Evaluation of Nanoscale Deformation Fields from Phase Field Crystal Simulations

Abstract

Different methods for evaluation of displacement and strain fields based on phase field crystal (PFC) simulations are shown. Methods originally devised for molecular dynamics (MD) simulations or analysis of high-resolution microscopy images are adapted to a PFC setting, providing access to displacement and strain fields for systems of discrete atoms, such as in MD, as well as to continuous deformation fields. The latter being achieved by geometrical phase analysis. As part of the study, the application of prescribed non-affine deformations in a 3D structural PFC (XPFC) setting is demonstrated as well as an efficient numerical scheme for evaluation of PFC phase diagrams, such as, for example, those required to stabilize solid/liquid coexistence. The present study provides an expanded toolbox for using PFC simulations as a versatile numerical method in the analysis of material behavior at the atomic scale.

1. Introduction

The Phase Field Crystal (PFC) method, introduced in [1], is a non-phenomenological continuum approach to simulating microstructure evolution in materials at the atomic scale. PFC models a time-averaged atomic number density field over long, diffusive, time scales while retaining atomic-level spatial resolution. This provides access to modeling regimes that fall beyond the reach of conventional techniques, such as Density Functional Theory (DFT) and Molecular Dynamics (MD), on the atomic scale, and phase fields or level sets on the mesoscale. The incorporation of atomic length scales means that dislocation mechanisms, as well as heterogeneities such as phase or grain boundaries in polycrystal aggregates, are automatically captured. PFC was introduced as an elaboration of standard phase field formulations, but it has later been shown that PFC can be derived directly from DFT, as discussed in [2,3]. PFC modeling has been employed for investigation of a wide range of phenomena, many of which involving solid–liquid transitions, including nucleation and growth, melting and glass formation [4,5]. However, solid-state phenomena have also been targeted, such as dislocation mechanics [6,7], grain boundary properties [8,9], grain boundary and particle interaction [10], precipitation [11] and atomic-scale fracture [12]. Various applications of PFC modeling are reviewed in, for example, [13,14].

In most PFC-based studies, lattice deformation and corresponding strains are not quantified, or done so by relatively crude measures. One example is the average local strain used in [15], which is a scalar quantity that, for example, does not distinguish shear deformation from a state of tension or compression. In the present study, different methods for evaluation of deformation and strain fields from PFC simulations are discussed. The investigated methods were originally introduced for use in either MD simulations or for analysis of experimental images obtained by High-Resolution Transmission Electron Microscopy (HRTEM). In this work, the methods are adapted to a PFC setting, taking advantage of PFC providing access to both a continuous density field as well as the discrete atom/particle positions through interpolation of the density maxima. Further, when PFC is employed for analysis of structures under mechanical load, the applied deformation is usually assumed to be homogeneous, for example in [16]. As part of the present study, it is demonstrated how displacement and strain field fields under non-homogeneous, or non-affine, deformation can be evaluated. In addition, control of the PFC model requires establishing a phase diagram to be able to stabilize different phases and to maintain phase coexistence. A numerical scheme for achieving this in a flexible and efficient manner is provided in the appendix. This study concerns analysis of simulation results provided by the wide range of PFC models based on [1], as well as those based on the structural PFC (XPFC) method introduced in [17]. An alternative approach, not covered here, is the amplitude expansion PFC (APFC) method, as discussed in [18], which offers other means for strain evaluation. While PFC and XPFC mainly differ in terms of how the direct pair correlation between particles or atoms is handled, APFC constitutes a more significant step in a different direction.

This paper is structured in such way that the PFC model is discussed first in Section 2. Various methods for evaluation of displacement and strain fields are discussed in Section 3. Different example applications of the methods are demonstrated in Section 4, along with a discussion of the results. Some concluding remarks close the paper in Section 5.

2. Phase Field Crystal Model

The starting point for the PFC model formulation is the definition of a time-averaged and normalized particle density field

$$n\left(\mathit{r}\right)=\frac{\rho \left(\mathit{r}\right)-{\rho }_l}{{\rho }_l}$$

(1)

where $\mathit{r}=\mathit{r}(x,y,z)$ is a position vector based on the spatial coordinates $(x,y,z)$ in a domain $\mathsf{\Omega }$, $\rho$ is the particle number density and ${\rho }_l$ a reference density in the liquid state. The equilibrium density of the solid phase can be approximated by the Fourier series

$$n\left(\mathit{r}\right)=n_0+\sum _j{A}_j{e}^{\frac{2\pi }{a}i{\mathit{k}}_j·\mathit{r}}$$

(2)

with $n_0$ denoting a reference density, $A_j$ is the amplitude of mode j, assumed to be identical for all lattice planes of family j, a is the lattice parameter and $k_j$ are reciprocal lattice vectors.

