Grain boundaries in ceramics
Grain boundaries
Grain boundary: interface between two crystals (grains) of the same phase but
different orientation.
Regions with:
- lower density
- different coordination of atoms/ions
- relaxation of atomic positions
- often different composition (segregation of impurities, dopants, lattice
defects)
- different properties (charge and mass transport, dielectric, optical, etc.)
Small angle tilt boundary: misfit accommodated
by formation of dislocations
Low energy tilt boundary: coincidence of lattice positions
Dislocations
Line defects originated by the relative shearing of two parts of a crystal (plastic deformation).
Non-equilibrium defects, can not be treated by thermodynamics.
Edge dislocation
Screw dislocation
Grain boundaries
Left. HRTEM image of the 5.4 [001] (010) symmetrical tilt grain boundary in SrTiO3. Right. Strain
field around the dislocation cores evaluated from the HRTEM image. The size of the lozenges
reflects the unit cell size in the respective area.
Imaging of oxygen sublattice and grain boundary structure in SrTiO3 by HRTEM
110
Tilt grain boundary of SrTiO3
The intensity profile along the g.b.
shows that the intensity of the O
column s is variable.
 Column with oxygen vacancies
Simulated image. Half oxygen atoms
were removed from one O column on
the g.b. (white arrow). The oxygen
deficiency produces a higher
brightness in this position ().
Central dotted lines connect CSL positions
(CSL: coincidence site lattice)
O Sr-O
Relaxation of atomic positions near the grain boundary
Expansion
Ti-Ti
Decreased
spacing
Sr(CSL)-Sr(CSL) compared to
Sr-Sr
perfect lattice
Increased
spacing
Differences of Ti-Ti spacing along
the direction perpendicular to the
g.b. The spacing closer to the g.b.
is smaller. Increased separation of
the two Ti columns facing each
other at the g.b.
Differences of Sr-Sr spacing along
the direction perpendicular to the
g.b. The spacing closer to the g.b.
is larger. Decreased separation of
the two Sr columns facing each
other at the g.b.
Differences normal to the g.b. of SrSr spacings located on CSL sites.
The first spacing is increased
meaning that there is an expansion
of the g.b. (0.043 nm, 1% lattice
parameter).
Observed displacements and grain boundary expansion in agreement with first-principle calculations (0.06
nm expansion). Expansion related to Ti-Ti repulsion and existence of O vacancies on the g.b.
Segregation at grain boundaries
dw
 G 
rev


Surface tension or surface energy  
dA  A T , P,n
i
Natural tendency to minimize the surface energy of a system by
redistribution of the components. Components which lower the
surface/interface
energy
tend
to
concentrate
at
the
surface/interface (adsorption, segregation). Oxygen is strongly
surface-active in liquid metals and non-oxide ceramics.
Segregation at surfaces and grain boundaries in ceramics is
determined by the different formation energies of defects at
interfaces than in bulk.
La2O3  LaAl  3OO
250 ppm MgO
H seg  EM ,sup  EM ,bulk
250 ppm CaO
1
'
2MgO  Mg Al
 2OO  VO
2
1000 ppm La2O3
Segregation in alumina ceramics as visualized by SIM
Segregation in alumina ceramics – atomistic simulation
Segregation modifies the energy of the different crystallographic surfaces and, consequently,
the equilibrium shape of crystals
Surface energies of pure and Y-doped -Al2O3
Basal
plane
Interfacial energies of pure and Y-doped -Al2O3
Equilibrium morphologies of undoped -alumina (a) front and
(b) top view. Equilibrium morphologies of 10 ppm Y-doped alumina at 1600°C seen again from (c) front and (d) top view.
Wulff’s theorem for the equilibrium shape
1
h1

