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We employ molecular dynamics simulations to investigate the self-assembly of amphiphilic Janus particles in a slit-pore consisting of two plane-parallel, soft walls. The Janus particles are modeled as soft spheres with an embedded unit vector pointing from the hydrophobic to the hydrophilic hemisphere. The structure formation is analyzed via cluster size distributions, density and polarization profiles, and in-plane correlation functions. At low temperatures and densities, the dominating structures are spherical micelles, whereas at higher densities we also observe wall-induced bilayer formation. Finally, we compare the MD results with those from a previous density functional study.

The term “Janus”-particles (named after the two-faced roman god) generally refers to particles composed of at least two chemically or physically distinctive surfaces. Significant experimental progress over the last years (see, e.g., [

In the present study we focus on amphiphilic spherical Janus particles, where one hemisphere has hydrophilic properties (

We note that there exists quite a number of experimental and theoretical studies on the related problem of the self-assembly of

Indeed, in a subsequent study of the

Against this background, the goal of the present MD study is twofold. First, by choosing a “representative” slit-pore confinement and some selected state points, we aim to explore the competition between interaction-induced micelle formation and surface-induced planar ordering (as suggested by the DFT). Second, to better understand the capabilities of the DFT approach we explicitly compare MD and DFT density profiles.

The remainder of this paper is organized as follows. The model and the method of investigation are briefly described in Section 2. In Section 3 we give a brief summary of our previous MD results for the bulk properties of the Janus particles [

To describe the system of amphiphilic Janus particles, we employ a model originally introduced by Tarazona and coworkers [_{i}

where _{ij}_{i}_{j}_{ij}_{ij}

where _{B}_{B} and

where _{1} (_{ij}

The free parameters in ^{−}^{1} in this study. The present model of amphiphilic Janus particles favors antiparallel side-by-side orientation, where the hydrophobic sides point towards one another. The opposite orientation (

Spatial confinement is introduced by the presence of two planar, structureless, soft walls at the Cartesian positions _{wall} = 0 and _{wall} = _{z}. Each of these walls leads to a particle-wall potential of the form

where we fix the wall density _{wall} and wall coupling strength _{wall} such that _{wall}^{3}_{wall} = _{z} = 10

In the present work, calculations are carried out using equilibrium molecular dynamics (MD) simulations involving ^{*}_{B}^{*}^{3}.

To analyze the structure formation in the slit-pore we calculate the usual number density profile, _{z}_{z}

where _{z}

Furthermore, to investigate the lateral structure within layers of particles formed parallel to the walls, we calculate the in-plane radial distribution function defined as

In _{layer} (_{jj}_{jj}_{jj}_{jj}_{layer} is the total number of particles in the layer, and _{layer} is the corresponding area density.

Before discussing the self-assembly in the slit-pore, we briefly summarize relevant results from our previous, extensive MD study [^{*}_{agg}(^{*}^{*}

As discussed in [_{C} (_{C} of size _{C} (^{*}_{agg}(^{*}^{*}^{*}

In the following we study the behavior of the spatially confined Janus-particle system, focusing on state points below the bulk aggregation line (see _{av} = _{z} ^{−}^{1} _{0}^{L}^{z} ^{*}_{av} = 0.1 and ^{*}_{av} = 0.65. We compare these systems to bulk systems at the same (bulk) density. It is well known that this way to “relate” bulk and confined systems is somewhat ambiguous since in the confined system, the regions close to each wall are effectively inaccessible to the particle’s centers of mass. The proper way to circumvent this problem is to work in the grand canonical ensemble, where the confined system can be uniquely related to the bulk by choosing the same _{av}). However, to get a first insight into the impact of confinement, here we rather choose the simplified way described above. We also note that, for the (soft) particle-wall potential given in

This choice takes into account that the region inaccessible by the particles at each wall has (in total) approximately a thickness of one particle diameter. With this formula, the average densities ^{*}_{av} = 0.1 and ^{*}_{av} = 0.65 correspond to ^{*}_{eff} = 0.111 and 0.722, respectively. In the subsequent two paragraphs we first present MD results for these two densities. Finally, we briefly compare the MD results to corresponding ones from our previous DFT study [

Given the relatively large wall separation considered in the present study (_{z} = 10_{C}(^{*}_{C}(^{*}_{C}(^{*}^{*}_{C}^{*}

In ^{*}^{*}_{agg}) reflect an essentially homogeneous density distribution (expect directly at the walls) and the absence of any preferred alignment. A reduction of ^{*}

We note here that these findings (micelle formation) somehow contradict those in our previous DFT study [

