Analysis of Cavitation Processes in Xylem

Cavitation in plants is caused by development of air bubbles, which is related to their equilibrium and development. There is a univariate cubic equation for bubble balance. New root formula of this kind of equation was proposed by Shenjin Fan, which is simpler than the Caldan’s. Using Shenjin formulas and taking water pressure l P as an independent variable, this paper gives the exact solution of the equation under certain conditions. The stability of the equilibrium of an air bubble in its different radius ranges is obtained by the way different from the previous. This kind of cavitation includes two types: First type may be caused by the growth of pre-existent air bubbles; Second type is air seeding, here defined as the sucking of air bubbles from already gas-filled conduits. For air seeding three ways of cavitation have been proposed. For the first type this paper puts forward that two ways of cavitation 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 first type, the range of water pressures is the same with that of the first way of air seeding. 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.

bolism formation via the process of air-seeding etc. [6] [7]. Whether the capillary failure is an appropriate physical model comes to be a question [7]. Then, from the experiments [8] [9] [10] [11], 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 [12].
Where does an air seeding event take place? Considering the potential importance of the rare pit hypothesis, Plavcová et al. [13] 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 [9] [11]. The development of nanobubbles snapped off at pit membranes can also cause cavitation events [14]. These all may involve the growth of pre-existent air bubbles in xylem.
Ponomarenko et al. [11] 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. [11].
Three ways of cavitation by air seeding have been proposed [15] (In the article [15] 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 [16] [17]. 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 [18]. A new formula for finding the root of univariate cubic equation was proposed by Fan [19], which is simpler than the Caldan's. Using Shenjin formula and taking absolute water pressure l P 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 [14], 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 [17]. Journal of Applied Mathematics and Physics 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.

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 o P , it is ignored. We also ignore some facts, including abundant hydrophobic surfaces and insoluble surfactants in xylem.
According to the ideal gas law g P nRT V = , the gas pressure P of the bubble of volume When a bubble is in an equilibrium, we have: The relationship among l P , atmospheric pressure o P and xylem pressure 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: Integrating Expression (4) gives ( ) ( ) Once Helmholtz function ( ) F r ( Figure 2) reaches an extreme, or ( ) 0 F r ′ = , the bubble will attain its equilibrium. Thus, from expression 4 we also have Equation (3).

Solution of Equation (3)
Letting the left side of Equation (3) be a function of r gives ( ) 2) If l 0 P ≠ , corresponding to the following equation   the analytic solution of Equation (3) can be gotten by Shenkin formula [19]. nRT r r σ ′ < π = . As xylem pressure l P is often negative, we do not pay more attention to it.
② When l 0 P < there are several situations as follows.
The values of r Ⅱ all are negative and should not be considered. iii) . When . Thus, r Ⅲ is 1 r in Figure 3, the values of which are in the range of o 1 * r r r < ≤ .
Therefore, in the range of * l l 0 P P To sum up, if bubble is o o 1 * 2 r r r r r ′ < < < < .

Stability of Bubble Equilibrium
The stability of an air bubble which is in equilibrium depends on Formula (6b). For

( )
F r reaches its minimum and the equilibrium of the bubble is stable. In turn, F r arrives at its maximum, the equilibrium of the bubble is unstable.

4) When
* l l P P < , a gas bubble could not be at any equilibrium. Every one of bubbles has its own nRT , also its own * l P , being called its Blake threshold pressure, and its * r , or Blake critical radius [20] [21].

First Type of Cavitation: Growth of Pre-Existent Air Bubbles in Conduits
Suppose that along with the decreasing of l P a bubble with n mole air in a conduit of radius c r enlarges stably. If its Blake radius * c r r > (or based on ways of cavitation all are the same with the first two ways of cavitation by air seeding [15] except of forming isolated embolized conduits without any reservoir.

Second Type of Cavitation: Air Seeding
When an air seed is sucked into a conduit of radius c r from atmosphere through a pore of radius p r in pit membrane, its initial radius equals p r and initial gas pressure o P P = [22]. In the range of pressure o l o 2P P P − < < its radius should be o r′ , o r or 1 r [17]. As l P 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 [15].
If a seed enters a conduit of radius c r through a pore of radius pc r in the conduit wall from atmosphere and will break up at * lc c there is a relationship  [22], the pressure l P at which the seed enters the conduit is ( ) However, at the moment the radius of the seed reaches c r . 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 [15] [17] the following conclusions are obtained.
1) In the range of lc l o P P P ≤ < and p pc r r ≥ , the first way of cavitation will form.
2) In the range of o l lc 2P P P − < < and p pc 0.487 μm r r < < the second way of cavitation will take place.
3) In l o 2 P P ≤ − and p 0.487 μm r ≤ , soon after an air seed is sucked into a conduit, as its radius is 2 r it will explode immediately and the conduit will be filled with the seed air instantly, presenting the third way of cavitation.
The experiments [8] [9] 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, o P in should be replaced by internal air pressure a P , causing some data to be recalculated.

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 l P (in bold). Also the values of corresponding nRT , o r , * r and * l P of the seeds of radii o r′ , o r and 1 r . For a seed of radius 1 r (or 2 r ), using the Journal of Applied Mathematics and Physics formula 8 the corresponding 2 r (or 1 r ) can be calculated. Note the two states of the bubble of radius 1 r and 2 r are at the same water pressure l P .
If a seed of radius pc r r = at l lc P P = enters a conduit of radius c 6.501 μm r = , from formulas 9 and 10, we got pc 2.740 μm r = and lc 0.04672 MPa P = (Table   1 line 2). In the range of lc l o P P P ≤ < the bubbles will expand gradually ( Table   1 line 1 and 2), presenting the first way of cavitation. In the range of o l lc 2P P P − < < , the bubbles of o r′ , o r and 1 r will expand to their respective * r , presenting the second way of cavitation (Table 1 line 3 → 5). For the seed of radius 1 r during the dropping of l P it will break up at * l P with * r r = before its radius reaches 2 r . Thus, in the range of o l 2 0 P P − < < the bubble of radius 2 r in the parentheses does not exist ( All air seeds of radius o r′ , o r , or 1 r (lines 1 → 5) by air seeding in Table 1 can also be regarded as pre-existent air bubbles. Occasionally the air seeds of ra-  Table 1 can be regarded as pre-existent. During the decreasing of l P they will expand. The bubbles of radius * c r r > in lines 1 → 2 will grow gradually. The others will expand to their respective radii * r at * l P to explode. The bubbles with the values in parentheses can't exist.  the second way will take place.
From Table 1 we can see that the more the amount nRT of a bubble, the larger its Blake critical radius * r and the higher its Blake threshold pressure.
This means that a bubble with more nRT 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  we can see that the smaller the σ , the smaller the absolute value of * l P , 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.

Conclusions
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 [17].
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. P : absolute water pressure at which an air bubble of radius * c r r = will burst pc r : radius of the pore through which an air seed enters a conduit of radius c r and will burst at * lc P lc P : absolute water pressure at which an air seed enters a conduit of radius c r and will burst at * lc P ( ) F r : Helmholtz function A: gas/water interface