Computational design of patterned interfaces using reduced order models

Patterning is a familiar approach for imparting novel functionalities to free surfaces. We extend the patterning paradigm to interfaces between crystalline solids. Many interfaces have non-uniform internal structures comprised of misfit dislocations, which in turn govern interface properties. We develop and validate a computational strategy for designing interfaces with controlled misfit dislocation patterns by tailoring interface crystallography and composition. Our approach relies on a novel method for predicting the internal structure of interfaces: rather than obtaining it from resource-intensive atomistic simulations, we compute it using an efficient reduced order model based on anisotropic elasticity theory. Moreover, our strategy incorporates interface synthesis as a constraint on the design process. As an illustration, we apply our approach to the design of interfaces with rapid, 1-D point defect diffusion. Patterned interfaces may be integrated into the microstructure of composite materials, markedly improving performance.


Atomistic simulations
Our atomic models are cylinders of radius R 0 =27 nm and height 2h=20 nm with free surfaces in all directions, as shown in Fig. S1. The models consist of an FCC and BCC layer, each of height h. The crystallographic planes at the interface are {111} FCC and {110} BCC . A relative twist of θ, measured with respect to the Nishiyama-Wasserman orientation relation, is imposed on the BCC side of the interface, as shown in Fig. S1.a).
To relax these models, neighboring FCC and BCC layers are first translated rigidly with respect to each other in directions parallel to the interface. The lowest energy configuration from such translations is subsequently relaxed by conjugate gradient potential energy minimization (PEM) with multiple restarts using the atomistic modeling code LAMMPS. During PEM, only atoms in an inner cylinder of radius R s =R 0 -2 nm, illustrated in Fig. S1.a), are allowed to move. The positions of atoms in the outer annular region (R s ≤ r ≤ R 0 ) are kept fixed to prevent sliding or rotation of the two layers relative to each other. PEM is terminated when the two-norm force on every atom within the inner cylinder is less than 0.1 pN. To avoid edge effects, interface energies and structures are characterized within a region of radius R a = R 0 -5 nm. Increasing R 0 , R 0 -R s , R 0 -R a , or h does not change the interface structures or energies reported in this work. ii The total interface energy γ is computed as Here, ! is the sum of atom energies in the cylinder of radius R a for a relaxed interface while ! is the sum of atom energies in the cylinder of radius R a when the FCC and BCC layers are separated. The free surface energies !!! !"" and !!" !"" are computed in a separate calculation. To enable comparison between the ROMM and atomistic calculations, all plots of interface energy as a function of the twist angle θ have been shifted so that the minimum energy occurs at 0 J/m 2 . The amounts by which the plots were shifted are given in Table SI. Fig. 4 in the text compares energies from the ROMM and atomistic simulations for Cu/Nb and Ag/V. Fig. S2 below shows similar plots for Cu/Fe and Cu/Mo. Values for ad hoc parameters P, Q, and R (Eqn. 4, 5, and 6 in the text) are also plotted. Filled circles indicate atomic models whose disregistry was analyzed. The ringed numbers next to them state the case that best matches the atomic disregistry. Ad hoc parameters P, Q, and R (Eqn. 4, 5, and 6) for b) Cu/Fe and d) Cu/Mo.

Disregistery analysis
Disregistries in the interfaces studied are superpositions of contributions from two sets of misfit dislocations. To compare with qFBE predictions, we must separate the atomistic disregistry into contributions from each individual set. A single set of parallel misfit dislocations with perfectly compact cores produces either a staircase-or sawtooth-shaped disregistry, depending on the configuration from which displacements are measured. If displacements are measured from the reference state in Fig. 2 (see text) to the natural state after all local interface relaxations have occurred, a staircase disregistry results. A sawtooth disregistry arises if displacements are measured from the unrelaxed natural state created by the mapping in Fig. 2 to the natural state after interface relaxations have occurred. In both approaches, discontinuities in the disregistry correspond to dislocation cores. Either approach may be used to analyze disregistries in atomistic simulations. We adopt the latter and therefore expect to see sawtooth-shaped disregistry plots associated with each set of misfit dislocations. However, since real dislocation cores are not perfectly compact, we expect rapid variations in disregistry-rather than discontinuities-around dislocations cores. Fig. S3.a) shows an illustrative disregistry plot, computed for a Cu/Nb interface at θ=1°. The disregistry is a position dependent vector field computed by subtracting the relative position of neighboring FCC and BCC atoms in the unrelaxed interface from the relative position between the same two atoms in the relaxed interface. Misfit dislocation Burgers lie in the interface plane for all cases studied here. Thus, we restrict attention to disregistry components in the interface plane only. We compare this disregistry with each of the three cases predicted by the qFBE. For example, Fig. S3.a) shows the pattern of misfit dislocations predicted by cases 1 and 3 (see Fig. 3 in text). To isolate the contribution of each dislocation set to the disregistry, we use an oblique coordinate system with axes and parallel to the dislocation directions, as in Fig. S3.a). We plot disregistry as a function of its position projected onto one of these axes. This way, all disregistries a given distance along that axis from the core of one set of misfit dislocations are plotted at the same abscissa. For example, any disregistry projected along plotted to the same abscissa corresponds to the same distance from a set 1 dislocation core.
Therefore, by plotting disregistry as a function of position projected onto for one set of misfit dislocations in the oblique coordinate system in Fig. S3.a), variations of disregistry due to the other set of misfit dislocations are clearly displayed. In the example above, variations in disregistry due to set 1 may be distinguished by projecting disregistry positions onto and vice versa. In addition to projecting disregistry positions, we also distinguish between screw and edge components of the disregistry itself. When the disregistry position is projected along , we decompose disregistry into components parallel and perpendicular to , i.e. the screw and edge components for set 1 misfit dislocations. Fig. S3.b) shows the screw and edge components of disregistry for set 1 as a function of position projected along for the disregistry in Fig. S3.a). The dislocation line directions correspond to case 3 from the qFBE. A definite sawtooth-shaped pattern is seen, as expected. Discrepancies between the predicted and computed desregistries in the vicinity of the predicted "sawtooth" discontinuities are due to the non-zero width of misfit dislocation cores in the atomistic model. A similar plot may be made for screw and edge disregistries of set 2 as a function of position projected along . The example in Fig. S3.b) compares the simulated atomic disregistry with the case 3 disregistry for a Cu/Nb interface at θ=1°. These two disregistries are in good agreement. Indeed, our ROMM predicts that case 3 has lowest γ elastic for this interface. For contrast, Fig. S3.c) compares atomic disregistry with case 1 for the same interface. The match between the predicted and simulated disregistries is visibly worse for this case.
To assess the discrepancy between predicted and simulated disregistries quantitatively, we compute the root mean square discrepancy (RMSD) between predicted and simulated disregistries: Here, y AS is the disregistery from atomistic simulations for either the screw or edge component along one dislocation line while ŷ qFBE is the corresponding desregistry predicted by the qFBE for the case of interest. k is the number of data points. We compute separate RMSD values for both the screw and edge components for both sets of dislocations for the case of interest and add them together. The case with lowest total RMSD is taken as the best match to the simulated disregistry.
For the Cu/Nb interface at θ=1° described above, RMSD is lowest for case 3. This case therefore best describes the misfit dislocation pattern, in agreement with the prediction of our ROMM.