# Droplets can enhance microcapsule deformation in channel flow

Dalei Jing, Ruixin Lu, Alexander Farutin, Ziyu Guo, Fan Wang, Wen Wang, Chaouqi Misbah, Yi Sui

► To cite this version:

Dalei Jing, Ruixin Lu, Alexander Farutin, Ziyu Guo, Fan Wang, et al.. Droplets can enhance microcapsule deformation in channel flow. Communications Physics, 2024, 7 (310), pp.1-10. 10.1038/s42005-024-01805-4. hal-05395768

HAL Id: hal-05395768

https://hal.science/hal-05395768v1

Submitted on 3 Dec 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
<figure><img src="image_2.png" /></figure>

# Droplets can enhance microcapsule deformation in channel flow

<figure><img src="image_3.png" /></figure>

Check for updates

Dalei Jing<sup>1,2</sup>, Ruixin Lu<sup>2</sup>, Alexander Farutin<sup>3</sup>, Ziyu Guo<sup>2</sup>, Fan Wang<sup>4</sup>, Wen Wang<sup>2</sup>, Chaouqi Misbah<sup>3</sup> & Yi Sui<sup>2</sup>

The dynamics of soft microparticles enclosed in a droplet flowing in a channel is an unexplored fundamental problem that lies at the heart of numerous applications, including droplet-based microfluidics, tissue engineering and smart material synthesis. Here we show that enclosing a flexible capsule into a droplet can amplify the capsule's deformation parameters in channel flow by up to two orders of magnitude. Previously unreported capsule equilibrium shapes in channel flow, including an oblate spheroid and a reversed bullet, have also been discovered. We propose two theoretical models to predict the equilibrium position of the capsule inside the droplet, and estimate the capsule deformation, respectively. The present study provides an effective but simple approach to enhance and control the deformation of soft particles in a flowing suspension, which may inspire widespread applications, from high-throughput single-cell mechanical phenotyping, enhanced cross-membrane drug delivery, to manufacturing shape-controlled non-spherical particles and artificial cells.

In the recent two decades, droplet-based microfluidics have emerged as ground-breaking technologies that are transforming wide disciplines including single-cell or single-molecule analysis<sup>1–8</sup>, tissue engineering<sup>9–12</sup>, smart materials synthesis<sup>13–17</sup>, soft robotics<sup>18–20</sup>, as well as sensing and monitoring<sup>21–24</sup>. These devices often employ two immiscible fluids in microchannels to generate sub-microliter droplets, at a very high-throughput rate, that precisely enclose biological cells, molecules, drugs, or chemical agents. The droplets can serve as isolated incubators or reaction vessels, and be subsequently manipulated for merging, splitting, sorting, or thermocycling<sup>25–27</sup>, and be analyzed using various analytical approaches such as fluorescence or electrochemical detection and mass spectrometry<sup>28–30</sup>.

In droplet-based microfluidic technologies, a fundamental problem that remains unexplored is the flow-induced deformation of soft particles enclosed in the droplets flowing in microchannels. Such particles can be living biological cells<sup>15,31,32</sup>, drug-carrying microcapsules, vesicles and liposomes<sup>13,17,33,34</sup>, or other soft matters manufactured within the droplets<sup>35–38</sup>. It is well known that in a straight microchannel, a droplet with a size that is comparable to the channel cross-sectional dimension deforms into a bullet shape. Counter-rotating vortices form within the droplet due to the opposite shear rate at the upper and lower droplet interface<sup>39–41</sup>. They are called Hill's vortices when the droplet is nearly spherical. This internal flow represents a scenario that is fundamentally different from the parabolic flow profile experienced by an isolated soft

particle in a straight channel. Then natural fundamental questions that arise are, how will a soft particle flow and deform within its enclosing droplet in a channel, and will the droplet interface serve as an "armor" and physically protect the particle? It should be noted that flow-induced deformation of soft particles has widespread applications, such as high-throughput cell mechanical phenotyping, enhanced drug delivery, and material manufacturing. Cell deformation under well-defined flow can be used to infer the mechanical properties of cells<sup>42–45</sup>, that are associated with the cell type and status. Flow-induced deformation of the membranes of cells or drug-carrying vesicles/liposomes can cause significant structural disruption, leading to enhanced intracellular delivery<sup>46</sup>. In microfluidics-based particle synthesis, flow-induced deformation enables controlling the shapes of the soft precursor particles, which are subsequently solidified to form non-spherical particles with desired electric, optical, and packing properties for the applications in self-assembling, photonics, pharmaceuticals, and drug delivery etc.<sup>47–50</sup>.

To address the open questions above, here we consider the deformation of a flexible microcapsule enclosed in a droplet flowing in a straight microchannel. The capsule consists of a liquid droplet enclosed by a thin elastic membrane that can resist shear deformation. Such microcapsules are not only used in food, cosmetic, textile, biomedical, and pharmaceutical industries as agents-carrying or -preserving vectors<sup>51,52</sup>, but also widely serve as simple mechanical models for biological cells. We use high-fidelity numerical simulations with an in-house developed computational

<footnote n="1">¹School of Mechanical Engineering, University of Shanghai for Science and Technology, Shanghai, China. ²School of Engineering and Materials Science, Queen Mary University of London, London, UK. ³Laboratoire Interdisciplinaire de Physique (LIPhy), Université Grenoble Alpes, Grenoble, France. ⁴School of Quality and Technical Supervision, Hebei University, Baoding, Hebei, China. ✉e-mail: chaouqi.misbah@univ-grenoble-alpes.fr; y.sui@qmul.ac.uk</footnote>
<caption>**Fig. 1 | Steady shapes and surrounding flow fields of an isolated droplet, an isolated capsule, and a capsule enclosed in a droplet in channel flow.**</caption>

a Schematic of the flow problem. The capsule is initially spherical with radius $R_c$ and is enclosed in a droplet with initial radius $R_d$. The channel has a square cross-section with a side length of $l$, and the flow is fully developed at the inlet. The steady profiles of **b** an isolated droplet with $R_d = 0.4l$ at $Ca_d = 0.05$, and **c** an isolated capsule with $R_c = 0.2l$ at $Ca_c = 0.2$. Both the droplet and the capsule have deformed to bullet shapes. **d–f** The steady profile of a capsule enclosed in a droplet with increasing surface tension coefficient (decreasing $Ca_d$ since $Ca_d = \mu U_{in}/\sigma$) transits from **d** a bullet shape inside a droplet with $Ca_d = 0.3$ to **e** an oblate spheroid at $Ca_d = 0.15$, and finally to **f** a reversed bullet shape at $Ca_d = 0.05$. In **b–f**, the radii of the droplet and capsule are $R_d = 0.4l$ and $R_c = 0.2l$, respectively. The capsule is initially aligned with the channel centerline. The streamlines are in the reference frames moving with the droplet or the capsule (in **c**), and the color contours are the pressure field $p^* = pl/\mu U_{in}$.

