Water ascends plants in a metastable state under tension, meaning that the xylem sap pressure is more negative than that of the vapor pressure of water . This is a physical effect. In the process cavitation, which is defined as void formation in xylem sap, often occurs in conduits . Cavitation of water columns within the xylem is deleterious for plant water relations because it results in embolism thereby reducing hydraulic conductivity . Cavitation is induced by air seeding, which can occur when air passes through the pores of pit membranes to form bubbles  . Cavitation is also caused by tiny air bubbles adhering to cracks in vessel walls . Zimmermann  studied the process of air seeding. He concluded that once a bubble formed by air seeding enters into a vessel, it immediately expands until the tension forces on the wall are released.
Nowadays some researchers pay their attentions to the facts: angiosperm xylem contains abundant hydrophobic surfaces; there are insoluble surfactants in xylem and bordered pit membranes play a crucial role in drought-induced embolism formation via the process of air-seeding etc.  . Whether the capillary failure is an appropriate physical model comes to be a question . Then, from the experiments    , it is obvious that the former hypothesis of air seeding is still effective to the xylem of some trees although cavitation in lipid bilayers has negative pressure stability limit .
Where does an air seeding event take place? Considering the potential importance of the rare pit hypothesis, Plavcová et al.  suggested that more attention should be paid to the structural irregularities, as those may represent the rare sites ultimately responsible for air-seeding.
Isolated conduit has been seen, which might be caused by another mechanism  . The development of nanobubbles snapped off at pit membranes can also cause cavitation events . These all may involve the growth of pre-existent air bubbles in xylem.
Ponomarenko et al.  distinguished two types of optical events. The first is the “nucleation” events, starting in a fully wet area, which might be caused by the growth of pre-existent air bubbles. The second is the “air-seeding” events, being defined as the appearance of bubbles near an already gas-filled conduit. The definition of types of cavitation in this paper follows that defined by Ponomarenko et al. .
Three ways of cavitation by air seeding have been proposed  (In the article  the word “way” was defined as word “type”). The types of cavitation by the growth of pre-existent air bubbles in xylem should be given more attention.
The two types of cavitation are all related to the equilibrium, stability and development of air bubbles in xylem. Analysis of bubble expansion by mechanism and by the equilibrium criterion of Helmholtz function has been made, which is based on the equation of bubble balance  . This is a univariate cubic equation. Taking mole number n of air in a bubble as an independent variable, its analytic solution has been made . A new formula for finding the root of univariate cubic equation was proposed by Fan , which is simpler than the Caldan’s. Using Shenjin formula and taking absolute water pressure as an independent variable, this paper gives the exact solution of the equation of bubble balance under certain conditions. As gas super-saturation is likely to occur in xylem sap almost daily , here the number n is regarded as a constant. And the stability of the equilibrium of an air bubble in its different radius ranges is obtained by the way different from our previous article .
For the first type this paper puts forward two ways of cavitation, which are the same with the first two ways of air seeding except of air reservoirs, etc. Then, the relationship of the two types of cavitation is given.
2. Air Babbles in Xylem
2.1. Equilibrium Equation of Air Bubbles
Provided there is a bubble of radius r with n mole air in xylem sap. In order to simplifying the problem, several assumptions were made. First, because the water saturation vapor pressure in a bubble is generally less than 0.0023 MPa at 20˚C, comparing with atmospheric pressure , it is ignored. We also ignore some facts, including abundant hydrophobic surfaces and insoluble surfactants in xylem.
According to the ideal gas law , the gas pressure P of the bubble of volume should be
When a bubble is in an equilibrium, we have:
The relationship among , atmospheric pressure and xylem pressure is . The point of intersection of the two curves and , or at , indicates temporary equilibrium of a bubble (Figure 1).
Figure 1. and at different values of .
Thus, the positive real roots of the following equation
is the radii of the bubble in an equilibrium.
Above system consists of three parts: an air bubble, the surrounding water and the interface between the air and the water. Corresponding to a fluctuation, the changes of its Helmholtz function are: for the air, for the water, and for the increase of the gas/water interface ( ). Thus, the total change of the Helmholtz function is , or
Integrating Expression (4) gives
Once Helmholtz function (Figure 2) reaches an extreme, or , the bubble will attain its equilibrium. Thus, from expression 4 we also have Equation (3).
2.2. Solution of Equation (3)
Letting the left side of Equation (3) be a function of r gives
Therefore, the real roots of Equation (3) are the intersections of the curve with r-axis and those the abscissa values of which are more than zero are the radii of the bubble in equilibrium (Figure 3).
1) When , .
