- Email: [email protected]
Thermal aspects of particle engulfment by a solidifying front
j. . . . . . . F C R Y S T A L GROWTH
Journal of Crystal Growth 128 (1993) 1130-1136 North-Holland
Thermal aspects of particle engulfment by a solidifying front M. Y e m m o u , M . A . A z o u n i , P. C a s s e s Laboratoire d'Adrothermique du CNRS, 4ter Route des Gardes, F-92190 Meudon, France
and G. P 4 t r 6 Chemical Physics, EP Department, CP 165, Universitd Libre de Bruxelles, 50 Avenue F.D. Roosevelt, B-1050 Brussels, Belgium
The solidification front under pushing conditions as well as the changes in shape that take place during the engulfment process are investigated for polyamide and steel spheres in an ice-water system. The examination of all subsequent stages of trapping make evident that thermal effects are of prime importance in the interface shape, when large spheres (size of the order of some millimeters) are considered.
1. Introduction The engulfment or rejection of solid particles by advancing solidification is important in metallurgy, materials science, cryobiology, cryoconcentration, soil mechanics and many other processes. It is either necessary to purify the solid phase and keep it free of inclusions or to achieve a controlled incorporation of particles yielding better mechanical or electrical properties of the solid matrix e.g composites. It has been known that the particle pushing occurs only below a certain velocity of the advancing solidification front called the critical velocity. If the velocity of the growth front is greater than the critical velocity, the particle is encapsulated by the front. All the theories of evaluation of the critical velocities are based on the concept of molecular surface forces repelling the particle from the growth front and thus facilitating the entry of the liquid into the gap between the growing solid and the particle. On the other hand, the interaction of the interface requires that there must be some force which brings the interface and the particle into contact, and some force preventing easy dis-
engagement [1-3]. A concise survey was presented by K6rber et al. [4]. From the experimental evidence, the rate of the advance of the freezing interface, the physical properties of the particle and the matrix together with the particle size and its surface state are decisive for the particle behaviour. In addition, one has to keep in mind that the critical velocity is closely linked to the shape of the solidification front, and hence to all the parameters that affect it. In a previous p a p e r [5], we determined experimentally the critical velocity for polyamide particle sizes ranging from 10-~ to 1 cm. In the present paper, we focus on the deformation of the interface and, using the same set-up and procedure, we extend our investigation to steel beads in order to contribute to the elucidation of the mechanism of engulfment.
2. Position of the problem The forces acting on a foreign particle near a solidification front have been recently developed
0022-0248/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved
1131
M. Yemmou et al. / Thermal aspects of particle engulfment by solidifying front
in ref. [6]. There are two attractive forces, one resulting from buoyancy F b and one from viscous drag due to the fluid flow around the particle F~. The repulsive force Fa derives for Van der Waals interactions and can be expressed in terms of surface free energy. The balance of these three forces leads to the expression of the critical velocity: Vb = -- 4'n'(Pp -- P e ) g R 3 e ,
67r/xR 2 V~ -
- -
vfPSe, Pe
h
lAIR F A-
6h 2
e,
(1) (2)
(3)
where A is the Hamaker constant, h the gap thickness, g the gravity, e the unit vector directed upwards, Vf the front velocity, p the density (p refers to particle, / to liquid and s to solid), and /~ the liquid viscosity. The expressions (1), (2) and (3) are found under the following general assumptions: the pushing of the particle is considered as a steady state phenomenon in which the particle and the solidification front move with the same velocity, and the gap width is maintained constant; - the material is free from soluble elements; - the particle is a smooth sphere and does not interact chemically with the solid interface; - there is no convection; the heat flow is unidirectional; - viscosity and thermal conductivity are uniform and temperature independent. One of the consequences of these restrictions is that the interface is perfectly plane. However, if it were not, one would have to determine the shape of the interface in order to calculate the viscous and the disjoining forces which both depend on the gap thickness h. In particular, for large particles and for macroscopically smooth front - i.e. the front follows the isotherms - the approximation of neglecting the differences in thermal properties between particles and matrix is actually never satisfied, since the perturbation produced by the particle in the local temperature gradient is not accounted for. -
-
Heat "Flow
Heat "Flow ,i!'
