1.School of Mechanical and Power Engineering, Tongji University, Shanghai 200433, China 2.Shanghai Institute of Applied Mathematics and Mechanics, Shanghai 200444, China 3.Air Conditioning Electronics Department, Pan Asia Technical Automotive Center Co., Ltd., Shanghai 201201, China
Fund Project:Project supported by the National Key R&D Program of China (Grant No. 2017YFE0101400).
Received Date:29 December 2018
Accepted Date:25 February 2019
Available Online:01 May 2019
Published Online:05 May 2019
Abstract:As one of high capacity electrode materials of lithium ion battery, silicon suffers significant stress effects, which further affects the voltage performance of battery. In this paper, a reaction-diffusion-stress coupled model is established, and the stress induced voltage hysteresis with consideration of diffusion induced stress, surface effects and interparticle compression under potentiostatic operation are investigated. It is found that stress and stress induced voltage hysteresis are dependent on particle size. For big particles, the diffusion induced stress is dominant and further aggravates the hysteresis of both stress and the overpotential consumed by it, indicating that more energy dissipates due to the stress effects. For small particles, especially ones with radius of a few nanometers, surface effects play a more prominent role than diffusion induced stress and the stress evolves into the state of compressive stress on the whole, leading the hysteresis of overpotential to be consumed by stress shrink and making the hysteresis plot of overpotential used to drive electrochemical reaction move downward. The electrode potential first reaches a cutoff voltage and finally the capacity of lithium ion battery decays. Therefore, too large or too small particle size in the electrode can both have a negative effect on the performance of lithium ion batteries, which indicates that an optimal size of the electrode particles must be designed in terms of electrode structure. Based on the calculation, particles with around 9 nm in radius are an appropriate option for electrode design in consideration of both diffusion induced stress and surface effect. In addition, for silicon electrodes, the silicon particles inevitably squeeze each other in a charge and discharge cycle. Therefore, interparticle compression is considered in this case. In detail, interparticle compression pushes the plot of stress hysteresis to the compressive state and leads to lower lithiation capacity, which makes the overpotential plot consumed by stress move downward and accordingly the overpotential plot used to drive the electrochemical reaction move upward. Denser electrode would strengthen this effect due to higher particle compression. It is indicated that for electrode design, the minimum of porosity ratio of electrodes should be adopted because higher interparticle compressive stress would reduce the battery capacity. Our results reveal that the voltage hysteresis of lithium ion batteries is related to the active particle size and the porosity ratio of the electrode, which is of great significance for guiding one in designing the lithium ion batteries. Keywords:lithium ion batteries/ surface effects/ interparticle compression/ voltage hysteresis
