pubs.acs.org/Langmuir © 2010 American Chemical Society
Smectic Foams Torsten Trittel,* Thomas John, and Ralf Stannarius Institut f€ ur Experimentalphysik, Otto-von-Guericke-Universit€ at Magdeburg, 39106 Magdeburg, Germany Received December 20, 2009. Revised Manuscript Received February 8, 2010 Because of their layered structure, thermotropic smectic mesogens can form stable foams. In this study, twodimensional foams of 8CB are prepared in the smectic A phase. We determine the structures of the foam cells and study the aging dynamics. Three stages of foam evolution are distinguished. The freshly prepared foam consists of multilayers of small cells. After several hours, a 2D foam with predominantly hexagonal cells develops. It takes several days until the foam reaches an asymptotic structure with a characteristic distribution of n-polygons and self-similar scaling behavior of the coarsening. The structural changes are essentially caused by gas exchange between cells; film rupture can be neglected. We confirm predicted distributions and asymptotic scaling laws quantitatively. In the nematic phase, stable foams could not be produced, but smectic foams survive a transition into the nematic state up to several degrees above the phase transition. The reason for that is obviously smectic ordering at the film surfaces. The nematic foams coarsen much faster than smectic foams; film rupture is the dominant contribution to the aging dynamics. With 5CB, which has no smectic phase, we were not able to prepare foams.
Introduction Foams play an important role in nature, in everyday life, and in technological processes. Since the first quantitative characterization of soap films by Joseph Plateau in the 19th century, foams have always been in the interest of scientific research. Foams are characterized by universal asymptotic structure properties and scaling behavior. While many properties are nowadays quite well investigated, there are still unresolved mathematical and theoretical questions connected with the structure and dynamics of foam cells.1-3 For example, different values have been reported for the width of the asymptotic foam cell distribution.3-7 Also, the role of different topological transformations of foam cells is not fully clarified. Drainage and coarsening are still not understand in full detail.8,9 In general, one distinguishes dry and wet foams by the content of liquid phase in the structures. In three-dimensional wet foams, the liquid content is larger than ≈26%, and the foam bubbles represent spherical inclusions of gas. Dry foams, with lower fluid content, consist of polyhedral cells. The walls separating the cells are formed by thin films which meet in so-called Plateau borders. Dry and wet foams differ from each other in many physical properties, most significantly in the aging dynamics. Aging is caused by gas exchange between neighboring foam cells. In wet foams, the mean radius of foam cells grows with the cubic root of time, while the scaling exponent in dry foams is 0.5. Smectic liquid crystals are ideal materials to produce foams similar to aqueous foam structures. Free-standing smectic films *To whom correspondence should be addressed. E-mail:
[email protected]. de. (1) Weaire, D.; Hutzler, S. The Physics of Foams; Clarendon Press: Oxford, 1999. (2) MacPherson, R. D.; Srolovitz, D. J. Nature 2007, 446, 1053. (3) Rutenberg, A. D.; McCurdy, M. B. Phys. Rev. E 2005, 73, 011403. (4) Herdtle, T.; Aref, H. J. Fluid Mech. 1997, 241, 233. (5) Weaire, D.; Lei, H. Philos. Mag. Lett. 1990, 62, 427. (6) Neubert, L.; Schreckenberg, M. Physica A 1997, 240, 491. (7) Chae, J. J.; Tabor, M. Phys. Rev. E 1997, 55, 598. (8) Saint-Jalmes, A.; Langevin, D. J. Phys.: Condens. Matter 2002, 14, 9397– 9412. (9) Saint-Jalmes, A.; Zhang, Y.; Langevin, D. Eur. Phys. J. E 2004, 15, 53–60. (10) Young, C. Y.; Pindak, R.; Clark, N. A.; Meyer, R. B. Phys. Rev. Lett. 1978, 40, 773–776.
