An Equivalent Time Approach for Scaling the Mechanical Alloying Processes

Intermetallics. Vol. 16. Núm. 3. 2008. Pag. 470-478 http://dx.doi.org/10.1016/j.intermet.2007.12.011 1 An equivalent time approach for scaling the mechanical alloying processes J. J. Ipus1, J. S. Blázquez1, V. Franco1, M. Millán1, A. Conde1, D. Oleszak2, T. Kulik2 1 Dpto. Física de la Materia Condensada, ICMSE-CSIC, Universidad de Sevilla, P.O. Box 1065, 41080, Sevilla, Spain. 2 Faculty of Materials Science and Engineering, Warsaw University of Technology, ul. Woloska 141, 02-507, Warsaw, Poland. ABSTRACT. Dynamics of a single ball into a planetary ball mill is analyzed leading to a cubic dependence of the power transferred during milling with the rotational speed, . This leads to the definition of an equivalent time to describe the state of ball milled powders independently of . Mechanical alloying of Fe75Ge20Nb5 composition is studied by a combination of experimental techniques (differential scanning calorimetry, scanning electron microscopy, energy dispersive x-ray spectrometry, x-ray diffraction, Mössbauer spectrometry and vibrating sample magnetometry) and results evidence a good agreement with the predictions of the equivalent time approach. Keywords: C. Mechanical alloying and milling; C. Nanocrystals 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] Intermetallics. Vol. 16. Núm. 3. 2008. Pag. 470-478 http://dx.doi.org/10.1016/j.intermet.2007.12.011 2 1. Introduction Ball milling has been shown as a very versatile technique for production of metastable systems: nanocrystalline, amorphous, supersaturated solid solutions, quasicrystals, etc [ 1 ]. Unlike ultra fast cooling techniques, which can freeze high temperature microstructures at low temperatures in a narrow compositional range around the eutectic, ball milling is applied to a wider compositional range. Mechanical evolution induced by ball milling is due to the energy transferred from the milling media to the powder particles, continuously submitted to fracture and cold welding processes which will define their final morphology. Among the different types of ball mills available, planetary ball mills are widely used to produce such metastable materials. In order to understand the dynamics of planetary ball mills, several authors have used the approach of single ball dynamics and have extrapolated their results to actual milling processes using several balls [ 2 , 3 , 4 , 5 , 6 , 7 , 8 ]. These studies analyze the energy transferred per collision between the ball and the vial wall and the dependence of the ball trajectory on different parameters such as ratio between the frequencies of main disk and vials, ball to powder mass ratio and others [1,2,3,4,5,6,7,8, 9 , 10 ]. Those results yield a parabolic dependence of the intensity of a single ball-powder interaction with the rotational speed. Moreover, the trajectory described by this single ball was found independent of the rotational speed [2]. These two features, as it will be shown in the following section of this paper, might lead to a cubic law for the estimated power transferred to the powder under a single ball approach. The reliability of this prediction is tested in the third section of the paper by studying the evolution of a mechanically alloyed Fe75Nb5Ge20 powder at different milling intensities during different times. Intermetallics. Vol. 16. Núm. 3. 2008. Pag. 470-478 http://dx.doi.org/10.1016/j.intermet.2007.12.011 3 2. Approach to the dynamics of the planetary ball mill 2.1 Movement of a single ball into the vial Single ball approximation has been used in the literature to predict the energy transfer to powder during milling. In this section, a brief description of the rotational speed dependence of both the energy involved in different ball-powder interactions and the frequency of these events is given. Figure 1 shows a scheme of the planetary ball mill, where  R is the position of the vial center and its modulus the radius of the main disk;  r , the position of the ball respect to the center of the vial, which modulus is the radius of the vial minus that of the ball;   rRp is the position of the center of the ball; = t+  0 is the angle rotated by the main disk at time t and  its angular speed; = t+  0 is the angle rotated by the vial at time t and  its angular speed. In this scheme, the rotations of the vial and that of the main disk are opposite. Three orthogonal reference systems have been defined to position the ball at point P; a Cartesian inertial system   ,,i j k and two polar and non-inertial systems: ,, R ee        , which follows the rotation of the main disk, and ,, r uu       , which follows the rotation of the vial. In the simplified studied system, the particle movement must fulfill some requirements, also considered by other authors [3,4]:  Only the movement of one ball is considered.  