- Email: [email protected]
Surface gradients in cemented carbides from first-principles-based multiscale modeling: Atomic diffusion in liquid Co
International Journal of Refractory Metals & Hard Materials 66 (2017) 174–179
Contents lists available at ScienceDirect
International Journal of Refractory Metals & Hard Materials journal homepage: www.elsevier.com/locate/IJRMHM
Surface gradients in cemented carbides from first-principles-based multiscale modeling: Atomic diffusion in liquid Co
MARK
Martin Walbrühla,⁎, Andreas Blomqvistb, Pavel A. Korzhavyia, C. Moyses Araujoc a b c
Department of Materials Science and Engineering, Royal Institute of Technology, SE-100 44 Stockholm, Sweden Sandvik Coromant R & D, SE-126 80 Stockholm, Sweden Materials Theory Division, Department of Physics and Astronomy, Uppsala University, SE-751 20 Uppsala, Sweden
A R T I C L E I N F O
A B S T R A C T
Keywords: ICME Diffusion Ab initio molecular dynamics Liquid co Cemented carbides DICTRA
The kinetic modeling of cemented carbides, where Co is used as binder element, requires a detailed diffusion description. Up to now, no experimental self- or impurity diffusion data for the liquid Co system have been available. Here we use the fundamental approach based on ab initio molecular dynamics simulations to assess diffusion coefficients for the liquid Co system, including six solute elements. Our calculated Co self-diffusion coefficients show good agreement with the estimates that have been obtained using scaling laws from the available literature data. To validate the modeling method, we performed one set of calculations for liquid Ni self-diffusion, where experimental data are available, showing good agreement between theory and experiments. The computed diffusion data were used in subsequent DICTRA simulations to model the gradient formation in cemented carbide systems. The results based on the new diffusion data allows for correct predictions of the gradient thickness.
1. Introduction Integrated Computational Materials Engineering (ICME) uses modeling tools that describe the material at several relevant structural levels ranging from atomic to macroscopic scale. This novel concept offers new opportunities to integrate fundamental research with industrially related computational engineering [1]. One such opportunity is to use theoretical modeling approaches based on quantum mechanics to investigate materials' properties that are difficult to measure experimentally. Among such properties are, for example, atomic diffusion coefficients in liquid metals [2]. Due to their good cutting performance, coated cemented carbides are of great interest for the metal cutting industry as materials for inserts. The base material for such inserts consists of a primary hard phase of hexagonal tungsten carbide WC and a secondary hard phase of a cubic carbide (or carbonitride). The hard particles are embedded into the metallic matrix of a binder phase (usually Co based), which provides the toughness to the cemented carbide system [3,4]. Industrial requirements concerning performance of cemented carbides are constantly increasing, and this calls for a better understanding of the influence of the binder phase on the cemented carbide properties [5,6]. Atomic diffusion in the binder phase, during manufacturing of the material, has important influence on the properties of the product. The
⁎
data from direct measurements of self-diffusion and impurity diffusion coefficients in liquid Co are unavailable, so the modeling can only be based on the values obtained from various estimates [3]. This work uses ab initio molecular dynamics (AIMD) simulations to obtain the lacking diffusion data for liquid Co and selected Co-based alloys. To increase cutting tool performance, cemented carbides are coated with hard coatings using physical vapor deposition (PVD) or chemical vapor deposition (CVD). A too large difference in thermal expansion coefficients between the base material and a CVD-type coating can cause cracking in the coatings after the deposition and the subsequent cooling. Due to the brittle nature of cemented carbides with a hard phase of cubic carbonitride, the cracks can propagate from the coating into the bulk material and cause failure. To prevent this, cemented carbides with a composition gradient (a tougher surface layer free from the cubic carbonitride phase) have been developed [7,8,9]. Sintering in a de-nitrogenizing atmosphere creates a gradient in the chemical potential of N. This causes the outward diffusion of N from the bulk material into the furnace atmosphere, which is the driving force for gradient formation in cemented carbide systems. Ti, which has a high affinity to N, diffuses inwards the bulk, to the remaining regions rich in N, causing the formation of a layer, free of the fcc phase. Looking from the materials design perspective, the gradient formation can be predicted using a software tool for modeling diffusion in
Corresponding author. E-mail address: [email protected] (M. Walbrühl).
http://dx.doi.org/10.1016/j.ijrmhm.2017.03.016 Received 24 November 2016; Received in revised form 24 February 2017; Accepted 24 March 2017 Available online 25 March 2017 0263-4368/ © 2017 Elsevier Ltd. All rights reserved.
International Journal of Refractory Metals & Hard Materials 66 (2017) 174–179
M. Walbrühl et al.
For assessing the diffusion by first-principles methods, the mean square displacement (MSD) method was used. In this method the displacement of atoms with time is calculated to obtain the diffusion coefficients for the observed elements using the Einstein relation [12,23,24]. By performing the AIMD simulations at different temperatures, the thermally activated process of diffusion can be fitted to an Arrhenius relation [25]. By using Eq. (1),
multicomponent alloys (such as DICTRA software [10]), which enables for the development of tailor-made materials. An accurate description of the kinetics behind the gradient formation requires that the databases behind the software contain accurate diffusion data for the elements dissolved in the binder phase. Since direct experimental data for liquid Co are not available, the purpose of the present work is to employ the computational modeling approach based on AIMD to obtain the missing diffusion data. Furthermore, we aim to demonstrate that combining atomistic ab initio simulations with continuum modeling, at larger length scales, has the potential to meet the industrial requirements for materials research and development.
⎛ −Q ⎞ ⎟ D = M0 exp ⎜ ⎝ RT ⎠
(1)
2. Methods
given in the form of a Arrhenius relation, it is possible to obtain both the frequency factor M0 and the activation energy Q which can be stored in a kinetic database for DICTRA [10].
2.1. AIMD simulations
2.2. DICTRA simulations
The supercell for the ab initio molecular dynamics simulations was chosen to contain 108 atoms which were randomly placed into the lattice sites. Since Co was used as the matrix phase, the initial lattice was chosen to have the face-centered cubic (fcc) structure for T > 661 K [11]. Two sets of alloy configurations were simulated: Co self-diffusion and binary Co alloy systems. The first simulation was performed for a configuration with 108 Co atoms (100 at.% Co) which is used to investigate the self-diffusion of Co. The binary diffusion simulations were carried out for Co-based binary alloy systems with W, C, N, Ti, Ta, or Nb to obtain the diffusion coefficient of the alloying element as well as to investigate how the diffusion coefficient of the Co host (with 97 atoms which equals 90 at.%) is affected by the alloying. The AIMD simulations were performed within the canonical NVT ensemble [12,13] using the projector augmented wave (PAW) method [14,15] as implemented in the Vienna ab-initio simulation package VASP [16]. The GGA-PBE was chosen to describe the exchange correlation functional. Initially, the simulations were run at 6000 K to ensure that the systems were completely molten without remains of the initial crystal structure. In total, four different temperatures T were considered for the evaluation: 1903 K, 2003 K, 2103 K and 2303 K. The temperature was controlled using a Nosé thermostat [13] and the subsequent Nosé frequencies were adjusted to be ~8 THz, which appears to be an adequate choice considering at the fcc-Co phonon spectrum [17,18]. To verify the results obtained with the thermostat, rescaling of the velocities [13] was also used in selected cases. For each composition and temperature, the supercell volume V was calculated using the Thermo-Calc software [19] and then imposed as a constraint on the supercell by fixing the lattice constants. The electronic temperature was controlled using the Fermi function smearing implemented in VASP. The electronic structure calculations were spin-polarized, with floating magnetic moments to take into account the magnetism of Co resulting in an average magnitude of ~ 1.6 μB/atom for the magnetic moment on the Co atoms in pure liquid Co. The moments were treated as collinear and were allowed to adjust their magnitude and orientation to the local atomic environment during the AIMD runs. No special treatment of the paramagnetic state was made in the simulations, relying on the experimentally well-established fact that magnetic ordering has no noticeable effect neither on self-diffusion, nor on impurity diffusion, coefficients in solid Co [20] (in contrast to those in solid Fe where the effect of magnetic ordering is strong [21]). The timestep in molecular dynamics simulations was, in general, kept to be 2 fs; the time averages were evaluated using the data from about 2 · 104 timesteps for each tested configuration. To balance the high computational cost of performing long simulations for large supercells, we had to evaluate the momentum-space integrals using a single k-point (Gamma-point), which will be discussed later in more details. More information regarding the setup of the present AIMD simulations can be found in [22].
