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An experimental and theoretical comparison of viscous and diffusive resistances to gas flow through amphistomatous leaves
Agricultural Meteorology - Elsevier Publishing Company, Amsterdam - Printed in The Netherlands
Research Papers AN EXPERIMENTAL AND THEORETICAL COMPARISON OF VISCOUS AND DIFFUSIVE RESISTANCES TO GAS FLOW T H R O U G H AMPHISTOMATOUS LEAVES P. G. JARVISI~ C. W. ROSE AND J. E. BEGG
Division of Land Research, Commonwealth Scientific and Industrial Research Organization, Canberra, A.C.T. (Australia) (Received April 20, 1966)
SU MMARY
A theory with general application has been developed enabling resistance to viscous (~) and diffusive gas flux (r) through amphistomatous leaves to be calculated from anatomical dimensions. The calculated resistances agreed very well with experimental values of ~ and r measured for a range of stomatal apertures in leaves of cotton (Gossypium hirsutum cv. POPE). The theory can be used to calculate stomatal aperture from either mass flow or diffusion porometer measurements, provided other relevant anatomical characteristics are known.
INTRODUCTION
For many years, mass flow porometers have been used to provide estimates of stomatal aperture (e.g., MASKELL,1928; PENMAN, 1942; SPANNERand HEATH, 1951; HEATH, 1959; SIJIMSHI, 1964; BIERHUIZENet al., 1965; RASCHKE, 1965; WAGGONER, 1965). In mass flow porometry, a pressure difference is applied across a leaf and the resulting viscous flow of air is taken as a relative measure of stomatal aperture. However, the exchanges of water vapour and carbon dioxide between leaf and atmosphere which occur in transpiration and photosynthesis are commonly assumed to be diffusive in character. Hence, if the facility of these diffusive exchanges is to be inferred from mass flow porometer measurements, a calibration of diffusive resistance in terms of the resistance to viscous flow of air is required. In the past, this calibration has been attempted empirically (e.g., MEIONER and SPANNER, 1959), and by a combination of experiment and theory of diffusive and viscous flow (e.g., MASKELL,1928; PENMAN, 1942). PENMAN (1942) derived such theory specifically for the resistance porometer i Present address: Department of Botany, University of Aberdeen, Aberdeen (Great Britain).
Agr. Meteorol., 4 (1967) 103-I 17
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P. G. JARVIS, C. W. ROSEAND J. E. BEGG
of GREGORY and PEARSE (1934), and for the diffusion porometer of GREGORY and ARMSTRONG (1936) and HEATH (1941). This theoryincluded the contribution to flow from outside the porometer cup. The theory was not developed in terms of anatomical dimensions. Recently WAGGONER(1965) has theoretically investigated the relation between viscous and diffusive resistances, and attempted a comparison between theory and experiment. The result of this comparison was somewhat inconclusive, partly because diffusive resistance was not measured. This paper presents theory for the calculated comparison between resistances to viscous flow of air and diffusive flux of nitrous oxide from one side of a leaf to the other. The calculated relationship between the viscous and diffusive flow resistances is compared with experimental data obtained on cotton leaves using a conventional mass flow porometer and the nitrous oxide diffusion porometer (SLATYER and JARVlS, 1966).
THEORY Analogous expressions are required for the viscous flow of air and the diffusive flux of gas from one side of a leaf to the other, through the stomatal pores and intercellular spaces in the mesophyll. As nitrous oxide was used to measure diffusive resistance, it is assumed that no diffusion could take place through the cuticle. The theory is developed for stomatal apertures with slit-like geometry, and for cylindrical intercellular spaces within the mesophyll, and the justification for this is considered later.
Viscous flow of air The basic equation of viscous flow is: dv = * / ~ (dyne cm -2)
(1)
where z is the shear stress, t/the coefficient of viscosity (g cm -1 sec-I) and dv/dz the velocity gradient (sec-1). Consider steady laminar flow through unit length of an infinitely long slit of width w (cm) and depth l (cm) in the direction of flow (the x-direction). An element of volume (shaded in Fig.l) of depth dx and width 2z normal to the surfaces of the slit, experiences equal and oppositely directed forces caused by viscous flow and by the pressure differences dp across it in the direction of flow. Equating these forces: (dp)2z = r/(dv/dz) 2 dx which on integration gives: v --
(dfl)
(z z)
~/ (dx)
1
(2)
+ C (cm sec-1)
(2)
Agr. Meteorol., 4 (1967) 103-117
VISCOUSAND DIFFUSIVERESISTANCESOF LEAVES
105 Direction of
flow
Wails of slit Fig.1. An infinitesimalelement of volume of unit length located in an infinitelylong slit of width w and depth I in the direction of mass flow of air caused by pressure difference(pl--p2) across the slit. where C is a constant of integration to be evaluated. When slit width is not large compared with the mean free path of the molecules the number of molecular collisions between successive contacts with the walls is relatively small. The effect of this on viscous flow can be accomodated by setting the velocity of air at the walls equal to a velocity of slip ( ~ 0), defined by: v = - - ~ (dv/dz) (era see-1)
(3)
where ~ is the coefficient of slip (cm), (KvNNARD, 1938). Substituting from eq.3 into eq.2 to evaluate C gives: 1
v -- 2r/
(dp)
w2
(dx) (z2
4
w ()
(cm sec-1)
(4)
The mass of air flowing per second through unit length of slit is: W
M = p 2
2 I
v dz = - - p
(dp) (w3) (1 ÷ 6 ~ ) (dx) (12~/) -w-
0 where p is the density of air. From the ideal gas law, p = p / R T where p is the mean pressure on the element [(pl + ps)/2] and R the gas constant per gram of air. Multiplying through by dx and integrating from 0 to 1 (the depth of the slit in the direction of flow) or from pl to/)2, gives: M--
w3p 12 r/l (1 + 6w0 (/Ol --p2)
(g cm-%ec -1) Agr. Meteorol., 4 (1967) 103-117
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P . G . JARVIS, C. W. ROSE AND J. E. BEGG
Thus the volume flow per unit surface area, Q, for a surface containing n identical rectangular slits per unit area, each of width w and length b normal to the direction of flow and depth l, is: Q
M bn P
6 r/l~ ( 12 '-k - --/ (pl - - p2) (cm3cm-2sec-a) -- 1w3bn
(5)
__ P l - - f 1 2
(6)
no where f~p, the resistance of the slit-shaped pores per unit area of surface, is: 12 ~/l (1) f~p = w3b( 1 + ~ ) ( n )
(g cm-Zsec-1)
(7a)
The corresponding equation for a surface containing n identical cylindrical pores with circular cross-section normal to the direction of flow, and each of radius a, is:
8 ,fl
F2p =
(1)
~ra4 (1 + _ ~ _ )
(g cm-Zsec-1)
(7b)
(n)
Assuming that flow through narrow slits is analogous to that through stomata, eq.7a may be applied to viscous flow through an amphistomatous leaf with n, and nl stomata per unit area on the upper and lower surfaces, respectively. Air passing from the upper to the lower side of the leaf experiences sequentially nu stomatal pore resistances f~p in parallel, in series with an internal resistance ~i resulting from the thickness and internal anatomy of the leaf, and nl resistances f~p in parallel. The total viscous resistance per unit area of leaf, f~, is thus:
12
=
t" 1
~n.
