Robust fabrication of ultra-soft tunable PDMS microcapsules as a biomimetic model for red blood cells

Microcapsules with liquid cores encapsulated by thin membranes have many applications in science, medicine and industry. In this paper, we design a suspension of microcapsules which flow and deform like red blood cells (RBCs), as a valuable tool to investigate microhaemodynamics. A reconfigurable and easy-to-assemble 3D nested glass capillary device is used to robustly fabricate water-oil-water double emulsions which are then converted into spherical microcapsules with hyperelastic membranes by cross-linking the polydimethylsiloxane (PDMS) layer coating the droplets. The resulting capsules are monodisperse to within 1% and can be made in a wide range of size and membrane thickness. We use osmosis to deflate by 36% initially spherical capsules of diameter 350 {\mu}m and a membrane thickness of 4% of their radius, in order to match the reduced volume of biconcave RBCs. We compare the propagation of initially spherical and deflated capsules under constant volumetric flow in cylindrical capillaries of different confinements. We find that only deflated capsules deform broadly similarly to RBCs over a similar range of capillary numbers (Ca) -- the ratio of viscous to elastic forces. Similarly to the RBCs, the microcapsules transition from a symmetric 'parachute' to an asymmetric 'slipper'-like shape as Ca increases within the physiological range, demonstrating intriguing confinement-dependent dynamics. In addition to biomimetic RBC properties, high-throughput fabrication of tunable ultra-soft microcapsules could be further functionalized and find applications in other areas of science and engineering


Introduction
Microcapsules with core-shell structures are widely used in industrial and biomedical domains, such as thermal energy storage 1 , enhanced oil recovery 2 , targeted drug delivery and controlled release 3,4 , and cell encapsulation and culture 5 .They also occur naturally in the form of, e.g., red blood cells (RBCs), bacteria and egg cells.Capsules that are used as carriers in confined media of complex geometry, e.g., enhanced oil recovery and biological delivery systems, have to be highly deformable and sufficiently robust to propagate efficiently and reach their targets.For example, the high deformability of RBCs enables them to flow through narrow pores and vessels of comparable size to deliver oxygen from the lungs to the rest of the body and carry carbon dioxide back to the lungs to be exhaled.Due to their high deformability, RBCs can adopt a variety of shapes in confined flow, including symmetrical parachute and symmetry-broken slipper-like shapes 6,7 (see Fig. 1) for sufficiently large capillary numbers Ca -a measure of the ratio of viscous to elastic forces acting on the capsule.Although the flow of isolated capsules (or RBCs) in confined vessels has been extensively studied 8 , the fluid dynamics of capsule suspensions is yet to be fully addressed, including their propagation in complex media 9 and rheological properties 10 .
Recently, numerical simulations have been applied to investigate the dynamics of RBC suspensions in some complex geometries.However, they are usually limited to low haematocrit (or RBC volume fraction) and small flow domains because of their computational expense 9 .The use of real RBCs in controlled experiments is also challenging because of their small size (typically less than 10 µm), fragility, biological variability, rapid ageing and limited availability 11,12 .Thus, various analogue models including capsules, elastic beads, vesicles and droplets have been de- veloped to mimic flow behaviours of RBCs [13][14][15] .Capsules have a core-membrane structure similar to RBCs and they typically exhibit larger deformations under flow than solid elastic beads 8,16 .Although vesicles 17 and droplets 18 can also deform considerably, they do not exhibit the shear elasticity of real RBCs 19 .Undesired coalescence of droplets further limits their use to dilute suspensions (equivalent to low haematocrit values).However, the geometrical and mechanical properties of microcapsules, such as membrane thickness and elastic modulus, need to be carefully chosen to access the deformations routinely observed in RBCs.Besides, to the best of our knowledge, only spherical objects have thus far been used in experiments to model RBCs, which adopt biconcave discoid shapes 20 at rest characterised by a reduced volume of 64% of that of a sphere with the same surface area.The reduced volume of RBCs has been shown both experimentally and numerically to considerably affect their ability of passing through the confined microchannels 21 and deformation in shear flow 22 .The deformation and buckling of a spherical capsule as a function of reduced volume has been modelled 23 and validated with experiments on beach balls 24 .However, characterisation of the motion and deformation of partially deflated capsules under shear flow is still lacking.Thus, there is a clear need for a robust approach to fabricate large populations of monodisperse microcapsules with ultra-thin soft membranes and a similar reduced volume to RBCs.
Droplet microfluidics is increasingly applied for capsule synthesis because this approach greatly simplifies the fabrication process by enabling precise control over the manufacturing conditions.The strategies typically encompass double emulsion templates, which consist of droplets encapsulated by an immiscible middle layer (drop-in-drop structure) and suspended in an outer liquid.The droplet coating is cured using either photo or thermally induced free-radical polymerisation 25,26 , complexation reactions 27 or solvent evaporation 28 .The membrane thickness is thus determined by the thickness of the coating layer.The axisymmetric glass capillary device firstly proposed by Utada et al. 29 is now most commonly used to produce double-emulsion droplets.This device is assembled with an array of nested glass capillaries that are pre-treated with silanes to achieve the required wettability distribution.Bandulasena et al. 30 and Levenstein et al. 31 optimised the device fabrication method to be low-cost, time-saving and reconfigurable.This microfluidic method also offers flexibility over coating materials and thus membrane properties 32 .Common materials used for capsule fabrication are UV-curable acrylates 26 , alginates 33 and proteins 34,35 .However, these mate-rials are either not sufficiently soft or exhibit rapid ageing, making them inappropriate for use in RBC analogues.In contrast, polydimethylsiloxane (PDMS) is a low-cost, transparent and flowable material which has been extensively characterised because of its ubiquitous use in microfabrication.It can be rapidly crosslinked under elevated temperature and exhibits very stable physical and chemical properties once cured, allowing the capsules to be stored for months.Besides, the mechanical properties of the membrane can be easily adjusted by mixing the PDMS base and the crosslinker in different ratios (see § 3.2).
The motion and deformation of single spherical capsules in confined flow have been extensively investigated both experimentally and numerically over the last two decades 8 .Risso et al. 36 showed experimentally that the steady propagation of a capsule in a capillary tube is governed by the capillary number Ca and the confinement parameter which is the ratio of capsule to tube diameter, and that capsule deformation is promoted by increase of either parameter.In a sufficiently confined geometry, the capsule extends into a barrel shape with spherical caps at both ends as Ca increases.Beyond a critical value of Ca, the rear of the capsule buckles inward to form a parachute-like shape, similar to the deformation of RBCs.The transport of capsules also depends on the viscosity ratio between internal and external liquids (µ int /µ ext ) and any pre-stress of the membrane.Although the viscosity ratio hardly affects steady capsule propagation, it can significantly influence transient behaviour by extending the time required for capsules with µ int /µ ext > 1 to reach a stable shape 37 .Numerical simulations performed by Lefebvre et al. 38 show that pre-inflation of the capsule suppresses the appearance of the parachute shape to larger values of Ca.The elastic modulus of the capsules can be measured by compression testing, for millimetric capsules 34 , and by atomic force microscopy or micropipette aspiration, for microcapsules, which requires skilled micro-manipulation 39 .In-flow measurements developed by Lefebvre et al. 40 are advantageous in that they circumvent size restrictions and enable high measurement throughput.However, they rely on a fit to numerical predictions from a membrane model based on a chosen constitutive behaviour.Individual capsules can also be trapped at the stagnation point of a microfluidic cross flow 35 to ensure sufficiently large deformations albeit with the need for skilled user intervention.Finally, capsules and RBCs also exhibit rich dynamical behaviours beyond steady deformation, e.g., relaxation upon exit from a single channel constriction 41 , or in complex channel geometries, such as T-junctions or networks 34,42 , where they transiently adopt highly deformed slipper-like shapes.
In this paper, we establish a robust methodology to manufacture large populations of highly monodisperse and stable microcapsules, which are partially deflated by osmosis to match the reduced volume of RBCs.The fabrication methods are described in § 2, while characterisation of capsule properties including size, membrane thickness, elastic modulus and deflation are discussed in § 3.In § 4 we compare the steady flow behaviour of spherical and partially deflated microcapsules in capillary tubes as a function of the viscous-elastic capillary number Ca and geometric confinement.Conclusions and outlook for using partially deflated microcapsules as RBC analogues are given in § 5.

