## Abstract

In a pure liquid, the behaviour of a gas or vapour microbubble is determined primarily by its size, the ambient pressure and the properties of the surrounding liquid. In practice, however, adsorption of a dissolved substance from the surrounding liquid onto the microbubble surface will often take place, producing a thin coating which can significantly affect both the microbubble's stability and its dynamic response. This can have important implications in a wide range of applications, including underwater acoustics, cavitation detection, medical imaging and drug delivery. The aim of this paper is to review the existing theoretical treatments of coated microbubbles and to present and discuss some recent developments. It will be shown that the presence of the coating can substantially modify the amplitude of microbubble volumetric oscillation, resonance characteristics and relative amplitude in tension and compression. Finally, the need for improved understanding of the dynamic behaviour of surface coatings at high frequencies will be discussed.

## 1. Introduction

The behaviour of bubbles is a subject which has long excited the interest of physicists, mathematicians and engineers, whether to explain the origins of cavitation damage (Besant 1859; Lord Rayleigh 1917), the sounds of the natural world (Minnaert 1933; Leighton & White 2004) or phenomena such as sonoluminescence (Lohse 2005). In a pure liquid, the motion of a gas- or vapour-filled cavity is determined primarily by its size, the ambient pressure and the properties of the surrounding liquid. Rarely, however, are such ‘clean’ bubbles encountered in practice. Adsorption of molecules of a dissolved substance from the surrounding liquid onto the gas–water interface results in the formation of a thin coating on the bubble surface. This may be due to the deliberate addition of surfactants to produce stabilized bubble suspensions and foams, or it may be incidental due to the natural occurrence of these substances, for example in biological tissue (Moore 1953; Pattle 1960). In either case, the presence of this film or coating can have a significant effect upon both the stability of the bubble and its dynamic behaviour, particularly if the bubble diameter is smaller than 1 mm.

A microbubble suspended in a liquid is inherently unstable due to the action of interfacial tension. This effect can be expressed in terms of the Laplace pressure acting on the microbubble surface(1.1)where *R* is the instantaneous radius of the microbubble and *σ* is the interfacial tension. For the sake of simplicity, this discussion will be restricted to the behaviour of gas microbubbles.

Under constant ambient pressure in an unsaturated liquid, the radius of a microbubble will decay exponentially as gas diffuses into the surroundings, with the rate of dissolution depending on the magnitude of the interfacial tension, the size of the microbubble, the ambient temperature and pressure, and the concentration and diffusivity of the gas in the liquid (Epstein & Plesset 1950; Readey & Cooper 1966). If the microbubble is subjected to a fluctuating pressure such as a sound field, however, it will undergo volumetric oscillations and, depending on its location and its size relative to the wavelength of the sound field, it may also experience surface oscillations and/or translation. These oscillations may be highly nonlinear on account of the microbubble's high compressibility and both the hydrodynamic and acoustic consequences of this motion may be of great significance. It is well known that cavitation can produce serious damage in hydraulic machinery (Young 1989), whilst, at lower levels, the erosion caused by microbubble oscillations can be exploited for cleaning and is being investigated as a means of enhancing drug uptake in certain types of therapy (Mitragotri *et al*. 1995). In underwater acoustics, the distinctive signature of bubbles is used in wake detection; and in diagnostic medical imaging, suspensions of microbubbles are used intravenously as contrast agents, where their ability to scatter ultrasound nonlinearly enables them to be distinguished from the surrounding tissue (Stride & Saffari 2003).

In the case of either a constant or a fluctuating pressure, the presence of a coating at the gas–liquid interface can significantly affect the microbubble's behaviour. Firstly, it can reduce the Laplace pressure (equation (1.1)) by lowering the interfacial tension. Secondly, it can alter the rate of diffusion of the gas across the interface and hence the rate of change of microbubble size (Fyrillas & Szeri 1995, 1996). Thirdly, it can impart additional resistance to the motion of the interface. The aim of this paper is to discuss the third of these phenomena and, in particular, the effect of a surface coating on the volumetric oscillations of a microbubble.

## 2. Review of existing models for bubble dynamics

The first equation of motion for a spherical (uncoated) cavity suspended in a liquid was derived by Besant (1859),(2.1)where *R* is the instantaneous radius of the cavity; and are the velocity and acceleration of the cavity wall, respectively; *ρ*_{L} is the density of the surrounding fluid; *p*_{L}(*R*) is the pressure in the liquid at the cavity surface; and *p*_{∞} is the pressure in the liquid at a large distance from the bubble.