The Helmholtz free energy for the system under consideration—scaled by ${\rho }_l{k}_{B}T$, with $k_B$ and T denoting the Boltzmann constant and the absolute temperature, respectively—can be written as a functional of the particle density n according to

$$F\left[n\right]=F_{id}\left[n\right]+F_{exc}\left[n\right]+F_{ext}\left[n\right]$$

(3)

where $F_{id}$ is related to the ideal gas behavior of non-interacting particles, $F_{exc}$ is the excess free energy due to particle interaction and $F_{ext}$ is a free energy component related to external forces. Based on the definition of the number density field in Equation (1), the Helmholtz free energy functional in Equation (3) describes the difference in energy with respect to a reference state at which $\rho ={\rho }_0$. The first two components on the right-hand side of Equation (3) are quite standard in PFC modeling and are taken, in non-dimensional form, as

$$F_{id}\left[n\right]={\int }_{\mathsf{\Omega }}d\mathit{r}\left[\frac{n{\left(\mathit{r}\right)}^2}{2}-\frac{n{\left(\mathit{r}\right)}^3}{6}+\frac{n{\left(\mathit{r}\right)}^4}{12}\right]$$

(4)

and

$$F_{exc}\left[n\right]=-\frac{1}{2}{\int }_{\mathsf{\Omega }}d\mathit{r}n\left(\mathit{r}\right)\left[C_2\ast n\left(\mathit{r}\right)\right]$$

(5)

where $C_2\left(\right|\mathit{r}-{\mathit{r}}^{\prime }|;\sigma )$ is a two-point correlation function, describing the interaction between two particles located at positions $\mathit{r}$ and ${\mathit{r}}^{\prime }$, respectively. The correlation function also depends on an effective temperature, denoted by $\sigma$. The convolution operation $C_2\ast n$, appearing in Equation (5), is defined by

$$C_2\ast n={\int }_{\mathsf{\Omega }}d{\mathit{r}}^{\prime }\left[C_2\left(\right|\mathit{r}-{\mathit{r}}^{\prime }|;\sigma )\right]n\left({\mathit{r}}^{\prime }\right).$$

(6)

Following [17], the correlation function can be approximated as a sum of Gaussian peaks in Fourier space by

$${\tilde{C}}_2\left(k\right)=\underset{j}{max}\left[exp\left(-\frac{{\sigma }^2{k}_j^2}{2{\rho }_j{\beta }_j}\right)exp\left(-\frac{{(k-k_j)}^2}{2{\alpha }_j^2}\right)\right]$$

(7)

for all reciprocal lattice vectors $\mathit{k}$, of length $k=\left|\mathit{k}\right|$. A particular lattice is constructed from a subset of j density waves, with wave numbers $k_j$. In the present case, wave numbers $k_1=\sqrt{3}{k}_0$ and $k_2=2k_0$, corresponding to the $\left[111\right]$ and $\left[200\right]$ planes, are used for FCC and wave numbers $k_1=\sqrt{2}{k}_0$ and $k_2=2k_0$, corresponding to the $\left[110\right]$ and $\left[200\right]$ planes, are used for BCC. With $k_0=2\pi$, a unit lattice parameter is defined by $a=2\pi /k_0$.

Appearing in Equation (7), ${\rho }_j$ is the atomic density and ${\beta }_j$ denotes the number of planes. For FCC, these parameters are $({\rho }_1,{\rho }_2,{\beta }_1,{\beta }_2)=(4/\sqrt{3},2,8,6)$ and for BCC they can be found as $({\rho }_1,{\rho }_2,{\beta }_1,{\beta }_2)=(2/\sqrt{2},1,12,6)$. The widths of the Gaussian peaks in Equation (7) are set by ${\alpha }_j$ and can be used to tune the elastic properties of the crystal structure to comply with a particular material behavior. For simplicity, ${\alpha }_j=1$ is used here.

The last term on the right-hand side of Equation (3), related to the external free energy, is included to permit prescribed deformation of the PFC system. Following [15], the external free energy is incorporated in the free energy functional as a penalty function, taken as

$$F_{ext}\left[n\right]={\int }_{\mathsf{\Omega }}d\mathit{r}\xi \left(\mathit{r}\right){\left[n\left(\mathit{r}\right)-n_{ext}(\mathit{r}+\Delta \mathit{r})\right]}^2.$$

(8)

The field $n_{ext}(\mathit{r}+\Delta \mathit{r})$ is based on an approximation of the density field, as shown in Equations (A3) and (A4). The external field $n_{ext}$ corresponds to the XPFC density field $n\left(\mathit{r}\right)$, but is subject to a prescribed change $\Delta \mathit{r}=(\Delta x,\Delta y,\Delta z)$ in the spatial coordinates whereby $\mathit{r}:=\mathit{r}+\Delta \mathit{r}$. Appearing in Equation (8), $\xi \left(\mathit{r}\right)$ is a modulation function which is zero outside the region in which the prescribed external field $n_{ext}$ is applied. Following [19], $\xi \left(\mathit{r}\right)$ is here taken as a normalized Gaussian of width ${\sigma }_{\xi }$, centered at ${\mathit{r}}^{\prime }$, according to

$$\xi \left(\mathit{r}\right)=\frac{1}{\sqrt{2\pi {\sigma }_{\xi }^2}}exp\left(-\frac{{(\mathit{r}-{\mathit{r}}^{\prime })}^2}{2{\sigma }_{\xi }^2}\right).$$

(9)

In order to accommodate both the diffusion time-scale governing the underlying XPFC dynamics, as well as the much faster relaxation of the elastic strains imposed by Equation (8), the modified PFC (MPFC) model proposed in [15,20] is adopted. The evolution of the normalized particle density field $n\left(\mathit{r}\right)$ is thus described by the damped wave equation

$$\gamma \frac{{\partial }^{2}n}{\partial t^2}+\beta \frac{\partial n}{\partial t}={\alpha }^2{\nabla }^2\frac{\delta F\left[n\right]}{\delta n}$$

(10)

where standard diffusion-based PFC dynamics are retrieved for the parameter choices $(\alpha ,\beta ,\gamma )=(1,1,0)$. PFC dynamics on the form described by Equation (10) is also adopted in [19,21]. The evolution of the XPFC system, as described by Equation (10), is implemented using the semi-implicit spectral scheme discussed in [10].

