PHYSICAL REVIEW B 1 JANUARY 1997-IIVOLUME 55, NUMBER 2
Molecular-dynamics calculations of thermodynamic properties of metastable alloys
Giorgio Mazzone* and Vittorio RosatoENEA Ente per le Nuove Tecnologie, I’Energia e I’Ambiente Centro Ricerche Casaccia, Divisione Nuovi Materiali, CP 2400, 0
Roma, Italy
Marco PintoreDipartimento di Scienze Agrarie ed Ambientali, Universita` di Sassari, Viale Italia 39, 07100 Sassari, Italy
Francesco Delogu, PierFranco Demontis, and Giuseppe B. SuffrittiDipartimento di Chimica, Universita` di Sassari, Via Vienna 2, 07100 Sassari, Italy
~Received 22 April 1996; revised manuscript received 24 July 1996!
In order to improve our current understanding of the microscopic structure of metastable alloys of immis-cible elements such as Ag-Cu and Co-Cu, the Helmholtz free energy of several microstructures based on an fccunit cell has been calculated and compared with that of a reference state. The microstructures considered forthe free energy calculations at fixed volume are~1! a structure formed by alternating layers of fixed thicknessof metal 1 and metal 2 separated by coherent interfaces;~2! an atomically disordered solid solution;~3! astructure comprising a random distribution of elemental cubic grains separated by coherent interfaces. Numeri-cal results show that the Helmholtz free energy of structure~3! decreases with increasing grain size and that itsvalue calculated for a sufficiently large grain size approaches the free energy of structure~1!. Furthermolecular-dynamics simulations for the Ag-Cu system have allowed the calculation of the enthalpy at theequilibrium volume of several microstructures including some of those listed above. A comparison of thecalculated values of the enthalpy with the heat release observed experimentally allows the advancement of anhypothesis concerning the reaction path and the structure of the equiatomic Ag-Cu alloy obtained by ballmilling. @S0163-1829~97!08901-7#
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I. INTRODUCTION
The formation of metastable alloys upon high-energy bmilling of mixtures of elemental powders having a larpositive enthalpy of mixing and, therefore, limited mutusolubility in the solid state, has been the subject of sevstudies concerned with both the structure and the thermonamics of these phases.1–4 Still, the nature of the drivingforce leading to the alloy formation and the atomic levstructure of these phases are the subject of an ongoingbate. In essence, in order to describe the alloying proctwo different explanations have been advanced. The firs3 isconcerned with the accumulation of structural defectsdishomogeneities during ball milling, and with their annihlation during the alloy formation once the free energy asciated with these defects exceeds the free energy of mixThe second4 is concerned with the occurrence of a transiesupersaturation of point defects and with the mechanicalergy supplied by the stress field of the plastic deformatiboth of which are supposed to allow nonequilibrium interdfusion of the atomic species.
On the basis of the above considerations it appearsticularly relevant to follow the evolution of the microstructure of the alloyed phase and to define its details atatomic level. To this purpose, in the case of the Cu-Fe stem, several techniques, such as transmission elecmicroscopy,4 Mossbauer spectroscopy,5 and extended x-rayabsorption fine structure6 have been employed, in an attemto complement the structural characterization by convtional x-ray diffraction, whose limits for investigating com
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positional dishomogeneities have been recently discuss7
The application of these techniques suggests that mixingthe two components occurs in the Cu-Fe system atatomic level, even if there is an indication of local compotional fluctuations5 and, to a limited extent, of modulatestructures with a periodicity of a few nm.4
No investigations at the same level of detail have beperformed for other immiscible systems such as Co-CuAg-Cu, so that in these cases the knowledge of the micscopic structure of the metastable alloys obtained bymilling is very limited and based only on conventional dfractometry which shows the existence of a single crystagraphic phase~within the limits mentioned above! and ondifferential calorimetry which evidences a heat releaseseveral kJ/mol when these alloys are heated to;1000 K.1–3,8
Another difficulty in understanding the mechanism of alloing in these systems arises from the lack of accurate inmation ~both experimental and calculated! concerning theenthalpy and the free energy of the different microstructuwhich could develop during the process of ball milling, icluding the ideal solid solution which, of course, is not eperimentally accessible in the whole range of compositiThe enthalpy and the free energy of the ideal solid solutcan be calculated2,3 with the calculation of the phase diagra~CALPHAD! method or with other more sophisticated aproaches such as the free-energy minimization meth9
~FEMM! or the second-order expansion10 ~SOE!, all ofwhich are of static nature and, therefore, less accurate thmolecular-dynamics~MD! calculation.