Capsule fabrication
We fabricated capsules by generating a train of coated droplets within a microfluidic nested-capillary device and collected the droplets by letting them fall from the end of a collection capillary bent at 90 • into a glass bottle, where they cured at rest; see Fig. 2(a).The double-emulsion generation was monitored in top-view with a monochrome CMOS high-speed camera (PCO 1200hs) fitted with long-distance magnifying optics, which consisted of a zoom lens (Navitar, 12× Zoom Lens System) coupled to a 5× microscope objective (Mitutoyo, M Plan APO).The maximum combined magnification therefore reaches 60×, and images were recorded with a maximum frame rate of 500 frames per second (fps).The microfluidic device was backlit with uniform, diffuse illumination of adjustable brightness provided by a custom-made white LED light box.
The nested glass capillary device shown in Fig. 2(a, b) comprised three glass capillaries of circular cross-section: injection and collection capillaries (O.D. = 1.0 mm, I.D. = 0.58 mm, World Precision Instruments) and an outer capillary (O.D. = 2.0 mm, I.D. = 1.8 mm, S Murray & Co, UK).The outer capillary was snapped into two pieces of approximately 2 cm length after scoring its midpoint with a ceramic tile (Sutter Instrument).The injection capillary was tapered with a micropipette puller (P-97, Sutter Instrument) and its tip was cut off and polished using the tile to a diameter of D i = 120 µm.A gas torch flame was used to melt one end of the collection capillary to form a constricted nozzle-like structure.We treated several capillaries and selected those with a nozzle diameter D c = 350 ± 20 µm to use in the nested capillary device.The 90 • bend of the collection capillary was applied to the initially straight capillary held in a horizontal position, by heating its midpoint under the gas torch flame until the heat-softened region allowed the end of the capillary to drop spontaneously under gravity.
We used a mixture of water, glycerol (Sigma Aldrich) and 2.0 wt% PVA (Polyvinyl alcohol, partially hydrolysed, MW approx.30000, Sigma-Aldrich) for both inner and outer phases, where water and glycerol were mixed in 36:64 by volume.Carminic acid (Sigma-Aldrich) was added at 0.1 wt% to the inner phase to dye the capsules red.The middle phase was PDMS (Sylgard 184, Dow Corning) which could be mixed with different ratios of base to crosslinker to vary the elastic properties of the cured material.The mixture was degassed in a vacuum chamber for 20 minutes to extract dissolved air before transferring it to a 1 mm syringe (Injekt-F) to be injected with a syringe pump.Because the PDMS progressively cured over time, thus increasing its viscosity, the mixture had to be used within 3 hours to ensure stable droplet formation with consistent size and shape.To achieve water-oilwater (W/O/W) double emulsions, the injection capillary had to be chemically treated with Sigmacote (Sigma-Aldrich), a solution of chlorinated organopolysiloxane dissolved in heptane which adsorbs to the glass surface to form a hydrophobic coating.The collection and outer capillaries were treated with oxygen plasma (HPT-100, Henniker Plasma) at 100% power for 3 minutes to enhance their hydrophilicity.
Immediately after plasma treatment, the injection and collec-tion capillaries were co-axially aligned inside the larger outer capillary aided by the magnified image from the camera.The outer capillary was held between two custom-made Teflon connectors (see the inset of Fig. 2(a)) which were accurately drilled through along their central axis; see Fig. S1 in ESI for the detailed design.The injection and collection capillaries were each pushed through a connector so that the tapered end of the injection capillary was separated from the nozzle of the collection capillary by a distance l 200 µm, as shown in Fig. 2(b).Fluid inlet tubes were connected at this stage and all joints were sealed and fixed with a UV curable glue consisting of a 70:30 (v/v) mixture of pentaerythriol triacrylate (PETA, Sigma-Aldrich) and Tri(propyleneglycol) diacrylate (TRPGDA, Sigma-Aldrich) with 5% of 1-Hydroxycyclohexylphenyl ketone (Sigma-Aldrich) as the photoinitiator.This glue solidified within a few seconds under UV irradiation, which enabled rapid assembly of the device.The entire device was positioned inside a well milled Perspex sheet, with an observation window cut out to enable imaging of the double emulsion formation.The entire design was focused on enabling rapid and reproducible dismantling and reassembly of the device, which typically took 20 minutes if all accessories were readily available.
To generate the double emulsion, the inner and middle phases were injected with constant volumetric rates (Q i and Q m ) into the injection capillary and the gap between the injection and outer capillaries through a tube into the leftmost connector, respectively.The outer phase was injected at Q o through the gap between the collection and outer capillaries through a tube into the rightmost connector (see Fig. 2(a, b)).The inner and outer phases were injected under flow rate control using a pressure controller (Elveflow Mk3+ 0-2 Bar, Elvesys) coupled to in-line Mass Flow Sensors (MFS, 0-80 µL/min, Elvesys).The middle phase was injected using a syringe pump (KD Scientific, Model 210) to avoid the occlusion of flow sensors because of the eventual curing of the middle phase.
The three phases came into contact in the region of length l between the injection and the collection capillaries (Fig. 2(b)), where the inner phase coated by the middle phase broke into double emulsion droplets suspended in the outer phase.In order to achieve the interface shapes necessary for the stable generation of coated droplets, the fluids had to be introduced in strict order.We first flowed the outer phase into the system and let it wet the entire device before introducing the middle phase.A stable cone-like interface was formed owing to the hydrophobic injection capillary and the hydrophilic outer capillary, resulting in the formation of a regular train of middle phase droplets.Finally, the inner phase liquid was introduced in order for it to be completely surrounded by the middle phase upon exit of the injection capillary, and the train of middle-phase droplets was thus replaced by droplets of inner phase encapsulated by the middle phase.The flow rate for each phase was adjusted to retain a stable dripping regime, as shown in Fig. 2(b).This capillary device could make in excess of one hundred capsules per minute under optimal operating conditions.Video 1 in ESI shows an example of doubleemulsion generation under such optimal conditions.Sub-optimal conditions are shown in Video 2, where mixing of the inner and outer phases occurs due to the misalignment of the injection and collection capillaries; in Video 3 the interface breaks up due to defective hydrophobic treatment on the injection capillary; and in Video 4 an irregular interface forms due to insufficient plasma treatment on the outer and collection capillaries.
We typically generated populations of tens of thousands of coated droplets to be cured.The bottle in which they were collected contained a buffer layer of the suspending liquid (outer phase) to avoid direct contact with the bottom wall upon impact.After collection, the bottle was placed in an oven at 70 • C for 4 hours to cure the PDMS coating and convert the droplets into capsules, as illustrated in Fig. 2(c).Curing the coating while the droplets were at rest ensured uniform membranes and avoided off-centre placement of the core 26 , in contrast with insitu UV-polymerisation where moving droplets were irradiated in the outlet capillary while subject to shear forces.Shrinkage of the PDMS coating by < 1.5% during curing resulted in a slight extensional pre-stress being applied to the membrane.Although physical and chemical properties remained stable over periods of months, regular stirring of the suspension was required to avoid capsules sticking together due to the drainage of the lubrication layer separating them.

