Analysis of Ice Crystal Growth Shape under High Pressure Using

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Analysis of Ice Crystal Growth Shape under High Pressure Using Molecular Dynamics Simulation Hiroki Nada* National Institute of Advanced Industrial Science and Technology (AIST), 16-1 Onogawa, Tsukuba 305-8569, Japan ABSTRACT: A molecular dynamics simulation was conducted to analyze the growth shape of an ice crystal from water near its melting point under high pressure. The simulation was performed using a constant-volume system in which a cylindrical ice crystal grew freely in all directions perpendicular to the crystal’s c-axis. As the ice crystal grew in the system, the pressure gradually increased to approximately 2000 atm. The growth rate gradually decreased to near zero because the increase in pressure caused melting point depression. As the pressure increased to 2000 atm, the ice crystal grew as a hexagonal plate with {1010} prismatic plane facets clearly visible. The simulation indicated a fluctuation of the growth front geometry in the a-axis direction; the formation of a molecularly sharp corner was repeatedly disrupted during growth. The molecularly sharp corner appeared transiently as a result of the layer-by-layer growth of prismatic plane layers toward the corner. When corner formation was disrupted, a transient, molecularly flat, {1120} secondary prismatic plane appeared, but it was not a stable plane facet. The features of the ice crystal growth shape observed in this simulation were then compared with real systems.

1. INTRODUCTION Ice crystal growth has been an important research subject for many years. It is essential for understanding many phenomena, such as the pattern formation of snow crystals,1
3 the freezing of water in biological systems,4
6 and the formation of ice particles in the atmosphere.7
9 The growth shape of ice crystals has been studied theoretically10
15 and experimentally.16
25 Recently, attention has been paid not only to the ice crystal growth shape from pure water but also to changes in the growth shape induced by biological molecules.22,25 It is well-known that the growth shape of an ice crystal at temperature T, just below the melting point Tm, and at pressure P of 1 atm is a thin circular disk that has wide {0001} basal planes as facets.26 The growth shape reflects a strong anisotropy in the growth rate R, which originates from the anisotropic growth mechanism of the ice plane. Thus, the anisotropic growth mechanism is crucial for understanding the growth shape of an ice crystal. However, the anisotropic growth mechanism and its relationship to the ice crystal growth shape are still poorly understood because elucidating them by experimental means is quite difficult. Computer simulations, such as molecular dynamics (MD), are helpful tools for investigating the anisotropic growth mechanism at the molecular level.27 To date, several MD simulation studies have been carried out to elucidate the ice crystal growth mechanism for several ice crystal crystallographic planes, including the {0001} basal, {1010} prismatic, and {1120} secondary prismatic planes.28
34 These studies showed a difference in the growth mechanism between the crystallographic planes, which qualitatively explained the anisotropic R of ice crystals in real systems. r 2011 American Chemical Society

However, these previous studies were not sufficient to clarify the relationship between the anisotropic growth mechanism and the ice crystal growth shape in real systems. Notably, these earlier studies used an ice
water interface system in which ice growth occurred for only a single crystallographic orientation. Hereafter, this system will be referred to as the directional growth system (DG-system, Figure 1a). Because the ice crystal growth shape is determined by the anisotropy in R for all crystallographic orientations of the ice crystal, a limited number of MD simulations of the DG-system for several different crystallographic planes are not sufficient for detailed determination of the growth shape. The simplest way to determine the ice crystal growth shape in an MD simulation is by directly observing the growth shape using a system in which an ice crystal grows freely in liquid water. If the same ice crystal growth shape is obtained in the simulation as in real systems, the simulation will significantly advance the understanding of the actual relationship between the molecular-scale anisotropic growth mechanism and the ice crystal growth shape. To the best of our knowledge, no one has yet performed such a challenging MD simulation of ice crystal growth from water near Tm. This paper presents an MD simulation study of ice growth in which an ice crystal grew freely in all directions normal to the caxis of an ice crystal. The study focused on the ice crystal growth shape near Tm at high P. In real systems, the ice crystal growth Received: March 28, 2011 Revised: May 10, 2011 Published: May 12, 2011 3130

