- Email: [email protected]
Coarsening kinetics of lamellar microstructures: Experiments and simulations on a fully-lamellar Fe-Al in situ composite
Acta Materialia 127 (2017) 230e243
Contents lists available at ScienceDirect
Acta Materialia journal homepage: www.elsevier.com/locate/actamat
Full length article
Coarsening kinetics of lamellar microstructures: Experiments and simulations on a fully-lamellar Fe-Al in situ composite X. Li a, F. Bottler a, R. Spatschek a, b, A. Schmitt c, M. Heilmaier c, F. Stein a, * a
Max-Planck-Institut für Eisenforschung GmbH, Düsseldorf, Germany IEK-2, Research Center Jülich, Jülich, Germany c Institute for Applied Materials (IAM), Karlsruhe Institute of Technology (KIT), Karlsruhe, Germany b
a r t i c l e i n f o
a b s t r a c t
Article history: Received 18 November 2016 Received in revised form 16 January 2017 Accepted 21 January 2017 Available online 23 January 2017
A very fine lamellar microstructure consisting of the intermetallic phases FeAl and FeAl2 forms in binary Fe-Al alloys due to the eutectoid reaction Fe5Al8 4 FeAl þ FeAl2, which takes place in the Al range between 56.0 and 64.4 at.% at 1095 C. A fully lamellar microstructure is obtained at the eutectoid composition 60.9 at.% Al. The initial lamellar spacing l0 of as-cast material is 200 ± 40 nm. In this study, the kinetics of coarsening of the FeAl þ FeAl2 lamellar microstructure is investigated at four different temperatures in the range 600 Ce1000 C with holding times from 10 min to 7000 h. It is found that the increase of the lamellar spacing can be described as l3 ¼ l30 þ kt. The value obtained for the activation energy proves that the lamellar coarsening is a volume-diffusion-controlled process. Besides the experimental investigations, phase-field modeling is used to simulate the lamellar coarsening. The results are in good agreement with the experimental observations with regard to the evolution of the lamellar morphology and the value of the coarsening exponent. © 2017 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
Keywords: Coarsening kinetics Intermetallics (iron-aluminides) Lamella Microstructure Phase field modeling
1. Introduction Fe-Al alloys with up to 50 at.% Al show excellent corrosion resistance and have a low density compared to steels, and hence, they became promising materials for structural applications [1e4]. When the Al content is above 50 at.%, the beneficial corrosion behavior persists [5] and the alloy density will further decrease. According to the Fe-Al phase diagram in Ref. [6], Fe-Al alloys in the range from 53.0 to 64.4 at.% Al contain the body-centered cubic phase Fe5Al8 (Cu5Zn8-type crystal structure [7]) at high temperature, which decomposes into cubic B2 FeAl and triclinic FeAl2 through a eutectoid reaction at 1095 C [6]. The orientation relationship between the two phases can be described by ð101ÞFeAl k ð114ÞFeAl2 and ½111FeAl k ½110FeAl2 [8]. The eutectoid transformation is a rapid reaction leading to the formation of a finescaled, fully lamellar microstructure at the eutectoid point (where the Al content is 60.9 at.% [6]), see, e.g., Fig. 1a showing the microstructure after 10 min holding at 1000 C. Similar lamellar microstructures are known from a2 þ g Ti-Al based alloys, where they were found to exhibit balanced properties with regard to
* Corresponding author. E-mail address: [email protected] (F. Stein). http://dx.doi.org/10.1016/j.actamat.2017.01.041 1359-6454/© 2017 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
creep, ductility and strength at elevated temperature [9]. In case of the Fe-Al system, there is no information in the literature about the mechanical behavior of such microstructures. When thinking about the capability for high-temperature structural applications, especially the stability of the lamellar structure is of high relevance. For this reason, the kinetics of lamellar coarsening in a fully lamellar FeAl þ FeAl2 alloy is studied in detail here. Another motivation to perform these investigations is that the extended, nearly perfect lamellar microstructures in the present samples offer an excellent possibility to obtain a broad and reliable database for evaluating existing theoretical models for the kinetics of lamellar coarsening. To this end, heat treatments were performed at 600, 700, 800, and 1000 C for various times ranging from 10 min to 7000 h. The effect of the colony size on lamellar coarsening was investigated as well. Finally, the experimental observations are complemented by simulation results on the lamellar coarsening behavior applying phase field modeling. 2. Overview of coarsening theories Eutectoid or eutectic lamellar microstructures can coarsen either continuously or discontinuously (see e.g., Ref. [10e12]). Models for the latter case are not included in the following
X. Li et al. / Acta Materialia 127 (2017) 230e243
231
Fig. 1. (a) Typical lamellar FeAl (light) þ FeAl2 (dark) microstructure of a heat-treated Fe-60.9 at.% Al sample (1000 C/10 min) showing characteristic terminations and branches of the FeAl phase (SEM micrograph obtained in back-scattered electron (BSE) mode); (b) fully lamellar microstructure of the as-cast alloy showing extended colonies (LOM micrograph; the dotted lines mark the colony boundaries).
overview as the samples investigated in the present study solely show continuous coarsening. The following brief overview will show that until today there exists no generally accepted model to describe the coarsening of lamellar microstructures. This is also related to the fact that there are only comparably few experimental data about lamellar coarsening in the literature, which do not allow verifying reliably the different approaches. A.J. Ardell states in his review [12] about coarsening of eutectic microstructures that “clearly much more work is needed on the coarsening of lamellar eutectics before any conclusion can be drawn regarding which theory is correct” [12]. This statement is still correct today. Eutectic or eutectoid reactions can produce a microstructure that consists of alternating lamellae of two phases or rods of the minor phase embedded in a continuous matrix of the second phase [12]. Lamellar microstructures appear in scanning electron microscope (SEM) micrographs as needle-like morphologies due to the two-dimensional imaging. An example is given in Fig. 1a, also showing the typical types of faults, i.e., branching and terminating lamellae. The extension of the lamellar colonies in the present Fe-Al alloy (Fig. 1b) corresponds to the grain size of the parent Fe5Al8 grains. In order to reach thermodynamic equilibrium, the total energy of a two-phase system can be minimized by decreasing the surface energy via reducing the interfacial area of the phases. This microstructure coarsening process is the basic concept of the theory of Ostwald ripening [13]. The driving force for this process is the curvature dependence of the chemical potential, which is
m ¼ m0 þ Vm gk;
(1)
which depends on several factors including the interfacial free energy and the diffusion coefficient. The most recent model for the kinetics of particle coarsening was developed by Ardell and Ozolins [16] and later elaborated by Ardell [17e19]. In contrast to the classical, Ostwald-ripening-based models, according to which coarsening is controlled by matrix diffusion, they assume that trans-interface diffusion processes control coarsening. The model predicts a r n growth law with an exponent n in the range of 2 and 3 depending on the width of the precipitate-matrix interface and the distribution of particle sizes [17,18]. In case of lamellar microstructures, the situation significantly differs from the above spherical growth. An ideal, i.e., faultless, extended lamellar microstructure would not coarsen as there is no driving force (diffusion of atoms from one to the next lamella or along the interface would not reduce the surface energy), i.e. lamellar structures are inherently (meta-)stable [20]. However, real lamellar microstructures contain faults like branches and terminations of lamellae, and it is well accepted in the literature that any coarsening mechanism must be based on the migration of such faults [10,20e30]. The underlying driving force for the fault migration is the interface curvature at the lamellae terminations, which leads to modifications of the local equilibrium conditions due to the Gibbs-Thomson effect [31,32]. At the interfaces, the concentrations locally deviate from the bulk equilibrium values by a contribution proportional to the local interface curvature. Since the signs of the curvature differ between a branch and a termination, there is a concentration difference between them. As the local curvature and the diffusional flux are directly related to the lamellar spacing l (yielding an inverse proportionality), it can be concluded that the shrinkage velocity n scales as (see, e.g. [23])
.
where k is the mean interfacial curvature, m0 stands for the chemical potential of an atom at a flat interface, Vm is the molar volume and g represents the surface energy [13]. Based on the concept of Ostwald ripening, Lifshitz and Slyozov [14] and Wagner [15] (LSW) established their nowadays wellestablished model for the coarsening of spherical precipitates (radius r) in a fluid matrix ending with the equation
Based on this idea of fault migration, Graham and Kraft were the first to develop a dedicated model for lamellar coarsening [10]. Assuming that the total number of lamellae remains constant and only lamellar length and spacing change by branch and termination cancellation, their model ends with a simple linear relation,
r 3 ¼ r03 þ k1 t:
l ¼ l0 þ k2 t;
(2)
Here, r0 stands for the initial particle radius and k1 is a constant,
n 1 l2 :
(3)
(4)
232
X. Li et al. / Acta Materialia 127 (2017) 230e243
where l0 stands for the initial lamellar spacing and k2 is a constant [10]. A few years later, Cline [21] made some changes to the above model taking into account the influence of fault-annihilation, which was neglected in the coarsening model of Graham and Kraft. The Cline model results in the equation
l2 ¼ l20 þ k3 t;
(5)
where k3 is another rate constant [21]. Brady [33] established a very similar coarsening model for exsolution lamellae in silicate minerals. He assumed the terminations of lamellae as ideal wedge-shaped ends and ended with the same equation as Cline did. The validity of the equation was additionally tested by comparing with experimental data of coarsening of mineral exsolution lamellae reported by McCallister [34], and Yund and Davidson [35], which in the original papers had been plotted according to a t 1=3 rate law. Even though Brady [33] stated that the square root law gives a good description of the data, McCallister [34] pointed out that fitting his data vs. t 1=2 , t 1=3 , or t 1=4 results in similar correlation coefficients not allowing a definite conclusion which exponent is the correct one. In a more recent study, Baker and Meng [36] measured the coarsening of B2 þ fcc lamellae in an Fe28Ni18Mn33Al21 alloy annealed at 900 C. They tested various growth exponents n in the range 2e3 finding that, although for n ¼ 2 they obtained the highest correlation coefficient, “n values from 2 to 3 could reasonably be used to describe the coarsening data” [36].
