Distinct aggregation patterns and fluid porous phase in a 2D model for colloids with competitive interactions
aa r X i v : . [ c ond - m a t . s o f t ] D ec Distinct aggregation patterns and fluid porous phase in a 2D model for colloidswith competitive interactions
Jos´e Rafael Bordin
Campus Cac¸apava do Sul, Universidade Federal do Pampa, Av. Pedro Anunciac¸ ˜ao, 111, CEP 96570-000, Cac¸apava do Sul, RS,Brazil
E-mail:[email protected]
Abstract
In this paper we explore the self-assembly patterns in a two dimensional colloidal system using ex-tensive Langevin Dynamics simulations. The pair potential proposed to model the competitive interactionhave a short range length scale between first neighbors and a second characteristic length scale betweenthird neighbors. We investigate how the temperature and colloidal density will affect the assembled mor-phologies. The potential shows aggregate patterns similar to observed in previous works, as clusters, stripesand porous phase. Nevertheless, we observe at high densities and temperatures a porous mesophase witha high mobility, which we name fluid porous phase, while at lower temperatures the porous structure isrigid. triangular packing was observed for the colloids and pores in both solid and fluid porous phases.Our results show that the porous structure is well defined for a large range of temperature and density, andthat the fluid porous phase is a consequence of the competitive interaction and the random forces from theLangevin Dynamics.
Keywords:
Competitive interactions, colloids, self-assembly, Langevin dynamics
1. Introduction
The study of chemical building blocks as amphiphilic molecules, block copolymers, colloids and nanopar-ticles have attracted the attention in soft matter physics in recent years due their properties of self-assembly [1,2, 3]. The variety of the length scales and geometry of the patterns arise from the different types of potentialenergies involved and from the shapes of the colloidal particles. Self-assembled nanomaterials have appli-cations in medicine, self-driven molecules, catalysis, photonic crystals, stable emulsions, biomolecules andself-healing materials [4, 5, 6, 7, 8, 9].Despite the fact that the competition between shape and interaction can influence the self-assembly ofmolecules and colloids [10, 11, 12, 13], experimental [14, 15] and simulational studies [16, 17, 18, 19,20, 21, 22, 23, 24] have show that spherical colloids with tunable competitive interactions can be usedto generate distinct aggregate patterns. The combination of a short attractive interaction and a long-rangerepulsion can describe the nucleation in these systems with competitive interactions [25, 26]. The repulsioncan be caused by a soft shell, as in the case of polymeric brushes [27, 28, 29], or by electrostatic repulsion
Preprint submitted to Elsevier June 16, 2018 n charged colloids and molecules [30, 31, 32, 33], while the attraction is caused by van der Walls forces orsolvent effects [34]. The patterns formation in two-dimensions (2D) has been extensively explored in theliterature by theoretical and computational works [35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49,50]. These patterns includes small clusters, lamellae phases, stripes and porous phases.In this paper we explore the phase diagram of a two dimensional colloidal system with a short rangeattraction and a long range repulsion (SALR). This potential is based in a well known ramp-like core-softened potential, extensively applied in studies of systems with water-like anomalies [51, 52]. Here, weparametrize the equation to reproduce a SALR potential with an attraction at a distance ≈ σ , the diameterof the disks, and a second length scale at a distance ≈ σ , with a energy barrier between these characteristicdistances. These distances were chosen to reproduce stripes with thickness ≈ σ , as observed in chargedglobular proteins and nanoparticles [26, 53]. Here, our continuum SALR system is studied using extensiveLangevin Dynamics simulations. In previous works [54, 55, 56] we have observed that Brownian forcesaffects the phase diagram of systems with pattern formation. These forces leaded to distinct self-assembledpatterns, distinct dynamical properties and even to a reentrant fluid phase. Therefore, we can expect thatnew phenomena arises due the Brownian effects in the SALR system with these characteristic distances.Our results show that the system has a rich variety of patters, with clusters, stripes and porous phases,similar to observed in lattice models [34, 36]. Yet, here we observe a curious fluid porous phase that wasnot observed. In this phase, the pores have a well defined structure but the mean square displacement showsthat the particles are diffusing. This indicates that the pores have a fixed size and also diffuses, in a similarway to bubbles moving in a fluid.Our paper is organized as follows. In the Section 2 we introduce our model and the details aboutthe simulation method. On Section 3 we show and discuss our results, and the conclusions are shown inSection 4.
