A METHOD AND A DEVICE FOR ACOUSTIC ESTIMATION OF BUBBLE PROPERTIES

Abstract

Acoustical methods and an associated device, to estimate one or more properties of bubbles in a liquid like medium are provided. Principally, the acoustical method comprises acoustically exciting one or more bubbles in a liquid like medium to oscillate at a resonant frequency, detecting a first signal emitted from an acoustical source arranged to acoustically excite the one or more bubbles and detecting a second signal produced from the one or more bubble oscillations, deriving at least a first and a second characteristic by performing frequency domain analysis on the detected first and second signals, the first characteristic comprising a frequency interference minimum f.sub.1min and the second characteristic comprising a bubble resonance fundamental frequency maximum f.sub.1max and estimating one or more bubble properties from at least the first and second characteristics. Further provided are acoustical methods to estimate the equilibrium size and location of one or more bubbles in a liquid-like medium.

Claims

1. An acoustical method to estimate one or more properties of bubbles in a liquid like medium, the acoustical method comprising: acoustically exciting one or more bubbles in a liquid like medium to oscillate at a resonant frequency; detecting a first signal emitted from an acoustical source arranged to acoustically excite the one or more bubbles and detecting a second signal produced from the one or more bubble oscillations; deriving at least a first and a second characteristic by performing frequency domain analysis on the detected first and second signals, the first characteristic comprising a frequency interference minimum f.sub.1min and the second characteristic comprising a bubble resonance fundamental frequency maximum f.sub.1max; and estimating one or more bubble properties from at least the first and second characteristics.

2. The method according to claim 1, further comprising deriving a third characteristic comprising a second harmonic resonance response frequency f.sub.2max.

3. The method according to claim 1, where the step of acoustically exciting the one or more formed bubbles to oscillate at a resonant frequency comprises driving the acoustical source to generate one of a pulsed signal, a tone burst signal, a chirp signal and a broadband acoustic source signal.

4. The method according to claim 1, where the bubble property includes the bubble equilibrium radius R.sub.0, and where R.sub.0 and is estimated from f.sub.1max and f.sub.1min using the relationship: $R_0=\frac{1}{2.Math.\pi }.Math.\sqrt{\frac{1}{f_{1.Math.\max }^2}-\frac{1}{f_{1.Math.\min }^2}}.Math.\sqrt{\frac{3.Math.\kappa .Math..Math.p_0+\frac{2.Math.{\sigma }_0}{R_0}.Math.\left(3.Math.\kappa -1\right)+\frac{4.Math.{\chi }_0}{R_0}}{{\rho }_L.Math.E.Math..Math.\Theta }};\left(f_{1.Math.\min }>f_{1.Math.\max }\right)$ where Θ is a dimensionless coefficient defined by the relationship: Θ =ζΔ+√{square root over (1+ζ.sup.2Δ.sup.2)}, ζ is a coefficient defined by the: $ϛ=\frac{1+{\lambda }^2}{1-{\lambda }^2};\lambda =\frac{f_{1.Math.\max }}{f_{1.Math.\min }},$ E is a dimensionless coefficient defined by: $E=R_0\left(\frac{1}{r}+\frac{1}{r_{\mathrm{SB}}}\right),$ and Δ is a dimensionless coefficient defined by: $\Delta =\frac{16.Math.{\left(\mu +\frac{{\kappa }_S}{R_0}\right)}^2}{R_0^2.Math.{\rho }_L\left[3.Math.\kappa .Math..Math.p_0+\frac{2.Math.{\sigma }_0}{R_0}.Math.\left(3.Math.\kappa -1\right)+\frac{4.Math.{\chi }_0}{R_0}\right].Math.E},$ where the gas polytropic index κ, ambient pressure p.sub.0, surface tension at equilibrium bubble radius σ.sub.o, elastic compression modulus χ.sub.0, liquid viscosity μ, liquid density ρ.sub.L, and encapsulating layer dilatational viscosity κ.sub.s are predetermined, the distance r between the bubble and receiver and distance r.sub.SB between the source and bubble are approximated, and where the bubble is either free or encapsulated.

5. The method according to claim 4, where the attached solids mass loading M.sub.s is estimated from f.sub.1max and f.sub.1min and R.sub.0 using the relationship: $M_S=\frac{R_0}{\delta }\left[\frac{3.Math.\kappa .Math..Math.p_0+\frac{2.Math.{\sigma }_0}{R_0}.Math.\left(3.Math.\kappa -1\right)+\frac{4.Math.{\chi }_0}{R_0}}{4.Math.{\pi }^2.Math.{\rho }_L.Math.f_{1.Math.\max }^2}-R_0^2+\frac{R_0^2}{2}.Math.E\left(1-\Theta \right)\right],$ where the solids density coefficient δ is defined $\delta =\frac{1}{4.Math.\pi }.Math.\left(\frac{1}{{\rho }_L}-\frac{1}{{\rho }_S}\right)$ and σ.sub.s is the solid density.

6. The method according to claim 4, where the attached solids mass loading M.sub.s is estimated from f.sub.1max and f.sub.1min and R.sub.0 using the relationship: $M_S=\frac{R_0}{\delta }\left[\left(\frac{3.Math.\kappa .Math..Math.p_0+\frac{2.Math.{\sigma }_0}{R_0}.Math.\left(3.Math.\kappa -1\right)+\frac{4.Math.{\chi }_0}{R_0}}{4.Math.{\pi }^2.Math.{\rho }_L}\right).Math.\left(\frac{\frac{1}{f_{1.Math.\max }^2}+\frac{1}{f_{1.Math.\min }^2}}{2}\right)-R_0^2\left(1-\frac{E}{2}\right)\right].$

7. The method according to claim 4, where the encapsulating layer dilatational viscosity κ.sub.s is estimated from f.sub.1max and f.sub.1min and R.sub.0 using the relationship: ${\kappa }_S=R_0.Math.\left\{\frac{R_0}{4}.Math.\sqrt{\left(\frac{{\Theta }^2-1}{2.Math.\mathrm{\Theta ϛ}}\right).Math.E.Math..Math.{\rho }_L\left[3.Math.\kappa .Math..Math.p_0+\frac{2.Math.{\sigma }_0}{R_0}.Math.\left(3.Math.\kappa -1\right)+\frac{4.Math.{\chi }_0}{R_0}\right]}-\mu \right\},$ where the dimensionless coefficient Θ is expressed as: $\Theta =\left[\frac{3.Math.\kappa .Math..Math.p_0+\frac{2.Math.{\sigma }_0}{R_0}+\frac{4.Math.{\chi }_0}{R_0}}{4.Math.{\pi }^2.Math.{\rho }_L.Math.{\mathrm{ER}}_0^2}\right].Math.\left(\frac{1}{f_{1.Math.\max }^2}-\frac{1}{f_{1.Math.\min }^2}\right).$

8. The method according to claim 1, where the bubble property includes the bubble equilibrium radius R.sub.0, and R.sub.0 is estimated from f.sub.1max using the relationship: $R_0=\frac{1}{2.Math.\pi .Math..Math.f_{1.Math.\max }}.Math.\sqrt{\frac{3.Math.\kappa .Math..Math.p_0+\frac{2.Math.{\sigma }_0}{R_0}.Math.\left(3.Math.\kappa -1\right)+\frac{4.Math.{\chi }_0}{R_0}}{{\rho }_L\left[1-\frac{E}{2}.Math.\left(1-\Theta \right)\right]}},$ where Θ is a dimensionless coefficient defined by the relationship: Θ=ζΔ+√{square root over (1+ζ.sup.2Δ.sup.2)}, ζ is a coefficient defined by the relationship: $ϛ=\frac{1+{\lambda }^2}{1-{\lambda }^2};\lambda =\frac{f_{1.Math.\max }}{f_{1.Math.\min }},$ E is a dimensionless coefficient defined by the relationship: $E=R_0\left(\frac{1}{r}+\frac{1}{r_{\mathrm{SB}}}\right),$ and Δ is a dimensionless coefficient defined by the relationship: $\Delta =\frac{16.Math.{\left(\mu +\frac{{\kappa }_S}{R_0}\right)}^2}{R_0^2.Math.{\rho }_L\left[3.Math.\kappa .Math..Math.p_0+\frac{2.Math.{\sigma }_0}{R_0}.Math.\left(3.Math.\kappa -1\right)+\frac{4.Math.{\chi }_0}{R_0}\right].Math.E},$ and where the gas polytropic index κ, ambient pressure p.sub.0, surface tension at equilibrium bubble radius σ.sub.o, elastic compression modulus χ.sub.o, liquid viscosity μ, liquid density σ.sub.L and encapsulating layer dilatational viscosity κ.sub.s are predetermined and the distance r between the bubble and receiver and distance r.sub.SB between the source and bubble are approximated, and where the bubble is a clean unloaded bubble.

9. The method according to claim 1, where the bubble property includes the bubble equilibrium radius R.sub.0, and R.sub.0 is estimated from f.sub.1max and f.sub.1min using the relationship: $R_0=\frac{1}{2.Math.\pi }.Math.\sqrt{\frac{1}{2}.Math.\left(\frac{1}{f_{1.Math.\max }^2}+\frac{1}{f_{1.Math.\min }^2}\right)}.Math.\sqrt{\frac{3.Math.\kappa .Math..Math.p_0+\frac{2.Math.{\sigma }_0}{R_0}.Math.\left(3.Math.\kappa -1\right)+\frac{4.Math.{\chi }_0}{R_0}}{{\rho }_L\left[1-\frac{E}{2}\right]}},$ where the gas polytropic index κ, ambient pressure p.sub.0, surface tension σ.sub.o, elastic compression modulus χ.sub.0, and liquid density σ.sub.L are predetermined and the distance r between the bubble and receiver and distance r.sub.SB between the source and bubble are approximated, and where the bubble is a clean unloaded bubble.

10. The method according to claim 1, where the encapsulating layer dilatational viscosity κ.sub.s is estimated from f.sub.1max and f.sub.1min and known or estimated equilibrium radius R.sub.0 using the relationship: ${\kappa }_S=R_0.Math.\left\{\frac{R_0}{4.Math.ϛ}.Math.\sqrt{\frac{{\rho }_L}{2}\left[3.Math.\kappa .Math..Math.p_0+\frac{2.Math.{\sigma }_0}{R_0}.Math.\left(3.Math.\kappa -1\right)+\frac{4.Math.{\chi }_0}{R_0}\right]\left[\frac{{\left(2-E\right)}^2-{ϛ}^2.Math.E^2}{2-E}\right]}-\mu \right\},$ where the liquid viscosity μ and density σ.sub.L, gas polytropic index κ and bubble surface tension parameters are predetermined, and r and r.sub.SB are approximated, and where the bubble is a ‘clean’ (unloaded) bubble.

11. (canceled)

12. The method according to claim 1, where R.sub.0 is estimated from f.sub.1max and f.sub.1min via the relationship: $R_0=\frac{1}{2.Math.\pi .Math..Math.f_{1.Math.\max }}.Math.\sqrt{\frac{3.Math.\kappa .Math..Math.p_0+\frac{2.Math.{\sigma }_0}{R_0}.Math.\left(3.Math.\kappa -1\right)}{{\rho }_L}}.Math..Math.\mathrm{or}.Math..Math.R_0=\frac{1}{2.Math.\pi .Math..Math.f_{1.Math.\min }}.Math.\sqrt{\frac{3.Math.\kappa .Math..Math.p_0+\frac{2.Math.{\sigma }_0}{R_0}.Math.\left(3.Math.\kappa -1\right)}{{\rho }_L\left(1-E\right)}},$ where $E=1-{\left(\frac{f_{1.Math.\max }}{f_{1.Math.\min }}\right)}^2,$ and where the gas polytropic index κ, ambient pressure p.sub.0, surface tension σ.sub.o, and density σ.sub.L are predetermined, and where there are nil attached solids, for a free bubble and negligible liquid viscosity effects on the bubble characteristics.

13. The method according to claim 12, wherein the attached solids mass loading M.sub.s is estimated in the case of a free bubble and negligible liquid viscosity effects on the bubble characteristics using the relationship: $M_S=\frac{R_0}{\delta }.Math.\left(\frac{3.Math.\kappa .Math..Math.p_0+\frac{2.Math.{\sigma }_0}{R_0}.Math.\left(3.Math.\kappa -1\right)}{4.Math.{\pi }^2.Math.{\rho }_L.Math.f_{1.Math.\max }^2}-R_0^2\right),$ where $\delta =\frac{1}{4.Math.\pi }.Math.\left(\frac{1}{{\rho }_L}-\frac{1}{{\rho }_S}\right)$ and where σ.sub.s, the density of a single solid particle attached to the bubble surface is predetermined.

14. The method according to claim 12, wherein M.sub.s is estimated using the relationship: $M_S=R_0.Math.\frac{\left[3.Math.\kappa .Math..Math.p_0+\frac{2.Math.{\sigma }_0}{R_0}.Math.\left(3.Math.\kappa -1\right)\right]}{4.Math.{\pi }^2.Math.{\rho }_L.Math.\delta }\left[\frac{1}{f_{1.Math.\max }^2}-\frac{1}{E}.Math.\left(\frac{1}{f_{1.Math.\max }^2}-\frac{1}{f_{1.Math.\min }^2}\right)\right],$ where $\delta =\frac{1}{4.Math.\pi }.Math.\left(\frac{1}{{\rho }_L}-\frac{1}{{\rho }_S}\right)$ and where σ.sub.s, the density of a single solid particle attached to the bubble surface is predetermined.

