Lattice Stability for Atomistic Chains Modeled by Local Approximations of the Embedded Atom Method
aa r X i v : . [ m a t h . NA ] A ug LATTICE STABILITY FOR ATOMISTIC CHAINS MODELED BY LOCALAPPROXIMATIONS OF THE EMBEDDED ATOM METHOD
XINGJIE HELEN LI AND MITCHELL LUSKIN
Abstract.
The accurate approximation of critical strains for lattice instability is a key criterionfor predictive computational modeling of materials. In this paper, we present a comparison of thelattice stability for atomistic chains modeled by the embedded atom method (EAM) with theirapproximation by local Cauchy-Born models. We find that both the volume-based local modeland the reconstruction-based local model can give O(1) errors for the critical strain since theembedding energy density is generally strictly convex. The critical strain predicted by the volume-based model is always larger than that predicted by the atomistic model, but the critical strain forreconstruction-based models can be either larger or smaller than that predicted by the atomisticmodel. Introduction
Predictive multiscale computational methods must be accurate near lattice instabilities thatcharacterize the formation and movement of cracks, dislocations, and grain boundaries. In thispaper, we present analytic results comparing the lattice instabilities predicted by an atomisticchain modeled by the embedded atom method (EAM) with the lattice instabilities predicted bylocal approximations of the atomistic model.Since it is not possible to compute large enough fully atomistic systems to accurately approxi-mate the interaction of local defects with long-range elastic fields, atomistic-to-continuum couplingmethods have been proposed [1, 2, 5, 11–13, 16, 18, 20]. For crystalline solids, the continuum regionis generally computed by coarse-graining a local approximation of the nonlocal atomistic model.An atom in the nonlocal atomistic region interacts with all of its neighbors within a cutoff radius.In the continuum region, the Cauchy-Born rule is used to derive a local model that approximatesthe interactions of atoms beyond their nearest neighbors by modified interactions of their nearestneighbors.To verify that an atomistic-to-continuum coupling method accurately reproduces the latticestability of the fully atomistic model, it is necessary to first verify that the local (continuum) modelitself reproduces the lattice stability of the fully atomistic model. Even if the local model reproducesthe lattice stability of the fully atomistic model, the atomistic-to-continuum coupling method maynot reproduce the lattice stability of the fully atomistic model because of the error introduced bythe coupling [3, 19].It has been proven in [3] that the Cauchy-Born local model reproduces the lattice stability for anatomistic chain modeled by Lennard-Jones type pair interaction (we note that the volume-based
Date : June 20, 2018.2000
Mathematics Subject Classification.
Key words and phrases. quasicontinuum, error analysis, atomistic to continuum, embedded atom model, quasi-nonlocal.This work was supported in part by DMS-0757355, DMS-0811039, the Institute for Mathematics and Its Applica-tions, and the University of Minnesota Supercomputing Institute. This work was also supported by the Departmentof Energy under Award Number de-sc0002085.
ATTICE STABILITY FOR ATOMISTIC CHAINS MODELED BY THE EAM 2 and reconstruction-based local models are equal for a pair potential interaction). And it has beenproven for multidimensional lattices that the set of stable uniform strains for the atomistic modelis a subset of the set of uniform strains for the Cauchy-Born volume-based local model [6, 8], butthe equality of these sets has not been demonstrated analytically. As a matter of fact, numericalexperiments in [8] suggest that the the inclusion is strict in some cases.In this paper, we prove for an atomistic chain that not only are the sets of stable uniform strainsdifferent for the atomistic model and the local models for a many-body potential, but the set ofstable uniform strains can be different for volume-based and reconstruction-based local models. Wewill focus our analysis on the embedded atom method [7, 9, 14], which is an empirical many-bodypotential that is widely used to model FCC metals such as copper and aluminum. We identify thecritical assumptions for the pair potential, electron density function, and embedding function tostudy the lattice stability of the atomistic and the different local models. We find that both thevolume-based local model and the reconstruction-based local model can give O(1) errors for thecritical strain since the embedding energy density is generally strictly convex.In Section 2, we present the notation used in this paper. We define the displacement space U and the deformation space Y F . We then introduce the norms we will use to estimate the modelingerror and the displacement gradient error. In Section 3, we briefly review the formulae of the fullyatomistic EAM model and the volume-based and the reconstruction-based local quasicontinuum(QCL) model, respectively.In Section 4, we give precise stability estimates for the fully atomistic model, the volume-basedand the reconstruction-based local models for a uniformly strained chain. We then compare thestability conditions of each model under different assumptions. We summarize our results anddiscuss extensions to multidimensional issues in the Conclusion.2. Notation
In this section, we present the notation used in this paper. We define the scaled reference lattice ǫ Z := { ǫℓ : ℓ ∈ Z } , where ǫ > Z is the set of integers. We then deform thereference lattice ǫ Z uniformly into the lattice F ǫ Z := { F ǫℓ : ℓ ∈ Z } where F > y F by ( y F ) ℓ := F ǫℓ for − ∞ < ℓ < ∞ . For simplicity, we consider the space U of 2 N -periodic zero mean displacements u = ( u ℓ ) ℓ ∈ Z from y F given by U := (cid:26) u : u ℓ +2 N = u ℓ for ℓ ∈ Z , and N X ℓ = − N +1 u ℓ = 0 (cid:27) , and we thus admit deformations y from the space Y F := { y : y = y F + u for some u ∈ U } . We set ǫ = 1 /N throughout so that the reference length of the periodic domain is fixed.We define the discrete differentiation operator, D u , on periodic displacements by( D u ) ℓ := u ℓ − u ℓ − ǫ , −∞ < ℓ < ∞ . ATTICE STABILITY FOR ATOMISTIC CHAINS MODELED BY THE EAM 3
We note that ( D u ) ℓ is also 2 N -periodic in ℓ and satisfies the zero mean condition. We will denote( D u ) ℓ by Du ℓ . We then define (cid:16) D (2) u (cid:17) ℓ := Du ℓ − Du ℓ − ǫ , −∞ < ℓ < ∞ , and we define (cid:0) D (3) u (cid:1) ℓ and (cid:0) D (4) u (cid:1) ℓ in a similar way. To make the formulas concise and morereadable, we sometimes denote Du ℓ by u ′ ℓ , D (2) u ℓ by u ′′ ℓ , etc., when there is no confusion in theexpressions.For a displacement u ∈ U and its discrete derivatives, we define the discrete ℓ ǫ norms by k u k ℓ ǫ := ǫ N X ℓ = − N +1 | u ℓ | ! / , k u ′ k ℓ ǫ := ǫ N X ℓ = − N +1 | u ′ ℓ | ! / , etc.Finally, for smooth real-valued functions E ( y ) defined for y ∈ Y F , we define the first and secondderivatives (variations) by h δ E ( y ) , w i := N X ℓ = − N +1 ∂ E ∂y ℓ ( y ) w ℓ for all w ∈ U , h δ E ( y ) v , w i := N X ℓ, m = − N +1 ∂ E ∂y ℓ ∂y m ( y ) v ℓ w m for all v , w ∈ U . The Embedded Atom Model and Its Local Approximations.