3. Experimental An Fe-Al alloy containing 60.9 at.% Al with a fully lamellar microstructure was produced by vacuum induction melting from high purity Fe (99.98 wt%) and Al (99.99 wt%) under argon and was cast in a cold copper mould with a diameter of 18 mm [6]. The impurity content of this alloy was determined by chemical analysis with P being measured by inductively coupled plasma-optical emission spectrometry (ICP-OES), C, O and S by infrared absorption spectroscopy and N by thermal conductivity measurement. The results are given in Table 1. The resulting rod was cut by electrical discharge machining (EDM) into several parts and a series of heat-treatments was performed, see Table 2. The samples heattreated for times of less than 24 h were water-quenched to room temperature, while all other samples heat-treated for longer times (24 h) were air-cooled. A 9 cm long piece of the as-cast rod was rotary-swaged at 1150 C, where the alloy is single phase Fe5Al8, in order to reduce the grain size and with that the colony size of the lamellar material. For protection of the brittle material from immediate breaking, the as-cast piece of the rod was wrapped in Mo foil and jacketed with a steel encapsulation before rotary swaging. This assembly was heated up to 1150 C, taken out of the furnace, and immediately processed in the rotary swaging machine. The diameter was reduced in seven passes (including intermediate re-heating to 1150 C) to about 50%. After dismantling, samples were cut from the swaged material and heat-treated for various times at 800 C (see Table 2) to study lamellar coarsening. Phase and overall chemical compositions were measured by
Table 1 Impurity contents in the as-cast Fe-60.9 at.% Al alloy as obtained by chemical analysis. Element
C
N
O
P
S
wt. ppm
37
16
200e370
<20
<10
Table 2 Heat treatment details of the Fe-60.9 at.% Al samples. All heat treatments were performed in Ar atmosphere. Temperature ( C) Time (h) 600 C 700 C 800 C 1000 C 800 C
Starting material
24, 1000, 3000, 5000, 7000 as-cast 24, 50, 100, 300, 1000, 2000, 3000, 5000 as-cast 0.5, 1, 3.5, 6, 18, 24, 48 as-cast 0.17, 0.5, 1, 3.5, 6, 18, 24, 36 as-cast 0.5, 1, 3.5, 6 cast and swaged
wavelength dispersive spectroscopy (WDS) that was carried out with an electron probe micro-analyser (EPMA) JOEL-JXA-8100. The instrument was operated at 15 kV and 20 nA. Pure elements were utilized as standards. For light optical microscopy (LOM), the samples were grinded up to 4000 grit followed by polishing with 3 mm and 1 mm diamond paste, and oxide polishing suspension (OPS), and finally were etched with Ti2 solution (68 vol % glycerine, 16 vol % 70% HNO3, 16 vol % 40% HF). For scanning electron microscopy (SEM), a JEOL JSM-6490 SEM and a JEOL JSM-6500F SEM both operated at 15 kV were used for low- and high-magnification microstructure imaging, respectively. FEI Helios NanoLab DualBeam 600i and 600 instruments combining focused ion beam (FIB) and SEM were used for milling and subsequent secondary electron (SE) SEM imaging. The working distance of the SEM was about 4 mm, 15 kV and 5.5e11 nA were applied for SEM and 30 kV and 21e62 nA were chosen for FIB milling. The depth of milling in the samples typically was about 10 mm. In order to study the kinetics of lamellar coarsening in a quantitative way, the true lamellar spacing ltrue in each sample was measured with the help of FIB milling and subsequent SEM observations as is explained below. The lamellar spacing observed on the sample surface generally does not correspond to the true lamellar spacing as the orientation of the lamellae below the surface is not known. Therefore, cuts perpendicular to the lamellar surface and to the lamellae direction on the surface were milled by FIB. From evaluating the lamellar spacing on the freshly cut surface by observation with SEM, the true lamellar spacing can be determined. The principle of this procedure is shown in Fig. 2. As the FIB milling results only in a small groove of about 10 mm in depth, it is not possible to observe the fresh surface perpendicularly but only under a certain angle b, which in the present case corresponds to the tilting angle necessary to allow FIB milling perpendicular to the surface (b ¼ 52 ). From a simple geometric consideration, it is possible to calculate the true lamellar spacing ltrue by the following formula:
ltrue
1 ¼ n
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 nlpro sina 2 þ nlpro cosa ; sinb
(6)
where lpro stands for the projected lamellar spacing, which could be easily measured on the SEM micrograph, a is the angle between milling edge and direction perpendicular to the lamellae on the projected milling cross section, and n means the number of counted lamellae (depending on coarsening state and chosen SEM magnification, between about n ¼ 20 and 100 lamellae were counted and the distance nlpro was measured). For each sample, this procedure was performed for at least 20 different cut surfaces. 4. Experimental results and discussion 4.1. Kinetics of lamellar coarsening Fig. 3 presents SEM-BSE images of the lamellar microstructures
X. Li et al. / Acta Materialia 127 (2017) 230e243
233
Fig. 2. (a) Schematic illustration of the procedure to determine the true lamellar spacing ltrue from 2D SEM images. The x-y-z coordinate system is defined by the original sample surface (x-y plane), the SEM detector (z direction), and the FIB source (in the x-z plane). By rotating the sample in the x-y plane, the lamellae are oriented parallel to the x-axis in a first step. After that, the sample is tilted around the y-axis by b ¼ 52 to allow FIB milling perpendicular to the surface and for the subsequent observation of the cut surface by SEM. The finally obtained SEM image is a projection (red, transparent area in the x-y plane) of the tilted, cut sample surface (marked in blue and designated as milling cross section). (b) SEM-SE image of a FIB-milled sample in the tilted position showing both the original and the vertically cut surface (i.e., both surfaces are projections on the x-y observation plane of the SEM. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
in both as-cast and various heat-treated states at 1000 C showing that the lamellae coarsen significantly with increasing heat treatment time. Concurrently, the amount of terminations and branches decreases clearly indicating that fault migration and annihilation contribute to the lamellar coarsening. Lamellae start shrinking and some of them even lose their initial shape and begin to spheroidize after only 36 h. During the later stages of the coarsening process, some lamellae merge with each other to form longer lamellae. Fig. 4 demonstrates the great difference of coarsening rates when the heat treatment temperature varies. A heat treatment at 600 C for 24 h has nearly no effect on the lamellar spacing whereas after a heat-treatment for the same time at 1000 C the microstructure has already coarsened so strongly that there are first indications of spheroidization of the FeAl phase. The strong effect of temperature is also clearly visible in Fig. 5, where the increase of average lamellar spacing l is shown as a function of time for the four different heat treatment temperatures. For instance, the l value of the sample that has coarsened at 600 C for 7000 h (590 ± 80 nm) is similar to that at 1000 C after only 10 min (570 ± 40 nm). The inset in Fig. 5 shows the values for the coarsening process at 1000 C on a customized time scale. The average lamellar spacing l0 of the as-cast alloy is 200 ± 40 nm, and has coarsened to 3400 ± 800 nm after only 18 h. After that time, spheroidization of the FeAl lamellae sets in and the determination of a reliable value for the average lamellar spacing is no longer possible. The systematically increasing values of the lamellar spacing l suggest that there is a common mechanism, which governs the coarsening process. For all temperatures, the lðtÞ plots look qualitatively similar resembling at first glance simple power law functions. In order to check if the data fit to one of the models of Graham & Kraft [10] and Cline [21] that had been introduced above in
section 2, all values were plotted as l vs: t and l2 vs: t. As an example, Fig. 6aeb shows the 1000 C data. It is obvious that both models fail to describe the experimental results. Instead we find that the best description of the data is achieved assuming a relation
l3 ¼ l30 þ kt;
(7)
as is presented in Fig. 6c. The same is true for the other temperatures as is shown by respective plots in Fig. 7. All data are very well fitted by a l3 law, as is also confirmed by the resulting high values of the correlation coefficient R2 (R2 > 0.98). As a consequence of this, a characteristic activation energy can be determined from the temperature dependence of the rate constant k. This activation energy Ea is obtained from the Arrhenius equation Ea
kðTÞ ¼ k0 eRT ;
(8)
where R is the gas constant, and k stands for the rate constants, which are taken from the slopes of linear fits in Fig. 7. Fig. 8 shows the Arrhenius plot and from its slope an activation energy Ea for the coarsening process of 265 ± 11 kJ/mol is obtained. This value is very similar to the activation energies of Fe and Al diffusion in Fe-50Al (B2) alloy, which were determined by radiotracer diffusion experiments to be 265 kJ/mol and 258 kJ/mol, respectively, for temperatures between 600 C and 1200 C [37]. Even though no respective values for diffusion in FeAl2 are known from the literature, the similarity of the values can be taken as a clear indication that the lamellar coarsening process is controlled by volume diffusion. 4.2. Volume fraction and thickening of FeAl lamellae From knowing the lattice parameters and compositions of the
234
X. Li et al. / Acta Materialia 127 (2017) 230e243
Fig. 3. Coarsening of the lamellar FeAl þ FeAl2 microstructure of the Fe-60.9 at.% Al after different times at 1000 C (SEM-BSE images, light phase: FeAl, dark phase: FeAl2).
two phases FeAl and FeAl2 [6], their volume fractions can be calculated. In doing so, one finds that in the investigated temperature range between 600 and 1000 C and up to the eutectoid reaction temperature of 1095 C, the volume ratios of the two phases vary only very slightly. One obtains volume fractions of 0.29 (±0.01) for FeAl and 0.71 (±0.01) for FeAl2 for an alloy of composition Fe60.9 at.% Al in the whole temperature range. Since the volume fractions stay constant during the coarsening process, it is to be expected that the thickness of the lamellae increases in the same way as the lamellar distance l. As an example, Fig. 9 shows a plot of the thickness h of the FeAl lamellae vs. the lamellar distance after various heat treatment times at 800 C. Obviously, there is a direct proportionality between h and l
hFeAl ¼ cl ðc ¼ 0:38±0:01Þ;
(9)
meaning the growth kinetics of the FeAl lamellae follows the same type of relation as found above (Eq. (7)) for the lamellar spacing, i.e.
h3 ¼ h30 þ k0t:
(10)
The proportionality constant c corresponds to the hFeAl =l ratio. Assuming ideal plate-like lamellae with constant thickness along
the whole plates and extending completely through the whole grain (i.e., neglecting the existence of terminations), this ratio of 0.38 should correspond to the volume fraction of FeAl phase. The difference between this clearly too high value and the above calculated volume fraction of 0.29 is explained by the existence of the large number of terminations of FeAl lamellae in the real microstructures. In contrast to an ideal stacking of perfect layers or lamellae of the two phases, the FeAl lamellae are frequently interrupted and the intermediate space instead consists of additional FeAl2 phase. 4.3. The effect of colony size on the lamellar coarsening kinetics Since the size of the colonies in the as-cast alloy is very large (>500 mm in length), it can be expected that possible diffusion processes along the colony boundaries do not affect the lamellar coarsening kinetics. In order to find out if and how the colony boundaries influence lamellar coarsening, samples with reduced colony size were produced by hot rotary swaging as described in the experimental chapter. This process results in mostly irregularlyshaped colonies with sizes reduced to less than 300 mm, roughly varying from 10 mm to 300 mm. For the investigation of the lamellar
X. Li et al. / Acta Materialia 127 (2017) 230e243
235
Fig. 3. (continued).
spacings, two groups of colonies with different size ranges were selected, with average diameters of 100e300 mm in the first and <80 mm in the second group. Samples of the swaged material were heat-treated at 800 C for identical times and under identical conditions as had been used for the as-cast material. Fig. 10 shows the measured lamellar spacings during heat treatment at 800 C for both groups of reduced colony size in comparison to the as-cast samples. As during cooling from the swaging temperature (1150 C) already some lamellar coarsening occurs, the starting lamellar spacing of the as-swaged samples is increased and corresponds to a value that is reached by the as-cast material after about 1.5 h holding at 800 C. Therefore, the curves for the two groups with reduced colony sizes were shifted correspondingly along the time scale allowing a better comparison. Even though the values for the colonies with reduced size seem to be slightly higher than those of the as-cast material, they completely agree within the experimental error bars and follow the same coarsening law. Thus, it appears that at least within the investigated range of colony sizes, the colony boundaries do not affect the kinetics of the lamellar coarsening process.
