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https://journals.aps.org/prb/abstract/10.1103/PhysRevB.55.735
DOI: 10.1103/PhysRevB.55.735
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Adiabatic switching applied to realistic crystalline solids:
Vacancy-formation free energy in copper
M. de Koning and A. Antonelli
Instituto de Fı´sica Gleb Wataghin, Universidade Estadual de Campinas, Unicamp 13083-970, Campinas, Sa˜o Paulo, Brazil
~Received 5 June 1996!
We study the application of the adiabatic switching molecular dynamics method to determine bulk and vacancy-formation Gibbs free energies as a function of temperature at zero pressure for copper. The bulk free energy has been determined through isochoric isothermal switching procedures in which a system consisting of 500 copper atoms interacting through a semiempirical tight-binding potential is turned into a system of 500 independent identical three-dimensional harmonic oscillators. The equilibrium volumes of these simulations were determined from equilibrium isobaric isothermal molecular dynamics simulations. The frequency of the oscillators is chosen to be of the order of a principal phonon frequency of copper in order to achieve com-petitive convergence. The resulting bulk free energy and entropy are in excellent agreement with experimental values. The vacancy-formation Gibbs free energy has been computed from isobaric isothermal switching procedures in which the interactions of a single copper atom are switched off. Considering the limited accuracy of the interatomic potential and the numerical noise present in the small energy differences measured, the estimated formation enthalpies and entropies agree remarkably well with experimental data. The method has shown to be computationally efficient. Typically, 6 h of CPU time on a Digital Alpha 3000/900 were required per data point for the bulk as well as the vacancy-formation parameters.@S0163-1829~97!07901-0#
I. INTRODUCTION
The process of formation of vacancies in crystalline ma-terials is of considerable interest. Besides the significant role vacancies play in the transport of matter within a crystal through diffusion mechanisms they affect mechanical prop-erties through their interaction with extended defects such as dislocations and stacking faults. In order to gain a better understanding of the influence of monovacancies on the structural properties of crystalline materials, knowledge of the thermodynamic parameters involved in their formation is indispensable.
During the last 15 years large scale computer simulations have become a powerful tool in the theoretical determination of thermodynamic and structural properties of solid and liq-uid materials.1 Among these properties, the thermal param-eters such as entropy and free energy which cannot be ex-pressed as ensemble averages are the most difficult to compute. Nevertheless, several methods have been devel-oped to overcome these difficulties. Methods based on har-monic approximations and thermodynamic integration have been used frequently.2–8
Within the harmonic approximation, the total cohesive en-ergy of a system is written as a harmonic expansion about the equilibrium positions of the atoms. The free energy can then be calculated in terms of the eigenfrequencies of the corresponding dynamical matrix. Recently, Zhao, Najafa-badi, and Srolovitz4 have carried out local harmonic and quasiharmonic studies of finite-temperature vacancy-formation thermodynamics for several fcc metals. Although the computed perfect crystal properties agree well with ex-perimental values, the defect formation parameters do not. For the case of copper, for example, they found the forma-tion enthalpy of a monovacancy to be about 1.29 eV, which
agrees well with the experimental value of (1.2860.03) eV.9 However, the formation entropy of 0.9kB (kB Boltzmann’s constant! they found is much lower than the experimental value (2.560.2)kB. The main reason for this discrepancy is probably the fact that anharmonic contributions to the forma-tion entropy are ignored. In the perfect crystal these anhar-monicities do not contribute significantly so that a harmonic description is sufficiently accurate. In the presence of a va-cancy, however, the local symmetry is disturbed causing an-harmonic effects that can no longer be neglected.
The thermodynamic integration method ~TI! is based on the idea of Kirkwood of coupling a system of interest with a reference system through a parameterl. The total energy of the coupled system is written as
H~l!5lH01~12l!Href, ~1.1!
where H0 and Href are the total energies of the systems of
interest and reference, respectively, and l varies between 0 and 1. The difference between the free energy of the systems of interest and reference can then be written as
DF5
E
0 1
K
]H]l
L
ldl, ~1.2!where the brackets with subscriptl indicate an average over the canonical ensemble corresponding with l. Accordingly, the free energy is obtained by evaluating the integrand in Eq.
~1.2! for a number of fixed values of l between 0 and 1
using standard Monte Carlo ~MC! or molecular dynamics
~MD! simulations and estimating the integral numerically. In
principle Eq. ~1.2! is exact so that anharmonic effects are included.
Foiles has employed TI to compute the free energies of several defects in embedded atom ~EAM! copper as a
func-PHYSICAL REVIEW B VOLUME 55, NUMBER 2 1 JANUARY 1997-II
55
tion of temperature.7 As a reference system Foiles used a system of harmonic oscillators with each oscillator centered near the equilibrium position of an atom. As Foiles pointed out, there is a fundamental problem in the evaluation of Eq.
