# Simulation of non-Newtonian biopolymer extrusion and fall in the centrifugal microencapsulation process

Matei Badalan, Giovanni Ghigliotti, Denis Roux, Guillaume Maîtrejean, Jean-luc Achard, Frédéric Bottausci, Guillaume Balarac

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Matei Badalan, Giovanni Ghigliotti, Denis Roux, Guillaume Maîtrejean, Jean-luc Achard, et al. Simulation of non-Newtonian biopolymer extrusion and fall in the centrifugal microencapsulation process. AIChE Journal, 2024, 70 (10), 10.1002/aic.18517. cea-04756982

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RESEARCH ARTICLE

Transport Phenomena and Fluid Mechanics

# Simulation of non-Newtonian biopolymer extrusion and fall in the centrifugal microencapsulation process

Matei Badalan<fnref n="1" />,<fnref n="2" />

Giovanni Ghigliotti<fnref n="2" /> ID

Denis Roux<fnref n="3" />

Guillaume Maîtrejean<fnref n="3" />

Jean-Luc Achard<fnref n="1" />,<fnref n="2" />

Frédéric Bottausci<fnref n="1" />

Guillaume Balarac<fnref n="2" />

**Correspondence**

Giovanni Ghigliotti, CNRS, Grenoble INP, LEGI, Univ. Grenoble Alpes, 38000 Grenoble, France.

Email: giovanni.ghigliotti@univ-grenoble-alpes.fr

**Abstract**

We present high-fidelity numerical simulations of the centrifugal microencapsulation process, that is of interest for biomedical applications as cell therapy. We provide first a comprehensive rheological characterization of high-molecular-weight calcium alginate, a commonly used material in microencapsulation. Building upon this, we employ a fluid model that accurately replicates the relevant non-Newtonian properties of the fluid. This model is applied to numerical simulations of the first three stages of the centrifugal microencapsulation process: capillary flow, ejection from the capillary, and fall through the air. The results are successfully compared with experiments. Furthermore, this model, which can be adapted to various centrifugal microencapsulation devices, effectively elucidates the physical factors contributing to different capsule shapes that can be achieved at the end of the process. This breakthrough opens the door to precise control of capsule shapes and production rates.

**KEYWORDS**

alginate microcapsules, centrifugal extrusion, centrifugal microfluidics, computational fluid dynamics, microencapsulation

# 1 | INTRODUCTION AND PROBLEM DEFINITION

Centrifugal microencapsulation (CME) is an encapsulation technique that has been mostly developed over the past decade as a response to strict constraints imposed by the biomedical field (precise size and shape control and monodispersity, controlled mechanical properties and porosity, aseptic production environment, high production rates, usage of highly viscous biopolymers), that are generally very difficult to fulfill simultaneously by conventional encapsulation equipment.<fnref n="1" />⁻<fnref n="3" /> CME was shown to be able to satisfy all these requirements, making it a promising technique for biomedical applications such as the

encapsulation of living cells and islets* for cell therapy, the encapsulation of organoids† for three-dimensional (3D) in vitro culture for personalized medicine and drug screening, the delivery of drugs, etc.

Previous works have focused on improving the CME technique and demonstrating its potential for various biomedical applications.⁶⁻³⁰ Other work was aimed at standardizing the experimental process and establishing a generic dimensionless phase diagram of the centrifugal encapsulation system for the prediction of capsule sizes and shapes.<fnref n="3" />,³¹

In this article, we consider a CME system based on standard laboratory equipment, introduced in References 3 and 31. A schematic representation of the device is shown in Figure 1A. A standard

<footnote n="1">¹CEA, LETI, Technologies for Innovation in Health division, Microfluidic Systems and Bioengineering Lab, Univ. Grenoble Alpes, Grenoble, France</footnote>

<footnote n="2">²CNRS, Grenoble INP, LEGI, Univ. Grenoble Alpes, Grenoble, France</footnote>

<footnote n="3">³CNRS, Grenoble INP, LRP, Univ. Grenoble Alpes, Grenoble, France</footnote>

<footnote>*Clusters of cells, notably within the pancreas, that produce hormones such as insulin.</footnote>

<footnote>†Miniaturized and simplified version of an organ produced in vitro in three dimensions that shows realistic micro-anatomy.⁴</footnote>
<figure><img src="image_2.png" /><figcaption>**FIGURE 1** (A) Schematic description of the centrifugal microencapsulation device (image reused from Reference 5). The domain of the centrifugal extrusion problem in the centrifugal microencapsulation (CME) system is delimited by a red dashed line. (B) Photo of actual needle, with a longitudinal cross-cut through the reservoir. (C) Technical drawing (longitudinal cross-section) of the ejection nozzle (source: Nordson EFD). The dimensions of the different needles used in this article are given in Table 1.</figcaption></figure>

laboratory tube contains gelation solution at its bottom and a needle placed vertically close to its opening. The alginate solution contained in the needle reservoir flows through the needle thanks to the centrifugal force, is then extruded in droplets at the needle tip. The droplets fall through air before impacting the gelation bath, where they sediment and gel thanks to a small density difference. The gelled capsules are collected with a pipette once centrifugation is finished.

In a previous work,⁵ we have defined for the first time a comprehensive model of the process including all its stages, using physical models to describe the transformations that occur at each stage. In that work, one important point that remained open was the lack of direct observation of the intermediate process stages: dynamics of drop formation at the capillary end, drop fall in air, drop impact on the gelation bath, and gelation dynamics.

The difficulty of visualization is due to the combination of high-speed rotating system with small submillimetric characteristic sizes and opaque centrifuge housing, imposed by the use of conventional lab material. So, most of the physical models used to describe the stages of the process had to be validated using only data collected at the end of the process, namely microscope images of the gelled products (capsules, but also fibers, filaments, satellite capsules) collected at

the bottom of the centrifuge tube after centrifugation and hardening. As a consequence, different questions remain to explain what happens in the intermediate stages of the process, even if some partial explanations have been proposed. As examples: (i) the reason of formation of satellite drops, filaments and fibers, ascribed to the ejection stage; (ii) the origin of flattened capsules; (iii) the nondeformation of droplets by aerodynamic forces; (iv) the formation of monodisperse tear-shaped capsules; (v) the formation of tear-shaped capsules polydisperse in shape. Moreover, an important question unanswered in our previous work⁵ is the effect of non-Newtonian rheology on drop-ejection.

Hence, in this article we propose a numerical investigation of the three first stages of the CME process, namely the capillary flow, the droplet ejection and the fall of the droplets in the air gap between the capillary outlet and the gelation solution.

The objectives of this investigation are multiple: (i) to gain a better physical understanding of the process during these key stages, notably accounting for non-Newtonian rheology; (ii) to corroborate some of the strongest assumptions made in previous work; (iii) to create a reliable computational model of the key process stages which can be used to validate analytical correlations
<table><caption>TABLE 1 Dimensions of the ejection needles used in this article. Data from Nordson EFD catalogue³² and Nordson EFD technical drawings.³³</caption><thead><tr><th>Gauge</th><th>D<sub>cap,ref</sub><br>mm</th><th>D<sub>cap,tech</sub><br>mm</th><th>D<sub>out,ref</sub><br>mm</th><th>L<sub>cap</sub><br>mm</th><th>L<sub>step</sub><br>mm</th><th>D<sub>step</sub><br>mm</th></tr></thead><tbody><tr><td>32</td><td>0.10</td><td>0.127</td><td>0.24</td><td>12.573</td><td>0.431</td><td>0.178</td></tr><tr><td>30</td><td>0.15</td><td>0.178</td><td>0.31</td><td>18.923</td><td>0.431</td><td>0.229</td></tr><tr><td>27</td><td>0.20</td><td>0.229</td><td>0.42</td><td>18.923</td><td>0.431</td><td>0.279</td></tr><tr><td>23</td><td>0.33</td><td>0.356</td><td>0.65</td><td>18.923</td><td>0.431</td><td>0.457</td></tr><tr><td>22</td><td>0.41</td><td>0.432</td><td>0.72</td><td>18.923</td><td>0.431</td><td>0.559</td></tr><tr><td>20</td><td>0.61</td><td>0.622</td><td>0.91</td><td>18.923</td><td>0.431</td><td>0.762</td></tr></tbody></table>

for droplet shape prediction and to optimize the production system.

## 1.1 | The centrifugal extrusion problem

The centrifugal device investigated here is presented schematically in Figure 1A, and it has been described in detail in previous work.³,⁵,³¹ It is composed of an ejection system, composed of a stainless steel needle (including a reservoir and a capillary, Figure 1B) and a holder, mounted in a standard centrifuge tube containing the gelation bath where the ejected droplets are collected for polymerization. The tube is placed in a commercial laboratory centrifuge with swinging buckets. This article focuses on the first three stages of the CME process (marked with a red dashed line in Figure 1A): the flow of the biopolymer (alginate) solution from the reservoir through the capillary, its ejection from the tip of the needle and its fall through the air before reaching the gelation solution.

During the operation of the CME device, the tube position becomes nearly horizontal thanks to the swinging bucket and the biopolymer solution is set in motion under the effect of the centrifugal force from the reservoir through the capillary and ejected at the needle outlet. A detailed drawing of the ejection system geometry is given in Figure 1C, with the corresponding dimensions listed in Table 1 for each type of needle used.

In this article, long-chained alginates are considered, due to their favorable characteristics for biomedical applications: better performance for cell encapsulation and implant applications.³⁴ Long-chained alginates are strongly non-Newtonian fluids, as thoroughly discussed in Section 2 of this article, and this cannot be overlooked for a quantitative study of capsule production.

In the literature, the centrifugal extrusion problem has been investigated analytically,³⁵,³⁶ under the assumption of Newtonian fluid behavior. More general works on droplet extrusion under gravity³⁷⁻³⁹ do not take into account centrifugation-specific effects or non-Newtonian properties.

Experimental visualization of the centrifugal extrusion has been attempted.¹,²,²⁸,⁴⁰,⁴¹ However, in most of these works, visualization was for demonstration purposes and no in-depth discussion on the physics of drop extrusion as a function of operating parameters has

been performed. In References 40 and 41 the authors observe dripping-jetting transitions and discuss the particle production in jetting regime, which produces polydisperse capsule populations. However, the influence of the nature of the fluid, and in particular its possible non-Newtonian fluid properties is not addressed. Indeed, in that study, short-chained alginates were used, having a behavior closer to Newtonian compared to long-chained ones.

More general numerical studies of droplet formation and ejection under gravity were undertaken for example in References 42–44 and notably in Reference 45, where also non-Newtonian fluid behavior was considered. However, a numerical simulation of centrifugal droplet extrusion under the conditions of CME devices (non-Newtonian fluid, centrifugal force) has not yet been performed to our knowledge.

This article aims therefore to complement the existing literature by investigating the flow of long-chained alginate solutions in CME devices. The material properties of the alginate solutions are determined through a thorough rheological characterization.

## 1.2 | Outline of this article

The first part of this article is dedicated to material characterization, description of the numerical method and its validation:

* **Section 2.** A constitutive relation is necessary to describe the non-Newtonian fluid behavior in the numerical model. In order to determine this relation, a rheological characterization of the alginate solutions is carried out, guiding the choice of a rheological model for these fluids.

* **Section 3.** The numerical methods that will be used to solve the equations describing the problem are specified: solver, boundary conditions, mesh, dimensionless problem formulation, parametrization.

* **Section 4.** In order to validate the numerical model, several series of experiments are performed, with both a Newtonian fluid (standard oil) and a non-Newtonian fluid (alginate). Measured values (flow rate, droplet volumes) are compared with numerical results.

The second part presents the main results of this article:

* **Section 5.** Different ejection regimes in the centrifugal extrusion problem (dripping, jetting, filament formation) are explored numerically. Capsule production experiments of the same configurations are performed. The combined analysis of the numerical and experimental results is used to understand the mechanisms of formation of different centrifugation products: capsules, fibers, filaments, satellite capsules.

* **Section 6.** We focus on the dripping ejection regime, and we investigate numerically the formation of various capsule shapes. To this end, both experiments and simulations are performed for different configurations. The comparison of the numerical and experimental results shows good agreement, and gives insight into the physical processes underlying the formation of different capsule shapes.
## 2 | RHEOLOGICAL CHARACTERIZATION OF THE WORKING FLUIDS

In previous work on the analytical modeling of the CME system, the single-phase flow in the capillary was studied, without the extrusion in air. It was shown that the shear-thinning behavior of the alginate solutions was crucial for the determination of the flow rate in the capillary.⁵

The goal of this article is an accurate description, not only of the capillary flow stage, but also of the ejection and fall in air of the alginate solution. Several additional phenomena increasing the complexity of the system (surface tension, possibly elastic or elongational effects) are involved in the latter two stages. There, the dependence of the results on the rheological properties can potentially be very strong. Hence, is a viscosity law like Carreau-Yasuda enough to describe the fluid behavior in all these stages with sufficient accuracy?

In order to answer this question, we first discuss the major classes of rheological models (constitutive equations) available for our problem (Appendix SM.10). Consequently, we perform an extended rheological characterization of the alginate solutions using different complementary measurement methods. After analyzing the results, we discuss the choice of the constitutive laws to be used in the numerical model.

### 2.1 | Experimental methods of rheological characterization

The biopolymer used for droplet ejection in this study is Na-alginate, which is obtained by diluting highly purified Sodium alginate PRO-NOVA SLG100 powder (Novamatrix) into an alginate buffer solution (containing water, NaCl and HEPES‡ buffer⁸). Three solutions with different concentrations are investigated here: 10g/L (1% w/v), 20g/L (2% w/v) and 40g/L (4% w/v). We will refer to this solution batch as “Alginate B” shortened Alg. B hereafter, in contrast to batch A (Alg. A), obtained with the same protocol and from the same components in previous studies,⁵ the density and surface tension of which were measured and reported in Reference 31. A thorough characterization of medium-to-high viscosity alginate solutions was performed in References 46 and 47. Here similar measurements in rotational (both steady and oscillation experiments) and elongational rheometry on these three alginate solutions have been carried out.

The characterization of the rheological behavior of the working fluids was performed using three different rheometers, in three sets of experiments, all carried out on solution samples from the same batch: first, a Kinexus Pro+ rotational rheometer (Malvern Instruments Ltd),

used for steady rotation and oscillatory measurements, both with a cone-plate geometry, then a TriPAV rheometer (TriJet Ltd), used for oscillatory experiments, and an elongational CaBER rheometer (Cambridge Polymer Group). More details and a motivation for the choice of these rheometers are given in Appendix SM.11.

In the following, we present the results of these experiments and the analysis performed to extract relevant rheological parameters from the data.