3. Evaluation of Nanoscale PFC Deformation and Strain Fields

Deformation and strain is not available directly from PFC (or XPFC) data, but must be evaluated based on the density field n. Tracing the evolution of n by Equation (10) gives access to a continuous density field and, as discussed in [8], interpolation of the density maxima can be employed to also provide the locations of the maxima, corresponding to the discrete “atom” positions. The latter makes it possible to adapt procedures introduced for evaluation of deformation and strain in MD into a PFC setting. As an initial example related to strain evaluation in PFC, it can be noted that the discrete atom positions are used in [15] to find a local average strain by considering the $\beta =1\dots N$ nearest neighbors of each atom $\alpha$. Knowledge of the initial distance $a^{\alpha \beta }$ between atoms $\alpha$ and $\beta$, in terms of the lattice parameter, the deformed distance $d{a}^{\alpha \beta }$ between the same atoms can be used to find the average local strain for each atom $\alpha$ according to

$${\overline{\epsilon }}^{\alpha }=\frac{1}{N}\sum _{\beta =1}^N\frac{d{a}^{\alpha \beta }}{a^{\alpha \beta }}$$

(11)

This is a simple approach to obtain a strain measure that conveys a local average strain magnitude and that gives a hint towards whether atom $\alpha$ is under an average state of compression (${\overline{\epsilon }}^{\alpha }<0$) or tension (${\overline{\epsilon }}^{\alpha }>0$). However, as Equation (11) is a scalar measure no information is provided on the strain directions nor is any distinction made between states of shear and tension/compression, i.e., the individual strain components are not provided.

In order to establish a more rigorous method for PFC strain analysis, the discrete atom positions are used in conjunction with a method devised for MD in Section 3.1 to evaluate a local deformation gradient and associated strain tensors. As an alternative approach, the continuous PFC density field is used in Section 3.2 to demonstrate how procedures originally devised for analysis of high-resolution microscopy images can be used to evaluate continuous, rather than discrete, displacement and strain fields.

3.1. Discrete Deformation Gradient and Related Strain Measures

The development of a discrete or local—in the sense of relating to an individual atom—deformation gradient can be motivated based on continuum mechanics. In continuum mechanics, a continuous body ${\mathsf{\Omega }}_0\subset {\mathbb{R}}^3$ can be defined in the material reference configuration, comprising particles positioned at $\mathit{X}\in {\mathsf{\Omega }}_0$. At a certain time, the nonlinear deformation map $\varphi \left(\mathit{X}\right)$ maps the reference configuration ${\mathsf{\Omega }}_0$ onto the current, spatial, configuration $\mathsf{\Omega }=\varphi \left({\mathsf{\Omega }}_0\right)\subset {\mathbb{R}}^3$. Accordingly, at a certain time a particle at $\mathit{X}\in {\mathsf{\Omega }}_0$ can be identified in the current configuration at $\mathit{x}=\varphi \left(\mathit{X}\right)$. The continuum mechanics deformation gradient $\mathit{F}$ is defined as linear mapping between tangent spaces, transforming tangent vectors in ${\mathsf{\Omega }}_0$ to tangent vectors in $\mathsf{\Omega }$, by

$$\mathit{F}=\frac{d\mathit{x}}{d\mathit{X}}.$$

(12)

The local volume change can be found from the determinant J of the deformation gradient, providing $J=det\left(\mathit{F}\right)\equiv V/V_0$ where $V_0$ and V is the local volume in the reference and in the current configuration, respectively. Different strain measures can be defined based on the deformation gradient. The Lagrangian (Green) strain tensor is defined entirely in terms of reference coordinates and is found as

$$\mathit{E}=\frac{1}{2}\left({\mathit{F}}^T\mathit{F}-\mathit{I}\right)$$

(13)

where $\mathit{I}$ is the second-order identity tensor and ${(·)}^T$ denotes a transpose. Alternatively, the Eulerian (Almansi) strain tensor can be defined in the current configuration by

$$\mathit{e}=\frac{1}{2}\left[\mathit{I}-{\left(\mathit{F}{\mathit{F}}^T\right)}^{-1}\right].$$

(14)

In the small strain limit, the Lagrangian and Eulerian strain tensors provided by Equations (13) and (14) will be approximately equal to each other and can be represented by the small strain tensor

$$\mathbf{\epsilon }=\frac{1}{2}\left[{\left(\nabla \mathit{u}\right)}^T+\nabla \mathit{u}\right]\equiv \frac{1}{2}\left({\mathit{F}}^T+\mathit{F}\right)-\mathit{I}$$

(15)

with the displacement field being defined by $\mathit{u}=\mathit{x}-\mathit{X}$ and where the last identity is based on the relation $\mathit{F}=\mathit{I}+\nabla \mathit{u}$. It can also be recalled that the volumetric dilatation is provided by the first strain tensor invariant

$$I_1=\mathrm{tr}\left(\mathbf{\epsilon }\right)$$

(16)

where $\mathrm{tr}\left(·\right)$ denotes the trace of a tensorial quantity. Further, based on the first equality in Equation (15), it is noted that the displacement gradient can be stated as $\nabla \mathit{u}=\mathbf{\epsilon }+\mathbf{\omega }$ with the skew-symmetric rotation tensor being defined by

$$\mathbf{\omega }=\frac{1}{2}\left[\nabla \mathit{u}-{\left(\nabla \mathit{u}\right)}^T\right].$$

