Effect of Cr:Fe ratio on the mechanical properties of (Cr,Fe)7C3 ternary carbides in abrasion-resistant white cast irons

(Cr,Fe)7C3 ternary carbides constitute the majority of eutectic carbides in abrasion-resistant white cast irons. Density functional theory models have predicted these carbides to have a combination of metallic, covalent and ionic bonding, in proportions depending on the carbide’s Cr:Fe ratio. However, experimental research to validate these predictions has been lacking. This study investigates the characteristics of the carbides as a function of Cr:Fe ratio, which was manipulated from Fe-rich to Cr-rich by varying the Cr:C ratio of the bulk alloy. The carbides’ crystalline structure, hardness, Young’s modulus, fracture toughness and abrasion performance have been assessed through techniques including nano-indentation, HR-TEM and the inner circumference abrasion test (ICAT). Fe-rich M3C formed at very low bulk Cr:C ratio was found to have an orthorhombic crystal structure. In all other alloys, with Cr:C ratios above 2.7, M7C3 was formed and found to have a hexagonal structure. Hardness, Young’s modulus and calculated fracture toughness of M7C3 all increase with Cr:Fe ratio, from (Fe5,Cr2)C3 up to a maximum for (Cr4,Fe3)C3 (in 18Cr–6.8Cr:C WCI). This gave the highest hardness (22.9 GPa) and Young’s modulus (315 GPa), but also the highest fracture toughness (4.5 MPa.m0.5). The peak fracture toughness at carbide composition of (Cr4,Fe3)C3 in this study is consistent with the prediction of DFT models in the literature; while the peak hardness at the same carbide composition shows a marginal deviation from the predictions. Abrasion performance generally increased with carbide hardness and fracture toughness, with one exception: (Cr4.3,Fe2.7)C3. Although (Cr4.3,Fe2.7)C3 showed marginally lower inherent fracture toughness than (Cr4.0,Fe3.0)C3, the higher Cr:Fe carbides imparted the highest abrasion performance, associated with modified eutectic morphology.

Early generations of NiHard class, such as NiHard-1 and NiHard-2, have M 3 C eutectic carbides (in the ledeburite morphology) in which M represents mostly Fe with only small concentrations of Cr and Mn [1,2]. These initial generations of NiHard have inferior abrasion performance and fracture toughness compared to NiHard-4 with M 7 C 3 eutectic carbides [1,2,7].
The literature indicates that the improved abrasion performance of NiHard-4 compared to NiHard-1 and NiHard-2 is due to the presence of M 7 C 3 , which has higher hardness and more convoluted morphology than M 3 C. This convoluted morphology is believed to deviate the crack path within the eutectic carbide network, thus increasing the fracture toughness of the bulk alloy. Together with the higher hardness of the carbide, this leads to an improved abrasion performance for NiHard-4 WCIs over previous generations [1,2].
Within successive NiHard generations, NiHard-2 has 1.4-4.0 wt% Cr while NiHard-4 has 8.0-11.0 wt% Cr [1,8]. As a result of this increased Cr content, M 3 C is largely replaced by M 7 C 3 . Further increases in Cr content lead to the ''high-Cr-Mo'' class of AR-WCIs, with 12-25 wt% Cr. This class again has M 7 C 3 eutectic carbides, but with higher Cr:Fe ratio [9][10][11]. It has now been demonstrated [9] that this results in superior abrasion performance compared to  The suggestion is that increasing Cr:Fe ratio in the M 7 C 3 alters the carbide's inherent properties, which in turn affects the abrasion performance of the AR-WCIs in which they are present. The primary M 7 C 3 have a high degree of anisotropy in hardness and fracture toughness properties [12,13].
Zum Gahr [14] measured hardness and fracture toughness (K c ) of massive (Cr 4 ,Fe 3 )C 3 , through the indentation method. He found hardness in the range 10.55-15.75 GPa, and fracture toughness in the range 1.99-4.68 MPa.m 0.5 . Franco and Sinatora [15] reported hardness of M 7 C 3 varying from 15.6 to 16.9 GPa, and fracture toughness 2. 29-3.99 MPa.m 0.5 . Studying wear-resistant cast iron, nickel and cobalt alloys, Berns [16] found the hardness of primary and eutectic M 7 C 3 to be in range 13.7-21.9 GPa. Using the Palmqvist indentation method, he found fracture toughness of Cr 7 C 3 to be on average 2.64 MPa.m 0.5 .
