# Effect of Geometric and Chemical Anisotropy of Janus Ellipsoids on Janus Boundary Mismatch at the Fluid–Fluid Interface

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^{†}

## Abstract

**:**

_{c}) required to preserve the pinned configuration was inversely proportional to the values of α and AR. From the numerical calculations, the empirical relationship of the parameter values of Janus ellipsoids was found; that is, $\lambda =\Delta {\beta}_{c}/\Delta \alpha \approx 0.61AR-1.61$. Particularly for the Janus ellipsoids with AR > 1, the β

_{c}value is consistent with the boundary between the tilted only and the tilted equilibrium/upright metastable region in their configuration phase diagram. We believe that this work performed at the single particle level offers a fundamental understanding of the manipulation of interparticle interactions and control of the rheological properties of particle-laden interfaces when particles are used as solid surfactants.

## 1. Introduction

## 2. Theoretical Basis

_{I}) and a particle immersed in the aqueous phase (E

_{w}) are

_{ij}is the surface area (i = P (polar) or A (apolar)) exposed to a fluid phase (j = w (water) or o (oil)), γ

_{ij}is the corresponding surface energy of surface i and fluid j, ${S}_{I}^{\left(2\right)}$ and ${S}_{I}^{\left(1\right)}$ are the surface area values of the oil–water interface when particles are present and absent, respectively, and ${S}_{I}={S}_{I}^{\left(2\right)}-{S}_{I}^{\left(1\right)}$ indicates the displaced area of the interface due to presence of the particle. The displaced area (S

_{I}) corresponds to the cross-sectional area of the particle at α

_{v}, which is the elevation angle measured from the major axis to the fluid interface (i.e., the three phase contact line), as shown in Figure 1a. ${S}_{A}^{tot}$ and ${S}_{P}^{tot}$ are the total surface area values of the apolar and polar regions of the particle, respectively. Substitution of the following Young’s Equations:

_{w}and E

_{o}); therefore, the shapes of the attachment energy profiles obtained from the equations should be identical. Since the use of either Equation (6) or (7) predicts the same configurational behavior, Equation (6) is used in this work. To calculate the surface area S

_{ij}, a numerical method using the hit-and-miss Monte Carlo method is employed [16]. The oil–water interfacial tension value is arbitrarily designated as γ

_{ow}= 50 mN/m, corresponding to the interfacial tension of a decane–water interface.

_{v}. To evaluate whether the Janus boundary of the particles is pinned or unpinned at the oil–water interface, the attachment energy is minimized to obtain the lowest energy minimum, $\Delta {E}_{att}^{min}\left({\alpha}_{v}\right)$, and the corresponding vertical position , α

_{v,min}.

_{v}) as well as the orientation angle (0° < θ

_{r}< 90°), in which θ

_{r}= 0° and 90° indicate the geometries of the Janus boundary parallel and perpendicular to the interface, respectively. In this case, the attachment energy is scanned by varying the α

_{v}values at a constant θ

_{r}. The minimum attachment energy is then calculated at the given θ

_{r}value, and the same procedure is repeated for different values of θ

_{r}to find a global energy minimum, $\Delta {E}_{att}^{min}\left({\alpha}_{v},{\theta}_{r}\right)$.

## 3. Results and Discussion

_{v,min}) and unpinning (α ≠ α

_{v,min}) behaviors of the Janus particles with respect to the interfaces can be determined through two competing factors: geometric and chemical anisotropy. To minimize the attachment energy in Equation (6), a Janus particle tends to be aligned toward increasing the displaced area (S

_{I}) at the oil–water interface, simultaneously increasing the surface area of the preferred wetting state (S

_{Ao}; note that $\mathrm{cos}{\theta}_{A}<0$). The state of preferred wetting indicates the configuration of the apolar and polar regions exposed to their favorable fluid phases, oil and water, respectively. Depending on the relative influence of these two factors, the particle is likely to preferentially adopt the pinned configuration when wettability effects are greater than geometric effects, whereas particles are unpinned from the interface when geometric effects are dominant. Notably, Janus prolates can adopt the two configurations, the upright and tilted one (e.g., when the AR value is high and wettability is moderate) [16]. In this particular case, the attachment energy may be further decreased by rotating the particle at the interface. We also examine the relationship between the pinning/unpinning and the upright/tilted configuration behaviors.

