Synthesis and Investigation of the nanocrystalline Li1.2Ni0.2Mn0.6O2 cathodes for Li-ion batteries by using ultrasonic/microwave-assisted co-precipitation method with different ultrasonic time
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摘要: 通过超声微波共沉淀法制备一系列纳米正极材料Li1.2Ni0.2Mn0.6O2。通过X射线衍射,扫描电镜,X射线光电子能谱和电化学方法研究超声时间对合成材料的影响。结果表明反应时间为2h时,材料表现出最优异的电化学性能,其在0.1C和2C倍率下的容量分别为265mAh.g-1 和180mAh.g-1。材料优异的电化学性能取决于其均一的颗粒粒径,理想的元素分布和高反应活性的氧化还原电对。超声微波系统可以在富锂材料的实际生产中起到良好的辅助作用,它使用方便,且能够节约时间。
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关键词:
- 共沉淀
Abstract: A series of nanocrystalline lithium-rich cathode materials (Li1.2Ni0.2Mn0.6O2) have been prepared via an ultrasonic/microwave-assisted co-precipitation method. The effects of the ultrasonic time are emphasis investigated by using X-ray diffraction, scanning electron microscopy, X-ray photoelectron spectroscopy and electrochemical measurements. The optimum reaction time is 2h, while the sample can exhibit the best electrochemical properties with an initial discharge capacity of 265mAh.g-1 at 0.1C and 180mAh-1 at 2C after 90 cycles, respectively. The superior electrochemical performance of this material can be attributed to the uniform particles, desired element distribution and high activities of the redox couples in the bulk material. This study also suggests that the ultrasonic/microwave system can be a good assistant in the practical Li-rich material production, while it can be easy to use and time-saving.-
Keywords:
- Co-precipitation
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设1p+1q=1(p>1),α, β ∈R,K(x, y)非负可测,若Lαp(0,+∞)={f(x)⩾0:‖f‖p,α=(∫+∞0xαfp(x)dx)1/p<+∞}, 则称不等式
∫+∞0∫+∞0K(x,y)f(x)g(y)dxdy⩽M‖f‖p,α‖g‖q,β 为Hilbert型积分不等式. 由于此类不等式与积分算子T:
T(f)(y)=∫+∞0K(x,y)f(x)dx 有密切的联系,故而Hilbert型积分不等式对于研究算子T的有界性与算子范数有重要意义.
1991年,XU和GAO[1]首次提出了研究Hilbert型不等式的权系数方法. 该方法的核心是:引入2个搭配参数a、b,利用Hölder不等式,可得到如下形式的不等式:
∫+∞0∫+∞0K(x,y)f(x)g(y)dxdy⩽W1/p1(b,p)W1/q2(a,q)(∫+∞0xα(a,b)fp(x)dx)1/p×(∫+∞0yβ(a,b)gq(y)dx)1/q. (1) 一般地,随意选取的搭配参数a、b并不能使式(1)的常数因子W11/p(b, p)W21/q(a, q)最佳. 已有的相关研究[2-13]基本上都是凭借丰富的经验和娴熟的分析技巧选取适当的搭配参数a、b,从而获得最佳的Hilbert型不等式.
若选取的搭配参数a、b能够使式(1)的常数因子最佳,则称其为适配参数或适配数. 文献[14]曾讨论了齐次核的Hilbert型级数不等式的适配参数问题,本文将对拟齐次核的Hilbert型积分不等式讨论搭配参数a、b成为适配数的充分必要条件,并讨论其应用.
1. 预备知识
设G(u, v)是λ阶齐次函数,λ1λ2>0,则称K(x, y)=G(xλ1, yλ2)为拟齐次函数. 显然K(x, y)为拟齐次函数等价于:对t>0,有
K(tx,y)=tλ1λK(x,t−λ1/λ2y),K(x,ty)=tλ2λK(t−λ2/λ1x,y). 下面给出本文证明过程中所需的引理.
