Growth of Spherical and Cylindrical Oxygen Bubbles at an Ice−Water

CRYSTAL GROWTH & DESIGN

Growth of Spherical and Cylindrical Oxygen Bubbles at an Ice-Water Interface

2008 VOL. 8, NO. 7 2108–2115

Kenji Yoshimura,† Takaaki Inada,*,‡ and Shigeru Koyama§ Fukuoka Industrial Technology Center, 3-6-1 Norimatsu, Yahatanishi-ku, Kitakyushu, Fukuoka 807-0831, Japan, National Institute of AdVanced Industrial Science and Technology, 1-2-1 Namiki, Tsukuba, Ibaraki 305-8564, Japan, and Faculty of Engineering Sciences, Kyushu UniVersity, 6-1 Kasugakoen, Kasuga, Fukuoka 816-8580, Japan ReceiVed March 14, 2007; ReVised Manuscript ReceiVed March 21, 2008

ABSTRACT: Structural features of oxygen gas bubbles incorporated into a growing ice crystal were experimentally and theoretically investigated at various ice growth rates and ambient pressures. Four patterns of the shapes of bubbles incorporated in ice were observed within the experimental conditions used in this study: (a) egg-shaped bubbles, (b) egg-shaped bubbles and cylindrical bubbles, (c) cylindrical bubbles, and (d) bifurcated cylindrical bubbles. These four patterns were mapped out as functions of the ice growth rate and ambient pressure. When cylindrical bubbles were incorporated, the average diameter and interval of the bubbles were measured as functions of the growth rate and ambient pressure. Both the measured diameter and interval decreased with increases in either the growth rate or ambient pressure. In an analytical model developed here, a one-dimensional (1D) approximation in the diffusion boundary layer was used to derive the diameter and interval of the cylindrical bubbles. The analytical results revealed that both the diameter and interval are inversely proportional either to the growth rate or to the square root of the ambient pressure. This analytical model well reproduced the measured diameter and interval. Introduction When a crystal grows from the melt, dissolved gases in the melt are redistributed at the solid-liquid interface. Because, in general, solubility of a gas in a solid phase is less than that in a liquid phase,1,2 the concentrations of the dissolved gases in the melt increase near the advancing interface. Gas bubbles then nucleate in the melt if the concentration surpasses a critical value. These bubbles are either incorporated into the crystal as spherical and cylindrical bubbles or released from the interface, as the crystal grows into the melt.3–9 The incorporation of gas bubbles in crystals is often encountered both in nature and in engineering applications. In nature, bubbles incorporated in rime, hailstones, and lake ice are well-known.3–5 In manufacturing processes, bubble incorporation in metals during casting or welding is a problematic issue, because it generally degrades the mechanical properties of metals.8–12 On the contrary, ordered cylindrical bubbles incorporated into metals are often used to improve the functional properties of the metals, such as impact energy absorption, acoustic energy absorption, air and water permeability, and low thermal conductivity.13–17 Another engineering application of ice crystals incorporating bubbles that contain ozone gas has been recently developed for food preservation and sterilization.18 The behavior of gas bubbles in crystals has therefore been extensively studied, especially by using ice crystals because of easy observation of bubbles in such transparent crystals.3–8,10–12,18–22 Such studies showed that several factors, such as the crystal growth rate and diffusion coefficient of the dissolved gas, affect how bubbles are incorporated into a crystal. Based on several experiments using metals, the ambient pressure is also known to affect the incorporation of bubbles into crystals.13–16 However, only a few studies using ice crystals have reported the effect of ambient pressure,5,20 and the details have not yet been clarified. * To whom correspondence should be addressed. Tel: +81-29-861-7272. Fax: +81-29-851-7523. E-mail: [email protected]. † Fukuoka Industrial Technology Center. ‡ National Institute of Advanced Industrial Science and Technology. § Kyushu University.

