Segregation in noninteracting binary mixture
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] A ug Segregation in noniteracting binary mixture.
Filip Krzy˙zewski and Magdalena A. Za luska–Kotur
Institute of Physics, Polish Academy of Sciences, Al. Lotnik´ow 32/46, 02-668 Warsaw, Poland
Process of stripe formation is analyzed numerically in a binary mixture. The system consists ofparticles of two sizes, without any direct mutual interactions. Overlapping of large particles, sur-rounded by a dense system of smaller particles induces indirect entropy driven interactions betweenlarge particles. Under an influence of an external driving force the system orders and stripes areformed. Mean width of stripes grows logarithmically with time, in contrast to a typical power lawtemporal increase observed for driven interacting lattice gas systems. We describe the mechanismresponsible for this behavior and attribute the logarithmic growth to a random walk of large particlesin a random potential due to the small ones.
PACS numbers: 68.35.Fx, 61.20.Ja, 64.60Qb, 5.50.+q
I. INTRODUCTION
Binary mixtures subject to an external driving forceare often found to segregate and form stripes of widthincreasing with time [1, 2, 3, 4]. Such phenomenon hasimportant practical applications in chemical or pharma-ceutical technology. Much theoretical and experimentaleffort has been devoted to understanding the main mech-anisms responsible for this process [5, 6, 7]. The simplestand best known model of stripe formation under externaldriving force is a simple lattice gas with attractive near-est neighbor interparticle interactions. It has been knownfor a long time that such system orders under an influ-ence of the external driving force [8, 9, 10]. Recently, thismodel has been studied intensively [11, 12, 13, 14, 15, 16]in a context of latest experimental results. We describebelow the process of stripe formation in a lattice modelwith two types of particles which do not interact via di-rect forces. The model is defined in such a way that largerparticles, which occupy five lattice sites each, can overlap.Hence they block smaller number of sites, when they areclose together, than when they are separated. Presenceof smaller particles induces effective interactions betweenlarge particles [17] and the system orders at high enoughdensities. In the ordering process, due to the presenceof smaller particles in the spaces between stripes, largeparticles realize random walk in a random potential. Ajump can occur only when there is enough free space ina chosen direction; the latter being a random event. Asa result kinetics of stripe growth in our model is differentthan typical power law temporal growth known in drivenlattice gas [9, 13] and a mean stripe width in the binarymixture studied here increases logarithmically with time.
II. STRIPE FORMATION IN THE BINARYSYSTEM
Stripe formation can be easily observed in a simplelattice gas model with nearest neighbor attractive inter-actions. It has been shown [8, 9, 10] by means of MonteCarlo simulations that such system orders successively
FIG. 1: Jump rules for large (at left) and for small (at right)particles. Shaded area sites can be shared by overlappinglarge particles. The external bias b affects the jump rates ofsmall particles only. in stripes under an influence of external driving force.When the evolution starts from some random configu-ration, stripes are formed: initially thin they graduallybecome thicker. This process, described and analyzed indetail in Refs 11, 12, 13, 14, 15, 16 occurs in two stages- stripe formation and then stripe growth. The meanwidth of stripes grows typically as a power of a simu-lation time t x (where time is measured in the numberof Monte Carlo steps). Typically two different powers x = 1 / x = 1 /
0 25t=5*10
0 25t=5*10
0 25t=3*10
0 25t=4*10 FIG. 2: Successive stages of separation of large (dark) andsmall (light) particles cles (large or small) do not block it. Small particles areaffected by external, biasing field characterized by a con-stant b . Jump rates of small particles in each directiondepend on b . When the particle moves in the directionof the bias the jump rate is p = 1. When it moves in thedirection perpendicular to it then p = b − . For a jumpopposite to the field we have p = b − . In most of ourcalculations we have used b = 5. Fig 1. illustrates thejump rules for large and small particles.There is no direct interaction between particles, butthey can partially or fully block sites, which then can-not be occupied by other particles. Large particles canoverlap; so they block fewer sites, when they are closertogether. When the large particles overlap, then the freedsites become immediately available to small particles.Once they are filled the large ones cannot move awayfrom each other. Therefore, a configuration of closelypacked large particles is more probable than other con-figurations. It is shown in Refs 17, 18 