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Deviations from cooperative growth mode during eutectoid transformation: Mechanisms of polycrystalline eutectoid evolution in Fe–C steels
Acta Materialia 97 (2015) 316–324
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Deviations from cooperative growth mode during eutectoid transformation: Mechanisms of polycrystalline eutectoid evolution in Fe–C steels Kumar Ankit a,b,⇑,1, Rajdip Mukherjee c, Britta Nestler a,b a b c
Institute of Materials and Processes, Karlsruhe University of Applied Sciences, Moltkestr. 30, 76133 Karlsruhe, Germany Institute of Applied Materials-Computational Materials Science, Karlsruhe Institute of Technology (Campus South), Haid-und-Neu-Str. 7, 76131 Karlsruhe, Germany Department of Materials Science and Engineering, Indian Institute of Technology, Kanpur 208016, UP, India
a r t i c l e
i n f o
Article history: Received 26 March 2015 Revised 10 June 2015 Accepted 21 June 2015 Available online 13 July 2015 Keywords: Non-cooperative transformation Divorced eutectoid Phase-field method Coarsening Polycrystalline austenite
a b s t r a c t Undercooling (below A1 temperature) and spacing between the preexisting cementite particles are known to be the factors that determine whether the isothermal eutectoid transformation in Fe–C proceeds in cooperative (resulting in lamellar pearlite) or non-cooperative mode (yielding divorced eutectoid). Typically, a divorced eutectoid microstructure consists of a fine dispersion of cementite in the ferritic matrix. Although, numerous experimental studies report a bimodal size distribution of cementite in the transformed eutectoid microstructure, the factors that facilitate the shift from a characteristic unimodal to bimodal size distribution have not been reported extensively. In the present work, we adopt a multiphase-field approach to study the morphological transition during isothermal eutectoid transformation which proceeds from an initial configuration comprising of a random distribution of cementite particles and grain boundary ferrite layers embedded in polycrystalline austenite. By conducting a systematic parametric study, we deduce the influence of preexisting arrangement of cementite, grain boundary ferrite thickness and prior austenite grain size on the mechanism by which eutectoid phases evolve. We also establish a synergy between the numerically simulated cementite morphologies and spatial configurations with those observed in experimental microstructures. Finally, we discuss the influence of the different factors that lead to the formation of mixed cementite morphologies (spheroidal and non-spheroidal) in the transformed microstructure and highlight the importance of 3D simulations. Ó 2015 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
1. Introduction Hypereutectoid steels are frequently annealed to develop a softer microstructure (as compared to pearlite), which consists of fine spheroidal cementite (h) particles embedded in a soft ferritic (a) matrix ([1,2] and references therein). Typically, two types of softening techniques are employed. (a) In the first method, the steel is annealed just below A1 temperature, during which, preexisting pearlitic lamellae break up into spheroids. Such a morphological transition is driven by a reduction in the total amount of h=a interfacial energy. (b) The second technique involves reheating ⇑ Corresponding author at: Institute of Applied Materials-Computational Materials Science, Karlsruhe Institute of Technology (Campus South), Haid-und-Neu-Str. 7, 76131 Karlsruhe, Germany. E-mail address: [email protected] (K. Ankit). 1 Present address: Institute of Applied Materials-Applied Materials Physics, Karlsruhe Institute of Technology (Campus North), Hermann-von-Helmholtz-Platz 1, 76344 Eggenstein-Leopoldshafen, Germany. http://dx.doi.org/10.1016/j.actamat.2015.06.050 1359-6454/Ó 2015 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
the steel so that it becomes almost fully austenitic (c) but with small amount of h remaining undissolved as spheroidal particles. On cooling through the eutectoid temperature, the preexisting h particles grow by absorbing the excess carbon that is partitioned into c, as the c=a transformation front evolves, thereby leading to a final structure of coarse h particles dispersed in a matrix of a. The main reason is attributed to presence of preexisting h particles in the parent c matrix facilitate pulling- away of the advancing a=c interface from h [3–5]. The final transformation product is known as the divorced eutectoid, since the evolving phases no longer grow cooperatively [6–8]. The focus of the present research is to analyze the mechanisms by which divorced eutectoid microstructure evolves in Fe–C steels. In the first part of this study [9], hereafter referred as Paper 1, we use a phase-field model based on the grand-potential formulation [10] to numerically simulate the non-cooperative evolution of the advancing a=c front during the isothermal eutectoid transformation in Fe–C alloy. By analyzing the simulated carbon
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redistribution mechanisms as the transformation proceeds from different ‘k’ and DT, we establish the criteria which govern the transition from cooperative to non-cooperative mode. In accordance with the earlier theories as well as the experimental studies [2,11,12], the numerical results presented in Paper 1 show that the mechanism by which an advancing a=c front evolves to form a lamellar or divorced morphology is strongly dependent on the initial interparticle spacing (k) of the preexisting cementite (h) and undercooling (DT) below the A1 temperature. We also identify the onset of a coarsening regime and amended the criteria for the non-cooperative eutectoid transformation. In this follow-up article, we study how the preexisting spatial arrangement of h particles influences the evolution of a=c transformation front (non-cooperative or otherwise). To begin with, we analyze the coarsening regime in detail, starting from an asymmetric configuration of preexisting h particles. Next, we investigate the influence of prior c grain boundaries and morphology of preexisting a layers on the temporal evolution of divorced eutectoid microstructure. Finally, by qualitatively comparing the simulated h morphologies with those observed in experimental microstructure, we deduce the mechanisms by which mixed morphologies (spheroidal as well as nonspheroidal) evolve. The remainder of the article is organized as follows: In Section 2, we briefly describe the multiphase-field model which is used for the present numerical simulations. In Section 3, we discuss the influence of introducing asymmetry in preexisting arrangement of h particles on the coarsening rate. We also examine the role of preexisting random arrangement of particles in facilitating a transition from cooperative to non-cooperative evolution of the advancing a=c transformation front. In Section 4, we analyze the non-cooperative eutectoid transformation mode in polycrystalline c and compare the simulated h morphologies to the experimental microstructures. Section 5 concludes the article. 2. Phase-field model In this section, we recount the multiphase-field equations used to study the evolution of eutectoid microstructure. The reader is referred to previous studies [10,13–15,9] for a detailed description. The evolution of phases is governed by the phenomenological minimization of the grand potential functional X,
XðT; l; /Þ ¼
Z
1
WðT; l; /Þ þ að/; r/Þ þ wð/Þ
V
dV;
ð1Þ
where T is the temperature, l is the chemical potential vector comprising of K 1 independent chemical potentials, / is the phase-field vector containing the volume fractions of the N-phases and is the length scale related to the interface. að/; r/Þ and wð/Þ represent the gradient and obstacle potential type energy density, respectively and V represents the domain volume. The grand potential density WðT; l; /Þ, which is the Legendre transform of the free energy density of the system f ðT; c; /Þ is written as an interpolation of individual grand potential densities
WðT; l; /Þ ¼
N X
Wa ðT; lÞha ð/Þ
a¼1
Wa ðT; lÞ ¼ f a ðca ðT; lÞ; T Þ
K 1 X
li cai ðT; lÞ;
ð2Þ
i¼1
where
ha ð/Þ
is
an
interpolation
function
of
the
form
ha ð/Þ ¼ /2a ð3 2/a Þ. The evolution equation for the N phase-field variables can be written as,
s
@/a ¼ @t
@að/; r/Þ @að/; r/Þ 1 @wð/Þ @ WðT; l;/Þ r K; @ r/a @/a @/a @/a ð3Þ
where K is the Lagrange parameter to maintain the constraint PN a¼1 /a ¼ 1. The concentration fields are obtained by a mass conservation equation for each of the K 1 independent concentration variables ci . The evolution equation for the concentration fields can be derived as, K 1 X @ci ¼r M ij ð/Þrlj @t j¼1
Mij ð/Þ ¼
!
ð4Þ
N X M aij g a ð/Þ;
ð5Þ
a¼1
where each Maij represents the mobility matrix of the phase a (related to the diffusivity). The diffusion coefficient (Di ) that accounts for the transport of solute inside the phase volumes (Dai ) and along the interphase interfaces (Diab ) is given by:
Di ¼
N;N N X 1X Dai /a þ Diab /a /b :
a¼1
ð6Þ
a
As the relaxed (stable) interface /ðxÞ contour has a sinusoidal profile, the effective diffusion coefficient through the interface can be determined using the values of Dai (diffusivity of ith component in phase volume) and Diab (diffusivity of ith component in interphase interfaces)
Z =2 =2
/a /b dx ¼
with /a ¼
Z =2 =2
/a ð1 /a Þdx ¼
8
p o 1n x : 1 þ sin 2
ð7Þ
On integrating Eq. (6) over the interfacial width (from =2 to =2), average diffusivity in the interphase interfaces is expressed as
Dab 1 1 Dinterface ¼ Dai þ Dai b : ¼ Dai þ i : i 8 8
ð8Þ
The thermodynamic data-fitting procedure to approximate the variation of the grand-potential of the respective phases as a function of chemical potential and the relation of the numerical simulation parameters with the corresponding quantities in the sharp-interface limit, are explained in the previous work [14]. Since, we are primarily interested in generic features of non-cooperative eutectoid transformation, the reported times are 2
normalized by l0 =Dc , where l0 ¼ r=ðRT=v m Þ is the capillary length and Dc is the diffusivity in austenite (c). T; R and v m denote transformation temperature, universal gas constant and the molar volume respectively. The numerical parameters used in the reported simulations are listed in Table 1. 3. Influence of arrangement 3.1. Divorced-coarsening transition in asymmetric arrangement According to the classical theory of non-cooperative eutectoid transformation [11], interparticle spacing (k) and undercooling (below A1 temperature) determine whether transformation proceeds in a cooperative manner or the advancing a=c front divorce the h particles. In the present section, we build upon the previous work of [9] and extend the numerical studies to investigate the influence of preexisting arrangement of h particles on the non-cooperative evolution of the a=c transformation front.
