1 Introduction

The intricate world of cast iron unveils an arsenal of mechanical marvels, where the interplay between composition, microstructure, and production methods crafts materials that withstand the harshest of environments. These alloys, recognized for their exceptional abrasive resistance, harbor secrets within their crystalline structures—secrets that hold the key to their remarkable mechanical prowess (Moustafa et al. 2000).

Within the heart of cast iron, concealed carbides and borides emerge as the unsung heroes, wielding the power to transform mundane iron into a formidable defender against wear and tear. These precipitated entities—Fe2C, Fe3C, Fe23C7, and their kin—bestow resilience upon the alloy. Yet, innovation has not ceased with tradition; a subtle infusion of chromium and boron, alchemical touchstones, reshapes these entities into (Fe, Cr)xC(B) formations (Feng et al. 2012). The outcome is a symphony of enhanced elasticity and tenacity, elevating the performance of these alloys above their ordinary counterparts. Unveiling the mystique behind this transformative alchemy requires more than just a discerning eye; it demands the precision of first principles calculations, grounded in the density functional theory (DFT).

This computational crucible illuminates the electronic tapestry, stability, and mechanical aptitude of these elusive compounds. As the veil of synthesis constraints lifts, ab initio calculations become the catalyst, enabling us to peer into the properties that define these alloys’ mettle. The symphony of cast iron’s mechanical performance echoes across the realm of abrasion-resistant applications. This hinged harmony pivots on the trinity of chemical composition, micro structural nuances, and crafting techniques. Fracture toughness and hardness—essential attributes—find their roots entwined with the crystalline choreography of carbides and borides (Uhlenhaut et al. 2006; Zhou et al. 2009a; Shein et al. 2006; Xie et al. 2005). Their intricate crystal structures may stand before us, but the underlying physical and chemical nuances, shrouded by fabrication limitations, persist as enigmas. In the realm of understanding, scientific pioneers have ventured forth. Shein’s voyage into the electronic and magnetic domains of X3C (where X dances with Fe, Co, Ni) unearthed the meta-stability of Fe3C and the allure of formation enthalpy. Meanwhile, Chen and colleagues, wielding empirical potentials, deciphered the stability and thermodynamics of X7C3 (Mn, Cr, Fe), while Music delved into Cr7C3’s electronic tapestry. Fe2B, a polymorph of intrigue, unveiled its unstable visage through nonequilibrium sorcery (Music et al. 2004).

Such revelations expanded further—multi-component carbides (Fe, Si)3C, (Fe, Cr)3C, Fe12Cr12W4C8B4—courted exploration. Si’s dalliance with Fe3C showcased solubility limitations, diverging from the script penned by Cr-doped Fe3C. These crystalline serenades played out under the watchful gaze of first principles calculations. In this ongoing odyssey, Xiao, and peers ventured into the mechanical choreography of X2B compounds, their insights intertwining with the grand tapestry of knowledge. And now, in this narrative, we embark upon a similar voyage, where EOSs calculations unveil the intricate dances of carbides and borides within cast iron’s realm. The mechanical properties of cast iron should consider the nano particle additive, which can greatly enhance the mechanical properties. (Xiao et al. 2011, 2010; Zhou et al. 2009b; Jang et al. 2009; Zuo and Liu 2021; Dixit et al. 2024).

The present study provided a different approach from convention existing models and the present approach is also shedding light on applicability of Equations of states. Various researchers have calculated the Gruneisen parameter from different equations of states, some of them have mentioned in the section of method of analysis. All the formulations of Gruneisen parameter have their own importance because they are applicable to various classes of materials (Srivastava et al. 2023b, c, g; Pandey et al. 2023a). Any single formulation of Gruneisen parameter cannot guarantee to be applicable on all class of materials even at high pressure range. So it become important to separate a good formulation of Gruneisen parameter from number of formulation existing, the present study provided such facility to the researchers. Various fundamental formulations to calculate Gruneisen parameter are given below…..