................. .,~
...............
~ . 1 . !J. i i ko>
-'ii i\\\,',,
'fl'
kx, k.
tt
))
\,>..~--7--L~.~............... ................. kp( kz, k.
Fig. 1. Expected interface shapes for the cases k o > k e and kp < k t.
Let, for instance, the particle thermal conductivity kp exceed the liquid thermal conductivity k e. Then the heat flow through the interface and the temperature near the particle are higher in comparison with these quantities in other regions of the interface, causing a deep cavity in the solidification front below the particle. On the contrary, the interface in the vicinity of a particle possessing lower heat conductivity (kp < k e) will have a convex shape, which applies particularly to the entrapment of gas bubbles. Fig. 1 shows schematically the shapes assumed for the growth front when k o > k e and kp < k e. Obviously, the shape of the solidification front is affected by the temperature distribution in the particle-matrix system and consequently, the critical velocity should depend on the thermal conditions. The importance of a difference in thermal conductivities was considered by Chernov and Mel'nikova theoretically [7], and supported experimentally by Zubko et al. [8] for the case of metallic particles in a metallic matrix: capture is favoured when the front is concave and hampered when it is convex. This idea was noted by Boiling and Ciss6 [2] and developed numerically by Aubourg [9], P6tschke and Rogge [10], and Sasikumar et al. [11] by introducing the interface curvature in the viscous drag only. They all come to the same conclusion: a deep cavity of the solidification front around the particle can increase the drag force by several orders of magnitude and decrease the critical velocity as compared with the
1132
M. Yemmou et al. / Thermal aspects of particle engulfment by solidifying front
thermally neutral particle (kp = ke). As pointed out by Temkin et al. [12], such a front dimple can also be due to a reduction of the melting point because of the enrichment of solute in the gap between the particle and the moving interface. So far, most experiments have been done with microscopic metallic, oxide or glass particles in water or in an organic matrix. To our knowledge, the only macroscopic particles explored were metallic particle of size R = 0.1-0.15 cm in a metallic matrix [8]. The purpose of the next sections is to provide an experimental demonstration of the front curving for very large individual particles of size 0.60.8 cm and of different thermal conductivities.
At the beginning of each run, water is at a uniform temperature, the upper and bottom copper plates are at the same positive temperature T i = Tu = T b = 0.4 + 0.1°C. Then a constant negative temperature T b is applied to the lower part, while the top is kept constant. For the set of experiments described here, T b ranges from - 2 to - 5 ° C and the i c e - w a t e r interface rate ranges from 3 x 10 -5 to 8 × 10 -4 c m / s . The different stages of p u s h i n g / t r a p p i n g are observed using a shadowgraph and the experiments are evaluated from video tape recordings. All pictures are taken by focusing the camera at the level of the largest cross-section of the particle, the sample is illuminated by a cold source light.
3. Experimental conditions 4. Results and discussion Freezing experiments in which an i c e - w a t e r front moves upward are performed in a plexiglas cell (8 cm x 6 cm X 2 cm) filled with doubly distilled and degassed water (fig. 2). The set-up used is identical to the one described previously [5], so the reader is referred to that p a p e r for details. The particles used are polyamide spheres (R = 0.4 cm, density p = 1.13 g/cm3; thermal conductivity k = 23 x 103 e r g / c m . °C. s) and steel spheres (R = 0.3 cm, density p = 8.89 g/cm3; thermal conductivity k = 46 x 105 e r g / c m • °C. s).
Fig. 3a shows a typical photograph of a convex ice-water interface getting close to a nylon particle of 0.8 cm size. This bending of the front behind the particle corresponds to the shape expected theoretically: the nylon thermal conductivity is lower than those of water and ice, k w a t e r = 56 x 103 e r g / c m . °C. s and kice = 221 × 103 e r g / c m . °C. s. Such a convex deformation of the interface is experimentally evidenced for the first time, and
Wu
Copperplates
lascell Coolingl i q u i ~ ~
Tb
Fig. 2. Schematic diagram of the freezing cell.
M. Yemmou et aL / Thermal aspects of particle engulfment by solidifying front
1133
12.