Besides experimental studies, several analytical models for the incorporation of bubbles into crystals have been proposed.3,8–10,14–16,22–24 However, most of those models consider only a single bubble or multiple bubbles generated intermittently at the same site on the advancing solid-liquid interface, and therefore do not reveal the interaction of multiple bubbles simultaneously existing at the interface. There are only a few analytical models that consider the interactions of multiple bubbles existing at the solid-liquid interface.14–16 In this study, we therefore theoretically and experimentally investigated the structural features of oxygen gas bubbles incorporated into a growing ice crystal in relation to the crystal growth rate and ambient pressure. Theory Concentration Distribution of Dissolved Gas. First, we introduce the concentration distribution of a dissolved gas during growth of a crystal from the melt, according to existing studies.1,2,11 We consider only a one-component gas. When the crystal has a flat surface, one-dimensional (1D) diffusion of the dissolved gas in the melt is expressed as

∂c ∂c ∂2c )D 2 +V ∂t ∂x ∂x

(1)

where c(x,t) is the molar concentration of the dissolved gas in units of mol/m3, D is the diffusion coefficient of the dissolved gas, V is the growth rate of the crystal, and the advancing solid-liquid interface is taken as the origin. Assuming a uniform initial concentration c0, the initial condition is expressed as

c(x, 0) ) c0

(2)

Assuming a constant concentration c0 at x ) ∞ and considering conservation of the gas concentration at the solid-liquid interface, the boundary conditions are represented as

10.1021/cg070251k CCC: $40.75  2008 American Chemical Society Published on Web 06/06/2008

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c(∞, t) ) c0

(3)

∂c(0, t) -D ) (1 - K)V c(0, t) ∂x

(4)

where K is the distribution coefficient, which is the ratio of the gas concentrations at the interface in the crystal and in the melt, respectively. Assuming K , 1 and no nucleation of gas bubbles in the melt, by using eqs 2, 3and 4, eq 1 is approximately solved as2

{

c 1-K Vx V(1 - K) (x + KVt) )1+ exp - exp c0 K D D

[ ( )

}]

φ)

cn ) ceq + cn0 )

p + cn0 H

(6)

where ceq is the equilibrium concentration of the dissolved gas in the melt, H is the Henry’s law constant in units of Pa · m3/ mol, and cn0 is the critical concentration of the dissolved gas for bubble nucleation when p approaches 0. Incorporation of Bubbles into a Crystal. Whether the bubbles generated at the advancing interface are released from the crystal or incorporated into the crystal is determined by several factors, such as the crystal growth rate, the buoyancy force acted on the bubbles, the interaction forces between the bubbles and the crystal, the difference in thermal conductivity between the melt and the bubbles, and the Marangoni effect at the liquid-gas interface.4,24,26,27 When the bubbles are incorporated into the crystal, they typically appear egg-shaped or elongated cylindrical. In the following analytical model, for simplification, all the bubbles generated in the melts are assumed incorporated into the crystal only as cylindrical. Amount of Incorporated Bubbles in the Steady State. In the steady state, the average concentration of the gas incorporated into the crystal should be equal to the concentration of the dissolved gas in the melt at x ) ∞, regardless of V and p, when compared by unit mass.5 Assuming that c0 is the equilibrium concentration ceq and that the amount of the gas dissolved in the crystal is neglected compared with that incorporated in the crystal as bubbles, namely, K , 1, then the gas concentration in the crystal cs in units of mol/m3 is expressed as

cs )

(1 - φ)Fs Fsp c0 ) (1 - φ) Fl FlH

(9)

where R is the gas constant and T is the temperature. Equation 9 shows that, in steady state, φ is constant under isothermal conditions, regardless of V and p. Shape of Cylindrical Bubbles. Figure 1 shows the arrangement of cylindrical bubbles incorporated into a growing crystal, used in this analytical model. Elongated cylindrical bubbles of diameter d are arranged at an interval w in the cylindrical coordinates (r, x). In this arrangement, φ can be simply expressed using d and w.

