A Model for Chemical Embrittlement in Materials.

Robb Thomson, Emeritus

Materials Science and Engineering Laboratory NIST Gaithersburg, MD 20899

A. E. Carlsson

Department of Physics University of Washington St. Louis, MO 63130 (February 29, 1996)

Abstract

We present a model for chemical embrittlement in materials based on the idea that the criteria for fracture can be couched as a balance between elastic driving forces and lattice resistance terms. This approach leads to a simple block construction for the lattice resistance term for blunting dislocation emission, which can be generalized to apply to first principles quantum mechanical bonding calculations for the crack. We have tested the model with a generic 2D hexagonal lattice with UBER pair force laws, and shown that the model gives quantitative results. The only exception is when the bonding relations at the crack tip lead to significant lattice trapping, for which the material is more ductile than predicted. With the generic system (which should be useful in a qualitative way for more realistic materials), we find that: 1) The crossover between ductile and brittle behavior is determined by the local bonding at the interface atoms, 2) The crossover occurs at λ ≈ 0.17 (in units of the lattice spacing), where λ is the range parameter in the UBER force law for a system emitting at an angle of about 60^{∘} to the cleavage plane, 3) the crossover is independent of the spring constants in the interface, but depends on the crystallography of the crack/emission system and on the elastic mismatch between the two sub lattices at the interface.

PACS numbers: 07.05.T, 62.20.M, 61.72.B, 61.72

Typeset using REVTEX

I. INTRODUCTION

A limiting problem in the ability to model fracture in materials is that the number of atoms necessary in a computer model of a crack is much larger than the number which can be encompassed in a full quantum mechanical treatment of the atomic bonding. Thus, at best, it is only possible to use semiempirical bonding relations in crack problems, which leaves one dissatisfied with the treatment of the bonding at the crack tip, where the distortion is very severe, or where special chemical effects may occur. This paper explores the possibility that important criteria for fracture and chemical embrittlement can be developed without a full scale treatment of the crack structure.

That such an approach is conceivable is suggested by the fact that the Griffith relation for cleavage of a brittle crack is a simple equation of balance between the elastic driving forces on the crack and the lattice resistance associated with the energy of the growing surface. The elastic driving force is given by the standard expression, G_{c} = (K_{I}^{2} + K_{II}^{2})/2μ′, where K refers to the stress intensity factor for Mode I tensile and Mode II shear opening. For a homogeneous isotropic medium in plane strain, μ′ = μ/(1 − ν), where μ and ν are the standard elastic constants. Then, if the intrinsic surface energy is γ_{s}, the Griffith relation is simply

G_{c} = 2γ_{s}.  (1)

Thus one can calculate the stress intensity required to propagate a crack by simply calculating the surface energy instead of doing the much more complex crack calculation. The Griffith relation is valid wherever the bonding forces on the crack tip are sufficiently smooth that negligible lattice trapping^{1} exists. Such a simple relation between purely elastic and purely nonlinear core terms follows from the fact that the very long ranged elastic field of the crack can be accurately represented by elastic expressions, while the bond breaking in the core is highly localized to the nonelastic core region near the crack tip.

Recently, Rice has shown^{2} that a Griffith–like relation can also describe the dislocation emission criterion for a crack in pure Mode II loading, in which the dislocation is emitted directly in front of the crack in a nonblunting configuration. (See also a closely related development by Schoeck^{3}.) The emission criterion is believed to be part of the criterion for intrinsic ductility of the material, because if the lattice fails in shear (dislocation emission) before it fails in tension (cleavage), then the material is ductile, and vice versa. In Rice’s picture, the driving force on the crack for dislocation emission is the Mode II part of the elastic driving force, G_{e} = K_{II}^{2}/2μ′, and the lattice resistance is simply the energy (per atom) maximum encountered under block-like sliding of adjacent planes. and called the unstable stacking fault energy, γ_{us}. Thus, the emission criterion becomes

G_{e} = R = γ_{us},  (2)

where we have introduced the quantity R to represent the lattice resistance to dislocation emission. Again, the calculation of γ_{us} is much simpler than a full crack calculation. Such a dislocation emission configuration does not blunt the crack, however, and leaves a sharp crack still capable of cleavage, if loaded appropriately. When a dislocation is emitted at a nonzero angle to the cleavage plane, the crack is blunted, and should be more difficult to cleave. Thus, the true intrinsic ductility criterion should relate to such a blunting emission configuration.

As the dislocation is emitted in the blunting configuration, however, the lattice resistance is not merely due to shearing the lattice, but must also involve true bond breaking which is associated with the formation of the ledge at the tip of the blunted crack. Since bond breaking is usually more difficult than bond shearing, one expects that the ledge formation during dislocation emission might dominate the unstable stacking fault term in Eqn. (2), at least in some cases^{4}. We have suggested an alternate construction which might be appropriate for describing the lattice resistance in this case^{5}, and the purpose of the present paper is to explore this prospect in detail for chemical embrittlement.

Specifically, we assume the crack lies on an interface which has absorbed, or segregated a chemical layer which is different from the remainder of the bulk lattice. Following Rice^{2,6}, we write a criterion for emission composed of the driving force for shear on the inclined shear plane of the emitting dislocation and a lattice resistance term derived from shearing a block of material at the crack tip along the slip plane. Then we investigate a series of crack simulations using a simple hexagonal 2D lattice with simple force laws to explore the accuracy of the prediction. We will show that the model works very well for a range of loadings and force laws, and for a variety of chemical differences in the bonding at the interface, but that limitations exist. We will first present the results for the base case, and then discuss an exception. Even though the systems studied do not represent realistic materials, we believe the essential physics of the model is given a severe test in this procedure, which is the main message of our paper. Needless to say, the study of more realistic cases is underway.

Although the main purpose of the paper is to explore chemical embrittlement, the homogeneous lattice is a special case with no “chemistry”, and the results will be applicable to the problem of the ductility of a homogeneous lattice. At several points in the paper, we will address this more restricted case, but our main object is chemical inhomogeneity, as stated in the title.

