Plant Physiol. (1980) 66, 250-253
0032-0889/80/66/0250/04/$00.00/0
Diffusion from a Circular Stoma through a Boundary Layer
A FIELD-THEORETICAL ANALYSIS
Received for publication August 8, 1979 and in revised form April 14, 1980
JAMES R. TROYER
Department of Botany, North Carolina State University, Raleigh, North Carolina 27650
ABSTRACT
The case of diffusion of a gas from a single circular stoma through an
unstirred boundary layer of finite thickness into a perfectly stirred atmosphere free of convective effects is examined theoretically, with the gas
assumed to be at constant concentration across the stoma. The analysis
employs a mathematical solution to an analogous problem in electrostatic
physics previously obtained by Kuz'min (1972 Sov Phys Tech Phys 17:
473-476). The diffusion flux is shown to be no more than 1% greater than
that into a perfectly unstirred atmosphere if the boundary layer is thicker
than 40 times the stomatal radius. Under the conditions assumed, for
realistic boundary-layer and stomatal dimensions, taking the diffusion flux
through the boundary layer to be Unear with the stomatal radius would
usually involve no significant error. This result may indicate that the
principal effect of wind velocity on mass exchange between leaf and
atmosphere may be exerted through influencing convection outside the
boundary layer rather than through determining the thickness of that layer.
Contemporary models of water transport in plants are usually
formulated in terms of various pathway resistances, that is, the
modeling approach is usually based on circuit theory. But circuit
theory, although extremely useful, treats three-dimensional phenomena with one-dimensional approximations which may be
inadequate in some cases, especially when the inclusion of circuit
elements with different characteristics introduces mathematical
discontinuities (3, 11). (Note, however, that the classical circuittheoretical approach of Brown and Escombe [2] has recently been
justified by Kelman [7] and Parlange and Waggoner [15].)
Much of modem physical science, on the other hand, is rooted
in the powerful principles of field theory, which treats generally
of fluxes in two or three dimensions (11, 12, 22). In addition to
retaining the higher dimensionality, field theory is appropriate to
a wide range of applications-electrostatics, electrodynamics,
magnetostatics, fluid flow, heat conduction, and diffusion, to name
a few. Since the treatments of all these phenomena involve the
definition of scalar potential functions, the mathematical developments used in field-theoretical approaches are similar. This
mathematical similarity and the wide applicability have the extremely important consequence that analogies can be deduced and
solutions transferred from one area of application to another. In
other words, if a problem is solved in one area, analogous problems
are thereby solved in the other areas to which field theory applies.
Cooke (3) has pointed out the relevance of these concepts to the
case of stomatal diffusion.
The gas diffusion problem is analogous, for example, to the
problem of the electrostatic field of a conductor. Thus (22), when
e is the dielectric constant, P is the electrostatic potential, D is the
diffusion coefficient, and C is the concentration or chemical
potential of the diffusing gas, the associated field vector for the
electrostatic field is
F =-e grad P
and that for the stationary diffusion field is
J =-D grad C.
The electric charge within a surface X, then, is in general
Q=-
e grad P dX
and, analogously, the total diffusion flux from a surface X is in
general
u = f D grad C dX.
The field transmittance for the electrostatic field is the capacitance,
whereas that for the diffusion field is the diffusional conductance.
The surface under consideration may be a circular or elliptical
disc. The electrostatic problem of the charge on such a circular or
elliptical conducting disc maintained at an electrostatic potential
with respect to infinity is analogous to the problem of the diffusion
of a gas through a single isolated circular or elliptical stoma into
a perfectly unstirred, homogeneous, and isotropic atmosphere, the
concentration of gas being constant across the stomatal opening.
For a circular conducting disc of radius a maintained at a potential
Po, the charge on one side only is
Qo = 2
7e
Ko PO
with the capacitance
Ko=-7T
the charge thus bears a linear relation to this capacitance and,
hence, to the radius of the disc (17, 22). Cooke (3) has solved the
cases of the diffusion of gas through circular or elliptical stomata.
For the circular stoma of radius a maintained at a gas concentration of CO (the concentration at the stoma minus that at infinity),
the diffusional flux is
UO = 2 r D Go CO
with the diffusional conductance
Go=
2a
the diffusion rate thus bears a linear relation to this conductance
and, hence, to the radius of the stoma. Stefan (6, 18) reached a
similar conclusion which is widely cited, although his development
was based on an incorrect form of a fundamental equation (3).
But how might these relations be modified if the atmosphere is
stirred? What is the consequence of a finite unstirred boundary
250
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Plant Physiol. Vol. 66, 1980
DIFFUSION THROUGH A BOUNDARY LAYER
layer, and what is the relation between the thickness of that
boundary layer, the radius of the stoma, and the diffusion rate?
Tranter (21) suggested a procedure by which the electrostatic
equivalent of this case might be approached but did not provide
a computational solution.