(5) Bubble Nucleation. When the concentration of the dissolved gas in the melt near the advancing solid-liquid interface reaches a certain critical value, gas bubbles spontaneously nucleate and grow. Assuming that the gas bubbles are generated by homogeneous nucleation, although they are generated heterogeneously at the solid-liquid interface in most cases,4,5,7,9,13 the critical concentration cn for the nucleation of gas bubbles is a linear function of the ambient pressure p, and is expressed as25

FsRT FlH + FsRT

φ)

πd2 ⁄ 4 π d ) 2 √3w ⁄ 2 2√3 w

2

()

(10)

In the present analytical model, we assume that within the diffusion boundary layer, the concentration c(r) of the dissolved gas in the melt is a function of r, but independent of x; there is gas diffusion only in the r direction in the boundary layer. From eq 5, the thickness of the diffusion boundary layer, or the diffusion length ld, is expressed as

ld )

D V

(11)

Based on this 1D diffusion model, in steady state, the diffusion rate of the dissolved gas in the r direction in the boundary layer is constant within d/2 e r e w/2, which is a control volume in this model. The diffusion rate n˙ in the r direction in the boundary layer per unit length is therefore expressed as

˙ n ) -2πrD

dc dr

(12)

To determine the boundary condition at r ) w/2 in this model, we first consider the actual concentration distribution c(r, x) of the dissolved gas. c(w/2,0) should be equal to the critical concentration cn for the nucleation of gas bubbles so that there is no bubble nucleation within d/2 < r < w/2, and c(w/2,∞) is equal to c0 () ceq). Therefore, the actual concentration distribu-

(7)

where Fs is the density of the bubble-free crystal, Fl is the density of the melt, and φ is the volume fraction of the gas bubbles incorporated in the crystal and is expressed as

φ)

csMg Fg

(8)

where Mg is the molecular weight of the gas, and Fg is the density of the gas. Assuming an ideal gas, by eliminating cs from eqs 7 and 8, φ can be expressed as

Figure 1. Arrangement of cylindrical bubbles incorporated into a crystal in the analytical model.

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tion of the dissolved gas at r ) w/2, schematically drawn as the broken line in Figure 2, is approximately expressed as

( VxD )

c(w ⁄ 2, x) ) ceq + (cn - ceq) exp -

(13)

By integrating eq 13, c in the boundary layer at r ) w/2 in the 1D diffusion model, schematically drawn as the solid line in Figure 2, is given by

c(w ⁄ 2) )

1 ld

∫0l c(w ⁄ 2, x)dx ) ceq + (1 - 1 ⁄ e)cn0 d

(14) The boundary condition at r ) d/2 in the 1D model is expressed as

c(d ⁄ 2) ) ceq

(15)

Integration of eq 12 yields n˙ by using eqs 14 and 15.

2π(1 - 1 ⁄ e)Dcn0 ˙ n)ln(w ⁄ d)

Figure 3. Experimental apparatus to observe oxygen bubbles incorporated into ice.

(16)

The time ∆t for the advancing solid-liquid interface to pass through the diffusion boundary layer is expressed as

∆t )

ld D ) V V2

(17)

Because, in steady state, n˙ given by eq 16 contributes only to the growth of the cylindrical bubble, the amount of dissolved gas supplied to the cylindrical bubble per unit length of the bubble is expressed as

n ) -n ˙∆t )

2π(1 - 1 ⁄ e)D2cn0 V2 ln(w ⁄ d)

(18)

Assuming an ideal gas, based on the arrangement of the cylindrical bubbles shown in Figure 1, φ can be expressed as

φ)

nMg ⁄ Fg

√3w2 ⁄ 2

)

4π(1 - 1 ⁄ e)D2cn0 R T

√3w2pV2 ln(w ⁄ d)

(19)

From eq 9, φ is constant, regardless of d, w, V and p. Therefore, eqs 10 and 19 can be rearranged for d and w as functions of p and V, as follows.

d)

C V√p

w ) 12-1⁄4

√ πφ VC√p

(20) (21)

π 2√3φ

(22)

