Foam Fix
Maria Mauri
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.
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.
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.
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:
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.
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]
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.
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]
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):
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.
where V \displaystyle V is the volume of the bubble, g \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
where γ is the surface tension, and r \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.
Examining this phenomenon from a capillarity viewpoint for a bubble that is being formed very slowly, it can be assumed that the pressure p \displaystyle p inside is constant everywhere. The hydrostatic pressure in the liquid is designated by p 0 \displaystyle p_0 . The change in pressure across the interface from gas to liquid is equal to the capillary pressure; hence,
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.[13]
The stabilization of a 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.
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 r \displaystyle r .
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/cms 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:
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.[16] With a curved interface, the pressure in one phase is greater than the pressure in another phase. The capillary pressure Pc is given by the equation of:
where γ \displaystyle \gamma is the surface tension. The bubble shown below is a gas (phase 1) in a liquid (phase 2) and point A designates the top of the bubble while point B designates the bottom of the bubble.
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 RA must be larger than the RB. 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 RA and RB too, which means the bubble deviates more from its shape the larger it grows.[13]
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 (T1 process) or collective (even of the "avalanche" type).
Being a multi-scale system involving many phenomena, and a versatile medium, foam can be studied using many different techniques. Considering the different scales, experimental techniques are diffraction ones, mainly light scattering techniques (DWS, see below, static and dynamic light scattering, X-rays and neutron scattering) at sub-micrometer scales, or microscopic ones. Considering the system as continuous, its bulk properties can be characterized by light transmittance but also conductimetry. The correlation between structure and bulk is evidenced more accurately by acoustics in particular. The organisation between bubbles has been studied numerically using sequential attempts of evolution of the minimum surface energy either at random (Pott's model) or deterministic way (surface evolver). The evolution with time (i.e., the dynamics) can be simulated using these models, or the bubble model (Durian), which considers the motion of individual bubbles.
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