Hard core-soft shell particles near repulsive interfaces: interplay between adsorption, aggregation and diffusion
aa r X i v : . [ c ond - m a t . s o f t ] F e b Hard core-soft shell particles near repulsiveinterfaces: interplay between adsorption,aggregation and diffusion
Murilo S. Marques † , ‡ and Jos´e Rafael Bordin ∗ , ¶ † Centro das Ciˆencias Exatas e das Tecnologias, Campus Reitor Edgard Santos,Universidade Federal do Oeste da Bahia, Rua Bertioga, 892, CEP 47810-059, Barreiras,Bahia, Brazil ‡ Instituto de F´ısica, Universidade Federal do Rio Grande do Sul, Caixa Postal 15051, CEP91501-970, Porto Alegre, Rio Grande do Sul, Brazil ¶ Departamento de F´ısica, Instituto de F´ısica e Matem´atica, Universidade Federal dePelotas. Caixa Postal 354, 96001-970, Pelotas, Brazil.
E-mail: [email protected]
Abstract
The behavior of colloidal particles with a hard core and a soft shell has attractedthe attention for researchers in the physical-chemistry interface not only due the largenumber of applications, but due the unique properties of these systems in bulk andat interfaces. The adsorption at the boundary of two phases can provide informationabout the molecular arrangement. In this way, we perform Langevin Dynamics simula-tions of polymer-grafted nanoparticles. We employed a recently obtained core-softenedpotential to analyze the relation between adsorption, structure and dynamic propertiesof the nanoparticles near a solid repulsive surface. Two cases were considered: flat orstructured walls. At low temperatures, a maxima is observed in the adsorption. It is elated to a fluid to clusters transition and with a minima in the contact layer diffusion- and is explained by the competition between the scales in the core-softened interac-tion. Due the long range repulsion, the particles stay at the distance correspondentto this length scale at low densities, and overcome the repulsive barrier as the pack-ing increases, However, increasing the temperature, the gain in kinetic energy allowsthe colloids to overcome the long range repulsion barrier even at low densities. Asconsequence, there is no competition and no maxima was observed in the adsorption. Introduction
A multitude of natural process take place at the boundary between two phases while othersare started at that interface. This is the case of adsorption - a universal phenomenon in col-loidal and surface science whose properties have been studied for a long time. Basically,it’s a surface effect which causes affluence (change in concentration) of atoms, molecules orions at two-phase interfaces drastically modifying its properties compared to bulk. Forspherical colloids and nanoparticles, macromolecules that often show competitive interac-tions, the adsorption and binding at interfaces associated with the various probabilities ofaggregation are fundamental for applications in biomedical, environmental, food, and mate-rials engineering.
In this context, the computational simulation of colloids at interfaceshave evolved enormously and nowadays the structure, dynamics, thermodynamics, phasetransitions, and reactivity of colloids in confined environments and interfacial geometrieshave been widely studied, mainly through attractive and repulsive competing interac-tions. Such interrelation had as their starting point the famous DVLO theory for chargedcolloids at 1940’s and the seminal work of Asakura and Oosawa (AO) at 1950’s: thesetwo approaches provide a framework by which attraction and repulsion between colloids maybe manipulated. Today the current stage of modeling has brought to us a variety of potentials that havebeing used to portray the competitive interactions between spherical colloids. The competi-2ions, usually between a Short range Attraction and a Long range Repulsion – the so-calledSALR colloids, arise once distinct conformations compete to rule the suspension behav-ior.
The short range attraction is caused by van der Walls forces or solvent effects, while the long range repulsion can be generated by many factors. For instance, it can becaused by electrostatic repulsion in charged colloids and molecules, or by soft shells ad inthe case of spherical colloids obtained by PEG aggregates or even by a polymeric brush,as metallic NP decorated with polymers and star polymers. From experimental and computational works, is well known that the SALR NPand colloids effective interaction can be depicted by core-softened potentials. In this way,many works have been devoted to study the behavior of competitive colloidal systems inbulk solutions.
