Towards a chemistry of cohesion and adhesion

Progress in Surface Science, Vol. 36, pp. 1-34 Printed in the U.S.A. All rights reserved. 0079-6816/91 $0.00 + .50 Copyright © 1991 Pergamon Press plc TOWARDS A CHEMISTRY OF COHESION AND ADHESION M.E. EBERHART Laboratory for Materials Synthesis, MassachusettsInstitute of Technology, Cambridge, MA 02139 M.M. DONOVAN Department ofChemist~ Harvard University, Cambridge, MA 02138 J.M. MACLAREN Department of Physics, Tulane University, New Orleans, LA 70118 and D.P. CLOUGHERTY Department of Physics, University of Cafifomia, Santa Barbara, CA 93106 Abstract Modem chemistry frequently describes the structure and reaction dynamics of molecules in terms of the general principle of "competition for bonds"; consequently, bonding forms the basis of the language of chemistry. The actual models used to represent these bonds are frequently system specific. Organic reactions are described in terms of bonds based on pairs of atomic valence electrons. Reactions of inorganic coordination complexes are described in terms of bonds based on a molecular orbital representation. In analogy to those chemistries, a representation for a bond and bond strength, suitable for describing the cohesive and adhesive properties of all classes of materials, is introduced. This representation proves to yield an explanation for the observed cohesive properties of a specific class of materials (cleavage in bcc metals), and it also provides a framework for exploring and analyzing the M.E. Eberhart et al. 2 more complex phenomena of cohesion and adhesion, such as environmentallyinduced embrittlement. A complete chemistry of cohesion and adhesion will require the demonstration that the specific bonding model used can form the basis for consistent interpretations for a wealth of experimental phenomena beyond environmentally-induced embrittlement; thus, as presented, this model does not provide a complete chemistry of cohesion and adhesion, but does embody the first steps in that direction. Confers Page 1. Introduction 3 2. The Molecular Structure Hypothesis 4 3. Bonds in Metals 4. 10 A. The Charge Density Between First Neighbors of FCC Metals 10 B. Second Neighbor Bonds 12 20 Fracture as Chemistry A. Cleavage of BCC Metals 20 B. IntergranularFailure 27 C. Transition Metal Aluminides 28 5. Conclusions 32 6. Acknowledgements 32 7. References 33 Abbreviations bee fcc LKKR MS-LDF MO Body centered cubic Face centered cubic Layer Korringa-Kohn-Rostoker Multiple-Scattering Local Density Functional Molecular orbital Chemistry of Cohesion and Adhesion 3 1. Introduction Increasingly sophisticated studies of condensed matter systems have led to a highly detailed understanding of the fundamental origins of bulk properties of materials. Those aspects of atomic structure which influence the electrical and thermal properties of matter are now well understood. Yet bulk properties represent just one aspect of a more complete and fundamental understanding of materials performance. In order to design and synthesize products which satisfy specified performance criteria, it is necessary to not only understand the origins of bulk properties, but also to understand how these properties change with the coexistence of different materials. These changes are determined by the properties of the surfaces between single phase bulk materials, i.e. the interfaces. When confronted with performance criteria requiring the development of a new material, it is possible to select from a large number of single phase materials from which to construct the desired end product. The next step in the materials development process, however, involves the consideration of a seemingly infinite number of interfaces which can be produced between the starting materials. It is therefore necessary to develop a methodology to limit the number of interfaces which must be explored experimentally or theoretically. Chemistry is one such methodology which can be applied when the properties under consideration are controlled by atomic scale interactions. In this paper we will discuss the progress which has been made in the development of a chemistry of the mechanical properties of surfaces and interfaces, as the first step in the development of a general chemistry of interfaces. There is a natural correspondence between a chemistry of interfacial mechanical properties and more traditional chemistries involving the identification and control of chemical reactions. This correspondence identifies the interface with the reactant, the atomic strain with the reaction path, and the surfaces formed as result of fracture with the products of a reaction. This identification will allow us to borrow strongly from the well developed methodology of chemical reaction theory in developing a chemistry of interfaces. In Section 2, we begin by briefly reviewing those concepts of chemistry and reaction theory which are needed for the approach developed in this paper. We will then demonstrate how these concepts have been successfully applied to the study of surfaces and interfaces in specific materials, and we will identify those areas where work is needed before these concepts can be generalized to the study of arbitrary interfaces. In Section 3, we will expand these chemical notions to understand the structure of metals; and in 4 M.E. Eberhart et al. Section 4, we will show how the methodology developed can be broadly applied to a study of the mechanical properties of surfaces and interfaces. 2. The Molecular Structure Hypmhesis All of chemistry is based on the molecular structure hypothesis, which simply stated is the following: a molecule or solid is a collection of atoms linked by a network of bonds, and all molecular properties result from this linked system of atoms. Accompanying this hypothesis are a number of models which seek to quantify these concepts of atoms and bonds and to correlate empirically-determined properties with the models. While each of the models represents these concepts differently, they have all had considerable success in explaining the observed properties of matter. Of the many models advanced in support of the molecular structure hypothesis, the most familiar is the Lewis model [1]. Within this model a bond is pictured as a pair of valence electrons shared by two atoms. This representation of a chemical bond, and its quantum mechanical extensions of directed valence [2-4] form the basis for understanding the chemistry of organic molecules. Structural, chemical, spectroscopic, and thermochemical data are interpreted using terms defined within the Lewis model of molecular structure. Other fields have embraced alternate representations of bonds. In coordination chemistry, for example, the bond is modelled through crystal field theory, ligand field theory, and molecular orbital theory [5-7]. Once again, empirically-derived information concerning coordination complexes is described using the terms of these models. For solid state systems, the bond is modelled within band extensions of the molecular orbital framework. Again, solid state properties are interpreted from the results of these extended models of molecular structure [8-11 ]. All of the chemistries which seek to provide a fundamental understanding of chemical reactivity adopt the "mechanistic approach" [12-13]. While the model of the bond may be different for each of these classes of chemistry, e.g. bio-, organic, inorganic, or coordination chemistry, the mechanistic approach to chemical reactivity is identical. Atoms are assumed to react along a reaction path, which is parameterized by the atomic positions. As the atoms move along this path their bonds are altered with an accompanying change in the molecular structure. This change in bonding is the reaction. Those atoms which actually experience bond redistribution as they move along this reaction path are referred to as the activated complex. The variation of the potential energy of the activated complex at each point along the reaction path is called the reaction potential Chemistry of Cohesion and Adhesion 5 and the highest energy along the path is the activation barrier. The arrangement of the atoms of the activated complex at the point of the activation barrier is called the "transition state." The goal of the mechanistic approach to chemical reactivity is to understand the source of changes in the reaction potential from the model of the bonding appropriate for the system of atoms undergoing reaction. To help make this point clearer, we will illustrate with two examples the mechanistic approach where different bonding models are evoked. As our first example we consider a conventional cohesion problem, the stresscorrosion cracking in silica glasses. Silica glass shows a decrease of strength with time under load in ambient environments. This is believed to be the result of slow growth at .. H~O~ ~i |i--O H I Si /1~ [AI \!