Tutorial . .

The Modeling of Interfaces: Atoms or Continua?

Rob Phillips

Interfaces are a central microstructural unit responsible for properties ranging from the capacity for void nucleation to enhanced diffuSion rates. Demand for materials with specific properties makes the tailoring of in­terfaces a central goal in materials modeling. In the consideration of detailed defect geom­etries or nanoscale effects, atomistic model­ing provides reliable insights. On the other hand, the response of materials to macro­scopic loads is ofi'en best treated within a continuum framework. This overview re­views both atomis ,tic and continuum models of interfaces with an eye to the possible rela­tionships between them. The problems of void nucleation and grain boundary segre­gation are used as examples to elucidate the progress that can be made on both the con­tinuum and atomistic fronts as well as the limitations of each. These approaches are further contrasted through consideration of the example of elastic wave propagation at interfaces. The treatment of waves localized at interfaces yields distinct predictions de­pending upon whether an atomistic or con­tinuum analysis is made.

INTRODUCTION

Interfaces have been invoked as the explanation for phenomena ranging from enhanced mass transport rates to premature fracture of materials under mechanical loading. Beyond this, their position as a basic microstructural unit has made them the subject of debate for wide classes of materials with an accom­panying interest in modeling them. In­terfacial modeling arises in a host of contexts. The aim of many of these ef­forts is to provide a mechanistic basis for the rich phenomenology that has been observed at interfaces. Though some hold out the hope of designing materials from first principles, at the moment, modeling usually plays the less ambi­tious role of providing insights into phe­nomena such as impurity-induced embrittlement at boundaries! or the ori­gins of the critical thickness for misfit dislocation formation in lattice-mismatch heteroepitaxial thin films.2

One of the intriguing features of in­terfaces is their relevance across a multi­tude ofIength scales. For example, metal­ceramic interfaces in composites often provide the nucleation site for the voids that lead to their ultimate demise.3 In this case, the void sizes can be as large as a micrometer. On the other hand, post-

1995 March • JOM

mortem Auger analysis of impurity con­centrations as a function of depth be­neath boundaries following failure by intergranular fracture indicate that the segregant population decays over length scales shorter than ten lattice spacings.4 Despite the small scales over which the segregants are localized, their presence leads to measurable reductions in the macroscopic fracture stress.s

Because of the hierarchy of length scales important to interfacial phenome­na, two basic approaches (atomistic and continuum) have arisen in the treatment of interfacial modeling. Conventionally, continuum models are used in cases for which the spatial variations in the quan­tities of interest take place over distances much larger than the interatomic spac­ing. On the other hand, atomistic model­ing is usually employed when specific details of atomic level geometry are of interest. In both atomistic and continuum models, the interface exhibits distinct structure and cohesion (Le., the strength

Interface

Figure 1. The partitioning of a solid with an interface into two regions. In one region, the constitutive response is given using conven­tional models such as linear elasticity. In the interfacial region, a separate interfacial con­stitutive relation is prescribed.

of the interfacial "bonds" differ from those of the bulk). When a continuum model is adopted, the presence of the interface is revealed via special condi­tions on the displacement fields and the tractions in the interfacial region. On the other hand, atomistic treatments of in­terfaces are characterized by a red uction in symmetry reflected in the positions of the atoms at the interface and changes in the resultant bonding.

The study of interfaces is a vast topic requiring input from a multitude of dis­ciplines, treatment of different classes of materials, and studies with reference to a number of different properties. Rather than attempting to be encyclopedic, this overview focuses on a few key examples of modeling that relate to the mechanical response of interfaces. In the end, one of the most powerful advances resulting from the flurry of activity in materials modeling is the possibility of now carry­ing out numerical experiments on real as well as hypothetical materials and their attendant defects.

CONTINUUM MODELING OF INTERFACES

Continuum models of material be­havior are stated as boundary value prob­lems. In the case of interfacial modeling, for example, one might ask, what are the displacements in the interfacial region given a particular externally applied load? The relevant degrees of freedom in a continuum model are the fields charac­terizing the body and quantities such as displacements, stresses, or temperature. Implementation of a continuum me­chanical formulation of a problem re­quires a few key ideas. First, it is nec­essary to specify the geometry of de­formation; this takes the form of kine­matic relations between displacements and strains, for example. Second, the governing equations for the continuum are identified with the balance laws of classical physics such as conservation of energy and linear momentum. Next, phenomenological relations between the stresses and strains must also be pro­vided. It is crucial to note that these constitutive relations are not derived from continuum mechanics. Rather, they are an input into a continuum model and in some cases, appeal may be made to atomistic models to determine them. Finally, the statement of the problem of

37

o NI2 3 4

Figure 2. Normalized normal component of traction vs. scaled opening displacement. interest requires boundary and initial conditions. For bodies with internal structure such as interfaces, additional information must be provided to charac­terize the response of the interface itself. One example of such analysis is the modeling of interfaces in composites.

Composites have been suggested as one possible answer to the call for mate­rials that are both light and strong. In­trinsic to the structure of these materials, however, is the presence of a variety of interfaces, some of which have been ar­gued to lead to low fracture toughness.6

One suggested explanation for this ef­fect is the idea that the metal-ceramic interfaces serve as the site of void nucle­ation by a mechanism of interfacial decohesion.3,6 Similar mechanisms are operative in other materials, such as

a

b O.2~m

Figure 3. (a) Finite-element mesh used in modeling AI-SiC interfacial decohesion. Initial void nucleation can be seen to have taken place near the sharp corner. (b) Observed void initiation at fiber end. For details see Reference 3. (Courtesy of A. Needleman and S. Nutt.)

38

steels, where interfacial debonding at inclusions initiates the void nucle­ation processes that lead to their ultimate ductile failure. These observa­tions (and a plethora of others) demonstrate that interfaces cannot be ig­noredincontinuummod­els of material response.

