Instantaneous growth approximation describing the nanocrystallization process of amorphous alloys: A cellular automata model

Journal of Non-crystalline Solids 354 (2008) 3597-3605. http://dx.doi.org/10.1016/j.jnoncrysol.2008.03.038 1 Instantaneous growth approximation describing the nanocrystallization process of amorphous alloys. A cellular automata model J. S. Blázquez, V. Franco, C.F. Conde, M. Millán, A. Conde* Dpto. Física de la Materia Condensada, ICMSE-CSIC, Universidad de Sevilla, P.O. Box 1065, 41080, Sevilla, Spain Abstract: A cellular automata simulation based on an instantaneous growth approximation is developed to model the nanocrystallization kinetics. Results yield Avrami exponent values near to 1 at low transformed volume fraction, which decrease as the crystalline fraction increases. The compositional dependence of the nanocrystalline microstructure achieved after primary crystallization is also modelled. Two different nucleation phenomena are considered: either in contact with an already formed crystallite, or in isolated regions. Results from the simulations are compared with experimental data for FeCoNbB alloys and a good qualitative agreement is found *Corresponding author: Prof. A. Conde Departamento de Física de la Materia Condensada. Universidad de Sevilla. Apartado 1065, 41080 Sevilla (Spain). Phone: (34) 95 455 28 85 Fax: (34) 95 461 20 97 E-mail: [email protected] Journal of Non-crystalline Solids 354 (2008) 3597-3605. http://dx.doi.org/10.1016/j.jnoncrysol.2008.03.038 2 1 Introduction Amorphous and nanocrystalline alloys have attracted much attention from the research community since the works of Duwez on AuSi [ 1 ] and Yoshizawa on FINEMET [ 2 ]. Besides the interest provoked by the fundamental physics underlying in such microstructures, technological applicability of amorphous and nanocrystalline alloys has been profusely explored due to the outstanding properties achieved, overcoming those of their conventional crystalline counterparts (e.g. mechanical, for Albased alloys [ 3 ]; magnetic, for Fe-based alloys [ 4 ], etc). Nanocrystalline alloys obtained from primary crystallization of a precursor amorphous alloy exhibit a microstructure formed by small crystallites (~10 nm in size) embedded in a residual amorphous matrix. Two big families can be identified: Fe-based [4] and Al-based alloys [3]. The former are interesting due to their magnetic properties and the latter, due to their mechanical ones. The following discussion will correspond to Fe-based alloys, although it could be easily extended to Al-based alloys and other families. The typical composition of Fe-based nanocrystalline alloys is Fe-M-ET-(Cu), where M is a metalloid and ET is an early transition metal. The addition of metalloids (B, P, Si) is necessary to obtain a precursor amorphous alloy by rapid quenching techniques. The early transition metals (Zr, Nb, Hf, Mo) have, if any, a very low solubility in the -Fe phase and, consequently, will be expelled out to the amorphous matrix. However, due to the very slow diffusivity of these elements in the amorphous phase, they pile up at the crystal–matrix interface constraining the growth of the crystalline phase to the nanocrystalline scale. This phenomenon is responsible for the development of a nanocrystalline microstructure by primary crystallization of a precursor amorphous alloy. Journal of Non-crystalline Solids 354 (2008) 3597-3605. http://dx.doi.org/10.1016/j.jnoncrysol.2008.03.038 3 The addition of Cu is not necessary to obtain nanocrystalline microstructure in some compositions named NANOPERM [ 5 ] but, generally, refines the microstructure through the formation of Cu-rich clusters previous to the nanocrystallization [ 6 ]. This phenomenon is also responsible for the fine nanocrystalline microstructure observed in FINEMET alloys [4]. Several compositional modifications have been proposed to improve the applicability of nanocrystalline alloys. For example, in 1998 [ 7 ], HITPERM alloys were obtained after partial Fe substitution for Co to extend up to higher temperatures the soft magnetic properties of nanocrystalline alloys. In such systems, Co concentration was found to be homogeneous throughout the amorphous matrix and the nanocrystals [ 8 , 9 ]. It was also found that Cu addition yields a refinement of the microstructure for Nb- HITPERM alloys [ 10 ] but was useless for Zr-HITPERM alloys [8]. The microstructure observed for Cu-free NANOPERM [ 11 ] and HITPERM [8,10] alloys (with a concentration of ET >5 at.