(17)

Turning to the present atom-scale representation of a material body, no continuous displacement field is available from which the deformation gradient can be evaluated. Instead, a discrete version of Equation (12) is sought. Here, the method proposed in [22,23] for evaluation of a discrete atom-scale deformation gradient in molecular dynamics simulations is considered. For this purpose, it is noted that an atom with index $\alpha$ will be positioned at ${\mathit{X}}^{\alpha }$ in the reference configuration and at ${\mathit{x}}^{\alpha }$ in the current configuration. The deformation in the local neighborhood of atom $\alpha$ is determined by the positions of its neighbors $\beta$, located at ${\mathit{X}}^{\beta }$ and ${\mathit{x}}^{\beta }$, respectively. The relative positions between the atom $\alpha$ and its neighbors $\beta$ are found as

$$\begin{array}{c}{\displaystyle \Delta {\mathit{X}}^{\alpha \beta }={\mathit{X}}^{\beta }-{\mathit{X}}^{\alpha }\in {\mathsf{\Omega }}_0}\hfill \\ {\displaystyle \Delta {\mathit{x}}^{\alpha \beta }={\mathit{x}}^{\beta }-{\mathit{x}}^{\alpha }\in \mathsf{\Omega }}\hfill \end{array}.$$

(18)

A linear mapping between these relative positions is defined by the deformation gradient ${\mathit{F}}^{\alpha }$ of atom $\alpha$, appearing as

$${\mathit{F}}^{\alpha }=\frac{\Delta {\mathit{x}}^{\alpha \beta }}{\Delta {\mathit{X}}^{\alpha \beta }}.$$

(19)

In contrast to the continuum deformation gradient $\mathit{F}$, defined in Equation (12), the discrete analogue ${\mathit{F}}^{\alpha }$ in Equation (19) is not a single unique mapping between the reference configuration ${\mathsf{\Omega }}_0$ and the current configuration $\mathsf{\Omega }$. Rather, evaluation of Equation (19) for atom $\alpha$ and each of its $\beta$ neighbors results in a system of linear equations. Following [22,23], a discrete local deformation gradient ${\widehat{\mathit{F}}}^{\alpha }$ is determined to minimize the mapping error between atom $\alpha$ and an individual neighbor $\beta$ in a least-squares sense. The squared mapping error is defined as

$${\eta }^{\alpha }=\sum _{\beta =1}^N{\left(\Delta {\mathit{x}}^{\alpha \beta }-{\widehat{\mathit{F}}}^{\alpha }\Delta {\mathit{X}}^{\alpha \beta }\right)}^T\left(\Delta {\mathit{x}}^{\alpha \beta }-{\widehat{\mathit{F}}}^{\alpha }\Delta {\mathit{X}}^{\alpha \beta }\right)$$

(20)

where the summation is performed over all N nearest neighbors $\beta$ of atom $\alpha$. Minimization of ${\eta }^{\alpha }$ with respect to the components of $\widehat{\mathit{F}}$, i.e., $\partial {\eta }^{\alpha }/\partial \widehat{\mathit{F}}=0$, yields a linear system of equations that can be written as

$${\widehat{\mathit{F}}}^{\alpha }{\mathit{D}}^{\alpha }={\mathit{A}}^{\alpha }\Rightarrow {\widehat{\mathit{F}}}^{\alpha }={\mathit{A}}^{\alpha }{\left({\mathit{D}}^{\alpha }\right)}^{-1}$$

(21)

where the matrices

$${\mathit{D}}^{\alpha }=\sum _{\beta }\Delta {\mathit{X}}^{\alpha \beta }{\left(\Delta {\mathit{X}}^{\alpha \beta }\right)}^T$$

(22)

and

$${\mathit{A}}^{\alpha }=\sum _{\beta }\Delta {\mathit{x}}^{\alpha \beta }{\left(\Delta {\mathit{X}}^{\alpha \beta }\right)}^T$$

(23)

were introduced. The discrete deformation gradient $\widehat{\mathit{F}}$, available from Equation (21), represents a local averaged deformation gradient, which can be used with Equation (13) and Equation (14) to evaluate the corresponding discrete strain tensors $\widehat{\mathit{E}}$, $\widehat{\mathit{e}}$ and $\widehat{\mathbf{\epsilon }}$, respectively. However, application of Equation (21) requires access to at least three neighbors $\beta$ in order to determine the nine components of $\widehat{\mathit{F}}$. In addition, these neighbors need to span ${\mathbb{R}}^3$ to render ${\mathit{D}}^{\alpha }$ invertible. This implies that the $\beta$ neighbors can be neither co-linear nor co-planar with atom $\alpha$.

The evaluation of $\widehat{\mathit{F}}$ by Equation (21), which was proposed in in [22,23] for strain evaluation in MD simulations, is here adopted for PFC simulations. To accomplish this, discrete atom positions are identified by performing a quadratic interpolation of the density maxima in the $n\left(\mathit{r}\right)$-field to provide the positions of the N nearest neighbors of each atom (8 in BCC and 12 in FCC) in the current configuration, described by $\Delta {\mathit{X}}^{\alpha \beta }$. In addition, the ideal lattice locations of the N nearest neighbors are used to provide $\Delta {\mathit{x}}^{\alpha \beta }$ in the reference configuration. The local, atomic-scale, deformation gradient $\widehat{\mathit{F}}$ is then evaluated for each atom (density maxima) using Equations (21)–(23).