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have been described already several decades ago,10,11 and they have been investigated extensively since (see, e.g., ref 12). In the literature, mainly planar films have been described, but there are also several studies of curved smectic films, e.g. in catenoid13-15 or spherical16-19 geometries. An interesting lyotropic system that has many features in common with foam has been described by Iwashita and Tanaka.20 In their system, a mixture of a surfactant with water produces biphasic regions in which the lamellar LR phase takes the appearance of walls (like the liquid phase in a conventional foam) separating cells filled with an isotropic L3 phase of the same material (like the gas phase in conventional foams). The coarsening of such structures occurs at a much faster time scale than in the smectic foams considered here, and the mechanism of coarsening is essentially different. In Iwashita’s system, material from the cell volume (sponge phase) can transform into wall material (lamellar phase) and vice versa. Our foams consist of two distinct phases of different materials, air and liquid crystal. The viscosity of smectics is usually much higher than that of water, and they often represent pure materials instead of binary or ternary mixtures. Because of the internal layer structure of the films, smectic foams are less vulnerable to drainage than soap films. We present here a quantitative study of foams of a smectic A liquid crystal. Two-dimensional foams are prepared between two glass plates, and their structure and coarsening dynamics are observed by means of optical microscopy and the analysis of sequences of optical images. The purpose of this study is the test of predicted structural properties and asymptotic scaling laws. (11) Rosenblatt, C.; Pindak, R.; Clark, N. A.; Meyer, R. B. Phys. Rev. Lett. 1979, 42, 1220–1223. (12) Oswald, P.; Pieranski, P. Smectic and Columnar Liquid Crystals: Concepts and Physical Properties Illustrated by Experiments; Taylor & Francis: Boca Raton, 2005. (13) Amar, M. B.; da Silva, P. P. Proc. R. Soc. London A 1998, 454, 2757–2765. (14) Amar, M. B.; da Silva, P. P. Eur. Phys. J. B 1998, 3, 197–202. (15) M€uller, F.; Stannarius, R. Europhys. Lett. 2006, 76, 1102. (16) Oswald, P. J. Phys. (Paris) 1987, 48, 897–902. (17) Stannarius, R.; Cramer, C. Liq. Cryst. 1997, 23, 371. (18) Sch€uring, H.; Thieme, C.; Stannarius, R. Liq. Cryst. 2001, 28, 241. (19) M€uller, F.; Kornek, U.; Stannarius, R. Phys. Rev. E 2007, 75, 065302(R). (20) Iwashita, Y.; Tanaka, H. Nat. Mater. 2006, 5, 147–152.
Published on Web 02/25/2010
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Figure 2. Nearly regular and monodisperse cells in a freshly prepared foam of 8CB in a 2 mm thick container. Figure 1. (a) Sketch of the microscopic cross section structure of lamellae in 2D dry smectic foam. (b) Chemical formula and phase sequence of the liquid crystal 8CB (bulk transition temperatures given by the manufacturer).
The smectic foams studied here are dry; their liquid crystal content is of the order of 10% and less. In the dry smectic foams, most of the liquid crystal material is contained in the Plateau borders. The geometrical properties of freshly prepared foam depend largely on the preparation conditions. Aging changes its structural features until an asymptotic state is reached. The gas exchange between the 2D cells is described by von Neumann’s law, according to which the number of edges of an n-polygonal cell determines the direction and quantity of gas exchange21 with neighboring cells. This law was recently extended to higherdimensional foam structures.2 The asymptotic state is characterized by universal statistical properties of the foam and specific scaling dynamics. For instance, there is a universal distribution of n-polygons, the Aboav-Weaire law describes the number of edges of neighboring cells for an n-polygonal cell, and Lewis’ hypothesis predicts the mean cell area of n-polygonal cells.
Liquid Crystal Material and Experimental Procedure Experimental Section. The preparation methods for twodimensional and three-dimensional foams of smectic liquid crystals are in principle similar. In this work, we report results for twodimensional foams, where data analysis from optical observations is much simpler than in 3D foams. The foams are prepared between two plane glass plates; they consist of smectic freestanding films connecting the upper and lower glass plates. A schematic view of the cross section of a single film in the quasi-2D foam geometry is shown in the top part of Figure 1. The distance between the glass plates is 2 mm in our setup. This distance is smaller than the typical lateral extensions of the individual foam cells in the late stage of coarsening, with diameters of few millimeters and larger. Strictly, the films are not exactly straight along the vertical coordinate; they have curvatures in both directions, in the cell plane and vertical to the cell plane. However, for all practical purposes we can consider the films as straight in the viewing direction normal to the glass plates. We can assume that their mean curvature, which is related to the Laplace pressure in the given cells, is given by the inverse of the radius of curvature in the cell plane. The liquid crystal consists of a single substance. We prepare the foams with the mesogen 4-octyl-40 -cyanobiphenyl (8CB) from Synthon. The material has a purity of 99.6% (GC).22 Chemical structure and mesomorphism are sketched in the bottom part of Figure 1. For our measurements, a careful temperature control of (21) von Neumann, J. Metal Interfaces; American Society for Metals: Cleveland, 1952; p 108. (22) http://www.synthon.com.
