The hydrogen diffusion in liquid aluminum alloys from ab initio molecular dynamics N. Jakse and A. Pasturel Citation: The Journal of Chemical Physics 141, 094504 (2014); doi: 10.1063/1.4894225 View online: http://dx.doi.org/10.1063/1.4894225 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/141/9?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Dynamic properties and local order in liquid Al-Ni alloys Appl. Phys. Lett. 105, 131905 (2014); 10.1063/1.4896403 Ab initio based understanding of diffusion mechanisms of hydrogen in liquid aluminum Appl. Phys. Lett. 104, 154101 (2014); 10.1063/1.4871469 Atomic structure and transport properties of Cu50Zr45Al5 metallic liquids and glasses: Molecular dynamics simulations J. Appl. Phys. 110, 093506 (2011); 10.1063/1.3658252 Erratum: “Ab initio molecular dynamics simulation of liquid Al 88 Si 12 alloys” [J. Chem. Phys.122, 034508 (2005)] J. Chem. Phys. 129, 089901 (2008); 10.1063/1.2970886 Ab initio molecular dynamics simulation of liquid Al 88 Si 12 alloys J. Chem. Phys. 122, 034508 (2005); 10.1063/1.1833355
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THE JOURNAL OF CHEMICAL PHYSICS 141, 094504 (2014)
The hydrogen diffusion in liquid aluminum alloys from ab initio molecular dynamics N. Jakse and A. Pasturel Sciences et Ingénierie des Matériaux et Procédés, INP Grenoble, UJF-CNRS, 1130, rue de la Piscine, BP 75, 38402 Saint-Martin d’Hères Cedex, France
(Received 4 May 2014; accepted 18 August 2014; published online 5 September 2014) We study the hydrogen diffusion in liquid aluminum alloys through extensive ab initio molecular dynamics simulations. At the microscopic scale, we show that the hydrogen motion is characterized by a broad distribution of spatial jumps that does not correspond to a Brownian motion. To determine the self-diffusion coefficient of hydrogen in liquid aluminum alloys, we use a generalized continuous time random walk model recently developed to describe the hydrogen diffusion in pure aluminum. In particular, we show that the model successfully accounts the effects of alloying elements on the hydrogen diffusion in agreement with experimental features. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4894225] I. INTRODUCTION
Increasing demands for ever lighter and hence more fuelefficient parts are boosting the use of cast aluminum alloys. In the same time, it is necessary to improve the casting specifications, as the presence of casting effects can compromise the use of aluminum alloys in critical applications like in the aerospace and automotive industries. Nowadays, one can hardly imagine such optimization without the help of numerical simulations and a corresponding physical modeling. However, the modeling task is often complicated by the lack of consistent, reliable physical property data. More particularly, the detrimental effects of hydrogen on the mechanical properties of cast aluminum alloys, reducing fatigue performance and total elongation, are known for many years.1, 2 They are mainly related to the formation of porosity during the solidification process and models of H-induced porosity rely on quantities such as hydrogen solubility and diffusion. However, a consensus on the physical mechanisms of the nucleation and growth kinetics of porosity and numerical values of hydrogen self-diffusion coefficients (DH ) in cast aluminum alloys, and even for pure aluminum, is still lacking. Indeed, published values of hydrogen diffusion in liquid aluminum and aluminum alloys are notoriously inconsistent due to experimental difficulties arising with reactivity of aluminum and its alloys above their melting temperatures and convection currents. Two main techniques have been used, namely, a modified Sievert’s technique3 and the capillarity technique4–6 to determine the self-diffusion coefficient of hydrogen in pure liquid aluminum but the four sets of experimental data display an important disparity, especially at low temperatures, i.e., around 1000 K. Anyalebechi7 proposes a linear regression of the experimental results and the hydrogen diffusivity in liquid aluminum follows the Arrhenius-type equation written as
DH (m2 /s) = 1.22 ∗ 10−4 exp
−54.5 (kJ/mol) . RT
0021-9606/2014/141(9)/094504/5/$30.00
(1)
However, Eq. (1) fails to reproduce experimental data obtained using the modified Sievert’s technique3 at low temperature. There are only two sets of published values of the hydrogen self-diffusion coefficient in liquid aluminum alloys, both obtained by the capillarity method. According to Vaschenko et al.,6 addition of copper reduces the hydrogen mobility up to 17% Cu while Chernega et al.5 show that addition of rareearth metals reduces also the mobility of hydrogen and in this case, the effect is more important. It comes from these few results that addition of alloying elements seems to profoundly affect the diffusivity of hydrogen in liquid aluminum and should be carefully studied. Let us mention that same effects occur at high temperatures in solid Al-Li alloys.8 In this case, the decrease of the hydrogen mobility has been attributed to a strong local binding energy between hydrogen and lithium atoms. In view of the technical difficulties involved in measuring diffusion coefficients accurately, theoretical models are essential to understand the physical mechanism underlying the hydrogen diffusion in liquid aluminum and its alloys. More particularly, at high temperatures, all the experimental studies indicate that the hydrogen diffusion in liquid aluminum is roughly two orders of magnitude faster than that of liquid aluminum.7 Let us mention that the fast diffusion or superdiffusivity of hydrogen is also observed in other liquid metals. To explain this superdiffusivity, our current knowledge is based on the Stokes-Einstein (SE) relation that relates the H self-diffusion coefficient, DH , to the viscosity of the molten aluminum,ηAl , namely, DH = CkB T /2π RηAl ,
(2)
