Article pubs.acs.org/JPCB
Hidden Asymmetry of Ice Mikhail V. Kirov*,†,‡ †
Institute of the Earth Cryosphere, Siberian Branch RAS, Tyumen 625000, Russia Tyumen State Oil and Gas University, Tyumen 625000, Russia
‡
S Supporting Information *
ABSTRACT: Ice is a very complex and fundamentally important solid. In the present article, we review a new property of the hydrogen-bonded network in ice structures: an explicit nonequivalence of some antipodal configurations with the opposite direction of all hydrogen bonds (H-bonds). This asymmetry is most pronounced for the structures with considerable deviation of the H-bond network from the tetrahedral coordination. That is why we have investigated in detail four-coordinated ice nanostructures with no outer “dangling” hydrogen atoms, namely, ice bilayers and ice nanotubes consisting of stacked nmembered rings. The reason for this H-bonding asymmetry is a fundamental nonequivalence of the arrangements of water molecules in some antipodal configurations with the opposite direction of all H-bonds. For these configurations, the overall pictures of deviations of the hydrogen bonds from linearity are qualitatively different. We consider the reversal of all H-bonds as an additional nongeometric operation of symmetry, more precisely antisymmetry. It is not easy to find the explicit breaking of the symmetry of hydrogen bonding (H-symmetry) in the variety of all configurations. Therefore, this asymmetry may be named hidden.
1. INTRODUCTION The crystal lattice of ordinary ice is formed only by oxygen atoms, and hydrogen atoms are located out of order.1 The number of hydrogen atom arrangements grows exponentially with the size of a system.2,3 The energy and other characteristics of the ice-like systems depend on the positions of the hydrogen atoms (protons) in the hydrogen bonds.4,5 Therefore, the results of computer simulation of ice and gas hydrates depend on the structure which was chosen as the initial one.6 This dependence reflects nonhomogeneity of the ice-like systems at the nanolevel. The complexity of the ice-like systems causes the utility of coarse-grained approaches such as the method of structural invariants.4,5 The other approach is based on the development of the discrete models for molecular interactions.7,8 These models take into account the most important peculiarities of Coulombic interaction of the first and more distant neighbors in H-bond network. Approaches of this sort have given us an insight into properties of the whole set of configurations with fixed position of the oxygen atoms and with different arrangements of the hydrogen atoms in the H-bonds. For the ice structures, we can separate the more subtle organizational level and the corresponding level of model description. This level reflects a difference between the antipodal configurations with opposite directions of all Hbonds. Earlier, we have suggested9,10 to consider a change of the direction of all H-bonds in ice-like systems as additional nongeometric operation of symmetry, more precisely antisymmetry.11,12 The black-and-white symmetry12 and the magnetic symmetry13,14 are different versions of antisymmetry. In addition to the usual symmetry operations, these generalized symmetry groups include the anti-identity operation (color © XXXX American Chemical Society
change, reversal of all spin orientations or reversal of all Hbonds). It was shown that in water clusters the antisymmetry of hydrogen bonding is approximate, although the distinction between the antipodal configurations with opposite direction of all H-bonds comprises only a small fraction of the total energy variation caused by the proton disorder.15 In physics, a new type of symmetry always arouses considerable interest as well as an unexpected symmetrybreaking. But in water clusters and finite fragments of extended ice-like systems the distinction between the configurations with the opposite direction of all H-bonds has a very simple explanation: they are characterized by different arrangement of the outer “dangling” hydrogen atoms. This raises the question of whether the free hydrogen atoms are the main and sole reason for breaking the hydrogen-bonding symmetry. In principle, the answer can be obtained in studies of extended three-dimensional systems. At the same time, the ice nanotubes16,17 and ice bilayers18−25 might be of particular interest, since here a considerable deviation of the H-bond network from the tetrahedral coordination is compensated by a large number of formed H-bonds and by the complete lack of the “dangling” hydrogen atoms. As these nanostructures are not strictly tetrahedrally coordinated, some properties of the Hbond network are most pronounced in them. Here, it should be stressed that neither of the crystal modifications corresponds exactly to the geometry of the water molecule. Special Issue: Physics and Chemistry of Ice 2014 Received: April 23, 2014 Revised: June 2, 2014
A
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The main aim of this article is to present explicitly nonequivalent antipodal configurations of ice-like systems with the opposite direction of all H-bonds and to give an explanation to this phenomenon. These configurations without “dangling” hydrogen atoms demonstrate a new property of the H-bond network in itself. It is not easy to find the explicit breaking of the symmetry of hydrogen bonding (H-symmetry) in the variety of all proton configurations. Therefore, this asymmetry may be named hidden.
