Overview 12: Fundamentals of dendritic solidification—I. Steady-state tip growth

Overview 12: Fundamentals of dendritic solidification—I. Steady-state tip growth

Copynphl OVERVIEW 0001-6160’81/0M701-I5S02.M)~0 e 1981 Pergamon Press Ltd 12 FUNDAMENTALS OF DENDRITIC SOLIDIFICATION-I. STEADY-STATE TIP GROWTH? ...

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0001-6160’81/0M701-I5S02.M)~0 e 1981 Pergamon Press Ltd


FUNDAMENTALS OF DENDRITIC SOLIDIFICATION-I. STEADY-STATE TIP GROWTH? S.-C. HUANGZ’ and M. E. GLICKSMAN2 ‘General Electric Company, Corporate Research & Development Center. Building 81. Room E102, Schenectady. NY 12345 2Materials Engineering Department, Rensselaer Polytechnic Institute, Troy. NY 12181, U.S.A. (Recrzued 20 Augusr 1980: in reuisedform

17 Noremher 1980)

Abstract-Systematic me~urements of dendrite tip radius and growth velocity in succinonitrile reveal that consideration of dendrite tip stability should be incorporated into the heat transfer theory to determine the steady-state dendritic growth condition. The dendritic stability criterion measured is 2ado/VR2 = 0.0195, where V is the dendritic growth velocity, R is the dendritic tip radius, a is the liquid thermal diffusivity, and do is a capillary length defined in the text. Several dendritic stability models are reviewed and discussed in comparison to the present experimental results. R&mm&-Les mesures systimatiques des rayons de courbure au sommet des dendrites et de leur vi&se de croissance dans le cas du succinonitrile revellent que la stabilite du sommet des dendrites doit itre prise en compte dan les mecanismes de transfert de chaleur pour determiner la condition de croissance dendritique suivant un regime stable. Le critire de stabilitt des dendrites a eti mesure egal a 2~ d&JR’ = 0.0195, 6u V est la vitesse de croissance dendritique. R le rayon de courbure au sommet des dendrites. a la conductivite thermique dans le liquide, et do un distance, caracteristique de l’energie de surface. definie dans le texte. Plusieurs modtles de stab&t des dendrites sont repris et confront& aux resultats experimentaux presentis. Znaamsnenfaasung---Systematische Messungen des Radius einer Dendritenspitze und seiner Wachstumsgeschwindigkeit im System Succinonitril deuten darauf hin, dass man die Stabilitit der Dendritenspitze in der Theorie des Warmeaustausches einfuhren sol], urn die Bedingung festzustellen, wobei eine Dendrite im stationaren Zustand wachsen kann. Die gemessene Stabilitltsbedingung liiutet 2a d&R2 = 0.0195, wobei i-’ die Dendriten wachstumsgeschwindigkeit ist. R das Radius der Dendritenspitze, 01die thermische Leitfahigkeit der Fliissigkeit, und do die im folgenden Artikel defmierte Kapiliarllnge. Einige Modelle der Dendritenstabilit~t werden analysiert und im Vergleich mit den vorliegenden versuchs ergebnissen besprochen.

INTRODUCTION growth is perhaps the most common form of solidification, especially in metals and other systems that freeze with relatively low entropies of transformation. Dendritic or branched growth in alloys generates microsegregation as well as other internal defects in castings, ingots. and weldments. More subtle effects introduced by the complex dendritic microstructure in solidified materials include crystallographic texturing hot cracking suboptimal toughness. and reduced corrosion resistance. Moreover. the dendritic microstructure and its effects may be modified by subsequent heat treatments, but they are seldom fully ‘erased.’ As such, the underst~ding and control of dendritic growth in solidification process-


+ Research supported by the Marshall Space Flight Center. through the Space Processing Applications Rocket (SPAR) Program. Office of Application. NASA. $ Formerly. graduate student at Rensselaer Polytechnic Institute.

ing is crucial in order to achieve specific material properties in final products. Dendritic growth is a coupling of two seemingly independent growth processes: the steady-state propagation of the dendrite main stem and the nonsteady-state evolution of dendrite branches. Until recently, the time-dependent features of dendrites were completely ignored and theoretical models of dendrites were limited to the mathematical description of a branchless, paraboloidal needle, which grew at a constant rate and in a shape-preserving manner. Furthermore, theoretical studies of dendritic growth have concentrated on the steady-state development of a one-component needle-dendrite growing in a pure melt, wherein the major transport process is heat conduction. Results of such theories, as reviewed in Ref. [I], express the axial dendritic growth velocity as a function of supercooling (defined as AT = T, - T,. where T, is the melting point and 7’, is the temperature of the melt far from the dendritic interface), viz., IL. = ~G(A~)b. 701