Geometrical properties
Healthy human RBCs typically exhibit low size variability with typical diameters of 7-8 µm, and thus any RBC analogue must also be highly monodisperse.Accurate control of capsule size and membrane thickness is also important to ensure reproducible experiments and meaningful upscaling from RBCs to capsules.The histogram in Fig. 3 The capsule fabrication method also enabled us to customise the suspension by applying different flow rate combinations during double emulsion generation.The steady-state generation of double emulsions means that Q m and Q i contribute uniformly to the capsule's inner core and coating layer volumes, respectively.Thus, mass conservation, based on an incompressible spherical capsule, predicts the ratio of membrane thickness to the outer capsule radius 2δ /d o that depends only on the ratio of the middle and inner phase flow rates Fig. 3(b) shows that Eq. ( 1) (the continuous dashed line in Fig. 3) accurately predicts the relative membrane thickness measured experimentally in the range from approximately 4% to 37% (solid circles).Rachik et al. 43 compared experimental characterisations of serum albumin-alginate capsules with model predictions to find that the thin shell assumption is only valid for relative membrane thickness of less than 5% beyond which it is necessary to consider bulk elastic effects.Thus, we kept the relative membrane thickness of our capsules at 4% to simplify the problem, as indicated with a red symbol in Fig. 3(b).
We also found that the variation of the outer phase flow rate Q o does not affect the relative membrane thickness but will change the capsule size, which provides an opportunity to customise the capsule diameter without perturbing the relative membrane thickness.Fig. 3(c) shows that the capsule size decreases with increasing Q o , while Q m and Q i are kept fixed to maintain a constant relative membrane thickness.We were able to vary the outer  44 .Vladisavljević et al. 45 analysed the double emulsion generation in a similar device based on mass conservation and found that the droplet diameter has a power-law dependence with an index of −1/3 to the droplet generation frequency that is directly determined by Q o .So, we can also get a power-law relation between the capsule diameter d o and Q o as ( Because the generation frequency itself is non-linear in Q o , the index b in this equation is different from −1/3 depending on the dimensions of the device and physical properties of the liquids.Michelon et al. 44 reported an index of −0.3 in their experiments.

