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A Continuum-on-Atomistic Framework with Bi-Stable Elements for the Computation of Minimum Free Energy Paths

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Abstract

The exploration of non-convex energy landscapes, as arising in phase transitions, is an important task in solid state mechanics. An often employed system in the literature for the study of phase transitions in deformable solids denotes the elastic bar with a non-monotone stress-strain curve. This setting is chosen and modelled by a continuum-on-atomistic model (molecular dynamics coupled with the finite element method). The rod’s material denotes a copper single crystal and undergoes a model phase transition. The resulting non-convex energy landscape is explored by the string method in collective variables. The string method allows for the computation of energy barriers between local minima and to identify minimum free energy paths. The novelty of the present work is the combination of a continuum-on-atomistic method with the string method applied to a problem in mechanics. Numerical examples demonstrate the performance of the numerical model.

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References

  1. E. Patoor, D. C. Lagoudas, P. B. Entchev, L. C. Brinson, et al., “Shape Memory Alloys, Part I: General Properties and Modeling of Single Crystals,” Mech. Mater. 38, 391–429 (2006).

    Article  Google Scholar 

  2. D. C. Lagoudas, P. B. Entchev, P. Popov, E. Patoor, et al., “Shape Memory Alloys, Part II: Modeling of Polycrystals,” Mech. Mater. 38, 430–462 (2006).

    Article  Google Scholar 

  3. J. Mohd Jani, M. Leary, A. Subie, M. A. Gibson, “A Review of Shape Memory Alloy Research, Applications and Opportunities,” Mater. Des. 56, 1078–1113 (2014).

    Article  Google Scholar 

  4. C. Cisse, W. Zaki, T. B. Zineb, “A Review of Constitutive Models and Modeling Techniques for Shape Memory Alloys,” Int. J. Plast. 76, 244–284 (2016).

    Article  Google Scholar 

  5. J. L. Ericksen, “Equilibrium of Bars,” J. Elasticity 5, 191–201 (1975).

    Article  MathSciNet  MATH  Google Scholar 

  6. G. Puglisi and L. Truskinovsky, “Mechanics of a Discrete Chain with Bi-Stable Elements,” J. Mech. Phys. Solids 48, 1–27 (2000).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  7. G. Puglisi and L. Truskinovsky, “Rate Independent Hysteresis in a Bi-Stable Chain,” J. Mech. Phys. Solids 50, 165–187 (2002).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  8. G. Puglisi and L. Truskinovsky, A Mechanism of Transformational plasticity,” Continuum Mech. Thermodyn. 14, 437–457 (2002).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  9. G. Puglisi and L. Truskinovsky, “Thermodynamics of Rate-Independent Plasticity,” J. Mech. Phys. Solids 53, 655–679 (2005).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  10. A. Vainchtein, P. Rosakis, “Hysteresis and Stick-Slip Motion of Phase Boundaries in Dynamic Models of Phase Transitions,” J. Nonlinear Sci. 9, 697–719 (1999).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  11. A. Vainchtein, “Dynamics of Phase Transitions and Hysteresis in a Viscoelastic Ericksen’s Bar on an Elastic Foundation,” J. Elasticity 57, 243–280 (1999).

    Article  MathSciNet  MATH  Google Scholar 

  12. A. M. Balk, A. V. Cherkaev, and L. I. Slepyan, “Dynamics of Chains with Non-Monotone Stress-Strain Relations. I. Model and Numerical Experiments,” J. Mech. Phys. Solids 49, 131–148 (2001).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  13. A. M. Balk, A. V. Cherkaev, and L. I. Slepyan, “Dynamics of Chains with Non-Monotone Stress-Strain relations. II. Nonlinear Waves and Waves of Phase Transition,” J. Mech. Phys. Solids 49, 149–171 (2001).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  14. Y. R. Efendiev and L. Truskinovsky, “Thermalization of a Driven Bi-Stable FPU Chain,” Continuum Mech. Thermodyn. 22, 679–698 (2010).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  15. T. Cohen and S. Givli, “Dynamics of a Discrete Chain of Bi-Stable Elements: A Biomimetic Shock Absorbing Mechanism,” J. Mech. Phys. Solids 64, 426–439 (2014).

