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Property models and theoretical analysis of novelsolid oxide fuel cell with triplet nano-compositeelectrode

Meina Chen, Ce Song, Zijing Lin*

Department of Physics & Collaborative Innovation Center of Suzhou Nano Science and Technology,

University of Science and Technology of China, Hefei 230026, China

a r t i c l e i n f o

Article history:

Received 25 October 2013

Received in revised form

4 January 2014

Accepted 6 February 2014

Available online xxx

Keywords:

Electric conductivity

TPB length

Hydraulic diameter

SOFC

Multi-physics modeling

Microstructure optimization

* Corresponding author. Tel.: þ86 551 636003E-mail address: [email protected] (Z. Lin)

Please cite this article in press as: Chentriplet nano-composite electrode, Ij.ijhydene.2014.02.036

0360-3199/$ e see front matter Copyright ªhttp://dx.doi.org/10.1016/j.ijhydene.2014.02.0

a b s t r a c t

Triplet nano-composite electrodes are actively examined experimentally, but there is a

shortage of theoretical study. Theoretical models are helpful for understanding the ex-

periments and provide guidance for design optimization of the novel electrode. Here new

models for computing the electrode electronic and ionic conductivities, TPB length and

hydraulic radius are presented. The novel properties determined by the models are used in

a multi-physics numerical model that couples the intricate interdependency among elec-

tric conductions, electrochemical reaction and gas transport in SOFC. The theoretical IeV

relations and hydraulic radius are in good agreement with the experiments, vali-

datingtheproposed property models. The property models are then used to examine the

influence of microstructure and material composition. The results show that: (i) Larger

core-particle size and smaller nano-particle size are helpful for improving electrode

properties; (ii) The required nano-particle loading is determined by the desired electronic

conductivity instead of the desired TPB length.

Copyright ª 2014, Hydrogen Energy Publications, LLC. Published by Elsevier Ltd. All rights

reserved.

Introduction

The concept of nano-particle composite electrode is seen as

an effective way to develop highly active and advanced elec-

trode for solid oxide fuel cells (SOFCs) and has attracted

increasing attention recently [1]. Various techniques such as

infiltration [2,3], electroless coating [4,5] and Pechini-type

polymerizable complex method [6,7] have been used to

fabricate nano-particle composite electrodes with coreeshell

structures. All of these nano-composite electrodes may yield

low polarization resistances for SOFCs operated at reduced-

temperature (<700 �C) due to the enlargement of triple-

45; fax: þ86 551 63606348.

M, et al., Property modenternational Journal

2014, Hydrogen Energy P36

phase boundaries (TPBs) [3,8e11]. However, the long-term

stability is still a potential problem especially for anode with

just one kind of metal nano-particles. The metal nano-

particles such as Ni are prone to coarsening that causes

reduced TPBs and increases polarization resistance [12] and

thereby causes the anode performance degradation.

To curb the anode degradation process and to further

improve the electrode performance, anodeswith binary nano-

particles such as nano-Cu/nano-CeO2/YSZ anode [13] and

nano-Ni/nano-YSZ/YSZ anode [6] have been developed.

Together with the backbone material such as YSZ, an elec-

trodewith binary nano-particles involves threematerial types

and is therefore named as a triplet nano-composite electrode.

.

ls and theoretical analysis of novel solid oxide fuel cell withof Hydrogen Energy (2014), http://dx.doi.org/10.1016/

ublications, LLC. Published by Elsevier Ltd. All rights reserved.

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i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n en e r g y x x x ( 2 0 1 4 ) 1e72

As reported [6], the typical triplet nano-composite anodes can

offer many advantages such as: (1) microstructural stability,

for example, aging of the nano-Ni/nano-YSZ/YSZ anode is

controlled by a fine YSZ skeleton and the coherence between

co-conjugatedNi and YSZ; and (2) TPB expansion by the binary

nano-conjugation. These advantages imply that the triplet

nano-composite anode has a great potential to become an

excellent and practical electrode in the future. Consequently,

the design of triplet nano-composite electrode is actively

examined experimentally [1].

