Solid State Ionics xxx (2013) xxx–xxx

SOSI-13192; No of Pages 4

Contents lists available at ScienceDirect

Solid State Ionics

j ourna l homepage: www.e lsev ie r .com/ locate /ss i

A theoretical model for the electrical conductivity of core–shell nano-composite electrode of SOFC

Meina Chen, Du Liu, Zijing Lin ⁎Department of Physics & Collaborative Innovation Center of Suzhou Nano Science and Technology, University of Science and Technology of China, Hefei 230026, China

⁎ Corresponding author. Tel.: +86 551 63606345.E-mail address: zjlin@ustc.edu.cn (Z. Lin).

0167-2738/$ – see front matter © 2013 Elsevier B.V. All rihttp://dx.doi.org/10.1016/j.ssi.2013.11.046

Please cite this article as: M. Chen, et al., Sol

a b s t r a c t

a r t i c l e i n f o

Article history:Received 7 May 2013Received in revised form 12 November 2013Accepted 16 November 2013Available online xxxx

Keywords:Electronic conductivityNi-YSZ anodePercolation thresholdI–V relationNumerical modeling

Anano-composite electrodewith a core–shell structure is superior to the traditional design in terms of performanceand durability. But the theory for the electrical conductivities influenced by the core–shell structure is lacking,hindering its theoretical analysis and modeling activity. Combining the physical picture of the core–shell structureand the percolation based conductivity theory, explicit expressions for the ionic and electronic conductivities of thecomposite electrode are derived. The newmodel reveals the relationship between the ionic and electronic conduc-tivities of the electrode and its compositions. Numerical results on the percolating threshold, electronic conductiv-ities and I–V curves for theNi-coatedYSZ anode are shown to be in good agreementwith the experimental data. Theproposed conductivitymodelmaybeused for in-depth analysis anddesign optimization of the core–shell electrode.

© 2013 Elsevier B.V. All rights reserved.

1. Introduction

Nano-scale engineering of electrode structures via metal salt solutionimpregnation or infiltration attracts increasing attention as an effectiveway to develop highly active and advanced electrodes for solid oxidefuel cells (SOFCs) [1]. Many experiments show that a nano-compositeelectrode with a core–shell structure is superior to the traditional com-posite electrode in terms of chemical, electrochemical and mechanicalperformances [2–10]. For example, as a result of the high density of atriple phase boundary (TPB), a nano-composite electrode with a core–shell structuremay show low activation polarization at a reduced operat-ing temperature (b700 °C) of SOFCs [6–8,11,12]. Such a nano-compositeelectrode allows for low Ni loading in the Ni-YSZ anode (Ni-coated YSZanode) and enhances the long-term stability of the unit cell enduringthermal and redox cycling [2,3,13,14]. In other words, the design of anano-composite electrode with a core–shell structure is promising forthe development of a high performance SOFC operated at an intermediatetemperature.

To fully exploit the potential of the core–shell composite electrode, anin-depth understanding of various parameters affecting the electrodeproperties is required so that the electrode may be optimally designed.For the most conventional Ni-YSZ anode, the effects of Ni loading andporosity on the electrochemical property (TPB length) and mechanicalproperty have been investigated experimentally and theoretically[15–19]. In particular, as the electrical and ionic conductivities are

ghts reserved.

id State Ionics (2013), http://

important factors affecting cell performance, the effect of Ni loading onanode conductivity has been examined extensively experimentally andthe results for the core–shell composite anode are distinctly differentfrom that for the traditional composite electrode [3,6,9]. However, atheoretical electrical conductivity model that properly accounts for theexperimental results is still lacking. The current theoretical treatment[20] for the electrical conductivity still uses the theory that was devel-oped from a random packing of spherical particles. The model with arandom packing of spherical particles [16,21,22] has been successfulfor modeling the conventional composite electrodes [21–23] and maysometime be extended to describe porous structures with unique fea-tures [24,25]. However, the current conduction path described by arandom packing of particles is very different from that of the core–shellstructure which we are interested in. Consequently, applying the modelof a random packing of spherical particles to the core–shell nano-composite electrode is bound to produce erroneous predictions for theelectrode conductivities. For example, the model with a random packingof spherical particles requires about 30 vol.% Ni to obtain electronic per-colation, while a percolation is obtained experimentally at around9 vol.% Ni or less [3,5]. Clearly, it is necessary to develop an electrical con-ductivity estimationmodel that properly reflects the core–shell structureof the nano-composite electrode.

