J. Am. Ceram. Soc., 83 [9] 2219 –26 (2000)
journal
Simulating Microstructural Evolution and Electrical Transport in
Ceramic Gas Sensors
Yunzhi Wang* and Yuhui Liu*
Department of Materials Science and Engineering, The Ohio State University, Columbus, Ohio
43210
Cristian Ciobanu* and Bruce R. Patton*
Department of Physics, The Ohio State University, Columbus, Ohio
An integrated computational approach to microstructural
evolution and electrical transport in ceramic gas sensors has
been proposed. First, the particle-flow model and the
continuum-phase-field method are used to describe the microstructural development during the sintering of a prototype
two-dimensional film. Then, the conductivity of the sintering
samples is calculated concurrently as the microstructure
evolves, using both resistor lattice models and effective medium theory for electrical transport in porous aggregates of
lightly sintered particles. This approach, when combined with
the modeling of resistivity at the grain– grain contacts as a
function of neck geometry, ambient gas concentration and
temperature, could facilitate the development and optimization of
novel microstructures for advanced ceramic gas sensors.
43210
The initial particle configurations at the green state were
simulated by the particle-flow model. Then, the microstructural
evolution during sintering was simulated using the continuumphase-field method. Starting with a realistic microstructure that has
been obtained from the phase-field simulation, the electrical
conductivity was calculated using a mesoscopic resistor-lattice
model. A phenomenological constitutive relation between quantitative microstructural features (such as the average grain size and
porosity) and the electrical conductivity was developed using an
effective media theory of electrical conduction in porous granular
ceramic materials. Our results show that the conductivity may be
predicted directly from the microstructural features of an inhomogeneous body. The simulation methods and their integration have
been described in section II. Practical applications are illustrated in
section III, using a prototype two-dimensional (2D) system. The
combination of this approach with other simulation methods for
the effect of ambient gas adsorption on the conductivity of
intergrain contacts27,28 also is discussed, in regard to potential
applications to the optimization of ceramic gas sensors.
I. Introduction
T
transport properties of electroceramics are critically dependent on microstructural inhomogeneities such as grain boundaries, pores, and second-phase particles. Thus, device optimization
requires information on both microstructural evolution and the
relation between microstructure and electrical transport.
Complex microstructural evolution during materials processing
is a nonequilibrium, nonlinear, and nonlocal multiparticle problem
that has analytic solutions only in a few simple cases. A similar
situation exists for calculations of the electrical conductivity for
realistic microstructures.1–3 However, advances in computer simulation techniques have provided numerical methods to characterize microstructural evolution during sintering4 –19 and calculate the
electrical properties of inhomogeneous media,2,20 –26 although
limited effort has been made to integrate microstructure simulation
with property calculations. A linked approach to modeling microstructure and transport is essential, because microstructural modeling alone cannot identify optimal electrical properties for a
particular application. In this paper, using the example of the
thermal processing of ceramic gas sensors, an integrated computational approach to determining the microstructure development
and the corresponding electrical conductivity has been proposed.
HE
II.
Principles of the Simulation Methods
(1) Simulation of Initial Microstructure Using the
Particle-Flow Model
The initial particle arrangement in the green state affects
microstructure development during firing; therefore, it is important
to start with realistic initial particle configurations. In the simulations, green microstructures that consist of circular particles with
different particle-size distributions and green densities have been
produced by the Particle Flow Code (PFC).29,30 They serve as the
initial configurations of the phase-field modeling.
(2) Continuum-Phase-Field Method and
Its Application to Sintering
Microstructure development during sintering involves multiple
kinetic events that lead to complicated morphologic patterns. Over
the past several decades, extensive efforts have been made in
developing theoretical and numerical models to describe the
sintering kinetics and microstructural evolution.4 –19,31–38 However, strong constraints on the geometry of particles and necks
between particles, as well as number of particles, were imposed in
most of these studies. The electrical conducting paths in ceramic
gas sensors are determined by the detailed geometry of particle–
particle interconnections;39,40 therefore, it is essential to consider a
realistic microstructure without any constraint on the morphology.
For this purpose, we have extended the phase-field method to
simulate the microstructural evolution during sintering, starting
from a powder compact without any a priori constraint on the
geometry and evolution path. In particular, neck formation and the
associated shape changes of pores and grains are examined.
(A) Description of Sintering Microstructure by Field Variables: In the phase-field approach, a complicated microstructure
is described by a set of spatially continuous and time-dependent
L.-Q. Chen—contributing editor
Manuscript No. 189371. Received May 20, 1999; approved March 3, 2000.