引理1 设1/p+1/q=1 (p>1),a, b, λ∈R,λ1λ2>0,G(u, v)是λ阶齐次非负函数,K(x, y)=G(xλ1, yλ2),aq/λ1+bp/λ2=1/λ1+1/λ2+ λ,记
W1(b,p)=∫+∞0K(1,t)t−bp dt,W2(a,q)=∫+∞0K(t,1)t−aq dt, 则W1(b, p)/λ1=W2(a, q)/λ2,且
ω1(b,p,x)=∫+∞0K(x,y)y−bp dy=xλ1(λ−bp/λ2+1/λ2)W1(b,p),ω2(a,q,y)=∫+∞0K(x,y)x−aq dx=yλ2(λ−aq/λ1+1/λ1)W2(a,q). 证明 由aq/λ1+bp/λ2=1/λ1+1/λ2+ λ,可得- λ1λ + λ1bp/λ2-λ1/λ2-1=-aq. 则有
W1(b,p)=∫+∞0tλ2λK(t−λ2/λ1,1)t−bp dt= λ1λ2∫+∞0K(u,1)u−λ1λ+λ1bp/λ2−λ1/λ2−1 du= λ1λ2∫+∞0K(u,1)u−aq du=λ1λ2W2(a,q), 故W1(b, p)/λ1=W2(a, q)/λ2.
作变换y=xλ1/λ2t,有
ω1(b,p,x)=∫+∞0xλ1λK(1,x−λ1/λ2y)y−bp dy= xλ1(λ−bp/λ2+1/λ2)∫+∞0K(1,t)t−bp dt= xλ1(λ−bp/λ2+1/λ2)W1(b,p). 同理可证ω2(a, q, y)=yλ2(λ-aq/λ1+1/λ1)W2(a, q). 证毕.
2. 适配参数的充分必要条件
定理1 设1/p+1/q=1 (p>1),a, b, λ∈R,λ1λ2>0,G(u, v)是λ阶齐次非负可测函数,K(x, y)=G(xλ1, yλ2),W1(b, p)与W2(a, q)如引理1所定义. 那么
(1) 若α=λ1[λ+1λ2+p(aλ1−bλ2)],β=λ2[λ+1λ1+p(bλ2−aλ1)], 则有
∫+∞0∫+∞0K(x,y)f(x)g(y)dxdy⩽W1/p1(b,p)W1/q2(a,q)‖f‖p,α‖g‖q,β, (2) 其中, f(x)∈Lαp(0,+∞),g(y)∈Lβq(0,+∞).
(2) 式(2)中的常数因子W11/p(b, p)W21/q(a, q)是最佳的,当且仅当aq/λ1+bp/λ2=1/λ1+1/λ2+ λ,W1(b, p)和W2(a, q)都收敛. 当aq/λ1+bp/λ2=1/λ1+1/λ2+ λ时,式(2)化为
∫+∞0∫+∞0K(x,y)f(x)g(y)dxdy⩽ W0|λ1|1/q|λ2|1/p‖f‖p,apq−1‖g‖q,bpq−1, (3) 其中, W0=|λ1|W2(a, q)= |λ2|W1(b, p).
证明 (i)选择a、b为搭配参数. 根据Hölder不等式和引理1,利用权系数方法,有
∫+∞0∫+∞0K(x,y)f(x)g(y)dxdy=∫+∞0∫+∞0(xaybf(x))(ybxag(y))K(x,y)dxdy⩽(∫+∞0∫+∞0xapybpfp(x)K(x,y)dxdy)1/p×(∫+∞0∫+∞0ybqxaqgq(y)K(x,y)dxdy)1/q=(∫+∞0xapfp(x)ω1(b,p,x)dx)1/p×(∫+∞0ybqgq(y)ω2(a,q,y)dy)1/q=W1/p1(b,p)W1/q2(a,q)×(∫+∞0xap+λ1(λ−bp/λ2+1/λ2)fp(x)dx)1/p×(∫+∞0ybq+λ2(λ−aq/λ1+1/λ1)gq(y)dx)1/q=W1/p1(b,p)W1/q2(a,q)‖f‖p,α‖g‖q,β, 故式(2)成立.