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the setup was necessary. The setup is sketched in Figure S1 of the Supporting Information. All quantitative experiments were performed in a container with a diameter of 12 cm. This container is placed inside an air temperature controlled box; the air temperature is kept uniform and constant with an accuracy of 0.5 K. The glass container is in contact with a temperature-regulated heating plate below. It keeps the container temperature constant with an accuracy of 0.2 K during the measurement. The temperatures at the top and bottom plates differ by less than 0.2 K. A digital camera (Nikon Coolpix 4500) automatically takes images of the glass container in regular intervals of 10 min, with a spatial resolution of 0.1 mm. For the presentation in this paper, we have selected a few representative experiments. All quantitative graphs presented here are based on four image sequences for 8CB foams at temperatures from 31 to 34 C. They are typical for all measurements. The 34 C data belong to a temperature just at the smectic-nematic phase transition in the sample. While the initial structure of the foams is essentially determined by the preparation conditions, aging processes lead to a universal foam structure in all experiments. The preparation procedure is actually not relevant for the final properties of the foam, but the initial foam structure determines how long one has to wait before reaching a “natural” foam. One can prepare rather monodisperse foams with hexagonal cells by injecting a constant air flow through a thin cylindrical needle into the smectic bulk material. A typical photograph of such a regular foam is shown in Figure 2. Such foam develops a polydisperse cell structure only very slowly. Since we are interested in the analysis of natural polydisperse foams, we have tried to avoid such uniform initial structures by preparing smaller and less regular cells with a modified needle shape (see next paragraph). In order to prepare the smectic foams for quantitative experiments, we tilt the container by 20 from the horizontal position at a temperature of 34 C, very close to the smectic-nematic phase transition. The foam is prepared near the phase transition because this works faster than deep inside the smectic phase. We inject air through the liquid crystal bulk material that is collected at the bottom of the container. Air displaced by the foam is allowed to escape through vent at the opposite side of the container (not shown in Figure 3). An air flow of 50 mL/h is chosen at the beginning and 25 mL/h at the end of the preparation; the foam forms at a rate of a few cells per second. The rate of gas injection is not crucial; this has been tested. It may be decreased, but then the filling time becomes much longer, or increased, but then many bubbles burst initially and the net production of foam cells is also lower. The rate we use is a compromise. With other techniques (multiple needles or a porous multichannel inlet) one can certainly optimize the preparation. Because of the extremely long observation times compared to filling times, we did not care about an optimized filling procedure. For injection, we use a flattened needle with inner diameter of 0.8 mm. Larger diameters of the injection needle or higher flow rates result in larger initially bubble diameters. The needle is Langmuir 2010, 26(11), 7899–7904
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Figure 4. (a) Snapshot of an 8CB foam at 34 C, 2 weeks after preparation. (b) Extracted cell geometries from the image in (a). Individual cells are randomly color labeled.
