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Resolving the Structure of Cellular Foams by the Use of Optical Tomography P. D. Thomas, R. C. Darton,*,† and P. B. Whalley Department of Engineering Science, University of Oxford, Parks Road, Oxford OX1 3PJ, U.K.
We present several methods which may be used to resolve the structure of liquid and solid foams by optical means only. One system is based on confocal imaging, and two others use variants of computerized axial tomography. In each case we show how the reconstruction accuracy depends on the details of the scanning system and show how much is possible with simple digital video technology housed in a personal computer. 1. Introduction The structure of cellular foams occurring in the real world has been the subject of much speculation and scientific interest for over a century. However, there are no simple theoretical or experimental methods for resolving the structure of a three-dimensional foam; the underlying physical laws are not well enough understood for a comprehensive theory of foaming to be described, and the experimental analysis of real foams suffers from many practical problems. The more interesting liquid foams are unstable and tend to collapse or rearrange if any kind of probe is brought into contact with any of the bubble surfaces. Foams are also generally shiny and reflect and refract light, making foam photography difficult. For these reasons, early experimental work by Plateau (1873) and Thomson (1887) was based mainly on properties of bubbles which could be deduced by eye. Later, Matzke (1946) drew bubble shapes seen toward the periphery of soap foams in glass containers. Morris and Morris (1965) used an X-ray tomography system to obtain two-dimensional cross-sectional views of an aqueous foam, the first such study. More recently, other researchers have used magnetic resonance imaging to produce three-dimensional models of foaming systems; see Gonatas et al. (1995) and Prause et al. (1995). Automatic analysis of video images of the top surface of an opaque foaming liquid has been described by Woodburn et al. (1994). This method gives no information on the internal cellular structure, however. There have been many attempts to correlate and predict the rate at which liquid foams drain and collapse. An important part of modeling these processes is a description of the foam structure, about which in most cases there is little information, so that a simplification is applied. Many authors, for example, Hartland and Barber (1974), assume equally-sized regular pentagonal dodecahedra even though it is known that this shape does not tesselate perfectly in space but leaves some 3% of the total volume unaccounted for. Some authors neglect the 3D structure altogether and approximate the foam drainage with flow in a number of vertical independent tubes, for instance, Verbist and † Telephone: +44 1865 273117. E-mail: richard.darton@ eng.ox.ac.uk.
Weaire (1994) and Fortes and Coughlan (1994). Knowledge of the actual structure of foams should enable more realistic modeling of the foam drainage and other properties. In this paper, we describe how the structure of threedimensional cellular foams may be resolved by optical tomographic methods, using visible light as the penetrating radiation and a camera as a plane detector. We have used three different techniques in our experiments which are described in this paper. 2. Confocal Imaging Real lenses have an important property known as the depth of field which is defined as the distance range of objects in a scene viewed by a lens, such that the lens produces a focused image of the objects in that range. It is possible to vary this property by altering various lens parameters such as the focal length (by this we mean the effective focal length of a lens comprised of many elements, not the focal length of a single element) and aperture. In particular, if a large aperture is used and the distance from the object to the lens is small, the depth of field will also be small. If the depth of field is small enough, it is possible to isolate foam features which are in the plane of focus and hence know their position in space. Then from a series of images a three-dimensional structure can be deduced. Our experimental apparatus is shown in Figure 1. An aqueous foam is created by bubbling air through a solution of water and detergent contained in a thin perspex vessel. Generally in this rig, the foam bubbles are on the order of several millimeters in size. It takes about 1 min to fill the box with foam, and then this is left for another minute to allow the foam to drain somewhat and stabilize. The resulting structure is then stable enough to photograph for a further minute or more. There are problems with the illumination of liquid foams as the bubble facets strongly reflect and refract specular light. The preferred way to take photographs is with the foam lit from behind by a diffuse light source. Provided the foams have a high gas fraction, this lighting causes the foam lamellae to be nearly invisible and the foam Plateau borders to cast dark shadows on the image plane. The camera used in these experiments is a Nikon F with Nikon 185 mm bellows and a Nikkor 135 mm lens.
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Figure 1. Experimental foam rig.
