Introduction

Minerals contain microstructures and much of what minerals can tell us about past geological processes, and about their own intrinsic properties, is related to microstructures. This balances the importance of microstructures with the actual crystallographic structure. Through the enormous progress in nanotechnology over the past decade, our perspective of materials in general and minerals in particular has shifted towards a much better understanding of microstructures. Microstructures cover a huge range of length scales from coarse twinning (mm scale), fine twins (typically on a micrometer scale) and tweed structures with repetition scales between 10 and 100 nm. On an even smaller scale we have structural disruptions, like kinks and domain wall bendings, so-called wobbles, inside these microstructures (Salje et al. 2017a; He et al. 2018; Wang et al. 2018; Nataf et al. 2020). These small disruptions appear as shifts of atomic positions and are typically measured on a pm scale (e.g. Van Aert et al. 2012 for the perovskite structure). The smallest range is reflected by electron microscopists who often use the term ‘nanoscopy’ to emphasise that the relevant scales for domains do not end at some micrometres. The range of length scales often covers some nine decades, which has led, in the field of correlated systems, to the hypothesis of ‘scale invariance’ indicating that certain aspects of microstructural physics are applicable over the full range of length scales.

A second development relates to the time scales on which microstructures change. Such changes are either induced by external forcing, like stress, electric or magnetic fields, oxygen fugacities, etc., or during creep experiments without external forcing (Salje et al. 2018). Geological processes do not always act on ‘geological’ time scales of longer than 103 years but they can also be very fast. Structural changes during radiation damage, for example, take only ca. 5 femto-seconds (10–15 s) and the propagation of a twin wall requires times between 10–8 s and 10–3 s in many cases. The fundamental question is then: what determines the origin of time scales? In this paper we argue that for avalanche processes there is not a ‘typical’ time scale but, instead, a large dynamic range of time scales.

Microstructures often evolve in a non-smooth manner. The shift of a domain boundary is virtually never continuous but occurs in a stop-and-go fashion. Cracks do not progress along straight trajectories but wobble, bifurcate and form complex patterns on an atomistic length scale. The appropriate description of such processes lies in the concept of avalanches. Their discovery, which was sometimes ignored in mineralogy, is probably the most important progress in the design and application of high-tech devices and covers a novel branch of scientific endeavour, referred to a ‘avalanche science’ with several books published in this field (e.g. Salje et al. 2017b). Historically, its importance stems from the intrinsic properties of microstructures, like holes, inclusions, twin boundaries, dislocation lines, twin junctions and so on. We know today that a transistor, as an example, does not need bulk materials to operate but is often localized in tiny areas inside twin boundaries or near junctions between boundaries. The same holds for ferroic memories and memristive conductors (Salje et al. 2017a; He et al. 2019; Bak et al. 2020; Lu et al. 2020a; Zhang et al. 2020; Salje 2021) where only a few atoms near domain boundaries move. The diameters or thicknesses of these functional regions are a few inter-atomic distances (Lu et al. 2019, 2020b; McCartan et al. 2020). Emerging properties such as ferroic memory elements are based inside twin boundaries while the surrounding crystal matrix is simply there to keep domain walls in place (Salje et al. 2016c; Salje 2020). Predesigned domain wall structures are constructed in the field of ‘domain boundary engineering’, which has become a very powerful approach in many recent applications (Salje 2010). We will allude to its relevance in mineral physics in this paper.

Most examples in this review are taken from the field of mineral physics. If the reader wishes to pursue the topic further for other minerals, we recommend consulting Salje and Dahmen (2014) as a reference paper which includes the major ideas of physical avalanche systems.

Avalanches

Crackling noise is encountered when a material is subjected to external forces with jerky responses spanning over a wide range of sizes and energies. The Barkhausen effect of pinned domain walls (Harrison et al. 2002; Robinson et al. 2002; Roberts et al. 2017, 2019) during magnetization processes (Durin and Zapperi 2006), martensitic transitions (Vives et al. 1994; Gallardo et al. 2010), plastic deformation in solids (Csikor et al. 2007; Weiss et al. 2007; Salje et al. 2009; Puchberger et al. 2017, 2018), or materials failure (Zapperi et al. 1997; Aue and De Hosson 1998) is well documented. Upon variation of an external field, avalanches show a spectacular absence of time and length scales. Crackling noise is often related to critical behaviour of avalanches, which stem from intrinsic inhomogeneities or by jamming of microstructures (Salje et al. 2011a). In all these cases one finds that the internal structures of the domain boundaries or the domain patterns display a high degree of complexity—very much in contrast to the early perception of Barkhausen noise (Barkhausen 1919; Tebble et al. 1950).