More recently Kuz'min (9) has analyzed a similar electrostatic
equivalent and offered a complete solution. This report analogizes
Kuz'min's results to the problem of diffusion from a circular
stoma through a finite boundary layer. This case is more simplified
and more abstract than those detailed by Parlange and Waggoner
(15) and Bange (1). Because it is relatively exact and a little more
realistic than that of the single circular stoma with an unstirred
atmosphere, it should be added to the theoretical arsenal available
to plant physiologists (1-3, 5, 7, 10, 15, 18).
THEORY
Consider the case (Fig. 1) of a single circular stomatal opening
S of radius a in an infinite planar cuticle E which is impermeable
to the diffusing gas. Assume the interior to be in a steady state
such that the gas is diffusing out through the stoma and that the
concentration of gas in the plane of the stoma is a constant value
C0 (the concentration at the stoma normalized with respect to the
concentration in the stirred portion of the atmosphere). Assume
the isotropic atmosphere M to be perfectly stirred up to distance
h from the stoma, with the concentration of the diffusing gas thus
constant throughout it. Assume that the layer of atmosphere B of
thickness h between the stoma and the stirred region is unstirred;
transport of the gas through this boundary layer is thus by
diffusion. This division of the atmosphere into a perfectly unstirred
boundary layer and a perfectly stirred region some distance from
the stoma omits from consideration the convective effects discussed by Parlange and Waggoner (15); this situation thus represents a simplified limiting case. Finally, assume that the diffusion
coefficient is constant. In cylindrical coordinates the boundary
conditions under these assumptions are thus
C = C0,
z =0,
0
-= 0,
z =0,
r > a;
C=0,
z h.
,
r
-
a;
--I
M
--
z
i
p-
in Figure 2. The stoma S in the impermeable cuticle E of Figure
I is analogous to a circular conducting disc of radius a, and the
perfectly stirred atmosphere M of Figure I is analogous to an
infinite conducting plane T at distance h from the disc. The disc
is assumed to be maintained at a constant electrostatic potential
Po with respect to that of the plane. The region between the disc
and the plane is, of course, analogous to the unstirred boundary
layer in the diffusion problem; the region beyond the plane is
disregarded because it is inside the conductor T.
Kuz'min (9) has solved this electrostatic problem, in fact, for
the more general circumstance in which the disc is opposite a
coaxial circular aperture in the plane. The present case is simply
that special one in which the aperture in the plane does not exist,
that is, in which the radius of the aperture is zero.
Kuz'min's approach (9) is as follows. After expression of the
solution in terms of Hankel integrals, the method of paired integral
equations is used to reduce the problem to a Fredholm integral
equation of the second kind. For various values of a, where
a=
a
hs
h
the ratio of the radius of the disc to the distance between the disc
and the plane, solutions to this Fredholm equation then are
obtained by successive approximations. From the Fredholm equation itself an expression for the charge on the disc is derived
2a C
Q = PO7ra I f(x) dx,
in which the symbols are as defined above. The charge then is
expressed in relative terms as a ratio of the charge on the disc to
Qo, the charge on a similar isolated disc not near a plane. (Kuz'min
[9] actually worked with the ratio of capacitances, but this is the
same as the ratio of charges when the same electrostatic potential
Po exists on each disc.) Such a ratio of charges is then equal to an
integral involving the solutions of the Fredholm equation,
QQ = f f(ay) dy,
These conditions would have to be applied in solving the fundamental equation for the diffusion flux through S.
The electrostatic equivalent of this diffusion problem is shown
/
251
/
/
and this integral is evaluated from the solutions previously obtained by successive approximation.
Kuz'min (9) presents the results as the values of the ratio of
charges (capacitances) for various values of a from 0 to 2. Because
of the analogy of the problems, these values of the ratio of charges
/
z
//
)
I
I,
V,
.1
FIG. 1. Circular stoma separated from a stirred atmosphere by an
unstirred boundary layer, in cylindrical coordinates. E: infinite impermeable cuticle, z = 0, r > a; S: circular stoma of radius a, z = 0, r a, at
concentration C = CO with respect to M; M: infinite perfectly stirred
atmosphere, z 2 h, at concentration C = 0 with respect to S; B: infinite
unstirred boundary layer of thickness h, 0 - z < h.
-
FIG. 2. Circular conducting disc near a conducting plane, in cylindrical
coordinates. R: circular conducting disc of radius a, z = 0, r < a, at
potential P = PO with respect to T; T: infinite conducting plane, z = h, at
potential P = 0 with respect to R.
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252
Plant Physiol. Vol. 66, 1980
TROYER
can be used as values of the ratio of diffusion fluxes in the problem
of interest. This latter ratio is U/Uo, where U is the diffusion flux
from the stoma through the finite boundary layer and UO is the
diffusion flux into an infinitely thick boundary layer from a stoma
having the same radius and the same concentration ofthe diffusing
gas.