where R is the radius of hydrogen atom and C is a numerical constant determined by the hydrodynamic boundary conditions.9 Unfortunately, Eq. (2) significantly underestimates the experimental values of the hydrogen self-diffusion coefficient in liquid aluminum. To go further, sometimes an empirical modification of the SE relation of the form
141, 094504-1
© 2014 AIP Publishing LLC
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094504-2
N. Jakse and A. Pasturel
D = Const/ηα is used, where the value of the exponent is typically fractional.10 The fractional dependence of D is often referred to as a microviscosity effect but its detailed microscopic understanding is not yet available. In a very recent contribution,11 we performed a comprehensive microscopic study of H diffusion in pure liquid aluminum by analyzing hydrogen trajectories calculated by means of ab initio molecular dynamics (AIMD). Our analysis showed that the hydrogen motion in liquid aluminum is divided into “sticking” intervals, in which the hydrogen performs localized vibrations around a site, and “jump” intervals in which the hydrogen executes jumps of various amplitudes to other sites. The striking feature is the broadness of the spatial distribution of jumps since hydrogen can perform jumps that may exceed 15 times the interatomic Al-H distance. These unusual long jumps do not correspond to a Brownian motion and the mean-square displacement of hydrogen presents large deviations from the linear time dependence. As a consequence, it cannot be used to compute the self-diffusion coefficient of hydrogen in liquid aluminum. We showed that the hydrogen motion can be described by a generalized continuous time random walk (CTRW) model, leading to computed self-diffusion coefficients of H in liquid aluminum in fair agreement with experimental data using the modified Sievert’s technique.3 Given that the hydrogen diffusion in pure liquid aluminum can be successfully described by the CTRW model, in this paper, we address the important question of the influence of alloying elements on hydrogen diffusion in liquid aluminum alloys. We choose simple systems but with elements (Cu, Li, rare-earth metals) entering the composition of commercial aluminum alloys. Our results highlight that the spatial distribution of jumps can be strongly affected by the presence of elements that present a strong affinity for hydrogen. In this case, the hydrogen diffusivity is reduced drastically. We believe that our results provide a quantitative explanation of the alloying effects on the H self-diffusion coefficients in liquid aluminum alloys but also demonstrate that ab initio molecular dynamics simulations can be used as an efficient tool to construct a database of the diffusion coefficient of hydrogen in liquid multicomponent commercial aluminum alloys.
J. Chem. Phys. 141, 094504 (2014)
sizes L, namely, L = 16.676 Å for Al90 Cu10 , L = 16.947 Å for Al90 Li10 , and L = 16.947 Å for Al90 La10 . The initial configurations were taken from a well-equilibrated liquid aluminum in which random substitution of some Al atoms with alloying elements was done. The AIMD simulations were run in the canonical ensemble with a time step of 1 fs. As indicated in our preceding work,11 the results are similar to those obtained with a time step of 0.5 ps. This reduces the computational effort by a factor of two, which still amounts approximately to 3 h run per picosecond.19 An equilibrium time of 10 ps was followed by 400 ps production runs. Temperature was controlled using a Nose-Hoover thermostat. The temperature is set to 1000 K to describe the liquid phase in Al90 Cu10 , Al90 Li10 and to 1300 K for Al90 La10 , which corresponds to temperatures just above the liquidus of their respective experimental phase diagram. At the studied temperatures, we consider that hydrogen can be treated as a classical particle,11 in agreement with another theoretical work dealing with hydrogen diffusion in solid aluminum at high temperature.20 III. RESULTS AND DISCUSSION A. Non-Gaussian distributions of hydrogen displacements
As mentioned above, we have studied very recently the dynamical properties of hydrogen in pure liquid aluminum.11 In particular, in analyzing in some detail the self-part of the van Hove function, we have found dynamic signatures typical of a non-Brownian motion. Let us recall here the main results. The self-part of the van Hove function is defined by21 GS (r, t) = δ(r − |ri (t) − ri (0)|),
(3)
where ri (t) denotes the position of particle i at time t. The function GS (r, t) measures the probability that a given particle has undergone a displacement r in a time interval of duration t and can be determined from molecular dynamics simulations as proposed by Rahman22 a long time ago. In Figure 1, we display GS (r, t) for H and Al obtained from our AIMD simulations11 for t = 3 ps and at T = 1000 K.
II. SIMULATION DETAILS
Ab initio molecular dynamics simulations were carried out using the Vienna ab initio Simulation Package (VASP) code.12 The local density approximation13 and the projected augmented wave method14 were used to describe the electronic structure of the system. The plane wave cutoff was set to 241 eV and only the -point sampling was used. We have shown that such approximations reproduce successfully self-diffusion coefficients of liquid aluminum15, 16 as well as the self-coefficient of hydrogen in liquid aluminum.11 We have considered systems of 257 atoms in a cubic box with periodic boundary conditions to describe liquid aluminum alloys, namely, Al90 Cu10 , Al90 Li10 , and Al90 La10 . For the three alloys, the simulation boxes contain 230 Al, 26 Cu, or Li, or La atoms, respectively, and 1 H atom. Experimental densities17, 18 were used to set simulation boxes
FIG. 1. Self-part of the van Hove function at a function of distance for H atoms for aluminum and alloys Al90 Cu10 , Al90 Li10 at T = 1000 K and Al90 La10 at T = 1300 K and for time t = 3 ps. Inset: self-part of the van Hove function at a function of distance for Al atoms at T = 1000 K.