2) If , corresponding to the following equation
Figure 2. Graphs of Helmholtz function F(r).
Figure 3. Graphs of .
the analytic solution of Equation (3) can be gotten by Shenkin formula .
For Equation (3), we obtained , and . Then, we got
If , , or . From Shenjin formula③ , , , . When , , . When , and . By other ways, both and ( ) have been gotten     except of considering water vapor pressure in the articles  . Thus, can be replaced by .
① When , using Shenjin formula ② and ④ , the analytical solution of can been obtained, knowing that . As xylem pressure is often negative, we do not pay more attention to it.
② When there are several situations as follows.
a) If , meaning , or , from Shenjin formula ④ , there are three real roots for Equation (3). Letting in Shenjin formula ④ . By calculating, and . Then we obtained the solutions of Equation (3) as follows.
i) . When and , then . When , . Therefore, is in Figure 3, the values of which are in the range of .
ii) . When , . If , . The values of all are negative and should not be considered.
iii) . When , we got . While , . Thus, is in Figure 3, the values of which are in the range of .
Therefore, in the range of ,
( ) (8)
are the solutions of Equation (3), which are none than the formulas
b) When , meaning , or , according to Shenjin formula ②  there are a conjugate pair of complex roots and one real root for Equation (3). The real root is negative. This means that a gas bubble could not be in any equilibrium when .
To sum up, if , Equation (3) has a positive real root ; If , its positive real solution is ( ); If , the positive real root is . In the range of , the positive real roots of Equation (3) are Formula (8). The relationship of radii of an air bubble is .
2.3. Stability of Bubble Equilibrium
The stability of an air bubble which is in equilibrium depends on Formula (6b). For , if , reaches its minimum and the equilibrium of the bubble is stable. In turn, , arrives at its maximum, the equilibrium of the bubble is unstable.
1) When , formula for all . and are the minima and the bubbles of radius or in xylem are stable.
2) When , there are two roots and ( ) for Equation (3). When , we have . Thus, reaches its extremum at and (Figure 3).
For , , . Therefore, is a minimum and the equilibrium of an air bubble of radius is stable.
For , , . Therefore, is a maximum and the equilibrium of an air bubble of radius is unstable.
3) If , we got and . Therefore, reaches its inflection point (Figure 2), leading the equilibrium of the bubble of is unstable.
4) When , a gas bubble could not be at any equilibrium.
Every one of bubbles has its own , also its own , being called its Blake threshold pressure, and its , or Blake critical radius  .
3. Two Types of Cavitation
3.1. First Type of Cavitation: Growth of Pre-Existent Air Bubbles in Conduits
Suppose that along with the decreasing of a bubble with n mole air in a conduit of radius enlarges stably. If its Blake radius (or based on , ), before its exploding at , it has become long shaped, leading the bubble only to expand and lengthen gradually. This is the first way of cavitation. Only if , or , can it explode at its to form a long bubble, and to lengthen gradually. This is the second way of cavitation, or the way of expanding—exploding, becoming a long bubble—lengthening gradually. Therefore, how a bubble develops depends on which of and is larger, or on which of and is larger. Thus, for a pre-existent air bubble the boundary of above two ways of cavitation is its at . The action of these two ways of cavitation all are the same with the first two ways of cavitation by air seeding  except of forming isolated embolized conduits without any reservoir.
3.2. Second Type of Cavitation: Air Seeding
When an air seed is sucked into a conduit of radius from atmosphere through a pore of radius in pit membrane, its initial radius equals and initial gas pressure . In the range of pressure its radius should be , or . As decreases, it will develop like the growth of a pre-existent air bubble in a conduit, presenting the first or second ways of cavitation but with air reservoirs .
If a seed enters a conduit of radius through a pore of radius in the conduit wall from atmosphere and will break up at with , there is a relationship . Therefore,
From and , the pressure at which the seed enters the conduit is
However, at the moment the radius of the seed reaches . Then, it should become a long shaped bubble. Thus, the exploding event might disappear.
Using formulas (9) and (10), and combining the results of the articles   the following conclusions are obtained.
1) In the range of and , the first way of cavitation will form.
2) In the range of and the second way of cavitation will take place.
3) In and , soon after an air seed is sucked into a conduit, as its radius is it will explode immediately and the conduit will be filled with the seed air instantly, presenting the third way of cavitation.
The experiments   show that as primary xylem conduits were directly connected to air-ﬁlled spaces within the pith, inter-conduit air seeding was the primary mechanism. Thus, in should be replaced by internal air pressure , causing some data to be recalculated.