15-18 (canceled)

19. A device to estimate one or more properties of bubbles in a liquid or liquid like medium, the device comprising: a chamber or vessel to contain or enable passage of a liquid or liquid like medium, the liquid or liquid like medium supporting one or more bubbles; at least one acoustic source configured to acoustically excite the one or more bubbles to oscillate at a resonant frequency; at least one broadband acoustic detector to detect a first signal emitted from the acoustic source and to detect a second signal produced from the bubble oscillations; and control means to (i) derive at least a first and a second characteristic by performing frequency domain analysis on the detected first and second signals, the first characteristic comprising a frequency interference minimum f.sub.1min and the second characteristic comprising a bubble resonance fundamental frequency maximum f.sub.1max; and (ii) estimate one or more bubble properties from at least the first and second characteristics.

20. The device according to claim 19 wherein the control means is operable to derive a third characteristic comprising a second harmonic resonance response frequency f.sub.2max

21. The device according to claim 19, wherein the control means is operable to perform frequency domain analysis on the detected first and second signals, or the first, second and third signals in order to determine the first f.sub.1min and second characteristic f.sub.1max, or first f.sub.1min, second f.sub.max, and third characteristics f.sub.2max.

22. The device according to claim 19, further comprising a plurality of acoustic sources configured to operate coherently in an array.

23. The device according to claim 19, wherein the or each acoustic source is situated either (i) on an interior wall of the chamber, (ii) on an exterior wall of the chamber or (iii) within the body of liquid containing bubbles.

24-32. (canceled)

33. An acoustical method to estimate the equilibrium size, and location of at least one unloaded bubble in a liquid-like medium, the acoustical method comprising: acoustically exciting one or more bubbles in a liquid like medium to oscillate at a resonant frequency; detecting a first signal emitted from an acoustic source and arranged to acoustically excite the one or more bubbles and detecting a second signal produced from the one or more bubble oscillations; deriving at least a first, a second and a third characteristic by performing frequency domain analysis on the detected first and second signals, the first characteristic comprising a frequency interference minimum f.sub.1min, the second characteristic comprising a bubble resonance fundamental frequency maximum f.sub.1max and the third characteristic comprising a second harmonic resonance response frequency f.sub.2max; estimating R.sub.o from each of the three characteristics based on a priori knowledge of the bubble surface dilatational viscosity, liquid viscosity (p) and the density of the liquid-like medium (ρ.sub.Sl); and estimating the location of the at least one bubble using R.sub.o.

34. (canceled)

35. An acoustical method to estimate the equilibrium size, attached solids mass loading and location of at least one loaded bubble in a liquid-like medium, the acoustical method comprising: acoustically exciting one or more bubbles in a liquid like medium to oscillate at a resonant frequency; detecting a first signal emitted from an acoustic source and arranged to acoustically excite the one or more bubbles and detecting a second signal produced from the one or more bubble oscillations; deriving at least a first, a second and a third characteristic by performing frequency domain analysis on the detected first and second signals, the first characteristic comprising a frequency interference minimum f.sub.1min, the second characteristic comprising a bubble resonance fundamental frequency maximum f.sub.1max and the third characteristic comprising a second harmonic resonance response frequency f.sub.2max; estimating R.sub.o from each of the three characteristics based on a priori knowledge of the bubble surface dilatational viscosity, liquid viscosity (μ), bubble surface tension (σ)), bubble gas polytropic index (κ) and the ambient pressure of the liquid-like medium (p.sub.0); estimating the attached solids mass loading M.sub.s using R.sub.o and using M.sub.sto estimate the location of said one or more bubbles.

Description

BRIEF DESCRIPTION OF DRAWINGS

[0136] In order that the present invention may be more clearly ascertained, embodiments will now be described, by way of example, with reference to the accompanying drawing, in which:

[0137] FIG. 1 is a schematic of the conceptual model for a single gaseous bubble surrounded by an encapsulating elastic layer coated with attached solids in a liquid, subject to active acoustic stimulation by a source and monitoring by a receiver;

[0138] FIG. 2 is a graph which illustrates example curves for the mass loading factor as a function of the fractional oscillation radius about the equilibrium bubble size;

[0139] FIG. 3 is a graph of the receiver pressure power spectra as a function of receiver frequency near the free bubble fundamental resonance frequency (1 mm equilibrium bubble radius and 0, 1 and 10 mg attached solids mass loading);

[0140] FIG. 4 is a graph of the receiver pressure power spectra as a function of receiver frequency near the free bubble second harmonic resonance frequency (1 mm equilibrium bubble radius and 0, 1 and 10 mg attached solids mass loading);

[0141] FIG. 5 is a graph of the receiver pressure power spectra as a function of receiver frequency near the encapsulated microbubble fundamental resonance frequency (1 μm equilibrium bubble radius and 0, 10 and 100 pg attached solids mass loading);

[0142] FIG. 6 is a graph of the receiver pressure power spectra as a function of receiver frequency near the encapsulated microbubble second harmonic resonance frequency (1 μm equilibrium bubble radius and 0, 10 and 100 pg attached solids mass loading);

[0143] FIG. 7 is a contour plot of the frequency of the maximum (near the resonance fundamental frequency) in the acoustic receiver average pressure power spectrum as a function of the equilibrium bubble radius and attached solids mass loading (0.25-2.5 mm equilibrium bubble radius and 0-10 mg attached solids mass loading);

[0144] FIG. 8 is a contour plot of the frequency of the minimum (near the resonance fundamental frequency) in the acoustic receiver average pressure power spectrum as a function of the equilibrium bubble radius and attached solids mass loading (0.25-2.5 mm equilibrium bubble radius and 0-10 mg attached solids mass loading);

[0145] FIG. 9 is a contour plot of bubble radius as a function of the frequencies of the resonance minimum and maximum received average acoustic power (1.8-5.0 kHz resonance minimum and 1.3-5.0 kHz resonance maximum);

[0146] FIG. 10 is a contour plot of bubble attached solids mass loading as a function of the frequencies of the resonance minimum and the difference between the frequencies of the maximum and minimum received average acoustic power (1.3-5.0 kHz resonance maximum and 0.0-0.6 kHz difference between resonance maximum and minimum);

[0147] FIG. 11a shows a graph of the power spectrum of the total acoustic response of a bubble (˜0.9 mm equilibrium radius and ˜0.85 mg attached solids) insonated by a sweep acoustic signal;

[0148] FIG. 11b shows a graph of the bubble response power spectrum of FIG. 6a, normalized by the background (bubble absent) power spectrum;

[0149] FIG. 12 is a graph of a waterfall plot of normalised power spectra of the total acoustic response of a single rising stream of bubbles of similar size insonated by an appropriate repeated burst acoustic signal. Power spectra as a function of frequency and bubble production rate;

[0150] FIG. 13 is a graph of the power spectrum of the total acoustic response of a cloud or swarm of rising bubbles of similar size insonated by an appropriate repeated burst acoustic signal.

[0151] FIG. 14 illustrates a schematic diagram of an acoustic spectrometer in accordance with one embodiment of the invention;

[0152] FIG. 15a schematically illustrates a top view of an acoustic spectrometer in accordance with a further embodiment of the invention;

[0153] FIG. 15b schematically illustrates a front elevation section A-A of the acoustic spectrometer shown in FIG. 4A; and

[0154] FIG. 16 schematically illustrates a top view of an acoustic spectrometer in accordance with a still further embodiment of the invention.

DESCRIPTION OF EMBODIMENTS

[0155] Embodiments generally relate to methods and apparatus utilising acoustic spectroscopy to measure various properties of bubbles in a liquid.

[0156] Throughout this specification, the term ‘free bubble’ refers to cavities filled with air or other gases or gas vapour from a surrounding liquid. Free bubbles have no artificial boundaries to prevent leakage of air or gas from the bubble itself, as a result they tend to be unstable. Free bubbles may float to the top of the liquid and disappear under the influence of gravitation force or may be dissolved into the liquid due to surface tension. A free bubble may be loaded/coated whether partially or fully with solid particles or it may be unloaded/uncoated. In contrast, the term ‘encapsulated bubble’ refers to a bubble with an encapsulating elastic shell which prevents fast gas dissolution and renders the bubble stable. An encapsulated bubble may similarly be loaded/coated with solid particles or it may be unloaded/uncoated. In the case where the encapsulated bubble is coated, the solid particles are typically embedded in the encapsulating elastic material. Technologies responsible for the formation of such encapsulating shells will be appreciated by those skilled the art. The term ‘clean bubble’ refers to a bubble which may be free or encapsulated, but which has no loading.

[0157] Embodiments generally operate on the principle that a gas bubble in a liquid, when insonated with acoustic energy from a source, will exhibit a resonant response which varies according to the frequency and magnitude of the insonant acoustic energy, and depends on various properties of the bubble, any encapsulating plastic layer, any solid particle loading and the surrounding liquid medium. The acoustic signal transmitted from the bubble, as well as the source signal, may then be detected by a receiver and analysed to determine certain properties of the insonated bubble. Fourier frequency power spectral analysis may be applied to the received acoustic signal, and peaks may be identified that are associated with the fundamental and second harmonic resonant frequencies of the bubble, and in some embodiments, also with the interference minimum which occurs due to the superposition of the source and bubble acoustic waves at the position of the receiver.

[0158] Once the signal analysis has determined the fundamental resonance frequency f.sub.1max, the interference minimum f.sub.1min, and the second harmonic maximum f.sub.2max, these values can be used to estimate certain properties of the insonated bubble, such as the equilibrium radius R.sub.0, the attached solids mass loading M.sub.s, and the encapsulating layer dilatational viscosity K.sub.s. Error bounds can be found on the estimates of the various bubble properties by a variety of methods. These include the use of error propagation formulae based on the known analytical solutions for each bubble property and variances of the estimates of each of the dependent variables in the same equations (receiver frequency characteristics, properties of the gas, any encapsulating plastic layer, solid particle loading, surrounding liquid medium and the system geometry). Another approach could be to use Monte Carlo methods to provide error estimates based on injecting random errors into the dependent variables for the various bubble properties. The accuracy of the estimates of the bubble properties can be improved by increasing the frequency resolution and acoustic power sensitivity of the receiver spectral data. Decreasing the distances between the source, bubble(s) and receiver(s) for smaller bubble sizes and increasing the power of the source signal (still maintaining mild acoustic excitation) can also increase the certainty of the bubble parameter estimates. An optimal form of source signal will provide uniform acoustic excitation across the range of frequencies for bubbles in the ranges of equilibrium size, attached solids mass loading and surface layer elasticity expected for the application of interest. The bubble properties can be estimated using the equations derived below from a theoretical model of the acoustic response of an insonated bubble.

[0159] Bubble Radius Non-Linear Forced Oscillation Model

[0160] Model Formulation

[0161] FIG. 1 is a schematic of the conceptual model 100 for a single gaseous encapsulated bubble 102 coated with attached solids in a liquid 104, subject to active acoustic stimulation by a source and monitoring by a receiver (not shown). A solids loaded bubble subject to a low driving sound field at resonance frequency undergoes simple harmonic oscillation about the mean radius R.sub.0 106, between the minimum bubble radius 108 and the maximum bubble radius 110. The boundary between the bubble gas 102 and surrounding pseudo-solid layer 112 consists of a thin, elastic monolayer shell 114 which encapsulates the bubble; for micron sized bubbles allowing it to persist for an extended period of time. An alternative viewpoint and that illustrated in FIG. 1 is that the pseudo-solid layer itself 112 is composed of solid particles embedded in encapsulating elastic material 114. As should be appreciated, in the case of a free bubble the boundary between the bubble gas and the surrounding pseudo-solid layer is a direct phase interface.

[0162] The bubble wall is at radius R(t) at any time (equilibrium radius R.sub.0) and is surrounded by a layer of pseudo-solid of thickness ε(t) and density σ.sub.att(t) containing solid particles of density σ.sub.s attached to the bubble surface and incompressible liquid of density σ.sub.L in the interstices. Some key model assumptions are as follows:

[0163] a) Attached solid particle size is significantly smaller than the bubble size.

[0164] b) The pseudo-solid layer is significantly smaller than the bubble size.

[0165] c) The attached solids are evenly spread over the surface of the bubble.

[0166] d) The mass of solids attached to the bubble is assumed to be constant at all times.

[0167] e) Outside of the pseudo-solid layer, the incompressible liquid extends to infinite distance.

[0168] f) There is a straight line from the acoustic source, through the bubble to the acoustic receiver, such that the acoustic ray path from source to receiver includes that from bubble to receiver. Hence the receiver may detect both the acoustic source and bubble response in transmission.

[0169] It should be noted that the effects of gravity and bubble motion relative to the slurry on the shapes of both the bubble and pseudo-solid layer are not considered in this analysis.

[0170] A bubble subject to a driving acoustic pressure field behaves as a non-linear oscillator. A modified form of Rayleigh-Plesset equation can be derived for non-linear spherical oscillations of a single bubble in the case of a possible encapsulating elastic shell, an attached mass of solids as particles in a surrounding pseudo-solid layer, and sinusoidal forcing by an acoustic source.