Casellas et al. [13] found the hardness of primary M 7 C 3 in tool steels to be 18 GPa. Fracture toughness was measured by the length of cracks initiated from a Berkovich indenter and the correlation developed by Laugier [17], giving an average of 2.2 MPa.m 0.5 . Coronado [12] examined primary M 7 C 3 for metallographic sections in two orientations, resulted in calculation of fracture toughness in the range 1.8-2.7 MPa.m 0.5 .
Contrary to the general belief that considers M 7 C 3 as a completely brittle phase with almost no capacity for plastic deformation, empirical studies showed significant plasticity for this compound [18][19][20]. Wang et al. [19] found that, under compressive loading, eutectic M 7 C 3 can undergo plastic deformation by mechanisms including dislocation glide, deformation twinning and sub-grain formation. The compressed M 7 C 3 showed higher hardness than undeformed carbide, indicating strain hardening capacity. Moreover, they noted that under shear and torsional deformation, twist grain boundaries can form, with twist angle of 8.2°, leading to grain refinement and grain boundary strengthening in the carbides.
Using hot compression tests, Pei et al. [18] found that the dominant fracture mechanism in primary M 7 C 3 is different at elevated and room temperatures. This temperature-dependent trait was found to be related to multidirectional stacking faults within the carbides. The anisotropy of hardness, and the strain hardening effect within M 7 C 3 with different orientations, were attributed to the size anisotropy of stacking faults within the carbides.
Other than demonstrating the possibility of plastic deformation in M 7 C 3 , these empirical studies do not elaborate on how the chemical composition might affect deformation capability and fracture toughness characteristics of the carbides.
Kagawa et al. [21] noted that the increase of Cr content in (Fe,Cr) 7 C 3 not only enhances the carbide's hardness but also improves the wear resistance of the bulk alloy. However, they did not investigate the carbide's fracture toughness.
Apart from experimental research, atomistic simulations and calculations have been performed on binary Fe 7 C 3 and Cr 7 C 3 and on ternary (Cr,Fe) 7 C 3 . Calculated thermodynamic formation energies for Cr 7 C 3 , Mn 7 C 3 and Fe 7 C 3 indicate that Cr 7 C 3 is the most stable of these [22]. Atomistic and thermodynamical calculations by Konyaeva and Medvedeva [23] have shown that increasing Cr content leads to a transition from (Fe,Cr) 3 C to (Cr,Fe) 7 C 3 and then to Cr 7 C 3 . Xiao et al. [24] stated that the most stable form of carbide in high-Cr white cast iron is Fe 4 Cr 3 C 3 (with pure Cr 7 C 3 being rare to non-existent). Thermodynamic calculations by Shi et al. [25] found that the most stable form occurs at a slightly higher Cr:Fe ratio, namely Cr 4 Fe 3 C 3 . Rather than a single stable ternary carbide form in AR-WCIs [24], the most stable composition is likely to vary as a function of the Cr:C ratio of the bulk cast iron [9,21].
In addition to thermodynamic stability, mechanical properties of these binary and ternary carbides have also been predicted from simulations. Using first principles calculations of bulk and shear moduli, Xiao et al. [24] predicted that the hardness of Fe 4-Cr 3 C 3 should be superior to that of Cr 7 C 3 . By contrast, Chong et al. [26] calculated higher hardness and elastic modulus for Cr 7 C 3 than for Cr 4 Fe 3 C 3 and Fe 7 C 3 . However, the majority of literature indicates that ternary carbides in the range Fe 4 Cr 3 C 3 -Cr 4 Fe 3 C 3 should have higher elastic modulus and hardness than either of the binary carbides Fe 7 C 3 or Cr 7 C 3 .
Ab initio simulation by Music et al. [27] indicates that Cr 7 C 3 has covalent-ionic (polar covalent) bonding between Cr and C atoms, but mostly metallic bonding between Cr-C-Cr chains. Li et al. [28], who considered only Cr-C bonds for hardness calculation, also proposed that Cr 7 C 3 has a mixture of metallic, covalent and ionic bonding. This mixture of bonding types is suggested to provide Cr 7 C 3 with relatively high hardness but also significant metallic behaviour or ductility.