how to map such amixture of large and small particles onto an Ising modelwith nearest-neighbor interactions. Strength of this ef-fective interaction depends on the density of the smallerparticles. The interaction is stronger when the densityis higher and, eventually, above certain critical density aphase transition occurs. This entropic interaction alongwith the biasing field causes a formation of stripes paral-lel to the field. The system structure at different evolu-tion times is shown in Fig. 2. Time for all presented datais measured in a number of Monte Carlo (MC) steps.Increasing fraction of free space is a signature of anordering process. Ratio m of the number of free sites totheir total number as a function of time t is plotted inFig. 3. At t > the curve shows a step like structure.Each step lasts a relatively long period of time (note log- m t(MC steps) FIG. 3: Number of free sites in the system as a function oftime for one sample. System size is (25 × . arithmic time scale in Fig.3) during which the number ofstripes does not change. The steps can be observed whenthere are only few stripes present in the system. A transi-tion from one step to the other means that the number ofstripes is reduced by one. Average number of stripes de-creases with time t and their average width l ( t ) increaseswith time, as seen in Fig. 4. We note that, as in Ref.[12] there are two stages of stripe width growth. The firststage starts when clusters of large particles coarsen andends at the time when stripes start to become as long asthe system width and a multistripe order becomes clearlyvisible. The second stage begins with well formed stripesand continues through the process of joining them to-gether. These two stages can be seen in Fig. 3. The firststage of stripe formation corresponds to a line of lowerslope and the second stage begins when the line bends upand becomes steeper. During the first stage the systemlooks like in the first left hand side panel in Fig 2, andthe second stage is represented in the next four panels.Number of unoccupied sites increases with time.We carried out simulations for various lattice sizes withdifferent width-to-height ratios. Results of all simula-tions were averaged over 100 realizations. Large particleswere distributed randomly at the beginning of every run.They could overlap. Then, given large fraction of remain-ing free sites was occupied randomly by small particles.Then, a biasing field was turned on and the simulationstarted. We have compared results obtained for closedsystems with fixed particle number with those for opensystems with fixed chemical potential controlling the par-ticle density. Formation of structures has been observedat a static bias field and at a field with periodically chang-ing orientation. The highest formation rate was noted fora constant field, and it decreased with an increasing fre-quency of the field variation. The results do not change d / a ta/L || d/ata/L ||
20 10 5 1
FIG. 4: Average width of stripes d = κl as a function of time,scaled by parameter κ . See text for explanations. qualitatively until a frequency of around 1 / (5 M C steps)is reached, above which stripes stop to form at all. Thus,most of the presented examples here are calculated for aconstant in time driving field.To find an average width of the stripes, a correlationfunction f c ( r ) for a given configuration of particles wascalculated. It is defined as f c ( r ) = (cid:26) N P [ i,j ] n i n j for | r | = 0 N P [ i,j ] n i n j for | r | 6 = 0 (1)where | r | is the distance between i-th and j-th site alongthe direction perpendicular to the external field; n i , n j =0 , n i = 1 when site i is occupied by one or more large particles and n i = 0when the site is empty. N is the number of large particles.Sums are over all sites whose coordinates perpendicularto stripes differ by | r | . Average width of the stripes issuch | r | for which of the correlation function has the firstminimum.Our simulations show that mean width of stripes growsas logarithm of time l = log( t/L k ), where L k is the sys-tem width, parallel to stripes ( L k = 25 in Fig. 2). Thecharacter of growth does not depend on the system size.This is illustrated in the Fig. 4, where seven data sets,are plotted in two groups for the binary mixture and topplot is for a one component, interacting driven lattice gas.The main panel shows the mean width of stripes l as a function of log( t ). Three lower data sets, plotted by > , +and × , represent results for systems of constant numberof smaller particles, and four data sets in the middle, plot-ted by (cid:4) , ◦ , • and △ represent results for open systems,with fixed chemical potential. For all these plots stripewidth l was rescaled by a parameter κ chosen in such away that each data set lies on one line log[ t/ ( αL k )] with α = 10 for closed and α = 1 . κ = 0 . , . . r = L k /L ⊥ equal, respectively to 2 , × × × N = 600 , κ = 0 . , , .