In this paper, a new theoretical model for the electrical conductivityof the specific core–shell structure of the nano-composite electrode isdescribed. Based upon the physical picture of the core–shell structure,combined with the useful portion of the traditional theory with arandom packing of spherical particles, explicit expressions for the elec-tronic and ionic conductivities are derived. Numerical results and discus-sion are presented to show that the model is capable of correctly

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

2 M. Chen et al. / Solid State Ionics xxx (2013) xxx–xxx

explaining the experimental results on the electronic conductivities ofthe nano-composite electrode with a core–shell structure as well as theassociated I–V relations.

2. Theoretical model

As shown experimentally [2,4,9,13], a nano-composite electrodewith core–shell structure such as Ni-coated YSZ anode may be viewedas a porous electrode composed of the random packing of equivalentsuper-particles. The equivalent super-particle consists of a large particle(YSZ) as its core, wrapped by some multi-layers of fine particles (Ni).The size of the core particle is often of the order of a few hundred nano-meters. The size of the fine particle is about an order of magnitudesmaller than the core particle. Such an electrode with the core–shellstructure is schematically illustrated in Fig. 1. The physical picturedepicted in Fig. 1 is the starting point for our theoretical model of theelectrode conductivity.

In principle, the core–shell structure allows for a variety of designs.For example, the core or coated particle may be an ionic or electronicconductor or a mixed conductor. Here we limit our discussion to thecases that the core particle is one purely ionic conducting materialsuch as YSZ and the coated particle is one purely electronic conductingmaterial such as Ni. A general model will be reported in near future.

As the core–shell structure is a random packing of the equivalentsuper-particles, the conventional percolation based theory of conductivitymay be used to calculate the electrical conductivity of the electrode, if theeffective conductivity of the equivalent super-particle is known. Denotingthe effective electronic and ionic conductivities of the equivalent super-particle as σe

eq and σieq respectively, the electronic (σe

ed) and ionic (σied)

conductivities of the electrode may be calculated as [16,21,22]:

σ ede ið Þ ¼ σ eq

e ið Þ 1−ϕeq� �ψeqPeq

h iγ: ð1Þ

Here ϕeq is the equivalent electrode porosity for the random packing ofthe equivalent super-particles and may be calculated using the overallelectrode porosity and thematerial composition of the core–shell struc-ture. ψeq is the volume fraction of the equivalent super-particles in thesolid structure and is 1 as there is no solidmaterial other than the equiv-alent super-particles. Peq is the percolation probability of the equivalentsuper-particles and is also 1 as the equivalent super-particles cannotfloat in pores and must be in contact with other equivalent super-particles in some way. γ is a Bruggeman factor reflecting the effects oftortuous conduction paths. γmay be viewed as an empirical parameterwith a typical value in the range of 1.5–3.5 [16].When the experimentaldata is available, γmay be determined by fitting. Therefore, according toEq. (1), the theoretical task of determining σe

ed and σied is the determi-

nation of σeeq and σi

eq.The equivalent super-particle of radius req consists of an ionic

conducting core particle of radius ri and an electronic conducting shellof thickness Dshell(=req − ri). For ionic conduction, a core particle hasto be connected to a neighboring core particle with a finite contactarea, or a nonzero contact angle α0, as shown in Fig. 1. The effective

Fig. 1. Schematic illustration of the core–shell str

Please cite this article as: M. Chen, et al., Solid State Ionics (2013), http://

value of α0 is dependent on the experimental technique. A value of15° is often used in theoretical modeling [16,21,22] and is adoptedhere in numerical calculation. The equivalent super-particle shouldtherefore be modeled as an incomplete spherical ball, with cuts inboth contacting ends, as shown in Fig. 1.

The electrical resistance of the equivalent super-particle for passingelectronic current may then be calculated as:

Reqe ¼

Zreq cos θ0

−req cos θ0

dzσ shell

e Se zð Þ

¼Zθ0

π−θ0

reqd cos θ

σ shelle πreq

2 sin2θ−πri2 sin2 α

� �

¼ 2ri cos α0

πσ shelle r2eq−ri

2� �

ð2Þ

where σeshell is the effective electronic conductivity of the shell layer to

be discussed later. Se(z) is a cross sectional area for the electronic cur-rent conduction that is contributed by the shell layer only. That is, Se(z) is the cross sectional area of the super-particle (πreq2 sin2θ) minusthe cross sectional area of the electronic insulating core particle (πri2 -

sin 2α). The geometrical relationship of req cos θ = ri cos α has beenused in the derivation.