The ingredients that are needed in the con­tinuum modeling of ma­terials with interfaces are shown schematically in Figure 1. Here, the body is divided into two re­gions, one of which is treated via standard bulk continuum analyses, while the formulation for

y

Free Surfac

x

Periodic

the interfacial region cen- Figure 4. Schematic illustrating the slip and opening displace-ters on the independent ment degrees offreedom treated by elastic potential $(0" Oy' ~). specification of a local 0, = 0Y. = 0 corresp~nds to pure decohesion, while ~ = 0 implies constitutive relation that _th_a_t _ol_sp;.,.l_ac_e_m_e_n_t_ls..;p;.,.u_r_e_s..;hp_. __________ _ reflects the coupling of the stresses at eters can, in some cases, be physically such an interface to the displacements motivated on the basis of atomistics. The they suffer. In this way, the bulk and tractions resulting from this type of po-interface are free to respond distinctly to tential have the form indicated sche-applied loads. One class of interfacial matically in Figure 2. constitutive models posits an elastic po- Interestingly, atomic-level processes tential <1>(6) which measures the re- are notthe only ones describable in these versible energy cost to displace adjacent terms. Suo and ShihB have discussed the planes across the interface by a relative way that crack-bridging mechanisms displacement 11.7 The resulting tractions such as ductile ligaments or distribu-are given by tions of voids lead to constitutive be­

T = a<l>(Il) n all

w here Tn is the traction (force / area) per­pendicular to the interfacial plane. Inter­facial constitutive relations of this ge­neric form have been postulated to de­scribe a range of processes and length scales.B When nanometer-length deco­hesion of adjacent atomic planes is the operative process, such decohesion has been seen to obey the universal binding­energy relation (UBER).9 By normaliz­ing the energy and length scales, the energetics of decohesion is found to sat­isfy a material-independent, universal form. In particular, this energy takes the generic form

(1)

Here I is a scaling length, and 00 is re­lated to the maximum traction that oc­curs during decohesion. Note that the parameters 00 and I are phenomeno­logical as far as the continuum model is concerned. On the other hand, the en­ergy vs. displacement relation embod­ied in Equation 1 has been found as a fit to the results of a series of quantum­mechanical calculations of the energy of decohesion,9 revealing that such param-

havior like that implied by Equation I, although in these cases the length scale (10-4-10-5 m) is considerably larger than that of the atomic scale decohesion en­visaged above. The qualitative feature shared by all tractions of this form is that for a range of displacements they in­crease, but beyond a critical length they decay until reaching zero at complete separation of the interface.

Once the relation between stress and strain has been advanced for the inter­faces and the bulk solid, the machinery is in place for a solution to the boundary value problem of interest. Finite-element calculations are often the numerical tech­nique of choice in the solution of such problems. One advantage of a finite­element formulation is its flexible treat­ment of different geometries and load­ing conditions, allowing for extension of models beyond the highly symmetric configurations normally required to make analytic headway. As an example of the continuum modeling of interfacial decohesion, consider the application of such models to interfacial debonding and subsequent void formation in AI­SiC composites. Experiments on this sys­tem indicate that void nucleation usu­ally occurs near sharp corners in the interface between the aluminum matrix and the SiC fibers.3,6 These observations pose a modeling challenge. For example,

JOM • March 1995

it is of interest to characterize the condi­tions under which an interface will debond completely or only in the vicin­ity of geometric singularities. It has been argued that the corners provide stress concentrations that lead to interfacial debonding.

N eedleman7 has considered this prob­lem by assuming that the cohesion of these interfaces is -characterized by a trac­tion vs. displacement relation similar to that derived from Equation 1. The con­tinuum analysis of this problem also requires a constitutive description of the bulk material, which may, for example, be treated as an elastic solid, although, in practice, more complex bulk constitu­tive behavior is adopted? In addition, to compute the deformation of the AI-SiC composite, boundary conditions must be prescribed. In this case,3 the axial tensile rate is prescribed. That is, the velocities of the boundary surfaces per­pendicular to the tensile loading direc­tion are given. Due to these prescribed strains, the solid responds with both elastic and plastic deformation. Because of inhomogeneities in the deformation and stress fields caused by the sharp corners in the AI··SiC interfaces, the inter­face is found to suffer initial decohesion near these corners, as shown in Figure 3a. This can be favorably compared to observed void initiation, shown for the AI-SiC interface in Figure 3b. Further, it is found that if the interface is suffi­ciently strong, the decohesion will re­main confined to the vicinity of the cor­ners. A second interesting conclusion is a noticeable absence of decohesion on interfaces paranel to the fiber axis (i.e., there is no fiber pullout). One important feature contained in the use of the inter­facial constitutive relation is that the decohesion is a natural outcome of the deformation history, rather than the re-

sult of some ad-hoc prescribed deco­hesion process.

The qualitative insights afforded by this type of modeling are as follows. First, in accordance with intuition, the modeling shows that the stress con­centrations due to the sharp corners lead to localized plastic strains and conse­quent void nucleation. These findings are in agreement with those observed experimentally. Second, depending upon the strength of the interface (re­lated to the parameter 0-0 in Equation I), the remainder of the interface mayor may not allow for decohesion. This ex­ample emphasizes the use of local con­stitutive relations to model the energet­ics of decohesion. However, the forego­ing has only considered displacements perpendicular to the interface; the slid­ing of adjacent planes can occur as well.

Interfaces and dislocations are in­extricably linked. For many grain bound­aries, their geometrical character is best described in terms of their dislocation content. On the other hand, despite their characterization as line defects, the ener­getics implied by the misfit accompany­ing dislocations may, in some cases, be modeled in terms of interfacial constitu­tive relations. The Peierls concept for characterizing the energetics of disloca­tions centers on the idea that the nonlinearity inherent in the dislocation is localized along a slip plane, which may be thought of as an interface be­tween regions subject to different slip.IO The modeling of the misfit transmitted across this slip plane is quite analogous to the treatment of decohesion described above. However, in the dislocation case, slip discontinuities (tangential to the in­terface) as well as cohesive disconti­nuities (normal to the interface) must be considered. As in the application to void nucleation, the idea is to partition the

~====:==============::, body into a region

Figure 5. Energy as a function of both opening (Li) and Slip (0) displacements in units of eV/A2. Opening displacements have been normalized by interplanar spacing d, and slip displace­ments are normalized by in plane periodiC length b. For this case, slip is on (001) plane of face-centered-cubic copper, with slip in the (110) direction. The line corresponding to pure slip (Li = 0) is highlighted.

