What is Contact Angle?
Contact angle is a fundamental metric used to quantify the wettability of a solid surface by a liquid. Understanding this behavior is essential for evaluating surface treatments, coating adhesion, and cleanliness across diverse manufacturing and research applications.
Static Contact Angle - Sessile Drop
When a liquid droplet is placed on a solid surface, it forms a contact angle θ determined by its surface tension γL, the solid's surface tension γS, and the interfacial tension γSL between the solid and the liquid. This relationship is known as Young's equation, and can be written as follows:
The contact angle is a straightforward and intuitive measure of wettability, making it a popular choice in industrial applications for assessing surface properties.
Sessile Drop – Function of Time
The contact angle measurement in the preceding paragraph is performed continuously to observe variations over time. Although there is no precise distinction between dynamic and static, we treat it as a type of dynamic contact angle for intervals of one second or fewer. This can also be used to follow absorption and other volatile situations.
Captive Bubble
Contact angle measurements typically measure the wettability of a solid by a liquid in a gaseous phase. Using a special kit, contact angle measurements of a liquid or an air bubble can be obtained in a surrounding bulk liquid phase.
The left image illustrates the measurement of a liquid droplet with a higher density than the bulk liquid phase. In contrast, the right image shows an air bubble or a liquid droplet with a lower density than the bulk phase. Here, an inverted needle deposits an air bubble or droplet beneath the solid sample.
Contact Angle Analysis Methods
Surface analysis relies on distinct mathematical models to calculate contact angles from droplet geometry. Choosing the appropriate evaluation method — such as the θ/2 method, the tangent method, curve-fitting algorithms, and the Young-Laplace fit — ensures precise data that match the substrate's specific topography and wetting characteristics.
θ/2 method
The θ/2, or height-width, method is commonly used to determine the contact angle. The equation below determines the contact angle, where r is the droplet radius and h is the droplet height.
⇓
In addition, the contact angle is defined as the angle formed by the straight lines connecting the three-phase points on the left and right sides of the droplet to its apex on a solid surface.
The θ/2 method shall be used only to measure small droplets, which are considered part of the imaginary circle's outline, thereby allowing it to ignore gravitational effects.
Tangent method
The droplet's shape is assumed to be part of the outline of an imaginary circle. This method determines the center of the imaginary circle and the contact angle, which is the angle between the two tangents to the circle. For example, from the figure below, the three arc points L1, L2, and L3 form the imaginary circle. The angle between the tangent line m and the drop baseline l is the left contact angle. The right contact angle can be measured in a similar manner using points R1, R2, and R3.
Therefore, the tangent method allows independent determination of the contact angle at the left and right endpoints. Thus, it is an effective measurement method for uneven surfaces with differing left- and right-hand contact angles.
Curve fitting methods
Assuming that the droplet's contour shape forms part of a true circle or ellipse, the least squares method is performed using the coordinates of all observations (fitting section) in a given interval. This calculation determines the optimal circle or ellipse parameters and computes the endpoint contact angle differential coefficients.
With the Tangent method, a perfect circle has been assumed as the contour shape. However, compared with the results of fitting the perfect circle, the second is the better-fitting perfect circle, as it uses more coordinates to achieve a smaller variation.
Young-Laplace fit
The Young-Laplace method is the most advanced and rigorous curve-fitting algorithm used in modern contact angle analysis. Unlike basic geometric approximations, this method analyzes the entire contour of a sessile drop by fitting it to the theoretical Young-Laplace equation, which balances the forces of surface tension and gravity.
Dynamic Contact Angle
On the assumption that the droplet is at rest on a solid surface and reaches equilibrium quickly relative to the differences at a specific time after droplet deposition, the contact angle data from the preceding paragraph are useful. However, this only applies if the data is not altered much. For example, in cases involving multiple states, such as coating or cleaning, where the liquid-solid interface is dynamic, sufficient data are not obtained.
This case simulates (with advancing and receding contact angles) a dynamic situation in which the liquid droplet's interface moves and increases.
Extension/contraction (captive drop) method
The contact angle of a droplet on a solid surface changes when a needle tip is inserted into the droplet, and the volume is controlled by the needle. This method measures the dynamic hysteresis of the contact angle by increasing or decreasing the droplet volume. The contact angles formed during volume increase and decrease are called the advancing and receding angles, respectively.
