3D NMR Imaging of Foam Structures

Transcription

1 JOURNAL OF MAGNETIC RESONANCE, Series A 118, (1996) ARTICLE NO D NMR Imaging of Foam Structures KATSUMI KOSE Institute of Applied Physics, University of Tsukuba, Tsukuba, Ibaraki 305, Japan Received June 13, 1995; revised October 9, Academic Press, Inc. INTRODUCTION Three-dimensional foam structures were measured using NMR imaging, and their 3D geometrical properties were analyzed. Eight bubble polyhedra of a polyurethane foam specimen were extracted from 3D NMR image data using a newly developed 3D geometrical structure analysis program, and quantitative geometrical data were measured for the first time for real foam systems. The results agreed well with the study by Matzke with soap bubbles but did not agree with the optimum solution by Weaire and Phelan for 3D space division into equal volume cells with minimum partitional area. The reason for this disagreement is not clear; however, improved foam preparation and more systematic measurements using the method developed here may clarify this difference equal volume bubbles made one by one with a syringe in a cylindrical dish. He then made observation using a binocular dissecting microscope and recorded the shapes of 600 bubbles in the central region of the dish. He observed many 12-, 13-, 14-, and 15-sided polyhedral bubbles but very few 11-, 16-, and 17-sided ones and he could not find Kelvin s tetrakaidecahedron. Although Matzke s paper gives us valuable knowledge on the 3D shapes of bubble polyhedra, even now we can- not obtain any quantitative data from it. Thus, quantitative measurements of 3D foam structures are highly desirable. The purpose of this study is, thus, to visualize 3D foam structures using NMR imaging, to measure 3D geometrical quantities using the image data, and to evaluate the observed structures. How to divide 3D space into equal volume cells with METHOD minimum partitional area is a classical mathematical problem (1). In 1887, Lord Kelvin proposed a solution based Two specimens were prepared by immersing two kinds on the 14-sided polyhedron ( tetrakaidecahedron) whose ( with different cell sizes) of commercially available polyure- faces are eight hexagons and six squares (2). He curved its thane foam into CuSO 4 -doped water in 20 mm diameter edges slightly to meet Plateau s rules which require that NMR sample tubes. Polyurethane foams, in which bubbles surfaces of cell polyhedra must meet at 120 and the edges of were produced by chemical reactions, were used because cell polyhedra must meet at the tetrahedral angle of the cell size seemed to be uniform and their foam structures Although there was no mathematical proof nor experimental were physically stable. For convenience, we denote the specimen verification, his tetrakaidecahedron has been thought of as with the smaller cell size as specimen I and that with an optimal solution for a long time. the larger cell size as specimen II. Recently, however, a better solution than Kelvin s tetrakaidecahedron In order to evaluate geometric distortion possibly present has been discovered by Weaire and Phelan in the NMR images, a phantom consisting of four concentric (3). Their unit cell consists of six 14-sided polyhedra and NMR sample tubes (with the outer diameters 5.0, 10.0, 15.0, two 12-sided polyhedra. The 14-sided polyhedron consists and 20.0 mm and the inner diameters 4.2, 9.0, 13.5, and of 12 pentagonal faces and two hexagonal faces while the 18.0 mm) filled with CuSO 4 -doped water was prepared. 12-sided polyhedron consists of 12 pentagonal faces. Al- Three-dimensional microscopic images [FOV, 19.2 mm 3 ; though they found this structure through a computational image matrix, 128 3, voxel size, 0.15 mm 3 ] were obtained search based on some crystal structures, this solution has with a homebuilt NMR imaging system using a 4.74 T, 89 not been verified experimentally. Since it is considered that mm vertical-bore superconducting magnet ( Oxford Instru- foam gives a division of 3D space with minimum partitional ments) and an actively shielded gradient coil ( Doty Scientific). area (2), measurements of 3D foam structures are essential The pulse sequence used was a conventional spinarea to this problem. echo 3D imaging sequence (TR Å 200 ms, TE Å 12 ms) to As for experimental research, in 1946, Matzke, American avoid susceptibility artifacts. The phase-encoding directions botanist, published an extensive study of three-dimensional were the x and y directions and the signal readout direction shapes of bubbles in foam (4). His foam consisted of about was the z direction. Since 4 to 16 signal accumulations were /96 $18.00 Copyright 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.

