Volume Diffusion in the Growth Kinetics of

Article pubs.acs.org/crystal

Roles of Surface/Volume Diffusion in the Growth Kinetics of Elementary Spiral Steps on Ice Basal Faces Grown from Water Vapor Harutoshi Asakawa,† Gen Sazaki,*,† Etsuro Yokoyama,‡ Ken Nagashima,† Shunichi Nakatsubo,† and Yoshinori Furukawa† †

Institute of Low Temperature Science, Hokkaido University, N19-W8, Kita-ku, Sapporo 060-0819, Japan Computer Centre, Gakushuin University, 1-5-1 Mejiro, Toshima-ku, Tokyo 171-8588, Japan

‡

W Web-Enhanced Feature * S Supporting Information *

ABSTRACT: We measured velocities Vstep of lateral displacement of individual elementary steps on an ice basal face, for the first time, by advanced optical microscopy, under various bulk water vapor pressure PH∞2O. Distances L between adjacent spiral steps exhibited considerable variation under constant PH∞2O. Hence, we analyzed Vstep as functions of L and PH∞2O. Then we found that (1) under a constant PH∞2O, Vstep decreased with decreasing distances L when L ≤ 15 μm and that Vstep remained constant when L int ∞ ≥ 15 μm. We named Vstep of L ≥ 15 μm (isolated steps) Vint step and analyzed dependencies of Vstep on PH2O. Then we found that int ∞ ∞ (2) the slope of the Vstep vs PH2O plot gradually decreased with increasing PH2O. We proposed a model that took into account both the volume diffusion of water vapor molecules and the surface diffusion of water admolecules on a terrace. Our model could explain result (1) mainly by the competition of adjacent spiral steps for water admolecules diffusing on a terrace but could not explain the result (2) satisfactorily. grown from supersaturated water vapor.12 Then we confirmed that ice crystal surfaces have a terrace−step−kink structure as in the case of flat faces of other crystals.12 We also found that there are two types of QLL phases that exhibit different morphologies and dynamics.13,14 These results clearly demonstrated that in situ observation by LCM−DIM can reveal the kinetics of displacement of elementary steps on ice crystal surfaces. The growth kinetics of elementary steps has been revealed by many studies on semiconductor crystals, such as Si15−23 and GaAs.24 In these studies, elementary steps grown under ultrahigh vacuum (UHV) were observed in situ by reflection electron microscopy, scanning electron microscopy, and lowenergy electron microscopy. Then, it is well acknowledged that surface diffusion of adatoms or admolecules on a terrace plays an important role in the growth kinetics of elementary steps

1. INTRODUCTION Ice is one of the most abundant materials on earth, and its growth and melting/sublimation govern a wide variety of phenomena, such as weather, environment-related issues, and life in a cryosphere. Ice crystals also play important roles in daily life: for example, they provide places for ice-skating and enable preservation of foods and organs. Hence an understanding of the kinetics of growth of ice crystals is important in a wide variety of fields. To reveal the kinetics of growth of ice crystals, many optical microscopy studies have been carried out using ordinary brightfield microscopy,1,2 differential interference contrast microscopy,3−6 two-beam interferometry,7−9 and laser reflection microscopy.10 However, all of those studies dealt with external forms and macroscopic surface morphologies of ice crystals. Recently, we have developed a technique of laser confocal microscopy combined with differential interference contrast microscopy (LCM−DIM).11,12 By using LCM−DIM, we succeeded in direct visualization of “elementary steps”, with a sufficient contrast level, on basal and prism faces of ice crystals © 2014 American Chemical Society

Received: October 2, 2013 Revised: April 30, 2014 Published: May 30, 2014 3210

dx.doi.org/10.1021/cg4014653 | Cryst. Growth Des. 2014, 14, 3210−3220

Crystal Growth & Design

Article

lower Cu plates were kept at Tsample = −15.0 and Tsource = −13.0 °C, respectively, for 2 h to nucleate sample ice crystals on the cleaved AgI crystal. Most of the ice crystals that appeared on the AgI crystal had the same orientation (inset of Figure S1B), indicating that they were grown heteroepitaxially. At Tsample = −15.0 and Tsource = −14.0 °C, the sample ice crystals were grown overnight until their lateral size and height reached 100−300 μm. As shown in Figure S2 of the Supporting Information, the total volume of the source ice crystals was significantly larger than that of the sample ice crystals on the AgI crystal. Hence, the supply of source ice crystals on the lower Cu plate did not run out during the observation experiments. In all measurements, the total pressure in the chamber was kept at 0.1 MPa (atmospheric pressure), and the partial pressure of water vapor was controlled by Tsource. (For details, see next section 2.3.) On a cleaved AgI substrate crystal, ice crystals were formed in serried ranks. Number density and area ratio of ice crystals, the basal faces of which were perpendicular to an optical axis, were 500 mm−2 and 94%, respectively. Average lateral size of basal faces was 150 μm. Among these crystals, one relatively large crystal (Figure 3A) was chosen and used for further in situ observation. In this study, all in situ measurements were performed on this ice crystal. Since the growth temperature (−8.4 °C) was much lower than the roughening transition temperature of prism faces (−1.3 °C),25 most of prism faces, which were parallel to optical axis, were well faceted. But also there were a small amount of nonfaceted (fast-growing) faces. As discussed in section 3.2, the existence of many crystals and a small amount of nonfaceted faces made precise analysis of a volume diffusion field of water vapor impossibly difficult. 2.3. Determination of Water Vapor Pressure. The temperatures of the sample ice crystals Tsample and the source ice crystals Tsource respectively determined water vapor pressures Pe and PH∞2O formed by evaporation of the sample and source ice crystals. When Pe < PH∞2O, the sample ice crystals grew. During growth of the sample ice crystals, water vapor pressure PHsurf2O in close vicinity of the sample ice crystals became lower than PH∞2O because of a bulk diffusion field of water vapor around the sample ice crystals. To evaluate Pe, the sample ice crystals kept growing (Pe < PHsurf2O) by observing the displacement of steps, and then Tsample was increased gradually. Melting of sample ice crystals was observed by LCM−DIM. The thermistor for the Tsample measurement was calibrated using the melting temperature of the sample ice crystals (0 °C). Then Pe was determined from Tsample and the solid−vapor equilibrium curve of water. In contrast, to evaluate PH∞2O, the source ice crystals located on the lower Cu plate could not be observed because of the structure of the observation chamber (Figure S1B, Supporting Information). Hence, the following observation of the sample ice crystals was performed. At a certain constant Tsample, Tsource was increased and decreased. Then the forward and backward movements of elementary steps and the growth and sublimation of crystal edges were observed in situ by LCM−DIM, as previously reported.26,27 From such observation, Tsource at which step velocities became zero (PHsurf2O = PH∞2O = Pe) was determined. Since Pe (= PH∞2O) could be calculated from Tsample and the solid−vapor equilibrium curve, the relationship between PH∞2O and Tsource could also be determined. For further details, see section S1 and Figure S3 of the Supporting Information. 2.4. Growth Temperature. It is well-known that with decreasing temperature, the external shape of ice crystals (snowflakes) grown freely in air (under a low-supersaturation range) changes repeatedly from a hexagonal plate (0 to −4.0 °C), to a hexagonal rod (−4.0 to −10.0 °C), and then to a hexagonal plate (−10.0 to −22.0 °C) and finally to a hexagonal rod (< −22.0 °C) (Figure 2).28−30 In this study, we tried to reveal the roles of surface diffusion and volume diffusion processes in the growth kinetics of an ice basal face, which is the easiest face for the in situ observation by our observation system. As a basal face grows faster, both diffusion processes become more ratedetermining. Then, we chose the growth temperature Tsample = −8.4

