6.2 Foam stability and drainage

Foam stability and drainage are central to understanding how foams behave over time and why they eventually collapse. These processes govern everything from the head on a beer to the performance of firefighting foams and enhanced oil recovery. This topic covers the factors that keep foams stable, how liquid drains out of them, how bubbles coarsen, and the experimental and modeling tools used to study all of it.

Factors affecting foam stability

Foam stability depends on the properties of the liquid phase and the gas-liquid interfaces. Several interconnected factors determine how long a foam persists before it thins, coarsens, and collapses.

Liquid viscosity impact

Higher liquid viscosity slows drainage through the foam's internal channels (called Plateau borders) and improves stability. Viscous liquids resist flow, which means the thin films between bubbles stay thicker for longer. This also reduces the rate of bubble coalescence and rupture.

Common high-viscosity additives used in foams include glycerol and polymeric thickeners like xanthan gum. Even small increases in viscosity can meaningfully extend foam lifetime.

Surface tension effects

Lower surface tension at the gas-liquid interface promotes foam stability. It reduces the driving force for bubble coalescence and lowers the energy needed to maintain the foam structure.

Surfactants are the main tool for controlling surface tension. They adsorb at the interface and lower the surface tension. For example, sodium dodecyl sulfate (SDS) is a widely used anionic surfactant in foam studies. Both the type and concentration of surfactant matter: too little won't stabilize the foam, while above the critical micelle concentration (CMC), additional surfactant goes into micelles rather than the interface.

Gibbs-Marangoni mechanism

The Gibbs-Marangoni effect is one of the most important stabilization mechanisms in foams. It works like this:

1. A thin liquid film between two bubbles gets stretched or disturbed (e.g., by drainage or mechanical stress).
2. Stretching thins the film locally, which dilutes the surfactant at that spot.
3. Lower surfactant concentration means higher local surface tension.
4. The resulting surface tension gradient pulls surfactant (and liquid along with it) back toward the thinned region.
5. This restoring flow (called Marangoni flow) opposes further thinning and helps maintain uniform film thickness.

This self-healing mechanism is why surfactant-stabilized foams are far more stable than foams made from pure liquids.

Disjoining pressure influence

Disjoining pressure arises from interaction forces between the two surfaces of a thin liquid film. Three main contributions matter:

• Van der Waals forces (usually attractive, destabilizing)
• Electrostatic double-layer forces (repulsive for like-charged surfaces, stabilizing)
• Steric forces (repulsive when polymers or particles are adsorbed, stabilizing)

A positive (repulsive) net disjoining pressure prevents the film from thinning to the point of rupture. The magnitude and sign depend on surface chemistry, ionic strength, and whether polymers or particles are present at the interfaces. At high salt concentrations, for instance, electrostatic repulsion is screened, which can destabilize the foam.

Foam drainage mechanisms

Foam drainage is the flow of liquid through the foam structure under gravity and capillary forces. As liquid drains, films thin, and the foam moves closer to collapse. Several distinct mechanisms govern this process.

Gravity-driven drainage

Gravity pulls liquid downward through the Plateau borders (the channels at the junction of three films) and nodes (the intersections where four Plateau borders meet). The drainage rate depends on liquid density, viscosity, and the permeability of the foam network.

In free drainage, no external pressure is applied. Liquid simply flows downward under its own weight. You can observe this directly: a freshly made foam is wetter at the bottom and drier at the top, and this gradient increases over time.

Capillary pressure gradients

Capillary pressure comes from the curvature of gas-liquid interfaces and is related to surface tension. In a foam, smaller bubbles have higher internal pressure than larger ones (a consequence of the Young-Laplace equation).

This creates pressure gradients that drive liquid from regions near smaller bubbles (high capillary pressure) toward regions near larger bubbles or the bulk liquid (low capillary pressure). Capillary-driven flow works alongside gravity and can either assist or oppose gravitational drainage depending on the geometry.

Plateau border suction

This mechanism occurs when liquid in the Plateau borders is drawn toward the nodes. The nodes have a larger radius of curvature than the Plateau borders, which means lower capillary pressure in the nodes. The resulting pressure difference "sucks" liquid from the borders into the nodes.

The effect contributes to thinning of the Plateau borders, which in turn thins the films between bubbles and can accelerate foam collapse.

