Modeling of the mechanical-alloying process

Journal of

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Journal of Materials Processing Technology 52 (1995) 539 546

Materials Processing Technology

Modeling of the mechanical-alloying process L. Lfi a'*, M.O. LaP, S. Z h a n g b aDepartment of Mechanical and Production Engineering, National University of Singapore, Singapore 0511, Singapore bGINT1C Institute of Manufacturing Technology, Nanyang Technological University, Singapore 2263, Singapore Received 23 February 1994

Industrial Summary New materials such as metastable phases, ordered intermetallic compounds and amorphous alloys that are difficult to fabricate using normal metallurgical techniques can be produced using the mechanical-alloying technique, this being a powder-processing technique consisting of repeated cold welding, fracturing and re-welding of powders in a dry, high-energy ball mill machine. Ultrafine microstructures with grain sizes down to the nanometer level can be produced using this method. At least four events of collision can be identified in the ball-milling process: (a) direct collision between balls; (b) collision with sliding between balls; (c) direct collision between balls and the inner surface of the rotating container; and (d) collision with sliding between balls and the inner surface of the rotating container. Since the balls normally move in the same direction, the most efficient impact event for welding is direct collision between the balls and the inner surface of the container. In the present study, a model based on dynamics and cold-welding theory is used in calculations related to collision events. Occurrence of cold welding in the following cases is considered: welding between two different alloys; between the same alloy; and between the same alloy, with the inclusion of another interposed alloy which is not cold-welded. Because cold welding is an essential condition for mechanical alloying, the critical deformation required to achieve cold welding is evaluated with the model. It is proposed that the minimum bonding strength of the powders to be cold welded be considered as a criterion for mechanical alloying. With this, the critical inner diameter of the milling container at a particular rotational speed can be calculated.

1. Introduction T h e m e c h a n i c a l - a l l o y i n g (MA) t e c h n i q u e was d e v e l o p e d a r o u n d the 1970s El, 2]. It is n o r m a l l y a dry, h i g h - e n e r g y b a l l - m i l l i n g process for the p r o d u c t i o n of c o m p o s i t e

* Corresponding author. 0924-0136/95/$9.50 ~;~ 1995 Elsevier Science S.A. All rights reserved SSDI 0 9 2 4 - 0 1 3 6 ( 9 4 ) 0 1 6 2 0 - G

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metal powders with controlled and extremely fine microstructures. Originally, it was developed to manufacture high-temperature alloys combining oxide dispersion and intermetallic compound strengthening [2]. In this process, two simultaneous processes are involved, namely, cold welding between powders and fracturing of the powders. Hence, the raw materials used in MA include at least one fairly ductile metal to act as a host or binder to hold together the other ingredients. The powders are deformed plastically under high energy collision between balls, and between balls and the wall of the container. With new clean surfaces, the powders are welded to each other to form new particles with different compositions. The welding and fracturing processes are repeated until an alloyed powder is finally formed. It can be seen therefore, that to ensure the progress of mechanical alloying, cold welding of the powders is essential. The weldability of the powders depends upon the material system, the collision energy of the balls and the temperature at collision. According to the principle of cold welding, the greater the deformation, the higher will be the bonding strength. In recent years, increasing attention has been devoted to the development of new alloys based on the MA technique [3 10]. Some models, based on flow dynamics and mathematics, have also been presented to describe the physics of the process [1 1 13]. The present study attempts to use cold-welding theory and a simple model to simulate the essential conditions for the application of MA using a horizontal ball mill.

2. Mechanical alloying MA is a process where powders are charged in the container of a ball mill and are then caused to be collided by high-energy moving balls. The process can be carried out using an attritor, a SPEX shaker mill, a planetary mill or a horizontal ball mill. Whichever mill is employed, however, the principles are the same. Since the powders are cold welded and fractured during MA, it is critical to establish a balance between cold welding and fi'acturing in order to mechanically-alloy successfully [2]. The ability of cold welding and fracturing of the powder depends on the alloy system and the milling conditions employed. Soft materials normally have good weldability but high toughness and hence are difficult to be broken. According to Gilman [-1] and Zhang [14], the process includes welding of the powders, formation of a layered structure, deformation of the layered structure and formation of the alloy. Fig. 1 illustrates the different stages of the process [2], from which illustration it is clear that Element A

Element B

Laminated powder

Alloyed powder

Fig. 1. Ew)lution of the microslructure during mechanical alloying.