+
wab( 1 q_~)
1 \
) --' ni (gcm-2sec -1)
(8)
n/
where f~p" and f~pt are the stomatal resistances to viscous flow in the upper and lower surfaces, respectively. The geometry assumed for air passages within the mesophyll is that they are cylinders of length equal to the thickness of the mesophyll. Denoting this thickness by Ii, and taking the number of air passages (of radii a;) per unit plan area of leaf to be hi, it follows from eq.7b that: ~'~i : =
8 tlli rc ai4(1 @ 4~')
(1)
(g cm-2sec-1)
(9)
(hi)
ai
Agr. Meteorol.,
4 (1967) 103-117
107
VISCOUS AND DIFFUSIVERESISTANCESOF LEAVES
Diffusive flux of nitrous oxide Measurements of diffusive resistance were obtained using nitrous oxide. The maximum difference in nitrous oxide concentration across the leaf was approximately 500 p.p.m, in these experiments. The resultant partial pressure gradients were therefore sufficiently small for the flux to be correctly calculated from a simple application of Fick's Law. From this law the volume flux of nitrous oxide by diffusion, q, through unit area of leaf surface containing n slit-like stomata of width w, length b, and depth lis: Ac
q = (wbn) Dz%o T
(cm3cm-Zsec--0
a c
(10)
rp where (A c/l) is the concentration gradient in nitrous oxide, and rp, the diffusive resistance of the slit-shaped pores per unit area of surface is: rp
--
/
(])
D~2° wb
(n)
(sec cm -1)
( 11a)
The corresponding equation assuming circular stomatal geometry is: (1) (n)
l
rp
Di~2o ~ a 2
(sec cm-1)
(1 l b)
Omitting the boundary layer resistances, the total resistance per unit leaf area experienced by nitrous oxide in diffusing from one side of a leaf to the other, r, is: r - - rp u j - rp l -~- ri
1 wb D~2o
(
1
@
\ nu
1
) @ ri
(sec cm -1)
(12)
nt
where rp" and r / a r e the stomatal pore diffusive resistances in the upper and lower leaf surfaces, respectively, and r~ the internal leaf diffusive resistance resulting from the thickness and anatomy of the leaf. Using the notation above, it follows from eq. 1lb that: ri
l~ DN20 rC ai 2
(1) (hi)
(see cm -1)
(13)
METHODS
Cotton plants (Gossypium hirsutum cv. PopE) were grown in solution culture on Agr. Meteorol.,
4 (1967) 103 117
108
P . G . JARVIS, C. W. ROSE AND J. E. BEGG
modified Hoagland's nutrient solution in a growth chamber with illumination from fluorescent tubes (Sylvania, Cool White V.H.O.) supplemented with tungsten globes, giving a radiation flux density of ca. 120 W m -~ (400-700 m/t) for 12 h each day. The photoperiod was extended to 14 h with the tungsten lamps alone. Air temperature was 35°C during the photosynthesis period and 24°C in the dark. Relative humidity was ca.65 ~ .
Experimental In the experiments the fifth emerged leaf from the apex was used when there were about ten leaves on the plant and before flowering commenced. The apparatus has been described in detail elsewhere (JARVIS and SLATYER, 1966). Opposed areas (31 cm z) of each surface of an attached cotton leaf were isolated in ventilated, air-conditioned chambers such that vapour and gas exchanges between each surface of the leaf and the ambient air could be followed for the identical surfaces used in porometry. Because of the large leaf surface within the porometer, the results obtained should be independent of porometer dimensions, and permit an accurate determination of r and ~ as defined in the theory. A range of stomatal apertures was obtained by varying the intensity of incident light and the ambient COz concentration in the leaf chamber. Diffusive resistance was calculated using an equation similar to eq. 10 from the measured volume flux by diffusion of nitrous oxide from one side of the leaf to the other in response to a concentration difference. Boundary layer resistance to diffusion was separately measured and allowed for in the calculation which is described in detail by SLATYERand JARVlS (1966). The viscous flow resistance was calculated using an equation similar to eq.6 from the flow of air from one side of the leaf to the other in response to a difference in pressure. The pressure difference across the leaf did not exceed 10 cm of water: if larger pressure differences were used, anomalous fluxes were obtained at small stomatal apertures (see also RASCHKE, 1965). The direction of the fluxes of air and nitrous oxide, though usually from upper to lower surface of the leaf, did not affect the result, confirming that the fluxes across the leaf were essentially one-dimensional and that possible errors due to lateral flow were insignificant. The N20 flux was monitored continuously: mass flow of air was recorded intermittently during periods of steady N~O flux.
Anatomical Stomatal dimensions and frequency were determined on leaves from the porometer using a light microscope. Using a freezing microtome, series of paradermal sections were obtained from
Agr. MeteoroL,4 (1967) 103-117
109
VISCOUS AND DIFFUSIVE RESISTANCES OF LEAVES
the fifth leaf of plants comparable to those used for the resistance measurements; 60 cellular and inter-cellular intercepts were measured on a series of transects taken in different directions across these sections. The resolution obtainable with a light microscope left some uncertainty as to stomatal slit length. Independent and more accurate evidence on this length was obtained using an electron microscope, for which the material was prepared by placing in 2% potassium permanganate in veronal buffer for 2 h, and Reynold's lead for 30 rain (REYNOLDS, 1963).
RESULTS
Experirnental resistances The relationship between experimentally determined values of diffusive resistance, 1,000
u
g m
~ 100 ° ~ o
I1)
o~
c o
~.@ro
o°-~
•
°
- - w = 1p
t_
o ::.
lO
~&~,,,v
= 1.5 ,u,
m
.&
es= 5.32~
10
100
viscous
1,000
10,000
100,000
resistance -Q_ ( g c m -2 sec-I.lO 4)
Fig.2. A doubly logarithmic plot o f diffusive resistance o f a c o t t o n leaf to nitrous oxide (r) against its viscous resistance to the m a s s flow o f air across the leaf (~). Experimental values f r o m four leaves s h o w n by small o p e n circles. Calculated values with different a s s u m p t i o n s fitted with curves.
Legend
nu no. m m -z nt no. m m -2 D i m e n s i o n s of stomatal aperture
Rectangular stomata • •
Circular stomata • o
57 92 138 209 b variable w variable
57 138
92 209
as = 5.32/1
Agr. Meteorol., 4 (1967) 103-117
110
p . G . JARVIS, C. W. ROSE AND J. E. BEGG
r, and the viscous flow resistance, fL is shown in Fig.2, At high fluxes of NzO and air, corresponding to the lowest resistances in Fig.2, the error in the measured resistances is low and they may be considered accurate to better than 0.5 sec cm -1 for r and 0.5 • 104 g cm-esec-1 for ~. At the lowest fluxes, the error in the resistances may be two orders of magnitude larger. Some of the spread in the points results from the effect of the operation of the mass flow porometer on stomatal aperture. Even at the low pressures used and with the short period of operation, the N 2 0 flux was often somewhat lower after the mass flow porometer had been used.
Anatomical The ranges of stomatal characteristics were as shown in Table I.
TABLE I RANGES OF STOMATAL CHARACTERISTICS
Mean stomatal frequency (n, number mm -2) Mean total length of stomatal slit (it) Mean depth of stomatal pore (l, It)
Upper
Lower
,~.rfaee
..,face
57-92 12.3-14.8 12.7-14.9
138-209 12.7-14.0 12.0-12.8
Electron micrographs, such as that shown in Fig.3, indicated that effective slit length (b) varied with slit width (w) for w -----0-2.5/1. For w ~ 2.5/t, b was at a maximum. The simplest model for the relation between b and w suggested by such micrographs was therefore to assume b proportional to w from w = 0 to w 2.5 ¢t, and b = constant for w ~ 2.5/t. In this model (Fig.4) the constant value taken for b in the calculations of resistance was 14.2/z, a value obtained on one of the leaves used in the experiment. This value is higher than the overall mean (13.4g) for all leaves investigated, and leads to about 5 % lower values in calculated resistances. The effect of this difference on the theoretical relationship between r and f~ in Fig.2 is insignificant. The thickness of mesophyll (li) was measured from transverse leaf sections and found to be 200 j= 10/1. The arithmetic mean of the intercellular spaces between mesophyll cells, obtained from paradermal sections, was found to be 1.8 -k 0.3 gc Since this mean was obtained from transects in many different directions, it could be used as an estimate of the mean diameter of a cylindrical air passage, or the mean side dimension of an air passage of square cross section. From eq.9 and 13 for cylindrical air passages it can be seen that resistances are inversely proportional to a power of the radius. Since the intercellular spaces cover a range of sizes from 0 to 7.2/1, use of the arithmetic
Agr. Meteorol., 4 (1967) 103 117
VISCOUS AND DIFFUSIVE RESISTANCESOF LEAVES
111
Fig.3. Electron micrograph showing partially open cotton stomata.