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J. Chem. Phys. 141, 094504 (2014)
B. Random walk analysis
To go further in the understanding of the hydrogen motion at the microscopic level, we considered that hydrogen undergoes a random walk process that enters the theoretical framework of the original CTRW model23 and added to the random process localized vibrations occurring between jumps.11, 24, 25 We have expressed the van Hove function as GS (r, t) =
∞ n=0
p(n, t)f (n, r),
(4)
where p(n, t) is the probability of making n jumps in a time t and f(n, r) the probability to move a distance r in n jumps. From our previous AIMD simulations, we have shown that distribution related to the vibrational process, fvib (r), as well as a distribution related to the jump process, fjump (r), are Gaussian. Then, hydrogen vibrations can be described by the following distribution, fvib (r) = (2π l 2 )3/2 exp(−r 2 /2l 2 ), l2 represents the variance of the amplitude of vibrations, while the spatial distribution of hydrogen jumps is distributed according to fjump (r) = (2π d2 )3/2 exp(−r2 /2d2 ), introducing d2 the variance in the amplitude of the jumps. To define the distribution of times between jumps we assumed that the time of the first jump is given by 1 (t) = (τ 1 )−1 exp (−t/τ 1 ) while the subsequent jumps occur with higher frequency using the distribution 2 (t) = (τ 2 )−1 exp(−t/τ 2 ), τ 1 and τ 2 being the first moments of the time distributions1 (t)and 2 (t), respectively. The van Hove can be then rewritten in the FourierLaplace space,24, 25 (r, t) → (q, s), namely, FIG. 2. (a) Trajectory of the H atom projected in the (x, y)-plane in liquid Al obtained from ab initio MD simulation at T = 1000 K. (b) Time evolution of the y-coordinate.
For both Al and H, the small r behavior of GS (r, t) is not far from a Gaussian distribution, corresponding to quasiharmonic vibrations in the cage formed by neighboring atoms. For Al shown in the inset of Figure 1, it is followed by tail at longer distances which decays as a Gaussian, characteristic of a Fickian diffusion whatever the time. For hydrogen, the large distance behavior is obviously non-Gaussian and therefore the hydrogen trajectory does not follow a Brownian motion at the timescale used in our simulations. As discussed previously,11 this large distance behavior can be related to the hydrogen motion characterized by a sequence of sticking-and-jumping processes as shown in Figure 2. The sticking process occurs when hydrogen is trapped by an aluminum atom because of its binding interaction with it. When another Al atom approaches the hydrogen atom, the hydrogen atom may break its binding interaction and makes a small jump to this other one or moves extremely fast in the cloud of aluminum atoms, due to its small mass. These long jumps that can exceed 15 times the interatomic Al-H distance are at the origin of the deviation of the hydrogen diffusion from a normal diffusion regime into a superdiffusive one.11
GS (q, s) = fvib (q)ϕ1 (s) + f (q)fvib (q)
1 (s)ϕ2 (s) (5) 1 − 2 (s)f (q)
with 1, 2 (s) = 1 − sϕ 1, 2 (s) and f (q) = fvib (q)fj ump (q). Introducing the specific forms of the space and time distributions, fvib (r), fjump (r), 1 (t), and 2 (t) in the above relation for GS (q, s) and expressed in the real space,26 the parameters τ 1 , τ 2 , l, and d are then fitted to reproduce the van Hove function obtained directly from the AIMD simulations. Finally, the diffusion coefficient of hydrogen can be obtained from the zero limit of Eq. (5) with respect to the Fourier and Laplace variables q and s and corresponding to the continuum limit11 DH = (l 2 + d 2 )/τ2 .
(6)
Finite size effects errors due to periodic boundary conditions were estimated27 to be of the order 0.07 Å2 /ps, whatever the alloy. As l is always small with respect to d,DH depends mainly on the variance in the amplitude of the jumps and the characteristic time for the occurrence of these jumps. As d and τ 2 depend implicitly on the strength of the Al-H interaction,11 Eq. (6) leads to an implicit dependence of the diffusion coefficient of hydrogen with the binding interaction of Al-H. Therefore, we can think that adding alloying elements with a strong affinity for hydrogen can modify drastically the hydrogen diffusion in liquid aluminum alloys.
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J. Chem. Phys. 141, 094504 (2014)
the number of process models that are currently used in the aluminum industry that require reliable hydrogen coefficient data. Among the different questions, an important point is the addition of alloying elements in liquid aluminum that presents a strong affinity for hydrogen like lithium or rare-earth metals, often used in liquid multicomponent commercial aluminum alloys. To provide a quantitative analysis of the influence of alloying elements on the hydrogen diffusion, we propose a systematic study using Cu, Li, and La with composition set to 10 at.%. These three alloying elements were selected in view of their technological importance in cast aluminum alloys. In Figure 3, we display the trajectory of hydrogen projected in the (x,y)-plane for the three alloys, namely, Al90 Cu10 , Al90 Li10 , and Al90 La10 . It clearly appears that the behavior of hydrogen in the alloys is similar to that obtained in pure liquid aluminum. We can see, however, that the regions of trajectory densification corresponding to the localized vibrations become more important in Al90 Li10 and Al90 La10 than in pure aluminum or in Al90 Cu10 . In Figure 1, we display the selfpart of the van Hove function of hydrogen for the three alloys and we compare them to that obtained in pure aluminum. For all the alloys, the van Hove function presents an exponential decay, characteristic of a non-Gaussian behavior. Nevertheless, for both Al90 Li10 and Al90 La10 , the extension at larger distances is less pronounced than in the case of pure liquid aluminum and Al90 Cu10 . The calculation of the diffusion coefficients of hydrogen in the three alloys is done within the CTRW model using Eq. (6), as described above. The values are reported in Figure 4 and compared to that in pure liquid aluminum. One can see that the most important reduction is obtained for the liquid Al90 La10 alloy since DH is reduced by roughly a factor of 3. The reduction is less pronounced for liquid Al90 Li10 alloy and the value obtained for Al90 Cu10 alloy is close to that of pure aluminum.