2. The Model and the Simulation details
In this paper we compute all the quantities using standard LJ units [57]. Distance, density of particles,time and temperature are given, respectively, by r ∗ ≡ r σ , ρ ∗ ≡ ρσ , and t ∗ ≡ t (cid:16) ε m σ (cid:17) / , and T ∗ ≡ k B T ε , (1)where σ , ε and m are the distance, energy and mass parameters, respectively. In this way, we will omit thesymbol ∗ to simplify the discussion. 2 r ij -6-4-202468 U-dU/drd U/dr Figure 1: Potential, force and second derivative of the potential between two particles as function of their separation.
The colloidal system consists of N = σ and mass m with a potential interactioncomposed of a short-range attractive Lennard Jones potential and a Gaussian therm centered in r , withdepth u and width c , to take in account the long-range repulsion U ( r i j ) = ε "(cid:18) σ r i j (cid:19) − (cid:18) σ r i j (cid:19) + u exp " − c (cid:18) r i j − r σ (cid:19) , (2)where r i j = | ~ r i − ~ r j | is the distance between two colloids i and j . This potential can be parametrized tohave a ramp-like shape, and this particular shape was extensively applied to study systems with water-likeproperties [51, 58]. In this work, we propose the parameters u = . c = r / σ = .
0. The interactionpotential is shown in figure 1. As showed by de Oliveira and co-workers [59],the potential equation (2) havetwo length scales associated with the minimum in their second derivative. In this way, the second derivativeof equation (2), dashed green line in the figure 1, indicates that the potential with the parameters proposedhere have a first length scale close to r i j ≈ . r i j ≈ .
0. Therefore, the pair interactionhave a short range first neighbors characteristic distance and a second length scale related to third neighbors.Usually colloids have interactions with a small range compared to their diameters, as extensively exploredin the literature [35, 42, 45, 24]. However, in our model the range is much bigger than the disk size. This isproposed to include the class of polymer-grafted colloids, where the polymer chain size and flexibility canlead to a long range in the interaction potential [60, 61].Colloids are usually immersed in a solvent. Then, Brownian effects are relevant for colloidal system,and an effective way to include solvent effects is by using Langevin Dynamics simulations. In this way,3angevin Dynamics simulations were performed using the ESPResSo package [62, 63]. Hydrodynamicsinteractions were neglected. Since the system is in equilibrium we do not expect that this will change thelong-time behavior. The number density is defined as ρ = N / A , where A = L is the area and L the size ofthe simulation box in the x - and y -directions. ρ was varied from ρ = .
05 up to ρ = .
80, and the size ofthe simulation box was obtained via L = ( N / ρ ) / . The cut off in the interaction, equation (2), is r cut = . L > × r cut . The temperature was simulated in the intervalbetween T = .
05 and T = .
50, with viscosity γ = .
0. The equations of motion for the fluid particles wereintegrated using the velocity Verlet algorithm, with a time step δ t = .
01, and periodic boundary conditionswere applied in both directions. We performed 1 × steps to equilibrate the system. The equilibrationtime was then followed by 5 × steps for the results production stage. To ensure that the system wasequilibrated, the pressure, kinetic and potential energy, number and size of aggregates were analyzed asfunction of time, as well several snapshots at distinct simulation times. Once two dimensional systems aresensitive to the number of particles in the simulation, we carried out simulations with 10000 colloids forsome points, and essentially the same results were obtained if T > .
10. Also, we carried out simulationswith up to 5 × steps, but the aggregates for most cases are stable and do not change after ∼ × steps.Nevertheless, even for the longer and bigger simulations, in some points at low temperatures, T < .
10, andhigher densities, ρ > .
