- Email: [email protected]
Topological phase transition in solidification of confined liquids
15 July 2002
Physics Letters A 299 (2002) 622–627 www.elsevier.com/locate/pla
Topological phase transition in solidification of confined liquids Shigeru Ishimoto a,∗ , Toshio Kobayashi b , Kimio Morimoto a , Izumi Nomura c , Akira Ozawa d , Shoji Suzuki a , Yutaka Takahashi e , Isao Tanihata d , Tsuneaki Tsuru a a High Energy Accelerator Research Organization (KEK), Tsukuba, Ibaraki 305-0801, Japan b Faculty of Science, Tohoku University, Sendai, Miyagi 980-8578, Japan c National Institute for Fusion Science, Toki, Gifu 509-5292, Japan d Institute of Physical and Chemical Research, Wako, Saitama 351-0198, Japan e RCNP, Osaka University, Ibaraki, Osaka 567-0047, Japan
Received 4 January 2002; accepted 29 May 2002 Communicated by J. Flouquet
Abstract We found a string-like object evolving in roll and entangled patterns in liquid hydrogen, argon, nitrogen and neon on cooling in the confined geometry. The propagating front was of vapor phase and followed by a solid tubule. The void space produced by the structure corresponds to shrinkage in solidification. 2002 Elsevier Science B.V. All rights reserved. PACS: 05.70.-a; 64.70.Dv; 81.30.Fb Keywords: Solidification; Topological phase transition
In the course of developing a solid hydrogen target for nuclear experiments we observed a topological solidification, in which a string-like object (hereafter called string for brevity) evolved in liquid simultaneously with crystal growth on the sidewall of a small copper cell cooled from the upper side. It made roll and entangled patterns in the liquid. The same phenomenon was observed also in liquid argon, nitrogen and neon [1].1
* Corresponding author.
The main part of the experimental apparatus is shown in Fig. 1. It was two small cells made of pure copper. The upper cylindrical cell was firmly attached to the bottom of the flat liquid helium reservoir of a continuous-flow type cryostat. The lower cell was a hole of 25 mm diameter bored in the center of a 10 mm thick copper plate which was thermally unified with the upper cell. Both faces were sealed with 5 mm thick silica glass windows. A straight hole of 7 mm diameter was drilled inside the flange to connect the cells. (In the initial stage of the experiment, thin Mylar sheets were used for windows and three holes of the
E-mail address: [email protected] (S. Ishimoto). 1 After completing the present manuscript we get information
that similar observations were made long ago. However, we present the manuscript in its original form since our experiment and analysis have been carried out for different materials under the different
point of view without knowing preceding works. See, S. Seki, Kagaku 18 (1948) 41; S. Seki, Kagaku 33 (1963) 424; G.L. Pollack, H.P. Broida, J. Chem. Phys. 38 (1963) 2012.
0375-9601/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 ( 0 2 ) 0 0 7 4 2 - 9
S. Ishimoto et al. / Physics Letters A 299 (2002) 622–627
Fig. 1. The main part of the experimental apparatus. Two thermally unified cells are attached to the bottom of the thin cylindrical liquid helium container of a continuous-flow type cryostat gas was introduced into the upper cell through a thin tube. Evolution of the string was observed in the lower cell through silica glass windows.
same diameter connected the two cells.) A Pt–Co thermometer was mounted in the upper part of the lower cell. The gas pressure of the upper cell was monitored during the experiment. The status of the liquid in the lower cell was digitally recorded by a camera and videotape with a time resolution of 1/30 s. We observed the solidification of normal hydrogen, argon, nitrogen and neon. The gas was fed from an outer vessel through a needle valve. The amount of the gas was so controlled that the liquid level was about in the middle of the upper cell. The gas was cooled down almost along the vapor–liquid line after supplying liquid helium, traversed in the neighborhood of the triple point and solidification in the lower cell started. Mode of solidification in liquid hydrogen changed according to the cooling rate. Continuous bubbling occurred to make accumulated frozen films in the lower cell at the cooling rate of 10−1 K/s. We observed the string mode in the range of 10−2 K/s. Only a transparent crystal grew on the copper sidewall remaining a void in the center bellow 10−3 K/s [1].
