Brice Saint-Michel

Valeria Garbin
email: v.garbin@tudelft.nl

Acoustic bubble dynamics in a yield-stress fluid

Yield-stress fluids naturally trap small bubbles when their buoyancy applies an insufficient stress to induce local yielding of the material. Under acoustic excitation, trapped bubbles can be driven into volumetric oscillations and apply an additional local strain and stress that can trigger yielding and assist their release. In this paper we explore different regimes of microbubble oscillation and translation driven by an ultrasound field in a model yield-stress fluid, a Carbopol microgel. We first analyse the linear bubble oscillation dynamics to measure the local, high-frequency viscosity of the material. We then use acoustic pressure gradients to induce bubble translation and examine the elastic part of the response of the material below yielding. We find that, at moderate pressure amplitude, the additional stresses applied by volumetric oscillations and acoustic radiation forces do not lead to any detectable irreversible bubble motion. At high pressure amplitude, we observe non-spherical shape oscillations that result in erratic bubble motion. The shape modes and critical pressures we observe differ from the predictions of a recent model of shape oscillations in soft solids. Based on our findings, we discuss possible reasons for the lack of bubble release in Carbopol and suggest other systems in which ultrasound-assisted bubble rise may be observed.

Introduction

Yield-stress fluids encompass a wide range of materials including foams, suspensions, emulsions and microgels 1,2 . These materials exhibit a threshold in applied stress, called the yield stress, below which the material behaves like a solid, and above which it flows like a liquid. A clear manifestation of the yield stress is the presence of trapped bubbles, when their buoyancy force is too small to yield the material. Trapped bubbles can be beneficial, for instance when they are used to impart texture to a food product, or they can be detrimental as they can negatively affect the thermal conductivity or optical transparency of a material. Strategies to control the amount and size distribution of trapped bubbles are therefore important in processing of formulations and advanced materials. There is some experimental evidence that driving bubbles into volumetric oscillations in yield-stress fluids can assist their removal 3 , but the effect of oscillations on yielding is poorly understood. This lack of understanding is particularly detrimental to the development of controlled bubble removal methods.

Understanding bubble dynamics in yield-stress fluids is particularly challenging since their rheology is not even fully understood in the case of simple shear. Yield-stress fluids are only well-understood in the limit of very small shear stresses a linear elastic behaviour is recovered, or for large, steady stresses for which their flow rheology usually obeys the Herschel-Bulkley equation 1 . For intermediate stresses, experimental results performed under steady or large amplitude oscillatory shear 4 have evidenced that yield-stress fluids exhibit non-linear 5 , time-dependent, cooperative 6 behaviour. Such features are only captured by the most recent microscopic 7 and continuum mechanics models 8 .

The capacity of a yield-stress fluid to entrap bubbles up to a critical effective radius R c can be expressed as the dimensionless number Y -1 c = 2ρ l gR c /3σ Y , where ρ l is the liquid density, g is the acceleration due to gravity and σ Y is the yield stress of the material 9 . Even with the most conservative estimate, a yield stress of only 10 Pa causes trapping of bubbles up to 2R c = 6 mm in diameter. Removing bubbles below the critical size can be achieved by centrifuging 10 , applying a vacuum, or by using low-frequency (∼ 100 Hz) vibrations to suppress the yield stress in fragile granular networks 11 . These techniques alter the physical parameters at play in the definition of Y -1 c rather than fundamentally altering this criterion.

Bubbles are however not passive under the application of vibrations and acoustic excitations, as the dynamic pressure field drives them into volumetric oscillations 12 . Oscillating bubbles apply a local strain field to the surrounding material, which in turn reacts by exerting a stress onto the bubble, altering the oscillation dynamics. Bubble dynamics in Newtonian liquids 12 and soft solids 13 is now a well-established topic, motivated e.g. by the direct role played by bubble collapse in therapeutic laser or ultrasound tissue ablation 14,15 . Bubble radius time profiles are now even used either in the linear regime 16 or the strongly non-linear, cavitation regime 17 to extract local rheological properties of soft solids.

In yield-stress fluids, oscillating bubbles may apply a strain that is sufficient to locally yield the material, defining a yielded, fluid region. Bubble rise may then proceed in this confined region even if their size is well below R c . The size and shape of this region has a key influence on the bubble rising velocity, and ultimately in the efficiency of the removal process.

While the shape of the yielded region has been investigated in great detail for passive bubble rise 9,18 , the case of oscillating bubbles has only been examined very recently 19,20 . Building upon the progress in modelling both bubble dynamics in soft materials 13 and the rheology of yield-stress fluids 21 , these articles confirm that bubble rise is indeed possible for bubbles below R c 19 ; they also compute the minimum oscillation amplitude required to initiate yielding 20 . To the best of our knowledge, these numerical and theoretical results have not yet been compared to experiments: experimental articles so far have focused on the case of bubble removal in a shear-thinning, viscoelastic surrounding fluid 22 and removal in yield-stress fluids for bubbles already close to the static rise radius at rest, R c 3 .

In this article, we conduct experiments to test the criterion for medium yielding and bubble removal that we previously derived 20 , using a Carbopol microgel as a model yield-stress fluid. We investigate the oscillation dynamics of initially spherical bubbles (100 -200 µm) excited by a standing-wave ultrasound field with controlled frequency (19 -30 kHz), acoustic pressure amplitude, and spatial distribution of pressure gradients. We measure the resonance curve of the bubbles, their mobility in a pressure gradient and the onset of non-spherical shape oscillations. We extract the viscosity and linear elastic modulus of the material, and compare these measurements to the predictions of the model 20 . We finally conclude on the efficiency of bubble removal through bubble oscillation in yield-stress fluids.

2 Bubble dynamics in yield-stress fluids

Governing equations for bubble oscillations

We briefly recall here the physics of the linear oscillations of a spherical bubble in a yield-stress fluid we derived in a previous article 20 . We will show in Section 3.4 that the assumptions of spherical bubble and linear dynamics are reasonable given the size of the bubbles and the rheological properties of the fluid that we use in the experiments.

A bubble with equilibrium radius R 0 is driven into volumetric oscillations under an acoustic excitation at a frequency f , i.e. a sinusoidal applied pressure p(t) = p sin(2π f t) far away from the bubble. The time-dependent radius, R(t), is:

R(t) = R 0 [1 + ζ (t)] . (1)
Applying the momentum and the mass conservation for the fluid between the spherical bubble surface r = R(t) and r → ∞ yields a generalised Rayleigh-Plesset equation valid for arbitrary fluids 23 . Previously our group has derived a model for bubble dynamics in yield-stress fluids by combining the generalised Rayleigh-Plesset equation 23 with the elasto-viscoplastic rheological model proposed by Saramito 21 . The details of the full model can be found in Ref. 20. We recall here that for small-amplitude oscillations and below the yield point, the rheological model reduces to a Kelvin-Voigt viscoelastic solid of linear elastic modulus G and solvent viscosity η s . A Taylor expansion of the momentum balance valid at order 1 in ζ may then be derived following the classical linear theory of bubble dynamics 24 :

ζ + 2β ζ + 4π 2 f 2 0 ζ = - p ρR 2 0 sin(2π f t) , (2)
in which ρ is the fluid density, assumed to be a constant, and β and f 0 are respectively the damping coefficient and the natural frequency of the bubble oscillations. These two quantities depend a priori on the rheology of the fluid.

Equation ( 2) is a standard second-order linear differential equation that we can reformulate in the frequency domain.