Authors YW and YL were supported by National Science Foundation, under Grant
No. DMR 9703044. Authors CC and BRP were supported by the National Science
Foundation Center for Industrial Sensors and Measurements, through Grant No.
EEC-9523358.
Presented at the 100th Annual Meeting of the American Ceramic Society,
Cincinnati, OH, May 4–6, 1998 (The Computational Modeling of Materials and
Processing Symposium, Paper No. SV-032-98).
Based in part on the thesis submitted by author YL for the M.S. Degree in
Materials Science and Engineering at The Ohio State University.
*Member, American Ceramic Society.
2219
2220
Journal of the American Ceramic Society—Wang et al.
field variables, such as local density, concentration, and crystallographic orientation or symmetry of the crystal structure (see
reviews by Chen and Wang41,42). As illustrated in Fig. 1(a), the
typical microstructural components of a green compact include
granular particles or grains and pores. The grains and pores will
rearrange themselves during sintering to eliminate surfaces and
grain boundaries (see Fig. 1(b)). In the phase-field approach, such
a heterogeneous microstructure can be described by a relative
density field ␳(r), which distinguishes the different microstructural
components in the system (grains, grain boundaries, pores, etc.),
and a set of long-range order (lro) parameter fields ␩p(r), which
describe the different crystallographic orientations of the grains.
As illustrated in Fig. 2, for example, ␳(r) assumes a value of 1
inside a grain, vanishes inside a pore, and takes on intermediate
values of 0 –1 at surfaces and grain boundaries. The lro parameter
field ␩p(r) is equal to ␩0 inside grains of orientation p, assumes
intermediate values of 0 –␩0 at the surface and grain boundaries of
the pth grain, and vanishes elsewhere. The use of a multicomponent lro parameter in the continuum-phase-field approach to
describe different structural domains can be found in the simulation studies of phase transformations with symmetry reductions,43– 45 microstructural evolution of ordered intermetallics with
multiple antiphase domains,46,47 and grain growth in polycrystalline materials.48,49
The sample volume will decrease as pores are eliminated during
sintering, whereas the shape of the sample surfaces will change to
maintain an equilibrium dihedral angle at the intersections of the
grain boundary and the free surface. This phenomenon complicates
the boundary condition. The problem can be simplified by including the surrounding vapor phase as part of the system, e.g.,
choosing a slightly larger computational cell with which to work
(see Fig. 1(a)). In this case, while the internal porosity decreases
during sintering, the volume of the surrounding vapor phase will
increase, which leaves the total vapor phase of the system—and,
hence, the total volume of the system— unchanged. The entire
system can be regarded as a two-phase mixture, with an equilibrium relative-density profile of ␳ ⫽ 1 for the solid phase and ␳ ⫽ 0
for the vapor phase.
(B) Field Kinetic Equations: Microstructural evolution in
the field method is described by temporal evolution of the field
variables, following the general nonlinear Cahn–Hilliard diffusion
equation for the relative-density field ␳(r,t)
冋 冉
␦F
⳵␳共r,t兲
⫽ ⵜ Mⵜ
⳵t
␦␳共r,t兲
冊册
(1)
and the time-dependent Ginzburg–Landau (TDGL) equations (also
called the Cahn–Allen equations) for the lro parameter fields ␩p(r,t):
⳵␩ p 共r,t兲
␦F
⫽ ⫺L
⳵t
␦␩ p 共r,t兲
共 p ⫽ 1, 2, . . . , n兲
(2)
Fig. 1. Typical microstructural evolution during sintering of a powder
compact ((a) the green compact and (b) the sintered body).
Vol. 83, No. 9
Fig. 2. Schematic drawings of the relative-density profiles and longrange-order (lro) parameter profiles across the grain boundary and the
surface.
Here, t is time, M and L are the kinetic coefficients that respectively characterize the diffusive and structural relaxation, n is the
total number of lro parameters needed to characterize the grain
orientations, and F is the total free-energy functional whose
variational derivatives, with respect to the field variables ␳(r) and
␩p(r) in the equation, form the thermodynamic driving forces for
the microstructure evolution. A recent review of the various forms
of the field kinetic equations has been given by Gunton et al.50
As one may notice, the geometry of the surface and grain
boundaries do not enter the field kinetic equations. They emerge as
solutions of these equations; e.g., the boundaries are defined by
those regions of sharp gradients of the field variables (see Fig. 2).
Therefore, no a priori assumptions are needed for the possible
morphology of the developing microstructure.
Note that, for microstructural evolution with the presence of
grain boundaries and free surfaces, relative rigid-body displacements (e.g., rotation and translation of entire particles) must be
considered.19 This rigid-body motion is not included in Eqs. (1)
and (2). In general, rigid-particle motion contributes to sample
shrinkage during sintering and, hence, affects the electrical properties. In the processing of ceramic gas sensors, however, samples
usually are sintered lightly to achieve a high surface-to-bulk ratio.