II. THE LATTICE RESISTANCE

As described above, R is the resistance to dislocation emission from the nonlinear lattice terms, such as the shear along the emission plane and the ledge effects. The total force on the emerging dislocation will be the sum of the lattice resistance and the elastic driving force, and at the critical emission point, these forces must balance to zero. Our derivation of the lattice resistance borrows the idea of the lattice misfit energy and stress from the Peierls theory of the dislocation core^{7}. In the Peierls construction, two semi-infinite blocks of atoms are rigidly sheared past one another, and the (nonlinear) misfit stress in the core of the actual dislocation is assumed to be the same as the stress for a rigid block displacement corresponding to the local shear strain. This same construction is used by Rice^{2}, who defined the maximum energy during the shear as the unstable stacking fault energy, γ_{us}. Such a scheme is approximate when applied to the dislocation core, due to the inhomogeneous strain field in the core of any actual dislocation. However, in spite of this intrinsic error, atomic simulations^{8} have verified that the rigid block construction incorporates much of the physics of the problem; the present results will extend this conclusion to Mode I emission.

The appropriate rigid block construction for blunting dislocation emission is shown in Fig. 1. This construction is identical to that of the bulk analysis^{5} except that the bonds along the cleavage plane correspond to a different “chemistry” from that of the bulk. The block is defined by the cleavage plane and the slip plane, which intersect at the crack tip. The bonds along the bottom of the block which cross the cleavage plane are cut. We will show that this uniform block geometry can provide information about the emission of an inhomogeneous dislocation distribution. This result should be very useful as a guide in performing ab-initio force calculations of fracture properties.

Our approach is to first calculate the configurational force on an outgoing inhomogeneous dislocation distribution, and then to connect it to the uniform-block construction. We obtain the configurational force in terms of the individual atomic forces in the presence of the emerging dislocation distribution. We denote the position of the center of gravity of the emerging dislocation distribution by ρ, measured along the emission plane. We assume that the profile of the distribution is frozen as it moves out. The displacements associated with the dislocation are u⃗_{i}, where i indexes the atoms; these are determined parametrically by ρ. Then the force on the dislocation (measured in the direction of the spur) is by definition

R = −dE/dρ = − ∑_{i}(∇_{i}E) · du⃗_{i}/dρ = ∑_{i} f⃗_{i} · du⃗_{i}/dρ,

where f⃗_{i} is the force on atom i and the du⃗_{i}/dρ are simply the values of the strain in the dislocation distribution at the atomic positions. Here the energies and the corresponding forces are taken to exclude the linear terms connecting the atoms along the emission plane with the “back” atoms in the bulk, so that R describes only the nonlinear lattice effects.

The atomic-level forces can be divided into a bulk shear part f⃗_{i}^{0}, giving the local forces that would have resulted from the given displacements in the absence of crack-tip broken bonds, and a remainder. The f⃗_{i}^{0} depend only on the local relative shear displacement across the emission plane, and correspond to the Peierls/Rice construction mentioned above. Separating off the bulk shear part is useful because it is straightforwardly calculated. In fact, as shown by Rice, under a virtual displacement δρ of the dislocation distribution, the change in the bulk shear energy along the slit simply consists of extending the faulted region by the distance δρ. This gives an energy change of Φ_{shear}^{0}δρ, where Φ_{shear}^{0} is the planar energy density correponding to the value of shear at the crack tip. This gives a contribution to the configurational force of

R_{shear} = Φ_{shear}^{0}  (3)

In order to evaluate the remainder, we separate off the bulk shear part with the help of Eqn. (3), as follows:

R = ∑_{i} f⃗^{0}(δ_{i}) · du⃗_{i}/dρ + ∑_{i}[f⃗_{i} − f⃗^{0}(δ_{i})] · du⃗_{i}/dρ  (4)
= Φ_{shear}^{0} + ∑_{i}[f⃗_{i} − f⃗^{0}(δ_{i})] · du⃗_{i}/dρ  (5)

where δ_{i} is the interplanar shear at atom i.

The rightmost part of the second equality is the ledge term. In most cases, it should be dominated by the regions in which the unstable stacking fault is highly disrupted, namely close to the crack tip. The values of f⃗ − f⃗^{0} for these atoms are probably not very sensitive to the inhomogeneity of the strain farther from the crack tip. Thus it is likely that these force differences can be calculated accurately from uniform block motions such as those shown in Fig. 1, and we have a procedure for calculating the lattice resistance R, given by Eq. (5), on the basis of ab-initio force calculations. The procedure is the following. First, one evaluates the planar shear energy Φ_{shear}^{0}; this can be done with comparatively small supercells, on the order of twenty atoms or less. Second, one evaluates the atomic-level forces on the right-hand side using a uniform-block construction. This can be done with on the order of a few hundred atoms. (In the uniform block geometry, we can work with the total forces on the atoms, since the displacements of the “back” atoms relative to the “slit” atoms vanish.) Finally, one needs an approximation to the strain du⃗_{i}/dρ, evaluated near the crack tip. As is shown later in the paper, this turns out to be a fairly universal dimensionless quantity. This procedure also provides a useful route for getting information from quicker, less sophisticated methods, such as the embedded-atom method^{10,11}. While full crack simulations can be done with these methods, Eq. (5) evaluated for uniform block motion provides a much faster way of getting results.

