US20040047758A1 - Method for enhancement of grain boundary cohesion in crystalline materials and compositions of matter therefor - Google Patents

Abstract

A method for identifying additive components for polycrystalline metals and materials that enhance grain boundary cohesion and compositions of such materials comprises calculation of an empirical value Δ

_B ^Awhich is dependent upon a summation of various energy values associated with the matrix and additives and identifying the additives having a negative value. Formulations of alloys having improved physical properties and their processing steps are also disclosed.

Description

CROSS REFERENCE TO RELATED APPLICATIONS
• This is a continuation in part utility application based upon pending utility application Ser. No. 09/755,821 filed Jan. 5, 2001 which is based upon provisional application Ser. No. 60/174,454, filed Jan. 5, 2000, entitled, “Method for Enhancement of Grain Boundary Cohesion in Crystalline Materials and Compositions of Matter Therefor”, and is also based upon pending provisional application Ser. No. 60/367,293 filed Mar. 25, 2002 entitled “Ultra High Strength Steels with Enhanced Grain Boundary Cohesion”, all of which are incorporated herewith by reference. [0001]
• [0002] This development was supported by the Office of Naval Research (Grant No. N00014-94-1-0188) and a grant of Cray-T90 computer time at San Diego Supercomputing Center and Cray-J90 computer time at the Artic Region Supercomputing Center.
BACKGROUND OF THE INVENTION
• Grain boundary cohesion is often the controlling factor in limiting the ductility of high strength metallic alloys. Understanding the influence of the transition metal alloying additions is of great importance in predicting and controlling grain boundary embrittlement (GBE) in metallic alloys since the complexity of GBE behavior is often correlated with the presence of substitutional alloying elements. Attempts to qualitatively explain GBE on the electron-atom level go back to the 1960's. Quantitatively, alloy designers, on the one hand, have built various atomistic theories such as thermodynamic and semi-empirical pair bonding models to understand and predict the influence of segregants on the mechanical properties of the grain boundary (GB). Electronic structure theorists, on the other hand, have employed both empirical and first-principles quantum-mechanical methods in their calculations to predict the influence of segregants. While the atomistic treatments have traditionally served as the starting point for materials design, the electronic level explorations, which are numerically more accurate, are more demanding in their complexity to predict precise effects in advanced materials. [0003]
• In spite of the complexity of the mechanical behavior and GB atomic structures, general trends in certain mechanical properties can be correlated with specific features of the electronic structure. First-principles computation have proved to be an accurate and powerful tool in attacking the problem of the mechanical properties of real materials such as GBE. An inherent advantage of the first-principles electronic theory is that it is independent of any adjustable perimeters. Therefore, the numerical results of first-principles calculations formed a basis or starting point for more advanced theories. Without such theories, one has to repeat the full procedure of calculations to predict quantitatively the effect of any substitutional element even on the same GB. In this case, the exact mechanism by which segregation elements cause embrittlement remains unclear, and in any event the desirability of any expedient way to predict the qualitative impact of segregants is deemed especially desirable for the custom design of alloys having desired properties such as workability. [0004]
• This work involves a new ultra-high strength, tough steel with enhanced stress corrosion cracking resistance using a systems approach to materials design. Special emphasis is given, in this design, to the integration of quantum mechanical principles for the enhancement of grain boundary cohesion. [0005]
• More specifically, two ultra-high strength steels are disclosed having enhanced grain boundary cohesion. Requirements for a transformation toughened alloy were found to be contrary to those for grain boundary cohesion enhancement. Therefore, transformation toughening was not incorporated in the disclosed steels and processing. One alloy, Fe-15Co-6Ni-3Cr-1.7Mo-2W-0.25C designated alloy QSW, is considered a commercially viable alloy. The second alloy, Fe-15Co-5Ni-3Cr-2.7Re-1.2W-0.18C designated alloy QSRe involves the effect of rhenium and tungsten, reducing site competition by eliminating molybdenum from the design. Both alloys are fully martensitic, M[0006] ₂C carbide strengthened without any precipitated austenite.
SUMMARY OF THE INVENTION
• Starting from first-principles quantum mechanical calculations on the strengthening and embrittling effects of alloying metals on the grain boundary cohesion of a crystalline matrix metal, there has been developed a method to predict quantitatively the effect of a substitutional alloying addition on grain boundary cohesion of the matrix metal or material. Thus, with the practice of the invention, using the bulk properties of the alloying material and the matrix material or metal utilized as input information, the mechanical behavior of a substitutional metallic element (A) near the grain boundary can be predicted without conduct of the first-principles calculations once the atomic structure of the corresponding clean grain boundary is determined. This model differs from the thermodynamic and atomistic theories in that it is not only based on first-principles and therefore displays an electronic level understanding of the grain boundary embrittlement, but also in that it takes the grain boundary volume expansion into account and hence yields a more precise treatment and result. Examples include the effect of Ru, W, and Re alloying metals on the Fe grain boundary and Ca alloying metal on the Ni grain boundary. Cohesion effects predicted by the method are confirmed by rigorous first-principles quantum-mechanical calculations. [0007]
• Thus, there has been developed a method and compositions of matter wherein grain boundary cohesion is predicted as directly enhanced for iron (Fe) and nickel (Ni) alloys by the addition of specific additive materials including tungsten, rhenium, osmium, niobium, iridium, technetium, molybdenum, ruthenium, platinum, tantalum, zirconium, hafnium, vanadium, and titanium in amounts which segregate to the grain boundaries as a result of alloy processing. Such additive materials have a cohesive effect regardless of other additive materials which, for example, may foster or may inhibit cohesion. Such cohesive enhancing materials may thus be used alone or in combination. Favored cohesion enhancing additives for iron (Fe) include tungsten, rhenium, niobium, and osmium. Specific iron (Fe) based examples are disclosed. Favored cohesion enhancing elements for nickel include molybdenum, rhenium and ruthenium. [0008]
• The method of the invention provides a means by which the embrittlement or cohesion potency of a substitutional atom in a grain boundary can be predicted without carrying first-principles calculations. The method is capable of prediction without solving the quantum-mechanical Schrodinger equation and makes use of atomic, or bulk quantities as inputs. This method has an electronic, rather than an atomistic or thermodynamic basis, in order to yield a quantum description of the mechanical behavior of a substitutional element near the grain boundary. [0009]
• Additionally, the method is considered applicable to any crystalline system including those comprised of metals and/or ceramics, wherein alloying elements or other substitute components are added to the matrix or base crystalline material in the polycrystalline matrix. Thus, the described method and the resulting compositions of matter and processing methods, though described in the context of certain specific metals and alloying elements, i.e., iron (Fe), nickel (Ni) and transition metal alloying elements as set forth in Tables 1 and 2, contemplates applicability in any crystalline material or composition. [0010]
• In general, therefore, as one feature of the invention, a methodology is disclosed which comprises a formula or protocol for efficiently predicting the desirable alloying elements (and their quantitative effect) for enhancement of grain boundary cohesion in crystalline materials, including metals and ceramics for structural and electronic applications. A first step undertaken in the development of the method is a rigorous quantum-mechanical (e.g. FLAPW) (full potential linearized augmented plane wave (FLAPW) method calculation of a high free volume (to be representative of weakest boundaries) and high symmetry (for calculational speed) grain boundary to define fundamental boundary parameters such as free volume. This is then applied in an approximate general model of the mechanical and chemical contributions to the effect of all alloying elements on boundary cohesion based on known properties of pure elements. The quantitative effect of predicted cohesion enhancers is then determined or verified by a pair of rigorous quantum-mechanical (e.g., FLAPW) calculations comparing the energy of segregation from crystal to boundary determining the efficiency with which the element can be specified for cohesion enhancement. [0011]
• The alloying element in the compositions proposed is thus substituted in the crystalline matrix at the grain boundaries. The method of the invention, as generally described above and for which various examples are provided hereinafter, determines the identity of alloying or additive materials which will enhance grain boundary cohesion. Various techniques may be utilized to deliver the alloying component to the grain boundaries or, in other words, cause the alloy or additive component to segregate to the grain boundaries. For example, low temperature heat treatment for various metals is a useful technique to effect such segregation and thereby enhance cohesion. The methodology used for delivery and the quantities of additive to be delivered to the grain boundaries may be within the scope of general materials knowledge or can be determined for specific conditions by limited testing or experimentation. However, the identity of such components is a result of the method of the invention, and the practice of the invention contemplates the identity and processing materials that result from the practice of the method. [0012]
• It is noted that the examples herein are directed to binary systems. However, the methodology is considered applicable to more complex systems and to systems wherein the additive material is counteracted by other additives, e.g., a cohesion enhancer is counteracted by hydrogen (H) embrittlement. Additionally, the amount of additive component or the range of the amount of additive component may be derived using techniques disclosed. [0013]
• The reference citations set forth in this application are incorporated by reference. [0014]
BRIEF DESCRIPTION OF THE DRAWING
• In the description, reference will be made to the drawing comprised of the following figures: [0015]
• FIG. 1 (FS) is a schematic representation of the atomistic array of a crystalline matrix material wherein the free surface energy is to be defined for iron; [0016]
• FIG. 1 (GB) is a schematic view representation of the atomistic array of an alloy material at a grain boundary in a matrix material for iron; [0017]
• FIG. 2 (FS) is analogous to FIG. 1 (FS) but is representative of nickel; [0018]
• FIG. 2 (GB) is analogous to FIG. 1 (GB) but is representative of nickel; [0019]
• FIG. 3 is a graph of the empirical calculation results using the method of the invention for iron and various additives; and [0020]
• FIG. 4 is a graph of the empirical calculation results using the method of the invention for nickel and various additives. [0021]
DESCRIPTION OF PREFERRED EMBODIMENTS
• Grain Boundary Cohesion Considerations [0022]
• A. Atomic Structure and Volume Effect [0023]
• As is the case of alloys and solid solutions, the volume effect of a segregant is of great importance for understanding many of its physical and mechanical properties near the GB. As a first-principles treatment, one has to fully relax the atoms near the GB. [0024]
• To address the problem of volume mismatch, the atomic size of the segregant and also the size of the GB hole should be well defined. Unfortunately, the geometric size of an atom has no absolute meaning and its definition depends on the physical, chemical, or mechanical problem under consideration. The problem of defining the size of an atom in metallic solid solutions has been discussed in detail by King, “AIME Symposium of the Alloying Behavior and effects in Concentrated Solid Solutions” (Gordon and Brach, New York 1965) in the study of substititional solid solutions, and by de Boer et al. “Cohesion in Metals” (North Holland, N.Y. 1988) in the study of energy effects in transition metal alloys. One of these definitions uses the atomic volume in the structure of the elemental crystals. In the case of metallic solid solutions, this definition is applicable when Vegard's law (i.e., the unit-cell volume equals the sum of elemental atomic-cell volumes) is observed. However, in the problem of GB segregation, there is no requirement to keep the bulk lattice symmetry. Therefore, such a definition of the atomic size is free from the limitation of Vegard's law. The atomic volume (VA) of each selected element in its elemental crystal is listed in Column 6, Tables 1 and 2. [0025]

  TABLE 1
  Model calculated embrittling effect ΔEB A (eV) for all of the transition
  elements on the Fe Σ3 grain boundary cohesion. Also listed are elemental
  cohesive energies $E_{\mathrm{Coh}}^A\mathrm{and}{\mathrm{\Delta E}}_{\mathrm{Coh}}^A\left(\mathrm{eV}\right),$ formation heat of alloy $\mathrm{AFe}{\Delta }_{\mathrm{Heat}}^A,$ atomic
  volumes VA (a.u.3), bulk moduli KA (×1011 N/m2), volume mismatch
  correction δΔ Ev and also the work needed to change the ground fcc
  (hcp is approximated by fcc) structure to bcc structure for an element.
  ${\mathrm{\Delta E}}_B^A=1\text{/}3\left({\mathrm{\Delta E}}_{\mathrm{Coh}}^A+{{\mathrm{\Delta E}}^A}_{\mathrm{Stru}}+{\mathrm{\Delta E}}_{\mathrm{Heat}}^A\right)+{\mathrm{\Delta E}}_V^A.$ ${\mathrm{\Delta E}}_{\mathrm{Coh}}^A=E_{\mathrm{Coh}}^A-E_{\mathrm{Coh}}^{\mathrm{Fe}}.$ GFe = 0.816 × 1011 N/m2.

  At-
  om	$-E_{\mathrm{Coh}}^A$	${\mathrm{\Delta E}}_{\mathrm{Coh}}^A$	${\mathrm{\Delta E}}_{\mathrm{Heat}}^A$	${{\mathrm{\Delta E}}^A}_{\mathrm{Stru}}$	VA	KA	ΔEv	ΔEB A

  Li	1.63	2.66	0.94	0.00	143.58	0.116	−0.04	1.16
  Be	3.32	0.97	−0.20	0.03	55.77	1.003	0.44	0.71
  Na	1.11	3.18	2.75	0.00	254.46	0.068	0.18	2.16
  Mg	1.51	2.78	0.67	0.01	156.93	0.354	0.19	1.34
  Al	3.39	0.90	−0.91	0.12	112.09	0.722	−0.07	−0.03
  K	0.93	3.36	4.80	0.00	481.33	0.032	0.33	3.05
  Ca	1.84	2.45	1.28	0.01	293.40	0.152	0.76	2.01
  Sc	3.93	0.36	−0.52	0.04	158.04	0.435	0.26	0.22
  Ti	4.86	−0.57	−0.74	0.02	119.22	1.051	−0.01	−0.44
  V	5.30	−1.01	−0.29	0.00	93.46	1.619	−0.11	−0.54
  Cr	4.10	0.19	−0.06	0.00	81.01	1.901	−0.02	0.02
  Mn	2.98	1.31	0.01	*	82.49	0.596	−0.07	0.37
  Fe	4.29	0.00	0.00	0.00	79.39	1.683	0.00	0.00
  Co	4.39	−0.10	−0.02	0.20	75.23	1.914	0.07	0.10
  Ni	4.44	−0.15	−0.06	0.06	73.83	1.86	0.09	0.04
  Cu	3.50	0.79	0.50	0.01	79.86	1.37	−0.01	0.42
  Zn	1.35	2.94	−0.14	0.07	103.02	0.598	−0.11	0.85
  Rb	0.85	3.44	4.75	0.00	587.83	0.031	0.47	3.20
  Sr	1.72	2.57	1.90	0.00	379.12	0.116	0.97	2.46
  Y	4.39	−0.10	−0.06	0.09	223.45	0.366	0.92	0.90
  Zr	6.32	−2.03	−1.17	0.01	157.30	0.833	0.47	−0.59
  Nb	7.47	−3.18	−0.70	0.00	121.37	1.702	0.05	−1.24
  Mo	6.81	−2.52	−0.09	0.00	105.11	2.725	−0.09	−0.96
  Tc	6.85	−2.56	−0.13	0.19	95.85	2.97	−0.11	−0.94
  Ru	6.62	−2.33	−0.20	0.53	91.68	3.208	−0.10	−0.77
  Rh	5.75	−1.46	−0.23	0.36	92.95	2.704	−0.10	−0.54
  Pd	3.94	0.35	−0.19	0.08	99.24	1.808	−0.11	−0.03
  Ag	2.96	1.33	1.23	0.00	115.35	1.007	−0.04	0.81
  Cd	1.16	3.13	0.42	0.04	145.43	0.467	0.15	1.35
  Cs	0.83	3.46	5.21	0.00	745.67	0.020	0.40	3.29
  Ba	1.86	2.43	2.12	0.00	421.77	0.103	1.04	2.56
  La	4.49	−0.20	0.25	0.11	249.93	0.243	0.84	0.89
  Hf	6.35	−2.06	−0.98	0.10	149.29	1.09	0.43	−0.55
  Ta	8.09	−3.80	−0.67	0.00	148.31	2.00	0.61	−0.88
  W	8.66	−4.37	0.00	0.00	107.11	3.232	−0.08	−1.54
  Re	8.10	−3.81	−0.01	0.27	99.24	3.72	−0.11	−1.29
  Os	8.10	−3.81	−0.17	0.85	94.51	4.18	−0.11	−1.15
  Ir	6.93	−2.64	−0.38	0.64	95.58	3.55	−0.11	−0.91
  Pt	5.85	−1.56	−0.59	0.16	101.93	2.783	−0.11	−0.77
  Au	3.78	0.51	0.37	0.00	114.37	1.732	−0.03	0.26
  Hg	0.69	3.60	0.69	*	158.41	0.382	0.22	1.65
  Tl	1.87	2.42	1.06	0.00	192.80	0.359	0.55	1.71
  Pb	2.04	2.25	0.95	0.00	204.49	0.430	0.81	1.88

• [0026]

  TABLE 2
  Model calculated embrittling effect ΔEB A (eV) for all of the transition
  elements on the NiΣ5 grain boundary cohesion. Also listed are elemental
  cohesive energies $E_{\mathrm{Coh}}^A\mathrm{and}{\mathrm{\Delta E}}_{\mathrm{Coh}}^A\left(\mathrm{eV}\right),$ formation heat of alloy
  $\mathrm{ANi}{\mathrm{\Delta E}}_{\mathrm{Heat}}^A,$ atomic volumes VA (a.u.3), bulk moduli KA (×1011 N/m2),
  volume mismatch correction δΔ Ev and also the work needed to change
  the ground bcc structure to fcc structure for an element.
  ${\mathrm{\Delta E}}_B^A=1\text{/}3\left({\mathrm{\Delta E}}_{\mathrm{Coh}}^A+{{\mathrm{\Delta E}}^A}_{\mathrm{Stru}}+{\mathrm{\Delta E}}_{\mathrm{Heat}}^A\right)+{\mathrm{\Delta E}}_V^A.$ ${\mathrm{\Delta E}}_{\mathrm{Coh}}^A=E_{\mathrm{Coh}}^A-E_{\mathrm{Coh}}^{\mathrm{Ni}}.$
  GNi = 0.839 × 1011 N/m2.

