Tree hysioogy 22, 655 65 2002 Heron ubishing Victoria, Canada hysica anaysis of the process of cavitation in xyem sap FANYI SHEN, 1,2 RONGFU GAO, WENJI LIU 4 and WENJIE ZHANG 1 1 Coege of Basic Sciences and Technique, Beijing Forestry University, Beijing 10008, R China 2 Author to whom correspondence shoud be addressed (fyshen@bjfueducn) Coege of Bioogy, Beijing Forestry University, Beijing 10008, R China 4 Department of Basic Courses, Zhengzhou Institute of Light Industry, Zhengzhou 450002, R China Received June 18, 2001; accepted October 8, 2001; pubished onine May 1, 2002 Summary Recent studies have confirmed that cavitation in xyem is caused by air bubbes We anayzed expansion of a preexistent bubbe adhering to a crack in a conduit wa and a bubbe formed by the passage of air through a pore of a pit membrane, a process known as air seeding We consider that there are two equiibrium states for a very sma air bubbe in the xyem: one is temporariy stabe with a bubbe radius r 1 at point s 1 on the curve (r) reating pressure within the bubbe () with bubbe radius (r); the other is unstabe with a bubbe radius r 2 at point s 2 on (r) (where r 1 < r 2 ) In each equiibrium state, the bubbe coapse pressure (2σ/r, where σ is surface tension of water) is baanced by the pressure difference across its surface In the case of a bubbe from a crack in a conduit wa, which is initiay at point s 1, expansion wi occur steadiy as water potentia decreases The bubbe wi burst ony if the xyem pressure drops beow a threshod vaue A formua giving the threshod pressure for bubbe bursting is proposed In the case of an air seed entering a xyem conduit through a pore in a pit membrane, its initia radius may be r 2 (ie, the radius of the pore by which the air seed entered the vesse) at point s 2 on (r) Because the bubbe is in an unstabe equiibrium when entering the conduit, it can either expand or contract to point s 1 As water vaporizes into the air bubbe at s 2, rises unti it exceeds the gas pressure that keeps the bubbe in equiibrium, at which point the bubbe wi burst and induce a cavitation event in accordance with the air-seeding hypothesis However, other possibe perturbations coud make the air-seeded bubbe contract to s 1, in which case the bubbe wi burst at a threshod pressure proposed for a bubbe expanding from a crack in a conduit wa For this reason some cavitation events may take pace at a xyem threshod pressure ( ) other than that determined by the formua, p = 2σ/r p, proposed by Sperry and Tyree (188), which is appicabe ony to air-seeded bubbes at s 2 The more genera formua we propose for cacuating the threshod pressure for bubbe breaking is consistent with the resuts of pubished experiments Keywords: air seeding, bubbe breaking, bubbe expansion, emboism, equiibrium, threshod pressure Introduction When water moves through the soi pant atmosphere continuum under a negative pressure, cavitation often occurs in xyem conduits (Tyree and Dixon 18, Oerti 1, ockman et a 15, Tyree 17) Since Miburn and Johnson (166) found that cavitation in pants can be detected by acoustic methods, cavitation and emboism have been intensivey studied (Tyree and Sperry 18, Miburn 1) Vunerabiity of xyem to cavitation is an important factor in the adaptation of pants to the environment (Tyree et a 1) Cavitation of xyem has been detected in stems, eaves and roots, and appears to imit the distribution of pant species (Tyree et a 1) Cavitation of water coumns within the xyem is deeterious for pant water reations because it resuts in emboism thereby reducing hydrauic conductivity (Jackson and Grace 16) Cavitation is induced by air seeding, which can occur when air passes through the pores of pit membranes to form bubbes (Cochard et a 12, Tyree 17) Cavitation is aso caused by tiny air bubbes adhering to cracks in vesse was (ickard 181, Miburn 1, Tyree et a 1) ickard (181) reviewed two possibe mechanisms of cavitation: bubbe formation in buk iquid; and nuceation at hydrophobic cracks Zimmermann (18) studied the process of air seeding He concuded that once a bubbe formed by air seeding enters into a vesse, it immediatey expands unti the tension forces on the wa are reeased Air seeding through a pore of an inter-vesse pit membrane is currenty considered the most common process resuting in cavitation Here we consider an air bubbe of radius r expanding from a crack in a conduit wa, which is in equiibrium so that the bubbe coapse pressure (2σ/r, where σ is surface tension of water) is baanced by the pressure difference across its surface When the water pressure in the conduit decreases, the gas pressure in the bubbe () drops as the radius of the bubbe increases There is no reason to beieve that is aways higher than the gas pressure that keeps the bubbe in equiibrium ( g ) It is possibe that the bubbe coud estabish a new equiibrium as it expands Therefore, we anayzed the process of bubbe expansion in detai We found that an air bubbe coud expand steadiy and wi burst ony if the xyem pressure drops beow
