Surface effects on buckling of nanowire electrode

Shang Shuai-Peng; Lu Yong-Jun; Wang Feng-Hui

doi:10.7498/aps.71.20211864

Abstract

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1. 引　言
2. 物理模型
2.1 扩散方程
2.2 表面效应
2.3 力学方程
2.4 屈曲准则
2.5 无量纲化
3. 结果与讨论
4. 结　论

References

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Abstract

Nanowire-based electrodes have attracted much attention due to their high surface energy, short distance for lithium insertion, and the ability to accommodate the enormous strain. However, the buckling behavior may occur during lithiation for such wire-like electrodes, which would lead the battery performance to deteriorate. Therefore, it is vital to quantitatively understand the mechanism about the bucking behavior of the nanowire-based electrodes. Although the buckling behavior of wire-like electrode has been extensively studied in the past few decades, the influence of surface effect on it has not yet been thoroughly explored. For this purpose, a theoretical model of surface effects on buckling of nanowire electrode is presented by taking into account the lithium diffusion, stress, and concentration-dependent elastic properties. Based on the established model, the effects of the residual surface tension and elastic hardening/softening coefficients on buckling are investigated. The results show that surface effects can improve the mechanical reliability, thus delaying the critical buckling time of nanowire electrode. In addition, it is indicated that the surface effects depend on the radius size and slenderness ratio of the nanowire electrode, specifically, the smaller the radius size and the larger the slenderness ratio, the greater the influence of the surface effect is. Furthermore, compared with elastic hardening, with the participation of surface effects, the larger the elastic softening coefficient, the longer it takes for the nanowire electrode to reach the buckled state, and the better the stability of the electrode is. The novelty of this work is that the proposed models highlight the importance of surface effects on buckling of nanowire electrode. These findings provide a prospective insight into the designing of higher structural reliability of electrode.

Keywords:

PACS:

31.15.em	(Corrections for core-spin polarization, surface effects, etc.)
82.47.Aa	(Lithium-ion batteries)
61.46.Km	(Structure of nanowires and nanorods (long, free or loosely attached, quantum wires and quantum rods, but not gate-isolated embedded quantum wires))
62.20.mq	(Buckling)

Authors and contacts

Bio-inspired and Advanced Energy Research Center, School of Mechanics, Civil Engineering and Architecture, Northwestern Polytechnical University, Xi’an 710129, China

Corresponding author: Wang Feng-Hui, fhwang@nwpu.edu.cn

Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 11572253, 11972302).

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1. 引　言

锂离子电池由于具有较高的功率密度和较长的循环寿命等优点, 成为电动汽车、便携式电脑等产品中储能设备的首选^[1,2]. 然而, 在电化学锂化/脱锂化过程中, 由电极膨胀引起的力学损伤和化学损伤问题, 使得锂电池行业不能大踏步地商业化发展^[3-6]. 例如, 由于“锂化诱导应力”的存在, 电极颗粒经常性地表现出开裂、容量衰减等现象. 为了解决这个问题, 科研人员尝试对电极进行结构化设计, 并取得了一定的效果. 最近的一些实验表明, 通过对纳米电极结构化设计, 如纳米线^[7]、纳米薄膜^[8]、中空纳米颗粒^[9]、纳米多孔颗粒^[10]等, 这些手段能够有效地减轻由锂化引起的力学损伤. 在这众多纳米结构类型中, 纳米线电极由于易于合成并且具有独特的特性(高表面/体积比使得电极能够与电解质充分接触)而被认为是最具有前途的纳米电极材料^[11-13]. Chan等^[7]的先驱性工作证实, 与其他薄膜或者球形颗粒结构相比, 纳米线电极不易被粉化.

对于线基电极, 当端部受到的压力超过其临界屈曲压力时, 电极将不能保持原有的平衡方式而丧失稳定性, 将导致屈曲现象的发生. 这种屈曲行为在Si纳米线电极^[7,14]、SnO₂纳米线电极^[15]的锂化实验中已有所观察, 并且发现屈曲会导致电极的局部断裂、位错的形成以及表面开裂, 进而造成电池容量的衰退^[16]. Chakraborty等^[17]首次将屈曲失稳引入线基电极的应力-扩散双向耦合公式中, 并对经典欧拉屈曲理论进行了修正. 在这之后, 线基电极的屈曲理论得到了国内学者的极大关注, 在双相锂化电极^[18,19]、核-壳纳米线电极^[20]、率相关电极^[21,22]、浓度影响弹性性质电极^[23]、复合材料梯度电极^[24]等方面有了开拓性工作. 最近, Chakraborty团队^[25]又一次研究了纳米线电极的屈曲情况, 不过这次他们考虑了粘结剂对电极包裹的影响, 这样的建模更加符合实际情况.

另一方面, 当电极尺寸进入小尺度范围(如10 nm)时, 其比表面积增加, 表面效应对电极的影响就必须加以考虑. 这是因为当粒子的特征尺寸低于几十纳米时, 表面原子的能量状态与本体内部原子不同, 从而导致单位面积自由能增加, 也就是所谓的“表面张力”^[26]. Gurtin和Murdoch^[27]、Miller和Shenoy^[28]、Cammarata^[29]对纳米颗粒的表面效应理论工作做出了奠基性的贡献, 强调了表面残余张力和表面模量对纳米颗粒建模的重要性. 随着对纳米电极的不断探索, 表面效应与之相关的理论也在不断地跟进发展. 已有的理论表明, 表面效应对扩散诱导应力有衰减作用, 降低了纳米颗粒中的拉应力, 甚至随着颗粒半径的减小, 拉应力会降至压应力^[30], 也因此有大量的研究将纳米电极颗粒呈现出较好的力学可靠性的原因归因于表面效应^[31-37]. 尽管表面效应对电极有较好的力学影响, 但是由于表面效应对颗粒表面的压缩会阻止颗粒对锂离子的吸收^[38], 最终会降低电池的充电容量^[39], 并且随着表面效应的增强, 电池的电化学性能下降得越来越大^[40].

早在数十年前, 基于经典的欧拉屈曲理论, 有学者提出了一种通过测量纳米线的轴向屈曲临界载荷来确定其弹性模量的新方法^[41,42]. 然而, Wang和Feng^[43]通过理论研究发现, 使用经典的欧拉屈曲模型计算的结果可能是不准确的, 这是因为经典欧拉屈曲模型并没有考虑表面效应的作用, 会使得临界屈曲压力的数值偏低. 换句话说, 含有表面效应的纳米线的抗屈曲性能要比不含表面效应的抗屈曲性能要强. Wang和Feng^[43]理论修正了经典的欧拉屈曲理论, 为准确表征纳米线或纳米梁等的力学性质做出了贡献. 基于Wang和Feng^[43]的理论, 笔者猜测在理论上有表面效应的纳米线电极应该比没有表面效应的纳米线电极的抗屈曲性能要强, 如图1所示. 此外, 大多数纳米电极的力学属性, 如弹性模量, 是随着锂化浓度变化而变化. 而浓度影响弹性性质又分为两种: 弹性软化(如硅电极材料^[44])和弹性硬化(如石墨电极材料^[45]). 力学性质的变化对纳米线电极的屈曲行为非常重要, 因为电极的抗屈曲性很大程度上取决于材料的力学性质. 由于弹性模量的变化和表面效应的引入, 如果还是使用经典欧拉屈曲公式来计算纳米线电极的临界屈曲性质, 所得到的结果显然是不准确的. 基于此, 本文通过构建扩散-应力耦合模型来研究表面效应对纳米线电极的屈曲影响, 以期对电极结构优化及性能改善有所指导.

