Mass production of shaped particles through vortex ring freezing

A vortex ring is a torus-shaped fluidic vortex. During its formation, the fluid experiences a rich variety of intriguing geometrical intermediates from spherical to toroidal. Here we show that these constantly changing intermediates can be ‘frozen' at controlled time points into particles with various unusual and unprecedented shapes. These novel vortex ring-derived particles, are mass-produced by employing a simple and inexpensive electrospraying technique, with their sizes well controlled from hundreds of microns to millimetres. Guided further by theoretical analyses and a laminar multiphase fluid flow simulation, we show that this freezing approach is applicable to a broad range of materials from organic polysaccharides to inorganic nanoparticles. We demonstrate the unique advantages of these vortex ring-derived particles in several applications including cell encapsulation, three-dimensional cell culture, and cell-free protein production. Moreover, compartmentalization and ordered-structures composed of these novel particles are all achieved, creating opportunities to engineer more sophisticated hierarchical materials.


Shape
Average Da Standard

Discussion on the concept of "freezing"
In materials science, "freezing" typically refers to a phase transition in which a liquid solidifies or turns into a solid when its temperature is decreased below its freezing point. In this work, we use the term of "freezing" in a more general sense; that is to "fix a shape". More specifically, we use the term to describe a process that fixes the unstable or flowable liquid vortex rings into stable or non-flowable hydrogel or solid microparticles of a defined shape. This process could be a chemical reaction, physical gelation (crosslinking) or precipitation.

Assisted-assembly of nanoclay hydrogel donut-VRP into a close-packed monolayer
Due to the unique anisotropic symmetry of the nanoclay hydrogel donut-VRP, when floating on the liquid surface, all the nanoclay hydrogel donut-VRP preserved the same orientation. By taking advantage of this interesting property, we designed a "Langmuir-Blodgett deposition -like" assembly strategy to assembly the nanoclay hydrogel donut-VRP into a close-packed monolayer. Firstly, we discover that the nanoclay hydrogel donut-VRP can float at the interface between Dulbecco's Modified Eagle Medium (DMEM) and Histopaque, a commonly used cell separation system based on the density difference ( Supplementary Fig. 3a). Then, this floating system was introduced into a "Langmuir-Blodgett deposition -like" device ( Supplementary Fig.   3b). Basically, two barriers were used to push the floating donut-VRP closer into a close-packed monolayer. Other nanoclay hydrogel VRP were assembled using the similar strategy.

Effect of the electrospray working distance on the microVRP size/shape
We have studied the effect of working distance (i.e. the distance between the electrospray tip and the surface of the collecting solution) on the microVRP size and shape ( Supplementary Fig. 4). The voltage was set at 7.5 kV and the same nanoclay solution was electrosprayed at different working distances ranging from 2 cm to 9 cm.
In general, we observed relatively uniform donut-microVRP with increasing outer diameter when the working distance varied from 2 cm to 5 cm. This is consistent with the trend we observed when we fixed the working distance but decreased the voltage (Fig. 2c). This was expected since increasing working distance has a similar effect on the electric field strength to decreasing the voltage. However, when the working distance was higher than 6 cm, the electrospray jet became unstable and resulted in nonuniform microVRP with various sizes and shapes. Further increase of the working distance to 9 cm, the electrospray jet became so unstable that no valid experiment could be performed. From these experiments, we conclude that it may be difficult to control the shape of microVRP by tuning the working distance alone. A better way to control the shape is through the adjustment of the viscosity of the solution as shown in Fig. 2d.

Model Overview, Assumptions and Geometry
The simulation uses an axisymmetric domain that is enclosed on all sides. A droplet with nanoclay bound at different ion concentrations (which leads to a different initial viscosities) starts slightly above the ion-bath. The droplet has an initial downward velocity while the bath is static. Supplementary Fig. 5 shows the axisymmetric model geometry and boundary and initial conditions of the simulation. The yellow is the bound nanoclay in water, white is air, and green is the ion water bath. The droplet radius was The surface tension between fluids was assumed to be (constant at 293 K) that of air/water mixture (0.0728 N/m) since there was only a 5% decrease in surface tension for a bound concentration of 4 eq m -3 when CaCl2 was initially added. The nanoclay particles were assumed to be spherical and have a particle diameter, dp, of 9.5 nm 5 . The reaction rate of bound nanoclay was iterated for until the structures in the simulation 23 were similar to those of the experiment. There was no assumed off reaction rate between the calcium and binding site. The dimensions were chosen based on the trade-off between computational time and boundary effects. Increasing domain sizes decreases the effect the wall has on the simulation but increases the computation time.

Fluid Flow -Navier Stokes
The fluids' velocity and pressure are solved for with the Navier-Stokes equation (1) and assuming an incompressible fluid for the continuity equation (Eq. 2). The fluids are assumed to be air, a, and water, l, with dispersed phases in the liquid as discussed below, ( The two body forces exerted on both fluids are from gravity, g, and surface tension, FST. The stress tensor has two components, the pressure and viscous stress where e(u) is the strain tensor.
The fluids properties are volume averaged based on the volume fraction of air, Vf,a.