    Article  ADS  Google Scholar 

  16. N. Nadkarni, C. Daraio, and D. M. Kochmann, “Dynamics of Periodic Mechanical Structures Containing Bistable Elastic Elements: From Elastic to Solitary Wave Propagation,” Phys. Rev. E 90, 023204 (2014).

    Article  ADS  Google Scholar 

  17. Q. Zhao and P. K Purohit, “Extracting a Kinetic Relation from the Dynamics of a Bistable Chain,” Modelling Simul. Mater. Sci. Eng. 22, 045004 (2014).

    Article  ADS  Google Scholar 

  18. I. Benichou and S. Givli, “Structures Undergoing Discrete Phase Transformation,” J. Mech. Phys. Solids 61, 94–113 (2013).

    Article  ADS  MathSciNet  Google Scholar 

  19. T. Blesgen, F. Fraternali, J. R. Raney, A. Arriendola, et al., “Continuum Limits of Bistable Spring Models of Carbon Nanotube Arrays Accounting for Material Damage,” Mech. Res. Commun. 45, 58–63 (2012).

    Article  Google Scholar 

  20. I. Benichou, E. Faran, D. Shilo, and S. Givli, “Application of a Bi-Stable Chain Model for the Analysis of Jerky Twin Boundary Motion in NiMnGa,” Appl. Phys. Lett. 102, 011912 (2013).

    Article  ADS  Google Scholar 

  21. Y. Gerson, S. Krylov, B. Ilic, and D. Schreiber, “Design Considerations of a Large-Displacement Multistable Micro Actuator with Serially Connected Bistable Elements,” Finite Elem. Anal. Des. 49, 58–69 (2012).

    Article  Google Scholar 

  22. R. Abeyaratne, “Story of F: the Driving Force on a Phase Boundary,” in Topics in Finite Elasticity, International Centre for Mechanical Sciences, Ed. by M. Hayes and G. Saccomandi (Springer, Vienna, 2001), Vol. 424, pp. 231–244.

    Chapter  Google Scholar 

  23. E. B. Tadmor, R. E. Miller, and R. S. Elliott, Continuum Mechanics and Thermodynamics (Cambridge University Press, Cambridge, 2012).

    MATH  Google Scholar 

  24. E. B. Tadmor, R. Phillips, and M. Ortiz, “Hierarchical Modeling in the Mechanics of Materials,” Int. J. Solids Struct. 37, 379–389 (2000).

    Article  MathSciNet  MATH  Google Scholar 

  25. W. K Liu, E. G. Karpov, S. Zhang, and H. S. Park, “An Introduction to Computational Nanomechanics and Materials,” Comput. Meth. Appl. Mech. Eng. 193, 1529–1578 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  26. A. Abdulle, W. E. B. Engquist, E. Vanden-Eijnden, “The Heterogeneous Multiscale Method,” Acta Numerica 21, 1–87 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  27. W. A. Curtin and R. E. Miller, “Atomistic/Continuum Coupling in Computational Materials Science,” Modelling Simul. Mater. Sci. Eng. 11, R33 (2003).

    Article  ADS  Google Scholar 

  28. R. E. Miller and E. B. Tadmor, “A Unified Framework and Performance Benchmark of Fourteen Multiscale Atomistic/Continuum Coupling Methods,” Modelling Simul. Mater. Sci. Eng. 17, 05300J (2009).

    Article  Google Scholar 

  29. X. Zeng and S. Li, “Recent Developments on Concurrent Multiscale Simulations,” in: Advances in Engineering Mechanics, Ed. by Q. H. Qin and B. Sun (Nova Science Publishers, Inc., 2010), Volume 1.

  30. H. Jonsson, G. Mills, and K W. Jacobsen, “Nudged Elastic Band Method for Finding Minimum Energy Paths of Transitions,” in: Classical and Quantum Dynamics in Condensed Phase Simulations, Ed. by B. J. Berne, G. Ciccoti, D. F. Coker (World Scientific, Singapore, 1998), pp. 385–404.

    Chapter  Google Scholar 

  31. D. Sheppard, R. Terrell, and G. Henkelman, “Optimization Methods for Finding Minimum Energy Paths,” J. Chem. Phys. 128, 134106 (2008).

    Article  ADS  Google Scholar 

  32. W. E, W. Ren, and E. Vanden-Eijnden, “Simplified and Improved String Method for Computing the Minimum Energy Paths in Barrier-Crossing Events,” J. Chem. Phys. 126, 164103 (2007).