This is mounted on a movable platform which can travel toward or away from the foam in increments of 0.25 mm. The aperture is set to the largest possible, f3.5, and the exposure time is 1/60 s. A film of 36 exposures can be taken before the foam collapses, and between each shot the camera is moved 0.25 mm toward the foam. This gives a volume foam sample 8.75 mm in depth and about 20 mm × 16 mm in width. 2.1. Image Processing. Each of the 36 negatives is printed, and the prints are scanned using a flatbed document scanner. The scanning resolution is 300 dots/ in. at a bit depth of 4 bits/pixel, but each digital image produced by this process is then reduced to 1/16 of its original size by averaging over a 16-pixel square neighborhood. The resolution of the resulting 8-bit images is then approximately 500 × 400 pixels. A problem with the use of photographic prints to produce the images of the foam is that these are often of inconsistent quality. Prints from the same film show small variations in brightness even with careful processing. Hence, each image is modified by adding or subtracting a small amount from each pixel until the average gray levels of the set of images are the same. Image registration is also nontrivial. Each photographic print is trimmed with the intent of minimizing the rotational alignment error between itself and its neighbors, but it is difficult to crop exactly so that each print has no translational mismatch. Moreover, since there is a time delay of over a minute between the first exposure and the last, the foam tends to sink very slightly during the capture period. This results in a vertical alignment error of a few millimeters between the first print and the last. These problems are fixed by cross-correlating each image with its neighbors to find the most likely alignment between all images. All images are then cropped to form a correctly-registered set which are all equally-sized. An edge detector is used to enhance the sections of foam in each image. This is composed of 127 × 7 convolution filters. The resulting enhanced image is then thresholded using the method Canny (1986) suggested. All pixels having a value above an upper threshold are retained, as are all those adjacent pixels which are above a lower threshold. All other pixels are deleted. Figure 2 shows how the process works on a single frame.
Figure 2. Same image before (top) and after segmentation (bottom).
Figure 3. Three-dimensional line model of aqueous foam.
Each segmented image is skeletonized using an algorithm by Arcelli et al. (1975) and then converted into a set of lines by an edge-following program. The whole set of lines forming the foam model is plotted in Figure 3. It is immediately clear from this diagram that the depth of field produced by the Nikkor lens is too great for accurate resolution in the depth direction. This should not be surprising, since photographers normally require as large a depth of field as possible and then lens has been designed with this in mind. But it does mean that parts of the foam Plateau borders remain in focus several millimeters away from the focal plane and their images extend across several frames. The way to correct this problem would be to use a more suitable lens, having a smaller depth of field. It is possible to proceed further with these results, though. If the skeletonization filter is applied to the whole model, treating the set of images as a voxel model with approximate dimensions 500 × 300 × 36, and the result of this operation is converted into a line model, we have the structure shown in Figure 4. (This is from the data
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Figure 6. Camera and foam sample arrangement for axial tomography. Figure 4. Foam line model after skeletonization.
Figure 5. Three-dimensional line model of aqueous foam after smoothing.
in Figure 3.) Each line in this model lies in one of the focal planes perpendicular to the camera axis. The z-direction (into the paper) has been stretched so that the original aspect ratio of the foam sample is maintained. The final step is to connect the segments in each focal plane with their nearest neighbors in other planes, so that Plateau borders which lie in different focal planes are continuous. The new line model is shown in Figure 5. The cellular structure of the foam is only partly visible, and there are no complete bubbles in the model. We concluded that substantially improved optics would be necessary for the confocal method to give unambiguous and detailed views of the foam structure. 3. Optical Axial Tomography I Unlike the confocal system, axial tomography will give best results using an optical system with a large depth of field, as will become apparent. Also, the circular scanning geometry eliminates much of the anisotropy which plagues the confocal method, since a foam sample is photographed from many different viewpoints. Our optical tomography system works in a fashion similar to an X-ray CT scanner except that the frequency of penetrating radiation is, of course, much
