# Mechanism of Dendritic Branching

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## Abstract

Theories of dendritic growth currently ascribe pattern details to extrinsic perturbations or other stochastic causalities, such as selective amplification of noise and marginal stability. These theories apply capillarity physics as a boundary condition on the transport fields in the melt that conduct the latent heat and/or move solute rejected during solidification. Predictions based on these theories conflict with the best quantitative experiments on model solidification systems. Moreover, neither the observed branching patterns nor other characteristics of dendrites formed in different molten materials are distinguished by these approaches, making their integration with casting and microstructure models of limited value. The case of solidification from a pure melt is reexamined, allowing instead the capillary temperature distribution along a prescribed sharp interface to act as a weak energy field. As such, the Gibbs-Thomson equilibrium temperature is shown to be much more than a boundary condition on the transport field; it acts, in fact, as an independent energy field during crystal growth and produces profound effects not recognized heretofore. Specifically, one may determine by energy conservation that weak normal fluxes are released along the interface, which either increase or decrease slightly the local rate of freezing. Those responses initiate rotation of the interface at specific locations determined by the surface energy and the shape. Interface rotations with proper chirality, or rotation sense, couple to the external transport field and amplify locally as side branches. A precision integral equation solver confirms through dynamic simulations that interface rotation occurs at the predicted locations. Rotations points repeat episodically as a pattern evolves until the dendrite assumes a dynamic shape allowing a synchronous limit cycle, from which the classic repeating dendritic pattern develops. Interface rotation is the fundamental mechanism responsible for dendritic branching.

## 1 Introduction

Crystalline dendrites appear everywhere throughout the natural world in myriad branched forms, as diverse as snow flakes and frost patterns, or as minerals crystallized from solutions and magmas. Dendritic crystals are also prevalent in metallurgical technologies, including alloy casting, primary metals production, welding, and soldering. Dendrites in metallurgy establish the initial microstructures of cast metals and alloys, and are responsible for setting patterns of chemical segregation, crystallographic texture, and grain size developed in these materials. Microstructures, in turn, strongly influence a material’s subsequent mechanical, physical, and chemical behaviors. Moreover, postsolidification treatments—the so-called “downstream processing” of many metallic materials—which ultimately yield both semifinished and final products, are chosen and are affected, to some extent, by the dendritic microstructure.

The descriptive term “dendrite” derives from “\(\delta \epsilon \nu \delta \rho \hbox{o} \nu\)”, a tree, with which its highly branched, arborescent appearance is aptly compared. A dendritic crystal will usually exhibit morphological hints of its underlying crystalline structure and symmetry, as it commonly consists of a primary stem, secondary arms, or side branches, with tertiary branches sprouting from the secondaries—all growing in selected crystallographic directions. It is the continuous generation of all these branches during dendritic solidification that establishes the ramified pattern, length scale, and spatial distributions of all the chemical components and impurities contained in a solidifying melt or solution. These features characterize, if not dominate, the microstructure of most cast alloys.

*in situ*during their active growth stage as shown in Figure 1. The most prominent features revealed in Figure 1 include what seems to be a “steady-state” tip, advancing at constant speed into its surrounding supercooled melt, followed closely by a repeating series of amplifying side branches growing in four directions—up, down, and in and out of the photograph’s focal plane. Farther back from the tip, these closely spaced branches interact and coarsen prior to complete solidification.

The history of dendrites specific to their application and interest in metallurgy is a long and interesting one, involving their visualization, description, and measurement in many cast materials, as well as the development of a number of hypotheses purporting to explain their origin.[1] A brief review of contemporary theories of dendritic growth now follows, to provide the reader a vantage point for understanding and appreciating the large disparity that has developed between theories of dendritic branching and growth kinetics and actual experimental findings on carefully developed test systems.

A local analysis of interface behavior will then be developed to provide some critically missing physics, which the author will show is essential to understanding the fundamental mechanism of dendritic branching.

## 2 Dendritic growth theories

### 2.1 Background

Metallurgists have long sought to predict and control alloy microstructures. \(\ldots\) cost effective manufacturing techniques ultimately depend on the precision with which we can solve this problem in non-equilibrium pattern formation. \(\ldots\) we would like to incorporate fundamental understanding of microstructures into computer codes that simultaneously help us design materials with made-to-order properties and optimize their manufacturability and performance.

James S. Langer,

Physics Today, October 1992.

Contemporary concepts of dendritic growth trace from the following two important early contributions: (1) the observations made by physicist A. Papapetrou,[2] who accurately described the geometry of a growing dendritic crystal and suggested modeling its shape as an *isothermal* paraboloid of revolution, the interface of which remains uniformly at its melting point, \(T_m;\) and (2) the follow-up analysis accomplished more than 60 years ago by G.P. Ivantsov, which describes quantitatively the nature of the thermal field surrounding an isothermal needle-like (branchless) dendrite undergoing steady-state growth.[3, 4, 5]

Traditionally, the subject of dendritic growth theory is divided into the following two independent components: (1) transport theory, which explains how the large-scale energetics of the \(\ell\rightarrow{s}\) transformation operates and controls the dendrite speed and size, which is mainly credited to Ivantsov and several other investigators who extended his findings to broader classes of dendritic geometries; and (2) interface physics, which captures the underlying microscopic phenomena that are responsible for the branching and directionality that characterize dendritic patterns in real materials. The two theoretical components of dendritic growth have both markedly different histories and status with respect to modern experiments conducted on dendritic growth kinetics and morphologies. We demonstrate in this article that the two components of dendrite theory—transport and interface physics—are not independent aspects at all but remain linked through a subtle effect involving energy transport along the interface.