< \]< I o H"~'~)H \O [Si I I H HI O I /I x /IlL [B] [el Fig. 1 Schematic representation of the mechanism of stresscorrosion cracking in silica glass the tip of existing flaws. Michalske and Freiman [14] proposed the following mechanism to account for the observed behavior. The silica glass structure is represented within the Lewis or valence-bond representation of molecular structure. In step [A], a water molecule from solution attacks the strained Si-O-Si bond at the crack tip. The water molecule is believed to be aligned by: (1) formation of a hydrogen bond with the O~dgi~ atom, and (2) interaction of the lone pair orbitals from O,,~, with the Si atom. The transition state is proposed to occur in step [B] with a concerted proton transfer to O~dg~r~ and electron transfer to the Si atom from Ow~e. In step [C], the reaction is completed through the rupture of the hydrogen bond between Ow~, and the transferred hydrogen to yield surface Si-O-H groups on each fracture surface. 6 M.E. Eberhart et al. To influence the rate of this reaction (i.e., to inhibit (promote) stress-corrosion cracking), one must destabilize (stabilize) the transition state. Conversely, those interactions which lower the energy of the transition state will favor the forward reaction, i.e. the stress-corrosion reaction. Those interactions which destabilize the transition state will favor the reverse reaction, inhibiting stress-corrosion cracking. As the transition state is thought to occur in step [B], an environment which provides lone pair donor sites opposite proton donor sites is required to promote stress-corrosion cracking in vitreous silica. Consequently, the rate of the stress-corrosion reaction should be controlled by two factors. The first factor is related to the efficiency with which the Si atom draws electron density from the lone-pair reacting environment (in this case, water). This efficiency will be determined by the relative electronegativity of the Si atom at the crack tip. The addition of elements to the glass which increase the electronegativity of the Si atoms will promote stress-corrosion cracking. The second factor which controls the rate of the stress-corrosion reaction is related to the strength of the hydrogen bond formed between Ob,dgmgand the proton of the corroding environment. The hydrogen bond strength is, in turn, controlled by the dipolar nature of the corroding environment. Those atoms which strongly withdraw electron density from the bound protons will lead to effective hydrogen bonding between the environment and Ob,dg~ng,lowering the activation energy and promoting the forward stress-corrosion reaction. In a series of experiments carefully designed to test the proposed mechanism [ 14], it was found that these predictions were correct. Water, ammonia, hydrazine, and formamide increase the rate of slow crack growth in silica glass while carbon monoxide, acetonitrile, and nitrobenzene have only a small effect (which can be explained by low level water contamination). These experimental results confirm the observation that both proton donor and electron donor sites are needed to stabilize the transition state. For the environments which promote the stress-corrosion reaction, their success is directly related to the efficiency with which they form hydrogen bonds. The corresponding reaction rates, as measured by the crack growth rate as a function of stress intensity, decrease through the series water, ammonia, hydrazine, and formamide. Finally, in a second set of experiments [ 15], it was shown that soda lime glass is less reactive than pure silica glass. One possible explanation of this observation is that the electron-withdrawing power of the Si atom is changed as a result of the Na additions. In the previous example, chemical reactivity was built around the Lewis model of molecular structure, and it was explained in terms of the efficiency of an atom to draw electron pairs and the availability of these electron pairs; the chemical reactivity is described as a competition for bonding electrons. Chemistry of Cohesion and Adhesion 7 Another example of the application of chemical reaction theory to a problem of cohesion and adhesion comes from the field of heterogeneous catalysis on metal surfaces. Here the bond is represented within a molecular orbital framework, but the reaction rate is once again explained in terms of a competition for bonding electrons. Consider the catalytic hydrogenation of CO to methane and other hydrocarbons: CO + 5H 2 ¢=~ CH 4 + 3H20 (1.1) This reaction proceeds over a transition metal catalyst, typically Ni. The reaction path is thought to involve surface carbon, resulting from the dissociation of CO to chemisorbed C and O. In a series of experiments by Araki and Ponce [16], the catalyst surface was covered with lsC before exposure to H 2 and 1R20. The initial methane yield consisted primarily of ~sCH4,and ~2H 4 was produced only when most of the surface lsC had been exhausted. It is believed that the dissociation of CO to C and O is the rate limiting step in the hydrogenation reaction; therefore, the transition state lies on the CO dissociation reaction path. The influence of poisons and promoters can be understood in terms of their effect on the dissociative reaction. Poisons, typically electronegative elements such as S, dramatically reduce the rate of product formation. Electronic structure calculations [17] show that chemisorbed S forms strong bonds to the substrate, hybridizing with states that would form CO-substrate bonds in the absence of S. One substrate state, which is no longer available to interact with CO, is the anti-bonding 2p* molecular orbital; this state in tum hybridizes with CO through back donation, promoting dissociation. This reduction in the states available for bonding interactions due to the presence of S will dramatically decrease the ability of the surface to respond to CO; consequently inhibiting chemisorption and dissociation. In effect, S competes much more effectively for surface bonds than CO, since, for example, the S-Ni surface bond energy is about 4 eV, while a CO-Ni bond energy is about 1.5 eV. These two mechanistic investigations have started with a representation of a bond and a measure of how atoms compete for bonding electrons. If we wish to develop a general chemistry with which to study cohesion and adhesion within and between materials, it is first necessary to evoke an appropriate representation of a bond and define a measure for the competition for bond formation in network, ionic, covalent, metallic, and van der Waals solids. A description of bonding which has the potential to describe 8 M.E. Eberhart et al. both molecular and solid state bonding has been outlined in a series of papers by Bader [ 18-22]. The framework proposed is empirical in nature, but it has faithfully reproduced conventional descriptions of molecular bonding; it produces the same topological features in the total charge density as the conventional valence bond or Lewis approach. As this method is based on features of the total charge density, there is no a priori reason to assume that it should not be fully generalizable to different types of bonding in different classes of materials. To our knowledge, there have been no investigations applying this approach to other classes of compounds, and, in particular, to bonding in condensed matter systems. The fact that bonds can be described by features of the total charge density is particularly appealing, as the total charge density is experimentally accessible through X-ray diffraction studies and one of the principal results of electronic structure calculations. In addition, the ground state charge density is in principle exactly calculable by techniques which solve the Kohn-Sham equations. From the results of many calculations [19-22], it has been observed that the electronic charge density can be accurately described by the relative positions and magnitudes of critical points: that is, maxima, minima, and saddle points. The only maxima in the charge density occur at atomic nuclei, and these are cusps rather than true maxima. At other positions, there are minima and saddle points. Atoms are modelled as space-filling objects defined by surfaces which satisfy the condition Vp = 0, and correspond to the conventional Wigner--Seitz cells in elemental solids. Furthermore, within this formalism, bonds are associated with the occurrence of saddle points in the charge density between two bound atoms. To be precise, the bond path is defined as the line of steepest descent in the charge density originating at a nucleus and terminating at a saddle point. There are only two such paths linking a saddle point to two of the neighboring nuclei, leading to a line joining these two nuclei along which the charge