The problem was reconsidered by Lamb (1879) and later by Lord Rayleigh (1917) in his analysis of a collapsing cavity. Equation (2.1) was subsequently reformulated for a vapour-filled bubble by Plesset (1949) and for a gas bubble by Noltingk and Neppiras (Noltingk & Neppiras 1950; Neppiras & Noltingk 1951). Finally, an additional term was included to account for viscous dissipation in the surrounding fluid. The result was the so-called Rayleigh–Plesset or Rayleigh–Plesset–Noltingk–Neppiras–Poritsky (RPNNP) equation (Lauterborn 1976),(2.2)where *p*_{v} is the vapour pressure; *κ* is the polytropic constant; *μ*_{L} is the viscosity of the surrounding liquid; *p*_{o} is the ambient pressure; *p*_{A}(*t*) is an imposed pressure field (for which it is assumed that the wavelength is large compared with the bubble radius); *R*_{o} is the initial bubble radius; and *σ* is the interfacial tension as described previously.

The effect of a surface coating was first considered by Fox & Herzfeld (1954) in their study of cavitation nuclei. They proposed a purely elastic membrane on the bubble surface, characterized by its Young's modulus, *E*, and Poisson's ratio, *ν*, and derived expressions for the changes in bubble resonance frequency and cavitation threshold due to the additional membrane resistance. Yount (1982) and Glazman (1983) subsequently derived models in which the ‘elastic’ properties of the membrane arose as a result of the variation in interfacial tension with surface concentration of adsorbed molecules; the latter deriving an equation of motion for the bubble interface which is similar in form to equation (2.2). The same approach was adopted by Marmottant *et al*. (2005), who also included an additional term to describe the viscous dissipation in the coating following Sarkar *et al*. (2005).

A different approach was taken by Avetisyan, who modelled the coating as a non-Newtonian liquid layer of finite thickness described by two viscoelastic parameters (Glazman 1983). Similarly, de Jong *et al*. (1992) defined ad hoc shell stiffness and friction parameters to describe the elastic resistance and damping effects. A rigorous derivation for a bubble coated with a viscoelastic shell of finite thickness was given by Church (1995) and this was shown to be mathematically equivalent to the de Jong model in the limit of a thin shell by Hoff *et al*. (2000). Alternative constitutive equations for the coating, e.g. hyperelastic materials, have also been considered (Allen & Rashid 2004) and which type of model represents the most appropriate treatment for a given coating is an important question which will be discussed subsequently in §7. In the following derivation, the bubble will be treated as having a coating consisting of a homogeneous insoluble molecular monolayer with interfacial tension, *σ*, and viscosity, *η*_{s}, that vary as a function of the surface molecular concentration, *Γ*.

## 3. Model for a spherical coated bubble

A spherical coated gas bubble of instantaneous radius *R* is considered, which is suspended in an infinite, incompressible, Newtonian liquid. Assuming spherical symmetry, conservation of momentum in spherical polar coordinates gives(3.1)where *ρ* is density; *u* is radial velocity; *t* is time; *r* is the radial distance measured from the bubble centre; *p* is pressure; and is the stress tensor with components *T*_{rr}, *T*_{θθ} and *T*_{ϕϕ}.

Similarly, conservation of mass for an incompressible liquid gives(3.2)

Since the thickness of the surfactant coating will be much smaller than the radius of the bubble, it is treated as a two-dimensional surface. Integrating equation (3.1) with respect to *r* from *R* to infinity gives(3.3)where *p*_{L}(*R*) and *T*_{rr,L}(*R*) are the pressure and stress in the liquid at the bubble surface as described previously. Assuming continuity of stress at the surface of the bubble provides the following boundary condition for *r*=*R*(3.4)where *η*_{s} is the surface viscosity of the bubble coating and *p*_{G} is the pressure of the gas inside the bubble. As explained above, the interfacial tension, *σ*, for an insoluble film depends upon the concentration of adsorbed surfactant molecules at the interface (Israelachvili 2003) and may be expressed as(3.5)where *σ*_{o} is the initial interfacial tension for *R*=*R*_{o}; *Γ*_{o} is the initial concentration of surfactant on the bubble surface; and *K* and *x* are respectively the constants of proportionality and exponent relating *σ* and *Γ* through a power law (Yount 1982; Atchley 1989). Similarly, the surface viscosity also depends upon as(3.6)where *η*_{s}_{o} and *Z* are constants for a given surfactant and *R*_{x} is the limiting bubble radius beneath which the surface buckles (Saccheti *et al*. 1993) and the interfacial tension will be reduced to zero (§7). The viscous stress in the surrounding liquid is given by(3.7)

Substituting the above into equations (3.3) and (3.4) gives(3.8)

