Image Anal Stereol 2012;31:79-87
Original Research Paper
doi:10.5566/ias.v31.p79-87
POROSIMETRY BY RANDOM NODE STRUCTURING IN VIRTUAL
CONCRETE
PIET STROEVEN1, NGHI LB LE,1, LAMBERTUS J SLUYS1 AND HUAN HE1,2
1
Faculty of Civil Engineering and Geosciences, Delft University of Technology, PO Box 5048, 2600 GA
Delft, the Netherlands; 2GeMMe, Minerals Engineering-Materials-Environment, University of Liège, Sart
Tilman B52, 4000 Liège, Belgium
e-mail: L.B.N.Le@tudelft.nl; P.Stroeven@tudelft.nl; L.J.Sluys@tudelft.nl; huan.he@hotmail.com
(Received September 6, 2011; revised February17, 2012; accepted April 24, 2012)
ABSTRACT
Two different porosimetry methods are presented in two successive papers. Inspiration for the development
came from the rapidly-exploring random tree (RRT) approach used in robotics. The novel methods are
applied to virtual cementitious materials produced by a modern concurrent algorithm-based discrete
element modeling system, HADES. This would render possible realistically simulating all aspects of
particulate matter that influence structure-sensitive features of the pore network structure in maturing
concrete, namely size, shape and dispersion of the aggregate and cement particles. Pore space is a complex
tortuous entity. Practical methods conventionally applied for assessment of pore size distribution may fail
or present biased information. Among them, mercury intrusion porosimetry and 2D quantitative image
analysis are popular. The mathematical morphology operator “opening” can be applied to sections and
even provide 3D information on pore size distribution, provided isotropy is guaranteed. However,
aggregate grain surfaces lead to anisotropy in porosity. The presented methods allow exploration of pore
space in the virtual material, after which pore size distribution is derived from star volume measurements.
In addition to size of pores their continuity is of crucial importance for durability estimation. Doublerandom multiple tree structuring (DRaMuTS), introduced earlier in this journal (Stroeven et al., 2012) and
random node structuring (RaNoS) provide such information.
Keywords: DEM, pore connectivity, porosimetry, star volume, virtual concrete.
nauskas (2006). Relevant discussion on this issue can
be found in Stroeven et al. (2009).
INTRODUCTION
Modern computer facilities have made studies on
virtual concrete simulated by discrete element method
(DEM) a liable alternative for experimental investigations. The two porosimetry methods introduced in
Stroeven et al. (2012) and herein are therefore intended
to be applied to virtual cement pockets between neighboring aggregate grains in concrete. They are produced
in an analogue version by the dynamic concurrent
algorithm-based physical DEM system HADES. This
is not a unique system, of course. Since the introduction
of physical methods for DEM in rock mechanics by
Cundall and Strack (1979), also many systems have
been developed world-wide in concrete technology; in
many cases however this was accomplished for specific
and thus restricted purposes. Reference can be given to
Jodrey and Torey (1981), Ansell and Dickinson (1986),
Mościński et al. (1989), Bentz et al. (1993), O’Connor
et al. (1997); Stroeven (1999), Tsunekawa and Iwashita
(2001), Puri and Uomoto (2002), Williams and Philipse
(2003), Li et al. (2006) and Markauskas and Kačia-
For (super) high performance concrete ((S)HPC)
the cement particle packing will be dense, with values
of water to cement ratio (w/c) as low as 0.2. The
fresh state is thereupon hydrated, a process that does
not change the position of the cement grains.
During hardening, pore space is generally accepted
to gradually de-percolate. This process has been demonstrated depending on the dispersion of the cement
particles and will thus be reflected in a biased way
by random generator-based (RG) simulation systems
popular in concrete technology. References to a selection of such RG systems can be found in Stroeven
et al. (2009). For specific examples, see Diekkämper
(1984), Roelfstra (1989), van Breugel (1991), Bentz
et al. (1993), Meakawa et al. (1999) and Bishnoi and
Scrivener (2009). Significant differences between the
de-percolation process characteristics obtained by
(analogue) DEM- and RG-based systems and with
the digitized model of Garboczi and Bentz (2001)
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STROEVEN P ET AL: Concrete porosimetry
nodes are built up by unobstructed straight line
segments connecting the nodes.
have been shown in Chen et al. (2006). For details
of the hydration algorithms in the case of spherical
cement grains, see Stroeven et al. (2011). This is
similar to hydration algorithms developed elsewhere
(Navi and Pignat, 1999; Stroeven, 1999).