3.2. Evaluation of Continuous Displacement and Strain Fields by Geometrical Phase Analysis

Geometrical phase analysis (GPA) was introduced in [24,25] as a method to evaluate displacement and strain fields from high resolution transmission electron microscopy (HRTEM) images of crystal lattices. Central to the method is evaluation of the phase components of the intensity peaks in the HRTEM image in Fourier space. The same approach is adopted here by recognizing the correspondence between the HRTEM intensity peaks and the maxima in the PFC density field. This permits evaluation of the PFC displacement and strain fields without prior interpolation of density peaks and use of discrete atom positions. Instead, the continuous PFC density field can be used directly.

GPA is based on formulating the intensity of a 2D crystal lattice image as the Fourier series

$$I\left(\mathit{r}\right)=\sum _g{A}_{g}exp\left(i{P}_g+2\pi i\mathit{g}·\mathit{r}\right)$$

(24)

where the periodicity of the intensities is defined by the vector $\mathit{g}$ in reciprocal lattice space. Further, $A_g$ and $P_g$ denote the amplitude and phase of the periodicity, respectively. With access to the intensity field in Fourier space, single $\mathit{g}$ vectors can be filtered out by masking the corresponding Bragg peaks. In the present implementation, a simple Gaussian mask is employed, taken as

$$M\left(\mathit{k}\right)=exp\left[-\frac{{\left(\mathit{k}-\mathit{g}\right)}^2}{2{\sigma }_M^2}\right]$$

(25)

with the width of the Gaussian being defined by the parameter ${\sigma }_M$. In the next step, the masked intensity image is brought back to real space by an inverse Fourier transform to provide the complex image $H_g$, which can be used to evaluate

$$P_g\left(\mathit{r}\right)=\mathrm{phase}\left[H_g\left(\mathit{r}\right)\right]-2\pi \mathit{g}·\mathit{r}.$$

(26)

This is the real-space phase difference between the masked image and the vector $\mathit{g}$, used to define the mask in Equation (25) and thereby choosing the reference lattice relative to which displacements and strains will be evaluated. If needed, the vector $\mathit{g}$ can be refined by evaluating the deviation $\Delta \mathit{g}$ from the reference lattice $\mathit{g}$. This is achieved by fitting the gradient the gradient of $P_g$ in a region of the original image in which the strain is assumed to be uniform, using

$$\Delta \mathit{g}=\frac{1}{2\pi }\nabla P_g.$$

(27)

To evaluate the full 2D displacement and strain fields, the steps involving Equations (25)–(27) can be repeated for two non-parallel vectors ${\mathit{g}}_1$ and ${\mathit{g}}_2$, corresponding to the real space vectors ${\mathit{a}}_1$ and ${\mathit{a}}_2$. Denoting the components of these vectors by subscripts x and y, the following matrices can be defined

$$\mathit{G}=\left[\begin{array}{cc}{g}_{1x}& g_{2x}\\ g_{1y}& g_{2y}\end{array}\right]\mathrm{and}\mathit{A}=\left[\begin{array}{cc}{a}_{1x}& a_{2x}\\ a_{1y}& a_{2y}\end{array}\right]$$

(28)

that satisfy ${\mathit{G}}^T\mathit{A}=\mathit{I}$, as shown in [25]. From the phase differences $P_{g1}$ and $P_{g2}$ corresponding to ${\mathit{g}}_1$ and ${\mathit{g}}_2$, the deviation from the local reciprocal reference lattice can be found using Equations (27) and (28), providing

$$\Delta \mathit{G}=\frac{1}{2\pi }\left[\begin{array}{cc}{\displaystyle \frac{\partial P_{g1}}{\partial x}}& {\displaystyle \frac{\partial P_{g2}}{\partial x}}\\ {\displaystyle \frac{\partial P_{g1}}{\partial y}}& {\displaystyle \frac{\partial P_{g2}}{\partial y}}\end{array}\right].$$

(29)

This makes it possible to formulate a local 2D deformation gradient $\tilde{\mathit{F}}\left(\mathit{r}\right)$, akin to Equation (12), between the reference lattice $\mathit{A}$ and the deformed lattice $\tilde{\mathit{A}}$ as

$$\tilde{\mathit{A}}\left(\mathit{r}\right)=\tilde{\mathit{F}}\left(\mathit{r}\right)\mathit{A}.$$

(30)

Inverting both sides of Equation (30) and using the relation ${\mathit{G}}^T={\mathit{A}}^{-1}$ provides

$${\tilde{\mathit{G}}}^T\left(\mathit{r}\right)={\mathit{G}}^T{\tilde{\mathit{F}}}^{-1}\left(\mathit{r}\right)\Rightarrow \tilde{\mathit{F}}\left(\mathit{r}\right)={\left[\mathit{A}{\tilde{\mathit{G}}}^T\left(\mathit{r}\right)\right]}^{-1}.$$

(31)

Finally, substituting Equation (29) for $\tilde{\mathit{G}}$ yields

$$\tilde{\mathit{F}}\left(\mathit{r}\right)={\left[\mathit{I}+\mathit{A}\Delta {\mathit{G}}^T\left(\mathit{r}\right)\right]}^{-1}.$$

(32)