<figure><img src="image_4.png" /></figure>

framework, that combines a well-tested level-set method$^{53,54}$ to track the droplet interface and an immersed boundary method$^{55–57}$ to resolve the flow-induced capsule deformation.

The present study discovers that, instead of physically protecting the enclosed capsule, a droplet significantly increases the capsule deformation, compared with an isolated capsule with the same properties under identical channel flow conditions. Within the droplet, we mainly find two equilibrium modes of capsule deformation. A small capsule in a nearly spherical droplet tends to move into the center region of the Hill's vortex, deforms into an asymmetrical ellipsoidal shape, and conducts a “tanking-threading” motion, where the capsule membrane rotates around the liquid inside. For a larger capsule in a more deformable droplet, the capsule migrates to the stagnation point of the extensional flow in the front region of the droplet, where the capsule is elongated along the direction perpendicular to the channel flow. Interestingly, in the second deformation mode, we find that the equilibrium shape of the capsule transits from a bullet to a reversed bullet, via an oblate spheroid shape, when the deformation of the enclosing droplet decreases. Neither the reversed bullet nor the oblate spheroid shape has been reported previously for isolated capsules in channel flow. We also propose simple theoretical models to explain the equilibrium deformation modes of the capsule inside the droplet, and quantitatively estimate the capsule deformation.

# Results

## New steady shapes

As illustrated in Fig. 1a, we consider the deformation of an initially spherical capsule with radius $R_c$ enclosed in a droplet with initial radius $R_d$ flowing in a three-dimensional (3D) square channel that has a cross-sectional side length of $l$. A Cartesian coordinate is defined at the center of the channel inlet cross-section, with an x-axis along the flow direction. The fluids in the capsule, the droplet, and the channel are incompressible Newtonian liquids with identical density $\rho$ and viscosity $\mu$. The capsule has a hyperelastic membrane satisfying the strain-hardening Skalak's (SK) law$^{58}$ (detailed in the “Methods” section), a widely used mechanical model for the membrane of biological cells and bioartificial microcapsules. In the limit of negligible inertia, the capsule deformation is governed by the following dimensionless parameters:

* the droplet capillary number $Ca_d = \mu U_{in}/\sigma$, which measures the relative importance of the fluid shear force and the droplet surface tension. The term $U_{in}$ is the average flow speed in the channel, and $\sigma$ is the surface tension coefficient of the droplet.

* the capsule capillary number $Ca_c = \mu U_{in}/G_s$, which measures the relative importance of the fluid shear force and the capsule membrane elastic force. The term $G_s$ is the membrane shear elastic modulus.

* the size ratios $R_d/l$ and $R_c/l$, which compare the size of the droplet and capsule relative to the channel cross-sectional dimension, respectively.

Our simulations are conducted using a well-tested computational framework, where we use a level-set method$^{53,54}$ to predict the dynamics of the droplet and an immersed boundary method$^{55–57}$ for the capsule deformation and its interaction with the surrounding fluid. More details of the methods can be found in the “Methods” section.

To provide a foundation for understanding the deformation of a capsule enclosed in a droplet, we first consider an isolated droplet and an isolated capsule flowing in a square channel and present their steady shapes with the surrounding flow fields in the symmetric x-z plane in Fig. 1b, c, respectively. Both the droplet and the capsule have reached steady symmetric bullet shapes where the front region has a higher curvature than that of the rear. This is due to the parabolic velocity profile in the channel cross-section where the fluid flows faster in the core. Consistent with previous studies$^{40}$, we observe that in the frame moving with the droplet, the streamline within the droplet forms two pairs of counter-rotating vortices (see Fig. 1b), symmetric about the channel centerline due to the opposite shear rate at the upper and lower droplet interface. The two pairs of vortices form an extensional-like flow, that shares some common features with the flow pattern generated by a four-roll mill, with a stagnation point formed at the channel centerline between the two pairs of vortices. For an isolated capsule, the fluid inside is largely in a translational motion with the same speed.

For a capsule enclosed in a droplet, we start with a baseline problem where the capsule and the droplet are initially concentric and aligned with the channel centerline. Figure 1d–f presents the steady shapes of a capsule with $R_c = 0.2l$ and $Ca_c = 0.2$ enclosed in droplets with increasing surface tension coefficient and a fixed radius $R_d = 0.4l$. Interestingly, the steady shapes of the capsule transit from a bullet to an oblate spheroid shape, which is characterized by comparable curvatures at the front and rear regions, and finally to a reversed bullet shape when the droplet has strong surface tension and little deformation. Both the oblate spheroid shape and the reversed bullet shape are novel ones that have not been reported in the literature for capsules in the channel flow.

Comparing the steady shapes of the capsule in Fig. 1d–f with that of an isolated capsule of the same size and properties in Fig. 1c, we can see that enclosing a capsule into a droplet can significantly increase its deformation
<figure><img src="image_5.png" /><figcaption>**Fig. 2 | Pressure along the channel centerline near the capsule.** The pressure variation between the front and rear regions of the capsule at $Ca_c = 0.2$ determines whether it adopts the bullet, oblate spheroid, or reversed bullet shape. Symbols mark the locations where the channel centerline intersects with the capsule membrane (◊ for the rear, ○ for the front). Radii of the droplet and capsule are $R_d = 0.4l$ and $R_c = 0.2l$, respectively.</figcaption></figure>

along the direction perpendicular to the flow. This can be understood by the evident extensional flow inside the droplet, which elongates both the front and rear regions of the capsule. The pattern is qualitatively different from that of an isolated capsule, where its front region is subject to compression along the transverse direction.

Regarding the formation of the two novel steady shapes, we find that it is mainly determined by the pressure drop across the capsule. Figure 2 presents the dimensionless pressure $p^* = pl/\mu U_{in}$ along the channel centerline near the capsule. It can be seen that for an isolated capsule, or a capsule enclosed in a droplet with a weak surface tension ($Ca_d = 0.3$), the pressure in the rear region is higher. This leads to the bullet shape of the capsule, with a larger curvature at the front, in order to balance the pressure drop outside the capsule along the flow direction. With the droplet surface tension coefficient increasing, a high capillary pressure is formed between the capsule front region and the droplet front interface, due to the curvature of the droplet interface and the strong surface tension. At $Ca_d = 0.05$, this capillary pressure is higher than the fluid pressure outside the capsule rear region, leading to a reversed bullet shape with a larger curvature at the rear. For the capsule inside a droplet with $Ca_d = 0.15$, the pressure at the front and rear regions of the capsule are comparable, resulting in the oblate spheroid shape with almost identical curvature values at the two regions.