The movement occurs in a plane. Intermetallics. Vol. 16. Núm. 3. 2008. Pag. 470-478 http://dx.doi.org/10.1016/j.intermet.2007.12.011 4  Only two forces are considered; the normal reaction of the vial wall over the ball,  N , and the friction force,   F .  The particle keeps fixed at a point of the vial wall since N>0.  When N=0, the ball detaches from the vial wall and moves freely until it impacts again with the vial wall.  After the collision, the ball moves again stuck to the wall. After analyzing the dynamics of the system and considering only the force component parallel to  r u an expression for N per mass unit of ball is found, )cos( 22   Rr m N (1) This expression is meaningful only for N>0, if N=0 the ball will detach from the vial wall, thus a detachment time, td, can be obtained as:                    00 2 2 arccos 1    R r td (2) Analogously to expression (1), considering the component parallel to   u , an expression for F  per mass unit of ball is obtained, )sin( 2    R m F (3) This solution will apply if the ball is assumed to be fixed at a point of the vial wall. Once the ball detaches from the vial wall it moves freely with constant velocity until it collides again with the vial wall. In order to calculate the time of flight, tf, the separation distance between the ball and the vial wall has been analyzed. In fact, the ball will Intermetallics. Vol. 16. Núm. 3. 2008. Pag. 470-478 http://dx.doi.org/10.1016/j.intermet.2007.12.011 5 collide when pR equals the radius of the vial, r. Figure 2 shows the curve of the difference between the square of pR and the square radius of the vial as a function of time from the detachment for the planetary mill used in section 3, with R=0.125 m, r=0.017 m (it has been considered that the ball has a radius of 0.005 m), at =150, 250 and 350 rpm, being the values of td and tf of the same order of magnitude. By using different values of , keeping constant the ratio /, it can be observed that both td and tf are proportional to 1/ (see expression (2) and inset of figure 2, respectively) and, therefore, the number of detachment and collision events remains constant along a period of rotation of the main disk. For t>td, N=0 (as well as F  ) and the ball is not in contact with the vial wall. After the ball collides with the wall, N will recover a non zero value. The trajectory of the ball, for both inertial and non-inertial reference systems are shown in figure 3. It can be observed that the point of collision, PC, is different to the position of the detachment point at the time of collision, PD’. Therefore, a phase angle,  , must be included in the argument of the cosine function of equation (1). After the collision, the process is repeated continuously and figure 4 shows the value of N, solid line, during a rotational period of the main disk along with the analytical solution of equation (1), dotted line, for comparison. 2.2 Estimation of the dependence of the energy transfer on the frequency The simple model described above can be used to propose a dependency of the energy transferred during milling from the balls and vial to the powder. Several Intermetallics. Vol. 16. Núm. 3. 2008. Pag. 470-478 http://dx.doi.org/10.1016/j.intermet.2007.12.011 6 mechanisms can be considered as candidates for energy transfer from the balls and vial to the powder. First, while the ball is in contact with the vial wall, the powder which is in between is compressed and sheared and the work done over the powder during this compression could be approximated to: 2 1kdNdxNWN (4) where d is a distance of the order of the powder particle size (micrometer) and k1 is a constant. For =150 rpm, the energy transferred by unit mass of ball can be estimated as ~10-4 J/kg for d~1 m. Analogously, it could be possible to estimate a similar order of magnitude for the shear work done by the friction force. On the other hand, when the ball, after being detached, impacts against the vial wall, a change in its velocity is necessary to allow the ball to move with the vial again. This change implies a reduction in the kinetic energy of the ball, which can be calculated as:   2 2 )( 2 1impdC vtvmE  (5) where v(td) is the speed of the ball at the detachment time and vimp is the modulus of the speed of the vial wall at PC:       2 2cossin)(         R r ttttRtv ddddd   (6)             2 2cossin         R r ttttttttRv fdfdfdfdimp   (7) The values of EC per unit mass of ball, as it occurs for the estimated work of compression, and as expected by expressions (6) and (7), follows a 2 law. For =150 rpm the order of magnitude of the transferred energy is 1 J/kg per collision event. Intermetallics. Vol. 16. Núm. 3. 