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Between these latter two modes a layered gentle bubbling-and-freezing mode, which proceeded from the top to the bottom, was observed. In the present Letter we describe the string mode and other details will be given elsewhere [2]. In the string mode the liquid in the upper cell was first solidified and naturally blocked the connecting hole resulting in confinement of the liquid in the lower cell (hereafter called simply cell). The pressure– temperature relation showed whether the block was complete or partial. The good thermal conductivity of silica glass produced a temperature gradient close to the threshold of string formation. A downward stream was seen during cooling down, but disappeared to leave a density fluctuation along the preexisting stream as approaching the triple point. Slightly after traversing near the triple point a transparent crystal started to grow on the sidewall. Simultaneously the front of the string appeared on the top of the cell and propagated downward. In liquid argon, after reaching the bottom the string went round along the sidewall once to three times, then it moved on a path passing through the side of the existing tail. Fine crystals grew on and around the string behind the front. As the liquid was occupied with the existing tail, the front moved round it becoming entangled. The initial roll in liquid argon is shown in Fig. 2. The leftand right-handed roll appeared with the same probability. The front propagation was stable. The string appeared metallic in a reflected light, but transparent in a transmitted light. In liquid hydrogen the propagation was not so stable as shown in Fig. 3. Only a partial roll was formed. Both the path, sometimes in the horizontal direction, and velocity fluctuated sometimes forming nodes on the string. No difference was observed between the normal and para-hydrogen. The nitrogen string was very stable and showed an enhanced up-and-down folding motion between rolls. In liquid neon, however, the usual string mode was observed only in the beginning of the initial cooling. Afterward an upper layer of the liquid gently bubbled and solidified accompanying a very thin threaded string moving around it. The process repeated from the top to the bottom. The typical radius and propagation velocity of the string were 0.50 ± 0.05 mm and 10.0 ± 0.5 mm/s in hydrogen and 0.70 ± 0.05 mm and 4.0 ± 0.4 mm/s in argon, respectively, at the cooling rate of 2 ×
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Fig. 2. The argon string. The photograph shows the front propagating counterclockwise along the sidewall in the reflected light. Fine crystals around the string are seen in white color due to random reflection. The radius and velocity of the string were 0.65 ± 0.05 mm and 4.2 ± 0.4 mm/s, respectively, at the cooling rate of 2.2 × 10−2 K/s. The growth rates of the transparent crystal on the sidewall and fine crystals were about 3.4 × 10−2 and 0.13 mm/s, respectively. The diameter of the window is 25 mm.
10−2 K/s. The propagation velocity was smaller in nitrogen for the same string diameter. An exceptionally small dp/dT observed in nitrogen indicated the incomplete confinement. The typical rates of crystal growth on the sidewall and around the string were 0.04 ± 0.01 mm/s and 0.1 ± 0.05 mm/s, respectively, in hydrogen. In the steady propagation the radius increased as the velocity decreased, although both sometimes fluctuated or turned back to their initial values. The product of the radius and velocity was found to be approximately constant as far as the propagation was stable (“constant product”). In order to see whether the string propagated along or cooperatively with the convecting stream we reduced the amount of hydrogen gas so that the liquid level was in the middle of the lower cell. (This experiment was done with Mylar windows.) The situation is shown in Fig. 4. The cooling rate was about 10−2 K/s. A 3 mm thick surface layer was immediately solidified upon cooling and confined the liquid bellow it. A thin string appeared beneath the layer. The picture clearly shows that the string moved along the convecting stream. The initial propagation was always downward as if the front moved along the
Fig. 3. The hydrogen string. The photograph was taken in the transmitted light. The string propagated on successive straight paths reflected on the boundary, although it generally showed a partial roll pattern. Fine crystals are black due to random reflection in this case. A not in the initial vertical part showed the clear influence of a local spiral motion of the liquid. Fluctuations of velocity made a structure in the tail. The radius and velocity were 0.53 ± 0.05 mm and 9.3 ± 0.5 mm/s, respectively, at 2.1 × 10−2 K/s. The growth rate of the transparent crystal and fine crystals were about 3.9 × 10−2 and 0.09 mm/s, respectively.
downward remnant stream. The propagation along the sidewall is similar to the roll in the Rayleigh–Bérnard convection. Another example is seen in Fig. 3 as a small spiral pattern in the initial vertical part of the string. However, we also observed an inclined straight path sometimes in liquid hydrogen, which showed a deviation from the fluid motion. In some cases the string melted on its way. Then, the second string started somewhere in the upper side of the cell. Only one string propagated except an occasion described bellow. No bifurcation was observed. The string moved in liquid without colliding, intersecting or sticking to the existing tail or any other objects on the way. It reflected on the boundaries such as the surface of the crystal or windows. A careful observation clarified that the front was bent or curved as if it were a slender balloon, when it reflected on the boundaries or moved round the existing tail. When the propagation became sufficiently slow, successive bubble nucleation was observed at the front in liquid hydrogen. It is the origin of nodes mentioned above. When the front stopped almost completely, a single bubble
S. Ishimoto et al. / Physics Letters A 299 (2002) 622–627
Fig. 4. The string evolution in liquid hydrogen in the case of the free liquid surface at the cooling rate about 10−2 K/s. After the immediate solidification of the surface layer, a thin string appeared beneath the undersurface of the layer on the right side but stopped near the bottom. Then, the second string with the radius about 0.5 mm started and moved along the sidewall to the left side. It crept under the solidified layer to the center, where it abruptly changed the path toward the bottom, and again moved along the sidewall. The picture clearly shows that the string moved along the convecting stream.
