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Crystal-isotropicity dependence of ionic conductivity enhancement at strained interfaces
Solid State Ionics 289 (2016) 168–172
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Solid State Ionics journal homepage: www.elsevier.com/locate/ssi
Crystal-isotropicity dependence of ionic conductivity enhancement at strained interfaces Weiqiang Lv a,1, Na Feng a,1, Yinghua Niu a, Fei Yang a, Kechun Wen a, Minda Zou a, Yupei Han a, Jiyun Zhao b, Weidong He a,⁎ a b
School of Energy Science and Engineering, University of Electronic Science and Technology of China, Chengdu, Sichuan 611731, PR China Department of Mechanical and Biomedical Engineering, City University of Hong Kong, Kowloon, Hong Kong
a r t i c l e
i n f o
Article history: Received 1 September 2015 Received in revised form 8 March 2016 Accepted 12 March 2016 Available online 24 March 2016 Keywords: Ionic conductivity Lattice strain Isotropicity Fuel cells
a b s t r a c t In the solid state ionic field, lattice strain is ubiquitous at heterostructural crystal interfaces and can lead to enormous variation in the transport efficiency of ions along the as-formed strained interfaces. In particular, substantial experimental work has been focused on Y-doped ZrO2 (YSZ), the O2− ionic conductivity of which can be enhanced ~104 times under a 4% lattice strain. Despite the consistence that theoretical modeling and computing finds with such an experimental enhancement, a fundamental investigation into the isotropicity dependence of ionic conduction in strained crystal interfaces is still lacking. In this report, we investigate the isotropicity dependence of ionic conduction in arbitrary orthorhombic crystals and reveal the intrinsic conduction enhancement upon applying strain at any designated crystal orientation. Our work provides fundamental basis for rational strain engineering of a crystal in general, the ionic conduction of which is subject to enhancement for energy applications. © 2016 Elsevier B.V. All rights reserved.
1. Introduction The transport efficiency of ionic species in solid-state materials is of pivotal importance for a number of applications. In particular, ensuring efficient electrical conduction of ions in solid crystals is the prerequisite to achieve fuel cell and battery systems with high energy density and fast charge/discharge [1,2]. In the field of solid oxide fuel cells (SOFCs), the advantage of high chemical–electrical conversion efficiency is compromised largely by the high operation temperatures, typically well above 500 °C [3–7]. Such an operation temperature is necessary to maintain an applicable ionic conductivity in the ion-conducting electrolytes including Y-doped ZrO2 (YSZ), CeO 2, etc. To mitigate this devastating drawback, a number of methods have been developed to improve the ionic conductivity of the aforementioned electrolyte crystals, including doping various elements in existing crystals and preparing multi-element nanocomposites [8–11]. Strain engineering, which is ubiquitous and known to lessen the energy barrier of ionic transport within a crystal, has exhibited unprecedented capability of enhancing the ionic conduction within an ion-conducting solid crystal [12–15]. For instance,
⁎ Corresponding author. E-mail address: [email protected] (W. He). 1 These authors contributed equally to this work.
http://dx.doi.org/10.1016/j.ssi.2016.03.011 0167-2738/© 2016 Elsevier B.V. All rights reserved.
by introducing 3%–5% lattice strain to YSZ, the ionic conductivity of the material is enhanced by nearly four orders of magnitude [16–21]; the ionic conductivity of Gd doped CeO2 (GDC), Sm doped CeO2 (SDC), Er doped Bi2O3 (ESB), are all efficiently enhanced by a few magnitudes by introducing certain extents of lattice strain [22–23]. As two materials with lattice mismatch are deposited to form nanostructures, the interface/bulk atom ratio increases significantly and the ionic conduction enhanced by the interface strain can be more dominant compared with the bulk ionic conduction [24,25]. Although substantial conductivity enhancement has been achieved frequently through strain engineering, few analytical models have been developed to investigate the isotropicity-dependence of such conductivity enhancement. In particular, the authors derived an analytical model to evaluate the conductivity enhancement versus strain correlation recently, but the model is based on pure isotropic crystals [26]. Indeed, understanding well the selectivity of lattice orientation is one of the keys to realizing high-performance heterostructural crystals. In this report, we employ a volume expansion-versus-conductivity enhancement model to investigate insightfully the isotropicitydependence of ionic conductivity enhancement for a general ionicconducting crystal. Our study provides an efficient fundamental basis for rational strain engineering towards improved applicability of electrolyte and electrode crystals in the areas of fuel cells and ion-based batteries.