  At-
  om	$-E_{\mathrm{Coh}}^A$	${\mathrm{\Delta E}}_{\mathrm{Coh}}^A$	${\mathrm{\Delta E}}_{\mathrm{Heat}}^A$	${{\mathrm{\Delta E}}^A}_{\mathrm{Stru}}$	VA	KA	ΔEv	ΔEB A

  Li	1.63	2.81	0.03	0.02	143.58	0.006	−0.04	0.99
  Be	3.32	1.12	−0.22	0.00	55.77	1.003	0.27	0.57
  Na	1.11	3.33	1.40	0.01	254.46	0.068	0.27	1.85
  Mg	1.51	2.93	−0.25	0.00	156.93	0.354	0.36	1.25
  Al	3.39	1.05	−1.39	0.00	112.09	0.722	0.06	−0.05
  K	0.93	3.51	2.35	0.00	481.33	0.032	0.40	2.35
  Ca	1.84	2.60	−0.37	0.00	293.40	0.152	0.91	1.65
  Sc	3.93	0.51	−1.79	0.00	158.04	0.435	0.46	0.03
  Ti	4.86	−0.42	−1.54	0.00	119.22	1.051	0.18	−0.47
  V	5.30	−0.86	−0.75	0.17	93.46	1.619	−0.05	−0.53
  Cr	4.10	0.34	−0.27	0.39	81.01	1.901	−0.05	0.10
  Mn	2.98	1.46	−0.33	0.00	82.49	0.596	−0.06	0.32
  Fe	4.29	0.15	−0.06	0.20	79.39	1.683	−0.05	0.05
  Co	4.39	0.05	−0.01	0.00	75.23	1.914	−0.01	0.00
  Ni	4.44	0.00	0.00	0.00	73.83	1.86	0.00	0.00
  Cu	3.50	0.94	0.14	0.00	79.86	1.37	−0.05	0.31
  Zn	1.35	3.09	−0.63	0.00	103.02	0.598	−0.02	0.80
  Rb	0.85	3.59	2.59	0.01	587.83	0.031	0.54	2.60
  Sr	1.72	2.72	−0.06	0.00	379.12	0.116	1.10	1.99
  Y	4.39	0.05	−1.62	0.00	223.45	0.366	1.17	0.65
  Zr	6.32	−1.88	−1.37	0.00	157.30	0.833	0.78	−0.30
  Nb	7.47	−3.03	−1.36	0.36	121.37	1.702	0.29	−1.05
  Mo	6.81	−2.37	−0.32	0.40	105.11	2.725	0.06	−0.70
  Tc	6.85	−2.41	0.03	0.00	95.85	2.97	−0.04	−0.83
  Ru	6.62	−2.18	0.02	0.00	91.68	3.208	−0.06	−0.78
  Rh	5.75	−1.31	−0.04	0.00	92.95	2.704	−0.05	−0.50
  Pd	3.94	0.50	0.00	0.00	99.24	1.808	−0.02	0.15
  Ag	2.96	1.48	0.68	0.00	115.35	1.007	0.12	0.84
  Cd	1.16	3.28	−0.24	0.00	145.43	0.467	0.33	1.34
  Cs	0.83	3.61	2.84	0.01	745.67	0.020	0.46	2.61
  Ba	1.86	2.58	0.01	0.00	421.77	0.103	1.16	2.02
  La	4.49	−0.05	−1.46	0.00	249.93	0.243	1.03	0.53
  Hf	6.35	−1.91	−2.04	0.00	149.29	1.09	0.76	−0.56
  Ta	8.09	−3.65	−1.33	0.19	148.31	2.00	1.07	−0.53
  W	8.66	−4.22	−0.14	0.52	107.11	3.232	0.10	−1.18
  Re	8.10	−3.66	0.10	0.00	99.24	3.72	0.00	−1.19
  Os	8.10	−3.66	0.06	0.00	94.51	4.18	−0.04	−1.24
  Ir	6.93	−2.49	−0.07	0.00	95.58	3.55	−0.04	−0.89
  Pt	5.85	−1.41	−0.22	0.00	101.93	2.783	0.02	−0.52
  Au	3.78	0.66	0.33	0.00	114.37	1.732	0.17	0.50
  Hg	0.69	3.75	0.04	0.00	158.41	0.382	0.41	1.67
  Tl	1.87	2.57	0.14	0.00	192.80	0.359	0.77	1.67
  Pb	2.04	2.40	0.08	0.00	204.49	0.430	1.08	1.91

• Volume expansion is an important property in the atomic structure of the GB. Its contribution to the space available for the substitutional element has a significant influence on the physical and mechanical behavior of this element. The existence of GB volume expansion makes the problem of the volume effect in the GB different from those in alloys and solid solutions, where the otherwise undisturbed atomic structure is perfect. A local measure for the GB expansion is the relative normal displacement of the two nearest atomic planes (M(2)) across the boundary plane. This gauge was adopted, e.g., by Lu et al. Phys. Rev. B59, 891 (1999), in Ni[0027] ₃AlΣ5 (210), and by Chen et al. J. Mater Reg. 5,955 (1990) in pure Ni and Al 5 (210 studies).
• This definition, however, is not appropriate for the investigation of the volume mismatch between the substitutional element and the GB hole, as the GB local environment is determined by not only M(2) but also by M(3) and M(4). The volume of the GB hole, V[0028] ^GB, can be taken as the summation of the displacements of M(2), M(3), and M(4). The calculated V^GB, for the Fe Σ3 (111) GB is 27.4 a.u., for the Ni Σ5 (210), it is 18.6 a.u.
• As pointed out by Geng et al., Phys. Ref. B62, 6208 (2000), not all the expanded volume near the GB is of the standard type available for the GB core atom. The GB core atom A/M (1) can form bonds only with M (2) and M (4), but not with M (3). This means that, near Fe Σ3 GB only two-thirds of the GB hole is available for A/M (1) and near the Ni Σ5 GB only 70% is available. [0029]
• Now the volume mismatch Δ[0030]

  

  ^A between a segregant and the GB can be defined,
• Δ

  

  ^A ≡

  

  ^A−(

  

  ^M +∝

  

  ^GB)
• where ∝ is 67% for Fe Σ3 and 70% for Ni Σ5. [0031]
• To address the volume effect of a substitutional addition A, one should compare the elastic energy of the clean and segregated GB. This elastic energy is associated with the volume mismatch between A/M (1) and the GB hole. Since the crystal lattice near the GB is not perfect even without the volume mismatch between A/M (1) and the GB hole, the bulk modulus and shear modulus (of M) near the GB no longer retain the perfect bulk values and are not well defined. As an approximation, these deviations are therefore neglected in the method. The elastic energy is then calculated in the framework of elasticity theory with the sphere and hole model developed by Eshelby and Friedel. In this model, a spherical hole with volume V[0032] ^M (atomic cell of metal M) in the matrix is partly filled by a sphere of another metal with volume V^A, (dissolved atomic cell). The remaining volume, (V^M−V^A), will disappear by elastic deformation of matrix and inclusion. If this is a state of purely internal stress, the total volume is unaffected. Both inclusion and hole are then subjected to a uniform hydrostatic pressure. The pressure on the inclusion is related to its bulk modulus K_A, while that on the hole is related to an effective bulk modulus equal to {fraction (4/3)} times the shear modulus of the matrix, G_M.
• The elastic energy yields [0033] $E_V^A=\frac{{K_A\left(\Delta V^A\right)}^2}{2V^A}+\frac{2{G^M\left(\Delta V^M\right)}^2}{3V^M}$

  
• Where ΔV[0034] ^A and ΔV^M are the volume changes of sphere and hole due to the internal stress. The pressures are adjusted such that they are continuous across the interface between matrix and inclusion, which leads to an expression for the elastic energy per mole of solute metal: $\stackrel{\stackrel{\_}{\_}}{E_V^A=\frac{2K_A{G_M\left(V^M-V^A\right)}^2}{3K_AV^M+4G_BV^A}}$

  
• In the GB environment, the hole cut from the matrix has a volume V[0035] ^M+V^GB rather than V^M. Therefore, to describe the GB case E_V ^A should take the form $\stackrel{\stackrel{\_}{\_}}{E_V^A=\frac{2K_AG_M\left(V^A-V^M-\frac{2}{3}V^{\mathrm{GB}}\right)}{3K_AV^M+4G_MV^A}}$

  
• The volume effect of a substantial addition A, is therefore[0036]
• ΔE _V ^A =E _V ^A −E _V ^M
• The calculated [0037] $\Delta E\frac{A}{V}$

  
• values for each alloying addition are listed in Column 8, Tables 1 and 2. [0038]
• B. Electronic Structure and Bonding Characters [0039]
• The other factor even more important in general, in determining the behavior of a segregant in the GB is its bonding character in both the GB and free surface (FS) environments. At the electronic level, a quantitative description of the chemical bonding will generally employ the concepts of change transfer, or electronegativity. Nevertheless, neither charge transfer nor electronegativity is well defined in a non-elemental crystal and therefore is not considered appropriate to be built into a unified theory. The macroscopic quantity that can be a measure of the bonding capacility of an element, as adopted in the thermodynamic or atomistic theory of GBE, is the elemental cohesive energy. It is also employed in the present method. [0040]
• According to the Rice-Wang thermodynamic theory (Mat. Sci. & Eng. A107 p. 23 (1989), the potency of a segregation impurity in reducing the ‘Griffith work’ of a brittle boundary separation is a linear function of the difference in binding energies for that impurity at the GB and the FS. For a substitutional addition, the above binding energies should be binding energy differences between the segregant and the host (GB core) atom. The first-principles FS and GB (Fe Σ3) chemical energies, defined as the work needed to remove the segregant while not permitting the hose (Fe) atoms to relax, have a relation: [0041] $E_{\mathrm{Chem}}^{\mathrm{FS}}\left(A\right)\stackrel{.}{=}\frac{2}{3}E_{\mathrm{Chem}}^{\mathrm{GB}}\left(A\right).$

  
• The difference in binding energies for A at the GB and the FS, [0042] $E_{\mathrm{Chem}}^A,$

  
• is [0043] $E_{\mathrm{Chem}}^A\stackrel{.}{=}\frac{1}{3}E_{\mathrm{Chem}}^{\mathrm{GB}}\left(A\right).\mathrm{Also}:E_{\mathrm{Chem}}^{\mathrm{GB}}\left(A\right)-E_{\mathrm{Chem}}^{\mathrm{GB}}\left(M\right)\stackrel{.}{=}E_{\mathrm{Coh}}^A-E_{\mathrm{Coh}}^M.$

  
• Combining equations: [0044] $\Delta E_{\mathrm{Chem}}^A\equiv E_{\mathrm{Chem}}^A-E_{\mathrm{Chem}}^M\stackrel{.}{=}\frac{1}{3}\left(E_{\mathrm{Coh}}^A-E_{\mathrm{Coh}}^M\right).$

  
• This means that the embrittlement potency of a substantial atom in the GB of FE is about ⅓ of the cohesive energy difference between that element and the host Fe atom if the volume effect is not significant. The factor ⅓ can be understood by the {square root}{square root over (Z)} theory. The simplest expression of band character is in the second-moment approximation to the tight-binding model, in which the cohesive energy per atom varies as {square root}{square root over (Z)}, where Z is the atomic coordination which can range from one (diatomic molecule) to 12 (fcc crystal). For the segregant in the Fe Σ3 (111) GB, Z[0045]

  

  8, and for that on the Fe (111) FS, Z

  

  

  4. Hence, by applying the {square root}{square root over (Z)} rule one will get $E_{\mathrm{Chem}}^{\mathrm{FS}}\left(A\right)\stackrel{.}{=}0.71\times E_{\mathrm{Chem}}^{\mathrm{GB}}\left(A\right).$

  
• “[0046]

  

  ”, rather than “=”, is used because in both FS and GB systems the bond lengths of M(1)−M(n) (n=2, 3, 4) differ from the bulk values. Taking the contributions from the volume effect and the bonding characters together, the embrittling effect, ΔE_B ^A, of an alloying addition A is $\Delta E_B^A=\frac{1}{3}\left(E_{\mathrm{Coh}}^A-E_{\mathrm{Coh}}^M\right)+\Delta E_V^A.$

  
• The heat of formation of metallic binary alloys must also be considered. The heat of formation for alloys can be viewed as a chemical shift of the bonding capability of the solute atoms. Comprehensive thermodynamic data for the heat of formation of all the A (A=Mo, Ru, Pd, and Re) in M (M=Fe and Ni) alloys is desired. The existing experimental data is far less than complete and the computational effort required for first-principles determination of these quantities is significant. Thus, as an alternative, the macroscopic atom model is employed, de Boer et al. “Cohesion in Metals”, (North Holland, N.Y., 1988) to estimate the heat of formation of alloys with a specific concentration which is determined by our slab model. [0047]
• In the macroscopic atom picture, the heat of formation of an ordered alloy A in M with a concentration c[0048] _A is $\begin{array}{c}\Delta E_{\mathrm{Heat}}^A=\left(1-C_A\right)\left[1+8{\mathrm{<figure-callout\; id="2"\; label="C\; A"\; filenames="US20040047758A1-20040311-D00000.png,US20040047758A1-20040311-D00002.png"\; state="\{\{state\}\}">C</figure-callout>}}_A^{}\times {\left(1-C_A\right)}^2\right]\times \Delta {\stackrel{\_}{H}}_{\mathrm{AinM}}^{°}\\ \mathrm{Where}C_A\mathrm{is}\\ C_A=\frac{{c_A\left(V^A\right)}^2/3}{{c_A\left(V^A\right)}^2/3+\left(1-c_A\right)\times {\left(V^M\right)}^2/3}\end{array}$

  
• and {overscore (H)}°[0049] _{A i n M} is the heat of formation of A in M in infinite dilution. In our first-principles calculation, C_A is {fraction (1/23)} for A in Fe, and {fraction (1/21)} for A in Ni. The calculated values of $\Delta E_{\mathrm{Heat}}^A$

  
• within the macroscopic atom model are listed in [0050] Column 4, Tables 1 and 2.
• The last term that should appear in the above equation is the one reflecting the preference for metallic elements to crystallize in one of the main crystallographic structures, namely bcc, fcc and hcp, depending on the number of their valence electrons. We use [0051] $\Delta E_{\mathrm{Stru}}^A$

  
• to denote the total energy difference of elemental crystal A between its ground state structure and that of the host. To make all (or, as many as possible,) contributions in our model to be found from the same basis, we carried out full GGA (generalized gradient approximation) calculations for all the elements under consideration. In doing so, we approximate the hcp by the fcc structure in order to save computational effort. This approximation will introduce an error to [0052] $\Delta E_{\mathrm{Stru}}^A$

  
• yielded by FLAPW-GGA are listed in Column 5, Tables 1 and 2. [0053]
• Taking the above two corrections into account, ΔE[0054] _B ^A becomes $\Delta E_B^A=\frac{1}{3}\left(E_{\mathrm{Coh}}^A-E_{\mathrm{Coh}}^M+\Delta E_{\mathrm{Heat}}^A+E_{\mathrm{Stru}}^A\right)+\Delta E_V^A).$

  
• The embrittlement potency of each substitutional addition calculated from this model is listed in Column 9, Tables 1 and 2. For Mo and Pd in Fe, the model calculated values for ΔE[0055] _B ^A are −0.96 and −0.03 eV, respectively. The first-principles results are −0.90 and +0.08 eV, respectively, (Geng et al. Phys. Ref. B 62, 6208 (2000)).
• C. Confirmation of the Model [0056]
• In order to verify the model, first-principles calculations were performed on the effects of Ru, W and Re segregation on the cohesion of the Fe Σ3 (111) GB and Ca on the Ni Σ3 (210) GB by using the same (FLAPW) method. As sketched in FIG. 1 for the Fe Σ3 GB case and FIG. 2 for the Ni Σ5 GB case, both the FS and GB were simulated by a slab model, which minimizes the impurity-impurity interactions inherent in the use of superlattice cells. [0057]
• In the FLAPW method, no shape approximations are made to the change densities, potentials, and matrix elements. For both host and alloying additions, the core states are treated fully relativistically and the valence states are treated semi-relativistically (i.e., without spin-orbit coupling). The GCA formulas for the exchange-correlation potential are from Perdew et al., Phys. Ref. Lett, 77, 3865 (1996). [0058]
• For Ru, W, and Re on the Fe Σ3 GB, first-principles results are −0.65, −1.31 and −1.31 eV, respectively. The values calculated with the model of the invention are −0.77, −1.54 and −1.29 eV, respectively. The largest discrepancy between the first-principles and the model invention results in about 0.2 eV in the case of W, whereas for Ca in the Ni Σ5 (210) GB, the model gives +1.65 eV and the first-principles results show an embrittlement potency of +1.4 (±0.2) eV. In general, the agreement between the semi-empirical invention method and first-principles is quite good. [0059]
• In review, starting from first-principles, a semi-empirical theory quantitatively predicts the mechanical behavior of a substitutional metallic element in the grain boundary without carrying out full first-principles calculations, once the atomic structure of the clean grain boundary is determined. This model displays an electronic level of understanding of the grain boundary embrittlement. Also, it takes the grain boundary expansion into account and hence yields a more precise treatment. [0060]
• From the results, it is concluded that the strongest cohesion enhancer in the Fe Σ3 (111) GB is W, followed by Re, Nb, and Os. The strongest cohesion enhancer in the Ni Σ5 (210) GB is Os, followed by Re, W, and Nb. For both Fe based and Ni based alloys (and therefore expected for most alloys based on transition metals) the effective grain boundary cohesion enhancing alloying elements are W. Re, Os, Nb, Ir, Tc, Mo, Ru, Pt, Ta, Zr, Hf, V and Ti as suggested by FIGS. 3 and 4. Referring to FIGS. 3 and 4, those segregants having negative and positive ΔE[0061] _B ^A are plotted as determined by the invention method. Negative value segregants are direct cohesion enhancers. Positive value segregants tend to cause embrittlement.
• Table 3 summarizes the rigorous FLAPW calculations for both ΔE[0062] _GB−ΔE_FS and the crystal/grain boundary segregation energy ΔE_GB for the alloying elements Pd, Mo, Ru, Re and W in Fe, validating the model predictions and showing significant negative ΔE_GB values promoting enhanced segregation to grain boundaries.

