Athermal Resistance to Interface Motion in the Phase-Field Theory of Microstructure Evolution

A method of introducing an athermal resistance to interface propagation for the Ginzburg-Landau (GL) approach to the first-order phase transformations (PTs) is developed. It consists of introducing oscillating fields of stresses (due to various defects or a Peierls barrier) or a jump in chemical energy. It removes some essential drawbacks in GL modeling: it arrests experimentally observed microstructures that otherwise converge to a single phase, and it reproduces rate-independent stress hysteresis. A similar approach can be applied for twinning, dislocations, and other PTs (e.g., electric and magnetic).

Figure 1

Evolution of randomly distributed ${\eta }_i$ fields for ${\sigma }_{ib}=10\mathrm{GPa}$. (a) and (b) Without a defects field, which finally converges to $M_2\text{}$. (c) Distribution of the stresses of defects ${\sigma }_{2d}=10\sin (16\pi x)\sin (16\pi y)=-{\sigma }_{1d}$ and the driving force $X_{1\to 2}^d=0.586{\sigma }_{2d}$ in a quarter of the sample; in black (white) regions variant $M_2\text{}$ ($M_1\text{}$) is promoted. (d) Corresponding stationary microstructure. (e) Stationary microstructure for ${\sigma }_{1d}=5\cos (16\pi x)\cos (16\pi y)$ and ${\sigma }_{2d}=5\sin (16\pi x)\sin (16\pi y)$ and consequently $X_{1\to 2}^d=-1.465\cos [16\pi (x+y)]$ (periodic function along the diagonal); everywhere blue (light gray) is $M_1\text{}$ and red (dark gray) is $M_2\text{}$. (f) Plot interface velocity $v$ vs the macroscopic driving force $X_{1\to 2}^b$ for four interfaces from microstructure in (e).

Figure 2

Evolution of initial embryo ${\eta }_i=0.1$ in a circle of the radius of 2 nm at the center of the sample for ${\sigma }_{ib}=15\mathrm{GPa}$; stationary microstructure is $M_2\text{}$. (a) and (b): Blue (light gray) outside of $M_2\text{}$ units is $A$ and red (dark gray) is $M_2\text{}$; distribution for $M_1$ is rotated by 90° distribution for $M_2\text{}$. (c)–(f): Blue (light gray) is $M_1$ and red (dark gray) is $M_2\text{}$.

Figure 3

The same problem as in Fig. 2 but with ${\sigma }_{2d}=10\sin (16\pi x)\sin (16\pi y)=-{\sigma }_{1d}$, i.e., with $X_{1\to 2}^d=0.586{\sigma }_{2d}$ from Fig. 1c. (a) Blue (light gray) outside of $M_2$ units is $A$ and red (dark gray) is $M_2$. (b)–(d): Blue (light gray) is $M_1$ and red (dark gray) is $M_2$. (d) is the stationary microstructure.

Figure 4

Similar problem as in Fig. 2 but with ${\sigma }_{2d}=-{\sigma }_{1d}$ and $X_{1\to 2}^d=0.586{\sigma }_{2d}$ shown in (a). Stress values are $-10$ (yellow, light gray), 0 (green, intermediate gray), and 10 GPa (blue, dark gray). (b) and (c) are the intermediate and stationary microstructures.

Figure 5

Two similar problems as in Fig. 2 but with ${\sigma }_{2d}=-{\sigma }_{1d}$ and $X_{1\to 2}^d=0.586{\sigma }_{2d}$ shown in (a) and (d). Stress values are $-10$ (yellow, light gray), 0 (green, intermediate gray), and 10 GPa (blue, dark gray). (b),(c) and (e),(f) are the intermediate and the stationary microstructures.