applied sciences

Article A Theoretical Model for the Triple Phase Boundary of Solid Oxide Cell Electrospun

Wei Kong *, Mengtong Zhang, Zhen Han and Qiang Zhang *

School of Energy and Power, Jiangsu University of Science and Technology, Zhenjiang 212003, Jiangsu, China; [email protected] (M.Z.); [email protected] (Z.H.) * Correspondence: [email protected] (W.K.); [email protected] (Q.Z.); Tel.: +86-511-84417398 (W.K.); Fax: +86-511-84404433 (W.K.)  Received: 31 December 2018; Accepted: 28 January 2019; Published: 31 January 2019 

Featured Application: Solid oxide fuel cells (SOFC) are one of the power-generated devices that have received much attention over the last decade due to advantages, such as a high efficiency, quietness, multiple , and the inexpensive catalyst nature. To improve the performance of SOFC, the electrospun is introduced to the SOFC, which can be achieved by impregnating nanoscale particles on the backbone surface of fibers. In this paper, a theoretical model was developed for the electrospun electrode. This model captures the key geometric parameters and their interrelationships, which can be used to design the microstructure parameters, such as the particle radius, fiber radius, and impregnation loading.

Abstract: Electrospinning is a new state-of-the-art technology for the preparation of electrodes for solid oxide fuel cells (SOFC). Electrodes fabricated by this method have been proven to have an experimentally superior performance compared with traditional electrodes. However, the lack of a theoretic model for electrospun electrodes limits the understanding of their benefits and the optimization of their design. Based on the microstructure of electrospun electrodes and the percolation threshold, a theoretical model of electrospun electrodes is proposed in this study. Electrospun electrodes are compared to fibers with surfaces that were coated with impregnated particles. This model captures the key geometric parameters and their interrelationship, which are required to derive explicit expressions of the key electrode parameters. Furthermore, the length of the triple phase boundary (TPB) of the electrospun electrode is calculated based on this model. Finally, the effects of particle radius, fiber radius, and impregnation loading are studied. The theory model of the electrospun electrode TPB proposed in this study contributes to the optimization design of SOFC electrospun electrode.

Keywords: solid oxide fuel cells; electrospinning; triple phase boundary; electrode

1. Introduction Solid oxide fuel cells (SOFC) have received much attention over the last decade due to advantages, such as a high efficiency, quietness, multiple fuels, and the inexpensive catalyst nature [1–4]. One of the major challenges related to SOFCs is the short lifetime introduced by the high operating temperature. The decrease of the working temperature is an effective method to increase the lifespan of SOFC. Therefore, lower or intermediate temperature SOFC have been extensively studied in recent years [5,6]. The electrochemical reactions in the SOFC electrode only take place at the so-called triple phase boundary (TPB), where oxygen , , and gaseous species cohere. The reaction activity of the TPB sharply decreases with decreasing temperature. Therefore, it is significant for lower or intermediate temperature SOFC to have a long TPB.

Appl. Sci. 2019, 9, 493; doi:10.3390/app9030493 www.mdpi.com/journal/applsci Appl. Sci. 2019, 9, 493 2 of 11