16
i 0
1.0
2.0 Slit
width
3o w (if)
4.0
Fig.4. Model for the relationship between stomatal slit length (b), and width (w) during stomatal opening.
Agr. Meteorol., 4 (1967) 103 117
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P . G . JARVIS~ C. W. ROSE AND J. E. BEGG
mean would overestimate the resistances. Therefore mean values of ai z (1.73/~2) and ai 4 (10.9/~4) were calculated for use in eq. 13 and 9, respectively. From the transect data the number of air passages in the mesophyll per unit leaf plan area (n,.), was calculated to be 71.5 • 104 cm -2. Calculated resistances
Substituting these values for l;, ai, and ni into eq. 13, ri was found to be 2.9 sec cm -1. A comparable value has been measured experimentally on the same material by an independent method (P. G. Jarvis and R. O. Slatyer, unpublished data, 1966). This is some justification for the use of the model which represents the intercellular spaces in the mesophyll as cylinders. Viscous resistance ~i was 1.2.104 g cm-2sec -1, calculated from eq.9. Substituting these values for fL" and r;, and the values for upper and lower surface stomatal frequencies (nu and nl), in eq.8 and 12 the viscous (~) and diffusive (r) resistances of cotton leaves were calculated for the anatomical characteristics given previously, and for w over the range 0.2-2.5/1. The value of ~ at 30°C was calculated as 0.0679 /~ from data by KENNARD (1938). The very good agreement between the calculated and experimental resistances is shown in Fig.2. Minimum stomatal resistance would be achieved for a circular stomatal pore. Assuming the maximum pore perimeter to be equivalent to that of a rectangular pore of length 14.2 iL and width 2.5 ¢t, the radius of a circular pore would be 5.32/~. Total resistances corresponding to minimum stomatal resistance were calculated replacing the stomatal terms of eq.8 and 12 by eq.7b and l l b lespectively, and are also shown in Fig.2. These points represent the lower limit for the relationship between r and ~.
DISCUSSION
Any difference between experimental and calculated results lies within the estimated experimental error mentioned in the section above. The calculated values of resistance depend on theory, anatomical dimensions, and the model for stomatal opening. If, instead of the model shown in Fig.4, stomata1 slit-length b was constant at its maximum value (14.2/0 for all slit widths, the relationship between the calculated resistances fall increasingly below those shown in Fig.2 as slit width decreases below I¢t. Viscous or diffusive stomatal resistance at a slit width of 2.5/~ may be twice as great as internal resistance depending on stomatal frequency. For circular stomatal pores of radius 5.32/t the internal resistance may be nearly three times as great as stomatal resistance, thus illustrating the importance of internal resistance to mass flow and diffusion at low total leaf resistance. In the high resistance region, on the other hand, internal resistance is negligible compared with stomatal resistance. The measured range in stomatal frequencies on the upper and lower leaf surfaces, whilst Agr. MeteoroL, 4 (1967) 103-117
113
VISCOUS AND DIFFUSIVE RESISTANCES OF LEAVES
considerably affecting the magnitude of the calculated resistances, has only a small effect on the relationship between resistances shown in Fig.2. The magnitude of the effect of molecular slip on calculated viscous resistance of stomata (f~p), is illustrated in Table II. If molecular slip were ignored it follows from Table II and Fig.2 that there would be a discrepancy between observed and calculated values when resistance is high.
T A B L E II THE EFFECT OF MOLECULAR SLIP ON VISCOUS RESISTANCE OF STOMATA (~'~p) AT DIFFERENT STOMATAL WIDTHS W
(nu = 5700 c m -e, nl = 13800 cm -2, / = 12.4/z, b = 14.2/t)
(11)
f~p (slip omitted) ( g c m -2 sec-1 • 10 4)
f~p (with slip) ( g e m -2 sec-1 . 10 4)
0.1 0.2 0.5 0.8 1.0
1,138,000 73,600 1,935 290 119.8
223,103 24,120 1,066 192 85.1
W
1.5 2.0 2.5
23.6 7.49 3.07
18.6 6.22 2.64
Another effect of molecular slip is that it increases the slope of theoretical relationship between log r and log f~ shown in Fig.2. Since slip is of greater significance at smaller stomatal widths where the contribution of internal resistance to the total is very small, internal resistance will be neglected in the considerations of the following two paragraphs, in which an expression for this slope is derived (eq. 17). Neglecting fL" and rl in eq.8 and 12 it follows that: r w2 (1 + 6~/w) ~-- = 12 t/D~2o
(14)
- f(w), a function of w. Taking logarithms: In r ~ In [f(w)] q- In Whence it follows that: d (In r) d(lnO) -- 1 +
f~ f(w)
d[f(w)] dw
F d f ~ q--1 t_dwA
(15)
Since d(ln r)/d(ln ~) = d(log r)/d(log ~), which is the slope of the theoretical relationship plotted in Fig.2, this slope is also given by eq. 15. Agr. Meteorol., 4 (1967) 103-117
114
P. G. JARVIS, C. W. ROSE AND J. E. BEGG
From eq. 8, 14 and 15, and a linear relation between b and w so that: b = kw
(16)
where k is a constant, it may be shown that d(log r) d(log~)
-
(1 + 3 ~ /w) 2(1 -5 4.5 ( / w )
1 -
(17)
It follows from eq.17 that if slit width were sufficiently large to neglect (i/w) in comparison with unity, the slope of the relation shown in Fig.2 would be 0.50. In .contrast, as the slit closes and w tends to zero the slope tends to the value 0.66. The increase in slope with decreasing slit width predicted by eq.17 is shown in Fig.5. It may be noted that eq.17 and therefore the slope of the calculated relation shown in Fig.2, is independent of the value of constant k in eq. 16. The full range of experimental data is well fitted by a straight line. Since the theory predicts an increase in slope with decreasing slit width (i.e., increasing resistance), possible evidence for a similar change in slope in the experimental data was sought by separate regression analyses for f~ < and > 30 • 104 g cm-Zsec -1. Mean slopes were 0.43 and 0.49 respectively, each with 95 % confidence intervals of :L 0.06. Thus the experimental data indicated a similar trend to the theory, although the increase in slope with increasing resistance was not significant. It may be seen from Fig.2 that the range of slit widths corresponding to the experimental data with ~q < 30 • 104g cm -2 sec -1 is from about 2.5 to 1.5 g. For this range the slope (0.43) for the experimental data falls below the curve for b ~ w in Fig.5, and above the curve for b : constant to be derived in the next paragraph. For f~ > 30 • 104 g cm -2 sec -1, slit widths range from about 1.5 to 0.5/t. The slope (0.49), is not significantly different from the curve for b ~ w. Performing a similar analysis to that used in deriving eq.17, but replacing
0.7 F
~ (3.6 bo~w
= CON St
O.B[
O
1 Sh't
2 width
3 w
(pL)
Fig.5. Theoretical relationship between the slope [d (log r)/d (log fD] and slit width (w) for slit length (b) constant, and proportional to slit width (w). Agr. Meteorol., 4 (1967) 103-117
VISCOUS AND DIFFUSIVE RESISTANCES OF LEAVES
11 5
eq.16 with assumption that slit-length, b, is constant and independent of slit-widths, it may be shown that: d(log r) d(logf~)
--
1--
2(1 + 3 l/w) 3(1 ~- 4 i/w)
(18)
