CHINESE JOURNAL OF PHYSICS
VOL. 42, NO. 1
FEBRUARY 2004
Equations of Motion and Calculations of Elastic Constants for a System under Anisotropic Loading Zicong Zhou∗ Department of Physics and Institute of Life Sciences, Tamkang University, Tamsui 251, Taiwan, R.O.C. (Received May 29, 2003) We derive a new set of equations of motion in conjunction with proper isothermalisotension and isoenthalpic-isotension ensembles to be used in molecular dynamics simulations for a system under anisotropic loading. The new equations of motion satisfy the requirement of invariance of physical properties with respect to the simulation cell transformation, and hence are more appropriate for use in studying a system undergoing a structural transformation. We also present the correct expressions for calculating the elastic constants in these ensembles. PACS numbers: 02.70.Ns, 05.20.Gg, 46.15.-x
I. INTRODUCTION
The behavior of a solid under the combined effects of external stress and temperature has considerable practical relevance. For a good understanding of this behavior, a detailed microscopic picture of these effects is very useful. The experience of the past four decades has shown that molecular dynamics (MD) simulations provide a valuable tool for investigating the microscopic picture of a solid. There are many ensembles of statistical mechanics used in MD simulations [1–16]. The MD methods operate most naturally in the microcanonical ensemble [1], i. e. the ensemble of constant energy E, constant volume V , and constant particle number N . The canonical (TVN) ensemble [2–4], however, is a more practical one, since the temperature T is kept constant in the ensemble. In order to compare with experiment more directly, Andersen [5] developed a method of carrying out MD simulations allowing a uniform dilation of the MD cell. His theory generates the isoenthalpic-isobaric (HPN) ensemble. Here H is the enthalpy and P the hydrostatic pressure. Treatments of the isothermal-isobaric (TPN) ensemble [1, 5, 6] appeared soon thereafter. Moreover, to deal with anisotropic solids of arbitrary shape and under arbitrary stress, Parrinello and Rahman extended Andersen’s theory to allow for changes in both the size and shape of the MD cell, first under hydrostatic pressure [7] and later under anisotropic applied stress [8], giving a new ensemble called the (HσN) ensemble [7–9], where σ is the applied stress tensor, which is kept constant in the simulation. Meanwhile, (EhN) and (ThN) ensembles [8, 10–13] were also proposed for simulating anisotropic systems, with h being the tensor constructed from the three vectors forming a parallelepiped, which is the periodically repeating MD cell. Furthermore, Ray
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and Rahman showed how the MD theory developed by Parrinello and Rahman can be put into accord with the theory of finite elasticity, developing the isoenthalpic-isotension (HtN) [14] and isothermal-isotension (TtN) [15] ensembles in MD theory, where t is the thermodynamic stress tensor [17, 18]. The (HtN) and (TtN) ensembles require a reference value for h, h0 ; Ray and Rahman suggested that h0 should be the average value of h in the stress-free state [14, 15]. However, we found that, with such a h 0 , the (HtN) and (TtN) ensembles so obtained may not work properly for a system with finite strain (i.e. ∆h ≡ h − h 0 is not small). Especially, the fluctuation formulae for calculating elastic constants in these ensembles are, in general, useless. Moreover, the Hamiltonian leading to the equations of motion for the two ensembles does not satisfy the requirement of invariance with respect to the transformation of the simulation cell, and so may be problematic for a system having structural transformations. We show that changing the definitions of h 0 and t, as well as a slight modification of the Hamiltonian, can remedy these problems and give appropriate (HtN) and (TtN) ensembles. The paper is organized as follows. We first present in section II the (TtN) ensemble proposed by Ray and Rahman, discuss the associated problems, and derive the proper equations of motion. Section III gives a brief discussion on the proper (HtN) ensemble. Section IV presents the correct expressions for the calculation of elastic constants in these ensembles. A summary concludes the paper.
II. EQUATIONS OF MOTION IN (TtN) ENSEMBLE
II-1. Ray and Rahman’s (TtN) ensemble The Hamiltonian in Ray and Rahman’s (TtN) ensemble for a central force system can be written [15] in the form H =
X Π0 (a)G−1 Π(a) 2ma
a
s2
+gkB T lns +
+
X
φ (x(a) − x(b))
ab
!
g(r(ab))xi (ab)xj (ab) .