In (Cr,Fe) 7 C 3 , the Cr:Fe ratio affects the nature of atomic bonds and the resulting properties of the carbide. The influence of Cr:Fe ratio was calculated and simulated by Zhang et al. [29]. Based on density functional theory (DFT) calculations between Cr 7 C 3 , Fe 7 C 3 and any combination of Cr and Fe, they predicted the highest hardness to occur at Fe 4 Cr 3 C 3 (consistent with the predictions of Xiao et al. [24]). However, Zhang et al. [29] predicted the highest metallicity or ductility to occur at the slightly higher Cr:Fe ratio of Cr 4 Fe 3 C 3 .
Regarding crystalline structure, Shi et al. [25] found that Cr 4 Fe 3 C 3 has a hexagonal structure. However, DFT calculations by Sobolev & Mirzoev [30] indicated that the ground state for Fe 7 C 3 is orthorhombic. By doping Cr into (Cr,Fe) 7 C 3 , they predicted that orthorhombic should be replaced by hexagonal at 60 at.% Cr (Cr:Fe ratio of 1.5).
The literature contains minimal experimental research that has empirically verified the above-reported simulation outcomes for eutectic M 7 C 3 with varying Cr:Fe ratios within AR-WCIs. To address this, the current study investigates the influence of Cr:Fe ratio on the crystalline structure and mechanical properties (including hardness and fracture toughness) of (Cr,Fe) 7 C 3 eutectic carbides in AR-WCIs, and the resulting abrasion performance of the bulk alloys.

Alloy preparation
Five hypoeutectic AR-WCIs alloys, with a wide range of bulk Cr:C ratios, were melted and cast. Carbide volume fractions (CVF), estimated through Maratray's formula [1,9,31], were held as nearly constant as possible.
Vol% CVF ¼ 12:33 Â ðC content in wt%Þ þ 0:55 Â ðCr content in wt%ÞÀ15:2 ð1Þ The measured chemical compositions of the resulting castings are shown in Table 1. In Fig. 1, the compositions of the experimental alloy series are plotted onto the 1000°C isothermal section of the Fe-Cr-C ternary phase diagram. Alloy classifications derived from ASTM A532 are shown by grey rectangles. The class and type labels (e.g. IIB, IIC) represent the finer subdivisions shown in the 1982 edition of the standard, but the minimum and maximum compositions are the broader ranges from the 2010 (reapproved 2019) edition.
From left (low-Cr) to right, the Cr:C alloy series spans: • From the two-phase austenite ? M 3 C field (between classes IA and ID); • Though the three-phase austenite ? M 3 C ? M 7 C 3 field (class ID, i.e. NiHard-4); • To the left side of the two-phase austenite ? M 7 C 3 field (class IIA); • To the middle region of the two-phase austenite ? M 7 C 3 field (boundary between classes IIB and IIE); • And then to the far-right of the two-phase austenite ? M 7 C 3 field (class IIIA) • For each alloy, a 6 kg melt was prepared in a 10 kHz induction furnace and poured into a sand mould. Each mould had 12 cavities of dimensions 60 mm 9 25 mm 9 22 mm.
After solidification the alloys were cooled to room temperature in the sand mould. All five alloys were used in the as-cast condition for the following analysis and evaluations. The cast blocks were cut and surface-ground to approximate dimensions of 50 mm 9 24 mm 9 20 mm for abrasion testing. Smaller pieces were cut, mounted, ground and metallographically polished for microstructural evaluation and macro-, micro and nano-hardness measurements.

Microstructural evaluation
The mounted and polished samples were etched with Villela's reagent. Optical micrographs were captured by Olympus Provis AX70 optical microscope. Scanning electron microscope (SEM) images, of the metallographic sections and the worn surfaces, were obtained using a Hitachi SU3500 microscope, equipped with Oxford X-ray energy dispersive spectrometer (EDS). Thin foils for transmission electron microscopy (TEM) were prepared by focused ion beam (FIB) technique using Ga ? ions. TEM images and selected area electron diffraction (SAED) patterns were captured using a field emission high-resolution Hitachi HF5000 microscope (HR-TEM), operating at 200 kV equipped with two Oxford EDS detectors. EDS results were analysed in Aztec software to indicate chemical compositions of regions of interest. HR-TEM images and SAED patterns were post-processed in DigitalMicrograph 3.5 software.