grown from a vapor phase.15,16,19−22,24 In contrast, ice crystals grow in a dense gas phase, such as air at 1 atm pressure. In such cases, volume diffusion of molecules in a dense gas phase should be also taken into account to reveal the growth kinetics of elementary steps because of a very short mean free path of gas molecules. In this study, we tried to clarify the growth kinetics of elementary steps grown in a dense gas phase, for the first time. We directly observed the lateral advancement of individual spiral steps on ice basal faces grown under atmospheric pressure by using LCM−DIM. Then we measured the dependencies of step velocities on distances between adjacent spiral steps and on water vapor pressures. Subsequently, we investigated the roles of surface diffusion and volume diffusion of water molecules in the growth kinetics of elementary steps of ice crystals grown under atmospheric pressures.

2. EXPERIMENTAL PROCEDURES 2.1. Observation System. The LCM−DIM system used in this study included all of the improvements made in our recent study12 for observation of elementary steps on ice crystal surfaces. A confocal system (FV300, Olympus Optical Co. Ltd.) was attached to an inverted optical microscope (IX70, Olympus Optical Co. Ltd.), as previously described11,12 (Figure S1A, Supporting Information). A super luminescent diode (Amonics Ltd., model ASLD68-050-B-FA, 680 nm) was used as an illumination light source. Figure 1 shows a sectional schematic illustration of an observation chamber. Details of the chamber are shown in Figure S1B of the

Figure 1. A sectional schematic drawing of the observation chamber. Sample ice crystals for in situ observation and other ice crystals for supply of water vapor to the sample ice crystals were grown on upper and lower Cu plates, respectively. Pe and PH∞2O indicate the water vapor pressures that were formed by evaporation of the sample and source ice crystals at Tsample and Tsource, respectively. PHsurf2O is water vapor pressure in close vicinity of the sample ice crystals. Details of the chamber are shown in Figure S1B of the Supporting Information. Supporting Information. The chamber had upper and lower Cu plates, and their temperatures Tsample and Tsource were measured by thermistors. Tsample and Tsource were separately controlled using Peltier elements with an accuracy of ±0.1 °C. At the center of the upper Cu plate, a cleaved AgI crystal (a kind gift from Emeritus Professor G. Layton of Northern Arizona University), known as an ice nucleating agent, was attached using heat grease. 2.2. Sample Preparation. In this section, preparation of sample ice crystals for in situ observation and preparation of other ice crystals for the supply of water vapor to the sample ice crystals are explained. Temperatures of the upper and lower Cu plates were first set at Tsample = 20.0 and Tsource = −15.0 °C, respectively. Then water vapor was supplied to the inside of the chamber by nitrogen gas bubbled through water (flow rate: 500 mL/min). After source ice crystals had grown on the lower Cu plate for 1 h, the water−vapor supply was stopped, and the chamber was kept airtight. Then the temperatures of the upper and 3211



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Figure 2. Habits of ice crystals grown from water vapor and growth conditions used in this study (open rhombuses). Tsample and PH∞2O denote the temperature of sample ice crystals and the water vapor pressure in a bulk vapor phase, respectively. The solid curve is the solid−vapor equilibrium curve of water. Under the conditions of a low supersaturation range, with decreasing Tsample, ice crystals show a hexagonal plate shape (0 to −4.0 °C), a hexagonal rod shape (−4.0 to −10.0 °C), and a hexagonal plate shape (−10.0 to −22.0 °C).28−30 °C (Pe = 298 Pa) at which temperature a basal face grows faster than a prism face (resulting in an ice crystal of a hexagonal rod shape). Rhombus plots in Figure 2 show supersaturations examined in this study.

