Interaction between encapsulated microbubbles: A finite element modelling study**Projects supported by the National Natural Science Foundation of China (Grant Nos. 11474161, 11474001, 116741731, 1774166, 11774168, 81527803, 81627802, and 81420108018), the Fundamental Research Funds for the Central Universities, China (Grant No. 020414380109, and the Qing Lan Project, China.

Abstract

Theoretical studies on the multi-bubble interaction are crucial for the in-depth understanding of the mechanism behind the applications of ultrasound contrast agents (UCAs) in clinics. A two-dimensional (2D) axisymmetric finite element model (FEM) is developed here to investigate the bubble–bubble interactions for UCAs in a fluidic environment. The effect of the driving frequency and the bubble size on the bubble interaction tendency (viz., bubbles’ attraction and repulsion), as well as the influences of bubble shell mechanical parameters (viz., surface tension coefficient and viscosity coefficient) are discussed. Based on FEM simulations, the temporal evolution of the bubbles’ radii, the bubble–bubble distance, and the distribution of the velocity field in the surrounding fluid are investigated in detail. The results suggest that for the interacting bubble–bubble couple, the overall translational tendency should be determined by the relationship between the driving frequency and their resonance frequencies. When the driving frequency falls between the resonance frequencies of two bubbles with different sizes, they will repel each other, otherwise they will attract each other. For constant acoustic driving parameters used in this paper, the changing rate of the bubble radius decreases as the viscosity coefficient increases, and increases first then decreases as the bubble shell surface tension coefficient increases, which means that the strength of bubble–bubble interaction could be adjusted by changing the bubble shell visco-elasticity coefficients. The current work should provide a powerful explanation for the accumulation observations in an experiment, and provide a fundamental theoretical support for the applications of UCAs in clinics.