$图 1 纳米线电极在锂化过程中屈曲的示意图　(a)有表面效应; (b)无表面效应\r\nFig. 1. Schematic illustration of nanowire electrode buckling during lithiation: (a) With surface effect; (b) without surface effect.$

图 1 纳米线电极在锂化过程中屈曲的示意图　(a)有表面效应; (b)无表面效应

Fig. 1. Schematic illustration of nanowire electrode buckling during lithiation: (a) With surface effect; (b) without surface effect.

下载: 全尺寸图片幻灯片

2. 物理模型

本节的目的是构造一个考虑表面效应的柱形结构扩散-应力的耦合模型, 以探讨表面效应对纳米线电极屈曲的影响. 为了简化模型, 有必要遵循一些假设: 1)不考虑电极在锂化阶段的相分离现象; 2)仅考虑径向的锂扩散, 沿轴向方向的扩散则予以忽略; 3)本模型只适用于小体积锂化膨胀的电极材料, 如钴酸锂(锂化膨胀量仅2%^[46])等, 因此所采用的变形条件为小变形条件. 对于硅(锂化膨胀量达263%^[46])等大体积锂化膨胀的电极材料则应采用大变形条件, 但不在本文的研究范围内; 4) 在与电极相关的材料属性中, 只有弹性模量被假定为与锂浓度有关. 应当注意的是, 尽管有资料表明泊松比也是随浓度变化的, 但变化范围非常小(0—0.12)^[46], 因此可以忽略.

2.1 扩散方程

柱形电极结构中锂离子的扩散方程有如下形式:

$\frac{\partial c}{\partial t}=-\frac{1}{r}\frac{\partial \left(rj\right)}{\partial r} ,$

(1)

其中 t和 $r$ 分别是时间和沿径向方向某一点的位置, $c$ 是锂的浓度, $j$ 是锂的扩散通量. 依据热动力学理论, 通量 $j$ 与化学势 $\mu$ 有如下关系^[40]:

$j=-\frac{{D}_{0}}{{R}_{\mathrm{g}}T}{c}_{\mathrm{m}}\tilde{c}\left(1-\tilde{c}\right)\frac{\partial \mu }{\partial r} ,$

(2)

其中, ${R}_{\mathrm{g}}$ 和T分别是通用气体常数和绝对温度, ${D}_{0}$ 是锂的扩散系数, ${c}_{\mathrm{m}}$ 是理论上颗粒所能吸收锂的最大量, $\tilde{c}=c/{c}_{\mathrm{m}}$ 是锂浓度的正则形式. 化学势 $\mu$ 表示的是溶质在溶剂中的驱动力^[47]:

$\begin{split} \mu =\;&{R}_{\mathrm{g}}T\left(1-2\tilde{c}+\mathrm{ln}\frac{\tilde{c}}{1-\tilde{c}}\right)-\varOmega {\sigma }_{\mathrm{h}}-\frac{1}{2{E}^{2}}\\ & \times \frac{\partial E}{\partial c}\left[9v{\sigma }_{\mathrm{h}}^{2}-\left(1+v\right)\left({\sigma }_{r}^{2}+{\sigma }_{\theta }^{2}+{\sigma }_{z}^{2}\right)\right] , \end{split}$

(3)

式中, 第一项表示由扩散引起的贡献, 第二项是静水应力( ${\sigma }_{\mathrm{h}}=\left({\sigma }_{r}+{\sigma }_{\theta }+{\sigma }_{z}\right)/3$ )的贡献, 第三项表示由锂化引起的弹性模量变化和应力的贡献. 其中 $\varOmega$ 和 $v$ 分别为锂在电极中的偏摩尔体积和泊松比. ${\sigma }_{r}$ , ${\sigma }_{\theta }$ 和 ${\sigma }_{z}$ 是电极的径向、切向和轴向应力. E为颗粒的浓度依赖弹性模量, 并且假设与浓度 $c$ 呈线性关系:

$E={E}_{0}\left(1+k\tilde{c}\right) ,$

(4)

其中 ${E}_{0}$ 是电极未锂化时的弹性模量, $k$ 是弹性硬化( $k > 0$ )/软化( $k < 0$ )系数.

锂在电极颗粒中扩散的边界条件假设为恒电流条件, 即颗粒表面的通量由恒定的电流密度 ${i}_{\mathrm{n}}$ 控制:

$j\left(t,r={r}_{0}\right)={j}_{0}= {{i}_{\mathrm{n}}}/{F} ,$

(5)

其中, ${r}_{0}$ 为纳米线电极半径, F为法拉第常数.

对于初始条件, 这里假设颗粒内部的初始浓度为零, 即

$c\left(t=0,r\right)=0 .$

(6)

2.2 表面效应

对于纳米线, 可以将其视为由“内芯”和零厚度“表膜”组成的结构. 从连续力学角度来看, 是“表膜”对“内芯”施加了作用力, 两者共同构成了一个平衡系统. 该作用力被称为表面应力, 它是由表面原子周围的电子密度重新分布引起的物理应力. 对于各向同性材料, 表面应力 ${\sigma }_{\alpha \beta }^{0}$ 与变形依赖的表面能 $\varLambda \left({\varepsilon }_{\alpha \beta }\right)$ 有关^[48]:

${\sigma }_{\alpha \beta }^{0}=\tau {\delta }_{\alpha \beta }+\frac{\partial \varLambda }{\partial {\varepsilon }_{\alpha \beta }} ,$

(7)

其中, $\tau$ 为表面残余张力, 等式右边第二项表示的是由表面弹性引起的项. 应当注意的是, 由于材料的分子结构或合成工艺等原因可能造成表面效应是各向异性的, 然而截至目前鲜有实验和理论模拟提到对纳米线电极各向异性表面效应的探讨, 因此, 为了模型的简便和易于计算, 这里对电极的表面采取和前人相同的处理方式^{[30,31,35,38]}, 即假设表面是各向同性的; 另一方面, 各向同性表面效应的研究具有一般性, 其模型可以推广到各向异性表面效应的研究, 这个将在未来的工作中做进一步的探究.

对“表膜”和“内芯”的界面处, 由 Young-Laplace方程可得如下平衡方程:

${\sigma }_{\alpha j}{n}_{j}+{\sigma }_{\alpha \beta ,\beta }^{0}=0 ,\tag{8a}$

${\sigma }_{ij}{n}_{i}{n}_{j}={\sigma }_{\alpha \beta }^{0}{\kappa }_{\alpha \beta } , \tag{8b}$

式中, ${n}_{j}$ 和 ${\kappa }_{\alpha \beta }$ 分别为表面处的单位法向矢量和曲率张量.

为了模型的简便忽略对表面弹性的探讨, 因此(7)式可以简化为 ${\sigma }_{\alpha \beta }^{0}=\tau$ . 那么根据(8)式, 对于无约束条件的纳米线, “内芯”在3个主方向的内部应力 ${\mathit{\sigma }}^{i}$ 有如下形式:

$\left\{\begin{aligned} &{\sigma }_{1}^{i}=- {2\tau }/{{r}_{0}},\\ &{\sigma }_{2}^{i}=- {\tau }/{{r}_{0}},\\ &{\sigma }_{3}^{i}=- {\tau }/{{r}_{0}},\end{aligned}\right.$

(9)

其中, ${\sigma }_{1}^{i}$ , ${\sigma }_{2}^{i}$ , ${\sigma }_{3}^{i}$ 分别为轴向、径向、切向的内部主应力.