density is a maximum with respect to any neighboring lines. This definition, while empirical, has been shown by many examples to produce a bond topology which is identical to that of usual valence bond description of the molecules studied. Bonds defined within this formalism do not have to be straight lines. For example, it has been shown that in cyclo-propane (which is known from calorimetric experiments to be destabilized by bond strain) the saddle points lie just outside the triangle whose vertices correspond to the carbon atoms of the three member ring; thus the bonds are curved, reflecting the strained nature of the C 3 ring. On the other hand in benzene, which is known to be resonance stabilized, the saddle points connecting the six carbon atoms are within the ring. These observations suggest that a characteristic of the total charge density, the saddle points, reflect the relative chemical reactivity. The saddle points in Chemistry of Cohesion and Adhesion 9 benzene are less accessible to attack than are those in cyclo-propane, reflecting benzene's greater structural stability. It has been shown through a number of examples [21], that structural and chemical instabilities which occur during the bond rearrangement of a chemical reaction can be associated with the crossing of bond paths at the transition state. As the reaction pathway is traversed, the reactant bonds distort continuously until the transition state is reached. At this point, the product bond paths cross those of the reactants and the product is formed. In another set of studies on diatomic molecules [18], it was shown that the curvatures of the charge density evaluated at the saddle point (measured by the Laplacian) provided quantitative information on the bond strength. This formalism of defining a bond from features of the total charge density has been applied primarily to organic systems and only to small molecules. We have begun the development of a chemistry of cohesion and adhesion by extending these concepts of a bond to condensed matter systems, in particular to metallic systems where there is no existing representation of a bond as a linkage between atoms. Any representation of a bond which is faithful to the molecular structure hypothesis must provide insight into the relationship between the bonding and the atomic arrangement from which it derives. Attempts to generate such descriptions of metals and alloys are have been restricted to the theories of Engel and Brewer [24-26] and Hume-Rothery [27, 28]; however, these theories do not evoke the concept of a bond as an atomic linkage. Instead, they attempt to correlate the observed atomic arrangements in metals and alloys with features of the electronic structure of the isolated constituent atoms. In this work, we will determine the nature of the total charge density and the corresponding bonding for two metallic structures: the lace and fcc structure. We will then attempt to correlate this bonding with the empirically-determined cohesive properties of these materials; thus, we will attempt to establish relationships between structure and properties where structure now refers to the combined structure of atoms and bonds rather than simply the typical metallurgical description of atomic structure. The calculations of the charge densities which follow were performed using the layer Korringa-Kohn-Rostoker (LKKR) method [29, 30] and the multiple-scattering local density functional (MS-LDF) cluster method [31, 32]. Any electronic structure technique could have been used, since charge density is a physical observable and should be basisindependent. The methods adopted here are particularly suited to the low-symmetry environments found at interfaces. 10 M.E. Eberhart et al. 3. Bonds in Metals A. The Charge Density Between First Neighbors of FCC Metals We began our investigation by performing LKKR calculations, obtaining the charge density for a series of metals at a point midway between first neighbor atoms (i.e., at the Wigner-Seitz boundary). If first-neighbor atoms in metals can be described as being bound in the sense required by the Bader formalism, we expect to find saddle points in the charge density at points midway between these atoms, with the charge normal to the internuclear axis decreasing away from the axis (negative curvature) and the charge parallel to the internuclear axis rising away from the midpoint towards the nuclei (positive curvature). These features have been found in all metals studied. Figure 2 shows the charge density in a (100) plane of fcc Ir centered midway between first neighbor atoms. There is a negative curvature perpendicular to the internuclear axis. The charge density on lines normal to and parallel to the internuclear axis are shown in Fig. 2. C B =. 0.0562 e~ c- ,~ 0.0558 03 c- £3 0.0554 t- B O A Parallel Direction B 0.05524 C .~ 0.05523 03 o tO 0.05522 C A Normal Direction Fig. 2 Charge density in the portion of the (100) plane offcc Ir shown in the upper left. Right: traces of this charge density on the lines normal and parallel to the internuclear axis C Chemistry of Cohesion and Adhesion 11 The calculated curvatures normal to and parallel to the intemuclear axis for a variety of fcc metals are compiled in Table 1. In the covalent materials studied by Bader, the curvature normal to the internuclear axis was negative and larger in magnitude than the curvature parallel to the axis. In the metals we studied, the normal curvature is negative, but it was one or two orders of magnitude smaller than the parallel curvature; however, both covalent and metallic bonds are successfully described in terms of features of their charge densities. Figure 3 shows a plot of the bulk modulus versus the curvature of the charge density at the saddle point parallel to the internuclear axis. The linear relationship between bulk modulus and curvature demonstrates the correlation between mechanical properties and features of the total charge density, identifying the bond strength with the charge density curvature. TABLE 1. Curvatures in Normal and Parallel Direction for Some FCC Metals Elemen~ A1 Cu Pd Ir Ag Rh Pt ._. 5 Parallel Curvature (in a.u.) Normal Corvature (in a.u.) 0.002150 0.053277 0.083035 0.207735 0.012748 0.169617 0.167651 -0.000197 -0.000504 -0.000794 -0.004578 -0.000510 -0.001696 -0.001552 0.3 t- °m 0.2 -I P "-I o n 0.1 m 0.0 . m 0 2 Bulk Modulus (dynes/cm) Fig. 3 Bulk modulus versus parallel curvature 12 M.E. Eberhart et al. B. Second Neighbor Bonds In fcc metals, second neighbor atoms sit at the vertices of regular octahedra. If second neighbor atoms are bound only when there is a saddle point in the charge density at the center of the octahedron, symmetry precludes the formation of second neighbor bonds in an elemental fcc material. The curvature along all the second neighbor axes must be equivalent by symmetry, and hence the curvature normal to a chosen axis must be equivalent to the curvature parallel to that axis. By definition, the curvature at such points can be only a minimum or a maximum -not the saddle point which identifies bond formation. Figure 4 is a plot of the charge density in a (100) plane centered on the octahedral hole of Al. The charge density displays a clear minimum between the second neighbor atoms. In bcc metals the equivalent hole is not a regular octahedron. Of the three pairs of atoms at the opposite vertices, one pair is second neighbors and the other two pairs are third neighbors; hence, the local symmetry is D4hand not Oh; hence in bcc metals, a second neighbor bond can be formed since the curvature parallel to the second neighbor axis need not be symmetry-equivalent to the curvature normal to this axis. Shown in Fig. 5 is a surface plot of the charge density in a (100) plane of bcc Ta. Note that it is very different from fcc A1 (Fig. 4), which displays a minimum. For Ta, there is a Fig. 4 Surface plot for the maximum at the midpoint between second neighcharge density in the bor atoms, and therefore the midpoint is a maximum shaded region of the on the line normal to the internuclear axis; however, (100) plane of A1. parallel to the second neighbor axis (Fig. 5), the charge density is a minimum at the midpoint, making this location a saddle point. The second neighbors in Ta are consequently bound. This is a general feature of all the bcc metals studied; all show second neighbor bond formation. The existence of second neighbor bonds in bcc metals could simply reflect the structure, i.e. all bcc geometries -whether or not they actually exist for a given metalwill exhibit second neighbor bonds. Conversely, second neighbor bond formation could be a requirement for the stability of the bcc structure, i.e. only those metals capable of Chemistry of Cohesion and Adhesion 13 forming second neighbor bonds will be stable in the bcc structure. If the latter is