Effectiveness of the Young-Laplace equation at nanoscale

Using molecular dynamics (MD) simulations, a new approach based on the behavior of pressurized water out of a nanopore (1.3–2.7 nm) in a flat plate is developed to calculate the relationship between the water surface curvature and the pressure difference across water surface. It is found that the water surface curvature is inversely proportional to the pressure difference across surface at nanoscale, and this relationship will be effective for different pore size, temperature, and even for electrolyte solutions. Based on the present results, we cannot only effectively determine the surface tension of water and the effects of temperature or electrolyte ions on the surface tension, but also show that the Young-Laplace (Y-L) equation is valid at nanoscale. In addition, the contact angle of water with the hydrophilic material can be further calculated by the relationship between the critical instable pressure of water surface (burst pressure) and nanopore size. Combining with the infiltration behavior of water into hydrophobic microchannels, the contact angle of water at nanoscale can be more accurately determined by measuring the critical pressure causing the instability of water surface, based on which the uncertainty of measuring the contact angle of water at nanoscale is highly reduced.


Pressure across water surface
In MD simulations, the piston (the bottom carbon plane) pressure applied on the reservoir (see Figure 2 in the main text) is expressed as the following equation: where l x , l y are the lateral sizes of piston, F is the total force applied on the piston, n p and n w are the numbers of the atoms of piston and water molecules inside the cut-off distance from piston, f ij , u ij are the interaction force and energy between piston atom i and water molecule j, respectively, which are the functions of the atomic distance r ij , and  and  are the L-J potential parameters. Figure S1 shows the relationship between the piston pressure and the density of water reservoir. The solid line in the figure is the equation of state calculated from the bulk water based on the SPC/E model. Thus, the piston pressure matches very well with the water pressure in reservoir.  After the system is equilibrated, there is a small fluctuation in P a (e.g., P pis /P pis < 10%), whereas the mean value of P pis matches very well with the reservoir pressure determined from the equation of state (P a ). In the present work, the pressure in reservoir (P a ) is determined by the equation of state based on the change of average density of reservoir. In the present model, the pressure outside reservoir is zero, and thus, the pressure across the water surface (the water molecules face the pore of top plane) is equal to the water pressure in reservoir (P = P a ). For simplification, the reservoir pressure is defined as P.

Curvature of water surface
For the equilibrium water conformation under a given pressure, two principal curvature radii R 1 (measured in xz plane) and R 2 (measured in yz plane) are defined for water surface, which are calculated from the average distribution of surface water molecules in xz and yz planes using MD simulations. In these two planes, the water surface boundary can be defined by two circles with the same origin (located in the axis line of nanopore) but different radius: all water molecules on the surface can be enclosed between these two circles. The average radius of these two circles is the curvature radius of water surface, shown as the red solid lines in Figure S2, and the uncertainty of the curvature radius is the half of the difference between the radii of those two circles, which is less than 2%. The curvature radii are the statistic results calculated from the distributions of the x, z coordinates (or x, y coordinates) of water molecules based on 3500 snapshots. Figure S2a shows the result of the carbon plane with the hydrophilic L-J parameters (set A, the reported contact angle based which is 65 1 ), in which the angle between the tangential line of the water surface and the carbon plate  <  c ( c is the contact angle). Figure S2b shows the result of the carbon plate with the hydrophobic L-J parameters (set B, the reported contact angle based which is 110 1 Figure S2. The curvature radius of water surface over nanopore. (a) the carbon plane with the hydrophilic L-J parameters (set A) and  <  c , ; (b) the carbon plane with the hydrophobic L-J parameters (set B),  = 90.

The uncertainty analysis of the surface tension
According to Equation (1)  After the pore material is changed from hydrophilic to hydrophobic (the surface contact angle is changed from 65 to 110), there is no noticeable change in the surface radius of curvature, which means that the surface tension determined is the intrinsic property of water (i.e., insensitive to the solid material property). In addition, when the pore shape is changed from the circular to elliptical, one of the principal radius (R 1 ) increases but the other decreases (R 2 ), which leads to the same average principal curvature radius. Figure S5 shows the calculated values of the water surface tension based on the data displayed in Figure S4 using Equation (1) (in the main text).   Figure S5. The water surface tension based on the data displayed in Figure S4 using Equation (1) (in the main text). 7

The conformation of water surface
The conformation of water structure can be typically described by the coordination number and the average near O-O distance of water molecules (representing the hydrogen bond structure).

The burst pressure of water out of nanopore
According to Equation (1) (in the main text), the curvature radius of water surface decreases with the increase of P, and the curvature radius cannot be less than the pore radius a, at which the water surface is an exact half spherical surface over the pore. There should be no water molecules outside of this half sphere when the water surface is stable, and thus, the number of water molecules outside of this half sphere (N out , also defined as free water molecules) can be used as a criterion for checking the water surface stability: N out will be rapidly increased when the applied pressure is higher than the critical value, which is defined as the burst pressure in the present work. Figure S8 shows the value of N out (a = 1 nm) varying with the MD simulation time under the different applied pressure. From the figure, the burst pressure of water with a = 1 nm is determined as 99.6  0.9 MPa.

The infiltration pressure of water into a rigid SWCNT segment
The water reservoir is connected with a rigid SWCNT segment (with the radius a = 1.0 nm), and the water cannot enter into SWCNT initially since the SWCNT is hydrophobic (using set B L-J parameter). In MD simulations, the reservoir pressure P a is calculated under the different piston displacements, which is calculated based on the same approach described in Section 1 (supporting material). With the increase of the piston displacement, the reservoir pressure P a increases in the initial stage and then P a converges to some critical value P in , which is defined as the infiltration pressure of water into SWCNT, as shown in Figure S9.