Experimental approaches can provide information
on porosity and pore size distribution. An obvious
method is quantitative image analysis, which is also
used in concrete technology. However, the most popular approach is definitely mercury intrusion porosimetry (MIP). The latter offers significantly biased
information, however, because of the unrealistic
schematization of the geometry of the pore channels
and the neglect of the so called bottle necks in the
pore system (Diamond, 2000). Insight into pore connectivity, which is of crucial importance for estimating
durability risks, therefore requires other approaches.
eliminated nodes
structured nodes
‘direct’ connections
‘indirect’ connections
The presented novel methods (see also Stroeven
et al., 2012) allow making a distinction between continuous pores (trunks) stretching over the full dimensions of the cement pocket, isolated pores and deadend pores branching off the trunks. Particularly the
latter have been demonstrated making the connections
between pore trees of neighboring interfacial transition
zones (ITZs), so that finite values of connected porosity
were found for bulk paste. This was demonstrated by
the earlier described DRaMuTS example in Stroeven
et al. (2012).
solid phases (particles)
Fig. 1. Structure of random nodes in RaNoS method.
When two nodes can be connected together by a
straight line that does not intersect with any part of
the solid phases, such two nodes have ‘direct connection’. This initiates the node-clustering process. A
cluster involves all nodes that are directly connected.
When two of such clusters during this process can be
directly connected, they merge into a new one.
Hence, once the clustering process is finished, there
are a number of structured clusters of nodes. In each
of these clusters, the nodes are mutually connected
reflecting the connectivity among pores in the virtual
cement paste. Based on the node structure, pore
characteristics such as porosity, degree of percolation,
pore location distribution and pore size distribution
are assessed.
Volume-based pore size distribution is derived
from star volume measurements in 3D, as demonstrated
also earlier. Of course, this information can be transferred to a traditional number-based pore size distribution.
RANDOM NODE STRUCTURING
ALGORITHM
Random node structuring (RaNoS) is a general
approach, which can be used for porosimetry in
hydrated virtual (blended) cement pastes or for other
purposes, such as porosimetry in packing of grains.
Fig. 1 presents a 2-D scheme for RaNoS. It starts by
generating with a pseudo-random generator algorithm a
system of nodes uniformly at random (UR) dispersed
in container space. Then, the nodes situated in solid
phase are eliminated from further consideration. This
yields nodes UR dispersed inside pores as a detection
system of the 3D capillary pore system in cement
paste. The next step is a structuring process for the
node system in which the relationships between the
IMPLEMENTATION
Elimination of nodes in solid phases
As mentioned above, to obtain nodes that are
uniformly at random distributed in pore space, an
elimination process is required of the nodes interfering
with solid phase. For particles simulated by spheres,
this can easily be accomplished. For particles simulated
by digital-image-based model (Bentz and Garboczi,
1991) this is also easily carried out by determining
the phase of the voxel in which the considered point
is situated.
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Image Anal Stereol 2012;31:79-87
For particles simulated by polyhedrons, this is
more complicated. Such particles have been employed
earlier for simulating crushed rock aggregate and for
cement grains (He, 2010). An efficient way to detect
if a point is inside a convex polyhedron was presented
in He (2010). If ni • (X – Vi) < 0 is satisfied for all
facets of the polyhedron, the considered point X is
inside. ni is the normal vector of facet i pointing
outside of the polyhedron; Vi is a vertex of facet i.
