Interaction of multiple drops and formation of toroidal-spiral particles

In the development of drug delivery technologies for treating complex diseases, encapsulating multiple compounds and manipulating their sustained-release kinetics independently (for optimal therapeutic effect) can be challenging. Toward this goal, we previously developed a fluid-dynamic technology based on multidrop interactions to produce solid toroidal-spiral (TS) particles. During sedimentation in a miscible, viscous liquid, polymeric drops self-assemble into a reproducible and controllable TS structure, which can be solidified into particles by photoinitiated cross-linking of the polymer. The goal of encapsulating multiple drops of different physical properties (such as size and density) generally requires complicated and time-consuming laboratory iteration on the starting conditions, because all satellite drops (containing drugs) must catch up and coalesce simultaneously with the main drop that forms the surrounding matrix upon solidification. In this paper we consider a model system for multidrop entrainment that features a main drop followed by three smaller satellite drops arranged in a horizontal, triangular array. Experiments visualized with a high-speed camera are used to validate computer simulations based upon a swarm-of-Stokeslets method. The simulations accurately track complex drop configurations involving intertwined interfaces. Replacing the actual starting drop shapes with suitably positioned, volume-equivalent spheres yields very similar configurations: the crucial deformations and interactions occur during sedimentation, as opposed to during the initial injection of the drops. The simulations are then used to formulate two robust “rules of thumb” by which further trial-and-error (whether in the laboratory or by computation) can be avoided toward encapsulating multiple satellite drops with different properties. The first rule applies to satellite drops of different properties but symmetric starting positions, and establishes the single-drop Hadamard-Rybczynski (HR) sedimentation velocity as the crucial parameter. The second rule makes use of a universal “entrainment map” by which three satellite drops of the same radius but different densities and asymmetric starting positions can all be encapsulated at an arbitrarily prescribed distance of sedimentation. Two final simulations demonstrate how both rules can be combined to successfully design an (asymmetric) injection geometry to encapsulate three satellite drops of different radii and densities, at an arbitrarily prescribed distance of sedimentation. Understanding fundamental hydrodynamics of interaction between multiple drops could lead to potential scale-up of production of TS particles and also impact applications of mixing and printing in general.

Figure 1

Starting configuration of four drops. (a) Experimental setup for drop sedimentation and entrainment. Central (main) and surrounding (satellite) drops were infused using syringe pumps with precisely controlled volume. (b) Volume-equivalent spheres were used to represent the droplets in some simulations. Radius $a_1$ refers to the main drop, whereas $a_2$, $a_3$, and $a_4$ refer to the (smaller) satellite drops. (c) Schematic diagram of the starting configuration of a drop simulation. The bottom-based vertical coordinates $z_{\alpha }$ are converted to center-based coordinates $z_{\alpha }^{\prime }$ for the simulation code. Horizontally projected center-to-center distances $d_{1\alpha }$ between the main and satellite drops match the experiments.

Figure 2

Single-drop sedimentation, experimental and simulated. (a) Velocity history of the experimental drop compared with a computer simulation of low-Reynolds-number sedimentation flow, starting from a numerical version of the same drop shape (inset, time zero). Velocity and time have been rendered dimensionless as described in the text. (b) Comparison of experimental versus simulated drop shapes at the corresponding (dimensionless) cumulative distances of sedimentation. These correspond to the dimensionless times listed [and identified with dotted lines in part (a)]. In these six images the Reynolds number is within the range 0.0903 to 0.0807.

Figure 3

Four-drop sedimentation, experimental and simulated. (a) Velocity history of four-drop experiment compared with a computer simulation of low-Reynolds number flow, starting from numerical versions of the same drop shapes for the main and satellite drops (inset, time zero). (b) Comparison of experimental versus simulated drop shapes at the corresponding (dimensionless) cumulative distances of sedimentation. These correspond to the dimensionless times listed [and identified with dotted lines in part (a)]. In these six images the Reynolds number is within the range 0.1091 to 0.0858.

Figure 4

Four-drop sedimentation in experiments and simulations: reproducibility and sensitivity to initial drop shape. (a) Comparison of high-speed-camera images from two nominally identical experiments. The two images at $t=0$ indicate reproducibility of the injection process to attain the starting configuration (initial stage), whereas the neighboring images indicate the similarity of the subsequent evolved shape and sedimentation time at the same distance of travel (50 mm). Scale bar represents 2 mm. (b) Simulated effect of initial drop shapes on the configurational evolution. In simulation I, all initial drop shapes were numerically converted from the laboratory image at left. In simulation II, only the main drop is numerically converted from the image, while the satellite drops are replaced with volume-equivalent spheres. In simulation III, all drops are replaced with volume-equivalent spheres. The numerical conversion process was used to estimate the volumes of the laboratory drops as well as their relative positions. The latter determined the horizontal ($d_{1\alpha }$) and vertical ($z_{\alpha }^{\prime }$) drop separations in the simulations. (c) Comparison of velocity and distance histories for the three simulations.