55, the energy was not well equilibrated. These points were not used to constructthe phase diagram. Five independent runs, with distinct and random initial positions and velocities for thecolloids, were performed to ensure that the patterns are not correlated to the initial configuration.The cluster size was analyzed based in the inter particle bonding [35]. Two colloids belong to the samecluster if the distance between them is smaller than the cutoff 1.5. This value ensures that the force betweenthe particles is near the minimum (as shown in the inset of the figure 1). The dependence of the clustersize with the simulation time is then evaluated. With this, we obtain the mean number of colloids in eachaggregate, < n c > , in each simulation run. If < n c > is smaller than the total number of colloids N , thenthe system is in a cluster phase. However, if any cluster has spanned the simulation box in at least oneof the directions, and if this did not change in time, we consider this as a percolating phase. As well, if < n c > = N then the system is in the percolating phase. Since this method is sensitive to the choice of thecutoff parameter, we also tested smaller values, 1.15 and 1.25, and essentially the same result was obtained.To ensure the correctness of the method, for some cases we calculated manually n c from the snapshot tocompare with the obtained from the cutoff parameter. This showed that a higher value for the cutoff distanceleaded to clusters bigger than the observed in the snapshots.4o characterize the “holes” in the porous mesophase we take a collection of 1000 snapshots of eachsimulation run and attempt to insert ghost particles with diameter σ in a square lattice with mesh size 0.25.If there is no overlap with the colloids or with ghost particles already inserted we insert a new ghost particlein the position. With the positions of the ghost particles, the same criteria used to characterize the colloidalaggregates is employed to characterize a pore, but with a distance criteria equal to 1.0. We consider as apore when the hole is filled with at least 4 ghost particles. Pores smaller than this was considered as smalldefects and did not account for analysis. With this, we can evaluate properties as the pore area A p and theradial distribution function g p − p between the center of mass of each pore. In figure 2 we show a example ofthis construction. Figure 2: Steps for the porous mesophase characterization. First, we select a snapshot of the system for a given temperature anddensity. Here, the colloids are the red disks. Then, we attempt to insert ghost particles in a square lattice with mesh 0.25. If thereis no overlap a ghost (blue) disk is inserted. With the positions of the ghost particles we can evaluate the porous properties.
To study the dynamical properties of the system we analyze the relation between the mean squaredisplacement (MSD) and time, h [ ~ r ( t ) − ~ r ( t )] i = h ∆ ~ r ( t ) i , (3)where ~ r ( t ) = ( x ( t ) + y ( t ) ) / and ~ r ( t ) = ( x ( t ) + y ( t ) ) / denote the coordinate of a colloid at a time t and at a later time t , respectively.
3. Results and Discussion
In the figure 3(a) we show the T × ρ phase diagram of the system. We should address that the phase di-agram can be considered qualitative in a first moment, since the system can be metastable in the coexistencelines. Therefore, the dashed lines indicates the separation between regions with distinct patterns. Never-theless, as we will show, there are indicatives that these lines represents the separation of the phases.Thedashed lines divides the phase diagram in regions. First, there is the fluid region, where the colloids are notarranged in clusters and the MSD indicates that the system diffuses. In this paper our focus is the aggre-gation patterns. In this way, we divided the aggregate phase in six regions in the T × ρ phase diagram, as5hown in the figure 3(a). In each region a specific patterns was observed.The first step to define the pattern is calculate the mean cluster size. In this way, the mean number ofparticles in each cluster normalized by the total number of particles, < n c / N > , was calculated and plottedas function of the density in figure 3(b) for three isotherms. We have observed that when < n c / N >< < n c / N > × ρ ,figure 3(b), is the discontinuity of the curves. In the clustering phase, region I, < n c / N > increases linearlywith ρ at low densities. Then, the curve have a discontinuity from the clustering densities ( < n c / N >< < n c / N > = < n c / N > = . < n c / N >< . P ( n c / N ) of find a cluster with size n c particles, is shown in thefigure 3(d) for some isochores and the isotherm T = .
50. As we can see, the probability of find a largecluster, and the mean size of the cluster, increases with ρ . The P ( n c / N ) for the percolating region is notshown since all the particles are in one single cluster in this region.Inside the percolation regions distinct patterns can be observed. To define the distinct patterns regionswe have analyzed the system snapshots. Since the mean size of the clusters can not be used to differentiatethe distinct patterns in the percolating phase, we have evaluated the mean potential energy by particle, u = U / N , and plotted it as function of the temperature. As an example, we show in the figure 3(c) the curveof u × T the isochore ρ = .
75, that cross the regions IV, V and VI of the phase diagram, figure 3(a). Aswe can see, there are and slope changes and discontinuities at certain points. These gaps in the energy arerelated to changes in the aggregate patterns.Also, we can see structural changes analyzing the radial distribution function between the colloids, g c − c ( r i j ) . Inside the region I, figure 3(e), the cluster phase, the increase of the density leads to a increasein the peak of the g c − c ( r i j ) near r i j ≈ .