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169
Fig. 1. 3D schematic of coherent solid ionic electrolyte interfaces with lattice mismatch, where Phase 1 has a lattice constant of a, b and c, and Phases 2 has a lattice constant of as. bs and cs. Among the parameters, c b cs, a b as and b b bs. Interfacial tensile strain for Phase 1 is induced due to the lattice mismatch.
2. Models and calculation methods Fig. 1 shows the 3D schematic of coherent solid ionic electrolyte interfaces between Phase 1 and Phase 2 with lattice mismatch. Phase 1 and Phase 2 are cubic, tetragonal or orthorhombic crystallographic ionic conductors. Phase 1 has lattice constants of a, b and c, and Phases 2 has lattice constants of as, bs and cs. If a b as and b b bs, tensile strain for Phase 1 is induced as the two phases form a coherent interface along xy plane. As a result, the unit cell parameters a and b expand to a1 and b1, respectively at the first atomic layer parallel to the interface, and to a2 and b2, respectively at the second atomic layer. According to the definition of lattice strain, we get Eqs. (1) and (2), a1 ¼ aðε1 þ 1Þ; b1 ¼ bðε1 þ 1Þ
ð1Þ
a2 ¼ aðε2 þ 1Þ; b2 ¼ bðε2 þ 1Þ
ð2Þ
where ε1 and ε2 are the lattice strain of the first atomic layer and the second atomic layer, respectively. ε1 and ε2 can be calculated by Eq. (3) [27–29], εðxÞ ¼ ε 1 exp½−αx
ð3Þ
where ε(x) is the lattice strain at the x position from the interface to the investigated lattice position, and α is a constant, which is equal to 4–5 nm−1 for YSZ and CeO2 [30]. The distance between the two atomic layers, c, changes to c1, which is resulted from ab plane stretching, and the correlation between c and c1 is expressed in Eq. (4) [31], 2vε1 c1 ¼ c 1− 1−v
ð4Þ
where v is the Poisson ratio of Phase 1. According to our previous work, the correlation between lattice cell volume expansion and the tensile strain can be expressed as Eq. (5) [26], ΔV ¼
2 ! 1 2vε1 2vε1 2vε1 ðε1 þ 1Þ2 þ 1 þ ε1 exp −αc 1− abc 1− −abc: þ ðε 1 þ 1Þ 1 þ ε 1 exp −αc 1− 3 1−v 1−v 1−v
ð5Þ
The activation energy of ion transport derived from the interface strain, ΔG, can be calculated based on the Zener and Keyes strain-energy model [32,33], ΔG ¼
−ΔVEð2 þ 2vÞ 7−10v−8ν 2
ð6Þ
where E is Young's modulus. The correlation between ionic conductivity and tensile strain is expressed in Eqs. (7)–(8) [26], ln
σt σ0
¼−
ΔG kB T
2 ! 1 2vε1 2vε1 2vε1 2 −abc abc 1− ε ð þ 1 Þ þ 1 þ ε exp −αc 1− þ ð ε þ 1 Þ 1 þ ε exp −αc 1− 1 1 1 1 3 1−v 1−v 1−v σt ln ¼ σ0 kB T 7−10v−8ν 2 =ð2E1 ð1 þ vÞÞ
ð7Þ
ð8Þ
170
W. Lv et al. / Solid State Ionics 289 (2016) 168–172
Fig. 2. The plots of tensile strain-versus-lattice volume expansion as the crystal cell varies (a) the ratio of x = a/b varies as a = c; (b) the ratio of y = a/c varies as a = b.