66 and 1 . × N = 600and 80%, 85%, 90% and 95% sites are occupied, respec-tively. The upper set printed by triangles ▽ represents astripe growth for one component interacting driven sys-tem at a temperature 0 . T c ∞ , where T c ∞ = 3 . J/k B (where J is interaction strength) and jump probabilities: p = 1 in the direction of field, p = 0 in the opposite direc-tion, and p = exp( − cJ/T ) in the direction perpendicularto the field, where c is a number of nearest neighbors.We can now compare results for a one-component in-teracting driven system with those for a non-interactingbinary mixture. Stripe formation is observed in bothcases. It can be seen, however, that time dependence ofthese processes has a different character. In contrast totypically noted power law in time stripe width growthfor one component interacting driven systems, in binarymixtures with entropic interactions we typically observeslower logarithmic time dependence. Ordering in the bi-nary system happens due to the indirect, effective at-tractive interaction between particles. Strength of thisentropic interaction decreases with increasing number offree sites [17]. In the closed system with a fixed parti-cle number the available free space expands with time(Fig. 3) and the effective interactions weaken. In orderto check if this weakening has influence onto the charac-ter of stripe growth, we studied also open systems withvarying small particle number and where mean numberof free sites was controlled by an external potential. Asillustrated in Fig. 4, the time dependence in open sys-tem has the same logarithmic character observed for theclosed system. We see that the decreasing with time in-teraction strength is not the main reason for the type ofobserved temporal width growth.Specific aspect of the system studied here is the ex-istence of particles of two types. In order to cross aninterstripe distance, large particle have to find their waybetween densely packed set of small ones. To execute ajump, the large particle has to wait until a passing streamof small particles creates a hole, large enough to fit in.As a waiting time for a jump in such case varies fromone event to the next, we can treat such process as arandom walk in a random potential. In the next sectionwe show that the logarithmic character of the temporalstripe width growth can be explained by such a descrip-tion of large particles kinetics. III. MECHANISM OF STRIPE GROWTH
Stripe growth is an anisotropic process that takes placein the driven systems. The main course of growth hap-pens along the direction perpendicular to stripes. Ex-istence of the second dimension controls relative proba-bilities of several mechanisms that compete in the stripegrowth process. This process in one component systemhas been analyzed and explained in details in Refs 12and 13. Ref. 13 describes two different competing mech-anisms: evaporation/condensation of particles from thesurface of the stripe and diffusion of particles/holes be-tween interfaces. The former one leads to l ∼ t / andcan be observed at earlier times or for shorter systems,whereas the latter leads to the l ∼ t / growth and isactivated at later stages of stripe formation or in longersystems.Let us consider a one component system with particlesattracting each other. System orders under influence ofa static bias field, initially forming many thin stripes.When the process continues some of stripes disintegratewhile the remaining ones become thicker. Stripe extinc-tion is a process consisting random actions of a singleparticle: the particle evaporates first from the stripe wall,then it walks randomly in an empty space until it read-sorbs at the same or the other wall. The process con-tinues until one of stripes disintegrates. Decay of one oftwo neighboring stripes is a problem similar to that ofthe gambler ruin. We are not asking, however, a stan-dard question about the probability of a ruin. Instead,we are interested in the mean time of ultimate decay ofthe first or the one of two neighboring stripes. This timeis proportional to the mean time of evaporation of oneparticle row across one stripe. Number of particles insuch row is equal to the width of the stripe and fluc-tuates as particles escape from and stick to the domainwalls. Emergence of a fluctuation of size l means that rowof such length disappears. Mean time of such an eventscales as l . Fluctuations occur independently in eachrow, so the time in which the entire stripe disappears isproportional to the number