1.10.12.2.1 FeCr

Steel is of particular importance to radiation damage. Stainless steel is based on FeCr alloys, which have been observed by first principles calculation to exhibit unusual energy of solution. For small Cr concentrations, the energy of solution is negative; however, once the concentration exceeds about 10%, it changes sign. Thus, the FeCr system has a miscibility gap, but even at 0 K, there is a finite Cr concentration in the Fe-rich region. The underlying physics of this is that it is favorable for a Cr atom to dissolve in ferromagnetic Fe, provided the Cr spin is opposite to the Fe. Two adjacent Cr cannot be anti­parallel to each other and to the Fe matrix. Thus, nearby Cr atoms suffer magnetic frustration, which leads to repulsion between Cr atoms in FeCr not seen in pure Fe or pure Cr. Reproducing this effect in a potential is a challenging problem.

In early work, EAM was regarded as being inap­propriate for bcc metals (this turned out to be due to the use of rapidly decaying functions). The original FS functional form stabilized bcc elements, but they were unable to obtain a good fit for the elastic constants in Fe and Cr without introducing further parameters.

The two-band model can be applied to the FeCr system46 by assuming that the material can be treated as ferromagnetic, and using s and d as the two bands. They adopted the functional form of the interactions from the iron potential by Ackland and Mendelev, scaling the Cr electron density by the ratio of the atomic numbers 24/26. The CrCr potential was refitted to elastic and point defect properties. As the previous Fe parameterization incorporated with effect of s-electrons in a single embedding function, the so — called s-band density ofthis model in fact depends only on the FeCr cross potential. It described the excess energy of alloying by a many-body rather than pairwise additive effect. By choosing values which favor Fe atoms with a single Cr neighbor, this potential gives the skew solubility. This is an ingenious solution: magnetic frustration is essentially a 2 + N-body effect. Cr atoms repel when in an Fe rich ferromagnetic environment; this is neatly captured by the long — ranged Slater orbital used for the s-electron. It is
debatable whether this term is really capturing phys­ics associated with the s-band.

A related approach47 created a potential in which the embedding function depends directly on the local Cr concentration. The skew embedding function readily reproduced the phase diagram, which was the intention of the work. However, the short-range ordering and the Cr-Cr repulsion which appears to underlie the physics of radiation damage are less well reproduced.

1.10.12.2.2 FeC

Carbon dissolves readily in iron, producing a strengthening effect that underlies all steel. The physics of this is rather complex: the solution energy is very high (6 eV), and carbon adopts an interstitial position in bcc Fe with a barrier of 0.9 eV to migration. It is attracted to tensile regions of the crystal and to vacancies. It is repelled from compressive regions, including interstitial atoms, although the asymmetry of the interstitial means there are some tensile sites at larger distances which are favorable. First principles calculation also shows that the carbon forms covalently bonded pairs in a vacancy site, and the energy gained from the bond more than compensated for the reduced space available to the second carbon atom. These criteria prove rather demanding for parameterizing FeC potentials, even though they only cover composi­tions with vey low carbon concentrations.

An early pair potential by Johnson48 proved extremely successful, and it was only once first prin­ciples calculation revealed the repulsion between C and interstitials that a major problem was revealed. Although interstitial atoms are specific to radiation damage applications, there is a strong implication that the binding to other overcoordinated regions such as dislocation cores may be wrong.

It appears to be very difficult to obtain the correct bonding of carbon in all the cases above with smooth EAM-type functions. Even in recent potentials,49 like those by Johnson, carbon binds chemically to the interstitial.

There is a qualitative explanation for this. Elec­tronic structure calculation50 shows that the electrons pile up between the two nearest neighbors in the octa­hedral configuration, essentially forming two FeC bonds. However, all the simple potentials described above obtain similar bonding from all six neighbors, stabilizing the octahedral site because the tetrahedral site has only four neighbors. This approach favors carbon bonding to highly coordinated defects, and underlies the bonding to interstitials. An EAM potential with a Tersoff-Brenner style saturation in the C cohesion has addressed this problem.51 This is tuned to saturate at two near neighbors, and so favors the octahedral site but not overcoordination. As a con­sequence, it does not bind carbon to the interstitial.

1.10.12.1 Austenitic Steel

Few potentials exist for fcc iron. Calculating high — temperature phase transitions is a subtle process invol­ving careful calculation of free energy differences, which makes it difficult to incorporate in the fitting process. Although the bcc-fcc transition has been reported for one EAM iron potential,52 it is probably fortuitous and has been disputed.53 In any case, it is at far higher temperature than experimentally observed. Worse, it is likely that magnetic entropy plays a significant role,54 and the magnetic degrees of free­dom are seldom included in potentials.

Some very recent progress has been made; an analytic bond order potential53 shows bcc-fcc-bcc transitions for iron and an MEAM parameterization by Baskes successfully reproduces the bcc-fcc-bcc phase transitions in iron on heating by using temper­ature-dependent parameters. It seems certain that the challenge of austenitic steel will be receiving more attention in the next few years.