2. SYMMMETRY ANALYSIS There are several different modifications of the bilayer ice. Some of them include pentagonal and other H-bonded cycles.22 Along with usual hexagonal bilayer (Figure 1a), we will Figure 2. Symmetry-distinct configurations of the rectangular unit cell for the usual ice bilayer. Antisymmetry elements (axes, plains, and center of antisymmetry) are shown by chain lines.
Symmetry of the bilayer ice is described by ‘layer groups of symmetry’. These are three-dimensional groups with twodimensional translations.28 So, the symmetry group of the usual bilayer ice is p6/mmm. However, it is convenient to use a rectangular face-centered unit cell (Figure 1a) with symmetry cmmm. With the exception of translations, the number of symmetry operations for this H-bonded framework νf is equal to 16. For each of the unsymmetrical proton configurations there are 16 absolutely equivalent structures, which may be obtained using all the symmetry operations. In the absence of any symmetry, the number of symmetry-distinct configurations n is equal to N/νf. In the general case,
Figure 1. Structure of ice nanolayers. (a) The “usual” hexagonal bilayer. (b) The shifted hexagonal bilayer. The rectangular unit cells are shown by the dashed lines.
consider the shifted hexagonal bilayer (Figure 1b), which was suggested in ref 26. These two modifications are of particular interest for theoretical analysis because of their structural homogeneity: all molecules are located in equivalent positions and all have an equivalent neighborhood. Each of these structures has eight molecules per unit cell. The nodes of Hbond network correspond to the oxygen atoms. We will indicate by arrows the direction of the H-bonds: from the proton donor to the acceptor. According to the well-known Bernal−Fowler ice rules,2 two arrows are incoming and two arrows are outgoing in each node. The exact total numbers of the proton configurations have been calculated by us in refs24, 26. In the unit cell of each of these layers this number is equal to 114, as in the orthorhombic unit cell of ordinary ice Ih.27 Usual Bilayer Ice. Using a special computer program, we have constructed all 114 proton configurations for the rectangular unit cell with periodic boundary conditions, and then we have selected the symmetry-distinct configurations which are shown in Figure 2. Here, the solid lines correspond to the basic H-bonds. The direction of the remaining bonds shown by dashed lines is defined by the periodic boundary conditions. Note that the true number of different configurations for the usual bilayer is 16. Two pairs of configurations in Figure 2 are symmetrically equivalent (see below). The point is that in the search of the nonisomorphic proton configurations we start from the symmetry of the rectangular cell. But this cell does not possess a three-fold symmetry which is characteristic for the bilayer as a whole. Therefore, it is required an additional check of equivalence of the configurations taking into account the missing symmetry. At the same time, the complete list of configurations (Figure 2) also makes sense. It allows us to check the completeness of the class of nonisomorphic structures according to the total number of configurations N = 114.
n
N=
∑ i=1
n
νf νc(i)
= νf ·∑ i=1
1 νc(i)
(1)
where νf and νc (i) are the number of symmetry operations of the H-bonded framework and of the concrete configuration, respectively. More precisely, they are the orders of the crystal classes. For clusters and finite fragments, they are merely the orders of the symmetry groups. Formula 1 is the relation of the symmetry balance. We have used it in ref 29. The symmetry of all configurations for the rectangular unit cell is given in Table 1. These groups are subgroups of the symmetry for the H-bonded framework (i.e., cmmm). For each configuration, the numbers of the symmetry operations are shown. They were established during the symmetry analysis. The number of configurations for each class of isomorphic structures is equal to νf /νc (i) (see formula 1). Since the sum of all these 18 numbers is 114, this proves the completeness of the class of configurations that is shown in Figure 2. However, it is necessary to check additionally the presence of skipped three-fold symmetry (C3) and, possibly, to make corrections. The configuration pairs 1−11 and 13−18 turned out to be equivalent, taking into account the skipped symmetry. For example, the clockwise rotation of configuration 1 by the angle 120° and subsequent reflection in the layer plane yields configuration 11. Note that the direction of external vertical Hbonds for rotated structure can be easily defined using the Bernal−Fowler rules. In Figure 2 the antisymmetry elements are shown by chain lines. Thus, the change of the direction of all H-bonds and the rotation by 180° about the vertical axis returns configuration 1 B
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Table 1. Symmetry of the Proton Configurations of Bilayer Ice no.