5. Phase field modeling of lamellar coarsening Phase field methods are frequently used for the modeling of microstructure evolution, see e.g. Refs. [38e42] for recent reviews. Concerning the formation of lamellar structures, in particular eutectic solidification has been studied frequently [43e52], but contrary to coarsening with spherical inclusions [53,54], the focus is typically on the growth instead of the coarsening regime. The experimental results of the present work are further supplemented by phase field modeling concerning the value of the coarsening exponent. For that, a generic and isotropic variational phase field alloy model was used, which assigns renormalized equilibrium concentrations c ¼ 0 (phase ‘a’ in our case: FeAl) and c ¼ 1 (phase ‘b’ in our case: FeAl2) for the bulk phases. It is based on the free energy functional
Z F¼
dr fgrad þ fdw þ fc ;
with the gradient square term
(11)
236
X. Li et al. / Acta Materialia 127 (2017) 230e243
Fig. 4. Effect of temperature on the coarsening of the lamellar microstructure after heat treatments for 24 h (SEM-BSE images).
x (dimension length) and the interfacial energy parameter g (dimension energy per volume). The double well potential reads
fdw ¼
6g
x
42 ð1 4Þ2 ;
(13)
which has minima at the bulk states 4 ¼ 0 and 4 ¼ 1. The concentration dependent contribution
fc ¼ hð4Þfb ðcÞ þ ½1 hð4Þfa ðcÞ
(14)
interpolates between the free energy densities of the two phases and determines the phase diagram. Since the present aim is the understanding of growth exponents, we exclude a temperature dependence, which is sufficient to simulate isothermal coarsening. Also, generic free energy densities are used, which are given as Fig. 5. Lamellar spacing l as a function of heat treatment time for different temperatures. The inset shows the 1000 C data with an expanded time scale.
fgrad ¼
3gx ðV4Þ2 ; 2
(12)
which contains the phase field 4, the interface thickness parameter
fa ðcÞ ¼
1 2 bc ; 2
fb ðcÞ ¼
1 bðc 1Þ2 ; 2
(15)
with a coefficient b with dimension energy per volume. According to the common tangent construction the two-phase region is located in the concentration range 0≪c1. The interpolation function to distinguish the phases by the phase field is chosen as
X. Li et al. / Acta Materialia 127 (2017) 230e243
237
Fig. 6. Increase of the lamellar spacing l at 1000 C plotted as ln vs. t according to (a) the Graham & Kraft model (n ¼ 1), (b) the Cline model (n ¼ 2), (c) the present findings (n ¼ 3).
hð4Þ ¼ 42 ð3 24Þ:
(16)
vc D 2 dF ¼ V ; vt b dc
(18)
The non-conserved phase field evolution equation is given by
v4 dF ¼ K vt d4
(17)
with a kinetic coefficient K. The concentration field evolves according to
where for simplicity the diffusion coefficient D is assumed to be phase independent. We assume translation invariance of the lamellar structure in a third direction. Hence, the description becomes effectively twodimensional, and periodic boundary conditions are used. This assumption is legitimate for the planar lamellar microstructures studied here. Spheroidization, in contrast, would require a three
238
X. Li et al. / Acta Materialia 127 (2017) 230e243
Fig. 7. Plots of l3 as a function of t for all investigated temperatures with linear fits l3 ¼ l30 þ kt yielding the rate constants k.
Fig. 9. The increase of the FeAl lamellar thickness hFeAl during coarsening at 800 C plotted vs. the simultaneous increase of lamellar spacing. Whereas the l values can simply be estimated by counting the number of lamellae over a certain distance, the h values must be determined by directly measuring the thickness of individual FeAl lamellae explaining the larger error bars for hFeAl.
Fig. 8. Arrhenius plot showing the temperature dependence of the rate constants k (k in m3/s).
dimensional description to capture the curvature in the third direction. Different regular and irregular lamellar structures are used as initial conditions to simulate the coarsening behavior. A GPGPU
X. Li et al. / Acta Materialia 127 (2017) 230e243
Fig. 10. Comparison of lamellar coarsening at 800 C for different colony sizes.
239
Fig. 13. Plot of the termination shrinkage velocity vs. the lamellar spacing showing 2 agreement with the expected scaling v 1=l according to Eq. (3). The simulations are carried out for fixed system size and varying number of lamellae. Parameters are L/ H ¼ 2 for a system of size L H, x/Dx ¼ 3, x/H ¼ 0.003, H/d0 ¼ 512 with the capillary length d0 ¼ g=b.
evolution equations on an equidistant grid. One of the central requirements for coarsening simulations is the proper representation of curvature effects. For a curved interface between the a and b phase, the equilibrium Gibbs-Thomson effect is correctly reflected by the phase field model, yielding supersaturations
ca ¼
2g*
b
k;
cb ¼ 1 þ
2g*
b
k
(19)
with the local sum of principal interface curvatures k. Without the concentration coupling fc , an equilibrium solution with a planar interface is
4ðxÞ ¼
Fig. 11. Renormalization of the interfacial energy g / g* through the coupling to the concentration field.
implementation is used to accelerate the code for coarsening studies, applying explicit forward Euler discretization of the
1 ð1 þ tanh x=xÞ; 2
(20)
and the corresponding interfacial energy is g. The chemical term fc perturbs the straight equilibrium interface profile and also renormalizes the interfacial energy g/g*, which is obtained numerically and shown in Fig. 11. The Gibbs-Thomson correction Eq. (19) is a central ingredient for the coarsening theories reviewed in section 2 (Eq. (1)). Based on this we verified that the phase field model correctly reproduces the shrinkage of a lamellar termination according to the scaling v 1=l2 (Eq. (3)). To show this, a termination is placed in
Fig. 12. Shrinkage of lamellae, as obtained from two-dimensional phase field simulations. The black (white) regions have 4 > 1/2 (4 < 1/2) and therefore correspond to the b (a) phase. The system size is L H with L ¼ 2H, and periodic boundary conditions are used in all directions. For the discretization, an equidistant grid with lattice spacing Dx was used. The scale separation is given by x=Dx ¼ 3 and l=x ¼ 75.9 with the lamellar spacing being l ¼ H/n with n ¼ 9. The left panel shows the initialization (t ¼ 0) with a truncated lamella, which shrinks in the course of time. The middle and right panel are at dimensionless times Dt=l2 ¼ 3.86 and Dt=l2 ¼ 6.95, respectively. For the parameters we use lb=g ¼ 56.9 and K g2 =Db ¼ 5, which correspond to the bulk-diffusion-limited regime.
240
X. Li et al. / Acta Materialia 127 (2017) 230e243
Fig. 14. Two-dimensional simulation of lamellar coarsening in a quadratic system of dimension L H with L ¼ H, using periodic boundary conditions. The parameters are x/Dx ¼ 3, K g2 /Db ¼ 5, L/x ¼ 5461, xb/g ¼ 1.5. a) Initially (t ¼ 0), 3200 lamellae with equal length L/16 and thickness 13.3x are placed at random positions and the same alignment in the matrix. Subsequent snapshots are taken at times b) t ¼ 53 x2 /D, c) t ¼ 107 x2 /D, d) t ¼ 160 x2 /D.
between perfect lamellae (see Fig. 12) and its shrinkage measured for different lamellar spacings l, as shown in Fig. 13. To investigate the coarsening behavior, different initial conditions were set up with straight parallel lamellae, characterized by sharp values of the concentration and phase field with 4(t¼0) ¼ c(t¼0) ¼ 0 or 1, using periodic boundary conditions. The simulations differ by system dimensions, initial lamellar density, length and distribution, as well as volume fractions of the phases. Computations with equal and random initial lengths of the lamellae were also performed. Essentially, all simulations lead to a similar scaling behavior. A representative case is shown in Fig. 14, where initially 3200 lamellae of equal length are present, which quickly merge and coarsen. For the extraction of the average lamellar spacing and thickness, the number of interfaces where 4 ¼ 1/2 was counted when
traversing through the system in vertical direction and averaging over the entire computational domain, which determines the average lamellar thickness l and height h. Scaling plots for l are shown in Fig. 15. The results support the scaling l3 l30 t as also found in the experiments, although a precise determination of the exponent remains difficult. Similarly to the experiments, the simulations also yield a linear relationship between the lamellar spacing and thickness.
6. Conclusions An Fe-Al alloy with 60.9 at.% Al was produced, which according to the phase diagram [6] undergoes a eutectoid transformation Fe5Al8 / FeAl þ FeAl2 at 1095 C during casting and cooling to room temperature. Due to this reaction, the as-cast alloy shows a
X. Li et al. / Acta Materialia 127 (2017) 230e243
241
Fig. 15. Measured averaged lamella spacing for the simulation in Fig. 14. The values of correlation coefficient R2 of the fitting lines indicate that the exponent n ¼ 3 gives the best fit.
fully lamellar microstructure with volume contents of about 30% FeAl and 70% FeAl2. The kinetics of coarsening of the lamellar microstructure was studied by both experiments and simulations yielding the following findings and conclusions: (1) According to the experimental results, the average lamellar spacing in the as-cast state l0 is 200 ± 40 nm. The isothermal coarsening of the lamellar microstructure can be described as
l3 ¼ l30 þ kt ð T ¼ const:Þ; where l is the lamellar spacing after time t and k is a rate constant.
