Foam physics: the simplest example of soft condensed matter
W. Drenckhan, S. Hutzler and D. Weaire
Physics Department, Trinity College Dublin, Dublin 2, Ireland
Abstract. A liquid foam is an easily accessible system that displays typical features of soft condensed matter. Here we
summarise recent theory and experiments regarding foam structure, drainage and rheology, including the flow of trains of
bubbles through specifically designed channels.
INTRODUCTION
In his Nobel acceptance speech in 1991 Pierre Gilles de Gennes highlighted the term “soft matter” when
speaking about polymers, surfactants and liquid crystals (de Gennes 1992). He concluded his speech by reciting a
poem on soap bubbles. Soap bubbles, aggregated in a foam, may be regarded as the simplest example of soft matter,
in terms of characterisation, theory and experimental accessibility (Weaire and Hutzler, 1999). This “possibility of
simple experiments” was indeed described by de Gennes as a “major feature” of the style of soft matter research.
Soft matter is now central to the description of a large variety of physical and biological systems, such as
colloids, emulsions and membranes. In 2000 a new journal, The European Physical Journal E - Soft Matter, was
introduced to cover all these aspects. A strict definition of this new type of matter, sometimes called “complex fluid”
or structured fluid does not seem to exist, but it is agreed that it should feature certain ingredients.
Amongst the physical properties arising out of these structures, and characterising soft matter, are non-linear
mechanical properties (e.g. shear thinning and shear bands), structural phase transitions and non-Newtonian flow
properties. Whether soft matter is solid or liquid will depend on circumstances: that is the very meaning of “soft”.
The length scales of the relevant structure may be mesoscopic (roughly, µm), thus between the atomic and
macroscopic scale, or in some cases macroscopic (as in most foams). An important difference between typical foam
and some more mesoscopic systems is the lack of significant thermal motion.
THE STATICS AND DYNAMICS OF FOAMS
A foam made by blowing air through a surfactant solution (washing-up liquid) is generally stable enough to be
described in purely static terms. What is the topology of a foam structure, what is its geometry, what can be said
about the distribution of bubble sizes, what is its elastic constant (shear modulus)? Observing such a foam for some
minutes reveals that liquid is draining from it, due to gravity. This will make the foam transparent.
The foam coarsens as gas diffuses from the smaller bubbles into the larger ones, leading to a significant change
in bubble size within an hour or so. Successful theories have been developed mainly in the limit of (quasi-)static
effects at low liquid fraction (figure). Challenges remain however in the description of fast dynamic effects,
ultimately linked to the detailed mechanism of sudden topological changes. These feature in coarsening, the
convective bubble motion in wet foams, and rearrangements of wet foams under shear. In wet foams they occur in
avalanches.
THE STRUCTURE OF FOAMS
A first detailed description of foam structure goes back to the Belgian scientist Joseph Plateau in the 19th century
(Plateau, 1873). From experiments he found that three soap films meet symmetrically in a line under angles of 120
degrees and that four such lines (later called Plateau borders) meet under the tetrahedral angle of 109.47 degrees. If
more than three films or more than four lines meet, the configuration is unstable. These results were later proven
mathematically, although it took nearly 100 years to do so strictly, to the highest standard of rigour.
The work of Plateau was well known at the time and consequently picked up by the scientific giant William
Thomson (later Lord Kelvin), in order to solve a theoretical puzzle that he considered in 1887 (Thomson, 1887).
Kelvin, like many other scientists at the time, was intrigued by the properties that were attributed to the ether, which
was associated with the propagation of light waves. He thought that its structure might be foam-like and spent a few
weeks pondering the question what shape bubbles would have to have to fill space and at the same time fulfill
Plateau’s equilibrium rules. Kelvin's solution for the structure of lowest surface area for equal-sized bubbles, his
tetrakaidecahedron, is shown in figure 1. It consists of six square and eight hexagonal faces, the latter showing a
subtle curvature which Kelvin approximated analytically. This cell which can readily be reproduced in the
laboratory within a confined foam in a cylindrical tube, was considered by him to be the one with the least surface
area for a given volume.
FIGURE 1. Space filling foam structures due to Kelvin and Weaire-Phelan.