[0171] The kinetic energy acquired by liquid surrounding a bubble of equilibrium radius R.sub.0 as it changes to radius R(t) due to an applied pressure field P(t) can be written as follows:

[00053]$\begin{array}{cc}{\phi }_k=\frac{1}{2}.Math.{\int }_R^{R+\mathrm{.Math.}}.Math.{\stackrel{.}{r}}^2.Math.{\rho }_{\mathrm{att}}.Math.4.Math.\pi .Math..Math.r^2.Math.\mathrm{dr}+\frac{1}{2}.Math.{\int }_{R+\mathrm{.Math.}}^{\infty }.Math.{\stackrel{.}{r}}^2.Math.{\rho }_L.Math.4.Math.\pi .Math..Math.r^2.Math.\mathrm{dr}& \left(1\right)\end{array}$

[0172] The thickness of the pseudo-solid layer ε(t) can be written as follows:

[00054]$\begin{array}{cc}\mathrm{.Math.}=\frac{M_S}{4.Math.\pi .Math..Math.{\rho }_S.Math.R^2.Math.{\delta }_S}& \left(2\right)\end{array}$

[0173] Here δ.sub.s is the attached solids volume fraction and σ.sub.att is the attached pseudo-solid density. The attached layer and pure solids densities are related by δ.sub.Sσ.sub.S=φ.sub.Sσ.sub.att where φ.sub.s is the attached solids mass fraction. They are also subject to the constituent relation σ.sub.att=σ.sub.Sδ.sub.S+σ.sub.L(1−δ.sub.S). Both the pseudo-solid and surrounding liquid layers are assumed incompressible. This leads to the following incompressibility condition equating the liquid flow at any radial position exterior to the bubble to the flow at the bubble wall:

[00055]$\begin{array}{cc}\frac{\stackrel{.}{r}}{\stackrel{.}{R}}=\frac{R^2}{r^2}& \left(3\right)\end{array}$

[0174] Introducing Eqns. (2) and (3) into Eqn. (1) leads to the following expression for the kinetic energy of the liquid surrounding the bubble in terms of the radius of the bubble and the thickness of the pseudo-solid layer at any time:

[00056]$\begin{array}{cc}{\phi }_k=2.Math.\pi .Math..Math.R^3.Math.{\stackrel{.}{R}}^2.Math.{\rho }_l\left(\frac{1+\lambda .Math..Math.\mathrm{.Math.}/R}{1+\mathrm{.Math.}/R}\right)& \left(4\right)\end{array}$

[0175] Here the ratio of pseudo-solid to liquid density is defined by:

[00057]$\begin{array}{cc}\lambda =\frac{{\rho }_{\mathrm{att}}}{{\rho }_L}& \left(5\right)\end{array}$

[0176] The assumption of a ‘thin’ pseudo-solid layer at any time (ε/R<<1) then leads to the following expression for the kinetic energy of the liquid surrounding the bubble:

φ.sub.k≈2πR.sup.3{dot over (R)}.sup.2σ.sub.LΓ.sub.p   (6)

[0177] Here the mass loading factor Γ.sub.p is defined as:

[00058]$\begin{array}{cc}{\Gamma }_p=1+\left(\lambda -1\right).Math.\frac{\mathrm{.Math.}}{R}& \left(7\right)\end{array}$

[0178] Equation (7) can also be written using Eqn. (5) and the constituent relation between attached pseudo-solid and purse solid densities as:

[00059]$\begin{array}{cc}{\Gamma }_p=1+\left(\frac{{\rho }_S}{{\rho }_L}-1\right).Math.\frac{M_S}{4.Math.\pi .Math..Math.{\rho }_S.Math.R^3}& \left(8\right)\end{array}$

[0179] The difference between the work done remote from the bubble by the pressure p.sub.∞=p.sub.0+P(t) (an applied forcing acoustic pressure P(t) and ambient pressure P.sub.0) and the work done by the pressure p.sub.L in the slurry just outside the bubble wall is given by:

[00060]$\begin{array}{cc}{\phi }_p={\int }_{R_0}^R.Math.\left(p_L-p_{\infty }\right).Math.4.Math.\pi .Math..Math.R^2.Math.\mathrm{dR}& \left(9\right)\end{array}$

[0180] At any time the kinetic energy acquired by the liquid is equal to the difference in work done during the process.

φ.sub.k=φ.sub.p   (10)

[0181] Two cases are considered for the effect of the bubble oscillatory radius on the properties of the attached solids loading layer: a) a monolayer of particles spherically symmetrically arranged on the bubble surface; and b) a ‘thin’ multilayer of particles packed on top of each other and spherically symmetrically arranged on the bubble surface.

[0182] In the case of (a) the attached solids volume fraction δ.sub.S varies as the bubble expands and contracts (δ.sub.S∝R.sup.−2), but the pseudo-solid layer thickness ε=2R.sub.p is a constant equal to twice the radius R.sub.p of the individual solid particles. This implies that:

[00061]$\begin{array}{cc}\frac{\partial \mathrm{.Math.}}{\partial R}=0& \left(11\right)\end{array}$

[0183] In the base of (b), a ‘thin’ multilayer of particles packed on top of each other and spherically symmetrically arranged on the bubble surface. In this case the pseudo-solid layer thickness ε varies as the bubble expands and contracts but the attached solids volume fraction δ.sub.s is a constant. This case may more closely model the situation of solids particles embedded in an elastic layer. Using Eqn. (2) and the assumption that the mass of the attached solids is a constant leads to the following relationship:

[00062]$\begin{array}{cc}\frac{\partial \mathrm{.Math.}}{\partial R}=-\frac{2.Math.\mathrm{.Math.}}{R}& \left(12\right)\end{array}$

[0184] In both cases however, it can be shown from Eqn. (7) that the dependence of the mass loading factor on the bubble oscillatory radius is the same. This can also be directly established from Eqn. (8), which applies in both cases. The derivative of the mass loading factor with respect to the bubble oscillatory radius can be written in both cases as:

[00063]$\begin{array}{cc}\frac{\partial {\Gamma }_p}{\partial R}=-\frac{3}{R}.Math.\left({\Gamma }_p-1\right)& \left(13\right)\end{array}$

[0185] Equations (6) and (9) are substituted into Eqn. (10), which is then differentiated with respect to bubble oscillatory radius. Taking into account Eqn. (13) leads to the following relationship which is valid for both monolayer and ‘thin’ multilayer attached solids loading:

[00064]$\begin{array}{cc}\frac{p_L-p_{\infty }}{{\rho }_L}={\Gamma }_p.Math.R.Math.\stackrel{\mathrm{.Math.}}{R}+\frac{3}{2}.Math.{\stackrel{.}{R}}^2& \left(14\right)\end{array}$

[0186] The liquid pressure p.sub.L, in the slurry just outside the bubble wall is found from the gas pressure inside the bubble via a boundary condition across the bubble wall interface. Taking into account the gas polytropic index κ, surface tension σ, liquid viscosity ν and the surface dilatational viscosity κ.sub.s of any elastic layer, the liquid pressure just outside the bubble may be written as:

[00065]$\begin{array}{cc}{p}_L=\left(p_0+\frac{2.Math.{\sigma }_0}{R_0}\right).Math.{\left(\frac{R_0}{R}\right)}^{3.Math.\kappa }-\frac{2.Math.\sigma \left(R\right)}{R}-\frac{4.Math.\mu .Math..Math.\stackrel{.}{R}}{R}-\frac{4.Math.{\kappa }_S.Math.\stackrel{.}{R}}{R^2}& \left(15\right)\end{array}$

[0187] where σ.sub.0=σ(R.sub.0) is the surface tension at equilibrium bubble radius.

[0188] Equation (15) and the expression p.sub.∞=p.sub.0=P(t) for the total local pressure (ambient plus acoustic forcing) are substituted into Eqn. (14). The result is the following modified form of Rayleigh-Plesset ordinary differential equation for non-linear spherical oscillations of a single bubble in a viscous liquid, in the case of an encapsulating elastic layer and an outer pseudo-solid attached layer, subject to acoustic excitation:

[00066]$\begin{array}{cc}{\Gamma }_p\left(R\right).Math.R.Math.\stackrel{\mathrm{.Math.}}{R}+\frac{3}{2}.Math.{\stackrel{.}{R}}^2=\frac{1}{{\rho }_L}\left[\left(p_0+2.Math.{\sigma }_0.Math.\text{/}.Math.R_0\right).Math.{\left(\frac{R_0}{R}\right)}^{3.Math.\kappa }-\frac{2.Math.\sigma \left(R\right)}{R}-\frac{4.Math.\mu .Math..Math.\stackrel{.}{R}}{R}-\frac{4.Math.{\kappa }_S.Math.\stackrel{.}{R}}{R^2}-p_0-P\left(t\right)\right]& \left(16\right)\end{array}$

[0189] Equation (16) in the case of nil attached solids mass loading and nil liquid and elastic layer viscosities reverts to the Rayleigh-Plesset equation, which is well known to be highly non-linear in the case of strong forcing.

[0190] The bubble radius dependent surface tension coefficient 6(R) in Eqn. (16) for an elastic layer encapsulated bubble is modelled as being linearly proportional to an elastic compression modulus χ.sub.0 and the difference in bubble radius from the R.sub.0 equilibrium value:

[00067]$\begin{array}{cc}\sigma \left(R\right)={\sigma }_0+2.Math.{\chi }_0\left(\frac{R}{R_0}-1\right)& \left(17\right)\end{array}$

[0191] Equation (17) is in accordance with an elastic regime surface tension model for small oscillations of an encapsulated bubble. In the case of a free bubble, Eqn. (17) reverts to a surface tension relevant to the equilibrium bubble size.

[0192] A monochromatic sinusoidal pressure forcing term can be written as:

P(t)=−p.sub.oη cos(Ωt)   18)

[0193] Here η is the amplitude of the acoustic forcing of the bubble relative to the ambient background pressure at the bubble location, n is the angular frequency of the pressure forcing and t is unscaled time. It should be noted that sinusoidal acoustic excitation is used in this analysis as it is a relatively tractable form of forcing (for perturbation analysis) that assists in demonstrating key results.

[0194] Model Solution by Perturbation Method

[0195] An approach to the solution of Eqn. (15) that can provide insight into key relationships between acoustic features and bubble physical characteristics for mild acoustic excitation is based on finding approximate analytical solutions for steady oscillations. A linearisation method is here employed to render the modified Rayleigh-Plesset equation tractable for analytical solution in the case of small amplitude excitation, resulting in the separation of the bubble response into fundamental and second harmonic steady oscillatory solutions.

[0196] The first step in the linearization procedure is an examination of the dependence of the attached solids mass loading nonlinearity factor Γ.sub.p on the amplitude of bubble oscillations. The radial oscillation of the bubble as a fraction x of the bubble equilibrium radius R.sub.0 is defined by:

R=R.sub.0(1+x)   (19)

[0197] FIG. 2 shows example curves for the mass loading factor Γ.sub.p (dimensionless) as a function of the fractional oscillation radius about the equilibrium bubble size. This example is relevant both for a bubble of size R.sub.0=1 mm loaded with solid particles of mass M.sub.s=1 mg or equivalently a microbubble of size R.sub.0=1 μm loaded with solid particles of mass M.sub.s=1 pg (see Eqn. (8)). Liquid and solid densities σ.sub.L=1000 kg m.sup.−3 and σ.sub.s=2200 kg m.sup.−3, respectively. The solid line 210 represents the variation of the mass loading nonlinearity factor (Eqn. (8)) with bubble oscillatory radius. The dotted line 220 is a characteristic (constant) value of the mass loading nonlinearity factor relevant to the equilibrium bubble radius. The characteristic value of the mass loading nonlinearity is only ˜5% larger than the unity value for an unloaded bubble. For large values of the oscillatory fractional radius (x˜1), the nonlinearity factor asymptotes towards unity (see Eqn. (8)). This means that as the bubble expands from its equilibrium size, the attached solids mass has less influence on the oscillatory dynamics of the bubble. On the other hand, as the bubble contracts from its equilibrium size (x<0), the mass loading factor quickly increases. This means that the bubble oscillatory dynamics are asymmetric with respect to the bubble equilibrium radius and are strongly affected by the solids mass loading during the contraction stage for large oscillations. The characteristic value for the mass loading nonlinearity factor relevant to the equilibrium bubble size is here adopted for subsequent analysis of the bubble oscillatory dynamics, with the caveat that this is a reasonable approximation only for small amplitude bubble oscillations (−0.25≦x≦0.25). This seems to be a reasonable first approximation for the mass loading non-linearity term associated with either an I(1 mm) radius bubble loaded with O(1 mg) solids mass or equivalently an O(1 μm) radius bubble loaded with O(1 pg) solids mass, both being situations of potential relevance to this study.

[0198] For the case of small amplitude forcing and bubble acoustic response, the modified Rayleigh-Plesset equation describing the non-linear oscillations of a single bubble of radius R in a viscous liquid with an encapsulating elastic layer and an outer pseudo-solid attached layer can be approximated as follows:

[00068]$\begin{array}{cc}{\Gamma }_p^{*}.Math.R.Math.\stackrel{\mathrm{.Math.}}{R}+\frac{3}{2}.Math.{\stackrel{.}{R}}^2=\frac{1}{{\rho }_L}\left[\left(p_0+2.Math.{\sigma }_0.Math.\text{/}.Math.R_0\right).Math.{\left(\frac{R_0}{R}\right)}^{3.Math.\kappa }-\frac{2.Math.\sigma \left(R\right)}{R}-\frac{4.Math.\mu .Math..Math.\stackrel{.}{R}}{R}-\frac{4.Math.{\kappa }_S.Math.\stackrel{.}{R}}{R^2}-p_0-P\left(t\right)\right]& \left(20\right)\end{array}$

[0199] Here Γ.sub.p* is the characteristic value of the mass loading nonlinearity factor relevant to the bubble equilibrium radius, as given by Eqn. (8) for Γ.sub.p*=Γ.sub.p(R.sub.0). Again, Eqn. (16) defines the surface tension for elastic oscillations of an encapsulated bubble.

