Instability Formation and Directional Dendritic Growth of Ice Studied by

CRYSTAL GROWTH & DESIGN

Instability Formation and Directional Dendritic Growth of Ice Studied by Optical Interferometry Michael F.

2001 VOL. 1, NO. 3 213-223

Butler†

Unilever Research, Colworth House, Sharnbrook, Bedfordshire MK44 1LQ, U.K. Received November 27, 2000

ABSTRACT: Mach-Zender optical interferometry was used to measure the concentration gradient at the icesolution interface, and the ice-solution interface morphology, during directional crystallization of ice in sucrose (disaccharide) and pullulan (polysaccharide) solutions. For sucrose, at all concentrations, the solute concentration decayed exponentially with distance from the interface. For pullulan, an exponential decay was only measured for dilute solutions, there being a substantial deviation from a single-exponential decay for concentrated, entangled solutions. Constitutional supercooling was shown to be responsible for the onset of interfacial instability in all solutions. The Mullins-Sekerka (MS) theory of interfacial instability was tested and provided a reasonable prediction of the instability wavelength for sucrose solutions over a range of velocities within which the partition coefficient varied considerably, but not for pullulan. The discrepancy for pullulan was explained in terms of the hindered diffusion mechanism caused by entanglement of the polymer chains, rendering the assumptions made in the MS theory inaccurate. Three different equations for predicting the dendrite primary spacing were tested for sucrose solutions. Reasonable agreement was obtained with two expressions at low solute concentrations but, owing to invalidity of the assumptions made in its formulation, one of the expressions (the Hunt and Lu model) became increasingly inaccurate at higher concentrations. Introduction Structures in a Temperature Gradient. An initially planar interface will develop a morphological instability under certain conditions leading to the development of dendritic or cellular structures.1-3 Cellular structures are formed at low and high velocities in the unstable region in which the growth conditions suppress the formation of side branches. Dendrites are formed in the intermediate regime and are characterized by a paraboloidal, or near-paraboloidal, profile and the presence of side branches.3 Although the shape (tip radius) of the main dendrite arms in undercooled solutions is fairly well understood in terms of the solvability theory,4-7 prediction of the morphology of a dendritic array during directional crystallization is more difficult. The quantities that define the morphology, namely the tip radius and the primary and secondary spacing, may be affected by the separation of the dendrite arms, since the solute and thermal diffusion fields may impinge.8 Additionally, interfacial attachment kinetics may become important at high growth rates.3 A planar interface may become unstable when the imposed temperature gradient in the liquid is smaller than the gradient of the local freezing point curve, since a region of liquid will exist ahead of the interface which has a temperature that is lower than the equilibrium freezing point. This phenomenon is known as constitutional supercooling. In the constitutionally supercooled region a small protrusion of solidified material experiences a greater driving force for growth than the planar front and, therefore, grows more rapidly. The protrusions become larger, and a cellular or dendritic morphology develops. Stabilizing influences, which act to † E-mail: [email protected]. Tel.: +44(0)1234 222958. Fax: +44(0)1234 222757.

preserve the planarity of the interface, are provided by the imposed temperature gradient and by the surface energy of the solid/liquid interface. By considering the competition between the stabilizing and destabilizing influences, Mullins and Sekerka developed a mathematical expression for the amplification rate of an arbitrary perturbation that could be used to determine the length scale of the instability.9,10 The resulting equation predicted that the amplification rate experienced a maximum for one particular wavelength:

{ {

δ˙ V ) Vω -2TMΓω2 ω* - (1 - k) δ D

[

] V V 2G[ω* - (1 - k)] + 2mG [ω* - ]}} D D / {2G(kk +- kk )[ω* - (DV(1 - k))] + 2ωmG } (1) C

s

l

s

l

C

ω* )

1/2 V 2 V + + ω2 2D 2D

[( )

]

(2)

where δ is the size of the perturbation, V is the growth velocity, TM is the equilibrium melting point, Γ is the ratio of the surface energy and the latent heat of fusion, D is the diffusion coefficient, k is the partition coefficient, G is the temperature gradient, GC is the solute concentration gradient at the interface, and ks and kl are the thermal conductivities of the ice and solution phases, respectively. Primary Dendrite Structure. Usually, the actual primary spacing is greater than the instability wavelength; i.e., some degree of coarsening of the structure occurs during growth. Observations have shown that the primary spacing first increases (as a function of the growth velocity) during cellular growth and then decreases slightly during dendritic growth.11-13 A variety

10.1021/cg005534q CCC: $20.00 © 2001 American Chemical Society Published on Web 04/07/2001