V,,, is the maximum growth velocity, G is a lumped material parameter, and fl and b are numerical coefficients and exponents, respectively, and are determined by and specific to each theoretical model. In a recent experiment, Glicksman et al. [l] measured the dendritic growth velocity, V, in succinonitrile (CN-C,H,-NC) over the range of supercooling from 1 to 10°C. The experiment was carefully designed for a critical test of the di~~ion-~ntrolle~ steady-state dendritic growth theory in the form of equation (1). The results of Glicksman ec al. revealed that, althou~ two of the theories considered precorrect power-law dicted relationship the (V x AT2.65), none of the theories predicted the correct growth velocity (i.e., the coefficient /I) to within two times that of the measured values. The present study extends the investigation of dendritic growth in succinonitrile to the regime of small supercooling from O.l”C to 1°C. Several new phenomena arise during dendritic growth at small supercoohngs. First, the dendrites grow relatively slowly (V < 0.1 cm/s) with concomitantly iarge dimensions (R 7 10V3cm)--a favorable situation for performing precise morphology studies. As to be seen in the Background section, simultaneous measurements of tip radius and growth velocity can provide a solid basis on which improvements may be introduced into the development of dendritic growth theory. This unique way to test dendritic growth theory cannot be extended easily to high supercooling situations as in Ref. Cl], where a detailed investigation of dendritic ~~omorpholo~ was prechsded by the then available techniques. Another morphological aspect of dendritic growth which could be revealed at small supercoolings and which is of great interest today is that of sidebranch evohttion. The nonsteady-state nature of the sidebranches controls their spacing, and the re!ated, technically important size-scale of segregation. Also, the branching mechanism itself might influence the steady-state growth of the dendrite main stem. In situ ob~rvation of branch formation at small supercoolings should allow us to explore to a fuller extent the branching mechanism and, thus, understand better the overall dendritic growth process. In addition, in situ morphological study is extended here to include the inv~tigation of the mech~ism responsible for the branch-spacing adjustment during dendritic growth. Second, in the range of small supercoohng investigated, the natural convective fluid tlow arising from the thermal gradients surrounding a dendrite will have a sizable influence on the dendritic growth velocity as well as on the dendrite morphology. Every aspect of the dendritic growth will then not only vary with su~r~o~ing but 3150 with the dendritic growth orientation relative to the fluid flow direction. For convenience, this paper is divided into two portions: Part I presents and discusses the results relevant to the steady-state aspect of dendritic growth, i.e.. the axial propagation of the dendrite tip. Part fI where

focuses on the nonsteady-state aspect of dendritic growth, i.e., the development of the sidebranch structure. The results on convection effects have been presented elsewhere [Z, 3,4] and will not be a subject of detailed discussion here. BACKGROUND A. Experimental tests of dendritic growth theory Attempts to achieve experimental measurements of the growth rate coefficient /I and exponent b in equation (1) have historically encountered great di~culties. One of the main experimental difficulties encountered in the past was the un~rt~nty in the material properties which constitute the parameter G in equation (1). For example, even in one of the most studied systems, ice/water, solid-liquid interfa~al energy had not been measured to adequate accuracy until just recently [S]. Before then, arbitrary vatues for the ice/water interfacial energy were assigned so that a “measured” fi value agreed with the prediction of a particular theory [6]. Another major experimental dif&ulty lies in the deveiopment of a dendritic growth technique to ensure a “free” growth condition. To be in accord with the theoretically required growth conditions, both the constraining effects of growth chamber walls and the proximity effects of neighboring dendrites must be avoided. In fact, disparate results were obtained for the ice/water system by different growth techniques [7-1 I]. Another serious experimental diEiculty worth mentioning is the complication introduced by convection and solute redistributionneither of which is included in any theory of the growth of thermal dendrites in pure systems. In the experiment of Glicksman et ai. El], the usual pitfalls were eliminated. Specifically, a succinonitrile specimen of 99.999% purity was used to help ensure the attainment of thermal diffusion conditions assumed in the theory; reliable rne~ur~~t techniques were employed to reduce the uncertainty in V and AT to within -r-W,; a special specimen configuration was designed to achieve free dendritic growth conditions; and, finally, the specimen properties were well-characterized, so that the lumped material parameter, G, appearing in ‘equation (1) was known to within 4 10%. In addition, the interfacial growth kinetic coefficient of succinonitrile was estimated to be about 20 cm/s/C Cl]. Such a rapid interfacial reaction process would have had an insignificant effect ,on the dendritic growth in the range of growth velocity studied (V < lOcm/s) and therefore could safely be ignored. Also, the anisotropy in solid-liquid interfacial energy, y, for succinonltrile is about only 104 (deduced from the equilibrium shape of the liquid droplets, as will be discussed in Part II). Such a small anisotropy should also have but a negligibie effect on the dendritie growth kinetics [l]. Thus, the theoretical assump tions of perfect isotropy and infinitely fast interfacial attachment kinetics used to obtain equation (1) are rather well-approximate in the case of pure succino-





nitrile freezing at AT < 10°C. In short, the careful techniques employed in Ref. [l] produced reliable experimental results. Consequently, the factors responsible for the failure of the dendritic growth theory to predict the correct growth kinetics must lie within the theory itself.