### 2.2 | Viscosity measurement in rotational rheometry

Two types of measurements were carried out on the rotational rheometer with a cone-plate geometry for the characterization of the flow behavior of alginate solutions: steady measurements, where a given constant shear rate is applied, and dynamic measurements (SAOS),⁹ where a frequency sweep is applied at a given deformation strain. The calibration of the rheometer was performed using a PSU2400 oil (Paragon Scientific Ltd.) used as a viscosity and density standard (UKAS 0649, UKAS 4589). The same oil was later used as a Newtonian fluid for validation experiments.

Results from the steady measurements $η=f(\dot{y})$ with the three Alg. B solutions are reported in Figure 2 (circle symbols). As expected, the solutions feature a Newtonian plateau (constant apparent viscosity) for low shear rates $\dot{y}$ and a shear-thinning behavior (decreasing viscosity) for high shear rates. The transition between the two regimes occurs as expected at lower shear rates as the polymer concentration increases.⁴⁶⁻⁴⁸ The shear viscosity $η$ decreased by nearly one order of magnitude for the highest shear rates and for the highest polymer concentration, while relative variations are smaller for lower concentrations. Modeling this behavior can have a significant impact on the flow rate in the centrifugation experiments.⁵

On the other hand, as expected, the standard oil PSU2400 presents a constant viscosity in the studied ranges of shear rate, indicating a Newtonian behavior. The error between the measured viscosity and the reference viscosity for this oil is less than 2%.

Dynamic (oscillatory) measurements on the same Alg. B solutions were used to determine the complex viscosity$^{#}$ $η^*$ (Pa s) as a function of the frequency $ω$ (rad/s). Superposing $η=f(\dot{y})$ and $η^*=f(ω)$ (Figure 2), shows a very good agreement between the oscillatory and the steady shear measurements, meaning that the Cox-Merz rule$^{||}$ is respected by the alginate solutions, in agreement with previous works.⁴⁶,⁴⁷,⁵¹

The Carreau-Yasuda model is used, as it has the ability to capture both the Newtonian and the shear-thinning regions (in contrast with simpler models as power law: $η = m\dot{y}^{n-1}$):

<footnote>‡N-2-Hydroxyethylpiperazine-N-2-Ethane Sulfonic Acid.</footnote>

<footnote>§The presence of NaCl and HEPES buffer is important when encapsulating cells, in order to ensure buffering capacity and osmotic balance, and thus biocompatibility of the solutions.²² Although this article does not present experiments of bioencapsulation, the solutions were formulated keeping in mind the requirements for bioencapsulation applications, which are the most common applications of the CME process.</footnote>

<footnote>||Small-amplitude oscillatory shear.</footnote>

<footnote>¶The frequency-dependent viscosity function determined for a viscoelastic fluid by subjecting it to oscillatory shear stress⁴⁹: $|η^*| = \sqrt{G^2 + G''^2}/ω$.</footnote>

<footnote>Stating that the shear-rate dependence of the steady-state viscosity equals the frequency dependence of the complex viscosity.⁵⁰</footnote>
**FIGURE 2** Results of the shear viscosity measurements with the Kinexus Pro+ rheometer in a cone-plate geometry, for the three Alg. B solutions (concentrations 1%, 2% and 4%), in steady regime (circles: plots of shear viscosity η vs. shear rate ˙γ) and in dynamic regime (triangles: plots of the real part of the complex viscosity η* vs. oscillation frequency ω [rad/s]). As reference, steady measurements were also performed for the standard oil PSU2400, for which the results are plotted as η vs. ˙γ and show a Newtonian behavior.

<figure><img src="image_3.png" /><figcaption>**TABLE 2** Values of experimentally fitted rheological parameters for the three working fluids PSU2400 (considered Newtonian), Alginate of batch B (this article) and Alginate of batch A (measured in previous work⁵).</figcaption></figure>

<table><thead><tr><th>Fluid</th><th>η<sub>0</sub> (Pa s)</th><th>λ<sub>C</sub> (s)</th><th>a</th><th>n</th></tr></thead><tbody><tr><td>PSU2400</td><td>1.45</td><td>—</td><td>—</td><td>—</td></tr><tr><td>Alg. B 1%</td><td>0.129</td><td>0.007</td><td>1.410</td><td>0.745</td></tr><tr><td>Alg. B 2%</td><td>1.129</td><td>0.014</td><td>1.128</td><td>0.507</td></tr><tr><td>Alg. B 4%</td><td>13.777</td><td>0.033</td><td>0.835</td><td>0.301</td></tr><tr><td>Alg. A 1%</td><td>0.104</td><td>0.008</td><td>2.589</td><td>0.817</td></tr><tr><td>Alg. A 2%</td><td>1.122</td><td>0.014</td><td>1.177</td><td>0.524</td></tr><tr><td>Alg. A 4%</td><td>15.170</td><td>0.040</td><td>0.798</td><td>0.295</td></tr></tbody></table>

$$ \frac{\eta - \eta_{\infty}}{\eta_0 - \eta_{\infty}} = [1 + (\lambda_C \dot{\gamma}(r))^a]^{\frac{n-1}{a}}, \quad (1) $$

where η₀ is the viscosity at zero-shear, η∞ the viscosity at infinite shear and λ_C, a and n are fitted parameters with the following physical signification: λ_C is the relaxation time, n is the viscosity exponent†† and a characterizes the curvature of the function in the transition region.

Since the shear viscosity at the typical shear rates occurring in the centrifugal extrusion system remains far from the lower Newtonian plateau of typical polymer solutions, we can consider for simplicity, without reducing the accuracy of the model in the studied ranges,

that η∞ = 0.⁴⁸ This simplifies the reliability of the fits, that pass from five to four parameters. The fitted parameters for three different alginate solutions measured in this article are listed in Table 2. The curves showing the fitted models are represented in Figure 2, demonstrating the ability of the chosen model to accurately represent the viscous behavior of the alginate solutions.

## 2.3 | Relaxation times in oscillatory measurements

The Carreau-Yasuda model is able to represent the shear-thinning of the alginate solutions studied here, as shown in the previous section. However, it does not account for any effects of the deformation history on the rheological properties. Are these effects present in the centrifugal extrusion problem, and have they any influence on process output quantities such as flow rate, capsule size or capsule shape?

To answer that question, we carry out an in-depth analysis of the relaxation times of the alginate solutions, which allow to assess the influence of history effects.

The dynamic measurements (SAOS) in rotational rheometry can be used to determine the storage and loss moduli G' and G'' of the alginate solutions.‡‡

SAOS measurements in rotational rheometry are limited to low and medium frequencies f due to inertial effects (proportional to f²), which become dominant at high frequencies and impede the measurement. Therefore, complementary oscillatory measurements of G' and G'' at high frequencies are preformed on the TriPAV rheometer.

<footnote>**Defined as the inverse of the critical shear rate ˙γ_c at which the transition from Newtonian to shear-thinning behavior occurs.</footnote>

<footnote>††Indicates the strength of the shear-thinning effect.</footnote>

<footnote>‡‡The storage modulus G' is an indicator of the elastic properties of a material, while the loss modulus G'' characterizes the viscous properties of a material.</footnote>
<figure><img src="image_4.png" /><figcaption>**FIGURE 3** Results of the rotational and TriPAV rheology measurements for the three Alg. B solutions (concentrations 1%, 2%, and 4%): measurement of storage modulus G' (circles) and of shear modulus G'' (triangles) vs. oscillation frequency. The two lines show the theoretical slope values 1 and 2 of G' and G'' at low frequency.</figcaption></figure>

<table><caption>**TABLE 3** Estimated values of the relaxation times for the three Alg. B solutions used in this article: $\lambda_C$ comes from the Carreau-Yasuda fit, $\tau_{TriPAV}$ from the TriPAV rheometer data analysis and $\lambda_{el}$ from the CaBER data analysis.</caption><thead><tr><th>Fluid</th><th>η<sub>0</sub><br>(Pa s)</th><th>λ<sub>C</sub><br>(ms)</th><th>τ<sub>TriPAV</sub><br>(ms)</th><th>λ<sub>el</sub><br>(ms)</th></tr></thead><tbody><tr><td>Alg. B 1%</td><td>0.129</td><td>7</td><td>0.206</td><td>0.12–0.27</td></tr><tr><td>Alg. B 2%</td><td>1.129</td><td>14</td><td>0.867</td><td>0.76–1.1</td></tr><tr><td>Alg. B 4%</td><td>13.777</td><td>33</td><td>7.38</td><td>3.2–4.3</td></tr></tbody></table>

In Figure 3 measured values of G' and G'' are plotted for the three Alg. B solutions, as a function of the oscillation frequency (expressed in rad/s) $\omega = 2\pi \cdot f$, using both the rotational ("rot") and the TriPAV ("tpv") rheometers. All the solutions, at low to medium frequencies, the loss modulus is greater than the storage modulus, indicating that the fluids have a predominantly viscous behavior. At higher frequencies, a cross-over between the G'' and G' is observed. Only for the 4% alginate solution, the cross-over point is captured both by the rotational and the TriPAV rheometers. For the other two solutions, only the TriPAV rheometer allows measurement up to the cross-over frequency $\omega_{G'=G''}$. However, generally a good overlap between the TriPAV and the rotational curves is observed, validating the TriPAV measurements.

The cross-over frequency holds significance, as it determines the transition from predominantly viscous to predominantly elastic behavior, indicating that the polymer coils are not able to relax completely.⁴⁷ The inverse value of the cross-over frequency gives the relaxation time $\tau_{G'=G''}$.

At low frequencies (1/f > $\tau_{G'=G''}$), the characteristic flow time 1/f is bigger than the characteristic material time, so that the studied

sample exhibits a fluid-like (viscous) behavior. Conversely, at high frequencies (1/f < $\tau_{G'=G''}$), the sample demonstrates a solid-like (elastic) behavior.⁵¹

The relaxation times computed from the inverse of the cross-over frequencies obtained with the TriPAV rheometer, $\tau_{TriPAV} = 1/\omega_{G'=G''}$, are given in Table 3. They correspond to the relaxation times that arise from the molecular reorganization at the polymer chain level. As the concentration of alginate increases, both the elastic and viscous moduli increase, leading to a corresponding increase in the relaxation time. This observation and modeling have been observed in linear polymers.⁵²⁻⁵⁴

## 2.4 | Elongational measurements

So far, we have determined the relaxation time $\lambda_C$ in steady shear flow, characterizing the onset of shear-thinning behavior, and the relaxation time $\tau_{TriPAV}$ in oscillatory flow, characterizing the viscoelasticity of the polymer close to its thermodynamic equilibrium. However, in the CME process, the fluid is subjected to an elongational flow during the ejection phase. Therefore, the estimation of the elongational relaxation times of the solutions is needed. Elongational rheometry measurements with the CaBER rheometer are then performed.

In the CaBER elongational rheometer measurements, the slow retraction method was used,⁵⁵⁻⁵⁷ where the upper plate of the rheometer is lifted at constant (and small) velocity. The evolution over time of the minimum diameter of the liquid filament was measured using a high-speed camera and image analysis software. The resulting data ($R_{min}(t)$) will be analyzed following indications from Reference 47.
<figure><img src="image_5.png" /><figcaption>FIGURE 4 Results of the extensional rheology measurements with the CaBER rheometer for the three Alg. B solutions (concentrations 1%, 2%, and 4% (w/v)). (A) Evolution of the minimum filament radius $R_{min}$ with the time. For each of the curves, the time $t = 0$ is chosen in the moment when the filament breaks up. The purple dashed line is set at a value of $R_{min} = 0.465$ mm, which was used as a threshold to plot the terminal values before rupture in (B). (B) Zoom in the elastocapillary thinning regime at late times, before breakup. The lines show the fit of an Oldroyd-B model on the elastocapillary thinning section of the curves. The relaxation time $\lambda_{el}$ can be determined from the slope of these lines as described in the text. For these curves, the time $t = 0$ was chosen when the liquid filament had a certain radius ($R = 0.465$ mm).</figcaption></figure>

The liquid ligament may go through several thinning regimes (inertial, viscous, elastic) before breakup. Notably, in the elastocapillary thinning regime, filament radius is expected to decrease exponentially, according to the following law$^{47,56,58}$:

$$R_{min}(t) = R_0 \left( \frac{GR_0}{2\gamma_{alg}} \right) \exp(-t/3\lambda_{el}), \quad (2)$$

where $R_0$ is the radius at the beginning of the filament thinning, G is the relaxation modulus and $\lambda_{el}$ is the stress-relaxation time of the polymer. In logarithmic scale, this represents a straight line of slope $-1/(3\lambda_{el})$. Thus, with a linear fit of Equation (2) on the experimentally measured curves $R_{min}(t)$ in logarithmic scale, the elongational relaxation time $\lambda_{el}$ can be determined. Values are reported in Table 3.

The measured values of the minimum filament radius $R_{min}$ over time for the three alginate solutions are reported in Figure 4A. There is currently no universal consensus in the literature on how to determine the region of elastocapillary thinning on the experimental data. As previously commented in the literature,$^{47,59}$ the characteristic time extracted from filament thinning CaBER experiments strongly depends on the choice of the equation and time regime to be used for data analysis. This hampers a straightforward estimate of the relaxation time at play during extension.

The authors in Reference 47 discuss two different approaches to interpret elongational experimental measurements. The first method considers that the filament successively follows a linear thinning, then a power law thinning and eventually and elastocapillary thinning. The exponential filament thinning (corresponding to the elastocapillary regime) takes place before the final extensional behavior of polymer chains and filament breakup. In the second approach, where viscous

effects might overcome elastic effects at early stages, it is considered that the elastocapillary regime occurs at later stages. So the fit is made on the data corresponding to the long-time regime, right before breakup.$^{47,59}$ In this article, the second approach is used to analyze the elongational rheometry data. As illustrated in Figure 3, it is observed that at the initial times (corresponding to low and intermediate angular frequencies), the samples are dominated by the viscous behavior ($G'' > G'$).

Figure 4B shows a zoom on the section of interest of the filament thinning curves at late times, when an elastocapillary regime is supposed to govern the evolution of the filament radius. By fitting a line on this region of the $R_{min} = f(t)$ curves in logarithmic scale, one can determine the relaxation time $\lambda_{el}$ from the slope of the fitted line, as shown in Figure 4B. The resulting values of $\lambda_{el}$ are given in Table 3. Since few points are available for the fit in the chosen region, the time regime chosen to compute the relaxation time $\lambda_{el}$ remains subjective, as commented in Reference 47. To account for this uncertainty, a range of values was given in Table 3 for $\lambda_{el}$, rather than unique values.

## 2.5 | Discussion on the elasticity in the rheological model

Dimensional analysis will be used here to estimate the relative magnitude of the driving processes in each stage of the extrusion, and notably to assess the importance of elastic effects on the centrifugal extrusion problem.