We conduct extensive studies of a capsule with $R_c = 0.2l$ enclosed in a droplet with $R_d = 0.4l$ in channel flow in a wide range of capsule shear elasticity and droplet surface tension coefficient, and summarize the steady shapes of the capsule in Fig. 3a. Both the two new steady capsule profiles and the transition from a bullet to a reversed bullet shape are found to be robust.

## Enhanced capsule deformation

We quantify the deformation of the capsule by using two deformation parameters. The first parameter is the Taylor shape parameter, which is widely used for ellipsoidal particles, defined as

$$D = \frac{|a - b|}{a + b}. \qquad (1)$$

We map the deformed capsule profile in the symmetric $x-z$ plane to an ellipse that shares the tensor of the moments of inertia, and the semi-major and semi-minor axes of the ellipse are respectively denoted by $a$ and $b$. The second parameter is the deformation index, which measures the

non-circularity

$$di = 1 - \frac{2\sqrt{\pi A}}{P}. \qquad (2)$$

where $A$ is the capsule surface projected area in the symmetric $x-z$ plane and $P$ is the capsule perimeter. Figure 3b, c presents the deformation parameters of the capsule in the phase diagram of Fig. 3a at $Ca_c = 0.05$ and $0.2$. Compared with an isolated capsule without an enclosing droplet, it is clearly seen that the existence of the droplet interface, which by intuition would act as an armor to physically protect the enclosed capsule, does not fulfill this function. On the opposite, the interface significantly increases capsule deformation. We find that the harder the droplet interface is, the larger the capsule deformation becomes. The deformation enhancement is particularly apparent for hard capsules, e.g., at $Ca_c = 0.05$, the steady capsule profile changes from having no visible deformation to a moderately deformed RB shape, as shown in Fig. 3a.

The observations are due to the extensional flow inside the droplet, whose intensity increases with the droplet surface tension coefficient. We investigate the field of the strain rate $dw/dz$, which is normalized by $U_{in}/l$, inside an empty droplet (see Supplementary Fig. 1). It can be found that a region of high elongational strain exists in the front region of the droplet, largely coinciding with the equilibrium position of a capsule, when enclosed into the same droplet (see Fig. 1f). We use the maximum $dw/dz$, defined as $(dw/dz)_{max}$ in the high elongational strain zone as the characteristic strain rate inside the droplet that determines the capsule deformation. Supplementary Fig. 2 presents the $(dw/dz)_{max}$ as a function of the droplet surface tension coefficient, which confirms that the stronger the surface tension (smaller $Ca_d$), the higher the $(dw/dz)_{max}$. In the limit of very small $Ca_d$, the droplet is almost spherical, the $(dw/dz)_{max}$ obtained from the present numerical simulation, approaches the theoretical results of Nadim and Stone$^{59}$, for a spherical droplet flowing in a straight channel (see the comparison in Supplementary Fig. 3). In a later section we will use this characteristic strain rate to build a simple model to estimate the capsule deformation.

With the enhancement of capsule deformation, as a result of the enclosing droplet, the maximum principal elastic tension in the membrane $\tau_{max} = \max[\tau_1, \tau_2]$, where $\tau_1$ and $\tau_2$ are calculated from Eq. (11) in the “Methods” section, also increases significantly, e.g., for more than three folds at $Ca_d = 0.01$, compared with the same capsule without an enclosing droplet, as can be seen from Fig. 3d. The maximum principal membrane tension is a relevant quantity to evaluate the likelihood of membrane rupture.

## Two equilibrium modes of capsule deformation

We test the robustness of the capsule deformation observed in Fig. 3. Extensive simulations are conducted for a capsule with various sizes and membrane properties within a droplet of different values of surface tension coefficient. The capsule is initially placed off the channel centerline by 0.02–0.15l along the $-z$ direction, to mimic practical experiments where it is hard to achieve perfect concentricity. Our simulation results suggest that there are mainly two equilibrium deformation modes. A relatively small capsule in a droplet with strong surface tension tends to migrate to the center region of the Hill’s vortex. An example is shown in Fig. 4a, where one can find that the capsule has deformed into an asymmetrical ellipsoidal shape with a high-curvature corner on the near-wall side and the long axis assuming a certain inclination with respect to the flow direction. The capsule carries out a tank-treading motion, in which its membrane is rotating around the fluid inside (see Fig. 4d). This deformation mode is due to the shear flow that the capsule experiences in the Hill’s vortex, with the shear stress decreasing towards the channel centerline.

When decreasing the droplet surface tension coefficient, or increasing the capsule size, the capsule will eventually migrate back to the channel centreline and reach one of the symmetric shapes that have been reported in the baseline problem, as shown in Fig. 4b, c, respectively. Along the centreline, the capsule undergoes translational motion, as evidenced in Fig. 4e, f. We also find that the equilibrium deformation mode of a capsule is not sensitive to its membrane elasticity, or a different initial off-center distance.
<figure><img src="image_6.png" /><figcaption>Fig. 3 | Phase diagram of the steady shapes and the significantly enhanced deformation of a capsule enclosed in a droplet in channel flow. a The increase of the droplet surface tension coefficient (decreasing $Ca_d$) not only results in the transition of the steady shape, from a bullet (B) to a reversed bullet (RB) via an oblate spheroid (OS) but also significantly increases the capsule deformation compared with an isolated capsule with the same size and properties subject to identical channel flow conditions (results are presented in the shaded zone). b, c Capsule</figcaption></figure>

deformation parameters can be increased by up to two orders of magnitude, and d the maximum capsule membrane principal tension can be increased by more than three folds by an enclosing droplet. Symbols are the b Taylor shape parameter, c deformation index, and d the maximum membrane principal tension of the capsule in (a) at $Ca_c = 0.05$ and 0.2. Horizontal lines mark the values for the same capsule without an enclosing droplet under identical flow conditions. Radii of the droplet and capsule are $R_d = 0.4l$ and $R_c = 0.2l$, respectively.

We summarize the capsule's equilibrium deformation modes as a function of the capsule size and droplet surface tension coefficient in a phase diagram in Fig. 4g. It is interesting to find that when the capsule size is larger than half of that of the droplet, a capsule will always reach the channel centerline. The equilibrium profiles of the capsules of $R_c = 0.2l$ and 0.25l in the phase diagram are presented in Fig. 4h. Compared with an isolated capsule, irrespective of the equilibrium mode, enclosing a capsule into a droplet always increases the capsule deformation, in particular when the droplet surface tension is strong. This is further confirmed by Fig. 4i, j, which provides a comprehensive summary of the Taylor shape parameter and deformation index of capsules with $R_c = 0.2l$ at $Ca_c = 0.05$, 0.2 enclosed in a droplet with various values of surface tension coefficient.