true, then metals which form the fcc structure should not show second neighbor bond formation, even when constrained to a bcc structure. To test this supposition, we performed MS-LDF cluster calculations on various metals using six-atom polyhedra. The polyhedra used were the perfect octahedron of the fcc structure and a distorted octahedron of the bcc structure. The interatomic distances were determined by the bulk atomic radius or by the distance between first neighbor atoms in the native crystal structure. In comparisons of cluster and LKKR charge densities, we have observed similar charge density topologies; although the magnitude of the curvatures varied between calculations, the sign of the curvature was independent of the method. The charge density in the plane containing second neighbor atoms (the Ov plane) of the bcc octahedron for Mo, Ir, and Cu has been calculated. One observation is that Mo possesses the saddle point characteristic of second neighbor bond formation, while Cu and Ir do not. Although Cu and Ir are constrained to the bcc geometry, the charge density at the center of the distorted octahedron is a minimum, but not as deep as that in the fcc octahedron. The curvatures computed at the centers of the octahedra decrease through the series Cu>Ir>0>Mo. This ordering suggests a relationship between bonding and properties. Simple structureproperty relationships suggest that fcc materials exhibit ductile behavior. Ir, though fcc, fails with a brittle morphology more characteristic of bcc materials. If the concept I -' t~ 0.0396 .=~ ~ 0.0394 'rn 0.0392 \ ¢.. o Normal to Shaded Plane Fig. 5 Surface plot for charge density in the shaded region of the (100) plane of Ta and trace of the charge density normal to this plane. 14 M.E. Eberhart et al. of structure is generalized to be consistent with the molecular structure hypothesis, as modelled by features of the total charge density, the Cu-Ir-Mo sequence defines a continuum of structures which pass smoothly from one arrangement of atoms (fcc) to another (bcc). The limitations of the standard metallurgical structure-property relationships do not affect the bonding-property relations expressed here. By associating a bond with a saddle point in the total charge density between two atoms, we have observed a relationship between second neighbor bonds and the bcc structure. In those systems where second neighbor bond formation is not possible, the fcc structure is preferred over the bcc structure. The occurrence of bonds in both the fcc and bcc structure constrains the distribution of charge throughout the Wigner-Seitz cell. Of particular importance are the relative positions of the critical points in the charge density: the maxima, minima, and saddle points. In Fig. 6, two octahedral holes of the fcc structure are pictured. Shown beneath these structures are possible charge density distributions along the line a-b-a. Point b is the midpoint between first neighbor atoms, and it therefore must correspond to a saddle point in the charge density; along the line a-b-a, it will appear as a maximum. Point a is the center of the octahedral hole, and the charge density must, by symmetry, correspond to a m i n i m u m with positive curvature. The charge distributions pictured in Fig. 6 are not however all consistent with the fcc structure. Curve 1 possesses a minimum at point b and a maximum (corresponding to a saddle a b a point) at point a. Such a charge distribution, in which there is no bond between first neighbor atoms, must correspond to a structural instability. 2 ~X4.J 3 O< .4 ~Z While each of the curves 2, 3, 4, and 5 is consistent with the fcc structure, these charge dis~z z~ tributions correspond to very different physical ! mO properties. In Fig. 7, the calculated charge densities along this line for the fcc metals A1, Cu, Pd, and Ir are compared. Moving through this series, there is an increase in the directional character of the Fig. 6 Possible charge densities bonding. Rigorously, an increase in directionality in the octahedral holes is associated with an increase in the curvature of the of the fcc structure. J Chemistry of Cohesion and Adhesion 15 0.055 t°_ O) t-- 0.045 Ir a 0) 0.035 Pd t'- 0.025 Cu AI b a b Fig. 7 Calculated values of charge density along b--a-b for selected metals. (Curves have been translated to allow for comparison.) charge density normal to the internuclear axis at the saddle point at point a, or equivalently with an increase in the ratio of the charge density at points a and b, as defined in the above diagram. The increase in the bond directionality gives rise to a greater resistance to deformation, and this directionality is frequently said to be the cause of the high shear modulus and anomalous mechanical properties of Ir. By associating the strength of the bond with the curvature of the charge density, one can conclude that the strength of the second neighbor interactions (mediated by the charge density at point a) and the strength of the first neighbor interactions are inversely coupled. As one moves through the series of curves 2 through 5, the second neighbor interactions are weakened with a corresponding increase in the curvature of the charge density normal to the intemuclear axis between first neighbor atoms. The variation of the physical properties of the fcc metals can be associated with the relative charge distribution between first neighbor and second neighbor atoms. Not surprisingly, we can also correlate ground state structures with the relative charge distributions between first and second neighbor atoms. Returning to Fig.6, the charge in the octahedral holes at point a increases along the line a-b-a through the series of curves 5, 4, 3, 2, to 1. Between curves 2 and 1, as we have mentioned, there is a point of structural instability, some first neighbor atoms must become second neighbor atoms to accommodate the change in the charge distribution. The structure consistent with this requirement is the bcc structure. We hypothesize that as the competition between first and second neighbor bonds reaches some critical value the fcc structure is no longer stable relative to the bcc structure and a phase transformation occurs. 16 M.E. Eberhart et al. To test this hypothesis, we have performed the set of calculations summarized in Table 2. The charge density of all of these metals has been calculated for the fcc structure. The ratios of the charge density at point a to that at point b are compared in Table 2. For those metals where the fcc structure is the stable ground state structure, this ratio is greater than 1.2. Likewise, for those metals where the stable ground state structure is bcc, this ratio is less than 1.2. We can also conclude that the bond directionality, as well as phase stability, results from competition between first and second neighbor bonding. TABLE 2. o(a)/o(b) for Several Elemental Metals. Element A1 Cu Pd Ir Ag Nb o(a)/o(b) 1.28 1.33 1.45 1.78 1.24 1.15 Element Rh Pt W Mo Ta o(a)/o(b) 1.62 1.60 1.20 1.19 1.16 Just as there is a correlation between empirically determined properties of fcc materials and the charge density distribution, we expect to find relations between the charge distribution in bcc materials and their empirically determined properties. Figure 8 shows possible charge distributions along the line a-b-a for the corresponding bcc structure. Points a andb are symmetry equivalent and the curvature at these points is negative in bcc materials with bound second neighbors. Curves 2, 3, 4, and 5 represent the only charge distributions which satisfy this requirement. The presence of saddle points of equal magnitude at points a and b necessitates a minimum in the charge density along this line at the point c. Point c is the tetrahedral hole of the bcc structure, although the local symmetry is not rigorously Td; rather, it is Dza. Figure 8 also shows the charge density along the line a-b-a for Ta. Note that the difference in magnitude between the peaks and the troughs is much smaller than in fcc metals. We will show later that the occurrence of this minimum has a major influence on the cohesive properties of bcc metals. The definition of a bond as a line between atoms which passes through a saddle point in the total charge density suggests relationships between structure and properties in the spirit of the molecular structure hypothesis. In the case of bcc and fcc metals, the Chemistry of Cohesion and Adhesion a c b C a a c b c a f 0.039 rID 0.037 t~ 0.035 tO I I I c b c 17 competition between first and second neighbor bonding appears to be a major influence on the physical properties. While the identification of features of the total charge density with bonds has revealed these relationships, we would like to predict when second neighbor bonds are likely to form. For this insight, we look to the one--electron