There are a plenty of works exploring the behavior of confined SALRsystems in recent times: Almarza et al have inspected the template-assisted pattern forma-tion in monolayers of particles by Monte Carlo simulations in a lattice gas generic model;
Litniewski and Ciach have analyzed the general features of adsorption phenomena in dilutesystems with particles self-assembling into small clusters; more recently, Panagiotopouloset al have scrutinized properties of lamellar structures formed by an SALR fluid in equilib-rium and non-equilibrium conditions by machine learning; Bilnadau et al investigated thecluster formation effect on adsorption phenomena they have laid down deviation in the shapeof the adsorption isotherm in comparison with simple fluids. Though, according to the au-thors’ knowledge, no relationship between aggregation, adsorption and dynamic managementwas listed in these SALR colloids at interfaces. In this work, we have looked over the behav-ior of a SALR system confined by two types of plates: rough and flat, and we’ve exploredthe distinctness of adsorption isotherms, lateral structuring (by means of the lateral radialdistribution function) and the dynamic behavior of the system in order to answer the ques-tion: how the surface smoothness influence the adsorption in systems shaped by competitiveinteractions?The answer to this inquiry is organized as follows. In Sec. II, we provide details of model3nd simulation details, while results and theoretical approach are presented and discussed inSec. III. Conclusions follow in Sec. IV.
The Model and Simulation Details (a) r ij U ( r ij ) r ij -404812 F ij r ij (b) Figure 1: (a) Schematic depiction of the polymer-grafted NP and simulations box. In theNP, the blue monomers stand for the polymers, the green is the monomer connected to thehard-core red bead. The simulation box with size L x × L y × L z in the x , y and z directionshas two walls in the z -extremes. The turquoise lines represents the density control volumeregion. (b) Effective potential U ( r ij ) employed to model the colloids as obtained in our recentwork. The inset is the product of the pair force F ij = − dU ( r ij ) /dr ij and the distance r ij as function of the distance.The fluid consists of spherical nanoparticles (NP) with a hard core and a soft corona, asschematically depicted in the figure 1(a). The effective core-softened interaction potential,obtained in a recent work by Marques and co-authors, is composed by a short-range at-tractive Lennard Jones potential and three Gaussian terms, each one centered in c j , withdepth h j and width w j : U ( r ij ) = 4 ǫ "(cid:18) σr ij (cid:19) − (cid:18) σr ij (cid:19) + X j =1 h j exp " − (cid:18) r ij − c j w j (cid:19) , (1)Here, r ij = | ~r i − ~r j | is the distance between two particles i and j . The Gaussian parameters4re given in table 1. With these parameters, the interaction potential has a shoulder-likeshape and two characteristic interaction length scales, as shown in the figure 1(b). Acloser length scale, located near 1 . σ , that stands for the hard-core, and a further secondscale r ij ≈ . σ from the soft-corona. Between them there is a entropic barrier - the scalesand the barrier are clear in the inset of the figure 1(b).Table 1: Interacting potential parameters in reduced units.Parameter Value h c w h c w h -3.8685 c w δt = 0 .
01 and γ = 1 . The simulation box has dimensions L x = 15 σ , L y = 30 σ and L z = 45 σ . The adsorption surfaces are confining plates placed in the limits of the z -direction. V F = [ L x × L y × ( L z − . x and y -directions.The confining plates can be flat or rough. In the rough case they are modeled as sphericalparticles with diameter σ distributed in a square lattice and fixed in space. The interac-tion between the wall and fluid particles is given by the purely repulsive Weeks-Chandler-Andersen (WCA) potential. The WCA interaction is a LJ interaction – first therm in Equa-tion 1 – cut at r ij = 2 σ and shifted by ǫ . In the flat case the wall was considered as a planethat repels the fluid by the projection of the WCA potential in the z -direction - interactingas a structureless flat surface.Simulations were performed with different bulk number density, ranging from ρ =5 /V F = 0 . σ to ρ = 0 . σ and temperatures from T = 0 . ǫ/k B to T = 0 . ǫ/k B .The initial number of particles in the system, obtained from N = ρ V F , were initially ran-domly distributed in the simulation box. However, considering that particles will adsorband get structured near to the wall, we create a Control Volume (CV) at the center of thesimulaton box to control the bulk density ρ , as depicted in the figure 1(a). The controlvolume has dimensions L x × L y × L z /