In this section, we will give a short description for the next-nearest neighbor atomistic EAMmodel and its approximations .3.1.
The Next-Nearest Neighbor Atomistic EAM Model.
Given deformations y ∈ Y F , thetotal energy per period of the next-nearest neighbor atomistic EAM model is E atot ( y ) := E a ( y ) + F ( y ) , (3.1)where E a ( y ) is the total atomistic energy and F ( y ) is the total external potential energy. The totalatomistic energy E a ( y ) is the sum of the embedding energy , ˆ E a ( y ) , and the pair potential energy ,˜ E a ( y ). The energy expression is E a ( y ) := ˆ E a ( y ) + ˜ E a ( y ) = ǫ N X ℓ = − N +1 (cid:16) ˆ E aℓ ( y ) + ˜ E aℓ ( y ) (cid:17) . (3.2)The embedding energy per atom (per atomistic reference spacing ǫ ) is defined as ˆ E aℓ ( y ) := G ( ¯ ρ aℓ ( y )) , where G (¯ ρ ) is the embedding energy function and ¯ ρ aℓ ( y ) is the total electron density at atom ℓ :¯ ρ aℓ ( y ) := ρ ( y ′ ℓ ) + ρ ( y ′ ℓ + y ′ ℓ − ) + ρ ( y ′ ℓ +1 ) + ρ ( y ′ ℓ +1 + y ′ ℓ +2 ) . The function ρ ( r/ǫ ) is the electron density contributed by an atom at distance r. The pair potential energy per atom (per atomistic reference spacing ǫ ) is˜ E aℓ ( y ) := 12 (cid:2) φ ( y ′ ℓ ) + φ ( y ′ ℓ + y ′ ℓ − ) + φ ( y ′ ℓ +1 ) + φ ( y ′ ℓ +1 + y ′ ℓ +2 ) (cid:3) , ATTICE STABILITY FOR ATOMISTIC CHAINS MODELED BY THE EAM 4 where φ ( r/ǫ ) is the pair potential interaction energy [7]. Our formulation allows general nonlinearexternal potential energies F ( y ) defined for y ∈ Y F , but for simplicity, we only consider the totalexternal potential energy for 2 N -periodic dead loads f F ( y ) := − N X ℓ = − N +1 ǫf ℓ y ℓ . The equilibrium solution y a of the EAM-atomistic model (3.1) then satisfies −h δ E a ( y a ) , w i = −h δ ˆ E a ( y a ) , w i − h δ ˜ E a ( y a ) , w i = h δ F ( y a ) , w i for all w ∈ U . (3.3)Here the negative of the embedding force of (3.3) is h δ ˆ E a ( y a ) , w i = ǫ N X ℓ = − N +1 G ′ (cid:16) ¯ ρ aℓ ( y a ) (cid:17) · h ρ ′ ( Dy aℓ ) w ′ ℓ + ρ ′ ( Dy aℓ + Dy aℓ − )( w ′ ℓ + w ′ ℓ − )+ ρ ′ ( Dy aℓ +1 ) w ′ ℓ +1 + ρ ′ ( Dy aℓ +1 + Dy aℓ +2 )( w ′ ℓ +1 + w ′ ℓ +2 ) i , the negative of the pair potential force of (3.3) is given by h δ ˜ E a ( y a ) , w i = ǫ N X ℓ = − N +1 h φ ′ ( Dy aℓ ) w ′ ℓ + φ ′ ( Dy aℓ + Dy aℓ − )( w ′ ℓ + w ′ ℓ − )+ φ ′ ( Dy aℓ +1 ) w ′ ℓ +1 + φ ′ ( Dy aℓ +1 + Dy aℓ +2 )( w ′ ℓ +1 + w ′ ℓ +2 ) i , and the negative of the external force is formulated as h δ F ( y ) , w i = N X ℓ = − N +1 ∂ F ∂y ℓ ( y ) w ℓ = − N X ℓ = − N +1 ǫf ℓ w ℓ . The Local EAM Approximations.
In this subsection, we will briefly review the idea ofthe two different local approximations, the volume-based and the reconstruction-based, and givetheir expressions respectively.3.2.1.
The Volume-Based Local EAM Approximation.
The idea of the volume-based local approxi-mation based on the Cauchy-Born rule was first proposed in [12, 15, 17]. We denote this energy by E c,v ( y ) , and we can formulate the local energy associated with each atom as E c,vℓ ( y ) := ˆ E c,vℓ ( y ) + ˜ E c,vℓ ( y ) = 12 G (cid:0) ¯ ρ c,vℓ ( y )) (cid:1) + 12 G (cid:0) ¯ ρ c,vℓ +1 ( y ) (cid:1) + 12 (cid:2) φ ( y ′ ℓ ) + φ (2 y ′ ℓ ) + φ ( y ′ ℓ +1 ) + φ (2 y ′ ℓ +1 ) (cid:3) , where the total local electron density at atom ℓ is¯ ρ c,vℓ ( y ) := 2 ρ ( y ′ ℓ ) + 2 ρ (2 y ′ ℓ ) . Then the total volume-based local energy is E c,vtot ( y ) := E c,v ( y ) + F ( y ) = ǫ N X ℓ = − N +1 E c,vℓ ( y ) − ǫ N X ℓ = − N +1 f ℓ y ℓ . (3.4)The equilibrium solution y c,v then satisfies −h δ E c,v ( y c,v ) , w i = −h δ ˆ E c,v ( y c,v ) , w i − h δ ˜ E c,v ( y c,v ) , w i = h δ F ( y c,v ) , w i for all w ∈ U . (3.5) ATTICE STABILITY FOR ATOMISTIC CHAINS MODELED BY THE EAM 5
The negative of the embedding force of (3.5) is h δ ˆ E c,v ( y c,v ) , w i = ǫ N X ℓ = − N +1 (cid:8) G ′ (cid:0) ¯ ρ c,vℓ ( y c,v ) (cid:1) · (cid:2) ρ ′ ( Dy c,vℓ ) + 2 ρ ′ (2 Dy c,vℓ ) (cid:3) w ′ ℓ + G ′ (cid:0) ¯ ρ c,vℓ +1 ( y c,v ) (cid:1) · (cid:2) ρ ′ ( Dy c,vℓ +1 ) + 2 ρ ′ (2 Dy c,vℓ +1 ) (cid:3) w ′ ℓ +1 (cid:9) , and the negative of the pair potential force of (3.5) is given by h δ ˜ E c,v ( y c,v ) , w i = ǫ N X ℓ = − N +1 (cid:8)(cid:2) φ ′ ( Dy c,vℓ ) + 2 φ ′ (2 Dy c,vℓ ) (cid:3) w ′ ℓ + (cid:2) φ ′ ( Dy c,vℓ +1 ) + 2 φ ′ (2 Dy c,vℓ +1 ) (cid:3) w ′ ℓ +1 (cid:9) . The Reconstruction-Based Local EAM Approximation.