The three relaxation times that were measured experimentally will be used to quantify the magnitude of elastic forces. They characterize the behavior of alginate solutions in different types of
<table><caption>TABLE 4 Dimensionless numbers characterizing the three stages of centrifugal droplet extrusion (index cap: capillary flow; ej: ejection; fall: fall).</caption><thead><tr><th>Fluid</th><th>De<sub>cap</sub></th><th>Ca<sub>ej</sub></th><th>Ec<sub>ej</sub></th><th>Oh<sub>ej</sub></th><th>Ec<sub>fall</sub></th><th>Oh<sub>fall</sub></th></tr></thead><tbody><tr><td>Alg. B 1%</td><td>4.61-31.7</td><td>0.02-0.57</td><td>0.21-2.37</td><td>0.86-1.90</td><td>0.39-0.67</td><td>0.88-1.15</td></tr><tr><td>Alg. B 2%</td><td>3.27-48.5</td><td>0.05-3.51</td><td>0.15-1.10</td><td>7.51-16.6</td><td>0.26-0.43</td><td>9.02-11.8</td></tr><tr><td>Alg. B 4%</td><td>1.29-32.2</td><td>0.11-11.2</td><td>0.05-0.61</td><td>91.7-203</td><td>0.23-0.38</td><td>123-161</td></tr></tbody></table>

$$Ec_{ej} = \frac{\lambda\gamma}{\eta R} \qquad (5)$$

$$Oh_{ej} = \frac{\eta}{\sqrt{\rho\gamma R}} \qquad (6)$$

flow: $\lambda_C$ describes the behavior of the fluid in continuous shear flow, $\tau_{TriPAV}$ describes the bulk elasticity in transient (oscillatory) flow and $\lambda_{el}$ describes the elasticity in elongational flow.

## 2.5.1 | Flow in the capillary

In the capillary, we have a continuous, established flow where the fluid is subjected to shear. The relevant relaxation time here is $\lambda_C$, obtained from rheology measurements in steady shear flow, characterizing the shear-thinning behavior of the polymer. This will be compared with the characteristic process time in the capillary for which we will use the inverse of the shear rate $1/\dot{\gamma}_W$ at the wall. The shear rate at the wall has been estimated using the method in Reference 5, with a Carreau-Yasuda viscosity law, for the twelve extrusion cases ER1-ER12 studied in this article (the parameters of these cases will be described in detail in Section 5).<fnref n="88" />

We define a Deborah number comparing the relaxation time $\lambda_C$ and the characteristic process time $1/\dot{\gamma}_W$:

$$De_{cap} = \frac{\lambda_C}{1/\dot{\gamma}_W} \qquad (3)$$

If $De_{cap} < 1$, the fluid undergoes shear in the Newtonian plateau region of its viscosity-shear rate curve and the shear-thinning effect can be neglected. On the contrary, if $De_{cap} > 1$ the shear-thinning effect is important and needs to be taken into account in the calculations.

The range of $De_{cap}$ corresponding to the extrusion cases studied have $De_{cap} > 1$, indicating that the shear-thinning effect in the capillary is important. Therefore, a Newtonian fluid behavior cannot be assumed in the simulations.

## 2.5.2 | Droplet ejection

During the ejection of droplets by centrifugal force, the main forces acting on the fluid are viscosity, surface tension, inertia and elasticity. In order to compare these forces, we use the following dimensionless numbers⁵⁹⁻⁶²:

$$Ca_{ej} = \frac{\eta U}{\gamma} \qquad (4)$$

Here, $\eta$ is the (zero-shear) viscosity; $U$ is the mean flow velocity in the capillary; $\rho$ is the density; $R$ is a characteristic system dimension (we will take $R_{cap}$); $\gamma$ is the surface tension; $\lambda$ is the relaxation time of the polymer. The capillary number $Ca_{ej}$ relates viscous forces to surface tension forces. The elastocapillary number $Ec_{ej}$ compares elastic and viscous effects. When $Ec_{ej} > 4.7$, the process is controlled by elastic effects. For $Ec_{ej} < 4.7$, viscosity dominates over elasticity in the ejection process. The Ohnesorge number $Oh_{ej}$ compares viscous and inertial effects. If $Oh_{ej} > 0.2077$, viscous effects will dominate over inertial effects.⁴⁷,⁵⁹,⁶¹

Since it is a priori not obvious to determine which elastic forces might come into play, both relaxation times $\tau_{TriPAV}$ and $\lambda_{el}$ describing bulk elasticity in transient and elongational flow will be considered for the definition of $Ec_{ej}$.

In Table 4, the ranges of these three dimensionless numbers are given for each of the alginate solutions characterized, in the extrusion cases studied in this article. The elastocapillary number $Ec_{ej}$ is lower than 4.7 in all cases (and even lower than unity in most cases) studied here. This indicates that in none of the cases studied here is the process of droplet detachment governed by elastic effects, and that in the conditions studied in this article, viscosity is dominant over elasticity. The Ohnesorge number $Oh_{ej}$ is higher than 0.2 for all the cases studied, indicating that viscosity dominates over inertial effects in our system. Finally, the capillary number $Ca_{ej}$ varies from values much lower than unity to values much greater than unity for the different extrusion cases. This indicates that in certain configurations the effect of surface tension will be fully damped by viscosity, whereas in other cases the surface tension will govern the fluid behavior.

As a conclusion, elastic effects are always dominated by viscosity during droplet detachment in our system, for the ranges of parameters explored in this article.

## 2.5.3 | Droplet fall

After ejection, the droplets fall through air. A priori, surface tension, viscosity and elasticity may act on the droplet shape during this fall. Thus, we will use again an elastocapillary number and an Ohnesorge number to estimate the relative magnitude of these effects. Since the droplet is already detached from the rest of the fluid in the capillary, no elongational flow is present inside the droplet. Therefore, for the

<footnote n="88">⁸⁸Cases ER1-ER12 represent a very wide range of values of the extrusion parameters and cover practically all the operating regimes encountered in practice for CME devices.⁵</footnote>
elastic effects only the relaxation time $\tau_{TriPAV}$, characterizing bulk elasticity, is relevant for this stage of the extrusion problem. The two numbers can be defined as follows:

$$Ec_{fall} = \frac{\lambda\gamma}{\eta R}. \qquad (7)$$

$$Oh_{fall} = \frac{\eta}{\sqrt{\rho\gamma R}}. \qquad (8)$$

Here, for R we use the droplet radius, estimated using the correlations in Reference 5. The ranges of values of these two dimensionless numbers in the configurations studied in this article are given in Table 4. Here, $Ec_{fall} < 1$ in all cases, indicating that viscosity dominates over elasticity in the behavior of the falling droplets. As well, $Oh_{fall} > 0.2$ for all cases, indicating that the droplet deformation will be governed by viscosity and surface tension.

### 2.5.4 | Conclusions

The dimensional analysis carried out in this section allowed to identify the dominant physical processes during the different stages of the centrifugal extrusion. The results show that for the operating regimes which are of interest for our application, elastic effects in the polymer are negligible compared to viscous and surface tension effects. At the same time, it was shown that shear-thinning is important in the range of shear rates expected to occur in our application.

Based on this analysis, using a rheological model which does not take into account elastic (history) effects is justified. Therefore, the Carreau-Yasuda model, that describes a Newtonian plateau and shear thinning effects –but not elastic ones– will be used in the numerical simulations as constitutive law to describe the long-chained alginate solutions. The computational cost of this model is low, and its implementation is simple.

## 3 | NUMERICAL MODEL OF THE CENTRIFUGAL EXTRUSION PROBLEM

In this section we present the mathematical definition of the fluid flow problem in our system and the numerical methods that will be employed for its resolution, using the rheological model determined in Section 2. The equations governing the problem and the numerical model used to solve these equations will be presented.

### 3.1 | Dimensional analysis

Before defining the numerical model used to solve the centrifugal extrusion problem, the problem has been rendered dimensionless. This procedure allows to identify the independent parameters, and also the dominant ones, controlling the extrusion system.

The input parameters of the centrifugal extrusion problem are listed in Table 5 and represented in Figure 5.

Since it is the centrifugal acceleration that governs the flow, the three parameters $\rho_{alg}$, $R_{cap}$ and $g_{tip}$ are chosen as repeating variables for the nondimensionalization.63,64 The obtained dimensionless numbers describing the problem are listed in Table 6. In the following, the range of variation and the problem sensitivity to each parameter are discussed. In this way, parameters that can be kept constant, dependent parameters, and dominant ones are differentiated.

The density ratio $r$ varies within a very small range in our experiments and will be considered constant.

The two-phase problem presents an asymptotic limit for $m \to \infty$, that is, the problem is insensitive to variations of viscosity ratio $m$ as long as it remains above a certain threshold, roughly $m = 500$ (more details in the Appendix SM.2). Since in our experimental device the value of $m$ is always much larger than 500 (typically around 5000 for the less viscous alginates), the value of $m$ can also be kept constant.

The value of $L_{cap}$ depends on the needle diameter in the experiments, since commercially available needles are used (see Table 1). Therefore, its independent variation will not be investigated in this study. The same is valid for $R_{out}$. Furthermore, since generally the interface of the liquid inside the capillary remains pinned at the inner edge of the capillary outlet during ejection, the outer needle radius $R_{out}$ is not expected to play a role.42 The reservoir geometry is identical for all the needles used, therefore $R_{res}^* = R_{res}/R_{cap}$ will only depend on the needle diameter and will not be independently varied in this study. The same is true for the value of $y_{tip}$ that depends on $L_{cap}$. As a consequence, $y_{tip}^*$ only depends on the needle diameter and will not be considered for a parametric study. The initial height $h_{res}(t=0) = h_0$ of the liquid column in the reservoir is variable in the experiments. Also, as the liquid is ejected, the height of the liquid in the reservoir $h_{res}$ decreases in time. In order to investigate the effect of this variation, the operating points will be simulated both in full and empty reservoir configurations.

<table><caption>TABLE 5 Summary of the input parameters in the centrifugal droplet ejection problem, with the common ranges of variation in our experiments.</caption><thead><tr><th>Name</th><th>Parameter</th><th>Minimum</th><th>Maximum</th></tr></thead><tbody><tr><td>Alginate density</td><td>ρ<sub>alg</sub> (kg/m<sup>3</sup>)</td><td>1002</td><td>1020</td></tr><tr><td>Alginate viscosity</td><td>η<sub>alg</sub> (Pa s)</td><td>0.1</td><td>20.0</td></tr><tr><td>Alginate surface tension</td><td>γ<sub>alg</sub> (mN/m)</td><td>56</td><td>72</td></tr><tr><td>Air density</td><td>ρ<sub>air</sub> (kg/m<sup>3</sup>)</td><td>1.2</td><td>1.2</td></tr><tr><td>Air viscosity</td><td>η<sub>air</sub> (Pa s)</td><td>1.8 · 10<sup>-5</sup></td><td>1.8 · 10<sup>-5</sup></td></tr><tr><td>Capillary inner radius</td><td>R<sub>cap</sub> (μm)</td><td>50</td><td>310</td></tr><tr><td>Capillary outer radius</td><td>R<sub>out</sub> (μm)</td><td>120</td><td>460</td></tr><tr><td>Capillary length</td><td>L<sub>cap</sub> (mm)</td><td>12.6</td><td>19.0</td></tr><tr><td>Reservoir radius</td><td>R<sub>res</sub> (mm)</td><td>2.015</td><td>2.015</td></tr><tr><td>Liquid height in reservoir</td><td>h<sub>res</sub> (mm)</td><td>0</td><td>7.5</td></tr><tr><td>Capillary tip position</td><td>γ<sub>tip</sub> (mm)</td><td>81</td><td>81</td></tr><tr><td>Centrifugal acceleration at tip</td><td>g<sub>tip</sub> (m/s<sup>2</sup>)</td><td>400</td><td>6000</td></tr></tbody></table>
<figure><img src="image_6.png" /><figcaption>FIGURE 5 Parameters of the centrifugal extrusion problem.</figcaption></figure>

<table><caption>TABLE 6 Dimensionless parameters governing the centrifugal droplet ejection problem.</caption><thead><tr><th>Parameter</th><th>Ga</th><th>Oh</th><th>r</th><th>m</th><th>R<sup>*</sup><sub>out</sub></th><th>L<sup>*</sup><sub>cap</sub></th><th>R<sup>*</sup><sub>res</sub></th><th>h<sup>*</sup><sub>res</sub></th><th>y<sup>*</sup><sub>tip</sub></th></tr></thead><tbody><tr><td>Expression</td><td>ρ<sub>alg</sub>g<sup>0.5</sup><sub>tip</sub>R<sup>1.5</sup><sub>cap</sub> / η<sub>alg</sub></td><td>η<sub>alg</sub> / √(ρ<sub>alg</sub>γ<sub>alg</sub>R<sub>cap</sub>)</td><td>ρ<sub>alg</sub> / ρ<sub>air</sub></td><td>η<sub>alg</sub> / η<sub>air</sub></td><td>R<sup>out</sup><sub>cap</sub></td><td>L<sup>*</sup><sub>cap</sub></td><td>R<sup>res</sup><sub>cap</sub></td><td>h<sup>*</sup><sub>res</sub></td><td>γ<sub>tip</sub> / R<sup>*</sup><sub>cap</sub></td></tr></tbody></table>

The two dominant dimensionless parameters of the centrifugal ejection problem are the Galilei number $Ga$ (representing the ratio between centrifugal and viscous forces) and the Ohnesorge number $Oh$ (representing the ratio between viscous and surface tension forces), which can both vary over several orders of magnitude in the experiments, due to the variations of $η_{alg}$, $g_{tip}$ and $R_{cap}$. Traditionally, droplet ejection problems are described by a set of three dimensionless parameters: either Weber number $We$, Bond number $Bo$ and Ohnesorge number $Oh$, or Reynolds number $Re$, capillary number $Ca$ and Ohnesorge number.$^{37,38}$ In the case of the centrifugal system, since the centrifugal acceleration is at the origin of both the capillary flow and the droplet ejection, it can be shown that both $We$ and $Bo$ are determined by a combination of $Ga$ and $Oh$ and some other geometric parameters (see Appendix SM.3). These geometric parameters are constant in our system, as discussed above, except for $h_{res}$. Therefore, each simulated case will be determined by a combination of $Ga$, $Oh$ and $h_{res}^*$.