The ability of thermodynamic methods to calculate t
enthalpy and the free energy of the metastable phase istricted to those structures for which some experimentalsults are available. This limitation can be overcome byuse, in an MD calculation, of interaction potentials havigood transferability, that is, depending only weakly on tparticular crystal structure whose properties have been ufor a fit of the free parameters appearing in the potential
In view of these considerations we have decided to calate the free energy and the enthalpy of a number of phaand microstructures of equiatomic Ag-Cu and Co-Cu allowhich could play a role in the alloying process of systewith a positive enthalpy of mixing, using an MD approabased on an-body empirical potential.11
II. ENERGY CALCULATIONS
A. Structural models
Two different types of MD calculations, relative to Helmholtz free energy and to enthalpy, have been performedthe first case, for which the Ag-Cu and the Co-Cu systehave been considered, the Bennet method12 has been used tocalculate the free energy difference between several mistructures of an equiatomic alloy, all of which are costrained to have the same density of the corresponding aobtained by ball milling. In the second case, relative onlythe Ag-Cu system, constant-pressure, constant-temperMD simulations have been performed to relax to their eqlibrium configuration some of the microstructures considefor the free-energy calculations, and other structureswhich the free energy could not be calculated using the Bnet method.
In order to apply Bennet prescription for the calculatiof the free energy, it is necessary to consider only thmicrostructures which are defined by a common set of codinates$Rj%, whereRj is the position vector of thej th atom,independent of the particular microstructure, so that the odifference between two microstructures tractable with tmethod is relative to the distribution of atomic specamong theRjs. With this limitation, the following micro-structures have been considered for the free-energy calctions:
~1! A cube containing 12312312 fcc unit cells, the uppehalf of which is composed of metal 1 and the lower halfmetal 2. The two halves are constrained to have the slattice parameter and are thus separated by a coherentface parallel to a~001! plane. Periodic boundary conditionhave been applied along the three coordinate axes so thasystem is composed of alternated slices, six unit cells thstacked alongz and extending indefinitely in thex,y direc-tions. This phase, labeled as structure 1, according torequirements of the Bennet method12 has been retained as threference phase for the free-energy calculations.
~2! An equiatomic fcc solid solution, structure 2, disodered at the atomic level and, therefore, characterized byabsence of any conventional interface, whose atoms areranged in the same lattice of those of structure 1.
~3! A coarse grained disordered structure made of pcubic grains of the two elements, the side of each grain beequal tod fcc unit cells. The individual grains, separatedcoherent interfaces, are distributed at random in a cubicume equal to that of the reference structure. Several diffe
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sizes with increasing values ofd ~d52,3,6! have been analyzed. These structures have been collectively labeled stture 3.
In addition to the free-energy calculations, enthalpy cculations have been performed for structure 1, structureand structure 3 withd56 andd58, for a three-dimensionallyordered arrangement of alternating cubic grains of the pelements, for the undercooled liquid alloy, for the orderL10 structure, and for a system with no periodic boundaconditions composed by two large grains of Cu and Aseparated by an incoherent~001! interface, whose dimensions along the directions parallel to the interface are vclose to a perfect match between the crystal edges. Usi2332338 Ag crystal and a 2632638 Cu crystal, one hasthat the edges of the square~001! faces of the two crystalsare equal to better than 0.1%. The reference state forenthalpy calculations is the weighted average of the consent elements.