In the present case of nearest-neighbor force laws, the effects come mainly from atom 0 not following along with the motion of the block, (cf. Fig. 1), leading to broken bonds. These effects change the forces only on atoms 1 and 3 (the even-numbered atoms always see a perfect atomic arrangement on the other side of the emission plane). We readily see that

f⃗_{1} − f⃗^{0}(δ_{1}) = −f⃗_{01}(δ_{1})

and

f⃗_{3} − f⃗^{0}(δ_{3}) = −f⃗_{03}(δ_{3})

In the notation of Fig. 1, f⃗_{01}(δ_{1}) is for an “B”-type bond, and f⃗_{03}(δ_{3}) is for a “C”-type bond. Assuming that δ_{1} and δ_{3} are roughly equal, we then have that

f⃗_{1} − f⃗^{0}(δ_{1}) + f⃗_{3} − f⃗^{0}(δ_{3}) = −f⃗_{B}(δ_{1}) − f⃗_{C}(δ_{1}) = −f⃗^{0}(δ_{1}) + f⃗_{A}(δ_{1}).  (6)

Since the 1-2 bond is an “A”-type bond, we have f⃗_{A}(δ_{1}) = f⃗_{12}. Furthermore, if we assume that the bulk shearing energies are much smaller than the bond-breaking energies, the second term on the right-hand side of Eq. (6) dominates. Labeling the associated strain at the crack tip by β(0), we obtain

R = Φ_{shear}^{0} + β(0)f_{12}.  (7)

In using this equation, we define f_{12} to be the component of the bond force along the slip plane. This is reduced relative to the magnitude of the total force, by a factor of roughly 1/2, although the magnitude of the reduction will be less if the magnitude of the opening displacement is large.

With these results in hand, the predicted condition for emission is

g_{IIe} = R,  (8)

where g_{IIe} is the Mode II part of the elastic driving force on the crack in the system of coordinates associated with the inclined slip plane, and R is the lattice resistance corresponding to the maximum force in the 1-2 bond as the dislocation emerges. The correct system of coordinates is essential here, because the driving force on the crack is that on the virtual crack which is branching onto the inclined plane with the emitting dislocation. Calculations by Hoagland^{13} have confirmed the validity of balancing an angle–dependent driving forcer with an inherent lattice resistance to emission. In terms of the virtual–crack coordinates, g_{IIe} = k_{II}^{2}/2μ′, where, according to Cotterell and Rice^{12},

k_{I}= \frac{1}{4}(3 cos \frac{θ}{2} + cos \frac{3θ}{2})K_{I} − \frac{3}{4}(sin \frac{θ}{2} + sin \frac{3θ}{2})K_{II}
k_{II}= \frac{1}{4}(sin \frac{θ}{2} + sin \frac{3θ}{2})K_{I} + \frac{1}{4}(cos \frac{θ}{2} + 3 cos \frac{3θ}{2})K_{II},  (9)

Here lower case k refers to the stress intensity factors for the branching virtual crack, and upper case K to the stress intensity factors for the parent crack. (We will sometimes refer to the coordinate system for the parent crack as the Lab system.) For the case of a crack on an interface between two elastically dissimilar sublattices, we have given prescriptions for similar local stress intensity factors valid for the inclined plane^{14}. Thus, the block construction provides a definite prediction for the emission loads for a crack either in a homogeneous lattice or on an interface, as well as for the case where there is chemical segregation at the interface. For the remainder of the paper, the specific construction in Fig. 1 is for the case of nearest neighbor forces and the hexagonal lattice, but the generalization for more realistic lattices and bonding relations is clear.

III. THE DUCTILITY PARAMETER AND UNIVERSAL BINDING ENERGY RELATION PREDICTIONS

The important issue for embrittlement is to describe the point where a material crosses over from ductile to brittle behavior, that is, the point where the emission criterion equals the cleavage criterion. Crossover can be most easily described in terms of the ductility parameter, defined as

D = \frac{G_{c}}{G_{e}} = \frac{2γ_{s}}{Y (θ)R},  (10)

where Eqn (10) is written in the coordinates of the parent crack via the geometric factor Y (θ) = G_{e}/g_{IIe}. For the homogeneous hexagonal lattice, with the slip plane at 60^{∘} to the cleavage plane, Y = 64/9. Crossover occurs at D = 1.

According to Eqn. (10), the cleavage/emission crystallography and geometry will be very important in determining the crossover point, because of the geometrical factor Y (θ). (The reader is warned that R does not have an explicit dependence on θ, because any such dependence is automatically built into the analysis of §II, and the conversion from the inclined plane coordinate system to the lab coordinate system is accounted for in Y.) Fig. 2 shows the angular dependence of 1/Y for a crack loaded in pure Mode I in a case where there is no elastic mismatch at the interface, and indicates a maximum near 70^{∘}. Thus for this case, a material will respond most easily in a ductile manner for an emission plane near 70^{∘} (which is also the angle of maximum shear for a Mode I crack).

0^{0.02}^{0.04}^{0.06}^{0.08}^{0.1}^{0.12}^{0.14}^{0.16} ^{0}  ^{10}  ^{20}  ^{30}  ^{40}  ^{50}  ^{60}  ^{70}  ^{80}  ^{90}^{\frac{1}{Y}}  θ

Figure 2. Plot of 1/Y (θ) as a function of θ when the parent crack is loaded in pure Mode I. The maximum appears near 70^{∘}, where the maximum shear stress occurs for a Mode I crack.

The simplicity of the construction of the last section and of the result Eqs. (7-8) which summarizes it means that it is possible to give analytic expressions for embrittlement for a simple force law. In this paper, we work with the Universal Binding Energy Relation (UBER) force law of Rose, Smith and Ferante^{15}, and it will be instructive to express D in terms of it.

The UBER force law is a pair potential of the form

f= −cue^{−u/λ}
E= −cλ(λ + u)e^{−u/λ},  (11)

where f is the radial force between the two atoms of the pair, u is the radial displacement from the stable distance, and E is the energy. The UBER has two parameters, the spring constant, c, and the range parameter, λ. These expressions are normalized to unit lattice spacing, and the spring constants are normalized to a standard lattice where c = 1. Since the UBER force law has only two independent parameters, the discussion of embrittlement can be couched in terms of critical values for these parameters. In the simple generic model we use, the appropriate spring constant in the emission and cleavage conditions is that between the interfacial atoms, c_{12}.

