Mercury Porosimetry - Alfred University

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Herbert Giesche*. Received: ... Herbert Giesche, NYSCC @ Alfred University, 2 Pine Street,. Alfred, NY ...... Correlation in Porous Media and its Effect on Mercury.
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Part. Part. Syst. Charact. 23 (2006) 1±11

Mercury Porosimetry: a General (Practical) Overview Herbert Giesche* Received: 28 September 2005; accepted: 9 March 2006 DOI: 10.1002/ppsc.200601009

Abstract The paper describes general concepts of mercury porosimetry measurements and provides an overview on the current status of pore-network analysis tools. Practical

aspects of the technique are described as well as emphasizing the need for testing of model pore structures and the status on pore network modeling software.

Keywords: Mercury Porosimetry, Overview, Pore network

1 Introduction Mercury porosimetry is an extremely useful characterization technique for porous materials. Pores between about 500 lm and 3.5 nm can be investigated. A complete analysis may take as little as half an hour of analysis time. Mercury porosimetry provides a wide range of information, e.g. the pore size distribution, the total pore volume or porosity, the skeletal and apparent density, and the specific surface area of a sample. No other porosity characterization technique can achieve this. However, one should realize that mercury porosimetry also has limitations. One of the most important limitations is the fact that it measures the largest entrance towards a pore (see Figure 1), but not the actual inner size

of a pore. For obvious reasons it can also not be used to analyze closed pores, since the mercury has no way of entering that pore. Through various software techniques an interpretation of the pore-network (cross-linking structure between pores) can be achieved. However, one should realize that numerous assumptions are made in that process and the final results are somewhat arbitrary.

2 Theory and Key-Parameters A key assumption in mercury porosimetry is the pore shape. Essentially all instruments assume a cylindrical pore geometry using a modified Young-Laplace equation, which is most of the time referred to as the Washburn equation.  DP ˆ c

Fig. 1. Schematic representation of pores.

*

Herbert Giesche, NYSCC @ Alfred University, 2 Pine Street, Alfred, NY 14802, +1 607 871 2677, e-mail: [email protected]

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1 1 ‡ r1 r2

 ˆ

2 c cos h rpore

…1†

It relates the pressure difference across the curved mercury interface (r1 and r2 describe the curvature of that interface) to the corresponding pore size (rpore) using the surface tension of mercury (c) and the contact angle (h) between the solid and mercury. The real pore shape is however quite different and the cylinder pore assumption can lead to major differences between the analysis and reality. As indicated in equation 1, we need to know surface tension and contact angle for the given sample and then http://www.ppsc.com

2 measure pressure and the intruded volume in order to obtain the pore size ± pore volume relation. Pressure is one of the key measurement variables and one has to realize that a wrong pressure reading will automatically result in the wrong pore size. The measurement spans 5 orders of magnitude. Consequently, the pressure also changes over an equally wide range. It would be very difficult to have a single transducer for the entire measurement. Thus, for practical reasons most instruments use several transducers. This set-up then provides us with a better resolution within each range, however, it can also cause artificial effects at the switchover points between the various transducers. When interpreting the analysis results one should always be aware of those switchover points, e.g. between the ªlowº and the ªhighº pressure run. Artificial effects at that specific point are not only caused by the switch between different pressure transducer, but also because of the different ªenvironmentº of the sample cell (air versus oil). It is difficult to detect problems with a pressure transducer unless a transducer totally fails or if the change is so severe that the results are obviously wrong. Running ªstandardº samples, which have several (!) well defined pore sizes, is one way to check the instrument. However, even then, it is usually up to the instrument manufacturer to correct for these problems. The instrument operator has no way of changing or correcting the pressure-transducer-readout. The operator can only reduce chances of a transducer failure by avoiding any sudden pressure changes, over-pressure exposure and major temperature swings. The second key measurement parameter is the pore volume. Essentially all instruments use capacitance measurement between a metal shield on the outside of a glass capillary and the length of the mercury column in the capillary. Obvious problems can arise if the inside or outside of the glass capillary is not uniform or in case there is any other external factor effecting the capacitance. Bad electrical contacts are one of the most frequent reasons for those problems. However, most of these problems are easily detected, since they will cause drastic spikes or other irregularities in the measurement. The operator should primarily check that the capillary is free of contaminations or obvious chips at the end of the glass. Checking the volume calibration factor with a known standard material is relative simple test and most instruments allow for any necessary correction factor. Nevertheless, one should contact the instrument manufacturer in case those deviations are detected, since it could be a sign of failed electronic components as well. The surface tension of mercury and the contact angle between mercury and the sample surface are additional factors, which have to be established.