Figure 5

Tableau of 15 simulations to test the hypothesis that the main determinant of configurational evolution is the HR sedimentation velocity of the satellite drops, irrespective (within reasonable limits) of how their size and force density conspire to produce that velocity. (a) Top view of the four-drop starting configuration (left panel) along with the sectional plane that defines the front view (right panel). The main drop has radius $a_1=1$ and force density $F_1=1$. Satellite drops are placed at same center-to-center height $z_{\alpha }^{\prime }$ and horizontally projected distance ($d_{1\alpha }=d$) from the central drop. The force density $F_{\alpha }$ of satellite drops increases in the progression from red ($\alpha =2$) to green ($\alpha =3$) to blue ($\alpha =4$). (b) Simulated TS shapes. In all cases the red drop ($\alpha =2$) has its force density ($F_2=2.4$) fixed at the value corresponding to configuration III in Fig. 4. In the first row, the radius ($a_2=0.47$) also matches the same configuration, which leads to an HR velocity relative to the main drop of $v_2/v_1=0.53$. The other two satellite drops have larger force densities and smaller radii. Both parameters are indicated along the bottom of the tableau, color coded to the images. For example, in the upper left image $F_3=2.6$ and $F_4=2.9$; the respective radii ($a_3=0.4481$, $a_4=0.4290$) are chosen to yield the same HR velocity: $v_2/v_1=v_3/v_1=v_4/v_1$. Progressing down the first column, the HR velocity of the red drop is increased by increasing its radius $a_2$ at fixed force density ($F_2=2.4$). The HR velocity and simulation time for each row is indicated at the left of the tableau. In a given image, all satellite drops have the same HR velocity irrespective of their (differing) sizes, and the distance traveled is indicated. Progressing across each row, we observe that the TS shape is relatively insensitive to the sizes of the satellite drops, provided that they have the same HR velocity. Progressing down each column, we see how the TS shape becomes distorted as the HR velocity increases above the advantageous value observed in the experiments and Fig. 4.

Figure 6

Entrainment map: curves of travel distance until TS channel formation versus initial horizontal distance of the satellite drops. (a) Initial four-drop configuration with satellite drops of the same radius ($a_2=a_3=a_4=0.47$) force density ($F=F_2=F_3=F_4$) and same center-to center distance ($d=d_{12}=d_{13}=d_{14}$). The main drop has radius $a_1=1$ and force density $F_1=1$. (b) Visual definition of the “final” stage of TS channel formation used for this comparison. For each value of the force density $F$, we define the endpoint of shape evolution as that time when the leading tip of the TS channel reaches the top of its curved arc in a sectional view and would subsequently wind further around. To aid in this visual identification, we observe that—irrespective of the initial distance $d$—this stage occurs when the leading tip comes back to roughly the same radial distance $r$ from the axis, indicated with a pair of vertical lines. Example final configurations (and the endpoint radial distances) are shown for two values of the force density: $F=2.4,3.6$. (c) Cumulative distance of travel $Z$ to attain the final TS shape [defined in part (b)] as a function of initial horizontal distance $d$ of the satellite drops for various force densities. The travel distances $Z_{\mathrm{I}}$, $Z_{\mathrm{II}}$, $Z_{\mathrm{III}}$, $Z_{\mathrm{IV}}$, indicated with horizontal lines and enumerated with Roman numerals, refer to the simulations depicted in Fig. 7.

Figure 7

Tableau of simulated drop configurations illustrates how the entrainment map from Fig. 6 can be used to determine a suitable starting configuration for satellite drops of different HR velocities to come together in a TS shape at an arbitrarily chosen sedimentation distance: $Z_{\mathrm{I}}$, $Z_{\mathrm{II}}$, $Z_{\mathrm{III}}$, $Z_{\mathrm{IV}}$. (a) Four-drop configuration for satellite drops of the same sizes ($a_2=a_3=a_4=0.47$) and vertical positions ($z_2^{\prime }=z_3^{\prime }=z_4^{\prime }=1.32$) but different horizontal distances ($d_{12}\ne d_{13}\ne d_{14}$) and force densities ($F_2\ne F_3\ne F_4$). The main drop has radius $a_1=1$ and force density $F_1=1$. (b) Simulated TS shapes successfully formed at four different sedimentation distances ($Z_{\mathrm{I}}$, $Z_{\mathrm{II}}$, $Z_{\mathrm{III}}$, $Z_{\mathrm{IV}}$) from satellite drops of different force densities. The leftmost image in each row is a side view from the direction indicated by the eye in part (a). Sectional views from three different angles are indicated in the subsequent three images: each angle emphasizes one of the three satellite drops.

Figure 8

Demonstration of both rules of thumb used simultaneously to arrive at suitable starting conditions for a case in which the satellite drops have different properties and asymmetric locations. We show simulated TS shapes successfully formed at the same sedimentation distance $Z_{\mathrm{I}}$ from Fig. 7. All satellite drops are placed at the same horizontal distances as before ($d_{12}=1.055$, $d_{13}=1.15$, $d_{14}=1.3$) and drops 3 and 4 have the same properties. However, the red satellite drop ($\alpha =2$) now has a smaller radius and commensurately larger body-force density to give the same HR velocity ratio $v_2/v_1=0.442$ compared with the red drop in Fig. 7. The main drop has radius $a_1=1$ and force density $F_1=1$.