2
h2
 .....
i
hi
i: surface tension of ith face
hi: distance from the center
  2 (surf .)   (int.)
Segregation in alumina ceramics – atomistic simulation
Predicted grain boundary structures
For highly symmetric (left) and more
general case (right).
Calculated grain boundary structure showing a regular La
pattern resulting from segregation.
Segregation in alumina ceramics – atomistic simulation
Enthalpy of segregation for La in alumina
H seg  EM ,sup  EM ,bulk
Calculated and experimental solubility limit of
MgO in alumina as a function of grain size
Grain boundary phases and films
In many cases, solid phases located at grain boundaries result from the solidification of a liquid phase
formed during sintering. The grain boundary phase can form a continuous film, pockets at the triple junctions
or discrete particles.
Wetting of a liquid on a solid
 LV cos   SV   SL
 > 90°: nonwetting
 < 90°: wetting
LV
SL
Necessary condition for spreading:
θ
SV
 LV   SV
 = 0°: spreading
Wetting of grain boundaries
 GB  2 SL cos

2
Liquid phase forming additives in ceramic oxides: SiO2, glass,
alkaline oxides (Li2O, Na2O, K2O), alkaline-earth oxides (CaO,
SrO, BaO), TiO2, B2O3, CuO, ZnO, V2O5
Distribution of liquid/amorphous phase at grain boundaries
1  GB
cos 
2 2  SL

Y2O3:ZrO2
CaO:Si3N4
(1)
(2)
AlN
(3)
(4)
(4)
(3)
AlN
Y2O3:ZrO2
(2)
Si3N4
(1)
Distribution of liquid/amorphous phase at grain boundaries
No grain boundary layer
“Special” grain boundaries show little segregation
and are free of an amorphous grain boundary layer
Criterion for film formation:
GBC
2 A   GBC
Grain boundaries in SrTiO3 ceramics
Special (A) +
random (B) grain
boundaries
(rotation)
2A: interfacial energy of a gb containing
a wetting amorphous phase
A
A
GBC: interfacial energy of a clean gb
Random grain
boundary
Distribution of secondary phase at grain boundaries
Ordered grain boundary phase in Ti-rich
BaTiO3 ceramics
Segregation and space charge at grain boundaries
The defect formation energies and defect chemistry at the grain boundaries is, in general, different from that
of the bulk. Preferential segregation of charged defects in ionic solids leads to net charge at the grain
boundary core which is compensated by a space charge cloud of opposite sign adjacent to the boundary,
with formation of an electrostatic Schottky barrier.
The thickness of the space charge layer is of the order of the Debye length.

 r  0 RT
2c z 2 F 2
r: relative dielectric constant
z: number of charges on defect
c: defect concentration in the bulk
2SrO  Fe2O3 SrTiO
3  2FeTi'  VO  2SrSr  5OO
Schematic diagram of a positively charged grain
boundary (segregation of oxygen vacancies) and
compensating space charge (acceptor impurity).
The region adjacent to the grain boundary will be
depleted in oxygen vacancies.
2


1
x

L


O
 exp 
  L is the width of the space-charge layer
cV  ()
 2     L = 2.5 nm in Y-doped ZrO2.
O
cV  ( x)
Segregation and space charge at grain boundaries
Transport through a polycrystal. Due to anisotropy of grain boundaries and their specific topology, different
situations are encountered: (a) parallel effects, (b) perpendicular effects and (c) e flux constriction.
The effect of grain boundaries
on the properties of ceramic
oxides
Ionic conductivity in oxides: the effect of grain boundaries and grain size
Y2O3 ZrO

2  2YZr  3OO  VO
Oxygen conduction in Y-doped ZrO2
The segregation of oxygen vacancies in acceptor-doped oxygen conducting electrolytes (Y:ZrO2, Gd:CeO2.
Fe:SrTiO3) leads to positively charged grain boundaries cores and a depletion of oxygen vacancies in the
adjacent space charge layer. The combined effect of the electrostatic potential barrier (Schottky barrier) and
the depletion layer determines a decrease of the oxygen conductivity at gbs (blocking gbs). In doped zirconia
ceramics with clean boundaries the resistivity of gbs is at least two orders of magnitude higher than the bulk
resistivity. A size effect is expected for grain dimensions in the nanoscale region (grain size <4λ).
[VO••]
[VO••]
[VO••]∞
+
+
+
+
+
+
D

