at the site where they attach. 400 380 360 y 340 320 FIG. 3. (a) Mean height of the upper surface, hm, as a function of the number of particles, M. The lines are as in figure 2a. (b) Blowup of thelarge-M region. 300 260 280 300 320 340 x The exponent φ is related to the codimensionality α = d D by φ = 1 and to the fractal dimension FIG. 4. Detail of a deposit for T∗ = 10−1. Sites whose − 1−α by D = d 1+φ−1. In figure 3 we plot our results for dipolesmakeanangle ofabsolute valuesmaller (larger) than h (M). T−he scaling law, equation (9), is known to be π/4 with thevertical axis are shown in black (grey). m valid in the limit M [16]. As in [16], we have per- → ∞ formedseverallinearregressionsforlargeM,intherange In order to make this idea more quantitative, we have M < M < M , for (M ,M ) = (0.5N,N), (0.25N,N), measuredtheanglesωbetweenthedirectionofthedipole 1 2 1 2 (0.1N,N) and (0.25N,0.5N) (recall that N = 50000). moments of all incoming particles and the direction of For every temperature and every range considered we growth at their point of attachment to the deposit. We 5 havedonesobyrecordingwhetheranewdipolebecomes the deposit. This is a consequence of the fact that the attached to the substrate due to a neighbour positioned lowest-energy configuration of two dipoles a fixed dis- to its left or to its right (lateralgrowth: the relevantan- tance apart is head-to-tail along the direction of the in- gle is that between the dipole and the horizontal axis), terdipole vector. These peaks are more pronounced at or above or below it (vertical growth: the relevant an- the lower temperature, and the peak for vertical growth gleis nowthatbetweenthe dipole andthe verticalaxis). isalittlehigherthanthatforlateralgrowthatbothtem- We didnottakeintoaccountparticlesthatattachtothe peratures, implying that growth in the vertical direction deposit having both vertical and horizontal neighbours. is more likely to happen with dipoles aligned vertically Once these angles were collected, for every deposit at than growth in the lateral direction with dipoles aligned eachtemperature,weconstructeda frequencyhistogram horizontally. by dividing the interval ( π,π) into 1000 sub-intervals. The curves exhibit some other, lower, peaks. There − In figure 5 we plot these results for lateral and vertical is a broad, low peak around ω = π/2 at T∗ = 10−1, growth at T∗ =10−1 and T∗ =10−4. which corresponds to lateral growth with vertically- aligned dipoles (‘black’ horizontal branches in figure 4), or to vertical growth with horizontally-aligned dipoles a) 0.008 (‘grey’ vertical branches in figure 4), at high temperatures. This can be explained by noting that the second- lowestminimumoftheinteractionenergyatfixedsepara- tion is for two antiparallel dipoles. These peaks seem to 0.006 disappearatT∗ =10−4,suggestingthatenergeticeffects become more important as the temperature is lowered. y c n There are several other peaks, occurring at the same ue0.004 q angles for both lateral and vertical growth, whose posi- e fr tions seem unaffected by changing the temperature. At the present stage of our work, we can only speculate as 0.002 totheirorigin. Webelievethesepeakscomefromacom- bination of lattice effects and the properties of the dipolar interaction. It is actually known that the minimum- 0 energyarrangementofn( 3)dipolesisobtainedbyplac- -0.5 -0.4 -0.3 -0.2 -0.1 ω0/π 0.1 0.2 0.3 0.4 0.5 ing them at the vertices≥of a regular n-sided polygon, tangent to the circumscribing circle. Thus four dipoles on a square lattice will minimse their energy by making b) 0.04 π/4 angles with the horizontal and vertical axes, which might explain the peaks