1995 March • JOM

treated by a conventional bulk constitutive relation such as linear elasticity, and a localized region across which the trac­tions are given as a func­tion of the slip disconti­nuity. The Peierls analy­sis has gained recent at­tention as the basis of models of dislocation nucleation at crack tipS.12,13 The central question here is the role of such nucle­ation in determining the outcome of the competi­tion between cleavage and crack-tip blunting. Armed withmicroscopi­cally based interfacial constitutive relations, it is possible to make ma­terial-specific predic­tions concerning the ten-

Figure 6. Trial grain boundary structure for ~5 tilt boundary with [001] tilt axis. Shaded and open circles correspond to different layers in the face-centered-cubic lattice in (001) pro­jection. The vacant spaces in the immediate boundary region are eliminated upon relax­ation.

dency for dislocation nucleation at crack tips, which is one ingredient in the ongo­ing struggle to theoretically characterize the features of a material that make it either brittle or ductile.

Central to the continuum analysis of such crack-tip dislocation nucleation and a point of contact with atomistic model­ing is the description of the misfit energy along the slip plane in terms of an inter­facial constitutive model. It is crucial to bear in mind that such interplanar constitutive relations do not admit of any simple experimental investigation. As a result, atomistic modeling can serve as a reliable basis for such constitutive phenomenology. Traditionally, the en­ergy for interplanar shear has been pro­vided by the Frenkel sinusoid,lO which for isotropic materials is given by

<1>(0)= ~b2 [1_cos(21t0)] (2) 41t 2d b

where b is the repeat distance along the slip direction, d is the spacing between the planes being sheared, 0 is the slip displacement, and ~ is the shear modu­lus. The prefactor in this equation is determined by insisting that Equation 2 agree with the linear elastic result in the limit of small displacements (o -t 0). Note that this model ignores any dis­placement jumps perpendicular to the slip plane.

Advances in the atomistic modeling of materials have made possible a new generation of interfacial constitutive re­lations based upon direct atomistic cal­culation.I4-16 These calculations find that the Frenkel sinusoid (Le., Equation 2), while qualitatively reasonable, lacks the quantitative features to represent the energetics of slip. A full continuum accounting of interfacial constitutive behavior should consider both opening and slip displacements simultaneously (so-called tension-shear couplingI7). To

39

Pt-3 at.% Ni Ni-3 at.% Pt of an interface through to its eventual debonding, may be mod­eled within the contin­uum framework . Per­haps surprisingly, it is also possible to argue that continuum models of dislocations employ a similar strategy to char­acterize the energetics of the misfit implied by the presence of a disloca tion. Recent work in this vein has addressed a spec­trum of interesting ap­plications. Examples in­clude refined estimates of the critical thickness for misfi t dislocation for­mation in heteroepitaxial systems? instability to crack branching in dy­namic fracture/o and extensions to the case in which the interfacial stresses depend upon the rates of opening as well as the opening displace­ments themselves.21

0.14 9=28.1·

i ~ 9=28.1·

0.08 ~ if • .- _e e ~. --

B ;:l 0.14 -0 !Zl

c: 0.08 0

'.;:l

! ~

~t \9= 33.9° 9 = 33.9°

. - _e .. ~

~ 0.14 U ·s

0 . 08 0 ~

9 = 36.9° f * ~ +\ 4

9 =36.9°

• - .... ... .. ~,

0.14

0.08

9 = 43.6° ~ I ~ ~

9 = 43.6°

t t i • ,. . ... o 8 16 24 32 8 16 24 32

(002) planes Figure 7. Concentration profiles of segregant in the grain boundary vicinity for the Pt-Ni system. Concentration is shown as a function of distance away from the boundary plane for a number of different twist angles. (From Reference 39).

compute the energy as a function of both slip and opening displacements, a crys­tal is divided into upper (z > 0) and lower (z < 0) half spaces, and the entire region of the crystal with z > 0 is sub­jected to the uniform displacement u = (ox' 0 , ~) as indicated schematically in Figu/e 4. The energy as a function of Ox and ° has been dubbed the gamma sur­face .l.)' Figure 5 shows the energy as a function of both sli P (0) and opening (~) displacements for the (001) plane of cop­per with slip along the (110) direction. These calculations have been performed using the embedded-atom method,lS which is a semi-empirical technique for trea ting the energetics of metals. Further details of the atomistic approach will be described below. The cross section at ° = o corresponds to the energetics of pure decohesion. Tractions like those shown in Figure 2 result upon differentiating this cut of the energy surface with re­spect to the opening displacement ~. A cut at ~ = 0 gives the periodic energy profile as a function of slip along the (110) direction.

The above consideration of interfacial slip and decohesion demonstrates the usefulness of local constitutive models in the continuum description of inter­faces . Through the device of exploiting atomistically based interfacial constitu­tive models, continuum analysis of ex­tended defects such as interfaces and dislocations can be built upon a sound microscopic foundation. It has also been shown that by using an interfacial con­stitutive relation, the entire deformation history, from initial elastic displacement

40

ATOMISTIC MODELING OF INTERFACES

As noted previously, for many inter­facial problems the operative lengths of the phenomena of interest demand a continuum analysis. On the other hand, detailed structural questions or issues surrounding grain boundary chemistry require an explicit consideration of the atomic degrees of freedom. One example in which the need for atomic-scale mod­eling is evident is that of grain boundary segregation.