layer groupa
νcb
νf/νcc
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 ∑
c1 (p1, #1) p11n (p11a, #5) p1211 (p2111, #9) pb11 (#12) p2an (pb2n, #34) p1a1 (pb11, #12) p1a1 (pb11, #12) p121/a1 (p21/b11, #17) p121/a1 (p21/b11, #17) p2an (pb2n, #34) c211 (#10) p1 (#1) p1̅ (#2) p2111 (#9) p1̅ (#2) p21/b11 (#17) p2/b11 (#16) c2/m11 (#18)
2 2 2 2 4 2 2 4 4 4 4 1 4 2 2 4 4 8
8 8 8 8 4 8 8 4 4 4 4 16 4 8 8 4 4 2 114
Figure 3. Symmetry-distinct configurations for the unit cell of the shifted bilayer. Layer symmetry groups are shown below, the numbers of symmetry operations νc are given in brackets.
symmetry operations νc (order of the crystal class) are shown in Figure 3. For this bilayer the total number of proton configurations per unit cell N is also equal to 114 (see ref 26), and the number of the symmetry operation of H-bonded network νf is 16 as before. Using eq 1, it is easy to prove the completeness of the class of nonisomorphic configurations which is shown in Figure 3. Again, the majority of proton configurations are antisymmetrical. In this case, pairs of antipodal configurations are completely equivalent, and again, only one pair of configurations (14, 15) are not antisymmetrical. This pair of the ordinary configurations is also of principal interest to us.
a
Layer symmetry group; standard setting of the symmetry groups; if different, serial numbers are given in parentheses. bOrder of symmetry class for the proton configuration. cNumber of symmetry equivalent (isomorphic) configurations, where νf is the order of symmetry class for the H-bonded framework.
to the initial state. It means that two antipodal configurations are completely equivalent for this structure. Only one pair of configurations (6, 7) are not antisymmetrical. That is of our main interest here. At the same time, these configurations are related by the two-fold axis of antisymmetry. The change of the direction of all H-bonds in configuration 6 and the rotation by the angle 180° in the plane of Figure 2 yields configuration 7. It is important to stress that all the proton configurations of ice-like systems can be divided into two types: antisymmetrical and nonantisymmetrical (which we will call ordinary) configurations.9,10 In the first case, a reversal of all H-bonds leads to absolutely equivalent configuration, which is related to the initial configuration by the operation of usual spatial symmetry (rotation, reflection). For the ordinary configurations, a reversal of all H-bonds gives us the other, symmetrically distinct configuration. For this reason, any list of symmetry-distinct proton configurations of ice-like systems includes both antisymmetrical configurations and pairs of ordinary configurations related by some antisymmetry operation (usual space symmetry plus the reversal of all H-bonds). Each configuration in Figure 2 presents a whole class of isomorphic configurations. Pairs of the ordinary configurations can be selected in such a way that they are different only by the direction of all H-bonds. For example, instead of configuration 7 we could choose the equivalent configuration 6′, which differs from configuration 6 only by the direction of all H-bonds (the direct antipode). Shifted Hexagonal Bilayer. For this bilayer we also have calculated all symmetry-distinct configurations of the unit cell (Figure 3). In this case, the unit cell is rectangular and completely reflects the symmetry of the system. Here, according to the classification of the layer groups,28 the symmetry of H-bonded framework is cmme. The symmetry of all the nonisomorphic proton configurations and the number of
3. RESULTS AND DISCUSSION Periodic replication of the unit cell generates the entire structure of the bilayers. So, the usual ice bilayers in Figure 4a
Figure 4. Pairs of nonantisymmetrical configurations with opposite direction of all H-bonds. (a, b) Configurations 6 and 7 for the usual bilayer ice. (c, d) Configurations 14 and 15 for the shifted bilayer ice. C
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Figure 5. Lattice binding energy and the dipole moment per molecule for optimized configurations of the usual bilayer ice (a) and the shifted bilayer ice (b). The data correspond to geometrical optimization with the TIP4P potential. The black diamonds correspond to the antipodal configurations with opposite directions of all H-bonds.