(ii) 充分性:设aq/λ1+bp/λ2=1/λ1+1/λ2+ λ,W1(b, p)和W2(a, q)收敛. 由引理1,有W1(b, p)/λ1=W2(a, q)/λ2,故
W1/p1(b,p)W1/q2(a,q)=(λ2λ1)1/qW1(b,p)=W0|λ1|1/q|λ2|1/p, 且α=apq-1,β=bpq-1,于是式(2)可化为式(3).
设式(3)的最佳常数因子为M0,则M0≤W0/(|λ11/q|λ2|1/p),且用M0取代式(3)中的常数因子后,式(3)仍然成立.
取充分小的ε>0及δ>0,令
f(x)={x(−apq−|λ1|ε)/p(x⩾1),0(0<x<1);g(y)={y(−bpq−|λ2|ε)/q(y⩾δ),0(0<y<δ). 则
‖f‖p,apq−1‖g‖q,bpq−1=(∫+∞1x−1−|λ1|εdx)1/p(∫+∞δy−1−|λ2|εdy)1/q=(1|λ1ε|)1/p(1|λ2|εδ−|λ2|ε)1/q=1ε|λ1|1/p|λ2|1/qδ−|λ2|ε/q,∫+∞0∫+∞0K(x,y)f(x)g(y)dxdy=∫+∞1x−aq−|λ1|ε/p(∫+∞δy−bp−|λ2|ε/qK(x,y)dy)dx=∫+∞1x−aq−|λ1|ε/p+λλ1(∫+∞δy−bp−|λ2|ε/qK(1,x−λ1/λ2y)dy)dx=∫+∞1x−1−|λ1|ε(∫+∞x−λ1/λ2δt−bp−|λ2|ε/qK(1,t)dt)dx⩾∫+∞1x−1−|λ1|ε(∫+∞δt−bp−|λ2|ε/qK(1,t)dt)dx=1|λ1|ε∫+∞δt−bp−|λ2|ε/qK(1,t)dt. 于是
1|λ1|∫+∞δt−bp−|λ2|ε/qK(1,t)dt⩽M0|λ1|1/p|λ2|1/qδ−|λ2|ε/q. 先令ε→0+,再令δ→0+,得
W1(b,p)=∫+∞0t−bpK(1,t)dt⩽M0|λ1|1/p|λ2|1/q. 再根据引理1,可得到W0/(|λ1|1/q|λ2|1/p)≤M0. 所以式(3)的最佳常数因子M0=W0/(|λ1|1/q|λ2|1/p).
必要性:设式(2)的常数因子W11/p(b, p)W21/q(a, q)是最佳的,则W1(b, p)和W2(a, q)是收敛的. 下证aq/λ1+bp/λ2=1/λ1+1/λ2+ λ.
