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Pore size measurement techniques

Porous materials are found throughout nature and are important for a multitude of industrial, medical, and natural processes. For example, the pores within catalysts increase the available surface for reactions to occur. In addition, reactants and products are directed to and from active sites through the porous structure. The size of the pores present in pharmaceutical tablets directly relates to their dissolution rate. The pores that travel through a filtration membrane define the size of particles which can physically pass through and those that will be removed from a fluid stream. This article provides information on classifying pores and the most commonly used techniques to accurately measure pore diameters in materials with different pore size distributions.

Figure 1: Pores are the pathways into and throughout porous materials as depicted here with hexagonal cylinder-like pores.

Figure 1: Pores are the pathways into and throughout porous materials as depicted here with hexagonal cylinder-like pores.

Introduction

Pores are the openings in solid surfaces which gases, liquids, or even foreign microscopic particles can occupy. Porous materials are made in a multitude of ways from the bottom-up to the top-down approach. The bottom-up approach builds pores from an architectural process as is the case with templated carbons or chemically arranged metal-organic frameworks (MOFs) or covalent organic frameworks (COFs). The top-down approach builds pores from non-porous materials, as in leaching/etching, sintering, or steam reforming. Both methods can be highly regulated to produce desired pore structures. Depending on the type(s) of pores present, different characterization methods or a combination of methods are used for characterization.

Pores come in a variety of sizes to address a very wide range of applications. When considering pore size, a set of standards approved by IUPAC has defined pore size ranges based on different size widths.[1] Pores with an internal width of less than 2 nm are referred to as micropores, those with an internal width between 2 nm and 50 nm as mesopores, and those larger than 50 nm are called macropores (See Figures 2 and 3).

Figure 2: A visual representation of micro-, meso-, and macropores arranged within a particle and macropores formed between particle packing known as interparticle pores

Figure 2: A visual representation of micro-, meso-, and macropores arranged within a particle and macropores formed between particle packing known as interparticle pores

Figure 3: Pore sizes and the techniques used to measure them

Figure 3: Pore sizes and the techniques used to measure them

Within the micropore domain, there is a further subdivision into narrow micropores (ultramicropores of less than 0.7 nm) and wide micropores (supermicropores of 0.7 nm to 2 nm).[2] These terms are used throughout different industries and in academia to easily compare and recognize material dimensions.

Pores are also defined by their accessibility to the surface of the material. A closed pore is inaccessible to its surfaces; a blind pore is accessible from the surface but does not travel completely from the upstream to the downstream surface, while a through pore travels from the upstream surface to the downstream surface of a material (See Figure 4). 

Figure 4: Various pore type accessibilities

Figure 4: Various pore type accessibilities

How to measure pore size

Closed porosity, the overall volume of closed pores contained within a material, is estimated by comparing true density results with expectations. This technique will not provide any pore size distribution information, but an estimate of the voids within a material may be generated with the use of a gas pycnometer. Because blind pores and through pores are accessible via the material surface, techniques such as gas adsorption, mercury intrusion, and capillary flow porometry can be used for their assessment. 

Gas adsorption

Gas adsorption experiments are used to characterize the surface area, pore size distribution, and pore volume of those pores accessible from the surface of porous materials. A wide range of pore sizes from 0.35 nm to over 100 nm can be analyzed with high accuracy using vacuum-volumetric or gravimetric adsorption techniques. Samples regularly evaluated using gas adsorption include zeolites, clays, activated carbons, templated materials, metal-organic frameworks, pharmaceuticals, catalysts, and many more.

