Fluidity and phase transitions of water in hydrophobic and hydrophilic nanotubes

We put water flow under scrutiny to report radial distributions of water viscosity within hydrophobic and hydrophilic nanotubes as functions of the water-nanotube interactions (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\epsilon }}_{sf}$$\end{document}ϵsf), surface wettability (θ), and nanotube size (R) using a proposed hybrid continuum-molecular mechanics. Based on the computed viscosity data, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\rm{\epsilon }}}_{{\rm{sf}}}/{\rm{\theta }}-{\rm{R}}$$\end{document}ϵsf/θ−R phase diagram of the phase transitions of confined water in nanotubes is developed. It is revealed that water exhibits different multiphase structures, and the formation of one of these structures depends on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\rm{\epsilon }}}_{{\rm{sf}}},\,{\rm{\theta }},$$\end{document}ϵsf,θ, R parameters. A drag of water flow at the first water layer is revealed, which is conjugate to sharp increase in the viscosity and formation of an ice phase under severe confinement (R ≤ 3.5 nm) and strong water-nanotube interaction conditions. A vapor/vapor-liquid phase is observed at hydrophobic and hydrophilic interfaces. A state of confinement is revealed at which water exhibits different multiphase structures under the same flow rate. The derived viscosity functions are used to accurately determine factors of flow enhancement/inhibition of confined water.


S1: Slip Boundary Conditions Should be replaced by a Radial Distribution of Viscosity
Nanoconfined water slips over a hydrophobic surface 1 , and it may stick to or leak through a hydrophilic surface 2 . Therefore, the conventional no-slip boundary conditions at the water-surface interfaces do not hold. Over the past years, various experimental and molecular dynamics (MD) models were proposed to report the slip boundary conditions of nanoconfined water 3 . Utilizing these slip boundary conditions at different water-surface interfaces, the classical continuum models of fluid mechanics were thought they can reflect the same results as experimental and MD models. However, many challenges are associated with this approach. Here, we discuss some of these challenges.
The slip boundary conditions of water in nanotubes were defined as follows 3,4 : where ( ) is the velocity profile function. and are slip correction parameters. is the slip velocity represents the velocity jump at the interface. is the slip length, which is the linear/nonlinear extrapolation of the velocity profile to a radius at which velocity would be zero.
The slip length was related to the slip velocity via a linear extrapolation as follows 3,4 : Here, we demonstrate that the slip parameters ( and ) of the boundary conditions are insufficient to describe accurately the nontraditional phenomena of nanoconfined water, and these parameters should be replaced by a radial distribution of the water viscosity.
Depending on the nanotube's size and wettability, the slip parameters attain values of ≥ 0 and ± ≥ 0. A non-zero slip velocity, > 0, is an indication of water slippage over the nanotube wall, and a zero slip velocity gives the conventional Hagen-Poiseuille model. Thus, the slip velocity is not a proper parameter to model the drag revealed in Fig.1 (see the main text) or the absorption 2 of water particles at the boundary with hydrophilic surfaces. The slip length, however, could be positive or negative. For water slippage over the surface, the slip length is positive. A negative slip length indicates a sticking of water particles to the surface and a drag in the water flow. Thus, the slip length is more efficient than the slip velocity to quantify the enhancement/inhibition of water flow in nanotubes. Therefore, different formulas were proposed to relate the flow enhancement/inhibition factor ( ) to the slip length ( ). These formulas are collected in Table S1. -The profile of water flow has a velocity jump at the interface followed by a parabolic flow.
is a fitting parameter.
is linearly extrapolated (Eq. (S2)). -The profile of water flow has a velocity jump at the interface followed by a parabolic flow. -Constant water viscosity within the nanotube. However, an effective water viscosity that depends on the nanotube radius was considered. -The effective viscosity is the weighted average of an arbitrary assumed interfacial viscosity (∼0.655 mPa·s) to the bulk water viscosity.
is the dynamic coefficient of friction. 0 (m/s) is a fitting parameter.
-The profile of water flow has a velocity jump at the interface followed by a parabolic flow.
-The profile of water flow has a velocity jump at the interface followed by a parabolic flow. -The slip length depends on the nanotube size.
-Distinguish between water viscosity at the interface ( ) and viscosity of water core ( ).
is a fitting parameter. surface contact angle.
is linearly extrapolated (Eq. (S2)). -The profile of water flow has a velocity jump at the interface followed by a parabolic flow. -Constant water viscosity within the nanotube. However, an effective water viscosity that depends on the nanotube radius was considered. -The effective viscosity is the weighted average of the interfacial viscosity to the bulk water viscosity. -The interfacial viscosity ( ) is related to the surface wettability: = 0 (−0.018 + 3.25) -The slip length depends on the surface wettability. 8 nanopores 3,9 . Discrepancies between the reported values of the slip length in the literature can be observed and have been discussed in previous studies 3,9 . Moreover, big discrepancies between slip lengths of experimental studies and slip lengths obtained from MD were discussed 3,9 .
Here, we demonstrate that the − relations in Table S1 are insufficient and this is the reason behind these discrepancies. These relations are only limited for a special case of water flow in nanotubes. These relations were derived assuming that the velocity jump is followed by a parabolic velocity profile within the water core and a constant distribution of water viscosity within the nanotube. However, because of the water drag at the first water layer (see Fig.1 in the main text), these relations cannot properly reflect the enhancement/inhibition of water flow in hydrophobic/hydrophilic nanotubes.
For further demonstration, we present in Fig. S1 two cases of water flow. In previous investigations, water flow was assumed like case A ( Fig.S1(a)). In this case, water particles were considered sliding over the wall (thick black line) where a jump in the velocity profile (blue) was considered at the water interface with the tube. The velocity profile, ( ), within the water core was assumed parabolic. According to these assumptions, the slope of the velocity (brown) linearly decreases within the water core because of the parabolic flow, and the slope of the velocity at the interface sharply decreases due to water slippage. According to the viscosity-slope relation (i.e. where is the pressure gradient), the viscosity of the water core is constant (green), and it sharply decreases at the interface. For Case A, the nonlinear extrapolation is more accurate than the linear extrapolation ( Fig.S1(a)). None of the − relations in Table S1 can accurately model water flows like Case A. All relations were derived based on = 1 + 4 , which assumes a linear extrapolation of the slip length. However, the accuracy of these relations can be modified via a nonlinear extrapolation of the slip length, and, in this case, these relations can give an accurate representation of water flow of Case A. However, because of the water-surface interactions, the actual water flow in nanotubes is similar to Case B ( Fig.S1(b)). Non-parabolic velocity profiles with a jump at the interface are usually observed when water flow in nanotubes (see Fig.1). The linear and (even) nonlinear extrapolations give wrong corrections of Hagen-Poiseuille flow, as shown in Case B. The viscosity, however, is a proper measure that can effectively reflect the changes in the velocity profiles due to water-tube interactions and the surface wettability. This non-parabolic velocity profile indicates that the viscosity of water radially varies as presented in Fig.S1(b).