We also consider a capsule that is initially off-centered along both the -z and -y directions in the square channel. Interestingly, we find that the capsule's trajectory is determined by the larger off-center distance

component. Along the other off-center direction, the capsule gradually moves back to the channel mid-plane and regains its symmetry. If the initial off-center distances along the two directions are equal, the capsule will move to the diagonal plane of the channel. However, the two equilibrium modes of capsule deformation reported in Fig. 4 are robust.

### A model to predict the capsule's equilibrium deformation mode

We propose a simple model to predict the equilibrium deformation mode of a small capsule enclosed in a droplet flowing in a channel. To the leading order, a small capsule can be approximated by a neutral buoyant solid particle, whose trajectory in an unbounded non-uniform flow can be determined by the following equation$^{60}$,

$$ \rho_c V_c \frac{dv_c}{dt} = 6\pi\mu R_c \left( \mathbf{v}_f - \mathbf{v}_c + \frac{1}{6} R_c^2 \nabla^2 \mathbf{v}_f \right). \quad (3) $$
The right-hand side of Eq. 3 represents the Stokes drag force with the Faxén correction due to the non-uniformity of the fluid flow inside the droplet. The term $ρ_c$ is the capsule density, $V_c$ is the capsule volume, $v_c$ and $v_f$ represent the capsule and local fluid velocities, respectively. When inertia is negligible, the capsule acceleration is instantaneous and Eq. 3 can be simplified to obtain the capsule velocity

$$v_c = v_f + \frac{1}{6} R_c^2 \nabla^2 v_f. \quad (4)$$

Using Eq. 4 and the unperturbed background flow in an isolated steadily deformed droplet in channel flow, that is obtained from numerical simulation, we find that the trajectories of small capsules are limit circles where a capsule circulates in the Hill's vortex within the droplet. We refine the model to take into account the cross-streamline migration of the capsule due to the hydrodynamic interactions with the droplet interface. It is shown that both the particle and interface deformability lead to hydrodynamic lift that pushes the particles further from the interface even in the absence of flow inertia (see a recent review⁶¹). Here we include the lift effect into the model by introducing a correction to the capsule velocity, based on a calculation of the lift velocity for a solid particle translating parallel to a fluid membrane⁶² (see Supplementary Note 1 for more details)

$$v_m = -\alpha Ca_d \left(\frac{R_c}{R_d}\right)^2 \gamma r, \quad (5)$$

where $α$ is a numerical parameter, $γ$ is the local shear rate of the unperturbed background flow in the droplet, $r$ is a vector pointing from the droplet mass center to the particle center. We find that the velocity correction facilitates the fixed-point equilibrium positions of the capsule in the droplet. When the droplet surface tension is weak, the droplet deformation is large and the strength of the Hill's vortex is not strong, the droplet interface-induced lift tends to drive the capsule out of the vortex, towards the stagnation point of the extensional flow in the front region of the droplet. Combining Eqs. 4 and 5 enables successful predictions of the phase diagram of Fig. 4g for small capsules. In Fig. 4g, the red solid lines mark the phase boundaries predicted by the simple model for small capsules with $R_c = 0.1l$ and $0.15l$, which are consistent with the results from the full numerical simulation. A value in the range of 3–4 for the numerical parameter $α$ yields good agreement between the theoretical predictions and the full simulations, as shown in Fig. 4g. These $α$ values are consistent with the order of magnitude estimations from our theoretical analysis, which is detailed in the Supplementary Note 1.

### Models to estimate capsule deformation

We also attempt to develop simple models to estimate the capsule deformation at the two equilibrium positions inside a droplet. As discussed earlier, for a capsule reaching the channel centerline, its deformation is largely due to the extensional flow inside the droplet, which has a characteristic strain rate of $(dw/dz)_{max}$. We borrow the small deformation theory of Barthes-Biesel and Rallison⁶³, for the steady deformation of an initially spherical elastic capsule in an axisymmetric elongational flow without inertia. To the first order, the steady Taylor deformation parameter of the capsule, with membrane following the SK law with hardness parameter $C=1$, is predicted to be

$$D = \frac{3}{2} \frac{25}{3} Ca'_c, \quad (6)$$

where $Ca'_c = μkR_c/G_s$ is the dimensionless shear rate. The term $k$ is the strain rate of the extensional flow. Here, we use the characteristic strain rate $(dw/dz)_{max}$ in the droplet to approximate $k$, and then the Taylor shape parameter of the capsule that migrates to the channel centerline within the

droplet can be predicted using

$$D = \frac{3}{2} \frac{25}{3} \frac{\mu (dw/dz)_{max} R_c}{G_s}. \quad (7)$$

For a capsule that migrates to the center region of the Hill's vortex within the droplet, its deformation is largely induced by the shear flow. For linear shear flow, in the limits of small capsule deformation and negligible inertia effect, the steady Taylor shape parameter of the capsule can be predicted by⁶⁴,⁶⁵

$$D = \frac{5}{4} \frac{5}{3} Ca''_c. \quad (8)$$

where $Ca''_c = μγR_c/G_s$ is the dimensionless shear rate, and $γ$ represents the shear rate. Since it is not straightforward to identify a characteristic shear rate in Hill's vortex, we use the average shear rate at the vortex center $ω = 1/2(|du_{vc}/dz| + |dw_{vc}/dx|)$. The Taylor deformation parameter of the capsule can then be predicted by

$$D = \frac{5}{4} \frac{5}{3} \frac{\mu \omega R_c}{G_s}. \quad (9)$$

Figure 5a, b presents the comparisons of the Taylor shape parameters obtained from the full numerical simulations and the two simple models, for all cases considered in the present study where the capsule has migrated to the channel centerline inside the droplet, or the center region of the Hill's vortex, respectively. It can be seen that the model estimations agree with the numerical simulations reasonably well.

### Discussion

This study represents the inaugural exploration into the deformation dynamics of a microcapsule enclosed within a droplet traversing a linear channel. A pivotal finding unveils that the deformation parameters of the capsule can undergo a substantial amplification of up to two orders of magnitude when subjected to the influence of an enveloping droplet, as opposed to an isolated capsule experiencing identical channel flow conditions.

Specifically, we find that in a droplet with density and viscosity comparable to those of the channel fluid, a capsule mainly has two equilibrium modes of deformation. When the droplet has strong surface tension and small deformation, its internal Hill's vortex forms a strong attractor to small capsules, which get trapped to the vortex center, deform into asymmetrical ellipsoidal shapes under the effect of fluid shear, and conduct tanking-threading motion where the capsule membrane rotates around the liquid inside. With the capsule size increasing, or the droplet surface tension getting weaker, the relative importance of the droplet interface-induced lift force applied to a capsule becomes stronger; as a result the capsule migrates to the stagnation point of the extensional flow in the front region of droplet, that is along the channel centerline, and deforms to a symmetric shape which is elongated along the direction perpendicular to the channel flow. For both deformation modes, the fluid stresses at the equilibrium positions of the capsule inside the droplet are stronger than that at the channel centerline, which is the equilibrium position of an isolated capsule, thus leading to significantly enhanced capsule deformation.