formed. It melted or, in some cases, from it the string started again. However, the most striking observation was that a few elongated bubbles were successively moving up inside the argon string when it melted on the top as shown in Fig. 5. It proved that the string was a solid tubule. Thus, the string consisted of a propagating vapor-phase front followed by the solid tubule. The front and its succeeding part were very smooth. It was not clear whether the parts were an elongated bubble or covered with a soft solid film. The termination of the string was plausibly on the surface of the transparent crystal grown on the sidewall. The resultant structure was stable at low temperatures, but about a half of fine crystals around the string was annealed and unified with the transparent crystal before melting after stopping liquid helium supply. Once on the melting process the surface layer solidified again and two thin strings propagated symmetrically in the melt showing the second mode of convection. Shrinkage in solidification in the confined geometry requires creation of a void space with an average rate of αV0 /τ , where α, V0 and τ are the shrink-
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Fig. 5. Bubbles moving inside the melting argon string. The bubbles partitioned by the leaked liquid were moving upward inside the string, of which front melted on the top of the cell. The picture clearly shows that the string was a tubule filled with vapor. Bubbles moving inside the melting string were also observed in the hydrogen string.
age factor, cell volume and solidification time, respectively. Since the vapor density and the wall thickness of the string are negligible, it must correspond to the average tubular volume produced per unit time πb2 v, where b and v are the radius and propagating velocity, respectively. The experimental value was πb 2 v/(αV0 /τ ) = 1.0 ± 0.15 in hydrogen and argon in the complete confinement. The tubular volume was just the void space required for solidification. Thus, the evolution of the string will be a general phenomenon in liquid, which shrinks in solidification, under confinement. Cooling from the upper side produced convection. Also solidification in the cell induced fluid motion. First it modified the temperature gradient by releasing the latent heat. However, the crystal growth was so uniform on the sidewall that the global temperature gradient was estimated to be |dT /dx| < 1.4 × 10−2 K/cm in liquid hydrogen. It was also estimated from the thermal diffusion in the copper plate, of which heat conductivity is very large at 14 K, to be in the region of 10−2 –10−3 K/cm depending on the experimental condition. Second it produced a fluid flow toward the boundary due to the density difference between solid and liquid. It was so small that the induced
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flow was negligible compared with that induced by the temperature gradient. However, it was important under the confined geometry because it would produce the pressure drop or negative pressure inside the liquid resulting in a metastable state [3]. The string formation was a continuous process of vaporization-and-solidification at the front region. Since supercooling was not so strong, vaporization due to the heat concentration could be hardly expected. The heat diffusion time was long (exceeding 1 second for the path of 0.5 mm) and the temperature gradient was small. The hydrodynamic effect required a velocity an order larger than the observed one. The tubular structure indicated that vaporization and solidification took place in the center and peripheral of the string, respectively. If the negative pressure induced by shrinkage was localized in the convecting stream, it caused vaporization in the central part of the stream, which so decreased the peripheral temperature to form a thin cylindrical solid wall. So, the evolution of the string would be the propagating decay of the metastable state along the convecting stream. If the rate of vaporization was too large, bubble formation rate exceeded the guidance of the stream and bubbles developed in any direction. On the contrary, if the convection was too weak, the gentle bubbling took place in the cooled layer. The string mode was only observed in rather weak supercooling state where the fluid flow could control the evolution of the phase separation. The aspect ratio of the cell, defined by the ratio of the horizontal to vertical scale, is 1 and small enough to characterize the fluid velocity by a parameter 1/2 a, vB = βg|dT /dx| (1)
thermal processes. The energetic balance between vaporization and solidification gives a thickness of the solid wall as thin as 5 µm. No modulation on the periodic motion such as the roll indicates that the front approximately commoved with the convecting stream in the complete confinement. The front pushed out the liquid that had occupied the volume before the propagation. In the steady state kinetic energy of the liquid was dissipated by the viscous friction. The situation is equivalent with that of a moving body in the liquid. If the front is bubble-like, it will be proportional to the propagating velocity and given by cπηbv, where η and c are the viscosity and proportional constant, respectively [5]. Then, we get the constant product as
where β, g, dT /dx and a are the volume expansion coefficient of the liquid, gravitational acceleration, temperature gradient and radius of the cell [4]. If we assume dT /dx = 1 × 10−2 K/cm, vB = 4.1 mm/s for hydrogen, which is nearly the observed velocity. Precise analysis of string dynamics will be quite difficult because convection is coupled with the phase separation. However, free energies involving in thermal processes are orders of magnitudes larger than the kinetic energy of convection. Even a slight imbalance would cause a drastic change in the motion of the string. Since we did not observe such a drastic change, we could assume that the propagation was governed by fluid dynamics at the front and not by the detail of
where, κ is the thermal diffusivity and Pr is the Prandtl number defined by Pr = ν/κ. From (2) and (3) we obtain the propagation velocity
bv = 2cν,
(2)
where ν is the kinetic viscosity defined by ν = η/ρ, ρ being the density of the liquid. The relation is in agreement with the experimental result obtained in the complete confinement. We get the constant c = 7.75 ± 0.50, which is compared with c = 12 for a spherical bubble. The relation means a constant Reynolds number, which is defined by Re = bv/ν, and really observed to be about 15. The constant Reynolds number may be attributed to the limited condition, under which the confinement was complete. When the confinement was incomplete, the constant product was still held but with different values. The viscous force gave a reaction to the liquid layer which moved with the front. Taking a balance with the buoyancy force, we get an estimation of the radius b ≈ 2κ(c Pr)1/2 vB , (3)
v ≈ (c Pr)1/2 vB .
