A systematic procedure is presented to determine the gradient energy coefficient and height of double-well potential (phase-field model parameters). The well-known Allen-Cahn equation is used to investigate cation interdiffusion in solid oxide fuel cell (SOFC). An established procedure that commonly applied in the study of solidification is adapted. The phase-field model parameters are determined under sharp as well as thin interface limits. The upper limit of the interface thickness is deduced for the thin interface limit. Effect of variation of interface thickness for a particular value of interfacial energy is evaluated on the cation interdiffusion in terms of interdiffusion length of the cations under thin interface limit and this is compared with that of sharp interface limit. Effect of variation of interfacial energy at the upper limit of the interface thickness is also studied under the thin interface limit. Further, the reason behind the variation of interdiffusion length is also presented under the thin interface limit for different values of interfacial energy. Under a large variation of interfacial energy, no significant changes are found in the interdiffusion length of the cations. From this study, it can be deduced that for a considerable range of interfacial energy, a wider range of model parameters can be selected for the phase-field model with reasonable accuracy.
IrshadMSirajKRazaR, et al.A brief description of high temperature solid oxide fuel cell’s operation, materials, design, fabrication technologies and performance. Appl Sci2016; 6: 75.
2.
RojasAREsparza-PonceHEFuentesL, et al.In situ X-ray Rietveld analysis of Ni-YSZ solid oxide fuel cell anodes during NiO reduction in H2. J Phys D Appl Phys2005; 38: 2276–2282.
3.
DwivediS. Solid oxide fuel cell: materials for anode, cathode and electrolyte. Int J Hydrogen Energy2020; 45: 23988–24013.
4.
SinghalSC. Advances in solid oxide fuel cell technology. Solid State Ionics2000; 135: 305–313.
5.
BansalNP (ed). Advances in solid oxide fuel cells II. 30th International Conference on Advanced Ceramics and Composites, January 22-27, 2006, Cocoa Beach, Florida. Hoboken, New Jersey: John Wiley & Sons, Inc., 2006.
6.
LarminieJDicksA. Fuel cell systems explained. England, UK: John Wiley & Sons, 2003.
7.
YokokawaHTuHIwanschitzB, et al.Fundamental mechanisms limiting solid oxide fuel cell durability. J Power Sources2008; 182: 400–412.
8.
DuongATMummDR. On the interaction of SSC and LSGM in composite SOFC electrodes. J Power Sources2013; 241: 281–287.
9.
PalcutMKnibbeRWiikK, et al.Cation inter-diffusion between LaMnO3 and LaCoO3 materials. Solid State Ionics2011; 202: 6–13.
10.
YangC-CTWeiW-CJRoosenA. Reaction kinetics and mechanisms between La0.65Sr0.3MnO3 and 8 mol% yttria-stabilized zirconia. J Am Ceram Soc2004; 87: 1110–1116.
11.
ChenMLiuYLHagenA, et al.LSM-YSZ reactions in different atmospheres. Fuel Cell2009; 9: 833–840.
12.
MalzbenderJBatfalskyPVaßenR, et al.Component interactions after long-term operation of an SOFC stack with LSM cathode. J Power Sources2012; 201: 196–203.
13.
AnanyevMVFarlenkovASEreminVA, et al.Degradation kinetics of LSM–YSZ cathode materials for SOFC. Int J Hydrogen Energy2018; 43: 951–959.
14.
GrosjeanASanséauORadmilovicV, et al.Reactivity and diffusion between La0.8Sr0.2MnO3 and ZrO2 at interfaces in SOFC cores by TEM analyses on FIB samples. Solid State Ionics2006; 177: 1977–1980.
15.
LiNMahapatraMKSinghP. Sintering of porous strontium doped lanthanum manganite-yttria stabilized zirconia composite in controlled oxygen atmosphere at 1400 °C. J Power Sources2013; 221: 57–63.
16.
HuJ-MLiangLJiY, et al.Interdiffusion across solid electrolyte-electrode interface. Appl Phys Lett2014; 104: 213907.
17.
KawadaTSakaiNYokokawaH, et al.Reaction between solid oxide fuel-cell materials. Solid State Ionics1992; 50: 189–196.
