_{1}

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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
*P*
_{1} 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.

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 [

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. [

Where does an air seeding event take place? Considering the potential importance of the rare pit hypothesis, Plavcová et al. [

Isolated conduit has been seen, which might be caused by another mechanism [

Ponomarenko et al. [

Three ways of cavitation by air seeding have been proposed [

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 [

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.

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 P o , it is ignored. We also ignore some facts, including abundant hydrophobic surfaces and insoluble surfactants in xylem.

According to the ideal gas law P = n R T / V g , the gas pressure P of the bubble of volume V g = ( 4 π r 3 ) / 3 should be

P = 3 n R T / ( 4 π r 3 ) (1)

When a bubble is in an equilibrium, we have:

P g = P l + 2 σ / r (2)

The relationship among P l , atmospheric pressure P o and xylem pressure P ′ l is P l = P o + P ′ l . The point of intersection of the two curves P ( r ) and P g ( r ) , or at P = P g , indicates temporary equilibrium of a bubble (

Thus, the positive real roots of the following equation

4 π P l r 3 + 8 π σ r 2 − 3 n R T = 0 (3)

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: d F g = − P d V g for the air, d F l = P l d V g for the water, and d F s = σ d A for the increase of the gas/water interface d A ( A = 4 π r 2 ). Thus, the total change of the Helmholtz function d F is d F = − ( 3 n R T / r ) d r + P l 4 π r 2 d r + σ 8 π r d r , or

F ′ ( r ) = d F / d r = 4 π P l r 2 + 8 π σ r − 3 n R T / r (4)

Integrating Expression (4) gives

F ( r ) = ( 4 π P l r 3 ) / 3 + 4 π σ r 2 − 3 n R T ln r + C (5)

Once Helmholtz function F ( r ) (

Letting the left side of Equation (3) be a function of r gives

f ( r ) = 4 π P l r 3 + 8 π σ r 2 − 3 n R T (6a)

and

f ′ ( r ) = 12 π P l r 2 + 16 π σ r (6b)

Therefore, the real roots of Equation (3) are the intersections of the curve f ( r ) with r-axis and those the abscissa values of which are more than zero are the radii of the bubble in equilibrium (

1) When P l = 0 , f ( r o ) = 8 π σ r o 2 − 3 n R T = 0 → r o = 3 2 n R T 2 π σ .

2) If P l ≠ 0 , corresponding to the following equation

a x 3 + b x 2 + c x + d = 0 (7)

the analytic solution of Equation (3) can be gotten by Shenkin formula [

For Equation (3), we obtained A = b 2 − 3 a c = ( 8 π σ ) 2 , B = b c − 9 a d = 3 × 4 × 9 π P l n R T and C = c 2 − 3 b d = 9 × 8 π σ n R T . Then, we got

Δ = B 2 − 4 A C = 3 2 × 4 2 π 2 × n R T ( 81 P l 2 n R T − 128 π σ 3 )

If Δ = 0 , 81 P l 2 n R T = 128 π σ 3 , or P l * = ± 8 π σ 9 2 π σ n R T . From Shenjin formula③ [

① When 0 < P l < P o , using Shenjin formula ② and ④ [

② When P l < 0 there are several situations as follows.

a) If Δ < 0 , meaning 81 P l 2 n R T < 81 ( P l * ) 2 n R T , or P l * < P l < 0 , from Shenjin formula ④ [

i) r Ⅰ = − 2 σ 3 p l ( 1 + 2 cos θ 3 ) . When P l = P l * → T = − 1 and θ = π , then r Ⅰ = − 4 σ 3 P l * = 3 2 n R T 2 π σ = r * . When P l → 0 , r Ⅰ → ∞ . Therefore, r Ⅰ is r 2 in

ii) r Ⅱ = − 2 σ 3 P l [ 1 − 2 sin ( π 6 + θ 3 ) ] . When P l = P l * , r Ⅱ = 2 σ 3 P l * . If P l → 0 , r Ⅱ → 8 σ 9 × 3 2 P l * . The values of r Ⅱ all are negative and should not be considered.

iii) r Ⅲ = − 2 σ 3 P l [ 1 − 2 sin ( π 6 − θ 3 ) ] . When P l = P l * , we got r Ⅲ = − 4 σ 3 P l * = 3 2 n R T 2 π σ = r * . While P l → 0 , r Ⅲ → − 8 σ 9 × 3 2 P l * → 3 2 n R T 2 π σ = r o . Thus, r Ⅲ is r 1 in

