Bubble dynamics for broadband microrheology of complex fluids
BBubble dynamics for broadband microrheology of complex fluids
Brice Saint-Michel and Valeria Garbin ∗ Department of Chemical Engineering, Delft University of Technology, Van der Maasweg 9, 2629HZ Delft, Netherlands
Bubbles in complex fluids are often desirable, and sometimes simply inevitable, in the processing of formu-lated products. Bubbles can rise by buoyancy, grow or dissolve by mass transfer, and readily respond to changesin pressure, thereby applying a deformation to the surrounding complex fluid. The deformation field around astationary, spherical bubble undergoing a change in radius is simple and localised, thus making it suitable forrheological measurements. This article reviews emerging approaches to extract information on the rheology ofcomplex fluids by analysing bubble dynamics. The focus is on three phenomena: changes in radius by masstransfer, harmonic oscillations driven by an acoustic wave, and bubble collapse. These phenomena cover a broadrange of deformation frequencies, from 10 − to 10 Hz, thus paving the way to broadband microrheology usingbubbles as active probes. The outstanding challenges that need to be overcome to achieve a robust technique arealso discussed.
INTRODUCTION
Bubbles are ubiquitous in the processing and structuringof complex fluids [1, 2]. They may be undesirable and leadto poor product performance, for instance contamination ofpersonal care products [3], or they may be key to impartingunique properties to advanced materials [4]. Bubbles are ex-traordinarily dynamic objects [5]: they can grow or shrinkby gas di ff usion, they can rise due to buoyancy, they can un-dergo break-up and coalescence. Owing to the compressibil-ity of their gas core, they respond to changes in pressure, andcan gently oscillate in response to acoustic waves or violentlycollapse in response to shock waves. Because the control oftheir number and size distribution is crucial in the processingand structuring of complex fluids, bubble dynamics have beenstudied extensively with a view to improve the stability andperformance of formulated products [6].A useful feature of bubble dynamics is that, provided thatthe bubble remains spherical, the deformation field in the sur-rounding fluid is purely extensional [7]. This feature presentsa unique opportunity to gain insights into the rheological prop-erties of the complex fluid surrounding a bubble, because thekinematics of the deformation is completely prescribed, sim-ple and localised – such controlled conditions are necessaryto perform a rheological measurement. A measurement ofthe applied stress, in combination with controlled material de-formation, is also required to perform a rheological measure-ment. Analysis of bubble dynamics provides information toindirectly obtain the stress in the surrounding medium, withsome assumptions on the constitutive behaviour [8]. In ad-dition, the deformation time scales of bubble dynamics covera very broad range that is normally not accessible to a sin-gle rheological technique: from the slow dynamics ( ∼ s)driven by gas di ff usion [9] to the extremely fast dynamics( ∼ − s) during bubble collapse [10].The idea of using spherical bubble deformation to probethe rheology of the surrounding material is in fact a classicalconcept [7], but its full potential has remained largely unmetdue to technical limitations in controlling bubble formationand subsequent deformation. Advances in experimental tech-niques to generate, control and deform single bubbles, and image their evolution in real time [11], as well as advancesin imaging of the microstructure of complex fluids, are nowbringing this new class of methods to the forefront of soft mat-ter and rheology [12].This article focuses on the recent emerging applications of bubble dynamics to the rheology of complex fluids, and high-lights the current gaps that still need to be addressed to make itan accessible technique for broadband microrheology of com-plex fluids. CONTROLLED DEFORMATION BY BUBBLE DYNAMICSDynamics of spherical bubble deformation
In this paper we focus on three dynamical phenomena thatcan result in the spherically symmetric deformation of a sta-tionary, isolated bubble: changes in radius by mass transfer,harmonic oscillations of the radius driven by an acoustic wave,and bubble collapse. We do not discuss the the techniquecalled “cavitation rheology”, based on bubble formation byinjection at the tip of a needle embedded in a soft material,which is in a more advanced state of development and has al-ready been widely adopted, as described in a recent review[12]. One advantage of the three phenomena reviewed hereis that the bubble is isolated (not attached to a needle) andtherefore its spherical deformation results in a purely exten-sional deformation of the surrounding medium, which will bedescribed in Section . The deformation is also localised toa region around the probe, decaying quickly away from thesurface of the bubble, thus satisfying a necessary criterion forapplications in microrheology [13].The rate of deformation of the material surrounding thebubble covers several orders of magnitude between the threephenomena considered. Firstly, changes in radius by masstransfer, that is bubble growth or dissolution by exchange ofgas with the surrounding medium, can occur at rates from1 s − to 10 − s − or even slower. The analysis of the change inradius of a bubble that is dissolving or growing by mass trans-fer, to extract information on the rheological properties of thesurrounding medium, will be described in Section . Secondly, a r X i v : . [ phy s i c s . f l u - dyn ] O c t harmonic oscillations of a bubble can be driven by an acousticfield that is close to the resonance frequency of the bubble. Inthe simplest case of a Newtonian fluid, and if surface tensione ff ects are negligible, this is given by the Minnaert frequency, ω M = R (cid:115) κ p ρ , (1)where R is the equilibrium bubble radius, ρ is the fluid den-sity, p the hydrostatic pressure, and κ the polytropic exponent(see Section ). Bubbles with radius ranging from 10 − m downto 10 − m therefore respond to acoustic waves with frequen-cies from 10 Hz to 10 Hz. The methods currently being de-veloped to exploit this phenomenon for rheological measure-ments are described in Section . Finally, bubble collapse is aviolent phenomenon where the bubble radius can change byroughly 10 times its equilibrium value on a time-scale of tensof microseconds. The extremely large strain rates associatedwith bubble collapse, up to 10 s − , can be used to probe thehigh-strain rate rheology of soft materials, as will be describedin Section . Di ff usive transport followed by cavitation havebeen shown to span 9 orders of magnitude of deformation rateof the surrounding material in a single experiment [14]. Therange of frequencies accessible by these phenomena thereforepaves the way to a new class of techniques for broadband, ac-tive microrheology of complex fluids. How a bubble deforms the surrounding fluid
A spherical bubble at a fixed position and with time-dependent radius R ( t ) generates a purely radial velocity field.With reference to a spherical coordinate system with origin atthe centre of the bubble, where r , θ , and φ are the radial, az-imuthal, and polar coordinates respectively, the deformation,or strain, at time t can be derived based on the continuity of thevelocity field at the bubble boundary in the ideal, spherically-symmetric case: v ( r ) = v r ( r ) e r = v r ( R ) R r e r = ˙ R R r e r . (2)where the dots denote derivatives with respect to time, v = ( v r , ,