Crackling noise avalanches, like the well-known snow avalanches, are collective motions, which follow well-defined statistical rules while their exact time-dependent behaviour of any part of the avalanche remains unknown. Collapse avalanches have been thoroughly analysed in porous minerals, like SiO2 based glass (Vycor) (Salje et al. 2011b), goethite (Salje et al. 2013), porous alumina (Castillo-Villa et al. 2013) and berlinite (Nataf et al. 2014b), to name just a few prototypic examples. Their statistical characteristics share many similarities with seismicity such as the Earth crust failure due to stresses originated from plate tectonics (Davidsen et al. 2007; Kawamura et al. 2012). These similarities go beyond the avalanche statistics and include the statistics of aftershocks and waiting times of acoustic emission or earthquakes (Baró et al. 2013). More specifically, it is shown that the Gutenberg–Richter law, the modified Omori’s law, the law of aftershock productivity and the universal scaling law for the waiting time distribution typically used in statistical seismology hold for all avalanches, often in a broad range of at least six decades of jerk energies with exponents similar to those obtained in earthquakes. Similar results were found in other collapsing minerals.

The following fundamental parameters are essential for our further discussions.

Amplitude A

The amplitude A(t) is a function of time t and captures the evolution of the conjugate parameter to the external force. In many cases, the force originates from the external stress (or strain), so that the amplitude parameter is the strain (or stress) in the sample due to the hole collapses. The time evolution of the amplitude is typically initiated by an incubation period where A(t) increases exponentially leading to the maximum amplitude, called Amax. It then decays with a long tail of strain signals until the avalanche terminates. Amplitudes can display very complex evolution patterns, in particular when several avalanches coincide. Sometimes they develop ‘eternal’ avalanches, which never fully end but just diminish and resurge. The obvious analogy to disease spreading mechanisms highlights the close similarity between these two areas of research.

Duration D

The duration is the time period over which an avalanche survives. Experimental time scales typically extend from a few microseconds to many milliseconds.

Energy E

The energy is the time integral over the local squared amplitude A(t)2, integrated over the full duration of the avalanche:

$$E = \int_{o}^{D} {A^{2} \cdot } {\text{dt}}$$

This means that for avalanches which represent a short δ-function excitation at the time tmax, A(t) = Amax δ (t–tmax) will always display a scaling E ~ Amax2. This is not true for long and smooth A(t) functions. Various scalings E ~ AX with 2 < x < 3 are discussed in literature (Casals et al. 2019, 2020, 2021a, b; McFaul et al. 2020).

Size S

The size of the amplitude indicates the number of particles that move during the avalanche. While this parameter appears intuitive in geometrical terms like a ‘patch’ of transformed material, this is not correct. If areas transform, they can do so in compact regions where every atom takes part in the transformation. They can also transform by selecting some of these atoms, forming some ‘sponge-like’ areas. The fractal dimension of these transformed areas becomes then paramount and while ‘size’ is popular in the general description of avalanches, the meaning of such ‘size’ parameter can be surprisingly complex. It is, therefore, recommended to explore the scaling of size with the amplitude or energy as a more fundamental parameter. As an example, if the movement relates to low-dimensional dynamical patterns, the relationship is linear S ~ A while in magnetic systems with high fractal dimensions we find S ~ A2. This already highlights that model calculations are often required to determine this S(A) scaling and that scaling depends sensitively on the fractal dimension of the domain patterns (Casals et al. 2019, 2021a; Nataf et al. 2020; Xu et al. 2020).

Waiting time or inter-event time tw

The two names are used interchangeably. They denote the time between avalanches, i.e. the time the system needs to recover after an avalanche has happened. In neural networks, these inter-event times are the ‘slee** periods’ after high neural activity. Their probability distributions are typically power laws with two different, approximate exponents for short and long times, P(tw) ~ tw−1 and P(tw) ~ tw−2, respectively. Note that in these scaling relationships the negative sign in the exponent is often included in the equation so that the term ‘exponent’ often means the value after the minus sign. These exponents represent the results in the simplest mean field (ML) theory (Salje and Dahmen 2014). Similar values have been observed experimentally and deviations from ML predictions are analysed in terms of specific physical models (Christensen et al. 1996; Corral and Paczuski 1999; Navas-Portella et al. 2016).

