# Modeling of soft impingement effect during solid-state partitioning phase transformations in binary alloys

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## Abstract

The soft impingement effect at the later stage of partitioning phase transformations has been modeled both for the diffusion-controlled growth model and for the mixed-mode model. Instead of the linear and exponential approximations for the concentration gradient in front of the interface used in the past, a general polynomial method of dealing with the diffusion field is proposed. The linear and exponential diffusion field approximations are two specific cases of the polynomial diffusion field approximation. The effect of soft impingement on the overall partitioning phase transformation is only related to the degree of the super-saturation in case of the diffusion-controlled growth model, while it is determined by both the growth mode and the degree of super-saturation in case of the mixed-mode model.

## Keywords

Ferrite Diffusion Length Austenite Phase Ferrite Phase Diffusion Profile## Introduction

During the partitioning phase transformation, two simultaneous processes take place: diffusion of alloying elements ahead of the interface and migration of the interface. In the past, much research has been done to investigate the dominant process during the growth process of the partitioning phase transformation, and two classical models have been developed: the diffusion-controlled growth model [1] and the interface-controlled growth model [2]. Recently, a number of publications [3, 4, 5] have shown that the growth kinetics of the partitioning phase transformation cannot be described correctly by the classical models. Consequently, a mixed-mode model [3, 4, 5] has been developed, which predicts that the growth mode of any partitioning phase transformation become more and more diffusion controlled as the phase transformation proceeds. The diffusion-controlled growth model and the interface-controlled growth model are the two extremes of the mixed-mode model.

Normally, the partitioning phase transformation can be divided into two stages [6, 7, 8]: (1) the first stage of the phase transformation in which the diffusion fields in front of opposing interfaces in the parent phase do not overlap; (2) the second stage in which the diffusion fields start to overlap, and the phase transformation slows down, the so-called soft impingement effect [6, 7, 8]. As diffusion-controlled growth models have been proposed for a very long time, the analytical diffusion-controlled growth model for the first, non-overlapping diffusion fields stage have been well developed and are widely applied to describe the kinetics of phase transformation [1, 9, 11]. For the second stage, initially a so called mean field approximation [12] was used to take the soft impingement effect into account in the diffusion-controlled growth models. Later, to treat the overlap of diffusion filed in a more strict way, a number of diffusion-controlled growth models [7, 8, 13], assuming a linear diffusion field in front of the interface, have been developed to describe the soft impingement effect more accurately.

In [3], assuming a linear diffusion field in front of the interface, an analytical mixed-mode model has been developed to indicate the mixed-mode character of the partitioning phase transformation. However, a recent study by Bos and Sietsma [14] has shown that the original mixed-mode model underestimates the partitioning phase transformation kinetics because of the linear diffusion field approximation. Also, the soft impingement effect at the later stage of partitioning phase transformations is not considered in the original mixed-mode model.

In this study, based on the polynomial method, a precise diffusion profile expression is introduced to reformulate the analytical diffusion-controlled growth model and the analytical mixed-mode model with considering soft impingement effect, and the newly reformulated analytical models are validated by a comparison with a fully numerical solution. Furthermore, the effect of soft impingement on the overall phase transformation kinetics is investigated for both the diffusion-controlled growth model and the mixed-mode model, and results are compared.

## Analytical models

*t*

_{2}as the time after which the diffusion fields start to overlap,

*x*

_{0}as the interface position,

*L*as the length of diffusion field, \( C_{\text{eq}}^{\beta \alpha } \) and \( C_{\text{eq}}^{\alpha \beta } \) as the equilibrium concentration in the β- and α-phase,

*C*

_{0}as the bulk concentration,

*C*

_{m}as the carbon concentration at the center of the β-phase, and 2

*X*as the thickness of the parent phase.

### Diffusion-controlled growth model

In the classical diffusion-controlled model for solid–solid partitioning phase transformations [1], local equilibrium is assumed to be maintained at the interface during the entire phase transformation, which means that chemical potential of all alloying elements is equal and there is no chemical Gibbs energy difference at the interface itself during the phase transformation. Local equilibrium can be maintained only when the interface mobility is infinitely fast.

#### Non-overlapping diffusion stage

*x*for the non-overlapping diffusion stage in the diffusion-controlled growth model is described in a quadratic form here:

*A*

_{1},

*A*

_{2}, and

*A*

_{3}are the pre-factors,

*C*(

*x*)is the solute concentration as a function of position.

*n*= 2, it would become linear approximation as applied in [7, 13], when

*n*= 3, the diffusion field would be quadratic.

*λ*is the diffusion growth coefficient,

*Ω*is the degree of super-saturation.

#### Overlapping diffusion stage

*C*

_{m}starts to increase beyond

*C*

_{0}is the start of soft impingement, and the

*C*

_{m}increases until the equilibrium concentration is approached at the final stage of partitioning phase transformation.

*C*

_{m}and interface migration velocity:

*L*(=

*X*−

*x*

_{0}).

### The mixed-mode model

In the mixed-mode model [3, 4], both the interface mobility and the finite diffusivity are considered to have effect on the kinetics of phase transformation, and the concentration of alloying elements at the interface does not evolve according to local equilibrium assumption but depends on the diffusion coefficient of alloying elements and interface mobility during the phase transformation. The mixed-mode model will also be reformulated in two stages here.