0) is the velocity field, and the unit vector in the ra-dial direction is e r = (1 , , R / c , where c is the speed of sound in the sur-rounding medium, and that the composition of the bubble ishomogeneous.The non-homogeneous nature of the velocity field in thesurrounding medium implies that changes in the bubble radius˙ R strain the surrounding medium at a rate ˙ (cid:15) = ( ∇ v + ∇ v T ) / ∇ v is the (tensor) gradient of the vector field v and( · ) T represents the matrix transposition operator. The spher-ical symmetry of the problem considerably simplifies the ex- pression of ˙ (cid:15) :˙ (cid:15) ( r ) = ∂ v r ∂ r v r r
00 0 v r r = ˙ R R r − (3)The fact that the strain rate ˙ (cid:15) is diagonal implies that the bub-ble applies a pure extension or compression of material ele-ments in the surrounding medium. This is in contrast withsimple shear flow, which combines pure extension and rota-tion. In the case of shear flow, the strain rate ˙ (cid:15) is called theshear rate, and it is used to compute the shear stress in liquids;in particular, they are proportional to each other in the case ofNewtonian media.In soft solids and yield-stress fluids, the stress is also a func-tion of the total strain applied to the material elements of thesurrounding medium. We can compute such a strain for anelement initially at a distance r from the centre of the bubbleat rest, i.e. for R = R . This quantity is also a tensor and mostcontinuum mechanics textbooks recommend using the Fingertensor B to quantify it [7]: B rr = B − θθ = B − φφ = + R ( t ) − R r − / (4)In the absence of deformation, the Finger tensor is not zerobut rather reduces to the identity tensor I . In the particularcase of pure extension, we can relate the components of theFinger tensor B to the linear stretch of the fluid elements. Wehave in particular, for the material elements of Figure 1(a) , B rr = [ l ( t ) / l ] . Figure 1(a) shows the properties of this strain field respec-tively for a bubble at rest (left panel), during its expansion(centre) and during its contraction (right panel). Both the par-ticle displacement and the net strain B − I are more sensitive tobubble expansion than compression and decay quickly awayfrom the surface of the bubble. Bubble dynamics: relating strain and stress fields
Both the local strain and the stress fields need to be knownfor bubble dynamics studies to qualify as a rheological mea-surement. While torque is easily measured on a rotationalrheometer and readily related to the local stress in the sample,we do not directly measure the stress field around bubbles.We instead rely on the momentum balance of the whole fluidsurrounding the bubble in the e r direction, which reads [8]: ρ (cid:32) R ¨ R +
32 ˙ R (cid:33) = (cid:90) ∞ R ( ∇ · σ ) r d r . (5)The tensor σ is the Cauchy stress tensor. As it includes pres-sure terms, its trace is non-zero. Far away from the bubble,we assume the material is at rest, which implies that the diag-onal terms of σ are all equal to − p ∞ ( t ), the total (ambient andacoustic) pressure applied by the operator far away from the R R r l (a) R ( t ) r ( t ) l ( t ) / / / l ( t ) /l ωt/ π G = 0 Pa G = 3000 Pa G = 30000 Pa . . ζ = 1 . ,G = 0 Pa ζ = 1 . ,G = 200 Pa ζ = 1 . ,G = 0 Pa ζ = 1 . ,G = 200 Pa (b) t/t R ( t ) / R t/t ∗ R ( t ) / R G = 0 Pa G = 3000 Pa G = 30000 Pa FIG. 1.
Kinematics and dynamics of bubbles in fluids and soft solids. (a)
Kinematics of a bubble (in white) surrounded by a fluid or a softsolid (light blue) when the former is at rest (left), expands (centre) and contracts (right). The bubble surface motion pushes and pulls tracers(small circles) in the radial direction. The tracer displacement is larger close to the bubble surface (as seen on the thick grey streak lines).Material elements (boxes around the tracers) are also strained and undergo biaxial (middle) or uniaxial extension (right) when the bubblegrows or shrinks. The colour of the boxes codes the stretch l ( t ) / l applied to the fluid particles in the radial direction. (b) Time evolution ofthe radius of a bubble with initial radius R due to mass transfer out of the bubble for a gas-saturated medium ( ζ =
1) and into the bubblefor a supersaturated medium ( ζ = . (c) Linear oscillations of the bubble radius, R ( t ), around the equilibrium radius R , under a small-amplitude acoustic excitation at angular frequency ω . (d) A bubble with initial radius 3 R undergoes violent collapse and subsequent rebounds,before reaching its equilibrium radius R . In (b-d) the surrounding fluid is assumed to be Newtonian when G = bubble. The spherical symmetry of the problem and the ab-sence of torque applied to the bubble implies that σ θθ = σ φφ .Equation 5 relates an integral of the non-homogeneousstress field (right hand side) to the bubble dynamics (left handside), yet the local stress distribution σ remains unknown, pre-venting any direct measurement of the material rheologicalproperties. We therefore need to choose a priori the relationbetween stress and strain in the surrounding medium and tryto accurately model the experimental bubble dynamics R ( t ).For Newtonian liquids and ideal (neo-Hookean) elastic solids,these relations read: σ = − p I + η ˙ (cid:15) (Newtonian liquid) , (6) σ = − p I + G ( B − I ) (neo-Hookean solid) , (7)where p is a pressure term that depends on r and matches theclassical definition of pressure for arbitrary strains in Newto-nian liquids, and only for small strains in neo-Hookean solids.The classical Newtonian viscosity η and linear elastic modu-lus G are considered to be free parameters of the models thatare fitted to the experimental data R ( t ). For more complex sur-rounding materials, such as visco-elastic liquids or solids, thestress and strain fields are related using more complex con-stitutive equations relating σ , ˙ (cid:15) and B and additional fittingparameters.In the specific case of Newtonian liquids and purely elasticsolids, all the physical quantities in Equation 5 are known or prescribed except for the pressure field p ( r ). Knowing thewhole pressure field is however not needed, as the right handside term of Equation 5 can be rewritten in the following way: (cid:90) ∞ R ( ∇ · σ ) r d r = − p ∞ ( t ) − σ rr ( R ) + (cid:90) ∞ R σ rr − σ θθ r d r . (8)In Equation 8, the pressure contribution cancels in σ rr − σ θθ and is hence only present in the radial stress at the bub-ble boundary σ rr ( R ). This quantity can be independentlysolved by considering the stress balance at the bubble inter-face, which reads: σ rr ( R ) = − p g + γ R , (9) γ being the surface tension between the gas inside the bub-ble and the surrounding material, and p g being the gas pres-sure in the bubble. The final step in solving Equation 5 thenconsists in choosing an equation of state for the gas pressureas a function of its volume 4 π R ( t ) /
3, its temperature and itscomposition. Bubbles injected using a syringe and containingair or other non-condensible gases are usually treated as idealgases with a homogeneous composition and are fairly simpleto model. In contrast, bubbles formed through laser vapor-isation of the medium are modelled as a mixture of vapourthat quickly condenses at the bubble surface and an inertgas, which leads to composition gradients between the centreand the bubble edge, which necessitates additional modellingsteps [15].