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Part. Part. Syst. Charact. 23 (2006) 1±11

In general, the surface tension of mercury is not of any great concern with respect to errors in the determination of the pore size distribution. A value of 0.485 N mÐ1 at 25 C is commonly accepted by most researchers. It is advisable to use fresh (triple distilled) mercury for every measurement. Although, using mercury, which was reclaimed through a siphon technique, is sufficient in many cases, it allows the possibility for a major and often undetected error. Reports by Allen [1] and Klobek [2] describe the effect of temperature and pressure on the surface tension value, but these corrections are not used on a general basis. The contact angle on the other hand is a parameter, which clearly affects the analysis results and numerous papers have demonstrated the wide range of contact angles between mercury and various different or even very similar solid surfaces. For example, contact angles of 128 to 148 degree have been determined for ªidenticalº systems of mercury on glass [3]. However, in most practical situations and out of convenience users often apply a fixed value irrespective of the specific sample material, e.g. 130 or 140. Several techniques are available to determine the contact angle, such as: placing a drop of mercury on the flat surface of the sample and either fitting the actual shape of the drop or measuring the maximum height, hmax, as the volume of mercury is increased. The contact angle can then be estimated using equation 2.

cos h ˆ 1

r g h2max 2 cHg; air

…2†

with g, gravity acceleration, and q, density of the liquid Alternatively a powder compact can be pressed in such a way that a well defined hole is created in a disk. Mercury is now placed on top of this disk and the contact angle can be calculated from the necessary pressure to force the mercury through this cylindrical pore. In addition, one should keep in mind possible errors due to a surface contamination by impurities and effects of microscopic surface roughness on the more macroscopic contact angle measurements. Further details are described elsewhere [4].

3 Sample Preparation The very first point in a ªgoodº analysis is having a welldefined unambiguous sample. One of the key parameters here is the sample weight. Porous materials are prone to adsorb water or other chemicals, which should be removed during the initial evacuation of the sample.

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Mercury porosimetry has no direct way of noticing if those impurities are removed and what the actual sample weight is. We can only indirectly judge from a low vacuum pressure and the absence of any possible ªleakratesº that the sample is in a clean and well defined starting condition. Other external measures might be needed to determine the ªclean sample weightº. Although advisable, it is usually not possible to apply any heat treatment to the sample during the evacuation process. Another source of possible problems is the creation of artificial pores due to packing of the sample inside the penetrometer cell. The inter-particle voids between (spray-dried) powder granules are an obvious example. However, even the voids between sheets of a sample (e.g. paper pieces) or the void between the sample and the glass wall of the penetrometer are possible error sources. Most of the time these errors will not be very significant since the actual sample porosity is much larger, but in case of nearly non-porous samples, the relative error can be quite significant. These artificial pores between sample pieces can be avoided by using a coarse (e.g. 1 mm diameter) stainless steel wire to keep the sample pieces separated. On the opposite side, a sample can be sealed in a thin walled rubber balloon, e.g. for determining the compressibility of the sample. Being aware of possible compressibility effects is especially important when analyzing ªsofterº sol-gel type samples or porous polymers. Alternatively the sample can be partially encased in epoxy, leaving only controlled areas of the sample free of epoxy, which allows one to study a specific penetration path into a sample.

4 Filling and Low Pressure System Initially the sample is evacuated to remove air and residual moisture or other liquids from the pore system. A complete evacuation is important in order to avoid possible air pockets and contamination issues. The sample cell is then filled with mercury as the entire system is still under reduced pressure. Slowly increasing the overall pressure then allows mercury to penetrate the largest pores in the sample or any void spaces between sample pieces. The first data point is usually taken at a pressure of 3000 to 4000 Pa (0.5 psi) or higher. Lower pressure readings are possible, but one has to understand that the height of a 1 cm mercury column would already correspond to a pressure of 1333 Pa. Thus, the top and bottom of a 1 cm tall sample would experience a pressure difference of 1333 Pa solely due to the weight of the mercury column. Each instrument uses different designs to keep

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this effect as small as possible. Nevertheless any reported pore diameter of more than 500 lm is extremely questionable due to the (head-pressure) effect as described before. Thereafter the pressure is increased up to several atmospheres (limited by safety factors of the set-up), which allows for a reasonable cross-over between the low and the high pressure part of the analysis. At the end of this ªlow pressureº-part of the analysis the weight of the penetrometer, filled with mercury and the sample, is determined, which allows one to calculate the bulk density of the sample (using corresponding blank-runs as a reference). The volume of intruded mercury is measured continuously through changes in the capacitance between the column of mercury in the dilatometer stem (capillary tube of know diameter connected to the sample cell) and a coaxial metal sheet surrounding the stem. Alternatively optical- and resistance- or contact-wire techniques have been used in the past.

