Microscopic Observation of the Light-Cone-Like Thermal Correlations in Cracking Excitations
11 Microscopic Observation of the Light-Cone-Like Thermal Correlations in Cracking Excitations
H.O. Ghaffari ,W.A.Griffith University of Texas at Arlington, Box 19049, Arlington, TX, 76019, USA
Many seemingly intractable systems can be reduced to a system of interacting spins . Here, we introduce a system of artificial acoustic spins – fictitious spins which are manipulated with ultrasound excitations associated with micro-cracking sources in thin sheets of crystals. Our spin-like system shows a peculiar relaxation mechanism after inducing an impulsive stress-ramp akin to splitting, or rupturing, of the system. Using real-time construction of correlations between spins states, we observe a clear emergence of the light cone effect. It has been proposed that equilibration horizon occurs on a local scale in systems where correlations between distant sites are established at a finite speed.
The observed equilibration horizon in our observations defines a region where elements of the material are in elastic communication through excited elementary excitations. We demonstrate that prior to the applied stress-ramp, the system is described by an algebraic correlation function, and above the critical point the correlation function shows exponential scaling characterized by the formation of kink-pairs. These results yield important insights into dynamic communication between failing elements in brittle materials during processes such as brittle fragmentation and dynamic stress triggering of earthquake-generating faults.
Introduction-
Cracking or splitting of a many body system is accompanied by excitations, interactions and recombination of many quasiparticles, and thus represents an extremely challenging problem [1-5]. The protocol of splitting, commonly referred to as rupture, in brittle solids is believed to be controlled in a process zone very near to the moving rupture tip. On the other hand, sudden injection of many emitted quasi-particles strongly induces a fast change in the state of the system yielding out-of-equilibrium evolution which eventually relaxes to a steady state [6-8]. To establish a steady state regime, the elements of the system must communicate with each other: the distant elements of the system must be correlated. Based on this requirement, then, the propagation of real-time correlations is a topic of utmost relevance in the study of fractures and fragmentation (multiple splitting) [9-10]. The spread of correlations in space-time is due to propagation of quasi-particles pairs forming light-cone-like propagation which establish the equilibration horizon [11-14]. In this work, we use a novel technique to study the emission of phonon excitations in relatively strong mechanical cracking sources (i.e., acoustic crackling noises) during indentation of thin films of natural occurring minerals. We use multi-array high frequency acoustic phonon sensors arranged in a ring and map their recorded excitations to a classical spin-like system with nanosecond resolution. We show that cracking-induced ultrasound excitations manifest themselves in flipping of the fictitious spins. We demonstrate that the system crosses a transition point below which the correlation function shows power law scaling decay, and above which the initial exponential function holds. Furthermore, our results establish that above the transition point, kink-pairs are formed, and these topological defects are associated with the exponential form of the correlation function. These results affirm that the relaxation of the system has already begun during the kink-pair generation. We also recognize slow fronts under slow stress-ramps which propagate at an order of magnitude slower than the fast fronts. We pay particular attention to the transitions from slow to fast fronts as well as the bifurcation and proliferation. Our results open horizons for understanding incipient plasticity in microscopic dynamic failure, and, more generally, non-equilibrium relaxation of many body systems.
Results-
Our experiments include micro-indentation of thin sheets of two different minerals (muscovite mica and graphite) at room temperature, and the corresponding material failure is recorded as multi-array ultrasound excitations (see Methods section for details of the sample preparations and data acquisition). Different loading protocols are employed to indent the thin films, and all data are recorded with resolution of 40MHz. In Fig.1, we show multi-array waves from a single cracking excitation parallel to the [001] face of a muscovite mica sheet. During a central-indentation test of the mica sheet, we were able to record up to 10 excitations with different excitation levels (i.e., cracking energy). By mapping this system to a classical spin system, we elucidate a new phenomenon in this relatively simple experiment: a clear spread of a light-cone-like thermal correlation front. To map our ring-like structure with finite nodes to spin-like particles, we use a thresholded-measure of each node’s activity (measured in mV ) to construct a network of nodes per each recorded data point (see Methods section for details of the algorithm). The constructed graph is characterized by the average of all nodes’ degree (
Scaled recorded multi-array acoustic waveforms in tens of μs for the highlighted event as the green-sphere. c) Evolution of average of all nodes’ degree
K-chain’s evolution. The system effectively is frozen around t=1000 ns for ~150 ns implying a non-adiabatic regime due to fast (local) stress ramp. f) The real-time correlation functions for the shown example. The correlation fronts, on average, spread with well-defined velocity of 5.05km/s (the arrows show the cross-over distances).