lower. Clearly, any foam sample must be transparent to some degree in order for an optical tomography analysis to work correctly. A camera can then record the attenuation of light along a set of ray paths through the foamseach image being one projectionsand given a sufficient number of images, reconstruction of the foam structure is possible. A high image data bandwidth is important for a successful tomographic reconstruction of a sample, so we have upgraded the imaging system to an electronic CCD camera and video digitizing board inside a 486 personal computer. The computer can capture six 160 × 120 pixel frames/s. The experimental arrangement is shown in Figure 6, from Darton et al. (1995). Liquid foams are supported by a glass tube of internal diameter 12 mm and wall thickness 1 mm. Solid foam samples are simply placed on the rotating turntable. The axis of rotation of the foam is vertical, but the camera views the sample at an angle of about 60° to this axis. This is essential because of the way shadows are cast by foam Plateau borders. Consider the X-ray tomography systemsradiation is attenuated almost in proportion to the amount of material it has passed through. If a ray travels through 10 mm of a material, it is attenuated much more than if it only travels through 5 mm of the same material. The optical case is different; while it might be expected that light rays are completely attenuated when hitting solid foam Plateau borders, the same is surprisingly true when light falls upon liquid Plateau borders. The foaming liquid may be transparent, but the Plateau border shadows are caused by the refraction of light from its path. Very little, if any, light passes through liquid Plateau borders and reaches the camera undeviated. Thus, Plateau borders act as though they absorb light completely. It is not possible for the reconstruction process to know how much material an individual light ray has passed through. Hence, any horizontal loops of liquid, such as the plateau borders surrounding a horizontal bubble face, will have an interior which is completely obscured to a camera whose axis is orthogonal to the vertical axis of rotation. An example of a voxel model of a Kelvin bubble scanned with this geometry is shown in Figure 7, from Thomas et al. (1995). The large horizontal plates are caused by the reconstruction process failing to determine the location of horizontal Plateau borders. This problem is eliminated by aligning the camera at an angle to the axis of rotation.
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Figure 7. Voxel model of a Kelvin bubble, viewed as an opaque isosurface.
3.1. Image Processing. A typical image dataset would be about 140 frames of resolution 160 × 120 pixels taken in the approximately 25 s period of one rotation of the turntable. There is some minor preprocessing to be completed before these images are used as projections in a tomographic reconstruction. First, the camera is only approximately aligned with the axis of rotation in the azimuthal direction. However, the images will be cropped to 120 × 120 pixels and this cropping window can be anywhere in the image. By marking the extents of any one feature in the foam as it rotates through one revolution and then bisecting, the center of rotation of the foam can be found and marked on the image. Images are then cropped so as to have this in the center of the frame. Second, foams with very thin Plateau borders, such as well-drained aqueous foams, benefit from the use of a line-enhancing filter applied to each image before reconstruction begins. A set of 7 × 7 convolution kernels is used to provide a filter sensitive to enhance even faint lines. Filtering has the added benefit of focusing Plateau borders which are on the periphery of the focal depth. Figure 8 shows this process. (To an experienced eye, Figure 8 shows a 14-sided bubble embedded within the foam structure.) Also, the video image in the figure has dark shadows where foam lamellae are oriented almost parallel to the camera axis, nearly side-on. The filter removes most of these unwanted shadows, leaving neighboring Plateau borders intact. 3.2. Reconstruction. The optical axial tomography system is an example of cone-beam tomography, where the penetrating radiation forms a beam which diverges in both directions perpendicular to its direction of travel. A reconstruction algorithm cannot therefore make use of the usual algebraic methods to speed up the backprojection process; see Swindell and Barrett (1977). Instead, a simple scheme, with a complexity depending on the number of voxels in the reconstructed model and the number of projection images, is used. Figure 9 shows the projection geometry. For each voxel, v, and image frame, F, it is a simple matter to calculate the pixel at (x, y) in the image which corresponds to the projection ray through the part of the liquid foam represented by v in the voxel model. The final value of v is the sum of the contributions from each image. In
this way the final structural model incorporates information from all images for every voxel. It follows that if an image is removed from the reconstruction process, the overall model quality will be slightly reduced. If an image was lost in the confocal method, there would be a large local loss of information resulting in a missing layer of the structure. This should be seen as an indication of the robustness of axial tomography. We have completed tomographic experiments on liquid and solid foams. The liquid foams are more interesting structurally since they form an optimal energy configuration, whereas the solid foams sometimes give better results because they remain completely motionless during the capture period and the glass tube which holds the liquid foams refracts light to some extent, slightly disrupting the reconstruction process. It is possible to correct for this effect be adding refraction calculations to the reconstruction software, but since these must be performed once for each voxel, the reconstruction time is enormously increased. Instead, we have decided to concentrate on resolving structure in the center of the tube which is least affected by light refraction. Fortunately, monodisperse foams which are easily created in a tubessee Weaire et al. (1992)smake excellent subjects for tomography experiments. The result of a liquid foam tomography experiment