3. RESULTS AND DISCUSSION 3.1. Dependencies of Step Velocities on Distances between Adjacent Steps and Supersaturation. We observed ice basal faces in situ by LCM−DIM. To obtain images, raw LCM−DIM images were processed according to the method explained in Figure S4 of the Supporting Information. Figure 3A shows a picture of an entire ice basal face. Edge angles of 120° certify that the crystal face observed was a basal face. As shown in Video S1, concentric elementary steps repeatedly emerged from the position marked by a black arrowhead in Figure 3A. All elementary steps laterally advanced in the left direction (white arrow). Although steps on the left side of the basal face (marked by a dashed curve) also show a concentric shape (the center of which is located on the left side), such a shape was formed by the nonuniformity in surface supersaturation (the Berg effect):31−33 supersaturation (water vapor pressure) in a peripheral region of a flat face of a polyhedral crystal is higher than that in a face center region. Because of the higher supersaturation in a peripheral region, elementary steps in the peripheral region moved faster than those in the central region, as shown in the area marked by the dashed curve. No steps emerged from the left side of the basal face. The existence of the nonuniformity in surface supersaturation clearly indicates that water vapor pressure in the vicinity of the basal face was smaller than that in a bulk vapor phase and hence that volume diffusion of water vapor from the source ice crystals to the sample ones played an important role in the growth of the ice crystal. Figure 3A also shows that distance between adjacent steps exhibited considerable variation, although steps generated by a stationary screw dislocation are usually equidistant.32,33 During the observation experiments, the appearance of nonequidistant steps from a single point could be observed ubiquitously on about 90% of the basal faces in the observation chamber (the remaining 10% of the basal faces were grown by the twodimensional nucleation growth mechanism.). Under our

Figure 3. LCM−DIM images of elementary spiral steps grown on an ice basal face at Tsample = −8.4 °C (Pe = 298 Pa) and PH∞2O = 323 Pa. (A) A photomicrograph (1024 × 1024 pixels) of an entire ice basal face. All elementary steps advanced laterally in the direction shown by a white arrow. A screw dislocation was located at the position shown by a black arrowhead. (B−D) Magnified views (456 × 124 pixels) in the white rectangle area shown in A. Images C and D were taken at 1.16 and 2.33 s after image B, respectively. White arrowheads in B−D show an identical elementary step. The line X−Y shown in B was used for preparation of Figure 4. Movies of the processes A and B−D are available as Videos S1 and S2, respectively, in the Supporting Information.

experimental conditions, no equidistant steps were observed. Since no growth mechanism other than spiral growth can explain such ubiquitous appearances of concentric steps, we concluded that we observed spiral steps. One possible reason for the nonequidistant spiral steps is ultrahigh growth temperature just below the melting point. At such a high temperature (T), the contribution of entropy (S) becomes important since G = H − TS (here G and H respectively denote Gibbs free energy and enthalpy of a system). We suppose that the fluctuation of step position may increase S and hence may decrease G. 3212



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As shown in Figure 3A, steps showed a circular shape, suggesting thermal roughening or kinetic roughening of steps on a basal face (here roughening does not mean roughening of a basal face but means that of only steps). To verify which type of roughening occurred, we gradually decreased water vapor pressure and then observed the process in which the direction of step movement changed from the forward direction (under supersaturated conditions) to the backward direction (under slightly undersaturated conditions). During this process, we observed steps under extremely near-equilibrium conditions; however, the shape of the steps was always circular. This result strongly suggests that the circular shape of the steps (Figure 3A) was caused not by kinetic roughening. For this issue, Furukawa and Kohata reported experimental results.2 They observed surfaces of negative ice crystals formed under reduced pressure by optical microscopy. Then they found that with increasing temperature, hexagonal steps observed on an ice basal face changed into circular steps at −20 °C. This result and our observation under extremely near-equilibrium condition strongly suggest that the circular shape of the steps at −8.4 °C was caused by thermal roughening. As mentioned above, we concluded that elementary steps grew by the spiral growth mechanism and that a screw dislocation was located at the position marked by the black arrowhead (Figure 3A). We have observed several hundred basal faces on which concentric steps emerged. However, screw dislocations were always located at the edges of basal faces, although its mechanism is still unclear. Hondo reported that the screw dislocations that emerge from basal faces do not have energetically stable configurations.34 We next tracked the advancement of individual elementary steps. The photomicrograph shown in Figure 3A was acquired over a 3.27-s scan time. LCM−DIM is a relatively slow observation technique. Hence, to obtain sufficient temporal resolution, we observed ice basal faces using much smaller scanning areas. Figure 3B shows an example of magnified views in a white rectangle area shown in Figure 3A. The photomicrograph shown in Figure 3B was acquired over a 0.29-s scan time. Images C and D were taken at 1.16 and 2.33 s after image B, respectively. White arrowheads in Figure 3B−D show an identical elementary step. Figure 3B−D shows that we succeeded in obtaining the temporal resolution of the image acquisition high enough to examine the kinetics of displacement of individual elementary steps (see Video S2). To efficiently analyze the relation between step velocities Vstep and distances L between adjacent spiral steps, we made a spatiotemporal image, a so-called time−space plot. We extracted pixels along a dedicated line (X−Y in Figure 3B) from sequential images and stacked them in a new x−y matrix35,36 (Figure 4). The horizontal axis of Figure 4 is composed of 150 images. Slopes of the trajectories of step contrasts correspond to Vstep. When the distance (vertical axis in Figure 4) from the edge of the basal face was smaller than 40 μm, Vstep showed constant values. However, when the distance became longer than 40 μm, Vstep gradually decreased with increasing distance. This decrease in Vstep demonstrates that the surface supersaturation decreases from X to Y in Figure 4. Since such the nonuniformity is too difficult to take into account using a simple model, we measured Vstep in the peripheral region, in which the decrease in surface supersaturation was negligibly small. As shown in Figure 4, we fitted a series of data using straight lines that started from the edge of the crystal. We

Figure 4. A time−space plot for the analyses of step velocities Vstep and distances between adjacent steps L. The pixels along a dedicated line (X−Y in Figure 3B) were extracted from sequential images (Figure 3B−D and Video S2) and were stacked in a new x−y matrix. The horizontal axis is composed of 150 images. To minimize the effect of nonuniformity in surface supersaturation, the trajectories of step contrasts were fitted using straight lines that started from the edge of the crystal (zero position). Vstep and L were calculated from the slopes of the white lines and the distances between the adjacent white lines in the vertical direction, respectively.

also obtained L from distances between adjacent straight lines in the vertical direction. Figure 5 shows Vstep as a function of L. As shown in Figures 3A and 5, L exhibited relatively wide dispersion. Error during