BUBBLE DISSOLUTION OR GROWTH BY MASSTRANSFER
The dissolution or growth of gas bubbles due to mass trans-fer of gas from the surrounding medium [
Figure 1(b) ] is arelatively slow process for which Equation 5 reduces to theclassical isostatic criterion, ∇ · σ = . The saturation concen-tration of the dissolved gas in the liquid phase, c s , is propor-tional to its partial pressure in the bubble following Henry’slaw: c s = k H p g . (10)Henry’s constant, k H , varies greatly between di ff erentgas / liquid pairs and depends on the temperature T . If we sup-pose that the liquid has been left long enough in contact withthe same gas before conducting the experiments, we can sup- pose an initial saturation condition, i.e. c ( r , t = = c = k H p , where p is the ambient pressure. For bubbles smallenough to be strongly a ff ected by surface tension e ff ects, thepressure in the gas bubble p g is greater than p , and the sat-uration concentration c s exceeds c at the bubble interface,driving bubble dissolution.For a su ffi ciently slow dissolution rate, the additional dis-solved gas simply di ff uses into the surrounding medium witha di ff usion coe ffi cient D . We then need to write the massbalance of the bubble and provide boundary conditions atthe bubble surface and at the outer edge of the surround-ing medium to derive the time evolution of the bubble ra-dius. We choose to work with an interface that is saturatedwith the solute based on the gas pressure in the bubble, i.e. c ( R , t ) = k H p g . This time, the infinite surrounding mediumcan be initially under-saturated or super-saturated with the dis-solved gas, meaning that c ( r → ∞ , t ) = c ( r , t = = ζ c with ζ a dimensionless constant being smaller (respectivelygreater) than 1 for under- (super-) saturation conditions. Fora neo-Hookean elastic surrounding medium of linear mod-ulus G , the time evolution of the bubble radius then satis-fies [6][16]: R Dk H R T d ˜ R d t = ζ − − δ/ ˜ R + δ G ( − / + / ˜ R + / R )1 + δ/ R + δ G (5 / − / R + /
12 ˜ R ) (cid:32) R + R √ π Dt (cid:33) , (11)in which ˜ R = R / R , δ = γ/ R p , δ G = G / p , T is thetemperature and R is the ideal gas constant. Equation (11) isa non-linear, ordinary di ff erential equation that can be solvednumerically. Interestingly, bubble stability against dissolutionmore simply depends on the sign of the numerator in Equa-tion (11). For an initially saturated medium ( ζ = R / d t = R + (cid:32) − γ R G (cid:33) R − = G , the left-hand side of Equa-tion (12) is negative for ˜ R = + ∞ for ˜ R → / R of elasticstresses for small bubbles. Consequently there always existsan equilibrium radius 0 ≤ R ∗ ≤ R in neo-Hookean solidsat which dissolution stops. For R = R ∗ , the gas pressure inthe bubble p g becomes equal to p , leading to an equal soluteconcentration c at the bubble interface and far away from it.Neo-Hookean solids then always arrest dissolution, as seen in Figure 1(b) . In contrast, stresses in a fluid medium do notdiverge for ˜ R → (cid:15) in the surrounding material, or, equivalently, in ˙ R . The gaspressure in the bubble may then always exceed p providedthat the bubble dissolution rate ˙ R / R is slow enough. Fluidsare then incapable of halting bubble dissolution even thoughvery viscous fluids may slow it down [6]. When the surrounding medium is initially super-saturatedwith the dissolved gas, ζ ≥
1, bubbles can grow by mass trans-fer from the surrounding medium. Elastic e ff ects slow downthis growth, and the resulting growth dynamics can be fitted toestimate both the super-saturation coe ffi cient ζ and the elasticmodulus of the medium, as shown in experiments by Andoand Shirota [17]. Bubbles in a soft, viscoelastic solid werefound to grow more slowly for increasing G despite similarsuper-saturation coe ffi cients ζ , as shown in Figure 2 .Recent experiments show that bubble dissolution can alsobe halted in yield-stress materials such as oleogels [18]. Thesituation is more complex in this case, as yield-stress materi-als behave as elastic solids for low applied stresses, yet flowabove it. In the context of spherical bubble dynamics, the cri-terion for the onset of flow behaviour reads ( σ rr − σ θθ ) ≥ σ [19], σ Y being a scalar quantity representing the yieldstrength of the material. Assuming the material behaves asan elastic solid below yielding [Equation 7], all of it remainssolid provided that[ σ rr ( R ) − σ θθ ( R )] = G (cid:18) R R (cid:19) − (cid:32) RR (cid:33) < σ (13)Bubble dissolution is then arrested provided that both Equa-tions (12) and (13) are satisfied. For sti ff materials for which γ/ R G (cid:28)
1, Equation (12) implies R ∗ (cid:39) R , and the no-yieldcriterion can be derived explicitly, leading to γ/ R ≤ σ Y / √ FIG. 2. E ff ect of medium elasticity on bubble growth by masstransfer . A laser-generated bubble grows in an elastic mediumsuper-saturated with air. The growth dynamics of the bubble radius R is used to estimate the super-saturation coe ffi cient ζ and the elasticmodulus G of the surrounding material. Reproduced with permissionfrom Ref. [17] Materials that can withstand a typical capillary stress √ γ/ R without yielding can then halt bubble dissolution. Injectingseveral bubbles of di ff erent sizes and examining the smallestbubble that does not dissolve could then be used to estimate insitu the yield stress of (gas-saturated) yield-stress materials. Inpractice, this picture is a bit more complicated, as most yield-stress fluids exhibit creep behaviour for applied stresses closeto σ Y , during which strain slowly accumulates over time.Attractive yield-stress fluids also often show thixotropic be-haviour, in which the structure of the fluid reinforces –and thematerial constants G and σ Y grow– over time at rest or undervery slow flow conditions. Creep and thixotropy will there-fore have antagonistic e ff ects on the ability of yield-stress flu-ids to stabilise bubbles, the former promoting dissolution byallowing additional material deformation for a given appliedstress, and the latter impeding dissolution when the flow isslow enough to result in an increase of σ Y . Surface activityof the fluid, while insu ffi cient to stabilise bubbles by itself inRef. [18], may also play a role in slowing down or halting dis-solution in conjunction with its bulk rheological properties. LINEAR BUBBLE OSCILLATIONSBubbles as harmonic oscillators