2 196 KATSUMI KOSE FIG. 1. Two-dimensional cross-sectional images of the concentric sample-tube phantom in a plane parallel to the xz plane (a) and in a plane parallel to the yz plane (b). For the 3D image, FOV, (19.2 mm) 3 ; image matrix, ; voxel size, (0.15 mm) 3. performed, the total measurement time for one 3D image Image distortion due to static field inhomogeneity can be was between 3.6 and 14.5 hours. To evaluate geometrical evaluated using Fig. 2. Figure 2a shows a 2D cross-sectional image distortion due to static magnetic field inhomogeneity, image cut from 3D image data of specimen II in a plane a pair of 3D images was acquired for specimen II with parallel to the xz plane. Figure 2b shows a difference between positive and negative readout gradients. two 3D image data sets at the same slice position which were measured with positive and negative readout (z) gradients. EXPERIMENTAL RESULTS Dark regions in Fig. 2a show edges or vertices of bubbles, and dark and bright pair regions in Fig. 2b demon- Figure 1 shows cross-sectional images of the water strate image distortion of the edges or vertices along the z phantom in a plane parallel to the xz plane (a) and in a direction due to static field inhomogeneity. Although plane parallel to the yz plane (b). The vertical direction whether this inhomogeneity is due to the magnet itself or in the images, which is parallel to the readout gradient, spatial variation of the magnetic susceptibility of the speciis the z direction. Since this phantom is axially symmetric, men cannot be identified, it is probably due to the magnet the axis is parallel to the static magnetic field and the itself because the dark and bright pair regions are absent image area is far enough from the tube ends so that the near the center of the image. In any event, this distortion or susceptibility-induced magnetic filed in the phantom can image shift is also less than one pixel length in the 15 mm be neglected. Thus, the geometric distortion along the x diameter central spherical region. and y directions observed in Fig. 1 is due to the nonlinear- From the above results, the geometrical distortions obity of the gradient field because static field inhomogeneity served in the 3D images are on the order of one pixel length distorts the images of the sample tubes only along the z in the central spherical region, though there are two factors direction. The maximum image distortion is about two which affect the image distortion. pixel lengths in the xz plane and about five pixel lengths Figures 3a and 3c show cross-sectional images of speciin the yz plane near the corners of the images. However, mens I and II. In these images, only cross sections of edges or within the central spherical region whose diameter is 15 vertices of bubble polyhedra are visible and the connections mm, the image distortion is less than one pixel length. between them are not clear. Thus, to visualize the network Image distortion due to nonlinearity of the z gradient, of the polyhedral edges, minimum intensity projection ( mip) which cannot be evaluated from images in Fig. 1, has images were computed from the 3D image data sets. already been confirmed to be less than one pixel length Figures 3b and 3d are mip images computed from 8 within the spherical region in earlier studies (5, 6). and 24 contiguous 2D slices, respectively, of the 3D image

3 3D IMAGING OF FOAMS 197 FIG. 2. (a) Two-dimensional cross-sectional image of specimen II cut in a plane parallel to the xz plane from the 3D volume data set. For the 3D image, FOV, (19.2 mm) 3 ; image matrix, ; voxel size, (0.15 mm) 3. (b) Difference at the same slice position between two 3D image data sets which were measured under positive and negative readout gradients. data. These images were made by taking a minimum value along each projection ray perpendicular to the projection plane, which was the xy plane in Fig. 3b and the xz plane in Fig. 3d. Since the thickness of one 2D slice is 0.15 mm, the thickness of the slabs for the mip calculation is 1.2 and 3.6 mm in Fig. 3b and Fig. 3d, respectively. In these figures, we can clearly see the connections or network of the polyhedral edges, so that we can realize the polyhedral faces. Although these mip images give no information along the projection direction which is perpendicular to the image plane, we can determine the x, y, and z coordinates of the vertices of the bubble poly- hedra by combining three series of mip images whose projection directions are the x, y, and z directions. In the next section, the methods to measure the x, y, and z coordinates for one polyhedron will be presented. Figure 4 is a surface-rendered 3D display of specimen II. Although we can realize the 3D structure of bubble polyhe- dra by seeing such displays from various viewing angles, it is difficult to measure the geometrical quantities of the polyhedra from these images. MEASUREMENTS OF 3D GEOMETRICAL QUANTITIES First, I shall explain why usual 3D volume-extraction techniques cannot be applied to the image data measured in this study. Such volume-extraction methods are based on image segmentation in 2D slices: specific regions such as brain, heart, or some other organ are cut by selecting appropriate threshold values in 2D cross-sectional slices, and 3D volume regions are defined by sets of the 2D regions. This algorithm is useful only for 3D objects of which exterior and interior regions can be distinguished from each other in 2D slices. In the 3D image data measured in this study, however, only polyhedral edges of bubbles are visualized as seen in Figs. 3a and 3c because their faces were already lost or too thin ( 100 mm) for detection. Thus, it is very difficult to distinguish between interior and exterior regions of bubble polyhedra in the 3D image data. Therefore, some other 3D volume-extraction technique must be developed to extract bubble polyhedra from the 3D image data set. I have devel- oped one technique utilizing series of mip images as shown in Figs. 3b and 3d. The computer program to measure geometrical quantities of the bubble polyhedra was developed on an X-window system running on a workstation (HP9000/712/60), and bubble polyhedra were interactively extracted on the CRT screen. However, geometrical quantities were mea- sured only for specimen II because the contrast-to-noise ratio of the mip images was not enough for specimen I. In all the mip images examined, little curvature was observed for the polyhedral edges. Thus, all bubble polyhedra were described as having straight edges. The procedure to measure the x, y, and z coordinates of vertices for one bubble polyhedron is as follows:

