A phase-field study on the peritectic phase transition in Fe-C alloys

Accepted Manuscript A phase-field study on the peritectic phase transition in Fe-C alloys

Shiyan Pan, Mingfang Zhu, Markus Rettenmayr PII:

S1359-6454(17)30345-2

DOI:

10.1016/j.actamat.2017.04.053

Reference:

AM 13740

To appear in:

Acta Materialia

Received Date:

12 January 2017

Revised Date:

20 April 2017

Accepted Date:

24 April 2017

Please cite this article as: Shiyan Pan, Mingfang Zhu, Markus Rettenmayr, A phase-field study on the peritectic phase transition in Fe-C alloys, Acta Materialia (2017), doi: 10.1016/j.actamat. 2017.04.053

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ACCEPTED MANUSCRIPT A phase-field study on the peritectic phase transition in Fe-C alloys Shiyan Pana,b, Mingfang Zhub,*, Markus Rettenmayrc aSchool

of Materials Science and Engineering, Nanjing University of Science and Technology, Nanjing, 210094, China

bJiangsu

Key Laboratory for Advanced Metallic Materials, School of Materials Science and Engineering, Southeast University, Nanjing, 211189, China

Chair of Metallic Materials, Otto Schott Institute of Materials Research, Friedrich Schiller

c

University, Löbdergraben 32, 07743 Jena, Germany * Corresponding author: E-mail: [email protected] Abstract A quantitative multi-phase-field model, with an anti-trapping current and involving diffusion in the solid, is proposed to simulate the peritectic phase transition in Fe-C alloys. An interface field method with a newly defined step function is adopted to formulate the governing equation of the multi-phase-field variables. The proposed model is applied to simulate the -platelet tip growth during the peritectic reaction and the subsequent peritectic transformation (-platelet thickness growth) in large computational domains of the experimental length scale. It is found that the local liquid concentration at the triple junction point L// is slightly higher than the liquid concentration in equilibrium with the -phase, but lower than the liquid concentration in equilibrium with the phase. This leads to slight melting of the -phase in the vicinity of the triple junction, while the platelet growth continues. Higher tip velocities at higher undercoolings produce a steeper carbon concentration gradient along the -platelet’s thickness direction, and thus yield a higher thickness growth velocity. It is also found that the ratio of tip and thickness growth velocity of the -platelets increases at higher undercoolings, leading to decreasing tip radius and platelet thickness. Good agreement between the simulations and the experimental data reported in literature is achieved, demonstrating the quantitative prediction capabilities of the proposed model, and confirming the diffusion control for the peritectic phase transition close to equilibrium conditions. Keywords: Phase-field modeling; Peritectic phase transition; Carbon diffusion; Triple junction; Fe– 1

ACCEPTED MANUSCRIPT C alloys 1. Introduction The peritectic phase transition, involving the sequence of peritectic reaction and peritectic transformation, is observed in various alloy systems, such as iron, copper, aluminium, titanium alloys and nickel-based superalloys. In the Fe-C system at 1768K, the peritectic reaction occurs by a reaction of ferrite () with liquid (L), to produce a new intermediate austenite () phase, namely,  + L. During the peritectic reaction, first a thin -platelet nucleates that subsequently grows along the /L interface until the -phase is completely separated from liquid by the advancing -phase. The subsequent growth of the -phase into the - and L-phases is referred to as the peritectic transformation that can obviously be distinguished from the peritectic reaction event [1]. Peritectic solidification plays a crucial role in the quality control for continuous casting of steels, and the production of magnetic and electronic materials. Owing to the importance of the scientific interest and practical application, the behavior of peritectic solidification has been an active research topic for many years. Numerous theoretical analyses [2-5] and experimental observations [1, 6-15] have been carried out to investigate the mechanisms of peritectic phase transition. Classical analytical models assuming diffusion control have been proposed to investigate the kinetics of the peritectic reaction [2-5] and peritectic transformation [3,4]. The Bosze-Trivedi (BT) model [2] and the Fredriksson-Nylén (FN) model [3] focused on the peritectic reaction, neglect the remelting of the -phase and the diffusion in the -phase, and do not deal with the motion of the triple junction explicitly. Fredriksson et al. [3,4] proposed an analytical model for predicting the growth rate of the -phase during the peritectic transformation; solute diffusion is only considered in the -phase, and the concentrations at the interfaces are assumed to be in equilibrium. Extensive experimental studies have been performed to investigate the peritectic transformation using a solid/liquid diffusion couple [6], and the peritectic reaction by the in-situ observations utilizing high temperature laser scanning confocal microscopy (HTLSCM) in combination with a concentric solidification technique [1,7-11]. Tip velocities during the peritectic reaction in Fe-C alloys under various cooling conditions were measured [1,7-11]. Griesser et al. [1,11] also 2

ACCEPTED MANUSCRIPT measured the tip radius of an advancing -platelet. It is found that the experimentally measured platelet tip velocities are much higher than the predictions of the analytical models [2,3]. With increasing peritectic undercooling and decreasing alloy composition, three modes of the peritectic phase transition were observed, including a diffusion controlled mode with the progression of a planar interface, a cellular/dendritic peritectic transformation mode, and a partitionless massive transformation mode [10]. Moreover, the velocity of the peritectic reaction is found to be also influenced by the alloy composition. Griesser et al. [10] reported that at an undercooling of 2 K below the equilibrium peritectic temperature, the reaction rate of an Fe-0.43 C alloy was 500 m/s with a planar interface, while an Fe-0.18C alloy yielded a much faster reaction rate of 6000 m/s, followed by cellular/dendritic growth of the -phase into the -phase. As stated above, the comparison between experimental observations and (over)simplifying analytical models led to incommensurate results [1]. Moreover, owing to the complexity of the peritectic phase transition and the difficulties in precisely controlling the experimental conditions, experimental investigations on the kinetics of the peritectic reaction are not entirely consistent [7,10]. Inconsistencies between experimental observations and analytical predictions, as well as between different experimental investigations, led to a variety of proposed governing mechanisms, such as carbon diffusion control [1], thermal diffusion control [8], and surface tension control [4]. Over the past decades, significant progress in the numerical simulation of microstructural evolution has been made with the advent of powerful computers and advanced numerical techniques. Without the need of explicitly tracking the moving interface, phase-field (PF) modeling coupled with external fields has become an important and powerful method to model arbitrary complex morphologies generated during solidification [16-25]. It is known, however, that the interface width in real materials ranges from a few angstroms to a few nanometers, and in PF simulations at least 5-10 grid points must be taken within the interfacial regions to maintain numerical stability. Thus, the diffuse interface feature of the PF approach poses the challenge to describe realistic solidification microstructures on experimentally relevant length and time scales. In order to increase simulation domain size and numerical efficiency, generally an extended 3