The case of linear oscillations of air bubbles [Figure 1(d)] isfairly simple to model. First, on the short time scales involvedand the limited oscillation amplitude, we can assume that airdoes not dissolve into the surrounding medium. Authors thenassume [20–23] a constant p g R κ in the gas phase, choosinga polytropic exponent 1 ≤ κ ≤ . κ =
1) nor adiabatic ( κ = . x ( t ) = R ( t ) / R −
1, we may derive the linear versionof Equation 5 [22]:¨ x + β ˙ x + ω x = − ρ R [ p ∞ ( t ) − p ] (14)The damping term β includes a viscous term proportional tothe surrounding medium viscosity η , a thermal loss term dueto thermal gradients present in the bubble, and a term account-ing for acoustic damping [24].The natural angular frequency ω also depends on the sur-rounding medium properties, ω = κ p + κ − γ/ R + G ρ R . (15)This frequency then deviates from the classical Minnaert res-onance frequency (see Equation 1) when bubbles are su ffi -ciently small bubbles: R ∼ γ/ p ∼ µ m in water at ambientpressure, and for elastic moduli G that are comparable with p , i.e. for sti ff gels. Measuring material properties in the Fourier domain
In the time domain, the damping parameter β and the natu-ral oscillation frequency ω can be fitted from the free bubbleoscillations, i.e. when p ∞ ( t ) = p and for initial conditions x ( t = (cid:44) x ( t = (cid:44)
0. An alternative approach isto study the steady-state oscillatory regime at an imposed an-gular frequency ω . Equation (14) can then be recast in theFourier domain: x = ρ R p ∞ − p (cid:113) ( ω − ω ) + ω β , (16)Equation 16 gives the resonance curve of an oscillating bub-ble in the linear regime, where it can be described as a linearharmonic oscillator [8]. It can be measured experimentally bysweeping the forcing frequency around the natural frequency ω , and recording the (maximum) oscillation amplitude x foreach value of the frequency. This approach was first demon-strated by Strybulevych et al. by measuring the acoustic re-sponse of millimetre-sized bubbles in an agar gel as a functionof frequency; the approach was termed “acoustic microrheol-ogy” [25]. In a similar approach termed “microbubble spec-troscopy”, resonance curves can be recorded by directly imag-ing with a high-speed camera the bubble dynamics duringacoustic forcing at di ff erent frequencies [26]. Jamburidze etal. [23] also used direct imaging in agarose gels and at mod-erate pressure amplitudes ( p ∞ ≤ R and natural frequency ω when the magnitude of theterm 2(3 κ − γ/ R in Eq. 15 is su ffi ciently small, i.e. forsu ffi ciently large bubbles. Resonance curves measured fromradius sweeps do not rely on a prior calibration of the trans-ducer response as a function of the frequency, making themparticularly attractive in experiments where the pressure fieldcannot be measured independently.In some experiments [23], the viscous contribution todamping β is dominant, and fitting the resonance curve di-rectly provides an estimate of the material parameters of theconstitutive model, in our case η and G . In the generalcase [22, 27], acoustic and thermal contributions have to be in-cluded in the total damping β ( ω, R ), which is not a constantin either frequency sweep or radius sweep experiments. Inpractice, reliable estimates of the viscous and thermal contri-butions have been determined by Prosperetti [24]. Resonancecurves are then fitted using the total damping β ( ω, R ), pre-scribing the viscous and thermal terms and letting the mediumviscosity η as the adjustable parameter of the fit. BUBBLE COLLAPSE
Bubble collapse combines the high-frequency descriptionof bubble oscillations of Section with the large deforma-tion framework derived in Section . Bubble collapse can beachieved for instance by applying a step change in the exter-nal pressure p ∞ (Rayleigh collapse), or by creating bubblesusing a laser pulse to vaporise the surrounding medium, andletting them relax (Flynn collapse). Regardless of the prepa-ration protocol and the medium properties, the complex re-laxation dynamics of the bubble assumes the shape of an ini-tial, extremely steep decrease in radius followed by a singleor multiple rebounds and an eventual relaxation to an equilib-rium radius (see, for instance, Figures 1d and 4). Conversely,the initial collapse time and the duration, number and am-plitude of the rebounds depend on the type of collapse andthe surrounding medium rheology, opening the way for highstrain-rate characterisation of soft materials [15, 28]. Giventhe violent nature of the process, elasticity e ff ects are onlyperceptible for sti ff er materials ( G ≥ Pa) whereas viscouse ff ects are almost always noticeable ( η ≥ − Pa.s) [15].Estrada et al. have studied the collapse dynamics of a rela-tively sti ff
10% polyacrylamide gel and compared it with pre-dictions from various rheological models including or com-bining neo-Hookean, Newtonian fluid and Maxwell fluid ele-ments. Figure 4 confirms that both elastic and viscous com-ponents must be included in the model –resulting in a neo-Hookean Kelvin-Voigt material– to accurately fit the bubblecollapse dynamics. Estrada et al. [15] have shown that in-creasing the complexity of the rheological model does notsignificantly improve the quality of the fit to the experimen-tal bubble collapse data. Nevertheless, Yang et al. [28] have suggested that including additional strain-sti ff ening terms inthe material elasticity reconciles the classical rheology mea-surements and the non-linear fits of bubble collapse, in agree-ment with the strain-sti ff ening behaviour generally observedin biopolymer gels [29].Bubble collapse is a violent, complex process and mod-elling e ff orts have been particularly cautious to evaluate allthe deviations from the ideal case of adiabatic oscillations inan incompressible model fluid. Indeed, the Mach number ˙ R / c during collapse is no longer small and first-order compress-ibility corrections in Equation (5) have to be included [15, 28].Barajas et al. has shown that heat transfer between the bubbleand the surrounding medium must be taken into account asit noticeably a ff ects the relaxation dynamics and the equilib-rium bubble size during collapse [30]. Lastly, Gaudron et al. have studied analytically and numerically the onset of non-spherical bubble oscillations during Rayleigh collapse [31].Their numerical results highlight that such e ff ects should notoccur for p ∞ ≤ G . They also show that working with a fi-nite deformation elastic framework –the neo-Hookean model–rather than linear elasticity –Hooke’s law– strongly promotesspherical bubble oscillations, in particular for sti ff materialsand high applied pressures p ∞ ∼ Pa.