Using the Cauchy-Born approxima-tion, one can also reconstruct the position of each atom [5] and compute the energy E c,r ( y ) by theapproximation E c,rℓ ( y ) = ˆ E c,rℓ ( y ) + ˜ E c,rℓ ( y ) = G (cid:0) ¯ ρ c,rℓ ( y ) (cid:1) + 12 (cid:2) φ ( y ′ ℓ ) + φ (2 y ′ ℓ ) + φ ( y ′ ℓ +1 ) + φ (2 y ′ ℓ +1 ) (cid:3) , where the reconstruction-based local electron density at atom ℓ is¯ ρ c,rℓ ( y ) := ρ ( y ′ ℓ ) + ρ (2 y ′ ℓ ) + ρ ( y ′ ℓ +1 ) + ρ (2 y ′ ℓ +1 ) . Thus, the total energy of the reconstruction-based local model is E c,rtot ( y ) := ˆ E c,r ( y ) + ˜ E c,r ( y ) + F ( y )= ǫ N X ℓ − N +1 n G (cid:2) ρ ( y ′ ℓ ) + ρ (2 y ′ ℓ ) + ρ ( y ′ ℓ +1 ) + ρ (2 y ′ ℓ +1 ) (cid:3) + 12 (cid:2) φ ( y ′ ℓ ) + φ (2 y ′ ℓ ) + φ ( y ′ ℓ +1 ) + φ (2 y ′ ℓ +1 ) (cid:3) o − ǫ N X ℓ = − N +1 f ℓ y ℓ . (3.6)The volume-based and reconstruction-based local energies have the same pair potential energy, buttheir approximations for the embedding energy are quite different.We compute the equilibrium solution of the reconstruction-based local model (3.6) from −h δ E c,r ( y c,r ) , w i = −h δ ˆ E c,r ( y c,r ) , w i − h δ ˜ E c,r ( y c,r ) , w i = h δ F ( y c,r ) , w i for all w ∈ U . (3.7)Here the negative of the embedding force of (3.7) is h δ ˆ E c,r ( y c,r ) , w i = ǫ N X ℓ = − N +1 G ′ (cid:0) ¯ ρ c,rℓ ( y c,r ) (cid:1) · (cid:2)(cid:0) ρ ′ ( Dy c,rℓ ) + 2 ρ ′ (2 Dy c,rℓ ) (cid:1) w ′ ℓ + (cid:0) ρ ′ ( Dy c,rℓ +1 ) + 2 ρ ′ (2 Dy c,rℓ +1 ) (cid:1) w ′ ℓ +1 (cid:3) , and the negative of the pair potential force of (3.7) is h δ ˜ E c,r ( y c,r ) , w i = ǫ N X ℓ = − N +1 (cid:8)(cid:2) φ ′ ( Dy c,rℓ ) + 2 φ ′ (2 Dy c,rℓ ) (cid:3) w ′ ℓ + (cid:2) φ ′ ( Dy c,rℓ +1 ) + 2 φ ′ (2 Dy c,rℓ +1 ) (cid:3) w ′ ℓ +1 (cid:9) . The pair potential energy of both local approximations are exactly the same, but the embeddingparts are quite different, which leads to different critical strains for lattice instability. We willanalyze the lattice stability for all of the models in the next section.
ATTICE STABILITY FOR ATOMISTIC CHAINS MODELED BY THE EAM 6 Sharp Stability Analysis of The Atomistic and Local EAM Models.
In this section, we analyze and compare the conditions for lattice stability of the atomisticmodel and the two local approximations for the next-nearest neighbor case. We will use techniquessimilar to those presented in [3] for the atomistic and quasicontinuum methods with pair potentialinteraction.4.1.
Stability of the Atomistic EAM Model.
