Sorting of capsules according to their stiffness: from principle to application

We assess experimentally the ability of a simple flow-based sorting device, recently proposed numerically by [Zhu et al., Soft Matter, 2014, 10, 7705-7711], to separate capsules according to their stiffness. The device consists of a single pillar with a half-cylinder cross-section which partially obstructs a flow channel so that initially centred, propagating capsules deform and circumvent the obstacle into an expanding channel (or diffuser). We perform experiments with millimetric capsules of fixed size which indicate that the deviation of the capsule in the diffuser varies monotonically with a capillary number - the ratio of viscous to elastic stresses - where the elastic stresses are measured independently to include the effects of pre-inflation, membrane thickness and material properties. We find that soft capsules with resistance to deformation differing by a factor of 1.5 can be reliably separated in the diffuser but that experimental variability increases significantly with capsule stiffness. We extend the study to populations of microcapsules with size polydispersity. We find that the combined effects of increasing capsule deformability and relative constriction of the device with increasing capsule size enable the tuning of the imposed flow so that capsules can be separated based on their shear modulus but irrespectively of their size.


I. INTRODUCTION
The sorting of cells from complex suspensions such as blood is key to many medical diagnostic and treatment strategies [1]. The propensity of cells to deform under mechanical loading is a biophysical marker which can help distinguish normal and cancerous cells.
Cancerous cells are usually softer than their healthy analogues, but they may also be stiffer in a few instances. In fact, normal to cancerous cell stiffness ratios between 0.7 and 32 have been measured by atomic force microscopy in a wide range of human tissues [2]. In malaria, the progressive reduction in red blood cell (RBC) deformability is associated with the growth and development of the parasite inside the cell as well as an approximately threefold increase in shear modulus of the encapsulating membrane, measured locally with pipette aspiration. These effects have been shown to result in an approximately tenfold increase in apparent shear modulus of the RBC [3]. RBC disorders such as sickle cell disease are also associated with increasing cell stiffness -an approximately threefold increase in membrane shear modulus [4] -and a broadening of the cell size distribution [5].
RBCs are examples of natural capsules consisting of a fluid core encapsulated by a semipermeable membrane [6]. In fact, the transport and flow-induced deformation of RBCs and other biological cells are commonly modelled with idealised capsules and soft beads [7,8].
The complex fluid-structure interaction flows of single particles or suspensions depend of the relative magnitude of hydrodynamic and elastic stresses. Flow-based separation of cells and capsules according to stiffness generally relies on channel geometries which promote extensional flows and thus capsule deformation. However, the resistance to deformation of a capsule is sensitive to both shear modulus of the membrane and capsule size if assuming constant membrane thickness. This means that robust sorting devices must either pre-sort cells according to size or be insensitive to size polydispersity. In this paper, we demonstrate experimentally the sorting of capsules according to their shear modulus irrespectively of their size.
Methods developed for the size-based separation of rigid, spherical particles such as deterministic lateral displacement (DLD) devices consisting of arrays of pillars [9,10] have been extended to sort particles according to their stiffness, as recently demonstrated with a two-dimensional numerical model of vesicle transport in a DLD [11]. In contrast, inertial separation techniques [12] have shown reduced efficiency when applied to deformable par-ticles [13]. The use of branched geometries has been investigated numerically in order to design a deformability-based sorting device [8] and explore the path selection of a capsule in the presence of inertia with a view to designing an enrichment device [14,15]. Moreover, sorting according to deformability was recently demonstrated experimentally in a T-junction for capsules of fixed size [16]. Moderate to high thoughput experimental cell-sorting methods are also emerging on microfluidic platforms. These include the deformation of single cells through generations of tapered constrictions driven by oscillatory flow [17], the flow through a periodic array of ridges oriented diagonally to the main stream [18] and rapid image analysis of individual cells deformed in a cross-flow with a throughput of up to 2000 cells per second [19].
In this paper, we focus on a simple sorting device recently introduced numerically by Zhu et al. [20] where a single pillar, with a half-cylinder cross-section, partially obstructs a flow channel; see figure 1. When capsules flowing along the centreline of the channel reach the pillar they are compressed along the flow direction and thus elongate tangentially to the curved obstacle. The deformation of the capsule as it circumvents the obstacle determines its trajectory in the downstream expanding channel (or diffuser) which it reaches after clearing the constriction created by the pillar. In the limit of Stokes flow, the deformation of the capsule is governed by an elastic capillary number, which corresponds to the ratio of viscous to elastic shear stresses. Zhu et al. [20] used three-dimensional boundary integral simulations to show distinct paths taken by capsules of different elasticity when circumventing the pillar.
