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Pour a bottle of beer. Restraining your thirst for the moment, admire its lively performance. One by one bubbles of gas are nucleated, rise and crowd together at the surface."

Denis Weaire, Stefan Hutzler, The Physics of foam; Claredon Press; Oxford, 1999, p. 1.


Physics of foams

Formation of bubbles

Vapor can be entrained into a liquid by stirring, vapor can be created by evaporation or released when the pressure is reduced. Whatever the source, as the vapor rises through the liquid its interface with the liquid can adsorb surface active solutes just as those solutes are adsorbed on the air/liquid surface.

As this schematic illustrates, the surfactant-covered-bubble touching a surfactant covered surface encounters the same repulsive forces as any other surfactant-covered-surfaces would; i.e. the surfactant-covered-surfaces of emulsion droplets and dispersed particles. The repulsive forces will be the same; the attractive forces even less; hence the bubble is stabilized just below the air/liquid surface, possibly raising it a little but not penetrating through the surfactant layers.

Bubbles at the surface are stabilized by the same mechanisms that stabilize emulsions and dispersions; and they are destroyed by the same mechanisms.


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Coalescence of bubbles into a foam

Separate bubbles at the air/liquid interface move across the surface freely. But when they approach each other (or a cluster of bubbles), the curvature between them creates a Laplace pressure. This reduced pressure in the liquid between them causes them to move toward each other.
The bubbles continue to approach one another until the slight curvature of the meniscus between them just matches the hydrostatic head of the liquid between them. The formation of a foam mass from individual bubbles is spontaneous. This process is easily seen when watching bubbles at the surface of milk, or coffee, or beer (although you have to be quick for the beer.)

The foam mass builds and builds as more and more bubbles rise to the surface and are pulled together by Laplace pressure.


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Bubble geometry

The morphology of all foams are determined by the minimization of the surface area of liquid films balanced against the compression of the bubbles.

Plateau's laws follow:

(1) Along an edge, three and only three liquid lamellae meet. They are equally inclined to one another. Hence the dihedral angles are 120o.

(2) At a point, four and only four of those edges meet. They are equally inclined to one another. Hence, the tetrathedral angle of just greater than 109o.

When a bubble ruptures in a foam, the entire foam re-arranges to satisfy Plateau's laws.

Clusters of a few bubbles demonstrate these laws most clearly. (Morrison and Ross, Chapter 23.)

Morrison, Fig. 23.1 (with correction at bottom left)

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Equation of state for foam?

The regularity of foam, its structure following such simple geometric laws, has led to an interesting speculation. The speculation arises from analyses of the properties of simple bubble clusters. Here is the derivation for just one bubble:

For a single bubble, the Laplace pressure is: <math>p-P=\frac{4\sigma }{r}\,\!</math>
For a sphere: <math>\frac{V}{A}=\frac{r}{6}\,\!</math>
For an ideal gas: <math>pV=nRT\,\!</math>
Combining gives: <math>PV+\frac{2}{3}\sigma A=nRT\,\!</math>

This last equation has been shown to be true for a few simple clusters, but never convincingly shown to be true for arbitrary clusters.

Nevertheless, it is a remarkable statement. For any foam, the external pressure is known, the surface tension is easily measured, the number of moles of gas contained can be measured by collapsing the foam (if necessary), and the temperature is known. Therefore the total internal surface area can be calculated from "The equation of state of a foam"!

If the foam collapses in a closed container, the following is useful and measurable.

A differential form is: <math>dA=-\frac{3V}{2\sigma }dP\,\!</math>

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Plateau borders and Gibbs angles

The Plateau borders are the thin lamelae next to "A" and "B". The Gibbs angles are indicated by "C". The pressure is reduced in the Gibbs angles by the Laplace equation, so that liquid flows from the Plateau borders into the Gibbs angles.
Exerowa and Kruglyakov, p.15
Gravity causes liquid to drain from the foam down connected Plateau borders. These thin until disjoining pressures balance hydrostatic pressures.
Weaire Fig. 1.14

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The geometry of foams

In a three-dimensional dry foam, lamellae meet at the Plateau borders with vertex angles of 120 degrees. Four lamellae meet at the Gibbs angle, in the limit about 109.5 degrees. (Weaire, p. 23-26).
Weaire, Fig. 2.4

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The Kelvin tetrakaidecahedron

A classic mathematical problem is the search for the shape that fills space (tesselation) with minimum surface area. Mathematics has not found the limit yet although the mathematicians cannot be far off.

Foams composed of equal sized bubbles do so spontaneously (it is asserted!).

Weaire 13.4
Lord Kelvin, from observations of bubbles, suggested the 14 sided figure photographed here; the tetrakaidecahedron (of course!)

Note that no foam films has any curvature (the bubbles all have the same pressure, but they are not flat!)

This is his original paper on the subject from Acta Mathematica. His notes are amusing, if you read them. Media:Kelvin_Cell.pdf

Unfortunately for Lord Kelvin, this structure is not the least area/volume ratio to tesselate space.