Hardness and fracture toughness
Macro-hardness was measured with a Leco LV800AT Vickers hardness tester at 30 kgf for 10 s. Microhardness was measured with a Struers-Duramin microhardness tester at 0.05 kgf for 12 s. Nano-indentation tests were performed using a Hysitron TriboScope system with Berkovich indenter tip. Nano-hardness and Young's modulus were calculated through the Oliver-Pharr technique [32]. The reported micro-and macro-hardness values are average of ten random readings, while nano-hardness values are average of five random readings on each area or sample. Fracture toughness (K c ) of eutectic carbides was calculated through the energy-based approach proposed by Gautham and Sasmal [33] and Cheng et al. [34]. Since eutectic carbides are smaller than primary carbides, the indentation load was kept relatively low (3 mN) to avoid influence from surrounding microconstituents. Consistent with the low loads, no cracks were observed emanating from the nano-hardness indents. Consequently the lateral crack length method [13,35] for estimation of K c was not possible. Instead, we used an energy-based approach based on the nano-indentation load-displacement data.
In this energy-based approach [33], the total applied energy (U t ) during nano-hardness indentation can be divided into two components: reversible or elastic energy (U e ); and irreversible energy (U ir ). Total energy is calculated from the area under the loading portion of load-displacement curve, while reversible energy is calculated from the area under the unloading portion. Subtracting U e from U t yields U ir , which includes energy for both pure plastic deformation (U pp ) and fracture (U crack ). By computing U pp from U t and subtracting it from U ir , energy for crack formation (hence fracture toughness) can be calculated.

Abrasion test
Relative abrasion performance of alloys was measured using the inner circumference abrasion test (ICAT), operated in low-stress sliding abrasion mode as reported in [9,10]. The abrasion test was performed using basalt gravel (3-6 mm) from Mount Marrow Blue Metal Quarries, Haigslea, Qld.
To calculate relative abrasion performance, the highest thickness loss among the five alloys (that of alloy 5Cr-1.6Cr:C) was considered as a reference point. The relative abrasion performance of each alloy is the reference thickness loss divided by that alloy's thickness loss.

Results
Morphology, crystalline structure and chemical composition 5Cr-1.6Cr:C For the 5Cr-1.6Cr:C alloy, optical, SEM and TEM micrographs, along with diffraction pattern of eutectic carbides, are shown in Fig. 2. EDS analysis results for this carbide are shown in Table 2.
The characteristic perforated morphology of ledeburite can be observed in the optical micrograph and also SEM images of this carbide (Fig. 2a, b and c). This morphology, along with the diffraction pattern (Fig. 2e), indicate that this carbide is cementite (M 3 C), with orthorhombic crystalline structure (space group Pnma). The EDS shows that this carbide comprises mostly Fe with only minor Cr, having a Cr:Fe ratio of 0.14 ( Table 2). 8Cr-2.7Cr:C The next higher Cr:C ratio in bulk composition is alloy 8Cr-2.7Cr:C. Its optical, SEM and TEM micrographs and diffraction pattern are shown in Fig. 3. EDS results for the eutectic carbides are shown in Table 2 and Table 3. The optical and SEM micrographs show the eutectic carbides to have a rod-andblade morphology, consistent with microstructural descriptions of the M 7 C 3 eutectic in AR-WCIs found in the literature [1,2,36]. The eutectic morphology in this alloy is markedly different from the perforated ledeburite morphology in 5Cr-1.6Cr:C (Fig. 2a, b and  c).
The SEM backscattered electron (composition contrast) image and corresponding EDS results (Fig. 3c) indicate that the carbides have a duplex structure, with significant variations in Cr:Fe ratio. The lighter-shaded region represents M 3 C, with lower Cr concentration and higher Fe, along with appreciable Mo content and a trace of Ni ( Table 2). The darker-shaded region represents M 7 C 3 , with higher Cr and lower Fe, as well as lower Mo (Table 3).