FIG. 3. Trajectory of the H atom projected in the (x, y)-plane in alloys Al90 Cu10 (a), Al90 Li10 (b) at T = 1000 K and Al90 La10 (c) at T = 1300 K.
C. Hydrogen diffusion in liquid aluminum alloys
As discussed in the Introduction, there are few published values of the hydrogen self-diffusion coefficient in liquid aluminum alloys. This is rather surprising, considering
FIG. 4. Self-diffusion coefficient of hydrogen in liquid aluminum and Al90 Cu10 , Al90 Li10 , Al90 La10 alloys as a function of inverse temperature. The dotted and dashed-double dotted lines are, respectively, the regressions for pure liquid Al (Ref. 15), and H in liquid Al AIMD results. The dasheddotted line corresponds to the regression of Eichenauer’s data (Ref. 3).
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N. Jakse and A. Pasturel
The decrease of the spatial extension of the van Hove function and consequently the reduction of the hydrogen diffusion for Al90 La10 and Al90 Li10 alloys can be understood as follows: First, in the sticking-jumping sequence that characterizes the hydrogen motion in liquid aluminum, the rate-determining process is related to the breaking process of the binding interaction11 of Al-H. The attempt frequency for breaking this interaction can be regarded as the frequency of the Al-H vibrations which has been estimated11 to have a mean value of 1.7 × 1013 s−1 . With the presence of alloying elements that present a strong affinity for hydrogen, the “sticking” intervals increase if the cage formed by neighboring atoms has at least one atom with a strong hydrogen affinity. It is the case with Li and La since we find that the frequency of the Li-H and La-H vibrations in liquid Al90 Li10 and Al90 La10 alloys is 2.07 × 1013 s−1 and 2.4 × 1013 s−1 , respectively, both values being greater than that of the Al-H vibrations. Let us mention that the strong affinity of lithium for hydrogen has also been used to explain the increase in the solubility of hydrogen when adding lithium to liquid aluminum.7 For Al90 Cu10 alloy, we find that the frequency of the Cu-H vibrations is equal to that of the Al-H vibrations. The second factor may be attributed to the increase of steric effects. It is known that icosahedral motifs provide the most compact local structures in metallic liquids15, 16 and consequently, an increase of the icosahedral short range order (ISRO) may reduce the hydrogen diffusion in the liquid phase by mainly reducing the length of long jumps. As discussed in Refs. 15 and 16, the degree of icosahedral ordering can be determined from the Honeycutt-Andersen pair analysis technique28, 29 and more particularly by the abundance of 15×× bonded pairs. We find that adding an alloying element like La increases the abundance of 15×× bonded pairs from 27% in pure aluminum to 48% in Al90 La10 alloy while Li and Cu modify only slightly ISRO. In Figure 5, we visualize this effect by plotting the distribution of jumps as a function of the alloying element. As a matter of fact, we find a strong reduction in the Al90 La10 alloy. The largest reduction of DH is explained by the combined effect of both factors.
FIG. 5. Spatial distribution of hydrogen atom in aluminum and Al90 Cu10 , Al90 Li10 alloys at T = 1000 K and Al90 La10 at T = 1300 K and for time t = 3 ps.
J. Chem. Phys. 141, 094504 (2014)
IV. CONCLUSION
In conclusion, we have shown that the hydrogen diffusion in liquid aluminum alloys can be described successfully by the generalized continuous time random walk model recently developed to study the hydrogen diffusion in pure liquid aluminum. This model is based on the analysis of the hydrogen trajectories in the liquid phase calculated by means of ab initio molecular dynamics. As for pure liquid aluminum, the hydrogen trajectories in liquid alloys are characterized by a sequence of “sticking” intervals in which the hydrogen performs localized vibrations around a site and “jumps” intervals in which the hydrogen executes jumps of various amplitudes. We highlight that the hydrogen diffusion can be modified either by the strong affinity of the alloying element for hydrogen like Li or La that favors localized vibrations in the hydrogen motion or by the increase of steric effects that favors the reduction of long jumps via the presence of icosahedral motifs known to be the most compact local structures. We think that our study is a first step for a better understanding of hydrogen diffusion in liquid multicomponent commercial aluminum alloys. ACKNOWLEDGMENTS
We acknowledge the CINES and IDRIS under Project No. INP2227/72914 as well as PHYNUM CIMENT for computational resources. 1 M.