%) consists on very irregular nanocrystals, which can be described as agglomerates of smaller and more regular units (~5 nm) with the same crystallographic orientation [10]. This microstructure can be understood as the result of the competition of two different nucleation phenomena: either in isolated regions, or in contact with a crystallite. The former has the advantage of nucleating in a region rich in Fe, but the interface energy between the nucleus and the surrounding amorphous is high. In the latter nucleation phenomenon, which could be also understood as epitaxial growth, the interface energy of the new nucleus is smaller but the region might have a lower Fe concentration. The size of the small units forming the crystalline agglomerates is almost constant along the crystallization process (e.g. ~5 nm for Nb-containing HITPERM alloys [10]). This fact, along with some recent results on the isothermal nanocrystallization kinetics [ 12 ], enables to consider an instantaneous growth Journal of Non-crystalline Solids 354 (2008) 3597-3605. http://dx.doi.org/10.1016/j.jnoncrysol.2008.03.038 4 approximation to describe the nanocrystallization process of such systems. This approximation implies that the time required for a new nucleus to grow up to ~5 nm in diameter is neglected in comparison with the time required for the full nanocrystallization process. Several computer simulations procedures have been applied to study the crystallization process, among them Montecarlo [ 13 , 14 ] and molecular dynamics [ 15 ] can be found. Cellular automata has also been applied to describe several crystallization processes [ 16 , 17 , 18 , 19 , 20 ] and results obtained have been compared to Johnson-Mehl- Avrami-Kolmogorov (JMAK) theory [ 21 , 22 , 23 ] predictions. Cellular automata modellizations consist on a discretization of the time and the space in order to apply evolution laws. Generally, the character of the evolution of the system is deterministic, i.e. the configuration of the system at a time step  is univocally determined by the configuration at the time step  -1. The application of these methods to crystallization processes is straightforward for growth processes but nucleation is a probabilistic phenomenon and this feature must be taken into account. Some authors choose an initial distribution of pre-nuclei, which can be active since the beginning of the process or retarded and being activated at a predetermined time step [16], neglecting the probabilistic character of the nucleation process and limiting the study to the deterministic effect of a growth process and its dependence with the nucleation density. Other authors [17,18] directly assume this probabilistic character of the nucleation process randomly choosing a set of sites to nucleate at each time step  among those which are able to nucleate (e. g. grain boundaries in recrystallization). The aim of this paper is to show that an instantaneous growth approximation can properly describe the nanocrystallization kinetics and the microstructure developed during this process. This will be done using a cellular automata simulation program: a Journal of Non-crystalline Solids 354 (2008) 3597-3605. http://dx.doi.org/10.1016/j.jnoncrysol.2008.03.038 5 three dimensional space is divided in cubic cells and the time is discretized in iteration steps. The process to be described in the present paper considers the nucleation phenomenon alone in a homogeneous amorphous matrix, neglecting the growth process. Therefore, every cell is suitable to nucleate but in order to do so it must fulfil not only deterministic requisites but the stochastic character of nucleation has to be taken into account. This is considered by randomly selecting a cell as candidate to develop a new crystalline nucleus. This nucleus will yield a nanocrystal only after fulfilling some deterministic and probabilistic requisites, which depend on the characteristic parameters of this cell and its neighbourhood: volume available for nucleation, surface energy developed between the eventual new nucleus and its environment, and composition of the region where the nucleus would be formed. 2 JMAK theory Crystallization of metallic amorphous alloys can be described by nucleation and growth processes [ 24 ]. These transformations are generally described by the JMAK theory, which describes the time evolution of the transformed fraction, X, as:      n ttkX 0 ·exp1  (1) where k is the frequency factor, t is the time, t0 is the induction time and n is the Avrami exponent. From the latter parameter, it is possible to extract information about the phenomena involved in the crystallization process as [24]: GI ndnn · (2) where nI is ascribed to nucleation process, being 1 for a constant nucleation rate, below 1 for a decreasing nucleation rate and above 1 for an increasing nucleation rate; d is the dimension of the growth; and nG is ascribed to the growth process, being ½ for diffusion controlled growth and 1 for interface controlled growth [24]. Journal of Non-crystalline Solids 354 (2008) 3597-3605. http://dx.doi.org/10.1016/j.jnoncrysol.2008.03.038 6 Although JMAK theory was developed for polymorphic transformations during isothermal regimes, its application has been successfully extended to transformations involving compositional changes [ 25 , 26 ] and non-isothermal regimes [ 