4 198 KATSUMI KOSE FIG. 3. Single-slice image (a) and minimum-intensity projection (mip) image (b) of specimen I. Single-slice image (c) and mip image (d) of specimen II. Images of specimen I are displayed in the xy plane and those of specimen II are displayed in the xz plane. 1. Compute three series of mip images whose projection for example, parallel to the xy plane. Click the mouse button directions are x, y, andz. The computation is repeated for slabs on a vertex of the polygon to determine the x and y coordi- of 24 contiguous slices while the slab volume is incrementally nates of the vertex. To measure the z coordinate, a mip moved from one edge to the other in the field of view to cover image perpendicular to the xy plane (parallel to the xz or the whole image area. Thus, three image series each consisting the yz plane) and containing the vertex is used. of 104 [128 0 (12 1 2)] mip images are obtained. 3. Click the mouse button on the next vertex which is on 2. Select a polygon (polyhedral face) in one mip image, the same polygon and has an edge connection with the first

5 3D IMAGING OF FOAMS 199 FIG. 5. Bubble polyhedra extracted from the mip images of specimen II: (a) 12-sided, (b) 13-sided, (c) 14-sided, and (d) 15-sided polyhedra. from the mip images. Although only the vertices and FIG. 4. Surface-rendered 3D display of specimen II. edges of the bubble polyhedra were measured from the images, surface areas, volumes, and flatness of the polyhedral faces were calculated under some approximations. vertex. Coordinates x, y, and z are determined in the same The results are listed in Table 1, and some of them are way as described above. In this way, the edge connecting shown in Fig. 5. the vertices is determined. In Table 1, surface areas and volumes of the polyhedra 4. Repeat operation 3 until the measured vertex reaches were calculated by taking averages over possible divisions the initial vertex. In this way, the x, y, and z coordinates of of the polyhedral faces into triangles because the faces vertices for the polygon are determined. were not always planar. Flatness, which represents flatness 5. Repeat operations 2 4 until the measured polygons of a polyhedral face, was calculated as follows: a polyhecover a polyhedron. dral face was divided into triangles, vectors normal to the triangles were calculated, angles which the normal vectors The above operations 2 5 were repeated, and eight ran- made with other normal vectors were calculated and averdomly chosen polyhedra in the central region, where image distortion was on the order of one pixel length, were measured. It took about an hour to extract one polyhedron aged over possible combinations of the normal vectors, and the averaged angles were further averaged over possible divisions of the polygon into triangles. The average TABLE 1 Geometrical Quantities of Polyhedra Extracted from Specimen II Number of faces Average Surface area Volume flatness Isometric Bubble no. Total Quadrilateral Pentagon Hexagon (mm 2 ) (mm 3 ) ( ) quotient Average