18.
WiikKSchmidtCRFaalandS, et al.Reactions between strontium-substituted lanthanum manganite and yttria-stabilized zirconia: I, Powder samples. J Am Ceram Soc1999; 82: 721–728.
19.
van RoosmalenJAMCordfunkeEHP. Chemical reactivity and interdiffusion of (La, Sr)MnO3 and (Zr, Y)O2, solid oxide fuel cell cathode and electrolyte materials. Solid State Ionics1992; 52: 303–312.
20.
TsogaANaoumidisAGuptaA, et al.Microstructure and interdiffusion phenomena in YSZ-CGO composite electrolyte. Mater Sci Forum1999; 308–311: 794–799.
21.
KlevelandKEinarsrudMASchmidtCR, et al.Reactions between strontium-substituted lanthanum manganite and yttria-stabilized zirconia: II, Diffusion couples. J Am Chem Soc1999; 82: 729–734.
22.
BrugnoniCDucatiUScagliottiM. SOFC cathode/electrolyte interface. Part I. Reactivity between La0.85Sr0.15MnO3 and ZrO2-Y2O3. Solid State Ionics1995; 76: 177–182.
23.
HongLHuJMGerdesK, et al.Oxygen vacancy diffusion across cathode/electrolyte interface in solid oxide fuel cells: an electrochemical phase-field model. J Power Sources2015; 287: 396–400.
24.
LeiYChengTLWenYH. Phase field modeling of microstructure evolution and concomitant effective conductivity change in solid oxide fuel cell electrodes. J Power Sources2017; 345: 275–289.
25.
MoelansNBlanpainBWollantsP. An introduction to phase-field modeling of microstructure evolution. Comput Coupling Phase Diagrams Thermochem2008; 32: 268–294.
26.
PlappM. Phase-field models. Multiphase Microfluidics: The Diffuse Interface Model. Vienna: Springer Vienna, 2012, pp. 129–175.
27.
ChenL-Q. Phase-field models for microstructure evolution. Annu Rev Mater Res2002; 32: 113–140.
28.
KumarMChakrabortyJDasPK. Interdiffusion across electrode-electrolyte interface in solid oxide fuel cell incorporating the finite size effect of the ions. J Electrochem Soc2018; 165: F1184–F1191.
29.
KumarMChakrabortyJDasPK. Combined effect of temperature difference across interface and finite size of the ions on interdiffusion in SOFC. Epub ahead of print 1 June 2023. DOI: 10.1177/09576509231176623.
30.
BoettingerWJWarrenJABeckermannC, et al.Phase-field simulation of solidification. Annu Rev Mater Res2002; 32: 163–194.
31.
OdeMSuzukiTKimSG, et al.Phase–field model for solidification of Fe–C alloys. Sci Technol Adv Mater2000; 1: 43–49.
32.
WangHZhangXLaiC, et al.Thermodynamic principles for phase-field modeling of alloy solidification. Curr Opin Chem Eng2015; 7: 6–15.
33.
LiQLiangLGerdesK, et al.Phase-field modeling of three-phase electrode microstructures in solid oxide fuel cells. Appl Phys Lett2012; 101: 033909.
34.
LiangLLiQHuJ, et al.Phase field modeling of microstructure evolution of electrocatalyst-infiltrated solid oxide fuel cell cathodes. J Appl Phys2015; 117: 065105.
35.
AbdullahTLiuL. Meso-scale phase-field modeling of microstructural evolution in solid oxide fuel cells. J Electrochem Soc2016; 163: F618–F625.
36.
SteinbachIPezzollaFNestlerB, et al.A phase field concept for multiphase systems. Phys D1996; 94: 135–147.
37.
ChenL-GYangW. Computer simulation of the domain dynamics of a quenched system with a large number of nonconserved order parameters: the grain-growth kinetics. Phys Rev B2000; 50: 15752–15756.
38.
KobayashiRWarrenJACraig CarterW. A continuum model of grain boundaries. Phys D2000; 140: 141–150.
39.
CarolloVGuillén-HernándezTReinosoJ, et al.Recent advancements on the phase field approach to brittle fracture for heterogeneous materials and structures. Adv Model Simul Eng Sci2018; 5: 8–29.