For thin films, samples are fired on top of a substrate.51 The rigid
substrate imposes a strong constraint on the shrinkage of the
sample during firing, especially when the film is only a few grains
thick. Therefore, rigid-body motion and the associated sample
shrinkage may not be relevant in thin-film gas-sensor processing
and is ignored in this 2D study. Instead, the focus has been on the
development of particle–particle contacts and the associated shape
changes of grains and pores during firing under the constraint that
no rigid-body motion occurs. Elsewhere, we have described a
generalized phase-field approach that includes rigid-body motion,
which describes pore elimination and sample shrinkage during
sintering.19
(C) Formulation of the Coarse-Grained Free Energy: Microstructural evolution during sintering is driven by free-energy
reduction that is associated with internal surfaces and grain
boundaries, which, in the phase-field approach, is described by a
coarse-grained free energy,50 which is formulated as a functional
of appropriately chosen field variables. For the typical heterogeneous microstructure shown in Fig. 1, the general form of such a
coarse-grained free energy can be written as
September 2000
F⫽
冕冉
1
2
V
Simulating Microstructural Evolution and Electrical Transport in Ceramic Gas Sensors
冘
can be formulated as
␣ ij ⵜ i ␳共r兲ⵜ j ␳共r兲
f⫽⫺
ij
冘冘
n
1
⫹
2
ij
p⫽1
冊
⫹
dV
(3)
⫹⬁
⌬f共␳,0, . . . ,␩,0, . . . , 0兲
⫺⬁
冉 冊 冉 冊册
␣ d␳
2 dx
⫹
2
⫹
␤ d␩
2 dx
冘冋
i⫽1
where ⵜi is the ith component of the vector operator ⵜ, xi is the ith
component of the coordinate vector r, ␣ij and ␤ij(p) are the
gradient energy-coefficient tensors, and ␳(r) and ␩p(r) are the
relative-density and lro parameter profiles, respectively. The variable f(␳,␩1,␩2, . . . ,␩n) represents the local free energy, and V is
the total volume of the system, including the vapor phase that
surrounds the sample.
The surface and grain-boundary energies in this model are
determined by the gradient coefficients as well as the local free
energy. An analytical expression of the interfacial energy resulting
from such a free-energy functional has been derived by Cahn and
Hilliard for a flat interface.52 A similar expression for the grainboundary energy can be found in the work of Chen and coworkers.48,49 Similarly, the energy of a flat surface of a grain can
be written as
冕冋
A1
A2
A3
共␳共r兲 ⫺ c 0 兲 2 ⫹ 共␳共r兲 ⫺ c 0 兲 4 ⫹ ␳ 4 共r兲
2
4
4
n
␤ ij 共 p兲ⵜ i ␩ p 共r兲ⵜ j ␩ p 共r兲
⫹f共␳,␩ 1 ,␩ 2 , . . . ,␩ n 兲
␥⫽
2221
2
dx
(4)
where ⌬f ⫽ f(␳,0, . . . ␩,0, . . . , 0) ⫺ min(f(␳,0, . . . ␩,0, . . . , 0))
is the potential barrier between the pure states of vapor and solid.
In general, the gradient coefficients in Eq. (3) should be dependent
on both the inclination of surfaces and grains boundaries and the
misorientation between grains. For simplicity, we have assumed
values of ␣ij ⫽ ␣␦ij and ␤ij(p) ⫽ ␤␦ij (where ␣ and ␤ are positive
constants) in Eq. (4) and throughout this study, which is equivalent
to an assumption of isotropic surface and grain-boundary energies.
The numerical values of ␣ and ␤ in Eq. (4) can be determined by
fitting the surface and grain-boundary energies to experimental
values for a given system.
The specific form of the local free energy, f (␳,␩1,␩2, . . . ,␩n),
can be approximated by a Landau-type polynomial expansion in
the order parameters, which usually is obtained by symmetry and
stability consideration.53,54 The general requirement for the description of sintering is that f(␳,␩1,␩2, . . . ,␩n) must provide an
equilibrium state that contains both the vapor phase and particles
of the solid phase with different crystallographic orientations.
