Origin of scale-free intermittency in structural first-order phase transitions

A salient feature of cyclically driven first-order phase transformations in crystals is their scale-free avalanche dynamics. This behavior has been linked to the presence of a classical critical point but the mechanism leading to criticality without extrinsic tuning remains unexplained. Here we show that the source of scaling in such systems is an annealed disorder associated with transformation-induced slip which coevolves with the phase transformation, thus ensuring the crossing of a critical manifold. Our conclusions are based on a model where annealed disorder emerges in the form of a random field induced by the phase transition. Such a disorder exhibits supertransient chaotic behavior under thermal loading, obeys a heavy-tailed distribution, and exhibits long-range spatial correlations. We show that the universality class is affected by the long-range character of elastic interactions. In contrast, it is not influenced by the heavy-tailed distribution and spatial correlations of disorder.

Figure 1

Description of a 2D crystalline solid using kinematically compatible elements. Each element represents a region at a mesoscopic scale where the deformation of the lattice is homogeneous, as shown by the insets. When an element is sheared to configurations such as ${\mathcal{S}}^{+}$ in Fig. 3, dislocations are created in the square lattice [8, 10, 47].

Figure 2

Panel (a) indicates the displacement fields $u_{i,j}^x,u_{i+1,j}^x$ and $u_{i,j+1}^x$ defining the strain variables ${\alpha }_{i,j}$ and ${\beta }_{i,j}$ shown in (b).

Figure 3

Piecewise parabolic function $f(\beta ,T)$ giving the periodic energy of a mesoscopic element. The insets on top show in red (gray) the shape of the elements sitting in the bottoms of the wells; the dashed lines indicate the associated lattice structure. The elementary shapes in the domain with $d=0$ correspond to elements in square austenite, $\mathcal{S}$, and two variants of oblique martensite, ${\mathcal{O}}_1$ and ${\mathcal{O}}_2$. The rest of shapes correspond to slipped lattices that are identical to those of the basic shapes (compare the dashed lines).

Figure 4

(a) Interaction $\mathbf{J}$ of an element at the center of a system of size $L=51$ with the rest of elements in the system ($c=0.2$). Red, white, and blue colors indicate positive, zero, and negative values of $\mathbf{J}$, respectively. (b) Decay of the absolute value of $\mathbf{J}$ with distance, $r$, from the element at the center of the system along the horizontal and vertical directions shown with dotted lines in (a). The dashed line indicates the $r^{-2}$ decay.

Figure 5

Spatial configuration of (a) the slip disorder, $\mathbf{h}$, (b) the field of phases, $\mathbf{s}$, and (c) the slip field, $\mathbf{d}$, in the low-temperature phase after ${10}^4$ cycles (blue and red correspond to negative and positive values) of a system with $L=51$ undergoing a close-to-reconstructive transition with $\epsilon =0.48$.

Figure 6

Fraction of elements transformed to martensite, $q\left(\tau \right)$, and standard deviation of the slip disorder, $\mathrm{\Delta }\left(\tau \right)$, for [(a) and (c)] $\epsilon =0.37$ and [(b) and (d)] $\epsilon =0.48$ in a series of 20 cycles. Different colors correspond to different cycles, as indicated by the colorbox in (c). Simulations correspond to systems of size $L=51$. The initial slip disorder is ${\mathrm{\Delta }}_0\left(\epsilon \right)={\left(\epsilon L^2\right)}^{-1}{\sum }_{i,j}{J}_{i,j,L/2,L/2}^2=7.6\times {10}^{-3}/\epsilon$.

Figure 7

Long chaotic transient followed by an abrupt transition to a periodic regime in the case of close-to-reconstructive transformation. The extreme values ${\mathrm{\Delta }}_{\text{M}}$ and ${\mathrm{\Delta }}_{\text{A}}$ in each cycle are presented. Data correspond to a system of size $L=51$ and $\epsilon =0.431$.

Figure 8

(a) Log-normal plot of the length $n_{\text{s}}$ of (super)transients vs. $\epsilon$ for systems of three different sizes marked by the legend. (b) Mean amplitude ${\overline{\mathrm{\Delta }}}_{\text{M}}-{\overline{\mathrm{\Delta }}}_{\text{A}}$ of the slip disorder vs. $\epsilon$ for the same system sizes as in (a).

Figure 9

Temperature dependence of the mean standard deviation of the slip disorder, $\overline{\mathrm{\Delta }}\left(\tau \right)$, during cooling runs (cycles 400–10${}^4$) for $L=51$ and several values of $\epsilon$, as marked by the legend.