0, while the figure 3(f) shows that when the system goes from thepercolating region II to percolating region III the g c − c ( r i j ) indicates a higher ordering in the region III. Thisordering becomes more clear with the analysis of the system snapshots. The first percolating structure is6 disordered stripe pattern, shown in the figure 4(c) and observed in the region II of the phase diagram,figure 3(a). The disordered stripe phase occurs at lower temperatures than the aligned stripe morphology,shown in figure 4(d), and enclosed by the region III in the phase diagram figure 3(a). The inter particledistance inside the stripe is equal to the disk diameter - the first length scale - and the distance betweenthe stripes is three times the disk diameter, or the second length scale. Increasing the density, the colloidsrearrange from the stripe morphology to a porous phase. The shape of the pores are temperature dependent.For lower temperatures, the particles do not have kinetic energy to overcome the repulsive part of thepotential to achieve the minimum in the energy. As consequence, the pores do not have circular symmetryand are randomly distributed. However, as usual for systems with two length scales [52, 56], a incrementin the temperature can lead to a more ordered state. As consequence, at higher temperatures the pores havea circular shape. The region IV in the phase diagram corresponds to the amorphous pore phase, while thecircular pore phase is inside the regions V and VI. A snapshot of these two distinct porous patterns areshown in the figures 4(e), (f) and (g), for regions IV, V and VI, respectively.The sequence of patterns is similar to observed in previous works, as in the lattice model by Almarzaand co-workers [34], Monte Carlo simulations by Imperio and Reatto [41] and theoretical approach byArcher [43]. An interesting difference arises when compared to the lattice model, ref. [34]. In their work,the disordered lamellae pattern is observed at higher densities than the oriented lamellae pattern. This is theopposite of the observed in our simulations. The difference came from the range of the potential. Whilein the work by Almarza et al. the second length scale is between second neighbors, in our case the secondcharacteristic distance corresponds to the interaction between third neighbors. Then, our stripes are thicker,with three particles side by side, while in the lattice model the stripes have a thickness of two particle. Asconsequence, the stripes at low densities are disordered, since is necessary a higher density to increase thestripe thickness and make it straight.The lines that separate the regions have the discontinuity in the energy, the jump from < n c / N >< . < n c / N > = .
0, and we can even observe patterns coexisting in the snapshots. For instance, in the figure 5we show a sequence of patterns for the isotherm T = .
50. As the plot in figure 3(b) indicates, at thisisotherm the system percolates for ρ > .
55. The first snapshot in figure 5, for ρ = .
50, have two patternscoexisting, large clusters and disordered stripes. Increasing the density to ρ = .
55 only disordered stripeswere observed, in agreement with the transition from the cluster region I to the percolating region II. Then,it is a indicative that at T = . ρ = .
50 the system changes from one pattern to another. Following thesame isotherm, and increasing the density to ρ = 0.60, we can observe the coexistence of disordered stripes7nd amorphous porous, what we have identified as the border between the regions II and IV. In the sameway, at ρ = 0.65 there is a coexistence of amorphous pores and circular pores. Finally, at ρ = 0.75 all thepores have approximately the same diameter, characterizing the region V. Despite the apparent metastabilityin these points due the coexistence of two patterns, we have indications that the system is well equilibrated.For instance, we plot the pressure p , the energy per particle u and the kinetic energy per particle k as functionof time for the point T = . ρ = .
