Impact of Volume Fraction and Size of Reinforcement Particles on the Grain Size in Metal–Matrix Micro and Nanocomposites

Abstract

In metal–matrix micro and nanocomposites (MMCs and MMNCs), the presence and interactions of various strengthening mechanisms are not well understood, but grain boundary strengthening is considered as one of the primary means of improving the yield strength of composites. Owing to the importance of grain size on mechanical properties, it is necessary to be able to describe how incorporation of nanoparticles (NPs) in both powder metallurgy (PM) and solidification processing (SP) affects this critical property. In the present work, we provide a basis for an empirical equation that relates particle fraction and particle size to MMNC grain size for both PM and SP synthesis methods. The model suggests that NPs retard grain coarsening in PM MMNCs and also seems to describe the effect of reinforcement concentration on grain size in SP MMCs and MMNCs.

Appendix

It is an underlying assumption of the model that metallic grains must be in contact to grow and that particles present at the grain surfaces reduce the area of contact between grains and thereby slows grain growth/coarsening. There are two relationships that likely influence the effectiveness of the particles to restrict grain growth and coarsening, i.e., the relative size of the grains and particles and the relative quantities of the grains and particles. In this Appendix, it will be shown from geometrical considerations and the assumption that grain growth rate is proportional to the grain/grain contact area that the refining potential, $p_{{{\text{f}}_{\text{p}} }}$, is inversely proportional to particle size, provided that (1) volume fractions of reinforcement are low, (2) reinforcement particles are uniformly dispersed over the grain surfaces, and (3) there is not a large difference between reinforcement size and initial grain size.

Particle volume fraction, f _p, depends on the average volume of particles, $\overline{V}_{\text{p}} ,$ and average volume of grains, $\overline{V}_{\text{g}}$, both of which depend on the number of particles, n _p , and number of grains, n _g , and the radius of each, $\overline{r}_{\text{p}}$ and $\overline{R}$.

$$ f_{\text{p}} = \frac{{\overline{V}_{\text{p}} }}{{\overline{V}_{\text{g}} + \overline{V}_{\text{p}} }} = \frac{{n_{\text{p}} \frac{4}{3}\pi \left( {\overline{r}_{\text{p}} } \right)^{3} }}{{n_{\text{g}} \frac{4}{3}\pi \left( {\overline{R} } \right)^{3} + n_{\text{p}} \frac{4}{3}\pi \left( {\overline{r}_{\text{p}} } \right)^{3} }} = \frac{{n_{\text{p}} \left( {\overline{r}_{\text{p}} } \right)^{3} }}{{n_{\text{g}} \left( {\overline{R} } \right)^{3} + n_{\text{p}} \left( {\overline{r}_{\text{p}} } \right)^{3} }} $$

(A1)

For low volume fractions, the volume of reinforcement particles contributes minimally to the total volume and Eq. [A1] can be approximated by Eq. [A2]

$$ f_{\text{p}} = \frac{{n_{\text{p}} }}{{n_{\text{g}} }}\left( {\frac{{\overline{r}_{\text{p}} }}{{\overline{R} }}} \right)^{3} $$

(A2)

The fraction n _p/n _g is the number of reinforcement particles per matrix grain. Therefore,

$$ \frac{{n_{\text{p}} }}{{n_{\text{g}} }} = f_{\text{p}} \left( {\frac{{\overline{R} }}{{\overline{r}_{\text{p}} }}} \right)^{3} $$

(A3)

Because these particles are at the surface of the grain, they cover or block a certain fraction of the surface area of the grain based on the projected area of each particle.

$$ \frac{{S_{\text{blocked}} }}{{S_{\text{total}} }} \cong \frac{{\frac{{n_{\text{p}} }}{{n_{\text{g}} }}\pi \left( {\overline{r}_{\text{p}} } \right)^{2} }}{{4\pi \left( {\overline{R} } \right)^{2} }} = \frac{1}{4}\left( {\frac{{n_{\text{p}} }}{{n_{\text{g}} }}} \right)\left( {\frac{{\overline{r}_{\text{p}} }}{{\overline{R} }}} \right)^{2} = \frac{{f_{\text{p}} }}{4}\left( {\frac{{\overline{R} }}{{\overline{r}_{\text{p}} }}} \right)^{3} \left( {\frac{{\overline{r}_{\text{p}} }}{{\overline{R} }}} \right)^{2} = \frac{{f_{\text{p}} }}{4}\left( {\frac{{\overline{R} }}{{\bar{r}_{\text{p}} }}} \right) $$

(A4)

Grain growth will take place where the grains are in contact. It is expected that the maximum rate of growth, $k_{{{\text{s}}_{\text{alloy}} }}$, will occur when the maximum contact surface area, $S_{{{\text{contact}}_{ \hbox{max} } }}$, is available. Any blocking of the contact area will result in slower growth, such that the growth rate, k _s, should be related to the unblocked or free contact surface area, S _contact, such that

$$ k_{{{\text{s}}_{\text{alloy}} }} \propto S_{{{\text{contact}}_{ \hbox{max} } }} $$

(A5)

$$ k_{\text{s}} \propto S_{\text{contact}} $$

(A6)

Given the relation of Eq. [6], it follows that

$$ \frac{{f_{\text{p}} }}{\kappa } = \frac{{k_{{{\text{s}}_{\text{alloy}} }} - k_{\text{s}} }}{{k_{\text{s}} }} = \frac{{S_{{{\text{contact}}_{ \hbox{max} } }} - S_{\text{contact}} }}{{S_{\text{contact}} }} $$

(A7)

If it is further assumed that maximum surface area of grains in contact ($S_{{{\text{contact}}_{ \hbox{max} } }}$) is proportional to total surface area of grain (S _total). Then,

$$ S_{{{\text{contact}}_{ \hbox{max} } }} \propto S_{\text{total}} $$

(A8)

$$ S_{\text{contact}} \propto \left( {S_{\text{total}} - S_{\text{blocked}} } \right) $$

(A9)

$$ \frac{{f_{\text{p}} }}{\kappa } = \frac{{S_{{{\text{contact}}_{ \hbox{max} } }} - S_{\text{contact}} }}{{S_{\text{contact}} }} = \frac{{S_{\text{blocked}} /S_{\text{total}} }}{{1 - S_{\text{blocked}} /S_{\text{total}} }} $$

(A10)

Solving for κ and substituting Eq. [A4] for S _blocked/S _total results in an expression requiring only the grain and particle radii.

$$ \kappa = \frac{{f_{\text{p}} }}{{S_{\text{blocked}} /S_{\text{total}} }} - f_{\text{p}} = 4\left( {\frac{{\bar{r}_{\text{p}} }}{{\bar{R}}}} \right) - f_{\text{p}} $$

(A11)

In the case of a low reinforcement volume fraction material in which the size of the reinforcement particle does not greatly differ from the initial size of the matrix grain Eq. [A11] simplifies to Eq. [A12], where κ will have an approximately linear dependence on the relative size of reinforcement and grain.

$$ \kappa \cong 4\left( {\frac{{\bar{r}_{\text{p}} }}{{\bar{R}}}} \right) $$

(A12)