the Gibbs-Thomson effect lowers the interface equilibrium temperature by T&K/,!_, which effectively reduces by the same amount the supercooling available to “drive” the thermal diffusion. This reduction in supercooling (as reflected in the nonzero [email protected], in equation (2) in both the modified Ivantsov model and in the more advanced Temkin model) prevents den0. A retest of the theory of capillary-limited dendritic drites from growing increasingly fast as R +O, as growth would be required by the “point effect.” Generally, the As introduced in the last section, steady-state dencapillary-limited theory in the form of equation (2) dritic growth theory confines itself to a needle-like only limits the growth velocity to being no larger than dendrite growing at a constant axial velocity in a an upper botmd value of Y (V,,,) at a given supercoolsupercooled, pure melt. The fully time-dependent dening [email protected] relationships for Ii vs AB and R vs [email protected] dritic growth problem is thus reduced to a steadyare therefore lacking in these models. To obtain a state heat diffusion problem around a smooth, shapeunique relationship for V vs AQ, it was convenient to preserving interface. The steady-state dendritic heatassume that dendrites grow at the maximum possible transfer problem is to be solved in the solid and liquid velocity allowed by the capillary effect, i.e., V,,,. The regions subject to two boundary conditions at the V,,, vs AT relationship so-generated [equation (l)]+ surface of the dendrite. The first condition is that the however, disagrees with the reliable experimental results obtained by Glicksman et al. [l J. temperature at each point on the solid-liquid interface In summary, dendritic growth theory may be equals the local equilibrium freezing temperature, which is determined by the local interface curvature K. divided into two portions: first is the main body of The second boundary condition requires that the the theory, in which the thermal diffusion problem is solved. Here, the existence of the dendrite sidelatent heat released at the interface during solidificabranches is neglected and the shape of the smooth tion is constantly conducted away through the adjoinneedle dendrite is constrained to some a priori geoing solid and liquid phases. In the case of succinonitrile. the thermal conductivities of solid and liquid are metry. Only recently, Nash and Glicksman [19] developed a nonlinear method to solve the theory of within 17; of being equal to each other. This circumstance permits use of certain simplified forms of the steady-state dendritic growth as a free boundary problem. In their theory, both the capillary and the heat transfer and stability theories. &IX conditions were applied rigorously to every point Note that both boundary conditions are incompletely defined, insofar as in the capillary condition as. on the dendrite surface, and a nonconstrained dendrite shape was determined as a part of the solution. well as the normal component of the temperature gradient in the energy conservation condition must be The maximum growth velocity predicted by this exact calculated from the dendrite geometry. The shape of theory, however, also disagrees with the experimental results-viz., it predicted velocities about 6.5 times the dendrite is not known a priori, however, and must larger than the velocities observed in Ref. [l]. Thus, be determined as part of the solution to the diffusion some remaining deficiency not associated with the problem. To eliminate the nonlinear aspects of this nature of the boundary conditions must be respon“free boundary problem,” the boundary conditions sible for the 65fold error remaining in the theory. given above were usually simplified or the dendrite Furthermore, a qualitative analysis [20] also led to shape predetermined by assumption. Various versions of the approximate theory have been reviewed in the conclusion that the 6.5fold predictive failure of the dendritic growth theory at AT > 1°C did not Ref. [l]; among them are the Ivantsov model [12,133, arise from the negiect of sidebranch effects in the m~ified Ivantsov model [14-161, and Temkin model theory. The second portion of dendritic growth theory [i7, IS], each yielding results in the form of is the “maximum growth velocity” hypothesis. which A0 = PC eK_?Z1(.PC) + AB,, (2) is used to select the operative growth state from among the possible (r! R, A0) coordinates to yield the where A6 is the dimensionless supercooling (ATCdL); C, is the heat capacity of the liquid phase; L is the ( Kli.X~A$) relation, as expressed by equation (1). The theoretical review presented above suggests that heat of fusion; E,(Pe) is the exponential integral funcalthough sidebranch effects are neglected and the cation &$e-“/u)dtc; Pe is the P&&et number at the dendrite tip (l’R/2u); R is the dendrite tip radius; and 118, pillary boundary condition is generally simplified, the is a term reflecting the influence of the capillary main body of the steady-state theory. i.e., equation (2). (Gibbs-Thomson) effect and is itself also a function of should have nonetheless been substantially correct. The failure of theory in predicting the relation f/ and R, which in turn is a function of the velocity V. between V and [email protected],i.e.. equation (1). is thus likely to Equation (2) reflects two major physical effects. The “point effect” (I/R = constant) is the solution to the be related to the use of the maximum growth velocity hypothesis. thermal diffusion model with an isothermal interface Experimentally. however, isolating the precise (Ivantsov model), in which y = 0 is assumed and contheoretical error remains uncertain when testing the sequently [email protected],= 0. On the other hand. if y # 0, then






theory only against the form of equation (I). Instead a full test of the theory against equation (2) would be capable of locating the defect in either the main body of the theory or the V,,, hypothesis. The latter approach requires the measurement of the tip radius, as well as the growth velocity, of a dendrite growing at a given supercooling. Based on the theoretical background reviewed above, it is expected that the (V, R) coordinate determined experimentally would fall on the solid line which describes equation (2), but not