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of approaches, inluding analytical solution of the growth equations,11,15 numerical solution,14 and dimensional analysis3 of the main length scales contributing to dendritic growth (the solute and thermal diffusion lengths and the capillarity length scale), have been adopted in order to predict the primary spacing. All of these analyses required various simplifying assumptions (such as a paraboloidal dendrite profile, no interaction between different dendrites in the array, constant partition coefficient, and straight liquidus curve) and regarded the primary spacing to be influenced mainly by solute diffusion. More recently, Makkonen16 derived an expression by treating the heat balance as the main factor. Generally, however, the expressions for the primary spacing adopt the following form in terms of the temperature gradient (G) and growth velocity (V):

λd ) AG-1/2V-1/4

(3)

with A being dependent on variables such as the diffusion coefficient, partition coefficient, and liquidus slope. Interestingly, some of the theories predict that a range of spacings are possible,14,15 while others predict a unique value of the spacing, for any given set of conditions.3,11,16 Experimentally, a range is usually observed, although the general dependence with growth speed of eq 3 is usually obeyed. The former discrepancy with the theories that predict a unique spacing can be reconciled by the fact that in reality dendrite profiles differ from the ideal paraboloidal form, and a range of profiles may actually form. Use of Optical Interferometry To Study Interfacial Instability and Dendritic Growth. Despite the importance of the solute concentration fields in determining crystal growth, only a relatively small number of experiments have been devoted to its measurement, and most authors have been content to use calculated steady-state values which may not provide an accurate description of the system. Early attempts were made to measure concentration profiles near the solid-liquid interface using spectophotometry and densitometric analysis of micrographs,17 although these techniques suffer from lack of spatial resolution and accuracy. Optical interferometry is, however, ideally suited to such studies because of its ability to measure accurately, in two dimensions, small relative changes in refractive index (which may subsequently be related to the solution concentration). A few authors have used this technique, in the form of Michelson,18-22 MachZender,23-25 wedge,26-30 and holographic31,32 interferometry, to study the concentration and temperature fields around isolated dendrites and single crystals to test theories of crystal growth. More recently, Nagashima and Furukawa used Mach-Zender interferometry to study the change in concentration gradient near a planar front during the development of interfacial instability.33 Their subsequent paper34 verified the predictions of the Mullins-Sekerka instability equation by showing that the instability wavelength calculated using the experimentally determined concentration gradient at the interface coincided with the directly measured wavelength.

Figure 1. Schematic diagram of the Mach-Zender optical interferometer.

Experimental Section Directional Crystallization of Ice. The ice structures that were examined using optical interferometry were grown in a temperature gradient microscope stage (TGMS),35 which consisted of two plates held at different temperatures and separated by a gap of 5 mm. The glass slide on which the sample was placed was positioned across the gap between two plates so that a temperature gradient existed across it. The slide was held in place by a movable sprung plate connected to a motor capable of moving the sample across the gap at speeds between 0.5 and 80 µm/s. The sample itself consisted of a drop of sucrose (a disaccharide with a molecular weight of approximately 360 amu) or pullulan (a polysaccharide consisting of maltotriose monomer units connected via a 1,6glycosidic linkage, with a molecular weight of about 240 000 amu) solution on a glass microscope slide. The drop was surrounded by a 75 µm spacer on top of which a standard 0.17 mm microscope coverslip was placed. The space between the edge of the coverslip and the microscope slide was filled with silicone grease to prevent sample evaporation. Coolant was passed through the plates of the TGMS to achieve temperatures (monitored by thermocouples inside the plates) of -15 and +5 °C on the “cold” and “hot” plates of the stage, respectively. A constant stream of dry nitrogen gas was passed over the sample to prevent condensation from forming on the top of the coverslip. Ice was allowed to nucleate and grow to the equilibrium position in the gap. Once formed, the ice-solution interface was allowed to stabilize for several hours until it became planar. Further ice growth was initiated by activating the motor and moving the sample toward the “cold” plate, thus cooling the solution ahead of the ice-solution interface. The morphology of the interface, from which the instability wavelength and the primary dendrite spacing were directly measured, and the sucrose concentration in the vicinity of the interface were observed using an optical interferometer. The sample was moved at speeds between 1 and 20 µm/s. Optical Interferometry. The interferometer that was constructed for the purpose of measuring the interface morphology and the solute concentration field was of the MachZender type. It was chosen because of its ability to simultaneously image the interference fringes from which the concentration field was calculated and the ice-solution interface from which the instability wavelength and primary dendrite spacing were measured. The layout of the components from which the interferometer was constructed is shown schematically in Figure 1. A 0.5 mW helium-neon laser produces a beam of coherent visible light (wavelength 632.8 nm) which is passed through a beam expander (microscope objective lens L1, 2.5× magnification, and condenser lens L2, focal length 250 mm) to produce a collimated beam with a diameter of approximately 3 mm. The collimated beam then passes through a beam splitter (BS1) that splits the beam into