Equation (32) permits evaluation of the local deformation gradient $\tilde{\mathit{F}}$ based on the phase gradients. Subsequently, also the corresponding (2D) Green and Almansi strain tensors $\tilde{\mathit{E}}$ and $\tilde{\mathit{e}}$ can be evaluated, based on GPA, using Equations (13) and (14). It is emphasized that the strain evaluated by GPA constitutes the lattice strain relative to a reference lattice, defined by the reciprocal lattice vector $\mathit{g}$. In the limit of small deformations, Equation (32) can be approximated by

$$\tilde{\mathit{F}}\left(\mathit{r}\right)=\left[\mathit{I}-\mathit{A}\Delta {\mathit{G}}^T\left(\mathit{r}\right)\right]$$

(33)

which can be recast into

$$\tilde{\mathit{F}}\left(\mathit{r}\right)=\mathit{I}+\nabla \mathit{u}\left(\mathit{r}\right).$$

(34)

The latter formulation provides the gradient of the displacement field as

$$\nabla \mathit{u}\left(\mathit{r}\right)=-\mathit{A}\Delta {\mathit{G}}^T\left(\mathit{r}\right).$$

(35)

Based on Equation (35), the small strain tensor $\tilde{\mathbf{\epsilon }}$ and rotation $\tilde{\mathbf{\omega }}$, corresponding to Equation (15) and Equation (17), are now available, based on GPA.

The phase difference maps $P_{g1}$ and $P_{g2}$, relating the deformed lattice to a chosen reference lattice, provide the displacement field components in the direction of ${\mathit{g}}_1$ and ${\mathit{g}}_2$, respectively, by

$$\begin{array}{c}{\displaystyle P_{g1}\left(\mathit{r}\right)=-2\pi {\mathit{g}}_1·\mathit{u}\left(\mathit{r}\right)}\hfill \\ {\displaystyle P_{g2}\left(\mathit{r}\right)=-2\pi {\mathit{g}}_2·\mathit{u}\left(\mathit{r}\right)}\hfill \end{array}.$$

(36)

Under the requirement that ${\mathit{g}}_1$ and ${\mathit{g}}_2$ are not co-linear, the two equations in Equation (36) can be written on matrix form and inverted to obtain displacement field by

$$\mathit{u}\left(\mathit{r}\right)=-\frac{1}{2\pi }{\left({\mathit{G}}^T\right)}^{-1}\left[\begin{array}{c}{\displaystyle P_{g1}\left(\mathit{r}\right)}\hfill \\ {\displaystyle P_{g2}\left(\mathit{r}\right)}\hfill \end{array}\right]\equiv -\frac{1}{2\pi }\mathit{A}\left[\begin{array}{c}{\displaystyle P_{g1}\left(\mathit{r}\right)}\hfill \\ {\displaystyle P_{g2}\left(\mathit{r}\right)}\hfill \end{array}\right]$$

(37)

where the identity ${\mathit{G}}^T\mathit{A}=\mathit{I}$ was used in the last step. Using Equation (37), the displacement field is also available from the PFC-based GPA.

4. Examples

To illustrate how the methods for deformation analysis, discussed in Section 3, perform in a PFC setting, some simulation examples are provided here. Two 3D simulation models are considered, as shown in Figure 1. Model 1, seen in Figure 1a, is a single crystal specimen with two notches, placed symmetrically at the top and bottom edges. The specimen is subject to tensile loading along the x-axis by an external force field defined according to Equation (8). The force field is applied by the Gaussian modulation function in Equation (9) being non-zero in the hatched regions in Figure 1a. The coordinates ${\mathit{r}}^{\prime }$ in Equation (9) are thus set to define lines along the y-axis through the center of each of the two hatched regions in Figure 1a. Periodicity of the simulation domain, as required by the spectral solution method, is maintained by defining liquid regions along the left and right domain boundaries. The same liquid state is defined in the cut-out regions in the notches. This approach to prescribing deformations in a PFC setting follows the method introduced in [20]. From the same study, the parameter values $(\alpha ,\beta ,\gamma )=(15,0.9,1)$ are adopted, cf. Equation (10). Working with a PFC formulation on non-dimensional form, a unit lattice parameter $a=1$ is used and model dimensions are in units of a. Following [20], the grid spacing is adjusted to be as close as possible to $\pi /8$ in all coordinate directions and a non-dimensional time increment of $\Delta t=1\times {10}^{-3}$ is used. To model the notches and to incorporate the liquid regions along the left and right domain edges, the PFC system must be in a state of stable solid/liquid coexistence. This requires identification of the appropriate reference density $n_0$ and amplitudes $A_j$ in Equation (2), as well as the corresponding value of the effective temperature $\sigma$ in Equation (7). Phase diagrams can be used to achieve this and an efficient numerical scheme for establishing the PFC phase diagrams, while avoiding common approximations on the amplitudes $A_j$, is discussed in Appendix A. For the BCC structure considered in Model 1, the density field approximation in Equation (A3) is used at the effective temperature $\sigma =0.05$ to find the amplitudes $[A_1,A_2]=[0.11594,0.053652]$ as well as the reference densities in the liquid and solid states $[n_l,n_s]=[-0.17069,-0.11550]$. These parameter settings place the PFC system in a stable state of solid/liquid coexistence, cf. Figure A1a.

Application of the deformation by MPFC is attractive as it allows simulation of a non-affine, or heterogeneous, deformation field. This is a more relevant approach compared to alternative methods used in other PFC studies in which a homogeneous, or affine, deformation is used when considering boundary value problems. Homogeneous deformation in a PFC setting is for example used in [16] based on a constant volume assumption and involve a continuous modification of the computational grid.