5 High Pressure System Once transferred to the high-pressure system, the sample-cell is surrounded by hydraulic fluid and pressures of up to 414 MPa (60,000 psi) are applied in an isostatic way. 5.1 Equilibration Most instruments can operate in a continuous or an incremental mode. The continuous mode offers the possibility to run an analysis in a very short time, 5 to 10 min for an entire analysis. However, this requires a careful consideration of a variety of correction factors, which are primarily the compressive heating effect and hindered flow of mercury through small pore channels. On the other hand, it allows for a large number of data points to be recorded and even small differences between samples can be observed, whereas those differences could occur between two data point during the incremental mode. In contrast, the incremental technique offers a better assurance that true equilibrium is reached for each data point as long as the equilibration time interval is chosen sufficiently long and temperature effects can be avoided, which is usually provided at equilibration times over 5 min. It is important to notice that those effects are primarily important for the high pressure / small pore size range. Volume readings taken before equilibrium has been reached may result in shifting of the distribution toward smaller pore sizes and smaller pore volumes during in-

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trusion and larger pore sizes as well as a larger amount of ªtrappedº mercury during the extrusion process. The advancement of the mercury interfaces in horizontal, cylindrical capillaries was computed and also verified experimentally by Wardlaw and McKellar [5]. For a horizontal cylindrical capillary mercury will enter the capillary at the threshold pressure, as given by the Laplace equation, but will not continue to advance. A finite rate of advance is dependent on an excess pressure (DP) above the threshold pressure and the distance to time relationship for the advancing mercury front is then given by tˆ

l2 4 g DP r2p

…3†

where t = time, l = distance, g = viscosity (1.536 centipoise), r = radius, and DP = pressure applied in excess of the injection pressure. The distance to time relationship for mercury in tubes of five different sizes is shown in Figure 2. For example, more than 100 seconds are needed to travel 3 cm in a tube of 0.5 lm radius. In practice those limitations apply primarily to large samples or to small (< 100nm) pores. Other side effects may require more attention, like the heating or cooling effects due to compression or expansion. A temperature increase of 10 to 15 C is not unusual when the system is pressurized at a fast rate. Corresponding negative temperature changes were observed during a fast depressurization. Temperature swings cor- Fig. 2. Advancing mercury in cylindrical pores of different radii; respond to volumetric expansion effects in the mercury applied pressure is 110% of injection pressure according to equation (3) [5]. and/or the sample cell, which consequently present themselves as artificial pore volume effects. Pore size as well as pore volume can be greatly influ- the intrusion (b) and the extrusion (c) at specific presenced by the intrusion rate settings as shown below. Five sure values. Prior to those individual tests, the sample samples of an alumina extrudate were analyzed with a was measured under ªnormalº conditions (300 s equiliMicromeritics Autopore 9420 using a so-called scanning mode (equilibration by time for 0 seconds), equilibration-interval settings of 2, 10, and 30 seconds, and an equilibration rate of 0.001 ll/g sec. The cumulative intrusion pore volume curves of the five experiments are shown in Figure 3. A difference of close to 10% in pore volume and 40 to 50% in pore size is noticeable between the ªfastestº and the ªslowestº analysis condition. Another aspect of equilibration time is shown in Figure 4. Here the intrusion volume was followed as a function of time for Fig. 3. Intrusion rate effects on pore size and pore volume [6]. http://www.ppsc.com

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at higher as well as lower pressure values (pore size) compared to the intrusion data. However, reaching equilibrium during the extrusion took substantially longer to reach equilibrium (over 2 hours for the data points evaluated in this example). It seems that other factors besides the flow of mercury through the porenetwork or temperature effects are responsible for the delay in reaching equilibrium. 5.2 Compressibility

2

2

1

→ 3 100 psi

1

1 100 psi → 2 200 psi (Dp = 100 nm)

1

Pressure

Pore Volume

(Dp = 70 nm)

1

1

0

0

1060

1090

1120

1150

1180

Time /min. 1

1

1

1

1

→ 700 psi (Dp = 310 nm)

Pressure

Pore Volume

10 000 psi → 2 000 psi (Dp = 109 nm)

1

→ 200 psi (Dp = 1090 nm)

0

0

0

1500

1560

1620

1680

1740

1800

Time /min.