### 2.2 Energy Transport

Ivantsov, as well as subsequent investigators of diffusion-controlled phase transformation in more complex geometries,[6] expressed the mathematical solutions for the thermal or diffusion fields that transport heat and/or solute during steady-state freezing in terms of a lumped dimensionless parameter, called the growth Péclet number.^{1}

*P*is conventionally defined in such transport problems as

*V*and

*R*are the speed of the interface’s advance and its radius of curvature, respectively, and \(\alpha\) is the diffusivity of the melt. Ivantsov’s transport solution relates the dendrite’s growth Péclet number to the melt’s dimensionless supercooling, or supersaturation, defined in this article for the case of a pure melt as \(\Updelta{\vartheta}=(T_{\infty}-T_{{m}})/(\Updelta{H}_{{f}}/C_p).\) Here, \(\Updelta{\vartheta}\) is the dimensionless temperature difference that transports the latent heat and, correspondingly, drives the overall phase transformation kinetics for a solid crystallizing from its pure melt. In the specific case of \(\ell\rightarrow{s}\) transformation, \(T_{{m}}\) is the material’s equilibrium freezing/melting temperature, and \(T_{\infty}\) is the supercooled melt temperature set at some large distance from the heat-releasing dendrite. The scale factor that nondimensionalizes the supercooling \((\Updelta{H}_{{f}}/C_p)\) is the characteristic temperature of the material, given by the ratio of the latent heat of fusion, \(\Updelta{H_{f}},\) to the melt’s specific heat, \(C_p.\) The characteristic temperatures for most metals are typically a few hundred Kelvins, but are much less for the organic test substances discussed subsequently in this article, which are often employed experimentally to check theories.

^{2}

*V*and

*R*, and the independent variable, \(\Updelta{\vartheta},\) namely

*V*or

*R*, which is based on physical principles beyond that of conservation and transport of the latent heat of the \(\ell\rightarrow{s}\) transformation. This irrefutable mathematical fact was fully appreciated by Ivanstsov himself and by numerous investigators who attempted to build upon, or modify, Ivantsov’s analysis of the fundamental heat transfer describing steady-state dendritic growth (Eq. [2]).

### 2.3 Interfacial Physics

- (a)
Maximum velocity

- (b)
Interfacial stability

- (c)
Shape self-consistency

- (d)
Minimum entropy production

- (e)
Marginal stability (maximum radius)

- (f)
Selective noise amplification

- (g)
Microscopic solvability

- (h)
Trapped wave theory

- (i)
Maximum entropy production

Take, for example, item (a), the concept of maximum velocity, which would occur, hypothetically, for a paraboloidal dendrite, the solid–liquid interface of which has interfacial energy, *i.e*., capillarity. As all interfaces have excess free energy, or surface tension, this idea could, conceivably, have provided a reasonable supposition about the operating state of dendrites: namely, that dendrites grow steadily when they achieve their maximum velocity, as allowed by the thermal conduction field and the level of supercooling specified in the melt. Crystal growth at the maximum velocity requires specifically that the dendritic tip radius be just twice the size of the critical radius required for homogeneous nucleation of the solid phase from the supercooled melt. Some early experiments designed to check the validity of this notion, however, showed convincingly that the tip radii of steadily growing dendrites were, in fact, far larger than that required for achieving the maximum velocity. Experiments found that dendrites actually grow with their tip radii more like 100 times the critical radii, and therefore advanced at steady speeds corresponding to only 1 pct of the predicted maximum velocity.[25,26]

### 2.4 Crystal-Melt Interfaces

Dendrites consist of crystal-melt interfaces only a few atomic or molecular spacings in thickness. This circumstance is treated conveniently with the “sharp-interface” approximation, in which the atomic or molecular transition between crystal and melt is reduced to a geometric surface of zero thickness, to which is assigned an excess energy, composition, and position, in a manner developed for classic models of heterophase interfaces by Gibbs.[32]

Diffuse interfaces, by contrast, are encountered in liquid-vapor systems near their critical points and during solid-state spinodal decomposition of phase-separating supersaturated alloys. Diffuse interfaces may be treated with theories and numerical models that relate to general free energy expansion methods developed originally by Ginzburg and Landau.[33] Free energy expansions deal explicitly with the gradient structures developed between related critical phases and with free energy descriptions that account for structural gradients.[34, 35, 36, 37]

Among the sharp interface descriptions postulated for dendritic growth, several approaches still prevail: Some sharp interface models, such as marginal stability,[13] remain in use, as does microscopic solvability, a steady-state theory that also incorporates interfacial energy anisotropy. Numerical simulations of diffuse interfacial structures, based on phase-field models, are now applied widely, and successfully, although approaching the limit of atomically thin interfaces remains computationally challenging.[38, 39, 40, 41, 42] Indeed, an array of different simulation methods, including the phase-field model, prove extremely useful for simulating a variety of solid–liquid and solid-state phase transformations.

Moreover, to test any specific hypothesis critically, one must establish a base of quantifiable observations. To date, the most reliable experiments capable of evaluating the predictions derived from theory are those that use test materials that are ultra-pure, stable, and transparent, such as \(\hbox{H}_{2}\hbox{O}\) ice,[25] and plastic crystalline compounds,[26,27, 28, 29, 30, 31] such as SCN and PVA. These particular experimental studies each involved observation of these well-characterized molecular substances solidifying from their melts in high-precision thermostats capable of setting small supercoolings accurately that control the kinetics of dendritic crystallization.

