Strains and defect-induced deformations have tremendous effects on the macroscopic properties of single and poly-crystalline materials.1 These effects have fostered a huge variety of studies for more than a century, starting with the first theories describing the elastic field generated by dislocations in solids.2,3

Deformation of crystal lattices, although involving changes in the positions of atoms, are crucial for understanding the behavior of systems defined on larger length scales.4 Continuum mechanics, and associated continuous elastic fields, are very useful for describing elastic effects on the mesoscopic and/or macroscopic scales. In this approach, a continuous representation of the displacement of atoms in a lattice with respect to a reference crystal is employed.5 It is useful for either relatively simple distortions, as the one given by pure elastic deformations and rotation, or for deformations induced by the presence of dislocations.5,6,7,8 Indeed, it can be exploited to provide in-depth studies of material properties allowing for direct comparisons with experiments and/or a priori predictions, as, e.g, for plasticity onset in complex heterostructures9,10 or elasticity effects on material transport mechanisms and morphological evolution.11,12

For some applications, however, continuum mechanics is not enough as neglecting the description of atoms leads to a crucial loss of information. For instance, this applies to contributions of the dislocation core to the elastic field13,14 and, in turn, to dislocation nucleation, motion, and reaction. In these cases, in order to describe material properties by elasticity theory, the elastic field must be described within mesoscale15,16 or atomistic approaches. Typically, severe restrictions are present for these methods in the description of long timescale and large-length scale.

An attempt to overcome the timescale limits of atomistic approaches, by focusing on diffusive timescales, lead to the development of the so-called phase-field crystal (PFC) model. It focuses on the dimensionless atomic density field difference, n, filtering out vibrations on lattice sites.17,18,19 It provides good descriptions of elasticity20 and dislocation dynamics21 even if it usually requires fine spatial discretizations. This latter limitation is overcome by the complex amplitude expansion (APFC)22,23,24,25 of the PFC model for which both long-time scales and large-length scales can be examined. It consists of a coarse-grained representation of the density n that is expressed by the sum of Fourier modes representing specific lattice symmetries. The slowly oscillating complex amplitudes of these modes, ηj, are then the variables used to characterize the crystalline lattice. Real amplitudes, which may be regarded as a special case of the APFC model, have been also considered,26,27 delivering long-range order parameters as in the classical phase-field approaches based on atomistic descriptions. They can be used to account for bridging-scale descriptions of elasticity effects by means of additional contributions as, e.g., in the presence of precipitates, alloys, or point defects.28,29,30,31,32 However, they do not directly encode rotational invariance and elasticity associated to the deformations of the crystal lattice.

Although some intrinsic limitations for large deformations and tilts exist,25 APFC has proved useful in the advanced modeling of materials as illustrated in studies of elasticity effects,20,25 compositional domains,33 binary alloys,34 dislocation dynamics,35,36 morphology and motion of dislocation networks at grain boundaries (GB),37 and control of material properties.38,39,40 However, the basic concept of APFC, namely the coarse-graining of an explicit lattice representation by focusing on the complex coefficients of Fourier modes, can be readily applied to any atomistic description as obtained, e.g, from theoretical modeling, atomistic simulations, or experimental imaging. Still, a direct connection to continuum elasticity (CE) is missing.

In this work, we first show how to exploit the representation delivered by the complex amplitudes expansion of the PFC model to derive expressions for elastic field components independent of lattice symmetry and system dimensionality. In practice, we describe how to reconstruct strain and rotation fields from the atomic density provided that the complex amplitudes functions of the corresponding Fourier modes are known. Then, we consider numerical simulations for some generic systems involving strained/tilted crystals and apply the new framework in order to depict and analyze the resulting deformations. To this purpose, we numerically solve the equations of the APFC model directly delivering the amplitudes function. Standard simulations as well as simulations which extend the current state of the art for APFC and PFC approaches are presented and discussed. The combination of using a coarse-grained approach as APFC with the detailed analysis of deformations proposed here results in a bridging-scale framework enabling the study of elasticity effects from the microscale to macroscale while describing microstructural evolution, at variance with many other methods focused only on some of these aspects at once. Moreover, using the APFC model allows us for its further assessment as a reliable coarse-grained method accounting for microscopic effects. The elastic field in presence of defects, derived from complex amplitudes as computed by APFC simulations, reproduce predictions of continuum mechanics as expected from PFC-based modeling.17,18,25 Moreover, the results also indicate that APFC includes some deviation from continuum mechanics that may be ascribed to atomistic structure at the dislocation cores, as it has also been observed and discussed in other continuum or atomistic approaches.13,14,41