According to Eqs. (7-8), the emission criterion is composed of two terms, γ_{us}, and a second one containing f_{12} = f_{max}/2, where f_{max} is the maximum value of the radial force. To anticipate, we find in the simulations to be presented shortly that γ_{us} can be neglected, so the emission criterion becomes G_{e} = Y (θ)βf_{max}/2. Since both the force law and energy from Eqn. (11) scale linearly in the interfacial spring constant, c_{12}, then the ductility parameter, D is independent of c_{12}.

What about D as a function of λ? From Eqn. (11), f_{max} = c_{12}λ/e, and the bond energy is −E(0) = E_{b} = γ_{s}, (recall that the lattice spacing is set to unity) so the Griffith energy release rate is

G_{c} = 2γ_{s} = 2c_{12}λ^{2} = \frac{2e^{2}f_{max}^{2}}{c_{12}} .  (12)

Thus, for case of UBER, the Griffith relation is a simple quadratic function of f_{max}. With these equations, the ductility parameter can be written analytically,

D = \frac{G_{c}}{G_{e}} = \frac{4e^{2}f_{max}}{c_{12}Y (θ)β(0)}.  (13)

More importantly, in terms of λ one gets

D= \frac{4eλ}{Y (θ)β(0)}
λ_{0}= \frac{Y (θ)β(0)}{2e} ≈ 0.17,  (14)

where λ_{0} is the critical value at crossover. The numerical value is obtained by setting Y = 64/9 and β(0) = .25 (a value we anticipate from the lattice simulations to be presented below when there is no elastic mismatch). Note that c_{12} does not appear in this equation. For λ > 0.2, the material should be ductile. This result confirms one’s physical intuition that longer ranged force laws will correspond to ductile materials.

The above has ignored the effect of the γ_{us} terms in the lattice resistance, since these are typically small in comparison with the bond-breaking terms. However, if the interfacial bonds become sufficiently weak, this term must come into play, yielding a brittle interface.

IV. LATTICE SIMULATIONS

We now turn to the promised lattice simulations for a check of the model. The simulations will be performed for a simple 2D hexagonal lattice with nearest neighbor central force laws of the UBER form^{15}, with a modification that the force is cut off smoothly at the next nearest neighbor position,

f = \frac{c}{A}u(e^{−u/λ} − e^{u_{0}/λ})  (15)

u_{0} = \sqrt{3}−1 is the cut–off distance at the second neighbor postion, and A = 1−exp(−u_{0}/λ). We assume that different force laws apply to atoms on the cleavage plane from those in the bulk lattice, to simulate chemical embrittlement of an interface. We first show results for the case where the two lattices are elastically the same, c = c_{12} = 1 everywhere, in order to explore the role of chemical inhomogeneity, separately from that of elastic inhomogeneity. The purpose here is to do rather simple generic simulations, in order to test the basic physics of the model, and then to change such things as the bonding on the “interface”, and the loading so that the embrittlement physics is thoroughly explored. As noted above, more realistic lattices and force laws will be addressed in separate work.

The simulations are performed by means of the lattice Green’s function methodology used by us in previous work, to which the reader is referred for details^{16}. The crack is 100 atoms in length, and is allowed to either cleave on the original plane, or emit a dislocation on a single inclined plane. The crack is loaded by tensile or shear forces applied to the atoms at the center of the crack. Figure 3 shows the crack, first (a), at the point where it must decide whether to cleave, (b), or to emit a dislocation, (c).

The result of the simulations is that there are two cases of interest. In the first, termed the base case, our model is obeyed. But there is one other case, associated with lattice trapping effects, where the model breaks down. We will present these in turn here, and discuss the results in the next section.

A. Base Case. Figure 4 summarizes the results of a number of simulations for the case where the bonds in the bulk are held fixed, but those on the interface are varied. It shows the critical G_{e} in the parent crack or “Lab” coordinates as a function of the maximum bond force in the “chemical” bond, f_{max}, in the crack tip. The specific value of λ for the bulk bond is chosen to be relatively high, and is held fixed for all the simulations plotted in each figure. For this λ value, the material will normally emit a dislocation before it will cleave. That is, the material is intrinsically ductile. Such a material will be embrittled by the segregation of weakly bonded atoms on the interface. The solid line in the figure is the emission line, and the dotted line is the Griffith line. The Griffith curve is obtained from the force law as twice the bond strength as a function of the maximum bond force. The curves meet precisely at that force law where the material begins to cleave for the given choice of chemical bond strength. The simulations to the right of the intersection were obtained under pure Mode I tensile loading. To the left of the intersection, the crack can be made to emit a dislocation instead of cleaving by adding shear Mode II loading, and the emission points represent increasing Mode II as one moves to the left from the intersection point. Finally, as the Mode II loading is increased, the lattice on the cleavage plane becomes unstable in shear, and dislocations begin to emit on the cleavage plane. This occurs for the left most point in the plot. The emission and cleavage curves overlap, of course when emission is induced by addition of Mode II loading.

0.04^{0.05}^{0.06}^{0.07}^{0.08}^{0.09}^{0.1}^{0.11}^{0.12}^{0.13}^{0.06} ^{0.07} ^{0.08} ^{0.09}  ^{0.1}  ^{0.11} ^{0.12} ^{0.13} ^{0.14} G_{e}  f^{max} ◇  ◇  ◇ ◇ ◇ ◇ ◇ ◇ ◇ +  +  +  + + +

Figure 4. Simulation results for the homogeneous lattice, Base Case. The critical emission G_{e} is plotted vs. f_{max} for a series of differing force laws on the cleavage plane. The bonding law in the matrix is held fixed. Results are plotted as the full line. G_{e} is computed from the loads for the parent crack, called the lab system. The dashed line is a plot of twice the bond strength of the atoms on the cleavage plane vs. f_{max}, and represents the Griffith condition for cleavage. For points to the right of the intersection of the emission and cleavage lines, the crack is loaded in pure Mode I. To the left of the intersection, emission is stabilized by increasingly large amounts of Mode II. The matrix bond corresponds to an unstable stacking fault energy (relaxed) of γ_{us} = 0.00774 in the normalized units.

