1

A REVIEW – FACTORS AND PROPERTIES AFFECTING IONIC

CONDUCTIVITY IN SOLID IONIC MATERIALS

Ian Rahmat Widi Perdana

School of Materials Engineering, Universiti Malaysia Perlis

Perlis, Malaysia

1 Introduction

Solid ionic materials are

categorized as electrolyte which may

conduct electricity by employing the

movement of ions inside the solid. They

have capability to conduct electric.

According to (Ruth, 2015), solid

electrolyte occur whilst there are existence

of defects especially Frenkel and Schottky

defects inside the crystal structure of solid.

Furthermore, solid electrolytes convey

current within the mobility of its ion and

their conductivity range much lower than

metals in terms of conducting electric. On

(Kong, Yao, Yu, & Li, 2015) research,

they stated that the ions move by

following modes of radial distribution

function on local dynamics and

configuration dependency.

Solid ionic materials are

convincing material on field of application

in batteries, fuel cells and semiconductors.

Solid ionic materials show dependency to

temperature in order to achieve their ionic

mobility to give rise on ionic conductivity

since solid ionic materials having defect

on their structure. According to (Mozalev,

Sakairi, Takahashi, Habazaki, & Hubálek,

2014; Okada, Ikeda, & Aniya, 2015), they

state that process to achieve ionic

conductivity of solid are mediated by

bond breaking which is caused by thermal

fluctuation (Arrhenius equation). On the

other hand, in solid material, the

oscillation of each ion in a potential well

performed by ions on surrounding and

may escape the potential well with certain

probability. Okada also mentioned that

when the mobile ion escape the potential

barrier by leaping in an adjacent site, the

broken bonds may occur between

connecting the oscillation to the

surrounding components.

Eq 1 : The mean residence time of the ion

placed in potential well (Okada et al.,

2015)

𝜏 = 𝑣−1exp(𝐸𝑍

𝑅𝑇)

𝜏 = mean residence time of the ion placed

in potential well

v = the frequency of oscillating ion

trapped in the potential well

Z = the coordination number of the mobile

ion

E = the bond energy between surrounding

ion and mobile ion

R = gas constant

Refer to the equation above, it

can be seen that mobile ion, the

coordination number and the bond energy

is very dependent with the mobile ion

occupies and vary relying on the changes

of temperature exposed. The higher

temperature changes tends to triggered

mobile ion faster out of their potential

well. As (Kang, Chung, Doh, Kang, &

Han, 2015) reported that as the decreasing

temperature applied is decrease on

polarization reaction, the frequency of the

bulk conduction on Li-ionic

proportionally decrease and the resistance

is increase inversely. As it depicted on .

2

In pursuance of (Zhang, 2006)

state that thermodynamically solid-oxide

fuel cell (SOFC) shows the appearance

between anode and cathode reaction

which is triggered by electrochemical

potential of oxide ions equivalence with

oxide-ion conducting electrolyte and

potential difference. Hence it generates

relationship to the net free energy change

of reaction, ΔG (Free-Gibbs).

Furthermore, the Free-Gibbs energy leads

to describe electrochemical reaction

kinetics of the SOFC. However Zhang on

the other hand also mentioned that

electrode kinetics may distinct from

kinetics in the influence potential drop in

Helmholtz double layer at the electrode

interface.

However, temperature gradient is

not the only major factor that affecting the

ionic conductivity of the solid ionic

materials. To reinforce that statement

(Hammoud, Vaucher, & Valdivieso,

2015), they reported that ionic

conductivity is widely depends on the type

of phases, volume of fraction, the

geometry of the pores, the grain

boundaries influence including their

distribution and the synthesis of solid

ionic materials itself. In accordance with

(Yao et al.), whilst discontinuities in

structural solid ionic material exists, it

creates a narrow charged zone is named

space charge zone at the interface.

Therefore, when certain critical

temperature are exceeded it generates

higher mobility of ion in the space charge

free zone.