4. Relationship of the Two Types of Cavitation
For the development of air seeds, Table 1 gives the values of radii (in bold) of some seeds, which are just sucked into conduits, and their corresponding pressures (in bold). Also the values of corresponding , , and of the seeds of radii , and . For a seed of radius (or ), using the formula 8 the corresponding (or ) can be calculated. Note the two states of the bubble of radius and are at the same water pressure .
If a seed of radius at enters a conduit of radius , from formulas 9 and 10, we got and (Table 1 line 2). In the range of the bubbles will expand gradually (Table 1 line 1 and 2), presenting the first way of cavitation. In the range of , the bubbles of , and will expand to their respective , presenting the second way of cavitation (Table 1 line 3 ® 5). For the seed of radius during the dropping of it will break up at with before its radius reaches . Thus, in the range of the bubble of radius in the parentheses does not exist (Table 1 line 5). When the air seeds of radius will explode soon after their entering, presenting the third way of cavitation (Table 1 lines 6 ® 9).
All air seeds of radius , , or (lines 1 ® 5) by air seeding in Table 1 can also be regarded as pre-existent air bubbles. Occasionally the air seeds of radii by air seeding can shrink to their corresponding . Thus, for this kind of bubbles of radii (lines 7 ® 9), during the decreasing of , they will expand and burst at their with radii . Thus, all bubbles of radii , , or , no matter how they formed before, can be regarded as pre-existent, including the bubble of radius in line 6. For a conduit of as long as ( ) during decreasing, except of the bubbles in the first two lines in Table 1, all other bubbles can develop from , or to , presenting the second way of cavitation.
This indicates that in the range of , for air seeding the third way of
Table 1. Development of air bubbles in a conduit of radius . After an air seed of , , or (in bold) with n mole air at enters the conduit at pressure (in bold), three ways of cavitation will take place (lines 1 ® 2, first way; lines 3 ® 5, second way and lines 6 ® 9, third way). On the other hand all the bubbles in Table 1 can be regarded as pre-existent. During the decreasing of they will expand. The bubbles of radius in lines 1 ® 2 will grow gradually. The others will expand to their respective radii at to explode. The bubbles with the values in parentheses can’t exist. , , .
cavitation will occur; but for pre-existent air bubbles of radius the second way will take place.
From Table 1 we can see that the more the amount of a bubble, the larger its Blake critical radius and the higher its Blake threshold pressure. This means that a bubble with more is prone to burst at higher pressure and only the nanobubbles with a small amount of air can exist steadily in larger ranges of water pressures. For example, in Table 1 at all bubbles of radii would burst at with . But in its last line, a bubble of radius at will break at with the radius . Thus, in the large range of the bubble can exist temporarily steadily.
The scale bars shown in the figures of the article  is 50 μm. The nanobubbles, the radius of which was shorter than 0.4867 μm (Table 1. line 6), might not be seen under light microscope before their bursting. Thus, the first type of cavitation proposed by Ponomarenko et al.  might be caused by nanobubbles.
There are insoluble surfactants in xylem sap. Because of surfactants effect, the values of may decreases. From we can see that the smaller the , the smaller the absolute value of , meaning that at higher water pressure an air bubble will burst and a cavitation event will occur easily. Thus, the values in Table 1 should be recalculated.
For the equation of bubble balance, using Shenjin formula, which is simpler than the Caldan’s, this paper gets its analytic solutions. The stability of equilibrium of air bubbles was made by the way different from the previous in the article . Two types of cavitation are analyzed further. For the first type of cavitation two ways can occur, which are the same with the first two ways of air seeding except of air reservoirs. Moreover, for the first way of the two types, the range of water pressures is the same. For the second way of the first type the range of water pressures is much wider, or the pressure range equals the pressure sum of the second and third ways of air seeding.
Through the specific data the relationship between the two types is given.
This work was financially supported by the National Natural Science Foundation of China (#30270343).
: absolute water pressure in a conduit
r: radius of a spherical air bubble
: radius of a spherical air bubble at
: radius of a spherical air bubble at
: bubble volume
R: gas constant
T: absolute temperature
n: molar number of air in a bubble
: atmospheric pressure
P: gas pressure of a bubble
: gas pressure of an air bubble in equilibrium
: surface tension of xylem water
: Blake threshold pressure of an air bubble
, Blake critical radius of an air bubble
: radius of a conduit
: radius of a pore in pit membrane
: absolute water pressure at which an air bubble of radius will burst
: radius of the pore through which an air seed enters a conduit of radius and will burst at
: absolute water pressure at which an air seed enters a conduit of radius and will burst at
: Helmholtz function
A: gas/water interface
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