These observed differences in Cr and Fe concentrations in the two carbide types are in agreement with literature for NiHard-4 AR-WCIs [1,7,10]. The bulk Cr:C ratio of 2.7 in alloy 8Cr-2.7Cr:C is a typical value for NiHard-4 [1,9,37]. When the Si content is below 1.9 wt% in NiHard-4, duplex carbides are commonly seen. Typically, during solidification a shell of M 3 C forms around a core of M 7 C 3 [1,7,10,38]. From the various optical and SEM micrographs of this alloy, it is evident that M 7 C 3 is      Figure 3d and e shows HR-TEM image and diffraction pattern from the M 7 C 3 region of the duplex carbides in this alloy. The diffraction pattern revealed a hexagonal crystalline structure for the M 7 C 3 region (darker shaded region in Fig. 3c). The EDS results from the M 7 C 3 region show higher Cr:Fe ratio (0.41) for this region (Table 3) than for the M 3 C in either this alloy or in 5Cr-1.6Cr:C alloy (Table 2).
14Cr-4.9Cr:C Alloy 14Cr-4.9Cr:C has Cr:C ratio comparable to ASTM A532 class II-B high-Cr-Mo AR-WCIs [1]. Optical, SEM and TEM micrographs and diffraction pattern for this alloy are shown in Fig. 4. EDS results for the eutectic carbides are shown in Table 3.
18Cr-6.8Cr:C and 24Cr-9.8Cr:C Alloy 18Cr-6.8Cr:C is shown in Fig. 5. This alloy's bulk Cr:C ratio is comparable to ASTM A532 class II-E [1]. EDS results for the eutectic carbides are reported in Table 3. The eutectic carbides (Fig. 5a, b and c) again have the rod-and-blade M 7 C 3 morphology and have homogenous chemical composition through the microstructure (Fig. 5c). The diffraction pattern (Fig. 5e) confirms the hexagonal crystalline structure. The Cr:Fe ratio is higher (1.3) than in the previous alloys.
Alloy 24Cr-9.8Cr:C, with Cr:C ratio comparable to ASTM A532 class III-A, is shown in Fig. 6. The eutectic carbide morphology, while still generally consistent with the rod-and-blade morphology of M 7 C 3 , is slightly different from that in 18Cr-6.8Cr:C. The differences are illustrated in Fig. 7.   The morphological difference relates to the length, aspect ratio and apparent connectivity of the blades or rods. In 18Cr-6.8Cr:C ( Fig. 7a and b) the carbides are elongated, and with a substantial degree of connectivity. By comparison, those in 24Cr-9.8Cr:C ( Fig. 7c and d) adappear shorter, but with comparable width, hence lower aspect ratio. Moreover, they appear (in the two-dimensional section) to be less connected, and with wider matrix strips between them. The intervening matrix consists of a mixture of retained austenite and martensite, as shown in Fig. 6c.
Despite the morphological change, the diffraction pattern for 24Cr-9.8Cr:C (Fig. 6e) again confirms the hexagonal crystalline structure, comparable to the other alloys. The backscattered SEM image (Fig. 6c) again indicates a homogenous composition. Consistent with the trend of increasing bulk Cr:C ratio, the EDS results (Table 3) confirm that the carbides in this alloy have the highest Cr:Fe ratio (1.6) in the experimental series.
In all alloys the matrix in the dendrites remained unetched, indicating retained austenite, as per the usual finding for AR-WCI in the as-cast condition [1,2]. However, the matrix immediately adjacent to the interface with the eutectic carbides is etched differently from the remaining matrix. This interfacial region is composed of martensite. The martensite is thought be formed as a result of solid-state growth of eutectic carbides during cooling, and consequent carbon depletion (hence self-destabilisation) of the austenite in this zone [1,2,9].

Measured mechanical properties
Bulk macro-hardness and matrix micro-hardness trends are shown in Fig. 8. Macro-hardness readings range from 460 to 520 HV 30 , typical of as-cast AR-WCIs [1]. Matrix micro-hardness readings are within the range of 310-345 HV 0.05 , confirming the austenitic matrix.
The trends of nano-hardness, Young's modulus and calculated fracture toughness for the eutectic carbides with increasing Cr:Fe ratio are shown in Table 4 and Fig. 9. All the measured hardness and calculated fracture toughness values determined in this study are within the range of results previously reported by other researchers based on a variety of techniques [12][13][14][15][16].