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Table 1 Numerical parameters used for the phase-field simulation of eutectoid transformation starting from polycrystalline c (‘r’ denotes grain radii) containing random distribution of h particles (‘kmin ’ denotes minimum inter-particle distance) and grain boundary a layers (of uniform thickness ‘L’). T E denotes the eutectoid transformation temperature of a binary Fe–C alloy, DT, the undercooling (below the eutectoid temperature) and r, surface energies (pairwise). Da ; Dh and Dc represents volume diffusivity in ferrite (a), cementite (h) and austenite (c), whereas Dac ; Dah and Dhc represent diffusivities along the corresponding transformation fronts. Symbol
Value
Units
TE DT
999 7.5 0:49
K K J/m2 m2 /s
rac ¼ rhc ¼ rah Da ¼ Dh Dc ac
D ¼D L kmin r
2 109 ah
hc
¼D
1 109 1000Da 0.1, 0.084, 0.0504, 0.0336 0.075, 0.084, 0.105, 0.147 1.2
m2 /s m2 /s lm lm lm
with the h phase. When the chosen k is small (kco ¼ 0:26 lm), the particles nearer to the pre-existing a=c transformation front (at odd positions), ripen while the ones farther (even position), shrink. As the eutectoid transformation proceeds, the h particles at even positions shrink and ultimately, dissolve. On analyzing the simulated chemical potential profile, we observe the carbon diffusion flux along the a=c transformation front, directed towards the h particles that are located at odd position positions. However, if the k is increased (kd ¼ 0:57 lm), coarsening of h is less pronounced and the transformation proceeds predominantly, in the noncooperative mode. For even larger k, curvature-driven coarsening of h is not observed. It is worth noting that the criterion of divorced-coarsening transition regime reported earlier in paper 1 holds for the case of asymmetric arrangements as well. However, the rate of coarsening, which influences the final microstructure, is dependent on the relative arrangement of preexisting h particles. 3.2. Mixed transformation products
In Fig. 1a, we compare the coarsening kinetics, as the transformation proceeds from a preexisting symmetric and asymmetric arrangements of two h particles. The evolution of the chemical potential and phase contours at representative times are shown for the two cases in Fig. 1b and c, respectively. It is observed that the boxed particle (shown by the schematic diagram in Fig. 1a) when placed at the symmetric position, shrinks at a faster rate, as compared to the latter. One may argue that the coarsening kinetics is expected to be much faster for the case of asymmetric arrangement, rather than the symmetric due to assistance from a=c interface diffusive flux (which is stronger than the volume diffusion flux) during the intermediate stages of the eutectoid transformation, as seen in Fig. 1c (at t ¼ 7:55 104 ). On the contrary, as the transformation proceeds from a symmetric arrangement, due to geometric considerations and the chosen value of k, the advancing a=c front does not get a chance to form an interface with the shrinking h particle, as clearly seen in the temporally evolving phase contours in Fig. 1b. Therefore, any assistance from a=c interface diffusion can be completely ruled out. Apparently, the question arises: In spite of availability of a faster diffusion path along the a=c interface, why is the coarsening rate slower when the isothermal eutectoid transformation proceeds from an asymmetric arrangement of preexisting h particles? The answer lies in analyzing the difference between the mean curvatures of the two h particles for the respective cases. Once the advancing a=c front forms an interface with the asymmetrically placed h particle, it continues to shrink in a lenticular morphology. This essentially means that the curvature difference between the h=c or h=a interfaces of the two particles depreciate, thereby, slowing down the curvature-driven coarsening rate. On the contrary, when the eutectoid transformation proceeds from a symmetric arrangement, the temporally shrinking particle retains the spheroidal morphology. On analyzing the temporal evolution of phases for both the cases (symmetric and asymmetric arrangements), it is apparent that the difference in curvatures of h=c or h=a interfaces of the two particles are greater when the shrinking particle retains the spheroidal morphology i.e. when symmetrically located with respect to the h particle closer to a=c front. Fig. 2 shows the influence of interparticle spacing (k) as the eutectoid transformation proceeds isothermally from asymmetric arrangement of preexisting h particles (undercooling, DT ¼ 10 K below A1 temperature). The results of the numerical simulations are illustrated for two different interparticle spacings (kco ¼ 0:26 lm and kd ¼ 0:57 lm). Starting from the initial configuration of phases. which comprise of asymmetrically arranged particles, the a=c transformation front advances to form an interface
It is typical to observe the non-spherical h particles co-existing with the spherical/well-rounded ones in a transformed eutectoid microstructure. As the presence of non-spheroidized h has an adverse effect on the machinability of steel, it is essential to establish the conditions of growth and process parameters which prevent the formation of non-spheroidized counterparts. In Fig. 3, we present the phase-field simulation results to demonstrate the influence of preexisting h particle in bringing about a shift from co-operative to non-cooperative evolution mode. For the case of single-layered arrangement, the preexisting particles that are nearer to the a=c transformation front, evolve non-cooperatively yielding a divorced eutectoid morphology. As the transformation front temporally evolves to form an interface with the farther set of particles, it pulls away and divorces the h particles in Level 1, as illustrated in the in Fig. 3a. Owing to an absence of any h particles ahead of the growth front, the transformation proceeds further in a cooperative mode. The apparent shift in the transformation mode (i.e. cooperative to non-cooperative) results in the evolution of lamellar/non-spheroidized eutectoid morphology (at t ¼ 5:01 103 s). On the contrary, when the transformation starts with a bi-layered arrangement of preexisting h particles (shown in Fig. 3b), a shift in the carbon redistribution mechanism is not observed. The temporally advancing a=c transformation front continues to pull-away from the h particles in Level 2 (in the first layer), which ultimately result in the formation of divorced eutectoid. Apparently, the present simulations suggest that the presence of preexisting h particles ahead of the growth front (in second layer) favor the non-cooperative evolution of the growth front. 4. Non-cooperative eutectoid transformation in polycrystalline austenite 4.1. Influence of preexisting a thickness (L) and configuration of h particles Polycrystalline c is modeled as an array of four hexagonal grains in a two-dimensional simulation domain with periodic boundary conditions. As the preexisting c grains are chosen to be equiaxed and of the same size, the ratio of the width (Nx grid-points) and height (Ny grid-points) of the numerical domain is restricted to pffiffiffi 3 : 2. This implies that a circle of radius ‘r’ which circumscribes the central grain relate to the numerical domain width as, pffiffiffi Nx ¼ 3r (or Ny ¼ 2 3r). For the present simulations, a computational box size of 500 433 grid-points is chosen, where the grid-spacings in the x and y directions (Dx and Dy respectively)
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6.0x10
-9
Symmetrical arrangement ( = 100 m) Asymmetrical arrangement ( = 100 m)
Volume fraction
5.0x10-9 4.0x10-9
3.0x10
-9
2.0x10
-9
1.0x10
-9
0
(a)
0.0x10 0.0x100
4.5x104
9.0x104
1.4x105
1.8x105
Time (non-dimensional) t = 1.71x104
t = 3.77x104
t = 4.39x104
5187.28
4378.63
4682.53
4382.45
t = 1.35x105
t = 7.55x104
4835.32
4353.11
4463.80
4353.81
4604.03
(c)
4345.38
Asymmetric
t = 1.43x103
4557.53
4386.03
4766.37
(b)
4183.92
Symmetric
t = 8.44x101
Fig. 1. (a) Temporal change in the volume fraction of the boxed h particle, as the a=c transformation front advances starting from a two different arrangements (symmetric and asymmetric). Corresponding evolution of the phase contours plotted over the chemical potential maps, as the transformation proceeds from (b) symmetric and (c) asymmetric arrangements. Green contours correspond to a=c interface while the white lines represent h=a and h=c interfaces.
are equal to 0.0042 lm. Therefore, the present choice of numerical domain size implies a c grain size (2r) that is equal to 1.2 lm. The four c grains in consideration are assigned distinct phase-fields and uniform equilibrium composition that is derived from the phase diagram. A spatial distribution of ‘N 0 ’ spherical h particles (initial radii equal to 8 grid-points) is algorithmically generated by using a numerical pre-processing technique. The minimum distance between two particles is constrained to be greater than kmin n grid-points, where kmin is the minimum inter-particle spacing and n is a random noise between 0 and 10 grid-points. The pre-processing algorithm iteratively fills the spherical particles inside the domain as per the imposed minimum inter-particle spacing (kmin ) criteria. Following every successful incorporation of a new particle in the numerical domain, the algorithm performs a number of iterations (lesser than the predefined maximum designated by imax ) to search for a suitable coordinate of the next particle that is to be added. For a limiting case, when the introduction of new particles is no longer possible for the chosen imax , the iteration exits. The maximum possible number of iterations (imax )
following every successful particle filling is chosen to be high enough to ensure that the algorithmically generated particle distribution is homogeneous. Finally, a layer of a (of uniform thickness ‘L’) is placed along the c grain boundaries and a parametric study is conducted by varying the simulation parameters as shown in Table 1. Figs. 4a–e and f–j show the temporal evolution of eutectoid phases staring from a polycrystalline configuration that is described above, for two different a layer thicknesses, L ¼ 0:0336 and 0:1 lm, respectively. Assuming other simulation parameters to be constant, numerical simulations are also carried out for intermittent thicknesses (0.0504 and 0:084 lm), to study the influence of initial grain boundary a layer thickness on the evolving as well as final eutectoid morphologies, in detail. It is observed that for a thickness of 0:0336 lm, the a layer that are in contact with cementite particle, pinch-off during the early stage of transformation and forms an island morphology to minimize the surface energy (Fig. 4b). Once the pinching-off stops, the a=c begins to temporally advance (Fig. 4c), in a predominantly non-cooperative mode. In the
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Fig. 2. Phase-field simulations showing the influence of k as the eutectoid transformation proceeds isothermally from a pre-existing asymmetric arrangement of h particles (DT ¼ 10 K). The phase contours are plotted on the corresponding chemical potential maps. At smaller k (kco ¼ 0:26 lm), onset of curvature-driven coarsening result in dissolution of h particle, marked at even positions. However, at larger k (kd ¼ 0:57 lm), the eutectoid transformation predominantly proceeds in non-cooperative mode.