ACCEPTED MANUSCRIPT interfacial width is adopted to construct a PF model. For the thin interface limit condition, Kim, Kim, and Suzuki (KKS) [16] introduced the condition of equal diffusion potential at both sides of the interface to create steady-state profiles of the phase-field in the interface region. Steinbach and Pezzolla [17] developed a multi-PF method for general multiphase transformations. By means of an interface field method, the complex multiphase problem is decomposed into a sum of dual phase changes. Moreover, the interface field method allows using an extended artificial interfacial width [18-20]. It is found, however, that the PF model in the thin interface limit, adopting an artificially extended width of the interface with different diffusivities in the parent phase and the new phase, suffers from the three spurious effects of diffusion potential jump across the interface, interface stretching, and surface diffusion along the arc length of the interface [21-25], which will destroy modeling precision. For eliminating the abnormal effects with an extended artificial interfacial width, Karma [21] devised an anti-trapping current that is added in the solute conservation equation. The PF mobility was determined based on the asymptotic analysis in the thin interface limit. Using these approaches, Karma developed a quantitative one-sided single-phase PF model that can be applied to simulate dendritic solidification with zero diffusivity in the solid [22, 23]. By extending Karma’s model to multi-phase systems, Folch and Plapp [24] proposed a quantitative one-sided multi-PF model for two-phase solidification processes. For Fe-C alloys, solute diffusion in solid is by no means negligible. Ohno and Matsuura [25] proposed a quantitative two-sided PF model with an anti-trapping current and a mobility that were obtained by an asymptotic analysis in the thin interface limit; the model is capable of simulating single-phase solidification involving solute diffusion in the solid. Various multi-PF models have been proposed for the simulation of peritectic solidification [20,26-35]. Many of them focus on simulating the evolution of peritectic morphologies [26-31]. Other authors performed quantitative simulations to study the kinetics of the peritectic phase transition [32-35]. Boussinot et al. [32] and Valloton et al. [33] adopted the quantitative one-sided multi-PF model proposed by Folch and Plapp [24] to simulate isothermal steady-state peritectic phase transitions in an Fe-Ni alloy [32], and peritectic coupled growth in Cu-Sn alloys [33]. Ohno 4

ACCEPTED MANUSCRIPT and Matsuura [34] incorporated mobility and anti-trapping current that they derived previously in the two-sided single-PF model [25] into Folch and Plapp’s one-sided multi-PF model [24], and optimized an added potential height for the mixture state of three phases. Based on this method, they developed a quantitative two-sided multi-PF model with arbitrary values of the solid diffusivities and interfacial energies. They applied the model to perform one-dimensional (1-D) calculations of peritectic transformations and two-dimensional (2-D) simulations of peritectic reactions in Fe-C alloys [35]. The calculated velocities of the planar /L and / interfaces during isothermal peritectic transformations precisely agree with the experimental data. The simulated rates of the peritectic reaction were fairly consistent with experimentally determined values. In addition, slight melting of the -phase near the triple junction was observed in their PF simulations. However, since the peritectic transformation and peritectic reaction were simulated in 1-D and 2-D separately, the interaction between the two processes could not be analyzed. Moreover, the simulation domain in Ohno and Matsuura’s work [34, 35] was limited to 6m3m. It is reported that tip radii and thicknesses of -platelets, measured from in-situ observations at low undercoolings, are around 36m and 1520m, respectively [1]. Accordingly, it is necessary to perform the PF simulations in a much large domain for a quantitative comparison of PF simulations with the experimental data obtained at low undercoolings. As summarized above, significant progress has been made regarding peritectic phase transition by theoretical analysis, in-situ experimental observations, and PF simulations. It is almost commonly agreed that the governing mechanism of the peritectic phase transition in Fe-C alloys at low undercoolings is carbon diffusion control [1, 3-4, 35]. Nevertheless, there are still some questions remaining unclear as specified in the following: (1) how do the local concentrations at the triple junction influence the melting of the -phase, the growth of the -phase, and the migration of the triple junction point? (2) How does the kinetics of the peritectic reaction (tip growth) influence the kinetics of the peritectic transformation (thickness growth) and the thickness of -platelets? (3) How does the simulated steady-state tip radius of -platelets compare with experimental results? It is pertinent to note that a fully quantitative evaluation of numerical results is essential to perform a 5

ACCEPTED MANUSCRIPT thorough comparison between simulations and experiments regarding the steady-state tip properties, including tip velocity and tip radius. In the present study, a quantitative two-sided multi-PF model is proposed for the simulation of peritectic phase transitions in Fe-C alloys. The model allows performing simulations in a large physical domain that is appropriate for the relevant experimental length scale. The microstructural evolution of the -platelet during peritectic reaction and subsequent peritectic transformation is analyzed in detail. Numerical tests are performed systematically regarding the computational convergence behavior and the elimination of an extra phase at the interfaces. The proposed model is applied to simulate peritectic phase transitions at various undercoolings. Extensive comparisons between the PF simulation results and experimental data regarding the -platelet morphology, tip velocities and radii, as well as the -platelet thickness are presented. 2. Governing equations and solving algorithm of the multi-PF model 2.1. Model description In the present multi-PF study, isothermal peritectic solidification (both peritectic reaction and transformation) is simulated for dilute Fe-C binary alloys, focusing on a single -platelet growing into the L- and -phases. The /, /L, and /L interfaces are assumed to be in local quasiequilibrium. The simulated peritectic reaction and subsequent peritectic transformation are governed by diffusional solute transport, and the effect of interface kinetics is not considered [36]. In addition, the present model neglects the effect of elasticity at the / interface [4]. The phase field variables, ,  and L, denote the volume fractions of the -, - and L-phases, respectively. The 2-D computation domain is divided into a uniform square mesh. Each mesh point is characterized by the composition, and the phase-field variables of the -phase ( =1), the -phase (=1), the L-phase (L=1), the /, /L, and /L interfaces, and the //L triple junction. At the beginning of simulation, the domain is divided into two equal areas consisting of the L- and phases in thermodynamic equilibrium. A seed of the -phase with an initial radius, R0, is assigned to the /L interface on the bottom of domain. The -phase grows along the /L interface and forms a platelet. After an initial transient, the -platelet reaches steady-state growth and continually 6

ACCEPTED MANUSCRIPT advances along the /L interface with steady-state tip velocity and radius. Meanwhile, the thickness of the -platelet behind the tip increases gradually according to the peritectic transformation, particularly, the -platelet growth laterally into the liquid (L) and into the -phase (), simultaneously. In order to quantitatively simulate the peritectic phase transition in a relatively large domain, the proposed multi-PF model with an anti-trapping current is constructed using the following scheme. An interface field method [17] is modified by introducing a new step function and then adopted to formulate the governing equation of the multi-PF variables. In the interface region, the condition of equal diffusion potentials [16] is utilized, ensuring steady-state profiles of the phasefields independently of the concentration fields. The governing equations of the multi-PF and concentration variables are reduced to single-PF models at the /L, /L and / interfaces. Thus, the PF mobility and anti-trapping current can be determined based on the available quantitative PF model for single phase solidification with two-side diffusion [25]. The governing equations and the solving algorithm of the present multi-PF model are described in detail below. 2.2. Governing equations In a 2-D Cartesian coordinate system (x, y), the Ginzburg-Landau-type free energy functional, describing the dynamic evolution of the peritectic phase transition, can be expressed as [17,37]:

F  FI  FC   ( f grad ({i })  f pot ({i }))  dV   f chem ({i },{ci }, T ))  dV ,

(1)

where FI is the interface energy functional and FC is the chemical free energy functional of the bulk phases. i characterizes the existence of phase i (i=, , L), and i satisfies the normalization condition of