Model 2, illustrated in Figure 1b, is the second simulation set-up under consideration. The simulation model comprises two crystals, defined by orientations ${\mathit{R}}_1$ and ${\mathit{R}}_2$, which are separated by two symmetrical tilt grain boundaries with tilt angle $\theta$. The grain boundaries are initiated as narrow liquid strips and over the course of the simulation the two crystals rapidly grow into the liquid strips to form the minimum energy grain boundary configuration. The steps involved in setting up this type of PFC simulation domain, for example in terms of ensuring periodicity, are detailed in [8]. For Model 2, no external force field is applied as deformation is caused by the presence of the grain boundaries. Consequently, the $F_{ext}$ component is omitted in the free energy functional in Equation (3) and the parameters $(\alpha ,\beta ,\gamma )=(1,1,0)$ are used in Equation (10). The time increment and the grid spacing are both set identical to Model 1.

4.1. Discrete Deformation Fields under Non-Affine Deformation

The notches are defined by the parameters $r_c=10$ and $l_c=10$, cf. Figure 1a, and the domain size is set to $\left[L_x,L_y,L_z\right]=\left[121,81,3\right]$. The external force field is applied over the hatched regions in Figure 1a, of width 10 and the liquid regions along the left and right domain boundaries also have a width of 10. A tensile deformation rate of $5\times {10}^{-2}$, without units in the present non-dimensional formulation, along the x-axis is used. As shown in Figure 1a, the single crystal BCC lattice is oriented with the $\left[100\right]$, $\left[010\right]$ and $\left[001\right]$ axes parallel to the x, y and z axes, respectively. The model comprises approximately 50,000 atoms (density peaks) which are localized by performing a quadratic interpolation in the computational grid at regular intervals. The intervals are chosen such that the magnitude of the movement $\Delta {\mathit{r}}^{\alpha }$ of an individual atom $\alpha$ between two interpolations is much less than the lattice parameter. The atom positions in the previously interpolated state are stored and each atom can be uniquely traced throughout the process by finding its incremental displacement relative to the position of its nearest neighbor among the previously stored position data set. The total accumulated displacement is also stored, along each coordinate direction, to provide the full displacement history for each atom by performing the update ${\mathit{u}}^{\alpha }:=bm{u}^{\alpha }+\Delta {\mathit{r}}^{\alpha }$. This is in line with the procedure outlined in [26], where the small strain tensor in Equation (15) is evaluated based on the atom displacement field in post-processing of PFC data. In the present study, a more general approach is taken as the discrete deformation gradient defined in Equation (19) is considered instead. For illustration purposes, however, the x-component $u_x$ of the displacement field obtained from the present PFC simulations using Model 1, cf. Figure 1a, is shown in Figure 2 and the y-component $u_y$ in Figure 3. The sequence of images in these figures shows how deformation is localized at the notch tips and the subsequent growth of a cleavage crack between the $\left[100\right]$ planes across the vertical direction of the simulation domain.

The same sequence of simulation steps as in Figure 2 is also shown in Figure 4, but in terms of the average strain $\overline{\epsilon }$ obtained using Equation (11). A gradually increasing concentration of a tensile strain state at the notch tips is evident.

Similar to the results in Figure 4, the x-component ${\widehat{E}}_{xx}$ of the Green–Lagrange strain tensor $\widehat{\mathit{E}}$, obtained by using the discrete deformation gradient $\widehat{\mathit{F}}$ from Equation (21) in Equation (13), is shown in Figure 5. However, while no additional information can be drawn from the average strain $\overline{\epsilon }$, the individual components of the strain tensor adds further possibilities to describe the strain state.

Figure 6 shows the strain ${\widehat{E}}_{yy}$ in the vertical y-direction, indicating a narrow zone of compressive strain in the vertical direction at the notch tips. This is in accord with the macroscale strain field in notched specimens, observed using digital image correlation in [27]. As a further example of the additional information residing in the full tensorial strain description, compared to the average strain $\overline{\epsilon }$, Figure 7 shows the shear strain component ${\widehat{E}}_{xy}$ which also traces the deformation localization at the notch tip throughout the simulated fracture process.

4.2. Continuous Deformation Fields Based on Geometrical Phase Analysis

Model 2, shown in Figure 1b, is used to illustrate the use of Geometrical Phase Analysis (GPA) on the PFC data. Two BCC crystals are defined with the $\left[100\right]$, $\left[010\right]$ and $\left[001\right]$ axes initially aligned with the x, y and z axes. The crystal orientations ${\mathit{R}}_1$ and ${\mathit{R}}_2$ are then defined by rotating the crystals through an angle of $\theta /2$ in opposite directions around the z-axis. In this study, a tilt angle of $\theta =12.{68}^{\circ }$ is used. This will result in low-angle tilt grain boundaries, cf. Figure 1b, comprising clearly distinguishable arrays of grain boundary dislocations. The domain size is taken as $\left[L_x,L_y,L_z\right]=\left[61.3,66.5,9.00\right]$, with the fractional numbers being due to the domain periodicity requirement.