[0200] Equation (20) (incorporating Eqns. (17) and (18)) is analytically solved for steady oscillatory motion of the bubble radius using a regular perturbation approach with second order accuracy. The same approach has previously been used to analyse induced non-linear oscillations of unloaded bubbles modelled via the Rayleigh-Plesset equation under a variety of types of low amplitude forcing. The regular perturbation linearisation results in analytical expressions for steady fundamental and second harmonic induced oscillations of the bubble wall.

[0201] The perturbation model for small fractional radial oscillations of the bubble wall (x<<1) about its equilibrium radius R.sub.0 is written as:

x=ξ(x.sub.0+ξx.sub.1)   (21)

[0202] Where x.sub.0 and x.sub.1 are the zeroth and first order perturbation solutions of the fractional radial oscillation amplitude (defined in Eqn. (18)), respectively. Here j is the perturbation parameter (ξ<<1) defined in terms of the acoustic forcing amplitude η as

[00069]$\begin{array}{cc}\xi =\left(1-\frac{2.Math.{\sigma }_0}{P_0.Math.R_0}\right).Math.\eta & \left(22\right)\end{array}$

[0203] And P.sub.0 is the sum of the ambient gas pressure and the equilibrium surface tension equivalent pressure, as defined by:

[00070]$\begin{array}{cc}{P}_0=p_0+\frac{2.Math.{\sigma }_0}{R_0}.& \left(23\right)\end{array}$

[0204] The linear ordinary differential equations derived from perturbation analysis of Eqn. (18) describe the time behaviour of the zeroth and first order perturbation bubble response to acoustic excitation (utilising the definitions in Eqns. (17) and (22)). The resulting coupled pair of forced, damped oscillator equations may be written (to second order accuracy in fractional radial oscillation) as:

[00071]$\begin{array}{cc}.Math.\frac{d^2.Math.x_0}{d.Math..Math.{\tau }^2}+{\omega }_p^2.Math.x_0+2.Math.\left(b_p+c_p\right).Math.\frac{{\mathrm{dx}}_0}{d.Math..Math.\tau }=\cos \left(\mathrm{\omega \tau }\right),\mathrm{and}& \left(24\right)\\ \frac{d^2.Math.x_1}{d.Math..Math.{\tau }^2}+{\omega }_p^2.Math.x_1+2.Math.\left(b_p+c_p\right).Math.\frac{{\mathrm{dx}}_1}{d.Math..Math.\tau }=-x_0.Math..Math.\cos \left(\mathrm{\omega \tau }\right)-\frac{3}{2.Math.{\Gamma }_p^{*}}.Math.{\stackrel{.}{x}}_0^2+\alpha .Math..Math.x_0^2+\left(4.Math.b_p+6.Math.c_p\right).Math.x_0.Math.{\stackrel{.}{x}}_0& \left(25\right)\end{array}$

[0205] where the scaled time τ and angular frequency ω, time and angular frequency scaling factor A , scaled angular eigenfrequency ω.sub.p of the system, liquid and shell viscosity damping terms b.sub.p and c.sub.p respectively, are defined as follows:

[00072]$\begin{array}{cc}\tau =\Lambda .Math..Math.t;.Math..Math.\omega =\Omega .Math.\text{/}.Math.\Lambda ;.Math..Math.\Lambda ={\left(\frac{P_0}{{\rho }_L.Math.{\Gamma }_p^{*}}\right)}^{1.Math.\text{/}.Math.2}.Math.R_0^{-1};.Math..Math.{\omega }_p^2=3.Math.\kappa -\frac{2.Math.{\sigma }_0}{P_0.Math.R_0}+\frac{4.Math.{\chi }_0}{P_0.Math.R_0};.Math..Math.b_p=\frac{2.Math.\mu }{{R_0\left({\rho }_L.Math.P_0.Math.{\Gamma }_p^{*}\right)}^{1.Math.\text{/}.Math.2}};.Math..Math.c_p=\frac{2.Math.{\kappa }_S}{{R_0^2\left({\rho }_L.Math.P_0.Math.{\Gamma }_p^{*}\right)}^{1.Math.\text{/}.Math.2}}& \left(26\right)\end{array}$

[0206] The time and frequency scalings are defined such that ωτ=Ωt.

[0207] The coupling coefficient a linking the amplitude of the second order solution to the square of the amplitude of the first order solution is defined by:

[00073]$\begin{array}{cc}\alpha =9.Math.\text{/}.Math.2.Math.\kappa \left(\kappa +1\right)-\frac{4.Math.{\sigma }_0}{P_0.Math.R_0}+\frac{8.Math.{\chi }_0}{P_0.Math.R_0}& \left(27\right)\end{array}$

[0208] The perturbation model oscillatory steady solutions for bubble fractional radius in the case of mild acoustic non-linear excitation of an elastic layer encapsulated, attached solids mass loaded bubble are as follows.

[00074]$\begin{array}{cc}O\left(\xi \right).Math.\text{:}.Math..Math.x_0=C.Math..Math.\cos \left(\mathrm{\omega \tau }+\phi \right);.Math..Math.C=\frac{1}{\left({\omega }_p^2-{\omega }^2\right).Math.\cos .Math..Math.\phi -2.Math.\left(b_p+c_p\right).Math.\omega .Math..Math.\sin .Math..Math.\phi };.Math..Math.\phi =\arctan \left[\frac{2.Math.\left(b_p+c_p\right).Math.\omega }{{\omega }^2-{\omega }_p^2}\right].& \left(28\right)\end{array}$

[0209] The first order perturbation solution given by Eqn. (28) describes the fundamental resonance response of an encapsulated, solids-loaded bubble to mild acoustic excitation. The amplitude of the solution is a function of the source scaled forcing angular frequency, scaled angular eigenfrequency of the system, and the amplitudes of the liquid and encapsulating shell viscous damping terms. The maximum amplitude C(ω*) of the first order perturbation response of the shell wall at angular frequency ω* is given by C(ω*)=1/[2(b.sub.p+c.sub.p)√{square root over (ω.sub.p.sup.2−(b.sub.p+c.sub.p).sup.2)}], where ω*.sup.2=ω.sub.p.sup.2−2(b.sub.p+c.sub.p).sup.2. O(ξ.sup.2):

x.sub.1=A.sub.1cos(2ωπ+φ)+A.sub.2cos(2ωπ+2φ)+A.sub.3.   (29)

[0210] The second order perturbation solution given by Eqn. (29) describes the second harmonic (non-linear) response of an encapsulated, solids-loaded bubble to mild acoustic excitation. The solution amplitude coefficients A.sub.1(ω, ω.sub.p,b.sub.p,c.sub.p,φ,C), A.sub.2(ω, ω.sub.p,b.sub.p,c.sub.p,φ,C) and A.sub.3(ω, ω.sub.p,b.sub.p,c.sub.p,φ,C) are functions of the source scaled forcing angular frequency, scaled angular eigenfrequency of the system, and the amplitude of the liquid and encapsulating shell viscous damping terms. It can be shown that the scaling factors may be written as follows.

[00075]$\begin{array}{cc}{A}_1=\frac{\begin{array}{c}\{\left[-2.Math.\left(b_p+c_p\right).Math.\omega .Math..Math.\cos .Math..Math.\phi -1.Math.\text{/}.Math.2.Math.\beta .Math..Math.\sin .Math..Math.\phi \right].Math.C+\\ \left[2.Math.\left(b_p+c_p\right).Math.\gamma -\left(2.Math.b_p+3.Math.c_p\right).Math.\beta \right].Math.\omega .Math..Math.C^2\}\end{array}}{\left[{\beta }^2+{\left(4.Math.\left(b_p+c_p\right).Math.\omega \right)}^2\right].Math.\sin .Math..Math.\phi },.Math.A_2=\frac{\left\{B_1.Math.C+B_2.Math.C^2.Math..Math.\sin .Math..Math.\phi -B_3.Math.C^2.Math..Math.\cos .Math..Math.\phi \right\}}{\left[{\beta }^2+{\left(4.Math.\left(b_p+c_p\right).Math.\omega \right)}^2\right].Math.\sin .Math..Math.\phi };& \left(30\right)\\ B_1=2.Math.\left(b_p+c_p\right).Math.\omega ;.Math..Math.B_2=\left[\frac{\mathrm{\beta \gamma }}{2}+4.Math.\left(b_p+c_p\right).Math.\left(2.Math.b_p+3.Math.c_p\right).Math.{\omega }^2\right];.Math..Math.B_3=\left[2.Math.\left(b_p+c_p\right).Math.\gamma -\left(2.Math.b_p+3.Math.c_p\right).Math.\beta \right].Math.\omega .Math..Math.\mathrm{and}.Math.& \left(31\right)\\ A_3=\frac{-C.Math.\text{/}.Math.2.Math..Math.\cos .Math..Math.\phi +1.Math.\text{/}.Math.2.Math.\left(\alpha -\frac{3.Math.{\omega }^2}{2.Math.{\Gamma }_p^{*}}\right).Math.C^2}{{\omega }_p^2}& \left(32\right)\end{array}$

[0211] Here the coefficients β and γ are defined as:

[00076]$\begin{array}{cc}\beta ={\omega }_p^2-4.Math.{\omega }^2;.Math..Math.\gamma =\alpha +\frac{3.Math.{\omega }^2}{2.Math.{\Gamma }_p^{*}}& \left(33\right)\end{array}$

[0212] Receiver Pressure Model

[0213] The pressure P.sub.B radiated by a spherical pulsating bubble in an incompressible liquid can be derived from Euler' s fluid dynamics equations. This leads to the following nonlinear ordinary differential equation:

[00077]$\begin{array}{cc}{p}_B=\frac{{\rho }_L.Math.R}{r}.Math.\left(\stackrel{\mathrm{.Math.}}{R}.Math.R+2.Math.{\stackrel{.}{R}}^2\right)-\frac{{\rho }_L.Math.{\stackrel{.}{R}}^2}{2}.Math.{\left(\frac{R}{r}\right)}^4& \left(34\right)\end{array}$

[0214] Here the variable r is the distance from the bubble to the receiver. It should be noted that as a consequence of the incompressibility assumption in the pressure field modelling, the sound speed in the liquid is infinite and therefore bubble oscillations are communicated instantly through the liquid to the receiver.

[0215] In this model both the bubble and the acoustic receiver are aligned in the beam of the excitation source, with the bubble between the source and the receiver. Hence the received signal is a superposition of components due to both the acoustic source and bubble response, detected in transmission. There is a distance r.sub.SB between the source and the bubble. Additionally, for a straight line path from the acoustic source through the bubble to the acoustic receiver, the source to bubble and bubble to receiver distances may simply be defined in terms of a fixed distance (r.sub.tot=r+r.sub.SB) between source and receiver.

[0216] Introducing the regular perturbation model (Eqn. (21)) for the bubble wall fractional oscillation into Eqn. (34) for the radiated bubble response pressure field leads to the following equation for acoustic pressure due to the bubble at the receiver:

[00078]$\begin{array}{cc}{P}_B={\rho }_L.Math.\left\{\frac{R_0^3}{r}\left[\xi .Math.{\stackrel{\mathrm{.Math.}}{x}}_0+{\xi }^2\left({\stackrel{\mathrm{.Math.}}{x}}_1+2.Math.x_0.Math.{\stackrel{\mathrm{.Math.}}{x}}_0+2.Math.{\stackrel{.}{x}}_0^2\right)\right]-\frac{{\xi }^2.Math.R_0^6.Math.x_0^2}{2.Math.r^4}\right\}+O\left({\xi }^3\right).& \left(35\right)\end{array}$

[0217] The perturbation solution to Eqn. (35) for the bubble response pressure field can be found by inserting the first and second harmonic solutions (Eqns. (28) and (29)) for the bubble fractional oscillation.

[0218] The total (measured) pressure at the receiver is the sum of the bubble (response) and source (imposed) pressure at that point. The total pressure P.sub.tot due to the acoustic source and the perturbation model solution for the bubble response fractional oscillation first and second harmonics may be written as follows:

P.sub.tot=A+P.sub.1cos(Ωt+φ)+P.sub.2cos(Ωt)+P.sub.3cos(2Ωt+φ)+P.sub.4cos[2(Ωt+φ)].   (36)

[0219] It can be shown that the amplitude coefficients of this receiver total pressure solution may be written as follows.