(a) Level 2 -Fe
-Fe3C Level 1
}
1st layer
t = 6.67 x 107
t=0 4432
4432
4317.11
4552.38
t = 5.5 x 108 4608.08
t = 1.4 x 109 4336.06
4544.50
4340.82
(b)
} }
2nd layer 1st layer
t = 6.67 x 107
t=0 4432
4432
4327.12
4552.97
t = 2.28 x 108 4319.89
4549.90
t = 6.2 x 108 4326.48
4524.34
Fig. 3. Temporal evolution of phase isolines (/ ¼ 0:5) and the corresponding chemical potential map for an asymmetric (a) single-layered and (b) bi-layered arrangement of h particles (initial spacing, k ¼ 0:84 lm), during an isothermal eutectoid transformation (undercooling, DT ¼ 7:5 K). The isolines plotted over the chemical potential map illustrate the corresponding sharp interfaces of the phases. Black lines correspond to a=c interface while the white lines represent h=a and h=c interfaces.
time elapsed (during the a layer pinch-off), the count of preexisting h particles plummets down as a result of curvature driven coarsening (the phenomena is discussed in the detail in a companion article [9]) which subsequently, lead to a cooperative growth of the a=c front during the final stages of eutectoid transformation (Fig. 4e). It is noteworthy that the a pinching-off is not observed, if the initial a layer thickness is assumed to larger (Fig. 4f–g). By conducting a parameter study, it is observed that the coarsening of the preexisting h particles is comparatively less, if L > 0:0336 lm. Apparently, the numerical simulations suggest that the pinching-off of grain boundary a-layer is strongly dependent on the surface energies and ceases once the equilibrium phase triple-junction angles develop. The temporal evolution of h volume fraction is plotted in Fig. 5a starting from different initial thicknesses (L) of grain boundary a layers. A significant deviation in the carbon redistribution mechanism is observed for the case of L ¼ 0:0336 lm, during the early stages of eutectoid transformation. Such a deviation can be attributed to curvature-driven coarsening regime, that is active during
the a layer pinch-off (in early stages of eutectoid transformation) and causes a net carbon diffusion flux from the h particles located near the grain center towards the a=c front as they start advancing from the prior c grain boundaries. Unarguably, for the case pertaining to L > 0:0336 lm, the influence of particle coarsening in bringing about a change in the transformation mechanism, cannot be denied. Apparently, a temporal decline in the number of preexisting h particles (located towards the grain center) is also observed in this case. It can be argued that the pinching-off of preexisting a layers that is observed in the present phase-field simulations is unrealistic from an experimental perspective. The result of the parametric study reported in Fig. 5a allows us to identify the minimum threshold (numerical value of L) below which, pinching-off occurs. To rule out such a possibility, we assume the a layer thickness to be sufficiently high (L ¼ 5:04 102 lm) for all the numerical simulations, that follow. In order to study the influence of arrangement and spacing of h particles on the resulting microstructure, we numerically simulate the eutectoid transformation staring from different kmin (minimum
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(a) t = 0
(b) t = 2.8 x 107
(c) t = 7.6 x 107
(d) t = 2.3 x 108
(e) t = 3.2 x 108
(f) t = 0
(g) t = 2.8 x 107
(h) t = 7.6 x 107
(i) t = 1.1 x 108
(j) t = 2.6 x 108
Ferrite ( )
Cementite ( )
Austenite ( )
Fig. 4. Phase-field simulation showing the temporal evolution of the mixed eutectoid morphologies during an isothermal transformation (DT ¼ 7:5 K below A1 temperature), starting from an initial configuration which comprise of grain boundary a layers of initial thickness L and h particles that are randomly dispersed (kmin ¼ 0:126 lm) in polycrystalline c (grain diameter = 1:2 lm). The effect of initial grain boundary a thickness on the temporal evolution of phases is observed in the present simulations. At a small thickness (L ¼ 3:36 102 lm), the grain boundary a pinches-off from h particles to minimize surface energy, at the early stages, as observed in (a)–(e). The temporal evolution shown in (f)–(j) suggests that a pinch-off do not take place at early stages, if the initial thickness is higher (L ¼ 0:1 lm).
(a)
(b)
Fig. 5. (a) The plot showing the effect of initial thickness of grain boundary a layer (L) on the temporal evolution of h volume fraction. Pinching-off of a layers is observed at early stages (at L ¼ 3:36 102 lm) which results in a deviation from the trend observed for higher L. (b) Starting with four different interparticle spacings, the eutectoid transformation is numerically simulated (kmin ¼ 0.075, 0.084, 0.105 and 0.147 lm) and the final particle distributions (obtained on completion of the phase transformation) are compared. For the sake of comparison, other simulation parameters such as grain size (1:2 lm), initial thickness of grain boundary a layers (L ¼ 5:04 102 lm) and undercooling (DT ¼ 7:5 K below A1 temperature) are kept constant for the four cases. A deviation from the bottle-shaped distribution profile is observed for kmin ¼ 0:105 lmðN0 ¼ 119Þ due to fragmentation of pearlitic lamellae.
inter-particle spacing). The criteria for kmin that is imposed by the numerical pre-processing algorithm already described above, yields various distribution of h particles (corresponding number of particles is designated as N0 ). Fig. 5b shows the h size distribution obtained at the end of eutectoid transformation, by initializing numerical simulations with different number of particles (or kmin ). A comparison of the final h size distribution suggests that the peak value decrease on increasing N 0 (or on decreasing kmin ). Due to an enhanced coarsening of h particles at smaller kmin (or larger N0 ), the number of pre-existing particles located towards the grain center decline temporally and subsequently, dissolve (in the later stages). As the transformation proceeds towards completion, due to a complete or near absence of preexisting h particles, the a=c transformation front starts to evolve co-operatively leading to the formation of elongated/bottle-shaped particles which are found to be
pointing towards the grain center. The supple reversal of the transformation mode (non-cooperative to co-operative) that results in the formation of spheroidal as well as elongated/bottle-shaped particles, explain the presence of more than one local maxima in the size distribution plots shown in Fig. 5b. It is noteworthy that the reported reversal in the transformation mode is not favored if the initial inter-particle spacing (kmin ) is large. The distribution plot pertaining to N0 ¼ 90 (kmin ¼ 0:105 lm) represents an intermediate case in which, non-cooperative as well as cooperative modes dominate during the early and later stages of the transformation respectively, as suggested by an absence of a unique distribution peak. These results essentially suggest that at smaller kmin , curvature-driven coarsening predominates. Although the influence of coarsening is less pronounced at larger values of kmin , the transformation for these cases proceed by a complex interplay of
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(a)
(b)
Fig. 6. Temporal evolution of the phases and the corresponding particle size distributions are plotted, by numerically simulating the isothermal eutectoid transformation (DT ¼ 7:5 K below the A1 temperature), starting from an initial configuration comprising of grain boundary a layers (initial thickness, L ¼ 5:04 102 lm) and h particles at minimum distances of (a) kmin ¼ 0:075 lm and (b) kmin ¼ 0:084 lm in polycrystalline c (grain diameter = 1:2 lm).
cooperative and non-cooperative modes, depending upon the local distribution of h particles in the proximity of advancing a=c front. For a more detailed understanding, we analyze the temporal evolution of eutectoid phases starting from (a) kmin ¼ 0:075 lm and (b) kmin ¼ 0:084 lm, by plotting the corresponding particle size distribution at the representative time-steps. In the case of smaller kmin (Fig. 6a), a temporal shift of size distribution peak towards the right-hand side, suggests the predominance of a characteristic curvature-driven coarsening regime during the intermittent as well as final stages (474:6 1559:4 ls). At the later stages of the eutectoid transformation, coalescence of particles results in formation of non-spheroidal morphologies. Interestingly, for the case of larger kmin (Fig. 6b), a bimodal size distribution of evolving h particles is obtained. To begin with (67:5 ls), the size distribution has a unique peak. However, as the a=c transformation front temporally advances in mixed-mode, spheroidal as well as non-spheroidal particles evolve. On comparing the size distribution and the number of particles (inversely dependent on kmin ) in the simulated microstructure (Fig. 6a and b), we conclude that the curvature-driven coarsening is not a dominant mode by which the eutectoid transformation proceeds, if the initial inter-particle spacing is larger.
4.2. Influence of prior c grain size (r) To understand the influence of prior c grain size on the final microstructure, we compare the size distributions of h simulated at different ‘r’ (1.2 and 1:8 lm) while assuming the k to be same (equal to 0:147 lm) for both the numerical test cases. A characteristic unimodal size distribution of h is obtained at larger grain size (1:8 lm), as shown in Fig. 7. On the contrary, a bimodal size distribution is favored, if the initial grain size is assumed to be smaller by 33%. It is imperative to note that the influence of introducing prior c grain boundaries on the size distribution of h is more pronounced at small grain size and diminishes, when increased.
0.3
Normalized number of particles (N/N0)
At this point, we would like to state that the simulated bimodal size distribution of h particles (for kmin ¼ 0:084 lm) is an inherent characteristic of a mixed-mode (cooperative/non-cooperative) eutectoid microstructures. As the transformation proceeds, the carbon redistribution is dependent on the preexisting arrangement as well as the distance of the nearest h particles ahead of the advancing a=c interface. Apparently, for a given random distribution of preexisting particles, the carbon redistribution mechanism is more complex when compared to the case of symmetric arrangement, for which theories are relatively well known [11,2,12]. The numerical results suggest an overlap of cooperative and non-cooperative transformation modes, which are primarily governed by interparticle spacing (from nearest neighbor) and spatial arrangement of h ahead of advancing the a=c front. As we demonstrate that the cooperative and non-cooperative evolution of the a=c front as well as curvature-driven coarsening of h particle are favored depending upon the initial interparticle spacing and arrangement, the cementite morphologies and distribution found in the transformed microstructure yields insights concerning the intermittent regimes that predominate during the isothermal transformation.
Grain Size = 1.2 µm (N0 = 60) Grain Size = 1.8 µm (N0 = 150)
0.225
0.15
4.3. Comparison with experimental microstructures 0.075
0 0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.11
0.12
Radius of particles ( m) Fig. 7. Comparison of h size distribution simulated at two different prior c grain sizes (r ¼ 1:2 and 1:8 lm). N0 denotes the number of h particles at the start of eutectoid transformation. k for both the test cases is assumed to be same (equal to 0:147 lm). Prior a=c boundaries are denoted by black lines in the schematic diagram.