can be seen in Figure 10, which is a rendering of a 1303 voxel model showing the advantage of low-pass filtering the model. In order to visualize the voxel model, a threshold value is chosen and all voxels greater than this value are rendered. The value of a voxel is some measure of material density at that point, but it does not directly correlate with any material property of the foam sample being studied. It is intended that the volume of the reconstructed model will match that of the sampled foam, but the best voxel threshold is chosen by a person viewing the model. We have not yet found an automatic procedure for doing this. The low-pass filter is simply an averaging function over a 27-voxel cubic neighborhood. This is applied once and it greatly reduces the high spatial frequency noise artifacts in the reconstructed model. Applying the filter more than once causes excessive blurring of the Plateau borders and no further noise rejection. Figure 10 shows a model of a 14-sided bubble found in the Weaire-Phelan minimal structure; see Weaire and Phelan (1994). This bubble, which has 2 hexagonal faces and 12 pentagonal faces (see also Figure 8), is seen in a column in the center of the foam tube if an approximately monodisperse bubble size distribution is used. (A monodisperse foam is easily created by allowing a steady stream of bubbles at a regular flow rate into the tube.) An example of the effect of varying the choice of voxel threshold can be seen in Figure 11, another 1303 voxel model. The 16-sided bubble in this figure has 2 horizontal 7-sided faces and 14 pentagonal faces. The effect of varying the voxel threshold is clear, but note that the increase in apparent liquid volume does not follow the effect of increasing the liquid fraction in a real foam. The model quality degrades and becomes unrepresentative of the cellular structure at low voxel threshold, and also if the threshold is set too high. The solid foams are made from polyurethane, and are open, since the production process has caused most of the lamellae to collapse, leaving just the Plateau
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Figure 8. (left) Unretouched video image of air/water foam. (right) Edge-enhanced video image of air/water foam.
Figure 9. Solving for x and y gives the value of the pixel in frame F, and hence the intensity for a ray passing through F to a voxel v.
borders. However, these foams are not generally in minimal geometric configurations, as liquid foams usually are, because the solidifying process causes deformations and agglomerations of Plateau borders in apparently random ways. Some examples of reconstructed solid foams are shown in Figure 12. The upper picture is of a small foam section reconstructed in a cubic voxel matrix of 1303 voxels. This is the result of using about 140 projections which had a resolution of 120 × 120 pixels. The other two pictures show more complex foam structures as voxel models having 2563 voxels. The projection data for these were taken from approximately 280 images having a resolution of 240 × 240 pixels. All three solid models in Figure 12 are recognizable as cellular structures, but the upper reconstruction of the small section of foam is the best. In the lower right quadrant of this image, it is possible to see a broken Plateau border which has either been damaged when the foam sample was cut from a larger piece or did not survive the production process. When the whole model is viewed rotating on a computer screen, the cellular structure is clearly visible.
Figure 10. Unfiltered (top) and filtered (bottom) Weaire-Phelan bubble.
4. Optical Axial Tomography II The axial tomography system described above has large computing resource requirements; a fast processor
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Figure 11. Varying thresholds for a 16-sided bubble model. Threshold values are (in rows, top to bottom) 90, 100, 110, 120, 130, and 140.
is needed to calculate the intersections between projection rays, and images and enough storage must be present to hold the voxel model, preferably in main memory. If we constrain the problem by considering only foams of very high gas fraction and stipulate that the computer model should be of the wire frame varietys set of points connected by straight, infinitely thin linessthe problem can be tackled with fewer computing resources in a much shorter time. The experimental arrangement will be the same as above (Figure 6). The image capture step is also the same, except that it will be more important to capture video frames at the highest possible resolution, so the number of images will be fewer. At the maximum resolution of 640 × 480 pixels, the system can capture about 1 frame/s. For the images of the foam system described here, edge detection is difficult. From each image, we would like to have a two-dimensional network of vertices connected by thin lines which would represent the Plateau borders in the image. There are no reliable algorithms which can cope with images as complex as those of even simple foams, at the resolutions with which we must work. Hence, a user must outline the Plateau borders by hand using a simple drawing program. Figure 13 shows one frame from a video sequence with Plateau borders marked with thick lines. After the lines have been marked, the reconstruction software takes a set of Plateau border networks in two dimensions, one corresponding to each projection image, and produces a wire frame model of the foam structure in three dimensions. This is done by first locating each vertex in space and then working back
Figure 12. Solid foam reconstructions at resolutions of 130 × 130 × 130 (top) and 256 × 256 × 256 (middle and bottom).
