# SPECIAL SECTION/INSTRUMENTATION: Mercury Intrusion Porosimetry

June 1, 2010

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While certain characteristics of a fired ceramic product are of utmost importance for specific applications of the finished product, the characteristics of the component materials, the composition and structure of the green ware, and the firing process are what ultimately determine those distinctive characteristics. The development of reliable analytical test methods to predict characteristics of the finished product is an important endeavor to avoid the waste of time, energy and materials that result if substandard products are produced.

Mercury intrusion porosimetry is an analytical technique that can determine many of the sought-after physical characteristics of raw materials and green ware. This analytical technique is most often used to determine percent porosity and pore size distribution of a material, but it also can determine the bulk (or envelope) and skeletal densities of a material, the particle size distribution, and the surface area of pore walls. The same data set also provides information about pore shape and, for some structures, permeability, tortuosity and compressibility.

The relationship between the pressure and the size of the pores into which mercury will enter was derived by Washburn.(1) The pore model for which the relationship was derived is that of a circular cylinder and is expressed as:

If pressure is changed in small increments, the intrusion or extrusion process is given sufficient time to equilibrate, and the volume (V) of mercury intruded or extruded is precisely measured at each step, then a plot of V vs. P describes the intrusion and extrusion curves (examples are shown in Figures 1, 2 and 3). Fine distinctions in the shape of the curves, particularly changes in slope and inflection, are indicative of changes in the pore-filling process caused by the pore structure and, to some extent, by the granularity and mechanical characteristics of the sample.

In its most basic function as an analytical instrument, the mercury porosimeter increases pressure gradually in a precisely controlled manner while accurately measuring the volume of mercury leaving the reservoir and entering voids in the sample. Typically, pressure is increased until all available voids in the sample are filled. The total amount of mercury removed from the reservoir is equal to the total pore volume of the sample. Dividing this volume by the sample weight yields the specific pore volume.

Another characteristic of the sample that can be readily determined with a mercury porosimeter is the bulk (or envelope) volume of the sample. Knowing this value permits the percent porosity to be calculated. For a density analysis, the penetrometer is used like a pycnometer to determine bulk or envelope density.

First, the weights of the sample material (Ws), the penetrometer filled with mercury (Wp), and the penetrometer with sample, along with the remaining volume filled with mercury (Wp+Ws), are measured. These known values, combined with the density of mercury, allow the determination of the volume of displaced mercury and, therefore, the volume of the sample to be calculated (bulk volume in the case of a powdered or granulated sample or envelope volume if the sample is monolithic). Obviously, these determinations do not require any function of the mercury porosimeter, but most of these steps are necessary prior to a porosity analysis, and the additional steps allow more information about the sample to be extracted than just the bulk or envelope density.

Further pressurization results in the filling of pores, cracks and crevices in the individual particles. At maximum pressure, 0 kpsi (414 MPa), all accessible pores of a diameter 0.003 µm and larger are filled. With all accessible pores filled, skeletal volume and density can be determined. These density determinations are applicable to ceramic powders, samples of greenware and fired products.

As mentioned in the discussion of the Washburn equation, the pore model is that of a cylinder. If the pore leads to a cavity of larger diameter, the volume recorded will be associated with the size of the opening or throat of the pore through which the mercury must be forced to fill the larger cavity. Therefore, the volume associated with a pore size is more exactly described as the volume accessible through pore throats of that size.

If it is known or assumed that the majority of pores can generally be described as cylindrical, then additional information can be extracted. For each size class in the size vs. volume distribution, a representative size value (average, maximum, minimum, etc.) can be assigned. Since the volume of all pores of that representative size is known and the shape of the pore is cylindrical, the surface area of the pore walls in each size class can be calculated. Since V and D are known for each class, the relationship of area to volume and size for a cylinder (A = 4V/D) expresses the total pore wall area in each size class.

In 1965, Mayer and Stowe(2) extended the work of Frevel and Kressley(3) on the mercury breakthrough pressure required to penetrate a bed of packed spheres and the subsequent filling of the interstitial void. This work related particle size to breakthrough pressure. Later work by Pospech and Schneider(4) resulted in a method for determining the size distribution of particles from the intrusion data in the range of interstitial filling.

An exceptional capability of this sizing technique compared to other particle sizing techniques is that the particles do not have to be dispersed. Even the size distribution of sintered particles can be evaluated by mercury porosimetry, provided there are accessible voids between the particles.

