# Dynamic boundary conditions for a coupled convection-diffusion model with heat effects: applications in cross-contamination control

• ## Authors

• N. Santatriniaina Mathematical Research Institute of Rennes
• J. Deseure
• T.Q. Nguyen
• H. Fontaine
• C. Beitia
• L. Rakotomanana
2015-01-16
• Cross-contamination, Coupling technique, Dynamic boundary conditions, Finite element methods, Mathematical model.
• ## Abstract

This work investigates the mass transfer of the Airborne Molecular cross Contamination (AMCs) between the Front Opening Unified Pod (FOUP) and wafer (silicon substrates) during the microelectronics devices manufacturing using dynamic boundary conditions. Such cross-contamination phenomena lead to detrimental impact on production yield in microelectronic industry and a predictive approach using modelling and computational methods is a very strong way to understand and qualify the AMCs cross contamination processes. The FOUP is made of polymeric materials and it is considered as a heterogeneous porous media, it can adsorb and desorb the contaminant, thus the modelled processes are the contamination of two-component in transient flow. Coupled diffusion and convection-diffusion model with heat effects are used to define the phenomena. The present methodology is, first using the optimization methods with the numerical solution in order to define the physical constants of various materials which have been studied experimentally and separately, and the second using the finite element methods including these physical constants and relevant interface condition in order to take into account the adsorption kinetics law. Numerical methods to solve the problem are proposed. The dynamics behaviour of the AMCs analysis was determined thanks to the switch of Dirichlet to Neumann condition. The mathematical model preserves the classical forms of the diffusion and convection diffusion equations and yields to consistent form of the Fick's law. The computed results are in correlation with the experimental measurements. Some numerical results are presented in this work.

• ## References

1. [1] T.Q.Nguyen, H.Fontaine and al. Identification and quantification of FOUP molecular contaminants inducing defects in integrated circuits manufacturing,

Microelectronic Engineering, Vol. 105, (2013), pp. 124--129.

[2] P.GonzÃ lez, H.Fontaine, C.Beitia and al. A comparative study of the HF sorption and outgassing ability of different Entegris FOUP platforms and materials, Microelectronic Engineering, Vol. 150, (2013), pp. 113--118.

[3] S.Hu, T.Wu, and al. Design and evaluation of a nitrogen purge system for the front opening unified pod, Applied Thermal Engineering, Vol. 27, (2007), pp. 1386--1393.

[4] H.Fontaine, H.Feldis and al. Impact of the volatile Acid Contaminant on Copper

Interconnects, Electrical Perform, Vol. 25, No: 5, (2009), pp. 78--86.

[5] N. Santatriniaina, J.Deseure, T.Q.Nguyen, H.Fontaine, C. Beitia, L.Rakotomanana. Mathematical modelling of the AMCs cross-contamination removal in the FOUPs: Finite element formulation and application in FOUP's decontamination, Inter. Journ. of Math., comput. sci. engrg, Vol. 8, No: 4, (2014), pp. 409--414.

[6] N. Santatriniaina, J.Deseure, T.Q.Nguyen, H.Fontaine, C. Beitia, L.Rakotomanana. Coupled system of PDEs to predict the sensitivity of the some material constituents of the FOUP with the AMCs cross-contamination, International Journal of Applied Mathematical Research, Vol. 3, No: 3, (2014), pp. 233--243.

[7] Alemayeuhu Ambaw, Randolph Beaudry, Inge Bulens, Mulugeta Admasu Delele, Q.Tri Ho, Ann Schenk, Bart M. Nicolai, Pieter Verboven, Modelling the diffusion adsorption kinetics of 1-methylcyclopropene (1-MCP) in apple fruit and non target materials in storage rooms, Journal of Food Engineering, Vol. 102, (2011), pp. 257--265.

[8] J.A.Boscoboinik, S.J. Manzi, V.D.Pereyra Adsorption-desorption kinetics of monomer-dimer mixture, Physics A, Vol. 389, (2010), pp. 1317--1328.

[9] Rico F. Tabor, Julian Eastoe, Peter J. Dowding, A two-step model for surfactant adsorption at solid surfaces, Journal of Colloid and Interface Science, Vol. 346, (2010), pp. 424.428.

[10] Anli Geng, Kai-Chee Loh, Effects of adsorption kinetics and surface heterogeneity on band spreading in perfusion chromatography-a network model analysis, Chemical Engineering Science, Vol. 59, (2004), pp. 2447--2456.

[11] Hiroki Nagaoka and Toyoko Imae, Analytical investigation of two-step adsorption kinetics on surfaces, Journal of Colloid and Interface Science, Vol. 264, (2003), pp. 335--342.

[12] J. Crank, The mathematics of diffusion, second edition, 1975 Clarendon Press, Oxford.

[13] R.Hirsch, C.C.Muller-Goymann, Fitting of diffusion coefficients in a three compartments sustained release drug formulation using a genetic algorithm, International Journal of Pharmaceutics, Vol. 120, (1995), pp. 229--234.

[14] K.J.Kuijlaars, C.R.Kleijin, H.E.A. van den Akker, Multi-component diffusion phenomena in multiple-wafer chemical vapour deposition reactors, The chemical Engineering Journal, Vol. 57, (2009), pp. 127--136.

[15] Koichi Aoki, Diffusion-controlled current with memory, Journal of electroanalytical Chemistry, Vol. 592, (2006), pp. 31--36.

[16] Shengping Ding, William T. Petuskey, Solutions to Fickâ€™s second law of diffusion with a sinusoidal excitation, Solid State Ionics, Vol. 109, (1998), pp. 101--110.

[17] Juergen Siepmann, Florence Siepmann, Modelling of diffusion controlled drug delivery, Journal of Controlled Release, Vol. 161, (2012), pp. 351--362.

[18] H.Denny Kamaruddin, William J.Koros,Some observation about the application of Fick's first law for membrane separation of multicomponent mixtures, Journal of Membrane Science, Vol. 1135, (1997), pp. 47--159.

[19] Ana Rita C. Duarte, Carlos Martins, Patricia Coimbra, Maria H.M. Gil, Herminio C. de Sousa, Catarina M.M. Duarte, Sorption and diffusion of dense carbon dioxide in a biocompatible polymer, Journal of Supercritical Fluids, Vol. 38, (2006), pp. 392--398.

[20] Wu Hai-jin, Lin Bai-quan, Yao Qian, The theory model and analytic answer of gas diffusion, Procedia Earth and Planetary Science, Vol. 1, (2009), pp. 328--335.

[21] Lagarias, J., Reeds, J., Wright, M., and Wright, Convergence Properties of the Nelder--Mead Simplex Method in Low Dimensions, P SIAM Journal on Optimization, Vol. 9, No: 1, (1998), pp. 12--147.

[22] Herv'e Fontaine, H. Feldis, A. Danel, S. Cetre, C. Ailhas, Impact of the volatile Acid Contaminant on Copper Interconnects, Electrical Performances. ECS Transactions, Vol. 25, No: 5, (2009), pp. 78-86.