S2: Drag of Water Flow at the First Water Layer
Majumder and Corry 10 carried out MD simulations of water flow in polarized CNTs to investigate effects of the electrostatic interactions between polar water particles and polar CNT walls. They added charges to the CNT wall to increase the interaction between water and CNT.
Because of the additional electrostatic potential, water-CNT interactions were increased from = 1.423 kJ/mol (for nonpolarized CNT) to = 6.95 kJ/mol (for polarized CNT with +/-0.5 e charge) and = 18 kJ/mol (for polarized CNT with +/-0.9 e charge). The flow of water in a nonpolarized CNT was obtained with a negligible drag (Fig.S2). The increase in the water-CNT interaction, however, resulted in a drag in the water flow at the first water layer. The increase in the water-CNT interaction increases the hydrophilicity in the system where water flow approaches to the parabolic flow of bulk water. In addition to the water-surface interactions, roughness of the confining surface causes a drag at the first water layer (see the case of Rough CNT in Fig. S2).
It should be mentioned that Majumber and Corry 10 did not refer to the drag of water flow at the first water layer. Instead, they demonstrated that, due to high electrostatic interactions or high roughness between water and the confining surface, the enhancement of water flow was affected and the flow converted to be a parabolic flow. Here, we show that the increase in the watersurface interaction and/or the roughness of the confining surface are accompanied with a drag at the first water layer, which is associated with a sharp increase in water viscosity at this layer.