Firstly, it is confirmed that M 3 C (in 5Cr-1.6Cr:C) has lower hardness compared to M 7 C 3 (all remaining alloys). Secondly, it is shown that the fracture toughness of M 3 C is substantially inferior to that of M 7 C 3 . Comparing 8Cr-2.7Cr:C with 5Cr-1.6Cr:C, M 7 C 3 has 19% higher hardness but 31% higher fracture toughness than M 3 C.
Within the M 7 C 3 , all of hardness, Young's modulus and fracture toughness generally increase with bulk Cr:C ratio and hence carbide Cr:Fe ratio. However, these properties reach a maximum at what appears to be an optimum Cr:C ratio of about 6.8 (alloy 18Cr-6.8Cr:C). The alloy with highest Cr:C ratio (9.8) has comparable hardness and Young's modulus, but significantly lower fracture toughness (Table 4). In the optimum alloy, the EDS results indicate a molecular formula of (Cr 4.0 ,Fe 3.0 )C 3 , which evidently provides the carbide with superior mechanical properties than those for (Cr 4.3 ,Fe 2.7 )C 3 .

Abrasion performance of carbides within AR-WCIs
For the ICAT basalt tests, the trends in thickness loss (and conversely the relative abrasion performance) are shown in Fig. 10. As Cr:C ratio increases, the alloys' abrasion resistance clearly increases, in a continuous and roughly linear manner.
To identify damage mechanisms which might help explain the quantitative trends, worn surfaces were examined by SEM. Figure 11 shows 5Cr-1.6Cr:C, with M 3 C eutectic, while Fig. 12 shows 18Cr-6.8Cr:C, with (Cr 4.0 ,Fe 3.0 )C 3 .
In both alloys, the matrix can readily be seen to being eroding faster than the eutectic carbides, as  evidenced by the protrusion of carbides highlighted in the secondary electron images (Fig. 11a and c,  Fig. 12a). Since the alloys were tested in the as-cast condition, the matrix is austenitic in all cases. Being much softer than the carbides (Fig. 8 and 9), the austenite erodes at a much higher rate [39][40][41]. Most importantly, since the austenitic matrix has almost constant hardness across all alloys (Fig. 8), any trends in relative abrasion performance must be entirely attributable to the characteristics of the eutectic carbide network. Comparison between Figs. 11 and 12 shows that the eutectic carbides in 5Cr-1.6Cr:C have larger numbers of cracks than those in 18Cr-6.8Cr:C. The  cracks in 5Cr-1.6Cr:C are predominantly oriented transverse to the abrasive flow direction (Fig. 11b and  d), as expected for brittle fracture under the influence of surface traction stresses. Depending on the abrasive environment, cracking of eutectic carbides is fairly common in AR-WCIs [12,42]; but the marked difference in density of cracks between these two alloys, under identical abrasive conditions, must be regarded as significant. A similar trend was observed in [9]. These qualitative damage mechanism observations imply higher fracture toughness of M 7 C 3 in 18Cr-6.8Cr:C than M 3 C in 5Cr-1.6Cr:C alloy. These qualitative observations are fully consistent with, and hence provide confirmation for, the relative fracture toughness values calculated from the nano-indentation measurements.

Discussion
The microstructural trends described above (''Morphology, crystalline structure and chemical composition'' section) show that, by increasing the bulk Cr:C ratio, a transition firstly occurs from M 3 C to M 7 C 3 . The alloy with 1.6Cr:C ratio contains a fully M 3 C network, while that with 2.7Cr:C ratio has a mixture of M 3 C and M 7 C 3 , appearing as duplex carbides, but with M 7 C 3 dominating. With further increase in bulk Cr:C ratio, the eutectic carbide network becomes fully M 7 C 3 , showing progressively increasing Cr:Fe ratios.
In all cases the M 7 C 3 ternary eutectic carbides, regardless of Cr:Fe ratio, were found to have a hexagonal crystalline structure (space group P63mc). These observations do not support the prediction by Sobolev and Mirzoev [30], who proposed an alteration of crystalline structure from orthorhombic to hexagonal at 1.5 Cr:Fe ratio in the (Cr,Fe) 7 C 3 .