With a view to establish a synergy between the numerically simulated and experimental microstructures, we compare the various morphologies of h observed in the experimental microstructure with the phase-field simulations. For the given set of growth conditions, we observe various non-spheroidized h morphologies, which lend insights into the evolution mechanism of a=c front during various stages of eutectoid transformation. Two such exemplary h morphologies namely elongated and bottle-shaped particles are shown in Figs. 8a and 8c respectively. The occurrence of elongated particles near the center of the prior-c grain in the
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800-30-710-10-7
(a) 800-30-710-10-8
Particle loop
(c)
840-60-750-15-3
Ferrite ( )
Cementite ( ) Austenite ( )
(b)
Fig. 8. Comparison of the h morphologies observed in experimental microstructures with the exemplary simulation results. The illustrated experimental microstructures are obtained by isothermally holding the austenized sample (T1 C, t1 minutes) at T2 C for t2 minutes (Fe-0.92C-0.66Si-1.58Mn-1.58Cr-0.12Ni-0.05Mo-0.178Cu (wt.%) alloy). (a) Elongated h particles aligned radially inwards and point towards the grain center. (b) As the a=c transformation front evolves non-cooperatively, the divorced h particles get aligned in a loop (T1 ¼ 810 C, t1 = 30 min, T2 = 1 = 710 °C, t2 = 10 min). (c) The bottle h morphology, which is representative of mixed co-operative/non-cooperative regime, active during the course of transformation. (T1 ¼ 840 C, t1 = 60 min, T2 = 1=750 °C, t2=15 min). The experimental microstructures are provided by Z.X. Yin and H.K.D.H. Bhadeshia.
t = 2.9 x 106
t = 1.9 x 107
t = 3.3 x 107
Fig. 9. 3-D numerical simulation showing the temporal evolution of eutectoid phases and the corresponding chemical potential map (central 2-D slice of the numerical domain) for a symmetric arrangement of preexisting h particles (DT ¼ 10 K; k ¼ 0:14 lm). The predominance of curvature driven coarsening at small k indicates the importance of accounting for the third dimension in numerical simulations.
transformed microstructure accentuate the present simulation results, which suggest that the a=c transformation front evolves cooperatively, owing to an absence of preexisting h ahead of it, during the final stages of eutectoid transformation. The temporal disappearance of h ahead of the growth front is an outcome of curvature-driven coarsening which predominates, if k is sufficiently small. On the contrary, bottle-shaped h morphologies evolve, if the transformation proceeds in a mixed mode, depending on the distance from the nearest preexisting particle. We also identify an interesting spatial arrangement of h particles in the transformed microstructure, which we prefer to call a ‘‘particle loop’’, as illustrated in Fig. 8b. Depending on the spatial arrangement and inter-particle spacing (kmin ¼ 0:147 lm), the a=c advancing transformation front drags the h particles as it evolves towards the prior-c grain center. For the present simulations, where it has been explicitly assumed that the preexisting a layers are aligned along the prior-c grain boundaries, the simulated h loop qualitatively agree with the experimental microstructure, illustrated along side.
At this point, it is worth clarifying that the illustrated experimental microstructures obtained by the inter-critical annealing of multi-component steel (Fe-0.92C-0.66Si-1.58Mn-1.58Cr-0.12Ni-0. 05Mo-0.178Cu wt.%) serve as exemplary cases for a qualitative comparison with the numerically simulated h morphologies and spatial arrangements. For a quantitative comparison, the influence of various alloying element on the non-cooperative eutectoid transformation, needs to be considered in the future numerical studies. As the current work focuses on the analysis of a complex carbon redistribution mechanism that result in the formation of characteristic h morphologies and arrangements in the transformed microstructure, the present numerical study is limited to the case of an isothermal transformation (DT ¼ 7:5 K) in binary Fe–C steel.
5. Concluding remarks Here, we have explained the influence of preexisting arrangement of h particles on the evolution of a=c transformation front.
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It is worth noting that the influence of arrangement of h, which is shown to have profound effects on the final microstructure, has been all together neglected in the previous theoretical, experimental and numerical studies. Using phase-field simulations, we also explain the influence of preexisting a layer thickness and c grain size on the evolution mechanisms in polycrystalline microstructures. It is shown that the criteria for the lamellar-divorced-coarsening transition reported earlier in paper 1 holds even if the preexisting arrangement of h particles is not symmetric. However, the coarsening rate is greatly influenced by the kmin and relative arrangement, which in turn determine the final microstructure. For the first time, we have reported the evolution of bimodal distribution of h particles in Fe–C alloys, which is in good agreement with the previous experimental findings [16–18]. By analyzing the eutectoid transformation which proceeds isothermally from an initial setting comprising of a random distribution of h particles and grain boundary a layers embedded in polycrystalline c, we explain the mechanism by which mixed h morphologies evolve. Such microstructures are typically used in bearing steels [19]. It is also shown that the changing the preexisting arrangement of h can trigger a change from unimodal to bimodal distribution of particles. We observe that the numerically simulated eutectoid microstructures possess many similarities with the experimental ones, as suggested by their comparison in Fig. 8 and a following discussion on their evolution mechanisms. The present numerical results suggest that the curvature-driven coarsening is a dominant regime at lower kmin values. If kmin is chosen to be larger, the transformation proceeds by complex interplay of cooperative and non-cooperative evolution modes. Using phase-field simulations, we have shown that the morphology of preexisting ferrite layers influences the mechanism by which eutectoid transformation proceeds. However, the influence of preferential grain boundary or triple-junction nucleation on the h size distribution has not been explored. In this context, a broader consensus on eutectoid nucleation can improve the microstructure predictive capability of the technique described in the present article. A significant limitation of this study is that the numerical simulations are limited to 2D. An exemplary simulation of the non-cooperative evolution in 3D is shown in Fig. 9. The addition of the third spatial dimension, essentially changes the dynamics of curvature-driven coarsening, which in turn, directly affects the final eutectoid microstructure. Therefore, the numerical studies need to be extended by analyzing the morphological evolution of phases during the divorced eutectoid transformation in 3D.
Acknowledgements The authors thank Z.X. Yin and H.K.D.H. Bhadeshia for providing the experimental microstructures. KA and BN acknowledge the financial support of DFG in the framework of Graduate School-1483 (Project No. B8). References [1] K.E. Thelning, Steel and its Heat Treatment, 2 ed., Butterworth and Co., London, 1984. [2] H. Bhadeshia, Steels for bearings, Prog. Mater. Sci. 57 (2012) 268–435. [3] T. Oyama, O.D. Sherby, J. Wadsworth, B. Walser, Application of the divorced eutectoid transformation to the development of fine-grained, spheroidized structures in ultrahigh carbon steels, Scr. Metall. Mater. 18 (1984) 799–804. [4] C.K. Syn, D.R. Lesuer, O.D. Sherby, Influence of microstructure on tensile properties of spheroidized ultrahigh-carbon (1.8 Pct C) steel, Metall. Mater. Trans. A 25 (1994) 1481–1493. [5] E.M. Taleff, C.K. Syn, D.R. Lesuer, O.D. Sherby, Pearlite in ultrahigh carbon steels: heat treatments and mechanical properties, Metall. Mater. Trans. A 27A (1996) 111–118. [6] K. Honda, S. Saito, On the formation of spheroidal cementite, J. Iron Steel Inst. 102 (1920) 261–269. [7] J. Whiteley, The formation of globular pearlite, J. Iron Steel Inst. 105 (1922) 339–357. Cited By (since 1996)11. [8] G. Lur’e, Y. Shteinberg, Microstructure of the surface layer of gray cast iron after hardening-finishing, Met. Sci. Heat Treat. 11 (1969) 351–353. [9] K. Ankit, R. Mukherjee, T. Mittnacht, B. Nestler, Deviations from cooperative growth mode during eutectoid transformation: insights from a phase-field approach, Acta Mater. 81 (2014) 204–210. [10] A. Choudhury, B. Nestler, Grand-potential formulation for multicomponent phase transformations combined with thin-interface asymptotics of the double-obstacle potential, Phys. Rev. E 85 (2012) 21602. [11] J.D. Verhoeven, E.D. Gibson, The divorced eutectoid transformation in steel, Metall. Mater. Trans. A 29 (1998) 1181–1189. [12] A.S. Pandit, H. Bhadeshia, Divorced pearlite in steels, Proc. R. Soc. A-Math. Phys. 468 (2012) 2767–2778. [13] D. Molnar, R. Mukherjee, A. Choudhury, A. Mora, P. Binkele, M. Selzer, B. Nestler, S. Schmauder, Multiscale simulations on the coarsening of Cu-rich precipitates in a-Fe using kinetic Monte Carlo, molecular dynamics and phasefield simulations, Acta Mater. 60 (2012) 6961–6971. [14] K. Ankit, A. Choudhury, C. Qin, S. Schulz, M. McDaniel, B. Nestler, Theoretical and numerical study of lamellar eutectoid growth influenced by volume diffusion, Acta Mater. 61 (2013) 4245–4253. [15] R. Mukherjee, A. Choudhury, B. Nestler, Composition pathway in Fe–Cu–Ni alloy during coarsening, Model. Simul. Mater. Sc. 21 (2013) 075012. [16] E. Brown, G. Krauss, Retained carbide distribution in intercritically austenitized 52100 steel, Metall. Trans. A 17 (1986) 31–36. [17] J.D. Verhoeven, The role of the divorced eutectoid transformation in the spheroidization of 52100 steel, Metall. Mater. Trans. A 31 (2000) 2431–2438. [18] N.V. Luzginova, L. Zhao, J. Sietsma, The cementite spheroidization process in high-carbon steels with different chromium contents, Metall. Mater. Trans. A 39A (2008) 513–521. [19] H.S. Hasan, M.J. Peet, H. Bhadeshia, Severe tempering of bainite generated at low transformation temperaturess, Int. J. Mater. Res. 103A (2012) 1319–1324.