The DICTRA simulations are performed using the homogenization model [26], which is suitable for performing diffusion simulations of alloy systems having dispersed phases inside a matrix phase, like cemented carbide systems. The kinetic database was assembled with the individual mobility values based on the AIMD simulations, and the thermodynamics used was according to the TCFE7 database [19]. 3. Assessment of diffusion coefficients The following section deals with the diffusion values presented in Fig. 1. The data on Co self-diffusion is compared with theoretical literature data for the Co system and experimental data on Ni selfdiffusion. Furthermore, the results of the assessed binary diffusion values are presented. The results of the present calculations of the liquid Co self-diffusion coefficient are presented in Fig. 1a where they are compared with the available literature data. To the authors' knowledge, no results of direct experimental measurement of the Co self-diffusion coefficient have hitherto been published. One general problem that complicates experimental investigations of diffusion in liquid metals is mass transfer by convection. The difficulty to distinguish between the diffusion and convection contributions to the measured mass transfer often invalidates the results of direct measurements [2]. Han et al. [27] performed experimental measurements of the surface tension for pure liquid Co, from which the Co self-diffusion coefficient was obtained as a by-product. Also available are calculated values of diffusion coefficients, which have been derived by more theoretical methods. A scaling law that relates the excess entropy and atomic diffusion coefficient in liquid metals (Rosenfeld [28] and Dzugutov [29]) was used in several studies (Yokoyama [30], Korkmaz and Korkmaz [31] and Yokoyama and Arai [32]) to describe the selfdiffusion in liquid Co. Chen et al. [33] used another approach, based on an Arrhenius relation derived from a modified Sutherland equation, for assessing the Co self-diffusion coefficient and further for assessing the impurity diffusion coefficients of Co-W and Co-Ta liquid alloy systems. LaBrosse et al. [34] used classical molecular dynamics (CMD) simulations, employing an original interatomic potential of embedded atom method (EAM) type, to describe the Co system. The results by Chen et al. and Yokoyama and Arai are in excellent agreement with the self-diffusion data for liquid Co obtained in this work, by means of AIMD calculations. The other values, obtained by either scaling laws or derived from experimental measurements of surface tension, are in good agreement with our results, which gives some confidence in the calculated values. Only the CMD calculated selfdiffusion coefficients are about one half of the value obtained in this work as well as of the values given by other authors. Since this work not only deals with the liquid Co self-diffusion, but also with the impurity diffusion in liquid Co, it is important to assess their accuracy. The only work to report results for liquid Co-based alloy 175
International Journal of Refractory Metals & Hard Materials 66 (2017) 174–179
M. Walbrühl et al.
Fig. 1. Diffusion data for the liquid Co system. Collection of self-diffusion data for liquid Co (a). Comparison of experimental values for liquid Ni self-diffusion with AIMD calculations for Co and Ni (b). Co self-diffusion and binary solute diffusion coefficients calculated by AIMD (c).
[39] are compared with the results of our AIMD simulations for liquid Co and Ni systems. As Fig. 1b shows, the diffusion coefficients of Co and Ni are similar to each other in the considered temperature range. In fact, the good agreement between the experimental and AIMD simulations results for liquid Ni is more important because it indicates that the simulation method yields correct diffusion data for liquid metal systems. The comparison between our results for the Co-W and Co-Ta liquid alloys and those by Chen et al. indicates that the two different theoretical approaches are in good agreement regarding the diffusion coefficients of Co (Table 2). At the same time, the diffusion coefficients of the solute/alloying elements, also given in Table 2, differ by a factor of about 1.4 between the two theoretical studies. In the present AIMD simulations, the concentration of 10 at.% corresponds to 11 solute
Table 1 Comparison between experimental data for liquid Fe and AIMD data for liquid Co. System
W in liquid Fe
C in liquid Fe N in liquid Fe
T [K]
1873 1813 1860 1903 1973 1823 1873
D [1E− 09 m2/s] This work (with diffusion in liquid Co)
Experimental
2.87 2.54 2.82 3.04 3.47 7.35 8.27
10 [35] 2.3 ± 0.07 [36] 2.5 ± 0.015 [36] 2.7 ± 0.07 [36] 3.2 ± 0.1 [36] 6.7–7.9 [37] 4.12 [38]
systems is that by Chen et al. In addition we compare the results of our calculations with experimental diffusion data for liquid Fe and Ni alloys to have an independent point of reference. The comparison is done to show that the values obtained through the AIMD simulations are in a reasonable range compared to a similar metal system where experimental data are available. The data of Table 1 show that the values of diffusion coefficients in the liquid Co system are quite comparable to the experimental values in the liquid Fe-based alloy system, taking into account the scatter of the experimental data (see [35,36]). The comparison with the two sets of experimental data indicates that AIMD simulations may be considered as a suitable option to obtain diffusion coefficients for liquid metal systems. This is further indicated by Fig. 1b where experimental results for the liquid Ni self-diffusion from Ref.
Table 2 Comparison solute diffusion of W and Ta in Co. System
Co-W
T [K]
1830 1865
Co-Ta
176
1830
Element
W Co W Co Ta Co
D [1E− 09 m2/s] This work
at.%
Chen et al. [33]
at.%
2.63 3.62 2.82 3.82 2.3 3.41
10 90 10 90 10 90
2–1.48 3.39–2.36 2 3.35 1.93–1.46 3.42–2.44
5.4–20 94.6–80 8.32 91.68 5.4–20 94.6–80
International Journal of Refractory Metals & Hard Materials 66 (2017) 174–179
M. Walbrühl et al.
The pure Co system was used for such an investigation and it has been shown the LDA exchange correlation function yields ~ 20% higher values. To balance the number of evaluated systems and the corresponding computational costs it was unavoidable to limit the system size and the used k-points. A bigger system would have allowed a larger number of solute atoms to keep the 10 at.%, which would result in overall better statistics – but looking at the R2 values of the MSD it appears to be sufficient high to run simulations without obvious computational anomalies. The reduction of the k-points is a drastic step but has a huge effect on the computational costs. The low k-point sampling introduces an error in the atomic forces at each time step, but since the time step is much smaller than the vibration of the atoms, the introduced error cancels out and does not affect the diffusion behavior. This was tested by performing simulations of a 1 × 1 × 1 and 2 × 2 × 2 k-point system for pure Co. To obtain the equivalent simulation length of 12 ps, the simulation had to run six times longer using the 2 × 2 × 2 k-points while the actual calculated diffusion value was nearly identical. For the investigated seven-component alloy system a 2 × 2 × 2 k-point mesh would have not been feasible with the available resources.
atoms per 97 atoms of the solvent (more information regarding the computational set-up are available in the method section). The MSD method to calculate the diffusion coefficients is based on the statistical averaging of the displacement over all atoms of each species in the simulation box. The number of Co atoms in the supercell is about 9 times greater than the number of solute atoms, which implies much better statistics for the solvent atoms. This could explain why the agreement with the results by Chen et al. is better for pure Co (Fig. 1a), as well as for the Co atoms in alloys, than for the solute elements. As both approaches only give a theoretical diffusion value, that cannot be compared to experimental data, we do not think it would add more value to the overall work to reproduce the exact systems from Chen et al. by the AIMD method. The results of diffusion simulations for binary alloy compositions are presented in Fig. 1c. The simulations show that, at the gradient sintering temperature of 1723 K, non-metallic elements C and N diffuse about two times faster than the metallic elements. At the temperature of 2300 K the difference increases to a factor of about 2.5. In the work reported by Garcia et al. [3], which was based on estimated values for the activation energy Q (65 KJ/mol) and the frequency factor M (9.34 · 10− 8 m2/s), the reductions by a factor of 2 of the mobility of the metallic solute elements resulted in an accurate description of the gradient formation. At the temperature of 1723 K, the self-diffusion of Co is calculated to be the fastest among the metallic elements considered in this work. The metallic solute elements have a gradually slower diffusion compared to the Co solvent, whereas Ta is the slowest element with a ~ 1.7 times slower diffusion. With increasing temperature the differences in diffusion coefficients decrease for the metallic elements. The calculated frequency factors and activation energies, assessed from the evaluation of the Arrhenius like diffusion behavior, are presented in Table 3. Looking at the activation energies no distinct difference between the metallic and non-metallic elements is evident. The difference is more obvious for the frequency factors with greater values for the non-metallic elements. Finally let us discuss about the accuracy and limitations related to this study. For each system four individual temperatures and the related MSD have been evaluated. As mentioned earlier, the MSD is a statistical average over the atomic displacement and thus gives us a chance to look at the linearity of the mean square displacement with the simulation time by linear regression. A linear displacement (high R2 value) would indicate the absence of simulation anomalies and thus a simulation can be considered as relevant for the assessment. The linearity for the diffusion values indicates whether or not the diffusion follows the expected linear Arrhenius behavior for the logarithm of diffusion coefficient as a function of temperature. The R2 values are presented in Table 3 and show generally values of ~0.94 or higher, which gives confidence in the applied methodology. In this work the GGA-PBE exchange correlation function was used for all simulations. It might be of general interest for the reader if the choice of the exchange correlation function has influenced the outcome.