and b are based on nonantisymmetrical configurations 6 and 7. Analogously, the structures in Figure 4c and d correspond to configurations 14 and 15 of the shifted bilayer ice. It is not difficult to see that the structures of the upper and lower monolayers are the same for both modifications of the bilayer (cf. a and c of Figure 4, as well as b and d). In the upper monolayers, all H-bonds of the horizontal zigzag chains are oriented to the right, but in the lower monolayers they are oriented alternately. The remaining rows of H-bonds are also oriented alternately, including the interlayer H-bonds. It would be recalled that a reversal of all H-bonds in configuration 6 of the unit cell for usual bilayer gives us the structure which differs only by 2-fold rotation around the vertical axis from configuration 7. Therefore, in Figure 4 the layer configuration in b is the antipode of configuration in a. To confirm this, it is sufficient to rotate the first configuration (Figure 4a) around the vertical axis. However, we must note here that this geometrically idealized representation of the structure for the antipodal configurations hinders the understanding of the distinctions between them, which have been revealed during more detailed analysis. Calculations with Pairwise-Additive Water Models. Geometrically optimized configurations were obtained using TINKER Molecular Modeling package.30 At the beginning, all calculations were performed using TIP4P potential for simulations of water.31 As we are interested in the existence of principal distinctions between the antipodal configurations more than exact values of physical magnitudes, we have not used any special methods for evaluating the long-range Coulombic interactions. For the usual bilayer ice, we used the rectangular unit cell (5 × 3) by size 23.815 Å × 24.750 Å and cutoff of interaction 11.5 Å. The energy and the dipole moment of the geometrically optimized configurations are shown in Figure 5. Note that these values correspond to all 114 proton configurations of the unit cells for both bilayers. For symmetry equivalent configurations we obtained very close values, which are indistinguishable in Figure 5. While the pairs of nonantisymmetrical configurations have pronounced differences in the binding energy and in the total dipole moment. It was verified by geometrical optimization with the TIP4P potential. The black diamonds correspond to the antipodal configurations with opposite directions of all H-bonds. Under reversal of all H-bonds the values for all antisymmetrical configurations remain unchanged, but the values in pairs of the ordinary configurations are swapped.
Table 2. Energy Differences between Antipodal Configurations of the Usual Bilayer Ice, Corresponding to Pairwise-Additive Water Models force field
SPC
TIP3P
TIP4P
TIP5P
TIP 3f
BNS
ΔE, kJ/mol
1.20
1.21
0.68
−0.05
0.62
0
The energy differences between the antipodal configurations (per molecule) are presented in Table 2 for different pairwise water potentials: SPC,32 TIP3P,31 TIP4P,31 TIP5P,33 and TIP 3f (Dang−Pettitt Flexible TIP3P 3-Site Water Model).34 For the BNS potential (the first version of the ST2 potential35) the H-bond symmetry is exact. The reason is the symmetry of this molecule model with respect to the interchange of positive and negative effective charges. A small value for the TIP5P potential is explained by the similarity of these models of water molecule. The energetically nonequivalent antipodal configurations have small but quite visible structural distinctions. The two configurations of the usual bilayer ice with the unit cells 6 and 7 (Figure 4a and b) are shown in Figure 6. Visualization is carried out using the ViewMol3D software.36 In the upper panel, we can see the structures obtained in the course of a preliminary short-time geometrical optimization of the artificially constructed configurations. They have been obtained during the first optimization step by varying only the orientations of all molecules, practically without the displacement of the oxygen atoms from the initial positions. It is natural to consider these structures as refined initial configurations. The subtle but crucial differences between the antipodal configurations are quite visible. Indeed, deflections of the hydrogen atoms from the vertical oxygen−oxygen lines are directed in different sides in configuration 6, but in configuration 7 the deflections are unidirectional. In the vertical projection xy, the hydrogen atoms of different monolayers form a zigzag line in configuration 6, and an angle is formed in configuration 7. In the side projection xz, on the contrary, the hydrogens of different monolayers form an angle in configuration 6, and a zigzag is formed in configuration 7. It is necessary to stress that the deflection of each atom from the oxygen−oxygen line (nonlinearity of hydrogen bonds) is of basic importance, whereas the zigzag lines in Figure 6 are formal indicators which show the positions of the hydrogen atoms in different monolayers. Nevertheless, the revealed geometrical peculiarities of the antipodal configurations prove the existence of a fundamental difference between them. More obvious distinctions of the antipodal configurations have revealed themselves during the complete geometrical D
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Figure 7. Antipodal configurations 14 and 15 for the shifted bilayer ice are shown at the right and left, respectively. The upper panels show different projections for the initial configurations. The lower panel shows the side xz-projections of optimized configurations obtained with the use of the TIP4P intermolecular potential. The inset panel shows the shifted bilayer in the direction of bisector XZ.
