Quantitative measurements of carbon using WDS are known to be difficult due to hydrocarbon contamination.21 We will for this reason begin by an examination of the accuracy of the measured carbon concentrations. A preliminary analysis of the concentration maps showed that there was a systematic drift in the measured carbon content from the first to the last row. Linear least squares regression suggested a mean slope of 1.015 × 10−3 wt%/μm. All carbon maps were for this reason corrected additively for this slope to eliminate the systematic drift, with the mean concentration over the maps conserved. While a systematic bias of the measured concentration may remain, the procedure makes sure that this bias is fairly uniform over a given map. Relative comparison of carbon concentrations measured in different regions of the map should for this reason be valid, but the accuracy of the absolute concentrations remains uncertain.
A useful point of reference for the carbon concentration in our specimens is the cementite phase found in the regions which were liquid before quenching. Examples of a particularly large cementite particle are shown in Figure 2, which shows a portion of the concentration map from the periphery of specimen C. A histogram of measurements over the particle indicates a peak centered on 6.73 wt% C. Considering that the stoichiometric carbon concentration of cementite (Fe3C) is 6.69 wt% (atomic weights taken from Reference 22), the corrected carbon concentrations seem to be reasonably accurate in this concentration range, with an estimated bias of + 0.04 wt%, despite the measured concentration exceeding the calibration range.
The concentration of carbon in γ does not obey a stoichiometric ratio and demands a closer investigation.
The Distribution of Carbon in Austenite
The aim of this section is to highlight the most important observations and discuss their implications. How representative the observed distribution of carbon is of its distribution before quenching and the influence of microsegregation on equilibrium concentrations of carbon will be discussed in later sections.
Figures 3 and 4 show concentration maps of carbon, where the color scale has been adjusted to capture the range of carbon concentrations found in regions which were γ prior to quenching (prior γ). SG and CG can be observed, embedded in the prior γ. Ledeburite regions, which were liquid prior to quenching, generally overshoot the color scale and are hence shown as white, as does graphite.
The highest carbon concentrations in prior γ are generally measured near the liquid, while lower concentrations are generally found near graphite. This is reasonable, considering that the carbon concentration of γ in equilibrium with liquid \( C_{\text{C}}^{\gamma /L} \) rises above the concentration in equilibrium with graphite \( C_{\text{C}}^{\gamma /G} \) at undercooling below the eutectic temperature, as illustrated using a binary Fe–C phase equilibrium diagram in Figure 5.
The carbon concentrations of γ are naturally lower than indicated in Figure 5 for our present multi-component alloy containing 1.88 wt% Si. For reference, the carbon concentration in γ at equilibrium with liquid \( C_{\text{C}}^{\gamma /L} \) is 1.85 wt% at 1140 °C according to the expression \( C_{\text{C}}^{\gamma /L} = \left( {1528.4 - T} \right) / 177.9 - 0.18 \times C_{\text{Si}} \),23 where \( C_{C}^{\gamma /L} \) and \( C_{Si} \) are concentrations in weight percent and T is the temperature in Celsius.
The skewed couple zone of cast irons may drive the carbon concentrations of liquid to exceed the eutectic concentration, which may in turn impact the carbon concentration in γ in local equilibrium with the liquid.24,25 Partitioning of carbon between martensite and γ during the quenching procedure may also have caused small zones of carbon depletion and supersaturation, as will be discussed in a later section of this paper.
We will now direct our attention toward the concentration of carbon in γ near graphite. Assuming uniform temperature, pressure and distribution of substitutional elements over the γ, \( C_{\text{C}}^{\gamma /G} \) is expected to also be uniform. On the contrary, Figures 3 and 4 indicate that the concentration of carbon in γ near graphite varies considerably. The lowest concentrations of carbon, around 1.0 wt%, are consistently found in γ in the interior of CG-γ eutectic cells, which can be observed as black and dark blue regions in Figures 3 and 4. Meanwhile, the measured carbon concentrations barely fall below 1.3 wt% near SG nor near the extremities of CG in the periphery of CG-γ cells.
The carbon concentration of γ in contact with graphite is by necessity either in local equilibrium or it is not. Hence, the observed variations in the carbon concentration of γ in contact with graphite likely reflect some combination of (1) departure from or (2) displacement of the carbon concentration in γ at equilibrium with graphite.
A departure from the equilibrium concentration could arise if the attachment kinetics of carbon to graphite is slow relative to the transport of carbon toward its surface. Displacement of the carbon concentration in γ at equilibrium with graphite could relate to local stresses or variations in chemical composition of γ due to microsegregation.
