Mathematics for the Modeling of Defects in Materials

Claude Le Bris

Introduction to manufacture tools. Without these mobile defects, ma- A famous quote1 says that “Crystals are like people, it is terials would deform elastically until they suddenly break, the defects in them which tend to make them interesting.” and bending even the thinnest metal bar would require a One of the best known examples of the importance of de- large amount of energy. Likewise, in a different domain, fects in materials is perhaps the fact that crystals, and more doped semiconductors, and therefore electronics, would generally materials in an ordered phase (as opposed to not exist without defects. Defects are useful for something! disordered or amorphous materials such as glasses), owe their plasticity to the existence and motion of dislocations, which are linear topological defects associated with an elastic stress field—a line of miscoordinated atomic sites indi- cating an irregularity within the periodic array of atoms (see Figure 1). As the stress increases and exceeds a certain threshold, the deformation is no longer reversible, as is the case in the elastic regime. The displacement of dislocations allows the material still to deform, but in a nonelas- tic, nonreversible manner. The density, the topology, the possible entanglements of these defects control this plastic behavior, in sharp contrast with the reversible, elastic behavior that is ruled by the geometry of the underlying, ide- alistically perfectly periodic, atomistic structure. Plasticity explains why one is able to shape metallic materials and

Claude Le Bris is a researcher at Ecole des Ponts and Inria, Paris, France. His Figure 1. Simple—called edge—dislocation at the microscopic email address is [email protected]. scale: an extra half-plane of atoms, the “defects,”is inserted The research of the author is partially supported by ONR under Grant N00014- in the periodic lattice (2-dimensional cut of a structure 15-1-2777, and by EOARD under Grant FA-9550-17-1-0294. periodic in the perpendicular direction). 1C. J. Humphreys, Stem Imaging of Crystals and Defects, in Introduction to Analytical Electron Microscopy, pp. 305–332, J. J. Hren et al. (eds.), 1979. Communicated by Notices Associate Editor Reza Malek-Madani. More generally, defects in media, not only materials, play For permission to reprint this article, please contact: a key practical role in several applications. One may have [email protected]. in mind all applications of computer vision and imaging DOI: https://doi.org/10.1090/noti2098 that aim to detect an unusual feature within an image and,

