Advances in Condensed Matter Physics

Volume 2019, Article ID 8767640, 8 pages

https://doi.org/10.1155/2019/8767640

## A Simplified Scaling Law of Cell-Dendrite Transition in Directional Solidification

^{1}Xi’an Aeronautical Polytechnic Institute, Xi’an 710089, China^{2}State Key Laboratory of Solidification Processing, Northwestern Polytechnical University, Xi’an, China^{3}Karlsruhe Institute of Technology, Karlsruhe 76131, Germany

Correspondence should be addressed to Yaochan Zhu; moc.liamtoh@8791nacoayuhz

Received 11 February 2019; Revised 11 May 2019; Accepted 21 May 2019; Published 2 June 2019

Academic Editor: Kiyokazu Yasuda

Copyright © 2019 Yaochan Zhu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

To describe the cell-dendrite transition (CDT) during directional solidification, a new simplified scaling law is proposed and verified by quantitative phase field simulations. This scaling law bears clear physical foundation with consideration of the overall effects of primary spacing, pulling velocity, and thermal gradient on the onset of sidebranches. The analysis results show that the exponent parameters in this simplified scaling law vary within different systems, which mediates the discrepancy of exponent parameters in previous experiments. The scaling law also presents an explanation for the destabilizing mechanism of thermal gradient in sidebranching dynamics.

#### 1. Introduction

As one of the typical problems of pattern formation, solidification microstructures are interesting for both scientific and technologic reasons [1–3]. Though remarkable progress in the study of microstructure evolution during solidification has been made, the mechanism of sidebranching dynamics is still unsatisfactorily understood [4–6]. During directional solidification, the onset of sidebranches means the occurrence of cell-dendrite transition (CDT). Over past few years, the identification of CDT has been widely performed experimentally [7–11], which showed that cell and dendrite coexist over a large range of control parameters. Given pulling velocity and thermal gradient, small primary spacing corresponds to cellular morphology in CDT region, while large primary spacing corresponds to dendrite. Accordingly, CDT significantly depends on the primary cellular/dendritic spacing, and a critical primary spacing corresponding to CDT should exist. In terms of the correlation between and control parameters, researchers have put much effort all along on finding the criteria of CDT by considering the spacing, pulling velocity , and thermal gradient . Based on abundant experiments [7–10], an empirical scaling law with the form of ∞ has been summarized to describe the critical primary spacing in which and are exponent parameters.

Although the form of scaling law has been proposed, some controversies still exist. On one hand, the values of the exponent and are inconsistent in different experiments. and were found in Gerogelin et al.’s experiments [7] and in Trivedi et al.’s experiments [8, 9]. On the other hand, this empirical scaling law was based on the data fitting without further physical foundation. Researchers have tried to expound the intrinsic physical foundation of the scaling law. In Trivedi et al.’s work [8], the critical primary spacing was given by the geometrical meaning of three characteristic lengths: solutal diffusion length , thermal length , and capillary length . Gerogelin et al. [7] also presented a self-similar asymptotic regime about , , and based on their experimental data. However, these analyses only focused on the assembly of characteristic lengths, not referring to the sidebranching dynamics, which characterizes the CDT. The exponent parameters selection and the physical foundation in the scaling law of CDT are still unclear and need further exploration.

The crossover of CDT is usually defined by the occurring of sidebranches. Accordingly, sidebranching should be one of the typical characteristics in CDT, and the connections between the scaling law and sidebranching dynamics should exist. However, previous scaling law did not take sidebranching dynamics into account. Therefore, it should be more reasonable to characterize CDT by the onset of sidebranching instability. Furthermore, the empirical scaling law indicates that, beyond CDT, increasing the thermal gradient will enhance the sidebranching dynamics. This destabilizing effect of thermal gradient on the sidebranching dynamics has been observed in [7–10]. Noise amplification theory failed in describing the effect of thermal gradient on sidebranching dynamics [7]. Therefore, to acquire a deep understanding of CDT and sidebranching, it is essential to propose more reasonable physical explanations on the scaling law and the destabilizing mechanism.