Compared to the intensive experimental activities, there

are very few theoretical studies. Tanner et al. [14] proposed a

model for an electrode with single-phase nano-particles as

consisting of regularly spaced corrugations of contiguous

electrolyte region, electro-catalyst, and porosity. Thismodel is

oversimplified with significant deviation from the actual

electrode structure with single-phase nano-particles and is

clearly inapplicable to the triplet nano-composite electrode.

Recently, Chen et al. [15e17] developed a theory for the elec-

trical conductivity that properly accounts for specific struc-

ture of the nano-composite electrode with one type of nano-

particles. The model yields quantitative agreement with the

experimental result on the dependence of the electronic

conductivity on the nano-particle loading. Though the basic

feature of the model of Chen et al. is reflective of the nano-

particle infiltrated electrode, a generalization of the model to

suit the specific structure of the triplet nano-composite elec-

trode is still required.

Here new expressions of electronic and ionic conductiv-

ities applicable to the triple phase nano-composite electrode

are described. The theory for computing the effective TPB

length is also given. In addition, a new model for calculating

the effective hydraulic radius that is essential for determining

the concentration polarization in nano-composite electrode is

proposed. These effective property theories are used in a

multi-physics numerical model to describe the overall IeV

performance of SOFC with nano-composite electrode. The

req ¼ ri cos a0

��1þ jnano

ð1� jnanoÞ1� fshell

nano

��1��

cos�2a0 � 13

�þ 13

�1=2

(6)

theoretical predictions are compared with the available ex-

periments and consistent quantitative agreements are found.

The new models are then used to discuss the dependences of

electrode properties on material compositions.

Theory

The triple phase nano-composite structure affects all the

essential electrode properties, including the electronic and

ionic conductivities, the effective TPB length and the hy-

draulic radius. Here models for these properties are discussed

first, followed by a summary of a multi-physics model that

utilizes these properties. The numerical method for solving

the multi-physics model is also briefly described.

Please cite this article in press as: Chen M, et al., Property modetriplet nano-composite electrode, International Journalj.ijhydene.2014.02.036

Effective properties

Effective conductivitiesUnlike a traditionalmicron-scale composite electrode thatmay

be viewedas a randompacking of spherical particles [18,19], the

nano-particles ina tripletnano-particlecomposite electrodeare

coated on the surface of core or backbone particles, as illus-

trated inFig. 1.However, if thecomplexofa coreparticleand the

nano-particles coated on the core particle surface is viewed as

an equivalent particle, the novel electrode may be viewed as a

random packing of equivalent particles. For that reason, the

traditional percolation theory based conductivity model is

applicable. For simplicity, the discussionhere is limited to cases

that the core particles are pure ionic conducting and one type of

thenano-particles is purely electronic conducting and the other

type of the nano-particles is purely ionic conducting.

Based on the physical picture of the coreeshell structure

and the percolation theory based conductivity model, math-

ematical analysis shows that the electronic conductivity sede

and the ionic conductivity sedi of the triple nano-composite

electrode may be expressed as:

sede ¼ s0

e

�r2i � r2eq

�ln½tan ðq0=2Þ�

reqri cos a0

�jshelle � PC

1� PCð1� feqÞ

�g(1)

sedi ¼ �2

ffiffiffiffiffiffiffiAB

pln tan ðq0=2Þ

r2eq lnffiffiffiffiffiffiA=B

pþcos q0ffiffiffiffiffiffi

A=Bp

�cos q0

� ð1� feqÞg (2)

A ¼ sshelli

�r2eq � r2i

�þ s0

i r2i (3)

B ¼ s0i r

2eq (4)

sshelli ¼ s0

i

�jshelli � PC

1� PC

�g

(5)