1.1 Factors That Affecting Ionic

Conductivity

Dielectric strength

Dielectric strength relates to the

permittivity, it has similar objective to

keep the material become insulator. High

dielectric strength leads to high energy

barrier that block ions hopping the gap

(X.-C. Wang, Lei, Ang, & Lu, 2013).

Which is hence the addition of dopant

concentration on the bulk material and

elevated are applied to overcome the

energy barrier and increase the kinetic

energy the ion in order to leap the gap and

creates ionic conductivity (Kong et al.,

2015).

The temperature difference

Temperature is major factors that

arise ionic conductivity inside the solid

ionic materials. In order to generate ionic

conductivity of the solid ionic

conductivity, elevated temperature need to

be applied. Since solid ionic materials are

a kind of dielectric materials, the external

force need to generate the ion mobility of

the ion, one or the other way temperature

is one of the way to generate the ion

hopping the energy barrier (Hammoud et

al., 2015).

(Bechibani, Litaiem, Ktari,

Garcia-Granda, & Dammak, 2014)

reported that the increasing temperature in

the solid oxide leads to transformation on

phase and generate potential difference in

it. Also according to (Lin, Kuo, & Chou,

2013), the possibility of increasing

structural defects may occur when higher

temperature is introduced. Elevated

temperature generates higher

concentration of lattice defects locally and

distorts the lattice structure, which one

way around leads to the average lattice

expansion followed by molar ratio

increment.

An elevated temperature also

corresponds to dopant behavior inside

solid oxide. In addition, (Le, Zhang, Zhu,

Zhai, & Sun, 2013) reported that the

capacity of dopant to the bulk component

at least couple orders magnitude lower

than grain boundary at lower temperature,

oppositely when the higher temperature

induced, the grain boundary between bulk

and dopant are no longer separated.

Concentration of dopant

Dopant plays important role to

achieve a ionic state of ionic conductivity

on solid ionic materials. Dopant creates a

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structural defects in microstructure of

solid oxides and create ions move and

mobile when it diffuse to bulk material (H.

Wang, Li, Ternström, Lin, & Shi, 2013;

Yuan et al., 2014). As (Cao et al.,

2015)reported that the increment

concentration of dopant may increase the

occupancy and inside the acceptor

microstructure and enhance the ionic

conductivity of materials. The increasing

occupancy site by the dopant may

introduce under high temperature as

reported by (Lin et al., 2013)

According to (Meng et al., 2014),

the doping process leads formation of

space charge region in vicinity of the

heterogeneous interface which generate

ionic conductivity and oxide ions

transportations. In pursuance of (Le et al.,

2013) reported that the doped alkaline

may exhibit high conductivity and good

chemical stability. On the other hands,

(Cao et al., 2015; Lin et al., 2013; Meng et

al., 2014) also reported that the increment

of concentration acceptor may reduce the

ionic conductivity by inhibit grain growth

of dopant which by the other line, grain

growth of dopants may change the

dielectric properties of the acceptor with

low dielectric loss (Yuan et al., 2014).

Change of Phases

Since the dopant are added to the

bulk materials, the dopants create

discontinuities, impurities inside bulk

materials which generate a space free

charge in heterogeneous interface of the

microstructure and eventually enlarge the

ionic conductivity of solid.

(Meng et al., 2014) reported that

continuous phase of the dopant to the

acceptor may electronic conductivity and

allows ion transport which enhance the

ionic conductivity of the solid ionic

materials.