_{v}at constant values of β and AR = 0.5 (oblate), 1 (sphere), or 2 (prolate). Since the supplementary wettability (β) is used and Janus particles with α ≤ 90° are considered, $\Delta {E}_{Iw}$ is calculated in the range of $0\xb0\le {\alpha}_{v}\le 90\xb0$. Note that similar results can be obtained in the case of Janus ellipsoids with α > 90°.

_{v,min}= 90°, as shown in Figure 2a–c. The result of α = α

_{v,min}indicates that the Janus boundary of the geometrically symmetric Janus ellipsoids is always pinned at the oil–water interface, regardless of the values of AR and β. The pinned configuration of the particles simultaneously satisfies both conditions; that is, the particles tend to possess maximum values of the displaced area (S

_{I}) and preferred wetting (S

_{Ao}) in Equation (6) when α = α

_{v,min}.

_{r}= 0.

_{v,min}at $\Delta {E}_{att}^{min}$ gradually decreases with β, and is not consistent with α if the value of β is not sufficiently high. The condition of α

_{v,min}= α indicates that a pinned geometry is found at a critical value of β, in which β

_{c}= 67°. Similar results are found for spheres with AR = 1 (Figure 3b) and prolates with AR = 2 (Figure 3c) when α = 60°, in which the critical wettability values are β

_{c}= 30° and 8°, respectively. The increase in β

_{c}for lower AR particles is due to the relatively large displaced area (S

_{I}) around the central regions of the particles; thus, the geometric effects become stronger than the wettability effects. Note that the value of β

_{c}= 30° for the Janus sphere obtained from the numerical calculation is consistent with the theoretical prediction of ${\beta}_{c}=90\xb0-\alpha $, validating the numerical method.

_{r}are shown in Figure 3e, and the corresponding energy minima are indicated by red and green arrows. Unlike the Janus prolates with α = 90°, it is interesting to note that the tilted only region is found at the conditions of relatively high AR and low β values. The presence of the tilted only region is likely due to the presence of the unpinning configuration of the particles; that is, when the prolate particles are unpinned, no energy minimum exists at θ

_{r}= 0, and consequently, the particles only adopt the tilted configuration. In contrast, when the Janus boundary of prolate particles are pinned to the interface in Figure 3c, they essentially possess an energy minimum at θ

_{r}= 0, leading to the upright/tilted coexisting regions (yellow and pink areas in Figure 3d) or the upright only region (light blue in Figure 3d). Note that the β

_{c}= 8° value for the Janus prolate with AR = 2 in Figure 3c shows a good agreement with the boundary between the light green and yellow regions in the Figure 3d.

_{v,min}consistently decreases as β increases, and does not match the value of α = 30° up to β = 50°. Due to large values of S

_{I}around the central regions of the oblate, geometric effects are likely to be dominant to wettability effects, resulting in particles with central regions located at the interface, adopting an unpinned configuration. For Janus ellipsoids with lower AR values, wettability effects gradually become significant. As shown in Figure 4b,c, the Janus boundaries of Janus ellipsoids with AR = 1 and 2 are pinned to the interface when the critical wettability value is β

_{c}= 60° or 22°. Note that the value of β

_{c}increases as the particles become more geometrically asymmetric (β

_{c}= 30° and 8° for particles (α = 60°) with AR = 1 and 2, respectively).

_{r}, representing each configuration region and the location of energy minima, are shown in Figure 4e. The tilted only region (light green area) in relatively high AR and low β values corresponds to the condition where the Janus boundary of particles is unpinned. For particles with AR = 2, for instance, the boundary between the light green and the yellow regions in Figure 4e is consistent with the critical wettability value, β

_{c}= 22° in Figure 4c. Except for the tilted only region, a portion of the particles always adopts the upright configuration due to the presence of an energy minimum at the pinned geometry in Figure 4c.

_{c}(due to wettability effects) is inversely proportional to the values of α and AR, which are due to geometric effects. As the values of AR decrease, the particles demonstrate reduced attachment energy when the interface is located at the central regions of the particles. When the value of α decreases (the particles become more geometrically asymmetric), a higher value of β is required to adopt a pinned configuration. More quantitatively, to find a relationship of β

_{c}, α, and AR, the β

_{c}values are plotted as a function of α, as shown in Figure 5b. Then, slopes ($\lambda =\Delta {\beta}_{c}/\Delta \alpha $) obtained from linear regression of each line representing different AR values are shown in the inset plot in Figure 5b, in which the slope ($\Delta \lambda /\Delta AR$) is found to be ~0.61 ± 0.06. Based on the values of $\lambda =-1$ when AR = 1, an empirical relationship for Janus ellipsoids is obtained, $\lambda =\Delta {\beta}_{c}/\Delta \alpha \approx 0.61AR-1.61$.