in the side xz-projection. However, there is no appreciable distortion here as in configuration 6 for the usual bilayer ice. The reason is the rigidity of the H-bonded framework for this type of bilayer. Note that the two types of the bilayer ice have much in common. In the direction between the x and z axes at angle 45°, the shifted bilayer is like the usual bilayer (Figure 7, inset). But the similarity of the obtained results for rectangular unit cells of the two bilayers has a deeper reason. The point is that these crystal lattices (see Figure 1) are equivalent in the combinatorial−topological sense, since they have the same fundamental finite graph (see ref 26, Figure 2e), and the two pairs of infinite quasi-two-dimensional configurations in Figure 4 (a−c, b−d) correspond to one pair of antipodal configurations of the common fundamental graph. Simple Proof of the Asymmetry. It is clear that high-level quantum-chemical calculations may provide more accurate estimation of the energy difference between the antipodal configurations with opposite direction of all H-bonds. This estimation is of unquestionable interest, but our main aim is to prove that the difference exists. In this section, we present a simple geometrical proof which is not related to any form of molecular interaction. First of all, we may notice that the antipodal configurations of the usual bilayer (6 and 7) do not have three-fold symmetry C3. Therefore, let us consider only the H-bonds which are vertically oriented in a and b of Figure 4 and in Figure 6. The unit cell configuration 6 is shown in Figure 8a. The orientation of each molecule may be determined by the direction of two outgoing arrows near the node. Now we note that the angle HOH is less than 120°, and then hydrogen atoms of the vertical H-bonds are deviated on different sides from the vertical line (Figure 8b). We can also use Figure 8a to build the antipodal configuration. In this case, the orientation of a molecule can be determined by the direction of two ingoing arrows. It is easy to verify that it gives the unidirectional deviation of hydrogen atoms located in the vertical H-bonds (Figure 8c). Taking into account the translational symmetry, we can see that in the first
Figure 6. Antipodal configurations 6 and 7 for the usual bilayer ice. Vertical (xy) and side (xz) projections for initial configurations (upper panel) and for configurations which were optimized using TIP4P intermolecular potential (lower panels).
optimization (Figure 6, lower panels). The crystal lattice of configuration 6 has been deformed, forming the rows of hexagons with alternate slopes as in ferroelastic materials. The reason for the deformation, as mentioned above, is the deflection of the hydrogen atoms from the oxygen−oxygen lines. It can be seen that it is profitable to shift the second (from the top) horizontal double-level zigzag chain to the left. Thus, the deflections of the hydrogen atoms in each monolayer are minimized, and H-bonds are straightened. Although in the side projection the interlayer hydrogen atoms are shifted from each other, but they refer to different H-bonds. As this takes place, the deflection of H-bonds from linearity is very small. On the contrary, the lattice configuration 7 remains practically unchanged in the course of geometrical optimization. Note that the deformation of crystal lattices as in configuration 6 was observed earlier in computer simulations of the bilayer ice with some arrangements of H-bonds.19 We have established that the same distortion is specific to the usual bilayer formed by configurations 3, 4, 6, 9, 14, and 16 (see Figure 2). The antipodal configurations 14 and 15 of the shifted bilayer ice are also nonequivalent (Figure 5b). Between the initial structures there is a slight distinction; it also takes place in the optimized ones (Figure 7). These distinctions are clearly visible E
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bonds. In addition, we rotate the obtained configurations in order to have similar comparable structures. These obtained configurations (Figure 9c, at the right) are very similar to the initial ones (Figure 9c, at the left). However, the configurations in these antipodal pairs have visible distinctions in the arrangements of some molecules (see ovals). Thus, we can conclude that if these configurations are really stable, they are not invariant under the reversal of all H-bonds. Hexagonal Ice. Ordinary hexagonal ice has tetrahedral geometry which is nearly perfect. It is reasonable to expect that the deviations of H-bonds from linearity are very small. However, these deviations are nonzero because the geometry of the water molecule does not correspond exactly to geometry of the H-bond network. The number of symmetry-distinct proton configurations for an orthorhombic unit cell with 8 water molecules is equal to 16.5,37 There is one pair of configurations with the opposite direction of all H-bonds. An