3.1. Dynamical regimes

Experiments were performed in the range of approximately $10 < {Ca} < 1000$ and $0.5 < {De} < 100$, whereas the majority of the simulations are in the range of $1 < {Ca} < 40$ and $1 < {De} < 10$. Simulations were performed for several vesicles matching the conditions in the experiments, as discussed in the following section (figures 2–4). It is possible to perform additional simulations at ${Ca} \approx 100$, but current results suggest that the vesicle dynamics do not significantly change at higher $Ca$ for quasi-spherical vesicles.

Figure 2.

Transient deformation parameter $D$ for vesicle dynamics in time-dependent LAOE from experiments and simulations. All vesicles have a viscosity ratio of $\lambda = 1.0$. The $\dot {\epsilon }/{Ca}$ line is the instantaneous strain rate of the external flow along the $x$-axis. A negative $\dot {\epsilon }/{Ca}$ value is compression along the $x$-axis. For each panel: (a) $Ca = 28.8$, $De = 6.40$, $\nu$ = 0.85; (b) $Ca = 10.9$, $De = 3$, $\nu$ = 0.88; (c) $Ca$ = 17.9, $De = 6$, $\nu$ = 0.91; (d) $Ca$ = 28.8, $De = 12$, $\nu$ = 0.85; (e) $Ca$ = 10.9, $De = 4.5$, $\nu$ = 0.88; (f) $Ca$ = 17.9, $De = 14.9$, $\nu$ = 0.91; (g) $Ca$ = 28.8, $De = 48$, $\nu$ = 0.85; (h) $Ca$ = 10.9, $De = 18.2$, $\nu$ = 0.88; (i) $Ca$ = 17.9, $De = 29.9$, $\nu$ = 0.91.

We observe three 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. The transitions between these regimes are continuous – in other words, there is no bifurcation between the regimes in the sense that the dynamics change suddenly. We define the regimes based on the deformation characteristics of vesicles in each case: symmetrical when the vesicle deforms to the same length in both orientations; pulsating when the vesicle's major axis stays along the same orientation; reorienting for the region between symmetrical and pulsating where the vesicle major axis changes orientation but does not deform to the same maximum length in both directions. Vesicles in all three regimes can experience significant nonlinear stress responses. Snapshots of vesicle shapes from simulations and experiments for each of these regimes over a full strain rate cycle are shown in figures 5 and 6.

We quantitatively compare the simulations and experiments by plotting the deformation parameter $D$ (defined in (2.17)) and instantaneous strain rate ${Ca}_x$ (defined in (2.16)) as a function of time, as shown in figures 2 and 3. 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.

Figure 3.

Lissajous-type curves of the deformation parameter $D$ versus the dimensionless instantaneous strain rate. All vesicles have a viscosity ratio of $\lambda = 1.0$. Black data points are experimental data; purple data points show numerical data. The oscillatory strain rate cycle is separated into four parts that have been noted with different markers, as shown in the legend in the bottom right-hand corner of each panel. For each panel: (a) $Ca$ = 28.8, $De = 6.4$, $\nu$ = 0.85; (b) $Ca$ = 10.9, $De = 3$, $\nu$ = 0.88; (c) $Ca$ = 17.9, $De = 6$, $\nu$ = 0.91; (d) $Ca$ = 28.8, $De = 12$, $\nu$ = 0.85; (e) $Ca$ = 10.9, $De = 4.5$, $\nu$ = 0.88; (f) $Ca$ = 17.9, $De = 14.9$, $\nu$ = 0.91; (g) $Ca$ = 28.8, $De = 48$, $\nu$ = 0.85; (h) $Ca$ = 10.9, $De = 18.2$, $\nu$ = 0.88; (i) $Ca$ = 17.9, $De = 29.9$, $\nu$ = 0.91.