Contents lists available at ScienceDirect
Acta Materialia journal homepage: www.elsevier.com/locate/actamat
Deviations from cooperative growth mode during eutectoid transformation: Mechanisms of polycrystalline eutectoid evolution in Fe–C steels Kumar Ankit a,b,⇑,1, Rajdip Mukherjee c, Britta Nestler a,b a b c
Institute of Materials and Processes, Karlsruhe University of Applied Sciences, Moltkestr. 30, 76133 Karlsruhe, Germany Institute of Applied Materials-Computational Materials Science, Karlsruhe Institute of Technology (Campus South), Haid-und-Neu-Str. 7, 76131 Karlsruhe, Germany Department of Materials Science and Engineering, Indian Institute of Technology, Kanpur 208016, UP, India
a r t i c l e
i n f o
Article history: Received 26 March 2015 Revised 10 June 2015 Accepted 21 June 2015 Available online 13 July 2015 Keywords: Non-cooperative transformation Divorced eutectoid Phase-field method Coarsening Polycrystalline austenite
a b s t r a c t Undercooling (below A1 temperature) and spacing between the preexisting cementite particles are known to be the factors that determine whether the isothermal eutectoid transformation in Fe–C proceeds in cooperative (resulting in lamellar pearlite) or non-cooperative mode (yielding divorced eutectoid). Typically, a divorced eutectoid microstructure consists of a fine dispersion of cementite in the ferritic matrix. Although, numerous experimental studies report a bimodal size distribution of cementite in the transformed eutectoid microstructure, the factors that facilitate the shift from a characteristic unimodal to bimodal size distribution have not been reported extensively. In the present work, we adopt a multiphase-field approach to study the morphological transition during isothermal eutectoid transformation which proceeds from an initial configuration comprising of a random distribution of cementite particles and grain boundary ferrite layers embedded in polycrystalline austenite. By conducting a systematic parametric study, we deduce the influence of preexisting arrangement of cementite, grain boundary ferrite thickness and prior austenite grain size on the mechanism by which eutectoid phases evolve. We also establish a synergy between the numerically simulated cementite morphologies and spatial configurations with those observed in experimental microstructures. Finally, we discuss the influence of the different factors that lead to the formation of mixed cementite morphologies (spheroidal and non-spheroidal) in the transformed microstructure and highlight the importance of 3D simulations. Ó 2015 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
1. Introduction Hypereutectoid steels are frequently annealed to develop a softer microstructure (as compared to pearlite), which consists of fine spheroidal cementite (h) particles embedded in a soft ferritic (a) matrix ([1,2] and references therein). Typically, two types of softening techniques are employed. (a) In the first method, the steel is annealed just below A1 temperature, during which, preexisting pearlitic lamellae break up into spheroids. Such a morphological transition is driven by a reduction in the total amount of h=a interfacial energy. (b) The second technique involves reheating ⇑ Corresponding author at: Institute of Applied Materials-Computational Materials Science, Karlsruhe Institute of Technology (Campus South), Haid-und-Neu-Str. 7, 76131 Karlsruhe, Germany. E-mail address: [email protected] (K. Ankit). 1 Present address: Institute of Applied Materials-Applied Materials Physics, Karlsruhe Institute of Technology (Campus North), Hermann-von-Helmholtz-Platz 1, 76344 Eggenstein-Leopoldshafen, Germany. http://dx.doi.org/10.1016/j.actamat.2015.06.050 1359-6454/Ó 2015 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
the steel so that it becomes almost fully austenitic (c) but with small amount of h remaining undissolved as spheroidal particles. On cooling through the eutectoid temperature, the preexisting h particles grow by absorbing the excess carbon that is partitioned into c, as the c=a transformation front evolves, thereby leading to a final structure of coarse h particles dispersed in a matrix of a. The main reason is attributed to presence of preexisting h particles in the parent c matrix facilitate pulling- away of the advancing a=c interface from h [3–5]. The final transformation product is known as the divorced eutectoid, since the evolving phases no longer grow cooperatively [6–8]. The focus of the present research is to analyze the mechanisms by which divorced eutectoid microstructure evolves in Fe–C steels. In the first part of this study [9], hereafter referred as Paper 1, we use a phase-field model based on the grand-potential formulation [10] to numerically simulate the non-cooperative evolution of the advancing a=c front during the isothermal eutectoid transformation in Fe–C alloy. By analyzing the simulated carbon
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redistribution mechanisms as the transformation proceeds from different ‘k’ and DT, we establish the criteria which govern the transition from cooperative to non-cooperative mode. In accordance with the earlier theories as well as the experimental studies [2,11,12], the numerical results presented in Paper 1 show that the mechanism by which an advancing a=c front evolves to form a lamellar or divorced morphology is strongly dependent on the initial interparticle spacing (k) of the preexisting cementite (h) and undercooling (DT) below the A1 temperature. We also identify the onset of a coarsening regime and amended the criteria for the non-cooperative eutectoid transformation. In this follow-up article, we study how the preexisting spatial arrangement of h particles influences the evolution of a=c transformation front (non-cooperative or otherwise). To begin with, we analyze the coarsening regime in detail, starting from an asymmetric configuration of preexisting h particles. Next, we investigate the influence of prior c grain boundaries and morphology of preexisting a layers on the temporal evolution of divorced eutectoid microstructure. Finally, by qualitatively comparing the simulated h morphologies with those observed in experimental microstructure, we deduce the mechanisms by which mixed morphologies (spheroidal as well as nonspheroidal) evolve. The remainder of the article is organized as follows: In Section 2, we briefly describe the multiphase-field model which is used for the present numerical simulations. In Section 3, we discuss the influence of introducing asymmetry in preexisting arrangement of h particles on the coarsening rate. We also examine the role of preexisting random arrangement of particles in facilitating a transition from cooperative to non-cooperative evolution of the advancing a=c transformation front. In Section 4, we analyze the non-cooperative eutectoid transformation mode in polycrystalline c and compare the simulated h morphologies to the experimental microstructures. Section 5 concludes the article. 2. Phase-field model In this section, we recount the multiphase-field equations used to study the evolution of eutectoid microstructure. The reader is referred to previous studies [10,13–15,9] for a detailed description. The evolution of phases is governed by the phenomenological minimization of the grand potential functional X,
XðT; l; /Þ ¼
Z
1
WðT; l; /Þ þ að/; r/Þ þ wð/Þ
V
dV;
ð1Þ
where T is the temperature, l is the chemical potential vector comprising of K 1 independent chemical potentials, / is the phase-field vector containing the volume fractions of the N-phases and is the length scale related to the interface. að/; r/Þ and wð/Þ represent the gradient and obstacle potential type energy density, respectively and V represents the domain volume. The grand potential density WðT; l; /Þ, which is the Legendre transform of the free energy density of the system f ðT; c; /Þ is written as an interpolation of individual grand potential densities
WðT; l; /Þ ¼
N X
Wa ðT; lÞha ð/Þ
a¼1
Wa ðT; lÞ ¼ f a ðca ðT; lÞ; T Þ
K 1 X
li cai ðT; lÞ;
ð2Þ
i¼1
where
ha ð/Þ
is
an
interpolation
function
of
the
form
ha ð/Þ ¼ /2a ð3 2/a Þ. The evolution equation for the N phase-field variables can be written as,
s
@/a ¼ @t
@að/; r/Þ @að/; r/Þ 1 @wð/Þ @ WðT; l;/Þ r K; @ r/a @/a @/a @/a ð3Þ
where K is the Lagrange parameter to maintain the constraint PN a¼1 /a ¼ 1. The concentration fields are obtained by a mass conservation equation for each of the K 1 independent concentration variables ci . The evolution equation for the concentration fields can be derived as, K 1 X @ci ¼r M ij ð/Þrlj @t j¼1
Mij ð/Þ ¼
!
ð4Þ
N X M aij g a ð/Þ;
ð5Þ
a¼1
where each Maij represents the mobility matrix of the phase a (related to the diffusivity). The diffusion coefficient (Di ) that accounts for the transport of solute inside the phase volumes (Dai ) and along the interphase interfaces (Diab ) is given by:
Di ¼
N;N N X 1X Dai /a þ Diab /a /b :
a¼1
ð6Þ
a
As the relaxed (stable) interface /ðxÞ contour has a sinusoidal profile, the effective diffusion coefficient through the interface can be determined using the values of Dai (diffusivity of ith component in phase volume) and Diab (diffusivity of ith component in interphase interfaces)
Z =2 =2
/a /b dx ¼
with /a ¼
Z =2 =2
/a ð1 /a Þdx ¼
8
p o 1n x : 1 þ sin 2
ð7Þ
On integrating Eq. (6) over the interfacial width (from =2 to =2), average diffusivity in the interphase interfaces is expressed as
Dab 1 1 Dinterface ¼ Dai þ Dai b : ¼ Dai þ i : i 8 8
ð8Þ
The thermodynamic data-fitting procedure to approximate the variation of the grand-potential of the respective phases as a function of chemical potential and the relation of the numerical simulation parameters with the corresponding quantities in the sharp-interface limit, are explained in the previous work [14]. Since, we are primarily interested in generic features of non-cooperative eutectoid transformation, the reported times are 2
normalized by l0 =Dc , where l0 ¼ r=ðRT=v m Þ is the capillary length and Dc is the diffusivity in austenite (c). T; R and v m denote transformation temperature, universal gas constant and the molar volume respectively. The numerical parameters used in the reported simulations are listed in Table 1. 3. Influence of arrangement 3.1. Divorced-coarsening transition in asymmetric arrangement According to the classical theory of non-cooperative eutectoid transformation [11], interparticle spacing (k) and undercooling (below A1 temperature) determine whether transformation proceeds in a cooperative manner or the advancing a=c front divorce the h particles. In the present section, we build upon the previous work of [9] and extend the numerical studies to investigate the influence of preexisting arrangement of h particles on the non-cooperative evolution of the a=c transformation front.