[00079]$\begin{array}{cc}\begin{array}{c}A=p_0-\frac{{\rho }_L}{4}.Math.{\left(R_0.Math.\Lambda .Math..Math.C.Math..Math.\omega .Math..Math.\xi \right)}^2.Math.{\left(\frac{R_0}{r}\right)}^4;\\ P_1=-{\rho }_L.Math.\xi .Math..Math.R_0^2.Math..Math.C.Math..Math.{\Lambda }^2.Math..Math.{\omega }^2.Math.\frac{R_0}{r};\\ P_2=-\frac{p_0.Math..Math.\eta .Math..Math.r_{\mathrm{SB}}}{\left(r_{\mathrm{SB}}+r\right)};\\ P_3=-4.Math.A_1.Math.{{\rho }_L\left(R_0.Math..Math.\Lambda .Math..Math.\omega .Math..Math.\xi \right)}^2.Math.\frac{R_0}{r};\\ P_4=-{{\rho }_L\left(R_0.Math.\Lambda .Math..Math.\omega .Math..Math.\xi \right)}^2+\frac{R_0}{r}\left[4.Math.A_2+2.Math.C^2-\frac{1}{4}.Math.{\left(\frac{R_0}{r}\right)}^3.Math.C^2\right].\end{array}& \left(37\right)\end{array}$

[0220] It should be noted that in this model the pressure amplitude P.sub.2 is due to the acoustic source (which excites the bubble response) but here with a value appropriate to the receiver location. The acoustic source is modelled as a point such that at the bubble location (r=0), the pressure disturbance amplitude is consistent with that used in Eqn. (17) to define the bubble forcing term. This implies that the pressure perturbation due to the excitation source is inversely proportional to the radial distance from the bubble along the line-of-sight to the receiver. Again, the wave speed is modelled as infinite so pressure phase differences due to a finite disturbance travel time between the source, bubble and receiver are not taken into account. The phase difference φ in Eq. (26) is purely due to a lag between the source pressure perturbation at the bubble and the bubble wall response, caused by the liquid viscosity (see Eq. (28)).

[0221] Receiver Pressure Average Power Model

[0222] The total average power in the pressure field at the receiver may be written as follows in terms of constant background (A.sup.2), source excitation frequency (P.sub.ω.sup.2) and double excitation frequency)(P.sub.2ω.sup.2) contributions:

P.sub.tot.sup.2=A.sup.2+P.sub.ω.sup.2+P.sub.2ω.sup.2.   (38)

[0223] The average pressure contributions at scaled angular frequencies ω and 2ω can be defined in terms of the amplitude coefficients of the total receiver pressure (Eqn. (37)) and the phase angle φ of the bubble wall oscillation response with respect to the forcing excitation as follows:

[00080]$\begin{array}{cc}{\stackrel{\_}{P}}_{\omega }^2=\frac{P_1^2}{2}+\frac{P_2^2}{2}+P_1.Math.P_2.Math..Math.\cos .Math..Math.\phi & \left(39\right)\\ \mathrm{and}& \\ {\stackrel{\_}{P}}_{2.Math..Math.\omega }^2=\frac{P_3^2}{2}+\frac{P_4^2}{2}+P_3.Math.P_4.Math..Math.\cos .Math..Math.\phi .& \left(40\right)\end{array}$

[0224] Equations (39) and (40) define the total average power in the pressure field at the receiver location in terms of orthogonal components at both the frequency of the excitation source and at double of the same frequency, due to an acoustic point source and the excitation response of a single bubble located between the source and receiver. They include interference effects due to the phase angle between the source pressure oscillations and the bubble response. Equations (39) and (40) can be used to identify the frequencies of the fundamental and second harmonics of the bubble resonance response to acoustic excitation and the frequency of maximum destructive interference between the acoustic source and bubble response average power pressure fields at the receiver location.

[0225] Location of Extrema in the Receiver Pressure Average Power as a Function of Frequency. The total average power in the receiver pressure field at the source excitation frequency

[0226] Analysis of the total average power in the receiver pressure field at the source excitation frequency (Eqn. (39)) was undertaken to determine the frequency locations of any maxima and minima in the receiver pressure power as a function of source frequency. The frequency of maximum pressure power response is identified with the fundamental of the forced resonance of the acoustically excited bubble. The frequency of minimum pressure power response is identified as the location of maximum destructive interference between the acoustic waves of the active beam and the bubble response as detected at the receiver location. The motivation for this analysis is to provide a solution to the forward problem of predicting the frequencies of the acoustic resonance maximum and interference minimum of a possibly encapsulated, solids-loaded bubble subject to mild acoustic excitation, as a function of bubble gas, liquid, solids and encapsulating elastic layer properties. A comparison between theoretical and experimental estimates of the frequencies of maxima and minima of the received pressure power associated with bubble forced acoustic resonance interference might be used to estimate bubble size and attached solids mass loading. This is a prelude to solution of the inverse problem for estimation of bubble parameters based on forced acoustic resonance interference monitoring features. Another approach that has also been investigated for solving the forward problem was based on finding analytical expressions for the ratio of model amplitudes of the pressure harmonics. However, it has been found that in many situations of interest this can lead to accuracy issues both for the theoretical estimates (violation of the bubble oscillation amplitude constraint implicit in the linearisation of the bubble oscillation equation) and for the experimental estimates (because of the interference between the activation beam and the fundamental resonance bubble response). Accordingly, analysis of the forward problem in this study has concentrated on relating frequency features of the acoustically excited bubble response to bubble characteristics.

[0227] Scaled angular frequency values ω* are here found for turning points in the total average power in the receiver pressure field as a function of receiver frequency, at the frequency of bubble acoustic excitation. The scaled angular frequencies of the turning points associated with Eq. (39) are given by ω* that satisfy the following expression:

[00081]$\begin{array}{cc}\frac{\partial P_1}{\partial \omega }.Math.\left(P_1+P_2.Math..Math.\cos .Math..Math.\phi \right)=P_1.Math.P_2.Math..Math.\sin .Math..Math.\phi .Math.\frac{\partial \phi }{\partial \omega }.& \left(41\right)\end{array}$

[0228] Equation (41) is evaluated by introducing the lowest order perturbation model solutions for bubble radius oscillation amplitude C and phase angle φ (Eqn. (28)) and the definitions of the pressure equation coefficients P.sub.1 and P.sub.2 (Eqn. (37)). This leads to a quadratic equation in ω.sup.2* which admits two positive solutions corresponding to the receiver frequencies associated with maximum and minimum total average power in the receiver pressure field (at the excitation frequency). This quadratic equation can be written as follows:

y.sup.2[ψ(1−2Φ)+P.sub.2(1−4Φ)]−yω.sub.p.sup.2(ψ+2P.sub.2)+P.sub.2ω.sub.p.sup.4=0   (42)

[0229] Here the variable y and coefficient ψ are defined by:

[00082]$\begin{array}{cc}\begin{array}{c}y={\omega }_{*}^2;\\ \psi =\frac{p_0.Math.\eta }{{\Gamma }_p^{*}}.Math.\frac{R_0}{r}\end{array}& \left(43\right)\end{array}$

[0230] The coefficient Φ (square of the relative strength of the total encapsulating shell and liquid viscosity damping effects to the scaled angular eigenfrequency of the system) is given by:

[00083]$\begin{array}{cc}\Phi =\frac{{\left(b_p+c_p\right)}^2}{{\omega }_p^2}& \left(44\right)\end{array}$

[0231] Equation (42) can be solved for the frequency locations of the extrema in the total average power in the receiver pressure field at the excitation frequency in the general case without any assumptions concerning forward model parameter values. This has been done and as expected, there is an exact match between the extrema frequencies predicted by solving Eqn. (42) in its entirety and the observed values of the extrema in the excitation frequency component (and also total) average power of the receiver pressure field computed from Eqn. (39) for all values for forward model parameters. General case solutions of the forward problem for the frequencies of the maximum and minimum of the total average power in the receiver pressure field as a function of relevant bubble, liquid, gas, solid and encapsulating elastic layer properties are given below. These solutions are appropriate for solids-loaded, encapsulated microbubbles used in biomedical applications and also for free, solids-loaded bubbles of all sizes. Solutions are also given for the forward problem in the special case Φ=0, corresponding to negligible liquid and encapsulating layer dilatational viscosities (and also nil elastic layer compression modulus), appropriate for relatively large, free bubbles of interest in many industrial applications.

[0232] It should be noted that additional information of value could also be found by undertaking similar analysis aimed at establishing the theoretical frequency locations of any extrema in the receiver pressure power curve at twice the frequency of bubble acoustic excitation.

[0233] General Case Solution for Encapsulated and Free Bubbles of all Sizes

[0234] The exact solution of Eqn. (42) leads to the following general case equations for the frequencies f*=(f.sub.1max, f.sub.1min) where f*=ω*A/(2π) of the maximum and minimum values for the total average power in the receiver pressure field at the excitation frequency:

[00084]$\begin{array}{cc}{f}_{1.Math.\max }^2=\hspace{1em}\left[\frac{3.Math.{\mathrm{\kappa p}}_0+\frac{2.Math.{\sigma }_0}{R_0}.Math.\left(3.Math.\kappa -1\right)+\frac{4_{{}_0}}{R_0}}{4.Math.{\pi }^2.Math.{\rho }_L.Math.{\Gamma }_p^{*}.Math.R_0^2}\right].Math.\left\{\frac{\psi +2.Math.P_2+\sqrt{{\psi }^2+8.Math..Math.\Phi .Math..Math.P_2\left(\psi +2.Math.P_2\right)}}{2\left[\psi +P_2-2.Math.\Phi \left(\psi +2.Math.P_2\right)\right]}\right\}& \left(45\right)\\ \mathrm{and}& \\ f_{1.Math.\min }^2=\hspace{1em}\left[\frac{3.Math.{\mathrm{\kappa p}}_0+\frac{2.Math.{\sigma }_0}{R_0}.Math.\left(3.Math.\kappa -1\right)+\frac{4_{{}_0}}{R_0}}{4.Math.{\pi }^2.Math.{\rho }_L.Math.{\Gamma }_p^{*}.Math.R_0^2}\right].Math.\left\{\frac{\psi +2.Math.P_2-\sqrt{{\psi }^2+8.Math..Math.\Phi .Math..Math.P_2\left(\psi +2.Math.P_2\right)}}{2\left[\psi +P_2-2.Math.\Phi \left(\psi +2.Math.P_2\right)\right]}\right\}& \left(46\right)\end{array}$

[0235] Equations (45) and (46) can be used to predict the frequencies of the fundamental resonance maximum and interference minimum total average power in a receiver pressure field for a mildly acoustically excited solids-loaded bubble. This applies for free or encapsulated bubbles of any size.

[0236] Specific Case Solution for a Free Bubble and Negligible Liquid Viscosity

[0237] The case of the acoustic response of a free or unencapsulated bubble (which may still be loaded with attached solids) for relatively negligible liquid viscosity is of interest in many industrial situations. This occurs if Φ, ψ=0 is substituted in Eqns. (45) and (46). A comparison between the extrema frequencies predicted by such a model ignoring liquid viscosity and those predicted by the general case solution of the full model for total average power in a receiver pressure field suggest that the liquid viscosity can only reasonably be ignored for R.sub.0≧50 μm .sub.in t.sub.he case of an air bubble in water. This is based on a relative error of less than 10% between the difference in zero liquid viscosity and full model predicted extrema frequencies scaled by the difference between the extrema frequencies themselves.

[0238] In the case of negligible liquid viscosity, the free bubble solutions for the frequencies of the maximum and minimum total average power in a receiver pressure field (associated with the fundamental bubble resonance maximum and interference minimum) for a mildly acoustically excited solids-loaded bubble can be written as follows:

[00085]$\begin{array}{cc}{f}_{1.Math.\max }^2=\frac{3.Math.{\mathrm{\kappa p}}_0+\frac{2.Math.{\sigma }_0}{R_0}.Math.\left(3.Math.\kappa -1\right)}{4.Math.{\pi }^2.Math.{\rho }_L.Math.{\Gamma }_p^{*}.Math.R_0^2}& \left(47\right)\\ \mathrm{and}& \\ f_{1.Math.\min }^2=\frac{3.Math.{\mathrm{\kappa p}}_0+\frac{2.Math.{\sigma }_0}{R_0}.Math.\left(3.Math.\kappa -1\right)}{4.Math.{\pi }^2.Math.{\rho }_L.Math.{\Gamma }_p^{*}.Math.R_0^2\left[1-\frac{E}{{\Gamma }_p^{*}}\right]}& \left(48\right)\end{array}$

[0239] The dimensionless coefficient E is here defined in terms of the distances of the bubble from the source and receiver by:

[00086]$\begin{array}{cc}E=R_0\left(\frac{1}{r}+\frac{1}{r_{\mathrm{SB}}}\right)& \left(49\right)\end{array}$

[0240] Provided the total distance from the source to the receiver is a priori known, E is dependent only on the bubble equilibrium radius and one of either the source to bubble or bubble to receiver distances.

[0241] Equations (47) and (48) predict the frequencies of maximum and minimum receiver pressure power response due to bubble fundamental resonance excitation, for a bubble at specified distances from both the acoustic source and receiver, as a function of bubble size, attached solids mass loading and surface tension. This is for the case of a free bubble with negligible liquid viscosity, and a straight-line of acoustic transmission from the source, through the bubble to the receiver. Equation (47) is the particulate solids mass loaded analogue of the classical Minnaert relationship between radius and acoustic resonance frequency for a ‘clean’ bubble, extended to include surface tension. It should be noted that if the acoustic source, bubble and receiver are not in a straight line, in principle the separate paths of the active beam and the bubble response beam to the receiver should be taken into account for the accurate estimation of the frequency of the interference minimum However, for the propagation of ˜3 kHz acoustic waves in water (Minnaert resonance frequency for R.sub.0˜1 mm bubbles), the wavelength is ˜0.5 metres. Hence if a receiver is positioned at a distance of ˜1-5 cm from a bubble then the phase differences between the source and response beams due to any acoustic path differences will be reasonably small regardless of where the source is placed with respect to a straight line between the bubble and the receiver. In this situation, Eq. (48) would remain valid regardless of the geometry of the source, bubble and receiver configuration.

