# Numerical Investigation of the Effect of Symmetry on Evaporation Triggered Elastocapillary Top-Gathering of High Aspect Ratio Micropillars

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## Abstract

**:**

## 1. Introduction

## 2. Numerical Method

## 3. Results and Discussion

## 4. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**The interaction energy between two micropillars. (

**a**) Schematic showing the capillary meniscus around two micropillars partially immersed in water, forming a contact angle of ‘$\theta $’. Various spacing of neighboring micropillars leads to the capillary rise on the sides of the micropillar of varying heights, resulting in an unequal force distribution. (

**b**) Schematics showing two partially immersed vertical micropillars in water. The dark blue circles around the micropillars are the effective zone of the capillary force and the top right one shows the capillary forces (${F}_{d1}$ and ${F}_{d2}$ ) acting on their opposite sides. Effective zone of the capillary force by neighbor micropillar is from 300° to 60°, considering the angles measured from the positive x-axis with positive values for counterclockwise. (

**c**) Meshing, and force distribution (of 100%–30%) along the circumference of the micropillar (diameter of $d$ = 0.75 μm and height of $h$ = 9 μm) used for simulating the deflection, due to the capillary forces. (

**d**) Shows the deflection of the micropillar for different number of segments indicating that the number of segments is negligible.

**Figure 2.**(

**a**) Schematic shows a water droplet placed on an array of micropillars in a Wenzel state (

**b**) Meshed array of micropillars in COMSOL Multiphysics. Height and diameter of each micropillar are 9 µm and 0.75 µm, respectively.

**Figure 3.**(

**a**) 2D (X-Y only) and (

**b**) 3D (X-Y-Z) are COMSOL simulation results. The micropillars height is $h$ = 9 µm and their diameter is $d$ = 0.75 µm. The top right of each micropillar displays the amount of its deflection in µm which are rounded to the hundredths place. (

**c**) Shows negligible effect of effective capillary force zone. The green line shows the deflection of the micropillar for effective zone of the capillary force extended from 285° to 75°, which is close to that of 300° to 60° (red one). (

**d**) Shows the increase of the micropillar deflection with an increase in asymmetric force distribution as well as negligible effect of the Z component of the capillary force on the micropillar deflection.

**Figure 4.**(

**a**) Plot showing the effect of elastic modulus on the micropillar deflection for partially immersed four micropillar configuration for different combinations of capillary force distribution percentage (${F}_{{d}_{1}}$–${F}_{{d}_{2}}$). (

**b**) The deflection values plotted against the force distribution (instead of elastic modulus, $E$). (

**c**) Plot showing the effect of micropillar geometry (diameter, $d=2r$ and pitch, $p$) on the capillary force percentage ($x$). The capillary interaction force decreases with increasing pillar separation distance ($p$ ). (

**d**) The zoomed in portion of the plot in (

**c**) for $p/d$ ranging between 1 and 5.

**Figure 5.**Comparison between numerical simulations of this study and experimental/theoretical work of others in prediction of collapsing behavior of a micropillar ($h$ = 150 µm, $d$ = 20 µm, and $E$ = 5.5 MPa with varying pitches). The simulation results agree with the experimental [11] and the theoretical data [24]. The green and red sections indicate stable (non-collapsed) and unstable (collapsed) micropillars, respectively.

**Figure 6.**The effect of symmetry on micropillar stability. (

**a**) Simulation results showing the progressive top-gathering phenomenon during water evaporation (

**b**) Top-view of the micropillars deflection. The micropillars height is $h$ = 9 µm and their diameter is $d$ = 0.75 µm. Max number next to the micropillars displays the maximum amount of the deflection in µm. The labeled numbers on the axis are in µm directing the coordinates of the micropillars. The amount of deflection is shown by the bar on the right side of the picture. As it is evident, at the onset of top-gathering, side micropillars initiate the top-gathering process which is due to asymmetric force distribution. As the evaporation progresses the final stage shows the complete top-gathering of the array which is compatible with the experimental data reported in [37].

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**MDPI and ACS Style**

Barghi Golezani, F.; Kishore Annavarapu, R.; Sojoudi, H.
Numerical Investigation of the Effect of Symmetry on Evaporation Triggered Elastocapillary Top-Gathering of High Aspect Ratio Micropillars. *Coatings* **2023**, *13*, 292.
https://doi.org/10.3390/coatings13020292

**AMA Style**

Barghi Golezani F, Kishore Annavarapu R, Sojoudi H.
Numerical Investigation of the Effect of Symmetry on Evaporation Triggered Elastocapillary Top-Gathering of High Aspect Ratio Micropillars. *Coatings*. 2023; 13(2):292.
https://doi.org/10.3390/coatings13020292

**Chicago/Turabian Style**

Barghi Golezani, Farshad, Rama Kishore Annavarapu, and Hossein Sojoudi.
2023. "Numerical Investigation of the Effect of Symmetry on Evaporation Triggered Elastocapillary Top-Gathering of High Aspect Ratio Micropillars" *Coatings* 13, no. 2: 292.
https://doi.org/10.3390/coatings13020292