# Micelle and Bilayer Formation of Amphiphilic Janus Particles in a Slit-Pore

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Model and Method

**û**

_{i}pointing from the hydrophobic to the hydrophilic hemisphere of each particle i. Following our former MD study of the bulk system [25], the pair interaction between two Janus particles then splits into a soft-sphere repulsion and an effective, solvent-mediated anisotropic pair interaction, that is

**r**

_{ij}=

**r**

_{i}–

**r**

_{j}is the distance vector between the particles, and r

_{ij}= |

**r**

_{ij}|. The soft sphere potential is defined as

_{B}T/ɛ = 1, with k

_{B}and T being Boltzmann’s constant and temperature, respectively. The anisotropic pair interaction potential reads, to lowest order (see [18,25])

_{1}(r

_{ij}) is a Yukawa potential given as

^{−}

^{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 (i.e., facing hydrophilic sides) is energetically most unfavorable, while parallel side-by-side and head-to-tail configurations are energetically neutral due to a vanishing anisotropic pair interaction (cf. Equation (3)). A more detailed discussion of the angle dependence and a comparison to other Janus models can be found in [23,25].

_{wall}= 0 and z

_{wall}= L

_{z}. Each of these walls leads to a particle-wall potential of the form

_{wall}and wall coupling strength ɛ

_{wall}such that ρ

_{wall}σ

^{3}ɛ

_{wall}= ɛ. In the present study we consider the case L

_{z}= 10σ. Also, we restrict ourselves to the discussion of neutral walls, thus focusing on the effect of confinement alone. In principle, however, it is straightforward to introduce surface potentials describing hydrophilic or hydrophobic walls; we have suggested corresponding potentials in our earlier DFT study [23] of confined Janus systems.

^{*}= k

_{B}Tσ/C and the reduced density ρ

^{*}= ρσ

^{3}.

_{z}〉 /(AΔz) (with A being the box area parallel to the walls and Δz being the thickness of the slice containing N

_{z}particles). In addition, we calculate an order parameter function characterizing the local “polarization”, that is

**ê**

_{z}is a unit vector pointing along the z-axis. It follows from Equation (6) that h (z) can vary between −1 and 1, with −1 (+1) meaning that the hydrophilic (hydrophobic) sides of all particles point towards the left wall.

_{layer}(r

_{jj},Δr

_{jj}) is the number of particles in a ring of width Δr

_{jj}at a distance r

_{jj}within the layer considered. Further, N

_{layer}is the total number of particles in the layer, and ρ

_{layer}is the corresponding area density.

## 3. Background: Aggregation in the Bulk System

^{*}

_{agg}(ρ

^{*}), which denotes (for each density ρ

^{*}) the temperature below which significant cluster formation occurs. Interestingly, we found no indication of a conventional condensation transition or coexistence between clustered phases (such as it has been predicted for another Janus model, see [16].

_{C}(s), i.e., the number of clusters N

_{C}of size s. For each temperature and density, this distribution is determined on the basis of a cluster search algorithm. At high temperatures, N

_{C}(s) decays essentially exponentially, indicating a random distribution of cluster sizes. Upon lowering the temperature, however, one observes the emergence of a minimum at s > 1, indicating the existence of bound clusters [27]. The bulk aggregation line T

^{*}

_{agg}(ρ

^{*}) derived from that criterion is plotted in Figure 1. At low densities (ρ

^{*}≤ 0.3), the typical cluster sizes are between 5 and 10 [25]. Specifically, the Janus particles aggregate into spherical micelles where the hydrophilic sides of each particle is facing away from the cluster center (see the left sketch in Figure 1). For higher densities (0.3 < ρ

^{*}≤ 0.8), the cluster size distributions exhibits a pronounced peak at s = 13. As shown in [25] (via an analysis of correlation functions), these structures correspond to icosahedrons, i.e., close-packed aggregates which cannot be periodically continued to give a translationally ordered lattice [28,29]. Interestingly, the formation of these icosahedrons is accompanied by hindered translational and orientational dynamics, as reflected by various MD time correlation functions [25].