Figure 5. Changes in step velocity Vstep as a function of distance between adjacent steps L at Tsample = −8.4 °C (Pe = 298 Pa) and PH∞2O = 329 Pa. The inset shows a sectional schematic illustration of a vicinal face and explanations for two types of plots. Open small squares show Vstep as a function of L between a step in question and an upstream adjacent step, while open rhombuses depict Vstep as a function of L between a step in question and a downstream adjacent step.

the determination of the values of Vstep was about ±0.3 μm/s. In the case of growth from vapor, the Ehrlich−Schwoebel (E− S) effect is well-known to be one of the causes of wide dispersion of L:37,38 at a step, the incorporation of molecules that came from an upstream terrace is slower than that from a downstream one. Hence, we first examined whether the vapor growth of ice crystals showed the E−S effect. The inset of Figure 5 shows a sectional schematic illustration of a vicinal face and explanations for two types of plots. Open small squares show Vstep as a function of L between a step in question and an upstream adjacent step. In contrast, open rhombuses depict 3213



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Vstep as a function of L between a step in question and a downstream adjacent step. As shown in Figure 5, the open small squares and open rhombuses overlapped closely. This result demonstrates that ice basal faces did not show a significant E−S effect, at least under our experimental conditions. Furthermore, ice basal faces did not exhibit an E−S effect under the condition of various values of PH∞2O

water admolecule can reach an upstream or downstream adjacent step with a probability of 100%. As a result, the adjacent steps compete with each other through the surface diffusion of admolecules on a terrace; i.e., the step−step interaction occurs. Hence, Vstep decreases with decreasing L when L < 2χS. In contrast, when L > 2χS, Vstep remained constant irrespective of L because of constant amount of water admolecules that can reach steps. Therefore, the step−step interaction through surface diffusion of water admolecules can explain the phenomena shown in Figure 6. In contrast, in the case of volume diffusion, diffusion of water vapor molecules is much faster than that of water admolecules on a basal face, although surface diffusion coefficient of water admolecules is still not determined experimentally. From the volume diffusion coefficient (DV = 2.22 × 10−5 m2/s)39 and mean free path (66 nm)40,41 of water vapor molecules in nitrogen gas at atmospheric pressure, speed of a water vapor molecule is estimated to be 336 m/s. This value is much faster than Vstep shown in Figure 6. In addition, as shown in Figure 3, temporal distribution of L looked random. Hence, it would be reasonable to assume that the temporal distribution of Psurf H2O (water vapor pressure in close vicinity of the sample ice crystals) was steady, although spatial distribution of PHsurf2O was nonuniform (Figure 3A). If PHsurf2O P is temporally steady, the decrease in Vstep with decreasing L (L ≤ 15 μm) (Figure 6) cannot be explained by volume diffusion. In section 3.2, we will discuss in detail the roles of processes for surface diffusion and volume diffusion (also solid curves in Figure 6). We have another observation that supports the abovementioned discussions. As shown in Figures 3A and 4, we could observe the nonuniformity in surface supersaturation significantly when the length scale was larger than 40 μm. In contrast, as shown in Figure 6, adjacent spiral steps competed with each other for water molecules when L ≤ 15 μm. From this discrepancy in the length scales, we assumed that PHsurf2O in close vicinity of the sample ice crystals was approximately homogeneous in the length scale smaller than 15 μm, suggesting that the decrease in Vstep with decreasing L (≤15 μm) was caused not by volume diffusion but by surface diffusion. Previous studies also showed the critical distance for competition of adjacent steps on ice basal faces.42−44 Hallet,42 Mayson,43 and Kobayashi44 measured the time course of distances between adjacent bunched steps on ice basal faces by conventional optical microscopy. At Tsample = −8.4 °C, they obtained 6 μm, which corresponds to our value of 15 μm that shows minimum L without competition of adjacent steps (Figure 6). It is interesting that the order of their value is almost the same as that of ours, although they observed “bunched steps”. As shown in Figure 6, when L ≥ 15 μm, Vstep shows a constant value. We named Vstep whose L ≥ 15 μm Vint step, and we ∞ determined Vint step as a function of PH2O (supersaturation). In Figure 7, open circles show Vint step determined from Vstep vs L plots (Figure 6). Two black rhombuses indicate PH∞2O at which we could observe only isolated elementary steps (L ≥ 15 μm) due to very small supersaturation. Hence, we obtained Vint step by just taking the average of Vstep measured. An open square depicts PH∞2O at which elementary steps advanced backward ∞ (Vstep < 0). The slope of the Vint step vs PH2Oplot gradually

(Figure S5 of the Supporting Information). Hence, hereafter we use an average of the upstream L and downstream L (as shown in Figure 6) for further discussion.

Figure 6. Changes in step velocity Vstep as a function of distance between adjacent steps L at Tsample = −8.4 °C (Pe = 298 Pa). The horizontal axis shows the averages of upstream L and downstream L explained in Figure 5. PH∞2O: 310 Pa (A), 329 Pa (B) and 368 Pa (C). Average values of Vstep when L ≥ 15 μm were named Vint step.

Figure 6 shows examples of Vstep measured under the condition of various values of PH∞2O as a function of averaged L. Error during the determination of the values of Vstep was also about ±0.3 μm/s. Regardless of the value of PH∞2O, Vstep decreased with decreasing L when L ≤ 15 μm. In contrast, Vstep remained constant when L ≥ 15 μm. These results show that when L ≤ 15 μm, adjacent spiral steps competed with each other for water molecules supplied and that mass transfer processes limited the advancement of elementary steps. The step velocity is determined by the following two elementary processes:32,33 (1) surface diffusion of water admolecules on the ice crystal surfaces, and (2) volume diffusion of water vapor molecules to the ice crystal surfaces. Migration distance χS of water admolecules diffusing on a basal face can explain the appearance of the critical distance for competition of adjacent steps. When a water molecule adsorbs on a terrace, if the distance from the water admolecule to an adjacent step is shorter than χS, the water admolecule can reach the step and be incorporated into the step. When L < 2χS, a 3214



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Figure 7. Changes in step velocity of isolated elementary steps Vint step as int functions of PH∞2O and (PH∞2O − Pe)/Pe. Open circles show Vstep determined from the Vstep vs L plots (Figure 6). Two black rhombuses indicate PH∞2O at which we could observe only isolated elementary steps (L ≥ 15 μm) due to very small supersaturation. Hence, we obtained Vint step by just taking the average of Vstep measured. An open square depicts PH∞2O at which elementary steps advanced backward (Vstep < 0). The black dashed line roughly shows a tangential line of the Vint step vs PH∞2O plot at Pe.

