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Keywords:

  • interface energy;
  • sessile-drop method;
  • solidification;
  • high-manganese steels;
  • phase-field method (PFM);
  • microstructure

Abstract

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Experimental Procedure
  5. 3 Model and Simulation Procedure
  6. 4 Results and Discussion
  7. 5 Conclusions and Future Prospects
  8. 6 Acknowledgements
  9. References

High manganese steels are able to deform by the TRIP effect, TWIP effect and microbanding formation. These steels are quite promising materials for mechanical construction, once they show an unusual combination of high ductility and high tensile strength. The casting of these steels represents a technological challenge, because they are extremely prone to macro- and microsegregation. Segregation, on its turn, may locally impair the desired mechanical properties. Simulations by the phase-field method may be utilized to investigate microstructure formation and the development of microsegregation patterns during solidification. Nevertheless, performing reliable microstructure simulations is only possible when reliable values for the solid–liquid interface energy are available. Through utilization of the sessile-drop method, first measurements of the interface energy in the Fe–Mn–C alloy system were performed. By utilizing the obtained values for the interface energy as an input, phase-field simulations were run aiming at investigating both the effect of the value for the interface energy and of the steel composition on the dendrite morphology.


1 Introduction

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Experimental Procedure
  5. 3 Model and Simulation Procedure
  6. 4 Results and Discussion
  7. 5 Conclusions and Future Prospects
  8. 6 Acknowledgements
  9. References

In the collaborative research center SFB 761 “Steel ab initio”, a new methodology for the development of materials is being worked out. Parallel to this, a new steel class based on Fe–Mn–C–Al system should be developed on the basis of this methodology. This new material should possess high formability and, at the same time, present a high tensile stress and toughness after sheet forming. This goal should be achieved through the right combination of the TRIP, TWIP or other additional effects. In this context, experimental and theoretical investigations of the microstructure, macrostructure, and segregation behavior of these steels during and after casting are of primary importance, because of their influence on the deformation mechanism.

Phase-field modeling and simulation may be utilized to investigate microstructure formation and the development of microsegregation patterns in these steels. Nevertheless, performing reliable simulations is only possible when reliable values for the Gibbs energy of the phases, for the diffusional mobilities and for certain thermophysical properties (e.g. for the solid–liquid interface energies) are available.

The measured contact angle between a liquid steel drop lying on a solid substrate made of the same steel composition may be used to indirectly determine a value for the corresponding solid–liquid interface energy. This is possible under the assumption that the values for the other interface energies acting at a triple point are known. The whole procedure is known as the “sessile-drop technique” for the measurement of interface energies. A critical issue in the utilization of the technique is the adjustment of the temperature measured in the region where liquid droplet and substrate coexist to a value slightly below the solidus temperature of the alloy.

Through utilization of the sessile-drop method, first measurements of the solid–liquid interface energy in the system Fe–Mn–C for carbon contents in the range from 0.0 to 0.7 wt% and manganese contents in the range from 12.0 to 25.0 wt% were performed. By utilizing the accessed values for the interface energy as an input, phase-field simulations were then run. These simulations aimed at investigating both the effect of the value for the interface energy and of the exact steel composition on the dendrite morphology, e.g. on the dendrite tip radius and on the sidebranching behavior. The consequences for the resultant final solidification microstructure are also discussed.This paper is structured as follows: In Section 'Introduction', the experimental procedure for the measurement of the interface energy is described. In Section 'Experimental Procedure', the phase-field model is described and the whole simulation procedure is explained in considerable detail. The different parameter settings utilized for the simulations are also described. In Section 'Model and Simulation Procedure', the results, both of sessile-drop experiments and of simulations are presented and discussed. In Section 'Results and Discussion', the conclusions and future prospects of the present work are delineated.