Experimental Section Figure 3 shows a schematic of the experimental apparatus used to observe oxygen gas bubbles incorporated into ice. A water vessel (30 mm deep, 70 mm wide, and 80 mm high) made of stainless steel was filled with water and oxygen gas. Two side walls (12 mm thick) of the vessel facing each other were made of acrylic resin for observation. In the bottom wall of the vessel, ethanol was circulated as a coolant through a constanttemperature bath, so that water froze starting from the bottom of the vessel. Prior to the experiments, distilled water was aerated with oxygen for 1 h in a glass vessel to remove other dissolved gases from water. Then, 150 mL of the aerated distilled water was transferred to the stainless-steel vessel. Oxygen pressure in the water vessel was controlled at a given pressure by using an oxygen cylinder connected to the top of the vessel, and then the water was left for 20 h to be sufficiently saturated with oxygen at the given pressure at 10 °C. After saturation, the water froze starting from the bottom by circulating ethanol at between -22 and -40 °C in the bottom of the vessel. The growth rate of ice depended on the distance from the bottom of the vessel and also on the ethanol temperature. Oxygen bubbles incorporated in the ice were observed through the acrylic side walls of the vessel at different growth rates of ice, i.e., at different vertical positions, by using a CCD camera (Keyence, VH-8000) and a zoom lens (Keyence, VH-Z25). The observed images revealed several structural features of egg-shaped bubbles and cylindrical bubbles. The average major axis dl and the average minor axis ds for egg-shaped bubbles and the average diameter d for cylindrical bubbles were determined directly from these observed 2D images. Assuming the arrangement of cylindrical bubbles as shown in Figure 1, the average interval w of adjacent cylindrical bubbles can be estimated as

w) Figure 2. Concentration distributions of dissolved gas at r ) w/2, as a function of x. Broken line represents the actual concentration distribution, and solid line represents the concentration distribution in the present 1D diffusion model.

( )

√

C ) 4D (1 - 1 ⁄ e)cn0RT ⁄ ln

( 34 ) √ 1⁄4

(df ⁄ 2)W0 N

(23)

where N is the equivalent number of cylindrical bubbles observed in the image, df is the depth of field in the image (7.0 × 10-4 m for the zoom lens used in this study), and W0 is the

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width of the image (typically 3.4 × 10-3 m for the images used in this study). The number of cylindrical bubbles was counted by applying a correction factor l/L0 for each bubble to evaluate the equivalent number of cylindrical bubbles N, where l is the length of a cylindrical bubble in the observed image and L0 is the length of the image (typically 2.8 × 10-3 m for the images used in this study). Here, dl, ds, d, and w were measured for a range of oxygen pressures p in the vessel between 1 × 105 and 11 × 105 Pa and for a range of ice growth rates V between 5 × 10-6 and 16 × 10-6 m/s. Results Figure 4 shows typical photographs of oxygen bubbles incorporated in ice at different crystal growth rates V and different oxygen pressures p. The shapes of oxygen bubbles were roughly classified into four patterns: (a) egg-shaped bubbles, (b) egg-shaped bubbles and cylindrical bubbles, (c) cylindrical bubbles, and (d) bifurcated cylindrical bubbles. Most bifurcated cylindrical bubbles in Figure 4d started to grow from inside the ice, not from the ice-water interface, whereas all the bubbles in Figures 4a-c nucleate at the ice-water interface and then grew toward inside the ice. Photographs in Figure 5 clearly show that bifurcated cylindrical bubbles started to grow from inside the ice. In patterns (a)-(c) in Figure 4, the ice-water interfaces were relatively flat, whereas in pattern (d), the ice-water interface had irregular roughness as shown in Figure 5. Figure 6 shows the shape patterns of oxygen bubbles as functions of V and p. Here, the bubble shape was classified by the ratio of the major axis to the minor axis of a bubble; if the major axis was smaller than 2.5 times the minor axis, this bubble was classified as egg-shaped and otherwise as cylindrical. At constant p, the egg-shaped bubbles became dominant with increasing V, whereas at constant V, the cylindrical bubbles became dominant with increasing p. The bifurcation of cylindrical bubbles was observed in this study only when V < 7 × 10-6 m/s and p g 6 × 105 Pa. Figure 7 shows the measured volume fractions φ of oxygen gas in ice as a function of V at various p, only when the shape pattern of bubbles was classified as “cylindrical bubbles”. For other shape patterns, estimating φ from the experimental data was difficult. The value of φ was estimated by substituting the measured average diameter d and the average interval w into eq 10, where w was estimated from eq 23. The solid line in Figure 7 was calculated using eq 9. The Henry’s law constant H of oxygen gas at 0 °C (H ) 4.6 × 104 Pa · m3/mol)28 was used in eq 9, because the thermal boundary layer is much larger than the diffusion boundary layer so that the temperature of water just beyond the diffusion boundary layer can be assumed to be 0 °C. Equation 9 yields φ, assuming that all bubbles generated at the ice-water interface are incorporated into ice at steady state; the solid line theoretically represents the maximum φ. Most of the values of φ estimated by using the measured d and w were lower than the theoretical maximum value. Figure 8 shows the average major axis dl and the average minor axis ds for egg-shaped bubbles and the average diameter d for cylindrical bubbles, as a function of V, when p is constant at 1 × 105 Pa. The data plots of dl and ds were adopted when the shape pattern of bubbles was classified as “egg-shaped bubbles” or as “egg-shaped bubbles and cylindrical bubbles”, and those for d were adopted when the shape pattern was classified as “egg-shaped bubbles and cylindrical bubbles” or