Only in 1993, and with the help of software called the Surface Evolver (Brakke, 1992), was a rival structure
computed which outperforms the Kelvin structure by a margin of 0.3 percent (Weaire and Phelan, 1994). It consists
of eight equal size bubbles in the unit cell, two dodecahedra and 6 fourteen sided bubbles made up of two hexagons
and 12 pentagons (figure 1). Several structures have been computed since that have less surface area per volume
than the Kelvin structure. However, the Weaire-Phelan structure is still the best (but unproven) solution of Kelvin's
problem.
The matter is equally unresolved from the experimental point of view, although fragments of the Weaire-Phelan
structure have been observed. In forms of soft matter on the mesoscopic scale, the thermal energy kT may be of the
order of the energy required to alter an element of the structure, and it will seek out its minimal energy structure, in
the same manner as an ordered crystal. In that case, fascinating ordered structures abound in the laboratory as well as
in the eye of the theorist. But our foams are resolutely disordered, except when confined by boundaries which assist
the ordering process, as shown in figure 2. The National Swimming Centre which will be built for the Beijing
Olympics in 2008 is based on the Weaire-Phelan structure, so this should remind us of the problem for some time to
come.
Monodisperse oil-water emulsions arrange in the same fashion as soap bubbles (Hutzler et al. 2004). These
might be particularly useful in the context of discrete fluidics, as outlined in section 6.
(e)
FIGURE 2. Confining foams (a-d) or emulsions (e) in cylinders with diameter comparable to the bubble diameter results in
ordered structures.
FOAM DRAINAGE
While the equilibrium structure of dry foams is well understood, wet foams, i.e. foams with a finite liquid
fraction, still offer many open questions. When a foam is formed by bubbling gas into a surfactant solution, the
rising bubbles carry an amount of excess liquid with them which is mainly contained in the Plateau borders. Foam
drainage refers to the mechanism by which this liquid drains out of the structure due to the forces of gravity and
capillarity. The now standard experiment called forced drainage, illustrated in figure 3, consists of injecting
surfactant solution at a constant rate into a dry foam (Weaire et al. 1997). This results in a rather sharp interface
between wet and dry foam moving downwards with a constant velocity. This velocity scales with the flow rate in the
form of a power law with an exponent that relates to the rigidity of the foam films. For completely rigid films, as for
example in protein foams, the exponent is 1/2; for fully moveable films, which may approximately be obtained with
certain surfactants, it is 1/3 (Koehler et al. 2000). Measurements of local velocities in Plateau borders have
confirmed the different boundary conditions imposed on bulk flow by different surfactants (Koehler et al., 2002).
Despite the difficulties in the description of the microscopic details of foam drainage, a generalised foam
drainage equation has been developed which is able to reproduce all the main features of drainage experiments. It is
given by
⎞
∂ ⎛ 1 ∂α k +1 / 2
∂α
− α k +1 ⎟⎟
= ⎜⎜
∂t ∂x ⎝ 2k + 1 ∂x
⎠
where x and t are (non-dimensionalised) vertical position and time, α is the liquid fraction, and the value of k (1
or ½) determines the type of flow in the Plateau borders, i.e. Poiseuille or plug flow. Note that this theory is also
applicable to oil-water emulsions, which appear to exhibit the non-slip boundary condition independent of the type
of surfactant (Hutzler et al. 2004).
Drainage theory assumes that the network of Plateau borders is essentially static, corresponding to a foam
structure with fixed topology, unaffected by the flow within it. The structural distortion is indeed small at low flow
rates. However, an unsolved problem is the occurrence of convective bubble motion, once a critical flow rate has
been exceeded (Hutzler et al. 1998, Vera et al. 1998). This convection can take on various forms, including a
cylindrically symmetric roll and is also prone to hysteresis. Its appearance poses a significant theoretical challenge,
combining drainage and rheology.
Any type of convective flow seems to imply a horizontal gradient in liquid fraction which according to drainage
theory can not be maintained for a long time. A possible explanation of this might be found in the non-linear
mechanical or rheological properties of foam, more specifically it might require the consideration of the effect of
dilatancy (a novelty in this subject), mentioned in the next section.
FIGURE 3. In forced drainage experiments surfactant solution is added onto a dry foam at a fixed flow rate. The observed
propagation of a drainage front through the foam characterises the drainage behaviour of the foam.
FOAM RHEOLOGY
Under a small applied load foam deforms elastically, as characterised by a shear modulus. Once a certain yield
stress is exceeded, plastic yielding sets in as the bubbles rearrange (figure 4). While such a behaviour is well
understood in the quasi-static limit, a series of questions is raised concerning the dependence on strain rate.