For the electronic current conduction, the equivalent super-particlemay also be viewed as a homogeneousmaterialwith an overall effectiveelectronic conductivity of σe

eq. Based on such a representation, its elec-tronic resistance is calculated to be:

Reqe ¼

Z θ0

π−θ0

reqd cos θ

σ eqe πreq

2 sin2θ¼ −2 ln tan θ0=2ð Þ

σeqe πreq

ð3Þ

where θ0 = arccos(ri cos α0/req). Comparing Eqs. (2) and (3) gives,

σeqe ¼ σ shell

er2i −r2eq ln tan θ0=2ð Þ½ �

reqri cosα0: ð4Þ

The shell electronic conductivity may be determined by an expres-sion analogue to Eq. (1),

σ shelle ¼ σ0

e 1−ϕshelle

� �ψshelle P shell

e

h iγ ð5Þ

where σe0 is the intrinsic electronic conductivity of the coating material

in the shell. Assuming there is no dramatic nano-scale effect on conduc-tivity, the electronic conductivity of the corresponding bulk materialmay be used for σe

0. ϕeshell is the porosity of the shell layer. ψe

shell is thevolume fraction of the electronic conducting particles in the solidstructure of the shell layer and is 1 in the case discussed here. Peshell

is the percolation probability of the fine particles in the shell. Peshell

should be close to 1, except for a shell with less than one layer of thecoated fine particles.

ucture of a Ni-coated YSZ composite anode.

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

Fig. 2. Comparison of the theoretical and experimental electronic conductivities of aNi-coated YSZ anode.

3M. Chen et al. / Solid State Ionics xxx (2013) xxx–xxx

Combining Eqs. (1), (4) and (5), the electronic conductivity of theelectrode is found to be,

σ ede ¼ σ0

e

ri2−r2eq

� �ln tan θ0=2ð Þ½ �

reqri cosα01−ϕshell

e

� �P shelle 1−ϕeq� �h iγ

: ð6Þ

Similar to the above derivation and considering the fact that only thecore particles are ionic conducting, the ionic conductivity of the elec-trode is found to be,

σ edi ¼ σ0

iri � ln tan θ0=2ð Þreq � ln tan α0=2ð Þ � 1−ϕeq� �γ ð7Þ

where σi0 is the intrinsic ionic conductivity of the core material.

3. Results and discussion

As the ionic conductivities of the novel nano-composite electrodewith a core–shell structure and the conventional electrode are notexpected to be vastly different, the experiment mainly focuses on mea-suring the electronic conductivity of the novel electrode. Therefore, wefocus here on comparing the theoretical and experimental electronic con-ductivity first. Specifically, the experimental results on Ni-coated YSZ an-odes reported in Ref.[2] are compared below.

To make the comparison, it is necessary to convert the experi-mental parameters into the model quantities. The electrode composi-tions such as the overall electrode porosity (ϕed), the volume fractions(ψi, ψe = 1 − ψi) and the radius (ri and re) of the core ionic conductingparticles and the coated electronic conducting fine particles are knownexperimentally. Based on these data and somemathematical derivation,some model parameters may be determined.

For example, the radius of the equivalent super-particle may beexpressed as:

req ¼ ri � cos α0 � δ cos−2α0−1.

3

� �þ 1

.3

� �1=2 ð8Þ

where δ = 1 + ψe[(1 − ψe)(1 − ϕeshell)]−1.ϕe

shell refers to the porosity ofthe coated electronic conducting layer. As reported in Ref [2,4], a porosityof 24.5% for the NiO-YSZ electrode is changed to a porosity of 40.5% afterNiO is reduced to Ni. Considering that the porosity for the NiO layer isabout 2% [18], it may therefore be estimated that ϕe

shell = ϕNishell = 41.4%.