A range of intriguing experiments in­dicated that the presence of segregants can alter the strength of interfaces sub­stantially5 and that the concentration of such segregants is enhanced consider­ably over a localized region near grain boundaries.4,22 Two conclusions can be drawn from these observations. First, the interfacial constitutive behavior de­scribed in the previous section requires amendment due to the presence of seg­regants. That is, the quantitative fea­tures of the traction vs. displacement relation shown in Figure 2 will vary in response to changes in the grain bound­ary composition. Said differently, the work of adhesion at such interfaces is affected by solute segregation.23 Prelimi­nary accounts of the coupling of segre­gation to the interfacial constitutive rela­tions addressed in the previous section have indeed found a suppression of the adhesive energy due to the presence of impurities.24 Second, the nanometer lengths over which the segregant con­centration profiles are observed to de­cay suggest that atomic-scale effects at

the grain boundary are critical in deter­mining the segregant profiles. In this section, the atomistic-approach to mod­eling the structure of grain boundaries and the related issue of susceptibility of a particular interface to segregation is described.

Atomistic modeling is based upon an explicit treatment of the microscopic degrees of freedom. This is in marked contrast with the continuum approach in which the relevant "structural" de­grees of freedom are the displacement fields. From the continuum point of view, the structure of a dislocation or grain boundary is contained in the specifica­tion of the associa ted displacement fields . The power of atomistic simulation is its ability to characterize the geometry of the system of interest (such as a grain boundary) explicitly in terms of atomic positions. Work over the last two de­cades has revealed a steady increase in the · complexity of the interfacial struc­tures that can be modeled. Progress has taken place on three main fronts .

First, with the advent of increasingly powerful computers, the size of the sys­tem tha t can be considered in direct simu­lation now exceeds 500 atoms for highly accurate first-principles calculations;25 when simpler methods are used, in ex­cess of 106 atoms can be simulated.26 A second advance that has made such simulations possible is the development of improved methods for computing the total energy of a collection of atoms. Progress has taken place both in the formulation of more reliable, semi-em­pirical total energy methods as well as in the algorithmic development of tech­niques for doing larger quantum me­chanical calculations. Finally, the exist­ence of experimental techniques such as high-resolution electron microscopy and scanning-tunneling microscopy (for con­sidering surfaces, for example) allow for direct comparison of predicted and ob­served atomic-scale geometries, provid­ing fresh impetus to the microscopic con­sideration of structural questions.

Prior to investigation of the equilib­rium segregation of impurities to grain boundaries, it is crucial to have confi­dence in the atomistic description of the structure of the grain boundaries them­selves. The susceptibility of a particular boundary to segregation is inherently dependent upon its structure, Hence, failure to accurately describe the grain boundary structure calls into question any subsequent conclusions concerning segregation.

Atomistic simulation centers on a few key ideas. It is often essential to provide an initial guess for the atomic positions. An example of a trial grain boundary structure is shown in Figure 6, which shows a low-angle tilt boundary with the misorientation evident across the boundary plane. Given initial atomic

JOM • March 1995

positions, the next step is to find the atomic coordinates that minimize the function E,o' ({Ri». E'a' is the total energy of the solid and is a function whose value depends explicitly on the posi­tions of every atom in the computational cell. There are a number of different methods for computing E'a" ranging from pair potentials to full-blown quantum mechanical analysis; for the moment, the particular choice is left unspecified. Given the total energy, the minimum energy structure is that for which the forces on all of the atoms vanish (Le., -aE'o/aRia = 0). Here, i labels the atom of interest and a labels Cartesian compo­nents. In practice, different methods such as conjugate gradient minimization,27 molecular dynamics, or Monte-Carlo schemes28 are adopted for different prob­lems. Each has its own advantages, though in simplest terms, they all achieve the same objective (Le., finding the atomic configuration that minimizes the total energy).

Thus far, this description has ignored a number of fine points. For example, there is no guarantee that a particular minimum energy structure is a global minimum of E,o" Also, there has been no discussion of the refinements needed to account for finite-temperature studies. For the purposes of this overview, it is sufficient to say that these issues can all be treated in turn.28 In the case of grain boundary structure, once the initialstruc­ture and a prescription for computing the function E,){RiJ) have been speci­fied, determination of the equilibrium coordinates becomes a computational matter. Often, it is of interest to compare the structures obtained via simulation with those found in experiment. One strategy involves an interactive proce­dure in which high-resolution images of

5,-----------rr----------,

4

:!:

: ~ : : :

o+.--~-,---~~--~_,--~~

X r y

observed boundaries are compared with simulated images obtained from the com­puted structure.29

As alluded to above, the state ofthe art in the atomistic modeling of materials takes a few different forms, each involv­ing some compromise. Highly accurate, quantum-mechanical calculations have been applied with success to point de­fects, dislocations, and interfaces.3o How­ever, this accuracy comes at a price. Namely, computational cells are gener­ally restricted to at most 500 atoms. For many problems, this constraint precludes a realistic treatment of the defect of inter­est. By way of contrast, simpler methods such as Lennard-Jones pair potentials have been applied to in excess of 2 x 106

atoms.26 Calculations of total energies using the latter models exact their own toll. In particular, semi-empirical poten­tials are notorious for lacking trans­ferability (i.e., they breakdown when applied to problems for which they were not explicitly fitted). For example, if a particular interatomic potential is fitted to bulk cohesive and elastic properties, there is no guarantee that it will predict the correct dislocation core or grain boundary structures. Transferability problems like this are especially evident in the study of covalent materials such as silicon.3!

The structure of grain boundaries has been considered using both first-prin­ciples techniques and semi-empirical methods in a number of different con­texts. For example, semi-empirical work on the copper-bismuth system yields structures in close accord with those observed experimentally.32,33 This is one of many representative examples where atomistic simulation using semi-empiri­cal techniques has yielded structural in­sights. First-principles structural stud-

5.-----------,,----------~

. .. .

4

a b

Figure 8. Vibrational spectra in gold for both (a) the perfect lattice and (b) the lattice with grain boundary defects. Note the splitting off of vibrational states at small frequencies for the interfacial case. (From Reference 49).

1995 March • JOM

ies have been advanced as well. Work on twist boundaries in germanium suggests the possibility that there exist multiple grain boundary structures that are nearly identical in energy.34 These studies, and others,35 indicate that atomistic simula­tion can serve as a useful gUide to de­termining the geometric structure and energetics of a variety of interfaces.