The TPB length is a critical geometric parameter of the electrode and dominates the SOFC and electrode performance [7,8]. Meanwhile, the TPB itself is strictly associated with the electrode micromorphology [9,10]. To further improve the electrode performance, many studies focused on deriving an optimal micromorphology from various fabricating processes, such as screen-printing [11,12], freeze-casting/phase inversion [13,14], co-precipitation [15], impregnation/infiltration [16,17], and electrospinning [18,19]. Several types of electrodes with distinct microstructures have been successfully prepared. Conventional electrodes (CE), which are fabricated by sintering composite powders, have been widely and deeply studied both experimentally and theoretically. The TPB length of CE can be extracted from the detailed microstructure obtained, by focused beam-scanning microscopy (FIB-SEM) [20,21] or X-ray diffraction [22]. Theoretical models for TPB calculation can be used to systematically study the impacts of the electrode properties, including the particle size, porosity, and component fraction on the TPB length. Among those models, the random sphere packing model has become popular [23–25]. However, to improve the electrode performance or lower the operation temperature, limitations of the CE, such as the incompatible thermal expansion coefficient of the (e.g., Ni and yttria-stabilized zirconia (YSZ)) and high polarization loss of the cathode [26–28], have to be overcome. To address those problems, a nanostructured electrode can be introduced to the SOFC, which can be achieved by impregnating nanoscale particles on the backbone surface of another phase [16,29]. The nanostructure of the impregnated electrode (IE) leads to a tremendous enhancement of the TPB length and the impregnation phase volume fraction (e.g., Ni) is much lower than that of the CE [30]. From a phenomenological viewpoint, there are two distinct IE constituents: backbone and coated impregnation nanoparticles. These features constitute the basis of theoretical models because the IE is modeled in many cases as a sphere-packed backbone with a surface that is coated with nanospheres of another phase [26,30]. Because the TPB length of the nanostructured electrode is directly related to the total backbone surface area [31], it is logical to explore new frameworks with increased surface areas to improve the electrode performance. The basic backbone formed by many one-dimensional nanostructures (e.g., nanofibers) possesses a high surface area and porosity [32], which implies a novel backbone structure. Because electrospinning is a simple and effective method to produce uniform one-dimensional nanofibers with a high specific area and porosity, it has become a new state-of-the-art technology for the preparation of electrodes for SOFCs [33]. Li et al. [34] studied fibrous Ni-coated YSZ , where the YSZ nanofibers were prepared via electrospinning and then electrolessly plated with a layer of Ni. Their results suggested that the Ni loading is only 30 wt.%, that is, lower than that of CE. Two anodes with the same Ni fraction were prepared by electrospinning and power processing, respectively. The test results showed that the peak power density of the cell with the electrospun anode is twice that obtained using the power-processing anode, which was attributed to the longer TPB of the former anode. Zhao et al. [33] compared La0.8Sr0.2Co0.2Fe0.8O3−δ (LSCF) nanorod/Ce0.8Gd0.2O1.9 (GDC) nanoparticle composite cathode with LSCF/GDC nanoparticle cathode. The polarization resistance of the former is five times smaller than that of the latter. Further studies indicated that the cathode performance is better when the diameter of the nanofibers is sufficiently small [33,35]. Other studies [36,37] confirmed the significant influence of the microstructure and intrinsic properties of the basic components on the cathode performance. Although the CE and IE have been widely studied, the electrospun electrode (EE) morphology differs from that of the CE or IE. The models and results for the CE and IE cannot be directly applied to the EE. The 3D reconstruction via FIB-SEM or X-ray diffraction is a powerful tool to quantitatively analyze the geometric properties of CEs. However, those methods cannot be used to reconstruct the impregnated microstructure due to the resolution gap [31]. Therefore, although the experimental activities are strong, as aforementioned, most experiments focused on the preparation and testing of the macro-performance; quantitative analyses of the geometric properties are rare. Contrary to in-depth Appl. Sci. 2019, 9, 493 3 of 11 experimental studies, there is little theoretical research, to the best of our knowledge. The work of Enrico, A. and Costamagna, P [38] was developed specifically for LSCF fibers. Being LSCF a MIEC (mixed ionic electronic conductor, embedding both electronic and ionic conduction simultaneously), a number of infiltrated particles below the percolation threshold were considered. Conversely, in the present work, the fibers are considered to be a pure ionic conductor, and percolation of the infiltrated electronic conducting particles is a mandatory feature for the electrochemical reaction to expand into the electrode bulk, i.e. for the infiltration/fiber interface to contribute to the TPB. In this paper, based on the microstructure of EE reported in literatures [18,19] and the percolation threshold, a theoretical model was developed for the EE by comparing the EE to fibers with surfaces that were coated with impregnated particles. This model captures the key geometric parameters and their interrelationship, which can be used to derive explicit expressions of several key electrode parameters. Furthermore, the TPB length of the EE was calculated based on this model. Finally, the effects of the particle radius, fiber radius, and impregnation loading were systematically studied.