  TABLE 3
  FLAPW Calculations of ΔEGB − ΔEFS and ΔEGB Alloying
  Elements in Fe based Alloys

  	X	ΔEGB − ΔEFS (eV/atom)	ΔEGB (eV/atom)

  	Pd	+0.08	−0.90
  	Mo	−0.90	−0.76
  	Ru	−0.65	−0.51
  	Re	−1.31	−0.49
  	W	−1.31	−0.68

• To summarize the calculated FLAPW value ΔE[0063] _GB−ΔE_FS must preferably be a negative number if the additive is to be effective as a cohesive material, and the FLAPW value ΔE_GB must also be negative since it represents the case with which the additive becomes transported to the grain boundary. When both numbers are negative in a system, then enhanced cohesion results. If one or both numbers are positive, then there is a tendency toward embrittlement.
• The method of the invention provides that a single number or value (ΔE[0064] _B ^A), if negative for an additive, represents a direct cohesion enhancer. The method employs approximations or inputs of various energy states of the additive (alloying element) and the matrix material including an energy value based upon the volume effect of the additive.
• The methodology is effective for any polycrystalline material and calculations for iron (Fe) and nickel (Ni) are born out by alternative (first-principles) calculations. The method is not limited to Fe and Ni alloys, however. Also, the method of the invention can be viewed as a process for identification of direct cohesion additives in general, followed by utilization of FLAPW calculations (See Table 3) to identify the separate ΔE[0065] _GB and ΔE_GB−ΔE_FS calculations which further identify subsets of additives that are especially effective as cohesion enhancers and will be easily employed as enhancers because they are identified as more easily diffused (e.g., tungsten).
EXAMPLES AND EXPERIMENTATION
• A. Alloy Design—General Background [0066]
• Alloys consisting of a Fe—Co—Ni martensitic matrix strengthened by a fine carbide dispersion were studied. These alloys are secondary hardening because the carbide dispersion can be changed from a coarse, soft cementite to a fine, hard carbide upon tempering at a proper temperature for a sufficient time. These alloys are generally used in applications that require high strength and toughness. Due to the environment that these alloys are typically used, resistance to hydrogen embrittlement is also needed. A flow block diagram describing the interplay between processing-structure-properties of this class of steel is shown in the following diagram: [0067]

  
• As shown in FIG. 2.[0068] 1 the Fe—Co—Ni lath martensitic matrix is controlled by the solution treatment and tempering of the material. Nickel additions are made to enhance cleavage resistance. Cobalt is necessary to maintain a dislocation network by hindering short-range order recovery. The dislocation forest is necessary to promote heterogeneous nucleation of M₂C carbides. Strength is determined by the M₂C carbide dispersion that is controlled by the tempering process. In addition, the M₂C carbides act as hydrogen traps which slow hydrogen from degrading grain boundary cohesion. Coarse carbide dispersions must be avoided because of their detrimental effect on toughness and to optimize hardness.
• Tempering can also allow an austenite dispersion to form. This dispersion will soften the alloy due to the inherent softness of face-centered cubic materials compared to martensite. However, such dispersions may be desirable because of their beneficial effects on toughness. Austenite dispersions can enhance toughness if the dispersion is present in sufficient quantity, has the proper stability, and has a large dilatation upon transformation from FCC to BCC. [0069]
• Solidification processing, hot working, and solution treatment will affect grain size. To prevent excessive grain growth during solution treatment, a grain refining dispersion is designed in the materials. A desirable grain refining dispersion is stable during solution treatment and is completely in solution before melting. Such dispersions will avoid coarse primary carbides while maintaining a fine dispersion to pin grain boundaries during solution treatment. In addition, the grain refining dispersion should have a high cohesive energy with the matrix to inhibit micro-void nucleation at the interface. [0070]
• Refining and deoxidation are processing steps to ensure grain boundary chemistry. Clean start material and clean processing will reduce tramp element impurities. In addition, small additions are added to getter any impurities found in the material. Grain boundary cohesion enhancers are added to the alloy to counteract hydrogen embrittlement. These elements segregate to the grain boundaries during solution and tempering treatments to enhance grain boundary cohesion. [0071]
• B. Martensitic Formation [0072]
• The lath martensitic microstructure has the most desirable combination of strength and toughness of any steel microstructure. Therefore, it is necessary to maintain a sufficiently high martensite start (M[0073] _S) temperature to ensure a lath martensitic structure and avoid retained austenite. The martensite transformation occurs when high temperature austenite (FCC) transforms to martensite (BCC). Minimum values for the M_S that ensure complete martensitic transformation upon quenching are approximately 200° C. As a general rule, alloying elements act to reduce the M_S temperature. Large amounts of alloying elements are needed to achieve the desired properties of ultra-high strength steels.
• Ghosh and Olson have developed a model to predict the compositional dependence of M[0074] _S temperature in steel. Ghosh and Olson have described the martensitic nucleus interface as a combination of coherency and anti-coherency dislocations that provide transformation strain and reduce the strain energy of the system, respectively. Growth of the nucleus is governed by the propagation of dislocations into the parent austenite. The motion of dislocations into the parent austenite is hindered by microstructural elements in the austenite. Microstructural obstacles can interact with the long-range stress field and the interfacial core of the martensite nucleus to hinder nucleus growth. The interactions with the martensite nucleus must be overcome by thermal activation and are, thus, highly temperature dependent. Conversely, the interactions with the long-range stress field are not temperature dependent. Therefore, the interfacial frictional work term can be described as the summation of two terms: a thermal term relating to interactions with the interfacial core and an athermal term relating to interactions with the long range stress field.
• The nucleation criterion for martensitic transformation can be described as the condition where the thermodynamic driving force of the transformation (FCC→BCC) is equal to a constant plus the interfacial frictional work term, as shown in the following equation:[0075]
• −ΔG _crit =K ₁ +W _μ(X _i)  2.1
• −ΔG[0076] _crit is equal to the chemical driving force for diffusionless transformation at a constant temperature. K₁ is a constant that accounts for strain, interfacial energy, and nucleating defect size. Wμ is the athermal interfacial frictional work term. The thermal term of the interfacial frictional work term is negligible for M_S temperatures greater than 450 K, those typically found in UHS steels. The composition dependence of Wμ is given by the following equation: $\begin{array}{cc}{W}_{\mu }=\sqrt{\sum _i^{}{\left(K_{\mu }^i{\mathrm{<figure-callout\; id="1"\; label="X\; i"\; filenames="US20040047758A1-20040311-D00000.png,US20040047758A1-20040311-D00002.png"\; state="\{\{state\}\}">X</figure-callout>}}_i^{}\right)}^2}+\sqrt{\sum _j^{}{\left(K_{\mu }^j{\mathrm{<figure-callout\; id="1"\; label="X\; j"\; filenames="US20040047758A1-20040311-D00000.png,US20040047758A1-20040311-D00002.png"\; state="\{\{state\}\}">X</figure-callout>}}_j^{}\right)}^2}+\sqrt{\sum _k^{}{\left(K_{\mu }^k{\mathrm{<figure-callout\; id="1"\; label="X\; k"\; filenames="US20040047758A1-20040311-D00000.png,US20040047758A1-20040311-D00002.png"\; state="\{\{state\}\}">X</figure-callout>}}_k^{}\right)}^2}+K_{\mu }^{\mathrm{Co}}{\mathrm{<figure-callout\; id="1"\; label="X\; Co"\; filenames="US20040047758A1-20040311-D00000.png,US20040047758A1-20040311-D00002.png"\; state="\{\{state\}\}">X</figure-callout>}}_{\mathrm{Co}}^{}& 2.2\end{array}$

  
• The Kμ terms represent values of the solid solution strengthening coefficient where i=C and N; j=Cr, Mn, Mo, Nb, Si, Ti, and V; k=Al, Cu, Ni, Re, and W. The values are separated by their relative strengths. Cobalt is treated separately because it lowers the interfacial frictional work term. [0077]
• Alloying elements will affect the M[0078] _S temperature in two ways. First, alloying additions will change the interfacial frictional work term as demonstrated above in 2.2 Secondly, alloying elements will change the chemical driving force for nucleation, as shown above in equation 2.1.
• C. Carbide Strengthening Dispersion [0079]
• The carbide dispersion in UHS steels dominates the strength of the alloy. The carbide dispersion controls plastic flow via Orowan dislocation bypass. The strength of the material is inversely proportional to the mean distance between strengthening particles. For a given particle volume fraction, finer particles will yield a smaller spacing between strengthening particles. However, if the particles are too small, dislocations will shear the particles instead of looping them. Therefore, the most desirable particle size is at the transition between particle shearing and Orowan looping. [0080]
• Jack and Jack provide a review of carbides in steel. There are four stages of it tempering in UHS steels. First stage tempering results in the formation of iron based epsilon carbides during quenching or tempering up to 200° C. Second stage tempering describes the decomposition of retained austenite. Cementite (Fe[0081] ₃C) forms during third stage tempering between 250° C. and 450° C. The fourth stage of tempering, between 450 and 700° C., results in the formation of alloy carbides (M₂C, M₂₃C₆, M₆C, and M₇C₃ where M=Cr, Mo, V, and W). The type of carbide found after stage four tempering depends on the kinetic and thermodynamic stability of the various carbides.
• The thermal and mechanical history of a UHS steel determines the nature of the carbide dispersion. The solution treatment and quench produces a super-saturated lath martensite structure. Prior to the fourth stage tempering; the carbide that forms is para-equilibrium cementite The composition of metal sites in para-equilibrium cementite is equal to the overall alloy composition because its formation occurs before metal diffusion can occur. Although this carbide is not the most thermodynamically stable, it is kinetically favorable because it only requires the diffusion of carbon. After sufficient time, para-equilibrium cementite dissolves due to formation of more energetically favorable carbides as carbide forming elements have time to diffuse. In an optimal case, all para-equilibrium cementite is dissolved at the point where peak carbide distribution is achieved. In practice, however, alloys need to be tempered beyond peak hardness to dissolve all para-equilibrium cementite. Dissolution of para-equilibrium cementite is necessary because of its detrimental effects on toughness. [0082]
• Of the carbides that form during [0083] stage 4 tempering, the M₂C is most desirable. M₂C carbides are metastable with respect to other carbides such as M₆C and M₂₃C₆. M₂C nucleates heterogeneously on dislocations, grain boundaries, lath boundaries, and martensite/cementite interfaces. This class of steel must maintain a large density of dislocations during tempering to ensure a fine dispersion of coherent carbides.
• Langer and Schwartz modeled precipitation at high supersaturation. They showed that the average particle size is always close to the critical particle size for growth or dissolution at high supersaturation. Further, they defined high supersaturation by the condition:[0084]
• δW*≦10kT  2.3
• where δW* is the work of formation of a critical nucleus and kT is the Boltzmann factor. At high supersaturation, the nucleation rate is very high causing the supersaturation to drop rapidly. Since the critical particle size is inversely proportional to the degree of supersaturation, the critical particle size increases greatly during precipitation. The decrease in supersaturation can cause smaller particles to dissolve because they are no longer larger than the critical nucleus size. The competition between nucleating particles and critical nucleus size inhibits growth of the particle distribution. The average particle size is initially governed by the nucleation process and smoothly transitions to a regime governed by coarsening. The growth stage is bypassed because all the supersaturation is consumed during nucleation and coarsening. This analysis shows the critical particle size and coarsening rate as the parameters that control particle size and, thus, strength. The critical particle size varies directly with particle/matrix interfacial energy and inversely with thermodynamic driving force. [0085]
• Montgomery, Olson and coworkers have completed a comprehensive study of M[0086] ₂C carbide precipitation in AF1410, a commercially available Co—Ni UHS steel. The alloy was austenized at 830° C. for one hour and tempered at 510° C. The following graphs (2.2) present the results from the AF 1410 studies.

  
• The top graph shows the M[0087] ₂C carbide particle size represented as the diameter of a sphere of equal volume to the rod-shaped carbides. The second plot shows that the aspect ratio of the carbides increase from 2 to 4 as the carbides evolve from nucleation to coarsening. The third plot shows the evolution of particle number density and volume fraction. The plot shows a smooth transition from nucleation to limited growth to coarsening. The number density remains constant from approximately 0.5 to 1.0 hr showing that nucleation has been completed and the particles are growing. From 1.0 hr to 2.0, the number density increases corresponding to re-nucleation. After re-nucleation, the carbide size increases due to coarsening. Volume fraction data show that precipitation is nearly complete after 10 hours. The fourth plot shows measured lattice parameters of the carbides. Coherency of the M₂C particles is maintained during early stages of tempering by carbide composition shifts to reduce the lattice parameter. The particles are coherent with the matrix up to about 10 hours and incoherent after 100 hours. The final plot shows the evolution of hardness. The material is hardest after 0.5 hours, corresponding to an equivalent particle size of approximately 3 nm.
• Similar studies have been conducted on AerMet100. Aermet100 is a commercially available Co—Ni UHS steel which has superior properties to AF1410. Yoo, et al. have shown the aspect ratio of M[0088] ₂C carbides to be nearly constant at 3:1 during tempering in AerMet100, as seen in the following graph 2.3.

  
• Results from a SANS study of the AerMet100 carbide dispersion, conducted by Jemian are shown in the following graphs 2.4. [0089]

  
• The alloy was tempered at 485° C. up to 20 hours. The results show similar behavior as those seen in the AF1410 study. One notable exception is the absence of a re-nucleation stage. The particle distribution is much finer than that found in AF1410 due to chemistry changes and lower tempering temperature. In addition, the carbide volume fraction is greater in AerMet100 compared to AF1410, accounting for the increase in strength in AerMet100. [0090]
• In order to accurately predict carbide evolution, a multi-component coarsening model is needed to predict tempering response. Lee has expanded the classical “LSW” binary coarsening theory to treat multi-component coarsening of shape preserving particles in a dilute solution matrix. The form of the rate equation has the same form as that from classical LSW theory, given in the following equation 2.4. [0091] $\begin{array}{cc}{\stackrel{\_}{R}}^3\left(t\right)-{\stackrel{\_}{R}}^3\left(0\right)=\frac{4}{9}K\left(t-t_0\right)& \left(2.4\right)\end{array}$

  
• The rate constant, K, is modified to account for the multi-component nature of carbide coarsening. Lee applied a model that is analogous to electrical resistance of series resistors. The inverse of the rate coefficient is equal to the sum of the inverses of the individual element rate constants, as shown in the following two equations 2.5, 2.6 [0092] $\begin{array}{cc}\frac{1}{K}=\sum _M^{}\frac{1}{K_M}& \left(2.5\right)\\ K_M=\frac{2\sigma V_m^B}{\mathrm{RT}\ln \left(2A_s\right)}\frac{D_M^{\alpha }}{\left(k_M-k_{\mathrm{Fe}}\right)\left(k_M-1\right)X_M^{\alpha }}& 2.6\end{array}$

  
• In the second equation, σ is the interfacial energy, V[0093] _m ^B is the molar volume of the dispersed phase, A_s is the particle aspect ratio, D is the bulk diffusivity of element M in BCC iron, k is the partitioning ratio of the various elements, and X is the mole fraction of element M in the matrix. In this model, the slowest diffusing element dominates the coarsening coefficient. The model is to be used as a guide to alloy design.
• Umantsev approached the problem of multi-component coarsening in non-dilute solutions using partial derivatives of chemical potentials with respect to mole fractions to derive his theory rather than equilibrium partitioning coefficients. This model addresses a minor error Lee made in assuming equal activities in both phases at equilibrium. In spite of the differences in the models, both calculate similar coarsening coefficients. [0094]
• Grain Refining Dispersion [0095]
• Small additions are added to alloys to form a carbide dispersion stable during austenization. The dispersion is necessary to pin grain boundaries and prevent excessive grain growth during solution treatment was described in general prior to this example. A small volume fraction of fine particles is optimal, as illustrated in the following graph 2.5. [0096]

  
• The graph shows the volume fraction of grain refiner needed at a particular particle size to maintain an average grain size of 10 μm. A volume fraction of large particles needed to maintain constant grain size is independent of solution temperature whereas a larger volume fraction of small particles is needed for higher solution temperatures due to thermally activated de-pinning. Additions of up to 0.04 wt. % titanium are added to form a desirable TiC grain refining dispersion. Such additions create dispersions stable at solution temperatures but soluble below the melting temperature. It is necessary to avoid primary carbide formation to ensure a fine distribution of carbides. [0097]
• E. Transformation Toughening Strategy [0098]
• Work on UHS steels has created a microstructure that is strong and tough. Improvements in composition and processing have eliminated previous fracture modes. For instance, brittle cleavage fracture has been inhibited by the addition of nickel. As one fracture mode is eliminated through progress, another fracture mode dominates and limits the uses of the material. [0099]
• Low toughness ductile fracture modes have been seen in nickel containing UHS steels. Garrison has explored the role of primary inclusions on primary void nucleation and coalescence. The resistance to primary void formation and coalescence is proportional to inclusion spacing. Therefore, it is desirable, for a given volume fraction of inclusion, to have very few, large particles. Similarly, it is desirable to reduce the volume fraction of inclusions. Clean processing and tight composition control have successfully eliminated primary void nucleation as the dominant fracture mode in UHS steels. [0100]
• Shear localization by microvoid nucleation is found to be one of the dominant fracture modes in clean steels. Microvoids nucleate on the grain refining dispersion, carbide strengthening dispersion, prior austenite structure, martensite lath and packet structure, and the dislocation structure. Studies have shown that fine particle dispersions with coherent interfaces are optimal for controlling microvoid nucleation. Inhibiting microvoid nucleation is difficult in UHS steels due to the microstructure needed to achieve other properties. The most promising microstructure modification is achieved by nucleating a transforming austenite dispersion. Certain austenite dispersions increase toughness by suppressing microvoid nucleation to higher strain levels [0101]
• Leal and Stavehaug examined the use of TRansformation Induced Plasticity (TRIP) to delay microvoid nucleation in fully austenitic steels. The flow behavior of TRIP steels is understood based on the interactions between stress and strain on the martensitic transformation. Applied stress initiates the martensitic transformation through the interaction of applied stress with the transformation strain. This mechanism is called stress-assisted transformation. A second mechanism for martensitic transformation is called strain-induced transformation. In this regime, plastic strain assists the transformation by creating heterogeneous nucleation sites at the intersections of shear bands. In the stress-assisted transformation, yielding of the material is governed by the transformation of the parent austenite and all initial plastic strain is the result of the transformation. In the strain-induced transformation, initial yield of the parent austenite is caused by slip resulting in the formation of martensite nuclei at the intersection of shear slip bands. [0102]
• Olson and Azrin have shown that the boundary between stress-assisted and strain-induced transformation can be determined by testing the temperature dependence of yield strength. The temperature at which transformation mechanism changes is typically called M[0103] _s ^σ shown qualitatively in the following graph 2.6.

