## Abstract

Considering the elastic response of the membrane of a lipid vesicle (artificial cell) in an arbitrary three-dimensional shear flow, we derive analytical predictions of vesicle shape and membrane tension for vesicles close to a spherical shape. Large amplitude deviations from sphericity are described using boundary integral numerical simulations. Two possible modes of vesicle rupture are found and compared favourably with experiments: (i) for large enough shear rates the tension locally exceeds a rupture threshold and a pore opens at the waist of the vesicle and (ii) for large elongations the local tension becomes negative, leading to buckling and tip formation near a pole of the vesicle. We experimentally check these predictions in the case of strong acoustic streaming flow generated near ultrasound-driven microbubbles, such as those used in medical applications.

## 1. Introduction

Small gas bubbles, when placed in a sound field, are powerful actuators: their vibration, displaying a displacement amplitude much higher than that in the surrounding liquid medium, leads to strong flow fields known as acoustic streaming. Oscillating bubbles are also known for being very efficient scatterers of sound, a property used to enhance the contrast of echographic images by injecting micron-sized bubbles as contrast agents in the blood stream (see Chang *et al*. 1993; Becher & Burns 2000). In the course of such applications, mechanical effects on nearby tissues were discovered, including efficient cell membrane permeabilization in the vicinity of these bubbles (Miller & Quddus 2000; Ward *et al*. 2000), due to the opening of small pores (Taniyama *et al*. 2002). Although the sound amplitudes in most of these experiments are very large, smaller driving pressure amplitudes below 1 bar have also proved sufficient to induce such effects (Greenleaf *et al*. 1998). This local sound-induced permeability (sonoporation) opens the perspective of *in vivo* drug delivery or gene transfer in a targeted area, where the sound is applied.

Acoustic streaming is a prime candidate for the aforementioned mechanical actuation effects. It was observed that acoustic streaming flow from gently oscillating micron-sized bubbles leads to shear forces that are sufficiently strong to rupture lipid membranes (Marmottant & Hilgenfeldt 2003). Vesicles composed of a single lipid bilayer were used as artificial cells of controlled mechanical properties in that study. The steady deformation of vesicles in simple shear flows has been well understood; it was studied numerically with boundary integral methods (Kraus *et al*. 1996) or particle dynamics methods (Noguchi & Gompper 2004), experimentally in a Couette apparatus (de Haas *et al*. 1997) or near a wall (Kantsler & Steinberg 2006; Mader *et al*. 2006), and analytically (Seifert 1999). In the present article, we will be interested in describing the vesicle membrane tension and in understanding the rupture mechanism of the membrane in *arbitrary* flow fields, including those set up in acoustic streaming.

This article is organized as follows. An analytical study of small vesicle deformation for arbitrary flow fields, focusing on the spatial dependence of membrane tension, is described in §2. The case of large deformation amplitudes is treated numerically in §3. The shear flow originating from a vibrating bubble is explained in §4. Experiments showing the elongation and rupture mechanisms of a lipid vesicle are described in §5.

## 2. Elongation and tension of a vesicle in a shear flow

The purpose of this section is to determine the response of the vesicle in a shear flow, in the case of arbitrary small amplitude deformations. We will focus on calculating the membrane tension and its spatial dependence along the deformed vesicle.

### (a) Elasticity of the membrane

Shear stress in a liquid leads to deformation of lipid vesicle membranes, which resist the shear through membrane elasticity. The elastic properties are parametrized by an extension modulus with typical value (Boal 2002) and a bending modulus (Rawicz *et al*. 2000). The lipid membrane is a quasi-two-dimensional liquid; its elastic shear modulus within the membrane plane vanishes.

The total lipid area *A* of the vesicle can be decomposed as , with *A*_{p} as the contour area of the mean shape and *A*_{fluct} as the excess area generated by thermal fluctuations. The fluctuation area at equilibrium for vanishing tensions is for a vesicle of radius *R*∼10 μm, *κ*∼10^{−19} J and *b*∼1 nm, a molecular scale that gives the minimum wavelength of the thermal undulations (de Haas *et al*. 1997). *A*_{p} provides a reservoir of area under shape changes.

In the shear flow, the vesicle adapts to the applied shear stress by developing a tension *σ* in its membrane and by deforming into an ellipsoidal shape. We define the deviation from sphericity by , with *A*_{s} as the area of a sphere of the same volume. At zero shear flow and near vanishing tension, the vesicle might already be deflated and have a non-vanishing Δ*A*_{p}. Under shear, this deviation will be augmented by the ironing out of fluctuations so that the global elastic response to a *homogeneous* tension *σ* is(2.1)The first term on the r.h.s. represents the initial area after ironing out, while the second one is due to the membrane elasticity itself.

