Phenomenology of bubble-collapse-driven penetration of biomaterial-surrogate liquid-liquid interfaces

The paper presents phenomenology of interaction and penetration of liquid-liquid material interfaces initiated by shock-driven collapse of single and multiple microbubbles situated near the material interface. Previous experimental studies have established such a generic setting as relevant for the investigation of sonoporation, i.e., the perforation of live cells by microbubble collapses. We consider a planar or spherical, single- or dual-layer, material interface between a gelatin material and water. A single or several ideal-gas microbubbles are positioned near the interface. Bubble collapse is initiated by a shock wave with a pressure profile specific to laser generation and is flat when hitting the gas-water interface. The interfacial acoustic impedance match singles out the collapse-induced re-entrant jet as main event. High-resolution sharp-interface numerical methods are employed to ensure that wave dynamics, hydrodynamics, and interface transporting are accurately resolved. Bubble configurations are varied between single and double and between attached and with standoff distance. Parameters varied are shock-wave peak pressure and viscosity ratio between single and double layers of gelatin and the surrounding water. For inertia-dominated cases, two regimes are observed, the first characterized by linear growth of the penetration depth and the second by a $t^{2/3}$ scaling. The latter range is affected by viscosity which reduces penetration speed. The results show that process parameters, in particular shock overpressure, control not only penetration depth but also the size of the interface perforation, indicating means to steer processes in biomedical applications.

Figure 1

(a) Considered configurations: (i) planar single interface, (ii) spherical interface, and (iii) spherical two-layer material. Gas bubble, tissue and cell material, cell core, and liquid are colored by white, yellow, red, and blue, respectively. The computational domain size is $L=80R_0,W=8R_0$; the bubble initial radius is $R_0=0.8\mathrm{mm}$ for (i) and $R_0=2.5\mu \mathrm{m}$ for (ii) and (iii). (b) Evolution of wave dynamics and hydrodynamics: top row for the planar interface; bottom row for the spherical interface; middle row shows enlarged subsections; wave dynamics is visualized by density-gradient contours. The overpressure for all cases is $p_s=163.2\text{MPa}$ and viscosity is zero.

Figure 2

Time history of the equivalent bubble radius $R\left(t\right)$ during the shock-bubble interaction near a tissuelike material for various overpressures $p_s$ (a) and the corresponding bubble collapse time (b). The collapse time, indicated by the minimal equivalent bubble radius, agrees well with the correlation $1.7t_c^R$, where $t_c^R$ is the Rayleigh collapse time [43]. See Movies S1 and S2 of the Supplemental Material [46].

Figure 3

(a) Time evolution of the penetration compared to the experimental data of Kodama and Takayama [8] for an overpressure of $p_s=10.2\text{MPa}$. (b) Normalized penetration depth $L_p/R_0$ vs $(t-t_p)/t_c^R$ for $10.2\text{MPa}\le p_s\le 163.2\text{MPa}$. $t_s$ indicates the transition between two scaling regimes.

Figure 4

Interface evolution during bubble-collapse-driven penetration processes for a single cell. Panels (a) and (b) show the single-liquid cell model for high $\left(p_s=163.2\mathrm{MPa}\right)$ and low $\left(p_s=40.8\mathrm{MPa}\right)$ overpressures, respectively. The results of the compound liquid cell model for high overpressure $\left(p_s=163.2\mathrm{MPa}\right)$ are shown in panel (c). See Sec. 3 and Movie S3 of the Supplemental Material [52].

Figure 5

(a) Penetration-depth evolution for the single-liquid-drop model for various viscosities. (b) Penetration-depth evolution for the compound liquid drop; the outer-layer viscosity is ${\mu }_1=10{\mu }_l$ and the core viscosity ranges from ${\mu }_2=2{\mu }_1$ to $10{\mu }_1$.

Figure 6

(a) Perforation diameter $L_w$ for the single drop model [(i)–(iv)] and the compound drop model [(v) and (vi)]. At $p_s=163.2\mathrm{MPa}$, the viscosities are ${\mu }_1=10{\mu }_l$ and ${\mu }_1=100{\mu }_l$ for (a, i) $(t=80$ ns) and (a, ii) $(t=100$ ns), respectively. Accordingly, panels (iii) and (iv) show the interface for low overpressure $p_s=40.8\mathrm{MPa}$. The pit diameter $L_w$ is shown in panel (b) for the single-drop model at $p_s=163.2\mathrm{MPa}$ (solid line) and $p_s=40.8\mathrm{MPa}$ (dashed line).

Figure 7

Interface evolution during bubble-collapse-driven penetration processes for a single cell and multiple microbubbles [two bubbles in panel (a) and three bubbles in panel (b)] under a large overpressue $\left(p_s=163.2\mathrm{MPa}\right)$. See Sec. 5 and Movie S4 of the Supplemental Material [58].

Figure 8

Time history of penetration depth for the shock-single-cell interaction problem for the single-liquid-drop model and a microbubble array. The red, green, and blue symbols refer to the penetration with single, double, and triple microbubbles. The solid lines indicate the power laws 1.0 and 2/3.