50 in the figure 6. As we can see, the energies and pressure did notvary with time, oscillating around the mean value. As well, the patterns observed in the snapshots did notchange with time. Then, we can assume that the system is well equilibrated.Next, comparing the patterns showed in the figures 4(f) and (g) we should assume in a first moment thatboth are in the same region, since the two patterns correspond to a porous phase with circular pores. Also,the pores are distributed in a triangular-like lattice in both cases, as the snapshots indicates. Nevertheless,the energy analysis figure 3(c) shows a discontinuity. Then, if the pattern is the same, what causes theenergy jump?To clarify this question, we have analyzed the colloids mean square displacement. The analysis of thedynamical properties indicates that the colloids are diffusing above a temperature threshold, as we show infigure 7(a). Therefore, the system is changing from a rigid porous structure to a fluid porous phase. Thefluid porous phase was designated as region VI in the phase diagram, figure 3(a), while the solid porousphase is the region V.The Langevin thermostat employed in this study includes two therms in the net force a the disk i . Thefirst is a drag force due the solvent viscosity, − m γ ~ v i . The second is a the random force ~ W i ( t ) due collisionsbetween solute and solvent. This second therm is modeled as a Wiener process and related to the systemtemperature T [57]. Therefore, as higher the temperature stronger is the random force. Is previous worksfor ramp-like core-softened fluids [55, 56], we have shown that the Langevin thermostat leads to a meltingand, as consequence, to a reentrant fluid phase at high T .Here, the entropic contribution from the random noise competes with the energy barrier in the potentialinteraction, figure 1, to melt the system. However, the energy barrier is high enough to ensure that the porouspattern will not be destroyed. Yet, we can see another changes in the colloidal system besides the increasein the MSD. For instance, analysis of the radial distribution function between the colloids, g c − c ( r i j ) , shownin figure 7(b), shows that for the solid ( T = .
00) and fluid ( T = .
50) porous phase the structure is similar,since the peaks are located at the same distances. But looking the the valley between the peaks we can seethat for the solid porous phase this valley goes down, becoming zero. However, for temperatures in the fluid8orous phase the valley did not go to zero – another indicative of a fluid porous phase.Using the construction discussed in the Section 2, we can evaluate the center of mass (CM) from theimaginary ghost particles (here, all ghost particles have the same mass m =
1, in LJ units) and evaluate theradial distribution function between the pores CM, g p − p ( r i j ) . In the figure 7(c) we show the g p − p ( r i j ) forsamples in the regions V and VI. We can see here the same behavior observed for the colloids: the peaksin the g p − p ( r i j ) are in the same distances for the solid and fluid porous phase. However, the valley goesto zero only in the solid case. Then not only the colloids have the same pattern in regions V and VI, butthe pores have the same distribution in both cases, regarding the presence or absence of diffusion. To seethe correlation between the colloid and pores radial distribution functions we rescaled the g c − c ( r i j ) curve.To do so, we consider that the first peak in the g p − p ( r i j ) curve occurs at r i j ≈ .
0. Note that this is twicethe thickness that we have expected when the model was proposed, see the discussion in Section 2, andobserved in the stripes of figure 4(d). The rescaled colloid-colloid radial distribution function is comparedwith the pore-pore radial distribution function in the figure 7(d). As we can see, both curves have peaksat the same distances. So colloids and pores have the same spatial distribution, with a triangular packing,as the snapshots in figure 4 indicates. The same triangular packing is observed in the fluid porous phase.Therefore, in region VI the system have a well defined porous structure (the peaks are well defined) buthave mobility (the valley did not go to zero) and diffuse (as the MSD curves show).Finally, we use the ghost particles to evaluate the mean area of each pore, A p . This information giveus a insight about the porous media properties to be used in technological applications, as filtration. Asexpected, at low temperatures the A p is larger once the system is in the region IV, with amorphous pores. Inthe solid region V and the liquid-crystal region VI the pore area did not vary with the temperature. However,there is a small dependence with the density. In figure 8 we show the curves for the densities ρ = .
65 and ρ = .
75. For the lower density the area for the circular pores is A p ≈ .
5. Using the equation for thearea of a circle, A = π a , where a is the radius of the circle, the pores have a mean radius of a ≈ .
91. Athigher density ρ = .
75, where A p ≈ . a ≈ .
65. Then, the difference in the pore radius due the increasein the density is small. As well, once inside the circular porous region, the pores radii did not vary withthe temperature. Therefore, the 2D porous membrane is well structured, with a well defined porous radius,inside a large range of densities and temperatures. 9 . Conclusion
To conclude, in this paper we have employed extensive Langevin Dynamics simulations to explore thephase diagram of a colloidal system with competitive interactions. The potential interaction has an attractionbetween first neighbors and a second length scale interaction between third neighbors. A large variety ofpatterns was observed. Despite the similarity with the structures observed in other works, the potentialshape used in this paper associated with Brownian Dynamics effects leads to a fluid porous phase. In thisfluid porous phase the structure is well defined, but the colloids have mobility and diffuses. The observedporous mesophase is stable, with a well defined size for the pores. Also, due the characteristics of thepotential – distance between the length scales and size of the energy barrier – the porous mesophase is welldefined in a long range of densities and temperatures. These findings shade light in the recent advances forcolloids engineered to spontaneously assemble in a desired nanostructure.