2. Theoretical Model Generally, EE is fabricated by combining electrospinning and impregnation methods. The nanofibers are firstly prepared via electrospinning, followed by casting/printing on the and co-sintering to generate the backbone. Subsequently, nanoparticles are coated on the surface of the backbone by impregnation. However, with increasing impregnation loading, the layers of the coated particles may vary from less than one layer to more than one layer. The multiple fabrication processes of EE lead to its distinctive microstructures. From a phenomenological viewpoint, the EE is a combination of the backbone and outer shell. The backbone closely resembles the packing of straight fibers because it is fabricated by sintering electrospinning fibers. Despite the sintering, the backbone still holds the straight fiber structure [33,38]. The outer shell consists of impregnated nanoparticles, which adhere to the backbone surface. Based on the microscopic images [33,34], the nanoparticles are spherical. To simulate the heat treatment, contact angles are allowed between the spheres and between fibers and spheres. Therefore, the backbone is considered to be a straight fiber structure in the theoretical model and the impregnated particles are considered to be spheres. In addition, all of the electrospun fibers and the impregnated particles in EE are of similar size. Thus, in order to simplify the model, the assumption that all of the electrospun fibers (the impregnated particles) have the same radii is adopted. The contact angle is widely used to describe the sintering process, which is generally set as 15◦ in CE and IE [29,35]. As a result, it is reasonable to assume the contact angle between particles or between fiber and particle is 15◦. As we all know, in the impregnating process, the nanoparticles randomly stick to the surface of the fibers irregularly [18]. Therefore, we assume that the particles are randomly dispersed on the fiber face. In fact, fibers possess three surfaces, including the top, side, and bottom surface where the nanoparticles can stick. The intersection between the fibers covers part of the surface of the fibers. However, it is a formidable challenge for the quantitative calculation of the reduction of the fibers surface due to the intersection between the fibers. Thus, we assume that the total area of the top and bottom surface of fiber equals to the reduction of the fibers surface due to the intersection between the fibers. The nanoparticles can only stick to the side surface of the fibers in this theoretical model. In a word, the assumptions include the following:

1. All electrospun fibers have constant radii. 2. All particles have constant radii. 3. The particles are coated on the fiber face; the contact degree is 15◦. 4. The contact degree between particles is 15◦. 5. If the outer shell is smaller than one layer, the particles will be randomly dispersed on the fiber face. 6. The intersecting region between the fibers is neglected. Appl. Sci. 2019, 9, 493 4 of 11

Appl. Sci.Figure 2019,1 9 shows, x FOR PEER the schematic REVIEW diagram of the developed theoretical model: Straight fibers (yellow4 of 11 cylinders)Appl. Sci. 2019 are, 9, randomlyx FOR PEER arrangedREVIEW in the electrode, acting as the backbone; nanoparticles (gray spheres)4 of 11 particlesare coated is 1. on However, the fiber if surfaces the shell in is three smaller possible than one configurations. layer, the particles Depending are randomly on the impregnationdispersed on shell theloading,particles fiber thefaceis 1. shell However,based may on bethe if smallerthe fifth shell assumption than is smaller one layer (Figurethan and one 2a). onelayer, In layer this the maycase,particles be the bigger areP randomly (Figuremust be1 ,dispersed fromdetermined left toon right). For a shell, which is not smaller than one layer, the percolation ratio Pshellshell of the particles is 1. viathe percolationfiber face based theory. on Several the fifth particles assumption cannot (Figure be connected 2a). In thisto any case, of thethe percolation P must clusters be determined (Figure 2a,However, red oval if frame), the shell and is smallerthus do thannot contribute one layer, to the th particlese TPB and are need randomly to be deleted, dispersed as shown on the in fiber Figure face via percolation theory. Several particles cannot be connected shellto any of the percolation clusters (Figure based on the fifth assumption (Figure2a). In this case, the P mustshell be determined via percolation 2b.2a, Inred this oval paper, frame), an andalgorithm thus do was not proposed contribute to to determine the TPB and the need P to of be the deleted, shell that as shown is smaller in Figure than theory. Several particles cannot be connected to any of the percolation clusters (Figure2a, red oval one layer. Pshell frame),2b. In this and paper, thus doan notalgorithm contribute was toproposed the TPB to and determine need to bethe deleted, of as the shown shell inthat Figure is smaller2b. In than this paper,one layer. analgorithm was proposed to determine the Pshell of the shell that is smaller than one layer.

Figure 1. The schematic diagram of the theoretical model under different impregnation loadings. The Figure 1. The schematic diagram of the theoretical model under different impregnation loadings. shell of the impregnated particles may be smaller than one layer (left), one layer (middle), and bigger TheFigure shell 1. The of theschematic impregnated diagram particles of the theoretical may be smallermodel under than onedifferent layer impregnation (left), one layer loadings. (middle), The thanandshell bigger oneof the layer thanimpregnated (right). one layer particles (right). may be smaller than one layer (left), one layer (middle), and bigger than one layer (right).