J. Couper, A. E. Neeson, and J. R. Griffiths, Fatigue Fract. Eng. Mater. Struct. 13, 213 (1990). 2 J. F. Major, AFS Trans. 105, 901 (1998). 3 W. Eichenauer and J. Z. Markopoulos, Metallkunde 65, 649 (1974). 4 K. I. Vaschenko, D. F. Chernega, G. A. Remizov, and O. M. Byalik, Izv. Vyssh. Ucheb Zavendenii 1, 50 (1972). 5 D. F. Chernega, M. K. Omurkulova, V. N. Kharchenko, Ya. Gotvyanskii Yu, and L. A. Vasil’eva, Sov. J. Non-Ferrous Met. Res. 9, 89 (1981). 6 K. I. Vaschenko, D. F. Chernega, D. Ivanchuk, O. M. Byalik, and G. A. Remizov, Izv. Vyssh. Ucheb. Zaved. Tsvet Metall 2, 48 (1975). 7 P. N. Anyalebechi, Light Metals 2003, edited by Paul N. Crepeau (The Minerals, Metals, and Materials Society, 2003), p. 857. 8 P. N. Anyalebechi, Met. Trans. B 21, 649 (1990). 9 T. A. Engh, Principles of Metal Refining (Oxford University Press, New York, 1992), p. 114. 10 G. L. Pollack, R. P. Kennan, J. F. Himm, and D. R. Stump, J. Chem. Phys. 92, 625 (1990). 11 N. Jakse and A. Pasturel, Appl. Phys. Lett. 104, 154101 (2014). 12 G. Kresse and J. Hafner, Phys. Rev. B 47, 558 (1993). 13 D. M. Ceperley and B. J. Alder, Phys. Rev. Lett. 45, 566 (1980); J. P. Perdew and A. Zunger, Phys. Rev. B 23, 5048 (1981). 14 G. Kresse and D. Joubert, Phys. Rev. B 59, 1758 (1999). 15 N. Jakse and A. Pasturel, J. Phys.: Condens. Matter 25, 285103 (2013). 16 N. Jakse and A. Pasturel, Sci. Rep. 3, 3135 (2013). 17 J. Brillo, I. Egry, and J. Westphal, Int. J. Mater. Res. 99, 162 (2008). 18 A. Inoue, Prog. Mater. Sci. 43, 365 (1998). 19 The runs were performed on 64 Intel Harpertown cores on Altix high performance computer with infiniband communications. 20 H. Gunaydin, S. V. Barabash, K. N. Houk, and V. Ozolins, Phys. Rev. Lett. 101, 075901 (2008). 21 L. van Hove, Phys. Rev. 95, 249 (1954). 22 A. Rahman, Phys. Rev. 130, 1334 (1963). 23 E. W. Montroll and G. H. Weiss, J. Math. Phys. 6, 167 (1965). 24 P. Chaudhuri, L. Berthier, and W. Kob, Phys. Rev. Lett. 99, 060604 (2007). 25 P. Chaudhuri, Y. Gao, L. Berthier, M. Kilfoil, and W. Kob, J. Phys.: Condens. Matter 20, 244126 (2008). 26 W. Kob, private communication (2013). 27 I.-C. Yeh and G. Hummer, J. Phys. Chem. B 108, 15873 (2004). 28 J. D. Honeycutt and H. C. Andersen, J. Phys. Chem. 91, 4950 (1987). 29 N. Jakse and A. Pasturel, Phys. Rev. Lett. 91, 195501 (2003).
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THE JOURNAL OF CHEMICAL PHYSICS 141, 094504 (2014)
The hydrogen diffusion in liquid aluminum alloys from ab initio molecular dynamics N. Jakse and A. Pasturel Sciences et Ingénierie des Matériaux et Procédés, INP Grenoble, UJF-CNRS, 1130, rue de la Piscine, BP 75, 38402 Saint-Martin d’Hères Cedex, France
(Received 4 May 2014; accepted 18 August 2014; published online 5 September 2014) We study the hydrogen diffusion in liquid aluminum alloys through extensive ab initio molecular dynamics simulations. At the microscopic scale, we show that the hydrogen motion is characterized by a broad distribution of spatial jumps that does not correspond to a Brownian motion. To determine the self-diffusion coefficient of hydrogen in liquid aluminum alloys, we use a generalized continuous time random walk model recently developed to describe the hydrogen diffusion in pure aluminum. In particular, we show that the model successfully accounts the effects of alloying elements on the hydrogen diffusion in agreement with experimental features. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4894225] I. INTRODUCTION
Increasing demands for ever lighter and hence more fuelefficient parts are boosting the use of cast aluminum alloys. In the same time, it is necessary to improve the casting specifications, as the presence of casting effects can compromise the use of aluminum alloys in critical applications like in the aerospace and automotive industries. Nowadays, one can hardly imagine such optimization without the help of numerical simulations and a corresponding physical modeling. However, the modeling task is often complicated by the lack of consistent, reliable physical property data. More particularly, the detrimental effects of hydrogen on the mechanical properties of cast aluminum alloys, reducing fatigue performance and total elongation, are known for many years.1, 2 They are mainly related to the formation of porosity during the solidification process and models of H-induced porosity rely on quantities such as hydrogen solubility and diffusion. However, a consensus on the physical mechanisms of the nucleation and growth kinetics of porosity and numerical values of hydrogen self-diffusion coefficients (DH ) in cast aluminum alloys, and even for pure aluminum, is still lacking. Indeed, published values of hydrogen diffusion in liquid aluminum and aluminum alloys are notoriously inconsistent due to experimental difficulties arising with reactivity of aluminum and its alloys above their melting temperatures and convection currents. Two main techniques have been used, namely, a modified Sievert’s technique3 and the capillarity technique4–6 to determine the self-diffusion coefficient of hydrogen in pure liquid aluminum but the four sets of experimental data display an important disparity, especially at low temperatures, i.e., around 1000 K. Anyalebechi7 proposes a linear regression of the experimental results and the hydrogen diffusivity in liquid aluminum follows the Arrhenius-type equation written as
DH (m2 /s) = 1.22 ∗ 10−4 exp
−54.5 (kJ/mol) . RT
0021-9606/2014/141(9)/094504/5/$30.00
(1)