[0242] Bubble Equilibrium Radius, Attached Solids Mass Loading and Encapsulating Layer Dilatational Viscosity as a Function of the Frequencies of Extrema in the Receiver Pressure Average Power

[0243] It is often the case that monitoring is performed in order to estimate some key parameters of a sample under consideration. This process often involves the solution of a model-based inverse problem for parameter estimation based on estimates of acoustic variables derived from the observations. In active acoustic resonance interference monitoring, the frequency positions of the extrema (f.sub.1max, f.sub.1min) in the total average power of the receiver pressure field in a transmission configuration, corresponding to the bubble fundamental resonance maximum and interference minimum, could be estimated from acoustic observations. This is provided that the bubble is stimulated sufficiently so as to induce a steady oscillatory response (but not excessively so as to result in transient cavitation) over a range of frequencies for sufficient time to allow the frequencies of the maximum and minimum receiver power response to be reliably estimated. Where the approximate bubble size is not a priori known, strategies of either pulsed, swept frequency or white noise excitation may be appropriate. The goal would then be to use the observed frequencies of the extrema in received AE power signal as input variables to solve the inverse problem of estimating bubble parameters such as bubble size, attached solids mass loading and encapsulating layer dilatational viscosity.

[0244] Closed-form analytical estimators are presented here for the bubble equilibrium radius, attached solids mass loading and encapsulating layer dilatational viscosity in terms of the variables of acoustic resonance fundamental maximum, interference minimum and in some instances the second harmonic maximum frequencies. General case solutions are presented which are valid for arbitrary bubble equilibrium size, liquid and encapsulating layer dilatational viscosity effects. Specific case solutions for bubble equilibrium radius and attached solids mass loading are also presented for large bubbles when total viscosity effects are negligible.

[0245] General Case Solutions for Bubble Size, Solids Mass Loading and Encapsulating Layer Dilatational Viscosity

[0246] Equations (45) and (46) are combined to provide an estimator for the bubble equilibrium radius (independent of the attached solids mass loading) in the general case of a free or encapsulated bubble of any size. It can be shown that the equilibrium bubble radius in the general case is given by the following expression:

[00087]$\begin{array}{cc}{R}_0=\frac{1}{2.Math.\pi }.Math.\sqrt{\frac{1}{f_{1.Math.\max }^2}-\frac{1}{f_{1.Math.\min }^2}}.Math.\sqrt{\frac{3.Math.\kappa .Math..Math.p_0+\frac{2.Math.{\sigma }_0}{R_0}.Math.\left(3.Math.\kappa -1\right)+\frac{4_{{}_0}}{R_0}}{{\rho }_L.Math.E.Math..Math.\Theta };}.Math.\left(f_{1.Math.\min }>f_{1.Math.\max }\right)& \left(50\right)\end{array}$

[0247] The dimensionless coefficient Θ in the denominator is defined by:

Θ=ζΔ+√{square root over (1+ζ.sup.2Δ.sup.2)}  (55)

[0248] The coefficient ζ is defined as follows, purely as a function of the frequencies of the resonance maximum and interference minimum of the total average power as a function of receiver frequency:

[00088]$\begin{array}{cc}\begin{array}{c}Ϛ=\frac{1+{\stackrel{\_}{\omega }}^2}{1-{\stackrel{\_}{\omega }}^2};\\ \stackrel{\_}{\omega }=\frac{f_{1.Math.\max }}{f_{1.Math.\min }}\end{array}& \left(52\right)\end{array}$

[0249] The dimensionless coefficient Δ is defined as follows, being finite only for either non-zero liquid viscosity or encapsulating layer dilatational viscosity:

[00089]$\begin{array}{cc}\Delta =\frac{16.Math.{\left(\mu +\frac{{\kappa }_S}{R_0}\right)}^2}{R_0^2.Math.{\rho }_L\left[3.Math.{\mathrm{\kappa p}}_0+\frac{2.Math.{\sigma }_0}{R_0}.Math.\left(3.Math.\kappa -1\right)+\frac{4_{{}_0}}{R_0}\right].Math.E}& \left(53\right)\end{array}$

[0250] The estimator for equilibrium bubble radius given by Eqns. (50)-(53) is dependent on knowledge of relevant gas, liquid and encapsulating layer properties (including dilatational viscosity), plus the frequencies of the fundamental resonance maximum and interference minimum of the total average power at the receiver. It should be noted that Eqn. (50) does not require a priori knowledge of the attached solids mass loading. The equation holds both in cases where attached solids and an encapsulating layer are present or absent.

[0251] The attached solids mass loading can be found from the following expression (for a given R.sub.0 estimated from Eqns. (50)-(53)):

[00090]$\begin{array}{cc}{M}_S=\frac{R_0}{\delta }\left[\frac{3.Math.{\mathrm{\kappa p}}_0+\frac{2.Math.{\sigma }_0}{R_0}.Math.\left(3.Math.\kappa -1\right)+\frac{4_{{}_0}}{R_0}}{4.Math.{\pi }^2.Math.{\rho }_L.Math.f_{1.Math.\max }^2}-R_0^2+\frac{R_0^2}{2}.Math.E\left(1-\Theta \right)\right]& \left(54\right)\end{array}$

[0252] where the solids density coefficient δ is defined by the following expression:

[00091]$\begin{array}{cc}\delta =\frac{1}{4.Math.\pi }.Math.\left(\frac{1}{{\rho }_L}-\frac{1}{{\rho }_S}\right)& \left(55\right)\end{array}$

[0253] Equation (54) is explicitly dependent on the Θ(μ, κ.sub.S) parameter and hence a known value for the encapsulating layer dilatational viscosity. An alternative expression for the attached solids mass loading which is not explicitly dependent on the values for the liquid and layer dilatational viscosities is:

[00092]$\begin{array}{cc}{M}_S=\frac{R_0}{\delta }\left[\left(\frac{3.Math.{\mathrm{\kappa p}}_0+\frac{2.Math.{\sigma }_0}{R_0}.Math.\left(3.Math.\kappa -1\right)+\frac{4_{{}_0}}{R_0}}{4.Math.{\pi }^2.Math.{\rho }_L}\right).Math.\left(\frac{\frac{1}{f_{1.Math.\max }^2}+\frac{1}{f_{1.Math.\min }^2}}{2}\right)-R_0^2\left(1-\frac{E}{2}\right)\right]& \left(56\right)\end{array}$

[0254] Equation (56) provides a general case estimator for the attached solids mass loading of a bubble of known equilibrium radius.

[0255] Equations (50)-(53) may themselves be inverted to provide an estimator for encapsulating layer dilatational viscosity as a function of bubble equilibrium radius and the frequencies of the fundamental resonance maximum and interference minimum in the receiver total average power. It can be shown by rearranging Eqns. (51) and (53) that the encapsulating layer dilatational viscosity can be written as:

[00093]$\begin{array}{cc}{\kappa }_S=R_0.Math.\left\{\frac{R_0}{4}.Math.\sqrt{\left(\frac{{\Theta }^2-1}{2.Math.\mathrm{\Theta ϛ}}\right).Math.E.Math..Math.{\rho }_L\left[3.Math.\kappa .Math..Math.p_0+\frac{2.Math.{\sigma }_0}{R_0}.Math.\left(3.Math.\kappa -1\right)+\frac{4.Math.{\chi }_0}{R_0}\right]}-\mu \right\}& \left(57\right)\end{array}$

[0256] Here Eqn. (50) is used to write the dimensionless coefficient Θ as

[00094]$\begin{array}{cc}\Theta =\left[\frac{3.Math.\kappa .Math..Math.p_0+\frac{2.Math.{\sigma }_0}{R_0}.Math.\left(3.Math.\kappa -1\right)+\frac{4.Math.{\chi }_0}{R_0}}{4.Math.{\pi }^2.Math.{\rho }_L.Math.{\mathrm{ER}}_0^2}\right].Math.\left(\frac{1}{f_{1.Math.\max }^2}-\frac{1}{f_{1.Math.\min }^2}\right)& \left(58\right)\end{array}$

[0257] Parameter estimation for a ‘clean’ bubble is of interest in many applications. Equation (54) may be rearranged for zero attached solids mass to provide an estimator for bubble equilibrium radius as:

[00095]$\begin{array}{cc}{R}_0=\frac{1}{2.Math.\pi .Math..Math.f_{1.Math.\max }}.Math.\sqrt{\frac{3.Math.\kappa .Math..Math.p_0+\frac{2.Math.{\sigma }_0}{R_0}.Math.\left(3.Math.\kappa -1\right)+\frac{4.Math.{\chi }_0}{R_0}}{{\rho }_L\left[1-\frac{E}{2}.Math.\left(1-\Theta \right)\right]}}& \left(59\right)\end{array}$

[0258] This is an extended form of the classical Minnaert relationship between the ‘clean’ bubble radius and the acoustic resonance frequency. In this case, even though the bubble is unloaded it may be encapsulated by an elastic layer. The bubble equilibrium radius in Eqn. (59) explicitly depends on the liquid viscosity and any surface layer dilatational viscosity.

[0259] Equation (56) may also be rearranged for the case of zero attached solids mass to give:

[00096]$\begin{array}{cc}{R}_0=\frac{1}{2.Math.\pi }.Math.\sqrt{\frac{1}{2}.Math.\left(\frac{1}{f_{1.Math.\max }^2}+\frac{1}{f_{1.Math.\min }^2}\right)}.Math.\sqrt{\frac{3.Math.\kappa .Math..Math.p_0+\frac{2.Math.{\sigma }_0}{R_0}.Math.\left(3.Math.\kappa -1\right)+\frac{4.Math.{\chi }_0}{R_0}}{{\rho }_L\left[1-\frac{E}{2}\right]}}& \left(60\right)\end{array}$

[0260] Equation (60) is an alternative estimator for the equilibrium radius of a ‘clean’ but possibly encapsulated bubble. It does not require a priori knowledge of any encapsulating layer dilatational viscosity.

[0261] In the case of a ‘clean’ bubble, the encapsulating layer dilatational viscosity can be found via the general case estimator (Eqn. (57)) for an equilibrium bubble radius itself estimated from

[0262] Eqn. (60). Alternatively, the encapsulating layer dilatational viscosity can also be found from the ratio of Eqns. (45) and (46) in the case Γ.sub.p*=1 (a ‘clean’ bubble), leading to the estimator:

[00097]$\begin{array}{cc}{\kappa }_S=R_0.Math.\left\{\frac{R_0}{4.Math.ϛ}.Math.\sqrt{\frac{{\rho }_L}{2}\left[3.Math.\kappa .Math..Math.p_0+\frac{2.Math.{\sigma }_0}{R_0}.Math.\left(3.Math.\kappa -1\right)+\frac{4.Math.{\chi }_0}{R_0}\right]\left[\frac{{\left(2-E\right)}^2-{ϛ}^2.Math.E^2}{2-E}\right]}-\mu \right\}& \left(61\right)\end{array}$

[0263] Equation (61) provides an estimate for the encapsulating layer dilatational viscosity of a ‘clean’ bubble of known equilibrium radius, for given frequencies of the fundamental resonance maximum and interference minimum, gas polytropic index, bubble surface tension parameters, monitoring system geometry, and liquid density ρ.sub.L, and viscosity.

[0264] Specific Case Solutions for Bubble Size and Solids Mass Loading for a Free Bubble and Negligible Liquid Viscosity

[0265] Equations (47) and (48) can be combined to provide an estimator for the bubble equilibrium radius (independent of the attached solids mass loading) in the case of a free bubble where there are negligible liquid viscosity and elastic layer compression modulus effects on the frequencies of the extrema of the total average power of the pressure at the receiver. This estimator can also be found directly from Eqn. (50) for nil values of liquid viscosity and shell layer dilatational viscosity (leading to Δ=0 via Eqn. (53)) plus nil shell elastic layer compression modulus. The equilibrium bubble radius in these circumstances is given by the following expression:

[00098]$\begin{array}{cc}{R}_0=\frac{1}{2.Math.\pi }.Math.\sqrt{\frac{1}{f_{1.Math.\max }^2}-\frac{1}{f_{1.Math.\min }^2}}.Math.\sqrt{\frac{3.Math.\kappa .Math..Math.p_0+\frac{2.Math.{\sigma }_0}{R_0}.Math.\left(3.Math.\kappa -1\right)}{{\rho }_L.Math.E}};\left(f_{\min }>f_{\max }\right)& \left(62\right)\end{array}$

[0266] This estimator for bubble equilibrium radius is also dependent on knowledge of the distance from the bubble to either the acoustic source or the (hydrophone) receiver (see Eqn. (49)). It should be noted that Eqn. (62) holds both in cases where attached solids are present or absent.

[0267] The attached solids mass loading in this case can be found from Eqns. (47) and (48) as either of the following equivalent expressions:

[00099]$\begin{array}{cc}{M}_S=\frac{R_0}{\delta }.Math.\left(\frac{3.Math.\kappa .Math..Math.p_0+\frac{2.Math.{\sigma }_0}{R_0}.Math.\left(3.Math.\kappa -1\right)}{4.Math.{\pi }^2.Math.{\rho }_L.Math.f_{1.Math.\max }^2}-R_0^2\right).Math..Math.\mathrm{and}& \left(63\right)\\ M_S=R_0.Math.\frac{\left[3.Math.\kappa .Math..Math.p_0+\frac{2.Math.{\sigma }_0}{R_0}.Math.\left(3.Math.\kappa -1\right)\right]}{4.Math.{\pi }^2.Math.{\rho }_L.Math.\delta }\left[\frac{1}{f_{1.Math.\max }^2}-\frac{1}{E}.Math.\left(\frac{1}{f_{1.Math.\max }^2}-\frac{1}{f_{1.Math.\min }^2}\right)\right]& \left(64\right)\end{array}$

[0268] Equation (63) for attached solids mass loading can be shown (utilising Eqn. (47) to estimate the unloaded bubble resonant response frequency) to be equivalent to Eqn. (14) which was derived to estimate monolayer attached high density solids mass loading from solids loaded bubble resonant response frequency for a priori known bubble equilibrium radius and resonant response frequency at nil attached solids mass loading.

