Modeling the influence of carbon spheres on the porosity of SOFC anode materials

Article abstract of DOI:10.1111/j.1551-2916.2011.05037.x

The influence of fabrication pressure and other ceramic processing variables, including volumetric loading fraction and the particle size ratio of pore-forming agents, on the porosity of fabricated ceramic anodes, was investigated using an integrated experimental approach with mathematical modeling to differentiate the impacts of each parameter. Despite historic observation of the properties of ceramic bodies, to date, there is a lack of available models to accurately interpret the ceramic properties as a function of the processing variables. Herein, we focus on the open porosity of the solid oxide fuel cell anode prepared from NiO/YSZ (nickel oxide/yttria stabilized zirconia) as the ceramic powder and using CMS (carbon microspheres) as a pore-forming agent. A range of pore-former volumetric blend ratios (4.4%-44.6%) and different particle size ratios between NiO/YSZ and CMS (11.27, 4.29, and 0.26) were used, whereas the influence of the applied uni-axial fabrication pressure on the open porosity of sintered anode was studied in a range of atmospheric pressure up to 40 MPa. A good agreement was observed between the proposed model and the experimental data, implying that the approach could be used to determine the significant processing parameters to fabricate ceramics with desired porosity. The model could also be used to interpret the physical basis of pore formation when using a pore-forming agent in a fabricated porous ceramic body.

Full text of DOI:10.1111/j.1551-2916.2011.05037.x

J. Am. Ceram. Soc., 1–8 (2012)
DOI: 10.1111/j.1551-2916.2011.05037.x
© 2012 The American Ceramic Society
ournal
J
Modeling the Influence of Carbon Spheres on the Porosity
of SOFC Anode Materials
Bahman Amini Horri, Cordelia Selomulya, and Huanting Wang†
Department of Chemical Engineering, Monash University, Clayton, Victoria, 3800, Australia
The influence of fabrication pressure and other ceramic pro-
cessing variables, including volumetric loading fraction and the
particle size ratio of pore-forming agents, on the porosity of
fabricated ceramic anodes, was investigated using an integrated
experimental approach with mathematical modeling to differen-
tiate the impacts of each parameter. Despite historic observa-
tion of the properties of ceramic bodies, to date, there is a lack
of available models to accurately interpret the ceramic proper-
ties as a function of the processing variables. Herein, we focus
on the open porosity of the solid oxide fuel cell anode prepared
from NiO/YSZ (nickel oxide/yttria stabilized zirconia) as the
ceramic powder and using CMS (carbon microspheres) as a
pore-forming agent. A range of pore-former volumetric blend
ratios (4.4%–44.6%) and different particle size ratios between
NiO/YSZ and CMS (11.27, 4.29, and 0.26) were used,
whereas the influence of the applied uni-axial fabrication pres-
sure on the open porosity of sintered anode was studied in a
range of atmospheric pressure up to 40 MPa. A good agree-
ment was observed between the proposed model and the experi-
mental data, implying that the approach could be used to
determine the significant processing parameters to fabricate
ceramics with desired porosity. The model could also be used
to interpret the physical basis of pore formation when using a
pore-forming agent in a fabricated porous ceramic body.
Tortuosity is another microstructural parameter in con-
trolling the electrochemical performance of the electrode in
SOFCs.11 The tortuosity affects the rate of diffusional trans-
port and thus concentration polarization in the porous cera-
mic electrodes.12,13 In an electrode-supported SOFC, the
electrode layer should be thick enough to mechanically sup-
port the cell. With a thicker electrode, the concentration
polarization is increased in the cell.14 For this reason, the
tortuosity of a porous ceramic should be statistically con-
trolled by tuning calcination and sintering conditions or by
changing the particle size of starting materials.15
Many ceramic bodies formed from powders may intrinsi-
cally contain some degrees of porosity due to incomplete
densification depending on both the fabrication process and
the selected parameters.5,16 In the fabrication of the ceramic
membranes, catalyst supports, or electrodes of the solid oxide
fuel cells, a conventional way to increase the porosity of
ceramic substrates is by adding a reasonable amount of a
pore-forming agent (PFA) e.g., synthetic polymers, natural
biopolymers, or carbon-based materials that could be burnt
out during the sintering steps.17,18 However, the ability to
fabricate the ceramic substrates with desired porosity and
controlled microstructural properties is still a challenge.19
Compaction of powders is a shape-forming technique that
has been widely used in ceramics and in other fields such as
metallurgy, pharmaceuticals, and civil engineering.20 Mathe-
matical equations describing compaction are often used to
adjust the required density of fabricated ceramic bodies in
both isostatic and uniaxial die pressing modes. Most of these
equations have been developed empirically to fit the experi-
mental data due to the limited theoretic basis.21 A summary
of more frequently used compaction equations is shown in
Table I. The first compaction model [Eq. (1)] proposed by
Walker correlated the relative volume of compacted powder
against the logarithm of applied axial pressure.22 Balshin
applied the concepts of fluid mechanics to justify Walker’s
equation by theoretic parameters, such as the pressing modu-
lus and the coefficient of deformability.23 Heckel24 and
Kawakita and Ludde25 models [Eqs. (3) and (4)] have been
frequently referred to in the pharmaceutical and metallurgy
fields due to their good compatibility with the elastic com-
pression phase, more practical relevance, and possibility to
measure the yield pressure. Bruch [Eq. (5)] was the first who
applied compaction curves to interpret ceramic (alumina
powder) consolidation behavior.26 A summary of early com-
paction equations up to 1983 has been listed by MacLeod.27
More recent overviews of compaction equations have been
discussed by Celik28 and Sonnergaard.29
I. Introduction
ONVENTIONALLY, the term “porosity” as a microstruc-
tural feature has been used to describe the degree of
C
porous nature and to quantitate the void volume of ceramic
bodies. Although a variety of factors such as particle size,
shape, nature (polycrystalline or composite), surface rough-
ness, chemical properties of crystals, fabrication processing
requirements, sintering parameters, morphology, and green
density impact the microstructure of fabricated-ceramic
bodies, porosity is the dominant factor affecting the proper-
ties of ceramic bodies.1 In general, the presence of porosity
degrades mechanical strength, elastic modulus, and ionic-,
electronic-, and thermal-conductivity of ceramic bodies.2–4 In
particular, the performance of electrodes in solid oxide fuel
cells depends on the availability of surface area for reaction
and the ability of reactants to reach the reaction sites.5,6 To
increase the performance of the ceramic substrate used as the
electrode in solid oxide fuel cells (SOFCs), an optimal poros-
ity level is required to provide sufficient gas permeability,
suitable electrical conductivity, enough mechanical strength,
high electrochemical catalytic activity, low Ohmic polariza-
tion (dielectric break-down voltage), and good compatibility
with other fuel-cell components.7–10
Comparably, the influence of particle size and particle size
distribution on the compaction properties has been given less
consideration in literature. Dalla Valle30 was the first who
introduced the concept of particle pressure diffusion, mean-
ing that small particles tend to diffuse into the void spaces
between large particles as a result of increase in applied pres-
sure. Cooper and Eaton31 used the same theory to propose
an empirical model [Eq. (6)] to describe the compaction
behavior of four different ceramic powders. The influence of
J. Stevenson—contributing editor
Manuscript No. 30243. Received August 29, 2011; approved November 28, 2011.
