E L S E V I E R Computational Materials Science 2 (1994) 137-142

COMPUTATIONAL MATERIALS SCIENCE

Whither computational materials science? Some thoughts from the mechanical properties front #

R o b b T h o m s o n

Laboratory for Materials Science and Engineering, National Institute of Standards and Technology, Gaithersburg, MD 20899, USA

(Received 21 June 1993)

Abstract

A claim is made that analysis will remain important and become a useful ally in helping computational materials science live up to its ultimate potential. Examples are given in the mechanical properties area where numerical simulations have been able to parameterize and mark out areas of validity for elasticity theory. The important role of developing asymptotic paths from one level or category of theory to another is commented on.

1. How clear is the view?

It is going to be interesting and perhaps illumi- nating to compare our various contributions in this issue, and see how much our opinions have in common. There will, of course, be complete una- nimity on the importance of computational mate- rials science, but that will apparently not deter any of us from giving our own versions of it. Mine goes something like this.

We are, I believe, witnessing a remarkable period which will lead in a decade, more or less, to a complete revolution in the way research in materials science is conducted. Up until now ma- terials science has been dominated by experimen- talists, for the simple reason that theory has been generally incapable of providing quantitative an-

# Supported in part by the Office of Naval Research.

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swers. There are a few exceptions to this blanket statement, such as calculations of the vacancy formation energy in simple metals, and so forth, but most of the important phenomena in materi- als science have been beyond the reach of quanti- tative treatment. Theory in mechanical properties has been one of the best examples of this, be- cause, generally, we have been able only to pro- vide qualitative insights and qualitative mecha- nisms about dislocations and cracks, and how they interact. This qualitative insight has been very useful, because of the thinking tools they provide, but that is a far cry from detailed predic- tions of mechanisms for work hardening, or brit- tle transitions, for example. The enormous con- troversies which have raged in materials science since its modern incarnation are noisy witnesses to the difficulty of deciding between conflicting mechanisms, when semi-quantitative theory is the best we can do. With the combination of new

138 Robb Thomson /Computational Materials Science 2 (1994) 137-142

mathematics and new computational tools, all that seems about to change dramatically, even if it does not disappear altogether.

Computational materials science probably will not become as influential in materials science as quantum chemistry has become in synthetic chemistry, because materials science is a long step more complex than molecular chemistry. However, I believe it will be possible to develop powerful collaborations between computational materials scientists and experimentalists, which will provide the truly significant advances of the future. Theory and modeling should become suf- ficiently powerful that, although we theorists may not be able to fly alone, we will make it impossi- ble for experimentalists to fly alone any more, either.

I believe the biggest road block to achieving this brave new world will not be limitations in computer hardware and software. The relevant developments in that area are coming so fast we probably need not worry. However, there is a major intellectual challenge related to the ques- tion of just what constitutes effective theory and modeling in materials science, and how to best develop it. I believe we need much greater ability to conceptualize the important issues in materials science. I am sure we cannot use the power of computers simply to do more complicated ver- sions of old problems, and hope to achieve the desired result. We cannot, for example, simply focus on solving the Schr6dinger equation for more and more complex systems, as the quantum chemists do. Our really challenging problems lie at a different level and are part of a different category. They have to do with the integration of the different levels and categories of theoretical description from the quantum to the macro- scopic. The self energies of single defects can now be computed with varying degrees of atomic bonding sophistication, but the most important problems facing us as materials scientists relate to how the various kinds of defects interact with one another, and how the ultimate material prop- erty is calculated from that. This is not a new idea, by any means, and has been commented on by many people before. My point is that we should not underestimate the difficulty or the

intellectual challenge in making the relevant progress. I do not believe that we have enough of the theoretical concepts, yet, which will be neces- sary before we can be very successful.