656 SHEN, GAO, LIU AND ZHANG a threshod vaue A formua for cacuating this threshod pressure is proposed Our anaysis is compatibe with the airseeding hypothesis Deveopment of a temporary equiibrium for a tiny air bubbe in xyem Based on Zimmermann (18), when an air bubbe passing through a pore of a pit membrane enters into metastabe water, its initia radius is the same as the radius of the pore (r p ) and its initia pressure is equa to atmospheric pressure, about 01 Ma Compared with the air pressure, the water saturation vapor pressure in the bubbe is generay ess than 0002 Ma at 20 C and is too sma to be significant Generay air pressure is determined by the idea gas aw: = nrt/v g, where n is moar number of the gas in voume V g, R is the gas constant and T is absoute temperature For a spherica bubbe, V g = 4πr /, the actua pressure of a gas in it is: = nrt/ 4π r (1) Equation 1 indicates that rapidy decreases with increasing radius (r) of the bubbe We know that after the refiing of emboized vesses there is the possibiity of tiny bubbes reforming and adhering to hydrophobic cracks in the conduit was (Grace 1a)A decrease in xyem pressure wi draw air out of the cracks to form bubbes again According to Henry s aw and Fick s aw, when the gas pressure in a bubbe is higher than atmospheric pressure the air in it must dissove in the surrounding water and diffuse For this reason the moar number (n) of the gas in it must change Based on Yang and Tyree s (12) empirica formua more than 10 h are required for compete recovery of conductivity when xyem pressure is 0 ka Moreover, when we injected water into a gass capiary, we found that if bubbes formed and adhered to the wa of the capiary, some remained there for at east days Therefore, in terms of the diurna cavitation repair cyce, we considered air dissoution of minor significance compared with bubbe expansion Furthermore, we concuded that air dissoution in water is unimportant as a mechanism of cavitation, ie, n was regarded as constant during bubbe expansion To simpify the anaysis (cf Oerte 1) bubbe expansion was aso assumed to be isotherma Hence, in our anaysis, is soey a function of r of the bubbe The coapse pressure 2σ/r of a bubbe is baanced by the pressure difference between the gas pressure ( g ) and the absoute pressure ( ) of xyem sap across its surface in the case of the bubbe in an equiibrium state This reationship can be expressed as: g = 1 +2σ/ r (2) Therefore, if the bubbe is in an equiibrium state, g is higher than the of the surrounding iquid The reationship among, atmospheric pressure o, and xyem pressure ( )is = o + If the bubbe is in an equiibrium state, shoud be equa to g, which keeps the bubbe in equiibrium However, changes in take pace frequenty, causing g to change Therefore, is not aways equa to g If is greater than g, the bubbe wi expand; otherwise it wi contract Consequenty, the change in the voume of a bubbe depends on how high or ow is with respect to g Figure 1 shows curves (r) and g (r) corresponding to Equations 1 and 2, respectivey Curve (r) is a third-order hyperboa, and when r, its asymptote is = 0 Curve g (r)is a first-order hyperboa, and when r, its asymptote is = The point of intersection of two curves indicates the equiibrium state for a bubbe, because = g at that point When 0, the two curves have ony one intersection point, s o, with abscissa r o (or s o with abscissa r o, not shown in Figure 1), representing a temporary equiibrium state for the bubbe If we assume that the water potentia remains constant, an increase in bubbe radius in response to a disturbance wi cause a drop in, eading to < g Thus, the bubbe wi contract, returning to s o Conversey, if the radius of the bubbe gets smaer than r o, it wi expand Therefore, the equiibrium of a bubbe at s o (or at s o ) is temporariy stabe Soving the equation (r) = g (r), we obtain: r = nrt/ 8πσ, for = 0 () o When < 0, the two curves wi intersect, be at a tangent or separate If is ess than 0, but higher than the absoute threshod pressure of xyem sap ( ), the curves wi intersect at two points, s 1 and s 2, corresponding to