In this manner, the two major components of dendritic growth theory could be separated and individually evaluated, as follows: (1) checking the thermal transport field responsible for energy flow during the \(\ell\rightarrow{s}\) phase transformation, *viz*., testing Ivantsov’s analysis by measuring a dendrite’s growth Péclet number, using the product of \(V\times{R}\) at known supercooling levels, \(\Updelta\vartheta\) (Section II–B); and (2) testing separately some key prediction derived from the interfacial physics that was added to make the theory unique and that allows explicit prediction of either *V* or *R* for a specified supercooling.

### 2.5 Testing Current Theories of Dendritic Growth

The following two theories of dendritic growth are accepted today: (1) marginal stability, developed in the mid-1970s,[13,14] which contends that a dendrite grows at its limit of stability, defined by stochastic fluctuations that continually perturb the dendrite’s tip; and (2) microscopic solvability, a steady-state approach developed in the 1980s that finds unique mathematical solutions (smooth tips) for specified anisotropic capillarity.[16, 17, 18, 19] These theories have been checked individually with experiment and were found incapable of predicting the correct scaling laws, growth kinetics, or branching patterns.

The first important experiment that tested microscopic solvability theory studied dendritic growth in SCN and PVA. The results were reported in the early 1990s by Muschol *et al*.[30] These investigators examined carefully several aspects of microscopic solvability theory. Muschol *et al*. concluded based on their studies that microscopic solvability was in severe disagreement with their experiments. Quoting their concluding remark, “MST [microscopic solvability theory] in its present form cannot realistically be viewed as being confirmed by experiment”.[30]

Marginal stability and solvability were again subject to an independent sequence of quantitative experiments known as the Isothermal Dendritic Growth Experiment (IDGE). The IDGE series of dendritic growth tests was flown in space by NASA three times, as semiautonomous and ground-based telemetry-controlled experiments aboard the U.S. Space Shuttle *Columbia*. Three space flights of the IDGE were launched successfully between March 1994, and December 1998 and were carried as part of an experiment complement comprising NASA’s United States Microgravity Payload (USMP) Missions. USMP-2, −3, and −4 collectively yielded growth kinetic and morphological data on ultra-pure SCN and PVA from more than 350 experiments.[43]^{3}

Low-earth orbit (LEO) provides a nearly ideal microgravity environment that eliminates all vestiges of buoyant, or natural, convection in the melt during solidification.^{4} Buoyant melt convection is induced in the presence of Earth’s gravity by any heat-emitting dendrite. Achieving slow dendritic growth speeds in either metals or pure test substances demands setting and measuring precisely extremely small supercooling levels in the melt, \(\Updelta{T}\ll1\,\hbox{K},\) which advantageously allows high resolution *in situ* microphotography, such as displayed in Figures 1 and 2. Convective melt flows alter and complicate the pattern of heat conduction during freezing of high-Prandtl number fluids, particularly at the slow dendritic growth speeds needed in these kinetic experiments. This interference precludes making terrestrial laboratory measurements of a dendrite’s growth Péclet number, which as explained, must be obtained under diffusive, or pure thermal conduction conditions achievable in fluids only in microgravity.^{5}

*V*increases rapidly, whereas the corresponding tip radius

*R*decreases more slowly. Their combined behavior allows the growth Péclet number, \(P=VR/2\alpha, \) to increase with supercooling, as shown by transport theory (Eq. [2]). However, marginal stability also predicts that the combination \(V\times{R^2}\) should remain constant and be independent of the melt supercooling. More specifically, marginal stability requires that \(2\alpha{d_0}/VR^2\approx1/4\pi^{2}.\) Here, \(\alpha\) is the thermal diffusivity of molten SCN and \(d_0\) is the capillary length of this crystal-melt system.

^{6}

*not*upheld by these measurements. Instead, one observes a well-resolved increase of the quantity \(VR^2\) with increased supercooling. Similar, carefully executed, and multiply repeated experiments, carried out both terrestrially and in microgravity on PVA in the third, and final, IDGE experiment, show more scatter than do the data for SCN and confirm a steady increase of \(VR^2\) with increasing melt supercooling. The conclusion drawn from those PVA dendritic growth data was stated by its authors as follows:

In spite of the large uncertainties in many of the measurements, the scaling parameter [

i.e., \(VR^2\) for PVA dendrites] does not appear to be a constant over the full supercooling range of these experiments, and does not appear to agree with current predicted scaling/selection rule values.[49]

### 2.6 Commonality Among Dendritic Growth Theories

A feature found in common among present-day dendritic growth models is that capillarity plays its role by providing the “inner” thermal boundary condition at the solid–liquid interface, to which the “outer” transport field conforms. This restrictive use of capillarity remains the case whether a model is based on noise-mediated pattern development, as in marginal stability theory, or seeks a solution based on steady-state solvability. Thus, the Gibbs-Thomson equilibrium boundary condition merely replaces the isothermal interface boundary condition originally assumed by Ivantsov. Moreover, application of capillarity as the temperature boundary condition on the solid–liquid interface comprises the identical methodology used since the earliest models of dendritic growth were considered over 50 years ago.[7,8]

In the next section, we show that the Gibbs-Thomson capillarity effect is much more than just a boundary condition on the surrounding transport field. Capillarity introduces temperature gradients along a dendritic interface—albeit extremely weak ones—so that the Gibbs-Thomson equilibrium temperature distribution acts as an *independent* energy field during crystal growth, producing, as we shall demonstrate, more important effects on the branching process than recognized previously.

