CRYSTAL GROWTH & DESIGN
Growth of Solutal Ice Dendrites Studied by Optical Interferometry
2002 VOL. 2, NO. 1 59-66
Michael F. Butler† Unilever Research, Colworth House, Sharnbrook, Bedfordshire MK44 1LQ, United Kingdom Received October 23, 2001
ABSTRACT: Optical interferometry was used to study ice dendrite growth in a homologous series of sugar solutions, chosen in order to probe the effect of solute diffusion coefficient on solutal ice dendrite growth. Solute concentration fields, three-dimensional dendrite morphologies, and dendrite growth velocities were measured, and the results were compared with predictions from the modified Ivantsov and solvability theories of dendrite growth. The dendrites were not axially symmetrical but had a parabolic profile parallel to the basal plane and were surrounded by a paraboloidal solute concentration field, in accordance with the Ivantsov predictions combined with anisotropic growth kinetics. Solvability theory, for a system governed by solute diffusion, accounted for the relation between the dendrite tip morphology and the growth velocity, with a value of 0.006 ( 0.001 measured for the stability parameter, σ*, from the basal tip radius. Different values were measured when the dendrites were formed in an imposed temperature gradient, which was ascribed to the increased importance of nonequilibrium effects in these cases. Introduction Dendrites are tree-like crystals formed from a solidifying melt or solution and provide one of the most familiar, widely studied, and complex examples of spontaneous pattern formation. Their growth is mainly controlled by the transport of latent heat and solute away from the solidifying interface but is also influenced by surface tension, interface attachment kinetics, and convection.1-3 A basic theoretical understanding of dendrite growth stems from the predictions of Ivantsov,4 who showed that the dendrite profile, the temperature field, and the solute concentration field in the vicinity of the dendrite tip should be paraboloidal for a shape-maintaining steady-state interface governed by heat and mass diffusion. Although detailed experiments have shown that this is not always precisely the case,5 it is a reasonably accurate approximation in practice. Using the theory of Ivantsov, with the addition of a stabilizing surface energy term, it is possible to relate the growth conditions (undercooling) to the thermal and solute Peclet numbers, (Pt and P, respectively) via the equation
∆T )
k0∆T0Iv(P) 2Γ ∆H Iv(Pt) + + cl R 1 - (1 - k0)Iv(P)
(1)
where ∆H is the latent heat of fusion, cl is the thermal conductivity of the liquid phase, k0 is the equilibrium partition coefficient, Pt is the thermal Peclet number, defined as Pt ) {VR}/{2a}, and P is the solute Peclet number, defined as P ) {VR}/{2D}, where V is the growth velocity, R is the dendrite tip radius, a is the thermal diffusivity and D is the solute diffusion coefficient. Iv(Pt) and Iv(P) are the Ivantsov parabolas calculated for the thermal and solute diffusion fields respectively, and can be approximated as1
Iv(P) ) a0 + a1P + a2P2 + a3P3 + a4P4 + a5P5 - ln P (2) † E-mail:
[email protected]. Tel. +44(0)1234 222958. Fax: +44(0)1234 222757.