We then obtain the second-order transfer function for the bubble oscillation amplitude ζ in the spirit of earlier works on bubble spectroscopy 16,25,26 :

ζ (t) = ζ sin(2π f t + φ ) (3a) ζ = p/ρR 2 0 4π π 2 f 2 0 -f 2 2 + β 2 f 2 (3b) φ = π 2 + arctan π β f ( f 2 0 -f 2 ) (3c)
The amplitude part of the transfer function [Equation (3b)] gives the resonance curve of the bubble. The phase lag between the bubble oscillation and the pressure field φ made explicit in Equation (3c) spans from π for f f 0 in the low frequency case to 0 for f f 0 in the high frequency case.

The natural oscillation frequency f 0 based on the model of Saramito 21 is derived in Ref. 20 :

f 2 0 = 3κ p 0 + 2(3κ -1)Γ/R 0 + 4G 4π 2 ρR 2 0 , (4)
in which Γ is the surface tension between the gas and the fluid and p 0 is the ambient atmospheric pressure . We also introduce here the polytropic exponent 1.0 ≤ κ ≤ 1.4 that indicates the nature of the thermodynamic process occurring in the bubble, from isothermal (κ = 1) to adiabatic (κ = 1.4) depending on the thermal Péclet number 24 .

For very soft materials for which G p 0 , and for sufficiently large bubbles, i.e. for R 0 Γ/p 0 = 1.0 µm, we recover the standard Minnaert frequency 27 for a given bubble radius R 0 :

f m = 1 2πR 0 3κ p 0 ρ . (5)
Equation ( 5) may be used as well to derive a resonant radius R m for a given oscillation frequency f . We also recall the predictions for the damping parameter β 20,26 :

β = 2η eff ρR 2 0 = 2 ρR 2 0 η s + π 2 ρ f 2 R 3 0 c + 3p 0 κ 8π f (6)
The three terms at the right hand side of Equation ( 6) respectively account for viscous dissipation proportional to the solvent viscosity η s in the Kelvin-Voigt model; acoustic scattering of the bubble, and thermal dissipation, in which the dimensionless quantity κ is related to the polytropic exponent κ introduced earlier 24 . Appendix B shows the relative magnitude of each contribution to β for our experiments. The relative uncertainty on these quantities is discussed in ESI Section 1.

Acoustic radiation forces

Gradients in a pressure field exert a force F = -V ∇ ∇ ∇p on objects of volume V . In a standing wave field p(x,t) = p(x) sin(2π f t), the average force F applied on an incompressible object of fixed volume V over one oscillation cycle is zero. Because bubbles expand and contract in response to oscillations in pressure, the same pressure gradient applies a larger net force on the object when its radius is large than when it is small. This leads to a net force over one oscillation period called Bjerknes force 28 :

F(x) = -2πR 3 0 ∇ ∇ ∇p(x)ζ cos(φ ) . (7)
For a driving frequency f and an equilibrium bubble size R 0 , small bubbles for which cos(φ ) = -1 will move towards high pressure areas (named anti-nodes) whereas large bubbles for which cos(φ ) = +1 will move towards low pressure areas (nodes), a classical result in Newtonian fluids 29 . Bjerknes forces are non-linear as both ∇ ∇ ∇p and ζ are proportional to the applied pressure. They are particularly efficient at pushing and pulling bubbles against gravity when the relative pressure gradient |∇ ∇ ∇p/p| is high.

Following Equation (3b) the pressure p required to obtain a constant oscillation amplitude ζ for all bubble radii R 0 is much higher far away from the resonance condition than at resonance. As a consequence, for an imposed oscillation amplitude ζ the pressure gradient ∇ ∇ ∇p in Equation ( 7) and the Bjerknes forces will also be stronger away from resonance.

We will use this strategy in Section 4.3 to apply strong Bjerknes forces while remaining in the linear range of the bubble oscillation amplitude ζ .

Recent articles have related the force applied to spherical objects and their displacement in purely elastic 30 or Kelvin-Voigt viscoelastic solids 31 , which can then be applied to yield-stress fluids for relatively small deformations. Assuming the pressure gradient ∇ ∇ ∇p at location x is directed alongside z we have:

∆z(x) R 0 = 1 R 0 F z (x,t) 4πGR(t) = - 1 3 R 0 G ∇ z p(x)ζ cos(φ ) (8)
Equation ( 8) remains valid as long as the oscillations do not alter the properties of the fluid. Interestingly, it provides a measurement of G that is unaffected by Γ and p 0 in contrast with Equation (4). We will use Equation 8 to measure G in Section 4.3.

Yielding criteria and impact on bubble dynamics

Yielding to oscillations When no pressure gradient is present, the centre of the bubble is not moving and the strain field is spherically symmetric. Its expression in the spherical reference frame (r, θ , ϕ) centred on the bubble reads [START_REF] Macosko | Rheology: Principles, Measurements and Applications[END_REF] :

ε rr (r,t) = 1 + R(t) 3 -R 3 0 r 3 -4/3 -1 -4ζ (t) R 3 0 r 3 (9)
The Kelvin-Voigt model, assumed to be valid below yielding, expresses the applied stress as a sum of an elastic stress Gε ε ε and a viscous stress η s ε ε ε. For sufficiently large oscillation amplitudes, the elastic stresses may satisfy the von Mises yield criterion 20,[START_REF] Hill | The mathematical theory of plasticity[END_REF] in a corona of fluid surrounding the bubble. The material then follows a Kelvin-Voigt rheology only outside of the yielded region, including at its edge, located at a distance r Y from the centre of the bubble:

r Y R 0 3 = 2 √ 3G σ Y |ζ (t)| (10)
Equation (10) defines the extent r Y of the yielded region as a function of time. Fluid yielding starts when the yielded region exceeds the bubble size at rest R 0 at least once during an oscillation cycle. This simplified yielding criterion reads

ζ ≥ ζ c = σ Y /2 √ 3G
and we hypothesise it is a necessary condition to initiate irreversible bubble rise.

In the yielded region, the purely elastic component of the Kelvin-Voigt model becomes a Maxwell element 21 , keeping its elastic modulus G and adding a non-linear plastic degree of deformation of viscosity η evp , traditionally defined as K εn-1 in rotational rheology. The elasto-plastic crossover time of the yielded material is (K/G) 1/n : the yielded material remains predominantly elastic below this time scale while plastic deformation dominates above it. Bubble oscillation dynamics is then only affected by yielding when the applied frequency satisfies 2π f (K/G) 1/n ≤ 1, in agreement with numerical simulations 20 .

Bubbles also apply a constant stress onto the fluid due to buoyancy or acoustic radiation forces. Hence, these forces will act on the yielded material during the whole time N/ f of the acoustic excitation. Irreversible bubble displacement may then be observed provided that f /N(K/G) 1/n ≤ 1.

Yielding to acoustic radiation forces

A second bubble release criterion can be computed from acoustic radiation forces, ignoring the contribution of the oscillatory stresses. We can compare the average acoustic radiation stress σ ac = F z /2πR 2 to the yield-stress in direct analogy with the yielding parameter Y -1 c used for gravity-driven bubble rise. This critical parameter varies between 1.1 for the most efficient, inverted teardrop shapes 34 to 5.1 for bubbles that are almost spherical 9 , which we consider in this article. Acoustic radiation forces then initiate bubble rise provided that:

1 σ Y F z (x,t) 2πR 2 (t) σ ac = 1 3 R 0 |∇ ∇ ∇p(x)| σ Y ζ | cos(φ )| ≥ 5.1 . ( 11
)
3 Materials and Methods

Carbopol microgel preparation and properties

The yield-stress fluid we use in this article is a Carbopol ETD 2050 microgel (Lubrizol Corporation, Wickliffe, Ohio, U.S.A.) of concentration 0.15% w/v that has been extensively studied in the literature 5,[35][36][37] . The Carbopol primary particles are made of crosslinked polyacrylic acid, which swells at high pH to form a jammed assembly of soft particles with a diameter of several microns 36 .

