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Keywords: phase transformations, nanowires, nonlocal models, nanostructures.

Summary

In this work we propose a simplified mesoscopic model for the analysis of nanowires of finite length. Our interest has been motivated by the fact that such nanowires can exhibit shape memory effects. This has been confirmed for a wide range of these structures, including ZnO, Cu, Ni, Al, Ag, and Au nanowires.

Here we focus on a two-dimensional analogue of what in the general three-dimensional case would correspond to cubic to tertragonal or tetragonal to orthorhombic transformations. Such a two-dimensional case deals with square to rectangle transformations which are easier to treat due to the fact that the energy as a function of lattice spacing exhibits two distinct minima corresponding to two different phases of the nanostructure.

Most results obtained up to date, both theoretical and computational, are concerned with infinitely long nanowires or infinitely large nanoplates for that matter. In this contribution, we develop a relatively simple and computationally inexpensive model to study phase transformations in finite nanostructures with our major focus given to nanowires of finite length. We note the models describing shape memory effects at the mesoscopic level such as those developed in [1] can be reduced to a two-dimensional case (and in the case of nanowires of infinite length, to the one-dimensional case). Our considerations are based on a modification of a coupled system of PDEs for the evolution of displacements and temperature [2,3]. One of the applications of the nanowires discussed in this paper stems from their integration in biomolecular technologies and we highlight the importance of the contributions of entropic elasticity in such cases.

Recently, we developed and demonstrated the application of several efficient methodologies to solve two-dimensional models describing square-to-rectangle phase transformations in materials with memory, in particular the finite volume methodology [3] and a numerical reduction procedure based on the proper orthogonal decomposition (POD) [2]. The methodology proposed in this contribution is different and is based on a combination of the Chebyshev pseudo-spectral technique combined with the domain decomposition method and second order backward differentiation. Based on this methodology, we analysed different types of nanowires and examplified our results here with Cu nanowires for different diameter-length ratio. Our results indicate that the initial pattern formation propagates towards the longitudinal boundaries of the structure when the nanowire width decreases. Based on our computational results, we provided an explanation of this phenomenon, confirming that in all cases studied here surface energy suppresses the pattern formation.

References

1: D.R. Mahapatra, R.V.N. Melnik, "Finite element approach to modelling evolution of 3D shape memory materials", Mathematics and Computers in Simulation, 76(1-3), 141-148, 2007. doi:10.1016/j.matcom.2007.01.014
2: L.X. Wang, R.V.N. Melnik, "Model reduction applied to square to rectangular martensitic transformations using proper orthogonal decomposition", Applied Numerical Mathematics, 57(5-7), 510-520, 2007. doi:10.1016/j.apnum.2006.07.004
3: L.X. Wang, R.V.N. Melnik, "Finite volume analysis of nonlinear thermo-mechanical dynamics of shape memory alloys", Heat and Mass Transfer, 43(6), 535-546, 2007. doi:10.1007/s00231-006-0129-3

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Front Page Browse CCP CSETS CTR IJRT Other Authors Search Purchase Guide FAQ Contact us	Civil-Comp Proceedings ISSN 1759-3433 CCP: 89 PROCEEDINGS OF THE SIXTH INTERNATIONAL CONFERENCE ON ENGINEERING COMPUTATIONAL TECHNOLOGY Edited by: M. Papadrakakis and B.H.V. Topping Paper 139 Phase Transformations in Finite Length Nanowires: Analysis with Mesoscopic Models L.X. Wang¹ and R. Melnik² ¹Faculty of Mechanical Engineering, Hangzhou Dainzi University, Xiasha, Hangzhou, P.R. China ² M²NeT Lab, Wilfrid Laurier University, Waterloo, Canada doi:10.4203/ccp.89.139 purchase the full-text of this paper Full Bibliographic Reference for this paper L.X. Wang, R. Melnik, "Phase Transformations in Finite Length Nanowires: Analysis with Mesoscopic Models", in M. Papadrakakis, B.H.V. Topping, (Editors), "Proceedings of the Sixth International Conference on Engineering Computational Technology", Civil-Comp Press, Stirlingshire, UK, Paper 139, 2008. doi:10.4203/ccp.89.139 Keywords: phase transformations, nanowires, nonlocal models, nanostructures. Summary In this work we propose a simplified mesoscopic model for the analysis of nanowires of finite length. Our interest has been motivated by the fact that such nanowires can exhibit shape memory effects. This has been confirmed for a wide range of these structures, including ZnO, Cu, Ni, Al, Ag, and Au nanowires. Here we focus on a two-dimensional analogue of what in the general three-dimensional case would correspond to cubic to tertragonal or tetragonal to orthorhombic transformations. Such a two-dimensional case deals with square to rectangle transformations which are easier to treat due to the fact that the energy as a function of lattice spacing exhibits two distinct minima corresponding to two different phases of the nanostructure. Most results obtained up to date, both theoretical and computational, are concerned with infinitely long nanowires or infinitely large nanoplates for that matter. In this contribution, we develop a relatively simple and computationally inexpensive model to study phase transformations in finite nanostructures with our major focus given to nanowires of finite length. We note the models describing shape memory effects at the mesoscopic level such as those developed in [1] can be reduced to a two-dimensional case (and in the case of nanowires of infinite length, to the one-dimensional case). Our considerations are based on a modification of a coupled system of PDEs for the evolution of displacements and temperature [2,3]. One of the applications of the nanowires discussed in this paper stems from their integration in biomolecular technologies and we highlight the importance of the contributions of entropic elasticity in such cases. Recently, we developed and demonstrated the application of several efficient methodologies to solve two-dimensional models describing square-to-rectangle phase transformations in materials with memory, in particular the finite volume methodology [3] and a numerical reduction procedure based on the proper orthogonal decomposition (POD) [2]. The methodology proposed in this contribution is different and is based on a combination of the Chebyshev pseudo-spectral technique combined with the domain decomposition method and second order backward differentiation. Based on this methodology, we analysed different types of nanowires and examplified our results here with Cu nanowires for different diameter-length ratio. Our results indicate that the initial pattern formation propagates towards the longitudinal boundaries of the structure when the nanowire width decreases. Based on our computational results, we provided an explanation of this phenomenon, confirming that in all cases studied here surface energy suppresses the pattern formation. References 1 D.R. Mahapatra, R.V.N. Melnik, "Finite element approach to modelling evolution of 3D shape memory materials", Mathematics and Computers in Simulation, 76(1-3), 141-148, 2007. doi:10.1016/j.matcom.2007.01.014 2 L.X. Wang, R.V.N. Melnik, "Model reduction applied to square to rectangular martensitic transformations using proper orthogonal decomposition", Applied Numerical Mathematics, 57(5-7), 510-520, 2007. doi:10.1016/j.apnum.2006.07.004 3 L.X. Wang, R.V.N. Melnik, "Finite volume analysis of nonlinear thermo-mechanical dynamics of shape memory alloys", Heat and Mass Transfer, 43(6), 535-546, 2007. doi:10.1007/s00231-006-0129-3 purchase the full-text of this paper (price £20) go to the previous paper go to the next paper return to the table of contents return to the book description purchase this book (price £95 +P&P)
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