Results
Figure 3. Image of a freshly generated polydisperse 8CB foam at 34 C. In the top view images, one cannot see the smectic films but the menisci of these films at the glass surfaces. The white ring is the distance holder of the cell; it has an opening for the needle and a vent on the opposite side (not shown in the image). flattened in order to obtain a polydisperse initial size distribution of foam cells. Without flattening of the injection needle, foam of predominantly uniform bubbles is obtained as shown in Figure 2. After about 2 h, the sample container is completely filled with foam; it is sealed and placed horizontally. Figure 3 shows an image of freshly prepared polydisperse foam at 34 C. This temperature is very close to the bulk transition temperature. From the coarsening time scales, we assume that the film material is still completely smectic; i.e., the smectic penetration length is of the order of the film thickness. Immediately after filling the cell, the temperature is changed from the preparation temperature close to the phase transition to the experimental temperature. The time for filling the cell is large compared to a preparation of soap foams but negligible on the time scale of foam aging. The vertical observation from top shows the Plateau borders on the glass plate above the films; they are much wider than the interior film thickness. The order of magnitude of the film thickness can be estimated from an observation of a sample from the side. Figure S2 in the Supporting Information shows an image of the outer foam cells in a 1 mm thick sample container, seen from the side. Already at 1 mm sample thickness, the dimensions of the Plateau borders can be considered small with respect to the container thickness. The colored reflections in the image result from the interference of the reflected light in the films. We cannot exactly determine the film thicknesses from these images since the exact angles of the film planes relative to the viewing direction are not known, but one can estimate that the thickness of the films is of the order of a few hundred nanometers. The blue and orange reflexes of a white light source, seen as two vertical stripes in the image, belong to the first order of interference. Image Processing and Data Analysis. In order to extract automatically the cell boundaries from the network of lines in the recorded images, we perform several image transformations using the MATLAB image processing toolbox. The transformations to a black and white image are conversion to gray scale, contrastlimited adaptive histogram equalization, noise removal using a Wiener filter, morphological reconstruction, and extended-maxima transformation. We extract a network of lines using the watershed transformation. Finally, the individual cells in the image are labeled, and properties of each foam cell are extracted. A representative color code labeled image is shown in Figure 4. In order to avoid boundary effects in the further analysis of foam properties, we consider only the central part of the foam, neglecting cells contacting the outer spacer. Langmuir 2010, 26(11), 7899–7904
Stability of Smectic Foams. The liquid-crystalline foams can be generated in the smectic phase and slightly above the smectic-nematic transition in the nematic phase. They are stable in the smectic phase, where almost exclusively gas exchange is responsible for coarsening. Although the foams survive heating into the nematic phase up to 40 C, it is practically impossible to prepare them at temperatures above 34.5 C. There is a plausible explanation for that. If the films are heated up from the smectic phase, they consist of a nematic core embedded in stabilizing smectic surface layers. If one tries to prepare foams from the nematic liquid, such surface layers do not exist and the bubbles burst immediately. Near the nematic isotropic phase transition, at about 40 C, the thin films become unstable and the foam collapses within a few minutes, as seen in Figure S3 of the Supporting Information. Our attempts to generate foams from LC material without a smectic mesophase, e.g., 4-cyano4-n-pentylbiphenyl (5CB; cryst 23 C N 35 C iso), were not successful at all. Aging of Smectic Foams. Our freshly prepared smectic foams in most cases consisted of multilayers of cells. Qualitatively, one can distinguish three phases of evolution of these foams23 (see Figure S4 of the Supporting Information). In the first, multilayer phase (I), coarsening increases the cell diameters and reduces the number of cells N. Finally, the cells reach a twodimensional geometry consisting of a monolayer of cells with polygonal shapes. In the second, transient phase (II), the selfsimilar scaling behavior is not yet reached. Depending on the initial situation, the foams are more or less ordered in that phase; there are predominantly 6-sided cells, whereas such with less than 5 sides or more than 7 sides are scarce. Figure 2 is indeed an ideal example of monodisperse foam; if one wishes, it can be considered as a special case of phase II. It was prepared in a thinner cell where the foam was monolayered from the beginning. We are not interested in such monodisperse structures in the present article. The foam characteristics in this ripening phase develop in time. In particular, the normalized probability distribution p(n) of n-polygons broadens. Unfortunately, no analytical prediction of p(n) in the scaling state is known, so that one has to define a certain criterion to decide whether the asymptotic state is reached. A good indication is the second moment of the distribution of n-polygons. It is defined as μ2 ¼
X
pðnÞðn - ÆnæÞ2
ð1Þ
n
where the brackets denote the ensemble average. One can assume that the asymptotic scaling state is reached when the second (23) Glazier, J. A.; Gross, S. P.; Stavans, J. Phys. Rev. A 1987, 36, 306–312.
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Figure 6. Evolution of the number of cells N in a smectic foam at 33 C. In the beginning of the scaling state (III) the foam consists of N ≈ 500 cells.
Figure 5. (a) Temporal evolution of the second moment μ2 of the
distribution p(n) of an 8CB foam at 32 C. The transitions between phases I-III are indicated by vertical lines. The inset shows the calculated distribution p(n) after 15 h with a second moment μ2 = 0.92. Representative parts of the 8CB foam at 32 C in phase II after 15 h (b) and in phase III after 250 h (c). Colors indicate the number of edges.