Figure 4. Photographs of typical shape patterns of oxygen bubbles incorporated in ice. (a) Egg-shaped bubbles (V ) 15.0 × 10-6 m/s, p ) 2 × 105 Pa). (b) Egg-shaped bubbles and cylindrical bubbles (V ) 10.0 × 10-6 m/s, p ) 1 × 105 Pa). (c) Cylindrical bubbles (V ) 7.5 × 10-6 m/s, p ) 2 × 105 Pa). (d) Bifurcated cylindrical bubbles (V ) 5.6 × 10-6 m/s, p ) 11 × 105 Pa).

as “cylindrical bubbles”. All three diameters decreased with increasing V. At the same V, the diameter increased in the order, d < ds < dl. This tendency held for all values of p in this study.

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Figure 7. Volume fraction φ of oxygen gas incorporated in ice as a function of ice growth rate V at various ambient pressures p. Experimental data φ were estimated using eq 10 with the measured diameters and intervals of cylindrical bubbles. Solid line represents the theoretical value calculated using eq 9.

Figure 5. Magnified photographs of oxygen bubbles incorporated in ice as bifurcated cylindrical bubbles (V ) 5.6 × 10-6 m/s, p ) 11 × 105 Pa), taken at intervals of 5 s. Arrows indicate a bubble grown from inside ice.

Figure 6. Typical patterns of oxygen bubbles incorporated in ice as functions of ice growth rate V and ambient pressure p.

Figure 9 shows d as a function of V at various p, when the shape pattern was classified as “cylindrical bubbles”. The curves in this figure show the fit of the measured d when p is 1 × 105, 2 × 105, 6 × 105, and 11 × 105 Pa by using eq 20 and assuming C ) 2.60 × 10-7. When p was constant, the measured d decreased with increasing V. When V was constant, the measured d decreased with increasing p.

Figure 8. Average major axis dl and average minor axis ds for eggshaped bubbles and average diameter d for cylindrical bubbles, as a function of ice growth rate V, when ambient pressure p is constant at 1 × 105 Pa.

Figure 9. Average diameter d for cylindrical bubbles, as a function of ice growth rate V at various ambient pressures p. Curves are the fits of eq 20 (C ) 2.60 × 10-7) to the measured d.