+
+
+
+
+
+
D


[VO••]
[VO••]∞

+
+
+
[VO••]∞
D
+
+
+
2
Specific bulk and grain boundary conductivity in 3 mol.%
Y2O3 doped ZrO2 (oxygen conductor)
Dopant segregation: decreasing effective bulk
dopant concentration
Space charge effect
At present, the minimum grain size (30-40
nm) of dense Y:ZrO2 ceramics is still >> 4λ
and strong size effects on ionic conductivity
are not observed.
Because of the high density of gbs in
nanoceramics, the total conductivity (not
shown) is dominated by the resistive grain
boundaries.
Mesoscopic fast ion conduction in thin-film heterostructures
σT versus 1/T
Parallel
ionic
conductivity
in
CaF2-BaF2
thin-film
heterostructures with overall thickness L comprising of N
layers of thickness d. The overall thickness (L) is
approximately the same in all cases.
d = L/N
Black lines: reference single phase films;
Green lines: semi-infinite space-charge zones (period >8)
Red lines: overlapping space-charge regions (period <8)
Variation of ionic conductivity with the
density of interfaces, N/L.
Nanosize effect. Loss of
individuality of the single
compounds.
d < 8λ
d
16nm
20nm
FF MF

2 F '  VF
430nm
d=50nm
d
interface effect
d > 8λ
Ionic conductivity in oxides: the effect of grain boundaries and grain size
Impedance spectroscopy
Time domain
v(t) = v0sin(ωt)
i(t) = i0sin(ωt+θ)
Fourier transform
v: voltage
i: current
: angular frequency
θ: phase difference
v(t)  V(ω)
i(t)  I(ω)
Frequency domain
V(ω) = I(ω) Z(ω)
Z(ω) = 1/(C ω j)
Z: impedance
C: capacitance
j=
1
V(ω), I(ω) and Z(ω) are complex quantities
Y axis
S
C 
S
S
  0 r
d
d
d
ω R1 C1 = 1
Simple RC circuit
ωp R1 C1 = 1
ω
R1
R0
R0+R1
Solid materials can be described by one (homogeneous single crystal) or more (ceramics, composites)
semicircle in the impedance plot. Each semicircle is described by one resistive and one capacitive component.
Ionic conductivity in oxides: the effect of grain boundaries and grain size
Proton conduction in Y-doped BaCeO3
Y2O3  2 BaO BaCeO
3  2YCe'  2YBa'  VO  5OO
VO  OO  H 2O BaCeO
3  2OH 
Sintered 1250°C/2h; gs: 0.38 μm
Sintered 1500°C/48h; gs: 5 μm
Intrinsic conductivity
grain interior
grain boundary
Total conductivity
Trivial size effect. The lower conductivity (left) of the fine grained ceramic is only due to the higher density of
resistive grain boundaries. The specific conductivities (right) are the same irrespective of grain size.
Ionic conductivity in oxides: the effect of the grain boundary phase
Continuous grain boundary phase
Impedance spectra
(a)
freq.
(a)
ZrO2: 3 mol % Y2O3
(b)
Lenticular grain boundary phase
+ clean boundaries
Bulk resistivity
Grain boundary resistivity
(b)
ZrO2: 6 mol % Y2O3
Colossal permittivity in CaCu3Ti4O12 : the role of interfaces
TiO6
octahedra
• Perovskite-like, non polar structure
• Not a ferroelectric relaxor
• Ab-initio calculations: r = 40
Ca
• Processing-dependent properties
Step-like behaviour of dielectric constant observed in
ceramics as well as in single crystals. Strong frequency
dispersion
Cu
The dielectric constant  (real part of
dielectric permittivity) is calculated from the
measured capacitance C taking into account
the sample geometry:
S
C 
S
S
  0 r
l
l
tanδ
Dielectric loss
Dissipation factor
l
 ac  0 r tan
Relative dielectric constant
Ceramic
102 Hz
106 Hz
Colossal permittivity in CaCu3Ti4O12 : the role of interfaces
Relative dielectric constant
Single crystal
20 Hz
106 Hz
Colossal permittivity in CaCu3Ti4O12 : the role of interfaces
Apparent colossal dielectric constant is of extrinsic oringin and is associated to the electrical
heterogeneity of the samples and the contribution of different interfaces:
- semiconductive grain interiors;
IBLC effect – only for ceramics
- more insulating grain boundaries and
Maxwell-Wagner relaxation
related interfacial polarization
- insulating layer at the electrode-ceramic interface;
also exist in single crystals
- insulating surface skin
}
}
Insulating gbs
Insulating skin
Semiconducting core
Semiconducting
grains
semiconducting ceramic
insulating layer
electrode
Brick layer model of a ceramic
Colossal permittivity in CaCu3Ti4O12 : the role of interfaces
The step-like behaviour of the dielectric constant and all other electrical properties can be reproduced by
using equivalent circuit models.
x gb 
Vgb
 1
V
V
xb  b  1
V
D
S
l
R
S
S
S
C     0 r
l
l
l
d
Rb  b
ρgb >> ρb; gb = b
Rgb 
1
 gb x gb
3
Cb depends only on composition
Cgb depends on microstructure
Cb   b
Cgb 
3 gb
xgb