Volume fraction

(Völker & McMeeking, 2012),

reported that the volume fraction

correlates to the ion and electron particle

size and the performance of solid ionic

fuel cell on their ionic conductivity. The

volume fraction (𝜑) of solid oxide is very

dependent to the particle size ratio (R)

between ions and electrons. it can be seen

as the equation below

Eq 2 : the relationship between particle

size ratio and volume fraction (Völker &

McMeeking, 2012)

𝑟𝑒𝑙 =𝑟𝑚𝑒𝑎𝑛

(1 − 𝜑𝑒𝑙𝑚𝑎𝑥)𝑅 + 𝜑𝑒𝑙

𝑚𝑎𝑥 𝑎𝑛𝑑𝑟𝑖𝑜𝑛

= 𝑅𝑟𝑒𝑙 Whereas,

R = 𝑟𝑖𝑜𝑛

𝑟𝑒𝑙

On the other hand,

𝑟𝑚𝑒𝑎𝑛 =𝜑𝑖𝑜𝑛𝑟𝑖𝑜𝑛 + 𝜑𝑒𝑙𝑟𝑒𝑙

Where,

𝑟𝑚𝑒𝑎𝑛 = mean particle radius

𝑟𝑒𝑙 = electron particle radius

𝑟𝑖𝑜𝑛 = ion particle radius

𝜑𝑖𝑜𝑛 = volume fraction of ion

𝜑𝑒𝑙 = volume fraction of electron

R = Particle size ratio

From the equation above, it can

be seen that the volume fraction plays

important role to the approximate the

performance of solid oxide fuel cell. By

achieving the electronic and ionic particle

size, their size ratio followed by the

volume fraction we can predict and

enhance the performance of fuel cell

(Völker & McMeeking, 2012). Volker and

Mcmeeking also reported that the volume

fraction arise maximum power density

that occur due to the activation and ohmic

loss during their trades-off. In brief, as

lower volume fraction of electron will

generates the higher ionic conductivity of

solid ionic materials.

On the other hand, in accordance

with (Meng et al., 2014), the low of

volume fraction also may induce non-

continuous phase which is hence create a

heterogeneous microstructure.

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Geometry of pores

The geometry of pores greatly

corresponds to the dielectric properties,

change in volume fraction which is

proportionally relates to the ionic

conductivity as reported (Naterer, Tokarz,

& Avsec, 2006).

By seeing (subchapter 2.3), it

can be seen that in micro-pores, the

Knudsen coefficient is affected by the

shape and geometry of the pores which

also related to the permittivity, diffusivity

rate and the velocity of the mobile ion with

the response of different pores geometry.

In addition, geometry of pores

greatly affects the mean radius of pores

which correspond to the volume fraction

of particles which creates characterization

in grain growth by difference of pores in

geometry(Mozalev et al., 2014; Naterer et

al., 2006; Sedighi & Thomas, 2014).

2 Effect Relationship of Solid Ionic

Materials Properties to Ionic

Conductivity

2.1 Permittivity

By terms, permittivity is another

name of dielectric constant. In dielectric

materials, materials are characterized as

insulator which have high dielectric

strength (Kotlan et al., 2015).

Furthermore, the permittivity also

indicates the gap between anion and

cation, the longer gap between anion and

cation, the lower permittivity of the

materials (X.-C. Wang et al., 2013).

However, the expression of ratio between

material permittivity and vacuum

permittivity is called relative permittivity.

Refer to (Bechibani, Litaiem, Ktari,

Zouari, et al., 2014), when the relative

permittivity decreases, the large ionic

conductivity may arise due to the fast

mobility of ions hopping through the

barrier (gap between anion and cation)

On the other hand, permittivity

has invers correlation to dielectric

strength. According to (Kotlan et al.,

2015), when the dielectric strength

decreases, high permittivity will arise and

allows ionic mobility that may induce

current which is produced by increasing

ionic conductivity. The effect of

increasing coarse grain generates

reduction on dielectric strength. This may

occurs at the elevated temperature. On the

other hand, the permittivity corresponds

the dissipation factor (tan𝛿) (Bechibani,

Litaiem, Ktari, Zouari, et al., 2014). Since

dissipation factor is one of main properties

of the dielectric materials, it associates

with the ion mobility when introduce to

the elevated temperature. The dissipation

factor acts similarly with permittivity, it

decreases when higher temperature

applied on reaction(Bechibani, Litaiem,

Ktari, Zouari, et al., 2014).

