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Ab-Initio Thermodynamics of Metal Alloys: From the Atomic to the Mesoscopic ScaleStefan M¨u llerUniversit¨a t Erlangen-N¨u rnberg,Lehrstuhl f¨u r Festk¨o rperphysikStaudtstr.7,91058Erlangen,GermanyAbstract.Density functional theory(DFT)based calculations allow us to study a number of metal alloy properties,as e.g.formation enthalpies,or electronic and elastic properties of intermetallic compounds.However,such so-called ab-initio cal-culations can only be applied as long as the alloy structure requires only small unit cells for the crystallographic description.It will be demonstrated how this limitation can be overcome without giving up the accuracy of DFT calculations by combin-ing them with so-called Cluster Expansion(CE)methods and Monte-Carlo(MC) simulations.This concept gives access to systems containing more than a million of atoms and therefore,permits to treat alloy properties which possess a delicate temperature-dependence.As examples,we discuss mixing enthalpies,short-range order,and precipitation without any empirical parameters as input,but with an accuracy that allows for the quantitative prediction of experimental results.1IntroductionThe increasing interest in metal alloys comes from their widefield of techni-cal applications ranging from lightweight cars and aircraft turbine blades to protective coatings in corrosive environments and magnetic devices.Beside experimental studies which are sometimes connected with a tremendousfi-nancial effort,more and more computer based approaches attract attention by many research groups.The set of all such theoretical methods forms a new discipline known as“computational materials science”.Indeed,this expres-sion includes a large number of different approaches reaching from contin-uum theory and semi-empirical accesses to so-calledfirst-principles methods, mostly based on density functional theory[1,2].Since the latter is strictly based on quantum-mechanical rules,it has a real predictive power,and–from a physical point of view–seems to be the most elegant solution to treat alloy problems.One should,however,keep in mind that we are already confronted with a many-body problem on a atomic scale and now,have tofind a way, how such calculations may be used to describe properties which are directly connected to the phase stability of an alloy,as e.g.short-range ordering.In principle,this demands to solve the following three problems:(i)The problem of bridging length scales:Many material properties take place on a mesoscopic or even macroscopic scale,i.e.model systems consisting of millions of atoms are necessary–a number far beyond the capacity of today’s computers.B.Kramer(Ed.):Adv.in Solid State Phys.44,415–426(2004)c Springer-Verlag Berlin Heidelberg2004416Stefan M¨u ller(ii)The problem of controlling huge configuration spaces:The validity of a pre-dicted ground-state diagram sensitively depends on the number of considered pared to the2N configurations of a binary alloy system con-taining N atoms,DFT calculations are restricted to a tiny part of this con-figuration space.So,normally a set of“intuitive candidates”is chosen and that with the minimum energy is postulated as ground-state.Naturally,this procedure fails to allow for surprises.(iii)The problem of treating configurational entropy:Ordering phenomena of alloys are temperature-dependent.This dependence cannot be treated by methods which only search the positional space.Instead,the configurational space must be taken into account allowing for exchange processes between individual atoms.It will be shown,how these problems can be overcome by combining DFT-based methods with concepts from statistical physics.Our focus will be on binary metal alloy systems.The structure of the present work is as follows: First the theoretical methods will be introduced(Section2).Then,the con-cept will be applied to two different alloy properties,namely precipitation (Section3)and short-range order(Section4).Finally,the limits and future possibilities will be briefly discussed in form of a summary(Section5).2MethodsThe concept of our theoretical approach is shown in Fig.1.The starting point builds DFT which is based on the Hohenberg-Kohn-theorem[1]stating that the energy of a system of interacting electrons in an external potential depends only on the ground state electronic density.In our case,namely