FigureFigure 2. TheThe schematicschematic diagram diagram of of the the case case in whichin whic theh shellthe shell is smaller is smaller than one than layer, one ( alayer,) the particles (a) the particlesareFigure randomly 2. are The randomly dispersedschematic dispersed on diagram the fiber on of facethe the fiber and case someface in andwhic particles someh the doparticles shell not percolateis dosmaller not inpercolate than the redone ovalin layer, the frame red (a )oval andthe frame(particlesb) shows and are the(b )randomly schematicshows the dispersed afterschematic deleting on after the the deletingfiber particles face th and thate particles some do not particles that percolate. do donot not percolate. percolate in the red oval frame and (b) shows the schematic after deleting the particles that do not percolate. 3.3. TPB TPB Length Length Calculation Calculation 3. TPBForFor Length the the TPB Calculation length calculation, the the fiber fiber and and part particleicle are are considered to to be ionic and electronic conductors, respectively. Exchanging their roles will not influence the TPB calculation because they conductors,For the respectively. TPB length calculation,Exchanging the their fiber roles and wi partll noticle influence are considered the TPB to calculation be ionic and because electronic they are pure (ionic or electronic) conductors. However, the influence on the TPB of the mixed ionic and areconductors, pure (ionic respectively. or electronic) Exchanging conductors. their However, roles wi llthe not influence influence on the the TPB TPB calculation of the mixed because ionic theyand electronic conductor (MIEC) is still under discussion [39]. electronicare pure (ionic conductor or electronic) (MIEC) isconductors. still under However,discussion the [39]. influence on the TPB of the mixed ionic and Figure1 shows that the impregnated fibers are randomly distributed in the electrode. They have electronicFigure conductor 1 shows that (MIEC) the impregnated is still under fibers discussion are ra [39].ndomly distributed in the electrode. They have different lengths, positions, and orientations. Because of the different volume fraction, the number differentFigure lengths, 1 shows positions, that the and impregnated orientations. fibers Becaus are rae ofndomly the different distributed volume in the fraction, electrode. the numberThey have of fibersof fibers also also differs. differs. For For the the TPB TPB calculation, calculation, a arepeating repeating unit unit is is first first abstracted abstracted by by tailoring tailoring the different lengths, positions, and orientations. Because of the different volume fraction, the numberv of impregnated fiber, as shown in Figure3. This way, the volume-specific effective TPB length λTPBλcanv impregnatedfibers also differs. fiber, asFor shown the TPB in Figure calculation, 3. This a way, repeating the volume-specific unit is first abstracted effective TPBby tailoringlength TPBthe be given by: λ v canimpregnated be given by: fiber, as shown in Figurev 3. This way, the volume-specific effective TPB length TPB λTPB = 2πrel sin α0 · Zio−elnv pio pel (1) can be given by: λπαv =⋅ TPB2sinrZnpp el 0 io-el v io el (1) λπαv =⋅ TPB2sinrZnpp el 0 io-el v io el α (1) where rel are the radii of the electron-conducting particles in the shell; 0 is the contact angle α betweenwhere r elthe are fibers the andradii partic of theles, electron-conducting which is 15° [40,41]; particles Zio-el is thein the coordination shell; 0 numberis the contact between angle the

between the fibers and particles, which is 15° [40,41]; Zio-el is the coordination number between the Appl. Sci. 2019, 9, 493 5 of 11

Appl. Sci. 2019, 9, x FOR PEER REVIEW 5 of 11 whereAppl.rel Sci.are 2019 the, 9 radii, x FOR of PEER the REVIEW electron-conducting particles in the shell; α0 is the contact angle5 of between 11 ◦ the fibersfibers and and particles, particles in which the repeating is 15 [40 unit;,41]; nZv io is− eltheis number the coordination of repeating number units per between unit volume; the fibers pio and fibers and particles in the repeating unit; nv is the number of repeating units per unit volume; pio particlesis the in percolation the repeating ratio unit; of thenv ion-conductingis the number offibers; repeating and p unitsel is the per percolation unit volume; ratiopio ofis the the electron- percolation is the percolation ratio of the ion-conducting fibers; and p is the percolation ratio of the electron- ratio ofconducting the ion-conducting particles. fibers; and pel is the percolationel ratio of the electron-conducting particles. conducting particles.