788 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 6 beyond, the many applications in inverse problems in the interaction potential 푉. By this we mean that a given parti- engineering sciences—localization of an oil reservoir in cle in 푋푘 only interacts with the particles 푋푘−1 and 푋푘+1 the underground, radars, sonars, etc. Interestingly, the in- (with the obvious adaptation when 푘 = −푁 or 푁). verse problem of detecting defects is also relevant in the When 푁 is finite, as in a finite size molecular system, the context of materials, for instance for crack detection in met- energy of the system is the sum of all interactions: als. However, before addressing the inverse problem, one first needs to adequately formalize the direct problem: the 퐸푁 = ∑ 푉(|푋푘+1 − 푋푘|). (1) modeling and study of defects are already critical issues for −푁≤푘≤푁−1 materials. In a nutshell, one may see a defect as a presumably Suppose now that the particles are clamped at integer posi- rare anomaly within an otherwise well-understood and of- tions, 푋푘 = 푘, which is typically the case if the pair interac- ten homogeneous background. It would be impossible tion potential 푉(푟), which is assumed to vanish at infinity, to cover the huge variety of defects in materials science; achieves its unique (negative) minimal value at 푟 = 1 and we refer to the many excellent monographs in mechanics- one considers the equilibrium, lowest energy, configura- related fields. The purpose of this mathematically oriented tion of the set of 2푁 + 1 particles. Next let 푁 grow to infin- presentation is to show a specific series of settings in ma- ity. The energy 퐸푁 = 2푁 푉(1) clearly diverges. This is not terials science, and more precisely multi-scale materials sci- unexpected, since the energy is an extensive quantity that ence, where mathematics plays a role in the modeling and scales linearly with respect to the amount of matter consid- thus in the understanding and numerical approximation ered. Only the energy per particle admits a limit, namely of defects. Even though our study can be motivated by practical concerns such as the modeling of plasticity, which 1 퐸 ⟶ 푉(1), (2) we mentioned at the opening of this article, we mostly 2푁 + 1 푁 consider here defects that are much simpler than dislocations and we consider them as mathematical objects inde- as 푁 → +∞. The latter value defines the energy, in this pendently of the meaning they can carry in a mechanical model, of the periodic system consisting of an infinity of context. Our purpose is also to present a set of theoretical periodically arranged atoms 푋푘 = 푘, 푘 ∈ ℤ. Let us now questions that, conversely, arise from this particular class present two ways to perturb this ideal periodic system. of applied problems. In the absence of defects, the math- First, we may modify the positions of all particles. This ematical problem is “nice and easy”—by the standards of may be achieved assuming now that the 푘-th particle has contemporary functional and differential equations analy- a random position 푋푘(휔) = 푘 + 푌푘(휔), slightly shifted sis. It derives from well-established techniques. It is also from 푘, where, say, all the random variables 푌푘, −푁 ≤ expected to be simple computationally. The presence of a 푘 ≤ 푁, are independent, identically distributed, and val- defect breaks this nice mathematical setting. The mathe- ued in (−1/2, 1/2). A similar argument as above, this time matical difficulty is due to the ensuing losses of finiteness, formalized by the law of large numbers, yields the expec- boundedness, or compactness. Numerical difficulties sim- tation value 피(푉(1 + 푌1 − 푌0)) as the energy per particle of ilarly follow. New tools are everywhere in order. the infinite system we have constructed. We may also mod- Of course, this article cannot cover the many existing ify the positions of particles deterministically: for instance, mathematical studies, in various areas of mathematics, the system that address the modeling of defects. It only presents an expository review of our approach to the problem, in col- 푘 + 1/2 for (2푛)2 < |푘| ≤ (2푛 + 1)2, 푋 = { (3) laboration with close colleagues and focused on the theory 푘 푘 for (2푛 + 1)2 < |푘| ≤ (2푛 + 2)2, of partial differential equations and related issues, along with connections with some other works that we find par- composed of periodic regions alternately shifted from 1/2 ticularly intimately related to ours. or not, clearly admits an energy per particle—which, in- Atomistic Models and Beyond cidentally, is identical to that of the original periodic A simple atomistic model. To begin with, we consider system—even though the system is by no means periodic a simplistic, 1-dimensional model of a material—at zero and the support of the defects extends to infinity. In other temperature, with only one atomic species—and we intro- terms, the defect is global, as in the random case, although duce the notion of defect in an atomic lattice at an elemen- entirely deterministic. tary level. A set of 2푁+1, eventually 푁 = ∞, point particles Another direction we may take is to leave all the particles unperturbed except only a few, say one, namely the or atoms 푋푘, 푘 ∈ {−푁, … , 푁}, with the convention 푋−푁 ≤ particle 푘 = 0, which we now clamp at 푋0 = 푎 ≠ 0. This 푋−푁+1 ≤ ⋯ ≤ 푋푁 interact via a nearest-neighbor pair is now our notion of local defect. The energy of the finite

JUNE/JULY 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 789 system now reads as such an ab initio derivation is however limited. In partic- −2 ular, in relation to our earlier discussion and in line with defect several existing works, it would be of major theoretical and 퐸푁 = ∑ 푉(|푋푘+1 − 푋푘|) 푘=−푁 practical interest to better understand, from models of de- + 푉(|1 + 푎|) + 푉(|1 − 푎|) fects at the atomistic scale, the often phenomenological 푁−1 models of plasticity.

