Surface energies in a two-dimensional mass-spring model for crystals

Florian Theil

ESAIM: Mathematical Modelling and Numerical Analysis (2011)

  • Volume: 45, Issue: 5, page 873-899
  • ISSN: 0764-583X

Abstract

top
We study an atomistic pair potential-energy E(n)(y) that describes the elastic behavior of two-dimensional crystals with n atoms where y ∈ ℝ 2 × n characterizes the particle positions. The main focus is the asymptotic analysis of the ground state energy as n tends to infinity. We show in a suitable scaling regime where the energy is essentially quadratic that the energy minimum of E(n) admits an asymptotic expansion involving fractional powers of n: min y E ( n ) ( y ) = n E bulk + n E surface + o ( n ) , n → ∞ . The bulk energy density Ebulk is given by an explicit expression involving the interaction potentials. The surface energy Esurface can be expressed as a surface integral where the integrand depends only on the surface normal and the interaction potentials. The evaluation of the integrand involves solving a discrete algebraic Riccati equation. Numerical simulations suggest that the integrand is a continuous, but nowhere differentiable function of the surface normal.

How to cite

top

Theil, Florian. "Surface energies in a two-dimensional mass-spring model for crystals." ESAIM: Mathematical Modelling and Numerical Analysis 45.5 (2011): 873-899. <http://eudml.org/doc/197544>.

@article{Theil2011,
abstract = { We study an atomistic pair potential-energy E(n)(y) that describes the elastic behavior of two-dimensional crystals with n atoms where $y \in \{\mathbb R\}^\{2\times n\}$ characterizes the particle positions. The main focus is the asymptotic analysis of the ground state energy as n tends to infinity. We show in a suitable scaling regime where the energy is essentially quadratic that the energy minimum of E(n) admits an asymptotic expansion involving fractional powers of n: $\{\rm min\}_y E^\{(n)\}(y) = n \, E_\{\mathrm\{bulk\}\}+ \sqrt\{n\} \, E_\mathrm\{surface\} +o(\sqrt\{n\}), \qquad n \to \infty.$ The bulk energy density Ebulk is given by an explicit expression involving the interaction potentials. The surface energy Esurface can be expressed as a surface integral where the integrand depends only on the surface normal and the interaction potentials. The evaluation of the integrand involves solving a discrete algebraic Riccati equation. Numerical simulations suggest that the integrand is a continuous, but nowhere differentiable function of the surface normal. },
author = {Theil, Florian},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Continuum mechanics; difference equations; continuum mechanics; two-dimensional crystals},
language = {eng},
month = {2},
number = {5},
pages = {873-899},
publisher = {EDP Sciences},
title = {Surface energies in a two-dimensional mass-spring model for crystals},
url = {http://eudml.org/doc/197544},
volume = {45},
year = {2011},
}

TY - JOUR
AU - Theil, Florian
TI - Surface energies in a two-dimensional mass-spring model for crystals
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2011/2//
PB - EDP Sciences
VL - 45
IS - 5
SP - 873
EP - 899
AB - We study an atomistic pair potential-energy E(n)(y) that describes the elastic behavior of two-dimensional crystals with n atoms where $y \in {\mathbb R}^{2\times n}$ characterizes the particle positions. The main focus is the asymptotic analysis of the ground state energy as n tends to infinity. We show in a suitable scaling regime where the energy is essentially quadratic that the energy minimum of E(n) admits an asymptotic expansion involving fractional powers of n: ${\rm min}_y E^{(n)}(y) = n \, E_{\mathrm{bulk}}+ \sqrt{n} \, E_\mathrm{surface} +o(\sqrt{n}), \qquad n \to \infty.$ The bulk energy density Ebulk is given by an explicit expression involving the interaction potentials. The surface energy Esurface can be expressed as a surface integral where the integrand depends only on the surface normal and the interaction potentials. The evaluation of the integrand involves solving a discrete algebraic Riccati equation. Numerical simulations suggest that the integrand is a continuous, but nowhere differentiable function of the surface normal.
LA - eng
KW - Continuum mechanics; difference equations; continuum mechanics; two-dimensional crystals
UR - http://eudml.org/doc/197544
ER -