3. Unlike previous works, were a Grand CanonicalMonte Carlo simulation was used to keep the chemical potential fixed in a CV, here weadopted a simpler and faster approach: we control the density in the CV at every 500 timesteps during the thermalization steps. If it deviated more than 2% from the initial value ρ we insert/remove particles to restore the desired density. We check for overlaps and new par-ticles are inserted with a initial velocity obtained from a Gaussian distribution at the properthermal energy. We observe that after 5 × thermalization steps the density in the CV donot deviates more than 2% from the mean value and, therefore, no more insertion/deletionmoves are performed. The thermalization steps are followed by 5 × steps to equilibratethe system. Finally, we run 2 × steps for the results production stage. To ensure that thesystem was thermalized, the pressure, kinetic and potential energy were analyzed as functionof time. The velocity-verlet algorithm was employed to integrate the equations of motion.5 independent simulations (distinct initial random positions and velocities) were performedand here we present the average of this 5 results – the errors bars are smaller than the pointsin the results shown in Section .The Gibbs adsorption isotherms were evaluated usingΓ( ρ ) = Z ∞ [ ρ ( z ) − ρ ] dz, (2)where ρ ( z ) is the density profile along the z -direction and ρ the density in the CV. Thelateral dynamics in the non-confined plane was analyzed by the relation between the lateral6ean square displacement (LMSD) and time, namely (cid:10) [ r ( t ) − r ( t )] (cid:11) = (cid:10) ∆ r ( t ) (cid:11) , (3)where r ( t ) = ( x ( t ) + y ( t )) / and r ( t ) = ( x ( t ) + y ( t )) / denote the coordinate of theparticle at a time t and at a later time t, respectively. The LMSD is related to the diffusioncoefficient D by D = lim t →∞ h ∆ r ( t ) i t . (4)The structure of the fluid was analyzed using the lateral radial distribution function (RDF) g || ( r ij ), g || ( r ij ) ≡ ρ V X i = j δ ( r − r ij )[ θ ( | z i − z j | ) − θ ( | z i = z j | − δz ] (5)where the Heaviside function θ ( x ) restricts the sum of particles pairs in a slab of thickness δz = 1 . σ .The clustering was analyzed based in the inter particle bonding. Two particles in alayer belong to the same cluster if the distance between two of them is shorter than a cutoff1.3 - a value slightly bigger than the first lenght scale. n c is the number of particles in eachcluster, and P ( n c ) the probability to find a cluster with size n c .In this work all the quantities are computed and presented in the standard Lennard Jones(LJ) reduced units, r ∗ ≡ rσ , ρ ∗ ≡ ρσ , and t ∗ ≡ t (cid:16) ǫmσ (cid:17) / , (6)for distance, density of particles and time , respectively, and p ∗ ≡ pσ ǫ and T ∗ ≡ k B Tǫ (7)for the pressure and temperature, respectively, where σ = 1 . ǫ core /k B = 10179 K, with7 B the Boltzmann constant, are the distance and energy parameters as previously works. m is the mass of a single NP. Since all physical quantities are defined in reduced LJ units,the ∗ will be omitted, in order to simplify the discussion. Results and discussion
To analyze the layering near the surfaces we have evaluated the density profile along theconfining direction. In figure 2 we show the densities profiles ρ ( z ) normalized by the bulkdensity ρ . To analyze the properties of distinct fluid layers we have sliced the system inslabs with thickness δz = 1 .
5. Then, we select three slabs: the contact slab (I), that rangesfrom z = 0 . z = 1 .
5, one slab in the bulk region (III), from z = 13 . z = 15 .
0, and aintermediate slab (II) from z = 6 . z = 7 .
5, as indicated by the vertical dotted lines inthe figure 2.As expected, the layering is influenced by the temperature - and so adsorption isothermwith distinct behaviors. System at the smaller temperature have a layering that can spamto the entire simulation box - as we can see for T = 0 .
10 and ρ = 0 .
100 for both surfaces.For higher temperature there is only the creation of a contact layers and a second layer athigh densities. Is also clear that the surface roughness do not affect drastically the layering,but make small changes in the occupancy in each layer. Analyzing only the contact layeris clear that, for all combinations of densities and temperatures, the shape of this peak inthe ρ ( z ) curve are distinct for each wall: in flat surfaces the adsorption peak are higher andthinner, while for the structured surfaces they are lower and thicker.Is remarkable the layering observed at T = 0 .
10 and ρ = 0 . T = 0 .
10 asfunction of the bulk density . As we can see, the density ρ = 0 .