Wzter
InterFace ,
[1.i[[ ...............
iiiii iiiiiiiiiiiiiiiiiiiiiii!ii!iiiiiiiii!ii iiiiiiiii !i i ::iii i
Fig. 3. Sequence of processed images representing the various phases of a nylon particle capture by a freezing front from below (R = 0.4 cm, Tu= +0.3°C, Tb= - 2°C).
M. Yemmou et al. / Thermal aspects of particle engulfment by solidifying front
1134
this might explain why in previous experiments [5], the results on critical velocities had a relatively large standard deviation. The main features of the nylon particle engulfment behaviour are shown in figs. 3b-3f. It is
worthwhile to notice the inversion of the interface curvature between figs. 3d and 3e where the interface is discernable by a thin straight black line, which is the projection of the interface at the rear of the particle. At advanced stage of
In~t"f~c~ /
//
iiiiiimiitiii|i|iiW|ii|iiiiiiiiiiiiiiiil iiiii!i!!!iiiiiii!!.
Fig. 4. Sequence of processed images representing the various phases of a steel particle capture by a freezing front from below (R = 0.3 em, Tu = + 0.3°C, Tb = - 5°C). Dark spots are air bubbles in ice.
M. Yemmou et al. / Thermal aspects of particle engulfment by solidifying front
encapsulation, the front cusping looks like an upside-down cone filled with water, and with a lateral wall made of ice (fig. 30. The engulfment velocity Ve (velocity of the front measured at the particle-interface junction) is nearly the same as the velocity of the front taken at the same time, on the horizontal part of the interface. They both decrease from approximately the value of 10 -4 to 3 x 10 - s c m / s during the engulfment described from fig. 3b to fig. 3d. Then, an inversion of the interface occurs, as shown in fig. 3e; Ve decreases quickly from 2.7 x 10 - s to 4.8 x ]0 -6 c m / s , and in the meantime Vf decreases from 2.3 x 10 s to 1.3 X 10 - s c m / s . 4.1. Steel particles
Other experiments were carried out with steel spheres which were much more conductive than the matrix in order to be compared with those done with less conductive polyamide spheres. Fig. 4 displays subsequent stages of the trapping of a steel particle by an ice front. When the front approaches the particle, it dimples behind the particle (the curvature of the front is made clearly visible in figs. 4a and 4b) and then, after entrapment, it bulges ahead of the particle (fig. 4d). Indeed, as the steel sphere touches the interface, its temperature becomes practically the same. At later stages of capture, the particle penetrates into the ice and its t e m p e r a t u r e changes to that of the deep part of the interface, i.e. a temperature below the melting point. This, in turn, will cause the overgrowth of the particle by the ice starting from the stage shown in fig. 4c. In fig. 4d, the particle is covered by a thin layer of ice which becomes thicker, leading to a dome-like cover of ice, as seen in fig. 4f. In contrast to the nylon particle case, the engulfment velocity V~ is higher than Vf. From fig. 4b to fig. 4c, the freezing front accelerates during the formation of the ice cap; it increases from 1.25 x 10 - s to 2.03 x 10 - s c m / s . Between, the two stages (figs. 4c-4d), Ve increases from 3.3 x 10 4 to 5.3 x ]0 4 c m / s , and Vf decreases from 1.6 x 10 -4 to 9.8 x 10 - s c m / s . Once the particle is encapsulated (fig. 4e), the notion of engulfment velocity as defined above does not have any sense;
1135
but if we consider the front velocity at the top of the particle, it eventually decreases to reach the value of the front velocity far from the particle (6.8 X 10 -s c m / s for t = 7222 s). It must be noted that the variability between different measurements of the engulfment velocity is large, perhaps as much as a factor of 2 or 3. In fact, engulfment velocities for large particles captured at low velocities (in our experiments, they are of the magnitude of 1 /~m/s) should also depend on the specific gravity and to some extent on the temperature gradient in ice, in addition to heat conductivity. The theory gives only the trend for engulfment velocities and there is no model which predicts very accurately the measured engulfment velocities.