One of the most interesting outcomes of studies of the coupling of the struc­ture and energetics of boundaries is the investigation of trends in grain bound­ary energy as a function of geometric parameters characterizing the bound­aries.36 For example, how does the en­ergy of symmetrical tilt boundaries vary with misorientation angle? Questions of this type are amenable to both exper­imental and theoretical analysis. For low­angle boundaries, Read and Shockley37

argued that the grain boundary energy could be thought of as resulting from the energetics of the dislocation array of which the boundary was composed. This continuum analysis yields a grain­boundary energy vs. misorientation angle of the form Egb(e) = e (a!-a2 In e), where a1 and a2 are constants related to the relevant dislocations. While the Read­Shockley analysis captures many of the essential features of the energetics of low-angle boundaries, both atomistic modeling and experimental analYSis of grain boundary energy exhibit more subtle effects as well. Note that the Read­Shockley result ignores the discrete ef­fects introduced by the presence of the underlying lattice. As a result, the en­ergy is a continuous function of the boundary dislocation spacing (or mis­orientation angle). Consequently, there are no dislocation spacings singled out as special in the continuum analysis. On the other hand, these lattice effects intro­duce cusps in the energy as a function of the misorientation angle corresponding to special, highly coherent boundaries.36

The strategy in carrying out atomistic simulation of grain boundary segrega­tion has two purposes: it is of interest to learn how the presence of segregants alters the grain boundary structure, and it is desirable to determine the equilib­rium distribution of segregants in the vicinity of the boundary. Experiments on the role played by segregrants in influencing intergranular fracture in steels and other intermetallics have mo­tivated a range of modeling activities aimed at addressing the types of ques­tions raised above. Experiments on steels reveal that the stress to initiate inter­granular fracture is a strong function of the thermal history of the sample as well as the chemical identity of the segregants. In particular, the presence of segregants is often found to lower the stress needed to initiate such fracture.s On the other hand, recent work on boron-doped NiAI alloys has exhibited the opposite effect

41

with a consequent increase in ductility due to the presence of segregants.38

For several metals, interatomic inter­actions have been modeled using the embedded atom method18 (also called Finnis/Sinclair potentials or the glue model) in which the total energy is given by

Etot = ~ L V(RiJ + Lf(p;} (3) l J l

The first term is a standard pair inter­action in which the interaction energy is a function of only the interparticle sepa­ration. The second term specifies the energy cost associated with depositing the ith atom into an "electron sea" of density Pi' where Pi is obtained by as­suming that neighboring atoms bleed charge into the region occupied by the atom of interest. There are subtleties as­sociated with the choice of the functions used for V, f, and P, References 18 and31 are recommended for greater detail.

Both atom probe field ion microscopy22 and Auger spectroscopt allow for in­vestigation of the concentration profiles for segregants in the vicinity of grain boundaries. One of the critical results of such experiments is the conclusion that the enhancement of impurity concentra­tion often has a decay length on the nanometer scale. The theoretical in­terpretation of these observations can be addressed numerically via Monte-Carlo simulation. Such simulations allow for determination of equilibrium at finite temperatures with respect to both the chemical and structural degrees of free­dom. The result of such a simulation is a set of optimal atomic positions and chem­ical identities.

To implement a Monte-Carlo simu­lation, the idea is first to posit a set of "moves" which alter the configuration of the atoms. In general, these moves can involve the motion of atoms as well as swapping of their chemical identities. Once the move has been made, the total energy of the new configuration is com­puted using Equation 3, for example. If the move results in a reduction of the energy, the new configuration is adopted. Alternatively, if the move results in an increase in energy, the move is accepted with probability given by the Boltzmann weight, exph~E/kT). After some num­ber of Monte-Carlo steps (usually> 105),

a state of equilibrium has been reached and thermal averages of quantities such as the concentration profile can be com­puted. Profiles for the Pt-Ni system com­puted in this way are shown in Figure 7.39 The figure shows concentration pro­files in the direction perpendicular to the boundary plane for a number of differ­ent misorientations for twist boundaries. Evidently, the segregant concentration dropped to its bulk equilibrium value within a distance of five lattice spacings.

42

As noted above, the rapid decay of seg­regant away from the boundary plane is observed experimentally as well.

Calculations of equilibrium segregant profiles are of interest in themselves, but really only serve as the first step in at­tempts to model interfaces. It is not enough to have determined the geomet­ric distribution of segregants. One of the central themes of materials modeling is that of the coupling of structure and properties. Given the geometric distri­bution of segregants as in the work de­scribed above, the next level of interfa­cial modeling is to begin to query the relation between structure and proper­ties. An interesting step in this direction has been made by Arias and Joanno­poulos,40 who have considered the effect of arsenic and gallium segregants on tilt boundaries in germanium. In particular, they find an electronically driven ten­dency for the segn~gation of n-type dopants such as arsenic and not for p­type dopants such as gallium. Their cal­culations provide initial insights into an asymmetry between electron and hole trapping at boundaries in germanium. These calculations demonstrate the spirit of current thrusts in the atomistic mod­eling of interfaces, and hold out the prom­ise of a true elucidation of the coupling of structure and properties.

Alhough this section emphasizes the role of atomistics in determining segre­gation effects, there remain interesting connections to continuum modeling as well. It was observed earlier that the interfacial constitutive relations used to treat decohesion require amendment in the presence of segregants. Another in­teresting point is consideration of the success with which equilibrium segre­gation profiles can be described using macroscopic thermodynamic argu­ments. Examination of these questions should shed light on the relative im­portance of kinetic and equilibrium fac­tors in governing the observed segregant distributions since it has been shown that the role of segregation in effecting embrittlement depends upon the anneal­ing history, indicating the important role of kinetic factors .

INTERFACIAL WAVES­ATOMISTIC AND CONTINUUM

APPROACHES

50 far, it has been argued that model­ing from both atomistic and continuum perspectives have yielded important in­sights into a range of interfacial prob­lems. This section considers a problem for which both models make definitive predictions and contrasts them. The im­portance of interfaces to problems in wave propagation is well known; ex­amples vary from the use of multilayer mirrors for x-ray lithography to the re­flection of acoustic waves in structures. 5toneley's 1924 analysis41 of the problem

of continuum elastic wave propagation at interfaces raises questions that can be addressed directly via atomistic simula­tion; this point is interesting since Stoneley's analysis was concerned with the propagation of geophysical waves.