contributions to the total charge density to identify the small number of orbitals which contribute density to the octahedral hole. MS-LDF calculations were performed on six-atom clusters of the 4d transition metals (excluding Tc), where the atoms were arranged at the vertices of the regular fcc octahedron. The cluster calculations give the charge contributions from each of the one-electron molecular orbitals (MO's) for the irreducible representations of the O h point group. The resulting cluster valence band MO's are consistent with the band structure calculations of Janak, Moruzzi, and Williams [33]. The only MO's which can contribute charge to the octahedral hole are those with a18 symmetry. Fig. 8 Possible charge densities There are two such orbitals for this cluster, coralong c-b--c in bcc strucresponding to the bottom of the s and d bands. The tures and the calculated s and dz 2 contributions to these MOs are shown in values for Ta. Fig. 9a and b. The actual molecular orbitals are hybridized combinations of these angular components, and the degree of hybridization will depend on the relative energies and radial extent of the s and d atomic orbitals. These orbitals contribute charge to both the first neighbor bonds, at point b, and second neighbor bonds, at point a. The contributions to the total charge density from these orbitals are plotted in Fig. 10 for both the first and second neighbor contributions (p(b) and p(a) respectively). The second neighbor charge density increases from Y (Z=39) to Mo (Z--42), and then drops sharply from Ru (Z=44) to Cd (Z=48). The first neighbor contribution follows the same trend, but much less dramatically. 18 M.E. Eberhart et al. Consistent with our analysis for the total charge density, the ratio of the orbital contributions to the charge density at point b and point a is a measure of second neighbor bonding efficiency. The a~g orbitals measure the total susceptibility to second neighbor bond formation for these systems, and consequently they are the only orbitals which directly determine the structure. This ratio reaches a m a x i m u m at the elements Nb and Mo which are both bcc, as conjectured. By understanding the factors which cause this charge density ratio to reach a maximum, we will be in a position to predict the occurrence of second neighbor bonds. Fig. 9 (a) s orbitals contribute to a MO which transforms as alg. (b) dz 2 orbitals also contributee an alg representation. (c) d(x2-y 2) orbitals make part of a t2grepresentation. (d) d , dyz orbitals transform as e u. Chemistry of Cohesion and Adhesion 19 0.012 [] [] O 0.010 t,.- .m °m t'"13 (1) (- [] 0.008 [] • p (a) 0.006 [] 0.004 . p(b) . D. 0.002 0.000 38 | | ! t | 40 42 44 46 48 Atomic number, Z Fig. 10 Total charge density for the 4d transition metal series decomposed into first and second neighbor contributions: p(a) and p(b). The peak in the second neighbor charge density at Mo can be explained by considering the effect of electron correlation on the filling of the valence band MO's. Inspection of Fig. 9a and b shows that the alg orbitals can potentially form both first and second neighbor bonds. As the bonding orbitals fill from Y to Mo, the amount of charge between the first neighbors increases, and electron-electron correlation promoted redistribution of the alg orbital charge towards the octahedral holes is observed. An example of one bonding orbital (shown in Fig. 9c) which contributes charge only between the firstneighbor atoms is the t~ orbital, which lies very close to the bottom of the d-band. The redistribution of charge from first to second neighbor bonds is seen in the increase of the p(a) component in Fig. 10. At Mo, with the electronic configuration 4d s 5s 1, the valence band is half-filled and additional electrons must occupy anti-bonding orbitals. An example of such an anti-bonding MO is the e u orbital in Fig. 9d, which flus between Mo and Ru. Anti-bonding orbitals will contribute nodes in the charge density between first neighbor atoms, and there will be no further driving force for electron redistribution to the octahedral hole. By Cd (Z=48), the charge from the alg orbitals is almost evenly distributed between first and second neighbor bonds. 20 M.E. Eberhart el al. This analysis suggests that a metal with a nearly half-filled band will form second neighbor bonds with a resulting bcc structure. A metal with bands that are more than half-filled will redistribute charge from the second to the first neighbor bonds, to compensate for the increasing nodal character between the first neighbors as anti-bonding orbitals are filled. For these metals, an fcc structure will be more stable. The simplest support of this explanation is offered by the Group I and II elements. The Group I elements (Li, Na,...) have only one s-valence electron and have a stable bcc structure, while the Group II elements (Ca and Sr) with two s-valence electrons have nearly-filled bands and have an fcc structure. Systems with very different orbital electronegativities, e.g. the transition metal aluminides or metalloid impurities at grain boundaries, will exhibit little or no hybridization between the Al(Metalloid)-s and TM-s or d bands. The Al(Metalloid)-s band will be nearly half-filled, regardless of the transition metal electronic configuration. In these systems, we consequently expect to see second neighbor A1-A1 (Metalloid-Metalloid) bond formation. The effect of these second neighbor bonds on the mechanical behavior is addressed in Section 4. 4. Fracture as Chemistry A. Cleavage of BCC Metals Having established a representation for bonds in metals and alloys, we turn now to understanding how these bonds respond to changes in geometry caused by an applied load which is the focus of this paper. In particular, we investigate the response of bcc materials to stress applied along the [100] and [110] directions. From this, we propose a mechanism for failure which can be applied to cases of environmentally-induced embrittlement and explain the intrinsically brittle behavior of transition metal-aluminides. The bcc metals cleave on (100) faces instead of (110) faces, although typically the close-packed planes are expected to be the planes of lowest surface energy and thus the cleavage planes. Griffith's criterion for brittle failure supports this prediction of (110) cleavage planes through a thermodynamic analysis of the energies of formation [34]. Griffith's criterion is an energy balance which states that the energy necessary to produce failure must be greater than or equal to the sum of the surface energies of the formed surfaces. In the case ofbcc metals cleaved in (100) and (110) planes Griffith's criterion Chemistry of Cohesion and Adhesion 21 gives: EF~oo~ -> 2~,~oo or (4.1) EF(,,0) -> 2r.0 where E F is the fracture energy, and 7 is the surface energy. This equality is believed to hold in ideally brittle materials and establishes a thermodynamic minimum for the energy necessary to induce failure for materials in general. In metals where there is potential for dislocation nucleation and propagation in advance of a growing crack, plastic deformation is an additional mechanism for energy storage. Rice and Thomson [35] have established a criterion for brittle failure in metals as an interplay between resolved shear strength and tensile strength. A crack will propagate if the resolved tensile stress at the tip of an atomically sharp crack exceeds the ideal tensile strength of the bonds at the crack tip before the resolved shear stress along slip directions exceeds the ideal shear strength of the bonds across the slip planes. If this condition is not reached, dislocations will nucleate, blunting the crack tip and lowering the stress concentration on the crack tip bonds. Such a material will fail in a ductile fashion. We wish to recast the Rice and Thomson criterion for brittle failure within a chemical nomenclature; that is, in terms of reaction paths, activated complexes, and transition states. By the Rice and Thomson criterion, the point of instability during the fracture reaction will occur when atoms displaced by the applied stress reach the maximum sustainable force on the crack tip bond, i.e. the ideal tensile strength of the material. This point of instability we will associate with the chemical instability of the transition state; the transition state for bond separation is the point of maximum sustainable force on the crack tip bond and the activation energy corresponds to the energy of maximum sustainable force or the point of inflection in an energy versus separation curve. The rate of the forward reaction, i.e. the susceptibility to failure, will be determined by the activation energy, and not the relative energy of the surfaces formed. The surface energy and the activation energy will be equivalent only in the case of a fully reversible process. The equality in the Griffith' s criterion represents a special case of reaction theory, where the transition state energy and product surface energies are identical. 