The same experimental data d as in Figure 9 are plotted as a function of p in Figure 10. The curves in this figure show the fit of the measured d when V is 6 × 10-6, 8 × 10-6, and 10 × 10-6 m/s by using eq 20 and assuming C ) 2.60 × 10-7. The fitted curves based on eq 20 using the same value of C in Figure 9 agree well with the experimental data. Figure 11 shows the average interval w of cylindrical bubbles as a function of V at various p, when the shape pattern was

Growth of Bubbles at Ice-Water Interface

Figure 10. Average diameter d for cylindrical bubbles, as a function of ambient pressure p at various ice growth rates V. Curves are the fits of eq 20 (C ) 2.60 × 10-7) to the measured d.

Figure 11. Average interval w for cylindrical bubbles, as a function of ice growth rate V at various ambient pressures p. Curves are the fits of eq 21 (C ) 2.60 × 10-7) to the measured w.

classified as “cylindrical bubbles”. The curves show the fit of the measured w when p is 1 × 105, 2 × 105, 6 × 105, and 11 × 105 Pa, by using eq 21 and assuming C ) 2.60 × 10-7 and φ ) 0.037, which is the average for the experimental data shown in Figure 7. When V was constant, the measured w decreased with increasing p. The fitted curves based on eq 21 using the same value of C in Figures 9 and 10 agree well with the experimental data. The same experimental data w as in Figure 11 are plotted as a function of p in Figure 12. The curves show the fit of the measured w when V is 6 × 10-6 and 8 × 10-6 m/s, by using eq 21 and assuming C ) 2.60 × 10-7 and φ ) 0.037. The fitted curves based on eq 21 using the same value of C in Figures 9–11 agree well with the experimental data. Discussion The observations of oxygen bubbles incorporated in ice at different V and different p revealed four patterns of bubble shape as shown in Figure 4: (a) egg-shaped bubbles, (b) egg-shaped bubbles and cylindrical bubbles, (c) cylindrical bubbles, and (d) bifurcated cylindrical bubbles. Figure 6 summarizes the changes in the patterns according to V and p. When p was constant, the pattern tended to change from “cylindrical bubbles” to “egg-shaped bubbles” with increasing V. This tendency agrees with previously reported observations.3,5,20 Our study here is the first report of the effect of p on the pattern of the incorporated bubbles in ice. When V was constant, the pattern tended to

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Figure 12. Average interval w for cylindrical bubbles, as a function of ambient pressure p at various ice growth rates V. Curves are the fits of eq 21 (C ) 2.60 × 10-7) to the measured w.

change from “egg-shaped bubbles” to “cylindrical bubbles” with increasing p. In the present analytical model, a bubble incorporated into ice extends infinitely as a cylindrical bubble along the growth direction of ice when the dissolved gas is continuously supplied from water. Actually, however, bubbles are incorporated in ice as egg-shaped bubbles and cylindrical bubbles of finite lengths as shown in Figures 4a-c. The finite length of the bubbles is probably due to the disturbance of the liquid-gas or the solid-liquid interface. Once the disturbance causes a slight decrease in the cap diameter of a bubble located at the solid-liquid interface, the internal pressure of the bubble increases and thus the concentration of the dissolved gas at the bubble surface increases according to Henry’s law. Thus, the diffusion of dissolved gas into the bubble decreases, and eventually the bubble stops extending, thus being completely incorporated in ice.6 As V increases, the decrease in the cap diameter becomes more rapid, and thus the bubble is more apt to be incorporated in ice as an egg-shaped or a cylindrical bubble of finite length. Namely, the pattern of the bubble shape tends to change from “cylindrical bubbles” to “egg-shaped bubbles” with increasing V. Another reason for the trigger of the decrease in the cap diameter of a bubble is unexpected bubble nucleation. If another bubble nucleates nearby at the ice-water interface, the supply of the dissolved gas to the already existing bubble is reduced, and then the already existing bubble stops extending.3–5,20 In the present analysis, the critical concentration for bubble nucleation is given by eq 6, and thus the bubbles incorporated into the solid phase are arranged at regular intervals as shown in Figure 1. Actually, however, unexpected bubble nucleation might frequently happen, thus shortening the length of already existing bubbles at the ice-water interface. Clear ice, which contains no visible bubbles, was not observed in this study, although it is often observed when V is decreased or water is agitated.3,5 In the present experiments, V was too high to produce such clear ice. Most bifurcated cylindrical bubbles started to grow from inside the ice, not from the ice-water interface, as shown in Figure 5. This observation indicates that a bubble nucleates in water remaining in grain boundaries of ice, and then the bubble grows along the complicated grain boundaries. When oxygen bubbles were incorporated as bifurcated cylindrical shape, the ice-water interface always had irregular roughness as shown in Figure 5, suggesting complicated grain boundaries of ice.