  
• Below M[0104] _s ^σ the transformation-controlled yield stress rises because the stability of the parent austenite increases as temperature rises and more stress is required for transformation. Above M_s ^σ, the slip-controlled yield stress decreases because of thermal activation of dislocation motion.
• Control of the austenite stability is critical to achieving optimal transformation toughening. Leal and Stavehaug demonstrated that the optimal toughening enhancement occurs at the M[0105] _s ^σ temperature for the crack-tip stress state. Leal controlled the stability of the austenite by varying test temperatures. Stavehaug conducted similar experiments controlling austenite stability by varying composition, as seen in the following graph 2.7;

  
• The driving force for transformation at the crack-tip is compared to the driving force at M[0106] _s ^σ. Stavehaug showed that toughening enhancements up to a factor of 5, compared to non-transforming austenite, were achieved when the crack-tip driving force equaled that at M_s ^σ.
• Haidemenopolous explored the effect of austenite dispersions on toughness in two martensitic steels. He examined the effect of retained austenite in 4340 steel. No toughening enhancement was seen due to insufficient austenite stability. The M[0107] _s ^σ was estimated to be approximately 150° C. for the crack-tip stress state so no toughening was observed at room temperature tests. No tests were conducted near the M_s ^σ due to its proximity to the tempering temperature.
• Haidemenopolous studied the effects of a precipitated austenite phase in AF1410. He determined that austenite dispersions could be stabilized by composition enrichment and size refinement. ThermoCalc calculations predicted an austenite dispersion nucleated at 510° C. would have adequate stability to provide a toughening enhancement at room temperature. Contrary to expectations, no toughening enhancement was seen in AF1410 after tempering at 510° C. for 8 hours. A non-equilibrium interlath austenite was found in the material instead of the predicted austenite dispersion. The interlath austenite lacked the needed stability because of its coarse size (˜200 nm films) and composition. The composition of the interlath austenite was Fe-14Ni-13.2Co-2.8Cr-1Mo compared to the calculated equilibrium composition of Fe-38.5Ni-4.1Co-1Cr-0.25Mo. The lack of nickel in the interlath austenite is the primary reason the austenite was too unstable at room temperature. [0108]
• Haidemenopolous was successful in achieving a toughening enhancement in AF1410 by altering the heat treatment. A two-stage heat treatment consisting of a brief high temperature step (15 minutes at 600° C.) followed by normal temperature step (8 hours at 510° C.) was used. Analysis of the austenite dispersion revealed fine particles (˜20 nm) of composition Fe-29.1Ni-10.5Co-4.2Cr-1.9Mo. Toughness of the two-stage condition was 40% higher than the normal condition at equivalent strength levels. [0109]
• Haidemenopolous applied the Olson-Cohen classical heterogeneous martensitic model to describe the transformation of the austenite particles. The result of this approach is given in the following equation 2.7:[0110]
• ΔG ^Ch +W _f =−[f(1nV _p)⁻¹ +ΔG ^σ +E ^Str]  (2.7)
• ΔG[0111] ^Ch is the transformation chemical free energy and W_f is the athermal frictional work term mentioned in section 2.2. ΔG^Ch is temperature and composition dependent while W_f is composition dependent. It is important to note that W_f will vary with tempering temperature due to changes in the austenite composition. The first term on the right side of the equation accounts for the particle size, defined by particle volume V_p, effect on stability. ΔG^σ is set by the stress state and E^str is elastic strain energy per unit volume associated with the semi-coherent nucleus structure. The optimal austenite dispersion for a given set of conditions or optimum service temperature for a given austenite dispersion can be determined from this relationship.
• Kuehmann conducted a study to thermally optimize AF1410, AerMet100, and a series of experimental alloys designated MTL. Multi-step tempering treatments were performed on all alloys in an attempt to nucleate a toughness enhancing austenite dispersion. Kuehtnann was unable to achieve high toughening enhancements in AF1410 due to the poor cleanliness of the particular heat. Toughness enhancements of 15-20% were seen in overaged AerMet100. Unfortunately, the high temperature short temper overaged the carbide strengthening dispersion and resulted in unacceptable decreases in hardness. Based on these results, Kuehmann determined that a ratio of austenite/carbide kinetics must be maintained to ensure proper microstructure. Carbide kinetics were reduced in two MTL alloys providing a kinetically favorable situation for both carbide and austenite dispersions. However, tramp element impurity segregation during tempering degraded grain boundary cohesion to the point where the fracture mode changed from ductile fracture to brittle intergranular fracture. No toughening enhancement was then seen due to the change in fracture modes. [0112]
• In addition to controlling austenite stability, it is desirable to maximize the volume change of transformation from austenite to martensite. While data is limited, Young has shown that the transformation toughening efficiency is proportional to the transformation volume change to the third power, as shown in the following graph 2.8. [0113]

  
• This graph shows the toughening increment as a function of the relative volume change. The two curves represent two levels of hardness difference between austenite and martensite. The difference in the two curves shows that the toughening effect due to strain hardening is minor compared to that from the transformation volume change. Based on this result, an expression for the toughening efficiency parameter (TEP) for dispersed austenite is given in the following equation 2.8. [0114] $\begin{array}{cc}\mathrm{TEP}={V_f\left(\frac{\Delta V}{V}\right)}^3& \left(2.8\right)\end{array}$

  
• V[0115] _f is the volume fraction of austenite present and ΔV/V is the relative volume change. The relative volume change is related to the lattice parameters of the BCC and FCC phase by the following equation 2.9: $\begin{array}{cc}\frac{\Delta V}{V}=2\left[\frac{{\alpha }_{\mathrm{BCC}}}{{\alpha }_{\mathrm{FCC}}}\right]-1& \left(2.9\right)\end{array}$

  
• Models to predict the composition dependence of the FCC and BCC lattice parameters have been developed. Ghosh has developed a model using known lattice parameters to predict the BCC lattice parameter at room temperature. The following equation 2.10 gives the model based on least squares fitting and linear superposition of Fe—X (X=C, Co, Cr, Mo, and Ni) binary systems. [0116] $\begin{array}{cc}\begin{array}{c}{\alpha }_{\mathrm{BCC}}\left(\mathrm{nm}\right)=0.34906X_C+0.28258X_{\mathrm{Co}}+0.28847X_{\mathrm{Cr}}+\\ 0.28665X_{\mathrm{Fe}}+0.31417X_{\mathrm{Mo}}+0.27801X_{\mathrm{Ni}}+\\ X_{\mathrm{Co}}X_{\mathrm{Fe}}\left[0.00346-0.00269\left(X_{\mathrm{Co}}-X_{\mathrm{Fe}}\right)\right]+\\ X_{\mathrm{Cr}}X_{\mathrm{Fe}}\left[0.00024-0.00224\left(X_{\mathrm{Cr}}-X_{\mathrm{Fe}}\right)\right]+\\ 0.00757X_{\mathrm{Fe}}X_{\mathrm{Mo}}+X_{\mathrm{Fe}}X_{\mathrm{Ni}}[0.00778+\\ 0.00743\left(X_{\mathrm{Fe}}-X_{\mathrm{Ni}}\right)]\end{array}& \left(2.10\right)\end{array}$

  
• The model expresses the lattice parameter in nm as a function of the mole fraction of each element. [0117]
• Prediction of the FCC lattice parameter is more difficult due to strongly non-linear compositional dependence arising from the INVAR magnetic anomaly in FCC Fe. [0118]
• The INVAR effect is problematic because the FCC lattice parameter is non-monotonic with its maximum near the composition of the austenite particles in the Co—Ni martensitic steels of interest. The effect will reduce the transformation dilatancy of the austenite and, therefore, the transformation toughening efficiency. The INVAR effect manifests itself in abnormal thermodynamic, magnetic, and thermal expansion behavior in certain alloys. Well-studied alloying elements that cause INVAR anomalies are nickel, palladium, and platinum [0119]
• The two-gamma states model assumes that there are two distinct magnetic states of FCC Fe. The two states have different thermodynamic quantities, magnetic moments, and atomic volumes. Applying the two-gamma states description of the INVAR effect is advantageous to predicting the FCC lattice parameter. The mathematical formalism can be extended to multi-component alloys using the thermodynamic model developed by Miodownik and Hillert. In addition, many binary and ternary iron based alloys have been modeled using the two-gamma states framework [0120]
• The INVAR anomaly is described by the relative amount of iron atoms in each state. The relative amounts of iron atoms in each state vary as composition or temperature are changed. The variations in iron atom distribution result in the unconventional physical properties. While the iron atom distribution is in dynamic equilibrium, a Schottky two-level excitation model can be used to describe the relative population of each state, as shown in the following equation 2.11. [0121] $\begin{array}{cc}\frac{f_1}{{\mathrm{<figure-callout\; id="2"\; label="f"\; filenames="US20040047758A1-20040311-D00000.png,US20040047758A1-20040311-D00002.png"\; state="\{\{state\}\}">f</figure-callout>}}_2}=\frac{g_1}{{\mathrm{<figure-callout\; id="2"\; label="g"\; filenames="US20040047758A1-20040311-D00000.png,US20040047758A1-20040311-D00002.png"\; state="\{\{state\}\}">g</figure-callout>}}_2}\exp \left(\frac{-\Delta E}{kT}\right)& \left(2.11\right)\end{array}$

  
• The fractions of elements in each state are given by f[0122] ₁ and f₂, g₁ and g₂ are the degeneracies of each state, ΔE is the energy difference between each state, and kT is the Boltzman factor. The energy difference between states does not vary with temperature but is dependent on composition. Miodownik determined the cobalt and chromium effects on the energy difference while Kuehmann modeled the effect of nickel using Redlich-Kister polynomials, as shown in the following two graphs 2.9 and 2.10.

  

  
• Kuehmann described the degeneracy in a functional form that accounts for the transition due to magnetic effects, expressed in the following equation 2.12. [0123] $\begin{array}{cc}\frac{g_1}{{\mathrm{<figure-callout\; id="2"\; label="g"\; filenames="US20040047758A1-20040311-D00000.png,US20040047758A1-20040311-D00002.png"\; state="\{\{state\}\}">g</figure-callout>}}_2}=\frac{A+B}{2}-\frac{A-\mathrm{<figure-callout\; id="2"\; label="B"\; filenames="US20040047758A1-20040311-D00000.png,US20040047758A1-20040311-D00002.png"\; state="\{\{state\}\}">B</figure-callout>}}{2}\tanh \left[D\left(\frac{T}{T_C}-1\right)\right]& \left(2.12\right)\end{array}$

  
• A is a fitting parameter above the Curie point, B is a fitting parameter below the [0124] Curie 1 point, D is related to the sharpness of the transition, and T_C is the Curie point. Using existing room temperature and high temperature data Kuehmann found values of A=1.05, B=3.10, and D=1.0 best fit the data. The following graph 2.11 presents the value of the degeneracy ratio from Kuehmann's model.

  
• The following equation 2.13 presents Kuehmann's model for predicting FCC lattice parameters in Fe—Co—Cr—Ni systems.[0125]
• α_FCC=(f ₁α_{γ 1} +f ₂αγ ₂ )X _Fe +X _Coα_Co +X _Crα_Cr +X _Niα_Ni  (2.13)
• The values of f[0126] ₁ and f₂ are described by the two equations preceding the above equation with the added constraint that f₁ and f₂=1. The following Table 3 presents properties of each state extrapolated from alloy data to the pure state.

  TABLE 3
  Properties of the two gamma states

  FCC Fe	Room Temperature	Magnetic Moment	Néel or Curie
  State	Lattice Parameter (Å)	per Atom (μB)	Temperature (K)

  γ1	3.57 ± 0.01	0.5	80
  γ2	3.64 ± 0.01	2.8	1800

• Cullity presented room temperature FCC lattice parameters of cobalt and nickel as 3.5440 Å and 3.5239 Å, respectively. The FCC Cr lattice parameter is inferred from high temperature Fe—Cr—Ni ternary data as 3.57 Å. The Cr FCC lattice parameter must be inferred because chromium is not stable in the FCC crystal structure. [0127]
• F. Thermodynamic Calculations [0128]
• ThermoCalc is a thermodynamic database and calculation package developed at the Royal Institute of Technology in Stockholm, Sweden The database is comprised of information from binary, ternary, and quaternary systems used to extrapolate to higher order systems. The Scientific Group Thermodata Europe (SGTE) database, included with ThermoCalc, contains data on more than 2000 condensed compounds. Solution temperature calculations used the SGTE database. [0129]
• Two databases, developed by the Steel Research Group at Northwestern University, were used in the alloy designs. The MART4 database is a fourth generation database used to calculate martensite start temperature. The database includes modified low temperature thermodynamic parameters for FCC and BCC phases for iron based systems. The database modifications are used to accurately predict martensite start temperature because the parameters in the SGTE database are based on high temperature data. The COHERENT3 database is used to predict the tempering response of the ultra-high strength steels. The database includes elasticity energy for coherent M[0130] ₂C carbides with an aspect ratio of 3.
• C. Grain Boundary [0131]
• The fracture resistance of ultra-high strength steels is greatly reduced in an aqueous environment due to hydrogen-assisted cracking. Toughness reductions of up to 80% are possible due to the transition of fracture mode from ductile to intergranular a brittle. The transition to intergranular fracture is promoted by impurity segregation to grain boundaries. [0132]
• Rice and coworkers developed a theoretical model describing the effect of grain boundary segregation on intergranular fracture. Their model postulates that embrittlement arises from a competition between crack tip blunting via dislocation emission and interfacial cleavage. According to this model, crack extension by dislocation emission will happen if the work for dislocation emission, G[0133] _dis1, is less than the Griffith work for intergranular fracture, 2γ_int. Ductile fracture is observed if G_dis1 is less than 2γ_int while brittle fracture is seen if 2γ_int is less. The crack extension force, G_dis1, is set by the strength of the material. However, the intergranular Griffith work term can be altered by the composition of the grain boundary, as described by the following equation 3.1
• 2γ_int=(2γ_int)₀−(Δg _b −Δg _s)Γ  (3.1)
• Equation 3.1 shows that the Griffith work is modified from an inherent value associated with the “clean boundary” by the, solute concentration, Γ, on the boundary. The embrittling effect of the solute, Δg[0134] _b−Δg_s, is the difference in segregation free energy of a solute atom on the grain boundary and free surface. If a solute atom has a lower free energy on the free surface compared to the grain boundary, the solute will reduce the Griffith work term and increase susceptibility to intergranular fracture. Conversely, it is possible to increase resistance to intergranular fracture by segregating a solute that has lower free energy on the grain boundary compared to the free surface. For equilibrium at a temperature T, the equilibrium concentration, Γ, of a solute is given in the following equation 3.2: $\begin{array}{cc}\frac{\Gamma }{{\Gamma }^0-\Gamma }=X_{\mathrm{bulk}}\exp \left(\frac{-\Delta g_b}{RT}\right)& \left(3.2\right)\end{array}$

  
• where Γ[0135] ⁰ is the number of sites at the grain boundary, X_bulk is the content of the solute in the bulk, and Δg_b is the grain boundary segregation energy of the solute. Since grain boundary segregation involves the transport of solute to the boundary, equilibrium may not be reached and kinetics must be considered. Equation 3.3 gives the concentration on the boundary as a function of time, as modeled by McLean $\begin{array}{cc}\frac{{\Gamma }_t-X_{\mathrm{bulk}}}{\Gamma -X_{\mathrm{bulk}}}=1-{\exp \left(\frac{2\sqrt{Dt}}{d\alpha }\right)}^2\mathrm{erfc}\left(\frac{2\sqrt{Dt}}{d\alpha }\right)& \left(3.3\right)\end{array}$