FigureFigure 3. The3. The schematic schematic diagram diagram of the the repeating repeating unit. unit. Figure 3. The schematic diagram of the repeating unit. v λ v BasedBased on Equation on Equation (1), the (1), primary the primary task of task determining of determiningλ is theTPB determination is the determination of the parametersof the Based on Equation (1), the primary task of determiningTPB λ v is the determination of the in theparameters equation, thatin the is, equation, the coordination that is, the number coordinati betweenon number the fibersbetweenTPB and the particles fibers and in particles the repeating in the unit parameters in the equation, that is, the coordination number between the fibers and particles in the Z repeating, number unit of repeating Zio-el , number units perof repeating unit volume unitsn per, percolation unit volume ratio nv , ofpercolation the ion-conducting ratio of the fibers ion- p , io−el Z v n io repeatingconducting unit fibers io-el p, number, and percolation of repeating ratio units of the per electron-conducting unit volume v , percolationparticles p ratio. of the ion- and percolation ratio of theio electron-conducting particles pel. el conducting fibers p , and percolation ratio of the electron-conducting particles p . The coordinationThe coordination numberio number between between the the fibers fibers and and particlesparticles in in the the repeating repeating unit: unit:el the thecoordination coordination numberThe coordinationZ between number the fibers between and particlesthe fibers in and the pa rerticlespeating in theunit repeating can be geometrically unit: the coordination derived. number Z − betweenio-el the fibers and particles in the repeating unit can be geometrically derived. numberio elZ between the fibers and particles in the repeating unit can be geometrically derived. FigureFigure4 illustrates 4 illustratesio-el the geometricthe geometric relations relations in in the the repeating repeating unit.unit. The coordination coordination number number is: is: Figure 4 illustrates the geometric relations in the repeating unit. The coordination number is: ο 360 ◦ Z = 2 360ο η (2) io-el = 360 ZioZ−el = 22 θ ηη (2) (2) io-el θθ where θ is the central angle of the fiber occupied by one particle (see Figure 4a). Notably, two layers where θ is the central angle of the fiber occupied by one particle (see Figure4a). Notably, two layers whereof particles θ is surround the central the angle fiber of in the the fibe repeatinr occupiedg unit, by oneas shown particle in (see Figure Figure 3. The4a). Notably,parameter two η layersis the of particles surround the fiber in the repeating unit, as shown in Figure3. The parameterη η is the ofimpregnation particles surround loading, the which fiber is in defined the repeatin as theg rati unit,o of as the shown total inimpregnated Figure 3. The particle parameter number, isto thethe impregnation loading, which is defined as the ratio of the total impregnated particle number, to the impregnationparticle number loading, of the whichone-layer is defined case. Therefore, as the rati oη of isthe 1 intotal the impregnated one-layer shell particle case. number,For shells to with the particle number of the one-layer case. Therefore, η isη 1 in the one-layer shell case. For shells with more particlemore than number one layer, of the η one-layer is still 1 case. because Therefore, the particles is beyond1 in the the one-layer first layer shell are case. not Forin contact shells withwith than one layer, η is still 1 becauseη the particles beyond the first layer are not in contact with the fibers, morethe fibers, than andone thuslayer, do not is influence still 1 because the TPB the length. particles However, beyond theη first is smaller layer thanare not 1 in in the contact shells with that and thus do not influence the TPB length. However, η is smallerη than 1 in the shells that are smaller theare fibers,smaller and than thus one do layer. not influence the TPB length. However, is smaller than 1 in the shells that than oneare smaller layer. than one layer.

Figure 4. The schematic illustration of geometric relations in the repeating unit. (a) the define of crucial parameters. (b) the define of h3. Appl. Sci. 2019, 9, 493 6 of 11

The central angle θ can be geometrically calculated by:

2h2 − h2 = 1 2 θ arccos 2 (3) 2h1 where h1 and h2 are schematically illustrated in Figure4, they are calculated as: q 2 2 h1 = rio − (rel sin α0) + rel cos α0 (4)

h2 = 2 · rel cos α0 (5) where rio is the radius of the ion conducting fiber. The number of repeating units per unit volume nv: The volume of the ion-conducting fiber in the repeating unit is: 2 Vfib = 2h3πrio (6) where h3 is schematically illustrated in Figure4, it is calculated as: √ ◦ h3 = 2rel cos α0 cos 30 = 3rel cos α0 (7)