Results and discussions

Strain and rotation fields from complex amplitudes

The complex amplitude functions, ηj, entering the APFC model (see Methods section) are connected to the deformation of a crystal u with respect to a reference lattice by the following equation:

$$\eta _j = \phi _j{\mathrm{exp}}\left( {i{\mathbf{k}}_j\cdot {\mathbf{u}}} \right),$$

with {kj} a specific set of reciprocal space vectors describing the lattice of an undeformed crystal. Equation (1) defines N independent equations. ϕj’s are the real values corresponding to the amplitudes in a relaxed, unrotated crystal. They can be computed by the minimization of the energy functional in Eq. (9) assuming constant, real amplitudes for each different length of kj. The quantity \({A^2} \equiv 2{\mathop {\sum}\nolimits_{j = 1}^N} {|\eta _j|^2}\) delivers an order parameter which is constant in the solid/ordered phase, decreases at defects and interfaces, and vanishes when approaching the liquid/disordered phase. Equation (1) can be rewritten as

$$\varphi _j = {\mathbf{k}}_j\cdot {\mathbf{u}},$$

with φj = arg(ηj) = arctan[Im(ηj)/Re(ηj)]. In order to determine the components of the deformation field ui from amplitudes (with i = x, y in 2D and i = x, y, z in 3D), Eq. (2) must be inverted, resulting in a system of d equations with d the dimensionality of the system. In 2D, by selecting d = 2 amplitudes labeled by generic indexes l and m, \(u_i \equiv u_i^{{\mathrm{2D}}}\) results in

$$u_i^{{\mathrm{2D}}} = \frac{{\epsilon _{ij}}}{{|{\mathbf{k}}_l \times {\mathbf{k}}_m|}}\left[ {k_m^j\varphi _l - k_l^j\varphi _m} \right],$$

with the two components of the displacement field obtained by index permutations on the group (i, j) = (x, y) and \(\epsilon _{ij}\) the 2D Levi–Civita symbol. In 3D, by selecting d = 3 amplitudes labeled by generic indexes l, m, and n, \(u_i \equiv u_i^{{\mathrm{3D}}}\) and we obtain

$$\begin{array}{l}u_{i}^{\mathrm{3D}} = \frac{1}{{\mathbf{k}}_{n}\cdot \left({\mathbf{k}}_{m} \times {\mathbf{k}}_{l} \right)} \left[ \varphi _{l}\left(k_{m}^{\kappa} k_{n}^{j} - k_{m}^{j}k_{n}^{\kappa} \right) + \varphi _{m}\left( k_{n}^{\kappa}\, k_{l}^{j} - k_{n}^{j}k_{l}^{\kappa} \right)\right.\\ \qquad\quad+\,\left.\varphi _{n}\left( k_{l}^{\kappa}\, k_{m}^{j} - k_{l}^{j}\,k_{m}^{\kappa} \right) \right],\end{array}$$

with the three components of the displacement field obtained by index permutations on the group (i, j, κ) = (x, y, z). Amplitudes must be chosen in order to have a non-vanishing denominator of the prefactor entering Eq. (4). Without loss of generality we fix l = 1, m = 2, and n = 3, referring to kj with the same length for each symmetry considered here. For small deformations, the strain tensor ε can be written ε = (1/2)[u + (u)T]. The strain components can then be explicitly computed from ηj by means of spatial derivatives of Eq. (3) (d = 2) or Eq. (4) (d = 3). This leads to ∂φj/∂xi terms. Notice that, φj are inherently discontinuous due to their functional form. However, ηj are, by definition, continuous complex functions in both their real and imaginary part and

$$\frac{{\partial \varphi _j}}{{\partial x_i}} = \frac{1}{{|\eta _j|^2}}\left[ {\frac{{\partial {\mathrm{Im}}(\eta _j)}}{{\partial x_i}}{\mathrm{Re}}(\eta _j) - \frac{{\partial {\mathrm{Re}}(\eta _j)}}{{\partial x_i}}{\mathrm{Im}}(\eta _j)} \right].$$