Figure 8. Schematic drawings of spatial distributions of water vapor pressure PH2O (A) and water admolecule concentration CH2O (B). Inplane positions X and Y show those in Figures 3B and 4. PHsurf2O and PH∞2O correspond to PH2O at a crystal surface and that in a bulk gas phase, respectively. Pe is equilibrium PH2O. δ shows the thickness of a volume diffusion layer.

decreased with increasing P H∞2 O . We will discuss this phenomenon in section 3.2. From the slope of the dashed line (in Figure 7), which ∞ roughly shows a tangential line of the Vint step vs PH2O plot at Pe, we L determined the step kinetic coefficient β ≅ 700 μm/s, although this value contained a large amount of uncertainty. To our knowledge, this is the first step kinetic coefficient that was directly obtained from vapor growth experiments. The value is comparable to β (100−1000 μm/s) of inorganic salts, such as ADP and KDP, grown from aqueous solutions.45 3.2. Roles of Surface Diffusion and Volume Diffusion. During the growth of ice crystals from supersaturated water vapor, it is acknowledged that two successive processes play important roles:40 one is a process for diffusing water vapor molecules through air from their sources toward the growing surface, and the other is the surface kinetic process during which water admolecules diffuse on a terrace and then are incorporated into kinks on steps. Before showing a model of the growth kinetics of elementary steps, we first summarize the situations that we assumed in Figure 8. When PHsurf2O < PH∞2O, water vapor molecules come out from source ice crystals and then diffuse to a basal face of a sample ice crystal. Hence, spatial nonuniformity in PH2O develops along the direction perpendicular to the sample crystal surface (the axis n in Figure 8A: normal distance from a sample crystal surface). As discussed in section 3.1, the diffusion speed of water vapor molecules is much faster than Vstep. In addition, as shown in Figures 3 and 4, temporal distribution of L looked random. Hence, it is reasonable to assume that the distribution of PHsurf2O is temporally steady. As shown in Figure 4, Vstep at the position X (at a crystal edge) was faster than that at the position Y (in a face center region) because of the nonuniformity in supersaturation along the surface. Hence, PHsurf2O exhibits a nonuniform distribution in the in-plane position direction of the crystal surface, as schematically shown in Figure 8A. However, Figure 4 shows that in the peripheral region (in particular, where distance from

the crystal edge was shorter than 40 μm), Vstep was almost constant. Hence, in such the peripheral region (“measurement field of Vstep” in Figure 8A), it is rational to presume that the spatial distribution of PHsurf2O is almost homogeneous. On the crystal surface, because of the surface diffusion of water admolecules on terraces and the subsequent incorporation of admolecules into steps (more precisely kinks on steps), admolecule concentration CH2O shows a nonuniform distribution between steps in the measurement field of Vstep (Figure 8B), as supposed in the theory proposed by Burton, Cabrera and Franck. 46,47 As time elapses, the spatial distribution of CH2O moves in the in-plane position direction according to the lateral growth of elementary steps. As shown in Figure 3, under our experimental conditions, steps were thermally roughened, and hence kink density on steps was significantly high. Therefore, we can neglect the diffusion of water admolecules along steps: i.e., water admolecules that reach steps are immediately incorporated into kinks on steps. In addition to the above-mentioned pictures, we make the following two assumptions: (1) the incorporation process of water admolecules into steps is significantly faster than the preceding surface diffusion process on terraces, i.e., CH2O = Ce the equilibrium concentration at the step, and (2) the process of diffusion in vapor and the process of diffusion on a surface are in series. Under UHV conditions, Yamaguchi et al.24 and Hibino et al.19 measured velocities of individual elementary steps on GaAs(111)A and Si(111) surfaces, respectively. Then they revealed that the incorporation process of admolecules was much faster than the surface diffusion process. Hence, it would be reasonable to assume that the assumption (1) is valid also in a dense gas phase, as the first step. In addition, since the speed of diffusion of water molecules in vapor is extremely faster than Vstep, it is also rational to presume that water vapor molecules 3215



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collide with a crystal surface “spatially equally”, suggesting that great majority of water vapor molecules first collide with terraces then diffuse to steps. Then, we believe that the assumption (2) is also valid. On the above-mentioned assumptions, in this study, we propose a model in which volume diffusion and surface diffusion processes are taken into account and then discuss our experimental results using the model. From the viewpoint of the volume diffusion of water vapor molecules to a crystal surface, the normal growth rate Vbasal of a basal face is proportional to the net flux of vapor into the surface. Hence, Vbasal can be expressed as follows:32,33 ⎛ ∂PH2O ⎞ Vbasal = D̃ V ⎜ ⎟ = D̃ V ⎝ ∂n ⎠surf

PH∞2O

− δ

Here, βL is the step kinetic coefficient (unknown) defined by eq 7, α is the sticking probability (unknown), m is the mass of one water molecule (3.0 × 10−26 kg), and χS is the migration distance of admolecules on a terrace (unknown). From eqs 5 and 6, indeterminate parameter PHsurf2O can be given as PHsurf 2O

Vstep =

PHsurf 2O (1)

(2)

