Supporting information for: Anomalous Stability of Two-Dimensional Ice. Confined in Hydrophobic Nanopore

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1 Supporting information for: Anomalous Stability of Two-Dimensional Ice Confined in Hydrophobic Nanopore Boxiao Cao, Enshi Xu, and Tianshu Li Department of Civil and Environmental Engineering, George Washington University, Washington, DC Phone: +1 (202) Fax: +1 (202) S1. Distribution of cubicity for an ideal confined 2D ice For an ideal confined 2D ice containing n basal ice planes, namely, no defects within the 2D plane, here we show that the distribution of cubicity displays exactly n peaks, evenly spaced between 0 and 1, i.e., 0, 1/n, 2/n,, and 1. Since an ice basal plane is puckered, namely, non-flat with respect to its neutral axis, with half of the water molecules above and the other half below, the water molecules within the lower half of one basal plane are vertically bonded to the water molecules within the upper half of another basal plane right below. The two water molecules connected by such vertical bond must both belong to either cubic or hexagonal simultaneously, since a particular stacking sequence needs to be defined by two adjacent ice basal planes. Therefore the lower half of one basal plane plus the upper half of another adjacent basal plane underneath form a stacking unit, within which all the To whom correspondence should be addressed S1

2 water molecules have the same identity, as shown in Fig. S1. A confined 2D ice with n basal planes thus contains n 1 such stacking unit, as the two ice basal planes in the immediate vicinity of the respective confining walls only contribute half stacking unit each. Since each stacking unit can only be either hexagonal or cubic, the overall cubicity for a confined 2D ice thus follows a binomial distribution with n possible outcomes, namely, with a possible total number of cubic stacking unit being 0, 1,, n 1. In Fig. 3A of the main text, the calculated distribution for quadlayer and pentalayer indeed displays strong peaks at k/n, where k [0, n]. The finite distribution in between these peaks is attributed to the occurrence of in-plane defects in the spontaneously crystallized 2D ice. S2. Role of interfacial stacking order in the stability of confined 2D ice To explicitly verify the overall stability of a confined 2D ice is fundamentally related to the intrinsically lower dynamic stability of cubic ice at interface, we examine the variation of vibrational free energy of a confined hexalayer (n = 6) with respect to the change of stacking sequence, by enumerating all possible stacking sequences in hexalayer and calculating the corresponding vibrational free energy F v. As shown in Fig. S2, the calculated F v can be naturally grouped into three categories, on the basis of the possible number (N c ) of cubic stacking units at the interface, i.e., N c = 0 (neither interface is cubic), 1 (one of the interfaces is cubic), and 2 (both interfaces are cubic). It is thus evident that the stability of confined hexalayer ice is primarily controlled by whether cubic stacking unit is present at ice-wall interface. Within each category, i.e., for a given N c, although F v still varies with stacking configuration, such dependence is found to be subtle thus secondary. Therefore stacking disorder is expected to still occur in the non-interfacial region of confined 2D ices. This is indeed suggested by the non-zero cubicity for 2D ices with n > 3, as shown in Fig. 3A of the main text. S2

3 S3. Layer resolved cubicity for ice nucleating on graphene A visual check on the structure of ice crystallized on graphene obtained in our previous simulation studies S1 S3 and from others S4 S6 shows the contact bilayer of ice appears to favor a hexagonal-like stacking, albeit that a cubic-like stacking is also found to present. To verify this observation quantitatively, we calculate the distribution of cubicity for the three bottom layers of ice forming on a graphene, over the ensemble of configurations (N = 116) obtained around the critical size ( 210 water molecules) through our forward flux sampling (FFS) S7 simulation of heterogeneous ice nucleation at 230 K. S1 The three bottom ice layers are respectively distinguished by the distance of a water molecule from graphene plane d, namely, 1st layer (d < 4.8 Å), 2nd layer (4.8 Å< d < 8.3 Å), and 3rd layer (8.3 Å< d < 11.8 Å). As shown in Fig. S3, the first contact layer displays a mean cubicity of χ c 0.032, significantly lower than those of second and third contact layer. molecules in 1st layer are only three-fold coordinated (see Fig. Since half of the water S1), and the concept of cubicity is sensible only for a truly ice-like water molecule (see Methods of main text), the layer-resolved cubicity for the 1st layer thus exactly reflects the cubicity of the first stacking unit (see section above), which is usually also referred as the first contact bilayer. S4. Lennard-Jones 12-6 Wall potential for phonon calculation The Lennard-Jones (LJ) 12-6 wall potential is used in phonon calculations to avoid the incommensurate issue. The LJ 12-6 wall potential takes the form: U w (z) = 4ε w [ ( σw (z D) ) 12 ( ) ] 6 σw, (1) (z D) where ε w, σ w, and D are the energy scale, length scale, and wall position, respectively. S3