The slope represented by eq.18 tends to 0.33 as slit width increases, and to 0.5 as w tends to zero, the entire relation also being given in Fig.5. Assuming the output of MEIDNER and SPANNER'S (1959) porometer to be proportional to diffusive conductance, it is of interest that the range of slopes they found for wheat stomata of rectangular shape was 0.32-0.37. An expression for [ d (log r)/d (log f~)] similar to eq. 18 may be derived including the effect of internal resistance. Since internal resistance affects this slope over only a limited resistance range, such an expression is of less interest than eq.17 and 18 and will not be considered here. A power-type relationship between viscous and diffusive resistance has commonly been assumed, corresponding to a linear relationship when these resistances are plotted logarithmically, as in Fig.2. The theory given in this paper predicts departure from a simple power-type relation for two reasons. Firstly, internal resistance adds a constant term to viscous and diffusive stomatal resistances (eq.8, 12). Secondly, molecular slip introduces a relationship more complex than a simple power of stomatal dimension (see eq.14) for stomata of slit-like geometry. Nevertheless a simple power-type relation may be adequate to fit existing experimental data, since, as shown in Fig.5, slope does not change rapidly with stomatal aperture, except when this is very low. One factor affecting diffusive resistance which has been omitted so far is the "end-correction" discussed by PENMAN and SCHOFIELD (1951) which increases the effective length of the stomatal pore. The effect of this end correction was calculated for the circular stomatal pore of minimum resistance already described. This correction was only made to the outer end of each stomatal pore since the method used to calculate ri includes the end correction for the inside of the pore. The end correction increased the calculated total diffusive resistance r from 4.1 to 4.5 sec cm -a, and 4.8 to 5.5 sec cm -a for leaves with high and low stomatal frequency respectively, thus hardly affecting the relationship shown in Fig.2. The relatively small effect of this end correction on r is partly the result of ri (2.9 sec cm -1) being the major component of r at this stomatal aperture. This dominance of internal resistance rapidly diminishes with decreasing stomatal aperture, but so does the relative importance of end correction to stomatal pore resistance. The effect of ti~is end correction on r is thus unlikely to be much larger than in the example given above. The resistance r defined in eq.12 differs from the total diffusive resistance for gas exchange; for example, r~ overestimates the internal resistance for transpiration and carbon dioxide uptake. Thus the effect of end correction can be more significant for gas exchange at large stomatal apertures (PENMAN and SCHOFIELD, 1951). Also Agr. Meteorol., 4 (1967) 103-117
116
P.G. JARVIS, C. W. ROSE AND J. E. BEGG
stomatal pore resistances are in series for r but effectively in parallel for gas exchange. B o u n d a r y layer resistance also affects gas exchange, whereas this resistance has been measured and subtracted f r o m the experimental data (see "experimental") and therefore has been excluded in the theory. A c o m m o n criticism o f the mass-flow porometer was that it did not provide an absolute measure o f stomatal aperture or diffusive resistance (e.g., HEATH, 1941). Using the theory given in this paper stomatal aperture can be calculated from the results o f suitably large mass-flow porometers, provided the internal a n a t o m y and the following stomatal dimensions o f the leaf are also determined. Where stomatal pores are approximately circular, stomatal frequencies (n) and depth (I) are required to calculate the radius (a) o f the stomatal pore from viscous resistance fL If stomatal pores are slit-like, slit length (b) is also required to calculate slit width (w). The theory m a y also be used to calculate stomatal aperture f r o m diffusion porometer measurements, provided again that other relevant anatomical dimensions are known. The theory given in this paper is in no way restricted to cotton, and has general application.
ACKNOWLEDGEMENTS The authors thank Dr. R. O. Slatyet for his encouragement, Dr. D. J. Goodchild of the C.S.1.R.O. Division o f Plant Industry for electron micrographs, Miss D. Jones and Miss J. Sheaffe for help with calculations and cultivation of the plants, and K. Rattigan for assistance with microtoming. The action o f Dr. P. E. Waggoner sending us his paper at the manuscript stage was very much appreciated.
REFERENCES BIERHUIZEN,R. F., SLATYER,R, O. and ROSE, C. W., 1965. A porometer for laboratory and field operation. J. Exptl. Botany, 46 : 182-191. GREGORY,F. G. and ARMSTRONG,J. I., 1936. The diffusion porometer. Proc. Roy. Soc. (London), Ser. B, 121 : 27-42. GREGORY,F. G. and PEARSE,H. L., 1934. The resistance porometer and its application to the study of stomatal movement. Proc. Roy. Soc. (London), Ser. B., 114 : 477-493. HEATH, O. V. S., 1959. The water relations to stomatal cells and the mechanisms of stomatal movements. In: F. C. STEWARD(Editor), Plant Physiology. Academic Press, New York, N.Y., 2 : 193-250. HEATH,O. V. S., 1941. Experimental studies of the relation between carbon assimilation and stomatal movement. IL The use of the resistance porometer in estimating stomatal aperture and diffusive resistance. Part 1. A critical study of the resistance porometer. With an appendix by H. L. PENMAN.Ann. Botany (London), 5 : 455-500. JARVIS,P. G. and SLATYER,R. O., 1966. A controlled environment chamber for studies of gas exchange by each surface of a leaf. C.S.L R. 0., Div. Land Res., Tech. Papers, in press. KENNARD,E. H., 1938. Kinetic Theory of Gases. McGraw-Hill, New York, N.Y., 483 pp.
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MASKELL, E. J., 1928. Experimental researches on vegetable assimilation and respiration. XVlII. The relation between stomatal opening and assimilation. A critical study of assimilation rates and porometer rates in leaves of cherry laurel. Proc. Roy. Soe. (London), Ser. B, 102 : 488-533. MEIDNER, H. and SPANNER,D. C., 1959. Differential transpiration porometer. J. Exptl. Botany, 10 : 190-205. PENMAN, H. L., 1942. The theory of porometers used in the study of stomatal movements in plant leaves. Proc. Roy. Soc. (London), Ser. B, 130 : 416-434. PENMAN,H. L. and SCHOFIELD,R. K., 1951. Some physical aspects of assimilation and transpiration. Symp. Soe. ExptL Biol., 5 : 115-129. RASCHKE, K., 1965. Der Seifenblasenporometer zur Messung der Stomaweite an amphistomatischen Bl~ittern. Planta, 66 : 113-120. REYNOLDS, E. S., 1963. The use of lead citrate at high pH as an electron opaque stain in electron microscopy. J. Cellular Biol., 17 : 208-212. SHIMSHI, D., 1964. The use of a porometer for the study of water stress in plants. Israel J. Agr. Res. 14 : 137-143. SLATYER, R. O. and JARVIS,P. G., 1966. A gaseous diffusion porometer for continuous measurement of the diffusive resistance of leaves. Science, 150 : 574--576. SPANNER, D. C. and HEATH, O. V. S., 1951. Experimental studies of the relation between carbon dioxide assimilation and stomatal movement. II. The use of the resistance porometer in estimating stomatal aperture and diffusive resistance. Part. 2. Some sources of error in the use of the resistance porometer and some modifications of its design. Ann. Botany (London), 15 : 319-331. WAGOONER, P. E., 1965. Calibration of a porometer in terms of diffusive resistance. Agr. Meteorol., 2 : 317-329.