TOWARDS BROADBAND MICRORHEOLOGY OFCOMPLEX FLUIDS USING BUBBLE DYNAMICSCurrent limitations of rheological measurements based onbubble dynamics
Here we identify the main limitations that need to be over-come to achieve a robust technique. The first practical is-sue is the limited control o ff ered by bubble microrheology interms of the applied strain or stress, at variance with classi-cal rheometers; the latter are indeed particularly e ffi cient atapplying a precise strain or stress history to the sample suchas step strain, shear startup, large amplitude oscillatory shearor creep tests. During bubble dissolution, neither the strainrate or the stress at the bubble boundary are fixed, as the dy-namics is rather governed by the di ff erence in chemical po-tential between the bubble and the fluid far from it. Linearbubble oscillations may be viewed as small amplitude oscilla-tory strain experiments, yet without an explicit control on theoscillation amplitude, which depends on the inertial terms inEquation (5) and the material properties. Lastly, bubble col-lapse experiments impose a fairly complex strain history tothe material which cannot be controlled by the operator. Thisprofile rather results from a balance between the bubble ther-modynamics, fluid inertia and material rheology.The second challenge is related to bubble formation, sta-bility and imaging. Injecting or embedding spherical bubblesof a desired radius in soft materials can be challenging, es-pecially if the material exhibits predominantly solid-like be-haviour, or if it is a yield-stress fluid. Bubbles may be in-jected while a gel is setting [23] but this is not always pos- FIG. 3.
Linear high-frequency rheology by acoustic bubble dynamics. (a)
Test to confirm the linearity of bubble oscillations and materialdeformation in agarose gels. The amplitude of oscillations ∆ R (normalised by the smallest oscillation amplitude ∆ R ) is plotted as a functionof the applied pressure amplitude ∆ p (normalised by the lowest applied pressure ∆ p ). Symbols are experimental data and lines are quadraticfits, with black (respectively red) data series shifted upwards by 1 (respectively 2) for clarity. (b) Impact of the surrounding material propertieson the resonance curves. Symbols are experimental data and solid lines represent a fit from Equation (16). As the gel concentration isincreased, the gel becomes sti ff er and the resonance frequency increases [see Equation (15)], and viscous damping also increases, broadeningthe resonance peak. Reproduced with permission from Ref. [23]FIG. 4. Collapse dynamics of a laser-generated bubble in a poly-acrylamide gel . The experimental data R ( t ) (orange squares) arecompared to the best fits assuming the gel behaves as a neo-Hookeanelastic solid (green dashed line), a Newtonian fluid (purple dashedline) and a neo-Hookean Kelvin-Voigt –viscoelastic solid– model(blue solid line; also see top-right schematic diagram). The fit to theviscoelastic model provides estimates of the its linear elastic modu-lus G and viscosity η (see right inset). Reproduced with permissionfrom Ref. [15] sible. Generation of bubbles by a focused laser pulse pro-vides the necessary spatio-temporal control on bubble forma-tion [15, 17, 22], although this is not a widely available tech-nique. Bubble dissolution will become an issue for linear os-cillation experiments at f ∼ R ∼ µ m, as the dissolution time ( ∼
10 ms in water)may then fall below the time needed to perform the measure-ments. Furthermore it is observed that shrinking bubbles inhydrogels can leave behind a pocket of water within a solid-like gel network that remains undeformed [22]. Fine-tuningthe saturation coe ffi cient ζ by letting samples equilibrate at adi ff erent temperature [22], or under an excess static pressure,can be used to slow down dissolution su ffi ciently to allow enough time for experiments, or to drive a slow bubble growthfor which solvent pockets are no longer an issue [17]. Imag-ing oscillating or collapsing bubbles around ∼ ∼ images per second. Com-mercial instruments now meet these requirements [32], andwill become more widely used as their price becomes morea ff ordable.A complex issue of linear bubble oscillations is the na-ture of the quantities G and η obtained from fitting the res-onance curve. Even though the two techniques are concep-tually equivalent, resonance curves obtained from frequencysweeps obviously contain information at multiple frequen-cies, whereas resonance curves obtain from radius sweeps aresingle-frequency measurements. It is therefore tempting toidentify the material properties G and η fitted from radiussweep data with the storage modulus G (cid:48) ( ω ) and loss mod-ulus G (cid:48)(cid:48) ( ω ) measured from classical oscillatory rheology atthe acoustic angular frequency ω . In this case, the e ff ectivefrequency of quantities fitted to frequency sweep resonancecurves become unclear. However, Equation (16) is derivedassuming a Kelvin-Voigt, linear rheology of the surroundingmedium. In classical rheology, this implies that the storagemodulus G (cid:48) ( ω ) = G and loss modulus G (cid:48)(cid:48) ( ω ) = ηω are pre-scribed at all frequencies. Fitting the resonance curves thenconstrains G (cid:48) and G (cid:48)(cid:48) at every frequency based on experi-mental data covering a limited frequency range –frequencysweep resonance curves– or even covering a single frequency–radius sweeps. This contrasts with classical oscillatory rhe-ology, for which measurements cover three decades in termsof frequency, allowing easy discrimination between di ff erentrheological models. Resonance curve data should then ide-ally be complemented by measurements from another tech-nique to validate (or question) the choice of the rheologicalmodel [23], keeping in mind that the rheological models pre-sented here (neo-Hookean, Maxwell, Kelvin-Voigt) are rathercrude approximations of the experimental behaviour of realsoft materials.The last limitation is related to the onset of shape oscil-lations, which break down the spherical symmetry assump-tion used to derive most of the equations derived previously.Shape oscillations have been observed both under moderateamplitude bubble oscillations and during cavitation events inboth Newtonian fluids and visco-elastic solids [22, 27, 33].Stability of spherical bubbles during during cavitation eventsand oscillations of moderate amplitude have been studied re-spectively by Gaudron et al. [31] and Murakami et al. [34].The latter article quantifies the shift in shape mode numberdue to medium elasticity and the impact of viscosity on thecritical oscillation amplitude above which shape oscillationsoccur. Agreement with the available, very limited experimen-tal data [22] was found to be good. More experimental datawould be valuable to validate the model and eventually useshape oscillation data to obtain rheological information fromsoft materials. Opportunities for high-frequency rheology