the investigation of solid structures,the external potential is the Coulomb poten-tial caused by the nuclei in a solid.Since there exists a number of excellent review articles[3,4]and books[5,6,7]about DFT,no further details are given here.In a second step,the accuracy of DFT is extended to huge configura-tion spaces by combining DFT with concepts from statistical mechanics.The basic idea by Sanchez,Ducastelle and Gratias[8]is called“Cluster Expan-sion”(CE),and sketched in Fig.2:For a given underlying lattice,the crystal structure is divided into characteristicfigures such as pairs,triangles,etc. Then,the energy of any configurationσon this lattice can be uniquely writ-ten[8]as linear combination of the characteristic energies J of each individual figure.In practice,the only error we make is that the sum must be truncated at some point.TheΠ’s in Fig.1are structure-dependent factors.In practice,the construction of the so-called effective interactions J is done by an inversion method[11].This demands the calculation of the en-ergies for a set(typically15-50)of geometrically fully relaxed structures via DFT which are then used tofind that set of interactions,{J},which mini-mize the deviation between energies resulting from CE and DFT for all struc-turesσ.This procedure has to result in effective cluster interactions J whichAb-Initio Thermodynamics of Metal Alloys417Fig.1.The combination of DFT plus MSCE[9]plus MC[10]allows for a descrip-tion of phase stability of metal alloys on a mesoscopic scale without any empirical parametersFig.2.The concept of cluster expansions:The crystal is separated in characteristic figures(here,shown for the fcc-lattice).The energy of any configuration can then be written as linear combination of the characteristic energies J f of thefigures on the average only causes an additional prediction error of1-2meV/atom compared to the direct DFT energy.For this,the set of DFT input struc-tures must be varied until the effective interactions becomes stable,i.e.will not longer change,if an input structure is removed or added to the set of structures(for details,see.e.g.Ref.[12]).Moreover,asfirst shown by Laks et al.[13],any CE in real space fails to predict the energy of long periodic coherent superlattices–a prerequisite for studying precipitation on a meso-scopic scale fromfirst-principles.In principle,the problem can be solved by transforming a group of interactions to the reciprocal space which is eas-iest to do for the pair interactions and considering the long-periodic limit separately[13].Finally,this leads to the Hamiltonian of the“Mixed-Space Cluster Expansion(MSCE)”[9,13].Here,any configurationσis defined by418Stefan M¨u llerspecifying the occupations of each of the N lattice sites by an A atom(spin-indexˆS i=−1)or a B atom(ˆS i=+1).The formation enthalpy of any configurationσat its atomically relaxed state is then given by∆H CE(σ)=k J pair(k)|S(k,σ)|2+MBfD f J f¯Πf(σ)+(1)+14x−1k∆E eqCS(ˆk,x)|S(k,σ)|2.Thefirst two terms represent the chemical energy,E chem(often refered to as interfacial energy).Here,thefirst sum describes all possible pairfigures. J pair(k)is the lattice Fourier transform of the real space pair interactions,and S(k,σ)are structure factors.The second sum describes many-bodyfigures, such as triangles,tetrahedra,etc.J f is the real-space effective many-body interaction offigure f,D f stands for the number of equivalent clusters per lattice site and¯Πf(σ)are spin products.The third term,the constituent strain,E CS(often referred to as coherency strain),describes the strain en-ergy necessary to maintain coherency between bulk element A and B along an interface with orientationˆk.It can be calculated by deforming the bulk elements from their equilibrium lattice constants a A and a B to a common lattice constant a perpendicular toˆk.The constituent strain is a function of composition x and directionˆk only,but does not include information about the strength of chemical interactions between A and B atoms.Although the CE ansatz allows to treat physical problems ranging from few up to a million atoms without giving up the accuracy of modern DFT calculations,the concept still needs to be extended tofinite temperatures. This can be done by using the MSCE Hamiltonian,Eq.