These experiments are performed by first cleaning the surfaces of the sample and then dosing the sample with an adsorbing gas, in a system typically held at the boiling point of the gas (e.g., N2 at 77 K, Ar at 87 K), and recording the resulting difference in volume or mass once thermodynamic equilibrium is achieved. This dosing procedure continues over a range of predefined pressures in order to generate a characteristic isotherm (a curve of the volume adsorbed as a function of pressure created at a constant temperature, see Figure 5) which is used to determine the pore size, pore volume, and surface area. Analysis of the pore size distribution is carried out by applying different theories and calculations. Classic methods exist to determine pore size distributions, for instance:

  • BJH – named for the scientists Barrett, Joyner, Helenda, it is used to describe mesopores (See Fig. 6).[3]
  • HK – named for the scientists Horvath and Kawazoe, it is generally applied to describe micropores.[4]

However, modern methods such as DFT (Density Functional Theory) or GCMC (Grand Canonical Monte Carlo) based molecular simulations are more regularly applied. These improved data fitting methods are very reliable for siliceous/oxidic materials using non-local DFT methods  and for carbonaceous materials using quenched solid DFT methods . 

Figure 5: Gas adsorption (gray circles) and desorption (red squares) isotherm generated on an alumina catalyst support with nitrogen at 77 K

Figure 5: Gas adsorption (gray circles) and desorption (red squares) isotherm generated on an alumina catalyst support with nitrogen at 77 K

Figure 6: BJH pore size distribution (gray circles) and cumulative pore volume (red diamonds) from the isotherm generated on an alumina catalyst support with nitrogen at 77 K

Figure 6: BJH pore size distribution (gray circles) and cumulative pore volume (red diamonds) from the isotherm generated on an alumina catalyst support with nitrogen at 77 K

Mercury intrusion

Measuring pores sized from 3.2 nm to larger than 400 μm accessible from the surface of a material is generally accomplished with the use of mercury intrusion porosimetry. 

Figure 7: The non-wetting behavior of liquid mercury ensures that the contact angle (θ) between it and just about any solid material will be between 130 degrees and 150 degrees.

Figure 7: The non-wetting behavior of liquid mercury ensures that the contact angle (θ) between it and just about any solid material will be between 130 degrees and 150 degrees.

This technique involves forcing the non-wetting liquid mercury (See Figure 7) into smaller and smaller pores with increased pressure. This is calculated using the Washburn equation[5]

Pr = -2γ cosθ

P= pressure
r = pore radius
γ = surface tension
θ = contact angle

The pore size distribution is obtained by monitoring the volume of intruded mercury into the pores as a function of applied pressure to produce a porosimetry curve (See Figure 8).

Figure 8: As mercury is intruded into smaller and smaller pores at increasing pressures the volume is recorded as a capacitance change along the stem of the sample cell.

Figure 8: As mercury is intruded into smaller and smaller pores at increasing pressures the volume is recorded as a capacitance change along the stem of the sample cell.

By relating the pressure at which intrusion is seen to the size of the pore being filled, a pore size distribution plot is calculated (See Figure 9). It is important to note that for powder samples, pores may exist within particles (intraparticle pores), but always between particles (interparticle pores). Therefore, it is often important to know about the physical nature of the material being measured for proper interpretation of the results.

Figure 9: Pore size distribution plot of the interparticle pores present in a non-porous silica with its distribution reported in microns (μm)

Figure 9: Pore size distribution plot of the interparticle pores present in a non-porous silica with its distribution reported in microns (μm)

It is worth noting that, unlike gas adsorption, mercury intrusion porosimetry is destructive and the sample is not recoverable upon measurement.

Capillary flow porometry

If the pores of interest are through pores, to predict the performance of an ultrafiltration or microfiltration media, or simply to better understand how a material will affect fluid flow, capillary flow porometry is the more appropriate method. Capillary flow porometry is used to define through pores sized from about 13 nm to more than 500 μm. It utilizes capillary effects (attractive forces between surface and fluid) to initially hold a completely wetting fluid within the pores of a material (See Figure 10).

Figure 10: The completely wetting nature of a fluid chosen for capillary flow porometry ensures that simple capillary forces will initially fill all of the pores of the sample.

Figure 10: The completely wetting nature of a fluid chosen for capillary flow porometry ensures that simple capillary forces will initially fill all of the pores of the sample.

This technique also uses the Washburn equation, but instead of a non-wetting liquid being forced into pores, a completely wetting fluid is filled into pores and expelled as increasing pressures of gas are applied to the upstream side of the sample and the capillary forces holding the fluid in the pores are overcome. Ultimately the pore size distribution can be derived from the measurement of the resulting gas flow through the sample (See Figures 11 and 12).