40.
LiangLStanMAnitescuM. Phase-field modeling of diffusion-induced crack propagations in electrochemical systems. Appl Phys Lett2014; 105: 163903.
LiangLQiYXueF, et al.Nonlinear phase-field model for electrode-electrolyte interface evolution. Phys Rev E2012; 86: 051609.
43.
GuyerJEBoettingerWJWarrenJA, et al.Phase field modeling of electrochemistry. I. Equilibrium. Phys Rev E2004; 69: 021603.
44.
GuyerJEBoettingerWJWarrenJA, et al.Phase field modeling of electrochemistry. II. Kinetics. Phys Rev E2004; 69: 021604.
45.
BhattacharyyaSBandyopadhyaySChoudhuryA. Phase-field modeling of electrochemical phenomena. J Indian Inst Sci2016; 96: 257–268.
46.
BazantMZ. Theory of chemical kinetics and charge transfer based on nonequilibrium thermodynamics. Acc Chem Res2013; 46: 1144–1160.
47.
EmmerichH. Advances of and by phase-field modelling in condensed-matter physics. Adv Phys2008; 57: 1–87.
48.
LesarR. Introduction to computational materials science. Cambridge, UK: Cambridge University Press, 2013.
49.
EmmerichH. What is an interface?The diffuse interface approach in materials science: Thermodynamic concepts and applications of phase-field models. Berlin, Heidelberg: Springer-Verlag Berlin Heidelberg, 2003, pp. 7–17.
50.
ChaP-RYeonD-HYoonJ-K. A phase field model for isothermal solidification of multicomponent alloys. Acta Mater2001; 49: 3295–3307.
51.
QinRSBhadeshiaHK. Phase field method. Mater Sci Technol2010; 26: 803–811.
52.
BellemansIMoelansNVerbekenK. Phase-field modelling in extractive metallurgy. Crit Rev Solid State Mater Sci2018; 43: 417–454.
53.
ProvatasNElderK. Phase-field methods in materials science and engineering. Weinheim, Germany: Wiley-VCH Verlag GmbH & Co. KGaA, 2010.
54.
EgglestonJJ. Phase-field models for thin film growth and ostwald ripening. Evanston, Illinois, USA: Northwestern University, 2001.
55.
ElderKRGrantMProvatasN, et al.Sharp interface limits of phase-field models. Phys Rev E2001; 64: 021604.
56.
CaginalpG. Stefan and Hele-Shaw type models as asymptotic limits of the phase-field equations. Phys Rev A1989; 39: 5887–5896.
57.
McfaddenGBWheelerAAAndersonDM. Thin interface asymptotics for an energy/entropy approach to phase-field models with unequal conductivities. Phys D2000; 144: 154–168.
CaginalpGChenX. Phase field equations in the singularlimit of sharp interface problems. On the evolution of phase boundaries. New York: Springer-Verlag, 1992, pp. 1–27.
60.
FinelALe BouarYDabasB, et al.Sharp phase field method. Phys Rev Lett2018; 121: 025501.
61.
KarmaARappelW-J. Phase-field method for computationally efficient modeling of solidification with arbitrary interface kinetics. Phys Rev E1996; 53: R3017–R3020.
62.
KarmaARappelW-J. Quantitative phase-field modeling of dendritic growth in two and three dimensions. Phys Rev E1998; 57: 4323–4349.
63.
KimSGKimWTSuzukiT. Interfacial compositions of solid and liquid in a phase-field model with finite interface thickness for isothermal solidification in binary alloys. Phys Rev E1998; 58: 3316–3323.
64.
KimWTKimSGLeeJS, et al.Equilibrium at stationary solid-liquid interface during phase-field modeling of alloy solidification. Metall Mater Trans A2001; 32: 961–969.
65.
KimSGKimWTSuzukiT. Phase-field model for binary alloys. Phys Rev E1999; 60: 7186–7197.
66.
AllenSMCahnJW. A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metall1979; 27: 1085–1095.
67.
BalluffiRWAllenSMCarterWC. Kinetics of materials. Hoboken, New Jersey, USA: John Wiley & Sons, 2005.