state growth. Based on this concept, several stability criteria for dendrite tips have been developed, all of which can be expressed mathematically as. VR2 = constant. (i) OIdJeld’s model. The stable size of a dendrite tip was determined, based on the idea that the destabilizing force due to thermal diffusion is equal to the stabilizing capillary force. In a simple dendrite geometry analysis, the following stability criterion was obtained (23):

at the point of V,,,. C. ~~abi~irycriteria for a steady-state dendrire The needle dendrite predicted by the steady-state model in conjunction with the maximum growth velocity optimization principle was regarded to be correct to first order by some authors [21]. Second-order corrections would assumedly be needed to include the effects of the nonsteady-state sidebranch growth, which is neglected completely in the steady-state theory. For instance, Kotler and Tiller [21] applied Mullins and Sekerka’s stability analysis [22] to the surface of the optimized dendrite predicted by Temkin’s modei. An important finding of this dendrite stability analysis was that the whole surface, including the tip of Ternkin’s optimized needle dendrite, is unstable. In particular, the greatest instability was found to lie in the region close to the dendrite tip. As a result of the nature of this instability associated with the optimized needle dendrite, branching perturbations appeared with their largest ~plitudes located only -0SR behind the dendrite tip [21]. The branch geometry predicted for an optimized Temkin dendrite is, however, contrary to that of a real dendrite, where large perturbations are not observed near the tip, but only away from the tip. In spite of this contradiction in the facts, their analytical result was viewed simply as evidence for a strong sidebranch inftuence on the dendrite tip [2x]. An alternative implication of the unrealistic branching perturbation predicted by dendrite stability analyses was that the basis solution (the optimized Temkin needle dendrite) used in the analyses was incorrect. In search of a new needle dendrite, Oldgeld [23] recognized the need for the dendrite tip to achieve a stable condition so that its steady-state growth could be ensured. According to this hypothesis (to be proved experimentally in this paper), the basis dendrite for a branching stabihty analysis should not be a needledend~te optimized to maximum growth velocity, but should be the one with a stable tip. Thus the interfacial stability, rather than providing only a secondary correction as previously thought, imposes a direct restriction on the steddy-

t It is stressed that planar growth in an adiabatic system is unlikely to occur [25], unless perhaps when the amount of supercooling exceeds L/C, [15]. The planar growth situation here is only invoked to simulate the heat-transfer condition at a dendrite tip.

GLrlp= 14?$,


where Gt,,i, is the temperature gradient normal to the dendrite tip, and w is a characteristic. length which describes the size of the dendrite. The relation between w and the dendrite tip radius R was evaluated by a numerical calculation; for a stable dendrite, Oldfield found that [23] w = 0.2R. The temperature gradient in front of a dendrite tip propagating at a steady velocity V can be expressed as GLtip

LV =





Therefore, the stability criteria of equation (3) becomes *a* = 2adoIVR2 = 0.02,


where cr* is the stability criterion constant, and do is a capillary length defined as QCJL’. (ii) Planar interface model. In this model [24], the dendrite tip is first considered as a point lying on a planar interfacet which propagates at the axial dendritic growth velocity, K see Fig. [l]. The stability condition for the dendrite tip is then assumed to be the same as that of the planar interface. According to Mullins and Sekerka [25], for a crystal with a planar s/l interface to grow stably the wavelength of the perturbation must remain smaller than i, = 2nJw+,


where Y*is known as the critical ~rturbation wavelength. If ys can be related to the size of the dendrite tip such that R c 1,,


then the criterion of equation (6) becomes a, = 2ad0iVR2 = -!- = 0.0253. 4x2


(iii) Spherical model. The above planar interface model simulates the steady-state heat-transfer condition at the dendrite tip; however, it may not account for the proper local curvature at the dendrite tip (see Discussion). In the present spherical model, based on Ref. [26], the stability condition of the dendrite tip is assumed to be identical to that of a sphere which describes the local total curvature of the den-








Fig. 1. Schematic &owing the planar interface model for estimating dendrite tip stability: (a) before perturbation is considered; (b) the marginally stable condition, at which li, = R and &, = a*&,. drite tip. 2/R. The absolute stability condition for a growing sphere [223 with an instantaneous radius R has been given by

GI,sphere =&$+2)(2n+



where n is the order of spherical harmonics. Note the similarity between this equation and equation (3). Note also that the growth of a sphere is indeed a nonsteady-state problem. To simulate the steady-state growth of a dendrite tip, steady-state conditions have to be imposed on the growing sphere. Under this “steady-state” condition, the temperature gradient around the sphere is again equal to (.LV/aCp). Now equation (9) becomes

u* =



2 (n + 2) (2n + 1).