(2) The lamellar coarsening process is influenced not only by the lamellar shrinkage and termination annihilation, but also by the merging of lamellae. The activation energy of the coarsening process was determined from an Arrhenius plot giving a value of 265 ± 11 kJ/mol. This result indicates that the lamellar coarsening is a volume-diffusion-controlled process. (3) The volume fractions of FeAl and FeAl2 and the ratio of the thickness of the FeAl lamellae hFeAl and the lamellar spacing l do not change during coarsening. Therefore, hFeAl follows a similar coarsening law as l, i.e., h3 ¼ h30 þ k0t. (4) The size of the lamellar colonies (within the investigated range of diameters from <80 to >500 mm) does not affect the
242
X. Li et al. / Acta Materialia 127 (2017) 230e243
coarsening kinetics indicating that colony boundary diffusion plays a negligible role. (5) The simulation results show a general agreement with the experimental findings, in particular concerning the evolution of the lamellar spacing and height. This is remarkable, as the model is rather simple and does not consider many effects, which may play a role for the experimental microstructure evolution, like the role of anisotropies, orientation relationships and strain effects, suggesting that the scaling behavior is rather generic. The experiments and numerical simulations suggest that the coarsening is different from a conventional LSW picture, although the same classical exponent n ¼ 3 seems to be obtained for the scaling of the lamellar spacing. Instead, a scaling theory should involve more than one length scale, at least the spacing l, the thickness h and the length L. Similar to Ref. [10,21] then the lamellar density scales as m ðlLÞ1 . Numerical simulations suggest the scaling m3 t, from which we conclude that the (effective) lamellar length remains essentially constant, similar to the assertions in Ref. [21] and the observed scaling l3 t. As mentioned above, this is only possible if the fast termination shrinkage is compensated by merging of lamellae. Consequently, the appearance of several length scales with different scaling behavior similar to Ref. [55,56] indicates that a LSW-like mean field behavior, which relies on a large separation between the inclusions, breaks down. Acknowledgements The authors would like to thank Mr. G. Bialkowski for cutting samples by EDM, Mrs. I. Wossack for EPMA analysis, and Mr. D. Kurz for chemical analysis. The authors are also grateful for the computing time granted on the supercomputer JURECA at Jülich Supercomputing Centre (JSC). The financial support by the Deutsche Forschungsgemeinschaft (DFG) within the project STE 1077/2 is gratefully acknowledged. In addition, R.S. acknowledges financial support by the DFG via the priority program 1713. References [1] I. Baker, P.R. Munroe, Mechanical properties of FeAl, Int. Mater. Rev. 42 (1997) 181e205. [2] N.S. Stoloff, Iron aluminides - present status and future prospects, Mater. Sci. Eng. A 258 (1998) 1e14. [3] M. Palm, Fe-Al materials for structural applications at high temperatures: current research at MPIE, Int. J. Mater. Res. 100 (2009) 277e287. [4] D.G. Morris, M.A. Morris-Munoz, Recent developments toward the application of iron aluminides in fossil fuel technologies, Adv. Eng. Mater. 13 (2011) 43e47. [5] A. Scherf, D. Janda, M. Baghaie Yazdi, X. Li, F. Stein, M. Heilmaier, Oxidation behavior of binary aluminium-rich FeeAl alloys with a fine-scaled, lamellar microstructure, Oxid. Met. 83 (2015) 559e574. [6] X. Li, A. Scherf, M. Heilmaier, F. Stein, The Al-rich part of the Fe-Al phase diagram, J. Phase Equilib. Diff 37 (2016) 162e173. [7] F. Stein, S.C. Vogel, M. Eumann, M. Palm, Determination of the crystal structure of the ε phase in the Fe-Al system by high-temperature neutron diffraction, Intermetallics 18 (2010) 150e156. [8] A. Scherf, A. Kauffmann, S. Kauffmann-Weiss, T. Scherer, X. Li, F. Stein, M. Heilmaier, Orientation relationship of eutectoid FeAl and FeAl2, J. Appl. Crystallogr. 49 (2016) 442e449. [9] P.J. Maziasz, C.T. Liu, Development of ultrafine lamellar structures in twophase g-TiAl alloys, Metall. Mater. Trans. A 29A (1998) 105e117. [10] L.D. Graham, R.W. Kraft, Coarsening of eutectic microstructures at elevated temperatures, Trans. Metall. Soc. AIME 94 (1966) 9. [11] J.D. Livingston, J.W. Cahn, Discontinuous coarsening of aligned eutectoids, Acta Metall. 22 (1974) 495e503. [12] A.J. Ardell, Coarsening of directionally-solidified eutectic microstructures, in: A. Pechenik, R.K. Kalia, P. Vashishta (Eds.), Computer-aided Design of Hightemperature Materials, Oxford University Press, Inc., New York, 1999, pp. 163e182. [13] P.W. Voorhees, The theory of Ostwald ripening, J. Stat. Phys. 38 (1985) 231e252.
[14] I.M. Lifshitz, V.V. Slyozov, The kinetics of precipitation from supersaturated solid solutions, J. Phys. Chem. Solids 19 (1961) 35e50. €sen (Ost[15] C. Wagner, Theorie der Alterung von Niederschl€ agen durch Umlo wald-Reifung), Zeitschrift für Elektrochemie, Ber. Bunsenges. Phys. Chem. 65 (1961) 581e591. [16] A.J. Ardell, V. Ozolins, Trans-interface diffusion-controlled coarsening, Nat. Mater. 4 (2005) 309e316. [17] A.J. Ardell, Quantitative predictions of the trans-interface diffusion-controlled theory of particle coarsening, Acta Mater. 58 (2010) 4325e4331. [18] A.J. Ardell, Trans-interface-diffusion-controlled coarsening of g’ precipitates in ternary Ni-Al-Cr alloys, Acta Mater. 61 (2013) 7828e7840. [19] A.J. Ardell, Trans-interface-diffusion-controlled coarsening in ternary alloys, Acta Mater. 61 (2013) 7749e7754. [20] Y.G. Nakagawa, G.C. Weatherly, The stability of lamellar structures, Metall. Trans. 3 (1972) 3223e3229. [21] H.E. Cline, Shape instabilities of eutectic composites at elevated temperatures, Acta Metall. 19 (1971) 10. [22] G.C. Weatherly, Y.G. Nakagawa, Coarsening of rod eutectics by fault migration, Scr. Metall. 5 (1971) 777e781. [23] Y.G. Nakagawa, G.C. Weatherly, E. Ho, Coarsening of lamellar structures at elevated temperatures, T. Jpn. I. Met. 15 (1974) 114e120. [24] L.Y. Lin, T.H. Courtney, Direct observations of lamellar fault migration in the Pb-Sn eutectic, Metall. Trans. 5 (1974) 513e514. [25] Y.Y.L. Tian, R.W. Kraft, Kinetics of pearlite spheroidizations, Metall. Trans. A 18 (1987) 1359e1369. [26] Y.L. Tian, R.W. Kraft, Mechanisms of pearlite spheroidization, Metall. Trans. A 18 (1987) 1403e1414. [27] M.F. Bartholomeusz, J.A. Wert, The effect of thermal exposure on microstructural stability and creep resistance of a two-phase TiAl/Ti3Al lamellar alloy, Metall. Mater. Trans. A 25 (1994) 2371e2381. [28] M.J. Whiting, P. Tsakiropoulos, Morphological evolution of lamellar structures: the Cu-Al eutectoid, Acta Mater. 45 (1997) 2027e2042. [29] G. Sharma, R.V. Ramanujan, G.P. Tiwari, Instability mechanisms in lamellar microstructures, Acta Mater. 48 (2000) 15. [30] R.J. El-Khozondar, H.J. El-Khozondar, Numerical simulations of coarsening of lamellar structures applications to metallic alloys, Egypt. J. Solids 27 (2004) 11. [31] J.S. Langer, Instabilities and pattern formation in crystal growth, Rev. Mod. Phys. 52 (1980) 1e28. [32] J.P. Gollub, J.S. Langer, Pattern formation in nonequilibrium physics, Rev. Mod. Phys. 71 (1999) S396eS403. [33] J.B. Brady, Coarsening of fine-scale exsolution lamellae, Am. Mineral. 72 (1987) 697e706. [34] R.H. McCallister, The coarsening kinetics associated with exsolution in an iron-free clinopyroxene, Contrib. Mineral. Petr. 65 (1978) 327e331. [35] R.A. Yund, P. Davidson, Kinetics of lamellar coarsening in cryptoperthites, Am. Mineral. 63 (1978) 470e477. [36] I. Baker, F. Meng, Lamellar coarsening in Fe28Ni18Mn33Al21 and its influence on room temperature tensile behavior, Acta Mater. 95 (2015) 124e131. [37] H. Mehrer, M. Eggersmann, A. Gude, M. Salamon, B. Sepiol, Diffusion in intermetallic phases of the Fe-Al and Fe-Si systems, Mater. Sci. Eng. A 239e240 (1997) 889e898. [38] L.Q. Chen, Phase-field models for microstructure evolution, Annu. Rev. Mater. Res. 32 (2002) 113e140. [39] W.J. Boettinger, J.A. Warren, C. Beckermann, A. Karma, Phase field simulation of solidification, Annu. Rev. Mater. Res. 32 (2002) 163e194. [40] I. Steinbach, Phase-field models in materials science, Modell. Simul. Mater. Sci. Eng. 17 (2009) 073001. [41] R. Spatschek, E. Brener, A. Karma, Phase field modeling of crack propagation, Phil Mag. 91 (2011) 75e95. [42] M. Militzer, Phase field modeling of microstructure evolution in steels, Curr. Opin. Solid State Mater. Sci. 15 (2011) 106e115. [43] F. Drolet, K.R. Elder, M. Grant, J.M. Kosterlitz, Phase-field modeling of eutectic growth, Phys. Rev. E 61 (2000) 6705e6720. [44] M. Plapp, A. Karma, Eutectic colony formation: a phase-field study, Phys. Rev. E 66 (2002) 061608. [45] A. Parisi, M. Plapp, Stability of lamellar eutectic growth, Acta Mater. 56 (2008) 1348e1357. [46] M. Asta, C. Beckermann, A. Karma, W. Kurz, R. Napolitano, M. Plapp, G. Purdy, M. Rappaz, R. Trivedi, Solidification microstructures and solid-state parallels: recent developments, future directions, Acta Mater. 57 (2009) 941e971. [47] A. Choudhury, M. Plapp, B. Nestler, Theoretical and numerical study of lamellar eutectic three-phase growth in ternary alloys, Phys. Rev. E 83 (2011) 051608. [48] K. Ankit, A. Choudhury, C. Qin, S. Schulz, M. McDaniel, B. Nestler, Theoretical and numerical study of lamellar eutectoid growth influenced by volume diffusion, Acta Mater. 61 (2013) 4245e4253. [49] K. Ankit, R. Mukherjee, T. Mittnacht, B. Nestler, Deviations from cooperative growth mode during eutectoid transformation: insights from a phase-field approach, Acta Mater. 81 (2014) 204e210. [50] K. Ankit, R. Mukherjee, B. Nestler, Deviations from cooperative growth mode during eutectoid transformation: mechanisms of polycrystalline eutectoid evolution in FeeC steels, Acta Mater. 97 (2015) 316e324. [51] S. Ghosh, A. Choudhury, M. Plapp, S. Bottin-Rousseau, G. Faivre, S. Akamatsu, Interphase anisotropy effects on lamellar eutectics: a numerical study, Phys.
X. Li et al. / Acta Materialia 127 (2017) 230e243 Rev. E 91 (2015) 022407. [52] S. Akamatsu, M. Plapp, Eutectic and peritectic solidification patterns, Curr. Opin. Solid State Mater. Sci. 20 (2016) 46e54. [53] D. Fan, S.P. Chen, L.-Q. Chen, P.W. Voorhees, Phase-field simulation of 2-D Ostwald ripening in the high volume fraction regime, Acta Mater. 50 (2002) 1895e1907.
243
[54] S.G. Kim, Large-scale three-dimensional simulation of Ostwald ripening, Acta Mater. 55 (2007) 6513e6525. [55] E.A. Brener, V.I. Marchenko, H. Müller-Krumbhaar, R. Spatschek, Coarsening kinetics with elastic effects, Phys. Rev. Lett. 84 (2000) 4914e4917. [56] E.A. Brener, H. Müller-Krumbhaar, R. Spatschek, Coarsening of cracks in a uniaxially strained solid, Phys. Rev. Lett. 86 (2001) 1291e1294.