5. Acknowledgments
We thank the Brazilian agency CNPq for the financial support.
ReferencesReferences [1] G. M. Whitesides, B. Grzybowski, Self-assembly at all scales, Science 295 (2002) 2418–2421.[2] K.-H. Roh, D. C. Martin, J. Lahann, Biphasic janus particles with nanoscale anisotropy., Nature Materials 4 (2005) 759.[3] S. H. L. Klapp, Collective dynamics of dipolar and multipolar colloids: From passive to active systems, Curr. Opin. in Coll.and Inter. Sci. 21 (2016) 76–85.[4] C. Casagrande, P. Fabre, E. Rapha¨el, M. Veyssi´e, Janus beads: Realization and behaviour at water/oil interfaces., Europhys.Lett. 9 (1989) 251–255.[5] D. M. Talapin, J. S. Lee, M. V. Kovalenko, E. V. Shevchenko, Prospects of colloidal nanocrystals for electronic and optoelec-tronic applications, Che. Rev. 110 (2010) 389–458.[6] J. Zhang, E. Luijten, S. Granick, Toward design rules of directional janus colloidal assembly, Annu. Rev. Phys. Chem. 66(2015) 581–600.[7] O. D. Velev, E. W. Kaler, Structured porous materials via colloidal crystal templating: From inorganic oxides to metals, Adv.Mater. 12 (2000) 531–534.[8] O. D. Velev, A. M. Lenhoff, Colloidal crystals as templates for porous materials, Current Opinion in Colloid & InterfaceScience 5 (2000) 56–63.[9] Y. Yin, Y. Lu, X. Xia, A self-assembly approach to the formation of asymmetric dimers from monodispersed sphericalcolloids, J. Am. Chem. Soc. 132 (2001) 771–772.[10] Z. Preisler, T. Vissers, F. Smallenburg, G. Muna`o, F. Sciortino, Phase diagram of one-patch colloids forming tubes andlamellae, The Journal of Physical Chemistry B 117 (32) (2013) 9540–9547.[11] X. Ye, J. Chen, M. Engel, J. A. Millan, W. Li, L. Qi, G. Xing, J. E. Collins, C. R. Kagan, J. Li, S. C. Glotzer, C. B. Murray,Competition of shape and interaction patchiness for self-assembling nanoplates, Nature Chemistry 5 (2013) 466–473.[12] J. R. Bordin, L. B. Krott, How competitive interactions affect the self-assembly of confined janus nanoparticles, J. Phys.Chem. B 121 (2017) 4308–4317.[13] P. O’Toole, G. Munao, A. Giacometti, T. S. Hudson, Self-assembly behaviour of hetero-nuclear janus dumbbells, Soft Matter13 (2017) –. doi:10.1039/C7SM01401E .URL http://dx.doi.org/10.1039/C7SM01401E
14] H. N. Lokupitiya, A. Jones, B. Reid, S. Guldin, , M. Stefik, Ordered mesoporous to macroporous oxides with tunableisomorphic architectures: Solution criteria for persistent micelle templates, Chemistry of Materials 28 (2016) 1653–1667.[15] K. Peters, H. N. Lokupitiya, D. Sarauli, M. Labs, M. Pribil, J. Rathousky, A. Kuhn, D. Leister, M. Stefik, , D. Fattakhova-Rohlfing, Nanostructured antimony-doped tin oxide layers with tunable pore architectures as versatile transparent currentcollectors for biophotovoltaics, Advanced Functional Materials 26 (2017) 6682–6692.[16] Y. Zhuang, P. Charbonneau, Recent advances in the theory and simulation of model colloidal microphase formers, J. of Phys.Chem. B 120 (2016) 7775–7782.[17] N. E. Valadez-Perez, R. Castaneda-Priego, Y. Liu, Percolation in colloidal systems with competing interactions: the role oflong-range repulsion, RSC Adv. 3 (2013) 25110–25119.[18] B. A. Lindquist, R. B. Jadrich, T. M. Truskett, Communication: Inverse design for self-assembly via on-the-fly optimization,J. Chem. Phys. 145 (2016) 111101.[19] B. A. Lindquist, S. Dutta, R. B. Jadrich, D. J. Milliron, T. M. Truskett, Interactions and design rules for assembly of porouscolloidal mesophases, Soft Matter 13 (2017) 1355.[20] W. D. Pi˜neros, T. M. Truskett, Designing pairwise interactions that stabilize open crystals: Truncated square and truncatedhexagonal lattices, J. Chem. Phys. 146 (2017) 144501.