Continuum Models

The starting point for the analysis of elastic waves from the continuum me­chanics perspective are the Navier equa­tions, which in the absence of body forces are given by

(4)

where CrkJ is the elastic modulus tensor, u·k,=a2U./aukau" and summation is im­phed ori the indices j, k, and I which label the three Cartesian directions. If a trial solution of the plane-wave form is adopted and substituted into Equation 4, there results a set of algebraic equa­tions in the displacements. The solu­tions of these equations give the allowed elastic wave speeds and corresponding propagation directions. In isotropic lin­ear elasticity, carrying through this calcu­lation leads to the conclusion that trans­verse waves propagate with velocity vt = (ll / p)l l2, while longitudinal waves propa­gate with velocity v, = «A + 21l)/p)1I2. Here, A and /J. are the familiar Lame constants of isotropic linear elasticity. Extension of this analysis to the case of an elastic medium with an interface re­quires the imposition of interfacial boundary conditions.41 ,42 It is assumed that the solid may be divided into two elastic half-spaces, one with the elastic modulus tensor C~k' and the other with elastic modulus tensor Cijkl' Plane wave trial solutions are then adopted for both half-spaces. Boundary conditions are im­posed by insisting on continuity of both displacements and tractions (related to the derivatives of the displacements) at the interface. This strategy is quite analo­gous to that used in the consideration of electromagnetic waves at interfaces. For the case of "perfect bonding" elucidated above, the continuum solution leads to the conclusion that propagation of waves within the interfacial plane is only al­lowed in certain directions and for a limited range of elastic mismatch be­tween the two half-spaces. In particular, for certain copper and gold bicrystals, propagation is forbidden perpendicular to the tilt axis.43 Furthermore, when so­lutions exist, such waves are "localized" in the sense that the displacements de­cay exponentially with increasing dis­tance from the interfacial plane. These observations can be contrasted with the results obtained by evaluating interfa­cial waves from an atomistic viewpoint.

Atomistic Models

In the atomistic case, the consider­ation of waves in a solid amounts to

(Continued on page 74.)

JOM • March 1995

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rials for nonlinear optical devices and optoelectronic circuits; high-temperature semiconductor devices; nonvolatile and dynamic memories; microelectro­mechanical devices; short-wavelength light emitters; photovo~aic devices; detectors; and displays.

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discussion of practical as well as fundamental as­pects of recycling of metals and engineered materi­als. In response to growing pressures to curb waste, many industries have established successful recy­cling loops. This has led to the development of sophisticated process technologies, which, in turn, have stimulated basic research in the field. Topics will include industrial practices and fundamental re­search on the recycling of aluminum; copper, nickel, and cobalt ; lead, zinc, and tin; precious metals and catalysts; aluminum consumer packaging; municipal waste; used beverage containers; automobile recy­cling; and aluminum by-product recovery.

Second International Symposium on Extraction and Processing lor the Treatment and Minimization 01 Wastes Sponsor: TMS Date: October 27-30,1996 Location: Scottsdale, Arizona Abstracts Due: September 15, 1995. Contact: V.

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Abstrac:s lor nvlS'spollsorea sympos ,a rT'LS: adhere to tile gJldellnes a~ page 78

Modeling Interfaces (Continued from page 42.) making a normal-mode analysis. Spe­cifically, the total energy is expanded about the equilibrium configuration to se<;Dnd order in the displacements (Le., Ri = R~ + u i '. where ui is the displace­ment of the i'h atom) on the grounds that small amplitude vibrations (I ui I I flo« I, where ao is the lattice parameter) are being considered. As a result, the in­crease in the energy relative to the equi­librium configuration may be written as

dE = ~ L L Vi~~Uia Uj~ (5) iJ ap

where v.a~ =azE ,IaR aRRistermed the lJ to HI JP-"

force constant matrix, ij label particular atoms, and a~ label Cartesian compo­nents. The resulting coupled equations of motion are

.. - ~~vn~ -muin - LL ij Uj~ j ~

(6)

Our interest in wave solutions motivates the ansatz uin = uin,o exp[i(k-r-rot)] . Sub­stitution of this solution into Equation 6 leads to a set of algebraic equations simi­lar to those described earlier in the con-

74

tinuum context.44 Solutions to this prob­lem again correspond to the allowed wave speeds and corresponding eigen­vectors that give the geometric character of the displacements associated with the mode of interest.

The computational strategy adopted by Alber et al.43 in conSidering copper and gold bicrystals is to set up a periodic array of grain boundaries that are pre­sumed to be far enough apart to prevent contamination of the results of interest by interboundary interactions. Given a particular boundary structure, they then proceed to solve the eigenvalue problem described above. The outcome of this analysis is a vibrational spectrum of the form shown in Figure 8. This plot shows the vibrational frequenCies associated with modes of wavevector where K

Ikl = 27t A

These modes maybe thought of as small­amplitude vibrations with sinusoidal spatial variation of wavelength

27t

lkI Further, given a particular wavevector,

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there is an associated set of allowed fre­quencies. It is expected that in the long wavelength limit (k-+O) the dispersion relation will correspond with that pre­dicted by continuum analysis. This claim is substantiated through an analysis of the elastic wave speeds of the low-fre­quency, k -+0 modes for the perfect crys­tal as shown in Figure 8a. These speeds are found to be in near perfect accord with those coming from the continuum analysis. Figure 8b reveals that in the interfacial case at small frequencies and wavevectors, certain vibrational modes are split off from the bulk states. These are the interfacial phonons that bear fur­ther consideration. In particular, it is of interest to contrast these waves with those resulting from the continuum analysis. Are such waves localized to the interfacial region? Are there restrictions on the allowed propagation directions?

