Background On Dendritic Growth Theory and Experiment

Dendritic growth exhibits both steady-state features, where the tip region of the dendrite grows at a constant speed in a shape preserving manner, and time-dependent features where side-branches periodically emerge, grow, and eventually coarsen. The quantitative study of dendrites began with the transport analysis of Ivantsov in 1946 [Ivantsov, 1947], and over the last fifty years a number of theories of steady-state dendritic crystal growth have been developed [Glicksman, 1993].

Ivantsov, based his transport analysis on a suggestion made by Papapetrou [Papapetrou, 1935], who, in 1935, first described the form of a dendritic crystal as a paraboloid of revolution with fixed radius of curvature R, growing at a constant velocity, V. Ivantsov solved the diffusion equation in co-moving paraboloidal coordinates, scaled by the only length scale in the problem, R. He solved the dimensionless energy diffusion equation

Equation #1:

where,

Equation #2:

is the non-dimensional scaled supercooling; Tm is the melting temperature of the material; T is the temperature of the melt far from the dendrite; DeltaHf is the molar latent heat, and Cp is the constant pressure molar specific heat. The coefficient appearing in Eq. (1), P=VR/2alpha , is the growth Péclet number, where V is the dendrite tip velocity, and alpha is the thermal diffusivity of the melt phase. By assuming that all the energy released by solidification diffuses away from the isothermal interface via the melt phase, Ivantsov obtained the following equation relating the supercooling and growth Péclet number for steady- state paraboloidal dendrites,

Equation #3:

where E1(P) is the first exponential integral function, and Iv(P) is defined here as the Ivantsov function. Inasmuch as the supercooling, theta, is the independent variable, a more useful transport relation is

Equation #4:

where Iv¯¹(theta,) is the formal inverse to Eq. (3). For a thorough discussion of the solution to the heat diffusion equation for dendrites see Langer's review [Langer, 1980] and Pelce's book [Pelce, 1961].

The heat transport solutions, such as Eq. (3), or Eq. (4), are well known to only provide an incomplete description of steady-state dendritic growth, insofar as they specify only the growth Péclet number, VR/2, as a function of the supercooling, and not the individual observable components V, and R. Just specifying the driving force, S, with the steady-state diffusion solution does not uniquely predict the operating state. Ivantsov's formulation is under-determined as it provides only one equation for the two unknowns. Bolling and Tiller [Bolling, 1961] were the first to ascribe the non-uniqueness of the diffusion solution as caused by a deficiency of physical length scales in the transport problem, viz., either the unknown diffusion length, V/alpha, or the unknown tip radius, R.

The original assumption used by Ivantsov that the interface temperature is the equilibrium melting point, Tm, precluded any role of interfacial physics, like capillary, in the problem formulation. Capillary effects in crystal growth dynamics have been well recognized since the study of morphological stability by Mullins and Sekerka [Mullins, 1963]. Moreover, most of the theoretical studies of dendritic growth produced since Ivantsov;s original paper have, with mixed success, reintroduced the effects of interfacial capillary and kinetics. Capillary, for example, provides another independent length scale, do, defined as

Equation #5:

where Omegas is the molar volume, Sf is the molar entropy of fusion, and gamma is the solid-liquid interfacial energy.

In the early 70's, Glicksman, Schaefer, and Ayers [Glicksman, 1976] developed the use of the BCC organic plastic crystal succinonitrile (SCN: NC(CH2)2CN), as a model system for studying dendritic growth. SCN solidifies similarly to the cubic metals, i.e., with an molecularly rough solid-liquid interface, yet SCN retains an advantage because it has a low melting temperature (~ 58.08 C), excellent chemical stability, optical transparency, and accurately known thermophysical properties. In fact, the use of SCN facilitated many dendritic growth studies over the past twenty years, permitting dendritic tip velocities and radii to be accurately measured and then used as a test of theory. During that period, several researchers found comparable results to those reported by Glicksman, Schaefer, and Ayers [Glicksman, 1976] using other pure or alloy model systems (see, for example, Tables 2 and 3 in reference [Glicksman, 1993]).

Since the mid-1970 s, some theoretical efforts within the physics community have been directed to answering the question as to whether, and under what conditions, a second independent equation or length scale exists, which, when combined with the Ivantsov diffusion solution, selects the observed dendritic operating state. Theoretical efforts to identify a second length scale were in large measure stimulated by Oldfield s numerical study [Oldfield, 1973] that first suggested that VR²=const., which is a consequence of the balance between the destabilizing interfacial effects of heat conduction or solute diffusion and the stabilizing effects of capillary. Oldfield's result was supported by subsequent experimental observations of unconstrained dendritic growth, confirming that VR², is approximately constant.