B. Method outline
For over a decade, computer simulations have handmetallic systems with satisfactory accuracy making useempirical n-body potentials.13,14 MD simulations based onthese potential schemes have been extensively used to orelevant information about the structure and the stabilitydifferent phases.15,16Of course, the equilibrium configuratioof a system can be investigated also by means of free-encalculations~this subject is extensively reviewed in Ref17–19!. To this purpose the equilibrium value of the freenergy of a system can be evaluated by a minimization ofGibbs free-energy functional. In this scheme the enthalpythe system can be calculated from a MD simulation, the cfigurational entropy can be evaluated with the use ofcluster variation method, and the vibrational entropy evaated in the quasiharmonic approximation.16
Another approach to the stability problem is based oncalculation of free-energy differences. These differencesusually calculated with respect to a state which can be eia phase whose free energy can be estimated by some me20
or simply a phase which represents a common referestate, even if its free energy is not known.12,21
As stated above, the present approach compares theenergy of systems which differ only in the ‘‘chemical oder,’’ i.e., in the arrangement of the atomic species instructure. In classical statistical mechanics, thermodynapotentials are determined from the configurational integwhich defines the partition function. MD simulations, however, are not able to evaluate accurately the 3N-dimensionalconfigurational integral so that the evaluation of the thermdynamic potentials is not reliable. Several methods hbeen proposed to overcome this difficulty. One of them, ually referred to as the method of ‘‘overlapping enerdistributions,’’12,21 makes use of a convenient manipulatioof the exponential probability distribution of states to obtaan equation determining the ratio of the configurational ingrals for systems with different potential-energy functions
With reference to the canonical~NVT! ensemble~number-density, volume, and temperature!, the Helmholtzfree energyF can be written in terms of the configurationintegralQ as,21
55 839MOLECULAR-DYNAMICS CALCULATIONS OF . . .
TABLE I. Potential parameters of the different atomic species used in the simulations@Eq. ~7!#.
The integration of Eq.~2! extends over the volumeV of thecoordinate space occupied by theN particles of the systemIf two systems are at the same temperature and volumefree-energy differenceDF can be written as
b~F12F2!5bDF5 lnSQ2
Q1D ~3!
with Q1 and Q2 integrated over the same volume of thcoordinate space. One can, in principle, define the enedifference distribution functionsh1~D! andh2~D! as
h1~D!5*Vd~U12U22D!e2bU1dqN
Q15^d~U12U22D!&1 ,
~4!
h2~D!5*Vd~U12U22D!e2bU2dq
N
Q25^d~U12U22D!&2 ,
~5!
whered(x) is Dirac’s delta,U1 andU2 represent the potential energy of phase 1 and phase 2, and the angular bra^ &i ~i51,2! as defined by Eqs.~4! and ~5!, represents theensemble average ofd(U12U22D) in the canonical en-semble generated byUi . By simple algebraic manipulationsone gets
Q1
Q25e2bD
h2~D!
h1~D!. ~6!
As the left-hand side of Eq.~6! is independent ofD, one canevaluate the right-hand side of Eq.~6! at any value ofD.Particularly convenient is the valueD8 such thath1(D8)5h2(D8). With this choice one obtainsQ1 /Q2
5e2bD85e2b(F12F2) so that, in practice, one interpolatethe numerical functions describingh1~D! andh2~D! in orderto find the intersection valueD8 which, as shown aboveequals the free-energy difference between the two states
C. Calculations
The model system used for the free-energy computatis an fcc structure with 6912 atoms with a lattice parame
he
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ket
sr
equal to the experimental value of the alloy that is 3.590for Cu-Co and 3.887 Å for Ag-Cu.