+ ∑ 푉(|푋푘+1 − 푋푘|). 푘=1 1 Of course, 퐸defect shares the same limit 2푁 + 1 푁 1 as 퐸 when 푁 grows to infinity. Our defect being 2푁 + 1 푁 localized, it does not affect the energy per particle in the limit of an infinite number of particles. In order to geta nontrivial effect—and “detect” the defect simply because the energy of the system differs from that of the periodic system—we have to proceed otherwise. To model the energy of the perturbed system differently, we consider the difference defect 퐸푁 − 퐸푁 = 푉(|1 + 푎|) + 푉(|1 − 푎|) − 2 푉(1), (4) which defines a quantity (the defect formation energy, in some sense) that, contrary to either of the two terms of the left-hand side taken separately, has a finite limit when 푁 → Figure 2. Computer simulation of a displacement cascade. +∞. Clearly, this limit may be seen as the energy of the periodic system with a defect at zero, being understood that In numerical simulations, atomistic models, possibly this energy is counted with reference to the ideal periodic embedding defects, can be used to better simulate the be- system. havior of real materials. An illustrative case is provided by Beyond atomistics. The above simplistic model can be the context of nuclear engineering. The aging of materials made more sophisticated in several directions. One may is studied by successively following in time (a) the evolu- consider an analogous discrete system of infinitely many tion of irradiation defects in the periodic structure—the atomic sites, in the ambiant 3-dimensional physical space, displacement cascades of Figure 2, (b) the aggregation of de- all interacting with one another via a potential. Think fects to form dislocations, (c) the evolution of dislocations, for instance of an interaction potential 푉 ∈ 퐿1(ℝ푛) ∩ and eventually (d) the effective dynamics of the density of 퐶0(ℝ푛\{0}), and it is then an easy exercise to generalize the the different categories of dislocations. Each stage in this formulae above and discover that the corresponding peri- string of modeling levels is a coarse-grained version of its predecessor. The defects, originally treated as discrete mi- odic potential 푉per = ∑ 푉(⋅ − 푘) plays a key role. Such 푘∈ℤ푛 croscopic objects, are monitored, at the macroscopic level, discrete models offer many cases of specific interest. as a density field of objects in interaction which move On the theoretical and modeling fronts, they can serve and transform. These techniques belong to the vast cate- as a starting point for a change of scale and the deriva- gory of molecular dynamics techniques—the “molecule” be- tion of models of continuum mechanics [9, 10]. Many ing, in this instance, the defects—although a more accu- of the latter models are indeed phenomenological, not to rate characterization would be computational statistical me- say ad hoc. Their mathematical study is a challenge, in chanics techniques; see e.g. [17]. A range of stochastic nu- particular when the material has indeed defects. As an ex- merical techniques is employed to achieve this, which of ample, one may think of the models of fracture and the course concurrently raises a series of interesting mathemat- works by Gilles Francfort and his collaborators. The ab ini- ical questions. The simulation of metastable trajectories and tio derivation of continuum models has been completed rare events are the central mathematical problematics of the for some models, for instance in elasticity, with various ap- field. proaches and in particular Γ-limit techniques—works by Atomistic models are also used in practice in compu- Andrea Braides and his collaborators; see [6] for a survey. tational multi-scale mechanics. Some approaches—an ex- Large-scale limits of dislocation dynamics, sometimes us- ample of which is the so-called quasi-continuum method— ing homogenization theory, have also been investigated. indeed use different models, either continuous or discrete, The range of macroscopic models that can be justified by

790 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 6 depending on the parameter 푁 that, up to mathematically irrelevant technicalities, fixes the total nuclear charge of the nuclei located at the points 푋푘 and the total electronic charge:

퐸푁 = inf {ℰ(휌, {푋푘}); 휌 ≥ 0 ∫ 휌 = 푁} . ℝ푛 The answer to this question is already challenging for periodic geometries in the absence of defects and may require subtracting the energy of a reference infinite system to make the limit finite, exactly as we did above for our simplistic atomistic system. A typical reference system popular in solid state physics is a jellium, that is, a homogeneously charged background. Proving that models for finite-size microscopic systems have a large size (푁 → +∞) limit Figure 3. Atomistic-to-Continuum simulation of a material. that coincides with models for the condensed phase is an Microcrack opening in a periodic lattice: at the vicinity of the important mathematical endeavor, with several outstand- fracture, where defects may appear and propagate, the ing contributions by Charles Fefferman, Elliott Lieb, Barry description is atomistic. Elsewhere, it is based on the Simon, etc. The question, known as the bulk or thermo- continuum approximation. The two levels of description are coupled seamlessly. dynamic limit problem, has been examined for models at zero, or at positive temperature, for models that are either in different regions of the material. The former category classical or quantum in nature, etc. We refer to [16] for a of models are—a priori or adaptively—selected in the re- survey. gions of the material where the strains and stresses are ex- Understanding how defects affect this process, and ac- pected to vary smoothly at the atomistic scale, while the counting for them, is an additional layer of difficulty, latter category of models are restricted to pathological re- which has only been attempted for some models, and for gions, typically containing atomistic defects of the crys- which various techniques of renormalization are useful. talline structure created by the large variations of stresses State-of-the-art works in this direction include works by and strains over atomistic distances; see Figure 3 for an il- Eric Cancès [7], Jianfeng Lu, Mathieu Lewin, and their re- lustration. The issues discussed above lay the groundwork spective collaborators. for the numerical analysis of such simulation approaches. The rationale. Although extremely, not to say overly, sim- The works of Mitchell Luskin, Christoph Ortner, and their plified, the above elementary atomistic model allows usto collaborators give more details on the mathematical un- illustrate some key difficulties related to the modeling of derstanding, the numerical analysis, and the latest devel- defects. In the absence of defects, we have a gentle situa- opments of the approach; see [18]. tion, where a geometric assumption, here periodicity, fol- An alternate direction—and eventually a complemen- lowed by a natural technique, here division of the energy tary direction when all levels of modeling are combined by the number of particles, allow us to model the infinite together—can be to consider the microscopic scale only, system as a finite one. Put differently, a problem apri- but refine the atomistic model using a model thatac- ori set on the—noncompact—whole space “miraculously” counts for the electronic structure of the material. Put dif- reduces to a problem on a unit cell. Formerly, possible ferently, classical mechanics is replaced by quantum me- difficulties were expected to prove the existence of mini- chanics. Models of the crystalline phase, used in prac- mizers, define the set of admissible functions—position, tice in computational solid state physics, can typically displacement, etc.—depending on their decay at infinity. be obtained by a bulk limit process as above. Density In contrast, every manipulation is now expected to be sim- functional theory type models—Thomas-Fermi, orbital- ple, and a setting using only classical functional spaces and free, Kohn-Sham-type models; wave-functions models— compactness techniques is sufficient. This in turn allows Hartree, Hartree-Fock type models; or even more sophis- us to anticipate, on the applied front, the use of standard ticated models, can all be derived in this manner. Math- discretization and numerical simulation techniques. ematically, the problem can be stated as identifying the When a defect is inserted, this idealistic landscape van- limit of the energy per unit volume, namely the analogous ishes, and we observe two phenomena. Firstly, depend- quantity to (2) suitably rescaled. The energy 퐸푁 is now ing on the nature of the defect—whether it is localized, a minimization problem set on functions, e.g. the den- that is, in some vague sense integrable, or it is global— sity 휌, describing the electronic state and parametrically there may be, or not, a nontrivial effect on the definition

