Multifractal analysis of ultrasonically machined surfaces of cylindrical quartz crystals: the effect of the abrasive grits

Since synthetic quartz is essential to produce 3D resonators for numerous applications in precision electronics, in this work the surface topography of cylindrical quartz bars is investigated using the multifractal technique. The cylindrical bars were manufactured with ultrasonic machining using five SiC grits ranging from 6 to 50 mu m. The machined surfaces were initially characterized by contact profilometry and scanning electron microscopy (SEM). The multifractality of the machined surfaces was scrutinized using a box-counting method applied to the images obtained with 500X magnification. The multifractal spectrum indicated that the fractal dimension f(alpha) and the width of the fractal spectrum Delta alpha are dependent on the grit size, but this dependence is not monotonic. The lowest (negative) value for Delta f(alpha) was found for 25 mu m grits indicating that for these grits the lower frequency events (grooves with tens mu m width occurring along the USM direction) control the surface topography much more than high-frequency events related to brittle microcracking. The abrasive wear due to the continuous slurry recycling in lateral tool-workpiece interfaces contributed to smooth the groove texture as well as the sharpness of microscopic indentations, which remained observed on the surfaces machined with 50 mu m grits. The opposite paths observed for the arithmetical mean deviation of the measured profile (R-a) and Delta f(alpha) parameters with the cutting rate measured for each grit size were valuable to differentiate flat-rough and unlevelled-rough topographies in quartz bars.