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FREE ENERGY CALCULATION VIA MD : METHODOLOGY AND APPLICATION TO
BICRYSTALS
J. Lutsko, D. Wolf, S. Yip
To cite this version:
J. Lutsko, D. Wolf, S. Yip. FREE ENERGY CALCULATION VIA MD : METHODOLOGY AND APPLICATION TO BICRYSTALS. Journal de Physique Colloques, 1988, 49 (C5), pp.C5-375-C5-379.
�10.1051/jphyscol:1988543�. �jpa-00228040�
Tome 49, octobre 1988
FREE ENERGY CALCULATION VIA MD : METHODOLOGY AND APPLICATION TO BICRYSTALS
*
J.F. LUTSKO, D. WOLF and S. YIP'^)
Materials Science Division, Argonne National Laboratory, Argonne, IL 60439, U.S.A.
ABSTRACT
Ttlree independent methods for the calculation of free energies via molecular dynamics are used to determine the free energies of crystalline solids. The relative accuracy and computational efficiency of the methods is compared and discussed. The free energies of two Cu bicrystals are then computed and compared over a wide range of temperatures and their relative stabilities are discussed.
I. INTRODUCTION
In principle, the free energy of a system contains most of the statistical mechanical information about the system. Its relative value in two different states determines which state is thermodynamically favored, while response functions may be determined from its derivatives. Furthermore, the difference between the free and internal energies is proportional to the entropy. In the study of interfaces, the free energy is of interest in such questions as the relative stability of symmetric and asymmetric grain boundaries (GBs) and segre- gation at interfaces as well as the characterization of phase transitions. For these reasons, attempts have been made recently1 to calculate the free energy of interfaces via molecular dynamics (MD) through the application of MD techniques developed for homogeneous systems. However, the calculation of excess free energies for GBs is much more difficult, numerically, than the bulk calculation because, typically, two large free energies must be subtracted to yield a small excess free energy. It thus seems appropriate to compare both the accuracy and computational efficiency of three useful MD techniques prior to their applica- tion, here, to the question of the relative stability of two GBs which have nearly the same energy at zero temperature.
11. BASIC TkEORY
The calculation of free energies from Ti requires specialized techniques which involve more inherent uncertainty than simple averages. This is due to the fact that the free energy, F, is not itself an average of a phase function but, rather, is directly related to the partition function, Z, through the fundamental relation;
where ? is the inverse temperature, H is the Hamiltonian of the system and dr is an infinitesimal volume in phase space. Several methods are now available for directly calculating the free energy from MD. Two in particular will be studied
*Work supported by the 1J.S. 1)epartment of h e s g y , 8ES-Materials Sciences, under Contract W-31-109-Eng-38.
tL)J?ernanent adciress: liect. of Nuclear Engineering, MIT, Cambridge, MA 02139
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1988543
C5-376 JOURNAL DE PHYSIQUE
here: energy sampling (ES), and a method described by Frenkel and I,add2 which we denote as FL. The free energies thus obtained will then be compared with lattice dynamics in the quasiharmonic approximation (QHA).
The QHA consists of equilibrating a system via MD to determine the equi- librium volume at a given temperature. The system is then quenched at constant volume and the resulting configuration used as the reference or input configura- tion of a lattice dynamics calculation. The free energy of the harmonic system is easily computed as;
where wn are the normal mode frequencies obtained by diagonalizing the dynamical matrix.
It is clear that while the QHA includes certain anharmonicities associated with the thermal expansion of the system, it does not treat all anharmonicity.
Thus, while computationally efficient, it must break down at high temperatures.
Energy sampling has been discussed in many *laces3. The form used here involves defining an auxiliary function, A(y,R), as;
A(Y,~) =
< ~(x-u)>
P = fdre -PH F(Y-u)/zp (2) where U is the potential energy of the system and d(...) is the Dirac delta function. This function, being a phase space average, is easily computed via MD. Its ratio at different temperatures is related to the difference in free energy of the system at the two temperatures;= ($13N/2 exp
-
(R'F'-
RF+
(8-st)x)thus implying that;
The QHA can be used, for example, to fix the absolute value of the free energy at some low temperature. We note that because the functions, A(x,R), are only known numerically, the temperatures R and R ' must be close enough that substantial overlap of the A-functions exists. Otherwise, some sort of interpolation tech- nique must be used, presumably increasing the error in the calculation.
Finally, the FL method2 relates a given system to a reference Einstein crystal by altering the potential U to U X according to
where
U , = i K 1 7 (R - R )2
n no ( 6 )
Here, K is an arbitrarily chosen constant,
%,
is the coordinate of the nth particle and %o is its equilibrium position in the Einstein crystal. It is then seen thatwhere FE is the free energy of the Einstein crystal which is known analyti- cally. The average is computed with MD and the integral performed numerically.
By choosing the spring constant appropriately, the integrand is smooth and requires few points for accurate integration.
