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Confinement effects on the properties of Janus dimers

Confinement has been suggested as a tool to tune the self-assembly properties of nanoparticles, surfactants, polymers and colloids. In this way, we explore the phase diagram of Janus nanoparticles using Molecular Dynamics simulations. The nanoparticle was modeled as a dimer made by one monomer that interacts by a standard Lennard Jones potential and another monomer that is modeled using a two-length scale shoulder potential. This specific design of nanoparticle exhibits in bulk distinct self-assembled structures and water-like diffusion anomaly. Our results indicate that besides the aggregates observed in bulk, new structures are observed under confinement. Also, the dynamic and thermodynamic behavior of the fluid phase are affected. The systems show a reentrant fluid phase and density anomaly. None of these two features were observed in bulk. Our results show that geometrical confinement leads to new structural, thermodynamical and dynamical behavior for this Janus nanoparticle.


I. INTRODUCTION
Colloids and molecules with anisotropic shapes and interactions play a significant role in condensed matter physics, specially in the design of self-assembled structures 1,2 . Particularly, Janus colloids are characterized as particles composed by at least two physically or chemically distinctive surfaces. They can have different shapes, as rods, spheres and dumbbells.
From the distinct Janus particles shapes, Janus dumbbells [12][13][14][15] are colloids formed by two spheres, each one with distinct characteristics linked together with a separation that varies from an almost total overlap to one or two monomer diameters. Due to the resemblance between Janus particles and competing interaction systems, Janus dumbbells behave as surfactant in water-based emulsions due its amphiphilic properties [16][17][18] . Self-assembly lamellae or micellae phases were observed on these systems due the competition between attractive and repulsive forces [19][20][21][22][23][24] .
Recent studies reported the production of silver-silicon (Ag-Si) 13 and silica-polystyrene (SiO 2 -PS) 25 Janus dimers. Silica and silicon are classified as anomalous fluids, and therefore have a set of properties that diverge from the observed for regular materials. For most part of the fluids, the diffusion coefficient decreases when the pressure (or density) increases.
However, materials as water 26 , silicon 27 and silica 28 show diffusion anomaly, characterized by a maximum in the diffusion coefficient at constant temperature. Besides diffusion (or dynamical) anomaly, water, silicon, silica and others fluids, the so-called anomalous fluids, also have other classes of anomalies, as structure and thermodynamic anomalies. Particularly, the density anomaly is characterized by the increase of density with the temperature at a fixed pressure.
Distinct computational models were proposed to study the anomalous behavior of fluids.
Among these models, effective two length scale (TLS) core-softened shoulder potential are an interesting tool to investigate systems with water-like anomalies. Particularly, the model proposed by Oliveira et. al. 29,30 reproduces qualitatively the diffusion, structural and thermodynamic anomalies, and was broadly used to study anomalous systems. This effective approach to describe anomalous fluids was used to study monomeric and dimeric systems of anomalous particles [29][30][31][32][33] . The TLS potential was used in our previous works to study the behavior of Janus dumbbells composed of one anomalous and one non-anomalous monomers in bulk 34,35 . Despite the presence of the non-anomalous monomer, the diffusion anomaly was preserved.
Confinement was proposed as a approach to tune the self-assembled morphologies. Controlling the confinement intensity it is possible to create micelles with distinct shapes. Computational and experimental studies have already explore the confinement effects if the self assembly of polyhedral nanoparticles 36 , patchy spherical colloids 37 , asymmetric and symmetric dumbbells 38,39 surfactants and polymers [40][41][42] . As well, the confinement affects the diffusivity of spherical Janus swimmers 11 . In fact, confinement strongly affects the behavior of fluids in general. For the case of anomalous fluids 43 , new anomalies can be observed due confinement 44 , and even a superdiffusive regime can be induced 45 . The anomalous region in the pressure × temperature (P T ) phase diagram is usually shifted due confinement.
This shift can be to higher or lower temperatures, regarding on the nature of the fluid-wall interaction 46 .
Therefore, the question that rises is how the confinement will affect not only the selfassembled structures, but the dynamical and thermodynamical behavior of the anomalous/nonanomalous Janus dumbbell system. In this way, we perform intensive Molecular Dynamics (MD) simulations of Janus nanoparticles composed of anomalous and non-anomalous monomers confined between two flat plates. In addition to the myriad of structures tuned by the confinement, we show how the confinement and the TLS potential lead the system to have not only diffusion anomaly, but also the density anomaly, not observed in bulk.
The paper is organized as follows: first we introduce the model and describe the methods and simulation details; next the results and discussion are given; and then we present our conclusions.