Following classical preparation protocols 5,37 , we first let the Carbopol flakes dissolve in MilliQ water (18.2 MΩ.cm)

for 1 hour under gentle agitation before adding 1% v/v 1M NaOH to adjust the pH to 7. The fluid is then stirred for 20 minutes by an overhead mixer (RW 20 fitted with a R1303 dissolver impeller, IKA, Staufen im Breisgau, Germany) at 2000 rpm. We then place the fluid in a vacuum chamber until all bubbles that have been incorporated during mixing are removed. The fluid is finally left to equilibrate overnight.

We characterise the rheology of the Carbopol microgel using a rotational rheometer (MCR 302, Anton Paar, Graz, Austria). We perform flow curves and oscillatory measurements, from which we deduce σ Y and G following standard fits 1 ; both data series are displayed in Appendix A. We measure the sound velocity in the fluid c using a separate acoustic setup. We assume that its density is equal to that of water at room temperature and we choose a surface tension Γ based on dedicated experiments eliminating the impact of elastic stresses 38 . We finally use the standard heat diffusivity D of air from classical sources [START_REF]CRC Handbook of Chemistry and Physics[END_REF] to compute the thermal dissipation coefficient κ from Section 2.1. The values of these parameters are compiled in Table 1.

Ultrasound excitation and high-speed imaging

Our experiments take place in a parallelepipedic container filled with the yield-stress fluid, as sketched in Figure 1. The walls of the containers are either made of glass or duralumin, ensuring total internal reflection of the incident acoustic wave. A lid fitted with needles partially dipped in the fluid is used at the top of the device to prevent sloshing while maintaining the total internal reflection with air.

We apply acoustic excitations using a Langevin transducer (Steminc, Doral, Florida, U.S.A.) oscillating between f = 19.45 and 29.2 kHz. We drive the transducer using a waveform generator (33210A, Agilent, Santa Clara, U.S.A.) coupled to a linear amplifier (AG 1021, T&C Power Conversion, Rochester, U.S.A.). The amplifier gain controls the voltage U applied to the transducer and ultimately the applied pressure amplitude p(x,t) during the experiment. We always work at The container dimensions L x = 10.2 cm, L y = 5 cm, and L z are adapted to produce a resonant standing wave pattern at the applied frequency f , where the pressure amplitude p(x) varies mostly alongside e z . This pattern, shown in Figure 1(a), corresponds to the (0, 0, 3/2) room mode of the container [START_REF] Morse | [END_REF] . Pressure measurements using a polyvinylidene fluoride hydrophone (RP 42s, RP Acoustics, Leutenbach, Germany) along the vertical line at the centre of the container [presented in Figure 1(b)] are compatible with the predicted room mode; they also show that the distortion level is small. We then define two locations named 1 and 2 (see Figure 1). The first location corresponds to the pressure anti-node at two-thirds of the cell height for which the pressure gradient ∇ ∇ ∇p(x 1 ) is zero. It is used in Sections 4.1 and 4.2 . The second location is chosen below the pressure node to achieve both a significant pressure and pressure gradient so as to maximise acoustic radiation forces, as explained in Section 2.2. At this location and for an applied frequency f = 19.45 kHz used throughout Section 4.3, we measure a relative pressure gradient |∇ ∇ ∇p(x 2 )/p(x 2 )| in the vertical direction equal to 82 m -1 . Given the efficiency of the resonant setup p(x 2 )/U = 0.13 kPa.V -1 at this location, the acoustic pressure gradient |∇ ∇ ∇p(x 2 )| exceeds the hydrostatic pressure gradient for voltages U ≥ 1 V.

We align the high-speed camera (Fastcam SA5, Photron, Tokyo, Japan) at location 1 or 2 by imaging the tip of the hydrophone. We then remove the hydrophone and inject a bubble of initial radius ≤ 300 µm with a small syringe in the frame of the camera before fine-tuning its position through careful manual pushing. Such a procedure inevitably modifies the internal stresses of the fluid around the bubble. Following Ref. 26, we consider the slow bubble dissolution (reported in ESI Section 2) as a sweep over the initial bubble radii R 0 and we then produce a resonance curve [Equation (3b)] for a constant frequency f and varying R 0 . At the start of the camera acquisition, a burst of N = 200 to 3000 sinusoidal cycles is sent by the waveform generator to the amplifier and the transducer. The camera records images up to 250 000 frames per second, corresponding to ∼ 10 images per oscillation period. We set the total acquisition time to measure both the bubble response to the acoustic excitation and its subsequent relaxation. We report in our acquisitions a small source of vibration at f = 130 Hz. It impacts bubble position measurements but does not affect the measured bubble radius and shape.

1

O 2 O • • ∼ p(x, t) L x z × L z O U (t) (a) (b) 0 2 4 6 0 20 40 60 1 O 2 O p(x) (kPa) z (mm)
Fig. 1 (a) Schematic diagram of the experiment. A Langevin transducer (in blue) provides an acoustic excitation to the yield-stress fluid container above it. The excitation frequency f matches the resonance of the transducer and that of a (0, 0, 3/2) standing wave pattern p(x,t) inside the container. Bubbles of interest are illuminated by an optical fibre facing the high-speed camera (right, in grey). (b) Vertical pressure profile in water at the centre of the setup for a frequency f = 21.5 kHz, an applied voltage U = 5.0 V and a fluid height L z = 5.3 cm. We report the vertical positions of locations 1 and 2 (dashed lines).

Bubble contour detection and decomposition into Legendre and Fourier modes

The data processing of video acquisitions is inspired by two recent works 16,41 . In short, our MATLAB routines normalise raw images and apply a luminosity threshold in order to retrieve the location of the bubble centroid x(t) and its mean radius

R(t) = R 0 [1 + ζ (t)] as a function of time.
We then perform a Fourier transform of the radius time series and pay a particular attention to ζ (ν) when ν is a multiple or a sub-multiple of the oscillation frequency f . Significant harmonic content indicates that we are no longer working in the linear bubble oscillation framework described in Section 2.

We also study whether bubbles remain spherical during the oscillations by examining the two-dimensional outline of the bubbles. To do so, we plot 360 lines originating at the bubble centroid, with equally spaced polar angles θ , defined from the vertical direction as shown in Figure 2(a). We define the local bubble radius R 0 [1 + ζ (θ ,t)] as the point where each line crosses the bubble edge. We then define the bubble orientation θ 0 (t) as the angle for which R(θ 0 + θ ,t) lies closest to R(θ 0θ ,t). In earlier studies 26,[41][42][43] , bubble outlines usually show a clear k-fold symmetry, which is empirically assumed to correspond to the degree, or mode, k of the spherical harmonics Y m k describing the three dimensional shape of the bubble. Following the same approach, we project the bubble shape outline ζ (θ ,t) on the Legendre polynomials of degree k, P k (cos(θ )) 41 . We may then define the instantaneous amplitude of a shape mode k, ζ k (t):

ζ k (t) = 2k + 1 2 1 -1 ζ (θ ,t)P k (u) du , (12)
using u = cos(θθ 0 ). This projection actually defines two integration paths, one for each bubble hemisphere. We choose to fit each hemisphere separately and define ζ k (t) as the average of the two. We finally compute the spectrograms ζk (ν,t)

of the bubble shape modes:

ζk (ν,t) = 2 ∆t t+∆t t-∆t ζ k (τ) exp(2iπντ) dτ (13)
We pay close attention to ζk (t) = ζk ( f /2,t), as f /2 is the frequency at which shape oscillations arise in Newtonian fluids and Kelvin-Voigt materials 29,44 . The time window size used to compute the spectrograms ∆t = 2/ f allows us to capture this component accurately.