(1),in Monte-Carlo (MC)simulations(Fig.1).We even can extend thefield of applications to non-equilibrium processes by switching from thermodynamic MC to kinetic MC[14].For this,activation barriers for exchange processes between neigh-bor atoms have to be considered,and the number of possible exchange pro-cesses between atoms must be restricted to small atomic distances.For phase-separating processes,as the formation and aging of precipitates which will be studied next,a MC code was created which is strongly related to the concept of the so-called“residence time algorithm”[15].Here,atoms are forced to perform exchange processes and the time correlated to this process is calcu-lated afterwards[10].This algorithm is especially efficient for simulations of long aging times(many hours or days).3PrecipitationSolid state decomposition reactions like phase separation of an alloy into its constituents,A1−x B x (1−x)A+xB,create so-called precipitates which define an important part of the microstructure of many alloy systems.TheAb-Initio Thermodynamics of Metal Alloys419 early stage of these reactions typically involves the formation of coherent pre-cipitates that adopt the crystallographic lattice of the alloy from which they emerge[16].Coherent precipitates have practical relevance,as they impededislocation motion,and thus lead to“precipitation-hardening”in many al-loys[16,17,18,19].So,their size vs.shape distribution as function of tempera-ture and aging time is of special interest.Despite their importance,precipitatemicrostructures were thus far not amenable tofirst-principles theories,since their description requires“unit cells”containing103−106atoms or more.In experiment,precipitates are formed by quenching a solid solution deep intothe two phase-region of the phase diagram,followed by sample aging as shown in Fig.3for fcc-based Al-rich Al-Zn alloys.The formed coherent precipitates can then be observed via transmission electron microscopy(TEM)[20].The black spots are coherent fcc-Zn precipitates.Following the classical textbook of Khachaturyan[18]the shape of a pre-cipitate is controlled by two competing energies:While the interfacial orchemical energy leads to a compact shape,the strain energy leads to aflatten-ing along the elastically soft direction of the precipitate.Our MSCE Hamilto-nian easily allows for a separation into these two characteristic energy parts: As already mentioned in Section2,thefirst two terms of Eq.(1)includes information about strength and importance of the individual pair and multi-body interactions and therefore,represents the chemical energy of the system. The last term,however,reflects the elastic properties of the alloy.We usedFig.3.Quenching an Al-rich Al-Zn solid-solution into the two phase region of the phase diagram(left),followed by sample aging,leads to the formation of Zn-precipitates which can then be observed via TEM[20]in form of black spots(right)420Stefan M¨u llerthis simple physical picture to analyze the ratio between chemical energy E chem and strain energy E CS,as function of the precipitate size for three different Al-rich fcc based binary alloy systems,namely Al-Li,Al-Cu,and Al-Zn.The precipitate distributions are shown in Fig.4,whereby only the Li,Cu,Zn atoms are shown,respectively.For the MC simulations model sys-tems with up to one million atoms were necessary in order to receive sufficient statistics.In the case of Al-Li,precipitates neverflatten,but always possess a spher-ical shape.This becomes clear by analyzing E chem/E CS for different precipi-tate sizes as shown below the precipitate distribution images in Fig.4:In the case of Al-Li,for all precipitate sizes the chemical energy E chem(white bars) clearly dominates over the strain energy E CS(black bars).The Li atoms seem to form a simple cubic lattice.This is due to the fact that the precipitates themselves show a Al3Li stoichiometry forming the L12structure sketched on the top left corner of Fig.4:While all corners of the unit cell are occupied by Li atoms,all faces are occupied by Al atoms.Since only the Li atoms are displayed in the real-space image,they form a simple cubic lattice.This ob-servation also makes clear that there are practically no anti-phase boundaries within the Al3Li precipitates which would demand the occupation of Al sites by Li atoms.Analyzing E chem/E CS for Al-Cu,wefind the opposite is true than for Al-Li:The platelet-stabilizing strain energy is now the dominating part.So,precipitatesflatten along the elastically soft direction of Cu