This, however, can not be used for the case of nonconvex polyhedrons.
is a surface-driven phenomenon this would be a next
step in simulating hydrating systems among other
things for porosimetry. Ultimately, also particles with
non-convex surface elements should be considered,
since this will influence the formation of hydration
products in a densely packed binder system. For this
purpose, the hydration algorithms will require significant adaptations. Yet, the HADES simulation system
should be advanced enough to render possible performing such investigations in the near future.
Detection of connection between two
nodes
Fig. 2 illustrates a more general way for this detection. A probing ray emanating from the considered
node in a random direction is generated. In fact, this
ray is a line segment (limited by two ends) and its
length must be long enough to possibly go through
the polygon. If the number of intersections of the ray
and the polyhedron’s surfaces (polygons) is odd then
the point is inside the polyhedron, otherwise it is
outside the polyhedron.
As mentioned above, the RaNoS method involves a
system of random nodes that are pair wise connected
by unobstructed line segments. The line segment will
be considered as being unobstructed if it does not
have any intersection with neighboring particles. In
case particles are simulated by spheres, the intersection
detection reduces to that of a line and a sphere. In
case particles are simulated by the digital-imagebased model (Bentz and Garboczi, 1991), a series of
points with constant spacing is distributed along the
line segment. Hence, the segment line is unobstructed
if there is no point situated in any voxel representing
a solid phase. In case particles are simulated by polyhedral shapes, the detection procedure is the same as
earlier described. Since two ends of the line segments
are outside particles, the number of intersections is
only either zero or an even number. If the number is
zero then there is no intersection. Otherwise, such an
intersection exists.
Localized and parallel computing
The RaNoS method centers on finding direct
connections between neighboring nodes. The cell
method is utilized for this purpose. Each cell contains a
list of particles that interfere with the cell and a list
of associated nodes. The cells render possible executing
localized operations to improve the speed of computation. So, the clustering process is first applied locally
in each cell. Thereupon, the clustering process is
applied in the whole simulated space. Moreover,
since local clustering processes can be implemented
independently and simultaneously, ‘parallel computing
processing’, the advanced feature of some programming languages, is also applied to speed up the computations.
node inside the polyhedron
node outside the polyhedron
probing ray
intersections of probe rays and
the polyhedron’s facets
Fig. 2. Determination whether a point is inside or
outside a polyhedron.
Although the example case is a simple one, whereby spherical cement grains are used, recent synchrotron-based X-ray imaging results by Garboczi
and Bullard (2004) revealed cement particles to
explicitly display convex as well as non-convex parts
in their surface. In our research, so far, mixtures of
ellipsoidal cement particles or of polyhedrons were
employed for simulating cement particle packing
(He, 2010). Hereby the surface to volume ratio is
similar to that found by Garboczi and Bullard (2004)
in the aforementioned experiments. Since hydration
Detection of pore connectivity
Considering a cubic sample pocket of cement
paste, two types of capillary pores can be distinguished. The percolated pores form continuous entities
connecting two opposite sides of the specimen. The
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STROEVEN P ET AL: Concrete porosimetry
Next, hydration was realized by the hybrid simulation approach described in Stroeven et al. (2011).
Finally, the pore structure of the samples was investigated by RaNoS at ultimate degree of hydration
(DOH). Due to shortage of water for chemical reactions, the hydration process of the samples stops at
ultimate DOH of 0.727. Note that the maximum
grain size of the simulated cement is about three
times smaller than in practice. Hence, this should be
accounted for in the evaluation of obtained results
(such as the extension of the boundary zone).
de-percolated pores or the isolated pores might be
connected together but are not connected to both
sides of the pocket. The connectivity of pores is
defined as the volume ratio of percolated pores to the
total of capillary pores. Since the system of UR
dispersed nodes can be considered as a discretized
system of pore space, the pore connectivity can be
estimated as the ratio of the number of nodes that are
connected to the two opposite sides, to the total
number of random nodes. For assessment of pore
connectivity, nodes should also be uniformly at
random distributed on the opposite outer surfaces of
the specimen and incorporated in the clustering
process. When a cluster of nodes contains at least
one node in each of the two opposite surfaces, this
cluster represents a percolated pore channel. In
analyzing characteristics of the pore system, e.g.,
porosity, connectivity, pore size distribution, the
nodes situated in the two end surfaces are not taken
into account, however.