记1λ1aq+1λ2bp−(1λ1+1λ2+λ)=c,a1=a−λ1cpq,b1=b−λ2cpq,则
α=λ1[λ+1λ2+p(a1λ1−b1λ2)]=α1,β=λ2[λ+1λ1+p(b1λ2−a1λ1)]=β1,W2(a,q)=∫+∞0K(t,1)t−aq dt=λ2λ1∫+∞0K(1,t)t−bp+λ2c dt. 于是可知式(2)等价于
∫+∞0∫+∞0K(x,y)f(x)g(y)dxdy⩽W1/p1(b,p)(λ2λ1∫+∞0K(1,t)t−bp+λ2c dt)1/q‖f‖p,α1‖g‖q,β1. 又经计算有a1q/λ1+b1p/λ2=1/λ1+1/λ2+ λ,α1=a1pq-1,β1=b1pq-1,故式(2)进一步等价于
∫+∞0∫+∞0K(x,y)f(x)g(y)dxdy⩽W1/p1(b,p)(λ2λ1∫+∞0K(1,t)t−bp+λ2c dt)1/q×‖f‖p,a1pq−1‖g‖q,b1pq−1. (4) 根据假设,式(4)的最佳常数因子是W11/p(b, p)×(λ2λ1∫+∞0K(1,t)t−bp+λ2cdt)1/q. 又由1λ1a1q+1λ2b1p=1λ1+1λ2+λ及充分性的证明,可知式(4)的最佳常数因子为
1|λ1|1/q|λ2|1/p(|λ2|∫+∞0K(1,t)t−b1p dt)=(λ2λ1)1/q∫+∞0K(1,t)t−bp+λ2c/q dt 于是得到
∫+∞0K(1,t)t−bp+λ2c/q dt=W1/p1(b,p)(∫+∞0K(1,t)t−bp+λ2c dt)1/q. (5) 对于1和tλ2c/q,应用Hölder不等式,有
∫+∞0K(1,t)t−bp+λ2c/q dt=∫+∞0tλ2c/qK(1,t)t−bp dt⩽(∫+∞01pK(1,t)t−bp dt)1/p(∫+∞0tλ2cK(1,t)t−bp dt)1/q=W1/p1(b,p)(∫+∞0K(1,t)t−bp+λ2c dt)1/q. (6) 根据式(5),可知式(6)取等号. 又根据Hölder不等式取等号的条件,可得tλ2c/q=常数,故c=0,即aq/λ1+bp/λ2=1/λ1+1/λ2+ λ1. 证毕.
注1 定理1表明: 当且仅当aq/λ1+bp/λ2=1/λ1+1/λ2+ λ时,搭配参数a、b是适配参数. 因此,只要选取a、b满足aq/λ1+bp/λ2=1/λ1+1/λ2+ λ,就可以得到各种各样的具有最佳常数因子的Hilbert型积分不等式.
推论1 设1/p+1/q=1 (p>1),λ1λ2>0,λ >0,1/r+1/s=1 (r>1),α=p(1- λλ1/r)-1,β=q(1- λλ2/s)-1,则
∫+∞0∫+∞0f(x)g(y)(xλ1+yλ2)λdxdy⩽1|λ1|1/q|λ2|1/p B(λr,λs)‖f‖p,α‖g‖q,β, (7) 其中的常数因子是最佳的,f(x)∈Lpα(0, +∞),g(y)∈Lqβ(0, +∞).
证明 记K(x, y)=G(xλ1, yλ2)=1/(xλ1+yλ2)λ,则G(u, v)是-λ阶齐次非负函数. 选取搭配参数a=1q(1−λλ1r),b=1p(1−λλ2s), 可得
1λ1aq+1λ2bp=1λ1(1−λλ1r)+1λ2(1−λλ2s)=1λ1+1λ2−λ, 故a、b是适配参数. 又因为apq-1=p(1- λλ1/r)-1=α,bpq-1=q(1- λλ2/s)-1=β,且
W0=|λ2|W1(b,p)=|λ2|∫+∞01(1+tλ2)λtλλ2/s−1 dt=∫+∞01(1+u)λuλ/s−1 du=B(λs,λ−λs)=B(λr,λs). 根据定理1,式(7)成立,且其常数因子是最佳的. 证毕.
3. 在求积分算子范数中的应用
根据Hilbert型不等式与相应积分算子的关系理论,由定理1可得如下定理.