5. Concluding remarks The present experiments have clearly shown the role of thermal conductivity in the ice front bend at a macroscopic distance from a particle. On approaching the particle, the front is concave when the thermal conductivity of the particle is higher than water, and conversely it is convex. At this point of research, the interface shape appears as a consequence of thermal effects. However, other effects - hydrodynamic and interfacial p h e n o m e n a - assumed to be of second-order, are now under way in our laboratory. On the other hand, for a curved interface, we intend to compare calculated and observed critical velocities. Moreover, an experimental investigation is still needed to establish the correlation between the interface deformation and the eng u l f m e n t / rejection process.
References [1] D.R. Uhlmann, B. Chalmers and K.A. Jackson, J. Appl. Phys. 35 (1964) 2986. [2] G.F. Boiling and J. Ciss6, J. Crystal Growth 10 (1971) 56. [3] A.A. Chernov and D.E. Temkin, in: Crystal Growth and Materials, Eds. E. Kaldis and H.J. Scheel (North-Holland, Amsterdam, 1977). [4] Ch. K6rber, G. Rau, M.D. Cosman and E.G. Carvalho, J. Crystal Growth 72 (1985) 649.
i136
M. Yemmou et al. / Thermal aspects of particle engulfment by solidifying front
[5] M.A. Azouni, W. Kalita and M. Yemmou, J. Crystal Growth 92 (1990) 201. [6] M.A. Azouni and W. Kalita, Advan. Space Res. 12 (1988) 219. [7] A.A. Chernov and A.M. Mel'nikova, Kristallografiya 10 (1965) 791. [8] A.M. Zubko, V.G. Lobanov and V.V. Nikonova, Kristallografiya 18 (1973) 385. [9] P. Aubourg, Interaction of second-phase particles with a
crystal growing from the melt, PhD Thesis, MIT, Cambridge, MA (1978). [10] J. P6tschke and V. Rogge, J. Crystal Growth 94 (1989) 726. [11] R. Sasikumar, T.R. Ramamohan and B.C. Pai, Acta Met. 37 (1989) 2085. [12] D.E. Temkin, A.A. Chernov and A.M. Mel'nikova, Soviet Phys.-Cryst. 23 (1977) 13.
Journal of Crystal Growth 128 (1993) 1130-1136 North-Holland
Thermal aspects of particle engulfment by a solidifying front M. Y e m m o u , M . A . A z o u n i , P. C a s s e s Laboratoire d'Adrothermique du CNRS, 4ter Route des Gardes, F-92190 Meudon, France
and G. P 4 t r 6 Chemical Physics, EP Department, CP 165, Universitd Libre de Bruxelles, 50 Avenue F.D. Roosevelt, B-1050 Brussels, Belgium
The solidification front under pushing conditions as well as the changes in shape that take place during the engulfment process are investigated for polyamide and steel spheres in an ice-water system. The examination of all subsequent stages of trapping make evident that thermal effects are of prime importance in the interface shape, when large spheres (size of the order of some millimeters) are considered.
1. Introduction The engulfment or rejection of solid particles by advancing solidification is important in metallurgy, materials science, cryobiology, cryoconcentration, soil mechanics and many other processes. It is either necessary to purify the solid phase and keep it free of inclusions or to achieve a controlled incorporation of particles yielding better mechanical or electrical properties of the solid matrix e.g composites. It has been known that the particle pushing occurs only below a certain velocity of the advancing solidification front called the critical velocity. If the velocity of the growth front is greater than the critical velocity, the particle is encapsulated by the front. All the theories of evaluation of the critical velocities are based on the concept of molecular surface forces repelling the particle from the growth front and thus facilitating the entry of the liquid into the gap between the growing solid and the particle. On the other hand, the interaction of the interface requires that there must be some force which brings the interface and the particle into contact, and some force preventing easy dis-
engagement [1-3]. A concise survey was presented by K6rber et al. [4]. From the experimental evidence, the rate of the advance of the freezing interface, the physical properties of the particle and the matrix together with the particle size and its surface state are decisive for the particle behaviour. In addition, one has to keep in mind that the critical velocity is closely linked to the shape of the solidification front, and hence to all the parameters that affect it. In a previous p a p e r [5], we determined experimentally the critical velocity for polyamide particle sizes ranging from 10-~ to 1 cm. In the present paper, we focus on the deformation of the interface and, using the same set-up and procedure, we extend our investigation to steel beads in order to contribute to the elucidation of the mechanism of engulfment.