Mechanical properties
To characterise the elastic properties of the membrane, we measured Young's modulus E by compression testing of a cylindrical sample and deduced the shear modulus theoretically in the limit of linear elasticity.Simply put, we measured the engineering stress as a function of the engineering strain, and the data is accurately captured by the Mooney-Rivlin model 46 , from which we deduced E in the limit of vanishing strain.The details of this method are discussed further in S3 of ESI.Experimental measurement of Young's modulus as a function of the mass ratio of the PDMS base to the crosslinker are shown in Fig. 4. Increasing the mixing ratio from 10:1 to 40:1 decreases Young's modulus by more than an order of magnitude from 1.4 MPa to 42 kPa.For isotropic polymeric materials, the bulk shear modulus G is related to Young's modulus E via Poisson's ratio ν, where ν 0.5 for PMDS which is assumed incompressible.The capsule membrane is treated as a thin sheet of a 3D homogeneous hyperelastic material with thickness δ , and the thin shell approximation is applied on the membrane mid-surface to compute the 3D effects 47 .Then, the surface shear modulus G s (N/m) is calculated based on the bulk shear modulus G by and the bending modulus κ is 8 : where ū is mean velocity of flow.

Controlled capsule deflation
With a typical surface area A 140 µm 2 and volume V 100 µm 3 , RBCs have a large surface area to volume ratio of A/V 1.56 times that of an equivalent sphere with the same surface area because of their biconcave shape at rest.A sphere with the same surface area as the RBC would have a radius R e such that 4πR 2 e = A, which yields R e 3.34 µm.Thus, the reduced volume of RBCs is α = V /V e = 0.64, where V e is the volume of the equivalent sphere 9 .This reduced volume can be matched in our analogue capsule model by deflating spherical capsules by 36%.We achieved capsule deflation through osmosis because the PDMS membrane is permeable to water but not to glycerol and thus, the osmotic pressure generated by a larger concentration of water inside the capsule core than in the suspending buffer solution in the collection bottle drives water across the membrane and out of the capsule until concentrations equilibrate.We started from a population of spherical capsules, a sample of which is visualised on a microscope glass slide in Fig. 5(a).Although both the inner phase, forming the core of the capsule, and the buffer solution in the bottle had the same mixing ratio of water to glycerol, the slightly lighter PDMS membrane meant that capsules spontaneously rose to the upper surface of the buffer solution in the bottle over a period of a few hours.Hence, we could easily remove the suspending liquid with a syringe and replace it with pure glycerol.The capsules were then carefully stirred into the pure glycerol with a glass bar taking utmost care to avoid trapping any air bubbles.The osmotic pressure generated by the difference in water concentration between the capsule core and pure glycerol in the bottle drove water across the membrane until the capsule core was mostly depleted of water because of the large volume of glycerol in the bottle.This process took about 30 minutes.The suspension was then left to rest for a few hours to allow the capsules to rise to the top surface again.We then repeated the process at least three times to ensure that all water in the capsule core had been removed.
By adjusting the mixing ratio of water to glycerol in the capsule core, we produced microcapsules with different levels of deflation.Fig. 5(b-d) show deflated capsules with volumes reduced by 80%, 60% and 36%, respectively.The capsules with 36% volume reduced match the reduced volume α of real RBCs.A 36:64 by volume solution of water and glycerol was used for the inner and outer phases to generate double emulsion required for these capsules.