JUNE/JULY 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 791 of the average—here, the energy per unit particle, say. In level of details—meaning in various functional spaces— any event, deciding whether there is or is not such an effect and characterizing it, exactly as the purpose of the limit requires us to a priori consider infinite systems that cannot performed in the previous section is to obtain the bulk be- be reduced to finite size systems as before. Secondly, in the havior of the system starting from its atomic description. case where there is no global effect, there might be a local Homogenization theory formalizes this question. In the pro- effect that can only be seen by zooming in locally, or by totypical and historical case where 푎휀 = 푎per(푥/휀) with 푎per ∗ subtracting the background, both techniques amounting a periodic function at scale one, the limit 푢 of 푢휀 is the so- to eliminating the problem at infinity—far away from the lution of the homogenized equation defect. −div(퐴∗∇푢∗) = 푓, (6) Note that the endeavor is subtle. On the one hand, let- ∗ ting the number 푁 of particles go to infinity is a first step where the homogenized coefficient 퐴 can indeed be com- ∗ in changing the scale at which materials are examined. In puted. The approximation of 푢휀 by 푢 holds strongly in 2 1 doing so, only phenomena that repeat themselves are sup- the 퐿 (Ω) sense and may be refined in the 퐻 (Ω) norm ∗ posed to survive. Small deviations from the average are upon using a periodic corrector, which adds to 푢 the pro- meant to disappear. The point of modeling at a given scale totypical oscillation of the original problem. This correc- is to keep from the finer scales only average information tor, or more precisely this 푛-tuple of correctors, is the solu- and neglect everything else that does not percolate to the tion to the periodic problem scale under consideration. The reason is that most macro- −div(푎per(푒푖 + ∇푤푝푒푟,푖)) = 0 (7) scopic phenomena are results of ensemble effects at a finer for 1 ≤ 푖 ≤ 푑 and 푒 the corresponding canonical basis vec- scale, not individual effects. On the other hand, not all 푖 tor of ℝ푛. It is enlightening to realize that equation (7), macroscopic phenomena are ensemble or macroscopic ef- set on the periodic cell of the periodic microstructure of fects. It is well known that the color of most gemstones the material encoded in 푎 , is obtained from the orig- is due to electronic bound states carried by impurities, a per inal equation (5) set on Ω by zooming in at the scale 휀 microscopic effect, which percolates throughout the scales and looking at variations around linear responses. The un- and never gets “averaged”. Likewise, we have seen that plas- derstanding of how 푢 converges to 푢∗ in topologies other ticity originates from defects. In sharp contrast to ensem- 휀 than 퐻1, like in all the Sobolev spaces 푊 푘,푝 or also in ble averaging, accounting for the presence of these small Hölder spaces 퐶푘,훼, is obtained by a careful analysis of scales or these defects precisely requires that we not omit the Green functions associated to the differential opera- the details. ∗ tors −div(푎per∇.) and −div(퐴 ∇.) at play in (5) and (6). Periodic Homogenization This was carried out in a series of seminal works [2] by Our next step is to move from a discrete description of a Marco Avellaneda and Fang-Hua Lin in the 1980s and re- material to a continuous description while keeping our at- cent works by Carlos Kenig and collaborators. The ques- tention focused on defects. Let us now change the setting tions are related to harmonic analysis and the theory of and consider partial differential equations. For simplicity Calderòn-Zygmund operators. All in all, the problem is again, we begin our discussion with the diffusion equation now well understood theoretically. The reader familiar with computational science readily