Fig. 4. How fast does the system (data points) equilibrate? The original data points collected are shown in a); graph b) shows the time-volume curves for intrusion {steps between the intrusion data points shown in a)}; and c) shows the time-volume curves for the blue extrusion data points.

bration time; Fig. 4a)). Interestingly reaching the equilibrium during intrusion (Fig. 4b)) was achieved much faster than during extrusion (Fig. 4c)) even so it was the same sample and the extrusion tests covered data points  2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Compressibility is another major effect, which has to be considered. Ideally this would be corrected by a corresponding blank-run using a non-porous sample of the same material. Unfortunately this is not always possible. Compressibility, b, is defined as the fractional change in volume per unit pressure. The linear equivalent is the Youngs Modulus. Most solids have a b-value of about 10Ð3 to 10 Ð4 (Pa)Ð1. Thus, a 1 cm3 sample will compress by about 0.006 to 0.06 cm3 at the final pressure of 400 MPa. Depending on the pore volume of the sample, this might be a minor effect. On the other hand, polymer or sol-gel materials usually have a substantially larger compressibility and vice versa a much greater effect on the analysis results. Compressibility effects can easily be detected when plotting the cumulative pore volume on a linear pressure scale. The combined compressibility effects have to be considered: mercury, sample cell as well as the sample itself. In case of highly non-compressible samples even a negative overall volume change might be noticed. Compressibility is primarily important at smaller pore sizes or higher pressures. However, this is also the region, which has the relative strongest effect on surface area calculations. Thus, the latter values can be significantly effected (increase as well as decrease) by compressibility. Using an encapsulated sample, e.g. a sample sealed in a thin rubber balloon, is one way of evaluating compressibility. However, one has to be aware that the compressibility of a porous sample, as measured when encapsulated, is not the same as the compressibility of the solid without porosity. Thus, the encapsulated sample simulates the sample behavior before any mercury has entered the pore space, whereas the ªsolid-compressibilityº is describing effects after mercury has filled the pore space and is now compressing the pore walls.

6 Data Interpretation and Analysis 6.1 Pore Size and Pore Volume Intrusion pressure values are directly converted into the corresponding pore size by using the Washburn equahttp://www.ppsc.com

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tion. This is a straightforward calculation. However, one should be aware that mercury porosimetry does not actually measures the internal pore size, but it rather determines the largest connection (throat or pore channel) from the sample surface towards that pore. Thus, mercury porosimetry results will always show smaller pore sizes compared with Scanning Electron Microscopy (SEM) or optical micrographs.

6.2 Density A simple pycnometry type calculation allows measuring the sample density. Especially for samples like spray dried granules density determination at the point when mercury has filled all the inter granular pores is of interest, since it will then describe the internal density of the granules. However, the precision of those measurements is rather rough (2 to 5% error) unless special care is taken with respect to temperature control.

6.3 Surface Area Rootare and Prenzlow [7] derived the following equation: A ˆ

cHg

1 cos h

ZV PdV

…4†

0

which allows converting the pore volume data into the corresponding surface area under the assumption of a reversible intrusion process. Even so, an interconnected pore network does not strictly follow this rule; many publications [7, 8] have reported good correlations between surface areas determined by mercury porosimetry and nitrogen adsorption measurements. Irrespective of this, one should be aware that minor measurement errors especially in the high pressure or small pore size range can effect the calculated surface area to a very large degree and one has to be especially careful with respect to ªgoodº data points in that part of the analysis. 6.4 Particle Size Data from mercury porosimetry can also be used to estimate the particle size of a powder material. Two possible methods are available for this purpose. On the one

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hand, the surface area as determined during the analysis can be converted into the equivalent sphere diameter (particle size) by assuming a specific density, q, of the material. r ˆ

3 rA

…5†

On the other hand, the pore size can be used to estimate the corresponding particle size by assuming a particular packing structure of the powder particles. The surface area technique calculates too small of a particle size if surface roughness, pores inside the particles, or a larger quantity of very small particles exists. The pore size conversion method on the other hand may lead to ªwrongº results if the particle shape is not spherical or when the degree of compaction changes, e.g. due to moisture acting as a lubricant between particles. On a rough scale the particle size is approximately 2 to 4 times larger compared with the measured pore size. In all of these situations the assumed packing structure is critical. One might use the measured pore volume as an indicator for the actual packing structure, but this can only be used as a vague approximation. Further details are described in a classical paper by Mayer and Stowe [9, 10] and later publications by Smith and Stermer [11]. 6.5 Multi-modal Pore Size Distributions Often analysis data show groups of pore sizes, which can either be attributed to the intentional structure of the materials or they could be caused by artificial measurement effects. For example, larger pores are frequently associated with the packing structure of powder particles or between sample pieces. In order to control the spacing between sample pieces one can prepare a stainless steel wire cage around individual sample pieces, e.g. membrane or filter plates, to create a minimum spacing (or pore space) between those pieces, which then shifts the size range of these artificial pores to larger sizes and thus, allows to distinguish porosity of the sample from artificial pores. Figure 5 shows an example of a multi-modal pore size distribution. The first step at 20 to 30 lm describes the inter granular porosity between spray-dried granules of about 60 to 100 lm in size. The second and third step (slope change below 0.07 lm) are part of the internal pore structure, which is caused by the various primary particles present within the granules.

