Diffusion behavior in diluted ( Fe,Cr ) alloys: An environment for H diffusion in ferritic steels
aa r X i v : . [ c ond - m a t . m t r l - s c i ] N ov Diffusion behavior in Nickel-Aluminium and Aluminium-Uraniumdiluted alloys
Viviana P. Ramunni
CONICET - Avda. Rivadavia 1917,Cdad. de Buenos Aires, C.P. 1033, Argentina. andDepartamento de Materiales, CAC-CNEA,Avda. General Paz 1499, 1650 San Martín, Argentina. ∗ (Dated: January 10, 2018) Abstract
Impurity diffusion coefficients are entirely obtained from a low cost classical molecular staticstechnique (CMST). In particular, we show how the CMST is appropriate in order to describe theimpurity diffusion behavior mediated by a vacancy mechanism. In the context of the five-frequencymodel, CMST allows to calculate all the microscopic parameters, namely: the free energy of vacancyformation, the vacancy-solute binding energy and the involved jump frequencies, from them, weobtain the macroscopic transport magnitudes such as: correlation factor, solvent-enhancementfactor, Onsager and diffusion coefficients. Also, we report for the first time the behavior of diffusioncoefficients for the solute-vacancy paired specie. We perform our calculations in diluted
N iAl and
AlU f.c.c. alloys. Our results are in perfect agreement with available experimental data for bothsystems and predict that for
N iAl the solute diffuses through a vacancy interchange mechanism, whilefor the
AlU system, a vacancy drag mechanism occursPACS number(s): Diffusion, Numerical Calculations, Vacancy mechanism, diluted Alloys, NiAl andAlU systems.
PACS numbers: ∗ Electronic address: [email protected]; This work was partially financed by CONICET - PIP 00965/2010. . INTRODUCTION The low enrichment of U − Mo alloy dispersed in an Al matrix is a prototype for new ex-perimental nuclear fuels [1]. When these metals are brought into contact, diffusion in the Al/U − M o interface gives rise to interaction phases. Also, when subjected to temperatureand neutron radiation, phase transformation from γU to αU occurs and intermetallic phasesdevelop in the U − Mo / Al interaction zone. Fission gas pores nucleate in these new phasesduring service producing swelling and deteriorating the alloy properties [1, 2]. An importanttechnological goal is to delay or directly avoid undesirable phase formation by inhibiting inter-diffusion of Al and U components. Some of these compounds are believed to be responsiblefor degradation of properties [3]. On the other hand, there is an experimental work [4], thatargues that these undesirable phases have not influence on the mobility of U in Al , based onthe results of the effective diffusion coefficients calculated from the best fit of their permeationexperimental curves.Another technique to study the diffusion of Uranium into Aluminum was based on the max-imum rate of penetration of uranium into aluminum as function of the temperature [5]. Fromthis perspective, the authors also report the activation energy values of Uranium mobility. Inavoiding interdiffusion, Brossa et. al. [6] studied the efficient diffusion barriers that should havea good bonding effect and exhibit a good thermal conductivity at the same time. In this work,deposition methods have been developed and the diffusion behavior of the respective couplesand triplets has been evaluated by metallographic, micro-hardness and electron microprobeanalyses. The practical interest of a nickel barrier is shown by several publications concerningto the diffusion in the systems AlN i , N iU and
AlN iU . The knowledge of the binary system isthe only satisfactory basis for the study of the ternary system, these binary systems are treatedbriefly before proceeding to the ternary. The study of the
N iAl binary system was, limitedto solid samples of the sandwich-type, clamped together by a titanium screw and diffusiontreatments have been carried out. Results from this work, have inspired as to also study the
N iAl together with the
AlU system.Therefore it is important to watch carefully and with special attention the initial microscopicprocesses that originate these intermetallic phases. In order to deal with this problem we startedstudying numerically the static and dynamic properties of vacancies and interstitials defectsin the Al ( U ) bulk and in the neighborhood of a (111) Al/ (001) αU interface using moleculardynamics calculations [7, 8]. Here, we review our previous works [7, 8], performing calculationof three diffusion coefficients, namely: the solvent self diffusion coefficient, the solute tracerdiffusion coefficient and one more, never before calculated in the literature for this alloy, of the2ranium-vacancy paired specie. With this purpose we use analytical expressions of the diffusionparameters in terms of microscopical magnitudes. We have summarized the theoretical toolsneeded to express the diffusion coefficients in terms of microscopic magnitudes as, the jumpfrequencies, the free vacancy formation energy and the vacancy-solute binding energy. Then westarts with non-equilibrium thermodynamics in order to relate the diffusion coefficients withthe phenomenological L -coefficients. The microscopic kinetic theory, allows us to write theOnsager coefficients in term of the jump frequency rates. At this point we follow the procedureof Okamura and Allnat [9], and Allnatt and Lidiard [10].The jump frequencies are identified by the model developed further by Le Claire in Ref. [11],known as the five-frequency model for f.c.c lattices. The method includes the jump frequencyassociated with the migration of the host atom in the presence of an impurity at a first nearestneighbor position. All this concepts need to be put together in order to correctly describe thediffusion mechanism. Hence, in the context of the shell approximation, we follow the techniquein Ref. [10] to obtain the corresponding transport coefficients which are related to the diffusioncoefficients through the flux equations. A similar procedure for f.c.c. structures was performedby Mantina et al. [12] for M g , Si and Cu diluted in Al but using density functional theory(DFT). Also, for b.c.c. structures, Choudhury et al. [13] have calculated the self-diffusion andsolute diffusion coefficients in diluted αF eN i and αF eCr alloys including an extensive analysisof the phenomenological L -coefficients using DFT calculations. Also the authors discuss aboutthe risk induced by radiation on based F eN i and
F eCr alloys.In the present work, we do not employ DFT, instead we use a classical molecular staticstechnique, the Monomer method [14]. This much less computationally expensive method allowsus to compute at low cost a bunch of jump frequencies from which we can perform averagesin order to obtain more accurate effective frequencies. Also, for the first time in the literature,we have calculated the diffusion coefficient of the paired solute-vacancy specie by exploring allthe possibilities of the solute mobility, either via direct exchange solute-vacancy mechanism orby a vacancy drag mechanism in which the solute-vacancy pair migrates as a complex defect.Although we use classical methods, we reproduce the migration barriers for
N iAl using theSIESTA code coupled to the Monomer method [15] using pipes of UNIX for the communications.We proceed as follows, first of all we validate the five-frequency model using the
N iAl systemas a reference case for which there are a large amount of experimental data and numericalcalculations [16, 17]. Since, the
AlU and
N iAl systems share the same crystallographic f.c.c.structure, the presented description is analogous for both alloys. The full set of frequenciesare evaluated employing the echonomic Monomer method [14]. The Monomer [14] is used tocompute the saddle points configurations from which we obtain the jumps frequencies defined3n the 5-frequency model. Here, the inter-atomic interactions are represented by suitable EAMpotentials [7] for the
AlU binary system. For the case of the
N iAl system, our results are inexcellent agreement with the experimental data for both, the solvent self-diffusion coefficientand the solute tracer diffusion coefficient [16, 17]. In this case we found that Al in N i at dilutedconcentrations migrates as free species, confirmed by a weak binding between Al with vacancies( V ). Comparison of the available experimental data of the diffusion coefficient of U diluted in Al , with the diffusion coefficient of the paired U + V complex, show an excellent agreement.From theoretical evidence here presented, and from experimental data in [4], we can infer thatin this alloy a vacancy drag mechanism is likely to occur. Magnitudes as, the strong uranium-vacancy binding, the values of the vacancy wind at high temperatures and negative values ofthe cross L -coefficient, (give us magnitudes that)lead us to this conclusion.The paper is organized as follows. In Section II we briefly introduce a summary of themacroscopic equations of atomic transport that are provided by non-equilibrium thermody-namics [10, 18]. In this way an analytical expression of the diffusion coefficients in binary alloysin terms of Onsager coefficients is presented. In section III, we describe the kinetic theory ofisothermal diffusion process with an emphasis on the magnitudes used later. This allows toexpress the Onsager coefficients in terms of the frequency jumps following the procedure ofAllnatt as in Ref. [10]. Section IV, is devoted to give the way to evaluate the Onsager phe-nomenological coefficients following the procedure of Okamura and Allnat [9] in terms of thejumps frequencies in the context of a multi-frequency model. In Section V, we present expres-sions to evaluate the self diffusion coefficient in terms of so called solvent enhancement factorat first order in the solute concentration ( c S ), and the solute diffusion coefficient is calculatedat zero order in c S . Finally, in Section VI we present our numerical results using the theoreticalprocedure here summarized and showing a perfect accuracy with available experimental data,that is, for the N iAl system. The last section briefly presents some conclusions.Readers trained in this theory, can directly jump to section V.
