Interplay Between Nucleation and Kinetics in Dynamic Twinning

Twinning and structural phase transformations are important phenomena in materials science and solid-state physics due to the significant role they play in influencing the properties and behavior of materials [1]. Some specific areas of applications are in shape memory alloys (SMAs) where the reversible twinning and detwinning of SMAs are responsible for their unique shape-changing behavior [2]. In other instances, structural phase transformations allow for tailoring material properties to specific applications, for instance in nanotwinned metals [3–6].

The microstructure of the materials in which twinning or phase transformations occur typically consists of homogeneously deformed regions separated by interfaces across which the deformation varies extremely rapidly. Many interesting material phenomena under extreme conditions are governed by the nucleation, motion, and response of the interfaces [2,7]. It is therefore important to be able to accurately model propagating interfaces in a dynamic setting, particularly when inertia plays a significant role since these interfaces move at velocities comparable to the sonic speeds.

Continuum models for dynamic twin interface nucleation and propagation fall into two broad classes: sharp interface models [8–13] and regularized interface models [14–19]. In the case of the former, numerical computations are extremely challenging as sharp interfaces require expensive tracking algorithm in a numerical discretization. This is essentially infeasible when one expects numerous evolving and interacting interfaces, and thus the sharp interface approaches are not widely applied.

Regularized interfaces on the other hand do not require explicit tracking of the interfaces, which makes the numerical methods much simpler and enables application to complex settings. Phase-field models, wherein an additional regularized phase-field variable is introduced to account for material transformations, have been widely applied to problems such as structural phase transformations, fracture, poromechanics, and dislocation dynamics, e.g., Refs. [20–30]. However, these methods also have significant shortcomings. Particularly, the dependence of the kinetics and nucleation of interfaces on the structure of the material model is completely opaque. Furthermore, the range of material responses that can be obtained is highly constrained. Furthermore, the nucleation and the kinetics of interfaces are physically distinct processes from the atomic perspective, but standard phase-field models are unable to transparently distinguish these processes. One consequence of these shortcomings is that even in the simplest 1D case, the precise critical condition at which nucleation occurs is completely opaque. Consequently, it is essentially impossible in practice to formulate a model to obtain a desired nucleation response. All of this is in sharp contrast to sharp interface models wherein the nucleation and kinetics of interfaces can be transparently specified in the model.

To retain the computational ease of phase-field models and at the same time obtaining the transparent physical structure of sharp interface models, we recently proposed a phase-field modeling approach that enables the transparent specification of interface nucleation and kinetics as input to the model [31–33]. The key elements of this approach are, first, formulating an energy density that separates nucleation from kinetics; and, second, formulating a geometrically motivated interface conservation principle to govern the nucleation and kinetics of interface. In this conservation principle, the kinetics of interfaces is governed by a transport term and the nucleation of interfaces is governed by a source term, and each of these contribution can be transparently and independently specified.

In this paper, we use the phase-field approach described above to study the interplay between kinetics and nucleation in dynamic twinning. Specifically, we study two distinct problems where it is essential to be able to explicitly specify the kinetic and nucleation behavior governing twin evolution.

First, we study twinning interfaces in 2D. When these interfaces are driven to move, we find that significant levels of twin nucleation occur ahead of the moving interface. Essentially, the finite interface velocity and the relaxation time of the stresses ahead of the interface allow for nucleation to occur before the interface is able to propagate to that point. Second, we study the growth of needle twins in antiplane elasticity. We show that both nucleation and anisotropic kinetics are essential to obtain predictions of needle twins. While standard regularized interface approaches do not permit the transparent specification of anisotropic kinetics, this is readily possible with the phase-field approach that we have used here.

We use $x0$ and $x(x0)$ to denote the reference and deformed configurations, and the displacement is denoted $u(x0):=x(x0)−x$⁠. We work completely in the Lagrangian setting for simplicity, while noting that there can be several advantages to an Eulerian approach in the large deformation setting [34–37]. We use $∇$ and $div$ to denote the gradient and divergence with respect to $x0$⁠, and superposed dots to denote time derivatives. The deformation gradient is denoted by $F:=∇x$⁠, the strain by $E:=(1/2)(FTF−I)$⁠, the spatial velocity gradient by $L=F˙F−1$⁠, and the first Piola–Kirchoff stress by $P$⁠. We define $D:=(1/2)(L+LT)$⁠, $J:=detF$⁠, and use $ϕ$ to denote the phase field.

We define a regularized indicator function $Hl(⋅)$ that smoothly transitions—over an interval of order $l$—from $0$ to $1$ when the argument changes from negative to positive. For specificity, we use the form $Hl(x)=(1/2)(1+tanh(x/l))$ and $l=0.1$⁠.