### (b) Deformation and distribution of membrane tension

#### (i) Arbitrary three-dimensional linear flow

To determine the location and triggering of membrane rupture, we now focus on the value and inhomogeneity of membrane tension, as well as the deformation. We begin with an analytical prediction at *small* deformation amplitudes, in the general case of an elongational and rotational flow. Indeed, in experiments we present, the flow is not a simple linear shear flow (where the elongation and vorticity rates are equal), but can be mainly elongational (near a stagnation point) or rotational (in a vortex centre).

We linearize an arbitrary flow field around the centre of mass location of the vesicle, i.e.at a position *x*_{i} away from the centre, where *G*_{ij} is the rate of strain tensor and *Ω*_{ij} is the antisymmetric rotational tensor, both evaluated at the centre of the vesicle. We choose the three-dimensional vector basis (*e*_{1}, *e*_{2} and *e*_{3}) with coordinates (*x*_{1}, *x*_{2} and *x*_{3}) so that *G*_{ij} can be written as diag(*g*_{1}, *g*_{2} and *g*_{3}); the rotation vector associated with *Ω*_{ij} is ** Ω**=

*Ω*

*e*_{3}and

*e*_{1}supports a larger elongational shear rate than

*e*_{2}, i.e.

*g*

_{1}>

*g*

_{2}.

We consider the vesicles whose internal liquid has the same viscosity *η* as the external liquid, leading to a simple tank-treading motion of the membrane, while viscosity contrast leads to complex dynamics termed tumbling (Rioual *et al*. 2004), breathing (Misbah 2006) or trembling and swinging (Kantsler & Steinberg 2005).

The behaviour of quasi-spherical vesicles in a flow was described by Seifert (1999). He studied the coupling between the deformations of the (incompressible) membrane and the hydrodynamics of the viscous motion of the fluid inside and outside the vesicles. The inhomogeneous tension of the membrane *σ*(*θ*, *ϕ*) (in spherical coordinates around the vesicle centre) and the local deviations from sphericity *r*=*R*(1+*u*(*θ*, *ϕ*)) create forces on the fluid. Using the Stokes equation, the force equilibrium prescribes the shape deviation and interface tension. The membrane tension and shape deviations can both be expanded in spherical harmonics.

Here, we explicitly derive the tension distribution in addition to the shape deviation. We use the fact that, for a (three-dimensional) linear flow, the normal and divergence Lamb components defined by Seifert (1999; for the velocity field on a sphere of radius *R*) are equal. Such a flow gives rise to mode 2 harmonics only, and thus we can use the expressions by Seifert to obtain the resulting mode 2 harmonics for the shape and tension. The vesicle outline in the *x*_{1}*x*_{2}-plane (orthogonal to the *x*_{3}-axis of vorticity) is then an elliptic curve whose major and minor semi-axes are(2.2)with . In the following, we assume large tension compared with the curvature-induced tension so that . Given the typical values cited above, this is an excellent assumption.

The major axis is oriented under an angle Δ*ϕ*_{0} with respect to the elongation *x*_{1}-axis, in the direction of the rotation ** Ω** (figure 1). The phase angle difference with respect to the main elongation axis is(2.3)

The vesicle is extended along the rotation *x*_{3}-axis, with a semi-diameter(2.4)When the rotation ratio is very small, the directions of the vesicle extension are oriented along the elongational axis and the extensions follow the elongational rates , and .

The average tension is derived from the area constraint. For small deviations, the increase in the outline area is written as(2.5)Equating this area increase to the elastic response (equation (2.1)) implicitly yields the average surface tension for a three-dimensional flow. Here, we assume that the average tension controls the total area extension.

Note that for vanishing rotation ratio , the area extension takes the symmetric form , using the flow incompressibility *g*_{1}+*g*_{2}+*g*_{3}=0.

The liquid shear forces on the membrane are not homogeneous and promote three membrane tension harmonics, *σ*_{2,0} and *σ*_{2,±2}, belonging to mode 2. In the *x*_{1}*x*_{2}-plane perpendicular to the rotation axis, the surface tension has maximum and minimum values given by(2.6)The surface tension minimum is located along an axis that makes an angle with the *x*_{1}-axis, and which is smaller than the shape orientation angle Δ*ϕ*_{0} of the largest elongationThe surface tension maximum is located along a perpendicular axis in the same plane (figure 1). On the *x*_{3}-axis, the surface tension is locally minimal if *g*_{3}>0.