†Author to whom correspondence should be addressed. e-mail: huanting.wang@
monash.edu
1

2
Journal of the American Ceramic Society—Horri et al.
Table I. Summary of Different Compaction Models and Their Applications in Original Form
Model
Correlation†
Application Remarks
Ref.
Walker (1923)
Fine powders (Calcium
Carbonate)
22
Vr ¼ a1 ꢀ b1:LogðRÞ
Balshin (1938)
Heckel (1961)
Metallic powders
23
24
LogðpÞ ¼ ꢀL:Vr þ c
ꢀ
ꢁ
1
Pharmaceutical and
metallurgy powders
Ln
¼ a2P þ b2
1 ꢀ qr
ꢀ
ꢁ
V0
1
P
Kawakita & Ludde (1971)
Pharmaceutical
powders
25
P
¼
þ
V0 ꢀ V
a3:b3 a3
Bruch (1967)
Ceramic powders
Ceramic powders
26
31
qr ¼ a4 ꢁ LnðPÞ þ b4
ꢀ
ꢁ
ꢀ
ꢁ
Cooper and Eaton (1962)
V0 ꢀ V
K1
P
K2
P
¼ a5Exp ꢀ
V0 ꢀ Vs
þ b5Exp ꢀ
ꢀ
ꢁ
ꢂ
ꢂ
ꢃꢃ
e0
e
a6
1 þ b6
Yu & Gu (1992)
Fine powder
(river sand)
37
Ln Ln
¼ Ln
þ ð1 þ b6Þ:LnðPÞ
†Symbols: a1, a2, a3, a4, a5, a6, b1, b2, b3, b4, b5, b6, k1, k2: Model coefficient (determined experimentally via curve fitting); c, The coefficient of deformability
(determined experimentally); L, The pressing modulus analogous to Young’s modulus (determined experimentally); P, Applied pressure in compression (MPa); R,
Resistance to compression (kg/cm2); V0, Volume of powder at zero (relative) pressure (m3); Vr, Relative volume of the powder (the ratio of apparent volume
to the skeletal volume, dimensionless); e, e0, Porosity and initial porosity at ambient pressure, respectively (dimensionless); ρr, Relative density of the powder
(the ratio of apparent density to the skeletal density, dimensionless).
particle size on the pressure modulus was modeled in other
works.32–34 The effect of particle size distribution by consid-
ering the packing efficiency as a function of forming pressure
was modeled by Zheng et al.35 The influence of particle size
range on the deformability of ceramic powder was modeled
in another paper, showing that an increase in the particle size
caused a decrease in the brittle fracture stress.36 Yu and
Gu37 investigated the influence of particle size distribution on
the compaction of cohesion-less powders and modeled the
densification process using a porosity-pressure correlation
[Eq. (8)]. The most recent work dealing with the compaction
modeling has used the discrete element method to describe
the behavior of aggregated ceramic powders.38 Despite vari-
ous studies concerning porosity and its effects on the proper-
ties of ceramic bodies, what is currently lacking is a reliable
mathematical model to accurately predict porosity as a func-
tion of the ceramic processing variables.39–43
II. Experimental Procedure
(1) Preparation and Characterization of Ceramic Powders
and Pore-Forming Agent
To prepare the ceramic powders for this study, nickel carbon-
ate-basic hydrate, BNC (NiCO3·2Ni(OH)2·xH2O, MW
= 304.12 g/mol; Sigma-Aldrich, Sydney, Australia) and yttria
stabilized zirconium (IV) oxide, YSZ (ZrO2,Y2O3, 99.9%,
MW = 123.22 g/mol; Sigma-Aldrich) were used as the source
of ceramic materials. About 300 g of BNC was thermally
decomposed at 500°C for 4 h. NiO/YSZ ceramic composite
was prepared by dispersing the prepared NiO and YSZ powders
into ethanol at the mass ratio of 60:40. The resulting suspension
was then ball-milled with zirconia balls (5 mm) for 24 h at
80 rpm. Subsequently, the suspension was dried at around 80°C
for 24 h. The prepared NiO/YSZ ceramic powder was charac-
terized using a X-ray diffraction method (Phillips 1140 diffrac-
tometer, Philips Co., Eindhoven, The Netherlands).
This work focused on developing a mathematical model
of the porosity of porous ceramic substrates, by quantitating
the influence of pore-forming agent and ceramic particle
properties such as blend ratio, size ratio, and the uni-axial
pressure. A range of pore-former blend ratio, particle size
ratio, and applied uniaxial pressure were selected as the pro-
cessing parameters. The model was developed using NiO/
YSZ (nickel oxide/yttria stabilized zirconia) as an example of
the ceramic powders widely used in the fabrication of solid
oxide fuel cells. The experimental results were obtained using
spherical carbon particles with three different diameters as
the pore-forming agent to minimize the effects of pore-
former shape factor on the final porosity. The proposed
equations can be specially used to assist in the fabrication of
porous ceramic substrates with a desired open porosity, by
adjusting these processing parameters. Also, this compaction
model can be generally used to understand the dominant
process parameters and the physical basis of pore formation
inside a porous ceramic body when a pore-forming agent is
used.
In this study, carbon micro-spheres (CMS) were used as
the pore-forming agent. They were prepared using hydrother-
mal treatment of sugar according to the method described
elsewhere.44 Here, processing parameters such as the hydro-
thermal treatment time and sugar concentration were used as
the main controlling factors affecting the spherules diameters.