(4)
This simple model gives correctly the front velocity in terms of the characteristic velocity. However, it should be noted that the above derivation yields only a rough estimation and also these formulae include vB or |dT /dx| as an essential parameter. Since the exact temperature gradient was unknown in our case, we estimate it by replacing b in (3) by the experimental value bexp . For the typical radius of the hydrogen string bexp = 0.5 mm we get |dT /dx| ≈
S. Ishimoto et al. / Physics Letters A 299 (2002) 622–627
4.6 × 10−3 K/cm in agreement with the previous estimation. In the case of argon the temperature gradient is slightly smaller. The propagation velocity calculated by (4) gives automatically the value in agreement with the experiment. The propagation velocity of the roll was always about 70% of the expected value. The reduction is due to the additional friction between the string and solid boundaries. We can consider the Rayleigh number, defined by Ra = βg|dT /dx|a 4 /(κν), as an effective one in our case. Using the string radius, we get Ra ≈ 4c(a/b)2.
(5)
The maximum radius is bmax ≈ a/(c Pr), which gives the critical Rayleigh number as Rac ≈ 4c(c Pr)2 . The estimated and observed maximum radius of the hydrogen string are bmax = 1.0 and 1.3 mm, respectively. The latter value gives Rac,exp ≈ 2900 for hydrogen, while Rac,exp ≈ 6000 for argon. The critical Rayleigh number depends on liquid because it is also related with the phase separation. Propagation of the string is, on one hand, a process giving the rigidity to the pattern produced in the dissipative system and, on the other hand, evolution of a chaotic pattern from the regular one by continuous formation of new solid boundaries. The front and tail of the string confine the symmetric phase (vapor) inside the symmetry-breaking phase (solid wall). Shrinkage in solidification induced the decay of the metastable liquid phase, in which the topological pattern was prepared, and produced such a structure. As the string is the frozen relic of the convecting stream that existed in the liquid generally with circulation around the string, it is plausible to stored the memory in the form of chirality. The tubular structure stands for this conjecture. Indeed the fluid motion has been found to break the chiral symmetry of a crystal [6]. The original wall made in the phase separation might be of an unconventional solid [7]. If the tubular solid has a definite chirality, the present string is thought to be a model of vortex lines observed in certain superconductors or superfluid helium. Its evolution associated with annealing of point defects produced around it is suggestive of the first order phase transitions under weak supercooling that might occur in the early universe [8].
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In summary, we found the topological phase transition with the string evolution in confined liquids, which will be the general phenomenon where shrinkage takes place in solidification. We clarified that the front and the following tail confine the symmetric phase (vapor) inside the symmetry-breaking phase (solid) with the possible chirality. The structure and evolution of such a string will give some insights into the microscopic line defects and the first order phase transitions in the early universe.
Acknowledgements We would like to thank Professors I. Imai, T. Nishikawa, H. Kuroda and Dr. H. Tokieda for their suggestions and warm encouragement. We are indebted to Professors T. Momose and Nakajima for information on crystal growth of hydrogen. One of the authors (K.M.) is grateful to Professor H. Ishimoto for helpful discussion and Mr. T. Morimoto for cooperation. We also thank Director S. Yamada and Professors M. Kobayashi and K. Nakamura for their support.
References [1] S. Ishimoto et al., KEK Preprint 2000-42/RIKEN-AF-NP-354, October, 2000; S. Ishimoto et al., Nucl. Instrum. Methods, in press. [2] S. Ishimoto et al., in preparation. [3] R. Kubo, Problems and Solutions in Thermodynamics and Statistical Mechanics, Shokabo Publishing Company, 1961, Chapter 1 (in Japanese); L.D. Landau, E.M. Lifshitz, Statistical Physics, 3rd edn., Pergamon Press, Oxford, 1980, Part 1. [4] C. Normand, Y. Pomeau, M.G. Velarde, Rev. Mod. Phys. 49 (1977) 581; S. Ostrach, Annu. Rev. Fluid Mech. 14 (1982) 313. [5] L.D. Landau, E.M. Lifshitz, Fluid Mechanics, 3rd edn., Pergamon Press, Oxford, 1966. [6] T. Buhse et al., Phys. Rev. Lett. 84 (2000) 4405. [7] G. Venkataraman, D. Sahoo, V. Balakrishnan, Beyond the Crystalline State, Springer, Berlin, 1989. [8] A. Vilenkin, E.P.S. Shellard, Cosmic String and Other Topological Defects, Cambridge University Press, 1994, Chapter 2.