This interesting discovery provides an effective but simple way to enhance the deformation of soft particles in flowing suspension, and may therefore inspire a wide range of applications. For example, in the context of microfluidic-based high-throughput single-cell mechanical characterization, the present work has set up the foundation for a new type of devices which enclose individual cells into droplets to significantly increase the cell deformation, in particular for hard cells, and reduce the likelihood of channel blockage. Conventional systems often use narrow channels that are comparable to or even smaller than the cell diameter in order to achieve sufficient cell deformation⁶⁶,⁶⁷. Channel blockage has been a challenge since cells always span
<figure><img src="image_7.png" /><figcaption>**Fig. 4** | Two equilibrium deformation modes of a capsule inside a droplet flowing in a channel. a An initially off-centered capsule with $R_c = 0.2l$ at $Ca_c = 0.1$ migrates to the center region of the Hill's vortex in a droplet with $R_d = 0.4l$ and strong surface tension coefficient ($Ca_d = 0.1$) and exhibits tank-treading motion in which its membrane rotates around the fluid inside as shown in (d) by a tracer particle attached to the membrane. b The same capsule of (a) reaches the channel centerline in a droplet that is of the same size as the one in (a) but has a weaker surface tension coefficient ($Ca_d = 0.4$), and carries out translational motion, as shown in (e). c Within the droplet of (a), a capsule of the same membrane properties but larger in size migrates back to the channel centerline, and then conducts translational motion which can be seen in (f). In a–c, the streamlines are in the reference frame moving with the droplet, and the color contours are the dimensionless pressure. g Phase diagram of a capsule's equilibrium deformation modes. A smaller capsule enclosed in a droplet with strong surface tension migrates to the center region of the Hill's vortex and conducts tank-treading motion (denoted by ♦); while a larger capsule in a droplet with weak surface tension tends to reach the channel centerline and</figcaption></figure>

undergoes translational motion along the flow (denoted by ○). All capsules have the same membrane elasticity with $Ca_c = 0.1$. The red lines represent the phase boundaries predicted by a theoretical model detailed in the following section.

h Equilibrium profiles of the capsules with $R_c = 0.2l$ and $0.25l$ in (g). The blue shaded zone marks the cases where the capsule migrates to the center region of the Hill's vortex. The “No droplet” column in the gray shaded zone presents the steady shapes of isolated capsules under the same flow condition. i, j Capsule deformation can be significantly increased by an enclosing droplet, irrespective of its equilibrium deformation mode. Symbols are the i Taylor shape parameter and j deformation index of the capsules with $R_c = 0.2l$ at $Ca_c = 0.05$, 0.2 enclosed in a droplet with $R_d = 0.4l$ and various values of surface tension coefficient. Solid and hollow symbols represent the cases where the capsule migrates to the center of the Hill's vortex, or to the channel centerline, respectively. Horizontal lines mark the deformation parameter values for the same capsule without an enclosing droplet under identical flow conditions.

a wide range of sizes even for the same type of cells¹³,⁶⁶,⁶⁸,⁶⁹. Size-based filtering of cells would lead to considerable cell loss. The present study suggests that droplet-based method can enable significantly enhanced cell deformation in wide channels, where the likelihood for blockage can be much lower.

In the context of intracellular drug delivery, it has been well established that mechanical deformation of the membranes of cells or drug-carrying capsules can cause membrane disruption, leading to higher permeability and enhanced cross-membrane mass transport. The present study may inspire new technologies, harnessing many merits of droplet-based microfluidics such as concentrating the drug in small confinement, and the controllable increased cell/capsule deformation, to control and enhance intracellular drug delivery, which may find applications in cell and tissue engineering.

Another major finding of the present study is the new equilibrium shapes of the capsule inside the droplet, including an oblate spheroid and a reversed bullet, which have not been reported in previous literature for isolated capsules in the channel flow. These new shapes are primarily observed for relatively large capsules that migrate to the front region of the droplet along the channel centerline. With the droplet surface tension increasing, the symmetric equilibrium shape of the capsule undergoes a transition from a bullet to reversed bullet, via an oblate spheroid shape, due to the combined effects of the flow extension in the droplet along the direction perpendicular to the channel flow, and the pressure difference between the front and rear regions surrounding the capsule.
**Fig. 5** | The simple models can predict the capsule deformation at the two equilibrium positions inside a droplet reasonably well. Comparisons of the Taylor shape parameters obtained from the full numerical simulations (symbols) and the two simple models (straight lines), for the capsules with $R_c = 0.15l$, $0.2l$, and $0.25l$ at $Ca_c = 0.05, 0.1$, and $0.2$ enclosed in a droplet with $R_d = 0.4l$ and various values of surface tension coefficient ($Ca_d$ changes from $0.05$ to $0.7$), where the capsule has migrated to a the channel centerline inside the droplet, or **b** the center of the Hill's vortex. The individual groups of identical symbols represent the Taylor shape parameters of the same capsule in a droplet with different surface tension coefficient.

<figure><img src="image_8.png" /></figure>

In the context of material synthesis, the new shapes found in this work may open a new avenue for manufacturing shape-controlled non-spherical particles, including artificial cells and drug delivery capsules, which have been shown to hold unique anisotropic properties compared to the spherical particles$^{47–50}$, and have wide applications in cosmetics, pharmaceutics, biotechnologies, and structural materials. The present study may also benefit tissue engineering, by providing a novel way to control the shape of cell aggregates, which can influence cellular behaviors such as collective migration, proliferation, and differentiation and gene expression$^{70,71}$, facilitating the development of functional tissues.

We have also proposed simple theoretical models to predict the equilibrium position of the capsule inside the droplet, and estimate the capsule deformation. The models have been tested against the results obtained from full numerical simulations, and reasonably good agreements have been demonstrated. The simple models provide convenient predictive tools to design future experiments, where one uses droplets in channel flow to enhance the deformation or control the shape of flexible microcapsules.