  
• Γ[0136] _t is the concentration at time t, and r is the equilibrium solute segregation at a given temperature, defined by equation 3.2. D is the diffusivity of the solute at the temperature of interest while d is the interplanar spacing. Finally, α is the ratio of equilibrium grain boundary concentration to the bulk concentration.
• Combination of equations 3.2 and 3.3 make it possible to predict the grain boundary concentration after processing. Lower temperatures promote grain boundary segregation by increasing the thermodynamic driving force for segregation. However, kinetics is reduced as temperature is lowered because of slower diffusivity. Therefore, grain boundary concentration exhibits a C-curve behavior when plotted as a function of time and temperature. A temperature, thus, exists where kinetic and thermodynamic effects are balanced to maximize grain boundary segregation at a given time. [0137]
• H. Grain Boundary [0138]
• Many studies have examined the effects of specific solutes on grain boundary cohesion. It is well known that elements, such as phosphorus, tin, sulfur, and antimony segregate to grain boundaries and degrade the cohesion of the grain boundaries [58]. Segregation is commonly measured by Auger electron micro-analysis that requires intergranular fracture in ultra-high vacuum. Boundary concentrations are compared to bulk concentrations to determine the partitioning ratio. Determining the bulk concentration and boundary concentration make it possible to calculate the free energy of segregation for a given temperature. There are discrepancies between data from different researchers, in part, because of different Auger data conversion methods. The segregation energy of phosphorus in iron at 500° C. is calculated to be approximately −50 kJ/mol based on data from Erhart and Grabke whereas, the segregation energy at 480° C. is calculated to be approximately −38 kJ/mol based on data from YuQing and McMahon. Briant determined the segregation enthalpy to be approximately −51.5 kJ/mol. Temperature dependence of the segregation free energy can be described by adding the segregation entropy term. The estimated segregation entropy term is between 0.02-0.03 kJ/molK. [0139]
• Control of sulfur and phosphorus content is very important in ultra-high strength steels. The Griffith work term for brittle fracture is reduced by 10% if phosphorus atoms occupy 25% of grain boundary sites. Reductions of the Griffith work term of 30% are possible if sulfur occupies one quarter of the grain boundary. Although many processing improvements have been made to increase the cleanliness of melts, it is desirable to reduce the amount of sulfur and phosphorus on grain boundaries to zero. Adding elements that form stable compounds with the impurities can further reduce bulk concentrations of impurity elements. Additions of manganese will form sulfides and reduce sulfur segregation to grain boundaries. Recent research has shown stable lanthanum oxy-sulfides and lanthanum phosphate compounds to form in rapidly solidified steels. [0140]
• Equilibrium segregation can increase as temperature is increased The abnormal behavior arises from an increase in bulk concentration due to dissolution of second phase particles. Going to more stable compounds such as the lanthanum oxysulfide, La[0141] ₂O₂S, minimizes this effect.
• A second way to reduce grain boundary impurity concentration is by site competition. Since there are a finite number of grain boundary sites, elemental additions can reduce the concentration of phosphorus and sulfur on the boundary by reducing the number of available sites. The beneficial effects of certain elements are two-fold because they provide a direct cohesion enhancement and limit phosphorus and sulfur concentrations. Carbon has been shown to compete with sulfur and phosphorus for grain boundary sites. In addition, carbon provides a direct cohesion enhancement [0142]
• The segregation free energy of carbon at 800° C. is approximately −80 kJ/mol. The necessity of carbon is apparent when one considers a common technique for inducing intergranular fracture is decarburizing samples in hydrogen gas, [0143]
• Liu and coworkers have examined the effects of boron on grain boundary segregation in high purity Fe—P alloys. Samples with bulk composition up to 5.4 ppm boron showed no boron grain boundary segregation and phosphorus grain boundary concentration remained constant. The concentration of dissolved boron is strongly dependent on the presence of carbon, nitrogen, and oxygen. The impurity levels in the alloys tested were sufficient to trap all boron, thus, reducing the effect bulk boron composition to zero. Alloys with boron concentration between 5.4 and 12.5 wt. ppm showed a decrease in phosphorus grain boundary concentration and an increase in boron grain boundary concentration. Maximum boron segregation, corresponding to minimum phosphorus segregation, was seen at 12.5 wt. ppm boron concentration. Quantities greater than 12.5 wt. ppm did not show added benefits due to formation of borides in the alloys. Liu and coworkers determined the segregation free energy of boron to be approximately −100 kJ/mol at 800° C. This value is significantly higher than that of carbon. [0144]
• Intergranular fracture was suppressed in all samples exhibiting boron segregation. Improvements to intergranular fracture were quantified by decreases in the intergranular ductile to brittle transformation temperature (DBTT). Two mechanisms are responsible for the reduction in DBTT. DBTT is reduced by the exchange of boron for phosphorus on the grain boundaries due to site competition. The reductions in DBTT are too great to be solely due to benefits from site competition. A direct grain boundary cohesion effect of boron is partly responsible for the improved fracture properties. Liu and coworkers concluded that the cohesion enhancing effect of boron is the predominant reason intergranular fracture was suppressed. [0145]
• In an effort to fully understand the embrittling effects of alloying additions, first principles full-potential linearized augmented plane-wave method (FLAPW) calculations have been performed. Initial calculations examined the effect of phosphorus and boron impurities on the FeΣ3[110]( 111) grain boundary and the corresponding Fe(111) free surface. shows a schematic of the Fe⊖3 grain boundary used in the calculation. [0146]

  
• substitutional grain boundary sites labeled #1, and interstitial sites are designated by the small, solid atom. [0147]
• The difference in energy between the two structures, ΔE[0148] _b−ΔE_s, was calculated to determine the embrittling effect. In the case of phosphorus, ΔE_b−ΔE_s was found to be +0.79 ev/atom, which is in good agreement with experimental data. The same study found boron to be a slight cohesion enhancer. Since the first principles calculations are demonstrated to predict embrittling effects accurately, more systems of interest have been examined. Experimental results show qualitatively that carbon is a cohesion enhancing element. The first qualitative theoretical determination of the cohesion enhancing effect of C was found to be ΔE_b−ΔE_s=−0.61 ev/atom by first principles calculations. Re-optimized structure calculations of the Fe—B system have found the enhancing effect of boron to be ΔE_b−ΔE_s=−1.30 ev/atom. Based on these results, boron is approximately twice as effective as carbon for cohesion enhancement.
• Most research on grain boundary embrittlement has focused on the effect of impurities. The impurity atoms typically occupy interstitial sites. Recently, work has examined the alloying effects of transition metals on grain boundary cohesion. Transition metal segregation is different from impurity segregation because the metal atoms occupy substitutional sites, instead of interstitial sites. Advanced FLAPW calculations have examined the effect of substitutional segregation of molybdenum and palladium on grain boundary cohesion in the Fe⊖3(111) grain boundary [70]. Molybdenum was found to be an effective cohesion enhancer (ΔE[0149] _b−ΔE_s=−0.90 ev/atom) primarily due to its strong bonding characteristics arising from its half-filled d band. Conversely, palladium was found to be a mild embrittler (ΔE_b−ΔE_s=+0.08 ev/atom) due to its weak bonding capability associated with its full d band. A model, based on the first-principles calculations of molybdenum and palladium, has been proposed to predict the alloying effect on grain boundary cohesion without carrying out complex first-principles calculations, for each case. Once the atomic structure of the clean grain boundary is determined, the embrittling potency can be predicted with simple inputs of “handbook” properties of the alloying element and matrix. Model predictions of the embrittling effect of metal alloy additions on the FeΣ3(111 ) grain boundary are shown in the following graph (3.2) which is the teaching set forth prior to the examples herein:

  
• Tungsten and rhenium are the most effective cohesion enhancing elements. Cobalt, an important alloying element, is predicted to be a weak embrittler. In order to test the accuracy of the model, first-principles calculations were conducted for cobalt, rhenium, ruthenium, palladium, and tungsten and compared to the values from the simple model prediction, as shown in the following Table 4: [0150]

  TABLE 4
  Predicted embrittling effects and segregation enthalpies
  from quantum mechanics calculations.

  				Grain Boundary
  		Embrittling Effect	Embrittling Effect	Segregation
  		(Model Prediction)	(1st Principles Calc.)	Enthalpy
  	Element	(ev/atom)	(ev/atom)	(ev/atom)

  	Co	+0.10	+0.05
  	Mo	−0.96	−0.90	−0.76
  	Pd	−0.03	+0.08	−0.90
  	Re	−1.29	−1.31	−0.49
  	Ru	−0.77	−0.65	−0.51
  	W	−1.54	−1.31	−0.68

• The largest discrepancy is 0.23 ev/atom in the case of tungsten. Based on these results, the model is effective in predicting the effect of transition metal alloying elements on grain boundary cohesion. [0151]
• Grain boundary segregation enthalpies relative to the bulk solution were also determined from first-principles calculations as shown in the above. Of elements that will act as significant cohesion enhancers, molybdenum has the highest segregation energy while rhenium has the lowest. From this information, one predicts that molybdenum will have the greatest driving force for segregation and displace other atoms, such as rhenium, that have lower segregation energies. [0152]
• I. Grain Boundary—Multi-Component Grain Boundary Cohesion Model [0153]
• A dynamic model to predict the grain boundary cohesion of multi-component alloys has been developed integrating the results of the first-principles calculations and prior research. A grain boundary cohesion model is desired in the design of material susceptible to hydrogen-assisted stress corrosion cracking. The model is able to predict total grain boundary cohesion by summing the effects of substitutional and interstitial atoms. In addition, the model is able to predict cohesion enhancement in non-equilibrium conditions by including kinetic parameters. Modeling the grain boundary with a two-site sub-lattice model separates substitutional and interstitial segregation. Simply stated, all elements that segregate to substitutional sites are treated separately from elements that segregate to interstitial sites. A system of equations, similar to the above equations, is used to predict the equilibrium composition of multi-component systems, as expressed in the following equation 3.4: [0154] $\begin{array}{cc}\frac{{\Gamma }_i}{1-\sum _i{\Gamma }_i}=X_{\mathrm{bulk}}^i\exp \left(\frac{-\Delta G}{RT}\right)& \left(3.4\right)\end{array}$

  
• Equation 3.4 accounts for site competition by multiple species and will be identical to equation 3.2 if only one element segregates. For the interstitial case, i=B, C, P, and S. A second system of equations, identical to equation 3.4, is solved for the substitutional elements. For the substitutional case, i=Mo, Re, and W. [0155]
• Equation 3.4 can also account for any “co-segregation.” Co-segregation is described as one segregant attracting (or repelling) another segregant to the grain boundary. A non-metal atom may be attracted to dissolved metal atoms and form compounds at the interface. This was first described for grain boundary segregation by Guttman and recently applied to surface segregation. Inclusion of co-segregation effects into the grain boundary model can be accomplished by viewing the attraction (or repulsion) effects as modifying the segregation free energy, as described in equation 3.5. [0156] $\begin{array}{cc}\Delta G=\Delta G_0+\sum _i{\alpha }_i{\Gamma }_i& \left(3.5\right)\end{array}$

  
• α is a binary interaction parameter between segregants, Γ[0157] _i is the same as above, and ΔG₀ I is the inherent segregation energy of the component of interest. Higher order interaction parameters could be added to equation 3.3 if necessary. Guttman originally assumed identical parameters as the bulk solution, but the predicted degree of co-segregation is not supported by experiment. Misra and Rama Rao found that segregation energies of carbon and phosphorus in a multi-component alloy were nearly the same as those found in simple binary alloys. Based on the results from Misra and Rama Rao and the lack of strong co-segregation observations, it is apparent that segregation energy data determined from simple binary systems is sufficient and no interaction parameters are needed to reasonably predict grain boundary segregation.
• The system of equations presented by equation 3.4 describes the equilibrium grain boundary composition, but does not address kinetics. Slow diffusers, such as tungsten, will not reach equilibrium during normal tempering treatments. No diffusivity information is available for rhenium in iron, but it is assumed that the diffusivity is similar to that of tungsten. Some enrichment of molybdenum will occur during tempering. Equation 3.3 is incorporated into the model to add kinetic parameters for each solute in order to predict the grain boundary segregation during tempering. [0158]
• Segregation during solution treatment must be addressed because substitutional segregation can only achieve equilibrium quantities at high temperatures. Description of interstitial segregation dynamics during solution treatment is not essential because near equilibrium conditions will be reached during tempering. FLAPW calculations of grain boundary segregation enthalpy in FCC iron have been limited to tungsten and found to be 0.80 ev/atom. Table 5 summarizes all data used in the grain boundary cohesion model. [0159]

  TABLE 5
  Grain boundary cohesion model data.

  	BCC	BCC Segregation	Cohesion
  	Diffusivity	Energy, ΔG	Enhancement (Δgb − Δgs)
  	(cm2/sec)	(J/mol)	(J/m2 * monolayer)

  B		−75,000 - 25T	−1.463
  C		−55,000 - 25T	−0.686
  P	1.38e5 exp	−34,300 - 21.5T	+0.214
  	(−332,000/RT)
  S	34.6 exp	−51,500 - 25T	+0.99
  	(−231,500/RT)
  Mo	0.44 exp	−73,203 - 25T	−1.013
  	(−238,000/RT)
  Re	25 exp	−47,197 - 25T	−1.474
  	(−298,000/RT)
  W	25 exp	−65,495 - 25T	−1.474
  	(−298,000/RT)

• The grain boundary cohesion model predicts the improvement in the Griffith work term, shown in equation 3.1. Spaulding has measured experimentally the effect of grain boundary cohesion on fracture strength in the Ni—Co secondary hardening steels considered here. Graph3.3 shows the effect of the Griffith work term on fracture strength at various steel hardness levels. [0160]

  
• It is important to note that more negative values of Griffith work are superior. In addition, the figure shows that improvements in grain boundary cohesion are needed to maintain constant fracture strength as hardness is increased. levels of approximately 16,000 J/mol for the M[0161] ₂C phase and 1,100 J/mol for the FCC phase. Based on this model, all design calculations here will include capillary energy additions in order to more accurately predict non-equilibrium alloy microstructure which arise from standard tempering practices.
• J. The Alloys Tested [0162]
• The objective of this design is an ultra-high strength alloy with enhanced toughness and superior hydrogen resistance. The desired microstructure is a fully martensitic, M[0163] ₂C carbide strengthened steel with a fine austenite dispersion. In addition, superior grain boundary cohesion enhancement is desired to limit impurity embrittlement. Properties of the design alloy include hardness levels greater than HRc 57, corresponding to an ultimate tensile strength greater than 325 ksi, fracture toughness of 75 ksi{square root}in, and K_ISCC/K_IC=0.5.
• The alloy design optimizes composition by considering M[0164] _s temperature, carbide fraction, austenite dispersion, and grain boundary cohesion. In order to achieve a fully martensitic microstructure, the M_s temperature is required to be at least 220° C., matching the M_s of AerMet100.
• Calculations, including capillary energy, of AerMet100 tempered at 455° C. predict a carbide fraction of 0.0277 and grain boundary cohesion enhancement of 1.656 J/m[0165] ².
• The carbon content of alloys is 0.25 wt. % that provides sufficient M[0166] ₂C carbide fraction. vanadium and molybdenum contents are fixed at 0.11 wt. % and 1.5 wt. %, respectively, to ensure high carbide volume fraction. Vanadium additions are limited by increases in solution temperature arising from the stability of VC carbides. Molybdenum content is set by constraints on segregation during processing. Chromium is added to ensure a minimum 2:1 mole ratio of carbide forming elements to carbon. In addition, tungsten additions enhance M₂C carbide formation. Limits on tungsten are set by dissolution of M₆C during solution treatment. Carbide fraction calculations with capillary energy corresponding to peak hardness microstructure, where M₂C particle size is equal to 3 nm, are used.
• Tungsten and rhenium are added to enhance grain boundary cohesion beyond levels found in AerMet100. Grain boundary cohesion in AerMet100 is due to the segregation of molybdenum and carbon to the grain boundaries. Rhenium and tungsten are considered because they have the greatest enhancing effect. Rhenium is considered because of its high solubility in BCC at tempering temperatures. Tungsten has low solubility in BCC because of its high carbide formation driving force. Molybdenum additions are necessary to saturate all substitutional grain boundary sites, as additions of tungsten and rhenium will not fill all sites. Carbon segregation can be reduced if large quantities of carbide forming elements are present, thus reducing its BCC solubility. Additions of boron can replace carbon on the boundary and provide greater cohesion enhancement. However, control of boron composition between heats is difficult because of its low solubility in iron. Excess additions of boron can form deleterious borides while a shortage of boron will not provide any segregation. The window of boron composition is between 5 and 15 ppm. [0167]
• Since substitutional segregation does not reach equilibrium conditions during tempering, tempering time is incorporated into the design to more accurately predict grain boundary segregation. Kuehmann and Olson have examined the correlation between precipitation half-completion time and the coarsening rate constant evaluated at half-completion. The half-completion coarsening rate constant is calculated from the thermodynamic properties, using the Lee model discussed previously the carbide fraction is half of the equilibrium value. The study examined a number of different 0.24 wt. % carbon alloys tempered at 510° C. and found the values to be correlated over two decades, as shown in the following graph 4.4: [0168]

  
• The austenite dispersion must have sufficient stability and proper formation kinetics to ensure optimum toughening enhancement. After satisfying primary considerations, toughening efficiency is maximized by optimizing dilatancy and austenite fraction. The stability of the austenite particle is related to its size and composition. Size is controlled by matching the ratio of austenite to carbide kinetic rate constants to that of prior alloys. The austenite rate constant is normalized by the carbide rate constant to predict final austenite precipitate size to final carbide size. Kinetic parameters are determined by calculating the Lee coarsening rate constants, shown in equations 2.5 and 2.6, for the predicted three-phase microstructure including capillary energy. The ratio of austenite to carbide rate constants of AerMet100 at 455° C. is calculated to be 3.6. Similar calculations for AF1410 tempered at 510° C. predicted a rate constant ratio of approximately 5. Since both conditions produced successful transformation toughened alloys, the target kinetic ratio is between 3 and 6. [0169]
• The composition dependence of austenite stability is manifest in a chemical and mechanical term. The chemical term-is evaluated by determining the difference in free energy between FCC and BCC at service temperature, generally room temperature. The composition used is determined from a three-phase calculation incorporating capillary energy at the tempering temperature. The mechanical term is identical to the athermal frictional work of interfacial motion represented by equation 2.2. The calculated austenite stability of AerMet100 tempered at 455° C. is 6.37 kJ/mol. It is important to note that equilibrium three-phase calculations predict a value of 4.35 kJ/mol. The large difference arises from the inclusion of capillary energy. Prior designs set a target of an increase in austenite stability of 0.5 kJ/mol to account for the increased strength level. The target austenite stability of the design alloy will have a lower bound of 6.87 kJ/mol, maintaining the same increment as prior designs. An upper bound will be set at 7.25 kJ/mol maintaining a constant percentage increase in austenite stability as the prior designs. [0170]
• The alloy uses the constraints presented previously to fix the alloy composition as Fe—Co—Cr—Ni—W-1.5Mo-1.5Re-0.11V-0.25C. Initial calculations examine two levels of chromium and tungsten content and 4 levels of cobalt to determine the effect on austenite stability, grain boundary cohesion, carbide fraction, and kinetics. Chromium content varies between 2 wt. % and 3 wt. %, tungsten content varies between 0.5 wt. % and 1.5 wt. %, and cobalt varies between 15 wt. % and 21 wt. %. Nickel content for each alloy composition is set by the M[0171] _s temperature constraint. All three-phase calculations were performed at 455° C. with capillary energy additions of 16,000 J/mol for M₂C and 1,100 J/mol for FCC using the COHERENT3 database.
• Graph 4.7 shows the alloy nickel content vs. cobalt content for each series of alloys that results in an M[0172] _s temperature of 220° C.