Thus, it should have a local minimum at
共␳ ⫽ 0, ␩ 1 ⫽ ␩ 2 ⫽ . . . ⫽ ␩ p ⫽ . . . ⫽ ␩ n ⫽ 0兲
(5a)
that characterizes the vapor phase (internal pores plus the surrounding atmosphere), together with n degenerate global minima
at
共␳ ⫽ ␳ g, ␩ 1 ⫽ ␩ 0 , ␩ 2 ⫽ ␩ 3 ⫽ . . . ⫽ ␩ p ⫽ . . . ⫽ ␩ n ⫽ 0兲
(5b)
共␳ ⫽ ␳ g, ␩ 2 ⫽ ␩ 0 , ␩ 1 ⫽ ␩ 3 ⫽ . . . ⫽ ␩ p ⫽ . . . ⫽ ␩ n ⫽ 0兲
(5c)
共␳ ⫽ ␳ g, ␩ n ⫽ ␩ 0 , ␩ 1 ⫽ ␩ 2 ⫽ . . . ⫽ ␩ p ⫽ . . . ⫽ ␩ n⫺1 ⫽ 0兲
(5d)
which characterizes grains of different orientations. The simplest
expansion polynomial that meets these topological requirements
⫹
A4
A5
共␳共r兲 ⫺ ␳ g兲 2 ␩ i2 共r兲 ⫺ ␳ 2 共r兲␩ i2 共r兲
2
2
A6 4
A7
␩ i 共r兲 ⫹
4
4
冘
n
␩ i2 共r兲␩ j2 共r兲
j⫽i
册
(6)
where A1, . . . , A7 and c0 are positive constants. In principle, the
phenomenological coefficients A1, . . . , A7 are dependent on temperature. In the simulations, a generic prototypical system has been
used, which is characterized by the free energy (Eqs. (3) and (6)),
using the following phenomenological parameters: ␣ ⫽ 1.5, ␤ ⫽
2.0, A1 ⫽ 2.6, A2 ⫽ 10.4, A3 ⫽ 1.3, A4 ⫽ 0.6, A5 ⫽ 1.2, A6 ⫽ 1.0,
A7 ⫽ 1.0, c0 ⫽ 0.5, and ␳g ⫽ 1.0. Here, A1–A7 are measured in
terms of a typical energy scale ⌬f0, which is defined by the energy
barrier between the solid phase and the vapor phase, and ␣ and ␤
are measured in terms of ⌬f0ᐉ2, where ᐉ is the length unit used in
the simulation. These parameters provide a qualitatively correct
topology for the free-energy hypersurface that characterizes a
sintering system as discussed above and yield a surface-energyto-grain-boundary-energy ratio of 0.85.
(D) Formulation of the Path-Dependent Kinetic Coefficient:
Many diffusion paths contribute to the creation of bonding
between particles; these paths include bulk diffusion, surface
diffusion, grain-boundary diffusion, and evaporation and condensation. In the phase-field method, these different paths result from
the dependence of the diffusion coefficient in Eq. (1) on the field
variables. If the evaporation and condensation mechanism, which
is not important in the solid-state sintering of ceramics, is
neglected, a general expression for the kinetic coefficient M can be
formulated in terms of ␳(r) and ␩(r):
M共r兲 ⫽
␰ sM s共r兲 ⫹ ␰ gbM gb共r兲 ⫹ ␰ gM g共r兲
␰ s ⫹ ␰ gb ⫹ ␰ g
(7a)
共ⵜ␳共r兲兲 2
max共共ⵜ␳共r兲兲 2 兲
(7b)
where
M s共r兲 ⫽
冘
p
M gb共r兲 ⫽
␩ i2 共r兲␩ j2 共r兲
冉冘
i, j⫽1
p
max
i, j⫽1
M g共r兲 ⫽
␳共r兲
max共␳共r兲兲
␩ 共r兲␩ 共r兲
2
i
2
j
冊
(7c)
(7d)
The variables ␰s, ␰gb, and ␰g are coefficients that characterize the
relative contribution of the surface-diffusion, grain-boundarydiffusion, and bulk-diffusion mechanisms to the transport. According to the definition given in Eqs. (7), M(r) assumes different
values at different regions (such as at the grain, grain boundary,
surface, and pore). In general, ␰s, ␰gb, and ␰g can be determined by
fitting the typical diffusion coefficients for each of these structural
components to experimental values.
(3) Calculation of Electrical Conductivity in
Granular Ceramic Materials
Although the conduction of charge carriers through a granular
material clearly is dependent on the microstructure, it also is
dependent on many details of the electronic structure of the grains
and the grain-boundary regions; these details include the carrier
density and conduction mechanisms in the grain, the nature of the
2222
Journal of the American Ceramic Society—Wang et al.
scattering at the grain boundary between grains, and possible
development of Schottky barriers and depletion regions near grain
surfaces. To separate the influence of different factors, in the
present work, we have focused on the importance of the microstructure and, thus, we will consider other variables to be fixed
during the sintering process. Thus, as in a boundary-conductionlimited material such as ZnO or TiO2, we will use relatively highly
conducting grains that are separated by grain-boundary regions of
constant high resistance. Then, the time evolution of the bulk
conductivity will be calculated directly from the detailed microstructure. This detailed microstructure is generated from the
phase-field simulation in two complementary ways: first, by
mapping the microstructure onto an equivalent mesoscopic resistor
lattice, and second, by developing an equivalent continuum effective medium model, which reproduces the results of the resistor
lattice calculation.