22 M.E. Eberhart et al. To interpret dislocation nucleation and propagation within the same context (i.e., as a chemical reaction), consider some complex of atoms subject to an applied stress. Taking the energy of the equilibrium position of these atoms as the zero of energy, application of a stress will result in shear along a slip plane, producing a displacement of the atoms. The resulting energy is stored in the bonds as elastic energy. When the elastic energy stored in the activated complex exceeds that necessary to nucleate a dislocation, the slip planes will be displaced relative to each other, and a dislocation will form and propagate out of the atomic complex. The atomic complex will then return to its original zero of energy, plus some small amount of elastic energy resulting from the long range interaction of the dislocation with the atomic complex. The point of maximum sustainable shear strength is the transition state for dislocation nucleation and propagation. We are now in a position to restate the Rice and Thomson condition for brittle versus ductile behavior in chemical terms. A material will behave in an intrinsically brittle fashion if an applied strain displaces the complex of atoms at the crack tip to the transition state of bond separation before reaching the transition state of dislocation nucleation and propagation. Furthermore, the fundamental hypothesis of chemical reaction theory is that in a set of similar reactions, the higher the activation energy, the later the transition state is reached in the reaction path. A comparison of where the transition state occurs along the reaction path for a number of similar reactions will give an estimate of the relative energies of those transition states. The reaction with the lowest energy will be the one where the transition state is reached earliest in the reaction path, and this reaction will be favored. In what remains of this paper, we will apply these concepts and those of bonding developed in Section 3 to describe the relation between bond strength and cohesion and adhesion. Our approach will be to analyze the bonding and the effect of an applied stress on this bonding in model polyhedra; for example, in the case of cleavage in bcc metals, one need only analyze the nature and response of the bonding around the octahedral and tetrahedral holes. This reduction to representative polyhedra is a general approach which can be applied to the study of more complex problems of intergranular or hetero-facial fracture. In these complex systems, a number of different boundaries can be constructed by the packing of a small number of polyhedra. Once the bonding and its response to stress in these individual polyhedra is understood, it will be possible to predict the response of a specific grain boundary from a knowledge of how these polyhedra are packed to form the boundary of interest [36-38]. Chemistry of Cohesion and Adhesion 23 100 STRAIN N~,~ j~" W Fig. 11 Octahedra of the bcc structure with strain. Figure 11 shows the octahedra of the bcc structure with strain vectors corresponding to an applied force in a [100] direction and a [110] direction. For [100] strain, the two half-spaces of the crystal will separate about the plane containing the tetrahedral holes (point c). This plane will also contain the saddle points of the charge density between the first neighbor bonds, the points d in Fig 11. It is the strength of these first neighbor bonds which is predominantly responsible for the cohesive strength of the material. As the two half-spaces of the crystal are rigidly separated around the shaded plane, the charge on this plane must go to zero and the charge between first neighbor atoms across this plane, and therefore their interaction energy, must also go to zero. However, in accordance with the Rice and Thompson condition for brittle failure it is not the bond energy which determines the transition state, but rather the first neighbor bond strength (i.e. the maximum sustainable force) which scales as the curvature of the charge density normal to the first neighbor internuclear axis at the saddle point. For the purposes of a simple qualitative argument, we will use the observation that the curvature of the charge density at the saddle point scales with changes in the ratio of the saddle point charge density to that at the center of the octahedral hole. We are consequently concerned with the changes in the charge density for small displacements of atoms responding to a [100] stress at three critical locations: the octahedral holes (points a), the tetrahedral holes (points c), and the saddle point between first neighbors across the separation plane (points d). For small displacements, the charge in the plane containing the tetrahedral holes will flow to the developing surfaces (from points c and d to points a and b); therefore, the charge density at the saddle points of the first neighbor bonds across the cleavage plane (and at the tetrahedral hole) will decrease, and the charge in the octahedral hole will increase. The ratio of the saddle point charge density and the octahedral hole density will decrease; therefore, the strength of the first neighbor cohesive bonds will decrease 24 M.E. Eberhart el al. rapidly. The increase in the charge density at the octahedral hole will result in an increase in the strength and energy of the developing second neighbor surface bonds. This will offset to some degree the loss in bond energy from the first neighbor bonds across the cleavage plane. In the case of [110] strain, the same set of special points can be considered for an understanding of the net flow of charge. In this case, however, the plane about which the two half-crystals rigidly separate passes through both the octahedral hole and the saddle points of the first neighbor bonds across the developing (110) surfaces. Small displacement of the atoms will decrease the charge at the octahedral hole and at the saddle point between first neighbor atoms. The ratio of these quantities and the corresponding bond strength need not decrease as rapidly as in the case of [100] strain, where these quantities change in opposite directions. From this analysis alone, we would predict that [100] strain in bcc metals would lead to a more rapid loss of strength than [110] strain. Hence, the transition state for cleavage along (100) planes will occur earlier (at smaller strains) than the transition state for (110) cleavage. All other factors being equal, this analysis predicts that (100) planes are the preferred cleavage planes (over (110) planes) in bec materials, in agreement with experimental observations. There is an alternate analysis which also predicts that the (100) transition state occurs earlier along the reaction path than the (110) transition state. If a transition state occurs early along the reaction path, it resembles the reactants more closely than the products; likewise, if it occurs late along the reaction path, it more closely resembles the products. In the case of (110) cleavage, the octahedral hole about which the two half-crystals separate is at a saddle point in the charge density. The charge density at these octahedral holes must be an absolute minimum in the cleaved products, since the half-spaces are infinitely separated from each other. The only way the octahedral hole can transform from a saddle point to a minimum is for the tetrahedral holes (where the charge is initially a minimum) to coalesce with the octahedral hole [21]. This coalescence can be associated with a structural instability, i.e. the transition state. The structure of the charge density in the transition state of [110] strain is therefore more like the products than the reactants, and it must occur late along the reaction path. For [ 100] strain, the charge density at the saddle point between broken first neighbor bonds will be zero for the products and the corresponding charge density at the octahedral holes will be finite and larger than that for the reactants. The transition state occurs when the charge density in the octahedral hole and at the saddle point are nearly equal (a total loss of strength); hence, the transition state for (100) cleavage more closely resembles the reactants than the products, and it must occur earlier along the reaction path. This argument again leads us to conclude Chemistry of Cohesion and Adhesion 25 that (100) cleavage is preferred over (110) cleavage, if all other factors are equal. We have discussed the geometrical factors which influence where the transition state occurs along the reaction path for cleavage along (100) and (110) planes. We have not, however, addressed those factors which influence