Perspectives and design considerations of capillary-driven artificial trees for fast dewatering processes

Recent progresses on nanocapillary-driven water transport under metastable conditions have substantiated the potential of artificial trees for dewatering applications in a wide pressure range. This paper presents a comprehensive performance analysis of artificial trees encompassing the principle for negative capillary pressure generation; impacts of structural, compositional, and environmental conditions on dewatering performance; and design considerations. It begins by delineating functionalities of artificial trees for evaporation (leaves), conduction (xylem), and filtration (root) of water, in the analogy to natural trees. The analysis revealed that the magnitude of (negative) capillary pressure in the artificial leaves and xylem must be sufficiently large to overcome the osmotic pressure of feed at the root. The required magnitude can be reduced by increasing the osmotic pressure in the artificial xylem conduits, which reduces the risk of cavitation and subsequent blockage of water transport. However, a severe concentration polarization that can occur in long xylem conduits would negate such compensation effect of xylem osmotic pressure, leading to vapor pressure depression at the artificial leaves and therefore reduced dewatering rates. Enhanced Taylor dispersions by increasing xylem conduit diameters are found to alleviate the concentration polarization, allowing for water flux enhancement directly by increasing leaf-to-root membrane area ratio.

S-2 Note S1. Modeling details of water transport in artificial trees 1

) Leaf pores
The near-equilibrium of water in the liquid and vapor phase at the meniscus (i.e., liquid-vapor interface in the leaf pore) requires equating the chemical potentials of water in the two phases. The chemical potential of water in the vapor phase (at z = H) under the ambient pressure (P a = 1 bar) is given as [1][2][3] : Here, R and T the universal gas constant and temperature, respectively; ' is the chemical potential of water at the reference state, i.e., at saturation vapor pressure ( 102 ' ( )) and for pure water, and only a function of temperature; the water activity ( . ) is expressed as . = .
5 . * = . 5 8 9 (:) 8 ; , where . 5 is the activity coefficient and . * is the mole fraction of water vapor in air ( . * ). " ($) is the saturation vapor pressure at the meniscus (z = H). At moderate pressure, the water vapor can be assumed an ideal gas, and hence . 5 ≈ 1. Then, Eq. (S1) can be rearranged to: For the liquid phase, the chemical potential can be expressed as following: S-3 where P ∞ (v) is the partial pressures of vapor in the ambient outside the boundary layer of water vapor (at z = z ∞ ), respectively; k leaf is the mass transfer coefficient for the water vapor, and estimated using Eq. (10) in the main text.
The leaf-to-root membrane area ratio is defined as: A R ≡ A leaf / A root and the mass conservation where J w is the water flux based on the root membrane area. Requiring " ($) = " (>) and combining with Eq. (S4), the hydraulic pressure in the leaf pore (P H ), which is negative in value is expressed as follows:

2) Stem (xylem conduit)
For a cylindrical tube with the diameter of xylem conduit (d X ), the momentum balance in zdirection shows: where r is the variable in the radial coordinate, u z is the fluid velocity in z-direction, and is the dynamic viscosity of liquid water. The integration of Eq. (S6) from r = 0 to r = d X /2 results in the average velocity: Integration of Eq. (S7) from z = 0 (xylem-root membrane interface) to z = H (leaf pore) leads to the following expression for the hydraulic pressure at z = 0: When the solute distribution reaches steady-state, the convective flux and diffusive flux need to be counterbalanced, which requires: Assuming a perfect solute rejection by the root membrane and perfect solute retention by the meniscus in the leaf pore, the total mass of the solute in the xylem conduit is conserved, which requires: S-4 . MNN (S12.)

3) Root membrane
From the Eq. (7) in the main text, the solution-diffusion mechanism can be written as: where A m is the water permeability of the membrane; the feed is exposed to the ambient pressure 0 ; Π mp and Π ' are the osmotic pressure at the feed-root membrane interface and the xylem-root membrane interface, respectively. Osmotic pressures are functions of solute concentrations, i.e., Π mp = Π mp C r,mp and Π ' = Π ' C F,' . The solute concentration at the feed-root membrane interface (C r,mp ) is elevated from the bulk concentration in the feed (C r,mO ) by concentration polarization 5 : The mass transfer coefficient in the feed (k root ) for turbulent flows is estimated as 43 : where ℎ stt2 ≡ stt2 / ' , where L is the characteristic length over which the boundary layer of solute concentration develops at the feed-root membrane interface, which is equated to be the length scale of root membrane (e.g., length scale of building foundation), and L = 50 m is used as S-5 a commonly observable length scale of urban buildings. Also, . = . / ' and {,mO = mO / . , where . is the kinematic viscosity of water. Feed flow velocity ( mO ) of 1 m s -1 is assumed for calculation.

