2.4 Wetting and contact angle

Wetting phenomena are central to colloid science because they govern how liquids interact with solid surfaces. The balance between adhesive forces (liquid-to-solid attraction) and cohesive forces (liquid molecules attracting each other) determines whether a liquid spreads into a thin film or beads up into droplets.

Contact angle measurement is the primary way to quantify this behavior. Young's equation connects the contact angle to the interfacial tensions at play, giving you a thermodynamic framework for predicting wetting. These concepts matter directly in applications like coating, printing, and microfluidics, where precise control of liquid-solid interactions drives performance.

Wetting phenomena

Wetting describes what happens when a liquid contacts a solid surface. If adhesive forces between the liquid and solid dominate, the liquid spreads. If cohesive forces within the liquid dominate, the liquid pulls itself into droplets to minimize contact area.

Wetting vs non-wetting

• Wetting occurs when a liquid spreads over a solid surface, forming a thin, uniform film. Water on clean glass is a classic example: the water molecules are strongly attracted to the glass surface.
• Non-wetting happens when a liquid forms discrete droplets, minimizing its contact area. Mercury on glass behaves this way because mercury's internal cohesive forces far exceed its adhesion to glass.

The wettability of any given system depends on the relative magnitudes of these adhesive and cohesive forces.

Contact angle

The contact angle ($\theta$) is the angle formed between the solid surface and the tangent to the liquid-vapor interface, measured at the three-phase contact point (where solid, liquid, and vapor all meet).

• $\theta < 90°$: the surface is wettable (hydrophilic for water). The liquid has a tendency to spread.
• $\theta > 90°$: the surface is non-wetting (hydrophobic for water). The liquid resists spreading.
• $\theta \approx 0°$: complete wetting (the liquid spreads into a thin film).
• $\theta \approx 180°$: perfect non-wetting (the droplet is nearly spherical with minimal contact).

Young's equation

Young's equation relates the equilibrium contact angle to the three interfacial tensions in the system:

$\gamma_{SV} = \gamma_{SL} + \gamma_{LV} \cos \theta$

• $\gamma_{SV}$: solid-vapor interfacial tension
• $\gamma_{SL}$: solid-liquid interfacial tension
• $\gamma_{LV}$: liquid-vapor interfacial tension
• $\theta$: equilibrium (Young's) contact angle

Rearranging gives you the contact angle directly:

$\cos \theta = \frac{\gamma_{SV} - \gamma_{SL}}{\gamma_{LV}}$

This tells you something intuitive: when the solid-vapor surface energy is much higher than the solid-liquid interfacial tension, $\cos \theta$ approaches 1 and the liquid spreads readily. When $\gamma_{SL}$ is large relative to $\gamma_{SV}$, the contact angle increases and wetting becomes unfavorable.

Note that Young's equation assumes an ideal surface: perfectly smooth, chemically homogeneous, rigid, and inert. Real surfaces deviate from this, which is why the models discussed later (Wenzel, Cassie-Baxter) are needed.

Factors affecting wetting

• Surface chemistry: The functional groups on the solid surface determine how strongly it interacts with the liquid. Hydroxyl groups on glass make it hydrophilic; fluorinated groups on Teflon make it hydrophobic.
• Surface roughness: Micro- and nanoscale texture changes the actual contact area between liquid and solid, amplifying the intrinsic wetting tendency (more on this in the Wenzel and Cassie-Baxter sections).
• Liquid properties: Surface tension, viscosity, and polarity all affect spreading. A low-surface-tension liquid like ethanol wets most surfaces more easily than water ($\gamma_{LV} \approx 22$ mN/m vs. $\approx 72$ mN/m).
• Environmental conditions: Temperature, humidity, and surface contaminants can shift wetting behavior. Even a thin layer of organic contamination can turn a high-energy metal surface into a poorly wetted one.

Measuring contact angle

Several techniques exist for measuring contact angles, each suited to different sample geometries and experimental goals.

Sessile drop method

This is the most common and straightforward approach.

1. Place a small liquid droplet on a flat, horizontal solid surface.
2. Capture the droplet profile using a camera or optical goniometer.
3. Fit a tangent line to the liquid-vapor interface at the three-phase contact point.
4. Read the angle between this tangent and the solid surface.

The method is fast, requires only small sample volumes, and works well for routine characterization of flat substrates.

Wilhelmy plate method

This technique measures the force exerted on a thin plate as it contacts a liquid.

1. Suspend a thin, vertical plate from a sensitive balance.
2. Bring the plate into contact with the liquid surface.
3. Measure the force acting on the plate due to the liquid's surface tension.

The contact angle is extracted using the Wilhelmy equation:

$F = p \, \gamma_{LV} \cos \theta$

• $F$: measured force on the plate
• $p$: wetted perimeter of the plate
• $\gamma_{LV}$: liquid-vapor surface tension
• $\theta$: contact angle

This method gives an average contact angle around the entire plate perimeter, making it useful for dynamic wetting studies and for measuring contact angles on fibers or powders.