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two paths. The reference path contains a mirror (M2) and compound microscope optics consisting of an objective lens (L3.1) that magnifies the beam and a collimating lens (L4.1) that collimates it. The sample path contains a mirror (M3), the sample, and the same compound microscope optics as in the reference path to provide a magnified image of the icesolution interface (lenses L3.2 and L4.2). The reference and sample paths are recombined in a second beam splitter (BS2). The presence of the sample introduces an extra optical path length in the sample path compared to that of the reference path. An interference pattern, consisting of dark and light areas where the light has destructively and constructively interfered, respectively, is therefore produced when the two beams are recombined and can be analyzed to obtain information on the refractive index (and hence solution concentration) variation throughout the sample. The interference pattern and superimposed magnified image of the sample is magnified after exiting BS2 by a biconcave lens (L5) and projected onto a screen. A CCD camera (Sony XC75CE) with a macro lens, connected to a PC with a frame grabber (Matrox Meteor) controlled by Zeiss KS400 software, was used to view and capture the image. Reference fringes were set up prior to introducing the sample into the sample path by tilting one of the mirrors. Any displacement of the fringes caused by the sample was related to the position of the reference fringes. Analysis of the optical path difference between the two beams with this geometry allows the following expression to be formulated, relating the solution concentration to the deviation in position of the fringes caused by variation in solution concentration throughout the sample:

C - C0 λ∆x ) C (n(C) - n0)hw

(4)

where C is the local solution concentration, C0 is the bulk concentration, λ is the laser wavelength (632.8 nm), ∆x is the deviation of the fringe position from that of the reference fringe, n(C) is the refractive index of the solution with concentration C, n0 is the refractive index of pure water (1.333), h is the sample thickness (75 µm), and w is the separation of the reference fringes. Measurement and Analysis of the Ice-Solution Interface Morphology. The ice-solution interface morphology was measured directly from the interference images obtained using the interferometer. The change of position of the interface as the planar interface developed instability was used to calculate the growth velocity at the point of instability. The instability wavelength and, when the morphology had reached a steady growth form, the primary dendrite spacing were also measured directly from the interference images. The theoretical instability wavelength was determined by plotting the amplification factor as a function of wavelength and taking the selected wavelength to be the one at which the amplification factor reached a maximum value. The value predicted using this function was then compared with the instability wavelength measured directly from the interference image for the range of growth velocities studied. Equations, representing examples of an analytical solution (Dexin and Sahm),11 a numerical solution (Hunt and Lu),14 and a dimensional expression (Kurz and Trivedi),3 that predicted the dendrite primary spacing were also plotted as a function of growth velocity. The experimentally measured spacing was superimposed on this plot in order to assess which, if any, of the models provided the most reasonable prediction of the primary spacing. To plot the functions described above, the following values of the relevant parameters were used: diffusion coefficient of sucrose, 3 × 10-10 m2 s-1; thermal conductivity of ice, 2.2 J m-1 s-1 K-1; thermal conductivity of solution, 0.56 J m-1 s-1 K-1; temperature gradient, 2500 K m-1; liquidus slope for ice/sucrose solution system, 0.0738 K (wt %)-1; surface energy, 2.8 × 10-2 J m-2; latent heat of water per unit volume, 3.33 × 108 J m-3, diffusion coefficient of pullulan (measured by dynamic light scattering from a dilute

Figure 2. Effective partition coefficient of sucrose (solid circles) and pullulan (open circles), for a range of growth velocities. solution), 3 × 10-11 m2 s-1, liquidus slope for ice-pullulan solution system, 0.0135 K (wt %)-1. The growth velocity and solute concentration gradient were measured from the interference images. The solute partition coefficient was measured experimentally by growing polycrystalline ice hemispheres at different rates (controlled by the temperature of the solution into which the ice was grown) in a solution of known starting concentration. Once the hemisphere had grown to a certain size, it was removed from the solution and melted. The water was evaporated in a vacuum oven held at 40 °C, and the solid residue was taken to be the solute that was present in the ice phase. Since the volume of the hemisphere and the total initial solute content of the system were known, the partition coefficient could be calculated. The values of partition coefficient for sucrose and pullulan, shown in Figure 2, were fitted to the following functions, shown as the solid lines, which were used to yield the partition coefficients at any given growth velocity:

ksucrose ) 1 - e-0.55V

(5a)

kpullulan ) 1 - 1.363e-4.865V

(5b)