moment is approximately constant.23-25 The evolution of p(n) in phase II is characterized by an increasing μ2(t) (see Figure 5). Times given in Figure 5 and following figures refer to the end of the preparation. One can roughly estimate the transition into phase III from the time when the second moment reaches the plateau value. Finally, in the scaling state (III), the number of foam cells N continuously decreases further as shown exemplarily in Figure 6, but all dimensionless distributions have reached their asymptotic form. We observe two basic types of topological transformations during foam coarsening. As shown in Figure 7, two cells can approach each other and push apart two other cells. This is called a T1 process or side swapping. During this process no cell vanishes, but the cells exchange their number of edges. The cell disappearance or T2 process is the second fundamental topological process. In theory for very dry 2D foams, three-sided (T2(3)), four-sided (T2(4)), and five-sided (T2(5)) cells can disappear directly. In our smectic foams only the direct disappearance of 3-sided cells is observed; 4- and 5-sided cells lose edges by T1 processes during coarsening (see Figure 7). In the vicinity of the nematic to isotropic phase transition near 40 C, rupture of individual films dominates the evolution (see Figure 7). This topological process occurs much faster than the diffusion-driven changes, practically instantaneously. The complete foam collapses on a time scale of minutes or seconds. After discussing the global properties of foams, we consider now the behavior of individual foam cells. Von Neumann’s law states that cells with seven and more edges grow and cells with six edges maintain their sizes, while cells with less than 6 edges shrink.21 In the smectic foams, we find that the growth rate dA(n)/dt of the area A(n) of n-sided cells is not the same for all individual cells, so the (24) Stavans, J.; Glazier, J. A. Phys. Rev. Lett. 1989, 62, 1318–1321. (25) Stavans, J. Phys. Rev. A 1990, 42, 5049–5051.
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Figure 7. (a) T1 process in an 8CB foam at 34 C after 250 h. The cells 1 and 2 lose one side, while the edge number of the cells 3 and 4 increases by one. (b) T2(3) process in a smectic foam at 34 C after 250 h. A three-sided cell (1) disappears, and the neighboring cells (2, 3, and 4) lose one side. (c) Rupture of films in a foam near the nematic to isotropic phase transition (40 C). From the cells 1, 2, and 3 emerges the larger cell 10 . The red lines guide the eye and visualize the polygons. They do not exactly follow the bent cell walls, therefore some polygon angles apparently differ from 120 deg.
von Neumann law21 fails for individual cells. However, it is fulfilled on average for the ensembles of n-sided cells. We find a linear dependence between the number of sides n and the mean growth rate ÆdA(n)/dtæ of n-sided cells, as shown in Figure 8. The experimentally determined growth rate can be described by
dAðnÞ ¼ Kðn - n0 Þ dt
ð2Þ
The parameter n0 defines where the linear fit function to the growth rate changes its algebraic sign. The experiments show that the linear dependence is well fulfilled at all temperatures. We find n0 ≈ 6 for all temperatures, as predicted by von Neumann’s law. The slope of the fit function, κ, increases with temperature, Langmuir 2010, 26(11), 7899–7904
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Article Table 1. Fitting Parameters for the Evolution of the Mean Cell Area ÆAæ in the Scaling State T [C]
tsb [h]
D [mm2/hβ]
t0 [h]
β
~ [mm2/h] D
t~0 [h]
31 32 33 34
124 105 85 50
0.044 0.075 0.145 0.169
-358 -169 -88 -44
0.98 ( 0.07 0.93 ( 0.06 0.91 ( 0.08 0.95 ( 0.06
0.037 0.044 0.079 0.125
-379 -207 -119 -56
Figure 8. Experimental test of von Neumann’s law in smectic foams at different temperatures; experimental data are averages over the foam cells. The straight lines show linear fits to the experimental data with the parameters given in Table S1 of the Supporting Information.