being the[100]direction[21].In the case of Al-Zn,the situation is more complex: Now,we see a strong size-dependence of the precipitate shape:Zn precipitates up to about2.5nm are more compact,i.e.chemically dominated,while larger precipitates becomes more and more ellipsoidal(strain dominated)[22].Since the most interesting size-shape relation is found for Al-Zn,we will now focus on this system.One remarkable feature of its coherent precipitates is the fact that their short axis is always along the[111](and symmetric equiv-alent)directions.Indeed,at afirst glance,this appears a bit as a surprise, because most fcc elements are elastically soft along the[100]direction,and consequently hard along[111].However,it should not be forgotten that we have to inquire about the stability and elastic properties of an unusual phase: fcc-Zn:While Zn is stable is the hcp structure,fcc-Zn shows an instability when deformed rhombohedrally along[111][23].As a consequent,fcc-Zn pre-cipitatesflatten along this direction[22].This feature allows the definition of a c/a ratio and therefore,a quantitative measure for the description of the precipitate shape as used in many experimental studies and schematically shown on the left side of Fig.5:While a represents the long axis of the ellip-soid(perpendicular to[111]),c is its thickness(parallel to[111]).The size is given by the radius of the associated sphere having the same volume as the corresponding precipitate.Figure5compares the experimental size-shape re-lation for two different aging times and concentrations[24,25,26,27,28]with those predicted from our method.For both temperatures,the agreement isAb-Initio Thermodynamics of Metal Alloys421Fig.4.Size-shape distribution of precipitates in Al-rich fcc based Al-Li,Al-Cu, and Al-Zn alloys(no Al atoms are shown)and their corresponding percentage of strain and chemical energy as function of the precipitate size.Precipitates of Al-Li form the L12structure as shown in the top left cornerFig.5.Shape(c/a)vs.size relation of Zn precipitates for two different tem-peratures.The lines denote the results from our calculations[29],the open points are taken from different experimental studies(exp.1-5correspond to Ref.[24,25,26,27,28])422Stefan M¨u llerexcellent,i.e.our ansatz allows for a quantitative prediction of the size versus shape versus temperature relation of coherent precipitates[29].4Short-Range Order and Mixing EnthalpiesIn general,all solid solutions of binary metal alloys show temperature-depen-dent short-range order(SRO).The MSCE Hamiltonian permit us to calculate SRO quantitatively by use of the Warren-Cowley SRO parameters given byαlmn(x)=1−P A(B)lmnx(2)where P A(B)lmn is the conditional probability that given an A atom at theorigin,there is a B atom at(lmn).For comparison with experimental data, the so-called“shells”lmn are introduced which are defined by the distancebetween A and B atoms in terms of half lattice parameters,(l a2,m a2,n a2),e.g.for an fcc-lattice the nearest-neighbor distance would be described by the shell(110),the second neighbor distance by(200)and so on.The sign ofαindicates whether atoms in a given shell prefer to order(α<0)or cluster(α>0).The SRO parameter may be written in terms of the cluster expansion pair correlations as[30]αlmn(x)= ¯Πlmn −q21−q2(3)where q=2x−1and ¯Πlmn are the mentioned structure-dependent factors (see Section2).In diffraction experiments the diffuse scattering due to SRO is proportional to the lattice Fourier transform ofαlmn(x)[31,32]α(x,k)=n Rlmnαlmn(x)e i·k·R lmn(4)where n R stands for the number of real space shells used in the transform. Equation(3)together with(4)opens the possibility to compare both,ex-perimental and theoretically predicted diffuse diffraction patterns(reciprocal space)and SRO-parameters(real space).As example,Fig.6compares experimental[33]and theoretical SRO pat-terns forα-brass[34].The patterns correspond to a sample with a Zn-concentration of31.1%at T=200◦C.Both patterns show SRO intensitiesaround k= 1140 .This reciprocal space vector is characteristic for the so-called D023structure which was earlier predicted as low-temperature ground state inα-brass[34].The structure belongs to the class of so-called long-periodic superlattices which are formed from the L12structure(which is sketched in Fig.4,top left corner)by anti-phase boundaries along the[001] direction.Their fundamental reciprocal