定理2 设1/p+1/q=1 (p>1),a, b, λ ∈ R,λ1λ2>0,α=apq-1,β=bpq-1,G(u, v)是λ阶齐次非负可测函数,K(x, y)=G(xλ1, yλ2),且
W1(b,p)=∫+∞0K(1,t)t−bp dt<+∞,W2(a,q)=∫+∞0K(t,1)t−aq dt<+∞, 则当aq/λ1+bp/λ2=1/λ1+1/λ2+ λ时,积分算子T:
T(f)(y)=∫+∞0K(x,y)f(x)dx,f(x)∈Lαp(0,+∞) 是从Lpα(0, +∞)到Lpβ(1-p)(0, +∞)的有界算子,且T的算子范数为
‖T‖=|λ2|W1(b,p)|λ1|1/q|λ2|1/p=(λ2λ1)1/q∫+∞0K(1,t)t−bp dt. 推论2 设1/p+1/q=1 (p>1),λ1λ2>0,-1 < λ < min{1±4/λ1, 1±4/λ2},α=p[1+ λ1(λ -1)/2]-1,β=p[1+ λ2(λ -1)/2]-1,则积分算子T:
T(f)(y)=∫+∞0|xλ1−yλ2|λmax{xλ1,yλ2}f(x)dx,f(x)∈Lαp(0,+∞) 是从Lpα(0, +∞)到Lpβ(1-p)(0, +∞)的有界算子,且T的算子范数为
‖T‖=1|λ1|1/q|λ2|1/p[ B(λ+1,1−λ2−2λ2)+ B(λ+1,1−λ2+2λ2)]. 证明 记K(x, y)=G(xλ1, yλ2)= |xλ1-yλ2|λ/max{xλ1, yλ2},则G(u, v)是λ -1阶齐次函数. 取a= 1q[1+λ12(λ−1)],b=1p[1+λ22(λ−1)],则
1λ1aq+1λ2bp=1λ1[1+λ12(λ−1)]+1λ2[1+λ22(λ−1)]= 1λ1+1λ2+λ−1, 故a、b是适配参数. 又apq−1=p[1+λ12(λ−1)]−1=α,bpq−1=q[1+λ22(λ−1)]−1=β. 则
(λ2λ1)1/q∫+∞0K(1,t)t−bp dt=(λ2λ1)1/q∫+∞0|1−tλ2|λmax{1,tλ2}t−[1+λ2(λ−1)/2]dt=1|λ1|1/q|λ2|1/p[ B(λ+1,1−λ2−2λ2)+B(λ+1,1−λ2+2λ2)]<+∞. 根据定理2,知推论2成立. 证毕.
推论3 设1/p+1/q=1 (p>1),1/r+1/s=1 (r>1),λ1λ2>0,α=p(1- λ1/r)-1,β=q(1- λ2/s)-1. 则积分算子T:
T(f)(y)=∫+∞0ln(xλ1/yλ2)xλ1−yλ2f(x)dx,f(x)∈Lαp(0,+∞) 是从Lpα(0, +∞)到Lpβ(1-p)(0, +∞)的有界算子,且T的算子范数为
‖T‖=1|λ1|1/q|λ2|1/p[ζ(2,1r)+ζ(2,1s)], 其中ζ(t, a)是Riemann函数.
证明 记
K(x,y)=G(xλ1,yλ2)=ln(xλ1/yλ2)xλ1−yλ2, 则G(u, v)是-1阶齐次非负函数.
取搭配参数a=1q(1−λ1r),b=1p(1−λ2s),则
1λ1aq+1λ2bp=1λ1(1−λ1r)+1λ2(1−λ2s)=1λ1+1λ2−1, 故a、b是适配参数. 又apq-1=p(1- λ1/r)-1=α,bpq-1=q(1- λ2/s)-1=β,且
(λ2λ1)1/q∫+∞0K(1,t)t−bp dt=(λ2λ1)1/q∫+∞0ln(t−λ2)1−tλ2tλ2/s−1 dt=1|λ1|1/q|λ2|1/p[ζ(2,1r)+ζ(2,1s)]<+∞ 根据定理2,知推论3成立. 证毕.
