VOLUME 67, NUMBER 18 PHYSICAL REVIEW LETTERS 28 OCTOBER 1991 / Surface and Bulk Tortuosity of Porous Ceramics near the Percolation Threshold Miguel Bernard and Gary A. Williams Physics Department, University of California, Los Angeles, California 90024 (Received 3 June 1991) The surface and bulk tortuosities of sintered A1203 ceramic samples have been measured using third- and fourth-sound propagation in superfluid He as a probe. Near the pore-space percolation threshold the index of refraction of both sound modes is found to increase rapidly as the porosity is lowered, to much higher values than previously observed. The bulk tortuosity exceeds the surface tortuosity for porosities below 30%, a reversal of the behavior observed at high porosity. PACS numbers: 81. 35. +k, 67.40. Hf, 67.40.Pm, 81. 60. Dq There is considerable interest in characterizing the structure of disordered materials. Transport properties of IIuids in porous media are of fundamental interest, and also have practical applications [1]. By using superfiuid He as an acoustic probe we have measured the surface and bulk tortuosities of porous ceramic materials. We observe a dramatic divergence in the index of refraction of both third and fourth sound as the pore space in the samples is reduced by sintering. This divergence indi- cates the approach to the percolation threshold, the point where there is no longer a connected path across the sam- ple. These techniques can allow an experimental test of recent theories of percolation in disordered systems [2]. Superfluid He acoustics is an important tool for study- ing the properties of porous materials because the zero viscosity of the superfluid component allows sound propa- gation in fine pores where a classical fluid would be im- mobilized by viscous forces. The measured speed of sound of He in the porous medium is reduced because of the tortuous path that the fluid is forced to follow through the connected pore spaces. The lowered speed is charac- terized by an index of refraction, which is the ratio of the theoretical speed in an ideal geometry (i.e. , straight capil- laries) to the experimentally measured speed. For the case of pores completely filled with superfluid He where the normal fluid is viscously locked, the propagating low- frequency acoustical mode is known as fourth sound [3]. The theoretical speed of fourth sound is c4=(p, /p)' c~, where p, /p is the bulk superfluid fraction and c ~ is the ve- locity of first sound. The fourth-sound index of refraction is then n4=c4/c4, „~1. A similar index can be defined for the case when only a thin film of He is adsorbed on the pore surface. The propagating mode in this case is third sound [3], a surface-height wave that is accompanied by temperature oscillations. The third-sound index of re- fraction is n3 =c3/c3 pt where r (1) 3 d4 with d the thickness of the He film, a the Van der Waals constant, and Do the thickness of the eAective nonsu- perIIuid layer [4]. The index of refraction of fourth sound is related to the bulk tortuosity parameter a3o of the Biot theory [5] of porous-media hydrodynamics by [6] n4 =a30. Similarly, a surface tortuosity parameter a20 can be defined [7] as n3 =a2D. In this Letter, we report the first systematic measurements of both a30 and a20 in the same sample as a function of the sample porosity p, where p is the ratio of pore volume to total volume. As remarked above, we find that both tortuosities increase rapidly close to the pore- space percolation threshold, to much higher values than have been observed previously. In the low-porosity re- gime, a3D is found to exceed a2D, a reversal of the behav- ior at high porosities [8, 9]. Our samples are sintered A1203 ceramics prepared us- ing a slip-casting [10] technique and firing in an oven. A1203 powder, of 500-A nominal grain size [11],and wa- ter are mixed for about 4 h in the ratio 2:1 by weight. A small amount of HC1 is added to maintain a pH of 3 for better mixing, yielding a homogeneous slurry that is then poured into a plaster of Paris mold. The mold removes the water, leaving behind a casting of the powder having a porosity between 52% and 60%. The cast is dried at 140'C for 12 h and then can be machined into several samples of regular shape (rectangular or cylindrical). Each sample is then sintered in a furnace for about 90 min at temperatures ranging from 1350 to 1650 C. This results in samples having a range of final porosities be- tween 50% and 6%. Porosity is measured by weighing the samples and measuring their dimensions, a technique which includes completely blocked-oA pore spaces. This was checked for several samples by measuring the volume of liquid helium needed to just fill the pores, and this yielded the same porosity to within a few percent. Elec- tron microscope pictures of the sample surfaces show an increasing grain size with decreasing porosity, as seen in Fig. 1. The grain size is about 0.5 pm in the 30% sample, increasing to -4 pm in the 6.7% sample. Each sintered sample is then epoxied with Stycast 2850 GT [12] into a brass cylinder whose length is machined to match the sample length. Each end of the cylinder is sealed (using an indium 0-ring) by a transducer cap that mounts flush to the ceramic surface. For third-sound measurements, the drive transducer is a heater film of resistive carbon paint of resistance about 50 Q. The pickup transducer is a current-biased 200-0 1] -W Allen-Bradley resistor with the insulating case partially 1991 The American Physical Society 2585