# Advanced Metal Matrix Nanocomposites
## 4 Dispersion
### Cavitation in Composite Systems
#### 4.2.2 Cavitation in Molten Metal-Nanoparticle Composite Systems
For example, the ultrasonic intensity (I) required for breaking up a nanoparticle agglomerate that is located at the front of a ultrasound wave in a molten metal, can be estimated by [116]:  
\[\mathrm{I}=2\mathrm{W}_{\mathrm{L}}\left(\frac{\sigma}{\mathrm{D}\omega\rho_{ \mathrm{L}}}\right)^{2} \tag{4}\]  
where is it assumed that the nanoparticle agglomerate is of spherical shape with a diameter of D. In Equation (4) \(\sigma\) is the tensile strength of an agglomerate beyond which agglomerate fractures. \(\mathrm{W}_{\mathrm{L}}\) is  
Figure 16: In situ ultrasonic processing setup on the Diamond–Manchester imaging branchline (I13-2) of the Diamond Light Source, UK. Schematic diagram on the right-hand side shows the furnace, crucible, and the sonotrode (inset is the view along the X-ray beam) [114, 118, 120]. the acoustic power introduced in the liquid and \(\rho_{\rm L}\) liquid metal density. \(\omega\) is the ultrasonic frequency. Obviously, with the reduction of agglomerate size D (below \(\sim\)40-50 \(\mu\)m), the deagglomeration force generated by certain frequency ultrasound (on the order of kHz) increases sharply [116]. In the case of aluminum melt with the introduction of Al\({}_{2}\)O\({}_{3}\) nanoparticles, modeling of the deagglomeration process using Equation (4) is in good agreement with experimental measurements [21; 116]. In order to simulate the dynamic process of breaking up of nanoparticle clusters [139; 140] during ultrasonic cavitation, mesoscopic-scale modeling based on multiphase (fluid and solid) granular flow model [141; 142; 143] have been developed. For example, modeling with the coupling of computational fluid dynamics (CFD) and discrete element method (DEM) has been carried out for a number of nanocomposite systems, including aluminum- and magnesium-based alloys with the addition of SiC or Al\({}_{2}\)O\({}_{3}\) nanoparticles [139]. The DEM-CFD modeling demonstrates that the high velocity pulses caused by the collapsing of cavitation gas bubbles during the ultrasonic treatment are capable of breaking up the agglomerates [139]. The deagglomeration is highly dependent on the interfacial energy between nanoparticles and liquid metals, the duration and the maximum value of the velocity pulse, as well as the size of the nanoparticle agglomerates [139; 140]. The deagglomeration rate of a nanoparticle clusters can be quantified [140] as the average distance from particles that initially belonged to the cluster to their geometrical center, scaled by the particle radius:  
\[\Delta=\frac{1}{R_{\rm np}N_{\rm np}}\sum_{\rm i}\left|\overline{\rm x_{i}}- \frac{1}{N_{\rm np}}\sum_{\rm i}\overline{\rm x_{i}}\right| \tag{5}\]  
where \(N_{\rm np}\) is the number of nanoparticles in a cluster, \(R_{\rm np}\) is the radius of nanoparticle, \(\rm x_{i}\) denotes the position and bar is for the vector notation. In addition, mesoscale modeling has been employed to simulate the effect of acoustic streaming (or ultrasonic stirring) on the deagglomeration process of nanoparticle clusters in molten metals. For example, CFD coupled with the dense discrete phase model (DDPM) was applied to simulate the dynamic process of nanoparticles injection into the cavitation region in the molten metals. In a simulation of injecting SiC or Al\({}_{2}\)O\({}_{3}\) nanoparticles (up to 1.0 wt%) into the molten A356 aluminum alloy, the nanoparticles are distributed well in the fluid after 3 s of ultrasonic melt treatment [144; 124]. Obviously, the intense flow convection induced by the ultrasonic cavitation plays an important role in the deagglomeration and dispersion of ceramic nanoparticles in molten metals [143; 144].