Langer and Müller-Krumbhaar [Langer, 1978] developed the first scaling analysis of dendritic growth by hypothesizing that dendrites grow at the margin of stability. They suggested that the steady-state tip radius matched a critical wavelength, lambda*. By coupling Ivantsov's solution with morphological stability, they showed that lambda* is proportional to the geometric mean between the thermal diffusion length, V/alpha, and the capillary length, do. If lambda* were equated with the operating tip radius R, Langer and Müller-Krumbhaar found that

Equation #6:

where sigma*, the theoretical scaling constant, is equal to 1/(4Pi²). In the twenty years since the suggestion of the marginal stability hypothesis, Langer, Levine, and others (see references in Langer article in Science [Langer, 1989] and in Brener and Mel'nikov [Brener, 1991]) adapted the method of microscopic solvability and developed steady-state theories which specify that the anisotropy in the interfacial energy is the main factor in selecting the dendritic operating state. A theory by Miyata et al. [Miyata, 1991] is also based on the idea that the dendrite operating state depends on an isotropy of the interfacial energy, but in a way quite different from microscopic solvability. Additionally, there are theories (see references in [Glicksman, 1993]) which suggest that thermal noise provides the additional physics needed for the second equation. In fact, similar scaling equations can be derived by starting from quite different dynamical considerations, although their results are expressible through a scaling constant as in Eq. (6). For example, solving Eqs. (3) and (6) for V and R yields,

Equation #7:

and,

Equation #8:

where V and R are the unique observables of steady-state dendritic growth depending only onthe supercooling and a few material constants. In practice, unfortunately, none of these theories actually provide a zero-parameter test of dendritic growth, because the value of the scaling constant is not calculated rigorously, but only estimated. Also, the experimental dendritic scaling constants are generally calculated from measured data only as a convenient way of classifying the results from a specific experiment.

In the case of pure SCN (4-9's or better), the data obtained by Glicksman, Scheafer, and Ayers [8] were consistent with a scaling constant of sigma*=0.019, which was inreasonable agreement with the theoretical value 0.025 estimated from marginal stability. That particular experimental sigma* value was obtained over a limited supercooling range, and the uncertainties in the observed tip radii were far too large to allow definitive conclusions. Huang and Glicksman [Haung, 1981] later performed similar experiments on high-purity SCN (better than 5-9's) at smaller supercoolings, where the velocities are slower and the tip radii are larger, permitting more precise measurements. At the larger supercoolings studied by Huang and Glicksman, their experimental results were consistent with both the scaling analysis and prior experiments. Detailed experimental work by Huang and Glicksman showed, however, that the direction with respect to gravity of the [100] growth axis of a primary SCN dendrite affected its steady-state velocity and radius [Glicksman, 1982]. In fact, Huang and Glicksman showed that gravity-induced convection can dominate the heat transport during dendritic growth, especiallyin the lower supercooling range of their experiments, which corresponds to typical conditions encountered in the solidification of industrial castings.

There have been a number of attempts to model the influence of natural or forcedconvection on dendritic growth by Saville [Saville, 1988], Ananth and Gill [Anath, 1988 1991], and more recently by Sekerka, McFadden, and Coriell [Glicksman, 1988]. All these hydrodynamic calculations that attempt to include gravitational effects are invariably coupled to as yet unproved elements of basic dendritic growth theory, and, consequently, can not provide an independent test of the theory. Experiments performed at higher supercoolings, where convective influences diminish in comparison to thermal conduction, do not assist much either. Unfortunately, the morphological scale of dendrites becomes too small to be resolved at the high growth speeds encountered. The experimental situation prior to the microgravity experiment reported onhere, was that there was too narrow a range of supercoolings in any dendritic system studied terrestrially that remained tolerably free of convection effects, and also permitted accurate determination of the tip radius of curvature. Thus, a microgravity dendritic growth experiment was proposed to NASA by one of the authors about ten years ago to measure definitively the kinetics and morphology of convection-free dendrites, and to determine the effects of gravity on dendritic growth [Schrieffer, 1987] [Glicksman, 1995].[20,21]. Such a test requires that a single, clearly defined experimental parameter the supercooling, fully determine the resulting steady-state dendritic growth velocity, V, and tip size, R.