100 corresponds to a maximain the adsorption. From the 2D phase diagram obtained in our previous study, we know8 z ρ ( z ) / ρ ρ = 0.200ρ = 0.175ρ = 0.150ρ = 0.125ρ = 0.100ρ = 0.075ρ = 0.050ρ = 0.025 Flat Repulsive Surface T = 0.10 (I) (II) (III) (a) z ρ ( z ) / ρ ρ = 0.200ρ = 0.175ρ = 0.150ρ = 0.125ρ = 0.100ρ = 0.075ρ = 0.050ρ = 0.025 Rough Repulsive Surface T = 0.10 (I) (II) (III) (b) z ρ ( z ) / ρ ρ = 0.200ρ = 0.175ρ = 0.150ρ = 0.125ρ = 0.100ρ = 0.075ρ = 0.050ρ = 0.025 T = 0.30 (I) (II) (III) (c) z ρ ( z ) / ρ ρ = 0.200ρ = 0.175ρ = 0.150ρ = 0.125ρ = 0.100ρ = 0.075ρ = 0.050ρ = 0.025 T = 0.30 (I) (II) (III) (d) z ρ ( z ) / ρ ρ = 0.200ρ = 0.175ρ = 0.150ρ = 0.125ρ = 0.100ρ = 0.075ρ = 0.050ρ = 0.025 T = 0.50 (I) (II) (III) (e) z ρ ( z ) / ρ ρ = 0.200ρ = 0.175ρ = 0.150ρ = 0.125ρ = 0.100ρ = 0.075ρ = 0.050ρ = 0.025 T = 0.50 (I) (II) (III) (f) z ρ ( z ) / ρ ρ = 0.200ρ = 0.175ρ = 0.150ρ = 0.125ρ = 0.100ρ = 0.075ρ = 0.050ρ = 0.025 T = 0.70 (I) (II) (III) (g) z ρ ( z ) / ρ ρ = 0.200ρ = 0.175ρ = 0.150ρ = 0.125ρ = 0.100ρ = 0.075ρ = 0.050ρ = 0.025 T = 0.70 (I) (II) (III) (h) z ρ ( z ) / ρ ρ = 0.200ρ = 0.175ρ = 0.150ρ = 0.125ρ = 0.100ρ = 0.075ρ = 0.050ρ = 0.025 T = 0.90 (I) (II) (III) (i) z ρ ( z ) / ρ ρ = 0.200ρ = 0.175ρ = 0.150ρ = 0.125ρ = 0.100ρ = 0.075ρ = 0.050ρ = 0.025 T = 0.90 (I) (II) (III) (j) Figure 2: Ramp-like colloids density profile along the z -direction near repulsive walls.that this point is located in the solid hexagonal phase. Despite the fact that going from2D to 3D or slab systems can change the phase diagram, each layer at this density isin a hexagonal lattice. Here is clear how the surface structure can affect the fluid layers9 .05 0.1 0.15 0.2 ρ -0.04-0.0200.020.04 Γ Flat surfaceRough Surface T = 0.10 (a) (b) r ij g || ( r ij ) ρ = 0.200ρ = 0.175ρ = 0.150ρ = 0.125ρ = 0.100ρ = 0.075ρ = 0.050ρ = 0.025 Flat Repulsive Surface I (c) r ij g || ( r ij ) ρ = 0.200ρ = 0.175ρ = 0.150ρ = 0.125ρ = 0.100ρ = 0.075ρ = 0.050ρ = 0.025 Rough Repulsive Surface I (d) ρ < n c > Flat surfaceRough surface (e) P ( n c ) Flat repulsive surfaceRough repulsive surface n c ρ = 0.100ρ = 0.150ρ = 0.200 (f) Figure 3: (a) Ramp colloids adsorption isotherm for T = 0 .
10. (b) Layers (I), (II) and (III)structure at the density ρ = 0 .
100 for flat and rough surfaces. LRDF of the contact layer(I) for all densities in (c) flat and (d) rough sufaces. (e) Mean number of particles in onecluster in the contact layer as function of bulk density and (f) probability of find a clusterwith size n c at distinct values of bulk densities.structure. In the flat surface even the contact layer do not a perfect hexagonal structure,as we show in the upper panel of the figure 3(b). As we move away from the wall more10acancies and defects can be seen. On the other hand, rough surfaces induces a well definedlayering without defects in the hexagonal lattice, as we can see in the bottom panel of thefigure 3(b).We can see the defects by looking at the LRDF of the contact layer (I), shown in figure 3(c)for flat walls and in figure 3(d) for rough walls. As we can see, the overall behavior issimilar: at low bulk densities the LRDF indicates no ordering, as in a gas-like phase. As ρ increases, the peak in the LRDF near the second length scale, at r ≈ .
2, grows and reacha maximum. At this point, the peak in the LRDF correspondent to the first length scale, at r ≈ .