(A) Resistor Lattice Model (RLM): In the resistor lattice
model (RLM), the electrical conductivity of an arbitrary microstructure is calculated by mapping the different microstructural
components onto a network of appropriate resistors.20,25,26 Increased scattering or activated hopping that is associated with
back-to-back Schottky barriers at the grain boundary gives the
grain boundary a much-lower conductivity than that of the interior.
Thus, a ceramic gas sensor has three components with very
different conductivities: grains, grain boundaries, and pores.
Finding the conductance of an arbitrary mixture of these three
components requires solving a general, three-value resistor network problem, for which an efficient iterative algorithm recently
was developed.26 Qualitatively, the spatial variation of the conductivity (and the values of the resistors) should be similar to that
of the relative density (see Fig. 2). Then, a conductivity field ␴(r)
can be defined as
␴共r兲 ⫽ ␴ g
␴共r兲 ⫽ ␭␴ g
␴共r兲 ⫽ 0
共if r is inside the grains兲
共if r is at the grain boundaries兲
共if r is inside the pores兲
(8a)
(8b)
(8c)
where ␭ is the ratio of grain-boundary conductivity (␴gb) to grain
conductivity (␴g). The value of ␭ may be calculated, for example,
from properties of the Schottky barrier at the grain– grain contacts;
we have used a value of ␭ ⫽ 0.001, which is typical for electronic
ceramics such as ZnO.55
In the computer simulations, ␴(r) is calculated on a discrete
mesh with a lattice spacing that is equal to that used in the
phase-field modeling, which is sufficiently small, in comparison
with the diffuse width of the grain boundary that is generated from
the phase-field calculation. The conductivity ␴(r) is constructed as
follows: at each mesh point, one of the conductivity values given
n
in Eq. (8) is assigned, according to the value of 兺i⫽1
␩2i (r) (which
is unity inside the grains and zero inside pores and has intermediate values inside the grain boundaries). Although grain boundaries characterized in this way are diffuse and have a finite
thickness, which may well exceed the values observed in the
experiments, the effective resistance of the grain boundary in the
model will reproduce a known experimental value for a particular
system via the proper choice of the phenomenological constant ␭
in Eq. (8). In the simulations, sites within the pores are removed
and the three-value regular resistor network is converted to a
two-value irregular resistor network. The relaxation method described by Ciobanu et al.26 has been used to calculate the electrical
conductivity of the network.
(B) Effective Medium Model (EMM): If electrical properties
such as the effective conductivities of the grains and grain
boundaries do not change during sintering, as we have assumed in
this paper, then the overall sample resistance is expected to
decrease as the grains sinter and grow in size. This expectation
suggests that the general features of the dependence of conductivity on the microstructure may be dependent on the relative volume
fractions of the various components, such as grains, voids, and
grain boundaries. In addition, although the previously discussed
Vol. 83, No. 9
RLM can incorporate minute details of an inhomogeneous porous
microstructure in the calculation of the electrical conductivity, it
may be useful to have a more continuum or coarse-grained
effective medium description, which provides a phenomenological
constitutive relation between the quantitative microstructural features of a given microstructure (such as average grain size and
porosity) and the electrical conductivity. In the effective medium
model (EMM),27,56 the microstructure is characterized by the
relative amount and spatial distribution of components of significantly different conductivities. For the microstructures that have
been considered in this paper, the components are pores, grains,
and grain boundaries. In regard to electrical transport, grains are
good conductors, grain boundaries are poor conductors, and pores
are insulators. The effective electrical conductivity of the sample is
dependent on the volume fractions and spatial arrangement of
these microstructural components, which can be obtained directly
from the phase-field modeling.