Here s0eðiÞ is the intrinsic electronic (ionic) conductivity of

the electrode material i. is the feature radius of the backbone

particle eq. is the radius of the equivalent particle, or the sum

of i and the thickness of the nano-particle shell. jshelleðiÞ is the

volume fraction of electronic (ionic) conducting nano-

particles in the shell 4. is the equivalent electrode porosity

for the random packing of equivalent particles and may be

calculated using the overall electrode porosity and the mate-

rial composition of the coreeshell structure. fshellnano refers to the

porosity for the infiltrated nano-particles shell jnano, is the

volume fraction of nano-particles in the solid phase of the

electrode q0. is the contact angle of equivalent particles and

may be determined by the geometrical relationship of

reqcosq0 ¼ ricosa0 [15], where a0 is the contact angle between

core particles aswell as between nano-particles and a0¼ 15o is

assumed [20,21] g. is a Bruggeman factor reflecting the effect




Fig. 1 e Schematic of triplet nano-composite electrode.

i n t e rn a t i o n a l j o u rn a l o f h y d r o g e n en e r g y x x x ( 2 0 1 4 ) 1e7 3

of tortuous conduction paths C. is the percolation threshold

which can be determined by the 2D and 3D percolation

thresholds P2C and P3

C:

PC ¼ P3C þ

P2C � P3

C

�$N�1=0:9 (7)

where N ¼ (req � ri)/2rnano is the number of nano-particle

layers coated on the core particle surface and rnano denotes

the average radius of nano-particles in the shell.

Effective triple-phase boundary lengthThe TPB length for the triplet nano-composite electrode ðlvTPBÞis the sum of the effective TPB length between the core and

nano-particles ðlvTPB;core�nanoÞ and the effective TPB length be-

tween the ionic and electronic nano-particles in the shell

ðlvTPB;nano�nanoÞ.The effective volumetric TPB lengths of the triplet nano-

composite electrode can be evaluated as:

lvTPB;core�nano ¼ 2prnanoðsin a0ÞnVcoreZcore;e�nanoPcorePe�nano (8)

lvTPB;nano�nano ¼ 2prnanoðsin a0ÞnVe�nanoZe�nano;i�nanoPe�nanoPi�nano

(9)

lvTPB ¼ lvTPB;core�nano þ lvTPB;nano�nano (10)

where nVcore ðnV

e�nanoÞ is the volumetric number density of core

particles (electronic nano-particles) in the electrode Zcor-

e,e_nano. is the coordination number between the core particle

and the electronic nano-particles in the shell Ze�nano,i�nano. is

the coordination number between the electronic nano-

particles and the ionic nano-particles Pcore. is the percolation

probability of core-particles and may be assumed to be 1 as


the core-particles are prepared as the backbone and likely to

be fully percolated Pe(i)�nano. is the percolation probability of

the electronic (ionic) nano-particles in the shell. For low nano-

particle loading, the network of nano-particles is approxi-

mately two-dimensional and one has [22]:

PeðiÞ�nano ¼ jshelleðiÞ � PC

1� PC

!5=36

(11)

Hydraulic radiusThe hydraulic radius (or pore radius), rg, is an important

parameter for gas transport in the porous electrode structure

[23]. The traditional theory for the hydraulic radius [21] as-

sumes a random packing of spherical particles. However, the

nano-particles are not randomly distributed with the core-

particles but instead coated on the core-particle surfaces.

Therefore, it is only reasonable to apply the traditional hy-

draulic radius model to the equivalent particles as the elec-

trode may be viewed as a random packing of the equivalent

super-particles (Fig. 1):

rg ¼ 2req3ð1� feqÞ (12)

Considering the LSMnano/YSZnano/YSZcore cathode re-

ported in Ref. [7], it involves the following experimental pa-

rameters: 39 nm for the radius of core-particle, 15 nm for the

radius of nano-particle jcore ¼ 42%, for the volume fraction of

the core particles and 4 ¼ 45% for the total porosity of the

electrode. Assuming the porosity of the shell layer to be

fshellnano ¼ 26%which corresponds to a close packing of the nano-

particles in the shell, the radius for the equivalent super

particle req is calculated to be 59 nm. The porosity for the

equivalent super-particles may be determined as

feq ¼ f� ð1� fÞ � ð1� jcoreÞ � fshellnano=ð1� fshell

nanoÞ and is calcu-

lated to be 34%. According to Eq. (12), the hydraulic diameter

(pore size) of the triplet nano-composite electrode is found to

be 119 nm. The result is in very reasonable agreementwith the

experimental data of 135 nme156 nm. In the other hand, the

hydraulic diameter of the LSMnano/YSZnano/YSZcore cath-

ode is calculated to be 49 nm by the traditional hydraulic

radius model [21], which is very different from the experi-

mental result [7]. The much improved agreement with the

experiment by using Eq. (12) strongly favors our hydraulic

radius model.