Steady flow and deformation of spherical and deflated capsules in capillaries
We now compare the steady flow and deformation of spherical and deflated capsules (with a reduced volume of 0.64 similar to RBCs) in cylindrical glass capillary tubes of inner diameters D = 0.4, 0.3, 0.2 mm.The experiments were performed by injecting a very dilute suspension of capsules into a capillary tube at constant volumetric rate using a syringe pump (KD Scientific, Model 210), which ensured that the capsules were sufficiently separated to avoid measurable interaction within the tube.We used the high speed camera fitted with long-distance magnifying optics (Navitar 12× Zoom Lens System coupled with a 10× microscope objective) to capture images of the capsule shapes.
Images were recorded at a frame rate between 125 to 1500 fps (depending on the flow rate) with an exposure time of 1/3000 second.The capillary tube was backlit with a powerful cold-light source (Karl Storz -Xenon Nova 300).A comparison between the spherical and deflated capsule properties is listed in Table 1.The effective diameter d eff , which refers to the diameter of a sphere with the same volume as the capsules, is reduced from d o = 350 µm to approximately 302 µm upon reduction of the capsule volume by 36%.This means that the confinement parameter β = d eff /D takes different values for spherical and deflated capsules in the same capillary tube.A 90:10 solution by volume of glycerol in water was used for both the suspending fluid and core of the spherical capsules, whereas pure glycerol was used for deflated capsules.Thus, there is no contrast between the viscosity of the capsule core and the suspending fluid.
Fig. 6 shows a comparison between steady state deformation of initially spherical capsules (Fig. 6(a,d,f)) and deflated capsules (Fig. 6(b,e,g)) in each of the three capillaries.For D = 0.4 mm, both types of capsules are unconfined (β = 0.875 for spherical capsules and β = 0.755 for deflated capsules), so that they remain undeformed in the absence of flow; see panels (a-i) and (b-i) of Fig. 6.The statically spherical capsule extends as Ca increases, consistent with previous reports in the literature 36 , and adopts a parachute shape seen in Fig. 6(a-iii).We quantify capsule deformation as the ratio of the maximum length of the capsule to its maximum width L/W , which is measured on a closely fitted rectangular bounding box, enclosing the imaged capsule contour, parallel to the flow direction; see an example in panel (a-iv) of Fig. 6.Fig. 7 shows that the deformation ratio L/W varies approximately linearly as a function of Ca for the initially spherical capsules (solid symbols) in all three capillaries.In contrast, the deflated capsules (open symbols) are buckled inward in the absence of confinement and flow (see Fig. 6(b-i)), due to compressive stresses in the membrane induced by the volume reduction 23 .In this case, they adopt random orientations (Fig. 6(b-i)), which results in a mean value of L 0 /W 0 1 at Ca = 0 despite the fact that their volume is less than that of spherical capsules.In weak flow (Fig. 6(b-ii)), these deflated capsules align with the shear flow so that their inward buckled region is situated at the rear and their shape resembles a parachute.However, they do not extend axially so that L/W remains approximately constant up to a threshold value of the capillary number Ca th = 0.18 (Fig. 6(b-iii)), where the shear forces on the capsule presumably balances the compressive pre-stress of the deflated capsule.Beyond this threshold, L/W increases linearly with Ca and the capsule retains its parachute-like shape (Fig. 6(b-iv)).For high capillary numbers (Ca ≥ 0.32), we observed both the symmetrical parachutelike and asymmetrical slipper-like shapes (Fig. 6(b-v)).The elongated shapes of these deflated capsules are consistent with the deformation of RBCs in an unconfined capillary tube (D = 10 µm, β ≈ 0.6) at comparable capillary numbers Ca 48 , as shown in panel (c) of Fig. 6.The values of β and Ca of RBCs are calculated in the same way as for capsules, using an effective diameter of 3 µm and a shear modulus of 5.4 µN/m 49 .RBCs (orange asterisks in Fig. 7) exhibit deformation ratios that are marginally smaller than the deflated capsule data for β = 0.755 (black open squares in Fig. 7) which is consistent with their smaller confinement parameter.
In capillaries with D = 0.3 mm and 0.2 mm, the spherical capsules are initially compressed into a cylindrical barrel shape with spherical end caps under tension (see panels (d-i) and (f-i) of Fig. 6), so that the static deformation ratio L 0 /W 0 exceeds unity (Ca = 0).For increasing Ca, the capsules extend considerably in the axial direction while only marginally narrowing (Fig. 6, panels (d-ii), (d-iii) and (f-ii)), so that the thickness of the lubrication layer between their surface and the wall of the tube remains approximately constant.We also observed axially oriented wrinkles around the circumference of the rear half of the capsule barrel (panels (d-ii), (d-iii) and (f-ii) of Fig. 6) due to the significant lateral compression exerted on the capsule membrane, as discussed by Hu et al. 50.The capsule only adopts a parachute shape (Fig. 6,  (d-iv) and (f-iii)) for sufficiently large values of Ca in excess of those required in the unconfined channel, because the confinement imposes axial tension on the capsule.
In contrast, the deformation of deflated capsules with increasing Ca is much closer to that observed in the absence of confinement (see Fig. 6(b,e,g)).This is because the deflated capsule has excess membrane that can accommodate the increased confinement without significantly stretching its membrane axially.It follows that the threshold Ca th beyond which the shear flow stretches the membrane decreases with increasing confinement β , as shown in Fig. 8(a), and we expect Ca th to tend to zero for sufficiently high β .However, when β decreases below unity, Ca th increases sharply.
Furthermore, we found that the spherical capsules typically break more easily under flow than the deflated capsules through rupture of their membrane.For spherical capsules, the threshold value of Ca for membrane rupture is always less than 0.1, as shown with a green dotted line in the inset of Fig. 7, where the vertical error bar indicates the Ca interval between intact and ruptured capsules.This suggests that the spherical capsules do not adequately model the transport of RBCs, which can sustain higher relative shear stresses and typically remain intact for Ca ≥ O(10 −1 ) 48,51 .In contrast, the deflated capsules deform within the same Ca range as RBCs (see the orange asterisks in Fig. 7) and only rupture at Ca values of approximately five times the value for the spherical capsules (purple dotted line in Fig. 7).Thus, the reduced volume and excess membrane of the deflated capsules make them a useful proxy for the RBCs.The results of Fig. 7 indicate that the elongation of both spherical and deflated capsules is mainly determined by Ca and β .In the absence of flow (Ca = 0), we use geometry to predict the dependence of the static capsule shape on β .If we assume both types of capsules are spherical for unconfined geometries (β ≤ 1), the initial capsule length and width approximately match their effective diameter L 0 = W 0 = d eff and L 0 /D = β .For β > 1, the capsule shape can be approximated by a cylinder with hemispherical caps at both ends (see inset schematic in Fig. 8(b)), and volume conservation implies which can be used to express L 0 /D in terms of the confinement parameter β = d eff /D.Hence, Eq. ( 7) gives Fig. 8(b) shows that this expression accurately captures the experimentally measured values L 0 /D for β > 1 for spherical capsules (green symbols), while it slightly underestimates data for deflated capsules (purple symbols) owing to their buckled initial shapes.We use Eq. ( 8) to rescale relative capsule elongation L/D in Fig. 8(d), leading to the engineering strain L/L 0 − 1 that is effectively zero at Ca = 0 for the spherical capsules and rises slightly above zero for the deflated capsules at Ca < Ca th .
Assuming that the cylindrical barrel region of a capsule is the primary source of its membrane surface extension in flow, and neglecting capsule width reduction (W = D) at increasing Ca, the relative surface extension is given by approximating the membrane surface strain with the capsule elongation strain L/L 0 −1.Further assuming that local membrane deformations are sufficiently small for the total elongation strain to be proportional to the applied shear stress (under steady flow in a confined channel), we have 9 where γw ∼ ū/D approximates the wall shear rate, based on the mean velocity ū and the channel diameter D, and k is a dimensionless proportionality coefficient.The stress-strain balance given by Eq. ( 10) captures the observed linear growth of the relative capsule elongation as a function of Ca (Fig. 7) and a linear relationship between the flow-induced elongation rate (L/L 0 − 1)/Ca and the confinement parameter β (Fig. 8(c)).
Mendez and Abkarian 51 rescaled the capillary number as (φ C) −1 β Ca to account for non-spherical capsule shapes and membrane pre-stress, introducing φ , a geometric quantity characterising the capsule deflation (i.e., related to the reduced volume ratio α), and C, a non-dimensional pre-stress constant.The inverse of the parameter k in Eq. ( 10) can therefore be interpreted as a measure of the combined effects of φ and C.
The compressive stress associated with larger surface-to-volume ratio of the deflated capsules contributes to the smaller slope coefficient in Eq. ( 11) compared to the spherical capsules, which corresponds to the parameter k in Eq. ( 10).Using Eq. ( 11) to account for the confinement, we rescaled the excess capillary number above its threshold value, based on Ca th (see Fig. 8(a)).Fig. 8(d) shows the scaled capsule elongation (engineering elongation strain) L/L 0 (β ) − 1 as a function of the effective capillary number (Ca −Ca th (β )) /λ (β ).Therefore, in Fig. 8(d), both spherical and deflated capsules are unstrained at zero effective capillary number.This scaling is sufficient to approximately collapse the data onto a master curve (orange line in Fig. 8(d)), which indicates that the deflated capsules exhibit similar flow-induced deformation to the spherical ones for Ca > Ca th .Fig. 8(d) also highlights that the deflated capsules can reach larger maximum elongation strains than the initially spherical capsules for a wide range of flow confinements.