Fig. 3 presents the random nodes distributed in
pore space of the two cement pockets. Based on these
systems, porosity of the C1 and C2 pockets was found
to amount 5.57 % and 5.59 %, respectively. The
associated connected pore fractions of the C1 and C2
pockets were 98.53% and 97.58%. The gradient
structure of pore volume as a function of distance
from the rigid (R) surface is displayed in Fig. 4 and
compared with observations on bulk paste (container
with six periodic (P) boundaries). Note the similarity
with the gradient structures shown in Fig. 6 in Stroeven
et al. (2012). Fig. 5 presents volume-based pore size
distribution functions assessed by the star volume
measurements.
RaNoS EXAMPLE
Two cubic pockets of simulated cement paste are
considered in this study. The characteristics of the
two pockets are shown in Table 1 (P = periodic, R =
rigid boundary). Fresh cement particles were generated
and packed with the HADES system. The initial size
of the pocket was eight times reduced in the dynamic
packing process. A rigid surface leads to the formation
of an interfacial transition zone (ITZ); an interesting
feature in concrete technology. However, it is the
inevitable result of the well-known wall effect, because
the cement particles will pile up differently close to
the wall, constituted by the aggregate grain’s surfaces.
In general, higher porosity and lower strength are
reported as in bulk (Ollivier et al., 1995; Maso,
1996; Scrivener, 1999; Diamond and Huang, 2001;
Stroeven and Stroeven, 2001). This paper does not
pursue discussing the phenomenon of ITZ development
in concrete, however.
Fig. 6 shows sensitivity analyses for the connected
fraction of porosity of the C1 pocket by RaNoS and
by the digital technique proposed by Hosen and
Kapelman and cited in Navi and Pignat (1999). In
order to check connectivity in the digital technique,
the microstructure is subdivided into a 3D regular
lattice of cubic volume-pixels (voxels), each of which
represents either a solid or a pore part. Then a clustering process for pore voxels is implemented to distinguish two types of pore voxels: voxel clusters that
can be connected to both top and bottom surfaces
and those which are isolated. The connected fraction
is then calculated as the ratio of the number of voxels
belonging to the first type to that belonging to the
second type.
Table 1. Characteristics of simulated cement samples.
Boundary
conditions
C1
6P
C2
4P + 2R
Diameter range Specific surface area Size of pocket
(cm2/g)
(m)
(m)
1~30
1838
100
82
Initial W/C
ratio
Number of
particles
0.304
4060
Image Anal Stereol 2012;31:79-87
Fig. 5. Effect of boundary conditions on volume-based
pore size distribution functions of Portland cement
pocket in concrete.
Fig. 3. System of nodes distributed uniformly at random
in pore space of cement pocket C1 (top) and C2
(bottom) (unit of axes is m).
Fig. 6. Sensitivity analyses of connected fraction of
porosity on a number of sampling nodes (C1 pocket)
by RaNoS versus ‘digital technique’ (cited in Navi
and Pignat, 1999).
Fig. 6 demonstrates that the connected pore fraction
becomes stable (accepting only a very small change)
with more than 1 million of random points in RaNoS.
In the digital technique a far higher number would be
required, revealing the efficiency of the new technique.
DISCUSSION
The two novel porosimetry methods introduced
herein (RaNoS) and in Stroeven et al. (2012)
(DRaMuTS) are somewhat similar but still have
different characteristics and potentialities. In RaNoS
a system of UR dispersed nodes is first generated and
then a clustering process starts, ultimately resulting in
groups of connected nodes. Contrary, in DRaMuTS a
Fig. 4. Density distribution of pore volume as function
of distance from the left rigid surface.