where σ0 is the bulk ionic conductivity, σt is the total ionic conductivity which includes the contribution of the tensile strain to ionic conductivity enhancement. kB is the Boltzmann constant, T is temperature, and E1 is the Young's modulus of Phase 1. YSZ is used as a model material with a lattice parameter of 5.14 Å. The Young's modulus, Poisson ratio and α are 200 GPa, 0.31, and 4.5 nm−1 [34,35]. 3. Results and discussion Ion conduction in solid materials relies on the hopping of ions through the crystal lattice. The expansion of lattice cell volume (Δ V) decreases significantly the mobile energy barrier of ions [16,19–21]. Introducing interfacial tensile strain is an efficient way to induce lattice expansion. Fig. 2 shows the plots of tensile train versus the lattice volume expansion as the crystal isotropicity changes. Tensile strains in the range of 0–6% are considered since interfacial strains above 6% induce lattice dislocations for most materials and are detrimental to ionic conductivity enhancement [26,35,36]. From Figs. 2a–b, in all cell parameters, cell lattice volume expansion increases as the tensile strain increases. Let x = a/b and y = a/c. In Fig. 2a, as x increases, the slope of ΔV versus ε decreases monotonously, indicating that as x increases, the strain-induced volume expansion decreases. Thus, solid cubic, tetragonal, and orthorhombic ionic conductors with larger surface area a × b per lattice unit cell along the strain direction, experience large lattice cell expansion via interfacial tensile strain. At a fixed tensile strain, ΔV increases as x decreases, and such an increase becomes more obvious as the tensile strain is larger. For instance, Δ V increases from 1.27 × 10−4 nm3 to 3.83 × 10−4 nm3 as x decreases from 1.5 to 0.5 with ε = 1%. With the same decrease in x, Δ V increases from 4.49 × 10− 4 nm3 to 13.47 × 10− 4 nm3 with ε = 4%. In Fig. 2b, the slope of ΔV versus ε increases as y increases, showing that as c decreases, the distance between the second atomic layer and the first atomic layer at the interface decreases and that the second layer experiences a larger strain. As a consequence, large ΔV is induced. At a fixed tensile strain, Δ V increases as y decreases, and such an increase becomes more
obvious if the tensile strain is larger. For instance, Δ V increases from 1.82 × 10−4 nm3 to 2.32 × 10−4 nm3 as y decreases from 1.5 to 0.5 as ε = 1%. With the same decrease in y, Δ V increases from 4.99 × 10−4 nm3 to 8.72 × 10−4 nm3 as ε = 4%. It is noted that as y is 0.5, Δ V does not change as ε is larger than 4%. This means that Δ V changes little with increasing ε as ε N 4% and c N 2a. Interfacial ionic conductivity can be greatly influenced by interfacial lattice strain and this influence is crystallographic-orientation dependent. Fig. 3 shows the plots of ionic conductivity versus tensile strain. In Fig. 3a, at a fixed tensile strain, ionic conductivity increases with decreasing x, and such an increase is more obvious as tensile strain is larger. For instance, as ε is 1%, Log(στ/σ0) only increases from 0.38 to 1.14 as x decreases from 1.5 to 0.5. As ε is 4%, Log(στ/σ0) increases dramatically from 1.33 to 3.99 as x decreases from 1.5 to 0.5. The result indicates that a larger tensile strain gives rise to significant ionic conductivity increases, especially as the lattice facet area per unit cell is larger. In Fig. 3b, as y is 0.5, Log(στ/σ0) increases from 0 to 1.28 quickly as ε increases from 0 to 3%, and then follows a slow increase as ε is larger than 3%. As y increases from 1 to 1.5, Log(στ/σ0) increases quasi-linearly as ε increases. The slopes of the plots of Log(στ/σ0) versus ε increases with increasing y. For instance, as y = 0.75, Log(στ/σ0) increases from 0.51 to 2.00 as ε increases from 1% to 6%. As y = 1.5, Log(στ/σ0) increases from 0.69 to 3.71 as ε increase from 1% to 6%. The result suggests that as y increases, the distance between first interfacial layer and the second atomic layer decreases, the tensile strain-induced ionic conductivity increases. Substantial ionic conductivity enhancement can be obtained as c is small and ε is adequately large, i.e. N4%.
Fig. 3. The plots of tensile strain-versus-logarithmic ionic conductivity as the crystal cell varies (a) the ratio of x = a/b varies with a = c; (b) the ratio of y = a/c varies with a = b and T = 1000 K.
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Fig. 4. The plots of lattice parameters-versus-logarithmic ionic conductivity (a) the ratio of x = a/b varies as a = c; (b) the ratio of y = a/c varies as a = b. Temperatures are varied from 400 K to 1200 K for each plot. The tensile strain is fixed at 2%.