References

top
  1. R. Alicandro, A. Braides and M. Cicalese, Continuum limits of discrete films with superlinear growth densities. Calc. Var. Par. Diff. Eq.33 (2008) 267–297.  
  2. S. Aubry, The twist map, the extended Frenkel-Kontorova model and the devil's staircase. Physica D7 (1983) 240–258.  
  3. X. Blanc, C. Le Bris and P.L. Lions, From molecular models to continuum mechanics. Arch. Rat. Mech. Anal.164 (2002) 341–381.  
  4. A. Braides and M. Cicalese, Surface energies in nonconvex discrete systems. Math. Models Meth. Appl. Sci.17 (2007) 985–1037.  
  5. A. Braides and A. DeFranchesi, Homogenisation of multiple integrals. Oxford University Press (1998).  
  6. A. Braides and M. Gelli, Continuum limits of discrete systems without convexity hypotheses. Math. Mech. Solids7 (2002) 41–66.  
  7. A. Braides, M. Solci and E. Vitali, A derivation of linear alastic energies from pair-interaction atomistic systems. Netw. Heterog. Media9 (2007) 551–567.  
  8. J. Cahn, J. Mallet-Paret and E. Van Vleck, Travelling wave solutions for systems of ODEs on a two-dimensional spatial lattice. SIAM J. Appl. Math.59 (1998) 455–493.  
  9. M. Charlotte and L. Truskinovsky, Linear elastic chain with a hyper-pre-stress. J. Mech. Phys. Solids50 (2002) 217–251.  
  10. W. E and P. Ming, Cauchy-Born rule and the stability of crystalline solids: static problems. Arch. Rat. Mech. Anal.183 (2005) 241–297.  
  11. I. Fonseca and S. Müller, A uniqueness proof for the Wulff theorem. Proc. Roy. Soc. Edinburgh Sect. A 119 (1991) 125–136.  
  12. G. Friesecke and F. Theil, Validitity and failure of the Cauchy-Born rule in a two-dimensional mass-spring lattice. J. Nonlinear Sci.12 (2002) 445–478.  
  13. G. Friesecke, R. James and S. Müller, A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity. Comm. Pure Appl. Math.55 (2002) 1461–1506.  
  14. D. Gérard-Varet and N. Masmoudi, Homogenization and boundary layer. Preprint available at www.math.nyu.edu/faculty/masmoudi/homog_Varet3.pdf (2010).  
  15. P. Lancaster and L. Rodman, Algebraic Riccati Equations. Oxford University Press (1995).  
  16. J.L. Lions, Some methods in the mathematical analysis of systems and their controls. Science Press, Beijing, Gordon and Breach, New York (1981).  
  17. J.A. Nitsche, On Korn's second inequality. RAIRO Anal. Numér.15 (1981) 237–248.  
  18. C. Radin, The ground state for soft disks. J. Stat. Phys.26 (1981) 367–372.  
  19. B. Schmidt, A derivation of continuum nonlinear plate theory from atomistic models. Multiscale Mod. Sim.5 (2006) 664–694.  
  20. B. Schmidt, On the passage from atomic to continuum theory for thin films. Arch. Rat. Mech. Anal.190 (2008) 1–55.  
  21. B. Schmidt, On the derivation of linear elasticity from atomistic models. Net. Heterog. Media4 (2009) 789–812.  
  22. E. Sonntag, Mathematical Control Theory. Second edition, Springer (1998).  
  23. L. Tartar, The general theory of homogenization. Springer (2010).  
  24. F. Theil, A proof of crystallization in a two dimensions. Comm. Math. Phys.262 (2006) 209–236.  

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.