  
• The calculations were performed using the MART4 database with an accuracy of ±25° C. for this class of steel. Increases in cobalt content allow higher nickel contents. Reductions of chromium from 3 wt. % to 2 wt. % allow an increase of approximately 1.5 wt. % nickel. Nickel content increases approximately 0.5 wt. % as tungsten is reduced from 1.5 wt. % to 0.5 wt. %. [0173]
• Austenite stability is determined by summing the chemical free energy term and a thermal frictional work term, as shown in Graph 4.8. [0174]

  
• Nickel partitions strongly to FCC and acts as an FCC stabilizer. Cobalt has no direct effect on FCC stability but increases in cobalt will yield more stable austenite due to the higher nickel content of the alloys. While a large percentage of chromium is expected to partition to the M[0175] ₂C, some of the excess chromium will partition to the FCC and decrease the chemical stability of FCC relative to BCC. Tungsten behaves in a similar manner as chromium but its effect is much milder. At high cobalt content, the 3 wt. % chromium series has slightly higher stability compared to the 2 wt. % series. While this result is unexpected, it can be explained by examining each component of the austenite stability, as shown in Graphs 4.9 and 4.10.

  

  
• The athermal frictional work term decreases as cobalt increases for both of the 2 wt. % chromium series. As cobalt content increases, the M[0176] ₂C fraction increases and consumes more chromium. The effect is not seen at higher chromium levels because M₂C fraction is maximized for all cobalt contents. Graph 4.11 shows the carbide fraction of each series.

  
• In order to improve stress-corrosion cracking resistance, the design alloy must have greater grain boundary cohesion than that of AerMet100. Grain boundary cohesion was calculated for each design series in two manners. Graph 4.12 shows the predicted grain boundary cohesion without boron additions and graph 4.13 shows predictions when 10 ppm boron is added. All calculations assumed sulfur content to be 10 ppm and phosphorus content to be 50 ppm. [0177]

  

  
• The interstitial site grain boundary cohesion behavior is dramatically different if boron is included. When boron is not included, increases in cobalt slightly reduce grain boundary cohesion because carbon solubility decreases. However, cohesion enhancement increases as cobalt increases when boron is added. The increase in cohesion enhancement is also due to a reduction in carbon solubility. As carbon solubility decreases, the boron grain boundary composition increases because there is less site competition. Boron is nearly twice as potent per atom as carbon in enhancing grain boundary cohesion. It is clear that boron must be added to the alloys to maximize grain boundary cohesion at the possible expense of ease of processability. [0178]
• Chromium and tungsten content are the primary factors controlling substitutional site grain boundary cohesion enhancement. Increasing tungsten content directly increases grain boundary cohesion by increasing the number of tungsten atoms in solution. Increasing chromium content has a similar effect because chromium and tungsten compete for sites in M[0179] ₂C carbides.
• All of the alloys have carbide fractions greater than AerMet100 suggesting that the strength requirement can easily be achieved. However, none of the alloy calculations yielded austenite stability in the target range. The 2Cr-0.5W series approached the target at high cobalt levels. Unfortunately, the same series has the lowest grain boundary cohesion enhancement when boron is added. None of the alloys examined in this study meet the criteria necessary to produce a transformation toughened alloy. [0180]
• A second study examined the feasibility of transformation toughened alloys of high cobalt content. It is clear from the previous results that high cobalt contents are needed to assure proper austenite stability. Cobalt content was fixed at 22 wt. % to maximize nickel content while maintaining reasonable material cost. Higher cobalt contents were not examined because of the high raw material cost of cobalt. Chromium, molybdenum, and tungsten contents were varied to examine the effect on grain boundary cohesion and austenite stability. Rhenium, vanadium, and carbon were fixed at the same content as the previous study. Nickel content was determined by the constraint that M[0181] _s temperature be 220° C. Table 6 presents the compositions of the designs examined.

  TABLE 6
  Composition of design alloys.

  	Cr content	Mo content	W content	Ni content
  Alloy	(wt. %)	(wt. %)	(wt. %)	(wt. %)
  A	3.0	1.5	1.8	9.52
  B	3.2	1.5	2.0	9.09
  C	3.2	0.5	2.0	9.38
  D	3.2	1.1	2.1	9.14
  E	3.2	1.1	2.3	9.02
  F	3.2	1.5	2.3	8.91
  G	3.3	1.5	2.3	8.76
  H	3.5	1.5	2.4	8.41

• Carbide fraction was not an issue with the second set of design alloys because each had at least 20% more carbide than AerMet100. Cohesion enhancement predictions included boron in order to achieve the highest possible values. Graph 4.14 shows a cross-plot of cohesion enhancement vs. austenite stability in order to determine the alloys that meet both criteria. [0182]

  
• All of the alloys have significant improvements in grain boundary cohesion compared to AerMet100. The highest enhancement corresponds to the alloy with the lowest molybdenum content. More tungsten and rhenium are able to segregate to the boundary because of reduced site competition. However, molybdenum must be present in order to saturate all substitutional sites. [0183]
• Five alloys meet or exceed the minimum austenite stability while exceeding grain boundary cohesion requirements. The ratio of FCC to M[0184] ₂C precipitation rate constant must be examined to determine if formation kinetics are favorable. In addition, FCC mole fraction is also examined to ensure that a large enough fraction is present to provide toughening enhancement. Graph 4.15 shows a property cross-plot of FCC mole fraction vs. kinetic ratio of the five alloys that met both criteria in FIG. 4.14.

  
• All alloys have at least 25% less austenite than that of AerMet100 indicating that the toughening enhancement will be less than that of AerMet100 unless the transformation volume change is enhanced by 10%. Typical transformation volume changes of 3% would have to be increased to 3.3%. [0185]
• In addition to the reduction in toughening efficiency, the kinetic ratio does meet specified goals. The crosshatched region of graph 4.15 represents the target region for kinetic ratio. The minimum kinetic ratio of the five alloys is approximately twice the target value. It is thus expected that the austenite dispersion of each alloy will exceed the desired size and not yield desired properties. The kinetic ratio is much too high because of slow carbide kinetics arising from the presence of tungsten. Removing tungsten from the system will lower the kinetic ratio to the desired values. However, removal of tungsten will result in inferior grain boundary properties. Based on these results, this study will not pursue the design of a transformation toughened alloy. [0186]
• K. Non-Transformation Toughened Alloy Design [0187]
• In order to ensure high strength in the alloy, no retained or precipitated austenite is desired in the alloys. Retained or precipitated austenite will not have properties leading to transformation toughening. Therefore, presence of austenite will lower strength without any toughening enhancement. The overall alloying additions will be limited by the constraint that M[0188] _s temperature be greater than 220° C. However, this constraint will not be used to determine nickel content, as in the previous design. Nickel content will be limited by the constraint that no precipitated austenite forms. Graph 4.16 shows a plot of predicted precipitated austenite fraction vs. alloy nickel content.

  
• The data plotted are from predictions from the previous alloy design. Added to the data are linear and second-order polynomial fits. The critical nickel content is predicted to be between 5 wt. % and 6.5 wt. %. [0189]
• L. Tungsten Steel [0190]
• The Fe—Co—Ni—Cr—Mo—W—C alloy system is examined as a candidate for an ultra-high strength steel. Carbon content is set at 0.25 wt. % to produce a carbide dispersion meeting the strength requirement. Cobalt is set at 15 wt. % in order to enhance M[0191] ₂C driving force and maintain dislocation recovery resistance. Nickel content is set at 6 wt. % to minimize austenite precipitation. Rhenium grain boundary composition is minimized in the presence of molybdenum and tungsten because of its low segregation energy compared to molybdenum and tungsten. Since rhenium has a very high cost and limited effectiveness, rhenium is excluded from this design. Chromium, molybdenum, and tungsten variations are examined to determine their effects on alloy properties. Chromium content is varied from 2 wt. % to 4 wt. %, molybdenum content is varied between 0 wt. % and 1.5 wt. %, and tungsten content is varied between 1 wt. % and 3 wt. %. A full-factorial design with three levels for each variable is examined. Among the properties examined are M_s temperature, solution temperature, grain boundary cohesion, M₂C fraction, and M₆C to M₂C driving force ratio.
• Calculations determining fraction M[0192] ₂C and cohesion enhancement were conducted at 455° C. using the COHERENT3 database. 16,000 J/mol was added to the M₂C phase to account for capillary energy at peak hardness. The SGTE database was used to determine solution temperature, and M₆C to M₂C driving force ratio. The MART4 database was used to predict the M_s temperature.
• Graph 4.17 presents the effect of composition on Ms temperature. [0193]

  
• All M[0194] _s temperatures in the design series exceed the minimum value of 220° C. Increases of 1 wt. % molybdenum reduce the M_s temperature by approximately 4° C. 1 wt. % increases of tungsten reduce the M_S temperature by nearly 10° C. while 1 wt. % increases of chromium reduce the M_s temperature by approximately 27° C.
• Proper solution treatment of the alloy is crucial to optimize strength, toughness, and stress-corrosion cracking resistance. Solution treatments must dissolve all large carbides, such as M[0195] ₂₃C₆ and M₆C, to limit void nucleation sites. In addition, all carbon must be in solution to maximize the M₂C strengthening dispersion. In addition, the majority of tungsten grain boundary segregation occurs during solution treatment. Therefore, solution temperature should be designed to maximize grain boundary segregation while dissolving all undesirable carbides. Solution time will be fixed at one hour to prevent excessive grain growth. Solution treatment tungsten segregation vs. temperature, for 1-hour treatment in an Fe-2 wt. % W alloy, is shown in FIG. 4.18 to predict optimum solution temperature. The BCC tungsten diffusivity is extrapolated to high temperature in the model.

  
• Graph 4.18 shows that solution temperatures above 900° C. will result in complete tungsten segregation after 1 hour. An upper limit on solution temperature is placed at 1100° C. due to processing concerns. Graph 4.19 shows the effect of composition on predicted solution temperature. [0196]

  
• Nearly all predicted solution temperature of all alloys falls within the desired range. As expected, solution temperature rises as additions are made to molybdenum and tungsten. Increases in chromium do not affect the solution temperature greatly because the low chromium M[0197] ₆C phase is the last carbide to dissolve. Typical solution treatments require an increase in temperature between 50° C. and 100° C. over the predicted equilibrium value to account for the kinetics of dissolution of the carbides. Including such a temperature allowance for kinetic effects causes alloys with high molybdenum and tungsten contents to exceed the desired solution treatment temperature.
• The majority of tungsten grain boundary segregation occurs during solution treatment while some molybdenum segregation will occur during tempering. Calculations assumed a solution treatment temperature of 1050° C. and a tempering temperature of 455° C. Tempering time was fixed at 8 hours. [0198]
• Graph 4.20. shows the predicted grain boundary cohesion enhancement of the design series. [0199]

  
• Increases in tungsten content enhance grain boundary cohesion because more tungsten is in solution. As chromium content increases, small increases in cohesion occur because tungsten solubility increases during tempering. Small tungsten enrichment (or depletion) occurs during tempering. As molybdenum is added to the alloy, the predicted cohesion enhancement decreases. The decrease arises from depletion of grain boundary tungsten content due to site competition with molybdenum. While it appears that additions of molybdenum are not beneficial to grain boundary cohesion, molybdenum additions are included in the design because it segregates to lath boundaries during tempering. Since lath boundaries form after quenching, boundary enhancement will only occur during tempering. Therefore, molybdenum must be included to enhance lath boundary cohesion because sufficient tungsten enrichment will not occur at these boundaries. [0200]
• Additions of tungsten will stabilize M[0201] ₆C carbides and increase the nucleation driving force to levels higher than those of the M₂C carbide. Fortunately, kinetic considerations favor M₂C carbides causing them to form before M₆C carbides. However, it is desirable to limit the M₆C stability to ensure that the M₂C carbide will remain stable over any possible tempering range. Graph 4.21 shows the dependence of M₆C/M₂C driving force ratio on composition.

  
• The driving force ratio does not vary greatly as molybdenum and chromium are changed. The carbide ratio increases as tungsten content is raised. However, the driving force for M[0202] ₆C formation is not significantly greater than that of M₂C formation and is not a concern in the examined composition range.
• As a final consideration, the M[0203] ₂C fraction must be high enough to ensure strength properties are achieved. The M₂C carbide fraction of each design series is shown in Graph 4.22.

  
• Other than alloys with low chromium and low tungsten, all design alloys have carbide fractions nearly 30% greater than that of AerMet100. [0204]
• Based on the results of the alloy design, an alloy composition of Fe—15Co—6Ni—3Cr-1.7Mo-2W-0.25C has been chosen. The alloy has a predicted M[0205] _S temperature of 273° C. and predicted solution temperature of approximately 975° C. The predicted grain boundary cohesion is approximately 2.3 J/m² that is approximately 12% improvement over AerMet100. In addition, the alloy will have sufficient molybdenum content to segregate to lath boundaries during tempering. The carbide dispersion is predicted to consist of fine M₂C precipitates and avoid M₆C carbides. This alloy is expected to meet minimum strength requirement while having good toughness. The stress-corrosion cracking resistance is predicted to be higher than existing alloys.
• M. Rhenium Steel [0206]
• A second alloy has been designed to quantify the effect of rhenium on grain boundaries. In order to enhance rhenium segregation, an alloy has been designed with reduced site competition and reduced solution temperature. Reducing the carbon content from 0.25 wt. % to 0.18 wt. % lowers solution temperature. Molybdenum is excluded from the design in order to minimize site competition. Cobalt and chromium content are fixed at the same levels as in the tungsten quantum steel of 15 wt. % and 3 wt. %, respectively. Nickel content is reduced from 6 wt. % to 5 wt. %. The reduction of nickel content will ensure no austenite forms and allow a qualitative determination of the importance of nickel on alloy properties. The rhenium content is fixed at 2.7 wt. % by cost constraints. Finally, 1.2 wt. % tungsten is added to enhance grain boundary cohesion by occupying sites that have not been occupied by rhenium. The addition of tungsten is limited by the constraint that predicted rhenium grain boundary site fraction is greater than 0.20. Graphs 4.23 and 4.24 show the predicted grain boundary site fraction of tungsten and rhenium. [0207]

  

  
• Two diffusivity estimates are used to bracket tungsten and rhenium grain boundary site fraction. The high diffusivity estimate extrapolates the tungsten BCC diffusivity to high temperature. The low diffusivity estimate applies the same FCC/BCC diffusivity ratio as that of chromium. Rhenium diffusivity is estimated to be equal to that of tungsten. The large dependence of diffusivity on tungsten site fraction illustrates the need for accurate diffusivity data for tungsten and rhenium. [0208]
• Tungsten segregation is limited by thermodynamics above 1000° C. for both diffusivity estimates. Rhenium segregation increases over the entire temperature range because of reduced site competition at higher temperatures. [0209]
• N. Materials [0210]
• The designed compositions are Fe-15Co-6Ni-3Cr-1.7Mo-2W-0.25C for alloy QSW and Fe-15Co-5Ni-3Cr-2.7Re-1.2W-0.18C for alloy QSRe. Allvac, an Allegheny Technologies Company, in Monroe, N.C., produced the alloys as 150 pound heats alloy, by vacuum induction melting. The heats were re-melted by vacuum arc re-melting to produce a 6″ round ingot. Table 7 shows the measured compositions of each alloy. [0211]

  TABLE 7
  Chemical compositions of design alloys.

  Composition
  (wt. %)	QSW	QSRe
  Fe	Balance	Balance
  Co	15.61	15.67
  Ni	6.01	4.92
  Cr	3.02	2.96
  Mo	1.72	—
  W	2.00	1.18
  Re	—	2.69
  Ti	0.04	0.04
  La	0.009	0.006
  C	0.249	0.179
  B	0.002	0.002
  S	0.0012	0.0010
  P	0.005	0.004
  O	0.0008	0.0008
  N	0.0004	0.0005

• small elemental additions were added for grain refinement and impurity gettering. 0.04 wt. % titanium was included in melt de-oxidation to form small amounts of TiC. TiC is stable during solution treatment and provides an effective grain refining dispersion with good interfacial adhesion for toughness.. A target value of 0.01 wt. % lanthanum was added as a late melt addition to getter oxygen and sulfur as La[0212] ₂O₂S. The boron content of the melts met the target of 15 ppm for grain boundary cohesion. A target value of 0.015 wt. % zirconium was added to each melt to getter phosphorus as ZrP. However, no residual zirconium was detected during chemical analysis.
• Each ingot was homogenized for 12 hours at 1190° C. The alloys were upset to 6.9″ round at a temperature of 1175° C. Alloys were re-heated if the temperature dropped below 1010° C. The alloys were press forged to 6″ round ingots at 1175° C. (minimum T=1010° C.). The alloys were upset a second time under identical conditions as the first upset. The alloys were press forged to 2.5″ square at 1093° C. The minimum temperature allowed during the forging was 927° C. The ingots were air cooled upon completion of press forging. [0213]
• O. Experimental Procedures [0214]
• 1. Dilatometry [0215]
• The dilatometry was performed on a computer controlled MMC Quenching Dilatometer. An induction furnace performs sample heating and jets of helium gas rapidly quench samples. Cylindrical samples 10 mm long and 3 mm wide are placed inside an induction coil and held in place by two low expansion quartz platens. The platens are spring loaded allowing the sample to expand or contract during thermal cycling. The length change is transmitted to an LVDT transducer via two quartz rods in contact with the platens. Temperature control is achieved with a thermocouple spot welded directly to the sample surface. The sample stage is enclosed in a vacuum chamber connected to a turbo-molecular pump and mechanical backing pump capable of quickly achieving a vacuum of 10[0216] ⁻⁴ torr.
• Dilatometry specimens were prepared by EDM machining 3 mm diameter rods from bar stock material. The 3 mm rods were cut to 10 mm length by EDM machining. The surfaces were sanded to remove surface oxide resulting from machining. The thermocouple was attached to the sample midway along its length. [0217]
• Under computer control, the samples were heated to 1050° C. at a rate of approximately 3° C./sec. and held for 5 minutes. Quench rate is controlled by a needle valve monitoring the flow of helium gas onto the sample. The samples were cooled from 1050° C. at approximately 15° C./sec. The test results in a data file containing time, temperature, and length information. Austenite start temperature was determined as the temperature at which the sample started contracting upon heating. Similarly, martensite start temperature was determined as the temperature at which the sample started expanding upon cooling. [0218]
• 2. Rockwell C Hardness Testing [0219]
• Rockwell C hardness testing determines the hardness of a material by the depth of penetration of the indenter under a constant load. The measurement consists of the additional depth the indenter travels upon application of a large load beyond the small indentation of the preload. A small preload is used to remove backlash in the loading apparatus and remove bias due to any surface layer effects. The Rockwell C scale uses a preload of 10 kg, a load of 150 kg, and a Rockwell C diamond indenter. The testing apparatus used was a standard Wilson Rockwell hardness tester. [0220]
• Before testing any samples, the machine was calibrated using R[0221] _c 45 and R_c 61 standard blocks. 5 to 7 indentations were randomly distributed along the sample. All indentations were at least 3 indentation diameters away from an edge or other indentation. Each indent consisted of preloading the sample, zeroing the indicator, and performing the test. Results were measured to the nearest 0.1 R_c and the average and standard deviation of each sample was determined.
• 3. Tensile Testing [0222]
• Round tensile specimens were cut from bar stock material parallel to the longitudinal direction according to ASTM standard E-8. Samples had a gage diameter of 0.25″ and a gage length of 1.00″. Samples were encapsulated in quartz under vacuum less than 5 mtorr. Samples were quenched in oil after solution treatment by breaking the quartz tube and quickly immersing in oil. Samples were cryogenically cooled in liquid nitrogen for two hours. The samples were encapsulated a second time in pyrex, under similar vacuum, and tempered. Upon completion of tempering, samples were air cooled in the pyrex tubes. [0223]
• Testing was conducted on a Sintech 20G tensile machine controlled by the Testworks software package at a rate of 1 mm/sec. An extensometer was attached during testing to generate load-displacement curves. Area reduction and extension were measured manually upon completion of the test. [0224]
• 4. Fracture Toughness Testing [0225]
• K[0226] _IC samples were prepared from bar stock material in the longitudinal configuration. Oversized blanks were abrasive cut from the billet and solution treated in air. Samples were cooled in liquid nitrogen for 2 hours after oil quenching. Final tempering was done in air. Upon completion of heat treatment, the surfaces of the blanks were ground parallel to final thickness. EDM machining was used to machine the final specimen shape.
• The method for determining K[0227] _IC is outlined by ASTM standard E-399 Sample geometry was identical to that used by Kuehmann Kuehmann performed combined J_IC and K_IC tests which require samples to be machined to specifications listed by ASTM standard E-813. Diagram 5.1 shows the compact tension specimen geometry used to measure fracture toughness.