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Figure 1. Schematics of the systems for simulations of ice growth from water. (a) The system that contains two flat ice
water interfaces (the DG-system). (b) The system in which an ice crystal grows freely in all directions normal to the c-axis (the FG-system). The dimensions of the FG system in the x, y, and z directions, lx, ly, and lz were 14.80, 15.65, and 2.21 nm, respectively.

shape near Tm changes from a circular disk to a hexagonal plate that has six {1010} prismatic plane facets when P is higher than 1600 atm.23 This growth shape transition suggests that the growth mechanism in directions normal to the c-axis of the ice crystal becomes strongly anisotropic at P > 1600 atm. Thus, it is important to investigate the ice crystal growth shape both at high P and at 1 atm to elucidate the relationship between the growth shape and the anisotropic growth mechanism.35 Our simulation successfully reproduced the same features of ice crystal shapes at high P as those obtained from experimental studies.23,24

2. SIMULATION METHOD 2.1. Simulation System. A schematic of the free growth system (FG-system) that we used in this study is shown in Figure 1b. A cylindrical ice crystal with a radius of 3 nm was placed at the center of the system so that the c-axis of the ice crystal corresponded to the z direction of the system. The remainder of the system corresponded to liquid water. The system contained 17 640 H2O molecules; the ice crystal and liquid water in the system contained 1930 and 15 710 H2O molecules, respectively, in the initial state. Periodic boundary conditions were imposed in the x, y, and z directions. The thickness of the ice crystal in the z direction was set at 2.21 nm, which was the same as that of the system in the z direction. Therefore, the ice crystal in the system was connected by the periodic boundary condition imposed in the z direction; in other words, basal plane growth was not taken into account. However, this system allows us to observe the shape of an ice crystal grown freely from water in all directions normal to the c-axis. The initial state, in which a cylindrical ice crystal coexists with the surrounding liquid water, was created by means of Monte Carlo (MC) simulation, as in our previous studies.31,32,36 Initially, all H2O molecules in the system were arranged on the ideal lattice sites of hexagonal ice. Then, an MC simulation for the NVT ensemble was carried out. The simulation was carried out at 600 K with trial translational and rotational movements of 1.571  106. In the simulation, the positions of the H2O molecules included in the cylinder with a 3-nm radius from the center of the system were fixed at the ideal lattice sites of hexagonal ice. A state in which a cylindrical ice crystal coexisted with liquid water

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was then created in the system. Next, the system was relaxed by performing an MC simulation for the NPT ensemble at 268 K and 1000 atm without fixing the positions of the H2O molecules. The final configuration obtained by this MC simulation for the NPT ensemble was used for the initial configuration of the MD simulation. Note that the MC simulation for the NPT ensemble was carried out in an undercooled state. Nevertheless, since the number of trial movements performed was not large, no obvious ice growth occurred in the system. Thus, the MD simulation started with a nonequilibrium state. However, as will be described in Section 2.3, the system gradually approached an equilibrium state during the MD simulation. 2.2. Potential Model. As with our previous studies,31,32,34 a six-site model of H2O was used for the estimation of the intermolecular potential between a pair of H2O molecules.37 All intermolecular interactions were smoothly truncated at intermolecular distances from 0.95 to 1 nm by means of a switching function.38 The six-site model was proposed for simulations of ice and water near the real Tm at 1 atm. Earlier MD simulation studies using the six-site model demonstrated that the model is suitable for MD simulations of ice crystal growth from water.31
34 The Tm of ice at 1 atm in the six-site model is higher than the actual Tm (273.15 K).32,39 In this study, the Tm of ice at 1 atm in the FG-system with the six-site model was assumed to be 287 K or slightly higher. There was no obvious ice crystal growth during a short MD simulation at 287 K, which was used to infer Tm in advance. This value of Tm is close to Tm = 289 K, which was estimated using the Ewald summation method,39 and also close to Tm = 281
285 K, which was previously estimated in our MD simulation study using the DG-system.32 A large difference in the size of the simulation system might cause the small difference in the estimation of Tm. 2.3. MD Simulation. The MD simulation was performed using an implicit leapfrog algorithm proposed by Fincham, with a time step of 1 fs.40 The total run was 20 ns. T was maintained at 268 K using the Berendsen’s thermostat with a coupling parameter of 0.5 ps.41 The volume of the system was kept constant at its initial value of 4.986  102 nm3. Therefore, P increases because the density difference between water and ice leads to expansion of the system as the ice crystal grows. However, the Tm of ice decreases as P increases. Therefore, as ice grows in the system, the system gradually approaches an equilibrium state at a given T. The strength of this simulation is that ice growth shape near Tm at high P can readily be determined, even if the exact value of Tm is not known in advance. Experimental studies of ice growth at high P were carried out using an Anvil cell in which P increased as the ice crystal grew, as in the simulation.23,24 The system reached a state near equilibrium when P was approximately 2000 atm, which is the P at which a hexagonal plate growth shape was observed experimentally.19,23,24 Notably, if the MD simulation is started at low P, such as 1 atm, the initial driving force for growth, that is, the initial supercooling might be too large for the system to approach an equilibrium state within the simulation. Ice growth under a large driving force over the whole simulation is not desirable for this study aimed at analyzing ice growth shape near Tm at high P. This is the reason why the present MD simulation started at a high P of 1000 atm. Owing to the high initial P, we successfully obtained the ice growth shape near Tm at high P within the present simulation. 3131

