For a tubular electrode with inner radius and outer radius , the radial stress on the free surface is zero, i.e., ( ) = 0and ( ) = 0. The radial and tangential stresses.
Supplemental Information for Aligned TiO2 Nanotube Arrays as Durable Lithium-Ion Battery Negative Electrodes Qing Liu Wu, Juchuan Li, Rutooj D. Deshpande, Navaladian Subramanian, Stephen E. Rankin, Fuqian Yang and Yang-Tse Cheng Chemical and Materials Engineering Department, University of Kentucky, 177 F.P. Anderson Tower, Lexington, KY 40506-0046, U.S.A.
Appendix A In the current work, TiO2 nanotubes with three different dimensions are studied, whose outer radii are 32, 53, and 70nm, respectively, and tube lengths are 355, 540, and 351 nm, respectively. For all three cases, the tubes are more prone to axial fractures, i.e., they fall under 'Case 1' above. DIS evolution in these tubes is modeled with a generalized plane strain condition. We assume that the electrode material is an isotropic, linearly elastic solid. Using the analogy between thermal stresses35,37 and DISs38,40, the stressstain relationships for the radial, tangential and axial components, are
− = − (
+ )
(A1)
Here and represent strain and stress in the ‘’ direction, respectively. , , and represent principle directions in a cylindrical coordinate system. is the coefficient of the volume expansion per lithium atom. Since mechanical equilibrium is established much faster than the atomic diffusion process in solids, we treat this as a static equilibrium
Q.L. Wu et al., Supplemental Information for “Aligned TiO2 Nanotube Arrays as Durable Lithium-Ion Battery Negative Electrodes”
problem. In the absence of any body force, the equation for static mechanical equilibrium in the bulk of a tube is given by35
+
=0
(A2)
Let denotes the radial displacement. For small values of u, strains are given as =
, and = .
For a tubular electrode with inner radius
and outer radius !, the radial stress on
the free surface is zero, i.e., ( ) = 0and (!) = 0. The radial and tangential stresses can be calculated as Ω
*&
+
= #($) %− & '* (()()( + +& *& (1 − & ) '* (()()(Ω
*&
+
= #($) % & '* (()()( + +& *& .1 + & / '* (()()( + (()-
(A3)
(A4)
We assume that the ends of the tube are free from external forces, i.e., 01 = +
41 '4 23( 1 )( = 0.Under this condition the axial stress is given by
1 = ( + ) + 5(*67 (!) − ((, 9)) :
(A5)
where *67 (() = & *& '* ((′)(′)( ′ is the average concentration within radius ( per unit length of the cylindrical electrode. The stresses can be changed to dimensionless form by using equation (7). Based on the concentration profiles, we compute all principle stresses at different radial positions as a function of time. It is found that the radial stresses are tensile only during lithiation, while hoop or axial stresses are tensile at the outer surface of the tube during 2
Q.L. Wu et al., Supplemental Information for “Aligned TiO2 Nanotube Arrays as Durable Lithium-Ion Battery Negative Electrodes”
lithiation and are tensile at the inner surface during delithiation. The maximum axial stress is several times larger than the maximum radial stress (Figure S1, a and f); hence the tubes are more prone to crack as a result of axial stress rather than radial stress. The maximum axial stress is four times larger during delithiation than it is during lithiation (Figure S1, b and f), which indicates that the tubular electrode would be more prone to cracking in delithiation than in lithiation. Appendix B Number of tubes per electrode substrate area = N ≅ =
(π ∙ 0.7: ) cm: area of a tube
(π ∙ 0.7: ) = 4.788 ∙ 104 π ∙ (32 ∙ 10P ):
Surface area of tubes for diffusion (inner surface area) = 2πahN = 2π ∙ 15 ∙ 10P ∙ 355 ∙ 10P ∙ 4.788 ∙ 104 = 16.01 cm: The real surface might be rougher than the ideal case. Real surface area of tubes for lithium conduction = =
calulated surface area roughness factor
16.01 cm: Vρ
Mass of titania in the tubes = ρπt(a + t)hN = 3.84 ∙ π ∙ 32 ∙ 10P ∙ 15 ∙ 10P ∙ 355 ∙ 10P ∙ 4.788 ∙ 104 = 0.161 mg Current = (C/3)m = 0.000161 g ∙ 70 mA/g = 11.3 ab ≅ 10 ab Effective current density =
current 10µA 0.62 = = µA cm: : surface area 16.01 cm Vρ
For ideal tubes, Vρ = 1 3
Q.L. Wu et al., Supplemental Information for “Aligned TiO2 Nanotube Arrays as Durable Lithium-Ion Battery Negative Electrodes”
Effective current density = 0.62µA cm: for the sample prepared at 10 V. Similarly the effective current densities for the other two tubes are calculated to be 5 .1 x 10-3,and 13 x 10-3mA cm-2 for the samples prepared at 20 V and 30 V anodization potentials.
Figure S1.
Comparison between the maximum stress that the tubular electrode
experiences (a) in the radial direction during delithiation (lithium extraction), and (b) in the tangential direction during lithiation (lithium insertion) and (c, d, e, and f) delithiation. The ratio of inner radius to outer radius of tube ( ⁄! ) for curves (a, b, and f) = 0.3, for curve (e) 0.4, for curve (d) 0.5, and for curve (c) 0.6.
4