27 , 28 , 29 ]. JMAK theory takes into account the geometrical overlapping between growing crystals as the only effect to slow down the crystallization process with the increase of transformed volume and, assuming a constant nucleation rate, n=2.5 for a three dimensional diffusion controlled growth and n=4 for a interface controlled three dimensional growth. However, in the case of nanocrystallization processes, this effect alone can not describe the transformation and values of Avrami exponent below 1 can be found [4,12,25,26,27, 30 ]. The responsible for these low values of n can be the blocking of the crystal growth as ET atoms pile up at the edge of the nanocrystals. Although this constraining effect is not considered in JMAK theory, a general qualitative description considers the presence of a strongly inhibited growth process and a progressive exhaustion of nucleation sites. The instantaneous growth approximation neglects the time a certain nucleus needs to grow up to its limit value, at which it is blocked by the wall of rejected ET atoms. Therefore, nG = 0, and expression (2) should be simplified to: I nn  (3) Under this approach, each formed nucleus would reach instantaneously its final size and following growth processes are banned. Previous experimental studies using this approach allowed to extract detailed information on the nanocrystallization kinetics of Nb-containing HITPERM alloys [12]. 3 Model program Journal of Non-crystalline Solids 354 (2008) 3597-3605. http://dx.doi.org/10.1016/j.jnoncrysol.2008.03.038 7 In terms of the different partitioning of the elements during nanocrystallization, a typical Fe-based composition should be FexSolyExc100-x-y; where Sol identifies the elements which are soluble in the -Fe (e.g. Co, Si, Ge…) and Exc identifies the elements which are excluded from the crystalline phase and expelled out to the residual amorphous matrix (Nb, Zr, B, …). The composition of the crystalline phase would be Fe100-ySoly and that of the residual amorphous phase could be obtained from the balance equation: FexSolyExc100-x-y = XC (Fe100-ySoly) + (1-XC) FeuSolyExcz (4) where XC is the crystalline volume fraction. If there is a preferential partitioning of an element to the crystals or to the amorphous phase, a convenient selection of the y index (the concentration of elements different to Fe in the crystalline phase) can take it into account. The model used in these experiments describes a three dimensional space formed by cubic cells. Following the microstructure observed for NANOPERM and HITPERM alloys and the two nucleation mechanisms considered to describe this microstructure (“in contact” and “isolated” nucleations), each cell is characterized by three parameters: occupation (Occ), composition (C) and order (Ord).  The occupation parameter takes three possible numeric values, indicating whether the cell is a crystalline unit (Occ=2), or it is amorphous but in contact with an already formed crystal (Occ=1), or isolated (Occ=0).  The composition indicates the Fe concentration in the cell.  The order is a number assigned to each nucleus formed in isolated regions by order of appearance. This nucleus will form an agglomerate of crystallites whose components will have a common value of the parameter Ord. Journal of Non-crystalline Solids 354 (2008) 3597-3605. http://dx.doi.org/10.1016/j.jnoncrysol.2008.03.038 8 At the beginning, all the cells of the space are set to isolated value for occupation, to the nominal value of Fe concentration (C = x) for composition and to zero for the order number, as no crystallite is formed and the initial composition is supposed to be homogeneous. A flow diagram corresponding to one iteration step of the program proposed to resemble an instantaneous growth crystallization process is schematically shown in figure 1. For each iteration step, a cell is randomly chosen and tested against several deterministic and probabilistic requisites. If all these requisites are fulfilled, the cell will nucleate leading to a new crystallite. However, if any requisite is not fulfilled, the cell will not nucleate and will be discarded for further considerations, starting the process again. 3.1 Considering the occupation of the cell The first stage of each iteration step of the process is to randomly choose a cell and to check its occupation value. If this value indicates that the cell is already nucleated (Occ=2), the cell is discarded for further considerations, an iteration step is counted and a new cell is chosen. Once the chosen cell fulfils the deterministic condition of being amorphous, the effect of a reduction in the surface energy for “in contact” nucleation compared to “isolated” nucleation is considered using the value of Occ and following expression:   RndOcc (5) where Rnd is a random number from 0 to 1 and  is a parameter related with the surface energy. A value of  =0 would indicate no energetic advantage for nucleation “in contact” with respect to “isolated” nucleation (every amorphous cell with Occ=0 or 1 will fulfil the requisite expressed in the previous equation). A value of  >1 would ban Journal of Non-crystalline Solids 354 (2008) 3597-3605. http://dx.doi.org/10.1016/j.jnoncrysol.2008.03.038 9 the nucleation in isolated regions. If the condition required in equation (5) is not fulfilled, the cell is discarded for further considerations, an iteration step is counted and a new cell is chosen. 