Abstract
One of the most successful ways to introduce samples in Serial Femtosecond Crystallography has been the use of microscopic capillary liquid jets produced by gas flow focusing, whose length-to-diameter ratio and velocity are essential to fulfill the requirements of the high pulse rates of current XFELs. In this work, we demonstrate the validity of a classical scaling law with two universal constants to calculate that length as a function of the liquid properties and operating conditions. These constants are determined by fitting the scaling law to a large set of experimental and numerical measurements, including previously published data. Both the experimental and numerical jet lengths conform remarkably well to the proposed scaling law. We show that, while a capillary jet is a globally unstable system to linear perturbations above a critical length, its actual and shorter long-term average intact length is determined by the nonlinear perturbations coming from the jet breakup itself. Therefore, this length is determined solely by the properties of the liquid, the average velocity of the liquid and the flow rate expelled. This confirms the very early observations from Smith and Moss 1917, Proc R Soc Lond A Math Phys Eng, 93, 373, to McCarthy and Molloy 1974, Chem Eng J, 7, 1, among others, while it contrasts with the classical conception of temporal stability that attributes the natural breakup length to the jet birth conditions in the ejector or small interactions with the environment.
Highlights
The shape, instability, and breakup of capillary jets have attracted scientific curiosity since long ago [1,2]
As occurs in the classical temporal stability model, the breakup mechanism assumed in this work relies on the growth of the most unstable capillary mode, which explains the size of the produced droplets and the distance between them
While the temporal stability approach assumes that the dominant capillary mode is triggered in the jet birth, the model adopted here presumes that the ultimate long-term source of excitation of this mode is the breakup region
Summary
The shape, instability, and breakup of capillary jets have attracted scientific curiosity since long ago [1,2]. To obtain a useful scaling law [28,30], one may assume that the most unstable (dominant) temporal mode is responsible for the breakup This mode is supposed to be triggered by a perturbation next to the jet inception region and is convected by the jet, which implies that the residence time in the jet scales as the inverse of the dominant mode growth rate. Using the temporal stability analysis, Ismail et al [31] derived two scalings for the breakup length of jets under the action of an axial electric field in the limits of small and large Reynolds numbers. An accurate study of the natural (not externally excited) capillary breakup of jets has been conducted by Umemura [32], who described the breakup mechanism as a “selfdestabilizing loop” In this loop, the energy of the perturbations responsible for the breakup comes exclusively from earlier breakup events. We make use of this view of the problem to rationalize our experimental observations
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