Slater gamma (Slater 1939) gave the formula for calculation of Gruneisen parameter which is given below……

$$\gamma = 0.5\left\{ {\frac{dK}{{dP}}} \right\} - \left( \frac{1}{6} \right)$$

Vashchenko and Zubarev derived free volume expansion for Gruneisen parameter (Srivastava et al. 2023f).

$$\gamma = \frac{{0.5\left\{ {\frac{dK}{{dP}}} \right\} - \frac{5}{6} + \frac{2}{9}\left( {\frac{P}{{K_{T} }}} \right)}}{{1 - \left( \frac{4}{3} \right)\left( {\frac{P}{{K_{T} }}} \right)}}$$

Borton and Stacey (Barton and Stacey 1985) gave the fallowing formula to calculate Gruneisen parameter…

$$\gamma = \frac{{0.5\left\{ {\frac{dK}{{dP}}} \right\} - \frac{1}{6} + \frac{2.35}{3}\left( {1 - \frac{P}{{3K_{T} }}} \right)}}{{1 - \left( \frac{4}{3} \right)\left( {\frac{P}{{K_{T} }}} \right)}}$$

2 Method of Analysis

As the general form of straight line equation is \(y = mx + c\), where m is slope of straight line from x axis. In the present analysis we have analyzed the Gruneisen parameter \(\gamma\) at high compression, so it is best suitable for us to replace y to \(\gamma\) and x to (V/V0).

Now the equation of straight line becomes

$$\gamma = m\left( {\frac{V}{{V_{0} }}} \right) + c$$
(1)

Since the graph of Gruneisen parameter to V/V0 is straight line and it passes through number of compression points. Taking only extreme points under consideration we will be able to analyze the slope of plots. Let at compression point \(\left( {\frac{V}{{V_{0} }}} \right)_{1}\) the value of Gruneisen parameter is \(\gamma_{1}\). While at compression point \(\left( {\frac{V}{{V_{0} }}} \right)_{2}\) the value of Gruneisen parameter is \(\gamma_{2}\).

Since all these points are within straight line so they will satisfy the equation of straight line.

From this we have obtained two different equation given fallowing.

$$\gamma_{1} = m\left( {\frac{V}{{V_{0} }}} \right)_{1} + c$$
(2)

and

$$\gamma_{2} = m\left( {\frac{V}{{V_{0} }}} \right)_{2} + c$$
(3)

From these two equations, we can easily obtain the value of slope of plot for Gruneisen parameter which is given by

$$m = \left[ {\frac{d\gamma }{{d\left( {\frac{V}{{V_{0} }}} \right)}}} \right] = \left[ {\frac{{\gamma_{2} - \gamma_{1} }}{{\left( {\frac{V}{{V_{0} }}} \right)_{2} - \left( {\frac{V}{{V_{0} }}} \right)_{1} }}} \right]$$
(4)

The EOS having minimum value of m will be the best suitable EOS for describing the thermo elastic properties of materials under consideration. For describing Gruneisen parameter we can obtain a straight line equation for it from Eqs. (1) and (2)

$$\gamma = \gamma_{1} + m\left[ {\left( {\frac{V}{{V_{0} }}} \right) - \left( {\frac{V}{{V_{0} }}} \right)_{1} } \right]$$
(5)

Now from Eqs. (4) and (5) we will obtain fallowing relationship.

$$\gamma = \gamma_{1} + \left[ {\frac{{\gamma_{2} - \gamma_{1} }}{{\left( {\frac{V}{{V_{0} }}} \right)_{2} - \left( {\frac{V}{{V_{0} }}} \right)_{1} }}} \right]\left[ {\left( {\frac{V}{{V_{0} }}} \right) - \left( {\frac{V}{{V_{0} }}} \right)_{1} } \right]$$
(6)

The Equations of states developed by Brennan–Stacey, Vinet–Rydberg, Birch–Murnaghan, and modified Lenard Jones are given below, respectively, by Eqs. (7–10)……..

$$\begin{gathered} P = \frac{{3K_{0} x^{ - 4} }}{{(3K_{0}{\prime} - 5)}}\left[ {\exp \left\{ {\frac{{(3K_{0}{\prime} - 5)(1 - x^{3} )}}{3}} \right\} - 1} \right] \hfill \\ \hfill \\ \end{gathered}$$
(7)
$$P = 3K{}_{0}x^{ - 2} (1 - x)\exp [\eta (1 - x)]$$
(8)
$$P = \frac{3}{2}K_{0} [x^{ - 7} - x^{ - 5} ][1 + \frac{3}{4}(K_{0}{\prime} - 4)(x^{ - 2} - 4)]\quad$$
(9)

where \(x = \left( {\frac{V}{{V{}_{0}}}} \right)^{\frac{1}{3}}\) and \(\eta = \frac{3}{2}(K^{\prime}{}_{0} - 1)\)

$$P = \left( {\frac{{K_{0} }}{n}} \right)\left( y \right)^{ - n} \left[ {y^{ - n} - 1} \right]$$
(10)

where \(n = \frac{{K_{0}{\prime} }}{3}\) and \(y = \left( {\frac{V}{{V_{0} }}} \right)\).