Morrison, Fig. 23.2

I found this very amusing: a paper cutout of a Kelvin cell. From this website. The bottom line: "Make as many copies as needed to fill all available space."


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Stability of foams

Electrostatic Stabilization of Foams.png
If the foam film is electrically charged, then as the film thins, electrical double layers overlap and the surfaces repel each other.

The same factors that reduce electrocratic dispersion stability reduces the stability of these lamallae.

Steric Stabilization of Foams.png
If the foam film has adsorbed polymer at its interfaces, then as the film thins, polymer molecules overlap and the surfaces repel each other.

The same factors that reduce steric dispersion stability reduces the stability of these lamallae.

Visual observations of draining lamallae show the refraction bands of the gradually thinning film. When the films are stable, the final film is too thin to refract light and appears "black". The upper portion of the film in the sixth frame is still intact, but thinner than the wave length of visible light, less than about 400 hundred nanometers.
Exerowa and Kruglyakov, p. 221
Wasan et al. discovered the thinning of black films was stepwise and not gradual. Over time they established that stable thicknesses are layers of close-packed micelles. The layers of micelles are stable until a few from one layer diffuse out of the film and then the entire layer moves away. This leads to a sudden change in the film thickness.
Wasan et al established that either micelles (upper structure) or, more likely, liquid crystal phases (lower structure) give long time stability to foam films. In this diagram the liquid is nonpolar so that hydrophobic chains are drawn in the liquid regions.

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Film measurements

Exerowa and Kruglyakov, p. 44
Thin films can be captured in a frame and kept in a controlled environment. The film thickness can be varied by increasing or reducing the liquid pressure outside the film. Since the film thickness can be measured optically, this provides a direct measurement of the disjoining pressure as a function of distance.
Elasticity of surface and thin films: ADSORPTION IS SLOW! IN THIN FILMS EVEN SLOWER!

Elasticity is the ratio of the increase in surface tension from a relative increase in surface area. (For a foam film.)

<math>E=\frac{2d\sigma }{d\ln A}\,\!</math>

When the surface is not in equilibrium (the common case) with the bulk, this is a Marangoni effect.

Ross & Suzin, Langmuir, 1985, 1, 145-9.
Dynamic foam stability is easily measured with a flow of gas creating a steady stream of bubble. A surprising discovery was that a preferred shape is conical. Small instabilities in the foam height are dampened: if the foam is slightly more stable for a time, the foam height increases, more surface is exposed, and the bubbles collapse sooner, reducing the foam height.

Exploring Foams

How do you explore a foam? You can't touch it, obviously, or you'll pop the bubbles and change the foam. Somehow, the researchers need a way to measure the traits of a foam without disturbing it. The answer is Light

In one method, called "diffuse-transmission spectroscopy," the scientists shine the beam through the foam and measure how much of the light reaches the point on the other side. In a foam with only a few, very large bubbles, most of the light will pass straight through with little interference; in a foam of many, tiny bubbles, the light will get scattered by the bubble membranes. Measuring how much light reaches the far side lets the scientists quantify the average bubble size.

The motion of the bubbles can also be detected using monochromatic (single-colored) light. As a laser beam passes through the foam, bubble membranes in motion cause a slight Doppler effect, shifting the frequency--and hence the color--of the light. Watching these ever-so-slight shifts in the light's frequency tells researchers how fast the bubbles are moving and in what direction. This technique is called "diffusing-wave spectroscopy."


Above: A schematic diagram of diffusing-wave spectroscopy.

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Foams and the phase diagram

Morrison and Ross, Fig. 16-1
Phase diagram of diethylene glycol and ethyl salicylate. The doted lines are the Gibbs excess concentrations (<math>{\mu m}/{m^{2}}\;\,\!</math>) of ethyl salicylate. The dotted lines are the cosorption contours.
Morrison and Ross, Fig. 16-2
Phase diagram and interpolated isaphroic lines of the two-component system diethylene glycol and ethyl salicylate. The average lifetime of a bubble, <math>\Sigma \,\!</math>, is measured in seconds.
Morrison and Ross, Fig. 23.7
The Ross-Nishioka effect in fortified bourbon whiskey on dilution with water. Reading from right to left, as the solutions approach a phase boundary, bubble stability increases until phase separation creates a foam inhibitor.

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Ross' Rule - Capillarity and the phase diagram

  • Adsorption precedes precipitation.
  • Dispersion stability suddenly changes.
  • Foaming can suddenly increase or disappear.
  • Foaming is an indication of some component ready to precipitate.
  • Surface and interfacial tensions change abruptly near phase boundaries.
  • The number and size of precipitates depend strongly on the position in the phase diagram.
  • Sudden changes in product behavior may indicate some component is near its solubility limit.