II. THE MODEL AND THE SIMULATION DETAILS
In this paper all physical quantities are computed in the standard Lennard Jones (LJ) units 47 , for distance, density of particles and time , respectively, and for the pressure and temperature, respectively, where σ, ǫ and m are the distance, energy and mass parameters, respectively. Since all physical quantities are defined in reduced LJ units, the * is omitted, in order to simplify the discussion.
The systems have N = 1000 dimeric particles, totalizing N = 2000 particles, confined between two smooth and parallel plates. The Janus dumbbells particles were modeled using two spherical core-softened particles, each one with mass m and effective diameter σ, linked rigidly at a distance λ. The dimers are formed by monomers of type A and type B.
The particles of type A present anomalous behavior and their interaction is given by a two length scales potential, potential AA, defined as 29,30 where r ij = | r i − r j | is the distance between two A particles i and j. The first term of the potential is a standard 12-6 LJ potential 47  The interaction between particles of type B, the potential BB, is given by a standard 12-6 LJ potential, like the first term of Eq. 3, cut and shifted at the cutoff radius r c , The BB potential has a cutoff radius of r c = 2.5. Meanwhile, the interaction for A-B particles is given by the Weeks-Chandler-Andersen (WCA) potential, defined by the equation 4 with r c = 2 1/6 is the cutoff. The interactions between dimers and walls are given by the projection of the WCA potential in the z-direction. The potentials are illustrated in Fig. 1. x and y-directions. The temperature was simulated in the interval between T = 0.05 and T = 0.60. The system temperature was fixed using the Langevin thermostat with γ = 1.0, and the equations of motion for the fluid particles were integrated using the velocity Verlet algorithm, with a time step δt = 0.01. We performed 1 × 10 6 steps to equilibrate the system.
These steps are then followed by 5 × 10 6 steps for the results production stage. To ensure that the system was equilibrated, the pressure, kinetic energy and potential energy were analyzed as function of time, as well several snapshots at distinct simulation times. Since confined systems can be sensitive to the number of particles in the simulation, in some points we carried out simulations with 5000 and 10000 particles, and essentially the same results were observed. As well, we run some points with a production time of 1 × 10 8 to test if the system was well equilibrated, and the same results were obtained.
The system dynamics was analyzed using the lateral mean square displacement (LMSD) as function of time, given by where r cm (t 0 ) = (x cm (t 0 ) 2 + y cm (t 0 ) 2 ) and r cm (t) = (x cm (t) 2 + y cm (t) 2 ) denote the parallel coordinate of the nanoparticle center of mass (cm) at a time t 0 and at a later time t, respectively. The LMSD is related to the lateral diffusion coefficient, D , by The pressure in confined systems by parallel plates is divided in parallel and perpendicular direction. The parallel pressure, P , was obtained from where σ xx and σ yy are the normal stress in the x and y direction.
The system structure was analyzed with the lateral radial distribution function (LRDF) g || (r), defined as 52 where δ(x) is the Dirac δ function and the Heaviside function θ(x) restricts the sum of particle pair in the same slab of thickness δz = σ. The lateral radial distribution function is proportional to the probability of finding a particle at a distance r from a referent particle inside the slab of thickness δz.
In order to check if the Janus system shows density anomaly we evaluate the temperature of maximum density (TMD). Using thermodynamical relations, the TMD can be characterized by the minimum of the pressure versus temperature along isochores, Confinement effects on the properties of Janus dimers The fluid, micellar and aggregated regions in the P T phase diagrams were defined analyzing the snapshots of the systems, the lateral diffusion coefficient, D , and the lateral radial distribution function, g (r). To define the nanoparticles in the same aggregate we defined a minimal distance equals to r min = 1.2.
If the distance between one monomer of one dimer and a monomer of a distinct dimer is smaller than r min then both dimers belong to the same cluster.