for a needle crystal within the range of Peclet numbers 0‚P‚1, which is the relevant range for the current study. The parameters a0-a5 take the values -0.577 215 66, 0.999 991 93, -0.249 910 55, 0.055 199 68, -0.009 760 04, and 0.001 078 57, respectively. Γ is a capillarity (surface tension) term, defined as Γ ) {γ∆V}/{∆S}, where γ is the solid liquid interfacial energy, ∆V is the molar volume of ice, and ∆S is the entropy of fusion and ∆T0 is defined as ∆T0 ) mC0(1 - k0)/k0, where m is the liquidus slope. Further developments were aimed at providing a means for independently predicting the dendrite tip radius and the growth velocity for given growth conditions and led to the “solvability theory”,1,6-9 which provides an independent equation that relates the dendrite tip radius, R, and the growth velocity, V, in terms of some material constants:
VR2 )
()
Ct 2ΓD σ*k0∆T0 C0
(3)
where Ct is the solute concentration at the dendrite tip, C0 is the bulk solute concentration, and σ* is a constant known as the “stability parameter”. For directional growth experiments the factor of 2 in the numerator is omitted.1 If eq 3 is valid, the quantity VR2 is therefore expected to be constant over a range of growth velocities. σ* has been measured to be approximately 0.02 for the majority of pure systems that grow as needle crystals1 but takes the value 0.075 for the pure ice/water system10,11 because of the anisotropy in surface energy and interface attachment kinetics between the basal and edge planes that causes the dendrites to lose cylindrical symmetry and assume an elliptical profile when viewed end-on.10,12,13 The dendrites therefore possess two tip radii, differing in size by an order of magnitude: the basal (larger) radius, R1, and an edge radius, R2. The value for σ* of 0.075 was obtained using the harmonic mean, Rm, of the two tip radii. Use of the basal tip radius in eq 3 has also been shown to yield a value of σ that is independent of undercooling10 (although different from the value obtained using Rm), but for the edge radius it
10.1021/cg0155604 CCC: $22.00 © 2002 American Chemical Society Published on Web 12/12/2001
60
Crystal Growth & Design, Vol. 2, No. 1, 2002
Butler
Table 1. Diffusion Coefficients (×10-10 m2 s-1) of the Various Solutes Used, for a Range of Concentrations diffusion coeff concn (M)
glucose
maltose
maltotriose
dextran
0.10 0.25 0.40 0.50 1.00
6.82 6.55 6.30 6.13 5.39
5.73 5.34 4.99 4.76 3.80
4.05 3.69 3.37 3.18 2.36
2.48 2.16 1.61
is a function of the undercooling. Interestingly, for the pure ice/water system the values of growth velocity were well predicted by the Ivantsov and solvability theories, assuming cylindrically symmetrical (needle crystal) dendrites, even though the dendrites were elliptical paraboloids. The reason for this agreement remains uncertain, however. Testing for this phenomenon in an ice/solution system (which has received much less attention that the pure ice/water system for quantitative studies) was therefore one of the motivations for the current study. Also, since it is known that the succinonitrile/acetone and succinonitrile/argon binary systems possess a different value of σ* than that measured for pure succinonitrile melts,14 another aim of the current study was to test for similar differences between an ice/solution system and the pure ice/water system. A homologous series of sugars was chosen for making the solutions, because it provided a well-characterized system in which the diffusion coefficient could be changed in a known way with both solute type and concentration while retaining chemical similarity of the solutes. Interferometry was used to study dendrite growth, because it was able to measure both the relative differences in concentration in the solution surrounding the dendrite tip15-17 and the threedimensional dendrite shape,10 therefore providing an effective way of testing dendrite growth theories that predict these quantities. Experimental Section Materials and Material Characterization. Solutions of glucose (Mw 180.16), maltose (Mw 360.32), maltotriose (Mw 504.44), and dextran (Mw 1500) with concentrations between 0.1 and 1.0 M were made by dissolving the sugar in distilled, deionized water. The solutions were stored in a refrigerator at 5 °C prior to use. The solute diffusion coefficient, D, was calculated using equations derived by Sano and Yamamoto18 from data on a series of sugars with molecular weights in the range 180-6100. The values obtained for the different sugars over the range of concentrations used are tabulated in Table 1. The liquidus slope, m, was obtained from tabulated values of the freezing point depression at different concentrations.19 The equilibrium partition coefficient was obtained by measuring the velocity, Vp-c, at which a planar interface became cellular during growth in a temperature gradient, G (7.2 K mm-1), and applying the equation1
Vp-c )
GD ∆T0
(4)
Values of 0.0067, 0.0018, and 0.0009 were obtained for glucose, maltose, and maltotriose, respectively. No value was obtained for dextran because measurement of Vp-c for the dextran solutions was not possible. Analysis of the results for dextran was therefore more limited than for the other sugars.