the position of the transition state for systems with different atoms but with the same bcc structure and direction of applied strain; for example, while the preferred cleavage plane for both Cr and Fe is the (100) plane, Cr is much more brittle than Fe. Part of the explanation must reflect a difference in dislocation nucleation and propagation transition state energies. This behavior must also originate from differences in local bonding and charge distribution, but it is not explained by the arrangement of atoms, since both Fe and Cr are bcc. We now seek to isolate those characteristics of the charge density and bonding which will alter the position of the transition state for a given deformation. We can define particular regions of an extended system of atoms to be either electronegative or electropositive. These are relative terms, reflecting whether these regions attract or repel electron density in the same sense as atoms and molecules. Regions which act electronegatively are characterized by a negative curvature in the charge density; thus saddle points and nuclei act electronegatively. Those regions which act electropositively are regions of positive curvature; thus, local minima and the regions parallel to the internuclear axis between bound atoms act to repel electron density. Returning to Fig. 11, we have described the reaction path for (100) cleavage as the process of charge transfer from the separation plane to the developing (100) surfaces. The electronegativity of this surface is a measure of the efficiency with which the electron density is transferred from the plane of separation to the surface. Those electronegative regions are the second neighbor bonds. The relative electronegativity of these second neighbor bonds can be ranked in terms of features of the total charge density. Those bonds which are more electronegative can accommodate the charge located in the first neighbor (cohesive) bonds across the cleavage plane for the most brittle materials where the cleavage transition state is reached early along the reaction path. It can be shown that the integral of the Laplacian of the charge density over the entire Wigner--Seitz cell must be zero [18]. When the bcc metal is separated about the (100) plane, the curvature in the tetrahedral hole will go from a positive value to zero. Since the Laplacian represents the sum of the curvatures in three orthogonal directions, there must be a corresponding negative change in curvature if the integral of the Laplacian is 26 M.E. Eberharl et al. to remain zero. This can be accomplished if the curvature between second neighbor bonds at the saddle point becomes less negative, i.e. flatter. Figure 12 [44] shows contour diagrams of an Fe grain boundary and a corresponding free surface. Note that the charge density of the surface varies less rapidly than that of the boundary. The regions of negative curvature at the grain boundary are flatter than Fig. 12 Contour diagram of Fe grain boundary (on left) and a free surface. at the surface in order to accommodate the electron density which had been confined to the first neighbor bonds and to maintain the zero value for the integral of the Laplacian. We conclude from these observations that those features of the total charge density which can accommodate the greatest changes in the charge density are the most electronegative. For two different second neighbor bonds, the one with the less negative curvature at the saddle point accommodates the greatest amount of charge, and hence this bond would correspond to the one with greatest electronegativity. The weakest bonds would consequently be the most electronegative bonds. An alternate analysis will again lead us to a consistent conclusion. In all systems that we have studied (and we believe this to be a general feature of surfaces and boundaries), the electron density of surfaces varies less rapidly than the density of a corresponding boundary; hence for two systems differing in the types of atoms but not in structure or magnitude and direction of applied stress, the system with the most slowly varying charge density across the cleavage plane will be the one that most resembles the reactants; this is the system where the transition state for cleavage will be reached first. Chemistry ol Cohesion and Adhesion 27 The presence of areas of weak negative curvature in the virtual surface are associated with the formation of weak bonds. Every bond in the (100) surface of a bcc material is a weak second neighbor bond, while only 1/3 of the bonds in the (110) surface are second neighbor bonds. Thus a (100) surface should be more reactive and more susceptible to brittle failure than the (110) surface, if all other factors are equal. This condition for bond separation, the presence of areas of weak negative curvature in the virtual surface, departs from any consideration of relative energetics of the boundaries and surfaces as the criterion for brittle failure. Rather it assigns the concept of reactivity to the surface, which is exactly analogous to the concepts used in molecular chemistry. B. Intergranular Failure The model of bond separation developed in the previous section can be applied to more complex problems of cohesion. In particular, we are interested in understanding how impurity segregation to grain boundaries can induce a transformation from ductile to brittle failure mechanisms; for example, Ni and Ni-base alloys typically behave in a ductile fashion, suggesting that the transition state for dislocation nucleation is reached before that for bond separation. When trace amounts of impurities like S, P and H segregate to alloy grain boundaries from the bulk or the environment, there is however frequently a rapid loss in toughness manifested by intergranular cleavage. The presence of these impurity elements at the grain boundaries rather than in the bulk is evidence that the mechanism for intergranular embrittlement results from a loss of strength across the boundary and not from a change in the stress necessary to propagate dislocations. These trace impurities at the grain boundaries of Ni must make the transition state for bond separation occur earlier along the reaction path than it does in the absence of these impurities by introducing areas of weak negative curvature. Grain boundary structure in bcc and fcc metals can be constructed from a packing of trigonal capped prisms [36-38] (shown in Fig. 13). Through slight distortions of this basic packing unit and adjustments of their relative orientations, it is possible to construct a large number of grain boundary geometries. The first neighbor bonds in these trigonal capped prisms are those between the cap and the prism atoms, depicted with light lines in Fig. 13. Those atoms which can form second neighbor bonds vary with the extent of the distortion of the capped trigonal prism. In general we expect those atoms parallel to the prism axis to form second neighbor bonds; these are shown with heavy lines in Fig. 13. If bonds existed between the cap and the opposite vertex of the prism or between caps, these would be second or higher order bonds, and they would thus possess the weak 28 M.E. Eberhart et al. negative curvature associated with weak bonds. In materials which do not show intrinsic grain boundary weakness, such as Fe, Ni, and Cu, our calculations have shown an absence of these second neighbor bonds in trigonal capped prisms. Thus the grain boundary strength is due entirely to the first neighbor bond across these boundaries, and the corresponding surfaces are unreactive. Metalloid atoms such as B, C, P, and S segregate to grain boundaries at the center of these trigonal capped prisms. The Fig. 13 Trigonal capped prism, bonds from the metalloid atoms to any of the trigonal-prism metal atoms will be first neighbor bonds, and consequently these bonds are not likely to introduce the areas of weak negative curvature which we associate with the onset of brittle behavior. Furthermore, if any second neighbor bonds exist in the uncontaminated grain boundary (i.e., between a cap and an opposite vertex of the prism), the segregation of a metalloid atom to the center of the prism might act to remove these second neighbor bonds by introducing a cusp in the charge density at the metalloid nucleus, removing the saddle point of the second neighbor bonds. For those metalloid atoms which cause intergranular embrittlement, the packing of the metalloid centered trigonal capped prisms must provide the environment for second neighbor bond formation. Two trigonal capped prisms can share a trigonal prism base, a geometry common in many grain boundaries. Metalloid atoms at the center of these prisms can form a bond across the trigonal face of these prisms. Such a bond, if it exists, would form an area of weak negative curvature, and thus the bond would