Node-dominated drainage

In some foams, the nodes (not the Plateau borders) are the primary bottleneck for liquid flow. This happens when nodes are small and constricted, creating a large pressure drop that limits throughput.

Node-dominated drainage is more common in foams with:

• High liquid fractions
• Small bubble sizes

The distinction between channel-dominated and node-dominated drainage matters for modeling, because the two regimes predict different scaling relationships between drainage rate and bubble size.

Foam coarsening processes

Foam coarsening is the gradual increase in average bubble size over time. The total number of bubbles decreases and the foam's specific surface area shrinks. Two mechanisms drive coarsening: coalescence and Ostwald ripening.

Coalescence of bubbles

Coalescence happens when the thin liquid film (lamella) between two adjacent bubbles ruptures, allowing the gas phases to merge into a single larger bubble. Factors that promote coalescence include:

• Very thin liquid films
• High capillary pressure
• Surface-active impurities that weaken film integrity

Coalescence is often the dominant coarsening mechanism in unstable foams or foams under mechanical stress.

Ostwald ripening phenomenon

Ostwald ripening is a subtler process driven by differences in Laplace pressure between bubbles of different sizes. Smaller bubbles have higher internal pressure, so gas dissolves into the surrounding liquid and diffuses toward larger bubbles where the pressure is lower.

The result: small bubbles gradually shrink and disappear, while large bubbles grow. Over time, the bubble size distribution shifts toward larger sizes and becomes narrower. This process is slow but relentless, and it dominates in foams where coalescence is suppressed (e.g., well-stabilized foams).

Laplace pressure differences

The Young-Laplace equation gives the pressure difference across a curved interface:

$\Delta P = \frac{2\gamma}{R}$

where $\gamma$ is the surface tension and $R$ is the bubble radius. A bubble with $R = 50 \, \mu m$ has twice the Laplace pressure of a bubble with $R = 100 \, \mu m$ (at the same surface tension). This pressure difference is the thermodynamic driving force behind Ostwald ripening.

Gas diffusion between bubbles

Gas molecules diffuse through the liquid films separating bubbles, moving down the concentration gradient created by Laplace pressure differences. The diffusion rate depends on:

• Gas solubility in the liquid (more soluble gases ripen faster; this is why $N_2$ foams are more stable than $CO_2$ foams)
• Gas diffusivity in the liquid
• Film thickness and composition (thicker films slow diffusion; adsorbed particles can act as barriers)

As gas transfers from small to large bubbles, the small bubbles vanish and the foam coarsens.

Methods to improve foam stability

Several strategies can extend foam lifetime by strengthening films, slowing drainage, or inhibiting coarsening.

Surfactant adsorption at interfaces

Surfactants stabilize foams by adsorbing at gas-liquid interfaces, lowering surface tension, and enabling the Gibbs-Marangoni mechanism. The choice of surfactant matters:

• Anionic surfactants like sodium laureth sulfate (SLES) are common in personal care foams
• Zwitterionic surfactants like cocamidopropyl betaine are often used in combination with anionics for synergistic stabilization
• Molecular structure, concentration, and the rate of adsorption all affect performance

Surfactant mixtures often outperform single surfactants because they pack more efficiently at the interface.

Increasing liquid viscosity

Adding viscosity-enhancing agents slows drainage and reduces coalescence. Common thickeners include:

• Glycerol and polyols (increase bulk viscosity)
• Polymeric thickeners like xanthan gum or hydroxyethyl cellulose (can also increase surface viscosity)

The increased viscosity retards flow through Plateau borders and nodes, preserving the foam structure for longer.

Incorporating solid particles

Solid particles can adsorb at gas-liquid interfaces and form a rigid shell around each bubble. These Pickering foams (named after S.U. Pickering) are often far more stable than surfactant-only foams.

The key parameter is the particle's contact angle at the interface. Particles with a contact angle near 90° anchor most strongly and are the most effective stabilizers. Silica nanoparticles and modified clays are commonly used. The particle layer acts as a physical barrier against both coalescence and gas diffusion, suppressing Ostwald ripening as well.

Polyelectrolyte complexation

Polyelectrolyte complexes (PECs) form when oppositely charged polymers (e.g., chitosan and sodium alginate) interact electrostatically at the gas-liquid interface. The resulting viscoelastic interfacial layer is thick and resilient, resisting deformation and film rupture.

PEC-stabilized foams combine the benefits of polymeric thickening with strong interfacial reinforcement, making them useful in applications requiring long-lived foams.