When rotation is small , the relative changes in the surface tension are , and similarly for and . The relative changes in the surface tension are opposite to those of the extensions, while the radius maximum is located at the minimum of tension (and vice versa).

#### (ii) Application to two-dimensional linear shear flow

The small amplitude deformation of a vesicle in a shear flow was explicitly derived in the specific case of a simple plane shear *u*_{x}=*Gy*, for the case of a vesicle with thermal fluctuations but without elasticity (de Haas *et al*. 1997; Seifert 1999). Recent experiments confirmed the prediction on deformation of the shape (Kantsler & Steinberg 2005), as well as the numerical simulations (Beaucourt *et al*. 2004). Here, we derive the equations for an *extensible* elastic membrane.

In a two-dimensional linear shear flow, *u*_{x}=*Gy*, *g*_{1}=−*g*_{2}=*G*/2, *g*_{3}=0, *Ω*=*G*/2 and the positive elongational rate eigenvector *e*_{1} makes an angle of *π*/4 with *Ox*. The deformation is described by the harmonics *u*_{2,±2} only, when the spherical axis *Oz* is perpendicular to the flow. The contour in the *xy*-plane is an ellipse, whose semi-axes are, according to equation (2.2),(2.7)while the shape is inclined by the angle with respect to the main elongation axis. The deformation is parametrized by only one parameter, known as the Taylor deformation parameter *D*=(*L*−*B*)/(*L*+*B*), defined with an ellipse major axis of length *L* and a minor axis of length *B*. This equation is similar to the deformation of a liquid droplet with constant interfacial tension *σ*_{interface} (Taylor 1932; see also the review article of Stone 1994 and the comparison between droplets and vesicles by Danker *et al*. in press). This extends the domain of applicability of the liquid drop model to fluctuating and extensible membranes by replacing the interfacial tension *σ*_{interface} with a variable membrane tension .

Rearranging the terms in equation (2.7), we note that the shear, expressed here by a shear *tension ηGR*, is counteracted by the product of the variable membrane tension and the deformation *D*.

The deformation implies an area increase through . Using the elastic response of area (equation (2.1)), we obtain(2.8)This expresses the global nonlinear ‘spring-like’ elastic behaviour of the vesicle, and gives the deformation *D* as a function of mean tension .

Rewriting equation (2.7), the homogeneous variable tension as a function of shear rate is then implicitly given by(2.9)where we have introduced the non-dimensional quantitywhich can be called a ‘capillary number’ since the extension modulus has the dimension of a capillary surface tension (Vitkova *et al*. 2004).

Thus we obtain both the tension and the deformation *D* as a function of *Ca*_{K}. Two regimes appear when solving equations (2.8) and (2.9): (i) a small tension regime when , with the deformation being and (ii) a large tension regime when elastic increase of the lipid membrane is predominant, with the deformation tending to be .

The inhomogeneity of membrane tension can be given as , with an amplitude of variation around the average of from equation (2.6), while the maximum and minimum tensions are(2.10)The maximum occurs near the narrowest part of the vesicle (where the membrane distance to the centre is the smallest). The minimum occurs at the angle with respect to the main elongation axis, different from the angle of the shape Δ*ϕ*_{0} (figure 1). From equation (2.10), we see that Δ*σ* is proportional to *D* to first order.

### (c) Predicted modes of breakup

We now consider the two possible scenarios for breakup, depending on the extremal values of membrane tension: either the maximum tension reaches a critical value or the minimum tension becomes negative.

#### (i) Membrane rupture

The lipid membrane cannot sustain an infinite elastic extension and above an elastic extension of approximately , the membrane ruptures (Needham & Nunn 1990; Boal 2002). The condition for breakup , which we approximate by for small deformations, provides a rupture criterion based on the capillary number when using equation (2.9) and substituting (2.11)

#### (ii) Membrane buckling

Extrapolating equation (2.10) to large amplitudes, we see that the tension might become negative near the pole, resulting in a *buckling* of the membrane. The situation *σ*^{min}<0 occurs when the deformation is larger than a critical value(2.12)which is approximately using equation (2.10). The deformation *D* is a growing function of *Ca*_{K}, starting from an initial value *D*_{0}, which is a function of the initial excess area. The onset of buckling is therefore expected to occur *before* rupture when the initial deformation is large enough. The purpose of §3 is to assess the validity of these predictions, which were based on small deformations (quasi-spherical shapes).