Conclusions and outlook
In this paper, we have developed a 3D nested capillary microfluidic device which can robustly fabricate a large number of monodisperse PDMS microcapsules with ultra-thin and soft membranes as a physical model for RBCs.The geometrical and mechanical parameters of these non-ageing capsules (e.g., size, membrane thickness and membrane elasticity) are accurately controlled by varying the flow conditions and the chemistry of the membrane.This means that polydisperse suspensions of capsules of different size and/or stiffness can also be obtained by mixing populations of capsules manufactured under slightly different flow conditions.We deflate our capsules using osmosis to accurately match the reduced volume of real RBCs.This enables our capsules to exhibit the large elastic deformations characteristic of RBCs in confined flow without rupturing.We compare the steady propagation of initially spherical and deflated capsules for a wide range of capillary numbers and confinement ratios.The presence of compressive stresses induced by capsule deflation delays the elongation of deflated capsules for Ca below a threshold value.Beyond this confinement-dependent threshold, capsule elongation increases approximately linearly with Ca.We show that only deflated capsules with the same reduced volume as RBCs exhibit comparable flow behaviour to RBCs over a similar range of capillary number (Ca ≥ O(10 −1 )).Although both spherical and deflated capsules can adopt the typical parachute-like shape of steady RBC flow, symmetry-breaking into the slipper-like shape only occurs for the deflated capsules at sufficiently high Ca.To our knowledge, this is the first time when an experimental model quantitatively reproduces the steady-state deformations of RBCs.
Similar to computational models of RBCs, our capsules provide a physical model for the deformation of RBCs in flow.Such a model can be advantageous compared with experimentation on real RBCs because control and robustness enable the systematic variation of parameters.However, our capsules remain idealised in that they do not exhibit the biconcave shape of RBCs 20 , nor do they match the viscosity ratio between internal and external fluids and the encapsulating membrane is hyperelastic rather than viscoelastic 51 .More generally, they bypass key physiological effects, such as RBC-specific membrane biochemistry, cytoskeleton effects 51 and aggregation phenomena 52 .Despite these simplifications, suspensions of these ultra-soft deflated capsules provide a powerful tool to explore the rheology of soft-particle suspension flows, with applications to haemodynamics and haemorheology in complex microvascular tissues, such as the human placenta 13 , as well as to other areas of biomedicine and industry, such as targeted drug delivery 3 and enhanced oil recovery 53 .