II. THEORY SUMMARY: THE FLUX EQUATIONS
Isothermal atomic diffusion in multicomponent systems can be described by the theory ofirreversible processes, in which the main characteristic is the rate of entropy production perunit volume S [10], T S = N X k ~J k . ~X k , (1)4here T is the absolute temperature, N the number of components in the system, ~J k describesthe flux vector density, while ~X k is the driving force acting on component k . A linear expressionfor the flux vector ~J k in terms of the driving forces, involves the Onsager coefficients L ij , ~J k = N X i L ki ~X i . (2)The second range tensor L ij is symmetric ( L ij = L ji ) and depends on pressure and temperature,but is independent of the driving forces ~X k . From (2) the st Fick’s law, which describe theatomic jump process on a macroscopic scale, can be recovered. On the other hand, on each k component, the driving forces may be expressed, in abscense of external force, in terms of thechemical potential µ k , so that [10], ~X k = − T ∇ (cid:16) µ k T (cid:17) . (3)Where the chemical potential µ k is the partial derivative of the Gibbs free energy with respectto the number of atoms of specie k , µ k = (cid:18) ∂G∂N k (cid:19) T,P,N j = k = µ ◦ k ( T, P ) + k B T ln( c k γ k ) . (4)with γ k , the activity coefficients, defined in terms of the activity a k = γ k c k and c k the concen-tration of specie k . For an isothermal diffusion process mediated by a vacancy mechanism, andby making use of the elimination of the dependent fluxes, N X i J i = 0 ⇒ N X k =1 L ki = 0 . (5)For the particular case of a binary diluted alloy ( A, S ) containing N A host atoms, N S , soluteatoms (impurities), N V vacancies after some algebra we arrive at the flux expressions, J A = L AA ( X A − X V ) + L AS ( X S − X V ) ; (6) J S = L SA ( X A − X V ) + L SS ( X S − X V ) ; (7) j V = − ( J A + J S ) . (8)Now we come back to the flux equations (6-8) where we will introduce the chemical potentialequations (4) in the driving forces (3). In this way we obtain the generalized st Fick’s law,which includes cross effects: J A = − (cid:18) L AA c A − L AS c S (cid:19) k B T (cid:18) ∂lnγ A ∂lnc A (cid:19) ∇ c A , (9) J S = − (cid:18) L SS c S − L AS c A (cid:19) k B T (cid:18) ∂lnγ S ∂lnc S (cid:19) ∇ c S , (10)5ence, for a binary system the diffusion coefficient of the solvent and the solute S are: D A = k B TN (cid:18) L AA c A − L AS c S (cid:19) φ A = D ⋆A φ A , (11) D S = k B TN (cid:18) L SS c S − L SA c A (cid:19) φ S = D ⋆S φ S . (12)and D V = k B Tc V ( L AA + L SS + 2 L AS ) . (13). D A , D S are commonly known as the intrinsic diffusion coefficients, while D ⋆A and D ⋆S are theisotopic tracers diffusion coefficients that are the magnitudes experimentally measured. D S isthe vacancy diffusion coefficient. In the spite of Gibbs-Duhem relation, X k N k X k = 0 , (14)the thermodynamic factors φ A , φ S are equal: φ A = (cid:18) ∂lnγ A ∂lnc A (cid:19) = φ S = (cid:18) ∂lnγ S ∂lnc S (cid:19) = φ . (15)We are interested in diluted alloys, that is, in the limit c S → where φ = 1 . The solutediffusion coefficient is calculated directly from the intrinsic one through the expression, D ⋆S = D S = 1 c S (cid:18) k B TN L SS (cid:19) ; c S → , (16)while for the solvent D ⋆A , is calculated from (11).In the next sections, we express these last Onsager coefficients in terms of microscopicalatomic jump frequencies. III. THE KINETIC EQUATIONS
In this section we present a brief description of the applicability of the master equation toatomic transport in metals in terms of the spatial distribution of atoms and defects [10]. Thetheory provides specific results to evaluate the atomic Onsager transport coefficients for systemsin which there is an attractive interaction between solute and vacancies. The solute-vacancypair is identified by the subscripts p, q . Where p, q denotes the sites in the lattice where thesolute and vacancy are locate respectivelly. By configuration we mean any distinct orientationof the pair .We suppose that the solute-vacancy defect changes from p to q by thermal activation at arate ω qp . These transition are taken to be Markovian, i.e, the ω qp depend on the initial and finalconfigurations but are independent of all previous transitions. We denote the number density6f defects which are in configuration p at time t by n p ( t ) . For a closed set of configuration p the rate equations for the densities n p ( t ) are, ∂n p ( x, t ) ∂t = − X q = p ω qp n p + X q = p ω pq n q . (17)The first term represents the rate at which the vacancy in p leave the site to all the otherconfigurations q ( q = p ) first neighbors of p . The second is the rate at which the vacancyreaches p from q . Here n p ( t ) = n d p p ( t ) where n d is the defect density independently of itsconfiguration and p p ( t ) is defined as the fraction of all defects that are in configuration p attime t . In matrix notation equation (17) is, dp p ( x, t ) dt = − P p p ( t ) , (18)where p p is a column matrix whose elements are the probabilities of occupation, p p ≡ { p , p , ... } ,while P is defined as follows: P qp = − ω qp ; ( q = p ) (19) P pp = X q = p ω qp . (20)One feature of this equation, that we will be used later, is that we can solve equation (17) interms of a reduced matrix Q , which can be obtained from P such that its matrix elements Q pq are Q pq = P pq − P pq , (21)now the indexes p and q only take positive values, that is, jumps that involve a drift in thepositively defined sense of the principal crystal axis, while jumps in the opposite direction aredenoted by overlineq . In this way, Q is a n × n dimension square matrix, where n is the numberof different configurations in the positive principal crystal axis minus 1, as we will see in nextsection. Under thermal equilibrium P ≡ P (0) , is given by statistical thermodynamics as, p (0) p = exp( − E (0) p /k B T ) P γ exp( − E (0) γ ) /k B T ; ( ∀ p ) (22)in which E (0) γ is the Gibbs energy of the system in state γ . Under the same conditions we writein the steady state ( dp p ( x,t ) dt = 0 ), the principle of detailed balance which gives us the usefulrelation, ω (0) qp p (0) p = ω (0) pq p (0) q ; ( ∀ p, q ) (23)that is, ω (0) qp ω (0) pq = exp( E (0) p − E (0) q ) . (24)7here supraindex (0) denote magnitudes in the thermodynamical equilibrium.From the master equations, it is possible to calculate mean values and second momenta of thebasic kinetic quantities, (ex. x coordinate of a tracer atom) in the thermodynamic equilibrium.Also it is very useful to perform averages in regions that are small in a macroscopic sense, butlarge enough to contain many lattice points. Then, solving the master equation in the linearresponse approximation it is possible to obtain formal expression for the transport coefficientsand to verify the Onsager relations. In this way, the macroscopic flux