4 The two body term of the Stillinger-Weber potential U sw (r) = Aε sw [B ( σsw r ) p ( σsw r ) q ] ( ) σsw exp. (2) r aσ sw has been parameterized for representing mw water-carbon interaction, S4 with the following parameters: A = , B = , p = 4, q = 0, a = 1.8, ε sw = ε 0 = 0.13 Kcal/mol, and σ sw = 3.2 Å. As it is impossible to exactly match LJ 12-6 wall potential U w (z) with graphene-water potential, we ensure the energy scale and the curvature of the potential at the minimum, 2 U/ z 2 z=z0, are equivalent between the two types of potentials, because phonon frequency is determined by force constant, i.e., curvature of the potential at equilibrium. To generate the effective graphene-water force field U Gr (z) based on water-carbon interaction U sw (r), an mw water molecule is placed at a distance z from graphene, with its x and y coordinates randomly sampled within the unit cell of graphene (since SW potential is short-ranged), and the total potential energy is calculated by summing all the water-carbon interactions within the cut-off E(z) = r<r cut U sw (r). The effective graphene-water potential U Gr (z) is then obtained by calculating the mean value through repeating the random sampling procedure for 10,000 times, i.e., U Gr (z) = E(z). Note that the well depth of U Gr (z) (0.136 Kcal/mol) is slightly deeper than that of water-carbon potential U sw (r) (0.13 Kcal/mol). Then the LJ 12-6 wall potential U w (z) is obtained by fitting Eqn. 1 through adjusting σ w and D while fixing ε w = Kcal/mol, which yields σ w = 6 Å and D = 3.65 Å. A comparison of the two potentials is shown in Fig. S4. S5. Dependence of F v on hydrophobicity and pressure To understand the generality of the identified preference of hexagonal ice over cubic ice at a hydrophobic surface, we examine the variation of F v = F cubic v F hex v with respect to the S4

5 change in both vertical pressure p zz and wall-water interaction strength. As shown in Fig. S5, F v is found to increase with vertical pressure, suggesting pressure can further enhance the preference of hexagonal ice over cubic ice at interface. On the other hand, such preference is found to be weakened by an enhanced wall-water interaction strength, but still remain positive even for a very large strength, e.g., 30ε w. As shown in previous studies, S2,S5,S8 an increase of wall-water interaction strength leads to a stronger binding of water. Indeed, as shown in S6, such a wide range of variation of wall-water strength leads to a substantial change in water contact angle, changing the wall from hydrophobic to hydrophilic. The fact that F v stays positive even for an extremely high wall-water interaction strength, e.g., 30 ε w, well indicates the preference of hexagonal ice over cubic ice at interface can be rather generic, irrespective of surface hydrophilicity. S6. Variation of water contact angle with wall-water interaction strength A common measure of surface hydrophobicity is contact angle θ c of water. Although surface hydrophilicity is expected to increase with wall-water interaction strength ε w, it is useful to know exactly how contact angle θ c varies with ε w. To understand this, we follow the procedure as described in Ref. S9 to compute the contact angle θ c of mw water droplet on a LJ 12-6 wall of different ε w using 36,035 mw water molecules at 300 K, as shown in Fig. S6. It can be seen that the fitted LJ wall, i.e., ε w = Kcal/mol, σ w = 6 Å, yields a water contact angle of 130, much larger than that (86 ) of graphene wall. S5 The difference can be attributed to that fact that the two types of potentials are indeed different except near the bottom of the well, particularly that the LJ potential is more repulsive than the SW potential (see Fig. S4) which is expected to enhance hydrophobicity. Nevertheless, regardless this difference, our results show the contact angle θ c of water on LJ wall exhibits S5