Therefore, in the range of P l * < P l < 0 ,

r 1 = − 2 σ 3 P l [ 1 − 2 sin ( π 6 − 1 3 arccos ( 1 − 2 P l 2 ( P l * ) 2 ) ) ] ( r o < r 1 < r * )

r 2 = − 2 σ 3 P l ( 1 + 2 cos 1 3 arccos ( 1 − 2 P l 2 ( P l * ) 2 ) ) ( r * < r 2 < ∞ ) (8)

are the solutions of Equation (3), which are none than the formulas 10 in the article [

b) When Δ > 0 , meaning 81 P l 2 n R T > 81 ( P l * ) 2 n R T , or P l < − 8 π σ 9 2 π σ n R T = P l * , according to Shenjin formula ② [

To sum up, if 0 < P l < P o , Equation (3) has a positive real root r ′ o ; If P l = 0 , its positive real solution is r o = 3 2 n R T 2 π σ ( r ′ o < r o ); If P l = P l * = − 8 π σ 9 2 π σ n R T , the positive real root is r * = 3 2 n R T 2 π σ . In the range of P l * < P l < 0 , the positive real roots of Equation (3) are Formula (8). The relationship of radii of an air bubble is r ′ o < r o < r 1 < r * < r 2 .

The stability of an air bubble which is in equilibrium depends on Formula (6b). For f ( r ) = 0 , if f ′ ( r ) > 0 , F ( r ) reaches its minimum and the equilibrium of the bubble is stable. In turn, f ′ ( r ) < 0 , F ( r ) arrives at its maximum, the equilibrium of the bubble is unstable.

1) When 0 ≤ P l < P o , formula f ′ ( r ) > 0 for all r > 0 . F ( r o ) and F ( r ′ o ) are the minima and the bubbles of radius r o or r ′ o in xylem are stable.

2) When P l * < P l < 0 , there are two roots r 1 and r 2 ( r 2 > r 1 ) for Equation (3). When f ′ ( r m ) = 0 , we have r m = − 4 σ 3 P l . Thus, f ( r ) reaches its extremum at r m and r 1 < r m < r 2 (

For r 1 < r m , Δ r 1 = r 1 − r m < 0 , f ′ ( r 1 ) = 4 π r 1 ( 3 P l r 1 + 4 σ ) = 4 π r 1 ( 3 P l Δ r 1 ) > 0 . Therefore, F ( r 1 ) is a minimum and the equilibrium of an air bubble of radius r 1 is stable.

For r 2 > r m , Δ r 2 = r 2 − r m > 0 , f ′ ( r 2 ) = 4 π r 2 ( 3 P l r 2 + 4 σ ) = 4 π r 2 ( 3 P l Δ r 2 ) < 0 . Therefore, F ( r 2 ) is a maximum and the equilibrium of an air bubble of radius r 2 is unstable.

3) If P l = P l * , we got F ′ ( r * ) = 0 and F ″ ( r * ) = 0 . Therefore, F ( r * ) reaches its inflection point (

4) When P l < P l * , a gas bubble could not be at any equilibrium.

Every one of bubbles has its own n R T , also its own P l * , being called its Blake threshold pressure, and its r * , or Blake critical radius [

Suppose that along with the decreasing of P l a bubble with n mole air in a conduit of radius r c enlarges stably. If its Blake radius r * > r c (or based on r * = 3 2 n R T 2 π σ , n R T = 8 π σ 9 r * 2 > 8 π σ 9 r c 2 ), before its exploding at P l * = − 4 σ / 3 r * , it has become long shaped, leading the bubble only to expand and lengthen gradually. This is the first way of cavitation. Only if r * < r c , or n R T < 8 π σ 9 r c 2 , can it explode at its P l = P l * 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 r * and r c is larger, or on which of n R T and 8 π σ 9 r c 2 is larger. Thus, for a pre-existent air bubble the boundary of above two ways of cavitation is its r * = r c at P l * = P lc * = − 4 σ / 3 r c . The action of these two ways of cavitation all are the same with the first two ways of cavitation by air seeding [

When an air seed is sucked into a conduit of radius r c from atmosphere through a pore of radius r p in pit membrane, its initial radius equals r p and initial gas pressure P = P o [