equations and thetransport coefficients may be expressed in terms of averaged microscopic variables. This isindeed a generalization of the Einstein relations for the Brownian motion. Also, it permits toexpress the Onsager coefficients in terms of the jump frequencies. This procedure is describedin details in [10].It is now very useful to introduce the expressions derived by Franklin and and Lidiard[22] for the Onsager coefficients and kinetic theory, in terms of the reduced Q matrix. Theauthors wrote equations for the fluxes J S and J D ( D can be vacancies, V , or interstitials, I ) interms of thermodynamical forces, which are precisely of the form required by non-equilibriumthermodynamics, then up to second-virial coefficients. The, let the Onsager coefficient obtaineddefined as, k B TN L KM = P p,q a Kqp a Mqp ω qp p (0) p + P d,p a Kdp a Mdp ω dp p (0) p + δ KM p (0) K (1 − z f p (0) K ) P r ( a Kr )) ω K (0) r − P (+) p,q v Kq ( Q − ) qp v Mp p (0) p . (25)The velocity function is defined as, v Mp = X u a Mup ω up + X p a Mdp ω dp . (26)The subscripts K, M each of which may be either A , S or D , where A, S represent the solventor solute atoms, while D stand for the defects that may be either vacancies or interstitials. Weuse the same labeling p , q and u for the paired species as before, and r for unpaired species thatcan be of type S (free solute) or D (free defects). While the label d (second term) takes intoaccount dissociative jumps, that is, it runs on sites that after the jump the defect is unpairedwith the solute. The assumption at which the species are regarded to be paired or free maybe fixed arbitrarily. The jump distance in a p → q transition are represented by a K,Mqp , theytake account the movement of the both, the K and M species. Similarly, a K (0) r are the jumpdistance of the free species, and a K,Mdp , stand for the distance of dissociative jumps.The first term on the right side in (25), is the uncorrelated contribution of transitions fromone paired configuration to another. The second term gives the sum of the two corresponding8ontributions from dissociation and association transition (equal by detailed balance, henceno factor / as in the first term), while the third term is the uncorrelated contribution fromthe free-free transition (corrected by the term in z f for the fact that some movements of theunassociated pair may result in the formation of an associated pair, the contribution of whichhave already been accounted for in the second term).Correlated movements are represented in the fourth term, which contain the Q matrix. K means not K (i.e., if K = V , then K = S and vice versa). Note that the summation onlyruns over those pair configurations in the (+) set, that is, jumps that involve a drift in thepositively defined sense of the principal crystal axis. Although the summations contained inthe velocity analogue v Mp are over all configurations which can be reached in one transitionfrom a configuration p lying in the (+) set. We note that the relevant point is to obtain thereduced Q matrix from Kinetic theory.Below we apply the formalism following the procedure described by Allnat and Lidiard [10]to calculate the Onsager coefficients L AA , L SS and L AS = L SA and therefore the tracer diffusioncoefficients D ⋆A and D ⋆S . IV. THE L -COEFFICIENTS IN THE SHELL APPROXIMATION Here we assume that the perturbation of the solute movement by a vacancy V , is limitedto its immediate vicinity, hence we adopt an effective five frequency model à la Le Claire [11]for f.c.c. lattices, to understand the effect of different vacancy exchange mechanisms on solutediffusion. In such a model, the frequencies jumps ω qp are now denoted only with one index ω i ( i = 0 , , , , ). In Figure 1 the jump rates are indicated as ω i ( i = 1 , , , ). We supposethat all gradient potential and concentration are along a particular crystal principal axis ˆ x .Considering only jumps between first neighbors, for them, w implies in the exchange betweenthe vacancy and the solute, w when the exchange between the vacancy and the solvent atomlets the vacancy as a first neighbor to the solute (positions denoted with circled 1 in figure 1).The frequency of jumps such that the vacancy goes to sites that are second neighbor of thesolute is denoted by ω (sites with circled 2). The model includes the jump rate ω for theinverse of ω . Jumps toward sites that are third and forth neighbor of the solute are denoted by ω ′ and ω ′′ respectively while ω ′ and ω ′′ are used for their respective inverse frequency jumps.The jump rate ω is used for vacancy jumps among sites more distant than forth neighborsof the solute atom. In this context, that enables association ( ω ) and dissociation reactions( ω ), i.e the formation and break-up of pairs, the model include free solute and vacancies tothe population of bounded pairs. It is assumed that a vacancy which jumps from the second9 IG. 1: The five-frequency model of a solute-vacancy pair in a f.c.c. lattice. to the third shell, with ω , never returns (or does so from a random direction). As in Ref. [13]we express ω ⋆ = 2 ω + 4 ω ′ + ω ′′ , (27)and ω ⋆ = 2 ω + 4 ω ′ + ω ′′ . (28)This procedure allows us to usufruct Boquet equations [19], and the technique develloped byAllnatt and Lidiard to evaluate the transport coefficients of dilute solid solutions [10]. Thesix symmetry types of vacancy sites that are in the first coordination shell (first neighbor ofthe solute) or the second coordination shell (sites accessible from the first shell by one singlevacancy jump) are listed in Table I (using the same notation as in Ref. [9]) and plotted inFigure 2. As usual [19], sites that are equally distant from the solute atom S at the origin,and that have the same abscissa (x-coordinate in Fig.2) share the same vacancy occupationprobability n i , n i . Table II resumes the here employed notation. Here, we denote the sitesprobability with n ij where for i = 0 there is only one index i that is given in crescent orderin the distance to the solute atom S . Also, non overlined indexes imply in a positive abscissa,while overlined ones i denote sites with negative x coordinate. For the sites in the x = 0 plane( i = 0 ), the sites are denoted with two subindexes n j , where the second index j is given increscent order of the distance to the solute atom S . Table II denotes the number of differenttypes of sites and the distance of them to the x axis. With this classification, the basic kineticequations (17) for the first coordinated shell approximation [9] in the steady state are written10 ABLE I: Symmetry types for f.c.c. lattice for vacancy-sites at the first 4 nearest − neighbor separationfrom the impurity S at the origin (Ref. [9]). The forth and fifth columns denote the velocity functionsof the solvent v ( A ) p and solute atoms v ( B ) p respectively