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[17]Z.H. Lu, J. R. Dahn. Understanding the Anomalous Capacity of Li / Li [ Ni x Li ( 1 / 3 ? 2x / 3 ) Mn ( 2 / 3 ? x / 3 ) ] O 2 Cells Using In Situ X-Ray Diffraction and Electrochemical Studies[J]. J. Electrochem. Soc. 149 (2002) A815-A822. DOI:10.1149/1.1480014
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[12]C.W. Lee, Y.K. Sun, J. Prakash. A novel layered Li [Li0.12NizMg0.32?zMn0.56]O2 cathode material for lithium-ion batteries. [J]. Electrochim. Acta 49 (2004) 4425-4432.DOI:10.1016/j.electacta.2004.04.033
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[14]C. J. Jafta, K. I. Ozoemena, M. K. Mathe,et al. Synthesis, characterisation and electrochemical intercalation kinetics of nanostructured aluminium-doped Li[Li0.2Mn0.54Ni0.13Co0.13]O2 cathode material for lithium ion battery [J].Electrochim. Acta 85 (2012) 411-422. DOI:10.1016/j.electacta.2012.08.074
[15]J. Wilcox, S. Patoux, M. Doeff. Structure and Electrochemistry of LiNi1/3Co1/3-yMyMn1/3O2 (M=Ti, Al, Fe) Positive Electrode Materials. [J]. J. Electrochem. Soc. 156 (2009) A192-A198. DOI:10.1149/1.3056109
[16]L. Li, B.H. Song, Y.L. Chang,et al. Retarded phase transition by fluorine doping in Li-rich layered Li1.2Mn0.54Ni0.13Co0.13O2 cathode material[J]. J. Power Sources 283 (2015) 162-170. DOI:10.1016/j.jpowsour.2015.02.085
[17]Z.H. Lu, J. R. Dahn. Understanding the Anomalous Capacity of Li / Li [ Ni x Li ( 1 / 3 ? 2x / 3 ) Mn ( 2 / 3 ? x / 3 ) ] O 2 Cells Using In Situ X-Ray Diffraction and Electrochemical Studies[J]. J. Electrochem. Soc. 149 (2002) A815-A822. DOI:10.1149/1.1480014
[18]M.G. Kim, M. Jo, Y.S. Hong, et al. Template-free synthesis of Li[Ni0.25Li0.15Mn0.6]O2 nanowires for high performance lithium battery cathode [J].Chem. Commun. 2 (2009) 218-220. DOI: 10.1039/b815378g
[19]R.R. Zhao, Z.J. Chen, Y. Zhang, et al. Ultrasonic/microwave-assisted co-precipitation method in the synthesis of Li1.1Mn0.433Ni0.233Co0.233O2 cathode material for lithium-ion batteries[J].Mater. Lett. 136 (2014) 160-163. DOI:10.1016/j.matlet.2014.08.060
[20]R.R. Zhao, I.M. Hung, Y.T. Li, et al. Synthesis and properties of Co-doped LiFePO4 as cathode material via a hydrothermal route for lithium-ion batteries [J] J.Alloys Compd. 513 (2012) 282-288. DOI:10.1016/j.jallcom.2011.10.037
[21]I. Belharouak, G.M. Koenig Jr., J. Ma, et al. Identification of LiNi0.5Mn1.5O4 spinel in layered manganese enriched electrode materials [J].Electrochem. Commun. 13 (201) 232-236. DOI:10.1016/j.elecom.2010.12.021
[22]L.J. Zhang, B.R. Wu, N. Li, et al. Rod-like hierarchical nano/micro Li1.2Ni0.2Mn0.6O2 as high performance cathode materials for lithium-ion batteries. [J].J. Power Sources, 240 (2013) 644-652. DOI:10.1016/j.jpowsour.2013.05.019 -
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