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6.6.1 Contact Angle Hysteresis

Cum. Pore Volume /ml/g

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.001

0.01

0.1

1

10

100

1000

Pore Diameter /µm

Fig. 5. Multimodal pore size distribution of a spray dried catalyst sample.

6.6 Hysteresis, Trapped Mercury, and Pore-connectivity Hysteresis between the intrusion (increasing pressure) and extrusion (decreasing pressure) is observed in essentially all samples during mercury porosimetry measurements. Several explanations have been proposed. Primarily those are: contact angle hysteresis, the ink bottle theory, and the percolation-connectivity model. A typical example is given in Figure 6, which shows results for a sample made from monodispersed silica spheres. The sample is of special interest, since the ordered arrangement of the spheres allows one to compare measured with predicted pore sizes. These model structures are of special interest, since they allow one to gain further insight into the intrusion and extrusion process.

Differences in advancing and receding contact angles are frequently observed, but the idea is somewhat questionable from a thermodynamic point of view. The surface roughness or impurities on the mercury or solid surface could certainly change the value of the contact angle. However, some observations cannot be explained by the contact angle hysteresis (see Figure 7). Such as: a) Contact angle hysteresis can't explain why some mercury remains trapped in the pore system after complete depressurization. b) Re-intrusion and extrusion curves should have a similar shape when plotted on a logarithmic pressure or pore size scale. However, literature data frequently show only a marginal fit between these curves. c) No volume changes should be observed, when scanning between the hysteresis branches (extrusion and intrusion curve). Yet, frequently those scans within the hysteresis-range aren't constant with respect to pore volume. d) Chemical changes on the sample surface (with no change in the pore structure) can lead to drastic changes in the extrusion curve, whereas the intrusion curve is frequently unaffected. A fact, which can not be explained by contact angle hysteresis. For further information on contact angle effects, the reader is referred to the publications by Lowell et al [12] and Salmas et al [13].

0.25

Extrusion 0.2

Fig. 7. Schematic drawing of theoretical (left) and experimental (right) observations with respect to contact angle hysteresis.

2nd extrusion run; partially filled run 2nd intrusion with mercury ∆) partially filled ((D)

0.15

0.1

6.6.2 Ink Bottle Theory

0.05

0

Intrusion Silica sample 1000 C 31 h

-0.05 0.01

0.1

1

10

Pore diameter /um

Fig. 6. Scanning within the mercury porosimetry hysteresis of silica samples calcined at 1000C for 31 hours [20].

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It is obvious that pores rarely are of uniform shape. The ªthroatº or entrance opening to a pore is smaller than the actual cavity. So, mercury will enter the pore cavity at a pressure determined by the entrance size and not the actual cavity size. During extrusion the mercury network would then break at all the throats (narrower connections) between pores, leaving a large amount of mercury trapped inside

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8 the sample. Figure 8 demonstrates this for a glass model sample with a carefully arranged pore structure. It is interesting to notice that more mercury is remaining in the structure as the ratio of inner pore-size and throatsize increases. The ink-bottle theory can, thus, easily explain trapped mercury, but it does not necessarily explain the pore size shift between intrusion and extrusion.

Fig. 8. Glass model of an artificial pore system. Trapped mercury is visible in black in the bottom picture after release of pressure [5].

