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Foams are materials formed by trapping pockets of gas in a liquid or solid.^[1][2][3]

A bath sponge and the head on a glass of beer are examples of foams. In most foams, the volume of gas is large, with thin films of liquid or solid separating the regions of gas. Soap foams are also known as suds.

Solid foams can be closed-cell or open-cell. 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. A bath sponge is an example of an open-cell foam: water easily flows through the entire structure, displacing the air. A sleeping mat is an example of a closed-cell foam: gas pockets are sealed from each other so the mat cannot soak up water.

Foams are examples of dispersed media. In general, gas is present, so it divides into gas bubbles of different sizes (i.e., the material is polydisperse)—separated by liquid regions that may form films, thinner and thinner when the liquid phase drains out of the system films.^[4] When the principal scale is small, i.e., for a very fine foam, this dispersed medium can be considered a type of colloid.

Foam can also refer to something that is analogous to foam, such as quantum foam.

Structure

A foam is, in many cases, a multi-scale system.

One scale is the bubble: material foams are typically disordered and have a variety of bubble sizes. At larger sizes, the study of idealized foams is closely linked to the mathematical problems of minimal surfaces and three-dimensional tessellations, also called honeycombs. The Weaire–Phelan structure is considered the best possible (optimal) unit cell of a perfectly ordered foam,^[5] while Plateau's laws describe how soap-films form structures in foams.

At lower scale than the bubble is the thickness of the film for metastable foams, which can be considered a network of interconnected films called lamellae. Ideally, the lamellae connect in triads and radiate 120° outward from the connection points, known as Plateau borders.

An even lower scale is the liquid–air interface at the surface of the film. Most of the time this interface is stabilized by a layer of amphiphilic structure, often made of surfactants, particles (Pickering emulsion), or more complex associations.

Mechanical properties of solid foams

Solid foams, both open-cell and closed-cell, are considered as a sub-class of cellular structures. They often have lower nodal connectivity^[jargon] 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.^[6][7]

The strength of foams can be impacted by the density, the material used, and the arrangement of the cellular structure (open vs closed and pore isotropy). 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.

Elastomeric foam

For elastomeric cellular solids, as the foam is compressed, first it behaves elastically as the cell walls bend, then as the cell walls buckle there is yielding and breakdown of the material until finally the cell walls crush together and the material ruptures.^[8] This is seen in a stress-strain curve as a steep linear elastic regime, a linear regime with a shallow slope after yielding (plateau stress), and an exponentially increasing regime. The stiffness of the material can be calculated from the linear elastic regime ^[9] where the modulus for open celled foams can be defined by the equation:

${\displaystyle \left({\frac {E^{*}}{E_{s}}}\right)_{f}=C_{f}\left({\frac {\rho ^{*}}{\rho _{s}}}\right)^{2}}$

where ${\displaystyle E_{s}}$ is the modulus of the solid component, ${\displaystyle E^{*}}$ is the modulus of the honeycomb structure, ${\displaystyle C_{f}}$ is a constant having a value close to one, ${\displaystyle \rho ^{*}}$ is the density of the honeycomb structure, and ${\displaystyle \rho _{s}}$ is the density of the solid. The elastic modulus for closed cell foams can be described similarly by:

${\displaystyle \left({\frac {E^{*}}{E_{s}}}\right)_{f}=C_{f}\left({\frac {\rho ^{*}}{\rho _{s}}}\right)^{3}}$

where the only difference is the exponent in the density dependence. However, in real materials, a closed-cell foam has more material at the cell edges which makes it more closely follow the equation for open-cell foams.^[10] The ratio of the density of the honeycomb structure compared with the solid structure has a large impact on the modulus of the material. Overall, foam strength increases with density of the cell and stiffness of the matrix material.

Energy of deformation

Another important property which can be deduced from the stress strain curve is the energy that the foam is able to absorb. The area under the curve (specified to be before rapid densification at the peak stress), represents the energy in the foam in units of energy per unit volume. The maximum energy stored by the foam prior to rupture is described by the equation:^[8]

${\displaystyle {\frac {W_{max}}{E_{s}}}=0.05\left({\frac {\rho ^{*}}{\rho _{s}}}\right)^{2}\left}$

This equation is derived from assuming an idealized foam with engineering approximations from experimental results. Most energy absorption occurs at the plateau stress region after the steep linear elastic regime.

Directional dependence

The isotropy of the cellular structure and the absorption of fluids can also have an impact on the mechanical properties of a foam. If there is anisotropy present, then the materials response to stress will be directionally dependent, and thus the stress-strain curve, modulus, and energy absorption will vary depending on the direction of applied force.^[11] Also, open-cell structures which have connected pores can allow water or other liquids to flow through the structure, which can also affect the rigidity and energy absorption capabilities.^[12]

Formation

Several conditions are needed to produce foam: there must be mechanical work, surface active components (surfactants) that reduce the surface tension, and the formation of foam faster than its breakdown. To create foam, work (W) is needed to increase the surface area (ΔA):

${\displaystyle W=\gamma \Delta A\,\!}$

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

${\displaystyle F_{b}=Vg(\rho _{2}-\rho _{1})\!}$

where ${\displaystyle V}$ is the volume of the bubble, ${\displaystyle g}$ 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

${\displaystyle F_{s}=2r\pi \gamma \!}$,

where γ is the surface tension, and ${\displaystyle r}$ 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.

${\displaystyle Vg(\rho _{2}-\rho _{1})>2r\pi \gamma \!}$

In addition, if the bubble is treated as a sphere with a radius of ${\displaystyle R}$ and the volume ${\displaystyle V}$ is substituted in to the equation above, separation occurs at the moment when

${\displaystyle R^{3}={\frac {3r\gamma }{2g(\rho _{2}-\rho _{1})}}\!}$

Examining this phenomenon from a capillarity viewpoint for a bubble that is being formed very slowly, it can be assumed that the pressure ${\displaystyle p}$ inside is constant everywhere. The hydrostatic pressure in the liquid is designated by ${\displaystyle p_{0}}$. The change in pressure across the interface from gas to liquid is equal to the capillary pressure; hence,

${\displaystyle p-p_{0}=\gamma \left({\frac {1}{R_{1}}}+{\frac {1}{R_{2}}}\right)\!}$

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 in account z, the distance from the top to the stem of the bubble. The new hydrostatic pressure at the stem of the bubble is p₀(ρ₁ − ρ₂)z. The hydrostatic pressure balances the capillary pressure, which is shown below:

${\displaystyle p-<!--\; -\; -->p_0-<!--\; -\; -->({\mathrm{\rho <!--\; \rho \; -->}}_2-<!--\; -\; -->{\mathrm{\rho <!--\; \rho \; -->}}_1)}$Zdroj:https://en.wikipedia.org?pojem=Foam
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