indicating a strongly increasing diffusion coefficient with increasing temperature (see Table S1 in the Supporting Information). When we assume that the smectic films have the same thicknesses on average, the permeation coefficient of air26,27 through 8CB films should be linearly proportional to κ. The reason for the failure of von Neumann’s law for individual cells can be explained readily. Von Neumann’s law calculates the pressure differences between adjacent cells under the assumption that all angles of the edges are 120 and that the gas flow through a given membrane depends upon the membrane area and Laplace pressure difference. This is correct when all films have the same thicknesses. We know that all smectic films in the foam have thicknesses in the submicrometer range, but individual films may differ in thickness within a factor of 2 at least. Then, a six-edged cell may shrink or grow depending upon the relative film thicknesses of its concave and convex sides. In the average over the foam, one can assume that film thicknesses are randomly distributed, and von Neumann’s law holds for the mean area change of n-polygons. The determination of growth exponent β of the mean cell area in the scaling state (phase III) is in the focus of interest in a number of studies of soap foams.25,28-30 Here, we report the scaling exponent for smectic foams. We use a method suggested by Glazier and Weaire.31 First we determine the time tsb when the second moment is consistent with self-similar scaling behavior. Then we fit our data in phase III with ÆAæðtÞ ¼ Dðt - t0 Þβ
ð3Þ
with the mean area ÆAæ per foam cell and fit parameters D, t0, and β. For all temperatures we obtain growth exponents β ≈ 1 (see Table 1). Thus, in smectic foams in the scaling state the mean cell area grows in good approximation linearly with time ~ - t~0 Þ ÆAæðtÞ Dðt
ð4Þ
(see Figure S5 in the Supporting Information). In Table 1, we have added fit parameters for the linear fit of A(t). Similar to the (26) Ishii, Y.; Tabe, Y. Eur. Phys. J. E 2009, 30, 257. (27) Li, J.-J.; Sch€uring, H.; Stannarius, R. Langmuir 2002, 18, 112. (28) Smith, C.S. Grain shapes and other metallurgical applications of topology; Metal Interfaces; American Society for Metals: Cleveland, 1952; p 65. (29) Aboav, D. A. Metallography 1970, 3, 383–390. (30) Durian, D. J.; Weitz, D. A.; Pine, D. J. J. Phys.: Condens. Matter 1990, 2, SA433–SA436. (31) Glazier, J. A.; Weaire, D. J. Phys.: Condens. Matter 1992, 4, 1867–1894.
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Figure 9. Smectic foams fulfill the Aboav-Weaire law; we find
a = 1.3 ( 0.3. The lines visualize the dependence between Æm(n)æn and n, with a = 1 (Aboav-Weaire, dashed) and a = 1.3 (best fit, solid), μ2 = 1.5. Experimental data are represented by symbols at integer n.
prefactor κ in von Neumann’s law, the diffusion parameter D~ is strongly temperature dependent. Properties in the Scaled State. The asymptotical state of the foam can be described by universal distributions and laws, which are typical for cellular structures like soap foams, lipid monolayers, or polycrystalline films. In detail we have studied the distribution of n-polygons p(n), the cell area distribution F(A), Lewis’ hypothesis ÆA(n)æ, and the Aboav-Weaire law Æm(n)æ. We are especially interested in the scaling state were all dimensionless distributions should be constant. For this reason we express the relative number of cells of area A, F(A), with the dimensionless parameter A~ = A/ÆAæ. Several publications about cellular structures deal with the distribution of n-polygons.23,32,33 For smectic foams this distribution p(n) in the scaling state (phase III) reaches a characteristic form, stationary and equal for all temperatures within experimental uncertainty (see Figure S6 in the Supporting Information). The long-term distribution has a second moment μ2 = 1.5 ( 0.2. It is centered around n = 5 and n = 6. Three-sided cells and cells with more than eight sides are rare. The Aboav-Weaire law predicts the mean number of sides m(n) of cells surrounding an n-sided cell.29,34 Aboav gives the expression ÆmðnÞæ ¼ 6 - a þ
6a þ μ2 n
ð5Þ
where Æm(n)æ is the average number of edges of all cells contacting an n-sided cell. For a = 1, eq 5 is known as the Aboav-Weaire law.35 We find a linear dependence between the average total number of edges of the neighbor cells, Æm(n)æn, and the number of sides n of the central cell. In the long-term limit, the smectic foams reach a = 1.3 ( 0.3 (see Figure 9). So smectic foam data are (32) (33) (34) (35)
Weaire, D.; Rivier, N. Contemp. Phys. 1984, 25, 59–99. Marder, M. Phys. Rev. A 1987, 36, 438–440. Aboav, D. A. Metallography 1980, 13, 43–58. Lambert, C. J.; Weaire, D. L. Metallography 1981, 14, 307–318.