approximate analysis of these configurations in idealized geometry that was carried out by Kuo and Singer,5 does not reveal any distinctions between them. Accurate calculations for the orthorhombic unit cell were performed by Hirsch and Ojamaë.37 The binding energy of all of these 16 distinct proton configurations was investigated by quantum-mechanical density functional theory calculations and compared by molecular-mechanics analytical potential models. These very accurate and detailed results can be used for our analysis. In the article of Hirsch and Ojamaë, two antipodal proton configurations are 12 and 13 (see ref 37, Figure 1). They have the same symmetry and identical set of H-bond dimer types. These configurations are very close in energy. The difference is most pronounced for pair-potential models: flexible SPC and COMPASS (about 0.08 kJ/mol). Though the energies of the antipodes are very close according to DFT methods, they are not equal. Nonequivalence of these configurations is visible in the parameters of the optimized unit cell (see ref 37, Table 3). In addition, these configurations have different mean HOH angles: 108.2° and 107.8°. Analogous quantum-chemical calculations for the orthorhombic unit cell of ice Ih are presented in ref 38. It is necessary to stress that these articles37,38 are devoted to the energetics of all possible proton configurations. The problem of invariance for the H-bond network is not discussed. Therefore, a quantum-chemical estimate of the energy difference for ordinary hexagonal ice remains an interesting and important challenge for the future. We note only that the unit cells obtained by extending the smallest orthorhombic unit cell are of no less interest. Thus, for accurate calculations it may be interesting that the unit cell (2 × 1 × 1) has antipodal
Figure 8. Nonantisymmetrical configurations of the unit cell for usual ice bilayer. (a,b) Configuration 6. (c) Directly antipodal (reversal of all H-bonds) configuration 6′ which is symmetrically equivalent to configuration 7. Only the molecules which form the vertical H-bonds are shown.
case (Figure 8b) the vertical H-bonds form a zigzag, but in the second case (Figure 8c) they form an angle. According to our symmetry analysis, the configurations 6′ and 7 are symmetry-equivalent (see above). At the same time one can build configuration 7 directly (see Figure 2). Thus, we have proved the qualitative nonequivalence of the geometric arrangements of water molecules in the antipodal configurations of the bilayer ice. Ice Nanotubes. Large deviations of the H-bond network from tetrahedral coordination in ice nanotubes that consist of stacked n-membered rings (Figure 9) are of a considerable interest. However, in small fragments of square, pentagonal, and hexagonal nanotubes with periodic boundary conditions all nonantisymmetrical proton configurations have an undesirable feature. We analyzed sequentially the fragments consisting of one, two, and three H-bonded rings with periodic boundary conditions. Nonantisymmetrical proton configurations were found only in three-section fragments. Two of such nonantisymmetrical configurations for square-cross-section nanotubes are shown in a and b of Figure 9. Each of these configurations represents a pair of nonantisymmetrical configurations with the opposite direction of all H-bonds. The undesirable feature, which was mentioned above, is the presence of molecules arranged lengthwise along the nanotube. In Figure 9a and b, the location of these molecules is depicted as ovals. Such a location of water molecules which was depicted by an arrow in Figure 9c is energetically rather unfavorable. Nevertheless, the two found pairs of nonantisymmetrical configurations of a square nanotube are stable with the use of the TIP4P potential. Here, we demonstrate only the qualitative nonequivalence of antipodes for these two pairs of configurations. In Figure 9c we can see the configurations in ball-andstick form. Their antipodes are obtained by reversal of all H-
Figure 9. Nonantisymmetrical configurations for the fragment of a square-cross-section nanotube. (a, b) Structure in graph-theoretical representation. (c) Initial configurations of antipodal pairs in ball-and-stick form. The ovals depict the location of the energetically unfavorable molecules (a, b) and the different fragments of the antipodal configurations (c). F
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Finally, note that there is an analogy between the Hsymmetry and fundamental discrete symmetries of physics: charge conjugation (C), space inversion (P), and time reversal (T). To a considerable degree, these symmetries and also their violations determine the laws of nature. Taking into account the paramount importance of water for human life,41 we can suggest that the H-symmetry of ice and water structures plays an key role for the biophysical problems such as the origin of homochirality in biological systems.