## 3. Deformation and distribution of tension at large amplitude: numerical calculations

### (a) The boundary integral method

A convenient way to compute the dynamics of a vesicle in a Stokes flow is provided by the boundary integral method (Pozrikidis 1992). For a vesicle, the force field along the membrane and the velocity field at a point ** r** of the membrane are related by(3.1)where

*f*_{mem}(

**) is the force field applied at point**

*r***of the membrane and is the Green tensor in free space (‘stokeslet’). The original three-dimensional problem can thus be reduced to a two-dimensional problem involving the membrane only.**

*r*This force field can be derived from the Helfrich curvature Hamiltonian (Helfrich & Servuss 1984),(3.2)where *c* is the local curvature of the membrane at point ** r** (

*c*=

*c*

_{1}+

*c*

_{2},

*c*

_{1}and

*c*

_{2}are the two principal curvatures at point

**of the membrane);**

*r**g*is the Gaussian curvature (

*g*=

*c*

_{1}

*c*

_{2}) that gives rise to a constant after integration since the topology does not change and we therefore do not take into account this term;

*σ*is the local surface tension;

*c*

_{0}is an eventual spontaneous curvature that we will not consider in the following; and

*κ*and are the curvature modulus and the Gaussian curvature modulus, respectively. The force field generated by this Hamiltonian can be calculated aswhere Δ

_{2D}is the Laplace–Beltrami operator on the membrane and ∇

_{2D}is the gradient operator on the membrane. The first term is the local curvature force and the second is the well-known Laplace force, while the last describes the adjustable force necessary to reduce the area towards the equilibrium. In the limit of small local area changes, the dynamics of the surface tension is given bywhere

*A*(

**) is the local area surrounding the point**

*r***(triangle in the computational mesh) and**

*r**K*

_{A}is the area extension modulus (large so that the extension remains small).

### (b) Mesh advection

We use a specific advection scheme to prevent entanglement of the mesh in the tank-treading regime due to the differential rotation between the points located in the equatorial region and the vertices located close to the rotation poles. Rather than remeshing the vesicle from time to time, which is the usual method, we prefer to modify the advection scheme so that the mesh reaches a steady state during the tank-treading motion. The mesh is advected to follow the dynamics of matter only in the normal direction, in the frame moving with the global shape of the vesicle.Where is the outgoing normal vector at point ** r** of the membrane,

*u*_{rot}is the solid body rotation velocity and

*v*_{cm}is the centre of mass velocity, we obtain the advection velocity in the laboratory frame asThis scheme can be considered as a continuous remeshing of the vesicle.

### (c) Parameters

The algorithm is implemented for a vesicle using a mesh with 642 points, obtained by refining an icosahedron and adding noise to the point positions to avoid instabilities. The vesicle is embedded in a simple two-dimensional shear flow *u*_{x}=*Gy*. The strain rate *G* was held constant in all computations, with a stretching ‘capillary’ number of *Ca*_{K}=*G*/(*K*_{A}/*ηR*)=10^{−4} (the code was unstable for higher values), while the bending capillary number was *Ca*_{K}=*G*/(*κ*/*ηR*^{3})=100, meaning that curvature effects are negligible when compared with viscous forces. The code was run with time steps of d*t*=10^{−6}/(*κ*/*ηR*^{3}), using a steady shear rate until a stationary shape was obtained (within 1%). The volume of the vesicle was regularly rescaled to ensure its conservation.

A given initial excess area Δ*A*_{0}/*A* is provided (with respect to the area, a sphere of equal volume). We prefer to express the initial excess area using , which directly provides the expected deformation for small strain rates in a two-dimensional flow in the small tension regime.

More commonly, the excess area is linked to the reduced volume , comparing the actual volume *V* to the maximum available volume for the same area (with a sphere shape). Its expression as a function of the excess area is . We vary the initial deformation from *D*_{0}=10^{−3} up to *D*_{0}=0.95, so that the initial reduced volumes *v*_{0} range from 0.9999992 to 0.55.

### (d) Results

Depending on the initial deflation *D*_{0} of the vesicle, the final steady shape of the vesicle is an elongated ellipsoid (figure 2). For small deflations, the main axis of the ellipsoid is along the positive elongational axis of flow, inclined at a 45° angle with respect to the fluid velocity. This angle decreases when deflation becomes appreciable (as shown by Kraus *et al*. 1996), in accord with the deviation angle Δ*ϕ*_{0} increasing with (equation (2.3)). The longest axis of the vesicle is not an axis of symmetry; the vesicle diameter is larger in the direction orthogonal to the plane of the image, where the elongational rate from the flow is absent. At very large deformation, the shape is far from ellipsoidal, with a long central part being close to cylindrical (‘cigar’ shape; figure 2*d*).

