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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]
2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
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
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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.
2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
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Part. Part. Syst. Charact. 23 (2006) 1±11
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
2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
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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Part. Part. Syst. Charact. 23 (2006) 1±11
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].
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2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
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Part. Part. Syst. Charact. 23 (2006) 1±11
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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Part. Part. Syst. Charact. 23 (2006) 1±11
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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Part. Part. Syst. Charact. 23 (2006) 1±11
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].
2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
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].
2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
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Part. Part. Syst. Charact. 23 (2006) 1±11
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
2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
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
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