111. MD RESULTS: COMPARISON OF I%TtlODS
The three methods have been applied to an ideal crystal of 108 particles interacting via a Lennard-Jones (LJ) potential and with periodic boundaries. The potential simulates Krypton and was cutoff at 1.4 lattice parameters (i.e.
between the third and fourth nearest neighbors) and was shifted to avoid cutoff effects. The runs were of approximately equal length allowing 1000 time steps for equilibration and 5000 steps to compute the free energies. The volumes were equilibrated at zero pressure using the
~arrinello- ahm man^
scheme and held fixed during the averaging.Figure 1 shows a comparison of the QHA, the FL method and the strictly harmonic approximation (HA) in which the reference system is the zero-temperature crystal. Figure 2 shows the FL, ES and QHA methods compared over a smaller temperature range.
The lack of smoothness in the curves is due, in part, to the short equilib- rations rather than to the methods. It is irrelevent for the comparisons, how- ever, since the equilibrated systems were identical at each temperature.
Several conclusions may be drawn from these figures. Most importantly, the various methods are remarkably self consistent, even with these relatively short
- 0 . 1 2 ~ I
0 2 0 4 0 6 0 8 0 100
TEMP(K)
Fig. 1. Comparison of QHA, FL and harmonic approximation (HA).
using a L-J potential for Kr.
-0.0934 .
.
-.
--
- * -.
- I 3 8 4 0 4 2 4 4 46 4 8 5 0TEMP(K)
Fig. 2. Comparison of ES, FL and the QUA using a L-J potential for Kr.
C5-378 JOURNAL DE PHYSIQUE
runs. It is interesting to note that the QHA, by far the computationally most efficient method, seems to work well up to about 75% of the melting tempera- ture (Tm
-
115 K). Further details of this study will appear elsewhere.IV. MD RESULTS: APPLICATION TO GBs
We have applied the YL method to the calculation of the free energies of bicrystals. Because it yields absolute values of the free energy, it allows one to map out the free energy of a system over a wide range of temperatures
efficiently and to thus determine the overall behavior of the free energy quickly. In these simulations, a shifted Lennard-Jones potential with a cutoff between fourth and fifth nearest neighbors was used to model Cu. The free energies of the symmetrical tilt boundaries on the (221) and (114) planes were determined. Their zero-temperature energies were determined to be 756 and 704 erg/cm2, respectively.
The simulations were performed as before with the exception that 2000 steps were allowed for the equilibrations. The simulation cell was chosen to be periodic in three dimensions and actually contained two identical GBs separated by 18 planes in both cases. The geometry of the unit cell is shown in Fig. 3.
Fig. 3. Geometry of a 3d periodic MD B unit cell containing two GBs, one at
the interface of regions A and B and a second GB at the image planes (IP).
The computational cell contains all atoms located within the two dashed lines. In the present case, the GB is A symmetrical so the lattice spacing in
regions A and B are actually the same.
Figure 4 shows the resulting excess free energy curves. We note that while the (114) boundary remains the lower-energy boundary, the energy difference becomes first larger and then smaller at higher temperatures as the excess energies must go to zero at the melting point.
V. CONCLUSIONS
Three methods of calculating free energies from MD have been compared. It is found that they are in good agreement for runs of similar length. The agree- ment of EL and ES with the QHA at moderate temperatures indicates the accuracy of the former as the QHA is expected to be valid up to moderate temperatures. Each method has its peculiar advantages and those of the FL method have been stressed as it is at present little used. Specifically, the QHA is computationally in- expensive due to the fact that only an MD equilibration of the system is neces- sary and appears to be accurate up to relatively high temperatures (-75% of melting). ES is valid at all temperatures and in all thermodynamic states.
however, it does not provide absolute values of the free energy and, in the form discussed here, leads to an accumulation of errors as the temperature is incre- mented. The EL method, like the QHA, is applicable only to solids although one
potential for Cu.
can imagine using, say, a hard sphere gas as a reference and extending it to liquid and gaseous states. Its primary advantage is that one obtains absolute values of the free energy up to melting. It should be noted that no MD technique samples configuration space effectively (i.e., the configurational entropy is not measured as well as the vibrational entropy). The free energies of two bicrys- tals were computed over a wide range of temperatures using FL and no change in their relative stability was found. These bicrystal calculations are preliminary and further calculations comparing the periodic system to one comprised of an isolated GB are underway as well as similar studies of the excess free energy of asymmetric boundaries.
ACKNOWUCDGEMENTS: The authors wish to acknowledge a grant of computer time on the Energy Research C W Y XMP at the Magnetic Fusion Computational Center at Livermore. Work supported by the U.S. Department of Energy, BES-Materials Sciences, under Contract W-31-109-Eng-38.
REFERENCES
1. A. J. C. Ladd, W. G. Hoover, V. Roisato, G. Kalonji, S. Yip, and
R. J. Harrison, Physics Letters lOOA (1984) 195; ti. Hasson, Ph.D. Thesis Universite de Paris VI (1972); P. Deymier and G. Kalonji, to be published.
2. D. Frenkel and A. J. C. Ladd, J. Chem. Phys.
2
(1984) 3188.3. G. Jacucci and N. Quirke, in Computer Simulation for Solids, Lecture Notes in Physics 166, C. K. A. Catlow and W. C. Mackrodt, eds. (Springer Verlag, Berlin, 1982) p. 38.
4. M. Parrinello and A. Rahman, Phys. Rev. Lett.% (1980) 1196; J. Appl. Phys.
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