Ionic conductivity is largely affected by temperature. The enhancement of ionic conductivity by interfacial strain at temperatures ranging from 300 K to 2000 K has been reported [36]. The temperature effect on the correlation between ionic conductivity and isotropicity is thus investigated, as shown in Fig. 4. In all the Figures, tensile strain is fixed at 2% along ab plane. In Fig. 4a, Log(στ/σ0) decreases abruptly as x increases from 0.5 to 1 at all temperatures, and then such decreases become rather gradual as x increases from 1 to 1.5. For instance, at 400 K, Log(στ/σ0) decreases from 5.45 to 2.73 as x increases from 0.5 to 1. As x increases from 1 to 1.5, Log(στ/σ0) decreases only from 2.73 to 1.82. The result indicates that the ionic conductivity is more sensitive to materials with a larger interfacial area per lattice unit cell. As x is fixed, Log(στ/σ0) decreases as temperature increases, the decrease becomes much slower as x is larger. For instance, at a fixed x of 0.5, the values of Log(στ/σ0) are 5.45, 3.64, 2.73, 2.18 and 1.82 at 400 K, 600 K, 800 K, 1000 K and 1200 K, respectively. As x is 1.5, the values of Log(στ/σ0) are 1.82, 1.21, 0.91, 0.73 and 0.61 at 400 K, 600 K, 800 K, 1000 K and 1200 K, respectively. The result suggests that the dependence of ionic conductivity on temperature becomes stronger as interfacial areas per unit cell along ab plane is larger. In Fig. 4b, the effect of lattice constant c, which is perpendicular to the interfacial ab plane, on the ionic conductivity at varied temperature is investigated. Log(στ/ σ0) decreases slightly as y increases, and reaches the minimum at around y ≈ 0.68, and then increases with further increasing y at all temperatures. The reason for such results is that, as y is smaller, the second interfacial ab plane interacts weakly with the first interfacial ab plane since the distance between the two layers is larger, Δ V is mainly induced by the distortion of the first ab plane, and ΔV and Log(στ/σ0) decreases as y increases. As y increases further, the interaction between the second interfacial ab plane and the first interfacial ab plane increases, Δ V is induced by both the distortion of the first interfacial layer and the second interfacial layer, and ΔV and Log(στ/σ0) thus increases as a/c increases. Therefore, there exist a critical y, at which ΔV and Log(στ/σ0) reach the minimum. At parameters set in this study, the ionic conductivities for tetragonal or orthogonal crystals with y ≈ 0.68 and along ab plane are insensitive compared with a/c of other values. At a fixed a/c, Log(στ/σ0) decreases with increasing temperature. For instance, as y = 0.5, the values of Log(στ/σ0) are 2.42, 1.61, 1.21, 0.97 and 0.81 as temperatures are 400 K, 600 K, 800 K, 1000 K and 1200 K, respectively. As y = 1.5, the values of Log(στ/σ0) are 3.37, 2.25, 1.68, 1.35 and 1.12 as temperatures are 400 K, 600 K, 800 K, 1000 K and 1200 K, respectively. The result indicates that at lower temperatures, ionic conductivity is more sensitive to tensile strains and the enhancement of ionic conductivity is more isotropicity-dependent. It is noted that the values of lattice mismatch and interfacial strain between two materials change with varying temperature since the thermal expansions of the two materials are different. The effect of thermal expansion must be considered while designing the strained interface between two materials.