devided by the spacing parameter a (see textbelow). Symmetry type i vacancy-sites (Ref. [19]) n.n.s. v ( A ) p /a v ( S ) p /a (1 , , , (1 , , , (1 , , , (1 , , (2 ω − ω ⋆ ) ω (2 , , ω ⋆ − ω ) 0 (2 , , , (2 , , , (2 , , , (2 , , ω ⋆ − ω ) 0 (1 , , , (1 , , , (1 , , , (1 , , ( ω ⋆ − ω ) 0(1 , , , (1 , , , (1 , , , (1 , , (2 , , , (2 , , , (2 , , , (2 , , ( ω ⋆ − ω ) 0 TABLE II: Probability of occurrence of the vacancy at a site of the subset n j . n ij (Ref. [19]) n n n n n n n n n n n n n of sites 4 8 4 1 4 4 4 4 4 1 4 8 4separation a a √ a √ a a √ a a √ a a √ a √ a as, ∂n ∂t = − (2 ω + ω + 7 ω ⋆ ) n + ω ⋆ n + 2 ω ⋆ n + 2 ω ⋆ n + ω ⋆ n + 2 ω n + ω n + ω n = 0 ,∂n ∂t = 4 ω ⋆ n − (8 ω + 4 ω ⋆ ) n + 4 ω n = 0 ,∂n ∂t = 2 ω ⋆ n + ω n − (10 ω + 2 ω ⋆ ) n + 2 ω n + ω n = 0 ,∂n ∂t = ω ⋆ n + ω n − (9 ω + 2 ω ⋆ ) n + ω n + ω ⋆ n + ω n + ω n = 0 ,∂n ∂t = ω ⋆ n + 4 ω n + 2 ω n − (8 ω + ω ⋆ ) n = 0 , (29) ∂n ∂t = 2 ω n + 2 ω ⋆ n − (4 ω + 7 ω ⋆ ) n + 2 ω ⋆ n + ω ⋆ n + 2 ω n + 2 ω ⋆ n = 0 ,∂n ∂t = ω n + 2 ω n + 2 ω n − (8 ω + 4 ω ⋆ ) n + ω n + 2 ω n = 0 ,∂n ∂t = 2 ω n + ω n + ω n − (11 ω + ω ⋆ ) n + 2 ω n = 0 , ...where the vertical dots denotes the analogous sets of equations for the overlined indexes ∂n i /∂t = 0 . Hence, the matrix P defined in (20) that stands from equation (29) is suchthat P ∈ R × . Then the reduced matrix Q , whose elements are Q pq , can be obtained from11 (cid:2) (cid:2)(cid:2) (cid:2) (cid:1) (cid:3) (cid:1) (cid:4) (cid:1) (cid:5) (cid:1) (cid:6) (cid:1) (cid:7)(cid:3) (cid:1) (cid:7)(cid:6) (cid:1) (cid:7) (cid:1) (cid:5) (cid:1) (cid:4) (cid:1) (cid:3) (cid:1) (cid:8) (cid:1) (cid:8) (cid:1) (cid:6) (cid:2) FIG. 2: The coordinated shell model in f.c.c. lattice (see Ref. [19]). The different types of symmetriesshown are detailed in Table II. In the figure, blue bullets are the first twelve neighbors sites to thesolute S at the origin. In green the 42 subsequent sites. In red, the third coordinated shell from whichthe vacancy never returns to the second shell. P as, Q pq = P pq − P pq , (30)now the indexes p and q take only positive values, such that Q is a five dimension square matrix,so that Q = (2 ω + 2 ω + 7 ω ⋆ ) − ω ⋆ ω ⋆ − ω ⋆ − ω ⋆ − ω ⋆ (4 ω ⋆ + ω ) − ω − ω ⋆ − ω (2 ω ⋆ + ω ) − ω − ω − ω ⋆ − ω (2 ω ⋆ + ω ) − ω − ω ⋆ − ω − ω ( ω ⋆ + ω ) (31)12he matrix element ( Q − ) of the inverse matrix of Q defines a factor named F , introducedby Manning [23], ( Q − ) = (2 ω + 2 ω + 7 ω ⋆ F ) − . (32)The quantity − F is the fractional reduction in the overall frequency of jumps from a first-shellsite to a second-shell site caused by returns of vacancy to first-shell sites, − F ) = 10 ǫ + B ǫ + B ǫ + B ǫ ǫ + B ǫ + B ǫ + B ǫ + B (33)where ǫ = ω ⋆ /ω and Table III shows the B i coefficients calculated by Koiwa [10, 24] and thatwill be employed in the present calculations. TABLE III: Coefficients in the expression for F for the fivefrequency model calculated by Koiwa [24]. B B B B B B B Ref. [24] 180.3 924.3 1338.1 40.1 253.3 596.0 435.3
For evaluating the L -coefficients, we shall need both the site fraction c p of solute atoms whicha vacancy among their z nearest-neighbor sites, also the fraction of unbounded vacancies c ′ V = c V − c p and of unbound solute atoms c ′ S = c S − c p . These are related through the mass actionequation, namely c p c ′ V c ′ S = z exp( − E b /k B T ) = ω ⋆ ω ⋆ . (34)With E b the binding energy of the solute vacancy pair related to ω ⋆ /ω ⋆ by the use of detailedbalance. Then, if the pairs and free vacancies are in local equilibrium and, since the fractionof solute c S will be much greater than c V and thus also c p , we can express the equilibriumconstant K as, c p c V − c p = zc S exp( − E b /k B T ) ≡ Kc S , (35)and equivalently c p = c V (cid:18) Kc S Kc S (cid:19) . (36)The Onsager coefficients can be entirely obtained from equation (25) in terms of the concen-tration of free and paired species, and in terms of the jump frequency rates ω i . For the case ofbinary alloys in f.c.c. lattices, symmetry arguments and spacial isotropy implies that the onlyneeded coefficients are L AA , L SS and L AS . In this respect, the velocity terms are depicted inthe forth and fifth column of Table I. For the case where the Onsager coefficients are expressedin terms of the five-frequency model, the only required elements of Q − is ( Q − ) all the other13lements appearing in L AS and L AA can be eliminated [10]. Hence, the Onsager coefficients(25) are [9] L AA = N s k B T n c ′ V (1 − c ′ S ) ω + c p ( A (0) AA + A (1) AA ) o (37) L AS = L SA = c p A (1) AS (38) L SS = N s c p ω k B T (cid:26) − ω Ω (cid:27) (39)where s = a A / √ is the jump distance, a A the lattice parameter of solvent A . The concentrationof free solute and vacancy defects are denoted with c ′ S and c ′ V respectively. While, Ω = 2( ω + ω ) + 7 ω ⋆ F. (40)We define the solute correlation factor f S for the bracket in (39) as, f S = 1 − ω ω + ω ) + 7 ω ⋆ F . (41)Completing the definitions in (37) and (38) with, A (0) AA = 4 ω + 14 ω ⋆ (42) A (1) AA = 1Ω (cid:2) − ω ⋆ − ω ) + 14 ω ⋆ (1 − F ) (cid:18) ω − ω ⋆ ω (cid:19) × (cid:26) (3 ω ⋆ − ω ) − ω + ω + 7 ω ⋆ / (cid:18) ω − ω ⋆ ω ⋆ (cid:19)(cid:27)(cid:21) (43) A (1) AS = ω Ω (cid:20) ω ⋆ − ω ) + 14 ω ⋆ (1 − F ) (cid:18) ω − ω ⋆ ω ⋆ (cid:19)(cid:21) . (44)In order to calculate the self diffusion coefficients D ⋆A and D ⋆S we must replace the L -coefficients expressions (37,38,39) in (11,12), that for the diffusion coefficients. V. EXPRESSIONS FOR D ⋆A , D ⋆S AND D ⋆p COEFFICIENTS
A comparison between experimental data and the present simulations are possible with theknowledge of the two tracer diffusion coefficients D ⋆A and D ⋆S . For D ⋆A or equivalently L AA itis necessary to consider the motion of the tracer atom A ⋆ via a vacancy mechanism causedby both, vacancies at first neighbors of S or at the unperturbed lattice sites. The tracer self-diffusion coefficient D ⋆A ( c S ) of the specie A in a diluted alloy with a concentration c S of soluteatoms S , can be written in terms of the self diffusion coefficient D ⋆A (0) , of the specie A in puref.c.c. lattice as, D ⋆A ( c S ) = D ⋆A (0)(1 + b A ⋆ c S ) , (45)14t first order in c S . The solvent enhancement factor, b A , is obtained in terms of the properties ofthe solute-vacancy model. On the other hand, for the pure solution, the self diffusion coefficient D ⋆A (0) is given by [11], D ⋆A (0) = a A c V f ω . (46)where a A is the solvent lattice parameter, f = 0 . is the correlation factor for the self-diffusion in f.c.c. lattices, and c V is the vacancy concentration at the thermodynamical equi-librium. This former is such that, c V = exp − E Vf k B T ! , (47)where T is the absolute temperature, E Vf is the formation energy of the vacancy in pure A . Theentropy terms are set to zero, which is a simplifying approximation. So that, inserting (47) weget D ⋆A ( c S ) = a A f ω exp − E Vf k B T ! . (48)We assume c S → then, we use pure lattice parameters for all our calculations. The solute-enhancement factor b A ⋆ , is obtained by replacing (37) in (11) up to first order in the soluteconcentration. In the particular case of the five-frequency model, the expressions for the On-sager coefficients are (37,38,39). Hence, as in Ref. [10, 11], we get, b A ⋆ = −