abscissa r 1 and r 2,respectivey For the same reason as given for the anaysis for a bubbe of 0 at point s o (or s o ), the equiibrium of a bubbe at point s 1 is temporariy stabe However, if a bubbe is at point s 2, it is unstabe This is because if the bubbe r is arger than r 2, then > g and the bubbe wi expand, whereas it wi contract if r < r 2 In either condition, the bubbe diverges from the equiibrium point s 2, ie, the bubbe is unstabe at s 2 Suppose an air bubbe just expanding from a crack of a conduit wa is temporariy in stabe equiibrium at point s 1 Asthe water potentia or xyem pressure drops as a resut of increasing transpiration and deveoping soi water deficit, g wi be ower than before, inducing the bubbe to expand and to decrease simutaneousy Subsequenty, wi be equa to g again at point s 1 and the bubbe wi be in a new stabe equiibrium temporariy Therefore, when the water potentia drops continuay, the radius of the bubbe wi grow steadiy and proceeds aong the curve (r) maintaining the bubbe in a temporary stabe equiibrium During this process, two intersecting points s 1 and s 2 graduay get coser to each other as shown in Figure 1 When has decreased to, the two points coincide at point s, where r = r and = g (at other r vaues, > g ) Thereafter, wi be ower than, and the two curves wi separate and there is no intersection for <, indicating that is aways higher than g when < For this reason, soon after becomes ower than, the bubbe wi burst and cavitation wi occur Because of the action of surface tension, the air takes the shape of a conduit or meniscus surface as wa- TREE HYSIOLOGY VOLUME 22, 2002
ANALYSES ON CAVITATION 657 Figure 1 Curves (r) (heavy soid ine) and g (r) at different vaues of (ight soid ines) Dashed ines denote negative vaues of g and therefore represent impossibe pressures ter recedes into the tapered ends of cavitated tracheid estabishing a new equiibrium between the air and water Therefore, a bubbe at s is at a critica stage and represents a type of threshod pressure for bubbe breaking and the actua pressure in the bubbe: = 4 σ 2πσ nrt (5b) New vaue of threshod pressure for bubbe breaking From the above anaysis, we know that the vaue of the threshod pressure of bubbe breaking is determined by the tangent point s of two curves At s, the sopes of the two curves are equa, ie, derivatives of the functions (r) and g (r) are equa From Equations 1 and 2: dg dr we obtain: d nrt = 2 σ and =, 2 4 r dr 4πr 2σ nrt =, r 2 4πr 4 which simpifies to: nrt r = r = 2 2πσ Substitution of r in Equations 1 and 2 with r in Equation 4, gives the pressure required to maintain equiibrium: g = + 4 σ 2πσ, nrt (4) (5a) At the tangent point s, g = : 4σ 2πσ 4σ 2πσ + = nrt nrt, which simpifies to: = 8 σ 2πσ nrt (6a) arameter is the threshod vaue of absoute pressure of xyem sap at which a bubbe breaks If we ignore the change in n, and based on Zimmermann s (18) study of the process of air seeding, et nrt = o V = o (4/)πr p, then: = 8 σ σ o p 2r Because = o + : 8σ σ = o, 2r o p (6b) where is the new threshod xyem pressure at which a bubbe breaks If we substitute the radius of a pore of a pit membrane as r p = 015 µm, T =20 Candσ = 007 Nm 1 in (7) TREE HYSIOLOGY ONLINE at http://heronpubishingcom
658 SHEN, GAO, LIU AND ZHANG Equation 7, we obtain = 1 Ma for bubbe breaking This vaue is within the range of xyem pressure for cavitation events (Grace 1b, Miburn 1), but ower than the vaue of 07 Ma cacuated by the threshod pressure formua p = 2σ/r p (Sperry and Tyree 188) for the process of air seeding The reationships between and r p and between p and r p are shown in Figure 2 Because of dissoution of air from the bubbe into the surrounding water when > o, nrt of a bubbe after a cavitation repair cyce must be ess than o V = o (4/)πr p, therefore the true xyem pressure for bubbe breaking wi be ower than the vaue cacuated with Equation 7 ockman et a (15) pointed out that different species have different cavitation threshods The reason may be that moar numbers, n, differ among species, perhaps because of differences in the diameters of pores in the pit membranes Anaysis of cavitation We know that cavitation is induced by air seeding, and that as soon as an air seed enters a xyem conduit, it expands immediatey, but we do not know why this process occurs From cacuations, we know that when a bubbe enters a xyem conduit, its initia radius is generay r 2 at point s 2 (it can aso equa r 1 ) because = 2σ/r p Because the water vapor in the bubbe does not reach saturation in this instant and water wi further vaporize