The effective partition coefficient, k, of glucose and maltose was taken to be unity above growth speeds of about 5 µm s-1.20,21 Although the values for maltotriose are not known, it was assumed that it also possessed an effective partition coefficient of 1 above 5 µm s-1. Values of 3.33 × 108 J m-3, 22 J mol-1 K-1, 1.96 × 10-6 m3, 4210 J kg-1 K-1, 1.34 × 10-7 m2 s-1, and 2.8 × 10-2 J m-2 were taken for ∆H, ∆S, ∆V, cl, a, and γ, respectively.19 Isothermal Dendrite Growth. Isothermal dendrite growth experiments were performed using a temperature-controlled microscope stage (Linkam MDS600) that was capable of controlling the sample temperature to a precision of 0.03 K. The stage was mounted in one arm of a Mach-Zender interferometer that was used as an interference microscope. An accurately measured volume of sample was placed in a sapphire crucible containing a 50 µm thick stainless steel spacing ring and covered with a 0.17 mm thick glass coverslip that acted as a lid which sealed the sample within the spacer. The microscope stage was sealed with a lid, and the volume containing the block and sample was purged with dry nitrogen to eliminate condensation from forming on the sample. Samples used for these experiments were glucose, maltose, and maltotriose solutions at concentrations of 0.1, 0.25, 0.4, 0.5, and 1.0 M and dextran solutions at concentrations of 0.1, 0.25, and 0.5 M. The sample was cooled to about -20 °C, which was sufficient to nucleate ice crystals that grew rapidly with a fine dendritic morphology, and then heated to a temperature in the vicinity of the melting point. By subjecting the sample to a thermal cycle of heating and cooling by a few tenths of a degree, it was possible to melt back most of the ice that had initially formed, leaving only a few, well-separated, disc-shaped crystals. Once a sufficiently small number of crystals were present, the sample was rapidly cooled to the temperature of interest and the ensuing dendrite growth was studied while the temperature was maintained at that value. The range of undercooling that could be studied was limited to the values for which the dendrites were large enough to be accurately measured. Directional Dendrite Growth. Directional dendrite growth experiments were performed using the temperature gradient stage described previously.20 The samples consisted of a drop of sugar solution on a glass microscope slide surrounded by a 50 µm spacer on top of which a standard 0.17 mm microscope coverslip was placed. The “cold” and “hot” plates of the stage were maintained at -15 and +5 °C, respectively, which gave a temperature gradient of 1000 K m-1. A constant stream of dry nitrogen gas was passed over the sample to prevent condensation from forming on the top of the coverslip. Once formed, the ice/solution interface was allowed to stabilize for several hours until it became uniform and planar. Further ice growth was initiated by moving the sample toward the “cold” plate at a controlled rate, thus cooling the solution ahead of the ice/solution interface. The morphology of the interface during steady-state dendrite growth, from which the tip radius was directly measured, was observed using optical microscopy. Glucose, maltose, and maltotriose concentrations of 0.5 M and maltose solutions of concentration 1.0 M were used in these experiments. Optical Interferometry and Measurement of SteadyState Dendrite Morphology. A Mach-Zender interferometer, described in detail previously,20 was used to measure the three-dimensional dendrite morphology and the solute concentration field around the dendrite tip. Interferometry was used in preference to microscopy because of its ability to simultaneously image the interference fringes, from which the concentration field and dendrite thickness were calculated where possible, and the lateral dendrite morphology, from which the dendrite tip radius parallel to the sample plane was obtained. To calculate the spatial variation of concentration the following formula was used:
C - C0 λ∆x ) C (n(C) - n0)hw
(5)
Growth of Solutal Ice Dendrites
Crystal Growth & Design, Vol. 2, No. 1, 2002 61
Figure 1. Growth of a disk crystal (a), in some cases exhibiting some faceting (b), until an instability is initiated around the crystal perimeter (c) that leads to dendritic growth (d). The fringes in the vicinity of the crystal become curved in the vicinity of the crystal (b, c), indicating the existence of a gradient in the solute concentration. where C is the local concentration, C0 is the initial bulk concentration, λ is the wavelength of the light used (633.8 nm in this case, from a 10 mW CW HeNe laser), ∆x is the deviation of the fringe from the reference position caused by the change in concentration ∆C ) C - C0, h is the sample thickness (50 µm), w is the separation of the reference fringes (set, by tilting one of the mirrors in the interferometer, to be 7.2 µm), n(C) is the refractive index of a solution with concentration C, and n0 is the refractive index of pure water (1.333 at 273 K). The concentration dependence of the refractive index was obtained from measurements of the refractive index (at 589.3 nm) of the solutions used in the isothermal experiment over a range of temperatures and extrapolated to 273 K. In all cases the concentration dependence of the refractive index was 2 orders of magnitude greater than the temperature dependence, which can therefore be treated as negligible. The spatial variation of the dendrite thickness, t, was calculated using the formula
t)
λ∆x (nice - n(C0))w
(6)
The refractive index of ice was taken to be 1.308, and it was assumed that the dendrite was surrounded by solution with a constant concentration equal to the initial bulk concentration. For steady-state dendrite growth this simplification is a reasonable one to make.1
Results Isothermal Dendrite Growth Morphology. In all cases, dendritic growth was initiated from small discshaped crystals that grew uniformly until they reached a certain radius (shown in Figure 1), whereupon a periodic perturbation appeared around the circumference that developed into a dendritic morphology. For low undercoolings the disk crystal became slightly faceted into hexagons (Figure 1b). The initial uniform
Figure 2. Steady-state dendritic morphology in a 0.5 M maltose solution dominated by primary dendrite growth. Note the curvature of the fringes in the solution near the tip.