However, Eq. (1) fails to reproduce experimental data obtained using the modified Sievert’s technique3 at low temperature. There are only two sets of published values of the hydrogen self-diffusion coefficient in liquid aluminum alloys, both obtained by the capillarity method. According to Vaschenko et al.,6 addition of copper reduces the hydrogen mobility up to 17% Cu while Chernega et al.5 show that addition of rareearth metals reduces also the mobility of hydrogen and in this case, the effect is more important. It comes from these few results that addition of alloying elements seems to profoundly affect the diffusivity of hydrogen in liquid aluminum and should be carefully studied. Let us mention that same effects occur at high temperatures in solid Al-Li alloys.8 In this case, the decrease of the hydrogen mobility has been attributed to a strong local binding energy between hydrogen and lithium atoms. In view of the technical difficulties involved in measuring diffusion coefficients accurately, theoretical models are essential to understand the physical mechanism underlying the hydrogen diffusion in liquid aluminum and its alloys. More particularly, at high temperatures, all the experimental studies indicate that the hydrogen diffusion in liquid aluminum is roughly two orders of magnitude faster than that of liquid aluminum.7 Let us mention that the fast diffusion or superdiffusivity of hydrogen is also observed in other liquid metals. To explain this superdiffusivity, our current knowledge is based on the Stokes-Einstein (SE) relation that relates the H self-diffusion coefficient, DH , to the viscosity of the molten aluminum,ηAl , namely, DH = CkB T /2π RηAl ,
(2)
where R is the radius of hydrogen atom and C is a numerical constant determined by the hydrodynamic boundary conditions.9 Unfortunately, Eq. (2) significantly underestimates the experimental values of the hydrogen self-diffusion coefficient in liquid aluminum. To go further, sometimes an empirical modification of the SE relation of the form
141, 094504-1
© 2014 AIP Publishing LLC
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094504-2
N. Jakse and A. Pasturel
D = Const/ηα is used, where the value of the exponent is typically fractional.10 The fractional dependence of D is often referred to as a microviscosity effect but its detailed microscopic understanding is not yet available. In a very recent contribution,11 we performed a comprehensive microscopic study of H diffusion in pure liquid aluminum by analyzing hydrogen trajectories calculated by means of ab initio molecular dynamics (AIMD). Our analysis showed that the hydrogen motion in liquid aluminum is divided into “sticking” intervals, in which the hydrogen performs localized vibrations around a site, and “jump” intervals in which the hydrogen executes jumps of various amplitudes to other sites. The striking feature is the broadness of the spatial distribution of jumps since hydrogen can perform jumps that may exceed 15 times the interatomic Al-H distance. These unusual long jumps do not correspond to a Brownian motion and the mean-square displacement of hydrogen presents large deviations from the linear time dependence. As a consequence, it cannot be used to compute the self-diffusion coefficient of hydrogen in liquid aluminum. We showed that the hydrogen motion can be described by a generalized continuous time random walk (CTRW) model, leading to computed self-diffusion coefficients of H in liquid aluminum in fair agreement with experimental data using the modified Sievert’s technique.3 Given that the hydrogen diffusion in pure liquid aluminum can be successfully described by the CTRW model, in this paper, we address the important question of the influence of alloying elements on hydrogen diffusion in liquid aluminum alloys. We choose simple systems but with elements (Cu, Li, rare-earth metals) entering the composition of commercial aluminum alloys. Our results highlight that the spatial distribution of jumps can be strongly affected by the presence of elements that present a strong affinity for hydrogen. In this case, the hydrogen diffusivity is reduced drastically. We believe that our results provide a quantitative explanation of the alloying effects on the H self-diffusion coefficients in liquid aluminum alloys but also demonstrate that ab initio molecular dynamics simulations can be used as an efficient tool to construct a database of the diffusion coefficient of hydrogen in liquid multicomponent commercial aluminum alloys.
J. Chem. Phys. 141, 094504 (2014)
sizes L, namely, L = 16.676 Å for Al90 Cu10 , L = 16.947 Å for Al90 Li10 , and L = 16.947 Å for Al90 La10 . The initial configurations were taken from a well-equilibrated liquid aluminum in which random substitution of some Al atoms with alloying elements was done. The AIMD simulations were run in the canonical ensemble with a time step of 1 fs. As indicated in our preceding work,11 the results are similar to those obtained with a time step of 0.5 ps. This reduces the computational effort by a factor of two, which still amounts approximately to 3 h run per picosecond.19 An equilibrium time of 10 ps was followed by 400 ps production runs. Temperature was controlled using a Nose-Hoover thermostat. The temperature is set to 1000 K to describe the liquid phase in Al90 Cu10 , Al90 Li10 and to 1300 K for Al90 La10 , which corresponds to temperatures just above the liquidus of their respective experimental phase diagram. At the studied temperatures, we consider that hydrogen can be treated as a classical particle,11 in agreement with another theoretical work dealing with hydrogen diffusion in solid aluminum at high temperature.20 III. RESULTS AND DISCUSSION A. Non-Gaussian distributions of hydrogen displacements
As mentioned above, we have studied very recently the dynamical properties of hydrogen in pure liquid aluminum.11 In particular, in analyzing in some detail the self-part of the van Hove function, we have found dynamic signatures typical of a non-Brownian motion. Let us recall here the main results. The self-part of the van Hove function is defined by21 GS (r, t) = δ(r − |ri (t) − ri (0)|),
(3)
where ri (t) denotes the position of particle i at time t. The function GS (r, t) measures the probability that a given particle has undergone a displacement r in a time interval of duration t and can be determined from molecular dynamics simulations as proposed by Rahman22 a long time ago. In Figure 1, we display GS (r, t) for H and Al obtained from our AIMD simulations11 for t = 3 ps and at T = 1000 K.