2. Position of the problem The forces acting on a foreign particle near a solidification front have been recently developed
0022-0248/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved
1131
M. Yemmou et al. / Thermal aspects of particle engulfment by solidifying front
in ref. [6]. There are two attractive forces, one resulting from buoyancy F b and one from viscous drag due to the fluid flow around the particle F~. The repulsive force Fa derives for Van der Waals interactions and can be expressed in terms of surface free energy. The balance of these three forces leads to the expression of the critical velocity: Vb = -- 4'n'(Pp -- P e ) g R 3 e ,
67r/xR 2 V~ -
- -
vfPSe, Pe
h
lAIR F A-
6h 2
e,
(1) (2)
(3)
where A is the Hamaker constant, h the gap thickness, g the gravity, e the unit vector directed upwards, Vf the front velocity, p the density (p refers to particle, / to liquid and s to solid), and /~ the liquid viscosity. The expressions (1), (2) and (3) are found under the following general assumptions: the pushing of the particle is considered as a steady state phenomenon in which the particle and the solidification front move with the same velocity, and the gap width is maintained constant; - the material is free from soluble elements; - the particle is a smooth sphere and does not interact chemically with the solid interface; - there is no convection; the heat flow is unidirectional; - viscosity and thermal conductivity are uniform and temperature independent. One of the consequences of these restrictions is that the interface is perfectly plane. However, if it were not, one would have to determine the shape of the interface in order to calculate the viscous and the disjoining forces which both depend on the gap thickness h. In particular, for large particles and for macroscopically smooth front - i.e. the front follows the isotherms - the approximation of neglecting the differences in thermal properties between particles and matrix is actually never satisfied, since the perturbation produced by the particle in the local temperature gradient is not accounted for. -
-
Heat "Flow
Heat "Flow ,i!'
................. .,~
...............
~ . 1 . !J. i i ko>
-'ii i\\\,',,
'fl'
kx, k.
tt
))
\,>..~--7--L~.~............... ................. kp( kz, k.
Fig. 1. Expected interface shapes for the cases k o > k e and kp < k t.
Let, for instance, the particle thermal conductivity kp exceed the liquid thermal conductivity k e. Then the heat flow through the interface and the temperature near the particle are higher in comparison with these quantities in other regions of the interface, causing a deep cavity in the solidification front below the particle. On the contrary, the interface in the vicinity of a particle possessing lower heat conductivity (kp < k e) will have a convex shape, which applies particularly to the entrapment of gas bubbles. Fig. 1 shows schematically the shapes assumed for the growth front when k o > k e and kp < k e. Obviously, the shape of the solidification front is affected by the temperature distribution in the particle-matrix system and consequently, the critical velocity should depend on the thermal conditions. The importance of a difference in thermal conductivities was considered by Chernov and Mel'nikova theoretically [7], and supported experimentally by Zubko et al. [8] for the case of metallic particles in a metallic matrix: capture is favoured when the front is concave and hampered when it is convex. This idea was noted by Boiling and Ciss6 [2] and developed numerically by Aubourg [9], P6tschke and Rogge [10], and Sasikumar et al. [11] by introducing the interface curvature in the viscous drag only. They all come to the same conclusion: a deep cavity of the solidification front around the particle can increase the drag force by several orders of magnitude and decrease the critical velocity as compared with the
1132
M. Yemmou et al. / Thermal aspects of particle engulfment by solidifying front
thermally neutral particle (kp = ke). As pointed out by Temkin et al. [12], such a front dimple can also be due to a reduction of the melting point because of the enrichment of solute in the gap between the particle and the moving interface. So far, most experiments have been done with microscopic metallic, oxide or glass particles in water or in an organic matrix. To our knowledge, the only macroscopic particles explored were metallic particle of size R = 0.1-0.15 cm in a metallic matrix [8]. The purpose of the next sections is to provide an experimental demonstration of the front curving for very large individual particles of size 0.60.8 cm and of different thermal conductivities.