## 3 Capillarity

### 3.1 Background

Interfacial energy effects during crystal growth, or capillarity, are such that the equilibrium temperature between a crystal and its melt, at a fixed pressure, varies slightly with the local interfacial curvature. Specifically, the equilibrium interface temperature is depressed imperceptibly (by a few milli-degrees below \(T_{{m}}\)) near a highly curved dendrite tip and is slightly elevated—and thus closer to \(T_{{m}}\)—at locations farther away from the tip, where the interface is somewhat flatter. As mentioned at the end of Section II–B, standard models of dendritic growth view such paltry capillary fields as energetically irrelevant because they appear, at least superficially, to be inconsequential when compared with the (Ivantsov) transport field that governs the overall energetics of the \(\ell\rightarrow{s}\) transformation. This view is certainly not unreasonable, as the Ivantsov field, acting normal to the growing crystal-melt interface, spans a temperature difference in the melt that is several orders of magnitude greater than the Gibbs-Thomson field along the interface. Furthermore, it may be argued that any thermal fluxes that happen to be associated with the gradients produced by the equilibrium temperature distribution itself are themselves confined to act *tangentially* along the interface. Thus, such weak gradients could not, in any event, assist in the energy transport directly affecting the rate of \(\ell\rightarrow{s}\) transformation. For these apparently cogent reasons, the Gibbs-Thomson temperature distribution remains today solely as a passive boundary condition in standard models of dendritic growth. As shown next, however, capillarity also plays a more subtle role, which has been overlooked, in the initiation and control of dendritic branching.

### 3.2 Gibbs-Thomson as an Energy Field

Conventional thinking notwithstanding, the Gibbs-Thomson temperature distribution was reconsidered as a full-fledged energy field after the present author analyzed experiments conducted on the melting kinetics of needle-like crystallites. These crystallites were suspended in their melt under microgravity conditions and exhibited the puzzling behavior of suddenly becoming more spherical as they decreased in volume to sufficiently small size prior to their complete extinction by melting.[50,51] It was shown recently that the observed onset of spheroidization was caused by the unexpected appearance during melting of heat currents generated *internally* to the crystallites. The “extra” energy for spheroidization arose through the action of interfacial capillarity when the needle-shaped crystallites melted down to sufficiently small sizes.[52,53] This unusual finding, which was discovered in experiments on *melting* crystals, prompted the present reexamination of the role played by the Gibbs-Thomson temperature distribution for the case of *freezing*.

A brief summary of this interface analysis and its further implications on understanding the mechanism of dendritic branching follow.

### 3.3 Tangential Fields

*x*-axis,

*a*, longer than its shorter semiminor

*y*-axis,

*b*. This provides a slender, finger-like starting shape suitable for a dendrite. Figure 5 shows the interfacial configuration.

*a*, so that the scaled coordinates \(x/a\equiv\mu,\) and \(y/a\equiv\eta\) define dimensionless coordinate axes for the starting shape. The equation of the ellipse transforms to

*a*, which bears the physical unit [K].

*a*= 3 and

*b*= 1. The interface potential distribution \(\vartheta(\mu),\) is plotted for this elliptical shape in Figure 6. The potential is negative everywhere and remains close to zero for almost half the distance along the semiellipse from the equator to tip. The potential then decreases rapidly as the tip at \(\mu=1\) is approached. One expects from this potential distribution relatively slowly increasing energy fluxes near \(\mu=0\) and stronger, variable fluxes toward the tip as energy flows down, and proportionately, to the local gradient. The vector gradient field caused by the Gibbs-Thomson potential distribution on an ellipse may be shown to be the following function of \(\mu{:}\)

^{7}

### 3.4 Interfacial Energy Conservation

The temperature variations caused by capillarity are extremely small and generally do not exceed a few millidegrees. This extreme weakness notwithstanding, the autonomous energy fluxes associated with capillarity must still be conserved locally at every point along the interface. The tangential fluxes along the 3:1 elliptical interface are shown in Figure 6-middle. A standard method for evaluating local energy conservation is to calculate the “convergence” of the tangential heat flux vectors on the interface, \(\hat\Upphi(\mu)\cdot\sigma,\) or equivalently, to calculate the divergence of the gradient at each point.

*local*balance

*tangential*flux or potential gradient; the second term, is proportional to the interfacial heat capacity, \(C_{p_{{int}}} \left[ \text{J}/\text{m}^2 - \text{K} \right]\) if the interfacial heat capacity is non-zero; and the third term is the capillary-induced

*normal*flux leaving the interface, directed either into the crystal or the melt. These energy terms must balance at every point along the crystal-melt interface, and account for the Gibbs-Thomson energy. The effect of the normal fluxes is to increase, or decrease, slightly, the large thermal gradients surrounding the interface from the Ivantsov transport field, which is primarily responsible for setting the local freezing rates.

*k*

_{int}Γ/a

^{3}), yields a fully dimensionless interface conservation equation for the Gibbs-Thomson energy field,

*increases*the local rate of freezing. The flux magnitude falls precipitously from its peak value near μ = 0.9, reversing sign just below μ = 0.95, where the energy becomes redirected toward the crystal and

*retards*the rate of freezing.