    Article  ADS  Google Scholar 

  33. M. Cameron, R. V. Kohn, and E. Vanden-Eijnden, “The String Method as a Dynamical System,” J. Nonlinear Sci. 21, 193–230 (2011).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  34. S. Fischer and M. Karplus, “Conjugate Peak Refinement: an Algorithm for Finding Reaction Paths and Accurate Transition States in Systems with Many Degrees of Freedom,” Chem. Phys. Lett. 194, 252–261 (1992).

    Article  ADS  Google Scholar 

  35. G. Henkelman and H. Jonsson, “A Dimer Method for Fnding Saddle Points on High Dimensional Potential Surfaces Using Only Frst Derivatives,” J. Chem. Phys. 111, 7010–7022 (1999).

    Article  ADS  Google Scholar 

  36. R. A. Miron and K A. Fichthorn, “The Step and Slide Method for Finding Saddle Points on Multidimensional Potential Surfaces,” J. Chem. Phys. 115, 8742–8747 (2001).

    Article  ADS  Google Scholar 

  37. H. B. Schlegel, “Geometry Optimization,” Wiley Interdiscip. Rev.-Comput. Mol. Sci. 1, 790–809 (2011).

    Article  Google Scholar 

  38. K. J. Caspersen and E. A. Carter, “Finding Transition States for Crystalline Solidsolid Phase Transformations,” Proc. Natl. Acad. Sci. U. S. A. 102, 6738–6743 (2005).

    Article  ADS  Google Scholar 

  39. P. Xiao, D. Sheppard, J. Rogal, and G. Henkelman, “Solid-State Dimer Method for Calculating Solid-Solid Phase Transitions,” J. Chem. Phys. 140, 174104 (2014).

    Article  ADS  Google Scholar 

  40. G.-R. Qian, X. Dong, X.-F. Zhou, Y. Tian, et al., “Variable Cell Nudged Elastic Band Method for Studying Solidsolid Structural Phase Transitions,” Comput. Phys. Commun. 184, 2111–2118 (2013).

    Article  ADS  Google Scholar 

  41. N. A. Zarkevich and D. D. Johnson, “Magneto-Structural Transformations Via a Solid-State Nudged Elastic Band Method: Application to Iron under Pressure,” J. Chem. Phys. 143, 064707 (2015).

    Article  ADS  Google Scholar 

  42. M. H. Ulz, “Coupling the Finite Element Method and Molecular Dynamics in the Framework of the Heterogeneous Multiscale Method for Quasi-Static Isothermal Problems,” J. Mech. Phys. Solids 74, 1–18 (2015).

    Article  ADS  MathSciNet  Google Scholar 

  43. P. Wurm and M. H. Ulz, “A Stochastic Approximation Approach to Improve the Convergence Behavior of Hierarchical Atomistic-to-Continuum Multiscale Models,” J. Mech. Phys. Solids 95, 480–500 (2016).

    Article  ADS  MathSciNet  Google Scholar 

  44. L. Maragliano, A. Fischer, E. Vanden-Eijnden, and G. Ciccotti, “String Method in Collective Variables: Minimum Free Energy Paths and Isocommittor Surfaces,” J. Chem. Phys. 125, 024106 (2006).

    Article  ADS  Google Scholar 

  45. L. Maragliano and E. Vanden-Eijnden, “On-the-fly string Method for Minimum Free Energy Paths 764 Calculation,” Chem. Phys. Lett. 446, 182–190 (2007).

    Article  ADS  Google Scholar 

  46. W. E and B. Engquist, “The Heterogeneous Multiscale Methods,” Commun. Math. Sci. 1, 87–132 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  47. W. E, B. Engquist, X. Li, W. Ren, et al., “Heterogeneous Multiscale Methods: A Review,” Commun. Comput. Phys. 2, 367–450 (2007).

    MathSciNet  MATH  Google Scholar 

  48. R. N. Thurston, “Wave Propagation in Fluids and Normal Solids,” in Physical Acoustics, Ed. by W. P. Mason (Academic Press, New York, 1964), Vol. 1 A.

    Google Scholar 

  49. L. E. Malvern, Introduction to the Mechanics of a Continuous Medium (Prentice-Hall, Inc., New Jersey, 1969).

    Google Scholar 

  50. J. Bonet, R. D. Wood, Nonlinear Continuum Mechanics for Finite Element Analysis (Cambridge University Press, Cambridge, 1997).