An example of the kind of new theoretical ideas we will need is related to the asymptotic relationships which exist between different levels of theory. In my own area of mechanical proper- ties, for example, elasticity is a powerful tool for handling dislocations and cracks (at least in 2D), because one can employ the standard armamen- tarium of the classical field theory of singulari- ties. Eshelby [1] has, for example, shown that the crack does not have a field inertial mass, like the classical electron has a field inertial mass, and that this property has to do with the kind of singularity which the crack exhibits in the elastic field. The most striking thing about theory in this area is that the concepts based on the elastic singularities which we use all the time are not easy to come by at the more basic quantum and atomic level, but our understanding would be incomplete and severely limited without them. That is, the analytic underpinning of elasticity provides a far richer and more complete under- standing of the world of dislocations and cracks than it could ever be if left entirely to the com- puter printout of atomic positions or a visualiza- tion of the time evolution of them. Working be- tween these two levels of theory is what I mean by the problem of the asymptotic relationship. It is sometimes possible to parameterize the elastic descriptions from the more basic atomic and quantum levels but, in the mathematical sense, there are often no clear asymptotic paths from one level to the other. Sometimes this leads to conceptual difficulties. I will return to some at- tempts to work in this area in the next section.

An even more serious difficulty, however, arises in the area of 3D interactions between defects, and of problems associated with microstructure. Some of the most active areas in computational materials science are in this area. I only offer the opinion that the most important challenge in that area is not the power of the computers, but in the inadequacy of our ideas for describing ensembles of defects, especially of those like dislocations, cracks and interfaces which have flexibility in 3D.

Robb Thomson /Computational Materials Science 2 (1994) 137-142 139

To return to the question posed as the title of this section. Our view is certainly murky, and scientists have been notoriously inept at project- ing the future. However, we certainly all perceive that something new and wonderful is happening in computational materials science. My principal comment is twofold: (1) we should not be too timid about the importance of what is happening, and (2) we probably do not adequately perceive the nature and difficulty of the true challenge! The implications of this prognosis are that many of us (both theorists and experimentalists) are in danger of going quickly out of date if we do not understand the imperatives of what may be hap- pening to us, and that the theoretical ideas we will be using in a few years are not likely to be our friends now. (So what 's new?!)

2. On the relation between elastic and atomic theories

In the next paragraphs, I will briefly review some recent work we have done on atomic calcu- lations of cracks and dislocations. I believe there is something of value in these calculations from the standpoint of learning more about the me- chanical properties of materials, and the tech- niques we use are unusual, as well. However, I wish to make the more general point noted above about one of the important uses of computations in materials science.

The general point is that bare simulation of materials, although often of great value in gaining insight about materials properties when pre- sented with imaginative visualization techniques, can be a much more powerful tool when coupled with an underlying analytic analysis. I shall illus- trate this point with three separate simulation studies with a generic model of a lattice bound with simple force laws.

The simulations are based on lattice Green 's function techniques, sometimes called lattice stat- ics [2]. Since the crack and dislocation strain fields are long ranged, it is very useful to be able to couple non-linear core regions, where purely numerical trial and error searches are required, to a very large linear lattice. The lattice Green 's

function techniques allow us to do this in such a way that the total system can be made very large, and so that the relaxation calculations of the final structure also can be carried out very fast. In our case, the super cell contains 4 × 10 6 atoms, and we find that a non-linear core region containing the crack and dislocation can be relaxed to within a 10 -4 to 10 -6 part of a lattice spacing of the true convergent solution in a matter of a minute or less. The Green 's function analysis is rigorous, so there are no uncertainties about satisfying boundary conditions anywhere in the problem. With this technique, it is then possible to carry out a large number of simulations to study a large parameter space with very nominal computer ca- pabilities.

We have studied three different problems, and our purpose in each case was to gain generic information about the validity of an elastic model of a crack interacting with one or more disloca- tions. It is thus not useful to make the model very sophisticated in terms of realistic lattices or ac- tual force laws. It is more important to choose a model to which we can rigorously apply isotropic elasticity in the continuum limit - the kind for which we can often get analytic results - and it is important to stay in 2D for the same reason. However, we do need to fully explore the role of non-linearity by using a variety of force laws, and the effects of discreteness. Also, we need to insist on realism in the sense that the structure must be in true equilibrium under the action of external forces and internal atomic forces. Also, we need to explore the effect of the mixing of tensile and shear modes of loading the system. In all these problems, the lattice of choice was the 2D hexag- onal lattice with nearest neighbor forces. I re- peat, we are after generic information here, it is soon enough to get more realistic after we have explored the generic physics.