Physics Letters A 299 (2002) 622–627 www.elsevier.com/locate/pla
Topological phase transition in solidification of confined liquids Shigeru Ishimoto a,∗ , Toshio Kobayashi b , Kimio Morimoto a , Izumi Nomura c , Akira Ozawa d , Shoji Suzuki a , Yutaka Takahashi e , Isao Tanihata d , Tsuneaki Tsuru a a High Energy Accelerator Research Organization (KEK), Tsukuba, Ibaraki 305-0801, Japan b Faculty of Science, Tohoku University, Sendai, Miyagi 980-8578, Japan c National Institute for Fusion Science, Toki, Gifu 509-5292, Japan d Institute of Physical and Chemical Research, Wako, Saitama 351-0198, Japan e RCNP, Osaka University, Ibaraki, Osaka 567-0047, Japan
Received 4 January 2002; accepted 29 May 2002 Communicated by J. Flouquet
Abstract We found a string-like object evolving in roll and entangled patterns in liquid hydrogen, argon, nitrogen and neon on cooling in the confined geometry. The propagating front was of vapor phase and followed by a solid tubule. The void space produced by the structure corresponds to shrinkage in solidification. 2002 Elsevier Science B.V. All rights reserved. PACS: 05.70.-a; 64.70.Dv; 81.30.Fb Keywords: Solidification; Topological phase transition
In the course of developing a solid hydrogen target for nuclear experiments we observed a topological solidification, in which a string-like object (hereafter called string for brevity) evolved in liquid simultaneously with crystal growth on the sidewall of a small copper cell cooled from the upper side. It made roll and entangled patterns in the liquid. The same phenomenon was observed also in liquid argon, nitrogen and neon [1].1
* Corresponding author.
The main part of the experimental apparatus is shown in Fig. 1. It was two small cells made of pure copper. The upper cylindrical cell was firmly attached to the bottom of the flat liquid helium reservoir of a continuous-flow type cryostat. The lower cell was a hole of 25 mm diameter bored in the center of a 10 mm thick copper plate which was thermally unified with the upper cell. Both faces were sealed with 5 mm thick silica glass windows. A straight hole of 7 mm diameter was drilled inside the flange to connect the cells. (In the initial stage of the experiment, thin Mylar sheets were used for windows and three holes of the
E-mail address: [email protected] (S. Ishimoto). 1 After completing the present manuscript we get information
that similar observations were made long ago. However, we present the manuscript in its original form since our experiment and analysis have been carried out for different materials under the different
point of view without knowing preceding works. See, S. Seki, Kagaku 18 (1948) 41; S. Seki, Kagaku 33 (1963) 424; G.L. Pollack, H.P. Broida, J. Chem. Phys. 38 (1963) 2012.
0375-9601/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 ( 0 2 ) 0 0 7 4 2 - 9
S. Ishimoto et al. / Physics Letters A 299 (2002) 622–627
Fig. 1. The main part of the experimental apparatus. Two thermally unified cells are attached to the bottom of the thin cylindrical liquid helium container of a continuous-flow type cryostat gas was introduced into the upper cell through a thin tube. Evolution of the string was observed in the lower cell through silica glass windows.
same diameter connected the two cells.) A Pt–Co thermometer was mounted in the upper part of the lower cell. The gas pressure of the upper cell was monitored during the experiment. The status of the liquid in the lower cell was digitally recorded by a camera and videotape with a time resolution of 1/30 s. We observed the solidification of normal hydrogen, argon, nitrogen and neon. The gas was fed from an outer vessel through a needle valve. The amount of the gas was so controlled that the liquid level was about in the middle of the upper cell. The gas was cooled down almost along the vapor–liquid line after supplying liquid helium, traversed in the neighborhood of the triple point and solidification in the lower cell started. Mode of solidification in liquid hydrogen changed according to the cooling rate. Continuous bubbling occurred to make accumulated frozen films in the lower cell at the cooling rate of 10−1 K/s. We observed the string mode in the range of 10−2 K/s. Only a transparent crystal grew on the copper sidewall remaining a void in the center bellow 10−3 K/s [1].