A microfluidic realisation of this device was used to demonstrate experimentally that elastic microcapsules differing by a factor of approximately three in shear modulus could travel along separate trajectories if the imposed flow was sufficiently strong [21].
The objective of this paper is to comprehensively assess the sorting ability of the halfcylinder device for capsules of fixed size and different stiffness but also for polydisperse capsule populations. Experiments performed on neutrally buoyant, millimetric capsules of controlled shape, size and stiffness are used to demonstrate that the trajectories of constantsize capsules are governed by an effective capillary number based on a measure of the force required to deform the capsule between parallel plates. This metric accounts for the effect of pre-inflation and different membrane thickness in addition to variations in the elastic properties of the membrane. Hence, they can be reliably separated but we find that the device performance decreases significantly with increasing capsule stiffness. We then turn 4 to the sorting of two populations of microcapsules with shear moduli differing by a factor of three, where each population comprises a wide range of capsule sizes between 20 µm and 120 µm. In addition to the increased propensity of the larger capsules to deform, the relative channel constriction which must be cleared by the capsule to reach the diffuser increases as the capsule size increase because the size of the device is fixed. We find that these two effects combine to separate capsules according to stiffness irrespective of the capsule size, provided that the capsules are larger than the width of this constriction.
The outline of the paper is as follows. The methods used to fabricate and test capsules in similar millimetric and micrometric devices are detailed in §II and the flow field in the microdevice is mapped out with rigid tracer particles in §II C. The paths of capsules in the two devices are compared in §III A for similar relative capsule sizes. The results of experiments on millimetric capsules of fixed size and polydisperse microcapsule populations are then presented in § §III B and III C, respectively. The significance of the results in terms of stiffness-based sorting is discussed in §IV.

II. MATERIALS AND METHODS
A. Capsule preparation and characterisation

Millimetric capsules:
The millimetric capsules consisted of a liquid core encapsulated in a cross-linked ovalbuminalginate membrane [22]. Their preparation and characterisation have been previously described in detail [16]. Briefly, we prepared spherical ovalbumin-alginate gel beads by dropwise addition of a solution in water of sodium-alginate (1 % w/v), propylene glycol alginate (2 % w/v) and ovalbumin (8 % w/v) to a solution of calcium chloride (10 % w/v).
We then cross-linked their outer-shell before re-liquefying the gel core of the bead. The manufactured capsules were stored in a saline solution (11 g/l NaCl) and reached equilibrium after approximately 24 hours. During this period, water permeated through their membrane, which resulted in the inflation of the capsules to a size larger than the initial bead radius by up to 25 %, depending on membrane elasticity and initial size. This implies the presence of a significant pre-stress in the capsule membrane. Solid beads were made in a similar manner to capsules, except that their core was not re-liquefied. 5 We selected four capsules with an average diameter D = 3.90 ± 0.03 mm and one elastic bead D bead = 3.83 mm for experimentation [16]. The capsules were from several batches manufactured under different experimental conditions in order to access different stiffness values. Hence, the capsules also had different values of inflation (1.15 ≤ D/D bead ≤ 1.24), membrane thickness h (0.18 ≤ 2h/D ≤ 0.23) and sphericity (ratio of the minimum to maximum diameter 0.88 ≤ D min /D max ≤ 0.93). The sphericity of the elastic bead was D min /D max = 0.85.
We characterised the elastic properties of each capsule by measuring the constitutive relation that governs capsule deformation by compression testing between parallel plates.
An Instron 3345 Single Column Testing System (5 N load cell, accuracy ±0.5 mN) was used to measure the force exerted by a top plate lowered quasi-statically to compress a capsule placed on an anvil within a saline bath [16]. Measurements were performed at most three days before conducting the flow experiments since the age of the capsule influences its mechanical properties significantly. The large deformations routinely observed when capsules were propagated in flow meant that a nonlinear form of the constitutive law was necessary to characterise their deformation. In addition, their wall thickness was significant which meant that a membrane model was not appropriate for these capsules and the stiffness due to pre-inflation is not captured by the surface shear modulus of the capsule membrane.
