Specific surface area of overlapping spheres in the presence of
obstructions
D. R. Jenkins
Citation: J. Chem. Phys. 138, 074702 (2013); doi: 10.1063/1.4790691
View online: http://dx.doi.org/10.1063/1.4790691
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THE JOURNAL OF CHEMICAL PHYSICS 138, 074702 (2013)
Specific surface area of overlapping spheres in the presence
of obstructions
D. R. Jenkinsa)
CSIRO Mathematics, Informatics and Statistics, Locked Bag 17, North Ryde, NSW 1670 Australia
(Received 3 December 2012; accepted 15 January 2013; published online 15 February 2013)
This study considers the random placement of uniform sized spheres, which may overlap, in the
presence of another set of randomly placed (hard) spheres, which do not overlap. The overlapping
spheres do not intersect the hard spheres. It is shown that the specific surface area of the collection
of overlapping spheres is affected by the hard spheres, such that there is a minimum in the specific surface area as a function of the relative size of the two sets of spheres. The occurrence of
the minimum is explained in terms of the break-up of pore connectivity. The configuration can be
considered to be a simple model of the structure of a porous composite material. In particular, the
overlapping particles represent voids while the hard particles represent fillers. Example materials are
pervious concrete, metallurgical coke, ice cream, and polymer composites. We also show how the
material properties of such composites are affected by the void structure. © 2013 American Institute
of Physics. [http://dx.doi.org/10.1063/1.4790691]
I. INTRODUCTION
Examples of composites can be found in nature as well
as manufactured materials. Here, a composite means a material whose internal structure consists of a set of different
components embedded in a homogeneous matrix. A porous
composite has, as one of its components, void space. The
simplest porous composite is one consisting of two phases,
the matrix and the void; examples include various types of
foams. An obstructed porous composite consists of the matrix, voids, and a third phase, most frequently being some sort
of solid particles. The term “obstructed” is used to indicate
that the third phase components preclude the location of voids
within them. Materials that can be considered as obstructed
porous composites are metallurgical coke1 (whose 3 components are fused coal components, inert components, voids),
ice cream2 (sugar syrup, ice crystals, and voids), various igneous rocks3 (amorphous material, crystals, voids), and pervious concrete4 (cement paste, aggregate, voids). In all of these
examples, a primary source of the voids is the development of
bubbles during the formation of the materials. Typically the
bubbles form/nucleate and grow during a stage when the matrix material is in some sort of liquid state. In the case when
there are many growing bubbles, there is a propensity for them
to coalesce. At some stage in the process, the matrix will become solid, effectively freezing in the bubble/void configuration. As a result, the configuration of the voids can be quite
complicated.
The configuration of the microstructure of a composite
material has a significant effect on its material properties, such
as elastic moduli, thermal and electrical conductivities, permeability, diffusivity, and reactivity. Of particular interest is
the configuration of the voids in a porous composite, which
can affect all of the above properties.5 Two key properties of
the voids are the porosity, φ, being their volume fraction, and
a) David.Jenkins@csiro.au.
0021-9606/2013/138(7)/074702/5/$30.00
the specific surface area, s, being their surface area per unit
volume of material. In this paper we are concerned with examining the effect of the obstructing particles upon the specific surface area of the voids, for a given porosity. In order to
do this in the simplest way, we consider the obstructing particles to be randomly located, non-overlapping (hard) spheres,
each having the same radius, Ro , and the voids to be comprised of a set of randomly located, overlapping (fully penetrable) spheres, each having radius, R. Moreover, we allow no
overlap between the obstructing particles and the voids.
There are several studies of the properties of the porosity
of a simple porous composite material6, 7 and in particular of
the variation of the specific surface area of voids comprising
overlapping spheres with the sphere size and porosity.6 However, there appears to be no particular study that considers the
effect of obstructing particles on the properties of the voids.
The specific surface area of overlapping spheres has interest from a geometrical point of view, in that it is a rough indicator of the amount of overlap of spheres, as well as from a
materials science perspective, as it is important in applications
dominated by chemical reactions at surfaces, or in charge storage devices (e.g., supercapacitors).
This paper describes an algorithm for formation of simulated microstructure of obstructed porous composites, then
presents calculated results of the variation of specific surface
energy with the relative size of obstructing and void spheres.
A simple analysis provides an explanation of the mechanism
that leads to the minimum in the specific surface energy, and
show how material properties of the material are affected by
the presence of obstructing spheres.