For the representation of the results, it is convenient to use the derived Reynolds number $Re_{th}$ in the capillary, because it compacts together the parameters $Ga$ and $h_{res}^*$. The derived Reynolds number in the capillary is defined as (demonstration in the Appendix SM.3):

$$Re_{th} = \frac{1}{8} \cdot \zeta \cdot Ga^2, \qquad (9)$$

with the composed geometric parameter $ζ$ accounting for the position and length of the needle, as well as for the height of the liquid in the reservoir:

$$ζ = \left(1 + \frac{h_{res}}{L_{cap}}\right) \left(1 - \frac{L_{cap} + h_{res}}{2γ_{tip}}\right). \qquad (10)$$

It is important to note that during the emptying of the reservoir, $ζ$ decreases due to the decrease of $h_{res}$ from the initial value $h_{res} = h_0$ to $h_{res} = 0$ when the reservoir is empty.

In conclusion, the results of the centrifugal extrusion problem in our system can be described in the space $Re_{th}-Oh$. The advantage of using the derived Reynolds number and not the Reynolds number $Re_{cap}$ (from Reference 5) based on the flow rate in the capillary, is that it can be derived immediately from the input parameters of the problem, without the need to compute nontrivial intermediate quantities such as the flow rate.

## 3.2 | Governing equations

The fluid flow problem is governed by the momentum and mass conservation equations for an incompressible two-phase flow with surface tension:

$$ρ(\partial_t u + u \cdot \nabla u) = -\nabla p + \nabla \cdot \tau + \gamma \kappa \delta_s n + \rho g_c(x), \qquad (11)$$

$$\nabla \cdot u = 0, \qquad (12)$$

where $ρ$ is the fluid density, $u = (u,v,w)$ is the velocity vector, $p$ is the pressure, $γ$ the surface tension coefficient, $κ$ is the local curvature of
<figure><img src="image_7.png" /><figcaption>FIGURE 6 (A) Simulation domain of the centrifugal extrusion problem in GERRIS. The green box represents the full model including the reservoir, whereas the violet box represents the simplified model without the reservoir. (B) Zoom in simulation domain at the tip of the ejection needle, that is, the region marked with a black dashed-line frame in (A). (C) Geometry and parametrization of the extrusion system used in Gerris. The numerical model is axisymmetric, therefore only half of the model presented in this figure is defined in Gerris. Here we show a full cross-section of the model for better visualization of the geometry.</figcaption></figure>

the interface, $\delta_s$ is a Dirac function representing the position of the interface, $\mathbf{n}$ is the normal vector at the interface and $\mathbf{g}_c$ is the centrifugal acceleration which depends, during stationary centrifuge rotation, on the position $\mathbf{x} = (x, y, z)$ as follows:

$$ \mathbf{g}_c = -\omega \times (\omega \times \mathbf{x}), \quad (13) $$

where $\omega = (0,0,\omega)$ is the rotation vector with constant rotation speed $\omega$. Centrifugal acceleration is taken into account as an external force, and the reference frame is inertial. This simplification of the way centrifugal acceleration is accounted for is needed for the use of two-dimensional axisymmetric simulations along the needle axis, that cannot include a non-axisymmetric force, as it is the centrifugal force in our system. The axisymmetric approximation is still representative for the quantities and phenomena analyzed in this article, as thoroughly discussed in Appendix SM.13.

The viscosity is described with the Carreau-Yasuda model in the formalism of a Generalized Newtonian Model (Appendix SM.10) to express the stress term:

$$ \tau = 2\eta(\ell_{2D})\tau = \eta(\ell_{2D})(\nabla\mathbf{u} + \nabla\mathbf{u}^T). \quad (14) $$

## 3.3 | The Gerris flow solver

In this article, we will use the open-source Gerris flow solver<fnref n="19" /> to solve the flow equations (11) and (12). This solver is described in more detail in Appendix SM.12.

## 3.4 | Simulation setup

After having defined the mathematical formulation of the flow problem and the general numerical framework used for the resolution, in this section several details are given with respect to the setup of the numerical model.

### 3.4.1 | Geometry and boundary conditions

The numerical domain considered is two-dimensional with an axisymmetry condition (Figure 6A, green dashed-line box) and includes the capillary, the reservoir, a region of surrounding air and a longer region downstream of the needle outlet where the droplet fall can be simulated over a sufficient fall height.

The geometry of the ejection system implemented in Gerris and its parametrization are represented in Figure 6C.## It is composed of:

* the capillary with inner radius $R_{\text{cap}}^*$, outer radius $R_{\text{out}}^*$ and length $L_{\text{cap}}^*$;

* a small entry region in the capillary with dimensions $R_{\text{step}}^*$ and $L_{\text{step}}^*$;

* the reservoir, assumed cylindrical with inner diameter $R_{\text{res}}^*$, height $h_{\text{res}}^*$ and wall thickness $e_{\text{res}}^*$.

Compared to the real system (Figure 1C), the geometry of the reservoir inner walls is simplified to a cylinder with a diameter equal to the mean diameter of the reservoir cone. Since the velocity in the

<footnote n="19">¹⁹https://gfs.sourceforge.net.</footnote>

<footnote>## Actually, only half of the model in Figure 6C is represented in Gerris, but the full cross-section was drawn in the figure for clearer visualization.</footnote>
<table><caption>TABLE 7 Summary of the experiments used as validation for the numerical results in this article, and their parameters.</caption><thead><tr><th>Exp. series</th><th>Exp. no.</th><th>Fluid</th><th>D<sub>cap,ref</sub> (mm)</th><th>ω (RPM)</th><th>h<sub>0</sub> (mm)</th><th>h<sub>fall</sub> (mm)</th></tr></thead><tbody><tr><td rowspan="5">Newtonian (NT)</td><td>1</td><td>PSU2400</td><td>0.330</td><td>705</td><td>7.4</td><td>—</td></tr><tr><td>2</td><td>PSU2400</td><td>0.330</td><td>1221</td><td>6.7</td><td>—</td></tr><tr><td>3</td><td>PSU2400</td><td>0.330</td><td>1865</td><td>6.7</td><td>—</td></tr><tr><td>4</td><td>PSU2400</td><td>0.410</td><td>1221</td><td>6.6</td><td>—</td></tr><tr><td>5</td><td>PSU2400</td><td>0.100</td><td>3925</td><td>6.7</td><td>—</td></tr><tr><td rowspan="9">Non-Newtonian (NN)</td><td>1</td><td>Alg. A 2%</td><td>0.330</td><td>1221</td><td>7.4</td><td>15</td></tr><tr><td>2</td><td>Alg. A 4%</td><td>0.330</td><td>1221</td><td>7.4</td><td>15</td></tr><tr><td>3</td><td>Alg. A 2%</td><td>0.330</td><td>705</td><td>7.4</td><td>15</td></tr><tr><td>4</td><td>Alg. A 2%</td><td>0.330</td><td>1931</td><td>7.4</td><td>15</td></tr><tr><td>5</td><td>Alg. A 2%</td><td>0.150</td><td>1221</td><td>7.4</td><td>15</td></tr><tr><td>6</td><td>Alg. A 2%</td><td>0.410</td><td>1221</td><td>7.4</td><td>15</td></tr><tr><td>7</td><td>Alg. A 4%</td><td>0.330</td><td>1221</td><td>2.6</td><td>15</td></tr><tr><td>8</td><td>Alg. A 4%</td><td>0.330</td><td>2442</td><td>7.4</td><td>15</td></tr><tr><td>9</td><td>Alg. A 1%</td><td>0.330</td><td>705</td><td>7.4</td><td>15</td></tr></tbody></table>

Notes: Alginate A is alginate from a batch used in previous experimental work by the authors, comprising the experimental series NN. Alginate B is alginate used in the experiments performed in this article, namely in the ER experimental series. $h_0$ is the initial height of the liquid column in the reservoir. $h_{fall}$ is the collecting distance in the experiments.

reservoir is much lower than in the needle (due to a diameter ratio of around 10 between reservoir and needle implying a factor $10^{-2}$ on the velocity ratio), the liquid in the reservoir is quasi-static and therefore only its hydrostatic pressure influences the flow at the time scale of the droplet formation and ejection. Therefore, the precise geometry of the reservoir walls does not play a role.

Furthermore, since the precise shape of the inner edge at the needle outlet is not known, a 90-degree edge was used in the numerical model.

As boundary conditions, we use an axisymmetric condition on the $x=0$ axis and free outlet conditions on the other three boundaries of the simulation domain.

### 3.4.2 | Parametrization of the numerical model

The parametrization of the simulations in Gerris (cf. Figure 6C) follows the guidelines from the dimensional analysis in Section 3.1. When experiments are reproduced in the simulation, the parameters (Table 7) are made dimensionless using $ρ_{alg} = 1008$, $||| g_{tip} = ω^2γ_{tip}$ and $R_{cap,ref}$.

The dimensionless lengths, radii and heights defining the geometry of the numerical model are obtained by dividing the system sizes from the experiments by the reference capillary radius in each experiment $R_{cap,ref}$. In particular, for the inner capillary radius, the value from the technical drawings from the manufacturer $R_{cap,tech}$ was used,³³*** so $R_{cap}^* = R_{cap,tech}/R_{cap,ref}$.

For the calculation of the dimensionless height of the reservoir $H_{reservoir}^*$, an approximated value of $H_{reservoir} = 9$ mm was used, a value with no influence on the solution as long as the liquid interface remains lower. For the same reason, the reservoir walls thickness has been maintained to a constant value $e_{res} = 0.1$ mm.

The following four dimensionless numbers:

$$ 
\begin{aligned}
Ga &= \frac{\rho_{alg} g_{tip}^{0.5} R_{cap,ref}^{1.5}}{\eta_{alg}} \\
Oh &= \frac{\eta_{alg}}{\sqrt{\rho_{alg} \gamma_{alg} R_{cap,ref}}} \\
r &= \rho_{alg} / \rho_{air} \\
m &= \eta_{alg} / \eta_{air}
\end{aligned}
$$

are used to determine the dimensionless fluid properties:

$$ \rho_l^* = 1. \qquad (15) $$

$$ \eta_{l,0}^* = 1/Ga. \qquad (16) $$

$$ \gamma^* = (\eta_{l,0}^*/Oh)^2. \qquad (17) $$

$$ \rho_g^* = 1/r. \qquad (18) $$

$$ \eta_g^* = \eta_{l,0}^*/m. \qquad (19) $$

Our experiments are††† in the asymptotic limit of $m > 1$, where the system is no longer sensitive to the value of $m$. In order to

<footnote>|||For simplicity, since the density varies very little among the alginate solutions in this article, we use a mean value of density of the alginate solutions (cf. Reference 31).</footnote>

<footnote>***This value was found in previous works to give more accurate predictions of flow rate than $R_{cap,ref}$. However, the uncertainty on the exact value of the capillary inner radius induces a significant uncertainty on the flow rate calculation, since the radius is present at the power four in the flow rate equation.⁵</footnote>

<footnote>†††The values in our experiments are typically of $m > 5000$, whereas a threshold of around 500 can been observed in the theoretical study in the Appendix SM.2.</footnote>
preserve the numerical simulation stability with the Gerris solver, m was set at 500 in all the simulations.

In the above equation, $\eta_{l0}^*$ represents the dimensionless zero-shear viscosity of the alginate solution. The viscosity function becomes thus:

$$ \eta_l^* = \eta_{l0}^* [1 + (\lambda_C \dot{\gamma})^a]^{\frac{n-1}{a}}, \quad (20) $$

where $\dot{\gamma}$ can be calculated from the dimensionless rate-of-strain tensor computed by the solver:

$$ \dot{\gamma} = \sqrt{2D_{ij}^* D_{ij}^*} \cdot \sqrt{\frac{g_{\text{tip}}}{R_{\text{cap}}}}. \quad (21) $$

In order to have a value of $g_{\text{tip}}^* = 1$ at the nozzle tip, the centrifugal force field will be imposed using the following function:

$$ g_y^*(y^*) = c \cdot (g_{\text{tip}}^* + (y^* - y_{\text{postip}}^*) \cdot \omega^{*2}), \quad (22) $$

where $y_{\text{postip}}^*$ is the position of the needle tip in the coordinate system of the simulation domain and $\omega^{*2} = \omega^2 R_{\text{cap,ref}} / g_{\text{tip}}$. By multiplying by the volume fraction c, the centrifugal acceleration is only applied to the liquid and not to the surrounding gas. This is done to avoid a spurious gas motion that is observed in the simulations when centrifugal acceleration is applied to the gas as well.

### 3.4.3 | Meshing

A dynamic mesh adaptation technique is used to refine the mesh around the interface between the two fluids (in cells with $0 < c < 1$) and at the solid boundaries, based on a quadtree strategy. A coarser mesh is used for cells only containing air, far away from the interface of the two fluids (see Figure 6B).

The standard mesh used for the validation simulations has a maximum refinement level 9 compared to the coarsest cells, and will be denoted “L9.” A mesh convergence study will be realized in order to ensure the independence of the numerical solution on the grid and to find the best compromise between accuracy and computational cost.

### 3.4.4 | Initialization

Since the transitory filling of the capillary with liquid is not the focus of this article, the numerical calculation is initialized in a configuration of filled reservoir and capillary, in order to reduce the simulated time. For numerical stability reasons, the centrifugal force field is imposed using a ramp function in time:

$$ g_y(y^*, t^*) = g_y(y^*) \cdot (t^* / t_{\text{ramp}}^*), \quad \text{if } t^* < t_{\text{ramp}}^*, \quad (23) $$

$$ g_y(y^*, t^*) = g_y(y^*), \quad \text{otherwise} \quad (24) $$

with $t_{\text{ramp}}^* \approx 100$.

## 4 | VALIDATION OF THE NUMERICAL MODEL

Before moving on to the simulation of centrifugal extrusion, a series of validation tests was carried out to evaluate the ability of the solver to accurately compute relevant flow quantities.

### 4.1 | Carreau-Yasuda viscosity model in Gerris

The first test is to validate the implementation of the constitutive equations implemented for alginate solutions in the numerical model. The studied configuration is a single-phase flow of a Carreau-Yasuda fluid in an infinite pipe (periodic boundary conditions in the streamwise direction) of radius $R = 165 \ \mu\text{m}$, with an applied constant volumetric force in the axial direction emulating the effect of a pressure difference between inlet and outlet for a finite-length pipe. The purpose of this test is to evaluate the ability of the Gerris solver to correctly describe the shear-thinning fluid behavior according to the Carreau-Yasuda model, in flow subjected to a volumetric gravity-like force $g_{\text{axial}} = 5297.4 \ \text{m/s}^2$. Several simulations are performed with the fluids Alg. B, at concentrations 1%, 2%, and 4% w/v (see Section 2 for properties), with water ($\rho = 997 \ \text{kg/m}^3$, $\eta = 1.005 \ \text{mPa}$) and with standard oil PSU2400 at $20^\circ$ ($\rho = 875 \ \text{kg/m}^3$, $\eta = 1.435 \ \text{Pa s}$).