6 the same trend as graphene wall. In particular, the calculation shows the LJ 12-6 potential with 6.5 ε w and σ w = 6 Å yields the same contact angle of water on graphene (Fig. S6). Importantly, the calculated F v as a function of wall-water interaction strength (Fig. S5) shows hexagonal ice is strongly thermodynamically favored across the entire wide range of wall-water strength ( [0.5 ε w, 30 ε w ]) thus a wide range of water contact angle. S6

7 Movie S1: Spontaneous freezing of water within a 10 wedge at 230 K Movie S2: Phonon mode in cubic trilayer: The 1st band at K Movie S3: Phonon mode in cubic trilayer: The 2nd band at K Movie S4: Phonon mode in cubic trilayer: The 3rd band at K Movie S5: Phonon mode in cubic trilayer: The 4th band at K Movie S6: Phonon mode in hexagonal trilayer: The 1st band at K Movie S7: Phonon mode in hexagonal trilayer: The 2nd band at K Movie S8: Phonon mode in hexagonal trilayer: The 3rd band at K Movie S9: Phonon mode in hexagonal trilayer: The 4th band at K S7

8 H C Stacking unit C H Figure S1: Stacking of ice basal planes in a confined pentalayer ice (n = 5). An ice basal plane is buckled with respect to its neutral plane (denoted by red dashed lines). The lower half of water molecules in one layer are vertically bonded to the upper half of water molecules in another ice basal plane right below, forming a stacking unit (represented by the shaped area). This pentalayer n = 5 thus contains n 1 = 4 stacking units, with a sequence of hexagonal (H), cubic (C), C, and H for each from the top to bottom, respectively. S8

9 N C =2-50 F v (J/mol) CHHHC CHCHC CCHHC CCCHC CCHCC CCCCC N C =1 HHHHC HHCHC HHHCC HCHHC CHCCH CCCHH CCHCH CCCCH N C = HHHHH HHCHH HHHCH HHCCH HCHCH HCCCH Figure S2: Calculated vibrational free energy F v for different stacking sequences in confined hexalayer (n = 6). The calculated F v can be naturally grouped on the basis of the number (N c ) of cubic stacking units at the immediate ice-wall interface. C and H denote cubic and hexagonal stacking units, respectively. S9

10 20 10 <χ c > = st layer f(χ c ) <χ c > = nd layer <χ c > = rd layer Cubicity χ c Figure S3: Layer-resolved cubicity distribution for critical ice nuclei crystallized on graphene at 230 K. The distribution is calculated from N = 127 configurations collected in our previous FFS study of heterogeneous ice nucleation on graphene. S1 S10

11 U Gr (z) U w (z) U (Kcal/mol) z (Å) Figure S4: A comparison between the graphene-water potential U Gr (z) based on the twobody term of Stillinger-Weber force field and the fitted Lennard-Jones 12-6 wall potential U w (z). S11

12 120 F v =F v cubic -Fv hex (J/mol) ε w 1ε w 1.5ε w 3ε w 5ε w 10ε w 20ε w 30ε w Pressure p zz (bar) Figure S5: Variation of F v for trialyer ice with the vertical pressure p zz and wall-water interaction strength. The original wall-water interaction strength ε w = Kcal mol 1 (red square) is identical to the energy scale of the graphene-water interaction U Gr (z) which is calculated based on the water-carbon potential U SW (r). S4 S12

13 140 Water contact angle θ c (degree) Wall-water interaction strength (ε w ) Figure S6: Variation of the calculated contact angles (black dots) of mw water θ c on LJ wall with respect to wall-water interaction strength for σ w =6 Å. The unit of wall-water strength is ε w = Kcal/mol. Based on the calculation, a LJ wall strength of 6.5 ε w is expected to lead to the same water contact angle of 86 on graphene, as denoted by the red star. S13

14 Figure S7: Configuration of a slit nanopore. The nanopore is created by placing two U- shaped graphene walls (light blue) back-to-back so that the flat region of both walls are parallel to each other. Water is filled both inside and outside of the pore, so when a periodic boundary condition is applied, a bulk water region can be created in contact with confined water. S14

15 z y x 14 nm 5 nm 20 nm A B C Figure S8: Configuration of a small-angle wedge. (a) Simulation cell for a 10-degree wedge (containing 3,248 mw water molecules), (b) an extended cell (containing 12,992 mw water molecules) along wedge contact line, and (c) the extended cell with water crystallized spontaneously inside the wedge. S15

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