Water and Solute Permeability of Plant Cuticles: Measurement and ...

4.5 Diffusion and Viscous Transport of Water 85
Table 4.4 Estimating pore size and pore number in MX-membranes of Citrus aurantium in Na +
form. Data for Pdiffusion were taken from Schönherr (1976a), and Pmaximum viscous
from Table 4.3
values were obtained
pH Pdiffusion P maximum
viscous Pvisc/Pdiff rpore Apore (m 2 ) Number of
(ms −1 ) (ms −1 ) (nm) pores/m 2
3.0 2.56 × 10 −7 0.74 × 10 −6 2.89 0.50 2.79 × 10 −4 3.55 × 10 14
6.0 4.16 × 10 −7 1.35 × 10 −6 3.25 0.54 4.16 × 10 −4 4.54 × 10 14
9.0 7.30 × 10 −7 1.92 × 10 −6 2.63 0.46 7.96 × 10 −4 11.97 × 10 14
rpore was calculated using (4.13); Apore is the total pore area per m 2 membrane calculated according
to (4.9) using D = 2.44 × 10 −9 m 2 s −1 and ℓ = 2.66 × 10 −6 m; number of pores = Apore/πr 2 pore .
reasons why we have no data for large solutes, even for the MX. Since Psolute for
cuticular membranes is at least 100 times smaller, it is clear that accurate measurements
in the steady state are not possible. The problem can be overcome using the
SOFP technique (Sect. 6.4).
From Pdiffusion and P maximum
viscous , equivalent pore radii can be calculated using (4.13).
The ratio Pviscous/Pdiffusion is larger than 2 (Table 4.4), indicating the presence of
aqueous pores (Nevis 1954). Their radii range from 0.46 to 0.54 nm. Since radii
are calculated from the ratio of two empirical permeances, the differences are not
significant (Schönherr 1976a). The pore radii did not depend on pH, and the mean
is 0.50 nm. The increase in permeance with pH can be attributed to an increase in
number of pores rather than to larger pore radii. The average pore size estimated
from diffusional and viscous permeability (0.5 nm) is larger than that calculated by
Schönherr (1976a), who obtained radii of 0.44–0.46 nm because he used Pviscous
obtained with raffinose, which is a little smaller than that estimated by curve fitting
(Table 4.3). This estimated pore radius of 0.5 nm is too small, since with
sucrose (rsolute = 0.555nm) and raffinose (rsolute = 0.654nm) viscous permeance
was smaller than P maximum
viscous . Clearly, the MX membranes were not totally impermeable
to these solutes. Some reasons for this discrepancy are considered next.
There are a number of assumptions inherent in the above calculations. The determination
of Pdiffusion and Pviscous is straightforward and no assumptions are needed.
However, in calculating the pore radius from the permeances the diffusion coefficient
and viscosity enter (4.11), and we used bulk properties (4.12). Activation
energies of diffusion of water and of viscous flow in MX membranes are 54 and
46kJmol −1 respectively, while for bulk liquid the level is only about 19kJmol −1
(Schönherr 1976a). The activation energy of a hydrogen bond is about 20kJmol −1
(Nobel 1983). Hence, when water molecules diffuse in bulk only one of the hydrogen
bonds is broken at a time, while in the pore liquid more than two H-bonds need
to be broken for a water molecule to move. This suggests that D in the pore liquids
is considerably smaller than in bulk water. This is not too surprising, since pores are
very narrow and the radius of a water molecule is about 0.19–0.197 nm, depending
on source (Durbin 1960; Renkin 1954).