4. Simulation of gradient formation In order to investigate the applicability of the obtained diffusion data we turn to cemented carbide systems which are in need of better kinetic description for the prediction of surface gradients. Diffusion simulations of gradient formation in the cemented carbide system have been treated using the homogenization model in DICTRA [26]. The WC and fcc carbide phases are in a solid state during the gradient sintering. Since the diffusion is assumed to take mainly place in the liquid binder phase, the solid phases are seen as obstacles which reduce the effective diffusion coefficients Deff by a so-called labyrinth factor λ(f) (Eq. (2)) with f being the volume fraction of the binder phase:
Deff = λ (f ) ∙D
In an early work by Ekroth [38] the labyrinth factor λ was taken as proportional to the volume fraction squared (λ ∝ f2) of the binder phase. In later studies [3,4], λ ∝ f was used. In the previous publications the labyrinth factor was fitted to the experimentally measured widths of the gradient zone, but the diffusion coefficients were evaluated based on different sets of estimated values [3,4,40]. Furthermore, the ramping (heating and cooling) during the gradient sintering have been neglected in the evaluations. The present simulations of gradient formation, which make use of the diffusion coefficients obtained in this work, show that neither λ ∝ f nor λ ∝ f2 as choice of the labyrinth factor gives a satisfactory description of the gradient formation. Our test calculations have shown that a λ ∝ f3.3 scaling for the labyrinth factor holds instead, yielding good agreement between the results of gradient simulations and the experimentally measured gradients. Fig. 2 shows the volume fraction of the WC, fcc carbide, and Co binder phases as functions of the depth. The width of the gradient zone measured by Garcia et al. [3] is 23 ± 1 μm after 2 h of sintering. The calculated gradient width, including the heating and cooling (a total sintering time of around 3 h), is 23 μm. By analyzing the effect of the labyrinth factor and the differences in the used diffusion coefficients between different publications, it is reasonable to conclude that the labyrinth factor expression has to be attempted in further research as no clear solution can be found. The values of diffusion coefficients used by Garcia et al. are ~6 times smaller than the values used in the present work. Because of the higher mobility values obtained in this work, the labyrinth factor needs to be smaller (which implies its stronger dependence on the volume fraction) to yield the same effective diffusion coefficient Deff and the same gradient width. Further reason for using a different labyrinth factor is the neglect of the ramping time in the previous studies. Taking into
Table 3 Frequency factors, activation energies and R2 values (avg. MSD and Diffusion) for the investigated systems.
Co Ti Nb Ta W C N
Frequency factor M0 [10− 7 m2/s]
Activation energy Q [kJ/ mol]
Configuration
1.15 1.46 1.28 2.72 1.21 6.60 7.60
52.65 60.62 56.34 72.59 58.28 68.17 70.40
Self-diffusion Binary diffusion " " " " "
R2 MSD
Diffusion
0.9996 0.9967 0.9983 0.9928 0.9959 0.9945 0.99735
0.972 0.937 0.9903 0.9796 0.9507 0.9784 0.9412
(2)
177
International Journal of Refractory Metals & Hard Materials 66 (2017) 174–179
M. Walbrühl et al.
Fig. 2. Width of an fcc-free gradient in cemented carbides. Comparison of the measured [3] (after 2 h of sintering) and the corresponding simulated width of the fcc-free zone with a labyrinth factor of λ ∝ f3.3 and the diffusion data from this work.
slow down the effective Ti diffusion additionally to the geometrical aspect. In turn it might be of interest to include an additional factor in the effective diffusion calculation that cannot be explained by the bulk diffusion and the pure geometrical aspects. As the important liquid diffusivities have been described now by fundamental methods they might be used as a non-adjustable parameter in the assessment of the effective diffusion in cemented carbide systems. In the authors' opinion a combination of carefully planned experiments and advanced 3D modeling, that investigates the geometrical effects that are caused by the realistic composite structure, are necessary to find a satisfying solution to model the effective diffusion. Only a straight forward experimental result allows to identify if, besides the geometry and diffusion, another factor has to be considered to predict the effective diffusion correctly. This knowledge can then be used to eventually give a full multiscale simulation approach for the surface gradient formation in the cemented carbide system.
account the additional sintering time during the ramping will yield a different gradient width, which should be compensated by the change in the labyrinth factor. More elaborated theories [41,42] are available to describe effective properties in composite materials and are thus favorable within the here presented multiscale approach. Hashin and Strikman [43] developed a theory, considering upper and lower bounds, which they used to calculate elastic properties in WC-Co cemented carbide systems. They used spherical shapes to model the two different phases and require the inherent properties of the two pure phases to express the extreme values for the bounds. Similarly Paul [44] described the Young's modulus for the WC-Co system but instead used a cubic shape to model the inclusions. His model requires as well the knowledge about the two phases in order to describe the minimum and maximum of the effective property. The underlying approach of the labyrinth factor, which has been previously used, is that the diffusion only occurs in the binder phases which leave the WC and fcc phase as non-diffusing obstacles. Using the more elaborated theories the maximum in diffusion can be expressed with the here presented binder diffusion values but it is not feasible to define the minimum, for the non-diffusion phases, as zero. Hashin and Strikman point out that their model is only able to estimate the effective property when the ratio between the different phase properties is not too large. Unfortunately, the models appear to be very sensitive to a change in the minimum phase property which does not allow to make reasonable predictions of the effective diffusion in the cemented carbide structure. In addition, the model approaches with spheres or cubes might be too simplified to represent the real prismatic shaped WC grains which will not allow to give an accurate description. It should be emphasized that the theories, like [43], suggest upper and lower bounds giving a theoretical range for the geometrical effect on the effective properties in composite materials. Several authors [41,42,45] indicated that the shape of the obstacle phase will have an influence on the effective transport properties and thus the shape consideration is necessary in order to create tighter bounds that allow more accurate predictions. The labyrinth factor, as previously used for cemented carbide systems, should only contain geometrical information but due the unknown diffusivities it was basically used a fitting parameter that might also contain non-geometrical information. Additional contributions, that do not arise from the geometrical aspects of the two phase structure, might alter the effective diffusion Deff. One of those, so far, not further investigated effects on the gradient formation might arise from additional chemical reactions, like the ability of Ti to form a few atomic layer thin film on the WC grain surface [46]. Due to its particular role in the gradient formation [9] those film formation might
5. Conclusions The use of methods of computational material design can help overcome limitations of empirically-based approaches in the design of new functionally graded materials with improved properties. Here we have demonstrated how different modeling methods that operate on different length scales (and use atomistic and continuum description) can be successfully combined. Ab initio molecular dynamical (AIMD) simulations were employed for calculating the kinetic properties (coefficients of atomic diffusion) in the liquid binder phase of cemented carbides. The results of the simulations were benchmarked against the available literature data and the agreement is found to be encouraging. In particular, the values of diffusion coefficients calculated in this work for Co-based alloys are in good agreement with the available literature data for liquid Co-based alloys, and in reasonable agreement with the experimental values available for the liquid Fe and Ni systems. Taking also into account the good agreement between the experimental AIMD simulated data for liquid Ni, we can conclude that the present simulations have demonstrated applicability of DFT-based modeling methods for solving industrially relevant materials problems. The diffusion simulations of gradient formation themselves bear a large degree of uncertainty. A number of thermodynamic and kinetic parameters have to be used as the input data in such simulations, but the data often come from different sources and have different accuracy. Modeling of diffusion in the cemented carbide systems is additionally complicated by the fact that the carbide particles are obstacles which reduce the effective diffusion coefficient by increasing the diffusion path. The labyrinth factor, only describing geometrical effects, which 178