In terms of mechanical properties, the higher hardness of M 7 C 3 than M 3 C is well known [1,21]. It has also been reported [1,21] that AR-WCIs containing M 7 C 3 have higher fracture toughness than those containing M 3 C. The higher hardness of M 7 C 3 is an inherent property of the carbide itself, ascribed to its crystalline structure and chemical bonds. By contrast, the higher fracture toughness of alloys containing M 7 C 3 is usually attributed to differences in the eutectic morphology, with the M 7 C 3 eutectic being more convoluted and less continuous than the M 3 C ledeburite eutectic. In the current work, however, the observed evidence indicates that the inherent properties of M 7 C 3 also play a role in the superior fracture toughness of alloys which contain this carbide. This superior inherent fracture toughness of M 7 C 3 compared to M 3 C is indicated both semiquantitatively (by carbide fracture toughness values calculated from nano-indentation load-displacement curves) and qualitatively (by SEM observations of carbide cracking on abraded surfaces).
Within M 7 C 3 ternary carbides, increasing Cr:C ratio results in increasing hardness, increasing Young's modulus and increasing fracture toughness, up to a certain optimum level ( Fig. 9 and Table 4). Since hardness and fracture toughness usually show a negative correlation, the clear positive correlation observed in this work is of great interest.
This unexpected positive correlation between the fracture toughness of the carbides and their hardness is postulated to be related to the influence of Fe and Cr co-ordination chemistry on the character of bonding within the ternary carbide. Using density functional theory (DFT), the chemical bonds in M 7 C 3 have been predicted to have a combination of metallic, covalent and ionic characteristics [27,43,44]. It has been proposed that this mixture of bonding characters imparts both high hardness and metallic behaviour to the ternary carbides [24,25,27,29].
Considering the co-ordination chemistry within the mutually intersoluble series of compounds from (Fe,Cr) 7 C 3 to (Cr,Fe) 7 C 3 to Cr 7 C 3 , it is inevitable that Fe-C bonds must progressively be replaced by Cr-C bonds. Similarly, as Cr:Fe ratio increases, Fe-C-Fe chains must be progressively replaced with Fe-C-Cr and Cr-C-Cr chains. In this context, the DFT calculations [27,29] predict the following characteristics: M-C bonds Although both Fe-C and Cr-C are polar covalent bonds [13], the ionicity of Fe-C is stronger than in the Cr-C bond. Conversely, Cr-C is more strongly covalent than Fe-C [16]. As a result of this, the hardness of the carbide tends to increase with increasing Cr:Fe ratio.
M-C-M chains The DFT calculations predict that the Fe-C-Cr and Cr-C-Cr chains are immersed in a Fermi electron gas, with high degree of delocalized electrons. Consequently, compounds in which the latter chains dominate are predicted to display a stronger degree of metallic behaviour [27,29,43], hence greater fracture toughness.
The current empirical results for the influence of Cr:Fe ratio on fracture toughness of M 7 C 3 ( Fig. 9 and Table 4) are in good agreement with the effect of Cr:Fe on degree of metallicity as calculated by Zhang et al. [29] using density functional theory (DFT). Zhang et al. computed the highest metallicity to occur for (Cr 4 ,Fe 3 )C 3 . This composition for maximum calculated metallicity agrees well with our maximum measured fracture toughness for (Cr 4 ,Fe 3 )C 3 in 18Cr-6.8Cr:C alloy.
For hardness, our empirical results indicate that the maximum hardness and Young's modulus also occur at (Cr 4 ,Fe 3 )C 3 . This measured maximum-hardness composition is somewhat shifted from that of (Fe 4 ,-Cr 3 )C 3 predicted by Xiao et al. [24] and Zhang et al. [29]. Further investigations would be required to determine which of the two findings is more reliable, but we would suggest that the empirical findings be regarded as more authoritative until proven otherwise. In any case the offset is relatively minor; both methods indicate an optimum somewhere in the vicinity of equal Cr and Fe stoichiometry in the carbide.