Symmetrical regime. Starting with the symmetrical regimes results, we find the symmetrical regime occurs under flow conditions where the vesicle deformation time scale is shorter or exactly equal to half of a strain rate cycle. Based on our simulations, this occurs approximately when ${Ca} \geq 3.33 {De}$ for a vesicle with a reduced volume $\nu = 0.80$. Our experiments and simulations show that vesicle dynamics in the symmetrical regime are described by two common characteristics (figure 4). First, the vesicle reaches approximately the same maximum length twice during one strain rate cycle, regardless of ${Ca}$. The observation of a maximum length is reasonable for quasi-spherical vesicles, as it has been shown that vesicles with $\nu > 0.75$ have a stable steady-state shape at infinite ${Ca}$, regardless of viscosity ratio (Narsimhan et al. Reference Narsimhan, Spann and Shaqfeh2014). Second, vesicle membranes exhibit transient wrinkling when vesicles are exposed to the compressional cycle of the oscillatory extensional flow. The transient wrinkling behaviour is examined later in this section. These features are illustrated in figure 4, where a characteristic time series of images of vesicle shape in LAOE is qualitatively compared with the equivalent numerical simulation. In general, vesicle shapes determined from experiments are in good agreement with those determined from numerical simulations. Turning to the deformation parameter plots (figures 2 and 3), we see the simulations and experiments agree well at the majority of the tested parameters. Some of the experimental datasets show fluctuations in the deformation over the strain rate cycles and disagreement between the simulations on the maximum deformation. These discrepancies likely occur due to challenges in imaging a three-dimensional (3-D) object in a 2-D 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 in transient flows.

Figure 4.

Comparison of the experimental and simulation vesicle shapes in the symmetrical regime over one flow cycle at the same conditions of $Ca$ = 10.9, $De = 3.0$, $\nu$ = 0.88, $\lambda$ = 1.00. The times in the figure are in seconds for the experimental movie. Shapes from the simulations at the same non-dimensional cycle times are shown below. Here $T$ is the non-dimensional period, defined as $T = 1/ {De}$.

Transient wrinkling dynamics were first reported by Kantsler et al. (Reference Kantsler, Segre and Steinberg2007) for a single cycle of suddenly reversed extensional flow and subsequently elaborated upon by Turitsyn & Vergeles (Reference Turitsyn and Vergeles2008). Wrinkling behaviour is caused by a negative surface tension created during vesicle compression. Moreover, a critical compression rate exists below which thermal fluctuations dominate the observed wrinkling. In our work, we study vesicle dynamics in an extensional flow with smoothly varying sinusoidal strain rate dependence, rather than an abrupt step-function reversal of compressional/extensional axes. We observe qualitatively the same membrane wrinkling features as those reported in prior work. In the experiments, we observe some wrinkling in the majority of the movies; it is unclear if this is from thermal fluctuations or the negative surface tension. In the simulations, we only observe significant wrinkling in the symmetrical regime. Our simulations do not take into account thermal fluctuations, therefore, we hypothesize that the critical wrinkling strain rate required for a given flow frequency is only reached in the symmetrical regime. Additional experimental snapshots of vesicles showing wrinkling dynamics are included in the supplementary materials (figures S2–S5).

Reorienting regime At lower ${Ca} / {De}$ ratios (when ${Ca} \approx 2{De}$ for $\nu =0.80$), the vesicle's major axis orientates along the $x$- and $y$-axes during the flow cycle, but the stretching along these axes will no longer be equal. The creates a deformation parameter that is negative during part of the cycle, but whose mean value is non-zero (figure 2d). We note that in prior work on droplets in oscillatory extensional flow we do not observe this behaviour, as only symmetrical deformation (i.e. equal deformation in the $x$- and $y$-orientations) has been reported regardless of flow strength and oscillatory frequency (Li & Sarkar Reference Li and Sarkar2005a,Reference Li and Sarkarb). Single polymers in LAOE also deform symmetrically between the two half-cycles for the range of Weissenberg and Deborah numbers studied in prior work (Zhou & Schroeder Reference Zhou and Schroeder2016a). The phenomenon of asymmetric stretching of vesicles along the two axes arises due to the enclosed membrane for fluid-filled vesicles. In particular, we posit that the asymmetrical deformation occurs because the energetically preferred shape for quasi-spherical vesicles at equilibrium is a prolate dumbbell (Seifert Reference Seifert1997). By deforming in this asymmetrical manner, the vesicle shape deviates less from the equilibrium shape over the strain rate cycle than it would if it deformed symmetrically.

Pulsating regime. At even lower ${Ca} / {De}$, the vesicle no longer reorients and simply pulsates along one axis during LAOE. We refer to this dynamical regime as the pulsating regime, which approximately occurs when ${Ca} \le 2 {De}$ for $\nu = 0.80$. Note that the strain in the pulsating regime is not necessarily infinitesimal. As shown in figure 2(g), the deformation parameter curve illustrates that vesicles are generally oriented along the $x$-axis and can deform significantly in this regime. It is possible to probe the small amplitude oscillatory extension regime by keeping the ${De}$ constant and reducing the ${Ca}$. In the small amplitude regime, vesicles do not deform appreciably, and the Lissajous curve approaches a constant value, thereby informing on the linear viscoelastic rheology of vesicle suspensions. Similar behaviour occurs when increasing the ${De}$ and keeping ${Ca}$ constant at small values. In this case, the membrane does not have appreciable time to reorient during the time at which the strain rate changes.