2, appears. However, for flat surfaces, we can see occupancy in the first length scaleat ρ = 0 .
100 while the rough surfaces has the higher occupancy in the second length scaleand no particles at the first length scale. The occupancy in the first length scale indicatesparticles aggregation. This is corroborated by the analysis of the mean number of particle inone aggregate cluster at the contact layer (I), shown in the figure 3(e). For flat surfaces and ρ = 0 .
100 the mean cluster size is bigger than 1.0 - indicating the formation of aggregates,while for rough surfaces at ρ = 0 . P ( n c ) shownin the upper panel of the figure 3(f) also shows this. Increasing the density, the cluster sizeincreases, and the occupancy in the first length scale increases as well.Increasing T in the 2D system melts the solid hexagonal phase. Here we observe thesame. However, at the intermediate temperatures T = 0 .
30 and T = 0 .
50, the adsorptionisotherms show an interesting behavior that can be related to the changes in the occupancyin each length scales. As we shown in figure 4(a), for T = 0 .
30 both surfaces have theanomalous behavior in the adsorption curve, that can be related to the competition betweenthe scales shown in figures 4(b) and (c). For the isotherms at T = 0 .
50 the anomaly remainsfor the rough surface, but it practically vanishes for the flat surface, as we can see in thefigure 4(a). This distinct behavior is also related with the LRDF of the contact layer, shownin figures 4(d) and (e). For the flat surfaces the first length scale is occupied even at lowdensities, since the thermal energy is high enough to overcome the ramp entalpic contribution11 .05 0.1 0.15 0.2 ρ -0.015-0.01-0.00500.005 Γ Flat surface, T = 0.30 Flat surface, T = 0.50 Rough Surface, T = 0.30 Rough Surface, T = 0.50 T = 0.30; 0.50 (a) r ij g || ( r ij ) ρ = 0.200ρ = 0.175ρ = 0.150ρ = 0.125ρ = 0.100ρ = 0.075ρ = 0.050ρ = 0.025 Flat Repulsive Surface I (b) r ij g || ( r ij ) ρ = 0.200ρ = 0.175ρ = 0.150ρ = 0.125ρ = 0.100ρ = 0.075ρ = 0.050ρ = 0.025 Rough Repulsive Surface I (c) r ij g || ( r ij ) ρ = 0.200ρ = 0.175ρ = 0.150ρ = 0.125ρ = 0.100ρ = 0.075ρ = 0.050ρ = 0.025 Flat Repulsive Surface I (d) r ij g || ( r ij ) ρ = 0.200ρ = 0.175ρ = 0.150ρ = 0.125ρ = 0.100ρ = 0.075ρ = 0.050ρ = 0.025 Rough Repulsive Surface I (e) Figure 4: Adsorption isotherms for T = 0 .
30 and T = 0 .
50 (a) and the LRDF for T = 0 . T = 0 .
50 and flat (d) and or (e)adsorption surfaces 12or the total energy. On the other hand, for the rough adsorption surface there is the extrapenalty due the friction with the surface beads. As consequence, the competition betweenthe scales and the anomaly is observed for the rough case. So, heating up the system theentropic contribution for the free energy overcome the penalties from the ramp and from thefriction, ending the competition, as we show in the figure 5(b) and (c), and the adsorptionanomaly, figure 5(a), for T = 0 . ρ -0.02-0.0175-0.015-0.0125-0.01-0.0075-0.005 Γ Flat Surface, T =0.70 Flat Surface, T = 0.90 Rough Surface, T = 0.70 Rough Surface, T = 0.90 T = 0.70; 0.90 (a) r ij g || ( r ij ) ρ = 0.200ρ = 0.175ρ = 0.150ρ = 0.125ρ = 0.100ρ = 0.075ρ = 0.050ρ = 0.025 Flat Repulsive Surface I (b) r ij g || ( r ij ) ρ = 0.200ρ = 0.175ρ = 0.150ρ = 0.125ρ = 0.100ρ = 0.075ρ = 0.050ρ = 0.025 Rough Repulsive Surface I (c) Figure 5: Adsorption isotherms for T = 0 .
70 and T = 0 .