[21] A. J. Archer, N. J. Wilding, Phase behavior of a fluid with competing attractive and repulsive interactions, Phys. Rev. E 76(2007) 031501.[22] D. F. Schwanzer, D. Coslovich, G. Kahl, Two-dimensional systems with competing interactions: dynamic properties of singleparticles and of clusters, J. Phys: Condens. Matter 28 (2016) 414015.[23] A. de Candia, E. D. Gado, A. Fierro, N. Sator, M. Tarzia, A. Coniglio, Columnar and lamellar phases in attractive colloidalsystems, Phys. Rev. E 74 (2006) 010403(R).[24] P. D. Godfrin, R. C. neda Priego, Y. Liu, N. J. Wagner, Intermediate range order and structure in colloidal dispersions withcompeting interactions, J. Chem . Phys. 139 (2013) 154904.[25] A. Stradner, H. Sedgwick, F. Cardinaux, W. C. Poon, S. U. Egelhaaf, P. Schurtenberger, Equilibrium cluster formation inconcentrated protein solutions and colloids., Nature 432 (2004) 492–495.[26] A. Shukla, E. Mylonas, E. D. Cola, S. Finet, P. Timmins, T. Narayanan, D. I. Sveergun, Absence of equilibrium cluster phasein concentrated lysozyme solutions, Proc. Natl. Acad. Sci. U.S.A. 105 (2008) 5075.[27] T. Lafitte, S. K. Kumar, A. Z. Panagiotoulos, Self-assembly of polymer-grafted nanoparticles in thin films., Soft Matter 10(2014) 786.[28] T. Curk, F. J. Martinez-Veracoechea, D. Frenkel, J. Dobnikar, Nanoparticle organization in sandwiched polymer brushes,Nano Letters 14 (2014) 2617–2622.[29] G. Nie, G. Li, L. Wang, X. Zhang, Nanocomposites of polymer brush and inorganic nanoparticles: preparation, characteriza-tion and application, Polym. Chem. 7 (2016) 753–769.[30] M. Quesada-P´erez, A. Moncho-Jord´a, F. Mart´ınez-Lopez, R. Hidalgo- ´Alvarez, Probing interaction forces in colloidal mono-layers: Inversion of structural data, J. Chem. Phys 115 (2001) 10897.[31] C. Contreras-Aburto, J. M. M´endez-Alcaraz, R. C. neda Priego, Structure and effective interactions in parallel monolayers ofcharged spherical colloids, J. Chem. Phys 132 (2010) 174111.[32] G. K. Ong, T. E. Williams, A. Singh, E. Schaible, B. A. Helms, D. J. Milliron, Ordering in polymer micelle-directed assem-blies of colloidal nanocrystals, Nano Letters 15 (2015) 8240–8244.[33] H. Montes-Campos, J. M. Otero-Mato, T. Mendez-Morales, O. Cabeza, L. J. Gallego, A. Ciach, L. M. Varela, Two-dimensional pattern formation in ionic liquids confined between graphene walls, Phys. Chem. Chem. Phys. 19 (2017) –. doi:10.1039/C7CP04649A .[34] N. G. Almarza, J. Pe¸kalski, A. Ciach, Periodic ordering of clusters and stripes in a two-dimensional lattice model. ii. resultsof monte carlo simulation, J. Chem . Phys. 140 (2014) 164708.[35] J. C. F. Toledano, F. Sciortino, E. Zaccarelli, Colloidal systems with competing interactions: from an arrested repulsive clusterphase to a gel, Soft Matter 5 (2009) 2390–2398.[36] J. Pe¸kalski, N. G. Almarza, A. Ciach, Periodic ordering of clusters and stripes in a two-dimensional lattice model. ice model.i. ground state, mean-field phase diagram and structure of the disordered p, J. Chem . Phys. 140 (2014) 114701.[37] N. G. Almarza, J. Pe¸kalski, A. Ciach, Effects of confinement on pattern formation in two dimensional systems with competinginteractions, Soft Matter 12 (2016) 7551–7563.[38] C. B. Muratov, Theory of domain patterns in systems with long-range interactions of coulomb type, Phys. Rev. E 66 (2002)066108.[39] S. Mossa, F. Sciortino, P. Tartaglia, E. Zaccarelli, Ground-state clusters for short-range attractive and long-range repulsivepotentials, Langmuir 20 (2004) 10756–10763.[40] J. Wu, J. Cao, Ground-state shapes and structures of colloidal domains, Physica A 371 (2006) 249–255.[41] A. Imperio, L. Reatto, Microphase separation in two-dimensional systems with competing interactions, J. Chem. Phys. 124 ρ T FLUID