II. MICROSTRUCTURE FORMATION
The creation of the microstructure used for this work
is based on consideration of a cube, consisting of N × N
× N volume elements (voxels), where typically we have used
138, 074702-1
© 2013 American Institute of Physics
074702-2
D. R. Jenkins
J. Chem. Phys. 138, 074702 (2013)
N = 320, a total of 32 768 000 voxels. The unit cell is considered to be periodic on all faces. Typically, we wish to construct a microstructure consisting of φ o volume fraction of
hard particles and φ volume fraction of voids. Particles are
laid down in two stages:
1. all voxels within the cube are initially considered to be
unoccupied.
2. a voxel is chosen randomly within the cube to act as the
centre of a hard particle. If the voxel is already occupied
by a hard particle, then another voxel is randomly chosen. The process is repeated until an unoccupied voxel
is located. If each voxel within a radius Ro of the centre
is unoccupied, then the hard particle is considered to be
centred at this point, and all the voxels within the radius
Ro are marked as occupied. This process is repeated until the volume fraction reaches φ o . Typically, up to about
35% volume fraction of hard particles can be located in
this simple fashion.
3. a voxel is chosen randomly within the cube to act as the
centre of a void sphere. If the voxel is already occupied
by a hard particle, then another voxel is randomly chosen. The process is repeated until an unoccupied voxel
is located. If each voxel within a radius R of the centre
is unoccupied by a hard particle, then the void sphere is
considered to be centred at this point, and all the voxels
within the radius R are marked as occupied by void. This
process is repeated until the volume fraction of voids
reaches φ. Because we pay no attention to the presence
of other voids, this allows for fully overlapping spheres.
Once this process is complete, each voxel is marked as
being either (a) hard particle, (b) void, or (c) unoccupied,
taken to be the matrix. For each voxel marked as a void, we
consider whether it has a neighbour (north, south, east, west,
up, down) which is not void, in which case it is considered
to be a surface voxel. The specific surface area is then the
total number of surface voxels divided by the total number
of voxels in the cell (N3 ). Calculated in this way, the specific
surface area is only approximate, although the accuracy of the
approximation will increase with N.
2.5
20%
30%
40%
50%
60%
2
1.5
s'R
1
0.5
0
0
0.5
1
1.5
2
2.5
3
3.5
R o /R
FIG. 1. Calculated specific surface area versus relative size of obstructing
particles and voids, for different values of the porosity. Each point on the
graph is the average of 50 simulations. In all of the cases shown here, the
obstructing spheres occupy 20% of the volume.
Similarly, Figure 2 shows the calculated specific surface
area for different amount of obstructing spheres. It demonstrates that adding obstructing spheres increases the value of
Ro /R where s R is a minimum.
Figure 3 shows images of the simulated configurations
having different values of Ro , when R, φ, and φ o are held constant. The most apparent effect of increasing the size of the
obstructing spheres is to yield a more “fractal” like surface
for the voids. This is relatively straightforward to understand:
as the particle size increases while holding the total volume
2
1.6
III. RESULTS
Figure 1 shows the calculated dimensionless specific surface area s R where s is the surface area per unit volume of
voids, for the case when 20% of the volume is occupied by
obstructing spheres. The quantity s is related to s by
s
s = .
φ
1.2
s'R
0.8
10%
0.4
It is plotted in this way to avoid overlap of the individual
curves. Note that, for each value of the porosity, there is a minimum in the specific surface area, and that the value of Ro /R
where the curve has a minimum increases with the porosity.
Note also that this minimum occurs when
Ro /R = O(1),
that is, when there is some similarity between the sizes of the
two sets of particles.
20%
30%
0
0
0.5
1
1.5
2
2.5
3
3.5
R o /R
FIG. 2. Calculated specific surface area versus relative size of obstructing
particles and voids, for different volume fractions of obstructing spheres
(shown in the legend). Each point on the graph is the average of 50 simulations. In all of the cases shown here, the voids occupy 40% of the volume.
074702-3
D. R. Jenkins
J. Chem. Phys. 138, 074702 (2013)
7
R o /R = 3.33
6
FIG. 3. Simulated configurations for different values of Ro /R being (a) 1, (b)
2, (c) 3. The black components are the obstructing spheres, the dark grey
components are the overlapping spheres, and the light grey is the matrix
phase.
relative volume
5
4
3
2
1
0
1
11
21
31
41
51
61
71
81
91 101 111 121 131 141 151 161 171 181 191
void number
24
R o /R = 0.769
22
20
18
16
relative volume
of particles constant, the average spacing between particles
increases. As a result, for sufficiently large particles, relative
to the void spheres, the particles have minimal effect upon the
location of the voids. However, for small particles, the spacing
between particles is smaller, and thus much more overlap of
voids is required in order to achieve the required porosity. As
a result, the surface of the voids is smoothed and the specific
surface area decreased. In essence, this explains the change in
specific surface area at the right hand end of Figures 1 and 2.