## 4. Confined Systems

_{av}= L

_{z}

^{−}

^{1}∈

_{0}

^{L}

^{z}dz ρ(z) takes the values ρ

^{*}

_{av}= 0.1 and ρ

^{*}

_{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 chemical potential (rather than the same ρ

_{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 Equation (5), one could define an effective density via

^{*}

_{av}= 0.1 and ρ

^{*}

_{av}= 0.65 correspond to ρ

^{*}

_{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 [23].

#### 4.1. Low Densities

_{z}= 10σ), one would expect the confinement effects to be relatively weak as long as the average density is low. That this is indeed the case can be seen, e.g., from the cluster size distributions N

_{C}(s). Results for this quantity in the confined system and in the bulk are given in Figure 2, where we consider some characteristic temperatures. At T

^{*}= 1.0, both systems are essentially homogeneous as indicated by the monotonic decay of N

_{C}(s). Bulk system starts to aggregate into spherical micelles at T

^{*}≈ 0.2 (see also Figure 1) as indicated by the appearance of a minimum in N

_{C}(s). This minimum is reproducible, as we have explicitly checked by performing several runs [25]. Interestingly, the corresponding function of the confined system is still monotonic at T

^{*}= 0.2, indicating that the confinement somewhat hinders the micelle formation. At the substantially lower temperature T

^{*}= 0.14, however, both functions clearly indicate the presence of (micellar) clusters of size 5 – 9, with an average of 〈N

_{C}〉 ≈ 7. Moreover, the main peak is of similar magnitude; this holds also for the even lower temperature T

^{*}= 0.1. To illustrate the structure of this strongly coupled, confined system we present in Figure 3 a corresponding simulation snapshot. It is seen that the particles tend to aggregate into micellar clusters in the middle of the pore, rather than at the surfaces.

^{*}= 1.0 ≫ T

^{*}

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

^{*}results in a shift of the contact peaks of ρ(z) towards the center of the pore (see Figure 4a). This is a consequence of the aggregation also seen in the snapshot (cf. Figure 3): As soon as micelles have formed, their centers of mass are located at larger distances from the wall than what one would find in a non-aggregating system. At the same time, the development of negative (left) and positive (right) peaks in the order parameter profile plotted in Figure 4b reflects that the hydrophilic sides of the particles tend to be close to the walls. This is consistent with the preferred order within the micelles (where the hydrophilic sides of the particles point outwards). In other words, under the dilute conditions studied here, there seems to be no real competition between the structures favored by the particle interactions and those favored by the wall.

#### 4.2. High Densities

^{*}

_{av}= 0.65 (ρ

^{*}

_{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 T

^{*}= 0.14), where the Janus particles form bilayers at the wall, contrary to the bulk system at the same density and temperature. In the latter, the preferred structure is the (spherical) icosahedron (see Figure 1). A snapshot of the confined system is shown in Figure 5a. From this snapshot one already sees that the bilayers are highly polarized in the sense that the particles in each single layer point outwards of the bilayer. These phenomena are also reflected by the density and polarization profiles, which are plotted (for various temperatures) in Figure 6. At the (somewhat higher) temperature T

^{*}= 0.3, where the bilayers are not yet formed, the density profile (cf. Figure 6a) has the typical oscillatory shape reflecting layer formation of a dense fluid in a slit-pore confinement. The corresponding polarization profile, on the other hand, already indicates a preferential orientation of the Janus particles close to the walls (cf. Figure 6b). Both profiles significantly change when we now consider the case T

^{*}= 0.14. Here, the formation of bilayers (see Figure 5a) leads to two sharp density peaks close to each wall. Further, we observe a clear depletion area at distances between the double layer and the pore center. Finally, the regular layered structure observed in the pore center at higher temperatures appears to be disturbed at T

^{*}= 0.14. The corresponding polarization profile supports our former statement that the bilayers are highly polarized with the hydrophilic particle sides pointing outwards (cf. Figure 6b). One also sees from h(z) that the particles related to the two density peaks around the pore center are highly polarized. We interpret this behavior as a signal of the formation of spherical clusters, consistent with what is seen in Figure 5a.