Whichever is the case, we must also consider at what stage the observed distribution of carbon developed. Is it representative of the distribution of carbon in γ prior to quenching, or was the distribution altered during the quenching process?
The potential influences of microsegregation and the quenching procedure on the observed distribution of carbon are discussed in the following sections, after which conclusions are summarized.
Consideration of Substitutional Elements
In cast irons, the silicon concentration is typically higher early compared to late crystallized γ.26,27,28,29,–30 This is due to the tendency of Si to partition at higher concentrations in γ than liquid during solidification, as well as its low diffusivity in γ making it resistant to back-diffusion.31 Since silicon is known to strongly affect the equilibrium concentrations of carbon in γ,23 its microsegregation in γ is important to consider when evaluating local carbon contents.
Figure 6 shows the measured map of Si concentrations corresponding to the region shown in Figure 4b). A comparison of the Si concentration in the low-carbon region in the interior of the CG-γ eutectic cell with the silicon concentration near the SG (dashed circles) shows that both display a color in the yellow range, indicating a silicon concentration of about 2.3 wt%.
The influence of Mn and Cu on equilibrium concentrations of carbon in γ is considerably weaker than of Si, and their concentrations were also found to be similar in the two regions. The concentration of P is very low in the compared regions.
As the composition of γ near SG and in the central region of CG-γ eutectic cells seems to be essentially the same aside from carbon, no displacement of the concentration of carbon at equilibrium with graphite can be expected on the basis of microsegregation. We can conclude that the observed differences in carbon concentration appear to have no relation to microsegregation of Si, Mn and Cu, which are the main substitutional alloying elements in the alloy.
Migration of Carbon During Quenching
There is a risk that the observed distribution of carbon differs from the distribution during solidification due to migration during the water quenching. To investigate how the quenching process influenced the distribution of carbon, a numerical simulation of the diffusion of carbon in γ around SG was performed. The simulation employed a control volume-based implicit finite difference method based on Fick’s law of diffusion for one dimension in a spherical coordinate system, essentially the same as described for heat conduction but for chemical rather than thermal diffusivity.32 The concentration- and temperature-dependent diffusivity was updated explicitly every time step according to an equation by Ågren.33 The contribution of concentration gradients of substitutional elements to the driving force for diffusion of carbon, i.e., thermodynamic factors, is neglected.34 This is fair as these gradients are small in early crystallized γ.
The simulation was performed in two steps: (1) an isothermal hold at 1140 °C for 60 s, allowing the concentration profile to reach steady state, and (2) a global temperature drop at a constant rate of −300 K/s, representing the quenching. The cooling rate during the quenching is uncertain, but experience from conducting the experiments suggests that the sound of boiling lasts only for a few seconds. The chosen cooling rate giving 3.5 s to reach 100 °C is thus reasonable.
As we are mainly interested in the carbon concentration near graphite, the concentration is kept constant at 1.84 wt% on the exterior boundary against the liquid.
At the interior boundary, against the SG, the consumption of carbon associated with growth of graphite is represented by a flux of carbon \( j \), which is assumed to be controlled by \( j = - K_{\text{C}} \left( {w_{\text{C}} - w_{\text{C}}^{\gamma /G} } \right) \), where \( K_{\text{C}} \) is a proportionality constant, \( w_{\text{C}} \) is the simulated boundary concentration and \( w_{\text{C}}^{\gamma /G} \) is the carbon concentration in γ at equilibrium with graphite. The flux condition was integrated into the implicit solver for concentrations. The authors were not able to find an adequate expression for \( C_{\text{C}}^{\gamma /G} \) as a function of T and Si in the literature. An expression was developed by fitting a bivariate second-order polynomial to the \( C_{\text{C}}^{\gamma /G} \) surface in the Fe–C–Si system at atmospheric pressure calculated using Thermo-Calc 2019a with the TCFE7 databank for the temperature range 400 to 1200 °C and 0 to 10 wt% Si. Using the coefficients given in Table 3, the expression (1) provides \( C_{\text{C}}^{\gamma /G} \) within a standard deviation of 4 × 10−4 wt% from the Thermo-Calc results. The carbon concentration in γ at equilibrium with graphite \( C_{\text{C}}^{\gamma /G} \) and the average silicon concentration of the system \( C_{\text{Si}}^{L} \) are given in weight percent, and the temperature T is given in °C.
