Measuring in Layers

A novel technology based on flow porometry has been developed to measure the pore structure of individual layers in multi-layered and functionally graded ceramics.

Multi-layered ceramics and functionally graded ceramic materials with gradually changing pore structures are being increasingly used for high-tech applications. The effectiveness of such materials depends on the pore structure of individual layers. Pore structures of ceramic materials are normally determined by porosimetry or flow porometry, in which the flow occurs parallel to the thickness of the sample. However, neither of these techniques is capable of determining the pore structure of individual layers of multi-layered composites.

Porosimetry measures the pore volume and pore volume distribution of the entire composite along the thickness direction of the material. Moreover, in many of the composite filters, one of the layers is very thin, and the low pore volume associated with this layer is not detectable by porosimetry.

Equation 1

Measuring Individual Layers

A new technology based on the principles of flow porometry has been developed to measure pore structure of individual layers of a composite. To test a sample using this new method, the sample’s pores are filled spontaneously with a wetting liquid for which the liquid/sample surface free energy is less than the gas/sample surface free energy. Pressure of a non-reacting gas is slowly increased on one side of the sample so as to displace the liquid from the pores and increase gas flow through the sample. When the wetting liquid is displaced from the pore, the gas/sample interfacial area increases at the expense of the liquid/sample area, and the free energy of the system increases. Gas can displace the liquid only when the work done by the gas is equal to the increase in surface free energy. Equating the two energy terms,1 the differential pressure, p, required for displacement of a low surface tension wetting liquid at a location in a pore is given by Equation 1 shown above.2

Equation 2
In Equation 1, g is the surface tension of the wetting liquid, and dS and dV are the increases in the gas/sample surface area and the volume of gas in the pore, respectively. The diameter of a pore at any location along the length of the pore is defined as the diameter, D, of a cylindrical opening, such that (dS/dV) of the pore at the location is equal to that of the opening. For a cylindrical opening, (dS/dV) is (4/D). Hence, Equation 1 is reduced as shown in Equation 2 above.

It follows from Equation 2 that the pressure required to empty a pore is the smallest for the largest pore. Consequently, gas flow rate, which is zero at the beginning, starts at the pressure required to empty the largest pore and increases with increasing pressure because the smaller pores are emptied.

Figure 1. Flow rate of gas through dry and wet samples as functions of differential gas pressure.
Typical results of tests carried out with dry and wet samples are shown in Figure 1. The figure also shows a half-dry curve that is calculated from the dry curve to yield half of the flow rate through the dry sample at any pressure. The indicated bubble point pressure is the pressure at which the largest pore is emptied and the flow starts. The pressure at which the half-dry and wet curves intersect is indicated as the mean flow pressure. All of these data are used to determine the pore characteristics.

Figure 2. Sketch of the two-layer composite.

The Technology

The sketch of the two layered composite used in this study is outlined in Figure 2. Layer 1 is thin and has small pores, while Layer 2 is thick and has large pores. The gas pressure required to empty pores in such a material filled with a wetting liquid is determined by pore size.

Figure 3. Pore showing variation of pore size along its length and the constricted pore size.
The pore diameter normally varies along the length of a pore (see Figure 3). Hence, the pressure required to displace the liquid (Equation 2) varies with location along the length of the pore and is a maximum at the most constricted part of the pore (Figure 3). When the pressure is equal to this maximum pressure, the pore becomes completely empty, gas flows through the pore, and the presence of the pore is detected by flow porometry. Thus, flow porometry measures the diameter of the constricted part of the pore.

This basic principle is used to determine the pore structures of the two layers of the ceramic composite (Figure 2). The pores are filled with a wetting liquid, and gas pressure under Layer 2 is increased. If gas is allowed to flow in the z-direction, flow porometry measures the size of the constricted parts of pores along the z-direction. Because the pores in the two layers are in series, the small pores present in Layer 1 act as constricted parts of z-direction pores. Therefore, the Layer 1 pores are measured. On the other hand, if gas is allowed to flow in the x-y plane, gas will tend to flow simultaneously in pores of both layers, which are in parallel. Flow porometry will first detect the large pores in Layer 2. At higher pressures, the small pores of Layer 1 will be detected. Thus, pore structures of both layers can be separately determined.