International Journal of Refractory Metals & Hard Materials 66 (2017) 174–179
M. Walbrühl et al.
[16] J. Hafner, Materials simulations using VASP—a quantum perspective to materials science, Comput. Phys. Commun. 177 (2007) 6–13. [17] O.P. Gupta, Crystal dynamics of face centered cubic cobalt, Solid State Commun. 42 (1982) 31–32. [18] Y.R. Wang, Temperature-dependent phonon dispersion relation in magnetic crystals, Sold State Commun. 54 (1985) 279–282. [19] Thermo-Calc Software, “TCS Steel and Fe - alloys Database”, Vers. 7.0. http://www. thermocalc.com, (2013). [20] Landolt-Börnstein, H. Mehrer (Ed.), New Series Group III, 26 Springer-Verlag, Berlin, 1990(p.51, 74, 131–132, 182). [21] Landolt-Börnstein, H. Mehrer (Ed.), New Series Group III, 26 Springer-Verlag, Berlin, 1990(p.49–51, 73, 124–130, 179–181). [22] M. Walbrühl, Diffusion in the Liquid Co Binder of Cemented Carbides – Ab Initio Molecular Dynamics and DICTRA Simulations, M.Sc. thesis Sweden, Royal Institute of Technology, 2014. [23] M. Hillert, J. Ågren, Diffusion and Equilibria an Advanced Course in Physical Metallurgy, Department of Materials Science and Engineering Royal Institute of Technology, Stockholm, 2002. [24] D.A. Porter, K.E. Easterling, Phase Transformations in Metals and Alloys, second ed., T.J. Press, Cornwall, 1992. [25] M. Hillert, Phase Equilibria, Phase Diagrams and Phase Transformations, Their Thermodynamic Basis, second ed., Cambridge University Press, 2007. [26] A. Borgenstam, A. Engström, L. Höglund, J. Ågren, DICTRA, a tool for simulation of diffusional transformations in alloys, J. Phase Equilib. 21 (2000) 269–280. [27] X.J. Han, N. Wang, B. Wei, Thermophysical properties of undercooled liquid cobalt, Philos. Mag. Lett. 21 (2002) 451–459. [28] Y. Rosenfeld, Relation between Transport-Coefficients and Internal Entropy of Simple Systems, Phys. Rev A 15 (1977) 2545–2549. [29] M. Dzugutov, A universal scaling law for atomic diffusion in condensed matter, Nature 381 (1996) 137–139. [30] I. Yokoyama, Self-diffusion coefficient and its relation to properties of liquid metals: a hard-sphere description, Phys. B Condens. Matter 271 (1999) 230–234. [31] S.D. Korkmaz, Ş. Korkmaz, Investigation of atomic transport and surface properties of liquid transition metals using scaling laws, J. Mol. Liq. 150 (2009) 81–85. [32] I. Yokoyama, T. Arai, Correlation entropy and its relation to properties of liquid iron, cobalt and nickel, J. Non-Cryst. Solids 293–295 (2001) 806–811. [33] W. Chen, W. Xie, L. Zhang, L. Chen, Y. Du, B. Huang, et al., Diffusion-controlled growth of fcc-free surface layers on cemented carbides: Experimental measurements coupled with computer simulation, Int. J. Refract. Met. Hard Mater. 41 (2013) 531–539. [34] M.R. LaBrosse, K. Johnson, A.C.T. van Duin, Development of a transferable reactive force field for cobalt, J. Phys. Chem. 114 (2010) 5855–5861. [35] K. Wiebking, H. Brantis, Das Auflösungsverhalten von Festkörpern aus Molybdän und Wolfram in flüssigem Eisen, Molybdän-Dienst 67 (1970). [36] P. Kubicek, B. Vozniakova, Study of tungsten in liquid iron diffusion coefficient dependency on temperature and concentration, Radioisotope Application in Metallurgy, 16 1982, pp. 231–235. [37] D.W. Morgan, J.A. Kitchener, Solutions in liquid iron. Part 3: - diffusion of cobalt and carbon, Trans. Faraday Soc. 50 (1954) 51–60. [38] E.A. Villegas, The Diffusion of Nitrogen in Liquid Iron Alloys at 1600 °C, PhD thesis Stanford University, USA, 1976. [39] S. Stüber, Diffusion Dynamics in Liquid and Undercooled Al-Ni Alloys, PhD thesis Technische Universität München, Germany, 2009. [40] M. Ekroth, Gradient Structures in Cemented Carbides, PhD thesis Royal Institute of Technology, Sweden, 2000. [41] D.K. Hale, Review – the physical properties of composite materials, J. Mater. Sci. 11 (1976) 2105–2141. [42] R.L. McCullogh, Generalized combining rules for predicting transport properties of composite materials, Compos. Sci. Technol. 22 (1985) 3–21. [43] Z. Hashin, S. Shtrikman, A variational approach to the theory of the elastic behavior of multiphase materials, J. Mech. Phys. Solids 11 (1963) 127–140. [44] B. Paul, Prediction of elastic constants of multi-phase materials, Technical report (Brown University, Division of Engineering), 1959, p. 3. [45] M.A. Elsayed, J.J. McCoy, Effect of fiber positioning on the effective physical properties of composite materials, J. Compos. Mater. 7 (1973) 466–480. [46] J. Weidow, H.-O. Andrén, Grain and phase boundary segregation in WC-Co with TiC, ZrC, NbC or TaC additions, Int. J. Refract. Met. Hard Mater. 29 (2011) 38–43.
accounts for the increased diffusion path has to be introduced and assessed, but the result of this assessment is very sensitive to the adopted values of the diffusion coefficients. As the diffusion values have now been closely investigated it opens up for further investigations regarding modeling of the effective diffusion in cemented carbide systems. Computer simulations and in-depth experimental studies may be an effective way of reducing the uncertainty of the labyrinth factor. Acknowledgments The authors would like to thank Sandvik Coromant Sverige AB for the technical and financial support. This work was performed within the VINNEX center Hero-m, financed by the Swedish Governmental Agency for Innovation Systems (VINNOVA), Swedish industry, and KTH Royal Institute of Technology. CMA acknowledges also financial support from the Swedish Research Council (VR). Computer resources for this study have been provided by the Swedish National Infrastructure for Computing (SNIC) and MATTER Network, at the National Supercomputer Center (NSC), Linköping. References [1] G.B. Olson, Genomic materials design: the ferrous frontier, Acta Mater. 61 (2013) 771–781. [2] P. Kubicek, T. Pepfica, Diffusion in molten metals and melts: application to diffusion in molten iron, Int. Metals Rev. 28 (3) (1983). [3] J. Garcia, G. Lindwall, O. Prat, K. Frisk, Kinetics of formation of graded layers on cemented carbides: experimental investigations and DICTRA simulations, Int. J. Refract. Met. Hard Mater. 29 (2011) 256–259. [4] R. Frykholm, M. Ekroth, B. Jansson, J. Ågren, H.-O. Andrén, A new labyrinth factor for modelling the effect of binder volume fraction on gradient sintering of cemented carbides, Acta Mater. 51 (2003) 1115–1121. [5] J. Garcia, Investigations on kinetics of formation of fcc-free surface layers on cemented carbides with Fe-Ni-Co binders, Int. J. Refract. Met. Hard Mater. 29 (2011) 306–311. [6] M. Walbrühl, J. Ågren, A. Borgenstam, Design of Co-free cemented carbides, Paper presented at the 3rd World Congress on Integrated Computational Materials Engineering (ICME 2015), USA, 2015. [7] H. Suzuki, K. Hayashi, Y. Taniguchi, The beta-free layer formed near the surface of vacuum-sintered WC-beta-co alloys containing nitrogen, Trans. Jpn. Inst. Metals 22 (1981) 758–764. [8] B.J. Nemeth, G.P. Grab. Preferentially binder enriched cemented carbide bodies and method of manufacture. US Patent 4,610,931 (1987). [9] M. Schwarzkopf, H.E. Exner, H.F. Fischmeister, W. Schintlmeister, Kinetics of compositional modification of (W, Ti)C–WC–Co alloy surfaces, Mater. Sci. Eng. A 105–106 (1988) 225–231. [10] J.O. Andersson, T. Helander, L. Höglund, P.F. Shi, B. Sundman, Thermo-Calc and DICTRA, computational tools for materials science, Calphad 26 (2002) 273-12. [11] B. Predel, Introduction, in: B. Predel (Ed.), SpringerMaterials - The LandoltBörnstein Database, 2012, http://dx.doi.org/10.1007/10793176_2 www. springermaterials.com. [12] A. Blomqvist, Insights into Materials Properties from Ab Initio Theory. Diffusion, Adsorption, Catalysis & Structure, PhD thesis Uppsala University, Sweden, 2010. [13] D. Frenkel, B. Smit, Understanding Molecular Simulation from Algorithms to Applications, second ed., Academic Press, San Diego, 2002. [14] P.E. Blöchl, Projector augmented-wave method, Phys. Rev. B 50 (1994) 17953–17979. [15] G. Kresse, D. Joubert, From ultrasoft pseudopotentials to the projector augmentedwave method, Phys. Rev. B 59 (1999) 1758–1775.