Hence, we chose the force required to deform a capsule to 50% of its original diameter, F 50% , as a direct experimental measurement of the capsule resistance to deformation. The measured values of F 50% for each capsule and the elastic bead are listed in Table I Ovalbumin microcapsules were prepared by interfacial cross-linking [24]. Briefly, a waterin-oil emulsion was formed by mechanical agitation using a 10 % ovalbumin solution in a phosphate buffer at pH 5.9 or pH 8 for the preparation of soft and stiff capsules, respectively. Cyclohexane added with 2 % (w/v) sorbitan trioleate was used as the external phase. Cross-linking of ovalbumin at the oil/water interface was induced by adding 2.5 % (w/v) terephtaloyl chloride to the organic phase. After 5 minutes, the chemical reaction was stopped by dilution with a solution of chloroform:cyclohexane (1:4, v/v). Capsules were then rinsed with an aqueous solution of polysorbate, followed by pure water. They were stored in water at 4 • C and suspended in glycerol when needed for the flow experiments. The viscosity of the encapsulated fluid (supernatant of the capsule suspension) was measured using a cone-plate viscometer (Haake) and found to vary in the range 0.8 ≤ µ ≤ 0.9 Pa s between experiments.
The elastic properties of the microcapsules were determined by inverse analysis techniques [25]. Capsules were imaged during steady propagation in a straight, fluid-filled capillary tube of circular cross-section to measure their two-dimensional contours in the mid-plane and velocity U . The viscosity µ of the suspending fluid was measured separately. The shape of an initially spherical capsule flowing through a narrow capillary of diameter d depends solely on two parameters: the confinement ratio D/d, which can be extracted from experimental data, and the capillary number, defined as the ratio of viscous to elastic forces Ca = µU/G s , where G s is the surface shear modulus of the capsule and U is the mean velocity of the suspending fluid [26]. The capillary number, and thus the value of G s , were determined for each capsule by comparison between the contour shape and a library of numerical profiles obtained for the appropriate confinement ratio and a range of Ca [25,27].
Values of the surface shear modulus G s = 0.030 ± 0.007 N/m and G s = 0.081 ± 0.026 N/m were obtained for the capsules fabricated at pH 5.9 and pH 8, respectively, with 5 min reticulation time. These results are consistent with previously published characterisation of capsules prepared according to the same protocol [25].
The polydispersity of the two microcapsule populations was quantified in sample of approximately 170 capsules, which were confined between a glass slide and a cover slip separated by a 150 µm thick spacer. Images from an inverted microscope (DMIL LED, Leica     Microsystems GmbH, Germany) were analysed with ImageJ to determine the diameter of each capsule using a circular fit. We found that the mean and standard deviation of the capsule diameter to be D = 60 ± 18 µm and D = 79 ± 21 µm for stiff and soft capsules, respectively.

B. Sorting devices
Similar sorting devices were used to separate millimetric and micrometric capsules according to their stiffness ( figure 1(a,b)). They consisted of a main channel of rectangular cross-section with width w and depth h listed in figure 1(d) alongside other key dimensions.
In We henceforth refer to this region of the device as the diffuser, which featured an opening angle of 45 • in both devices. We now describe the experimental details pertaining to each sorting device in § §II B 1 and II B 2.

Millifluidic experiments:
The millimetric sorting device ( figure 1(a) During operation, the sorter was placed on the stage of an inverted optical microscope (DMIL LED, Leica Microsystems GmbH, Germany) equipped with a high speed camera (Fastcam SA3, Photron, USA). Images were acquired at 5000 frames per second. A 10× magnification was chosen so that propagating capsules could be tracked from the obstacle to the diffuser. Images were analysed using the ImageJ software.

C. Flow field in the microfluidic sorter
The role of the diffuser is to amplify the stiffness-dependent displacement of the capsule imposed by the obstacle by increasing the separation between capsules of different stiffness.
Prior to introducing capsules to the device, we mapped typical trajectories past the obstacle and through the diffuser by propagating Lycopodium spores (Sigma Aldrich). These are rigid particles with a diameter of approximately 20 µm, which is less than half the width of the narrowest passages in the device between the half-cylinder and the side walls of the main channel, hereafter referred to as the constriction. Therefore, they could be made to propagate through the narrow passages at different distances from the obstacle.
By injecting a very dilute suspension of Lycopodium powder in glycerol through the crossbranches of the flow focusing module (figure 1(e)) at (P ext , P int ) = (4500 mbar, 1200 mbar), we captured a range of different trajectories, which are shown in figure 2(a)  Note that we expect a similar value in the millifluidic device because the size ratio w/λ is the same and so is the opening angle of the diffuser. Values of β > β max correspond to particles travelling through the constriction closer to the channel wall than to the obstacle wall. When capsules are propagated along the centre line of the main channel, as discussed in §II B 2, they deform as they approach the obstacle and circumvent it adjacent to its surface. Also, the capsules that flow past the obstacle necessarily have a local width that is less than the constriction width (w − λ)/2. Hence, they are associated with a centroid position δ ≤ δ max , which yields β ≤ β max . All the capsules propagate along approximately straight lines in the diffuser when they are more than two capsule diameters from its extremities (in the streamwise x-direction).