If we consider, for the moment, a single-phase polycrystalline
material without pores, the system can be treated as a twocomponent medium with low-conductivity regions (grain boundaries plus associated depletion regions) surrounding highconductivity interior regions (grains). The effective conductivity of
an inhomogeneous medium in which one phase coats the other can
be described by the unsymmetrical effective medium theory:56
共␴ eff ⫺ ␴ g兲 d
共␴ gb ⫺ ␴ g兲 d
⫽ 共1 ⫺ f g兲 d
␴ eff
␴ gb
(9)
Here, ␴eff is the effective conductivity, ␴g and ␴gb are the
respective conductivities of the grain and grain boundary (including the depletion region), fg is the volume fraction of the highconductivity component (the nondepleted regions of the grain),
and d is the dimensionality of the sample (d ⫽ 3 for bulk material
and d ⫽ 2 for a thin film whose thickness is comparable to the
grain size). The latter dimensionality value—d ⫽ 2—is used in the
following discussion, for comparison with the 2D microstructure
simulations. In sharp contrast to a random composite, the important feature of Eq. (9) is that, due to the correlation (coating)
between the two components, the effective resistivity is related to
the series resistance of the highly conducting grain and the
high-resistance grain boundary; this observation is the manifestion
of the grain-boundary-controlled conduction mechanism in these
materials. Although it is difficult to improve systematically on the
EMM, many features of percolative conduction in random media
are well described qualitatively by the approximation, which is
exact to fourth order.57
When pores are present, the simple nonsymmetric model or
coated EMM previously described cannot be correct as given to
this point. However, the effect of pores can be included in a
straightforward fashion by considering the modifications that they
induce in the parameters in Eq. (9): i.e., the reduction of the
effective grain conductivity by pores inside the grains, and the
reduction of conduction between grains by pores at the grain
boundary. From the phase-field simulation, we can determine, at
each time moment, the interior grain volume (Vg), the grainboundary or interface volume (Vgb) (including the depletion layer),
pore volumes inside grains (Vpg) and at grain boundaries (Vpgb),
and the volume of the surrounding vapor phase (Vpe). The total
volume (V0) is the sum of all these volumes and is fixed during a
simulation. Then, the effect of pores can be incorporated into the
EMM in Eq. (9) by appropriately renormalizing the volume
fractions and conductivities of the grains and grain boundaries. If
a sample volume is defined as Vs ⫽ V0 ⫺ Vpe, then the relative
fractions that are associated with grains, grain boundaries (plus
depletion layer), pores inside grains, and pores at grain boundaries
may be defined as fg ⫽ Vg/Vs, fgb ⫽ Vgb/Vs, fpg ⫽ Vpg/Vs, and fpgb
⫽ Vpgb/Vs, respectively. Then, we may include the contribution of
pores inside grains by defining an effective grain volume fraction
(fge ⫽ fg ⫹ fpg) and using the renormalized value ␴ge ⫽ ␴g(1 ⫺
fpg)d/(d⫺1) (which comes from applying Eq. (9) to a conducting
September 2000
Simulating Microstructural Evolution and Electrical Transport in Ceramic Gas Sensors
material (grain) that is coating a pore with zero conductivity) for
the effective grain conductivity, rather than the original grain
conductivity ␴g.
Similarly, the effective insulator fraction fgbe can be identified
as the sum of the grain-boundary and grain-boundary-pore fractions (fgbe ⫽ fgb ⫹ fpgb) and use an appropriately renormalized
effective grain-boundary conductivity ␴gbe. To determine the
effective conductivity that is associated with the grain boundary,
we note that pores introduce regions of zero conductivity between
adjoining grains. Because the grain-boundary regions are in series
with the conducting grains, the pores at grain boundaries effectively reduce the grain-boundary conductivity in proportion to the
cross-sectional area of the sample that is occupied by pores at the
grain boundaries. Thus, the grain-boundary region can be described by a renormalized effective conductivity ␴gbe, which is
related to the grain-boundary depletion-layer conductivity ␴gb by
the relation ␴gbe ⫽ ␴gb(1 ⫺ f (d⫺1)/d
).27
pgb
The nonsymmetric effective medium theory from Eq. (9),
together with the equations for the volume fractions fge and fgbe
and the effective conductivities ␴ge and ␴gbe, constitute a mapping
of the mesoscopic results of the phase-field modeling onto the
EMM at the macroscopic level. This theory may provide a
constitutive relation between the quantitative microstructural parameters and the electrical conductivity through the dependence of
fge and fgbe and ␴ge and ␴gbe on the average grain size, porosity,
and pore distribution in the sample. The effective conductivities
␴ge and ␴gbe also will change when a reducing gas, the amount of
which may be determined using chemical kinetic methods, as
described separately,27 is absorbed.
III.
2223
Fig. 3. Simulated microstructural evolution during early stages of sintering, showing neck formation and shape changes of grains and pores in a
sample with 90% 2D green density.
Results and Discussion
Although the phase-field model may be applied in any dimension, the computer simulations have been performed in two
dimensions, because numerical solutions of the phase field (Eqs.