Table sugar (Merryfield white sugar, 100% Australian pure)
as the carbon precursor was dissolved into distilled water as
the solvent. Three different CMSs (referred to as CMS1, 2,
and 3) with different diameters were prepared using sugar
solutions with different concentrations of 2.055, 1.233, and
0.616 mol/L. The sugar solutions were filled in separate
Teflon autoclaves and placed in an oven at the same temper-
ature of 190°C for 6.5, 2, and 2.5 h, respectively.
The morphology of the prepared CMSs and NiO/YSZ were
examined using scanning electron microscopy (JEOL
JSM-840A, JEOL Ltd., Tokyo, Japan). The particles size dis-
tributions of the prepared powders were measured using laser
scattering method (Mastersizer 2000; Malvern Instruments

Porosity of SOFC Anode Materials
3
(a)
(b)
Fig. 1. XRD pattern of prepared NiO/YSZ.
Ltd., Worcestershire, U.K.). The particle density was deter-
mined using a helium pycnometric technique (AccuPyc-1330
pycnometer; Micromeritics Instrument Corp., Norcross, GA).
(c)
(d)
(2) Sample Preparation and Characterization
To study the effects of volumetric fraction of the pore-
forming agent in the ceramic powder, five different samples
containing 1, 2.5, 5, 10, and 15 wt% of the prepared CMSs
with three different average diameter ranges (CMS1:
11.54 lm, CMS2: 4.39 lm, and CMS3: 0.27 lm) were mixed
with the prepared NiO/YSZ (a total of 15 samples). Wet
ball-milling method in ethanol medium (with zirconia balls
of 3 mm and running time of 5 min) was used to mix the
resulting suspension. To ensure enough mechanical strength
of the system for compression, polyvinyl alcohol, 20 wt% of
PVA (-CH2CH(OH)-, MW = 31000–50000 g/mol; Sigma-Aldrich)
in deionized water was used as the binder. The PVA solution
was added to the prepared ceramic suspensions in a ratio of
0.5 wt% of the total ceramic powders. The final suspensions
were finally dried in an oven at around 60°C for 48 h.
Fig. 2. SEM image of prepared ceramic powder and pore-former.
(a) NiO/YSZ, (b) CMS1, (c) CMS2, and (d) CMS3.
The prepared ceramic powders were compressed under a
constant pressure of 5 MPa to make similar disks with diame-
ters of 10 mm, using a desktop powder pressing machine
(model: FY-24-A; MTI Corporation). To study the effect of
applied uniaxial compression pressure on the porosity, seven
additional samples were prepared without the pore-former
under different pressures of 1, 2.5, 5, 10, 20, 30, and 40 MPa for
comparison. The porosity of powder was measured separately.
All prepared samples were sintered in air at 1400°C for
5 h and then reduced at 800°C for 3 h using a hydrogen
stream (at around 40 cm3/min). The CMSs in the prepared
samples were totally burnt out during the sintering step. The
porosity and pore-size distribution of the produced disks
were investigated using a mercury intrusion porosimeter
(AutoPore III; Micrometric Instrument Corporation).
Fig. 3. Particle size distribution for the prepared ceramic and pore-
former powders.
Table II. Particle Size Distribution Data
Particle size parameter† (lm)
NiO/YSZ
CMS1
CMS2
CMS3
D(0.1)
D(0.5)
D(0.9)
Surface weighted
mean diameter
Volume weighted
mean diameter
0.566
0.972
1.563
0.887
8.267
11.245
15.103
10.935
2.891
4.240
6.106
4.047
0.137
0.242
0.433
0.222
1.024
11.539
4.390
0.266
†The particle size parameters reported using Malvern instrument has less
than 1% error.
(3) Data Analysis Method
MS Excel was used to fit and plot the curves. To relate the
parameters of each model to the sample properties and data,
stepwise multiple linear and nonlinear regression analyses
were performed based on the Levenberg–Marqurdt (L–M)
algorithm. For each model, the goodness of fit was statisti-
cally determined using the coefficient of determination (R2)
and the Root of Mean Squared Errors (RMSE).
size distribution for the prepared powders. Each powder was
dispersed in distilled water before particle size measurement,
and the results were presented in term of equivalent volume-
averaged diameters. The prepared powders showed unimodal
distributions in the particle size range (Fig. 3), with detailed
data shown in Table II.
III Results and Discussion
(1) Characterization and Microstructure of the Prepared
Ceramic and Pore-Former Powders
Figure 1 shows the XRD pattern of the prepared NiO/YSZ
powder. The SEM images of the prepared NiO/YSZ and also
CMSs are shown in Fig. 2(a)–(d). Figure 3 shows the particle
(2) Porosity Measurement Results
After sintering and reduction steps, all prepared samples
were subjected to the volumetric porosity and the pore-size
distribution measurements. The experimental values of sam-
ples porosities due to the applied uniaxial pressure (without

4
Journal of the American Ceramic Society—Horri et al.
structure, porosity, and pore-size distribution of ceramic sub-
strates. To study the effects of the pore-former, the following
process parameters were investigated in this work:
Table III. Volumetric Porosity of NiO/YSZ
Ceramic Samples (Without Pore-Former) as a Function
of Applied Uniaxial Pressure
Volumetric porosity†, e (%)
1. The applied uniaxial pressure (P) as used in the powder
compaction shape-forming technique.
Applied uniaxial pressure (MPa)
0
1
59.25
31.89
26.87
23.52
22.62
22.12
22.02
21.81
2. The blend ratio (ξ) which is the volumetric fraction of
the pore-forming agent in the total ceramic material and
is expressed as:
2.5
5
10
20
30
40
Vp
Xp
n ¼
¼
(8)
qp
qc
Vp þ Vc
Xp þ
where Vp, and Vc are the skeletal volumes of pore-former
and ceramic powder, respectively; Xp is the weight ratio of
†Porosity measurement average error: 1%.
mp
pore-former in ceramic powder Xp ¼ , and ρp, and ρc are
the skeletal densities of pore-former mac nd ceramic powder,
respectively. In this study, the measured values of ρp and ρc
are 1.373 and 6.253 g/cm3, respectively.