3 Results and Discussion

The important input parameters used in the analysis of Gruneisen parameter are given in the Table 1:

Table 1 Table for Bulk modulus and pressure derivative of bulk modulus at zero pressure
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The values of Gruneisen parameter obtained from different EOS are illustrated in the Figs. 1, 2, 3 and 4. The formulation predicted by Borton and Stacey (Barton and Stacey 1985) for Gruneisen parameter have been used in the analysis of carbides and bromides in cast irons. Four different EOSs viz. modified Lenard Jones (Sun 2005), Brennan–Stacey (Stacey et al. 1981; Pandey et al. 2023b), Vinet–Rydberg (Vinet et al. 1986; Pandey et al. 2023c), and Birch–Murnaghan EOS (Birch 1947; Pandey et al. 2023e; Srivastava et al. 2023a) have been used. The values of pressure and Gruneisen parameter at high compression obtained from different EOS are given in Tables 2, 3, 4, and 5.

Fig. 1
figure 1

Gruneisen parameter of Fe12Cr12W4C12 at high compressions

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Fig. 2
figure 2

Gruneisen parameter of Fe2B at high compressions

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Fig. 3
figure 3

Gruneisen parameter of Fe3C at high compressions

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Fig. 4
figure 4

Gruneisen parameter of Fe12Cr12W4C8B4 at high compressions

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Table 2 Calculation of Pressure and Gruneisen parameter at high compression for Fe2B
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Table 3 Calculation of pressure and Gruneisen parameter at high compression for Fe3C
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Table 4 Calculation of pressure and Gruneisen parameter at high compression for Fe12Cr12W4C12
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Table 5 Calculation of pressure and Gruneisen parameter at high compression for Fe12Cr12W4C8B4
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From Fig. 1 and Table 4, we can observe that the modified Lenard Jones EOS has minimum slope while Vinet–Rydberg EOS has maximum slope. For calculation of slope of plots we can use Eq. (4)

Slope \(\frac{{{\text{d}}\gamma }}{{d\left( {\frac{V}{{V_{0} }}} \right)}} = \left[ {\frac{{\gamma_{2} - \gamma_{1} }}{{\left( {\frac{V}{{V_{0} }}} \right)_{2} - \left( {\frac{V}{{V_{0} }}} \right)_{1} }}} \right]\).

So the slope for modified Lenard Jones EOS is \(\left[ {\frac{{{\text{d}}\gamma }}{{d\left( {\frac{V}{{V_{0} }}} \right)}}} \right]_{{M - L\quad {\text{Jones}}}} = \frac{1.882657 - 1.995}{{0.91 - 1}} = 0.80381\).

and the slope for Vinet–Rydberg EOS is \(\left[ {\frac{{{\text{d}}\gamma }}{{d\left( {\frac{V}{{V_{0} }}} \right)}}} \right]_{{{\text{Brennan}} - {\text{StaceyEOS}}}} = \frac{1.747357 - 1.955}{{0.91 - 1}} = 2.307\).

So from these calculations we can observe that the graph of Gruneisen parameter obtained from modified Lenard Jones EOS makes an angle 38.79° while an angle of 66° formed in case of Vinet–Rydberg EOS from (V/V0) axis. So from these observations we can see that the values obtained from modified Lenard Jones EOS are decreasing but very slowly in comparison with other. The modified Lenard Jones EOS is useful to calculate Gruneisen parameter up to compression range 0.91 for Fe12Cr12W4C12.

Slope \(\frac{d\gamma }{{d\left( {\frac{V}{{V_{0} }}} \right)}} = \left[ {\frac{{\gamma_{2} - \gamma_{1} }}{{\left( {\frac{V}{{V_{0} }}} \right)_{2} - \left( {\frac{V}{{V_{0} }}} \right)_{1} }}} \right]\).