Table 3 Fitted Coefficients for Eqn. 1 $$ \begin{aligned} C_{\text{C}}^{\gamma /G} & = a_{1} + a_{2} C_{\text{Si}}^{\gamma } + a_{3} \left( {C_{\text{Si}}^{\gamma } } \right)^{2} \\ & \quad + \left( {a_{4} + a_{5} C_{\text{Si}}^{\gamma } + a_{6} \left( {C_{\text{Si}}^{\gamma } } \right)^{2} } \right)T \\ & \quad + \left( {a_{7} + a_{8} C_{\text{Si}}^{\gamma } + a_{9} \left( {C_{\text{Si}}^{\gamma } } \right)^{2} } \right)T^{2} \\ \end{aligned} $$
(1)
The value of \( K_{\text{C}} \) is unknown; however, it has been suggested that, just like diffusivity, interface mobility has an Arrhenius type relationship to temperature.35 A clue can be derived from the fact that, under conventional cooling conditions, precipitation of graphite is known to continue by desaturation of γ as \( C_{\text{C}}^{\gamma /G} \) drops with temperature.36,37 Further precipitation of graphite also occurs at lower temperatures when γ transforms to ferrite.36,38 This indicates that interface kinetics are still considerable at temperatures well below the freezing temperature. Trial and error showed that \( K_{\text{C}} = 10^{ - 11} {\text{ms}}^{ - 1} {\text{wt}}\%^{ - 1} \) was sufficient to maintain an interface concentration near \( C_{\text{C}}^{\gamma /G} \), while \( K_{\text{C}} = 10^{ - 12} {\text{ms}}^{ - 1} {\text{wt}}\%^{ - 1} \) allowed for notable departure from \( C_{\text{C}}^{\gamma /G} \) at steady state. For simplicity, \( K_{\text{C}} \) is kept constant during the simulation.
Simulation results for \( K_{C} = 10^{ - 11} \) and \( K_{\text{C}} = 10^{ - 12} \) are presented in Figures 7 and 8, respectively, in the form of concentration profiles as a function of distance from the SG. In the vicinity of the graphite, calculated steady-state concentration profiles do not fall below the theoretical \( C_{\text{C}}^{\gamma /G} \) of 1.58 wt%. The measured carbon concentrations near SG approaching 1.3 wt% are thus considerably lower than expected for the steady-state concentration profile at 1140 °C.
Despite the relatively rapid cooling, the numerical calculations suggest that the concentration profile changes substantially during the quenching process. As temperature drops, \( C_{\text{C}}^{\gamma /G} \) quickly falls, causing depletion of carbon in the vicinity of the graphite. Despite the parallel fall of diffusivity (from 3.1 × 10−10 m2/s at 1140 °C, to 6.6 × 10−22 m2/s at 192 °C), carbon migrates enough for the depleted zone to extend about 20 μm away from the graphite interface.
According to Figures 7 and 8, the carbon concentration at the interface tends to catch up with \( C_{\text{C}}^{\gamma /G} \) during quenching, resulting in carbon concentrations far below the 1.3 wt% measured around SG. The portion of the profile which is below 1.3 wt% extends about 3–4 μm which is near the resolution of the microprobe. In other words, it seems possible that the lowest concentrations closest to the graphite were not captured accurately with EPMA due to bleeding of information from the nearby graphite. It is also important to consider that the choice of \( K_{\text{C}} \) is probably unrealistic and is likely in part responsible for the shape of the calculated concentration profiles after the quench. That carbon concentration seems to catch up with \( C_{\text{C}}^{\gamma /G} \) near the graphite during the quench, even when a departure is observed at steady state (\( K_{\text{C}} = 10^{ - 12} \)), is most certainly a consequence of the fall of diffusivity with temperature relative to \( K_{\text{C}} \) which remains constant. In reality, \( K_{\text{C}} \) likely drops with temperature along with diffusivity, and it is not clear whether quenching promotes departure from or convergence toward \( C_{C}^{\gamma /G} \).
The considerably higher carbon concentration near SG compared to the central region of CG-γ eutectic cells is difficult to explain. It may, for example, relate to differences in the graphite at various scales, such as the local access to graphite interface, the topology of the graphite interfaces or its crystallographic structure.
An additional numerical calculation was performed for \( K_{\text{C}} = 10^{ - 11} \) and a quench rate of -3000 K/s to investigate to what extent an increase in the quench rate suppresses migration of carbon. The result shown in Figure 9 indicates a carbon-depleted zone around SG limited to a distance of about 7 μm from the graphite interface, leaving more of the concentration profile representative of its state before quenching. Such a quench rate may be achieved, for example, by quenching a smaller piece of metal.