Results Instability Wavelength. Figure 3 shows a sequence of interference images during the development of structure at the ice-solution interface for 5% sucrose, during growth at a pulling speed of 5 µm/s. The solution is on the left side of the image and the ice on the right. At the initial condition (t ) 0, where t is the growth time measured from the start of movement) the interface between the phases was planar and the interference fringes in the solution were parallel and horizontal, revealing the uniformity of the concentration of the solution. The fringes in the ice were largely horizontal except near the interface, where a slight curvature indicated that the ice-solution interface was curved in the thickness direction of the sample. At t > 0, curved interference fringes were observed in both the solution and the ice. The fringes in the solution became increasingly curved, with the interface remaining planar, until a critical degree of curvature was reached. At this point (t ) 260 s) the interface developed the wavy perturbation (clearly visible by t ) 320 s) from which the dendritic morphology emerged (t ) 420 s). Similarly, in the ice phase a significant degree of fringe curvature developed, while the interface remained planar. Once the dendritic growth began, the fringe curvature in solution decreased significantly to a constant amount

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Figure 3. Interference images showing the development of the solute diffusion field in the vicinity of the ice-solution interface during instability formation in the thickness direction (upper four images) and during instability formation in the lateral direction (lower four images).

ahead of the tips as a steady state was reached. The fringe curvature increased into the interstices between the dendrites, however. The growth velocity of the dendrites was also greater than that of the planar front,

and the dendrite tips eventually grew with the same speed as the pulling speed of the slide. Similar results were obtained for crystallization in all of the pullulan solutions, except for 1% and 5% pullulan



Figure 4. Development of the concentration profile near the ice-solution interface for 5% sucrose solution.

drawn at the lowest speed of 2 µm s-1, for which a concentration developed but no instability formed. In comparison to sucrose, the pullulan concentration at the interface increased more rapidly, and the interfacial instability developed sooner. For example, for 5% pullulan drawn at 5 µm s-1 the instability occurred about 30 s after the slide began to move. The other difference between sucrose and pullulan solutions was that a cellular rather than a dendritic morphology developed in the situations where the planar front did become unstable. Using eq 4, the concentration profile in the vicinity of the interface was calculated. Figure 4 demonstrates some results for a sucrose solution with an initial bulk concentration of 5%, and Figure 5 shows some results for a series of pullulan solutions. During growth the solute diffusion field developed, with a maximum concentration of solute at the interface and an exponentially decreasing concentration away from the interface. The relative increase in concentration at the interface was lower for solutions with higher bulk concentrations and for interfaces growing at higher speeds. Figure 6 shows the equilibrium melting temperature, Teq, of ice in the 5% sucrose solution as a function of distance from the interface, calculated using the phase diagram of the ice-sucrose solution system and the concentration profiles shown in Figure 4. The temperature profile of the imposed gradient is also shown. The instability in the thickness direction (see Figure 3) occurred when the slope of the concentration profile caused by the solute concentration profile at the interface exceeded the imposed temperature gradient: i.e., when constitutional supercooling was present. Parts a and b of Figure 7 show the equilibrium melting temperature for pullulan solutions in the cases where the front remained planar and where it became unstable, respectively. In the former case constitutional supercooling did not occur, whereas in the latter it did. The time at which constitutional supercooling occurred also coincided with the appearance of the instability. The instability wavelength measured directly from the images is compared with the instability wavelength calculated using the values of concentration gradient at the interface in parts a-c of Figure 8 for 5%, 10%, and 20% sucrose solution, respectively. Parts a-c of Figure 9 show the same comparison for 5%, 10%, and 20% pullulan. The error bars are estimated from the uncertainty in the values of the variables inserted in

Figure 5. Development of the concentration profile near the ice-solution interface for (a) 1% pullulan, (b) 10% pullulan, and (c) 20% pullulan grown at 5 µm s-1. The inset graph in (c) demonstrates the deviation of the curve from the nearest single-exponential fit.

the Mullins-Sekerka equation. The values of the diffusion coefficient and growth velocity (hence temperature gradient) were found to have the largest influence on the value of the instability wavelength calculated from the Mullins-Sekerka equation. The partition coefficient was slightly less significant, and the surface energy much less significant under the conditions used. Both calculated and directly measured instability wavelengths agreed well for 5% sucrose and reasonably well for 10% and 20% sucrose (with the predicted values tending to be slightly higher than the measured ones for the last two cases). For the pullulan samples, however, there was no agreement between calculated and directly measured instability wavelength for any of the samples. It should be noted that the concentration plots shown in Figures 4 and 5 can also be used to provide information on the diffusion process and its influence on the dendritic morphology through calculation of the diffu-

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Figure 6. Demonstration of constitutional supercooling for 5% sucrose. The solid lines are the actual temperatures from the imposed temperature gradient.

Figure 7. Demonstration of constitutional supercooling in a pullulan solution where the instability developed but not when the front remained planar. Part a shows the lack of constitutional supercooling in the planar 1% solution grown at 2 µm s-1, whereas part b shows its occurrence in the 5% solution grown at 10 µm s-1 which formed a cellular morphology.