^{*}= 0.14. Indeed, as shown in part (b) of the snapshot in Figure 5, the particles arrange into a highly ordered (yet not perfect) configuration with hexagonal-like symmetry. In this two-dimensional crystal-like structure, the positions of the particles in the two single layers are shifted relative to one another. These features are also reflected by the corresponding in-plane correlation functions, which are plotted as a function of the lateral distance in Figure 7. Specifically, we show the functions g

_{2D}(r

_{jj}) for each of the individual layers forming the bilayer, as well as for the complete bilayer. Considering first the two single-layer correlations, we find from Figure 7 that these are essentially indistinguishable. In other words, the internal structure within each layer is identical. To interpret the shape of these correlations, we note that the function g

_{2D}(r

_{jj}) of a perfect hexagonal lattice would display peaks at 1σ, $\sqrt{3}\sigma $ and 2σ. In the present, thermal system, we observe one peak at approximately 1.1σ and a broad maximum between approximately 2σ to 2.3σ (for each single layer). This “softening” (as compared to the perfect lattice) reflects the presence of thermal motion and lattice defects, which are also apparent from the snapshot in Figure 5b. To complete the discussion of the lateral structure, we note from Figure 7 that the g

_{2D}(r

_{jj}) evaluated for the entire bilayer clearly differs from that within each layer. This difference reflects the relative shift of the two layers along a direction parallel to the wall.

^{*}= 0.14. We stress again that these structures are purely surface-induced in the sense that the corresponding bulk system forms not planar, but rather spherical, specifically icosahedron structures. Interestingly, it turns out that this behavior is recovered also in the confined system, when we further lower the temperature (we recall in this context that our dimensionless temperature T

^{*}measures the coupling strength of the anisotropic particle interactions). This change becomes apparent from Figure 8a, where we show a snapshot of the entire confined system at T

^{*}= 0.1. Instead of bilayers (or any clearly defined layers at all), we find that the particles indeed arrange into icosahedrons (see circle in Figure 8a). Further, the density profile changes such that the peaks close to the walls become somewhat smaller and give rise to two additional peaks reflecting the different aggregation (cf. Figure 6a). We note that the density profile does not possess depletion areas at T

^{*}= 0.1. An additional order parameter (G

_{cluster}

^{101}) [25] indicating the orientation of hydrophilic sides relatively to the cluster center shows a significantly smaller value than 1 at T

^{*}= 0.14 and a value of 0.77 at T

^{*}= 0.1. This means that aggregates at T

^{*}= 0.1 are more spherical in average. We note that G

_{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 Figure 8b.

^{*}= k

_{B}Tσ/C ≈ 0.14). Indeed, when the particle-particle interactions become even stronger against the thermal energy (C/k

_{B}Tσ ≥ 10), we observe “reentrant” bulk behavior, that is, formation of micelles.

#### 4.3. Comparison to Density Functional Theory

^{*}

_{av}= 0.1 (as before) whereas in the DFT, ρ

^{*}

_{av}= 0.107 (0.096) (corresponding to ρ

^{*}

_{bulk;DFT}= 0.12 (0.1)) for T

^{*}= 1.0 (0.1). We note in this context that, in order to approximately match the positions of the density peaks close to the walls, the DFT calculations have been performed with a wall separation of L

_{z}= 9σ rather than 10σ as in the MD calculations.

^{*}= 1.0, there is rather good agreement between MD and DFT calculations. Both methods predict an essentially homogeneous distribution of particles in the pore, accompanied by a nearly vanishing polarization profile (not shown). However, pronounced differences appear at low temperatures such as T

^{*}= 0.1. Here, the density peak positions predicted by MD are shifted (relative to the high-temperature case) towards the center of the system, reflecting the formation of spherical clusters preferably in the middle of the pore (see also Figures 3 and 4 discussed in Section 4.1). Contrary to the MD profiles, the DFT profiles at T

^{*}= 0.1 are characterized by a double peak close to each wall (located at z = 1.2σ and 2σ), possibly indicating the beginning of formation of a bilayer. We stress, however, that our DFT calculations are purely one-dimensional and thus do not allow for detection of a spherical clusters. In view of the MD results this is clearly a severe limitation of the DFT calculations when applied to low density Janus systems.