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Between these latter two modes a layered gentle bubbling-and-freezing mode, which proceeded from the top to the bottom, was observed. In the present Letter we describe the string mode and other details will be given elsewhere [2]. In the string mode the liquid in the upper cell was first solidified and naturally blocked the connecting hole resulting in confinement of the liquid in the lower cell (hereafter called simply cell). The pressure– temperature relation showed whether the block was complete or partial. The good thermal conductivity of silica glass produced a temperature gradient close to the threshold of string formation. A downward stream was seen during cooling down, but disappeared to leave a density fluctuation along the preexisting stream as approaching the triple point. Slightly after traversing near the triple point a transparent crystal started to grow on the sidewall. Simultaneously the front of the string appeared on the top of the cell and propagated downward. In liquid argon, after reaching the bottom the string went round along the sidewall once to three times, then it moved on a path passing through the side of the existing tail. Fine crystals grew on and around the string behind the front. As the liquid was occupied with the existing tail, the front moved round it becoming entangled. The initial roll in liquid argon is shown in Fig. 2. The leftand right-handed roll appeared with the same probability. The front propagation was stable. The string appeared metallic in a reflected light, but transparent in a transmitted light. In liquid hydrogen the propagation was not so stable as shown in Fig. 3. Only a partial roll was formed. Both the path, sometimes in the horizontal direction, and velocity fluctuated sometimes forming nodes on the string. No difference was observed between the normal and para-hydrogen. The nitrogen string was very stable and showed an enhanced up-and-down folding motion between rolls. In liquid neon, however, the usual string mode was observed only in the beginning of the initial cooling. Afterward an upper layer of the liquid gently bubbled and solidified accompanying a very thin threaded string moving around it. The process repeated from the top to the bottom. The typical radius and propagation velocity of the string were 0.50 ± 0.05 mm and 10.0 ± 0.5 mm/s in hydrogen and 0.70 ± 0.05 mm and 4.0 ± 0.4 mm/s in argon, respectively, at the cooling rate of 2 ×
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Fig. 2. The argon string. The photograph shows the front propagating counterclockwise along the sidewall in the reflected light. Fine crystals around the string are seen in white color due to random reflection. The radius and velocity of the string were 0.65 ± 0.05 mm and 4.2 ± 0.4 mm/s, respectively, at the cooling rate of 2.2 × 10−2 K/s. The growth rates of the transparent crystal on the sidewall and fine crystals were about 3.4 × 10−2 and 0.13 mm/s, respectively. The diameter of the window is 25 mm.
10−2 K/s. The propagation velocity was smaller in nitrogen for the same string diameter. An exceptionally small dp/dT observed in nitrogen indicated the incomplete confinement. The typical rates of crystal growth on the sidewall and around the string were 0.04 ± 0.01 mm/s and 0.1 ± 0.05 mm/s, respectively, in hydrogen. In the steady propagation the radius increased as the velocity decreased, although both sometimes fluctuated or turned back to their initial values. The product of the radius and velocity was found to be approximately constant as far as the propagation was stable (“constant product”). In order to see whether the string propagated along or cooperatively with the convecting stream we reduced the amount of hydrogen gas so that the liquid level was in the middle of the lower cell. (This experiment was done with Mylar windows.) The situation is shown in Fig. 4. The cooling rate was about 10−2 K/s. A 3 mm thick surface layer was immediately solidified upon cooling and confined the liquid bellow it. A thin string appeared beneath the layer. The picture clearly shows that the string moved along the convecting stream. The initial propagation was always downward as if the front moved along the
Fig. 3. The hydrogen string. The photograph was taken in the transmitted light. The string propagated on successive straight paths reflected on the boundary, although it generally showed a partial roll pattern. Fine crystals are black due to random reflection in this case. A not in the initial vertical part showed the clear influence of a local spiral motion of the liquid. Fluctuations of velocity made a structure in the tail. The radius and velocity were 0.53 ± 0.05 mm and 9.3 ± 0.5 mm/s, respectively, at 2.1 × 10−2 K/s. The growth rate of the transparent crystal and fine crystals were about 3.9 × 10−2 and 0.09 mm/s, respectively.
downward remnant stream. The propagation along the sidewall is similar to the roll in the Rayleigh–Bérnard convection. Another example is seen in Fig. 3 as a small spiral pattern in the initial vertical part of the string. However, we also observed an inclined straight path sometimes in liquid hydrogen, which showed a deviation from the fluid motion. In some cases the string melted on its way. Then, the second string started somewhere in the upper side of the cell. Only one string propagated except an occasion described bellow. No bifurcation was observed. The string moved in liquid without colliding, intersecting or sticking to the existing tail or any other objects on the way. It reflected on the boundaries such as the surface of the crystal or windows. A careful observation clarified that the front was bent or curved as if it were a slender balloon, when it reflected on the boundaries or moved round the existing tail. When the propagation became sufficiently slow, successive bubble nucleation was observed at the front in liquid hydrogen. It is the origin of nodes mentioned above. When the front stopped almost completely, a single bubble
S. Ishimoto et al. / Physics Letters A 299 (2002) 622–627
Fig. 4. The string evolution in liquid hydrogen in the case of the free liquid surface at the cooling rate about 10−2 K/s. After the immediate solidification of the surface layer, a thin string appeared beneath the undersurface of the layer on the right side but stopped near the bottom. Then, the second string with the radius about 0.5 mm started and moved along the sidewall to the left side. It crept under the solidified layer to the center, where it abruptly changed the path toward the bottom, and again moved along the sidewall. The picture clearly shows that the string moved along the convecting stream.