−div(푎휀∇푢휀) = 푓, (5) understands that the approach provides a natural numerical approximation of 푢휀. which, set on a domain Ω ⊂ ℝ푛 and supplied with suit- Homogenization of periodic problems goes far beyond able boundary conditions on 휕Ω, models a large variety of the simple diffusive linear setting described above; see physical phenomena. For example, if the coefficient 푎휀 de- the two reference textbooks by major contributors to the notes the heat conductivity of the medium, and 푓 models field: A. Bensoussan, J.-L. Lions, and G. Papanicolaou [4] the heat source, then the solution 푢휀 is thought of as the and V. V. Jikov, S. M. Kozlov, and O. A. Ole˘ınik[15]. A temperature field within the domain Ω. Equivalently, with large portfolio of linear partial differential equations and vector-valued solutions 푢휀 and loadings 푓, the same equa- systems of partial differential equations has been success- tion models the linear elasticity of a material that has elas- fully addressed in homogenization problems, along with ticity tensor 푎휀. In our notation, fields carry a subscript 휀 to the associated eigenproblems, time-dependent variants, indicate that they presumably vary at a tiny lengthscale 휀 etc. Many nonlinear problems have also been addressed, much smaller than the typical scale of the domain Ω. The should the nonlinearity arise from the terms of the equa- question, similar to that of considering 푁 → ∞ in the pre- tion themselves or from the geometry of the domain on vious section, concerns the behavior of the solution 푢휀 in which the equation is set: linear equation on a perforated the asymptotic limit 휀 → 0. Taking the limit aims at cap- domain, on a domain with an unknown or moving bound- turing the average behavior of 푢휀, possibly within a certain ary, .... State-of-the-art results include elaborate equations

792 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 6 two ingredients together allow for envisioning a practical numerical approach for the computation of the homogenized limit that is certified by a numerical analysis. The question arises to find sufficiently general settings that still allow for the quality of results of the periodic setting. This brings us back to our question of modeling defects, if defects are seen as violations of periodicity. The re- cent decade has witnessed several mathematical endeavors in this direction. Definite progress along this line concerns random homogenization. This setting has been originally addressed, in the linear case, in seminal works by George Pa- panicolaou, Srinivasa Varadhan, and Sergei Kozlov, and, in the nonlinear case, by Gianni Dal Maso and Luciano Modica, and by Luis Caffarelli and Panagiotis Souganidis. Remarkable progress has recently been made in a series of significant works by Felix Otto, Antoine Gloria, Scott Arm- Figure 4. Defects in a periodic structure. In the unperturbed strong, Charles Smart, Jean-Christophe Mourrat, and their periodic environment, the inclusions are periodic. The elimination of some of these inclusions are the defects coworkers; see e.g. [13] for an example of such a contribu- considered. The elimination may be deterministic or random. tion. The “partial differential equation” version of our ran- One may also consider small probabilities of elimination and dom defects 푋푘(휔) = 푘+푌푘(휔), and of much more general construct the corresponding mathematical setting. random distributions of defects, is now well understood:

−div(푎(푥/휀, 휔)∇푢휀(푥, 휔)) = 푓, (8) such as fully nonlinear equations, Hamilton-Jacobi type equations, etc. Several important questions remain in the with 푎(., 휔) ergodic stationary. Most qualitative properties periodic setting, in particular for hyperbolic equations, but of periodic homogenization are proven to survive in ran- the challenges we wish to now return to concern settings dom homogenization, with some differences though (the other than periodic. nature and rates of convergences may be substantially different). Of course, an important point is that these prop- Homogenization Problems with Defects erties are now established on average, that is, mathemati- We return to the diffusion equation (5) but do not assume cally, in expectation, in law, or at best almost surely. Put differently, nothing is known in full generality on a particular any longer that 푎휀 is a rescaled periodic function 푎per(푥/휀). Under quite general and mild assumptions on the (pos- realization of the random defects. On the other hand, studies regarding periodic settings perturbed by specific types sibly matrix-valued) diffusion coefficient 푎휀, presumably varying at the tiny scale 휀, the equation admits a homoge- of defects, again in the spirit of our simple atomistic model nized limit, which is indeed of the form (6). Celebrated above, have been conducted by the author in collaboration results along these lines are due to Ennio De Giorgi, Ser- with Xavier Blanc and Pierre-Louis Lions [5]. We now give gio Spagnolo, and Luc Tartar and their respective collabo- one prototypical example of such a setting, where we illus- rators; see [1, Chapter 1], [20] for reviews. The correspond- trate what we believe is the novelty of the mathematical ing theories carry the names of Γ-limit, 퐺-convergence, 퐻- questions involved. convergence. The strength of such results is their generality. Return to (5) and assume that 푎휀 = 푎(./휀) where the They are essentially obtained by a compactness argument. coefficient 푎 models a periodic material perturbed by a lo- In essence, for our particular setting (5), the sequence of calized defect. This setting, mathematically, may be en- 푝 푛 −1 coded in 푎 = 푎per + ̃푎 for ̃푎∈ 퐿 (ℝ ) for some 푝 < +∞. inverse operators [−div(푎휀∇.)] is weakly compact in the suitable topology, it converges, up to an extraction, and The presence of this defect does not affect the macroscopic behavior, that is, the homogenized equation (6). The ho- its limit can be proven to be an operator of the same type, ∗ −1 mogenized coefficient 퐴 remains the same, because it actu- namely [−div(퐴∗∇.)] . On the other hand, and precisely ally depends only on averages of 푎 over large volumes for because of the generality, not much is known on the limit which the addition of a function such as ̃푎 does not mat- when 푎 is generic. This contrasts with periodic homoge- 휀 ter. On the other hand, when it comes to making this limit nization which is both explicit—the limit coefficient 퐴∗ is more precise, one intuitively realizes, zooming in locally known by a formula, amenable to computation, as a func- in the material, that the corrector equation that describes tion of the, also known, corrector—and precise—the rate ∗ the microscopic response of the material reads as of convergence of 푢휀 to 푢 is known for a large variety of norms. Besides their theoretical interest per se, the above −div(푎(푒푖 + ∇푤푖)) = 0, (9)