Contents lists available at ScienceDirect
Acta Materialia journal homepage: www.elsevier.com/locate/actamat
Full length article
Coarsening kinetics of lamellar microstructures: Experiments and simulations on a fully-lamellar Fe-Al in situ composite X. Li a, F. Bottler a, R. Spatschek a, b, A. Schmitt c, M. Heilmaier c, F. Stein a, * a
Max-Planck-Institut für Eisenforschung GmbH, Düsseldorf, Germany IEK-2, Research Center Jülich, Jülich, Germany c Institute for Applied Materials (IAM), Karlsruhe Institute of Technology (KIT), Karlsruhe, Germany b
a r t i c l e i n f o
a b s t r a c t
Article history: Received 18 November 2016 Received in revised form 16 January 2017 Accepted 21 January 2017 Available online 23 January 2017
A very fine lamellar microstructure consisting of the intermetallic phases FeAl and FeAl2 forms in binary Fe-Al alloys due to the eutectoid reaction Fe5Al8 4 FeAl þ FeAl2, which takes place in the Al range between 56.0 and 64.4 at.% at 1095 C. A fully lamellar microstructure is obtained at the eutectoid composition 60.9 at.% Al. The initial lamellar spacing l0 of as-cast material is 200 ± 40 nm. In this study, the kinetics of coarsening of the FeAl þ FeAl2 lamellar microstructure is investigated at four different temperatures in the range 600 Ce1000 C with holding times from 10 min to 7000 h. It is found that the increase of the lamellar spacing can be described as l3 ¼ l30 þ kt. The value obtained for the activation energy proves that the lamellar coarsening is a volume-diffusion-controlled process. Besides the experimental investigations, phase-field modeling is used to simulate the lamellar coarsening. The results are in good agreement with the experimental observations with regard to the evolution of the lamellar morphology and the value of the coarsening exponent. © 2017 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
Keywords: Coarsening kinetics Intermetallics (iron-aluminides) Lamella Microstructure Phase field modeling
1. Introduction Fe-Al alloys with up to 50 at.% Al show excellent corrosion resistance and have a low density compared to steels, and hence, they became promising materials for structural applications [1e4]. When the Al content is above 50 at.%, the beneficial corrosion behavior persists [5] and the alloy density will further decrease. According to the Fe-Al phase diagram in Ref. [6], Fe-Al alloys in the range from 53.0 to 64.4 at.% Al contain the body-centered cubic phase Fe5Al8 (Cu5Zn8-type crystal structure [7]) at high temperature, which decomposes into cubic B2 FeAl and triclinic FeAl2 through a eutectoid reaction at 1095 C [6]. The orientation relationship between the two phases can be described by ð101ÞFeAl k ð114ÞFeAl2 and ½111FeAl k ½110FeAl2 [8]. The eutectoid transformation is a rapid reaction leading to the formation of a finescaled, fully lamellar microstructure at the eutectoid point (where the Al content is 60.9 at.% [6]), see, e.g., Fig. 1a showing the microstructure after 10 min holding at 1000 C. Similar lamellar microstructures are known from a2 þ g Ti-Al based alloys, where they were found to exhibit balanced properties with regard to
* Corresponding author. E-mail address: [email protected] (F. Stein). http://dx.doi.org/10.1016/j.actamat.2017.01.041 1359-6454/© 2017 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
creep, ductility and strength at elevated temperature [9]. In case of the Fe-Al system, there is no information in the literature about the mechanical behavior of such microstructures. When thinking about the capability for high-temperature structural applications, especially the stability of the lamellar structure is of high relevance. For this reason, the kinetics of lamellar coarsening in a fully lamellar FeAl þ FeAl2 alloy is studied in detail here. Another motivation to perform these investigations is that the extended, nearly perfect lamellar microstructures in the present samples offer an excellent possibility to obtain a broad and reliable database for evaluating existing theoretical models for the kinetics of lamellar coarsening. To this end, heat treatments were performed at 600, 700, 800, and 1000 C for various times ranging from 10 min to 7000 h. The effect of the colony size on lamellar coarsening was investigated as well. Finally, the experimental observations are complemented by simulation results on the lamellar coarsening behavior applying phase field modeling. 2. Overview of coarsening theories Eutectoid or eutectic lamellar microstructures can coarsen either continuously or discontinuously (see e.g., Ref. [10e12]). Models for the latter case are not included in the following
X. Li et al. / Acta Materialia 127 (2017) 230e243
231
Fig. 1. (a) Typical lamellar FeAl (light) þ FeAl2 (dark) microstructure of a heat-treated Fe-60.9 at.% Al sample (1000 C/10 min) showing characteristic terminations and branches of the FeAl phase (SEM micrograph obtained in back-scattered electron (BSE) mode); (b) fully lamellar microstructure of the as-cast alloy showing extended colonies (LOM micrograph; the dotted lines mark the colony boundaries).
overview as the samples investigated in the present study solely show continuous coarsening. The following brief overview will show that until today there exists no generally accepted model to describe the coarsening of lamellar microstructures. This is also related to the fact that there are only comparably few experimental data about lamellar coarsening in the literature, which do not allow verifying reliably the different approaches. A.J. Ardell states in his review [12] about coarsening of eutectic microstructures that “clearly much more work is needed on the coarsening of lamellar eutectics before any conclusion can be drawn regarding which theory is correct” [12]. This statement is still correct today. Eutectic or eutectoid reactions can produce a microstructure that consists of alternating lamellae of two phases or rods of the minor phase embedded in a continuous matrix of the second phase [12]. Lamellar microstructures appear in scanning electron microscope (SEM) micrographs as needle-like morphologies due to the two-dimensional imaging. An example is given in Fig. 1a, also showing the typical types of faults, i.e., branching and terminating lamellae. The extension of the lamellar colonies in the present Fe-Al alloy (Fig. 1b) corresponds to the grain size of the parent Fe5Al8 grains. In order to reach thermodynamic equilibrium, the total energy of a two-phase system can be minimized by decreasing the surface energy via reducing the interfacial area of the phases. This microstructure coarsening process is the basic concept of the theory of Ostwald ripening [13]. The driving force for this process is the curvature dependence of the chemical potential, which is
m ¼ m0 þ Vm gk;
(1)
which depends on several factors including the interfacial free energy and the diffusion coefficient. The most recent model for the kinetics of particle coarsening was developed by Ardell and Ozolins [16] and later elaborated by Ardell [17e19]. In contrast to the classical, Ostwald-ripening-based models, according to which coarsening is controlled by matrix diffusion, they assume that trans-interface diffusion processes control coarsening. The model predicts a r n growth law with an exponent n in the range of 2 and 3 depending on the width of the precipitate-matrix interface and the distribution of particle sizes [17,18]. In case of lamellar microstructures, the situation significantly differs from the above spherical growth. An ideal, i.e., faultless, extended lamellar microstructure would not coarsen as there is no driving force (diffusion of atoms from one to the next lamella or along the interface would not reduce the surface energy), i.e. lamellar structures are inherently (meta-)stable [20]. However, real lamellar microstructures contain faults like branches and terminations of lamellae, and it is well accepted in the literature that any coarsening mechanism must be based on the migration of such faults [10,20e30]. The underlying driving force for the fault migration is the interface curvature at the lamellae terminations, which leads to modifications of the local equilibrium conditions due to the Gibbs-Thomson effect [31,32]. At the interfaces, the concentrations locally deviate from the bulk equilibrium values by a contribution proportional to the local interface curvature. Since the signs of the curvature differ between a branch and a termination, there is a concentration difference between them. As the local curvature and the diffusional flux are directly related to the lamellar spacing l (yielding an inverse proportionality), it can be concluded that the shrinkage velocity n scales as (see, e.g. [23])
.
where k is the mean interfacial curvature, m0 stands for the chemical potential of an atom at a flat interface, Vm is the molar volume and g represents the surface energy [13]. Based on the concept of Ostwald ripening, Lifshitz and Slyozov [14] and Wagner [15] (LSW) established their nowadays wellestablished model for the coarsening of spherical precipitates (radius r) in a fluid matrix ending with the equation
Based on this idea of fault migration, Graham and Kraft were the first to develop a dedicated model for lamellar coarsening [10]. Assuming that the total number of lamellae remains constant and only lamellar length and spacing change by branch and termination cancellation, their model ends with a simple linear relation,
r 3 ¼ r03 þ k1 t:
l ¼ l0 þ k2 t;
(2)
Here, r0 stands for the initial particle radius and k1 is a constant,
n 1 l2 :
(3)
(4)
232
X. Li et al. / Acta Materialia 127 (2017) 230e243
where l0 stands for the initial lamellar spacing and k2 is a constant [10]. A few years later, Cline [21] made some changes to the above model taking into account the influence of fault-annihilation, which was neglected in the coarsening model of Graham and Kraft. The Cline model results in the equation
l2 ¼ l20 þ k3 t;
(5)
where k3 is another rate constant [21]. Brady [33] established a very similar coarsening model for exsolution lamellae in silicate minerals. He assumed the terminations of lamellae as ideal wedge-shaped ends and ended with the same equation as Cline did. The validity of the equation was additionally tested by comparing with experimental data of coarsening of mineral exsolution lamellae reported by McCallister [34], and Yund and Davidson [35], which in the original papers had been plotted according to a t 1=3 rate law. Even though Brady [33] stated that the square root law gives a good description of the data, McCallister [34] pointed out that fitting his data vs. t 1=2 , t 1=3 , or t 1=4 results in similar correlation coefficients not allowing a definite conclusion which exponent is the correct one. In a more recent study, Baker and Meng [36] measured the coarsening of B2 þ fcc lamellae in an Fe28Ni18Mn33Al21 alloy annealed at 900 C. They tested various growth exponents n in the range 2e3 finding that, although for n ¼ 2 they obtained the highest correlation coefficient, “n values from 2 to 3 could reasonably be used to describe the coarsening data” [36].