Multi-physical numerical model

The above models are used to calculate the new properties of

triplet nano-composite electrodes. Parameters required for

computing the new properties are chosen to correspond to the

experimental data [6] as much as possible. When the required

parameters are not specified in the experiment, e.g., the

electrode porosity and the sizes of LSM and YSZ particles in

the cathode [6], the same data set used in Ref. [20] of a typical

standard cell is adopted here.

These property data are used in a multi-physics numerical

model for the triplet nano-composite electrodes. The multi-

physics model couples the intricate interdependency among

ionic conduction, electronic conduction, electrochemical





reaction and gas transport. For the brevity of this paper, the

governing equations, boundary conditions and model geom-

etry settings as well as other required technical details are

omitted here, but may be found in Refs. [24e26]. The finite

element commercial software COMSOL MULTIPHSICS�

Version 3.5a [27] is used in the present study to solve the

coupled partial differential equations of electronic, ionic and

gas transport macro-models. Structured mesh elements were

used and consisted of 6800 rectangles with 109347 degrees of

freedom. The direct solver (UMFPACK) was used to solve the

coupled partial differential equations with a relative conver-

gence tolerance of 1e�6.

Results and discussion

This section first describes the validation of the proposed

models. The validated models are then used to discuss some

effects of microstructure and material compositions on the

properties of the triplet nano-composite electrode.

Model validation

Direct comparisons of the proposed models with the experi-

mental data are generally difficult due to the lack of the

experimental measurements. Other than the above

mentioned hydraulic radius, only indirect validation of the

models is possible.

Fig. 2 shows a comparison of the theoretical and experi-

mental IeV curves. The theoretical IeV curves are obtained

using our multi-physics numerical model with the electrode

propertydata computedby the abovedescribed expressions.As

shown in Fig. 2, very good agreement is obtained between the

theoretical and experiment IeV curves at different tempera-

tures. It should be pointed out that the IeV curve is strongly

influenced by the electrochemical activation polarization that

is highly dependent on the TPB length and the temperature.

Similarly, the ionic conductivities of theYSZ electrolyte and the

triple nano-composite electrode are also strongly temperature

dependent. The temperature dependence of the ionic ohmic

Fig. 2 e Comparison of theoretical and experimental IeV

curves at different temperatures.


polarization is very different from that of the activation polar-

ization. The IeV relation is also strongly affected by the ionic

ohmic polarization [28]. The good agreement between the

theoretical and experimental IeV curves at different tempera-

tures may be obtained only if the TPB length and the ionic

conductivity of the electrode are properly determined. In other

words, the comparison provides an indirect yet strong support

for our ionic conductivity and TPB models.

The electronic conductivity of a single phase nano-

composite electrode and its relatively weak temperature

dependence obtained based on a model similar to Eq. (1) have

been verified previously [15]. The difference between the

electronic conductivities of the single- and binary-phase

nano-composite electrodes is that the former

(w1000 S cm�1) is larger than the latter (w100 S cm�1) by one

order ofmagnitude. Consequently, the electronic conductivity

of a triplet nano-composite electrode may also substantially

influence its IeV curves. The good agreement between the

theoretical and experimental IeV curves implies that the

electronic conductivity is reasonably estimated by our model.

In fact, even if the concentration polarization that is often

relatively small is not properly determined, significant de-

viations from the experiments may be obtained, as to be

shown below. Therefore, the high quality agreement of the

theoretical and experimental IeV curves serves as a strong

support of our models.

