## 1 Introduction
[37] have also previously produced a 3D serpentine micro-mixer in order to examine the dynamics of carbamates. The serpentine geometry they employed was again found to significantly enhance the mixing efficiency, enabling them to achieve a high synthesis throughput of 5.3 g/h with continuous flow. [PERSON] et al. [38] also used the same mixer design incorporating a nozzle for mixing-and-spraying a protein solution onto a copper grid for electron microscopy measurements. Using this approach, they were able to capture different conformations of RecA-ssDNA filaments using cryo-electron microscopy. Their mixer was able to achieve high mixing efficiencies (up to 90%) on millisecond mixing time scales (2-5 ms) using nanolitre volumes of the sample. Although this work is the first simulation study examining chaotic mixing in integrated mix-and-inject devices, these types of effects in isolated micromixers have been reported on previously [17; 19; 20; 21; 22]. For example, [PERSON] et al. [39] have carried out a parametric study of advection in order to investigate phenomena such as the stretching of interfaces and the chaotic stirring that occurs in micromixers incorporating twisted pipes. They reported that passive mixing utilising the chaotic effect enhances the mixing efficiency (stirring quality) which can be obtained without the need for adding any additional energy into the system. Whilst the influence of passive mixing on micromixers in isolation has been reported before in the literature, here, we are interested in investigating the range of parameters and mixing times that can be achieved using integrated mix-and-inject devices. Understanding and optimising these characteristics is critical to achieving the maximum time-resolution for molecular dynamics experiments. Such devices are currently used on both synchrotron and XFEL sources to study dynamic changes in proteins, and also have the potential to be deployed in a range of different molecular sensing applications. The two key aims with the present work are (1) to determine the influence of the device geometry on mixing performance under conditions of stable jetting and (2) to predict the minimum time required for homogeneous mixing, which will determine the limit on the timescales for the dynamic processes which can be studied. Previously, we experimentally demonstrated the application of a new approach to fabricating planer microfluidic mix-and-inject devices based on high-resolution photolithography using SU8 on glass [40; 41]. These devices have the advantage of being relatively straightforward to fabricate whilst having very high mechanical rigidity and chemical inertness. One of the primary applications for these devices is the investigation of molecular dynamics using X-ray diffraction at XFEL sources. In our previous experimental work, we used fluorescence intensity analysis to demonstrate millisecond and sub-millisecond combined mixing and jetting. To date, the new field of integrated mix-and-inject microfluidic devices has received much less attention in terms of numerical simulation studies than standalone micromixers. Integrated devices enable both rapid mixing and the stable formation of a free-standing liquid microfluidic jet for sample delivery, which facilitates time-resolved molecular imaging studies. Here, we present micromixer simulations that are specifically conducted under conditions that are consistent with the formation of a stable jet. We determine the characteristics of these devices through numerical simulation to predict the minimum mixing time, using (partially) chaotic mixing, which is achieved with a serpentine micromixer. Characterising the mixing performance of these devices is an essential element in sample delivery in the context of imaging molecular dynamics at synchrotrons and XFELs. We also attempt to provide a guide for the optimisation of the mixing component for future molecular imaging studies to further reduce the mixing time whilst still maintaining stable liquid jet delivery.