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Table 1 Numerical parameters used for the phase-field simulation of eutectoid transformation starting from polycrystalline c (‘r’ denotes grain radii) containing random distribution of h particles (‘kmin ’ denotes minimum inter-particle distance) and grain boundary a layers (of uniform thickness ‘L’). T E denotes the eutectoid transformation temperature of a binary Fe–C alloy, DT, the undercooling (below the eutectoid temperature) and r, surface energies (pairwise). Da ; Dh and Dc represents volume diffusivity in ferrite (a), cementite (h) and austenite (c), whereas Dac ; Dah and Dhc represent diffusivities along the corresponding transformation fronts. Symbol
Value
Units
TE DT
999 7.5 0:49
K K J/m2 m2 /s
rac ¼ rhc ¼ rah Da ¼ Dh Dc ac
D ¼D L kmin r
2 109 ah
hc
¼D
1 109 1000Da 0.1, 0.084, 0.0504, 0.0336 0.075, 0.084, 0.105, 0.147 1.2
m2 /s m2 /s lm lm lm
with the h phase. When the chosen k is small (kco ¼ 0:26 lm), the particles nearer to the pre-existing a=c transformation front (at odd positions), ripen while the ones farther (even position), shrink. As the eutectoid transformation proceeds, the h particles at even positions shrink and ultimately, dissolve. On analyzing the simulated chemical potential profile, we observe the carbon diffusion flux along the a=c transformation front, directed towards the h particles that are located at odd position positions. However, if the k is increased (kd ¼ 0:57 lm), coarsening of h is less pronounced and the transformation proceeds predominantly, in the noncooperative mode. For even larger k, curvature-driven coarsening of h is not observed. It is worth noting that the criterion of divorced-coarsening transition regime reported earlier in paper 1 holds for the case of asymmetric arrangements as well. However, the rate of coarsening, which influences the final microstructure, is dependent on the relative arrangement of preexisting h particles. 3.2. Mixed transformation products
In Fig. 1a, we compare the coarsening kinetics, as the transformation proceeds from a preexisting symmetric and asymmetric arrangements of two h particles. The evolution of the chemical potential and phase contours at representative times are shown for the two cases in Fig. 1b and c, respectively. It is observed that the boxed particle (shown by the schematic diagram in Fig. 1a) when placed at the symmetric position, shrinks at a faster rate, as compared to the latter. One may argue that the coarsening kinetics is expected to be much faster for the case of asymmetric arrangement, rather than the symmetric due to assistance from a=c interface diffusive flux (which is stronger than the volume diffusion flux) during the intermediate stages of the eutectoid transformation, as seen in Fig. 1c (at t ¼ 7:55 104 ). On the contrary, as the transformation proceeds from a symmetric arrangement, due to geometric considerations and the chosen value of k, the advancing a=c front does not get a chance to form an interface with the shrinking h particle, as clearly seen in the temporally evolving phase contours in Fig. 1b. Therefore, any assistance from a=c interface diffusion can be completely ruled out. Apparently, the question arises: In spite of availability of a faster diffusion path along the a=c interface, why is the coarsening rate slower when the isothermal eutectoid transformation proceeds from an asymmetric arrangement of preexisting h particles? The answer lies in analyzing the difference between the mean curvatures of the two h particles for the respective cases. Once the advancing a=c front forms an interface with the asymmetrically placed h particle, it continues to shrink in a lenticular morphology. This essentially means that the curvature difference between the h=c or h=a interfaces of the two particles depreciate, thereby, slowing down the curvature-driven coarsening rate. On the contrary, when the eutectoid transformation proceeds from a symmetric arrangement, the temporally shrinking particle retains the spheroidal morphology. On analyzing the temporal evolution of phases for both the cases (symmetric and asymmetric arrangements), it is apparent that the difference in curvatures of h=c or h=a interfaces of the two particles are greater when the shrinking particle retains the spheroidal morphology i.e. when symmetrically located with respect to the h particle closer to a=c front. Fig. 2 shows the influence of interparticle spacing (k) as the eutectoid transformation proceeds isothermally from asymmetric arrangement of preexisting h particles (undercooling, DT ¼ 10 K below A1 temperature). The results of the numerical simulations are illustrated for two different interparticle spacings (kco ¼ 0:26 lm and kd ¼ 0:57 lm). Starting from the initial configuration of phases. which comprise of asymmetrically arranged particles, the a=c transformation front advances to form an interface
It is typical to observe the non-spherical h particles co-existing with the spherical/well-rounded ones in a transformed eutectoid microstructure. As the presence of non-spheroidized h has an adverse effect on the machinability of steel, it is essential to establish the conditions of growth and process parameters which prevent the formation of non-spheroidized counterparts. In Fig. 3, we present the phase-field simulation results to demonstrate the influence of preexisting h particle in bringing about a shift from co-operative to non-cooperative evolution mode. For the case of single-layered arrangement, the preexisting particles that are nearer to the a=c transformation front, evolve non-cooperatively yielding a divorced eutectoid morphology. As the transformation front temporally evolves to form an interface with the farther set of particles, it pulls away and divorces the h particles in Level 1, as illustrated in the in Fig. 3a. Owing to an absence of any h particles ahead of the growth front, the transformation proceeds further in a cooperative mode. The apparent shift in the transformation mode (i.e. cooperative to non-cooperative) results in the evolution of lamellar/non-spheroidized eutectoid morphology (at t ¼ 5:01 103 s). On the contrary, when the transformation starts with a bi-layered arrangement of preexisting h particles (shown in Fig. 3b), a shift in the carbon redistribution mechanism is not observed. The temporally advancing a=c transformation front continues to pull-away from the h particles in Level 2 (in the first layer), which ultimately result in the formation of divorced eutectoid. Apparently, the present simulations suggest that the presence of preexisting h particles ahead of the growth front (in second layer) favor the non-cooperative evolution of the growth front. 4. Non-cooperative eutectoid transformation in polycrystalline austenite 4.1. Influence of preexisting a thickness (L) and configuration of h particles Polycrystalline c is modeled as an array of four hexagonal grains in a two-dimensional simulation domain with periodic boundary conditions. As the preexisting c grains are chosen to be equiaxed and of the same size, the ratio of the width (Nx grid-points) and height (Ny grid-points) of the numerical domain is restricted to pffiffiffi 3 : 2. This implies that a circle of radius ‘r’ which circumscribes the central grain relate to the numerical domain width as, pffiffiffi Nx ¼ 3r (or Ny ¼ 2 3r). For the present simulations, a computational box size of 500 433 grid-points is chosen, where the grid-spacings in the x and y directions (Dx and Dy respectively)
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6.0x10
-9
Symmetrical arrangement ( = 100 m) Asymmetrical arrangement ( = 100 m)
Volume fraction
5.0x10-9 4.0x10-9
3.0x10
-9
2.0x10
-9
1.0x10
-9
0
(a)
0.0x10 0.0x100
4.5x104
9.0x104
1.4x105
1.8x105
Time (non-dimensional) t = 1.71x104
t = 3.77x104
t = 4.39x104
5187.28
4378.63
4682.53
4382.45
t = 1.35x105
t = 7.55x104
4835.32
4353.11
4463.80
4353.81
4604.03
(c)
4345.38
Asymmetric
t = 1.43x103
4557.53
4386.03
4766.37
(b)
4183.92
Symmetric
t = 8.44x101
Fig. 1. (a) Temporal change in the volume fraction of the boxed h particle, as the a=c transformation front advances starting from a two different arrangements (symmetric and asymmetric). Corresponding evolution of the phase contours plotted over the chemical potential maps, as the transformation proceeds from (b) symmetric and (c) asymmetric arrangements. Green contours correspond to a=c interface while the white lines represent h=a and h=c interfaces.
are equal to 0.0042 lm. Therefore, the present choice of numerical domain size implies a c grain size (2r) that is equal to 1.2 lm. The four c grains in consideration are assigned distinct phase-fields and uniform equilibrium composition that is derived from the phase diagram. A spatial distribution of ‘N 0 ’ spherical h particles (initial radii equal to 8 grid-points) is algorithmically generated by using a numerical pre-processing technique. The minimum distance between two particles is constrained to be greater than kmin n grid-points, where kmin is the minimum inter-particle spacing and n is a random noise between 0 and 10 grid-points. The pre-processing algorithm iteratively fills the spherical particles inside the domain as per the imposed minimum inter-particle spacing (kmin ) criteria. Following every successful incorporation of a new particle in the numerical domain, the algorithm performs a number of iterations (lesser than the predefined maximum designated by imax ) to search for a suitable coordinate of the next particle that is to be added. For a limiting case, when the introduction of new particles is no longer possible for the chosen imax , the iteration exits. The maximum possible number of iterations (imax )
following every successful particle filling is chosen to be high enough to ensure that the algorithmically generated particle distribution is homogeneous. Finally, a layer of a (of uniform thickness ‘L’) is placed along the c grain boundaries and a parametric study is conducted by varying the simulation parameters as shown in Table 1. Figs. 4a–e and f–j show the temporal evolution of eutectoid phases staring from a polycrystalline configuration that is described above, for two different a layer thicknesses, L ¼ 0:0336 and 0:1 lm, respectively. Assuming other simulation parameters to be constant, numerical simulations are also carried out for intermittent thicknesses (0.0504 and 0:084 lm), to study the influence of initial grain boundary a layer thickness on the evolving as well as final eutectoid morphologies, in detail. It is observed that for a thickness of 0:0336 lm, the a layer that are in contact with cementite particle, pinch-off during the early stage of transformation and forms an island morphology to minimize the surface energy (Fig. 4b). Once the pinching-off stops, the a=c begins to temporally advance (Fig. 4c), in a predominantly non-cooperative mode. In the
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Fig. 2. Phase-field simulations showing the influence of k as the eutectoid transformation proceeds isothermally from a pre-existing asymmetric arrangement of h particles (DT ¼ 10 K). The phase contours are plotted on the corresponding chemical potential maps. At smaller k (kco ¼ 0:26 lm), onset of curvature-driven coarsening result in dissolution of h particle, marked at even positions. However, at larger k (kd ¼ 0:57 lm), the eutectoid transformation predominantly proceeds in non-cooperative mode.