0.05^{0.06}^{0.07}^{0.08}^{0.09}^{0.1}^{0.11}^{0.12}^{0.13}^{0.14}^{0.07} ^{0.08} ^{0.09}  ^{0.1}  ^{0.11} ^{0.12} ^{0.13} ^{0.14} ^{0.15} G_{e}  f^{max} ◇  ◇ ◇  ◇  ◇  ◇ +  + +  +  +  +

Figure 5. Same as Fig. 4, except that the crack lies on an interface between two dissimilar sublattices. The elastic constant for the lower lattice is twice that of the upper, and the spring constant between the two lattices is the same as in the upper sublattice (sublattice 2). γ_{us}(2) = 0.00774.

Figure 5 shows similar results for a crack on an interface between two elastically different materials. The ratio of the spring constants for the lower to the upper sublattices is c_{1}/c_{2} = 2, which represents a modest mismatch. This choice of mismatch corresponds to emission into the softer material, which is physically expected. Again, the bonds between the upper and lower sublattices are different from either, representing chemical segregation at the interface. The spring constant for the interface bonds is the same as the upper sublattice, c_{12} = c_{2} = 1.

These results conform generally to the base case, but are shown as an example of embrittlement of an elastically mismatched interface. The emission line has a small anomaly near its intersection with the cleavage line where some curvature is evidenced, but otherwise follows the expected behavior.

B. Lattice Trapping. Figure 6 shows a similar set of simulations for a relatively weak bulk force law, in the sense of having a short length scale. For this force law, the material will cleave before it emits a dislocation. This material can be strengthened on the cleavage plane, however, so that cleavage is suppressed, and the crack will then emit a dislocation for a range of force laws. The plot is the same as before, but now the Griffith line overlaps the emission line so extensively that no prediction can be made. However, it turns out that this case corresponds to significant lattice trapping, and when the lattice trapping curve is plotted on the same graph, the intersection of the trapping curve occurs, again, precisely where the crack begins to emit dislocations under pure Mode I load. Again, the crack can be stabilized to emit a dislocation by the addition of Mode II loading, and the points to the left of the intersection correspond to this stabilized emission.

0.08^{0.1}^{0.12}^{0.14}^{0.16}^{0.18}^{0.2}^{0.22}^{0.09} ^{0.1} ^{0.11} ^{0.12} ^{0.13} ^{0.14} ^{0.15} ^{0.16} ^{0.17} ^{0.18} G_{e}  f^{max} ◇ ◇ ◇ ◇ ◇ ◇ ◇  ◇  ◇ ◇ ◇ + + + + + + +  +  +  + + □ □ □ □ □ □ □  □  □  □ □

Figure 6. Same as Fig. 4, except for a different force law in the matrix. Here the matrix bond is weaker than in Fig. 4, γ_{us} = 0.00483, but the crack exhibits strong trapping. The dashed line represents the Griffith law and the square points are the simulation results. The full line shows the actual load for cleavage of the lattice trapped crack, and lies appreciably above the Griffith line until crossover occurs.

V. COMPARISON OF UBER PREDICTIONS WITH SIMULATIONS

A. The Base Case. The cross-over from ductile to brittle behavior is encapsulated in the ductility parameter, D, defined in Eqn. (10), and the prediction made for it with the UBER law is given in Eqn. (13). We first consider the case shown in Fig. 4, where the emission criterion is accurately portrayed over its entire Mode I range by a linear relation between the critical driving forece, G_{e}, and f_{max}, and cleavage occurs precisely at the intersection of the emission and cleavage lines.

The crossover point is, of course, not a fully unique “point” because in all cases, for the lower f_{max} values, the emission can be stabilized by adding Mode II loading. For this discussion, then, we will take the crossover to be that associated with the more restrictive pure Mode I loading, which does define a unique point at crossover. In Fig. 4, this point is accurately determined by the intersection of the upper linear portion of the emission curve with the Griffith line. Then, to the left of the crossover point, the emission curve changes slope, and follows the Griffith line until emission ultimately occurs on the cleavage plane, when the Mode II load is sufficiently large.

The contribution of γ_{us} to the emission criterion is negligibly small, because the emission line in Fig. 4, when extended, goes through the origin with zero intercept. Apparently the reason is because emission takes place before Φ_{shear}^{0} has become appreciable. We find that the shear at the crack tip, which represents the total Burgers vector emitted, is in the neighborhood of Δu = 0.2, well below the peak in the Φ_{shear}(Δu) function at Δu = 0.5. We believe that this occurs because the maximum-force point in our force laws corresponds to bond strains considerably less than 0.5, which gives crack-tip shears that are even smaller. Another factor, of course, is that γ_{us} ≪ 2γ_{s} for these force laws. Thus, for these reasons, we expect an emission criterion dominated by the ledge term in Eqn (7-8), as happens in all our simulations. We believe this may be a general conclusion for most force laws, because for realistic bonding functions in metals, the maximum in the bond force will occur well before the strain reaches the nominal maximum in the shear function, Φ_{shear}. On the other hand, for bonds with strong angular components, the shear function is expected to develop its maximum early, and perhaps compete with the f_{max} criterion.

Even though we have argued that the γ_{us} does not contribute to the emission in our simulations, the actual value of the unstable stacking energy fault used in the bulk, γ_{us} = .0077, converts to a driving force in the lab system of G_{e} ≈ .05, which is at the lower end of the graph, but not really negligible. But γ_{us} corresponds to the maximum contribution to Φ_{shear}, reachable only for values of displacement in the core of Δu = .5, which we have shown is never reached. Hence, the mere fact that γ_{us} may seem non negligible compared to f_{max} is not a sole reason for including it in the emission criterion. To know if the γ_{us} is a dominant factor in the emission criterion, one must check that Δu at the critical emission configuration is about 0.5. (Partial dislocation considerations lead, of course, to slightly different values of these numbers.)