This is illustrated in figure 3(a) where the trajectory of the capsule centroid is shown with blue squares and a superposed solid blue line highlights the region of the diffuser where the trajectory is linear. As in §II C, we quantify the deviation of the capsules from the centre line with the angle β spanning the arclength from the centre line of the device to the capsule trajectory, which is independent of the measurement location along the diffuser.
We also define the capsule displacement ζ as the distance separating the centroid of the capsule from the centreline at the end of the diffuser. Figure 3 shows that β decreases for increasing flow rate or pressure head. The use of different types of capsules and different capsule confinement ratios in the two devices suggests that this behaviour is robust.  Table I This is because capsule deformation on the obstacle increases with Q as previously shown in figure 3. For flow rates Q ≤ 5 ml/min, β decreases more steeply the softer the capsule.
This decrease becomes gentler for Q ≥ 5 ml/min and a clear separation emerges between the trajectory angles of capsules of different stiffness, indicating that for a given flow rate Q, the trajectory angle β increases with capsule stiffness.
In contrast with capsule behaviour, the trajectory angle of the elastic bead (C5) remains approximately constant over the entire range of flow rates investigated. This is because the deformation of the elastic bead is small within this parameter range, so that its trajectory is approximately independent of the flow rate applied. Hence, the elastic bead, which is at least 50 % stiffer than the stiffest capsule (C4) and 8 times stiffer than the softest capsule (C1), behaves approximately like a rigid sphere in this device. The trajectories of the elastic bead were therefore used to estimate the trajectory angle for a rigid particle, β max = 17.6 • ± 0.5 • , which also corresponds to the maximum angle of a capsule in the limit of vanishing deformation. In contrast, the smallest trajectory angle recorded for the softest capsule (C1) is β min = 8.6 • ± 0.1 • for Q 25 ml/min.
In figure 4(b), the trajectory angle is shown as a function of the elastic capillary number based on the force F 50% required to compress the capsule between parallel plates to half of its diameter (see section II A 1), where µ the viscosity of the fluid, hw the cross-sectional area of the main channel and D the capsule diameter [16]. The main source of error on the capillary number is the ±0.5 mN uncertainty on the value on F 50% , which is propagated into horizontal error bars in figure   4(b).
The data collapse onto a monotonically decreasing master curve as a function of Ca 50% to within experimental uncertainty. After a steep initial decrease for Ca 50% ≤ 0.01, β appears to tend towards an approximately constant value at large capillary numbers (Ca 50% > 0.03). The occurrence of a non-zero plateau value is expected in the limit of large Ca 50% (corresponding to infinitely soft objects) because the distance between the capsule centroid and the obstacle must remain greater than its membrane thickness in order to ensure the conservation of fluid within the capsule. However, we remain far from this idealised limit (see figure 3(b)), which would in practice be superseeded by capsule rupture. The experimental measurements previously shown in figure 4 demonstrate the separation on average of capsules of fixed size in the millifluidic device. However, in order to achieve reliable sorting, the trajectory angles of capsules with different stiffness must remain different in every repetition of the experiment. Figure 5 shows all the values of β obtained at a fixed flow rate of Q = 10 ml/min. The data is grouped by particle and presented as a function of initial offset κ, using the same colour scheme as in figure 4. The initial offset of capsules, following their passage through the centring module (see figure 1(a)) was | κ | ≤ 0.5 mm (i.e. < 3 % of the width of the channel) for the capsule experiments and | κ | ≤ 1.2 mm (i.e. < 7.5 % of the width of the channel) for the bead.
The flow behaviours of capsules C2 and C3, which have values of F 50% differing by less than 16 %, are indistinguishable because of the experimental scatter of the trajectory angles of approximately 10 %. Figure 4 also shows that there is no overlap between the trajectory angles measured for capsules C1 and C2/C3, but that the largest value measured for C1 is only marginally smaller than for the smaller value obtained with C2. In fact, the minimum difference in trajectory angle between capsules C1 and C2 is ∆β = 0.8 • , and thus the minimum distance separating the centroids of the capsules at the outlet of the diffuser is ∆ζ = 0.4 mm (10 % of diameter), in contrast with ∆β = 1.5 • (∆ζ = 1 mm) on average.
This means that the factor of approximately 1.5 between the stiffness of C1 and C2/C3 is the smallest contrast that can be resolved by the millimetric device.