The interactions between the atomic species have bdescribed with an-body potential derived from a secondmoment approximation of a tight-binding Hamiltonian.11 Inthis scheme the cohesive energy can be written as
Ec5(i51
N H (j51
N
Aabe2pab~2r i j /rab
021!
2A(j51
N
jab2e22qab~2r i j /rab
021!J , ~7!
whereAab , jab , pab , andqab are potential parameters referring to the interactions between pairs of atoms of speca andb. The potential cutoff has been extended up to fineighbors for Ag-Cu and up to ninth neighbors for Co-CWhile the potential parameters for the pure species are aable in the literature,11 the cross parameters~Ag-Cu and Co-Cu! have been evaluated using a fitting procedure slighmodified with respect to that currently used. In fact, the eperimental quantities commonly used for the fitting procdure of cross terms are the cohesive energy and the elconstants, with the boundary condition of vanishing exterpressure at the equilibrium volume of the intermetallic phaat a given stoichiometry. In the absence of any intermetacompound~as it happens for systems with positive heatmixing!, we have used the values of the enthalpy of mixifor liquid alloys reported by Hultgrenet al.,22 that isDHm~Ag50Cu50!54.58 kJ/mol at T51423 K andDHm~Co95Cu5!51.565 kJ/mol atT51473 K. Assuming avanishingly small volume change upon mixing of the liqumetals, we have assigned arbitrary initial values to the crparameters. These values have been adjusted in succeconstant-volume MD simulation runs until the cohesive eergy of the liquid alloy at zero external pressure reacheddesired value. The cross parameters deduced for the lialloy using both an attractive and a repulsive term have bused for the solid-state calculations. Table I reports therameters used in Eq.~7! for the computations.
In the free-energy calculations, the simulations have bperformed in the canonical~NVT! ensemble ~number-density, pressure, and temperature!, as prescribed by thetheoretical layout. For the enthalpy calculations the systhas been relaxed in the constant pressure, constant temture ~NPT! ensemble. All calculations have been performatT5300 K, with a time step for the integration of the equtions of motion of 10215 s.
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840 55GIORGIO MAZZONEet al.
III. RESULTS AND DISCUSSION
TheDF values calculated between the reference struc~which in the case of Co-Cu contains Co in the higtemperature fcc phase! and the other structures are shownFigs. 1 and 2 as a function of grain sized. In order to includein these figures the results relative to the solid solution,have assigned to this structure the valued50 as a reasonablapproximation to the effective value of the ‘‘grain size’’ ithis case. Actually, Figs. 1 and 2 show that the calculaDF ’s can be interpolated by a smooth decreasing functionthe grain size. The deviations of the actualDF values fromthe interpolated curve are due to the small number of graused in the simulations which does not exactly reproducrandom distribution. With respect to structure 1 the valuesthe free energy of structure 3 calculated ford56 are negativeby a few tenths of kJ/mol, which is of the order of the stistical fluctuations affecting each calculated value. Consering that the free energy of structure 1 and of structurmust approach the same asymptotic value for increalayer thickness and for increasing grain size, it appearsin the case of both Ag-Cu and Co-Cu the free energy osystem composed of coherent domains with size.2 nm isonly weakly dependent on the geometrical details of thecrostructure. On the other hand, the free energy of the ssolution is considerably higher than that of the referenstructure.
FIG. 1. Calculated values ofDF between structure 1 and structures 2 and 3 for equiatomic Ag-Cu. Dots refer to structure 3 wd52,3,6; the square to random solid solution.
FIG. 2. Same as in Fig. 1 for Co-Cu.