3.2. quasi-spherical initial shape and orientation

The simulations discussed up to this point (including results in figures 5 and 2) were performed using a prolate-like initial shape, because it is the global equilibrium shape for reduced volumes $\nu \geq 0.652$ (Seifert Reference Seifert1997). 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 figure 7, which shows that vesicle dynamics during the steady limit cycle for the oblate initial condition (figure 7a) are the same as that observed from the prolate-like initial condition (figure 7b). We repeated these simulations at several other capillary numbers and Deborah numbers, observing no change in the dynamics.

Figure 5.

Snapshots of vesicle shapes from simulations over a flow cycle for the three dynamical regimes. The values under the figures are fractions of a strain rate period defined as $T = 1/{De}$.

Figure 6.

Snapshots of vesicle shapes from experiments over a flow cycle for the three dynamical regimes. The values under the figures are fractions of cycle time $T$ in seconds. Scale bar is 20 $\mathrm {\mu }$m. False colour is applied to the greyscale images for enhancing the resolution.

Figure 7.

Lissajous-type deformation parameter curves from an oblate shape initial condition and a prolate shape initial condition. The legend at the top right in each panel indicates colour coding for the strain rate cycle. The black circle marks the deformation parameter of the initial shape. The panels are (a) oblate initial; (b) prolate initial.

We additionally examined different starting orientations of the prolate initial shape. Aligning the prolate vesicle with the $y$-axis instead of the $x$-axis does not change the dynamics significantly. The symmetrical regime remains unchanged, while the pulsating and reorienting regimes preferentially stretch along the $y$-axis instead of the $x$-axis. The observed dynamics change when aligning the prolate vesicle along the $z$-axis – i.e. orthogonal to the flow plane. At lower ratios of $Ca/De$, the vesicle deforms symmetrically while maintaining the major axis orientation along the $z$-axis. At higher ratios of $Ca/De$, the dynamics become the same as those observed in the symmetrical regime with other starting orientations. We also simulated vesicles at other out-of-plane orientations and found they can maintain their orientation at low ratios of $Ca/De$ over 15 flow cycles. Simulation movie files of vesicle dynamics starting from the $z$-axis orientation and angled at 70 degrees between the $x$- and $z$-axes are included in the supplementary materials (movies 8–10).

Experimentally, we have not observed any of the dynamics suggested by the simulations with alternative starting orientations. It is unclear if these orientations are unstable to perturbation or if the experimental methods limit the possible orientations of the vesicles. One would need to use a microscopy method that can obtain $z$-axis information to better understand the effect of starting orientation.

3.3. LAOE analysis considerations

In regards to the application of LAOE for vesicle analysis, we note that it may be possible to extract some material properties of the vesicle by LAOE analysis. One could fit the deformation parameter over time of an experimental run to that of a simulation to approximate an unknown parameter, such as the reduced volume or capillary number. There are significant error margins when approximating experimental parameters, such as reduced volume, so confirmation with LAOE could be beneficial. We have not tested the feasibility or accuracy of such a process in this study, however.

3.4. quasi-spherical phase diagrams

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 are plotted in figure 8

Figure 8.

Phase diagrams for the low to medium to high deformation regimes for vesicles of reduced volume $\nu = 0.80$ and $\nu = 0.90$. Lines in the diagrams are from the semianalytical theory presented near the end of § 3.1. Due to uncertainty in determining the $D_{o}$ value, a 5 % error has been included on the lines.

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, ${Ca}/{De}=\dot {\epsilon }_0/\omega$. Here, we derive the phase boundaries in the limit of a quasi-spherical vesicle (Vlahovska & Gracia Reference Vlahovska and Gracia2007). For small excess area ($\varDelta =4{\rm \pi} (\nu ^{-2/3}-1)\ll 1$), the vesicle shape is characterized by a perturbation series in terms of spherical harmonics (Vlahovska & Gracia Reference Vlahovska and Gracia2007). 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 (2.17), one obtains

(3.1)\begin{align} &D(t) = (L_{\infty}-1)\nonumber\\ &\quad \left(1-2\left(1+\left(\frac{1+A_o}{1-A_o}\right)\exp\left(\frac{60}{{\rm \pi}(32+23\lambda)}\frac{{Ca}}{{De}}\frac{1}{L_{\infty}-1}\left[\cos(2{\rm \pi} {De}\, t)-1\right]\right)\right)^{{-}1}\right), \end{align}

where parameters $L_{\infty }=1+\sqrt {15/8}(\nu ^{-2/3}-1)^{1/2}$ 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 material.