Figure 8. Comparison between the instability wavelength predicted using the measured concentration gradient at the ice-solution interface in the Mullins-Sekerka theory with the directly measured wavelength for (a) 5% sucrose, (b) 10% sucrose, (c) 20% sucrose.

sion length. The diffusion length is defined as the distance over which the exponential concentration profile decays to 1/e of the starting value. Using this value, a diffusion length of 380 µm was measured for sucrose. This value compares reasonably with the value that may be calculated from the diffusion coefficient and growth velocity, which was 399 µm. For 1% pullulan (for which the concentration profile was a single-exponential decay), however, the diffusion length measured from the concentration profile was about 10 µm, whereas the one calculated from the diffusion coefficient was about 30 µm.

Primary Spacing. During the development of the dendritic structure from the initial instabilities the following observations were made. (1) A dendritic structure, i.e., side branches were present on the main branches, was formed for the growth conditions studied except for the lowest growth velocity of 1 µm s-1. (2) After the instabilities had formed, growth of the dendritic structure was initially rapid and then slowed as the dendrite growth velocity matched the pulling speed of the slide. (3) Once the growth velocity of the dendrites had reached a steady value, no further change in the structure occurred. (4) While the structure was reaching



tively. The Dexin and Sahm equation clearly does not provide a reasonable fit to the data points, whereas the Hunt and Lu model and the dimensional model reported by Trivedi and Kurz appears to provide a closer fit to the data for the 5 and 10 wt % sucrose solution, although it is less accurate for 20 wt % sucrose solution. Suitable values of the prefactor, A, that enable the line generated by the Kurz and Trivedi equation to lie in the vicinity of the experimental data points are 4.5, 4.0, and 3.5 for the 5, 10, and 20 wt % sucrose solutions, respectively. A power law could also be fitted to the data points, since, for a given concentration, the primary spacing was a function of the growth velocity. From this result a scaling relationship could be formulated to relate the primary spacing to the instability wavelength, which could also be fitted in a similar manner. For 5 wt % sucrose, the following relationship was determined between the instability wavelength and the primary spacing:

λdendrite ) 5.17λinstabilityV0.078

(6)

Discussion

Figure 9. Comparison between the instability wavelength predicted using the measured concentration gradient at the ice-solution interface in the Mullins-Sekerka theory with the directly measured wavelength for (a) 1% pullulan, (b) 5% pullulan, and (c) 10% pullulan.

the steady-state velocity, i.e., between instability formation and steady-state dendrite growth, the dendritic morphology changed rapidly. Coarsening of the structure occurred, such that approximately every other instability grew into a fully fledged dendrite while its neighbor decayed, and the radius of the dendite tips became larger as their growth velocity decreased from the initial rapid growth rate to the steady-state value. Parts a and b of both Figures 10 and 11 are micrographs of the steady-state dendritic morphology for ice in 10% and 20% sucrose solutions, respectively, growing at a rate of 10 and 20 µm s-1, to indicate how the dendritic morphology is altered by growth velocity and solute concentration. Parts a-c of Figure 12 plot the dendrite primary spacing calculated using the equations reported by Dexin and Sahm (eq 4), Hunt and Lu (eq 6) and Trivedi and Kurz (eq 14), with the experimentally measured values for 5%, 10%, and 20% sucrose, respec-

Mullins-Sekerka Instability. The evolution of the interference fringes in the solution and in the ice prior to the development of the instability reveals the physical processes occurring in the sample at times when cursory examination using simple optical microscopy would reveal no more than a planar ice-solution interface. The increasing curvature of the fringes in the solution occurred because the solute was unable to diffuse away from the advancing interface quickly enough to prevent the concentration from building up. The lower diffusion coefficient of pullulan accounted for the faster accumulation of solute at the interface than for sucrose, leading to earlier formation of the interfacial instability. An increase of the growth speed was also associated with an increase in the amount of solute trapped in the ice, as shown by Figure 2. This fact explains why the curvature of the fringes in solution was smaller when the faster growing cellular or dendritic structure had formed; the solute was trapped in the growing ice and could not accumulate in the solution near the interface. It also explains why the relative increase in concentration at the interface was lower for the samples pulled at higher speeds. However, the fact that the fringes continued to curve into the interstices between the cell or dendrite arms provides direct evidence for the rejection of solute from the tips of the dendrites into the spaces between them. Fringe curvature in the ice phase indicates a varying thickness of the ice, since the dependence of the optical path on the ice crystal thickness is much greater than the dependence on the concentration change in the solution.33 The increasing curvature of the ice in the thickness direction of the sample prior to the development of the instability parallel to the slide reveals that instability occurs first in the thickness direction and indicates that even in a sample of 75 µm thickness three-dimensional effects may be important. The reason for the initial curvature in the thickness direction is uncertain at present. The most likely possibility is that a temperature gradient exists across the thickness of

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Figure 10. Dendrite array morphology for 10% sucrose growing steadily at (a) 10 µm s-1 and (b) 20 µm s-1.