    MATH  Google Scholar 

  51. P. Deuflhard, “Newton Methods for Nonlinear Problems: Affine Invariance and Adaptive Algorithms,” in Springer Series in Computational Mathematics, Vol. 35, 2011.

  52. H. C. Andersen, “Molecular Dynamics Simulations at Constant Pressure and/or Temperature,” J. Chem. Phys. 72, 2384–2393 (1980).

    Article  ADS  Google Scholar 

  53. M. Parrinello and A. Rahman, “Polymorphic Transitions in Single Crystals: A New Molecular Dynamics Method,” J. Appl. Phys. 52, 7182–7190 (1981).

    ADS  Google Scholar 

  54. J. R. Ray and A. Rahman, “Statistical Ensembles and Molecular Dynamics Studies of Anisotropic Solids. II,” J. Chem. Phys. 82, 4243–4247 (1985).

    Article  ADS  Google Scholar 

  55. S. Nose, “A Unified Formulation of the Constant Temperature Molecular-Dynamics Methods,” J. Chem. Phys. 81, 511–519 (1984).

    Article  ADS  Google Scholar 

  56. P. Podio-Guidugli, “On (Andersen-) Parrinello-Rahman Molecular Dynamics, the Related Metadynamics, and the Use of the Cauchy-Born Rule,” J. Elasticity 100, 145–153 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  57. M. H. Ulz, “Comments on a Continuum-Related Parrinello-Rahman Molecular Dynamics Formulation,” J. Elasticity 113, 93–112 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  58. M. H. Ulz, A Multiscale Molecular Dynamics Method for Isothermal Dynamic Problems Using the Seamless Heterogeneous Multiscale Method,” Comput. Methods Appl. Mech. Eng. 295, 510–524 (2015).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  59. E. B. Tadmor and R. E. Miller, Modeling Materials: Continuum, Atomistic and Multiscale Techniques (Cambridge University Press, Cambridge, 2011).

    Book  MATH  Google Scholar 

  60. C. Truesdell and R. A. Toupin, “The Classical Field Theories,” in Encyclopedia of Physics, ed. by S. Flügge, (Springer, Berlin, 1960), Volume III/1.

    MATH  Google Scholar 

  61. J. R. Ray and A. Rahman, “Statistical Ensembles and Molecular Dynamics Studies of Anisotropic Solids,” J. Chem. Phys. 80, 4423–4428 (1984).

    Article  ADS  Google Scholar 

  62. W. G. Hoover, “Canonical Dynamics: Equilibrium Phase-Space Distributions,” Phys. Rev. A 31, 1695–1697 (1985).

    Article  ADS  Google Scholar 

  63. P. H. Hunenberger, “Thermostat Algorithms for Molecular Dynamics Simulations,” Adv. Polym. Sci. 173, 105–149 (2005).

    Article  Google Scholar 

  64. J. H. Weiner, Statistical Mechanics of Elasticity, 2nd ed., (Dover Publications, Inc., New York, 2002).

    MATH  Google Scholar 

  65. F. Reif, Fundamentals of Statistical and Thermal Physics (Waveland Press Inc, Long Crove, 2008).

    Google Scholar 

  66. W. E, Principles of Multiscale Modeling (Cambridge University Press, Cambridge, 2011).

    MATH  Google Scholar 

  67. T. F Miller III, E. Vanden-Eijnden, and D. Chandler, “Solvent Coarse-Graining and the String Method Applied to the Hydrophobic Collapse of a Hydrated Chain,” Proc. Natl. Acad. Sci. U. S. A. 104, 14559–14564 (2007).

    Article  ADS  Google Scholar 

  68. A. C. Pan, T. M. Weinreich, Y. Shan, D. P. Scarpazza, et al., “Assessing the Accuracy of Two Enhanced Sampling Methods Using EGFR Kinase Transition Pathways: The in Uence of Collective Variable Choice,” J. Chem. Theory Comput. 10 (2014) 2860–2865.

    Article  Google Scholar 

  69. B. Hashemian, D. Millan, and M. Arroyo, “Charting Molecular Free-Energy Landscapes with an Atlas of Collective Variables,” J. Chem. Phys. 145 174109 (2016).