Agr. MeteoroL, 4 (1967) 103-117
Research Papers AN EXPERIMENTAL AND THEORETICAL COMPARISON OF VISCOUS AND DIFFUSIVE RESISTANCES TO GAS FLOW T H R O U G H AMPHISTOMATOUS LEAVES P. G. JARVISI~ C. W. ROSE AND J. E. BEGG
Division of Land Research, Commonwealth Scientific and Industrial Research Organization, Canberra, A.C.T. (Australia) (Received April 20, 1966)
SU MMARY
A theory with general application has been developed enabling resistance to viscous (~) and diffusive gas flux (r) through amphistomatous leaves to be calculated from anatomical dimensions. The calculated resistances agreed very well with experimental values of ~ and r measured for a range of stomatal apertures in leaves of cotton (Gossypium hirsutum cv. POPE). The theory can be used to calculate stomatal aperture from either mass flow or diffusion porometer measurements, provided other relevant anatomical characteristics are known.
INTRODUCTION
For many years, mass flow porometers have been used to provide estimates of stomatal aperture (e.g., MASKELL,1928; PENMAN, 1942; SPANNERand HEATH, 1951; HEATH, 1959; SIJIMSHI, 1964; BIERHUIZENet al., 1965; RASCHKE, 1965; WAGGONER, 1965). In mass flow porometry, a pressure difference is applied across a leaf and the resulting viscous flow of air is taken as a relative measure of stomatal aperture. However, the exchanges of water vapour and carbon dioxide between leaf and atmosphere which occur in transpiration and photosynthesis are commonly assumed to be diffusive in character. Hence, if the facility of these diffusive exchanges is to be inferred from mass flow porometer measurements, a calibration of diffusive resistance in terms of the resistance to viscous flow of air is required. In the past, this calibration has been attempted empirically (e.g., MEIONER and SPANNER, 1959), and by a combination of experiment and theory of diffusive and viscous flow (e.g., MASKELL,1928; PENMAN, 1942). PENMAN (1942) derived such theory specifically for the resistance porometer i Present address: Department of Botany, University of Aberdeen, Aberdeen (Great Britain).
Agr. Meteorol., 4 (1967) 103-I 17
104
P. G. JARVIS, C. W. ROSEAND J. E. BEGG
of GREGORY and PEARSE (1934), and for the diffusion porometer of GREGORY and ARMSTRONG (1936) and HEATH (1941). This theoryincluded the contribution to flow from outside the porometer cup. The theory was not developed in terms of anatomical dimensions. Recently WAGGONER(1965) has theoretically investigated the relation between viscous and diffusive resistances, and attempted a comparison between theory and experiment. The result of this comparison was somewhat inconclusive, partly because diffusive resistance was not measured. This paper presents theory for the calculated comparison between resistances to viscous flow of air and diffusive flux of nitrous oxide from one side of a leaf to the other. The calculated relationship between the viscous and diffusive flow resistances is compared with experimental data obtained on cotton leaves using a conventional mass flow porometer and the nitrous oxide diffusion porometer (SLATYER and JARVlS, 1966).
THEORY Analogous expressions are required for the viscous flow of air and the diffusive flux of gas from one side of a leaf to the other, through the stomatal pores and intercellular spaces in the mesophyll. As nitrous oxide was used to measure diffusive resistance, it is assumed that no diffusion could take place through the cuticle. The theory is developed for stomatal apertures with slit-like geometry, and for cylindrical intercellular spaces within the mesophyll, and the justification for this is considered later.
Viscous flow of air The basic equation of viscous flow is: dv = * / ~ (dyne cm -2)
(1)
where z is the shear stress, t/the coefficient of viscosity (g cm -1 sec-I) and dv/dz the velocity gradient (sec-1). Consider steady laminar flow through unit length of an infinitely long slit of width w (cm) and depth l (cm) in the direction of flow (the x-direction). An element of volume (shaded in Fig.l) of depth dx and width 2z normal to the surfaces of the slit, experiences equal and oppositely directed forces caused by viscous flow and by the pressure differences dp across it in the direction of flow. Equating these forces: (dp)2z = r/(dv/dz) 2 dx which on integration gives: v --
(dfl)
(z z)
~/ (dx)
1
(2)
+ C (cm sec-1)
(2)
Agr. Meteorol., 4 (1967) 103-117
VISCOUSAND DIFFUSIVERESISTANCESOF LEAVES
105 Direction of
flow
Wails of slit Fig.1. An infinitesimalelement of volume of unit length located in an infinitelylong slit of width w and depth I in the direction of mass flow of air caused by pressure difference(pl--p2) across the slit. where C is a constant of integration to be evaluated. When slit width is not large compared with the mean free path of the molecules the number of molecular collisions between successive contacts with the walls is relatively small. The effect of this on viscous flow can be accomodated by setting the velocity of air at the walls equal to a velocity of slip ( ~ 0), defined by: v = - - ~ (dv/dz) (era see-1)
(3)
where ~ is the coefficient of slip (cm), (KvNNARD, 1938). Substituting from eq.3 into eq.2 to evaluate C gives: 1
v -- 2r/
(dp)
w2
(dx) (z2
4
w ()
(cm sec-1)
(4)
The mass of air flowing per second through unit length of slit is: W
M = p 2
2 I
v dz = - - p
(dp) (w3) (1 ÷ 6 ~ ) (dx) (12~/) -w-
0 where p is the density of air. From the ideal gas law, p = p / R T where p is the mean pressure on the element [(pl + ps)/2] and R the gas constant per gram of air. Multiplying through by dx and integrating from 0 to 1 (the depth of the slit in the direction of flow) or from pl to/)2, gives: M--
w3p 12 r/l (1 + 6w0 (/Ol --p2)
(g cm-%ec -1) Agr. Meteorol., 4 (1967) 103-117
106
P . G . JARVIS, C. W. ROSE AND J. E. BEGG
Thus the volume flow per unit surface area, Q, for a surface containing n identical rectangular slits per unit area, each of width w and length b normal to the direction of flow and depth l, is: Q
M bn P
6 r/l~ ( 12 '-k - --/ (pl - - p2) (cm3cm-2sec-a) -- 1w3bn
(5)
__ P l - - f 1 2
(6)
no where f~p, the resistance of the slit-shaped pores per unit area of surface, is: 12 ~/l (1) f~p = w3b( 1 + ~ ) ( n )
(g cm-Zsec-1)
(7a)
The corresponding equation for a surface containing n identical cylindrical pores with circular cross-section normal to the direction of flow, and each of radius a, is:
8 ,fl
F2p =
(1)
~ra4 (1 + _ ~ _ )
(g cm-Zsec-1)
(7b)
(n)
Assuming that flow through narrow slits is analogous to that through stomata, eq.7a may be applied to viscous flow through an amphistomatous leaf with n, and nl stomata per unit area on the upper and lower surfaces, respectively. Air passing from the upper to the lower side of the leaf experiences sequentially nu stomatal pore resistances f~p in parallel, in series with an internal resistance ~i resulting from the thickness and internal anatomy of the leaf, and nl resistances f~p in parallel. The total viscous resistance per unit area of leaf, f~, is thus:
12
=
t" 1
~n.