Figure 11. Dendrite array morphology for 20% sucrose growing steadily at (a) 10 µm s-1 and (b) 20 µm s-1.

the sample. Variation in solution concentration would also cause the curvature in the thickness direction. However, since the sample was mounted so that the thickness direction was horizontal, i.e., not parallel to the direction of gravity, this reason can probably be discounted. The difference between the shape of the concentration profiles measured in sucrose solutions compared to those in the majority of the pullulan solutions may be explained by the different nature of the solutions. In sucrose and dilute pullulan (1%) solutions, the solute molecules may be regarded as separate entities, with their motion described by straightforward translational diffusion. The concentration profile ahead of the interface is therefore a single-exponential function, characterized by one length scale. In the more concentrated pullulan solutions, its polymeric properties become important. At concentrations exceeding the critical overlap concentration the pullulan molecules become entangled, which acts to severely hinder diffusion.36 Molecules that have become entangled move more slowly, since chain segments must disentangle before they can diffuse further. In fact, translational diffusion is limited in all directions except parallel to the chain axis. Variations in concentration in the entangled solution cause variations in the local translation diffusion coefficient with the result that there will no longer

be one characteristic length scale and therefore no longer a single-exponential concentration profile. It should also be noted that polydispersity in molecular weight could also contribute to a departure from singleexponential behavior; smaller chains diffuse more rapidly, according to the theory of reptation that describes polymer motion in concentrated solutions36,37 which would cause a broadening of the concentration profile.35 Regardless of the exact shape of the concentration profile, conversion of the local concentration into the equilibrium freezing temperature provided direct evidence for the occurrence of constitutional supercooling leading to the formation of interfacial instability. In all of the cases where the planar front became unstable, constitional supercooling was observed, whereas it was absent in the situations where the front remained planar. Two main considerations, demonstrated by the results reported in this paper, are considered to be pertinent to the validity of the Mullins-Sekerka equation for the prediction of the instability wavelength. First, in the calculation of the instability wavelength it is essential to use the values of the growth velocity, partition coefficient, and concentration gradient for the time at which the instability first develops. Many previous analyses, which have provided poor predictions of the instability wavelength, have used the steady-state


Figure 12. Experimentally measured primary spacing and theoretical predictions for (a) 5% sucrose, (b) 10% sucrose, and (c) 20% sucrose.

values for the growth velocity, with the corresponding values of partition coefficient and concentration gradient.34 The instability wavelength calculated using the steady-state values for sucrose, for example, is about an order of magnitude lower than the experimentally measured ones. Second, the limits of validity of the Mullins-Sekerka analysis are revealed by the use of different solutes with vastly different diffusion rates and mechanisms. The Mullins-Sekerka analysis assumes that the system is growing at a constant rate at the time of instability formation in a steady solute diffusion field, whereas in reality the rate is increasing and the diffusion field is changing. A comparison of the relative rates of change of the solute diffusion field compared to the time over which instability formation occurred reveals the reason for the success of the Mullins-Sekerka equation for sucrose but not for pullulan. For sucrose, the time over which the perturbation developed, given by (δ˙ /δ)-1, was a few seconds, whereas


the time over which the concentration gradient developed was about 200 s. Therefore, although rejection of solute was likely to be enhanced at the tip of the perturbation once lateral instability had occurred, sucrose diffused quickly enough to prevent the concentration profile from altering too rapidly while wavelength selection occurred. To a first approximation, therefore, the solute diffusion field was in a quasi steady state for the time over which the instability wavelength was selected, and the steady-state analysis was valid. For pullulan, however, the time over which the perturbation developed and the time over which the concentration gradient developed were roughly equal. Therefore, the quasi-steady-state assumption was not valid and the Mullins-Sekerka analysis was not suitable for the accurate prediction of the instability wavelength. A rigorous prediction of the instability wavelength would require a non-steady-state analysis to be performed. Further limitations were possibly revealed by the results for sucrose solutions showing a trend toward overestimation of the instability wavelength with increasing concentration. Although several factors may contribute to a discrepancy between measured and predicted wavelength (such as the accuracy of the values of partition coefficient and diffusion coefficient, and the possibility of freeze concentration of the bulk solution), it should also be noted that the Mullins-Sekerka equation was formulated for dilute systems. Although it may accurately represent the 5% sucrose solution, some account of concentration may be necessary at higher concentrations. Primary Spacing. Formation of the dendritic/cellular morphology follows the expected pattern of behavior. At very low growth velocities the front is planar, at slightly higher ones it is cellular, and then at even higher ones a cell-dendrite transition occurs, with a decrease in the dendrite primary spacing with increasing growth velocity. During the transition from the initial perturbation to the steady-state structure the primary spacing changed via tip-splitting and the growth of tertiary branches. However, although qualitatively conforming with prediction, description in quantitative terms remains more elusive. For the steady-state dendritic morphology, the results show that the theoretical analysis by Dexin and Sahm11 using a simplified version of the solvability theory can be discounted as a reasonable model for describing directionally crystallized dendritic ice structures in sucrose solutions. Reasons for this probably include the simplistic nature of some of the assumptions regarding the geometry of the dendrites and the omission of terms known to be important for describing dendritic growth such as the surface energy anisotropy. The Hunt and Lu model also contains assumptions, such as the constant partition coefficient over all growth velocities, although it takes into account anisotropy of the surface energy.14,38 Importantly, it also takes into account that the dendrites grow in an array and are not isolated. In the case of the systems studied in this report the diffusion length of the solute is between 100 and 200 µm, whereas the dendrite primary spacing is in the range 50-100 µm. Consequently, the dendrites cannot be regarded as isolated entities and definitely influence each other’s growth (this may be particularly important