2.3 力学方程

将化学膨胀应变与热膨胀应变做类比^[49], 那么描述电极颗粒在锂化过程中引起的变形和应力的本构方程有如下形式:

$\left\{\begin{aligned} &{\varepsilon }_{r}=\frac{{\sigma }_{r}-v\left({\sigma }_{\theta }+{\sigma }_{z}\right)}{E}+{\varepsilon }^{\mathrm{s}\mathrm{w}},\\ &{\varepsilon }_{\theta }=\frac{{\sigma }_{\theta }-v\left({\sigma }_{r}+{\sigma }_{z}\right)}{E}+{\varepsilon }^{\mathrm{s}\mathrm{w}},\\ &{\varepsilon }_{z}=\frac{{\sigma }_{z}-v\left({\sigma }_{r}+{\sigma }_{\theta }\right)}{E}+{\varepsilon }^{\mathrm{s}\mathrm{w}},\end{aligned}\right.$

(10)

其中, ${\varepsilon }_{r}$ , ${\varepsilon }_{\theta }$ 和 ${\varepsilon }_{z}$ 分别为纳米线电极的径向、切向和轴向应变. ${\varepsilon }^{\mathrm{s}\mathrm{w}}=\varOmega c/3$ 为由锂化引起的化学膨胀应变.

此外, 总应变 ${\varepsilon }_{ij}$ 和径向位移u有如下关系:

${\varepsilon }_{r}=\frac{\partial u}{\partial r},\;\;{\varepsilon }_{\theta }=\frac{u}{r} .$

(11)

在不考虑体力的情况下, 轴对称柱形结构的平衡方程可有如下表达:

$\frac{\partial {\sigma }_{r}}{\partial r}+\frac{{\sigma }_{r}-{\sigma }_{\theta }}{r}=0 .$

(12)

同时, 在表面效应的影响下, 总的轴向应变 ${\varepsilon }_{z}^{\mathrm{t}}$ 是“内芯”的应变 ${\varepsilon }_{z}$ 与“表膜”的应变 ${\varepsilon }_{z}^{\mathrm{s}}$ 的等效之和:

${\varepsilon }_{z}^{\mathrm{t}}=\frac{{\varepsilon }_{z}\cdot {A}^{\mathrm{i}}+{\varepsilon }_{z}^{\mathrm{s}}\cdot {A}^{\mathrm{s}}}{{A}^{\mathrm{i}}+{A}^{\mathrm{s}}} ,$

(13)

式中, ${A}^{\mathrm{i}}$ 和 ${A}^{\mathrm{s}}$ 分别为“内芯”和“表膜”所占的面积. 但是对于金属等硬度比较大的材料, 由于材料的高强度, 其表面应力引起的变形非常小, 往往可以忽略^[50,51], 并且表面原子所占截面的面积也非常小, 因此在这里规定 ${\varepsilon }_{z}^{\mathrm{s}}=0$ , ${A}^{\mathrm{s}}=0$ . 应当注意的是, 不同于硬材料, 如今在材料领域比较火热的水凝胶等软物质, 其纳米尺寸的材料引起的变形一般不能忽略^[52].

于是, 在平面应变的条件下 ${\varepsilon }_{z}^{\mathrm{t}}=0$ , 也即

${{\varepsilon }_{z}=\varepsilon }_{z}^{\mathrm{t}}=0 .$

(14)

将(14)式代入(10)式中, 那么应力由应变表达的关系式如下:

${\sigma }_{r}=\frac{E\{(1-v)[{\varepsilon }_{r}-(1+v){\varepsilon }^{\mathrm{s}\mathrm{w}}]+v[{\varepsilon }_{\theta }-(1+v){\varepsilon }^{\mathrm{s}\mathrm{w}}]\}}{(1+v)(1-2v)} , \tag{15a}$

${\sigma }_{\theta }=\frac{E\{(1-v)[{\varepsilon }_{\theta }-(1+v){\varepsilon }^{\mathrm{s}\mathrm{w}}]+v[{\varepsilon }_{r}-(1+v){\varepsilon }^{\mathrm{s}\mathrm{w}}]\}}{(1+v)(1-2v)} , \tag{15b}$

${\sigma }_{z}=v\left({\sigma }_{r}+{\sigma }_{\theta }\right)-E{\varepsilon }^{\mathrm{s}\mathrm{w}} .\tag{15c}$

相应地, 初始和边界条件可定义如下:

$u\left(t=0,r\right)=0 ,\tag{16a}$

$u\left(t,{r}_{0}=0\right)=0 , \tag{1bb}$

${\sigma }_{r}\left(t,r={r}_{0}\right)=-\frac{\tau }{{r}_{0}} , \tag{16c}$

其中, (16b)式表示的是颗粒中心处的位移为零, 这是显而易见的, 因为电极结构为轴对称结构; (16c)式表示的是切向方向表面效应对纳米线本体内部的作用, 由(9)式的第二项推导而出. 应当注意的是, 由于本模型的约束条件为平面应变条件, 因此(9)式的第一项对轴向方向的边界条件将不再适用.

2.4 屈曲准则

当纳米线两端受到约束时, 由锂化引起的膨胀可能会导致结构的不稳定和屈曲的发生, 如果对表面效应加以考虑, 经典的欧拉屈曲公式将不再适用, 必须对其加以修正. 根据已有的研究结果^[43,53,54], 锂化纳米线的临界屈曲压力有如下表达:

${P}_{\mathrm{c}\mathrm{r}}=\chi \frac{{\mathrm{\pi }}^{2}{\left(EI\right)}^{*}}{{l}_{0}^{2}}+H \text{, }$

(17)

式中, $H=4\tau {r}_{0}$ 是表面张力对轴向载荷的贡献影响, ${l}_{0}$ 为纳米线电极的长度. $\chi$ 是一个无量纲常数用来确定纳米线的约束条件, 如 $\chi =1/4$ 对应的是一端固定, 另一端自由的情况; $\chi =1$ 对应的是两端铰支情况; $\chi =2$ 对应的是两端固定情况. ${\left(EI\right)}^{*}$ 是纳米线的等效柔性刚度, 有如下形式:

${\left(EI\right)}^{*}=\frac{1}{4}\mathrm{\pi }{E}'{r}_{0}^{4}=\frac{\mathrm{\pi }{r}_{0}^{2}}{2}\int _{0}^{{r}_{0}}E\left(\tilde{c}\right)r\mathrm{d}r ,$

(18)

其中, ${E}'=\dfrac1{\mathrm{\pi }{r}_{0}^{2}}{\displaystyle\int _{0}^{{r}_{0}}2\mathrm{\pi }rE\left(\tilde{c}\right)\mathrm{d}r}$ 是颗粒体的等效杨氏模量.