2 Experimental Procedure

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Experimental Procedure
  5. 3 Model and Simulation Procedure
  6. 4 Results and Discussion
  7. 5 Conclusions and Future Prospects
  8. 6 Acknowledgements
  9. References

In spite of its technological importance, direct measurements of the solid–liquid interface energy are relatively rare. In the scope of his experimental studies on the homogeneous nucleation of the solid phase forming from a pure undercooled metallic melt, Turnbull[1] determined indirectly the solid–liquid interface energy (σsl) of various metals (including iron) by measuring the critical undercooling values for nucleation and determining σsl as a function of it. In the eighties of last century, Gündüz and Hunt[2] determined the σsl-values for the two solid phases of a binary eutectic by analyzing the shape of the solid/liquid boundary of a sample annealed in a thermal gradient. Another common method for metallic alloys is the assessment of an approximate value for σsl by analyzing the ripening kinetics of dendrites grown under well-defined conditions.[3] A well-known method for directly determining σsl is the so-called “sessile-drop” method. In this method, a small droplet of a liquid phase with a pre-defined chemical composition is carefully deposited on a solid substrate consisting of the same or of another material.[4] The system is then kept under well-defined thermodynamic conditions for a sufficient length of time. By analyzing the geometry of the drop and of the substrate next to the vapor–liquid–solid triple line, it is possible to calculate a value for σls from the values of the two other interface energies for the remaining phase interfaces in the system.

In the scope of his Diploma-thesis, one of the authors[5] measured the solid–liquid interface energy in the Fe–Mn–C system by utilizing the sessile-drop method. The equipment, which is realized in an autoclave and had been originally built in the scope of a previous Diploma-thesis[6] is schematically depicted on Figure 1. The experiment was performed as follows: at first the autoclave was filled with Argon. Secondly, the solid steel substrate with a defined chemical composition was heated through conductive heating to attain a temperature just below the solidus temperature. This substrate temperature was then maintained (Figure 2). Thirdly, an electric arc was generated between the steel specimen (with the same chemical composition as the substrate) and the tungsten electrode, both situated slightly above the substrate. The heat generated by the electric arc caused the steel specimen to melt and a liquid droplet was then deposited on the substrate. After this last event the substrate was maintained at the envisaged temperature for approx. 30 s, in order for a homogeneous temperature distribution to be attained in the substrate region in the immediate neighborhood of the droplet. Subsequently, the substrate heating was turned off. When the sample attained room temperature, it was cut and examined by usual metallographic methods in order to define its geometric details on a section perpendicular to the solid–liquid–vapor triple line. A typical section is shown on Figure 3.

image

Figure 1. Schematic representation of the equipment for the sessile-drop experiments.

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image

Figure 2. Conductive heating of the substrate.

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image

Figure 3. Typical metallographic section of the sample after cooling and solidification.

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The typical experimental result of Figure 3 can be represented schematically as in Figure 4. It is clear that this situation cannot correspond to equilibrium between the three phases in the system – for instance, a correct equilibrium between the forces acting on a triple point can only be obtained if σls possesses a component pointing downwards, which in the present situation is not the case.

image

Figure 4. Definition of the contact angle in the sessile-drop experiment.

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Additionally, the equilibrium, e.g. between the solid and liquid phases can only be attained when the two phases are allowed to change their compositions so as to bring the system to a state of chemical equilibrium, of course also taking into account the effect of the curved interfaces on this equilibrium. Nevertheless, experiments were performed with seven different steel compositions with experiment replication for experimental error assessment. In most of these experiments, the measured contact angle for a certain steel composition was reproducible (see Section 'Model and Simulation Procedure'). This guarantees that following hypothesis hold:

  1. Once there was no visible movement of either the solid–vapor or the liquid–solid interfaces, the wetting conditions correspond to that of inert wetting.[4]
  2. Thermal equilibrium was attained in the region of the substrate next to the droplet, as evidenced by the infrared camera used to monitor the temperature of the sample during the experiment.
  3. The droplet changed its shape next to the triple point in order to form an approximately constant contact angle θ, for the same setting of experimental conditions, i.e. chemical composition of the steel.