If a seed enters a conduit of radius r c through a pore of radius r pc in the conduit wall from atmosphere and will break up at P lc * = − 4 σ / 3 r c with r * = r c , there is a relationship n R T = 8 π σ 9 r * 2 = 8 π σ 9 r c 2 = P o × 4 π r pc 3 3 . Therefore,

r pc = ( 2 σ r c 2 / 3 P o ) 1 / 3 (9)

From P l = P o + P ′ l and P ′ l = − 2 σ / r pc [

P l = P lc = P o − 2 σ / r pc = P o − ( 2 σ r c ) 2 / 3 ( 3 P o ) 1 / 3 (10)

However, at the moment the radius of the seed reaches r c . 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 [

1) In the range of P lc ≤ P l < P o and r p ≥ r pc , the first way of cavitation will form.

2) In the range of − 2 P o < P l < P lc and 0.487 μm < r p < r pc the second way of cavitation will take place.

3) In P l ≤ − 2 P o and r p ≤ 0.487 μm , soon after an air seed is sucked into a conduit, as its radius is r 2 it will explode immediately and the conduit will be filled with the seed air instantly, presenting the third way of cavitation.

The experiments [

For the development of air seeds,

If a seed of radius r = r pc at P l = P lc enters a conduit of radius r c = 6.501 μm , from formulas 9 and 10, we got r pc = 2.740 μm and P lc = 0 .04672 MPa (

All air seeds of radius r ′ o , r o , or r 1 (lines 1 ® 5) by air seeding in

This indicates that in the range of P l ≤ − 2 P o , for air seeding the third way of

n R T | P l | r ′ o | r o | r 1 | r 2 | r * | P l * | |
---|---|---|---|---|---|---|---|---|

1 | 52.36 | 0.07080 | 5.000 | (9.253) | (16.03) | −0.00607 | ||

2 | 8.617 | 0.04672 | 2.740 | 3.754 | (6.501) | −0.01497 | ||

3 | 3.351 | 0.02700 | 2.000 | 2.341 | 4.054 | −0.02401 | ||

4 | 1.304 | 0 | 1.460 | 2.529 | −0.03849 | |||

5 | 0.4190 | −0.04600 | 0.8276 | 1.000 | (2.919) | 1.433 | −0.06790 | |

6 | 0.04828 | −0.2000 | 0.2810 | 0.4867 | 0.4867 | 0.4867 | −0.2000 | |

7 | 0.04349 | −0.2106 | 0.2667 | 0.4541 | 0.4700 | 0.4619 | −0.2107 | |

8 | 0.002058 | −0.7588 | 0.05801 | 0.07392 | 0.1700 | 0.1005 | −0.9687 | |

9 | 0.0004189 | −1.360 | 0.02617 | 0.03104 | 0.1000 | 0.04533 | −2.147 |

cavitation will occur; but for pre-existent air bubbles of radius r ≤ 0.4867 μm the second way will take place.

From

The scale bars shown in the figures of the article [

There are insoluble surfactants in xylem sap. Because of surfactants effect, the values of σ may decreases. From P l * = − 8 σ 9 2 π σ n R T we can see that the smaller the σ , the smaller the absolute value of P l * , meaning that at higher water pressure an air bubble will burst and a cavitation event will occur easily. Thus, the values in

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 [

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).

None declared.

Shen, F.Y. (2020) Analysis of Cavitation Processes in Xylem. Journal of Applied Mathematics and Physics, 8, 1767-1778. https://doi.org/10.4236/jamp.2020.89133

P l : absolute water pressure in a conduit

r: radius of a spherical air bubble

r o : radius of a spherical air bubble at P l = 0

r ′ o : radius of a spherical air bubble at P l > 0

V g : bubble volume

R: gas constant

T: absolute temperature

n: molar number of air in a bubble

P o : atmospheric pressure

P: gas pressure of a bubble

P g : gas pressure of an air bubble in equilibrium

σ : surface tension of xylem water

P l * : Blake threshold pressure of an air bubble

r * , Blake critical radius of an air bubble

r c : radius of a conduit

r p : radius of a pore in pit membrane

P lc * : absolute water pressure at which an air bubble of radius r * = r c will burst

r pc : radius of the pore through which an air seed enters a conduit of radius r c and will burst at P lc *

P lc : absolute water pressure at which an air seed enters a conduit of radius r c and will burst at P lc *

F ( r ) : Helmholtz function

A: gas/water interface