However, in a low temperature

environment, there is no clue that ion may

passes through the permittivity (high

permittivity) which means the dielectric

strength applied energy called barrier

energy (Kong et al., 2015). Hence the

hopping possibility of atom is relying on

the kinetic energy arise by the surrounding

atom. Thus simply as (Kong et al., 2015)

conclude that the probability of hopping

may be achieved if the kinetic energy is

bigger than the barrier energy. In addition,

elevate temperature to higher temperature

and adding concentration of dopant is one

of variables that may increase the kinetic

energy.

2.2 Polarization

In ionic solid, covalent character

may produce after polarization of the ionic

bond inside the solid ionic materials. The

electron clouds of anions are distorted

toward the cation by influence of external

forces. There are 3 types of polarization

may arise in solid ionic materials which

are activation, ohmic and concentration

polarization (Zhang, 2006).

Activation polarization

In electron transfer process

within the solid electrolyte there is

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existence of electrode reaction. In

accordance with (Okada et al., 2015;

Zhang, 2006), the reaction of kinetic

movement of the charge-transfer reaction

to leap the obstacle at the

electrode/electrolyte interface is called

activation polarization. It is the slowest

step where the activation energy in the

form of potential difference is required to

proceed the reaction.

Beside of that, the activation

polarization has significance influence to

the whole reaction and the rate of the

reaction as well. It is related to Tafel

equation which can be seen as it followed

(Zhang, 2006):

Eq 3 : The Tafel equation (Zhang, 2006)

η = ß log (i/io)

where, η = over potential

ß = tafel slopes

log I = linear for both anodic and

cathodic polarization

io = exchange current density

In addition, (Shen, Yang, Guo, &

Liu, 2014) also mentioned on their

research polarization model for a SOFC

with a mix ionic electronic conductor

(MIEC) that whilst the current density

increases is proportionally linear to the

electrolyte ionic conductivity the over

potential of the reaction shift inversely

which decreases.

Ohmic polarization

Ohmic polarization impedes or

resist the motion of electrical charge

between electrolyte, electrode materials,

current collectors and contact between

particles. It is described by Ohm’s Law,

when resistance is high, the current is

inversely reduced. In pursuance of

(Völker & McMeeking, 2012), the ohmic

polarization induce a loss of mechanism

of ionic reaction, the activation loss occurs

for the hydrogen oxidation reaction in the

anode and reduction reaction in cathode.

Hence the ohmic polarization may

indicates the performance of the solid

oxide fuel cell.

According to (S. J. Kim, Kim, &

Choi, 2015), the large ohmic polarization

that obstructive to high electrolysis

current is showed by the presence of thick

electrolyte (≥-300μm). Hence, this leads

to the separation of anodic from cathodic

polarization under high electrolysis

current is difficult. In addition, (Joneydi

Shariatzadeh, Refahi, Abolhassani, &

Rahmani, 2015; S. J. Kim et al., 2015), to

overcome the ohmic loss, high operating

temperature shall be necessarily applied to

the process reaction (9000 C-10000C).

Concentration polarization

Concentration polarization is

definitively related to the existence of

slow and fast diffusion process occur. The

concentration of mobile ion will decide

the ionic conductivity of that induce

electrical charge. In line with that (Zhang,

2006) stated that the limited mass

transport capability of diffusion of active

ion and from the electrode surface to

sustain the reaction by replace the reacted

materials that generate concentration

polarization. In line with that, a voltage

drops generating by diffusing mass

transport is required to achieve

concentration polarization (Naterer et al.,

2006). Briefly, concentration polarization

occur when Helmholtz double-layer exist

on the electrode by electrolyte (Kang et

al., 2015).

Furthermore, in research

reported by (Naterer et al., 2006), in SOFC

when the electrical current undergo

through electrodes, the gas steam bulk

pressure is greater than the partial pressure

at the catalyst layer reaction, whereby

leading to tendency of diffusive

irreversibility as so called concentration

polarization. Based on Arrhenius Law that

stated by (Kang et al., 2015) the

conductivity of solid ionic materials are

greatly related to the concentration of the

electrode with a function of temperature

6

(Kang et al., 2015; Naterer et al., 2006;

Okada et al., 2015).