It is interesting and important to note that the capillary-mediated modulation of the local freezing rates from point to point along an interface, as demonstrated in Figure 6, occurs autonomously by tangential heat fluxes arising from the gradients of the Gibbs-Thomson energy field.

### 3.5 Le Chatelier-Braun Responses

Now, a crystal-melt interface on which heat energy is being added or withdrawn from point to point is an example of what Van’t Hoff termed mobile equilibrium.[55] The local response of such a system to capillary-induced autonomous energy changes is predicted by the Le Chatelier-Braun effect, the fundamental basis for which is the combined 1st and 2nd laws of thermodynamics.[56, 57, 58, 59] The Le Chatelier-Braun effect posits[55,60] that mobile systems respond by negative feedback to imposed extensive variable changes. In this instance, the changes involve the autonomous addition or loss of energy from the Gibbs-Thomson field.

Viewed the other way round, the Le Chatelier-Braun effect also states that mobile systems must respond to intensive variable changes (rising or falling potential) in a manner so as to increase the corresponding change in the conjugate *extensive* variable, which is the energy or entropy added or released along the interface. These countervailing descriptions cause confusion about the Le Chatelier-Braun effect and, consequently, were distinguished by Paul Ehrenfest in his explanation of this peculiar dichotomy, as either *Widerstandfähigkeit* (capable of resisting) or *Passungsfähigkeit* (capable of adopting).[61] Both of these descriptions are, however, thermodynamically correct and equivalent, and will be used to determine the direction of the energy fluxes normal to the interface added by capillarity.

## 4 Kinematic Rotation

## 5 Dynamic Verification of Kinematic Rotation

The dynamic behaviors of the 3:1 ellipse discussed previously, as well as that for a 2:1 ellipse, were checked independently using a low-noise integral equation solver developed by Lowengrub and Li.[62] This solver can evolve accurately simulated diffusion-limited dendrites and other interesting patterns such as fluid-fluid interpenetrating Saffman-Taylor “viscous fingers”[63,64] observed in Hele Shaw cells.[65, 66, 67, 68] In the current study, the solver provides numerical solutions to Laplace’s equation tracking the thermal field surrounding the evolving dendritic pattern and follows the interface as it develops over time with the main transport field. The boundary condition applied on the solid–liquid interface is the Gibbs-Thomson equilibrium temperature distribution Eq. [5] which reflects both the instantaneous value of the interface curvature and the presence of interfacial energy.

The predicted location of the rotation points, which are determined analytically on these starting shapes, may be compared with their dynamically evolved positions as observed with the solver. The analytically determined value for the initial root position and corresponding rotation point for a 3:1 semi ellipse with isotropic interface energy is \(\mu_{{local}}^{\star}=0.9878. \) The actual rotation point observed independently with the dynamical solver occurs at \(\mu_{{global}}^{\star}=0.9877. \) These rotations correspond to the dimensional location \(x^{\star}\approx2.96, \) which occurs close to the tip position at *x* = 3.

In a second case tested with a 2:1 ellipse and isotropic interface energy, the analytically determined root and rotation point position was \(\mu_{{local}}^{\star}=0.9688, \) which compares well with the first rotation point observed through dynamic evolution at \(\mu_{{global}}^{\star}=0.9687. \) These positions correspond, respectively, to the dimensional interface position at \(x^{\star}\approx1.94, \) which is again near the ellipse’s tip at *x* = 2. We note that with isotropy of the interface energy, the initial rotation points of those elliptical shapes occur within approximately 2 to 3 pct of the tip positions of the ellipses.

## 6 Summary and Conclusions

This study reviewed theories of dendritic crystal growth and, citing pertinent experimental studies, showed that current dendrite theories fail to predict observed phenomena properly. The common thread among all theories of dendrite formation is their restricted use of capillarity, namely, how the Gibbs-Thomson effect is applied to the interface. The Gibbs-Thomson temperature distribution on a curved dendritic interface consists of an extremely weak variation from the curved tip region to the flatter regions away from the tip. Specifically, the equilibrium temperature variation spans only a few millidegrees in totality. This tiny range of temperature—compared with the more robust Ivantsov transport field that surrounds the dendrite—has traditionally relegated the Gibbs-Thomson temperature distribution as an interface boundary condition on the external field. The new view taken here is that the equilibrium temperature distribution is an *active* interface energy field, albeit it an extremely weak one. As such, capillarity acting on a crystal interface, which is well away from from its equilibrium configuration, produces energy gradients and fluxes and, consequently, divergences of those weak vector fields.

A reexamination of the consequences of a weak energy field along the crystal-melt interface revealed that additional capillary heat fluxes either increase or retard the local freezing rate, and where they vanish interface rotation occurs. The Le Chatelier-Braun principle—a consequence of the laws of thermodynamics—requires a response from the mobile interface to the weak fluxes induced *autonomously* by the Gibbs-Thomson energy field. We find that curved regions near the tip of an elliptical interface will warm, and be retarded, so they respond by flattening, whereas flatter regions will cool, and accelerate, so they respond by sharpening. Where the incipient processes of flattening and sharpening become adjacent, the interface undergoes a rotation, by tilting and enhancing the exterior local temperature gradients in the melt. This action couples the interface to the surrounding transport field and results in branch formation.

