1 The criterion of planar instability in alloy solidification under varying conditions A viewpoint from free energy Feng yi Yu

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The criterion of planar instability in alloy solidification under varying

conditions: A viewpoint from free energy

Fengyi Yu

CAS Key Laboratory of Mechanical Behavior and Design of Materials, Department of Modern Mechanics,

University of Science and Technology of China, Hefei 230026, China

E-mail address: fengyi.yu.90@gmail.com

ABSTRACT

In alloy solidification, the transport processes of heat and solute result in morphological instability of the

interface, forming different patterns of solidification structure and determining the mechanical properties of

components. As the first observable phenomenon of the morphological instabilities, the planar instability

influences the subsequent stages significantly, deserving in-depth investigations. In this paper, the planar

instability in alloy solidification under varying conditions is studied. Firstly, the dynamical evolution of the

planar instability is performed by the theoretical model and phase-field model, respectively. Secondly, to

represent the history-dependence of solidification, the varying parameters are adopted in the simulations.

Then the criterion of the planar instability under the varying conditions is discussed. This paper considers the

critical parameters of the planar instability are the excess free energy at the interface and corresponding

interfacial energy. Finally, to validate the criterion, the comparisons between the phase-field model and

theoretical model are carried out, showing good consistency. Moreover, solidification processes with

different preferred crystallographic orientations are performed, demonstrating the effect mechanism of the

excess free energy and interfacial energy on the planar instability. The idea of the interfacial energy

influencing the planar instability could be applied to investigating other patterns induced by interfacial

instability.

KEYWORDS

Planar instability; Varying solidification conditions; Free energy; Interfacial energy; Phase-field model.

I. INTRODUCTION

The solidification structures dominate the mechanical properties of as-solidified parts. An accurate prediction

of the solidification structures could provide a theoretical basis for optimizing the parameters. To achieve the

accurate prediction, the solidification dynamics need to be revealed. Due to the different characteristics of

physical processes at different scales, the investigation of solidification dynamics has been a long-standing

challenge. [1,2] From the viewpoint of mesoscale, solidification structures are dominated by the interactions

between interfacial processes and transport processes of heat and solute. [3-5] The diffusive nature of the

transport processes, including spatial and temporal evolution, gives rise to morphological instability of the

solid/liquid (S/L) interfaces, resulting in different patterns of solidification structure.

As the first observable phenomenon in the evolution of solidification patterns, the planar instability

affects the subsequent solidification stages significantly [6,7], deserving in-depth investigations. Chalmers et

al. [8] analyzed the heat and solution balance at a moving S/L interface. They gave the idea of Constitutional

Supercooling (CS) and established the CS criterion of the morphological instability by G/VP<ΔT0/DL. In the

expression, G is the thermal gradient, VP is the pulling speed, ΔT0 is the solidification temperature range in

the phase diagram, and DL is the solute diffusion coefficient in the liquid. Although it reveals the

thermodynamic essence of the interfacial instability, the CS theory does not account for the transport

processes of heat and solute. Mullins and Sekerka [9,10] analyzed the stability of a crystal, based on a dynamic

approach in which the equations governing heat flow and solute diffusion are solved simultaneously while

allowing for a change of shape due to a perturbation, known as the MS theory. Compared with the CS theory,

the MS theory is based on a dynamic approach, considering the interplay of the diffusion transport and

interfacial energy, representing the spatial and temporal evolution of solidification patterns. However, the

MS theory is performed assuming a steady-state planar interface, neglecting the time dependence of diffusion

transport. By assuming the solute concentration evolves with time, Warren and Langer (WL) [11] extended the

MS theory to non-steady-state dynamics. Their analysis indicates the solidification evolution is history-

dependent, depending on the detailed way in which the sample is prepared and set in motion. The interfacial

instability predicted by the WL model agrees well with the experimental observations of real-time

synchrotron X-ray radiography [12], demonstrating the validity of the WL model. Recently, by combining the

time-dependent linear stability analysis in the WL model with the Fourier synthesis, Wang et al. [13] developed

a simple model to predict the morphological evolution of the S/L interface directly at the initial stage. The

model is verified by the experimental observations under the steady-state conditions [14]. Subsequently, Dong

et al. [15,16] modified this model from steady-state conditions to non-steady-state conditions, i.e., the model

can represent the time-dependent G and VP. The onsets of the planar instability under the same G and VP,

with different increasing rates of VP, were carried out. The results illustrate the increasing rate of VP does

affect the solidification evolution, including the incubation time and average wavelength of the planar

instability. This study demonstrates the microstructure evolution depends on the detailed way the

solidification conditions are achieved. To date, considerable investigations of the interfacial instability have

been made. However, these theoretical models involve many approximations and simplifications, resulting

from the constraint that the solutions of analytical and semi-analytical models can only apply under simple

conditions. As a result, these theoretical models could hardly handle the complex morphologies of the S/L

interfaces and the corresponding interfacial effects. Moreover, the as-simplified solidification conditions are

far from the realistic processes, limiting the application of these theoretical models.

