Jian-Jun Xu

Dynamical theory of dendritic growth in convective flow

Preface. 1: Introduction. 1. Interfacial Pattern Formations in Dendritic Growth. 2. Dendritic Growth Interacting with Convective Flow. 3. Mathematical Formulation of the Problem. 2: Interfacial Wave Theory of Dendritic Growth with No Convection. 1. Steady State of Dendritic Growth with Zero Surface Tension Ivantsovs Solution. 2. The Basic State for Dendritic Growth with Nonzero Surface Tension. 3. Regular Perturbation Expansion of Axi-symmetric, Basic State of Dendritic Growth. 4. Global Interfacial Wave Instability. 5. Three-Dimensional, Linear Perturbed States around the Axi-symmetric Basic State of Dendritic Growth. 6. Outer Solution in the Outer Region away from the Singular Points. 7. The Inner Solutions near the Singular Point xic. 8. Tip Inner Solution to the Tip Region. 9. Global Trapped-Wave (GTW) Modes and Quantization Condition. 10. The Comparison of Theoretical Predictions with Experimental Data. 3: Steady Dendritic Growth from Melt with Convective Flow. 1. Mathematical Formulation of the Problem with Navier Stokes Model. 4: Steady Viscous Flow Past a Slender Paraboloid of Revolution. 1. Mathematical Formulation of the Problem. 2. The Oseen Model Problem. 3. Uniformly Valid Asymptotic Solution for Steady Viscous Flow Past a Slender Paraboloid of Revolution. 5: Asymptotic Solution of Dendritic Growth in External Flow (I). 1. Mathematical Formulation of the Problem. 2. Laguerre Series Representation of Solutions. 3.Asymptotic Expansion Form of the Solution as epsilon2--> 0. 6: Asymptotic Solution of Dendritic Growth in External Flow (II). 1. Laguerre Series Representation of Solutions. 2. Asymptotic Expansion Forms of the Solution for the Flow Field. 3. Leading-Order Asymptotic Solutions of Flow Field. 4. Asymptotic Expansion Solution of the Temperature Field. 5. A Brief Summary. 7: Steady Dendritic Growth with Natural Convection (I). 1. Mathematical Formulation of the Problem. 2. Laguerre Series Representation of Solutions. 3. Asymptotic Expansion Solution with Small Buoyancy Effect. 4. Summary. 8: Steady Dendritic Growth with Natural Convection (II). 1. Laguerre Series Representation and Asymptotic Forms of Solutions. 2. Leading Order Asymptotic Expansion Solutions. 3. First Order Asymptotic Expansion Solutions. 4. Summary of the Results. 9: Stability and Selection of Dendritic Growth with Convective Flow. 1. Basic Steady State Solution. 2. Linear Perturbed System around the Basic Steady State Solution. 3. Outer Expansion Solution. 4. Stability Criterion and Selection Condition of Tip Velocity. 5. Some Special Cases. 6. A Summary. 10: Concluding Remark. References.

GLOBAL NEUTRAL STABILITY OF DENDRITE GROWTH AND SELECTION CONDITION OF TIP VELOCITY

The present paper is concerned with the long-standing selection problem of the tip velocity in dendritic growth. Based on the global stability analysis of the system, a global neutral stability (GNS) condition of dendritic growth is proposed, which determines the tip velocity and the pattern formation. The theoretical predictions show a good agreement with the experimental investigations.

Stability and selection of dendritic growth with anisotropic kinetic attachment

In the investigations of free dendritic growth, the effect of kinetic attachment at the interface on pattern formation and selection has been an important issue attractinga g reat deal attention of researchers in the field of dynamics of pattern formation duringthe last decade. Particularly, the role of the effect of anisotropic kinetics played in the stability mechanisms and the selection of dendrite tip’s velocity has become a crucial element for further understandingof the behaviors of dendritic growth of many important materials. In this paper, we attempt to study this subject. We assume that the system has both anisotropic interfacial energy and kinetic attachment at interface. In terms of the approach developed in the interfacial wave theory, we explore the effect of kinetic attachment on the stability mechanisms and the selection criterion of the limiting state of dendritic growth, by solving the related eigenvalue problem. It is found that the effect of kinetic attachment may render a significant effect on the global travelling wave (GTW) instability mechanism, as well as the low-frequency (LF) instability mechanism. The kinetic attachment may considerably stabilize the GTW mechanism by reducingthe correspondingstability criterion e * : However, the kinetic attachment some times stabilizes, and some times destabilizes the LF instability mechanism. On this aspect, the ratio of anisotropy parameters of kinetic attachment and surface tension, r4; plays the important role. r 2002 Published by Elsevier Science B.V.