formed. It melted or, in some cases, from it the string started again. However, the most striking observation was that a few elongated bubbles were successively moving up inside the argon string when it melted on the top as shown in Fig. 5. It proved that the string was a solid tubule. Thus, the string consisted of a propagating vapor-phase front followed by the solid tubule. The front and its succeeding part were very smooth. It was not clear whether the parts were an elongated bubble or covered with a soft solid film. The termination of the string was plausibly on the surface of the transparent crystal grown on the sidewall. The resultant structure was stable at low temperatures, but about a half of fine crystals around the string was annealed and unified with the transparent crystal before melting after stopping liquid helium supply. Once on the melting process the surface layer solidified again and two thin strings propagated symmetrically in the melt showing the second mode of convection. Shrinkage in solidification in the confined geometry requires creation of a void space with an average rate of αV0 /τ , where α, V0 and τ are the shrink-
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Fig. 5. Bubbles moving inside the melting argon string. The bubbles partitioned by the leaked liquid were moving upward inside the string, of which front melted on the top of the cell. The picture clearly shows that the string was a tubule filled with vapor. Bubbles moving inside the melting string were also observed in the hydrogen string.
age factor, cell volume and solidification time, respectively. Since the vapor density and the wall thickness of the string are negligible, it must correspond to the average tubular volume produced per unit time πb2 v, where b and v are the radius and propagating velocity, respectively. The experimental value was πb 2 v/(αV0 /τ ) = 1.0 ± 0.15 in hydrogen and argon in the complete confinement. The tubular volume was just the void space required for solidification. Thus, the evolution of the string will be a general phenomenon in liquid, which shrinks in solidification, under confinement. Cooling from the upper side produced convection. Also solidification in the cell induced fluid motion. First it modified the temperature gradient by releasing the latent heat. However, the crystal growth was so uniform on the sidewall that the global temperature gradient was estimated to be |dT /dx| < 1.4 × 10−2 K/cm in liquid hydrogen. It was also estimated from the thermal diffusion in the copper plate, of which heat conductivity is very large at 14 K, to be in the region of 10−2 –10−3 K/cm depending on the experimental condition. Second it produced a fluid flow toward the boundary due to the density difference between solid and liquid. It was so small that the induced
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flow was negligible compared with that induced by the temperature gradient. However, it was important under the confined geometry because it would produce the pressure drop or negative pressure inside the liquid resulting in a metastable state [3]. The string formation was a continuous process of vaporization-and-solidification at the front region. Since supercooling was not so strong, vaporization due to the heat concentration could be hardly expected. The heat diffusion time was long (exceeding 1 second for the path of 0.5 mm) and the temperature gradient was small. The hydrodynamic effect required a velocity an order larger than the observed one. The tubular structure indicated that vaporization and solidification took place in the center and peripheral of the string, respectively. If the negative pressure induced by shrinkage was localized in the convecting stream, it caused vaporization in the central part of the stream, which so decreased the peripheral temperature to form a thin cylindrical solid wall. So, the evolution of the string would be the propagating decay of the metastable state along the convecting stream. If the rate of vaporization was too large, bubble formation rate exceeded the guidance of the stream and bubbles developed in any direction. On the contrary, if the convection was too weak, the gentle bubbling took place in the cooled layer. The string mode was only observed in rather weak supercooling state where the fluid flow could control the evolution of the phase separation. The aspect ratio of the cell, defined by the ratio of the horizontal to vertical scale, is 1 and small enough to characterize the fluid velocity by a parameter 1/2 a, vB = βg|dT /dx| (1)
thermal processes. The energetic balance between vaporization and solidification gives a thickness of the solid wall as thin as 5 µm. No modulation on the periodic motion such as the roll indicates that the front approximately commoved with the convecting stream in the complete confinement. The front pushed out the liquid that had occupied the volume before the propagation. In the steady state kinetic energy of the liquid was dissipated by the viscous friction. The situation is equivalent with that of a moving body in the liquid. If the front is bubble-like, it will be proportional to the propagating velocity and given by cπηbv, where η and c are the viscosity and proportional constant, respectively [5]. Then, we get the constant product as
where β, g, dT /dx and a are the volume expansion coefficient of the liquid, gravitational acceleration, temperature gradient and radius of the cell [4]. If we assume dT /dx = 1 × 10−2 K/cm, vB = 4.1 mm/s for hydrogen, which is nearly the observed velocity. Precise analysis of string dynamics will be quite difficult because convection is coupled with the phase separation. However, free energies involving in thermal processes are orders of magnitudes larger than the kinetic energy of convection. Even a slight imbalance would cause a drastic change in the motion of the string. Since we did not observe such a drastic change, we could assume that the propagation was governed by fluid dynamics at the front and not by the detail of
where, κ is the thermal diffusivity and Pr is the Prandtl number defined by Pr = ν/κ. From (2) and (3) we obtain the propagation velocity
bv = 2cν,
(2)
where ν is the kinetic viscosity defined by ν = η/ρ, ρ being the density of the liquid. The relation is in agreement with the experimental result obtained in the complete confinement. We get the constant c = 7.75 ± 0.50, which is compared with c = 12 for a spherical bubble. The relation means a constant Reynolds number, which is defined by Re = bv/ν, and really observed to be about 15. The constant Reynolds number may be attributed to the limited condition, under which the confinement was complete. When the confinement was incomplete, the constant product was still held but with different values. The viscous force gave a reaction to the liquid layer which moved with the front. Taking a balance with the buoyancy force, we get an estimation of the radius b ≈ 2κ(c Pr)1/2 vB , (3)
v ≈ (c Pr)1/2 vB .