(13)
In Eq. (9), a reference value h0 , which comes from the definition of η (Eq. (5)), is required. Ray and Rahman suggested that h0 should be the average value of h when the stress is zero [14, 15]. II-2. On the definitions of h0 and t However, there are some problems relevant to such a definition of h 0 . First, as shown by Parrinello and Rahman (Appendix A of ref. [8]), the η given in Eq. (5) can indeed be identical to the Lagrangian strain in the theory of elastic continuum if ∆h is small. However, for finite ∆h, the existence of such a correspondence is not clear. Meanwhile, h has nine independent components but η has only six. This is because some changes in dh represent an infinitesimal rotation of the whole system, which can be described by introducing the
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EQUATIONS OF MOTION AND . . .
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antisymmetric rotation tensor [19, 22] 1 −1 T −1 T )T · dhT · h · h−1 dω = ((h−1 0 − (h0 ) · h · dh · h0 ). 2 0
(14)
ω is independent of η. Note that only in the limit of infinitesimal deformation does the tensor ω represent an infinitesimal rotation of the whole system, which is negligible. However, in general, a finite ω (due to a finite ∆h) does not give a pure rotation, but includes some strain, and so cannot be ignored anymore. In other words, η alone is not enough to describe the deformation of the system. Such finite strain effects are often important in the theory of the elasticity of stressed solids. Consequently, to choose h 0 to be the stress-free one may be not work properly for a system under finite strain, since ω is not considered in the equations of motion. Moreover, as we shall point out in section IV, in general we have no way to find significant elastic constants for a system under finite strain with such a h 0 . Another minor problem is, in such an ensemble, not only the sizes and shape of the system fluctuate, but σ also fluctuates, and so the results obtained from this ensemble may be difficult to compare directly with experiments. In fact, in the theory of an elastic continuum, the choice of the reference state for η can be arbitrary, either a stress-free one or a stressed one [17–21]. There is no special reason to choose h0 as the stress-free one. But even with an arbitrary h 0 , all the above problems are still not solved. It has been shown that, to find significant elastic constants, the reference parameters in computer simulations must be the ones of the current state [17–21]. Therefore, a solution for the above problems is to choose h 0 to be the current one, i.e. h0 =< h >, where < ... > denotes the ensemble average. However, simply letting h0 =< h > is not enough, because from Eq. (6), we cannot get t = σ, in contrast to the theory of an elastic continuum [17, 18, 20]. To solve this problem, one can modify the definition of t. Bearing in mind that the derivation of the relationship between t and σ ([17, 18]) is based on the elastic theory, which is a macroscopic theory, we can also use the macroscopic form of h, i.e. the ensemble average of h, in the definition of t so t =
h0 < h >−1 σ < h0 >−1 h00 . V0
(15)
h0 can be still arbitrary in Eq. (15). However in simulations we should set < h >= h 0 , so t = σ. With this choice of h0 and t, we change neither the equations of motion, i.e. Eqs. (7)– (9), nor the conclusion that the average calculated using the trajectories generated by these equations is equal to the (TtN) ensemble average [15], since t is still a constant in the derivation. Now Eq. (9) can be reduced to ¨ = PA − hΓ0 , Wh
(16)
where Γ0 =< V >< h >−1 t < h0 >−1 . The reference value h0 does not appear in the equations of the motion any more, and both t and σ are constants.