~1.2! at high temperatures when defects are present. At high
temperatures, defects generally cause diffusive behavior in the system. For l near 0 this diffusive behavior is inhibited by the harmonic reference potential. However, forl near 1 when the reference potential becomes weak, diffusion may occur so that the energy associated with the reference system becomes very large. As a result, the integrand in Eq. ~1.2! varies rapidly complicating the evaluation of the integral. To circumvent this problem the free energy is computed by TI for a relatively low temperature, after which the free energy at higher temperatures can be extrapolated computing the enthalpy H at various temperatures and integrating the ther-modynamic relation d dT
S
G TD
52 H T2, ~1.3!where G is the free energy and T is the absolute temperature. Besides these most frequently utilized methods, two ‘‘me-chanical’’ ~i.e., not based on the ensemble formulation of statistical mechanics! methods have been proposed. In 1991, Rickman and Phillpot10introduced a method to calculate the absolute free energy and the entropy of a classical, crystal-line solid at a fixed temperature T by calculating the allowed volume of phase space from the motion of the solid’s par-ticles. The method is based on the work of Ma11in which the entropy of a system can be determined by counting coinci-dences of states along its trajectory in phase space. Unfortu-nately, application of this method to situations in which de-fects are present is far from straightforward.
The other mechanical method, the subject of interest in this paper, is the adiabatic switching method which has been proposed by Watanabe and Reinhardt12 and is based on the adiabatic invariants of slowly changing dynamical systems. Hertz13 showed that a constant energy shell of an ergodic, deterministic, Hamiltonian energy conserving dynamics is precisely mapped upon a constant energy shell of another dynamical system satisfying the same conditions if both sys-tems are connected through an adiabatic transformation. During this adiabatic transformation the total phase-space volume of the instantaneous energy shells is conserved. Wa-tanabe and Reinhardt recognized that adiabatic microcanoni-cal MD transformations connecting the dynamics of a system of interest to the dynamics of a system of reference can be used to determine the entropy-energy relation of the system of interest. An important feature of the adiabatic switching method is the fact that in principle, only one switching tra-jectory is sufficient to obtain the desired information. Thus, the method is computationally much more efficient than TI where several trajectories with different values for the cou-pling constantl need to be generated.
Very recently the present authors have applied the adia-batic switching method to calculate the Helmholtz free en-ergy of a collection of identical independent harmonic oscil-lators~Einstein crystal!.14In the procedure, the characteristic frequency of the oscillators was slowly changed under con-stant temperature conditions created by the massive Nose´
Hoover chain dynamics ~MNHC!.15 The free energy differ-ence obtained deviated less than 1% from the analytical value indicating the possibility of application of the adiabatic switching method to realistic solids using the Einstein crystal as a reference system.
In this paper it is our purpose to evaluate the performance of the adiabatic switching method applied to determine ther-mal quantities in realistic solid materials. We present results of the application of the method to determine bulk and vacancy-formation free energies and entropies in copper as a function of temperature. We have chosen copper as our test system since it is a material studied extensively, both theo-retically and experimentally, and thus provides a good benchmark for measuring the performance of the adiabatic switching method.
The switching procedures were established through iso-choric isothermal (VT) and isobaric isothermal ( pT) MD simulations. The VT conditions were created using the MNHC dynamics whereas the pT conditions were estab-lished by a combination of the MNHC and Andersen16 tech-niques. We have used the tight-binding ~TB! potential of Cleri and Rosato17 as a representation of the interactions of copper atoms in the solid. We have chosen this potential since it has reproduced quantities such as elastic constants, phonon dispersion curves, and vacancy-formation volumes very successfully. The bulk free energies are determined by a
VT switching procedure in which the interaction between all
copper atoms is slowly switched off and the atoms are turned into noninteracting three-dimensional harmonic oscillators. The vacancy-formation parameters are obtained by a pT switching procedure switching off the mutual interactions of one single copper atom, turning it into an independent three-dimensional harmonic oscillator.
An important observation regarding this pT switching procedure is that possible diffusion processes occurring dur-ing the transformation do not interfere with the results. This is an important advantage of adiabatic switching over TI since it enables a direct determination of the vacancy free energy at any desired temperature not requiring the indirect extrapolation procedure mentioned earlier.
The paper has been organized in the following way. In Sec. II, we discuss the theory of the adiabatic switching method. First, we investigate the properties of perfectly adia-batic processes, after which we consider small deviations from adiabaticity and derive the expression for the energy dissipation using the fluctuation-dissipation theorem. Fur-thermore, we outline how to choose the switching function in order to be able to completely eliminate dissipation effects in a simple way. In Sec. III we show how the adiabatic switch-ing method can be applied to determine the Gibbs free en-ergy of formation of a vacancy and describe how the method is implemented in the case of copper. In Sec. IV we present and discuss the numerical results of the simulations. We end with conclusions in Sec. V.
II. ADIABATIC SWITCHING METHOD
In this section we discuss the details of the adiabatic switching method. In part A we discuss perfectly adiabatic theory. In part B we consider small deviations from adiaba-ticity and use the fluctuation-dissipation theorem to calculate
the corresponding energy dissipation. Based on this analysis we show how dissipation effects can be eliminated by choos-ing a suitable switchchoos-ing function.