The tension has minimum values near the main axis ‘pole’ of the vesicle and maximum values at the equator and in the plane of shear. This can be understood graphically; the shape is flattened due to the additional tension and extra area bulges out of the initial sphere due to decreased tension.

The position of the minimum tension deviates slightly from the position of the maximal values radius (figure 2*d*, the tension minimum being oriented closer to the 45° axis), a tendency predicted by the small amplitude predictions of the previous section, the deviation angle Δ*ϕ*′_{0} being smaller than Δ*ϕ*_{0}.

The actual steady deformation was defined here using , where the vesicle and excess areas are evaluated by fitting the vesicle mesh to an ellipsoid. We did *not* use the Taylor *D*=(*L*−*B*)/(*L*+*B*) definition that has the inconvenience of saturating to the value 1 for large elongations. The steady deformation of vesicles in a given shear flow is found to depend strongly on initial deformation. At large *D*_{0}>10^{−1}, the steady deformation *D* is close to *D*_{0}, as expected. However, all initial deformations *D*_{0}, no matter how small, give rise to a finite value for steady deformation (figure 3*a*). This effect can be understood from equation (2.9); small deformations induce strong tensions, which in-turn increase the elastic area and the actual deformation until a balance is reached, with the deformation reaching the value predicted by the large tension regime of equation (2.9). The deformation obtained from the numerical computations is slightly higher than the predicted value.

The computed average tension is inversely proportional to the actual deformation and closely follows the low-amplitude prediction (figure 3*b*), except at large deformations where the nonlinearities develop fully, resulting in a larger tension than predicted. The assumption of uniform tension, used in the liquid drop model, fails at this point.

The minimum tension tends towards the vanishing values for increasing deformation (figure 3*b*). For initial deflation parameters *D*_{0} larger than 0.95 (*v*_{0} smaller than 0.55), the vesicle shape becomes numerically unstable. We expect a tip-streaming-like instability to occur at this point, since the numerical shape develops a growing tip before breakdown of the calculation. Similar buckling instabilities were found on solid capsules (Lac *et al*. 2004) and on liquid droplets (Stone 1994). According to the small amplitude deformation theory, the minimum tension departs from the average with growing deformations (equation (2.10)). Extrapolating this prediction to finite values surprisingly provides a correct estimate of *σ*^{min}, and predicts a vanishing tension for close to the numerical stability limit.

## 4. Shear flow generated near an ultrasound-driven microbubble

We now consider the description of the strong shear flow induced by the vibration of a bubble, prior to the introduction of lipid vesicles into this flow.

The set-up and the materials and methods used in the experiments are described in detail by Marmottant & Hilgenfeldt (2003, 2004). Briefly, we consider air bubbles in liquid, of radius *a*=10–100 μm, fixed by capillarity on the wall of a quartz cuvette (Hellma, Germany, 10×10×50 mm). The quartz cuvette, completely filled with liquid, is excited by a piezoelectric transducer, glued to one of the cuvette walls, establishing a standing ultrasound wave field in the liquid (figure 4).

### (a) Recirculating acoustic streaming flow

The fast vibration of the bubble wall, even though perfectly time-periodic, generates a steady flow; this is the acoustic streaming phenomenon. It has its origin in the inertial nonlinearity of the Navier–Stokes equations where a boundary vibrating with a displacement *ϵ* cos *ωt* will give a velocity response whose components *v*_{i} can be expanded in *ϵ* as . While the first term is time-periodic and averages out in time, the second quadratic term containing gives a steady flow, on average.

In response to sound, the bubble gently vibrates with an oscillating radius . Owing to the presence of the wall, the centre of mass is also translating with a position (Marmottant *et al*. 2006*b*). The acoustic streaming flow resulting from this combination of radial and translational vibrations was described near a boundary by Marmottant & Hilgenfeldt (2003). Using this prediction, we plot streamlines near a wall in figure 5*a*.

The velocity amplitude predicted by the acoustic streaming is at most , near the bubble wall. From the measured oscillation amplitude (Hansen *et al*. submitted) and the radius and frequency parameters *a*=35 μm and *f*=67 kHz, we expect the maximum velocity to be of the order 6 mm s^{−1} (if sin *Φ* is of the order 1, in reality sin *Φ* was observed to be 0.22, Marmottant *et al*. 2006*b*). The far-field dependence of the acoustic streaming velocity, decaying as *r*^{−3} due to the closeness of the wall, is consistent with the experimental trend. The motion is thus much localized around the bubble.