3. Experimental An Fe-Al alloy containing 60.9 at.% Al with a fully lamellar microstructure was produced by vacuum induction melting from high purity Fe (99.98 wt%) and Al (99.99 wt%) under argon and was cast in a cold copper mould with a diameter of 18 mm [6]. The impurity content of this alloy was determined by chemical analysis with P being measured by inductively coupled plasma-optical emission spectrometry (ICP-OES), C, O and S by infrared absorption spectroscopy and N by thermal conductivity measurement. The results are given in Table 1. The resulting rod was cut by electrical discharge machining (EDM) into several parts and a series of heat-treatments was performed, see Table 2. The samples heattreated for times of less than 24 h were water-quenched to room temperature, while all other samples heat-treated for longer times (24 h) were air-cooled. A 9 cm long piece of the as-cast rod was rotary-swaged at 1150 C, where the alloy is single phase Fe5Al8, in order to reduce the grain size and with that the colony size of the lamellar material. For protection of the brittle material from immediate breaking, the as-cast piece of the rod was wrapped in Mo foil and jacketed with a steel encapsulation before rotary swaging. This assembly was heated up to 1150 C, taken out of the furnace, and immediately processed in the rotary swaging machine. The diameter was reduced in seven passes (including intermediate re-heating to 1150 C) to about 50%. After dismantling, samples were cut from the swaged material and heat-treated for various times at 800 C (see Table 2) to study lamellar coarsening. Phase and overall chemical compositions were measured by
Table 1 Impurity contents in the as-cast Fe-60.9 at.% Al alloy as obtained by chemical analysis. Element
C
N
O
P
S
wt. ppm
37
16
200e370
<20
<10
Table 2 Heat treatment details of the Fe-60.9 at.% Al samples. All heat treatments were performed in Ar atmosphere. Temperature ( C) Time (h) 600 C 700 C 800 C 1000 C 800 C
Starting material
24, 1000, 3000, 5000, 7000 as-cast 24, 50, 100, 300, 1000, 2000, 3000, 5000 as-cast 0.5, 1, 3.5, 6, 18, 24, 48 as-cast 0.17, 0.5, 1, 3.5, 6, 18, 24, 36 as-cast 0.5, 1, 3.5, 6 cast and swaged
wavelength dispersive spectroscopy (WDS) that was carried out with an electron probe micro-analyser (EPMA) JOEL-JXA-8100. The instrument was operated at 15 kV and 20 nA. Pure elements were utilized as standards. For light optical microscopy (LOM), the samples were grinded up to 4000 grit followed by polishing with 3 mm and 1 mm diamond paste, and oxide polishing suspension (OPS), and finally were etched with Ti2 solution (68 vol % glycerine, 16 vol % 70% HNO3, 16 vol % 40% HF). For scanning electron microscopy (SEM), a JEOL JSM-6490 SEM and a JEOL JSM-6500F SEM both operated at 15 kV were used for low- and high-magnification microstructure imaging, respectively. FEI Helios NanoLab DualBeam 600i and 600 instruments combining focused ion beam (FIB) and SEM were used for milling and subsequent secondary electron (SE) SEM imaging. The working distance of the SEM was about 4 mm, 15 kV and 5.5e11 nA were applied for SEM and 30 kV and 21e62 nA were chosen for FIB milling. The depth of milling in the samples typically was about 10 mm. In order to study the kinetics of lamellar coarsening in a quantitative way, the true lamellar spacing ltrue in each sample was measured with the help of FIB milling and subsequent SEM observations as is explained below. The lamellar spacing observed on the sample surface generally does not correspond to the true lamellar spacing as the orientation of the lamellae below the surface is not known. Therefore, cuts perpendicular to the lamellar surface and to the lamellae direction on the surface were milled by FIB. From evaluating the lamellar spacing on the freshly cut surface by observation with SEM, the true lamellar spacing can be determined. The principle of this procedure is shown in Fig. 2. As the FIB milling results only in a small groove of about 10 mm in depth, it is not possible to observe the fresh surface perpendicularly but only under a certain angle b, which in the present case corresponds to the tilting angle necessary to allow FIB milling perpendicular to the surface (b ¼ 52 ). From a simple geometric consideration, it is possible to calculate the true lamellar spacing ltrue by the following formula:
ltrue
1 ¼ n
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 nlpro sina 2 þ nlpro cosa ; sinb
(6)
where lpro stands for the projected lamellar spacing, which could be easily measured on the SEM micrograph, a is the angle between milling edge and direction perpendicular to the lamellae on the projected milling cross section, and n means the number of counted lamellae (depending on coarsening state and chosen SEM magnification, between about n ¼ 20 and 100 lamellae were counted and the distance nlpro was measured). For each sample, this procedure was performed for at least 20 different cut surfaces. 4. Experimental results and discussion 4.1. Kinetics of lamellar coarsening Fig. 3 presents SEM-BSE images of the lamellar microstructures
X. Li et al. / Acta Materialia 127 (2017) 230e243
233
Fig. 2. (a) Schematic illustration of the procedure to determine the true lamellar spacing ltrue from 2D SEM images. The x-y-z coordinate system is defined by the original sample surface (x-y plane), the SEM detector (z direction), and the FIB source (in the x-z plane). By rotating the sample in the x-y plane, the lamellae are oriented parallel to the x-axis in a first step. After that, the sample is tilted around the y-axis by b ¼ 52 to allow FIB milling perpendicular to the surface and for the subsequent observation of the cut surface by SEM. The finally obtained SEM image is a projection (red, transparent area in the x-y plane) of the tilted, cut sample surface (marked in blue and designated as milling cross section). (b) SEM-SE image of a FIB-milled sample in the tilted position showing both the original and the vertically cut surface (i.e., both surfaces are projections on the x-y observation plane of the SEM. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
in both as-cast and various heat-treated states at 1000 C showing that the lamellae coarsen significantly with increasing heat treatment time. Concurrently, the amount of terminations and branches decreases clearly indicating that fault migration and annihilation contribute to the lamellar coarsening. Lamellae start shrinking and some of them even lose their initial shape and begin to spheroidize after only 36 h. During the later stages of the coarsening process, some lamellae merge with each other to form longer lamellae. Fig. 4 demonstrates the great difference of coarsening rates when the heat treatment temperature varies. A heat treatment at 600 C for 24 h has nearly no effect on the lamellar spacing whereas after a heat-treatment for the same time at 1000 C the microstructure has already coarsened so strongly that there are first indications of spheroidization of the FeAl phase. The strong effect of temperature is also clearly visible in Fig. 5, where the increase of average lamellar spacing l is shown as a function of time for the four different heat treatment temperatures. For instance, the l value of the sample that has coarsened at 600 C for 7000 h (590 ± 80 nm) is similar to that at 1000 C after only 10 min (570 ± 40 nm). The inset in Fig. 5 shows the values for the coarsening process at 1000 C on a customized time scale. The average lamellar spacing l0 of the as-cast alloy is 200 ± 40 nm, and has coarsened to 3400 ± 800 nm after only 18 h. After that time, spheroidization of the FeAl lamellae sets in and the determination of a reliable value for the average lamellar spacing is no longer possible. The systematically increasing values of the lamellar spacing l suggest that there is a common mechanism, which governs the coarsening process. For all temperatures, the lðtÞ plots look qualitatively similar resembling at first glance simple power law functions. In order to check if the data fit to one of the models of Graham & Kraft [10] and Cline [21] that had been introduced above in
section 2, all values were plotted as l vs: t and l2 vs: t. As an example, Fig. 6aeb shows the 1000 C data. It is obvious that both models fail to describe the experimental results. Instead we find that the best description of the data is achieved assuming a relation
l3 ¼ l30 þ kt;
(7)
as is presented in Fig. 6c. The same is true for the other temperatures as is shown by respective plots in Fig. 7. All data are very well fitted by a l3 law, as is also confirmed by the resulting high values of the correlation coefficient R2 (R2 > 0.98). As a consequence of this, a characteristic activation energy can be determined from the temperature dependence of the rate constant k. This activation energy Ea is obtained from the Arrhenius equation Ea
kðTÞ ¼ k0 eRT ;
(8)
where R is the gas constant, and k stands for the rate constants, which are taken from the slopes of linear fits in Fig. 7. Fig. 8 shows the Arrhenius plot and from its slope an activation energy Ea for the coarsening process of 265 ± 11 kJ/mol is obtained. This value is very similar to the activation energies of Fe and Al diffusion in Fe-50Al (B2) alloy, which were determined by radiotracer diffusion experiments to be 265 kJ/mol and 258 kJ/mol, respectively, for temperatures between 600 C and 1200 C [37]. Even though no respective values for diffusion in FeAl2 are known from the literature, the similarity of the values can be taken as a clear indication that the lamellar coarsening process is controlled by volume diffusion. 4.2. Volume fraction and thickening of FeAl lamellae From knowing the lattice parameters and compositions of the
234
X. Li et al. / Acta Materialia 127 (2017) 230e243
Fig. 3. Coarsening of the lamellar FeAl þ FeAl2 microstructure of the Fe-60.9 at.% Al after different times at 1000 C (SEM-BSE images, light phase: FeAl, dark phase: FeAl2).
two phases FeAl and FeAl2 [6], their volume fractions can be calculated. In doing so, one finds that in the investigated temperature range between 600 and 1000 C and up to the eutectoid reaction temperature of 1095 C, the volume ratios of the two phases vary only very slightly. One obtains volume fractions of 0.29 (±0.01) for FeAl and 0.71 (±0.01) for FeAl2 for an alloy of composition Fe60.9 at.% Al in the whole temperature range. Since the volume fractions stay constant during the coarsening process, it is to be expected that the thickness of the lamellae increases in the same way as the lamellar distance l. As an example, Fig. 9 shows a plot of the thickness h of the FeAl lamellae vs. the lamellar distance after various heat treatment times at 800 C. Obviously, there is a direct proportionality between h and l
hFeAl ¼ cl ðc ¼ 0:38±0:01Þ;
(9)
meaning the growth kinetics of the FeAl lamellae follows the same type of relation as found above (Eq. (7)) for the lamellar spacing, i.e.
h3 ¼ h30 þ k0t:
(10)
The proportionality constant c corresponds to the hFeAl =l ratio. Assuming ideal plate-like lamellae with constant thickness along
the whole plates and extending completely through the whole grain (i.e., neglecting the existence of terminations), this ratio of 0.38 should correspond to the volume fraction of FeAl phase. The difference between this clearly too high value and the above calculated volume fraction of 0.29 is explained by the existence of the large number of terminations of FeAl lamellae in the real microstructures. In contrast to an ideal stacking of perfect layers or lamellae of the two phases, the FeAl lamellae are frequently interrupted and the intermediate space instead consists of additional FeAl2 phase. 4.3. The effect of colony size on the lamellar coarsening kinetics Since the size of the colonies in the as-cast alloy is very large (>500 mm in length), it can be expected that possible diffusion processes along the colony boundaries do not affect the lamellar coarsening kinetics. In order to find out if and how the colony boundaries influence lamellar coarsening, samples with reduced colony size were produced by hot rotary swaging as described in the experimental chapter. This process results in mostly irregularlyshaped colonies with sizes reduced to less than 300 mm, roughly varying from 10 mm to 300 mm. For the investigation of the lamellar
X. Li et al. / Acta Materialia 127 (2017) 230e243
235
Fig. 3. (continued).
spacings, two groups of colonies with different size ranges were selected, with average diameters of 100e300 mm in the first and <80 mm in the second group. Samples of the swaged material were heat-treated at 800 C for identical times and under identical conditions as had been used for the as-cast material. Fig. 10 shows the measured lamellar spacings during heat treatment at 800 C for both groups of reduced colony size in comparison to the as-cast samples. As during cooling from the swaging temperature (1150 C) already some lamellar coarsening occurs, the starting lamellar spacing of the as-swaged samples is increased and corresponds to a value that is reached by the as-cast material after about 1.5 h holding at 800 C. Therefore, the curves for the two groups with reduced colony sizes were shifted correspondingly along the time scale allowing a better comparison. Even though the values for the colonies with reduced size seem to be slightly higher than those of the as-cast material, they completely agree within the experimental error bars and follow the same coarsening law. Thus, it appears that at least within the investigated range of colony sizes, the colony boundaries do not affect the kinetics of the lamellar coarsening process.