Table IV. Volumetric Porosity of NiO/YSZ
Ceramic Samples (with Pore-Former) as a Function of
Particle Size Ratio and the Blend Ratio of Pore-Former at
a Pressure of 5 MPa
3. The size ratio (k) which is the ratio of the volumetric
average particle size of pore-forming agent and ceramic
particles.
Volumetric percentage porosity† at
Pore-former loading fraction
different mean particle size ratio
(A) Effect of Fabrication Pressure:
A
predictive
wt%
vol% (Blend ratio, ξ)
k = 11.268
k = 4.287
k = 0.260
porosity model must satisfy two important boundary condi-
tions. First, the predicted porosity must approach the pow-
der porosity as the applied pressure decreases to atmospheric
pressure. Secondly, the porosity predicted by a model must
approach zero as the applied pressure approaches infin-
ity.25,37,49 These conditions can be expressed as:
0
1
2.5
5
10
15
0
23.52
33.01
37.49
42.41
51.05
61.18
23.52
31.87
35.18
39.53
47.24
55.46
23.52
29.02
31.07
33.85
39.17
45.4
4.40
10.46
19.33
33.60
44.56
lim e ¼ e0
(9)
†Porosity measurement average error: 1%.
P!Pa
lim e ¼ 0
(10)
P!1
where e and e0 are porosity and powder porosity at ambient
pressure, respectively. All powders and fabricated porous
materials contain both solid and void fractions. The dimen-
sionless relationship between porosity (void volume fraction)
and solid volume fraction (φ) is usually expressed as:
Vs
e ¼ 1 ꢀ / ¼ 1 ꢀ
(11)
V
where Vs and V are the skeletal (apparent) volume and
the bulk (envelop) volume of the porous solid (or powder),
respectively.50 For a compaction process, application of the
work done in a cylinder-piston system according to thermo-
dynamic definition of work can be written as:
dW ¼ v:dP ꢀ Pdv
(12)
In Eq. (12), W is the total work required for a piston dis-
placement, v is the specific volume of the material inside the
cylinder, and P is the applied uniaxial pressure. By dividing
the volumes in Eq. (11) by the total mass of solid, a relation-
ship for the bulk specific volume can be written as:
Fig. 4. Pore-size distribution measured by mercury porosimeter for
prepared samples at 5 MPa and 15 wt% of added CMS).
pore-former) and the effects of both particle size and loading
fraction of the pore-former (CMSs) are presented in
Tables III and IV, respectively. Porosity was shown to
decrease with lower particle size ratio or the loading fraction
of the added CMSs as the pore-former. Figure 4 shows the
pore-size distribution of the prepared samples with and with-
out pore-former. According to Fig. 4, the pore-size formed
in each sample is compatible with particle size of the applied
pore-former. For the sample made by CMS3, the pores
formed inside the fabricated ceramic body shows a bigger
size (around 1 lm) due to presence of bigger particles and
agglomeration of CMS particles in the ceramic substrate.
vs
v ¼
(13)
1 ꢀ e
where v and vs are the bulk and skeletal specific volumes,
respectively. By substituting Eq. (13) to Eq. (12) and inte-
grating the resulting equation for any piston displacement
during a reversible and adiabatic compaction process at a
constant pressure, the following equation can be expressed
as:
ꢄ
ꢅ
1
1
Wp ¼ P:vs:
ꢀ
(14)
1 ꢀ e0 1 ꢀ e
where WP is the work done by exerting the uniaxial pres-
sure for compaction of the solid particles. Equation (14) can
be rearranged as Eq. (15) for porosity, by introducing a cor-
relation for WP:
(3) Porosity Modeling and Analysis
As discussed in previous studies,7,45–48 the size, shape, and
quantity of the pore-formers significantly influence the micro-

Porosity of SOFC Anode Materials
5
the root mean square error (RMSE = 0.0115). Table V
shows the predictive capability of the proposed model [Eq.
(18)] for the rest of the experimental data from this work
using these predetermined values of A and B. It shows that
the proposed equation could predict the experimental values
of open porosities by an average error of 3.09%.
1
e ¼ 1 ꢀ
(15)
Wp
P:vs
1
ꢀ
1ꢀe0
Compaction process of solid particles in a cylinder-piston
system involves several densification sequences that caused
some difficulties in getting a direct theoretic expression for
WP. Depending on the powder types and properties, those
sequences can be summarized as elastic deformation, plastic
deformation, and particle fragmentation.28,51,52
To find a suitable correlation for the pressure work (WP),
Eq. (14) was used to calculate the values of WP according to
the experimental data reported in Table III (porosity versus
the applied uniaxial pressure). Curve fitting analysis was used
to correlate the equivalent pressure work in terms of both
porosity (e) and the powder porosity (e0), as a function of
To understand the porosity-pressure (uniaxial pressure)
behavior of the applied ceramic powder (NiO/YSZ) at higher
pressures, the proposed model [Eq. (18)] was applied using
the predetermined values of A and B with the results shown
in Fig. 6. Accordingly, three different regimes could be iden-
tified during the compaction process. At the first regime
starting from the atmospheric pressure, a small increase in
the applied uniaxial pressure caused a rapid quasi-linear
decrease in porosity until the critical porosity (e0) was
reached. Critical porosity corresponded to a small uniaxial
pressure referred to here as the critical pressure (Pc). Beyond
the critical pressure and by applying higher uniaxial pres-
sures, the second regime was observed with a nonlinear
dependency of porosity to pressure (uniaxial strength) until
reaching a specific pressure called the transitional pressure
(PT). The transitional pressure for the proposed model corre-
sponded to the inflection point of the curve (by logarithmic
scale), equaling the value of the parameter B (B = PT). Thus,
pressure. Figure 5 shows the plot of the experimental values
of the pressure work in terms of
Wp
P:vs
(as used in Eq. 15) ver-
sus pressure difference (P ꢀ Pa). The following equation was
found to satisfy the experimental data from this work:
ꢄ
ꢀ
ꢀ
ꢁꢁꢅ
Wp
P ꢀ Pa
P ꢀ Pa
¼ C A
þ ð1 ꢀ AÞ 1 ꢀ Exp ꢀ
P:vs
P
B
PꢀPa
(16)
the model used two dimensionless pressure groups (i.e.,
and
P
PꢀPa
) as independent parameters to predict the porosity
as a dBimensionless dependent parameter, confirming that the
model was supported by a valid dimensional analysis. In
addition, it was found that the value of parameter A approx-
imately corresponded to the estimated porosity at the transi-
where A, B, and C are the model coefficients. To express a
model for porosity, Eq. (16) was substituted into Eq. (15),
and consequently the resulting equation must satisfy the
boundary conditions for general porosity models, i.e., Eqs.