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. $\ln (({1+A_o})/({1-A_o}))\approx {1}/{(L_{\infty }-1)}\ln (({1+D_o})/({1-D_o}))$,

(3.2)\begin{gather} \frac{{Ca}}{{De}}=\frac{{\rm \pi}(32+23\lambda)}{120}\log\left(\frac{1+D_o}{1-D_o}\right)\quad \text{for pulsating}/\text{reorienting phases}, \end{gather}

(3.3)\begin{gather} \frac{{Ca}}{{De}}=\frac{{\rm \pi}(32+23\lambda)}{60}\log\left(\frac{1+D_o}{1-D_o}\right) \quad \text{for reorienting}/\text{symmetrical phases}. \end{gather}

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 ${Ca}$ and ${De}$ numbers. Based on the quasi-spherical 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 (Vlahovska & Gracia Reference Vlahovska and Gracia2007). Figure 8(a) shows the phase boundaries are accurately calculated by using (3.2) and (3.3) 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 (figure 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 versus analytical solutions) and $\lambda = 10$ results in the supplementary materials for brevity.

3.5. Stress response and dilute suspension rheology

For dilute vesicle suspensions where the macroscopic length scale is large in comparison with 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 BI formulation, we calculate the particle stresslet (Pozrikidis Reference Pozrikidis1992)

(3.4)\begin{equation} \tilde{S}_{ij}^P = \int_D {\frac{1}{2} ([[f_i]] x_j + [[f_j]] x_i)}\,\textrm{d}S - \int_D {(1 - \lambda) \mu_{{out}} (v_i n_j + v_j n_j)}\,\textrm{d}S, \end{equation}

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

(3.5)\begin{equation} S_{ij} = \frac{\tilde{S}_{ij}^P}{\dot{\epsilon} \mu_{{out}} V_p}, \end{equation}

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

(3.6)$$\begin{gather} N_1 = S_{xx} - S_{yy}, \end{gather}$$

(3.7)$$\begin{gather}N_2 = S_{yy} - S_{zz}. \end{gather}$$

Comparing the normal stress differences with the strain rate, we can derive the rheological characteristics of a dilute vesicle suspension, such as the effective viscosity and bulk normal stresses (Danker, Verdier & Misbah Reference Danker, Verdier and Misbah2008). 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

(3.8)\begin{equation} \eta_E = \frac{\sigma_{11} - \sigma_{22}}{\dot{\epsilon}}. \end{equation}

The planar Trouton ratio is

(3.9)\begin{equation} \frac{\eta_E}{\eta} = 4 + \phi * N_1, \end{equation}

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 nonlinear viscoelasticity.

We examine the stress response for a vesicle that starts off oriented along the $x$-axis. To link the single vesicle stress response to the expected bulk response for a suspension of randomly oriented vesicles, one needs to average over all possible orientations. Therefore, the following results would instead be indicative of a suspension of vesicles all initially oriented along the $x$-axis. However, in the symmetrical regime, all of the starting orientations we tested lead to the same dynamics for the limit cycle behaviour. It is possible that the bulk stress response in the symmetrical regime at the limit cycle does not depend on starting orientation. In the pulsating and reorienting regimes on the other hand, the stable orientations depend on several parameters that we have not examined in detail in this study – as reported in § 3.2.

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 figure 9, 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 nonlinear viscoelasticity, the normal stress differences will display more complex behaviours.

Figure 9.

Normal stress differences versus time for simulations in the pulsating, reorienting and symmetrical regimes. Data over two strain rate cycles is plotted. The $\dot {\epsilon }/{Ca}$ dotted line is the strain rate of the external flow; it is used to show the directionality of the flow. Parameters used are included in the figure legends. The panels are (a) symmetrical regime; (b) reorienting regime; (c) pulsating regime.