(a) Level 2 -Fe
-Fe3C Level 1
}
1st layer
t = 6.67 x 107
t=0 4432
4432
4317.11
4552.38
t = 5.5 x 108 4608.08
t = 1.4 x 109 4336.06
4544.50
4340.82
(b)
} }
2nd layer 1st layer
t = 6.67 x 107
t=0 4432
4432
4327.12
4552.97
t = 2.28 x 108 4319.89
4549.90
t = 6.2 x 108 4326.48
4524.34
Fig. 3. Temporal evolution of phase isolines (/ ¼ 0:5) and the corresponding chemical potential map for an asymmetric (a) single-layered and (b) bi-layered arrangement of h particles (initial spacing, k ¼ 0:84 lm), during an isothermal eutectoid transformation (undercooling, DT ¼ 7:5 K). The isolines plotted over the chemical potential map illustrate the corresponding sharp interfaces of the phases. Black lines correspond to a=c interface while the white lines represent h=a and h=c interfaces.
time elapsed (during the a layer pinch-off), the count of preexisting h particles plummets down as a result of curvature driven coarsening (the phenomena is discussed in the detail in a companion article [9]) which subsequently, lead to a cooperative growth of the a=c front during the final stages of eutectoid transformation (Fig. 4e). It is noteworthy that the a pinching-off is not observed, if the initial a layer thickness is assumed to larger (Fig. 4f–g). By conducting a parameter study, it is observed that the coarsening of the preexisting h particles is comparatively less, if L > 0:0336 lm. Apparently, the numerical simulations suggest that the pinching-off of grain boundary a-layer is strongly dependent on the surface energies and ceases once the equilibrium phase triple-junction angles develop. The temporal evolution of h volume fraction is plotted in Fig. 5a starting from different initial thicknesses (L) of grain boundary a layers. A significant deviation in the carbon redistribution mechanism is observed for the case of L ¼ 0:0336 lm, during the early stages of eutectoid transformation. Such a deviation can be attributed to curvature-driven coarsening regime, that is active during
the a layer pinch-off (in early stages of eutectoid transformation) and causes a net carbon diffusion flux from the h particles located near the grain center towards the a=c front as they start advancing from the prior c grain boundaries. Unarguably, for the case pertaining to L > 0:0336 lm, the influence of particle coarsening in bringing about a change in the transformation mechanism, cannot be denied. Apparently, a temporal decline in the number of preexisting h particles (located towards the grain center) is also observed in this case. It can be argued that the pinching-off of preexisting a layers that is observed in the present phase-field simulations is unrealistic from an experimental perspective. The result of the parametric study reported in Fig. 5a allows us to identify the minimum threshold (numerical value of L) below which, pinching-off occurs. To rule out such a possibility, we assume the a layer thickness to be sufficiently high (L ¼ 5:04 102 lm) for all the numerical simulations, that follow. In order to study the influence of arrangement and spacing of h particles on the resulting microstructure, we numerically simulate the eutectoid transformation staring from different kmin (minimum
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(a) t = 0
(b) t = 2.8 x 107
(c) t = 7.6 x 107
(d) t = 2.3 x 108
(e) t = 3.2 x 108
(f) t = 0
(g) t = 2.8 x 107
(h) t = 7.6 x 107
(i) t = 1.1 x 108
(j) t = 2.6 x 108
Ferrite ( )
Cementite ( )
Austenite ( )
Fig. 4. Phase-field simulation showing the temporal evolution of the mixed eutectoid morphologies during an isothermal transformation (DT ¼ 7:5 K below A1 temperature), starting from an initial configuration which comprise of grain boundary a layers of initial thickness L and h particles that are randomly dispersed (kmin ¼ 0:126 lm) in polycrystalline c (grain diameter = 1:2 lm). The effect of initial grain boundary a thickness on the temporal evolution of phases is observed in the present simulations. At a small thickness (L ¼ 3:36 102 lm), the grain boundary a pinches-off from h particles to minimize surface energy, at the early stages, as observed in (a)–(e). The temporal evolution shown in (f)–(j) suggests that a pinch-off do not take place at early stages, if the initial thickness is higher (L ¼ 0:1 lm).
(a)
(b)
Fig. 5. (a) The plot showing the effect of initial thickness of grain boundary a layer (L) on the temporal evolution of h volume fraction. Pinching-off of a layers is observed at early stages (at L ¼ 3:36 102 lm) which results in a deviation from the trend observed for higher L. (b) Starting with four different interparticle spacings, the eutectoid transformation is numerically simulated (kmin ¼ 0.075, 0.084, 0.105 and 0.147 lm) and the final particle distributions (obtained on completion of the phase transformation) are compared. For the sake of comparison, other simulation parameters such as grain size (1:2 lm), initial thickness of grain boundary a layers (L ¼ 5:04 102 lm) and undercooling (DT ¼ 7:5 K below A1 temperature) are kept constant for the four cases. A deviation from the bottle-shaped distribution profile is observed for kmin ¼ 0:105 lmðN0 ¼ 119Þ due to fragmentation of pearlitic lamellae.
inter-particle spacing). The criteria for kmin that is imposed by the numerical pre-processing algorithm already described above, yields various distribution of h particles (corresponding number of particles is designated as N0 ). Fig. 5b shows the h size distribution obtained at the end of eutectoid transformation, by initializing numerical simulations with different number of particles (or kmin ). A comparison of the final h size distribution suggests that the peak value decrease on increasing N 0 (or on decreasing kmin ). Due to an enhanced coarsening of h particles at smaller kmin (or larger N0 ), the number of pre-existing particles located towards the grain center decline temporally and subsequently, dissolve (in the later stages). As the transformation proceeds towards completion, due to a complete or near absence of preexisting h particles, the a=c transformation front starts to evolve co-operatively leading to the formation of elongated/bottle-shaped particles which are found to be
pointing towards the grain center. The supple reversal of the transformation mode (non-cooperative to co-operative) that results in the formation of spheroidal as well as elongated/bottle-shaped particles, explain the presence of more than one local maxima in the size distribution plots shown in Fig. 5b. It is noteworthy that the reported reversal in the transformation mode is not favored if the initial inter-particle spacing (kmin ) is large. The distribution plot pertaining to N0 ¼ 90 (kmin ¼ 0:105 lm) represents an intermediate case in which, non-cooperative as well as cooperative modes dominate during the early and later stages of the transformation respectively, as suggested by an absence of a unique distribution peak. These results essentially suggest that at smaller kmin , curvature-driven coarsening predominates. Although the influence of coarsening is less pronounced at larger values of kmin , the transformation for these cases proceed by a complex interplay of
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(a)
(b)
Fig. 6. Temporal evolution of the phases and the corresponding particle size distributions are plotted, by numerically simulating the isothermal eutectoid transformation (DT ¼ 7:5 K below the A1 temperature), starting from an initial configuration comprising of grain boundary a layers (initial thickness, L ¼ 5:04 102 lm) and h particles at minimum distances of (a) kmin ¼ 0:075 lm and (b) kmin ¼ 0:084 lm in polycrystalline c (grain diameter = 1:2 lm).
cooperative and non-cooperative modes, depending upon the local distribution of h particles in the proximity of advancing a=c front. For a more detailed understanding, we analyze the temporal evolution of eutectoid phases starting from (a) kmin ¼ 0:075 lm and (b) kmin ¼ 0:084 lm, by plotting the corresponding particle size distribution at the representative time-steps. In the case of smaller kmin (Fig. 6a), a temporal shift of size distribution peak towards the right-hand side, suggests the predominance of a characteristic curvature-driven coarsening regime during the intermittent as well as final stages (474:6 1559:4 ls). At the later stages of the eutectoid transformation, coalescence of particles results in formation of non-spheroidal morphologies. Interestingly, for the case of larger kmin (Fig. 6b), a bimodal size distribution of evolving h particles is obtained. To begin with (67:5 ls), the size distribution has a unique peak. However, as the a=c transformation front temporally advances in mixed-mode, spheroidal as well as non-spheroidal particles evolve. On comparing the size distribution and the number of particles (inversely dependent on kmin ) in the simulated microstructure (Fig. 6a and b), we conclude that the curvature-driven coarsening is not a dominant mode by which the eutectoid transformation proceeds, if the initial inter-particle spacing is larger.
4.2. Influence of prior c grain size (r) To understand the influence of prior c grain size on the final microstructure, we compare the size distributions of h simulated at different ‘r’ (1.2 and 1:8 lm) while assuming the k to be same (equal to 0:147 lm) for both the numerical test cases. A characteristic unimodal size distribution of h is obtained at larger grain size (1:8 lm), as shown in Fig. 7. On the contrary, a bimodal size distribution is favored, if the initial grain size is assumed to be smaller by 33%. It is imperative to note that the influence of introducing prior c grain boundaries on the size distribution of h is more pronounced at small grain size and diminishes, when increased.
0.3
Normalized number of particles (N/N0)
At this point, we would like to state that the simulated bimodal size distribution of h particles (for kmin ¼ 0:084 lm) is an inherent characteristic of a mixed-mode (cooperative/non-cooperative) eutectoid microstructures. As the transformation proceeds, the carbon redistribution is dependent on the preexisting arrangement as well as the distance of the nearest h particles ahead of the advancing a=c interface. Apparently, for a given random distribution of preexisting particles, the carbon redistribution mechanism is more complex when compared to the case of symmetric arrangement, for which theories are relatively well known [11,2,12]. The numerical results suggest an overlap of cooperative and non-cooperative transformation modes, which are primarily governed by interparticle spacing (from nearest neighbor) and spatial arrangement of h ahead of advancing the a=c front. As we demonstrate that the cooperative and non-cooperative evolution of the a=c front as well as curvature-driven coarsening of h particle are favored depending upon the initial interparticle spacing and arrangement, the cementite morphologies and distribution found in the transformed microstructure yields insights concerning the intermittent regimes that predominate during the isothermal transformation.
Grain Size = 1.2 µm (N0 = 60) Grain Size = 1.8 µm (N0 = 150)
0.225
0.15
4.3. Comparison with experimental microstructures 0.075
0 0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.11
0.12
Radius of particles ( m) Fig. 7. Comparison of h size distribution simulated at two different prior c grain sizes (r ¼ 1:2 and 1:8 lm). N0 denotes the number of h particles at the start of eutectoid transformation. k for both the test cases is assumed to be same (equal to 0:147 lm). Prior a=c boundaries are denoted by black lines in the schematic diagram.