A. Adiabatic processes
An adiabatic switching procedure is based upon the intro-duction of a dynamics D@l(t)# which depends on a time-dependent parameter l(t). This means that the equations of motion propagating the microscopic state of a physical sys-tem through phase space contain explicitly time-dependent elements. A simple example is the case in which a dynamics of the form
D@l~t!#5l~t!D01@12l~t!#D1 ~2.1!
is introduced. Here, D0and D1represent two different
~time-independent! sets of equations of motion and the parameter
l(t) is a smooth, monotonical function of time with l(t50)51 and l(t5ts)50 (ts the switching time! estab-lishing the coupling of the two sets. It is important to note that D0 and D1 are not necessarily Hamiltonian.
14
The Hertz theorem states that if D@l(t)# is ergodic and conserves a Hamiltonian energy E5K1V (K and V the ki-netic and potential energy, respectively! for any fixed l and if the switching is so slow that it may be considered adia-batic, any initial constant energy shell s(E0) of D@l51#
with phase-space volume V0 is precisely mapped upon a
constant energy shell s(E1) of D@l50# with exactly the
same phase-space volume V0. The theorem was
demon-strated by Hertz on a statistical-mechanical basis.13,14 Adopting Gibbs’ definition of entropy
S5kBlnV, ~2.2! the invariance of phase-space volume implies that the total entropy is conserved during the switching procedure. This result, derived on a microscopic basis, agrees with the ther-modynamic definition of a macroscopic reversible adiabatic process in which reversible work is done without heat ex-change.
An example of a different type of switching procedure is
D@l~t!#5Ds@l~t!#1Db. ~2.3! Here, Ds@l(t)# represents the time-dependent equations of motion describing the physical system of interest, not neces-sarily of the form ~2.1!, and Db describes the dynamics of external degrees of freedom which have been coupled to those of the physical system to subject the switching proce-dure to boundary conditions. An example of such a boundary condition is the condition of constant temperature. In this case Db represents the dynamics of the degrees of freedom of the thermal bath at temperature T coupled to the physical system.
If the switching procedure represented by Eq.~2.3! satis-fies the conditions required for the validity of the Hertz theo-rem, it guarantees that the total phase-space volume of an initial constant energy shell of D@l(t)# is conserved along the transformation. However, in this case, the total phase-space volume can no longer be related directly to the entropy of the physical system since the external degrees of freedom are included.
Watanabe and Reinhardt showed that if we choose Db to be the canonical ensemble dynamics of Nose´ the Helmholtz free energy difference between the physical systems de-scribed by Ds@l51# and Ds@l50# at temperature T is equal to the total internal energy difference between the composite systems represented by D@l51# and D@l50#. They have shown this explicitly by evaluating the microca-nonical partition function corresponding with the dynamics
~2.3! at fixed l. As was shown by the present authors,14this
result can also be deduced considering the process as an isothermal adiabatic process in the thermodynamic sense. This kind of analysis is illuminating since it does not con-sider the microscopic details of the equations of motion in-volved but rather concentrates on the macroscopic thermo-dynamic system they represent. Following this analysis, any canonical ensemble dynamics Db of which the microscopic properties are compatible with the macroscopic thermody-namic conditions of isothermal processes should give the same results when applied in an adiabatic switching proce-dure.
Another example of adiabatic transformations subject to boundary conditions are switching processes under constant temperature and pressure ( pT) conditions. In this case the time-dependent dynamics would be of the form
D@l~t!#5Ds@l~t!#1DT1Dp, ~2.4! where DT and Dp represent the constant temperature and pressure conditions, respectively. Analogous to the constant temperature case, we consider pT adiabatic processes de-scribed by~2.4! from the thermodynamic point of view. For this purpose, we consider a closed system A composed of the physical system of interest A1, a thermal bath A2 at tempera-ture T, and a pressure ‘‘reservoir’’ A3 at pressure p. The
structure of A has been illustrated schematically in Fig. 1. The physical system A1 can only exchange heat with the
thermal bath A2 and perform mechanical work exclusively
on the pressure reservoir A3. System A is closed so that its
internal energy remains constant. At a certain instant, we open system A and switch on an external work source which performs reversible work on A1 at an adiabatic rate without
exchanging heat. During this process, A1 exchanges heat
with A2 and performs mechanical work on A3 in order to
maintain its temperature T and pressure p constant. To see what happens with the internal energies (E1,E2,E3), the
en-tropies (S1,S2,S3), and the Gibbs free energies
(G1,G2,G3) we consider the change of these quantities as a FIG. 1. Schematic representation of a thermodynamic system subjected to an isobaric isothermal transformation.
result of an infinitesimal amount of work dWext done on
A1. According to the first law of thermodynamics we have
dE15TdS12pdV1dWext, dE25TdS2, dE35pdV.
~2.5!
Obviously, dS350 since A3 absorbs mechanical work
with-out heat exchange. The corresponding changes in the Gibbs free energies of the components are
dG15dE11pdV2TdS1, dG25dE22TdS2,
dG35dE32pdV. ~2.6!
Adding the expressions in ~2.6! including the condition of adiabaticity
dS11dS250, ~2.7!
we get
dG11dG21dG35dE11dE21dE3. ~2.8!
Substituting the expressions of~2.5! into ~2.6! we find
dG25dG350, ~2.9!
leading to
dG15dE11dE21dE3. ~2.10!
After the work source has been switched off, the total change of the Gibbs free energy of the system of interest will be
DG15DE11DE21DE3. ~2.11!