179
Contents lists available at ScienceDirect
International Journal of Refractory Metals & Hard Materials journal homepage: www.elsevier.com/locate/IJRMHM
Surface gradients in cemented carbides from first-principles-based multiscale modeling: Atomic diffusion in liquid Co
MARK
Martin Walbrühla,⁎, Andreas Blomqvistb, Pavel A. Korzhavyia, C. Moyses Araujoc a b c
Department of Materials Science and Engineering, Royal Institute of Technology, SE-100 44 Stockholm, Sweden Sandvik Coromant R & D, SE-126 80 Stockholm, Sweden Materials Theory Division, Department of Physics and Astronomy, Uppsala University, SE-751 20 Uppsala, Sweden
A R T I C L E I N F O
A B S T R A C T
Keywords: ICME Diffusion Ab initio molecular dynamics Liquid co Cemented carbides DICTRA
The kinetic modeling of cemented carbides, where Co is used as binder element, requires a detailed diffusion description. Up to now, no experimental self- or impurity diffusion data for the liquid Co system have been available. Here we use the fundamental approach based on ab initio molecular dynamics simulations to assess diffusion coefficients for the liquid Co system, including six solute elements. Our calculated Co self-diffusion coefficients show good agreement with the estimates that have been obtained using scaling laws from the available literature data. To validate the modeling method, we performed one set of calculations for liquid Ni self-diffusion, where experimental data are available, showing good agreement between theory and experiments. The computed diffusion data were used in subsequent DICTRA simulations to model the gradient formation in cemented carbide systems. The results based on the new diffusion data allows for correct predictions of the gradient thickness.
1. Introduction Integrated Computational Materials Engineering (ICME) uses modeling tools that describe the material at several relevant structural levels ranging from atomic to macroscopic scale. This novel concept offers new opportunities to integrate fundamental research with industrially related computational engineering [1]. One such opportunity is to use theoretical modeling approaches based on quantum mechanics to investigate materials' properties that are difficult to measure experimentally. Among such properties are, for example, atomic diffusion coefficients in liquid metals [2]. Due to their good cutting performance, coated cemented carbides are of great interest for the metal cutting industry as materials for inserts. The base material for such inserts consists of a primary hard phase of hexagonal tungsten carbide WC and a secondary hard phase of a cubic carbide (or carbonitride). The hard particles are embedded into the metallic matrix of a binder phase (usually Co based), which provides the toughness to the cemented carbide system [3,4]. Industrial requirements concerning performance of cemented carbides are constantly increasing, and this calls for a better understanding of the influence of the binder phase on the cemented carbide properties [5,6]. Atomic diffusion in the binder phase, during manufacturing of the material, has important influence on the properties of the product. The
⁎
data from direct measurements of self-diffusion and impurity diffusion coefficients in liquid Co are unavailable, so the modeling can only be based on the values obtained from various estimates [3]. This work uses ab initio molecular dynamics (AIMD) simulations to obtain the lacking diffusion data for liquid Co and selected Co-based alloys. To increase cutting tool performance, cemented carbides are coated with hard coatings using physical vapor deposition (PVD) or chemical vapor deposition (CVD). A too large difference in thermal expansion coefficients between the base material and a CVD-type coating can cause cracking in the coatings after the deposition and the subsequent cooling. Due to the brittle nature of cemented carbides with a hard phase of cubic carbonitride, the cracks can propagate from the coating into the bulk material and cause failure. To prevent this, cemented carbides with a composition gradient (a tougher surface layer free from the cubic carbonitride phase) have been developed [7,8,9]. Sintering in a de-nitrogenizing atmosphere creates a gradient in the chemical potential of N. This causes the outward diffusion of N from the bulk material into the furnace atmosphere, which is the driving force for gradient formation in cemented carbide systems. Ti, which has a high affinity to N, diffuses inwards the bulk, to the remaining regions rich in N, causing the formation of a layer, free of the fcc phase. Looking from the materials design perspective, the gradient formation can be predicted using a software tool for modeling diffusion in
Corresponding author. E-mail address: [email protected] (M. Walbrühl).
http://dx.doi.org/10.1016/j.ijrmhm.2017.03.016 Received 24 November 2016; Received in revised form 24 February 2017; Accepted 24 March 2017 Available online 25 March 2017 0263-4368/ © 2017 Elsevier Ltd. All rights reserved.
International Journal of Refractory Metals & Hard Materials 66 (2017) 174–179
M. Walbrühl et al.
For assessing the diffusion by first-principles methods, the mean square displacement (MSD) method was used. In this method the displacement of atoms with time is calculated to obtain the diffusion coefficients for the observed elements using the Einstein relation [12,23,24]. By performing the AIMD simulations at different temperatures, the thermally activated process of diffusion can be fitted to an Arrhenius relation [25]. By using Eq. (1),
multicomponent alloys (such as DICTRA software [10]), which enables for the development of tailor-made materials. An accurate description of the kinetics behind the gradient formation requires that the databases behind the software contain accurate diffusion data for the elements dissolved in the binder phase. Since direct experimental data for liquid Co are not available, the purpose of the present work is to employ the computational modeling approach based on AIMD to obtain the missing diffusion data. Furthermore, we aim to demonstrate that combining atomistic ab initio simulations with continuum modeling, at larger length scales, has the potential to meet the industrial requirements for materials research and development.
⎛ −Q ⎞ ⎟ D = M0 exp ⎜ ⎝ RT ⎠
(1)
2. Methods
given in the form of a Arrhenius relation, it is possible to obtain both the frequency factor M0 and the activation energy Q which can be stored in a kinetic database for DICTRA [10].
2.1. AIMD simulations
2.2. DICTRA simulations
The supercell for the ab initio molecular dynamics simulations was chosen to contain 108 atoms which were randomly placed into the lattice sites. Since Co was used as the matrix phase, the initial lattice was chosen to have the face-centered cubic (fcc) structure for T > 661 K [11]. Two sets of alloy configurations were simulated: Co self-diffusion and binary Co alloy systems. The first simulation was performed for a configuration with 108 Co atoms (100 at.% Co) which is used to investigate the self-diffusion of Co. The binary diffusion simulations were carried out for Co-based binary alloy systems with W, C, N, Ti, Ta, or Nb to obtain the diffusion coefficient of the alloying element as well as to investigate how the diffusion coefficient of the Co host (with 97 atoms which equals 90 at.%) is affected by the alloying. The AIMD simulations were performed within the canonical NVT ensemble [12,13] using the projector augmented wave (PAW) method [14,15] as implemented in the Vienna ab-initio simulation package VASP [16]. The GGA-PBE was chosen to describe the exchange correlation functional. Initially, the simulations were run at 6000 K to ensure that the systems were completely molten without remains of the initial crystal structure. In total, four different temperatures T were considered for the evaluation: 1903 K, 2003 K, 2103 K and 2303 K. The temperature was controlled using a Nosé thermostat [13] and the subsequent Nosé frequencies were adjusted to be ~8 THz, which appears to be an adequate choice considering at the fcc-Co phonon spectrum [17,18]. To verify the results obtained with the thermostat, rescaling of the velocities [13] was also used in selected cases. For each composition and temperature, the supercell volume V was calculated using the Thermo-Calc software [19] and then imposed as a constraint on the supercell by fixing the lattice constants. The electronic temperature was controlled using the Fermi function smearing implemented in VASP. The electronic structure calculations were spin-polarized, with floating magnetic moments to take into account the magnetism of Co resulting in an average magnitude of ~ 1.6 μB/atom for the magnetic moment on the Co atoms in pure liquid Co. The moments were treated as collinear and were allowed to adjust their magnitude and orientation to the local atomic environment during the AIMD runs. No special treatment of the paramagnetic state was made in the simulations, relying on the experimentally well-established fact that magnetic ordering has no noticeable effect neither on self-diffusion, nor on impurity diffusion, coefficients in solid Co [20] (in contrast to those in solid Fe where the effect of magnetic ordering is strong [21]). The timestep in molecular dynamics simulations was, in general, kept to be 2 fs; the time averages were evaluated using the data from about 2 · 104 timesteps for each tested configuration. To balance the high computational cost of performing long simulations for large supercells, we had to evaluate the momentum-space integrals using a single k-point (Gamma-point), which will be discussed later in more details. More information regarding the setup of the present AIMD simulations can be found in [22].