将(18)式代入(17)式, 可得到 ${P}_{\mathrm{c}\mathrm{r}}$ 新的表达式:

${P}_{\mathrm{c}\mathrm{r}}=\frac{\chi {\mathrm{\pi }}^{3}}{2}{\left(\frac{{r}_{0}}{{l}_{0}}\right)}^{2}\int _{0}^{{r}_{0}}E\left(\tilde{c}\right)r\mathrm{d}r+4\tau {r}_{0} .$

(19)

那么, 当纳米线电极受到的屈曲压力等于内部轴向应力的面积分与表面应力的线积分之和时, 系统达到临界屈曲状态, 即

${P}_{\mathrm{c}\mathrm{r}}={-F}_{z} .$

(20)

其中,

${F}_{z}=\int _{0}^{{r}_{0}}{\sigma }_{z}\left(2\mathrm{\pi }r\right)\mathrm{d}r+\tau \left(2\mathrm{\pi }{r}_{0}\right) .$

(21)

2.5 无量纲化

为了方便求解, 引入以下无量纲变量:

$\begin{split} & \tilde{r}=\frac{r}{{r}_{0}} , \; ~~ \tilde{t}=\frac{{D}_{0}}{{r}_{0}^{2}}t , \; ~~ {\tilde{\sigma }}_{r,\theta ,z,e}=\frac{{\tilde{\sigma }}_{r,\theta ,z,e}}{{E}_{0}} , \\ & \tilde{u}=\frac{u}{{r}_{0}} , ~~~~ \tilde{c}=\frac{c}{{c}_{\mathrm{m}}} ,~~{\tilde{P}}_{\mathrm{c}\mathrm{r}}=\frac{{P}_{\mathrm{c}\mathrm{r}}}{{E}_{0}{r}_{0}^{2}} , \\ & {\tilde{F}}_{z}=\frac{{F}_{z}}{{E}_{0}{r}_{0}^{2}} ,\;~~~~ \tilde j = \frac{{{r_0}j}}{{{D_0}{c_m}}}. \end{split}$

(22)

相应地, 可得到化学势的无量纲形式:

$\begin{split} \tilde{\mu }=\;&1-2\tilde{c}+\mathrm{ln}\frac{\tilde{c}}{1-\tilde{c}}-\frac{\varOmega {E}_{0}}{{R}_{\mathrm{g}}T}{\tilde{\sigma }}_{\mathrm{h}}-\frac{{E}_{0}k}{2{\left(1+k\tilde{c}\right)}^{2}{c}_{\mathrm{m}}{R}_{\mathrm{g}}T}\\ &\times\left[9v{{\tilde{\sigma }}_{\mathrm{h}}}^{2}-\left(1+v\right)\left({{\tilde{\sigma }}_{r}}^{2}+{{\tilde{\sigma }}_{\theta }}^{2}+{{\tilde{\sigma }}_{z}}^{2}\right)\right] .\\[-10pt]\end{split}$

(23)

扩散通量的无量纲形式:

$\tilde j = - \tilde c\left( {1 - \tilde c} \right)\frac{{\partial \tilde \mu }}{{\partial \tilde r}}.$

(24)

扩散方程的无量纲形式:

$\frac{{\partial \tilde c}}{{\partial \tilde t}} + \frac{{\partial \tilde j}}{{\partial \tilde r}} + \frac{{\tilde j}}{{\tilde r}} = 0.$

(25)

临界压力的无量纲形式:

${\tilde{P}}_{\mathrm{c}\mathrm{r}}=\chi {\mathrm{\pi }}^{2}{\left(\frac{{r}_{0}}{l}\right)}^{2}\left[\frac{\mathrm{\pi }}{2}\int _{0}^{1}\left(1+k\tilde{c}\right)\tilde{r}\mathrm{d}\tilde{r}\right]+\frac{4\tau }{{E}_{0}{r}_{0}} .$

(26)

轴力的无量纲形式:

${\tilde{F}}_{z}=\int _{0}^{1}{\tilde{\sigma }}_{z}\left(2\mathrm{\pi }\tilde{r}\right)\mathrm{d}\tilde{r}+\frac{2\mathrm{\pi }\tau }{{E}_{0}{r}_{0}} .$

(27)

以及力学平衡方程的无量纲形式:

$\frac{\partial {\tilde{\sigma }}_{r}}{\partial \tilde{r}}+\frac{{\tilde{\sigma }}_{r}-{\tilde{\sigma }}_{\theta }}{\tilde{r}}=0 .$

(28)

其他边界条件如下:

$\tilde j\left( {\tilde t,\tilde r = 1} \right) = {\tilde j_0} , ~~ \tilde j\left( {\tilde t,\tilde r = 0} \right) = 0 ,$

(29)

${\tilde{\sigma }}_{r}\left(\tilde{t},\tilde{r}=1\right)=-{\tau }/({{E}_{0}{r}_{0}}) .$

(30)

3. 结果与讨论

本文以钴酸锂(LiCoO₂)作为纳米线电极的活性材料, 如非特别说明, 相关的材料及热力学参数如表1所列. 同时, 屈曲模式的选择只考虑纳米线被约束在两个无摩擦平行板之间的情况(两端铰支), 即 $\chi =1$ . 此外, 无量纲通量的边界条件设为 ${\tilde j_0} = 0.01$ , 并且当纳米线电极表面浓度达到理论最大浓度 $\left({c}_{\mathrm{m}}\right)$ 时, 锂化终止.

表 1 材料参数

Table 1. Material parameters^[55].

物理参数	符号/单位	值
未锂化的弹性模量	${E}_{0}$ /GPa	70
泊松比	v	0.2
偏摩尔体积	$\varOmega /({\mathrm{m} }^{3}{\cdot}{\mathrm{m}\mathrm{o}\mathrm{l} }^{-1})$	1.92 × 10^–6
锂的最大浓度	${c}_{\mathrm{m} }/(\mathrm{m}\mathrm{o}\mathrm{l}{\cdot} {\mathrm{m} }^{-3})$	2.33 × 10⁴
扩散系数	${D}_{0}/({\mathrm{m} }^{2}{\cdot} {\mathrm{s} }^{-1})$	1.76 × 10^–15
表面残余张力	$\tau /(\mathrm{J}{\cdot}{\mathrm{m} }^{-2})$	2
长细比	${l}_{0}/{r}_{0}$	50
电极半径尺寸	${r}_{0}$ /nm	10

下载: 导出CSV

| 显示表格

图2(a)给出了临界屈曲压力随时间的变化情况, 可以看出, 有表面效应的临界屈曲应力总是要大于无表面效应的临界屈曲应力, 这反映了表面张力对纳米线电极抗屈曲失稳的积极性贡献. 此外, 当锂化逐步深入时, 弹性硬化/软化系数(k)深刻地影响了临界屈曲压力的走向. 如 $k=0.5$ (弹性硬化)时, 临界屈曲压力随着时间的增大而线性增大; $k=-0.5$ (弹性软化)时, 临界屈曲压力却随着时间的增大而线性减小, 这也能从(26)式中窥得端倪. 应当注意的是, 尽管拥有弹性硬化性质的纳米线电极比弹性软化性质的纳米线电极具有更高的临界屈曲压力, 但这并不意味着其抗屈曲性能要好, 这是因为决定力学可靠性的另一因素, 即轴力(图(3)), 也在变化. 临界屈曲压力与轴力的比值才最终决定了哪种性质的纳米线电极具有更好的抗失稳性. Zhang等^[23]曾研究了弹性软化的电极, 结果与本模型的理论预测一致, 但是他们并没有对弹性硬化的电极进行考究, 图2(a)的结果证明了弹性硬化和弹性软化的临界屈曲压力随时间的演化行为是不一样的, 应该分开讨论, 这也是本文研究浓度依赖弹性性质的意义所在. 图2(b)和图2(c)分别为有表面效应 ${(\tilde{P}}_{\mathrm{c}\mathrm{r}})$ 与无表面效应 $({\tilde{P}}_{\mathrm{c}\mathrm{r}, 0})$ 的初始临界屈曲压力的临界屈曲压力之比 ${(\tilde{P}}_{\mathrm{c}\mathrm{r}}/{\tilde{P}}_{\mathrm{c}\mathrm{r}, 0})$ 随纳米线电极半径 $\left({r}_{0}\right)$ 和长细比 $({l}_{0}/{r}_{0})$ 的变化情况, 从中很明显地看到表面效应的影响随着纳米线电极的尺寸增大而减小, 但是却随着纳米线电极的长细比增大而增大, 该结果与Wang和Feng^[43]的理论预测一致.