(7) and (10). The porosity equation satisfies the first bound-
ary condition, showing that as the pressure difference
approaches zero, the porosity approaches the powder poros-
ity (e0). The second boundary condition can be applied using
the following:
tional pressure divided by the skeletal specific volume of
eðPT
solid (A ꢂ
Þ). By increasing the applied uniaxial pressure
vs
beyond the transitional pressure, the third pressure-porosity
regime observed displayed another quasi-linear behavior. At
the third pressure-porosity regime, a small change in porosity
required a significant change in pressure, especially at the
lower porosities. The outcomes (Fig. 6) were in agreement
with the idealized uniaxial strength versus porosity configura-
tion.53,54
lim e ¼ lim 1
P!1
P!1
1
Â
À
À
ÁÁÃ
ꢀ
ꢀ C A PꢀP þ ð1 ꢀ AÞ 1 ꢀ Exp ꢀ
PꢀPa
1
a
1ꢀe0
P
B
¼ 0
Table V. Prediction Ability of the Proposed Model [Eq. (18)]
(17)
e
Applied
which implies that C takes the value of
. By substitut-
ing the corresponding expression for C, th1eꢀep0 roposed equa-
tion for porosity becomes:
uniaxial
Experimental
Predicted
pressure (MPa)
porosity, e (%)
porosity, e (%)
Error (%)
0
1
59.25
31.89
26.87
23.52
22.62
22.12
22.02
21.81
59.25
29.88
26.19
24.74
23.73
22.71
21.92
21.21
0.00
ꢀ6.28
ꢀ2.53
+5.17
+4.91
+2.66
ꢀ0.43
ꢀ2.74
1 ꢀ e0
Â
À
À
ÁÁÃ
e ¼ 1 ꢀ
(18)
PꢀPa
1 ꢀ e0 A PꢀP þ ð1 ꢀ AÞ 1 ꢀ Exp ꢀ
a
P
B
2.5
5
10
20
30
40
The parameters A and B can be determined from experi-
mental data using curve fitting analysis. Application of the
curve fitting analysis to the experimental data (Table III)
based on this work provided the values of A and B of 0.7858
and 254.39, respectively, with an acceptable value of the
coefficient of determination (R2 = 0.9931) and a low value of
Fig. 5. Plot of experimental data in terms of porosity versus the
applied pressure difference (P ꢀ Pa) according to Eq. (15).
Fig. 6. Predicted uniaxial strength versus porosity for NiO/YSZ at
higher pressures.

6
Journal of the American Ceramic Society—Horri et al.
(B) Effect of Pore-Former Blend Ratio and Size
Ratio: For the fabricated ceramic bodies, parameters such
as blend ratio and particle size ratio of the pore-forming
agent are the main factors affecting the final porosity. In gen-
eral, adding a given amount of pore-forming agent creates
additional porosity on top of the available intrinsic porosity
of ceramic bodies. The intrinsic porosity is the porosity of
the sintered ceramic material without the use of any pore-
former. So, the final porosity of the sintered ceramic body
can be expressed as:
e ꢀ ei
1
¼
À
Á
(26)
n2 þ n ꢀ km LnðnÞ
K
1 ꢀ ei
Equation (26) was used to predict the experimental data
of this work (Table IV), with the results shown in Fig. 7.
The model coefficients and the goodness of fit statistic are
reported in Table VII. The proposed model (Eq. 26) showed
an acceptable consistency with the experimental data
(R2 > 0.9975 and RMSE < 0.0075). The model also complied
with the physical boundary conditions for the blend ratio
and the size ratio, for example, as the porosity took the value
of intrinsic porosity (ei) when the size ratio or the blend ratio
approached zero, i.e., when no pore-forming agent was
added . For finite size ratios, the porosity took the value of 1
when the blend ratio approached 1, meaning that the solid
mixture was entirely composed of pore-former (with no cera-
mic solid). For the case of similar particle size for pore-
former and ceramic particles, parameter k should equal 1
and consequently, Eq. (26) would take the form of Eq. (24).
(C) Applications: We also applied Eqs. (18) and (26)
using appropriate data in literature covering different ranges
of fabrication pressure, pore-former fractions, and particle
size ratio. Tables VIII and IX summarized the goodness of
fit and the predicted coefficients for different ceramic and
nonceramic powders, demonstrating the versatility of the
proposed models to predict the porosity for different ceramic
and nonceramic powders. The reported values of R2 and
RMSE (averagely R2 > 0.99 and RMSE < 0.015) in
Tables VIII and IX demonstrates a relatively good predictive
capability for the proposed models (Eqs. (18) and (26)).
Determining the significance of the processing parameters
as investigated here would depend on the selected fabrication
method. For example, if a fabricated ceramic body is made
using the compression method (without pore-former), the
main parameter is the applied uni-axial pressure, and Eq.
(18) can be used to predict the final open porosity. On the
other hand, if a ceramic body is made by compression or
other methods using a pore-forming agent, at first, the intrinsic
e ¼ ei þ ec
(19)
where ei and ec are the intrinsic- and the created-porosity,
respectively. The blend ratio of pore-former is the primary
influencing parameter to change the final porosity. Physi-
cally, adding any pore-former to increase the porosity must
satisfy two boundary conditions. First, the final porosity
must equal the intrinsic porosity as the blend ratio
approaches zero, and secondly, the final porosity must equal
1 if the blend ratio approaches 1. These boundary conditions
can be expressed as:
lim e ¼ ei
(20)
(21)
n!0
lim e ¼ 1
n!1
Curve fitting analysis of the experimental data (Table IV)
showed a relationship between the created porosity (ec) and
the pore-former blend ratio (ξ) as follows:
C1
n2 þ ðn ꢀ C2ÞLnðnÞ
ec ¼
(22)
where C1, and C1 are the model coefficients. The coefficient
C1 can be estimated by substituting Eq. (22) into Eq. (19)
and then using the second boundary condition [Eq. (21)] as
follows:
C1
n2 þ ðn ꢀ C2ÞLnðnÞ
lim e ¼ ei þ
¼ 1
(23)
n!1
Equation 23 implies that the value of parameter C1 is
equal 1 ꢀ ξi, and consequently the final porosity can be
expressed as a function of the intrinsic porosity and the
blend ratio by the following equation:
e ꢀ ei
1 ꢀ ei
1
¼
(24)
n2 þ ðn ꢀ C2ÞLnðnÞ
Thus, Eq. (24) was used to fit the experimental data of
this work (Table IV, all sets of data), with the goodness of
fit summarized in Table VI. Equation (24) involved one
model coefficient (C2) that could be expressed as a function
of the particle size ratio (k). The size ratio as applied in the
experiments and the physical boundary conditions were used
to find the experimental correlation for coefficientC2:
K
km
C2 ¼
(25)
Fig. 7. Plots of porosity as a function of blend ratio and size ratio
of the pore-former, fitted with Eq. (26).