19 + 4 ω + 14 ω ⋆ ω (cid:18) ω ⋆ ω ⋆ (cid:19) + 1( ω + ω + 7 ω ⋆ F/ {− − F )( ω − ω ⋆ ) × ( ω + ω + 7 ω ⋆ / ω ω ⋆ (49) + ω ⋆ (3 ω ⋆ − ω ) + 14 ω ⋆ (1 − F ) × (3 ω ⋆ − ω )( ω − ω ⋆ ) ω ω ⋆ (cid:27) . In the diluted limit ( c S → ), D ⋆S is identical to the intrinsic diffusion coefficient D S given by(16) D S = D ⋆S = k B Tn S L SS . (50)Introducing L SS in (39) and the detailed balance equation (34) in (50), we obtain an expressionfor the tracer solute diffusion coefficient, D ⋆S = a ω (cid:18) c p c S (cid:19) × (cid:26) ω + 7 ω F/ ω + ω + 7 ω F/ (cid:27) = a ω (cid:18) c p c S (cid:19) × f S . (51)In (51) we introduce the solute correlation factor f S as, f S = (cid:26) ω + 7 ω F/ ω + ω + 7 ω F/ (cid:27) . (52)15here F was previously defined in (32). In the Le Claire description, D ⋆S can also be expressedas, D ⋆S = a A f S ω exp − E Vf + E b k B T ! . (53)For the drift of solutes in a vacancy flux we shall make contact with the alternative phenomenol-ogy offered by Johnson and Lam [29]. In terms of thermodynamic forces, which are preciselyof the form required by non-equilibrium thermodynamics, up to second-virial coefficients, theflux of solute atoms J B is expressed as J S = − D p ∇ C p + σ V c ′ S D V ∇ c ′ V , (54)The coefficients D p and D V are interpreted as diffusion coefficients of pairs and free vacancies,respectively, while σ V is a sort of cross section for vacancies to induce solute motion. Whenwe insert the appropriate chemical potential gradients (see Franklin and Lidiard [30]) into thethermodynamic flux equation (7), we find that (7) is equivalent to (54) if D p = k B TN c p L SS (55)and σ V D V = k B TN L AS + 2 L SS c p (cid:18) ω ⋆ ω ⋆ (cid:19) . (56)We see that for a vacancy mechanism, solute atoms may only move when they are paired witha vacancy and it is reasonable therefore that D S should be equal to ( c p /c S ) as (12) and (55)require. To obtain σ V from (56), we need the full expressions for L AA and L AS in (37) and (38).If we take this to be the vacancy diffusion coefficient in the perfect solvent lattice, i.e. a Al ω ,we then obtain σ V = 2 ω ω × [(3 + 7 F ) ω ⋆ + 7(1 − F )( ω − ω ⋆ )] ω + ω + 7 ω ⋆ F . (57)We proceed to show the results obtained by direct application of the previous theory, to thestudy of the diffusion of impurities in dilute alloys mediated by a vacancy mechanism.
VI. RESULTS
We present our numerical results for
N iAl and
AlU systems. The interatomic interactionsare represented by suitable EAM potentials [7, 25, 26] for binary systems. For
AlU , the crosspotential has been fitted taking into account the available first principles data [25, 26]. Latticeparameters, formation energies and bulk modulus for each intermetallic compound are wellreproduced. We obtain the equilibrium positions of the atoms by relaxing the structure viathe conjugate gradients technique. The lattice parameters that minimize the crystal structure16nergy are respectively a Ni = 3 . Å and a Al = 4 . Å for
N i and Al solvents. For all the cal-culations we used a christallyte of 2048 atoms, eventually including one substitutional Al atomin N i and one substitutional U atom in Al bulk and a single vacancy in both defective systems.The current calculations have been performed at T = 0 K . In this case, the entropic barrier isignored. Our calculations are carried out at constant volume, and therefor the enthalpic barrier ∆ H = ∆ U + p ∆ V is equal to the internal energy barrier ∆ U .In Table IV, we present our results for the vacancy formation energy ( E Vf ) in pure hosts of N i and Al calculated as, E Vf = E ( N −
1) + E c − E ( N ) , (58)where, E ( N ) for the perfect lattice of N atoms, E ( N − is the energy of the defective system,and E c the cohesion energy. The migration barrier of the vacancy in perfect lattice ( E Vm ), iscalculated with the Monomer method [14], and the activation energy E Q as, E Q = E Vf + E Vm . (59) TABLE IV: Energies and lattice parameters for the pure Al and N i f.c.c. lattices. The first columnspecifies the metal, vacancy formation energy E Vf ( eV ) are shown in the second column. The thirdcolumn displays the migration energies E Vm , calculated from the Monomer method [14]. In the forthcolumn we show the lattice parameter a A (Å). The last column displays the activation energy E Q ( eV ) .Reference Latt. E Vf ( eV ) E Vm ( eV ) a A (Å) E Q ( eV ) Present work Al Al N i
N i
For the case of a diluted alloy, we may consider the presence of solute vacancy complexes, C n = S + V n in which n = 1 st , nd , rd , . . . (see the insets in Fig. V) indicates that the vacancyis a n − nearest neighbors of the solute atom S . The binding energy between the solute and thevacancy for the complex C n = S + V n in a f.c.c. matrix of N atomic sites is obtained as, E b = { E ( N − , C n ) + E ( N ) } − { E ( N − , V ) + E ( N − , S ) } , (60)where E ( N − , V ) and E ( N − , S ) are the energies of a crystallite containing ( N − ) atomsof solvent A plus one vacancy V , and one solute atom S respectively, while E ( N − , C n ) isthe energy of the crystallite containing ( N − ) atoms of A plus one solute vacancy complex17 n = S + V n . With the sign convention used here E b < means attractive solute-vacancyinteraction, and E b > indicates repulsion.We calculate the migration energies E m using the Monomer Method [14], a static technique tosearch the potential energy surface for saddle configurations, thus providing detailed informationon transition events. The Monomer computes the least local curvature of the potential energysurface using only forces. The force component along the corresponding eigenvector is thenreversed (pointing “up hill"), thus defining a pseudo force that drives the system towards saddles.Both, local curvature and configuration displacement stages are performed within independentconjugate gradients loops. The method is akin to the Dimer one from the literature [28], butroughly employs half the number of force evaluations which is a great advantage in ab-initiocalculations.Binding energies in Tables V and VI are displayed, respectively for N iAl and
AlU . Relativeto the