The DICTRA simulations are performed using the homogenization model [26], which is suitable for performing diffusion simulations of alloy systems having dispersed phases inside a matrix phase, like cemented carbide systems. The kinetic database was assembled with the individual mobility values based on the AIMD simulations, and the thermodynamics used was according to the TCFE7 database [19]. 3. Assessment of diffusion coefficients The following section deals with the diffusion values presented in Fig. 1. The data on Co self-diffusion is compared with theoretical literature data for the Co system and experimental data on Ni selfdiffusion. Furthermore, the results of the assessed binary diffusion values are presented. The results of the present calculations of the liquid Co self-diffusion coefficient are presented in Fig. 1a where they are compared with the available literature data. To the authors' knowledge, no results of direct experimental measurement of the Co self-diffusion coefficient have hitherto been published. One general problem that complicates experimental investigations of diffusion in liquid metals is mass transfer by convection. The difficulty to distinguish between the diffusion and convection contributions to the measured mass transfer often invalidates the results of direct measurements [2]. Han et al. [27] performed experimental measurements of the surface tension for pure liquid Co, from which the Co self-diffusion coefficient was obtained as a by-product. Also available are calculated values of diffusion coefficients, which have been derived by more theoretical methods. A scaling law that relates the excess entropy and atomic diffusion coefficient in liquid metals (Rosenfeld [28] and Dzugutov [29]) was used in several studies (Yokoyama [30], Korkmaz and Korkmaz [31] and Yokoyama and Arai [32]) to describe the selfdiffusion in liquid Co. Chen et al. [33] used another approach, based on an Arrhenius relation derived from a modified Sutherland equation, for assessing the Co self-diffusion coefficient and further for assessing the impurity diffusion coefficients of Co-W and Co-Ta liquid alloy systems. LaBrosse et al. [34] used classical molecular dynamics (CMD) simulations, employing an original interatomic potential of embedded atom method (EAM) type, to describe the Co system. The results by Chen et al. and Yokoyama and Arai are in excellent agreement with the self-diffusion data for liquid Co obtained in this work, by means of AIMD calculations. The other values, obtained by either scaling laws or derived from experimental measurements of surface tension, are in good agreement with our results, which gives some confidence in the calculated values. Only the CMD calculated selfdiffusion coefficients are about one half of the value obtained in this work as well as of the values given by other authors. Since this work not only deals with the liquid Co self-diffusion, but also with the impurity diffusion in liquid Co, it is important to assess their accuracy. The only work to report results for liquid Co-based alloy 175
International Journal of Refractory Metals & Hard Materials 66 (2017) 174–179
M. Walbrühl et al.
Fig. 1. Diffusion data for the liquid Co system. Collection of self-diffusion data for liquid Co (a). Comparison of experimental values for liquid Ni self-diffusion with AIMD calculations for Co and Ni (b). Co self-diffusion and binary solute diffusion coefficients calculated by AIMD (c).
[39] are compared with the results of our AIMD simulations for liquid Co and Ni systems. As Fig. 1b shows, the diffusion coefficients of Co and Ni are similar to each other in the considered temperature range. In fact, the good agreement between the experimental and AIMD simulations results for liquid Ni is more important because it indicates that the simulation method yields correct diffusion data for liquid metal systems. The comparison between our results for the Co-W and Co-Ta liquid alloys and those by Chen et al. indicates that the two different theoretical approaches are in good agreement regarding the diffusion coefficients of Co (Table 2). At the same time, the diffusion coefficients of the solute/alloying elements, also given in Table 2, differ by a factor of about 1.4 between the two theoretical studies. In the present AIMD simulations, the concentration of 10 at.% corresponds to 11 solute
Table 1 Comparison between experimental data for liquid Fe and AIMD data for liquid Co. System
W in liquid Fe
C in liquid Fe N in liquid Fe
T [K]
1873 1813 1860 1903 1973 1823 1873
D [1E− 09 m2/s] This work (with diffusion in liquid Co)
Experimental
2.87 2.54 2.82 3.04 3.47 7.35 8.27
10 [35] 2.3 ± 0.07 [36] 2.5 ± 0.015 [36] 2.7 ± 0.07 [36] 3.2 ± 0.1 [36] 6.7–7.9 [37] 4.12 [38]
systems is that by Chen et al. In addition we compare the results of our calculations with experimental diffusion data for liquid Fe and Ni alloys to have an independent point of reference. The comparison is done to show that the values obtained through the AIMD simulations are in a reasonable range compared to a similar metal system where experimental data are available. The data of Table 1 show that the values of diffusion coefficients in the liquid Co system are quite comparable to the experimental values in the liquid Fe-based alloy system, taking into account the scatter of the experimental data (see [35,36]). The comparison with the two sets of experimental data indicates that AIMD simulations may be considered as a suitable option to obtain diffusion coefficients for liquid metal systems. This is further indicated by Fig. 1b where experimental results for the liquid Ni self-diffusion from Ref.
Table 2 Comparison solute diffusion of W and Ta in Co. System
Co-W
T [K]
1830 1865
Co-Ta
176
1830
Element
W Co W Co Ta Co
D [1E− 09 m2/s] This work
at.%
Chen et al. [33]
at.%
2.63 3.62 2.82 3.82 2.3 3.41
10 90 10 90 10 90
2–1.48 3.39–2.36 2 3.35 1.93–1.46 3.42–2.44
5.4–20 94.6–80 8.32 91.68 5.4–20 94.6–80
International Journal of Refractory Metals & Hard Materials 66 (2017) 174–179
M. Walbrühl et al.
The pure Co system was used for such an investigation and it has been shown the LDA exchange correlation function yields ~ 20% higher values. To balance the number of evaluated systems and the corresponding computational costs it was unavoidable to limit the system size and the used k-points. A bigger system would have allowed a larger number of solute atoms to keep the 10 at.%, which would result in overall better statistics – but looking at the R2 values of the MSD it appears to be sufficient high to run simulations without obvious computational anomalies. The reduction of the k-points is a drastic step but has a huge effect on the computational costs. The low k-point sampling introduces an error in the atomic forces at each time step, but since the time step is much smaller than the vibration of the atoms, the introduced error cancels out and does not affect the diffusion behavior. This was tested by performing simulations of a 1 × 1 × 1 and 2 × 2 × 2 k-point system for pure Co. To obtain the equivalent simulation length of 12 ps, the simulation had to run six times longer using the 2 × 2 × 2 k-points while the actual calculated diffusion value was nearly identical. For the investigated seven-component alloy system a 2 × 2 × 2 k-point mesh would have not been feasible with the available resources.
atoms per 97 atoms of the solvent (more information regarding the computational set-up are available in the method section). The MSD method to calculate the diffusion coefficients is based on the statistical averaging of the displacement over all atoms of each species in the simulation box. The number of Co atoms in the supercell is about 9 times greater than the number of solute atoms, which implies much better statistics for the solvent atoms. This could explain why the agreement with the results by Chen et al. is better for pure Co (Fig. 1a), as well as for the Co atoms in alloys, than for the solute elements. As both approaches only give a theoretical diffusion value, that cannot be compared to experimental data, we do not think it would add more value to the overall work to reproduce the exact systems from Chen et al. by the AIMD method. The results of diffusion simulations for binary alloy compositions are presented in Fig. 1c. The simulations show that, at the gradient sintering temperature of 1723 K, non-metallic elements C and N diffuse about two times faster than the metallic elements. At the temperature of 2300 K the difference increases to a factor of about 2.5. In the work reported by Garcia et al. [3], which was based on estimated values for the activation energy Q (65 KJ/mol) and the frequency factor M (9.34 · 10− 8 m2/s), the reductions by a factor of 2 of the mobility of the metallic solute elements resulted in an accurate description of the gradient formation. At the temperature of 1723 K, the self-diffusion of Co is calculated to be the fastest among the metallic elements considered in this work. The metallic solute elements have a gradually slower diffusion compared to the Co solvent, whereas Ta is the slowest element with a ~ 1.7 times slower diffusion. With increasing temperature the differences in diffusion coefficients decrease for the metallic elements. The calculated frequency factors and activation energies, assessed from the evaluation of the Arrhenius like diffusion behavior, are presented in Table 3. Looking at the activation energies no distinct difference between the metallic and non-metallic elements is evident. The difference is more obvious for the frequency factors with greater values for the non-metallic elements. Finally let us discuss about the accuracy and limitations related to this study. For each system four individual temperatures and the related MSD have been evaluated. As mentioned earlier, the MSD is a statistical average over the atomic displacement and thus gives us a chance to look at the linearity of the mean square displacement with the simulation time by linear regression. A linear displacement (high R2 value) would indicate the absence of simulation anomalies and thus a simulation can be considered as relevant for the assessment. The linearity for the diffusion values indicates whether or not the diffusion follows the expected linear Arrhenius behavior for the logarithm of diffusion coefficient as a function of temperature. The R2 values are presented in Table 3 and show generally values of ~0.94 or higher, which gives confidence in the applied methodology. In this work the GGA-PBE exchange correlation function was used for all simulations. It might be of general interest for the reader if the choice of the exchange correlation function has influenced the outcome.