   1 . ci is the composition of phase i, and the mean solute composition c is i i

calculated by the mixture rule, c   i i ci . f grad ({i })  ( 2 / 4) i (i ) 2 is the gradient energy term, where  is the gradient energy coefficient, and  is the gradient operator. f pot ({i })   ({i })   i i2 (1  i ) 2

is the potential barrier between the bulk phases, and

 ({i })  ( j i iji2 j2   tri  (i  j k )n ) / (2 j i i2 j2 )

determines

the

potential

height,

where

tri=L+L+. The exponent n exerts an influence on the extra phase appearing at the i/j 7

ACCEPTED MANUSCRIPT interface. Ohno and Matsuura [34] optimized the value as n=1.4 to reduce the formation of an extra L-phase at the / interface. f chem ({i },{ci }, T )   i g (i ) fi (ci , T ) is the contribution of the free energies of the bulk phases, where g(i)=i is an interpolation function, and fi(ci,T) is the free energy density of phase i as a function of the concentration of phase i, ci, and the system temperature, T. The free energy functional given in Eq. (1) and an interface field method [17] are adopted to formulate the governing equation of the multi-PF variables, in which the multiphase problem is decomposed into a sum of dual phase transitions. The time evolution of the phase field variables is thus expressed as [17-19]

i 2  t N

s s M j i

i j

ij

 F  F  2    N  i  j 

s s M j i

i j

ij

  F  F    F  F    I  I    C  C   ,  i  j   i  j  

(2)

where t is time,  FI i   FI  j is the driving force related to the i/j interface energy,  FI i   1 2   2 2i  2 {i }  i 1  i 1  2i    {i } i   i i2 1  i  ,  FC i   FC  j 2

is the chemical driving force for the phase transformation between the phases i and j, Mij is the multi-PF mobility of the i/j interface, N   i si , and si is a step function. The interface field method was originally proposed by Steinbach and Pezzolla [17] for simulating multiphase transformations, in which the step function was defined as si=1 if i > 0, and si=0 otherwise. We found, however, that for the simulation of the peritectic solidification, the numerical treatment of the step function, si, following the original definition suppresses the propagation of the phase-field variables in the region initialized by sisj = 0 (namely, i = 0 or j = 0). In the present model, we propose a new approach to evaluate the step function si. In a 2-D coordinate system (x, y) at time t, the definition of si is expressed as a function of the parameter, i, for each phase i (i=, , L) by si ( x, y, t ) 

1  0

i ( x, y, t )  0 

or i ( x, y, t )  0 and i (xnb , ynb , t )  i  otherwise

,

(3)

where (xnb, ynb) represents any of the four nearest and four next-nearest neighbor grid points of the grid point (x, y). Using the definition given by Eq. (3), the phase-field variable in the grid points that are initialized by a single phase and that neighbor the i/j interfacial region, obtain si = 1 and propagate smoothly. It is found that the parameter, i, influences the convergence behavior and the 8

ACCEPTED MANUSCRIPT formation of an extra phase at the interface. A systematic test regarding the effect of the i values on the steady-state tip velocity of -platelets and the extra phase appearing at the i/j interface is presented in Sec. 3.2 to determine the appropriate i values. In the present model, the relation between the concentrations ci and cj at the i/j interface is determined by assuming equal diffusion potentials, fi / ci  f j / c j , as proposed in the KKS model [16]. Thus, the partition coefficient, kij  ci / c j  cij,e / cij ,e , can be determined by the phase equilibrium between the phases i and j, where cij,e is the concentration of phase i in equilibrium with phase j. The functional derivative of the chemical free energy functional, FC, in Eq. (2), with respect to the PF variable, i, is written as  FC i  f i  ci f i ci which is considered to be the diffusion potential of the solvent in phase i (i=, , L). Taking the liquid phase as reference phase and approximating  FC i in the dilute solution limit for the -, - and L-phases, respectively, the expressions of the diffusion potential of solvent in phase i (i=, , L),  FC / i , are written as

 FC   f  c f c  Vm1 RT (cL ,e  cL,e  c ) ,

(4a)

 FC   f  c f c  Vm1 RT (cL ,e  cL,e  c ) ,

(4b)

 FC L  f L  cL f L cL  Vm1 RT (cL ) ,

(4c)

where Vm is the molar volume and R is the gas constant. Thus, the chemical driving force for the phase transformation between phases i and j is calculated as Vm1 RT (1  k L )(cL ,e  cL ) for L, and

Vm1 RT (1  k L )(cL ,e  cL ) for L. To perform a quantitative PF simulation with non-zero diffusivity in the solid, an anti-trapping current is added in the solute conservation equation to eliminate three non-physical effects. The time evolution of the concentration field, c, can be generally expressed as [21,24] c t    ( J c  J at ) ,

(5)

where J c   i Dii ci is the solute flux, Di is the diffusion coefficient of the solute atom in phase i, Jat is the anti-trapping current. Combining c   i i ci and ci  kiL cL , the composition of phase i,

   k  that is    k  into the

ci , can be expressed as a function of the mean solute composition c, ci  kiL c used to evaluate the chemical driving force in Eq. (2). Substituting ci  kiL c 9

i i iL

i i iL

ACCEPTED MANUSCRIPT solute flux Jc, and rewriting leads to  c J c   i Dii ci   i kiL Dii    k i i iL 

  . 

(6)

To determine the PF mobility and the anti-trapping current, the governing equations of the multi-PF and concentration variables are reduced to single-PF models at the /L, /L, and / interfaces, respectively. For example, at the i/j interface (i + j =1, si = sj =1, and N = 2), the governing equation of the phase field variables can be written as   F  F    F  F   i   M ij   I  I    C  C   . t  i  j   i  j  

(7)

Since the potential height  ({i })  ( j i iji2 j2   tri  (i  j k )n ) / (2 j i i2 j2 ) satisfies  ({i })   ij / 2 and  {i } i  0 at the i/j interface,  FI i and  FI  j can be expressed as  FI i   1 2   2 2i   ij  i 1  i 1  2i  ,

(8a)

 FI  j   1 2   2 2 j   ij   j 1   j 1  2 j 

  1 2   2 2 1  i    ij  1  i i  1  2 1  i   .