Since \({|\eta _j|^2} = {\phi _0^2} {\,>\,} 0\) almost everywhere in the crystal phase, the terms ∂φj/∂xi and then ε can be readily computed. j|2 only vanishes exactly at the dislocation core position for some indexes, j, consistent with CE theory. By exploiting the components of the displacement field, rotations of the crystal structure with respect to the reference orientation can also be evaluated as ω =  × u. ωij corresponds to the rotational angle in the xixj plane. For d = 2, ωij ≡ ω, describing the rotation in the two-dimensional domain. For the sake of readability, the expressions of ε and ω for d = 2, 3, as functions of ∂φj/∂xi, are reported in the Supplementary Information S2. The resulting deformation fields deliver descriptions similar to advanced continuum theories, as e.g. in refs. 41,42,43, directly connected here to atomic arrangements via ηj functions.

Deformations induced by dislocations

Let us consider pairs of dislocations in a 2D triangular lattice that form at the interface between layers of different atomic spacing in order to accommodate a misfit strain. As performed in ref. 39, the dislocations can be described by APFC by setting an initial condition for ηj in order to reproduce opposite deformations (namely initial strains) ±ε. This can be done using Eq. (1) with an in-plane displacement field u(r) defined by ux = ±(atri/Lx)x and uy = 0, where \({\mathbf{r}} = x\widehat {\mathbf{x}} + y\widehat {\mathbf{y}}\) (\(+ z\widehat {\mathbf{z}}\) in 3D). atri is the distance between maxima of the density as in Eq. (8) for triangular symmetry and Lx is the size of the computational domain, both evaluated along the x-axis. These opposite deformations are imposed as illustrated in Fig. 1a.

Fig. 1
figure 1

Dislocations and strain field in 2D for a crystal having triangular symmetry. a A2 (left) and n (right), showing a schematics of the position and orientation of defects, as well as atomic planes at and close to one defect. b Magnification of n and Burgers vector. c εxx as computed from ηj. Comparison between results from APFC (black circles) and from continuum elasticity (CE) (solid red line) along specific directions as in panel (c) are shown: (d) εxx along \(\ell _1\), (e) εxx along \(\ell _2\), (f) εyy along \(\ell _1\), (g) εxy along \(\ell _3\)

The relaxation of such an initial condition until the defect shape is stationary in a square system of linear dimension Li = 40atri is illustrated in Fig. 1a (details about simulations are reported in the Methods section and references therein). A2 is shown on the left and the reconstructed density n from Eq. (8) is shown on the right. Two pairs of dislocations form within the simulation domain where A2 is constant in the solid and decreases at defects. Although the APFC approach does not allow for the exact representation of atoms at the dislocation-core, the lattice distortion is well described as shown by the illustration of atomic planes by solid lines, highlighting the presence of one specific defect. The deformation of the lattice can be quantified at the atomic level by evaluating the Burgers vector, b, as shown in Fig. 1b. Notice that it corresponds to a lattice spacing in \(\widehat {\mathbf{x}}\) direction, i.e. \(|{\mathbf{b}}| = 4\pi /\sqrt 3\). εxx is shown in Fig. 1c while, for the sake of completeness, other components are shown in the Supplementary Information S3.

