Nonisothermal effects on water potential measurement in a simple geometry

In this paper, we investigate quantitatively the coupling between gradients of temperature and of chemical or water potential under steady-state conditions in the vapor phase. This coupling is important for the measurement and modeling of the dynamics of water in unsaturated environments such as soils and plants. We focus on a simple nonequilibrium scenario in which a gradient of temperature exists across an air-filled gap that separates two aqueous phases with no net transfer of water. This scenario is relevant for measurements of the water potential in environmental and industrial contexts. We use a tool, a microtensiometer, to perform these measurements. We observed variations of the water potential with a difference of temperature across the air gap of $-7.9\pm 0.3\mathrm{MPa}{\mathrm{K}}^{-1}$, in agreement with previous measurements. Our result is close to a first-order theoretical prediction, highlighting that most of the effect comes from the variation of saturation pressure with temperature. We then show that thermodiffusion (Soret effect) coupled to natural convection could occur in our experiment and discuss how these effects could explain the small discrepancy observed between measurements and first-order theoretical predictions.

Figure 1

Context of interest. (a) Unsaturated porous medium. Within the porous matrix, pockets of capillary condensed liquid water coexist with pockets of vapor in air. In the presence of a temperature gradient, a gradient of water potential is established which drives a net water flux $\vec{j}$ through the medium. The gradient of water potential $\nabla \mathrm{\Psi }$ is proportional to the gradient of temperature $\nabla T$ [see Eq. (4)]. (b) Simplified situation studied here. Aqueous phases are kept separated by a vapor gap, in a closed system ensuring that there is zero net flux of mass ($\vec{j}=\vec{0}$). An imposed stationary temperature gradient $\nabla T$ results in a steady water potential gradient $\nabla \mathrm{\Psi }$. We focus on the special case in (b) in this study.

Figure 2

Pictures of the tensiometer. (a) Global view of a packaged tensiometer with a copper strip. See Figs. 3 and 3 for a schematic diagram of the packaged device. (b) Details of the tensiometer. The top view shows the piezoresistors (an example circled in red) that constitute the strain gauge to measure deformations of the diaphragm and the platinum wire used as PRT. The bottom view shows the cavity containing bulk water and the nanoporous membrane (dark on the picture) connecting it with the outside. Microfluidic veins are etched in the membrane without linking directly the cavity and the outside in order to decrease the response time of the device. The image of the nanopores has been acquired by scanning electron microscopy and have an average pore radius $r_{\text{p}}=1.7\mathrm{nm}$. Details on microfabrication are given in the text. [Image credit (b): Antoine Robin.]

Figure 3

Schematic diagrams of the experimental setup (not to scale). (a) Cross-sectional view of the experimental setup. The tensiometer (in black) is glued to a copper strip (in orange) and packaged in urethane (in gray). The device is capped by a glass tube (in white) closed by a porous hydrophobic mesh (dashed line) and dipped in an osmotic solution of known water potential (in blue) through a thermal insulation (in light yellow). Temperature of the tensiometer is controlled by a Peltier module (in pink) attached at the emerging end of the copper strip and powered through an Arduino-based feedback loop imposing a constant temperature difference with the solution. (b) Expanded view of the vapor gap. While the temperature of the solution $T_2$ is imposed by a thermostat, that of the tensiometer $T_1$ is controlled with a Peltier element connected to the copper strip on which the tensiometer is glued with thermal paste. When a temperature difference is imposed, there is a difference between the water potential measured by the tensiometer ${\mathrm{\Psi }}_1$ and that of the solution ${\mathrm{\Psi }}_2$. The water potential in the cavity is obtained by measuring the deformation of the diaphragm with the strain gauge, giving a voltage $U$ proportional to ${\mathrm{\Psi }}_1$. (c) Expanded view of the nanoporous membrane. When in contact with an unsaturated vapor, curvature of the menisci at the pore mouth decreases the pressure of the pore liquid to achieve mechanical equilibrium.

Figure 4

Evolution of the voltage $U$ across the Wheatstone bridge of the tensiometer with the measured difference of temperature $\mathrm{\Delta }{T}^{*}$ between the tensiometer and pure liquid water. The red lines are linear regressions to determine the offset between the measured difference $\mathrm{\Delta }{T}^{*}$ and the true difference $\mathrm{\Delta }T=\mathrm{\Delta }{T}^{*}+\theta$.

Figure 5

Variation of bridge voltage $U$ as a function of the (corrected) temperature difference $\mathrm{\Delta }T$ across the air gap for several solutions of different water potential ${\mathrm{\Psi }}_{\text{ref}}$ (pure water ${\mathrm{\Psi }}_{\text{ref}}=0\mathrm{MPa}$; urea solutions ${\mathrm{\Psi }}_{\text{ref}}=-1.16$, $-3.10$, $-3.94$, $-4.06$, $-6.13$, and $-8.88\mathrm{MPa}$; PEG solution ${\mathrm{\Psi }}_{\text{ref}}=-1.2\mathrm{MPa}$). Lighter colors correspond to increasing absolute water potential $|{\mathrm{\Psi }}_{\text{ref}}|$. Lines are linear fits.

Figure 6

Absolute water potential of the solution $-{\mathrm{\Psi }}_{\text{ref}}$, determined by a chilled mirror hygrometer, as a function of the bridge voltage $U(\mathrm{\Delta }T=0)$ measured by the tensiometer in isothermal conditions. The red line corresponds to a linear fit according to Eq. (9).

Figure 7

Variation of water potential measured by the tensiometer relative to the value in isothermal conditions, ${\mathrm{\Psi }}_{\text{ref}}-\mathrm{\Psi }$, for several solutions of different ${\mathrm{\Psi }}_{\text{ref}}$ (pure water ${\mathrm{\Psi }}_{\text{ref}}=0\mathrm{MPa}$; urea solutions ${\mathrm{\Psi }}_{\text{ref}}=-1.16$, $-3.10$, $-3.94$, $-4.06$, $-6.13$, and $-8.88\mathrm{MPa}$; PEG solution ${\mathrm{\Psi }}_{\text{ref}}=-1.2\mathrm{MPa}$). Lighter colors correspond to increasing absolute water potential $|{\mathrm{\Psi }}_{\text{ref}}|$. The red line is a linear fit.