Equation (3.8) can be compared with those derived by Glazman (1983) and Marmottant *et al*. (2005) which, although derived by different means, can be shown to represent special cases of equation (3.8) for particular values of *x* and *Z*. It should be noted that damping due to thermal conduction and acoustic radiation has been neglected in this derivation as the discussion is focused on the effect of the coating; and it has been shown by Stride (2005) that both these types of damping are negligible compared with viscous effects for bubbles smaller than 50 μm radii and excitation pressures for which the coating would be expected to remain intact. For larger bubbles and higher amplitudes of oscillation, however, these effects may not be negligible and additional terms must be included in equation (3.8), which must then be solved simultaneously with the equations for conservation of mass and energy. This has been demonstrated for a free bubble by Prosperetti (1977; Prosperetti *et al*. 1988) and a coated bubble by Stride (2005).

## 4. Influence of the coating on the dynamic response

Equation (3.8) may be rearranged as(4.1)where *f*_{P}, *f*_{I}, *f*_{LV}, *f*_{SV} and *f*_{E}, respectively, represent the contributions to the acceleration of the bubble wall due to pressure, liquid inertia, liquid viscosity, coating viscosity and effective elasticity due to the varying interfacial tension. By putting the equation in this form, an analysis analogous to that applied by Flynn (1964) for free gas bubbles can be performed to identify which components have the most significant effect on the motion of the bubble. Figure 1 compares the amplitudes for each term from a numerical solution to equation (4.1) for a free bubble and surfactant-coated bubble. Calculations were performed in Matlab Release 2006b (The Mathworks, Natick, MA, USA). The parameter values used are given in appendix A unless otherwise specified.

As shown in figure 1, the radial acceleration is dominated by the coating until the negative pressure amplitude is sufficiently high and the inertial term (*f*_{I}) dominates during the compression phase. In Flynn's analysis, this is the criterion for the onset of inertial cavitation. At this point, the motion becomes similar to that of a free bubble but is initiated by a much higher incident pressure. Qualitatively similar results are seen for different coating parameters, bubble sizes and frequencies. This indicates that, while surfactant-coated bubbles will provide stable nuclei to initiate cavitation, the presence of the coating will also affect the nature of the cavitation process. This concurs with the finding by Postema & Schmitz (2007) that there is poor correlation between the theoretical threshold for inertial cavitation (Blake threshold) and the pressure at which fragmentation of surfactant-coated bubbles (in their case ultrasound contrast agents Sonovue and Albunex) is observed experimentally. This has important implications for understanding the mechanisms underlying enhanced cell permeability in ultrasound-mediated drug delivery, where ultrasound exposure is frequently carried out in the presence of surfactant solutions (Mitragotri *et al*. 1995).

## 5. Amplitude of oscillation

Figure 2*a* demonstrates the constraining effect that the surface coating can have upon bubble oscillations. From equation (3.8), it is clear that the degree of constraint depends upon the nature of the surfactant (*K*, *x*, *η*_{so} and *Z*) and the surface concentration (*Γ*_{0}). This may be important for interpreting acoustic emissions from bubble populations (Leighton *et al*. 2004) and in applications involving bubble destruction, e.g. drug delivery using microbubbles, where it is important to know the ultrasound pressure amplitude required for destruction. An additional consideration for drug delivery applications is the effect of the drug itself, which is normally incorporated into the bubble to form an additional surface layer that may be up to 500 nm thick (Harvey *et al*. 2002). For a Newtonian viscous surface layer, equation (3.8) becomes(5.1)where *R*_{1} is the radius of the gas core; *R*_{2}−*R*_{1} is the layer thickness; *V*_{s} is the layer volume; and *μ*_{s} its viscosity. The solution to equation (5.1) shown in figure 2*b* indicates that the additional layer has a damping effect upon the bubble oscillations, which would increase the incident pressure amplitude required for bubble destruction. It would also reduce the amplitude and harmonic content of the scattered signal from the bubbles making them more difficult to detect after injection, as well as the ‘ring-down’ for pulsed excitation, which may be important for certain imaging protocols. Further damping effects may be encountered depending on the solubility of the coating. The problem of a soluble surfactant was considered by Fyrillas & Szeri (1995, 1996) in the context of rectified diffusion and also by Glazman (1984), who found that mass transport between the surface layer and the surrounding liquid could significantly reduce the amplitude of oscillation.

## 6. Resonance frequency

Linearization of equation (3.8) enables an analytical expression for the resonance frequency to be obtained, from which the dependence upon the coating parameters can be clearly seen,(6.1)

As it is the nonlinear bubble behaviour that is frequently of most interest, however, it is appropriate to look at the frequency response of the numerical solution to equation (3.8). Figure 3 compares the maximum amplitude for different excitation frequencies of a free and coated bubble at different excitation pressures. As may be seen, the frequency at which the maximum amplitude occurs is significantly affected by the coating and also changes with increasing excitation pressure as the bubble oscillation becomes more nonlinear. Both of these effects represent important considerations in interpreting acoustic emissions from bubbles and optimizing excitation protocols.