As complex fluids are structured from intermediate ( ∼ µ m)to molecular ( ∼ Å) sizes, they also show a broad distributionof relaxation time scales which can be probed through lin-ear oscillatory rheology. In polymer solutions and wormlikemicelles, these time scales are directly related to spatial phys-ical quantities of the polymers such as the chain persistencelength and the entanglement or crosslink density. In par-ticulate suspensions, gels and emulsions, the linear oscilla-tory spectra contain valuable information on the onset of theglass transition or rigidity percolation [35]. Commercial ro-tational rheometers cannot provide high frequency measure-ments ( ≥
50 Hz) due to inertial e ff ects of the rheometer head.Time-temperature superposition has then been widely used inpolymer solutions and melts to fill this particular gap in ex-perimental data [35]; yet, the technique cannot be appliedfor out-of-equilibrium (arrested or glassy) systems or thoseundergoing phase transitions. Dedicated piezoelectric andmicroelectromechanical systems (MEMS) o ff er direct high-frequency rheological measurements, as recently reviewed bySchroyen et al. [36]. Traditional microrheology techniques,relying on the motion of passive or active tracers embedded inthe fluid [37], have also been developed to complement relax-ation spectra both at very small and large frequencies. Bubbledynamics o ff er an opportunity to extend the range of availabletechniques for active microrheology.Many industrial flows, such as jetting, injection moulding,or lubrication, and natural flows, such as sneezing [38], applystrains to complex fluids at high frequency and beyond theirlinear deformation, that is, at high strain rates. Such transfor-mations break and reorient their microstructure, usually lead-ing to shear-thinning behaviour. The associated, very highstrain rates ˙ (cid:15) ≥ s − once again exceed the capabilities ofrotational rheometers due to sample expulsion, particle migra- tion and edge fracture. Pressure-driven flows, such as capil-lary rheometers, are a viable alternative to rotational geome-tries as they do not present any free surface in the fluid regionof interest, yet they require careful pressure corrections dueto end e ff ects and potential wall slip [7]. Bubble microrheol-ogy, in particular based on bubble collapse, is una ff ected bywall slip, since the strain is applied by a movement normal tothe bubble-fluid interface, and meets both the high-frequencyand high-strain criteria to provide insights on high-strain raterheology of many complex fluids.Lastly, bubble dynamics, collapse and cavitation can gener-ate shear waves in viscoelastic materials when the bubbles arelocated close to a boundary [39–41], because the strain profilearound the bubble is no longer symmetric. These waves havebeen observed using echography [39], particle tracking [40]or birefringence [41] techniques. The propagation speed andattenuation of shear waves can provide a direct measurementof the high-frequency material properties G and η of the ma-terial [42], which have been compared to low-frequency mea-surements obtained using a rheometer in [40]. Opportunities for extensional deformation of complex fluids
Extensional deformation is the main mode of deformationduring spinning, rolling, convergent or divergent die injectionand spraying in industrial processes [7] but also in biologicalflows, e.g. during sneezing [38]. Extensional deformation, incontrast with simple shear, strongly imposes the orientationof the fluid elements, leading for instance to the well-knowncoil-stretch transition and extensional thickening of polymersolutions and the high extensional viscosity of rigid fibre sus-pensions [7]. In yield-stress fluids, the yield stress in exten-sional flow is known to di ff er from the one obtained in sim-ple shear [43, 44] due to the non-linear normal stress di ff er-ences arising in the material even below yielding [45, 46].As solid boundaries in a geometry impose shear flows, trueextensional flows work with stress-free boundary conditions,for instance by examining the thinning and breakup dynamicsof fluid filaments, following either the separation of two endplates [47] or the Rayleigh-Plateau instability in jet flows [48].As the strain in the filament leads to an exponential decreaseof their cross-section, these techniques are limited in terms ofmaximum applied strain. Bubble microrheology is then par-ticularly interesting to locally apply very strong extensionalstrains, given by (cid:15) = B θθ ( R ) − = ( R / R ) − − a priori and depends on the fluid rheology and themagnitude of wall slip. As a consequence, these geometriesare usually complemented with local flow birefringence [49]or microparticle image velocimetry [46] to precisely monitorthe time-dependent, non-uniform strain applied to the fluid el-ements. Direct imaging of the deformation field and evolutionof the microstructure is also possible in combination with bub-ble dynamics experiments. For instance, the reorganisation ofa microfibre network around an expanding bubble has beendirectly visualised for di ff erent deformation rates, as shownin Figure 5a-d , evidencing a strong reorganisation of the fi-bre network close to very slowly, quasi-statically expandingbubbles, which progressively disappears as the expansion rateincreases. The change in the microstructure of a bicontinuousinterfacially jammed emulsion gel (bijel) due to bubble mo-tion during centrifugation has been imaged by confocal mi-croscopy (
Figure 5e-f ). These examples support the notion ofsimultaneous imaging of the evolution of the microstructureof a complex fluid, in combination with rheological measure-ments based on the bubble dynamics phenomena described inthis paper. ex p a n s i on ( µ m / s ) ex p a n s i on ( µ m / s ) ce n t r i f ug a t i on ( g ) before deformation after deformation(a) (b)(c) (d)(e) (f) FIG. 5.