Thus, the change of the Gibbs free energy of the physical system of interest after a pT adiabatic process is equal to the change of the total internal energy of the composite system
A.
The application of adiabatic switching to determine ther-mal quantities of physical systems is based on simulation of the adiabatic procedures considered above. In practice, this is established by MD simulations integrating the corresponding adiabatic time-dependent equations of motion. In all cases, the dynamics of the system of interest is transformed into the dynamics of which the required thermal quantity is known. If the entropy is desired, the transformation is not subject to any boundary conditions and a simple microcanonical en-semble MD simulation with time-dependent equations of motion is used. However, when the Helmholtz or Gibbs free energy is to be calculated, the transformation is subject to constant temperature or constant temperature and pressure conditions. These circumstances are created by extending the microcanonical equations of motion with extra degrees of freedom and their corresponding equations of motion. Ex-amples of these extended systems are the MNHC dynamics,15 which is used in canonical ensemble MD simu-lations and the Andersen algorithm,16 which is capable of keeping the pressure constant during a simulation.
B. Dissipation in nearly adiabatic processes: Fluctuation-dissipation theorem
The processes discussed in the previous subsection are adiabatic only if the switching is infinitely slow or ts→`. In
MD simulations of these processes the switching times are necessarily finite making the processes essentially nonadia-batic. The effect of this nonadiabaticity is energy dissipation which causes the breakdown of the validity of the Hertz in-variant. Essentially, this breakdown is reflected by the loss of the pointwise mapping property of constant energy shells as well as the invariance of the phase-space volume. Although MD simulations of switching processes are never perfectly adiabatic, it is not very difficult to generate nearly adiabatic transformations. For these nearly adiabatic processes, the re-sults obtained using the relations of perfectly adiabatic theory are still valuable in the sense that the errors involved remain small. Quantitatively, the deviations of the results of a nonadiabatic process with respect to the perfectly adiabatic transformation are measured in terms of the systematic and statistical error which denote, respectively, the errors in the conservation of phase-space volume and the pointwise map-ping property. Estimation of the statistical error is rather straightforward computing the standard deviation of the re-sults obtained from a number of independent switching tra-jectories.
The systematic error, which is directly related to the en-ergy dissipation in the process, is more difficult to determine. Previously, Tsao, Sheu and Mou18 have derived an expres-sion for the average energy dissipation considering specifi-cally processes in which Ds@l(t)# is of the form ~2.1!. For this kind of process, in which the characteristics of the entire system are changed, the adiabatic energy curve is necessarily convex due to the thermodynamic stability requirement. Their derivation explicitly uses this convexity to calculate the energy dissipation and show it is positive definite as pre-dicted by the second law of thermodynamics.
This derivation will fail, however, for processes in which, for example, the interactions of one single atom are switched off leaving the rest of system unaltered. As we will see later in this paper, such a process will result in a concave adiabatic energy curve, which is a result of the tendency of the system to remain thermodynamically stable upon the removal of an atom.
Below, we derive the dissipation formula of Tsao, Sheu, and Mou based on the fundamental fluctuation-dissipation theorem of nonequilibrium statistical mechanics. This deri-vation does not rely on the specific nature of the switching procedure and the only assumption made is that during the switching process, the nonequilibrium states occurring be-cause of the nonadiabaticity of the transformation always remain close to equilibrium in the sense that the theory of linear response is valid. This assumption is reasonable since we can approach adiabaticity as close as we wish decreasing the switching rate, of course within the computational limits. We consider a system described by an ergodic dynamics
Ds@l0# where l0 is a fixed parameter. Along a trajectory
generated by Ds@l0#, the energy H(l0) is a constant of
mo-tion. In the distant past an infinitesimal perturbation dl was invoked, changing the dynamics of the system slightly to
Ds@l01dl# and its energy to H(l01dl). The perturbation
is maintained for a sufficiently long time for the system to reach equilibrium. The corresponding equilibrium distribu-tion isr(l01dl). Then, at t50 the perturbation is suddenly
turned off so that the original dynamics Ds@l0# and energy
of statesr(l01dl) established under the effect of the
per-turbation becomes a nonequilibrium distribution. The prob-lem now is to find out how this nonequilibrium system re-laxes towards equilibrium and how much energy is absorbed by the system during this relaxation.
The total energy of the perturbed system is given by
H~l01dl!5H~l0!1dl]H
]l
U
l0
. ~2.12!
At t50 the nonequilibrium average of the phase function
]H/]lul
0 equals the equilibrium ensemble average over the
equilibrium distribution function r(l01dl) established in
the presence of the perturbation:
S
]H ]lU
l 0D
l 0 ~t50!5K
]]HlU
l0L
l 01dl . ~2.13!Here, the bar indicates a nonequilibrium average whereas the brackets represent equilibrium ensemble averaging. After
t50, the nonequilibrium average starts relaxing towards the
equilibrium average corresponding with the equilibrium dis-tributionr(l0) under the influence of Ds@l0#. Since the
per-turbation to the total energy in ~2.12! is small, linear re-sponse theory may be applied and the relaxation of the nonequilibrium average can be written as19
S
]H ]lU
l 0D
l 0 ~t!5K
]]HlU
l0L
l 0 2 dl kBT Cl 0~t!, ~2.14!where the equilibrium averages were evaluated for the ca-nonical ensemble. Here, Cl
0(t) is the equilibrium
autocorre-lation function Cl 0~t!5
K
dS
]H ]lU
l 0D
l0 ~0!dS
]]HlU
l0D
l0 ~t!L
l0 . ~2.15!Relation~2.14! is the mathematical representation of Onsag-er’s regression hypothesis which states that when a nonequi-librium system is not too far from equinonequi-librium, the relaxation of any mechanical property can be described in terms of the proper equilibrium autocorrelation function.