Advancing angle by increasing the sessile drop volume
Receding angle by decreasing the sessile drop volume
Compared to measuring the static contact angle, the advancing and receding angles provide more valuable information for evaluating dynamic wetting in coating processes such as spin-coating and roll-coating, as they are closer to real coating conditions.
Sliding method – measurement of roll-off angles
When a droplet is placed on a flat solid surface, and the surface is tilted, it will start to roll or slide off at a specific angle. This angle is known as the sliding or roll-off angle α. The contact angle at the lower end is called the advancing angle, θa, while at the upper end it is the receding angle, θr.
The liquid's roll-off angle largely depends on droplet volume, making cross-volume comparisons impractical. Our FAMAS software uses Wolfram's experimental formula to determine the adhesion energy between the liquid and solid at the critical tilt angle at which the droplet begins to roll off. In this formula, E represents the adhesive energy, calculated from the droplet's mass m, gravitational acceleration g, roll-off angle α, and contact radius r.
Adhesion Energy Equation
The left graph illustrates how the roll-off angle varies with droplet volume, while the right graph indicates that the calculated adhesion energy remains nearly constant regardless of volume, making it a valuable metric for comparing roll-off angles across different droplet sizes.
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Dynamic Contact Angle using Surface Tensiometers
Dynamic contact angles reveal how a liquid front moves across a surface. Using high-precision surface tensiometers to measure advancing and receding angles allows researchers to quantify hysteresis, providing critical insights into surface roughness, chemical heterogeneity, and real-world coating behavior.
Wilhelmy method
When the measuring unit (Wilhelmy plate) contacts the liquid surface, the liquid will wet the plate upward. In this case, surface tension acts along the plate's perimeter, pulling on it. This method detects the pulling force and determines the surface tension. At this time, the surface tension can be determined by using a plate material that has a contact angle of 0º with the liquid to the measuring unit (plate). Platinum is commonly used as the plate material.
Conversely, in the Wilhelmy method contact angle measurement, we measured the force on the solid sample as the liquid is brought into contact with the suspended sample. The contact angle between the liquid and the solid is determined by the difference between the surface tensions, and the pulling force is less than the liquid's surface tension. The contact angle is calculated from the relationship between the surface tension and the decline in force.
| F | Forces acting on the solid (measured value) |
| γ | Surface tension of the liquid (must be known for contact angle calculation) |
| L | Circumference of the solid sample perimeter (must be known) |
| θ | Contact angle of the liquid with the solid (assumed to be 0º) |
| S | Cross-sectional area of the solid sample |
| h | Immersion depth of the solid sample |
| ρ | Density of the liquid sample |
| g | Gravitational acceleration |
A solid sample is suspended from the balance and immersed in the liquid. Raising the stage will immerse the solid sample deeper into the liquid, and the advancing contact angle θA can be measured. Lowering the stage will pull the solid out of the liquid, and the receding contact angle θR can be measured.
The more the solid sample is immersed in the liquid, the greater the buoyant force, causing the force acting on the balance to decrease. Therefore, for both the advancing contact angle θA and the receding contact angle θR, the obtained weight curves (see the figure on the right) must be determined by linear regression, and the wetting forces must be extrapolated to zero-depth immersion.
Powder Contact Angle
Measuring the wettability of particulate matter poses unique challenges due to void spaces and capillary action. By applying specialized force or pressure-based penetration techniques, it is possible to accurately determine the contact angle of powders, an essential parameter for optimizing pigment dispersion, pharmaceutical formulations, and composite material manufacturing.
Infiltration rate method
The Washburn equation below explains how liquid penetrates into the powder.
| l | Height of the liquid front |
| t | Time |
| r | Capillary radius of the compressed powder |
| γ | Surface tension of the liquid |
| θ | Contact angle of the liquid with the powder |
| η | Viscosity of the liquid |
In the actual measurement, a liquid sample penetrates a column filled with powder, and the change in weight w is tracked over time t. Ideally, the linear relationship is obtained by plotting t for w2.
The contact angle and infiltration rate are calculated from the slope of this line. In addition, to calculate the contact angle, the capillary radius of the powder-filled column is required, along with the liquid surface tension and viscosity. That capillary radius value is determined experimentally by measuring the sufficiently wet liquid for the powder and regarding the contact angle as 0º.
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