得到. (5)式表明, 电极电势同时取决于反应速率和应力状态. 甚至在开路状态下(in等于0), 电极中应力状态的变化也同样可以引起电极电势的变化. 依据定义, 平衡电势Eeq指的是电极在不考虑应力状态下的开路电势[12], 而实际情况中, 一方面由于基底约束将不可避免地产生应力, 另一方面电极颗粒之间的挤压也会产生应力, 因此无应力状态的电极并不存在. 本文计算电极的平衡电势采用Sethuraman等[19]的多项式拟合结果, 拟合结果同时考虑了锂化和去锂化的过程, 如(6)式和图2所示. 为尽可能地减小应力和其他副反应导致的误差, 实验中将电池进行了长时间的静置. 图 2 Sethuraman等[19]硅电极开路电势的拟合结果和实验结果, 其中红色的数据点是实验测得的锂化数据, 蓝色数据点是实验测得的去锂化数据, 实线是在C/8恒流充放电条件下的实验曲线, 虚线是依据实验数据拟合的结果 Figure2. Fitting results and experimental results of open-circuit potential of silicon electrode proposed by Sethuraman et al.[19]. The red data point is the lithium data measured in the experiment, and the blue data point is the dilithiated data measured in the experiment. The solid line was obtained under a C/8 constant current charge-discharge operation, and the dashed line is the fitting function
表1活性材料的材料参数[13,37-40] Table1.The material parameters of active materials[13,37-40]
图3(a)是700 nm的电极颗粒在恒压条件下的浓度演化图, x轴代表无量纲的半径$\overline r = r/R$, 变化范围由颗粒中心0至颗粒表面1; y轴是颗粒中的无量纲浓度$\overline c = c/{c_{\max}}$; 其中实线和虚线分别代表锂化和去锂化过程的浓度分布. 可以看出, 锂化阶段活性颗粒的浓度由外而内减少. 因此, 颗粒表层的膨胀量更大, 在颗粒内层的约束作用下外层受压. 同理, 在去锂化过程中, 浓度梯度为负, 导致颗粒表层受拉. 变化趋势上, 颗粒表层的浓度梯度先增大后减小, 因此应力的演化趋势也相应地先增加后减少. 从图3(b)可以看出, 不同尺寸的颗粒在锂化过程中扩散诱导应力为压应力, 而去锂化过程中其为拉应力, 从而形成了迟滞环. 图 3 扩散诱导应力对电压迟滞的影响 (a)电极颗粒在锂化和去锂化过程中锂离子的浓度分布; (b)一次充放电循环中, 不同尺寸的颗粒扩散诱导应力演化 Figure3. Effect of diffusion induced stress on voltage hysteresis: (a) Distribution of concentration of lithium ions in a electrode particle during lithiation and delithiation; (b) diffusion-induced stress evolution diagrams of particles with different sizes in primary charge-discharge cycle
为了进一步明确扩散诱导应力部分的过电势在总过电势中所占比重, 图5为锂化和去锂化过电势差值在不同颗粒尺寸下的变化关系. 此处的差值指的是在一次锂化和去锂化过程中最大值与最小值之差. 由于是恒压充放电, 所以总过电势的差值一直保持为750 mV. 随着颗粒尺寸从10 nm增加到700 nm, 扩散诱导应力部分的过电势差值由58.55 mV增加到300 mV, 所占百分比由7.8%增加到40%. 因此对于大颗粒来讲, 扩散诱导应力对电压迟滞的影响较为显著. 图 5 应力分担过电势的差值、总过电势差值及其所占百分比在不同颗粒尺寸下的变化, 其中差值是指过电势回线中最大值与最小值之差 Figure5. Dependence of overpotential gap consumed by stress, total overpotential gap and the corresponding percentage on different particle sizes. The gap refers to the difference between the maximum value and the minimum value in the overpotential loop
24.2.表面效应对电压迟滞的影响 -->
4.2.表面效应对电压迟滞的影响