In order to better understand the reason for the existence
of the minimum in the curves, however, we considered the
number of isolated (i.e., non-connected) voids in the simulations. Because, for most cases considered, the void fraction exceeds the percolation threshold for uniform overlapping spheres of about 29%,8 almost all of the void spheres are
connected. In general, the void space consists of a single connected void that comprises virtually all of the porosity, along
with a small number of isolated voids. In the situation where
Ro /R is large, these isolated voids are mostly single spheres,
but when Ro /R is small, there is a larger percentage of isolated
voids comprised of multiple overlapping spheres, as shown in
the graphs of Figure 4.
Moreover, if we consider the number of isolated voids
(i.e., the number of voids excluding the largest, highly connected void) per unit volume, nB , then we see that there is
a steep increase around Ro /R ≈ 1, as shown in the graph of
Figure 5. Consequently, the large connected void comprises a
lower percentage of the total porosity.
The reason can again be related to the spacing between
obstructing particles. When Ro /R > 1, there is sufficient space
between the obstructing particles for overlapping voids to create a single connected void that occupies most of the voidage.
However, at low values of Ro /R, the spacing between obstructing particles is too small for the largest void to occupy as
much of the total porosity, so more isolated voids, comprising
multiple void spheres, are created. An increase in the number
of isolated voids will generally increase the overall specific
surface area, thus accounting for the increase in s R at the left
hand end of Figures 1 and 2.
In order for the obstructing particles to cause the breakup
of the major connected void, there needs to be regions of the
volume in which the obstructing particles form an effective
“cage” for the void spheres. This is more likely as the size of
the obstructing particles decreases, for a given value of φ o ,
since the average spacing between the particles decreases. If,
instead of locating the obstructing spheres randomly, they are
14
12
10
8
6
4
2
0
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20 21 22 23
void number
FIG. 4. Graphs showing the volume of isolated voids, relative to the volume
of a single void sphere, for two different values of Ro /R. In each case there
is 20% volume fraction of obstructing spheres and 30% volume fraction of
voids.
located on the vertices of a cubic lattice, then it is straightforward to show that no void spheres can overlap if
1
Ro
<
,
R
α−1
where
2π
.
α3 = √
3 2φo
In the examples shown in Figure 5, where φ o = 0.2, this yields
Ro
< 1.054,
R
which is shown by the dotted line in the graph. Even
though the obstructing particles are randomly placed in these
074702-4
D. R. Jenkins
J. Chem. Phys. 138, 074702 (2013)
0.3
35
2
20%
30%
30
40%
50%
0.25
1.7
relative permeability
25
nB
20
15
s'R
0.2
1.4
0.15
1.1
10
5
0.1
0.8
0
0.5
1
1.5
2
2.5
R o /R
0
0.5
1
1.5
2
2.5
3
3.5
R o/R
FIG. 5. Graph showing the number of isolated voids per unit volume, as
a function of Ro /R, for different values of φ o , as shown in the legend. The
vertical dotted line marks the point Ro /R = 1.054, indicating the point at
which obstructing spheres on a cubic lattice would preclude overlap of void
spheres.
examples, there is a clear change in the number of isolated
voids in the vicinity of this point.
IV. EFFECT ON MATERIAL PROPERTIES
We have shown that the presence of obstructing particles
causes a significant change in the nature of the packing of
overlapping spheres. In particular, as the size of the obstructing particles reduces (for a given volume fraction), they initially lead to more overlap, as evidenced by the reduction in
the specific surface area of the voids, but then there is a change
in the structure as the particle size is further reduced, as the
single connected void breaks up due to the obstacles. While
this is of interest itself in terms of the geometrical structures
formed, it is most relevant to materials science applications,
since it is likely that the nature of the structure thus formed
will affect the properties of this type of composite material.
As examples of this, we have calculated the effective permeability and Young’s modulus of the structures simulated here,
using a commercial finite element tool (MesoProp9 ). In the
implementation used by us, the size of the unit cell was limited to 80 × 80 × 80 voxels. Calculated permeabilities from
the simulations are shown in Figure 6. Note that, in these simulations, only the voids are assumed to be permeable.
The figure shows that there is a pronounced drop in permeability, starting at the point at which there is a minimum
in the specific surface area, as Ro /R is decreased. We conjecture that this is related to the reduction in the amount of voids
available for percolation, due to the breakup of the main void.
Similarly, we used MesoProp to determine the effective
elastic properties of the composite material. Figure 7 shows
the calculated (relative) Young’s modulus as a function of
FIG. 6. Calculated effective permeability (left hand axis) and specific surface
area (right hand axis) versus Ro /R, when φ o = 0.2. Each point on the graph
represents an average of 25 simulations.