4. Conclusions In this work, the isotropic dependence of interfacial tensile strain induced enhancement of ionic conductivity is studied. Our results show that a larger lattice unit cell area along the interface experiences larger distortion and thus results in higher enhancement of ionic conductivity. There exists a critical value for a/c, at which the crystal lattice experiences the smallest distortion at a given tensile strain and temperature. The result indicates that strain induced ionic conductivity is isotropicity dependent. Strain and the isotropicity effects on ionic conductivity enhancement are more pronounced at lower temperatures. Although the huge change of tensile strain at the interface is frequently undesirable for the ion transport in solid-state batteries, the ionic conduction parallel to the orientation of stable interfaces and at the tensile strain side can considerably enhance the ionic conductivity. Our study facilitates the rational design of efficient heterogeneous interfaces for a variety of solid ionic conductor, to gain the most dramatic enhancement of ionic conductivity. Acknowledgments The work is supported by UESTC new faculty startup fund, the National Natural Science Foundation (grant no. 21403031 and grant no. 51501030) and the Fundamental Research Funds for the Chinese Central Universities (grant no. ZYGX2014J088 and grant no. ZYGX2015Z003). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]
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Contents lists available at ScienceDirect
Solid State Ionics journal homepage: www.elsevier.com/locate/ssi
Crystal-isotropicity dependence of ionic conductivity enhancement at strained interfaces Weiqiang Lv a,1, Na Feng a,1, Yinghua Niu a, Fei Yang a, Kechun Wen a, Minda Zou a, Yupei Han a, Jiyun Zhao b, Weidong He a,⁎ a b
School of Energy Science and Engineering, University of Electronic Science and Technology of China, Chengdu, Sichuan 611731, PR China Department of Mechanical and Biomedical Engineering, City University of Hong Kong, Kowloon, Hong Kong
a r t i c l e
i n f o
Article history: Received 1 September 2015 Received in revised form 8 March 2016 Accepted 12 March 2016 Available online 24 March 2016 Keywords: Ionic conductivity Lattice strain Isotropicity Fuel cells
a b s t r a c t In the solid state ionic field, lattice strain is ubiquitous at heterostructural crystal interfaces and can lead to enormous variation in the transport efficiency of ions along the as-formed strained interfaces. In particular, substantial experimental work has been focused on Y-doped ZrO2 (YSZ), the O2− ionic conductivity of which can be enhanced ~104 times under a 4% lattice strain. Despite the consistence that theoretical modeling and computing finds with such an experimental enhancement, a fundamental investigation into the isotropicity dependence of ionic conduction in strained crystal interfaces is still lacking. In this report, we investigate the isotropicity dependence of ionic conduction in arbitrary orthorhombic crystals and reveal the intrinsic conduction enhancement upon applying strain at any designated crystal orientation. Our work provides fundamental basis for rational strain engineering of a crystal in general, the ionic conduction of which is subject to enhancement for energy applications. © 2016 Elsevier B.V. All rights reserved.
1. Introduction The transport efficiency of ionic species in solid-state materials is of pivotal importance for a number of applications. In particular, ensuring efficient electrical conduction of ions in solid crystals is the prerequisite to achieve fuel cell and battery systems with high energy density and fast charge/discharge [1,2]. In the field of solid oxide fuel cells (SOFCs), the advantage of high chemical–electrical conversion efficiency is compromised largely by the high operation temperatures, typically well above 500 °C [3–7]. Such an operation temperature is necessary to maintain an applicable ionic conductivity in the ion-conducting electrolytes including Y-doped ZrO2 (YSZ), CeO 2, etc. To mitigate this devastating drawback, a number of methods have been developed to improve the ionic conductivity of the aforementioned electrolyte crystals, including doping various elements in existing crystals and preparing multi-element nanocomposites [8–11]. Strain engineering, which is ubiquitous and known to lessen the energy barrier of ionic transport within a crystal, has exhibited unprecedented capability of enhancing the ionic conduction within an ion-conducting solid crystal [12–15]. For instance,
⁎ Corresponding author. E-mail address: [email protected] (W. He). 1 These authors contributed equally to this work.
http://dx.doi.org/10.1016/j.ssi.2016.03.011 0167-2738/© 2016 Elsevier B.V. All rights reserved.
by introducing 3%–5% lattice strain to YSZ, the ionic conductivity of the material is enhanced by nearly four orders of magnitude [16–21]; the ionic conductivity of Gd doped CeO2 (GDC), Sm doped CeO2 (SDC), Er doped Bi2O3 (ESB), are all efficiently enhanced by a few magnitudes by introducing certain extents of lattice strain [22–23]. As two materials with lattice mismatch are deposited to form nanostructures, the interface/bulk atom ratio increases significantly and the ionic conduction enhanced by the interface strain can be more dominant compared with the bulk ionic conduction [24,25]. Although substantial conductivity enhancement has been achieved frequently through strain engineering, few analytical models have been developed to investigate the isotropicity-dependence of such conductivity enhancement. In particular, the authors derived an analytical model to evaluate the conductivity enhancement versus strain correlation recently, but the model is based on pure isotropic crystals [26]. Indeed, understanding well the selectivity of lattice orientation is one of the keys to realizing high-performance heterostructural crystals. In this report, we employ a volume expansion-versus-conductivity enhancement model to investigate insightfully the isotropicitydependence of ionic conductivity enhancement for a general ionicconducting crystal. Our study provides an efficient fundamental basis for rational strain engineering towards improved applicability of electrolyte and electrode crystals in the areas of fuel cells and ion-based batteries.