In Eq. (25), k, and m are the model coefficients. Substitut-
ing Eq. (25) to Eq. (24) results in the final porosity correla-
tion for the fabricated ceramic body as follows:
Table VII. Model Coefficients and Goodness of Fit Data for
the Proposed Model for Porosity [Eq. (26)]
Table VI. Model Coefficients and Goodness of Fit Data for
the Fitted Curve on Volumetric Data [Eq. (24)]
Particle size ratio
Model coefficients
Particle size ratio
Model coefficients
and Goodness of fit
k = 11.268
k = 4.287
k = 0.2598
and Goodness of fit
k = 11.268
k = 4.287
k = 0.2598
K
m
10.791
0.5707
0.9975
0.0075
8.6451
0.6946
0.9985
0.0049
3.5300
0.1961
0.9985
0.0033
C2
2.7083
0.9975
0.0067
3.1450
0.9985
0.0044
4.5981
0.9985
0.0029
R2
RMSE
R2
RMSE

Porosity of SOFC Anode Materials
7
Table VIII. Model Evaluation and Curve Fitting Results for Eq. (18) Using Literature Data
Ref.
Powder
Pressure range (MPa)
e0
A
B
R2
RMSE
55
Electrolytic Copper
Stainless Steel
Tungsten Carbide
Zink Powder
Mixture of Yttrium, Copper, and
Barium oxides
0–1000
0–926.1
0–864
0–450
0–350
0.700
0.626
0.799
0.620
0.910
0.6376
0.5847
0.7302
0.7692
0.9653
204.26
496.22
800.04
176.50
304.62
0.9965
0.9961
0.9859
0.9948
0.9963
0.0130
0.0122
0.0152
0.0162
0.0231
56
57
58
Calcium Phosphate cement PL2
Calcium Phosphate cement PL3
SiC/a-Alumina/Graphite
Ammonium Nitrate
Potassium Chloride
Potassium Nitrate
0–175
0.575
0.445
0.4085
0.2340
0.9736
0.7626
0.8905
0.6267
253.91
413.31
584.78
10.156
27.173
31.879
0.9905
0.9976
0.9988
0.9979
0.9908
0.9881
0.0112
0.0040
0.0012
0.0064
0.0158
0.0164
59
22
30–90
1–19.6
1.2–76.9
0.6–76.4
0.9723†
0.7076†
0.8866†
0.6617†
†Predicted value.
Table IX. Model Evaluation and Curve Fitting Results for Eq. (26) Using Literature Data
Ref.
Ceramic powder
Pore-former
k
ei
K
m
R2
RMSE
45
60
42
YSZ
Alumina
a-Alumina
Graphite
PVAc
~40
20†
18
23.5
81.6
23.33
33.33
17.5
0.431
0.05
14.079
12.321
13.439
15.410
18.060
14.409
16.220
14.009
0.3601
0.4382
0.4884
0.4433
0.4053
0.4626
0.4275
0.4722
0.9988
0.9928
0.9992
0.9975
0.9998
0.9958
0.9954
0.9955
0.0045
0.0127
0.0084
0.0119
0.0028
0.0139
0.0137
0.0095
Corn starch
Wheat starch
Potato starch
Corn starch
Wheat starch
Corn starch
0.1107†
0.1636†
0.1446†
0.1235†
0.1165†
0.1869†
61
Alumina
62
Alumina (78 wt%)
†Predicted value.
4R. W. Rice, “Evaluating Porosity Parameters for Porosity-Property Rela-
tions,” J. Am. Ceram. Soc., 76 [7] 1801–8 (1993).
porosity should be measured experimentally (by mercury po-
rosimetry or Archimedes technique), before applying Eq. (26)
to predict the final porosity. It should be noted that as
shown from Eq. (26), the blend ratio of a pore-former was
more important compared with the size ratio in determining
the final porosity of the ceramic bodies.
5J. J. Haslam, A. Q. Pham, B. W. Chung, J. F. Di Carlo, and R. S. Glass,
“Effects of the Use of Pore Formers on Performance of an Anode Supported
Solid Oxide Fuel Cell,” J. Am. Ceram. Soc., 88 [3] 513–8 (2005).
6J. Kim, A. V. Virkar, K. Fung, K. Mehta, and S. C. Singhal, “Polarization
Effects in Intermediate Temperature, Anode-Supported Solid Oxide Fuel
Cells,” J. Electrochem. Soc., 146 [1] 69–78 (1999).
7A. Sanson, P. Pinasco, and E. Roncari, “Influence of Pore Formers on
Slurry Composition and Microstructure of Tape Cast Supporting Anodes for
SOFCs,” J. Eur. Ceram. Soc., 28 [6] 1221–6 (2008).
IV. Conclusions
8R. M. C. Clemmer and S. F. Corbin, “Influence of Porous Composite
Microstructure on the Processing and Properties of Solid Oxide Fuel Cell
Anodes,” Solid State Ionics, 166 [3–4] 251–9 (2004).
This work detailed the development of porosity models for
fabricated ceramic anode, phenomenologically developed as a
function of the ceramic processing variables of the applied
uni-axial pressure, the pore-former volumetric blend ratio,
and the particle-size ratio. Experiments were conducted by
preparing ceramic/pore-former blends comprising NiO/YSZ
as the ceramic powder and CMSs (carbon microspheres) as
the pore-forming agent, using a range of solid volume frac-
tions of the pore-former and diameters of the CMSs. We also
investigated the effects of fabrication pressure, by measuring
the open porosities of the prepared samples to determine the
porosity behavior of the sintered ceramic substrates. The
applied uni-axial pressure as the main fabrication parameter
and the blend ratio of CMS significantly influenced the
porosity of the final ceramic body, whereas the proposed
models could potentially be used to predict the porosity of
porous bodies from ceramic and nonceramic powders.