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Figure 13. Variation of ice growth rate V with time when coolant temperature is -22 °C. Table 1. Time Constant To Reach Steady State Concentration Profile Compared with Time Required To Decrease Ice Growth Rate W by 10% at Various W V (m/s)a 6 7 8 9

× × × ×

-6

10 10-6 10-6 10-6

t0 (s)b

∆t (s)c

t0/∆t

640 470 360 280

1140 660 410 270

0.56 0.71 0.88 1.04

a Ice growth rate. b Time constant to reach steady state. quired to decrease V by 10%.

c

Time re-

Roughness of the ice-water interface increased with increasing the pressure and with decreasing the growth rate. As shown in Figure 7, the volume fraction φ of cylindrical bubbles incorporated in ice seems to decrease with increasing V, and most of the measured values of φ are smaller than the theoretical value calculated from eq 9. This discrepancy is not due to the bubble escaping from the ice-water interface by buoyancy force, because no bubbles escaping from the ice-water interface were observed in the present experiments. The reason for the discrepancy is that V varied slightly with time in the experiments, although steady state was assumed in the analytical model. Figure 13 shows the variation of V with time in the experiments, when the coolant temperature was -22 °C. All the data in Figure 7 were obtained at the coolant temperature of -22 °C. In Figure 13, V decreased with time due to the increasing thermal resistance of ice. When V decreases, from eq 5, c(x ) 0) would decrease at a position where no bubble exists, compared with that of steady state. Therefore, measured φ would be smaller than the theoretical value of eq 9, until c(x ) 0) is restored to steady state. From eq 5, the time constant t0 to reach steady state is expressed as

t0 )

D KV2

(24)

Table 1 shows t0 at various V compared with the time ∆t, which is required to decrease V by 10% in Figure 13. t0 was estimated assuming K ) 0.048 and D ) 1.1 × 10-6 m2/s, both for oxygen in water at 0 °C.19,29 As V decreases, the ratio of t0 to ∆t decreases, and thus deviation from the steady state assumption will decrease. This tendency corresponds to the results in Figure 7. However, several measured values of φ in Figure 7 are larger than the theoretical value of eq 9, and these data cannot be explained by the failure of the steady state assumption. The data above the theoretical value might reflect some experimental errors.

Comparison of the size between egg-shaped bubbles and cylindrical bubbles shown in Figure 8 indicates that the minor axis ds of egg-shaped bubbles was approximately twice the diameter d of cylindrical bubbles at the same V conditions. Previously reported experimental results showed the same tendency.5 The measured d and w of cylindrical bubbles as functions of V and p can be fitted well with those calculated using eqs 20 and 21 and the same value of the fitting parameter C in eq 22, as shown in Figures 9–12. Although several analytical models for investigating the incorporation of bubbles into a crystal during crystal growth from the melt have been proposed,3,8–10,14–16,22–24 most of these models consider only a single bubble incorporated into the crystal at the solid-liquid interface, assuming uniform diffusion of the dissolved gas from the melt around the single bubble cap. In actual cases, however, multiple bubbles at the solid-liquid interface interact with each other to affect the diffusion of the dissolved gas in the melt, and thus the arrangement of the bubbles in the crystal should be determined by this interaction. There are a few analytical models that consider the interaction of multiple bubbles to solve the 2D concentration distribution of the dissolved gas.14–16 These analytical models can strictly predict the influences of V and p on d and w. However, the solution of the 2D concentration distribution is too complicated to offer physical interpretation of the effects of V and p. In the analytical model proposed in this study, we divided the melt into the diffusion boundary layer and the bulk part, and then approximated the 2D diffusion problem by a 1D problem within the boundary layer by assuming that the concentration of the dissolved gas c in the boundary layer is uniform in the direction perpendicular to the solid-liquid interface. This assumption allows simple solutions to be derived as shown in eqs 20–22 and enables sufficient physical interpretation of the effects of V and p on the shape and arrangement of cylindrical bubbles incorporated into a crystal. In eqs 20–22, the only unknown factor in the experiments is cn0. Equation 22 can be rearranged as