N iAl system, a weak energy interaction, E b , between that vacancy and solute can beobserved in almost all the nearest neighbor configurations. Also, a weak attractive interactionexists between the vacancy an the Al solute atom only at st and th nearest neighbor configu-rations, while is repulsive for the rest of the pairs. The same behavior is observed for the AlU system but in this case, the binding energy of the pair ( U + V ) at first neighbor position, isstrongly attractive. Tables V and VI also display the different type of solute vacancy complex C n = S + V n with its binding energies E b . Also, the same tables, depict the possibles configu-rations and jumps that involve the corresponding C n = S + V n complex with the correspondingjump frequencies. This jumps imply in a migration of the vacancy whose energies are shownfor the direct as well as for the reverse jumps. Relative to the migration barriers, we see that,for N iAl and from Tables V, the vacancy migration barriers E ← m are close to that in the perfectlattice E Vm = 0 . eV . 18 ABLE V: Jumps and frequencies in
N iAl . The first column denotes C n = S + V n where V n meansthat the vacancy is n nearest neighbor of the solute. Binding energy E b is shown in the second column.The jumps are depicted in the third column, while the forth column describes the jump frequency ω i and the configurations involved in each jump. Migration energies E m for direct and reversed jumpsare written in the fifth and sixth column respectively. C n = S + V n E b ( eV ) Config.( F n ) ω i E → m ( eV ) E ← m ( eV ) C -0.06 C ω / / C ω o o C S -0.06 C S ω / / C Sω o o C C ω / / C ω o o C C ω ′ / / C ω ′ o o C -0.001 C ω ′′ / / C ω ′′ o o C − . C ω ⋆ / / C ω ⋆ o o C − . C ω ⋆ / / C ω ⋆ o o C − . C ω / / C ω o o ABLE VI: Jumps and frequencies in
AlU . The columns description is the same as in Table V. C n = S + V n E b ( eV ) Config.( F n ) ω i E → m ( eV ) E ← m ( eV ) C -0.139 C ω / / C ω o o C S -0.139 C S ω / / C Sω o o C C ω / / C ω o o C C ω ′ / / C ω ′ o o C C ω ′′ / / C ω ′′ o o C C ω ⋆ / / C ω ⋆ o o C -0.003 C ω ⋆ / / C ω ⋆ o o AlU the migration barriers are quite different from that in perfect lattice for associ-ation jumps, except for c . In comparison with the N iAl case, the jump C ω ′′ / / C ω ′′ o o , involvesmore than one atom, i.e. is a multiple jump as indicated in the figure in Table VI. In tableVII, we show the migration barriers for more distant neighbors pairs than the forth, with thepurpose to find out from where the jump frequency are similar to that of the perfect crystal ω .In order to obtain the jump frequencies, we assume that the jumps are thermally activated TABLE VII: Jumps beyond the second coordinated shell. The columns denotes the same notation asin Table V, binding energies are shown in the second column. The third column denoted the frequencyrate, where the supra indexes ( ⊥ , ∓ ) on ω implies vacancy jumps perpendicular to, backward or forward ∓ ˆ x respectively. Migration energies for the direct and backward jumps ares shown in column four andfive respectively C n = S + V n E b ( eV ) ω i E → m ( eV ) E ← m ( eV ) C ω ⊥ C → C C ω − C → C C ω +0 C → C ( ⊥ , ∓ ) on ω impliesvacancy jumps perpendicular to, backward or forward ∓ ˆ x respectively. and then the frequencies ω i can be expressed as, ω i = ν exp( − E → m /k B T ) . (61)where E → m are reported in Tables V and VI for both systems. For the prefactor in (61), we usea constant attempt frequency ν = 6 × Hz , taken from Ref. [31] for pure Al . We also use,in terms of the Wert model [32], a temperature dependent attempt frequency [27] as, ν ( T ) = k B Th , (62)21here h is the Planck constant. Also in Tables V and VI, the migration barriers and thecorresponding rate frequency for each jump are shown. For both N iAl and
AlU , Table VIIIpresents the calculated frequencies at two different temperatures. We adopt the Wert model asin Ref. [17, 27], i.e, a temperature dependent pre-exponential factor from(61).
TABLE VIII: Vacancy jump frequencies rate ω i calculated with a temperature dependent attemptfrequency ν ( T ) , at two different temperatures in N iAl and
AlU alloys. The symbol ( ⋆ ) indicates thatwe are calculating the effective frequencies ω ⋆ and ω ⋆ . N iAl AlUT = 800 K T = 1700 K T = 300 K T = 700 Kω i ω i ( Hz ) ω i ( Hz ) ω i ( Hz ) ω i ( Hz ) ω . × . × . × . × ω . × . × . × − . × ω . × . × . × . × ω ⋆ . × . × . × . × ω ⋆ . × . × . × . × It is clear that the inclusion of U in Al has significant influence on the solvent frequencyjumps that the inclusion of Al in N i . This fact may be a consequence of the marked differencebetween the solute and solvent atomic numbers, Z U − Z Al = 92 −
13 = 79 for U diluted in Al ,while it is Z Al − Z Ni = 13 −
28 = − for Al in N i .Once we have calculated the jump frequencies, then the solute correlation factors f S andthe solvent enhancement factors b A can be obtained. We present our results in Table IX, wherewe show for different temperatures, the solvent-enhancement factor b ⋆A calculated from (49),with f = 0 . , and the solute-correlation factor f S from (41), for both F = 1 and F = 1 approximations. Also, table IX, resumes the jump frequencies ratios calculated according tothe five-frequency model of solute-vacancy interaction for pre-exponential factor depending onthe temperature.The solute-correlation factor ( f S ) with T and calculated from (52) and (41) in the F = 1 and F = 1 approximations. They are shown in Table IX and Figures 4 and 5, for N iAl and
AlU respectively. The factor F obtained from equation (33) is also shown.Concerning to the solvent-enhancement factors, ( b A ), calculated from (49), the results areshown together with f S in Table IX, also in Figures 6, 7, respectively for N iAl and
AlU , as afunction of the temperature. In the Le Claire approximations and for F = 1 , b Ni and b Al arepositive with T . For F = 1 , b ⋆A decrease for both N i and Al solvents with respect to the Le22 ABLE IX: Solvent enhancement and solute correlated factors for
N i, Al and
Al, U at different tem-peratures, for both F = 1 and F = 1 approximations. For the solvent enhancement factor b A (columnstwo and three), and for the solute correlated factor f S (columns four and five). The last tree columnsdescribe the jump frequency ratios of the solute − vacancy interaction.Alloy T /K b F =1 Ni ∗ b F =1 Ni ∗ f F =1 Al f F =1 Al ω ω ω ⋆ ω ω ⋆ ω N iAl
700 30.22 24.96 0.78 0.68 3.78 3.79 3.28800 31.24 25.11 0.79 0.69 3.69 3.69 3.26900 25.47 21.05 0.79 0.70 3.19 3.19 3.861000 21.38 18.05 0.79 0.71 2.84 2.83 2.571100 18.34 15.75 0.79 0.72 2.58 2.57 2.361200 16.00 13.93 0.80 0.72 2.38 2.37 2.201300 14.16 12.46 0.80 0.73 2.23 2.22 2.071400 12.66 11.25 0.80 0.73 2.11 2.09 1.961500 11.43 10.24 0.80 0.73 2.01 1.99 1.881600 10.40 9.37 0.80 0.73 1.92 1.91 1.811700 9.52 8.63 0.80 0.74 1.85 1.84 1.74Alloy
