Studies on the Tempo of Bubble Formation in Recently Cavitated Vessels: A Model to Predict the Pressure of Air Bubbles

A cavitation event in a vessel replaces water with a mixture of water vapor and air. A quantitative theory is presented to argue that the tempo of filling of vessels with air has two phases: a fast process that extracts air from stem tissue adjacent to the cavitated vessels (less than 10 s) and a slow phase that extracts air from the atmosphere outside the stem (more than 10 h). A model was designed to estimate how water tension (T) near recently cavitated vessels causes bubbles in embolized vessels to expand or contract as T increases or decreases, respectively. The model also predicts that the hydraulic conductivity of a stem will increase as bubbles collapse. The pressure of air bubbles trapped in vessels of a stem can be predicted from the model based on fitting curves of hydraulic conductivity versus T. The model was validated using data from six stem segments each of Acer mono and the clonal hybrid Populus 84K (Populus alba × Populus glandulosa). The model was fitted to results with root mean square error less than 3%. The model provided new insight into the study of embolism formation in stem tissue and helped quantify the bubble pressure immediately after the fast process referred to above.Vulnerability curves (VCs) have been viewed as a good measure of the drought resistance of woody stems (Cochard et al., 2013). Increasing drought increases the xylem tension (T) and eventually induces cavitation of the water in conduits when the T exceeds a certain threshold (Sperry and Tyree, 1988; Sperry et al., 1996). A cavitated vessel first fills with water vapor and eventually fills with air at atmospheric pressure because of Henry’s law, which describes gas equilibrium at the water/air interface. The time required for the progress mainly depends on the penetration rate of air into the recently cavitated vessel lumen via diffusion through the liquid phase.Previous studies were made about how fast bubbles disappear in embolized stems because of the solubility of air in water when water pressure exceeds atmospheric pressure, and the process takes 10 to 100 h depending on conditions (Tyree and Yang, 1992; Yang and Tyree, 1992). The tempo of bubble disappearance was measured by following the rise in stem hydraulic conductivity (k_h) versus time. The theory of Yang and Tyree (1992) relied on the same principles used in this article (Henry’s law, Fick’s law, and the ideal gas law), but modeling and experiments were done at pressures between 1 and 3 times atmospheric pressure rather than subatmospheric pressure (negative pressure). However, much less is known about the tempo of bubble formation in recently cavitated vessels (Brodersen et al., 2013). If the progress of embolus formation takes several minutes, then no changes in conductivity could be observed with available techniques, but if it takes hours, then the tempo of bubbles can be studied by rapidly inducing cavitation with increasing T and after cavitation induction measuring the influence of T on stem k_h as T is reduced gradually to zero. If air bubbles are at a pressure (bubble pressure [P_b*]) lower than a threshold near atmospheric pressure, bubbles ought to collapse when T decreases according to the ideal gas law and Henry’s law (see theory below). The consequence of bubble collapse will be partial filling of vessels with water and the rest with air bubbles. The partial filling of water in a recently cavitated vessel ought to increase the lumen conductivity from zero and connect the embolized vessel to adjacent conductive vessels and, hence, ought to increase the conductivity of the stem by an additional flow pathway (Wheeler et al., 2005; Hacke et al., 2006). The vascular system of stems is a complicated network with vessels of different lengths, diameters, and orientation (Evert, 2006), and the complex vessel network makes the additional pathway possible. Therefore, bubble collapse could be detected through the impact of T on the k_h of the stems in a way that is very similar to the methods used by Yang and Tyree (1992) but requires a more sophisticated centrifuge technique to induce embolism.Many studies have assumed that the bubble pressure in newly cavitated vessels ought to be near atmospheric pressure, and no corrections for bubble pressure have been taken in measuring percentage loss of conductivity (PLC) when T is lower than a critical threshold (Li et al., 2008; Wang et al., 2014a). As a result of bubble collapse, the measured k_h under a mild T should be higher than that under high T (greater than 0.5 MPa). And the lower the initial bubble pressure, the more bubbles collapse with decreasing T.The aim of this study is to construct a model that estimates average bubble pressure in partly embolized stems from the functional dependence of k_h on T, and with this model, we can further our understanding of the tempo of bubble formation in stems. Here, we will argue that the tempo of bubble formation is in two phases: an initial rapid phase (seconds to minutes to complete) followed by a much slower phase (many hours to complete). Since there is no method for measuring the rapid phase, the rapid phase will be described theoretically below. Next, a theory will be developed that allows the estimation of the pressure of air in recently formed bubbles in vessels during the slow phase. An experimental validation of the model will follow that will yield values of bubble pressure within the first 1 to 2 h following the fast phase of embolism formation in vessels.