2008. Pag. 470-478 http://dx.doi.org/10.1016/j.intermet.2007.12.011 7 Some authors have estimated the energy transferred to the powder, assuming a Hertzian collision approximation and after estimating the amount of powder in between the ball and the vial wall [4,6,8]. Several parameters affect the final results, as ball to powder mass ratio and elastic properties of the milling media as well as of the powder, etc, which were constant along the experiments performed in this work. Therefore, the  dependence of the energy transferred during the milling process is preserved. As it has been shown in figure 4, during a period of rotation of the main disk, a constant number of compression events and impacts occur. Assuming the work done during each compression is described by equation (4) and that of each impact is described by equation (5), the average power could be estimated, for long times, as the whole work done during a period of rotation of the main disk divided by this time:   3 2  k T EWn PCN (8) where n is the constant number of events per period and k2 is a constant. Therefore, the energy transferred after a time t would be proportional to 3 and, if several values of  are used (keeping constant the ratio /) an equivalent time, teq, can be defined as:   3 0 / tteq (9) where 0 is a reference frequency (in the following study 0=150 rpm, which is the lowest frequency used). As milling time increases, energy is transferred from milling media to powder yielding microstructure evolution and different transformations [1]. Quantifying experimentally this energy is a difficult task as its acquisition is not completely reversible. In order to overcome these obstacles, some authors have proposed energy maps to approximately compare microstructures obtained from different milling conditions [3,8]. In this work, it has been assumed that the state of the powder is Intermetallics. Vol. 16. Núm. 3. 2008. Pag. 470-478 http://dx.doi.org/10.1016/j.intermet.2007.12.011 8 univocally determined by the amount of energy transferred to the powders. Therefore, powder of Fe75Nb5Ge20 composition has been characterized by using different structural and magnetic techniques after milling at different frequencies during different times, keeping constant the / ratio. Results from different experimental techniques have been rescaled using an equivalent time defined by equation (9) with 0=150 rpm. 3. Application to experimental results 3.1 Experimental techniques Fe75Nb5Ge20 composition was produced from pure powders (purity99 %) by ball milling in a Fritsch Pulverisette 5 planetary ball mill using hardened steel balls (10 mm diameter) and vials. The initial powder mass was 5 g and the ball to powder ratio 10:1. The ratio between the rotational speed of the vial () and that of the main disk () was fixed to /=-2. Three different values of  were used: 150, 250 and 350 rpm. Some powder was taken out after selected times (from 1 h to 150 h), opening and closing the vials in argon atmosphere to avoid oxygen and humidity contamination. Size and morphology of the powder particles were studied by scanning electron microscopy (SEM) in a Jeol JSM-6460 LV and energy dispersive X-ray (EDX) analyses were performed using an Incax-sight of Oxford Instruments. The crystalline structure was studied by X-ray diffraction (XRD) using Cu-K radiation in a Bruker D8I diffractometer and the local environment of Fe atoms was analyzed by Mössbauer spectrometry (MS). Mössbauer spectra were recorded at room temperature in a transmission geometry using a 57Co(Rh) source. The values of the hyperfine parameters were obtained by fitting with NORMOS program [ 11 ]. Thermal stability of the samples was studied by differential scanning calorimetry (DSC) using a Perkin-Elmer DSC7 Intermetallics. Vol. 16. Núm. 3. 2008. Pag. 470-478 http://dx.doi.org/10.1016/j.intermet.2007.12.011 9 under argon flow. Specific saturation magnetization,  S was measured in a Lakeshore 7407 vibrating sample magnetometer (VSM), applying a maximum field of 1.5 T. 3.2 Calorimetry Figure 5 shows the DSC scans of the alloy after milling 150 h at different rotational speeds. Both samples milled at 250 and 350 rpm exhibit similar curves, showing a broad exothermic maximum generally ascribed to relaxation and crystal growth effects [1]. This fact implies that the energy released after heat treatment is not proportional to the amount of energy supplied by the milling media, as milling at 350 rpm must be more energetic than milling at 250 rpm. On the other hand, the sample milled at 150 rpm exhibits, along with the broad exotherm, an endothermic peak. This endotherm is ascribed to heterogeneities observed in the powder; as it will be shown later by XRD and MS techniques, several crystalline phases coexist in the powder formed after 150 h milling at 150 rpm. A detailed discussion on the thermal evolution of milled samples as a function of the milling time will be reported elsewhere [ 12 ] as it is out of the scope of the present paper. 