+
wab( 1 q_~)
1 \
) --' ni (gcm-2sec -1)
(8)
n/
where f~p" and f~pt are the stomatal resistances to viscous flow in the upper and lower surfaces, respectively. The geometry assumed for air passages within the mesophyll is that they are cylinders of length equal to the thickness of the mesophyll. Denoting this thickness by Ii, and taking the number of air passages (of radii a;) per unit plan area of leaf to be hi, it follows from eq.7b that: ~'~i : =
8 tlli rc ai4(1 @ 4~')
(1)
(g cm-2sec-1)
(9)
(hi)
ai
Agr. Meteorol.,
4 (1967) 103-117
107
VISCOUS AND DIFFUSIVERESISTANCESOF LEAVES
Diffusive flux of nitrous oxide Measurements of diffusive resistance were obtained using nitrous oxide. The maximum difference in nitrous oxide concentration across the leaf was approximately 500 p.p.m, in these experiments. The resultant partial pressure gradients were therefore sufficiently small for the flux to be correctly calculated from a simple application of Fick's Law. From this law the volume flux of nitrous oxide by diffusion, q, through unit area of leaf surface containing n slit-like stomata of width w, length b, and depth lis: Ac
q = (wbn) Dz%o T
(cm3cm-Zsec--0
a c
(10)
rp where (A c/l) is the concentration gradient in nitrous oxide, and rp, the diffusive resistance of the slit-shaped pores per unit area of surface is: rp
--
/
(])
D~2° wb
(n)
(sec cm -1)
( 11a)
The corresponding equation assuming circular stomatal geometry is: (1) (n)
l
rp
Di~2o ~ a 2
(sec cm-1)
(1 l b)
Omitting the boundary layer resistances, the total resistance per unit leaf area experienced by nitrous oxide in diffusing from one side of a leaf to the other, r, is: r - - rp u j - rp l -~- ri
1 wb D~2o
(
1
@
\ nu
1
) @ ri
(sec cm -1)
(12)
nt
where rp" and r / a r e the stomatal pore diffusive resistances in the upper and lower leaf surfaces, respectively, and r~ the internal leaf diffusive resistance resulting from the thickness and anatomy of the leaf. Using the notation above, it follows from eq. 1lb that: ri
l~ DN20 rC ai 2
(1) (hi)
(see cm -1)
(13)
METHODS
Cotton plants (Gossypium hirsutum cv. PopE) were grown in solution culture on Agr. Meteorol.,
4 (1967) 103 117
108
P . G . JARVIS, C. W. ROSE AND J. E. BEGG
modified Hoagland's nutrient solution in a growth chamber with illumination from fluorescent tubes (Sylvania, Cool White V.H.O.) supplemented with tungsten globes, giving a radiation flux density of ca. 120 W m -~ (400-700 m/t) for 12 h each day. The photoperiod was extended to 14 h with the tungsten lamps alone. Air temperature was 35°C during the photosynthesis period and 24°C in the dark. Relative humidity was ca.65 ~ .
Experimental In the experiments the fifth emerged leaf from the apex was used when there were about ten leaves on the plant and before flowering commenced. The apparatus has been described in detail elsewhere (JARVIS and SLATYER, 1966). Opposed areas (31 cm z) of each surface of an attached cotton leaf were isolated in ventilated, air-conditioned chambers such that vapour and gas exchanges between each surface of the leaf and the ambient air could be followed for the identical surfaces used in porometry. Because of the large leaf surface within the porometer, the results obtained should be independent of porometer dimensions, and permit an accurate determination of r and ~ as defined in the theory. A range of stomatal apertures was obtained by varying the intensity of incident light and the ambient COz concentration in the leaf chamber. Diffusive resistance was calculated using an equation similar to eq. 10 from the measured volume flux by diffusion of nitrous oxide from one side of the leaf to the other in response to a concentration difference. Boundary layer resistance to diffusion was separately measured and allowed for in the calculation which is described in detail by SLATYERand JARVlS (1966). The viscous flow resistance was calculated using an equation similar to eq.6 from the flow of air from one side of the leaf to the other in response to a difference in pressure. The pressure difference across the leaf did not exceed 10 cm of water: if larger pressure differences were used, anomalous fluxes were obtained at small stomatal apertures (see also RASCHKE, 1965). The direction of the fluxes of air and nitrous oxide, though usually from upper to lower surface of the leaf, did not affect the result, confirming that the fluxes across the leaf were essentially one-dimensional and that possible errors due to lateral flow were insignificant. The N20 flux was monitored continuously: mass flow of air was recorded intermittently during periods of steady N~O flux.
Anatomical Stomatal dimensions and frequency were determined on leaves from the porometer using a light microscope. Using a freezing microtome, series of paradermal sections were obtained from
Agr. MeteoroL,4 (1967) 103-117
109
VISCOUS AND DIFFUSIVE RESISTANCES OF LEAVES
the fifth leaf of plants comparable to those used for the resistance measurements; 60 cellular and inter-cellular intercepts were measured on a series of transects taken in different directions across these sections. The resolution obtainable with a light microscope left some uncertainty as to stomatal slit length. Independent and more accurate evidence on this length was obtained using an electron microscope, for which the material was prepared by placing in 2% potassium permanganate in veronal buffer for 2 h, and Reynold's lead for 30 rain (REYNOLDS, 1963).
RESULTS
Experirnental resistances The relationship between experimentally determined values of diffusive resistance, 1,000
u
g m
~ 100 ° ~ o
I1)
o~
c o
~.@ro
o°-~
•
°
- - w = 1p
t_
o ::.
lO
~&~,,,v
= 1.5 ,u,
m
.&
es= 5.32~
10
100
viscous
1,000
10,000
100,000
resistance -Q_ ( g c m -2 sec-I.lO 4)
Fig.2. A doubly logarithmic plot o f diffusive resistance o f a c o t t o n leaf to nitrous oxide (r) against its viscous resistance to the m a s s flow o f air across the leaf (~). Experimental values f r o m four leaves s h o w n by small o p e n circles. Calculated values with different a s s u m p t i o n s fitted with curves.
Legend
nu no. m m -z nt no. m m -2 D i m e n s i o n s of stomatal aperture
Rectangular stomata • •
Circular stomata • o
57 92 138 209 b variable w variable
57 138
92 209
as = 5.32/1
Agr. Meteorol., 4 (1967) 103-117
110
p . G . JARVIS, C. W. ROSE AND J. E. BEGG
r, and the viscous flow resistance, fL is shown in Fig.2, At high fluxes of NzO and air, corresponding to the lowest resistances in Fig.2, the error in the measured resistances is low and they may be considered accurate to better than 0.5 sec cm -1 for r and 0.5 • 104 g cm-esec-1 for ~. At the lowest fluxes, the error in the resistances may be two orders of magnitude larger. Some of the spread in the points results from the effect of the operation of the mass flow porometer on stomatal aperture. Even at the low pressures used and with the short period of operation, the N 2 0 flux was often somewhat lower after the mass flow porometer had been used.
Anatomical The ranges of stomatal characteristics were as shown in Table I.
TABLE I RANGES OF STOMATAL CHARACTERISTICS
Mean stomatal frequency (n, number mm -2) Mean total length of stomatal slit (it) Mean depth of stomatal pore (l, It)
Upper
Lower
,~.rfaee
..,face
57-92 12.3-14.8 12.7-14.9
138-209 12.7-14.0 12.0-12.8
Electron micrographs, such as that shown in Fig.3, indicated that effective slit length (b) varied with slit width (w) for w -----0-2.5/1. For w ~ 2.5/t, b was at a maximum. The simplest model for the relation between b and w suggested by such micrographs was therefore to assume b proportional to w from w = 0 to w 2.5 ¢t, and b = constant for w ~ 2.5/t. In this model (Fig.4) the constant value taken for b in the calculations of resistance was 14.2/z, a value obtained on one of the leaves used in the experiment. This value is higher than the overall mean (13.4g) for all leaves investigated, and leads to about 5 % lower values in calculated resistances. The effect of this difference on the theoretical relationship between r and f~ in Fig.2 is insignificant. The thickness of mesophyll (li) was measured from transverse leaf sections and found to be 200 j= 10/1. The arithmetic mean of the intercellular spaces between mesophyll cells, obtained from paradermal sections, was found to be 1.8 -k 0.3 gc Since this mean was obtained from transects in many different directions, it could be used as an estimate of the mean diameter of a cylindrical air passage, or the mean side dimension of an air passage of square cross section. From eq.9 and 13 for cylindrical air passages it can be seen that resistances are inversely proportional to a power of the radius. Since the intercellular spaces cover a range of sizes from 0 to 7.2/1, use of the arithmetic
Agr. Meteorol., 4 (1967) 103 117
VISCOUS AND DIFFUSIVE RESISTANCESOF LEAVES
111
Fig.3. Electron micrograph showing partially open cotton stomata.