The different steps in the GPA analysis are illustrated in Figure 8. The starting point is the PFC density field, shown in Figure 8a, where it is noted that the density maxima correspond to the intensity peaks seen in HRTEM images of crystal lattices. This observation conveniently makes GPA directly applicable to the analysis of PFC density fields. A Fourier transform (FFT) of the density field provides the intensity map in Figure 8b and the two peaks used for the subsequent analysis are indicated by red circles, showing the reciprocal lattice vectors ${\mathit{g}}_1=\left[100\right]$ and ${\mathit{g}}_2=\left[010\right]$. A mask is placed, in turn, at ${\mathit{g}}_1$ and ${\mathit{g}}_2$ using Equation (25). The mask is multiplied by the intensity data in Fourier space (a convolution in real space) and the result is then brought back to real space by a reverse FFT. Figure 8c shows the real part of the resulting complex $H_{g1}$ map after masking by ${\mathit{g}}_1$. The same procedure is repeated using ${\mathit{g}}_2$ instead, providing $H_{g2}$. Using Equation (26), the phase difference maps $P_{g1}$ and $P_{g2}$ related to ${\mathit{g}}_1$ and ${\mathit{g}}_2$, respectively, are evaluated. As an example, Figure 8d shows the phase difference map $P_{g1}$, normalized to the range $[-\pi ,\pi ]$. A phase difference of $2\pi$, indicated by an abrupt discontinuous shift from black to white in the figure, corresponds to the presence of additional lattice planes. Using Equation (37), the 2D displacement field can be evaluated based on the phase difference maps $P_{g1}$ and $P_{g2}$. Figure 8e,f show the displacement components along the x and y axis, respectively. It is emphasized, however, that in contrast to the discrete atom displacements shown in Figure 2 and Figure 3, the GPA-based displacements in Figure 8e–f are measured relative to the chosen reference lattice, defined by ${\mathit{g}}_1$ and ${\mathit{g}}_2$.

Based on Equation (32), a deformation gradient $\tilde{\mathit{F}}$ can be obtained by GPA and corresponding strain tensors can be evaluated by using Equations (13)–(15). It is emphasized that just as for the displacements also the strains obtained from GPA will be relative to the chosen reference lattice. Figure 9 shows some strain and rotation components obtained from GPA applied to the PFC data provided by Model 2. Characteristic alternating tension/compression regions at the individual dislocations are clearly distinguishable in the ${\tilde{E}}_{xx}$ field shown in Figure 9a. In addition, Figure 9b shows the concurrent ${\tilde{E}}_{yy}$ strain component, considerably smaller in magnitude than ${\tilde{E}}_{xx}$, and Figure 9c shows the shear strain component ${\tilde{E}}_{xy}$. The PFC-based strain fields evaluated by GPA are in qualitative agreement with the analytical elastic strain fields expected for edge dislocations [28]. The rotation component ${\tilde{\omega }}_{xy}$, cf. Equation (17), is shown in Figure 9d and agrees with the crystal rotations described by $\pm \theta /2$. The rotation magnitude in Figure 9d is evaluated as $tan\left({\tilde{\omega }}_{xy}\right)$ and is shown in degrees.

The GPA method, introduced in [25] and described in Section 3.2, provides the displacement and strain fields over 2D surfaces. Unfortunately, the extension to 3D is not straightforward as the selection of Bragg peaks for masking of the intensity field becomes quite involved in the 3D analogue to Figure 8b. A multitude of density peaks are present due to the symmetries of the lattices. In addition, the GPA results are highly sensitive to the selection of the reciprocal lattice vectors $\mathit{g}$ and manual refinement using Equation (27) is usually required for the individual vectors. Moreover, the size ${\sigma }_M$ of the mask in Equation (25) can in general not be held constant, but must be manually adjusted for each $\mathit{g}$-vector. Taken together, these factors defy an autonomous algorithmic implementation of GPA for the general 3D case. As a compromise, however, different planar 2D sections through the 3D PFC data can be analyzed by GPA to provide the strain components on these planes. Figure 10 gives some examples of how strain components can be evaluated over orthogonal planes through a point in the 3D PFC data. This procedure is, due to the manual steps involved, mainly useful to access the full strain tensor in singular points or small regions of particular interest.

5. Conclusions

The present study demonstrates how numerical methods for the evaluation of deformation fields, originally intended for molecular dynamics simulations or for analysis of high-resolution microscopy images, can be adapted to a PFC setting. Two main strategies are identified and demonstrated: either evaluation of discrete deformation fields based on the interpolated positions of individual PFC density maxima (identified as atoms) or evaluation of continuous deformation fields, based on a spectral analysis of the full density field using Geometrical Phase Analysis (GPA). It is shown that both the discrete and the continuous field approach can be used to post-process PFC data to obtain access to the full displacement and strain fields. Working with discrete points, 3D data sets can be conveniently handled and this method is also attractive for large-scale simulations where it becomes unfeasible to store the full PFC density field in comparison to just storing data at the discrete density maxima. Alternatively, working with the continuous data using GPA, more highly resolved deformation fields can be evaluated in 2D. GPA-based strain analysis becomes restricted in 3D as it is not straightforward to perform the required phase analysis in three dimensions. A compromise is demonstrated in the present work, however, evaluating the full deformation field by its individual components on orthogonal 2D sections through the 3D PFC density field. Extension of 3D PFC strain analysis based on spectral methods will be a target for future work. In addition to the different approaches to evaluation of PFC deformation fields, the present study also demonstrates application of prescribing non-affine deformations in a 3D structural PFC (XPFC) setting (to the author’s knowledge only 2D implementations exist in the literature) and an efficient and general numerical scheme for evaluation of PFC phase diagrams is provided in Appendix A. Taken together, the findings of this study expand the toolbox available for using PFC as a versatile method for numerical simulations of material behavior at the atomic scale.