Observation of the Kibble–Zurek Mechanism in Microscopic Acoustic Crackling Noises

Characterizing the fast evolution of microstructural defects is key to understanding “crackling” phenomena during the deformation of solid materials. For example, it has been proposed using atomistic simulations of crack propagation in elastic materials that the formation of a nonlinear hyperelastic or plastic zone around moving crack tips controls crack velocity. To date, progress in understanding the physics of this critical zone has been limited due to the lack of data describing the complex physical processes that operate near microscopic crack tips. We show, by analyzing many acoustic emission events during rock deformation experiments, that the signature of this nonlinear zone maps directly to crackling noises. In particular, we characterize a weakening zone that forms near the moving crack tips using functional networks, and we determine the scaling law between the formation of damages (defects) and the traversal rate across the critical point of transition. Moreover, we show that the correlation length near the transition remains effectively frozen. This is the main underlying hypothesis behind the Kibble-Zurek mechanism (KZM) and the obtained power-law scaling verifies the main prediction of KZM.

In this Supplementary Information, we provide further results and discussions on the proposed ideas in the main text.

Mapping Multiple-Acoustic Excitations to Network Space
To evaluate recorded multiple acoustic emissions (multiple time series for an occurred event), we use a previously published functional network algorithm on waveforms from our reordered acoustic emissions [1][2][3]. The main steps of the algorithm are as follows [1]: (1) The waveforms recorded at each acoustic sensor are normalized to the maximum value of the amplitude in that station.
(2) Each time series is divided according to maximum segmentation (i.e., each segment includes only one data point). The amplitude of the jth segment from ith time series (1 i N ≤ ≤ ) is denoted by , ( ) i j x t (with units of mV). N is the number of nodes or acoustic sensors. With considering the length of each segment as a unit, we considers the high temporal resolution of the system's evolution, smoothing the raw signals with 20-40 time windows (~400-800ns). 1 below for the definition of this property. In [4], we showed that selecting the threshold level by using the minimum variation of B.C versus ζ is equivalent to finding the most stable structures in the networks. Each node is characterized by its degree i k representing the number of links connected to that node, and its betweenness centrality (B.C) [5]: , in which hj ρ is the number of shortest paths between h and j , and ( ) i hj ρ is the number of shortest paths between h and j that pass through i.
In general, the modularity of a network measures the degree of division of that network into modules: if a network has high modularity, the connectivity in individual modules is strong, whereas the connectivity between modules is not. The network's modularity characteristic is addressed as the quantity of densely-connected nodes relative to a null (random) model. The main diagnostic in this work is the Q-profile. The modularity is the result of some optimization of the cluster structure of a given network. The modularity Q is defined as [6]: (2) in which M N is the number of modules (clusters), We use the Louvian algorithm [7] to optimize Eq.2, which has been used widely to detect communities in different complex networks. Then, in each time step during the evolution of waveforms (here over observation windows of ~200 µs), we obtain a Q value. The temporal evolution of Q values in the monitored time interval forms the Q-profile. Three main stages for a single recorded acoustic excitation event are (Fig.S1) [1][2]8]: (1) S-phase: main deformation phase in the form of initial strengthening (2) W-Phase: a fast-slip or weakening phase , and (3) D-phase: a slow slip or decelerating stage (Fig. 2a). Comparing the evolutionary phases of micro-cracks to dynamic stress change from macro-sliding events (recorded by strain gauges) reveals similar time characteristics and trends ( Fig.S1b- [1,[9][10]). In using the reciprocal of Qprofiles (which we refer to as R-profiles), we magnify the initial strengthening phase.

Brief review of Experiments
Lab.EQ1: a saw-cut fault, which is embedded in a cylindrical rock sample, is perturbed slowly with a strain rate control feed-back. The detailed experimental results have been reported in [9]. Our main data set includes the recorded discrete and continuous waveforms (i.e., acoustic emissions-AEs) using 16 piezoelectric transducers from a saw-cut sample of Westerly granite under triaxial loading [9]. The saw cut was at a 60 degree angle and polished with silicon carbide 220 grit (Fig.S.2). Each triggered event had a duration of 204.8 µs (recorded at 10 MHz), while the three main stick-slip events occurred. The experiment was servo-controlled using an axial strain rate of MPa for three reported main stick-slip events, producing 109 located-rupture fronts events.

Lab.EQ2:
The second data set (LabEQ2) are the results of the two main cycles of loadingunloading (stick-slip) of Westerly granite on a preexisting natural fault by loading at constant confining pressure. A natural rough fault was created using a triaxial loading system at the constant confining pressure of 50 MPa and with acoustic emission feedback control [10].