Similarly, the equivalent electrode porosity for the random packing ofthe equivalent super-particles ϕeq is calculated to be 23.7%. Choosing theexperimental estimate data of ri = rYSZ = 150 nm, re = rNi = 10 nmand Ni vol.%:YSZ vol.% = 40%:60%, one gets req = 200 nm. As the shellhas a thickness of req − ri = 50 nm that is 2.5 times the Ni particle sizeof 2rNi = 20 nm, the shell contains at least 2.5 layers of coated Ni parti-cles. Therefore, it is reasonable to use Pe

shell = 1.Based on the above data and the electronic conductivity of bulk Ni

metalσNi0 (S/m) = 3.27 × 106 − 1065.3 T [22], Fig. 2 shows the compar-

ison between the theoretical and experimental [2] results on the elec-tronic conductivity of the anode. The model used the Bruggeman factoras the only adjustable parameter. A value of 2.7 is usedwhich is quite rea-sonable considering the proposed range of 1.5–3.5 [16]. The excellentagreement between the theoretical and experimental conductivities atdifferent temperatures gives another support for the proposed model.

Even though it is reasonable to use Peshell = 1 for the above experi-

ment, it should be pointed out that the value of Peshell may be wellbelow onewhen the nano-particle loading is low and the shell containsless than one layer of Ni particles. In such cases, the percolation thresh-old should be determined based on a 2D scenario. Considering the per-colation threshold of 0.5 for a close packing 2D hexagonal lattice [26],the threshold of Ni vol.% for forming percolating Ni particles may be es-timated to be 10% for an anode with rYSZ/rNi = 15 and α0 = π/12. Theestimated threshold of Ni volume fraction is close to the experimental

Please cite this article as: M. Chen, et al., Solid State Ionics (2013), http://

observation of 9% [3], providing strong support for the reasonability ofour theoretical conductivity model. It should also be noted that thethreshold of Ni volume fraction is dependent on the value of rYSZ/rNi.The larger the value of rYSZ/rNi is, the smaller the threshold of Ni volumewill be. This theoretical observation is in line with the experimentalfindings [27] that the electronic conductivity for the same Ni volumefraction is higher for larger rYSZ/rNi.

The conductivity model is further used in a 2D multi-physics numer-ical model to verify our electrical model at another level. The numericalmodel couples the intricate interdependency among ionic conduction,electronic conduction, electrochemical reaction and gas transport. Thegoverning equations, boundary condition settings for electrochemicaland electrical processes and gas transport may be found in Ref. [28].Model parameters are chosen to correspond to the available experimen-tal data [2].When the required parameters are not specified in the exper-iment, e.g., the porosity and particle sizes of LSM and YSZ in the cathode,the same data set used in Ref. [22] of the optimized cell is used here. Thefinite element commercial software COMSOLMultiphysics®Version 3.5awas used to solve the coupled partial differential equations of electronic,ionic and gas transport macro-models. Structured mesh elements wereused and consisted of 6270 rectangles with 100,893 degrees of freedom.The direct solver (UMFPACK)was used to solve the coupled partial differ-ential equations with a relative convergence tolerance of 1E−6.

Fig. 3 shows the comparison of the theoretical and experimental I–Vcurves [2]. Clearly, very good agreement between the theoretical andexperimental I–V curves is also obtained, indicating that the electrodeionic conductivity is also correctly determined by our model.

4. Summary

A theoreticalmodel for calculating the ionic and electronic conductiv-ities of an electrode composed of ionic conducting core particles coatedwith electronic conducting nano-particles is proposed. The derivedexplicit expressions show the dependence of the conductivity on thecomposition of the electrode and explain naturally the experimentalphenomenon that a reduced amount of electronic conducting materialis required for providing adequate conductivity in a nano-composite elec-trode. The estimated percolating threshold for the electronic conductionis in agreement with the experimental observation. Themodel is also ca-pable of providing results that are in excellent agreementwith the exper-imental data on the electrode electronic conductivity. The electricalconductivity model is further used in a multi-physics numerical modeland very good agreement with the experimental I–V curves is obtained,demonstrating that the electrode ionic conductivity is also properly de-termined by our model. The conductivity model is therefore useful bothexperimentally and theoretically by allowing for in-depth analysis and

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

Fig. 3. Comparison of the theoretical and experimental I–V curves of an SOFC with aNi-coated YSZ anode.

4 M. Chen et al. / Solid State Ionics xxx (2013) xxx–xxx

design optimization of the nano-composite electrode for improvedperformance.

Acknowledgments

The financial support of the National Basic Research Program ofChina (973 Program Grant No. 2012CB215405), the National NaturalScience Foundation of China (Grant Nos. 11074233 & 11374272) andthe Specialized Research Fund for the Doctoral Program of Higher Educa-tion (Grant Nos. 20113402110038 & 20123402110064) are gratefullyacknowledged.