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for consideration of the secondary structure). However, although the Hunt and Lu model is suitably able to predict the range of primary spacing observed in 5 wt % sucrose solution, it is not effective for 10 wt % and 20 wt % sucrose solutions, becoming less reasonable as the concentration increases. Again, the discrepancy may be due to invalidity of the assumptions, or it may be due to the fact that the values for the partition coefficient and diffusion constant differ with concentration. However, increasing the concentration might be expected to lower the diffusion coefficient, which has the effect of decreasing the primary spacing predicted by the Hunt and Lu model. It should be noted that most of the reported experimental data that have been successfully predicted by the Hunt and Lu model have been made with relatively dilute systems38-40 and there are cases in systems that are not dilute where it fails to predict the observed spacing.11 That the Trivedi and Kurz model provides a reasonable agreement between the theoretical and experimental results is perhaps not too surprising, since this model contains all of the relevant physics of the processes dominating the expressed length scales, assuming that interface attachment kinetics are unimportant. However, it is also the least satisfying, because it incorporates an empirical prefactor, the value of which is presumably determined by complicating factors such as the interaction between the solute and temperature fields of neighboring dendrites. The existence of a scaling relationship between the instability wavelength and primary spacing is also reasonable. For the succinonitrile-acetone system, Trivedi and Somboonsuk found that the instability wavelength was the same as the dendrite secondary spacing, which could be related to the dendrite primary spacing via a scaling law.41 Trivedi and Kurz give an expression, obtained via a dimensional argument as for their expression for the dendrite primary spacing, for the instability wavelength and secondary spacing:3

λsecondary ∝ G-1/3V-1/3

(7)

Therefore, the scaling relationship between the primary spacing, given by eq 3, and the instability wavelength is

λdendrite ) BλsecondaryV1/12

(8)

This equation is very similar to the experimentally determined scaling relationship (eq 6). In eq 8 the exponent of the growth velocity is 0.083, whereas in the experimental case it is 0.078. Such an agreement further supports the claim that diffusion and capillarity terms adequately describe the growth and that no modifications are required due to other factors such as interface attachment kinetics. Conclusions Mach-Zender interferometry has been used to measure, in situ, the solute concentration profile in the vicinity of the ice-solution interface for the ice-sucrose solution and ice-pullulan solution systems. It was demonstrated that constitutional supercooling caused instability formation. Furthermore, using the values of