growth was accompanied by bending of the interference fringes in the solution around the crystal. The steady-state growth morphology depended upon the amount of undercooling applied. At low undercoolings the main branches of the dendrites were large and rounded and the side branches were not well-developed. At slightly higher undercoolings, fully formed dendrites grew (as shown in Figure 2) with a regular array of welldeveloped secondary branches. At even higher undercoolings there was a rapid growth of tertiary branches parallel to the primary ones. In some cases tertiary branches emerged that initially grew faster than the parent primary branch. Consequently, when the distance between the parent primary tip and the first
62
Crystal Growth & Design, Vol. 2, No. 1, 2002
Butler
Figure 3. (a) Measured dendrite tip profile for 0.5 M maltose at 0.9 K undercooling with parabolic fit. Points 1-3 indicate the positions at which the dendrite thickness, shown in (b), was measured from the interference image, assuming the dendrite to be symmetrical about the horizontal plane through the center.
secondary branch was sufficiently small, the tertiary branches overwhelmed the primary ones and became themselves the next generation of primary dendrites. At the highest undercoolings studied, growth of the primary dendrites was so rapid that the tertiary branches never grew fast enough to overwhelm them. Occasionally, at all undercoolings studied, the primary tip split into two branches. In some cases both branches continued to grow apace but in others one of the branches became dominant. In all cases the tip profile was measured to be parabolic in the plane of the sample, demonstrated in Figure 3a for 0.5 M maltose samples, undercooled by 0.9 K. Away from the tip, in the vicinity of the side branches, the interface shape deviated substantially from being parabolic. Interferometric measurement of the dendrite thickness, via the fringe curvature in the ice, showed that the dendrites possessed a flattened profile. Figure 3b shows the thickness profiles through the dendrite at different distances from the tip for the same sample for which the lateral tip profile was shown in Figure 3a. Although there was insufficient data for
Figure 4. Plots of the experimentally measured growth velocity for dendrites grown in (a) glucose, (b) maltose, and (c) maltotriose solutions. The lines are theoretical predictions for the growth speeds of dendrites calculated using eqs 1-3.
accurate measurement of the edge tip radius, the edge tip radius was approximately 1 order of magnitude smaller than the basal tip radius. Growth Velocity. The growth velocities measured for glucose, maltose, and maltotriose, at all concentrations, are shown in parts a-c of Figure 4, respectively. The lines are the predicted values using eqs 1-3, using the experimentally determined value for σ*. For all of the samples the growth velocity at a given undercooling decreased with increasing solute concentration, whereas the predicted velocities passed through a maximum. The predicted values tended to understimate the velocities for low concentration and overestimate them at high
Growth of Solutal Ice Dendrites
Crystal Growth & Design, Vol. 2, No. 1, 2002 63
Figure 5. Steady-state concentration profile in the vicinity of a dendrite tip in an undercooled 0.5 M dextran solution.