9J. Hu, Z. Lu, K. Chen, X. Huang, N. Ai, X. Du, C. Fu, J. Wang, and W.
Su, “Effect of Composite Pore-Former on the Fabrication and Performance of
Anode-Supported Membranes for SOFCs,” J. Membr. Sci., 318 [1–2] 445–51
(2008).
10A. C. Muller, D. Herbstritt, and E. Ivers-Tiffe
´
e, “Development of a Multi-
¨
layer Anode for Solid Oxide Fuel Cells,” Solid State Ionics, 152–153, 537–42
(2002).
11S. C. Singhal and K. Kendall, High Temperature Solid Oxide Fuel Cells:
Fundamentals Design and Applications. Elsevier, Oxford, 2003.
12A. V. Virkar, J. Chen, C. W. Tanner, and J. Kim, “The Role of Electrode
Microstructure on Activation and Concentration Polarizations in Solid Oxide
Fuel Cells,” Solid State Ionics, 131 [1–2] 189–98 (2000).
13H. Yakabe, M. Hishunuma, M. Uratani, Y. Matsuzaki, and I. Yasuda,
“Evaluation and Modeling of Performance of Anode-Supported Solid Oxide
Fuel Cell,” J. Power Sources, 86 [1–2] 423–31 (2000).
14C. L. Tsai and V. H. Schmidt, “Tortuosity in Anode-Supported Proton
Conductive Solid Oxide Fuel Cell Found from Current Flow Rates and
Dusty-Gas Model,” J. Power Sources, 196 [2] 692–9 (2011).
15A. Konno, H. Iwai, M. Satio, and H. Yoshida, “A Corrugated Mesoscale
Structure on Electrode–Electrolyte Interface for Enhancing Cell Performance
in Anode-Supported SOFC,” J. Power Sources, 196 [18] 7442–9 (2011).
16R. W. Rice, Mechanical Properties of Ceramics and Composites: Grain and
Particle Effects. Marcel Dekker, New York, 2000.
Acknowledgments
The authors acknowledge the technical support from Monash Center for Elec-
tron Microscopy. This work was supported by the Australian Research Coun-
cil. Huanting Wang thanks the Australian Research Council for a Future
Fellowship (FT100100192).
17E. Chevalier, D. Chulia, C. Pouget, and M. Viana, “Fabrication of Porous
Substrates: A Review of Processes Using Pore Forming Agents in the Bioma-
terial Field,” J. Pharm. Sci., 97 [3] 1135–54 (2008).
18E. Gregorova
´
and W. Pabst, “Porous Ceramics Prepared Using Poppy
Seed as a Pore-Forming Agent,” Ceram. Int., 33 [7] 1385–8 (2007).
19A. I. Ratko, A. I. Ivantes, and S. M. Azarov, “Effect of Additives on the
Pore Structure of Ceramics Based on Crystalline SiO2,” Inorg. Mater., 44 [7]
778–84 (2008).
References
1R. W. Rice, Porosity of Ceramics. Marcel Dekker, New York, 1998.
2R. McPherson, “A Review of Microstructure and Properties of Plasma
Sprayed Ceramic Coating,” Surf. Coat. Technol., 39/40 [1] 173–81 (1989).
3R. W. Rice, “Effects of Inhomogeneous Porosity on Elastic Properties of
Ceramics,” J. Am. Ceram. Soc., 58 [9–10] 58–9 (1975).
20W. Chen and S. G. Malghan, “Investigation of Compaction Equations for
Powders,” Powder Technol., 81 [1] 75–81 (1994).
21P. J. Denny, “Compaction Equations: A Comparison of the Heckel and
Kawakita Equations,” Powder Technol., 127 [2] 162–72 (2002).

8
Journal of the American Ceramic Society—Horri et al.
22E. E. Walker, “The Properties of Powders. Part VI. The Compressibility
44Q. Wang, H. Li, L. Chen, and X. Huang, “Monodispersed Hard Carbon
Spherules with Uniform Nanopores,” Carbon, 39 [14] 2211–4 (2001).
45M. Boaro, J. M. Vohs, and R. J. Gorte, “Synthesis of Highly Porous Yt-
tria-Stabilized Zirconia by Tape-Casting Methods,” J. Am. Ceram. Soc., 86 [3]
395–400 (2003).
of Powders,” Trans. Faraday Soc., 19, 73–82 (1923).
23R. Panelli and F. A. Filho, “A Study of a New Phenomenological Com-
pacting Equation,” Powder Technol., 114 [1–3] 255–61 (2001).
24R. W. Heckel, “Density-Pressure Relationships in Powder Compaction,”
Trans. Metall. Soc. AIME, 221, 671–5 (1961).
46E. Gregorova, W. Pabst, and I. Bohacenko, “Characterization of Different
Starch Types for Their Application in Ceramic Processing,” J. Eur. Ceram.
Soc., 26 [8] 1301–9 (2006).
25K. Kawakita and K. H. Ludde, “Some Considerations on Powder Com-
¨
pression Equations,” Powder Technol., 4 [2] 61–8 (1971).
26C. A. Bruch, “Problems in Die-Pressing Submicron Size Alumina Pow-
der,” Ceram. Age, 83 [10] 44–53 (1967).
47S. W. Jung, J. M. Vohs, and R. J. Gorte, “Preparation of SOFC Anodes
by Electrodeposition,” J. Electrochem. Soc., 154 [12] B1270–5 (2007).
48M. Kim, J. Lee, and J. H. Han, “Fabrication of Anode Support for Solid
Oxide Fuel Cell Using Zirconium Hydroxide as a Pore Former,” J. Power
Sources, 196 [5] 2475–82 (2011).
27H. M. MacLeod, “Compaction of Ceramics”; pp. 241–76 in Enlargement
and Compaction of Particulate Solids, Edited by N. G. Stanley-Wood. Butter-
worths, London, 1983.
28M. Celik, “Overview of Compaction Data Analysis Techniques,” Drug
Dev. Ind. Pharm., 18 [6–7] 767–810 (1992).