Visualisation of bubble-induce deformation of complexfluids. (a-d)
Quasi-static expansion of a bubble in a sparse microcel-lulose yield-stress fluid. Reproduced with permission from Ref. [51].In (a-b), very slow expansion rates allow a relative motion betweenthe solvent and the fibre network and changes in the microstructureare visible close to the expanded bubble. In (c-d), for higher ex-pansion rates, no relative motion is possible and the microstructurearound the bubble remains similar before and after inflation. (e-f)
Deformation and destruction of a bijel due to bubble displacementduring centrifugation. Reproduced with permission from Ref. [52].
CONCLUSION
Spherical bubbles are versatile active probes to characterisethe local rheology of soft materials thanks to the purely exten-sional, localised nature of the deformation field they apply tothe material. In this paper we discussed three emerging micro-rheological measurements based on recording the time evolu-tion of the radius of a bubble, during three distinct processesthat provide complementary information on the surroundingmaterial. The first process is the slow, quasistatic bubble dis-solution dynamics and its potential arrest, which can be usedto evaluate the linear elastic modulus G of a material for neo-Hookean or Kelvin-Voigt materials, or to obtain an estimatethe critical stress σ Y of yield-stress fluids. A second techniqueexploits linear bubble oscillations driven by an acoustic wave.Examining the bubble response in the time or in the frequencydomain provides an estimate from high-frequency data of both G and the solvent viscosity η , for an equivalent Kelvin-Voigtmaterial. Lastly, the violent bubble collapse process o ff ers ad-ditional, deeper insights on the finite-strain rheology of softmaterials at extremely high strain rates (typically 10 s − ). In-formation is obtained by fitting experimental collapse timeseries to fairly complex, non-linear, finite strain models ofbubble dynamics. These modelling steps are needed to esti-mate the stress field σ in the material, which is otherwise un-available, in contrast with classical rheology measurements.Dissolution and collapse processes impose high extensionalstrains close to the bubble boundary and may be a power-ful technique to examine extreme phenomena such as the ex-tensional thickening of polymer solutions due to individualchain uncoiling; in particular since the direct bubble imagingcan easily be combined with particle image velocimetry tech-niques or flow birefringence. We also discussed outstandingchallenges of bubble-based microrheology, for instance con-trolling the stress history and the process of bubble injection.More work is also needed to combine more realistic models todescribe soft materials with the governing equations of bubbledynamics, especially under high-frequency, high-strain ratedeformation. Under such conditions, the use of simple mod-els, for instance neo-Hookean Kelvin-Voigt solids, should atleast be better justified given the numerous deviations fromsuch a behaviour observed in classical rheology [29]. CONFLICT OF INTEREST STATEMENT
The Authors declare no conflicts.
ACKNOWLEDGEMENTS
This work was supported by a European Research CouncilStarting Grant [grant number 639221] (V.G.).0 ∗ Corresponding Author:[email protected][1] S. Everitt, O. Harlen, H. Wilson, and D. Read, J. Non-Newtonian Fluid Mech. , 83 (2003).[2] G. Pagani, M. J. Green, P. Poulin, and M. Pasquali, Proc. Natl.Acad. Sci. , 11599 (2012).[3] T. J. Lin, J. Soc. Cosmet. Chem , 323 (1970).[4] S. Fujii and Y. Nakamura, Langmuir , 7365 (2017).[5] D. Lohse, Phys. Rev. Fluids , 110504 (2018).[6] W. Kloek, T. van Vliet, and M. Meinders, J. Colloid Interf. Sci. , 158 (2001), Note: derivation of the dynamics of quasi-static bubble dissolution in the presence of surface and bulkelasticity .[7] C. W. Macosko,
Rheology: Principles, Measurements and Ap-plications (Wiley-VCH New York, 1994).[8] B. Dollet, P. Marmottant, and V. Garbin, Ann. Rev. Fluid Mech. , 331 (2019).[9] P. S. Epstein and M. S. Plesset, J. Chem. Phys. , 1505 (1950).[10] M. P. Brenner, S. Hilgenfeldt, and D. Lohse, Rev. Mod. Phys. , 425 (2002).[11] M. Versluis, Exp. Fluids , 1 (2013).[12] C. W. Barney, C. E. Dougan, K. R. McLeod, A. Kazemi-Moridani, Y. Zheng, Z. Ye, S. Tiwari, I. Sacligil, R. A. Rig-gleman, S. Cai, et al., Proc. Natl. Acad. Sci. U.S.A. , 9157(2020).[13] E. M. Furst and T. M. Squires, Microrheology (Oxford Univer-sity Press, 2017).[14] M. A. Bruning, M. Costalonga, J. H. Snoeijer, and A. Marin,Phys. Rev. Lett. , 214501 (2019).[15] J. B. Estrada, C. Barajas, D. L. Henann, E. Johnsen, andC. Franck, J. Mech. Phys. Solids , 291 (2018), ISSN 0022-5096,