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Figure 2. Potential energy U and pressure P of the system as functions of time t.

3. RESULTS AND ANALYSIS 3.1. Growth Shape. Figure 2 shows the system’s potential energy, U, and P as a function of time t. A decrease in U indicates the growth of ice. It can be seen that U decreased as P increased and remained almost constant when P increased to approximately 2000 atm (t > 16 ns), indicating that R gradually decreased to near zero as P increased. This suggests that the Tm of ice at P = 2000 atm in the system was very close to T = 268 K, which is lower by approximately 19 K than the Tm of ice at P = 1 atm in the six-site model (287 K). This decrease in the Tm because of the increase in P from 1 to 2000 atm in the six-site model is comparable to the decrease in Tm by 22 K caused by the same increase in P in real systems: Tm =
22 °C at P = 2000 atm in real systems. Therefore, the six-site model provides a reasonable description of the P dependence of Tm, even up to 2000 atm. Figure 3 shows time sequence snapshots of H2O molecules in the system. The snapshots were created from the time-averaged coordinates of H2O molecules over periods of 1.9
2, 5.9
6, and 15.9
16 ns. The snapshots of all H2O molecules are shown on the left, and those of ice-like H2O molecules (ILMs) are shown on the right. The ILMs were defined as H2O molecules that were captured in the ice crystal lattice sites and connected by a hydrogen bond with at least three nearest neighboring H2O molecules. The ILMs were also analyzed using the time-averaged coordinates that were used for creating the snapshots. In the analysis of the ILMs, if the distance between a H2O molecule and a lattice site was less than 0.055 nm, the H2O molecule was defined as captured in the lattice site. Whether a pair of H2O molecules was connected by a hydrogen bond was judged using an energetic definition.28 The growth shapes at 2 and 6 ns were generally rounded rather than faceted. The prismatic plane appeared only transiently at parts of the ice crystal growth front. However, the prismatic plane appeared as a facet much more clearly at 16 ns than at 2 and 6 ns. Moreover, some of the corners in the a-axis direction became distinct (arrows in the snapshots at 16 ns, Figure 3). Thus, the growth shape at 16 ns resembled a hexagonal plate. This hexagonal plate growth shape gradually appeared as P increased and became distinct, particularly at P > 1800 atm (t > 12 ns), although a hexagonal plate growth shape with complete 6-fold symmetry was not observed. 3.2. Interface Structure. If R is smaller at the prismatic interface than at the interfaces for other crystallographic orientations, the ice crystal growth shape is a hexagonal plate. Normally, R at a molecularly flat interface is smaller than that at a molecularly rough interface, because growth at a molecularly flat interface requires a nucleation process. Thus, the hexagonal plate growth shape at high P implies that as P increased the prismatic

Figure 3. Snapshots of H2O molecules in the system at 2, 6, and 16 ns. In the right-hand snapshots, only ILMs are shown. The a-axis directions of the ice crystal are shown in the left-hand snapshots. The positions of the prismatic plane are shown by red dashed lines.