4. Simulation of gradient formation In order to investigate the applicability of the obtained diffusion data we turn to cemented carbide systems which are in need of better kinetic description for the prediction of surface gradients. Diffusion simulations of gradient formation in the cemented carbide system have been treated using the homogenization model in DICTRA [26]. The WC and fcc carbide phases are in a solid state during the gradient sintering. Since the diffusion is assumed to take mainly place in the liquid binder phase, the solid phases are seen as obstacles which reduce the effective diffusion coefficients Deff by a so-called labyrinth factor λ(f) (Eq. (2)) with f being the volume fraction of the binder phase:
Deff = λ (f ) ∙D
In an early work by Ekroth [38] the labyrinth factor λ was taken as proportional to the volume fraction squared (λ ∝ f2) of the binder phase. In later studies [3,4], λ ∝ f was used. In the previous publications the labyrinth factor was fitted to the experimentally measured widths of the gradient zone, but the diffusion coefficients were evaluated based on different sets of estimated values [3,4,40]. Furthermore, the ramping (heating and cooling) during the gradient sintering have been neglected in the evaluations. The present simulations of gradient formation, which make use of the diffusion coefficients obtained in this work, show that neither λ ∝ f nor λ ∝ f2 as choice of the labyrinth factor gives a satisfactory description of the gradient formation. Our test calculations have shown that a λ ∝ f3.3 scaling for the labyrinth factor holds instead, yielding good agreement between the results of gradient simulations and the experimentally measured gradients. Fig. 2 shows the volume fraction of the WC, fcc carbide, and Co binder phases as functions of the depth. The width of the gradient zone measured by Garcia et al. [3] is 23 ± 1 μm after 2 h of sintering. The calculated gradient width, including the heating and cooling (a total sintering time of around 3 h), is 23 μm. By analyzing the effect of the labyrinth factor and the differences in the used diffusion coefficients between different publications, it is reasonable to conclude that the labyrinth factor expression has to be attempted in further research as no clear solution can be found. The values of diffusion coefficients used by Garcia et al. are ~6 times smaller than the values used in the present work. Because of the higher mobility values obtained in this work, the labyrinth factor needs to be smaller (which implies its stronger dependence on the volume fraction) to yield the same effective diffusion coefficient Deff and the same gradient width. Further reason for using a different labyrinth factor is the neglect of the ramping time in the previous studies. Taking into
Table 3 Frequency factors, activation energies and R2 values (avg. MSD and Diffusion) for the investigated systems.
Co Ti Nb Ta W C N
Frequency factor M0 [10− 7 m2/s]
Activation energy Q [kJ/ mol]
Configuration
1.15 1.46 1.28 2.72 1.21 6.60 7.60
52.65 60.62 56.34 72.59 58.28 68.17 70.40
Self-diffusion Binary diffusion " " " " "
R2 MSD
Diffusion
0.9996 0.9967 0.9983 0.9928 0.9959 0.9945 0.99735
0.972 0.937 0.9903 0.9796 0.9507 0.9784 0.9412
(2)
177
International Journal of Refractory Metals & Hard Materials 66 (2017) 174–179
M. Walbrühl et al.
Fig. 2. Width of an fcc-free gradient in cemented carbides. Comparison of the measured [3] (after 2 h of sintering) and the corresponding simulated width of the fcc-free zone with a labyrinth factor of λ ∝ f3.3 and the diffusion data from this work.
slow down the effective Ti diffusion additionally to the geometrical aspect. In turn it might be of interest to include an additional factor in the effective diffusion calculation that cannot be explained by the bulk diffusion and the pure geometrical aspects. As the important liquid diffusivities have been described now by fundamental methods they might be used as a non-adjustable parameter in the assessment of the effective diffusion in cemented carbide systems. In the authors' opinion a combination of carefully planned experiments and advanced 3D modeling, that investigates the geometrical effects that are caused by the realistic composite structure, are necessary to find a satisfying solution to model the effective diffusion. Only a straight forward experimental result allows to identify if, besides the geometry and diffusion, another factor has to be considered to predict the effective diffusion correctly. This knowledge can then be used to eventually give a full multiscale simulation approach for the surface gradient formation in the cemented carbide system.
account the additional sintering time during the ramping will yield a different gradient width, which should be compensated by the change in the labyrinth factor. More elaborated theories [41,42] are available to describe effective properties in composite materials and are thus favorable within the here presented multiscale approach. Hashin and Strikman [43] developed a theory, considering upper and lower bounds, which they used to calculate elastic properties in WC-Co cemented carbide systems. They used spherical shapes to model the two different phases and require the inherent properties of the two pure phases to express the extreme values for the bounds. Similarly Paul [44] described the Young's modulus for the WC-Co system but instead used a cubic shape to model the inclusions. His model requires as well the knowledge about the two phases in order to describe the minimum and maximum of the effective property. The underlying approach of the labyrinth factor, which has been previously used, is that the diffusion only occurs in the binder phases which leave the WC and fcc phase as non-diffusing obstacles. Using the more elaborated theories the maximum in diffusion can be expressed with the here presented binder diffusion values but it is not feasible to define the minimum, for the non-diffusion phases, as zero. Hashin and Strikman point out that their model is only able to estimate the effective property when the ratio between the different phase properties is not too large. Unfortunately, the models appear to be very sensitive to a change in the minimum phase property which does not allow to make reasonable predictions of the effective diffusion in the cemented carbide structure. In addition, the model approaches with spheres or cubes might be too simplified to represent the real prismatic shaped WC grains which will not allow to give an accurate description. It should be emphasized that the theories, like [43], suggest upper and lower bounds giving a theoretical range for the geometrical effect on the effective properties in composite materials. Several authors [41,42,45] indicated that the shape of the obstacle phase will have an influence on the effective transport properties and thus the shape consideration is necessary in order to create tighter bounds that allow more accurate predictions. The labyrinth factor, as previously used for cemented carbide systems, should only contain geometrical information but due the unknown diffusivities it was basically used a fitting parameter that might also contain non-geometrical information. Additional contributions, that do not arise from the geometrical aspects of the two phase structure, might alter the effective diffusion Deff. One of those, so far, not further investigated effects on the gradient formation might arise from additional chemical reactions, like the ability of Ti to form a few atomic layer thin film on the WC grain surface [46]. Due to its particular role in the gradient formation [9] those film formation might
5. Conclusions The use of methods of computational material design can help overcome limitations of empirically-based approaches in the design of new functionally graded materials with improved properties. Here we have demonstrated how different modeling methods that operate on different length scales (and use atomistic and continuum description) can be successfully combined. Ab initio molecular dynamical (AIMD) simulations were employed for calculating the kinetic properties (coefficients of atomic diffusion) in the liquid binder phase of cemented carbides. The results of the simulations were benchmarked against the available literature data and the agreement is found to be encouraging. In particular, the values of diffusion coefficients calculated in this work for Co-based alloys are in good agreement with the available literature data for liquid Co-based alloys, and in reasonable agreement with the experimental values available for the liquid Fe and Ni systems. Taking also into account the good agreement between the experimental AIMD simulated data for liquid Ni, we can conclude that the present simulations have demonstrated applicability of DFT-based modeling methods for solving industrially relevant materials problems. The diffusion simulations of gradient formation themselves bear a large degree of uncertainty. A number of thermodynamic and kinetic parameters have to be used as the input data in such simulations, but the data often come from different sources and have different accuracy. Modeling of diffusion in the cemented carbide systems is additionally complicated by the fact that the carbide particles are obstacles which reduce the effective diffusion coefficient by increasing the diffusion path. The labyrinth factor, only describing geometrical effects, which 178