## 7. Amplitude in tension and compression

The presence of the coating can also affect the relative amplitude of oscillation in tension and compression (i.e. *R*_{max} and *R*_{min}). The oscillations of a free gas bubble are inherently nonlinear (equation (2.2)) and the difference between *R*_{max} and *R*_{min} becomes more pronounced with increasing excitation pressure. For a coated bubble, this effect can be enhanced by the additional nonlinearity in the relationship between interfacial tension and surface concentration. Both the interfacial tension and surface viscosity terms in equation (3.8) may be nonlinear depending on the values of *x* and *Z*. In addition, there may exist upper and lower concentration limits for which the interfacial tension is respectively zero (corresponding to the buckling radius *R*_{x}) and equal to the ‘clean’ liquid–gas interfacial tension (Israelachvili 2003; Marmottant *et al*. 2005) and at which there will be sharp changes in the resistance to oscillation. Figure 4*a* shows such an asymmetric oscillation.

The significance of this behaviour is illustrated in figure 4*b*, which shows the frequency spectrum for the pressure radiated by an oscillating bubble for (*R*_{max})/(*R*_{min})=1 and <1. As may be seen, the harmonic content for the latter is higher. This may be usefully exploited in applications such as contrast-enhanced ultrasound imaging where the greater the harmonic content of the radiated signal, the better the signal-to-noise ratio. Deliberately selecting or engineering bubbles that display this type of behaviour will give increased contrast enhancement at low incident pressures. This is desirable in terms of both minimizing the risk of damage to the surrounding tissue and preserving the bubbles, especially if they are also being used for drug delivery. It should also be considered in applications where the frequency spectrum of the radiation from a bubble is used to infer its dynamic response such as cavitation detection.

This type of asymmetric behaviour has been observed experimentally (de Jong *et al*. 2007) although relating the results to theoretical models through parameters that can be measured independently still represents a considerable challenge. In principle, the relationship between interfacial tension and surface concentration from (*Γ*=0, *σ*=*σ*_{gas/liquid}) to (*Γ*=*Γ*_{critical}, *σ*=0) can be defined for a given surfactant using well-established experimental techniques from surface chemistry (e.g. Langmuir trough, micropipette aspiration). However, measurements of the relevant quantities (*x*, *K*, *η*_{so}, *Z*, *Γ*_{critical}, etc.) for surface films are normally carried out at forcing rates that are low compared with the rates considered here, particularly for dynamic behaviour, and the frequency dependence of these parameters is poorly understood. More direct approaches such as fitting data from acoustic measurements made in microbubble suspensions to theoretical curves are unsatisfactory because there are normally multiple solutions as well as high uncertainty (e.g. in the bubble size distribution). This results in a wide range of parameter values even for simple models (Gorce *et al*. 2000). Optical measurements of individual bubble radius–time curves also suffer from similar problems, although recent advances in simultaneous optical and acoustic measurements from single bubbles may overcome these (van der Meer *et al*. 2007).

The need for accurate experimental data as inputs for theoretical models raises a further question as to which type of model constitutes the most appropriate treatment for a given surface coating. The most obvious response is that it will depend upon the type of coating being considered, for example whether it is thick or thin in comparison with the bubble radius, it consists of a molecular monolayer (equation (3.8)), a more rigid polymer (Church 1995) or hyperelastic material (Allen & Rashid 2004), a viscous fluid (Sarkar *et al*. 2005), or it is multilayered (e.g. equation (5.1)). As noted by Hoff *et al*. (2000), however, the various forms of these models can be shown to be mathematically equivalent, e.g. in the limit of a thin shell or for a given set of parameter values. Moreover, while it is clear from experimental studies that a coating will impart both elastic and dissipative resistance, the precise nature of these effects are unlikely to relate, for example, to the behaviour of a bulk polymer or the low-frequency response of a thin film. It must therefore be concluded that there is a pressing need for improved understanding of the dynamic behaviour of surface coatings, particularly at high frequencies, not only to determine the inputs for theoretical models but also in formulating the models themselves.

## Acknowledgments

This work was supported by the Royal Academy of Engineering and the Engineering and Physical Sciences Research Council. The author would also like to thank Prof. Tony Harker, Dr Nader Saffari and Dr Alex Livshics for many helpful discussions.

## Footnotes

One contribution of 11 to a Theme Issue ‘New perspectives on dispersed multiphase flows’.

- © 2008 The Royal Society