concentrations. Reasonable agreement between measured and predicted values was attained at concentrations of approximately 0.5 M for glucose, 0.25 and 0.4 M for maltose and 0.1 and 0.25 M for maltotriose. The gradient of the line made by the data points agreed with the predicted gradient for glucose and maltose solutions, and for maltose, the increase in gradient that was observed in the 1.0 M sample was also predicted by the theoretical values. The predicted gradient was, however, lower than than the actual one for the maltotriose samples. No agreement was found between the experimental data and the growth velocity predicted using a value of 0.02 for σ*. Solute Concentration Field. Figure 5 shows the steady-state concentration field in the vicinity of a dendrite in a 0.5 M dextran solution. The iso-concentration lines were parabolic near the tip, with the influence of the side branches (visible at the top of the image) manifested as a region of enhanced concentration in their vicinity. For the smaller molecular weight sugars, the distance between the tip and the first side branch was too small to enable the two-dimensional concentration field to be mapped satisfactorily. It was, however, possible to measure the concentration profile away from the tip for the 0.5 M and 1.0 M maltose and maltotriose solutions (for the other samples the concentration resolution of the interferometer was not sufficient to provide accurate data). The concentration profile with the greatest tip concentration is shown in Figure 6. A single-exponential fit was placed on these data, from which the diffusion length was calculated (taken as the distance over which the concentration decayed to 1/e of the initial value). The ratio of the theoretical diffusion length (V/D) to the measured diffusion length was as follows: 0.5 M maltose, 2.0; 1.0 M maltose, 3.2; 0.5 M maltotriose, 2.4; 1.0 M maltotriose, 3.9. The discrepancy between the expected and the measured diffusion length therefore apparently increased as both the solute concentration and molecular weight increased. The difference between the tip concentration and the bulk concentration, shown in Figure 6, was generally fairly small, which is consistent with the measured effective partition coefficient being close to unity, and therefore the ratio Ct/C0 was taken to be approximately equal to 1. Tip Selection Criterion. Equation 3 was tested by plotting VR2 as a function of D/(k0∆T0) for glucose, maltose, and maltotriose solutions, as shown in Figure 7 for the isothermal dendrite growth data. The data for
Figure 6. Concentration measured away from the dendrite tip in an undercooled 1.0 M maltose solution.
Figure 7. Test of the selection criterion for ice dendrites grown in glucose, maltose, and maltotriose solutions.
dextran could not be analyzed, since the value of k0 was not known. Within the limits of experimental uncertainty, all of the data points lay on a common line. A straight line passing through the origin could be fitted to these data, leading to the following value of the stability parameter, σ*, for isothermal dendrites in sugar solutions: 0.006 ( 0.001. The product VR2 is shown as a function of growth velocity in Figure 8 for dendrites formed in the directional growth experiments. Figure 8a shows the results for 0.5 M glucose, maltose, and maltotriose, and Figure 8b shows the results for the 1.0 M glucose solution. The value of VR2 was constant, within error, for each solute concentration, as predicted by eq 3, and shown by the solid lines in Figure 8. However, although the values of VR2 were in the correct order (i.e., they decreased with increasing solute molecular weight/decreasing solute diffusion coefficient), they did not agree with the values of VR2 calculated using the values of σ* measured from the isothermal growth experiments. Furthermore, the stability parameter varied with both solute type and solute concentration. The values of σ* (subject to an uncertainty of about 20%) extracted from the data in Figure 8 were as follows: 0.5 M glucose, 0.0008; 1.0 M glucose, 0.0001; 0.5 M maltose, 0.0002; 0.5 M maltotriose, 0.0003.
64
Crystal Growth & Design, Vol. 2, No. 1, 2002
Figure 8. VR2 for directionally crystallized dendrites: (a) 0.5 M glucose, maltose, and maltotriose solutions; (b) 1.0 M glucose and maltose solutions. The solid lines are the average of the VR2 values for each sample, to demonstrate that VR2 was constant over the range of growth velocities studied.
Discussion Isothermal Dendrite Growth Morphology. The qualitative results of dendrite growth in sugar solutions agree with findings from previous studies of the pure ice/water system, where several stages in the growth of dendritic structures were observed in the undercooling range from 0.06 to 0.29 K. These were categorized as the growth of an initially smooth disk crystal, development of instabilities, and the final establishment of a shape-preserving steady state.22 The more complicated morphologies that were observed at higher undercoolings in the present system were probably a result of a temporary reduction in growth velocity of the primary dendrite during the formation of each secondary dendrite.23 If the distance between the primary dendrite tip and the first secondary dendrite was sufficiently small, and if the tertiary dendrite grew sufficiently rapidly, the possibility exists that it could overtake the temporarily slower primary dendrite and itself become the new primary dendrite. Tip splitting at low undercoolings (