5. Phase field modeling of lamellar coarsening Phase field methods are frequently used for the modeling of microstructure evolution, see e.g. Refs. [38e42] for recent reviews. Concerning the formation of lamellar structures, in particular eutectic solidification has been studied frequently [43e52], but contrary to coarsening with spherical inclusions [53,54], the focus is typically on the growth instead of the coarsening regime. The experimental results of the present work are further supplemented by phase field modeling concerning the value of the coarsening exponent. For that, a generic and isotropic variational phase field alloy model was used, which assigns renormalized equilibrium concentrations c ¼ 0 (phase ‘a’ in our case: FeAl) and c ¼ 1 (phase ‘b’ in our case: FeAl2) for the bulk phases. It is based on the free energy functional
Z F¼
dr fgrad þ fdw þ fc ;
with the gradient square term
(11)
236
X. Li et al. / Acta Materialia 127 (2017) 230e243
Fig. 4. Effect of temperature on the coarsening of the lamellar microstructure after heat treatments for 24 h (SEM-BSE images).
x (dimension length) and the interfacial energy parameter g (dimension energy per volume). The double well potential reads
fdw ¼
6g
x
42 ð1 4Þ2 ;
(13)
which has minima at the bulk states 4 ¼ 0 and 4 ¼ 1. The concentration dependent contribution
fc ¼ hð4Þfb ðcÞ þ ½1 hð4Þfa ðcÞ
(14)
interpolates between the free energy densities of the two phases and determines the phase diagram. Since the present aim is the understanding of growth exponents, we exclude a temperature dependence, which is sufficient to simulate isothermal coarsening. Also, generic free energy densities are used, which are given as Fig. 5. Lamellar spacing l as a function of heat treatment time for different temperatures. The inset shows the 1000 C data with an expanded time scale.
fgrad ¼
3gx ðV4Þ2 ; 2
(12)
which contains the phase field 4, the interface thickness parameter
fa ðcÞ ¼
1 2 bc ; 2
fb ðcÞ ¼
1 bðc 1Þ2 ; 2
(15)
with a coefficient b with dimension energy per volume. According to the common tangent construction the two-phase region is located in the concentration range 0≪c1. The interpolation function to distinguish the phases by the phase field is chosen as
X. Li et al. / Acta Materialia 127 (2017) 230e243
237
Fig. 6. Increase of the lamellar spacing l at 1000 C plotted as ln vs. t according to (a) the Graham & Kraft model (n ¼ 1), (b) the Cline model (n ¼ 2), (c) the present findings (n ¼ 3).
hð4Þ ¼ 42 ð3 24Þ:
(16)
vc D 2 dF ¼ V ; vt b dc
(18)
The non-conserved phase field evolution equation is given by
v4 dF ¼ K vt d4
(17)
with a kinetic coefficient K. The concentration field evolves according to
where for simplicity the diffusion coefficient D is assumed to be phase independent. We assume translation invariance of the lamellar structure in a third direction. Hence, the description becomes effectively twodimensional, and periodic boundary conditions are used. This assumption is legitimate for the planar lamellar microstructures studied here. Spheroidization, in contrast, would require a three
238
X. Li et al. / Acta Materialia 127 (2017) 230e243
Fig. 7. Plots of l3 as a function of t for all investigated temperatures with linear fits l3 ¼ l30 þ kt yielding the rate constants k.
Fig. 9. The increase of the FeAl lamellar thickness hFeAl during coarsening at 800 C plotted vs. the simultaneous increase of lamellar spacing. Whereas the l values can simply be estimated by counting the number of lamellae over a certain distance, the h values must be determined by directly measuring the thickness of individual FeAl lamellae explaining the larger error bars for hFeAl.
Fig. 8. Arrhenius plot showing the temperature dependence of the rate constants k (k in m3/s).
dimensional description to capture the curvature in the third direction. Different regular and irregular lamellar structures are used as initial conditions to simulate the coarsening behavior. A GPGPU
X. Li et al. / Acta Materialia 127 (2017) 230e243
Fig. 10. Comparison of lamellar coarsening at 800 C for different colony sizes.
239
Fig. 13. Plot of the termination shrinkage velocity vs. the lamellar spacing showing 2 agreement with the expected scaling v 1=l according to Eq. (3). The simulations are carried out for fixed system size and varying number of lamellae. Parameters are L/ H ¼ 2 for a system of size L H, x/Dx ¼ 3, x/H ¼ 0.003, H/d0 ¼ 512 with the capillary length d0 ¼ g=b.
evolution equations on an equidistant grid. One of the central requirements for coarsening simulations is the proper representation of curvature effects. For a curved interface between the a and b phase, the equilibrium Gibbs-Thomson effect is correctly reflected by the phase field model, yielding supersaturations
ca ¼
2g*
b
k;
cb ¼ 1 þ
2g*
b
k
(19)
with the local sum of principal interface curvatures k. Without the concentration coupling fc , an equilibrium solution with a planar interface is
4ðxÞ ¼
Fig. 11. Renormalization of the interfacial energy g / g* through the coupling to the concentration field.
implementation is used to accelerate the code for coarsening studies, applying explicit forward Euler discretization of the
1 ð1 þ tanh x=xÞ; 2
(20)
and the corresponding interfacial energy is g. The chemical term fc perturbs the straight equilibrium interface profile and also renormalizes the interfacial energy g/g*, which is obtained numerically and shown in Fig. 11. The Gibbs-Thomson correction Eq. (19) is a central ingredient for the coarsening theories reviewed in section 2 (Eq. (1)). Based on this we verified that the phase field model correctly reproduces the shrinkage of a lamellar termination according to the scaling v 1=l2 (Eq. (3)). To show this, a termination is placed in
Fig. 12. Shrinkage of lamellae, as obtained from two-dimensional phase field simulations. The black (white) regions have 4 > 1/2 (4 < 1/2) and therefore correspond to the b (a) phase. The system size is L H with L ¼ 2H, and periodic boundary conditions are used in all directions. For the discretization, an equidistant grid with lattice spacing Dx was used. The scale separation is given by x=Dx ¼ 3 and l=x ¼ 75.9 with the lamellar spacing being l ¼ H/n with n ¼ 9. The left panel shows the initialization (t ¼ 0) with a truncated lamella, which shrinks in the course of time. The middle and right panel are at dimensionless times Dt=l2 ¼ 3.86 and Dt=l2 ¼ 6.95, respectively. For the parameters we use lb=g ¼ 56.9 and K g2 =Db ¼ 5, which correspond to the bulk-diffusion-limited regime.
240
X. Li et al. / Acta Materialia 127 (2017) 230e243
Fig. 14. Two-dimensional simulation of lamellar coarsening in a quadratic system of dimension L H with L ¼ H, using periodic boundary conditions. The parameters are x/Dx ¼ 3, K g2 /Db ¼ 5, L/x ¼ 5461, xb/g ¼ 1.5. a) Initially (t ¼ 0), 3200 lamellae with equal length L/16 and thickness 13.3x are placed at random positions and the same alignment in the matrix. Subsequent snapshots are taken at times b) t ¼ 53 x2 /D, c) t ¼ 107 x2 /D, d) t ¼ 160 x2 /D.
between perfect lamellae (see Fig. 12) and its shrinkage measured for different lamellar spacings l, as shown in Fig. 13. To investigate the coarsening behavior, different initial conditions were set up with straight parallel lamellae, characterized by sharp values of the concentration and phase field with 4(t¼0) ¼ c(t¼0) ¼ 0 or 1, using periodic boundary conditions. The simulations differ by system dimensions, initial lamellar density, length and distribution, as well as volume fractions of the phases. Computations with equal and random initial lengths of the lamellae were also performed. Essentially, all simulations lead to a similar scaling behavior. A representative case is shown in Fig. 14, where initially 3200 lamellae of equal length are present, which quickly merge and coarsen. For the extraction of the average lamellar spacing and thickness, the number of interfaces where 4 ¼ 1/2 was counted when
traversing through the system in vertical direction and averaging over the entire computational domain, which determines the average lamellar thickness l and height h. Scaling plots for l are shown in Fig. 15. The results support the scaling l3 l30 t as also found in the experiments, although a precise determination of the exponent remains difficult. Similarly to the experiments, the simulations also yield a linear relationship between the lamellar spacing and thickness.
6. Conclusions An Fe-Al alloy with 60.9 at.% Al was produced, which according to the phase diagram [6] undergoes a eutectoid transformation Fe5Al8 / FeAl þ FeAl2 at 1095 C during casting and cooling to room temperature. Due to this reaction, the as-cast alloy shows a
X. Li et al. / Acta Materialia 127 (2017) 230e243
241
Fig. 15. Measured averaged lamella spacing for the simulation in Fig. 14. The values of correlation coefficient R2 of the fitting lines indicate that the exponent n ¼ 3 gives the best fit.
fully lamellar microstructure with volume contents of about 30% FeAl and 70% FeAl2. The kinetics of coarsening of the lamellar microstructure was studied by both experiments and simulations yielding the following findings and conclusions: (1) According to the experimental results, the average lamellar spacing in the as-cast state l0 is 200 ± 40 nm. The isothermal coarsening of the lamellar microstructure can be described as
l3 ¼ l30 þ kt ð T ¼ const:Þ; where l is the lamellar spacing after time t and k is a rate constant.