$图 2 临界屈曲压力的结果　(a)临界屈曲压力随时间的变化情况, 其中实线表示考虑了表面效应, 点虚线表示没有考虑表面效应, 且红色线表示弹性硬化($ k=0.5 $), 靛蓝色线表示无弹性硬化/软化($ k=0 $), 黑色线表示弹性软化($ k=-0.5 $); (b)有表面效应与无表面效应的初始临界屈曲压力之比随电极半径的变化情况; (c)有表面效应与无表面效应的初始临界屈曲压力之比随电极长细比的变化情况\r\nFig. 2. Results of critical buckling load: (a) Evolution of critical buckling load with time (The solid lines represent considering the surface effect, dotted lines represent without considering the surface effect, and red lines denote $ k=0.5 $, indigo lines denote $ k=0 $, black lines denote $ k=-0.5 $); (b) ratio of initial critical buckling load between with and without surface effect as a function of radius; (c) ratio of initial critical buckling load between with and without surface effect as a function of slenderness ratio.$

图 2 临界屈曲压力的结果　(a)临界屈曲压力随时间的变化情况, 其中实线表示考虑了表面效应, 点虚线表示没有考虑表面效应, 且红色线表示弹性硬化(

$k=0.5$

), 靛蓝色线表示无弹性硬化/软化(

$k=0$

), 黑色线表示弹性软化(

$k=-0.5$

); (b)有表面效应与无表面效应的初始临界屈曲压力之比随电极半径的变化情况; (c)有表面效应与无表面效应的初始临界屈曲压力之比随电极长细比的变化情况

Fig. 2. Results of critical buckling load: (a) Evolution of critical buckling load with time (The solid lines represent considering the surface effect, dotted lines represent without considering the surface effect, and red lines denote

$k=0.5$

, indigo lines denote

$k=0$

, black lines denote

$k=-0.5$

); (b) ratio of initial critical buckling load between with and without surface effect as a function of radius; (c) ratio of initial critical buckling load between with and without surface effect as a function of slenderness ratio.

下载: 全尺寸图片幻灯片

同样地, 表面效应、电极尺寸和弹性硬化/软化系数对轴力的影响也进行了探讨, 结果如图3所示. 从(27)式可以看到, 当弹性硬化/软化系数 $\left(k\right)$ 固定时, 表面应力的参与相当于给了纳米线电极一个初始的拉应力, 因此从图3(a)可以观察到在初始时刻有表面效应的轴力总是要大于无表面效应的轴力. 但随着锂化的进行, 由于轴向方向的约束, 圆柱体在轴向的膨胀会产生一个压缩应力, 来抵消表面效应的贡献, 因此有表面效应的轴力随着时间逐步降至为零, 再反向增加. 而无表面效应的轴力则在整个锂化阶段一直为负的轴向压力, 这与Chakraborty等^[17]和Zhang等^[56]的理论结果一致. 另一方面, 尽管弹性硬化和弹性软化引起的轴力随时间变化的趋势是一致的, 但是两者的变化速率是不同的, 如对于无表面效应的轴力, 弹性硬化引起轴力的变化速率随着时间不断变大, 而弹性软化引起轴力的变化速率随着时间不断减小, 这意味着对于某些弹性软化系数更大( $k < -0.5$ )的纳米线电极, 可能直至锂化结束电极内部的轴力仍为正的轴向拉力, 这反映了纳米线电极表面效应引起的轴力与扩散诱导轴力之间的竞争. 最后, 图3(b)给出了有表面效应的初始轴力随电极半径尺寸变化的曲线图, 可以看到初始轴力随着电极尺寸的增加而不断地减小. 这是显而易见的, 从(27)式可以看到初始轴力与电极半径是成反比例关系, 这从侧面也印证了表面效应背后的机制, 即随着尺寸的不断减小, 表面效应带来的影响却愈加不能忽视.

$图 3 纳米线电极轴力的结果　(a)轴力随时间的变化情况, 其中实线表示考虑了表面效应, 点虚线表示没有考虑表面效应, 且红色线表示弹性硬化($ k=0.5 $), 靛蓝色线表示无弹性硬化/软化($ k=0 $), 黑色线表示弹性软化($ k=-0.5 $); (b) 初始轴力随纳米线电极半径的变化情况\r\nFig. 3. Results of axial force $ {\stackrel{~}{F}}_{z} $: (a) Variation of axial force $ {\stackrel{~}{F}}_{z} $ with time $ \stackrel{~}{t} $ (The solid lines represent considering the surface effect, dotted lines represent without considering the surface effect, and red lines denote $ k=0.5 $, indigo lines denote $ k=0 $, black lines denote $ k=-0.5 $); (b) initial axial force with surface effect as a function of radius$

图 3 纳米线电极轴力的结果　(a)轴力随时间的变化情况, 其中实线表示考虑了表面效应, 点虚线表示没有考虑表面效应, 且红色线表示弹性硬化(

$k=0.5$

), 靛蓝色线表示无弹性硬化/软化(

$k=0$

), 黑色线表示弹性软化(

$k=-0.5$

); (b) 初始轴力随纳米线电极半径的变化情况

Fig. 3. Results of axial force

${\stackrel{~}{F}}_{z}$

: (a) Variation of axial force

${\stackrel{~}{F}}_{z}$

with time

$\stackrel{~}{t}$

(The solid lines represent considering the surface effect, dotted lines represent without considering the surface effect, and red lines denote