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STROEVEN P ET AL: Concrete porosimetry
system of virtual trees consisting of vertices (nodes)
and edges (line connecting a pair of nodes) are first
generated in pore space. Note that the vertices are
distributed randomly but not uniformly as a result of
introduced improvements to speed up exploration of
pore space. So, this cannot be used to analyze pore
characteristics. For that purpose, a second system of
UR dispersed nodes is generated. A common point of
the two methods is therefore that pore characteristics
(such as pore size distribution) are analyzed on the
basis of a system of UR dispersed nodes. The tree
systems in DraMuTS additionally render possible
exploring pore space with respect to topological
properties. Trunks of the trees can be defined as pore
network parts connected to external sides of the
pocket, whereas others parts of the trees branch off
the trunks, basically form-ing dead-end pores unless
coalescing with those of a neighboring tree. Finally,
isolated pores can be distinguished. This topological
exploration is impossible in RaNoS.
(Zheng et al., 2011b). Of course, further extension
and generalization as to grain shape would readily be
possible by HADES simulation.
The porosimetry study by Chen et al. (2004;
2006), whereby use was made of serial sectioning
and 3D reconstruction, led to the conclusion that (for
low w/c-ratios) continuous porosity was zero outside
a narrow zone adjacent to the aggregate surface.
Hence, diffusivity of concrete would rely on a spatial
pore network structure whereby a significant degree
of ITZ percolation should provide for the connection
in the continuous pore channels circumventing the
aggregate grains. This would not be in agreement
with the fore-going.
For instance, with a chosen ITZ thickness of
t = 30/3 = 10 μm in the virtual material, a value of
1.98 is obtained for the ratio of ITZ porosity to bulk
porosity, whereas with t = 50/3 17 μm this value is
1.32. Hence, based on data in Table 2 and Fig. 4, the
global paste porosity p can be calculated for Vagg =
0.6:
As an example, for a Fuller distribution with
maximum aggregate size of 16 mm, ITZ thickness is
taken as t = 30 μm or 50 μm (selected from literature).
Zheng et al. (2011a) provide numerical data for ITZ
volume fraction (VITZ) at spherical aggregate volume
fractions (Vagg) of respectively 0.6 and 0.75 as listed
in Table 2. Fig. 4 allows approximately estimating
average porosity over an ITZ scaled down by a factor
of 3.
Table 2. ITZ volume fraction for different values of
aggregate volume fraction and ITZ thickness.
VITZ
Vagg
t = 30 μm
t = 50 μm
0.6
0.045
0.085
0.75
0.060
0.105
pbulkVbulk 1.98 pbulkVITZ
t 30 m ; p
V paste
pbulk (0.4 0.045) 1.98 pbulk 0.045
1.110 pbulk
0.4
t 50 m ; p pbulkVbulk 1.32 pbulkVITZ
V paste
pbulk (0.4 0.085) 1.32 pbulk 0.085 1.068 p
bulk
0.4
Instead, the porosimetry applications of the presented methods demonstrate that pore continuity is realized outside the aforementioned narrow zones with
high degree of continuous porosity. The significant
bulk value displayed in Fig. 4 is most probably due
to dense network structures of (basically dead-end)
pores branching off the continuous channels in the
narrow zones. Even without ITZ percolation, these
pore network structures can connect to form
transport routes in bulk paste.
and for Vagg = 0.75:
pbulkVbulk 1.98 pbulkVITZ
t 30 m ; p
V paste
pbulk (0.25 0.06) 1.98 pbulk 0.06
1.235 pbulk
0.25
t 50 m ; p pbulkVbulk 1.32 pbulkVITZ
V paste
pbulk (0.25 0.105) 1.32 pbulk 0.105 1.134 p
bulk
0.25
Data in Fig. 4 render possible roughly estimating in an effective medium approach the impact
of higher ITZ porosity on global porosity, and thus
on the expected ITZ contribution to durability issues
such as chloride diffusion. Of course, ITZs partly
overlap in a dense random packing of aggregate.
Garboczi and Bentz (1997) analytically approached
this problem for spherical aggregate grains. Zheng et
al. (2011a) extended the analytical approach to RGsimulated dispersions of ellipsoidal aggregate and
applied it for estimating chloride diffusivity in concrete
in which V and p denote, respectively, the volume
fraction and porosity.