interface became molecularly flat, compared with the interfaces for the other crystallographic orientations. The appearance of entirely rounded growth shapes, which were observed at 2 and 6 ns, might be because of kinetic roughening at the prismatic interface.42 Because the driving force for growth was large, H2O molecules near the prismatic interface attached readily to the interface lattice sites, which resulted in a molecularly rough prismatic interface. If the roughness of the prismatic interface was equal to the roughness of the interfaces for other orientations, growth would be isotropic, and the growth shape would be rounded. Therefore, the anisotropy in the time sequence of the interface roughness is essential for understanding the P dependence of the ice growth shape observed in the simulation. The roughness of the interface was investigated by analyzing the number of hydrogen bonds, NHB, which each H2O molecule formed with its neighboring H2O molecules, using the ILM coordinates. In bulk ice NHB = 4, and at the interface it is 1800 atm, which agrees with this experimental result. However, it is not clear whether the transition of the ice growth shape from a circular disk to a hexagonal plate is also reproduced in the simulation. Extensive MD simulations over a wide range of T are needed to verify this observation. Cahoon et al. observed both the growth and melt shapes of an ice crystal near Tm at high P.24 They found that at 2000 atm, the melt shape became asymmetric compared with the growth shape. Although the melt shape was a hexagonal plate, it was covered with a plane rotated by 30° from the prismatic plane. They also found that the transition between the growth and melt shapes occurred via the reversible, transient formation of a 12-sided state, where the ice crystal had six prismatic plane facets and six planes that were rotated by 30° from the prismatic plane. They concluded that the planes rotated by 30° from the prismatic plane had a molecularly rough structure. In this simulation, a molecularly flat secondary prismatic plane appeared during the disruption of the corner in the a-axis direction. However, it appeared only transiently, not as a stable plane facet. At almost all times, planes that appeared during the disruption of the corner in the a-axis direction had a molecularly rough structure. This result agrees with the experimental result by Cahoon et al. who found that molecularly rough planes rotated by 30° from the prismatic plane appeared in the a-axis direction during melting.24 The disruption process observed in the present simulation was essentially the same as the interpretation of the melting process at the corner by Cahoon et al.24 Thus, we concluded that the features of the ice shape at high P observed in our simulation were qualitatively consistent with those in real systems. 4.2. Comparison with Theory. According to the theory proposed by Jackson,43 whether the structure of the ice crystal’s prismatic plane is molecularly rough or flat can be predicted by the parameter R, which is defined as R = (3/4)ΔH/kBTm, where kB is the Boltzmann constant. Jackson’s theory provides a prediction that the roughness of a plane structure increases as R decreases. The experimental value of R at 1 atm is 2.00.19 However, the value of R at 1 atm in the six-site model was estimated as 2.51, which is much larger than the experimental value. For the estimation of R in the six-site model, ΔH/kBTm = 3.35 was used. This value was obtained from separate MD simulations of bulk ice and water at 287 K and 1 atm. Thus, Jackson’s theory suggests that the prismatic plane in the six-site model is more molecularly flat than in real systems. This might be reflected in the transient appearance of the prismatic plane as a facet at 1 atm. However, Jackson’s theory may not be sufficient to explain ice crystal shapes in real systems. The experimental studies by Maruyama et al.19,23 and our simulation suggest that the roughness of the prismatic interface decreased as P increased. Maruyama et al. found a decrease in R from 2.00 to 1.56 as P increased from 1 to 1960 atm,19 which contradicts Jackson’s theory that predicts an increase in the roughness of the prismatic interface as R decreases. Our simulation also showed a decrease in R from 2.51 to 2.36 as P increased from 1 to 2000 atm. More detailed theoretical studies are needed to clarify the faceting of the prismatic plane at high P. Because the six-site model overestimates Tm and R, our results should be confirmed using other potential models that reproduce Tm and R quantitatively. Moreover, the six-site model was