Characteristic quantities and dimensionless groups

The small bubbles (75 µm ≤ R 0 ≤ 300 µm) we consider in this article correspond to very small Bond-Eötvös numbers,

Bo = ρ l gR 2 0 Γ ≤ 0.01 , (14)
and modest elasto-capillary numbers,

El = GR 0 Γ ≤ 0.17 . (15)
We then expect bubbles to remain spherical at rest, as hypothesised in Section 2. The yielding parameter for such bubbles is also small,

Y -1 = 2ρ l gR 0 3σ Y = 0.26 , (16)
since the critical value Y -1 c needed to initiate the rise of spherical bubbles is 5.1 9 . We also provide a numerical estimate of the critical oscillation amplitude ζ c required to initiate Carbopol yielding following Equation (10):

ζ c = 1 2 √ 3 σ Y G = 0.043 (17)
We can finally compute the ratio of the elasto-plastic crossover time scale in the yielded material to that of the bubble oscillations, as defined in Section 2.3. We refer to it as the Deborah number of our experiments:

De = 2π f K G 1/n = 590 1. (18)
Hence, even if the material has yielded, it will remain predominantly elastic, and we do not expect the bubble oscillations dynamics to be affected by yielding. However, if the material has yielded due to bubble oscillations, irreversible bubble displacement may occur due to buoyancy and acoustic radiation forces, which are applied continuously during N ≥ 1000 cycles, resulting in a time scale ratio f /N(K/G) 1/n = De/2πN below unity.

We may lastly define the Péclet number comparing heat diffusion in the air to its advection due to bubble oscillations:

Pe = 2π f R 2 0 /D = 160.
For this range of Péclet numbers, thermal dissipation is the dominant contribution to the damping term β (see Appendix B) and the polytropic exponent is κ = 1.30 24 .

4 Experimental Results

Linear Response : General Observations

We first show, for a typical experiment, the criteria we use to define spherical and linear bubble oscillations. Figure 2(a)

shows an image sequence of bubble oscillation after the end of the transient regime at location 1 . We also show in In all our experiments, spherical bubble oscillations are also linear in the time domain.

Resonance Curve

We then measure the resonance curve [Equation (3b)] of bubbles in the linear regime, sweeping over their initial radius R 0 . Figure 3 highlights the excellent agreement between the measured oscillation amplitude ζ from a series of experiments conducted at a constant pressure amplitude and the prediction of Equation (3b). We first verify that the fitted pressure amplitude p = 1.73 kPa matches independent pressure measurements using the hydrophone (data not shown).

All experiments show neither any significant shape oscillation nor any non-linear behaviour in the time domain. By (10). Squares represent experimental data. The solid black line is a fit of the linear data following Equation (3b), with two free parameters, the solvent viscosity η s (included in the damping term β ) and the applied pressure p. The 95% confidence interval region is smaller than the size of the markers.

other experiments. Material yielding has therefore no impact on the bubble dynamics. This result confirms the prediction made in Sections 2.3 and 3.4 that elastic stresses do not have time to relax in the yielded material and on the time scale of the oscillations.

More surprisingly, we note that none of the experiments for which yielding is expected shows any noticeable displacement of the centre of the bubble .

Response to acoustic radiation forces

In Section 4.2 we did not observe any significant displacement of bubbles driven into oscillations at location 1 , where they are subject only to the buoyancy force. Next we test the effect of acoustic radiation forces [Equation (11)] on bubble displacement, by looking at bubbles positioned at location 2 where they are subject also to acoustic pressure gradients. Bubble trajectories are non-trivial: they cannot be fitted by a simple exponential law related to the Kelvin-Voigt solid visco-elastic relaxation time f t KV = η s /G, which amounts to less than one oscillation cycle, nor to the time needed for the transient regime to die out, which corresponds to around 100 cycles, or even the typical elasto-plastic relaxation time of the yielded material, given by De/2πN = 1, also close to N = 100 cycles. (see Figure 2). We can rule out viscous or plastic responses of the fluid as the bubble centroid does not reach a constant, finite velocity dz/dt. They are however not long enough to be completely conclusive regarding more complex, non-linear responses of the fluid, such as creep 5 . experimental data superposes well with the theoretical expression for the average stress applied onto the bubble σ ac /p 2 from Equations (3b), (3c) and (7), which suggests a linear relation between bubble stress and strain. bubble shape oscillations.

f t (z -z 0 )/R 0 R 0 = 0.80R m R 0 = 0.95R m R 0 = 1.20R m ∆z ∆z ∞ σ ac /p 2 (a.u.) 0.8 1 1.2 -2 0 2 •10 -9 R 0 /R m ∆z/R 0 p 2 (Pa -
σ ac /σ Y (∆z -∆z ∞ )/∆z (d)

Shape oscillations

Critical pressure and observed modes

In Newtonian fluids and soft solids, shape oscillations of mode number k may grow when the applied pressure p exceeds a critical value p c,k , which depends on R 0 , the applied frequency f and the material properties 44,45 . For a fixed driving frequency f , these predictions define regions in the (R 0 , p) plane in which bubble oscillations either remain spherical, allow the growth of a single shape mode k, or allow multiple shape modes. Linear instability predictions for shape oscillations in Kelvin-Voigt soft solids have recently been derived 44 ; they are recalled in Appendix C. In Newtonian fluids, experimental results match the linear instability predictions fairly well 42,46,47 . In contrast, numerical 48 and experimental 26 data on the onset of bubble shape oscillations in non-Newtonian fluids are scarce and not yet conclusive. Phase diagram for bubble shape oscillations as a function of the applied pressure p/p 0 and the initial radius of bubbles R 0 /R m for an excitation frequency f = 22.5 kHz. Experimental data have been acquired throughout the dissolution of seven bubbles, identified with roman numerals i to vii from earliest to latest data series. As the applied pressure p is kept constant for each bubble, the seven data series form horizontal series of points, starting from high values of R 0 and ending for low values of R 0 , following the direction of the grey arrows. Individual acquisitions for each data series are shown as symbols, The modes that we could define without any ambiguity are plotted as red rightwards pointing triangles (k = 8), orange upwards pointing triangles (k = 7), yellow circles (k = 6), light green diamonds (k = 5) and green squares (k = 4). Most acquisitions with shape oscillations have a non-clear mode [e.g. last row of Figure 5]; they are plotted as crossed circles. The small white squares represent stable spherical oscillations. The critical pressure for each shape mode, computed from Ref. 44 and Equation (3b), is shown as a line with the same colour coding as the experiments. Above these lines lie coloured regions where only one oscillation mode k can grow, and a broader grey region where multiple shape modes may grow. The black solid line depicts the threshold for fluid yielding defined combining Equations (3b) and (10).

close to R 0 = R m ; further away from R m , it quickly grows and ultimately exceeds the maximum pressure achieved in our setup for R 0 ≤ 0.8R m and R 0 ≥ 1.3R m . Our data indicates that the shape number k in our experiments increases with R 0 , in qualitative agreement with models 44,45 and experiments in Newtonian fluids 42,47 .

We overlay in Figure 6

Impact on net bubble motion

In Newtonian fluids, bubble shape oscillations are closely related to an unpredictable, dancing 49 motion of their centre of gravity. This motion stems from a non-linear interaction with both spherical oscillations and other shape modes 49 . In the context of bubble removal, we wish to understand the impact of shape oscillations and dancing motion on the ability of ultrasound devices to push and pull bubbles irreversibly in yield-stress fluids.