  
• Specimen surfaces were polished to 6 μm finish in order to observe crack propagation during pre-cracking. Pre-cracking was accomplished on a servo-hydraulic MTS load frame. Pre-cracking was accomplished at 20 Hz with a stress ratio of 0.1. Load control was accomplished by constraining maximum strain levels. Load is controlled by strain because as crack propagation occurs, stress levels decrease. If constraints are placed on maximum stress, samples may fracture prematurely because specified peak stress may exceed fracture stress of the sample after the crack has propagated. Initial peak load was 1000 pounds. Samples underwent 10,000 cycles at the initial load. If no crack initiation is observed, the peak stress level is increased 200 pounds. The process is increased until crack initiation is observed. Pre-cracking is terminated when the notch and fatigue crack are between 0.5″ and 0.55″ depth. Pre-cracked samples were fractured on a Sintech 20G tensile machine controlled by the Testworks software package generating a load vs. crack opening displacement curve. Crack opening displacement was measured with an MTS model 632.03E-30 clip gage with a gage length of 0.1″ attached to the knife edges of the sample. [0228]
• K[0229] _IC was determined from fractured samples according to ASTM standard E-399. First, the K_Q value is determined by equation 5.1. $\begin{array}{cc}{K}_Q=\left(\frac{P_Q}{B{\mathrm{<figure-callout\; id="1"\; label="W"\; filenames="US20040047758A1-20040311-D00000.png,US20040047758A1-20040311-D00002.png"\; state="\{\{state\}\}">W</figure-callout>}}^{1/2}}\right)·f\left(\frac{a}{W}\right)& \left(5.1\right)\end{array}$

  
• B is the sample thickness, W is the sample width, and a[0230] ₀ is the initial crack length. Measuring the crack length of five points evenly spaced along the sample and taking their average determines initial crack length. The functional form represents a geometry dependent calibration factor. A 95% secant construction on the load vs. crack opening displacement curve determines the value of P_Q, as shown in graph 5.2.

  
• The K[0231] _Q values are equal to K_IC if the relation in equation 5.2 is met. $\begin{array}{cc}2.5{\left(\frac{K_Q}{{\sigma }_{\mathrm{YS}}}\right)}^2<B\mathrm{and}a_0& \left(5.2\right)\end{array}$

  
• K[0232] _Q is determined by equation 5.1, σ_YS is the 0.2% offset yield stress, B is the sample thickness, and a₀ is the initial crack length. separate the overlapping peaks, and the background-subtracted integrated intensities were obtained using Desktop Spectrum Analyzer (DTSA 2.01) software from NIST.
• Samples were ground to a thickness of approximately 50 microns. Transmission sections were polished using a Twin Jet electropolisher at a temperature below −50° C. The electrolyte used was 20% percholoric acid in methanol. [0233]
• 5. Electron Microscopy [0234]
• A Hitachi 3500 scanning electron microscope with a tungsten wire filament was used to investigate the fracture surfaces of the compact tension specimens. The microscope uses Quartz PCI Image Management Software through a Windows 95 interface. Fractured samples were mounted using graphite tape and examined in the scanning electron microscope with a 20 kV electron beam. Examination was conducted at a vacuum level of 10[0235] ⁻⁴ torr. Fracture surfaces were examined and micrographs of characteristic features were taken at 500× and 1000×.
• Auger electron microanalysis was conducted at Oak Ridge National Laboratory on a Physical Electronics PHI G80 Auger Nanoprobe. Voltage was set at 10 kV and current was set at 10 nA. Specimens were examined under vacuum of 1.78×10[0236] ⁻⁹ torr. Samples were fractured under vacuum on a cold fracture stage. The fracture stage was capable of quickly reaching −100° C. by liquid nitrogen cooling. Auger spectra were taken on all features of the fracture surface.
• Auger specimen rods were machined from heat-treated material by EDM machining. Samples were cut to length by abrasive saw. A grinding wheel was used to machine required notches into the samples. [0237]
• Analytical electron microscopy was performed in a Hitachi HF-2000 cold-field emission gun equipped with a Gatan 666 parallel electron energy-loss spectrometry detector, an ultra-thin window Link EDS detector and processor, and a Gatan charge-coupled device camera for high-resolution imaging. The electron microscope was operated at 200 kV with a beam size of 3 nm. The x-ray spectra were deconvoluted, to [0238]
• P. Test Results [0239]
• The final aspect of this study attempted to examine the grain boundaries of the Two experimental methods were used to study the composition of the grain boundaries. First, Auger electron microanalysis of fracture surfaces was conducted to determine if any enrichment of boron, tungsten, and rhenium is present. Samples were fractured under high vacuum (10[0240] ⁻⁹ torr) at −100° C. Samples were fractured at low temperature in order to induce brittle fracture. The second method used high-resolution, analytical electron microscopy to examine prior austenite grain boundary composition. The composition of the grain boundary is compared to the bulk composition to determine if any enrichment is present.
• 1. Dilatometry [0241]
• A dilatometry study was conducted to determine the martensite start temperature of the design alloys. Graph 6.1 presents the heating and cooling curves showing the relative length of the sample vs. temperature for alloy QSW. [0242]

  
• In addition to the experimental curve, polynomial fits of the heating and cooling curves are presented. The martensite start temperature is determined by comparing experimental sample length to that predicted by the cooling curve. Experimental length values do not differ from the polynomial fit by more than 1.9% at temperatures greater than 300° C. The martensite start temperature is chosen as the temperature at which experimental values differ from the polynomial fit by more than 5%. The martensite start temperature of alloy QSW is 296° C. The Ghosh-Olson model predicts a martensite start temperature of 273° C. [0243]
• Graph 6.2 presents the dilatometry trace for alloy QSRe. [0244]

  
• The martensite start temperature of alloy QSRe is 353° C. The Ghosh-Olson model predicts a martensite start temperature of 350° C. [0245]
• 2. Solution Treatment Study [0246]
• A study was conducted on both design alloys in order to determine the optimal solution treatment condition and confirm ThermoCalc predictions. 8″ bars of stock material were treated in a gradient heat furnace, at Allvac research laboratories, up to a temperature of 1125° C. As-quenched hardness measurements were taken along the length of the bar. Bars were treated for 1 hour, 1.5 hours, and 2 hours to determine any effect of solution treatment time. [0247]
• Solution treatment behavior of alloy QSW is predicted by ThermoCalc calculations. The calculation allowed M[0248] ₂₃C₆, M₆C, M₇C₃, M₂C, and MC carbides to form Graph 6.3 shows the predicted equilibrium carbide fraction as a function of temperature.

  
• Graph 6.4 shows the results of the gradient heat treat study for alloy QSW. [0249]

  
• A rapid increase in hardness is seen over the temperature range of 750° C. to 825° C. corresponding to dissolution of M[0250] ₂₃C₆ carbides. The 1-hour sample has lower hardness than that of the longer treated samples indicating that 1 hour is not sufficient time to reach equilibrium. Over the temperature range of 825° C. to 1000° C., a gradual increase in hardness is observed in samples treated for 1.5 and 2 hours. This behavior is attributable to the gradual dissolution of M₆C carbides. Once again, 1 hour is not sufficient to reach equilibrium. At temperatures above 1000° C., hardness does not vary as solution time is increased suggesting that complete solution treatment is accomplished within 1 hour. No drop in hardness is observed as solution temperature increases. This indicates that 0.04 wt. % addition of Ti forms an effective grain refining dispersion preventing excessive grain growth. The peak solution hardness approaches HR_c 55. Any 1 hour solution treatment between 1000° C. and 1050° C. will achieve this condition. The experimentally determined minimum solution temperature of 1000° C. is similar to the predicted solution temperature of 975° C.
• In addition to optimizing solution treatment, the gradient heat treat study provided information about anneal softening. In order to improve machinability, material is generally shipped in a softened condition. A 2-hour heat treatment at 725° C. drops the hardness of alloy QSW below HR[0251] _c 35.
• The results of the ThermoCalc solution treatment calculations on alloy QSRe are shown in Graph 6.5. [0252]

  
• Alloy QSRe shows a rapid increase in hardness over the temperature range 780° C. to 870° C. corresponding to dissolution of carbides. The temperature at which carbide dissolution occurs drops 30° C. as solution time is increased from 1 to 2 hours corresponding to kinetic effects. The hardness remains relatively constant over the temperature range 900° C. to 1100° C. indicating that excessive grain growth is prevented by an effective grain refining dispersion. The solution treatment condition for alloy QSRe is 1 hour at 870° C. The predicted solution temperature is approximately 800° C. [0253]

  
• The predicted equilibrium carbides are M[0254] ₆C carbides and M₇C₃ carbides.
• The results of the gradient heat treat study conducted on alloy QSRe are shown in Graph 6.6. that is nearly identical to the experimentally determined solution temperature of the 2-hour sample. [0255]
• 3. Isothermal Tempering Study [0256]
• An isothermal tempering study was conducted on both design alloys to determine their tempering characteristics. Alloy QSW was solution treated at 1050° C. for 1 hour while alloy QSRe was solution treated at 870° C. for 1 hour. Samples were tempered at 482° C. and 510° C. for times up to 40 hours to determine the tempering response of each alloy. Graph 5.7 shows the tempering response of alloy QSW. [0257]

  
• The solution treated hardness of the samples was approximately HR[0258] _c 52.5. Alloy QSW reaches a peak hardness of HR_c 58.9 after tempering for 20 hours at 482° C. Upon tempering 40 hours at 482° C., the hardness shows a small drop to HR_c 58. The alloy shows strong coarsening resistance as expected from the presence of tungsten. As shown in Chapter 2, one slow diffusing element, such as tungsten, will impart significant coarsening resistance.
• Tempering at 510° C. allows the alloy to reach peak hardness after 2 hours. However, the maximum hardness achieved at 510° C. is HR[0259] _c 57.9. In addition, a reduction of approximately HR_c 1.5 is seen after tempering 4 hours. The hardness drops to values comparable to the solution hardness after 10 hours.
• The reduction in peak hardness is expected because the driving force for M[0260] ₂C nucleation is inversely related to tempering temperature. Lower tempering temperature will increase the maximum peak hardness because finer particles will nucleate. However, tempering times will dramatically increase because of the relatively high activation energy of tungsten-diffusion-controlled precipitation. Time to peak hardness in alloy QSW increases from 2 hours to 20 hours, as tempering temperature is reduced 28° C. While superior properties are attainable as tempering temperature is reduced, the increase in tempering time is undesirable because of added processing time. An optimal compromise should be attainable at an intermediate tempering temperature.
• Graph 6.8 presents the results of the isothermal tempering study of alloy QSRe. [0261]

  
• The tempering response of alloy QSRe is similar to that of alloy QSW. Tempering at 482° C. yields higher peak hardness, but takes longer to achieve. The peak hardness of HR[0262] _c 53.3 occurs after 12.5 hours at 482° C. This value represents an increase of approximately HR_c 6.5 over the solution treated condition. The hardness does not drop significantly after tempering for 24 hours at 482° C. Tempering at 510° C. reduces the peak hardness to HR_c 52.2 and the time to peak hardness to 4.5 hours. Tempering treatments of 10 hours reduce the hardness to HR_c 50.5.
• 4. Fracture Toughness Testing [0263]
• Fracture toughness testing was conducted on both alloys to determine the optimum combination of strength and toughness. Alloy QSW was solution treated at 860° C. for 2 hours and 1050° C. for 1 hour. One sample was tempered at 200° C. for 1 hour to complete stage one tempering. Other samples were tempered at 482° C. between 5 and 160 hours. Graph 6.9 presents the results of the fracture toughness study of alloy QSW by showing the fracture toughness vs. hardness trajectory during tempering. [0264]

  
• The fracture toughness of alloy QSW follows the anticipated pattern. At short tempering times, an increase in hardness is accompanied by a significant decrease in fracture toughness. The decrease is the result of precipitation of coarse para-equilibrium cementite. As tempering continues, para-equilibrium cementite is dissolved as M[0265] ₂C carbides form. At the longest tempering times, toughness is increased as all para-equilibrium cementite is dissolved. Hardness drops as M₂C carbides coarsen to particle sizes exceeding optimum size. Points in parentheses are estimates of samples that failed during pre-cracking due to load spikes. The max load during the spike was used to estimate the fracture toughness. Results from a prior fracture toughness test, denoted by a triangle marker on FIG. 6.9, suggest that the fracture toughness after tempering 20 hours is approximately 35 ksi{square root}in. This information suggests that the estimates err on the low side.
• Samples undergoing solution treatment for 1 hour at 860° C. were also tested. The solution treated strength and toughness of these samples were lower than that of the sample solution treated at 1050° C. The decrease in strength and toughness is due to an incomplete solution treatment. Un-dissolved carbides reduce the strength by decreasing the amount of carbon in solution. Fracture toughness decreases because the relatively coarse carbides acts as microvoid nucleation sites. One sample aged 12 hours fractured during pre-cracking. The premature fracture is the result of inferior fracture toughness arising from a combination of para-equilibrium cementite and undissolved alloy carbides. A sample tempered for 24 hours has a fracture toughness of approximately 30 ksi{square root}in. at a hardness of HR[0266] _c 55.4. This combination of properties is inferior and confirms that incomplete solution treatment will result in poor strength and toughness.
• Micrographs were taken of the fracture surfaces to determine the mode of fracture. Ductile fracture was observed in the stage one tempered (200° C./1 Hr), 1050° C. solution treated condition, as shown in graph 6.10. [0267]

  
• The fracture surfaces of samples tempered between 5 and 20 hours are quite different from the surface of the solution treated sample, as shown in a representative micrograph in graph 6.11. [0268]

  
• The fracture surface exhibits faceting on a fine scale indicating that the failure mode is quasi-cleavage. Quasi-cleavage is similar to normal cleavage except the length scale is greatly reduced due to the lath martensite microstructure. Notably absent from the figure are large facets indicative of intergranular fracture. [0269]
• As tempering time increases, hardness drops and fracture toughness increases. Fracture surfaces of samples tempered between 40 and 160 hours show a combination of quasi-cleavage and ductile microvoid fracture. The majority of the 40-hour fracture surface is quasi-cleavage with small regions of ductile fracture, as shown in graph 6.12. After 160 hours, the majority of the fracture surface shows ductile fracture, similar to the stage one tempered sample, as shown in graph 6.13. [0270]

  

  
• Fracture toughness samples of alloy QSRe were solution treated at 1050° C. for 1 hour. One sample was tempered at 200° C. for 1 hour to complete stage one tempering. Samples were tempered at 510° C. for up to 8 hours. In addition, one sample was double-austenized at 1050° C. for 1 hour and 860° C. for 2 hours. The double-austenized sample was tempered at 510° C. for 8 hours. Graph 6.14 shows fracture toughness vs. hardness for alloy QSRe. [0271]

  
• The fracture toughness drops dramatically after tempering at 510° C. for 1 hour due to precipitation of para-equilibrium cementite. Hardness increases at relatively constant fracture toughness as M[0272] ₂C carbides precipitate at the expense of para-equilibrium cementite after tempering up to 4 hours. Hardness decreases after tempering 8 hours due to carbide coarsening. Double austenizing increases fracture toughness by at approximately 12%, as seen in samples tempered at 510° C. for 8 hours. The increase in fracture toughness is attributed to finer grain size resulting from re-crystallization during the 860° C. austenizing step.
• Unlike alloy QSW, fracture toughness does not increase as hardness drops in the over aged condition. The lack of fracture toughness enhancement in the over aged condition indicates that fracture mode changes as tempering time increases. Graph 6.15 shows ductile fracture of alloy QSRe in the stage one treated condition. Graph 6.16 shows quasi-cleavage with large facets, possibly indicating intergranular fracture, after tempering 8 hours at 510° C. [0273]

  
• Fracture surface studies show that the primary mode of fracture is quasi-cleavage in both alloys. Alloy QSW has superior fracture toughness compared to alloy QSRe, indicating better resistance to quasi-cleavage. Improved quasi-cleavage resistance is likely due to the higher nickel content in alloy QSW. The nickel content in alloy QSW is 6 wt. % compared to 5 wt. % in alloy QSRe. The alloy nickel contents were set in order to maintain high strength by avoiding precipitated austenite. Comparing fracture toughness of the two samples suggests that maximizing solution nickel content will have strong positive effects on fracture toughness. [0274]
• Intergranular fracture is suppressed in alloy QSW, but present in alloy QSRe. Molybdenum is included in alloy QSW but excluded in alloy QSRe. Tungsten and rhenium are predicted to segregate to austenite boundaries during solution treatment. Molybdenum is predicted to segregate to boundaries during solution and tempering treatments. Lath martensite boundary cohesion is not enhanced in alloy QSRe because no molybdenum is present to segregate during tempering. The lack of lath martensite boundary cohesion enhancement is likely responsible for the intergranular fracture contribution limiting the fracture toughness in alloy QSRe. [0275]
• 5. Tensile Properties [0276]
• Tests were performed on both alloys to determine tensile properties. Heat treat conditions were chosen based on the results of the hardness and fracture toughness study. Alloy QSRe was solution treated at 860° C. for 2 hours and aged at 482° C. for 20 hours. Tensile bars of alloy QSW were solution treated at 1050° C. for 1 hour and [0277]

  

  tempered at 482° C. Specimens tempered 20 and 40 hours were examined. The load-displacement curve of alloy QSW tempered for 40 hours is shown in graph 6.17.