We first consider the fully atomistic model. Theuniform deformation y F is an equilibrium of the atomistic model (3.2) without external force. Wecall y F stable in the atomistic model if and only if h δ E a ( y F ) is positive definite, that is, h δ E a ( y F ) u , u i = h δ ˆ E a ( y F ) u , u i + h δ ˜ E a ( y F ) u , u i > u ∈ U \ { } . (4.1)We computed h δ ˜ E a ( y F ) u , u i in [3] to obtain h δ ˜ E a ( y F ) u , u i = ˜ A F k D u k ℓ ǫ − ǫ φ ′′ F k D (2) u k ℓ ǫ , (4.2)where ˜ A F := φ ′′ F + 4 φ ′′ F for φ ′′ F := φ ′′ ( F ) and φ ′′ F := φ ′′ (2 F ) (4.3)is the continuum elastic modulus for the pair interaction potential . Thus, we focus on h δ ˆ E a ( y F ) u , u i ,which can be formulated as h δ ˆ E a ( y F ) u , u i = ǫ N X ℓ = − N +1 ( G ′′ F (cid:2) ρ ′ F ( u ′ ℓ + u ′ ℓ +1 ) + ρ ′ F ( u ′ ℓ − + u ′ ℓ + u ′ ℓ +1 + u ′ ℓ +2 ) (cid:3) + G ′ F (cid:2) ρ ′′ F ( u ′ ℓ ) + ρ ′′ F ( u ′ ℓ + u ′ ℓ − ) + ρ ′′ F ( u ′ ℓ +1 ) + ρ ′′ F ( u ′ ℓ +1 + u ′ ℓ +2 ) (cid:3) ) , (4.4)where we use the simplified notation ρ ′ F := ρ ′ ( F ) , ρ ′′ F := ρ ′′ ( F ) , ρ ′ F := ρ (2 F ) , ρ ′′ F := ρ ′′ (2 F ) ,G ′ F := G ′ (¯ ρ aℓ ( y F )) = G ′ (¯ ρ c,vℓ ( y F )) = G ′ (¯ ρ c,rℓ ( y F )) ,G ′′ F := G ′′ (¯ ρ aℓ ( y F )) = G ′′ (¯ ρ c,vℓ ( y F )) = G ′′ (¯ ρ c,rℓ ( y F )) . We define the continuum elastic modulus for the embedding energy to beˆ A F := 4 G ′′ F (cid:0) ρ ′ F + 2 ρ ′ F (cid:1) + 2 G ′ F (cid:0) ρ ′′ F + 4 ρ ′′ F (cid:1) , (4.5)and we define A F := ˆ A F + ˜ A F , B F := − h φ ′′ F + G ′′ F (cid:16) ( ρ ′ F ) + 20( ρ ′ F ) + 12 ρ ′ F ρ ′ F (cid:17) + G ′ F (cid:0) ρ ′′ F (cid:1)i ,C F := G ′′ F (cid:0) ρ ′ F ) + 2 ρ ′ F ρ ′ F (cid:1) , and D F := − G ′′ F (cid:0) ρ ′ F (cid:1) . (4.6)Then (4.1) becomes h δ E a ( y F ) u , u i = A F k D u k ℓ ǫ + ǫ B F k D (2) u k ℓ ǫ + ǫ C F k D (3) u k ℓ ǫ + ǫ D F k D (4) u k ℓ ǫ , (4.7)where the detailed calculation can be found in the paper [10].We will analyze the stability of h δ E a ( y F ) u , u i by using the Fourier representation [8] Du ℓ = N X k = − N +1 k =0 c k √ · exp (cid:18) i k ℓN π (cid:19) . (4.8) ATTICE STABILITY FOR ATOMISTIC CHAINS MODELED BY THE EAM 7
We exclude k = 0 since D u must satisfy the mean zero condition P Nℓ = − N +1 Du ℓ = 0.It then follows from the discrete orthogonality of the Fourier basis that h δ E a ( y F ) u , u i = N X k = − N +1 k =0 | c k | · ( A F + B F (cid:20) (cid:18) kπ N (cid:19)(cid:21) + C F (cid:20) (cid:18) kπ N (cid:19)(cid:21) + D F (cid:20) (cid:18) kπ N (cid:19)(cid:21) ) . (4.9)We see from (4.9) that the eigenvalues λ ak for k = 1 , . . . , N of h δ E a ( y F ) u , u i with respect to the k D u k ℓ ǫ norm are given by λ ak = λ aF ( s k ) for s k = 4 sin (cid:18) kπ N (cid:19) where λ aF ( s ) := A F + B F s + C F s + D F s . The energy and electron densities figures in [7] and [14] satisfy the following conditions whichwe shall assume in our analysis φ ′′ F > , φ ′′ F < ρ ′ F ≤ , ρ ′ F ≤ ρ ′′ F ≥ , ρ ′′ F ≥
0; and G ′′ F ≥ . (4.10)We can derive from the assumption (4.10) that C F > , D F < , and 8 | D F | ≤ C F . (4.11)Since (4.11) implies that | D F s | ≤ | D F | ≤ C F / , for 0 ≤ s ≤ , we have that λ aF ′ ( s ) = B F + 2 C F s + 3 D F s ≥ B F + C F s for all 0 ≤ s ≤ . (4.12)We note from (4.12) that the condition B F ≥ φ ′′ F + G ′′ F h(cid:0) ρ ′ F (cid:1) + 20 (cid:0) ρ ′ F (cid:1) + 12 ρ ′ F ρ ′ F i + G ′ F ρ ′′ F = − B F ≤ , (4.13)implies that λ aF ( s ) is increasing for 0 ≤ s ≤ . We thus conclude that if B F ≥ , then h δ E a ( y F ) u , u i ≥ λ aF ( s ) k D u k ℓ ǫ ≥ (cid:16) ˆ A F + ˜ A F (cid:17) k D u k ℓ ǫ for all u ∈ U . (4.14)This result is summarized in the following theorem: Theorem 4.1.
Suppose that the hypotheses (4.10) and B F ≥ hold. Then the uniform deformation y F is stable for the atomistic model if and only if λ aF ( s ) = A F + B F h (cid:16) π N (cid:17)i + C F h (cid:16) π N (cid:17)i + D F h (cid:16) π N (cid:17)i = ˆ A F + ˜ A F − (cid:16) π N (cid:17) n φ ′′ F + G ′′ F h(cid:0) ρ ′ F (cid:1) + 20 (cid:0) ρ ′ F (cid:1) + 12 ρ ′ F ρ ′ F i + G ′ F ρ ′′ F o + 4 sin (cid:16) π N (cid:17) G ′′ F h η (cid:0) ρ ′ F (cid:1) + 2 ρ ′ F ρ ′ F i − sin (cid:16) π N (cid:17) G ′′ F (cid:0) ρ ′ F (cid:1) > . We note that the differences between s k and s k − and between λ aF ( s k ) and λ aF ( s k − ) are oforder O (cid:16) kπ N (cid:17) = O (cid:0) kǫ (cid:1) for k = 1 , . . . , N. When the number of atoms N is sufficiently large,min ≤ s ≤ λ aF ( s ) can be used to approximate min ≤ k ≤ N λ aF ( s k ) with 1 ≤ k ≤ N with error at mostof order O ( ǫ ) since N ǫ = 1 . ATTICE STABILITY FOR ATOMISTIC CHAINS MODELED BY THE EAM 8
When B F < N is sufficiently large, the minimum eigenvalue of δ E a ( y F ) is no longer λ aF ( s ) and is given by the following theorem. Theorem 4.2.
Suppose that the hypotheses (4.10) and B F < hold, and the number of atoms N is sufficiently large. Then λ aF ( s ) defined in Theorem 4.1 will no longer be the minimum eigenvalueof the second variation h δ E a ( y F ) u , u i . Instead, the minimum eigenvalue will be given by λ aF ( s k ∗ ) for some s k ∗ , < k ∗ ≤ N , that is either equal to or close to s ∗ := C F − q C F − B F D F − D F with difference of order O (cid:0) k ∗ ǫ (cid:1) . Proof.
