Foam

Foams are two-phase material systems where a gas is dispersed in a second, non-gaseous material, specifically, in which gas cells are enclosed by a distinct liquid or solid material. Foam "may contain more or less liquid according to circumstances", although in the case of gas-liquid foams, the gas occupies most of the volume.
In most foams, the volume of gas is large, with thin films of liquid or solid separating the regions of gas.

Etymology

The word derives from Old English fām, from Proto-Germanic faimaz, ultimately related to Sanskrit phéna.

Structure

One scale is the bubble: material foams are typically disordered and have a variety of bubble sizes. The Weaire–Phelan structure is reported in one primary philosophical source to be the best possible unit cell of a perfectly ordered foam, while Plateau's laws describe how soap-films form structures in foams.
Foams are examples of dispersed media. In general, gas is present, so it divides into gas bubbles of different sizes —separated by liquid regions that may form films, thinner and thinner when the liquid phase drains out of the system films. When the principal scale is small, i.e., for a very fine foam, this dispersed medium can be considered a type of colloid.

Formation

Several conditions are needed to produce foam: there must be mechanical work, surface active components that reduce the surface tension, and the formation of foam faster than its breakdown. To create foam, work is needed to increase the surface area :
where γ is the surface tension.
One of the ways foam is created is through dispersion, where a large amount of gas is mixed with a liquid. A more specific method of dispersion involves injecting a gas through a hole in a solid into a liquid. If this process is completed very slowly, then one bubble can be emitted from the orifice at a time as shown in the picture below.
One of the theories for determining the separation time is shown below; however, while this theory produces theoretical data that matches the experimental data, detachment due to capillarity is accepted as a better explanation.
The buoyancy force acts to raise the bubble, which is
where is the volume of the bubble, is the acceleration due to gravity, and ρ₁ is the density of the gas ρ₂ is the density of the liquid. The force working against the buoyancy force is the surface tension force, which is
where γ is the surface tension, and is the radius of the orifice. As more air is pushed into the bubble, the buoyancy force grows quicker than the surface tension force. Thus, detachment occurs when the buoyancy force is large enough to overcome the surface tension force.
In addition, if the bubble is treated as a sphere with a radius of and the volume is substituted in to the equation above, separation occurs at the moment when
Examining this phenomenon from a capillarity viewpoint for a bubble that is being formed very slowly, it can be assumed that the pressure inside is constant everywhere. The hydrostatic pressure in the liquid is designated by. The change in pressure across the interface from gas to liquid is equal to the capillary pressure; hence,
where R₁ and R₂ are the radii of curvature and are set as positive. At the stem of the bubble, R₃ and R₄ are the radii of curvature also treated as positive. Here the hydrostatic pressure in the liquid has to take into account z, the distance from the top to the stem of the bubble. The new hydrostatic pressure at the stem of the bubble is p₀z. The hydrostatic pressure balances the capillary pressure, which is shown below:
Finally, the difference in the top and bottom pressure equals the change in hydrostatic pressure:
At the stem of the bubble, the shape of the bubble is nearly cylindrical; consequently, either R₃ or R₄ is large while the other radius of curvature is small. As the stem of the bubble grows in length, it becomes more unstable as one of the radius grows and the other shrinks. At a certain point, the vertical length of the stem exceeds the circumference of the stem and due to the buoyancy forces the bubble separates and the process repeats.

Stability

Stabilization

The stabilization of foam is caused by van der Waals forces between the molecules in the foam, electrical double layers created by dipolar surfactants, and the Marangoni effect, which acts as a restoring force to the lamellae.
The Marangoni effect depends on the liquid that is foaming being impure. Generally, surfactants in the solution decrease the surface tension. The surfactants also clump together on the surface and form a layer as shown below.
For the Marangoni effect to occur, the foam must be indented as shown in the first picture. This indentation increases the local surface area. Surfactants have a larger diffusion time than the bulk of the solution—so the surfactants are less concentrated in the indentation.
Also, surface stretching makes the surface tension of the indented spot greater than the surrounding area. Consequentially—since the diffusion time for the surfactants is large—the Marangoni effect has time to take place. The difference in surface tension creates a gradient, which instigates fluid flow from areas of lower surface tension to areas of higher surface tension. The second picture shows the film at equilibrium after the Marangoni effect has taken place.
Curing a foam solidifies it, making it indefinitely stable at STP.