Ro /R. Note that, in these calculations, both the obstructing
particles and the matrix material are assumed to have the same
elastic properties, while the voids are assumed to have the
properties of air. Accordingly, the variation of Young’s modulus is due entirely to the configuration of the voids in this case.
The figure shows that there is a significant variation in the
Young’s modulus of the composite as Ro /R is varied. Moreover, there appears to be a change in the rate of variation of
Y at around the point where s R is a minimum, showing that
it is important to consider the size of the obstructing particles
when determining material properties of composite materials.
The relevance of these results to materials science applications needs to be considered. First, we note that the
Young's modulus
0
0.8
2
0.6
1.7
s'R
0.4
1.4
0.2
1.1
0
0.8
0
0.5
1
1.5
2
2.5
3
R o/R
FIG. 7. Calculated effective Young’s modulus (left hand axis) and specific
surface area (right hand axis) versus Ro /R, when φ o = 0.2. Each point on the
graph represents an average of 25 simulations.
074702-5
D. R. Jenkins
minimum in specific surface energy occurs over a relatively
wide range of porosity, φ, and across a range of volume fractions of the obstructing particles. Next, the minimum in specific surface area occurs when Ro and R are of similar size. If
R Ro , then the obstructing particles are not likely to affect
the formation of the porosity. As an example, the formation
of a vesiculated volcanic rock.10 Once the bubbles/vesicles
become considerably larger than the crystals within the melt,
then the crystals are likely to move within the melt and not
contribute to obstructing the final porosity in the way considered here. Conversely, if Ro R then there is minimal obstruction of the voids in any case. Finally, the accuracy of the
method is reduced at large and small values of Ro /R, due to
the effect of voxel size relative to the size of the different elements of the microstructure.
A key issue not considered here is the mechanism of
formation of porous composites. In particular, the growth,
collapse and coalescence of bubbles within a composite are
likely to be impacted by the structures within the material.
For example, the presence of small inert particles in the melt
comprising the bridges between expanding bubbles may increase the likelihood of the bridges breaking, thus causing
the bubbles to coalesce.10 Thus it may be possible to use the
approach described here to find an optimal configuration for
a microstructure for a specific material requirement, but the
challenge of creating such a microstructure remains.
V. CONCLUSION
The results shown in this paper have an intrinsic interest
in terms of the properties of the formation of random structures, as well as relevance to the properties and behaviour of
real composite materials. While the work presented here has
J. Chem. Phys. 138, 074702 (2013)
considered spheres of a uniform size for both the overlapping
and hard components, it is natural to examine whether similar
behaviour occurs in real materials, where the voids are likely
to have a distribution of sizes and hard particles are rarely
spheres. In fact, in at least one example, that of simulations
of metallurgical coke microstructure,1 which motivated this
work, it appears that a minimum in specific surface area exists in the same way as considered here. In that study, the hard
particles were considered to be cylinders whose length and diameter were both chosen from a random distribution, and the
voids were considered to be overlapping spheres whose radius was chosen from a size distribution. This result indicates
that the behaviour presented here is robust in terms of the size
and shape of components of composites and hence likely to
be relevant to a range of porous composite materials.
ACKNOWLEDGMENTS
The author acknowledges valuable discussion with
Dr. M. J. Buckley, CSIRO Mathematics, Informatics and
Statistics.
1 D.
R. Jenkins, M. R. Mahoney, and A. D. Miller, Proceedings of the 6th
International Congress on the Science and Technology of Ironmaking, Rio
de Janeiro, Brazil, October 2012.
2 C. Clarke, Phys. Educ. 38(3), 248–253 (2003).
3 D. Zandomeneghi, M. Voltolini, L. Mancini, F. Brun, D. Dreossi, and M.
Polacci, Geosphere 6(6), 793–804 (2010).
4 O. Deo and N. Neithalath, Mater. Sci. Eng., A 528, 402–412 (2010).
5 S. Torquato, Int. J. Solids Struct. 37, 411–422 (2000).
6 P. A. Rikvold and G. Stell, J. Chem. Phys. 82, 1014 (1985).
7 S. B. Lee and S. Torquato, J. Chem. Phys. 89, 3258–3263 (1988).
8 C. D. Lorenz and R. M. Ziff, J. Chem. Phys. 114, 3659 (2001).
9 A. A. Gusev and H. R. Lusti, Adv. Mater. 13(21), 1641–1643 (2001).
10 A. A. Proussevitch, D. L. Sahagian, and V. A. Kutolin, J. Volcanol.
Geotherm. Res. 59, 161–178 (1993).