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consistent with the Aboav-Weaire law within the experimental uncertainty. No significant temperature dependence could be observed. Icaza et al.36 suggested that the log-normal distribution is the best approximation for the cell area distribution in soap foams. We compare the cumulative distribution function (cdf) of normalized cell areas ~ ¼ RðAÞ
Z
A~
~ dA~ FðAÞ
0
in Figure S7 of the Supporting Information. The black line represents the integral of a log-normal distribution fitted to the 31 C data. For small cells with areas A < 2ÆAæ, the normalized ~ is roughly similar to a log-normal disarea distribution F(A) tribution, but for larger cell areas this correlation fails. Very large cells are under-represented in the distribution. Lewis in 192837,38 supposed a linear relation between the mean cell area ÆA(n)æ of n-polygons and their number of sides n in a twodimensional network. As in soap foams, Lewis’ hypothesis can be confirmed for more than 4-sided cells in smectic foams; this is seen in Figure S8 of the Supporting Information. The deviation for n = 9 in one of the data sets is due to a large statistical error in this foam. We find the relation
Aðn > 4Þ ÆAæ
¼ c1 n þ c2
ð6Þ
with c1 ≈ 0.52 and c2 ≈ -2.1 for all temperatures in the scaling state, as given in Table S2 in the Supporting Information. For cells with 4 and less sides, the linear relation is no longer appropriate; note that eq 6 yields zero for n ≈ 4. A similar characteristic has been described before, e.g., in studies of two-dimensional soap foams in a flat container23 and in a microfluidic structure.39
Summary Two-dimensional smectic A foams have been prepared and investigated at different temperatures. We have determined important structural features in the asymptotic scaling state and confirmed distribution functions known for soap films. Coefficients determining the shrinkage or growth of individual foam cells as well as global aging dynamics have been obtained. We have quantitatively measured the distribution of n-polygons, the Aboav-Weaire law, the distribution of normalized cell areas, and Lewis’ hypothesis. Smectic A foams behave essentially like soap (36) De Icaza, M.; Jimenez-Ceniceros, A.; Casta~no, V. M. J. Appl. Phys. 1994, 76, 7317–7321. (37) Lewis, F. T. Anat. Rec. 1928, 38, 341–376. (38) Lewis, F. T. Anat. Rec. 1928, 50, 235–265. (39) Marchalot, J.; Lambert, J.; Cantat, I.; Tabeling, P.; Jullien, M.-C. Europhys. Lett. 2008, 83, 64006.
7904 DOI: 10.1021/la904779a
foams when they have reached the scaling state. Irrespective of the structural differences between aqueous soap films and free-standing smectic films, the dynamical properties are similar in both types of foams. The aging dynamics of smectic foams is determined by the gas exchange between cells. This process is slow enough so that the hydrodynamic properties of the smectic material are not relevant for structural evolution of the foams. The non-Newtonian character of the liquid crystalline material of the cell walls is therefore not reflected in the coarsening dynamics. An open question is the dramatic change of the prefactors κ and D in eqs 2 and 3 with temperature. The temperature dependence of the permeation of air through smectic films has been determined earlier27 for a different smectic material. In that study, an increasing gas permeation toward the smectic-nematic transition was reported. Since gas permeation in ref 27 was measured for other substances, these data cannot be directly compared with the measurements in the present study. It is not clear whether the difference in gas diffusion coefficients is the only reason for faster foam aging toward the smectic-nematic transition. Possibly the smectic films in the foams are on average thinner at higher temperatures. In order to test this hypothesis, it would be necessary to determine gas permeation through 8CB films in an independent experiment. Finally, we would like to mention an unpublished smectic foam study by Buchanan40 that comes to quite different conclusions. This article describes a three-dimensional foam with higher liquid content. A coarsening exponent of R = 0.2 for the mean cell diameter was reported. It seems that that study evaluated only bubbles near the container front wall. While we have no final explanation of the quantitative discrepancies, there are several possible reasons. First, it seems that the scaling state was not yet reached at least in part of the measurements. Second, the bubbles near the container wall might follow different scaling characteristics than the bulk cells. Third, the scaling characteristics of a wet smectic foam may be different from dry one. Acknowledgment. The authors acknowledge financial support by DFG within project STA 425/20 and by BMWT within project OASIS-CO. V. Aksenov is acknowledged for participating in some of the experiments. Supporting Information Available: A sketch of the experimental setup, images of foams in different states, graphs of the evolution of the mean cell area, the distribution of n-polygons in the scaling state, the cumulative cell area distribution function and the test of Lewis’ hypothesis, and tables with the parameters for von Neumann’s law and Lewis’ hypothesis. This material is available free of charge via the Internet at http://pubs.acs.org. (40) Buchanan, M. http://arxiv.org/abs/cond-mat/0206477, unpublished.
Langmuir 2010, 26(11), 7899–7904