Consequently, it was assumed that a local equilibrium of forces existed (along the horizontal x-axis) acting on the triple point, as expressed by the Young's equation below:

  • display math(1)

This equation was used to calculate the liquid–solid interface energy for the following steel compositions (in weight percentages):

  1. S1: Fe–0.302% C–12.74% Mn
  2. S2: Fe–0.403%C–23.39% Mn
  3. S3: Fe–0.708%C–20.98% Mn

From Equation (1), it becomes clear that σsl can be determined from the value of the contact angle only if σvs and σlv can be determined by some other experimental or theoretical means. The details of the calculation procedure for determining σsl will be explained in Section 'Model and Simulation Procedure'.

3 Model and Simulation Procedure

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Experimental Procedure
  5. 3 Model and Simulation Procedure
  6. 4 Results and Discussion
  7. 5 Conclusions and Future Prospects
  8. 6 Acknowledgements
  9. References

In the last two decades, the phase-field method (PFM) has become the method of choice for simulating microstructure formation in materials. The method is based on approaches from the general theory of phase transitions and from the theory of irreversible thermodynamics.[7-9] With this method, it is possible to perform investigations, e.g. on the evolution of the microstructure during solidification under conditions of diffusion control,[10-14] to study the influence of elastic/plastic effects on liquid–solid and solid–solid phase transformations[15-17] and to investigate the interactions between convection and microstructure evolution during solidification.[18, 19]

In this study, a model for multicomponent dendritic solidification is employed which is based on the work by Cha et al.[20] The model equations are the following ones:

  • display math(2)
  • display math(3)

Equation (2) describes the evolution in space and time of the phase-field variable (φ). f is the free-energy density, which is designed to ensure local chemical equilibrium at the solid–liquid interface. ϵ is a parameter related to the interface energy and its anisotropy, defined as:

  • display math(4)

Equation (3) describes the evolution of the concentration field. ck is the concentration of the k-species. The summations in the equation are carried over the (n − 1) independent chemical elements in the system, where n is the total number of chemical elements in the system. Mki is an atomic mobility.

Atomic mobilities for the solid phases were obtained through the software DICTRA®, MOB2-Database. All thermodynamic data to construct the function f=f(φ, c1, c2, …, cn−1) were obtained from THERMOCALC®, TCFE6-Database and references therein. For the liquid phase, atomic mobilities were set so as to reproduce the diffusivity values from reference.[21]

Simulations were performed for two steel compositions, which were defined for being of interest in the SFB 761 project:

  1. C1: Fe–23%Mn
  2. C2: Fe–23%Mn–0.3%C

The domain temperature was always fixed to a value which corresponded to an equilibrium solid fraction of Ω = 0.82. This corresponds to a highly undercooled regime.

Table 1 and 2 contain the values of all physical parameters that are necessary to define the simulation parameters. In the case of the liquid phase, diffusion coefficients were supposed to assume constant values. In the case of the solid phase, diffusion coefficients are really variable, the values shown on Table 1 and 2 being only reference values for the corresponding equilibrium compositions of the solid. Values of diffusion constants which are not given were assumed to be identically zero, e.g. for the case of the cross-coupling terms in the liquid phase for the ternary alloy case.

Table 1. Physical parameters for phase-field simulation of the solidification for alloy C1
ParameterValue used
L (domain size)8.35 × 10−5 m
T (domain temperature)1687 K
2λ (interface thickness)3.00 × 10−7 m
inline image(diffusion coefficient)2.00 × 10−13 m2 s−1
inline image(diffusion coefficient)2.50 × 10−9 m2 s−1
Table 2. Physical parameters for phase-field simulation of the solidification for alloy C2
ParameterValue used
L (domain size)2.50 × 10−4 m
T (domain temperature)1640 K
2λ (interface thickness)3.86 × 10−7 m
inline image (diffusion coefficient)5.21 × 10−10 m2 s−1
inline image (diffusion coefficient)−1.14 × 10−11 m2 s−1
inline image(diffusion coefficient)−5.83 × 10−14 m2 s−1
inline image(diffusion coefficient)1.20 × 10−13 m2 s−1
inline image(diffusion coefficient)1.00 × 10−8 m2 s−1
inline image(diffusion coefficient)2.50 × 10−9 m2 s−1

If the value for the interface energy is also defined, it is possible, on the basis of the values shown on Table 1 and 2 and of the data extracted from THERMOCALC, from DICTRA and from reference,[21] to calculate all parameters[20] that are needed to perform numerical FDM simulations based on Equation (2) and (3) for both alloys. The values for interface energy were defined on the basis of the sessile-drop experiments, see Section 'Model and Simulation Procedure'.