The first problem deals with the intrinsic sta- bility of a crack in a lattice against shear break- down at the tip and subsequent dislocation emis- sion [3]. This problem goes back to a proposition stated some twenty years ago [4] that a material will be brittle or ductile in the intrinsic sense if a crack can exist in the lattice without shear break- down at the crack tip, given a particular kind of

140 Robb Thomson / Computational Materials Science 2 (1994) 137-142

bonding between the atoms. Recently, Rice [5] has proposed a very useful criterion for the dislo- cation formation in terms of the theoretical shear strength of the lattice, expressed as an energy, and called by him the unstable stacking fault energy,

(1- v)KZe 2/x =Y~" (1)

The material will be ductile or brittle if the stress intensity factor in shear, KII e, is greater than the Grifith stress intensity factor in tension, K I, given by

(1 - v ) K ~ = 2ys. (2)

In these equations,/x and v are the usual elastic constants. Yus is the unstable stacking fault en- ergy and ys is the intrinsic surface energy. The usefulness and power of Rice's criterion is that one can calculate the critical quantities Yus and y~ from bulk lattice theories, and it is not necessary to solve the crack problem at the same time. It is important to know the conditions under which this ducti le/bri t t le criterion is valid.

In our simulations [3], we showed that his criterion is remarkably well satisfied for the case where the dislocation is emitted ahead of the crack onto the cleavage plane - what we have called Mode II emission. When the dislocation is emitted at a finite angle to the cleavage plane (Mode I emission), however, we find that his criterion is not very accurate, and that a different kind of physics seems to be involved [6]. In the second case, the critical state for emission is quite different from the first. The size of the incipient Burgers vector in Mode II is close to b / 2 , where b is Burgers vector, in agreement with Rice's elastic modeling, while in Mode I, the critical Burgers vector is more like b / l O . We can see evidence of strong interactions between the ten- dency of the crack to cleave and its tendency to emit a dislocation, an interaction which is not so strong in Mode II. The simulations point to a different kind of elastic modeling using dis-

tributed dislocations than was done in order to fit the physics.

The second problem involved the use of the same atomic model to investigate non-local con- tributions to the shielding of a crack by an exter- nal dislocation, which can be a mechanism for changing the ducti le/bri t t le criterion just de- scribed [7]. In this case, we had a suggestion that such a mechanism might be possible from very qualitative elastic ideas. After doing the atomic modeling, we developed a much more realistic elastic model with which we could develop quali- tative understanding of the effect. Further, with the use of the atomic simulations, we were able to find values for parameters in the elastic theory. Then the elastic theory could be used to calculate the effects of large numbers of dislocations - much larger than would be possible to simulate on the atomic level. Practical applications will involve these large numbers of shielding disloca- tions, so the parameterized elastic theory is es- sential. We believe such a ducti le/bri t t le transi- tion may be possible in the case of nanostructure materials.

Finally, we have used the same atomic model to study interfacial cracking, and dislocation for- mation from cracks on interfaces [8]. Our prime purpose here was to investigate an old quandary in the elastic theory of cracks on interfaces. The elastic theory of such cracks predicts oscillatory displacements in the vicinity of the tip of the crack, which have been argued over in the me- chanics community for decades. We have been able to show that such oscillations in displace- ment are not possible if the crack is assumed to be in equilibrium - that is, if the crack is assumed to exist at all! On the other hand, we demon- strated that the "mode conversion" which the elastic theory also predicts is present. The mode conversion involves a "rotat ion" of the complex loading stress intensity factor, K = I K[ exp(i0) = K I + iKii from, say, pure tension at the point of loading to partial shear at the crack tip. If one defines a different stress intensity factor which characterizes the mode of loading at the tip by 2 f = K exp(i~), where the real part of og( corre- sponds to tensile load at the tip, then the elastic solution predicts that ~ = e l n ( r / L ) , where e is a