(1) and (2)) in three dimensions,58 – 60 using a uniform-grid
finite-difference method, are still computationally intensive. Simulation results on microstructural evolution are shown later in Figs.
3 and 5(a). Two cases with 2D green densities of 90% and 84%
have been considered. The subsequent microstructural evolution
during sintering was obtained using the phase-field model. The
system sizes used for these simulations were 512 ⫻ 512 (Fig. 3)
and 256 ⫻ 256 (Fig. 5(a)). Strictly speaking, the 2D simulations
apply to disks or parallel cylindrical grains in three dimensions;
however, we expect them to be also applicable to thin granular
films on a substrate, with a thickness on the order of the average
grain size. The simulated microstructures are presented by shades
of gray that correspond to numerical values of the relation
n
兺i⫽1
␩2i (r), where the larger values appear brighter. Therefore,
white regions represent grains, dark-gray lines represent grain
boundaries, and black regions represent pores.
Figure 3 shows the detailed morphologic changes of the powder
compact during early stages of sintering in a sample with a 2D
green density of 90%. A total of two hundred particles, of four
different sizes, are present in the system. Thirty-six crystallographic orientations have been assigned randomly to the particles.
According to Chen and Wang,48 such a number provides a
reasonably good description of a polycrystalline microstructure.
Initially, the particles have sharp surfaces (Fig. 3(a)). When the
sintering starts, the sharp interfaces gradually relax to diffuse
interfaces. Meanwhile, necks between adjacent particles start to
form via bulk, grain-boundary, and free-surface diffusion considered in the simulation. For simplicity, the diffusion coefficients
have been assumed to be the same within grains, grain boundaries,
and surfaces in these simulations.
As the sintering proceeds, necks grow and the initially circular
grains change their shapes to polygons, with a clear “undercutting”
at the neck regions, which can be observed in the contour plot
shown in Fig. 4. Pores become isolated by the grains, and their
shapes change from concave to convex (compare the microstructures at the reduced times of 28 and 316 in Fig. 3). Depending on
Fig. 4. Relative-density contour plot of a microstructure at the reduced
time of 5, starting from the initial powder compact shown in Fig. 3. The
three types of necks—“A,” “B,” and “C”—shown in the inset may be
identified from the detailed neck geometry (see, for example, the arrows).
Shaded areas in the inset represent the depletion layers.
the initial coordination number of the surrounding particles, the
particle–particle contacts and the pores may have different shapes.
The geometrical shapes of the neck are determined by the
requirement that the equilibrium dihedral angle at the grainboundary/free-surface intersections must be maintained. According to Beekmans61 and McAleer et al.,62 the geometry of the
intergrain contacts has a dramatic effect on the electrical conducting paths through a ceramic semiconductor. Depending on the
thickness of the depletion layer, the necks can be divided into three
different types: open neck (type “A”), closed neck (type “B”), and
2224
Journal of the American Ceramic Society—Wang et al.
Schottky barrier (type “C”) (see inset in Fig. 4). Each type of neck
has a different dependence of the activation energy for electrical
conduction on the partial pressures of oxygen and the reducing
gases. When the depletion length is comparable to the neck width,
the conductance becomes very sensitive to variations in the surface
charge.63 Obviously, all three types of necks may exist in the
simulated microstructure (see, e.g., the arrows marked “A,” “B,”
and “C” in Fig. 4). Therefore, any model that is based on only one
type of intergranular contact may not be able to describe an
arbitrary, lightly sintered microstructure; some bypass around any
given high-resistance boundary would always exist.
We notice that some adjacent particles happen to have exactly
the same crystallographic orientation. These particles coalesce and
form a big open neck. This situation can be avoided by introducing
a constraint in the simulation that no two neighboring particles
should have the same lro parameter. In most cases, however, the
adjacent particles have different grain orientations, and grain
boundaries are formed between them.
The simulated micrographs clearly indicate that the initial
microstructure of the green compact, e.g., the sizes of the starting
particles and their spatial distribution, have a strong influence on
the subsequent microstructural evolution during sintering which, in
turn, affects the electrical conductivity of the sample. For example,
there is a dramatic change in the electrical resistivity of the sample
that accompanies the microstructural evolution, as illustrated in
Fig. 5, where the electrical resistivity of the sample is calculated
simultaneously as the microstructure evolves using the RLM. As
stated previously, the ␴gb/␴g ratio is considered to be constant and
a numerical value of 0.001 has been used in the calculations. In
this simulation, one hundred particles of three different sizes have
been considered, and the initial particle configuration is generated
by the PFC program.
At the earlier stages of sintering, the microstructural evolution
is characterized by neck formation and growth, which convert the
initial poor contacts between particles to better conducting paths.