The total energy dWl
0 absorbed after the complete
relax-ation of the nonequilibrium state is given by
dWl 05dl lim t→` 1 t
E
0 tS
]H ]lU
l 0D
l 0 ~t!dt. ~2.16! Inserting ~2.14! we find dWl 05dlF
K
]H ]lU
l 0L
l0 2 dl kBT lim t→` 1 tE
0 t Cl 0~t!dtG
5dlK
]]HlU
l0L
l0 , ~2.17!where we have assumed that the integral converges. This result implies that if the system is allowed to relax com-pletely, no energy is dissipated so that the energy change of the system is precisely equal to the reversible work done.
Now we consider a nearly adiabatic procedure in which the dynamics Ds@l(t)# is continuously transformed by the time-dependent monotonical parameter l(t) for which
l(0)51 and l(ts)50. The total energy absorbed by the system during the transformation is
DW5
E
1 0
dldWl
dl . ~2.18!
Substituting the first expression for dW in~2.17! this integral turns into DW5
E
1 0 dlK
]H ]lL
l 2E
1 0dl kBTH
dl t~l!E
0 t~l! Cl~t!dtJ
. ~2.19!The first term represents the reversible work done during the transformation. In the second term, we may no longer take the limit for t→` since the nonequilibrium states are not allowed to relax completely due to the finite switching rates. Consequently, it does not vanish as in ~2.17! but rather ac-counts for the dissipation caused by the nonadiabaticity. The local relaxation timet(l) is related to the switching rate by
dl t~l!;u dl dtu52 dl dt . ~2.20!
Furthermore, we assume that the equilibrium autocorrela-tion funcautocorrela-tion ~2.21! decays exponentially as
Cl~t!5Cl~0!exp~2t/tl!5 var
S
]H]l
D
lexp~2t/tl!,~2.21!
where tl represents the characteristic correlation time and ‘‘var’’ indicates the equilibrium variance. Inserting ~2.20! and ~2.21! in ~2.19! and assuming t(l)@tl since we are dealing with nearly adiabatic processes, we find
DW5
E
1 0 dlK
]H ]lL
l1 1 kBTE
1 0 dldl dtU
l tlvarS
] H ]lD
l. ~2.22!Transforming to integrals over the switching time interval we find the expression
DW5
E
0 ts dtdl dtU
tK
]H ]lL
t 1k1 BTE
0 ts dtS
dl dtU
tD
2 ttvarS
]H ]lD
t 5DE1Ediss. ~2.23!Here, DE denotes the reversible work associated with the transformation and Ediss represents the energy dissipation
caused by the nonadiabatic switching rates. The integrand of the latter integral coincides precisely with the one given by Tsao, Sheu, and Mou, interpreting the correlation timesttas the time lags of the configurations. Obviously, it is a positive-definite function assuring the energy dissipation to be positive.
In an application of the adiabatic switching method, three aspects have to be considered. First, we have to choose a suitable reference system and assign appropriate values to its parameters. Second, we need to consider the question of how to choose a proper switching function. Finally, it is necessary to estimate the energy dissipation generated during the pro-cess quantitatively, in order to be able to correct for system-atic errors. Obviously, when making particular choices, one should always pursue competitive computational efficiency and quantitative reliability.
In general, few options are available for the choice of a reference system. Considering cases in which the system of interest is solid material, the Einstein crystal is the most ap-propriate choice. The main argument for this is that using this system, no phase transitions will appear during the switching. This, for example, is in contrast to a situation in which the ideal gas is employed as a reference and such transitions cannot be avoided. In order to decrease dissipa-tion effects and speed up convergence it is best to choose the vibrational properties of the Einstein crystal such that they remain ‘‘close’’ to those of the system of interest. In prac-tice, this is accomplished by choosing the characteristic fre-quency of the harmonic oscillators to be of the order of a principal phonon mode of the system of interest.
Aiming at a rapid convergence of the switching process, it would be best to find the particular switching function
lm(t) that minimizes the energy dissipation. In principle, it is possible to determine this function. Having a closer look at the dissipation integral in~2.23!, we see it is a functional of
l and dl/dt. Thus, the problem is to find the function lm(t) subject to fixed boundary conditions for lm(0) and lm(ts), that minimizes this functional. Obviously, this is a variational problem and its solution is contained in the Euler-Lagrange equation corresponding with it. Despite the rapid convergence, this procedure would be computationally ex-pensive. Besides the computational effort to set up and solve the Euler-Lagrange equation, one would have to evaluate the dissipation integral in~2.23! numerically, which requires the time lag and the equilibrium variances as a function of l.