Robb Thomson / Computational Materials Science 2 (1994) 137-142 141

constant which depends on the elastic mismatch between the two materials, L is the total length of the crack and r is the distance along the cleavage plane to a reference point ahead of the crack. The phase shift at the crack tip where r = 0 is thus singular, but it can be defined for some finite distance from the tip, such as the core size of the crack. The atomic simulations show that, indeed, there is a phase shift at the core of the crack, relative to the load point, but the quantitative prediction of elasticity is not a good description of it. Also, the phase shift depends on the force law, as is suggested by a core size effect. Moreover, the simulations show that the limiting amount of mode conversion at the crack tip is determined by the emission of dislocations at the crack tip. That is, for a crack growing under a tensile load, as the crack lengthens, the stresses at the crack core gradually change from tension to shear, and ultimately dislocation emission in- tervenes. After that, the phase angle at the crack tip is determined by dislocation shielding as well as by the external load. The simulations show that the emission criterion is not well described by Rice's criterion (properly extended to the in- terface case) even for the Mode II emission case. On the other hand, the Griffith relation at the crack tip could be very accurately generalized to an opening criterion, which takes account of the observed phase shift dependence at the crack tip.

Thus, in the case of interracial fracture, the atomistic studies were again used to give validity to a very flawed elastic theory, and to parameter- ize it for useful applications.

In future work, we will continue to pursue the generic studies to gain understanding of the basic physics operating at interfaces between brittle and ductile materials, and to study ductility prob- lems in multilayers, where the size scale is much altered from what we are familiar with in bulk materials. Along the way, of course, we will be exploring what surprises await when we increase the sophistication of the modeling to include more realistic anisotropic lattices and atomic bonding. Currently, we are developing the Green 's func- tion technique so that it can be used for a rela- tively wide assortment of lattices and force laws. Hopefully, at some stage, the programs will be-

come sufficiently robust to be generally available to others beside ourselves.

Coming back to the general point raised at the start of the section. We have certainly demon- strated that discrete atomic techniques can be particularly powerful when used to gain sharper and more complete elastic analytic descriptions. In part this power comes from our ability to validate the elastic concepts which are very rich thinking tools, and in part it comes from our success in parameterizing a theory which can operate in a range of variables (such as number of dislocations or number of atoms) where no computer programmed for discrete atoms would be able to operate. However, beyond this, our experience also suggests a serious mathematical question: how does one develop a set of asymp- totic mathematical statements about the transi- tion between the lattice and the continuum? This question comes up in a particularly interesting way in regard to the singular phase shift in the interface problem. The Green 's functions for de- fects are found by solving a Dyson equation for the "defect space" [2]. The solution of the Dyson equation is not easily seen as the asymptotic limit of the procedures one uses in the solution of the crack problem in an elastic medium. However, if we could find an analytic prescription for obtain- ing the continuum limit in the Dyson equation, we could make a very useful connection between the two levels of theory, which now seem so unlike each other.

I believe that finding such asymptotic rules and prescriptions will pose one of the most chal- lenging and useful results of computational mate- rials science. I also believe that combining com- puterized versions of theory with analytic reason- ing will be far more powerful than relying solely on the computer by itself. The reason is simply that we need ways of thinking which are more human in form than number crunching, or even of computer visualization. Of course, it remains to be seen if sophisticated forms of computer visualization will ever supplant analytic reasoning

- I think not - because analysis is built on a mathematical rationalization of mechanisms and relationships. Rather, I think it will be a combi- nation of the two which will ultimately prove the

142 Robb Thomson / Computational Materials Science 2 (1994) 137-142

most powerful use of computional techniques in material science.

3. References

[1] J.D. Eshelby, in: Physics of Strength and Plasticity, ed. A. Argon (MIT, Cambridge, MA, 1969) p. 263.

[2] R. Thomson, S. Zhou, A. Carlsson and V. Tewary, Phys. Rev., Phys. Rev. B46 (1992) 10613.

[3] S. Zhou, A. Carlsson and R, Thomson, Phys. Rev. B47 (1993) 7710.

[4] J. Rice and R. Thomson, Philos. Mag. 29 (1974) 73. [5] J. Rice, private communications. [6] S. Zhou, A. Carlsson and R. Thomson, to be published. [7] P. Anderson and R. Thomson, to be published. [8] R. Thomson and S. Zhou, to be published.