Part. Part. Syst. Charact. 23 (2006) 1±11

6.6.3.1 Energy Barrier or ªSnap-offº Factor, and Pore Geometry

Ideally intrusion into and extrusion out of uniformly sized pores of cylindrical shape should happen at the same pressure. However, in most real samples we do not have that type of ideal pore geometry. During the extrusion process new mercury interfaces have to be created as the mercury retracts from the pore system. This process requires additional energy, thus, an ªenergy-barrierº is to be expected. Alternatively the energy barrier can be expressed as a ªsnap-offº factor; describing to what degree the extrusion pressure (an additional pull) has to be lowered relative to the corresponding intrusion pressure until the mercury network breaks apart and mercury can retreat from a specific pore. Several simulated ideas have been presented trying to calculate this snap-off factor for different pore geometries. For example, the snap-off factor for long cylindrical pores of uniform diameter is relative small. However, the factor becomes larger as the pore length is decreased (see Figure 9). In a similar way the snap-off factor is affected by the opening angle of conical cylinder pores (as shown in Figure 10) or the contact angle between mercury and the solid. Moreover, Tsakiroglou & Payatakes [17] simulated the snap-off factor for lenticular throats. The authors then combined their calculations with model pore structures etched in glass slides and compared the simulated and experimental intrusion and extrusion curves. It is interesting to notice that a pore will empty not only according to their own size and the size of the connecting throats, but the specific geometric arrangement of the throats, which are still filled with mercury, is important as well. For example a pore, which is con-

Cylindrical Pore

6.6.3 Connectivity Model The connectivity model uses a network of pores. It is kind of an extension of the ink-bottle theory. Yet it has an added component in terms of considering the connection effects between the pores. In order for a pore to become filled with mercury it must be equal to or larger than the corresponding ªpore sizeº at the applied pressure, but it also requires a continuous path of mercury leading to that pore. Large internal voids, which are surrounded by smaller pores, will not be filled unless the pressure is sufficient to fill a pathway towards that pore. During the extrusion process, the reverse process occurs, and certain pores or islands of pores will remain filled with (trapped) mercury, once they no longer have a continuous mercury path towards the sample surface. Various studies [5, 13±20] have taken this idea and studied the effects in model pore structures as described in the following section.

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1 µ m diameter;

θ

= 140°; γ = 0.48 N/m

Length

Intrusion

Extrusion

µm

MPa

MPa

PI/PE

100

0.735

0.726

1.01

10

0.735

0.650

1.13

5

0.735

0.566

1.31

2

0.735

0.309

2.78

1.5

0.735

0.166

7.58

1.4

0.735

0.050

14.66

Fig. 9. Calculation of ªEnergy-Barriersº (ªsnap-offº factors) in cylindrical pores of different aspect ratios [19, 20].

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10 ” 10 ” 10 array of cubes (pores) joined by a network of cylinders (throats) in the 3-D network. The software program, commercially distributed as ªPore-Corº, uses a similar ªsnap-offº as well as a connectivity (pore blocking) factor to optimize the fit between simulated and experimental data. The model is somewhat crude by using only a 10 ” 10 ” 10 pore matrix, but it is a very interesting approach to extract and simulate pore network information from the mercury porosimetry measurements and to use that information thereafter to calculate and predict permeability or tortuosity of a sample. Similar network simulations, which are not specifically mentioned here, have been published by S. Rigby [30±33].

7 Conclusions Fig. 10. Calculation of ªEnergy-Barriersº (ªsnap-offº factors) in conical cylinder pores [19, 20].

nected to the mercury network by only one filled throat (and 3 empty throats), is drained at a higher pressure (earlier) during extrusion compared to a pore, which is connected via 2 or 3 ªfilledº throats. Similar effects were observed in an ordered packed sphere structure as demonstrated in Figure 6. On partial intrusion (as compared to a complete 100% intrusion) the mercury network contained numerous ªbreakagepointsº. Thus, extrusion from the smaller tetrahedral pores could be observed separately from the octahedral pores during the extrusion, in case they were connected to empty channels or throats. On the other hand the completely filled pore network did not have those existing breakage points and a much lower pressure was required to create these breakage points. The additional pull for the breakage point then resulted in an extrusion pressure, which was too low to distinguish octahedral and tetrahedral pores (as visible in the extrusion of the partially filled sample in Figure 6). The hysteresis finestructure was no longer detectable. G. Mason et al [21±26] published simulations on the intrusion/extrusion process as related to different pore geometries. They studied rods of equal or different diameters, rods in contact with a flat plate and so on. He also performed detailed calculations of the exact surface curvature and energy minimization and calculated the influence between adjacent pores with respect to drainage and extrusion. 6.6.3.2 Network Simulation Software

Another approach was taken by P. Matthews and his coworkers [27±29]. They simulated the pore system by a

