Numerical modeling of dissolution process of spherical precipitates in alloys by differential quadrature method (DQM)

Subject Areas : journal of New Materials

nozar anjabin ¹ (Department of materials science and engineering, Shiraz university)

Received: 2021-04-16 Accepted : 2021-06-20 Published : 2021-05-22

Keywords: Numerical Modeling, precipitate dissolution, differential quadrature method, front-fixing method, moving boundary problem,

Abstract :

Prediction of the dissolution kinetics of precipitates is important in various metallurgical processes such as welding, homogenization, and preheating of the age hardenable alloys. The problem of spherical particle dissolution is a moving boundary problem, which has no exact solution yet. In the present study, a numerical model based on the differential quadrature method is presented to solve the problem of precipitate dissolution with spherical geometry in a matrix with finite dimensions. In the proposed model, the dissolution kinetics is expressed as a function of the volume fraction of the precipitate, the concentration of the alloying element in the matrix, and precipitate, equilibrium concentration at the precipitate /matrix interface, and the annealing temperature. The convergence of the presented numerical model in solving the dissolution problem is evaluated by examining the effect of time step size and number of grid points on the numerical solution results. The accuracy of the proposed model is also evaluated by comparing the model results with the results of an approximate analytical model as well as experimental data. The results show that the proposed model converges even with a low number of grid points and is in good agreement

References:

[1]          L. Hu, S.-J. Zhao, and Q. Liu, "The effect of size of Cu precipitation on the mechanical properties of microalloyed steel," Materials Science and Engineering: A, vol. 556, pp. 140-146, 2012.

[2]          J. Kai, G. Yu, C. Tsai, M. Liu, and S. Yao, "The effects of heat treatment on the chromium depletion, precipitate evolution, and corrosion resistance of INCONEL alloy 690," Metallurgical transactions A, vol. 20, no. 10, pp. 2057-2067, 1989.

[3]          W. Yang, S. Ji, Q. Zhang, and M. Wang, "Investigation of mechanical and corrosion properties of an Al–Zn–Mg–Cu alloy under various ageing conditions and interface analysis of η′ precipitate," Materials & Design, vol. 85, pp. 752-761, 2015.

[4]          N. Anjabin, A. K. Taheri, and H. Kim, "Crystal plasticity modeling of the effect of precipitate states on the work hardening and plastic anisotropy in an Al–Mg–Si alloy," Computational materials science, vol. 83, pp. 78-85, 2014.

]5 [م. میرزایی، م. روشن، س. جوادپور، "افزایش شدید خواص مکانیکی آلیاژ آلومینیوم 2024 با اعمال یک پاس نورد سرد"، مجله مواد نوین، جلد 4، شماره 3، ص 67-78، بهار 1393.

[6]          M. Nicolas and A. Deschamps, "Characterisation and modelling of precipitate evolution in an Al–Zn–Mg alloy during non-isothermal heat treatments," Acta Materialia, vol. 51, no. 20, pp. 6077-6094, 2003.

[7]          T. Marlaud, A. Deschamps, F. Bley, W. Lefebvre, and B. Baroux, "Evolution of precipitate microstructures during the retrogression and re-ageing heat treatment of an Al–Zn–Mg–Cu alloy," Acta materialia, vol. 58, no. 14, pp. 4814-4826, 2010.

[8]          D. Bratland, Ø. Grong, H. Shercliff, O. Myhr, and S. Tjøtta, "Overview No. 124 Modelling of precipitation reactions in industrial processing," Acta Materialia, vol. 45, no. 1, pp. 1-22, 1997.

[9]          D. Zhang, X. Wang, Y. Pan, Sh. Hou, J. Zhang, L. Zhuang, L. Zhou, "Friction stir welding of novel T-phase strengthened Zn-modified Al–Mg alloy," Journal of Materials Science, vol. 56, no. 8, pp. 5283-5295, 2021.

[10]       F. Vermolen and K. Vuik, "A numerical method to compute the dissolution of second phases in ternary alloys," Journal of computational and applied mathematics, vol. 93, no. 2, pp. 123-143, 1998.

[11]       I. Sadeghi, M. A. Wells, and S. Esmaeili, "Effect of particle shape and size distribution on the dissolution behavior of Al2Cu particles during homogenization in aluminum casting alloy Al-Si-Cu-Mg," Journal of Materials Processing Technology, vol. 251, pp. 232-240, 2018.

[12]       O. Gopkalo, X. Liu, F. Long, M. Booth, A. Gerlich, and B. Diak, "Non-isothermal thermal cycle process model for predicting post-weld hardness in friction stir welding of dissimilar age-hardenable aluminum alloys," Materials Science and Engineering: A, vol. 754, pp. 205-215, 2019.