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541

the essential condition for MA is cold welding, which binds different materials together and hence enables diffusion process to occur at the later stage of MA. At an intermediate stage of processing, the welded powders are deformed plastically to produce a laminated structure. Thereafter, dissolution of the solute elements and formation of areas of solid solution throughout the powder matrix are facilitated by slight heating due to collision, lattice defects and a short diffusion distance. The deformation also creates a high defect density, decreases the diffusion distance and increases the powder temperature, which latter enhances the diffusion rate. The powders are further refined as fracturing and cold-welding continue until, finally, fully-alloyed powers are obtained.

3. Modeling of the mechanical-alloying process 3.1. Kinetics of the balls during milling Consider a MA process using a conventional horizontal ball mill, illustrated in Fig. 2. In this system, the balls or rods and powders are placed in a container which rotates about its horizontal axis at a speed of co. The balls and powders will move along the inner surface of the container if the container rotates sufficiently quickly. If it is assumed that the balls and powders have the same speed as that of the container, the normal accelerate of the ith ball, as shown in Fig. 3, can be written as a i --

D-d 2

(D 2 ,

(1)

Fig, 2. Configuration of a horizontal ball mill.

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L. Li~ et al. /Journal o f Materials Processing Technology 52 (1995) 539 546

03

l

Fig. 3. Force balance of a ball during rotation in a horizontal ball mill.

where D is the inner diameter of the container and d is the diameter of the ball. According to N e w t o n ' s law, the ball should satisfy the following condition during m o v e m e n t (15): N

j=l

f i j = mi ai,

(2)

w h e r e f j is the force acting on the ith ball, mi is the mass of the ith ball and ai is the acceleration in the n o r m a l direction of the ith ball. Three forces, namely gravity P, friction force F between the inner wall and the ball, and n o r m a l force N are exerted on a ball during milling. F r o m Fig. 3, the resultant force in the radial direction can be written as N + P cos q).

(3)

Substituting Eq. (3) into Eq. (2), it follows that N +Pcos~o =

P

g

ai,

(4)

where g is the gravitational acceleration. Substituting Eq. (1) into Eq. (4) the latter can be written as N --

PD-d

,q

2

(o 2

P c o s (#,

(5)

If e) is constant during milling, N is a function of (p. The essential condition for milling is that the balls must fall d o w n to impact the powders. This critical condition can be described by N = 0 when the balls separate from the inner surface of the container and

L. Lfi et al./Journal of Materials Processing Technology 52 (1995) 539-546

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fall down. The critical angle tpc at which a ball separates from the inner surface can be obtained from Eq. (5): cos ~0c -

(D -- d ) o 2 2g

(6)

It can been seen from Eq. (6) that the critical angle is a function only of the inner diameter and the rotational speed of the container. After the ball has separated from the wall, it travels along a parabolic line that can be expressed as

D-d x = Vo t cos ~0c - - 2

sin q~c,

D--d y = Vo t sin ~o~ - 1 gt 2 + ~ COS ~Pc,

(7)

(8)

where Vo is the speed when the ball just separates from the wall and t is the time after the separation. The ball travels until it impacts the wall (or other balls). The height h through which the ball falls can be calculated by solving Eqs. (7) and (8) as well as the following:

The speed of the ball before impact is vx = Vo cos ~p~,

(10)

vy = Vo sin ~oc - gt,

(11)

V = ~2x2 + Vy, 2

(12)

where vx, vy and v are the speed components of the ball in the x, y and resultant directions respectively. The collision energy, Ec, is given by E c = ½ m y 2.