using the two-dimensional network information to decide which vertices are connected by Plateau borders in the foam. Vertices are located by backprojecting a ray through their location in a projection and looking for intersections with rays of other vertices in other images. Exact intersections never occur because of the inaccuracy of
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Figure 13. User selection of Plateau borders in an image; in this case, a vertical stack of Kelvin cells in an air/water/soap foam in a glass tube.
the experimental system, so “near misses” where rays cross less than a predetermined small distance apart are counted as intersections. This unfortunately leads to many false intersections, but all the actual vertex locations are detected as well. A clustering algorithm looks for high densities of intersection points and averages them to form a vertex. This has the effect of positioning a vertex more accurately and also rejecting false intersections. As vertices are clustered, the software takes account of which Plateau borders in the two-dimensional projection networks should be connected to the new clusters. In this way, the wire model can be completed, although there may be some errors. Occasionally, Plateau borders are attached between incorrect vertices. This can happen when a false intersection is accidentally included as part of a vertex cluster, so that its Plateau border connection information is automatically and erroneously added to the cluster. However, if we are trying to model liquid foams only, we can reject any Plateau borders and vertices which do not correspond to the geometric rules which these foams must satisfy. In particular, each Plateau border must meet its neighbors at a junction at the tetrahedral angle of 109.47°, and hence there must be a maximum of four Plateau borders joining at each vertex. Although in a real foam there cannot be less than four, our model may have fewer Plateau borders at the edges of the imaged region. A model postprocessing program looks for vertices with too many Plateau borders and throws away those Plateau borders which meet at an impossible angle to the remaining ones. This is a majority vote process; Plateau borders which have less than two joining Plateau borders at angles of approximately 109.47° at both ends are rejected. We must allow for a range of angles centered on the tetrahedral angle because the curvature of Plateau borders can mean that locally the angle constraint is satisfied, but a straight line joining the Plateau border end points, as modeled, may have a large angle error.
Figure 14. Kelvin foam bubbles; a single image of the foam (top) and a view of the reconstruction (bottom).
An example model is shown in Figure 14. Here, a column of Kelvin bubbles is the subject of the tomography operation. Using 12 480 × 480 projection images, a near perfect model of three adjacent Kelvin bubbles is produced. There are errors in the reconstruction at the upper and lower extremities of the model, though. 5. Discussionsthe Potential of Optical Tomography The confocal imaging method has the advantage of being easy to set up next to an existing foam vessel, since the camera only requires one direction of view. But the resolution in the depth direction depends greatly on the quality of the lens used, in particular how small a depth of field can be obtained. Also, it seems that a relatively large number of images will be needed if the depth resolution is to approach the resolution in the plane of the image. This is important, as the accuracy of measuring foam properties will depend on the lowest directional resolution of the model. Axial tomography in its raw form removes the dependence of reconstruction accuracy on high-quality optics. The price for this is that the foam sample being studied must be viewable from all points on at least a planar circular arc. While this may be ideal for studying small
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liquid or solid foam samples, it is not practical for use next to a large foaming vessel. Also, a large number of projections with as high a resolution as possible are needed to create an accurate reconstruction of the foam. The second axial tomography variant reduces the computational cost of tomography, but the resulting wire model contains less information about the foam. In some cases this may be exactly what is required, resolving very high gas fraction liquid soap foams, for example. However, locating Plateau borders by hand is a potentially time-consuming process. Application of optical tomography methods to larger samples, perhaps where the foam is unstable, is uncertain. Possible problems fall into two categoriessfirst, there is the question of the computational power required to reconstruct detailed models of the foam; second, there are fundamental physical limits to the use of light as the imaging radiation. Computational requirements depend on both the type of foams being resolved and the time delay which is acceptable before results are available. As an example of an application which is demanding but perhaps feasible eventually, consider the use of an optical system which provides feedback control for a distillation column. Suppose the objective is to generate a foam on the distillation trays which provides a large surface area for mass transfer but not an excessive volume which would flood the column. As an element in the control loop, an optical tomography system would be required to process large volumes of data with very short delay so that a control decision could be made. A successful system would have a large video bandwidth and possibly specialized hardware for tomographic reconstruction as used by medical scanners. Given the availability of fast and cheap hardware and software, possible applications in process monitoring and control could include froth flotation, foam fractionation, and sparged reactors. On the other hand, the requirements of foam research are much less demanding. The resolution of foam structure might not need to be completed in real time, and video images grabbed at a sufficiently high speed could be cheaply stored and processed more slowly by considerably less complex general purpose workstations or personal computers. Tomographic reconstruction software similar to that described here may, in time, be offered by suppliers of image processing software as part of a standard package. The use of visible light in tomography experiments does have limitations. Clearly, foams which obstruct or greatly refract large amounts of light are not going to be amenable to optical study. However, it is still impossible to estimate the complete range of gas fractions for which tomographic analysis might prove to be possible. So much depends on the properties of the foam system being studied, especially whether solid or liquid based. It seems safe to say that most mixtures at the high gas fraction end of the scale should be suitable to optical tomographic study, whether static or dynamic.