3 b
xgb
d Cb  gb

D Cgb  b
Cgb>>Cb
Intrinsic
behaviour
Colossal permittivity in CaCu3Ti4O12 : the role of interfaces
Inhomogeneous conduction probed by atomic force microscopy (AFM)
conducting
tip
ceramic
V
electrode
Insulting grain
boundaries (brown)
Current image
Strongly nonlinear
current-voltage
properties
Topograpic
image
Fracture
surface
Influence of grain size on the dielectric constant of ferroelectric BaTiO3 ceramics
Progressive depression of the dielectric constant with decreasing grain size when d1< 1 micron
εi ≡ Ki’
grain
1
 eff

x1
1
g
x2
2
grain boundary
“clean” boundary
The microstructure of BaTiO3 ceramics
corresponds to ferroelectric grains with
high dielectric constant (ε1 = 3000-5000)
separated by non ferroelectric (ε 2  100)
grain boundaries (“dead layer”). The
NFE gbs do not necessarily imply a
second phase grain boundary layer.
d2 = 1-3 nm depending on ceramic
preparation method.
ε 1 >> ε 2
ε 2 ε eff  ε 1
d1
ε2
ε1
d2
Influence of grain size on the dielectric constant of ferroelectric BaTiO3 ceramics
Relative dielectric constant (298 K, 10 KHz) of dense BaTiO3 ceramics, 1998-2006
Arlt et al., HPS
Frey & Payne, IP
Randall et al., CSM
Randall et al., HPS
Takeuchi et al., SPS
Zhao et al., SPS
Buscaglia et al., SPS
Deng at al., SPS
Zhu et al., SPS
Wang, 2SS
Relative dielectric constant
6000
5000
4000
3000
2000
1000
Dead layer effect
Domain size and
mobility effect
HPS: hot pressing
IP: pseudo isostatic pressing in a
multi-anvil cell
CSM:combined sintering method
SPS: spark plasma sintering
2SS: two-step sintering
0
10
100
1000
10000
Grain size (nm)
Dispersion of experimental values related to processing (purity and stoichiometry of powders, sintering
method) and microstructure (porosity, second phase grain boundary layer)