(8b)

 1 2   2 2i   ij  1  i i  2i  1    FI i

Substituting  FC i  f i  ci f i ci ,  FC  j  f j  c j f j c j , fi ci  f j c j ,  FI i in Eq. (8a) and  FI  j in Eq. (8b) into Eq. (7), we obtain i  M ij   2 2i  2 iji 1  i 1  2i   f j  fi  f j c j   c j  ci   . t

(9)

Eq. (9) is the governing equation of the phase field variable for the single-PF model. Thus, we can take advantage of the available PF mobility and anti-trapping current, derived via a second-order asymptotic analysis in the thin interface limit for single phase solidification with two-side diffusion[25]. When the peritectic transition is carbon diffusion control, interface kinetics effects can be neglected [36] and are eliminated by neglecting the kinetic coefficient. The concentrations at the

/L, /L, and / interfaces maintain nearly equilibrium. Thus, the mobility of the i/j interface in the thin interface limit, Mij, can be expressed as [25,34] 15a2 2 RTm ,ij D 1 1  (1  kij )(c ij ,e  cij,e )[1  (1  kij i )  ij ] , M ij 4 ij D jVm 2 Dj

(10)

where a2=0.6276 and Tm,ij is the transition temperature between phases i and j in the pure substance, 10

ACCEPTED MANUSCRIPT ij is a parameter associated with the composition field at the i/j interface and taken as 0 [25]. The gradient energy coefficient, , and the potential height, ij, are expressed as   3 2 ijWij and  ij  3 2( ij / Wij ) , where ij and Wij are the interface energy and width of the i/j interface,

respectively. The multi-phase anti-trapping current, Jat, originally proposed by Karma [21], and extended by Ohno and Matsuura [25,34] for arbitrary values of the solid diffusivities (D>0 and D>0), is expressed as J at  2 (ni  n j )  aij j i

  i    c c j  ci  i ni  2 (ni  n j )  aij k jL  kiL   n,      k  t i t  ij  ij j i  i i iL 

(11)

where aij  (1 / 2 2)( D j  kij Di )[1  (1 / 2)(1  kij ( Di / D j ))  ij ] , and ni  i / | i | is the unit vector perpendicular to the contour lines of the phase field i. 2.3. Solving algorithm of the multi-PF model The simulation system is initialized with domain length, grid size, and the selected parameters. The thickness of the /L interface, WL, is set to three grid points, WL=3x, where x is the grid size. The thicknesses of the /L and / interfaces are calculated by WL=WLL/L and W=WLL/, respectively. Eqs. (2) and (5) are solved using an explicit finite difference scheme with the time step t being determined by t  x 2 / (4.5DL ) . Zero-flux (Neumann type) boundary conditions are applied to calculate both the phase-field and concentration field. The physical and thermodynamic properties of Fe-C alloys used in the present work are given in Table 1 [34]. For the thermodynamic data listed in Table 1, the corresponding linearized phase diagram of Fe-C around the peritectic equilibrium is plotted in Fig. 1. The PF simulations of isothermal peritectic phase transitions are performed at an undercooling T=Tp-T0 below the peritectic temperature, Tp. For calculating the diffusion potential of solvent and the multi-PF mobility using Eqs. (4) and (10), the   L equilibrium concentrations, cL ,e , cL,e , c ,e , c ,e cL ,e and c ,e , at T0, are obtained from the phase

diagram shown in Fig. 1.

11

ACCEPTED MANUSCRIPT 3. Results and discussion 3.1. Steady-state properties of an advancing -platelet The multi-PF model is applied to simulate isothermal peritectic solidification of Fe-C alloys in a 2-D domain of a 625×1250 mesh with x=0.08m. The parameters, i, in the step function are taken as =0.05 and L==0.07. The determination of i will be discussed in Sec. 3.2 in detail. Fig. 2 shows the evolution of the concentration fields in the L-, - and -phases at T=Tp-T0=3K. As shown in Fig. 2 (a), at the beginning of the simulation, the - and L-phases coexist and exhibit a planar interface in thermodynamic equilibrium. The initial compositions of the - and L-phases at T0=Tp-3K are calculated to be cL0  cL ,e  2.50 mol.% and c0  cL,e  0.45 mol.%. A single  nucleus with a radius of R0=4 m and an initial composition c0  cL,e  0.85 mol.% is assigned to the /L interface on the domain bottom. After the simulation is initiated, the emerging -phase grows along the /L interface. As shown in Fig. 2 (b)-(e), in the L region solute (carbon) is enriched at the /L interface, because solute is rejected from the -phase into the liquid during  solidification. The enriched carbon concentration in the L-phase is slightly lower near the tip (2.52mol.%) than that near the root (2.54mol.%) of the -platelet. Conversely, in the -phase region the solute is depleted at the / interface, due to the fact that the -phase absorbs carbon from the -phase during the transformation . The carbon concentration in the -phase shows a slightly higher value near the

-platelet tip (0.44mol.%) than near the -platelet root (0.41mol.%). In the -phase region, there is a concentration gradient from the /L interface to the / interface (Figs. 2 (d) and (e)). It is also noted from Fig. 2 that after a short time of 15 ms, the shape of the -platelet tip and the carbon concentrations of the L-, -, and -phases at the triple junction point are nearly unchanged, indicating the steady-state of the peritectic reaction. Fig. 3 shows the advancing -platelet contour taken from Fig. 2. In the figure, the black solid lines indicate the -phase boundaries with the level set of =0.5 at different times. The -phase boundaries, including the /L and / interfaces, are indicated by the blue dot-dashed line (x=0) at the initial time, and the red dashed lines at t=15.7, 41.7, 95.2, and 148.7 ms, respectively. It is noted that during the -platelet growth, the triple junction point of the L// interface deviates from the 12

ACCEPTED MANUSCRIPT initial L/ interface (the blue dot-dashed line at x=0), and slightly deflects to the right-hand side. This implies that melting of the -phase occurs at the front of the growing -platelet during the peritectic reaction. Fig. 4 shows a magnification of the  tip region at time of 95.2 ms. Same as those in Fig. 3, the blue dot-dashed line located at x=0 indicates the initial L/ interface; the red dashed line indicates the -phase boundaries across from the / interface to the /L interface, while the black open circles represent the -platelet boundary from the / interface to the /L interface. Apparently, the hatched liquid region (L region) is formed by the melting of -phase. During the peritectic reaction process, the growing -phase rejects carbon, leading to an increase of liquid concentration at the L/ interface (Fig. 2). According to the phase diagram (Fig. 1), the equilibrium liquid concentration at the /L interface, cL ,e is lower than that at the /L interface, cL ,e . As shown in Fig. 2, the local liquid concentration, cL , at the triple junction is 2.52 mol.% that is higher than cL ,e (2.50 mol.%), but lower than cL ,e (2.55 mol.%), respectively. According to Eq. (4), the liquid concentration

difference,

cL  cL ,e  cL  0

and

cL  cL ,e  cL  0 ,

leads

to

the

phase

transformations of L and L, respectively. Therefore, melting of the -phase in the vicinity of the triple junction occurs to reduce the local supersaturation with respect to this phase, while the growth of the -platelet is enhanced. Fig. 3 also displays that the thickness of the -platelet growing into the -phase is thicker than that into the L-phase, producing an asymmetrical shape of the advancing -platelet with respect to the initial /L interface (the blue dot-dashed line at x=0). This simulated feature is identical to the experimental observations [1,6], 1-D PF simulation [35] and the analytical analysis [3,4] for the peritectic transformation. As shown in Figs 2 and 3, the simulated tip shape of the growing -platelet remains essentially unchanged during the steady-state peritectic reaction. The tip radius is measured based on a parabolic fit to the -platelet shape using a nonlinear least squares method. In Fig. 4, the black solid line shows the parabolic fit to the points at the -phase boundary represented by the black open circles in the region near the -platelet tip. The fitted second order polynomial can be expressed as  ( x)  ax 2  bx  c , and the vertex of the parabola is marked as (x, y), where x=-b/(2a) and y=13

ACCEPTED MANUSCRIPT (b2-4ac)/(4a), determined through  ( x )  0 . Therefore, the curvature at the vertex (x, y) of the parabola can be evaluated as  ( x )   ( x) (1  ( x) 2 ) 3/ 2 |x  x   ( x )  2 a with  ( x )  0 . 