00 (a) and the LRDF for T = 0 . The diffusion anomaly is characterized by the increase in the self-13iffusion constant D as the pressure or density increases. When confined or near surfaces,this anomalous behavior is affected by the surface. To see how the colloids diffuses atthe interface or in the bulk we have evaluated the lateral mean square displacement (LMSD)for distinct layer. With this, we obtain the lateral diffusion constant D l for each layer, with l ranging from 1, the contact layer, to 15, the layer exactly at the simulation box centerwhere the density was fixed in ρ . Here we show in the figure 6 the contact diffusion D divided by the bulk diffusion D for all the temperatures. Interesting, for all cases thecontact layer diffuses faster at the interface when the colloidal density is small. This iscounter-intuitive, once we should expect a smaller diffusion due the friction with the wall.However, core-softened fluids can show a anomalous increase in D near solvophobic surfaces,as we have show previously. As ρ increases, we can see that the isotherms have distinctbehaviors. At high T , were no adsorption anomaly was observed, the diffusion decreaseswith the density. However, for the cases where the system has adsorption anomaly, weobserve a diffusion anomaly. The ratio D /D decay with ρ , indicating that the contactlayer is diffuse slower than the bulk layers up to a threshold, where the curve has a mininumand increases as ρ increases - similar to the water-like diffusion anomaly. This minima inthe diffusion corresponds to the maxima observed for the adsorption, indicating that thesequantities are related. This allow us to correlate the adsorption not only to the structure ofthe adsorbed colloids, but to their dynamic as well. Conclusions
In this paper we have explored the behavior of SALR particles near solid surfaces. Twospecies of surfaces were simulated: a flat, smooth surface and a rough, structured one. Bycontrolling the bulk density, we were able to see how the colloids adsorb at the surface atdistinct temperatures. As usual for this system, we observe a layering as the density increasesat low temperatures. Curiously, the system has more layers at the intermediary simulated14 .05 0.1 0.15 0.2 ρ D / D Flat - T = 0.10 Rough - T = 0.10 Flat - T = 0.30 Rough - T = 0.30 Flat - T = 0.50 Rough - T = 0.50 Flat - T = 0.70 Rough - T = 0.70 Flat - T = 0.90 Rough - T = 0.90 Figure 6: First layer diffusion divided by the central layer (bulk-like) diffusion for distinctbulk densities.density than at the higher density. This was related with the observation of a triangularlattice in the contact layer. This structural conformation was observed for all layers at T = 0 .
10 and ρ = 0 .
100 - with defects and vacancies in the structure of the layers in thecase of flat walls.This well defined layering at T = 0 .
10 and ρ = 0 .
100 leads to a maxima in the NPadsorption. After that, the colloids start to aggregate. SALR interactions are characterizedby the existence of two length scales, the short range attraction and the long range repulsion.This clustering is related to the occupancy in the first length scale - what was not observedprior to the adsorption maxima. Essentially, as ρ increases the packing allow the colloids toovercome the energetic penalty from the long range repulsion. With this, they became closerand start to aggregate. Similarly, the temperatures T = 0 .
30 and T = 0 .
50 have changedin the adsorption curve when the particles starts to occupy the first characteristic distance.However, when T increases, the gain in kinetic energy allow the colloids to overcome therepulsive barrier even at low densities. With this, the first length scale is occupied even atlow densities and no maxima is observed in the adsorption.To check how it is related to the dynamical properties, we evaluated the layer’s diffusionand compare the diffusion of the contact layer and the central (bulk-like) layer. We could see15hat the maxima in the adsorption corresponds to a minima in the diffusion - this diffusionminima as function of density is similar to the waterlike diffusion anomaly, which is relatedto competition between the length scales observed in the water molecule interaction.With this, we could connect adsorption, clustering and diffusion with the competition be-tween the length scales of SALR. In fact, the correlation between maximum in the adsorptionand clustering agrees with the recent work by Bildanau and co-workers for 2D systems, and our results provide information regarding the connection with the diffusion behavior atthe interface and the competition between the scales. Acknowledgement
Without public funding this research would be impossible. MSM thanks the Brazilian Agen-cies Conselho Nacional de Desenvolvimento Cient´ıfico e Tecnol´ogico (CNPq) for the PhDScholarship and Coordena¸c˜ao de Aperfei¸coamento de Pessoal de N´ıvel Superior (CAPES)for the support to the collaborative period in the Instituto de Qu´ımica Fisica Rocasolano.JRB acknowledge the Brazilian agencies CNPq and Funda¸c˜ao de Apoio a Pesquisa do RioGrande do Sul (FAPERGS) for financial support. JRB is greatly indebted to AlexandreDiehl for illuminating discussions. All simulations were performed in the SATOLEP Clusterof the Group of Theory and Simulation in Complex Systems from UFPel.
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