Here we will briefly discuss the role of the coefficient B F and leave the rigorous discussionof min ≤ s ≤ λ aF ( s ) under the condition B F < B F ≥ u ′ ℓ = sin( ǫℓπ ) is the eigenfunction corresponding to theminimum eigenvalue of δ E a ( y F ) with respect to the norm k D u k ℓ ǫ . In fact, when B F <
0, we have λ aF ′ (0) < λ aF (0) will be strictly larger than λ aF ( s ∗ ).We note that the condition B F ≥ G ′′ F ( ρ ′ F ) > G ′ F < F < B F ≥ F < (cid:3) Remark . We would like to point out that when N is small, λ aF ( s ) may be still the minimumeigenvalue of δ E a ( y F ) even if B F <
0. This is because λ aF ( s k ) is defined on the discrete domain1 ≤ k ≤ N , so the continuous function λ aF ( s ) is not a good approximation unless N is sufficientlylarge.4.2. Stability of the Volume-Based and the Reconstruction-Based Local EAM Models.
In this subsection, we will give stability estimations for the volume-based and the reconstruction-based local models, respectively.4.2.1.
Stability of the Volume-Based Local EAM Model.
We focus on the stability of the volume-based local model under a uniform deformation y F . Using the equilibrium equation (3.5), we obtainthe second variation δ E c,v ( y F ) for any u ∈ U \ { }h δ E c,v ( y F ) u , u i = (cid:16) ˆ A F + ˜ A F (cid:17) k D u k ℓ ǫ = A F k D u k ℓ ǫ , where ˆ A F and ˜ A F are defined in (4.5) and (4.3), respectively. It follows that y F is stable in thevolume-based local model if and only if A F := ˆ A F + ˜ A F >
0. We summarize this result in thefollowing theorem.
Theorem 4.3.
Suppose that the hypotheses (4.10) holds. Then the uniform deformation y F isstable in the volume-based local model (3.4) if and only if A F := ˆ A F + ˜ A F > .Remark . Comparing the conclusions in Theorem 4.1 and Theorem 4.3, we observe that when thehypothesis (4.13) is satisfied, the difference between the minimum eigenvalues of the fully atomisticand the volume-based local models is of order O ( ǫ ). This result is the same as for the pair potentialcase [4]. However, when the assumptions fails, the volume-based local model will be strictly morestable than the fully atomistic model, which will be discussed in the next remark. ATTICE STABILITY FOR ATOMISTIC CHAINS MODELED BY THE EAM 9
Remark . The assumption (4.13) is necessary for the validity of Theorem 4.1. We now give anexplicit example showing that the uniform deformation can be strictly more stable for the volume-based local model (3.4) than for the fully atomistic model when (4.13) fails. We consider thecase φ ′′ F + G ′′ F (cid:0) ρ ′ F + 2 ρ ′ F (cid:1) + G ′ F ρ ′′ F > . (4.15)Then (4.13) does not hold since it follows from (4.10) that φ ′′ F + G ′′ F h(cid:0) ρ ′ F (cid:1) + 20 (cid:0) ρ ′ F (cid:1) + 12 ρ ′ F ρ ′ F i + G ′ F ρ ′′ F = h φ ′′ F + G ′′ F (cid:0) ρ ′ F + 2 ρ ′ F (cid:1) + G ′ F ρ ′′ F i + 8 G ′′ F (cid:16) (cid:0) ρ ′ F (cid:1) + ρ ′ F ρ ′ F (cid:17) > . We define an oscillatory displacement ˜ u , corresponding to the k = N eigenmode in the Fourierexpansion (4.8), by ˜ u ℓ := ( − ℓ ǫ/ (2 √ . Therefore, ˜ u ′ ℓ = ( − ℓ / ( √ , k D ˜ u k ℓ ǫ = 1 , ˜ u ′′ ℓ = ( − ℓ ( √ /ǫ. From (4.2) and (4.4) we can get h δ E a ( y F )˜ u , ˜ u i = h δ ˜ E a ( y F )˜ u , ˜ u i + h δ ˜ E a ( y F )˜ u , ˜ u i = ǫ N X ℓ = − N +1 G ′ F ρ ′′ F
12 + (cid:0) φ ′′ F + 4 φ ′′ F (cid:1) k D ˜ u k ℓ ǫ + ( − ǫ φ ′′ F ) k D (2) ˜ u k ℓ ǫ = G ′ F ρ ′′ F + (cid:0) φ ′′ F + 4 φ ′′ F (cid:1) − φ ′′ F = φ ′′ F + G ′ F ρ ′′ F . (4.16)Thus, we can obtain inf u ∈U\{ } , k D u k ℓ ǫ =1 h δ E a ( y F ) u , u i ≤ φ ′′ F + G ′ F ρ ′′ F . On the other hand, from Theorem 4.3 we have thatinf u ∈U\{ } , k D u k ℓ ǫ =1 h δ E c,v ( y F ) u , u i ≡ ˜ A F + ˜ A F = 4 h φ ′′ F + G ′′ F (cid:0) ρ ′ F + 2 ρ ′ F (cid:1) + G ′ F ρ ′′ F i + φ ′′ F + G ′ F ρ ′′ F . Therefore, from (4.15) and (4.16) we haveinf u ∈U\{ } , k D u k ℓ ǫ =1 h δ E c,v ( y F ) u , u i > φ ′′ F + G ′ F ρ ′′ F ≥ inf u ∈U\{ } , k D u k ℓ ǫ =1 h δ E a ( y F ) u , u i . This inequality indicates that when the assumption (4.13) fails, the uniform deformation y F canbe unstable for the atomistic model, but still stable for the volume-based local model.4.2.2. Stability of the Reconstruction-Based Local EAM Model.