Then, the radius of the -platelet tip is calculated as   1/  ( x )  1/ (2 a ) . It is seen that the shape of the -platelet close to the tip is slightly asymmetrical, and deviates from both the //L triple junction point and the vertex of the fitted parabola. As discussed above, this deviation is caused by melting of the -phase in front of the advancing -platelet. To obtain the steady-state growth velocity of the -platelet tip, the tip velocity varying with time for the case of Fig. 2 is recorded and presented in Fig. 5. The tip velocities are calculated by the derivative of the recorded tip position at the level set of =0.5 with respect to time, t, from t=0 to t=25 ms. As shown, the tip velocity starts from a large value and then rapidly decreases. After a transient period, the tip velocity approaches a steady-state level. It is known that for dendrite growth of single phase solidification there also exists an initial transient before reaching steady-state, and it is of the order of DL Vn2 , where Vn is the steady-state tip velocity of dendrite growth [14,38]. In the case of Fig. 5, the steady-state velocity of the -platelet tip is obtained as Vtip=465µm/s when the growth time is around 15 ms. Accordingly, the initial transient for reaching a steady-state of the peritectic reaction is also in the order of magnitude as DL Vtip2 that is about 80 ms, calculated by DL=1.7210-8m2/s and Vtip =465µm/s. 3.2. Convergence performance As described in Sec. 2.2, in the present multi-PF model the step function, si, involves the parameters i (i=, , L). To determine suitable  i values, the first numerical test is performed regarding the steady-state tip velocity of a -platelet with various i values. For this simulation, i is set to be 0.01…0.1, and the other simulation conditions are identical with those used for Fig. 2. It is found that the tip velocity tends to stabilize at 465µm/s (T =3K) when i is smaller than 0.07. However, when i is smaller than 0.04, an extra -phase appears at the L/ interface in front of the

-platelet as shown in Fig. 6 (a). When i is taken as larger than 0.04, the extra -phase at the L/ interface disappears, and a reasonable morphology of peritectic reaction is obtained (Fig. 6 (b)). Accordingly, considering the two aspects regarding the convergence of the steady-state tip velocity 14

ACCEPTED MANUSCRIPT and the suppression of an extra -phase appearing at the L/ interface, the parameter should be restricted to the region 0.04i0.07. The formation of the extra phase at the L/ and / interfaces is tested with various i values. It is found that if the parameters ==L=0.05 are equal, there is still a tiny amount of extra L- and

-phases appearing at the / and L/ interfaces, respectively. When L==0.07 is used, the extra Land -phases at the / and L/ interface can be sufficiently suppressed. In addition, the values of L and  are found to have less influence on the steady-state tip velocity of -platelets. Thus, in the following simulations, the parameters are taken as =0.05, and L==0.07. In addition, it is found that the optimized parameter, i, decreases with decreasing mesh size. The second test regarding the convergence behavior is carried out with respect to the mesh size for various undercoolings. The simulations are performed in the domains with different mesh point numbers and mesh sizes. After the -platelet reaches steady-state, the -platelet tip velocity, Vtip, and radius, ρ, are evaluated using the approach described in Sec. 3.1. It is found that to reach convergence, a larger undercooling of T=5K requires a smaller mesh size of x=0.04 m, whereas a smaller undercooling of 1 K allows using a larger mesh size of 0.2 m. Based on the test data, the mesh size for convergence, xc, as a function of undercooling, T, is fitted as xc0.2/T, which is used to determine the mesh sizes for the quantitative PF simulations at various undercoolings performed in Sec. 3.3. We have also tested the effect of domain size on the tip velocity and thickness of the -platelet. It is found that when the domain sizes of 50m100m and 100m100m are used for the cases of T>0.85 K and T0.85 K, respectively, the effect of finite domain size could be nearly eliminated. 3.3. Effect of undercooling on -platelet growth Utilizing the experimental technique of HTLSCM, Griesser et al. [1,11] performed in-situ observations of the peritectic reaction of Fe-0.43 wt.% C under conditions close to chemical and thermal equilibrium. The specimens were machined to a thickness of 0.18 mm and bulk effects were minimized. Thus, the in-situ HTLSCM experiments by Griesser et al. [1,11] could be 15

ACCEPTED MANUSCRIPT considered to be of quasi-2D character. PF simulations are carried out and compared with the experimental investigations. As shown in Fig. 7 (a) and (c), the scale of the experimental morphologies is 50 μm100 μm, and the steady-state velocities of -platelets were measured as 36 μm/s and 510 μm/s for the given two cooling conditions, respectively [1,11]. The PF simulations are performed in a domain with a 500500 mesh with x=0.2μm at T=0.85K in Fig. 7 (b), and a 6251250 mesh with x=0.08μm at T=3.17K in Fig. 7 (d). The undercoolings for the simulations are determined by matching the experimental tip velocities. The initial compositions are calculated to be cL0 =2.38 mol.%, c0 =0.43 mol.%, c0 =0.80 mol.%, for T=0.85K, and cL0 =2.50 mol.%, c0 =0.45 mol.%, c0 =0.85 mol.%, for T=3.17K. Other conditions are identical with those used for Fig. 2. Figs. 7 (b) and (d) show the -platelet morphologies presented by the concentration field at t=1848.5 and 133.1 ms for the two undercoolings. The simulated steady-state tip velocities are Vtip =36.5μm/s for T=0.85K and Vtip =521μm/s for T=3.17K, which are close to the experimentally measured data in Figs. 7 (a) and (c), respectively [1,11]. As shown, the PF simulation and experimental observation show the same tendency that the -platelet thickness decreases with increasing tip velocity. Figs. 7 (b) and (d) also indicate that the concentration distribution is influenced by the advancing -platelet velocity. To investigate the mechanism behind the -platelet thickness varying with tip velocity, we analyzed and compared the thicknesses and carbon concentrations at the positions of the black dotdashed lines with y=37.5 m from the bottom (the length of 25m from the bottom has been cut off) in Figs 7 (b) and (d). For the peritectic transformation controlled by diffusion, it is expected that the thickness of the -platelet is proportional to the square root of time. However, in the present work, the simulations are performed for the peritectic reaction and the subsequent peritectic transformation simultaneously. The growth kinetics of the -platelet thickness is significantly influenced by the peritectic reaction. Thus, we use the average velocity to evaluate the growth velocity of the -platelet thickness. It is recorded that the -platelet thicknesses at the height of y=37.5 m are d=17.7 m and 8.44 m, and the growth times are t-t0= 1028.5 ms and 72.0 ms for T=0.85K and T=3.17 K, 16

ACCEPTED MANUSCRIPT respectively, where t0 is the initial time when d = 0 at the height of 37.5 m. Thus, the average thickness growth velocities, calculated by Vthickness= d/(t-t0), at the position of y=37.5 m are Vthickness=17.2 m/s for T=0.85K and Vthickness=117.2 m/s for T =3.17K. According to an analytical analysis for the peritectic transformation [3,4], the -platelet thickness growth velocity is proportional to the concentration gradient along the -platelet’s thickness direction, a relation that is obviously not accessible in the experiments. As shown in Fig. 7, the simulated local concentrations at the /L and / interfaces, intersecting with the dot-dashed line (y=37.5 m), are cL =0.80 mol.% and c =0.78 mol.% for T=0.85K, and cL =0.85 mol.% and c =0.77 mol.% for

