A modified Marmottant model to study the effects of a shell rupture on the subharmonic threshold of encapsulated microbubbles

Document Type : Research Article


Faculty of Mechanical Engineering, Shahid Rajaee Training Teacher University, Tehran, IRAN


This study considers the radial behavior of a coated microbubble after a shell rupture using the Marmottant model. The surface tension of the encapsulated microbubble should equal the free bubble in the rupture state of the Marmottant model. Despite the assumption that the bubble is considered free in the third state, dilatational interfacial viscosity is constant in the equation in this model. This paper assumes that dilatational interfacial viscosity decreases gradually after shell rupture until it becomes zero. The decrease of dilatational interfacial viscosity caused by the shell rupture significantly affects radial behavior and the nonlinear response of the encapsulated microbubble, such as subharmonic response. Because the subharmonic response is extensively used in ultrasound imaging, the effect of a decrease in dilatational interfacial viscosity on the subharmonic threshold needs to be investigated. In figures showing the radius versus time and the frequency response of the coated microbubble, it is observed that at high excitation pressure, the proposed model is more nonlinear than the Marmottant model, resulting in a lower subharmonic threshold.

Graphical Abstract

A modified Marmottant model to study the effects of a shell rupture on the subharmonic threshold of encapsulated microbubbles


  • A modified model for the Marmottant model was investigated.
  • κS in this model gradually decreases after rupturing the shell until it becomes zero. 
  • The model offers a lower subharmonic threshold than the Marmottant model.


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