Simulations were always performed in two dimensions and for symmetric boundary conditions, i.e. derivatives normal to the domain boundaries for both phase-field and concentrations fields were set equal to zero. It was verified that the concentrations on both sides of the interface always satisfied local equilibrium conditions.

4 Results and Discussion

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Experimental Procedure
  5. 3 Model and Simulation Procedure
  6. 4 Results and Discussion
  7. 5 Conclusions and Future Prospects
  8. 6 Acknowledgements
  9. References

4.1 Experimental Results

As mentioned in Section 'Experimental Procedure', the determination of σsl is only possible if, at first, σvs and σlv are determined by another method. For a ternary alloy, it is possible to calculate σlv as a function of the system temperature and phases compositions by using the so-called Butler equation.[22] For the calculation of the σvs-value, there is no such thermodynamics-based procedure available. Semi-empirical approaches seem to indicate that for pure metals there is an approximate direct proportionality between σvs and σlv.[23] So let the quotient between these two values be defined as:

  • display math(5)

It was shown that, for many metals, kσ is approximately equal to 1.18.[23] For other metals, this factor was found to be in the range [1.09, 1.33].

This proportionality was also assumed in the presented work for the case of the investigated Fe–Mn–C alloys. To define the kσ-value in the present case, measurements of the contact angle were performed at first for Armco iron. For the case of Armco iron, it was then assumed that σsl was equal to 0.204 J m−2, the classical value for pure iron obtained by Turnbull.[1] Following the work by Kingery and Humenik[24] the value σlv= 1.240 J m−2 was taken for Armco iron. By using Equation (1), it was then possible to calculate kσ as being equal to 1.098.

For the alloys S1, S2, and S3 (see Section 'Introduction'), it was possible to determine the corresponding values for σlv of each alloy by using the Butler equation. By applying the calculated proportionality factor of 1.098, the σlv-values were then calculated for each alloy. Through the measured values of the contact angle, one then finally determines the values for the liquid–solid interface energy, which are listed on Table 3 for each alloy.

Table 3. Measured values for the solid–liquid interface energy
Alloyσsl [J m−2]Experimental error [J m−2]
S10.461+0.205/−0.092
S20.358+0.026/−0.025
S30.280+0.042/−0.036

In the near future, new evaluations of the solid–liquid interface energy will be conducted at IEHK. These evaluations will also include the effect of additional alloy elements (e.g. aluminum) on this physical property.

As reported on the literature, the interface free energy for pure iron may be different from the original value determined by Turnbull (see, e.g. the theoretical investigations by Waseda and Miller[25] or the approach by Granasy and Tegze[26] whose results were checked against experiments using the grain boundary groove method). This would lead to a change of the present estimates for the solid–liquid interface energy in the case of Fe–Mn–C alloys.

4.2 Simulation Results

By observing the values of the solid–liquid interface energy from Section 3, it is concluded that the maximum value that was measured is σsl = 0.461 J m−2. It is important to notice that the compositions of the phases on the liquid and solid sides of the moving solidification interface are not equal to one another (as it was the case during the performed sessile-drop experiments), being instead of this related to one another by the local equilibrium condition. In order to investigate the effect of changing the interface energy on the simulation results, simulations were performed for both steel compositions C1 and C2, by varying, in each case, the interface energy from σsl = 0.204 J m−2 (value for pure iron) to σsl = 0.461 J m−2. The value of the anisotropy coefficient (γe), which controls the anisotropy of the interface energy was set equal to 0.05 for all simulations. The matrix for the different simulation set-ups depicted in Table 4 can then be constructed.