T=3.17K. Thus, the thickness concentration gradient can be evaluated as 1.1 mol.%/mm for T=0.85K, and 9.7 mol.%/mm for T=3.17K. The higher thickness concentration gradient at T=3.17K produces a higher thickness growth velocity of the -platelet than that at T=0.85 K. Yet, as described above, a slower tip velocity provides a longer time for thickness growth to increase the -platelet thickness. The ratios of tip and average thickness growth velocity, Vtip/Vthickness, are calculated to be 2.12 for T=0.85K, and 4.45 for T=3.17K, quantifying the higher ratio of tip and thickness growth velocity, Vtip/Vthickness, of the -platelet at a higher undercooling. As a result, the -platelet with the lower tip velocity is thicker than that with the higher tip velocity, as shown in Fig. 7. As shown in Fig. 7 (a) for the experimentally observed shape with Vtip = 36 μm/s, in the vicinity of the -platelet tip there is a short liquid groove. However, the morphology obtained by the PF simulation does not have this feature (Fig. 7 (b)). Ohno and Matsuura [34,35] also did not produce the shape with a liquid groove behind the triple junction in their PF simulations. It was experimentally observed that the liquid groove appeared only within a certain region of peritectic reaction velocities [1]. When the tip velocity is smaller than 5 µm/s or larger than 36 µm/s, there is no liquid groove, but a leading triple junction at the top of the -platelet as displayed in Fig. 7 (c). Moreover, as shown in Fig. 7 (a), even when there is a liquid groove, the triple point L// is close to the -platelet tip, and there is no liquid film separating the entire - and - phases. In this case, the growing -phase releases the carbon atoms into the L-phase, which are subsequently absorbed by 17

ACCEPTED MANUSCRIPT the melting /L interface that is close to the /L interface. As a result, the carbon diffusion length is very short around the moving platelet tip region. Accordingly, the experimental -platelet shape, even having a liquid groove in the vicinity of the -platelet tip, is still formed by the peritectic reaction. It is considered that the morphology difference between the experimental observation (Fig. 7 a) and PF simulation (Fig. 7 b) might be mainly caused by the fact that the thermal conditions of the two cases are not exactly identical. In the HTLSCM experiment [1,7-11], the specimen was heated by a focused laser beam in the center, leading to a radial temperature gradient across the specimen. A cooling rate was effectuated by adjusting the temperature of the heat source in the center. It can also be expected that the growing -phase releases latent heat that may promote the melting of phase near the -tip [8] or increase the local temperature. However, the present PF simulation adopts an isothermal temperature field, and neglects the effect of latent heat generated during the peritectic reaction. Tracking the liquid groove formation near the triple junction as shown in Fig. 7 (a) might need to couple the concentration and temperature fields. However, such conditions are beyond the focus of the present study. We will investigate this issue in future work. Fig. 8 presents the -platelet thickness, d, as a function of the distance from the platelet tip, y', measured from the simulated -platelets for the two tip velocities shown in Figs. 7 (b) and (d). The experimental data measured by Griesser [11] for the cases of Figs. 7 (a) and (c) are also plotted in Fig. 8 for comparison. It can be seen that for the simulated and experimental profiles, in the region close to the tip, the -platelet thickness increases rapidly as y' increases. When y' is larger than 10…15 m, the -platelet thickness increases only slightly. For the lower tip velocity, the platelet thickness is obviously thicker than that with a higher tip velocity. When y' is less than 25m, the simulated and experimental data agree well. When y'25m, the simulated profiles are slightly higher than the experimental data. Nevertheless, the relative errors between the PF simulations and experimental measurements for the platelet thickness at y'=50 m are less than 10% and 20% for Vtip = 36 m/s and 510 m/s, respectively. Thus, the agreement between PF simulation and experiment is considered to be good. 18

ACCEPTED MANUSCRIPT The steady-state velocities of the -platelet tip during the peritectic reaction are simulated with respect to various undercoolings of 0.21K…12K. As discussed in Sec. 3.2, a relatively large mesh size is used for small undercoolings to increase the computational efficiency. Table 2 presents the mesh sizes and the mesh node numbers for various undercoolings. The initial compositions of the

-, -, and L-phases are calculated according to the thermodynamic equilibrium condition at the relevant

undercooling,

T,

by

cL0  cL ,e  Tp  T  Tm , L  m L ,

c0  cL,e  k LcL ,e

and

c0  cL,e =k L Tp  T  Tm , L  m L . Other simulation conditions are identical with those used for Fig. 2.

The simulated steady-state -tip velocities at various undercoolings are presented in Table 2 and Fig. 9 (a). The experimental data obtained by Griesser et al. [10] for an Fe-0.43 wt.% C alloy are also included in Fig. 9 (a) for comparison. Fig. 9 (a) obviously confirms that the -tip velocity increases with increasing undercooling. The PF simulation and the experimental data obtained by Griesser et al. [10] again agree reasonably well. Fig. 9 (b) presents the ratio of steady-state -tip velocity and radius, obtained by the PF simulations and experimental measurements [1]. The simulation data are taken from Table 2 for the undercoolings of 0.21…3.17K. As shown, while the tip radii obtained from the experiment are somewhat larger than those from the PF simulations, the agreement between the PF simulations and the experimental data is quite good, considering that the two methods used for measuring the tip radius might be prone to some uncertainties. It is evident that the steady-state -tip radius, , is associated with the thickness of the -platelet. As discussed above about Fig. 7, the -platelet thickness decreases with increasing tip velocity, which goes along with a decreasing tip radius. The comparisons between PF simulations and experimental data in Fig. 9 demonstrate that the present multi-PF model can quantitatively simulate the diffusion controlled peritectic reaction. 4. Summary A quantitative two-sided multi-PF model is proposed to simulate the peritectic phase transition (peritectic reaction and subsequent peritectic transformation) in Fe-C alloys for carbon diffusion control. The model exhibits a good convergence performance considering the extended mesh size. An extra phase at the interfaces is effectively suppressed by properly determining the parameters i 19