Note: comparison of constitutive models to probe thehigh-strain rate rheology of soft materials .[16] Note1, the sign of the elastic component on the numerator hasbeen corrected from Ref. 6.[17] K. Ando and E. Shirota, Phys. Fluids , 111701 (2019), Note:estimating linear elastic modulus from bubble growth bymass transfer .[18] S. Saha, B. Saint-Michel, V. Leynes, B. Binks, and V. Garbin,Rheol. Acta , 255–266 (2020).[19] M. De Corato, B. Saint-Michel, G. Makrigiorgos, Y. Di-makopoulos, J. Tsamopoulos, and V. Garbin, Phys. Rev. Fluids , 073301 (2019).[20] X. Yang and C. C. Church, J. Acoust. Soc. Am. , 3595(2005), Note: theoretical framework combining linear bub-ble dynamics with linear material deformation .[21] R. Gaudron, M. Warnez, and E. Johnsen, J. Fluid Mech. , 54(2015),
Note: theoretical framework combining non-linearbubble dynamics with finite material deformation .[22] F. Hamaguchi and K. Ando, Phys. Fluids , 113103 (2015), Note: linear oscillatory rheology using ultrasound-drivenbubbles; radius sweep method .[23] A. Jamburidze, M. De Corato, A. Huerre, A. Pommella, andV. Garbin, Soft Matter , 3946 (2017), Note: linear oscilla-tory rheology using ultrasound-driven bubbles; frequencysweep method .[24] A. Prosperetti, L. A. Crum, and K. W. Commander, J. Acoust.Soc. Am. , 502 (1988).[25] A. Strybulevych, V. Leroy, M. G. Scanlon, and J. H. Page, in Proceedings of Symposium on Ultrasonic Electronics (2009), vol. 30, pp. 395–396.[26] S. M. van der Meer, B. Dollet, M. M. Voormolen, C. T. Chin,A. Bouakaz, N. de Jong, M. Versluis, and D. Lohse, J. Acoust.Soc. Am. , 648 (2007).[27] B. Saint-Michel and V. Garbin, Soft Matter (2020), in Press .[28] J. Yang, H. C. Cramer III, and C. Franck, Extreme Mech. Lett.p. 100839 (2020).[29] C. Storm, J. J. Pastore, F. C. MacKintosh, T. C. Lubensky, andP. A. Janmey, Nature , 191 (2005).[30] C. Barajas and E. Johnsen, J. Acoust. Soc. Am. , 908 (2017).[31] R. Gaudron, K. Murakami, and E. Johnsen, Journal of the Me-chanics and Physics of Solids , 104047 (2020).[32] R. Kuroda and S. Sugawa,
The Micro-World Observed by Ul-tra High-Speed Cameras (Springer International Publishing,2018), chap. Cameras with On-chip Memory CMOS ImageSensors.[33] E.-A. Brujan and A. Vogel, J. Fluid Mech. , 281 (2006).[34] K. Murakami, R. Gaudron, and E. Johnsen, Ultrasonics Sono-chemistry , 105170 (2020).[35] R. G. Larson, The Structure and Rheology of Complex Fluids (Oxford University Press, 1999).[36] B. Schroyen, D. Vlassopoulos, P. Van Puyvelde, and J. Vermant,Rheol. Acta , 1 (2020).[37] T. G. Mason, Rheol. Acta , 371 (2000).[38] B. E. Scharfman, A. H. Techet, J. W. M. Bush, andL. Bourouiba, Exp. Fluids , 24 (2016).[39] S. Montalescot, B. Roger, A. Zorgani, R. Souchon, P. Grasland-Mongrain, R. Ben Haj Slama, J.-C. Bera, and S. Catheline,Appl. Phys. Lett. , 094105 (2016).[40] M. Tinguely, M. G. Hennessy, A. Pommella, O. K. Matar, andV. Garbin, Soft Matter , 4247 (2016).[41] J. Rapet, Y. Tagawa, and C.-D. Ohl, Appl. Phys. Lett. ,123702 (2019).[42] S. Catheline, J.-L. Gennisson, G. Delon, M. Fink, R. Sinkus,S. Abouelkaram, and J. Culioli, J. Acoust. Soc. Am. , 3734(2004).[43] K. Niedzwiedz, H. Buggisch, and N. Willenbacher, Rheol. Acta , 1103 (2010).[44] X. Zhang, O. Fadoul, E. Lorenceau, and P. Coussot, Phys. Rev.Lett. , 048001 (2018).[45] H. De Cagny, M. Fazilati, M. Habibi, M. M. Denn, andD. Bonn, J. Rheol. , 285 (2019).[46] S. Varchanis, S. J. Haward, C. C. Hopkins, A. Syrakos, A. Q.Shen, Y. Dimakopoulos, and J. Tsamopoulos, Proc. Natl. Acad.Sci. U.S.A. (2020).[47] G. H. McKinley and A. Tripathi, J. Rheol. , 653 (2000).[48] B. Keshavarz, V. Sharma, E. C. Houze, M. R. Koerner, J. R.Moore, P. M. Cotts, P. Threlfall-Holmes, and G. H. McKinley,J. Non-Newtonian Fluid Mech. , 171 (2015).[49] S. J. Haward, M. S. Oliveira, M. A. Alves, and G. H. McKinley,Phys. Rev. Lett. , 128301 (2012).[50] T. J. Ober, S. J. Haward, C. J. Pipe, J. Soulages, and G. H.McKinley, Rheol. Acta , 529 (2013).[51] J. Song, M. Caggioni, T. M. Squires, J. F. Gilchrist, S. W.Prescott, and P. T. Spicer, Rheologica Acta , 231 (2019), Note: method using bubble inflation to estimate the yieldstress and long-time relaxation time scale in a yield-stressmaterial .[52] K. A. Rumble, J. H. J. Thijssen, A. B. Schofield, and P. S.Clegg, Soft Matter12