### (b) Strain rate generated by the flow

The local deformation of fluid particles is characterized by the rate of strain tensor . After diagonalization, we find the elongation rate in three directions (*g*_{1}, *g*_{2}, *g*_{3}). Taking the steady streaming component of *v*_{i} as the velocity *u*_{i}, the direction of the positive elongation axis is plotted with an elongated ellipse (figure 5*a*). The sign of the transverse elongation rate is indicated by the filling of the ellipse. When a fluid particle is approaching the symmetry axis, the transverse elongation rate *g*_{3} is negative. Since the other elongation axes maintain a negative and a positive elongation rate, the signs of the elongation rates are (+, − and −): an initially spherical fluid particle is deformed into a cigar-like (prolate) ellipsoid. On the other hand, when the fluid particle is going further away from the axis, the elongation rates have the signs (+, + and −): a sphere is deformed into a disc-like (oblate) ellipsoid. A fluid particle is thus found to alternately deform towards a prolate and an oblate ellipsoid during a revolution on its orbit, giving rise to the deformations observed and quantified in figure 6.

The strain rate magnitude is evaluated by the Frobenius norm of strain rate tensor, . The strain rate is maximal near the bubble, where it reaches a value of approximately 10*v*_{0}/*a* (figure 5*b*). The corresponding strain rate is very high, G∼*ϵ*^{2}*ω*∼3×10^{3} s^{−1} for typical experimental values.

## 5. Experimental observations with vesicles

In this section, we present a variety of phenomena that are observed when the vesicles are in the vicinity of oscillating bubbles. These results qualitatively compare with and validate the predictive approach of the previous sections.

### (a) Materials and methods

Lipid vesicles are injected into the liquid contained in the cuvette. We chose giant unilamellar lipid vesicles owing to their well-known mechanical properties (Rawicz *et al*. 2000; Boal 2002). Vesicles of radii 10–100 μm were grown with the electroformation method (Dimitrov & Angelova 1987) with the lipid 1,2-dioleoyl-sn-glycero-3-phosphocoline (DOPC). We fill the vesicles with a liquid different from the outer liquid, to create an optical index contrast that results in a high contrast for phase microscopy. For this purpose, the vesicles were grown in a 0.3 M sucrose solution, plus 1 mM of the bactericide sodium azide and then transferred into an equi-osmolar glucose solution (Sandre *et al*. 1999). Their interior appears dark with phase contrast optics. Because the interior density is 4% higher, they slowly sediment and reach the bottom of the cuvette, and thus the plane of observation for an inverted microscope (Zeiss Axiovert 25CFL, Germany). For fluorescence microscopy, either the membrane or the interior liquid was marked. The membrane dye was a florescent lipid, 2-(4,4-difluoro-5,7-dimethyl-4-bora-3a, 4a-diaza-s-indacene-3-pentanoyl)-1-hexadecanoyl-sn-glycero-3-phosphocholine (*β*-BODIPY FL C5-HPC, Molecular Probes, The Netherlands), chosen for the similarity of its structure with DOPC. They are both glycerophospholipids, with the same phosphocholine head, and their carbon chains have comparable lengths. The liquid interior was dyed with a large weight fluorescent molecule that does not permeate through the membrane (5,6 carboxyfluorescein, Sigma Chemicals). The viscosity of the medium was adjusted by mixing the water solution with different concentrations of glycerol, providing higher viscosity solutions, up to 20 times more viscous than water.

The interaction of the bubble with vesicles was observed through the inverted microscope using phase contrast or fluorescence, and images were recorded on a digital still picture camera (Sony DSC-S75), a digital video camera (Sony DCR-PC120E) or a high-speed camera (Kodak Ektapro CR Imager, up to 2000 frames s^{−1}).

### (b) Vesicle deformation

Looking perpendicularly through the glass wall on which the bubble is attached, along an axis that we call *Oz*, we observe an apparent periodic ‘bouncing’ of the vesicle on the bubble, due to the projection of the vesicle orbit. The vesicle is deformed in this process: elongated when it is approaching, compressed when it is receding (see the high-speed image sequence in figure 6 where the vesicle is out of focus in the last two images).

Seen along the axis *Oz*, the projected shape is roughly elliptical. This ellipse has a symmetry axis along the streamline, its extent is along this direction and *l*_{⊥} in the transverse one. The aspect ratio of the vesicle seen along *Oz* is estimated through the parameter (figure 6*a*). The parameter *D*_{app} is positive when the vesicle looks elongated along the trajectory and negative when it looks compressed. It changes sign when the vesicle is nearest to the bubble and when the vesicle is receding from the bubble (figure 6*c*).