The electrical resistivity of the sample decreases sharply. After
conducting paths have been established among the particles, the
decrease of the resistivity is controlled by the relative amount of
grain boundaries and pores in the conducting paths. The microstructural evolution at these stages is characterized mainly by
shape reconstruction, in which pores change their shapes from
concave to convex and grains from convex to concave; therefore,
the fractions of grain boundaries and pores in the conducting paths
change slowly, which leads to a relatively stable regime for the
sample resistivity.
At later stages, the migration of internal pores into the surrounding vapor phase results in a decrease of the total volume fraction
of internal pores and the grain coarsening results in a significant
decrease in the volume fraction of grain boundaries. Both effects
are responsible for the increasing decrease in the resistivity in the
middle portion of Fig. 5(b). Eventually, the pores diffuse out of the
sample, which leads to densification, as illustrated in Fig. 5(a). The
pores close to the sample surface escape faster than the pores deep
inside the sample. As the pores are eliminated, the sample shrinks
and significant grain growth occurs during the later stages of
sintering. The average pore sizes also increase during this coarsening stage. However, the results will not be the same if the
rigid-body motion of grains, which contributes considerably to the
sample shrinkage, is considered.19 Corresponding to these microstructural changes, the resistivity attains a second plateau (see, for
example, Fig. 5(b)). A similar resistivity change has been observed
for a sample with a density of 65%.
The electrical resistances that have been calculated from RLM
and EMM agree well with each other. A comparison is given in
Fig. 6. RLM is more sensitive to local microstructural changes
than EMM, in particular, for the small samples that are considered
in the simulation; therefore, there are larger fluctuations in
resistance in the calculated results from RLM than that from
EMM. The advantage of RLM is its ability to consider detailed
microstructural features in the electrical-conductivity calculations.
The results obtained from RLM can be used to validate the
constitutive relations derived from EMM. However, the gross
Vol. 83, No. 9
Fig. 5. (a) Simulated microstructural evolution during sintering of a
sample with a 2D relative density of 84%; (b) corresponding changes in the
normalized electrical resistivity of the sample, calculated by RLM.
features of the dependence of the conduction on the microstructure
seem to be understandable, in terms of boundary-limited conduction and the role that pore and grain-boundary fractions have as the
microstructure evolves. Corresponding simulations are underway
to characterize the constitutive relations between the electrical
conductivity in granular ceramics with different microstructural
characteristics.
Figure 5(a) shows that the shape of the solid–vapor surfaces is
dependent strongly on the orientation of the grain boundaries that
intersect the surfaces. The dihedral angles of the surface grooves
and the internal pores are ⬃110°, which agrees well with the value
predicted from the ratio of the interfacial energy to the grainboundary energy (108°). The detailed shapes of the surface
grooves that are connected to grain boundaries of different
orientations (see Fig. 5(a) at the reduced time of 2250) are in good
agreement with those obtained from the variational analyses16 and
September 2000
Simulating Microstructural Evolution and Electrical Transport in Ceramic Gas Sensors
2225
the morphology of the sintering microstructure, which, in turn, has
a strong influence on the electrical-transport properties. This work
has illustrated the possibility of using computer simulations to
investigate the electrical-transport behavior of ceramic gas sensors.
Acknowledgments
The authors thank J. Lannutti, L.-Q. Chen, A. Kazaryan, B. Chwieroth, C. Shen,
S. A. Akbar, P. K. Dutta, M. J. Mill, and P. I. Gouma for useful discussions.
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Fig. 6. Comparison of the electrical resistivity of the sample shown in
Fig. 5(a), calculated using EMM and RLM. Resistivity values at different
sintering times in the plots are normalized by the final value at ␶ ⫽ 2250
in both cases.
from boundary integral analyses.64 The angles between two grain
boundaries at the triple junctions are ⬃120°, which agrees with the
isotropic grain-boundary energy that is assumed in the simulation.
The 2D simulations performed in this work are directly relevant
to thin granular films on a substrate, with a thickness that is on the
order of the average grain size. However, they provide only
qualitative insight into three-dimensional (3D) systems such as
thick films or bulk materials. More efficient algorithms are critical
for the success of large-scale, 3D simulations. Encouraging
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clearly affects the grain-boundary resistance during microstructural evolution. However, the above-discussed numerical methods
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ceramic gas sensors. In particular, it could be possible to calculate
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profiles near the grain surfaces and their effect on the grainboundary conductivities, this method may enable a comprehensive
description and possible design of gas sensors with controlled
properties.
IV.
Conclusion
An initial formulation for an integrated computational approach
to microstructural evolution and electrical transport in ceramic gas
sensors has been developed. The simulations using the phase-field
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