A cylindrical pore will completely empty when the pressure applied to the mercury is reduced slightly below the filling pressure. Pores with an opening leading to a cavity of larger diameter than that of the opening were considered previously in regard to their effect on volume data associated with pore size. For simplicity, this type of pore is modeled as two stacked cylinders, the smaller one being the connection between the surface and the larger cylinder; these are called "ink-bottle" pores.

In this arrangement, the smaller pore at the surface impedes filling of the larger pore until pressure is sufficient to force mercury to enter and pass through the smaller pore. During extrusion, only the smaller surface pore empties while the larger diameter pore below remains filled, until pressure is lowered sufficiently to cause it to empty. If ink-bottle pores predominate, there will be a significant difference in volume associated with the intrusion and extrusion processes at the same pressure (or pore size).

In 1966, Reverberi, Ferraiolo and Peloso(5) developed a method to quantify the distribution of uniform cylindrical pores and ink-bottle pores in a material. The method yields a three-dimensional array of cavity size and throat size vs. volume. The method requires an intrusion curve to be generated to a pressure that encompasses the size range of interest, followed by a series of partial extrusion steps, each step extending to increasingly lower pressures and each followed by intrusion back to maximum pressure.

In 1986 and 1987, Katz and Thompson(6,7) introduced an empirical correlation relating permeability to a critical pore diameter determined by mercury porosimetry. The method was developed for rocks, particularly those associated with petroleum deposits, and the results agreed closely with other methods. The method has been applied to other materials with mixed results and is generally believed not to be applicable to cementitous materials,(8) but it has been shown applicable to other porous materials.(9)

A key to obtaining valid data from the Katz-Thompson treatment is identifying the first inflection point on the steeply rising range of the intrusion curve and relating this to the corresponding pressure. This point is referred to as the threshold pressure and corresponds closely to the pressure (and pore size) at which mercury first finds a path spanning the sample. A detailed explanation of the Katz-Thompson method is located in the referenced papers.

If the extrusion curve follows the intrusion curve in the uptake region, the material is demonstrating restitution or elasticity, returning to its original volume. If the extrusion curve fails to retrace the intrusion curve, the material is permanently deformed (to some extent). In the former case, the shape of the curve usually can be fitted to a quadratic equation from which the coefficient of elasticity can be extracted

A pore filled with mercury applies pressure to the pore walls with essentially the same pressure as that applied by the bulk mercury surrounding the sample. Therefore, structural collapse is not likely caused by collapse of open pore structure, but more likely is due to voids that are inaccessible to the mercury.

However, the benefit of mercury porosimetry in respect to these determinations is that all these additional dimensions of the sample are extracted from the same data set as the pore size and pore volume data. Depending on the sample, the data can be of satisfactory quality to eliminate the need for further analysis. In applications where the quality of the supplementary data is insufficient to serve as a final qualifier, mercury intrusion porosimetry offers a means of screening that comes at little or no additional analytical effort.

Mercury intrusion porosimetry is an analytical technique that can determine many of the sought-after physical characteristics of raw materials and green ware. This analytical technique is most often used to determine percent porosity and pore size distribution of a material, but it also can determine the bulk (or envelope) and skeletal densities of a material, the particle size distribution, and the surface area of pore walls. The same data set also provides information about pore shape and, for some structures, permeability, tortuosity and compressibility.

## Theory

All information in mercury intrusion porosimetry is derived from a plot of the volume of mercury intruded into or extruded from voids as a function of changing pressure. The intrusion/extrusion volume vs. pressure relation results from mercury being a non-wetting liquid for essentially all solid materials. This relationship between the liquid and solid results in the liquid tending to bridge certain-sized openings rather than entering them, unless forced to do so.The relationship between the pressure and the size of the pores into which mercury will enter was derived by Washburn.(1) The pore model for which the relationship was derived is that of a circular cylinder and is expressed as:

where P is the applied pressure, γ the surface tension of the mercury, θ the contact angle between the mercury and solid surface, and D the smallest diameter of an opening into which mercury will penetrate at pressure P. Since γ and θ are constant through an analysis, the affected pore size is inversely proportional to the applied pressure; therefore, as pressure is increased, smaller pores are filled. As with many other analytical techniques, assuming a regular particle shape or pore shape is necessary to be able to interpret the raw data.

If pressure is changed in small increments, the intrusion or extrusion process is given sufficient time to equilibrate, and the volume (V) of mercury intruded or extruded is precisely measured at each step, then a plot of V vs. P describes the intrusion and extrusion curves (examples are shown in Figures 1, 2 and 3). Fine distinctions in the shape of the curves, particularly changes in slope and inflection, are indicative of changes in the pore-filling process caused by the pore structure and, to some extent, by the granularity and mechanical characteristics of the sample.