I IIIII IV VVI(a) ρ < n c > / N T = 0.10 T = 0.50 T = 1.0 (b) T < u > IV V VI ∆ U V-VI ∆ U IV-V (c)
Figure 3: (a) T × ρ phase diagram for the system. The gray dots are the simulated points. The dashed black lines divides thedistinct aggregation regions. Region I is the cluster region, while regions II to VI are the percolating regions. These regions weredivide accordingly with the observed structures and energy analysis. (b) Mean number of colloids in a aggregate as function of thesystem density for distinct temperatures, showing the discontinuity when the system percolates. Inset: zoom in the region at lowdensities. (c) Mean potential energy by particle, u = U / N , as function of the temperature for density ρ = .
75. At this density thesystem is in the percolation region, but changes in the structure leads to gaps in the u × T curve. (d) Probability P ( n c / N ) of finda cluster with size n c particles for the isotherm T = .
50, showing the distribution of cluster sizes in the non-percolating region.(e) Radial distribution function between colloids for two distinct point inside the region I. (f) Radial distribution function betweencolloids for one point inside the region II and one point inside region III. igure 4: Aggregate patterns observed in distinct regions. (a) Small clusters in region I, T = .
25 and ρ = .
20, (b) big clustersin region I, T = .
25 and ρ = .
34, (c) disordered stripes in region II, T = .
70 and ρ = .
55, (d) straight stripes in region III, T = .
00 and ρ = .
50, (e) porous mesophase with disordered pores in region IV, T = .
10 and ρ = .
75, (f) porous mesophaseregion V, T = .
00 and ρ = .
75, (g) porous liquid-crystal in region VI, T = .
10 and ρ = . igure 5: Sequence of patterns for the isotherm T = .
50 and densities ρ = 0.50, ρ = 0.55, ρ = 0.60, ρ = 0.65, and ρ = 0.75 (leftto right). The first snapshot is at the border of regions I and II, and have patterns from both regions. The third snapshot show thecoexistence patterns observed in regions II and IV, while the fourth snapshot have a pattern that is a mixture of the regions IV andV. The second and fifth snapshots are points inside the regions II and V, respectively. p u time k Figure 6: Pressure p , energy per particle u and the kinetic energy per particle k as function of time for the point T = . ρ = . t M S D Regions IV and V:
T < 1.00
Region VI: 1.10 <
T < 1.50 (a) r ij g c - c ( r ij ) T = 0.10T = 1.50 (b) r ij g p - p ( r ij ) T = 1.00 T = 1.50(c) r ij g ( r ij ) g p-p g c-c (d) Figure 7: (a) Mean square displacement as function of time for the density ρ = .
75 and distinct temperatures, showing thetransition from a solid to a liquid-crystal phase. (b) Radial distribution function for the colloids at ρ = .
75 and for temperaturesranging from T = .
10 (red line) to T = .
50 (green line). (c) Radial distribution function between the pores center of mass at ρ = .
75 and for densities T = .
00 (region V) and T = .
50 (region VI). (d) Comparison between the pores g p − p (black line) andthe rescaled radial distribution function of the colloids g c − c (red line) at T = .
00 and ρ = .
65, showing that the pores packingfollows the colloid packing. T A p IV V VI (a) 0 0.5 1 1.5 T A p IV V VI (b)
Figure 8: Mean area of the pores as function of the the temperature for the isochores (a) ρ = .
65 and (b) ρ = .
75. The dashedlines indicates the separation between the regions IV, V and VI in the T × ρ phase diagram.phase diagram.