4) Solving by the iterative method
The Newton-Raphson iterative method was used to solve the coupled Eqs. (S1 -S15). 7 Specifically, in Eq. (13), P 0 is substituted by Eq. (S5) and (S8); Π mp = Π mp C r,mp and Π ' = Π ' C F,' are substituted by Eq. (S11) and (S14). Through iterations, the water flux J w is obtained, as well as C X,0 , C X,H , P 0 , P H , and other parameters. Accordingly, The concentration polarization factor in the xylem conduit is determined (Fig. 4B). The saturation vapor pressure at the meniscus, P H (v) is then determined, which is a function of the temperature, solute concentration (C X,H ), and hydraulic pressure (P H ), known as the Kohler equation 8,9 : where Π " = Π " ( F," ) . Accordingly, the vapor pressure depression is determined (Fig. 4C).
Knowing P H , the contact angle between the meniscus and the leaf pore wall (q) is also determined from the Young-Laplace equation (Eq. (1)), which is required to be larger than the receding contact angle (q min = 10° for hydrophilic surfaces considered here). S-6

Note S2. Heterogeneous nucleation for different negative pressures and xylem hydrophobicity
Surface hydrophobicity has a determining role for the propensity of cavitation, as a more hydrophobic surface reduces the free energy barrier to form bubble size larger than critical size.
The probability of bubble formation from homogeneous nucleation is quite small. 10 Therefore, we only consider the heterogeneous nucleation here. The probability of the occurrence of a cavitation event following the formation of a vapor nucleus on a solid surface can be calculated using the classical nucleation theory. The nucleation rate, I, that is, the number of nucleation events per unit solid surface area per unit time is given as 10, 11 : where I 0 is a kinetic prefactor and k is the Boltzmann constant. DG cav is the free energy barrier for cavitation given as: Here, is the surface tension of liquid; F is the contact angle between the liquid and the solid surface (e.g., inner surface of artificial xylem conduit); Δ is the pressure difference between the liquid phase and the vapor phases (i.e., inside the bubble) across the liquid-vapor interface. The pressure in the vapor phase is essentially saturation vapor pressure determined by the temperature, curvature and solute concentration in the liquid phases 8,9 (also in Eq. S16), of which magnitude is generally less than 1 bar. The pressure in the liquid phase can be negative for both natural and artificial trees, and large negative pressure will greatly reduce the free energy barrier for cavitation. Compared to a perfectly hydrophilic surface ( F = 0°), a highly hydrophobic surface ( F = 150°) S-8 has ~100 times lower free energy barrier, which greatly elevates the risk of cavitation. This again emphasizes that not only the xylem conduit surface needs to be hydrophilic, but also a very tight root membrane would be necessary to prevent any permeation of hydrophobic contaminants, onto which bubble formation may occur.

Note S3. Evolution of solute concentration profile in artificial xylem conduit
Due to low flow velocities in the artificial xylem conduits expected (<~O(10 -5 m s -1 ), Fig. 4 and   5), it takes a rather long time duration to reach the steady-state, solute concentration distribution.
Neglecting radial distribution, the cross-section averaged, solute concentration distribution ( F OE ) in the xylem conduit along the z-direction can be determined based on the convection-diffusion equation: When the time scale for convection is much larger than that for diffusion, i.e., H/ F >> H 2 /D eff , the convection time scale (H/ F ) may be taken to non-dimensionalize the time variable t. Taking , the Eq. (S19) can be non-dimensionalized as following: We note from Eq. (S19) and Fig. 5 that when Pe H >> 1, the flow is dominated by convective flow.
Eq. (S19) was numerically solved using the implicit Euler method for time integration and central difference for spatial discretization, 7 with the results for Pe H = 10 shown in Fig. S2. S-10