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Image Anal Stereol 2012;31:79-87
simulations of fresh cementitious materials by a dynamic algorithm-based DEM system, HADES. During
maturation, a process of pore de-percolation takes
place that can be numerically assessed. To do so, the
developing pore network structure is analyzed by
modern methods of which development is inspired
by experiences in robotics. The first of the two presented methods (Stroeven et al., 2012) renders possible
separating between continuous pores (trunks), dead-end
pores branching off such pores, and isolated dead-end
pores. Tree-like pore network structures are obtained
of which density is highest in the ITZ. As to this
aspect they confirm results obtained by the far more
laborious approach of serial sectioning and 3D reconstruction.
This demonstrates that the accuracy of estimating
the thickness of the ITZ can be low. The aggregate
grain surfaces increased overall porosity in the paste
as a consequence by about 9% in the case of more
dispersed aggregate grains and by 19% in the dense
aggregate case. In this effective media approach, the
porosity in the aggregate is neglected.
The presented method as well as the earlier introduced method, DraMuTS (Stroeven et al., 2012), offers
new information on the pore network structure in
hardening cement paste. This should have influence
on models nowadays used for prediction of transport
through porous concrete. Such methods were discussed
in Hu (2004). A new approach to durability estimation
might however require setting sensitivity on a somewhat lower level than in the plateau range of Fig. 6.
This would modify the pore network structure in Fig.
3 into a partly de-percolated one, which is more in
accordance with the commonly accepted concept in
this field.
Aggregate grains have been demonstrated coming
close enough to cause neighboring ITZs to overlap or
percolate (at least) partly. This is the location where
probability is highest for pore trees associated with
neighboring aggregate grains to connect through
their branches of dead-end pores. The result is that
the dense random packing of the aggregate leads to
connected pathways (trunks) over the full extension
of concrete specimens or elements. However, it is
demonstrated by application of the developed porosimetry methods that also bulk cement paste between
somewhat more remote aggregate grains contains
continuous pores through mutual connection of deadend pores branching off the trunks, although density
is lower that in the ITZ. As to this aspect, results
deviate from those obtained by serial sectioning and
3D reconstruction.
Moreover, Fig. 5 already demonstrates that introduction of ITZs leads to pore refinement. In Le and
Stroeven (2012) it is shown that this phenomenon
primarily takes place in the ITZ. For durability
estimation in the case that mechanical influences
(i.e., deformation and micro cracks developed due to
loading in the ITZ) can be neglected, the effect of
ITZ percolation can be expected not significant, which
is in agreement with Zheng et al. (2011b).
The relative potentialities of the two methods
should still be explored as to efficiency, reliability
and capabilities for various applications (not necessarily limited to porosimetry in hydrated systems).
At the moment we do so, but it is likely that both
methods will have their own merits under different
circumstances. Generally speaking, although DRaMuTS is more advanced for pore space exploration,
RaNoS is cheaper and simpler.
A simple effective medium approach demonstrates that overall porosity is as a consequence not
dominated by higher (continuous) porosity in the ITZ.
Moreover, the effect of somewhat higher average
ITZ porosity on global hydraulic properties can be
expected even further reduced because of the relative
fineness of ITZ’s pore system. Hence, transport of
harmful substances (including water) through concrete
can be expected not significantly influenced by the
degree of ITZ percolation in case mechanical influences by loading are neglected.
The new methods concern physical DEM producing an analogue representation of concrete. Not
discussed herein are stochastic DEM that have been
introduced in Stroeven et al. (2009), whereby among
other things reference was made to Dequiedt et al.
(2001), who claim such models to be even more
promising than the physical ones.
The complex and tortuous pores in the network
structure can easily be analyzed as to size distribution by application of star volume measurements.
This is a well-known experimental approach in life
sciences. Its usage is shown straight forward when
dealing with the 3D virtual pore structure. However,
it can equally be applied to 2D sections (physical
experiments) from which an unbiased estimate of
local pore size can be obtained. Such data (in 2D and
CONCLUSIONS
Novel approaches to porosimetry in virtual
concrete are presented that combine reliability and
economy. These methods rely on realistic analogue
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