16
i 0
1.0
2.0 Slit
width
3o w (if)
4.0
Fig.4. Model for the relationship between stomatal slit length (b), and width (w) during stomatal opening.
Agr. Meteorol., 4 (1967) 103 117
112
P . G . JARVIS~ C. W. ROSE AND J. E. BEGG
mean would overestimate the resistances. Therefore mean values of ai z (1.73/~2) and ai 4 (10.9/~4) were calculated for use in eq. 13 and 9, respectively. From the transect data the number of air passages in the mesophyll per unit leaf plan area (n,.), was calculated to be 71.5 • 104 cm -2. Calculated resistances
Substituting these values for l;, ai, and ni into eq. 13, ri was found to be 2.9 sec cm -1. A comparable value has been measured experimentally on the same material by an independent method (P. G. Jarvis and R. O. Slatyer, unpublished data, 1966). This is some justification for the use of the model which represents the intercellular spaces in the mesophyll as cylinders. Viscous resistance ~i was 1.2.104 g cm-2sec -1, calculated from eq.9. Substituting these values for fL" and r;, and the values for upper and lower surface stomatal frequencies (nu and nl), in eq.8 and 12 the viscous (~) and diffusive (r) resistances of cotton leaves were calculated for the anatomical characteristics given previously, and for w over the range 0.2-2.5/1. The value of ~ at 30°C was calculated as 0.0679 /~ from data by KENNARD (1938). The very good agreement between the calculated and experimental resistances is shown in Fig.2. Minimum stomatal resistance would be achieved for a circular stomatal pore. Assuming the maximum pore perimeter to be equivalent to that of a rectangular pore of length 14.2 iL and width 2.5 ¢t, the radius of a circular pore would be 5.32/~. Total resistances corresponding to minimum stomatal resistance were calculated replacing the stomatal terms of eq.8 and 12 by eq.7b and l l b lespectively, and are also shown in Fig.2. These points represent the lower limit for the relationship between r and ~.
DISCUSSION
Any difference between experimental and calculated results lies within the estimated experimental error mentioned in the section above. The calculated values of resistance depend on theory, anatomical dimensions, and the model for stomatal opening. If, instead of the model shown in Fig.4, stomata1 slit-length b was constant at its maximum value (14.2/0 for all slit widths, the relationship between the calculated resistances fall increasingly below those shown in Fig.2 as slit width decreases below I¢t. Viscous or diffusive stomatal resistance at a slit width of 2.5/~ may be twice as great as internal resistance depending on stomatal frequency. For circular stomatal pores of radius 5.32/t the internal resistance may be nearly three times as great as stomatal resistance, thus illustrating the importance of internal resistance to mass flow and diffusion at low total leaf resistance. In the high resistance region, on the other hand, internal resistance is negligible compared with stomatal resistance. The measured range in stomatal frequencies on the upper and lower leaf surfaces, whilst Agr. MeteoroL, 4 (1967) 103-117
113
VISCOUS AND DIFFUSIVE RESISTANCES OF LEAVES
considerably affecting the magnitude of the calculated resistances, has only a small effect on the relationship between resistances shown in Fig.2. The magnitude of the effect of molecular slip on calculated viscous resistance of stomata (f~p), is illustrated in Table II. If molecular slip were ignored it follows from Table II and Fig.2 that there would be a discrepancy between observed and calculated values when resistance is high.
T A B L E II THE EFFECT OF MOLECULAR SLIP ON VISCOUS RESISTANCE OF STOMATA (~'~p) AT DIFFERENT STOMATAL WIDTHS W
(nu = 5700 c m -e, nl = 13800 cm -2, / = 12.4/z, b = 14.2/t)
(11)
f~p (slip omitted) ( g c m -2 sec-1 • 10 4)
f~p (with slip) ( g e m -2 sec-1 . 10 4)
0.1 0.2 0.5 0.8 1.0
1,138,000 73,600 1,935 290 119.8
223,103 24,120 1,066 192 85.1
W
1.5 2.0 2.5
23.6 7.49 3.07
18.6 6.22 2.64
Another effect of molecular slip is that it increases the slope of theoretical relationship between log r and log f~ shown in Fig.2. Since slip is of greater significance at smaller stomatal widths where the contribution of internal resistance to the total is very small, internal resistance will be neglected in the considerations of the following two paragraphs, in which an expression for this slope is derived (eq. 17). Neglecting fL" and rl in eq.8 and 12 it follows that: r w2 (1 + 6~/w) ~-- = 12 t/D~2o
(14)
- f(w), a function of w. Taking logarithms: In r ~ In [f(w)] q- In Whence it follows that: d (In r) d(lnO) -- 1 +
f~ f(w)
d[f(w)] dw
F d f ~ q--1 t_dwA
(15)
Since d(ln r)/d(ln ~) = d(log r)/d(log ~), which is the slope of the theoretical relationship plotted in Fig.2, this slope is also given by eq. 15. Agr. Meteorol., 4 (1967) 103-117
114
P. G. JARVIS, C. W. ROSE AND J. E. BEGG
From eq. 8, 14 and 15, and a linear relation between b and w so that: b = kw
(16)
where k is a constant, it may be shown that d(log r) d(log~)
-
(1 + 3 ~ /w) 2(1 -5 4.5 ( / w )
1 -
(17)
It follows from eq.17 that if slit width were sufficiently large to neglect (i/w) in comparison with unity, the slope of the relation shown in Fig.2 would be 0.50. In .contrast, as the slit closes and w tends to zero the slope tends to the value 0.66. The increase in slope with decreasing slit width predicted by eq.17 is shown in Fig.5. It may be noted that eq.17 and therefore the slope of the calculated relation shown in Fig.2, is independent of the value of constant k in eq. 16. The full range of experimental data is well fitted by a straight line. Since the theory predicts an increase in slope with decreasing slit width (i.e., increasing resistance), possible evidence for a similar change in slope in the experimental data was sought by separate regression analyses for f~ < and > 30 • 104 g cm-Zsec -1. Mean slopes were 0.43 and 0.49 respectively, each with 95 % confidence intervals of :L 0.06. Thus the experimental data indicated a similar trend to the theory, although the increase in slope with increasing resistance was not significant. It may be seen from Fig.2 that the range of slit widths corresponding to the experimental data with ~q < 30 • 104g cm -2 sec -1 is from about 2.5 to 1.5 g. For this range the slope (0.43) for the experimental data falls below the curve for b ~ w in Fig.5, and above the curve for b : constant to be derived in the next paragraph. For f~ > 30 • 104 g cm -2 sec -1, slit widths range from about 1.5 to 0.5/t. The slope (0.49), is not significantly different from the curve for b ~ w. Performing a similar analysis to that used in deriving eq.17, but replacing
0.7 F
~ (3.6 bo~w
= CON St
O.B[
O
1 Sh't
2 width
3 w
(pL)
Fig.5. Theoretical relationship between the slope [d (log r)/d (log fD] and slit width (w) for slit length (b) constant, and proportional to slit width (w). Agr. Meteorol., 4 (1967) 103-117
VISCOUS AND DIFFUSIVE RESISTANCES OF LEAVES
11 5
eq.16 with assumption that slit-length, b, is constant and independent of slit-widths, it may be shown that: d(log r) d(logf~)
--
1--
2(1 + 3 l/w) 3(1 ~- 4 i/w)
(18)