The spherical harmonic. n, is a positive integer, and, in a model that assumes material isotropy, the choice should be independent of crystallographic symmetry. We note here that the wavelength of a perturbation wave around a sphere of radius R is a function of n. i.e., 7” = 2nRjn. n = 6 is selected here to be consistent with the assumption used in the planar model that the

dendritic tip radius approximates the perturbation wavelength, or equivalently for n = 6 1 2n, R = 0.95&,


see Fig. 2. The stability criterion now reduces to u* = 2ad,,JVR2 = 0.0192.


(iv) Longer and Miiller-Krumbhaar (L&f-K) theory. In this theory, rigorous mathematics and extensive numerical computation were used to study the interface behavior of the whole dendrite [24]. The procedure used by LM-K started with the derivation of a linear integro-differential equation to describe the displacement of the dendrite surface away from the unperturbed base paraboloid. The equation of motion was derived by making a quasi-stationary approximation for the time-dependent diffusion field. Also, in the derivation, thermal diffusion in the solid phase was neglected and perfect material isotropy assumed. At the limit P&-+0, an equation of motion was obtained which contains the stability parameter u as the only systemdependent quantity. The equation of motion had no exact solution when (T+ 0 and was studied numerically by an eigenva~ue analysis. If the dendrite shape displacement had an exponential time-







Fig. 2. Schematic showing the spherical interface model at a marginal stability condition with harmonics

n = 6 but 6, # 0. dependence, then the eigenstate found was such that when Q* = 2rdojVR2 = 0.025 f 0.007,


ali the unstable modes at the dendrite tip vanished.

Note that the +30”/, uncertainty in r~* shown in equation (13) reflects only the uncertainty in the numerical procedures used by LM-K to solve the equations. All the stability criteria discussed above are in the same mathematical form, namely, VR2 = constant. Stability criteria are thus capable of providing the conditions needed to dissociate the V-R relationship predicted by steady-state models. equation (2). Moreover. the stability constants u* all cluster tightly near the value 0.02, quite independently of the geometry assumed or the level of sophistication employed in the analyses. A test of the stability criteria can be performed simultaneously in a test of equation (2) in which both V and R will be measured. EXPERIMENTAL


A. Specimen purijication and characterization A specimen of high purity study of dendritic growth in Succinonitrile supplied by pure) was therefore purified

is required in the present a one-component system. Eastman Kodak (-99% further in a series of dis-

tillation and zone refining steps [l, 271. The purity of the specimen at each stage was characterized quanti-

tatively by a liquidus temperature, 7i, and a solidus temperature, T,. In order to correlate these temperatures with the final specimen purity, a preliminary series of experiments were carried out to measure r, Table 1. Characterization

and T for succinonitrile samples extracted from various positions along a zone-refined column. These preliminary experiments allowed the calculation of some the~~yn~ic properties of the specimen material [27], as summarized in Table 1. When I; of the final specimen for thermal dendritic growth studies was measured, no meaningful deviation from T, could be resolved with our measuring equipment. As discussed in the next section, the temperature resolution was limited to 10-40C. From the material characteristics given in Table 1, this temperature resolution corresponds to a maximum detectable active impurity level of 5 x lo-’ mole fraction which corresponds to a purity of 99.99995% (6-9’s plus). B. Temperature measurement and control All the temperatures were monitored with a platinum resistance thermometer, the output from which was measured with a Smith thermometer bridge. This thermometer, which has a tem~rature-r~istan~ COeflicient of lO*C/o~, was capable of a resolution of O.OGGV’C, and was calibrated by the National Bureau of Standards to an accuracy of -+_O.O02”C. Additionally, the bridge, a double Kelvin type, was calibrated by the manufacturer (Leeds & Northrup), and was capable of resolving loss ohms. Temperature fluctuations in the observation tank were detected continuously by a null detector. The tem~rature controlled ob~rvation tank was constructed from double-wall sheet glass with a well insulated exterior. A Tronac-40 temperature control system employing a nickel thermohm was used to provide nonlinear, proportioning control signals to a 220 watt, low-impedance, stainless steel, tubular heat-

of specimen material (succinonitrile)

Segregation Coefficient

Liquidus Slope





Solidus Slope MU

- 1735.1

Melting Point WC)

58.083 k 0.002






Fig. 3. Schematic of specimen configuration and support stage. A and 3 are control heaters; C is the crystal growth chamber; D is the tilting and secondary’rotating device; E is the primary rotation and X-Y translation stage; and F is the tank cover. The diameter of the crystal growth chamber C is about 4 cm.