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169
Fig. 1. 3D schematic of coherent solid ionic electrolyte interfaces with lattice mismatch, where Phase 1 has a lattice constant of a, b and c, and Phases 2 has a lattice constant of as. bs and cs. Among the parameters, c b cs, a b as and b b bs. Interfacial tensile strain for Phase 1 is induced due to the lattice mismatch.
2. Models and calculation methods Fig. 1 shows the 3D schematic of coherent solid ionic electrolyte interfaces between Phase 1 and Phase 2 with lattice mismatch. Phase 1 and Phase 2 are cubic, tetragonal or orthorhombic crystallographic ionic conductors. Phase 1 has lattice constants of a, b and c, and Phases 2 has lattice constants of as, bs and cs. If a b as and b b bs, tensile strain for Phase 1 is induced as the two phases form a coherent interface along xy plane. As a result, the unit cell parameters a and b expand to a1 and b1, respectively at the first atomic layer parallel to the interface, and to a2 and b2, respectively at the second atomic layer. According to the definition of lattice strain, we get Eqs. (1) and (2), a1 ¼ aðε1 þ 1Þ; b1 ¼ bðε1 þ 1Þ
ð1Þ
a2 ¼ aðε2 þ 1Þ; b2 ¼ bðε2 þ 1Þ
ð2Þ
where ε1 and ε2 are the lattice strain of the first atomic layer and the second atomic layer, respectively. ε1 and ε2 can be calculated by Eq. (3) [27–29], εðxÞ ¼ ε 1 exp½−αx
ð3Þ
where ε(x) is the lattice strain at the x position from the interface to the investigated lattice position, and α is a constant, which is equal to 4–5 nm−1 for YSZ and CeO2 [30]. The distance between the two atomic layers, c, changes to c1, which is resulted from ab plane stretching, and the correlation between c and c1 is expressed in Eq. (4) [31], 2vε1 c1 ¼ c 1− 1−v
ð4Þ
where v is the Poisson ratio of Phase 1. According to our previous work, the correlation between lattice cell volume expansion and the tensile strain can be expressed as Eq. (5) [26], ΔV ¼
2 ! 1 2vε1 2vε1 2vε1 ðε1 þ 1Þ2 þ 1 þ ε1 exp −αc 1− abc 1− −abc: þ ðε 1 þ 1Þ 1 þ ε 1 exp −αc 1− 3 1−v 1−v 1−v
ð5Þ
The activation energy of ion transport derived from the interface strain, ΔG, can be calculated based on the Zener and Keyes strain-energy model [32,33], ΔG ¼
−ΔVEð2 þ 2vÞ 7−10v−8ν 2
ð6Þ
where E is Young's modulus. The correlation between ionic conductivity and tensile strain is expressed in Eqs. (7)–(8) [26], ln
σt σ0
¼−
ΔG kB T
2 ! 1 2vε1 2vε1 2vε1 2 −abc abc 1− ε ð þ 1 Þ þ 1 þ ε exp −αc 1− þ ð ε þ 1 Þ 1 þ ε exp −αc 1− 1 1 1 1 3 1−v 1−v 1−v σt ln ¼ σ0 kB T 7−10v−8ν 2 =ð2E1 ð1 þ vÞÞ
ð7Þ
ð8Þ
170
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Fig. 2. The plots of tensile strain-versus-lattice volume expansion as the crystal cell varies (a) the ratio of x = a/b varies as a = c; (b) the ratio of y = a/c varies as a = b.