$k=0.5$

, indigo lines denote

$k=0$

, black lines denote

$k=-0.5$

); (b) initial axial force with surface effect as a function of radius

下载: 全尺寸图片幻灯片

图4(a)和图4(b)分别为电极细长比( ${l}_{0}/{r}_{0}$ )取不同值时, 纳米线电极有表面效应条件下轴力与临界屈曲压力之比( ${\tilde{F}}_{z}/{\tilde{P}}_{\mathrm{c}\mathrm{r}}$ )与无表面效应条件下轴力与临界屈曲压力之比( ${\tilde{F}}_{z, 0}/{\tilde{P}}_{\mathrm{c}\mathrm{r}, 0}$ )随时间的变化, 其中点虚线为临界屈曲线, 值为 ${\tilde{F}}_{z}/{\tilde{P}}_{\mathrm{c}\mathrm{r}}=-1$ 或 ${\tilde{F}}_{z, 0}/{\tilde{P}}_{\mathrm{c}\mathrm{r}, 0}=-1$ , 在临界屈曲线之上表示没有屈曲, 而临界屈曲线之下表示已经屈曲. 由图4(a)和图4(b)可知, 随着细长比的增大, 纳米线电极达到临界屈曲所需的时间变小, 表明电极长细比越大纳米线电极的力学稳定性越差, 结果与文献[18, 19, 22-24]的理论预测一致. 值得注意是, 通过对比图4(a)和图4(b)可以发现, 当电极长细比固定时, 有表面效应的纳米线电极达到临界屈曲所需的时间总是比无表面效应的纳米线电极要长, 这说明有表面效应的纳米线电极的力学稳定性要好于无表面效应的纳米线电极. 此外, 图4(c)给出了达到临界屈曲条件( ${\tilde{F}}_{z}/{\tilde{P}}_{\mathrm{c}\mathrm{r}}=-1$ )时有表面效应所需的时间( ${t}_{\mathrm{c}\mathrm{r}}$ )与无表面效应所需的时间( ${t}_{\mathrm{c}\mathrm{r}, 0}$ )之比随电极长细比( ${l}_{0}/{r}_{0}$ )的变化情况, 其趋势表明表面效应的影响随着长细比的增大而增大. 用同样的方法分析电极尺寸的影响情况, 结果如图4(d)—图4(f)所示, 通过对比图4(d)和图4(e)可以发现, 当电极尺寸固定时, 有表面效应的纳米线电极达到临界屈曲所需的时间总是比无表面效应的纳米线电极要长. 对于不同尺寸之间的电极比较, 有表面效应的小尺寸电极可能比无表面效应的大尺寸电极的抗屈曲性更好, 如有表面效应的6 nm尺寸的电极所需临界屈曲时间为1 s, 而无表面效应的10 nm尺寸的电极所需临界屈曲时间为0.35 s. 另外, 从图4(d)可以看到, 当有表面效应的纳米线电极尺寸小于4 nm时, 电极将永远不能达到屈曲条件; 相反, 对于无表面效应的纳米线电极(如图4(e)), 几乎任意尺寸的电极均达到了屈曲条件, 背后的原因可以通过图4(f)得到. 图4(f)为达到临界屈曲条件时有表面效应所需的时间( ${t}_{\mathrm{c}\mathrm{r}}$ )与无表面效应所需的时间( ${t}_{\mathrm{c}\mathrm{r}, 0}$ )之比随电极尺寸( ${r}_{0}$ )的变化情况, 可以看到, 表面效应对屈曲的影响随着电极半径尺寸的增加而减小, 这说明当电极尺寸越来越大时, 纳米线尺寸对屈曲的影响将回归经典的欧拉屈曲方式中, 即在经典的欧拉屈曲方式中, 杆件尺寸越大其抗屈曲性能越好. 但是, 当杆件尺寸非常小时, 其屈曲条件可能违背日常的认知, 如图4(d)中反映的尺寸小于4 nm时, 纳米线将永远不会屈曲, 这是因为在这个尺度范围内, 表面效应起到了绝对的主导作用, 这也从侧面解释了以往实验中纳米电极颗粒相对大尺寸电极颗粒表现出更好的力学可靠性的现象^[57].

$图 4 纳米线电极屈曲的结果情况　(a)有表面效应的条件下, 轴力与临界屈曲压力之比在不同的电极长细比下随时间的变化情况; (b)无表面效应的条件下, 轴力与临界屈曲压力之比在不同的电极长细比下随时间的变化情况; (c)有表面效应与无表面效应的纳米线电极达到临界屈曲所需的时间之比随电极长细比的变化情况; (d)有表面效应的条件下, 轴力与临界屈曲压力之比在不同的电极尺寸下随时间的变化情况; (e)无表面效应的条件下, 轴力与临界屈曲压力之比在不同的电极尺寸下随时间的变化情况; (f)有表面效应与无表面效应的纳米线电极达到临界屈曲所需的时间之比随电极尺寸的变化情况\r\nFig. 4. Results of nanowire electrode buckling: (a) Under the condition of with surface effect, the ratio of axial force to critical buckling load as a function of time for different slenderness ratio; (b) under the condition of without surface effect, the ratio of axial force to critical buckling load as a function of time for different slenderness ratio; (c) ratio of the time required for nanowire electrodes with surface effect and without surface effect to reach critical buckling varies with slenderness ratio; (d) under the condition of with surface effect, the ratio of axial force to critical buckling load as a function of time for different radius; (e) under the condition of without surface effect, the ratio of axial force to critical buckling load as a function of time for different radius; (f) ratio of the time required for nanowire electrodes with surface effect and without surface effect to reach critical buckling varies with radius.$

图 4 纳米线电极屈曲的结果情况　(a)有表面效应的条件下, 轴力与临界屈曲压力之比在不同的电极长细比下随时间的变化情况; (b)无表面效应的条件下, 轴力与临界屈曲压力之比在不同的电极长细比下随时间的变化情况; (c)有表面效应与无表面效应的纳米线电极达到临界屈曲所需的时间之比随电极长细比的变化情况; (d)有表面效应的条件下, 轴力与临界屈曲压力之比在不同的电极尺寸下随时间的变化情况; (e)无表面效应的条件下, 轴力与临界屈曲压力之比在不同的电极尺寸下随时间的变化情况; (f)有表面效应与无表面效应的纳米线电极达到临界屈曲所需的时间之比随电极尺寸的变化情况

Fig. 4. Results of nanowire electrode buckling: (a) Under the condition of with surface effect, the ratio of axial force to critical buckling load as a function of time for different slenderness ratio; (b) under the condition of without surface effect, the ratio of axial force to critical buckling load as a function of time for different slenderness ratio; (c) ratio of the time required for nanowire electrodes with surface effect and without surface effect to reach critical buckling varies with slenderness ratio; (d) under the condition of with surface effect, the ratio of axial force to critical buckling load as a function of time for different radius; (e) under the condition of without surface effect, the ratio of axial force to critical buckling load as a function of time for different radius; (f) ratio of the time required for nanowire electrodes with surface effect and without surface effect to reach critical buckling varies with radius.

下载: 全尺寸图片幻灯片

最后, 图5给出了弹性硬化/软化系数(k)对纳米线电极屈曲的影响. 如图5(b)所示, 在无表面效应的情况下, 弹性硬化和弹性软化对纳米线电极的屈曲行为基本上是没有区别的, 这个结果与Zhang等^[23]的理论预测结果完全符合. 相反, 在有表面效应的情况下, 如图5(a)所示, 弹性硬化系数( $k > 0$ )越大纳米线电极屈曲的时间越早, 而弹性软化系数( $k < 0$ )越大纳米线电极屈曲的时间却越晚, 甚至当 $k < -0.4$ 时, 纳米线电极将不可能屈曲. 其背后原因可以从(26)式和(27)式对比得到, 当无表面效应时, 只有扩散诱导应力对纳米线屈曲起作用, 轴力和临界屈曲压力之比可能对弹性硬化/软化系数不那么敏感, 造成图5(b)中两者近似无差别的现象; 但当有表面效应参与时, 即(26)式和(27)式两者的第二项, 拉大了弹性硬化与弹性软化之间的区分度. 因此从图5(a)和图5(b)观察到, 相对于弹性硬化, 具有弹性软化性质的纳米线电极具有更好的抗屈曲性. 但是如图5(c)所示, 这种表面效应的影响随着弹性硬化系数的增大而越来越弱, 却随着弹性软化系数的增大而越来越强. 总之, 图5(a)—图5(c)表明在纳米线电极的选择上, 拥有浓度依赖弹性软化性质的纳米线电极是最适合做电极材料的.