Table 4. Matrix describing the simulation set-ups
σsl [J m−2]Alloy composition
Fe–23%MnFe–23%Mn–0.3%C
0.204C1E1C2E1
0.461C1E2C2E2

On Figure 5a and b, simulations results are shown for the alloy Fe–23%Mn. All simulations shown in this contribution were performed for a domain, which corresponded to the upper right quadrant of the corresponding depicted box. The picture as a whole was then obtained by reflection. For the input parameters that were utilized in this case, dendritic sidebranches did not appear. Nevertheless, the interface shows already instabilities at the opening of the groove situated at a direction of 45° to the horizontal line. For a longer simulation time and a larger domain, sidebranches would then probably develop. Another important point is, although sidebranches did not appear, the dendrite main stem showed stable growth, in the sense that it grew with a stable velocity and a stable tip radius. This fact can be used to quantify and compare this result with the result of other simulations.

image

Figure 5. Simulation results for the iron-manganese alloy a) σsl = 0.204 J m−2, b) σsl = 0.461 J m−2. Both areas are 160 µm × 160 µm large. The figures show only a cut-out of the original simulation domain.

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When the main dendrite stems of Figure 5a and b (simulation C1E1) (simulation C1E2) are compared to one another, it is clear that dendrites are slenderer in simulation C1E1 (low interface energy) than in simulation C1E2 (high interface energy). This is an expected result, once a higher value for the interface energy tends to not only stabilize the interface against the growth of instabilities but also to increase the energetic cost of curving an interface, producing then larger curvature radii.

On Figure 6a and b, simulations results are shown for the alloy Fe–23%Mn–0.3%C. Results of simulation C2E1 (low interface energy) showed a much more ramified structure than simulation C2E2 (high interface energy). In addition to this, simulation C2E1 showed also a slenderer main stem (with apparently a smaller dendrite tip radius) than simulation C2E2. Notice that the breath of the region free from liquid on both sides of the symmetry axis of the vertical main stem is smaller in C2E1 than in C2E2.

image

Figure 6. Simulation results for the iron-manganese carbon alloy a) σsl = 0.204 J m−2, b) σsl = 0.461 J m−2. Both areas are 360 µm × 360 µm large. Figures show only a cut-out of the original simulation domain.

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All these facts can be related to the higher energetic cost for creating or bending interfaces in the case of higher interface energy.

In order to compare simulation results, it is important to define a common parameter that can be calculated for all simulations. One important characteristic of the dendrite is its radius at the tip (Rtip). This value was calculated from the simulations for each one of the simulation set-ups. The calculation procedure involved approximating the tip region by a parabolic function, y(x) = ax2 + bx + c, by employing a non-linear regression with the least-squares method. In all simulations, only the region of the tip was included in the polynomial regression. A correlation coefficient for the regression of over 0.99 was obtained in all four cases. Once the polynomial coefficients are known, the determination of the tip curvature is trivial. It is important to point out that it was shown in the literature that a functional relationship exists between the Rtip-value and the value of the secondary dendrite arm spacing established during growth in a region near to the dendrite tip (initial secondary dendrite spacing).[27] Table 5 shows the calculated values for Rtip in a highly undercooled regime, which expectedly generates radii values much smaller than the ones that would be observed in a conventional solidification experiment.

Table 5. Results of dendrite tip radius for the various simulation set-ups
 Simulation set-up
C1E1C1E2C2E1C2E2
Rtip [µm]3.24.04.14.2

It is noticeable that, for the case of the Fe–Mn–C alloy, an increase of the interface energy from 0.204 to 0.460 J m−2 caused an increase of the tip radius of only 2.5%. Nevertheless, there was a considerable change on the microstructure (see Figure 6). In the case of the Fe–Mn alloy, there was, as expected, also an increase of the radius as a consequence of the increase of the interface energy value. But in this last case, the radius variation was much larger, of about 20%.

For the same value of the interface energy, the alloy with carbon showed a larger dendrite radius than the binary one, but the difference between radii was larger for the lower value of the interface energy (compare data for C1E1 and C2E1 in Table 5). Carbon diffuses in the melt much quicker than manganese and once the dendrite radius increases with the diffusion length, it is reasonable that by alloying with carbon, the dendritic radius should become larger.