  
• Table 8 summarizes the results of tensile testing. [0278]

  TABLE 8
  Tensile properties of design alloys.

  		Ultimate		Reduction
  	Yield Strength	Tensile	%	in
  Alloy	(ksi)	Strength (ksi)	Elongation	Area (%)

  QSRe	240	274	17	59
  QSW-20 Hr	297	335	15	53
  QSW-40 Hr	293	328	12.8	55.6

• Both alloys show reasonable ductility in all tested conditions. Tensile properties of alloy QSW compare favorably to typical ultra-high strength alloys. Graph 6.18 shows a tensile property comparison of alloy QSW to several commercial ultra-high strength steels [84]. [0279]

  
• The ultimate tensile strength of alloy QSW (Quantum 335 on chart) is greater than that of any commercial alloy. In addition, the yield strength of alloy QSW is greater than the ultimate tensile strength of most commercial alloys. [0280]
• 6. Auger Electron Microanalysis [0281]
• Composition analysis of fracture surfaces from alloy QSRe was conducted by Auger electron microanalysis. The goal of this study was to determine if predicted rhenium and tungsten enrichment occurs on prior austenite grain boundaries by examining intergranular fracture surfaces. Alloy QSRe was chosen because its fracture surface exhibited large facets indicative of possible intergranular fracture. Alloy QSW was not examined because it did not exhibit intergranular fracture. [0282]
• Two solution treatments and two tempering treatments were examined by Auger electron microscopy. All samples were solution treated 1 hour at 1050° C. and tempered 8 hours at 510° C. In addition, two samples were solution treated 2 hours at 860° C. One sample from each solution treatment condition was post-tempered at 370° C. for 1 hour. The heat treatment conditions were chosen to determine the effect of solution temperature on grain boundary segregation. The post-temper treatment was designed to enhance boron grain boundary concentration. [0283]
• Samples were fractured under vacuum at −100° C. All samples exhibited similar fracture surfaces, as shown in graph 6.19. [0284]

  
• Three different regions are evident on the fracture surface. The majority of the fracture surface exhibits fine faceting indicative of quasi-cleavage. A dimpled region indicative of ductile fracture is present in small quantities. Finally, a large facet suggests intergranular fracture. Auger spectra were taken from each region to determine the composition along the fracture surface. The spectra from each region were nearly identical, as shown in graph 6.20. [0285]

  
• Iron, cobalt, nickel, carbon, and oxygen were present on all regions of the fracture surface. No tungsten, rhenium, or boron peaks were discernible from the background noise. The absence of tungsten, rhenium, and boron in the Auger spectra suggest that the facets seen on the fracture surface are not from prior austenite intergranular fracture. The large facets are more likely from fracture along lath boundaries. Since tungsten and rhenium are only expected to segregate during solution treatment, lath boundary cohesion was not enhanced in alloy QSRe. Auger analysis on the fracture surface was unable to provide any evidence of grain boundary segregation because fracture could not be achieved on prior austenite grain boundaries. [0286]
• Evidence of successful oxygen and sulfur gettering suggest the presence of oxygen on the fracture surface is most likely due to contamination in the Auger chamber. This may have limited the detectability of expected segregants such as boron. Graph 6.21 shows the fracture surface and Auger spectrum indicating the presence of an La[0287] ₂O₂S particle.

  
• The area indicated by the arrow has an apparent lanthanum concentration of 23%, sulfur concentration of 12.6%, and oxygen concentration of 27.9%. The remainder of the area consists of iron, cobalt, and nickel from the matrix. The La:O:S ratio is nearly identical to the expected stoichiometric ratio of La[0288] ₂O₂S.
• 7. Analytical Electron Microscopy [0289]
• Analytical electron microscopy was used to analyze prior austenite grain boundaries. A prior austenite grain boundary was found in alloy QSRe solution treated 1 hour at 1050° C. and tempered 1 hour at 200° C., as shown in graph 6.22. [0290]

  
• Since many lath packets terminate at the boundary, it is concluded that the boundary is a prior austenite grain boundary. [0291]
• X-ray spectra were taken at seven points along the grain boundary and two points in the bulk. For each point, W M radiation peaks were normalized by the Fe K, Cr K, Co K, and Ni K radiation. The grain boundary points were averaged and compared to the average of the bulk points to determine if any grain boundary enrichment occurred. The spectrum from each grain boundary point consists of contributions from the grain boundary region and bulk region due to beam size. The analysis assumes a grain boundary thickness of 0.1 nm and a beam diameter of 5 nm. Based on these assumptions, the grain boundary region will comprise approximately 27% of the total beam area. The relation given in equation 6.1 determines the grain boundary partition ratio. [0292] $\begin{array}{cc}\frac{{\mathrm{Spectrum}}_{G.B.}}{{\mathrm{Spectrum}}_{\mathrm{bulk}}}=\frac{\alpha X_{\mathrm{bulk}}^iA_{G.B.}+X_{\mathrm{bulk}}^i\left(\pi R^2-A_{G.B.}\right)}{X_{\mathrm{bulk}}^i\pi R^2}& \left(6.1\right)\end{array}$

  
• R is the beam radius, X[0293] ⁱ _bulk is the alloy composition of element i, A_G.B. is the area of the grain boundary region, and α is the grain boundary partition ratio. Values greater than 1 correspond to grain boundary enrichment. Grain boundary composition is determined by multiplying bulk composition by the partition ratio. Table 9 gives the site fraction of tungsten and rhenium when normalized by each element.

  TABLE 9
  Site fraction data from analytical electron microscopy.

  Normalization Element	Tungsten Site Fraction	Rhenium Site Fraction

  Fe	0.0416	0.186
  Co	0.0384	0.179
  Cr	0.0354	0.165
  Ni	0.0362	0.171

• The results presented in table 9 show that enrichment of rhenium is approximately 60% of the model predicted value of 0.29 shown in [0294] chapter 4. The grain boundary site fraction of tungsten is approximately 0.04 which corresponds to an enrichment of approximately 10 times the bulk. However, this site fraction is significantly lower than the model predicted value of 0.66.
• Based on the assumption that grain boundary segregation is not kinetically limited at 1050° C., segregation free energies and enthalpies were determined for tungsten and rhenium based on the experimental data. The segregation free energy of tungsten at 1050° C. was determined to be approximately −27,300 J/mol while the segregation free energy of rhenium at 1050° C. was −36,500 J/mol. [0295]
• Graph 6.23 shows tungsten grain boundary segregation energy as a function of temperature in FCC and BCC iron. [0296]

  
• The figure shows the quantum mechanics estimates of grain boundary segregation energy at 0 K. The straight line connects the FCC quantum mechanics estimate with the experimental data point, corresponding to a segregation entropy of −37.6 J/molK. This segregation entropy is of similar magnitude but opposite sign to that used in the original grain boundary cohesion model, which was based on interstitial element data rather than substitutional elements. BCC segregation energy is calculated from experimental site fraction data from Lee, et al. over the temperature range 773 K to 873 K. The dashed line represents a fit through the Lee data using the same segregation entropy of −37.6 J/molK estimated for FCC. The experimental data are consistent with the quantum mechanics prediction that the energy of tungsten segregation is larger in FCC than BCC. [0297]
• Q. Toughened Tungsten Steel [0298]
• Composition modifications should enhance cleavage resistance because the fracture mode at peak hardness was quasileavage. The first alloy design limited nickel content to 6 wt. % to avoid precipitated austenite. The re-design will not place any constraints on nickel content related to austenite precipitation. Since the carbide dispersion in alloy QSW meets the requirements, chromium, molybdenum, and tungsten compositions will not change in the modified alloy. In addition, cobalt is held constant to that of alloy QSW. Carbon content will increase from 0.25 wt. % to 0.27 wt. %. Carbon is increased to provide a slight boost in carbide fraction. The increased carbide fraction is intended to maintain strength levels comparable to alloy QSW by offsetting possible softening from precipitated austenite. Nickel content is maximized within the constraint that martensite start temperature must not drop below 225° C. in order to limit retained austenite. The re-designed composition (in wt. %) is Fe-15Co-3Cr-1.7Mo-7.3Ni-2W- 0.27C. The predicted martensite start temperature is 232° C., which allows for slight composition variations (±0.1 wt. %) that may arise during melting. [0299]
• In order to increase ultimate tensile strength from 335 ksi to 375 ksi, alloy carbon content must be raised to increase M[0300] ₂C carbide fraction. Graph 7.1 plots ultimate tensile strength vs. M₂C fraction for both quantum steels and commercial AerMet100. Carbide fraction is calculated with additions of 16,000 J/mol added to the M₂C phase for each alloy to account for capillary of 3 nm particle.

  
• Three fits were used to extrapolate to higher strength levels. First, a liner fit was placed through the data. Two fits were used based on previous models. One assumed a relationship based on carbide fraction to the V[0301] ₂ power. The other assumed a y-intercept of 81.6 ksi an fit an exponential dependence on carbide fraction. The three equations are shown on graph 7.1. An ultimate tensile strength of 375 ksi will require a minimum carbide fraction of 4.16% based on the linear fit or 4.24% based on the other two fits. The three nearly identical over the strength range of interest and can be used interchangeably.
• ThermoCalc calculations using the composition of alloy QSW and varying carbon content were performed to determine the minimum alloy carbon content needed to ensure required carbide fraction. Calculations added 16,000 J/mol to the coherent M[0302] ₂C phase to account for capillary energy at the optimal particle size of 3 nm. Graph 7.2 shows the effect of alloy carbon content on M₂C fraction tempered at 482° C.

  
• The minimum alloy carbon content in alloy QSW needed to ensure sufficient M[0303] ₂C carbide fraction is 0.298 wt. %. The carbon content is set at 0.32 wt. % for the alloy re-design of Quantum 375. Excess carbon was intentionally set to allow significant overaging of the alloy in order to fully dissolve cementite.
• In addition, graph 7.2 shows that alloy QSW has an excess of M[0304] ₂C carbide forming elements relative to alloy carbon contents up to 0.4 wt. %. Since alloy QSW has ample carbide forming elements at a carbon level of 0.32 wt. %, the Quantum 375 re-design maintains the same chromium, molybdenum, and tungsten contents to maintain grain boundary cohesion enhancements designed in alloy QSW.
• The design examined the effect of cobalt compositions between 15 wt. % and 20 wt. % on allowed nickel content, solution temperature, and carbide fraction. Nickel content was set by the constraint that martensite start temperature be at least 225° C. Graph 7.3 shows the effect of cobalt on nickel content, carbide fraction, and solution temperature. [0305]

  
• Increasing cobalt has positive effects on all three properties. Increasing cobalt causes a minor increase in carbide fraction. Solution temperature drops nearly 20° C. as cobalt is increased from 15 wt. % to 20 wt. %. As cobalt content is increased, allowable nickel content is increased. The design alloy will have a cobalt content of 19 wt. % to lower solution temperature to that of alloy QSW and increase the alloy nickel content to 7.1 wt. %. The design composition is then Fe-19Co-3Cr-1.7Mo-7.1Ni-2W-0.32C. [0306]
• Tungsten and rhenium primarily segregate during solution treatment while molybdenum and interstitial elements segregate during tempering. [0307]
• A study of the heat treatment optimization has investigated the response of both design alloys to various solution and tempering treatments. The solution treatment study examined the effect of solution temperature and time on as-quenched hardness. The optimum solution treatment of alloy QSW is 1 hour at 1050° C. corresponding to predicted complete dissolution of Cr-rich and W-rich carbides. The optimum solution treatment of alloy QSRe is 2 hours at 860° C., also consistent with model predictions. [0308]
• The tempering study examined the effect of two tempering temperatures on strength. Alloy QSW reached a peak ultimate tensile strength of 335 ksi (HR[0309] _c 58.8) after 20 hours at 482° C. The fracture toughness in the peak strength condition is 35 ksi{square root}in. Tempering 40 hours at 482° C., to more fully dissolve cementite, increases fracture toughness to 47 ksi{square root}in with minimal reductions to strength. Alloy QSRe reached peak ultimate tensile strength of 274 ksi (HR_c 53.3) after 20 hours at 482° C. Samples tempered for 8 hours at 510° C. achieved hardness of HR_c 51.6 and fracture toughness up to 44 ksi{square root}in. Fracture surfaces of alloy QSW showed no evidence of intergranular fracture suggesting effective cohesion enhancement.
• Analytical electron microscopy examined a prior austenite grain boundary in alloy QSRe. Results show a grain boundary site fraction of rhenium of approximately 0.175 corresponding to an enrichment of nearly 20 times the bulk. The model predicted rhenium site fraction is 0.29. Grain boundary tungsten site fraction is approximately 0.0375 corresponding to an enrichment of approximately 10 times the bulk. The experimental value differs significantly from predicted tungsten site fraction 0.664. The large discrepancy between experimental and predicted tungsten site fraction emphasize the need for accurate tungsten diffusion and segregation information. [0310]
• Low toughness in alloy QSRe arises from a combination of low nickel content and intergranular fracture along lath boundaries. Since molybdenum was excluded from the design, no lath boundary cohesion enhancement is expected in alloy QSRe. Auger microanalysis did not reveal any evidence of tungsten or rhenium enrichment along intergranular fracture surfaces. Since analyticaI electron microscopy showed tungsten and rhenium enrichment along prior austenite grain boundaries, it is concluded that intergranular fracture does not occur along prior austenite grain boundaries. [0311]
• Two alloy compositions have been proposed to improve mechanical properties of the prototype alloys. One alloy composition seeks to maximize nickel content relative to alloy QSW within the constraint that martensite start temperature be greater than 225° C. Carbon content is increased slightly to anticipate potential reduction in strength that may arise from precipitated austenite. [0312]
• The second alloy composition is designed to reach an ultimate tensile strength of 375 ksi. The carbon, nickel, and cobalt contents of alloy QSW are modified to achieve this goal. Carbon content is raised from 0.25 wt. % to 0.32 wt. % in order to raise carbide fraction. Cobalt and nickel content are optimized within the constraint that martensite start temperature be greater than 225° C. Cobalt content is raised from 15 to 19 wt. % to reduce solution temperature and increase alloy nickel content to over 7 wt. %. [0313]
• While a preferred embodiment of the method and compositions have been set forth, the invention is limited only by the following claims and equivalents thereof. [0314]

Claims (7)

What is claimed is:

1. An alloy, comprising in combination:

a crystalline matrix of at least one metal selected from the group consisting of iron and nickel, wherein said matrix includes crystals with grain boundaries, and at least an alloying element selected from the group consisting of tungsten (W), rhenium (Re), osmium (Os), niobium (Nb), iridium (Ir), technetium (Te), ruthenium (Ru), platinum (Pt), tantalum (Ta), zirconium (Zr), hafnium (Hf), vanadium (V) and titanium (Ti) in an amount that enhances cohesion at said boundaries characterized by segregation of said alloying element to the grain boundaries and the term ΔE_B ^A being negative, said alloy being processed by low temperature heat treatment to segregate said alloying element to a grain boundary.

2. The alloy of claim 1 wherein said matrix is a nickel compound and said alloying element is selected from the group consisting of osmium (Os), rhenium (Re), ruthenium (Ru), tungsten (W) and niobium (Nb).

3. The alloy of claim 1 wherein said matrix is an iron compound and said alloying element is selected from the group consisting of tungsten (W), rhenium (Re), niobium (Nb), and osmium (Os).

4. A generally solution heat treated alloy composition consisting essentially of a formulation in weight percent of

0.18-0.40 C

15.0-20.0 Co

6.0-7.5 Ni

2.0-4.0 Cr

0-1.7 Mo

1-3.0 W

0-3.0 Re

Balance Fe

and one or more additives in weight percent selected from the group consisting of up to 0.05 % Ti, 0.010 % La, 0.02 % Zr, 10-20 ppm B, combinations thereof and impurities, said alloy solution heat treated in the range of 800° C. for at least one hour to form carbides characterized primarily as M₂C and to effect migration of Mo, W and/or Re to grain boundaries.

5. The alloy of claim 4 subjected to tempering in the range of 450° C. to 550° C. for at least one hour to increase hardness.

6. The alloy of claim 4 having an M_S temperature in the range of about 225° C. to 400° C.

7. The alloy of claim 4 wherein the mole percent of M₂C carbides is greater than 0.030.

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