由(17a)式可知表面效应引起的静水应力由两部分组成, 一部分取决于颗粒的平均锂离子浓度cav(R), 表明扩散与表面效应是耦合的, 另一部分是不依赖于浓度的静水应力. 因此, 在充放电循环中由表面效应引起的表面张力与归一化浓度存在线性的变化关系且没有迟滞效应. 图6为不同尺寸的颗粒在一个充放电循环中由表面效应引起的静水应力演化图及相应的应力变化范围. 3张子图都表明表面应力随归一化浓度线性变化且无迟滞效应. 此外, 当颗粒尺寸小于100 nm时平均应力和应力的变化幅值增长很快. 从黑色曲线可以看出, 对于尺寸在100 nm以上的颗粒表面效应可以忽略. 图 6 由表面效应引起的表面张力及其分担过电势的演化曲线, 其中黑色曲线的应力值为一次充放电循环中的平均应力值, 竖直曲线为各自的变化范围; 3张子图为对应的表面张力演化曲线 Figure6. Evolution curve of surface tension due to surface effect and the corresponding overpotential. The dark line represents the mean stress in a cycle and the bar defines the range. The three subplots are evolutions of surface stress due to surface effects in a cycle
现在, 将扩散诱导应力和表面效应耦合在一起并讨论二者的相互竞争关系. 如图7(a)为不同颗粒尺寸下扩散诱导应力和表面效应耦合后的总应力演化曲线. 可以看出, 随着颗粒尺寸的减小, 应力迟滞效应在逐渐地减弱, 尤其在颗粒尺寸达到4 nm时, 应力迟滞环几乎消失. 另外, 随着颗粒尺寸的减小, 表面效应增强, 应力曲线整体上向压应力状态演化, 由对称状态向非对称状态演化. 这一现象预示着扩散诱导应力和表面效应之间的竞争机制. 依据图3(b)和图6, 当颗粒尺寸大于100 nm时, 扩散诱导应力占据主导而表面效应可以忽略, 进而导致明显的、对称的应力迟滞环. 然而, 当颗粒尺寸小于100 nm时, 表面效应变得显著而扩散诱导应力逐渐变得不明显, 原因为颗粒由外到内更短的扩散路径导致锂离子浓度趋于均匀. 图 7 (a)不同颗粒尺寸下, 表面张力和扩散诱导应力共同作用下的表面静水应力演化图; (b), (c), (d), (e)分别是颗粒尺寸为4, 10, 100和400 nm时的各部分过电势演化图 Figure7. (a) Evolution of surface hydrostatic stresses in consideration of both diffusion induced stress and surface effects under different particle sizes; (b), (c), (d), (e) evolutions of all parts of overpotential in the particles of 4, 10, 100 and 400 nm radius, respectively
图7(b)—(e)为不同颗粒尺寸下总过电势、总应力所占过电势和驱动电化学反应过电势的演化图. 可以看出, 随着颗粒尺寸的减少, 总应力所占过电势的回线在缩小, 使更高的过电势用于驱动实际的电化学反应. 同时由于颗粒表面效应的增强, 一方面使总应力所占过电势回线下移, 回线逐渐地失去对称性, 另一方面颗粒的锂化容量减少, 这是因为锂化过程表面效应产生的压应力阻碍了锂化过程. 综上所述, 在锂离子电极设计时, 需要考虑电极颗粒的最优尺寸, 如果颗粒尺寸过大, 扩散诱导应力所占过电势的回线将加大, 电压迟滞效应增强导致更多的能量损失. 如果颗粒尺寸太小, 表面效应将导致更高的表面张力, 阻碍锂化反应. 基于这两种因素, 需要将电极的颗粒尺寸控制在一个合理的范围内. 图8为基于上述计算结果, 不同颗粒尺寸下扩散诱导应力和由表面效应引起的静水应力对比图. 其中扩散诱导应力取为一次充放电循环中最大应力值与最小应力值之差的一半, 表面效应引起的静水应力取为一次充放电循环中应力的最大值的绝对值. 在以上参数选取下, 恒压充放电时, 表面效应和扩散诱导应力随颗粒尺寸的演化关系图如图8所示. 综合考虑表面效应和扩散诱导应力间的竞争机制, 应该选择表面效应和扩散诱导应力之和的最小值处对应的颗粒半径为最优尺寸. 两种效应之和的演化曲线如图9所示, 由图9可知, 半径为10 nm的颗粒对应的表面效应和扩散诱导应力之和最小, 即半径为10 nm的硅颗粒表现出较为均衡的综合性能. 因此, 从应力及其过电势回线的角度出发, 对于电极设计而言, 颗粒尺寸选取在10 nm左右是较为合理的选择. 图 8 不同颗粒尺寸下扩散诱导应力和表面效应引起的静水应力的绝对值对比 Figure8. Absolute values of the hydrostatic stress due to surface effects and the surface stress due to diffusion induced stress under different particle sizes.
图 9 不同颗粒尺寸下扩散诱导应力与表面效应之和的演化 Figure9. Evolution diagram of the sum of diffusion induced stress and surface effect under different particle sizes
24.3.电极中颗粒间挤压对电压迟滞的影响 -->
4.3.电极中颗粒间挤压对电压迟滞的影响
在充放电循环过程中锂离子电池的电极颗粒发生膨胀变形, 不可避免地要与周围的电极颗粒发生挤压, 从而产生颗粒间的压应力. 众所周知, 对于孔隙率较低的电极, 颗粒间的挤压力尤其显著. 因此, 本节讨论分析电极内颗粒间的挤压力对电压迟滞的影响. 考虑到通常情况下实际电极的颗粒尺寸为微米量级, 此部分颗粒尺寸取为1 μm, 充放电条件为恒压充放电, 锂化时电极电势保持为0.24 V, 去锂化时电极电势保持为0.51 V. 图10表明颗粒间的挤压作用显著影响了颗粒内的应力. 图10(a)为4种情况下即单独一个球颗粒和3种不同孔隙率电极内的球颗粒, 表面应力演化图. 可见由于颗粒间的挤压, 使应力迟滞回线逐渐失去对称性, 朝着压应力状态演化. 即使在较低的孔隙率条件下, 仍然十分明显. 图 10 颗粒间挤压对电压迟滞的影响 (a) 不同孔隙下电极的应力迟滞回线图; (b)?(e) 不同孔隙率下电极各部分过电势的回线图; 其中p为电池结构的孔隙率 Figure10. Impacts of interparticle compression on the voltage hysteresis: (a) Stress hysteresis for electrodes with different porosity ratios; (b)?(e) loop diagram of all parts of overpotential in the electrode with different porosity ratios. p is the porosity of the electrode structure.