Instead of focusing all attention on finding the analytic form of the switching function minimizing dissipation, it is much better to choose a function having certain global sym-metry properties. We will show now that if we employ a switching function of which the time derivative is symmetric with respect to a mirror placed at t5ts/2, it is very simple to estimate the dissipation in an adiabatic switching procedure. Although it will generally be larger than in the ideal case, it will be much easier to correct for it and achieve quantitative reliability with much less computational effort. To see this, we consider adiabatic switching processes established by a switching function having the mentioned symmetry property. An example of such a function is C1(t) ~Refs. 14 and 18!
~witht5t/ts), which has been shown in Fig. 2. Suppose we
perform a closed adiabatic process in which we transform the dynamics of the system of interest to the system of reference, followed by the opposite procedure. The energy differences measured after both stages ~including dissipation! are DW1 andDW2, respectively. Due to the fact that the time
deriva-tive of the switching function has the symmetry property mentioned above, it is easy to show that
DW15DE1Ediss, DW252DE1Ediss. ~2.24!
Thus, the energy hysteresis DW11DW2 of the closed
adia-batic switching process is precisely equal to 2Ediss.
Al-though the dissipation might be significant for both stages, we can easily eliminate it from the final result by simply addingDW1andDW2. This procedure is very efficient since,
in principle, only two MD trajectories are required.
III. DETERMINATION OF VACANCY FORMATION FREE ENERGY USING ADIABATIC SWITCHING
The Gibbs free energy of formation gf of a monovacancy is defined as
gf5G~N,p,T,n51!2G~N,p,T,n50!. ~3.1!
Here N represents the number of atoms present in the system which is supposed be large, p the pressure, T the absolute temperature, and n the number of vacant lattice sites. In or-der to compute this formation energy we have to design an adiabatic switching procedure transforming the thermody-namic state of the system according to
~N,p,T,0!→~N,p,T,1!. ~3.2!
In practice it is not feasible to construct such processes suit-able for MD simulation. The reason for this is that the intro-duction of a vacant lattice site, maintaining the fixed number of atoms, results in symmetry alterations that cannot be rep-resented in a computational cell subject to periodic boundary conditions. Because of this we cannot determine the vacancy
FIG. 2. Example of a switching function of which the time derivative is symmetric with respect to a mirror placed at t5ts/2.
formation parameters directly. Instead, one defines adiabatic switching procedures that represent transformations
~N,p,T,0!→~N21,p,T,1!, ~3.3!
in which the interactions of the atom associated with a cer-tain lattice site are turned off thus reducing the number of atoms in the system by 1. Performing such pT transforma-tions adiabatically, the Gibbs free energy difference
DGS5G~N21,p,T,1!2G~N,p,T,0! ~3.4! between the respective thermodynamic states can be readily determined. Note thatDGScan be identified as the chemical potentialm( p,T). The relation between the formation energy
gf and DGS can be seen realizing that definition ~3.1! is established in the thermodynamic limit of large N. Thus, we may also write the definition as
gf5G~N21,p,T,1!2G~N21,p,T,0!. ~3.5!
Combining ~3.1!, ~3.4!, and ~3.5! it is easy to see that
gf5DGS1g0, ~3.6!
where
g05G~N,p,T,0!2G~N21,p,T,0! ~3.7!
represents the Gibbs free energy of an atom in bulk material. In this work we have determined the vacancy formation Gibbs free energy of copper at zero pressure as a function of temperature by MD simulation of adiabatic transformations of the type ~3.3!. For this purpose, we introduced coupling parameters li into the interatomic forces defined by the tight-binding potential of Cleri and Rosato17 writing the re-pulsive and band energies of atom i as
ER i5l i
(
jÞi N ljAexp@2p~ri j/r021!# ~3.8! and EBi52H
li(
jÞi N ljj2exp@22q~ri j/r021!#J
1/2 , ~3.9! respectively. Here, ri j represents the distance between atoms i and j , r0 is the nearest-neighbor distance 0 K, j is anef-fective hopping integral, and p and q are parameters. The total cohesive energy of the N atoms is given by
Ec5
(
i51 N ~ER i1E B i!. ~3.10!The coupling parameterli, which may vary between 0 and 1, determines to what extent atom i interacts with the other atoms. When li51 atom i interacts fully with the other at-oms while if li50 it does not interact at all and may be considered a free particle.
The MD simulations were performed in a cubic computa-tional cell containing 500 atoms subject to convencomputa-tional pe-riodic boundary conditions. In this case, using the fifth-nearest-neighbor cutoff for the interatomic potential, the size of the computational cell is sufficiently large to avoid first-order interactions between the vacancy and its periodic
im-ages. The transformation ~3.3! was established by slowly switching off the interactions of one atom. To achieve this the coupling parameter l corresponding to the atom was made time-dependent through the switching function
l~t!5C1~t!, ~3.11!