spatial orientation with respect to the axis of observation was necessary to account for the effect of stereographic projection. More specifically, a dendrite growing in a random orientation was observed from two different directions to determine the true growth velocity and the true growth orientation with respect to the gravity vector. Since the microscope was fixed, this required rotation of the specimen chamber. Similarly, to measure the tip radius of a dendrite, a dendrite was positioned in such a way that it was observed in one of the (lOO> directions. This was based on our observation, shown in Fig. 4, that the branchC. Specimen con~gurat~n and measurements of growth less tip region as well as the branched region disvelocity and tip radius played different profiles depending on the angle of Dendritic growth studies were carried out in the observation. As shown in the figure, the smooth specimen chamber shown schematically in Fig. 3. As region of the succinonitrile dendrite tip developed into a three-dimensional shape resembling the head of discussed in detail in Ref. Cl], this specimen configuaa Phillip’s screwdriver. Four fin-like protrusions, tion was capable of achieving the “free” dendritic growth conditions over the range of supercooling we which grew in the {IOO~ planes, represent the true shape of the growing dendritic tip. To illustrate explored. further the nature of this anisotropic shape deveiopAs also shown in Fig. 3, the specimen was mounted ment, the profiles of the dendrite tip viewed from a on a special supporting stage which allowed full rota(100) direction, Fig. 4(a), and from a (110) direction, tiorl and some degree of tilt and translation. The manipulation of a growing dendrite into a desired* Fig. 4(c), were superimposed on Fig. 5. In this figure,

ing coil immersed in the tank. To offset any excessive heat input from the control heater, a continuous, steady flow of water (rn~nt~n~ about 8°C below the control set point) was cirblated through the tubular heating coil. This system provided a differential “source/sink” capable of rapidly responding to either demands for extra heating or extra cooling. Shortterm temperature stability of ~0.0004”C was achieved routinely; moreover, thermal drift of the system was found to be less than O.OOl”C over a period of one week.








Fig 4. Influence of crystalline anisotropy on the configuration of a succinonitrile dendrite tip, which is viewed from: (a) [lOO] direction, (b) a direction between [lCO] and [llO], and (c) [llO] direction; (d) shows the sequential cross sections of these dendrites. illustrating the development’of sidebranches and the rotational symmetry about the growth axis close to the tip.

it is clear that the dendrite

profile observed from a (100) direction best fits the standard parabolic curve (shown as black dots) until branch oscillations appeared. Consequently, all tip radii measurements were taken from profiles viewed from a (100) direc-

tion. Comparison of Figs 4 and 5 also shows that the tip regions of succinonitrile dendrites are bodies of revolution for which the total tip curvature is 2/R. t The speed of dendritic growth increased rapidly with supercooling and the size-scale of the morphology decreased. Consequently both the time for observation and the resolution required for photography quickly exceeded the capabilities of available microscope systems. AT = 2’C was the practical limit. :This limit occurs because the characteristic diffusion length of the dendritic heat-transfer process became comparable to the size of the growth chamber, and therefore latent heat passed through the chamber walls into the thermostat.

RESULTS AND DISCUSSION Dendritic growth kinetics studies were carried out at 17 supercoolings. The supercooling ranged from a maximum of 2°C where a technical limit? for morphological studies was met, to a minimum of 0.043”C. where the required adiabatic isolation of the growing crystals was no longer maintained.$ At each’supercooling studied, the dendritic growth velocity was measured as a function of the true growth angle relative to the gravity vector. The orientation dependence of the dendritic growth velocity is evident when the supercooling level is less than about 1°C. The orientation dependence reflects the influence of the convective fluid flow, and has been discussed in some detail in papers previously published [2+]. In the present paper, discussion is limited to diffusion-controlled dendritic growth. To avoid confusion with the spatial orientation effect, only downward-growing dendrites






Fig. 5. Superimposition of Figs 4(a) and (c), showing fit to common parabolic curve (black dots). Dendrites of succinonitrile are bodies of revotution only in the neigh~rh~d of the tip; progressive deviations from the paraboloidal shape occur as the fourfold symmetric branching ‘sheets’ and the oscillatory branching ‘bumps’ develop.

(i.e., dendrites growing parallel to gravity) will be considered here. The tip radii of dendrites growing at 16 supercoolings from 0.043”C to 1.8OO”Cwere measured. Above a supercooling of approximately 2°C the high growth velocity combined with the small size of the dendrites precludes acquisition of useful photographs of dendritest. Thus, the tip radius analysis at and above AT = 2°C could not be carried out with the desired accuracy. Again, only the tip radii for downward growing dendrites is included in the discussion which follows.

t The speed of dendritic growth increased rapidly with supercooling and the size-scale of the morphology decreased. Consequently both the time for observation and the r~olution required for photo~aphy quickly exceeded the capabilities of available microscope systems. AT = 2°C was the practical limit.

A. A new steady-srate dendritic growth theory Now that both the growth velocity and the tip radius of a dendrite have been measured simultaneously. a rigorous test becomes possible of steadystate dendritic growth theory in the form of equation (2). To illustrate, the coordinate (V, R) determined experimentally at AT = 1.2”C is plotted in Fig. 6 for comparison with three of the diffusional theories. The experimental point lies at a position remote from the maxima of the curves and in a region in which the various theories tend to converge. Briefly, the following conclusions can be drawn from this result, all of which are consistent with those presented in the Theoretical Background section: 1. The simplest theory. based on the solution of the isothermal diffusion problem of Ivantsov. is substantially correct in describing the kinetics of dendritic tip growth. Thus a steady-state dendrite is essentially isothermal at T, and effectively paraboloidal in shape.