Our system behaves with a universal pattern characterized by the following steps: (1) the system is pulled to a state where all spins polarize in the outward direction. This is the first, rising part of
1( )
N ii m SN where N is the number of spins. In Fig.1e, we show the effect of the stress-ramp on decay of m where around the peak of
Snapshots of real-time dynamics of two-point correlation matrix evolution with time (1 to 8) for a single acoustic emission event in indentation of a graphite film. In evolution from snapshots (7) to (8), we observe domain wall dynamics resulting combination and annihilation event. The blue domain is “melted” with combination of two red domains. (b) Shows the source mechanisms of domains.
The arrows represent the in-plane components.
This finding in our cracking experiments implies a new understanding of microscopic dynamic failures: let us assume that the ramp profiles (
Figure 4| a,b)
Fast crossing from the low energy to high energy level is governed by tuning parameter
A reverse ramp (right-hand of the triangle pulse in inset of Fig.3a) with much slower rate results the growth of correlation fronts with slow prorogation rate. In this example, we observe proliferation of three fronts with transient velocity from ~1500m/s to ~625m/s in transition from front I to II ( b ). c) Another slow ramp with bifurcating fronts as shown in (e) at ~329.3µs and ~332.5 µs. A gliding-like dislocation is evident at ~330.5 µs. In this case, the bifurcation event is induced by sudden acceleration in the ramp. The evolution of the K-chain with color assisted code as local time clearly shows the splitting the front (pink arrow). In (d) we have shown the correlation function in each time step manifested in space-time correlation in (e).
Discussion-
In summary, we studied the dynamics of fictitious spins constructed from ultrasound waves generated by cracking excitations due to indentations of thin sheets of materials. The pseudo-spin systems were globally swept from a degenerate state to another degenerate state with higher energy. By monitoring the real-time evolution of correlation functions, we found that the dynamics of the correlation functions show a clear “ light cone ”-like behavior: it takes a finite time for the correlations to travel, proportional to the distance between points. The finite travel velocity of the equilibration horizon moves with a well-defined velocity, which are different for the employed thin-films and are at or below the range of the longitudinal phonon velocity. The observed equilibration horizon defines a region (in a given time step) where elements of the material are in elastic communication through elementary excitations. We showed that the transition from algebraic to exponential scaling is the result of the generation of kink pairs and the subsequent the relaxation path to a (quasi) stable state in our system is accompanied by removal of kinks. Furthermore, under the slower stress ramp (i.e.,
Experimental Procedures:
We indented three different crystalline materials (Muscovite Mica, Graphite, Gypsum (CaSO ·2H O)) and an amorphous glass film. Our focus on the reported tests in the main text was on mica and graphite films (from Princeton Scientific Corp). The indentation on mica was a (001)-oriented mica sheet. We first suspended a thin sheet of mica on a ring–like structure of ultrasound sensors and carefully glued on sensors. Then, we mechanically exfoliated by peeling sheets from the top of the specimen similar to making graphene from graphite [41]. We repeated this process 10 to 15 times to achieve a thin layer of the mineral sheet suspended on the sensors, and we repeated this procedure on large samples of natural Gypsum. The central indentation was performed using a micro-indentation instrument (LM300AT). The minimum and maximum employed indentation loads were 90mN and 490mN, respectively. In order to detect ultrasound excitations we utilized piezoelectric sensors. In the primairy set-up (Fig.1), we used 8 nodes (sensors), while to confirm the results we also employed a second set-up with 16 sensors. The acoustic excitation signals are first pre-amplified at 60 dB, before being received and digitized. The p-wave velocities of specimens are measured prior to indentation tests (and prior to the exfoliation of the samples when the initial thickness was ~500 µm) with pre-defined a 100 V high-frequency (0.5 µs rise-time) pulse which excites each channel while the other channels are recording. Knowing the onset time of the excitation pulse and the sensors positions, the measurement of the travel times yields the average velocity along each ray path.