6. Conclusions It is possible to resolve many features of the structure of liquid and solid foams by purely optical means using cheap and widely available hardware. Optical tomography is thus a highly promising technique, offering the further advantage that the improvements in hardware capability will lead to superior performance even with the same reconstruction software. Acknowledgment Financial support from Shell Research BV is gratefully acknowledged. Literature Cited Arcelli, C.; Cordella, L.; Levialdi, S. Parallel Thinning of Binary Pictures. Electron. Lett. 1975, 11, 148. Canny, J. A Computational Approach to Edge Detection. Trans. PAMI 1986, PAMI-8, 679. Darton, R. C.; Thomas, P. D.; Whalley, P. B. Optical Tomography. In Frontiers in Industrial Tomography; Scott, D. M., Williams, R. A., Eds.; Engineering Foundation: New York, 1995. Fortes, M. A.; Coughlan, S. Simple Model of Foam Drainage. J. Appl. Phys. 1994, 76, 4029. Gonatas, C. P.; Leigh, J. S.; Yodh, A. G.; Glazier, J. A.; Prause, B. Magnetic Resonance Images of Coarsening Inside a Foam. Phys. Rev. Lett. 1995, 75, 573. Hartland, S.; Barber, A. D. A Model for a Cellular Foam. Trans. Inst. Chem. Eng. 1974, 52, 43. Matzke, E. B. The Three-Dimensional Shape of Bubbles in Foamsan Analysis of the Role of Surface Forces in ThreeDimensional Cell Shape Determination. Am. J. Bot. 1946, 33, 58. Morris, R. M.; Morris, A. The Measurement of Interfacial Areas of Foams and Froths by a Radiographic Technique. Chem. Ind. 1965, 1902. Plateau, J. Statique Experimentale et Theoretique des Liquides Soumis aux Seulles Forces Mole´ culaires; Gauthier-Villars: Ghent, Paris, 1873. Prause, B. A.; Glazier, J. A.; Gravina, S.; Montemagno, C. D. Three-dimensional Magnetic Resonance Imaging of a Liquid Foam. J. Phys.sCondens. Matter 1995, 7, L511. Swindell, W.; Barrett, H. H. Computerized Tomography: Taking Sectional X-Rays. Phys. Today 1977, 32. Thomas, P. D.; Darton, R. C.; Whalley, P. B. Liquid Foam Structure Analysis by Visible Light Tomography. Chem. Eng. J. 1995, 56, 187. Thomson, W. (Lord Kelvin) On the Division of Space with Minimum Partitional Area. Philos. Mag. 1887, 24, 503. Verbist, G.; Weaire, D. A Soluble Model for Foam Drainage. Europhys. Lett. 1994, 26, 631. Weaire, D.; Phelan, R. A Counter-Example to Kelvin’s Conjecture on Minimal Surfaces. Philos. Mag. Lett. 1994, 69, 107. Weaire, D.; Hutzler, S.; Pittet, N. Cylindrical Packings of Foam Cells. Forma 1992, 7, 259. Woodburn, E. T.; Austin, L. G.; Stockton, J. B. A Froth-Based Flotation Kinetic Model. Trans. Inst. Chem. Eng. A 1994, 72, 211.
Received for review February 17, 1997 Revised manuscript received May 8, 1997 Accepted May 20, 1997 IE9701433