(13)

3.2. Collision event During MA, four possible impact events may take place: (a) normal impact between balls and the inner surface of the container; (b) impact with sliding displacement between balls and the inner surface of the container; (c) normal impact between balls; and (d) impact with sliding displacement between balls. In the case of collision between balls, since the balls normally move in the same direction, the more efficient collision is the direct impact between balls and the inner surface of the container rather than between balls themselves. Because the powder-particle aggregates trapped between colliding bodies are much smaller than the colliding bodies themselves, the deformation of powders between two balls or between a ball and the inner wall of the

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L. Li~ et al. /Journal of Materials Processing Technology 52 (1995) 539 546

container during collision events can be likened to that in an upset forging between two parallel plates. The deformation strain per collision event can then be written as

~max =

In H

HA ~ '

(14)

where H is the instantaneous height of the powder cluster, and AH is the change in height of the powder cluster during one collision. The mean stress needed for the deformation of a cluster of powders is:

(15) where a is the mean stress during upset forging, k is the shear stress at yield, q is the diameter of a cluster of powders, and l~ is the friction parameter between powder and balls. Because of work-hardening of the powders, the yield shear stress is actually a function of deformation strain. For simplicity, however, the yield shear stress is assumed to be constant during MA. Hence, if the diameter of a cluster of powders is considered constant, the deformation energy, E can be written as

E-krl2~(AHr//LElnH-ln(H-AH)]) 2

3

'

(16)

where AH can obtained by substituting Eq. (16) into Eq (13). 3.3. Cold-welding event

The welding process during collision may be approximated as the cold welding o.f two plates under pressure. The increase in pressure during collision increases the real area of contact between the powders. According to cold-welding theory, cold welding is associated with a particular range of deformation strain of powders. The deformation strain required by cold welding depends on the material system employed and can occur in the following cases: (a) welding between two different alloys; (b) welding between the same alloy; and (c) welding between the same alloy with the inclusion of another alloy which is not cold welded. The bond strength between the powders is a function of the deformation strain. Fig. 4 shows the relationship between the bond strength of different materials and the deformation reduction. It is proposed that the value of the deformation strain at an intersection between an extrapolating line of the experimental data with the x axis is the minimum deformation value for cold welding to occur during MA. Hence, the deformation reduction of powders per collision event should be greater than this critical value. It should be noted that the softer the material, the better is the weldability. At the very early stage of MA, since the powders are normally soft, an increase in powder size due to welding has been observed. However, the powder size decreases at a later stage of MA because of the increase in the hardness of the powders. To increase the milling efficiency, either the powders have to be annealed at an intermediate stage or the milling speed has to be increased.

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1.4 ~

1.2

~

1.0

~

0.8

zx Cu o AI •

Zn ..

~ o.6 rm e.-,

0.4 0.2

o

0 o

20

40

60

80

100

Deformation Strain (%) Fig. 4. Relationship between bond strength and deformation strain (after [17]).

In the case of high-speed milling, more heat will be generated, the latter can be predicted by [12]:

AT_troVer ~ 2 KopC p

(17)

where A T is the increase in temperature, Cp is the specific heat of the powder, a0 is the initial flow stress, Ko is the thermal conductivity of the powder and p is the powder density. Powders can be dynamically annealed under high-speed milling conditions, where under these conditions, because of the increase in milling temperature, welding also becomes easier.

4. Conclusions The present model presents a criterion of welding for mechanical alloying when at least one soft material is employed. Mechanical alloying takes place if any of the following mechanisms occurs: welding between different alloys; welding between the same alloy; and welding between the same alloy with the inclusion of another alloy which is not cold-welded. Because cold welding is an essential condition for mechanical alloying, the critical deformation required to achieve cold welding is evaluated in the model. It is proposed that the minimum bonding strength of cold-welded powders be considered as a criterion for mechanical alloying. With this, the critical inner diameter of the milling container at a particular rotational speed can be calculated.

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