ACCEPTED MANUSCRIPT in the step function. The proposed model allows performing the simulations in a large computational domain of the experimental length scale. The morphology and growth kinetics of the advancing -platelet and the concentration fields in the -, -, and L-phases, varying with time during the peritectic phase transition, are simulated and analyzed in detail. The -platelet tip radius is also evaluated based on a parabolic fit. The results show that after an initial transient that is of the order of DL Vtip2 , a steady-state of the peritectic reaction is reached with essentially stable tip velocity and radius. The triple junction point L// is found to slightly deflect towards the -phase region, indicating that melting of the -phase occurs in the vicinity of the triple junction. This is due to the fact that the local liquid concentration at the triple junction is slightly higher than the equilibrium liquid concentration with respect to the phase, but lower than the equilibrium liquid concentration with respect to the -phase. Moreover, unequal growth lengths at the / and L/ interfaces during the peritectic transformation are obtained by the simulation. PF simulations have been performed to investigate the interactions among the kinetics of the tip and thickness growth, the local concentrations, and the -platelet morphology at various undercoolings. It is found that higher tip velocities at higher undercoolings produce a steeper carbon concentration gradient along the -platelet’s thickness direction, leading to a higher thickness growth velocity. However, a slower tip velocity provides a longer time for thickness growth to increase the -platelet thickness. It is quantified that a higher undercooling yields a higher ratio of the tip and thickness growth velocity, Vtip/Vthickness, of the -platelet. Therefore, both -platelet tip radius and thickness decrease with increasing tip velocity. The simulation results are compared with in-situ experimental observations reported in the literature. Good agreement between the simulations and the experimental data is obtained, demonstrating the quantitative capabilities of the proposed model. The agreement between the PF simulations and experimental data verifies that diffusion control is the basic mechanism for the peritectic phase transition under the given conditions. The simulation results also provide insight into the complicated relationship between the kinetics of the peritectic reaction (-platelet tip growth) and transformation (-platelet thickness 20

ACCEPTED MANUSCRIPT growth). Acknowledgments We thank Dr. Stefan Griesser, University of Wollongong, Australia, for the helpful discussions. This work was supported by the NSFC (Grant Nos. 51501091 and 51371051), Jiangsu Key Laboratory of Advanced Metallic Materials (Grant No. BM2007204) and the Fundamental Research Funds for the Central Universities (Grant No. 2242016K40008). References [1] S. Griesser, C. Bernhard, R. Dippenaar, Mechanism of the peritectic phase transition in Fe – C and Fe – Ni alloys under conditions close to chemical and thermal equilibrium, ISIJ Int. 54 (2014) 466–473. [2] W.P. Bosze, R. Trivedi, On the kinetics expression for the growth of precipitate plates, Met. Trans. 5 (1974) 511–512. [3] H. Fredriksson, T. Nylén, Mechanism of peritectic reactions and transformations, Met. Sci. 16 (1982) 283–294. [4] H. Nassar, H. Fredriksson, On peritectic reactions and transformations in low-alloy steels, Metall. Mater. Trans. A. 41 (2010) 2776–2783. [5] G. Boussinot, C. Hüter, R. Spatschek, E. A. Brener, Isothermal solidification in peritectic systems, Acta Mater. 75 (2014) 212–218. [6] K. Matsuura, H. Maruyama, Y. Itoh, M. Kudoh, K. Ishii, Rate of peritectic reaction in ironcarbon system measured by solid/ liquid diffusion couple method, ISIJ Int. 35 (1995) 183–187. [7] H. Shibata, Y. Arai, M. Suzuki, T. Emi, Kinetics of peritectic reaction and transformation in FeC alloys, Metall. Mater. Trans. B. 31 (2000) 981–991. [8] D. Phelan, M. Reid, R. Dippenaar, Kinetics of the peritectic reaction in an Fe-C alloy, Mater. Sci. Eng. A. 477 (2008) 226–232. [9] S. Griesser, M. Reid, C. Bernhard, R. Dippenaar, Diffusional constrained crystal nucleation during peritectic phase transitions, Acta Mater. 67 (2014) 335–341. [10] S. Griesser, C. Bernhard, R. Dippenaar, Effect of nucleation undercooling on the kinetics and 21

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ACCEPTED MANUSCRIPT radiography of the initial transient during directional solidification of an Al–Cu alloy, Acta Mater. 60 (2012) 199–207. [23] H. Xing, X. Dong, H. Wu, G. Hao, J. Wang, C. Chen, et al., Degenerate seaweed to tilted dendrite transition and their growth dynamics in directional solidification of non-axially oriented crystals: a phase-field study, Sci. Rep. 6 (2016) 26625. [24] R. Folch, M. Plapp, Quantitative phase-field modeling of two-phase growth, Phys. Rev. E. 72 (2005) 011602. [25] M. Ohno, K. Matsuura, Quantitative phase-field modeling for dilute alloy solidification involving diffusion in the solid, Phys. Rev. E. 79 (2009) 031603. [26] J. Tiaden, Phase field simulations of the peritectic solidification of Fe–C, J. Cryst. Growth. 198-199 (1999) 1275–1280. [27] T.S. Lo, S. Dobler, M. Plapp, a. Karma, W. Kurz, Two-phase microstructure selection in peritectic solidification: from island banding to coupled growth, Acta Mater. 51 (2003) 599–611. [28] M. Ode, S.G. Kim, W.T. Kim, T. Suzuki, Numerical simulation of peritectic reaction in Fe-C alloy using a multi-phase-field model, ISIJ Int. 45 (2005) 147–149. [29] D. Phelan, M. Reid, R. Dippenaar, Kinetics of the peritectic phase transformation : in-situ measurements and phase field modeling, Metall. Mater. Trans. A, 37 (2006) 985–994. [30] A. Choudhury, B. Nestler, A. Telang, M. Selzer, F. Wendler, Growth morphologies in peritectic solidification of Fe-C: A phase-field study, Acta Mater. 58 (2010) 3815–3823. [31] J. Kundin, R. Siquieri, H. Emmerich, A quantitative multi-phase-field modeling of the microstructure evolution in a peritectic Al–Ni alloy, Phys. D Nonlinear Phenom. 243 (2013) 116– 127. [32] G. Boussinot, E.A. Brener, D.E. Temkin, Kinetics of isothermal phase transformations above and below the peritectic temperature: phase-field simulations, Acta Mater. 58 (2010) 1750–1760. [33] J. Valloton, J.A. Dantzig, M. Plapp, M. Rappaz, Modeling of peritectic coupled growth in Cu– Sn alloys, Acta Mater. 61 (2013) 5549–5560. [34] M. Ohno, K. Matsuura, Quantitative phase-field modeling for two-phase solidification process involving diffusion in the solid, Acta Mater. 58 (2010) 5749–5758. 23

ACCEPTED MANUSCRIPT [35] M. Ohno, K. Matsuura, Diffusion-controlled peritectic reaction process in carbon steel analyzed by quantitative phase-field simulation, Acta Mater. 58 (2010) 6134–6141. [36] M. Rettenmayr, Melting and remelting phenomena, Int. Mater. Rev. 54 (2009) 1–17. [37] J. Eiken, B. Böttger, I. Steinbach, Multiphase-field approach for multicomponent alloys with extrapolation scheme for numerical application, Phys. Rev. E. 73 (2006) 066122. [38] M. Zhu, D. Stefanescu, Virtual front tracking model for the quantitative modeling of dendritic growth in solidification of alloys, Acta Mater. 55 (2007) 1741–1755. Captions of figures and tables Fig. 1 Fe-rich part of the Fe-C phase diagram with linearized phase boundary lines. Fig. 2