configurations with zero dipole moment. At the same time, the evident distinction of the energies for a very approximate SPC potential that was mentioned above is a weighty argument in support of the noninvariance of the H-bond network under reversal of all H-bonds. Finally, note that our calculations of the energy difference yield 0.10 kJ/mol. We used the flexible potential TIP 3f,34 Ewald summation of electrostatic interactions,30 and the atomic coordinates from ref 37.
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4. CONCLUDING REMARKS The finding of essentially nonequivalent proton configurations with the opposite direction of all H-bonds in bilayer ice and ice nanotubes allows us to come to a conclusion about the existence of inherent asymmetry in ice-like systems. It is necessary to stress that this asymmetry is an intrinsic property of the systems. It is not related to spontaneous breaking of the H-bonding symmetry. The reason is a fundamental nonequivalence of the arrangements of water molecules in some antipodal configurations. An accurate calculation of the energy difference between antipodes is of great interest, but this difference exists, and it is irremovable, since the structural distinctions of some antipode configurations are qualitative. We think that the revealed asymmetry is a new fundamental property of the H-bonded frameworks in ice-like systems. It can be the reason for subtle self-organization processes, which are still unstudied. Thus, in contact with other objects, the ice-like systems and the hydrated water layers can initiate a new type of asymmetry. For example, we believe that the H-bonding asymmetry of water and ice systems may be the initial cause of the origin and saving of the mirror asymmetry for living systems, i.e. homochirality of biopolymers.39 A certain link between the revealed asymmetry of H-bonding and the mirror asymmetry can be seen in Figure 4. The antipodal configurations of each bilayer type differ only by a reflection of lower monolayers in the plane yz. Therefore, under the fixed direction of all the remaining H-bonds, the mirrorsymmetrical configurations of lower monolayers are not energetically equivalent. Actually, it is sufficient to fix only the electrically polarized state of upper monolayers, since the direction of intermediate H-bonds will remain because of the Bernal−Fowler ice rules. In this case, the hidden H-bonding asymmetry of bilayers breaks the mirror symmetry of lower layers. This induced symmetry breaking is also unexpected, since the configurations of the lower layers look completely equivalent, even taking into account the preferential direction of electric polarization of upper layers. It is important to note that the periodic boundary conditions increase the symmetry and the antisymmetry. In other words, a fully crystalline state destroys the H-bonding asymmetry. At the same time, the energy relaxation (transition to more stable proton configurations) also destroys this asymmetry. The point is that the fraction of the symmetrical configurations40 and, what is more important here, the antisymmetrical configurations15,29 is essentially greater among the set of optimized structures. Note also that the number of the ordinary (nonantisymmetrical) proton configurations grows rapidly by the size of unit cells. Therefore, in the orthorhombic unit cell of ordinary ice Ih (2 × 1 × 1), having 16 water molecules, there are 28 pairs of the ordinary proton configurations. Some of them have zero total dipole moment. In the large system limit, the majority of the proton configurations are completely unsymmetrical and also nonantisymmetrical (that is, ordinary configurations).
ASSOCIATED CONTENT
S Supporting Information *
Potential energy, dipole x,y,z-components, dipole moment magnitude (TIP4P potential), and the initial coordinates for all proton configurations of the bilayer ice unit cell. This material is available free of charge via the Internet at http://pubs.acs.org.
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected] Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS I would like to thank Emil S. Medvedev for his constructive criticism and Sotiris S. Xantheas for helpful discussion. This study was supported in part by Interdisciplinary integration projects of SB RAS No. 144.
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