The Smaller the Harder: Theorization of a Threshold Size below which Fluid Inclusions do not Decrepitate

Fluid inclusions are known to be formed at pressures reaching some tens of kilobars. The solid matrix encompassing the fluid filled cavity experiences decompression as a consequence of uplift processes such as eruptions. This event may prompt the mechanical failure of the host-mineral matrix through either stretch or decrepitation, depending on a ductile or brittle mechanism of matrix failure, respectively. Laboratory experiments performed on synthetic inclusions show that the decrepitation temperature is strongly size dependent, with smaller cavities observed to decrepitate at higher temperatures. On the other hand, natural inclusions which undergo migration through a pressure gradient are always found intact below a critical size. In this paper, we model fluid inclusions as spherical cavities in a continuous elastic medium. Under these conditions, the tangential stress applied in the matrix has a cubic dependence on 1/r. This means that the maximum tensile stress concentrates at the cavity/matrix interface, and, as can be demonstrated, this stress is independent on cavity size. This means that if a fracture criterion based on the maximum stress concentration is adopted, there is no way for accounting for the size dependence of fracturing. Here, we address this problem by adopting a non-local stress approach to fracturing. Our mechanical model is based on two parameters: d, a characteristic distance which is material-dependent and accounts for the brittleness of the matrix, and T0, the uniaxial tensile strength of the matrix. We show that the model calculation well approximates experimental datasets relating internal pressure to cavity size and we demonstrate the fundamental prediction that the decrepitation phenomenon is characterized by a threshold size, Dth, and a threshold internal pressure of the cavity, below which decrepitation would not be allowed. The order of magnitude of the decrepitation threshold size is 1 µm for the analysed datasets of quartz and olivine inclusions. This means that unperturbed inclusions are likely to be observed, even though the submicrometric and nanometric ones would preserve, if accessible with non-destructive methods, the real world of Earth interior.