The slope represented by eq.18 tends to 0.33 as slit width increases, and to 0.5 as w tends to zero, the entire relation also being given in Fig.5. Assuming the output of MEIDNER and SPANNER'S (1959) porometer to be proportional to diffusive conductance, it is of interest that the range of slopes they found for wheat stomata of rectangular shape was 0.32-0.37. An expression for [ d (log r)/d (log f~)] similar to eq. 18 may be derived including the effect of internal resistance. Since internal resistance affects this slope over only a limited resistance range, such an expression is of less interest than eq.17 and 18 and will not be considered here. A power-type relationship between viscous and diffusive resistance has commonly been assumed, corresponding to a linear relationship when these resistances are plotted logarithmically, as in Fig.2. The theory given in this paper predicts departure from a simple power-type relation for two reasons. Firstly, internal resistance adds a constant term to viscous and diffusive stomatal resistances (eq.8, 12). Secondly, molecular slip introduces a relationship more complex than a simple power of stomatal dimension (see eq.14) for stomata of slit-like geometry. Nevertheless a simple power-type relation may be adequate to fit existing experimental data, since, as shown in Fig.5, slope does not change rapidly with stomatal aperture, except when this is very low. One factor affecting diffusive resistance which has been omitted so far is the "end-correction" discussed by PENMAN and SCHOFIELD (1951) which increases the effective length of the stomatal pore. The effect of this end correction was calculated for the circular stomatal pore of minimum resistance already described. This correction was only made to the outer end of each stomatal pore since the method used to calculate ri includes the end correction for the inside of the pore. The end correction increased the calculated total diffusive resistance r from 4.1 to 4.5 sec cm -a, and 4.8 to 5.5 sec cm -a for leaves with high and low stomatal frequency respectively, thus hardly affecting the relationship shown in Fig.2. The relatively small effect of this end correction on r is partly the result of ri (2.9 sec cm -1) being the major component of r at this stomatal aperture. This dominance of internal resistance rapidly diminishes with decreasing stomatal aperture, but so does the relative importance of end correction to stomatal pore resistance. The effect of ti~is end correction on r is thus unlikely to be much larger than in the example given above. The resistance r defined in eq.12 differs from the total diffusive resistance for gas exchange; for example, r~ overestimates the internal resistance for transpiration and carbon dioxide uptake. Thus the effect of end correction can be more significant for gas exchange at large stomatal apertures (PENMAN and SCHOFIELD, 1951). Also Agr. Meteorol., 4 (1967) 103-117
116
P.G. JARVIS, C. W. ROSE AND J. E. BEGG
stomatal pore resistances are in series for r but effectively in parallel for gas exchange. B o u n d a r y layer resistance also affects gas exchange, whereas this resistance has been measured and subtracted f r o m the experimental data (see "experimental") and therefore has been excluded in the theory. A c o m m o n criticism o f the mass-flow porometer was that it did not provide an absolute measure o f stomatal aperture or diffusive resistance (e.g., HEATH, 1941). Using the theory given in this paper stomatal aperture can be calculated from the results o f suitably large mass-flow porometers, provided the internal a n a t o m y and the following stomatal dimensions o f the leaf are also determined. Where stomatal pores are approximately circular, stomatal frequencies (n) and depth (I) are required to calculate the radius (a) o f the stomatal pore from viscous resistance fL If stomatal pores are slit-like, slit length (b) is also required to calculate slit width (w). The theory m a y also be used to calculate stomatal aperture f r o m diffusion porometer measurements, provided again that other relevant anatomical dimensions are known. The theory given in this paper is in no way restricted to cotton, and has general application.
ACKNOWLEDGEMENTS The authors thank Dr. R. O. Slatyet for his encouragement, Dr. D. J. Goodchild of the C.S.1.R.O. Division o f Plant Industry for electron micrographs, Miss D. Jones and Miss J. Sheaffe for help with calculations and cultivation of the plants, and K. Rattigan for assistance with microtoming. The action o f Dr. P. E. Waggoner sending us his paper at the manuscript stage was very much appreciated.
REFERENCES BIERHUIZEN,R. F., SLATYER,R, O. and ROSE, C. W., 1965. A porometer for laboratory and field operation. J. Exptl. Botany, 46 : 182-191. GREGORY,F. G. and ARMSTRONG,J. I., 1936. The diffusion porometer. Proc. Roy. Soc. (London), Ser. B, 121 : 27-42. GREGORY,F. G. and PEARSE,H. L., 1934. The resistance porometer and its application to the study of stomatal movement. Proc. Roy. Soc. (London), Ser. B., 114 : 477-493. HEATH, O. V. S., 1959. The water relations to stomatal cells and the mechanisms of stomatal movements. In: F. C. STEWARD(Editor), Plant Physiology. Academic Press, New York, N.Y., 2 : 193-250. HEATH,O. V. S., 1941. Experimental studies of the relation between carbon assimilation and stomatal movement. IL The use of the resistance porometer in estimating stomatal aperture and diffusive resistance. Part 1. A critical study of the resistance porometer. With an appendix by H. L. PENMAN.Ann. Botany (London), 5 : 455-500. JARVIS,P. G. and SLATYER,R. O., 1966. A controlled environment chamber for studies of gas exchange by each surface of a leaf. C.S.L R. 0., Div. Land Res., Tech. Papers, in press. KENNARD,E. H., 1938. Kinetic Theory of Gases. McGraw-Hill, New York, N.Y., 483 pp.
Agr. Meteorol., 4 (1967) 103-117
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MASKELL, E. J., 1928. Experimental researches on vegetable assimilation and respiration. XVlII. The relation between stomatal opening and assimilation. A critical study of assimilation rates and porometer rates in leaves of cherry laurel. Proc. Roy. Soe. (London), Ser. B, 102 : 488-533. MEIDNER, H. and SPANNER,D. C., 1959. Differential transpiration porometer. J. Exptl. Botany, 10 : 190-205. PENMAN, H. L., 1942. The theory of porometers used in the study of stomatal movements in plant leaves. Proc. Roy. Soc. (London), Ser. B, 130 : 416-434. PENMAN,H. L. and SCHOFIELD,R. K., 1951. Some physical aspects of assimilation and transpiration. Symp. Soe. ExptL Biol., 5 : 115-129. RASCHKE, K., 1965. Der Seifenblasenporometer zur Messung der Stomaweite an amphistomatischen Bl~ittern. Planta, 66 : 113-120. REYNOLDS, E. S., 1963. The use of lead citrate at high pH as an electron opaque stain in electron microscopy. J. Cellular Biol., 17 : 208-212. SHIMSHI, D., 1964. The use of a porometer for the study of water stress in plants. Israel J. Agr. Res. 14 : 137-143. SLATYER, R. O. and JARVIS,P. G., 1966. A gaseous diffusion porometer for continuous measurement of the diffusive resistance of leaves. Science, 150 : 574--576. SPANNER, D. C. and HEATH, O. V. S., 1951. Experimental studies of the relation between carbon dioxide assimilation and stomatal movement. II. The use of the resistance porometer in estimating stomatal aperture and diffusive resistance. Part. 2. Some sources of error in the use of the resistance porometer and some modifications of its design. Ann. Botany (London), 15 : 319-331. WAGOONER, P. E., 1965. Calibration of a porometer in terms of diffusive resistance. Agr. Meteorol., 2 : 317-329.
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