We now consider confined systems at ^{*}_{av} = 0.65 (^{*}_{eff} = 0.722). As expected, confinement effects at this high density are much more pronounced compared to the dilute case studied before. In particular, we find that there is a small range of temperatures (around ^{*}^{*}^{*}^{*}

Apart from the vertical ordering, it is also interesting to inspect the ^{*}_{2D}(_{jj}_{2D}(_{jj}_{2D}(_{jj}

So far we have concentrated on the bilayer formation at ^{*}^{*}^{*}^{*}_{cluster}^{101}) [^{*}^{*}^{*}_{cluster}^{101} is even larger for a bulk system of icosahedrons, which leads to the assumption that micelles close to the wall are somewhat deformed with respect to an icosahedral local structure. The strong preference of this cluster type, which involves 13 particles, is also indicated by the corresponding cluster size distribution. The latter is shown in

We conclude from this section that the surfaces are capable of influencing the self-assembly of the Janus particles only in a small window of reduced temperatures (around ^{*}_{B}_{B}

As mentioned before, we have previously investigated the self-assembly at planar surfaces and in slit-pores via classical density functional theory [

Examples are plotted in ^{*}_{av} = 0.1 (as before) whereas in the DFT, ^{*}_{av} = 0.107 (0.096) (corresponding to ^{*}_{bulk;DFT} = 0.12 (0.1)) for ^{*}_{z} = 9

At high temperatures above the aggregation line, such as ^{*}^{*}^{*}

A comparison of the density profiles at a higher density is given in ^{*}_{av} = 0.65 (^{*}_{av} = 0.64). For each method, we focus on temperatures where bilayer formation occurs. Indeed, given that our DFT involves a mean-field approximation for the anisotropic interactions and that we allow for planar structures alone, it is not surprising that this approach predicts bilayers already at much larger reduced temperatures than the MD. Specifically, the DFT results in ^{*}^{*}

Inspection of

The main purpose of this paper was to present MD simulation results for the self-assembly of amphiphilic Janus particles in a slit pore. We focused on the case of a relatively wide pore, characterized by a wall separation of ten (effectively nine) particle diameters, and neutral walls. At small average densities, the impact of the surfaces is found be quite minor. Indeed, we observe essentially the same structures, that is, spherical micelles, as in the bulk system at the same density and temperature (

Contrary to the low-density case, confined Janus-particles systems at high average densities can display structures strongly different from their bulk counterpart. Indeed we found that, within a small range of reduced temperatures, the surfaces stabilize bilayer structures characterized by a high degree of polarization relative to the walls. Such structures are absent in a bulk system at the same temperatures and (average) density. At even lower temperatures, however, the anisotropic particle-particle interactions become again dominant, and one observes formation of icosahedral micelles, consistent with the bulk. This reentrant behavior indicates that dense, confined Janus-particle systems are subject to a strong competition between planar structures preferred by the surfaces and non-planar (micellar) structures induced by particle-particle interactions.

In the last part of the paper, we have compared MD density profiles of the confined systems to density profiles obtained by classical DFT [

Clearly, the present MD study has rather exploratory character in the sense that we considered only one wall separation, and only one type of surfaces here. Regarding the wall separation, it would be interesting to explore, on the one hand, the case of larger _{z} where both bilayers at the walls and micelles in between can be formed. On the other hand, for wall separations of the order of the particle diameter, our previous DFT study [

We gratefully acknowledge financial support from the DFG via the International Research Training Group 1524 “Self-Assembled Soft Matter Nano-Structures at Interfaces” (project B1.1). Gerald Rosenthal also thanks Professor Keith E. Gubbins for the kind hospitality during his stay at the North Carolina State University.

Aggregation line of the bulk system determined from the cluster size distribution. The line connecting the data points is a guide for the eye. The inserted sketches indicate typical cluster shapes at low and higher densities, respectively.

Cluster size distribution _{C}(^{*}_{av} = 0.1 (^{*}_{eff} = 0.111); and (^{*}

Snapshot from MD simulations at ^{*}_{av} = 0.1 and ^{*}

(^{*}_{av} = 0.1 and three temperatures ^{*}

Snapshots from MD simulations involving soft walls (separation _{z} = 10^{*}_{av} = 0.65 (^{*}_{eff} = 0.722) and ^{*}

(^{*}_{av} = 0.65 for various temperatures ^{*}

In-plane radial distribution function _{2D}(_{jj}^{*}_{av} = 0.65 and ^{*}

(^{*}_{av} = 0.65 (^{*}_{eff} = 0.722) and ^{*}_{C}(^{*}_{av} = 0.65 and ^{*}

Comparison of DFT and MD density profiles at (^{*}_{av} = 0.1 (MD); and (^{*}_{av} = 0.65 (MD). For the DFT parameters, see main text.