In this case, we do a similar calcu-lation for the reconstruction-based local model and derive the second variation δ E c,r ( y ) from the ATTICE STABILITY FOR ATOMISTIC CHAINS MODELED BY THE EAM 10 equilibrium equation given by (3.7) h δ E c,r ( y F ) u , u i = ǫ N X ℓ = − N +1 n G ′′ F (cid:0) ρ ′ F + 2 ρ ′ F (cid:1) (cid:0) u ′ ℓ + u ′ ℓ +1 (cid:1) + G ′ F (cid:0) ρ ′′ F + 4 ρ ′′ F (cid:1) h(cid:0) u ′ ℓ (cid:1) + (cid:0) u ′ ℓ +1 (cid:1) io + ǫ N X ℓ = − N +1 n φ ′′ F h(cid:0) u ′ ℓ (cid:1) + (cid:0) u ′ ℓ +1 (cid:1) i + φ ′′ F h(cid:0) u ′ ℓ (cid:1) + 4 (cid:0) u ′ ℓ +1 (cid:1) io = h G ′′ F (cid:0) ρ ′ F + 2 ρ ′ F (cid:1) + 2 G ′ F (cid:0) ρ ′′ F + 4 ρ ′′ F (cid:1) + φ ′′ F + 4 φ ′′ F i k D u k ℓ ǫ − ǫ G ′′ F (cid:0) ρ ′ F + 2 ρ ′ F (cid:1) k D (2) u k ℓ ǫ = A F k D u k ℓ ǫ + ǫ ˜ B F k D (2) u k ℓ ǫ , where A F is defined in (4.3) and ˜ B F is defined to be˜ B F := − G ′′ F (cid:0) ρ ′ F + 2 ρ ′ F (cid:1) ≤ . (4.17)We recall that for the EAM-atomistic model, the coefficient B F of k D u k ℓ ǫ in (4.7) is defined as B F = − h φ ′′ F + G ′′ F (cid:16)(cid:0) ρ ′ F (cid:1) + 20 (cid:0) ρ ′ F (cid:1) + 12 ρ ′ F ρ ′ F (cid:17) + G ′ F (cid:0) ρ ′′ F (cid:1)i . Comparing B F with ˜ B F defined in (4.17), we find that B F = ˜ B F − h φ ′′ F + G ′′ F (cid:16) (cid:0) ρ ′ F (cid:1) + 8 ρ ′ F ρ ′ F (cid:17) + G ′ F (cid:0) ρ ′′ F (cid:1)i . The assumption (4.10) that φ ′′ F < B F can be positive while ˜ B F is always negative.We similarly use the Fourier representation Du ℓ = N X k = − N +1 k =0 c k √ · exp (cid:18) i k ℓN π (cid:19) to analyze the stability of δ E c,r ( y F ) . Again, we exclude k = 0 because of the mean zero conditionof D u . From the discrete orthogonality of the Fourier basis we have h δ E c,r ( y F ) u , u i = N X k = − N +1 k =0 | c k | · ( A F + ˜ B F (cid:20) (cid:18) kπ N (cid:19)(cid:21) ) . (4.18)The eigenvalues λ c,rk of h δ E c,r ( y F ) u , u i with respect to the k D u k ℓ ǫ norm are given by λ c,rk = λ c,rF ( s k ) for k = 1 , . . . , N, where s k = 4 sin (cid:18) kπ N (cid:19) and λ c,rF ( s ) := A F + ˜ B F s. The assumption (4.10) implies that ˜ B F ≤ λ c,rF ( s ) is decreasing for 0 ≤ s ≤ δ E c,r ( y F ) is achieved at k = N , i.e, s N = 4:min ≤ k ≤ N λ c,rF ( s k ) = λ c,rF (4) = A F + 4 ˜ B F = 2 G ′ F (cid:0) ρ ′′ F + 4 ρ ′′ F (cid:1) + φ ′′ F + 4 φ ′′ F . ATTICE STABILITY FOR ATOMISTIC CHAINS MODELED BY THE EAM 11
The minimum eigenmode is given by the oscillatory displacement ˆ u ′ ℓ = − ˆ u ′ ℓ +1 since h δ E c,r ( y F )ˆ u , ˆ u i = (cid:2) G ′ F (cid:0) ρ ′′ F + 4 ρ ′′ F (cid:1) + φ ′′ F + 4 φ ′′ F (cid:3) k D ˆ u k ℓ ǫ . We thus have the following stability result for the reconstruction-based local model.
Theorem 4.4.
Suppose that the hypotheses (4.10) holds. Then the uniform deformation y F isstable in the reconstruction-based local model (3.6) if and only if G ′ F (cid:0) ρ ′′ F + 4 ρ ′′ F (cid:1) + φ ′′ F + 4 φ ′′ F > . Remark . Comparing the conclusions in Theorem 4.1, Theorem 4.3, and Theorem 4.4, we notethat when the assumption (4.13) holds, i.e., B F ≥
0, the fully atomistic model is strictly more stablethan the reconstruction-based local model. The difference between their minimum eigenvalues is O (1), not O ( ǫ ) as for the volume-based approximation. When the assumption (4.13) fails, i.e. B F <
0, the conclusion will be different, and we will rigorously analyze this case in section 5.5.
Comparison of the Stability of the Atomistic and Local EAM Models
In this section, we would like to give a full discussion of the sharp stability estimates for all ofthe EAM models. Recall that the eigenvalue function of δ E a ( y F ) is λ aF ( s k ) := A F + B F s k + C F s k + D F s k for s k = 4 sin (cid:18) kπ N (cid:19) , k = 1 , . . . , N, where the coefficients A F , B F , C F and D F are given in the equation (4.6).To simplify the following analyses, the number of atoms N is assumed to be sufficiently large.Thus, we use the global minimum of the continuous function λ aF ( s ) := A F + B F s + C F s + D F s for 0 ≤ s ≤ ≤ k ≤ N λ aF ( s k ). We note that their difference is at most of order O (cid:0) kǫ (cid:1) ≤ O ( ǫ ).We recall that min ≤ s ≤ λ aF ( s ) = λ aF (0) if B F ≥ . (5.1)To find min ≤ s ≤ λ aF ( s ) when B F < , we first evaluate λ aF ( s ) at s = 0 , λ aF (0) = A F = 4 G ′′ F (cid:0) ρ ′ F + 2 ρ ′ F (cid:1) + 2 G ′ F (cid:0) ρ ′′ F + 4 ρ ′′ F (cid:1) + φ ′ F + 4 φ ′′ F ,λ aF (4) = φ ′′ F + 2 G ′ F ρ ′′ F . We next compute the first and second derivatives of λ aF ( s ), which are λ aF ′ ( s ) = B F + 2 C F s + 3 D F s ,λ aF ′′ ( s ) =2 C F + 6 D F s. Since λ aF ′ ( s ) is a quadratic function, we thus have two critical points of λ aF ( s ) when the coefficientssatisfy C F − B F D F ≥ φ ′′ F + 2 G ′ F ρ ′′ F ≤ G ′′ F (cid:0) ρ ′ F − ρ ′ F (cid:1) . We can summarize the case when B F < C F − B F D F ≤ ≤ s ≤ λ aF ( s ) = λ aF (4) if B F < C F − B F D F ≤ . (5.2)In the case C F − B F D F > s = C F − q C F − B F D F − D F and s = C F + q C F − B F D F − D F . ATTICE STABILITY FOR ATOMISTIC CHAINS MODELED BY THE EAM 12
Since D F < , λ aF ( s ) will then have a local minimum at s ∗ = s and a local maximum at s . Thecorresponding local minimum value is λ aF ( s ∗ ) = A F + B F s ∗ + C F ( s ∗ ) + D F ( s ∗ ) = A F + s ∗ (cid:18) B F − D F s ∗ ) (cid:19) , where we use λ aF ′ ( s ∗ ) = 0 to get the last equality. We can thus summarize all of the cases bymin ≤ s ≤ λ aF ( s ) = λ aF (0) if B F ≥ , min ≤ s ≤ λ aF ( s ) = λ aF (4) if B F < C F − B F D F ≤ , min ≤ s ≤ λ aF ( s ) = min { λ aF ( s ∗ ) , λ aF (4) } if B F < C F − B F D F > . (5.3)We note that the minimum eigenvalues of the volume-based and the reconstruction-based localmodels are separately given by the following expressionsmin ≤ s ≤ λ c,vF ( s ) = λ aF (0) = A F = 4 G ′′ F (cid:0) ρ ′ F + 2 ρ ′ F (cid:1) + 2 G ′ F (cid:0) ρ ′′ F + 4 ρ ′′ F (cid:1) + φ ′′ F + 4 φ ′′ F , min ≤ s ≤ λ c,rF ( s ) = λ c,rF (4) = A F − G ′′ F (cid:0) ρ ′ F + 2 ρ ′ F (cid:1) = 2 G ′ F (cid:0) ρ ′′ F + 4 ρ ′′ F (cid:1) + φ ′′ F + 4 φ ′′ F . (5.4)5.1. The Volume-based Local EAM Model versus the Fully Atomistic EAM Model.