T /K b F =1 Al ∗ b F =1 Al ∗ f F =1 U f F =1 U ω ω ω ⋆ ω ω ⋆ ω AlU . × . × . × − . × − . × . × . × . × . × . × − . × − . × . × . × . × . × . × − . × − . × . × . × . × . × . × − . × − . × . × . × . × . × . × − . × − . × . × . × . × . × . × − . × . × . × . × . × . × − . × . × . × . × . × . × − . × . × . × . × . × . × − . × . × . × . × . × . × . × . × Claire approximation (i.e., F = 1 ). It must be taking into account that this difference will behighly diminished in the diffusion coefficient because the enhancement factor is multiplied bythe solute concentration c S , which is low for diluted alloys.The Onsager and Diffusion coefficients were calculated assuming a solute mole fraction of23 IG. 4: Solvent correlation factor f Al ⋆ in the N iAl system as a function of the temperature for both, F = 1 (circles) and F = 1 (squares) approximations. The F factor is denoted with up triangles.FIG. 5: Same as figure 4 for the AlU system. c S = 4 . × − , for both alloys, which corresponds to n Al = 4 . × cm − atoms/cm for N iAl and n U = 3 . × cm − atoms/cm for AlU system. Once calculated L AS and L SS , and following the reasoning in Ref. [13], we also calculate the vacancy wind coefficient G = L AS /L SS = − (1 + L V S /L SS ) . The results are presented in Figures 8 and 9, for N iAl and
AlU systems respectively. We see that if
G < − , L V B is positive, then the vacancy and thesolute diffuse in the same direction as a complex specie [13]. This transport phenomena couldoccur in the
AlU case, due to the strong binding of the U + V pair, while is unlikely to occurfor Al in N i by the opposite argument. The vacancy wind parameter verifies
G > − for N iAl in both, F = 1 and F = 1 approximations, while for Al, U the behavior changes drasticallydepending on the case. If F = 1 , G remains positive, but for F = 1 , G > − as shown inFig. 9 in the temperature range [300 − ◦ C , this being an indication that a vacancy drag24 IG. 6: Solvent-enhancement factor b Ni in the N iAl system as a function of the temperature for both, F = 1 (circles) and F = 1 (squares) approximations.FIG. 7: Solvent-enhancement factor b Al in the AlU system as a function of the temperature for both, F = 1 (circles) and F = 1 (squares) approximations. mechanism can occurs for AlU .The full set of L -coefficients for F = 1 , are displayed in Figs. 10 and 11, against theinverse of the temperature for the N iAl and
AlU respectively. We see that for the
N iAl casethe L -coefficients follow an Arrhenius behavior, which implies a linear relation between thelogarithm of L -coefficients against the inverse of the temperature (see Fig. 10). While for the AlU case, at high temperatures, we can appreciate a slight deviation from the Arrhenius law(see Fig. 11). In Figure 11, the cross L AlU = L UAl coefficient is negative for all the range oftemperature. Now, we are in position to obtain the diffusion coefficients D⋆ A (0) , D⋆ B (0) and D p , for the paired specie. First, we present the ratio of calculated tracer diffusion coefficients D ⋆S /D ⋆A as a function of the inverse of the temperature for the N iAl and
AlU in Figures 1225
IG. 8: Ratio of the vacancy-Onsager coefficients of Al in N i calculated from eqs.(38,39) vs /T forboth, F = 1 (circles) and F = 1 (squares).FIG. 9: Ratio of the vacancy-Onsager coefficients of U in Al calculated from eqs.(38,39) vs /T forboth, F = 1 (circles) and F = 1 (squares). and 13, respectively.The calculated D ⋆A and D ⋆S for F = 1 , using the equations (48) and (51), are shown in Figures14 and 15 respectively for N iAl and
AlU . The diffusion coefficient of the paired specie, D p ,calculated from (55) is also shown. It is import to perform a comparison between theoreticalresults obtained in present work with reliable experimental data. We have verified that thetracer self diffusion coefficient D ⋆A ( c S ) for a diluted alloy is practically equal to that for the puresolvent D ⋆A (0) (i.e., D ⋆A ( c S ) ≃ D ⋆A (0) ). Hence, we can test our results for D ⋆A ( c S ) with the bestestimative of the diffusion parameter for pure solvent, D ⋆L ( A ) , taken from Campbell et al. [16].The authors, have been used weighted means statistics to determine consensus estimators whichrepresents best the available experimental data. They use a Gaussian distribution to represent26 IG. 10: Vacancy-Onsager coefficients vs /T for the N iAl system for F = 1 . Squares denote L AlAl ,empty circles denote L NiNi while L NiAl is described with filled circles.FIG. 11: Vacancy-Onsager coefficients vs /T for the AlU system for F = 1 . Squares denote L UU ,empty circles denote L AlAl while L UAl is described with filled circles. the experimental error used to determine the best estimates of the parameters common to allof the included studies in the parameter D ⋆L ( A ) , the self-diffusivity of species A in pure A givenin cm s − . The best estimate is given through an expression of the form, D ⋆L ( A ) = D A exp( − Q A /RT ) , (63)where D A and Q A from Ref. [16] are dysplayed in Table X for pure N i and Al , R = 8 . J/mol K is the ideal gas constant and T is the absolute temperature and represented by solidlines in Figures 14 and 15.As can be observed, D ⋆L ( A ) fits perfectly with the values of D ⋆A calculated in the present work.For the case of N iAl alloys, in Fig. 14 experimental data of the solute diffusion coefficient areplotted with stars and cruxes respectively for T = [914 − ◦ C [36] and T = [1372 − ◦ C IG. 12: Ratio of the tracer diffusion coefficient D ⋆S /D ⋆A in N iAl vs /T for both, F = 1 (circles) and F = 1 (squares) approximations.FIG. 13: Ratio of the tracer diffusion coefficient D ⋆S /D ⋆A in AlU ) vs /T for both, F = 1 (circles) and F = 1 (squares) approximations. [37]. As we can observe the accuracy with the calculated solute diffusion coefficient D ⋆Al isastonishing, showing that the here employed procedure gives excellent results for calculatingthe diffusion coefficients in diluted f.c.c. alloys. The diffusion coefficient for the paired specie Al + V in N i, Al is also shown.A little more attention we devote to
AlU system. In the literature, we have found experi-mental values for the U diffusion coefficient at infinite dilution in Al [4]. The authors fit theirown experimental results solving numerically the diffusion equation ∂C ( x, t ) ∂t = D ∂ C ( x, t ) ∂x , (64)with boundary condition x = 0; C (0 , t ) = S , where S is the maximum solubility of the28 ABLE X: Parameters involved in the expression for the self-diffusion consensus fit D ⋆L ( A ) , where theparameter A indicates N i or Al hosts. The first column denotes the reference where the values weretaken from. The solvent lattice is indicated in the second column. The third and fourth columnsdenote the preexponential factor D A and the activation energy Q A for equation (63) respectively. Therange of temperatures of the description is referred in column five, while the relative error of the selfdiffusion coefficient is shown in column six. The last column stands for experimental or theoreticalresults. The values were taken from Campbell work [16] and references therein.Ref. Lattice D A ( cm s − ) Q A ( KJ/mol ) T ( ◦ C ) error type[33, 34] N i . . − exp.[35] Al .