where σ0 is the bulk ionic conductivity, σt is the total ionic conductivity which includes the contribution of the tensile strain to ionic conductivity enhancement. kB is the Boltzmann constant, T is temperature, and E1 is the Young's modulus of Phase 1. YSZ is used as a model material with a lattice parameter of 5.14 Å. The Young's modulus, Poisson ratio and α are 200 GPa, 0.31, and 4.5 nm−1 [34,35]. 3. Results and discussion Ion conduction in solid materials relies on the hopping of ions through the crystal lattice. The expansion of lattice cell volume (Δ V) decreases significantly the mobile energy barrier of ions [16,19–21]. Introducing interfacial tensile strain is an efficient way to induce lattice expansion. Fig. 2 shows the plots of tensile train versus the lattice volume expansion as the crystal isotropicity changes. Tensile strains in the range of 0–6% are considered since interfacial strains above 6% induce lattice dislocations for most materials and are detrimental to ionic conductivity enhancement [26,35,36]. From Figs. 2a–b, in all cell parameters, cell lattice volume expansion increases as the tensile strain increases. Let x = a/b and y = a/c. In Fig. 2a, as x increases, the slope of ΔV versus ε decreases monotonously, indicating that as x increases, the strain-induced volume expansion decreases. Thus, solid cubic, tetragonal, and orthorhombic ionic conductors with larger surface area a × b per lattice unit cell along the strain direction, experience large lattice cell expansion via interfacial tensile strain. At a fixed tensile strain, ΔV increases as x decreases, and such an increase becomes more obvious as the tensile strain is larger. For instance, Δ V increases from 1.27 × 10−4 nm3 to 3.83 × 10−4 nm3 as x decreases from 1.5 to 0.5 with ε = 1%. With the same decrease in x, Δ V increases from 4.49 × 10− 4 nm3 to 13.47 × 10− 4 nm3 with ε = 4%. In Fig. 2b, the slope of ΔV versus ε increases as y increases, showing that as c decreases, the distance between the second atomic layer and the first atomic layer at the interface decreases and that the second layer experiences a larger strain. As a consequence, large ΔV is induced. At a fixed tensile strain, Δ V increases as y decreases, and such an increase becomes more
obvious if the tensile strain is larger. For instance, Δ V increases from 1.82 × 10−4 nm3 to 2.32 × 10−4 nm3 as y decreases from 1.5 to 0.5 as ε = 1%. With the same decrease in y, Δ V increases from 4.99 × 10−4 nm3 to 8.72 × 10−4 nm3 as ε = 4%. It is noted that as y is 0.5, Δ V does not change as ε is larger than 4%. This means that Δ V changes little with increasing ε as ε N 4% and c N 2a. Interfacial ionic conductivity can be greatly influenced by interfacial lattice strain and this influence is crystallographic-orientation dependent. Fig. 3 shows the plots of ionic conductivity versus tensile strain. In Fig. 3a, at a fixed tensile strain, ionic conductivity increases with decreasing x, and such an increase is more obvious as tensile strain is larger. For instance, as ε is 1%, Log(στ/σ0) only increases from 0.38 to 1.14 as x decreases from 1.5 to 0.5. As ε is 4%, Log(στ/σ0) increases dramatically from 1.33 to 3.99 as x decreases from 1.5 to 0.5. The result indicates that a larger tensile strain gives rise to significant ionic conductivity increases, especially as the lattice facet area per unit cell is larger. In Fig. 3b, as y is 0.5, Log(στ/σ0) increases from 0 to 1.28 quickly as ε increases from 0 to 3%, and then follows a slow increase as ε is larger than 3%. As y increases from 1 to 1.5, Log(στ/σ0) increases quasi-linearly as ε increases. The slopes of the plots of Log(στ/σ0) versus ε increases with increasing y. For instance, as y = 0.75, Log(στ/σ0) increases from 0.51 to 2.00 as ε increases from 1% to 6%. As y = 1.5, Log(στ/σ0) increases from 0.69 to 3.71 as ε increase from 1% to 6%. The result suggests that as y increases, the distance between first interfacial layer and the second atomic layer decreases, the tensile strain-induced ionic conductivity increases. Substantial ionic conductivity enhancement can be obtained as c is small and ε is adequately large, i.e. N4%.
Fig. 3. The plots of tensile strain-versus-logarithmic ionic conductivity as the crystal cell varies (a) the ratio of x = a/b varies with a = c; (b) the ratio of y = a/c varies with a = b and T = 1000 K.
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Fig. 4. The plots of lattice parameters-versus-logarithmic ionic conductivity (a) the ratio of x = a/b varies as a = c; (b) the ratio of y = a/c varies as a = b. Temperatures are varied from 400 K to 1200 K for each plot. The tensile strain is fixed at 2%.
