Foams are two-phase
material
A material is a matter, substance or mixture of substances that constitutes an Physical object, object. Materials can be pure or impure, living or non-living matter. Materials can be classified on the basis of their physical property, physical ...
systems where a gas is dispersed in a second, non-gaseous material, specifically, in which
gas
Gas is a state of matter that has neither a fixed volume nor a fixed shape and is a compressible fluid. A ''pure gas'' is made up of individual atoms (e.g. a noble gas like neon) or molecules of either a single type of atom ( elements such as ...
cells are enclosed by a distinct
liquid
Liquid is a state of matter with a definite volume but no fixed shape. Liquids adapt to the shape of their container and are nearly incompressible, maintaining their volume even under pressure. The density of a liquid is usually close to th ...
or
solid
Solid is a state of matter where molecules are closely packed and can not slide past each other. Solids resist compression, expansion, or external forces that would alter its shape, with the degree to which they are resisted dependent upon the ...
material.
[ Note, this source focuses only on liquid foams.][ Note, this source also focuses on liquid foams.] Foam "may contain more or less liquid
r solid
R, or r, is the eighteenth Letter (alphabet), letter of the Latin alphabet, used in the English alphabet, modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is English alphab ...
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
Gas is a state of matter that has neither a fixed volume nor a fixed shape and is a compressible fluid. A ''pure gas'' is made up of individual atoms (e.g. a noble gas like neon) or molecules of either a single type of atom ( elements such as ...
is large, with thin films of liquid or solid separating the regions of gas.
Etymology
The word derives from the
medieval German and otherwise obsolete ''veim'', in reference to the "frothy head forming in the glass once the beer has been freshly poured" (cf. ''ausgefeimt'').
Structure
A foam is, in many cases, a multi-scale system.
One scale is the bubble:
material
A material is a matter, substance or mixture of substances that constitutes an Physical object, object. Materials can be pure or impure, living or non-living matter. Materials can be classified on the basis of their physical property, physical ...
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 surface
In mathematics, a minimal surface is a surface that locally minimizes its area. This is equivalent to having zero mean curvature (see definitions below).
The term "minimal surface" is used because these surfaces originally arose as surfaces that ...
s and three-dimensional
tessellations
A tessellation or tiling is the covering of a surface, often a plane (mathematics), plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to high-dimensiona ...
, also called
honeycombs
A honeycomb is a mass of hexagonal prismatic cells built from beeswax by honey bees in their nests to contain their brood (eggs, larvae, and pupae) and stores of honey and pollen.
Beekeepers may remove the entire honeycomb to harvest honey. H ...
. The
Weaire–Phelan structure
In geometry, the Weaire–Phelan structure is a three-dimensional structure representing an idealised foam of equal-sized bubbles, with two different shapes. In 1993, Denis Weaire and Robert Phelan found that this structure was a better solutio ...
is reported in one primary philosophical source to be the best possible (optimal)
unit cell
In geometry, biology, mineralogy and solid state physics, a unit cell is a repeating unit formed by the vectors spanning the points of a lattice. Despite its suggestive name, the unit cell (unlike a unit vector
In mathematics, a unit vector i ...
of a perfectly ordered foam, while
Plateau's laws
Plateau's laws describe the structure of soap films. These laws were formulated in the 19th century by the Belgian physicist Joseph Plateau from his experimental observations. Many patterns in nature are based on foams obeying these laws.
Laws ...
describe how soap-films form structures in foams.
At lower scale than the bubble is the thickness of the film for
metastable
In chemistry and physics, metastability is an intermediate energetic state within a dynamical system other than the system's state of least energy.
A ball resting in a hollow on a slope is a simple example of metastability. If the ball is onl ...
foams, which can be considered a network of interconnected films called
lamellae
Lamella (: lamellae) means a small plate or flake in Latin, and in English may refer to:
Biology
* Lamella (mycology), a papery rib beneath a mushroom cap
* Lamella (botany)
* Lamella (surface anatomy), a plate-like structure in an animal
* Lame ...
. 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
In chemistry, an amphiphile (), or amphipath, is a chemical compound possessing both hydrophilic (''water-loving'', polar) and lipophilic (''fat-loving'', nonpolar) properties. Such a compound is called amphiphilic or amphipathic. Amphiphilic c ...
structure, often made of
surfactant
Surfactants are chemical compounds that decrease the surface tension or interfacial tension between two liquids, a liquid and a gas, or a liquid and a solid. The word ''surfactant'' is a Blend word, blend of "surface-active agent",
coined in ...
s, particles (
Pickering emulsion
A Pickering emulsion, sometimes named Ramsden emulsion, is an emulsion that is stabilized by solid particles (for example colloidal silica) which adsorb onto the interface between the water and oil phases. Typically, the emulsions are either wate ...
), or more complex associations.
Foams are examples of
dispersed media
A dispersion is a system in which distributed particles of one material are dispersed in a continuous phase of another material. The two phases may be in the same or different states of matter.
Dispersions are classified in a number of diff ...
. In general, gas is present, so it divides into gas
bubbles
Bubble, Bubbles or The Bubble may refer to:
Common uses
* Bubble (physics), a globule of one substance in another, usually gas in a liquid
** Soap bubble
* Economic bubble, a situation where asset prices are much higher than underlying fundame ...
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
A film, also known as a movie or motion picture, is a work of Visual arts, visual art that simulates experiences and otherwise communicates ideas, stories, perceptions, emotions, or atmosphere through the use of moving images that are gen ...