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Pore size distribution data generated by mercury porosimetry are primarily useful in comparative studies of similar materials. This is true because the absolute accuracy of the data depends on various assumption and experimental factors. Some of these factors cancel out for the relative comparisons of similar materials and thus, it is not quite as critical to have perfectly correct values in these cases. In general, pore size and pore volume are repeatable to better than 1% standard deviation, but the results are also limited a priori in three ways: l Mercury porosimetry determines the largest entrance to a pore, but not the actual ªpore sizeº. l The smallest pore size, which can be filled with mercury, is limited by the maximum pressure, which can be achieved by the instrument, e.g. 3.5 nm diameter at 400 MPa assuming a contact angle of 140. l The largest measurable pore size is limited by the height of the sample, which determines a minimum ªhead-pressureº, e.g. a 1cm sample height is approximately equivalent to a pore of 1 mm diameter. In addition properties of the sample may affect the reproducibility and create difficulties in giving an unambiguous interpretation of the result: l A loosely packed powder might become further compacted due to the pressure exerted on the sample before the mercury actually penetrates the pore spaces. Erroneously the compaction effect then might be interpreted as porosity. l Likewise elastic or permanent structural changes can occur in highly porous or ªsoftº materials due to the applied pressure. A second intrusion run can only partially reveal those changes. The concept of a connected pore network and the existence of a snap-off factor, during the extrusion process, has been studied and demonstrated by different re-

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10 search groups. These concepts provide the most powerful explanation at the moment for hysteresis effects and the fact that mercury remains trapped inside the sample after depressurization. However, these concepts are still in their early stages and more detailed studies of modelpore-structures are needed to fully understand the effects of individual factors in those models. Ordered packed-sphere structures are a very useful model system, since they have very well defined pore sizes and connections between the pores. Additional studies should include the theoretical calculations combined with experimental intrusion/extrusion data of samples of etched channels in glass, which will further enhance our understanding of these ªnetwork effectsº. Despite all these limitations mercury porosimetry still is an extremely useful analysis technique. It provides exceedingly important information about the porosity of samples, not at least due to the fact that it covers pore sizes over a range of 5 orders of magnitude from 0.4 mm to less than 4 nm.

8 References [1] T. Allen, Particle Size Measurement, Vol. 2, Chapman & Hall, 1997, 162. [2] J. Klobek, Hysteresis in Porosimetry, Powder Technol. 1981, 29, 63±73. [3] A. W. Adamson, Physical Chemistry of Surfaces, John Wiley & Sons, New York, 1982, 349. [4] H. Giesche, Mercury Porosimetry, in Handbook of Porous Solids Vol. 1, (Eds.: F. Schüth, K. S. W. Sing, J. Weitkamp), Wiley-VCH, Weinheim, 2002, 309±351. [5] N. C. Wardlaw, M. McKellar, Mercury Porosimetry and the Interpretation of Pore Geometry in Sedimentary Rocks and Artificial Models, Powder Technol. 1981, 29, 127±143. [6] P. A. Webb, C. Orr, Analytical Methods in Fine Particle Technology, Micromeritics, Norcross, 1997, 165 & 171. [7] H. M. Rootare, C. F. Prezlow, Surface Areas from Mercury Porosimetry Measurements, J. Phys. Chem. 1967, 71 (8), 2733±2736. [8] H. Giesche, K. K. Unger, U. Müller, U. Esser, Hysteresis in Nitrogen Adsorption and Mercury Porosimetry on Mesoporous Model Adsorbents Made of Aggregated Monodisperse Silica Spheres, Colloids Surfaces 1989, 37, 93±113. [9] R. P. Mayer, R. A. Stowe, Mercury Porosimetry ± Breakthrough Pressure for Penetration Between Packed Spheres, J. Colloid Sci. 1965, 20, 893±911. [10] R. P. Mayer, R. A. Stowe, Mercury Porosimetry; Filling of Toroidal Void Volume Following Breakthrough Between Packed Spheres, J. Phys. Chem. 1966, 70 (12), 3867±3873.

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Part. Part. Syst. Charact. 23 (2006) 1±11