observed in figure 4 at that angle. The remaining peaks may likewise correspond to other arrangements of dipoles realising other minima of 0.03 the energy of sets of dipoles on a square lattice. y c n ue0.02 q IV. CONCLUDING REMARKS e fr 0.01 We have simulated the deposition of dipolar particles ona1dsubstrateusingalatticemodel. Ourfindingssug- gestthatthefractaldimensionofthedepositsisthesame as for DLD and hence unaffected by the dipolar interac- 0 -0.5 -0.4 -0.3 -0.2 -0.1 ω0/π 0.1 0.2 0.3 0.4 0.5 tions,butalsothatthereisacrossoverfromtemperature- dependent to temperature independent behaviour which can be very broad. A fuller characterisation in terms of FIG. 5. Frequency histogram of the angle ω between the height-height correlation function, tree size distribu- dipole orientation and the local direction of growth, for (a) tions and density scaling with system size, is in progress T∗ =10−1 and (b) T∗ =10−4. The solid lines correspond to and will be published elsewhere. These quantities would lateral growth and thedotted lines tovertical growth. provide additional routes to the fractal dimension, thus allowing us further to verify (or falsify) our preliminary Because all curves have period π and are even, only conclusions. Growth and roughness exponents will also the interval ( π/2,π/2) is shown. The main feature of be calculated. − all the curves are the strong peaks at ω = 0, imply- We have now started work on the off-lattice version ing that most dipoles align in the direction of growth of of the present model, so as to be free from any possible 6 artifacts arisingfromthe discretisationof space. Results allows us to express the total energy as sofarsuggestthatthe latticehasaverysmalleffect, but wearecurrentlysomewhatlimitedbytheveryhighcom- N E = µ µ ψ(r ) putational cost of evaluating the interactions between a · i i particle and all its periodic images at every step. More Xi=1 efficient algorithms are being developed to tackle this. N +3 µ·∇c µi·∇c θ(ri,c)|c=0. (A4) Xi=1(cid:16) (cid:17)(cid:16) (cid:17) ACKNOWLEDGEMENTS To calculate ψ(r) and θ(r,c) we use the identities: Funding from the Fundaçaõ para a Ciência e Tecnolo- x−2u = 1 ∞tu−1e−xt2dt, (A5) gia (Portugal) is gratefully acknowledged in the form Γ(u)Z0 of post-doctoralfellowships nos.SFRH/BPD/5654/2001 1/2 (F. de los Santos) and SFRH/BPD/1599/2000 (M. e−t|r+n|2−ic·(r+n) = π eik·r tL2 Tasinkevych)andpartialsupport(P.I.C.Teixeira). We n (cid:18) (cid:19) k X X thank M. M. Telo da Gama for helpful discussions and (k+c)2 exp . (A6) J. J. Weis for advice on how to treat the Ewald sums. × − 4t (cid:18) (cid:19) k=2π(k,0)/L with k integer is a reciprocal-lattice vec- APPENDIX A: EWALD SUMS FOR THE tor and r and n are as above. Equation (A5) is a direct DEPOSITION PROCESS consequence of the definition of the Gamma function, while equation (A6) is a form of the Poissonsummation formula for d = 1, which is the dimensionality of the We havegeneralisedtoarbitrarydimensionsamethod lattice formed by repeating the box. proposed by Grzybowski et al. [18] for evaluating Ewald For ψ(r) we set u=3/2, leading to sums. Here for simplicity we just present results for our cwaeseimdp=os1e(pwehreioredidcibsothuendnaurmybceornodfitdiiomnse)n.sioOnusrinsiwmhuilcah- ψ(r)= 1 ∞t1/2e−|r+n|2tdt. (A7) Γ(3/2) tion box consists of a rectangle with a base of length L n Z0 X and any height that can accommodate N dipoles. The The sum over direct-lattice vectors converges fast for dipolesµ arealwaysthree-dimensionalvectors,withpo- i large t, while that over reciprocal-lattice vectors does so sition vectors R , i=1,2,...,N. The simulation box is i for small t. We therefore