[13]       J. Van de Langkruis, N. Kuijpers, W. Kool, F. Vermolen, and S. Van der Zwaag, "Modeling Mg₂Si dissolution in an AA6063 alloy during preheating to the extrusion temperature," in proceedings of international aluminum extrusion technology seminar, 2000, vol. 1, pp. 119-124: Citeseer.

[14]       B. A. Chen, L. Pan, R. H. Wang, G. Liu, P. M. Cheng, L. Xiao, J. Sun, "Effect of solution treatment on precipitation behaviors and age hardening response of Al–Cu alloys with Sc addition," Materials Science and Engineering: A, vol. 530, pp. 607-617, 2011.

[15]       M. Whelan, "On the kinetics of precipitate dissolution," Metal Science Journal, vol. 3, no. 1, pp. 95-97, 1969.

[16]       H. B. Aaron, D. Fainstein, and G. R. Kotler, "Diffusion‐limited phase transformations: a comparison and critical evaluation of the mathematical approximations," Journal of applied physics, vol. 41, no. 11, pp. 4404-4410, 1970.

[17]       R. Tanzilli and R. Heckel, "Numerical solutions to finite diffusion-controlled 2-phase moving-interface problem (with planar cylindrical and spherical interfaces)," Transactions of the Metallurgical Society of AIME, vol. 242, no. 11, pp. 2313, 1968.

[18]       D. L. Baty, R. A. Tanzilli, and R. W. Heckel, "Solution kinetics of CuAl₂ in an Al-4Cu alloy," Metallurgical Transactions, vol. 1, no. 6, pp. 1651-1656, 1970.

[19]       U. H. Tundal and N. Ryum, "Dissolution of particles in binary alloys: Part I. computer simulations," Metallurgical Transactions A, vol. 23, no. 2, pp. 433-444, 1992.

[20]       Q. Zuo, F. Liu, L. Wang, C. F. Chen, and Z. H. Zhang, "An analytical model for secondary phase dissolution kinetics," Journal of Materials Science, vol. 49, no. 8, pp. 3066-3079, 2014.

[21]       Z. Xu and P. Meakin, "Phase-field modeling of two-dimensional solute precipitation/dissolution: Solid fingers and diffusion-limited precipitation," The Journal of chemical physics, vol. 134, no. 4, p. 044137, 2011.

[22]       G. Wang, D. S. Xu, N. Ma, N. Zhou, E. J. Payton, R. Yang, M. I. Mills, Y. Wang, "Simulation study of effects of initial particle size distribution on dissolution," Acta Materialia, vol. 57, no. 2, pp. 316-325, 2009.

[23]       I. Kovačević and B. Šarler, "Solution of a phase-field model for dissolution of primary particles in binary aluminum alloys by an r-adaptive mesh-free method," Materials Science and Engineering: A, vol. 413, pp. 423-428, 2005.

[24]       C. W. Bert and M. Malik, "Differential quadrature method in computational mechanics: a review," 1996.

[25]       G. Meral, "Differential quadrature solution of heat-and mass-transfer equations," Applied Mathematical Modelling, vol. 37, no. 6, pp. 4350-4359, 2013.

[26]       B. Kaya, "Solution of the advection-diffusion equation using the differential quadrature method," KSCE Journal of civil engineering, vol. 14, no. 1, pp. 69-75, 2010.

[27]       P. Malekzadeh and H. Rahideh, "Two-dimensional nonlinear transient heat transfer analysis of variable section pin fins," Energy Conversion and Management, vol. 50, no. 4, pp. 916-922, 2009.

[28]       A. Vosoughi, N. Anjabin, and S. Amiri, "Thermal post-buckling analysis of moderately thick nanobeams," Iranian Journal of Science and Technology, Transactions of Civil Engineering, vol. 42, no. 1, pp. 33-38, 2018.

[29]       A. R. Vosoughi, P. Malekzadeh, U. Topal, and T. Dede, "A hybrid DQ-TLBO technique for maximizing first frequency of laminated composite skew plates," Steel and Composite Structures, vol. 28, no. 4, pp. 509-516, 2018.

[30]       C. Shu, Differential quadrature and its application in engineering. Springer Science & Business Media, 2012.

[31]       U. H. Tundal and N. Ryum, "Dissolution of particles in binary alloys: part II. experimental investigation on an Al-Si alloy," Metallurgical Transactions A, vol. 23, no. 2, pp. 445-449, 1992.