## Rudimentary Volume and Density Determinations

The sample holder used in mercury porosimetry is called a penetrometer or dilatometer. In addition to containing the sample, the device also serves as a mercury reservoir and is a critical component in the measurement of the volume of mercury entering or exiting the sample during the test.In its most basic function as an analytical instrument, the mercury porosimeter increases pressure gradually in a precisely controlled manner while accurately measuring the volume of mercury leaving the reservoir and entering voids in the sample. Typically, pressure is increased until all available voids in the sample are filled. The total amount of mercury removed from the reservoir is equal to the total pore volume of the sample. Dividing this volume by the sample weight yields the specific pore volume.

Another characteristic of the sample that can be readily determined with a mercury porosimeter is the bulk (or envelope) volume of the sample. Knowing this value permits the percent porosity to be calculated. For a density analysis, the penetrometer is used like a pycnometer to determine bulk or envelope density.

First, the weights of the sample material (Ws), the penetrometer filled with mercury (Wp), and the penetrometer with sample, along with the remaining volume filled with mercury (Wp+Ws), are measured. These known values, combined with the density of mercury, allow the determination of the volume of displaced mercury and, therefore, the volume of the sample to be calculated (bulk volume in the case of a powdered or granulated sample or envelope volume if the sample is monolithic). Obviously, these determinations do not require any function of the mercury porosimeter, but most of these steps are necessary prior to a porosity analysis, and the additional steps allow more information about the sample to be extracted than just the bulk or envelope density.

If the sample is fine-grained or powdered, the bulk volume described above is the sum of the volumes of the particles, the inter-particle space, and the pores in the particles. As pressure is increased, the inter-particle voids (being larger) are first to be filled with mercury, followed by the filling of pores within the particles. The conclusion of inter-particle void filling and the beginning of pore filling can be determined from the intrusion curve. Thus, the amount of inter-particle volume can be determined and, subtracting this from the bulk volume, reveals the volume of the particles and their voids.

Further pressurization results in the filling of pores, cracks and crevices in the individual particles. At maximum pressure, 0 kpsi (414 MPa), all accessible pores of a diameter 0.003 µm and larger are filled. With all accessible pores filled, skeletal volume and density can be determined. These density determinations are applicable to ceramic powders, samples of greenware and fired products.

## Percent Porosity, Total Pore Volume and Pore Volume Distribution

Thus far, determining the cited physical characteristics of the sample depends only on a few data points on the intrusion curve. Pore volume distribution utilizes many points and pore size classes, with the number of data points determining the resolution of the distribution. When analyzing pore volume distribution, the x-axis is typically expressed in size units rather than pressure. Thus, the intrusion curve represents the cumulative intrusion volume. Being a cumulative curve, the first derivative yields the incremental volume, which is the volume of all pores in each size class.As mentioned in the discussion of the Washburn equation, the pore model is that of a cylinder. If the pore leads to a cavity of larger diameter, the volume recorded will be associated with the size of the opening or throat of the pore through which the mercury must be forced to fill the larger cavity. Therefore, the volume associated with a pore size is more exactly described as the volume accessible through pore throats of that size.

If it is known or assumed that the majority of pores can generally be described as cylindrical, then additional information can be extracted. For each size class in the size vs. volume distribution, a representative size value (average, maximum, minimum, etc.) can be assigned. Since the volume of all pores of that representative size is known and the shape of the pore is cylindrical, the surface area of the pore walls in each size class can be calculated. Since V and D are known for each class, the relationship of area to volume and size for a cylinder (A = 4V/D) expresses the total pore wall area in each size class.

## Particle Size Distribution

For particle size distribution determinations from the mercury intrusion curve, it is assumed that the system is composed of uniform solid spheres packed in a regular manner. The spherical particle assumption is common to most particle sizing techniques, so it is not inconsistent with other sizing methods.In 1965, Mayer and Stowe(2) extended the work of Frevel and Kressley(3) on the mercury breakthrough pressure required to penetrate a bed of packed spheres and the subsequent filling of the interstitial void. This work related particle size to breakthrough pressure. Later work by Pospech and Schneider(4) resulted in a method for determining the size distribution of particles from the intrusion data in the range of interstitial filling.

An exceptional capability of this sizing technique compared to other particle sizing techniques is that the particles do not have to be dispersed. Even the size distribution of sintered particles can be evaluated by mercury porosimetry, provided there are accessible voids between the particles.