As discussed in Section 2.1.3, our model for the hydraulic

radius of a triple nano-composite electrode provides a result

in good agreement with the available experiment. Even

though the concentration polarization is often small for

conventional electrode [28], a correct determination of the

hydraulic radius and the resulting concentration polariza-

tion is crucial for predicting the IeV relation of a cell with a

triplet nano-composite electrode. Fig. 3 shows a comparison

of the theoretical and experimental IeV curve at 700 �C. Thetheoretical model for Fig. 3 is the same as that for Fig. 2,

except that the blue theoretical curve in Fig. 3 is obtained by

Fig. 3 e Comparison of theoretical and experimental IeV

curves. Except for the hydraulic radius, all other

parameters are the same for the traditional model and our

model. (For interpretation of the references to color in this

figure legend, the reader is referred to the web version of

this article).





using the traditional theory for the hydraulic radius. Clearly,

the theory with the traditional hydraulic radius model pro-

vides a poor prediction of the IeV curve. The reason is that

the traditional model predicts a very small hydraulic radius

for the triplet nano-composite electrode, as discussed above

in Section 2.1.3. As a result, the traditional model signifi-

cantly overestimates the concentration polarization for the

triplet nano-composite. Consequently, it is important to use

the improved model for the hydraulic radius of triplet nano-

composite electrode.

Influence of microstructure and material composition onelectrode properties

There are a large number of adjustable parameters in a triplet

nano-composite electrode, e.g., the core particle radius, the

electrode porosity, the nano particle radius, the nano-particle

loading, the relative quantity of electronic and ionic nano

particles. Here we only discuss the influence of some typical

parameters on the electrode properties and leave a systematic

investigation on the parametric space for future studies.

The most commonly triplet nano-composite electrode is

manufactured by infiltrating both electronic and ionic nano-

particles onto an established porous electrode backbone

with a porosity of 60%w65% in the electrode [2]. We focus here

on discussing Ninano/YSZnano/YSZcore anodes with the

porosity of the initially porous YSZ backbone of 65%. The

volume fraction ratio of the of nano Ni and nano YSZ particles

in the shell is fixed at 6:4, a ratio that is beneficial for perco-

lation of both nano Ni and nano YSZ networks [21].

Fig. 4 shows the dependences of electronic conductivity

and TPB length on the nano Ni loading for two core-particle

radii of 150 nm and 500 nm. The radius of the nano particle

is 15 nm. As shown in Fig. 4a), the conductivity for ri ¼ 500 nm

is generally higher than that for ri ¼ 150 nm. A none-zero

conductivity is obtained for ri ¼ 500 nm at a Ni loading of a

little over 2.5%, while it requires almost 10% Ni loading for

ri ¼ 150 nm. Adequate conductivity (�100 S cm�1) is obtained

for ri ¼ 500 nm with a Ni loading of about 8%, but an almost

twice as much Ni loading is required for ri ¼ 150 nm. The re-

sults are quite understandable. For a given Ni volume fraction

in the electrode, the number of nanoNi particles coated on the

surface of a core particle is larger for a larger core particle due

Fig. 4 e Dependences of electronic conductivity and effective TP

electronic conductivity, b) TPB length.


to the given volume ratio of the core and nano particles.

Consequently, the percolation threshold of the coated Ni

particle network is reached with a smaller Ni loading for a

larger core particle. Similarly, the conductivity for a given Ni

loading is higher for a core particle backbone of larger size due

to the presence of more layers of conducting nano Ni layers.

A larger core particle radius is also beneficial for obtaining

longer effective TPB length, as shown in Fig. 4b). However, the

difference between the TPB lengths for different core particle

radii is relatively small when the Ni loading is above the

percolation threshold for the electrode with smaller core

particle. Themost important difference occurs when the nano

Ni loading is below or near the threshold required for the

smaller core particle electrode.