Here, D̃ V is the parameter defined by eq 2, (∂PH2O/∂n)surf is the normal derivative of the water vapor distribution at a position on the surface, Ω is the volume occupied by one water molecule in an ice crystal (3.25 × 10−29 m3),48 kB is the Boltzmann constant (1.381 × 10−23 J/K), Tsample is the absolute temperature of the sample ice crystals (264.8 K), and DV is the volume diffusion coefficient (2.22 × 10−5 m2/s)39 of water vapor molecules at 0.1 MPa and at 273.2 K. The thickness δ of a volume diffusion layer (unknown) is defined by the distance between the crystal surface and the point at which the tangent line of PH2O at a surface and the extrapolate line of P∞ H2O are intersected (Figure 8A):49 ⎛ ∂PH2O ⎞ ⎜ ⎟ δ ≡ PH∞2O − PHsurf 2O ⎝ ∂n ⎠surf

R VD =

(3)

(4)

where L is the distance between adjacent spiral steps (experimentally determined) and h is the height of spiral steps (0.37 nm).50 The law of conservation of matter in a surface indicates that eqs 1 and 4 are equivalent. Therefore, the step velocity Vstep is expressed by Vstep = D̃ V

PH∞2O − PHsurf 2O L h δ

(5)

On the other hand, the step velocity Vstep can be determined by solving the two-dimensional diffusion equation of water admolecules between steps including the net flux from vapor.32,33,46 When the kink density at the steps is large, Vstep is expressed by32,33 Vstep = β βL =

P surf L H 2O

− Pe

Pe α ΩPe

2πmkBTsample

(6)

⎛ L ⎞ ⎟⎟ tanh⎜⎜ h ⎝ 2χS ⎠

(8)

PH∞2O − Pe L /h R SK + R VD Pe 2πmkBTsample L 1 α ΩPe 2χS tanh(L /2χS ) δ kBTsample D V ΩPe

(9)

(10)

(11)

Here, RSK shows the resistance of the surface kinetics, RVD is the resistance of the volume diffusion. In eqs 9−11, unknown parameters are three: α, χS, and δ. Yokoyama and Kuroda40 so far reported the model equations of normal growth rate of ice crystals grown from vapor, in which model both the surface kinetics and volume diffusion were taken into account. However, with respect to the step velocity, eq 9 in this study is the first, as far as we know. First, we tried to fit the Vstep vs L plots (Figure 6) under various P∞ H2O to eq 9. We performed the curve fittings under three fixed values of δ = 20, 200, and 2000 μm (Figure 9). Although the value of δ has not yet been determined experimentally, it is reasonable to assume that the thickness of the volume diffusion layer is in the order of the size of the sample crystal (∼200 μm in this study: Figure 3).49,51 Hence, the actual value of δ was probably in the range of 20−2000 μm. Blue, red, and green solid curves in Figure 9 show the results of the curve fittings. Figure 9 demonstrates that irrespective of the value of δ, the Vstep vs L plots could be fitted to eq 9 well. These results suggest that the volume diffusion process was not significantly dominant under our experimental conditions (later we will discuss this issue quantitatively using Figure 11). From the curve fittings shown in Figure 9, we obtained two fitting parameters α and χS. Figure 10 shows the dependencies of α and χS on PH∞2O. Because of relatively large error bars of α and χS, it is not clear whether α and χS remained constant irrespective of PH∞2O or decreased with increasing PH∞2O. At this moment, we do not have any physical picture that can reasonablly explain the dependencies in α and χS on PH∞2O. Hence, we assumed that the values of α and χS were constant regardless of the value of PH∞2O, and obtained averaged values of α = 7.7 × 10−5 and χS = 4.5 × 10−6 m. The changes in α and χS with increasing PH∞2O are subjects of future investigation. With respect to the sticking probability, two papers have been reported. Beckmann and Lacmann measured negative normal growth rate of ice crystals during sublimation under atmospheric pressure,52 and then they obtained α = 0.14−0.16. Sei and Gonda also measured normal growth rate of ice crystals during growth under reduced pressure (40 Pa),4 and then they obtained α = 0.14−0.17. The former case measured

As shown in Figure 3, the local slope of the vicinal surface was sufficiently small. Hence, the normal growth rate is nearly equal to the growth rate along the c axis of the ice crystal, i.e., Vbasal. Then Vbasal is expressed by using the step velocity Vstep:

h Vstep L

β Lh/L + PeD̃ V /δ

Substituting eq 8 into eq 5 or 6 yields the following equations:

R SK =

Ω D̃ V = DV kBTsample

Vbasal =

= Pe

β Lh/L + PH∞2OD̃ V /δ

2χS

(7) 3216



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Figure 10. Changes in sticking probability α (A) and mean migration distance χS (B) as functions of PH∞2O and (PH∞2O − Pe)/Pe. The values of α and χS were calculated from the curve fitting (solid curves shown in Figure 9) using eq 9.

UHV. Bowker and King55 also experimentally measured a typical preexponential factor of surface diffusion (1.0 × 10−3 cm2/mol) but at 1041−1470 K under UHV. Hence, regarding the value of χS, so far no data were reported at temperatures near the melting point and under atmospheric pressure. In the near future, we will perform the similar measurements of χS under various partial pressure of nitrogen in the future. We tried to discuss the roles of the surface kinetics (surface diffusion) and the volume diffusion under various conditions. Figure 11A shows the dependencies of RSK (open circles) and

Figure 9. Changes in step velocity Vstep as a function of distance between adjacent steps L at Tsample = −8.4 °C (Pe = 298 Pa). The plot shown in Figure 6 was redrawn. PH∞2O: 310 Pa (A), 329 Pa (B) and 368 Pa (C). Solid curves show fitted curves using eq 9, when the thickness of a volume diffusion layer δ was 20, 200, and 2000 μm, respectively.