. When the principal scale is small, i.e., for a very fine foam, this dispersed medium can be considered a type of
colloid
A colloid is a mixture in which one substance consisting of microscopically dispersed insoluble particles is suspended throughout another substance. Some definitions specify that the particles must be dispersed in a liquid, while others exte ...
.
Formation
Several conditions are needed to produce foam: there must be mechanical work,
surface active components (surfactants) that reduce the
surface tension
Surface tension is the tendency of liquid surfaces at rest to shrink into the minimum surface area possible. Surface tension (physics), tension is what allows objects with a higher density than water such as razor blades and insects (e.g. Ge ...
, and the formation of foam faster than its breakdown. To create foam,
work
Work may refer to:
* Work (human activity), intentional activity people perform to support themselves, others, or the community
** Manual labour, physical work done by humans
** House work, housework, or homemaking
** Working animal, an ani ...
(W) is needed to increase the
surface area
The surface area (symbol ''A'') of a solid object is a measure of the total area that the surface of the object occupies. The mathematical definition of surface area in the presence of curved surfaces is considerably more involved than the d ...
(ΔA):
:
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
Buoyancy (), or upthrust, is the force exerted by a fluid opposing the weight of a partially or fully immersed object (which may be also be a parcel of fluid). In a column of fluid, pressure increases with depth as a result of the weight of t ...
force acts to raise the bubble, which is
:
where
is the volume of the bubble,
is the acceleration due to gravity, and ρ
1 is the density of the gas ρ
2 is the density of the liquid. The force working against the buoyancy force is the
surface tension
Surface tension is the tendency of liquid surfaces at rest to shrink into the minimum surface area possible. Surface tension (physics), tension is what allows objects with a higher density than water such as razor blades and insects (e.g. Ge ...
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 R
1 and R
2 are the radii of curvature and are set as positive. At the stem of the bubble, R
3 and R
4 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 ''p''
0(''ρ''
1 − ''ρ''
2)''z''. 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 R
3 or R
4 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.
[Bikerman, J.J. "Formation and Structure" in ''Foams'' New York, Springer-Verlag, 1973. ch 2. sec 24–25]
Stability
Stabilization
The stabilization of foam is caused by
van der Waals force
In molecular physics and chemistry, the van der Waals force (sometimes van der Waals' force) is a distance-dependent interaction between atoms or molecules. Unlike ionic or covalent bonds, these attractions do not result from a chemical elec ...
s between the molecules in the foam,
electrical double layer
Electricity is the set of physical phenomena associated with the presence and motion of matter possessing an electric charge. Electricity is related to magnetism, both being part of the phenomenon of electromagnetism, as described by Maxwel ...
s created by
dipolar
In physics, a dipole () is an electromagnetic phenomenon which occurs in two ways:
* An electric dipole deals with the separation of the positive and negative electric charges found in any electromagnetic system. A simple example of this system ...
surfactants, and the
Marangoni effect
The Marangoni effect (also called the Gibbs–Marangoni effect) is the mass transfer along an Interface (chemistry), interface between two phases due to a gradient of the surface tension. In the case of temperature dependence, this phenomenon may ...
, 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. ρ
1 and ρ
2 is the density for a gas and liquid respectively in units of g/cm
3 and ῃ
1 and ῃ
2 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/s
2.
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
The Marangoni effect (also called the Gibbs–Marangoni effect) is the mass transfer along an Interface (chemistry), interface between two phases due to a gradient of the surface tension. In the case of temperature dependence, this phenomenon may ...
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 R
1 and R
2. 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 (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.

At the top of the bubble at point A, the pressure in the liquid is assumed to be p
0 as well as in the gas. At the bottom of the bubble at point B, the hydrostatic pressure is:
:
:
where ρ
1 and ρ
2 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''(''ρ''
2 − ''ρ''
1). 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
In physics, gravity (), also known as gravitation or a gravitational interaction, is a fundamental interaction, a mutual attraction between all massive particles. On Earth, gravity takes a slightly different meaning: the observed force b ...
causes drainage of liquid to the foam base, which Rybczynski and Hadamar include in their theory; however, foam also destabilizes due to
osmotic pressure
Osmotic pressure is the minimum pressure which needs to be applied to a Solution (chemistry), solution to prevent the inward flow of its pure solvent across a semipermeable membrane.
It is also defined as the measure of the tendency of a soluti ...
causes drainage from the lamellas to the Plateau borders due to internal concentration differences in the foam, and
Laplace pressure
The Laplace pressure is the pressure difference between the inside and the outside of a curved surface that forms the boundary between two fluid regions. The pressure difference is caused by the surface tension of the interface between liquid and ...
causes diffusion of gas from small to large bubbles due to pressure difference. In addition, films can break under
disjoining pressure
In surface chemistry, disjoining pressure (symbol ) according to an IUPAC definition arises from an attractive interaction between two surfaces. For two flat and parallel surfaces, the value of the disjoining pressure (i.e., the force per unit are ...
, 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).
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 (open vs closed and pore isotropy). To characterize the
mechanical properties
A material property is an intensive property of a material, i.e., a physical property or chemical property that does not depend on the amount of the material. These quantitative properties may be used as a metric by which the benefits of one mate ...
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
An elastomer is a polymer with viscoelasticity (i.e. both viscosity and elasticity) and with weak intermolecular forces, generally low Young's modulus (E) and high failure strain compared with other materials. The term, a portmanteau of ''elas ...
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.
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 where the
modulus for open celled foams can be defined by the equation:
where
is the modulus of the solid component,
is the modulus of the honeycomb structure,
is a constant having a value close to one,
is the density of the honeycomb structure, and
is the density of the solid. The elastic modulus for closed cell foams can be described similarly by:
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. 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: