Vesicle dynamics in large amplitude oscillatory extensional flow
Charlie Lin, Dinesh Kumar, Channing M. Richter, Shiyan Wang, Charles M. Schroeder, Vivek Narsimhan
DDynamics of vesicles in large amplitude oscillatoryextensional flow
Charlie Lin, ∗ Dinesh Kumar,
2, 3, ∗ Channing M. Richter, ShiyanWang, Charles M. Schroeder,
2, 3, 5, † and Vivek Narsimhan ‡ Davidson School of Chemical Engineering, Purdue University, West Lafayette, IN 47907 Department of Chemical and Biomolecular EngineeringUniversity of Illinois at Urbana-Champaign, Urbana, IL, 61801 Beckman Institute for Advanced Science and TechnologyUniversity of Illinois at Urbana-Champaign, Urbana, IL, 61801 Department of Chemical and Biomedical Engineering,University of Illinois at Urbana-Champaign, Urbana, IL 61801 Department of Materials Science and EngineeringUniversity of Illinois at Urbana-Champaign, Urbana, IL, 61801 (Dated: February 19, 2021)Although the behavior of fluid-filled vesicles in steady flows has been extensively studied, far lessis understood regarding the shape dynamics of vesicles in time-dependent oscillatory flows. Here,we investigate the nonlinear dynamics of vesicles in large amplitude oscillatory extensional (LAOE)flows using both experiments and boundary integral (BI) simulations. Our results characterize thetransient membrane deformations, dynamical regimes, and stress response of vesicles in LAOE interms of reduced volume (vesicle asphericity), capillary number (Ca, dimensionless flow strength),and Deborah number (De, dimensionless flow frequency). Results from single vesicle experiments arefound to be in good agreement with BI simulations across a wide range of parameters. Our resultsreveal three distinct dynamical regimes based on vesicle deformation: pulsating, reorienting, andsymmetrical regimes. We construct phase diagrams characterizing the transition of vesicle shapesbetween pulsating, reorienting, and symmetrical regimes within the two- dimensional Pipkin spacedefined by De and Ca. Contrary to observations on clean Newtonian droplets, vesicles do not reacha maximum length twice per strain rate cycle in the reorienting and pulsating regimes. The distinctdynamics observed in each regime result from a competition between the flow frequency, flow timescale, and membrane deformation timescale. By calculating the particle stresslet, we quantify thenonlinear relationship between average vesicle stress and strain rate. Additionally, we present resultson tubular vesicles that undergo shape transformation over several strain cycles. Broadly, our workprovides new information regarding the transient dynamics of vesicles in time-dependent flows thatdirectly informs bulk suspension rheology.
I. INTRODUCTION
In recent years, fluid-filled vesicles have been used in a wide array of technological applications ranging from foodproducts to bioinspired microreactors, and reagent delivery applications in functional materials [18]. Moreover, giantvesicles are widely regarded as a model membrane system in various biophysical and biochemical processes [6, 13].In these applications, precise characterization of the membrane shape dynamics in response to a fluid flow is offundamental importance. Despite the increasing prevalence of vesicles in biophysics and materials science, we lack acomplete understanding of how time-dependent flows influence the membrane shape dynamics and overall rheologicalresponse of vesicle suspensions [1, 51]. Lipid vesicles consist of a small amount of fluid enclosed by a bilayer membraneof thickness ≈ ∗ These authors contributed equally † To whom correspondence must be addressed: [email protected] ‡ [email protected] a r X i v : . [ c ond - m a t . s o f t ] F e b transition regime between tumbling and tank treading [2, 31, 54], and characterization of tank-treading, vacillating-breathing (trembling), and tumbling motion with increasing viscosity ratio between the interior and the exterior ofthe vesicle [11, 36, 50]. Knowledge of single vesicle dynamics has been essential for interpreting the bulk rheologicalresponse for dilute vesicle suspensions. For instance, it is now known that the tank-treading to tumbling behavior ofvesicles directly affects the bulk viscosity of the suspension, where tumbling results in a higher bulk viscosity withthe minimum bulk viscosity occurring at the tank-treading to tumbling transition [51].Compared to the vast body of experiments in shear flows, vesicle dynamics in hyperbolic flows even for the canonicalcase of steady elongational flow are more challenging to understand. In extensional flow, fluid elements separateexponentially in time [30], and it is generally not possible to observe a single vesicle in flow for long periods of timein the absence of feedback controllers. Automation in flow control techniques using sophisticated feedback algorithmshas recently enabled the precise characterization of vesicle dynamics in elongational flows [28, 29, 45–47]. In a steadyextensional flow, it is known that highly deflated tubular vesicles undergo a conformation change to a symmetricdumbbell shape [21, 26, 37, 38] while moderately deflated vesicles transition to an asymmetric dumbbell shape [9, 26].Precise control over the center-of-mass position of single vesicles led to detailed studies of the transient and steady-state stretching dynamics of membranes [26], and direct observation of the double-mode relaxation following highdeformation [27]. Prior work in unsteady flows has been limited to a one-time reversal of elongational flow andreported membrane wrinkling shapes for quasi-spherical vesicles [20].Extensional flows are commonly encountered in microfluidic devices that utilise contractions or expansions, porousmedia, and other complex channel geometries. Moreover, in vivo capillaries and complex microfluidic devices that havemany bifurcations and sharp directional changes routinely encounter time-dependent pulsatile flows. The biomedicalcommunity has created several biomimetic capillary designs that contain several rows of bifurcations and contractionswith small angle zigzags in between, resulting in improved flow control and lower fluid flow resistance [15, 34]. Ingeneral, elastic particles traversing through these fluidic systems experience spatially dependent external flows and willnot reach a steady-state conformation. From this view, there is a need for comprehensive studies on how microscopicstretching and compression of vesicles in complex, time-dependent oscillatory flows will affect their shape and bulkrheology.Recently, the shape dynamics of elastic capsules were studied numerically in large amplitude oscillatory extensional(LAOE) flow [7]. However, the non-equilibrium stretching and compression dynamics of lipid vesicles in LAOEflows is largely unexplored. Vesicle dynamics are strongly governed by membrane bending elasticity; therefore, weanticipate that vesicles will exhibit qualitatively different behavior than capsules in time-dependent extensional flow.In this paper, we study the dynamics of single vesicles in LAOE using a combination of microfluidic experiments andboundary integral (BI) simulations. LAOE experiments are performed using the Stokes trap [28, 29, 45, 46], whichis a new method for controlling the center-of-mass position, orientation and trajectories of freely suspended singleand multiple vesicles using only fluid flow. We find that single vesicles experience periodic cycles of compression andextension in LAOE with membrane dynamics governed by the dimensionless flow strength Capillary number (Ca),reduced volume (measure of vesicle asphericity, ν ) and flow frequency Deborah number (De). Experimental resultsare compared to BI simulations without thermal fluctuations, and our results show that BI simulations accuratelycapture the dynamics of single quasi-spherical vesicles over a wide range of parameters. In addition, we identify threedistinct dynamical regimes for vesicle dynamics, including the pulsating, reorienting, and symmetrical regimes, basedon the amount of deformation occurring in each half cycle of the LAOE flow. The qualitatively different dynamicsobserved in each regime results due to a competition between the flow frequency, flow time scale, and membranedeformation timescale. We further construct precise phase diagrams characterizing the transition of vesicle shapesbetween pulsating, reorienting, symmetrical regimes. We find that the relationship between average vesicle stress andstrain rate is nonlinear, which is discussed in the context of bulk suspension rheology. Finally, we present results onthe shape dynamics of long tubular vesicles in LAOE which exhibit markedly different behavior in flow comparedto their quasi-spherical analogues. Taken together, our results provide new insights into the direct observationof membrane dynamics during time-dependent oscillatory flows, which opens new avenues for understanding bulksuspension rheology in unsteady flows. II. METHODSA. Vesicle preparation
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 togenerate giant unilamellar vesicles (GUVs) with the electroformation process described by Angelova et al. [3]. Forelectroformation of GUVs, a stock lipid solution in chloroform is prepared with 25 mg/mL DOPC and 0.04 mg/mL (a)(b)
Fluid ReservoirPressure RegulatorObjective LensPDMS Coverslip ε x (t) ε y (t) Time
TT/2 S t r a i n R a t e ε x (t)<0, ε y (t)>0 ε x (t)>0, ε y (t)<0 FIG. 1. Stokes trap for studying vesicle dynamics in large amplitude oscillatory extensional (LAOE) flow. (a) Schematic of theexperimental setup used to generate planar extensional flow. Inlet/outlet channels in the microfluidic device are connected tofluidic reservoirs containing the vesicle suspension and pressurized by regulators controlled by a custom LabVIEW program,thereby generating pressure-driven flow in the cross-slot. (b) Schematic of the sinusoidal strain rate input function for one fullcycle. Inset: schematics showing the oscillatory extensional flow profile in the microfluidic cross-slot device during the first half(0 < t < T /
T / < t < T ) of the cycle. DOPE-Rh for fluorescent imaging. Next, 10 µ L of the lipid solution in chloroform is spread on a conductive indiumtin oxide (ITO) coated glass slide (resistance 5 Ω, 25 × × ≈ ◦ C. An alternating current (AC) electric fieldof 2 V/mm is then applied at 10 Hz for 120 min at room temperature (22 ◦ C). Under these conditions, DOPC lipidremains in the fluid phase [23]. Most of the vesicles prepared by this method are quasi-spherical and unilamellar withfew defects in the size range of 5–25 µm in radius. B. Stokes trap for large amplitude oscillatory extension
It is challenging to observe vesicle dynamics in time-dependent extensional flow for long periods of time whilesimultaneously imposing precisely controlled flow rates. To achieve this, we used the Stokes trap [28, 46] to preciselyposition the center-of-mass of single vesicles near the center of a cross-slot microfluidic device for long times usingmodel predictive control (Fig. 1a). Briefly, the centroid of a single vesicle is determined in real-time using imageprocessing and fluorescence microscopy and is communicated to the controller. The controller determines the optimalflow rates through four-channels of the device to maintain a fixed vesicle position with desired strain rate. Theflow rates are then applied through four independent pressure regulators (Elveflow). During this process, the deviceoperates at a net positive pressure so that each of the four ports can act as inlet or outlet. This whole procedurerequires ≈
30 ms in a single cycle, as previously described [46, 57, 58]. In this work, a sinusoidal strain rate input isimposed (Fig. 1b) while simultaneously trapping a single vesicle such that:˙ (cid:15) x ( t ) = − ˙ (cid:15) sin (cid:18) πT t (cid:19) ˙ (cid:15) y ( t ) = ˙ (cid:15) sin (cid:18) πT t (cid:19) where T is the period of the sinusoidal cycle and ˙ (cid:15) is the maximum strain rate in one cycle. During the first half-cyclefor 0 < t < T /
2, the x-axis is the compressional axis and y-axis is the elongational axis ( ˙ (cid:15) x ( t ) < , ˙ (cid:15) y ( t ) > T / < t < T , the direction of flow reverses, and fluid is delivered by the two vertical portsin the cross-slot device as shown in Fig. 1b. We note that during vesicle trapping, the correctional pressure requiredfor controlling the vesicle’s position is small compared to the magnitude of the base pressure used to generate theoscillatory extensional flow [46]. Thus, the strain rate is well defined during the LAOE cycle, which is determinedas a function of the input pressure using particle tracking velocimetry (PTV) as previously described [26]. We alsodetermined the characteristic response time for actuating fluid flow in the microfluidic device in response to a stepchange in pressure. For an extreme change in pressure from 0 to 4 psi (strain rate jump from 0 to ∼ s − ), the risetime and settling time are ∼
20 ms and ∼
300 ms respectively(Fig. S1). However, the maximum value of pressure usedin our experiments is 0.4 psi, which is continuously varied with small incremental changes during the LAOE cycle, forwhich we generally expect much smaller characteristic response times. Nevertheless, the lowest cycle time T in ourexperiments is 2 seconds which is much larger than the maximum characteristic response time for actuating flow inthe 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 allowfor equilibration, followed by LAOE flow for at least 2 strain rate cycles. During the equilibration step, the vesiclereduced volume ν and equivalent radius a are determined, as previously described [9, 26]. Reduced volume ν is adimensionless quantity that measures the amount of osmotic deflation, and is described as: ν = 3 V √ πA / (1)where V and A are the vesicle volume and surface area, respectively. The equivalent radius a of the vesicle isobtained as a = (cid:112) A/ π . Specifically, ν is a measure of vesicle asphericity such that ν = 1 represents a perfectlyspherical shape. For the experiments in this paper, the typical range of reduced volume is 0 . < ν < (cid:15) experienced by a vesicle in a half-cycle is non-dimensionalized to define a capillarynumber Ca = µ out ˙ (cid:15) a /κ where µ out is the suspending medium viscosity, a is the equivalent vesicle radius, and κ is the membrane bending modulus. Prior to vesicle experiments in LAOE flow, we determined the average bendingmodulus of nearly spherical vesicles to be κ = (22 . ± . 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 De = µ out a /κT .Single vesicle experiments are generally performed in the range 10 < Ca < . < De <
100 by adjusting theinput pressures and strain rate cycle periods. Only vesicles near the center plane of the microchannel (with respect tothe z-direction) are considered during experiments. Single vesicle trajectories are analyzed using a custom MATLABprogram that allows for determination of the vesicle deformation parameter in flow.
C. Numerical methods
1. Governing equations and non-dimensionalization
The system is modeled as a droplet surrounded by a two-dimensional incompressible membrane with a bendingresistance. At the length scale of a GUV ( a ≈ µ m) with a strain rate at ˙ (cid:15) ≈ s − the Reynolds number isRe = ˙ (cid:15)ρa /µ ≈ − , allowing us to model the inner and outer velocity fields using the Stokes equations. Due tothe nature of the time-dependent flow, it is also important to check the Womersley number to assess whether thetime-dependent Stokes equations are required. At a flow frequency of ω = 10 s − , the Womersley number is α = (cid:112) ωρa /µ ≈ .
03. In this work, the flow frequencies are ω < s − , therefore the time-dependent Stokes equationsare not necessary. The Stokes equations are: ∇ · u = 0 , ∇ p = µ ∇ u . (2)where u is fluid velocity, p is the pressure, and µ is the fluid viscosity ( µ in for the inner fluid and µ out for theouter fluid). The system is subject to continuity of velocity across the interface and a traction balance acrossthe phospholipid bilayer. The short timescales and low deformation rates used in previous studies makes membranedilatation negligible [17, 42]. Vesicles are also known to have negligible shear rigidity as they do not have a cytoskeletalnetwork or an actin cortex. We therefore use the Helfrich model [16] for the membrane: H = (cid:73) κ H ) dA + (cid:73) σdA. (3)In Eq. (3), H represents the elastic energy of the vesicle membrane, κ is the membrane bending modulus, H isthe mean curvature, and σ is the surface tension. The surface tension is a spatially varying Lagrange multiplier thatensures local area conservation. The surface tension enforces ∇ s · u = 0 on the interface, where ∇ s = ( I − nn ) · ∇ .We note that the original Helfrich model includes spontaneous curvature, a parameter to describe a membrane’scurvature preference when the sides of the bilayer are chemically different. Although biological vesicles may havemultiple lipid components or chemical differences between the inner and outer fluids [5, 12, 14], our experimentsfocus on simple vesicles with only a viscosity difference between the inner and outer fluids, prompting a negligiblespontaneous curvature. We further neglect contributions from thermal fluctuations, membrane viscosity, and bilayerfriction [39, 44].The force balance at the membrane surface is: [[ f ]] = [[ T · n ]] = f t + f b (4) f t = (2 Hσ n − ∇ s σ ) (5) f b = κ (4 KH − H − ∇ s H ) n (6)where [[ f ]] is the jump in viscous traction across the interface which can be decomposed to the bending ( f b ) andtension ( f t ) contributions, n is the outward-pointing unit normal vector, and K is the Gaussian curvature of theinterface. The mean curvature H is defined to be one for the unit sphere.The vesicle is placed in a time-dependent extensional flow field described by u ∞ = ∇ u ∞ · x and defined as: ∇ uuu ∞ = ˙ (cid:15) − sin(2 πωt ) 0 00 sin(2 πωt ) 00 0 0 where ω is the frequency of the oscillatory flow and ˙ (cid:15) is the maximum strain rate.The membrane area ( A ) is maintained constant by the incompressibility constraint while the low permeability ofthe membrane allows us to assume that the volume ( V ) of the vesicle is constant during the timescale of experiments(minutes). Therefore, we non-dimensionalize distances by the equivalent radius a = (cid:112) A/ (4 π ), time scales by κ/a µ out ,velocities by κ/a µ out , stresses by κ/a , and surface tensions by κ/a . We obtain four relevant dimensionless groupsfrom the non-dimensionalization: Ca ≡ µ out ˙ (cid:15) a κ , De ≡ ωa µ out κλ ≡ µ in µ out , ν ≡ V πa These parameters were previously described in Section II B and are elaborated upon here. The base capillary num-ber (Ca) compares the viscous stress to the bending stress and corresponds to the non-dimensionalized, maximumextension rate experienced by the vesicle during the flow cycle. De is the flow frequency non-dimensionalized by thebending timescale. When De (cid:29)
1, the fluid flow will have a short cycle time compared to the membrane’s bendingtime. The viscosity ratio ( λ ) is the ratio of inner and outer fluid viscosities. Cellular systems such as red blood cells(RBCs) commonly have a more viscous inner fluid, and this parameter can be tuned to more closely model the systemof choice. The reduced volume ( ν ) is a measure of the asphericity of the vesicle, corresponding to its osmotic deflation.For example, a reduced volume of ν = 1 corresponds to a perfectly spherical vesicle shape, while a value of ν = 0 . ∇ uuu ∞ = Ca − sin(2 π De t ) 0 00 sin(2 π De t ) 00 0 0 (7)where all parameters are assumed to be non-dimensional from this point forward.
2. Boundary integral formulation
The Stokes flow assumption enables the use of the boundary integral (Green’s function) formulation to simulatevesicle shape dynamics. The Stokes equations are recast into a boundary integral form:1 + λ u j ( x ) = u ∞ j ( x ) − π (cid:90) S G ij ( x , x )[[ f i ]]( x ) dA ( x )+ 1 − λ π (cid:90) S T ijk ( x , x ) u i ( x ) n k ( x ) dA ( x ) (8)where u ∞ i is the external velocity field, x is the singularity point, and [[ f i ]] is the jump in viscous traction across theinterface, given in Eq. (6). The kernels G ij ( x , x ) and T ijk ( x , x ) are the Stokeslet (point force) and stresslet (pointdipole) solutions to Stokes flow: G ij ( x , x ) = δ ij r + ˜ x i ˜ x j r (9) T ijk ( x , x ) = − x i ˜ x j ˜ x k r (10)where ˜ x = x − x and r = | ˜ x | . Repeated indices are assumed to be summed in the above equations. These equationsare also subject to the membrane incompressibility constraint: ∇ s · u = 0 (11)Implementation details for the simulations are similar to prior work [35]. Here, we highlight a few key differences.In this paper, we start with an icosahedron and subdivide the mesh into 1024 elements for a quasi-spherical vesicle;5024 elements for the tubular vesicles. We then use a scaling transformation to deform the mesh into a prolatespheroid with the desired reduced volume ν , followed by relaxing the mesh to its equilibrium (no flow) configuration.In this way, the vesicle has a prolate spheroid-like shape at the start of any cycle. It is possible to start with an oblatespheroid or any arbitrary ellipsoid-like shape, but it has been shown that the global minimum energy state for a vesiclewith reduced volume greater than 0.652 is of the prolate shape family[44]. Choosing an alternative starting shapedoes not affect the steady limit cycle behavior. Vesicle dynamics are simulated in oscillatory flow with a timestep of10 − strain units.The majority of the analysis in this study is focused on vesicle behavior that has reached a steady limit cyclein time-dependent flow, such that the dynamics are the same regardless of the number of additional strain ratecycles. The startup dynamics have been simulated but are not elaborated on in this paper. We simulate vesicles ofreduced volumes between 0 . < ν < .
90 and viscosity ratios λ = 0 . , .
0, and 10 for flows with capillary numbers0 . < Ca <
80 and Deborah numbers 0 . < De <
10. Significantly higher capillary numbers (Ca (cid:39) x ( t ) ≡ − Ca sin(2 π De · t ) (12)which represents the time-dependent capillary number in the x-direction. This will be the measure used for theinstantaneous strain rate. We also define a deformation parameter: D ≡ l x − l y l x + l y (13)where l x and l y are the x- and y-axis lengths of the vesicle respectively. The deformation parameter ( D ) providesa measure of vesicle shape distortion. For D values near zero, the vesicle shape projected in the x-y plane will becircular. Positive values of D ≈ III. RESULTS AND DISCUSSIONA. Dynamical regimes
Experiments were performed in the range of approximately 10 < Ca < . < De < < Ca <
40 and 1 < De <
10. Simulations were performed for severalvesicles matching the conditions in the experiments, as discussed in the following section (Fig. 2, Fig. 3 and Fig. 4).It is possible to perform additional simulations at Ca ≈ D (defined inEq. (13)) and instantaneous strain rate Ca x (defined in Eq. (12)) as a function of time, as shown in Figs. 2 and 3.Experimental trajectories are generally limited to 2–4 strain rate cycles due to the photobleaching of the vesiclemembrane during fluorescence imaging experiments. Observing vesicle deformation over more strain rate cycles isexperimentally 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 vesicledynamics over at least 10 strain rate cycles. Symmetrical regime:
Starting with the symmetrical regimes results, we find the symmetrical regime occursunder flow conditions where the vesicle deformation timescale is shorter or exactly equal to half of a strain rate cycle.Based on our simulations, this occurs approximately when Ca ≥ .
33 De for a vesicle with a reduced volume ν = 0 . ν > .
75 have a stable steady-state shape at infinite Ca, regardless of viscosity ratio [37].Second, vesicle membranes exhibit transient wrinkling when vesicles are exposed to the compressional cycle of theoscillatory extensional flow. The transient wrinkling behavior is examined later in this section. These features areillustrated in Fig. 4, where a characteristic time series of images of vesicle shape in LAOE is qualitatively compared tothe equivalent numerical simulation. In general, vesicle shapes determined from experiments are in good agreementwith those determined from numerical simulations. Turning to the deformation parameter plots (Fig. 2 and Fig. 3),we see the simulations and experiments agree well at the majority of the tested parameters. Some of the experimentaldatasets show fluctuations in the deformation over the strain rate cycles and disagreement between the simulationson the maximum deformation. These discrepancies likely occur due to challenges in imaging a three-dimensionalobject in a two-dimensional plane and because the experiments are limited to a few strain rate cycles. Nevertheless,we generally observe good agreement between simulations and experiments in terms of the deformation parameter intransient flows.Transient wrinkling dynamics were first reported by [24] for a single cycle of suddenly reversed extensional flow andsubsequently elaborated upon by [48] and [33]. Wrinkling behavior is caused by a negative surface tension createdduring vesicle compression. Moreover, a critical compression rate exists below which thermal fluctuations dominate theobserved wrinkling. In our work, we study vesicle dynamics in an extensional flow with smoothly varying sinusoidalstrain rate dependence, rather than an abrupt step-function reversal of compressional/extensional axes. We stillobserve qualitatively the same membrane wrinkling features as those reported in prior work. Additional experimental
Symmetrical time ( t / a ) D ( l x l y l x + l y ) exp.sim./ Ca (a) Ca = 28.8, De = 6.40, ν = 0.85 time ( t / a ) D ( l x l y l x + l y ) exp.sim./ Ca (b) Ca = 18.2, De = 3, ν = 0.88 time ( t / a ) D ( l x l y l x + l y ) exp.sim./ Ca (c) Ca = 17.9, De = 6, ν = 0.91 Reorienting time ( t / a ) D ( l x l y l x + l y ) exp.sim./ Ca (d) Ca = 28.8, De = 12, ν = 0.85 time ( t / a ) D ( l x l y l x + l y ) exp.sim./ Ca (e) Ca = 18.2, De = 4.5, ν = 0.88 time ( t / a ) D ( l x l y l x + l y ) exp.sim./ Ca (f) Ca = 17.9, De = 14.9, ν = 0.91 Pulsating time ( t / a ) D ( l x l y l x + l y ) exp.sim./ Ca (g) Ca = 28.8, De = 48, ν = 0.85 time ( t / a ) D ( l x l y l x + l y ) exp.sim./ Ca (h) Ca = 18.2, De = 18.2, ν = 0.88 time ( t / a ) D ( l x l y l x + l y ) exp.sim./ Ca (i) Ca = 17.9, De = 29.9, ν = 0.91 FIG. 2. Transient deformation parameter D for vesicle dynamics in time-dependent LAOE from experiments and simulations.All vesicles have a viscosity ratio of λ = 1 .
0. The ˙ (cid:15)/
Ca line is the instantaneous strain rate of the external flow along the x-axis.A negative ˙ (cid:15)/
Ca value is compression along the x-axis.
Symmetrical
20 0 20 Ca x ( a ) D ( l x l y l x + l y ) C a (a) Ca = 28.8, De = 6.4, ν = 0.85
10 5 0 5 10 Ca x ( a ) D ( l x l y l x + l y ) C a (b) Ca = 18.2, De = 3, ν = 0.88
10 0 10 Ca x ( a ) D ( l x l y l x + l y ) C a (c) Ca = 17.9, De = 6, ν = 0.91 Reorienting
20 0 20 Ca x ( a ) D ( l x l y l x + l y ) C a (d) Ca = 28.8, De = 12, ν = 0.85
10 5 0 5 10 Ca x ( a ) D ( l x l y l x + l y ) C a (e) Ca = 18.2, De = 4.5, ν = 0.88
10 0 10 Ca x ( a ) D ( l x l y l x + l y ) C a (f) Ca = 17.9, De = 14.9, ν = 0.91 Pulsating
20 0 20 Ca x ( a ) D ( l x l y l x + l y ) C a (g) Ca = 28.8, De = 48, ν = 0.85
10 5 0 5 10 Ca x ( a ) D ( l x l y l x + l y ) C a (h) Ca = 18.2, De = 18.2, ν = 0.88
10 0 10 Ca x ( a ) D ( l x l y l x + l y ) C a (i) Ca = 17.9, De = 29.9, ν = 0.91 FIG. 3. Lissajous-type curves of the deformation parameter D versus the dimensionless instantaneous strain rate. All vesicleshave a viscosity ratio of λ = 1 .
0. Black data points are experimental data; purple data points show numerical data. Theoscillatory strain rate cycle is separated into four parts that have been noted with different markers, as shown in the legend inthe bottom right hand corner. t = 00.000s t = 02.066s t = 03.400s t = 10.800s t = 16.200s t = 18.066s t = 31.666 t = 27.066s 0 t = 32.534s 0.036T 0.105T 0.332T 0.499T 0.556T 0.667T 0.833T 1.002T FIG. 4. Comparison of the experimental and simulation vesicle shapes in the symmetrical regime over one flow cycle at thesame conditions of Ca = 39.5, De = 1.7, ν = 0.83, λ = 1.00. The times in the figure are in seconds for the experimental video.Shapes from the simulations at the same non-dimensional times are shown below. T is the non-dimensional period, defined as T = 1 / De. snapshots of vesicles showing wrinkling dynamics are included in the supplementary materials (Fig. S2, Fig. S3, Fig.S4 and Fig. S5).
Reorienting regime:
At lower Ca / De ratios, vesicles no longer reach the same length twice during a strain ratecycle, and the dynamics are no longer described by the symmetrical regime. The reorienting deformation regime occurswhen the vesicle deformation rate and flow frequency are comparable (Ca ≈ ν = 0 . Pulsating regime:
At even lower Ca / De, the vesicle no longer reorients and simply pulsates along one axis duringLAOE. We refer to this dynamical regime as the pulsating regime, which approximately occurs when Ca ≤ ν = 0 .
80. Note that the strain in the pulsating regime is not necessarily infinitesimal. As shown in Fig. 2g,the deformation parameter curve illustrates that vesicles are generally oriented along the x-axis and can deformsignificantly in this regime. It is possible to probe the small amplitude oscillatory extension regime by keeping the Deconstant and reducing the Ca. In the small amplitude regime, vesicles do not deform appreciably, and the Lissajouscurve approaches a constant value, thereby informing on the linear viscoelastic rheology of vesicle suspensions. Similarbehavior occurs when increasing the De and keeping Ca constant at small values. In this case, the membrane doesnot have appreciable time to reorient during the time at which the strain rate changes.
B. Quasi-spherical initial shape considerations
Overall, the simulations discussed up to this point (including results in Fig. 5 and Fig. 2) were performed using aprolate-like initial shape, because it is the global equilibrium shape for reduced volumes ν ≥ .
652 [44]. These resultssuggest that the unequal stretching observed in the pulsating and reorienting regimes occurs during the steady limitcycle, for this particular initial shape. However, there are other local minimum energy shapes for vesicles, such as theoblate shape family. To determine whether the pulsating and reorienting regimes are possible with a different initialcondition, 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 throughoptical microscopy would show a circle. The oblate initial condition simulations test if the anisotropic deformationswill 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 duringthe steady limit cycle for the oblate initial condition (Fig. 7a) are the same as that observed from the prolate-likeinitial 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.1
T/4 T/2 3T/4 T0 (a) Symmetrical regime
T/4 T/2 3T/4 T0 (b) Reorienting regime
T/4 T/2 3T/4 T0 (c) Pulsating regime
FIG. 5. Snapshots of vesicle shapes from simulations overa flow cycle for the three dynamical regimes. The valuesunder the figures are fractions of a strain rate period de-fined as T = 1 / De.
Scalebar – 20 umMain Figure for all regime comparison with simulations
T/2 3T/4 T (a) Symmetrical regime (b) Reorienting regime(c) Pulsating regime
FIG. 6. Snapshots of vesicle shapes from experimentsover a flow cycle for the three dynamical regimes. Thevalues under the figures are fractions of cycle time T inseconds. Scale bar is 20 µ m. False color is applied to thegrayscale images for enhancing the resolution. Ca x ( a ) D ( l x l y l x + l y ) De = 1.000Ca = 1.000 = 0.800 = 1.000 Ca (a) oblate initial Ca x ( a ) D ( l x l y l x + l y ) De = 1.000Ca = 1.000 = 0.800 = 1.000 Ca (b) prolate initial FIG. 7. Lissajous type deformation parameter curves from an oblate shape initial condition and a prolate shape initial condition.Top right legend indicates color coding for the strain rate cycle. The black circle marks the deformation parameter of the initialshape. De ( a / ) C a ( a / ) = 0.80, = 1.00pulsatingreorientingsymmetrical1 22468 (a) De ( a / ) C a ( a / ) = 0.90, = 1.00pulsatingreorientingsymmetrical1 224 (b) FIG. 8. Phase diagrams for the low to medium to high deformation regimes for vesicles of reduced volume ν = 0 .
80 and ν = 0 .
90. Lines in the diagrams are from the semi-analytical theory presented near the end of Section III A. Due to uncertaintyin determining the D o value, a 5% error has been included on the lines. C. Quasi-spherical phase diagrams
By comparing the deformation parameter results for each simulation, we can plot a phase diagram of differentdynamical regimes observed during oscillatory flows. Which regime a vesicle experiences can be quantitatively de-termined by assessing the minimum and maximum deformation parameter over a cycle. If both the minimum andmaximum deformation parameter are positive, the vesicle dynamics are classified as the pulsating regime, reflectingthat 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 eachother, the vesicle is in the symmetrical regime, since the vesicle reaches the same maximum length twice a cycle. Thisthreshold value was chosen heuristically to reflect the discretization accuracy. If the magnitudes are not within thisthreshold value, vesicle dynamics are classified as the reorienting regime. Results from this analysis is plotted in Fig. 8The phase boundaries appear to be mostly linear, suggesting that the dynamics result from a simple interactionbetween the flow frequency and the strain rate, Ca / De = ˙ (cid:15) /ω . Here, we derive the phase boundaries in the limit ofa quasispherical vesicle [50]. For small excess area (∆ = 4 π ( ν − / − (cid:28) D ( t ) = ( L ∞ − (cid:32) − (cid:18) (cid:18) A o − A o (cid:19) exp (cid:18) π (32 + 23 λ ) CaDe 1 L ∞ − π De t ) − (cid:19)(cid:19) − (cid:33) , (14)where parameters L ∞ = 1 + (cid:112) / (cid:0) ν − / − (cid:1) / and A o = ( ν − / l maxx / − / ( L ∞ − l maxx is the maximumx-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 in3the limit of A (cid:28)
1, i.e. ln (cid:16) A o − A o (cid:17) ≈ L ∞ − ln (cid:16) D o − D o (cid:17) ,CaDe = π (32 + 23 λ )120 log (cid:18) D o − D o (cid:19) for pulsating/reorienting phases , (15)CaDe = π (32 + 23 λ )60 log (cid:18) D o − D o (cid:19) for reorienting/symmetrical phases (16)In the above equations, D is the maximum deformation parameter during the LAOE cycle. Note that the valueof D o is determined by our numerical runs at the highest Ca and De numbers. Based on the quasispherical vesicletheory, the deformation phase boundaries depend on the viscosity ratio, where the factor (23 λ + 32) − is related tothe relaxation time of the quasi-spherical vesicle [50]. Fig. 8a shows the phase boundaries are accurately calculatedby using Eq. (15) and Eq. (16) when the reduced volume is ν = 0 .
8. Increasing ν from 0.80 to 0.90 shifts the phaseboundaries downwards, but maintains a similar linear relation (Fig. 8b). We also simulated viscosity ratio λ = 10and found that higher viscosity ratios shift the boundaries to higher capillary numbers. We include the dynamicsevolution of l x and l y (simulations vs. analytical solutions) and λ = 10 results in the supplementary materials forbrevity. D. Stress response and dilute suspension rheology
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 andthe particle stresslet: σ Pij = n ˜ S Pij . Using the boundary integral formulation, we calculate the particle stresslet [41]:˜ S Pij = (cid:90) D
12 ([[ f i ]] x j + [[ f j ]] x i ) dS − (cid:90) D (1 − λ ) µ out ( v i n j + v j n j ) dS (17)where [[ f ]] is the surface traction, λ is the viscosity ratio, µ out is the outer viscosity, v is the velocity, and n is thenormal vector. We define the dimensionless particle coefficient of stresslet as: S ij = ˜ S Pij ˙ (cid:15)µ out V p (18)where V p is the vesicle volume and ˙ (cid:15) is the strain rate. Similarly the normal stress differences are defined as: N = S xx − S yy (19) N = S yy − S zz (20)Comparing the normal stress differences to the strain rate, we can derive the rheological characteristics of a dilutevesicle suspension, such as the effective viscosity and bulk normal stresses [10]. For extensional flow rheology, a keyquantity of interest is the extensional viscosity of a solution. Extensional viscosity is often characterized using aquantity known as a Trouton ratio (ratio of extensional to shear viscosity), which for a planar extensional flow is amultiple of N . For a planar flow, the extensional viscosity is η E = σ − σ ˙ (cid:15) . (21)The planar Trouton ratio is η E η = 4 + φ ∗ N , (22)where φ is the volume fraction of vesicles in the suspensions, and N is the first normal stress difference. Oursimulations have focused on rather large deformations of the vesicle shape, therefore the stress response analysis willreflect the non-linear viscoelasticity.Using the definitions of the particle stresslet and normal stress differences, we determine the vesicle stress asa function of time in extensional flow. In Fig. 9, we show the stress response over two cycles for three sets ofparameters; one from each of the three dynamical regimes discussed before. A linearly viscoelastic material will show4 time ( t / a ) n o r m a l s t r e ss d i ff e r e n c e ( P ij a V p ) De = 1.000Ca = 4.000 = 0.800 = 1.000 N N (a) Symmetrical regime time ( t / a ) n o r m a l s t r e ss d i ff e r e n c e ( P ij a V p ) De = 1.000Ca = 3.000 = 0.800 = 1.000 N N (b) Reorienting regime time ( t / a ) n o r m a l s t r e ss d i ff e r e n c e ( P ij a V p ) De = 1.000Ca = 2.000 = 0.800 = 1.000 N N (c) Pulsating regime FIG. 9. Normal stress differences versus time for simulations in the pulsating, reorienting, and symmetrical regimes. Data overtwo strain rate cycles is plotted. The ˙ (cid:15)/
Ca dotted line is the strain rate of the external flow; it is used to show the directionalityof the flow. Parameters used are included in the figure legends. purely sinusoidal normal stress differences for this type of plot, as there is a simple linear relation between the strainrate and the stress. On the other hand, for non-linear viscoelasticity, the normal stress differences will display morecomplex behaviors.Fig. 9 shows that vesicle dynamics in the three regimes (symmetrical, reorienting, and pulsating) have non-linearcharacteristics. To analyze these stress responses, we re-plot the data from Fig. 9 into a Lissajous-type form with theinstantaneous strain rate (Ca x ) on the x-axis and the stress response on the y-axis (Fig. 10). For this type of plot,a purely viscous material would display a straight line, whereas a purely elastic material would produce an ellipticalcurve. For example, the first and second normal stress difference for Newtonian flow around a rigid sphere correspondsto the lines: N = 10 · Ca x / Ca and N = − · Ca x / Ca. Here, we focus on N because it is related to the extensionalviscosity of the solution (Trouton ratio). We also discuss the N stress differences for completeness.In the symmetrical regime (Fig. 10a), we observe that N is symmetric across the origin and that the lines forincreasing and decreasing strain rate are nearly the same for − < Ca x <
2. On the other hand, N differs significantly5 Ca x n o r m a l s t r e ss d i ff e r e n c e ( P ij a V p ) De = 1.000Ca = 4.000 = 0.800 = 1.000 N N Ca (a) Symmetrical regime Ca x n o r m a l s t r e ss d i ff e r e n c e ( P ij a V p ) De = 1.000Ca = 3.000 = 0.800 = 1.000 N N Ca (b) Reorienting regime Ca x n o r m a l s t r e ss d i ff e r e n c e ( P ij a V p ) De = 1.000Ca = 2.000 = 0.800 = 1.000 N N Ca (c) Pulsating regime FIG. 10. Lissajous-type normal stress difference versus strain rate (Ca x ) curves for simulations in the pulsating, reorienting,and symmetrical regimes. The strain rate cycle is separated into four periods demarcated by the line formatting. Parametersused are included in the figure legends. depending on the directionality of the flow. The N curve is mostly linear in the region − < Ca x < − < Ca x < − < Ca x < N 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. 10b and 10c), the N curves are no longer symmetric across theorigin, and the stress responses for increasing and decreasing strain rate are distinct. The maximum N response islarger in magnitude than the minimum for both regimes; this is likely due to the unequal amounts of deformationbetween the two strain rate period halves (Fig. 3). For this analysis, qualitative differences between the shape of the6reorienting and pulsating regime curves correspond to the extent of asymmetry in the N response. Moreover, weobserve vesicles in the pulsating regime can have a non-zero normal stress difference when the time-dependent strainrate is zero, as seen in Fig. 10c.For a more quantitative analysis, we decomposed the stress responses into a Fourier series. This decomposition iscommonly 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 shearstress response in the non-linear regime [19, 52]. The computation is straightforward and relies on taking the Fouriertransform of the N or N stress difference: f ( k ) = (cid:90) ∞−∞ N , ( t ) e − πitk dt, (23)In this way, the periodic stress signal is transformed into frequency space. Because the external flow field is sinusoidal,the strain rate ( ˙ (cid:15) ) and strain ( (cid:15) ) are proportional to sine and cosine functions. Therefore, the Fourier transformeddata provide a description of how the stress depends on different orders of the strain and strain rate. If the stressresponse was purely linear order, the Fourier transformation would show a single peak at the first mode. A non-linearstress response would have additional peaks at higher modes.The Fourier decompositions for both N and N are shown in Fig. 11, where it is clear that all three regimes showhigher order behavior. For all regimes, we observe the expected behavior of the linear order mode being the highestamplitude with the higher order modes decreasing monotonically for N . On the other hand, the highest amplitudemode for N is not the linear order mode, with the highest generally being the second or third mode. Comparing the N decompositions between the dynamical regimes, we observe that the symmetrical regime does not have even ordermodes, whereas the reorienting and pulsating regimes have even higher order modes. This change in FT rheology isconsistent with the phase boundary defined in Section III A, and this transition can be used instead of the deformationparameter analysis to demarcate the phase boundary.In large amplitude oscillatory shear (LAOS), the typical macroscopic stress response shows that the stress is anodd 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 thesymmetrical regime, both the vesicle stress response and deformation are time symmetric, leading to only odd orderFourier modes. The time symmetry does not hold for the reorienting or pulsating regimes, allowing for even ordermodes. 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 arean interesting example of how anisotropic microstructural deformations can lead to complex rheology. E. Transient dynamics of tubular vesicles in large amplitude oscillatory extension
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 ofthe flow cycle similar to quasi-spherical vesicles. However, we occasionally observe buckling instabilities that induceunexpected shape changes. In these situations, the vesicle’s initial, tubular shape is not recovered at the end of theflow cycle.Fig. 12a shows experimental snapshots of a tubular vesicle with reduced volume ν = 0 . ± .
02 exposed to asinusoidal strain rate at Ca = 21 . .
7. In this situation, the vesicle exhibits pulsating motion along thex-axis with buckles during the compressional part of the flow cycle. The vesicle’s starting, tubular shape is recoveredat the end of the LAOE cycle. To further demonstrate this behavior, we construct single vesicle Lissajous curves(Fig. 13(d)) defined as plot of deformation parameter as a function of Ca, and deformation parameter as a function oftime (Fig. 13(a)). These plots show the vesicle reaches the same value of deformation parameter D ≈ . . y axis in both the simulations and experiments, as shown in Fig. 13b,e. Surprisingly,the experimental results show the vesicle deformation parameter reducing with each subsequent LAOE cycle. Thedeformation at the end of first cycle is D ≈ . D ≈ . D ≈ . A m p li t u d e N Frequency / De A m p li t u d e De = 1.000Ca = 4.000 = 0.800 = 1.000 N (a) Symmetrical regime A m p li t u d e N Frequency / De A m p li t u d e De = 1.000Ca = 3.000 = 0.800 = 1.000 N (b) Reorienting regime A m p li t u d e N Frequency / De A m p li t u d e De = 1.000Ca = 2.000 = 0.800 = 1.000 N (c) Pulsating regime FIG. 11. Fourier decompositions of the stress responses for indicative parameter sets in each of the dynamical regimes. vesicle did not recover its original tubular shape even when relaxed for ≈ x axis to y axis, undergoes a wrinkling instability during compressionand 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 similardynamics as those observed for quasi-spherical vesicles. This behavior is more apparent in Fig. 13c,f which shows aslight reduction in deformation at the end of first cycle. We observe a large difference in deformation between thesimulations and experiments at these parameters. Where the simulations predict the vesicle stretching to D ≈ .
63, theexperiments only reach D ≈ .
25. Additionally, the simulations show that the vesicle does not deform symmetricallyat these parameters, reaching D ≈ − . D ≈ .
6. The experiments were performed sequentially from the higherto lower De on the same vesicle in the experiments, and it seems that the gradual change in vesicle deformation carriedover from the previous experiments.8
T/2
T/4 T/2
T/2 (a) Pulsating regime, T= 4 s(b) Pulsating regime, T= 8 s(c) Symmetric full reorientation, T= 15 s
FIG. 12. Dynamics of a tubular vesicle with reduced volume ν = 0 . ± .
02 in LAOE. (a) Snapshots showing pulsatingdynamics of a vesicle over one sinusoidal strain rate input cycle with time period T = 4 s at Ca = 21 . De = 17 .
7. (b)Snapshots showing pulsating dynamics with wrinkles of a vesicle over one sinusoidal strain rate input cycle with time period T = 8 s at Ca = 21 . De = 8 .
9. (c) Snapshots showing change in 2D shape of a vesicle over one flow cycle with timeperiod T = 15 s at Ca = 21 . De = 4 .
7. Scale bar is 20 µ m. False coloring is applied to the grayscale images for resolutionenhancement. In summary, the experimental data in Fig. 12 and Fig. 13 shows that the maximum deformation of tubular vesiclesmay 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 theobservation of shape transition from prolate tubular to oblate spheroid during LAOE deformation in Fig. 12b,c canbe explained in the context of the area-difference elasticity model [44]. Briefly, the negative membrane tension on thevesicle membrane during the compressional phase of LAOE flow leads to a decrease in area per lipid which reducesthe preferred monolayer area difference [4, 43]. The decrease in monolayer area difference triggers the shape transitionfrom a prolate tubular shape to an oblate spheroid in accordance with the ADE model [44, 59]. This hypothesisis consistent with prior observations where the prolate to oblate transition was triggered by chemical modificationof the ambient environment of vesicles [25]. Resolving what exactly is occurring during compressional flow requiresadditional experiments, likely with 3D confocal microscopy to obtain the full three dimensional vesicle shape.Additional experimental data on dynamics of highly deflated vesicles ( ν = 0 .
35) is included in the SupplementaryInformation (Fig. S6 and Fig. S7).In steady extensional flow with De = 0, the critical capillary number required to trigger dumbbell shape transitionis a function of reduced volume and the comprehensive phase diagram in Ca − ν space has been reported in an earlierwork [26]. Fig. 14 qualitatively demonstrates how oscillatory extensional flow alters these shape instabilities. AtDe = 1 .
2, we observe that the critical capillary number Ca required to induce asymmetric dumbbell is much highercompared to steady extensional flow at De = 0. For instance, the critical Ca required to generate asymmetric dumbbellin steady extension for ν = 0 .
69 is ≈ .
2, the transition to dumbbell shapeoccurs at Ca = 52 . T and inverse of the predicted growthrate of asymmetric instability from linear stability analysis [38]. Briefly, the presence of flow oscillations (De > T . Thus, a large Ca is neededto 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 dumbbellon Ca − De space for the entire range of reduced volumes using the Stokes trap, the parameter space is vast and itremains a ripe area for future numerical simulations.9 time ( t / a ) D ( l x l y l x + l y ) exp.sim./ Ca (a) Ca = 21.3, De = 17.7, ν = 0.64 time ( t / a ) D ( l x l y l x + l y ) exp.sim./ Ca (b) Ca = 21.3, De = 8.9, ν = 0.64 time ( t / a ) D ( l x l y l x + l y ) exp.sim./ Ca (c) Ca = 21.3, De = 4.7, ν = 0.64
20 10 0 10 20 Ca x ( a ) D ( l x l y l x + l y ) C a (d) Ca = 21.3, De = 17.7, ν = 0.64
20 10 0 10 20 Ca x ( a ) D ( l x l y l x + l y ) C a (e) Ca = 21.3, De = 8.9, ν = 0.64
20 10 0 10 20 Ca x ( a ) D ( l x l y l x + l y ) C a (f) Ca = 21.3, De = 4.7, ν = 0.64 FIG. 13. Experimental and simulation single vesicle Lissajous curves and deformation plots for ν = 0 . Scalebar – 10 um
FIG. 14. Asymmetric dumbbell formation in a vesicle with reduced volume ν = 0 .
69 exposed to LAOE flow at Ca = 52 . .
2. Scale bar is 10 µ m. IV. CONCLUSIONS
In this work, we examined the dynamics of vesicles in large amplitude oscillatory extensional (LAOE) flow usingboth experiments and numerical simulations. The experiments were carried out using the Stokes trap experimentaltechnique while the simulations were done with the boundary element method. For quasi-spherical vesicles, the simu-lations are found to capture the transient wrinkling dynamics as well as the overall vesicle shapes from experiments.We have identified three dynamical regimes based on their deformation characteristics and named them the symmet-rical, reorienting, and pulsating regimes. Based on these results, we generated a phase diagram in capillary number0and Deborah number space for the dynamical regimes; our data suggest that the phase boundaries are linear. Theunique deformation observed in the pulsating and reorienting regimes also has interesting effects on the stress responsein that the time symmetry of the stress does not hold. Additional analysis of the stress response and confirmationby experimental studies is required for a better idea of the dynamics. Finally, we presented results on highly deflatedtubular vesicles which shows that lower reduced volume vesicles tend to undergo a shape change following repeatedLAOE deformation. From a broad perspective, we have shown through experiments and simulations that the vesiclesystem shows interesting dynamics in extensional oscillatory flows. We have also shown how microstructural changesfrom extensional and compression of a cell-like suspension can affect the overall rheology. Similar dynamics might beobserved in other cell-like systems such as red blood cells or single-celled organisms, prompting additional study intotime dependent flows for these systems.
V. ACKNOWLEDGEMENTS
This work was funded by the National Science Foundation (NSF) through grant CBET PMP 1704668 for C.M.S.and by a PPG-MRL Graduate Research Fellowship from the Illinois Materials Research Lab and Almar T. WidigerFellowship for D.K.
DECLARATION OF INTEREST
The authors declare no conflict of interest. [1]
Abreu, David, Levant, Michael, Steinberg, Victor & Seifert, Udo
Advances in colloidand interface science , 129–141.[2]
Abreu, David & Seifert, Udo
Phys. Rev. Lett. (23).[3]
Angelova, M. I., Sol´eau, S., M´el´eard, Ph., Faucon, F. & Bothorel, P.
Trends in Colloid and Interface Science VI (ed. C. Helm, M. L¨osche& H. M¨ohwald), p. 127–131. Darmstadt: Steinkopff.[4]
Avital, Yotam Y & Farago, Oded
The Journal of chemicalphysics (12), 03B619 1.[5]
Bagatolli, Luis & Sunil Kumar, P. B.
Soft Matter , 3234–3248.[6] Boal, David
Mechanics of the cell Cambridge . UK: Cambridge University Press.[7]
Bryngelson, Spencer H & Freund, Jonathan B
European Journal of Mechanics-B/Fluids , 171–176.[8] Callens, N., Minetti, C., Coupier, G., Mader, M.-A., Dubois, F., Misbah, C. & Podgorski, T.
EPL (Europhysics Letters) (2), 24002.[9] Dahl, Joanna B., Narsimhan, Vivek, Gouveia, Bernardo, Kumar, Sanjay, Shaqfeh, Eric S. G. & Muller, Su-san J.
Soft Matter (16), 3787–3796.[10] Danker, G., Verdier, C. & Misbah, C.
Journal of Non-Newtonian Fluid Mechanics (1), 156–167, 4 th International workshop on NonequilibriumTheromdynamics and Complex Fluids.[11]
Deschamps, J., Kantsler, V. & Steinberg, V.
Phys. Rev. Lett. , 118105.[12]
Deuling, HJ & Helfrich, W
Journalde Physique (11), 1335–45.[13] Dimova, Rumiana & Marques, Carlos
The Giant Vesicle Book . CRC Press.[14]
Dobereiner, Hans-G¨unther, Selchow, Olaf & Lipowsky, Reinhard
European Biophysics Journal (2), 174–178.[15] Domachuk, Peter, Tsioris, Konstantinos, Omenetto, Fiorenzo G & Kaplan, David L
Advanced materials (2), 249–260.[16] Helfrich, W.
Zeitschrift f¨ur NaturforschungC , 693–703.[17] Henriksen, J. & Ipsen, J.
The European PhysicalJournal E (2), 149–167. [18] Huang, Shao-Ling & MacDonald, Robert C
Biochimica et Biophysica Acta (BBA)-Biomembranes (1-2), 134–141.[19]
Hyun, Kyu, Wilhelm, Manfred, Klein, Christopher O, Cho, Kwang Soo, Nam, Jung Gun, Ahn, Kyung Hyun,Lee, Seung Jong, Ewoldt, Randy H & McKinley, Gareth H
Progress in Polymer Science (12), 1697–1753.[20] Kantsler, Vasiliy, Segre, Enrico & Steinberg, Victor
Physical review letters (17), 178102.[21] Kantsler, Vasiliy, Segre, Enrico & Steinberg, Victor
Phys. Rev. Lett. (4).[22]
Kantsler, V., Segre, E. & Steinberg, V.
EPL (Europhysics Letters) (5).[23] Kantsler, Vasiliy & Steinberg, Victor
Phys. Rev. Lett. , 258101.[24] Kantsler, Vasiliy & Steinberg, Victor
Phys. Rev. Lett. , 036001.[25] Kodama, Atsuji, Morandi, Mattia, Ebihara, Ryuta, Jimbo, Takehiro, Toyoda, Masayuki, Sakuma, Yuka,Imai, Masayuki, Puff, Nicolas & Angelova, Miglena I
Langmuir (38), 11484–11494.[26] Kumar, Dinesh, Richter, Channing M & Schroeder, Charles M
Soft Matter (2), 337–347.[27] Kumar, Dinesh, Richter, Channing M & Schroeder, Charles M
Physical Review E (1), 010605.[28]
Kumar, Dinesh, Shenoy, Anish, Deutsch, Jonathan & Schroeder, Charles M
Current Opinion in Chemical Engineering , 1–8.[29] Kumar, Dinesh, Shenoy, Anish, Li, Songsong & Schroeder, Charles M
Physical Review Fluids (11), 114203.[30] Leal, L Gary
Laminar flow and convective transport processes: scaling principles and asymptotic analysis .Butterworth-Heinemann Boston:.[31]
Levant, Michael, Deschamps, Julien, Afik, Eldad & Steinberg, Victor
Phys. Rev. E , 056306.[32] Li, Xiaoyi & Sarkar, Kausik
Physicsof fluids (2), 027103.[33] Li, Xiaoyi & Sarkar, Kausik
Journal of non-newtonian fluid mechanics (2-3), 71–82.[34]
Lim, Daniel, Kamotani, Yoko, Cho, Brenda, Mazumder, Jyotirmoy & Takayama, Shuichi
LabChip , 318–323.[35] Lin, Charlie & Narsimhan, Vivek
Physical ReviewFluids (12), 123606.[36] Mader, M., Vitkova, V., Abkarian, M., Viallat, A. & Podgorski, T.
The European Physical Journal E (4), 389–397.[37] Narsimhan, Vivek, Spann, Andrew P. & Shaqfeh, Eric S. G.
Journal of Fluid Mechanics , 144–190.[38]
Narsimhan, Vivek, Spann, Andrew P. & Shaqfeh, Eric S. G.
Journal of Fluid Mechanics , 1–26.[39]
Noguchi, H & Gompper, G
Physical Review E (1).[40] Podgorski, Thomas, Callens, Natacha, Minetti, Christophe, Coupier, Gwennou, Dubois, Frank & Misbah,Chaouqi
Microgravity Science and Technology (2),263–270.[41] Pozrikidis
Boundary integral and singularity methods for linearized viscous flow . Cambridge University Press.[42]
Rawicz, W., Olbrich, K.C., McIntosh, T., Needham, D. & Evans, E.
Biophysical Journal (1), 328–339.[43] Sakashita, Ai, Urakami, Naohito, Ziherl, Primoˇz & Imai, Masayuki
Soft Matter (33), 8569–8581.[44] Seifert, Udo
Advances in Physics (1), 13–137.[45] Shenoy, Anish, Kumar, Dinesh, Hilgenfeldt, Sascha & Schroeder, Charles M
Physical Review Applied (5), 054010.[46] Shenoy, Anish, Rao, Christopher V. & Schroeder, Charles M.
Proceedings of the National Academy of Sciences
Shenoy, Anish, Tanyeri, Melikhan & Schroeder, Charles M
Microfluidics and Nanofluidics (5-6), 1055–1066.[48] Turitsyn, KS & Vergeles, SS
Physical review letters (2), 028103.[49] Vitkova, Victoria, Mader, Maud-Alix, Polack, Benoˆıt, Misbah, Chaouqi & Podgorski, Thomas
Biophysical Journal (6), L33–L35.[50] Vlahovska, Petia M. & Gracia, Ruben Serral
Phys. Rev. E. .[51]
Vlahovska, Petia M., Podgorski, Thomas & Misbah, Chaouqi
Comptes rendus - Physique (8), 775–789.[52] Wilhelm, Manfred
Macromolecular materials and engineering (2), 83–105.[53]
Yu, Miao, Lira, Rafael B, Riske, Karin A, Dimova, Rumiana & Lin, Hao
Physical review letters (12), 128303.[54]
Zabusky, Norman J., Segre, Enrico, Deschamps, Julien, Kantsler, Vasiliy & Steinberg, Victor
Physics of Fluids (4).[55] Zhao, Hong, Spann, Andrew P. & Shaqfeh, Eric S. G.
Physics of Fluids (12).[56] Zhou, Hernan, Gabilondo, Beatriz Burrola, Losert, Wolfgang & van de Water, Willem
Physical Review E (1), 011905.[57] Zhou, Yuecheng & Schroeder, Charles M
Physical Review Fluids (5), 053301.[58] Zhou, Yuecheng & Schroeder, Charles M
Macromolecules (20), 8018–8030.[59] Ziherl, Primoˇz & Svetina, Saˇsa
EPL(Europhysics Letters)70