Therefore, the number of repeating units per unit volume nv can be derived by:

Vio nv = (8) Vfib where Vio is the volume fraction of the ion conducting fibers in electrode. Percolation ratio of ion-conducting fibers pio: Based on the fabrication process and microscopic images, all ion-conducting fibers are interconnected. Therefore, the percolation ratio of the ion-conducting fibers is set to pio = 1. Percolation ratio of electron-conducting particles pel: The percolation ratio of the electron-conducting particles depends on the impregnation layers of the shell. In shells with equal or more than one layer, all particles percolate; therefore, pel = 1. However, if the shell is smaller than one layer, the pel is obtained based on percolation theory. An algorithm was developed to automatically identify percolating particles. Because the percolated cluster must start from the top particle layer to the bottom particle layer, the algorithm starts from the top layer particles, like the green ones in Figure5a. A loop is used to identify the clusters starting from every top layer particle. In this way, some isolated clusters will be automatically filtered out, such as the C4 and C5 in Figure5a. The result will be all of the clusters starting from the top, as the C1, C2 and C3 shown in Figure5b. However, not every identified cluster can stretch to the bottom of the cylinder. If a cluster contains no bottom layer particle, for example, the C2, it will be deleted. Thus, the remaining clusters are all percolated, shown as C1 and C3 in Figure5c. The amount of the particles in those clusters divided by the total particleAppl. Sci. 2019 number, 9, x FOR in shellsPEER REVIEW with one layer will be pel. 7 of 11

Figure 5. TheThe schematic schematic illustration illustration of of the the algorithm algorithm of of ppelel.(. a(a)) all all of of the the clusters.clusters. ((bb)) allall ofof the clusters starting from the top. (c) all the percolatedpercolated clusters.

4. Results and Discussion The TPB length of the EE was firstly calculated using our proposed theoretical model by Office Excel. Subsequently, a study was conducted to determine the effects of the impregnation loading, particle radius, and fiber radius on the TPB length. Baseline conditions were set for the parametrical study: the volume fraction of the ion-conducting fibers Vio is 0.45, the fiber radius is 150 nm, the = particle radius is 30 nm, and pel 1. Unless otherwise stated, the baseline conditions were adopted.

Figure 6 shows the dependence of the percolation ratio pel on the impregnation loading with different radii of the electron conducting particles rel . Note that for an impregnation loading below the threshold, pel = 0 due to no percolating particles existing. Furthermore, the threshold of the η impregnation loading increases with the increase of the rel . This is because the smaller rel , the more particles around the fiber, which will provide more options from the top to bottom of the fiber and form the percolated cluster. However, for an impregnation loading higher than the threshold, the percolation ratio pel steeply increases. When the impregnation loading is larger than 0.66, all particles are connected for different rel , that is, pel = 1.

Figure 6. The dependence of the percolation ratio pel on the impregnation loading.

Figure 7 shows the dependence of the TPB length on the impregnation loading. At a fixed Vio , particles do not percolate when the impregnation loading is too low. Thus, the effective TPB length is 0. However, when the impregnation loading reaches the threshold value of 0.48, an effective TPB begins to form and reaches for η in the range from 0.48 to 0.6, which can be attributed to the increase <<η in the percolation ratio pel and coordination number. Compared with 0.48 0.6 , the effective TPB length enhances relatively slowly for η >0.6 . This is due to the fact that the percolation ratio η > pel reaches its maximum (1) at 0.6 and remains constant. Appl. Sci. 2019, 9, x FOR PEER REVIEW 7 of 11

Figure 5. The schematic illustration of the algorithm of pel . (a) all of the clusters. (b) all of the clusters

Appl. Sci.starting2019, 9from, 493 the top. (c) all the percolated clusters. 7 of 11