6 200 KATSUMI KOSE FIG. 7. Distribution of intersection angles at the vertices of the bubble polyhedra. The mean value is , which is close to the tetrahedral angle of voids. To attain a larger isometric quotient, polyhedra which can fill 3D spaces should be as close to a sphere as possible. However, as shown in Fig. 5, the measured polyhedra have structures elongated along the vertical direction in the figure. Since this direction is the same for all polyhedra, this elongation of the structures is presumed to be caused by some tension or other physical mechanism during the formation of the polyurethane foam. Thus, to minimize the surface area and obtain a better isometric quotient, the structures of the measured polyhedra were shrunk by a factor of 0.60, which was optimal for this foam, along the vertical direction in FIG. 6. Distributions of surface areas for (a) quadrilaterals, (b) penta- Fig. 5. The isometric quotients are tabulated in Table 1. To gons, and (c) hexagons in the bubble polyhedra. the best of the author s knowledge, this is the first such result for real foams. The average value of the isometric quotients for the eight polyhedra was improved from flatness for one polyhedron is the average of the flatness to by this correction. We will discuss this quotient in of polyhedral faces over the polyhedron. Thus, if a polyhe- more detail in the next section. dron consists of planar faces only, the average flatness Figure 6 shows distributions of surface areas of the faces is 0. in the measured polyhedra. There is a definite correlation The isometric quotient, which is shown in the ninth column of Table 1 and represents a figure of merit for surface shows a distribution of intersection angles at the vertices between surface areas and the number of sides. Figure 7 area minimization, is defined by 36pV 2 /A 3 (7), where V of the polyhedra. These angles were calculated after the and A are the volume and surface area of the cell. Its value is for a cube which can completely fill a 3D space and 1.0 for a sphere which cannot fill a 3D space without correction described above were made. Most of them ranged from 70 to 150 and their average was , which was very close to the tetrahedral angle of This result TABLE 2 Parameters of Space-Filling Polyhedra Ratio of numbers of Ratio of numbers of Average number of polyhedral faces polyhedra faces per Isometric (4:5:6) (12:13:14:15) polyhedron quotient Kelvin (2) 43:0:57 0:0:100: Weaire and Phelan (3) 0:89:11 25:0:75: Matzke (4) 11:67:22 12:30:36: This study 9:70:21 13:38:25:

7 3D IMAGING OF FOAMS 201 shows that the intersection angles at vertices are randomly agreement between the experiments and Kelvin s tetrakaidecahedron distributed around the tetrahedral angle: the distribution is is, however, not important because a more optimal nearly Gaussian and its standard deviation is about 15. solution has been already proposed by Weaire and Phelan. To summarize, the measured results are as follows: Only A remarkable difference between the experiments and the 12-, 13-, 14-, and 15-sided polyhedra were observed. Polyhe- solution is that the latter does not contain any quadrilateral dra with less than 12 faces and more than 15 faces were not faces nor any 13-sided and 15-sided polyhedra. In other observed. The average number of faces per one polyhedron words, the unit cell of the theory contains only eight polyhedra was Only quadrilateral, pentagonal, and hexagonal but the unit cell in real foam, if it exists, seems to contain faces were observed and triangular faces and faces with more a much larger number of polyhedra. However, since the than six edges were not observed. The ratio of numbers of isometric quotient obtained in this experiment is much quadrilaterals, pentagons, and hexagons was 9.2:69.7:21.1. smaller than that of the theory and the experimental precision The cell size was nearly uniform: most cells fell within is limited, we cannot explain this disagreement quantita- {10% of the mean value. The average of isometric quotients tively. The method developed here is, however, very powerful for polyhedra was improved to by shrinking them for studying 3D structure of foams, so more careful foam along one direction. There was a positive correlation be- preparation and more systematic research using this technique tween the surface area and the number of sides of the polyhedral may clarify this contradiction. faces. Intersection angles at vertices of the bubble polyhedra were distributed around the tetrahedral angle. ACKNOWLEDGMENTS DISCUSSION The author thanks Dr. E. Fukushima for critical reading of the manuscript. He also thanks Dr. T. Hashimoto for cooperation at the early stage of this study, Professor D. Weaire for useful comments on this study, and Professor Table 2 summarizes parameters of space-filling polyhedra T. Ogawa for kind guidance to this field and encouragement. This work by theoretical studies of Kelvin (2) and Weaire and Phelan was supported in part by Grant-in-Aid for Scientific Research from the Ministry of Education, Science, and Culture in Japan. (3) and experimental studies by Matzke (4) and in the present paper. The results by Matzke and those presented here gave good agreement. Therefore, the same physical mechanism REFERENCES must determine the structure of those foams, although 1. J. Gray, Nature 367, 598 (1994). Matzke made soap bubbles one by one and the polyurethane 2. W. Thomson (Lord Kelvin), Philos. Mag. 24, 503 (1887). foam used in this study was made through some chemical 3. D. Weaire and R. Phelan, Philos. Mag. Lett. 69, 107 (1994). reactions which were accompanied by gas production. 4. E. Matzke, Am. J. Botany 32, 58 (1946). Although the agreement between experimental results is 5. K. Kose, Phys. Rev. Lett. 72, 1467 (1994). satisfactory, as described above, there is a significant differ- 6. K. Kose, Phys. Fluids 6, S4 (1994). ence between experimental and theoretical studies. The dis- 7. S. Ross, Am. J. Phys. 46, 513 (1978).