cn0 )

( )

π C2 ln 16(1 - 1 ⁄ e)D2RT 2√3φ

(25)

Substituting C ) 2.60 × 10-7 obtained in Figures 9–12, the diffusion coefficient of oxygen in water D ) 1.1 × 10-9 m2/s at 0 °C,29 and the average measured φ ) 0.037 into eq 25 yields cn0 ) 7.8, and therefore, from eq 6, the critical saturation ratio for bubble nucleation is cn/ceq ) 4.6. The critical saturation ratio cn/ceq can be estimated also from the periodicity length L of the bubble nucleation.3,7,19 Periodic nucleation of gas bubbles is often observed as shown in Figure 4a. Because the concentration of the dissolved gas at the ice-water interface should be ceq just after the incorporation of a layer of egg-shaped bubbles into ice, the time t () L/V) at which the next bubble nucleation occurs at the ice-water interface can be estimated by substituting x ) 0 and c ) cn into eq 5, and the result rearranged for cn/ceq is then expressed as

cn 1 1-K K (1 - K)VL ) exp ceq K K D

(

)

(26)

L was roughly estimated to be 6 × 10-4 m at V ) 25 × 10-6 m/s and p ) 1 × 105 Pa in the present experiments, although correctly estimating L is difficult due to the small number of experimental data classified into the pattern of “egg-shaped

Growth of Bubbles at Ice-Water Interface

bubbles”. Therefore, assuming K ) 0.048 for oxygen in water at 0 °C,19 cn/ceq can be estimated to be 10 from eq 26. This value is on the same order as cn/ceq ) 4.6 obtained from eq 25 based on the analysis proposed in this study, indicating the validity of the present analysis. Experimental data of cn/ceq based on the measurement of L have been previously reported, although it is meaningless to compare the present value of cn/ceq with these data due to the strong dependence of cn/ceq on the impurities in the water. For references, cn/ceq was estimated to be between 12 and 31 by Carte (assuming K ) 0.01),3 about 7 by Geguzin and Dzuba (assuming K ) 0.1),7 and about 20 by Lipp et al. (assuming K ) 0.48).19 Conclusions Oxygen gas bubbles incorporated into a growing ice crystal were experimentally investigated, when the ice has a flat surface in water. Observations of the bubbles revealed four patterns of the shape of bubbles incorporated in ice, within the experimental conditions used in this study: (a) egg-shaped bubbles, (b) eggshaped bubbles and cylindrical bubbles, (c) cylindrical bubbles, and (d) bifurcated cylindrical bubbles. These patterns depended on the ice growth rate V and ambient pressure p. When the bubbles were incorporated as cylindrical bubbles, the average diameter d and interval w of the bubbles were measured as functions of V and p. Both the measured d and measured w decreased with either increasing V or p. The incorporation of cylindrical bubbles into a crystal was also investigated theoretically. In the analytical model, 1D approximation in the diffusion boundary layer was used to derive simple expressions of d and w as functions of V and p. The analytical results showed that both d and w are inversely proportional to V and to the square root of p. This analytical model well reproduced the measured d and w as functions of V and p. Acknowledgment. We thank Tomohiko Hatakeyama for helpful discussions.

References (1) Tiller, W. A.; Jackson, K. A.; Rutter, J. W.; Chalmers, B. Acta Metall. 1953, 1, 428–437.

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