S3: VPR Function
The velocity-to-pressure gradient ratio (VPR) is plotted as a function of the nanotube radius based on the results of MD simulations of water flow in CNTs 5,9,[11][12][13][14][15][16] . It should be mentioned that the first author previously used this approach to identify the interfacial viscosity of water flow in CNTs 17 . Figure S3: Determination of the VPR function. Figure S4: Relation between water-surface interaction energy ( ) and contact angle ( ) as a measure of the surface wettability.

S5: Model Validation
To validate the proposed hybrid continuum-molecular mechanics (HCMM), tests were carried out by comparing the results obtained by the proposed HCMM to the results of experimental and MD models available in the literature.
Comparison to Whitby et al. 4 Using a nanoporous membrane with 43 ± 3 nm CNTs, Whitby et al. 4 reported the flow characteristics of water through the membrane. The pressure was measured for different imposed flow rates through the membrane. The density of the CNTs in the membrane was determined by 1.07 × 10 10 #/cm 2 . Table S2 shows the flow rates as determined by the proposed HCMM in comparison to the experimentally determined ones. Comparison to Majumder et al. 13 The experimental setup in Majumder et al. 13   Comparison to Holt et al. 11 Holt et al. 11 reported water flow measurements through membranes with aligned double-walled carbon nanotubes DWCNTs serve as pores with diameters of less than 2 nm. They measured the permeability of three membranes with 20, 3, and 2.8 μm thicknesses and compared it to the permeability of polycarbonate membrane. The density of pores was estimated by ≤ 0.25 × 10 12 #/cm 2 for the DWCNTs membranes and 6 × 10 8 #/cm 2 for polycarbonate membrane. Table S4 shows the measured permeability of the membranes in comparison to the calculated one using the proposed HCMM. Comparison to Qin et al. 14      They reported a decrease in the diffusivity with a decrease in the nanopore diameter, which indicated an enhancement in the flow of water due to water confinement 18,19 . The energy parameters used in the MD simulations carried out by Milischuk and Ladanyi 18 were = 1.912 kJ/mol and = 0.27 nm. Table S5 shows the results of the performed analyses using the proposed HCMM.   Results of the proposed HCMM are presented by solid curves.

S6: Modified Navier-Stokes Equation
Here, a detailed formulation of the model presented in Methods is given. Consider a water particle at a position in the space. This water particle is surrounded by a set of other water particles and is located close to a solid surface, as shown in Fig.S10. Interactions between this water particle and other water particles are generated. The strength of each one of these interactions is (this interaction force is conjugate to the water-water interactions). In addition, to , a residual interaction force is generated between the water particle and the nearest particle of the solid surface ( ), as shown in Fig.S10. The strength of the residual interaction depends on the distance between the water particle and the solid particle. Now, Newtonian's balance conditions can be applied to the water particle to give: where ̇ is the momentum of the water particle, which is balanced by the water-water interactions and water-solid interactions. Figure S10: A schematic of a water particle under water-water interactions and water-surface interactions.
For a nanoconfined water occupies a volume and bounded by a surface , according to Eq.(S3), the continuity, balance of momentum, balance of moment of momentum equations can be, respectively, obtained as follows: where is the mass density. denotes a body force vector. where is the unit normal vector and is the unit tensor. denotes the combined momentumflux tensor. is the applied pressure field.
In Eq.(S7), denotes the viscous stress tensor conjugate to the water-water interactions while is the viscous stress tensor that is conjugate to the water-solid interactions. It should be noted that the momentum-flux tensor and the viscous stress tensor presented in Eqs.(S7) are general tensors, which can be decomposed into symmetric and skew-symmetric parts.
According to Eq.(S7) and the divergence theorem, the continuity and balance equations of a water continuum can be written as follows: where the gravitational force, , is introduced as a body force.
Equation (S10) indicates that the skew-symmetric parts of the viscous stress tensors, and , vanish. Thus, the balance equations (Eq.(S9)) can be rewritten as follows: where the balance equations depend on the symmetric part of the viscous stress tensors, and .
It should be mentioned that Eq.(S11) represents a modified Navier-Stockes equation for the water-surface interactions. Thus, if water-surface interactions are neglected, Eq.(S11) reduces to the conventional Navier-Stockes equation.
The viscous stress tensors can be formed based on Newton's law of viscosity as follows: = − 0 ( + ) (S12) = − ( )( + ) (S13) where 0 denotes the viscosity of bulk water. is a newly introduced viscosity to account for water-solid interactions.