It is expected that the simulation set-up C1E1 will show much slenderer dendrites than all other alloys/set-ups, which were studied in this work, because its dendrite tip radius is much smaller than the other ones. A simulation for alloy C1E1 and for a larger domain will probably show this and it will be reported on this aspect in a near future. The relative absence of sidebranches in the simulations C1E1 and C1E2 can be explained by the synergy of three important factors: (i) the high supersaturation that was imposed in this case, (ii) the low diffusion rate of manganese in the liquid compared to the one of carbon for the composition C2, (iii) the narrower solidification interval for the iron-manganese alloy (ΔT = 21 K) compared to the one for the iron-manganese-carbon alloy (ΔT = 66 K).

Another question that will be answered in future works is the one related to the roughening of the structure when carbon is added to the alloy: alloying the Fe–23%Mn steel with 0.3% carbon leads to a larger dendrite radius. From this fact, it does not necessarily follow that the initial secondary dendrite spacing will get larger through carbon addition, once the stability of the interface is related not only to the diffusion rate in the liquid but also to the details of the phase diagram. Those details are much different for a Fe–Mn and for a Fe–Mn–C alloy (see chapter 3 of the book by Kurz and Fisher[28] for a discussion of the binary case).

Although simulations results were checked by changing the interface thickness by a factor of 0.5, new simulations will be performed by including the corrections which are necessary to make the results concerning the kinetics and the growth morphology really independent of the interface thickness.[29, 30] The present results can nevertheless be considered to supply a reasonable estimate of the dendrite tip radii and its dependence on the value for the interface energy. In addition to this, the effect of a even higher value for the solid–liquid interface energy on simulation results will also be investigated in detail in future works (see Section 'Experimental Results').

These simulations did not include the effect of the change in the interface energy on the dynamics of the Ostwald-ripening. According to Kurz and Fisher,[28] the interface energy influences the Ostwald ripening of secondary dendrite arms like following:

  • display math

where M is defined as

  • display math

and where ΔT is the solidification interval for the alloy, dT/dt is the cooling rate, Δsf is the fusion entropy, Dl is the liquid diffusion coefficient, inline image is the concentration of the liquid at the end of solidification, C0 is the nominal composition of the alloy, m is the liquidus slope and k is the equilibrium distribution coefficient.

These equations show that, for the binary case, the exact value of the interface energy will have an important effect on the secondary dendrite spacing after the Ostwald ripening. Quite recently, the Ostwald ripening of dendrites in a high-alloyed steel was investigated by utilizing both the PFM and an analytical model.[31]

5 Conclusions and Future Prospects

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Experimental Procedure
  5. 3 Model and Simulation Procedure
  6. 4 Results and Discussion
  7. 5 Conclusions and Future Prospects
  8. 6 Acknowledgements
  9. References

In this work, it was shown that the investigated increase of the interface energy from 0.204 to 0.461 J m−2 has a considerable effect on the solidification microstructure of a Fe–23%Mn–0.3%C alloy. The effect of this change in the interface energy for a binary Fe–23%Mn alloy is potentially even greater, as the values for dendrite tip radii seem to indicate.

Next investigations will include investigating the effects of the interface energy on the long-time dynamics of dendritic growth for this class of alloys both in the isothermal case (including Ostwald ripening) and for the case of directional solidification.

6 Acknowledgements

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Experimental Procedure
  5. 3 Model and Simulation Procedure
  6. 4 Results and Discussion
  7. 5 Conclusions and Future Prospects
  8. 6 Acknowledgements
  9. References

The authors gratefully acknowledge the financial support of the Deutsche Forschungsgemeinschaft (DFG) within the collaborative research center (SFB) 761 “Steel ab initio”.

References

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Experimental Procedure
  5. 3 Model and Simulation Procedure
  6. 4 Results and Discussion
  7. 5 Conclusions and Future Prospects
  8. 6 Acknowledgements
  9. References