**1. Introduction**

Vibration treatment of a solidifying melt is one of the known methods of influencing the casting quality of aluminum alloys through reducing the grain size and obtaining a more uniform microstructure. An amplitude up to 5 cm and a frequency up to 10 kHz can be achieved with mechanical vibrations. At a higher frequency, e.g., above 16 kHz, ultrasonic treatment of melts can be performed [1]. Ultrasonic melt treatment of an A390 hypereutectic Al–Si alloy is shown to enhance the homogeneity of the microstructure and to increase the yield strength of the as-cast alloy by12% [2].

There are numerous theoretical and experimental works devoted to the influence of acoustic and mechanical vibrations on liquid metals during solidification (see for example [1] and [3–9]). It has been shown that vibration can modify the microstructure by multiplication of solidification substrates [5–7]. Study of the mechanical properties of A356 alloy after vibration treatment with frequencies of 100 Hz and 150 Hz showed the tensile and yield strengths improved by 20% and 10%, respectively, but the use of vibration frequencies of 200 Hz caused the formation of a high porosity microstructure and caused major defects [8]. The multiplication of grains is achieved through mechanical and thermal fragmentation of dendritic crystals by elastic stresses, microflows (thermal and solutal convection in a liquid) and cavitation [9–14]. Generally, the refinement of crystalline grains and the increasing soundness of a casting lead to improved mechanical properties and quality of cast products.

A mathematical model describing the vibration effect on liquid metals as a function of cavitation and turbulent flows in the bulk melt and in the solid-liquid zone has been proposed in [15]. Numerous prior studies showed the positive effect of both phenomena (cavitation and turbulence in the melt bulk) on the quality of casting microstructures [1,3,5]. These conditions can be implemented under high-intensity processes with relatively large amplitudes (more than 1 cm) and frequencies (more than 60 Hz but less than the ultrasonic range). In this case, cavitation in the liquid metal may occur in the melt saturated with gases (e.g., hydrogen in aluminum) during the solidification ("pseudocavitation") or under vibration treatment.

It was experimentally found [16] that the effect of grain refinement increases up to a certain frequency depending on the liquid alloy properties with an increase in the pre-ultrasonic vibration frequency. A further increase in the frequency reduces the effect as compared to an optimal vibration frequency (about 50 Hz with an amplitude of 0.49 mm in [16]). In our earlier work [15], the optimal frequency for high-intensity vibration treatment of melts was explained by the simultaneous occurrence of pseudocavitation and turbulent flows. The work [17], investigated the crystal behaviors under vibration using a transparent NH4Cl-70%H2O alloy (frequency from 20 to 1 000 Hz and acceleration from 10 to 100 m/s2). The optimal frequency of 50 Hz and acceleration of 100 m/s2 were found when the grain refinement effect was strengthened.

However, even in a more general case with low-intensity vibration, a small amount of gases in a melt, and a lack of cavitation and turbulence, vibration treatment of a solidifying alloy can also lead to a positive result [2,5]. In this case, there should also be an optimal frequency and amplitude under which the improvement of casting microstructure is most pronounced.

The aim of this study is to examine the mechanisms for improving the as-cast microstructures by vibration during alloy solidification; and based on a numerical solution of the Stefan problem to determine the optimal conditions (frequency and amplitude) for the vibration treatment without cavitation and turbulence.
