# The physics of mechanical alloying: A first report

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## Abstract

In this paper, we present a first attempt to define the basic geometry, mechanics, and physics of the process of mechanical alloying. The geometry of the collision events which lead to particle fragmentation and coalescence is modeled on the basis of Hertzian contacts between the grinding media which entrap a certain amount of material volume between the impacting surfaces. This geometry essentially defines the volume of material affected per collision, and from this information and characteristics of the specific mill and the material being processed, impact times, powder strain rates and strains, powder temperature increase, powder cooling times, and milling times can be approximated.

## Keywords

Metallurgical Transaction Nusselt Number Powder Particle Impact Velocity Mechanical Alloy## List of symbols

*a*deceleration of particle due to factional drag

*A*particle surface area

*C*_{p}specific heat

*C*_{R}charge ratio (mass of grinding balls/mass of powder)

*D*_{m}diameter of SPEX mill

*E*Young’s modulus

*E*_{eff}effective Young’s modulus of impacting media having different Young’s moduli

*f*fluid mechanics friction factor

*f*_{p}fractional powder volume (powder volume/ (mill volume - tool volume))

*F*_{D}fluid mechanics factional drag force

*g*gravitational constant

*g*_{p}geometrical constant relative to pressure developed in a Hertzian collision

*g*_{r}geometrical constant relative to Hertzian radius

*g*_{T}geometrical constant relative to collision time

*h*height of powder cylinder impacted during collision

*h*_{o}initial height of powder cylinder impacted during collision

*h*_{t}heat transfer coefficient

*h*_{f}height of fluid above reference line as a result of rotational velocity

*k*thermal conductivity of powder

*K*coefficient in constitutive plasticity equation for powder aggregate

*l*interlamellar thickness in powder particle

*l*_{o}initial interlamellar thickness in powder particle

*L*length of SPEX mill

*m*mass of colliding grinding media

*n*strain-hardening coefficient of powder aggregate

*n*_{B}number of balls in SPEX mill

*n*_{c}number of powder collision events

- Nu
Nusselt number

*p*pressure

*p*_{max}maximum pressure generated in a Hertzian collision

- Pr
Prandtl number

*r*effective powder particle radius

*r*_{d}radius of horizontal ball mill

*r*_{h}Hertz radius

*r*_{o}radius of attritor

*R*radius of grinding balls

- R
_{1,2} radii of curvature of impacting spherical bodies

*R*_{o}distance from center of attritor tank

- Re
Reynolds number

*t*time

*t*_{o}time between powder particle collisions

*t*_{cb}time between grinding media collisions

*t*_{f}time for completion of mechanical alloying

*t*_{p}processing time

*T*temperature

*T*_{a}ambient mill temperature

*T*_{b}bulk temperature of powder particles

- T
_{s} postimpact temperature of powder particles (

**=T**_{ a }+**ΔT**)*T*_{s}surface temperature of powder particles

*u*relative velocity of two contacting particles

*u*_{p}plastic work per unit volume on powder during impaction

*U*_{E}elastic strain energy associated with a Hertzian collision

*U*_{p}plastic work on powder during impaction

*v*precollision relative velocity of impacting media

- v
instantaneous relative velocity of impacting media

*v*_{a}ball velocity in attritor

*U*_{air}air velocity in SPEX mill

*v*_{g}ball velocity component due to gravity in a horizontal ball mill

*u*_{t}transverse velocity component of ball in a horizontal ball mill

*V*particle volume

*V*_{B}total ball volume in mill

*V*_{c}powder volume impacted during collision

*V*_{M}mill volume

*V*_{p}total powder volume in mill

- V
_{s} sound velocity

*V*_{s}volume swept by grinding balls between collisions

- V
_{∞} bulk stream velocity

*x*distance

*y*vertical drop distance of ball in horizontal ball mill

- α
thermal diffusivity

*β*=*−R*_{2}/*R*where

*R*_{2}is magnitude of radius of curvature of impacted surface- γ
proportionality constant

*δ*(linear) measure of deformation of impacting media

- δ
_{max} maximum center of mass displacement in a Hertz collision

- ε
powder strain during collision

- ε
_{max} maximum powder strain per collision

- ε
powder strain rate during collision

- E
accumulated strain during mechanical alloying

- E
_{f} critical accumulated strain for alloying completion

- η
viscosity

*μ*coefficient of friction

*p*density

*σ*stress experienced by powder during collision

*σ*_{o}coefficient in plasticity constitutive equation of powder aggregate

- τ
half-duration of impact during Hertzian collision

*v*Poisson’s ratio

*ϕ*proportionality constant

*ω*angular velocity of attritor or ball mill

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