S1 Structure design of the Teflon end caps
The Teflon end caps are fabricated according to the design reported by Levenstein et al. 1 , as shown in Fig. S1.Each of them has a large circular recess (2.0 mm) at the centre of one end of the Teflon cylinder.A smaller recess (1.5 mm) containing a hole (1.0 mm) through the centre of the cylinder is manufactured to allow the injection or collection capillary to insert into.A radial hole (1.5 mm) is drilled into the small recess to connect the tubing that supplies either the middle or outer liquid.

S3 Measurement of Young's modulus
We employ a commonly accepted compression method that has been described in the reference 2 to measure the Young's modulus of cured PDMS with the mixing ratio of PDMS base to the crosslinker ranging from 10:1 to 40:1.Liquid PDMS is moulded into cylindrical samples of 2.0 cm in diameter and 3.0 cm in height, followed by a uniaxial compression test with an Instron 5569 machine (Instron, High Wycombe, UK), where the loaddisplacement relationship is recorded, as shown in Fig. S2  where L (N) is the force exerted on the top surface of the test cylinder, A 0 is the initial area of the top surface, ∆h is the displacement of the top surface of the cylinder and h 0 is its initial height.We apply a thin lubricating layer of Vaseline to the top and bottom surfaces of the test cylinders to ensure they are deformed uniaxially.The Young's modulus is then obtained by fitting the experimental stress-strain data with the theoretical models of elastic materials.Here, we consider three common models: Hookean model, neo-Hookean model and the two-term Mooney-Rivlin model 3 .By assuming the material to be isotropic, homogenrous and incompressible under uniaxial deformations, a relationship between the axial stress and one-dimensional strain is derived from the basic equations.Then, the nominal stress-strain relationships for these three models are expressed as below 2 .Hookean model: Neo-Hookean model: (S4) Mooney-Rivlin model: In Eq.S3 to S5, E (Pa) is the Young's modulus, C 1 (Pa) and C 2 (Pa) are the parameters obtained by fitting the experimental data with corresponding equations.For small deformations (ε 1), the neo-Hookean and Mooney-Rivlin models reduce to σ = 6C 1 ε. (S6) Then, the Young's modulus modulus is calculated as Table S1 The Young's modulus E obtained by fitting Hookean, neo-Hookean and Mooney-Rivlin models to the experimental stress-strain data (unit: kPa).The error comes from the standard deviation of results performed at different compression rates (from 0.01 mm/s to 1.00 mm/s).Fig. S2 (Right) shows an example of the fittings of Eq.S3 (red), S4 (blue) and S5 (green) to the experimental stress-train data (black +).In this case, PDMS and its crosslinker are mixed in 40:1, the compression rate is 0.01 mm/s.The results show that Hookean model gives a poor fitting to the experimental data, which indicates that the generally accepted linear elasticity of