3.3 Powder particle size and composition SEM images were used to measure the average powder particle size, <d>, (statistic over an average of 200 particles per sample) as a function of time and rotational speed. Figure 6.a shows the values of <d> as a function of time (upper panel) and the equivalent time defined by equation (9) (lower panel). For samples milled at 150 rpm, powder particle size keeps almost constant (~15 m) for low milling times to decrease for longer times down to a constant value of ~5 m. For samples milled at 250 rpm, <d> starts decreasing at very low milling times and the same constant value, ~5 Intermetallics. Vol. 16. Núm. 3. 2008. Pag. 470-478 http://dx.doi.org/10.1016/j.intermet.2007.12.011 16 Table 2 Milling time ranges for the three types of MS fittings performed: a) single site at HF=33 T, b) site at 33 T plus two magnetic hyperfine field distributions and c) using a single magnetic hyperfine field distribution.  (rpm) [/0]3 a) single HF=33 T b) HF=33 T +HF distribution c) single HF distribution t (h) teq (h) t (h) teq (h) t (h) teq (h) 150 1 up to 10 20 to 150 >150 250 4.6 up to 1 4.6 5-20 23-92 50 230 350 12.7 none 2-5 25.4-63.5 10 127 Intermetallics. Vol. 16. Núm. 3. 2008. Pag. 470-478 http://dx.doi.org/10.1016/j.intermet.2007.12.011 17 Figure captions Figure 1. Scheme of the three reference systems used. Figure 2. Detachment distance between ball and vial wall during flight for 150, 250 and 350 rpm, the crosses show the moment of collision. The inset shows the linear relationship between the time of flight, tf, and the rotational period of the main disk, T. Figure 3. Left; inertial reference system: solid line, ball trajectory, dashed line, trajectory of the centre of mass of the vial, dotted circles show the vial position at the detachment and collision events. Right, non-inertial system: Circle represents the vial; thick line, ball trajectory; straight solid lines indicate the detachment point of the ball from the wall at the detachment time, PD, and at the collision time, PD’; PC indicates the collision point of the ball;  is the angle between PD’ and PC. Figure 4. Normal force exerted over the ball (continuous line) and analytical solution of equation 6 (dotted line) during one period of the main disk rotation calculated for =150 rpm. Figure 5. DSC scans of Fe75Ge20Nb5 samples after 150 h milling at different frequencies. Figure 6. a) Powder particle size as a function of actual time and equivalent time. b) Average Cr content as a function of actual time and equivalent time. Figure 7. XRD patterns of samples after different milling times at different milling intensities. Crosses, bcc-Fe; square, fcc-Ge; circles, bcc-Nb; triangle, intermetallic. Figure 8. a) Angular position, 2, and b) full width at half maximum, FWHM, of the (110) diffraction maximum of the -Fe(Ge,Nb) phase as a function of actual time (above) and equivalent time (below). c) Area fraction of fcc Ge and bcc Fe phases Figure 9. Mössbauer spectra of samples after different milling times at different frequencies. Intermetallics. Vol. 16. Núm. 3. 2008. Pag. 470-478 http://dx.doi.org/10.1016/j.intermet.2007.12.011 18 Figure 10. a)Average hyperfine magnetic field, <HF>, and b) saturation magnetization of the different studied samples as a function of actual time (above) and equivalent time (below). Intermetallics. Vol. 16. Núm. 3. 2008. Pag. 470-478 http://dx.doi.org/10.1016/j.intermet.2007.12.011 19 Figure 1     e eR ur u R r P j i p Intermetallics. Vol. 16. Núm. 3. 2008. Pag. 470-478 http://dx.doi.org/10.1016/j.intermet.2007.12.011 20 Figure 2 0.0 0.5 1.0 0.0 0.1 0.2 0.00 0.02 0.04 0.06 -4 -3 -2 -1 0 1 2 3 4 5 350 rpm 250 rpm |R-p|2-r2 (10-4 m2) time (s) 150 rpm tf (s) T (s) Intermetallics. Vol. 16. Núm. 3. 2008. Pag. 470-478 http://dx.doi.org/10.1016/j.intermet.2007.12.011 21 Figure 3. 0.00 0.05 0.10 0.15 0.00 0.05 0.10 0.15 Y (m) X (m) -0,02 0,00 0,02 -0,02 0,00 0,02 PD'PC PD Y (m) X (m)  Intermetallics. Vol. 16. Núm. 3. 2008. Pag. 470-478 http://dx.doi.org/10.1016/j.intermet.2007.12.011 22 Figure 4. 0.0 0.5 1.0 0.0 0.1 0.2 0.3 0.4 -10 0 10 20 30 40 50 60 N/m (N/kg) time (s) time/T Intermetallics. Vol. 16. Núm. 3. 2008. Pag. 470-478 http://dx.doi.org/10.1016/j.intermet.2007.12.011 23 Figure 5. 400 500 600 700 800 900 350 rpm Temperature (K) 250 rpm Exo (2 W/g) 150 rpm dH/dt Intermetallics. Vol. 16. Núm. 3. 2008. Pag. 470-478 http://dx.doi.org/10.1016/j.intermet.2007.12.011 24 Figure 6. 110 100 1000 0 5 10 15 110 100 1000 0 5 10 15 150 rpm 250 rpm 350 rpm <d> (m) milling time (h) a) <d> (m) t·[]3 (h) 110 100 1000 0.0 0.1 0.2 0.3 0.4 0.5 0.6 110 100 1000 0.0 0.1 0.2 0.3 0.4 0.5 0.6 b) Cr content (%) milling time (h) Cr content (%) t·[]3 (h) Intermetallics. Vol. 16. Núm. 3. 2008. Pag. 470-478 http://dx.doi.org/10.1016/j.intermet.2007.12.011 25 Figure 7. 30 40 50 60 70 1 h 2 (degrees) 5 h 10 h 20 h 50 h 100 h 150 rpm 150 h 30 40 50 60 70 1 h 2 (degrees) 5 h 10 h 20 h 50 h 100 h 250 rpm 150 h 30 40 50 60 70 2 h 2 (degrees) 5 h 10 h 20 h 50 h 100 h 350 rpm 150 h