of rows in one stripe, L k andto the time τ needed for a single particle to pass thedistance from one stripe to another. Thus we have τ = νL k τ l , (2)where ν is time scale parameter.Time τ is mean first-passage time of a distance be-tween stripes by a particle. In a general case of an in-homogeneous potential the first-passage time is given by[19, 20] τ = s − X n =0 p n s − X k = n k Y j = n q j p j (3)where s = l (1 − ρ ) /ρ is the distance between stripes, ρ adensity of large particles in the system, p j is jump ratefrom site j in the direction pointing from the initial site 0 to the final site s , and q j is jump rate in the oppositedirection. For the one component system we can assumethat p j = q j in the Eq. (3) and that p j are the same for all j = 1 , ...s except when j = 0 for a jump originating at asite neighboring to the stripe. The rate p is a probabilityrate for a particle jumping out of other particles. Theparticles attract each other, so this rate is smaller thanall others: p < p . The interaction and so the value of p depend on the number of neighboring particles. Wecan write τ ∼ sp + s p (4)and treat p as an effective rate averaged over manyjumps. The above formula is correct when the densityof particles between stripes is low, not higher than onefree particle per row. If s > p /p then the second term of(4) dominates. In this case, however, the particle densityis higher, than one particle per row so proper expressionfor the time τ is obtained by dividing by a number ofparticles that reach the wall per time unit. This numberis proportional to the distance s , so for s > p /p we get τ ∼ sp . (5)Equation (5) is valid if the density of particles betweenstripes is higher than one particle per row but is still quitelow. For higher densities, however, pair interactions inthe empty space start to play a role, causing the entireprocess to slow down. Using Eq. (2) we obtain dldt = lτ = 1 L k lτ . (6)Solution of this equation for τ given in (4) is(1 − ρ ) l p ρ + l (1 − ρ ) p ρ ∼ tL k , (7)the exponent of the power law growth of l changes be-tween 1 / /
4. When (5) is used, we get l ∼ ( tL k ) / . (8)i.e. the power law time dependence with a single expo-nent x = 1 / x , usually smaller than 1 /
3, are observed. Still, l ∼ t / is a dominant behavior for wide range of tem-peratures and system geometry parameters.When the system consists of two different types of par-ticles, random walk from one stripe to another is not free.Each particle has to wait until there is enough space forit jump. We can treat the process of particle motion in adense medium as a random walk in a random potentiallandscape. A jump to the left with rate p l and jump tothe right with rate p r are in this approach independentevents, occurring according to the same probability dis-tribution. Such a model leads to the following expressionfor the mean first-passage time [19] τ ∼ γ ( γ s − γ − ∼ e λl (9)where γ = h p l ih p r i > hi being an average over random variable realiza-tions. Thus all linear in s terms in expression (9) are forlarge s irrelevant and we get λ = log( γ )(1 − ρ ) /ρ . Usingnow (2) and (9) we get the following equation dldt ∼ e − λl L k l . (11)Its solution for large l and t can be written as l ∼ log( t/L || ) (12)and, indeed such character of the time dependence is ob-served in Fig 4 for binary systems. It can be seen in theinset of Fig 4., that power law cannot be fitted to the datasets for binary mixtures. The character of stripe growthis the same for closed system, where number of free sitesincreases as it is for an open system with constant densityof small particles controlled by external potential. IV. SUMMARY
We have investigated binary mixture system driven byan external force. Particles in this binary system do notinteract with each other directly but they effectively doso via indirect entropy interaction. The system ordersforming stripes, similarly like in driven single componentsystem with attractive forces.The existence of two different particle types leadsto the logarithmic temporal growth of the mean stripewidth. Such time dependence is slower than the powerlaw temporal growth in an interacting one componentsystem. In binary systems large particles travel amongdensely packed small particles, which effectively slowdown their wandering. We have attributed the logarith-mic growth process to a random walk of large particlesin an effectively random potential.
Acknowledgments
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