International Journal of Refractory Metals & Hard Materials 66 (2017) 174–179
M. Walbrühl et al.
[16] J. Hafner, Materials simulations using VASP—a quantum perspective to materials science, Comput. Phys. Commun. 177 (2007) 6–13. [17] O.P. Gupta, Crystal dynamics of face centered cubic cobalt, Solid State Commun. 42 (1982) 31–32. [18] Y.R. Wang, Temperature-dependent phonon dispersion relation in magnetic crystals, Sold State Commun. 54 (1985) 279–282. [19] Thermo-Calc Software, “TCS Steel and Fe - alloys Database”, Vers. 7.0. http://www. thermocalc.com, (2013). [20] Landolt-Börnstein, H. Mehrer (Ed.), New Series Group III, 26 Springer-Verlag, Berlin, 1990(p.51, 74, 131–132, 182). [21] Landolt-Börnstein, H. Mehrer (Ed.), New Series Group III, 26 Springer-Verlag, Berlin, 1990(p.49–51, 73, 124–130, 179–181). [22] M. Walbrühl, Diffusion in the Liquid Co Binder of Cemented Carbides – Ab Initio Molecular Dynamics and DICTRA Simulations, M.Sc. thesis Sweden, Royal Institute of Technology, 2014. [23] M. Hillert, J. Ågren, Diffusion and Equilibria an Advanced Course in Physical Metallurgy, Department of Materials Science and Engineering Royal Institute of Technology, Stockholm, 2002. [24] D.A. Porter, K.E. Easterling, Phase Transformations in Metals and Alloys, second ed., T.J. Press, Cornwall, 1992. [25] M. Hillert, Phase Equilibria, Phase Diagrams and Phase Transformations, Their Thermodynamic Basis, second ed., Cambridge University Press, 2007. [26] A. Borgenstam, A. Engström, L. Höglund, J. Ågren, DICTRA, a tool for simulation of diffusional transformations in alloys, J. Phase Equilib. 21 (2000) 269–280. [27] X.J. Han, N. Wang, B. Wei, Thermophysical properties of undercooled liquid cobalt, Philos. Mag. Lett. 21 (2002) 451–459. [28] Y. Rosenfeld, Relation between Transport-Coefficients and Internal Entropy of Simple Systems, Phys. Rev A 15 (1977) 2545–2549. [29] M. Dzugutov, A universal scaling law for atomic diffusion in condensed matter, Nature 381 (1996) 137–139. [30] I. Yokoyama, Self-diffusion coefficient and its relation to properties of liquid metals: a hard-sphere description, Phys. B Condens. Matter 271 (1999) 230–234. [31] S.D. Korkmaz, Ş. Korkmaz, Investigation of atomic transport and surface properties of liquid transition metals using scaling laws, J. Mol. Liq. 150 (2009) 81–85. [32] I. Yokoyama, T. Arai, Correlation entropy and its relation to properties of liquid iron, cobalt and nickel, J. Non-Cryst. Solids 293–295 (2001) 806–811. [33] W. Chen, W. Xie, L. Zhang, L. Chen, Y. Du, B. Huang, et al., Diffusion-controlled growth of fcc-free surface layers on cemented carbides: Experimental measurements coupled with computer simulation, Int. J. Refract. Met. Hard Mater. 41 (2013) 531–539. [34] M.R. LaBrosse, K. Johnson, A.C.T. van Duin, Development of a transferable reactive force field for cobalt, J. Phys. Chem. 114 (2010) 5855–5861. [35] K. Wiebking, H. Brantis, Das Auflösungsverhalten von Festkörpern aus Molybdän und Wolfram in flüssigem Eisen, Molybdän-Dienst 67 (1970). [36] P. Kubicek, B. Vozniakova, Study of tungsten in liquid iron diffusion coefficient dependency on temperature and concentration, Radioisotope Application in Metallurgy, 16 1982, pp. 231–235. [37] D.W. Morgan, J.A. Kitchener, Solutions in liquid iron. Part 3: - diffusion of cobalt and carbon, Trans. Faraday Soc. 50 (1954) 51–60. [38] E.A. Villegas, The Diffusion of Nitrogen in Liquid Iron Alloys at 1600 °C, PhD thesis Stanford University, USA, 1976. [39] S. Stüber, Diffusion Dynamics in Liquid and Undercooled Al-Ni Alloys, PhD thesis Technische Universität München, Germany, 2009. [40] M. Ekroth, Gradient Structures in Cemented Carbides, PhD thesis Royal Institute of Technology, Sweden, 2000. [41] D.K. Hale, Review – the physical properties of composite materials, J. Mater. Sci. 11 (1976) 2105–2141. [42] R.L. McCullogh, Generalized combining rules for predicting transport properties of composite materials, Compos. Sci. Technol. 22 (1985) 3–21. [43] Z. Hashin, S. Shtrikman, A variational approach to the theory of the elastic behavior of multiphase materials, J. Mech. Phys. Solids 11 (1963) 127–140. [44] B. Paul, Prediction of elastic constants of multi-phase materials, Technical report (Brown University, Division of Engineering), 1959, p. 3. [45] M.A. Elsayed, J.J. McCoy, Effect of fiber positioning on the effective physical properties of composite materials, J. Compos. Mater. 7 (1973) 466–480. [46] J. Weidow, H.-O. Andrén, Grain and phase boundary segregation in WC-Co with TiC, ZrC, NbC or TaC additions, Int. J. Refract. Met. Hard Mater. 29 (2011) 38–43.
accounts for the increased diffusion path has to be introduced and assessed, but the result of this assessment is very sensitive to the adopted values of the diffusion coefficients. As the diffusion values have now been closely investigated it opens up for further investigations regarding modeling of the effective diffusion in cemented carbide systems. Computer simulations and in-depth experimental studies may be an effective way of reducing the uncertainty of the labyrinth factor. Acknowledgments The authors would like to thank Sandvik Coromant Sverige AB for the technical and financial support. This work was performed within the VINNEX center Hero-m, financed by the Swedish Governmental Agency for Innovation Systems (VINNOVA), Swedish industry, and KTH Royal Institute of Technology. CMA acknowledges also financial support from the Swedish Research Council (VR). Computer resources for this study have been provided by the Swedish National Infrastructure for Computing (SNIC) and MATTER Network, at the National Supercomputer Center (NSC), Linköping. References [1] G.B. Olson, Genomic materials design: the ferrous frontier, Acta Mater. 61 (2013) 771–781. [2] P. Kubicek, T. Pepfica, Diffusion in molten metals and melts: application to diffusion in molten iron, Int. Metals Rev. 28 (3) (1983). [3] J. Garcia, G. Lindwall, O. Prat, K. Frisk, Kinetics of formation of graded layers on cemented carbides: experimental investigations and DICTRA simulations, Int. J. Refract. Met. Hard Mater. 29 (2011) 256–259. [4] R. Frykholm, M. Ekroth, B. Jansson, J. Ågren, H.-O. Andrén, A new labyrinth factor for modelling the effect of binder volume fraction on gradient sintering of cemented carbides, Acta Mater. 51 (2003) 1115–1121. [5] J. Garcia, Investigations on kinetics of formation of fcc-free surface layers on cemented carbides with Fe-Ni-Co binders, Int. J. Refract. Met. Hard Mater. 29 (2011) 306–311. [6] M. Walbrühl, J. Ågren, A. Borgenstam, Design of Co-free cemented carbides, Paper presented at the 3rd World Congress on Integrated Computational Materials Engineering (ICME 2015), USA, 2015. [7] H. Suzuki, K. Hayashi, Y. Taniguchi, The beta-free layer formed near the surface of vacuum-sintered WC-beta-co alloys containing nitrogen, Trans. Jpn. Inst. Metals 22 (1981) 758–764. [8] B.J. Nemeth, G.P. Grab. Preferentially binder enriched cemented carbide bodies and method of manufacture. US Patent 4,610,931 (1987). [9] M. Schwarzkopf, H.E. Exner, H.F. Fischmeister, W. Schintlmeister, Kinetics of compositional modification of (W, Ti)C–WC–Co alloy surfaces, Mater. Sci. Eng. A 105–106 (1988) 225–231. [10] J.O. Andersson, T. Helander, L. Höglund, P.F. Shi, B. Sundman, Thermo-Calc and DICTRA, computational tools for materials science, Calphad 26 (2002) 273-12. [11] B. Predel, Introduction, in: B. Predel (Ed.), SpringerMaterials - The LandoltBörnstein Database, 2012, http://dx.doi.org/10.1007/10793176_2 www. springermaterials.com. [12] A. Blomqvist, Insights into Materials Properties from Ab Initio Theory. Diffusion, Adsorption, Catalysis & Structure, PhD thesis Uppsala University, Sweden, 2010. [13] D. Frenkel, B. Smit, Understanding Molecular Simulation from Algorithms to Applications, second ed., Academic Press, San Diego, 2002. [14] P.E. Blöchl, Projector augmented-wave method, Phys. Rev. B 50 (1994) 17953–17979. [15] G. Kresse, D. Joubert, From ultrasoft pseudopotentials to the projector augmentedwave method, Phys. Rev. B 59 (1999) 1758–1775.
179