Compared with analytical methods, numerical methods could solve the equations under complex

conditions, having the advantage of representing relatively realistic processes. As a representative, the Phase-

Field (PF) method combines the insights of thermodynamics and the dynamics of transport processes, having

solid physical foundations. [17-20] Moreover, since it avoids the shape error caused by tracking interfaces

during computation, the PF method has high numerical accuracy. [18-21] By introducing a phenomenological

“Anti-Trapping Current” (ATC) term [22], the PF model can simulate alloy solidification quantitatively. The

quantitative PF model has been applied to increasingly complex conditions, from isothermal solidification

[22], directional solidification [23,24] to melt pool solidification. [25,26] Based on the PF simulations, the

mechanisms of solidification evolution are studied, including the planar to cellular transition [14,15,27] the

selection of growth direction [28-30], the competitive growth [31-33], the columnar to equiaxed transition [34,35],

and the sidebranching dynamics [36-38], etc. The PF simulations agree well with the experimental observations,

indicating the accuracy of the PF method. In one word, the PF model avoids the limitations caused by the

simplifications in the analytical models. Moreover, since it can capture the complex morphologies and

characteristic parameters of the interfaces, the PF model can represent the interplay between the interfacial

processes and transport processes accurately, which is suitable for investigating the interfacial instability

systematically.

In this paper, the planar instability in alloy solidification under varying conditions is studied. Firstly, the

evolution of the planar instability is represented by the WL model and PF model, respectively. Secondly, to

represent the history-dependence of solidification, the dynamic parameters are adopted in the simulations.

Based on the simulations, the criterion of the planar instability under varying conditions is discussed. This

paper considers the critical parameters of the instability are the excess free energy at the S/L interface and

the corresponding interfacial energy. Finally, to validate the criterion, the comparisons between the PF model

and theoretical model are carried out. Moreover, solidification processes with the different Preferred

Crystallographic Orientations (PCOs) are performed, demonstrating the influences of the excess free energy

and interfacial energy on the planar instability.

II. MODELS AND METHODOLOGY

In directional solidification, the so-called “frozen temperature approximation” is adopted, given by:

( ) ( )

( )

T z t T G z z V t dt=+ −−

∫

(1)

where T0 is the melt temperature of the pure material, the pulling direction is along the z axis, and z0 is the

position of the interface. G is the temperature gradient, and VP is the pulling speed. This approximation is on

the basis of the assumptions: (1) The latent heat is ignored, i.e., the temperature field is undisturbed by the

evolution of the S/L interface. It is essentially a statement concerning the relative magnitudes of the terms in

the Stefan condition, ρsLfv*n<<ks,l∇Ts,l·n. In the expression, ρs is the density of the solid phase, Lf is the latent

heat, and v*n is the rate of solidification. ks,l means the thermal conductivity. ∇Ts,l is the thermal gradient, and

n is the normal direction of the S/L surface. The rate of alloy solidification (v*n) is limited by solute diffusion.

Since the coefficient of thermal conductivity is much larger than solute diffusion, the crystallization latent

heat can be released quickly through heat conduction. As a result, the effect of latent heat on the temperature

field could be ignored. [39] (2) There is no flow in the liquid, consistent with the assumption that the densities

of the solid and liquid are equal. [3] It should be pointed out, the frozen temperature approximation is just for

the heat transport. The thermodynamic model for solidification contains the latent heat, both in the theoretical

model and the PF model.

A. Theoretical model

1. Description of the model

The theoretical model is based on the linear instability analysis under the non-steady-state conditions,

the detailed derivations can be found in literature [11,13,14]. The following are the key equations.

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1Thecriterionofplanarinstabilityinalloysolidificationundervaryingconditions:AviewpointfromfreeenergyFengyiYuCASKeyLaboratoryofMechanicalBehaviorandDesignofMaterials,DepartmentofModernMechanics,UniversityofScienceandTechnologyofChina,Hefei230026,ChinaE-mailaddress:fengyi.yu.90@gmail.comABSTRACTInallo...

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1 The criterion of planar instability in alloy solidification under varying conditions A viewpoint from free energy Feng yi Yu

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