Figure 9 shows that vesicle dynamics in the three regimes (symmetrical, reorienting and pulsating) have nonlinear characteristics. To analyse these stress responses, we replot the data from figure 9 into a Lissajous-type form with the instantaneous strain rate (${Ca}_x$) on the $x$-axis and the stress response on the $y$-axis (figure 10). 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 $N_1 = 10 \times {Ca}_x/{Ca}$ and $N_2 = -5 \times {Ca}_x / {Ca}$. 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.

Figure 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. Parameters used are included in the figure legends. The panels are (a) symmetrical regime; (b) reorienting regime. (c) pulsating regime.

In the symmetrical regime (figure 10a), 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 $-2 < {Ca}_x <2$. 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 $-2 < {Ca}_x < 2$ 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 with the stress response. From the simulation movie and the Lissajous-type deformation parameter curve (figure 3), we know that the vesicle retains a prolate spheroid-like shape and only changes marginally for the ${-2 < {Ca}_x < 2}$ region. The relatively small amount of deformation that occurs in the $-2 < {Ca}_x < 2$ 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 (figure 10b,c), 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 (figure 3). 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 figure 10(c).

For a more quantitative analysis, we decomposed the stress responses into a Fourier series. A similar decomposition was performed by Farutin & Misbah (Reference Farutin and Misbah2012a) to analytically examine the stress over vesicles over all orientations in small amplitude oscillatory shear with a background constant shear rate under the quasi-spherical assumption. This decomposition is commonly applied to large amplitude oscillatory shear (known as LAOS) experiments and is known as Fourier transform (FT) rheology. The FT rheology is commonly performed using oscillatory shear flows on polymeric liquids to probe the shear stress response in the nonlinear regime (Wilhelm Reference Wilhelm2002; Hyun et al. Reference Hyun, Wilhelm, Klein, Cho, Nam, Ahn, Lee, Ewoldt and McKinley2011). The computation is straightforward and relies on taking the FT of the $N_1$ or $N_2$ stress difference,

(3.10)\begin{equation} f(k) = \int_{-\infty}^{\infty} {N_{1, 2}(t) \textrm{e}^{{-}2 {\rm \pi}\textrm{i} t k}}\,\textrm{d}t. \end{equation}

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 of a linear order, the Fourier transformation would show a single peak at the first mode. A nonlinear stress response would have additional peaks at higher modes.

The Fourier decompositions for both $N_1$ and $N_2$ are shown in figure 11, where it is clear that all three regimes show higher-order behaviour. For all regimes, we observe the expected behaviour 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 § 3.1, and this transition can be used instead of the deformation parameter analysis to demarcate the phase boundary.

Figure 11.

Fourier decompositions of the stress responses for indicative parameter sets in each of the dynamical regimes. The panels are (a) symmetrical regime; (b) reorienting regime; (c) pulsating regime.

In large amplitude oscillatory shear, the typical macroscopic stress response shows that the stress is an odd function of the direction of shearing (Hyun et al. Reference Hyun, Wilhelm, Klein, Cho, Nam, Ahn, Lee, Ewoldt and McKinley2011). 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 (Li & Sarkar Reference Li and Sarkar2005b). Broadly speaking, our results show that membrane-bound vesicles are an interesting example of how anisotropic microstructural deformations can lead to complex rheology.

3.6. Transient dynamics of tubular vesicles in large amplitude oscillatory extension

We also investigated the transient dynamics of tubular vesicles in large-amplitude oscillatory extensions (figure 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.

Figure 12.

Dynamics of a tubular vesicle with reduced volume $\nu = 0.64 \pm 0.02$ in LAOE. (a) Snapshots showing pulsating dynamics of a vesicle over one sinusoidal strain rate input cycle with time period $T= 4$ s at $Ca=21.3$ and $De=17.7$. (b) Snapshots showing pulsating dynamics with wrinkles of a vesicle over one sinusoidal strain rate input cycle with time period $T= 8$ s at $Ca=21.3$ and $De=8.9$. (c) Snapshots showing change in the 2-D shape of a vesicle over one flow cycle with time period $T= 15$ s at $Ca=21.3$ and $De=4.7$. Scale bar is 20 $\mathrm {\mu }$m. False colouring is applied to the greyscale images for resolution enhancement.

Figure 12(a) shows experimental snapshots of a tubular vesicle with reduced volume $\nu =0.64\pm 0.02$ exposed to a sinusoidal strain rate at ${Ca} = 21.3$ and ${De} = 17.7$. 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 behaviour, we construct single vesicle Lissajous curves (figure 13d) defined as plots of the deformation parameter as a function of ${Ca}$, and the deformation parameter as a function of time (figure 13a). 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.