JUNE/JULY 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 793 thus is different from (7). In sharp contrast with (7), where for both equations the notion of solution is that of the equation (9) does not reduce to an equation set on a viscosity solutions. For a periodic potential 푉, the problem bounded domain with periodic boundary conditions. In is now standard, but introducing an additional local de- essence, it is posed on the entire ambient space ℝ푛, a re- fect makes it atypical. The historical contribution on the flection of the fact that, at the microscopic scale, thede- periodic setting is a landmark article by P.-L. Lions, G. Pa- fect has broken the periodicity of the environment: the panicolaou, and S. R. S. Varadhan, while the random set- local response is affected by the defect and depends on ting was first considered by Fraydoun Rezakhanlou and the state of the whole material. A considerable mathe- James Tarver, and by Panagiotis Souganidis. Furthermore, matical difficulty follows. The classical toolbox for the on equations such as (10), we see another feature arising. study of the well-posedness of—here linear—equations on Equation (5), as a diffusive equation, is in some loose bounded domains, namely the Lax-Milgram lemma in the sense “forgiving”: a local microscopic defect is forgotten elliptic case, the Fredholm alternative, etc., all techniques at the macroscopic scale, that is, in the homogenization that one way or another rely upon the boundedness of the limit (6). Diffusion smooths things over, and unless one domain or the compactness of the setting, are now ineffec- looks at things locally, as in the corrector problem (9), the tive. Should 푎 be random, then equation (9) would ad- defect is harmless. In equations of the type (10), where mit an equivalent formulation on the abstract probability other types of phenomena, such as propagation of waves, space, and this would temporarily make up for compact- or fronts, can be dominant, this is no longer true: waves ness, even though other significant complications indeed see defects as obstacles in their propagation, and the so- arise. Here, the difficulty must be embraced. A related lution is modified. A local microscopic defect may affect difficulty is to define the set of admissible functions for the macroscopic, homogenized limit. This may already be solutions, or the variational space in an energetic formu- seen upon considering simple 1-dimensional equations: ′ lation of the problem. The decay of these functions away take 푢휀 + |(푢휀) | = 푏(푥/휀) for 푏 smooth, compactly sup- from the defect is key in the mathematical study. In that ported, nonpositive, and attaining its minimum 푏(0) = −1 respect, we recover here a classical difficulty in the more at zero; interpret the right-hand side as a defect superim- general context of the modeling of defects. For instance, posed to the null function; and check that 푢휀 converges works by John Ball and his collaborators underlined the to − exp(−|푥|), a function macroscopically different from the importance of the selection of the functional variational null function, the unique (bounded) solution, in the vis- space in all problems of the calculus of variation and in cosity sense, to 푢 + |푢′| = 0. particular when modeling defects in liquid crystals [3]. In Various other cases of defects may be considered for ho- 푝 푛 the specific case 푎 = 푎per + ̃푎 with ̃푎∈ 퐿 (ℝ ), one seeks mogenization problems that are otherwise “simple.” They the solution to (9) under the form 푤푖 = 푤푝푒푟,푖 + ˜푤푖, that is, may formally decay at infinity, like the “localized” func- with reference to the periodic solution 푤푝푒푟,푖 to (7)—a trick tions ̃푎 manipulated above, or not. In the former case, the that, since our first section, the reader is not a stranger to. problem at infinity (that is, the problem obtained upon Equation (9) rewrites as translating the equation far away from the defect) is identical to the underlying periodic problem. In the latter case, −div(푎 ∇ ˜푤 ) = div(푓)˜ 푖 the situation may sensitively depend upon what the prob- where 푓˜ ∈ 퐿푝(ℝ푛). This suggests that the suitable func- lem “at infinity”—in the same sense as above—looks like. tional space for ∇ ˜푤 is 퐿푝(ℝ푛). The question then reduces There may even exist several such problems. This is the to whether [∇] [div(푎 ∇ .)]−1 [div] operates continuously case for our simple atomistic system (3) above. Whether 푝 푛 in 퐿 (ℝ ). In the above specific setting, this holds true for one sits close to an 푋푝2 for 푝 integer, or far away from such all 1 < 푝 < +∞ (see [5]). points, one does not see the same environment asymptoti- The procedure above is not restricted to the linear dif- cally. Another prototypical example is related to the mod- fusion problem (5). One may consider semilinear equa- eling of grain boundaries in materials science: two differ- tions, quasi-linear equations, systems, etc. The study gets ent, periodic structures are connected across an interface. all the more delicate as the complexity of the equation in- The defect is, say, the plane separating the two structures, creases. In order to show the extent of the complications, and at large distances from this interface, different peri- we temporarily consider the homogenization of an equa- odic structures are present, depending upon which side of tion not strictly related to materials, a viscous Hamilton– the interface is considered. The corresponding mathemat- Jacobi equation such as ical problem is theoretically interesting and practically relevant. In all cases, the challenge is to identify the homog- −휀 Δ푢 + 푢 + 퐻 (∇푢 ) − 푉(푥/휀) = 0, (10) 휀 휀 휀 enized, macroscopic limit, while, in the meantime, retain- which leads to a corrector problem of the form ing some of the microscopic features that make the problem relevant. −Δ푤 + 퐻 (∇푤) − 푉(푥) = 퐻, (11)