The slope of the emission line is a measure of the effective dislocation content at the crack tip, and from Fig. 4, β(0) = 0.25 (β, being a strain, is dimensionless). We find that β(0) is remarkably constant in all our simulations. Even when the matrix force law is changed from one value of γ_{us} or λ to another, β(0) remains in the vicinity of 0.3. Thus, our earlier proposal^{5}, based on standard Peierls theory that β(0) should be proportional to γ_{us} in the matrix, does not apply to these simulations. But β(0) appears as a critical parameter in D, and in the crossover prediction, and we do not know enough about what determines its value. For the moment, we must leave that question partially unanswered in spite of our results, simply because intuitively one expects β(0) to change when the generic form of the force law changes. For example, one type of force law which we believe will very likely lead to a change in β(0) is one which has strong angular dependence, as noted above. (The slope of the emission line does depend, of course, upon the elastic mismatch when the crack lies on a true interface, as will be discussed in a later paragraph.)

The fact that the emission line is (nearly) linear as a function of f_{max} and meets the Griffith line in Fig. 5 precisely where emission is no longer possible in pure Mode I is confirmation of the block construction. The small curvature in the emission line at the crossover to cleavage is apparently due to the interaction between events on the slip and cleavage planes. That is, when cleavage is imminent then the core is extended along the cleavage plane, and the effective incipient dislocations associated with the cleavage process interact with the incipient slip dislocations in the core on the slip plane. This nonlinear interaction between the two parts of the core can be expected to modify the effective value of β_{0}, which is a measure of the slope of the emission line. Fortunately, this effect of imminent cleavage on β_{0} does not extend beyond the immediate vicinity of the crossover. The critical value of λ for which crossover occurs in the simulations is λ = 0.33. This value is about twice that obtained in §III for the unmodified UBER force law. Since the force law is modified to go to zero in the simulations at the second neighbor, it is not expected that the two predictions are the same.

In the case of Fig. 5, for the elastically mismatched interface, the slope of the emission line is predicted to be somewhat higher than for the homogeneous case because of the dependence of Y (θ) on the elastic mismatch, and this should lead to a somewhat more brittle material, with λ_{0} larger than for the homogeneous case. Instead, we find that λ_{0} = 0.28, which is somewhat smaller than for the case in Fig. 4. But the increase in Y for the interface with c_{1}/c_{2} = 2 is only about 10%, so the expected effect is small. Thus, the observed value is probably an indication of the approximations made in the model. When we increase the elastic mismatch significantly to c_{1}/c_{2} = 10, then the crossover does move in the brittle direction, as predicted. For modest mismatches, these results show that the elastic mismatch does not lead to significant changes from the homogeneous case.

However, in Fig. 5, the emission line shows some curvature in the vicinity of the intersection, which is more of a concern than the small deviation from the predicted λ_{0}. The curvature is not large, but serves a slight warning that some unexplained physics still lurks in the problem. A much smaller curvature may be just discernible at the crossover in Fig.

4. We suspect the crossover area may be disturbed in a way leading to curvature (and the altered value of β(0) which it suggests), by an interaction in the core between the dislocation distributions associated with blunting emission, and the “crack dislocations” on the cleavage plane associated with the cleavage event which is almost able to take place. That is, the crystal is in a configuration where both emission and cleavage are very nearly activated at the same time, and the two incipient events interact nonlinearly with each other.

In Fig. 7, we expand the range of the parameters, to show the complete cleavage function, and its relation to the emission line. This diagram shows that the cleavage line is quadratic as predicted for the UBER in Eqn. (12), for the smaller values of f_{max}. The cleavage line straightens out to a linear relation for the larger values because of the modification made to the UBER law in Eqn. (15) for simulation purposes.

0^{0.02}^{0.04}^{0.06}^{0.08}^{0.1}^{0.12}^{0.14}^{0.16} ^{0}  ^{0.02} ^{0.04} ^{0.06} ^{0.08}  ^{0.1}  ^{0.12} ^{0.14} ^{0.16} G_{e}  f^{max} ◇ ◇  ◇ ◇ ◇ ◇◇◇ ◇

Figure 7. Same as Fig. 4, except for expanded range of axes. Full line is the Griffith function, and points are simulation results from Fig. 4.

The linearity of the emission line to the right of the crossover point, the precision of the prediction of the crossover point, and the correct prediction of the critical λ_{0} shows that the general model of the paper is beautifully obeyed in a particular region of physical space. We now turn to a case which illustrates an important limitation of the base model.

B. Lattice Trapping. The primary limitation to the model arises in cases where lattice trapping is important. Then, from Fig. 6, it is the intersection of the lattice trapping line with the emission line which determines the crossover. Unfortunately, lattice trapping is a phenomenon which depends sensitively on the nonlinear discrete response of the lattice to the form of the force law, because lattice trapping is correlated with a short ranged force law with strong “snapping” character^{17}. Thus, in this case, the trapping line can only be determined by a detailed simulation of the crack in the particular lattice, and any attempt to develop a physical prescription for embrittlement which is independent of the details of crack structure is doomed to failure. We do feel that this case is unlikely to occur in experimental systems, since if the interface bonds are stronger than the bulk ones the crack will likely move off the interface. Nevertheless, we include it because it usefully delimits the range of accuracy of the simple model.

VI. GENERAL CONCLUSIONS

A material may be embrittled by any agent which weakens a path in the material for fracture. Typically, this will take place on some sort of interface in the material, where weakening occurs because of the special bonding situation associated with the interface, or due to the segregation of minority chemical species at the interface. Such segregation can strengthen the interface, as well as weaken it, of course, and cosegregation effects at interfaces are common^{18}. In brittle materials, modifiying the strength of an interface can cause crack path deviation, which can be source of strengthening^{19}, but the primary issue for us is the dramatic change in toughness which can occur in ductile materials when the interfaces in the material are embrittled. Embrittlement can also occur due to external chemical attack at a crack tip, such as occurs in liquid metal embrittlement^{20} or for silica glass in the presence of water^{21}. In all of these cases, the bonding at the crack tip is different from that in the matrix, and the considerations of this paper apply, at least in a generic sense. We thus consider a crack which is somehow created on such a weak “interface”. The point of the paper is that the crack may still behave in a ductile manner, or cleave, depending on the criteria given in Eqn. (10). In other words, the simple presence of a weak path in the material is not a sufficient condition for a crack to cleave there.

