^{1}, A. A. Hemeda

^{1}, M. M. Amrei

^{1}, A. Luzar

^{2}, M. Gad-el-Hak

^{1}and H. Vahedi Tafreshi

^{1,a)}

### Abstract

A mathematical framework is developed to predict the longevity of a submerged superhydrophobic surface made up of parallel grooves. Time-dependent integro-differential equations predicting the instantaneous behavior of the air–water interface are derived by applying the balance of forces across the air–water interface, while accounting for the dissolution of the air in water over time. The calculations start by producing a differential equation for the initial steady-state shape and equilibrium position of the air–water interface at t = 0. Analytical and/or numerical solutions are then developed to solve the time-dependent equations and to compute the volume of the trapped air in the grooves over time until a Wenzel state is reached as the interface touches the groove's bottom. For demonstration, a superhydrophobic surface made of parallel grooves is considered, and the influence of the groove's dimensions on the longevity of the surface under different hydrostatic pressures is studied. It was found that for grooves with higher width-to-depth ratios, the critical pressure (pressure at which departure from the Cassie state starts) is higher due to stronger resistance to deflection of the air–water interface from the air trapped in such grooves. However, grooves with higher width-to-depth ratios reach the Wenzel state faster because of their greater air–water interface areas.

This research has been supported in part by the VCU Presidential Research Incentive Program (PRIP) as well as the National Science Foundation CMMI 1029924 and CHE 1213814 programs.

I. INTRODUCTION

II. PROBLEM FORMULATION

A. Calculating the critical pressure and its corresponding interface profile

B. Calculation of failure time in regime I

1. Before interface detachment

2. After interface detachment

C. Calculation of failure time in regime II

III. RESULTS AND DISCUSSIONS

A. Effects of groove dimensions

B. A case study

IV. CONCLUSIONS

### Key Topics

- Hydrostatics
- 41.0
- Interfacial properties
- 32.0
- Dissolution
- 24.0
- Water vapor
- 22.0
- Gas liquid interfaces
- 16.0

## Figures

(a) Schematic of a superhydrophobic surface with ridges and grooves. (b) Schematic of the air–water meniscus on one of the grooves. The shape of the meniscus is calculated by applying balance of forces between hydrostatic and trapped air pressures, and capillarity.

(a) Schematic of a superhydrophobic surface with ridges and grooves. (b) Schematic of the air–water meniscus on one of the grooves. The shape of the meniscus is calculated by applying balance of forces between hydrostatic and trapped air pressures, and capillarity.

Schematics of the transient behavior of the air–water meniscus at (a) regime I and (b) regime II. Note that t cr is the critical time at which the interface is separated from the sharp edges of the groove, and that t f is the failure time at which the interface touches the bottom of the groove.

Schematics of the transient behavior of the air–water meniscus at (a) regime I and (b) regime II. Note that t cr is the critical time at which the interface is separated from the sharp edges of the groove, and that t f is the failure time at which the interface touches the bottom of the groove.

(a) Contour plot presenting normalized critical pressure P cr /P ∞ vs. groove width w and groove depth h. Hydrostatic pressures smaller (greater) than this pressure correspond to regime I (regime II). Note that higher critical pressures are obtained when w is large and h is small (larger contributions from the entrapped air at h/w ≪ 1). Conditions at which the air–water interface touches the groove's bottom before reaching the critical time (h = h min ) are omitted from the contour plot. (b) Minimum depth h min of grooves with different widths and Young–Laplace contact-angles.

(a) Contour plot presenting normalized critical pressure P cr /P ∞ vs. groove width w and groove depth h. Hydrostatic pressures smaller (greater) than this pressure correspond to regime I (regime II). Note that higher critical pressures are obtained when w is large and h is small (larger contributions from the entrapped air at h/w ≪ 1). Conditions at which the air–water interface touches the groove's bottom before reaching the critical time (h = h min ) are omitted from the contour plot. (b) Minimum depth h min of grooves with different widths and Young–Laplace contact-angles.

The curve in the h–w plane corresponds to the groove widths and depths at which the derivative of the critical pressure with respect to the w, , vanishes. In the region below the curve ( ), compression of the trapped air is the dominant effect, while in the region above the curve ( ), capillarity is dominant.

The curve in the h–w plane corresponds to the groove widths and depths at which the derivative of the critical pressure with respect to the w, , vanishes. In the region below the curve ( ), compression of the trapped air is the dominant effect, while in the region above the curve ( ), capillarity is dominant.

The effects of groove width depth on failure time (in minutes) subjected to hydrostatic pressures of 0.4 m ((a) and (b)), and 2.5 m ((c) and (d)) of water.

The effects of groove width depth on failure time (in minutes) subjected to hydrostatic pressures of 0.4 m ((a) and (b)), and 2.5 m ((c) and (d)) of water.

Comparison between performance of two grooves with different widths of 20 and 100 μm but otherwise identical under two different hydrostatic pressures of 14.7 kPa (the critical pressure for the narrower groove) and 100.1 kPa (critical pressure for the wider groove) in (a) and (b), respectively.

Comparison between performance of two grooves with different widths of 20 and 100 μm but otherwise identical under two different hydrostatic pressures of 14.7 kPa (the critical pressure for the narrower groove) and 100.1 kPa (critical pressure for the wider groove) in (a) and (b), respectively.

(a) Air–water interface at different times, for a groove with w = 100 μm, h = 30 μm, and θ = 115°, subject to a hydrostatic pressure of 10 cm. (b) Location of the deepest point of the interface (x = 0) vs. time. (c) Surface transitioning from the Cassie state to the Wenzel state is shown with contour plots.

(a) Air–water interface at different times, for a groove with w = 100 μm, h = 30 μm, and θ = 115°, subject to a hydrostatic pressure of 10 cm. (b) Location of the deepest point of the interface (x = 0) vs. time. (c) Surface transitioning from the Cassie state to the Wenzel state is shown with contour plots.

Failure and critical times for a groove with w = 100 μm, h = 30 μm, and θ = 115°, subject to a wide range of hydrostatic pressures from 10 cm to 5 m. The critical pressure for the groove is 3.5 m of water.

Failure and critical times for a groove with w = 100 μm, h = 30 μm, and θ = 115°, subject to a wide range of hydrostatic pressures from 10 cm to 5 m. The critical pressure for the groove is 3.5 m of water.

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