So the slope for modified Lenard Jones EOS is \(\left[ {\frac{{{\text{d}}\gamma }}{{d\left( {\frac{V}{{V_{0} }}} \right)}}} \right]_{{M - L\quad {\text{Jones}}}} = \frac{1.249495 - 1.27}{{0.91 - 1}} = 0.227833\).

and the slope for Brennan–Stacey EOS is \(\left[ {\frac{{{\text{d}}\gamma }}{{d\left( {\frac{V}{{V_{0} }}} \right)}}} \right]_{{{\text{Brennan}} - {\text{StaceyEOS}}}} = \frac{1.134989 - 1.27}{{0.91 - 1}} = 1.5\)

The above mathematical analysis explains distinct characteristics among different equations of states (EOS), while calculating the slopes derived from calculations using Eq. (4), as shown in Fig. 2 and Table 2. Particularly, the modified Lenard Jones EOS displays a minimal slope, while the Brennan–Stacey EOS having the maximum slope. This observation is consistent with the graphical representation and numerical data. By applying Eq. (4) to determine the slope of the plots, it becomes evident that the graph depicting the Gruneisen parameter derived from the modified Lenard Jones EOS forms an angle of 12.80 with respect to the (V/V0) axis. Conversely, the Brennan–Stacey EOS graph forms an angle of 56°. Such findings underscore the notable difference in behavior, indicating that values obtained from the modified Lenard Jones EOS remain relatively constant at higher compression levels in comparison with other models. Further analysis employing the Birch–Murnaghan EOS demonstrates that the Gruneisen parameter graph derived from this equation forms an angle of 16.30. This suggests that the Birch–Murnaghan EOS and the modified Lenard Jones EOS perform similarly in predicting the Gruneisen parameter up to a compression range of 0.95. However, as compression increases beyond this point, the modified Lenard Jones EOS emerges as the more reliable model for accurately predicting the behavior of Fe2B at higher compression ratios.

Slope \(\frac{d\gamma }{{d\left( {\frac{V}{{V_{0} }}} \right)}} = \left[ {\frac{{\gamma_{2} - \gamma_{1} }}{{\left( {\frac{V}{{V_{0} }}} \right)_{2} - \left( {\frac{V}{{V_{0} }}} \right)_{1} }}} \right]\).

So the slope for modified Lenard Jones EOS is \(\left[ {\frac{{{\text{d}}\gamma }}{{d\left( {\frac{V}{{V_{0} }}} \right)}}} \right]_{{M - L\quad {\text{Jones}}}} = \frac{1.159798 - 1.175}{{0.91 - 1}} = 0.16891\).

and the slope for Brennan–Stacey EOS is \(\left[ {\frac{{{\text{d}}\gamma }}{{d\left( {\frac{V}{{V_{0} }}} \right)}}} \right]_{{{\text{Brennan}} - {\text{StaceyEOS}}}} = \frac{1.048555 - 1.175}{{0.91 - 1}} = 1.40\).

By analyzing Fig. 3 and Table 3, it is apparent that the modified Lenard Jones EOS has the least slope, while the Brennan–Stacey EOS exhibits the highest slope. The slope calculations, derived from Eq. (4), reveal that the Gruneisen parameter graph derived from the modified Lenard Jones EOS forms an angle of 9.50 degrees with the (V/V0) axis. Further, the Brennan–Stacey EOS graph creates an angle of 54.460 degrees with the same axis. These observations suggest that the values obtained from the modified Lenard Jones EOS remain relatively constant at higher compression levels compared to others. Further calculations for the Birch–Murnaghan EOS show that its Gruneisen parameter graph forms an angle of 14.70 degrees. This indicates that, up to a compression ratio of 0.95, both the Birch–Murnaghan EOS and the modified Lenard Jones EOS are equally effective in calculating the Gruneisen parameter. However, at higher compression ratios, the modified Lenard Jones EOS emerges as the superior choice among all EOSs for Fe3C, offering more accuracy and stability.

Slope \(\frac{{{\text{d}}\gamma }}{{d\left( {\frac{V}{{V_{0} }}} \right)}} = \left[ {\frac{{\gamma_{2} - \gamma_{1} }}{{\left( {\frac{V}{{V_{0} }}} \right)_{2} - \left( {\frac{V}{{V_{0} }}} \right)_{1} }}} \right]\).