wheret5t/ts. The coupling parameters corresponding with the other atoms were maintained fixed at a value of 1. The
pT conditions during the transformation were obtained using
the MNHC-Andersen15,16 technique. Its set of equations of motion is given by q˙i5 pi mi1 1 3qi pV M V, p˙i52 ]Ec~l! ]qi 2pi
S
ph i,1 Qi 1 1 3 pV M VD
, h˙i,15phi,1 Qi , h˙i,25 ph i,2 Qi , p˙h i,15 pi2 mi2kB T2ph i,1 ph i,2 Qi , p˙h i,25 ph i,1 2 Qi 2kBT, V˙5 pV M , p˙V5 1 VS
1 3i(
51 n pi2 mi1WD
2pext, ~3.12!where qi and pi describe the n one-dimensional degrees of freedom of the copper atoms, Ecis the total cohesive energy,
hi,1, hi,2, phi,1, and phi,2represent the coordinates and mo-menta of the 2n degrees of freedom of the thermostat, V is the volume of the physical system, and pVis the momentum of the piston controlling the volume variations. The Qiact as moments of inertia for the motion of the thermostat coordi-nates whereas M represents the moment of inertia of the piston. Furthermore, W is the virial, T is the desired constant absolute temperature, and pext is the desired constant pres-sure. The values of Qi are chosen to be of the order of kBT/v2, where v is some characteristic frequency of the physical system, to establish near resonance between the physical system and the thermal bath components. Parameter
M is chosen such that the average frequency of the volume
fluctuations is of the order of the average fluctuation fre-quency of the pressure under constant volume conditions. For fixed l the MNHC-Andersen dynamics conserves the energy H~pW,qW, pWh,hW, pV,V,l!5Ec~qW,l!1
(
i51 n pi2 2mi 1(
i51 n p hi,1 2 1p hi,2 2 2Qi 1(
i51 n kT~hi,1 1hi,2!1 pV2 2 M1pextV. ~3.13! During the transformation, the atom which is being switched off is slowly attached to a spring of which the origin is fixed at the original lattice site of the atom to prevent it from escaping from the computational cell.The Gibbs free energy g0 of a copper atom in bulk
mate-rial, needed to calculate the formation free energy from
DGS, has been determined in the following way. First, the MNHC-Andersen technique has been used for bulk copper to
determine the thermal expansion as a function of temperature for zero pressure. Then, to determine g0, the equilibrium
volumes obtained from these equilibrium simulations were used in a set of MNHC VT adiabatic switching procedures similar to the one used in our previous paper, turning the copper atoms into independent harmonic oscillators. In prin-ciple, we might avoid this indirect determination by directly carrying out the procedure under pT conditions. However, the origins of the harmonic oscillators can no longer be maintained fixed and should follow the volume fluctuations inherent to an isobaric process. This is computationally in-convenient and can be circumvented by the procedure de-scribed above. Obviously, the MNHC-Andersen process, in which the vacancy is created, does not suffer from such problems because of the absence of an absolute reference coordinate system. Consequently, the method will not break down at high temperatures where diffusion of the vacancy becomes likely.
IV. RESULTS AND DISCUSSION
The equations of motion of the MNHC and MNHC-Andersen techniques have been integrated with the leapfrog algorithm with a time step Dt51 fs. Observing the vibra-tions of the atoms from MNHC equilibrium simulavibra-tions, the phonon frequencyv519.6 THz was utilized to calculate the value for the Qi. By trial and error, the moment of inertia of the piston in the MNHC-Andersen dynamics was chosen
M53.3231010kg m4.
In Fig. 3 we have plotted the equilibrium lattice parameter of copper as determined from pT MNHC-Andersen equilib-rium simulations as a function of temperature at zero pres-sure. The solid line represents the corresponding experimen-tal data.9,20 The Cleri-Rosato TB potential yields systematically higher values for the lattice parameter. At
T51300 K the discrepancy is about 2%. The thermal
expan-sion coefficient predicted by the TB model deviates
consid-erably more from the experimental observations. At
T5600 K the coefficient predicted by the model is
a52.231025K21 whereas the experimental value a51.931025K21is about 15% lower. These results are in
agreement with the observations of Cleri and Rosato in their paper.
The equilibrium lattice parameters determined by the TB model have been utilized in the VT adiabatic switching pro-cedure in which the copper atoms were transformed into in-dependent harmonic oscillators with their origins situated at the fcc lattice sites. All 500 oscillators had the same charac-teristic frequency v519.6 THz. By this choice the vibra-tional characteristics of the reference system remain ‘‘close’’ to those of the system of interest with the purpose of achiev-ing a reasonable convergence. To estimate the amount of dissipation involved, we simulated five independent closed adiabatic processes for each temperature. The systematic er-rors involved were found to be of the order of 1% relative to the absolute energy differences. The corresponding absolute systematic errors in the bulk free energy were so small that a correction of the results was not necessary.
The bulk Gibbs free energy and entropy computed in the simulations have been depicted in Figs. 4 and 5. Here, the free energy has been plotted with respect to the standard state at T5298 K. The entropy has been computed directly using the Gibbs free energy resulting from the switching processes and the enthalpies from the equilibrium simulations per-formed to determine the equilibrium volumes. The data points for the free energy were calculated as averages of five independent switching trajectories with a switching time
ts58.65 ps for each temperature. No error bars ~representing the statistical error! have been included since they were found to be smaller than the symbols used in the plot. Every single trajectory required about 70 min of CPU time on a Digital Alpha 3000/900 computer. The agreement with ex-perimental values21 is excellent over the entire temperature
FIG. 3. Lattice parameter of copper at zero pressure as a func-tion of temperature. The open circles are data points obtained from the simulations. The solid line represents experimental data taken from Refs. 9 and 20.