4. Results and Discussion 4. Results and Discussion The TPB length of the EE was firstly calculated using our proposed theoretical model by Office Excel.The Subsequently, TPB length ofa study the EE was was conducted firstly calculated to dete usingrmine our the proposedeffects of theoreticalthe impregnation model byloading, Office particleExcel. Subsequently, radius, and fiber a study radius was on conductedthe TPB length. to determine Baseline theconditions effects ofwere the set impregnation for the parametrical loading, particlestudy: the radius, volume and fraction fiber radius of the on ion-conducting the TPB length. fibers Baseline Vio conditions is 0.45, the were fiber set radius for the is parametrical150 nm, the study: the volume fraction of the ion-conducting= fibers Vio is 0.45, the fiber radius is 150 nm, the particle particle radius is 30 nm, and pel 1. Unless otherwise stated, the baseline conditions were adopted. radius is 30 nm, and pel = 1. Unless otherwise stated, the baseline conditions were adopted. Figure 6 shows the dependence of the percolation ratio pel on the impregnation loading with Figure6 shows the dependence of the percolation ratio pel on the impregnation loading with different radii of the electron conducting particles rel . Note that for an impregnation loading below different radii of the electron conducting particles rel. Note that for an impregnation loading below the threshold, pel = 0 due to no percolating particles existing. Furthermore, the threshold of the the threshold, pel = 0 due to no percolating particles existing. Furthermore, the threshold of the η r r impregnation loadingloading η increasesincreases with with the the increase increase of theof therel . Thisel . This is because is because the smallerthe smallerrel, the el more, the particlesmore particles around around the fiber, the which fiber, willwhich provide will provid more optionse more fromoptions the from top to the bottom top to of bottom the fiber of andthe formfiber andthe percolated form the percolated cluster. However, cluster. for However, an impregnation for an impregnation loading higher loading than the higher threshold, than thethe percolationthreshold, theratio percolationpel steeply increases.ratio pel When steeply the increases. impregnation When loading the impregnati is larger thanon loading 0.66, all particlesis larger arethan connected 0.66, all particlesfor different are rconnectedel, that is, pforel =different 1. rel , that is, pel = 1.

FigureFigure 6. 6. TheThe dependence dependence of of the the percolation percolation ratio ratio pelel on on the the impregnation impregnation loading. loading.

Figure7 shows the dependence of the TPB length on the impregnation loading. At a fixed Vio, Figure 7 shows the dependence of the TPB length on the impregnation loading. At a fixed Vio , particles do not percolate when the impregnation loading is too low. Thus, the effective TPB length particles do not percolate when the impregnation loading is too low. Thus, the effective TPB length is 0. However, when the impregnation loading reaches the threshold value of 0.48, an effective TPB is 0. However, when the impregnation loading reaches the threshold value of 0.48, an effective TPB begins to form and reaches for η in the range from 0.48 to 0.6, which can be attributed to the increase begins to form and reaches for η in the range from 0.48 to 0.6, which can be attributed to the increase in the percolation ratio pel and coordination number. Compared with 0.48 < η < 0.6, the effective in the percolation ratio p and coordination number. Compared with 0.48<<η 0.6 , the effective TPB length enhances relativelyel slowly for η > 0.6. This is due to the fact that the percolation ratio p η > el TPBreaches length its maximum enhances (1)relatively at η > 0.6slowly and for remains 0.6 constant.. This is due to the fact that the percolation ratio η > pel reachesFigure7 itsshows maximum that the (1) effective at TPB0.6 and length remains increases constant. with increasing Vio when the impregnation loading is larger than the threshold value of 0.48. Moreover, the difference in the effective TPB length due to the change of Vio increases with increasing η. For example, when η is 0.5, the effective TPB 12 -1 length for Vio of 0.5 is 2.1 × 10 m higher than that for Vio of 0.4. However, when η is 0.6, the effective 12 -1 TPB length for Vio of 0.5 is 13.8×10 m higher than that for Vio of 0.4. Appl. Sci. 2019, 9, x FOR PEER REVIEW 8 of 11

Appl. Sci. 2019, 9, 493 8 of 11 Appl. Sci. 2019, 9, x FOR PEER REVIEW 8 of 11

Figure 7. The dependence of triple phase boundary (TPB) length on impregnation loading.

