Abstract
By incorporating the contribution of solute atoms to the Helmholtz free energy of solid solution, a linear relation is derived between Young’s modulus and the concentration of solute atoms. The solute atoms can either increase or decrease Young’s modulus of solid solution, depending on the first-order derivative of the Helmholtz free energy with respect to the concentration of solute atoms. Using this relation, a closed-form solution of the chemical stress in an elastic plate is obtained when the diffusion behavior in the plate can be described by the classical Fick’s second law with convection boundary condition on one surface and zero flux on the other surface. The plate experiences tensile stress after short diffusion time due to asymmetrical diffusion, which will likely cause surface microcracking. The results show that the effect of the concentration dependence of Young’s modulus on the evolution of chemical stress in elastic plates is negligible if the change of Young’s modulus due to the diffusive motion of solute atomsis is not compatible in magnitude with Young’s modulus of the pure material. Also, a new diffusion equation is developed for strictly regular binary solid solution. The effective diffusivity is a nonlinear function of the concentration of solute atoms.
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Abbreviations
- C :
-
concentration of solute atoms
- i reaction :
-
anodic or cathodic current density at the surface of the plate
- n :
-
number of participating electrons
- u i (i=1, 2, 3):
-
components of the displacement vector
- C ijkl :
-
stiffness tensor
- D 0 :
-
diffusivity of solute atoms
- D eff :
-
effective diffusivity of solute atoms
- E :
-
Young’s modulus
- F j (i=1, 2, 3):
-
components of body force
- J :
-
vector of the diffusion flux
- M :
-
mobility of the diffusing component
- R :
-
gas constant
- S :
-
entropy
- T :
-
absolute temperature
- U I :
-
internal energy
- U strain :
-
strain energy density
- β :
-
charge transfer barrier for the anodic or cathodic reaction
- ν :
-
Poisson’s ratio
- µ:
-
chemical potential
- γ :
-
activity coefficient
- σ :
-
hydrostatic stress
- σ ij (i, j=1, 2, 3):
-
components of stress tensor
- ɛ ij (i, j=1, 2, 3):
-
components of strain tensor
- ℑ:
-
Holmholtz free energy
- Ω:
-
the coefficient of the volume change per mole of solute atoms
- ϒ:
-
Faraday’s constant
References
Abbaschian R, Abbaschian L, Reed-Hill R E. Physical Metallurgy Principles. 4th ed. Stamford: Cengage Learning, 2009
Chen D, Xiong S, Ran S H, et al. One-dimensional iron oxides nanostructures. Sci China Ser G-Phys Mech Astron, 2011, 54(7): 1190–1199
Yang F Q. Effect of local solid reaction on diffusion-induced stress. J Appl Phys, 2010, 107(10): 103516
Lee H T, Chen M H, Jao H M, et al. Influence of interfacial intermetallic compound on fracture behavior of solder joints. Mater Sci Eng A, 2003, 358(1–2): 134–141
Zhang N, Shi Y W, Guo F, et al. Study of the impact performance of solder joints by high-velocity impact tests. J Electron Mater, 2010, 39(12): 2536–2543
Prussin S. Generation and distribution of dislocation by solute diffusion. J Appl Phys, 1961, 32(10): 1876–1881
Li J C M. Physical chemistry of some microstructural phenomena. Metall Trans A, 1978, 9(10): 1353–1380
Li J C, Dozier A K, Li Y, et al. Crack pattern formation in thin film lithium-ion battery electrodes. J Electrochem Soc, 2011, 158(6): A689–A694
Sze S M. Physics of Semiconductor Devices. 1st ed. New York: Wiley, 1969. Chapter 2
Chu J L, Lee S B. Chemical stresses in composite circular-cylinders. J Appl Phys, 1993, 73(5): 2239–2248
Ko S C, Lee S B, Chou Y T. Chemical stresses in a square sandwich composite. Mater Sci Eng, 2005, 409(1–2): 145–152
Lin H Y, Ko S C, Lee S B. Chemical stresses in boundary layer diffusion. J Appl Phys, 2004, 96(11): 6183–6187
Yang F Q, Li J C M. Diffusion-induced beam bending in hydrogen sensors. J Appl Phys, 2003, 93(11): 9304–9309
Chu J L, Lee S B. The effect of chemical stresses on diffusion. J Appl Phys, 1994, 75(6): 2823–2829
Zhang T Y, Ko S C, Lee S B. Effects of absorption and desorption on the chemical stress field. J Appl Phys, 2002, 91(4): 2002–2008
Kandasamy K. Influences of self-induced stress on permeation flux and space-time variation of concentration during diffusion of hydrogen in a palladium alloy. Int J Hydrogen Energy, 1995, 20(6): 455–465
Yang F Q. Interaction between diffusion and chemical stresses. Mater Sci Eng A, 2005, 409(1–2): 153–159
Zhang X C, Shyy W, Sastry A M. Numerical simulation of intercalation-induced stress in Li-ion battery electrode particles. J Electrochem Soc, 2007, 154(10): A910–A916
Kalnaus S, Rhodes K, Daniel C. A study of lithium ion intercalation induced fracture of silicon particles used as anode material in Li-ion battery. J Power Sources, 2011, 196(19): 8116–8124
Soma T, Takashima S, Kagaya H M. Elastic moduli of Al-Si and Al-Ge solid solutions. J Mater Sci, 1992, 27(5): 1184–1188
Soma T, Ishizuka M, Kagaya H M. Solid solubility of Cu in Al under pressure and elastic moduli. Phys Stat Sol, 1994, 186(1): 95–100
Qi Y, Guo H, Hector L G, et al. Threefold increase in the Young’s modulus of graphite negative electrode during lithium intercalation. J Electrochem Soc, 2010, 157(5): A558–A566
Shenoy V B, Johari P, Qi Y. Elastic softening of amorphous and crystalline Li-Si Phases with increasing Li concentration: A first-principles study. J Power Sources, 2010, 195(19): 6825–6830
Deshpande T, Qi T, Cheng Y T. Effects of concentration-dependent elastic modulus on diffusion-induced stresses for battery applications. J Electrochem Soc, 1020, 157(8): A967–A971
Dewey J M. The elastic constants of materials loaded with non-rigid fillers. J Appl Phys, 1947, 18(6): 578–581
Eshelby J D. The continuum theory of lattice defects. Solid State Phys, 1956, 3(1): 79–144
Yang F Q. Size-dependent effective modulus of elastic composite materials: Spherical nanocavities at dilute concentrations. J Appl Phys, 2004, 95(7): 3516–3520
Yang F Q. Effect of interfacial stresses on the elastic behavior of nanocomposite materials. J Appl Phys, 2006, 99(5): 054306
Kondepudi D, Prigogine I. Modern Thermodynamics. New York: John Wiley & Sons, 1998. 314
Bard A J, Faulkner L R. Electrochemical Methods and Applications. 2nd ed. New York: Wiley, 2000
Newman J, Thomas-Alyea K E. Electrochemical Systems. 3rd ed. Hoboken: Wiley, 2004
Boley A, Weiner J H. Theory of Thermal Stresses. Mineola: Dover Publications, Inc., 1997
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Yang, F. Diffusion-induced stress in inhomogeneous materials: concentration-dependent elastic modulus. Sci. China Phys. Mech. Astron. 55, 955–962 (2012). https://doi.org/10.1007/s11433-012-4687-8
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DOI: https://doi.org/10.1007/s11433-012-4687-8