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The disadvantage of this method is that < h > appears the in equations of motion, but we in general don’t know < h > a priori, so a self-consistent calculation is necessary. But this is not a big problem, since in simulations, < h > converges rapidly; in general a few thousands time steps is enough to obtain a good < h >, even in the critical regime. II-3. Invariance under the MD cell transformation The equation of motion developed in the last section may still not work properly for a system undergoing a structural transformation, since it is inconsistent with the invariance of the physical properties under the MD cell transformation. In a system with translational symmetry, two different choices of h are related to a transformation matrix B with det(B)= 1 such that h 2 = h1 B and h˙ 2 = h˙ 1 B [23–25]. Any dynamic or structural variable in such a system must be invariant with respect to the transformation [23–25]. It is therefore required that the Hamiltonian or Lagrangian must be invariant with respect to the transformation. However, the MD cell kinetic-energy term Kcell , with the choice of the MD cell momentum Π h = W h˙ in Eq. (2), does not satisfy this requirement. Such a choice of Kcell tends to stabilize the initial shape of the MD cell in simulations and may produce some artificial symmetry for a system [24]. We can correct this problem by defining K cell to be W 2 ˙ −1 h0−1 h˙ 0 ) V 3 Tr(hh 2 1 0 0 = 2 Tr(Πh h hΠh ), 3 2W V
Kcell =
(17)
with the new momentum conjugate to h being 2 ˙ −1 h0−1 . Πh = W V 3 hh
(18)
2
The factor V 3 here is necessary to ensure that Z Z 0 −Kcell /kB T e d{Πh } = e−(Ph Ph )/(2W kB T ) d{Ph }
(19)
does not depend on the volume, so that the ensemble average on theQreal phase space 1 produces the (T tN ) ensemble again [15], where P h ≡ V − 3 Πh h0 , {dA} = 3i,j=1 dAij . With this new MD cell kinetic-energy, it is easy to see that the equations of motion for q and s, i.e. Eqs. (7) and (8), still hold. The only change in the equations of the motion is the term related to Kcell and Πh . From 2 2 ∂Kcell ˙ −1 )0 (hh ˙ −1 )h0−1 − Kcell (h0 )−1 = W V 3 (hh ij , ∂hij 3 ij
(20)
˙ h )ij = − ∂H , we can get by using (Π ∂hij ˙ −1 )0 (hh ˙ −1 )h0−1 + 2 Kcell (h0 )−1 . ˙ h = PA − hΓ0 − W V 32 (hh Π 3
(21)
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Furthermore, from Eq. (18) we can find the equations of motion for h: ˙ −1 h˙ ¨ = V − 32 PA − hΓ0 h0 h − W h0−1 h˙ 0 h˙ + 2 V − 32 Kcell h + W hh Wh 3 ˙ −1 h0−1 h˙ 0 h − 2 W hTr( ˙ ˙ −1 ). hh +W hh 3 For a two-dimensional system, the equations of motion for h should be replaced by ¨ = V −1 PA − hΓ0 h0 h − W h0−1 h˙ 0 h˙ + V −1 Kcell h + W hh ˙ −1 h˙ Wh ˙ −1 h0−1 h˙ 0 h − W hTr( ˙ ˙ −1 ), +W hh hh
(22)
(23)
where V is the area of the system.
III. EQUATIONS OF MOTION FOR THE (HtN) ENSEMBLE
In the same way, by using the Hamiltonian H=
X Π(a)0 G−1 Π(a) a
2ma
+
X
φ (x(a) − x(b)) + Kcell + V0 Tr(tη),
(24)
a. Furthermore, if Kcell is given by Eq. (17), i.e. if we consider the invariance under the MD cell transformation, the equations of motion for h, i.e. Eq. (26), should be replaced by Eq. (22) for a three-dimensional system or by Eq. (23) for a two-dimensional system.