Capillary rise method

This method exploits the spontaneous rise (or depression) of liquid in a narrow tube.

When a wettable liquid contacts the inside of a capillary, surface tension pulls it upward until gravity balances the capillary force. The equilibrium height is described by Jurin's law:

$h = \frac{2 \gamma_{LV} \cos \theta}{\rho g r}$

• $h$: height of the liquid column
• $\rho$: liquid density
• $g$: gravitational acceleration
• $r$: capillary radius

By measuring $h$ and knowing the other parameters, you can solve for $\theta$. This approach is particularly useful for porous materials and confined geometries where sessile drop measurements aren't practical.

Advantages vs disadvantages

Surface free energy

Surface free energy is the excess energy at a material's surface compared to its bulk. It quantifies how "eager" a surface is to interact with other phases, and it directly governs wetting, adhesion, and other interfacial phenomena. You can't measure solid surface free energy directly, but you can estimate it from contact angle data.

Solid-liquid interactions

Solids with high surface free energy (metals, metal oxides, clean glass) tend to be easily wetted by most liquids. Solids with low surface free energy (polymers like PTFE, waxes) resist wetting.

The work of adhesion ($W_A$) between a solid and a liquid quantifies the energy needed to separate the two phases. It depends on the surface free energies of both materials and their mutual interfacial tension. Higher work of adhesion corresponds to stronger solid-liquid interaction and better wetting.

Dispersive vs polar components

Surface free energy can be decomposed into two contributions:

• Dispersive (non-polar) component ($\gamma^d$): arises from London dispersion forces, which are present in all materials due to temporary fluctuations in electron density.
• Polar component ($\gamma^p$): arises from permanent dipole interactions, hydrogen bonding, and acid-base interactions.

The relative magnitudes of these components determine how a surface interacts with different liquids. A surface with a large polar component will be wetted well by polar liquids like water; a surface dominated by dispersive interactions will be wetted more readily by non-polar liquids like hexane.

Owens-Wendt approach

This widely used method estimates solid surface free energy by splitting it into dispersive and polar parts:

$\gamma_S = \gamma_S^d + \gamma_S^p$

The work of adhesion between a solid and a liquid is expressed as:

$W_A = 2\left(\sqrt{\gamma_S^d \, \gamma_L^d} + \sqrt{\gamma_S^p \, \gamma_L^p}\right)$

To apply this method:

1. Measure contact angles on the solid using at least two probe liquids with known $\gamma_L^d$ and $\gamma_L^p$ values (commonly water and diiodomethane).
2. Use the Young-Dupré relation ($W_A = \gamma_{LV}(1 + \cos\theta)$) to calculate $W_A$ for each liquid.
3. Set up a system of equations and solve for $\gamma_S^d$ and $\gamma_S^p$.

Fowkes theory

Fowkes theory takes a similar approach, proposing that the work of adhesion is the sum of dispersive and polar contributions:

$W_A = W_A^d + W_A^p$

The dispersive component is given by:

$W_A^d = 2\sqrt{\gamma_S^d \, \gamma_L^d}$

The polar component is often approximated using the geometric mean:

$W_A^p = 2\sqrt{\gamma_S^p \, \gamma_L^p}$

Fowkes theory was historically important because it established the idea that you could treat interfacial interactions as separable contributions. The Owens-Wendt approach builds directly on this foundation, and in practice the two give equivalent results when the geometric mean approximation is used for the polar term.

Wetting on real surfaces

Real surfaces are neither perfectly smooth nor chemically uniform. Surface roughness and chemical heterogeneity cause the observed (apparent) contact angle to differ from the intrinsic (Young's) contact angle. Two key models address these deviations.

Surface roughness effects

Roughness increases the actual solid-liquid contact area relative to the projected (geometric) area. This amplifies whatever wetting tendency the surface already has:

• On an intrinsically hydrophilic surface ($\theta < 90°$), roughness makes it more hydrophilic (lower apparent contact angle).
• On an intrinsically hydrophobic surface ($\theta > 90°$), roughness makes it more hydrophobic (higher apparent contact angle).

Wenzel model

The Wenzel model applies when the liquid fully penetrates the surface texture, maintaining complete contact with the rough features.

$\cos \theta^* = r \cos \theta$

• $\theta^*$: apparent contact angle on the rough surface
• $r$: roughness factor, defined as the ratio of actual surface area to projected (flat) area. For any real surface, $r \geq 1$; a perfectly smooth surface has $r = 1$.
• $\theta$: intrinsic contact angle on a smooth surface of the same material

Because $r > 1$, the Wenzel equation amplifies the cosine of the intrinsic angle. If $\theta < 90°$, then $\cos\theta > 0$ and roughness pushes $\theta^*$ lower. If $\theta > 90°$, then $\cos\theta < 0$ and roughness pushes $\theta^*$ higher.

Cassie-Baxter model

The Cassie-Baxter model applies when the liquid sits on top of the surface features, trapping air pockets in the valleys beneath the droplet. The liquid contacts a composite interface of solid and air.