improve the reactivity of the resulting surface. Both cluster and LKKR calculations have shown that when P and S segregate to the grain boundaries of Fe and Ni, respectively [30, 39], an impurity band is formed consistent with the existence of P-P or S-S bonds. We are currently calculating the total charge density for these systems in an attempt to quantify the nature of the impurity interactions in terms of the curvature of the charge density. C. Transition Metal Aluminides Another complex problem which can be addressed with the model for bond formation developed here is the brittle failure of the transition metal aluminides, materials with tremendous technological potential. Their low density, high melting points, and their tendency to form a passive alumina surface layer make them attractive as high tempera- Chemistry of Cohesion and Adhesion 29 ture structural materials. This class of materials, however, is characterized by brittle behavior, showing failure mechanisms very different from those materials which they may replace, such as the Ni-base superalloys. Much theoretical and experimental effort has been extended in explaining the observed mechanical properties of these aluminides. Two of these studies [42, 43] noted that for the general ordered intermetallic compound A BL_x, there is a correlation between certain atomic properties of the constituent elements A and B and the empirically determined mechanical properties of the alloy. In particular, it was noted that brittle behavior could be associated with compounds where the elements A and B came from different groups with the difference between their representative group numbers giving a semi-quantitative measure of susceptibility to brittle behavior. A more quantitative basis for these observations was provided [42] when it was noted that the same partitioning was observed when the s-orbital electronegativities of elements A and B were used; those alloys where the constituent elements have similar s-orbital electronegativities exhibit ductile behavior, while those with very different s-orbital electronegativities are more prone to brittle behavior. It was argued that this partitioning of mechanical properties resulted from the degree of isotropy of the alloy charge density with the difference in the s-orbital electronegativity being one predictor of this behavior. The transition metal aluminides would then fall into the class of brittle materials, since AI has a very different s-orbital electronegativity from all the transition metals. As was argued in Section 3B, the existence of half-filled valence bands can be associated with second neighbor bond formation. In the transition metal aluminides where there is poor hybridization between the Al-s orbitals and the TM-s and d orbitals, the A1 s-band will be half-full. The very directional first neighbor interactions between the TM-d and Alp orbitals will result, through electron correlation, in the build up of charge between A1 atoms through the A1-A1 bonding interaction at the bottom of the AI s-band. The corresponding weak second neighbor bond formed will be very reactive, and it will lead to brittle behavior. To test this possibility, we begin by examining the nature of the charge density in the L1 o alloy TiAI. The unit cell of the L1 o structure is shown in Fig. 14 for a model AB alloy. Although this structure is based on the fcc lattice, the available bonding is quite different. The octahedral hole no longer has the O h symmetry of the fcc lattice, but it now has the D4h symmetry of the bcc lattice; second neighbor bond formation is not precluded by symmetry. Figure 14 also shows two octahedral holes of the L1 o lattice: one in the A-atom plane, and the other in the B-atom plane. Shown beneath these structures are possible representations of the charge density along the line a-b-a (A-B-A). As the two planes in the L1 o structure are not identical, any combination of the charge densities corresponding to curves 1-5 on the left with curves 1-5 on the right is possible. Each of the two curves labelled I is associated with second neighbor bond formation. Figure 14 also illustrates the computed 30 M E Eberhart et al charge densities along the line a-b-a (A-B-A) for both the Ti-atom plane and the Al-atom plane 0f TiA1. Note that there is maximum in the charge density at point a in the Ti-atom plane, indicating an A1-AI second neighbor bond. It is this second neighbor bond between L10 Structure B Atom A Atom 0 a 0 a Possible L10 Charge Densities A A B b i, C Or¢)0 ~5~ o o~ <~ <'E m~ __ta~ \Jr --~ ~> \ ~7 g., 4 5 5_= (/) ~_-q o~ (,9 A B t$~ O3 m (b -n A a =i a b a Charge Density of TiAI ~i 0.030 -.i ~, °°°° 0.028 • • 0.026 • • • • 0.024 •0°•• 0o 0.032 0.030 ~ 0.028 ~ ...-"-... . •= = ••••oQoo• •••• ,~ •• 0.026 "~" %000 ° ;-v B A AI plane B b a b Ti plane Fig. 14 L10 unit cell with possible charge densities along the lines a-b-a and A-B-A and the calculated values for TiA1. Chemistry of Cohesion and Adhesion 31 AI atoms and its associated area of weak negative curvature, in part, which we believe is responsible for the brittle failure mechanisms of TiAI. The surfaces in TiAI which will be the most reactive are the ones with the greatest density of A1-AI second neighbor bonds (i.e., the (100) surfaces). This suggests that the cleavage planes of TiA1 will be the (100) planes, if we consider only the reactivity of the surfaces formed, and we do not consider the ease of dislocation nucleation and propagation. There is an interesting experimental observation in the Ni-aluminide system which can be readily explained by this model for failure. The Ni-aluminides include NiA1, which has the B2 structure, and Ni3A1, with the L12 structure (shown in Fig. 15). NiAI is both transgranularly and intergranularly brittle, while Ni3A1 is only intergranularly brittle. In NiA1, second neighbor bonds can be formed between the AI atoms and between the Ni atoms. Just as the existence of second neighbor bonds are essential to the stability of the bcc structure over the fcc structure, our preliminary calculations have shown that the B2 structure is characterized by second neighbor bonds, and this accounts for its stability over the L10 structure. The existence of these second neighbor bonds in NiAI would explain the observed transgranular failure. B2 L12 Inspection of Fig. 15 demonstrates that Ni second neighbor bond formation in Ni3AI is precluded by symmetry, as second neighbor Ni atoms now sit at the vertices of a regular octahedron; thus, there must be a greatly reduced density of second neighbor bonds in Ni3A1 compared with NiA1. Fig. 15 The L12 and B2 structures. If there are no A1-AI second neighbor bonds, there will be a total absence of reactive sites in the L12 Ni3A1, explaining the observed absence of transgranular failure mechanisms. In the reduced symmetry of grain boundaries, there must be orientations in which the Ni and A1 atoms are capable of forming second neighbor bonds, accounting for the intergranular failure. 32 M.E. Eberhart el al. 5. Conclusions This work has demonstrated that bonds in solids can be described by a saddle point in the charge density at a position which is between the two bound atoms, in analogy to Bader' s formalism previously applied to small molecules. Electronic structure calculations have demonstrated that bonds are in fact formed between second neighbor atoms in bcc metals, and these bonds do not exist in fcc metals even when symmetry constraints are removed. The ratio of the charge density at points midway between first and second neighbor bonds is driven in part by electron correlation. One-electron MO calculations provide evidence that metals with nearly half-filled valence bands will form second neighbor bonds; metals with nearly-filled bands will tend to redistribute second neighbor bonding density toward the first neighbor bonds. The bond strength (i.e., the maximum sustainable force of a bond) and the directionality of the bond, which is a measure of its resistance to deformation, can be quantified through the curvature of the charge density at the saddle point which defines the existence of the bond. It was observed that this curvature scales with the ratio of the saddle point charge density to that at the octahedral hole, the same ratio which determines the phase stability. These definitions of bonds in terms of features of the total charge density have been combined with the concepts of chemical reaction theory to give an understanding of bond separation which plays a large role in describing problems of cohesion and adhesion. This formalism provides an explanation of cleavage in bcc materials which is consistent with experimental observations. Additionally, this formalism overcomes some of the limitations of previously proposed models for cleavage. 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