The elastic field as computed from the complex amplitudes matches very well with CE as shown in Fig. 1d–g. The latter can be computed provided that the dislocation character, namely edge, screw or mixed,8 as well as b and elastic constants are known. Within the APFC model the elastic constants are determined by the parameters entering the energy (see Methods section and Eq. (9)) as discussed in ref. 20. They do not appear explicitly in the equations reported above but affect amplitude values and their spatial derivatives. As reported in the Supplementary Information S4, the defects in Fig. 1c can be modeled as a 2D array of edge dislocations and the strain field according to CE εce can be computed as superposition of the elastic field of single dislocations. The elastic constant entering these equations is the Poisson ratio ν set here to 1/3 as in refs. 17,18 (see also Supplementary Information S4). Figure 1d–g show εce by solid red lines along \(\ell _1\), \(\ell _2\), and \(\ell _3\) as defined in Fig. 1c. The strain components computed from ηj’s are shown by black circles. An almost perfect agreement is found far away from dislocation cores as observed along \(\ell _2\) for εxx. The same holds true for the other cases except for regions close to the dislocation cores, namely at \(\ell_{i}{\sim} {\pm} {10a}_{\mathrm{tri}}\) where, however, a continuous description of lattice deformations is not well-posed.

Deeper insight can be obtained by focusing on the analysis of the elastic field as in refs. 13,14. Therein, the components of the strain field in the presence of a straight dislocation, \(\varepsilon _{ij}^{\mathrm{d}}\), are decomposed as \(\varepsilon _{ij}^{\mathrm{d}} = \varepsilon _{ij}^{{\mathrm{ff}}} + \varepsilon _{ij}^{{\mathrm{core}}}\), with \(\varepsilon _{ij}^{{\mathrm{ff}}}\) a far-field predictor of the elastic field and \(\varepsilon _{ij}^{{\mathrm{core}}}\) a correction due to the core effects. A decay \(\varepsilon _{ij}^{{\mathrm{ff}}}\sim r^{ - 1}\), with r the distance from the dislocation core, is expected in agreement with CE theory. The dislocation core effects are characterized by a decay \(\varepsilon _{ij}^{{\mathrm{core}}}\sim r^{ - 2}\) independently on the lattice symmetry, except for some high-symmetry nominal position of the core where \(\varepsilon _{ij}^{{\mathrm{core}}}\sim r^p\) with p < −2.13,14 \(\varepsilon _{ij}^{{\mathrm{ce}}}\) corresponds to an explicit expression of \(\varepsilon _{ij}^{{\mathrm{ff}}}\). The contribution of the dislocation core can then be analyzed by evaluating \(\varepsilon _{ij}^{{\mathrm{core}}} = \varepsilon _{ij}^{{\mathrm{apfc}}} - \varepsilon _{ij}^{{\mathrm{ce}}}\), with \(\varepsilon _{ij}^{{\mathrm{apfc}}}\) the elastic field components computed from ηj. The scaling of these quantities is illustrated in Fig. 2 where a system as in Fig. 1 is considered with Li = 320atri. \(\varepsilon _{xx}^{{\mathrm{apfc}}}\) (red squared) and \(\varepsilon _{xx}^{{\mathrm{ce}}}\) (green triangles) are shown along the line \(\ell _1\) (see Fig. 1c) with r = 0 the position of one of the defects. In the large r limit the strain shows a decay that is very close to ~r−1, but some small deviations are observed. By evaluating the difference, \(\varepsilon _{xx}^{{\mathrm{apfc}}} - \varepsilon _{xx}^{{\mathrm{ce}}}\), (blue circles) we observe a faster decay that scales as ~rq with q = −2.05 ± 0.15 recalling the more localized correction given by the dislocation core. We can then conclude that the elastic field description proposed here not only matches the description of CE far from dislocation cores but also includes a correction that may be ascribed to atomic-scale effects.

Fig. 2
figure 2

Scaling of εxx along the direction \(\ell _1\) as in Fig. 1c with r = 0 the position of a dislocation. The value computed by APFC (red squared) and by continuum elasticity (green triangles) are shown along with their difference (blue circles)