## Methods

### Capsule mechanical model

In the present study, the capsule consists of a hyperelastic membrane enclosing Newtonian fluid. The membrane is infinitely thin and follows the strain-hardening Skalak's (SK) law$^{58}$, with a strain energy function of

$$W = \frac{1}{4} G_s (I_1^2 + 2I_1 - 2I_2) + \frac{1}{4} C G_s I_2^2. \quad (10)$$

The term $W$ is the strain energy density per unit undeformed surface area, $G_s$ is the surface shear elasticity modulus, $I_1 = \lambda_1^2 + \lambda_2^2 - 2$ and $I_2 = (\lambda_1\lambda_2)^2 - 1$ are two strain invariants, where $\lambda_1$ and $\lambda_2$ are the principal extension ratios. The membrane area dilatation modulus is $K_s = (1 + 2C) G_s$, where $C$ is the hardness parameter, and is set to 1 in the present study. It has been widely shown that the SK law can account for large deformation of capsules and biological cells$^{52,55–57}$. The principal membrane tensions $\tau_1$ and $\tau_2$ in the membrane plane are given by

$$\tau_1 = \frac{G_s \lambda_1}{\lambda_2} (\lambda_1^2 - 1 + C \lambda_2^2 I_2), \quad \tau_2 = \frac{G_s \lambda_2}{\lambda_1} (\lambda_2^2 - 1 + C \lambda_1^2 I_2), \quad (11)$$

The bending stiffness of the capsule membrane is modeled using Helfrich's bending energy formulation$^{72}$

$$E_b = \frac{k_c}{2} \int_A (2\kappa - \kappa_0)^2 dS, \quad (12)$$

where $k_c$ is the bending modulus, $\kappa$ is the average membrane curvature, $\kappa_0$ is the membrane spontaneous curvature and $S$ is the capsule surface area. A small bending stiffness $k_c = 0.0008G_s l^2$ is used to prevent the formation of membrane wrinkles.

### Immersed boundary method

The capsule deformation and its interaction with the surrounding fluid are solved using an extensively tested immersed boundary method (IBM) for capsule dynamics in various flow geometries$^{52,55–57,73}$. This method employs two different coordinate systems: the capsule membrane is represented by Lagrangian coordinates and the surrounding fluid is represented by Eulerian coordinates. Under the no-slip velocity boundary condition, the material elements of the capsule membrane move at the same velocity of the fluid around it. This motion causes capsule deformation and generates membrane stresses. The membrane forces are calculated from the membrane constitutive laws (e.g., Eqs. 10 and 12), and are distributed to the Eulerian fluid mesh points near it by means of a three-dimensional Dirac function. More details can be found from pertinent literatures$^{55–57}$. We validate the immersed boundary method of the present study by considering the deformation of an isolate capsule in simple shear flow, and in channel flow. Our results agree very well with previous studies, as seen in Supplementary Figs. 4 and 5.

### Level-set method

We model the droplet deformation in the channel flow using a well-tested level-set method$^{53,54}$. The level-set function $\varphi$ is a signed distance function from the interface, with the sign used to distinguish between the two fluids, and the zero level set $\varphi = 0$ corresponding to the droplet interface. The level-set function follows the advection equation

$$\frac{\partial \varphi}{\partial t} + \mathbf{U} \cdot \nabla \varphi = 0. \quad (13)$$

Since the equation only holds at the droplet interface, a 'reinitialization' step is employed to ensure that the value of the level-set function adjacent to the interface keeps approximately the signed distance function$^{74}$. The fluid flow in the channel is governed by the Navier-Stokes equations. Surface tension is represented as a singular force density term introduced into the momentum equation as $f = \sigma\kappa_c n \nabla H(\varphi)$, where $\sigma$ is the surface tension coefficient, $\kappa_c$ is the droplet interfacial curvature, $n$ is the outward-pointing unit normal at the droplet interface, and $H$ is the smoothed Heaviside function to smoothly distribute the physical properties across the interface over a thickness of $\xi$, which is set to be 1.5 times of the fluid mesh size in the present study.

In the present implementation, a finite-volume method based on a uniform marker-and-cell (MAC) mesh is employed. The velocity components are defined at cell faces and scalar variables such as pressure and level-set function are defined at the cell centers. A fifth-order weighted essentially non-oscillatory (WENO) scheme$^{75}$ is employed in the discretization of the advection term in Eq. (13) using the local flow velocity as the up-winding direction. The Navier-Stokes equations are solved with a standard projection method$^{76}$. We validate the level-set method by considering the deformation of an isolated droplet in a straight channel. Excellent agreement with previous studies has been achieved as shown in Supplementary Fig. 6.

For the dynamics of a capsule enclosed in a droplet flowing through a channel, since it takes long for the capsule to reach an equilibrium state, we
use a moving mesh method$^{77}$, where the computational domain moves with the capsule at the same velocity, which enables efficient simulations with a much smaller computational domain. Our computational tests suggest that the moving mesh method generates identical results to the standard static mesh method using a longer computational domain (see Supplementary Fig. 7). We have also conducted extensive mesh convergence tests for both the fluid and capsule membrane grids, and found that a mesh size of $\Delta x = \Delta y = \Delta z = l/80$ for the fluid domain, with the capsule membrane discretized into 8192 flat triangular elements connecting 4098 nodes, is sufficient to accurately capture the capsule deformation in the present study.

## Data availability

The data that support the findings of this study are available upon reasonable request to the corresponding authors.

## Code availability

All numerical codes in this paper are available upon reasonable request to the corresponding authors.

Received: 25 March 2024; Accepted: 16 September 2024;
Published online: 20 September 2024

## References

1. Mazutis, L. et al. Single-cell analysis and sorting using droplet-based microfluidics. *Nat. Protoc.* **8**, 870–891 (2013).

2. Klein, A. M. et al. Droplet barcoding for single-cell transcriptomics applied to embryonic stem cells. *Cell* **161**, 1187–1201 (2015).

3. Zilionis, R. et al. Single-cell barcoding and sequencing using droplet microfluidics. *Nat. Protoc.* **12**, 44–73 (2017).

4. Terekhov, S. S. et al. Microfluidic droplet platform for ultrahigh-throughput single-cell screening of biodiversity. *Proc. Natl Acad. Sci. USA* **114**, 2550–2555 (2017).

5. Gerard, A. et al. High-throughput single-cell activity-based screening and sequencing of antibodies using droplet microfluidics. *Nat. Biotechnol.* **38**, 715–721 (2020).

6. Bowman, E. K. et al. Sorting for secreted molecule production using a biosensor-in-microdroplet approach. *Proc. Natl Acad. Sci. USA* **118**, e2106818118 (2021).

7. Madrigal, J. L. et al. Characterizing cell interactions at scale with made-to-order droplet ensembles (MODEs). *Proc. Natl Acad. Sci. USA* **119**, e2110867119 (2022).

8. Chen, K. Y. et al. High-throughput droplet-based analysis of influenza A virus genetic reassortment by single-virus RNA sequencing. *Proc. Natl Acad. Sci. USA* **120**, e2211098120 (2023).