Although sidebranching dynamics has received considerable attentions in free dendritic growth [1, 4–6], only some basic understanding about the sidebranching dynamics has been obtained in directional solidification [7, 10–15]. Gerogelin et al [7] presented a model to describe the sidebranching dynamics, in which the noise amplitude at the cellular tip was controlled by the feedback of sidebranches. However, it is difficult to determine the growth factor in their model. On the other hand, experimental results and phase field simulations indicated that the diffusion instability of dendritic trunk is the most possible reason for sidebranching dynamics [14–16] and revealed that the initial sidebranching spacing only depends on the pulling velocity, but the sidebranch amplitude is determined not only by pulling velocity but also by the primary spacing and the thermal gradient . With consideration of the overall effects of primary spacing, pulling velocity , and thermal gradient on the onset of sidebranches, the scaling law of CDT may gain more physical foundation.

In this article, firstly, we briefly review the factors in determining the onset of sidebranches from previous experiments and quantitative phase field simulations. Then a new scaling law of CDT is proposed based on the sidebranching dynamics and the physical foundation of this new scaling law is presented. The effects of pulling velocity and thermal gradient on CDT will be analyzed according to this new scaling law. Finally, the quantitative phase field simulations will be used to testify this proposed scaling law.

#### 2. Factors Determining Sidebranches during Directional Solidification

It has been widely accepted that the primary spacing , pulling velocity , and thermal gradient play an important role in determining the sidebranches dynamics. The effects of , , and on the dendritic growth have also been well studied. Here the related results and how the sidebranching dynamics is affected by the controlled parameters are recalled briefly.

Primary spacing affects the sidebranch dynamics significantly. In experiments, the microstructures near the CDT show dendritic array with large spacing and cellular array with small spacing [7–10]. Quantitative phase field simulations revealed the intrinsic reason [15, 17] that the smaller spacing suppresses the sidebranch growth due to the strong interdendritic solutal interaction. However, after the appearance of sidebranch, the location of first sidebranch and the initial sidebranch spacing are almost independent of primary spacing. Therefore, the primary spacing only influences the amplitude of the sidebranch but does not change the initial sidebranch spacing. Accordingly, the primary spacing only supplies spacing for sidebranch growth.

Pulling velocity is an important control parameter during directional solidification. According to experimental investigations and simulation, the initial sidebranch spacing has a scaling law with the pulling velocity [7, 14, 15, 18]. Dendrite trunk also depends on the pulling velocity greatly. The width of dendrite trunk decreases as pulling velocity increases. Within the same primary spacing, the decrease of pulling velocity enlarges the diffusion length, which enhances the interdendritic interaction and suppresses the sidebranch growth. As aforementioned, the interdendritic interaction does not change the initial sidebranch spacing. Therefore, the variation of initial sidebranch spacing is mainly attributed to the variation of pulling velocity. In previous investigations, the variation of initial sidebranch spacing with pulling velocity satisfies [7, 14, 18].

The role of thermal gradient in sidebranch growth is a little bit complex. Previous investigation on the cell-dendrite transition indicated that positive thermal gradient promotes the generation of sidebranch [7], but the initial sidebranch spacing was independent of thermal gradient [14]. It is attributed to the remarkable interdendritic solutal interaction near CDT, where the thermal gradient significantly affects the mush zone in directional solidification. However, the details on the thermal gradient effects are still absent and some confusion still exists. For example, the effects of thermal gradient on dendritic morphologies are relatively weaker for dendritic array growth, where the sidebranch amplitude is almost independent of thermal gradient [15].

#### 3. Scaling Law of CDT from Sidebranching

From the above review, we can found that previous investigations have presented lots of details about sidebranching dynamics during directional solidification However, the intrinsic sidebranching mechanism during directional solidification is still not fully revealed. Compared with free growth, the interdendritic interaction will suppress the sidebranch growth, so the interdendrite interaction plays an important role in determining the sidebranches. It is still difficult to reveal CDT by directly deriving the sidebranching amplitude evolution from the basic diffusion equation and interface condition.

Here we focus on the interdendritic solutal interaction to reveal the CDT and derive the scaling law of CDT in the following. Sketch of the dendrite with sidebranches and definition of parameters during the derivations are shown in Figure 1, where is the dendritic tip radius, is the tip position along -axis, is primary spacing, is the sidebranch spacing, and and are groove width. The parameter definitions are as follows: solutal diffusion length , thermal length , and capillary length , where is the liquid solutal diffusion coefficient, , Γ is the Gibbs-Thomson coefficient [3], is the initial alloys composition, is the liquid slope, and is the equilibrium solute-partition coefficient. In the interdendrite region, we define the groove width and lateral diffusion length , where is the lateral interface velocity.