Simple experiments can be performed on soap bubbles that are trapped between two glass plates. Pushing a
staggered array of such bubbles around a bend (in a set-up similar to the ones shown in figure 5 and 6) reveals that a
some critical flow rate the bubbles will rearrange themselves in a completely ordered and reproducible fashion.
Computer simulations have shown that the inclusion of a viscous drag force due to the movement of the films over
the surface is enough to to achieve this behaviour (Kern et al. 2004).
The formation of shear bands (localised regions of topological changes) poses another challenge (Debregeas et
al., 2001, Kabla and Debregeas, 2003). It is linked to the existence of a small maximum of the stress around the
yield stress, allowing for the coexistence of sheared and unsheared regions.
Based on computer simulations and theory the effect of dilatancy has recently been introduced to the physics of
foams (Weaire and Hutzler, 2003). This is well known from granular materials: it is the expansion of a material due
to shear. It has been suggested that this mechanism would be able to account for gradients in liquid fraction in the
convective instabilities described above.
Work is currently under way to describe dilatancy within the context of continuum mechanics and design
appropriate experiments to measure it directly.
FIGURE 4. The rheological properties of a liquid foam may be summarised in a diagram showing the variation of yield stress
and elastic modulus.
THE PROMISE OF CONFINED FOAMS
While monodisperse confined foams and emulsions offer many challenges from a theoretical point of view, their
regularity and stability also offer the promise of practical applications concerning the manipulation of gas and liquid
samples (Hutzler et al., 2002). Provided the confinement is of the order of only a few bubble diameters, the
geometry will give rise to an ordered arrangement of the bubbles (see figure 2 for foam confined in tubes). We have
exploited this to construct a scenario of controlled transport of foam through channels. While these channels are of
width of one to two bubble diameters, their small depth effectively renders the foams two-dimensional.
Two typical examples can be seen in figure 5. Pushing a double row of bubbles through a Y-shaped channel
junction leads to the splitting of the row, reversing the direction of flow allows for the merging of two single rows. A
surprising result is found in an S-shaped channel where two rows of bubbles may be swapped. Other types of
junctions can be used, for instance, to add, remove or replace individual bubbles in a bubble train (see figure 6).
a)
b)
foam flow
FIGURE 5. Two experimentally demonstrated examples of channel geometries manipulating the structure of a flowing
quasi two-dimensional foam. The typical length scale is a few millimeters. a) Two rows of bubbles can be separated or
merged at a Y-shaped junction. b) Two rows of bubbles can be flipped by passing through an S-shaped neck in the
channel.
FIGURE 6. A cross-junction of two channels may be used to replace a bubble (light gray) in the main bubble train with one from
a side channel (dark bubble). This is experimentally realized by applying a pressure pulse in the side channel.
We see this manipulation of trains of bubbles as having great technological potential for the analysis and
manipulation of gaseous or liquid samples. Reduction of length scale will be required to compete with current
developments in Lab-on-a-Chip technologies. However, the plethora of designs of the above kind provides a vast
number of well-controlled and reliable sample operations, which can be combined in a network of channels to
perform a specific task; for instance the sorting of samples according to specific chemical criteria. Combination of
these networks with sample detection, storage and analysis presents a novel concept of an Integrated Analysis
System.
Sample detection, for instance, can be realised by making use of the different physical properties (e.g. electrical
conductivity or optical properties) of the dispersed and continuous phase of the foam. Figure 7 shows an example of
conductivity measurements of a bubble train passing between two small electrodes (Drenckhan et al., 2003). The
sharp peaks corresponding to the detection of the films can be exploited for a “clocking system” of a network.
FIGURE 7. Conductivity measurement of a bubble train passing between two small electrodes. The sharp peaks
associated with the conductivity of the films separating the bubbles can be used for a "clocking system" for a bubble
network.
Another dimension of remote manipulation of the structures can be added by the application of magnetic field
gradients to ferrofluid foams. This allows for example for a very precise control of bubble volume and can also
trigger structural transitions (Hutzler et al., 2002, Drenckhan et al., 2003).
ACKNOWLEDGMENTS
This work was funded by Enterprise Ireland (Basic Research Grant SC/2002/011) and the European Space
Agency (contract 14308/00/NL/SH). W. Drenckhan receives a scholarship from the German National Merit
Foundation.
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