In conclusion, concentrations measured in prior γ near graphite tend to be considerably lower than the theoretical value for γ in equilibrium with graphite at the quenching temperature of 1140 °C. The numerical calculations indicate that this is likely related to depletion of carbon in a zone around graphite during quenching. A feasible approach to better preserve the distribution of carbon through the quenching is to increase the quench rate, for example by reducing the amount of material to quench. The higher concentrations measured in the vicinity of SG compared to the central region of CG-γ eutectic cells are difficult to explain.
The Influence of Martensite on the Distribution of Carbon
The regions which were γ prior to quenching display relatively large variations in carbon concentration over distances of a few pixels. Comparison of the distribution of carbon to micrographs indicates that these variations relate to martensite which formed during the water quenching. Figure 10 shows that streaks of lower carbon concentrations correlate with martensite plates, while no martensite is visible in carbon-rich regions.
This is an indication that carbon has partitioned from martensite into the surrounding retained γ, as has been observed for steels.39,40 This means that the distribution of carbon appears to have changed to an extent after martensite transformation. That pale regions between darker needles in dendrite arms correspond to retained γ has been verified previously using EBSD in research work on a similar quenched material.41
Nital etching was applied to the materials after completion of the EPMA to better observe the distribution of martensite in the material. Unfortunately, the etching failed in the mapped regions, presumably due to local hydrocarbon contamination of the surface. However, inspection of the remainder of the cross-sectional area revealed that the distribution of martensite followed a similar pattern regardless of location. The distribution of martensite in the mapped regions can thus be inferred qualitatively by extrapolation from successfully etched regions. Figure 11 includes micrographs which are representative this pattern.
Figure 11a shows that prior γ which is remote from graphite, such as γ dendrite arms, generally displays a larger area fraction of retained γ (pale regions), traversed by a few thick martensite plates (dark regions).
Moreover, Figure 11b, c shows that regions near CG and SG particles feature more martensite and smaller area fractions of retained γ.
Given that carbon appears to have partitioned out from the martensite plates into the surrounding γ, there seems to be a risk that martensite transformation contributed to further migration of carbon. We will for this reason take a closer look at how far carbon is likely to have diffused after martensite formation, using simple diffusion calculations.
We assume that a 4-μm-thick plate of martensite forms in γ with a uniform carbon content 1.5 wt% at the temperature for start of martensite transformation MS. All carbon in the martensite plate is assumed to instantaneously migrate to the interface of the nearby γ, after which carbon diffuses away from the interface into the γ at constant diffusivity. The problem can then be approximated as diffusion along one dimension from an instantaneous point source. The evolution of the carbon profile in the γ can be approximated using Eqn. 2:
$$ C_{\text{C}} \left( {z,t} \right) = C_{\text{C}}^{0} + \frac{M}{{\sqrt {4\pi D_{\text{C}}^{\gamma } t} }}\exp \left( { - \frac{{z^{2} }}{{4D_{\text{C}}^{\gamma } t}}} \right) $$
(2)
where \( C_{\text{C}} \) is the concentration of carbon as a function of space and time, \( C_{\text{C}}^{0} \) is the initial uniform mass fraction carbon in γ, z is the distance from the martensite plate in meters, t is the time after martensite formation, \( D_{\text{C}}^{\gamma } \) is the constant diffusivity of carbon in γ in units of m2/s and M is the amount of carbon displaced from the martensite plate to the γ-martensite interface.42
Figure 12 shows the temperature for start of martensite transformation MS according to a number of models. The predicted MS diverges strongly for carbon concentrations beyond 0.7 wt% because of the limited calibration range for the models. To avoid extrapolation and underestimation of \( D_{\text{C}}^{\gamma } \), we assume a generous MS of 200 °C, corresponding to around 0.7 wt%. The carbon concentration of γ varies with time and distance from the martensite. For simplicity, a constant \( D_{\text{C}}^{\gamma } \) of 8 × 10−16 m2/s is chosen, corresponding to a carbon content of 5 wt% and 200 °C according to Ågren.33
The result of the diffusion calculations using Eqn. 2 is presented in Figure 13. The evolution of the carbon concentration profile indicates that considerable diffusion of carbon in γ over distances beyond a few micrometers is unlikely.
It seems safe to conclude that the martensite transformation did not contribute considerably to the observed depletion of carbon in the vicinity of graphite. As is indicated in Figure 12, the large area fraction of martensite found near graphite is more likely a consequence of carbon depletion than a cause of it. Sufficient carbon content of γ suppresses martensite transformation, resulting in retained γ.43,46,47