As the percolation threshold is linked to both the electronic

conductivity and the effective TPB length, overall it may be

simply said that a larger radius of core-particle is beneficial for

obtaining a desirable electronic conductivity at a low nano Ni

loading. However, it is worthy pointing out that there is a

theoretical upper-limit for the core-particle size to make the

model conclusion valid. That is, the thickness of the electrode,

H, should be larger than the correlation length x, of the

percolated packing of equivalent particles [29]:

H > x ¼ ri ji � pc

�0:9. Here ji is the volume fraction of the core-

particles in the electrode and pc ¼ 0.294 is the percolation

threshold of the electrode with a random packing of spherical

particles [21,30]. In other word, the above analysis is valid only

when the core-particle radius is smaller than its upper-limit,

rmaxi ¼ H

ji � pc 0:9.

Fig. 5 shows the dependences of electronic conductivity

and effective TPB length on the radius and loading of nano Ni.

The discussion here tries to correlate with the experiments in

Refs. [2,6,31]. Hence, the volume fraction of core particles in

the electrode is set at 58.33% as used experimentally. The

radius of nano Ni particle in the experiments may vary from

10 nm to 15 nm. Considering the agglomeration of Ni particles

that is often limited to an increase of Ni particle size of 50%

[32e35], larger radii of nano Ni particles up to 25 nm are also

included.

As shown in Fig. 5, at a given Ni volume fraction, both the

electronic conductivity and the effective TPB length decrease

with the increase of the Ni particle size. The trend is opposite

to the trend for the core particle radius, but the underlying

B length on the core-particle radius and nano Ni loading: a)




Fig. 5 e Dependences of electronic conductivity and effective TPB length on the radius and loading of nano Ni: a) electronic

conductivity, b) TPB length.


mechanism is quite similar. That is, due to the fixed volume

ratio of the core and nano particles for a given Ni volume

fraction, the number of nano Ni particles coated on the sur-

face of a core particle is smaller for larger Ni particles. As a

result, the conductivity as well as the TPB length for a given Ni

loading are lower for larger Ni particles.

As may be seen in Fig. 5, a sufficiently high TPB length

(�1013 m$m�3) may be obtained with a nano Ni loading

smaller than that required for obtaining an adequate elec-

tronic conductivity (�100 S cm�1). In other words, the required

Ni loading is determined by the desired electronic conduc-

tivity. Even though an adequate conductivity may be obtained

at a Ni loading of 9% for rnano¼ 10e15 nm, a nano Ni loading of

11% or more is required for long term operation that involves

Ni agglomeration. The extra nano-particle loading should be

considered when manufacturing practical cells.

Summary

A triplet nano-composite electrode may be viewed as a

random packing of equivalent particles with the equivalent

particle consists of a core particle and a wrapping shell of

nano-particles coated on the core particle surface. Based on

this physical picture, new theoretical expressions for

computing the electrode electronic and ionic conductivities,

effective TPB length and hydraulic radius are proposed. These

new models are used in a multi-physics numerical model to

predict the performance of SOFC with triplet nano-composite

electrode. The multi-physics model couples the intricate

interdependency among ionic conduction, electronic con-

duction, electrochemical reaction and gas transport. Numer-

ical simulation based on the proposed propertymodels shows

good agreement with the available experimental data,

demonstrating the validity of the models.

The verified models are further used to discuss the influ-

ence of microstructure and material composition on the

electrode properties. The analysis shows that: (i) The larger

the core-particle size or the smaller the nano-particle size is,

the higher the electronic conductivity and TPB length will be,

when all other factors being equal; (ii) An adequate electronic

conductivity requires a higher nano-particle loading than that


required for obtaining a sufficiently high TPB length. The

requiredNi nano-particle loading is determined by the desired

electronic conductivity.

Acknowledgments

The financial support of the National Basic Research Program

of China (973 Program Grant No. 2012CB215405), the National

Natural Science Foundation of China (Grant Nos. 11074233 &

11374272) and the Specialized Research Fund for the Doctoral

Program of Higher Education (Grant Nos. 20113402110038 &

20123402110064) are gratefully acknowledged.

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http://dx.doi.org/10.1016/j.ssi.2013.11.046


















































































property models and theoretical analysis of novel solid oxide fuel cell with triplet nano-composite...

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