Advancing -platelet shape and concentration field varying with time during peritectic

phase transition for an Fe-C alloy with cL0  2.50 mol.%, c0  0.45 mol.% and c0  0.85 mol.% at T =3K: (a) 0 ms; (b) 15.7 ms; (c) 41.7 ms; (d) 95.2 ms; (e) 148.7 ms (numbers in the figures show the local carbon concentrations). Fig. 3 -platelet contour at different times (t=0, 15.7, 41.7, 95.2, and 148.7 ms), taken from Fig. 2. Fig. 4 Magnified tip region of the -platelet taken from Fig. 2 (d), where L represents the original liquid region, and L represents the liquid region formed due to the melting of the -phase. Fig. 5 Tip velocity varying with time of an advancing -platelet under the conditions used for Fig. 2. Fig. 6 Extra -phase at the /L interface displayed by the  level set with (a) i=0.03; (b) i=0.05. The region in red color is 0.95, the region in yellow color is 0.50.95, the region in green color is 0.050.5, the region in light blue color is 0.05 in the L-phase, and the region in dark blue color is 0.05 in the -phase. Fig. 7 Comparison of the -platelet morphologies with different tip velocities obtained by the experiment [1,11] and PF simulation: (a) experiment, Vtip = 36 μm/s; (b) PF simulation, Vtip = 36.5 μm/s with T=0.85K; (c) experiment, Vtip = 510μm/s; (d) PF simulation, Vtip = 521μm/s with T=3.17K (numbers in the figures show the local carbon concentrations).

24

ACCEPTED MANUSCRIPT Fig. 8 Comparison of PF simulations and experimental measurements [11] regarding the -platelet thickness as a function of the distance from the -platelet tip with different tip velocities for the cases of Fig. 7. Fig. 9 Comparison of PF simulations and experimental measurements [1,10] regarding the tip velocity of the -platelet vs. (a) undercooling and (b) tip radius during the peritectic reaction.

Table 1 Physical and thermodynamic properties used in the present work [34] Symbol

Definition and units

Value

(m3/mol)

Vm

Molar volume

R

Gas constant (J/(mol K))

kL

Partition coefficient of the /L interface

7.710-6 8.314 0.179

kL

Partition coefficient of the /L interface

0.334

k

Partition coefficient of the / interface

L L 

Interfacial energy of the /L interface (J m-2)

kL/kL 0.204

Interfacial energy of the /L interface (J m-2)

0.319

Interfacial energy of the / interface (J m-2)

0.370

m L

Liquidus slope of the -phase (K

(mol.%)-1)

-1828

m L

-1399

Tm,L

Liquidus slope of the -phase (K Melting temperature of the pure bcc-Fe (K)

Tm,L

Melting temperature of the pure fcc-Fe (K)

1801

Tm, Tp DL

/ Transition temperature of pure Fe (K)

1667

Temperature of the peritectic point (K) Diffusion coefficient in the L-phase (m2 s-1)

5.210-7exp(-5.0104/(RT))

D

Diffusion coefficient in the -phase (m2 s-1)

1.2710-6exp(-8.3104/(RT))

D

Diffusion coefficient in the -phase (m2 s-1)

7.6110-6exp(-13.7104/(RT))

(mol.%)-1)

1811

1768.4

Table 2 Domain parameters, and simulated steady-state tip velocities and radii for various undercoolings T (K)

0.21

0.32

0.85

1.58

3.17

4

5

8

10

12

Mesh size (m)

0.2

0.2

0.2

0.08

0.08

0.04

0.04

0.02

0.02

0.015

Nx

500

500

500

625

625

1250

1250

2500

2500

3333

Ny

500

500

500

1250

1250

2500

2500

5000

5000

6667

Tip velocity (m/s)

2.82

5.90

36.5

128

521

896

1360

3730

5990

8780

Tip radius (m)

3.58

2.69

0.94

0.60

0.29

0.22

0.19

0.097

0.066

0.058

25

ACCEPTED MANUSCRIPT

Temperature, T (K)

Tm,L 1800 1770

L

mL 

mL



1740

Tp T0

1710 L

1680 Tm,

c,e 

c,ec,e

L

c,e



cL,e



cL,e

2 4 Carbon concentration, c (mol.%)

6

Fig. 1 Peritectic portion of the linearized Fe-C binary phase diagram.

Fig. 2 Advancing -platelet shape and concentration field varying with time during peritectic phase transition for an Fe-C alloy with cL  2.50 mol.%, c  0.45 mol.% and c0  0.85 mol.% at T =3K: (a) 0 ms; (b) 0

0

15.7 ms; (c) 41.7 ms; (d) 95.2 ms; (e) 148.7 ms (numbers in the figures show the local carbon concentrations).

ACCEPTED MANUSCRIPT 80 70 L

60





y (m)

50 40 30 20 10

R0

0

-10

0 x (m)

10

Fig. 3 -platelet contour at different times (t=0, 15.7, 41.7, 95.2, and 148.7 ms), taken from Fig. 2.

51.0

 boundary Fitting parabola Tip radius

50.5

y (m)

50.0

L'

L

(x, y )





49.5



49.0 48.5 -0.5

0.0

0.5

1.0 1.5 x (m)

2.0

2.5

3.0

Fig. 4 Magnified tip region of the -platelet taken from Fig. 2 (d), where L represents the original liquid region, and L represents the liquid region formed due to the melting of the -phase.

Tip velocity of -platelet, Vtip (m/s)

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1000 Steady-state tip velocity Vtip =465 m/s

800 600 400 0.000

0.005

0.010 0.015 Time, t s)

0.020

0.025

Fig. 5 Tip velocity varying with time of an advancing -platelet under the conditions used for Fig. 2.

Fig. 6 Extra -phase at the /L interface displayed by the  level set with (a) i=0.03; (b) i=0.05. The region in red color is 0.95, the region in yellow color is 0.50.95, the region in green color is 0.050.5, the region in light blue color is 0.05 in the L-phase, and the region in dark blue color is 0.05 in the -phase.

ACCEPTED MANUSCRIPT

Fig. 7 Comparison of the -platelet morphologies with different tip velocities obtained by the experiment [1,11] and PF simulation: (a) experiment, Vtip = 36 μm/s; (b) PF simulation, Vtip = 36.5 μm/s with T=0.85K; (c) experiment, Vtip = 510μm/s; (d) PF simulation, Vtip = 521μm/s with T=3.17K (numbers in the figures

Thickness of -platelet, d (m)

show the local carbon concentrations).

30

PF simulation, Vtip = 36.5 m/s Experiment, Vtip = 36 m/s [11] PF simulation, Vtip = 521m/s

25 20

Experiment, Vtip = 510 m/s [11]

15 10 5 0

0

10 20 30 40 50 Distance from -platelet tip, y' (m)

Fig. 8 Comparison of PF simulations and experimental measurements [11] regarding the -platelet thickness as a function of the distance from the -platelet tip with different tip velocities for the cases of Fig. 7.

Tip velocity of -platelet, Vtip ( m/s)

ACCEPTED MANUSCRIPT

10

Tip velocity of -platelet, Vtip (m/s)



Experiment [10] PF simulation

8

(a)

6 4 2 0 0

2

4 6 8 10 Undercooling,  (K)

3

10

12

(b)

2

10

1

10

0

10

Experiment [1] PF simulation 0

10 Tip radius of -platelet,  m)

1

10

Fig. 9 Comparison of PF simulations and experimental measurements [1,10] regarding the tip velocity of the

-platelet vs. (a) undercooling and (b) tip radius during the peritectic reaction.