(4)
This simple model gives correctly the front velocity in terms of the characteristic velocity. However, it should be noted that the above derivation yields only a rough estimation and also these formulae include vB or |dT /dx| as an essential parameter. Since the exact temperature gradient was unknown in our case, we estimate it by replacing b in (3) by the experimental value bexp . For the typical radius of the hydrogen string bexp = 0.5 mm we get |dT /dx| ≈
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4.6 × 10−3 K/cm in agreement with the previous estimation. In the case of argon the temperature gradient is slightly smaller. The propagation velocity calculated by (4) gives automatically the value in agreement with the experiment. The propagation velocity of the roll was always about 70% of the expected value. The reduction is due to the additional friction between the string and solid boundaries. We can consider the Rayleigh number, defined by Ra = βg|dT /dx|a 4 /(κν), as an effective one in our case. Using the string radius, we get Ra ≈ 4c(a/b)2.
(5)
The maximum radius is bmax ≈ a/(c Pr), which gives the critical Rayleigh number as Rac ≈ 4c(c Pr)2 . The estimated and observed maximum radius of the hydrogen string are bmax = 1.0 and 1.3 mm, respectively. The latter value gives Rac,exp ≈ 2900 for hydrogen, while Rac,exp ≈ 6000 for argon. The critical Rayleigh number depends on liquid because it is also related with the phase separation. Propagation of the string is, on one hand, a process giving the rigidity to the pattern produced in the dissipative system and, on the other hand, evolution of a chaotic pattern from the regular one by continuous formation of new solid boundaries. The front and tail of the string confine the symmetric phase (vapor) inside the symmetry-breaking phase (solid wall). Shrinkage in solidification induced the decay of the metastable liquid phase, in which the topological pattern was prepared, and produced such a structure. As the string is the frozen relic of the convecting stream that existed in the liquid generally with circulation around the string, it is plausible to stored the memory in the form of chirality. The tubular structure stands for this conjecture. Indeed the fluid motion has been found to break the chiral symmetry of a crystal [6]. The original wall made in the phase separation might be of an unconventional solid [7]. If the tubular solid has a definite chirality, the present string is thought to be a model of vortex lines observed in certain superconductors or superfluid helium. Its evolution associated with annealing of point defects produced around it is suggestive of the first order phase transitions under weak supercooling that might occur in the early universe [8].
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In summary, we found the topological phase transition with the string evolution in confined liquids, which will be the general phenomenon where shrinkage takes place in solidification. We clarified that the front and the following tail confine the symmetric phase (vapor) inside the symmetry-breaking phase (solid) with the possible chirality. The structure and evolution of such a string will give some insights into the microscopic line defects and the first order phase transitions in the early universe.
Acknowledgements We would like to thank Professors I. Imai, T. Nishikawa, H. Kuroda and Dr. H. Tokieda for their suggestions and warm encouragement. We are indebted to Professors T. Momose and Nakajima for information on crystal growth of hydrogen. One of the authors (K.M.) is grateful to Professor H. Ishimoto for helpful discussion and Mr. T. Morimoto for cooperation. We also thank Director S. Yamada and Professors M. Kobayashi and K. Nakamura for their support.
References [1] S. Ishimoto et al., KEK Preprint 2000-42/RIKEN-AF-NP-354, October, 2000; S. Ishimoto et al., Nucl. Instrum. Methods, in press. [2] S. Ishimoto et al., in preparation. [3] R. Kubo, Problems and Solutions in Thermodynamics and Statistical Mechanics, Shokabo Publishing Company, 1961, Chapter 1 (in Japanese); L.D. Landau, E.M. Lifshitz, Statistical Physics, 3rd edn., Pergamon Press, Oxford, 1980, Part 1. [4] C. Normand, Y. Pomeau, M.G. Velarde, Rev. Mod. Phys. 49 (1977) 581; S. Ostrach, Annu. Rev. Fluid Mech. 14 (1982) 313. [5] L.D. Landau, E.M. Lifshitz, Fluid Mechanics, 3rd edn., Pergamon Press, Oxford, 1966. [6] T. Buhse et al., Phys. Rev. Lett. 84 (2000) 4405. [7] G. Venkataraman, D. Sahoo, V. Balakrishnan, Beyond the Crystalline State, Springer, Berlin, 1989. [8] A. Vilenkin, E.P.S. Shellard, Cosmic String and Other Topological Defects, Cambridge University Press, 1994, Chapter 2.