[0269] The case of bubble size for nil attached solids is again of interest. Equations (47) and (63) may both be rearranged for the case of zero attached solids mass as follows:

[00100]$\begin{array}{cc}{R}_0=\frac{1}{2.Math.{\mathrm{\pi f}}_{1.Math.\max }}.Math.\sqrt{\frac{3.Math.\kappa .Math..Math.p_0+\frac{2.Math.{\sigma }_0}{R_0}.Math.\left(3.Math.\kappa -1\right)}{{\rho }_L}}& \left(65\right)\end{array}$

[0270] Again, this is the classical Minnaert relationship, between the equilibrium radius of a ‘clean’, unencapsulated bubble and the fundamental frequency of acoustic resonance (usually used to describe a freely oscillating bubble), here extended to include surface tension.

[0271] Introducing Eqn. (65) into (62) leads to the following expression for the equilibrium radius of the bubble in the nil attached solids case:

[00101]$\begin{array}{cc}{R}_0=\frac{1}{2.Math.{\mathrm{\pi f}}_{1.Math.\min }}.Math.\sqrt{\frac{3.Math.\kappa .Math..Math.p_0+\frac{2.Math.{\sigma }_0}{R_0}.Math.\left(3.Math.\kappa -1\right)}{{\rho }_L\left(1-E\right)}}& \left(66\right)\end{array}$

[0272] Combining Eqns. (65) and (66) leads to the following expression for the dimensionless coefficient E in the case of a ‘clean’, free bubble in a liquid with negligible viscosity:

[00102]$\begin{array}{cc}E=1-{\left(\frac{f_{1.Math.\max }}{f_{1.Math.\min }}\right)}^2& \left(67\right)\end{array}$

[0273] Introducing Eqn. (67) into Eqns. (62) and (66) results in the extended Minnaert relationship of Eqn. (65) as expected.

[0274] Equation (62) can be used to estimate the bubble equilibrium radius for arbitrary attached solids mass loading in the specific case of an unencapsulated bubble and negligible liquid viscosity effect on bubble size. It requires reliable experimental estimates of the frequencies of the fundamental resonance maximum and interference minimum of the acoustic receiver average power response. Either of Eqns. (63) or (64) can then be used to estimate bubble attached solids mass loading. These expressions are valid for a bubble at known distances from a point acoustic source and a suitable acoustic receiver when subject to low amplitude forced acoustic excitation at frequencies near the fundamental resonance frequency. Equations (62), (65) and (66) can all be used to estimate the equilibrium size of an unloaded bubble.

[0275] Additional Estimators Based on Resonance Response at Double the Excitation Frequency

[0276] The use of two acoustic features (frequencies of the fundamental resonance maximum and interference minimum in receiver total average power) as inputs to the bubble parameter inverse problem only permits two distinct parameters of a single bubble to be uniquely estimated. Hence bubble equilibrium radius and attached solids mass loading can be estimated for known values of the encapsulating layer dilatational viscosity and the distance of the bubble from the source or receiver. Alternatively, bubble radius and encapsulating layer dilatational viscosity can be uniquely estimated for nil attached solids mass loading. However, it should be noted that the frequency of the resonance maximum associated with receiver total average acoustic power at double the excitation frequency could also provide an alternative estimator for bubble size and attached mass in the case of O(1-10) micron-sized bubbles. A turning points analysis of Eqn. (40) may lead to closed-form analytical expressions for the frequency of the second harmonic acoustic response maximum as a function of bubble and monitoring system properties. Alternatively, Eqn. (40) can be used to plot the receiver total average power at double the excitation frequency as a function of receiver frequency. This analysis reveals that the frequency of the second harmonic maximum is close to double the first harmonic peak resonance frequency associated with a bubble of the same equilibrium size and attached solids mass loading bubble but with negligible encapsulating layer and liquid viscosity, regardless of whether the bubble is actually free or has an encapsulating layer. This result suggests that the peak frequency f.sub.2max of the maximum total average power in the receiver pressure field associated with the second harmonic of bubble resonance for a mildly acoustically excited solids-loaded (and possibly encapsulated) bubble can be written as follows:

[00103]$\begin{array}{cc}{f}_{2.Math.\max }^2\approx \frac{3.Math.\kappa .Math..Math.p_0+\frac{2.Math.{\sigma }_0}{R_0}.Math.\left(3.Math.\kappa -1\right)+\frac{4.Math.{\chi }_0}{R_0}}{{\pi }^2.Math.{\rho }_L.Math.{\Gamma }_p^{*}.Math.R_0^2}& \left(68\right)\end{array}$

[0277] Equation (68) leads to the following additional estimator for the attached solids mass loading in terms of the second harmonic peak frequency f.sub.2max, bubble equilibrium radius and other system properties:

[00104]$\begin{array}{cc}{M}_S\approx \frac{R_0}{\delta }.Math.\left(\frac{3.Math.\kappa .Math..Math.p_0+\frac{2.Math.{\sigma }_0}{R_0}.Math.\left(3.Math.\kappa -1\right)+\frac{4.Math.{\chi }_0}{R_0}}{{\pi }^2.Math.{\rho }_L.Math.f_{2.Math.\max }^2}-R_0^2\right)& \left(69\right)\end{array}$

[0278] It should be noted that f.sub.2max≠2f.sub.1max for O(1-10) micron equilibrium radius bubbles in water. However, as bubble size increases further the frequency of the second harmonic resonance maximum converges on double the frequency of the fundamental resonance maximum and does not provide any additional information to assist in bubble parameter estimation. Equation (69) thus reverts to Equation (63). The implications of this result are that only two bubble properties (e.g. R.sub.0 and M.sub.S) can be simultaneously determined for larger (macro)bubbles from either of the two characteristics f.sub.1min and f.sub.1max or f.sub.1min and f.sub.2max.

[0279] The estimators for attached solids mass loading in Eqns. (56) and (69) can now be equated to obtain an additional general case estimator for bubble equilibrium radius at arbitrary attached solids mass loading and encapsulating layer dilatational viscosity. This bubble equilibrium radius estimator can be written as:

[00105]$\begin{array}{cc}{R}_0\approx \frac{1}{2.Math.\pi }.Math.\sqrt{2\left[\frac{4}{f_{2.Math.\max }^2}-\frac{1}{2}.Math.\left(\frac{1}{f_{1.Math.\max }^2}+\frac{1}{f_{1.Math.\min }^2}\right)\right]}.Math.\sqrt{\frac{3.Math.\kappa .Math..Math.p_0+\frac{2.Math.{\sigma }_0}{R_0}.Math.\left(3.Math.\kappa -1\right)+\frac{4.Math.{\chi }_0}{R_0}}{{\rho }_L.Math.E}}& \left(70\right)\end{array}$

[0280] The equilibrium bubble radius in Eqn. (70) is a function of three features of the receiver total power associated with forced acoustic resonance interference spectroscopy. These features are the frequencies of the fundamental resonance maximum, interference minimum and second harmonic maximum in receiver total average power. As expected, inserting f.sub.2max=2f.sub.1max in Eqn. (70) simply reduces it to a reduced form of Eqn. (50) for the case Θ=1 (nil liquid viscosity and shell dilatational viscosity). Equations (57) and (69) can now be used to uniquely estimate the dilatational viscosity of any encapsulating layer and attached solids mass loading, respectively.

[0281] Simulation Results - Example Solutions of the Forward Problem for Acoustic Response as a Function of Bubble Characteristics

[0282] Receiver Total Average acoustic Power as a function of frequency

[0283] Free Bubble—loaded

[0284] Simulation results are presented here for the response of a millimetre sized unencapsulated but attached solids mass loaded bubble to mild acoustic excitation. FIGS. 3 and 4 show model results for receiver total average acoustic power associated with first and second forced harmonic responses respectively, as a function of receiver frequency. In each plot separate curves are shown for three different attached solids mass loadings (0, 1 and 10 mg of solid particles of density 2200 kgm.sup.−3 (simulating glass balloting). In this case the bubble is 1 mm radius and the distances from the bubble centre to both the point acoustic source and hydrophone receiver are both 10 mm (in opposite directions). The acoustic excitation has pressure amplitude of 100 Pa at the bubble location and is sinusoidal with frequencies over the range 2.5-7.5 kHz. The surface tension is 7.2e-2 kgs.sup.−2, appropriate for air bubbles in water. The gas polytropic index is 1.4, as appropriate for air bubbles. The liquid viscosity is 8.94e-04 kgm.sup.−1 s.sup.−1, as appropriate for water. Smaller values of liquid viscosity do not change the profiles of total receiver pressure power as a function of excitation frequency.

[0285] Peaks in the pressure power response spectrum are readily apparent at the fundamental frequency and second harmonic of bubble forced resonance. The frequency of the peaks clearly decreases with attached solids mass loading (and bubble equilibrium size). Hence the resonance peak frequencies could potentially be used to estimate bubble parameters such as size, attached solids mass loading and encapsulating layer dilatational viscosity (additionally the distance between the bubble and either the source or receiver). Minima in receiver pressure power just above both the fundamental and second harmonic resonance frequencies are also very clear. Again, these minima are due to destructive interference between the source acoustic beam and the bubble acoustic response as detected at the receiver. Both the location of these minima could be related to bubble parameters such as size, attached solids mass loading and encapsulating layer dilatational viscosity. In this analysis only the frequency location of the minimum just above the fundamental resonance maximum response are used for bubble parameter estimation. It should be noticed that in practical application, broadband excitation of bubbles could lead to overlapping first harmonic and second harmonic responses due to the simultaneous excitation at multiple frequencies. In this case, the total average acoustic power at any frequency within the range of overlapping first and second harmonic frequencies would actually consist of incoherently summed excitation frequency and double excitation pressure contributions.

[0286] The fundamental and second harmonic peaks due to resonant excitation in this case actually correspond to bubble wall radial oscillations that exceed the bounds of validity of the regular perturbation model. Hence the (relative) strength of the receiver pressure signal at the fundamental and second harmonic frequencies may not be reliable for accurately determining the solids mass loading of a bubble. However, the analytical solutions for frequencies of the fundamental and second harmonic maxima and related interference minima are unaffected by the strengths of both the source and the bubble resonant response and vary strongly with both the attached solids mass loading and bubble size. This has been confirmed by examination of simulation results for the fundamental and second harmonic total average acoustic power as a function of receiver frequency, varying the source amplitude over a 1-10.sup.5 Pa range. It is anticipated that estimating bubble parameters from the frequencies of features in the forced acoustic spectrum at the receiver is valid and accurate over a much broader range of source amplitudes than any model based on the strengths of the harmonics at the receiver.

[0287] Encapsulated Microbubble—loaded

[0288] Simulation results are presented here for the response of a micron sized encapsulated and attached solids mass loaded microbubble to mild acoustic excitation. FIGS. 5 and 6 show model results for total receiver pressure power associated with first and second forced harmonic responses respectively, as a function of receiver frequency. In each plot separate curves are shown for three different attached solids mass loadings (0, 10 and 100 pg) for an attached layer composed entirely of solids of density 1100 kgm.sup.−3, simulating the mass effect of a lipid encapsulating layer. The encapsulating layer dilatational viscosity is 2.4×10.sup.−9 kgs.sup.−1 and the shell elastic layer compression modulus is 0.38 kgs.sup.−2, as appropriate for the Definity™ ultrasound contrast agent. In this case the bubble is 1 μm radius in accordance with an optically determined mean size of a population of Definity ultrasound contrast agent. The distances from the bubble centre to both the point acoustic source and hydrophone receiver are both 100 μm (in opposite directions). The acoustic excitation has pressure amplitude of 1000 Pa at the bubble location and is sinusoidal with frequencies over the range 3-12 MHz. The equilibrium surface tension is taken as 7.2e-2 kgs.sup.−2. The gas polytropic index is 1.06, as appropriate for the C.sub.3F.sub.8 bubble core used in Definity ultrasound contrast agent. The liquid viscosity is 8.94e-04 kgm.sup.−1s.sup.−1, as appropriate for water.

[0289] Peaks are again apparent in the first harmonic (˜5-7 MHz) and second harmonic (˜11-15 MHz) total average acoustic power response spectra. However, the peaks are much broader and lower amplitude in the case of an encapsulated microbubble in comparison to a millimetre sized free bubble. This is the result of relatively strong total viscosity acoustic damping effects for micron sized bubbles. This viscous damping is due both to the liquid viscosity and the encapsulating layer dilatational viscosity. Despite this, the frequency location of the peaks can still be used to estimate bubble parameters such as size, attached solids mass loading and encapsulating layer dilatational viscosity (additionally the distance between the bubble and either the source or receiver). It should be noted that as expected, the frequency location of the peaks is strongly increased by the encapsulating layer elastic layer compression modulus [see Eqn. (45) and Eqn. (68)]. A secondary maximum at ˜5-7 MHz is apparent in FIG. 6) in the second harmonic responses. This is at frequencies near (but not the same as) those associated with the first harmonic (excitation frequency) response. An interference minimum in receiver pressure power is clear in FIG. 5) above the fundamental resonance frequency. The location of this minimum can also be related to bubble parameters such as size, attached solids mass loading and encapsulating layer dilatational viscosity. There is no local minimum in FIG. 6) at frequencies above the second harmonic maximum response.