Figure 4. Arrangement for flow along the z-direction.
Figure 4 shows the arrangement for flow in the z-direction,

Figure 5. Arrangement for flow along the x-y directions.
and Figure 5 shows the arrangement for flow along the x-y directions.

Figure 6. Capillary flow porometer.
Figure 6 shows the flow porometer used in this study, a completely automated instrument capable of giving highly accurate and reproducible data.2

Figure 7. Flow rates along the z-direction.

Results and Discussion

Pore Structure of Layer 1
Layer 1 had smaller pores. To find its pore structure, a sample saturated with silwick (g = 20.1 dynes/cm) was placed between two O-rings in the sample chamber (Figure 4), and the air pressure under Layer 2 was slowly increased. The air flowed only along the z-direction and escaped to the atmosphere because the O-rings prevented air from flowing in the x and y directions. The flow rates are shown in Figure 7. The largest pore diameter was calculated from the bubble point pressure. The mean flow pore diameter was calculated from the mean flow pressure.

Table 1. Pore diameters.
The mean flow pore diameter showed that half of the flow through the sample is through pores larger than the mean flow pore diameter. The pore diameters are listed in Table 1.

Figure 8. Pore size distribution in Layer 1.
The pore size distribution function, f, is defined as shown in Equation 3. In this equation, Fw and Fd are flow rates through wet and dry samples, respectively, at the same differential pressure. The leading negative sign on the right hand side of the equation is due to the fact that (Fw /Fd) increases with decreases in pore diameter.

Equation 4
The pore distribution is presented in Figure 8.

The area under the curve in any size range gives the percentage flow in that range (see Equation 4). The amount of flow is determined by pore diameter and pore size. The pore distribution suggests that most of the pores are in the range of about 0.7 to 0.1 microns.

Pore Structure of Layer 2
Layer 2 had large pores. To analyze the pore structure, the sample was soaked in silwick, sandwiched between two non-porous plates and loaded in the sample chamber (Figure 5). A small central hole in the bottom plate (next to Layer 2) allowed entry of air to the sample. Air could not escape in the z-direction because of the non-porous top plate, but could escape to the atmosphere by flowing in the x and y directions in the sample.

The flow rates through wet and dry samples are shown in Figure 1. The largest pore diameter and the mean flow diameter are listed in Table 1. As expected, the pore diameters in Layer 2 are much larger than those in Layer 1.

Figure 9. Pore distribution for flow in x and y directions.
The pore distribution for flow through x and y directions are shown in Figure 9. When air pressure is increased, the large radial pores in Layer 2 open up at low pressures. The group of large pores detected in the pore size range of about 0.5 to 4 microns is in Layer 2. Had the pressure been increased to a high enough value, the small radial pores in Layer 1 would have been detected. Thus, it was possible to measure both groups of pores in the two layers.

Mercury Porosimetry

The composite was also examined by mercury intrusion porosimetry. The results simply showed the pore volume of Layer 2. No indication of the pore volume of Layer 1 was obtained.

Accurately Determining Pore Structure

Using a new technique based on flow porometry, the largest pore diameter, the mean flow pore diameter and the pore size distribution of each layer of a two-layered ceramic composite were determined. In comparison, porosimetry could not measure any property of Layer 1.

As multi-layered ceramics and functionally graded ceramic materials are increasingly used for high-tech applications, this new technique can be used to accurately determine the pore structures of the individual layers of these composites, thus ensuring the quality of the final product.

For More Information

For more information about this new measurement technology, contact PMI at 83 Brown Road, Building 4, Ithaca, NY 14850; (800) 825-5764; fax (607) 257-5639; e-mail;

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