into the bubbe, causing to be greater than g, the ikeihood of the bubbe expanding wi be greater than the ikeihood of the bubbe contracting Under these conditions the bubbe bursts at s 2, inducing cavitation events in accordance with the air-seeding hypothesis Therefore, the pubished formua, p = 2σ/r p (Sperry and Tyree 188), where p is xyem threshod pressure for bubbe breaking, is ony appicabe to air-seeded bubbes that burst at s 2 However, there are other possibe perturbations that make the air-seeded bubbe contract to s 1, in which case the bubbe wi burst at a new threshod pressure as proposed for a bubbe drawn from a crack in a conduit wa For this reason some Figure 2 Threshod of xyem pressure and p for bubbe breaking at s 1 and s 2 arameter p is the threshod pressure for bubbe breaking at s 2 and was cacuated by Equation 7 cavitation events may take pace at different xyem threshod pressures,, from that determined by the formua p = 2σ/r p (Sperry and Tyree 188) The more genera formua of Equation 7 that we propose for cacuating the threshod pressure for bubbe breaking coincides with the resuts of pubished experiments A cavitation event induced by air seeding woud not occur unti reaches 2σ/r p However, when xyem pressure decreases to beow zero, but is sti higher than, an air bubbe in the xyem sap, either being sucked out of a crack in the conduit wa or resuting from contracting air seeds, coud be in temporary stabe equiibrium at s 1 Inresponse to a further decrease in xyem pressure, it may burst at a different xyem threshod pressure than that of the initia air seed bursting When water oss from eaves of pants is suddeny increased as the resut of increased soar irradiance, the water potentia, or the xyem pressure in the conduits of the pants wi rapidy decrease, causing air seeds to enter vesses and more bubbes in a temporary stabe equiibrium to burst, thereby inducing cavitation events If evaporation is controed by encosing a pant or pant organ in a poyethyene bag, the cavitation rate wi diminish This is because the water potentia wi remain more or ess constant and some bubbes wi remain in temporary stabe equiibrium at s 1, and cavitation events wi resut ony from air seeding References Cochard, H, Cruiziat and MT Tyree 12 Use of positive pressures to estabish vunerabiity curves: further support for the air-seeding hypothesis and impications for pressure voume anaysis ant hysio 100:205 20 Grace, J 1a Refiing of emboized xyem In Water Transport in ants Under Cimatic Stress Eds M Borghetti, J Grace and A Raschi Cambridge University ress, Cambridge, pp 52 62 Grace, J 1b Consequences of xyem cavitation for pant water deficits In Water Deficits: ant Responses from Ce to Community Eds JAC Smith and H Griffiths Bios Scientific, Oxford, pp 10 128 Jackson, GE and J Grace 16 Fied measurement of xyem cavitation: are acoustic emissions usefu? J Exp Bot 47:164 1650 Miburn, JA 1 Cavitation A review: past, present and future In Water Transport in ants Under Cimatic Stress Eds M Borghetti, J Grace and A Raschi Cambridge University ress, Cambridge, pp14 26 Miburn, JA and RC Johnson 166 The conduction of sap II Detection of vibrations produced by sap cavitation in Ricinus xyem anta 66:4 52 Oerti, JJ 1 Effect of cavitation on the status of water in pants In Water Transport in ants under Cimatic Stress Eds M Borghetti, J Grace and A Raschi Cambridge University ress, Cambridge, pp 27 40 ickard, WF 181 The ascent of sap in pants rog Biophys Mo Bio 7:181 22 ockman, WT, JS Sperry and JW O Leary 15 Sustained and significant negative water pressure in xyem Nature 78:715 716 Sperry, JS and MT Tyree 188 Mechanism of water stress-induced xyem emboism ant hysio 88:581 587 Tyree, MT 17 The cohesion tension theory of sap ascent: current controversies J Exp Bot 48:175 1765 TREE HYSIOLOGY VOLUME 22, 2002
ANALYSES ON CAVITATION 65 Tyree, MT and MA Dixon 18 Cavitation events in Thuja occidentais L? Utrasonic acoustic emissions from the sapwood can be measured ant hysio 72:104 10 Tyree, MT and JS Sperry 18 Vunerabiity of xyem to cavitation and emboism Annu Rev ant hysio ant Mo Bio 40:1 8 Tyree, MT, S Sao, A Nardini, MAL Guo and R Mosca 1 Refiing of emboized vesses in young stems of aure Do we need a new paradigm? ant hysio 120:11 21 Yang, S and MT Tyree 12 A theoretica mode of hydrauic conductivity recovery from emboism with comparison to experimenta data on Acer saccharum ant Ce Environ 15:6 64 Zimmermann, MH 18 Xyem structure and the ascent of sap Springer-Verag, Berin, pp 47 TREE HYSIOLOGY ONLINE at http://heronpubishingcom