On the scaling of fragmentation and energy dissipation in collisions of dust aggregates

Abstract

Abstract Fragmentation of granular clusters may be studied by experiments and by granular mechanics simulation. When comparing results, it is often assumed that results can be compared when scaled to the same value of $$E/E_{\mathrm{sep}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>E</mml:mi> <mml:mo>/</mml:mo> <mml:msub> <mml:mi>E</mml:mi> <mml:mi>sep</mml:mi> </mml:msub> </mml:mrow> </mml:math> , where E denotes the collision energy and $$E_{\mathrm{sep}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>E</mml:mi> <mml:mi>sep</mml:mi> </mml:msub> </mml:math> is the energy needed to break every contact in the granular clusters. The ratio $$E/E_{\mathrm{sep}}\propto v^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>E</mml:mi> <mml:mo>/</mml:mo> <mml:msub> <mml:mi>E</mml:mi> <mml:mi>sep</mml:mi> </mml:msub> <mml:mo>∝</mml:mo> <mml:msup> <mml:mi>v</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:math> depends on the collision velocity v but not on the number of grains per cluster, N . We test this hypothesis using granular-mechanics simulations on silica clusters containing a few thousand grains in the velocity range where fragmentation starts. We find that a good parameter to compare different systems is given by $$E/(N^{\alpha }E_{\mathrm{sep}})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>E</mml:mi> <mml:mo>/</mml:mo> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>N</mml:mi> <mml:mi>α</mml:mi> </mml:msup> <mml:msub> <mml:mi>E</mml:mi> <mml:mi>sep</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , where $$\alpha \sim 2/3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>∼</mml:mo> <mml:mn>2</mml:mn> <mml:mo>/</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:math> . The occurrence of the extra factor $$N^{\alpha }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>N</mml:mi> <mml:mi>α</mml:mi> </mml:msup> </mml:math> is caused by energy dissipation during the collision such that large clusters request a higher impact energy for reaching the same level of fragmentation than small clusters. Energy is dissipated during the collision mainly by normal and tangential (sliding) forces between grains. For large values of the viscoelastic friction parameter, we find smaller cluster fragmentation, since fragment velocities are smaller and allow for fragment recombination. Graphic abstract