9. Griffin, D. R. et al. Accelerated wound healing by injectable microporous gel scaffolds assembled from annealed building blocks. *Nat. Mater.* **14**, 737–744 (2015).

10. Petit-Pierre, G. et al. In vivo neurochemical measurements in cerebral tissues using a droplet-based monitoring system. *Nat. Commun.* **8**, 1239 (2017).

11. de Rutte, J. M., Koh, J. & Di Carlo, D. Scalable high-throughput production of modular microgels for in situ assembly of microporous tissue scaffolds. *Adv. Funct. Mater.* **29**, 1900071 (2019).

12. Griffin, D. R. et al. Activating an adaptive immune response from a hydrogel scaffold imparts regenerative wound healing. *Nat. Mater.* **20**, 560–569 (2020).

13. Wang, J. T., Wang, J. & Han, J. J. Fabrication of advanced particles and particle-based materials assisted by droplet-based microfluidics. *Small* **7**, 1728–1754 (2011).

14. Bawazer, L. A. et al. Combinatorial microfluidic droplet engineering for biomimetic material synthesis. *Sci. Adv.* **2**, e1600567 (2016).

15. Huang, H. et al. Generation and manipulation of hydrogel microcapsules by droplet-based microfluidics for mammalian cell culture. *Lab Chip* **17**, 1913–1932 (2017).

16. Weiss, M. et al. Sequential bottom-up assembly of mechanically stabilized synthetic cells by microfluidics. *Nat. Mater.* **17**, 89–96 (2018).

17. Giannitelli, S. M. et al. Droplet-based microfluidic synthesis of nanogels for controlled drug delivery: tailoring nanomaterial properties via pneumatically actuated flow-focusing junction. *Nanoscale* **14**, 11415–11428 (2022).

18. Liu, X. et al. Reconfigurable ferromagnetic liquid droplets. *Science* **365**, 264–267 (2019).

19. Fan, X. et al. Reconfigurable multifunctional ferrofluid droplet robots. *Proc. Natl Acad. Sci. USA* **117**, 27916–27926 (2020).

20. Fan, X. et al. Scale-reconfigurable miniature ferrofluidic robots for negotiating sharply variable spaces. *Sci. Adv.* **8**, eabq1677 (2022).

21. Agresti, J. J. et al. Ultrahigh-throughput screening in drop-based microfluidics for directed evolution. *Proc. Natl Acad. Sci. USA* **107**, 4004–4009 (2010).

22. Gielen, F. et al. Ultrahigh-throughput-directed enzyme evolution by absorbance-activated droplet sorting (AADS). *Proc. Natl Acad. Sci. USA* **113**, E7383–E7389 (2016).

23. Colin, P.-Y. et al. Ultrahigh-throughput discovery of promiscuous enzymes by picodroplet functional metagenomics. *Nat. Commun.* **6**, 10008 (2015).

24. Terekhov, S. S. et al. Ultrahigh-throughput functional profiling of microbiota communities. *Proc. Natl Acad. Sci. USA* **115**, 9551–9556 (2018).

25. Teh, S. Y. et al. Droplet microfluidics. *Lab Chip* **8**, 198–220 (2008).

26. Schneider, T., Kreutz, J. & Chiu, D. T. The potential impact of droplet microfluidics in biology. *Anal. Chem.* **85**, 3476–3482 (2013).

27. Leman, M. et al. Droplet-based microfluidics at the femtolitre scale. *Lab Chip* **15**, 753–765 (2015).

28. Zhu, Y. & Fang, Q. Analytical detection techniques for droplet microfluidics-a review. *Anal. Chim. Acta* **787**, 24–35 (2013).

29. Liu, W. W. & Zhu, Y. "Development and application of analytical detection techniques for droplet-based microfluidics"-a review. *Anal. Chim. Acta* **1113**, 66–84 (2020).

30. Moragues, T. et al. Droplet-based microfluidics. *Nat. Rev. Methods* **Prim.** **3**, 32 (2023).

31. Clausell-Tormos, J. et al. Droplet-based microfluidic platforms for the encapsulation and screening of mammalian cells and multicellular organisms. *Chem. Biol.* **15**, 427–437 (2008).

32. Jang, M., Yang, S. & Kim, P. Microdroplet-based cell culture models and their application. *BioChip J.* **10**, 310–317 (2016).

33. Dimitriou, P. et al. Droplet microfluidics for tumor drug-related studies and programmable artificial cells. *Glob. Chall.* **5**, 2000123 (2021).

34. Shao, C. et al. Droplet microfluidics-based biomedical microcarriers. *Acta Biomater.* **138**, 21–33 (2022).

35. Seiffert, S. Microgel capsules tailored by droplet-based microfluidics. *Chemphyschem* **14**, 295–304 (2013).

36. Martino, C. & deMello, A. J. Droplet-based microfluidics for artificial cell generation: a brief review. *Interface Focus* **6**, 20160011 (2016).

37. Staufer, O. et al. Bottom-up assembly of functional intracellular synthetic organelles by droplet-based microfluidics. *Small* **16**, e1906424 (2020).

38. Wischke, C. Concepts for efficient preparation of particulate polymer carrier systems by droplet-based microfluidics. *Int J. Pharm.* **584**, 119401 (2020).

39. Guido, S. & Preziosi, V. Droplet deformation under confined Poiseuille flow. *Adv. Colloid Interface Sci.* **161**, 89–101 (2010).

40. Che, Z., Yap, Y. F. & Wang, T. Flow structure of compound droplets moving in microchannels. *Phys. Fluids* **30**, 012114 (2018).

41. Tiribocchi, A. et al. The vortex-driven dynamics of droplets within droplets. *Nat. Commun.* **12**, 82 (2021).

42. Fregin, B. et al. High-throughput single-cell rheology in complex samples by dynamic real-time deformability cytometry. *Nat. Commun.* **10**, 415 (2019).
43. Nawaz, A. A. et al. Intelligent image-based deformation-assisted cell sorting with molecular specificity. *Nat. Methods* **17**, 595–599 (2020).

44. Gossett, D. R. et al. Hydrodynamic stretching of single cells for large population mechanical phenotyping. *Proc. Natl Acad. Sci. USA* **109**, 7630–7635 (2012).

45. Williams, D. C. & Wood, D. K. High-throughput quantification of red blood cell deformability and oxygen saturation to probe mechanisms of sickle cell disease. *Proc. Natl Acad. Sci. USA* **120**, e2313755120 (2023).

46. Karaz, S. & Senses, E. Liposomes under shear: structure, dynamics, and drug delivery applications. *Adv. NanoBiomed Res.* **3**, 2200101 (2023).

47. Shum, H. C. et al. Droplet microfluidics for fabrication of non-spherical particles. *Macromol. Rapid Commun.* **31**, 108–118 (2010).

