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Brownian Motion Effects on Particle Pushing and Engulfment During Solidification in Metal-Matrix Composites

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Abstract

Particle pushing and/or engulfment by the moving solidification front (SF) is important for the uniform distribution of reinforcement particles in metal-matrix composites (MMCs) synthesized from solidification processing, which can lead to a substantial increase in the strength of the composite materials. Previous theoretical models describing the interactions between particle and moving SF predict that large particles will be engulfed by SF while smaller particles including nanoparticles (NPs) will be pushed by it. However, there is evidence from metal-matrix nanocomposites (MMNCs) that NPs can sometimes be engulfed and distributed throughout the material rather than pushed and concentrated in the last regions to solidify. To address this disparity, in this work, an analytical model has been developed to account for Brownian motion effects. Computer simulations employing this model over a range of the SF geometries and time steps demonstrate that NPs are often engulfed rather than pushed. Based on our results, two distinct capture mechanisms were identified: (i) when a high random velocity is imparted to the particle by Brownian motion, large jumps allow the particle to overcome the repulsion of the SF, and (ii) when the net force acting on the particle is insufficient, the particle is not accelerated to a velocity high enough to outrun the advancing SF. This manuscript will quantitatively show the effect of particle size on the steady state or critical velocity of the SF when Brownian motion are taken into consideration. The statistical results incorporating the effects of Brownian motion based on the Al/Al2O3 MMNC system clearly show that ultrafine particles can be captured by the moving SF, which cannot be predicted by any of classical deterministic treatments.

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Abbreviations

\( a \) :

effective atomic/molecular size in the liquid (e.g., the diameter of an atom of the liquid metal)

\( f_{\rm{max}} \) :

upper limit of the fraction of the particles that have a tendency toward capture

\( F_{\rm{Drag}} \) :

viscous drag force acting on the particle

\( F_{\rm{Net}} \) :

net force acting on the particle

\( F_{{{\rm{Net}}_{t} }} \) :

net force acting on the particle for current iteration

\( F_{\rm{SF}} \) :

force exerted on the particle by the solidification front

\( h \) :

distance between particle and solidification front

\( h_{t = o} \) :

distance between particle and solidification front at start of simulation

\( I_{t} \) :

impulse acting of particle during current iteration

\( k_{\rm{B}} \) :

Boltzmann’s Constant

\( m_{\rm{fp}} \) :

mass of liquid atom/molecule

\( m_{\rm{p}} \) :

mass of particle

\( n \) :

number of iterations

\( P_{\rm{A}} \) :

size-dependent probability of particles that have the tendency to be captured by the solidification front

\( P_{\rm{B}} \) :

size-dependent probability of “capturable” particles that are actually captured by the solidification front

\( R \) :

radius of the particle

\( R_{\rm{A}}^{*} \) :

critical particle size

\( R_{\rm{B}}^{*} \left( n \right) \) :

critical “capturable” particle size

\( \Delta t \) :

time step of iteration

\( T \) :

temperature of the liquid

\( v_{\rm{fp}} \) :

velocity of liquid atom/molecule

\( v_{\rm{p}} \) :

velocity of particle

\( v_{\rm{SF}} \) :

velocity of the solidification front

\( \bar{v}_{t} \) :

average velocity of particle during current iteration

\( \Delta v_{t} \) :

change in velocity during current iteration

\( v_{\rm{ss}} \) :

steady-state particle pushing velocity

\( v_{{{\rm{ss}}_{\rm{Brown}} }} \) :

steady-state particle pushing velocity taking into consideration Brownian motion

\( v_{t = o} \) :

velocity of particle at start of simulation

\( v_{t - 1} \) :

velocity of particle at the end of the previous iteration

\( v_{{t_{\rm{o}} }} \) :

velocity of particle at start of iteration after adding velocity change due to Brownian motion

\( \Delta v_{\rm{B}} \) :

change in particle velocity due to Brownian motion

\( \Delta v_{\rm{p}} \) :

change in velocity of the particle

\( W_{\rm{A}} \) :

parameter controlling the shape of \( P_{\rm{A}} \)

\( W_{\rm{B}} \left( n \right) \) :

parameter controlling the shape of \( P_{\rm{B}} \)

\( X_{{{\rm{SF}}_{t} }} \) :

distance traveled by the solidification front at time t

\( X_{{{\rm{p}}_{t} }} \) :

position of particle at time t

\( \Delta X_{\rm{p}} \) :

change in position of particle

\( \alpha \) :

ratio of radius of curvature of particle to radius of curvature of the solidification front

\( \sigma_{\rm{sl}} \) :

solid/liquid interfacial energy

\( \Delta \sigma_{\rm{cls}} \) :

complex interfacial energy

\( \eta \) :

viscosity of liquid metal

\( \rho_{\rm{p}} \) :

density of particle

\( \theta \) :

angle between direction of particle velocity and direction of solidification front

\( \theta_{t} \) :

angle between direction of particle velocity and direction of solidification front for current iteration

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Acknowledgments

The authors wish to thank Dr. Robert T. McSweeney and Dr. Dev Venugopalan for their review and suggestions. This material is based upon work supported by the U.S. Army Research Laboratory under Cooperative Agreement No. W911NF-08-2-0014. The views, opinions, and conclusions made in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of Army Research Laboratory or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation herein.

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Authors and Affiliations

  1. Materials Science and Engineering Department, University of Wisconsin-Milwaukee, 3200 N. Cramer St., Milwaukee, WI, 53211, USA

    J. B. Ferguson (Postdoctoral Research Associate), Benjamin F. Schultz (Postdoctoral Research Associate), Pradeep K. Rohatgi (Professor) & Chang-Soo Kim (Senior Engineer)

  2. Department of Nanotechnology, University of Miskolc, Egyetemvaros, Miskolc, 3515, Hungary

    George Kaptay (Professor)

  3. U.S. Army Research Laboratory, Weapons and Materials Research Directorate, Aberdeen Proving Ground, Aberdeen, MD, 21005, USA

    Kyu Cho (Senior Engineer)

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  1. J. B. Ferguson
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  2. George Kaptay
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  4. Pradeep K. Rohatgi
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Correspondence to Chang-Soo Kim.

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Manuscript submitted February 2, 2014.

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Ferguson, J.B., Kaptay, G., Schultz, B.F. et al. Brownian Motion Effects on Particle Pushing and Engulfment During Solidification in Metal-Matrix Composites. Metall Mater Trans A 45, 4635–4645 (2014). https://doi.org/10.1007/s11661-014-2379-x

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