Non-Newtonian Viscosity-Cross Model
Based on experimental data, the viscosity (Supplementary Fig. 6) was modeled as a function of the bound concentration of nanoclay and volume fraction of clay particles The viscosity (Eq. 8) of the liquid is taken as the sum of the two fluids and the density (Eq. 9) is based on the volume fraction (Eq. 10) of nanoclay particles in the liquid. Thus, the nanoclay solution is not neutrally buoyant and will sink as the density of nanoclay was 2530 kg m -3 5 .

Phase Tracking -Cahn-Hilliard
The governing equation for tracking the fluid phases (air and water) and their interface is the Cahn-Hilliard equation 8 (Eq. 11) with ψ as the chemical potential (Eq. 12). The variable, ϕ, is the phase field variable and is directly related to the volume fractions (Eqs. 13 and 14).
The variables ζ, ξ, and ε are mobility, mixing energy density, and the interface thickness, respectively. The mobility is assumed to be 50ε 2 , and the mixing energy (where σ is the surface tension of air/water at 20 ºC) while the interface thickness is half the maximum element size. The surface tension force (Eq. 15) is then calculated from the phase field by multiplying the chemical potential times the gradient of the phase field variable.

Mass Transfer -Diffusion, Convection and Reaction
To track the dispersed phases of calcium ions, nanoclay particles, unbound binding sites, and bound complexes (calcium ions with nanoclay), a diffusion-convectionreaction equation (Eq. 16) was used (i represents ion, nanoclay particles, unbound sites and bound ion-nanoclay complexes). The binding site concentration is expressed as equivalent (eq) of hydrogen per m 3 . The CaCl2 is in units of mol m -3 . The diffusivity was based on literature for the calcium chloride (7.54×10 -10 m 2 s -1 6 ) ions and the diffusivity of other species was set to zero because they consisted of large bound molecules with negligible diffusivity.
The velocity, uc (Eq. 17) was set to a combination of the fluid velocity as calculated by Navier-Stokes, a slip velocity uslip (Eq. 18), and a corrected velocity for near interface particles. The nanoclay particle diameter used for the slip velocity,   The total number of binding sites, unbound sites and bound sites is conserved

Boundary and Initial Conditions
The boundary conditions are no slip and no flux everywhere with a reference point at the top right as the pressure constraint. The initial velocity is zero everywhere except the droplet has an initial downward velocity, v, which was ranged from 0.3 to 1.7 m s -1 .
Initially, there is 2 weight percentage of nanoclay particles with an initial concentration of binding sites, C, in the droplet. The initial bound concentration of Ca 2+ was either 0.5, 1, or 2 mol m -3 to vary the initial droplet viscosity.

Numerical Implementation
The equations are solved using a commercial finite element package, COMSOL processor.

Simulation Results
Supplementary Table 1 shows the Dahmköhler numbers, Da, of all five shapes.
There are three important ranges: a high (above 0.03), a medium (0.015-0.03), and a low (below 0.015) range. In the high range, the reaction rate dominates over fresh interface rate leading to the nanoclay quickly bonding and forming compact teardrop shape. In the low range, the nanoclay does not react quickly enough and the rate of new interface dominates leading to a fractured structure. While in the middle range, neither force dominates and the nanoclay reacts on a time scale similar to the interfacial creation rate such that complex structures can form. The cap has a high Dahmköhler number preventing the stalk, observed in the jellyfish, from forming. The fracture is simply formed by the extremely high shear stress dispersing the nanoclay such that it cannot react.

Average Shear Rate with Time
Supplementary Fig. 8 shows the weighted average shear rate with time of the nanoclay drop. The two noticeable features are the initial high shear rate peak followed by a possible second peak. The second peak represents the off-center vortices that makes the jellyfish and donut shape. The fracture has much higher peaks than all the other shapes. The teardrop and cap do not have a secondary vortices after impact

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As convective mixing is increased by lowering the droplet viscosity, the Da decreases and cap-VRP are observed. Further increases in the mixing rate (all else equal) leads to jellyfish-and donut-VRP, respectively.
Alternatively, Figure 3b could be read at a constant viscosity ratio with increasing We . Increasing We represents an increase in the impact inertia of the droplet, and thus an increase in the rate of mixing of the droplet within the bath at any given time.
Therefore, an increase in We with constant reaction rate leads to a decrease in Da and the same progression of VRP shapes is observed.
Interesting is the reversal of gelation of the jellyfish-and cap-shapes with The experimental Damköhler number (Eq. 28) was estimated based on the simulation reaction rate (kC=16.2 s -1 ) divided by an intuitive, theoretical shear rate (Eq. 29). An assumption was made that the unknown viscosity function, f, was equal to the viscosity ratio (Eq. 30) raised to an unknown exponent, n. This assumption is based on the fact that as the droplet viscosity increased, the shear within the droplet would decrease upon impact. Hence favoring a lower shear rate and a higher Damköhler