The shape of the vesicle evolves along its trajectory, experiencing the heterogeneous liquid strain rate distribution mapped in figure 5. We predict the vesicle shape, first by determining the streamline on which the vesicle is orbiting (solid line in figure 6*b*), then by using the theoretical principal strain rate elongations along this streamline in equation (2.5). We determine the average tension for two cases using an incompressible membrane (with an excess area of 4%) or elastic membrane with equation (2.1). From the average tension, we can determine the principal axis lengths of the ellipsoidal vesicle (*R*_{1}, *R*_{2}, *R*_{3}), with *R*_{3} the axis transverse to the trajectory plane. We project them to obtain the apparent axes *l*_{⊥}=*R*_{3} and , with the axis tilted by *θ*, to obtain the apparent deformation parameter *D*_{app}. The predicted deformation reproduces the changes in sign and is quantitatively correct when the vesicle is close to the bubble (figure 6*c*). Using the elasticity of the membrane (solid line) instead of a fixed membrane excess (dashed line) obtains a better estimate of the vesicle shape at large distances, i.e. small strain.

### (c) Membrane rupture

The deformations are enhanced in a more viscous solution and can be amplified to the point where breakup of the membrane becomes possible. In all the experiments that we present in this section, the solution is a 1 : 2 (by volume) mixture of water and glycerol, which increases the dynamic viscosity by a factor of approximately 20. Several regimes leading to membrane breakup could be observed, depending on the parameters governing the bubble oscillation dynamics.

#### (i) Vesicle approaching the bubble

A large vesicle in the viscous liquid undergoes large deformations even far away from the bubble, and its membrane eventually ruptures when the vesicle is close enough (figure 7). The shearing motion of the liquid breaks the membrane in several pieces (last image).

#### (ii) Tank-treading vesicle trapped in a vortex centre

Some vesicles can be trapped in the centre of the recirculation vortex; their position remains stationary and they only display a tank-treading motion. This attraction to the stationary line happens after a few revolutions in orbits of decreasing amplitude around the line.

The rotation frequency and deformation increase with the sound amplitude. A tip pointing towards the bubble is formed on large vesicles (figure 8), and observations with the microscope at different focusing elevations show that the tip is drawn between the bubble and the wall, as if the vesicle was aspirated there. Smaller vesicles are just elongated. Above a critical sound amplitude, the membrane cannot sustain the deformation and breaks up (not shown).

#### (iii) Reversible rupture: pore opening and closure

The opening of a hole in a lipid membrane is not irreversible and pores can spontaneously close. As the lipid membrane is liquid in its plane, a line tension acts on the edge of a pore, effected by the exposed hydrophobic tails of the lipids. Hole opening and closure have been observed in the case of a lipid vesicle tensed by an intense illumination or adhesion to a substrate (Sandre *et al*. 1999; Brochard-Wyart *et al*. 2000). The ejection of liquid through the pore then releases the tension and the hole closes.

In our experiments, vesicles can be subject to a transient shear when an oscillating bubble, initially attached to the wall, is pushed away by acoustic pressure when the sound amplitude is suddenly increased. In this fashion, a vesicle rotating on its trajectory near a bubble experiences a strong transient shear flow that ruptures the membrane (figure 9). Some internal liquid is ejected out of this pore and dissipates quickly. The hole appears to close completely after a few seconds, as the observed leakage of dye ceases. The volume ejection can be inferred from the size reduction of the vesicle. In figure 9, we measure a volume loss of 33%.

Using equation (2.11), we estimate the capillary number for breakup to be , using and an initial excess area . In our experiments for small vesicles in pure water (*Ca*_{K}∼10^{−4}−10^{−3}), the vesicle indeed never ruptured. In the glycerol/water mixture with *η*=20*η*_{water} and large vesicles *R*∼20 μm, this breakup criterion is fulfilled at large sound amplitudes, where we had *G*∼3×10^{3} s^{−1}. All the observed ruptures occurred in experiments with such a large capillary number.

### (d) Strong localized deformation: tip and tether formation

The membrane deformation can be localized to a small part of the vesicle. Experiments with bubbles positioned on a substrate designed with shallow holes (Marmottant *et al*. 2006*a*) showed cusp shapes and tip streaming of the vesicles (figure 10*a*). This appears to be a case where breakup is preferred through negative tension and buckling localized at the tip of the vesicle.

At high acoustic pressures, negative pressure (tension) in the liquid results in spontaneous formation of small bubbles (cavitation). These bubbles travel fast in the bulk and abruptly change directions. Exposing a solution of lipid vesicles to cavitation produces a different kind of vesicle deformation and their shape is turned into long tubes (figure 10*b*). The remaining initial vesicle can be seen on one end of these tubes (figure 10*c*), where the tube is a tether pulled out of the vesicle by the large hydrodynamic shear stresses around the moving bubbles.