794 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 6 Related Problems be useful for a large spectrum of problems in the engineer- It is interesting to notice that the questions we discuss on ing sciences and prepares the design, the analysis, and the the corrector equations (7), (9), (11), although arising in improvements of the numerical approaches. Multiscale the specific context of homogenization, are genuine ques- materials science is indeed a heavy user of various numer- tions on partial differential equations and have their own ical techniques: multiscale finite elements methods, het- interest in the modeling of materials, for a large category erogeneous multiscale method, and many others; see [11]. of equations set at one given scale. Localization of classical There is a definite interest in certifying these approaches or quantum waves in periodic materials owing to the pres- in a variety of situations, especially when materials have ence of defects [19] requires the study of eigensystems of defects. The theoretical analysis that has been overviewed the type above may constitute a first step toward this goal. 1 −div( ∇푢) = 휆푢 References 휀(푥) [1] G. Allaire, Shape optimization by the homogenization method, or Applied Mathematical Sciences, 146, New York, Springer, −Δ푢 + 푉(푥)푢 = 휆푢, 2002. MR1859696 respectively, for dielectric constants 휀(푥) or potentials 푉(푥) [2] M. Avellaneda and F.-H. Lin, Compactness methods in the that may be modeled as the diffusion constants 푎(푥) above. theory of homogenization, Parts I & II, Comm. Pure Appl. As is well known, the spectrum of a periodic operator has Math. 40 (1987), no. 6, 803–847, 42 (1989), no. 2, 139– 172. MR910954 a band structure and generalized eigenfunctions that are [3] J. Ball, Liquid crystals and their defects, Mathematical ther- periodic up to phases. Defects in the periodic background modynamics of complex fluids, Lecture Notes in Math., may create eigenvalues associated with localized, nonperi- 2200, 1–46, Springer, 2017. MR3729353 odic, eigenfunctions. The understanding of such phenom- [4] A. Bensoussan, J. L. Lions, and G. Papanicolaou, Asymp- ena is key for the modeling of material defects at the micro- totic analysis for periodic structures, Studies in Mathemat- scopic scale. The famous problem of Anderson localization, ics and its Applications, 5, North-Holland Publishing Co., related to the presence of random defects, is yet another ex- Amsterdam-New York, 1978. MR503330 ample of the theoretical and practical importance of such [5] X. Blanc, C. Le Bris, and P.-L. Lions, On correctors for linear issues; see [19]. elliptic homogenization in the presence of local defects, Comm. Partial Differential Equations 43 (2018), no. 6, 965–997. An example of a practically important application con- MR3909031 sists of the modeling of waveguides, as in the studies con- [6] A. Braides and M. S. Gelli, From discrete systems to conducted by Patrick Joly, Sonia Fliss, and their collaborators; tinuous variational problems: an introduction, Topics on con- see e.g. [12]. Such problems, and in particular the scatter- centration phenomena and problems with multiple scales, ing properties of the device, are sensitive to the periodic Lect. Notes Unione Mat. Ital. 2, Springer, Berlin, 2006, 3– structure of the waveguide, and to the possible presence of 77. MR2267880 defects therein. The typical modeling equation reads as [7] E. Cancès and C. Le Bris, Mathematical modeling of point defects in materials science, M3AS 23 (2013), 1795–1859. 2 2 −Δ푢 − 푛 (푥) 휔 푢 = 푓, MR3078676 where the refraction index 푛 encodes the nature of the ma- [8] P. Cardaliaguet, C. Le Bris, and P. Souganidis, Perturba- tion problems in homogenization of Hamilton-Jacobi equations, terial and may be a perturbation by some defects of an oth- J. Math. Pures Appl. 117 (2018), 221–262. MR3841511 erwise periodic matrix. Several of the options considered [9] A. V. Cherkaev and R. V. Kohn, Topics in the mathematical above for modeling various categories of defects can be use- modelling of composite materials, reprint of the 1997 edition, ful in this particular framework. Modern Birkhäuser Classics, 2018. MR3822094 In closing, we emphasize that the many options for [10] G. Dal Maso, A. DeSimone, and F. Tomarelli, Variational modeling defects, depending on their nature, geometry, problems in material science, Progress in Nonlinear Differ- random character, etc., which we have approached above, ential Equations and Their Applications, 68, Birkhäuser, have a manyfold interest. They provide a selection of inter- Basel, 2006. MR2222648 esting problems in the theory of partial differential equa- [11] B. Engquist and P. E. Souganidis, Asymptotic and numerical homogenization, Acta Numerica (2008), 147–190. tions which are all relevant practically and challenge our MR2436011 mathematical understanding of the equations considered. [12] S. Fliss and P. Joly, Wave propagation in locally perturbed pe- They allow us to explore how sensitively the properties of riodic media (case with absorption): numerical aspects, J. Com- a given equation depend on the nature of its coefficients. put. Phys. 231 (2012), no. 4, 1244–1271. MR2876453 Put formally, we indeed study the linear tangent operator [13] A. Gloria and F. Otto, Quantitative results on the correc- of the map associating the solution to a given coefficient tor equation in stochastic homogenization, J. Eur. Math. Soc. in the equation. There also is, of course, a practical and (JEMS) 19 (2017), no. 11, 3489–3548. MR3713047 numerical interest. Our study provides a toolbox that may