The Models Contrasted

As noted above, the continuum analy­sis allows for the propagation of interfa­cial waves only in certain special in­plane directions. In particular, for the case of the copper and gold bicrystals considered by Alberet al., the continuum

JOM • March 1995

analysis yields the result that for certain symmetric copper and gold bicrystals, propagation is forbidden for k vectors perpendicular to the tilt axis ofthe bound­ary. In stark contrast with the perfect bonding continuum results, the atomistic analysis leads to solutions for propa­gation in directions both parallel and perpendicular to the tilt axis. A second point of contention between the atomistic and continuum results is in the predic­tion of the extent to which interfacial waves are localized. Despite the nearly identical elastic anisotropy of copper and gold, atomistic analysis finds that the modes are highly localized for gold while for copper they are not. These conclu­sions are a dramatic example in which the continuum and atomistic pictures yield different results. Alber et al. at­tribute the discrepancy between the atomistic and continuum results to a failure of the continuum model to ac­count for inhomogeneity of the elastic response in the vicinity of the grain boundary. The linear elastic analysis upon which the Stoneley description is predicated assumes that the elastic modulus tensor is constant (though dif­ferent) in the two half-spaces adjacent to the interfacial plane. However, byevalu­ating the atomic-level moduli,45 it has been shown that the nature of the moduli in the vicinity of the interfacial plane differ considerably from their bulk val­ues. The conclusion is that one cannot simply borrow elastic properties from the bulk solid and treat them as rep­resentative of the elastic response in the near interface region. These calculations, and others, argue that the near interfa­cial region must be assigned its own elastic identity,4&-48 in keeping with the distinct cohesive properties elucidated in earlier sections.

In summary, under the restrictive con­dition of perfect bonding (continuity of tractions and displacements across the interface), continuum wave solutions exist only for special propagation direc­tions. Atomistic models, in contrast, yield solutions for propagation directions both parallel and perpendicular to the tilt axis. Reconciliation of these models is made possible by attributing distinct elastic properties to the interfacial region, a re­sult made plausible by the earlier claim that such interfaces are marked by unique structure and cohesion as well.

CONCLUSIONS

The demand for materials with prop­erties tailored for specific thermo­mechanical or electrical applications coupled with advances that now make reliable modeling efforts possible have resulted in redoubled efforts to produce predictive models of material behavior. These advances hinge, in part, on the treatment of interfacial phenomena. Grain boundaries have been shown to

1995 March • JOM

mediate diverse phenomena; examples range from the enhancement of diffu­sion rates to the reduction in the fracture stress needed to initiate intergranular fracture. Beyond this, surfaces, stacking faults, and heterophase interfaces, to name a few other interfacial defects, play a role in processes as diverse as catalysis and device fabrication. All told, inter­faces are a key player in driving many material properties. One of the intrigu­ing features presented by interfaces is the hierarchy of length scales associated with them. This article has shown that continuum analyses can be modified to account for the distinct cohesive proper­ties of interfaces with the effect that in­terfacial deformation (and ultimate deco­hesion) are the predicted outcome of applied loading and subsequent re­sponse rather than an initial condition prescribed by fiat. This calculational strategy has been applied with benefit to problems involving both the slip and decohesion of interfaces. On the other hand, atomistic modeling aims to eluci­date the geometrical features of defects and has provided a variety of insights into interfacial structure, energetics, and chemistry.

The article also emphasized the ben­efits of considering atomistic and con­tinuum approaches to interfacial model­ing in conjunction. It has been shown, for example, that continuum models of interfacial decohesion can be improved as a result of the type of microscopic information provided by the VBER, in turn providing a sound atomistic basis for constitutive models of atomic-level decohesion. Similarly, a reassessment of the idea of elastic moduli from the atomic level allows for a reconciliation of atomistic and continuum theories in the treatment of elastic wave propagation at interfaces. The examples presented, though interesting in themselves, serve the more important role of illustrating the wide scope of interfacial modeling and to point with high expectation to future efforts in this direction.

ACKNOWLEDGEMENTS

I am grateful to C. Briant, J. Bassani, R. Miller, M. Mills, A. Needleman, and A. Schwartzman for comments and discussion. All of them have made thinking about inter­faces more interesting. This work was sup­ported in part by National Science Founda­tion grant CMS-9414648.

References 1. L. Goodwin, RJ. Needs, and V. Heine, f. Phys. Condens. Matter, 2 (1990), p. 351. 2. These efforts are reviewed in L.B. Freund, Adv. Appl. Mech., vol. 30, ed. J.W. Hutchinson and T.Y. Wu (San Diego, CA: Academic Press, 1993), p. 1. 3. S.R Nutt and A Needleman, Scripta Met., 21 (1987), p. 705. 4. For Auger concentration vs. depth profiles see Figure 5 of D.Y. Lee et aI., Metall. Trans., 15A (1984), p. 1415. 5. W. YU-Qing and CJ. McMahon, Jr., Mat. Sci. and Technol­ogy, 3 (1987), p. 207; J. Kameda and CJ. McMahon,]r., MetaU. Trans., 11A (1980), p. 91. 6. S.R Nutt and J.M. Duva, Scripta Met., 20 (1986), p. 1055. 7. A Needleman, Ultramicroscopy, 40 (1992), p. 203. 8. Z. Suo and CF. Shih,Fundamentals of Metal-Matrix Compos-