“sublimation of rough surfaces” and the latter measured under “reduced pressure”. In contrast, we measured growth of individual elementary steps on a flat surface under atmospheric pressure. Although our value was significantly smaller than the values reported so far, at this moment, there is no value of α that can be directly compared with ours. The migration distance χS = 4.5 × 10−6 m obtained from the curve fitting using eqs 9−11 gives tanh(L/(2χS)) ∼ 0.76 in eq 10 when L = 2χS. This is equivalent to the critical value of L = 15 μm obtained by visual estimation in Figure 6, since tanh(L/ (2χS)) ≈ 1 when L = 15 μm. With respect to the surface diffusion of water admolecules on ice crystal surfaces, Hale et al.53 theoretically calculated an activation energy of surface diffusion (2.5 kcal/mol), using the H2O−H2O intermolecular potential on ice basal faces. In contrast, Brawn and George54 experimentally measured an activation energy of desorption (12.7 kcal/mol) and a vibrational frequency of water admolecules (3.2 × 1031 cm−2 s−1) but at 180−210 K under

Figure 11. Changes in resistivities of the surface kinetics RSK (open circles) and volume diffusion RVD (filled circles) as functions of the thickness of a volume diffusion layer δ (A) and the distance between adjacent spiral steps L (B). Resistivities were calculated when L = 20 μm (A) and when δ = 200 μm (B). Solid lines and numbers located beside open circles show the total resistance (RSK + RVD) and the ratio of RSK(RSK/(RSK + RVD) in %), respectively. 3217



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RVD (filled circles) on δ when L = 20 μm, and Figure 11B depicts the dependencies of RSK and RVD on L when δ = 200 μm (the size of the sample crystal). To calculate the values of RSK and RVD, we used the averaged values of α and χS. In Figure 11, the solid lines and numbers located beside open circles represent the total resistance (RSK + RVD) and the ratio of RSK (RSK/(RSK + RVD) in %), respectively. Since the value of δ is expected to be in the order of the size of the sample crystal (∼200 μm),49,51 Figure 11A demonstrates that RSK played a dominant role under our experimental conditions. Hence, the result that the volume diffusion process was not significantly dominant in the curve fitting of the Vstep vs L plots (Figure 9) is reasonable. Figure 11B also demonstrates that RSK was dominant when L ≅ 15 μm, at which value Vstep started to decrease with decreasing L irrespective of PH∞2O (Figures 6 and 9). When the distance between the adjacent spiral steps is long enough (when L/2χS ≫ 1), in eqs 6 and 10, tanh[L/(2χS)] ≅ 1. Hence, from eq 9, the velocity of such isolated steps Vint step (Figure 7) is expressed as int Vstep =

PH∞2O − Pe

L /h 2πmkBTsample L α ΩPe 2χS

+

conditions (Figure 11), the large error of a numerator and that of a denominator in eq 12 canceled each other out. We substituted the values of Lav (Figure 12A) into L of eq 12 and ∞ then calculated Vint step as a function of PH2O (δ = 200 μm), using averaged values of α and χS. In Figure 12B, the data shown in Figure 7 were replotted. The solid line in Figure 12B depicts the result of the calculation. Sizes of error bars in calculated int values of Vint step were 0.004−0.2 μm/s. The solid line shows Vstep whose values are in the same order of those measured experimentally (open circles). However, the line cannot ∞ reproduce the decrease in the slope of the Vint step vs PH2O plot ∞ with increasing PH2O. In this study, we performed in situ measurement of Vstep of individual elementary steps on an ice basal face under atmospheric pressure, for the first time. The dependencies of Vstep on L were roughly explained by the surface kinetics. But ∞ the dependencies of Vint step on PH2O could not be fully explained by the model that includes both surface kinetics and volume diffusion. Therefore, it is essential to understand the effects of other elementary processes, such as adsorption/desorption of water molecules and environment gas (nitrogen) molecules on crystal surfaces, in addition to elementary processes taken into account in this study.

δ kBTsample D V ΩPe

Pe (12)

4. CONCLUSIONS In this study, we tried to clarify the growth kinetics of elementary steps grown in a dense gas phase, for the first time. We directly observed the lateral advancement of individual spiral steps on ice basal faces at Tsample = −8.4 °C by LCM− DIM. Then we measured step velocities of elementary steps Vstep under various bulk water vapor pressure PH∞2O. We obtained the following key results. (1) Ice basal faces did not show a significant Ehrlich− Schwoebel effect under our experimental conditions. (2) Unexpectedly, distance L between adjacent spiral steps exhibited considerable variation under constant PH∞2O. Vstep decreased with decreasing L when L ≤ 15 μm, whereas Vstep remained constant when L ≥ 15 μm. (3) The slope of the Vint step (Vstep of isolated steps when L ≥ 15 μm) vs PH∞2O plot gradually decreased with increasing PH∞2 O . The step kinetic coefficient obtained at Pe (equilibrium pressure) was βL ≅ 700 μm/s. (4) We proposed the model that took into account both the volume diffusion of water vapor molecules and the surface diffusion of water admolecules on a terrace. Our model could explain the key result (2) mainly by the competition of adjacent spiral steps for water admolecules diffusing on a terrace, but could not explain the key result (3) satisfactorily.

As shown in Figure 3A, L exhibited considerable variation under the same supersaturation. Then we calculated the average values of L (≥15 μm) that were measured experimentall, and named the average values Lav. To evaluate Vint step of isolated steps, we eliminated values of L ≤ 15 μm from the calculation of Lav. According to the theory proposed by Burton, Cabrera and Franck,46,47 the value of Lav decreases with increasing PH∞2O. However, as shown in Figure 12A, such tendency is unclear because of significantly large error bars of Lav. Despite this, the large variation in Lav did not provide significant error in the values of Vint step fortunately. Since surface kinetics dominated the growth kinetics (RSK ≫ RVD) under our experimental

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ASSOCIATED CONTENT

* Supporting Information S

Determination of Pe and PH∞2O, schematic drawings of the experimental setups, sample and source ice crystals grown on the upper and lower Cu plates of the observation chamber, relation between Tsample and Tsourcemeasure at which temperatures step velocity became zero, image processing performed to obtain LCM−DIM images, changes in step velocity Vstep as a function of distances L between adjacent spiral steps under