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At constant volume, the contribution to the free-enerdifference between structures 1 and 3 arising from the pence of coherent interfaces can be thought of as composetwo parts: the first is of chemical origin and for systemhaving a positive heat of mixing is positive; the second isentropic nature and, for a coherent interface, contains almexclusively a vibrational contribution, which, in the preseconditions, depends very little on the topology of the inteface. Accordingly, the free-energy differences at 300should be dominated by the chemical contribution. Tqualitative feature is apparent in Fig. 3, which shows a pof the free energy of Ag-Cu as a function of the densitys ofinterface area defined as the ratio between the sum of almetal 1–metal 2 interface areas actually present in a gistructure ~irrespective of the interface topology! and thenumber of atoms. These considerations, apart from a rtively small configurational term easy to evaluate in tBragg-Williams approximation, apply also to the randosolid solution for whichs has been defined asS/4, whereSis the surface area of the Wigner-Seitz cell of the fcc strture.
For the enthalpy calculations of the Ag-Cu system, soof the structures described above~namely structure 1, structure 2, and structure 3 withd56 and 8! have been relaxed aT5300 K and vanishing external pressure. The thrdimensionally ordered arrangement of alternating cugrains with d56 ~three-dimensional chess board!, the or-deredL10 phase, the undercooled liquid, and the two-grasystem have been also relaxed at the same temperaturcomparison purposes. In the case of the two-grain systemhave not calculated the enthalpy of the whole structuwhich is bounded by free surfaces, but only the energy osmall inner core. The dimensions of the core, comprising tadjoining blocks of 93934 Cu unit cells and 83834 Agunit cells are such that the free surfaces of the two gracontribute a negligible amount to the enthalpy of the coatoms. On the other hand, in order to be sure to includethe perturbation arising from the interface, the thicknessthe core~four unit cells on each side of the interface! hasbeen chosen to be about twice as large as the range ointeraction potential.
The enthalpy, the molar volume, and the density of intface area calculated for each of the above structures arported in Table II. Listed in Table II are also the results o
FIG. 3. SameDF values of Fig. 1, shown as a function ofs.The open circle corresponds to structure 1.
nstituent
55 841MOLECULAR-DYNAMICS CALCULATIONS OF . . .
TABLE II. Thermodynamic data for different structures of Ag-Cu~see text!. The densitys of metal1–metal 2 interfaces has been given~wherever available! before ~unrel! and after~rel! the relaxation atvanishing external pressure. The reference structure corresponds to the weighted average of the coelements.
Structure ^H& ~kJ/mol! DH ~kJ/mol! V ~cm3/mol! sunrel ~srel!
first-principles calculation for theL10 structure23 performed
at T50 K in reasonable agreement with the present Mvalues.
In accordance with the arguments advanced with regarfree energy, Fig. 4 shows the enthalpy difference betweach of the above microstructures and the reference statefunction ofs, including the inner core of the two-grain sytem and the undercooled liquid for which we have useddensity of interface area of the solid solution scaled totwo-thirds power of the volume. Notwithstanding the diffeence between the structures considered in Fig. 4 the exenthalpy appears to depend only slightly on the detailseach microstructure.
In the case of the solid solutions we have repeatedcalculations for a number of configurations in order to haan estimate of the statistical fluctuation. The resulting erbar is of the order of 0.1 kJ/mol and, if reported in the figuwould be smaller than the dot size.
Comparing with the reference state the enthalpy of strture 1 and of the inner core of the two-grain system,enthalpy of the~001! interface has been calculated as 1.J/m2 for the incoherent case and 1.89 J/m2 for the coherent
FIG. 4. Calculated values ofDH for equiatomic Ag-Cu. Eachdot is numbered as the corresponding structure in the first colof Table II.
tons a
ee
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eer,
-e
one. The energy of a certain microstructure will therefodepend on both the type and the density of interface area.instance, in the case of the inner core of the two-grain stem, the substitution of the incoherent interface with a cohent one would increase the energy of the inner core by ab1.5 kJ/mol.