The domain and its boundary are denoted by $Ω$ and $∂Ω$⁠, respectively.

The evolution equation for $ϕ$ was proposed in Ref. [38] and was shown to correspond to a balance law for interfaces [32,39]. $G$ originates as a source term in that balance law, and in this work is used to induce the nucleation of an appropriate phase when the system evolves to access nonphysical regions of the energy landscape [33]. The specific forms of $G$ will be described in later sections.

References [31–33] provide conditions that $G$ and $v^(f)$ must satisfy for consistency with the second law of thermodynamics [22,27].

The boundary conditions for momentum balance (6a) are the standard displacement or traction prescribed, and the boundary condition for phase evolution (6c) is the standard variational condition $∇ϕ⋅n^=0$ [18] on the entire boundary.

The momentum balance and the phase evolution are evolved in time using the implicit Newmark method and the implicit Euler method, respectively.

We apply the model described in Sec. 2 to predict the motion of a flat twin interface in a 2D nonlinear elastic setting, and specifically study the role of phase nucleation when inertial effects are significant.

We assume that the elasticity tensor $C$ is isotropic and identical in both phases, and set the extensional modulus to 206 GPa and the Poisson ratio to 0.3.

To setup the initial condition for dynamic calculations described later on, we obtain a stress-free compatible interface by setting $ϕ=1$ on the right of the domain and $ϕ=0$ on the left side, and then minimize the energy in (1). The resulting static twin interface is shown in Fig. 1.

We first find the hyperplane in strain space that separates the twinning phases. Considering strains corresponding to the stretch tensors given in (7), the hyperplane that is the set of points that are of equal distance from $E1$ and $E2$ has the equation $|E*−E2|=|E*−E1|$⁠. Solving for $E*$ gives $E11*=E22*$ as the equation of the hyperplane. On one side of the hyperplane, we are closer to $E1$ and $ϕ$ should be less than $0.5$⁠; on the other side, we are closer to $E2$ and require that $ϕ$ is greater than $0.5$ (Fig. 2).

Since the role of kinetics is not the central focus here, we use the simple linear form $v^(f)=κf$⁠. We use $κ=1$ for the numerical calculations.

We perform dynamic calculations using the boundary conditions shown in Fig. 3. The boundary conditions are setup to minimally constrain the deformation of the body, and this allows a rigid vertical translation mode, but this mode does not interfere with our observation of the twinning deformation. We use the static equilibrium interface as the initial conditions (Fig. 1).

Figure 4 provides a representative example that highlights the effect of the nucleation term. The highlight of the calculations shown in Fig. 4 is that even at subsonic velocities, it is apparent that accounting for nucleation has a very significant effect and dominates over interface kinetics. Furthermore, it shows that nonphysical regions in the energy landscape were accessed when nucleation is not accounted for.

We also note that the calculations in Fig. 4 do not have viscosity, but the calculations with viscosity (⁠$η=1×10−9s$⁠) are essentially identical. This overall conclusion holds for all the calculations described in this section.

For greater insight into the role of nucleation, Fig. 5 plots the strain fields in a specimen where the nucleation term is inactive, i.e., the twinning transformation occurs exclusively due to interface motion and not nucleation. Looking at a time at which the calculation in Fig. 4 that allows nucleation had nucleated several new domains, Fig. 5 shows clearly that $E11>E22$ in the right part of the specimen. We would thus expect the $ϕ=1$ phase to nucleate in that region.

Figure 6 shows the driving force for nucleation $G$ as a function of position, at a time when nucleation is active. We observe that the value of $G$ is very high in the lower right part of the specimen.

Comparing systems with and without nucleation, we observe in Fig. 7 that the driving force for nucleation $G$ drove the nucleation of the $ϕ=1$ phase in the right of the specimen, which corresponds to the position where the value of $G$ was very high several time-steps ago. This suggests that the driving force for nucleation has driven the system to a physical region of the energy landscape.

A final point is that when we attempt to drive the interface to propagate faster (via higher loading rates), we encounter pinning—essentially, very slow evolution—of the interface where it meets the boundary of the specimen. As the twin phases are related by a shear, the driving force for the interface to propagate is related to the shear traction on the plane that is parallel to the interface. The boundaries are traction-free on the top and bottom of the domain, resulting in zero shear stress at the twin interface on the top and bottom boundaries. This effectively pins the interface at the boundaries by causing it to evolve very slowly compared to the interior. This is further discussed in Ref. [31].