Figure 7 shows the strong impact of shape oscillations on bubble motion. The first three bubbles [Figure 7 (a-f)] show motion towards an antinode in agreement with their initial size R 0 ≤ R m . The presence of a clear shape mode enhances bubble mobility, as shown in Figure 7(c-d) for k = 4. We also observe spurious motion in the direction transverse to the pressure gradient when a single bubble shape mode k is no longer clearly identified [as seen in Figure 7(e-f)]. We also have observed reversals of the bubble direction of motion following the onset of shape oscillations. [Figure 7(g-h)]. In general we conclude that while shape oscillations increase bubble displacement, the direction of motion can no longer be controlled.

10 0 k = 2 k = 4 k = 6 k = 8 k = 3 k = 5 k = 7 k = 9 (a) (b) (c) (d

Discussion

Precision and relevance of the solvent viscosity measurement

The experimental bubble oscillation amplitude ζ in the linear regime may be fitted to the theoretical resonance curve to extract the oscillation damping parameter β . After carefully subtracting from β the dominant acoustic and thermal contributions, we measure a fluid viscosity η s = 1.3 mPa.s. An analysis of the fitting procedure and the uncertainties on β shows that this value is not statistically different from the viscosity of water used as the solvent here, and in agreement with the rheological model 20,21 . This low value seems surprising considering the bulk oscillatory rheology of Carbopol (see Appendix A) which rather suggests a viscosity G /2π f 1 Pa.s in the linear regime for low frequencies f = 1 Hz, while the Kelvin-Voigt model we use assumes a constant viscosity below yielding for all frequencies. Indeed, real hydrogels and yield-stress fluids under oscillatory shear do not show a constant viscosity as a function of f : classical rheological measurements show that their loss modulus G behaves as a constant or as slowly increasing power laws of f 50 leading to a decreasing viscosity G /2π f . These power law scalings may be reproduced by fractional derivative models [START_REF] Jaishankar | Proc. Roy. Soc. A[END_REF] but their microscopic origin remain insufficiently understood 7 . The value of the viscosity deduced from G in oscillatory rheology may therefore not be particularly meaningful. In contrast, viscous or close-to-viscous scaling of the stress has been experimentally observed in yield-stress fluids at high frequencies and strain rates [START_REF] Mason | [END_REF]52 . At such frequencies, dissipation due to the solvent, scaling as η s f , may become the dominant contribution to G , and the material could then recover a Kelvin-Voigt rheology. Our results suggest that bubble oscillations experiments fit into this high-frequency limit and allow a proper measurement of the solvent viscosity.

Linear response to Bjerknes forces

We have used in Section 4.3 the constant (or zero-frequency) part of the acoustic radiation force to perform an equivalent of step-stress tests, but at a local scale R 0 . For moderate acoustic stresses σ ac ≤ σ Y , we measure a linear strain-stress relation at the end of oscillations, from which we deduce an independent measurement of the local linear elastic modulus of the fluid below yielding, G = 44.4 ± 3.5 Pa, comparable to that obtained using bulk rheology, G = 36 Pa. All quantities used to derive G are either directly measured or estimated from the resonance curve: hence, in contrast with previous works 53 , our measurement is truly independent from bulk rheology. The complex time dependence of the displacement shown in Figure 4(a) is reminiscent of creep behaviour 5 . Creep is however usually associated to irreversible strain and a non-linear stress-strain relation in bulk rheology experiments, both of which are not observed here. Interestingly, fully reversible creep motion up to the yield point has also been reported in experiments in which acoustic radiation forces are used to push small spheres 53 . The relatively small pressure gradients applied in our experiments according to Equation (11) then cannot alone initiate bubble rise. Performing experiments of longer duration may reveal whether the response to acoustic radiation forces indeed follows a power law or an exponential profile with time, which could be helpful to validate the recent, advanced models of yield-stress fluids 7,8 .

Absence of irreversible rising motion

Several experiments satisfy the bubble oscillation yielding criterion ζ ≥ ζ c and apply acoustic radiation stresses σ ac comparable with the yield stress for a sufficiently long time to let elastic stresses relax. Yet, they do not suffice to induce irreversible bubble motion and we do not observe the finite average rising speed predicted in the recent numerical simulations of Ref. 19. The yielding criterion ζ c = 0.043 we have derived is then a necessary condition, but not sufficient, to induce bubble rise at a useful rate for removal applications.

One explanation for this lack of irreversible motion is that the steady-state bubble rise velocity is too small to be observed. Firstly, the yielded region remains under 1.25 times the size of the bubble radius, increasing drag by a factor 40 compared to the unconfined case 54 . Secondly, the plastic viscosity in the yielded material stays significantly higher than the solvent viscosity. The corresponding rising velocities may therefore be too small to be resolved in experiment.

Additional factors may prevent irreversible rising motion. For instance, the von Mises yield criterion [Equation (10)] has been shown to fail for bulk yielding in extension, as already reported in other simple yield-stress fluids [55][56][57] . Another possibility lies in finite-size effects given the relatively small size of the bubble compared to the constitutive elements of Carbopol. Local restructuration around slowly-growing bubbles has been recently evidenced in sparse networks of microfibrillated cellulose, which impacts their bubble retention capacity 58 . It is difficult to know at the moment whether this scenario applies in our case, since Carbopol is soft-jammed and isotropic and the strain rates at play are high. We may finally question the relevance of the very notions of yielding and unyielding in our experiment since the oscillation timescale 1/ f can be below that of the microscopic plastic rearrangements used in yield-stress fluids models 7,8 .

Nature and onset of shape oscillations

The critical pressure p k c above which we experimentally observe shape oscillations is significantly higher in Carbopol thant what we expect from a linear instability analysis in Kelvin-Voigt materials 44 if we use the fluid properties we derived in Sections 4.2 and 4.3. Yield-stress fluids are known to exhibit residual stresses at rest, with unknown spatial distribution.

We expect non-homogeneous residual stresses around the bubble to impact bubble shape oscillations by altering the critical pressure p c,k depending on the compatibility between the geometry of the shape modes and that of the residual stresses. Further analysis of the bubble shapes, conducted in ESI Section 4, shows that residual stresses induce a very small (0.5%) residual deformation of the bubble at rest. Under acoustic excitation, the bubble shape modes do neither respect the orientation nor the symmetry of these residual deformations. Hence we do not observe any direct impact of residual stresses on bubble shape oscillations even though we cannot rule out their influence. Using a solvent with a higher viscosity in experiments would be particularly helpful to either reconcile experimental data with the linear instability model 44 or to prove that it is not applicable to yield-stress fluids.

Consequences on acoustic bubble removal performance

Shape oscillations imply unpredictable bubble motion that inevitably reduces the efficiency of any directed motion induced by acoustic radiation forces or bubble buoyancy. We notice that the window of operation for bubble removal, lying above the black line and below the coloured symbols of Figure 6, is limited especially since the yielding criterion of Equation (10) does not warrant bubble rise. Bubble removal using acoustic excitation in Carbopol could then be performed using stronger pressure gradients, for instance using focused ultrasound beams. We suspect Carbopol is particularly resistant to the removal process due to its very wide linear elastic regime, as shown in bulk rheology (Appendix A). Bubble removal should be easier in almost any other yield-stress fluid as they break down under much smaller strains 59,60 . Increasing η s could also improve bubble removal by raising the critical pressure at which shape oscillations arise.

Conclusion

In this article, we have investigated how a small bubble oscillating at a high frequency interacts with Carbopol, a model yield-stress fluid. Bubbles of different sizes allow us to perform bubble spectroscopy 25,26 and extract a viscosity η s = 1.3 mPa.s of the fluid at high frequency and for a finite extensional deformation, in agreement with the solvent viscosity of water and as expected from a previous numerical study 20 . We have also used pressure gradients to apply acoustic radiation Experiments performed at higher pressure always resulted in non-spherical shape oscillations. As shape oscillations result in an unpredictable bubble motion in all directions, acoustic bubble removal is quite inefficient in Carbopol. Future studies should explore the applicability of acoustic bubble removal in more fragile networks, corresponding to a wide range of attractive colloidal and athermal yield stress fluids in which spherical bubble oscillations largely beyond the yield point are possible, resulting in a strong decrease of both bubble confinement and plastic viscosity during its assisted motion.