Modeling foam drainage

Mathematical models of foam drainage help predict stability and guide formulation design. The models differ in which drainage mechanism they emphasize and how they treat the foam geometry.

Verbist model for free drainage

The Verbist model is the standard framework for free drainage. It describes liquid flow through Plateau borders and nodes under gravity, assuming laminar flow. The governing equation relates the liquid fraction $\varepsilon$ to height $z$ and time $t$:

$\frac{\partial \varepsilon}{\partial t} = -\frac{\partial}{\partial z}\left(K(\varepsilon) \frac{\partial \varepsilon}{\partial z}\right)$

Here, $K(\varepsilon)$ is a permeability coefficient that depends on the liquid fraction and foam structure. The model captures the basic drainage profile: liquid accumulates at the bottom while the top of the foam dries out.

Forced drainage under pressure

In forced drainage, an external pressure gradient drives liquid through the foam (for example, by adding liquid at the top at a controlled rate). This setup is widely used experimentally because it produces a well-defined drainage front that moves through the foam at a measurable velocity.

Models for forced drainage balance the applied pressure gradient against gravity and viscous resistance. The liquid velocity is typically higher and the liquid distribution more uniform than in free drainage.

Channels vs. nodes approach

Drainage models fall into two categories based on where the dominant flow resistance lies:

• Channel-dominated models assume the Plateau borders control drainage, with negligible resistance at the nodes. These predict drainage rate scales as $\varepsilon^{1/2}$.
• Node-dominated models treat the nodes as the primary bottleneck. These predict different scaling behavior.

Which approach is appropriate depends on the foam's liquid fraction, bubble size, and interfacial mobility. Experiments with rigid (immobile) interfaces tend to match channel-dominated predictions, while mobile interfaces favor node-dominated behavior.

Incorporating Marangoni flows

Surface tension gradients along the interfaces generate Marangoni flows that can oppose gravitational drainage. These gradients arise from local variations in surfactant concentration (caused by stretching or compression of the interface) or from temperature differences.

Advanced drainage models couple the liquid flow equations with surfactant transport equations along the interface. Including Marangoni effects is important for accurately predicting drainage in surfactant-stabilized foams, because these flows effectively stiffen the interface and slow drainage beyond what viscosity alone would predict.

Experimental techniques for studying foams

A range of experimental methods probe foam structure, drainage kinetics, and mechanical properties. These techniques are essential for validating models and understanding real foam behavior.

Multiple light scattering methods

Multiple light scattering (MLS) techniques, including diffusing-wave spectroscopy (DWS), analyze temporal fluctuations in scattered light intensity from a foam sample. Because light scatters many times within the foam, these methods are sensitive to very small bubble motions and rearrangements.

DWS can measure the mean square displacement of bubbles, providing information about viscoelastic properties and aging dynamics. It's a non-invasive technique that works well for opaque, concentrated foams where direct imaging is difficult.

Conductivity measurements of liquid fraction

The electrical conductivity of a foam depends on how much conductive liquid is present and how it's distributed. By placing electrodes at different heights in a foam column, you can measure the liquid fraction profile and track how it evolves over time as drainage proceeds.

This technique gives direct, quantitative data on drainage kinetics and is relatively simple to implement. The relationship between conductivity and liquid fraction is typically calibrated using known standards.

Confocal microscopy imaging

Confocal microscopy provides high-resolution 3D images of foam microstructure. Fluorescent dyes that selectively stain the liquid phase or the interfaces reveal bubble arrangement, film thickness, and Plateau border connectivity.

This technique enables quantitative measurements of bubble size distributions, local liquid fraction, and structural rearrangements. It's particularly valuable for studying drainage and coarsening at the single-bubble level, though it requires relatively transparent or thin foam samples.

Rheological characterization of foams

Rheology probes the mechanical response of foams, which connects directly to stability and application performance.

• Oscillatory shear rheology measures the storage modulus $G'$ (elastic response) and loss modulus $G''$ (viscous response), characterizing the foam's viscoelastic behavior.
• Steady shear rheology reveals flow behavior under continuous deformation, including shear-thinning and the presence of a yield stress (the minimum stress needed to make the foam flow).
• Time-dependent measurements track how foam properties evolve during aging, drainage, or coarsening.

These measurements are critical for applications where the foam must maintain its structure under mechanical loads, such as in construction materials or food products.