3.2 Considering the composition of the environment When all the previous requisites have been satisfied, the composition of the chosen cell and that of the neighbourhood must be considered. In order to build a crystalline cell, its Fe concentration must increase to reach a certain value (100-y). Therefore, Fe must replace the elements Exc (non soluble in the -Fe phase). A first step would be to check if the neighbourhood of the chosen cell is rich enough in Fe to supply the concentration needed by the chosen cell. At this point, it is worth noticing that the instantaneous growth approach used throughout this paper is based on the assumption that the distances for diffusion are very short (due to the blocking effect of ET atoms). Therefore, the volume available to the forming nucleus to extract Fe from the surrounding cells might be limited. In the present study, a maximum volume is considered as 6 times that of the cell (corresponding to the six neighbour cells, assuming Von Neumann neighbourhood [16]) but a parameter, vol, is defined to take into account the possibility of a reduction of the affected volume by the formation of a new crystallite. As an example, it can be considered the volume of the smallest sphere in which the cubic cell is contained, which has a radius 2/3 times the length of the side of the cubic cell. The value of vol in such a case would be 1.72, which is the difference between the volume of the sphere and that of the cubic cell. This volume would be equally divided among the six neighbour cells. Journal of Non-crystalline Solids 354 (2008) 3597-3605. http://dx.doi.org/10.1016/j.jnoncrysol.2008.03.038 16 crystalline volume fraction, the Avrami exponent is close to 1. Its increase at this stage can be ascribed to the triggering of “in contact” nucleation mechanism by the increase in number of crystallites able to supply sites for this heterogeneous nucleation. As crystalline volume fraction increases, the Avrami exponent decreases due to the decrease in the number of available places for new nuclei. No explanation in the frame of Johnson-Mehl-Avrami-Kolmogorov theory was previously found for these low values of Avrami exponent, close to 1, observed for the nanocrystallization processes and some modifications were proposed to explain them [25,26,32, 37 , 38 ]. However, an instantaneous growth approximation can yield these low values, as it was shown above, the Avrami exponent would be n = nI. A value of nI > 1 would indicate an increasing nucleation frequency and nI < 1 a decreasing nucleation frequency. Properly using the parameter Ord ascribed to each cell, it is possible to build compositional maps and simulate two-dimensional images which could be compared with experimental images of nanocrystalline alloys obtained by transmission electron microscopy (TEM). Figure 7 and 8 show such compositional maps as a function of the number of iterations, for a selected 10x10 cells plane of the 1003 cells space used in the simulation, for two FexSol78-xExc22 alloys (x= 60 and 18, respectively). These two modelled alloys could be compared to the real Fe60Co18Nb6B16 and Fe18Co60Nb6B16 alloys, respectively. TEM images of such alloys, obtained at different times of annealing at 35 K below the onset of crystallization are also shown in figures 7 and 8, respectively. Both simulated and experimental images agree describing a continuous nucleation process (the number of crystalline units continuously increases). In the case of the alloy with 60 at. % Fe, agglomerates appears since the very beginning and although their size increases, the size of their crystalline units is almost independent of Journal of Non-crystalline Solids 354 (2008) 3597-3605. http://dx.doi.org/10.1016/j.jnoncrysol.2008.03.038 17 the annealing time. In the case of 18 at. % Fe alloy, there are no agglomerates and the only difference as the nanocrystallization progresses is the increase in number of small regular crystallites ~5 nm in size. Conclusions The cellular automata simulation program proposed in the present paper is based on an instantaneous growth approximation. In this frame, the time required for a nucleus to grow up to its maximum size is neglected in comparison with the time required for the crystallization process. This simulation yields to the well known and experimentally observed two slopes behaviour of the JMAK plots. The local Avrami exponent obtained from the simulations resembles the