A low-noise dynamic solver verifies the locations of the initial rotation points, which depends sensitively on both the interface shape and the anisotropy of the interfacial energy. The full details of these dynamic tests will be reported elsewhere. Dynamic solver studies also allow the observation of subsequent rotation points that occur episodically at locations near the tip, the shape of which changes over time. (The location of subsequent rotation points cannot be found just using the current local analytical theory.) Eventually, the tip develops an appropriate “steady” tip shape that leads to a synchronicity between the occurrence of kinematic rotations and the tip advancement. These act in concert as a nonlinear limit cycle. Once established, the dendrite’s limit cycle produces the classic branching pattern.

Thus, we conclude, capillarity-induced rotation provides the fundamental *deterministic* mechanism responsible for dendritic branching. Specifically, rotation occurs where the surface Laplacians of the Gibbs-Thomson potential and the interface curvature vanish. These are equivalent statements when the interfacial energy is isotropic. Selective noise amplification, marginal stability, or other stochastic phenomena do not enter the process at this early stage or seem to play any direct role in pattern-forming dynamics, per se. Some aspects concerning how the rotation couples with the exterior transport field have yet to be resolved and are part of an ongoing analysis of global energy conservation on near-equilibrium interfaces. Certainly more computer-based dynamic testing and new experiments on well-characterized materials are called for to test these ideas more critically and perhaps to achieve a deeper understanding of their implications for predicting cast microstructures.

## Footnotes

- 1.
The mathematical expressions for the transport fields surrounding dendrites in pure materials and alloys are identical; only the transport coefficients defining the Péclet number differ.

- 2.
The characteristic temperatures, \((\Updelta{H}_{f}/C_p)\) for body centered cubic (bcc) succinonitrile (SCN) and face centered cubic (fcc) pivalic acid anhydride (PVA) are 23 K, and 11 K, respectively.

- 3.
Interested readers may access NASA’s official archives for the IDGE-USMP series, available at http://pdlprod3.hosc.msfc.nasa.gov. To locate associated NASA Technical Reports for the IDGE-USMP series please go to http://naca.larc.nasa.gov/index.jsp?method=aboutntr.

- 4.
The gravitational acceleration in low-earth orbit, \(g_{{LEO}},\) affecting the IDGE experiments on NASA’s USMP missions was reduced to a quasi-static level of \(g_{{LEO}}\approx10^{-7}g_0,\) where \(g_0=9.807\,\hbox{m/s}^{2}\) is the average, or standard terrestrial value of the gravitational acceleration.

- 5.
The higher the Prandtl number of a melt, the more that hydrodynamic flow affects heat transfer. The Prandtl number of a fluid

*Pr*is the ratio of its kinematic viscosity, or momentum diffusivity \(\nu [\hbox{m}^2/\hbox{s}],\) to its thermal diffusivity, \(\alpha [\hbox{m}^2/\hbox{s}].\) Plastic crystals, such as succinonitrile and pivalic acid anhydride, are stable and conveniently transparent for microphotography but have relatively large Prandtl numbers, \(Pr=\nu/\alpha>10,\) whereas molten metals, which suffer from opaqueness, reactivity, and much higher melting temperatures, exhibit small Prandtl numbers, \(Pr\ll 1.\) - 6.
The capillary length, \(d_0=2.82\pm0.17\) nm, is defined from marginal stability as \(d_0\equiv(C_{p}T_{{m}}\Upomega\gamma_{s\ell})/\Updelta{H_{{f}}}^2.\) All the constituent thermo-physical constants for \(d_0\) for SCN are fully documented,[45] including, its equilibrium melting point, \(T_{{m}}=331.233\pm0.001\,\hbox{K};\) as well as the molar specific heat of the melt, \(C_p=160.91\pm1.6\,\hbox{J/mol-K};\) the molar volume of the melt, \(\Upomega=0.816\pm0.006\times10^{-4}\,\hbox{m}^3/\hbox{mol};\) the interfacial energy density, \(\gamma_{s\ell}=8.94\pm0.5\,\hbox{mJ/m}^{2};\) and the molar heat of fusion, \(\Updelta{H_{{f}}}=3.704\pm0.002\,\hbox{kJ/mol}.\)

- 7.
Conventional,

*i.e*., bulk, thermal conductivities bear System International (SI) units of [watts/m-K]; however,*surface*, or interfacial, thermal conductivities must carry SI units of [watts/K] in order that the interface flux exhibits proper units of [watts/m].

## Notes

### Acknowledgments

The author is honored by the ASM International for his selection as the 2011 Edward DeMille Campbell Lecturer at MS&T Columbus, OH, October 18, 2011, prompting preparation of this paper. Thanks are extended to colleagues Professor John Lowengrub, Mathematics Department, University of California, Irvine, CA, and Professor Shuwang Li, Department of Mathematics, Illinois Institute of Technology, Chicago, IL, for applying their integral equation solver that provided the initial independent dynamical checks on the local-response theory presented here; and to Professor Markus Rettenmayr and Mr. Klemens Reuther, Friedrich Schiller University, Jena, Germany, for their independent checks using their unstructured grid dynamic solver. The author acknowledges helpful discussions held on this subject with Dr. Geoffrey McFadden, NIST, Gaithersburg, MD; Dr. Alexander Chernov, Lawrence Livermore National Laboratory, Livermore, CA; Professor Bernard Billia, Faculté des Sciences et Techniques, University of Marseille, France. The author is grateful for the encouragement received from Professor Emerita Jean Taylor, Rutgers University, Cream Ridge, NJ, and the Courant Institute of Mathematical Sciences, New York University, New York City, and from Professor John W. Cahn, University of Washington, Seattle, WA.