So the slope for modified Lenard Jones EOS is \(\left[ {\frac{{{\text{d}}\gamma }}{{d\left( {\frac{V}{{V_{0} }}} \right)}}} \right]_{{M - L\quad {\text{Jones}}}} = \frac{1.178339 - 1.195}{{0.91 - 1}} = 0.18512\).

and the slope for Brennan–Stacey EOS is \(\left[ {\frac{d\gamma }{{d\left( {\frac{V}{{V_{0} }}} \right)}}} \right]_{{{\text{Brennan}} - {\text{StaceyEOS}}}} = \frac{1.066672 - 1.195}{{0.91 - 1}} = 1.42586\).

Based on Fig. 4 and Table 5, it is evident that the modified Lenard Jones equations of states (EOS) demonstrates the lowest slope, while the Brennan–Stacey EOS exhibits the highest slope. Using Eq. (4) for slope calculations of the plots, we can observe that the graph derived from the modified Lenard Jones EOS forms an angle of 10.470 degrees with the (V/V0) axis, whereas the Brennan–Stacey EOS forms an angle of 54.950 degrees. These calculations highlight that the values obtained from the modified Lenard Jones EOS remain relatively constant at higher compression levels compared to others. Similarly, upon analyzing the Birch–Murnaghan EOS, the Gruneisen parameter obtained forms an angle of 14.890 degrees. The whole story indicates that the Birch–Murnaghan EOS and the modified Lenard Jones EOS are equally effective in calculating the Gruneisen parameter up to a compression ratio of 0.95. However, as we move compression ratios, the modified Lenard Jones EOS seems as the very best choice among the EOSs considered for Fe12Cr12W4C8B4 due to its consistency and reliability.

4 Conclusions

Based on the provided results and observations, we can draw the following conclusions: Comparison of EOS Angles: The Gruneisen parameter graph obtained from the modified Lenard Jones EOS makes an angle of approximately 38.790, 12.80, 9.50, and 10.470 with the (V/V0) axis for different materials (Fe12Cr12W4C12, Fe2B, Fe3C, and Fe12Cr12W4C8B4, respectively). The Vinet–Rydberg EOS graph forms an angle of approximately 660 for Fe12Cr12W4C12. The Brennan–Stacey EOS graphs form angles of approximately 560, 54.460, and 54.950 for Fe12Cr12W4C12, Fe2B, and Fe12Cr12W4C8B4, respectively. The Birch–Murnaghan EOS graph forms angles of approximately 16.30, 14.70, and 14.890 for Fe12Cr12W4C12, Fe2B, and Fe12Cr12W4C8B4, respectively.

4.1 Behavior of Modified Lenard Jones EOS

The values obtained from the modified Lenard Jones EOS exhibit a trend of decreasing, but very slowly, as the compression increases. This is observed for different materials. The modified Lenard Jones EOS values remain almost constant at high compression levels in comparison with other EOS models (Brennan–Stacey and Birch–Murnaghan).

4.2 EOS Suitability for Gruneisen Parameter Calculation

The modified Lenard Jones EOS is stated to be useful for calculating the Gruneisen parameter up to a compression range of 0.91 for Fe12Cr12W4C12, 0.95 for Fe2B, and 0.95 for Fe3C. At higher compression ratios, the modified Lenard Jones EOS is suggested to be the best overall EOS for calculating the Gruneisen parameter for Fe2B, Fe3C and Fe12Cr12W4C8B4.

4.3 Birch–Murnaghan EOS Comparison

The Birch–Murnaghan EOS is noted to be equally useful as the modified Lenard Jones EOS in calculating the Gruneisen parameter up to a compression range of 0.95 for different materials.

4.4 High Compression Behavior

The modified Lenard Jones EOS values are observed to be almost constant at high compression, making it potentially more suitable for these conditions compared to other EOS models. In summary, the modified Lenard Jones EOS exhibits specific advantages in terms of its behavior at high compressions and its suitability for calculating the Gruneisen parameter for various materials. However, the Birch–Murnaghan EOS also performs well in certain cases. The specific choice of EOS model would depend on the material being studied and the compression range of interest (Srivastava et al. 2023d, e).