Comparison is made with analytical solutions obtained by numerical integration of the flow equations in radial direction, as described in Reference 5, using a standard ODE solver.‡‡‡ In Figure 7, the velocity profiles from the CFD simulation are compared with analytical solutions. An excellent agreement is achieved, confirming the ability of Gerris to accurately incorporate shear-thinning fluid behavior described by the Carreau-Yasuda model.

### 4.2 | Validation experiments on the CME device

Further validation of the numerical centrifugal extrusion model is performed by comparison with results of capsule production experiments. Here we describe briefly the different series of validation experiments realized in this study.

The experimental method for microcapsule production has been described in detail in previous work by the authors,³,³¹ and here only an overview will be given. The biopolymer solution is placed in the reservoir. During centrifugation, it flows through the capillary and is ejected under the form of droplets (or jet depending on the operating

<footnote>‡‡‡Adapted from Reference 65.</footnote>
<figure><img src="image_8.png" /><figcaption>FIGURE 7 Velocity profiles in a capillary with imposed pressure gradient, containing a non-Newtonian Carreau-Yasuda fluid with the properties of Alg B (1%, 2% and 4%) or a Newtonian fluid (water or PSU2400 oil). Comparison between numerical simulations with Gerris and theoretical calculations.</figcaption></figure>

parameters), falling through the air gap and entering into the gelation solution, where the biopolymer solution cross-links by a chemical reaction, resulting in the formation of capsules (or fibers) which sediment on the bottom of the centrifuge tube.

A summary of the parameters used in each experiment is given in Table 7.

A first set of experiments is devoted to the validation of simulations representing a Newtonian fluid. Experiments NT1-NT5$^{§§§}$ were performed with the aforementioned PSU2400 oil, having a Newtonian behavior in the operating ranges. In these experiments, the flow rate was measured, but droplet sizes could not be measured, since visualization is not possible in the experimental device and the oil droplets do not cross-link to form capsules.

A second set of experiments will be used to validate the simulations of centrifugal extrusion with a non-Newtonian fluid (alginate). For this, we will use results from experiments NN1-NN9,$^{¶¶¶}$ that were performed in previous work by the authors (Appendix S.1 from$^{5}$), where both flow rates and droplet volumes were measured. The procedure for the measurement of the flow rate has been described in the supporting information S.1 of.$^{5}$ The measurement of droplet sizes was performed according to the procedure described in Reference 5, where diameters of the gelled capsules are measured by microscopic image analysis at the end of the centrifugation process (after a hardening time of 10 min) and converted into the droplet diameters using the “shrinkage factor” $k_{SF}$, which depends on the concentrations of the alginate and gelation solutions, to compensate for the capsule shrinking due to polymerization and subsequent water ejection.

In the following, the comparison of the numerical and experimental results for these cases is discussed.

## 4.3 | Droplet ejection with Newtonian fluids

A second series of tests is performed on the geometry of the centrifugal droplet ejection system, using the parameters of experiments NT 1-5 (centrifugal ejection of Newtonian fluid) described in Section 4.2. The parametrization of the numerical model is performed according to section *Parametrization of the numerical model*, with the simplification that here the viscosity of the alginate solution is constant and equal to $\eta_l^*$.

The standard mesh refinement of level L9 is used. Unlike the most common cases in fluid dynamics simulations, in our simulations we do not impose a flow rate since it is not a priori known. Rather, we need to impose a volumetric force representing the centrifugal acceleration. Knowing the flow rate is important because it allows an estimation of the quantity of biopolymer that can be processed in a given amount of time. We need then to compute the flow rate of the alginate solution in the simulation. In this article, we compute the flow rate by integrating the velocity profile in the capillary, at an axial position close to the capillary outlet (at a distance of about $L_{cap}/25$ before the outlet):

$$Q^* = \int_{0}^{R_{cap}} u_{y}^{*} \cdot 2\pi x dx. \quad (25)$$

In Figure 8A, numerically computed flow rates are compared with experimental measurements. The values are normalized by the

<footnote>$^{§§}$NT stands for “Newtonian.”</footnote>

<footnote>$^{¶¶}$NN stands for “non-Newtonian.”</footnote>
<figure><img src="image_9.png" /><figcaption>FIGURE 8 Results of the validation computations with GERRIS. (A) Flow rates in centrifugal extrusion with a Newtonian fluid. Comparison with experimental values. (B) Droplet volumes and time evolution of flow rate in centrifugal extrusion with a Newtonian fluid. Comparison with analytical estimations. (C,D) Flow rates and droplet volumes in centrifugal ejection with a non-Newtonian fluid. Comparison with experimental values.</figcaption></figure>

measured flow rates in each experiment. The relatively large error bars on the flow rate calculation are caused by the uncertainty on the inner capillary diameter, as discussed in detail in the Appendix SM.1, since flow rate depends on the fourth power of the capillary diameter. A series of measurements of the inner diameter of several needles with the same reference diameter were performed and it was found that the values are disperse between $D_{cap.tech}$ and $D_{ref.tech}$, but closer to the technical diameter $D_{cap.tech}$. That is why all the simulations in this article were performed using the value of $D_{cap.tech}$ as capillary inner diameter (extracted from CAD drawings available from the manufacturer, EFD Nordson, General Purpose Stainless Steel Tips³³). Since we chose the higher limit in the range of possible diameters, this is probably the reason why the numerical calculation in most cases falls above the experimental mean flow rate. The relative error between simulation and experiment lies below 15% for all cases.

Furthermore, a test simulation is run using the same conditions as experiment NN9 (centrifugal ejection of non-Newtonian fluid, i.e., alginate), but with a Newtonian fluid behavior (the viscosity of the liquid is constant and equal to the zero-shear viscosity of the alginate used in the original experiment NN9, $η = 0.104 Pa.s$. For this case, we compare numerical results with analytical estimations of flow rate.

The flow rate in the capillary over time for the same case is represented in Figure 8B and compared with an analytical solution calculated with the Poiseuille equation for the capillary inlet region (subscript “step”) and the capillary (subscript “cap”):

$$Q_v = \frac{\Delta p \cdot \pi}{128\eta \left( \frac{L_{step}}{D_{step}^4} + \frac{L_{cap}}{D_{cap}^4} \right)}, \quad (26)$$
with $Δp$ being the difference between the pressure at the capillary inlet $p_1$ and the pressure in the liquid at the capillary outlet $p_2$, and it can be estimated according to Reference 5 as follows:

$$ \Delta p = \rho_{\text{alg}} \omega^2 (y_1 - h_{\text{res}}(t)/2) \cdot h_{\text{res}}(t) - \frac{\gamma_{\text{alg}}}{R_{\text{cap}}}, \quad (27) $$

with $h_{res}(t)$ the height of the liquid column in the reservoir as a function of time, $\omega$ the centrifuge rotation speed and $y_1$ the position of the inlet of the capillary in the coordinate system of the centrifuge.

In the numerical solution computed with Gerris, the typical evolution of the flow rate in the CME system can be noticed when ejecting droplets (Figure 8B): the short ascending region $t^* < t^*_{ramp}$ at the beginning corresponds to the transient part in the simulation due to the ramp-like initialization of the centrifugal acceleration ($t^*_{ramp} \approx 100$). After this initial transient, periodic alternations of (smooth) growth and (sudden) drop regions, corresponding to the formation of droplets at the capillary tip and to their detachment, respectively, can be seen.

A rolling average can be extracted from the fluctuating flow rate and compared with analytical estimations. From this comparison it can be concluded that Gerris predicts correctly the pressure drop through the emptying of the reservoir and thus the decrease in flow rate, as predicted by the analytical solution.

## 4.4 | Droplet ejection with NN fluids

A further series of validation simulations was carried out on the configurations NN1-9 (centrifugal extrusion of non-Newtonian fluid, i.e., alginate), using a Carreau-Yasuda fluid model with the parameters corresponding to the batch A alginates in Table 2. An L9 mesh is used for these simulations.

In Figure 8C,D, CFD and experimental flow rates and droplet volumes### are compared, whereas each time normalization is performed using the experimental values. Generally good agreement is observed, whereas experimental uncertainty is relatively high due to the manufacturing tolerances on the inner needle diameter (Appendix SM.1). The slight overestimation of the quantities in the CFD simulations is also probably due to the usage of the technical diameter from the manufacturer CAD drawings, as discussed in Section 4.3.

These results confirm the validity of the non-Newtonian model and its ability to quantitatively describe the centrifugal extrusion problem.

## 4.5 | Mesh convergence study

The previously discussed validation simulations were run using a L9 refinement level. A grid convergence study is carried out using three different mesh refinement levels: L8, L9, and L10. A L11 refinement level would need a computing time of around 16–20 weeks and has

<table><caption>**TABLE 8** Characteristic sizes of the three meshes used in the grid convergence study, and resulting maximum timesteps for the simulation of case NN1.</caption><thead><tr><th>Mesh</th><th>L8</th><th>L9 (standard)</th><th>L10</th></tr></thead><tbody><tr><td># cells / R<sub>cap</sub></td><td>~ 6</td><td>~ 12</td><td>~ 24</td></tr><tr><td># cells total</td><td>~ 11K</td><td>~ 29K</td><td>~ 89K</td></tr><tr><td>dt<sub>max</sub></td><td>1.25e-2</td><td>4.4e-3</td><td>1.6e-3</td></tr></tbody></table>

<table><caption>**TABLE 9** Results of the grid convergence study on case NN1.</caption><thead><tr><th>Mesh</th><th>V<sup>*</sup><sub>drop</sub></th><th>Q<sup>*</sup><sub>V</sub></th><th>T<sup>*</sup><sub>drop</sub></th><th>SF<sub>y5</sub></th><th>Δy<sub>f5</sub> (mm)</th></tr></thead><tbody><tr><td>L8</td><td>14.20</td><td>71.9e-3</td><td>197.8</td><td>0.094</td><td>1.52</td></tr><tr><td>L9</td><td>11.84</td><td>69.3e-3</td><td>173.5</td><td>0.138</td><td>1.96</td></tr><tr><td>L10</td><td>11.23</td><td>69.9e-3</td><td>161.4</td><td>0.121</td><td>1.81</td></tr></tbody></table>

Abbreviations: $Δy_{f5}$, axial position of the fourth droplet front when the ligament radius equals 5% of the capillary radius.; $Q_V^*$, average flow rate after the initialization ramp; $SF_{y5}$, sphericity factor of fourth drop; $T_{drop}^*$: period between third and fourth drop; $V_{drop}^*$, volume of the fourth ejected droplet.

therefore not been considered in the present work. This mesh study is performed on the NN1 configuration, that corresponds to the centrifugal extrusion of a non-Newtonian fluid (Alginate A, 20g/L). Resulting numbers of cells and maximum timestep values in the simulations are reported in Table 8.

Quantitative results of the mesh convergence study are given in Table 9. The quantities that are considered in the comparison are the mean flow rate $Q_V^*$, the volume of the fourth### ejected droplet $V_{drop}^*$, the production period $T_{prod}^*$ (time between third and fourth droplet), the sphericity factor of the fourth ejected droplet $SF_{y5}$ when the droplet front reaches a distance of 5 mm from the needle tip, and the position $Δy_{f5}$ of the droplet front at ejection when the radius of the fluid ligament is 5% of the capillary inner radius. In numerical simulations, global quantities as the flow rate are rather easy to be predicted accurately, while local ones—as interface position or droplet shape—are much more challenging.

If the simulation with the finest mesh L10 is taken as reference, we can observe that both L8 and L9 meshes predict the flow rate within a few percent (less than 1% for L9) of the value calculated with the L10 mesh. For droplet sizes, L8 overpredicts by more than 25%, whereas L9 is still within around 5% accurate with respect to the reference L10. Since the flow rate is equal to the product between droplet volume and production frequency (cf. Reference 5), the L8 calculation shows a large error on droplet production frequency, whereas L9 is still within about 7.5% accurate. The droplet shape is more delicate to assess quantitatively. The variation of both parameters $SF_{y5}$ and $Δy_{f5}$ is not monotonous with mesh refinement, but seems to converge toward the value obtained with the L10 mesh. Although the results obtained with the L9 mesh are closer to the reference values obtained with L10 than the results

<footnote>### Droplets were not obtained in all the simulations because of the very long simulation time. Especially in the simulations with low $g_{tip}$, high alginate viscosity and low capillary diameter $R_{cap}$, the simulation time needed until the formation of droplets is very long. For practical reasons, the simulations were stopped before the ejection of droplets.</footnote>

<footnote>### The first droplets may be sensitive to the initialization of the problem. We use the fourth droplet in order to make sure that we are in a periodic regime. In practice, in our simulations drop volumes are stable starting from the third generated droplet.</footnote>
<figure><img src="image_10.png" /><figcaption>FIGURE 9 (A-C) Numerical results of the mesh convergence study: detachment of second droplet from the capillary in case NN1 with the three mesh refinements (A) L8, (B) L9 and (C) L10. (D) Experimental results of case NN1 (products observed in the gelation solution at the end of the process and hardening time).</figcaption></figure>

obtained with the L8 mesh, the deviation from the reference values remain relatively high (14% and 8%, respectively). This could mean that the mesh appears to be converged at the level L9 with respect to the calculation of flow rate and droplet sizes, but not entirely converged with respect to the calculation of droplet shapes.

A qualitative comparison of the droplet ejection with the three different meshes for case NN1 (centrifugal ejection of alginate droplets) is given in Figure 9, including a comparison with experimental results. While mesh L9 produces a droplet of a very similar shape and size to that produced using L10, the satellite droplet produced with the mesh L9 is at the limit of numerical resolution (only around 3-4 cells of pure liquid inside the droplet, compared to around 15 for the L10 mesh). Also, the refinement in mesh L8 is not enough to capture the formation of satellite droplets. Their presence in the gelation solution has been confirmed by the experiments. According to the simulation with the refined mesh L10, they are formed during the detachment of the main droplet and the rupture of the fluid ligament connecting it to the capillary.

The following conclusions can be retained from these results:

* All the studied properties converge with mesh refinement and the differences between L9 and L10 meshes are small.

* Mesh L8 is sufficient for a relatively accurate flow rate calculation, but the prediction of ejected droplet properties, as volume or sphericity factor, is affected by significant errors, over 10%. No satellite droplets formation is captured using an L8 refinement level.

* Mesh L9 provides accurate flow rate calculation and a relatively accurate prediction of droplet volume and production period (errors of around 5% and 7.5%, respectively). However, the prediction of droplet and interface shapes in affected by higher deviation. Satellite droplets formation is hardly captured using L9 refinement, at the limit of mesh resolution.

* Mesh L10 was used as reference for the calculation of flow rate and droplet properties. The formation of satellites, confirmed by

experimental results, is well resolved (15 or more cells inside a satellite droplet) using this mesh refinement.