III. RESULTS AND DISCUSSION
We start our discussion showing the distinct micelles observed for the confined system.
Regarding the temperature and density, the Janus dimers aggregate in clusters with distinct number of nanoparticles per cluster, n c . At lower densities, trimeric clusters, with n c = 3 nanoparticles in each aggregate and tetrahedral clusters, with n c = 4, are more common.
Increasing the density, hexahedral clusters (n c = 6) are observed, as well spherical and elongated micelles with distinct n c . The shape of each aggregate is shown in figure 2. For densities up to a threshold a coexistence of two or three of these micelles was observed. Above the threshold, one single rippled lamellae cluster with all the nanoparticles is observed. This lamellae phase can have a disoriented structure, as shown in figure 2(viii), or an oriented structure, figure 2(ix).
The region of micelles coexistence and the threshold to the lamellae phase depend on the temperature, so lets take the example of T = 0.10. In figure 3, we show the mean number of dimers in each cluster, < n c >, as function of the system density. We can establish a relation between < n c > and the type of aggregates. For 0.05 < ρ ≤ 0.10, < n c > varies between 4 and 5. Analyzing the system snapshots, for these densities we can see a coexistence of trimeric, tetrahedral and hexahedral clusters. As ρ increases, more hexahedral aggregates and less trimeric aggregates are observed in the solution. This region, with the coexistence For the densities inside the region II we observe a mixture of tetrahedral and hexahedral clusters, in region III hexahedral and small spherical clusters and in region IV a coexistence of small spherical and elongated clusters. All these previous aggregates where observed in the bulk simulation, with a similar < n c > 34 . However, the confinement frustrates the selfassembly as the density increases. Therefore, if for lower densities the system aggregates in the same micelles observed for the bulk case, for higher densities new kinds of self-assembled aggregates should be induced by the confinement. In this way, the spherical and elongated micelles can have a higher < n c > compared to the bulk case. Basically, the limited space induced by the confinement leads two or more smaller micelles to merge in a large cluster.
This region, with spherical and elongated micelles formed by more nanoparticles than the In the figure 5 we show the qualitative P || × T phase diagram for the confined system.
The distinct micellae and lamellae regions are indicated. As discussed previously, new selfassembled structures are induced by the confinement. However, the aggregation region does not shift to higher or lower temperatures. Curiously, at the temperatures were we observe the oriented lamellae structure, for densities above ρ = 0. 45  The fluid phase also shows interesting properties, distinct from the bulk case. The first one is the density anomaly. As we can see, the isochores in figure 5 show a minimum. The dashed blue line, the so called TMD line, connects these minimum points. The density anomaly was observed for pure anomalous monomer and dimers (AA dimers) 29,31 , however was not present in the bulk Janus system 34 . The reentrant region, that occurs at higher densities, is located where the TMD line ends and splits the lammelar phase VI-B in two regions.
The bulk system has only diffusion anomaly, and it is preserved in the confined case.
However, the diffusion extrema line for the confined system, defined by the red dot-dashed red line in the phase diagram, figure 5, have a temperature range larger than in bulk.
Studies for confined anomalous fluids indicate that the TMD moves to lower temperatures for solvophobic confinement and higher temperatures for solvophilic confinement 46,59,60 . For the  bulk case, our hypothesis was that the TMD line was absorbed by the micellae region. Hence, since our confinement is solvophobic (the WCA potential is purely repulsive), is surprising that the TMD appears, moving to higher temperatures. This leads to the question: why the anomalies lines are shifted to higher temperatures?
Gavazzoni and co-authors 32 showed that the anisotropy can shift the solid-fluid phase boundary of dumbbells systems made only by A monomers. More than this, the work argues that the kinetic energy and, therefore, the temperature, has two contribution: the translational temperature and the non-translational temperature. Hence, the contribution from the dimer rotations to the kinetic energy plays a significant role in this system behavior.
In bulk, our Janus dumbbell can rotate freely in any direction -the only limitation are collisions with others dimers. However, the confinement imposes a constriction to the system.
In our strongly confined system, with L z = 4.0 and λ = 0.8, not only the translation in zdirection is limited, but the combination of confinement with the competing interactions of Janus systems lead to a interesting phenomena. As the figure 6 shows, the diffusion coefficient is distinct from zero -so, the particles are moving in the parallel direction. In the figure 8(i) we show a frontal snapshot of the nanoparticles arrangement. As we can see, the particles are disordered -as expected for fluids. Notwithstanding, the side snapshot shows that in the z-direction the particles have a preferential position: the attractive B particles stay at the center, and the repulsive A particles are near the wall. Then, due the confinement and the Janus characteristics, the dimers are translating in the xy-plane, but without rotation. This places the A monomers side by side in the xy-plane, with a internal layer of attractive B monomers. This internal layer will act similar to a solvophilic wall, with the B monomers pulling one another. As consequence, the behavior is similar to water in hydrophilic confinement, and the anomalous region shifts to higher temperatures.

IV. CONCLUSION
We report the study of confined Janus nanoparticles. This system has special interest in the design of new material using the confinement to control the self-assembled structures.
We have found a rich variety of aggregates and micelles, including large micelles not observed in the bulk system. As well, two lamellae phases, with distinct orientations, where induced by the confinement. The oriented lamellae phase region in the P T phase diagram is splitted by a reentrant fluid phase. This melting, induced by increasing the density, was not present in the bulk system as well.
Another feature that was not observed in the bulk system is the density anomaly. The combination of confinement effects with the competing interactions of Janus dimers shifts the TMD line to higher temperatures, rising the anomaly that was hide inside the aggregation region for the bulk case. As well, the diffusion anomaly region increases, reaching higher temperatures.
Our results show that materials composed by the association of a monomer which can be modeled by a two length scale potential and a standard LJ monomer have an interesting and peculiar behavior.
V. ACKNOWLEDGMENTS