Artificial Acoustic spins with ultrasound waves of crackling noises:
To construct an artificial acoustic spin, we map the recorded multi-array acoustic emission (multiple time series for each event) to a mathematical graph in a way that linked sites form a surface with the same sign (+/-) of the evolution rate. This can be interpreted as the surface of constant phase in analogy with “wavefronts” and defects of this surface (i.e., network structure) sets in parallel with
Nye-Berry’s theory on dislocations in waves [31]. We use a well-established algorithm to construct the mathematical graphs from our reordered acoustic emissions with a fixed number of nodes [16-17, 35]. The main steps of the algorithm are as follows: (1) The waveforms recorded at each acoustic sensor are normalized to the maximum value of the amplitude at that node. (2) Each time series is divided according to maximum segmentation, such that each segment includes only one data point. The amplitude of the j th segment from the i th time series ( i N ) is denoted by , ( ) i j u t (in units of mV). N is the number of nodes or acoustic sensors. We set the length of each segment as a unit with a resolution of 25ns. (3) , ( ) i j u t is compared with , ( ) k j u t to create links between the nodes. If , , ( ( ), ( )) i j k j d u t u t (where is the threshold level discussed in the following point) we set ( ) 1 ik a j otherwise ( ) 0 ik a j where ( ) ik a j is the component of the connectivity matrix and , , ( ) ( ) ( ) i j k j d u t u t is the employed similarity metric . With this metric, we simply compare the amplitude of sensors in the given time-step. The employed norm in our algorithm is the absolute norm. (4) Threshold level ( ): To select a threshold level, we use a method introduced in [35 and references therein] that uses an adaptive threshold criterion to select . The result of this algorithm is an adjacency matrix with components given by , , ( ( ), ( )) ( ( ) ( ) ) i j k ji k a x t x t u t u t . Here (...) is the Heaviside function. The constructed graph is characterized by the average of all nodes’ degree (
Supplementary information accompanies this paper.
Acknowledgements
This material is based upon work supported in part by the U. S. Army Research Laboratory and the U. S. Army Research Office under contract/grant number _W911NF1410276.
Author Contributions
All authors contributed to the analysis the results and reviewed the manuscript. H.O.G and W.A.G co-wrote the manuscript. H.O.G designed the main tests and performed the calculations. W.A.G supervised the research and helped to analyze the results.
Competing financial interests
The authors declare no competing financial interests.
References [
1] Dickinson, J.T., Donaldson, E.E. and Park, M.K., The emission of electrons and positive ions from fracture of materials.
Journal of Materials Science , 16(10), pp.2897-2908 (1981). [2] Theofanis, P.L., Jaramillo-Botero, A., Goddard III, W.A. and Xiao, H. Nonadiabatic study of dynamic electronic effects during brittle fracture of silicon . Physical review letters , 108(4), p.045501(2012). [3] Langen, T., Erne, S., Geiger, R., Rauer, B., Schweigler, T., Kuhnert, M., Rohringer, W., Mazets, I.E., Gasenzer, T. and Schmiedmayer, J. Experimental observation of a generalized Gibbs ensemble.
Science ,348(6231), pp.207-211 (2015). [4] Enomoto, Y. and Hashimoto, H. Emission of charged particles from indentation fracture of rocks.
Nature , 346(6285), pp.641-643 (1990). [5] Camara, C.G., Escobar, J.V., Hird, J.R. and Putterman, S.J. Correlation between nanosecond X-ray flashes and stick–slip friction in peeling tape.
Nature , 455(7216), pp.1089-1092 (2008). [6] Papenkort, T., Axt, V. M. and Kuhn, T. Coherent dynamics and pump-probe spectra of BCS superconductors.
Phys. Rev. B
76, 224522 (2007). [7]Krull, H., Manske, D., Uhrig, G. S. and Schnyder, A. P. Signatures of nonadiabatic BCS state dynamics in pump-probe conductivity.
Phys. Rev. B
90, 014515 (2014). [8] Matsunaga, R. et al. Higgs amplitude mode in the BCS superconductors Nb1−xTixN induced by terahertz pulse excitation.
Phys. Rev. Lett . 111, 057002 (2013). [9] Kibble, T.W. Topology of cosmic domains and strings.
Journal of Physics A: Mathematical and General , 9(8), p.1387(1976). [10] Grady, D.E. Length scales and size distributions in dynamic fragmentation.
International Journal of Fracture , 163(1-2), pp.85-99(2010). [11] Langen, T., Geiger, R., Kuhnert, M., Rauer, B. & Schmiedmayer, J. Local emergence of thermal correlations in an isolated quantum many-body system.