Figure 11: Beginning with a completely wetted sample an increasing gas pressure is applied to the upstream side until capillary forces are overcome and gas flow can be measured. At higher and higher pressures, smaller and smaller pores empty and the resulting increases in gas flow are measured.

Figure 11: Beginning with a completely wetted sample an increasing gas pressure is applied to the upstream side until capillary forces are overcome and gas flow can be measured. At higher and higher pressures, smaller and smaller pores empty and the resulting increases in gas flow are measured.

Figure 12: The largest pore will empty first, defining the Max Pore Size and the Bubble Point. The Minimum Pore Size is defined at the point where the wet curve meets the dry curve. The Mean Pore Size is defined as the point at which the amount of flow through the sample on the wet curve is exactly 50 percent of the amount of flow at the same pressure when the sample is dry. Note: The wet curve monitors the applied gas pressure and the flow of gas when fluid is being expelled; the dry curve monitors the sample without fluid in its pores.

Figure 12: The largest pore will empty first, defining the Max Pore Size and the Bubble Point. The Minimum Pore Size is defined at the point where the wet curve meets the dry curve. The Mean Pore Size is defined as the point at which the amount of flow through the sample on the wet curve is exactly 50 percent of the amount of flow at the same pressure when the sample is dry. Note: The wet curve monitors the applied gas pressure and the flow of gas when fluid is being expelled; the dry curve monitors the sample without fluid in its pores.

Figure 13: The calculated pore size distribution, in microns, from the experiment performed in Figure 12

Figure 13: The calculated pore size distribution, in microns, from the experiment performed in Figure 12

Unlike gas adsorption and mercury intrusion, this technique does not quantify the pore volume, only the diameters of the through pores present in a sample; and only the smallest diameter present along the pathway of each through pore. The technique provides a means to understand how many pathways of a particular pore size exist and therefore how many particles of a certain size will be removed from a fluid stream before cleaning or replacement of the membrane must occur. Even though these pores are included in gas adsorption and mercury intrusion measurements their actual volume is oftentimes too small to be sufficiently discernible by those techniques. However, for materials containing only through pores which have uniform pore structures, such as nuclear track-etched membranes, pore size distribution results from a capillary flow porometer will be similar to gas adsorption or mercury intrusion results (depending upon the size range of the pores present).

Conclusion

There is a wide range of analytical methods available in the laboratory to conduct pore size analysis. To find the proper method to measure the pore size(s) within a material, it is best to consider a few factors. First, what pore size range will the materials possess? Second, is the material to be recovered? What type(s) of pores are to be measured? Finally, how representative is the measurement for the entire sample? Answering these questions will guide the user to select the best method to perform successful pore size analysis. 

References

  1. Sing, S. W., Everett, D. H., Haul, R. A. W., Moscou, L., Pieroti, R. A., Rouquerol, J., Siemieniewska, T. (1985). Reporting physisorption data for gas/solid systems with special reference to the determination of surface area and porosity, Pure Appl. Chem. 57, 603
  2. Thommes, M., Kaneko, K., Neimar, A. V., Olivier, J. P., Rodriguez-Reinoso, F., Rouquerol, J., Sing. K. S.W. (2015). Physisorption of gases, with special reference to the evaluation of surface area and pore size distribution (IUPAC Technical Report), Pure and Applied Chemistry, Volume 87, Issue 9-10, pp 1051–1069.
  3. Barrett, E. P., Joyner, L. G., Halenda, P. R. (1951). The determination of pore volume and area distributions in porous substances. I. Computations from nitrogen isotherms. J. Am. Chem. Soc., 73(1), pp. 373–380
  4. Horvath, G., Kowazoe, K. (1983). Method for the calculation of effective pore size distribution in molecular sieve carbon. J. Chem. Eng. Jpn., 16(6), pp. 470–475.
  5. Washburn, E. W. (1921). Note on a method of determining the distribution of pore sizes in a porous material. Proc. Natl. Acad. Sci. USA, 7(4), pp. 115–116.