(2) The lamellar coarsening process is influenced not only by the lamellar shrinkage and termination annihilation, but also by the merging of lamellae. The activation energy of the coarsening process was determined from an Arrhenius plot giving a value of 265 ± 11 kJ/mol. This result indicates that the lamellar coarsening is a volume-diffusion-controlled process. (3) The volume fractions of FeAl and FeAl2 and the ratio of the thickness of the FeAl lamellae hFeAl and the lamellar spacing l do not change during coarsening. Therefore, hFeAl follows a similar coarsening law as l, i.e., h3 ¼ h30 þ k0t. (4) The size of the lamellar colonies (within the investigated range of diameters from <80 to >500 mm) does not affect the
242
X. Li et al. / Acta Materialia 127 (2017) 230e243
coarsening kinetics indicating that colony boundary diffusion plays a negligible role. (5) The simulation results show a general agreement with the experimental findings, in particular concerning the evolution of the lamellar spacing and height. This is remarkable, as the model is rather simple and does not consider many effects, which may play a role for the experimental microstructure evolution, like the role of anisotropies, orientation relationships and strain effects, suggesting that the scaling behavior is rather generic. The experiments and numerical simulations suggest that the coarsening is different from a conventional LSW picture, although the same classical exponent n ¼ 3 seems to be obtained for the scaling of the lamellar spacing. Instead, a scaling theory should involve more than one length scale, at least the spacing l, the thickness h and the length L. Similar to Ref. [10,21] then the lamellar density scales as m ðlLÞ1 . Numerical simulations suggest the scaling m3 t, from which we conclude that the (effective) lamellar length remains essentially constant, similar to the assertions in Ref. [21] and the observed scaling l3 t. As mentioned above, this is only possible if the fast termination shrinkage is compensated by merging of lamellae. Consequently, the appearance of several length scales with different scaling behavior similar to Ref. [55,56] indicates that a LSW-like mean field behavior, which relies on a large separation between the inclusions, breaks down. Acknowledgements The authors would like to thank Mr. G. Bialkowski for cutting samples by EDM, Mrs. I. Wossack for EPMA analysis, and Mr. D. Kurz for chemical analysis. The authors are also grateful for the computing time granted on the supercomputer JURECA at Jülich Supercomputing Centre (JSC). The financial support by the Deutsche Forschungsgemeinschaft (DFG) within the project STE 1077/2 is gratefully acknowledged. In addition, R.S. acknowledges financial support by the DFG via the priority program 1713. References [1] I. Baker, P.R. Munroe, Mechanical properties of FeAl, Int. Mater. Rev. 42 (1997) 181e205. [2] N.S. Stoloff, Iron aluminides - present status and future prospects, Mater. Sci. Eng. A 258 (1998) 1e14. [3] M. Palm, Fe-Al materials for structural applications at high temperatures: current research at MPIE, Int. J. Mater. Res. 100 (2009) 277e287. [4] D.G. Morris, M.A. Morris-Munoz, Recent developments toward the application of iron aluminides in fossil fuel technologies, Adv. Eng. Mater. 13 (2011) 43e47. [5] A. Scherf, D. Janda, M. Baghaie Yazdi, X. Li, F. Stein, M. Heilmaier, Oxidation behavior of binary aluminium-rich FeeAl alloys with a fine-scaled, lamellar microstructure, Oxid. Met. 83 (2015) 559e574. [6] X. Li, A. Scherf, M. Heilmaier, F. Stein, The Al-rich part of the Fe-Al phase diagram, J. Phase Equilib. Diff 37 (2016) 162e173. [7] F. Stein, S.C. Vogel, M. Eumann, M. Palm, Determination of the crystal structure of the ε phase in the Fe-Al system by high-temperature neutron diffraction, Intermetallics 18 (2010) 150e156. [8] A. Scherf, A. Kauffmann, S. Kauffmann-Weiss, T. Scherer, X. Li, F. Stein, M. Heilmaier, Orientation relationship of eutectoid FeAl and FeAl2, J. Appl. Crystallogr. 49 (2016) 442e449. [9] P.J. Maziasz, C.T. Liu, Development of ultrafine lamellar structures in twophase g-TiAl alloys, Metall. Mater. Trans. A 29A (1998) 105e117. [10] L.D. Graham, R.W. Kraft, Coarsening of eutectic microstructures at elevated temperatures, Trans. Metall. Soc. AIME 94 (1966) 9. [11] J.D. Livingston, J.W. Cahn, Discontinuous coarsening of aligned eutectoids, Acta Metall. 22 (1974) 495e503. [12] A.J. Ardell, Coarsening of directionally-solidified eutectic microstructures, in: A. Pechenik, R.K. Kalia, P. Vashishta (Eds.), Computer-aided Design of Hightemperature Materials, Oxford University Press, Inc., New York, 1999, pp. 163e182. [13] P.W. Voorhees, The theory of Ostwald ripening, J. Stat. Phys. 38 (1985) 231e252.
[14] I.M. Lifshitz, V.V. Slyozov, The kinetics of precipitation from supersaturated solid solutions, J. Phys. Chem. Solids 19 (1961) 35e50. €sen (Ost[15] C. Wagner, Theorie der Alterung von Niederschl€ agen durch Umlo wald-Reifung), Zeitschrift für Elektrochemie, Ber. Bunsenges. Phys. Chem. 65 (1961) 581e591. [16] A.J. Ardell, V. Ozolins, Trans-interface diffusion-controlled coarsening, Nat. Mater. 4 (2005) 309e316. [17] A.J. Ardell, Quantitative predictions of the trans-interface diffusion-controlled theory of particle coarsening, Acta Mater. 58 (2010) 4325e4331. [18] A.J. Ardell, Trans-interface-diffusion-controlled coarsening of g’ precipitates in ternary Ni-Al-Cr alloys, Acta Mater. 61 (2013) 7828e7840. [19] A.J. Ardell, Trans-interface-diffusion-controlled coarsening in ternary alloys, Acta Mater. 61 (2013) 7749e7754. [20] Y.G. Nakagawa, G.C. Weatherly, The stability of lamellar structures, Metall. Trans. 3 (1972) 3223e3229. [21] H.E. Cline, Shape instabilities of eutectic composites at elevated temperatures, Acta Metall. 19 (1971) 10. [22] G.C. Weatherly, Y.G. Nakagawa, Coarsening of rod eutectics by fault migration, Scr. Metall. 5 (1971) 777e781. [23] Y.G. Nakagawa, G.C. Weatherly, E. Ho, Coarsening of lamellar structures at elevated temperatures, T. Jpn. I. Met. 15 (1974) 114e120. [24] L.Y. Lin, T.H. Courtney, Direct observations of lamellar fault migration in the Pb-Sn eutectic, Metall. Trans. 5 (1974) 513e514. [25] Y.Y.L. Tian, R.W. Kraft, Kinetics of pearlite spheroidizations, Metall. Trans. A 18 (1987) 1359e1369. [26] Y.L. Tian, R.W. Kraft, Mechanisms of pearlite spheroidization, Metall. Trans. A 18 (1987) 1403e1414. [27] M.F. Bartholomeusz, J.A. Wert, The effect of thermal exposure on microstructural stability and creep resistance of a two-phase TiAl/Ti3Al lamellar alloy, Metall. Mater. Trans. A 25 (1994) 2371e2381. [28] M.J. Whiting, P. Tsakiropoulos, Morphological evolution of lamellar structures: the Cu-Al eutectoid, Acta Mater. 45 (1997) 2027e2042. [29] G. Sharma, R.V. Ramanujan, G.P. Tiwari, Instability mechanisms in lamellar microstructures, Acta Mater. 48 (2000) 15. [30] R.J. El-Khozondar, H.J. El-Khozondar, Numerical simulations of coarsening of lamellar structures applications to metallic alloys, Egypt. J. Solids 27 (2004) 11. [31] J.S. Langer, Instabilities and pattern formation in crystal growth, Rev. Mod. Phys. 52 (1980) 1e28. [32] J.P. Gollub, J.S. Langer, Pattern formation in nonequilibrium physics, Rev. Mod. Phys. 71 (1999) S396eS403. [33] J.B. Brady, Coarsening of fine-scale exsolution lamellae, Am. Mineral. 72 (1987) 697e706. [34] R.H. McCallister, The coarsening kinetics associated with exsolution in an iron-free clinopyroxene, Contrib. Mineral. Petr. 65 (1978) 327e331. [35] R.A. Yund, P. Davidson, Kinetics of lamellar coarsening in cryptoperthites, Am. Mineral. 63 (1978) 470e477. [36] I. Baker, F. Meng, Lamellar coarsening in Fe28Ni18Mn33Al21 and its influence on room temperature tensile behavior, Acta Mater. 95 (2015) 124e131. [37] H. Mehrer, M. Eggersmann, A. Gude, M. Salamon, B. Sepiol, Diffusion in intermetallic phases of the Fe-Al and Fe-Si systems, Mater. Sci. Eng. A 239e240 (1997) 889e898. [38] L.Q. Chen, Phase-field models for microstructure evolution, Annu. Rev. Mater. Res. 32 (2002) 113e140. [39] W.J. Boettinger, J.A. Warren, C. Beckermann, A. Karma, Phase field simulation of solidification, Annu. Rev. Mater. Res. 32 (2002) 163e194. [40] I. Steinbach, Phase-field models in materials science, Modell. Simul. Mater. Sci. Eng. 17 (2009) 073001. [41] R. Spatschek, E. Brener, A. Karma, Phase field modeling of crack propagation, Phil Mag. 91 (2011) 75e95. [42] M. Militzer, Phase field modeling of microstructure evolution in steels, Curr. Opin. Solid State Mater. Sci. 15 (2011) 106e115. [43] F. Drolet, K.R. Elder, M. Grant, J.M. Kosterlitz, Phase-field modeling of eutectic growth, Phys. Rev. E 61 (2000) 6705e6720. [44] M. Plapp, A. Karma, Eutectic colony formation: a phase-field study, Phys. Rev. E 66 (2002) 061608. [45] A. Parisi, M. Plapp, Stability of lamellar eutectic growth, Acta Mater. 56 (2008) 1348e1357. [46] M. Asta, C. Beckermann, A. Karma, W. Kurz, R. Napolitano, M. Plapp, G. Purdy, M. Rappaz, R. Trivedi, Solidification microstructures and solid-state parallels: recent developments, future directions, Acta Mater. 57 (2009) 941e971. [47] A. Choudhury, M. Plapp, B. Nestler, Theoretical and numerical study of lamellar eutectic three-phase growth in ternary alloys, Phys. Rev. E 83 (2011) 051608. [48] K. Ankit, A. Choudhury, C. Qin, S. Schulz, M. McDaniel, B. Nestler, Theoretical and numerical study of lamellar eutectoid growth influenced by volume diffusion, Acta Mater. 61 (2013) 4245e4253. [49] K. Ankit, R. Mukherjee, T. Mittnacht, B. Nestler, Deviations from cooperative growth mode during eutectoid transformation: insights from a phase-field approach, Acta Mater. 81 (2014) 204e210. [50] K. Ankit, R. Mukherjee, B. Nestler, Deviations from cooperative growth mode during eutectoid transformation: mechanisms of polycrystalline eutectoid evolution in FeeC steels, Acta Mater. 97 (2015) 316e324. [51] S. Ghosh, A. Choudhury, M. Plapp, S. Bottin-Rousseau, G. Faivre, S. Akamatsu, Interphase anisotropy effects on lamellar eutectics: a numerical study, Phys.
X. Li et al. / Acta Materialia 127 (2017) 230e243 Rev. E 91 (2015) 022407. [52] S. Akamatsu, M. Plapp, Eutectic and peritectic solidification patterns, Curr. Opin. Solid State Mater. Sci. 20 (2016) 46e54. [53] D. Fan, S.P. Chen, L.-Q. Chen, P.W. Voorhees, Phase-field simulation of 2-D Ostwald ripening in the high volume fraction regime, Acta Mater. 50 (2002) 1895e1907.
243
[54] S.G. Kim, Large-scale three-dimensional simulation of Ostwald ripening, Acta Mater. 55 (2007) 6513e6525. [55] E.A. Brener, V.I. Marchenko, H. Müller-Krumbhaar, R. Spatschek, Coarsening kinetics with elastic effects, Phys. Rev. Lett. 84 (2000) 4914e4917. [56] E.A. Brener, H. Müller-Krumbhaar, R. Spatschek, Coarsening of cracks in a uniaxially strained solid, Phys. Rev. Lett. 86 (2001) 1291e1294.