The most striking feature of figure 5 is the significant increase in the experimental scatter of β with increasing stiffness of the particle, which does not appear to be strongly correlated to the initial centring of the capsules, see also §IV. The scatter varies from approximately 1 • for the three softest capsules to 2 • for capsule C4 and 10 • for the elastic bead C5. This means that the elastic bead cannot be reliably separated from capsule C4. However, capsule C4 can easily be separated from the softer capsules tested. We attribute this behaviour to the propensity of the softer particles tested to deform in flow (away from the obstacle). If the position of a capsule in a channel is perturbed, its shear-induced deformation generates lift forces that restore it to the centreline of the channel [31]. This means that the value of β is only weakly sensitive to small positional errors and non-uniformities in the sphericity or membrane thickness of these softer capsules. In contrast, the absence of lift forces in the flow of elastic beads over the range of flow rates investigated renders their pathway highly sensitive to small variations in experimental conditions. Hence, our measurements indicate that the decreasing level of fluctuation in particle position as the particle becomes softer is a key factor in making sorting realizable in practice.
Although capsules C1 and C2 are clearly separated in terms of the trajectory angle β, their actual separation distance measured at the outlet of the 27 mm diffuser amounts to ∼ 1 mm, which is less than the capsule diameter of 3.9 mm. Hence, the millimetric device provides a proof of concept of the sorting of capsules according to their stiffness, but segregating capsules into different channels would require a longer diffuser. This would enhance the separation distance at the cost of at least a tenfold increase in transit time, because a diffuser length ≥ 108 mm would be required to achieve a separation of 4 mm.     Zhu et al. [20] in that it operates for capsules larger than the constriction and relies on the constriction to divert the stiffer capsules to large trajectory angles, while the softer capsules acquire low trajectory angles by deforming on the half cylinder obstacle in the usual way.

IV. DISCUSSION AND CONCLUSION
We have tested the sorting device proposed in the numerical study by Zhu et al. [20] with two complementary experimental approaches. The device consists of a half-cylinder centred at the end of a wide channel, so that a capsule propagating along the centreline of the channel is compressed onto the cylinder, resulting in its elongation tangentially to the cylinder's surface. The deformation of the capsule as it circumvents the obstruction determines its trajectory in the downstream, linearly-expanding channel. Thus, this device enables the sorting of capsules according to their stiffness. Using accurately manufactured millimetric capsules of fixed size and different stiffness, we have shown that the angle of deviation of the capsule trajectory from the centreline of the channel depends solely on an elastic capillary number, based on the force required to deform the capsule to 50% in compression tests. This measure indicates the resistance to deformation of the capsule and thus, includes the effects of pre-stress and different membrane thickness which are significant 23 in our millicapsules. We find that we can reliably separate capsules with a stiffness constrast down to a factor of 1.5, but that the resolution of the device degrades as the capsule stiffness increases.
The second experiment used a microfluidic device to separate polydisperse populations of microcapsules according to their shear modulus. The fixed size of the device meant that the confinement of the capsules in the constriction increased with capsule size. However, their resistance to deformation decreases with size, so that larger capsules were more easily deformed. We demonstrated that two polydisperse populations of capsules with a factor of 2.7 between their shear moduli could be separated in the sorter provided that their diameter exceeds the width of the constriction by the side of the cylinder. This separation method relies on the stiff capsules filling the constriction as they squeeze through so that they adopt a large trajectory angle close to the maximum of 24 • . In contrast, the soft capsules deform sufficiently on the obstacle so that they can adopt a low trajectory angle which remains constant at approximately 10 • because the width of the capsules as they circumvent the half-cylinder does not increase with capsule size.
The large difference in trajectory angles achieved between the two populations is key to the robust sorting of these microcapsules because of the considerable fluctuations in the data from the microfluidic device. Although these may in part stem from imperfections in individual capsules, a key experimental parameter influencing the trajectory angle is the initial offset of the capsule from the centreline of the channel. In millicapsules, we found that for moderate offsets of up to 12.5% of the capsule diameter variations in the trajectory angle β (and therefore also the displacement ζ) of up to ±5% were weakly correlated with κ, implying the influence of other experimental variabilities. Increasing the offset to 60% of the capsule diameter yielded an approximately linear variation of the trajectory angle with offset, which increased by a factor of approximately 1.5. A similar trend was measured in the microfluidic device but the sensitivity of the capsule trajectories to the offset was considerably enhanced with an offset of 10% of the capsule diameter resulting in a 30% increase in ζ. Hence, the majority of the scatter in the data from the microfluidic device can be attributed to variations in initial capsule offset within that range.
The ability to robustly sort polydisperse suspensions according to stiffness with a single device, without the need for sifting the particles according to size before perfusing them into the microchannel, opens new opportunities for high-throughput separation of cells, with