Wefirst compare the minimum eigenvalues of the volume-based local and the fully atomistic models.Combining the results of Theorem 4.1 and Theorem 4.2, we have the following theorem.
Theorem 5.1.
The relation of the stability of the volume-based local model and the fully atomisticmodel depends on the sign of B F := − h φ ′′ F + G ′′ F (cid:16) ( ρ ′ F ) + 20( ρ ′ F ) + 12 ρ ′ F ρ ′ F (cid:17) + G ′ F (cid:0) ρ ′′ F (cid:1)i and can be summarized as follows: min ≤ s ≤ λ aF ( s ) = λ aF (0) = λ c,vF if B F ≥ , min ≤ s ≤ λ aF ( s ) = min { λ aF ( s ∗ ) , λ aF (4) } < λ aF (0) = min ≤ s ≤ λ c,vF if B F < . This observation indicates that the set of stable uniform strains for the volume-based local modelalways includes that for the fully atomistic EAM model.5.2.
The Reconstruction-based Local EAM Model versus the Fully Atomistic EAMModel.
The relation of the minimum eigenvalues for δ E a ( y F ) and δ E c,r ( y F ) is more complicated.We note that assumption (4.10) implies λ aF (0) = A F = 4 G ′′ F (cid:0) ρ ′ F + 2 ρ ′ F (cid:1) + 2 G ′ F (cid:0) ρ ′′ F + 4 ρ ′′ F (cid:1) + φ ′′ F + 4 φ ′′ F ≥ G ′ F (cid:0) ρ ′′ F + 4 ρ ′′ F (cid:1) + φ ′′ F + 4 φ ′′ F = min ≤ s ≤ λ c,rF ( s ) , and we have λ aF (4) − min ≤ s ≤ λ c,rF ( s ) = − (cid:0) φ ′′ F + G ′ F · ρ ′′ F (cid:1) . We thus conclude that if φ ′′ F + G ′ F · ρ ′′ F ≤ , (5.5) ATTICE STABILITY FOR ATOMISTIC CHAINS MODELED BY THE EAM 13 then λ aF (4) ≥ min ≤ s ≤ λ c,rF ( s ) . The equal sign is achieved if and only if φ ′′ F + G ′ F · ρ ′′ F = 0. We also have the identity φ ′′ F + G ′ F · ρ ′′ F = − B F − h G ′′ F (cid:16) ( ρ ′ F ) + 20( ρ ′ F ) + 12 ρ ′ F ρ ′ F (cid:17)i . We next compare λ aF ( s ∗ ) and min ≤ s ≤ λ c,rF ( s ). The difference of these two is λ aF ( s ∗ ) − min ≤ s ≤ λ c,rF ( s )= 4 G ′′ F (cid:0) ρ ′ F + 2 ρ ′ F (cid:1) + s ∗ (cid:18) B F − D F s ∗ ) (cid:19) = 4 G ′′ F (cid:0) ρ ′ F + 2 ρ ′ F (cid:1) (5.6)+ C F − q C F − B F D F − D F · B F D F − C F + C F (cid:16) C F + q C F − B F D F (cid:17) D F = 4 G ′′ F (cid:0) ρ ′ F + 2 ρ ′ F (cid:1) + B F C F − D F + 2 (cid:0) C F − B F D F (cid:1) (cid:16) C F − q C F − B F D F (cid:17) D F ≥ G ′′ F (cid:0) ρ ′ F + 2 ρ ′ F (cid:1) + B F C F − D F . According to the assumption (5.5), we can use the definition of C F and D F (4.6) to get B F = − (cid:2) φ ′′ F + G ′ F · ρ ′′ F + G ′′ F (cid:0) ( ρ ′ F ) + 12 ρ ′ F ρ ′ F + 20( ρ ′ F ) (cid:1)(cid:3) ≥ − G ′′ F (cid:0) ( ρ ′ F ) + 12 ρ ′ F ρ ′ F + 20( ρ ′ F ) (cid:1) . We thus can obtain from the above inequality and the assumption (4.10) that λ aF ( s ∗ ) − min ≤ s ≤ λ c,rF ( s ) ≥ G ′′ F (cid:0) ρ ′ F + 2 ρ ′ F (cid:1) + B F C F − D F ≥ G ′′ F (cid:0) ρ ′ F + 2 ρ ′ F (cid:1) + G ′′ F (8 ρ ′ F + 2 ρ ′ F ) (cid:0) ( ρ ′ F ) + 12 ρ ′ F ρ ′ F + 20( ρ ′ F ) (cid:1) − ρ ′ F = 2 G ′′ F ( ρ ′ F + 2 ρ ′ F ) (2 ρ ′ F − ρ ′ F ) ρ ′ F ≥ . Therefore, we have thatmin ≤ s ≤ λ aF ( s ) = min { λ aF (0) , λ aF ( s ∗ ) , λ aF (4) } = min ≤ s ≤ λ c,rF ( s ) if φ ′′ F + G ′ F · ρ ′′ F = 0 , min ≤ s ≤ λ aF ( s ) = min { λ aF (0) , λ aF ( s ∗ ) , λ aF (4) } > min ≤ s ≤ λ c,rF ( s ) if φ ′′ F + G ′ F · ρ ′′ F < . (5.7)Now let us turn to the case that the assumption (5.5) fails, which means φ ′′ F + G ′ F · ρ ′′ F > . (5.8) ATTICE STABILITY FOR ATOMISTIC CHAINS MODELED BY THE EAM 14
