Superhydrophobe - Theory

Theory

In 1805, Thomas Young defined the contact angle θ by analyzing the forces acting on a fluid droplet resting on a solid surface surrounded by a gas.

where

= Interfacial tension between the solid and gas

= Interfacial tension between the solid and liquid

= Interfacial tension between the liquid and gas

θ can be measured using a contact angle goniometer.

Wenzel determined that when the liquid is in intimate contact with a microstructured surface, θ will change to

where r is the ratio of the actual area to the projected area. Wenzel's equation shows that microstructuring a surface amplifies the natural tendency of the surface. A hydrophobic surface (one that has an original contact angle greater than 90°) becomes more hydrophobic when microstructured – its new contact angle becomes greater than the original. However, a hydrophilic surface (one that has an original contact angle less than 90°) becomes more hydrophilic when microstructured – its new contact angle becomes less than the original.

Cassie and Baxter found that if the liquid is suspended on the tops of microstructures, θ will change to

= φ(cos θ + 1) – 1

where φ is the area fraction of the solid that touches the liquid. Liquid in the Cassie-Baxter state is more mobile than in the Wenzel state.

It can be predicted whether the Wenzel or Cassie-Baxter state should exist by calculating the new contact angle with both equations. By a minimization of free energy argument, the relation that predicted the smaller new contact angle is the state most likely to exist. Stated mathematically, for the Cassie-Baxter state to exist, the following inequality must be true.

cos θ < (φ-1)/(r - φ)

A recent alternative criteria for the Cassie-Baxter state asserts that the Cassie-Baxter state exists when the following 2 criteria are met: 1) Contact line forces overcome body forces of unsupported droplet weight and 2) The microstructures are tall enough to prevent the liquid that bridges microstructures from touching the base of the microstructures.

Contact angle is a measure of static hydrophobicity, and contact angle hysteresis and slide angle are dynamic measures. Contact angle hysteresis is a phenomenon that characterizes surface heterogeneity. When a pipette injects a liquid onto a solid, the liquid will form some contact angle. As the pipette injects more liquid, the droplet will increase in volume, the contact angle will increase, but its three phase boundary will remain stationary until it suddenly advances outward. The contact angle the droplet had immediately before advancing outward is termed the advancing contact angle. The receding contact angle is now measured by pumping the liquid back out of the droplet. The droplet will decrease in volume, the contact angle will decrease, but its three phase boundary will remain stationary until it suddenly recedes inward. The contact angle the droplet had immediately before receding inward is termed the receding contact angle. The difference between advancing and receding contact angles is termed contact angle hysteresis and can be used to characterize surface heterogeneity, roughness, and mobility. Surfaces that are not homogeneous will have domains which impede motion of the contact line. The slide angle is another dynamic measure of hydrophobicity and is measured by depositing a droplet on a surface and tilting the surface until the droplet begins to slide. Liquids in the Cassie-Baxter state generally exhibit lower slide angles and contact angle hysteresis than those in the Wenzel state.

A simple model can be used to predict the effectiveness of a manmade micro- or nano-fabricated surface for its conditional state (wenzel or cassie-baxter), contact angle and contact angle hysteresis. The main factor of this model is the contact line density, Λ, which is the total perimeter of asperities over a given unit area.

The critical contact line density Λ_c is a function of body and surface forces, as well as the projected area of the droplet.

where

ρ = density of the liquid droplet

g = acceleration due to gravity

V = volume of the liquid droplet

θ_a = advancing appearant contact angle

θ_a,0 = advancing contact angle of a smooth substrate

γ = surface tension of the liquid

w = tower wall angle

If Λ > Λ_c, drops are suspended in the cassie-baxter state. Otherwise, the droplet will collapse into the wenzel state.

To calculate updated advancing and receding contact angles in the cassie-baxter state, the following equations can be used.

with also the wenzel state:

where

λ_p = linear fraction of contact line on the asperities

θ_r,0 = advancing contact angle of a smooth substrate

θ_air = contact angle between liquid and air (typically assumed to be 180°)