# Contact Angle: A Guide to Theory and Techniques

## What is a Contact Angle?

A contact angle gives us an indication of how well (or how poorly) a liquid will spread over a surface. While formulating an ink, contact angles provide a useful indicator of how a modification to the ink will affect its spreading.

A contact angle can be large or small, depending on the physical properties of the materials being investigated. Figure 1 shows three different droplets on a surface. The left-most droplet has a large contact angle, as it does not spread over the solid surface. The right-most droplet has a low contact angle, as it has spread well. This spreading is know as ‘wetting’, and a droplet either ‘wets’ or ‘dewets’ when deposited on a surface.

Figure 2 shows a two-dimensional cross section of a droplet on a solid surface. Locate the point at which the droplet outline intersects the solid surface. The angle between the droplet outline and the solid surface is the contact angle.

If we wanted a solution to spread more easily over a substrate, we could alter the solvents used in the formulation, and test them to check if this had increased its wetting capabilities. In this situation, a low contact angle would be the desired outcome.

Alternatively, we might be developing a waterproof coating for an item of clothing. In this case, a high contact angle would be desirable. We would alter the coating formulation, and use water droplets to determine which coating was more resistant to wetting.

## Background Theory

### Surface Tension

The surface tension of the droplet is determined by the interactions between its constituent molecules. The molecules in a droplet of liquid are shown in Figure 3. In the bulk of the droplet, intermolecular forces act upon a molecule from all sides. However, at the surface of the droplet, there is an absence of liquid molecules on the external side.

The molecules at the surface are therefore more strongly bound to each other than the molecules in the bulk, as they are not being pulled from all sides. This means that it is more difficult for an object to penetrate the surface than it is for an object to move within the bulk once submerged.

### Balanced Forces

There are three boundaries to consider when a droplet is in contact with a solid surface: the solid, the liquid, and the vapour (usually air) surrounding them.

Figure 4 shows a force diagram the point at which a droplet edge meets a solid surface. The three arrows represent the forces exerted by the surface tensions at three interfaces: liquid-surface, liquid-vapour, and solid-vapour.

Each force is pulling away from the equilibrium point, so if the droplet is in equilibrium, then the forces are balanced and can be described by the following equation:

Where cos θ gives the x-component of the liquid-vapour surface tension. This can be re-arranged to give:

This equation provides some useful information.

If γsv < γls, then cos θ will be negative, and θ is therefore < 90o (and the droplet wets). This can occur with a high surface-energy solid (such as a metal), or a low surface-tension liquid.

If γsv > γls, cos θ is positive, and θ is > 90o (and the droplet dewets). This can occur with a low surface-energy solid, or a high surface-tension liquid (such as water).

This may raise the question of how we equate surface energy density (units J/m2) with surface tension (units N/m).

Surface energy density and surface tension can therefore be equated.

## Practical Measurement

The method of measurement detailed here involves 2 steps:

### Step 1) Image Capture

The equipment required to perform a contact angle measurement can be very simple. The most common method uses three basic components:

• A light source
• A camera
• A tilting stage

There are then three stages to a contact angle measurement:
1. A droplet of liquid is deposited on to the tilting stage.
2. The droplet is illuminated from behind, and an image is recorded by the camera.
3. The image is analysed using code or software, and a contact angle measurement is determined.

### Step 2) Polynomial Fit Method

There are several methods for measuring the contact angle from an image. This guide will focus on the polynomial fitting method, which fits a curve to the droplet edge.

Step 1. Use the previously discussed experimental setup to obtain an image of a droplet. The droplet should have a high contrast with its back ground.

Step 2. Crop the image to a region of interest. The bottom line should intersect the 'pointy' parts of the droplet image, as this is where the droplet meets the substrate surface and is reflected.

Step 3. Use a thresholding method to get rid of any background, such as the section of substrate behind the droplet.

Step 4. Find the edge points of the thresholded image. If the treshold has been performed correctly, the first non-zero element in each row of the image will be the edge of the droplet.

Step 5. Fit a polynomial line to the points on this edge.

Step 6. Differentiate the polynomial to find the gradient at the intersection with the baseline. This gradient is the tangent to the droplet edge at the surface. Calculate the angle between this tangent and the baseline. This is the contact angle of the droplet.