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
Suppose a liquid droplet is deposited onto a solid surface. In that case, it will form a contact angle that depends on its surface tension, the solid substrate's surface tension, and the interfacial tension between the solid and the liquid.
This equation is known as the "Young equation", and the angle θ formed by the solid surface and the droplet's tangent is called the "contact angle".
The "contact angle" is intuitive and easy to understand as an indicator of wettability, and has been widely adopted in industrial fields as an evaluation method for surfaces.
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 specific topography and wetting characteristics of your substrate.
θ/2 method
The θ/2 or height-width method is generally used to determine the contact angle. The equation given below determines the contact angle. Where radius r and height h of the droplet.
In addition, the contact angle is measured as the angle formed by the straight lines connecting the left and right three-phase points of the droplet to its apex on a solid surface.
In the θ/2 method, since the shape of the droplet is assumed to be part of the outline of an imaginary circle, measurement is done with a tiny droplet that can ignore the effect of the gravitational force.
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 similarly using points R1, R2, and R3.
The contact angle is determined as the average of the left and right endpoints. Conversely, the Tangent method allows one to independently determine the contact angle at the left and right endpoints. Thus, it is an effective measurement method for uneven surfaces with differing left- and right-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 deposited on a flat solid surface, and the surface is tilted, it will start to roll or slide off at a certain angle. This angle is defined as the sliding or roll-off angle α.
The contact angle at the lower end is the advancing angle θa, and the one on the upper end is the receding angle θr.
θa: Advancing angle
θr: Receding angle
α: Sliding angle
The roll-off angle of a liquid largely depends on its droplet volume, making it impossible to compare the roll-off angles of liquids with different volumes. Here, the adhesive energy between the liquid and the solid's surface at the time the roll-off angle is determined can help, as it is unlikely to depend on droplet volume, as the graphs below show.
E: Adhesive energy
r: Radius of the droplet
m: Mass of the droplet
g: Acceleration of gravity
α: Sliding angle
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- Versatile Contact Angle Meter
DMo-WA series
- Wafer Contact Angle Meter
DMs-402
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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 perimeter of the plate, and the liquid pulls 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 more the buoyant force increases, 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.
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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
It is known that the liquid penetration rate into the powder is described by the Washburn equation.
ｌ… Height of the liquid front
ｔ… Time
ｒ… Capillary radius of the compressed powder
γ… Surface tension of the liquid
η… Viscosity of the liquid
θ… Contact angle
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 capillary 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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