FIG. 4. Gibbs free energy g0with respect to the standard state at T5298 K of an atom in bulk copper at zero pressure as a function
of temperature. The circles represent data obtained from the simu-lations; the solid line represents experimental data taken from Ref. 21.
range. The largest discrepancy is about 1.5% for the free energy and less than 1% for the entropy at T51300 K.
Next, we computedDGS as a function of temperature by MNHC-Andersen pT switching processes. In order to esti-mate an appropriate switching time ts and to see to what extent dissipation effects needed to be corrected, we aver-aged the results of 20 individual closed trajectories for dif-ferent switching times ts using C1(t). The time needed for
convergence of the process was found to be of the order of 2 ps, which is relatively short. The systematic error was found to be negligibly small. The values of the energy hysteresis for the 20 trajectories were randomly positive and negative indicating that the systematic error was immersed in the much larger numerical noise in the results. This numerical noise was present due to the small absolute energy differ-ences we were considering (;1 eV for the total system con-taining 500 atoms!.
The differencesDGSas a function of temperature at zero pressure have been determined by averaging the results of 20 individual switching trajectories with a switching time
ts52.5 ps. The necessary CPU time on a Digital Alpha 3000/ 900 for a single trajectory is about 16 min. In Fig. 6 we have shown a typical H(l) curve generated by the adiabatic switching procedures in which the interactions of a single atom are slowly turned off. Contrary to the switching pro-cesses in which a complete system is changed,14,18the curve is concave.
In Fig. 7 we present the Gibbs free energy of formation of a vacancy in copper at zero pressure as a function of time which has been calculated using relation ~3.6!. The dashed line represents a linear fit to experimental data9according to the relation
gf5hf2Tsf, ~4.1! where hf and sf are, respectively, the enthalpy and the en-tropy of formation. The single triangle in the plot represents
an experimental data point which has been included to indi-cate the order of the errors in the experimental data. Because of the linear temperature dependence of gf, hf and sf are temperature independent. Despite the scatter of the data and the large error bars due to numerical noise, the adiabatic switching results agree well with the experimental data. The estimated enthalpy and entropy of formation are (1.2760.07) eV and (2.960.8)kB, respectively, which is quite close to the experimental estimates @(1.2860.03) eV and (2.560.2)kB, respectively#. From Fig. 7, we observe that all the estimates of gf are systematically lower than the experimental values. Yet this is not alarming considering the
FIG. 5. Entropy S of an atom in bulk copper at zero pressure as a function of temperature. The circles represent data obtained from the simulations; the solid line represents experimental data taken from Ref. 21.
FIG. 6. Typical H(l) curve generated in adiabatic switching processes in which the interactions of a single atom are turned off.
FIG. 7. Gibbs free energy of formation of a vacancy in copper at zero pressure as a function of temperature. The dashed line is a linear fit from Ref. 9 to experimental data. The solid line is a linear fit to the simulation results. The triangle represents an experimental data point indicating the order of the experimental errors involved.
small absolute energy discrepancies which are of the order of 0.05 eV and the limited accuracy of the semiempirical inter-action potential.
V. CONCLUSIONS
We have determined the vacancy formation Gibbs free energy in copper as a function of temperature at zero pres-sure. For this purpose we first determined the bulk free en-ergy g0 of a copper atom in bulk copper as a function of
temperature by VT adiabatic switching procedures where the copper atoms were slowly turned into independent three-dimensional harmonic oscillators. The frequency of the os-cillators was chosen to be of the order of a principal phonon frequency of copper to achieve competitive convergence. The agreement with experiment was found to be excellent over the entire temperature range. The free energy and the entropy agreed with experimental data within 1% and 1.5%, respectively, showing the good quantitative perfor-mance of the method for bulk material.
The vacancy formation Gibbs free energy for copper as a function of temperature at zero pressure was determined in a
pT switching procedure in which the interactions of a single
copper atom were slowly switched off. The agreement with experimental values is remarkably good. The formation
en-thalpy was found to be (1.2760.07) eV, which is very close to the experimental value (1.2860.03) eV. The discrepancy between the adiabatic switching entropy estimate (2.960.8)kB and the experimental value (2.560.2)kB is somewhat larger. Nevertheless, this is still very reasonable when one realizes that very small energy differences are in-volved and recognizes that the description of the mutual atomic interactions by the semiempirical potential has a lim-ited accuracy.
Evaluating the method from the computational point of view, its shows to be efficient. Every data point required approximately 6 h of CPU time on a Digital Alpha 3000/900 for a system of 500 particles.
Considering these results we conclude that the adiabatic switching method is a competitive tool for determination of thermal properties in realistic materials both from the quan-titative and computational point of view.
ACKNOWLEDGMENTS
The MD simulations were performed on a Digital Alpha 3000/900 of the CENAPAD-SP Computer Center. The au-thors gratefully acknowledge the financial support granted by the Brazilian funding agencies CAPES, FAPESP, and CNPq.
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