choose an arbitrary separation repeated in the x (horizontal) direction, giving rise to a parameter α2 for the t integration and decompose the regularlatticewhosesitesarelocatedatn=(n,0)L. Let lattice sum into two terms: µandRbethedipolemomentandpositionofanincom- ing particle. Then, the distance between the incoming ψ(r)= 2 ∞t1/2e−t|r+n|2dt preniaer≡rtgicRyleb−ien(tRwtheiee+nontrhi)g,einiin=cceo1lml,ai2nn,g.d.p.aa,nrNotit.chlTeerhainendtaotnthaeilmNinatgpeeararctceitclilloeinss √+π2Xn Zα2 α2t1/2e−t|r+n|2dt. (A8) in the box and their infinite replicas is √π n Z0 X N ∞ µ µ [µ (r +n)][µ (r +n)] Takingintoaccountthat tr+n2 = tx+n2 ty2and E = · i 3 · i i· i . − | | − | | − Xi=1Xn (|ri+n|3 − |ri+n|5 ) uarsrinivgetahtePoissonsummationformula,equation(A6),we (A1) ψ(r)= 2 ∞t1/2e−t|r+n|2dt According to the geometry of the system ri = (xi,yi,0) √π n Zα2 wherexi (yi)isthehorizontal(vertical)distancebetween 2X√π α2 k2 the incomingparticle anda particlein the deposit. Note + e−ikx exp ty2+ dt. (A9) thatthe incomingparticledoesnotinteractwithits own L k Z0 (cid:18)− 4t(cid:19) X images. Introducing the notation In the case of θ(r,c) we set u=5/2 with the result ψ(r)= 1 , r=0, (A2) θ(r,c)= 1 e−ic·(r+n) ∞t3/2e−t|r+n|2dt (A10) Xn e|n−i+c·(rn|+3r) 6 Γ(5/42)Xn Zαα22 k2 θ(r,c)= , r=0. (A3) + eikx−iycy texp ty2+ dt. n+r5 6 3L − 4t n | | k Z0 (cid:18) (cid:19) X X 7 We nowsubstitute ψ(r)andθ(r,c) into equation(A4) I (α,β)= ∞tνe−tβ2dt, (B4) ν for E: Zα2 α2 E = √2π i n (cid:26)(µ·µi)I1/2(α,β) Jν(α,a,b)=Z0 tνe−at−b/tdt. (B5) XX WeareinterestedinI andI . Theycanbeevaluated 2 µ (r +n) µ (r +n) I (α,β) 1/2 3/2 − · i i· i 3/2 with the help of I( 1/2), which is easy: h ih i (cid:27) − 2 + µ µ +µ µ eikxiJ (α,y ,k) √π y iy z iz 0 i L I (α,β)= erfc αβ , (B6) Xi Xk (cid:16) (cid:17) −1/2 β 4 (cid:16) (cid:17) µ µ y2eikxiJ (α,y ,k) − L y iy i 1 i where we have made the change of variable t = z2/β2. Xi Xk Now the cases ν = 1/2 and ν = 3/2 can be easily found 1 + µ µ k2eikxiJ (α,y ,k) (A11) by integrating by parts: x ix −1 i L Xi kX=6 0 αe−β2α2 √π 2 +i k µxµiy+µixµy yieikxiJ0(α,yi,k), I1/2(α,β)= β2 + 2β3erfc αβ , (B7) L Xi kX=6 0 (cid:16) (cid:17) αe−β2α2 3 (cid:16) (cid:17)3√π I (α,β)= α2+ + erfc αβ , (B8) where β = r +n and the integrals are given by (see 3/2 β2 2β2 4β5 Appendix B|foir det|ails) (cid:18) (cid:19) (cid:16) (cid:17) having resorted to the changes of variables u=t1/2 and αe−|ri+n|2α2 u′ = e−β2t. Turning next to J , the values of ν we are I1/2 α,|ri+n| = r +n2 intere−sted in depend on the sysνtem dimensionality. For i (cid:16) (cid:17) | | d = 1, ν = 1,0,1 and no analytical results are avail- + √π erfc αr +n , (A12) able, in cont−rast with the d = 2 case. One of the three 2r +n3 | i | αe−||rii+n|2α|2 (cid:16) 3 (cid:17) itnwtoe:grals,however,canbeexpressedintermsoftheother I α, r +n = α2+ 3/2 | i | r +n2 2r +n2 (cid:16) (cid:17) +| i 3√π| (cid:18)erfc αr|+i n |, (cid:19) k42J−1+J0−y2J1 =α2exp −y2α2− k42t . (B9) 4r +n5 | i | (cid:18) (cid:19) i | | (cid:16) (cid:17) α2 Jν(α,yi,k)= tνe−tyi2−k2/4tdt. (A13) Z0 Since E is real, equation (A11) can be further simpli- fiedbyreplacingexp(ikx )bycos(kx )+isin(kx ). Note i i i also that it is even in k and that the case k = 0 can be [1] G.C.Hadjipanayis(editor),MagneticStorageSystemsBe- treated analytically. Therefore, no distinction is made yond 2000, NATO Science Series II, Vol. 41 (Kluwer Aca- in the code between simulationbox imageswith positive demic Publishers, Dordrecht, 2001). and negative k, and the case k = 0 is considered sepa- [2] R. E. Rosensweig, Ferrohydrodynamics (Cambridge Uni- rately. versity Press, New York,1985). 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