# American Institute of Mathematical Sciences

September  2014, 9(3): 453-476. doi: 10.3934/nhm.2014.9.453

## Continuum surface energy from a lattice model

 1 Department of Applied Mathematics, University of Crete, Heraklion 70013, Greece

Received  January 2014 Revised  June 2014 Published  October 2014

We investigate connections between the continuum and atomistic descriptions of deformable crystals, using certain interesting results from number theory. The energy of a deformed crystal is calculated in the context of a lattice model with general binary interactions in two dimensions. A new bond counting approach is used, which reduces the problem to the lattice point problem of number theory. The main contribution is an explicit formula for the surface energy density as a function of the deformation gradient and boundary normal. The result is valid for a large class of domains, including faceted (polygonal) shapes and regions with piecewise smooth boundaries.
Citation: Phoebus Rosakis. Continuum surface energy from a lattice model. Networks and Heterogeneous Media, 2014, 9 (3) : 453-476. doi: 10.3934/nhm.2014.9.453
##### References:
 [1] I. Bárány and D. G. Larman, The convex hull of the integer points in a large ball, Mathematische Annalen, 312 (1998), 167-181. doi: 10.1007/s002080050217. [2] A. Barvinok and J. E. Pommersheim, An algorithmic theory of lattice points in polyhedra, New Perspectives in Algebraic Combinatorics, 38 (1999), 91-147. [3] X. Blanc, C. Le Bris and P.-L. Lions, From molecular models to continuum mechanics, Arch. Rational Mech. Anal., 164 (2002), 341-381. doi: 10.1007/s00205-002-0218-5. [4] M. Beck and S. Robins, Computing the Continuous Discretely: Integer Point Enumeration in Polyhedra, Undergraduate Texts in Mathematics, Springer-Verlag, 2007. [5] A. Braides and M. Cicalese, Surface energies in nonconvex discrete systems, Mathematical Models and Methods in Applied Sciences, 17 (2007), 985-1037. doi: 10.1142/S0218202507002182. [6] J. G. Van der Corput, Over Roosterpunten in het Platte vlak (de Beteekenis van de Methoden van Voronoi en Pfeiffer), Noordhoff, 1919. [7] B. Dacorogna and C.-E Pfister, Wulff theorem and best constant in Sobolev inequality, Journal de mathématiques pures et appliquées, 71 (1992), 97-118. [8] A. Eichler, J. Hafner, J. Furthmüller and G. Kresse, Structural and electronic properties of rhodium surfaces: an ab initio approach, Surface Science, 346 (1996), 300-321. doi: 10.1016/0039-6028(95)00906-X. [9] I. Fonseca, The Wulff theorem revisited, Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences, 432 (1991), 125-145. doi: 10.1098/rspa.1991.0009. [10] C. Herring, Some theorems on the free energies of crystal surfaces, Physical Review, 82 (1951), 87-93. doi: 10.1103/PhysRev.82.87. [11] M. N. Huxley, Exponential sums and lattice points III, Proceedings of the London Mathematical Society, 87 (2003), 591-609. doi: 10.1112/S0024611503014485. [12] A. Ivic, E. Krätzel, M. Kühleitner and W. G. Nowak, Lattice points in large regions and related arithmetic functions: Recent developments in a very classic topic,, preprint, (). [13] C. Mora-Corral, Continuum limits of atomistic energies allowing smooth and sharp interfaces in 1D elasticity, Interfaces and Free Boundaries, 11 (2009), 421-446. doi: 10.4171/IFB/217. [14] G. A. Pick, Geometrisches zur Zahlenlehre, Sitzenber. Lotos Naturwissen Zeitschrift (Prague), 19 (1899), 311-319 [15] J. E. Reeve, On the volume of lattice polyhedra, Proc. London Math. Soc., 3 (1957), 378-395. [16] P. Rosakis, Surface and Interfacial Energy in Three Dimensional Crystals, in progress, (2014). [17] J. D. Sally and P. Sally, Roots to research: A vertical development of mathematical problems, American Mathematical Society, (2007). [18] A. V. Shapeev, Consistent energy-based atomistic/continuum coupling for two-body potentials in one and two dimensions, Multiscale Modeling and Simulation, 9 (2011), 905-932 doi: 10.1137/100792421. [19] F. Theil, Surface energies in a two-dimensional mass-spring model for crystals, ESAIM Math. Model. Numer. Anal., 45 (2011), 873-899 doi: 10.1051/m2an/2010106. [20] K.-M. Tsang, Counting lattice points in the sphere, Bull. London Math. Soc., 32 (2000), 679-688. doi: 10.1112/S0024609300007505.