In this case we have the opposite conclusion that the fully atomistic model E a ( y ) is strictly lessstable than the reconstruction-based local model E c,r ( y ). From the condition (5.8), we have λ aF (4) − min ≤ s ≤ λ c,rF ( s ) = − (cid:0) φ ′′ F + G ′ F · ρ ′′ F (cid:1) < , i.e., λ aF (4) < min ≤ s ≤ λ c,rF ( s ) . Before comparing λ aF ( s ∗ ) and min ≤ s ≤ λ c,rF ( s ), we recall that s ∗ exists if and only if φ ′′ F + 2 G ′ F ρ ′′ F ≤ G ′′ F (cid:0) ρ ′ F − ρ ′ F (cid:1) . Thus, we actually consider the case that0 < φ ′′ F + 2 G ′ F ρ ′′ F ≤ G ′′ F (cid:0) ρ ′ F − ρ ′ F (cid:1) . We substitute B F = − (cid:2) φ ′′ F + G ′ F · ρ ′′ F + G ′′ F (cid:0) ( ρ ′ F ) + 12 ρ ′ F ρ ′ F + 20( ρ ′ F ) (cid:1)(cid:3) into the inequality(5.6) and get λ aF ( s ∗ ) − min ≤ s ≤ λ c,rF ( s ) ≥ G ′′ F (cid:0) ρ ′ F + 2 ρ ′ F (cid:1) + B F C F − D F = 4 G ′′ F (cid:0) ρ ′ F + 2 ρ ′ F (cid:1) + φ ′′ F + G ′ F · ρ ′′ F − G ′′ F ( ρ ′ F ) + G ′′ F (cid:0) ( ρ ′ F ) + 12 ρ ′ F ρ ′ F + 20( ρ ′ F ) (cid:1) − G ′′ F ( ρ ′ F ) ! C F = ( φ ′′ F + G ′ F · ρ ′′ F ) (8 ρ ′ F + 2 ρ ′ F ) − ρ ′ F + 2 G ′′ F ( ρ ′ F + 2 ρ ′ F ) (2 ρ ′ F − ρ ′ F ) ρ ′ F . Since φ ′′ F + 2 G ′ F ρ ′′ F ≤ G ′′ F ( ρ ′ F − ρ ′ F ) , therefore λ aF ( s ∗ ) − min ≤ s ≤ λ c,rF ( s ) ≥ ( φ ′′ F + G ′ F · ρ ′′ F ) (8 ρ ′ F + 2 ρ ′ F ) − ρ ′ F + 2 G ′′ F ( ρ ′ F + 2 ρ ′ F ) (2 ρ ′ F − ρ ′ F ) ρ ′ F ≥ G ′′ F ( ρ ′ F − ρ ′ F ) (8 ρ ′ F + 2 ρ ′ F ) − ρ ′ F + 2 G ′′ F ( ρ ′ F + 2 ρ ′ F ) (2 ρ ′ F − ρ ′ F ) ρ ′ F = 13 G ′′ F (cid:0) ρ ′ F − ρ ′ F (cid:1) ρ ′ F + 4 ρ ′ F ρ ′ F ≥ . Hence, when s ∗ exists and the assumption (5.8) holds, we havemin ≤ s ≤ λ aF ( s ) = min { λ aF (0) , λ aF ( s ∗ ) , λ aF (4) } = λ aF (4) < min ≤ s ≤ λ c,rF ( s ) . When s ∗ does not exist, we can immediately get thatmin ≤ s ≤ λ aF ( s ) = min { λ aF (0) , λ aF (4) } = λ aF (4) < min ≤ s ≤ λ c,rF ( s ) . We now combine this result with (5.7) and summarize the stability relation between the fullyatomistic model and the reconstruction-based local model by the following theorem.
ATTICE STABILITY FOR ATOMISTIC CHAINS MODELED BY THE EAM 15
Theorem 5.2.
The relation between the stability of the reconstruction-based local model and theatomistic model depends on the sign of φ ′′ F + G ′ F · ρ ′′ F and is given by min ≤ s ≤ λ aF ( s ) = λ aF (4) < min ≤ s ≤ λ c,rF ( s ) if φ ′′ F + G ′ F · ρ ′′ F > , min ≤ s ≤ λ aF ( s ) = min { λ aF (0) , λ aF ( s ∗ ) , λ aF (4) } = min ≤ s ≤ λ c,rF ( s ) if φ ′′ F + G ′ F · ρ ′′ F = 0 , min ≤ s ≤ λ aF ( s ) = min { λ aF (0) , λ aF ( s ∗ ) , λ aF (4) } > min ≤ s ≤ λ c,rF ( s ) if φ ′′ F + G ′ F · ρ ′′ F < . (5.9)We note from the theorem that the reconstruction-based local model can be less stable than thefully atomistic model, which might cause stability problems when constructing a coupling method.6. Conclusion.
In this paper, we give precise estimates for the lattice stability of atomistic chains modeled by thefully atomistic EAM model and the volume-based and the reconstruction-based local approxima-tions. We identify the critical assumptions for the pair potential, the electron density function, andthe embedding function to study lattice stability. We find that both the volume-based local modeland the reconstruction-based local model can give O(1) errors for the critical strain. The criticalstrain predicted by the volume-based model is always larger than that predicted by the atomisticmodel, but the critical strain for reconstruction-based models can be either larger or smaller thanthat predicted by the atomistic model.Further research is needed to determine the significance of these results for multidimensionallattice stability and for atomistic-to-continuum coupling methods that couple an atomistic regionwith a volume-based local region through a reconstruction-based local region [5, 18].
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