II. SIMULATION DETAILS
Ab initio molecular dynamics simulations were carried out using the Vienna ab initio Simulation Package (VASP) code.12 The local density approximation13 and the projected augmented wave method14 were used to describe the electronic structure of the system. The plane wave cutoff was set to 241 eV and only the -point sampling was used. We have shown that such approximations reproduce successfully self-diffusion coefficients of liquid aluminum15, 16 as well as the self-coefficient of hydrogen in liquid aluminum.11 We have considered systems of 257 atoms in a cubic box with periodic boundary conditions to describe liquid aluminum alloys, namely, Al90 Cu10 , Al90 Li10 , and Al90 La10 . For the three alloys, the simulation boxes contain 230 Al, 26 Cu, or Li, or La atoms, respectively, and 1 H atom. Experimental densities17, 18 were used to set simulation boxes
FIG. 1. Self-part of the van Hove function at a function of distance for H atoms for aluminum and alloys Al90 Cu10 , Al90 Li10 at T = 1000 K and Al90 La10 at T = 1300 K and for time t = 3 ps. Inset: self-part of the van Hove function at a function of distance for Al atoms at T = 1000 K.
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N. Jakse and A. Pasturel
J. Chem. Phys. 141, 094504 (2014)
B. Random walk analysis
To go further in the understanding of the hydrogen motion at the microscopic level, we considered that hydrogen undergoes a random walk process that enters the theoretical framework of the original CTRW model23 and added to the random process localized vibrations occurring between jumps.11, 24, 25 We have expressed the van Hove function as GS (r, t) =
∞ n=0
p(n, t)f (n, r),
(4)
where p(n, t) is the probability of making n jumps in a time t and f(n, r) the probability to move a distance r in n jumps. From our previous AIMD simulations, we have shown that distribution related to the vibrational process, fvib (r), as well as a distribution related to the jump process, fjump (r), are Gaussian. Then, hydrogen vibrations can be described by the following distribution, fvib (r) = (2π l 2 )3/2 exp(−r 2 /2l 2 ), l2 represents the variance of the amplitude of vibrations, while the spatial distribution of hydrogen jumps is distributed according to fjump (r) = (2π d2 )3/2 exp(−r2 /2d2 ), introducing d2 the variance in the amplitude of the jumps. To define the distribution of times between jumps we assumed that the time of the first jump is given by 1 (t) = (τ 1 )−1 exp (−t/τ 1 ) while the subsequent jumps occur with higher frequency using the distribution 2 (t) = (τ 2 )−1 exp(−t/τ 2 ), τ 1 and τ 2 being the first moments of the time distributions1 (t)and 2 (t), respectively. The van Hove can be then rewritten in the FourierLaplace space,24, 25 (r, t) → (q, s), namely, FIG. 2. (a) Trajectory of the H atom projected in the (x, y)-plane in liquid Al obtained from ab initio MD simulation at T = 1000 K. (b) Time evolution of the y-coordinate.
For both Al and H, the small r behavior of GS (r, t) is not far from a Gaussian distribution, corresponding to quasiharmonic vibrations in the cage formed by neighboring atoms. For Al shown in the inset of Figure 1, it is followed by tail at longer distances which decays as a Gaussian, characteristic of a Fickian diffusion whatever the time. For hydrogen, the large distance behavior is obviously non-Gaussian and therefore the hydrogen trajectory does not follow a Brownian motion at the timescale used in our simulations. As discussed previously,11 this large distance behavior can be related to the hydrogen motion characterized by a sequence of sticking-and-jumping processes as shown in Figure 2. The sticking process occurs when hydrogen is trapped by an aluminum atom because of its binding interaction with it. When another Al atom approaches the hydrogen atom, the hydrogen atom may break its binding interaction and makes a small jump to this other one or moves extremely fast in the cloud of aluminum atoms, due to its small mass. These long jumps that can exceed 15 times the interatomic Al-H distance are at the origin of the deviation of the hydrogen diffusion from a normal diffusion regime into a superdiffusive one.11
GS (q, s) = fvib (q)ϕ1 (s) + f (q)fvib (q)
1 (s)ϕ2 (s) (5) 1 − 2 (s)f (q)
with 1, 2 (s) = 1 − sϕ 1, 2 (s) and f (q) = fvib (q)fj ump (q). Introducing the specific forms of the space and time distributions, fvib (r), fjump (r), 1 (t), and 2 (t) in the above relation for GS (q, s) and expressed in the real space,26 the parameters τ 1 , τ 2 , l, and d are then fitted to reproduce the van Hove function obtained directly from the AIMD simulations. Finally, the diffusion coefficient of hydrogen can be obtained from the zero limit of Eq. (5) with respect to the Fourier and Laplace variables q and s and corresponding to the continuum limit11 DH = (l 2 + d 2 )/τ2 .
(6)
Finite size effects errors due to periodic boundary conditions were estimated27 to be of the order 0.07 Å2 /ps, whatever the alloy. As l is always small with respect to d,DH depends mainly on the variance in the amplitude of the jumps and the characteristic time for the occurrence of these jumps. As d and τ 2 depend implicitly on the strength of the Al-H interaction,11 Eq. (6) leads to an implicit dependence of the diffusion coefficient of hydrogen with the binding interaction of Al-H. Therefore, we can think that adding alloying elements with a strong affinity for hydrogen can modify drastically the hydrogen diffusion in liquid aluminum alloys.