The behavior ofDH as a function ofs can be easilyreproduced, at least for the geometry of structure 3, onbasis of a simple model based on a pair interaction scheConsider a large cube of sideL comprisingn3 elementalcubic grains of sidel arranged as in structure 3. Let us asume that at eachA/B interface the interaction extends ova rangeD l so that the enthalpy increase of each grain, duethe interaction with the adjoining grains of a different atomspecies, is proportional to the volume fraction of each grcomprised within a distanceD l from the interfaces. The calculation of the relevant volume fraction is particularly eawith the conditionD l< l , in which case each grain interaconly with its nearest neighbors. With this restriction, ifH isthe total enthalpy due to the interfaces andN is the totalnumber of atoms of the large cube, one can write
h5H
N}n3
@ l 32~ l2D l !3#
N, ~8!
where the quantity between square brackets is proportioto the volume fraction of each grain defined above. Fromdefinition ofn asL/ l one has
h}@ l 32~ l2D l !3#
l 353
D l
l23S D l
l D 21S D l
l D 3. ~9!
DefiningS as the total interface area contained in the lacube one has
s5S
N}n3l 2}
1
l~10!
so that if we definex5D l / l we have thatx is proportional tos. In summary,
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842 55GIORGIO MAZZONEet al.
with 0<x<1. Recalling thatx is proportional tos, one cancompare the parabolic behavior exhibited by Eq.~11! withthe behavior ofDH as a function ofs shown in Fig. 4.
The essence of the model, as shown by Eq.~8!, is thatwhen a volume element belongs to more than one interfacis counted only once. Even admitting that this assumptiontoo simplistic, it appears nonetheless that the hypothesisnonadditivity of the interface effects gives a clue for modeing the results of the enthalpy calculations.
The computed values of the enthalpy can be compawith the heat released by the Ag-Cu alloys upon heaticorrected for the enthalpy of formation of the terminal soltions and for the heat release due to recovery and grgrowth. These corrected values are reported in Ref. 9 afor the equimolar composition, the heat release amounts;5 kJ/mol. Adding to this value the interface enthalpy~es-timated as 1 J/m2 according to the present computation!calculated for a grain size ranging between 20 and 40 none obtains that the experimental value of the enthalpy dference between the Ag-Cu alloy and the reference state ithe order of 6 kJ/mol.
A comparison of this result with the values shown in Fi4 suggests that the Ag-Cu alloys obtained by high-eneball milling are not atomically disordered solid solutions bcontain coherent regions which, at least in one directiohave a thickness of the individual layers of the order of a fenm. This result is independent of the uncertainties affectthe corrected values of the heat release which, in any cassmaller by a factor of 2 than the computed enthalpy diffeence between the reference state and the solid solutioncourse, this conclusion does not rule out the possibility tthis partially decomposed structure is the result of the tra
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formation of a higher-energy microstructure which cougrow during the short-time interval typical of the ball impaand then relax quickly to a less energetic state. Actuallyhas been already pointed out with reference to the Fecase4 that the mechanical energy supplied by the stress fiand the transient enhancement of the defect concentraduring the plastic deformation may be able to induce tformation of a solid solution. In the same case it was aobserved that the density of extended defects was relativlow, thus suggesting that some recovery of the defectsated by the plastic deformation had occurred. All theseservations point to the presence of some excess mobilitytherefore to the occurrence of a microstructural evolutionthe samples for a certain time after the deformation.
The above remarks may be generalized saying thattransformations undergone by the system after the primalloying process must correspond to a free-energy decre
As a consequence it appears that, in the Ag-Cu systwhile a transformation from the solid solution to a modulatcoherent structure containing more or less pure layers ofelements is energetically possible, the reverse transformais not allowed unless the refinement of the coherent molated microstructure continues down almost to atomic levIn this case, however, the formation of the solid solutifrom the pure elements as a result of the injection in tsystem of a considerable amount of mechanical energypears much more realistic.
ACKNOWLEDGMENTS
The authors acknowledge Dr. Massimo Celino~ENEA!for helping with the calculations. Dr. M. Vittori Antisari isalso acknowledged for helpful discussions and comment
e-ol.
K.ary
s.
*Also at Istituto Nazionale di Fisica della Materia, Unita` di Peru-gia, Perugia, Italy.
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