The evaluation of the elastic field can be readily provided also in 3D regardless of the lattice symmetry (see also Supplementary Information S2). We consider now a configuration with peculiar 3D features as shown in Fig. 3. An initial condition mimicking layers with opposite, biaxial strain along the \(\widehat {\mathbf{x}}\) and \(\widehat {\mathbf{y}}\) directions is considered. ηj’s are initialized as in Fig. 1 with uy = ux39 and, without loss of generality, we select a crystal having bcc lattice symmetry (by setting kj accordingly, see Supplementary Information S1). The size of the domain is set to Li = 60abcc, with abcc the lattice constant for bcc arrangements. Interfaces between layers are (001) planes. APFC simulations account for the formation of dislocation networks at the interface as shown in Fig. 3a. In this figure, the gray structure corresponds to the region where A2 < 0.8 max(A2), i.e. to the defects as A2 significantly decreases. The initial interfaces between layers are higlighted by blue (top) and green (bottom) planes. The atomic structure at the defects as constructed by Eq. (8) and the Burgers vector (here \(|{\mathbf{b}}| = 2\pi \sqrt 2\)) are illustrated in Fig. 3b in a small 2D region around a defect lying on the yz-plane Π highlighted in Fig. 3a. The strain field computed from amplitudes is illustrated by means of εyz in Fig. 3c. By accounting for the parallel dislocations forming along both the in-plane directions, the elastic field can be approximated by CE. A comparison of εyz as obtained from amplitudes (APFC), computed by the simulations illustrated in Fig. 3a, and from CE, by adapting the equations of Supplementary Information S2 (accounting then for two sets of dislocations oriented along \(\widehat {\mathbf{x}}\) and \(\widehat {\mathbf{y}}\) directions and having perpendicular Burgers vector as in Fig. 3b), is reported in Fig. 3d showing a general agreement similar to Fig. 1. Large-length scale decays as in Fig. 2, not explicitly addressed here, are expected also in this case.13,14

Fig. 3
figure 3

Dislocations and strain field in 3D for a crystal having bcc lattice symmetry. a Defect network from APFC simulations for a multilayer configuration. Π is the (orange) yz-plane on which next panels focus. b Reconstructed density at a defect shown in panel (a) and Burgers vector. c εyz on Π. d Comparison between εyz computed from APFC (black circles) and from continuum elasticity (CE) (solid red line) along the the line parallel to the \(\widehat {\mathbf{y}}\) direction connecting the defects on Π

Lattice rotations and polycrystalline systems

We focus here on the analysis of deformations and rotations in polycrystalline systems, which involve the evolution of small-angle GBs. We study first a simple system made of a straight GB forming between two crystals with a symmetric tilt. In particular, a rectangular domain, Lx × Ly with \(\widehat {\mathbf{x}} = [10]\) and \(\widehat {\mathbf{y}} = [01]\), is considered with a straight vertical GB at the center. The relative tilt angle between the two crystals, namely 2θ, is set by initializing the ηj functions as

$$\eta _j = \phi _j\exp \left( {i\delta {\mathbf{k}}_j(\theta )\cdot {\mathbf{r}}} \right),$$

with δkj(θ) = kj · R(θ) − kj and R(θ) the counterclockwise rotation matrix. A ±θ tilt is imposed for the left and the right part of the simulation domain, respectively (see also Fig. 4a). By using periodic boundary conditions a GB with infinite extension is considered. A second GB is also expected, that is shared between the left and right periodic boundary of the simulation domain. Lx can be chosen arbitrarily while Ly has to match the periodicity of amplitudes along \(\widehat {\mathbf{y}}\). Specific details about this simulation setup can be found in ref. 39.

Fig. 4
figure 4

Deformation and rotation field at a symmetric-tilt GB in 2D. A (small) portion of the entire simulation domain is reported, illustrating: a A2, b ω, and c εxy

Figure 4a illustrates the result of relaxing the aforementioned initial condition by means of the A2 field in a small region at the GB, formed by an array of dislocations, showing three defects as minima of A2. ω is reported in Fig. 4b. A constant value of this field is obtained when moving away from the GB, while a modulation at the defects and at the GB is observed, reproducing the effect of the dislocation on the local orientation of the crystal lattice. The values at which ω saturates in the crystals correspond to ±θ imposed in the initial condition. A measure of the tilt angle is then obtained by exploiting the deformation field. It should be noted, however, that it is not possible to directly obtain the tilt angle by inverting Eq. (6) due to the phase term. Figure 4c also illustrates the strain component εxy, calculated from amplitudes ηj as discussed previously. In this instance a significant superposition of the strain lobes is obtained, due to the proximity of the dislocations.