Destabilization

Witold Rybczynski and Jacques Hadamard developed an equation to calculate the velocity of bubbles that rise in foam with the assumption that the bubbles are spherical with a radius.
with velocity in units of centimeters per second. ρ₁ and ρ₂ is the density for a gas and liquid respectively in units of g/cm³ and ῃ₁ and ῃ₂ is the dynamic viscosity of the gas and liquid respectively in units of g/cm·s and g is the acceleration of gravity in units of cm/s².
However, since the density and viscosity of a liquid is much greater than the gas, the density and viscosity of the gas can be neglected, which yields the new equation for velocity of bubbles rising as:
However, through experiments it has been shown that a more accurate model for bubbles rising is:
Deviations are due to the Marangoni effect and capillary pressure, which affect the assumption that the bubbles are spherical. For laplace pressure of a curved gas liquid interface, the two principal radii of curvature at a point are R₁ and R₂. With a curved interface, the pressure in one phase is greater than the pressure in another phase. The capillary pressure P_c is given by the equation of:
where is the surface tension. The bubble shown below is a gas in a liquid and point A designates the top of the bubble while point B designates the bottom of the bubble.
At the top of the bubble at point A, the pressure in the liquid is assumed to be p₀ as well as in the gas. At the bottom of the bubble at point B, the hydrostatic pressure is:
where ρ₁ and ρ₂ is the density for a gas and liquid respectively. The difference in hydrostatic pressure at the top of the bubble is 0, while the difference in hydrostatic pressure at the bottom of the bubble across the interface is gz. Assuming that the radii of curvature at point A are equal and denoted by R_A and that the radii of curvature at point B are equal and denoted by R_B, then the difference in capillary pressure between point A and point B is:
At equilibrium, the difference in capillary pressure must be balanced by the difference in hydrostatic pressure. Hence,
Since, the density of the gas is less than the density of the liquid the left hand side of the equation is always positive. Therefore, the inverse of R_A must be larger than the R_B. Meaning that from the top of the bubble to the bottom of the bubble the radius of curvature increases. Therefore, without neglecting gravity the bubbles cannot be spherical. In addition, as z increases, this causes the difference in R_A and R_B too, which means the bubble deviates more from its shape the larger it grows.
Foam destabilization occurs for several reasons. First, gravitation causes drainage of liquid to the foam base, which Rybczynski and Hadamar include in their theory; however, foam also destabilizes due to osmotic pressure causes drainage from the lamellas to the Plateau borders due to internal concentration differences in the foam, and Laplace pressure causes diffusion of gas from small to large bubbles due to pressure difference. In addition, films can break under disjoining pressure, These effects can lead to rearrangement of the foam structure at scales larger than the bubbles, which may be individual or collective.

Mechanical properties

Liquid foams

Solid foams

In closed-cell foam, the gas forms discrete pockets, each completely surrounded by the solid material. In open-cell foam, gas pockets connect to each other. Solid foams, both open-cell and closed-cell, are considered as a sub-class of cellular structures. They often have lower nodal connectivity as compared to other cellular structures like honeycombs and truss lattices, and thus, their failure mechanism is dominated by bending of members. Low nodal connectivity and the resulting failure mechanism ultimately lead to their lower mechanical strength and stiffness compared to honeycombs and truss lattices.
The strength of foams can be impacted by the density, the material used, and the arrangement of the cellular structure. To characterize the mechanical properties of foams, compressive stress-strain curves are used to measure their strength and ability to absorb energy since this is an important factor in foam based technologies.