In addition, there is a multitude of secondary scaling laws, in particular those describing aftershock activities (Baró et al. 2013, 2018; Nataf et al. 2014a) and inter-correlations of times (Baró et al. 2016a, 2018) with several important practical extensions for mineral behaviour, as described in (Jiang et al. 2017).

Acoustic emission (AE) spectroscopy

During 100 years of research many experimental methods were developed to quantify avalanches. They range from magnetic measurements to electrical depolarization currents in ferroelectrics and optical observations of crack patterns and the determination of fractal dimensions (Lung and Zhang 1989; **. Phys Rev E 96:042122. https://doi.org/10.1103/PhysRevE.96.042122 " href="/article/10.1007/s00269-021-01138-6#ref-CR110" id="ref-link-section-d50308837e2004">2017c) for detail]. The horizontal dashed lines indicate the PDF slopes and hence the exponents. The ML curves of bio-cemented sand samples show increases with increasing Emin indicating that the AE signals are damped by absorption or scattering of the acoustic signals (Salje et al. 2017c).

Fig. 10
figure 10

a Distribution of AE energies, b shows the ML-fitted exponent as a function of a lower threshold Emin for the three experiments during the full experiment. c ML-fitted exponent in different time windows (Wang et al. 2021)

Full size image

Grains show excellent plateaus with ε = 1.4 remarkably close to the theoretical MF result 4/3. The energy exponent for sands is 1.7, which is near another MF prediction value of 5/3. MICP ceramics show an overall energy exponent 1.46. The most important observation is that in MICP ceramics, but not in sand or gains, different time windows show variable values (Fig. 10c) between 1.35 and 1.6. These avalanches are a combination of the AE energies of sands and grains with variable proportions. This observation allows us to conclude that the collapse mechanism is the breaking of the bio-mineralized bridges between the hard grains. This process starts at very low stress and is not visible by the macroscopic shape change. Compression leads to a ‘rubble’ of grains mixed with larger bio-mineralized segments. The grains are constrained by neighbours, which hinders their rotations. This makes the materials harder for compression than sand. Once the sample is compressed further it transforms back to sand with a dusting of MICP particles.

Conclusion

Changes of microstructures often progress in a wild, non-smooth manner. Experimental evidence rules out simple catastrophic events, like one big step when a twin wall moves. Instead, we find universal behaviour with multitudes of small ‘jerks’ which can cut down the big step into millions of small steps. This phenomenon appears in many systems, and only three of them were briefly reviewed here. The overall behaviour of the totality of the jerks follows very strict rules. These rules are the same as what is theoretically expected for avalanches, which establishes a close link between avalanches and microstructural evolution. As the probability to find a ‘jerk’ with an energy E follows a power law with well-defined exponents, which appear to be universal for many systems. The power law is important because it is ‘scale invariant’. To illustrate the scale invariance, let us consider the energy probability P(E) = E−ε. Consider an energy interval between E and 2E; then the probabilities are between E−ε and (2E)−ε. We now scale the energy by a factor x. The interval is now from xE to 2xE and the probabilities change to (x E)−ε and 2−ε (x E)−ε. The common numerical prefactor x−ε is irrelevant for the functional form, which remains exactly the same as before. This proves that the power-law distributions are scale invariant. Note that this is a special property of the power law and that other functions are not scale invariant. Furthermore, combinations of power laws are not power laws and hence not scale invariant.

The scale invariance in avalanches is not restricted to energies but holds equally for the amplitudes, durations and, with some modification, for waiting times. In practical terms, anything we see in a space (or time) interval is exactly the same as in any other. The limits are given by cut-offs, such as the atomic diameter or the sample size, but the region between these cut-offs can reach many orders of magnitude. AE allows us to estimate this range and we find that 6–8 orders of magnitude of energy are not uncommon for microstructural changes. The question on which length scale do structural changes happen is hence ill-posed: there is a large interval of length scales and wherever we situate our experiment we will see the same change.

This powerful approach is important for minerals where defects and lattice imperfections favour avalanches. The induced behaviour is then independent of these obstacles and significant similarities are found in a multitude of different minerals. For reasons alluded to in the introduction, the full power of this method has been used to solve several problems in solid-state physics and metallurgy, but much less in mineral physics. There is a wide range of mineralogical research waiting to be done in future.