## Pore Shape

The pore filling and emptying processes are somewhat different even for uniform cylindrical pores, due primarily to the advancing and receding contact angles differing by a few degrees between the processes. This difference alone results in a lag in the volume extruded vs. the volume intruded at the same pressure, forming a hysteresis loop in the intrusion and extrusion curves. Other effects also contribute to hysteresis, but pore shape and interconnectivity have the greatest influence on the shape of the extrusion curve and hysteresis loop.A cylindrical pore will completely empty when the pressure applied to the mercury is reduced slightly below the filling pressure. Pores with an opening leading to a cavity of larger diameter than that of the opening were considered previously in regard to their effect on volume data associated with pore size. For simplicity, this type of pore is modeled as two stacked cylinders, the smaller one being the connection between the surface and the larger cylinder; these are called "ink-bottle" pores.

In this arrangement, the smaller pore at the surface impedes filling of the larger pore until pressure is sufficient to force mercury to enter and pass through the smaller pore. During extrusion, only the smaller surface pore empties while the larger diameter pore below remains filled, until pressure is lowered sufficiently to cause it to empty. If ink-bottle pores predominate, there will be a significant difference in volume associated with the intrusion and extrusion processes at the same pressure (or pore size).

In 1966, Reverberi, Ferraiolo and Peloso(5) developed a method to quantify the distribution of uniform cylindrical pores and ink-bottle pores in a material. The method yields a three-dimensional array of cavity size and throat size vs. volume. The method requires an intrusion curve to be generated to a pressure that encompasses the size range of interest, followed by a series of partial extrusion steps, each step extending to increasingly lower pressures and each followed by intrusion back to maximum pressure.

## Permeability

Permeability is a basic property of a permeable medium that, unlike porosity, cannot be defined apart from fluid flow. It corresponds to the proportionality factor between the fluid flow rate and an applied pressure.In 1986 and 1987, Katz and Thompson(6,7) introduced an empirical correlation relating permeability to a critical pore diameter determined by mercury porosimetry. The method was developed for rocks, particularly those associated with petroleum deposits, and the results agreed closely with other methods. The method has been applied to other materials with mixed results and is generally believed not to be applicable to cementitous materials,(8) but it has been shown applicable to other porous materials.(9)

A key to obtaining valid data from the Katz-Thompson treatment is identifying the first inflection point on the steeply rising range of the intrusion curve and relating this to the corresponding pressure. This point is referred to as the threshold pressure and corresponds closely to the pressure (and pore size) at which mercury first finds a path spanning the sample. A detailed explanation of the Katz-Thompson method is located in the referenced papers.

## Compressibility and Structural Integrity

For some materials, the intrusion curve at near maximum pressure takes a sudden upward swing. In cases where the implied pore volume is inconsistent with the fine pore structure at higher pressure, the apparent uptake of mercury is likely caused by mercury filling the volume within the sample cup that becomes available when the sample material either collapses or compresses. The extrusion curve can help reveal which has occurred.If the extrusion curve follows the intrusion curve in the uptake region, the material is demonstrating restitution or elasticity, returning to its original volume. If the extrusion curve fails to retrace the intrusion curve, the material is permanently deformed (to some extent). In the former case, the shape of the curve usually can be fitted to a quadratic equation from which the coefficient of elasticity can be extracted

A pore filled with mercury applies pressure to the pore walls with essentially the same pressure as that applied by the bulk mercury surrounding the sample. Therefore, structural collapse is not likely caused by collapse of open pore structure, but more likely is due to voids that are inaccessible to the mercury.

## Conclusion

Mercury porosimetry is primarily used to determine pore size and pore volume information. Its value as an analytical tool is based largely on its capability to analyze pores from the micropore to macropore size range and to do so much faster than other techniques, such as physical adsorption. For other determinations discussed above, different analytical techniques may be better suited.However, the benefit of mercury porosimetry in respect to these determinations is that all these additional dimensions of the sample are extracted from the same data set as the pore size and pore volume data. Depending on the sample, the data can be of satisfactory quality to eliminate the need for further analysis. In applications where the quality of the supplementary data is insufficient to serve as a final qualifier, mercury intrusion porosimetry offers a means of screening that comes at little or no additional analytical effort.

*For additional information regarding mercury intrusion porosimetry, contact Micromeritics Instrument Corp. at (770) 662-3633, e-mail**ussales@micromeritics.com**or visit**www.micromeritics.com**; or Micromeritics Analytical Services at (770) 662-3630, e-mail**mas@particletesting.com**or visit**www.particletesting.com**.*## Links

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