Please cite this article as: M. Chen, et al., Solid State Ionics (2013), http://

References

[1] S.P. Jiang, Int. J. Hydrog. Energy 37 (2012) 449–470.[2] S.-D. Kim, H. Moon, S.-H. Hyun, J. Moon, J. Kim, H.-W. Lee, Solid State Ionics 177

(2006) 931–938.[3] T. Klemensø, K. Thydén, M. Chen, H.-J. Wang, J. Power Sources 195 (2010)

7295–7301.[4] S.-D. Kim, H. Moon, S.-H. Hyun, J. Moon, J. Kim, H.-W. Lee, Solid State Ionics 178

(2007) 1304–1309.[5] Z. Zhan, D.M. Bierschenk, J.S. Cronin, S.A. Barnett, Energy Environ. Sci. 4 (2011)

3951–3954.[6] X. Liu, X. Meng, D. Han, H. Wu, F. Zeng, Z. Zhan, J. Power Sources 222 (2013) 92–96.[7] Z. Jiang, C. Xia, F. Chen, Electrochim. Acta 55 (2010) 3595–3605.[8] J.-H. Myung, H.-J. Ko, J.-J. Lee, J.-H. Lee, S.-H. Hyun, Int. J. Hydrogen Energy 37 (2012)

11351–11359.[9] S.K. Pratihar, A. Dassharma, H.S. Maiti, Mater. Res. Bull. 40 (2005) 1936–1944.

[10] H. Da, X. Liu, F. Zeng, J. Qian, T. Wu, Z. Zhan, Sci. Rep. 2 (2012) 462–466.[11] M. Shah, S. Barnett, Solid State Ionics 179 (2008) 2059–2064.[12] Y. Huang, K. Ahn, J.M. Vohs, R.J. Gorte, J. Electrochem. Soc. 151 (2004)

A1592–A1597.[13] S.-D. Kim, H. Moon, S.-H. Hyun, J. Moon, J. Kim, H.-W. Lee, J. Power Sources 163

(2006) 392–397.[14] A. Buyukaksoy, V. Petrovksy, F. Dogan, ECS Trans. 45 (2012) 509–514.[15] H. Zhu, R.J. Kee, J. Electrochem. Soc. 155 (2008) B715–B729.[16] J. Sanyal, G.M. Goldin, H. Zhu, R.J. Kee, J. Power Sources 195 (2010) 6671–6679.[17] J. Li, W. Kong, Z. Lin, J. Power Sources 232 (2013) 106–122.[18] J. Wilson, W. Kobsiriphat, R. Mendoza, H. Chen, J. Hiller, D. Miller, K. Thornton,

P. Voorhees, S. Adler, S. Barnett, Nat. Mater. 5 (2006) 541–544.[19] D. Sarantaridis, A. Atkinson, Fuel Cells 7 (2007) 246–258.[20] Y. Zhang, C. Xia, J. Power Sources 195 (2010) 4206–4212.[21] D. Chen, Z. Lin, H. Zhu, R.J. Kee, J. Power Sources 191 (2009) 240–252.[22] D.H. Jeon, J.H. Nam, C.J. Kim, J. Electrochem. Soc. 153 (2006) A406–A417.[23] M. Koyama, H. Tsuboi, N. Hatakeyama, A. Endou, H. Takaba, M. Kubo, C.A. Del Carpio,

A. Miyamoto, ECS Trans. 7 (2007) 2057–2064.[24] M. Koyama, K. Ogiya, T. Hattori, H. Fukunaga, A. Suzuki, R. Sahnoun, H. Tsuboi, N.

Hatakeyama, A. Endou, H. Takaba, M. Kubo, C.A. Del Carpio, A. Miyamoto, J. Comput.Chem. Jpn. 7 (2008) 55–62.

[25] T. Hattori, A. Suzuki, R. Sahnoun, M. Koyama, H. Tsuboi, N. Hatakeyama, A. Endou, H.Takaba, M. Kubo, C.A. Del Carpio, Appl. Surf. Sci. 254 (2008) 7929–7932.

[26] D. Stauffer, A. Aharony, Introduction to Percolation Theory, revised 2nd ed. Taylor &Francis, London, 1994.

[27] S.K. Pratihar, A.D. Sharma, H. Maiti, Mater. Chem. Phys. 96 (2006) 388–395.[28] S. Liu, C. Song, Z. Lin, J. Power Sources 183 (2008) 214–225.

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