Figure 10

Log-normal plot of the distribution $D\left(h\right|\tau )$ for evolving disorder at three values of $\tau$ (marked by dot-dashed vertical lines in Fig. 9). Data correspond to cooling runs (cycles 400–10${}^4$) in a system with $L=51$ and $\epsilon =0.48$.

Figure 11

Temperature dependence of the excess kurtosis, $K_{\text{e}}$, corresponding to slip disorder during cycles 400–10${}^4$ in a system of size $L=51$ and several values of $\epsilon$.

Figure 12

Spatial correlation function of disorder, $C_{\text{h}}\left(r\right)$, in systems of size $L=51$ in regime ${\mathrm{II}}^{\prime }$. $C_{\text{h}}\left(r\right)$ decreases in a similar way for all values of $\tau$; the dashed line illustrates the $r^{-2}$ decay.

Figure 13

Avalanche size distribution at given temperature, $D\left(s\right|\tau )$, for cooling runs in a system with $\epsilon =0.48$ and $L=51$. Note that $\tau$ decreases from left to right while $\mathrm{\Delta }$ is increasing as shown in Fig. 9.

Figure 14

Temperature dependence of (a) the power-law exponent $\kappa$ and (b) the cut-off parameter $\lambda$ of the avalanche size distribution for heating runs. Different symbol types correspond to different values of $\epsilon$, as marked by the legend. Dashed circles show the parameters of fits to $D\left(S\right|\tau )$ in systems with Gaussian, quenched, and uncorrelated random variables, $\mathbf{h}$. Systems of size $L=51$.

Figure 15

Avalanche size distribution $D\left(s\right|\tau )$ for three different values of $\epsilon$ corresponding to the value of $\tau$ at which the cut-off parameter $\lambda$ assumes its minimum value [see Fig. 14]. The continuous lines show the results of the maximum likelihood fit for each distribution. The distributions for $\epsilon =0.46$ and $\epsilon =0.48$ have been shifted vertically for clarity. Systems of size $L=51$.

Figure 16

Temperature dependence of (a) the power-law exponent, $\kappa$, and (b) the cutoff, $\lambda$, of the avalanche size distribution for heating runs. Different symbol types correspond to different values of $\epsilon$, as marked by the legend. The vertical dashed line indicates our estimate for the critical driving parameter for heating runs.

Figure 17

Mean fraction of elements in martensite for cooling runs in systems of size $L=51$. Data are obtained by averaging the value of $q$ at given $\tau$ over ${10}^5$ cycles during supertransients. Different symbols correspond to different values of $\epsilon$ in the regime with supertransient behavior, as marked by the legend.

Figure 18

Probability density functions for the stability fields (a) ${\mu }_{\text{AM}}$, (b) ${\mu }_{\text{MA}}$, and (c) ${\mu }_{\text{S}}$ in a system with $L=51$ and $\epsilon =0.48$. Different symbol types correspond to different values of the driving parameter, $\tau$, as marked by the legend.

Figure 19

Probability density functions (a) $D_{\text{AM}}\left({\mu }_{\text{AM}}\right|\tau )$ and (b) $D_{\text{S}}\left({\mu }_{\text{S}}\right|\tau )$ for stability variables at ${\tau }_{\text{c}}=-0.385$ during cooling runs in systems of size $L=51$. The inset shows the exponent ${\alpha }_{\text{S}}$ obtained by fitting a power law $D_{\text{S}}\left({\mu }_{\text{S}}\right|\tau )\sim C_{\text{S}}\left(\tau \right){\mu }_{\text{S}}^{{\alpha }_{\text{S}}\left(\tau \right)}$ for small values of ${\mu }_{\text{S}}$.

Figure 20

Mean maximum number of elements becoming unstable with respect to the (a) AM and (b) S stability limits as a consequence of the phase transition from austenite to martensite of any element in the system. Note the log-normal scales in (a). The system size is $L=51$.

Figure 21

Contributions to the mean reproductive number of elements reaching the (a) AM and (b) S stability limits as a consequence of the phase transition from austenite to martensite of any triggering elements in a system of size $L=51$.

Figure 22

Mean reproductive number $\overline{R}\left(\tau \right)$ for individual avalanches during cooling runs in long series of thermal cycles for a system of size $L=51$ with $\epsilon =0.48$. The shaded region indicates the standard deviation of $\overline{R}\left(\tau \right)$.