    Article  ADS  Google Scholar 

  70. W. E, W. Ren, and E. Vanden-Eijnden, “String Method for the Study of Rare Events,” Phys. Rev. B 66, 052301 (2002).

    Article  ADS  Google Scholar 

  71. W. E, W. Ren, and E. Vanden-Eijnden, “Finite Temperature String Method for the Study of Rare Events,” J. Phys. Chem. B 109, 6688–6693 (2005).

    Article  Google Scholar 

  72. W. Ren, E. Vanden-Eijnden, “A Climbing String Method for Saddle Point Search,” J. Chem. Phys. 138 134105 (2013).

    Article  ADS  Google Scholar 

  73. W. G. Nöhring and W. A. Curtin, “Dislocation Cross-Slip in FCC Solid Solution Alloys,” Acta Mater. 128, 135–148 (2017).

    Article  Google Scholar 

  74. Z. P. Pi, Q. H. Fang, B. Liu, Y. Liu, et al., “Effect of a Generalized Shape Peierls Potential and an External Stress Field on Kink Mechanism in a Continuum Model,” Int. J. Plast. 90, 267–285 (2017).

    Article  Google Scholar 

  75. S. Saroukhani, L. D. Nguyen, K. W. K. Leung, C. V. Singh, et al., “Harnessing Atomistic Simulations to Predict the Rate at which Dislocations Overcome Obstacles,” J. Mech. Phys. Solids 90, 203–214 (2016).

    Article  ADS  MathSciNet  Google Scholar 

  76. S. Saroukhani and D. H. Warner, Investigating Dislocation Motion through a Field of Solutes with Atomistic Simulations and Reaction Rate Theory,” Acta Mater. 128 77–86 (2017).

    Article  Google Scholar 

  77. D. Frenkel, “Free Energy Computation and First-Order phase Transitions,” in Molecular-Dynamics Simulation of Statistical-Mechanical Systems: Proceedings of the 97th International School of Physics “Enrico Fermi”, ed. by G. Ciccotti, W G. Hoover(North-Holland, Amsterdam, 1986), pp. 151–188.

    Google Scholar 

  78. M. E. Tuckerman, Statistical Mechanics: Theory and Molecular Simulation (Oxford University Press, Oxford, 2010).

    MATH  Google Scholar 

  79. W. F. van Gunsteren, X. Daura, and A. E. Mark, “Computation of Free Energy,” Helv. Chim. Acta 85, 3113–3129 (2002).

    Article  Google Scholar 

  80. C. Chipot and A. E. Pohorille, Free Energy Calculations (Springer-Verlag, Berlin, 2007).

    Book  Google Scholar 

  81. M. Brokate and J. Sprekels, “Hysteresis and Phase Transitions,” in Applied Mathematical Sciences (Springer, New York, 1996), Vol. 121.

    Google Scholar 

  82. D. Frenkel and A. J. C. Ladd, “New Monte Carlo Method to Compute the Free Energy of Arbitrary Solids. Application to the FCC and HCP Phases of Hard Spheres,” J. Chem. Phys. 81, 3188–3193 (1984).

    Article  ADS  Google Scholar 

  83. D. Frenkel and B. Smit, Understanding Molecular Simulation: From Algorithms to Applications (Academic Press, San Diego, 2002).

    MATH  Google Scholar 

  84. X. W. Zhou, H. N. G. Wadley, R. A. Johnson, D. J. Larson, et al. “Atomic Scale Structure of Sputtered Metal Multilayers,” Acta Mater. 49 (19), 4005–4015 (2001).

    Article  Google Scholar 

  85. M. P. Allen and T. J. Tildesley, Computer Simulation of Liquids (Clarendon Press, Oxford, 2009).

    MATH  Google Scholar 

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Acknowledgement

The author would like to express his sincere gratitude to Prof. Eric Vanden-Eijnden, Courant Institute of Mathematical Sciences (NYU), for his invaluable suggestions.

All of the computational results presented have been achieved using the Vienna Scientific Cluster (VSC).

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  1. Institute of Strength of Materials, Graz University of Technology, Kopernikusgasse 24/I, 8010, Graz, Austria

    M. H. Ulz

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Ulz, M.H. A Continuum-on-Atomistic Framework with Bi-Stable Elements for the Computation of Minimum Free Energy Paths. Mech. Solids 54, 975–994 (2019). https://doi.org/10.3103/S0025654419060128

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