These conclusions justify the following strategies used in the simulations in this article:

* For the validation cases NT1-5 and NN1-9, where only quantities like flow rate and droplet volume were considered, the standard mesh refinement L9 is sufficiently accurate.

* For the study of ejection regimes it is necessary to precisely capture the interface during droplet rupture in order to conclude on the formation of satellite droplets or filaments. Therefore, mesh refinement L10 is more appropriate for simulations ER1-12.

* For simulations that are computationally very costly (high viscosity, implying long physical simulation time, and low capillary diameter, implying high spatial resolution), we use an L8 refinement in the capillary to compute the flow rate and pendant drop formation, and only apply an L10 refinement at the interface when the pendant droplet is formed and during detachment and ligament rupture. This allows a significant reduction of the simulation time.

## 4.6 | Simplified geometry

Simulations with smaller capillary diameters, low centrifugal acceleration and/or high alginate viscosities require very long computational times.

For these cases, a simplified geometry (Figure 6A, violet box) is defined, containing only the capillary, a small lateral space with surrounding air and the fall height of the droplet. Eliminating the reservoir reduces the mesh size, thus significantly reducing the computational cost. In this model, the pressure at the capillary inlet is imposed via a hydrostatic
<table><caption>TABLE 10 Summary of the cases ER01-ER12 for the study of centrifugal ejection regimes.</caption><thead><tr><th>Exp. series</th><th>Exp. no.</th><th>Fluid</th><th>D<sub>cap,ref</sub> (mm)</th><th>ω (RPM)</th><th>Experimental product</th></tr></thead><tbody><tr><td rowspan="12">Ejection regime (ER)</td><td>1</td><td>Alg. B 1%</td><td>0.610</td><td>1221</td><td>Thick fiber, few irregular polydisperse capsules</td></tr><tr><td>2</td><td>Alg. B 1%</td><td>0.410</td><td>1221</td><td>Medium-thick fiber, polydisperse capsules</td></tr><tr><td>3</td><td>Alg. B 1%</td><td>0.200</td><td>1221</td><td>Monodisperse ellipsoidal capsules and satellites</td></tr><tr><td>4</td><td>Alg. B 1%</td><td>0.100</td><td>1221</td><td>Monodisperse ellipsoidal capsules</td></tr><tr><td>5</td><td>Alg. B 2%</td><td>0.610</td><td>1931</td><td>Medium-thick fiber, polydisperse capsules with and without filament</td></tr><tr><td>6</td><td>Alg. B 2%</td><td>0.410</td><td>1931</td><td>Thin fiber, polydisperse capsules with filament</td></tr><tr><td>7</td><td>Alg. B 2%</td><td>0.200</td><td>1931</td><td>Monodisperse ellipsoidal capsules with tiny tip and satellites</td></tr><tr><td>8</td><td>Alg. B 2%</td><td>0.100</td><td>1931</td><td>Monodisperse ellipsoidal capsules</td></tr><tr><td>9</td><td>Alg. B 4%</td><td>0.610</td><td>2730</td><td>Fiber with periodic “bumps”, polydisperse capsules with and without filament</td></tr><tr><td>10</td><td>Alg. B 4%</td><td>0.410</td><td>2730</td><td>Capsules with filament</td></tr><tr><td>11</td><td>Alg. B 4%</td><td>0.200</td><td>2730</td><td>Monodisperse (slightly) tear-shaped capsules with satellites</td></tr><tr><td>12</td><td>Alg. B 4%</td><td>0.100</td><td>2730</td><td>Monodisperse ellipsoidal capsules with tiny tip</td></tr></tbody></table>

Notes: The initial height of the liquid column in the reservoir $h_0$ was equal to 6.7 mm in all the 12 experiments. The collecting distance $h_{fall}$ was equal to 5 mm in all the 12 experiments. The last column in the table lists the centrifugation products obtained experimentally.

assumption to take in account the reservoir. All other boundary conditions remain unchanged.

Validation of the simplified geometry is performed by comparison with the full geometry for one of the simulated configurations (case NN1 at refinement level L9). The results, summarized in Appendix SM.5, show very good agreement (less than 5% difference for all the analyzed indicators) between the simplified and full models, thus validating the use of the simplified geometry.

## 5 | NUMERICAL RESULTS – EJECTION REGIMES

The CFD model being validated, it can be used to investigate some major hypotheses made in previous analytical and experimental work³,⁵ regarding the formation of microcapsules by centrifugation, and to gain a better insight into the fluid dynamics of the process.

A first issue concerns the identification of the process stage in which secondary products, that is, products other than capsules: fibers, filaments, satellite droplets, are formed.

A second issue is the occurrence in a single experiment of different centrifugation products, for certain values of the control parameters. Common combinations are: capsules + satellites, fiber + capsules, capsules + filaments + satellites.

In order to investigate these issues, we will use the following approach: both experiments and simulations are performed for 12 cases, denoted ER01 to ER12.⁴⁻⁶ The parameters of these cases are summarized in Table 10. The variation of the three parameters $n_{alg}$, $D_{cap}$, and $\omega$ results in the variation of two dimensionless numbers, $Re_{th}$ and $Oh$. The combined analysis of these results will allow to verify previous hypotheses and to assess the ability of the numerical simulation to reproduce the different ejection regimes. For an accurate

comparison between experiments and simulations, the experiments are realized with the same alginate solutions which have been previously characterized rheologically.

In the following sections, the link between the formation of different types of centrifugation products and the ejection regimes is discussed.

### 5.1 | Formation of fibers

Fibers are elongated and continuous volumes of extruded polymer, generally with a locally cylindrical aspect. They are observed in a gelled condition in the gelation solution, most often as a long entangled thick “wire.” Experimentally, the formation of fibers is characteristic for high $Ga^2 \cdot \zeta$ values, which can be obtained by using high rotation speeds, low viscosity alginates or high capillary diameters. In the analysis of the experimental results, fibers are always accompanied by some other products that are present in the gelation solution, such as (mostly polydisperse) capsules with or without an attached filament, satellites or fiber fragments, as it can be seen in subfigures (A)–(C) of the Figures 10–12. The fiber obtained in experiment ER09 presents significant variations of the thickness of the fiber can be observed, under the form of pulsations along the fiber longitudinal axis (see Figure 12A,C).

For all the three cases ER1, ER5, and ER9, numerical simulations were performed in two configurations: with a full ($\zeta \approx 0.94$) and with an empty ($\zeta \approx 0.77$) reservoir. Those two cases will be referred to as full reservoir configuration, frc, and empty reservoir configuration, etc.

For case ER01, the results are presented in Figure 10D,E.††† In the frc, it can be seen that the polymer solution is extruded under the

<footnote>†††If not mentioned otherwise, in all the numerical results reported here, the droplet shown corresponds to a periodic extrusion regime starting two or more drops after the simulation initialization, that can impact the first few generated drops.</footnote>

<footnote>***ER stands for “ejection regime.”</footnote>
<figure><img src="image_11.png" /><figcaption>**FIGURE 10** Results for the ER1 case. (A–C) Experimental. The fiber in (A) is the predominant product, whereas a few polydisperse and irregular capsules (B) and (C) are found along with the fiber in the gelation solution. (D,E) Numerical results showing the interface between the polymer solution and air at different times (starting from the beginning of the stationary condition $g_{tip}(t) = g_{tip}$), full reservoir configuration and empty reservoir configuration (E). The gelation bath is not part of the simulation and was drawn during post-processing for purpose of comparison with the experimental results.</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>**FIGURE 11** Results for the ER5 case. (A–C) Experimental. The fiber in (A) is the predominant product, whereas some capsules with filament, polydisperse smaller droplets and satellites are also found along with the fiber in the gelation solution (B,C). (D,E) Numerical results showing the interface between the polymer solution and air at different times (starting from the beginning of the stationary condition $g_{tip}(t) = g_{tip}$), full reservoir configuration (D) and empty reservoir configuration (E). The gelation bath is not part of the simulation and was drawn during postprocessing for purpose of comparison with the experimental results.</figcaption></figure>
<figure><img src="image_13.png" /><figcaption>**FIGURE 12** Results for the ER9 case. (A)–(C) Experimental. The fiber in (A) is the predominant product, but some capsules with filament, fiber fragments, polydisperse smaller capsules and satellites are also found along with the fiber in the gelation solution (B). The fiber presents variations of thickness along the chord (C). Numerical results showing the interface between the polymer solution and air at different times (starting from the beginning of the stationary condition $g_{tip}(t) = g_{tip}$), full reservoir configuration (D) and empty reservoir configuration (E). The gelation bath is not part of the simulation and was drawn during postprocessing for purpose of comparison with the experimental results.</figcaption></figure>

form of a jet that has a stable diameter over time at the level of virtual contact with the gelation solution. This expected result agrees with the hypothesis made previously in the literature$^{7,19}$ that the mechanism of fiber formation is the ejection of a jet which gels upon entry in the gelation solution, resulting in a long continuous fiber. The relative stability of the jet diameter over time is in good agreement with the characteristics of the fiber obtained experimentally, which features a roughly constant or a very gradually varying diameter. In the erc, first a jet with a smaller diameter than in the frc is obtained. This is probably due to a decrease in the flow rate by the emptying of the reservoir. At some point in time, air bubbles start to enter the capillary from the emptying reservoir. When the air bubbles reach close to the tip of the ejection needle, the liquid is no longer ejected under the form of a jet, but under the form of irregular-sized droplets, as the air bubbles together with liquid on the inner walls of the capillary reach the needle tip. The simulation offers a unique insight into this highly nontrivial and unexpected phenomenon, and helps to understand the formation of these by-products in jetting regimes and to attribute their origin to the transient regime at the very end of the capillary emptying. For regime classification purposes, we still consider that the ejection regime in the erc is a jetting regime.

The numerical results for case ER05 (Figure 11D,E) show a similar ejection behavior as in case ER01. In the frc, the polymer is extruded under the form of a stable jet, thinner than the one produced in case ER01, as is the case in the experimental results too. In the erc, when air bubbles enter the capillary, a succession of different droplets are ejected, among which droplets with filament (the red contour in

Figure 11E) and polydisperse droplets without filament (the blue contour in Figure 11E). These results are in very good agreement with the experiments, where both capsules with filament, capsules without filament of different sizes and satellites are observed in the gelation solution besides the fiber (Figure 11B,C).

Case ER09 shows a pulsating oscillation of the jet thickness for both frc and erc: regions with significantly higher jet thickness are formed at the needle tip and then advected downstream (Figure 12D,E). Nevertheless, both in the frc and erc the jet is continuous and there is no rupture between the ejection tip and the virtual position of the interface of the gelation solution. In the frc, the amplitude of the thickness oscillations decreases over time, and the diameter of the jet stabilizes. The experimental results of case ER09 (Figure 12) are particularly interesting, because they present a wide variety of different features (irregular fibers with varying thickness, polydisperse capsules with and without filament). Although the experimental results present more complexity than what is captured in the simulations (non-axisymmetric bumps, curved filaments, etc., reasonably due to the Coriolis force and to the impact with the gelation solution), all the features of the numerical results that were discussed are in very good agreement with the experimental observations. The erc simulation of case ER09 (Figure 12E) shows at first the ejection of a thin jet, similar to the one found at the end of the simulation with full reservoir. Then, as air bubbles enter the capillary and reach close to the outlet, the ejection regime changes. First, individual droplets with a long attached filament are ejected (e.g., red contour in Figure 12E). Later, individual droplets without filament are ejected in a quasi-
<figure><img src="image_14.png" /><figcaption>**FIGURE 13** Results for the ER10 case. (A,B) Experimental. Abundant capsules with filament are observed in the gelation solution. The length of the filaments varies randomly, but the overall aspect of the capsules is relatively uniform. Some filament fragments or satellites are also noticed in the gelation solution. (C,D) Numerical results showing the interface between the polymer solution and air at different times (starting from the beginning of the stationary condition $g_{tip}(t) = g_{tip}$), with a full reservoir (C) and an empty reservoir (D). The gelation bath is not part of the simulation and was drawn during postprocessing for purpose of comparison with experimental results.</figcaption></figure>

periodic regime, for the whole remaining simulation time (the simulation did not reach a complete emptying of the capillary). This ejection regime found at the end of the emptying of the reservoir is probably the explanation for the individual capsules found experimentally in the gelation solution.

It is important to notice that the formation of fibers is sensitive to the collecting distance, that is, the distance between the gelation bath and the ejection needle. If this distance is increased, jet breakup might occur before reaching the gelation solution and a different type of products might be obtained (fiber fragments and highly polydisperse capsules, see Reference 3). However, focusing on the ejection regimes, this topic is not investigated in more depth here.

## 5.2 | Formation of capsules with filament

Capsules with filament differ from fibers in that they represent discrete (discontinuous) products where the capsules are clearly identifiable as the bulk of the extruded polymer volume, featuring an attached gelled cord (filament) with a certain length. Experimentally, the region of filament formation is comparatively smaller than that of fiber or satellite formation (Figure 18), and it occurs generally at medium to large viscosities of the alginate solution (although very rare cases of filament formation with low-viscosity alginates at 1% concentration have been reported in Reference 31), at $Ga^2 \cdot \zeta$ values lying between those of fiber formation and those of satellite formation.

In Figure 13A,B we present experimental results for case ER10, where capsules with filament are found as products of the centrifugal extrusion, accompanied by filament fragments or satellites.

In this case, no other major centrifugation product is found, meaning that the regime of the centrifugal extrusion remains constant throughout the experiment.

The numerical results show a very good agreement with the experimental data. In both situations frc and erc for case ER10 (Figure 13C,D), we observe the formation of distinct droplets which detach from the capillary tip and form a long and thin ligament that still connects them to the bulk of the fluid in the capillary, until they reach the gelation solution. In the absence of the gelation solution, this filament breaks at several points and forms a polydisperse population of fragments and satellite droplets. By comparison with the experimental results we can conclude that the effect of the gelation bath is to preserve the droplet and the attached lower part of the ligament, whereas the upper parts of the ligaments break up and fall as fragments into the gelation solution. This explains the formation of capsules with filament accompanied by filament fragments and satellites.

Finally, we mention that here also, the formation of capsules with filament is sensitive to the collecting distance in the experiments. If the gelation bath were located at a larger distance from the ejection needle, the ligament might break and form satellites instead.