But in a general way, if a path in a ductile material is weakened sufficiently, then a crack on it will surely cleave. There are two reasons. The first is that as f_{max} decreases, intersection between the emission line and the cleavage line always occurs as in Fig. 4. But, also in Eqn. (7) the unstable stacking fault energy in the matrix may be sufficiently high to inhibit emission, and again cleavage will occur.

Our simulations ascribe the entire effect of chemical embrittlement to a “chemical” bonding change at the tip, and neglect any other structural defects which will also generally be present. For example, an incoherent grain or interface boundary will possess such structural deffects, as well as special chemistry effects. Also, size effects associated with the misfit of an external molecule attacking a crack tip can be important^{22}. All such effects are neglected in our treatment, and in any practical case, the impact of these additional factors will have to be assessed in order to arrive at a final conclusion regarding the observed embrittlement. But our results are presented in the hope that the purely “chemical” effects may be dominant in many cases.

Our main results are:

1) The critical physical factor for ductile/brittle crossover is the range parameter, λ, and we find in our generic lattice that embrittlement occurs for approximately λ < 0.17 for the simple UBER pair force law. Embrittlement corresponds to an increasing ratio of the maximum bond force, which determines the emission lattice resistance, to the surface energy, which determines the cleavage lattice resistance.

2) In addition, the crossover depends on the geometry of the lattice through Y (θ), with higher values of θ leading to greater ductility.

3) Y also depends on the elastic mismatch at the interface, and with a stiffened substrate, the upper layer is moved in the brittle direction. But the effect is small for modest elastic mismatch.

3) The crossover from ductile to brittle behavior is independent of the spring constants in the interface.

4) For very small interfacial bond strengths, the bulk shear terms dominate the emission energetics, and the interface becomes brittle.

5) Trapping effects violate the predictions for the critical λ_{0}, and make the material more ductile than predicted.

Our modeling is predicated on nearest neighbor force laws, and as such, highlights local bonding effects at the segregated interface. Indeed, except for the elastic mismatch effects acting on Y (θ), embrittlement in our model is independent of the bulk lattice properties, except that for very weak interfacial bonding, the unstable stacking fault of the matrix lattice (Eqn. (7)) would become important. But in none of our simulations was this point reached. (Also, of course the bulk lattice does exert an effect by means of the elastic constants which appear in G_{c}, etc.) When one looks at the kinds of changes which appear when the force law is extended beyond nearest neighbors, rather undramatic things happen, even though the criteria become much more complex. We do not know the extent to which the bonding of real segregant atoms can be approximated by the simple UBER laws, but suspect there will be cases of interest where they can. In these cases, then the embrittlement predictions can be used directly. But if the segregant atoms are strongly covalent, then the results of the present paper are not expected to be quantitatively applicable, and further investigation of such force laws is in order. This is to say that so long as the general form of the bonding is not drastically different from UBER, the general predictions of this paper should be reasonably robust. In fact, recent simulations based on the embedded-atom (EAM) force methodology^{23} have indicated that for blunting emission, the required stress intensity greatly exceeds the “bulk shear” value alluded to above, suggesting that our main conclusion of the dominance of the ledge term may also hold for this more realistic force law. In addition, several analyses^{24,25,11} have shown that the type of many-atom effects described by EAMtype methods can be modeled, in a restricted range of configuration space, by an effective pair potential.

The demonstrated role of trapping as a general limit for the predictions of the base case is of considerable interest. The simulations carried out in the paper were done with a very nonphysical case, where the interface was stonger than the matrix. In such a case, of course, the crack will simply move off the interface, and the action will take place in the matrix material. For the purposes of the simulations, of course, we can specify any conditions we please, even if nonphysical, if they illustrate something important, as they did in this case. But what are the realistic expectations for lattice trapping to be important? We find that trapping at the interface can be induced by elastic effects in the bulk. For example, if the spring constants on the interface are weaker than in the bulk lattice, then the stiff matrix dominates the crack opening at the tip atom, and can generate lattice trapping. Likewise, elastic mismatch effects which have the effect of stiffening the lattice relative to the interface atoms will have the same effect. Since weakening the spring constants of the interface bonds will also usually go hand in hand with general weakening in the interface, the embrittlement to be expected will be not so great as expected, since lattice trapping effects move the material in the ductile direction.

In previous work^{4,5}, we have noted that the crossover is independent of γ_{s}, and have suggested that the crossover condition can be expressed in terms of a critical γ_{us}. But these results were for homogeneous lattices without chemical segregation, and were carried out with a wider variety of force laws, where an additional parameter was introduced. With the additional parameter (now three independent parameters), the unstable stacking fault and bond strength can be separated more than in the UBER case, and the γ_{s} was shown to cancel out of the ductility parameter, D. In thinking over the role of the third parameter in the present context, however, we are not persuaded that going beyond the UBER form in a pair force has any particular merit, from the standpoint of what might be happening in real materials. So we are content here to restrict ourselves to the simple UBER force laws, but with different laws for the bulk and the chemically segregated atoms on the interface. Within the context of the UBER law, however, since the spring constants scale out of the crossover problem, then any variable which depends on λ will contribute to the crossover. In particular, in this work, the crossover does depend on both γ_{us} and on γ_{s}. That is, if λ is eliminated from the expression Eqn. (14), then D can be expressed in terms of γ_{s}. In fact, one gets

D= \frac{4e}{Y (θ)β(0)}\sqrt{\frac{γ_{s}}{c_{12}}}
γ_{s0}= \frac{c_{12}Y^{2}(θ)β^{2}(0)}{16e^{2}}  (16)

where γ_{s0} is a critical bond strength at crossover.