experimental results obtained for the kinetics of nanocrystallization, consisting on an Avrami exponent ~1 for low values of the transformed fraction, which decreases as the crystalline volume fraction increases. In the frame of an instantaneous growth approximation, results can be described by the JMAK theory. The Avrami exponent, n=nI, accounts only for nucleation processes as nG=0. Therefore, n=1 indicates a constant nucleation frequency, n>1 indicates an increasing nucleation frequency and n<1 a decreasing nucleation frequency. Compositional dependence of the microstructure achieved in nanocrystalline alloys is reproduced considering two different nucleation phenomena: in contact with an already formed crystallite and in isolated regions. Model predictions were compared with experimental results on FeCoNbB alloys and a good qualitative agreement was found for the compositional dependence of nanocrystallization kinetics and the microstructure developed during the process. Journal of Non-crystalline Solids 354 (2008) 3597-3605. http://dx.doi.org/10.1016/j.jnoncrysol.2008.03.038 18 Acknowledgements This work was supported by the Spanish Government and EU FEDER (Project MAT 2004-04618) and by the PAI of the Regional Government of Andalucía (Project P06-FQM-01823). J.S.B. acknowledges a research contract from this Regional Government. 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[ 37 ] Kristiakova K, Svec P, Phys Rev B 2001 ; 64 : 184202 [ 38 ] Hermann H, Mattern N, Roth S, Uebele P, Phys Rev B 1997; 56: 13888. Journal of Non-crystalline Solids 354 (2008) 3597-3605. http://dx.doi.org/10.1016/j.jnoncrysol.2008.03.038 21 Figure captions Figure 1: Flow diagram of the simulation program. Figure 2. (a) Crystalline fraction versus iteration steps, (b) JMAK plot and (c) Avrami exponent calculated for a Fe60Sol18Exc22 alloy:  =0.5 and vol=1.72 (blue) and 6 (red). Figure 3. (a) Crystalline fraction versus iteration steps, (b) JMAK plot and (c) Avrami exponent calculated for a Fe60Sol18Exc22 alloy: vol=1.72 and  =0, 0.5 and 0.9. Figure 4. (a) Crystalline fraction versus iteration steps, (b) JMAK plot and (c) Avrami exponent calculated for a FexSol78-xExc22 alloy (x= 60 and 18):  =0.5, vol=1.72. Figure 5. (a) Average size of the agglomerates versus the iteration step number and (b) agglomerate size distribution after 3·106 iteration steps for a FexSol78-xExc22 alloy (x= 60 and 18). In both,  =0.5, vol=1.72. Figure 6. (a) Crystalline volume fraction versus time minus the induction time, (b) JMAK plot and (c) Avrami exponent obtained for a Fe60Co18Nb6B16 alloy during isothermal annealing at 743 K. The thick line in (b) and (c) marks the values obtained for t-t0>2t0 (t0=85 s). Figure 7. Plane images simulated by the model for a Fe60Sol18Exc22 alloy after different iteration steps compared with TEM images of nanocrystalline samples of Fe60Co18Nb6B16 alloy annealed different times at 716 K. In the simulated image obtained after 106 iteration steps, the number indicate the order of formation of the different agglomerates. Figure 8. Plane images simulated by the model for a Fe18Sol60Exc22 alloy after different iteration steps compared with TEM images of nanocrystalline samples of Fe18Co60Nb6B16 alloy annealed different times at 686 K. Journal of Non-crystalline Solids 354 (2008) 3597-3605. http://dx.doi.org/10.1016/j.jnoncrysol.2008.03.038 22 Figure 1 The Fe content available is calculated A probability associated with Fe available is calculated The cell nucleates END of the iteration step A cell is randomly chosen YES YES NO NO NO NO Is the cell amorphous? Is the surface energy overcome? Is there enough Fe? Is the probability overcome? START of the iteration step YES YES Journal of Non-crystalline Solids 354 (2008) 3597-3605. http://dx.doi.org/10.1016/j.jnoncrysol.2008.03.038 23 Figure 2 0 0.5 1 1.5 2 2.5 3 x 106 0 0.2 0.4 0.6 iteration steps Crystalline fraction 910 11 12 13 14 15 -6 -5 -4 -3 -2 -1 0 1 ln(t) ln(-ln(1-X)) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 0.5 1 1.5 Crystalline fraction Avrami index vol=6 vol=1.7 2 vol=6 vol=1.7 2 vol=6 vol=1.7 2 (a) (b) (c) Journal of Non-crystalline Solids 354 (2008) 3597-3605. http://dx.doi.org/10.1016/j.jnoncrysol.2008.03.038 24 Figure 3 910 11 12 13 14 15 -8 -6 -4 -2 0 ln(t) ln(-ln(1-X)) 0 0.5 1 1.5 2 2.5 3 x 106 0 0.1 0.2 0.3 0.4 0.5 0.6 iteration steps Crystalline fraction 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 0.5 1 1.5 2 Crystalline fraction Avrami index (a) (b) (c)  =0  =0  =0  =0.5  =0.5  =0.5  =0.9  =0.9  =0.9 Journal of Non-crystalline Solids 354 (2008) 3597-3605. http://dx.doi.org/10.1016/j.jnoncrysol.2008.03.038 25 Figure 4 0 0.5 1 1.5 2 2.5 3 x 106 0 0.1 0.2 0.3 0.4 0.5 0.6 iteration steps Crystalline fraction 910 11 12 13 14 15 -6 -5 -4 -3 -2 -1 0 ln(t) ln(-ln(1-X)) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Crystalline fraction Avrami index (a) (b) (c) x =18 x =18 x =18 x =60 x =60 x =60