^{*}

_{av}= 0.65 (ρ

^{*}

_{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 Figure 9b pertain to T

^{*}= 0.28, whereas those from MD correspond to T

^{*}= 0.14 (see Figure 5 for a snapshot).

## 5. Conclusions

_{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 [23] suggests interesting frustration effects. Regarding the type of surface, it would be very interesting to consider the case of additional surface fields, such as surfaces with hydrophilic or hydrophobic coatings [20]. Indeed, based on our previous DFT study of dense suspensions at such surfaces [23], we would expect that hydrophilic surfaces strongly stabilize bilayer formation, whereas hydrophobic walls could enforce totally different structures. Another open question concerns the impact of curved surfaces. An example is the inner surfaces of the (typically cylindrical) pores of an ordered porous material. This question has already also been investigated in the context of the self-assembly of amphiphilic molecules [32,33]. We expect a similar competition between surface- and interaction-induced ordering for Janus particles at curved surfaces. Finally, it would be very interesting to investigate the role of the size of the hydrophilic versus that of the hydrophobic part of the Janus particles for their self-assembly in confinement. That these particle-related details could be important has already been suggested in a previous theoretical investigation for bulk systems, yet on the basis of a different model [15].

## Acknowledgments

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**Figure 1.**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.

**Figure 2.**Cluster size distribution N

_{C}(s) (with s being the cluster size) and various temperatures for (

**a**) the confined system at ρ

^{*}

_{av}= 0.1 (ρ

^{*}

_{eff}= 0.111); and (

**b**) the bulk system at ρ

^{*}= 0.1.

**Figure 4.**(

**a**) Density profiles; and (

**b**) polarization profiles at ρ

^{*}

_{av}= 0.1 and three temperatures T

^{*}.

**Figure 5.**Snapshots from MD simulations involving soft walls (separation L

_{z}= 10σ) at ρ

^{*}

_{av}= 0.65 (ρ

^{*}

_{eff}= 0.722) and T

^{*}= 0.14. (

**a**) Side view of the entire confined system; (

**b**) top view onto one bilayer.

**Figure 6.**(

**a**) Density profiles ρ(z); and (

**b**) order parameter functions h(z) at ρ

^{*}

_{av}= 0.65 for various temperatures T

^{*}.

**Figure 7.**In-plane radial distribution function g

_{2D}(r

_{jj}) evaluated for the single layers and the complete bilayer at ρ

^{*}

_{av}= 0.65 and T

^{*}= 0.14.

**Figure 8.**(

**a**) MD simulation snapshot of the confined system (side view) at ρ

^{*}

_{av}= 0.65 (ρ

^{*}

_{eff}= 0.722) and T

^{*}= 0.1. The black circle indicates an icosahedron; (

**b**) Cluster size distribution N

_{C}(s) at ρ

^{*}

_{av}= 0.65 and T

^{*}= 0.1.

**Figure 9.**Comparison of DFT and MD density profiles at (

**a**) ρ

^{*}

_{av}= 0.1 (MD); and (

**b**) ρ

^{*}

_{av}= 0.65 (MD). For the DFT parameters, see main text.

© 2012 by the authors; licensee Molecular Diversity Preservation International, Basel, Switzerland. This article is an open-access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

## Share and Cite

**MDPI and ACS Style**

Rosenthal, G.; Klapp, S.H.L.
Micelle and Bilayer Formation of Amphiphilic Janus Particles in a Slit-Pore. *Int. J. Mol. Sci.* **2012**, *13*, 9431-9446.
https://doi.org/10.3390/ijms13089431

**AMA Style**

Rosenthal G, Klapp SHL.
Micelle and Bilayer Formation of Amphiphilic Janus Particles in a Slit-Pore. *International Journal of Molecular Sciences*. 2012; 13(8):9431-9446.
https://doi.org/10.3390/ijms13089431

**Chicago/Turabian Style**

Rosenthal, Gerald, and Sabine H. L. Klapp.
2012. "Micelle and Bilayer Formation of Amphiphilic Janus Particles in a Slit-Pore" *International Journal of Molecular Sciences* 13, no. 8: 9431-9446.
https://doi.org/10.3390/ijms13089431