## 5.3 | Formation of satellites

Satellite droplets can be described as droplets with a diameter much smaller in size than the main capsules (typically five times smaller or less). In gravitational dripping systems, the formation of satellite droplets has been visualized experimentally and modeled analytically, and
<figure><img src="image_15.png" /><figcaption>**FIGURE 14** Results for the ER3, ER7, and ER11 cases. (A–C) Experimental. In all three cases, monodisperse capsules accompanied by satellite droplets are observed in the gelation solution. In all three cases, the number of satellites observed is comparable with the number of main capsules. (D,E) Numerical results for case ER3 with a full reservoir (D) and an empty reservoir (E). (F,G) Numerical results for case ER7 with a full reservoir (F) and an empty reservoir (G). (H) Numerical results for case ER11 with a full reservoir (H).</figcaption></figure>

is attributed to the retraction of the fluid ligament resulting from the detachment of a droplet from the bulk fluid in the capillary and subsequent capillary instability.⁶⁶ Eral et al. also proposed an analytical modeling of the formation of satellite droplets in centrifugal extrusion and validated the model with experimental centrifugal capsule formation results.³⁶ Experimentally, the region of satellite formation spreads across all the *Oh* numbers tested, for $Ga^2 \cdot \zeta$ values lower than those of fiber or filament formation (Figure 18).

The formation of satellites in cases ER3, ER7, and ER11 is shown in Figure 14. Experimentally, in each of the three cases a monodisperse population of main capsules is obtained, accompanied by a roughly equal number of satellites. The shape of the capsules changes gradually from an ellipsoidal shape (Figure 14A) to a tear-like shape (Figure 14B,C) as the viscosity of the polymer solution increases.

In the corresponding numerical simulations, we can observe in detail the mechanism of satellite formation at droplet detachment. The pending droplet detaches from the capillary tip and forms a very thin and relatively short ligament connecting it to the bulk fluid in the

capillary. This thin ligament breaks in two or several points, both on the side of the droplet and on the side of the capillary and retracts into one or several small droplets. A very similar behavior is found for the frc (Figure 14D,F,H) and the erc (Figure 14E,G). The simulations also capture the evolution of the droplet shape with increasing polymer viscosity. While in case ER3 the main droplet is perfectly spherical when reaching the gelation solution, in simulation ER7 the droplet is slightly elongated on the back side, and in simulation ER11 the droplet is clearly tear shaped when reaching the interface of the gelation bath.

## 5.4 | Formation of capsules without satellites

In cases with sufficiently low rotation speed or low capillary diameters, the formation of capsules without secondary products (particularly without satellites) is observed experimentally. Examples of experimental results from cases ER4, ER8, and ER12, where capsules
<figure><img src="image_16.png" /><figcaption>**FIGURE 15** Results for the ER4, ER8, and ER12 cases. (A–C) Experimental. In all three cases, monodisperse capsules are observed in the gelation solution, with a very low number of satellites in case ER4 and no satellites in cases ER8 and ER12. In case ER12, the dark dots seen in the background are due to impurities and scratches in the plastic material of the Petri dish. (D,E) Numerical results for case ER4 with a full reservoir (D) and an empty reservoir (E). (F,G) Numerical results for cases ER8 and ER12 respectively, both with a full reservoir.</figcaption></figure>

without satellites are formed, are shown in Figure 15A-C. Again, the shape of the capsules changes gradually from an ellipsoidal shape to a tear-like shape with increasing polymer viscosity.

For all three cases ER4, ER8, and ER12, not according to expectations, the corresponding numerical simulations with full and empty reservoir (Figure 15D-G) show the formation of very small satellite droplets. Most of the time, these satellites present less than 10 cells with *c* = 1 inside the droplet, for instance four cells in the case of ER8. This is different from the simulations ER3, ER7, and ER11 presented in Section 5.3, where 15 or more cells are present inside each satellite droplets, indicating a better resolution of these droplets. One possible explanation for the formation of satellites in simulations ER4, ER8 and ER12 is that these satellites are numerical artefacts due to the spatial resolution which is close to the size of the satellite droplets. It could be imagined that a higher resolution would allow the filament to break differently, possibly avoiding the formation of satellite droplets in these regimes. Also, tiny satellite capsules may be present in the experiments, but difficult to visualize. This point deserves further investigation, which is beyond the scope of the numerical study presented here.

## 5.5 | Transitions between regimes

As mentioned in Section 3.1, the height $h_{res}$ of liquid in the reservoir changes during one centrifugation experiment due to the emptying of the reservoir, which results in a change of the ejection conditions:

$Re_{th}$ will decrease with the decrease in the geometric parameter $ζ$. In our experiments, it is not possible to dissociate the products generated at the beginning from those generated at the end of one same centrifugation run. However, several experiments featured results combining two different types of products, which are generated in different ejection regimes according to the results discussed in Sections 5.1, 5.2, and 5.3, but which are found together in abundance in the gelation bath at the end of the centrifugation.†††† For example, in case ER6 (Figure 16A-C), a thin fiber is obtained experimentally along with numerous capsules with filament. As well, in case ER2 (Figure 17A-C), a fiber is obtained along with numerous capsules and satellites of various sizes.

Numerically, it is possible to dissociate between the ejection regime at the beginning of the centrifugation run (frc) and toward the end of the centrifugation run (erc), and this can represent a valuable tool to validate the hypothesis that the different centrifugation products obtained experimentally in the same run result from a change in ejection regime during centrifugation.

Figure 16D,E shows results of numerical simulations for case ER6 (simulation of centrifugal ejection of alginate solution of medium concentration, with a relatively high nozzle diameter) with frc and erc, respectively. At the beginning of the centrifugation, a transient regime is observed where few capsules with filament are

<footnote>††††Unlike in the cases presented in Section 5.1, where only few capsules are found along with the fiber in the gelation solution.</footnote>
<figure><img src="image_17.png" /><figcaption>**FIGURE 16** Results for the ER6 case. (A–C) Experimental. Along with the long fiber in (A), relatively abundant capsules with filament are observed in the gelation solution, along with some satellite droplets (B,C). The sizes of the capsules with filament vary, but remain roughly in the same order of magnitude. (D,E) Numerical results showing the interface between the polymer solution and air at different times (starting from the beginning of the stationary condition $g_{\text{tip}}(t) = g_{\text{tip}}$), with a full reservoir (D) and an empty reservoir (E). The gelation bath is not part of the simulation and was drawn during postprocessing for purpose of comparison with experimental results.</figcaption></figure>

<figure><img src="image_18.png" /><figcaption>**FIGURE 17** Results for the ER2 case. (A–C) Experimental. Along with the long fiber in (A), relatively abundant polydisperse capsules are observed in the gelation solution, along with some satellite droplets (B,C). (D,E) Numerical results showing the interface between the polymer solution and air at different times (starting from the beginning of the stationary condition $g_{\text{tip}}(t) = g_{\text{tip}}$), with a full reservoir (D) and an empty reservoir (E). The gelation bath is not part of the simulation and was drawn during postprocessing for purpose of comparison with experimental results.</figcaption></figure>

produced. Later, a quasi-stationary regime of (thin) jet extrusion is established. This part of the centrifugation accounts for the formation of the fiber observed experimentally. In the erc, a quasi-periodic regime is established from the beginning, where capsules with a long thin ligament are ejected. This regime was observed steadily

throughout the simulation. This part of the centrifugation is at the origin of the numerous capsules with filament obtained experimentally. Thus, the simulation validates the hypothesis that a change in the ejection regime takes place at a certain point during the emptying of the reservoir, which accounts for the formation of the two
<figure><img src="image_19.png" /><figcaption>**FIGURE 18** Positioning of the cases ER1-12 as a function of Oh (vertical lines) and Re<sub>th</sub> (oblique iso-lines), with Re<sub>th</sub> = 0.125 · Ga<sup>2</sup> · ζ. The colored regions correspond to the convex hulls of the experimental results from cases ER1-29. The triangle symbols represent the simulations for the twelve cases. In the cases where both simulations frc (higher Re<sub>th</sub> and ere (lower Re<sub>th</sub>) were conducted, two distinct symbols are used to represent both simulations. Both the experimental and numerical results use the same color code.</figcaption></figure>

different centrifugation products. We make a distinction between this case, where the change in ejection regime takes place during the emptying of the reservoir, generating a nonnegligible amount of products of the new regime, and the cases shown in Section 5.1, where only a very short transient regime was observed at the end of the emptying of the capillary. Only the former situation will be called hereafter a transition in the ejection regime.

Due to long computational times, the simulation for the ere was not conducted until the full emptying of the reservoir, so that the final transient was not explored. However, from the previous numerical results presented in Section 5.1, we infer that polydisperse droplets, not necessarily with a filament, may result from this short transient phase. This would account for the few polydisperse capsules without filament observed experimentally in the gelation solution.

Figure 17D,E shows numerical results for case ER2 (simulation of centrifugal ejection of alginate solution of low concentration, with a relatively high nozzle diameter), with frc and ere, respectively. In the frc, the polymer is extruded as a jet with stable thickness. This is in agreement with the formation of a fiber observed experimentally. In the ere, individual droplets are ejected, with the formation of a thin ligament connecting the droplet to the bulk of the liquid in the capillary. The ligament breaks at both ends while the droplet is still in the air, and then further fragments into multiple parts of variable volumes. The filament fragments then retract while still in air and form satellite droplets of variable volumes. This ejection regime—droplet ejection with multiple polydisperse satellites—is in very good agreement with the experimental results, where numerous capsules are obtained, accompanied by a polydisperse population of satellite droplets. Again for this case, the two simulations corroborate the hypothesis that the different products obtained

experimentally are due to a transition in the ejection regime which occurs during the emptying of the reservoir.

## 5.6 | Summary of ejection regime simulations

After comparing in detail the formation of each type of centrifugation products, the previous results can be summarized on a diagram based on the dimensionless numbers introduced in Section 3.1. Figure 18 shows the considered centrifugal extrusion cases in the Re<sub>th</sub> – Oh diagram.

In order to better represent the experimental regions, a complementary series of experiments for the ejection regime (ER13–ER29) was performed. The parameters of these latter experiments are given in the Appendix SM.6. The colored regions in the diagram correspond to the convex hulls of the experimental results from cases ER1–29, where each color represents a different class of centrifugation products obtained (fiber, filament, satellites and capsules without secondary products, labeled as “no satellites”). If in one experiment several types of products were observed in relevant quantities (excluding the case of very few capsules accompanying a fiber), the experimental point was assigned to both classes corresponding to those products. That is why some regions representing the experimental results overlap in the diagram.

The numerical results for the frc and ere ER1–12 are plotted with individual markers in the diagram and colored according to the same color code. In the cases where both frc (higher Re<sub>th</sub> and ere (lower Re<sub>th</sub>) were conducted, two distinct symbols are used to represent both simulations. The diagram shows the slight decrease in Re<sub>th</sub> for ere, compared to frc. This illustrates the fact that the extrusion operating point is shifted toward lower Re<sub>th</sub> values during centrifugation, as the reservoir is emptied. For a given parameter set, in the ere, when the simulation exhibits a different ejection regime than the simulation of the frc for the same parameters, this indicates the cases where a transition between ejection regimes during centrifugation was captured by the numerical model (as discussed in Section 5.5).

As discussed in detail in Section 5, the numerical results are in good agreement with the experimental observations, which validate the assumption of negligible elastic effects in this system. In addition to the finer understanding of the underlying phenomena and transitions during centrifugal extrusion, the numerical results allow to better localize the transitions between the different regimes in the diagram, since numerically we can dissociate between the frc and ere, which is difficult to achieve experimentally.

Moreover, in Figure 18, the horizontal lines represent iso-Re<sub>th</sub> · Oh<sup>1.6</sup> lines. We can see that the transitions between the ejection regimes are roughly parallel to these lines, which is in agreement with the results in previous experimental and theoretical work by the authors,<sup>5</sup> where transitions between ejection regimes were discussed in detail.§§§§

<footnote>§§§§The difference is that here the derived Reynolds number Re<sub>th</sub>, based on the proportionality between Re<sub>th</sub> and Ga<sup>2</sup> from (9), is used for convenience, whereas in<sup>5</sup> the Reynolds number based on the analytically calculated flow rate Re<sub>cap</sub> was used.</footnote>
<table><caption>TABLE 11 Summary of the cases S21, S24, S26, and S29 for the study of the formation of different capsule shapes.</caption><thead><tr><th>Exp. series</th><th>Exp. no.</th><th>Fluid</th><th>D<sub>cap,ref</sub> (mm)</th><th>ω (RPM)</th><th>h<sub>0</sub> (mm)</th><th>h<sub>fall</sub> (mm)</th><th>Experimental product</th></tr></thead><tbody><tr><td rowspan="4">Capsule shapes (S)</td><td>21</td><td>Alg. B 4%</td><td>0.410</td><td>1115</td><td>6.7</td><td>5</td><td>Tear-shaped capsules and few satellites.</td></tr><tr><td>24</td><td>Alg. B 1%</td><td>0.150</td><td>705</td><td>6.7</td><td>4</td><td>Polydisperse tear-shaped capsules.</td></tr><tr><td>26</td><td>Alg. B 2%</td><td>0.330</td><td>1221</td><td>6.7</td><td>15</td><td>Oblate ellipsoidal capsules and satellites.</td></tr><tr><td>29</td><td>Alg. B 4%</td><td>0.410</td><td>1931</td><td>6.7</td><td>25</td><td>Elongated tear-shaped capsules.</td></tr></tbody></table>

Notes: $h_0$ is the initial height of the liquid column in the reservoir. $h_{fall}$ is the collecting distance in the experiments.

## 6 | NUMERICAL RESULTS – DROPLET SHAPES

In Section 5 it has been shown that the CFD model is accurate in representing the interface phenomena during the ejection and fall of the bio-polymer solution, whereby different ejection regimes could correctly be reproduced, in good agreement with the experimental results. Here we will focus on the drop-by-drop ejection regime and study the shape of the droplets during their fall and before entering the gelation solution. The purpose of this analysis is to validate different hypotheses made in previous work⁵ on the origin of capsule shapes, notably tear-shaped capsules, polydisperse tear-shaped capsules, flattened ellipsoidal capsules and elongated tear-shaped capsules.<fnref n="19191" />

A similar approach as in the previous section will be used. For each category of capsule shapes, one case is selected, according to the capsule shapes produced experimentally. The same case is then simulated numerically (frc) and the results are compared to experiments and discussed. The experiments S21, S24, S26, and S29 have been carried out for this purpose, and their operating parameters are summarized in Table 11.

### 6.1 | Formation of tear-shaped capsules

Tear-shaped capsules are one of the most common shapes that can be obtained with centrifugal encapsulation.³ Their formation compared to spherical capsules has been explained by an insufficient relaxation of viscous droplets between the moment of their detachment from the capillary and the moment they penetrate into the gelation solution and begin to gel. Visualization of this phenomenon has been performed for gravitational dripping systems,⁶⁷ but not for centrifugal systems. Moreover, the existence of a subclass “elongated tear-shaped capsules” and of a different class “polydisperse tear-shaped capsules” imposes a finer modeling of the formation of each of these capsule types, in order to validate the differences in the formation mechanisms that were assumed in previous work⁵: deformation due to high inertia at the impact on the gelation bath for the elongated ones, and temporary attachment to the interface of the gelation bath for polydisperse ones.

