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As illustrated in figure, normal to the surface of the liquid thread, the pressure and shear stress difference of the fluids are balanced by the interfacial tension force, |
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Using Microbubbles for Improving Sensitivity in Molecular Imaging |
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Theory |
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Microbubble Production |
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where po, μo, pw, and μw are the pressures and viscosities of the oil and water phase, respectively, σ the interfacial tension of the water–oil interface, and rt is the radius of the liquid thread. The thread radius decays due to the continuous disturbances initiated by the extension of liquid thread. The head of the partially formed droplet passes the orifice while the neck of the droplet continues to decrease until a sharp point is developed during which the droplet detaches from the thread. The breaking of the thread occurs on a finite time governed by the shear rate and flow rates. During this time, a finite amount of liquid volume travels through the orifice into the expanding nozzle and the tip of the thread. Within the liquid thread, due to the difference in shear stresses at the orifice and in the expanded part of the nozzle, a pressure force is created in the direction of flow. This force is combined with the tangential shear stress exerted by the oil phase to separate the droplet from the liquid thread. |
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Gas composition is a major factor in determining the length of time a microbubble lasts in the circulation. The diffusivity of a gas is described by the Ostwald partition coefficient |
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Filled Gas |
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which is equal to the ratio of the amount concentrations C in the liquid and in the gas. Nitrogen has a high Ostwald coefficient (L = 14 480 at 35 uC) and therefore a higher water solubility compared with a PFC gas such as n-C3F8 (L = 530 at 35 uC). The following formula describes the rate of bubble shrinkage by the dissolution of gas in the bloodstream, |
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where D is the gas diffusivity in water, L is the Ostwald partition coefficient, Patm is atmospheric pressure, P* is an excess pressure term impacted by the blood pressure and gas metabolism, c is the interfacial tension, r is the bubble radius, and t is time: |
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The Laplace pressure is the main mechanism responsible for the disappearance of a bubble. The use of high molecular weight gases such as perfluorocarbons reduces diffusion out of the microbubble core and enhances bubble stability and circulation lifetime by counterbalancing the Laplace and blood pressures. |