Figure 7 shows that the effective TPB length increases with increasing Vio when the impregnation loading is larger than the threshold value of 0.48. Moreover, the difference in the η η effective TPB length due to the change of Vio increases with increasing . For example, when 12 -1 is 0.5, the effective TPB length for Vio of 0.5 is 2.1 × 10 m higher than that for Vio of 0.4. However, η 12 -1 when is 0.6, the effective TPB length for Vio of 0.5 is 13.8×10 m higher than that for Vio of 0.4. Figure 7. The dependence of triple phase boundary (TPB) length on impregnation loading. Figure 8 shows the TPB length as a function of the electron-conducting particle radius rel and Figure8 shows the TPB length as a function of the electron-conducting particleV radius r and14 ion-conductingFigure 7 showsfiber radius that ther witheffective the ηTPB=1 . Thelength TPB increaseslength of EEwith is highincreasing on the orderio ofwhen 110×el the, io 14 ion-conductingimpregnation loading fiber radius is largerrio with than the theη =threshol1. Thed TPB value length of 0.48. of EE Moreover, is high on the the difference order of 1 ×in10 the, which is close to the value of IE [30]. Note that the TPB increases with decreasing rel or rio . The TPB which is close to the value of IE [30]. Note thatV the TPB increases with decreasingη rel or rio. The TPBη effective TPB length due to the change of io increases with increasing . For example, when= length at a rel value of 30 nm only accounts for 37% of that at a rel value of 10 nm for rnmio 150 . length at a rel value of 30 nm only accounts for 37% of12 that-1 at a rel value of 10 nm for rio = 150 nm. is 0.5, the effective TPB length for Vio of 0.5 is 2.1 × 10 m higher than that for Vio of 0.4. However, However, the influence influence of of rrel on on the the TPB TPBlength length varies.varies. ForFor instance, at a smaller rrel,, thethe TPBTPB lengthlength η el 12 -1 el when is 0.6, the effective TPB length for Vio of 0.5 is 13.8×10 m higher than that for Vio of drops moremore rapidly;rapidly; atat aa larger largerr elr,el the, the influence influence is weakened.is weakened. Although Although finer finer impregnation impregnation particles particles are 0.4. favorableare favorable for thefor TPBthe length,TPB length, they canthey also can increase also incr theease aggregation the aggregation risk and risk directly and directly lead to a lead decrease to a Figure 8 shows the TPB length as a function of the electron-conducting particle radius r and ofdecrease the TPB of lengththe TPB and length bad electrodeand bad electrode performance. performance. In practice, In thepracti sizece, of the the size impregnation of the impregnation particlesel η × 14 mustparticlesion-conducting be carefullymust be fiber carefully controlled radius controlled torio maximizewith the to maximize the=1 electrode. The the TPB electrode performance. length of performance. EE is high on the order of 110, which is close to the value of IE [30]. Note that the TPB increases with decreasing rel or rio . The TPB = length at a rel value of 30 nm only accounts for 37% of that at a rel value of 10 nm for rnmio 150 .

However, the influence of rel on the TPB length varies. For instance, at a smaller rel , the TPB length drops more rapidly; at a larger rel , the influence is weakened. Although finer impregnation particles are favorable for the TPB length, they can also increase the aggregation risk and directly lead to a decrease of the TPB length and bad electrode performance. In practice, the size of the impregnation particles must be carefully controlled to maximize the electrode performance.

FigureFigure 8. 8. TheThe dependence dependence of of TPB TPB length length on on relel andand riorio. . 5. Conclusions 5. Conclusions A theoretical model was established for the electrospun electrode, which considers the fibers as cylinders and the impregnated particle as spheres. Various layers of spheres are coated on the cylinders based on different impregnation loadings. Based on this model, the TPB length of the electrospun electrode was calculated, and the effects of particle radius, fiber radius, and impregnation loading were also investigated. The results indicate that the effective TPB length difference due to Figure 8. The dependence of TPB length on rel and rio . the change of Vio increases with increasing impregnation loading η. Furthermore, thinner particle

5. Conclusions Appl. Sci. 2019, 9, 493 9 of 11 and fiber sizes are favorable for the enhancement of the TPB length. An algorithm is developed to calculate the percolation ratio of the electron conducting particles pel. It is found that the threshold of the impregnation loading η increases with the increase of the rel. For an impregnation loading higher than the threshold, the percolation ratio pel steeply increases. When the impregnation loading is larger than 0.66, all particles are connected for different rel, that is, pel = 1.

Author Contributions: Research articles with four authors. Methodology, W.K. and Q.Z.; software, M.Z.; validation, Z.H.; formal analysis, Z.H.; investigation, Q.Z.; resources, W.K.; data curation, M.Z.; writing—original draft preparation, Q.Z.; writing—review and editing, W.K.; visualization, M.Z.; supervision, W.K.; project administration, W.K.; funding acquisition, W.K. Funding: This work was financially supported by the National Science Foundation of China (21406095) and the Natural Science Foundation through the Jiangsu Province General Program (BK20151325). Conflicts of Interest: The authors declare no conflict of interest.

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