The importance of extracting the rotation of grains by means of a scalar field defined everywhere becomes more evident when looking at poly-crystalline systems, as illustrated in Fig. 5 (and Supplementary Videos 13). Therein a system with Li = 320atri is considered. The initial condition contains 10 crystal seeds, namely circular regions with |ηj| > 0 and random positions, random radii ranging from 10π to 20π, and uniformly distributed values of θ (−5°, 5°). According to the parameters of the free energy, the crystal phase has the lowest energy so that the crystals grow and merge forming a complex network of small-angle GBs made of dislocations. Afterward, these GBs evolve resulting in the shrinkage of some grains and the annihilation of dislocations. This is illustrated in detail in Fig. 5 by means of A2, ω, and εxy starting from the merged crystal, as well as by an analysis of the tilt-angle distribution over time. Faceted GBs are obtained (here in 2D corresponding to closed polygonal chains). Moreover, the motion of dislocations leading to the shrinkage of grains occurs along preferred directions related to the crystal lattice and the local tilt, revealing the accurate description achieved by APFC despite its coarse-grained nature. All these dynamic features can be better observed in the Supplementary Videos 1 and 2. In addition, the velocity of a specific defect (within the white and black circles) is highlighted in Fig. 5. ω is nearly constant within the grains and varies at the GBs, thus it can be used to identify single grains as they are characterized by different tilts. Moreover, ω accounts for the contribution of single dislocations. Indeed, the features of the extracted rotation field allows for the analysis of grains as reported in Fig. 5d. Here, the relative volume of regions having the same tilt angle within bins of Δθ ≈ 0.2° is shown. The top panel illustrates the initial condition where the crystal seeds cover just a portion of the entire system. The panel at the bottom shows the analysis of the three stages shown in Fig. 5a–c. By this analysis (see also the Supplementary Video 3) important information can be extracted, such as the volume fraction (V) occupied by grains in a specific orientation. Peaks in V correspond to the grains with different orientations as shown in Fig. 5d. Their broadening is related to the presence of defects, missing in the first stages as the crystals are separated with no dislocations. During the evolution, the relative volume of grains changes and some of them eventually disappear. Notice that before vanishing some peaks shift to larger tilts pointing out a rotation due to the proximity of the dislocations (see also Supplementary Video 3). This is in agreement with 2D atomistic calculations showing an increase of the interface energy and a rotation of the grains during similar processes.20,44,45

Fig. 5
figure 5

Deformation and rotation in a polycrystalline system (2D, triangular symmetry). Ten crystals with θ [−5°, 5°) evenly spaced, random initial position, and radius are considered. The rows illustrate the coarsening dynamics with three stages by means of (a) A2, (b) ω, (c) εxy. (d) Relative volume of regions having similar tilt angle with uniform binning Δθ ≈ 0.2°: the initial condition (“Ini”, top panel) and stages reported in (a)–(c), labeled as left, center, and right, respectively (bottom panel), are shown. See also Supplementary Videos 13

A three-dimensional system involving rotations and consisting of a single spherical crystal that is rotated with respect to the larger single crystal surrounding it is shown in Fig. 6. Li = 50afcc with afcc the lattice constant for fcc arrangements. A rotation of 5° about the [111] direction is set. The initial condition for ηj is set by exploiting Eq. (6) (see also ref. 37). The resulting dislocation network after a first relaxation phase is shown in Fig. 6a. As in Fig. 3a, the gray structure corresponds to the dislocations. A closed network of defects forms, having the peculiar structure of defects at twisted GBs when the surface normal of the spherical inclusion approaches the rotational axis. As expected by classical grain growth theory46 and 3D simulations37 this defect network evolves leading to the shrinking of the rotated inclusion. As noticed first in ref. 47, where PFC simulations of a similar configuration with bcc lattice symmetry are reported, an anisotropic shrinkage of the grain occurs, that is faster along the rotational axis as a result of the dynamics and reaction of dislocations. However, an almost linear scaling of the grain boundary extension is expected.37,47 The extraction of the rotation field on the yz-plane Π highlighted in Fig. 6a is illustrated in Fig. 6b at different stages during the evolution. ωyz is zero outside the inclusion and almost constant inside. Similarly to Fig. 5a, a short wavelength modulation is observed at defects. From the ωyz color map, a rotation of the grain is observed.20,44,45,47 The corresponding εyz field is shown in Fig. 6c.