[11] D. M. Smith, D. L. Stermer; Theoretical and Experimental Characterization of Random Microsphere Packings, J. Colloid Interface Sci. 1986, 111 (1), 160±168. [12] S. Lowell, J. E. Shields, M. A. Thomas, M. Thommes; Characterization of Porous Solids and Powders: Surface Area, Pore Size and Density, Kluwer, 2004. [13] C. Salmas and G. Androutsopoulos; Mercury Porosimetry: Contact Angle Hysteresis of Materials with Controlled Pore Structure, J. Colloid Interface Sci. 2001, 239 (1), 178±189. [14] I. Kornhauser, S. Cordero, C. Felipe, F. Rojas, A. J. Ramirez-Cuesta, J. L. Riccardo; On Comparing Pore Characterization Results from Sorption and Intrusion Processes, Proc. Fundamentals of Adsorption 7, Eds. Kaneko et al., 2000, 1030±1037. [15] F. A. L. Dullien, G. K. Dhawan, Characterization of Pore Structure by a Combination of Quantitative Photomicrography and Mercury Porosimetry, J. Colloid Interface Sci. 1974, 47, 337±349. [16] F. A. L. Dullien, G. K. Dhawan, Bivariante Pore-Size Distribution of Some Sandstones, J. Colloid Interface Sci. 1975, 52 (1), 129±135. [17] C. Tsakiroglou, A. Payatakes, Mercury Intrusion and Retraction in Model Porous Media; Adv. Colloid Interface Sci., 1998, 75 (3), 215±253. [18] C. D. Tsakiroglou, G. B. Kolonis, T. C. Roumeliotis, A. C. Payatakes, Mercury Penetration and Snap-off in Lenticular Pores, J. Colloid Interface Sci. 1997, 193 (2), 259±27. j [19] H. Giesche, Interpretation of Hysteresis ªFine-Structureº in Mercury-Porosimetry Measurements, in Advances in Porous Materials, Mat. Res. Soc. Symp. Proc. Vol. 371, Eds. S. Komarneni et al., MRS, Pittsburgh, 1995, 505±510. [20] H. Giesche, Within the Hysteresis: Insight into the Bimodal Pore-Size Distribution of Close-Packed Spheres, in Characterization of Porous Solids IV, Eds. B. McEnaney, T. J. Mays, J. Rouquerol, F. Rodriguez-Reinoso, K. S. W. Sing, K. K. Unger, The Royal Society of Chemistry, Cambridge, 1997, 171±179. [21] S. Bryant, G. Mason, D. Mellor, Quantification of Spatial Correlation in Porous Media and its Effect on Mercury Porosimetry, J. Colloid Interface Sci. 1996, 177, 88±100. [22] G. Mason, D. W. Mellor, Simulation of Drainage and Imbition in a Random Packing of Equal Spheres, J. Colloid Interface Sci. 1995, 176, 214±225. [23] G. Mason, N. R. Morrow, Effect of Contact Angle on Capillary Displacement Curvatures in Pore Throats Formed by Spheres, J. Colloid Interface Sci. 1994, 168, 130±141. [24] G. Mason, N. R. Morrow, Capillary Behavior of a Perfectly Wetting Liquid in Irregular Triangular Tubes, J. Colloid Interface Sci. 1991, 141, 262±274. [25] G. Mason, N. R. Morrow, Coexistence of Menisci and the Influence of Neighboring Pores on Capillary Displacement Curvatures in Sphere Packings, J. Colloid Interface Sci. 1984, 100, 519±535.

 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Part. Part. Syst. Charact. 23 (2006) 1±11

[26] G. Mason, M. D. Nguyen, N. R. Morrow, Effect of Contact Angle on the Meniscus between Two Equal Contacting Rods and a Plate, J. Colloid Interface Sci. 1983, 95, 494±501. [27] G. P. Matthews, M. C. Spearing, Measurement and Modeling of Diffusion, Porosity and Other Pore Level Characteristics of Sandstones, Marine and Petroleum Geology 1992, 9, 146±154. [28] G. P. Matthews, A. K. Moss, M. C. Spearing, F. Voland, Network Calculation of Mercury Intrusion and Absolute Permeability in Sandstone and Other Porous Media, Powder Technol. 1993, 76, 95±107. [29] G. P. Matthews, C. J. Ridgway, M. C. Spearing, Void Space Modeling of Mercury Intrusion Hysteresis in Sandstone, Paper Coating, and Other Porous Media, J. Colloid Interface Sci. 1995, 171, 8±27.

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11 [30] S. P. Rigby, S. Daut, A Statistical Model for the Heterogeneous Structure of Porous Catalyst Pellets, Adv. Colloid Interface Sci. 2002, 98 (2), 87±119. [31] S. P. Rigby, R. S. Fletcher, S. N. Riley, Characterization of Macroscopic Structural Disorder in Porous Media Using Mercury Porosimetry, J. Colloid Interface Sci. 2001, 240 (1), 190±210. [32] S. P. Rigby, .F. Gladden, Deconvolving Pore Shielding Effects in Mercury Porosimetry Data Using NMR Techniques, Chem. Eng. Sci. 2000, 55 (23), 5599±5612. [33] S. P. Rigby, A Hierarchical Structural Model for the Interpretation of Mercury Porosimetry and Nitrogen Sorption, J. Colloid Interface Sci. 2000, 224 (2), 382±396.

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