space vector is k= 112M0 with MAb-Initio Thermodynamics of Metal Alloys423Fig.6.SRO-pattern for a Cu-Zn solid solution(T=200◦C,x Zn=0.31)resulting from neutron diffraction experiments[33]and from our calculation[34].In bothcases,SRO peaks around 1140 are visible favoring D023as low-temperature groundstate.Lower part:Corresponding SRO parameters and real space images Chains of Zn-atoms along[001]can be seenbeing the modulation wavelength.So,we see the highest intensity for the LPS structure with M=2being the D023structure.The SRO parameters can now be determined by a Fourier transformation of these SRO patterns.In experiment,they were used to construct a real space image of the alloy by using them as input for an inverse MC approach in order to obtain characteristic interactions[33].Figure6compares the real space structure deduced from experiment[33]and from our calculation.In both cases,chains of Zn atoms are visible along[001],indicating that short-range order is essential for a quantitative correct description of the physical properties of the disordered solid-solution ofα-brass.For a quantitative comparison of the computed SRO with experiment, Fig.6also gives the Warren-Cowley SRO parameterαlmn(Eq.(3))for the first5shells.Considering the fact that the experimental error ofα000amounts to be as large as8%(sinceα000=1.000by definition),the predicted and experimentally determined values agree very well.We see thatα110is nega-tive,indicating that Zn atoms prefer Cu atoms as nearest neighbors.Since SRO plays an important role inα-brass,it should also influence its energet-ics,i.e.the mixing enthalpy of the system should possesses a temperature-dependence.To clarify this,Fig.7compares calculated mixing enthalpies,∆H mix(x,T),for different temperatures with experimental data[35].We424Stefan M¨u llerFig.7.Calculated mixing enthalpies ofα-brass for different temperatures[34]and comparison with experimental data[35](bold line)start with the random alloy which is realized by an unphysically high tem-perature(T=105K),and go down to temperatures where SRO sets in. Comparing the energy curves for the random and the disordered alloy,we see that the calculation neglecting SRO leads to much higher mixing enthalpies. For higher Zn concentrations a good agreement between experiment and cal-culated mixing enthalpies can only be reached,if SRO is taken into account. 5SummaryThe applied combination of quantum mechanics(DFT)and statistical physics (MSCE,MC)certainly represents one of the most successful approaches to study binary alloy properties without any empirical parameters.At the mo-ment,the limitation of the presented access is given by the underlying lattice which does,for example,not permit to study melting processes.Regarding ordering phenomena in the solid phase,the presented results allow for a quan-titative prediction of experimental data.We have focused on two important alloy properties namely precipitation and short-range order(SRO).For the latter,it could be seen that a quantitative experiment-theory agreement can only be achieved,if the influence of SRO on the energetics of the disordered alloy is taken into account.This was demonstrated forα-brass,where an excellent agreement between experimental and calculated mixing enthalpies could be reached.As an important example for a decomposition reaction, precipitation in different fcc-based Al-rich solid solution were studied.Here, we profit by the fact that the approach applied allows to study configurationsAb-Initio Thermodynamics of Metal Alloys425 consisting of millions of atoms.In the case of Al-Zn,it was demonstrated that our prediction even goes through a quantitative comparison of the size-shape-temperature relation with experimental results.The presented method is by no means restricted to the two characteristic ordering phenomena,but can also be applied to e.g.surface problems as adsorbate systems[36]or surface segregation[37].Its application potential is up to now not yet clear.So,for example,after some further developments,it should be possible to describe properties as e.g.nucleation or island growth at surfaces.AcknowledgementsSpecial thanks go to Chris Wolverton,Alex Zunger,Lin-Wang Wang,Walter Wolf,Raimund Podloucky,and Long-Qing Chen for helpful discussions.Part of this work was supported by the US Department of Energy. 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