As an example of tear-shaped capsule production we select experiment S21 (high alginate concentration, low rotation speed and medium-large nozzle diameter). Experimental results for this case are shown in Figure 19A,B, where we observe the production of monodisperse tear-shaped capsules. The numerical results for this same case are presented in Figure 19C, showing that droplets form a liquid ligament while detaching from the capillary. When this ligament breaks, the droplet keeps initially a tear-like shape. During the fall of the droplet through air, the shape of the droplet relaxes toward spherical due to surface tension. However, the high viscosity renders this relaxation slow, so that the droplet does not reach the spherical shape before entering the gelation solution, but a partially relaxed tear-like shape. This can be quantified by the sphericity factor of the droplet in the simulation, which is a measure of the droplet elongation:

$$SF_{\text{drop}} = \frac{A - B}{A + B'} \quad (28)$$

where A and B are the droplet's major and minor axes, respectively. A droplet having $SF_{drop} < 0.05$ is considered to be spherical.⁶⁷ The time evolution of the sphericity factor of a droplet in case S21 is plotted in Figure 20. We notice the decrease of SF corresponding to the relaxation of the droplet during its fall through the air. A value of SF > 0.05 upon entry in the gelation solution predicts the formation of tear-shaped capsules, since the droplet cross-links in a partially relaxed shape.

These results are qualitatively in good agreement with the model introduced in previous work⁵ explaining the formation of monodisperse tear-shaped capsules by an insufficient relaxation of the droplet shape before entering the gelation bath. Although in the experiments a pointed tip is present on many capsules, in the numerical simulations the droplets do not feature a pointed tip upon arrival at the virtual surface of the gelation solution. In these simulations, this tip is present on the droplets during their fall through the air (cf. Figure 19C, contour at 3133.29 ms), but is reabsorbed by surface tension before reaching the gelation solution (Figure 19C, contour at 3141.69 ms). Indeed, the pointed tip may be due to the volume-decreasing polymerization in the gelation bath, in analogy with what happens at water droplets during solidification,⁶⁸ where a pointed tip appears due to the density change of water during the solidification process.

In the simulation the droplets have a sphericity factor $SF_{drop} \approx 0.267$ before entry in the gelation solution. In the experiment capsules

<footnote n="19191">¹⁹¹⁹¹ The detailed definitions and classifications of capsule shapes are given in previous work by the authors.³

⁹⁹⁹S stands for “shape.”</footnote>
<figure><img src="image_20.png" /><figcaption>**FIGURE 19** Experimental and numerical results for cases where tear-shaped (case S21) and polydisperse tear-shaped (case S24) capsules are formed. (A–C) Case S21, where tear-shaped capsules are produced: (A,B) experimental results; (C) numerical simulation with full reservoir. (D–G) Case S24, where polydisperse tear-shaped capsules are produced: (D–F) experimental results; (G) numerical simulation with full reservoir.</figcaption></figure>
<figure><img src="image_21.png" /><figcaption>**FIGURE 20** Evolution of the sphericity factor (SF) of a droplet as a function of the traveled distance from the needle tip, in the simulations of cases S21, S24, S29, and S26. The dashed line indicates the value SF<sub>drop</sub> = 0.05 below which droplets can be considered spherical.</figcaption></figure>

feature a sphericity SF<sub>caps</sub> ≈ 0.24-0.27, indicating a good quantitative agreement between numerical and measured values. |||||

## 6.2 | Formation of polydisperse tear-shaped capsules

Experiment ER24 is investigated as an example of polydisperse tear-shaped capsule production. This particular class of capsule shapes has been identified in the parametric study in Reference 3 and its characteristics are recalled in the following. Their formation was explained in previous modeling work,⁵ by comparison with the literature results and by defining an adequate dimensionless number characterizing the phenomenon of interface crossing by the droplets, the impact Froude number Fr<sub>imp</sub>. The experimental results for this case are shown in Figure 19D-F. Typical features of this class of capsule shapes can be noticed: the tear-shaped capsules are polydisperse in terms of shape and size of the protuberance; only a fraction of the capsules present a protuberance, whereas the others are of ellipsoidal shape. Another specificity of polydisperse tear-shaped capsules is that they tend to form from lower-viscosity alginates, contrary to the more classical monodisperse tear-shaped capsules which form usually from higher viscosity alginate solutions.

The numerical results for this case are represented in Figure 19G. The striking observation here is that the droplet has a perfectly spherical shape before reaching the interface of the gelation bath. This can be measured quantitatively using the SF<sub>drop</sub> factor, as illustrated in Figure 20. Indeed, SF<sub>drop</sub> is much lower than the sphericity threshold 0.05 before the droplets impact the gelation bath. Unlike the previous case of tear-shaped capsule formation, here the droplet relaxation process is finished before the droplet enters the gelation solution. Therefore, the origin of the protuberance (tear-like shape) in this case is not related to the process stages of ejection or fall in air, but it is to be attributed to the latter stages of impact and gelation. This corroborates the hypotheses made in previous work,⁵ where the polydisperse tear-shaped capsule formation was attributed to the impact phase (by attachment of the droplet to the interface).

## 6.3 | Formation of elongated tear-shaped capsules

Elongated tear-shaped capsules are a particular sub-category of tear-shaped capsules. They are formed in the same ejection conditions as tear-shaped capsules, but at high impact Froude numbers Fr<sub>imp</sub>, which is generally equivalent to high collecting distances h (cf. Reference 5). In Figure 21A,B we show examples of elongated tear-shaped capsules formed in experiment S29. In this experiment, the shapes of all the capsules are monodisperse.

Numerical results for the case S29 are shown in Figure 21C. Like in the case of tear-shaped capsules, we notice the tear-like shape of the droplet at the moment of

<footnote>|||||| A direct comparison of the SF<sub>drop</sub> value (before impact) with the SF<sub>caps</sub> value (after impact and gelation) may not be fully relevant, since capsules observed under microscope after the encapsulation experiment have undergone further stages of transformation, including impact, gelation, and sedimentation, as opposed to the numerically simulated droplet before impact. However, in Reference 5 it is suggested that high-viscosity alginate droplets (such as in the case of experiment S21) deform very little upon impact, and thus comparison of shapes before and after impact and gelation might be possible in these cases.</footnote>
<figure><img src="image_22.png" /><figcaption>**FIGURE 21** Experimental and numerical results for cases where different capsule shapes are generated. (A)–(C) Case S29, where elongated tear-shaped capsules are produced: (A,B) experimental results; (C) numerical simulation with full reservoir; (D)–(F) Case S26, where oblate ellipsoidal capsules are produced: (D,E) experimental results; (F) numerical simulation with full reservoir.</figcaption></figure>

detachment from the capillary and a progressive relaxation toward a spherical shape. The evolution of the sphericity factor SF<sub>drop</sub> is plotted in Figure 20. Since the collecting distance here is relatively large, the relaxation of the droplet advances more than in cases like S21. Since the final shape of the capsule is more elongated than the droplet at the impact with the gelation solution, the

elongation of droplets is not produced during the stages of ejection and fall through air.

These numerical results represent a confirmation of the hypothesis made in previous theoretical work, and provide a solid basis for the assumption that the droplet elongation takes place during the impact and/or sedimentation of the droplet.
## 6.4 | Formation of flattened ellipsoidal capsules

Flattened ellipsoidal capsules (oblate) are obtained using low-to-medium viscosity polymer solutions. Since oblate capsules deposit horizontally in the solution under the microscope, they are not immediately identifiable using single two-dimensional images, on which they look spherical. A complementary analysis is therefore performed under the microscope in order to investigate the 3D-aspect of the produced capsules, based on a by-hand pipette manipulation to rotate individual capsules and capture images of the same capsule from different angles.³ A hint of the presence of these capsules is given by the visualization of few ones stuck upwards being surrounded by other capsules that hinder their movement. An example of this is shown in Figure 21D,E, from experiment S26. The 3D-aspect of the capsule shapes in experiment S26 was analyzed according to the above by-hand technique, which confirmed the flattened ellipsoidal shape of the capsules.

In Figure 21F we show the numerical results for case S26. The simulation shows that the ejected droplets quickly regain a spherical shape after detachment from the capillary and maintain a spherical shape during their fall (cf. plot of SF<sub>drop</sub> in Figure 20). Hence, the simulation confirms that the flattened aspect of the capsules obtained experimentally is not due to the ejection and fall stages of the process, so it is reasonably supposed that it is due to the later stages of impact and sedimentation. These results again corroborate the hypotheses made in the previous theoretical work.⁵

## 6.5 | Summary on capsule shapes

In general, an excellent agreement was found between the numerical results and the assumptions made in previous theoretical work by the authors on the origin of the different capsule shapes.⁵ The only minor disagreement is the origin of a sharp tip of Figure 19A,B, compared to the simulation in Figure 19C (rounded tip). However, pointed tips may form in the solidification step,⁶⁸ which is not captured in the simulations performed in this article.

## 7 | CONCLUSIONS

In the CME technique, the prediction and control of capsule shapes is crucial for a reliable and repeatable production. The numerical model developed in this article represents a major improvement in the understanding and prediction of capsule shape formation mechanisms in the key stages of the CME process, completing simpler analytical models developed in previous work.

We have reported the construction and validation of a CFD model of centrifugal extrusion of a shear-thinning fluid, corresponding to the stages of capillary flow, droplet ejection and droplet fall. Comparison with experimental data shows that the model can accurately predict flow rates, droplet sizes and shapes, and also reproduce all the ejection regimes encountered in practice.

The obtained results represent an insight into the physics of the intermediate stages of droplet formation and fall through air of the CME process, which is difficult to obtain experimentally due to visualization issues on standard centrifugation devices. Thus, these results corroborate and complete previous analytical models that had been developed for the CME process:

1. Careful rheological investigation of alginate solutions at different concentrations indicates shear-thinning as the dominant behavior and justifies the usage of a Carreau-Yasuda model for the description of the fluid behavior. History effects are not significant in the behavior of alginate solutions under the experimental conditions of centrifugal extrusion.

2. As one would expect, the formation of satellite droplets, filaments and fibers is confirmed to be a result of the ejection regime and is governed mainly by the Ga and Oh numbers. But surprisingly, the regions of formation of each of these products in the Ga-Oh space are shifted slightly during the emptying of the reservoir, under the decreasing hydrostatic pressure in the extrusion needle.

3. Several conclusions can be drawn on the formation of tear-shaped capsules.

The formation of monodisperse tear-shaped capsules is confirmed numerically to be the result of the ejection stage and of an insufficient relaxation of the droplet during the fall through the air.

The formation of polydisperse tear-shaped capsules has been shown not to be a result of the centrifugal extrusion. Indeed, simulated droplet shapes were spherical before reaching the gelation solution. The formation of elongated tear-shaped capsules has also been shown not to be a result of the centrifugal extrusion. Finally, note that the same conclusion can be drawn for oblate ellipsoidal capsules.

4. The robust CFD model proposed here can be used for optimization of the operation of CME devices, allowing a precise determination of capsule production frequency and of the nature and sizes of the by-products.

Future work could be aimed at the modeling of the subsequent stages of the process, namely the impact and gelation stages, that are equally important for determining the final shape of the capsule. These stages imply more complex physics, such as three-phase hydrodynamics, and the diffusion of Calcium ions inside the alginate and the chemical cross-linking reaction triggering solidification, acting simultaneously on the droplet.

### AUTHOR CONTRIBUTIONS

Matei Badalan: conceptualization (equal); data curation (lead); investigation (equal); methodology (equal); visualization (lead); writing – original draft (lead). Giovanni Ghigliotti: conceptualization (equal); investigation (supporting); methodology (equal); supervision (lead); validation (equal); writing – review and editing (equal). Denis Roux: conceptualization (equal); investigation (equal); methodology (equal); resources (lead); validation (supporting). Guillaume Maîtrejean: methodology (equal); resources
(equal); software (equal). **Jean-Luc Achard**: conceptualization (equal); supervision (equal); writing - review and editing (supporting). **Frédéric Bottausci**: conceptualization (supporting); methodology (supporting); resources (supporting); validation (supporting); writing - review and editing (supporting). **Guillaume Balarac**: conceptualization (supporting); project administration (supporting); supervision (equal).

## ACKNOWLEDGMENTS

The authors thank Patrick Bégou for the valuable help in installing and running Gerris CFD software on the LEGI computing facilities, and Vincent Verdoot and Lydia Esteban for the valuable advice and help in carrying out the rheology experiments.

## DATA AVAILABILITY STATEMENT

The data and source codes that support the findings of this study are available in the Appendix S1 of this article. The technical drawing from Figure 1C was extracted from 3D STEP drawings provided by the manufacturer.³³ The numerical values of the different dimensions, measured from the 3D models using CAD visualization software eDrawings, were tabulated in Table 1. The numerical data from Figures 2,3,8,18, and 20 are tabulated and described in the Appendix S1. The numerical values from Figure 4 are provided in a separate csv file, in the "data.zip" archive attached to this article, and described in the Appendix S1. The source codes for the numerical simulations (using Gerris) shown in Figures 7,9D, 10,11,12,13,14,15,16,17,19, and 21 are provided in the "data.zip" archive and described in the Appendix S1. The latest official version of Gerris Flow Solver for CentOS mentioned on the Gerris website was used for running these simulations. The MATLAB code used to calculate the analytical velocity profiles from Figure 7 are also provided in the attached "data.zip" archive. The calculation of the error bars on the experimental and numerical values is described in the Appendix S1. The experiments of flow rate measurement have been performed according to the method described in Reference 5 and have been verified with analytical calculations in that same work. A sufficient number of points (between 4 and 6) were taken for each case, to reduce the error on the slope calculation. The experiments of alginate capsules production have been performed in previous work⁵ and the results were used here for validation of the simulations. Capsule production experiments with that same device and procedure had been realized in triplicate and their repeatability had been demonstrated in previous work.³¹ Extensive validation of the numerical model with analytical calculations and with experimental data from this and previous work has been performed before using the numerical model for ejection regimes and droplet shape predictions. These validations are discussed in Section 4 of this article.

## ORCID

Giovanni Ghigliotti https://orcid.org/0000-0001-5389-8668

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## SUPPORTING INFORMATION

Additional supporting information can be found online in the Supporting Information section at the end of this article.

**How to cite this article:** Badalan M, Ghigliotti G, Roux D, et al. Simulation of non-Newtonian biopolymer extrusion and fall in the centrifugal microencapsulation process. *AIChE J.* 2024; 70(10):e18517. doi:10.1002/aic.18517