Fig. 6
figure 6

Deformation and rotation in the presence of a spherical rotated inclusion with a rotation of θ = 5° about the [111] direction. Fcc lattice symmetry is considered. a Emerging dislocation network at the interface between the rotated and the unrotated crystal. Π is the (orange) yz-plane where the quantities shown in the following panels are evaluated. (b) ωyz and (c) εyz

A 3D ploycrystalline system is shown in Fig. 7, where the growth of 30 crystal seeds having fcc lattice symmetry with random rotation θ (−10°, 10°) about the [111] direction, random position and size is simulated. Li = 100afcc while the seeds are generated within a distance 40afcc from the center. The evolution is illustrated in Fig. 7a by means of three snapshots showing the region where A2 is significantly larger than zero. The rotation field in the Π plane is shown in Fig. 7b. Figure 7c illustrates the analysis of the local rotations in terms of the volume of regions with the same angles within bins of Δθ = 0.1° normalized with respect to the larger one. It delivers similar features as discussed in Fig. 5b. In particular, the initial configuration exhibits a smaller angular dispersion than later stages, while peaks broaden due to the formation of dislocations. The capability of the adopted framework can be further appreciated in Fig. 7d, where the complex dislocation network forming in a central (spherical) region of the growing polycrystal (last step of Fig. 7a) is shown. Features of the dislocation network observed in Fig. 6a can be here recognized, as hexagonal arrangements and elongated defects according to direction of the normal of the interface between grains. Similar dislocation networks with different spacing (e.g. see the presence of hexagons with different sizes at GB highlighted as blue shaded region) are obtained within the polycrystal due to the different relative rotations between grains.

Fig. 7
figure 7

Growth of randomly tilted crystal seeds (3D, fcc lattice symmetry). Thirty crystals with random tilt θ (−10°, 10°) about the [111] direction are considered with random initial position and radius. a Three stages during the growth of the polycrystalline system. b ωyz in the plane Π illustrated in panel a. c Normalized volume fraction of crystal with similar rotation of the first (“Ini”) and last (“End”) stages in panel a within bins of Δθ = 0.1°. d Defects (yellow network) within a spherical region at the center of the growing polycrystal (last stage in panel a)

The analysis of polycrystalline systems allows us to summarize the main findings of this work. A continuous description of lattice deformations exploiting the complex amplitude of Fourier modes representing the periodicity of crystal lattices has been derived. Deformation fields can then be readily computed from an atomistic representation of crystals, provided that amplitude functions can be extracted, without any ad hoc post-processing procedure and independently of system dimensionality and lattice symmetry. This framework has been shown to achieve its full potential combined with the APFC approach. Other than trivially providing a direct access to amplitudes and allowing for describing either strain or rotated crystals, the APFC model easily allows for large-length scale simulations approaching the ones typical of continuum theories, still retaining essential microscopic features. This has been further demonstrated by showing that APFC encodes dislocation-core contributions to the elastic field, which are missing in standard CE and usually requires more refined theories.41 The connection between amplitudes and displacements or strains enables the detailed study of the effect of any deformation which are encoded by continuous deformation fields, such as displacements or strain/stress due to single dislocations or due to external loads. We explicilty illustrated that extracting the local orientation as continuous field directly enables microstructural analysis. In addition, it will enable the development of optimized numerical method as, e.g., orientation-based meshing criterion for APFC.48