$图 5 弹性硬化/软化系数对屈曲的影响　(a)在有表面效应的条件下, 轴力与临界屈曲压力之比在不同的电极弹性硬化/软化系数下随时间的变化情况; (b)无表面效应的条件下, 轴力与临界屈曲压力之比在不同的电极弹性硬化/软化系数下随时间的变化情况; (c)有表面效应与无表面效应的纳米线电极达到临界屈曲所需的时间之比随电极弹性硬化/软化系数的变化情况\r\nFig. 5. Effect of elastic hardening/softening coefficient on buckling: (a) Under the condition of with surface effect, the ratio of axial force to critical buckling load as a function of time for different hardening/softening coefficient; (b) under the condition of without surface effect, the ratio of axial force to critical buckling load as a function of time for different hardening/softening coefficient; (c) ratio of the time required for nanowire electrodes with surface effect and without surface effect to reach critical buckling varies with hardening/softening coefficient.$

图 5 弹性硬化/软化系数对屈曲的影响　(a)在有表面效应的条件下, 轴力与临界屈曲压力之比在不同的电极弹性硬化/软化系数下随时间的变化情况; (b)无表面效应的条件下, 轴力与临界屈曲压力之比在不同的电极弹性硬化/软化系数下随时间的变化情况; (c)有表面效应与无表面效应的纳米线电极达到临界屈曲所需的时间之比随电极弹性硬化/软化系数的变化情况

Fig. 5. Effect of elastic hardening/softening coefficient on buckling: (a) Under the condition of with surface effect, the ratio of axial force to critical buckling load as a function of time for different hardening/softening coefficient; (b) under the condition of without surface effect, the ratio of axial force to critical buckling load as a function of time for different hardening/softening coefficient; (c) ratio of the time required for nanowire electrodes with surface effect and without surface effect to reach critical buckling varies with hardening/softening coefficient.

下载: 全尺寸图片幻灯片

4. 结　论

通过建立一个包含锂扩散、应力、浓度影响弹性模量的理论模型, 本文研究了表面效应对纳米线电极屈曲失稳的影响. 首先探讨了纳米线电极的半径、长细比、硬化/软化系数对有/无表面效应的临界屈曲压力和轴力的影响, 然后研究了这些变量对临界屈曲时间的影响. 主要结论如下: 1)与无表面效应相比, 表面效应提高了电极的力学稳定性, 延缓了纳米线电极的临界屈曲时间; 2)表面效应依赖于纳米线电极的半径尺寸和长细比, 即半径尺寸越小、长细比越大时表面效应的影响越大; 3)相对于弹性硬化, 如果考虑表面效应, 具有弹性软化性质的纳米线电极拥有更好的抗屈曲性.

由于本模型只适用于小体积锂化膨胀模型, 对于像硅(Si)等大体积锂化膨胀材料, 模型还需进行修正以考虑非线性变形等因素. 此外, 纳米线在锂化的过程中可能发生表面开裂行为, 颗粒表面的开裂失效会影响纳米线电极的临界屈曲时间, 这在今后的研究中也应该加以考虑. 本文的重要意义在于给出了在表面效应影响下的纳米线电极临界屈曲的理论模型, 证明了如果不考虑表面效应的影响, 理论预测结果是不准确的, 为电极的结构化设计和力学可靠性优化提供了帮助.

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Cited By

期刊类型引用(2)

1.	李清禄，张海坤，王思瑶，张靖华. 表面效应对多孔纳米梁后屈曲行为的影响分析. 力学与实践. 2023(01): 83-89 . 百度学术
2.	席智强，肖云华，席智勇，白静，谢林君. 水热纳米杆制备及屈曲性能分析. 山东化工. 2023(17): 42-44 . 百度学术

其他类型引用(1)

图 1 纳米线电极在锂化过程中屈曲的示意图　(a)有表面效应; (b)无表面效应

Figure 1. Schematic illustration of nanowire electrode buckling during lithiation: (a) With surface effect; (b) without surface effect.

DownLoad: Full-Size Img PowerPoint

图 2 临界屈曲压力的结果　(a)临界屈曲压力随时间的变化情况, 其中实线表示考虑了表面效应, 点虚线表示没有考虑表面效应, 且红色线表示弹性硬化( $k=0.5$ ), 靛蓝色线表示无弹性硬化/软化( $k=0$ ), 黑色线表示弹性软化( $k=-0.5$ ); (b)有表面效应与无表面效应的初始临界屈曲压力之比随电极半径的变化情况; (c)有表面效应与无表面效应的初始临界屈曲压力之比随电极长细比的变化情况

Figure 2. Results of critical buckling load: (a) Evolution of critical buckling load with time (The solid lines represent considering the surface effect, dotted lines represent without considering the surface effect, and red lines denote $k=0.5$ , indigo lines denote $k=0$ , black lines denote $k=-0.5$ ); (b) ratio of initial critical buckling load between with and without surface effect as a function of radius; (c) ratio of initial critical buckling load between with and without surface effect as a function of slenderness ratio.

DownLoad: Full-Size Img PowerPoint

图 3 纳米线电极轴力的结果　(a)轴力随时间的变化情况, 其中实线表示考虑了表面效应, 点虚线表示没有考虑表面效应, 且红色线表示弹性硬化( $k=0.5$ ), 靛蓝色线表示无弹性硬化/软化( $k=0$ ), 黑色线表示弹性软化( $k=-0.5$ ); (b) 初始轴力随纳米线电极半径的变化情况

Figure 3. Results of axial force ${\stackrel{~}{F}}_{z}$ : (a) Variation of axial force ${\stackrel{~}{F}}_{z}$ with time $\stackrel{~}{t}$ (The solid lines represent considering the surface effect, dotted lines represent without considering the surface effect, and red lines denote $k=0.5$ , indigo lines denote $k=0$ , black lines denote $k=-0.5$ ); (b) initial axial force with surface effect as a function of radius

DownLoad: Full-Size Img PowerPoint

Figure 4. Results of nanowire electrode buckling: (a) Under the condition of with surface effect, the ratio of axial force to critical buckling load as a function of time for different slenderness ratio; (b) under the condition of without surface effect, the ratio of axial force to critical buckling load as a function of time for different slenderness ratio; (c) ratio of the time required for nanowire electrodes with surface effect and without surface effect to reach critical buckling varies with slenderness ratio; (d) under the condition of with surface effect, the ratio of axial force to critical buckling load as a function of time for different radius; (e) under the condition of without surface effect, the ratio of axial force to critical buckling load as a function of time for different radius; (f) ratio of the time required for nanowire electrodes with surface effect and without surface effect to reach critical buckling varies with radius.

DownLoad: Full-Size Img PowerPoint

Figure 5. Effect of elastic hardening/softening coefficient on buckling: (a) Under the condition of with surface effect, the ratio of axial force to critical buckling load as a function of time for different hardening/softening coefficient; (b) under the condition of without surface effect, the ratio of axial force to critical buckling load as a function of time for different hardening/softening coefficient; (c) ratio of the time required for nanowire electrodes with surface effect and without surface effect to reach critical buckling varies with hardening/softening coefficient.

DownLoad: Full-Size Img PowerPoint

表 1 材料参数

Table 1. Material parameters^[55].