The extension of a vesicle into a tube was observed by Shahidzadeh *et al*. (1998), but for low (0.5 s^{−1}) constant shear rates. The tube can be a stable shape for the membrane if it is tensionless, especially if its radius is close to the spontaneous curvature radius of the membrane (Rossier *et al*. 2003). Here, a tension is still present in the membrane and results in a pearling instability (Bar-Ziv & Moses 1994) that is visible in figure 10*b*.

Compared with the previous experimental situation that never resulted in such a tether formation, the difference is the size of the cavitation bubbles, which are much smaller than the vesicle (their apparent diameter when they travel fast in front of the objective is less than 10 μm), the high intensity of their oscillation and the speed of their translation. This tether formation process is similar to the local application of a point force with optical tweezers that is used to pull out the membrane out of an initially spherical vesicle (Fygenson *et al*. 1997; Powers *et al*. 2002).

From theory, the tension is expected to vanish at a specific location of the membrane when the deformation is large enough (see §2). In practice, vesicles are not sufficiently deflated initially to reach the large critical deformations determined by the numerical calculation. Tip formation is, however, expected to occur more easily when the shear flow is not homogeneous but is stronger near one end of the vesicle. In this case, the flow cannot be linearized and the previous analysis does not hold any more. This occurs when the vesicle is very close to a bubble where the flow field is inhomogeneous on scales smaller than the vesicle size (figure 10*a*). The same small-scale argument applies to the more extreme case of tether-pulling.

## 6. Conclusions and perspectives

Gentle oscillations of microbubbles attached to a wall generate intense acoustic streaming flows. Soft objects, such as lipid vesicles, experience a high shear stress in the streaming, which deforms their outline and generates membrane tension that varies in time and space. The theoretical analysis in a linearized shear flow shows that the membrane tension has a maximum near the minor axis (the ‘waist’ of the vesicle) and a minimum near the major axis. The average tension determines the overall shape. The membrane is expected to rupture at the waist when the maximum tension reaches a critical value (controlled by the capillary number *Ca*_{K}). On the other hand, the minimum tension can become so small that it nearly vanishes (which occurs for large enough deformation *D*), leading to buckling and a tip-streaming-like process of local membrane extension. Tip streaming is expected to occur before breakup for deflated vesicles, i.e. those with sufficient initial excess surface area or in sufficiently inhomogeneous shear flows.

Upon breakup of vesicles, pores are nucleated and interior and exterior fluids are exchanged. In the case of living cells, these pores are believed to be at the origin of enhanced drug delivery or gene transfer when diagnostic ultrasound is applied (sonoporation). However, the acoustic streaming flows are much more controlled and have much smaller peak stresses than cavitating bubbles in ultrasound diagnostics, which often undergo violent collapses and damage cells indiscriminately. Note that the ultrasound amplitudes used in medical ultrasound are typically in the MPa range, while the typical amplitudes in our streaming experiments are below 10 kPa. Nevertheless, we demonstrated pore opening in lipid membranes, and preliminary experiments have shown that the membrane of HeLa cancer cells can also be lysed in the streaming flow near microbubbles. Further tests will determine whether the pore opening can be gentle enough to allow cell repair and survival. The effect of low-intensity ultrasound on vesicles and cell rupture indicates that more controlled experiments are possible *in vitro* and *in vivo*. We have shown that violent bubble collapses are not necessary for the rupture of membranes, and thus for transfection.

The shear flow is focused on very small volumes because the acoustic streaming velocity of the bubble decays rapidly as *r*^{−3}. Consequently, the shear stresses decay even faster. The focusing turns the energy in a mm-wavelength ultrasound driving into an incisive tool on the micron scale, effectively establishing microacoustic effects with low-frequency ultrasound. Micrometre-scale forces can thus be exerted on small objects such as cells, to push, deform or break them. With microacoustic tools, new potential applications in microfluidic devices arise, such as a single-cell manipulation, acoustic probing and permeabilization.

## Acknowledgments

We would like to thank Detlef Lohse for his support, and Claus-Dieter Ohl and Rudiger Toegel for their help while the experimental work was performed at the University of Twente. We enjoyed fruitful discussions with Gerrit Danker, David Hansen and Manouk Abkarian.

## Footnotes

↵† Present address: Engineering Sciences and Applied Mathematics and Department of Mechanical Engineering, Northwestern University, 2145 Sheridan Road, Evanston, IL 60208, USA.

- Received December 7, 2007.
- Accepted February 28, 2008.

- © 2008 The Royal Society