#### Non-overlapping diffusion stage

*M*is the effective interface mobility, Δ

*G*is the driving force for interface migration and dependent on the solute concentration at the interface in the parent phase [3].

*M*, which is temperature-dependent, can be expressed as

*M*

_{0}is a pre-exponential factor,

*Q*

_{ G }is the activation energy for the atomic motion.

*G*, can be expressed as

*p*is the number of alloying elements in the system, \( C_{i}^{\alpha } \) is the concentration of the alloying element

*i*in the α-phase, \( \mu_{i}^{\beta } \) and \( \mu_{i}^{\alpha } \) are the chemical potential of the alloying element

*i*in the β- and α-phase, respectively.

*G*, can be approximated to be proportional to the deviation of the mobile alloying element concentration in the parent phase at the interface from the equilibrium concentration, and can be expressed as

The equation for the interface concentration is in the same form as that in the original mixed-mode model in which a linear diffusion field is assumed, however, the parameter *Z* in the original mixed-mode model is just one case of that present in this study. When *n* = 2, the mixed-mode model presented here is the same as the original mixed-mode model.

#### Overlapping diffusion stage

## Numerical solutions

*i*at time

*t*, \( c_{i}^{t + \Updelta t} \) is the concentration at grid point

*i*at time

*t*+ Δ

*t*. The second term on the right side of Eq. 33 accounts for the time dependence of the grid points. The diffusion-controlled growth model and the mixed-mode model will be combined with the Murray–Landis method to simulate the partitioning phase transformation here.

## Results and discussion

To illustrate the effect of soft impingement here, the austenite to ferrite transformation in the binary Fe-1.0 at.%C alloys at 1050 K is investigated here. At the given temperature, Thermo-Calc gives χ = 110 J/(at.%), and the equilibrium carbon concentration in the austenite phase and ferrite phase is 2.05 and 0.09 at.%, respectively. The diffusion coefficient of carbon in austenite is \( 1.14 \times 10^{ - 12} \;{\text{m}}^{2} / {\text{s}} \), and the interface mobility *M* is taken to be \( 5.4 \times 10^{ - 8} \;{\text{mol m/Js}}. \) In order to consider the phase transformation in a finite medium, the finite thickness of the austenite phase 2*X* is assumed to be 20 μm, and the specific volumes of both the phases are taken equal.

*n*− 1) order polynomial diffusion profile, in which

*n*parameters have to be determined, which means

*n*boundary conditions are needed to solve the problem. Except Eqs. 2 and 3, (

*n*− 2) extra boundary conditions can be written as

*C*

_{0}affects the evolution of diffusion length during the austenite to ferrite phase transformation at a certain temperature, and it is concluded that the smaller the bulk concentration

*C*

_{0}, the shorter the diffusion length. However, at different temperatures, the equilibrium concentrations in the austenite and ferrite phase are different, which also affects the diffusion length evolution. Therefore, it is necessary and meaningful to summarize all the effect factors into one factor to obtain a general law. In Fig. 5, the diffusion lengths as a function of the thickness of ferrite phase for different degrees of super-saturation are calculated by the diffusion-controlled growth model. The value of diffusion length is only affected by the degree of super-saturation, in which the bulk concentration and the equilibrium concentration in both the austenite and the ferrite phase are included. It shows that the magnitude of the diffusion length decreases with increasing the degree of super-saturation. Actually, decreasing the bulk concentration at a certain temperature discussed in the original mixed-mode model [3] is just one specific case of increasing the degree of super-saturation according to Eq. 13, and Fig. 5 can be considered as a master curve for estimating the diffusion length.

*C*

_{0}for all values of

*M*

_{0}when the ferrite fraction is zero. However, in the initial stage, the interfacial carbon concentration increase extremely fast, this steep increase is not properly reflected in Fig. 10a. In Fig. 10b, the newly defined growth mode parameter \( H\left( { = {\frac{{C_{\text{eq}}^{\beta \alpha } - C^{\beta } }}{{C_{\text{eq}}^{\beta \alpha } - C_{\text{eq}}^{\alpha \beta } }}}} \right) \)[17] as a function of fraction of ferrite phase for different interface mobilities is presented. For a certain interface mobility, the

*H*decreases as the phase transformation proceeds, which indicates that the growth mode become more diffusion controlled. As the soft impingement effect is corrected in the present mixed-mode model, the growth mode parameter would decrease to 0 when the thermodynamic equilibrium is established. Also, the growth mode is changed correspondingly with the variation of interface mobility, and the growth mode is approaching to the pure diffusion-controlled growth with increasing the ratio of interface mobility and the diffusion coefficient. In other words, it can also be stated that the dominance of soft impingement on the transformation kinetics is determined by the growth mode if the degree of super-saturation is fixed.

## Conclusions

Applying the polynomial diffusion field and considering the diffusion field overlap quantitatively at the later stage of phase transformation, the mixed-mode model and diffusion-controlled growth model are reformulated in an analytical form to be more accurate and physically reasonable in this study. The effect of soft impingement on the overall partitioning phase transformation kinetics is solely determined by the super-saturation according to diffusion-controlled growth, and it decreases with increasing the super-saturation. However, in the mixed-mode model, the effect of soft impingement on the overall phase transformation kinetics is determined by both the degree of the super-saturation and the growth mode.

## Notes

### Open Access

This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

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