Ionic conductivity is largely affected by temperature. The enhancement of ionic conductivity by interfacial strain at temperatures ranging from 300 K to 2000 K has been reported [36]. The temperature effect on the correlation between ionic conductivity and isotropicity is thus investigated, as shown in Fig. 4. In all the Figures, tensile strain is fixed at 2% along ab plane. In Fig. 4a, Log(στ/σ0) decreases abruptly as x increases from 0.5 to 1 at all temperatures, and then such decreases become rather gradual as x increases from 1 to 1.5. For instance, at 400 K, Log(στ/σ0) decreases from 5.45 to 2.73 as x increases from 0.5 to 1. As x increases from 1 to 1.5, Log(στ/σ0) decreases only from 2.73 to 1.82. The result indicates that the ionic conductivity is more sensitive to materials with a larger interfacial area per lattice unit cell. As x is fixed, Log(στ/σ0) decreases as temperature increases, the decrease becomes much slower as x is larger. For instance, at a fixed x of 0.5, the values of Log(στ/σ0) are 5.45, 3.64, 2.73, 2.18 and 1.82 at 400 K, 600 K, 800 K, 1000 K and 1200 K, respectively. As x is 1.5, the values of Log(στ/σ0) are 1.82, 1.21, 0.91, 0.73 and 0.61 at 400 K, 600 K, 800 K, 1000 K and 1200 K, respectively. The result suggests that the dependence of ionic conductivity on temperature becomes stronger as interfacial areas per unit cell along ab plane is larger. In Fig. 4b, the effect of lattice constant c, which is perpendicular to the interfacial ab plane, on the ionic conductivity at varied temperature is investigated. Log(στ/ σ0) decreases slightly as y increases, and reaches the minimum at around y ≈ 0.68, and then increases with further increasing y at all temperatures. The reason for such results is that, as y is smaller, the second interfacial ab plane interacts weakly with the first interfacial ab plane since the distance between the two layers is larger, Δ V is mainly induced by the distortion of the first ab plane, and ΔV and Log(στ/σ0) decreases as y increases. As y increases further, the interaction between the second interfacial ab plane and the first interfacial ab plane increases, Δ V is induced by both the distortion of the first interfacial layer and the second interfacial layer, and ΔV and Log(στ/σ0) thus increases as a/c increases. Therefore, there exist a critical y, at which ΔV and Log(στ/σ0) reach the minimum. At parameters set in this study, the ionic conductivities for tetragonal or orthogonal crystals with y ≈ 0.68 and along ab plane are insensitive compared with a/c of other values. At a fixed a/c, Log(στ/σ0) decreases with increasing temperature. For instance, as y = 0.5, the values of Log(στ/σ0) are 2.42, 1.61, 1.21, 0.97 and 0.81 as temperatures are 400 K, 600 K, 800 K, 1000 K and 1200 K, respectively. As y = 1.5, the values of Log(στ/σ0) are 3.37, 2.25, 1.68, 1.35 and 1.12 as temperatures are 400 K, 600 K, 800 K, 1000 K and 1200 K, respectively. The result indicates that at lower temperatures, ionic conductivity is more sensitive to tensile strains and the enhancement of ionic conductivity is more isotropicity-dependent. It is noted that the values of lattice mismatch and interfacial strain between two materials change with varying temperature since the thermal expansions of the two materials are different. The effect of thermal expansion must be considered while designing the strained interface between two materials.
4. Conclusions In this work, the isotropic dependence of interfacial tensile strain induced enhancement of ionic conductivity is studied. Our results show that a larger lattice unit cell area along the interface experiences larger distortion and thus results in higher enhancement of ionic conductivity. There exists a critical value for a/c, at which the crystal lattice experiences the smallest distortion at a given tensile strain and temperature. The result indicates that strain induced ionic conductivity is isotropicity dependent. Strain and the isotropicity effects on ionic conductivity enhancement are more pronounced at lower temperatures. Although the huge change of tensile strain at the interface is frequently undesirable for the ion transport in solid-state batteries, the ionic conduction parallel to the orientation of stable interfaces and at the tensile strain side can considerably enhance the ionic conductivity. Our study facilitates the rational design of efficient heterogeneous interfaces for a variety of solid ionic conductor, to gain the most dramatic enhancement of ionic conductivity. Acknowledgments The work is supported by UESTC new faculty startup fund, the National Natural Science Foundation (grant no. 21403031 and grant no. 51501030) and the Fundamental Research Funds for the Chinese Central Universities (grant no. ZYGX2014J088 and grant no. ZYGX2015Z003). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]
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