[0290] The Frequencies of the Fundamental Maximum and Interference Minimum as a function of Bubble Size and Solids Loading

[0291] An example is provided of the frequencies of the fundamental resonance maximum and interference minimum of the receiver total average acoustic power as a function of bubble equilibrium radius and attached solids mass loading. These model predictions are based on Eqns. (47) and (48) in the case of a free bubble and negligible liquid viscosity effect on the acoustic power response.

[0292] FIG. 7 is a filled contour plot of the frequency of the maximum (near the bubble resonance fundamental frequency) in the acoustic receiver average pressure power spectrum as a function of the equilibrium bubble radius and attached solids mass loading in the case of a free bubble and negligible liquid viscosity effect on bubble resonance frequencies. The plot axis boundaries are R.sub.0=0.25−2.5 mm and M.sub.S=0−10 mg.

[0293] As expected, the frequency of maximum acoustic power varies strongly with both bubble equilibrium radius and attached solids mass loading. The maximum possible value for f.sub.1max is on the R.sub.0 axis near the origin. A feature of considerable interest is a ‘ridge’ of maximal frequency at any given solids mass loading that extends in a positive direction in terms of both bubble equilibrium radius and attached solids mass loading from the R.sub.0 axis near the origin of the plot. The significance of this ridge is that for any given solids mass loading there may be two values of bubble radius that result in the same frequency of maximal acoustic response. Even if the amount of solids attached to the bubble is a priori known, two possible sizes of bubble could produce the same maximal acoustic power response frequency. This non-uniqueness is a result of the solids mass attached to the bubble. In situations of bubbles loaded with even a known amount of solids, it is necessary to measure both f.sub.1max and f.sub.min in order to uniquely estimate the bubble size.

[0294] An equation can be found that describes the ‘ridge’ of maximal frequency response near the fundamental resonance frequency. The position of maximal f.sub.1max for any given M.sub.S can be found by finding the turning points R.sub.0,f1max of Eq. (47) with respect to bubble equilibrium radius. This leads to the cubic equation:

[00106]$\begin{array}{cc}{R}_{0,f.Math..Math.1.Math.\max }^3+\frac{\sigma }{3.Math.\kappa .Math..Math.p_0}.Math.\left(3.Math.\kappa -1\right).Math.R_{0,f.Math..Math.1.Math.\max }^2-\frac{\delta .Math..Math.M_S}{2}=0& \left(71\right)\end{array}$

[0295] An analytical solution can be found to Eqn. (71). However, more insight can be obtained by considering the case of negligible surface tension. In this case, the ‘ridge’ of acoustic maximal frequency response is described by the line:

[00107]$\begin{array}{cc}{R}_{0,f.Math..Math.1.Math.\max }={\left(\frac{\delta .Math..Math.M_S}{2}\right)}^{1.Math.\text{/}.Math.3}& \left(72\right)\end{array}$

[0296] A bifurcation appears from a unique to dual solution for bubble equilibrium radius (as derived from the acoustic maximal response frequency alone) as soon as solids mass is attached to the bubble. The solutions for bubble equilibrium size based on the frequency of the fundamental resonant response alone are unique only in the case of ‘clean’ bubbles.

[0297] FIG. 8 is a filled contour plot of the frequency of the minimum (near the resonance fundamental frequency) in the acoustic receiver average pressure power spectrum as a function of the equilibrium bubble radius and attached solids mass loading. Again, non-unique solutions for bubble radius (in the case of a solids loaded bubble) are possible if the prediction is based only on the frequency of the interference minimum near the fundamental resonant response.

[0298] Example Solutions of the Inverse Problem for Bubble Size and Attached Solids Mass Loading as a Function of Frequencies of Extrema in the Receiver Total Average Acoustic Power

[0299] An example is provided of the estimated equilibrium bubble size and attached solids mass loading associated with the frequencies of the fundamental resonance maximum and interference minimum of the receiver total average acoustic power. These model predictions are based on Eqns. (62) and (63) in the case of a free bubble and negligible liquid viscosity effect on the acoustic power response. The Newton-Raphson iterative method is used to solve Eqn. (62) for R.sub.0 (quartic in bubble equilibrium radius). This is increasingly important at small bubble radii (O(1) μm or less) where surface tension has an important effect on bubble acoustic oscillations.

[0300] FIG. 9 is a filled contour plot of bubble radius as a function of the frequencies of the fundamental resonance maximum and interference minimum received total average average acoustic power. The plot axis boundaries are f.sub.1min=1.8−5.0 kHz and f.sub.1max=1.3−5.0 kHz. Contours of bubble equilibrium radius are over the range 0.25-2.5 mm for attached solids mass loading over the range 0-10 mg. The solutions for bubble equilibrium radius are unique for this range of maximal and minimal first harmonic (excitation frequency) acoustic power response frequencies.

[0301] FIG. 10 is a filled contour plot of bubble attached solids mass loading as a function of the frequencies of the first harmonic resonance maximum and the difference between the frequencies of the first harmonic interference minimum and resonance maximum received total average acoustic power. The plot axis boundaries are f.sub.1max=1.3−5.0 kHz and f.sub.1min−f.sub.1max=0.0−0.6 kHz. Contours of bubble attached solids mass loading are over the range 0-10 mg for bubble equilibrium radius over the range 0.25-2.5 mm

[0302] The solutions for bubble attached solids mass loading are unique for this range of maximal and minimal first harmonic acoustic power response frequencies. However, the range of f.sub.1min and f.sub.max for which there are solutions is restricted to a band and the contours of attached solids loading are tightly packed in some regions of (f.sub.min, f.sub.1max) space. Experimentally, a higher accuracy in frequency resolution of the minimum and maximum first harmonic acoustic pressure power response would be needed for the same relative error in estimation of attached solids mass loading in comparison to bubble equilibrium radius.

[0303] As shown in FIG. 10, plotting the attached solids mass as a function of (f.sub.1max, f.sub.1min−f.sub.1max) clearly demonstrates the complexity of the relationship between attached solids mass and the acoustic resonance interference first harmonic extrema frequencies. At relatively low resonance maximum frequency and low difference between the frequencies of the acoustic power extrema the high attached solids mass is a slowly varying function. The triangular region outside the contours at lower values of f.sub.1maxis associated with attached solids mass loadings slightly above the 10 mg upper limit of the contour plot. Relatively large changes in the frequency of the acoustic power resonance maximum and difference between the minimum and maximum frequency are associated with relatively small changes in the attached solids mass in this region. In these circumstances, observations of the resonance frequency and difference between resonance and interference minimum frequency would lead to a robust prediction of attached solids mass. However, at very low values for the difference between the acoustic response extrema frequencies and also at relatively low resonance frequency but high frequency difference, the contours of added mass are densely packed. In these circumstances, there would be much larger error margins on predictions of attached solids mass loading.

[0304] Estimation of the Properties of a Liquid-Like Medium Based on Bubble Active Acoustic Response

[0305] The inventor has additionally determined that the active acoustic response of a gaseous bubble in a liquid-like medium, can in turn be used to estimate certain properties of the liquid-like medium. Estimators for the properties of a liquid-like medium based on the active acoustic response of a ‘clean’ (nil attached solids) bubble are here presented for two cases representative of negligible and significant viscosities for the combined liquid and any bubble surface layer. [0306] 1. A ‘Clean’ Bubble (Nil Attached Solids) and Negligible Liquid and Bubble Surface Dilatational Viscosities

[0307] Previously presented equations (49) and (67) can be combined to derive the following estimator for bubble equilibrium radius based purely on the first two characteristics (frequencies of the resonance fundamental maximum and interference minimum responses) and the geometry of the active monitoring system:

[00108]$\begin{array}{cc}{R}_0=\frac{\left[1-{\left(\frac{f_{1.Math.\max }}{f_{1.Math.\min }}\right)}^2\right]}{\left[\frac{1}{r}+\frac{1}{r_{\mathrm{SB}}}\right]}.& \left(73\right)\end{array}$

[0308] Equation (65) then allows the density of the surrounding liquid-like medium (denoted here as σ.sub.Sl) to be estimated from R.sub.0, f.sub.1max, the medium ambient pressure p.sub.0, surface tension at equilibrium bubble radius σ.sub.o and the gas polytropic index κ as follows (assuming nil elastic compression modulus for an unencapsulated bubble):

[00109]$\begin{array}{cc}{\rho }_{\mathrm{SI}}=\frac{3.Math.\kappa .Math..Math.p_0+\frac{2.Math.{\sigma }_0}{R_0}.Math.\left(3.Math.\kappa -1\right)}{4.Math.{\pi }^2.Math.R_0^2.Math.f_{1.Math.\max }^2}.& \left(74\right)\end{array}$

[0309] Assuming that the slurry (liquid-like medium) between the bubble and the acoustic receiver is two phase, consisting of solid (particle) and pure liquid phases with densities ρ.sub.s and ρ.sub.L, respectively, the following equation for φ.sub.s, the solids volumetric fraction, is readily derived:

[00110]$\begin{array}{cc}{\phi }_S=\frac{{\rho }_{\mathrm{SI}}-{\rho }_L}{{\rho }_S-{\rho }_L}.& \left(75\right)\end{array}$

[0310] Equations (73), (74) and (75) can be used to estimate the bubble equilibrium size, liquid-like medium density and solids volumetric fraction of the medium respectively, from the bubble active acoustic response characteristics f.sub.1max and f.sub.1min. These equations apply in the case of a ‘clean’ (nil attached solids), unencapsulated bubble when there is negligible influence of the viscosity of the medium on the bubble active acoustic response characteristics. [0311] 2. A ‘Clean’ Bubble (Nil Attached Solids) and Significant Liquid or Bubble Surface Dilatational Viscosities

[0312] Equation (68) allows the density of the surrounding liquid-like medium in the case of a ‘clean’ bubble to be estimated from R.sub.0, f.sub.2max, the medium ambient pressure p.sub.0, surface tension at equilibrium bubble radius σ.sub.o, encapsulating layer elastic compression modulus χ.sub.0 and the gas polytropic index κ to be written as follows:

[00111]$\begin{array}{cc}{\rho }_{\mathrm{SI}}\approx \frac{3.Math.\kappa .Math..Math.p_0+\frac{2.Math.{\sigma }_0}{R_0}.Math.\left(3.Math.\kappa -1\right)+\frac{4.Math.\mathrm{\chi 0}}{\mathrm{R0}}}{{\mathrm{\pi 2f}}_{2.Math.\max }^2.Math.R_0^2}.& \left(76\right)\end{array}$

[0313] Equation (76) can be inserted into Eqn. (70) to derive the following equation for the bubble equilibrium radius based purely on the first three characteristics (frequencies of the resonance fundamental maximum, interference minimum and second harmonic maximum responses) and the geometry of the active monitoring system:

[00112]$\begin{array}{cc}{R}_0\approx \frac{\left[2-\frac{f_{2.Math.\max }^2}{4}.Math.\left(\frac{1}{f_{1.Math.\max }^2}+\frac{1}{f_{1.Math.\min }^2}\right)\right]}{\left[\frac{1}{r}+\frac{1}{r_{\mathrm{SB}}}\right]}.& \left(77\right)\end{array}$

[0314] Equation (77) is applicable for arbitrary medium viscosity and bubble encapsulating layer dilatational viscosity. It should be noted that in cases where the liquid and bubble dilatational viscosities are negligible, f.sub.2max=2f.sub.1max, resulting in Eqn. (77) being reduced to Eqn. (73) as expected.

[0315] A net viscosity μ of the combined medium and the encapsulating layer around the bubble can be defined in terms of the shear viscosity μ of the medium and the surface dilatational viscosity κ.sub.S of any elastic layer encapsulating the bubble by:

[00113]$\begin{array}{cc}{\mu }^{\prime }=\mu +\frac{{\kappa }_S}{R_0}.& \left(78\right)\end{array}$

[0316] The net viscosity can then be estimated by a rearrangement of Eqn. (61) as follows:

[00114]$\begin{array}{cc}{\mu }^{\prime }=\frac{R_0}{4.Math.ϛ}.Math.\sqrt{\frac{{\rho }_{\mathrm{SI}}}{2}\left[3.Math.\kappa .Math..Math.p_0+\frac{2.Math.{\sigma }_0}{R_0}.Math.\left(3.Math.\kappa -1\right)+\frac{4.Math.{\chi }_0}{R_0}\right]\left[\frac{{\left(2-E\right)}^2-{ϛ}^2.Math.E^2}{2-E}\right]}.& \left(79\right)\end{array}$

[0317] Here the dimensionless coefficient E defined by Eqn. (49) is defined by use of Eqn. (77) as

[00115]$\begin{array}{cc}E=2-\frac{f_{2.Math.\max }^2}{4}.Math.\left(\frac{1}{f_{1.Math.\max }^2}+\frac{1}{f_{1.Math.\min }^2}\right).& \left(80\right)\end{array}$

[0318] The coefficient ζ, is defined by Eqn. (52) as follows:

[00116]$\begin{array}{cc}ϛ=\frac{1+{\stackrel{\_}{\omega }}^2}{1-{\stackrel{\_}{\omega }}^2};.Math..Math.\stackrel{\_}{\omega }=\frac{f_{1.Math.\max }}{f_{1.Math.\min }}.& \left(81\right)\end{array}$