物理参数	符号/单位	值
未锂化的弹性模量	${E}_{0}$ /GPa	70
泊松比	v	0.2
偏摩尔体积	$\varOmega /({\mathrm{m} }^{3}{\cdot}{\mathrm{m}\mathrm{o}\mathrm{l} }^{-1})$	1.92 × 10^–6
锂的最大浓度	${c}_{\mathrm{m} }/(\mathrm{m}\mathrm{o}\mathrm{l}{\cdot} {\mathrm{m} }^{-3})$	2.33 × 10⁴
扩散系数	${D}_{0}/({\mathrm{m} }^{2}{\cdot} {\mathrm{s} }^{-1})$	1.76 × 10^–15
表面残余张力	$\tau /(\mathrm{J}{\cdot}{\mathrm{m} }^{-2})$	2
长细比	${l}_{0}/{r}_{0}$	50
电极半径尺寸	${r}_{0}$ /nm	10

DownLoad: CSV

[1]	Larcher D, Tarascon J 2015 Nat. Chem. 7 19 Google Scholar
[2]	Scrosati B, Garche J 2010 J. Power Sources 195 2419 Google Scholar
[3]	Zhang S, Zhao K, Zhu T, Li J 2017 Prog. Mater. Sci. 89 479 Google Scholar
[4]	Gür T 2018 Energy Environ. Sci. 11 2696 Google Scholar
[5]	Edge J, O’Kane S, Prosser R, Kirkaldy N, Patel A, Hales A, Ghosh A, Ai W, Chen J, Yang J, Li S, Pang M, Bravo D, Tomaszewska A, Marzook M, Radhakrishnan K, Wang H, Patel Y, Wu B, Offer G 2021 Phys. Chem. Chem. Phys. 23 8200 Google Scholar
[6]	Zhang W 2011 J. Power Sources 196 13 Google Scholar
[7]	Chan C, Peng H, Liu G, McIlwrath K, Zhang X, Huggins R, Cui Y 2008 Nat. Nanotechnol. 3 31 Google Scholar
[8]	Yu C, Li X, Ma T, Rong J, Zhang R, Shaffer J, An Y, Liu Q, Wei B, Jiang H 2012 Adv. Energy Mater. 2 68 Google Scholar
[9]	Yao Y, McDowell M, Ryu I, Wu H, Liu N, Hu L, Nix W, Cui Y 2011 Nano Lett. 11 2949 Google Scholar
[10]	Kim H, Han B, Choo J, Cho J 2008 Angew. Chem. 120 10305 Google Scholar
[11]	Dasgupta N P, Sun J, Liu C, Brittman S, Andrews S C, Lim J, Gao H, Yan R, Yang P 2014 Adv. Mater. 26 2137 Google Scholar
[12]	Chen L, Lu N, Xu C, Yu H, Wang T 2009 Electrochim. Acta 54 4198 Google Scholar
[13]	Chan C, Zhang X, Cui Y 2008 Nano Lett. 8 307 Google Scholar
[14]	Liu, X, Fan F, Yang H, Zhang S, Huang J, Zhu T 2013 ACS Nano 7 1495 Google Scholar
[15]	Huang J, Li Z, Chong M, John P, Wu X, Li Q, Scott X, Nicholas S, Xiao H, Arunkumar S 2010 Science 330 1515 Google Scholar
[16]	Wang X, Fan F, Wang J, Wang H, Tao S, Yang A, Liu Y, Beng C, Mao S, Zhu T, Xia S 2015 Nat. Commun. 6 8417 Google Scholar
[17]	Chakraborty J, Please C, Goriely A, Chapman S 2015 Int. J. Solids Struct. 54 66 Google Scholar
[18]	Zhang K, Li Y, Wu J, Zheng B, Yang F 2018 Int. J. Solids Struct. 144 289 Google Scholar
[19]	Shen X, Wan Y 2021 Meccanica 1 1 Google Scholar
[20]	Li Y, Zhang K, Zheng B, Yang F 2016 J. Phys. D: Appl. Phys. 49 285602 Google Scholar
[21]	Zhang K, Chen J, Li Y, Liu D, Zheng B, Kai Y 2020 Results Phys. 16 103018 Google Scholar
[22]	Li Y, Mao W, Zhang K, Jia Y, Yang F 2019 J. Phys. D: Appl. Phys. 52 435502 Google Scholar
[23]	Zhang Y, Zhan S, Zhang K, Zheng B, Lyu L 2021 Eur. J. Mech. A/Solids 85 104111 Google Scholar
[24]	Xing H, Liu Y, Wang B 2019 Acta Mech. 230 4145 Google Scholar
[25]	Yeerella R, Boddeda H, Sengupta A, Chakraborty J 2020 J. Appl. Phys. 128 234901 Google Scholar
[26]	Herring C 1951 Phys. Rev. 82 87 Google Scholar
[27]	Gurtin M, Murdoch A 1975 Arch. Ration. Mech. Anal. 57 291 Google Scholar
[28]	Miller R, Shenoy V 2000 Nanotechnology 11 139 Google Scholar
[29]	Cammarata R 1994 Prog. Surf. Sci. 46 1 Google Scholar
[30]	Cheng Y, Verbrugge M 2008 J. Appl. Phys. 104 083521 Google Scholar
[31]	Deshpande R, Cheng Y, Verbrugge M 2010 J. Power Sources 195 5081 Google Scholar
[32]	Hao F, Gao X, Fang D 2012 J. Appl. Phys. 112 103507 Google Scholar
[33]	Sengupta A, Chakraborty J 2019 Acta Mech. 231 999 Google Scholar
[34]	Gao Y, Zhou M 2011 J. Appl. Phys. 109 014310 Google Scholar
[35]	Gao X, Fang D, Qu J 2015 Proc. R. Soc. London, Ser. A 471 20150366 Google Scholar
[36]	Jia N, Peng Z, Wang S, Li J, Yao Y, Chen S 2020 Sci. China Ser. E: Technol. Sci. 63 2413 Google Scholar
[37]	Zhang X, Chen H, Fang D 2020 Int. J. Mech. Sci. 169 105323 Google Scholar
[38]	Stein P, Zhao Y, Xu B 2016 J. Power Sources 332 154 Google Scholar
[39]	Bucci G, Swamy T, Bishop S, Sheldon B, Chiang Y, Carter W 2017 J. Electrochem. Soc. 164 A645 Google Scholar
[40]	Lu Y, Zhang P, Wang F, Zhang K, Zhao X 2018 Electrochim. Acta 274 359 Google Scholar
[41]	Young S, Ji L, Chang S, Fang T, Hsueh T, Meen T, Chen I 2007 Nanotechnology 18 225603 Google Scholar
[42]	Riaz M, Nur O, Willander M, Klason P 2008 Appl. Phys. Lett. 92 103118 Google Scholar
[43]	Wang G, Feng X 2009 Appl. Phys. Lett. 94 141913 Google Scholar
[44]	Shenoy V, Johari P, Qi Y 2010 J. Power Sources 195 6825 Google Scholar
[45]	Qi Y, Guo H, Hector J, L G, Timmons A 2010 J. Electrochem. Soc. 157 A558 Google Scholar
[46]	Qi Y, Hector L, James C, Kim K 2014 J. Electrochem. Soc. 161 F3010 Google Scholar
[47]	Swaminathan N, Balakrishnan S, George K 2015 J. Electrochem. Soc. 163 A488 Google Scholar
[48]	Sharma P, Ganti S, Bhate N 2003 Appl. Phys. Lett. 82 535 Google Scholar
[49]	Zhang X, Shyy W, Marie S 2007 J. Electrochem. Soc. 154 A910 Google Scholar
[50]	Tian L, Rajapakse R 2007 Int. J. Solids Struct. 44 7988 Google Scholar
[51]	He L, Li Z 2006 Int. J. Solids Struct. 43 6208 Google Scholar
[52]	Wang S, Li X, Yi X, Duan H 2021 Extreme Mech. Lett. 44 101211 Google Scholar
[53]	Wang G, Feng X 2007 Appl. Phys. Lett. 90 231904 Google Scholar
[54]	He J, Lilley C 2008 Nano Lett. 8 1798 Google Scholar
[55]	Song X, Lu Y, Wang F, Zhao X, Chen H 2020 J. Power Sources 452 227803 Google Scholar
[56]	Zhang K, Li Y, Zheng B, Wu G, Wu J, Yang F 2017 Int. J. Solids Struct. 108 230 Google Scholar
[57]	Lee S, McDowell M, Berla L, Nix W, Cui Y 2012 Proc. Natl. Acad. Sci. U. S. A. 109 4080 Google Scholar

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