_{c}is unpinned, they rotate and adopt the tilted configuration to further decrease their attachment energy by increasing the displaced area (S

_{I}) [16,17] and, consequently, all portions of the particles would adopt the tilted configuration. In this case, the β

_{c}value corresponds to the boundary between two configuration regions—the tilted only and the tilted equilibrium/upright metastable regions.

## 4. Conclusions

_{c}corresponded to the boundary between the tilted only and the tilted equilibrium/upright metastable regions in their configuration phase diagram. In the case of non-supplementary wettability where $90\xb0-{\theta}_{P}\ne {\theta}_{A}-90\xb0$, the pinning and unpinning behaviors are likely similar to the case of supplementary wetting when the orientation angle is not considered (AR ≤ 1). For Janus prolates that can adopt tilted configurations, it was reported that the relative strength of θ

_{P}and θ

_{A}significantly affected the tilted angle and the configuration behaviors [17]. We believe that this work offers fundamental ideas with regard to the attachment and configurational behaviors of nonspherical Janus particles that consequently enable manipulation of interparticle interactions and control of rheological properties of interfaces when used as solid surfactants.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**Schematics of a Janus ellipsoid for attachment energy calculations. (

**a**) Geometric relationships of the Janus ellipsoid; (

**b**) Three-phase contact angles of homogeneous polar and apolar spheres at the oil–water interface; (

**c**) Schematic illustration of Janus ellipsoid attachment to the oil–water interface.

**Figure 2.**Attachment energy profiles of Janus ellipsoids with α = 90° and (

**a**) Aspect ratio (AR) = 0.5 (oblate); (

**b**) 1 (sphere); or (

**c**) 2 (prolate). In all cases, the Janus boundary is pinned at the oil–water interface; (

**d**) Configuration phase diagram of Janus prolates as functions of the AR and β values; (

**e**) The attachment energy profiles of Janus prolates as a function of the orientation angle (θ

_{r}).

**Figure 3.**Attachment energy profiles of Janus particles with α = 60° (vertical dashed line) and (

**a**) AR = 0.5; (

**b**) 1; or (

**c**) 2. Pink and blue regions in each plot represent unpinned and pinned configurations, respectively. Green circles denote the lowest energy minimum ($\Delta {E}_{att}^{min}$) and the corresponding value of α (α

_{v,min}). The effect of β on α

_{v,min}is shown on the right; (

**d**) Configuration phase diagram of Janus prolates as functions of the AR and β values; (

**e**) The attachment energy profiles of Janus prolates as a function of the orientation angle (θ

_{r}).

**Figure 4.**Attachment energy profiles of Janus particles with α = 30° (vertical dashed line) and (

**a**) AR = 0.5; (

**b**) 1; or (

**c**) 2. Green circles denote the values of $\Delta {E}_{att}^{min}$ and α

_{v,min}. The relationship between β and α

_{v,min}is shown on the right; (

**d**) Configuration phase diagram of Janus prolates as functions of the AR and β values; (

**e**) The attachment energy profiles of Janus prolates as a function of the orientation angle (θ

_{r}).

**Figure 5.**Critical wettability values (β

_{c}) as functions of AR and α. (

**a**) Plot of β

_{c}versus AR. The regions above and below each curve correspond to the pinned and unpinned configurations, respectively; (

**b**) Plot of β

_{c}versus α. The inset indicates the slope of each line in the plot (β

_{c}versus α) as a function of AR.

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**MDPI and ACS Style**

Kang, D.W.; Ko, W.; Lee, B.; Park, B.J. Effect of Geometric and Chemical Anisotropy of Janus Ellipsoids on Janus Boundary Mismatch at the Fluid–Fluid Interface. *Materials* **2016**, *9*, 664.
https://doi.org/10.3390/ma9080664

**AMA Style**

Kang DW, Ko W, Lee B, Park BJ. Effect of Geometric and Chemical Anisotropy of Janus Ellipsoids on Janus Boundary Mismatch at the Fluid–Fluid Interface. *Materials*. 2016; 9(8):664.
https://doi.org/10.3390/ma9080664

**Chicago/Turabian Style**

Kang, Dong Woo, Woong Ko, Bomsock Lee, and Bum Jun Park. 2016. "Effect of Geometric and Chemical Anisotropy of Janus Ellipsoids on Janus Boundary Mismatch at the Fluid–Fluid Interface" *Materials* 9, no. 8: 664.
https://doi.org/10.3390/ma9080664