At the beginning of each run, water is at a uniform temperature, the upper and bottom copper plates are at the same positive temperature T i = Tu = T b = 0.4 + 0.1°C. Then a constant negative temperature T b is applied to the lower part, while the top is kept constant. For the set of experiments described here, T b ranges from - 2 to - 5 ° C and the i c e - w a t e r interface rate ranges from 3 x 10 -5 to 8 × 10 -4 c m / s . The different stages of p u s h i n g / t r a p p i n g are observed using a shadowgraph and the experiments are evaluated from video tape recordings. All pictures are taken by focusing the camera at the level of the largest cross-section of the particle, the sample is illuminated by a cold source light.
3. Experimental conditions 4. Results and discussion Freezing experiments in which an i c e - w a t e r front moves upward are performed in a plexiglas cell (8 cm x 6 cm X 2 cm) filled with doubly distilled and degassed water (fig. 2). The set-up used is identical to the one described previously [5], so the reader is referred to that p a p e r for details. The particles used are polyamide spheres (R = 0.4 cm, density p = 1.13 g/cm3; thermal conductivity k = 23 x 103 e r g / c m . °C. s) and steel spheres (R = 0.3 cm, density p = 8.89 g/cm3; thermal conductivity k = 46 x 105 e r g / c m • °C. s).
Fig. 3a shows a typical photograph of a convex ice-water interface getting close to a nylon particle of 0.8 cm size. This bending of the front behind the particle corresponds to the shape expected theoretically: the nylon thermal conductivity is lower than those of water and ice, k w a t e r = 56 x 103 e r g / c m . °C. s and kice = 221 × 103 e r g / c m . °C. s. Such a convex deformation of the interface is experimentally evidenced for the first time, and
Wu
Copperplates
lascell Coolingl i q u i ~ ~
Tb
Fig. 2. Schematic diagram of the freezing cell.
M. Yemmou et aL / Thermal aspects of particle engulfment by solidifying front
1133
12.
Wzter
InterFace ,
[1.i[[ ...............
iiiii iiiiiiiiiiiiiiiiiiiiiii!ii!iiiiiiiii!ii iiiiiiiii !i i ::iii i
Fig. 3. Sequence of processed images representing the various phases of a nylon particle capture by a freezing front from below (R = 0.4 cm, Tu= +0.3°C, Tb= - 2°C).
M. Yemmou et al. / Thermal aspects of particle engulfment by solidifying front
1134
this might explain why in previous experiments [5], the results on critical velocities had a relatively large standard deviation. The main features of the nylon particle engulfment behaviour are shown in figs. 3b-3f. It is
worthwhile to notice the inversion of the interface curvature between figs. 3d and 3e where the interface is discernable by a thin straight black line, which is the projection of the interface at the rear of the particle. At advanced stage of
In~t"f~c~ /
//
iiiiiimiitiii|i|iiW|ii|iiiiiiiiiiiiiiiil iiiii!i!!!iiiiiii!!.
Fig. 4. Sequence of processed images representing the various phases of a steel particle capture by a freezing front from below (R = 0.3 em, Tu = + 0.3°C, Tb = - 5°C). Dark spots are air bubbles in ice.