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developed by assuming that all intermolecular interactions are truncated at the intermolecular distance of 1 nm. Therefore, the simulation results should be also verified using the Ewald summation method for estimating the Coulomb interactions. The development is underway of a modified six-site model, which is parameterized by assuming the use of the Ewald summation. This modified model will be used to check our current results. 4.3. Simulation Methodology. In this study, the FG-system was introduced. A similar system in which a cylindrical ice crystal was placed at the center of the system was used in an MD simulation of the equilibrium and melt shapes of an ice crystal in vapor.44 The FG-system was also introduced in our previous preliminary MD simulation of ice growth at T much less than Tm.45 However, to the best of our knowledge, this study is the first to introduce the FG-system for an MD simulation of ice crystal growth near Tm. Although the present simulation successfully reproduced features of ice shapes near Tm at high P in real systems, there are several problems. The basal plane growth is neglected in the simulation, whereas in real systems an ice crystal grows in three dimensions in liquid water. Studies of ice crystal pattern formation suggest that three-dimensional growth is important for the growth shape.15,17,18 Thus, ideally, basal plane growth should be taken into account when investigating ice crystal growth shape. This would require a much larger simulation system than was used in this study. A further problem is that the latent heat, which is released from the interface during growth, was immediately removed from the system; T was kept constant. In an MD simulation with a constant T algorithm, when T deviates, the molecular velocities are instantly rescaled to maintain a steady T. In real systems, the latent heat released at the interface creates a temperature gradient around the crystal. The temperature gradient influences R and also the growth shape of the crystal.15,17 Therefore, the effect of the latent heat on the growth shape should not be neglected. An alternative algorithm is the NPH ensemble, in which enthalpy, H, is kept constant instead of T. If an MD simulation of crystal growth is carried out using the NPH ensemble, the latent heat released by the growth is not removed instantly from the system, and T is not kept constant. A comparison of the results from the constant T and constant H simulations would be interesting to perform in future studies. Although the present methodology may need modifications, the results suggest that it is sufficient to analyze features of ice growth shape at high P in real systems. The methodology presented herein contributes to the understanding of the relationship between the molecular-scale anisotropic growth kinetics and ice crystal growth shape in real systems.

5. CONCLUSIONS The shape of an ice crystal grown from water was investigated by an MD simulation using the FG-system, in which an ice crystal grew freely in all directions perpendicular to the c-axis. Since the volume of the FG-system was kept constant in the simulation, P increased and R simultaneously decreased as the ice crystal grew. The system reached a state near equilibrium when P was approximately 2000 atm. Under these conditions, the shape of the growing ice crystal was a hexagonal plate. To the best of our knowledge, this study is the first to directly observe the time evolution of the shape of an ice crystal grown from water near Tm in an MD simulation. 3135

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Crystal Growth & Design The hexagonal plate shape of the ice crystal at high P was attributed to the decrease in roughness of the prismatic interface relative to the other crystallographic planes. The simulation indicated that the prismatic plane layer grew in a layer-by-layer mode. As a result of the layer-by-layer growth of the prismatic plane layer toward the corner, in the a-axis direction, a molecularly sharp corner formed transiently in the a-axis direction. However, its formation was immediately disrupted, and a molecularly flat secondary prismatic plane appeared momentarily. The features of the ice shape were qualitatively consistent with those observed in experimental studies by Maruyama et al.19,23 and Cahoon et al.24 Thus, we confirmed that an MD simulation with the FG-system enables us to analyze features of the ice growth shape at high P in real systems, even though the size of the ice crystal in the simulation is much smaller than in real systems. In conclusion, the formation mechanism of ice crystal shape at high P applies to real systems. We demonstrated that an MD simulation using the FG-system has the potential to contribute to the understanding of the anisotropic growth mechanism and its relationship to ice crystal shape in real systems. The method can be applied to studies on the alternation in ice growth shapes induced by antifreeze proteins,4,5,46
48 if a much larger FGsystem is used. Similarly, the method could be applied widely to studies of crystal growth shape and its relationship to the binding of impurities to crystal surfaces, which is of fundamental importance for understanding the control of crystal growth shape by organic molecules in biomineralization.49

’ AUTHOR INFORMATION Corresponding Author

*Tel. þ81-29-861-8231. Fax. þ81-29-861-8722. E-mail: hiroki. [email protected].

’ ACKNOWLEDGMENT This work was supported by Grant-in-Aid for Scientific Research (C) (No. 21540423) from the Japan Society for the Promotion of Science. A part of this work, the development of a new simulation methodology, was supported by a Grant-in-Aid for Scientific Research on Innovative Areas of “Fusion Materials: Creative Development of Materials and Exploration of Their Function through Molecular Control” (No. 2206) from the Ministry of Education, Culture, Sports, Science and Technology, Japan (MEXT). The computation in this work was done using facilities of the Super Computer Center, the Institute of Solid State Physics, The University of Tokyo. ’ REFERENCES (1) Nakaya, U. Snow Crystals
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