Dynamics of vesicles in large amplitude oscillatory extensional flow

A mixture of 1,2-dioleoyl-sn-glycero-3-phosphocholine (DOPC, Avanti Polar Lipids) and 0.12 mol% of 1,2-dioleoyl-sn-glycero-3-phosphoethanolamine-N-(lissamine rhodamine B sulfonyl) (DOPE-Rh, Avanti Polar Lipids) is used to generate giant unilamellar vesicles (GUVs) with the electroformation process described by Angelova et al. [3]. For electroformation of GUVs, a stock lipid solution in chloroform is prepared with 25 mg/mL DOPC and 0.04 mg/mL DOPE-Rh for fluorescent imaging. Next, 10 $\mu$L of the lipid solution in chloroform is spread on a conductive indium tin oxide (ITO) coated glass slide (resistance 5 $\Omega$, 25$\times 50\times$1.1 mm, Delta Technologies) and dried under vacuum overnight. The pair of ITO slides are sandwiched together using a 1.5 mm Teflon spacer, forming a chamber with a volume of $\approx$ 2.4 mL and coupled to a function generator (Agilent 33220 A). The electroformation chamber is filled with a mixture of 100 mM sucrose solution (Sigma-Aldrich), and glycerol-water is added to achieve a total viscosity of 0.030 Pa-s measured using a benchtop viscometer (Brookfield) at 22^∘C. An alternating current (AC) electric field of 2 V/mm is then applied at 10 Hz for 120 min at room temperature (22^∘C). Under these conditions, DOPC lipid remains in the fluid phase [23]. Most of the vesicles prepared by this method are quasi-spherical and unilamellar with few defects in the size range of 5–25 $\mu m$ in radius.

It is challenging to observe vesicle dynamics in time-dependent extensional flow for long periods of time while simultaneously imposing precisely controlled flow rates. To achieve this, we used the Stokes trap [46, 28] to precisely position the center-of-mass of single vesicles near the center of a cross-slot microfluidic device for long times using model predictive control (Fig. 1a). Briefly, the centroid of a single vesicle is determined in real-time using image processing and fluorescence microscopy and is communicated to the controller. The controller determines the optimal flow rates through four-channels of the device to maintain a fixed vesicle position with desired strain rate. The flow rates are then applied through four independent pressure regulators (Elveflow). During this process, the device operates at a net positive pressure so that each of the four ports can act as inlet or outlet. This whole procedure requires $\approx$ 30 ms in a single cycle, as previously described [46, 58, 57]. In this work, a sinusoidal strain rate input is imposed (Fig. 1b) while simultaneously trapping a single vesicle such that:

	$\displaystyle\dot{\epsilon}_{x}(t)=-\dot{\epsilon}_{0}\sin\left(\frac{2\pi}{T}t\right)$
	$\displaystyle\dot{\epsilon}_{y}(t)=\dot{\epsilon}_{0}\sin\left(\frac{2\pi}{T}t\right)$

where $T$ is the period of the sinusoidal cycle and $\dot{\epsilon}_{0}$ is the maximum strain rate in one cycle. During the first half-cycle for $0<t<T/2$, the x-axis is the compressional axis and y-axis is the elongational axis ($\dot{\epsilon}_{x}(t)<0,\dot{\epsilon}_{y}(t)>0$), and the fluid is delivered from the two horizontal inlets of the microfluidic device by the pressure regulators (Fig. 1b). During the second half-cycle for $T/2<t<T$, the direction of flow reverses, and fluid is delivered by the two vertical ports in the cross-slot device as shown in Fig. 1b. We note that during vesicle trapping, the correctional pressure required for controlling the vesicle’s position is small compared to the magnitude of the base pressure used to generate the oscillatory extensional flow  [46]. Thus, the strain rate is well defined during the LAOE cycle, which is determined as a function of the input pressure using particle tracking velocimetry (PTV) as previously described  [26]. We also determined the characteristic response time for actuating fluid flow in the microfluidic device in response to a step change in pressure. For an extreme change in pressure from 0 to 4 psi (strain rate jump from 0 to $\sim$ 30 $s^{-1}$), the rise time and settling time are $\sim$20 ms and $\sim$300 ms respectively(Fig. S1). However, the maximum value of pressure used in our experiments is 0.4 psi, which is continuously varied with small incremental changes during the LAOE cycle, for which we generally expect much smaller characteristic response times. Nevertheless, the lowest cycle time $T$ in our experiments is 2 seconds which is much larger than the maximum characteristic response time for actuating flow in the device corresponding to a step input pressure.

For all experiments, single vesicles are first trapped and imaged for 10–30 s under zero flow conditions to allow for equilibration, followed by LAOE flow for at least 2 strain rate cycles. During the equilibration step, the vesicle reduced volume $\nu$ and equivalent radius $a$ are determined, as previously described [9, 26]. Reduced volume $\nu$ is a dimensionless quantity that measures the amount of osmotic deflation, and is described as:

$$\nu=\frac{3V\sqrt{4\pi}}{A^{3/2}}$$

(1)

where $V$ and $A$ are the vesicle volume and surface area, respectively. The equivalent radius $a$ of the vesicle is obtained as $a=\sqrt{A/4\pi}$. Specifically, $\nu$ is a measure of vesicle asphericity such that $\nu=1$ represents a perfectly spherical shape. For the experiments in this paper, the typical range of reduced volume is $0.75<\nu<1$.

The maximum strain rate $\dot{\epsilon}_{0}$ experienced by a vesicle in a half-cycle is non-dimensionalized to define a capillary number $\mathrm{Ca}=\mu_{out}\dot{\epsilon}_{0}a^{3}/\kappa$ where $\mu_{out}$ is the suspending medium viscosity, $a$ is the equivalent vesicle radius, and $\kappa$ is the membrane bending modulus. Prior to vesicle experiments in LAOE flow, we determined the average bending modulus of nearly spherical vesicles to be $\kappa=(22.3\pm 0.5)k_{B}T$ using contour fluctuation spectroscopy [26]. Similarly, the cycle period is rendered dimensionless by the bending time scale to define the Deborah number $\mathrm{De}=\mu_{out}a^{3}/\kappa T$. Single vesicle experiments are generally performed in the range $10<\mathrm{Ca}<1000$ and $0.1<\mathrm{De}<100$ by adjusting the input pressures and strain rate cycle periods. Only vesicles near the center plane of the microchannel (with respect to the z-direction) are considered during experiments. Single vesicle trajectories are analyzed using a custom MATLAB program that allows for determination of the vesicle deformation parameter in flow.

Experiments were performed in the range of approximately $10<\mathrm{Ca}<1000$ and $0.5<\mathrm{De}<100$, whereas the majority of the simulations are in the range of $1<\mathrm{Ca}<40$ and $1<\mathrm{De}<10$. Simulations were performed for several vesicles matching the conditions in the experiments, as discussed in the following section (Section III.1, Section III.1 and Fig. 4). It is possible to perform additional simulations at $\mathrm{Ca}\approx 100$, but current results suggest that the vesicle dynamics do not significantly change at higher Ca for quasi-spherical vesicles.

We observe three distinct dynamical regimes of vesicle dynamics based on the ratio between Capillary number and Deborah number. We refer to these regimes as symmetrical, reorienting, and pulsating based on the deformation characteristics of vesicles in each case. Vesicles in all three regimes can experience significant non-linear stress responses. Snapshots of vesicle shapes from simulations and experiments for each of these regimes over a full strain rate cycle are shown in Fig. 6 and Fig. 6.

We quantitatively compare the simulations and experiments by plotting the deformation parameter $D$ (defined in Eq. 12) and instantaneous strain rate $\mathrm{Ca}_{x}$ (defined in LABEL:eq:cax) as a function of time, as shown in Sections III.1 and III.1. Experimental trajectories are generally limited to 2–4 strain rate cycles due to the photobleaching of the vesicle membrane during fluorescence imaging experiments. Observing vesicle deformation over more strain rate cycles is experimentally feasible, however, we generally opted to observe dynamics under different experimental parameters (Ca, De) for the same vesicle in a series of subsequent experiments. For the numerical data, we simulated vesicle dynamics over at least 10 strain rate cycles.

Reorienting

Pulsating

Reorienting

Pulsating

Overall, the simulations discussed up to this point (including results in Fig. 6 and Section III.1) were performed using a prolate-like initial shape, because it is the global equilibrium shape for reduced volumes $\nu\geq 0.652$ [44]. These results suggest that the unequal stretching observed in the pulsating and reorienting regimes occurs during the steady limit cycle, for this particular initial shape. However, there are other local minimum energy shapes for vesicles, such as the oblate shape family. To determine whether the pulsating and reorienting regimes are possible with a different initial condition, we performed simulations using an oblate shape such that the initial deformation parameter was set to zero. We examined this initial condition because vesicle shape is isotropic in the x-y plane, where an image obtained through optical microscopy would show a circle. The oblate initial condition simulations test if the anisotropic deformations will still occur if the vesicle starts with a shape isotropic in the x-y plane rather than an initially anisotropic shape. Simulation results for the oblate initial condition are plotted in Fig. 7, which shows that vesicle dynamics during the steady limit cycle for the oblate initial condition (Fig. 7a) are the same as that observed from the prolate-like initial condition (Fig. 7b). We repeated these simulations at several other capillary numbers and Deborah numbers, observing no dependence of the dynamics on the initial conditions.

By comparing the deformation parameter results for each simulation, we can plot a phase diagram of different dynamical regimes observed during oscillatory flows. Which regime a vesicle experiences can be quantitatively determined by assessing the minimum and maximum deformation parameter over a cycle. If both the minimum and maximum deformation parameter are positive, the vesicle dynamics are classified as the pulsating regime, reflecting that the vesicle does not change orientation. If the vesicle has a positive maximum $D$ and a negative minimum $D$, we check if the differences in magnitudes are within a threshold value of 0.01. Should they be within 0.01 of each other, the vesicle is in the symmetrical regime, since the vesicle reaches the same maximum length twice a cycle. This threshold value was chosen heuristically to reflect the discretization accuracy. If the magnitudes are not within this threshold value, vesicle dynamics are classified as the reorienting regime. Results from this analysis is plotted in Fig. 8

The phase boundaries appear to be mostly linear, suggesting that the dynamics result from a simple interaction between the flow frequency and the strain rate, $\mathrm{Ca}/\mathrm{De}=\dot{\epsilon}_{0}/\omega$. Here, we derive the phase boundaries in the limit of a quasispherical vesicle [50]. For small excess area ($\Delta=4\pi(\nu^{-2/3}-1)\ll 1$), the vesicle shape is characterized by a perturbation series in terms of spherical harmonics [50]. In a planar extensional flow, there are only two modes excited for the leading order correction to the vesicle shape. When one solves for the deformation parameter as defined in Eq. 12, one obtains

	$\displaystyle D(t)$	$\displaystyle=(L_{\infty}-1)$		(13)
		$\displaystyle\left(1-2\left(1+\left(\frac{1+A_{o}}{1-A_{o}}\right)\exp\left(\frac{60}{\pi(32+23\lambda)}\frac{\mathrm{Ca}}{\mathrm{De}}\frac{1}{L_{\infty}-1}\left[\cos(2\pi\mathrm{De}~{}t)-1\right]\right)\right)^{-1}\right),$

where parameters $L_{\infty}=1+\sqrt{15/8}\left(\nu^{-2/3}-1\right)^{1/2}$ and $A_{o}=(\nu^{-1/3}l_{x}^{max}/2-1)/(L_{\infty}-1)$; $l_{x}^{max}$ is the maximum x-axis length of the vesicle. For the detailed derivation of these results, one can refer to the supporting information.

Following the definitions of the phase boundaries discussed previously, we can derive the two phase boundaries in the limit of $A_{0}\ll 1$, i.e. $\ln\left(\frac{1+A_{o}}{1-A_{o}}\right)\approx\frac{1}{(L_{\infty}-1)}\ln\left(\frac{1+D_{o}}{1-D_{o}}\right)$,

$$\frac{\mathrm{Ca}}{\mathrm{De}}=\frac{\pi(32+23\lambda)}{120}\log\left(\frac{1+D_{o}}{1-D_{o}}\right)\quad\text{for pulsating/reorienting phases}$$

(14)

$$\frac{\mathrm{Ca}}{\mathrm{De}}=\frac{\pi(32+23\lambda)}{60}\log\left(\frac{1+D_{o}}{1-D_{o}}\right)\quad\text{for reorienting/symmetrical phases}$$

(15)

In the above equations, $D_{0}$ is the maximum deformation parameter during the LAOE cycle. Note that the value of $D_{o}$ is determined by our numerical runs at the highest $\mathrm{Ca}$ and $\mathrm{De}$ numbers. Based on the quasispherical vesicle theory, the deformation phase boundaries depend on the viscosity ratio, where the factor ${(23\lambda+32)}^{-1}$ is related to the relaxation time of the quasi-spherical vesicle [50]. Fig. 8a shows the phase boundaries are accurately calculated by using Eq. 14 and Eq. 15 when the reduced volume is $\nu=0.8$. Increasing $\nu$ from 0.80 to 0.90 shifts the phase boundaries downwards, but maintains a similar linear relation (Fig. 8b). We also simulated viscosity ratio $\lambda=10$ and found that higher viscosity ratios shift the boundaries to higher capillary numbers. We include the dynamics evolution of $l_{x}$ and $l_{y}$ (simulations vs. analytical solutions) and $\lambda=10$ results in the supplementary materials for brevity.

For dilute vesicle suspensions where the macroscopic length scale is large in comparison to the size of the vesicles, the extra stress (the bulk stress contribution from the particles) is the product of the number density of particles and the particle stresslet: $\sigma_{ij}^{P}=n\>\tilde{S}_{ij}^{P}$. Using the boundary integral formulation, we calculate the particle stresslet [41]:

$$\tilde{S}_{ij}^{P}=\Int{\frac{1}{2}([[f_{i}]]x_{j}+[[f_{j}]]x_{i})}{S,D}-\Int{(1-\lambda)\mu_{\mathrm{out}}(v_{i}n_{j}+v_{j}n_{j})}{S,D}$$

(16)

where $[[\bm{f}]]$ is the surface traction, $\lambda$ is the viscosity ratio, $\mu_{\mathrm{out}}$ is the outer viscosity, $\bm{v}$ is the velocity, and $\bm{n}$ is the normal vector. We define the dimensionless particle coefficient of stresslet as:

$\displaystyle S_{ij}=\frac{\tilde{S}_{ij}^{P}}{\dot{\epsilon}\mu_{\mathrm{out}}V_{p}}$

(17)

where $V_{p}$ is the vesicle volume and $\dot{\epsilon}$ is the strain rate. Similarly the normal stress differences are defined as:

	$\displaystyle N_{1}=S_{xx}-S_{yy}$		(18)
	$\displaystyle N_{2}=S_{yy}-S_{zz}$		(19)

Comparing the normal stress differences to the strain rate, we can derive the rheological characteristics of a dilute vesicle suspension, such as the effective viscosity and bulk normal stresses [10]. For extensional flow rheology, a key quantity of interest is the extensional viscosity of a solution. Extensional viscosity is often characterized using a quantity known as a Trouton ratio (ratio of extensional to shear viscosity), which for a planar extensional flow is a multiple of $N_{1}$. For a planar flow, the extensional viscosity is

$\displaystyle\eta_{E}=\frac{\sigma_{11}-\sigma_{22}}{\dot{\epsilon}}.$

(20)

The planar Trouton ratio is

$\displaystyle\frac{\eta_{E}}{\eta}=4+\phi*N_{1},$

(21)

where $\phi$ is the volume fraction of vesicles in the suspensions, and $N_{1}$ is the first normal stress difference. Our simulations have focused on rather large deformations of the vesicle shape, therefore the stress response analysis will reflect the non-linear viscoelasticity.

Using the definitions of the particle stresslet and normal stress differences, we determine the vesicle stress as a function of time in extensional flow. In Section III.3, we show the stress response over two cycles for three sets of parameters; one from each of the three dynamical regimes discussed before. A linearly viscoelastic material will show purely sinusoidal normal stress differences for this type of plot, as there is a simple linear relation between the strain rate and the stress. On the other hand, for non-linear viscoelasticity, the normal stress differences will display more complex behaviors.

Section III.3 shows that vesicle dynamics in the three regimes (symmetrical, reorienting, and pulsating) have non-linear characteristics. To analyze these stress responses, we re-plot the data from Section III.3 into a Lissajous-type form with the instantaneous strain rate ($\mathrm{Ca}_{x}$) on the x-axis and the stress response on the y-axis (Section III.4). For this type of plot, a purely viscous material would display a straight line, whereas a purely elastic material would produce an elliptical curve. For example, the first and second normal stress difference for Newtonian flow around a rigid sphere corresponds to the lines: $N_{1}=10\cdot\mathrm{Ca}_{x}/\mathrm{Ca}$ and $N_{2}=-5\cdot\mathrm{Ca}_{x}/\mathrm{Ca}$. Here, we focus on $N_{1}$ because it is related to the extensional viscosity of the solution (Trouton ratio). We also discuss the $N_{2}$ stress differences for completeness.

In the symmetrical regime (LABEL:fig:Lissajous_stress_reponse_sym), we observe that $N_{1}$ is symmetric across the origin and that the lines for increasing and decreasing strain rate are nearly the same for $-2<\mathrm{Ca}_{x}<2$. On the other hand, $N_{2}$ differs significantly depending on the directionality of the flow. The $N_{1}$ curve is mostly linear in the region $-2<\mathrm{Ca}_{x}<2$ and is approximately equal to zero when the strain rate is zero, suggesting that the vesicle contributes a purely viscous response in that region. We further examine this region in more detail by comparing the vesicle deformation to the stress response. From the simulation video and the Lissajous-type deformation parameter curve (Section III.1), we know that the vesicle retains a prolate spheroid like shape and only changes marginally for the $-2<\mathrm{Ca}_{x}<2$ region. The relatively small amount of deformation that occurs in the $-2<\mathrm{Ca}_{x}<2$ region suggests that the vesicle acts like a rigid particle there, explaining the close to linear stress response for $N_{1}$ in the region. In the other strain rate regions, the stress differences shift rapidly in accordance to the vesicle’s large deformations and reorientation.

In the reorienting and pulsating regimes (Figs. 10c and 10b), the $N_{1}$ curves are no longer symmetric across the origin, and the stress responses for increasing and decreasing strain rate are distinct. The maximum $N_{1}$ response is larger in magnitude than the minimum for both regimes; this is likely due to the unequal amounts of deformation between the two strain rate period halves (Section III.1). For this analysis, qualitative differences between the shape of the reorienting and pulsating regime curves correspond to the extent of asymmetry in the $N_{1}$ response. Moreover, we observe vesicles in the pulsating regime can have a non-zero normal stress difference when the time-dependent strain rate is zero, as seen in Fig. 10c.

For a more quantitative analysis, we decomposed the stress responses into a Fourier series. This decomposition is commonly applied to large amplitude oscillatory shear (LAOS) experiments and is known as Fourier transform (FT) rheology. FT rheology is commonly performed using oscillatory shear flows on polymeric liquids to probe the shear stress response in the non-linear regime [19, 52]. The computation is straightforward and relies on taking the Fourier transform of the $N_{1}$ or $N_{2}$ stress difference:

$$f(k)=\Int{N_{1,2}(t)e^{-2\pi itk}}{t,-\Infinity,\Infinity},$$

(22)

In this way, the periodic stress signal is transformed into frequency space. Because the external flow field is sinusoidal, the strain rate ($\dot{\epsilon}$) and strain ($\epsilon$) are proportional to sine and cosine functions. Therefore, the Fourier transformed data provide a description of how the stress depends on different orders of the strain and strain rate. If the stress response was purely linear order, the Fourier transformation would show a single peak at the first mode. A non-linear stress response would have additional peaks at higher modes.

The Fourier decompositions for both $N_{1}$ and $N_{2}$ are shown in Section III.4, where it is clear that all three regimes show higher order behavior. For all regimes, we observe the expected behavior of the linear order mode being the highest amplitude with the higher order modes decreasing monotonically for $N_{1}$. On the other hand, the highest amplitude mode for $N_{2}$ is not the linear order mode, with the highest generally being the second or third mode. Comparing the $N_{1}$ decompositions between the dynamical regimes, we observe that the symmetrical regime does not have even order modes, whereas the reorienting and pulsating regimes have even higher order modes. This change in FT rheology is consistent with the phase boundary defined in Section III.1, and this transition can be used instead of the deformation parameter analysis to demarcate the phase boundary.

In large amplitude oscillatory shear (LAOS), the typical macroscopic stress response shows that the stress is an odd function of the direction of shearing[19]. Such a restriction is not necessarily expected in an extensional flow, but would be related to whether the microstructure of the fluid stretches symmetrically during these flows. In the symmetrical regime, both the vesicle stress response and deformation are time symmetric, leading to only odd order Fourier modes. The time symmetry does not hold for the reorienting or pulsating regimes, allowing for even order modes. Based on the currently available results, we do not expect droplets to have even order Fourier modes in LAOE, regardless of flow rate or flow frequency [33]. Broadly speaking, our results show that membrane-bound vesicles are an interesting example of how anisotropic microstructural deformations can lead to complex rheology.

We also investigated the transient dynamics of tubular vesicles in large amplitude oscillatory extension (Fig. 12). In general, we find that tubular vesicles undergo wrinkling/buckling instabilities during the compression phase of the flow cycle similar to quasi-spherical vesicles. However, we occasionally observe buckling instabilities that induce unexpected shape changes. In these situations, the vesicle’s initial, tubular shape is not recovered at the end of the flow cycle.

Fig. 12a shows experimental snapshots of a tubular vesicle with reduced volume $\nu=0.64\pm 0.02$ exposed to a sinusoidal strain rate at $\mathrm{Ca}=21.3$ and $\mathrm{De}=17.7$. In this situation, the vesicle exhibits pulsating motion along the x-axis with buckles during the compressional part of the flow cycle. The vesicle’s starting, tubular shape is recovered at the end of the LAOE cycle. To further demonstrate this behavior, we construct single vesicle Lissajous curves (Section III.5(d)) defined as plot of deformation parameter as a function of $\mathrm{Ca}$, and deformation parameter as a function of time (Section III.5(a)). These plots show the vesicle reaches the same value of deformation parameter $D\approx 0.7$ at the end of each of the three repeated flow cycles, implying that the vesicle conformation is fully recovered after deformation. There is decent agreement in the qualitative dynamics between the simulations and experiments in this region, but the simulated deformation parameters appear to be lower than the ones measures experimentally.

When the same vesicle is exposed to a flow cycle at a lower frequency ($\mathrm{De}=8.9$), the membrane has more time to deform in response to the flow. Here, the vesicle undergoes pulsating motion with wrinkles (Fig. 12b) and we observe appreciable deformation along $y$ axis in both the simulations and experiments, as shown in Section III.5b,e. Surprisingly, the experimental results show the vesicle deformation parameter reducing with each subsequent LAOE cycle. The deformation at the end of first cycle is $D\approx 0.7$ and it decreases to $D\approx 0.6$ at the end of second cycle, and further to $D\approx 0.5$ at the end of third cycle. Experimentally, it seems that the vesicle conformation changes over each LAOE cycle while our simulations predict no change over the strain rate cycles. By the end of third repeated cycle, we experimentally observe that the 2D shape of vesicle appears to be more spheroidal than tubular. Interestingly, the vesicle did not recover its original tubular shape even when relaxed for $\approx$ 2 min. It is noteworthy that we did not observe any reduction in deformation parameter at the higher flow frequency discussed previously (De = 17.7). These observations suggest that for a given $\mathrm{Ca}$, there appears to be a critical $\mathrm{De}$ below which the change occurs.

Finally, the same vesicle is exposed to LAOE flow cycle with an even lower frequency (De = 4.7). We observe that the vesicle undergoes full reorientation from the $x$ axis to $y$ axis, undergoes a wrinkling instability during compression and the initial spheroidal shape changes to a more spherical shape at the end of the first periodic cycle (Fig. 12c). The deformation behavior seen experimentally during the second repeated cycle is symmetric and follows similar dynamics as those observed for quasi-spherical vesicles. This behavior is more apparent in Section III.5c,f which shows a slight reduction in deformation at the end of first cycle. We observe a large difference in deformation between the simulations and experiments at these parameters. Where the simulations predict the vesicle stretching to $D\approx 0.63$, the experiments only reach $D\approx 0.25$. Additionally, the simulations show that the vesicle does not deform symmetrically at these parameters, reaching $D\approx-0.5$ and $D\approx 0.6$. The experiments were performed sequentially from the higher to lower $\mathrm{De}$ on the same vesicle in the experiments, and it seems that the gradual change in vesicle deformation carried over from the previous experiments.

In summary, the experimental data in Fig. 12 and Section III.5 shows that the maximum deformation of tubular vesicles may decrease in repeated LAOE cycles and the initial tubular shape may not be recovered. In contrast, the quasi-spherical vesicles always recover a prolate shape following repeated LAOE deformation cycles. We conjecture that the observation of shape transition from prolate tubular to oblate spheroid during LAOE deformation in Fig. 12b,c can be explained in the context of the area-difference elasticity model [44]. Briefly, the negative membrane tension on the vesicle membrane during the compressional phase of LAOE flow leads to a decrease in area per lipid which reduces the preferred monolayer area difference [4, 43]. The decrease in monolayer area difference triggers the shape transition from a prolate tubular shape to an oblate spheroid in accordance with the ADE model [59, 44]. This hypothesis is consistent with prior observations where the prolate to oblate transition was triggered by chemical modification of the ambient environment of vesicles [25]. Resolving what exactly is occurring during compressional flow requires additional experiments, likely with 3D confocal microscopy to obtain the full three dimensional vesicle shape.

Additional experimental data on dynamics of highly deflated vesicles ($\nu=0.35$) is included in the Supplementary Information (Fig. S6 and Fig. S7).

In steady extensional flow with $\mathrm{De}=0$, the critical capillary number required to trigger dumbbell shape transition is a function of reduced volume and the comprehensive phase diagram in $\mathrm{Ca}-\nu$ space has been reported in an earlier work [26]. Fig. 14 qualitatively demonstrates how oscillatory extensional flow alters these shape instabilities. At $\mathrm{De}=1.2$, we observe that the critical capillary number $\mathrm{Ca}$ required to induce asymmetric dumbbell is much higher compared to steady extensional flow at $\mathrm{De}=0$. For instance, the critical $\mathrm{Ca}$ required to generate asymmetric dumbbell in steady extension for $\nu=0.69$ is $\approx$ 5.3 [26]. However, in LAOE flow at $\mathrm{De}=1.2$, the transition to dumbbell shape occurs at $\mathrm{Ca}=52.5$ which is approximately ten times higher than the critical $\mathrm{Ca}$ for steady flow. This observation can be rationalized by considering the competition between flow cycle time $T$ and inverse of the predicted growth rate of asymmetric instability from linear stability analysis [38]. Briefly, the presence of flow oscillations ($\mathrm{De}>0$) prevents any instability formation which requires a time scale larger than cycle time $T$. Thus, a large $\mathrm{Ca}$ is needed to reduce the time scale of instability sufficiently to observe the dumbbell formation within the flow cycle time $T$. While it is possible to explore the phase diagram describing conformation change to asymmetric/symmetric dumbbell on $\mathrm{Ca}-\mathrm{De}$ space for the entire range of reduced volumes using the Stokes trap, the parameter space is vast and it remains a ripe area for future numerical simulations.