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Three-phase foams, From the “Activated Sludge Pages”
  • Powders with finite (receding) contact angles sit at the air/liquid surface.
  • Particles may stabilize thin films if they have low contact angles, holding the two interfaces apart.
  • The finer the particles, the better the stability; lead, silica, ferric oxide are examples.
  • "Collectors" are sometimes added to aid this dewetting of particles.
  • The particles move with the bubble - flotation.
Thomas Kostakis, Rammile Ettelaie, and Brent S. Murray, Langmuir 2006, 22, 1273-1280
Fraction (F) of bubbles remaining as a function of time (t) formed in dispersions of 1wt%of 33% SiOR particles at different NaCl concentrations: 3 mol dm-3 ([), 2 mol dm-3 (0), 1 mol dm-3 (2), and 0.5 mol dm-3 (4).

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Foams and antifoams

Mechanisms of antifoaming are:

  • Contact with a hydrophobic interface, such as Teflon/water, siliconized solid/water.
  • Addition of an insoluble, low-surface-tension liquid to a standing foam. Typically, naturally occurring, oils, lard, fatty acids and alcohols, silicone oils,
  • Presence of vapor of a volatile liquid.
  • Contact with a hot source, such as an electrically heated wire.
  • Destruction of a foaming agent by precipitation or heat. e.g. Soap added to a protein (as in distillation of whiskey, etc.) or acid added to a soap solution or cationic agent added to an anionic agent.
  • Combating the Marangoni effect by a rapid attainment of static surface tension on addition of low molecular weight amphipaths.

With an antifoam drop adsorbed on one surface, electrostatic or steric stabilization is lost. The practical procedure is to spray the antifoam on the top surface. Each drop of antifoam breaks foam films as it falls through the foam.
(a) Antifoam drop

(b) Entering the surface

(c ) Leading to rupture of the film.

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Silicone antifoams

  • Silicone oil is emulsified into water with about HLB = 8 dispersant. Silicones are "activated" by the addition of 3-4% silica. Hydrophilic silica is heated in the oil.
  • The PDMS spreads, but is retarded by the silica leading to a reasonable sized weakness in the lamella.
  • Hypothesis: it is the silica particle that is the defoamer! The silicone oil is only the carrier.

“Silicone antifoams” by Kulkarni et al. in Prud'homme and Khan, Chapter 14.

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Lung surfactant

Exerowa and Kruglyakov, p. 753
The alveolar surface in the lung.

Lung surfactant, a lipo-protein complex, is a highly surface-active material found in the fluid lining the air-liquid interface of the alveolar surface. Surfactant plays a dual function of preventing alveolar collapse during breathing cycle and protection of the lungs from injuries and infections caused by foreign bodies and pathogens.

Pulmonary surfactant is essential for normal breathing, alveolar stability and host defense system in the lungs. Basically, three very interesting biophysical properties of pulmonary surfactant underlie its physiological and immunological functions: 1) Once secreted to the alveolar spaces, surfactant adsorbs rapidly to the air-liquid interface (this happens during a newborn baby’s first breath). 2) Once at the interface, surfactant films reduce surface tension to extremely low values when compressed during expiration (this means that our lungs don’t collapse when we breath out). 3) Surfactant proteins recognize bacterial, fungal and viral surface oligosaccharides and thus can opsonize these pathogens.

The surface tension of the alveolar air-water interface provides the retractive force opposing lung inflation. The presence of surfactant in the fluid film can lower air-water surface tensions to near zero values. This ensures that the alveolar space is open during the whole respiratory cycle preventing intra-pulmonary shunts resulting in inadequate oxygenation of the blood. Thus, the net benefit is reduced work of breathing

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Carbon nanofoam is an allotrope of carbon. An allotrope is a variant of a substance composed of only one type of atom. The best-known allotropes of carbon are graphite and diamond. Carbon nanofoam, the 5th allotrope of carbon, was discovered in 1997 by Andrei V. Rode and his team at the Australian National University in Canberra, in collaboration with Ioffe Physico-Technical Institute in St Petersburg. Its molecular structure consists of carbon tendrils bonded together in a low-density, mistlike arrangement.

Carbon nanofoam is similar in some respects to carbon and silicon aerogels produced before, but with about 100 times less density. Carbon nanofoam has been extensively studied under electron microscope by John Giapintzakis and team at the University of Crete. Its production and study has primarily been pioneered by Greek, Russian, and Australian scientists.


The carbon nanofoam is produced by firing a high-pulse, high-energy laser at graphite or disordered solid carbon suspended in some inert gas such as argon. Like aerogels, carbon nanofoam has extremely high surface area and acts as a good insulator, capable of being exposed to thousands of degrees Fahrenheit before deforming. It is practically transparent in appearance, consisting of mostly air, and fairly brittle.

One of the most unusual properties displayed by carbon nanofoam is that of ferromagnetism; it is attracted to magnets, like iron. This property vanishes a few hours after the nanofoam is made, though it can be preserved by cooling the nanofoam to extremely low temperatures, about -183° Celsius (-297° Fahrenheit). Other allotropes of carbon, such as fullerenes at high pressure, display some properties of magnetism, but not at the level carbon nanofoam does. The magnetic properties of carbon nanofoam remind scientists that the magnetism of a substance cannot be determined simply by the type of substance, but by its allotrope and temperature as well.

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