49P. Lamy, G. Thomas, L. Brunet, and R. Erre, “Modelling the Porosity
Evolution of a Powder Under Uniaxial Compression,” Propellants, Explosives,
Pyrotechnics, 30 [6] 397–403 (2005).
29J. M. Sonnergaard, “A Critical Evaluation of the Heckel Equation,” Int.
J. Pharm., 193 [1] 63–71 (1999).
50P. A. Webb, Volume and Density Determinations for Particle Technologists.
Micromeritics Instrument Corporation, Norcross, Georgia, 2001.
51J. Van der Zwan and C. A. M. Siskens, “The Compaction and Mechani-
cal Properties of Agglomerated Materials,” Powder Technol., 33 [1] 43–54
(1982).
30J. M. Dalla Valle, Micromeritics. Pitman Publishing Corp., New York, 41
–67, 1948.
31A. R. Cooper Jr. and L. E. Eaton, “Compaction Behavior of Several
Ceramic Powders,” J. Am. Ceram. Soc., 45 [3] 97–101 (1962).
32C. L. Huffine and C. F. Bonilla, “Particle-Size Effects in the Compression
of Powders,” AIChE J., 8 [4] 490–3 (1962).
52M. Duberg and C. Nystrom, “Studies on Direct Compression of
Tablets: XVII. Porosity Pressure Curves for the Characterisation of Volume
Reduction Mechanisms in Powder Compression,” Powder Technol., 46 [1]
67–75 (1986).
33J. T. Fell and J. M. Newton, “Effect of Particle Size and Speed of Com-
paction on Density Changes in Tablets of Crystalline and Spray-Dried Lac-
tose,” J. Pharm. Sci., 60 [12] 1866–9 (1971).
53L. Li and M. Aubertin, “A General Relationship Between Porosity and
Uniaxial Strength of Engineering Materials,” Can. J. Civ. Eng., 30 [4] 644–58
(2003).
34P. York, “Particle Slippage and Rearrangement During Compression of
Pharmaceutical Powders,” J. Pharm. Pharmacol., 30 [1] 6–10 (1978).
35J. Zheng, W. B. Carlson, and J. S. Reed, “Dependence of Compaction
Efficiency in Dry Pressing on the Particle Size Distribution,” J. Am. Ceram.
Soc., 78 [9] 2527–33 (1995).
54M. Aubertin, L. Li, R. Simon, and B. Bussiere, A General Plasticity and
Failure Criterion for Materials of Variable Porosity. Technical Report EPM-
´
RT-2003-11, Ecole Polytechnique de Montreal (2003).
´
36M. Djuric, R. M. Neducin, J. Ranogajec, and M. Radeka, “Particle Size
Range as a Factor Influencing Compressibility of Ceramic Powder,” Ceram.
Int., 21 [4] 227–30 (1995).
55G. Rong-De, “A new Powder Compaction Equation,” Int. J. Powder Me-
tall., 27 [3] 211–6 (1991).
56P. J. James, “Particle Deformation During Cold Isostatic Pressing of
Metal Powders,” Powder Metall., 20, 199–204 (1977).
37A. B. Yu and Z. H. Gu, “Characterization of the Porosity-Pressure Rela-
tion of Cohesionless Powders,” Adv. Powder Technol., 4 [3] 199–216 (1993).
38A. Balakrishnan and C. L. Martin, “Modeling of Compaction and Green
Strength of Aggregated Ceramic Powders,” J. Am. Ceram. Soc., 94 [4] 1046–
52 (2011).
57S. Salib and C. Vipulanandan, “Property-Porosity Relationships for Poly-
mer-Impregnated Superconducting Ceramic Composite,” J. Am. Ceram. Soc.,
73 [8] 2323–9 (1990).
58K. Ishikawa and K. Asaoka, “Estimation of Ideal Mechanical Strength
and Critical Porosity of Calcium Phosphate Cement,” J. Biomed. Mater. Res.,
29 [12] 1537–43 (1995).
39D. M. Liu, “Influence of Solid Loading and Particle Size Distribution on
the Porosity Development of Green Alumina Ceramic Mouldings,” Ceram.
Int., 23 [6] 513–20 (1997).
59S. Ding, S. Zhu, Y. P. Zeng, and D. Jiang, “Fabrication of Mullite-
Bonded Porous Silicon Carbide Ceramics by In Situ Reaction Bonding,” J.
Eur. Ceram. Soc., 27 [4] 2095–102 (2007).
40J. Divisek, R. Wilkenhoener, and Y. Volfkovich, “Structure Investigations
of SOFC Anode Cermets. Part I: Porosity Investigations,” J. Appl. Electro-
chem., 29 [2] 153–63 (1999).
60T. Isobe, Y. Kameshima, A. Nakajima, and K. Okada, “Preparation and
Properties of Porous Alumina Ceramics with Uni-Directionally Oriented Pores
by Extrusion Method Using a Plastic Substance as a Pore Former,” J. Eur.
Ceram. Soc., 27 [1] 61–6 (2007).
41S. F. Corbin, J. Lee, and X. Qiao, “Influence of Green Formulation and
Pyrolyzable Particulates on the Porous Microstructure and Sintering Charac-
teristics of Tape Cast Ceramics,” J. Am. Ceram. Soc., 48 [1] 41–7 (2001).
42E. Gregorova, Z. Zivcova, and W. Pabst, “Starch as a Pore-Forming and
Body-Forming Agent in Ceramic Technology,” Starch/Sta¨rke, 61 [9] 495–502
(2009).
61E. Gregorova and W. Pabst, “Porosity and Pore Size Control in Starch
Consolidation Casting of Oxide Ceramics-Achievements and Problems,” J.
Eur. Ceram. Soc., 27 [2–3] 669–72 (2007).
43L. Mingyi, Y. Bo, X. Jingming, and C. Jing, “Influence of Pore Formers
on Physical Properties and Microstructures of Supporting Cathodes of Solid
Oxide Electrolysis Cells,” Int. J. Hydrogen Energy, 35 [7] 2670–4 (2010).
62E. Gregorova and W. Pabst, “Process Control and Optimized Preparation
of Porous Alumina Ceramics by Starch Consolidation Casting,” J. Eur.
Ceram. Soc., 31 [12] 2073–81 (2011).
h