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##### References:
 [1] I. Bárány and D. G. Larman, The convex hull of the integer points in a large ball, Mathematische Annalen, 312 (1998), 167-181. doi: 10.1007/s002080050217. [2] A. Barvinok and J. E. Pommersheim, An algorithmic theory of lattice points in polyhedra, New Perspectives in Algebraic Combinatorics, 38 (1999), 91-147. [3] X. Blanc, C. Le Bris and P.-L. Lions, From molecular models to continuum mechanics, Arch. Rational Mech. Anal., 164 (2002), 341-381. doi: 10.1007/s00205-002-0218-5. [4] M. Beck and S. Robins, Computing the Continuous Discretely: Integer Point Enumeration in Polyhedra, Undergraduate Texts in Mathematics, Springer-Verlag, 2007. [5] A. Braides and M. Cicalese, Surface energies in nonconvex discrete systems, Mathematical Models and Methods in Applied Sciences, 17 (2007), 985-1037. doi: 10.1142/S0218202507002182. [6] J. G. Van der Corput, Over Roosterpunten in het Platte vlak (de Beteekenis van de Methoden van Voronoi en Pfeiffer), Noordhoff, 1919. [7] B. Dacorogna and C.-E Pfister, Wulff theorem and best constant in Sobolev inequality, Journal de mathématiques pures et appliquées, 71 (1992), 97-118. [8] A. Eichler, J. Hafner, J. Furthmüller and G. Kresse, Structural and electronic properties of rhodium surfaces: an ab initio approach, Surface Science, 346 (1996), 300-321. doi: 10.1016/0039-6028(95)00906-X. [9] I. Fonseca, The Wulff theorem revisited, Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences, 432 (1991), 125-145. doi: 10.1098/rspa.1991.0009. [10] C. Herring, Some theorems on the free energies of crystal surfaces, Physical Review, 82 (1951), 87-93. doi: 10.1103/PhysRev.82.87. [11] M. N. Huxley, Exponential sums and lattice points III, Proceedings of the London Mathematical Society, 87 (2003), 591-609. doi: 10.1112/S0024611503014485. [12] A. Ivic, E. Krätzel, M. Kühleitner and W. G. Nowak, Lattice points in large regions and related arithmetic functions: Recent developments in a very classic topic,, preprint, (). [13] C. Mora-Corral, Continuum limits of atomistic energies allowing smooth and sharp interfaces in 1D elasticity, Interfaces and Free Boundaries, 11 (2009), 421-446. doi: 10.4171/IFB/217. [14] G. A. Pick, Geometrisches zur Zahlenlehre, Sitzenber. Lotos Naturwissen Zeitschrift (Prague), 19 (1899), 311-319 [15] J. E. Reeve, On the volume of lattice polyhedra, Proc. London Math. Soc., 3 (1957), 378-395. [16] P. Rosakis, Surface and Interfacial Energy in Three Dimensional Crystals, in progress, (2014). [17] J. D. Sally and P. Sally, Roots to research: A vertical development of mathematical problems, American Mathematical Society, (2007). [18] A. V. Shapeev, Consistent energy-based atomistic/continuum coupling for two-body potentials in one and two dimensions, Multiscale Modeling and Simulation, 9 (2011), 905-932 doi: 10.1137/100792421. [19] F. Theil, Surface energies in a two-dimensional mass-spring model for crystals, ESAIM Math. Model. Numer. Anal., 45 (2011), 873-899 doi: 10.1051/m2an/2010106. [20] K.-M. Tsang, Counting lattice points in the sphere, Bull. London Math. Soc., 32 (2000), 679-688. doi: 10.1112/S0024609300007505.
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