### References

- 1.M.E. Glicksman:
*Principles of Solidification*, Springer, New York, NY, 2011, Ch. 13, pp. 305–12.Google Scholar - 2.A. Papapetrou:
*Z. Kristallographie*, 1935, vol. 92, p. 89.Google Scholar - 3.G.P. Ivantsov:
*Dokl. Akad. Nauk*, USSR, 1947, vol. 58, p. 567.Google Scholar - 4.G.P. Ivantsov:
*Dokl. Akad. Nauk*, USSR, 1951, vol. 81 p. 179.Google Scholar - 5.G.P. Ivantsov:
*Dokl Akad Nauk USSR*, 1952, vol. 83, p. 573.Google Scholar - 6.G. Horvay and J.W. Cahn:
*Acta Metall*., 1961, vol. 9, p. 695.CrossRefGoogle Scholar - 7.D.E. Temkin:
*Dokl. Akad. Nauk USSR*, 1960, vol. 132, p. 1307.Google Scholar - 8.G.F. Bolling and W.A. Tiller:
*J. Appl. Phys*., 1961, vol. 32, p. 2587.CrossRefGoogle Scholar - 9.E.G. Holtzmann:
*J. Appl. Phys*., 1970, vol. 41, p. 1460.CrossRefGoogle Scholar - 10.E.G. Holtzmann:
*J. Appl. Phys*., 1970, vol. 41, p. 4769.CrossRefGoogle Scholar - 11.W. Oldfield:
*J. Mater. Sci. Eng*., 1973, vol. 11, p. 211.CrossRefGoogle Scholar - 12.G.E. Nash and M.E. Glicksman:
*Acta Metall*., 1974, vol. 22, p. 1283.CrossRefGoogle Scholar - 13.J.S. Langer and H. Müller-Krumbhaar:
*J. Cryst. Growth*, 1977, vol. 42, p. 11.CrossRefGoogle Scholar - 14.J.S. Langer and H. Müller-Krumbhaar:
*Acta Metall*., 1978, vol. 26, p. 1681.CrossRefGoogle Scholar - 15.J.S. Langer:
*Rev. Mod. Phys*., 1980, vol. 52, p. 1.CrossRefGoogle Scholar - 16.D. Kessler and H. Levine:
*Phys. Rev. Lett*., 1986, vol. 57, p. 3069.CrossRefGoogle Scholar - 17.D. Kessler and H. Levine:
*Phys. Rev. A*, 1987, vol. 36, p. 2693.CrossRefGoogle Scholar - 18.D. Kessler and H. Levine:
*Acta Metall*., 1987, vol. 36, p. 2693.Google Scholar - 19.A. Barbieri and J.S. Langer:
*Phys. Rev. A*, 1987, vol. 39, p. 5314.CrossRefGoogle Scholar - 20.J.J. Xu:
*Advances in Mechanics and Mathematics*, vol 1, Springer, New York, NY, 2002, p. 213.Google Scholar - 21.J.J. Xu:
*Introduction of Dynamical Theory of Solidification Interfacial Instability*, Chinese Academy Press, Beijing, China, 2007.Google Scholar - 22.X.J. Chen, Y.Q. Chen, J.P. Xu, and J.J. Xu:
*Front. Phys*. China, 2008, vol. 3, p. 1.Google Scholar - 23.Y.Q. Chen, X.X. Tang, and J.J. Xu:
*Chinese Phys. B*, 2009, vol. 18, p. 686.CrossRefGoogle Scholar - 24.J.A. Sekhar,
*J. Mater. Sci*. in press.Google Scholar - 25.T. Fujioka and R.F. Sekerka,
*J. Cryst. Growth*, 1974, vol. 24–25, p. 84.CrossRefGoogle Scholar - 26.M.E. Glicksman, R.J. Schaefer, and J.D. Ayers:
*Metall. Trans. A*, 1976, vol. 7A, p. 1747.Google Scholar - 27.S.C. Huang and M.E. Glicksman:
*Acta Metall*., 1981, p. 701.Google Scholar - 28.S.C. Huang and M.E. Glicksman:
*Acta Metall*., 1981, p. 717.Google Scholar - 29.H. Esaka and W. Kurz:
*Z. Metall*., 1985, vol. 76, p. 127.Google Scholar - 30.M. Muschol, D. Liu, and H.Z. Cummins:
*Phys. Rev. A*, 1992, vol. 46, p. 1038.CrossRefGoogle Scholar - 31.W. Losert, B.Q. Shi, and H.Z. Cummins:
*Proc. Natl. Acad. Sci. USA*, 1998, vol. 95, p. 431.CrossRefGoogle Scholar - 32.J.W. Gibbs:
*Trans. Conn. Acad*., 1877–78, vol. III, p. 343.Google Scholar - 33.V.L. Ginzburg and L.D. Landau:
*Sov. Phys. JETP*, 1950, vol. 20, p. 1064.Google Scholar - 34.J. Cahn and J. Hilliard:
*J. Chem. Phys*., 1958, vol. 28, p. 258.CrossRefGoogle Scholar - 35.H. Emmerich:
*Phase Field Methods*, Springer, Berlin, Germany, 2003.Google Scholar - 36.N. Provatas, N. Goldenfeld, J. Dantzig, J.C. LaCombe, A. Lupulescu, M.B. Koss, M.E. Glicksman, and R. Almgren:
*Phys. Rev. Lett*., 1999, vol. 82, p. 4496.CrossRefGoogle Scholar - 37.G.B. McFadden, A.A. Wheeler, R.J. Braun, S.R. Coriell, and R.F. Sekerka:
*Phys. Rev. E*, 1993, vol. 48, p. 2016.CrossRefGoogle Scholar - 38.B.J. Spencer and H.E. Huppert:
*J. Cryst. Growth*, 1995, vol. 148, p. 305.CrossRefGoogle Scholar - 39.A. Karma and W.-J. Rappel:
*J. Cryst. Growth*, 1997, vol. 174, p. 54.CrossRefGoogle Scholar - 40.O. Penrose and P.C. Fife:
*Physica D*, 1990, vol. 43, p. 44.CrossRefGoogle Scholar - 41.A.A. Wheeler, B.T. Murray, and R.J. Schaefer:
*Physica D*, 1993, vol. 66, p. 243.CrossRefGoogle Scholar - 42.B.T. Murray, A.A. Wheeler, and M.E. Glicksman:
*J. Cryst. Growth*, 1995, vol. 154, p. 386.CrossRefGoogle Scholar - 43.M.E. Glicksman, M.B. Koss, L.T. Bushnell, J.C. LaCombe, and E.A. Winsa:
*Adv. in Space Res*., 1995, vol. 16, p. 181.CrossRefGoogle Scholar - 44.M.E. Glicksman, M.B. Koss, and E.A. Winsa:
*Phys. Rev. Lett*., 1994, vol. 73, p. 573.CrossRefGoogle Scholar - 45.M.B. Koss, J.C. LaCombe, L.A. Tennenhouse, M.E. Glicksman, and E.A. Winsa:
*Metall. Mater. Trans. A*, 1999, vol. 30A, p. 3177.CrossRefGoogle Scholar - 46.J.C. LaCombe, M.B. Koss, and M.E. Glicksman:
*Phys. Rev. Lett*., 1999, vol. 83, p. 2997.CrossRefGoogle Scholar - 47.D.P. Corrigan, M.B. Koss, J.C. LaCombe, K.D. de Jager, L.A. Tennenhouse, and M.E. Glicksman:
*Phys. Rev. E*, 1999, vol. 60, p. 7217.CrossRefGoogle Scholar - 48.M.E. Glicksman:
*ASM Handbook*, 2008, vol. 15, pp. 398–401.Google Scholar - 49.J.C. LaCombe, M.B. Koss, and M.E. Glicksman:
*Metall. Mater. Trans. A*, 2007, vol. 38A, p. 116.CrossRefGoogle Scholar - 50.M.E. Glicksman, A. Lupulescu, and M.B. Koss:
*J. Thermophys. Heat Trans*., 2003, vol. 17, pp. 69–76.CrossRefGoogle Scholar - 51.A. Lupulescu, M.E. Glicksman, and M.B. Koss:
*J. Cryst. Growth*, 2005, vol. 276, pp 549–65.CrossRefGoogle Scholar - 52.M.E. Glicksman:
*Proc. of the 4th International Symposium on Physical Sciences in Space*, Bonn-Bad Godesburg, Germany, July 2011.Google Scholar - 53.A.G. Cheong and A.D. Rey:
*Phys. Rev. E*, 2002, vol. 66, p. 021704.CrossRefGoogle Scholar - 54.J.W. Cahn and W.C. Carter:
*Metall. Mater. Trans. A*, 1996, vol. 27A, p. 1431.CrossRefGoogle Scholar - 55.J.H. Van’t Hoff:
*Etudes de Dynamique Chimique*, Frederik Müller & Co., Amsterdam, the Netherlands, 1884.Google Scholar - 56.A.L. LeChatelier:
*Compt. Rend*., 1884, vol. 99, p. 786.Google Scholar - 57.F.Z. Braun:
*Z. Physik. Chem*., 1887, vol. 1, p. 269.Google Scholar - 58.F.Z. Braun:
*Ann. Physik*, 1888, vol. 33, p. 337.Google Scholar - 59.P.S. Epstein:
*Textbook of Thermodynamics*, John Wiley & Sons, Ch. XXI, New York, NY, 1939.Google Scholar - 60.J. DeHeer:
*J. Chem. Educ*., 1957, vol. 34, p. 375.CrossRefGoogle Scholar - 61.P. Ehrenfest:
*Z. Physik. Chem*., 1911, vol. 77, p. 227.Google Scholar - 62.M.E. Glicksman, J. Lowengrub, S. Li, and X. Li:
*J. Metals*, 2007, vol. 59, pp. 27–34.Google Scholar - 63.G. Saffman and G.I. Taylor:
*Proc. R. Soc. London Ser. A*, 1958, vol. 245, p. 312.CrossRefGoogle Scholar - 64.D.A. Kessler and H. Levine:
*Phys. Rev. A*, 1986, vol. 33, p. 2621.CrossRefGoogle Scholar - 65.H.J.S. Hele Shaw:
*Nature*, 1898, vol. 58, p. 34.Google Scholar - 66.R.A. Wooding:
*J. Fluid Mech*., 1969, vol. 39, p. 477.CrossRefGoogle Scholar - 67.P. Van Meurs:
*Trans. A.I.M.E*., 1957, vol. 210, p. 295.Google Scholar - 68.L. Paterson:
*J. Fluid Mech*., 1981, vol. 113, p. 513.CrossRefGoogle Scholar