137 123 . − exp.[16] N i . .
35 [769 − − D ⋆L ( N i ) [16] Al .
292 129 . − − D ⋆L ( Al ) diffusing specie in the alloy ( U → Al ). They propose a solution for equation (64) as, C ( x, t ) = S [1 − erf ( x/ √ Dt ] . (65)In Table XI we show the results of the fit of experimental data taken from [4], against we willcompare our theoretical results. Also, the authors argue that at infinite dilution the dissolutionof precipitates do not disturb the U process diffusion in Al . In Figure 15, we establish a TABLE XI: Solubility and diffusion of U in Al . D × cm s − ( S × at). T ( ◦ C ) 1 . W t . W t . W t × − % W t
620 1 . ± .
20 1 . ± .
15 1 . ± .
15 1 . ± . ( ± ) ( ± ) ( ± ) ( ± )
600 0 . ± .
08 0 . ± .
07 0 . ± .
15 0 . ± . ( ± ) ( ± ) ( ± ) ( ± )
580 0 . ± .
12 0 . ± .
12 0 . ± .
15 0 . ± . ( ± ) ( ± ) ( ± ) ( ± )
560 0 . ± .
10 0 . ± .
10 0 . ± .
10 0 . ± . ( ± ) ( ± ) ( ± ) ( ± ) comparison with experimental data in Table XI for an Uranium dilution of × − % W t ,which corresponds to C U = 6 . × − . We see that, experimental values are in perfect29greement with D p , contrarily to the N iAl for which our calculations and calculations in Ref.[17] reveal a weak Aluminium-vacany binding, then experimental values of solute diffusion goeswith D ⋆Al .The diffusion of Uranium into Aluminum was also calculated in a study of the maximum rateof penetration of uranium into aluminum in the temperature range − ◦ C as describedin a report from the literature [5]. The maximum values calculated in [5] for the penetrationcoefficient was, K T = x /t = 0 . , . , . × − inch /hr at temperatures of , and ◦ C , respectively. The activation energy Q from the expression K = K exp − Q/RT in thetemperature range T = [200 − is Q = 14 . in calories per mole, R the gas constantin calories per / ◦ C per mole, and T the absolute temperature. K O is the proportionalityconstant. The plot lnK vs /T provides a convenient basis for expressing and comparingpenetration coefficients.Not shown here, but also performed, we recalculate all the microscopical parameters for acrystallite containing atoms using classical molecular static technique, including one soluteatom and a vacancy at first neighbor sites of the solute. We reproduce all the migration barriersand therefore the jump frequency rates.In summary, for pure N i and Al materials, a large amount of experimental data are availablein the literature, which have been summarized by Campbell [16] in a best confidence estimationof the self diffusion coefficient. Our calculations for pure hosts match perfectly well with thisbest estimation when a temperature dependent ν is assumed, although results for a constantvalue of ν , also gives accurate results.Concerning with diluted alloys, our results are in excellent agreement with experiments[17, 33–35] for the tracer diffusion coefficient in N iAl . For the diffusion behavior in
AlU , weonly found in the literature the work by Housseau et al. [4]. Our results when compared withthe experimental data [4], suggest that the diffusion behavior is mainly due to a vacancy dragmechanism.
VII. CONCLUDING REMARKS
In summary, We propose a general mechanism based on first principles for obtaining diffusioncoefficients.The flux equations permits to relates the diffusion coefficients with the Onsager tensor.Non equilibrium thermodynamics allows to write this Onsager coefficients in terms of jumpfrequencies. In this way we could write expressions for the diffusion coefficients only in terms ofmicroscopic magnitudes, i.e. the jump frequencies. This last ones have been calculated thanks30
IG. 14: Tracer diffusion coefficients of Al ( D ⋆Al in open squares) and N i ( D ⋆Ni in open circles) in thealloy. Solid line represents the best estimative of the pure N i self-diffusion coefficient D ⋆L ( N i ) takenfrom Campbell work [16]. Available experimental data, for the Al diffusion coefficient in the alloy, aredisplayed with stars [36] and cruxes [37].FIG. 15: Tracer diffusion coefficients of U ( D ⋆U in open squares) and Al ( D ⋆Al in open circles) in thealloy. Solid line represents the best estimative of the pure Al self-diffusion coefficient D ⋆L ( Al ) , takenfrom Campbell work [16]. Available experimental data, for the U diffusion coefficient in the alloy, aredisplayed with triangles [4]. to to the economic static molecular techniques namely the monomer method.The five frequency model has also been of great utility in order to discriminate the relevantjump frequencies for both the the Le Claire approximation ( F = 1 ) and one more accurate when F neq is considered. Hence, we have calculated the full set of phenomenological coefficientsfrom which the full set of diffusion coefficients are obtained through the flux equation.Although in this work we have performed the treatment for the case of f.c.c. latices where31he diffusion is mediated by vacancy mechanism, a similar procedure can be adopted for othercrystalline structures or different diffusion mechanism (for example, interstitials).We have exemplified our calculations for the particular cases of binary N iAl and
AlU f.c.c.diluted alloys.When a temperature dependent attempt frequency is considered the agreement betweenexperimental data and numerical calculations is excellent while, when we assume a constantattempt frequency is also in very good agreement, but under estimate the experimental value.Negative enhancement factor as observed Al solvent, this could promote an enhancement ofthe solvent diffusion coefficient for less diluted alloys.Finally, the F = 1 and F = 1 approximations yield practically to the same results forthe L ij and D ⋆ in both systems here studied. Differences were observed for the ratios D ⋆B /D ⋆A , L AB /L BB , f B and b A ⋆ , evidently not reflected in the self and solute diffusion coefficients, despitethe notorious differences observed for the strong/weak attractive interaction between the solute U/Al -vacancy diluted in
Al, N i hosts respectively.The vacancy tracer diffusion coefficient for the
N iAl and
AlU system were compared withavailable experimental data obtaining an excellent agreement with the here described theory.Calculations for the diffusion coefficient of the paired specie, shows that a vacancy drag mech-anism could occur for
AlU when F = 1 , but is unlikely to occur for N iAl in both, F = 1 and F = 1 .This opens the door for future works in the same direction where similar procedure will beused that includes interstitial defects. Acknowledgements
I am grateful to A.M.F. Rivas and Joaquín Guillén, for comments on the manuscript. Also, Iam grateful to Martín Urtubey for the Figure 2. This work was partially financed by CONICETPIP-00965/2010 and the CNEA/CAC - Gerencia Materiales.
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