TIP RADIUS &ml Fig. 6. Comparison of the dendrite-tip operating state measured (solid circle) to theoreticai predictions. Open circles designate the maximum growth velocities predicted by two nonisothermal models. Broken line locates the operating state predicted by stability criterion, which agrees with experiment.

The inclusion of capillary effects (in all the nonisothermal models) into the isothermal theory requires only relatively small adjustments. This circumstance occurs here because the dendrites actually observed are all relatively large in size. For example, the interfacial temperature depression at the dendrite tip in question is only 6 x 10b4 K, which is only 0.5% of the total supercooling. This temperature depression was calculated from rhe well known Gibbs-Thomson relationship and from the known curvature at the dendrite tip, namely 2 x lo3 cm-‘. Therefore, the capillary corrections to the interracial temperature and the shape deviations from Ivantsov’s isothermal paraboloidal dendrite are essentially trivial. Various versions of dendritic growth theory developed over the last 20 years to include capiallary effects thus result in only minor differences among themselves in the operating region of real dendrites. The success of the. Ivantsov theory also suggests that sidebranch growth has a negligible effect in as far as the growth kinetics at a dendrite tip are concerned. 2. The “maximum growth velocity” principle is incorrect. It is in fact due to adopting this incorrect hypothesis that the substantially correct (but incomplete) diffusional growth theories all fail to predict the right V - AT relationship. It therefore appears certain that a new physical criterion other than the “maximum velocity” principle is needed for a correct and complete theory of steady-state dendritic growth in pure materials. 3. As also shown in Fig. 6, such a criierion can be based on the stability of a dendrite tip as discussed in the Theoretical Background section (i.e., u* = Zrrd,,/ VR” = constant). The “stability” criterion constant

measured in the present study is o* = 0.0195 (see Fig. 7), which is within rhe limits of the ~rnputation~ uncertainty in LM-K theory, equation (13). The results of the experimental test of the “maximum growth velocity” hypothesis vs the “stability” criterion is further demonstrated in Fig. 8. In this figure, we plot as a function of supercooling the dimensionless P&let number (I% = VR,&) measured in the present experiments, along with those predicted by various dendritic growth theories which evoke either the “maximum growth velocity” principle or the “stability” criterion. When using the maximum growth velocity principle, the capillary-limited dendritic growth theories lose their predictive accuracy which is retained only when the “stability” criterion is employed. In summary, an acceptable steady-state dendritic growth theory should, in the main, fulfill the heat diffusion condition around the whole dendrite and the marginal interfacial stability condition at the dendrite tip. Predictions of the new theory, subject to only minor adjustment from capillary effects, are shown in Fig. 9. In one case (Temkin’s model with the stability criterion), agreement with experimental results is within 10%. without any adjustable parameters. The predidivity of the new theory for the dendritic tip radius is shown in Fig. 10 and essentially perfect agreement results where convection effects are small (AL\T O.f”C). B. Further

discussion on the dendrite

tip stability


It has been shown above that a stability criterion in the form of VR’ = constant is successful in selecting























Fig, 7. Comparison of experimentally determined stability criterion constant 2ado/VR2 with that predicted from the LM-K theory. Shaded band indicates the range of uncertainty within the theory. Measurements show a systematic increase in the stability constant at small supercoolings possibly due to convection effects.

nature of that model, the stability constant predicted by the model cannot be taken too seriously. The stability constant predicted by the simple planar interface model agrees to 25?& As discussed previously, planar interface solidification is thermally unfavorable for a one-component system at supercoolings smaller than (I&,). Also, the reasonable assumption that R = is

the correct

steady-state tip growth condition from the dendritic heat transfer solutions. The stability constant measured (a* = 2ado/VR2 = 0.0195) lies near the lower limit of the LM-K theory (o* = 0.025 + 0.007). Note again that the 35% uncertainty in the LM-K theory is only due to the

uncertainty in the numerical computation used to solve the equation of interface motion. Similarly, the stability criterion constant measured in the present experiment also agreed with the stability criteria predicted by each of the three stability models presented in the Theoretical Background section. For example, the criterion given by Oldfield’s model agrees to within 2.5%. However, due to the overly simple ID-1














shown in Fig. 11. This might have accounted for the inaccuracy of the planar interface model. However, if the planar perturbation wave is to describe the dendrite tip curvature, the arbitrary amplitude 6, of the critically stable perturbation wave would have to be


09QI 9 ** I


remains to be justified rigorously. Actually, the above assumption underestimates the R values by 150/:, as




Fig. 8. PkcIet number vs supercooling for pure succinonitrile.






SUCCINONITRILE Qlick8m8n. Schmlor ( 99.999% Present


& Ayers



( pwmll*l to a J




[ ‘C]

Fig. 9. Growth velocity vs supercooling. The solid curve shows prediction of Ivantsov model and the dashed curve of Temkin model, both with stability criterion (a* = 0.0195) applied. Systematic deviation from theoretical curves at small supercooling is due to convection effects.

confined to 6, = b*&


See Fig. 2. It should be noted here that the amplitude of the perturbation wave is not actually predicted by the linear theory, but is merely a random quantity determined by the intensity of physical disturbances surrounding the tip. The limitation of equation (14) thus stands as a serious deficiency in applying the planar interface model.