ites, ed. S. Suresh, A. Mortensen, and A. Needleman (Stone­ham, MA: Butterworth-Heinemann, 1993). The use of inter­facial constitutive laws of the form described by Suo and Shih in the fracture context is well described in B. Lawn, Fracture of Brittle Solids (Cambridge, MA: Cambridge Uni­versity Press, 1993). 9. j.H. Rose, J, Ferrante, and j.R Smith, Phys. Rev. Letters, 47 (1981), p. 675. 10. J.P. Hirth and j. Lothe, Theory of Dislocations (Malabar FL: Krieger Publishing Co., 1992). 12. J.R. Rice, J. Mech. Phys. Solids, 40 (1992), p. 239. 13. J.R Rice and G.E. BeJtz,J. Mech. Phys. Solids, 42 (1994), p. 333. 14. G. Bozzolo, J. Ferrante, and J.R Smith, Scripta Met., 25 (1991), p. 1977. 15. E. Kaxirasand MS. Duesbery,Phys. Rev. Letters, 70(1993), p.3752. 16. R Miller and R Phillips, unpublished. 17. Y. Sun, G.E. Beltz, and J.R Rice, Mat. Sci. and Eng., A170 (1993), p. 67. 18. M.s. Daw, S.M. Foiles, and M.1. Baskes, Materials Sci. Repts., 9 (1993), p. 251; M.W. Finnis and J.E. Sinclair, Phil. Mag., A50 (1984), p. 45; F. Ercolessi, M. Parrinello, and E. Tosati, Phi/' Mag., A58 (1988), p. 213; KW. Jacobsen, j.K. Norskov, and MJ Puska, Phys. Rev., B35 (1987), p. 7423. 19. V. Vitek, Phil. Mag., 18 (1968), p. 773. 20. X.-P. Xu and A Needleman, J. Mech. Phys. Solids, 42 (1994), p. 1397. 21. L.-O. Fager et al., Int. Journal of Fracture, 52 (1991), p. 119. 22. S.-M. Kuo et aI., Phys. Rev. Letters, 65 (1990), p. 199. 23.j.R Riceandj.-S. Wang, Mat. Sci. and Eng., A 107 (1989), p. 23. 24. T. Hong,J,R Smith, and D.j. Srolovitz, Phys. Rev. Lett., 70 (1993), p. 615. 25. I. Stich et aI., Phys. Rev. Lett., 68 (1992), p. 1351; K.D. Brommer~ M. NeedeIs~ B.E. Larson, and lD. ]oannopoulos, Phys. Rev. Lett., 68 (1992), p. 1355. 26. F.F. Abraham et aI., Phys. Rev. Lett., 73 (1994), p. 272. 27. The conjugate gradient method is discussed clearly in Numerical Recipes, ed. W.H. Press et al. (New York: Cam­bridge University Press, 1986). 28. Monte Carlo methods and molecular dynamics are de­scribed in Computer Simulation of Liquids by M.P. Allen and D.J. Tildesley (New York: Oxford University Press, 1987). 29. Implementation of this iterative strategy for consid­eration of grain boundaries in Al is given in M.J. Mills et al., Ultramicroscopy, 40 (1992), p. 247. 30. Two interesting first-principles applications to the dislo­cation problem are T.A. Arias and j.D. joannopoulos, Phys. Rev. Lett., 73 (1994), p. 680;J.R.K. Biggeret al.,Phys. Rev. Lett., 69 (1992), p. 2224. Applications of first-principles techniques to the investigation of interfaces may be found in J. Crain et al.,Phys. Rev. Lett., 70, (1993)as well as Refs. 34 and 40 below. These are just a few representative examples. 31. A.E. Carlsson, Solid State Physics: Advances in Research and Applications, ed. H. Ehrenreich and D. Turnbull, vol. 43 (New York: Academic Press, 199]), pp. 1-91. 32. D.E. Luzzi et aI., Phys. Rev. Lett., 67 (199]), p. 1894. 33. M. Yan et aI., Phys. Rev., B47 (1993), p. 5571. 34. E. Tarnow et aI., Phys. Rev., B42 (1990), p. 3644. 35. A few other interesting atomistic studies of grain bound­ary structure are U. Wolfet al.,Phil. Mag.,A66 (1992), p. 991; V. Vitek et al., Scripta Met., 17 (1983), p. ]83; G.H. Campbell et aI., Phys. Rev. Lett., 70 (993), p. 449; S.P. Chen, D.J. Srolovitz, and AF. Voter, J. Mater. Res., 4 (1989), p. 62. 36. The relation between grain boundary structure and ener­getics is discussed in the article by D. Wolf and K.L. Merkle in Materials Interfaces, ed. D. Wolf and S. Yip (New York: Chapman and Hall, 1992). 37. W.T. Read and W. Shockley,Phys. Rev., 78 (1950), p. 275. 38. CT. Liu, CL. White, andJ.A Horton, Acta Met., 33 (1985), p. 213; E.P. George, CT. Liu and D.P. Pope, Scripta Met., 28 (1993), p. 857. 39. D.N. Seidman, B.W. Krakaver, and D. Udler, J. Phys. Chern. Solids, 55 (1994) p. 1035. Additional information may be found in S.M. Foiles and D.N. Seidman in Materials Interfaces, ed. D. Wolf and S. Yip (New York: Chapman and Hall, 1992); A. Seki et aI., Acta Met., 39 (991), p. 3167. 40. T.A Arias and J.D. joannopoulos, Phys. Rev. Lett., 69 (1992), p. 3330. 41. R Stoneley, Proc. Roy. Soc., AI06 (1924), p. 416. 42. D.M. Barnett et aI., Proc. Roy. Soc., A402 (1985), p. 153. 43. E.s. Alber et aI., Modelling simul. Mater. Sci. Eng., 2 (1994), p.455. 44. Phonons are reviewed in P. Bruesch~ Phonons: Theory and Experiments I (Berlin, Germany: Springer-Verlag, 1982). 45. J.L. Bassani, V. Vitek, and I. Alber, Acta Met., 40 (1992), p. S307. 46.1. Alber et aI., Phil. Trans. Roy. Soc., A339 (1992), p. 555. 47. M.D. Kluge et aI., J. Appl. Phys., 67 (1990), p. 2370. 48. J.B. Adams, W.G. Wolfer, and S.M. Foiles, Phys. Rev., 640 (1989), p. 9479. 49. V. Vitek et al.,/. Phys. Chem. Solids, 55 (1994), p. 1147.

ABOUT THE AUTHOR

Rob Phillips earned his Ph.D. in physics at Washington University in 1989. He is cur­rently an assistant professor in the Division of Engineering at Brown University.

For more information, contact R. Phillips, Divi­sion of Engineering, Box D, Brown University, Providence, Rhode Island 02912; e-mail phlllips@alpha1,engin.brown.edu.

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