Lab.EQ3:
The third data set (LabEQ2) slightly faster loading of intact Westerly granite rock samples with similar loading condition (confinement loading) of Lab.EQ1 and 2. The details of the experiment have been reported in details in [11].
Lab.EQ4: Samples of basalt from Mt. Etna volcano (a basalt rock of approximately 3.8% porosity) were deformed using a standard triaxial deformation apparatus installed at University College London (UK). Cylindrical samples 40 mm in diameter and 100 mm in length were isolated from a confining medium (silicone oil) via an engineered rubber jacket containing inserts for mounting piezoelectric sensors in order to detect AE events. AE event signals (voltages) are first pre-amplified 40 dB, before being received and digitized at 10 MHz sampling rate (as well as Lab.EQ1,2,3). The sample was dry. The details of the test can be found in [12][13].

Kibble-Zurek mechanism (KZM) -
The Kibble-Zurek mechanism (KZM) describes the spontaneous formation of defects (symmetry breaking) in systems crossing a second-order (continuous) phase transition at a finite rate [14][15]. Experimental evidence of the KZM has been observed in superfluid 3 He [16], in superconducting films [17], in ion chains [18] and magnets or ferroelectrics with discrete microscopic symmetries [19]. Based on the KZM, in the vicinity of the critical point, the dynamics can be subdivided into the three main stages [14]: adiabatic, frozen, and again adiabatic (see Fig.S.3). Note that based on universality classes for second-order phase transitions, this mechanism provides a prescription for estimating a correlation length ξ and then, from this, the density of defects. For a second-order phase transition that evolves quasi-statically, ξ diverges at the critical point and no defects are formed [20].
The idea behind the KZM is to compare the relaxation time (or healing time of the system) with the timescale of change of the control parameter (i.e., ε). We assume a linear change of the control parameter in the vicinity of the critical point ε(t) = t/τ s , where τ s is the ramp time. The relaxation or healing time  = − ∂ where we mapped acoustic emissions to K-strings and then the spin-system. c) Crossing critical point is interpreted to transition between two degenerate states. (Here we have half of the (a) where the system "pressure-quenched" from ordered phase to a disordered state).
To identify the signature of defects, we use the concept of undulating "K-strings". In Fig.S.4b-d, we show the evolution of K max . The trends of evolution of K max and R(t) are similar and include two main sub-stages separated by an inflection point: the first sub-stage is the accelerating portion and the second (nucleation) portion with negative concavity (Fig.S5-S.6).

Figure S.4| a)
Evolution of maximum degree of the constructed networks (K max is the scaled value) and the modularity index (Q(t)) ;b) evolution of K max around the impulse-zone. S-phase includes two sub-stages: accelerated/loading-zone (stage I) and nucleation zone (stage II). c) R-profile around the critical point. Inset shows the second derivative of R(t). (d) Evolution of node degree for an event from Lab.EQ2.

K-strings (chains):
To visualize and study defects in the S-W transition, we use plot the node degree (k i as the number of links attached to i th node) in a polar coordinate system  The formation and merging of the kinks can be followed in Fig S.5. Interestingly, prior to the failure (defined as the point at which <k> reaches its maximum) and during nucleation of kinks, is nearly constant (Fig.S.6d). Then the maximum and minumum   Figure S7. a-c) A 3d array of sensors on a surface of a cylindrical sample is mapped to a 2d configuration with assigning a direction (out or in) per each site . Furthermore, in a true 3D configuration (d-f) with using true triaxial tests [1] we can find out a 3D vector 'components per each site.
According to KZM theory, the rate of transition to the critical regime where we observe the first defect (here the rate of "stress (pressure)-quench") controls defect density. To measure the ramp rate, we carefully measure the slope of R(t). This will yield the arte of R(t) and from Fig.S1 we have ( ) ( ) R t t σ ∝ and then we obtain (local) stress rate. In this study, we assume a linear ramp.
Next, we measure the number of flipped nodes and then density of kinks for typical events (Fig.S.8-10). Interestingly, faster ramp (shorter ramp-time) shrinks the precipitate-out time (which is the time that takes to complete the phase transition) and increases the density of defects (Fig.S.9) which decreases ξ (coherence length)-also see Fig     order parameter shows an impulsive trend and remains effectively frozen [27]. Furthermore, we have calculated a correlation function for the system including all nodes while it approaches the critical point (Fig.S13 a, b) . The correlation function has the form ( ) (1 ) exp( ) L is the total number of nodes, x is distance and ξ is the correlation length. Correlation length ξ is the cut-off length of the correlation function where for cases with shorter distance than the correlation length, a power law function can be fitted [20].   The size of the system is 300 nodes .  The evaluated events in our study do satisfy the condition of the second-order phase transition: under this class of phase transition, order parameter m s = < > rises (or declines) from zero (or 1) continuously as shown in Fig.S.13-15 and Fig.S17. Nevertheless, we could recognize some rare events that do not obey this classification (Fig.S.18): The order parameter jumps (drops) discontinuously to a non-zero value below R max as the critical point. We will investigate the properties of this first-order crackling noises in future work.