the growth velocity and concentration gradient at the interface at the time of instability formation, the Mullins-Sekerka theory of instability formation was verified for the case of a small molecule (sucrose) over a range of values of partition coefficient. The limitations of the theory were demonstrated, however, by the use of an entangled polymer system (pullulan), in which molecular entanglement significantly slowed the solute diffusion rate. In the case that the solute concentration field changes over the same time scale as the time taken for the instability to develop, the Mullins-Sekerka theory no longer supplies an accurate prediction of the instability wavelength. Three different equations for predicting the final dendrite primary spacing were tested for sucrose solutions. The Hunt and Lu model worked well for dilute systems but decreased in accuracy as the solutions became more concentrated. Reasonable agreement was obtained with an equation that contained terms relating to the diffusion length scales (thermal and solute) and a capillarity term but also contained one freely varying parameter, suggesting that since these terms are sufficient to describe dendrite growth interfacial attachment kinetics must play only a minor role. A scaling relationship was also found between the instability wavelength and the dendrite primary spacing, with the same exponent as one predicted by dimensional arguments. Acknowledgment. Julia Telford, of Unilever Research, Colworth, is thanked for performing the solute partition coefficient measurements. References (1) Tiller, W. A. in The Science of Crystallization; Macroscopic Phenomena and Defect Generation; Cambridge University Press: Cambridge, U.K., 1991; pp 231-364. (2) Pimpinelli, A.; Villain, J. In Physics of Crystal Growth; Cambridge University Press: Cambridge, U.K., 1998; pp 156-180. (3) Trivedi, R.; Kurz, W. Int. Mater. Rev. 1994, 39, 49. (4) Barbieri, A.; Hong, D. C.; Langer, J. S. Phys. Rev. A 1987, 35, 1802. (5) Langer, J. S.; Hong, D. C. Phys. Rev. A 1986, 34, 1462. (6) Barbieri, A.; Langer, J. S. Phys. Rev. A 1989, 39, 5314. (7) Kessler, D. A.; Koplik, J.; Levine, H. Adv. Phys. 1988, 37, 255. (8) Laxmanan, V. Phys. Rev. E 1988, 57, 2004. (9) Mullins, W. W.; Sekerka, R. F. J. Appl. Phys. 1964, 35, 444. (10) Sekerka, R. F. J. Appl. Phys. 1965, 36, 264. (11) Dexin, M. A.; Sahm, P. R. Metall. Mater. Trans. A 1998, 29, 1113. (12) Somboonsuk, K.; Mason, J. T.; Trivedi, R. Metall. Trans. A 1984, 15, 967. (13) Trivedi, R.; Somboonsuk, K. Mater. Sci. Eng. 1984, 65, 65. (14) Hunt, J. D.; Lu, S.-Z. Metall. Mater. Trans. A 1996, 27, 611. (15) Warren, J. A.; Langer, J. S. Phys. Rev. E 1993, 47, 2702. (16) Makkonen, L. J. Cryst. Growth 2000, 208, 772. (17) Korber, C. Q. Rev. Biophys. 1988, 21, 229. (18) Raz, E.; Lipson, S. G.; Polturak, E. Phys. Rev. A 1989, 40, 1088. (19) Onuma, K.; Tsukamotom, K.; Nakadate, S. J. Cryst. Growth 1993, 129, 706. (20) Kostianovski, S.; Lipson, S. G.; Ribak, E. N. Appl. Opt. 1993, 32, 4744. (21) Miyashita, S.; Komatsu, H.; Suzuki, Y.; Nakada, T. J. Cryst. Growth 1994, 141, 419. (22) Notcovich, A. G.; Lipson, S. G. Physica A 1998, 257, 454. (23) Mantani, M.; Sugiyama, M.; Ogawa, T. J. Cryst. Growth 1991, 114, 71. (24) Braslavsky, I.; Lipson, S. G. Physica A 1998, 249, 190.

Directional Dendritic Growth of Ice (25) Notcovich, A. G.; Braslavsky, I.; Lipson, S. G. J. Cryst. Growth 1999, 198/199, 10. (26) Tanaka, A.; Sano, M. J. Cryst. Growth 1992, 125, 59. (27) Emsellem, V.; Tabeling, P. Europhys. Lett. 1992, 25, 277. (28) Emsellem, V.; Tabeling, P. J. Cryst. Growth 1996, 166, 251. (29) Bedarida, F. J. Cryst. Growth 1986, 79, 43. (30) Bedarida, F.; Zefiro, L.; Boccacci, P.; Aquilano, D.; Rubbo, M.; Vaccari, G.; Mantovani, G.; Sgualdino, G. J. Cryst. Growth 1988, 89, 395. (31) Yu, Y.; Yue, X.; Gao, H.; Chen, H. J. Cryst. Growth 1990, 106, 690. (32) Shen, Y. B.; Poulikakos, D. J. Heat Transfer 1996, 118, 249. (33) Nagashima, K.; Furukawa, Y. J. Phys. Chem. B 1997, 101, 6174. (34) Nagashima, K.; Furukawa, Y. J. Cryst. Growth 2000, 209, 167.

Crystal Growth & Design, Vol. 1, No. 3, 2001 223 (35) Hunt, J. D.; Jackson, K. A.; Brown, H. Rev. Sci. Instrum. 1966, 37, 805. (36) Strobl, G. In The Physics of Polymers; Springer-Verlag: Berlin, 1996; pp 277-296. (37) Doi, M.; Edwards, S. F. In The Theory of Polymer Dynamics; Oxford University Press: New York, 1986; pp 188-288. (38) Hunt, J. D.; Han, Q.; Wan, X. J. Mater. Sci. Technol. 1997, 13, 161. (39) Feng, J.; Hunag, W. D.; Lin, X.; Pan, Q. Y.; Li, T.; Zhou, Y. H. J. Cryst. Growth 1999, 197, 393. (40) Ding, G. L.; Huang, W. D.; Huang, X.; Lin, X.; Zhou, Y. H. Acta Mater. 1996 44, 3705. (41) Trivedi, R.; Somboonsuk, K. Acta Metall. 1984, 33, 106.

CG005534Q

Instability Formation and Directional Dendritic Growth of Ice Studied by

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