Further examinations of dendritic growth theories

Abstract Two analytical theories of free dendrite growth, the microscopic solvability condition (MSC) theory and the interfacial wave (IFW) theory have been proposed during the past decade, attempting to resolve the problem of selection of dendrite growth, and explain the essence of the pattern formation. This article attempts to clarify the differences and commonalities between these two theories and compare the predictions of these theories with some latest numerical evidence and experimental data. Since the MSC theory is most well-developed for the two-dimensional case, the comparisons of the theories with the numerical simulations are made mainly by using, but not restricted to, the two-dimensional, numerical solutions for dendrite growth with anisotropy of surface tension. Such kinds of numerical simulations have been lately carried out by Wheeler et al. (Physica D 66 (1993) 243), Provatas et al. (Phys. Rev. Lett. 80 (15) (1998) 3308; 82 (22) (1999) 4496) and Karma et al. (Phys. Rev E 53 (1996) 3071; Phys. Rev. Lett. 77 (1996) 4050; J. Crystal Growth 174 (1997) 54) with the phase field model, and by Ihle and Muller-Krumbhaar with the free-boundary problem model (1994). It is seen that in a region where the anisotropy parameter is not too small, the numerical simulations yield steady needle solutions, whose side-branching structures are not self-sustaining. These results support the conclusions drawn by both the MSC and IFW theories. However, the numerical simulations also showed that there exists ‘a smallest value of the anisotropy parameter’, less than which ‘no steady needle solution was found’ (refer to Wheeler et al. Physica D 66 (1993) 243). This numerical evidence appears to be in agreement with the IFW theory and contradict the MSC theory. The prediction of the IFW theory is also compared with the latest experimental data obtained by Glicksman et al. in the microgravity of space and an excellent overall agreement is found.

Interfacial wave theory of dendritic growth from a binary mixture : a comparison with experiments

This paper presents a wave theory for dendritic growth from a dilute binary mixture and its comparison with experiments. Our study shows that the global instability mechanism, the so-called global trapped wave (GTW) mechanism, that we first explored for dendritic solidification from a pure substance is valid for the present systems in a very similar fashion. Due to the presence of this instability mechanism, the present system also permits a discrete set of unstable global modes, which we call the global trapped wave (GTW) modes, and a unique global neutrally stable (GNS) mode. This GNS mode solution is related to the material parameters and the growth conditions. It determines the micro-structure of the dendrite and selects its tip-velocity. We have completed the numerical computations of the GNS mode solution for a large range of various physical parameters, and compared the theoretical results with the available experimental data, without any adjustable parameters. The theory shows good agreement with the experiments.

Dendritic growth with external flow: interfacial wave theory and its comparison with experiments

During the past years, the approach of the interfacial wave (IFW) theory has been applied to various systems of dendritic growth and viscous fingering with success [Xu, Interfacial Wave Theory of Pattern Formation, Springer, Berlin, 1997]. In this paper, we apply this approach to the case of dendrite growth in an external flow. We derived the global mode solutions of the system and explored the selection criteria of dendrites tip velocity. The comparison of theoretical predictions with the available experimental data has been made. It is found that both are in good overall agreement.

Uniformly valid asymptotic solution for steady dendrite growth in external flow

Abstract This work deals with dendritic growth with external flow. Assuming that the Prandtl number Pr, based on the tip velocity and the thermal length is large, we obtain a uniformly valid asymptotic solution to the steady state in the whole growth region for the case of zero surface tension.

Selection and resonance of dendritic growth with oscillatory external sources

The present paper attempts to study the selection and related resonance phenomenon of dendritic growth with oscillatory external sources. As a prototype problem, we consider dendritic growth from a pure undercooled melt with the interference of a laser beam. The laser beam is assumed to act at the tip of the dendrite during dendrite growth, which introduces an external point heat source at the tip, and, as a consequence, creates a tiny hot spot there. The strength of laser beam is easy to control. Hence, the effective area of the nose spot created at the tip is also controllable. Moreover, this nose spot can be made to be steady, or oscillatory with a given frequency. For the case of steady laser beam, we derive the quantization condition and selection mechanism of dendrite growth depending on the effective area of the nose spot. We find that the steady laser beam may considerably affect the growth speed of dendrite. For the case of an unsteady laser beam, we deduce that the modulation frequency of the laser beam may induce a resonance with the dendritic growth.

Examinations of dendritic growth theories with some latest numerical simulations and experimental data

Two analytical theories of free dendrite growth, the microscopic solvability condition (MSC) theory and the interfacial wave (IFW) theory were consecutively proposed during the past decade. This article attempts to clarify the differences and commons between these two theories and examine these theories with some latest numerical evidences and experimental data. We shall show that these numerical solutions do not support the MSC theory for the cases of anisotropy parameter smaller than a critical number, while they are consistent with the IFW theory. Moreover, the IFW theory is found to be in an excellent agreement with the experimental results.

Regular perturbation expansion solution for steady nonclassical needle crystal growth

The present paper is concerned with the steady, dendrite growth from a pure melt with arbitrary undercooling parameter (−1<T∞<0). We formulated and discussed the so-called steady nonclassical needle crystal growth problem, described the regular perturbation expansion (RPE) solution in terms of the small surface tension parameter e2 and also carried out extensive computation for their numerical results.