IV. CALCULATION OF THE ELASTIC CONSTANTS
It has been shown [8, 10, 16] that in either isothermal-isotension or isoenthalpicisotension ensembles, kB T ∂ηij = hηij ηkl i − hηij i hηkl i . V0 ∂tkl
(27)
In the linear regime, we have tij = t0ij + Cijkl ηkl ,
(28)
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ZICONG ZHOU
27
where t0 is the value of t in the reference state. Therefore the thermodynamic stiffnesses (Cijkl , also referred as the “elastic constants”), either adiabatic or isothermal, can be calculated by kB T −1 C = hηij ηkl i − hηij i hηkl i V0 ijkl 1 = (hMij Mkl i − hMij i hMkl i) , 4
(29)
if Eq. (28) is valid. However, for a finite strain, Eq. (28) is in general invalid, the nonlinear terms and the effects from ω have to be considered. In this case, Eq. (27) is useless since it gives nothing relevant to Cijkl . In fact, to describe elasticity correctly, the reference state in calculating the elastic constants must be taken as the current one [19–21], i.e. in Eq. (29) we must take h0 =< h >. We should point out that for a stressed system, there are different kinds of “elastic constants”, but only the stress-strain stiffnesses (also referred as elastic stiffness coefficients) which govern the stress-strain relations can correctly describe the elasticity and mechanical stability of the system [19–21]. The relationship between the stress-strain stiffnesses and thermodynamic stiffnesses is [19–21] 1 cαβστ ≡ Cαβστ − (2σαβ δστ − σασ δβτ − σατ δβσ − σβτ δασ − σβσ δατ ). 2
(30)
V. CONCLUSIONS
In summary, we modified the equations of motion developed by Ray and Rahman for the (TtN) and (HtN) ensembles. For the (TtN) ensemble, they are Eqs. (7), (8), and (22) for a three-dimensional system, or Eqs. (7), (8), and (23) for a two-dimensional system. For the (HtN) ensemble, they are Eqs. (25) and (22) for a three-dimensional system, or Eqs. (25) and (23) for a two-dimensional system. The key point is that to describe the elasticity correctly, the reference state in the simulation must be taken as the current one [19–21], i.e. h0 =< h >. The new equations of motion can work properly under high anisotropic applied stress, and are consistent with the theory of finite elasticity, as well as satisfying the requirement of invariance of the physical quantities with respect to the MD cell transformation. The corresponding expressions for the calculation of the elastic constants are also presented in Eq. (29). These expressions will be useful for simulating the system under strong loading and helpful in avoiding some artificial stable states, which often occur in computer simulations, especially in the critical regime.
Acknowledgments
The author thanks the Physics Division at the National Center for Theoretical Sciences of the Republic of China for its hospitality during the writing of the paper. This work
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has been supported by the National Science Council of Republic of China under grant No. NSC 91-2112-M-032-006.
References Electronic address:
[email protected] D. Brown and J. Clarke, Mol. Phys. 51, 1243 (1984). L. Woodcock, Chem. Phys. Lett. 10, 257 (1970). Y. Hiwatari, E. Stoll, and T. Schneider, J. Chem. Phys. 68, 3401 (1978). W. Hoover, Pysica A118, 111 (1983). H.C. Andersen, J. Chem. Phys. 72, 2384 (1980). D. Evans and G. Morriss, Chem. Phys. 77, 63 (1983). M. Parrinello and A. Rahman, Phys. Rev. Lett. 45, 1196 (1980). M. Parrinello and A. Rahman, J. Appl. Phys. 52, 7182 (1981). M. Parrinello and A. Rahman, J. Chem. Phys. 76, 2662 (1982). J.R. Ray, J. Appl. Phys., 53, 6441 (1982). J.R. Ray, M.C. Moody, and A. Rahman, Phys. Rev. B 32, 733 (1985). J.R. Ray, M.C. Moody, and A. Rahman, Phys. Rev. B 33, 895 (1986). T. Cagin and J.R. Ray, Phys. Rev. B 38, 7940 (1988). J.R. Ray and A. Rahman, J. Chem. Phys. 80, 4423 (1984). J.R. Ray and A. Rahman, J. Chem. Phys. 82, 4243 (1985). Z. Zhou, J. Chem. Phys. 114, 8769 (2001). D. C. Wallace, Thermodynamics of Crystals, (Wiley, New York, 1972). J.H. Weiner, Statistical Mechanics of Elasticity, (John Wiley & Sons, Toronto, 1983). T.H.K. Barron and M.L. Klein, Proc. Phys. Soc., 85, 523 (1965). Z. Zhou and B. Jo´ os, Phys. Rev. B 54, 3841 (1996). Z. Zhou and B. Jo´ os, Phys. Rev. B 66, 54101 (2002)). R.N. Thurston, in Physical Acoustics, edited by W. P. Mason(Academic, New York, 1964) Vol. I., Part A. [23] C.L. Cleveland, J. Chem. Phys. 89, 4987 (1988). [24] R.M. Wentzcovitch, Phys. Rev. B44, 2358 (1991). [25] Mo Li and W.L. Johnson, Phys. Rev. B46, 5237 (1993). ∗
[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]