In the spherical interface model, the stability constant is predicted to agree with the experimental result to within 1.5%. The steady-state thermal condition still has to be imposed at the spherical interface in order to simuiate the growth condition of a dendrite tip. The dendrite tip radius is stil1 assumed to be given approximately by the perturbation wavelength, as in the case of the planar interface model. This condition, in fact, is equivalent to selecting n = 6 as the dominant mode of the critical perturbation waves.
















Dendrite tip radius versus supercooling, comparison to diffusion-controlled steady-state theories.

However, the spherical model can account for the dendrite tip radius readily when no actual perturbation is excited. as when 6, = 0,


where 6, is the amplitude of the spherical perturbation wave. Equation (15) describes the basic mode of growth, equivalent to the eigenstate solution in LM-K theory. However, even when some disturbance to growth is present and the dendrite tip is perturbed, the tip can still grow in a steady-state because of the marginal stability condition persisting there (&‘S= 0). In the latter growth mode, the excited dendrite tip propagates at the same velocity as an unexcited one, but at a different (constant) tip radius, see Fig. 2. The disturbance must at the same time perturb the thermal diffusion condition and thus the growth velocity. These two types of perturbations counterbalance each other and should result in continuous adjustments in NORMALIZED

Fig. 11. Normalized

the tip growth condition. The growth state variations should be microscopic in scale, assuming that the disturbance is small. On a macroscopic scale, however, the perturbed dendrite tip remains in a “steady-state,” which may differ from the state of an unperturbed tip, The excited mode of dendritic growth should be the one actually observed as disturbances of some sort are inevitable. However, it should be noted that, for analytical reasons, the solution for an unperturbed tip should be regarded as the fundamental tip growth state. One example which might lead to the excited mode of tip growth is the original version of the spherical model 1261. In that version of the model, the temperature gradient was taken as r

?.‘ 1


dendrite tip radius vs supercooling.

showing R = 1.2i., over a wide range of
















Fig. 12. Dendrite tip radius normalized to critical nucleation radius, R/R*, as a function of supercooling. which is the actual temperature gradient around a growing sphere. Also, the relative stability condition @/S = d/R) was used [28]. Both conditions were inadequate to describe the steady-state situation at a dendrite tip. If the absolute stability condition (b/a = 0) is used, equations (9) and (16) combine to give R = R* [l + f(n + 2)(2n + l)],


where R* is the critical nucleation radius defined as R* = 2T,yIASAT. The above equation compares favorably with the experimental result of R/R* z 100 when n = 9 is used, Fig 12. However, a relation between R and R* can be obtained only when equation (16) is used, which fails to simulate the thermal condition at a steady-state dendrite tip. Therefore, the original spherical model, although resulting in a convenient relation between R and R*, cannot be considered correct. An explanation of dendritic growth processes in terms of periodic “nucleation” of protuberances at the dendrite tip [29] is consequently of doubtful validity for freely growing dendrites.

CONCLUSIONS I. A complete steady-state dendritic growth theory should in the main include (a) heat transfer and (b) consideration of dendrite tip stability. 2. Ivantsov’s simple solution to the dendritic heattransfer problem is substantially correct. Accordingly, the capillary effects must be trivial (since dendrites grow with tip dimensions about 100 times larger than the critical radius for nucleation). Also, the assump tion of a needle dendrite shape seems consistent with the experimentally observed axial growth kinetics at the tip. 3. The hypothesis of maximum growth velocity, conventionally used in conjunction with dendritic heat-transfer theories to predict the V-AT relationship, is incorrect and should be replaced by a morphological stability criterion.

4. Stability criteria developed in several models all assume the mathematical form of VR’ = constant. Among the models considered, LM-K theory is the most sophisticated one, but a modified spherical stability model developed in this study yields the criterion that provides the best agreement to our measurements in succinonitrile, i.e., 2ado/VR2 = 0.0192. 5. Thefundamental growth state of a dendrite tip, as revealed in the modified spherical interface model in accordance with the eigenstate solution of the LM-K theory, is critically stable (&, = 0) and unexcited (6” = 0). 6. The physical justification of the assumption R = I, used in the planar interface model and n = 6 in the spherical interface model remains uncertain. Acknowledgements-The authors gratefully acknowledge the assistance of Mr Keith Matuszyk in conducting the dendrite tip radius measurements and of Mr James Pan for providing technical advice on the photography. In addition, we thank Dr Robert J. Schaefer for providing guidance in designing the thermostatic observation tank and in making precision temperature measurements. Finally, the authors thank Dr John Carruthers, Dr Robert J. Naumann, and Dr Lewis Lacy for the support and encourage-

ment provided by the Marshall Space Flight Center and the Materials Processing in Space Office of the National Aeronautics and Space Administration.

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