η therm = 3p 0 κ 8π f η acoust = π 2 ρ f 2 R 3 0 4c (19)
and we plot η therm , η acoust and η s for a solvent viscosity η s = 1.3 mPa.s deduced from Section 4.2. While acoustic damping is smaller than the viscous term in our operating range, thermal damping dominates them both and is up to 50 times higher than the viscous contribution. As we fit η s by subtracting thermal dissipation η therm and acoustic dissipation η acoust from the total damping term β in the resonance curves, the solvent viscosity η s fluctuates greatly for relatively small relative changes in η therm and, to a lesser extent, in η acoust . Obtaining a reliable value of the viscosity η s then necessitates very high-quality resonance curve data and precise values of all the physical quantities present in thermal and acoustic damping, which are: p 0 , R 0 , ρ, c and D through the Péclet number in κ . Uncertainties on both η therm and η acoust have been estimated in ESI Section 1. The lack of precise measurements on the thermal diffusion coefficient D results in a significant uncertainty on η therm , of the same order as η s , while the uncertainty on η acoust remains negligible.

C Threshold for shape oscillations in soft materials

We recall here the predictions of Ref. 44, who derived the critical bubble oscillation amplitude above which shape oscillations may be observed in a neo-Hookean, Kelvin-Voigt viscoelastic solid. Defining intermediate quantities: The critical amplitude ζ c,k above which a shape mode k develops may be expressed as: Since our experiments show that bubble oscillations up to the shape oscillation threshold are linear in the time domain, we may combine Equations (3b) and (24) to derive explicitly the critical pressure p c,k for all modes k. One surprising consequence of Equation ( 24) is that, despite modelling the three-dimensional growth of spherical harmonics Y m k generally defined by two shape modes k and m, the critical pressure of the model is independent of m.

λ 1 = 4(k -1)(k + 1)(k + 2) Γ 4π 2 f 2 ρR 3 0 (20)
Close to R 0 = R m , the critical pressure p c,k reaches a minimum for all modes k because it corresponds to the resonance condition of spherical oscillations. In addition, shape modes have a natural oscillation frequency, given by:

π 2 f 2 = G ρR 2 0 (k + 1) 4 + k + k(k + 1) 3 + Γ ρR 3 0 (k + 1)(k -1)(k + 2) . ( 25
)
When f is imposed, Equation 25defines a radius at which a given shape mode k resonates, corresponding to the minima of the coloured tongues in Figure 6. In some particular cases (here, for k = 5 and 6), both the spherical mode and the shape mode resonate around R m , resulting in particularly low critical pressures p c,5 and p c,6 , as observed in Figure 6 and in the experiments.

Notes and references

Fig. 2

 2 Fig. 2 Linear oscillation of a bubble in a yield-stress fluid under acoustic excitation at a frequency f = 21.5 kHz. (a) Snapshots of the bubble oscillation during two oscillation cycles. The red scale bar is 250 µm. (b) Time series of the relative departure from the initial radius ζ (t) as a function of time. Acoustic excitation starts at f t = 0 and stops at f t = 1000, marked by a dotted line. (c) Spectrogram ζk (t) of the bubble shape modes for k ≤ 10, showing no notable shape oscillation content at f /2. (d) Complex modulus of the Fourier transform of ζ .

Figure 2 (

 2 Figure 2(b) the bubble oscillation amplitude ζ (t), which highlights the typical time scale ∼ 100/ f needed for the transient state to vanish.The bubble radius at rest before and long after the oscillations are equal, ruling out any significant gas diffusion into or out of the bubble. Figure2(c) confirms that bubbles remain spherical as the shape oscillation modes ζk remain at a very low level before, during and after the acoustic excitation. We then set the noise threshold for shape oscillations to 10 -3 for the rest of this article. Figure2(d) confirms the linearity in time of the bubble response. The bubble oscillation spectrum ζ (ν) shows a peak at ν = f and harmonic content is almost absent aside from a small peak at ν = 2 f .

Fig. 3

 3 Fig.3Resonance curve of a bubble in Carbopol obtained for an oscillation frequency of 21.5 kHz and an acoustic pressure amplitude p = 1.73 kPa. The number of cycles has been set to 1000. The Minnaert resonance radius is R m = 148 µm. The dashed horizontal line represents the onset of Carbopol yielding deduced from Equation(10). Squares represent experimental data. The solid black line is a fit of the linear data following Equation (3b), with two free parameters, the solvent viscosity η s (included in the damping term β ) and the applied pressure p. The 95% confidence interval region is smaller than the size of the markers.

Figure 4 (

 4 Figure 4(a) shows the vertical position of the bubble centroid z(t) for three experiments. Bubbles smaller (respectively larger) than the resonant radius R m show a net downwards (respectively upwards) motion towards the pressure anti-node (respectively pressure node), in line with the change of sign of cos(φ ) in Equation (3c). The inset of Figure 4(a) highlights the zero-average oscillatory part of the acoustic radiation forces [averaged out in Equation (7)], clearly noticeable and superposed with the slower displacement related to the Bjerknes force.

Figure 4 (

 4 Figure 4(b) examines the sensitivity of bubbles to acoustic radiation forces as a function of their size, defined as the normalised displacement ∆z/R 0 p 2 as Bjerknes forces are quadratic in pressure amplitude (see Section 2.2). Our

Fig. 4

 4 Fig. 4 Displacement of the bubble centroid in the vertical direction for spherical bubble oscillation experiments conducted at f = 19.45 kHz. The Minnaert resonance radius is R m = 163 µm. (a) Normalised bubble vertical displacement as a function of time for three experiments: R 0 1.2R m (red), R 0 0.95R m (light grey) and R 0 0.8R m (blue). Data have been averaged over an oscillation period. Inset: raw temporal profile of the bubble vertical position highlighting the oscillating part of the signal. (b) Typical strain applied by the bubble net motion for an applied pressure of 1 Pa, ∆z/R 0 p 2 (markers), superposed with the expected acoustic radiation stress applied to the bubbles σ ac /p 2 (solid line, see main text) also for p = 1 Pa. Note the separate y axes. (c) Local rheology of the fluid: typical measured strain ∆z/R 0 plotted as a function of the acoustic radiation stress normalised by the yield stress σ ac /σ Y . The solid line is the best linear fit of the data for |σ ac /σ Y | < 0.5 and the dashed lines show the linear fits from the boundaries of the 95% confidence interval. (d) Recovery after strain of the bubble: the amount of recovered strain (∆z -∆z ∞ )/∆z long after the end of acoustic excitation is plotted as a function of the normalised acoustic radiation stress. In Panels (b), (c) and (d), the red, light grey and red markers correspond to the three lines of Panel (a).

Figure 4 (

 4 Figure4(c) directly plots the acoustic radiation strain ∆z/R 0 as a function of the corresponding stress, normalised here by the yield stress σ Y . We compute here the stress using experimental values of ζ , R 0 , p and |∇ ∇ ∇p| and we choose cos(φ ) based on Equation (3c). The data confirms the linear trend suggested from Figure4(b) at low applied stresses and shows a noticeable non-linear deviation for higher stresses. As yielding due to the oscillation amplitude has no impact on the bubble mobility (see ESI Section 3), this deviation may only stem from a non-linear behaviour of the emission setup or non-linear elasticity of the Carbopol. We measure the slope of the linear trend at low stress in Figure4(c) to extract an estimate of the linear elastic modulus of the surrounding medium G = 44.4 ± 3.5 Pa, following Equation (8). This value is in fair agreement with that obtained from bulk oscillatory rheology, G = 36.0 Pa.