Eq 4 : Arrhenius Law (Okada et al., 2015;

Zhang, 2006)

𝜎 = (𝑧𝑒)2(𝐶)(𝑙)2𝑣0

6𝑘𝑇exp(

−𝐸𝑎𝑘𝑇

)

The Arrhenius Law above shows

that the concentration difference generate

changes to conductivity of electrode (𝜎). While on the other hand, e and l are

indicate the electronic charge that jump

over distance between two stable diffusion

sites. In addition v0 and Ea denote the

attempting frequency of the ion and the

activation energy for diffusion,

respectively. Even more (Kong et al.,

2015) stated that the lower temperature

changes with large amount of activation

energy leads to the increases rate of

diffusivity. It can be expressed on the

equation below.

Eq 5 : Einstein Equation (Kong et al.,

2015)

𝐷 = lim𝑡→∞

(𝑟2(𝑡))

2 dim(𝑡)

Where,

D = Diffusivity

𝑟2(𝑡) = The mean square displacement

of ion over time

Dim = Dimension of diffusion

t = Time

2.3 Ionic porosity

As we discussed it previously

above, the porosity is one of elementary

factor that affecting ionic conductivity of

solid ionic materials. In pursuance of

(Hammoud et al., 2015; E. S. Kim, Park,

& Yoon, 2003; Shan & Yi, 2015) stated

that the porosity is very dependent to the

sintering temperature. It is reported that

the sample they were taken have

significantly decreasing of porosity during

high temperature approached on sintering.

Kim et al. also mentioned that theoretical

permittivity (dielectric constant) could be

achieved from the measured permittivity

by porosity modification

Eq 6 : Wiener’s equation (E. S. Kim et al.,

2003)

Kw = Kmea (1 + 1.5P)

Whereas

Kw = theoretical dielectric constant

Kmea = Measured dielectric constant

P = Porosity

For dielectric constant of

polycrystalline and spherical pores with 3-

0 connectivity, Kim at al. used equation

follows:

Eq 7: Maxwell’s Equation (E. S. Kim et

al., 2003)

𝐾𝑀 = 𝐾2(1 +3𝑉𝑓(𝐾1−𝐾2)

𝐾1+2𝐾2−𝑉𝑓(𝐾1−𝐾2))

Whereas

Km K1 K2 = Dielectric constant for mixture

Vf = Volume fraction of dispersed

phase

On the other hand, if K2 >> K1

(=1), use the equation as follows:

Eq 8 : Rearrangement Maxwell’s

Equation (E. S. Kim et al., 2003)

𝐾𝑀 = 𝐾2(2 + 𝑉𝑓 − 3𝑉𝑓

2 + 𝑉𝑓)

However, if the square porosity

occur we may neglect the fraction of

dispersed phase square (Vf2), it can be

expressed as

7

Eq 9 : Rearrangement Maxwell equation

corresponded to square porosity (E. S.

Kim et al., 2003)

𝐾𝑚 = 𝐾2(4−2𝑉𝑓−4𝑉𝑓+4𝑉𝑓

2

4−𝑉𝑓2 ) ≅ 𝐾2(

2−3𝑉𝑓

2)

𝐾𝑀 =𝐾𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑(2

2 − 3𝑃)

Where Km and K2 are

corresponded to K measured (dielectric

constant measurement). On the other

hand, KM is the theoretical dielectric

constant. In brief that the higher porosity

size will leads to lower quantities of the

permittivity. (Naterer et al., 2006)

reported that porous electrode involved on

the diffusion rate of the reaction, the

maximum diffusion rate and high current

densities, it may reduce concentration at

the reaction site. From the (eq. 1), since

we obtained the mean residence of time of

the ion placed in potential well, it may

generate the expression of diffusivity that

related to the diffusion coefficient (Okada

et al., 2015) as it follows:

Eq 10 : Diffusion Coefficient

(Diffusivity) As a Function of 𝜏(Okada et

al., 2015)

𝐷 =𝑔𝑙2

𝜏

Where,

D = Diffusity

g = Geometrical factor

l = Jump distance

Whereby it also can be expressed

by Nernst-Einstein since it has a

correlation point on D (Okada et al., 2015)

Eq 11 : Ionic conductivity by Nernst –

Einstein equation (Okada et al., 2015)

𝜎 = (𝑧𝑒)2(𝑛)(𝐷)

𝑓𝑘𝐵𝑇

Where,

f = the correlation factor

n = the mobile ion

it can be seen theoretically that

the diffusion and ion mobility may effect

proportionally to the ionic conductivity.

However, the diffusion and the ion

mobility have invers correlation. (Naterer

et al., 2006) also reported that the

polarization increase which is

proportionally to ion mobility whilst the

electrode pores decreases.

In line with that, according to

(Sedighi & Thomas, 2014), porosity is

divided into two major categories which

are macro-porosity and micro-porosity.

Macro-porosity consist between

aggregates, on the other hand micro-

porosity consist between particles.

Generally, diffusion that

occurred due to porous electrode affect the

catalytic reaction which involves both in

the macro-porosity as it is called ordinary

diffusion and in the micro-porosity which

is called Knudsen diffusion (Naterer et al.,

2006). Both diffusion may simultaneously

occur within the fuel cell as long as the

porous electron exist. Beside of that,

(Zhang, 2006) also reported several types

of diffusion in small porosity which are

Knudsen, Fickian, transition and surface

diffusion.

Knudsen diffusion

In Knudsen diffusion, higher

frequency of molecules collide with the

pore walls than other molecules are

obtained (Naterer et al., 2006; Zhang,

2006). This phenomena leads to reduction

of mass transfer rates. In line with that, the

kinetic theory by means of associating the

molecule mean free path with diameter of

the pore may derive the Knudsen

coefficient. It can be express by the

equation below:

Eq 12 : Knudsen coefficient (Naterer et

al., 2006)

8

Dk = 𝑣𝑟

6

Where,

v = the molecular velocity

r = the mean pore radius

Eq 13 : the velocity for round, small,

straight pores (Naterer et al., 2006)

v = 587√𝑇

𝑀𝐴

where,

T = exergy destruction due to ohmic

heating

MA = the molecular mass of the porous

solid

Eq 14 : the Knudsen coefficient for

porous solid (Naterer et al., 2006)

Dk = 97r√𝑇

𝑀𝐴

Eq 15 : the approximation mean pore

radius (Naterer et al., 2006)

𝑟 = 2휀𝐴𝑆𝐴𝜌𝐴

where,

휀𝐴 = porosity of material A

𝑆𝐴 = surface area of the porous solid on

material A

ρA = bulk density of the solid electrolyte

A

From the equations above it can

be seen that the diffusion only occurs in

pores space. The Knudsen coefficient

indicates for the porosity of the electrode

and the tortuous path of molecular motion.

However, for the effective Knudsen

coefficient it can be approximate by the

ration solid porosity and the tortuosity of

the molecule path (Kong et al., 2015;

Naterer et al., 2006) which can be seen

from the equation bellow:

Eq 16 : the effective Knudsen coefficient

(Naterer et al., 2006)

Dk(eff) = Dk (𝜀

𝜉)

where,

𝜉 = The tortuosity of molecule path

Moreover, (Zhang, 2006) stated

that Knudsen diffusion occurs in diameter

range of pores of 2-50 nm and the mean

free path is relatively longer compared to

size of pore.