M. Yemmou et al. / Thermal aspects of particle engulfment by solidifying front
encapsulation, the front cusping looks like an upside-down cone filled with water, and with a lateral wall made of ice (fig. 30. The engulfment velocity Ve (velocity of the front measured at the particle-interface junction) is nearly the same as the velocity of the front taken at the same time, on the horizontal part of the interface. They both decrease from approximately the value of 10 -4 to 3 x 10 - s c m / s during the engulfment described from fig. 3b to fig. 3d. Then, an inversion of the interface occurs, as shown in fig. 3e; Ve decreases quickly from 2.7 x 10 - s to 4.8 x ]0 -6 c m / s , and in the meantime Vf decreases from 2.3 x 10 s to 1.3 X 10 - s c m / s . 4.1. Steel particles
Other experiments were carried out with steel spheres which were much more conductive than the matrix in order to be compared with those done with less conductive polyamide spheres. Fig. 4 displays subsequent stages of the trapping of a steel particle by an ice front. When the front approaches the particle, it dimples behind the particle (the curvature of the front is made clearly visible in figs. 4a and 4b) and then, after entrapment, it bulges ahead of the particle (fig. 4d). Indeed, as the steel sphere touches the interface, its temperature becomes practically the same. At later stages of capture, the particle penetrates into the ice and its t e m p e r a t u r e changes to that of the deep part of the interface, i.e. a temperature below the melting point. This, in turn, will cause the overgrowth of the particle by the ice starting from the stage shown in fig. 4c. In fig. 4d, the particle is covered by a thin layer of ice which becomes thicker, leading to a dome-like cover of ice, as seen in fig. 4f. In contrast to the nylon particle case, the engulfment velocity V~ is higher than Vf. From fig. 4b to fig. 4c, the freezing front accelerates during the formation of the ice cap; it increases from 1.25 x 10 - s to 2.03 x 10 - s c m / s . Between, the two stages (figs. 4c-4d), Ve increases from 3.3 x 10 4 to 5.3 x ]0 4 c m / s , and Vf decreases from 1.6 x 10 -4 to 9.8 x 10 - s c m / s . Once the particle is encapsulated (fig. 4e), the notion of engulfment velocity as defined above does not have any sense;
1135
but if we consider the front velocity at the top of the particle, it eventually decreases to reach the value of the front velocity far from the particle (6.8 X 10 -s c m / s for t = 7222 s). It must be noted that the variability between different measurements of the engulfment velocity is large, perhaps as much as a factor of 2 or 3. In fact, engulfment velocities for large particles captured at low velocities (in our experiments, they are of the magnitude of 1 /~m/s) should also depend on the specific gravity and to some extent on the temperature gradient in ice, in addition to heat conductivity. The theory gives only the trend for engulfment velocities and there is no model which predicts very accurately the measured engulfment velocities.
5. Concluding remarks The present experiments have clearly shown the role of thermal conductivity in the ice front bend at a macroscopic distance from a particle. On approaching the particle, the front is concave when the thermal conductivity of the particle is higher than water, and conversely it is convex. At this point of research, the interface shape appears as a consequence of thermal effects. However, other effects - hydrodynamic and interfacial p h e n o m e n a - assumed to be of second-order, are now under way in our laboratory. On the other hand, for a curved interface, we intend to compare calculated and observed critical velocities. Moreover, an experimental investigation is still needed to establish the correlation between the interface deformation and the eng u l f m e n t / rejection process.
References [1] D.R. Uhlmann, B. Chalmers and K.A. Jackson, J. Appl. Phys. 35 (1964) 2986. [2] G.F. Boiling and J. Ciss6, J. Crystal Growth 10 (1971) 56. [3] A.A. Chernov and D.E. Temkin, in: Crystal Growth and Materials, Eds. E. Kaldis and H.J. Scheel (North-Holland, Amsterdam, 1977). [4] Ch. K6rber, G. Rau, M.D. Cosman and E.G. Carvalho, J. Crystal Growth 72 (1985) 649.
i136
M. Yemmou et al. / Thermal aspects of particle engulfment by solidifying front
[5] M.A. Azouni, W. Kalita and M. Yemmou, J. Crystal Growth 92 (1990) 201. [6] M.A. Azouni and W. Kalita, Advan. Space Res. 12 (1988) 219. [7] A.A. Chernov and A.M. Mel'nikova, Kristallografiya 10 (1965) 791. [8] A.M. Zubko, V.G. Lobanov and V.V. Nikonova, Kristallografiya 18 (1973) 385. [9] P. Aubourg, Interaction of second-phase particles with a
crystal growing from the melt, PhD Thesis, MIT, Cambridge, MA (1978). [10] J. P6tschke and V. Rogge, J. Crystal Growth 94 (1989) 726. [11] R. Sasikumar, T.R. Ramamohan and B.C. Pai, Acta Met. 37 (1989) 2085. [12] D.E. Temkin, A.A. Chernov and A.M. Mel'nikova, Soviet Phys.-Cryst. 23 (1977) 13.