Nature Phys . 9, 640–643 (2013) [12] Mathey, L. & Polkovnikov, A. Light cone dynamics and reverse Kibble-Zurek mechanism in two-dimensional superfluids following a quantum quench.
Phys. Rev. A
81, 60033 (2010). [13] Cheneau, M. et al. Light-cone-like spreading of correlations in a quantum many-body system.
Nature
Rev. Mod. Phys . 83, 863–883 (2011). [15] Newman, M. E. J.
Networks: An Introduction (Oxford University Press, 2010). [16] Ghaffari, H. O., W. A. Griffth, P. M. Benson, K. Xia, and R. P. Young. Observation of the Kibble–Zurek mechanism in microscopic acoustic crackling noises.
Scientific Reports
Scientific Reports , 3. (2013). [18] Gao, Z.K., Yang, Y.X., Fang, P.C., Jin, N.D., Xia, C.Y. and Hu, L.D. Multi-frequency complex network from time series for uncovering oil-water flow structure. Scientific reports , 5, p.8222 (2015). [19] Collins, D.R., Stirling, W.G., Catlow, C.R.A. and Rowbotham, G. Determination of acoustic phonon dispersion curves in layer silicates by inelastic neutron scattering and computer simulation techniques.
Physics and Chemistry of Minerals , 19(8), pp.520-527(1993). [20] Maultzsch, J., Reich, S., Thomsen, C., Requardt, H. and Ordejón, P., Phonon dispersion in graphite.
Physical review letters , 92(7), p.075501(2004). [21] Jurcevic, P. et al. Quasiparticle engineering and entanglement propagation in a quantum many-body system.
Nature
Physical Review A , 73(4), p.043614 (2006). [24] Dziarmaga, J. Dynamics of a quantum phase transition and relaxation to a steady state.
Advances in Physics , 59(6), 1063-1189 (2010). [25] Zener, C. Non-adiabatic crossing of energy levels.
Proc. R. Soc. London, Ser.
A 137, 696–702 (1932). [26] Landau, L. D. On the theory of transfer of energy at collisions II.
Physik. Z. Sowjet . 2, 46–51 (1932). [27] Frimmer, M. and Novotny, L. The classical Bloch equations.
American Journal of Physics , 82(10), 947-954(2014). [28] Braun, O. M. & Kivshar, Y. S.
The Frenkel–Kontorova Model: Concepts, Methods, and Applications (Springer, 2004). [29] Eshelby, J.D. The interaction of kinks and elastic waves.
Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences , 266(1325), 222-246,(1962). [30] Vilenkin, A. and Shellard, E.P.S.
Cosmic strings and other topological defects (Cambridge University Press,2000). [31] Nye J F and Berry M V . Dislocations in wave trains.
Proc. R. Soc. Lond. A
336 165–90(1974). [32] Rubinstein, S., Cohen, G. & Fineberg, J. Detachment fronts and the onset of dynamic friction.
Nature . 430, 1005–1009 (2004). [33] Nielsen, S., Taddeucci, J. & Vinciguerra, S. Experimental observation of stick-slip instability fronts.
Geophys. J. Int . 180, 697 (2010). [34] Ohnaka, M. & Shen, L. F. Scaling of the shear rupture process from nucleation to dynamic propagation: Implications of geometric irregularity of the rupturing surfaces.
J. Geophys. Res. B
Nonlinear Processes Geophys.
21, 4 (2014). [36] Barmettler, P., Punk, M., Gritsev, V., Demler, E. and Altman, E., Relaxation of antiferromagnetic order in spin-1/2 chains following a quantum quench.
Physical review letters , 102(13), p.130603(2009). [37] Podolsky, D., Felder, G.N., Kofman, L. and Peloso, M. Equation of state and beginning of thermalization after preheating.
Physical Review D , 73(2), p.023501(2006). [38] Lieb, E. & Robinson, D. The finite group velocity of quantum spin systems . Commun. Math. Phys . 28, 251–257 (1972) [39] Hauke, P. & Tagliacozzo, L. Spread of correlations in long-range interacting quantum systems.
Phys. Rev. Lett . 111, 207202 (2013) [40] Richerme, P., Gong, Z.X., Lee, A., Senko, C., Smith, J., Foss-Feig, M., Michalakis, S., Gorshkov, A.V. and Monroe, C. Non-local propagation of correlations in quantum systems with long-range interactions.