JUNE/JULY 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 795 [14] R. D. James, Materials from mathematics, Bull. Amer. Math. Soc. N.S. 56 (2019), 1–28. MR3886142 [15] V. V. Jikov, S. M. Kozlov, and O. A. Ole˘ınik, Homogeniza- tion of differential operators and integral functionals, Springer- Verlag, Berlin, 1994. MR1329546 [16] C. Le Bris and P.-L. Lions, From atoms to crystals: a mathematical journey, Bull. Amer. Math. Soc. N.S. 42 (2005), 291– 363. MR2149087 [17] T. Lelièvre, M. Rousset, and G. Stoltz, Free energy compu- tations. A mathematical perspective, Imperial College Press, London, 2010. MR2681239 [18] M. Luskin and Ch. Ortner, Atomistic-to-continuum- coupling, Acta Numerica (2013), 397–508. MR3038699 [19] P. Stollmann, Caught by disorder. Bound states in random media, Progress in Mathematical Physics, Vol. 20, 2001. MR1935594 [20] L. Tartar, The general theory of homogenization. A person- alized introduction, Lecture Notes of the Unione Matemat- ica Italiana, 7, Springer-Verlag, Berlin; UMI, Bologna, 2009. MR2582099

Claude Le Bris Credits Opening image is courtesy of Getty. Figure 1 is courtesy of Claude Le Bris. Figure 2 is courtesy of Kai Nordlund. Original publica- tion: Nordlund et al., Nature Communications 9 (2018), 1084. Creative Commons license: creativecommons .org/licenses/by/4.0/. Figure 3 is courtesy of Alexander Shapeev. Figure 4 is courtesy of SIAM MMS 9 (2011), no. 2, 513– 544. ©2011 Society for Industrial and Applied Mathemat- ics. Reprinted with permission. All rights reserved. Photo of Claude Le Bris is courtesy of Pierre & Gilles.

796 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 6