With a view to establish a synergy between the numerically simulated and experimental microstructures, we compare the various morphologies of h observed in the experimental microstructure with the phase-field simulations. For the given set of growth conditions, we observe various non-spheroidized h morphologies, which lend insights into the evolution mechanism of a=c front during various stages of eutectoid transformation. Two such exemplary h morphologies namely elongated and bottle-shaped particles are shown in Figs. 8a and 8c respectively. The occurrence of elongated particles near the center of the prior-c grain in the
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800-30-710-10-7
(a) 800-30-710-10-8
Particle loop
(c)
840-60-750-15-3
Ferrite ( )
Cementite ( ) Austenite ( )
(b)
Fig. 8. Comparison of the h morphologies observed in experimental microstructures with the exemplary simulation results. The illustrated experimental microstructures are obtained by isothermally holding the austenized sample (T1 C, t1 minutes) at T2 C for t2 minutes (Fe-0.92C-0.66Si-1.58Mn-1.58Cr-0.12Ni-0.05Mo-0.178Cu (wt.%) alloy). (a) Elongated h particles aligned radially inwards and point towards the grain center. (b) As the a=c transformation front evolves non-cooperatively, the divorced h particles get aligned in a loop (T1 ¼ 810 C, t1 = 30 min, T2 = 1 = 710 °C, t2 = 10 min). (c) The bottle h morphology, which is representative of mixed co-operative/non-cooperative regime, active during the course of transformation. (T1 ¼ 840 C, t1 = 60 min, T2 = 1=750 °C, t2=15 min). The experimental microstructures are provided by Z.X. Yin and H.K.D.H. Bhadeshia.
t = 2.9 x 106
t = 1.9 x 107
t = 3.3 x 107
Fig. 9. 3-D numerical simulation showing the temporal evolution of eutectoid phases and the corresponding chemical potential map (central 2-D slice of the numerical domain) for a symmetric arrangement of preexisting h particles (DT ¼ 10 K; k ¼ 0:14 lm). The predominance of curvature driven coarsening at small k indicates the importance of accounting for the third dimension in numerical simulations.
transformed microstructure accentuate the present simulation results, which suggest that the a=c transformation front evolves cooperatively, owing to an absence of preexisting h ahead of it, during the final stages of eutectoid transformation. The temporal disappearance of h ahead of the growth front is an outcome of curvature-driven coarsening which predominates, if k is sufficiently small. On the contrary, bottle-shaped h morphologies evolve, if the transformation proceeds in a mixed mode, depending on the distance from the nearest preexisting particle. We also identify an interesting spatial arrangement of h particles in the transformed microstructure, which we prefer to call a ‘‘particle loop’’, as illustrated in Fig. 8b. Depending on the spatial arrangement and inter-particle spacing (kmin ¼ 0:147 lm), the a=c advancing transformation front drags the h particles as it evolves towards the prior-c grain center. For the present simulations, where it has been explicitly assumed that the preexisting a layers are aligned along the prior-c grain boundaries, the simulated h loop qualitatively agree with the experimental microstructure, illustrated along side.
At this point, it is worth clarifying that the illustrated experimental microstructures obtained by the inter-critical annealing of multi-component steel (Fe-0.92C-0.66Si-1.58Mn-1.58Cr-0.12Ni-0. 05Mo-0.178Cu wt.%) serve as exemplary cases for a qualitative comparison with the numerically simulated h morphologies and spatial arrangements. For a quantitative comparison, the influence of various alloying element on the non-cooperative eutectoid transformation, needs to be considered in the future numerical studies. As the current work focuses on the analysis of a complex carbon redistribution mechanism that result in the formation of characteristic h morphologies and arrangements in the transformed microstructure, the present numerical study is limited to the case of an isothermal transformation (DT ¼ 7:5 K) in binary Fe–C steel.
5. Concluding remarks Here, we have explained the influence of preexisting arrangement of h particles on the evolution of a=c transformation front.
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It is worth noting that the influence of arrangement of h, which is shown to have profound effects on the final microstructure, has been all together neglected in the previous theoretical, experimental and numerical studies. Using phase-field simulations, we also explain the influence of preexisting a layer thickness and c grain size on the evolution mechanisms in polycrystalline microstructures. It is shown that the criteria for the lamellar-divorced-coarsening transition reported earlier in paper 1 holds even if the preexisting arrangement of h particles is not symmetric. However, the coarsening rate is greatly influenced by the kmin and relative arrangement, which in turn determine the final microstructure. For the first time, we have reported the evolution of bimodal distribution of h particles in Fe–C alloys, which is in good agreement with the previous experimental findings [16–18]. By analyzing the eutectoid transformation which proceeds isothermally from an initial setting comprising of a random distribution of h particles and grain boundary a layers embedded in polycrystalline c, we explain the mechanism by which mixed h morphologies evolve. Such microstructures are typically used in bearing steels [19]. It is also shown that the changing the preexisting arrangement of h can trigger a change from unimodal to bimodal distribution of particles. We observe that the numerically simulated eutectoid microstructures possess many similarities with the experimental ones, as suggested by their comparison in Fig. 8 and a following discussion on their evolution mechanisms. The present numerical results suggest that the curvature-driven coarsening is a dominant regime at lower kmin values. If kmin is chosen to be larger, the transformation proceeds by complex interplay of cooperative and non-cooperative evolution modes. Using phase-field simulations, we have shown that the morphology of preexisting ferrite layers influences the mechanism by which eutectoid transformation proceeds. However, the influence of preferential grain boundary or triple-junction nucleation on the h size distribution has not been explored. In this context, a broader consensus on eutectoid nucleation can improve the microstructure predictive capability of the technique described in the present article. A significant limitation of this study is that the numerical simulations are limited to 2D. An exemplary simulation of the non-cooperative evolution in 3D is shown in Fig. 9. The addition of the third spatial dimension, essentially changes the dynamics of curvature-driven coarsening, which in turn, directly affects the final eutectoid microstructure. Therefore, the numerical studies need to be extended by analyzing the morphological evolution of phases during the divorced eutectoid transformation in 3D.
Acknowledgements The authors thank Z.X. Yin and H.K.D.H. Bhadeshia for providing the experimental microstructures. KA and BN acknowledge the financial support of DFG in the framework of Graduate School-1483 (Project No. B8). References [1] K.E. Thelning, Steel and its Heat Treatment, 2 ed., Butterworth and Co., London, 1984. [2] H. Bhadeshia, Steels for bearings, Prog. Mater. Sci. 57 (2012) 268–435. [3] T. Oyama, O.D. Sherby, J. Wadsworth, B. Walser, Application of the divorced eutectoid transformation to the development of fine-grained, spheroidized structures in ultrahigh carbon steels, Scr. Metall. Mater. 18 (1984) 799–804. [4] C.K. Syn, D.R. Lesuer, O.D. Sherby, Influence of microstructure on tensile properties of spheroidized ultrahigh-carbon (1.8 Pct C) steel, Metall. Mater. Trans. A 25 (1994) 1481–1493. [5] E.M. Taleff, C.K. Syn, D.R. Lesuer, O.D. Sherby, Pearlite in ultrahigh carbon steels: heat treatments and mechanical properties, Metall. Mater. Trans. A 27A (1996) 111–118. [6] K. Honda, S. Saito, On the formation of spheroidal cementite, J. Iron Steel Inst. 102 (1920) 261–269. [7] J. Whiteley, The formation of globular pearlite, J. Iron Steel Inst. 105 (1922) 339–357. Cited By (since 1996)11. [8] G. Lur’e, Y. Shteinberg, Microstructure of the surface layer of gray cast iron after hardening-finishing, Met. Sci. Heat Treat. 11 (1969) 351–353. [9] K. Ankit, R. Mukherjee, T. Mittnacht, B. Nestler, Deviations from cooperative growth mode during eutectoid transformation: insights from a phase-field approach, Acta Mater. 81 (2014) 204–210. [10] A. Choudhury, B. Nestler, Grand-potential formulation for multicomponent phase transformations combined with thin-interface asymptotics of the double-obstacle potential, Phys. Rev. E 85 (2012) 21602. [11] J.D. Verhoeven, E.D. Gibson, The divorced eutectoid transformation in steel, Metall. Mater. Trans. A 29 (1998) 1181–1189. [12] A.S. Pandit, H. Bhadeshia, Divorced pearlite in steels, Proc. R. Soc. A-Math. Phys. 468 (2012) 2767–2778. [13] D. Molnar, R. Mukherjee, A. Choudhury, A. Mora, P. Binkele, M. Selzer, B. Nestler, S. Schmauder, Multiscale simulations on the coarsening of Cu-rich precipitates in a-Fe using kinetic Monte Carlo, molecular dynamics and phasefield simulations, Acta Mater. 60 (2012) 6961–6971. [14] K. Ankit, A. Choudhury, C. Qin, S. Schulz, M. McDaniel, B. Nestler, Theoretical and numerical study of lamellar eutectoid growth influenced by volume diffusion, Acta Mater. 61 (2013) 4245–4253. [15] R. Mukherjee, A. Choudhury, B. Nestler, Composition pathway in Fe–Cu–Ni alloy during coarsening, Model. Simul. Mater. Sc. 21 (2013) 075012. [16] E. Brown, G. Krauss, Retained carbide distribution in intercritically austenitized 52100 steel, Metall. Trans. A 17 (1986) 31–36. [17] J.D. Verhoeven, The role of the divorced eutectoid transformation in the spheroidization of 52100 steel, Metall. Mater. Trans. A 31 (2000) 2431–2438. [18] N.V. Luzginova, L. Zhao, J. Sietsma, The cementite spheroidization process in high-carbon steels with different chromium contents, Metall. Mater. Trans. A 39A (2008) 513–521. [19] H.S. Hasan, M.J. Peet, H. Bhadeshia, Severe tempering of bainite generated at low transformation temperaturess, Int. J. Mater. Res. 103A (2012) 1319–1324.