Fig. 1
Fig. 1 Symmetrical parachute-like (left) and symmetry-broken slipperlike (right) shapes of flowing RBCs at steady state in a cylindrical glass capillary tube 6 .

Fig. 2
Fig. 2 (a) Schematic diagram of the experimental setup for capsule fabrication.The inset shows the labelled image of the 3D nested glass capillary device mounted in a perspex holder.(b) Schematic illustration of the capillary device to make W/O/W double emulsions.The optical microscope image shows the formation of double emulsion droplets at typical dripping flow regime (Q i = 24 µL/min, Q m = 3 µL/min and Q o = 150 µL/min for the inner, middle and outer phases respectively).(c) Schematic diagram of microcapsule cured from double emulsion templates.
(a) shows the distribution of outer and inner capsule diameters measured from a sample of 70 capsules manufactured with Q i = 35 µL/min, Q m = 10 µL/min and Q o = 300 µL/min.The mean outer and inner diameters are d o = 373.7 ± 1.8 µm and d i = 343.9±1.4 µm, respectively, with very small standard deviations of 0.4% and 0.5% of the mean values, respectively.

Fig. 3
Fig. 3 (a) Capsule diameter distribution determined for 70 PDMS capsules fabricated at Q i = 35 µL/min, Q m = 10 µL/min and Q o = 300 µL/min.(b) Relative membrane thickness 2δ /d o as a function of the middle to inner phase flow rate ratio Q m /Q i .The errors are the standard deviation of at least 50 capsule samples.Dashed line indicates the theoretical value predicted by Eq. 1 according to the mass conservation.The dots marked red indicate our red blood cell models.(c) Power-law fitting of the capsule diameter (d o ) as a function of the flow-rate ratioQ o /(Q i +Q m ).The relative membrane thickness is keeping constant while changing the outer phase flow rates: red circles -9.14%, purple squares -5.90%, blue diamonds and green triangles -3.85%.

Fig. 3 (
c) shows that our capsule diameter also agrees well with Eq. 2 (the continuous dashed line), giving a = 583 ± 7 µm and b = −0.23 ± 0.01.For our scaled-up RBC model, the capsule size is kept at approximately 350 µm.

)
For 350 µm capsules with a membrane thickness of 4% of the radius, the surface shear elastic modulus G s = 0.098 N/m and the bending modulus κ = 1.6 × 10 −12 N•m.We also define the capillary number (Ca) based on the viscosity of the suspending J o u r n a l N a me , [ y e a r ] , [ v o l .] , 1-S2 | 5 fluid µ ext and the surface shear modulus G s as

Fig. 4
Fig. 4 Young's modulus as a function of the mixing ratio of PDMS base to crosslinker predicted by Mooney-Rivlin model 46 .Each point corresponds to the mean value of five replicate measurements, and the error bars come from errors of the fitting parameter.

Fig. 7
Fig. 7 Variation of the capsule deformation ratio (L/W ) with Ca.The solid dots and hollow squares correspond to the spherical and deflated capsules, respectively, where the different colour indicates different capillary diameters (black -0.4 mm, red -0.3 mm and blue -0.2 mm).The error bar comes from the standard deviation of the parameters for at least 8 capsules.The dashed lines are the linear fittings of the data.The orange asterisks denote relative deformation of the RBCs in a 10 µm silica capillary tube (β ≈ 0.6) measured by Lanotte et al. 48.The green and purple dotted lines indicate the threshold Ca for membrane rupture for spherical and deflated capsules, respectively, summarised in inset as a function of the confinement β .The labels refer to the capsule snapshots shown in Fig. 6.

Fig. 8
Fig. 8 (a) The threshold value of Ca for deflated capsules starting to be stretched as a function of β .The dotted line here is only used to guide the eyes.(b) The initial capsule length scaled by the tube diameter L 0 /D as a function of confinement β for β > 1.The black line indicates the theoretical values predicted by Eq. (8).(c) The linear relations between the capsule elongation rate 1/λ and β .The dash dotted lines are the linear fittings and the error bars come from the errors of the fitting.(d) Variation of the scaled capsule elongation as a function of scaled Ca.

Fig. S1
Fig. S1 Structure of the Teflon connectors (Left).The measured load-displacement data is then converted into the nominal stress σ (Pa) and train ε (mm/mm) according to σ =

Table 1
Comparison between key capsule parameters.All measurements were taken at (20 ± 1) • C.