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N. Jakse and A. Pasturel
J. Chem. Phys. 141, 094504 (2014)
the number of process models that are currently used in the aluminum industry that require reliable hydrogen coefficient data. Among the different questions, an important point is the addition of alloying elements in liquid aluminum that presents a strong affinity for hydrogen like lithium or rare-earth metals, often used in liquid multicomponent commercial aluminum alloys. To provide a quantitative analysis of the influence of alloying elements on the hydrogen diffusion, we propose a systematic study using Cu, Li, and La with composition set to 10 at.%. These three alloying elements were selected in view of their technological importance in cast aluminum alloys. In Figure 3, we display the trajectory of hydrogen projected in the (x,y)-plane for the three alloys, namely, Al90 Cu10 , Al90 Li10 , and Al90 La10 . It clearly appears that the behavior of hydrogen in the alloys is similar to that obtained in pure liquid aluminum. We can see, however, that the regions of trajectory densification corresponding to the localized vibrations become more important in Al90 Li10 and Al90 La10 than in pure aluminum or in Al90 Cu10 . In Figure 1, we display the selfpart of the van Hove function of hydrogen for the three alloys and we compare them to that obtained in pure aluminum. For all the alloys, the van Hove function presents an exponential decay, characteristic of a non-Gaussian behavior. Nevertheless, for both Al90 Li10 and Al90 La10 , the extension at larger distances is less pronounced than in the case of pure liquid aluminum and Al90 Cu10 . The calculation of the diffusion coefficients of hydrogen in the three alloys is done within the CTRW model using Eq. (6), as described above. The values are reported in Figure 4 and compared to that in pure liquid aluminum. One can see that the most important reduction is obtained for the liquid Al90 La10 alloy since DH is reduced by roughly a factor of 3. The reduction is less pronounced for liquid Al90 Li10 alloy and the value obtained for Al90 Cu10 alloy is close to that of pure aluminum.
FIG. 3. Trajectory of the H atom projected in the (x, y)-plane in alloys Al90 Cu10 (a), Al90 Li10 (b) at T = 1000 K and Al90 La10 (c) at T = 1300 K.
C. Hydrogen diffusion in liquid aluminum alloys
As discussed in the Introduction, there are few published values of the hydrogen self-diffusion coefficient in liquid aluminum alloys. This is rather surprising, considering
FIG. 4. Self-diffusion coefficient of hydrogen in liquid aluminum and Al90 Cu10 , Al90 Li10 , Al90 La10 alloys as a function of inverse temperature. The dotted and dashed-double dotted lines are, respectively, the regressions for pure liquid Al (Ref. 15), and H in liquid Al AIMD results. The dasheddotted line corresponds to the regression of Eichenauer’s data (Ref. 3).
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The decrease of the spatial extension of the van Hove function and consequently the reduction of the hydrogen diffusion for Al90 La10 and Al90 Li10 alloys can be understood as follows: First, in the sticking-jumping sequence that characterizes the hydrogen motion in liquid aluminum, the rate-determining process is related to the breaking process of the binding interaction11 of Al-H. The attempt frequency for breaking this interaction can be regarded as the frequency of the Al-H vibrations which has been estimated11 to have a mean value of 1.7 × 1013 s−1 . With the presence of alloying elements that present a strong affinity for hydrogen, the “sticking” intervals increase if the cage formed by neighboring atoms has at least one atom with a strong hydrogen affinity. It is the case with Li and La since we find that the frequency of the Li-H and La-H vibrations in liquid Al90 Li10 and Al90 La10 alloys is 2.07 × 1013 s−1 and 2.4 × 1013 s−1 , respectively, both values being greater than that of the Al-H vibrations. Let us mention that the strong affinity of lithium for hydrogen has also been used to explain the increase in the solubility of hydrogen when adding lithium to liquid aluminum.7 For Al90 Cu10 alloy, we find that the frequency of the Cu-H vibrations is equal to that of the Al-H vibrations. The second factor may be attributed to the increase of steric effects. It is known that icosahedral motifs provide the most compact local structures in metallic liquids15, 16 and consequently, an increase of the icosahedral short range order (ISRO) may reduce the hydrogen diffusion in the liquid phase by mainly reducing the length of long jumps. As discussed in Refs. 15 and 16, the degree of icosahedral ordering can be determined from the Honeycutt-Andersen pair analysis technique28, 29 and more particularly by the abundance of 15×× bonded pairs. We find that adding an alloying element like La increases the abundance of 15×× bonded pairs from 27% in pure aluminum to 48% in Al90 La10 alloy while Li and Cu modify only slightly ISRO. In Figure 5, we visualize this effect by plotting the distribution of jumps as a function of the alloying element. As a matter of fact, we find a strong reduction in the Al90 La10 alloy. The largest reduction of DH is explained by the combined effect of both factors.
FIG. 5. Spatial distribution of hydrogen atom in aluminum and Al90 Cu10 , Al90 Li10 alloys at T = 1000 K and Al90 La10 at T = 1300 K and for time t = 3 ps.
J. Chem. Phys. 141, 094504 (2014)
IV. CONCLUSION
In conclusion, we have shown that the hydrogen diffusion in liquid aluminum alloys can be described successfully by the generalized continuous time random walk model recently developed to study the hydrogen diffusion in pure liquid aluminum. This model is based on the analysis of the hydrogen trajectories in the liquid phase calculated by means of ab initio molecular dynamics. As for pure liquid aluminum, the hydrogen trajectories in liquid alloys are characterized by a sequence of “sticking” intervals in which the hydrogen performs localized vibrations around a site and “jumps” intervals in which the hydrogen executes jumps of various amplitudes. We highlight that the hydrogen diffusion can be modified either by the strong affinity of the alloying element for hydrogen like Li or La that favors localized vibrations in the hydrogen motion or by the increase of steric effects that favors the reduction of long jumps via the presence of icosahedral motifs known to be the most compact local structures. We think that our study is a first step for a better understanding of hydrogen diffusion in liquid multicomponent commercial aluminum alloys. ACKNOWLEDGMENTS
We acknowledge the CINES and IDRIS under Project No. INP2227/72914 as well as PHYNUM CIMENT for computational resources. 1 M.
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