Physical impact of a surfactant on the nonlinear oscillations of a microbubble considering a dynamic surface tension and subject to an external acoustic field

The dynamics of microbubbles under the action of external acoustic forces has become particularly important in several applications. In this work, we are particularly interested in studying the transport of surfactant molecules to the surface of an oscillating microbubble, considering the impact that the dynamic surface tension and temporal evolution of the radius of the microbubble has when an acoustic pressure as a driving force is used to promote the nonlinear oscillations. The resulting governing equations to predict the radius of the microbubble and the evolution of the surfactant at the surface are written in dimensionless form. For these equations, we identify two fundamental dimensionless parameters: the Gibbs elasticity $E$, and the cohesive (or repulsive) parameter $K$. Using the physical domain $0\le E\le 10$ and $-13.2\le K\le 13.2$, and considering that the diffusive Péclet number is large, as occurs in some applications, the surfactant concentration equation is solvable by using a similarity transformation, whereas the Rayleigh-Plesset-type equation that includes the influence of the previous parameters $E$ and $K$ is solved by the fourth-order Runge-Kutta method. When the numerical predictions are compared with the well-known cases $E=K=0$, strong deviations reveal that the oscillation mechanisms can be significantly altered.

Figure 1

(a) Schematic sketch of an encapsulated bubble with surfactant molecules under an acoustic excitation, with the initial and boundary conditions. (b) Close-up of the wall of the encapsulation, composed of a monolayer of surfactant molecules and considering a zero thickness because of the difference of magnitude between the radius of the bubble and the molecule size.

Figure 2

The distribution of the dimensionless bulk surfactant concentration $\psi$ and its derivative $\frac{d\psi }{d\xi }$ as a function of the dimensionless coordinate $\xi$, for five different values of the parameter $\psi \left(0\right)$, according to Eqs. (24) and (25).

Figure 3

Comparison between the model developed in this work and the Marmottant model [14]. Dimensionless radius $a$ as a function of the dimensionless time $\tau$ for ${\beta }_0=0.321$, ${\beta }_A=1.1257$, $\mathrm{Re}=17.322$, $\mathrm{We}=0.468$, and $\delta =0.88$.

Figure 4

Dimensionless radius $a$ as a function of the dimensionless time $\tau$ for three different values of the initial surface concentration, ${\psi }^s\left(0\right)$, ${\beta }_A=1.1257$, $E=0.19$, $K=13.2$, and $\delta =4.244$.

Figure 5

Dimensionless surface concentration ${\psi }^s$ as a function of the dimensionless time $\tau$ for four different values of the initial surface concentration, ${\psi }^s\left(0\right)$, ${\beta }_A=1.1257$, $E=0.19$, $K=13.2$, and $\delta =4.244$.

Figure 6

Dimensionless surface pressure ${\pi }^{*}$ as a function of the dimensionless time $\tau$ for four different values of the initial surface concentration, ${\psi }^s\left(0\right)$, ${\beta }_A=1.1257$, $E=0.19$, $K=13.2$, and $\delta =4.244$.

Figure 7

(a) Dimensionless radius $a$ versus the dimensionless time $\tau$ for five different values of the elasticity parameter $E$: ${\psi }^s\left(0\right)=0.5$, ${\beta }_A=1.1257$, $K=13.2$, and $\delta =4.244$. (b) Close-up, showing the oscillation for $E=5$ around the bubble radius $a\approx 0.708$.

Figure 8

Dimensionless surface pressure ${\pi }^{*}$ versus the dimensionless time $\tau$ for five different values of the elasticity parameter $E$; ${\psi }^s\left(0\right)=0.5$, ${\beta }_A=1.1257$, $K=13.2$, and $\delta =4.244$.

Figure 9

Dimensionless radius $a$ as a function of the dimensionless time $\tau$ for three different values of the molecular interaction parameter $K$: ${\psi }^s\left(0\right)=0.5$, ${\beta }_A=1.1257$, $E=0.19$, and $\delta =4.244$.

Figure 10

Dimensionless surface concentration ${\psi }^s$ as a function of the dimensionless time $\tau$ for three different values of the molecular interaction parameter $K$: ${\psi }^s\left(0\right)=0.5$, ${\beta }_A=1.1257$, $E=0.19$, and $\delta =4.244$.

Figure 11

Mean values of the dimensionless radius $a_{\mathrm{mean}}$ and the surface concentration ${{\psi }^s}_{\mathrm{mean}}$ as functions of (upper) the elasticity parameter $E$, and (lower) the molecular interaction parameter $K$, for ${\psi }^s\left(0\right)=0.5$. The values of the rest of the parameters are indicated in each graphic.