In real materials, especially the fcc materials, the dislocations are split. Although we have not considered split dislocations directly in our work, one can make a qualitative statement about its expected effect. The partial Burgers vector will be smaller, so the initial emission event (the appearance of the first partial) should be easier. Not only is γ_{us} smaller, but much more important for all the cases considered by us, the ledge will also be smaller. Thus, emission will be easier, and the material will move in the ductile direction, relative to our prediction. Simulations of the fcc metals in the light of our results are indicated.

It appears that there is some basic correlation between the length parameter λ defined above and the observed brittle vs. ductile behavior of the metallic elements. Values for the UBER length scale parameter for a large variety of metals have been given in Ref. ( 15). (We use the ”surface” length scale defined in this paper, because we are examining the formation of surfaces). In order to avoid lattice-specific effects, we focus on trends within a particular crystal-structure class. For the fcc elements, the values of λ range from 0.16 and 0.23. The smallest value corresponds to brittle Iridium, and the highest to ductile Aluminum, suggesting some validity for λ as a basic ductility parameter. However, the value of 0.18 for Au, which is ductile, is only slightly higher than that for Iridium. In the bcc metals, λ ranges from 0.19 for Tungsten to 0.34 for Lithium. The alkali metals have higher values in general than the transition metals, again consistent with their higher ductility. However, we cannot attribute all of these differences to the λ parameter. For example, the bcc transition metals have significant angular forces^{11,26} which should modify the predictions significantly. The hcp metals display roughly the same range of values as the fcc metals; we are not aware of well-defined trends in their ductility so we do not comment further

It remains to be seen if these correlations hold for the case when the material has an interface with a different chemistry. In that case (the main object of the paper), it is assumed the matrix is ductile, and it is the property of the chemical bond across the interface which is crucial to embrittlement, e.g. the value of the “chemical λ_{0}”. Clearly, this bonding is specific to the particular chemical species at the interface, and the lattice within which it is embedded. Here, we are not comfortable using the UBER- or EAM-type force laws because the bonding possibilities are so much richer than in the elemental case. So, confirmation of the theory presented here will have to wait upon the performance of suitable ab-initio calculations.

Our results are presented for Mode I load (except when the system falls below the crossover). Mixed mode loading changes the results in a predictable manner^{14}, but we have chosen to keep the discussion closely focussed on the essential physics, and not introduce the additional complexity which the mixed modes brings. Mixed mode loading is an essential part of the discussion of any elastic mismatch interfacial cracking, but for the case described here where the mismatch is modest, the effects are not large. We will return to this issue in a separate paper, where a stability diagram for the crack in mixed loading will be introduced to guide one’s thinking.

We believe our results are sufficiently specific to offer a suggestion that the proposed block construction should be explored by means of any of the more sophisticated bonding relations for materials, especially for the case of chemical embrittlement in ductile metals like Aluminum or Copper. In such materials, we believe physical insight into the factors controlling embrittlement can be gained by the study of this construction, perhaps in conjunction with calculations of the unstable stacking fault energy and the γ–surface.

VII. ACKNOWLEDGEMENTS

We are grateful to Richard Hoagland and S. J. Zhou for informative conversations. The work at Washington University was supported by the Office of Naval Research under Grant number N00014-92-J-4049.

REFERENCES

1 R. Thomson, Solid State Physics 39, 1 (1986). 2 J. R. Rice, J. Mech. Phys. Solids 40, 239 (1992). 3 G. Schoeck, Philos., Mag. A 63, 111 (1991). 4 S. J. Zhou, A. E. Carlsson and R. Thomson, Phys. Rev. Ltr. 72, 852 (1994). 5 R. Thomson and A. E. Carlsson, Philos. Mag. A 70, 893 (1994). 6 J. R. Rice, Z. Suo and J.-S. Wang in Metal-Ceramics Interfaces, ed. M. Ruhle, A. G. Evans, M. F. Ashby and J. Hirth, Acta Scripta Met. Proceedings Series 4 (Pergamon, New York, 1990) p. 269. 7 J. Hirth and J. Lothe, Theory of Dislocations, (J. Wiley, New York 1982). 8 S. J. Zhou, A. E. Carlsson, and R. Thomson, Phys. Rev. B 47, 7710 (1993). 9 This corresponds to the case where the bulk shear energy includes only the energy of the atoms above the cleavage plane. If atoms below are included as well, this causes a correction proportional to γ_{us}, which is small compared to the other terms discussed here.

10 M. S. Daw and M. I. Baskes, Phys. Rev. B29, 6443 (1984). 11 A. E. Carlsson, in Solid State Physics: Advances in Research and Applications, Vol. 43,

edited by H. Ehrenreich and D. Turnbull (Academic, New York, 1990), p. 1. 12 B. Cotterell and J. R. Rice, Int. J. Fracture 16, 155 (1980). 13 R. G. Hoagland, Proceedings of the Symposium on New Techniques for Characterizing

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Figures

Figure 1. Block construction for Mode I emission configuration. The shaded block of atoms is cut from the lower lattice on the cleavage plane up to the crack tip, and then sheared rigidly on the inclined slip plane. The rigid displacement is in the direction shown by the arrow. The atoms whose energy is modified by the displacement are numbered. The lettered atoms correspond to bonds whose energy is altered by the standard unstable stacking fault construction when two semifinite blocks of lattice are homogeneously sheared past one another.

Figure 3. A series of simulations showing a) a crack which has grown on its cleavage plane to a point where it may follow several paths: b) The crack continues to cleave on its initial cleavage plane. c) The crack emits a dislocation on an inclined slip plane. In these simulations, the crack half length is 50 lattice spacings, and is loaded at its center by a combined tensile and shear force pair. The crack is embedded in an infinite periodic lattice with repeat cell size 2000 in each direction. The degree of shading shows the magnitude of the force on a particular atom exerted by atoms across the cleavage or slip plane from itself. The outer circle around each atom shows the cut–off distance of the force law.