Figure 13.

Experimental and simulation single vesicle Lissajous curves and deformation plots for $\nu =0.64$. For each panel: (a) $Ca = 21.3$, $De = 17.7$, $\nu$ = 0.64; (b) $Ca = 21.3$, $De = 8.9$, $\nu$ = 0.64; (c) $Ca = 21.3$, $De = 4.7$, $\nu$ = 0.64; (d) $Ca = 21.3$, $De = 17.7$, $\nu$ = 0.64; (e) $Ca = 21.3$, $De = 8.9$, $\nu$ = 0.64; (f) $Ca = 21.3$, $De = 4.7$, $\nu$ = 0.64.

When the same vesicle is exposed to a flow cycle at a lower frequency (${De}=8.9$), the membrane has more time to deform in response to the flow. Here, the vesicle undergoes pulsating motion with wrinkles (figure 12b) and we observe appreciable deformation along $y$-axis in both the simulations and experiments, as shown in figure 13(b,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 the third repeated cycle, we experimentally observe that the 2-D 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 ${Ca}$, there appears to be a critical ${De}$ below which the change occurs.

Finally, the same vesicle is exposed to a 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, undergoing a wrinkling instability during compression, and the initial spheroidal shape changes to a more spherical shape at the end of the first periodic cycle (figure 12c). The deformation behaviour seen experimentally during the second repeated cycle is symmetric and follows similar dynamics as those observed for quasi-spherical vesicles. This behaviour is more apparent in figure 13(c,f) which shows a slight reduction in deformation at the end of the 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 ${De}$ on the same vesicle in the experiments, and it seems that the gradual change in vesicle deformation is carried over from the previous experiments.

In summary, the experimental data in figures 12 and 13 show 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 figure 12(b,c) can be explained in the context of the area-difference elasticity model (Seifert Reference Seifert1997). 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 (Avital & Farago Reference Avital and Farago2015; Sakashita et al. Reference Sakashita, Urakami, Ziherl and Imai2012). 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 (Seifert Reference Seifert1997; Ziherl & Svetina Reference Ziherl and Svetina2005). This hypothesis is consistent with prior observations where the prolate to oblate transition was triggered by chemical modification of the ambient environment of vesicles (Kodama et al. Reference Kodama, Morandi, Ebihara, Jimbo, Toyoda, Sakuma, Imai, Puff and Angelova2018). Resolving what exactly is occurring during compressional flow requires additional experiments, likely with 3-D confocal microscopy to obtain the full 3-D vesicle shape.

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

In steady extensional flow with ${De}=0$, the critical capillary number required to trigger dumbbell shape transition is a function of reduced volume and the comprehensive phase diagram in ${Ca}$–$\nu$ space has been reported in an earlier work (Kumar et al. Reference Kumar, Richter and Schroeder2020a). The dumbbell-like shape has also been observed in simulations of a reduced volume $\nu =0.60$ vesicle in a steady shear flow (Farutin & Misbah Reference Farutin and Misbah2012b). Figure 14 qualitatively demonstrates how oscillatory extensional flow alters these shape instabilities. At ${De}=1.2$, we observe that the critical capillary number ${Ca}$ required to induce an asymmetric dumbbell is much higher compared with steady extensional flow at ${De}=0$. For instance, the critical ${Ca}$ required to generate an asymmetric dumbbell in steady extension for $\nu =0.69$ is $\approx 5.3$ (Kumar et al. Reference Kumar, Richter and Schroeder2020a). However, in LAOE flow at ${De}=1.2$, the transition to a dumbbell shape occurs at ${Ca}=52.5$ which is approximately 10 times higher than the critical ${Ca}$ for steady flow. This observation can be rationalized by considering the competition between flow cycle time $T$ and the inverse of the predicted growth rate of asymmetric instability from linear stability analysis (Narsimhan et al. Reference Narsimhan, Spann and Shaqfeh2015). Briefly, the presence of flow oscillations (${De} > 0$) prevents any instability formation which requires a time scale larger than cycle time $T$. Thus, a large ${Ca}$ 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 ${Ca}$–${De}$ 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.

Figure 14.

Asymmetric dumbbell formation in a vesicle with reduced volume $\nu =0.69$ exposed to LAOE flow at ${Ca}=52.5$ and ${De}=1.2$. Scale bar is 10 $\mathrm {\mu }$m.