It is worth recalling that the limitation of the APFC model in describing large deformations25 poses some constraints to the numerical approach adopted here, but not on the derivation of the deformation fields and on the possible analysis of deformations extracted from other atomistic frameworks (e.g. the PFC model itself or experiments). Moreover, some attempts to overcome this APFC limit have been recently proposed.49 Notice that several PFC type models, which are based on a continuous probability density can be easily converted in the proper set of amplitudes, and are therefore naturally compatible with the descriptions illustrated in this work (see e.g. refs. 50,51). In addition, work on extensions of the PFC model can be done in order to derive the corresponding amplitude expansions as done, for instance, in ref. 52 with the so-called XPFC model.53


Amplitude expansion of PFC model

The PFC model accounts for the lattice structure by means of a continuous periodic field n describing the dimensionless atomic probability density.17,18,19 It is based on a free-energy functional, Fn, that reads

$$F_n = {\int_\Omega} {\left[ {\frac{{\Delta B_0}}{2}n^2 + \frac{{B_0^x}}{2}n(1 + \nabla ^2)^2n - \frac{t}{3}n^3 + \frac{v}{4}n^4} \right]} {\mathrm {d}}{\mathbf{r}},$$

where ΔB0, \(B_0^x\), v, and t are parameters as in ref. 54. In the crystalline state n can be generally approximated as sum of plane waves as

$$n({\mathbf{r}}) = n_0({\mathbf{r}}) + \mathop {\sum}\limits_{j = 1}^N {\left[ {\eta _j({\mathbf{r}}){\mathrm {e}}^{i{\mathbf{k}}_j\cdot {\mathbf{r}}} + \eta _j^ \ast ({\mathbf{r}}){\mathrm {e}}^{ - i{\mathbf{k}}_j\cdot {\mathbf{r}}}} \right]} ,$$

with n0(r) the average density, ηj(r) the amplitude of each plane wave, and kj the reciprocal space vector representing a specific crystal symmetry (see Supplementary Information S1). In the so-called amplitude expansion of the PFC model (APFC),22,23,24 these amplitudes are the variables used to describe a given crystalline system. Lattice symmetries are described by means of a fixed set of vectors kj. Complex amplitude functions ηj(x) allow for distortions and rotations of the crystal lattice with respect to a reference state accounted for by kj vectors. The free energy, expressed in terms of ηj’s, reads

$$F = {\int_{\Omega}} \left[ \frac{\Delta B_{0}}{2}A^{2} + \frac{3v}{4}A^{4} + \mathop {\sum}\limits_{j = 1}^{N} {\left( {B_{0}^{x}|{\cal{G}}_j\eta _j|^{2} - \frac{3v}{2}|\eta _j|^{4}} \right)} \vphantom{\mathop {\sum}\limits_{j = 1}^N}+ f^{\mathrm{s}}(\{ \eta _j\} ,\{ \eta _{j}^{\ast} \} ) \right]{d\mathbf{r}},$$

with \({\cal{G}}_j \equiv \nabla ^2 + 2i{\mathbf{k}}_j\cdot \nabla\) and \(A^2 \equiv 2\mathop {\sum}\limits_{j = 1}^N {|\eta _j|^2}\). The term \(f^{\mathrm{s}}(\{ \eta _j\} ,\{ \eta _j^ \ast \} )\) corresponds to a complex polynomial of ηj and \(\eta _j^ \ast\) and depends on the specific crystalline symmetry as reported in ref. 39 for triangular, body-centered cubic (bcc) and face-centered cubic (fcc) lattices. The evolution laws for ηj’s read

$$\frac{{\partial \eta _j}}{{\partial t}} = - |{\mathbf{k}}_j|^2\frac{{\delta F}}{{\delta \eta _j^ \ast }}.$$


The simulations reported in this work are performed with the aid of high performance computing facilities55 exploiting the finite element toolbox AMDiS56,57 with a semi-implicit integration scheme and mesh adaptivity as reported in ref. 39. Large, three-dimensional simulations as reported in Fig. 7 have been obtained thanks to an improvement of the numerical approach described elsewhere.48 Periodic boundary conditions are used for all the boundaries of the simulation domains. To describe crystalline phases, the parameters entering the free energy are set to favor the crystal phase as follows: Bx = 0.98, v = 1/3, t = 1/2, and ΔB = 0.02.34,37,39