Figure 12. Changes in averaged value Lav of the distance between adjacent spiral steps L (A) and step velocity of isolated elementary ∞ ∞ steps Vint step (B) as functions of PH2O and (PH2O − Pe)/Pe. A solid line in int B shows the calculated value of Vstep using eq 12 and the averaged values of α and χS. 3218



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PH∞2O of 310 Pa (A) and 368 Pa (B), descriptions of Videos S1 and S2. This material is available free of charge via the Internet at http://pubs.acs.org.

optical microscopy. Proc. Natl. Acad. Sci. Unit. States. Am. 2010, 107 (46), 19702−19707. (13) Sazaki, G.; Zepeda, S.; Nakatsubo, S.; Yokomine, M.; Furukawa, Y. Quasi-liquid layers on ice crystal surfaces are made up of two different phases. Proc. Natl. Acad. Sci. U.S.A. 2012, 109 (4), 1052− 1055. (14) Sazaki, G.; Asakawa, H.; Nagashima, K.; Nakatsubo, S.; Furukawa, Y. How do quasi-liquid layers emerge from ice crystal surfaces? Cryst. Growth Des. 2013, 13 (4), 1761−1766. (15) Latyshev, A.; Aseev, A.; Krasilnikov, A.; Stenin, S. Transformations on clean Si (111) stepped surface during sublimation. Surf. Sci. 1989, 213 (1), 157−169. (16) Ichikawa, M.; Doi, T. Study of Si (001) 2× 1 domain conversion during direct current and radiative heatings. Appl. Phys. Lett. 1992, 60 (9), 1082−1084. (17) Métois, J. J.; Stoyanov, S. Impact of the growth on the stability− instability transition at Si (111) during step bunching induced by electromigration. Surf. Sci. 1999, 440 (3), 407−419. (18) Yagi, K.; Minoda, H.; Degawa, M. Step bunching, step wandering and faceting: self-organization at Si surfaces. Surf. Sci. Rep. 2001, 43 (2−4), 45−126. (19) Hibino, H.; Hu, C.-W.; Ogino, T.; Tsong, I. Decay kinetics of two-dimensional islands and holes on Si (111) studied by low-energy electron microscopy. Phys. Rev. B 2001, 63 (24), [245402−1]− [245402−8]. (20) Métois, J.-J.; Heyraud, J.-C.; Stoyanov, S. Step flow growth of vicinal (111) Si surface at high temperatures: step kinetics or surface diffusion control. Surf. Sci. 2001, 486 (1), 95−102. (21) Pierre-Louis, O.; Métois, J.-J. Kinetic step pairing. Phys. Rev. Lett. 2004, 93 (16), [165901−1]−[165901−4]. (22) Hibino, H. Studies on formation mechanism and control of step arrangement on Si(111) surface. Waseda University, 2006. (23) Hibino, H.; Kageshima, H.; Uwaha, M. Instability of steps during Ga deposition on Si(111). Surf. Sci. 2008, 602 (14), 2421− 2426. (24) Yamaguchi, H.; Homma, Y. Imaging of layer by layer growth processes during molecular beam epitaxy of GaAs on (111)A substrates by scanning electron microscopy. Appl. Phys. Lett. 1998, 73 (21), 3079−3081. (25) Elbaum, M. Roughening transition observed on the prism facet of ice. Phys. Rev. Lett. 1991, 67 (21), 2982−2985. (26) Fujiwara, T.; Suzuki, Y.; Sazaki, G.; Tamura, K. Solubility measurements of protein crystals under high pressure by observation of steps on crystal surfaces. J. Phys.: Conf. Ser. 2010, 215 (1), 012159− 1−012159−5. (27) Fujiwara, T.; Suzuki, Y.; Takahashi, H. Solubility measurements by in situ observation of the apex region formed by the (110), (110) and (101) faces of tetragonal lysozyme crystals. J. Cryst. Growth 2011, 334 (1), 134−137. (28) Nakaya, U. Snow Crystals: Natural and Artificial; Harvard University Press: Cambridge, 1954. (29) Kobayashi, T. Experimental researches on the snow crystal habit and growth by means of a diffusion cloud chamber. J. Meteorol. Soc. Jpn. 1957, 35, 38−44. (30) Furukawa, Y.; Wettlaufer, J. S. Snow and ice crystals. Phys. Today 2007, 60 (12), 70−71. (31) Berg, W. Crystal growth from solutions. Proc. R. Soc., A 1938, 164 (916), 79−95. (32) Chernov, A. A.; Givargizov, E. I. Modern Crystallography III: Crystal Growth; Springer-Verlag: Berlin, 1984. (33) Markov, I. V. Crystal Growth for Beginners: Fundamentals of Nucleation, Crystal Growth and Epitaxy; World Scientific: Singapore, 2003. (34) Hondo, T. An Overview of Microphysical Processes in Ice Sheets: Toward Nanoglaciology; Institute of Low Temperature Science. Hokkaido University: Sapporo, 2009; Vol. 68, pp 1−23. (35) Dold, P.; Ono, E.; Tsukamoto, K.; Sazaki, G. Step velocity in tetragonal lysozyme growth as a function of impurity concentration

W Web-Enhanced Features *

Videos S1 and S2 in AVI format.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Phone and fax: +81-11706-6880. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors thank Y. Saito and S. Kobayashi (Olympus Engineering Co., Ltd.) for their technical support of LCM− DIM, G. Layton (Northern Arizona University) for the provision of AgI crystals, M. Sato (Kanazawa University), Y. Suzuki (The University of Tokushima), H. Katsuno (Ritsumeikan University), and N. Watanabe (Hokkaido University) for valuable discussion. G.S. is grateful for the partial support by JSPS KAKENHIs (Grant Nos. 23246001 and 24656001).

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ABBREVIATIONS LCM-DIM, laser confocal microscopy combined with differential Interference contrast microscopy; QLL, quasi-liquid layer; UHV, ultrahigh vacuum; E−S effect, Ehrlich−Schwoebel effect

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REFERENCES

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Volume Diffusion in the Growth Kinetics of

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