Figure 4 (Fig. 5

 45 Figure 4(d) shows the recovered strain 2000 cycles after the end of the acoustic excitation. The recovery is close to 100% for all experiments, which confirms the elastic nature of the deformation shown in Figure 4(c) expected for experiments conducted for σ ac /σ Y ≤ 5.1. Irreversible bubble motion can then only be achieved for higher applied pressure and oscillation amplitude ζ . As we will see in Section 4.4, we could not perform such experiments due to the onset of

Figure 5 Fig. 6

 56 Figure 5 highlights four shape oscillation modes 4 ≤ k ≤ 7 that have been clearly identified in experiments at location 1 . Less than half of the experimental data is sufficiently clear to define unambiguously a shape mode number k. Several experiments (see last row of Figure 5) instead show a complex outline, which likely results from the projection in the imaging plane of a three-dimensional mode Y mk with m = {0, k, -k} and a random orientation. In all cases, the frequency of the shape oscillations is f /2, confirming that shape oscillations also result from a sub-harmonic instability in yield-stress fluids.

  the predicted critical pressure p c,k derived in Appendix C by combining Equation (3b) and the critical bubble oscillation amplitude ζ c,k above which spherical oscillations are linearly unstable. We choose the value of the viscosity we fitted in Section 4.2, η s = 1.3 mPa.s and the elastic modulus measured in Section 4.3, G = 44.4 Pa. A large amount of experiments shows stable spherical oscillations whereas the model predicts they are linearly unstable with respect to shape oscillation modes 4 to 8. The model however correctly predicts that modes 5 and 6 are favoured for R R m in agreement with the low values of p c,5 and p c,6 in this region.

Fig. 7

 7 Fig. 7 Analysing experiments for which one or several shape oscillation modes are observed and steady. Panels (a)-(c)-(e)-(g), spectrograms of the recorded shape mode intensity as a function of time for an experiment with no shape oscillation (a), a clearly identified mode k = 4 (c) and two experiments (e,g) for which the shape modes are not clearly identified. The acoustic excitation starts at t = 0 and stops at f t = 1000, marked by a dotted line. Only the four most prominent projection modes are shown. Panels (b)-(d)-(f)-(h), net bubble motion corresponding to the spectrograms of Panels (a)-(c)-(e)-(g). The grey line shows the relative displacement along the x (horizontal) axis while the black line shows its z (vertical) counterpart. The white square and the black circle respectively represent the position of the bubble in the x and z axes at f t 3100, long after the oscillations have stopped.

  forces on bubbles, from which we measure the local linear shear modulus of the fluid G = 44.4 Pa, in fair agreement with bulk rheology. As long as the oscillations remain spherical, bubble motion is fully reversible given the range of acoustic radiation stresses |σ ac | ≤ σ Y achieved in our experiment. In particular, motion reversibility appears unaffected by the oscillatory yielding criterion derived by De Corato et al. 20 .

Fig. 8

 8 Fig. 8 Rheology of the Carbopol microgel. (a) Flow curve. Circles, decreasing shear rates, and squares, increasing shear rates, performed right after the blue circles. The black line represents the best fit to a Herschel-Bulkley law. (b) Oscillatory rheology: amplitude sweep starting from low strain amplitude at f = 1 Hz. Circles represent the storage modulus G while squares show the loss modulus G . The black dashed line represents the average value of the storage modulus in the elastic plateau, G = 36 Pa.

Fig. 9

 9 Fig.9Thermal, viscous and acoustic damping under linear bubble oscillation plotted as effective viscosities for an applied frequency f = 22.5 kHz. The thermal and acoustic contributions are directly plotted from Equation (6), while the value of η s has been fitted to the resonance curve in Section 4.2. The greyed out region is our usual operating range.

λ 2 =

 2 2(k + 2)(2k + 1) η s 2π f ρR 2 0 (21) λ 3 = 4(k + 1) G 4π 2 f 2 ρR 2 0 (22)λ 4 = 12k(k + 2)η s 2π f ρR 2 0(23)

1 - 1 ) 3 2 + 4λ 2 2 (2k + 1 )

 11321 + λ 3 4 + 4k/3 + k 2 /-3/2λ 1 + 2λ 2 2λ 3 (18 + 19k/3 + k 2 /3)

Table 1

 1 Physical parameters of bubble oscillation in Carbopol. Source of the data: n, K, σ Y , G and c have been measured by the authors. The surface tension Γ and the heat diffusivity D and taken from Refs. 38 and 39 respectively. The polytropic index κ is computed following Ref. 24.

	Name	Fluid	Symbol	Value Unit
	Polytropic Index		κ	1.30
	Ambient Pressure Heat diffusivity Viscosity Specific gravity Sound velocity Carbopol Air Air Water Water Surface Tension Carbopol	p 0 D η 0 ρ c Γ	1.013 10 5 Pa 1.9 10 -5 m 2 .s -1 1.0 10 -3 Pa.s 9.98 10 2 kg.m -3 1.495 10 3 m.s -1 6.2 10 -2 N.m -1
	Flow Index Carbopol	n	0.36
	Flow Consistency Carbopol	K	5.0 Pa.s n
	Yield Stress Carbopol	σ Y	5.3 Pa
	Shear Modulus Carbopol	G	36.0 Pa

relatively low input voltage and amplifier gain to prevent non-linear distortion of the amplifier or transducer response.

1-23 J o u r n a l N a me , [ y e a r ] , [ v o l . ] ,

J o u r n a l N a me , [ y e a r ] , [ v o l . ] , 1-23 | 9

Acknowledgements

The authors wish to thank M. De Corato, J. Tsamopoulos and Y. Dimakopoulos for stimulating discussions and their critical reading of the paper. They also thank D. Baresch for his help with the design of the experimental setup. This work is supported by European Research Council Starting Grant No. 639221 (V.G.).

Conflicts of Interest

There are no conflicts to declare.

Appendix A Carbopol Rheology

We characterise the rheology the Carbopol microgel using a standard rotational rheometer (MCR 302) working with a cone-plate geometry fitted with sandpaper discs (grit P1500) to suppress wall slip. Before every test, we apply a pre-shear step at γ = 1500 s -1 for 30 s and a rest step at σ = 0 Pa for 20 s. We perform two consecutive flow curves for decreasing and increasing shear rates γ between 0.001 s -1 and 1500 s -1 , choosing 5 s steps and 10 points per decade. Amplitude sweeps are conducted at a frequency of 1 Hz for increasing shear strains γ between 0.01% to 1000 %, and we choose to acquire 15 points per decade and average over 10 oscillation cycles. Our data are presented in Figure 8. We also notice that the storage modulus is rather insensitive to the applied frequency f in the range accessible to the rheometer, 0.1 Hz to 10 Hz (data not shown).