RESULTS AND DISCUSSION
To exhibit characteristics of the relationship which are of special
interest with regard to the stomatal diffusion problem, the ratio of
diffusion rates U/Uo is plotted here against h/a, the ratio of
boundary-layer thickness to stomatal radius (the inverse of
Kuz'min's [9] a) (Fig. 3). Enough points to render this curve
meaningful in the region of interest were obtained by plotting
Kuz'min's results (9) against a on a large scale and then interpolating graphically. The relation is such that the interpolation can
be made accurately and no more sophisticated transformation is
necessary.
As the boundary-layer thickness increases without bound, so
does the ratio h/a, and the ratio of fluxes approaches one-that
is, if the boundary layer is infinitely thick, the entire atmosphere
is unstirred and the diffusion rate is UO. On the other hand, as the
boundary layer becomes very thin, the ratio h/a tends toward zero
and the diffusion rate increases greatly, becoming equal to 1.821
UO when the boundary-layer thickness is one-half the stomatal
radius (the smallest value of h/a for which a result is plotted in
Figure 3, the equivalent of a value of 2 for Kuz'min's [9] a). The
interesting feature is that as the boundary-layer thickness increases
from zero, the diffusion to flux ratio decreases very sharply. The
diffusion rate is only 10% greater (U/U0 = 1.1) with a boundarylayer thickness only 3.5 times the radius of the stoma. The
diffusion rate is just 1% greater (U/Uo = 1.01) with a boundarylayer thickness only 40 times the stomatal radius. And with still
thicker boundary layers, the diffusion rate is essentially equal to
that into an infinitely thick boundary layer (U is only 1.008 or
1.004 times UO for boundary-layer thicknesses 50 or 100 times the
stomatal radius, respectively).
Using the factor of 40 mentioned above, for stomatal radii in
the range of those of real stomata, say 3 to 10 ,um (4, 8), the
diffusion rates would be no more than 1% larger than those into
an unstirred atmosphere if the boundary layer were thicker than
120 to 400 ,um. Actual boundary-layer thicknesses vary with a
number of factors, of course, especially with wind velocity and
leaf size but also with turbulence intensity and leaf shape ( 13, 16).
A boundary-layer thickness of 400 ,um is smaller than most of the
values calculated by Nobel (14) for a number of wind velocities
and leaf dimensions. (The values over the infinite planar cuticle
of the present model would be even greater than those derived by
Nobel [ 14].) Thus, for a single stoma under the most usual
circumstances, the diffusion rate through a finite boundary layer
would be no more than 1% greater than that into an unstirred
atmosphere, a difference which may be regarded as insignificant.
Even for the most extreme case assumed by Nobel (14) (a wind
velocity of 1,000 cm s-' and a leaf dimension of only 0.2 cm), the
resulting boundary-layer thickness of 57 ,im is only 5.7 times the
largest radius mentioned above and the diffusion flux only 6%
greater (U/U0 = 1.06). If one assumes for plants in general a more
usual stomatal radius of 5 t,m, then a boundary layer as thin as 57
,tm would result in a diffusion flux less than 4% greater (U/U0 =
1.035).
Thus this theory predicts, within the applicability of the assumptions, that with boundary layers of realistic thickness the
diffusion rate will usually not be significantly different from that
into a perfectly unstirred atmosphere. Thus also, again within the
restrictions imposed by the assumptions, treating the diffusion flux
through the boundary layer from a single stoma as linear with the
stomatal radius will usually introduce no significant error even if
the atmosphere is stirred. Since wind velocity, which affects among
other things boundary-layer thickness, is known to have major
effects on mass exchange between leaf and atmosphere (14), this
prediction would seem to conflict with observation. The present
case is idealized in that it considers the atmosphere to be perfectly
stirred beyond the boundary-layer; real boundary layers are, of
course, not so sharply delimited. As has been pointed out by
Parlange and Waggoner (15), in real situations, convective effects
occur in the region beyond the boundary layer (their "region 2")
which are influenced by wind and which would, in turn, affect the
concentration field. The prediction of the present model suggests
that in fact the major effect of wind velocity may be exerted
outside the boundary layer, rather than in determining the thickness of that layer.
Leaves possess many stomata, and so there arises the problem,
not addressed here, of stomatal interaction (15, 19, 20). Parlange
and Waggoner (15) concluded that such interaction is negligible
if the stomata are spaced at a distance which is at least three times
the larger dimension of a stoma, even for noncircular pores.
Acknowledgment-This paper is dedicated to the memory of Dr. Henry Lawrence
Lucas, Jr., late William Neal Reynolds Professor of Statistics and Director of the
Biomathematics Program, North Carolina State University; mens invicta manet.
LITERATURE CITED
l'
U)
I.
h
I
FIG. 3. Diffusion to flux ratio as a function of the ratio of boundarylayer thickness to stomatal radius for the system of Fig. 1. U: diffusion
flux from a circular stoma of radius a through an unstirred boundary layer
of thickness h; Uo: diffusion flux into an infinitely thick boundary layer
from a circular stoma of the same radius and at the same concentration.
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Plant Physiol. Vol. 66, 1980
DIFFUSION THROUGH A BOUNDARY LAYER
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