Molecular or Fickian diffusion

In pursuance of (Zhang, 2006),

the ion transport that occur when pore size

is relatively longer than the mean free path

which can be described by fick’s law

below:

Eq 17 : Fick’s law equation (Zhang, 2006)

J = -Dt∇𝑐

where,

J = the mass flux

Dt = Fickian diffusions coefficient

∇𝑐 = the concentration gradient

The Fick’s law is clearly shows

that the macroscopic flux of molecule

relates to the transport diffusivity in the

system by the force extracted from the

concentration. The movement of particles

are random and independent on its

previous motion (Zhang, 2006).

Transition diffusion

According to (Zhang, 2006), the

transition diffusion is the blend behavior

and properties of both Knudsen and

Fickian deffusion. The pore generates to a

reduction in self-diffusivity, yet it does not

affect the molecule transport diffusivity.

Surface diffusion

Commonly, surface diffusion is

used to explain the type of pore diffusion

in which solutes adsorb on top of surface

of pore which describe the interaction

between the surface and molecules which

9

occur whilst ion hope from one site to

other site, stated by (Zhang, 2006).

2.4 Grain size

Obtaining from (Faryna,

Bobrowski, Pędzich, & Bućko, 2015), the

bulk diffusion and the grain boundary

diffusion of oxygen ions are two modes to

exhibit ionic conductivity. The research

that they have done showed that the

reduction of grain boundary density is

obtained at the higher sintering

temperature. It reinforced by (Yao et al.)

that stated the increasing of grain

boundary with the elevated temperature.

Furthermore the coarser

formation of microstructure also are

obtained during the elevated temperature

due to higher migration of grain boundary

during increasing temperature (Faryna et

al., 2015). This phenomena occur because

at higher temperature, driving force and

diffusion rate arise the migration of the

grain boundary. However, more uniform

microstructure also occur when sintering

process undergoes higher temperature

(Faryna et al., 2015). On the other words,

there are lesser fraction of largest grains

and smallest grains during high

temperature sintering. In addition, (Yao et

al.) mentioned that the reduction of

fraction occurred at higher temperature

due to the increasing interaction between

phases in the structure solid. The grain

boundaries densities are calculated as the

ratio of the grain boundaries size (area) to

the analyzed volume, whereas the grain

boundaries size are approximated by the

total areas within the analyzed solid

materials (Faryna et al., 2015).

In spite of that, the grain size

affects the grain boundary conductivity

(fig 5), as the size decrease the grain

boundary decrease which correlate to

reduction of grain boundary respectively

at elevated temperature which are

inversely correlated to the component of

ionic conductivity(Faryna et al., 2015). In

brief, it simply concluded that the elevated

temperature the grain density reduce

which is lowered its dielectric strength

that may exhibit the increasing ionic

conductivity (Kotlan et al., 2015).

2.5 Impurities

When solids having an impurities

inside their structure, meaning that the

solid materials having interstitial sites

inside their structure or we can name it as

discontinuities or defects which is hence

ion may move or hopped around the

structure of solid ionic materials. Since the

breakdown of symmetry matrix which is

caused by discontinuities, ionic

conductivity of electrode-electrolyte may

be determined by their interfaces (Lu, Xia,

Lemmon, & Yang, 2010; Meng et al.,

2014). As reported by (Yao et al.), at the

interface, the space charge zone which is

narrow exist and on the other hand the

defects concentration in space charge zone

much higher near the boundary than in

bulk. On the other words, the high

conductivity pathways for ionic

transportations are supplied by the

interfaces.

According to (Meng et al., 2014),

much higher concentration of dopant

concentration may indicate the defect

association to the strength of ionic

conductivity. Since the dopant exhibit a

mobile ion which is creating unstable

formation, by the increasing concentration

of dopant may increase the ionic

conductivity of the solid materials(Meng

et al., 2014; Shan & Yi, 2015).

Meanwhile, the common defects that

produce is point defects which are named

as Schottky and Frenkel defects. When the

one of electrode whether anion or cation is

missing from the lattice is categorized as

Frenkel defect and when both electrode

are missing at the sampe lattice at the same

time is categorized as Schottky

defects(Oda, Weber, & Tanigawa, 2016).

10

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