$\cos \theta^* = f_1 \cos \theta_1 - f_2$

• $f_1$: area fraction of the solid-liquid interface
• $\theta_1$: intrinsic contact angle on the solid
• $f_2$: area fraction of the liquid-air interface (where $f_1 + f_2 = 1$)

The $-f_2$ term reflects the fact that the contact angle on air is 180° ($\cos 180° = -1$). This trapped air dramatically increases the apparent contact angle and is the mechanism behind superhydrophobicity. The lotus leaf is the classic natural example: its hierarchical micro- and nanostructure traps air beneath water droplets, producing contact angles above 150°.

Heterogeneous surfaces

For chemically heterogeneous (but flat) surfaces with patches of different composition, the Cassie equation applies:

$\cos \theta^* = f_1 \cos \theta_1 + f_2 \cos \theta_2$

• $f_1$, $f_2$: area fractions of the two surface types
• $\theta_1$, $\theta_2$: intrinsic contact angles on each patch

This is a weighted average of the cosines, not the angles themselves. Designing surfaces with controlled chemical patterning allows you to tune the apparent contact angle to a desired value.

Dynamic wetting

So far, the discussion has focused on equilibrium contact angles. In practice, liquids are often in motion on surfaces, and the contact angle depends on whether the liquid front is advancing or retreating.

Advancing vs receding angles

• Advancing contact angle ($\theta_A$): the maximum stable angle observed as liquid is slowly added to a droplet, pushing the contact line outward.
• Receding contact angle ($\theta_R$): the minimum stable angle observed as liquid is slowly withdrawn, pulling the contact line inward.

The advancing angle is always greater than or equal to the receding angle: $\theta_A \geq \theta_R$.

Contact angle hysteresis

Contact angle hysteresis is the difference between the advancing and receding angles:

$\Delta \theta = \theta_A - \theta_R$

This quantity reflects how strongly the contact line is pinned to the surface. High hysteresis means the droplet resists motion; low hysteresis means droplets slide easily. Surfaces designed for self-cleaning or anti-fouling need both high contact angles and low hysteresis.

Factors influencing hysteresis

• Surface roughness: Surface asperities pin the contact line, increasing hysteresis.
• Chemical heterogeneity: Local variations in surface composition create regions of different wettability that resist contact line motion.
• Liquid properties: Viscosity, surface tension, and the presence of surfactants all affect how readily the contact line moves.
• Droplet size: Smaller droplets are more sensitive to local surface heterogeneities and may show greater hysteresis.

Measurement techniques

• Tilted plate method: Slowly tilt the surface until a droplet begins to slide. Measure $\theta_A$ at the downhill edge and $\theta_R$ at the uphill edge just before sliding begins.
• Sessile drop with volume change: Use a syringe to slowly add liquid (measure $\theta_A$) or withdraw liquid (measure $\theta_R$) from a sessile droplet while recording the contact angle.
• Wilhelmy plate with immersion/emersion cycles: Dip the plate into the liquid (advancing) and pull it out (receding), measuring the force at each stage to extract both angles.
• Capillary bridge method: Form a liquid bridge between two surfaces and measure the advancing and receding angles as the surfaces are separated or brought together.

Applications of wetting

Adhesion and bonding

Good wetting is a prerequisite for strong adhesion. If an adhesive doesn't spread on the substrate, it can't form intimate molecular contact, and the bond will be weak. Contact angle measurements help predict adhesive performance: a low contact angle between the adhesive and substrate indicates favorable thermodynamic conditions for bonding. This matters in applications ranging from dental composites to pressure-sensitive adhesives and structural bonding in aerospace.

Printing and coating

In printing, ink must spread uniformly on the substrate to produce sharp, high-quality images. In coating processes, the liquid must wet the surface evenly to achieve consistent thickness and good adhesion. Controlling substrate surface energy (through plasma treatment, primers, or surface roughness modification) is a standard strategy for optimizing wetting in inkjet printing, gravure printing, and paint application.

Microfluidics and lab-on-chip

In microfluidic devices, capillary forces often dominate over gravity and inertia. The contact angle directly determines the magnitude and direction of these capillary forces in microchannels. By patterning surfaces with regions of different wettability, you can guide liquid flow, control mixing, and confine reactions to specific zones. This is the basis for point-of-care diagnostic chips and high-throughput drug screening platforms.

Superhydrophobicity and self-cleaning

Superhydrophobic surfaces have water contact angles above 150° and very low contact angle hysteresis (typically below 10°). Water droplets on these surfaces are nearly spherical and roll off at slight tilts, picking up dirt and contaminants as they go.

Achieving superhydrophobicity requires two ingredients:

1. Low surface energy material (fluorinated coatings, hydrophobic silanes).
2. Hierarchical roughness at both the micro- and nanoscale, which promotes the Cassie-Baxter wetting state and traps air beneath droplets.

The lotus leaf combines waxy surface chemistry with papillae (microscale bumps) covered in nanoscale hair-like structures. Synthetic superhydrophobic surfaces mimic this architecture for self-cleaning windows, stain-resistant textiles, and anti-icing coatings.