48. Paulsen, K. S., Di Carlo, D. & Chung, A. J. Optofluidic fabrication for 3D-shaped particles. *Nat. Commun.* **6**, 6976 (2015).

49. Wang, W. et al. Controllable microfluidic fabrication of microstructured materials from nonspherical particles to helices. *Macromol. Rapid Commun.* **38**, 1700429 (2017).

50. Kim, J. W., Larsen, R. J. & Weitz, D. A. Synthesis of nonspherical colloidal particles with anisotropic properties. *J. Am. Chem. Soc.* **128**, 14374–14377 (2006).

51. Bhujbal, S. V., de Vos, P. & Niclou, S. P. Drug and cell encapsulation: alternative delivery options for the treatment of malignant brain tumors. *Adv. Drug Deliv. Rev.* **67-68**, 142–153 (2014).

52. Barthès-Biesel, D. Motion and deformation of elastic capsules and vesicles in flow. *Annu Rev. Fluid Mech.* **48**, 25–52 (2016).

53. Sui, Y. & Spelt, P. D. M. An efficient computational model for macroscale simulations of moving contact lines. *J. Comput. Phys.* **242**, 37–52 (2013).

54. Sui, Y. & Spelt, P. D. M. Non-isothermal droplet spreading/dewetting and its reversal. *J. Fluid Mech.* **776**, 74–95 (2015).

55. Sui, Y. et al. A hybrid method to study flow-induced deformation of three-dimensional capsules. *J. Comput. Phys.* **227**, 6351–6371 (2008).

56. Wang, Z. et al. Path selection of a spherical capsule in a microfluidic branched channel: towards the design of an enrichment device. *J. Fluid Mech.* **849**, 136–162 (2018).

57. Lu, R. X. et al. Transient deformation of a viscoelastic capsule in a cross-slot microchannel: effects of inertia and membrane viscosity. *J. Fluid Mech.* **962**, A26 (2023).

58. Skalak, R., Tozeren, A., Zarda, R. P. & Chien, S. Strain energy function of red blood cell membranes. *Biophys. J.* **13**, 245–264 (1973).

59. Nadim, A. & Stone, H. A. The motion of small particles and droplets in quadratic flows. *Stud. Appl. Math.* **85**, 53–73 (1991).

60. Maxey, M. R. & Riley, J. J. Equation of motion for a small rigid sphere in a nonuniform flow. *Phys. Fluids* **26**, 883–889 (1983).

61. Bureau, L., Coupier, G. & Salez, T. Lift at low Reynolds number. *Eur. Phys. J. E* **46**, 111 (2023).

62. Rallabandi, B. et al. Membrane-induced hydroelastic migration of a particle surfing its own wave. *Nat. Phys.* **14**, 1211–1215 (2018).

63. Barthès-Biesel, D. & Rallison, J. M. The time-dependent deformation of a capsule freely suspended in a linear shear flow. *J. Fluid Mech.* **113**, 251–267 (1981).

64. Barthès-Biesel, D., Diaz, A. & Dhenin, E. Effect of constitutive laws for two-dimensional membranes on flow-induced capsule deformation. *J. Fluid Mech.* **460**, 211–222 (2002).

65. Lac, E. et al. Spherical capsules in three-dimensional unbounded Stokes flows: effect of the membrane constitutive law and onset of buckling. *J. Fluid Mech.* **516**, 303–334 (2004).

66. Urbanska, M. et al. A comparison of microfluidic methods for high-throughput cell deformability measurements. *Nat. Methods* **17**, 587–593 (2020).

67. Kollmannsperger, A. et al. Live-cell protein labelling with nanometre precision by cell squeezing. *Nat. Commun.* **7**, 10372 (2016).

68. Duchene, C. et al. Clogging of microfluidic constrictions by monoclonal antibody aggregates: role of aggregate shape and deformability. *Soft Matter* **16**, 921–928 (2020).

69. Belling, J. N. et al. Lipid-bicelle-coated microfluidics for intracellular delivery with reduced fouling. *ACS Appl. Mater. Interfaces* **12**, 45744–45752 (2020).

70. North, P. Collective cell migration. *Annu Rev. Cell Dev. Biol.* **25**, 407–429 (2009).

71. O'Connor, J. W. & Gomez, E. W. Cell adhesion and shape regulate TGF-beta1-induced epithelial-myofibroblast transition via MRTF-A signaling. *PLoS ONE* **8**, e83188 (2013).

72. Zhong-can, O. Y. & Helfrich, W. Bending energy of vesicle membranes: general expressions for the first, second, and third variation of the shape energy and applications to spheres and cylinders. *Phys. Rev. A* **39**, 5280–5288 (1989).

73. Peskin, C. S. The immersed boundary method. *Acta Numer.* **11**, 479–517 (2003).

74. Russo, G. & Smereka, P. A remark on computing distance functions. *J. Comput. Phys.* **163**, 51–67 (2000).

75. Liu, X. D., Osher, S. & Chan, T. Weighted essentially non-oscillatory schemes. *J. Comput. Phys.* **115**, 200–212 (1994).

76. Spelt, P. D. M. A level-set approach for simulations of flows with multiple moving contact lines with hysteresis. *J. Comput. Phys.* **207**, 389–404 (2005).

77. Wang, L., Guo, Z. L. & Mi, J. C. Drafting, kissing and tumbling process of two particles with different sizes. *Comput. Fluids* **96**, 20–34 (2014).

## Acknowledgements

D.J. was supported by a Newton International Fellowship provided by the Royal Society (NIF/R1/191510) and the National Natural Science Foundation of China (12272236). Y.S. and C.M. acknowledge a Royal Society research grant under the international exchange scheme (IES/R2/212075). R.L. and Z.G. thank the PhD studentships provided by the Queen Mary University of London (QMUL) and the Chinese Scholarship Council. The simulations were performed using the high-performance computer clusters of QMUL (funded by the UK EPSRC grants EP/K000128/1, EP/P020194/1 and EP/T022213/1).

## Author contributions

Y.S. and C.M. designed the research. D.J., R.L., Z.G., F.W., and W. W. conducted the numerical simulations. D.J., A.F., C.M., and Y.S. developed the theory. All authors analyzed the results, and wrote and reviewed the manuscript.

## Competing interests

The authors declare no competing interests.

## Additional information

**Supplementary information** The online version contains supplementary material available at https://doi.org/10.1038/s42005-024-01805-4.

**Correspondence** and requests for materials should be addressed to Chaouqi Misbah or Yi Sui.

**Peer review information** *Communications Physics* thanks the anonymous reviewers for their contribution to the peer review of this work. Handling Editors: Jacopo Fregoni.

**Reprints and permissions information** is available at http://www.nature.com/reprints

**Publisher's note** Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
**Open Access** This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

© The Author(s) 2024