B Contributions to damping of bubble oscillations

We compare in Figure 9 the relative magnitude of the three contributions to dissipation detailed in Equation ( 6). We respectively note:


There are only rarely attempts to attend to the different senses and design museums to promote the multisensory experience. This is important because, whether addressed explicitly or not, modern neuroscience teaches us that basically all experience of the world is multisensory. Therefore, museum curators and directors ought to understand the importance of explicitly designing multisensory museum experiences. Essentially, the museum experience will be multisensory, whether we want it or not-thus it is better to pay attention to achieve desired effects rather than allowing for incidental and potentially undesirable effects » (Levent & Pascual-Leone, 2014, 18)

the reality of museums

On a case-bycase basis

Interviews with museum professionals : understand the pragmatic or political reasons behind the choices made in terms of sound mediation (creation / dissemination ) There are always pragmatic explanations A web-app rather a dynamic binaural device… because of network coverage

The periodic abandonment of headsets or earphones during pandemics for health reasons plan : from origins to issues

State of the art

Understanding acoustics

Understanding the psychophysics of listening

Understanding the visitor experience

An identified problem

A problem

An observation

The challenges of my research

Goals

An example of an experimental protocol

State of the art

For beginning : how do we listen to museum ? Visitor-environment dynamics (Forrest, 2014)

• " (...) Noise can be synonymous with conviviality, excitement or interactivity... it can also be synonymous with confusion, chaos, overcrowding and the sabotage of contemplation and dialogue"

(Sterne & De Luca, 2019, 301) • Créneau-Cressound
Noise causes mental tiredness, negative impact on concentration (Persinger, Tiller & Koren, 1999), psychological distress (Evans & Stecker, 2004) "The interference between what one hears in the headset and what one sees in the environmentthe "visio-phonic node" mentioned by Thibaud -is reinforced through contextual narratives that amplify the process of projection of the imaginary linked to what one listens to in the places traversed (and vice versa). Thus, what one listens to rewrites the context, as much as the context (in continuous evolution) transforms what one listens to » (Biserna, 2015, 193-213) Aesthetisation of space : articulating the links between stimuli and consciousness (ambient noise, smell, lighting, colour, traffic flow, etc.)

New auditoriums

• What produces and generates an auditorium (listening space) ? (Joy, 2015)

• "The auditorium is a place built according to acoustic standards and arranged for listening to a speaker or to musical or theatrical works" (Wikipedia)

•

The auditorium is a listening space, a place and a common moment for listening and interaction between listeners (concert hall, networked auditoriums, etc.)

• Understand the influence of distances and temporal "delays" on broadcasting and listening, to optimise listening by controlling trajectories, dynamics and positions in order to create an impactful architectural and scenographic context Cloisons MBA France pour l'exposition « Benoît Clarys, le passé comme si vous y étiez ? » au Musée de Préhistoire de Solutré © SMGS.

Music Museum -

  Paris ©Emmanuelle Giuliani Acoustically • Historically, modern museum and galleries as unisensory visitor experience (Lacey et Sathian, 2014, p 3) • In reality, museums are noisy... omnipresence of sounds in museal environment ! (CRESSON) • Niche : intersection between a sound message and a context • Flutter echo : in the reverberation of a room, the emergence of a particular frequency and its harmonics which remain between two parallel and reflecting walls • Reverberation : propagation effect whereby auditory sensations persist after the sound emission has stopped • Ubiquity : it is « linked to the spatiotemporal conditions of propagation, and marks the difficulty or impossibility of locating a sound source ». An « open door to the metaphysical dimension of sound » « Le salon carré au musée du Louvre » (1883) Louis Béroud -Musée Fabre -Montpellier Salomé Voegelin (2014) / « Sound Walk 1 » (Tate Britain) Start here (on Millbank) • Stand at the bottom of the steps and listen for approximately two minutes to yourself, then listen for another three minutes to yourself in the environment. • Walk up the stairs, listening to the people who come toward you. • Enter the museum. • (…) • Walk all the way in a straight line to the very back of the gallery, all the while listening to yourself and other visitors inside the space. • Go to the first piece of work you find ; listen to it. • Go to the next piece of work you find ; look at it… Listening questions • How do you listen to the museum ? • Does the gallery space invite listening ? • Does that listening, or not listening, have an influence on how you perceived the work displayed ? • Do you, when looking at artworks in the space, also hear the space? • (…) • Where would you have installed a sound work ? • How would you work with sound in this museum ? • What would the ideal sound museum look and sound like ?

Hearing versus listening

• Listening is a complex activity that suppresses or diverts sound information to a lower level

• CRESSON / Cressound • Delocalization : in other words, the recognition of an illusion as to the origin of the phenomenon, « implies the recognition of an error in the location of the sound source »

• Synecdoche : « The ability to select one element or another » in a « complex sound environment ».

•

The experience of listening to the museum is above all a human science before being an exact science (acoustics), « [...] each man selects what he wants to hear -or what his culture has taught him to hear » (Lortat-Jacob, 2003, 2) Increase listening

• Binaural listening : physiological discrimination allowing relief and spatialization • New generations of headphones : SoundGuide (Sicomore, 2019), the Confident (RSF Studio), the connected Confident (AKKEN)... « So in both cases, we say that we have to focus on listening through an acoustic tool linked to the ear, and this goes beyond listening with the ears alone. It is said that we must improve, increase human listening, and we talk about techniques to achieve this : methods of isolating oneself in a sound world, methods of participating in meaning and of paying attention to things in a different sound way » (Between 3m16 and 5m04. Jonathan Sterne, A l'origine des sound studies, https://www.franceculture.fr/emissions/la-suite-dans-les-idees/l-origine-des-sound-studies)

Left : Robust and light, the Soundguide semi-closed headset developed by Sycomore enhances the value of immersive "hands-free" visits ©Sycomore Right : The Confident (here, a 3D simulation) is the basis of the digital mediation system at the Hôtel de la Marine. It provides a sound path that follows the visitor's progress through the monument ©RSF

The heyday of listening in museums Sound system for museum spaces

• Why sound in a museum space ? (Salmet dir., 2016)

• Delineate the different thematic areas of the tour route • Improve the accessibility of the exhibition : accompany the visitor in the perception of a universe, double the textual information, reinforce the comfort of use of interactive and audiovisual devices)

• Promoting immersive cognitive environments • Helping visitors find their way (fixed devices along the route or on-board devices)

Expositions et parcours de visite accessibles (2016), p. 199.

Sonography

• Sonography (Leboeuf, 1999) : development of sensitive environments where emotions become the keys to a more intuitive and immediate understanding of collections

An identified problem

Museum policies

• Maximisation of the experience in a positivist logic

• Devices inspired by expertise in cultural marketing and design : « wow » effect, sensationalism, to increase museum attendance...

• External service providers relying solely on empirical expertise

• Innovations close to cultural consumption habits (networks, interfaces, headphone listening, applications, etc.)

• Only quantitative studies : surveys, satisfaction surveys, statistics, etc.

• How effective are these proposals in transmitting knowledge and learning ? What do we mean by « immersive », « participative » or « interactive » ?

Jewish Museum Berlin ©La Terre est un Jardin

An observation

Artists, researchers and museum professionals carry out research in their own way but no joint research work

There are few or no links between museums, service companies and academic research

The challenges of my research

The objectives

Measuring the efficiency of these systems would make it possible to : • Sterne, J. A l'origine des sound studies, https://www.franceculture.fr/emissions/la-suite-dans-les-idees/l-origine-des-sound-studies, consulté le 8 janvier 2022

• Tisseron & Bass, 2011

• Thibaud, J-.P. (1994). Composer l' espace : les territoires du pas chanté. Les faces cachées de l' urbain, 183-195 • Truax, B. (2021). R. Murray Schafer (1933-2021) and the World Soundscape Project. Organised Sound, 26 (3), 419-421

• Varutti, M. (2020). Vers une muséologie des émotions. Culture & Musées,(36),[171][172][173][174][175][176][177] • Vecchione, B. (1992). La recherche musicologique aujourd'hui : Questionnements, intersciences, métamusicologie. Interface, 21 (3-4), 1992, 281-312 • Voegelin, S. (2014). Soundwalking the Museum : A Sonic Journey through the Visual Display. The Multisensory museum: cross-disciplinary perspectives on touch, sound, smell, memory, and space, 119-130