When twins initially nucleate, they grow in a characteristic needle-like geometry, e.g., the experimental observations in Ref. [45]. The growth of needle twins was studied using a sharp interface model using theoretical approaches in several works by Rosakis [46–48]. A key finding in that body of work was that anisotropic kinetics is essential to capture needle-like twins. As discussed in Ref. [47], the incompatible direction should have more twinning dislocations than the compatible directions to accommodate the deformation, and hence the kinetics will be significantly different for twin motion along compatible and incompatible directions. This micromechanical effect gives rise to mesoscale anisotropy in the kinetics.

We aim to develop a phase-field description that can capture the phenomenon of needle twin growth. A key challenge for standard phase-field models is the inability to incorporate anisotropic kinetics. We develop a phase-field model here that has anisotropic kinetics and show that it is able to capture needle-like twin growth, and also show that incorporating nucleation is essential to prevent nonphysical broadening. Following Ref. [13], we work in the antiplane elastic setting for simplicity.

We work in the 2D antiplane setting. The independent space variable is denoted $x=(x1,x2)$ in Cartesian coordinates; and the displacement has a single non-zero component in the out-of-plane $3$-direction, denoted $u(x,t)$ where $t$ is the time. The strain is denoted $ε$⁠, and has only two non-zero components: $ε1=(1/2)∂x1u$ and $ε2=(1/2)∂x2u$⁠.

We use displacement boundary conditions all around and use this to simulate dynamic loading. We start with a single phase and have a small nucleus (of size $0.05$⁠) of the second phase near one of the vertical edges to drive the growth of a single needle twin in a convenient location.

The stress-free strains are $ε01=0$ and $ε02={0.0,1.0}$⁠. Consequently, the compatible direction is $n=e2={0,1}$⁠.

For viscous dissipation, we use $ηε˙$ with $η=0.005$⁠.

We begin by examining the effect of anisotropic kinetics, but do not account for nucleation to enable us to precisely understand their separate roles.

Figure 8 shows the evolution of the twin needle in time, comparing isotropic kinetics to anisotropic kinetics. We see that anisotropy is essential to obtain needle-like twin growth, consistent with the work of Ref. [47]. However, we also notice that the twin slowly gets wider in the thin direction beginning at the base.

Building on the observation that anisotropic kinetics appears essential to predict realistic needle twins, we next consider the effect of further accounting for viscous stress and nucleation. Figure 9 compares the effect of accounting for nucleation and viscous stresses in addition to anisotropic kinetics. We see that viscous stresses attenuate the leading elastic waves but do not significantly affect the needle twin microstructure. In contrast, the nucleation term serves the important purpose of suppressing the nonphysical broadening of the twin.

We further plot the driving forces for kinetics and nucleation in Fig. 10. First, we notice that both driving forces are of comparable magnitude. Second, we highlight that the nucleation term is active at the growing tip as expected, and also at the base of the needle twin where it acts to prevent nonphysical twin broadening.

We next look at the role of nucleation and viscous stress in supersonic twinning. To prevent the use of overly large stresses, we achieve supersonic twinning by increasing the kinetic coefficient $κ$ by a factor of 10. Figure 11 compares the twin microstructure in a model with only anisotropic kinetics to a model that also accounts for nucleation and viscous stress. As can be inferred from the Mach cone structure, both twins grow supersonically, with the model with viscous damping showing faster twin growth.

We next plot the kinetic driving force for both models in Fig. 12, and notice that there are no significant differences. As expected in our class of models, the driving force is localized around the moving interface [31,33].

Figure 13 shows the nucleation driving force and the viscous stress. We notice that both are highly localized at the tip of the growing twin, and that the largest value of the nucleation driving force is comparable to the kinetic driving force.

We have used a recent phase-field modeling framework proposed in Refs. [31–33] to study the interplay between kinetics and nucleation in the dynamic evolution of twins. The ability of this framework to explicitly and transparently specify nucleation and kinetic behavior is essential to model phenomena such as needle twin growth. For instance, anisotropic interface kinetics is essential to model needle twin growth, but it is unclear how—or if it is even possible—to specify anisotropic kinetics in standard phase-field approaches or alternatives such as peridynamics.

For the problems studied in this paper, we find that viscous stresses do not play an important role. This is a significant difference compared to studies in other settings, e.g., Refs. [33,49]. In contrast to those works, the process of twinning in higher dimensions is dominated by geometric compatibility, which leads to this significant difference.

We thank NSF (2108784 and 2012259) and ARO (MURI W911NF-19-1-0245) for support; NSF for XSEDE computing resources provided by Pittsburgh Supercomputing Center; Maryam Khodadad for reviewing the code; and Tony Rollett and Phoebus Rosakis for useful discussions.

There are no conflicts of interest.

The code developed for this work and the associated data are available.²