For the first four alloys, the relative abrasion performance trend displayed in Fig. 10 shows a reasonable degree of similarity to the hardness and fracture toughness trends of eutectic carbides in the first four points in Fig. 9. From this it can be deduced that the improvement in abrasion performance with increasing bulk Cr:C ratio across these four alloys is attributable to a combination of increasing hardness and enhanced fracture toughness of the eutectic carbides, resulting from their increased Cr:Fe ratio. Carbide hardness is a well-known factor, but the significant contributory role of fracture toughness is highlighted by the differential cracking observations in Fig. 11 and Fig. 12. These qualitative damage mechanism observations corroborate the nano-indentation fracture toughness measurements, which in turn are fully consistent with the DFT predictions described earlier.
The only weakening in the correlation between abrasion performance and carbide mechanical properties is the plateau at the highest Cr:C ratio in Fig. 9. Despite the M 7 C 3 in 24Cr-9.8Cr:C having comparable hardness and slightly lower fracture toughness than those in 18Cr-6.8Cr:C, the 24Cr alloy has the best abrasion performance. The superior performance of this last alloy is postulated to be attributable to its altered eutectic morphology. Its less elongated and apparently less continuous eutectic carbides (Fig. 7) can reasonably be proposed to further enhance the alloy's resistance to micro-fracture damage mechanisms [1,4,45]. Micro-fracture damage mechanisms are well understood to affect abrasion performance [5,[46][47][48], especially under high-stress abrasion conditions [49,50]. This has evidently compensated for the marginally lower inherent fracture toughness of these carbides.

Conclusions
In a series of abrasion-resistant white cast irons in the as-cast condition, it has been shown that increasing the bulk Cr:C ratio from 1.6 to 9.8 results in a steady improvement in abrasion performance. The performance improvements are solely attributable to the characteristics of the eutectic carbides, since all alloys had an austenitic matrix and near-constant matrix micro-hardness.
With increasing bulk Cr:C ratio, initially there was a transition from orthorhombic M 3 C to hexagonal M 7 C 3 ternary eutectic carbides. The first transitional alloy, with a bulk Cr:C ratio of 2.7, had a mixture of M 3 C and M 7 C 3 . Thereafter (Cr:C ratio from 4.9 upwards) they were fully M 7 C 3 with hexagonal crystal structure. On increasing Cr:C ratio from 2.7 to 9.8, the Cr:Fe ratio within the M 7 C 3 was confirmed to increase continuously, from 0.41 to 1.60.
Nano-indentation measurements were used to determine both the hardness of the eutectic carbides and their fracture toughness, using energy-based analysis from the nano-indentation load-displacement curves. Increasing Cr:Fe ratio not only increased the carbides' hardness (from 15.2 to 22.9 GPa), but also their fracture toughness (from 2.9 to a maximum of 4.5 MPa.m 0.5 ). In this alloy series, the peak for both hardness and fracture toughness occurred at the second-highest Cr:Fe ratio, with carbides measured at (Cr 4.0 ,Fe 3.0 )C 3 in 18Cr-6.8Cr:C.
The findings of the energy-based fracture toughness assessment are corroborated by two independent sources of information. Firstly, the observed peak toughness at a carbide composition of (Cr 4.0 ,-Fe 3.0 )C 3 agrees with the predictions of density functional theory sourced from literature. The DFT analysis predicts the M-C-M chains are immersed in a metallic Fermi free electron gas, providing the highest metallicity (hence maximal fracture toughness) for this compound. Secondly, worn surface SEM observations showed that micro-fracture damage mechanisms were much more frequent in an alloy with low Cr:C ratio than in the higher Cr:C, peak-performance alloy. These findings clearly emphasise the importance of carbide fracture toughness.
At the very highest Cr:C ratio (9.8), the (Cr 4.3 ,-Fe 2.7 )C 3 carbides showed slightly lower fracture toughness than the (Cr 4.0 ,Fe 3.0 )C 3 in the 6.8Cr:C alloy. Despite this, the abrasion performance continued to rise. This apparent breakdown in the correlation between abrasion performance and the inherent mechanical properties of the carbides was explained in terms of the eutectic morphology. At the highest Cr:C ratio, the eutectic carbides showed less elongation and reduced connectivity. This is likely to lead to improved resistance to fracture-related damage mechanisms, compensating for the marginally lower inherent fracture toughness of the carbides.

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Declarations
Conflict of interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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